Superconducting Charge Qubits Denzil Anthony RodriguesSuperconducting Charge Qubits Denzil Anthony Rodrigues H.H.WillsPhysicsLaboratory UniversityofBristol AthesissubmittedtotheUniversityofBristolin

Superconducting

Charge Qubits

Denzil Anthony Rodrigues

H. H. Wills Physics Laboratory

University of Bristol

A thesis submitted to the University of Bristol in

accordance with the requirements of the degree of

Ph.D. in the Faculty of Science

November 2003

Word Count: 36, 000

Abstract

In this thesis we discuss a particular type of superconducting qubit, the charge qubit.

We review how the Hamiltonian for a single qubit can be constructed by quantising the

Josephson relations as if they were classical equations of motion.

We examine the charging energy of various qubit circuits, in particular focussing on

the effect of connecting a superconducting reservoir to small islands. We establish the

correct form for the charging energies and show that the naive method of constructing

a quantum Hamiltonian by adding the charging energy of the circuit and the tunnelling

energy for each junction leads to the correct Josephson relations for each junction.

We describe a coupled two-qubit system and describe the operations necessary to test

a Bell inequality in this system. We show how passing a magnetic flux through the circuit

leads to oscillations as a function of both time and flux and show that these oscillations can

be considered as being caused by interference between virtual tunnelling paths between

states.

We describe the strong coupling limit of the BCS Hamiltonian, where the sums over

electron creation and annihilation operators can be replaced by operators representing

large quantum spins. We then solve this Hamiltonian in both the exact and mean-field

limits. We show that the spin solutions are equal to the full solutions in the limit where

the interaction energy is much larger than the cutoff energy.

We show how this simple model can be used to easily investigate various phenomena.

We describe an analogue of the quantum optical effects of superradiance where the cur-

rent through an array of Cooper pair boxes is proportional to the square of the number

of boxes. We also describe an analogue of quantum revival, where the coherent oscilla-

tions of a Cooper pair box coupled to a reservoir decay, only to revive again later. We

describe a similar two-qubit effect, where the oscillations of two entangled Cooper pair

boxes decay and revive. If the entanglement between the boxes is calculated, we find that

the entanglement also decays and then revives.

We describe how quantum spins can be represented using a wavefunction that is a

function of phase. This representation is used to find a microscopic derivation of the

charge qubit Hamiltonian in terms of phase. We find that in the limit of large size, the

Hamiltonian found by quantising the Josephson equations is recovered. Corrections to

this phase Hamiltonian are described.

To my family.

Acknowledgments

I have been lucky enough to have two supervisors to whom I could turn in times of

distress; I would like to thank Balazs for instructing me in basic physics as if I was stupid,

and Tim for reminding me that I’m not. As well as technical advice and encouragement,

both have provided instruction by example in the appropriate attitude to research, and in

how to fit research in to a (relatively) normal life.

My fellow PhD students have been of great importance to me over the last three years.

I would like to thank Ben and Mark for proving that it was possible to survive a PhD

and to thank Andy for giving me something to aim for. I thank my roommate Danny for

asking me stupid questions, both ones that I knew the answer to and those I did not. I

thank David and Maria for stopping me when I was talking rubbish and Jamie and Emma

for not stopping me when I was talking rubbish.

Finally I thank Liz giving me something to look forward to at the end of the day.

Authors Declaration

I declare that the work in this thesis was carried out in accordance with the regulations of

the University of Bristol. The work is original except where indicated by special reference

in the text and no part of the thesis has been submitted for any other degree. Any

views expressed in the thesis are those of the author and in no way represent those of

the University of Bristol. The thesis has not been presented to any other University for

examination either in the United Kingdom or overseas.

“You can never take your badge off.”

Jamie Walker.

Contents

1 Superconductivity and The Josephson Effect 1

1.1 Introduction to the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction to Superconductivity . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 The Ginzburg-Landau Equations . . . . . . . . . . . . . . . . . . . . 6

1.3.2 The Josephson Effect in Ginzburg-Landau Theory . . . . . . . . . . 9

1.4 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . 10

1.5 The Fermi Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 BCS and BdG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.8 The Pair Tunnelling Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 20

1.9 Small Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.9.1 The Parity Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.9.2 The Richardson Exact Solution . . . . . . . . . . . . . . . . . . . . . 25

2 The Superconducting Charge Qubit 31

2.1 Experimental Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 The Josephson Equations for Small Circuits . . . . . . . . . . . . . . . . . . 33

2.2.1 The Classical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2 The Classical Hamiltonian and the Conjugate Variables . . . . . . . 36

2.2.3 Quantising the Classical Josephson Equations . . . . . . . . . . . . . 37

2.3 Tuneable Josephson Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 The Two-Level System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Qubit Dynamics and the NOT Gate . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Decoherence and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xiii

xiv CONTENTS

2.6.1 Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6.2 Discrete Noise in the Josephson Energy . . . . . . . . . . . . . . . . 52

3 The Josephson Circuit Hamiltonian 57

3.1 The Capacitance of Qubit Circuits . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Single Junction with Gate Voltage . . . . . . . . . . . . . . . . . . . 58

3.1.2 Single Junction with Gate Voltage and Self-Capacitance . . . . . . . 60

3.1.3 The 2 Qubit Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.1.4 Multiple Islands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Quantising the Josephson Relations . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.1 Single Junction with Gate Voltage . . . . . . . . . . . . . . . . . . . 65

3.2.2 Single Junction with Gate Voltage and Self-Capacitance . . . . . . . 67

3.2.3 The 2 Qubit Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 The Dynamics of a Two Qubit System 71

4.1 The Two Qubit Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1.1 Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.2 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 A Bell Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Virtual Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 Path Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Finite Superconductors as Spins 93

5.1 The Strong Coupling Approximation . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Quantum Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3 The Majorana Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.1 Spin l2 in terms of Spin 1

2 . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3.2 The Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.3 The Majorana Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Mean Field Spin Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4.1 Rotation of Spin Operators . . . . . . . . . . . . . . . . . . . . . . . 104

5.4.2 The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.3 Self Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4.4 Comparison to the BCS Theory . . . . . . . . . . . . . . . . . . . . . 109

CONTENTS xv

5.5 Solving the Spin Hamiltonian Exactly . . . . . . . . . . . . . . . . . . . . . 110

5.5.1 Broken Symmetry: Finite size . . . . . . . . . . . . . . . . . . . . . . 112

5.5.2 Broken Symmetry: Infinite Size . . . . . . . . . . . . . . . . . . . . . 114

5.6 Expanding the Richardson Solution to First Order . . . . . . . . . . . . . . 115

6 The Super Josephson Effect 119

6.1 The Cooper Pair Box Array . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2 Superradiant Tunnelling Between Number Eigenstates . . . . . . . . . . . . 122

6.3 Superradiant Tunnelling: Number State to Coherent State . . . . . . . . . . 124

6.4 Superradiant Tunnelling Between Coherent States . . . . . . . . . . . . . . 126

7 Revival 129

7.1 The Superconducting Analogue of the Jaynes-Cummings Model . . . . . . . 129

7.1.1 Quantum Revival of the Initial State . . . . . . . . . . . . . . . . . . 132

7.2 Revival of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8 An Electronic Derivation of The Qubit Hamiltonian 141

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.2 The Phase Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.2.1 Phase Representation of Spins . . . . . . . . . . . . . . . . . . . . . 143

8.2.2 The Limit of Large Size . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.2.3 Phase Representation of BCS States . . . . . . . . . . . . . . . . . . 148

8.3 Island - Reservoir Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.4 Inter - Island Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9 Conclusion 157

A Eigenstates of the Two Qubit Hamiltonian 161

A.1 Exact Eigenstates with Real T12 . . . . . . . . . . . . . . . . . . . . . . . . 161

A.2 Perturbation in T12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B Limit of BCS Self Consistency Equations 165

xvi CONTENTS

List of Figures

1.1 The Ginzburg-Landau free energy as a function of the order parameter ψ. . 7

1.2 The Temperature dependence of the superconducting gap ∆(T ) measured

in units of the zero-temperature gap ∆(0). . . . . . . . . . . . . . . . . . . . 16

2.1 A simple circuit diagram for a charge qubit consisting of two regions of

superconductor connected by a Josephson junction with critical current IC

whose capacitance is represented by a capacitor of capacitance C . . . . . . 34

2.2 A simple circuit diagram for a charge qubit with a gate voltage applied

and the single Josephson junction replaced by a magnetic flux dependent

double-junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 A double junction consisting of a ring of superconductor broken by two

junction and threaded by a flux Φ. The two paths from point 1 to point 2

are labelled ca and cb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 The charging energy as a periodic function of ng. . . . . . . . . . . . . . . . 43

2.5 The qubit energy levels as a periodic function of ng. . . . . . . . . . . . . . 44

2.6 The avoided crossing at half-integer values of ng. . . . . . . . . . . . . . . . 45

2.7 A not gate takes a state |0〉 and returns a state |1〉, and vice versa. . . . . 47

2.8 Oscillations in the probability of the qubit being in state 1〉 (solid line) or

state |0〉 (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.9 The probability of a Cooper pair box being in the state |1〉 as a function of

time, with Gaussian noise in the Josephson energy. . . . . . . . . . . . . . . 52

2.10 The probability of a Cooper pair box being in the state |1〉 as a function of

time with discrete noise in the Josephson energy. . . . . . . . . . . . . . . . 55

3.1 Single Voltage with Gate Voltage Circuit Diagram . . . . . . . . . . . . . . 59

xvii

xviii LIST OF FIGURES

3.2 Single junction with gate voltage and self-capacitance, which is represented

by a capacitive coupling to a third region, labelled w (for ‘world’). . . . . . 60

3.3 Two Qubit circuit diagram. The system consists of two islands (1, 2) con-

nected to a reservoir (r), with gate voltages applied to each. We wish to

solve the problem for a reservoir of general size, and so a further region, w

is included to represent the rest of the world. . . . . . . . . . . . . . . . . . 63

4.1 The energy levels of a two qubit system as a function of n1g = n2g, when

the tunnelling between the islands is zero. The parameters of the system

are EC1 = EC2 = 100, EC12 = 10, T1 = 10, T2 = 10 . . . . . . . . . . . . . 73

4.2 The energy levels of a two qubit system as a function of n1g = n2g, when the

tunnelling between the islands is included. The parameters of the system

are EC1 = EC2 = 100, EC12 = 10, T1 = 10, T2 = 10, T12 = 30 . . . . . . . . 74

4.3 Oscillations between state |01〉 (solid line), |10, 〉 (dashed line) and the states

|00〉 and |11〉 (dotted lines). Halfway between oscillations, the system is in

a maximally entangled state (vertical lines). . . . . . . . . . . . . . . . . . . 75

4.4 The bases for measurements that violate a Bell inequality. . . . . . . . . . . 76

4.5 A two qubit circuit with tunnelling between the qubits and a magnetic flux

passed through the circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6 The Probability of the Two Qubit system being in the state |00〉 (upperplot) and |01〉 (lower plot). The horizontal axis is time and the vertical axis

is the flux through the loop in units Φ02π . White represents high probability

of occupation and black represents low probability. . . . . . . . . . . . . . . 82

4.7 The Probability of the Two Qubit system being in the state |10〉 (upperplot) and |11〉 (lower plot). The horizontal axis is time and the vertical axis

is the flux through the loop in units Φ02π . White represents high probability

of occupation and Black represents low probability. . . . . . . . . . . . . . . 83

4.8 The Probability of the Two Qubit system being in each of the four charging

states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.9 The change in frequency of the oscillations in the probability of being in

the state |00〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.10 The two second order (a, b) and third order (c, d) tunnelling processes

involved in the oscillation |00〉 → |11〉. . . . . . . . . . . . . . . . . . . . . . 87

LIST OF FIGURES xix

4.11 The three third order tunnelling processes involved in the oscillation |01〉 →|10〉. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.12 The oscillations of the two qubit system when the system is not in the limit

T ¿ EC12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.13 The Probability of the Two Qubit system being in the state |01〉 or |11〉 i.e.the probability that the first qubit is occupied regardless of the occupation

of the second. The z axis represents increasing probability. . . . . . . . . . 91

5.1 The Majorana Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 Each point on the Majorana sphere can be represented by its projection

onto the complex plane, z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 Rotation of basis of spin operators. . . . . . . . . . . . . . . . . . . . . . . . 104

6.1 An array of lb superconducting islands, or Cooper Pair Boxes, that are

individually coupled to a superconducting reservoir, r, through capacitive

Josephson tunnel junctions (with no inter-box coupling). . . . . . . . . . . . 120

7.1 Spin Coherent State Revival for Nr = 10, lr = 50. . . . . . . . . . . . . . . . 133

7.2 Decay and revival of a single Cooper pair island state, showing the degree

of linear entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.3 Analytic asymptotic form for decay oscillations in the probability against

time in units of 2π/EJ(solid line) compared with the numeric evolution

(dashed line) Nr = 10, lr = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.4 Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. . . 137

7.5 Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50,

showing the degree of entanglement. . . . . . . . . . . . . . . . . . . . . . . 138

8.1 Energy levels of a charge qubit as a function of ng including the finite size

correction to the tunnelling energy. The dashed lines indicate the uncor-

rected energy levels. The parameters are EC = 10 EJ = 1 and l = 100 . . . 154

xx LIST OF FIGURES

Chapter 1

Superconductivity and The

Josephson Effect

We discuss the basic principles of quantum computing and list the requirements of a phys-

ical system used to construct a quantum computer. We discuss the advantages of using

superconductors to construct qubits. We describe both phenomenological and microscopic

theories used to describe superconductivity and the Josephson effect. We also examine how

bulk superconductivity is modified in a superconductor of finite size.

1.1 Introduction to the Thesis

At the beginning of the Twentieth Century, a new description of the world appeared:

Quantum Mechanics. Since then it has developed into the most accurate physical theory

ever constructed, and explained many phenomena that would otherwise remain complete

mysteries.

Despite the overwhelming success of quantum mechanics, an understanding of the

theory is far from complete, and new avenues of exploration are continually opening. One

such avenue that is currently attracting much interest among physicists, mathematicians

and computer scientists is the field of Quantum Computing. This field involves finding

a quantum mechanical description of not only physical objects, but also the information

that they represent and manipulate [1, 2]. Just as quantum mechanics tells us more about

the world than classical mechanics, a quantum computer could do calculations that would

be impossible on a classical computer.

1

2 Superconductivity and The Josephson Effect

The first theoretical demonstration that a quantum mechanical algorithm could out-

perform anything that is possible classically was made by David Deutsch and Richard

Jozsa [3]. Whilst this sparked interest in the field of quantum computing, it was after

the discovery that quantum computing could factorise large numbers exponentially more

quickly than any known classical algorithm [4], and could give a√N improvement in the

time needed for a database search [5] that the field really took off. A great boost was given

to the credibility of quantum computing when the discovery of quantum error correction

codes [6, 7, 8] showed that despite the fact that a quantum state cannot be determined by

a single measurement, errors occurring below a certain rate can be corrected algorithmi-

cally. This means the number of gate operations performable is no longer limited to that

which can be performed within the ‘bare’ decoherence time of the system (as long as the

decoherence time is long enough to allow the error correction codes to operate).

The basic idea that underlies all modern (classical) computation is binary logic. This

is the idea that information is stored in switches, or ‘bits,’ that are either on or off (1

or 0) and never anywhere in between. The bits take many forms, from the voltages

on a transistor, to the direction of magnetic moments on a hard drive, to the presence

or absence of pits on a CD. The underlying idea of quantum computation is that these

switches behave according to the rules of quantum mechanics - in particular, they can be

placed in a superposition state. If a bit is placed in a superposition of its two states, then

it can be thought of as being in both the ‘on’ and ‘off’ states at the same time. In terms

of the binary number stored, the qubit (quantum bit) represents both the number ‘0’ and

‘1’ simultaneously, and a series of n such bits (a register), each placed in a superposition,

can represent all the numbers from 0 to 2n − 1 simultaneously. Any operation performed

on this register of qubits then performs a calculation on all of the numbers stored by the

qubits in one operation. It is this ‘quantum parallelism’ that gives quantum computing

its power [9, 10].

At present there is no single preferred way of constructing quantum bits, and in fact

there is a vast range of different systems that have been suggested as possible qubits,

including the energy levels of ions [11] or atoms [12] held in electromagnetic traps or

cavities, impurities embedded in a substrate [13], or quantum dots formed from silicon

nanostructures [14].

With such a large array of possibilities, a set of criteria by which these potential qubits

could be assessed was needed, and the so-called ‘DiVincenzo checklist’ [15] has become

1.1 Introduction to the Thesis 3

the standard for determining the viability of different proposals. The five main criteria

outlined in this checklist are:

• Well characterised qubits, scalable in number.

• The ability to initialise these qubits in a simple ‘starting state.’

• Long decoherence times, compared with the gate operation time.

• A universal set of quantum gates.

• The ability to measure qubits.

As is to be expected, some realisations of quantum bits are more successful at meeting

different criteria than others. Two of the most successful realisations to date are ion traps

and NMR, in which multiple sequential gates have been applied [16, 17] and, in the case of

NMR, even Shor’s factoring algorithm has been performed [18]. Whilst these realisations

have well-defined qubits upon which quantum gates can be accurately performed, they

suffer from the problem of scalability - it is difficult to see how the techniques can be

applied to the hundreds, thousands or millions of qubits that would be required in a

working quantum computer.

By contrast, realisations of qubits in the solid state, such as implanted ions or quantum

dots, are intrinsically scalable in that a micro-fabricated array of qubits could be produced,

each of which is well defined and is distinguishable due to its position.

The main problem that solid-state realisations face is the problem of decoherence.

Whilst ions in a trap are isolated to a large degree from the external world, solid state

qubits are manufactured devices subject to noise from, for example, the movement of

charges in the substrate. Although quantum error-correction schemes have been developed

[6, 7, 8], these require that a large number of gate operations (∼ 104) can be performed

within the decoherence time of a qubit. If this limit can be obtained then, in principle, very

long calculations can be performed in a fault tolerant manner, as long as the additional

qubits required for error-correction can be provided.

One possible way of constructing a solid-state qubit that is less vulnerable to decoher-

ence is to make it from a superconductor, as superconductors are known to posses some

remarkable quantum coherent features that may be of value in the construction of a qubit.


The ability to describe the phenomenon of superconductivity is one of quantum me-

chanics’ great successes. Superconductivity was discovered, experimentally, around the

start of the last century, when it was discovered that when certain metals are cooled down

to close to absolute zero, their electrical resistance vanished. One of the key features of

superconductivity is the existence of an energy gap: a minimum amount of energy is re-

quired to produce excitations above the ground state. This means the superconducting

state is protected against low-energy excitations, and to some extent the superconductor

is isolated from the external world. A further problem for many qubits is the difficulty of

measurement. Qubits constructed from ‘fundamental’ objects (such as ions) may produce

only a small signal, which may be difficult to isolate from the other qubits in the system.

Solid state qubits in general have the problem that a measurement device connected to

them will cause decoherence during the operation of the gates. Qubits constructed from

qubits take advantage of the macroscopic quantum coherence of the electrons to provide a

large signal that may be easier to measure, and also benefit from decades of research into

Josephson junctions and other superconducting circuits.

These factors suggest that a superconductor might be the ideal material from which

to construct a quantum computer. In this thesis we discuss one of the ways of making a

qubit out of a superconducting circuit, known as the superconducting charge qubit, and

various other interesting phenomena that could possibly be observed in such systems.

1.2 Introduction to Superconductivity

In 1901, Kammerlingh-Onnes developed a technique for liquifying Helium, and proceeded

to investigate the properties of materials at the low temperatures that this allowed.

In 1911, he noticed that the electrical resistance of Mercury became very low below

4.2K [19]. Further investigation revealed two things: that the resistance was not just

low, but so low that it could not be measured, and that the resistance did not smoothly

decrease as the temperature was reduced, but dropped off suddenly at 4.2K.

Kammerlingh-Onnes named the state of the Mercury below this temperature the super-

conducting state. A century later, experimental and theoretical work on superconductivity

continues apace, and it is a major research topic in condensed matter physics. Supercon-

ductivity turned out to be a common phenomenon, and most metals will superconduct

if their temperature is reduced enough. Recently, many other systems have been shown

1.3 Ginzburg-Landau Theory 5

to exhibit superconductivity, including organic superconductors [20], magnesium diboride

[21], and carbon nanotubes [22] among others. The superconducting state was found to

have several distinctive characteristics other than zero resistance. In particular, the Meiss-

ner effect [23], in which magnetic fields are excluded from a superconductor, indicates that

a superconductor is not merely a perfect conductor (which would trap any flux present

before the sample was cooled to its superconducting state) but a perfect diamagnet as

well. Another important characteristic of superconductors is an exponential specific heat

capacity at low temperatures, which suggests a finite energy gap between the

Several phenomenological theories of superconductivity were developed in the first

half of the last century, notably the London - London equations [24] which describe both

the infinite conductivity and the exclusion of the magnetic field. A more complete phe-

nomenological description is given by Ginzburg-Landau theory [25] which treats the onset

of superconductivity as a second-order phase transition with a complex order parameter.

Despite the success of the various phenomenological theories, it was half a century

after Kammeling-Onnes’ discovery before a microscopic theory of superconductivity was

formulated. A key insight came when it was shown [26] that a small attractive interaction

between electrons could lead to a bound pair of electrons (a Cooper pair), explaining the

energy gap. This insight led to the formulation of a microscopic theory of superconductiv-

ity by Bardeen, Cooper and Schrieffer now known as BCS theory [27]. Excitations above

the BCS ground state are described using Bogoliubov-de Gennes (BdG) theory [28].

In BCS theory, the ground state of a superconductor is a state where all the Cooper

pairs have a common phase. This is reminiscent of Ginzburg-Landau theory, which uses

a complex order parameter with both an amplitude and a phase to describe the collective

state of the electrons as a whole. In fact, it can be shown that the Ginzburg-Landau order

parameter is proportional to the BCS energy gap ∆ [29]. The truly macroscopic nature

of this phase is illustrated in the Josephson effect [30], where the zero-voltage current

between two superconductors is dependent on the difference between the phases of the

superconductors.

1.3 Ginzburg-Landau Theory

Ginzburg-Landau theory [25] is a phenomenological description of superconductivity. An

expression for the free energy is given in terms of an order parameter, which must be


minimised with respect to the order parameter and the electromagnetic vector potential.

This leads to the Ginzburg-Landau equations, which give a clear, intuitive picture of

the appearance of the superconducting order parameter with temperature, and allow the

treatment of spatially inhomogeneous situations, such as the Meissner effect.

1.3.1 The Ginzburg-Landau Equations

The Ginzburg-Landau equations can be derived from a spatially inhomogeneous micro-

scopic theory [29], but in this section we shall treat them in the way they were originally

derived, from the phenomenological theory of second-order phase transitions.

A first-order phase transition shows a discontinuity in the entropy of a system. A

second order transition shows a discontinuity in the derivative of the entropy, and we need

to construct a theory that replicates this behaviour.

The Ginzburg-Landau term for free energy can be viewed as an expansion of the free

energy density, f , in terms of a complex order parameter ψ. The free energy must of

course be real, so the first two allowed terms are the quadratic and quartic terms:

f = α|ψ|2 + β|ψ|4 (1.3.1)

The order parameter is then chosen to minimise the free energy. From this we see that

β must be positive, otherwise the minimum of free energy would be at |ψ| =∞. For there

to be a non-zero minimum of free energy, α must be negative. Finally, we want |ψ| to be

zero above some temperature TC , and non-zero below this temperature. Thus, α must be

a function of temperature, going from negative to positive at T = TC . We expand α to

first order in (TC − T )/TC , and obtain:

f = −a(TC − T )TC

|ψ|2 + β|ψ|4 (1.3.2)

Where a and β are positive real constants. Minimising this equation (which has

only two parameters) gives the behaviour of ψ around the critical temperature. Above

this temperature, the free energy is minimised by an order parameter of zero, i.e. no

superconductivity. Below this temperature, the order parameter has a non-zero value

|ψ∞|2 = a2β

(TC−T )TC

. The subscript denotes the fact that this is the value for a bulk, or

infinite, superconductor.


Figure 1.1: The Ginzburg-Landau free energy as a function of the order parameter ψ, for

the cases when: a) α is positive (above TC) and b) α is negative (below TC). Above TC

the minimum in free energy is at ψ = 0, whereas below TC , the free energy has a minimum

at a non-zero value ψ = α/2β.

Note that as 1.3.2 depends only on the magnitude of ψ, the phase of the parameter

can take any value, and the free energy is independent of this.

The above expression is for a perfectly homogenous superconductor of infinite size with

no currents or magnetic fields. In general, the free energy is an integral over space, with

a position dependent order parameter, ψ(r). We add a kinetic energy term, 12m∗ p

2, where

the canonical momentum operator p = −i~∇− e∗

c A(r) and A(r) is the vector potential.

We also add the contribution of the magnetic field, H(r) = ∇×A(r), to the energy.

f =

∫dr3α|ψ(r)|2 + β|ψ(r)|4 + 1

2m∗

∣∣∣∣(~i∇− e∗A(r)

)ψ(r)

∣∣∣∣2

+µrµ02

H(r)2 (1.3.3)

Note that eq. 1.3.3 contains the expressions m∗ and e∗, the effective mass and effective

charge. These replace the usual electron mass and charge because the charge carriers are

Cooper pairs rather than individual electrons. The free energy must now be minimised

with respect to ψ(r) and A(r). Discarding a surface term, the minimisation condition

with respect to ψ(r) leads to:

0 = αψ + 2β|ψ|2ψ − ~2

2m∗

∣∣∣∣(∇− ie∗

~A

)∣∣∣∣2

ψ (1.3.4)

This is the first Ginzburg-Landau equation, and from this we see a natural length


scale for the variation of the order parameter, ζ2(T ) = ~22m∗|α| . Minimising eq. 1.3.3 with

respect to A(r) gives:

0 = ∇×∇×A− e∗

2m∗(ψ∗(−i~∇− e∗A)ψ + ψ(i~∇− e∗A)ψ∗) (1.3.5)

Inserting Ampere’s law, ∇×∇×A = µrµ0J, we find an expression for the current J:

J =e∗

2m∗(ψ∗(−i~∇− e∗A)ψ + ψ(i~∇− e∗A)ψ∗)

= −i~ e∗

2m∗(ψ∗∇ψ − ψ∇ψ∗)− e∗2

m∗|ψ|2A (1.3.6)

This is the second Ginzburg-Landau equation.

We can use eq. 1.3.6 to derive the Meissner effect and screening currents at the surface

of a superconductor. In the limit where (ψ∗∇ψ − ψ∇ψ∗) is negligible, taking the curl of

both sides of the equation, we find:

∇×∇×H = − 1

λ2|ψ|2H (1.3.7)

with λ2 = m∗

µrµ0e∗2|ψ|2 . First we consider the component of the field (HZ) perpendicular to

the surface of a superconductor. We note that no current can flow in the z direction which,

using J = ∇ ×H implies that the left hand side of eq. 1.3.7 is zero, so that H = 0, i.e.

there can be no field perpendicular to the surface of a superconductor. Considering the

field parallel to the surface, H = H(z)x, we use the identity ∇×∇×H = ∇(∇·H)−∇2H

to obtain:

∂2Hx

∂z2=

1

λ2Hx (1.3.8)

Which give us the expression for the field parallel to the field of a superconductor:

Hx(z) = Hx(0)e−z/λ (1.3.9)

We see that the magnetic field at the surface decays exponentially inside the super-

conductor with a characteristic length λ. This also implies the existence of a surface

current:


Jy(z) =c

4πλHx(0)e

−z/λ (1.3.10)

This current is known as the screening current, and it is this that excludes (or screens)

the magnetic field from the interior of the superconductor.

Despite their apparent simplicity and phenomenological basis, the Ginzburg-Landau

equations can be used to describe many of the important phenomena in superconductivity,

and the 2003 Nobel Prize in physics was awarded for such work [31, 25, 32]. Phenomena

that can be described include the proximity effect, the flux lattice in the mixed state of

type-II superconductors and, of particular interest to us, the Josephson effect.

1.3.2 The Josephson Effect in Ginzburg-Landau Theory

In 1962, Brian Josephson made the remarkable discovery that a zero-voltage current could

flow between two superconductors separated by a barrier of insulator or normal metal [30].

Before discussing the microscopic theory, following [33] we shall use Ginzburg-Landau

theory to deduce the existence of the effect between two bulk superconductors separated

by a ‘weak link’ consisting of a short, one-dimensional bridge of superconductor. If we

introduce a normalised order parameter ψ′ = ψ/ψ∞, then in the absence of magnetic

fields, the condition for the minimisation of the free energy in one dimension is:

0 = ψ′ − |ψ′|2ψ′ + 1

2m∗ζ2(T )

d2ψ′

dx2(1.3.11)

We choose the boundary conditions that in both bulk superconductors (where we

consider the gradient of ψ to be zero), the magnitude of the order parameter is at the

value, ψ = ψ∞, i.e. the normalised wavefunction |ψ′| = 1 at either end of the bridge. The

Ginzburg-Landau free energy is independent of phase in the absence of gradients, and so

we set ψ′ = eiφ1 at one end of the bridge and ψ′ = eiφ2 at the other.

If the coherence length ζ is much greater than the length of the bridge, L, then the

kinetic term is much greater than the other terms and we have 0 = d2ψ′

dx2, i.e. ψ′ ∝ x. The

form for the order parameter that satisfies eq. 1.3.11 and the boundary conditions at each

end of the bridge is:

ψ′(x) =(1− x

L

)eiφ1 +

x

Leiφ2 (1.3.12)


If we now calculate the current using the standard Ginzburg-Landau current expression

eq. 1.3.6, we find:

I = IC sin (φ2 − φ1) (1.3.13)

Where IC = 2~e∗|ψ∞|2m∗

AL is the critical current, the maximum supercurrent that can be

sustained by a junction of area A.

Although this is a highly simplistic derivation done for the particular case of a constric-

tion weak link, we see a central feature of the Josephson effect - the sinusoidal dependence

of the current on the phase difference between the two superconducting regions.

1.4 Creation and Annihilation Operators

One of the reasons that a microscopic theory was so elusive is that superconductivity is

intrinsically a coherent collective phenomenon, and that any description must be a true

many-body description that describes the behaviour of the many-body wavefunction of

the electrons.

Whilst the wavefunction of a system of identical Bosons must be symmetric, the wave-

function of a system of identical Fermions (such as electrons) must be anti-symmetric

under particle exchange:

ψ(r1, r2, r3 · · · ri) = −ψ(r2, r1, r3 · · · ri) (1.4.1)

This fact alone has many deep consequences, the most obvious of which is the Pauli

exclusion principle stating that two fermions cannot occupy the same quantum state. Any

many-body calculation must ensure that the wavefunction is at all times anti-symmetric.

For two fermions, with single electron states k1 and k2, the two-particle state of the system

is of the form:

ψ(r1, r2) = ψk1(r1)ψk2(r2)− ψk1(r2)ψk2(r1) (1.4.2)

Performing calculations with wavefunctions that have been symmetrised ‘by hand’ such

as eq. 1.4.2 will obviously be very laborious, especially considering the vast number of

1.5 The Fermi Sea 11

electrons present in a condensed matter system. To deal with this we use a notation known

as ‘second quantisation’ in which we have creation and annihilation operators c†k↑, ck↑ that

add electrons to the wavefunction in such a way as to ensure that the symmetrisation

properties are at all times taken care of (see eg. [34]). Thus, if |0〉 is the vacuum state,

then c†k1↑|0〉 is a state with one electron in the state k1 with spin up, and c†k2↓c†k1↓|0〉 is

a state with one electron in the state k1 and one electron in the state k2 (with correct

anti-symmetry as given in eq. 1.4.2). Rather than describing the action of the operators

in terms of their commutators [A, B] = AB − BA as we would for Bosons, we describe it

in terms of the anti-commutation relations [A, B]+ = AB + BA:

c†kσc†k′σ′ + c†k′σ′c

†kσ = 0

ckσck′σ′ + ck′σ′ckσ = 0

c†kσck′σ′ + ck′σ′c†kσ = δk,k′δσ,σ′ (1.4.3)

It is clear from eq. 1.4.3 that c†kσc†kσ = 0, i.e. two electrons cannot be put into the

same state. Thus the commutation relations of the creation and annihilation operators

embody, in a very direct way, the Pauli exclusion principle.

1.5 The Fermi Sea

The operators defined in section 1.4 can be used to formulate the solutions to problems

in many-body physics in a clear way.

Perhaps the simplest problem of condensed matter physics is that of a free-electron solid

(see eg. [34]). In a free electron solid, the potential that the electrons see is considered

constant - the atoms in the lattice are ‘smeared out’ to form a constant background

of positive charge or ‘Jellium.’ The Coulomb interaction between the electrons is also

neglected, although of course the Pauli exclusion principle in not, and is taken into account

by the properties of the creation and annihilation operators (eq. 1.4.3). The Hamiltonian

of the system is just the sum of the single-electron energy levels:

Hfe =∑

k,σ

εkc†kσckσ (1.5.1)


In the zero temperature state of a free-electron solid, the electrons in the system occupy

the lowest energy single-electron states with energy εk. The maximum occupied energy

level is known as the Fermi energy, εF . This state can be written as:

|ψ〉 =∏

k′,σ

c†k′σ|0〉 (1.5.2)

where the product k′ runs over the single electron states with the lowest energy.

It is by no means obvious that this free-electron description will be adequate; in fact it

seems unlikely given that the interactions between electrons have been neglected. When

interactions are accounted for, the elementary excitations of the system are not excitations

of a single electron, but instead are collective excitations of all of the electrons. The reason

that Fermi-liquid theory is successful is that (close to the Fermi energy) these elementary

excitations are long-lived and behave like independent particles [35, 36]. Both momentum

and spin are good quantum numbers, and so the system of interacting electrons can be

described by a system of non-interacting excitations.

An important part of a microscopic understanding of superconductivity is the forma-

tion of pairs. In 1956, Cooper considered the behaviour of two electrons above a filled

Fermi sea [26]. The Fermi sea is assumed to have no effect on the two electrons other

than excluding them from the occupied levels. Cooper found that if there is an attractive

potential between the electrons, they will form a bound state with opposite momenta.

The bound state is formed no matter how weak the interaction between the electrons is,

in contrast to the situation without the presence of the Fermi sea where bound states are

only formed when the interaction is greater than some minimum value.

It is perhaps surprising that there should be an attractive interaction between two

charged electrons. The attraction arises from interaction of the electrons with the positive

ions in the solid [37]. Although there is always a repulsive Coulomb interaction between

two electrons, this is screened by the other electrons, and so the phonons (which describe

the quantised vibrational modes of the ions - see [34]) can provide an attraction that

outweighs the screened Coulomb repulsion. A simple picture of how this occurs is that

the first electron disturbs the lattice, which in turn affects the second electron, leading to

a net interaction between electrons.

In the BCS treatment of superconductivity, this interaction is assumed to be attractive

for electrons within a small energy region around the Fermi energy, |εk − εF | < ~ωc, and

1.6 BCS and BdG 13

zero otherwise.

The discovery that the Fermi sea was unstable to the formation of bound pairs paved

the way for Bardeen, Cooper and Schrieffer to formulate a microscopic theory of super-

conductivity, which we discuss in the next section.

1.6 BCS and BdG

The reduced BCS Hamiltonian [27] describes a system with interactions between paired

electrons:

HBCS =∑

k,σ

(εk − µ)c†kσckσ −∑

k,k′

Vk,k′c†k↑c

†−k↓c−k′↓ck′↑ (1.6.1)

The difficulty in solving this Hamiltonian comes from the interaction term; this term is

quartic rather than quadratic. Following [28], we eliminate this problem by making the

mean-field approximation. This means that each electron only feels an interaction due

to the average of all the other electrons around it. Each electron only interact with the

average of all the other electrons. This is a valid approximation when the operator c†k↑c†−k↓

does not deviate much from its average value, i.e. when c†k↑c†−k↓ − 〈c

†k↑c

†−k↓〉 ¿ 〈c

†k↑c

†−k↓〉.

We write c†k↑c†−k↓ = 〈c

†k↑c

†−k↓〉−(〈c†k↑c

†−k↓〉−c

†k↑c

†−k↓) and discard terms in the Hamiltonian

that are quadratic in (〈c†k↑c†−k↓〉 − c

†k↑c

†−k↓).

HBCS =∑

k,σ


k,k′

Vk,k′(〈c†k↑c†−k↓〉c−k′↓ck′↑ + c†k↑c

†−k↓〈c−k′↓ck′↑〉

−〈c†k↑c†−k↓〉〈c−k′↓ck′↑〉) (1.6.2)

We now introduce the pairing parameter ∆k =∑k′Vk,k′〈c−k′↓ck′↑〉 and write the mean-

field Hamiltonian as:

HBCS =∑

k,σ


k

∆∗kc−k↓ck↑ + c†k↑c†−k↓∆k − |∆∗k|2 (1.6.3)

We wish to diagonalise this Hamiltonian, i.e. to find operators γk,σ so that the Hamil-

tonian can be written∑k,σ

Ekγ†k,σγk,σ


We assume that these operators can be written as a linear combination of creation

and annihilation operators of time reversed single-electron states, i.e. γ−k,↓ = ukc−k↓ +

vk↑c†k↑, where uk, vk are unknown parameters. We require the elementary excitations of the

system to be Fermions, and so we demand that the operators γ obey the Fermionic anti-

commutation relations (eq. 1.4.3). This requirement implies that |uk|2 + |vk|2 = 1 (from

[γ−k↓, γ†−k↓]+ = 1) and also implies that γ†k↑ = ukck↑ − v∗kc

†−k↓ (from [γ−k↓, γ

†k↑]+ = 0).

We find the value uk, vk by noting that if the γ operators diagonalise the Hamiltonian,

then:

[∑

k

Ekγ†k,σγk,σ, γ−k,↓] = [HBCS , ukc−k↓ + vk↑c

†k↑] (1.6.4)

Equating these commutators leads directly to the equations for uk and vk:

|uk|2 =1

2

(1 +

εkEk

)

|vk|2 =1

2

(1− εk

Ek

)

Ek =√

(εk − µ)2 + |∆k|2 (1.6.5)

The operators γ create and destroy elementary excitations of the system. Each γ†k,σ

creates a combination of particle and hole, which we call a quasiparticle. The form of Ek

means that all quasiparticles have an excitation energy of at least ∆ so that there is a

gap in the energy spectrum. The energy of the whole system is given by the sum over

the energies of all the quasiparticles. From this we see that the lowest energy state, the

ground state, is the state with no quasiparticles present. We find this state by noting that

if there are no quasiparticles in the ground state, then it will be destroyed by any of the

quasiparticle annihilation operators γk,σ. The state defined by this property is the famous

BCS state:

|BCS〉 =∏

k

(uk + vkeiφc†k↑c

†−k↓)|0〉 (1.6.6)

where uk and vk are now considered real and we have explicitly included the phase of vk.

In the mean-field Hamiltonian 1.6.3, each electron sees an average pairing field ∆k. If

we calculate the behaviour of each electron in this field and find that averaging over all

electrons returns ∆k, the solution is self-consistent.

1.6 BCS and BdG 15

∆k =∑

k′

Vk,k′〈BCS|c−k′↓ck′↑|BCS〉

=∑

k′

Vk,k′uk′vk′eiφ

=∑

k′

Vk,k′∆k′

2Ek′(1.6.7)

If we make the approximation that the interaction potential Vk,k′ is constant for

|εk − εF | < ωc, then ∆k is also constant with this region, and zero outside. The gap

equation becomes:

1 =V

2

εF+ωc∑

εF−ωc

1√(εk − µ)2 + |∆|2

(1.6.8)

The chemical potential must be chosen so that the average number in the ground state

is correct N = 〈BCS|N |BCS〉. This leads to the equation:

N =1

2

εF+ωc∑

εF−ωc1− (εk − µ)√

(εk − µ)2 + |∆|2(1.6.9)

These two equations must be solved self-consistently. At zero temperature, this can

be done analytically (see Appendix B). The zero temperature value for the gap is:

∆ =~ωc

sinh(1/N (0)V )≈ 2~ωce

− 1N (0)V (1.6.10)

This expression for the zero-temperature gap reveals one of the reasons that a theory of

superconductivity proved so hard to formulate; this expression cannot be expanded about

V = 0 i.e. superconductivity can not be treated as a perturbation about the normal state.

We can introduce temperature dependence by noting that the quasiparticles are non-

interacting fermions. This means that they obey the statistics of a free Fermi gas. The

thermally averaged expectation values of the quasiparticle occupations are given by the

Fermi occupation factor, 〈γ†k, σγk, σ〉 = fk, σ, where fk, σ = 1/(eEk/KBT + 1). We have

a temperature dependent pairing parameter which is the thermally averaged expectation

value∑k′Vk,k′〈c−k′↓ck′↑〉. This leads to the two temperature self-consistency equations:


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.2 0.4 0.6 0.8 1 1.2 1.4T/Tc

Figure 1.2: The Temperature dependence of the superconducting gap ∆(T ) measured in

units of the zero-temperature gap ∆(0).

1 =V

2

εF+ωc∑

εF−ωc

tanh EkKBT

Ek(1.6.11)

and:

N =1

2

εF+ωc∑

εF−ωc1−

(εk − µ) tanh EkKBT

Ek(1.6.12)

Although these temperature-dependent self consistency equations cannot be solved

exactly, we can approximate the sum by an integral over ε and solve the equations numer-

ically (figure 1.2). We find that the gap drops to zero at a critical temperature, TC , which

when calculated is given by ∆(0) = 1.76TC .

Solving the self-consistency equations means that we have a value for ∆, and thus

for the parameters uk, vk. With the parameters of the ground state (eq. 1.6.6) found,

we examine this state in more detail. One of the most noticeable things about the BCS

state is that it does not contain an exact number of electrons, but is a superposition over

terms with different number eigenvalues. This is to be expected, as the Hamiltonian eq.

1.6.3 does not commute with the number operator. This is due to the fact that we have

made a mean field approximation; the Hamiltonian 1.6.1 does commute with the number

operator. Although the BCS state is not an exact number eigenstate, we have chosen

µ so that the expectation value of the number operator is correct, and furthermore, the

1.7 The Josephson Effect 17

fluctuation in the number operator (〈N2〉 − 〈N〉2)/〈N〉2 goes to zero as the size of the

superconductor goes to infinity, meaning that for bulk superconductors the mean field

treatment is adequate.

A number eigenstate |BCS,N〉 can be extracted from the BCS state (which we denote

|BCS, φ〉 for clarity) by integration over the phase φ. We can reverse the process by

summing over all N:

|BCS,N〉 =

2π∫

0

|BCS, φ〉e−iφNdφ

=

(∏

k

uk

)(S+BCS)

N |l, 0〉

|BCS, φ〉 =∑

N

|BCS,N〉eiφN (1.6.13)

Where the operator S+BCS =

∑k

vkukc†k↑c

†−k↓ increases the number of Cooper pairs on the

superconductor by one.

By integrating over φ, i.e. making the phase completely uncertain, we obtain an exact

number eigenstate. Conversely, by making the number completely uncertain, we regain

the usual BCS state where the phase is well-defined. While this is not important for

a bulk superconductor, where the number of Cooper pairs is of order 109 or more, this

uncertainty relation between the phase and number will come into play when we consider

superconductors of finite size.

1.7 The Josephson Effect

The Josephson effect [30] can be derived from a microscopic theory, which we shall sum-

marise here.

We begin with a Hamiltonian that describes two regions of superconductor, and the

tunnelling between them.

H = H1 + H2 + Ht (1.7.1)

H1 and H2 are the BCS Hamiltonians (eq. 1.6.1) for each superconductor, and Ht is

a Hamiltonian that describes the tunnelling of single electrons between two metals [38]:


Ht = −∑

k,q

tk,q(c†kc−q + c−kc

†q) (1.7.2)

The single electron levels in the two separate superconductors are labelled by k and q,

with a single electron tunnelling matrix element tk,q.

The tunnelling term is treated as a perturbation on H1 + H2. We assume H1 + H2

to be solved in the sense that we assume that the Hamiltonian for each superconductor

can be written in the form∑k,σ

Ekγ†k,σγk,σ where the γ’s are the quasiparticle operators.

We shall use the interaction picture to represent the time dependence. In the interaction

picture, the time dependence of the operators is given by the commutator with the un-

perturbed Hamiltonian (see eg. [39]). The time evolution of the annihilation operator ck↑

is determined by:

dck↑dt

=i

~[H1, ck↑] (1.7.3)

We wish to incorporate a voltage difference V across the junction. We do this by

replacing H1 with H1 − eV∑k,σ

c†k,σck,σ where∑k,σ

c†k,σck,σ measures the number of electrons

in superconductor 1. This is equivalent to setting the chemical potentials µ1, µ2 on each

side to the junction to differ by eV . If we represent the annihilation operator at zero

voltage by ck↑ and the operator at voltage V by ck↑, the time evolution of ck↑ is given by:

dck↑dt

=i

~[H1, ck↑]−

i2eV

~[N1, ck↑]

=i

~[H1, ck↑] +

i2eV

~ck↑

(1.7.4)

Substituting the trial solution ck↑ = ck↑eiφ2 (φ/2 is chosen so that the phase of a Cooper

pair will turn out to be φ).

eiφ2dck↑dt

+i

2

dφ

dtck↑ =

i

~ei

φ2 [H1, ck↑] +

i2eV

~ck↑

(1.7.5)

Comparing eq. 1.7.3 with eq. 1.7.4, we see that ck↑ = ck↑eiφ2 is a solution if:

1.7 The Josephson Effect 19

dφ

dt=

2eV

~(1.7.6)

Thus we find that the rate of change of the phase across the junction is directly

proportional to the voltage difference across it. This equation for the time dependence of

the phase across the junction is known as the AC Josephson effect [30]. We now move on

to discuss the current across the junction.

In the discussion of tunnelling current, we are considering the rate of change of the

number of electrons on each superconductor. To allow us to deal with this, we modify the

quasiparticle operators so that each operator creates or destroys exactly one electron on

each side.

γ†k↑ = ukc†k↑ − vkS+c−k↓

γ†−k↓ = ukc†−k↓ + vkS

+ck↑ (1.7.7)

The current is the rate of change of electrons on one side of the junction, the expectation

value of which is given by the commutator of the number operator with the Hamiltonian

[(H1 + H2), N1] =∑k,q

tk,q(c−kc†q − c†kc−q). Time dependent perturbation theory gives the

state of the system at time τ :

|ψ(t)〉 ≈

1− i

~

t∫

−∞

eητ′HIt(t

′)dt′

|ψ(0)〉 (1.7.8)

where HIt is the tunnelling Hamiltonian in the interaction picture. The expectation value

of the current operator can then be taken with the state at time t, using the form for

the quasiparticles given in eq. 1.7.7. The expectation value contains terms relating to

the tunnelling of quasiparticles. Rather than get bogged down in details, we shall merely

quote the term relating to the Cooper pair tunnelling [40, 41]:

I =2e|t|2~N1(0)N2(0)

∞∫

−∞

dε1dε2∆1∆2

ε1ε2

f(ε1)− f(ε2)ε1 − ε2 − eV12

sinφ

= IC sinφ (1.7.9)


where |t|2 is the square of the single-electron tunnelling matrix element, which we have

approximated as being independent of k, q and φ is the phase difference across the junction.

IC is known as the critical Josephson current, and is the maximum supercurrent the

junction can support. We see we have derived an expression proportional to eq. 1.3.13.

The equations eq. 1.7.6 and eq. 1.7.9 are the AC and DC Josephson effects respectively,

and together determine the behaviour of a Josephson junction. The current does not vanish

when the voltage is reduced to zero if the superconductors on either side of the junction

have different phases. Thus a current can be passed through the junction by a current

source, with no voltage appearing across the junction, up to the critical current IC .

Alternatively, if the voltage across the junction is held constant, eq. 1.7.6 gives the

evolution of phase as φ(t) = φ(0)+2 eV~ t, and the current I = IC sin(φ(0) + 2eV

~ t)oscillates

as a function of time with frequency 2eV~ .

In what follows we shall be interested in the pair tunnelling current. Rather than

continue to work with a Hamiltonian describing the single-electron tunnelling from which

pair tunnelling can be derived, we shall follow [42] and instead show that a pair tunnelling

Hamiltonian can be derived from a single electron tunnelling Hamiltonian.

1.8 The Pair Tunnelling Hamiltonian

The Hamiltonian 1.7.1 describes two superconductors and the tunnelling between them.

We wish to extract the tunnelling of Cooper pairs from this Hamiltonian, so we make a

canonical transformation to eliminate the first order terms, which correspond to single-

electron tunnelling.

Hcan = e−K(H1 + H2)eK = (H1 + H2) + [(H1 + H2), K]

+1

2[[(H1 + H2), K], K] + · · · (1.8.1)

First order terms are eliminated by setting Ht + [(H1 + H2), K] = 0. If we take the

expectation value of this condition with a complete set of states of H0, |n〉, we find:

〈n|K|m〉 = 〈n|Ht|m〉/(Em − En) (1.8.2)

From this we find the matrix elements of the second order term 12 [Ht, K], which are:

1.8 The Pair Tunnelling Hamiltonian 21

1

2

∑

p

〈n|Ht|p〉〈p|Ht|m〉(

1

Em − Ep− 1

Ep − En

)(1.8.3)

If a pair is transferred from one side of the junction to the other, the final state has

no quasiparticle excitations and therefore has the same energy as the initial state. In

the intermediate state |p〉, an electron has been removed from state −q on one side and

placed in state k on the other, creating a broken pair on either side. The energy of the

intermediate state is then given by Ep − En = Ek + Eq, where Ek/q are the quasiparticle

energies Ek/q =√

(εk/q − µ)2 +∆k/q:

∑

p

〈n|Ht|p〉〈p|Ht|m〉Ek + Eq

(1.8.4)

As states |n〉 and |m〉 are both states with no broken pairs and state |p〉 is a state

with an extra electron in state k and extra hole in state −q, with 〈p|Ht|m〉 the matrix

element for the creation of this state, then 〈n|Ht|p〉 is the matrix element for the transfer

of an electron from the state q to the state −k, and eq. 1.8.4 is the matrix element for the

transfer of a pair from q,−q on one side to k,−k:

Tk,q = − tk,qt−k,−qEk + Eq

= − |tk,q|2

Ek + Eq(1.8.5)

This derivation justifies the use of the pair tunnelling Hamiltonian used in the rest

of the thesis. Note that this matrix element is proportional to the square of the single-

electron tunnelling element, as is expected - the single-electron tunnelling element has to

act twice to transfer a pair across the junction.

We note that this tunnelling element is not independent of k, q. However we shall

approximate by an independent tunnelling element in the same way as we have approx-

imated the BCS interaction Vk,k′ by one that is k - independent, V . We now take the

Hamiltonian for the junction to be:

H = H1 + H2 + HT (1.8.6)


Where H1,2 are the Hamiltonians describing each superconductor (eq. 1.6.1) and the

pair tunnelling is described by HT :

HT = −T∑

k,q

c†kc†−kc−qcq + c†qc

†−qc−kck (1.8.7)

1.9 Small Superconductors

When superconductivity is generally discussed (eg. the BCS wavefunction), the standard

condensed matter assumption of infinite size is made. However, at very small sizes of

island, various effects come into play due to the departure from bulk behaviour.

Recent fabrication techniques have allowed the fabrication of extremely small particles

of aluminium, with radii of order 5nm. This allowed Ralph, Black and Tinkham [43] to

study the nature of superconductivity in ultrasmall grains. The grains were studied by

constructing a Single Electron Transistor (SET) by placing the grain between two leads,

and applying a gate voltage to the grain. The current through the grain can then be

studied as a function of the bias voltage between the two leads for different applied gate

voltages.

These particles were so small that the single-electron energy level spacing d = 1/N (εF )

needed to be regarded as discrete; in particular, d becomes comparable to the bulk super-

conducting gap, ∆. Furthermore, as the number of electrons on the island was well-defined,

a canonical rather than grand-canonical description was required. An extensive review of

the theory of finite size effects can be found in [44]

The spectroscopic gap of the grains in these experiments is particularly significant.

For grains with radii & 5nm, a spectroscopic gap could be observed when an even number

of electrons was present on an island, but could not be observed for islands with an odd

number of electrons. This rather striking parity effect is a clear indication that super-

conducting pairing correlations are taking place in these grains. For smaller grains, there

was no observed spectroscopic gap, even for particles with an even number of electrons.

Thus, these experiments show both the existence of superconducting pairing correlations

in small grains, and the disappearance of a spectroscopic gap as the size of the grains is

reduced even further and so a description of these grains requires an understanding of how

superconductivity changes from the bulk to grains that are so small that superconductivity

is suppressed.

1.9 Small Superconductors 23

1.9.1 The Parity Effect

One of the more startling effects in small metallic grains is the parity effect; despite the

fact that the number of electrons is of order 109, there are measurable differences in the

properties of grains with odd and even numbers of electrons. The main difference is

that the condensation energy of the two is different. This can be simply understood in

the bulk limit; in an odd grain there will always be one unpaired electron, and so the

difference in the condensation energies will be ∆, the energy of a single quasiparticle. In

fact, this energy difference is observed in Coulomb blockade phenomena [45, 46], where

the periodicity of the Coulomb oscillations in the gate voltage changes from e to 2e.

As well as a change in condensation energy, the pairing parameter (∆P , where the

P = 0, 1 indicates the parity of the grain) is also parity dependent. The bulk limit (first

order in d/∆) of the parity dependence has been studied by Janko et al. [47] and von

Delft et al. [48, 49].

Starting with the BCS Hamiltonian for a finite system:

H =∑

k,σ

εkc†k,σck,σ − V

∑

k,k′

c†k′↑c†−k′↓c−k↓ck↑ (1.9.1)

where the labels k,−k refer to generic time-reversed single electron states. In ultrasmall

grains, the number of electrons is well defined, and so the canonical distribution should

really be used. However, the grand-canonical distribution is used to approximate the

canonical distribution, and can in fact be regarded as a saddle-point approximation to

it. Von Delft et al. treated the problem using a parity-projected grand canonical distri-

bution. In this, the exact number of electrons present can vary, but the parity cannot.

This technique allows the use of mean-field techniques, but ensures that parity effects are

included.

ZGP (µ) ≡ TrG1

2[1± (−)NL ]e−β(HL−µNL) (1.9.2)

The procedure followed by von Delft et al. was to use the parity-projected partition

function (eq. 1.9.2) to calculate the finite-temperature properties of the superconducting

grain. Essentially, the grains are described by two gap equations, one for even grains and

one for odd grains.


1

λ= d

∑

q<ωc/d

1

2EP(1− 2fqP ) (1.9.3)

where the Fermi distribution function fqP = 〈γ†qγq〉P also depends on the parity of the

grains. The chemical potential must be chosen to give either an odd or even number

of electrons on the grain, which determines the Fermi function. At zero temperature,

fqp = δj0δP0 i.e. there are no quasiparticles in the even case and only one, at the chemical

potential, in the odd case. Equation 1.9.3 can be solved numerically at finite temperatures,

but at T = 0 it can be evaluated exactly:

1

λd=

ωc∫

0

dω

Eeven,ωtanh(

πEe,ωd

) (1.9.4)

for even, and:

1

λd=

ωc∫

0

dω

Eodd,ω

1

tanh(πEodd,ω

d )− d

Eodd,ω(1.9.5)

for odd. In the limit where the level spacing d is much smaller than the bulk gap, ∆, the

form for the parity-projected pairing parameters is:

∆even(d, T = 0) = ∆(1−√

∆/de−2π∆/d) (1.9.6)

∆odd(d, T = 0) = ∆− d/2 (1.9.7)

We have two different values for the pairing parameter ∆P depending on whether there

are an odd or even number of electrons on the island, which also leads to two different

values for the condensation energy. We find that the forms for ∆P described in eqs.

1.9.6, 1.9.7 lead to the intuitive result that, in the limit d/∆ → 0, the difference in the

condensation energies is equal to the energy caused by having one unpaired quasiparticle.

We also see from these results that the pairing parameter for an odd grain is always

smaller than for an even grain, implying that there is a size region in which an even grain is

superconducting, but an odd grain is not. In fact, as the level spacing becomes larger, the

mean-field approach is no longer adequate. Nevertheless, these results serve to show that

there can be a significant difference in the pairing parameters and condensation energies

of grains with odd or even number of electrons.


Whilst these results were originally derived to treat ultrasmall superconducting grains,

they are present at larger grain sizes, just smaller in magnitude. In terms of supercon-

ducting qubits, the parity effect means that there may be fluctuations in the values of EC

and EJ between the values for odd and even grains. Some possible effects of this kind of

fluctuation are discussed in section 2.6.2.

Although a range of techniques for treating ultrasmall superconducting grains have

been developed, there is in fact an exact solution of the BCS Hamiltonian for finite islands,

known as the Richardson solution.

1.9.2 The Richardson Exact Solution

Recently it has come to the attention of the condensed matter community that the reduced

BCS Hamiltonian for a finite superconductor:

H =∑

k,σ

εkc†k,σck,σ − V

∑

k,k′

c†k′↑c†−k′↓c−k↓ck↑ (1.9.8)

has an exact solution, which reproduces the BCS results in the bulk limit [50, 51]. This

solution was discovered by Richardson as far back as 1963 in the context of nuclear physics

[52]. The existence of an exact solution to this Hamiltonian came as a great surprise to

the community and we will take a while to discuss it.

We can write the above Hamiltonian in terms of the Nambu spins [53]. The creation,

annihilation and number operators for each electron pair k are represented as spin half

operators. These are defined as:

σZk =1

2(c†k↑ck↑ + c†−k↓c−k↓ − 1)

σ+k = c†k↑c†−k↓

σ−k = c−k↓ck↑ (1.9.9)

(1.9.10)

The labels k are generic electron spin labels, rather than wavevectors. We also note

that σZk = σ+k σ−k − 1/2, and write the reduced BCS Hamiltonian:

∑

k,k′

(2εkδk,k′ − V )σ+k σ′k (1.9.11)


Our task is to diagonalise this Hamiltonian. We write the diagonalised Hamiltonian

and eigenstates in terms of the operators B+Jν , and our job is now to find these operators.

H =

N∑

ν=1

EJνB+JνB

−Jν

|ψN 〉 =N∏

ν=1

B+Jν |0〉 (1.9.12)

Note that these eigenstates of the Hamiltonian are number eigenstates, and there are

many different eigenstates that all have the same number of Cooper pair on the island, N .

We shall see that the form for the operators B+Jν that diagonalises the Hamiltonian is:

B+Jν =

∑

k

σ+k2εk − EJν

(1.9.13)

Our task is now to find the parameters EJν . To do this we note that, if the form for the

eigenstates is correct, then H

(N∏ν=1

B+Jν |0〉

)=

N∑ν=1

EJν

(N∏ν=1

B+Jν |0〉

)= ε

(N∏ν=1

B+Jν |0〉

)

HN∏

ν=1

B+Jν |0〉 =

N∑

ν=1

EJν

N∏

ν=1

B+Jν |0〉

[H,N∏

ν=1

B+Jν ] |0〉+

N∏

ν=1

B+JνH|0〉 =

N∑

ν=1

EJν

N∏

ν=1

B+Jν |0〉

[H,N∏

ν=1

B+Jν ] |0〉 = ε

N∏

ν=1

B+Jν |0〉 (1.9.14)

We need to calculate the commutator of the Hamiltonian with the product over all the

B+J operators. Calculating the commutator with just a single operator:

[H, B+Jν ] = EJνB

+Jν + S+

(1− V

∑

k

1

2εk − EJν

)

+S+V∑

k

2σ+k σ−k

2εk − EJν(1.9.15)

Following [54] we notice that this is a generalisation of the boson case. Our Nambu spin

operators (sometimes referred to as hard-core boson operators) differ from true bosons by

one of the three commutation relations. In true bosons, we find [bk, b†k] = 1, whereas for


our hard-core bosons, we have [σ−k , σ+k ] = 1− 2σ+k σ

−k = −2σZk . If we were discussing true

bosons, then the commutator above (1.9.15 would only consist of the first two terms. We

could eliminate the second term by setting the condition for the EJν parameters:

0 =

(1− V

∑

k

1

2εk − EJν

)

(1.9.16)

This would mean that the commutator would become [H, B+Jν ] = EJνB

+Jν and therefore

that

HN∏

ν=1

B+Jν |0〉 = [H,

N∏

ν=1

B+Jν ] |0〉

=∑

µ

∏

ν 6=µ[H, B+

Jν ] |0〉

=

(∑

µ

EJµ

)N∏

ν=1

B+Jν |0〉 (1.9.17)

That is, in the true boson case, choosing the parameters EJν so that the condition

1.9.16 is satisfied means that the operators B+Jν diagonalise the Hamiltonian.

We, however, wish to consider the full Nambu spin case. This means that the final

term in 1.9.15 is not zero, and must be taken into account. Returning to the commutator

of the Hamiltonian with the product over B+Jν operators:

[H,N∏

ν=1

B+Jν ] =

N∑

ν=1

ν−1∏

η=1

B+Jη

[H, B+

Jν ]

N∏

µ=ν+1

B+Jµ

(1.9.18)

Examining our expression (eq. 1.9.15) for [H, B+Jν ], we see that the first two terms

(EJνB+Jν and S+

(1− V ∑

k

12εk−EJ

)) commute with the products over the remaining B+

J

operators, and so we can simply move these terms past the products in eq. 1.9.20. We

need to commute the third term of eq. 1.9.15 past the products as well. To do this, we

first need to know its commutator with the individual operators:


[S+V∑

k

2σ+k σ−k

2εk − EJν, B+

Jµ] = S+2V∑

k

1

2εk − EJνσ+k

2εk − EJµ

=1

EJν − EJµ

(σ+k

2εk − EJν− σ+k

2εk − EJµ

)

= S+2VB+Jν −B+

Jµ

EJν − EJµ(1.9.19)

We can now move the third term of eq. 1.9.15 past the products:

N∑

ν=1

ν−1∏

η=1

B+Jη

(S+V

∑

k

2σ+k σ−k

2εk − EJν

)N∏

µ=ν+1

B+Jµ =

N∑

ν=1

ν−1∏

η=1

B+Jη

N∑

µ=ν+1

µ−1∏

ρ=ν+1

B+Jρ

(S+2V

B+Jν −B+

Jµ

EJν − EJµ

)ν−1∏

τ=µ+1

B+Jτ

(1.9.20)

All the B+Jν and S+ operators commute, so we can collect up all the products, re-order

the sums in the second term and then interchange the dummy indices µ, ν in the sums in

the first term:

=N∑

ν=1

N∑

µ=ν+1

2S+VN∏η 6=µ

B+Jη

EJν − EJµ−

N∑

ν=1

N∑

µ=ν+1

2S+VN∏η 6=ν

B+Jη

EJν − EJµ

=N∑

ν=1

N∑

µ=ν+1

2S+VN∏η 6=µ

B+Jη

EJν − EJµ−

N∑

µ=1

µ−1∑

ν=1

2S+VN∏η 6=ν

B+Jη

EJν − EJµ

=N∑

µ=1

N∑

ν=ν+1

2S+VN∏η 6=ν

B+Jη

EJµ − EJν−

N∑

µ=1

µ−1∑

ν=1

2S+VN∏η 6=ν

B+Jη

EJν − EJµ

=N∑

µ,ν 6=µ

2S+VN∏η 6=ν

B+Jη

EJµ − EJν(1.9.21)

We now have the third term of eq. 1.9.20. Writing out the commutator of the Hamil-

tonian with the product over B+J operators in full, we get:


[H,N∏

ν=1

B+Jν ] =

N∑

ν=1

EJν

N∏

ν=1

B+Jν

+N∑

ν=1

S+

(1− V

∑

k

1

2εk − EJν

)N∏

η 6=νB+Jη

+N∑

ν=1

S+

N∑

µ6=ν

2V

EJµ − EJν

N∏

η 6=νB+Jη (1.9.22)

From eq. 1.9.14, we know that the B+J operators diagonalise the Hamiltonian if the

second two terms above vanish. That is, we have the correct diagonal form for the Hamil-

tonian if, for all ν:

1− V∑

k

1

2εk − EJν+∑

µ6=ν

2V

EJµ − EJν= 0 (1.9.23)

Chapter 2

The Superconducting Charge

Qubit

The progress in constructing superconducting qubits to date is reviewed. A phenomenologi-

cal theoretical description of superconducting charge qubits is formulated, and the dynamics

of such systems examined. We look at the effect of noise in the qubit parameters on the

evolution of the system, and describe new work on how discrete noise in the qubit param-

eters affects the system.

In section 1.1 we mentioned briefly why small superconducting circuits might be well

suited for use as quantum bits. As with all solid-state realisations, each qubit is well-

defined as each is physically separated, and techniques of circuit fabrication suggest that

large numbers of interacting qubits could be manufactured. In addition, superconducting

qubits have the advantage over other solid-state implementations in that the supercon-

ducting gap protects the condensate from the environment to some extent.

Whilst these arguments are suggestive, we require a physical theory of superconducting

circuits in order to describe how they can be used as qubits, and where any possible

problems arise.

2.1 Experimental Progress

Although far behind such implementations as ion trap [11, 16, 17] and NMR-based [55,

56, 57, 18] quantum computing realisations, superconducting charge qubits have developed

31

32 The Superconducting Charge Qubit

quickly in recent years, as have other types of superconducting qubit. As an indication of

the rapid growth of this field, superconducting charge qubits were first suggested in 1997

by Shnirman et al. [58], and by 2001 coherent oscillations were being observed [59].

A superconducting charge qubit consists of a small island of superconducting material

(usually aluminium) connected via a Josephson tunnel junction to a superconducting

reservoir. The island is small enough (the capacitance low enough) that the charging

energy of a single Cooper pair becomes significant - it costs an appreciable amount of

energy to add extra Cooper pairs to the island. The two qubit states |0〉 and |1〉 are the

states where there are 0 or 1 excess Cooper pairs on the island. The separation of the

two charging energy states is controlled by a gate voltage, which gives a diagonal term in

the Hamiltonian (σZ in the spin-half notation where |0〉 = ( 01 ), |1〉 = ( 10 ) and σX , σY , σZ

are the Pauli matrices). The Josephson coupling to the reservoir allows Cooper pairs to

tunnel on and off the island, and provides an off-diagonal term (σX).

Oscillations between the two charge states were observed in 1999 by Nakamura et

al. [59]. These oscillations are an indication that the system is behaving quantum me-

chanically and can be used to infer the existence of a superposition of charge states at a

midpoint of the oscillation. Similar oscillations have since been seen by several groups in

various superconducting systems [60, 61, 62], and spectroscopic measurements made on a

range of others including those based on flux [63, 64] and phase [65, 66]

Coupling between two qubits can be by use of capacitive or Josephson coupling between

the superconducting islands, or through a common (or ‘bus’) inductance [67] or LC-

oscillator circuit [68].

Two-qubit oscillations have now been observed in charge qubits [69] and there is spec-

troscopic evidence for coupling between superconducting flux qubits separated by a dis-

tance of 0.7mm [70]. Recently, the first implementation of a controlled not gate (CNOT)

was performed between two coupled charge qubits [71]

As well as demonstrations of basic quantum behaviour, experiments illustrating the

possibility of more sophisticated manipulations such as spin-echo have been performed

[72]. This experiment in particular gives a clear indication of phase coherent evolution,

and suggests that NMR-style techniques to reduce the effects of decoherence and noise

may be viable.

If superconducting qubits are to be used in quantum computers, they must also satisfy

the final DiVincenzo criteria, which requires that the qubits be individually measurable.

2.2 The Josephson Equations for Small Circuits 33

In particular, in most of the current experiments, an ensemble measurement is made of the

state of the qubit. The Nakamura experiments measure the charge on the qubit by placing

a readout lead close to the qubit island. If the qubit is in the state with an extra Cooper

pair on the island, the Cooper pair will decay through quasiparticle tunnelling to the

lead. Individual Cooper pair decays cannot be detected in this manner, so each oscillation

is repeated many times during one measurement so that there are enough quasiparticle

decays to be detected as a current. The current on the readout lead is then proportional to

the probability of the state |1〉 being occupied. A further problem with this measurement

scheme is that it is ‘always on’; the tunnelling to the leads is present during the evolution

of the qubit, and relies on the fact that the quasiparticle decay time is much longer than

the oscillation time to ensure that the evolution is unaffected. Whilst this is adequate for

demonstrations of coherent oscillations, over a long period of evolution a coupling to a

measurement device will cause decoherence. Ideally, we would like to be able to decouple

(or ‘turn off’) the measurement whilst the quantum evolution of the qubit takes place.

A quantum computer will require single-shot measurements, i.e. measurements re-

turning 1 or 0, rather than a probability. In theory these sorts of measurements could be

done using a single electron transistor (SET) [58, 73]. In such a scheme, the presence of

a Cooper pair on the qubit modifies the voltage on the central island of the transistor,

and allows current to flow. A further advantage of this readout scheme is that if no bias

voltage is applied to the SET, no measurement takes place, and thus the qubit dynamics

are not affected.

Single shot measurements are being performed in ‘quantronium’ qubit systems [60],

and experiments using SETs performed [61], but as yet SETs have not been used to

perform single-shot experiments on charge qubits.

2.2 The Josephson Equations for Small Circuits

Figure 2.1 shows a simple schematic of a circuit that could be used as a charge qubit. A

small island of superconductor is connected via a capacitive Josephson tunnel junction to

a superconducting reservoir. If the island was a bulk region, then the evolution of the

phase and voltage across the Junction would be adequately described by the Josephson

equations ( 1.7.6, 1.7.9 ). Although the Josephson effect is an intrinsically quantum effect,

derived from the quantum behaviour of electrons, the equations of motion 1.7.6, 1.7.9


q

-q

C IC

Figure 2.1: A simple circuit diagram for a charge qubit consisting of two regions of super-

conductor connected by a Josephson junction with critical current IC whose capacitance

is represented by a capacitor of capacitance C

are classical - the superconductors on either side of the junction are considered infinitely

large, and the capacitance of the junction can be ignored. The superconductors can be

in a superposition of different number eigenstates and therefore the classical equations

for phase are adequate. As the size of the island becomes small, the capacitance of the

junction becomes small and the charging energy large. A large charging energy means

that states with too many or too few Cooper pairs are unfavourable. We can no longer

sum over all number states, and so the phase is not a well-defined variable, and we need

to take into account its quantum mechanical nature. In particular, we need to account for

the conjugate nature of the phase and number operators.

As has been stated, the Josephson equations describe a large junction that can be

adequately described classically. To describe non-classical effects, there are two possible

approaches we could take: a microscopic, or ‘bottom up’ approach, and a semiclassical, ‘top

down’ approach. The microscopic approach is to return to the full quantum description

of the superconductors, and try to re-derive a Josephson equation without making the

assumption of infinite size made in 1.7.

In this chapter, and in chapters 3 and 4, we consider the semiclassical approach. The

motivation for this is the fact that the classical Josephson equations describe very well the

behaviour of large Josephson junctions, and that effects due to the finite size of the circuit

will be a modification to these. In the semiclassical approach, we ignore the quantum


origin of the Josephson equations, and treat them as purely classical equations of motion.

We then quantise these ‘classical’ equations in the same way as we would for any other

classical equations.

As this semiclassical treatment, and its differences from the microscopic theory will

feature heavily in later chapters of this thesis, we shall spend some time on quantising the

simplest quantum Josephson circuit, shown in figure 2.1. Later we shall show (chapter 8)

how these semiclassical results can be derived as a limiting form of a microscopic theory.

2.2.1 The Classical Lagrangian

We would like a quantum description of the circuit shown in figure 2.1, which consists

of two regions of superconductor connected by a Josephson junction of critical current

IC and capacitance C. The charge on the capacitor is equal to the excess charge on the

island, q, and so the voltage across the capacitor is given by V = q/C.

The first stage in quantising any classical equations of motion is to write the equations

in terms of Lagrangian mechanics (see eg. [74]). That is, we need to find a classical

Lagrangian that leads to the equations of motion we wish to quantise. The Josephson

equations derived in 1.7 are:

I = IC sinφ (2.2.1)

dφ

dt=

2eV

~(2.2.2)

We note that the voltage across the junction is given by V = q/C, and that the current

across the junction is the rate of change of charge on the island. This leads to an equation

of motion for the phase difference:

d2φ

dt2=

2e

~CIC sinφ (2.2.3)

We need to find a Lagrangian that corresponds to this equation. The Euler - Lagrange

equations of the desired Lagrangian give eq. 2.2.3. Using educated guesswork, we try the

Lagrangian:

L =1

2

~2C4e2

(dφ

dt

)2

+~IC2e

cosφ (2.2.4)


Putting this form for the Lagrangian into the Euler-Lagrange equations for φ, ∂∂t

(∂L∂φ

)−

∂L∂φ = 0 indeed returns eq. 2.2.3.

2.2.2 The Classical Hamiltonian and the Conjugate Variables

If we choose φ to be the canonical position variable, then we find the canonical momentum,

π to be:

π =∂L∂φ

=~2C4e2

φ (2.2.5)

From eqs. 2.2.4 and 2.2.5 we get the classical Hamiltonian:

H = πφ− L

= EC1

~2π2 − EJ cosφ (2.2.6)

Let us examine this Hamiltonian and the conjugate variables further. We know the

physical meaning of the canonical position, φ, is the phase difference between the two

superconducting condensates on either side of the junction. From eq. 2.2.2, we find that

the canonical momentum π can be rewritten:

π =~2C4e2

φ =~ CV2e

=~q2e

= ~N (2.2.7)

The phase and the number of Cooper pairs are conjugate variables. EC = 4e2

2C is the

charging energy of a single Cooper pair, i.e. the energy required to increase the number

of Cooper pairs on the island by one. EJ = ~IC2e is the Josephson tunnelling energy.

We can calculate the equations of motion for φ and π (or N) from the Hamiltonian

directly, just as we did from the Lagrangian. However, as we intend to quantise this

system, let us calculate the equations of motion in another way - in terms of Poisson

brackets.


Poisson brackets in classical mechanics are analogous to commutators in quantum

mechanics. The Poisson bracket,A,B, of two functions of the canonical positions and

momentums of the system, A,B, is defined as:

A,B =∑

i

(∂A

∂qi

∂B

∂πi− ∂A

∂πi

∂B

∂qi

)(2.2.8)

If we assume that A,B have no explicit time dependence, then the equations of motion

can be derived simply using:

dA

dt= A,H (2.2.9)

Substituting A = φ,A = π and the Hamiltonian in eq. 2.2.6 into eq. 2.2.9 gives back

the Josephson relations (2.2.1,2.2.2) and confirms that we have the correct Hamiltonian

for the system.

2.2.3 Quantising the Classical Josephson Equations

Now we have the correct classical Hamiltonian that described the circuit, we obtain a

quantum description by promoting the conjugate variables (φ, π) to be operators (φ, π).

H = ECN2 − EJ cos φ (2.2.10)

As these operators are conjugate, they have the commutation relation [φ, π] = i~.

Using the number operator N = π/~, we have the relation [φ, N ] = i (see also [75, 76]).

We describe the state of the system by a wavefunction ψ(φ) which is a complex function

of the canonical position φ. The operators are represented by differential operators that

give the correct commutation relations:

φ = φ

N = id

dφ(2.2.11)

This leads to a differential form for the Hamiltonian:


H = EC

(id

dφ

)2

− EJ cosφ (2.2.12)

Having found our quantum Hamiltonian, we note that we can recover the classical

equations of motion for the circuit through the use of Ehrenfest’s theorem, which gives

the expectation value of the rate of change of an operator O in terms of its commutation

relation with the Hamiltonian:

i~

⟨dO

dt

⟩=⟨[H, O]

⟩(2.2.13)

This leads to a set of equations of motion for the expectation values of current and

phase difference across the junction:

〈I〉 = 2e

⟨dN

dt

⟩=

2e

~EJ〈sin φ〉

⟨dφ

dt

⟩=

2e

~2e

C〈N〉 (2.2.14)

where 2eC 〈N〉 is identified as the voltage across the capacitor, V .

Having found the correct classical Hamiltonian, quantised it, and found a convenient

differential form for the operators, we go on to investigate how a two-level system can be

extracted.

2.3 Tuneable Josephson Energy

In the previous section we formulated a quantum mechanical description of our super-

conducting circuit. For this system to be used as a qubit, it must behave as a two level

system, and we must be able to adjust the relative charging and coupling energies in order

to perform controlled operations.

In order to be able adjust the charging and coupling energies EC and EJ , we replace

the circuit shown in figure 2.1 with a slightly more complicated one (figure 2.2). This

circuit can be described by a Hamiltonian with a gate voltage dependent charging energy,

and a Josephson energy that depends on the flux. We shall leave the dependence on the

gate voltage until chapter 3, where it will be discussed in some detail. For the present, we

2.3 Tuneable Josephson Energy 39

Vg

Island

Reservoir

Cg

IC ICC

Figure 2.2: A simple circuit diagram for a charge qubit with a gate voltage applied and

the single Josephson junction replaced by a magnetic flux dependent double-junction.

shall examine how a double-junction through which a magnetic flux is passed can act like

a single junction with a controllable Josephson energy.

The line integral of the vector potential A around a closed contour is equal to the flux

passing through that contour:

Φ =

∫

c

A.ds (2.3.1)

We write our integral around the double junction 2.3 as a sum of the contribution of

two paths ca and cb. Note that the contour cb is in the opposite sense to contour ca and

thus its contribution is negative.

Φ =

∫

ca

A.ds−∫

cb

A.ds (2.3.2)

The contour passes through the centre of the superconducting electrodes and crosses

the junction. We can make the junctions infinitely thin, so that their contribution to the

integral is essentially zero. The path goes through the centre of the electrodes, where the

supercurrent density given in eq. 1.3.6 is zero, i.e. , where (∇φ − 2πA/Φ0) = 0, with φ

the phase of the order parameter ψ = |ψ|eiφ. We can write the integral as:


1 2

ca

cb

Figure 2.3: A double junction consisting of a ring of superconductor broken by two junction

and threaded by a flux Φ. The two paths from point 1 to point 2 are labelled ca and cb.

Φ =

∫

ca

Φ0

2π∇φ.ds−

∫

cb

Φ0

2π∇φ.ds (2.3.3)

Doing the integrals, we find that the difference in phase acquired by going from 1 to

2 by following path ca instead of cb is determined by the magnetic flux threading the two

paths:

2πΦ

Φ0= ∆φca −∆φcb (2.3.4)

We can now calculate the current from point 1 to point 2, passing through both paths.

I = Ica sin

(φca +

πΦ

Φ0

)+ Icb sin

(φca −

πΦ

Φ0

)(2.3.5)

If we assume the junctions are equal, Ica = Icb, then:

I = 2Ica sin

(∆φca +

πΦ

Φ0

)cos

(πΦ

Φ0

)(2.3.6)

If we consider the double junction as a whole, we see that we have a critical Josephson

current that is dependent on the magnetic flux through the loop Ic = 2Ica cos(πΦΦ0

).

2.4 The Two-Level System 41

Substituting this into the expression for Josephson energy and assuming the inductance of

the circuit is negligible, we have a Josephson energy that can be adjusted by an external

control parameter, the magnetic field.

EJ(Φ) =~Icae

cos

(πΦ

Φ0

)(2.3.7)

Note that while we have derived the flux dependence of the current using Ginzburg

- Landau theory for a particular junction configuration, this dependence is essentially

nothing more than the Aharonov-Bohm effect [77], and as such is a much more general

phenomena than may be suggested by the above derivation. The phase of an electron

wavefunction is changed by:

exp

(ie

~c

∫A.ds

)(2.3.8)

The charge carrier in a superconductor is a Cooper pair, and so the phase changes by

twice this amount.

This dependence of the behaviour of the qubit on the applied magnetic field has been

impressively demonstrated in the experiments of Nakamura et al. [59, 69].

2.4 The Two-Level System

The Hamiltonian of the system as a function of gate voltage and magnetic flux can be

written:

H = EC

(id

dφ− ng

)2

− EJ(Φ) cosφ (2.4.1)

Where Φ is the magnetic flux through the double-junction, and we have redefined

the origin of phase compared to 2.3.7 so that the cosine term has no flux dependence.

For now, we note that the interaction energy between a charge −2eN and an external

voltage Vg will be of the form −2eNVg = −EcNng and assume that a gate voltage can be

incorporated into the Hamiltonian as above with −2eng = CgVg defining a ‘gate charge’;

this shall discussed in detail in chapter 3.


Adjusting the gate voltage allows us to control the ‘zero position’ of charge on the

circuit; the energy depends on how far N is from ng. In effect, ng determines at which

value of N the island will be electrically neutral.

In a superconducting charge qubit, EC À EJ and (for most values of ng) the charge

states are a good approximation to the energy eigenstates.

If we consider an island with a fixed number of Cooper pairs N , then we see that the

charging energy EC(N−ng)2 varies quadratically with ng; a plot of EC against ng produces

a parabola centred on N . This is only for a fixed number of Cooper pairs, however, and

the Josephson junction allows Cooper pairs to tunnel on and off the island. If ng is set

close to N , then the charging energy is small and the state |N〉 will be the ground state.

As ng increases, however, the charging energy goes up, until the state |N +1〉 has a lower

energy and becomes the ground state. At this point, if the system is allowed to relax, a

Cooper pair will tunnel from the reservoir to the island.

This tunnelling means that the charging energy of the ground state is a periodic func-

tion of ng. If we plot the charging energy for all states |N〉 against ng (figure 2.4), the

charging energy for each is a parabola centred on ng = N , and the parabolas cross at

half-integer values of ng, where the states N and N + 1 are degenerate.

The effect of the Josephson energy on this picture is to break the degeneracy of the

charge states N,N + 1 at the points ng = N + 1/2 (figure 2.5).

Around these points, the two lowest energy eigenstates will be the symmetric and anti-

symmetric combinations of the N and N + 1 states (figure 2.6), and the other states will

be separated in energy by EC . If EC is large compared to EJ , then only the N , N + 1

states will be occupied and we can regard the Cooper pair box as a two level system. That

is, if we write the Hamiltonian 2.2.12 in terms of the number eigenstates:

H = EC∑

N

|N〉(N − ng)2〈N | −EJ2

∑

N

(|N〉〈N + 1|+ |N + 1〉〈N |) (2.4.2)

We keep only the states N , N + 1 as the other are all suppressed by the charging

energy and obtain a two-level system with the Hamiltonian:

H = EC(1/2− ng) (|1〉〈1| − |0〉〈0|)

+EJ2

(|1〉〈0|+ |0〉〈1|) (2.4.3)


0

0.2

0.4

0.6

0.8

1

Ec

1 2ng

Figure 2.4: The charging energy as a periodic function of ng. The plot is a series of

parabolas centred on ng = N + 0, ng = N + 1, etc. At half-integer values of ng, the

parabolas cross and the ground state changes from N to N + 1.


0

5

10

15

20

3 4 5 6

Figure 2.5: The qubit energy levels as a periodic function of ng. The degeneracy of the

charging energies at half-integer values of ng is broken by the tunnelling energy EJ . The

solid line represents the ground state energy, whilst the first and second excited states are

represented by the dashed and dotted lines respectively.


–1

–0.5

0

0.5

1

Ec

0.2 0.4 0.6 0.8 1ng

Figure 2.6: The avoided crossing at half-integer values of ng. The degeneracy of the

charge eigenstates (dotted lines) is broken by the Josephson tunnelling, and the energy

eigenstates become the symmetric and anti-symmetric combinations of the charge states

(solid lines).


Where we have discarded a term proportional to the identity. We can conveniently

write this in matrix form, or in terms of the Pauli matrices:

H =

Ech/2 −EJ/2−EJ/2 −Ech/2

=Ech2

σZ +EJ2σX

Where Ech = EC(1 − 2ng) is the effective charging energy, as determined by the gate

voltage.

The eigenvalues E± and eigenvectors |v±〉 of this Hamiltonian are:

E± = ±1

2

√E2ch + E2

J

|v±〉 =

(1

Ech−2E±EJ

)(2.4.4)

Our gate voltage allows us to manipulate the Hamiltonian and change the eigenstates

and eigenvalues. Two cases of particular interest are the case where we set Ech À EJ ,

and the case where Ech ¿ EJ (but EC À EJ , so the two-level approximation is still

valid). If the charging energy EC À EJ , then as long as we set ng away from 1/2, then

the Hamiltonian will be diagonal and the eigenstates will be the charge states(10

)and

(01

)with eigenvalues ±Ech. If, however, we set ng = 1/2, then the two charge states are

degenerate. This degeneracy is broken by the Josephson coupling, and the eigenstates are

1√2

(11

)and 1√

2

(1−1)with eigenvalues ±EJ/2.

This ability to manipulate the Hamiltonian by adjusting the gate voltage (and also

the flux through the double junction, discussed later in section 2.3) allows us to perform

controlled quantum operations, or quantum logic gates, by applying a series of pules to

the gate capacitor. We shall discuss an example of this in section 2.5.

2.5 Qubit Dynamics and the NOT Gate

From a quantum algorithmic point of view, a not gate is trivial. This is a gate that takes

a qubit that is initially in the state |0〉 =(01

)and returns the state |1〉 =

(10

)(figure 2.7).

This is perhaps the simplest quantum gate, acting on single qubit alone, and from a

algorithmic perspective merits no more than a mention. However, the notion of a ‘gate’ is

2.5 Qubit Dynamics and the NOT Gate 47

1

0

0

1

Figure 2.7: A not gate takes a state |0〉 and returns a state |1〉, and vice versa.

useful precisely because it is an abstract concept independent of the physics of the process

that implements it. In order to even consider discussing ‘gates,’ we must understand the

dynamics of our system thoroughly. Only when we have a complete understanding of the

physics can we ignore it!

In terms of the dynamics of our two-level system, implementing a not gate is not

as trivial as figure 2.7 would suggest, and is certainly challenging experimentally, as the

coherent quantum oscillation of a qubit must be observed. Perhaps the most famous

experimental paper on superconducting charge qubits is the 1999 paper by Nakamura et

al. [59] in which the coherent evolution of a charge qubit was first demonstrated. The

procedure described in this paper is essentially that required to perform a not gate.

In order to illustrate the difficulties and subtleties involved in even the simplest quan-

tum operation, we shall describe this experiment in some detail.

The operation can be broken down into three separate parts: Initialisation, evolution

and readout. It will turn out that the complete operation can be achieved by applying a

single voltage pulse to the capacitor.

In the first part of the operation, initialisation, we wish to set the state of the system to

be a charge eigenstate. This is done by adjusting the parameters of the Hamiltonian 2.4.3

so that one of the charge eigenstates, |0〉 is the ground state. We do this by setting the

gate charge far from the degeneracy point, ng = 1/2. If EC À EJ then the Hamiltonian

will be essentially diagonal in the charge basis and the state |0〉 will be the ground state.

At zero temperature, if we wait long enough the system will decay into this state. At low

but finite temperature, the system will decay to an equilibrium state that is close to the

ground state. A true NOT gate must invert any general input state, but for demonstration

purposes it is easier to use a charge eigenstate as an input state.

We now wish to control the evolution of the system so that the state of the system

oscillates between the states |0〉 and |1〉. We do this by applying a voltage pulse that takes


the gate charge to ng = 1/2 for the period t0 < t < t0 +∆t. If the gate voltage is ramped

rapidly enough, we can use the sudden approximation and assume that the system remains

in the state |0〉 at the start of the pulse.

The time evolution of a quantum system is governed by a unitary operator that follows

directly from the Schrodinger equation:

|ψ(t)〉 = exp

− i

~T

t∫

t0

H(t′)dt′

|ψ(t0)〉 (2.5.1)

If the Hamiltonian is constant, we can ignore the time-ordering operator T and do the

integral:

|ψ(t)〉 = e−i~ H(t−t0)|ψ(t0)〉 (2.5.2)

With ng set to 1/2, the diagonal parts of the Hamiltonian vanish, and the eigenstates

are the states 1√2

(11

)and 1√

2

(1−1). We write the state of the system at t0 in terms of these

states. Each of these eigenstates then picks up a phase factor when acted on by the time

evolution operator. This phase factor depends on the energy eigenvalues which in this

case are ±EJ/2:

|ψ(t)〉 = e−i~ H(t−t0)

((1

1

)−(

1

−1

))

= e+i~EJ2(t−t0)

(1

1

)− e− i

~EJ2(t−t0)

(1

−1

)

=

(e+

i~EJ2(t−t0) − e− i

~EJ2(t−t0)

e+i~EJ2(t−t0) + e−

i~EJ2(t−t0)

)(2.5.3)

At the end of a pulse, at time t = t0 +∆t, the gate voltage is again moved away from

ng = 1/2, and the Hamiltonian is again diagonal in the charge basis.

The final stage of the operation is the readout stage. This is implemented simply

by keeping the Hamiltonian in the charge basis, so that the probability of being in each

charge state does not change. If the box is in the state with an extra Cooper pair, then

this Cooper pair will decay after a time tqp via quasiparticle tunnelling to the lead, which

can be detected. This decay also has the advantage of re-initialising the system to the

state |0〉.

2.5 Qubit Dynamics and the NOT Gate 49

0

0.2

0.4

0.6

0.8

1

Prob

1 2 3 4(Ej t) /(2 h)

Figure 2.8: Oscillations in the probability of the qubit being in state 1〉 (solid line) or state

|0〉 (dotted line). The times required to perform a not (π/2) and to prepare a superposition

(π/4) are indicated by vertical dashed lines.

In theory the individual quasiparticle decays could be detected, for instance by means

of a single-electron transistor and of course, a single-shot measurement like this is essential

for a quantum computing implementation. If instead we are interested in demonstrating

coherent quantum oscillations, we only wish to know the probability that a Cooper pair

was present, and furthermore would like the measurement to be as simple (experimentally)

as possible.

With this in mind, the oscillation is repeated many times leading to many quasiparticle

decays, to get a (macroscopic) current proportional to the probability of there being an

extra Cooper pair on the box. The beauty of this experiment is that a whole sequence

of measurements can be performed by applying a sequence of voltage pulses of length ∆t

and separated by a time trest > tqp, and that the probability of a Cooper pair being on

the box can be found by making a current measurement on the lead.

At the end of each pulse, the probabilities of the system being in the states |0〉 and|1〉 are given by |〈0|ψ(t0 +∆t)〉|2 and |〈1|ψ(t0 +∆t)〉|2 which evaluate to cos2(EJ2~ ∆t) and

sin2(EJ2~ ∆t).

These probabilities oscillate with an angular frequency EJ~ (figure 2.8.) In Nakamura’s

experiment [59], the value of EJ was altered by manipulating the magnetic flux through

the double junction (as in section 2.3), and the change in the oscillation frequency was

observed.


To perform a not gate, we just need to set the length of the pulse to ∆t = π~/EJ . From

eq. 2.5.3, the state of a system starting in state |0〉 =(01

)is then |1〉 =

(10

)and vice versa.

It is also worth noting that the system can be put into an equal superposition of the charge

states by setting ∆t = π~/2EJ , in which case the state of the system becomes 1√2

(i1

)if it

was in the state(01

)at the start of the pulse. This can be thought of as the square root

of a NOT gate (√NOT ) in the sense that performing this gate twice implements a NOT

gate. There is no analogue to this gate in classical computing [78].

The oscillations between the two charge states can be taken as evidence that the system

at some point passes through a superposition of the two states [79].

2.6 Decoherence and Noise

We discussed the DiVincenzo checklist in section 1.1, and mentioned that one of the ad-

vantages of solid-state qubits is that they are easily identifiable by their location and

should be scalable to large numbers by means of microfabrication. Their main disadvan-

tage is that they are particularly prone to decoherence. Although the superconducting

energy gap goes some way to reducing this, it does not eliminate decoherence entirely.

One source of error that is intrinsic to many types of qubit is the truncation of the state

space. Many realisations of qubits approximate a two level system by ensuring that there

is a large separation between two low-lying energy levels (the qubit states or computa-

tional subspace) and other states of the system. In atomic or ionic implementations, two

low-lying electronic energy levels are chosen from the entire spectrum. In the case of our

superconducting charge qubits, the computational subspace is the two charge states clos-

est to the gate charge, and the other states are those with greater (or smaller) numbers of

Cooper pairs on the island. The effect of this approximation has been considered in detail

in [80], in which the number of one-qubit gates feasible before decoherence is around 4000

when the ratio of the Josephson and charging energies is 0.02. For two-qubit gates, the

number of operations possible is less, but measurement on the non-computational space

may reduce this error. Some techniques to reduce this problem are considered in [81].

Other sources of decoherence in charge qubits include the fact the voltage pulses applied

to perform the gates will not be perfectly square but will have a finite rise time [82],

coupling to charge noise in the environment, and non-zero inductances [58, 83, 84, 85, 86].

A further effect that needs consideration is the effect of noise in the control parame-

2.6 Decoherence and Noise 51

ters. In subsection 2.6.1 we review the effect of Gaussian noise, which could arise due to

imperfect control of the external control parameters, and in subsection 2.6.2 we present

a calculation demonstrating the effect of discrete jumps in the tunnelling energy, which

could occur due to physical processes such as the parity effect (1.9.1) leading to jumps in

the tunnelling energy with quasiparticle tunnelling.

2.6.1 Gaussian Noise

A simple example of the effect that noise in the parameters has on the dynamics of a qubit

is the effect of Gaussian noise in the Josephson energy EJ . When the gate voltage is set

to the degeneracy point (which is the relevant voltage to demonstrate oscillations, or to

perform a not gate), the evolution in the presence of noise can be calculated analytically.

From eq. 2.5.3, the probability of a system initialised in the state |0〉 evolving to the

state |1〉 after a time ∆t is given by P (1) sin2(EJ2~ ∆t). For this calculation, we consider

a Josephson energy that fluctuates on an longer timescale that of the voltage pulse, i.e.

tfluc > ∆t, so that the Josephson energy can be considered constant for each evolution.

We can then find the current measured by the lead by calculating the evolution for a

particular EJ and then averaging over the ensemble of all possible values of EJ , which we

take to have a Gaussian distribution.

The time evolution of a charge qubit set at the degeneracy point ng = 1/2 is given in

eq. 2.5.3, and the probability |〈1|ψ(t)〉|2 that the system will be in state |1〉 is given by

P (1) = sin2(EJ2~ ∆t). To calculate the evolution in the presence of noise, we average the

probability over all EJ :

Pav(1) =1√

2π∆EJ

∞∫

−∞

e− (EJ−EJ )

2

2(∆EJ )2 sin2

(EJ2~

∆t

)dEJ (2.6.1)

where we have assumed that EJ has a Gaussian distribution of width ∆EJ centred on EJ .

We do the integral by completing the square and find:

Pav =1

2

(1− e−

(∆t∆EJ

~

)2cos

EJ∆t

~

)(2.6.2)

After averaging, we have oscillations with a frequency EJ/~, determined by the average

Josephson energy, that decay exponentially with a time constant ∆EJ/~, determined by

the amount of noise.


0

0.2

0.4

0.6

0.8

1

Prob

2 4 6 8 10 12 14(Ej t) /(2 h)

Figure 2.9: The probability of a Cooper pair box being in the state |1〉 as a function of

time, with Gaussian noise in the Josephson energy. The exponential decay envelope is

plotted as dotted lines.

Although these analytic results are only for the case ng = 1/2, an exponential decay

over time is a generic feature of the time evolution when the parameters have a Gaussian

distribution over their average value. Numeric analysis of the effect of Gaussian noise in

EJ and EC can be found in [87].

2.6.2 Discrete Noise in the Josephson Energy

Whilst Gaussian noise is a good generic model of noise in the control parameters, there

are other types of noise which may have different effects on the dynamics. in this section,

we present new work on a simple model of discrete noise.

In this model we consider the case where the Josephson energy can fluctuate, but only

between two discrete values. Again, we consider the case when the gate charge is set to

ng = 1/2.

We model the fluctuation in EJ by assuming that the distribution in the number of

jumps between the two Josephson energy values follows the Poissonian distribution, i.e.

that over a period of time t, the probability of there being j jumps in the Josephson energy

is given by e−βt (βt)j

j! .

The calculation of the evolution of the system is found by calculating the evolution for

a particular number of jumps j, and then averaging over all j.

For this calculation it will be helpful to use a density matrix formulation ρ(t). The


time evolution of the density matrix is given by ρ(t) = U(t)ρ(0)U †(t) where U(t) is the

time evolution operator that operates on the wavefunction in eq. 2.5.1. If ρch is the density

matrix in the charge basis, then we can make a transformation to the energy eigenbasis,

ρeg:

ρeg =1

2

1 1

1 −1

ρch

1 1

1 −1

ρeg =1

2

1 1

1 −1

ρ00 ρ01

ρ10 ρ11

1 1

1 −1

(2.6.3)

As before, we make the sudden approximation, so that for a given series of jumps at

times τ1, τ2 etc. the time evolution of the density matrix in the energy eigenbasis is given

by:

ρeg(t) = U(t)ρeg(0)U†(t)

= Uj(t− τj) · · ·U1(τ2 − τ1)U0(τ1)

ρeg(0) U†0(τ1)U

†1(τ2 − τ1) · · ·U †j (t− τj) (2.6.4)

The off-diagonal elements become:

ρ+/−(t) = ρ+/−(0)e−iEJ0τ1~ e−i

EJ1(τ2−τ1)

~ · · · e−iEJj(t−τj)

~ (2.6.5)

and its complex conjugate.

We have specified the number of jumps, and need to average over all possible times τ

of those jumps by integrating over all possible times for the jumps, and then dividing by

the integration time:

ρav1+/−(t) = ρ+/−(0)

t∫

0

dτ1t

t∫

τ1

dτ2t− τ1

· · ·t∫

τj

dτjt− τj

e−iEJ0τ1

~ e−iEJ1(τ2−τ1)

~ · · · e−iEJj(t−τj)

~

ρav+/−(t) = χ(t)ρ+/−(0) (2.6.6)

where χ denotes the series of integrals over t. This means that we can write the density

matrix in the charge basis as:


ρch =1

2

1 1

1 −1

ρ+/+(0) ρ+/−(0)χ(t)

ρ−/+(0)χ∗(t) ρ−/−(0)

1 1

1 −1

(2.6.7)

The problem of the time averaged evolution is now reduced to the problem of calcu-

lating the integral χ(t).

For instance, in the case of a single jump.

χ(t) =1

t

t∫

0

dτ1e−iEJ0τ1~ e−i

EJ1(t−tau1)

~

=

(e−i

EJ0t

~ − e−iEJ1t

~

−i(EJ0 − EJ1)t/~

)(2.6.8)

If we started the system in the state |0〉 =(01

), we find that the average probability of

being in the state |1〉 after a time t given that a single jump has occurred is:

ρav111 (t) =1

2− ~

4(EJ0 − EJ1)t

(sin

2EJ0t

~− sin

2EJ1t

~

)(2.6.9)

So the overall time averaged evolution, including the case where there is no jump, and

the case with only one jump (we assume 2 or more jumps are unlikely, i.e. β ¿ 1) becomes

(figure 2.10):

ρav0,111 (t) = e−βt(1− sin2

EJ0~

)

+(e−βtβt)

(1

2− ~

4(EJ0 − EJ1)t

(sin

2EJ0t

~− sin

2EJ1t

~

))(2.6.10)

There are two timescales in this averaging; 1/β, which determines the time between

jumps, and 1/(EJ0 − EJ1)/~. At times longer than the latter timescale, the average

evolution reduces to the exponential decay seen in section 2.6.1.

Further terms can be found by calculating the integrals χ(t) (eq. 2.6.6) over more

jumps, and the evolution of the density matrix calculated. The evolution of the density

matrix element ρ11, which represents the probability that the system will be found in the

state |1〉, is given by:


10 20 30 40

0.2

0.4

0.6

0.8

1

Figure 2.10: The probability of a Cooper pair box being in the state |1〉 as a function of

time with discrete noise in the Josephson energy. One jump is included, with parameters

EJ0 = 1, EJ1 = 0.5, β = 0.1. The Gaussian decay is plotted as dashed lines with the

ideal, non-decaying oscillations denoted by dotted lines.

ρav11(t) =∑

j

e−βt(βt)j

j!

(1

2− 1

2(ρ+/−(0)χ(t) + ρ−/+(0)χ

∗(t))

)(2.6.11)

Chapter 3

The Josephson Circuit

Hamiltonian

In this chapter we examine in greater detail the construction of the qubit Hamiltonian for

superconducting charge qubits. We first examine the circuit diagrams of individual and

multiple qubits, and consider the capacitive charging energy of these circuits. We confirm

that the capacitive term found is correct by deriving the Josephson equations of motion for

each junction, and finally produce a quantum Hamiltonian for each circuit.

3.1 The Capacitance of Qubit Circuits

In section 2.2 we derived a Hamiltonian for a single Josephson junction (2.2.10). This

Hamiltonian was written in terms of phase. In order to derive it, we took the Josephson

junction relations, i.e. the equations of motion for the phase difference between the two

superconductors, found a classical Hamiltonian and a set of conjugate variables from which

these equations of motion could be derived using Poisson brackets, and then elevated the

conjugate variables to operators.

A realistic superconducting charge qubit will be much more complicated than a single

Josephson junction. For instance, a single qubit consisting of an island coupled to a

reservoir will not only contain a Josephson junction, but will also need a gate voltage

applied to the island in order to adjust the relative energies of the two charge states. The

island we are discussing is small, and so we will have to consider its self-capacitance as

well as the capacitance between it and the reservoir.

57

58 The Josephson Circuit Hamiltonian

When quantising the Josephson junction relations, we write down equations for motion

of phase and voltage difference. In order to derive the Josephson relations from single-

electron operators, it will be useful to have a charging term which is written in terms

of total charge (number of Cooper pairs) on an island, rather than the charge stored on

a capacitor. In this, we are following the spirit of [68, 47] in calculating the effective

capacitance of a quantum circuit. However, we go further in that we show that this leads

to a quantum Hamiltonian that reproduces the correct Josephson equations for the circuit

as its equations of motion. We also show a general method of performing such calculations.

Finally, a single qubit consists of an island coupled to an infinite reservoir. We would

like to be sure we are taking the correct limit as we allow the size of the reservoir to

become large.

3.1.1 Single Junction with Gate Voltage

We wish to consider more complicated circuits including gate voltages and self-capacitance.

As the circuits become more complex, it will become convenient to solve the circuit equa-

tions using a computer. As with any calculation done on computer, we need to be especially

clear of the procedure used, and so we shall go through a simpler problem, in order to

illustrate the details of the calculation.

After a single Josephson junction, the most simple case to consider is that of two

regions of superconductor coupled by a capacitive Josephson junction, with a gate voltage

between the two regions. For the present, we ignore any self capacitance.

The regions of superconductor are labelled 1 · · · i for the small islands, and r for the

reservoir (in this case 1, r). The charges on these regions are labelled with a single index

qi. The capacitors between the regions i, j have capacitances Cij . The charge on the

capacitor Cij is qij (with a double index) and the voltage across it is Vij . The gate voltage

and their capacitances are labelled with a g. The sign of the charges on the capacitors

and voltage sources is arbitrary, but needs to be consistent within each calculation.

We wish to write the capacitive energy of the circuit in terms of the charges on the

superconductors. We start with Kirchoff’s laws and have one equation (3.1.1) representing

the fact that the sum of the voltages around the circuit must sum to zero, and another

(3.1.2) that indicates that the charge on each island is given by the sum of the charges on

the capacitors connected to it.

3.1 The Capacitance of Qubit Circuits 59

q1

qr

q1r-q1r

q1g

-q1g

q1g

-q1gV1g

Figure 3.1: Single Voltage with Gate Voltage Circuit Diagram

q1rC1r

+q1gC1g

+ V1g = 0 (3.1.1)

q1r − q1g = q1 (3.1.2)

Note that we have no equation for the charge on the reservoir. In this calculation we

are assuming that the circuit is electrically neutral overall, and so q1 = −qr. We shall

relax this restriction in the next calculation.

We solve equations 3.1.1 and 3.1.2 for q1r, q1g:

q1r =

(C1rC1g

C1r + C1g

)(q1C1g− V1g

)

q1g = −(

C1rC1g

C1r + C1g

)(q1C1r

+ V1g

)(3.1.3)

We write the energy due to the charges in the circuit in terms of the potential energy

of the charges on the voltage source and the work done in placing the charges on the

capacitors:

EC =q21r2C1r

+q21g2C1g

+ V1gq1g (3.1.4)

We now substitute the expressions for the charges on the capacitor (eqns. 3.1.3) into

the above expression for the charging energy:


q1

qr

q1r-q1r

q1g

-q1gq1g

-q1g V1g

q1w-q1w

qrw -qrw

qw

Figure 3.2: Single junction with gate voltage and self-capacitance, which is represented

by a capacitive coupling to a third region, labelled w (for ‘world’).

EC =

(C1rC1g

C1r + C1g

)2( 1

2C1r(q1/C1g − V1g)2 +

1

2C1g(q1/C1r − V1g)2

)+ V1gq1g

=1

2(C1r + C1g)(q1 − V1gC1g)

2 −V 21gC1g

2(3.1.5)

This result shows that the gate voltage has had the effect we expected; the junction

can be described with an effective capacitance, given by 1/Ceff = 1/(C1r + C1g), and a

term that effectively resets the zero of capacitance. This term we call the gate charge, and

denote qg = V1gC1g.

3.1.2 Single Junction with Gate Voltage and Self-Capacitance

As mentioned earlier, in the previous section we have made the restriction that our circuit

is electrically neutral, i.e. that q1 + qr = 0. As we want to consider the region r as a

reservoir, this is an undesirable assumption. In order to deal with this, we introduce

a third region that represents ‘the rest of the world.’ This means that we can use the

restriction that the entire circuit (including the ‘world,’ w), is neutral, and our qubit

circuit itself (1+ r) does not have to be. The self-capacitance of the regions is represented

by their coupling to w.

This time we have two voltage loops around the circuit, and two charges on the regions

1, r. Again, the charge on w gives us no new information as we take the entire circuit to


be charge neutral.

q1rC1r

+q1gC1g

+ V1g = 0

q1wC1w

+qrwCrw

− q1rC1r

= 0

q1g − q1r − q1w = q1

q1r + qrw − q1g = qr (3.1.6)

We have four equations, which is enough to give the four charges on the capacitors in

terms of q1, qr. Solving four simultaneous equations by hand would be time consuming,

but can easily be solved on a computer, using a package such as Maple or Mathmatica

(as long as we are careful - this is why we have gone through the previous calculations in

such detail). As an example, the expression found for q1g is:

q1g = −C1g

C2Σ

(C1w(qr + V1gC1r) + Crw(−q1 + C1rV1g) + V1gC1wCrw) (3.1.7)

where C2Σ is defined by 1/C2

Σ = 1/(C1wC1g+C1wC1r+CrwC1g+CrwC1r+C1wCrw). There

are similar expressions for the other capacitor charges. We can substitute these into the

equation for charging energy, and then rearrange to get a charging energy in terms of q1

and qr.

EC =q21r2C1r

+q21w2C1w

+q2rw2Crw

+q21g2C1g

+ V1gq1g

=(C1r + C1g + Crw)

2C2Σ

(q1 − V1gC1g)2 +

(C1r + C1g + C1w)

2C2Σ

(qr + V1gC1g)2

+(C1r + C1g)

C2Σ

(q1 − V1gC1g)(qr + V1gC1g) +K (3.1.8)

where K represents constant terms. We see that we now have three terms, a term pro-

portional to q21, a term proportional to q2r , and a cross term. Again, the ‘zero position’ is

determined by the gate charge.

We have specifically included a region, w, representing the external world, allowing us

to consider the case where the charge on the circuit is non-zero. Now this has been done

with general capacitances, we can take the limit where C1w → 0 and Crw → ∞, i.e. the


limit where the island is totally unconnected to the outside world, and the reservoir, r is

strongly connected.

In this limit, the coefficients of q2r and q1qr go to zero. The limit of the q21 coefficient

is:

(C1r + C1g + Crw)

2C2Σ

→ Crw2(CrwC1g + CrwC1r + CrwC1w)

→ 1

2(C1g + C1r)(3.1.9)

By comparing this result with eq. 3.1.5 we see that we have regained the previous

result. This confirms that eq. 3.1.5 is indeed a valid expression for the charging energy of

an isolated island connected via a capacitive tunnel junction to a reservoir.

We have one further task remaining; the energy is given in terms of the charge on the

island, q1, and the Josephson equations are given in terms of phase and voltage differences.

This was unimportant in the previous case, as we were only considering two isolated

islands, and therefore q1 = −qr = 1/2(q1 − qr). To express our charging energy for this

system in terms of charge differences, we substitute qs = (q1 + qr)/2 and q1r = (q1− qr)/2into the expression for charging energy (eq. 3.1.8) before we take the limit.

3.1.3 The 2 Qubit Circuit

Now we have confirmed the form for the charging energy of a single junction, we can move

onto a larger system. We wish to find the charging energy of a two-qubit system. This

consists of two islands coupled to a large superconductor. As before, we wish to find a

result when the reservoir has a finite size and is coupled to the outside world, and only

then take the limit of an infinitely large reservoir. We also include a gate voltage on each

of the two islands.

Examining the circuit diagram (fig. 3.3), we see that we can define five loop equations

and three equations for the charges on the islands. This gives us enough information to

calculate the eight unknown capacitor charges, qij .

As before, inserting these values for the capacitances into the expression for charging

energy (eq. 3.1.10) gives us the charging energy in terms of the charges on the islands q1,

q2, qr.


q1 q2

qr

q2w

qw

q12

q1g

q1w

qrw

q1r

q2gq2r

-q1w -q2w

-qrw

q1g

-q1g

-q1g

-q1r

-q2r

-q2g

q2g

-q2g

V2gV1g

-q12

Figure 3.3: Two Qubit circuit diagram. The system consists of two islands (1, 2) connected

to a reservoir (r), with gate voltages applied to each. We wish to solve the problem for a

reservoir of general size, and so a further region, w is included to represent the rest of the

world.


EC =∑

i,j 6=i

q2ij2Cij

+∑

i6=gVigqig (3.1.10)

As before, we are considering region r to be a large reservoir, and so we now wish

to take the limit where q1w, q2w → 0 and qrw → ∞. We find, as before, that the terms

containing the charge on the reservoir qr vanish. We find that the charging energy only

depends on the charges on the two islands.

EC =(q1 − q1g)22Ceff 1

+(q2 − q2g)22Ceff 2

+(q1 − q1g)(q2 − q2g)

2Ceff 12(3.1.11)

We have a term containing only q1, one containing only q2, and a cross term which

is the capacitive coupling between the qubits. Again, the gate charges qg modify the

zero-position of charge. The effective capacitances are given by:

1

Ceff 1=

(C2g + C2r + C12)

C2Σ

1

Ceff 2=

(C1g + C1r + C12)

C2Σ

1

Ceff 12=

2C12

C2Σ

C2Σ = C2gC12 + C2gC1g + C2gC12 + C12C2r

+C2rC1g + C1rC2r + C1gC12 + C1rC12 (3.1.12)

3.1.4 Multiple Islands

The same process as above can be applied to three or more islands. We find very similar

results; there is a quadratic term dependent on the charge on each island alone, and a

cross term between any two charges:

EC =∑

i

(qi − qig)2Ceff i

+∑

i,j 6=i

(qi − qig)(qj − qjg)Ceff ij

(3.1.13)

We examine some of the effective capacitances of a three-island system, in particular

the cross terms between islands:

3.2 Quantising the Josephson Relations 65

1

Ceff 12=

2C12(C3g + C23 + C3r)

C3Σ

1

Ceff 13=

2C12C23

C3Σ

(3.1.14)

where C3Σ is a term with units of capacitance cubed. Every island in the system has an

interaction with every other. However, if we consider the interaction capacitances to be

much smaller than the capacitances between the islands and the reservoir, we find that

effective capacitances between islands that are not adjacent are correspondingly reduced.

3.2 Quantising the Josephson Relations

In section 2.2 we quantised the equations of motion of a single capacitive Josephson junc-

tion between two islands. In this section we wish to ensure that this procedure is still

valid when we consider more complicated circuits. We expect the Hamiltonian of a circuit

to consist of a capacitive term and a Josephson tunnelling term. If we have the correct

Hamiltonian and conjugate variables, we should find that the equations of motion given

by the Poisson brackets (classically) or commutators (quantum mechanically) are just the

Josephson equations for that circuit. Again, we shall consider the simplest case in detail

first before we examine more complicated circuits.

3.2.1 Single Junction with Gate Voltage

In order to find a classical Hamiltonian for the circuit in 3.1, we shall guess a Hamiltonian

based on the charging energy found in 3.1.1 and a Josephson tunnelling energy, and a

pair of conjugate variables. We shall then check if this Hamiltonian is indeed correct by

comparing the equations of motion derived from it to the Josephson equations for the

circuit.

We consider the Hamiltonian:

H =(q1 − V1gC1g)

2

2(C1r + C1g)− 2T cos (φ1 − φr) (3.2.1)

The conjugate variables are the phase φ1, and the conjugate momentum to this, N1 =

~q1/2e. The Poisson brackets for these two variables give:


φ1, H =dH

dπ

=2e

~(q1 − C1gV1g)

2(C1r + C1g)(3.2.2)

Next we note that the voltage across the junction is given by V1r = q1r/C1r:

V1r =C1rC1g

C1r + C1g

q1/C1g − V1gC1r

=q1 − V1gC1g

C1r + C1g(3.2.3)

Inserting this value for the voltage into equation 3.2.2, we find:

d (φ1 − φr)dt

= φ1, H

=1

~(2e)V1r (3.2.4)

This is the expected Josephson equation for rate of change of phase difference, i.e. that

it is proportional to the voltage across the junction.

Similarly, the rate of change of charge is given by its Poisson bracket:

d q1dt

=2e

~π1, H

=−2T (2e)

~− dH

dφ1

=−2T (2e)

~sin(φ1 − φr) (3.2.5)

That is, the rate of change of charge on the island is given by the current onto the

island.

We have found that the Hamiltonian (eq. 3.2.1) for a single junction with a gate

charge leads to the equations of motion for the charge and phase difference given by the

Josephson relations for that junction. This confirms that this indeed the correct classical

Hamiltonian. We obtain the quantum Hamiltonian by elevating the variables N1 and φ1 to

be operators with the commutation relation [φ1, N1] = i. We can now use the commutation

relations to find the equations of motion for the expectation values of the operators.


d 〈(φ1 − φr)〉dt

=2e

~2e〈N1〉 − C1gV1g

C1r + C1g

d 2e〈N1〉dt

=−2T (2e)

~〈sin(φ1 − φr)〉 (3.2.6)

We now move on to more complicated circuits.

3.2.2 Single Junction with Gate Voltage and Self-Capacitance

In section 3.1.2 we considered two regions of superconductor coupled to each other and to

a third region representing the rest of the world. We found the charging energy in terms

of the charges on the islands, and then took the limit C1w → 0, Crw → ∞. We can also

check that the Hamiltonian gives the correct equations of motion before the limit is taken.

Using the Hamiltonian H = EC−2T cos(φ1−φr) (we drop the hats on operators Ni, φi

for convenience) with the EC found in section 3.1.2, we find the commutator for the phase

difference:

[H, (φ1 − φr)] =i (2e)(C1w2eNr − Crw2eN1 − CrwC1gV1g − C1wC1gV1g)

(C1rCrw + C1rC1w + C1wCrw + C1wC1g + CrwC1g)(3.2.7)

Again, we can compare this with the voltage across the junction, V1r = q1r/C1r.

V1r =C1r(C1w2e〈Nr〉 − Crw2e〈N1〉 − CrwC1gV1g − C1wC1gV1g)

(C1rCrw + C1rC1w + C1wCrw + C1wC1g + CrwC1g)

1

C1r(3.2.8)

This leads to the relation we had in the last section, i.e. :

d 〈φ1 − φr〉dt

=1

i~〈[H,φ1]〉

=2e

~V1r (3.2.9)

The same calculation as done in section 3.2.1 confirms the second Josephson equation

for this system.

3.2.3 The 2 Qubit Circuit

We can now check the Hamiltonian for the 2 qubit circuit described in section 3.1.3. We

have gate voltages on islands 1, 2 and include a region w to represent the self-capacitance


of the other regions. The Hamiltonian consists of the charging energy (eq. 3.1.11) and

terms describing the tunnelling between the regions 1,2 and r.

H = EC − 2T1r cos (φ1 − φr)− 2T2r cos (φ2 − φr)− 2T12 cos (φ1 − φ2) (3.2.10)

The three tunnelling junctions are the junctions between the reservoir (r) and the

islands 1, 2, and the junction between the two islands. If we check (again, using a computer

program as before) we find that the rate of change of phase difference across each of these

junctions is indeed proportional to the voltage difference. This is true before we take the

limits C1w, C2w → 0, Crw → ∞. The full results are unwieldy, so the results after the

limit has been taken shall be presented here. When the limit is taken, the expression for

the charging energy (eq. 3.1.11) does not contain qr and so the phase φr commutes with

the Hamiltonian. The commutation relations for the three phase differences are:

[H,φ1 − φr] = 2e−(C12 + C2r + C2g)

C2Σ

(2e〈N1〉 − V1gC1g)−C12

C2Σ

(2e〈N2〉+ V2gC2g)

[H,φ2 − φr] = 2e−(C12 + C2r + C2g)

C2Σ

(2e〈N1〉 − V1gC1g)−C12

C2Σ

(2e〈N2〉+ V2gC2g)

[H,φ1 − φ2] = 2e−(C2r + C2g)

C2Σ

(2e〈N1〉 − V1gC1g)

+(C1r + C1g)

C2Σ

(2e〈N2〉+ V2gC2g) (3.2.11)

where C2Σ is as defined as in eq. 3.1.12. If these are compared to the voltages across each

of the three junctions (V1r = q1r/C1r etc.) we see that the equations of motion for phase

calculated using the commutators are those given by the Josephson relations.

We also calculate the rate of change of charge on the islands. The rate of change of

the charge on island one is given by:

d 2e〈N1〉dt

=2e

i~[H,N1]

= −2T1r2e

i~[〈cos(φ1 − φr)〉, N1]−

2T122e

i~[〈cos(φ1 − φ2)〉, N1]

= −2T1r(2e)

~〈sin(φ1 − φr)〉 −

2T12(2e)

~〈sin(φ1 − φ2)〉 (3.2.12)

This is a sum of the currents across the two junctions onto island one, as expected.


The rate of change of charge on island one is given by the sum of the currents onto

the island.

We find a similar expression for island two:

d 2e〈N2〉dt

= −2T2r(2e)

~〈sin(φ2 − φr)〉+

2T12(2e)

~〈sin(φ1 − φ2)〉 (3.2.13)

The sign on the current across the junction 1, 2 is reversed as a current onto island one

is also a current off island two.

In this chapter we have shown in detail that there is a consistent procedure quantising

Josephson circuits with arbitrarily many gate capacitors, junctions, etc. and that the

simplistic technique of adding the charging energy of the circuit to the tunnelling energy

for each junction produces a Hamiltonian that generates equations of motion for the phase

differences across each junction and the charges on each island that are equivalent to the

Josephson equations for each junction.

Whilst we have been dealing in this chapter with the macroscopic quantisation of

the circuits in terms of phase, in later chapters we will go beyond this to a microscopic

derivation of these equations.

Chapter 4

The Dynamics of a Two Qubit

System

Below we examine a system consisting of two interacting charge islands, each of which is

used as a qubit. In this chapter we consider the qubits as purely two-level systems and

investigate their energy level structure and dynamics. We show how oscillations in the

behaviour of the system with the magnetic flux through the circuit can be considered in

terms of virtual tunnelling of the Cooper pairs.

4.1 The Two Qubit Circuit

In the previous chapter we found the Hamiltonian of a system of two superconducting

islands coupled to a reservoir. We included gate voltages on each of the two islands, and

allowed for fluctuations in the number of Cooper pairs on the reservoir by considering a

capacitive coupling to the rest of the world before taking the limit.

H = EC1(n1 − n1g)2 + EC2(n1 − n1g)2 + EC12(n1 − n1g)(n2 − n2g)

−T1 cos (φ1 − φr)− T2 cos (φ2 − φr)− T12 cos (φ1 − φ2) (4.1.1)

Here we have changed notation so that EC1, T1 are the effective charging energy and

Josephson tunnelling energy respectively between island one and the reservoir, and EC12,

T12 are the charging and tunnelling energies between the two islands. As before, n1 and

φ1 are conjugate variables that have been elevated to quantum mechanical operators.

71

72 The Dynamics of a Two Qubit System

For a system to behave as a true qubit, it should be well described by a two-level

system. For systems that are not intrinsically two level systems (as opposed to, say,

the spin of an electron in a quantum dot) this means that the two lowest energy levels

of each qubit are well separated in energy from the others, making transitions to other

levels unlikely. For a charge qubit, the energies are well within the charging regime, i.e.

T1, T2 ¿ EC1, EC2. In this case, the two lowest levels for each qubit are the charge states

closest to n1g,n2g. Any other states have an extra charge of at least 2e and are therefore

unlikely to be occupied (note that the finite probability of occupation of other levels leads

to decoherence - see [80]).

If we discard all but the lowest lying levels, we can write the Hamiltonian as a four

by four matrix acting on the charge basis. The single qubit (i.e. uncoupled) part of the

Hamiltonian can be written as:

HS =

−Ech1 − Ech2 −T1 −T2 0

−T1 Ech1 − Ech2 0 −T2−T2 0 −Ech1 + Ech2 −T10 −T2 −T1 +Ech1 + Ech2

(4.1.2)

where Ech1 = EC1(1/2− n1,g) is a charging energy term that depends on the gate charge

n1,g = C1gV1g. The four basis states are the charge states |00〉, |01〉, |10〉, |11〉 respectively.Note that if both charging energies are set to 1/2 the four charge states are degenerate.

The interaction between the qubits is:

HI =

0 0 0 0

0 −EC12n2g −T12 0

0 −T12 −EC12n1g 0

0 0 0 EC12(1− n1g − n2g)

(4.1.3)

4.1.1 Energy Levels

The energy levels and of the Hamiltonian HS + HI are plotted in for the case when

the inter-qubit tunnelling is zero (figure 4.1) and when it is included (figure 4.2). The

parameters are chosen to be experimentally reasonable, considering those quoted in [69].

When the gate voltages are set far from the degeneracy point, the eigenstates are close to

4.1 The Two Qubit Circuit 73

–100

–50

0

50

100

0 0.2 0.4 0.6 0.8 1

Figure 4.1: The energy levels of a two qubit system as a function of n1g = n2g, when the

tunnelling between the islands is zero. The parameters of the system are EC1 = EC2 = 100,

EC12 = 10, T1 = 10, T2 = 10

the charge states. The energies of the states |00〉 and |11〉, where either both islands have

an excess Cooper pair or neither do, are well separated in energy. The states with only

one island having an extra Cooper pair are degenerate if the boxes are identical.

When the gate charges are set to the degeneracy position, n1g = n2g = 1/2, all four

charge states of the non-interacting system would be degenerate. The degeneracy between

|00〉, |11〉 and the two states |01〉 and |10〉 is broken by the tunnelling energies T1, T2. The

additional degeneracy between |01〉 and |10〉 is broken by the charging interaction EC12.

The degeneracy in figure 4.2 is due to the fact that we have treated the islands as

identical. Any difference in the tunnelling energies T1, T2 will break the degeneracy.

4.1.2 Two-Qubit Gates

Universal quantum computation can be achieved if we have a complete set of one-qubit

gates and a two-qubit gate that can produce a maximally entangled state. In this section

we show how an entangling two-qubit gate can be performed. As in section 2.5, the

parameters of the qubits are set in such a way that the time evolution of the system

performs the gate. We are considering a system with no tunnelling between the islands,


–100

–50

0

50

100

0 0.2 0.4 0.6 0.8 1

Figure 4.2: The energy levels of a two qubit system as a function of n1g = n2g, when

the tunnelling between the islands is included. The parameters of the system are EC1 =

EC2 = 100, EC12 = 10, T1 = 10, T2 = 10, T12 = 30

i.e. T12 = 0, and equal tunnellings from the two islands to the reservoir T1 = T2. The

evolution is plotted in figure 4.3.

If the system is initialised in the state |01〉 then it will oscillate to the state |10〉 aftera time determined by the differences in the eigenenergies (denoted e2 and e4 in appendix

B).

Halfway through the oscillation, the state will be:

|ψ〉 = 1

2((1 + i)|01〉 − (1− i)|10〉) (4.1.4)

Which is a maximally entangled state. This is an alternative to the square root of

imaginary swap, (√iSWAP ) gate described in [88].

4.2 A Bell Inequality

For a small superconducting island to function as a true qubit, we must be able to perform

many complicated operations on it. One set of operations of particular interest is a set

4.2 A Bell Inequality 75

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2 1.4t

Figure 4.3: Oscillations between state |01〉 (solid line), |10, 〉 (dashed line) and the states

|00〉 and |11〉 (dotted lines). Halfway between oscillations, the system is in a maximally

entangled state (vertical lines).

of operations to test a Bell inequality [89]. As an example of the level of control that we

must have over the charge qubit, we shall go through the operations in detail.

An example of an experiment to test a Bell inequality that is particularly easy to

visualise is that of a pair of spin-half particles being emitted from a source to widely

separated locations. The particles are in a spin singlet state |s〉 that has zero total angular

momentum:

|s〉 = 1√2(| ↑〉A| ↓〉B − | ↑〉B| ↓〉A) (4.2.1)

where | ↑〉 represents the state when a particle is in the spin-up eigenstate of the σZ

operator, and | ↓〉 represents spin down.

At the two locations (A and B), there are devices to measure the spin along a given

direction (SA(θA), SB(θB)). We measure the spin along two directions, θ, θ′ at each

location and take the correllators, C(θA, θB), between them.


z

y

z+yz-y

Figure 4.4: The Bell inequality is violated if a pair of particles in the singlet state are

measured along the bases shown. Note that in three of the combinations of measurements,

the bases are at 45 deg to each other, and in one of the measurements (y, (z−y)/√2) they

lie at 135 deg.

C(θA, θB) = 〈SA(θA)SB(θB)〉

= PθA,θB (↑↑) + PθA,θB (↓↓)− PθA,θB (↑↓)− PθA,θB (↓↑) (4.2.2)

where PθA,θB (↑↑) is the probability that when the spins of the two particle are measured

along the directions θA, θB, both will be found in the up state.

A Bell inequality is a statement about the values of correlations in measurements made

by separated observers. Any local realistic theory predicts that the measurements made

at two separated locations must obey the condition:

|C(θA, θB) + C(θ′A, θB) + C(θA, θ′B)− C(θ′A, θ′B)| ≤ 2 (4.2.3)

The measurements at A are made in the perpendicular directions z and y (with the

corresponding operators SZA and SYA ) , and the measurements atB are made along 1√2(z+y)

and 1√2(z − y), (figure 4.4).


C(z,1√2(z + y)) = 〈s|SZA ,

1√2(SZB + SYB )|s〉

= −1/√2

C(z,1√2(z − y)) = −1/

√2

C(y,1√2(z + y)) = −1/

√2

C(y,1√2(z − y)) = +1/

√2 (4.2.4)

We find that these correllators violate the Bell inequality:

C(z,1√2(z + y)) + C(z,

1√2(z − y)) + C(y,

1√2(z + y))− C(y, 1√

2(z − y))

= | − 1/√2− 1/

√2− 1/

√2− (+1/

√2)|

£ 2 (4.2.5)

The violation of the Bell inequality shows in a very clear and simple way that quantum

mechanics produces results that are fundamentally different from any possible classical

results. As such, a demonstration of a violation of a Bell inequality in a system is an

indication of quantum behaviour. Furthermore, recent experiments have offered hints of

entanglement in superconducting qubit systems, [69, 62], and the ultimate proof of the

existence of entanglement is the violation of a Bell inequality.

Although a coupled pair of charge qubits is a very different physical system from the

two spin-half particles described above (in particular, they are not spatially separated in

the relativistic sense and so cannot violate relativistic locality) we can use Bell inequality

demonstration as an example of the kind of control necessary to implement quantum

computing protocols.

In the above discussion, we stated that the particles began in a spin singlet state. We

need to describe how we can produce such a state. We initialise the system in the state |01〉by setting the gate charges to n1g = 1, n2g = 0 so that Ech1 = EC1(1/2− n1g) = −EC1/2and Ech2 = EC2(1/2−n2g) = EC2/2. With the tunnelling energies much smaller than the

charging energies, the Hamiltonian of the system becomes:


HS =

EC1−EC22 0 0 0

0 −EC1−EC22 0 0

0 0 EC1+EC22 0

0 0 0 −EC1+EC22

(4.2.6)

The ground state of the system is now |01〉 if we wait long enough, the system will decay

to this state. If we now suddenly set n1g = n2g = 1/2 and set the individual tunnelling

energies T1, T2 to zero, then only the tunnelling between the two islands remains. If we

allow the system to evolve (under the evolution operator U = e−iHt~ ) for time t = π~

4T12the

system evolves to the state (|01〉−i|10〉)√2

. We need to perform a further pulse to change this

state to the singlet state. If we suddenly set n1g = 1/2 and n2g = 1, and evolve for a time

t = π~Ech2

then the system evolves to be in the singlet state.

We now wish to make measurements of the four operators SZA , SYA ,

SZB+SYB

2 andSZB−SYB

2 .

In a system of superconducting charge qubits however, we can only measure in the SZ

(charge) basis. To get round this we must ‘rotate’ the state of each qubit so that measuring

SZ is equivalent to measuring in some other basis. We achieve this by setting the tunnelling

energies to the values given in eq. 4.2.7.

SZA : T1t~ = 0

SYA : T1t~ = −π

4SZB + SYB√

2: T2t

~ = −π8

SZB − SYB√2

: T2t~ = +

π

4(4.2.7)

For example, if we wish to measure qubit A in the y basis, and qubit 2 in the zB+yB√2

basis, we would set the tunnelling energies to T1t~ = −π

4 and T2t~ = −π

8 and evolve for a

time t.

After this rotation, the system will be in one of four states:


U(z,1√2(z + y))|s〉 =

−i(1−√2)

1

−1i(1−

√2)

1

2√

2−√2

U(z,1√2(z − y))|s〉 =

i(1−√2)

1

−1−i(1−

√2)

1

2√

2−√2

U(z,1√2(z + y))|s〉 =

i(1−√2)

1

−1−i(1−

√2)

1

2√

2−√2

U(z,1√2(z − y))|s〉 =

−i−(1−

√2)

(1−√2)

i

1

2√

2−√2

(4.2.8)

We now measure both qubits in the z basis. The probabilities of each result are given

by the squares of the elements in the vectors representing the states after rotation given

in eq. 4.2.8, eg. P (00) = |〈00|U(z, 1√2(z + y))|s〉|2. These, when inserted into 4.2.2 lead

to the values of the correllators given in eq. 4.2.4. Again, we find that the Bell inequality

is violated: 2√2 £ 2.

We have given a brief description of how a Bell inequality violation could be observed

in superconducting charge qubits. It is apparent from even this simple analysis that the

series of gates that need to be applied is complicated. Current experiments are now at the

point where a single 2-qubit gate is possible, and so a simple signature of 2-qubit quantum

behaviour would be useful experimentally. In particular, it would be useful if we could

observe a phenomenon that is intrinsically quantum mechanical but does not require the

complex control of a demonstration of the violation of a Bell inequality.


T1

EC12

EC1

T12

EC2 T2

Φ

1 2

Figure 4.5: A two qubit circuit with tunnelling between the qubits and a magnetic flux

passed through the circuit

4.3 Virtual Tunnelling

In section 4.1.2 we considered a two-qubit system with no qubit-qubit tunnelling, and

looked at oscillations between states that occurred when the gate voltages were set to the

degeneracy positions (figure 4.3)

An simple extension to the investigation of these oscillations is to consider the oscilla-

tions when tunnelling between the qubits is allowed and a magnetic field is passed through

the circuit (figure 4.5).

When a Cooper pair completes a circuit it picks up a phase equal to the line integral of

the vector potential A around that circuit (which is equal to the flux through the circuit

- see section 2.3). In terms of our Hamiltonian describing the two two-level systems, we

ensure that the correct phase is picked up by including a phase factor ei 2πΦ0

∫c

A.ds= e

i 2πΦΦ0

in one of the tunnelling matrix elements, T12. Note that it does not matter which of the

tunnelling elements the phase is included in as long as the correct phase is picked up on

4.3 Virtual Tunnelling 81

completing the loop - the overall phase is unimportant.

For simplicity, we consider the gate charges to be set to 1/2, where the charge states

are close to degeneracy.

H =

0 −T1 −T2 0

−T1 −EC12/2 −T ∗12 −T2−T2 −T12 −EC12/2 −T10 −T2 −T1 0

(4.3.1)

We wish to investigate the evolution of the system over time, and in particular, how

this evolution is affected by the magnetic field. As usual, we calculate the eigenvectors of

the Hamiltonian, write the initial state in terms of these eigenvectors, and multiply each

eigenvector by a phase factor dependent on the eigenenergy. Unfortunately, the above

Hamiltonian cannot be solved exactly (unless we force T12 to be real) and so the time

evolution must be calculated numerically.

Starting in the initial state |00〉, the evolution of the system is as shown in Figs. 4.6 -

4.7:

We note that not only do the probabilities oscillate in time as seen in recent papers

(eg. [69]), but also with the magnetic flux through the loop.

Whilst it is easy to calculate the evolution of the system numerically, we would like to

find a framework for understanding these oscillations.

In order to gain an understanding of this system, we set the charging interaction

between the qubits EC12 to be much larger than the tunnelling either between the islands

and reservoir, or between the two islands themselves. In this case, the charge states form

two pairs, each of which is degenerate. The states where only one of the island has a

charge on (|01〉 and |01〉) have lower energy, −EC12/2. The other pair of degenerate states(|00〉 and |11〉) is higher in energy.

The tunnelling elements break the degeneracies, and so the eigenstates of the Hamil-

tonian are then (to first order) given by the symmetric and anti-symmetric combinations

of each set of two states. That is:

|v1,4〉 =|00〉 ± |11〉

2

|v2,3〉 =|01〉 ± |10〉

2(4.3.2)


0

1

2

3

4

5

6

Flux

Time

0

1

2

3

4

5

6

Flux

Time

Figure 4.6: The Probability of the Two Qubit system being in the state |00〉 (upper plot)and |01〉 (lower plot). The horizontal axis is time and the vertical axis is the flux through

the loop in units Φ02π . White represents high probability of occupation and black represents

low probability.


0

1

2

3

4

5

6

Flux

Time

0

1

2

3

4

5

6

Flux

Time

Figure 4.7: The Probability of the Two Qubit system being in the state |10〉 (upper plot)and |11〉 (lower plot). The horizontal axis is time and the vertical axis is the flux through

the loop in units Φ02π . White represents high probability of occupation and Black represents

low probability.


0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3t

Figure 4.8: The Probability of the Two Qubit system being in each of the four charging

states. The solid line is the probability of being in |00〉, the dashed line state |11〉, andthe probability of occupation of the other two states are just visible above the time axis.

The parameters are: EC12 = 500, T1 = T2 = T12 = 10

It is easy to see that if we start the system in the charge eigenstate |01〉 it will oscillatebetween this state and the state |10〉. Similarly, if the system starts in the state |00〉, theoscillations will be between |00〉 and |11〉. This oscillation is plotted in figure 4.8.

It is interesting to note two facts about this oscillation: the system never has a signif-

icant probability of being in state |01〉 or |10〉, and there is no tunnelling element in the

Hamiltonian between the states |00〉 and |11〉. We have an apparent contradiction - if we

were to think classically, we would say that the system must move through either |01〉 or|10〉, but at no time is there any probability of being in either state.

This is an example of the quantum mechanical phenomenon of virtual tunnelling where

a system ‘passes through’ a state without occupying that state. If we think of this process

physically, the system starts with no charge on either island i.e. the two excess Cooper

pairs are both in the reservoir. The system tunnels to the state where there is a Cooper

pair on each island without passing through the intermediate step of having only one

island occupied by a Cooper pair.

The frequency of these oscillations can be calculated, and is given by the difference of


the two energy eigenvalues e1, e4. Because the Hamiltonian (eq. 4.3.1) cannot be solved

analytically, we have to calculate this numerically in general. We do find an analytic

solution, however, when all the tunnelling elements are real. This means the eigenvectors

and eigenvalues can be found when the phase on T12 is 0 or π, i.e. when there is an integer

or half-integer number of flux quanta through the circuit. The eigenvalues are:

e1,4 = −1

4EC12 ∓

1

2T12 +

1

4

√(EC12 ± 2T12)2 + 16(T1 ± T2)2

e2,3 = −1

4EC12 ∓

1

2T12 −

1

4

√(EC12 ± 2T12)2 + 16(T1 ± T2)2 (4.3.3)

To find the frequency of the oscillations between |00〉 and |11〉, we need to find e1− e4.To first order in T/EC12, this is zero. To third order, we find:

e1 − e4 = −8T1T2EC12

− 8T12(T21 + T 2

2 )

E2C12

+O(

T

EC12

)4

(4.3.4)

The oscillation between |01〉 and |10〉 has a frequency that is non-zero to first order in

T/EC12:

e2 − e3 = −2T12 +8T1T2EC12

+O(

T

EC12

)3

(4.3.5)

These differences give a good estimate of the frequencies of the oscillations, as is shown

in figure 4.9.

Examining the terms in the oscillation frequencies, we can see that they give an in-

dication of the physical processes at work. There is a direct tunnelling matrix element

between |01〉 and |10〉, and T12 appears to first order in the oscillation frequency. There

is no direct tunnelling between |00〉 and |11〉, and so the first non-zero term is 8T1T2EC12

. This

can be thought of representing the virtual tunnelling processes of a Cooper pair tunnelling

through one of the intermediate states |01〉 or |10〉.

a) |11〉 = T2|01〉 = T2(T1|00〉)

b) |11〉 = T1|10〉 = T1(T2|00〉)

where T1 represents that part of the Hamiltonian describing tunnelling from the reser-

voir to island 1. The labels a and b correspond to the diagrams fig. 4.10. We can interpret


0

0.2

0.4

0.6

0.8

1

1 2 3 4 5t

Figure 4.9: The change in frequency of the oscillations in the probability of being in

the state |00〉. The solid line is the oscillation with T12 = +10 and the dashed line is

the oscillation with T12 = −10. The vertical lines are the periods of these oscillations

calculated from the energy difference in eq. 4.3.4.

4.4 Path Interference 87

1 2 2 1

1

2

3 1

2

3

a)

d)c)

b)

Figure 4.10: The two second order (a, b) and third order (c, d) tunnelling processes in-

volved in the oscillation |00〉 → |11〉.

the other terms in the frequencies in a similar manner; the third order term in e1 − e4 is8T12(T 2

1+T22 )

E2C12

. This represents the two third-order tunnelling processes whereby the state

|00〉 can tunnel to |11〉:

c) |11〉 = T1|10〉 = T1(T12|01〉) = T1(T12(T1|00〉))

d) |11〉 = T2|01〉 = T2(T12|10〉) = T2(T12(T2|00〉))

These processes can be illustrated graphically (fig. 4.10):

We see that the oscillation frequencies consist of terms representing all possible tun-

nelling paths. This observation can give us some insight into the behaviour of the oscilla-

tions as a function of the magnetic flux through the circuit.

4.4 Path Interference

We can view the interference fringes in fig. 4.6, 4.7 as being due to interference between

all possible paths the Cooper pairs could take to go from the initial to the final state.

Unfortunately, the Hamiltonian can not be solved analytically for a general phase, and

so we must go to perturbation theory. Here again we have a problem, as the charging

Hamiltonian on its own has doubly-degenerate energy eigenvalues, and so considering the

tunnelling as a perturbation on the charging-only Hamiltonian will not be simple! Instead,


we can take our unperturbed Hamiltonian as including both the charging interaction

and the tunnelling to the reservoir from the islands, and then considering the tunnelling

between islands as a perturbation on this.

H = H0 +H ′

=

0 −T1 −T2 0

−T1 −EC12/2 0 −T2−T2 0 −EC12/2 −T10 −T2 −T1 0

+

0 0 0 0

0 0 −T ∗12 0

0 −T12 0 0

0 0 0 0

(4.4.1)

The unperturbed energy eigenvalues are:

e01,4 = −1

4EC12 +

1

4

√(EC12)2 + 16(T1 ± T2)2

e02,3 = −1

4EC12 −

1

4

√(EC12)2 + 16(T1 ± T2)2 (4.4.2)

The eigenvectors can be simply written in terms of the eigenenergies, for example:

|v01〉 =

1

−e01/(T1 + T2)

−e01/(T1 + T2)

1

1(2 + 2

(e01)2

(T1+T2)2

)1/2 (4.4.3)

We can now do perturbation theory in T12. This allows us to investigate the oscillations

as a function of phase on T12, i.e. as a function of magnetic flux through the circuit. We

can, for instance, find the oscillations by expanding the energy differences in all three tun-

nelling energies. Using perturbation theory to first order in T12 and expanding to second

order in T1, T2, we find that the eigenvalues, including the secound order perturabtions

e′, are:

(e01 − e04) + (e′1 − e′4) = −8T1T2EC12

− 8T12(T21 + T 2

2 )

E2C12

(e02 − e03) + (e′2 − e′3) =8T1T2EC12

− (T12 + T ∗12)− 8(T12 + T ∗12)T1T2E2C12

(4.4.4)


2

1

1

3

2 1

3

2

a)

c)b)

3

Figure 4.11: The three third order tunnelling processes involved in the oscillation |01〉 →|10〉.

These expressions agree with those given above (eqns. 4.3.4, 4.3.5) for the case when

T12 is real. It should be noted we have not done a full expansion to third order in T/EC12,

but have instead expanded to different orders in the different matrix elements. To get a

full third order expansion, we would have to expand to third order in T1, T2 and do third

order perturbation theory in T12.

Nevertheless, these calculations indicate that we can indeed think of the oscillations

as interference between different tunnelling paths, and also suggests that a diagrammatic

approach may be used to ‘keep track’ of the terms in the expansion. For instance, we

know that the energy difference e1 − e4 = (e01 − e04) + (e′1 − e′4) above (eq. 4.4.4) is correct

to third order in all tunnelling matrices as there are no other possible diagrams going from

the state |00〉 to |11〉. We also know that the correction to e2 − e3 = (e02 − e03) + (e′2 − e′3)is incomplete, as there is a (T12)

3 diagram missing (figure 4.11).

It is also worth noting that this idea of using diagrams to keep track of terms does

not only apply to the degeneracy point (n1g, n2g = 1/2), but would work for any pertur-

bation theory about the charging-only Hamiltonian. Away from the degeneracy point, of

course, we also have the advantage that the charge states are not degenerate, making the

perturbations easier to calculate.

When the condition T ¿ EC12 is relaxed, more and more paths contribute to the

frequencies, but the flux dependence can still be observed. We also observe oscillations


0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1t

Figure 4.12: The oscillations of the two qubit system when the system is not in the limit

T ¿ EC12. The solid line is probability of state |00〉, the dashed line is the probability

of |11〉 and the grey lines are the probability of being in state |01〉 or |10〉. The vertical

lines are the periods of these oscillations calculated from the energies in eq. 4.3.3. The

parameters are: EC12 = 50, T1 = T2 = T12 = 10


0

1

2

3

4

5

6

00.2

0.40.6

0.81

1.21.4

00.20.40.60.8

1

Figure 4.13: The Probability of the Two Qubit system being in the state |01〉 or |11〉 i.e.the probability that the first qubit is occupied regardless of the occupation of the second.

The z axis represents increasing probability.

from |00〉 into the states |01〉 and |10〉, with a frequency given by e1 − e2 (figure 4.12).

These oscillations are in the probabilities of the system being in the two-qubit states.

To calculate them, all four states must be distinguished from each other. This would

require instantaneous measurements of the charge on each island at the same time, for a

range of times over the evolution.

These single shot measurements may be difficult to perform at present, so the tun-

nelling current off each island may be measured instead. This measures the probability

of a Cooper pair being on each island independently. In effect these currents will be mea-

suring P (01) + P (11) and P (10) + P (11), where P (00) is the probability of finding the

system in state |00〉, etc.The interference fringes can still be observed, as in fig. 4.13. The importance of this

is that the interference fringes are a signature of coherent quantum tunnelling that can

be detected with an ensemble (current) measurement on an individual qubit, and as such

may be much a more experimentally accessible method of inferring quantum behaviour

than, for example, measuring a Bell inequality.

Chapter 5

Finite Superconductors as Spins:

The Strong Coupling Limit

In this chapter we discuss a microscopic model of a finite region of superconductor. This

model is known as the strong coupling approximation where the interaction potential is

much larger than the cutoff energy. This allows us to write the well-known BCS Hamilto-

nian in terms of large but finite quantum spins. After recapping the quantum mechanics

of spins, we look at the solution of this Hamiltonian, both with and without the mean field

approximation, which we compare to the standard BCS solution and to the Richardson

solution.

5.1 The Strong Coupling Approximation

We wish to consider the behaviour of small islands of superconductor, which are small

enough for finite size effects to come into play. Eventually we will wish to consider the

interactions between such islands, and interactions of these islands with a superconducting

reservoir, as has already been done in chapter 3, by quantising the Josephson equations

for each circuit. We can describe a single isolated island using the BCS Hamiltonian [27]:

H =∑

k, σ

εkc†k, σck, σ −

∑

k, k′

Vk, k′ c†k↑c

†−k↓c−k′↓ck′↑ (5.1.1)

As we are discussing a finite superconducting island, the label k does not refer to a

free electron wavevector but to a generic single-electron eigenstate.

93

94 Finite Superconductors as Spins

A common approximation to this equation is made by assuming the pairing potential

Vk, k′ is equal for all k, k′ in a region around the Fermi energy determined by the cutoff

energy ~ωc and zero outside this region. That is, Vk, k′ = V for |εk−εF | < ~ωc and Vk, k′ = 0

otherwise. This greatly simplifies the above equation whilst retaining the essential physics.

H =∑

k, σ

εkc†k, σck, σ − V

εF+~ωc∑

k, k′

εk=εF−~ωc

c†k↑c†−k↓c−k′↓ck′↑ (5.1.2)

To render the problem tractable, we make a further approximation; we take all the

single electron energy levels εk within the cutoff region around the Fermi energy to be

equal to the Fermi energy εF . This crude approximation turns out to be equivalent to

taking the strong coupling limit [90] as will be discussed further in 5.4.4. We can write

the kinetic energy term as:

Ho =∑

k, σεk<εF−~ωc

εkc†k, σck, σ +

εF+~ωc∑

k, σεk=εF−~ωc

εkc†k, σck, σ +

∑

k, σεk>εF+~ωc

εkc†k, σck, σ (5.1.3)

where k is no longer a wavevector, but is a generic label for the single electron energy

levels, with −k representing the time-reversed state. Outside the cutoff region, there is no

scattering and the Hamiltonian is diagonal in k. Considering only the terms within the

cutoff region (the dashes on the sums indicate they are taken over the region εF − ~ωc <εk < εF − ~ωc), we have:

Ho =′∑

k, σ

(εF + (εk − εF )) c†k, σck, σ (5.1.4)

The term (εk − εF ) never has a magnitude greater than ~ωc. In the strong coupling

limit, ~ωc ¿ V , so (εk − εF ) can be discarded. We are left with a Hamiltonian acting on

terms only within the cutoff region:

H = εF

′∑

k

c†k, σck, σ − V′∑

k, k′

c†k↑c†−k↓c−k′↓ck′↑ (5.1.5)

Having made this approximation, we now define operators that correspond to the sums

in the Hamiltonian:

5.2 Quantum Spins 95

SZ =1

2

′∑

k

c†k↑ck↑ + c†−k↓c−k↓ − 1

S+ =′∑

k

c†k↑c†−k↓

S− =′∑

k

c−k↓ck↑ (5.1.6)

These sums obey the usual commutation relations for quantum spins (see section 5.2)

and are essentially Cooper pair number, creation and annihilation operators. Therefore,

we can express our approximated Hamiltonian solely in terms of quantum mechanical spin

operators. Discarding constants, the Hamiltonian becomes:

Hsp = 2εFSZ − V S+S− (5.1.7)

We have now rewritten our Hamiltonian in a vastly simplified form. The strong cou-

pling limit has been studied since early days of BCS theory [91, 92, 93], for materials such

as lead. More recently it has been studied, in particular, as a treatment of high tempera-

ture superconductivity (see eg. [94]) and as a description of nanoscopic superconducting

grains [95] where the symmetry of the grain can cause the degeneracy of the energy levels

to be very large.

Obviously, we need to check the validity of this approximation and its relation to well

known existing results. Before doing this, however, we shall recap the theory of quantum

spins.

5.2 Quantum Spins

We begin our discussion of quantum spins with the commutation relations of our three

operators (5.1.6). All the properties of quantum spins can be derived from these relations

without reference to their physical ‘spin’ nature - i.e. once we have the commutation

relations, we do not need any further physical intuition to find their properties. It is

important to emphasise this as we are discussing a system which is not a real quantum

spin, but merely shares the same commutation relations.

From our superconducting perspective, these commutation relations follow directly the

definition of the operators in terms of electron creation and annihilation operators.


[SZ , S−] =′∑

k, k′

[1

2(c†k↑ck↑ + c†−k↓c−k↓ − 1), c−k′↓ck′↑]

=

′∑

k, k′

1

2[c†k↑ck↑, c−k′↓ck′↑] +

1

2[c†−k↓c−k↓, c−k′↓ck′↑]

=′∑

k, k′

1

2δk, k′(−c−k′↓ck′↑ − c−k′↓ck′↑)

= −′∑

k

c−k↓ck↑

= −S− (5.2.1)

Similar calculations give the other two commutation relations.

[SZ , S+] = S+

[SZ , S−] = −S−

[S+, S−] = 2SZ (5.2.2)

Next we define another operator, S2:

S2 = (SZ)2 +

(S+ + S−

2

)2

+

(S+ − S−

2i

)2

(5.2.3)

In the interpretation of these operators as quantum spin operators, this operator mea-

sures the total spin l2(

l2 + 1) and can be written S2 = (SZ)2 + (SX)2 + (SY )2, with the

Hermitian operators SX = (S+ + S−)/2 and SY = (S+ − S−)/2i. We do not need to

make this physical interpretation yet however, and instead note that [S2, SZ ] = 0. This

means that the operators share a common set of eigenvectors:

S2|l, N〉 = α|l, N〉

SZ |l, N〉 = β|l, N〉 (5.2.4)

We have given the eigenstates these labels because we are anticipating their physical

meaning. The label l will turn out to be the number of levels within the cutoff region

around the Fermi energy εF ± ~ωc. In terms of physical spins, l2 gives the size of the spin,

5.2 Quantum Spins 97

sometimes denoted J . N will turn out to be the number of Cooper pairs on the island and

relates to the eigenvalue of SZ through N − l2 = m. For now though, l and N are merely

labels of the eigenstates with unknown eigenvalues α and β. We take the expectation

value of S2 with one of these common eigenstates:

〈l, N |S2|l, N〉 = 〈l, N |(SZ)2|l, N〉+ 〈l, N |(SX)2|l, N〉+ 〈l, N |(SY )2|l, N〉 (5.2.5)

We have defined SX and SY to be Hermitian, and therefore the expectation values of

these operators are the magnitudes of the vectors SX |l, N〉 and SY |l, N〉. The magnitude

of a vector is always ≥ 0, and so we get:

〈l, N |S2|l, N〉 ≥ 〈l, N |(SZ)2|l, N〉

α ≥ β

This means that there is a state with a minimum SZ eigenvalue. We call this state |l, 0〉and define − l

2 as its eigenvalue. Again, the eigenvalue has been chosen in anticipation of

its physical meaning.

As well as the minimum SZ eigenstate, we wish to find the other eigenstates of SZ .

We use a proof by induction to show that the states (S+)N |l, 0〉 are eigenstates of the

operator SZ . If |l, N − 1〉 is an eigenstate with eigenvalue (N − 1)− l2 , then:

SZ |l, N〉 = SZ(S+)N |l, 0〉

= ([SZ , S+] + S+SZ)(S+)N−1|l, 0〉

= (S+ + S+SZ)(S+)N−1|l, 0〉

= (1 + (N − 1)− l

2)(S+)N−1|l, 0〉 (5.2.6)

As the state |l, 0〉 is an eigenstate of SZ with eigenvalue − l2 , we have proved:

SZ |l, N〉 = (N − l

2)|l, N〉 (5.2.7)

Similarly, we can use another proof by induction to find the correct normalisation

constants. Consider the action of the S− operator on the state |l, N + 1〉. Assuming that


S−(S+)N−1|0〉 = ((N−1)l−(N−2)(N−1))(S+)N−2|0〉, which is trivially true for N = 1,

we find:

S−(S+)N |0〉 = ([S−, S+] + S+S−)(S+)N−1|l, 0〉

= (−2SZ + S+S−)(S+)N−1|l, 0〉

= (l − 2N + 2)(S+)N−1|l, 0〉

+S+((N − 1)l − (N − 2)(N − 1))(S+)N−2|l, 0〉

= [Nl −N(N − 1)](S+)N−1|l, 0〉 (5.2.8)

So the final line is true for all N . We use this to find the magnitude of (S+)N |0〉.

〈l, 0|(S−)N (S+)N |l, 0〉 = (Nl −N(N − 1))〈l, 0|(S−)N−1(S+)−1N |l, 0〉

= (Nl −N(N − 1))l!(N − 1)!

(l − (N − 1))!

= (l − (N − 1)))(l!N !)

(l − (N − 1))!(5.2.9)

Thus, the assumption 〈l, N−1|l, N−1〉 = l!(N−1)!(l−(N−1))! coupled with the observation that

〈l, 0|l, 0〉 = 1 proves that the magnitudes of the un-normalised states is given by:

〈l, 0|(S−)N (S+)N |l, 0〉 = (N)!l!

(l −N)!(5.2.10)

With this value for the normalisation constant, we can define the orthonormal eigen-

states of the SZ operator:

|l, N〉 = (S+)N |l, 0〉√

(l −N)!

l!N !(5.2.11)

and the previous calculations also allow us to calculate the action of the spin operators:

SZ |l, N〉 = (N − l

2)|l, N〉

S+|l, N〉 =√

(N + 1)(l −N)|l, N + 1〉

S−|l, N〉 =√N(l − (N − 1))|l, N − 1〉

S2|l, N〉 =l

2(l

2+ 1)|l, N〉 (5.2.12)

5.3 The Majorana Representation 99

In summary, we have a set of operators that act on the state space |l, N〉. For each

eigenvalue of S2, l2 , we have l states |l, N〉 where N runs from zero to l, with eigenvalues

N − l2 .

It is worth taking a moment to re-connect these results, which we have been thinking

of in terms of quantum spins, with the original problem of superconductivity in terms of

single-electron creation and annihilation operators. An intuitive way of doing this can be

done be relating our problem to a representation of quantum spins due to Majorana. This

is the topic of the next section.

5.3 The Majorana Representation

5.3.1 Spin l2in terms of Spin 1

2

In his 1932 paper [96], Majorana noted that the Hilbert space of a spin l2 system could be

represented by the product of l spin 12 systems. In the description of the system above,

this has a very clear meaning.

The spin operators defined in 5.1.6 can be viewed as a sum of operators labelled k,

where k runs from 1 to l:

σZk =1

2(c†k↑ck↑ + c†−k↓c−k↓ − 1)

σ+k = c†k↑c†−k↓

σ−k = c−k↓ck↑ (5.3.1)

(5.3.2)

Each of the k sets of operators acts like a set of spin-half operators, e.g.:

σ+k |0〉k↑|0〉−k↓ = c†k↑c†−k↓|0〉k↑|0〉−k↓

σ+k | ↓〉k = | ↑〉kσZk|0〉k↑|0〉−k↓ =

1

2(c†k↑ck↑ + c†−k↓c−k↓ − 1)

= −1

2|0〉k↑|0〉−k↓

σZk| ↓〉k = −1

2| ↓〉k (5.3.3)


The vacuum state |l, 0〉 refers to the product of the l vacuum states |0〉k↑|0〉−k↓, i.e.the Hilbert space of our spin J operators can be represented as the product of l spin-half

down spaces, where each spin-half space refers to the absence or presence of a pair in a

particular energy level.

The action of the S+ operator is to add a Cooper pair to the system, in such a way

that there is an equal amplitude of the pair being in any of the l levels. In spin language,

this gives:

|l, 0〉 = | ↓↓↓ · · · ↓↓〉

S+|l, 0〉 = | ↓↓↓ · · · ↓↑〉+ | ↓↓↓ · · · ↑↓〉+ · · ·+ | ↑↓↓ · · · ↓↓〉

(S+)2|l, 0〉 = | ↓↓↓ · · · ↑↑〉+ | ↓↓↑ · · · ↑↓〉+ · · ·+ | ↑↑↓ · · · ↓↓〉

(S+)N |l, 0〉 = N !| ↓↓ · · · ↓︸︷︷︸ ↑ · · · ↑︸︷︷︸〉sym (5.3.4)

N (l −N)

where the sym subscript refers to a symmetric sum of all possible states with N up-spins,

divided by N!. This last equation gives a description of the |l, N〉 states in terms of spin-

half particles. A general state will be a superposition of such states. It turns out that a

general state can also be represented by l spin-half states, but with general orientation,

instead of just up or down, i.e.:

|ψ〉 = aN

l∑

N=0

|l, N〉

= [| ↓ · · · ↓〉sym + (∑

z∗k)| ↓ · · · ↓↑〉sym + · · ·+ (∏

z∗k)| ↑ · · · ↑〉sym]

= |(↓ +z∗1 ↑)(↓ +z∗2 ↑) · · · (↓ +z∗l ↑)〉sym= | ↓← · · · 〉sym (5.3.5)

The complex numbers z∗k are the coefficients of spin-up for each of the spin-half states.

They will be of particular interest in a special set of states: coherent states.

5.3.2 The Coherent State

When considering the harmonic oscillator we define a state as the eigenstate of the de-

struction operator a†. This is the coherent state (see eg. [97]):

5.3 The Majorana Representation 101

|α〉 =∞∑

N=0

αN (a†)N |0〉N !

(5.3.6)

We can define a similar state for our system. This is usually referred to as a (spin)

coherent state [97], but it should be noted that whilst the harmonic oscillator coherent

state (eq. 5.3.6) is defined as an eigenstate of the destruction operator, the analogous spin

state is not an eigenstate because the sum does not run to infinity.

|l, z〉 = 1√2l

l∑

N=0

(z∗S+)N |l, 0〉N !

(5.3.7)

The coherent state is a state where all the component spin-half particles are pointing in

the same direction, i.e. all the points on the Majorana sphere (see section 5.3.3) converge

to one location. The coherent state with z = cot(θ2

)eiφ is the state with maximal spin l

2

along the axis θ, φ. That is, if we define an operator that gives the component of spin in

the direction (θ, φ), then:

Sθ,φ|l, z〉 =l

2|l, z〉 (5.3.8)

We can easily see that this is true for the axis z, i.e. θ = 0 or θ = π. These cases

correspond to the limits |z| → 0 and |z| → ∞ respectively, and examining the definition

above (eq. 5.3.7) show that in these cases the coherent state tends to |l, 0〉 or |l, l2〉.In a real spin, these states are important states; they are the maximal eigenstates of

spin in each direction. The states |l, 0〉 and |l, l2〉 are only two such states, determined by

the definition of the z-axis, and they can be transformed into any other coherent state by

a rotation of co-ordinate basis.

5.3.3 The Majorana Sphere

The values zk in eq. 5.3.5 have a geometric interpretation. Consider the Bloch Sphere, as

a representation of a spin-half particle. Here every pure state |l = 1, ψ〉 (i.e. |j = 12 , ψ〉)

can be uniquely represented by a point on the sphere, with the north pole corresponding

to spin up, the south pole to spin down, and all other states lying on the surface of the

sphere.

Each state can also be specified by a complex number z. If the complex plane passes

through the equator of the sphere, and a line is passed from a point on the sphere to the


a) b)

Figure 5.1: The Majorana Sphere. Each state is represented by a series of points on the

surface of a sphere. In the coherent state, all the points coincide (a), whereas for ZZ

eigenstates, they are distributed around the sphere at a constant ‘latitude.’

north pole, then it crosses the plane at a single point z. Thus, a complex number can

represent a point on the sphere, and thus a spin-half state.

|l = 1, ψ〉 = a1| ↑〉+ a0| ↓〉

= cosθ

2e−i

φ2 | ↑〉+ sin

θ

2ei

φ2 | ↓〉 (5.3.9)

z =1

tan θ2

eiφ

=a0a1

(5.3.10)

In the Majorana representation [96, 98], a spin l2 state is represented by l spin-half

states, i.e. l points on the sphere (figure 5.1), or l projections, zk, onto the complex plane

(figure 5.2).

Any state, then, can be represented by a series of points on a sphere. In particular,

for the coherent state, all the zk are equal, and all the points on the Majorana Sphere

coincide. This point corresponds to the direction in which the state is a maximal spin

state.

As mentioned above, in the case of the maximal and minimal eigenstates of the SZ

operator, the value of z is zero or infinity. The point z = 0 on the complex sphere

corresponds to the south pole, and z =∞ corresponds to the north pole. Thinking about

the states in this way makes it very clear that |l, 0〉 and |l, l2〉 are merely coherent states,

and occupy a special position only in reference to a chosen co-ordinate basis.

5.4 Mean Field Spin Solution 103

z

Figure 5.2: Each point on the Majorana sphere can be represented by its projection onto

the complex plane, z.

5.4 Mean Field Spin Solution

We wish to ensure that our approximation to eq. 5.1.2 is valid. One way to increase

our confidence in the model is to compare its solutions to solutions to the full model.

In particular, we will find the mean-field solution of 5.1.7 and compare it to the mean-

field solution of 5.1.2. When making the mean field approximation, we assume that the

operators S± are close to their expectation values. Writing S± = 〈S±〉+(S± − 〈S±〉), wecan rewrite the Hamiltonian 5.1.7. The mean field Hamiltonian is derived by discarding

terms to second order or higher in (S± − 〈S±〉).

Hsp = 2(εF − µ)SZ − V(〈S+〉+

(S+ − 〈S+〉

)) (〈S−〉+

(S− − 〈S−〉

))(5.4.1)

≈ 2(εF − µ)SZ − V(〈S+〉S− + 〈S−〉S+ − 〈S+〉〈S−〉

)

= 2(εF − µ)SZ −∆S+ −∆∗S− +|∆|2V

(5.4.2)

The term 〈S+〉〈S−〉 is a constant, and will be unimportant for most purposes, and we

shall discard it below when we find the mean field solution. We will need to take note of

it later, however, when we compare the mean field solution to the exact solution (section

5.5).

HMF = 2(εF − µ)SZ −∆S+ −∆∗S−

= 2ξf SZ − SX(∆ +∆∗)− SY i(∆−∆∗) (5.4.3)

Now the use of this spin-model becomes evident. Observing the above Hamiltonian,


ZoZn

Yn

Xn

Yo Xo

o n

Figure 5.3: Rotation of basis of spin operators. The operators SZo , S+o , S

−o refer to a

particular basis o. We can consider another basis, n with three operators SZn , S+n , S

−n . The

operators corresponding to any basis can be written as a linear combination of any other

basis.

we see it is a linear combination of SZ , SY and SX operators. This means that the whole

Hamiltonian is proportional to a spin operator in a different direction. We have reduced

the problem of finding the eigenvalues of eq. 5.4.3 to the problem of rotating a quantum

mechanical spin.

5.4.1 Rotation of Spin Operators

The operators SZ , S+, S− respectively represent a measurement, raising and lowering of

angular momentum with respect to a particular frame of reference, say o. For clarity, we

call these three original operators SZo , S+o , S

−o .

Other unit vectors, n, define other frames of reference, and therefore each direction n

has three operators, SZn , S+n , S

−n , associated with it. An operator in one frame of reference

can be expressed as a combination of the three operators in any other frame of reference.

We want to find a diagonal representation for our Hamiltonian; we want to find a

direction n, such that HMF = γSZn . If we can do this, the eigenstates of the Hamiltonian

are the spin states |l, Nn〉.We need to find the raising and lowering operators for this new reference frame. Again


we express the new operator S+n in terms of the old operators SZo , S

+o , S

−o , with d, e, f the

unknown parameters.

S+n = dSZo + eS+

o + fS−o (5.4.4)

Inserting the expressions for SZn and S+n (5.4.4, 5.4.3) into the commutation relation

for spins will allow us to find d, e and f :

[SZn , S+n ] = S+

n

[1

γ(2ξFS

Zo −∆S+

o −∆∗S−o ), (dSZo + eS+

o + fS−o )] = dSZo + eS+o + fS−o

(2∆∗e− 2∆f)SZo + (2ξF e+∆d)S+o + (−∆∗d− 2ξF f)S

−o = γ(dSZo + eS+

o + fS−o )

(5.4.5)

Equating the coefficients in the commutation relation gives three simultaneous equa-

tions:

(2∆∗e− 2∆f) = γd

(2ξF e+∆d) = γe

(−∆∗d− 2ξF f) = γf (5.4.6)

Solving these gives:

S+n = d(SZ +

−∆2ξF − γ

S+ +−∆∗

2ξF + γS−)

γ =√

(2ξF )2 + 4|∆|2 (5.4.7)

Where we note that γ is equal to twice the quasiparticle energy at the Fermi level, EF .

A similar process gives an expression for S−n (the Hermitian conjugate of S+n ). Finally, we

can use the commutation relation [S+n , S

−n ] = 2SZn to find the parameter d.

Summarising, we found that we could express a mean-field Hamiltonian in terms of

spin operators SZo , S+o , S

−o . This Hamiltonian is then recognized as a spin operator in

another direction, n. The spin commutation relations allow us to find the raising and

lowering operator as well.


γSZn = 2ξFSZo − S+

o ∆− S−o ∆∗

S+n = dSZo + eS+

o + fS−o

|d| = |∆|/EFe =

−d∆2ξF − 2EF

f =−d∆∗

2ξF + 2EF(5.4.8)

We can write the above transformation as a matrix. Using the definitions for d, e, and

f given in eq. 5.4.8, we have:

2ξF /γ ∆/γ ∆∗/γ

d e f

d∗ f∗ e∗

SZo

S+o

S−o

=

SZn

S+n

S−n

(5.4.9)

Note that this matrix is not unitary. This is due to the fact the matrix acts on the

operators SZ , S+, S− rather than on SZ , SX , SY . The inverse of the above matrix is

given by:

2ξF /γ d∗/2 d/2

2∆∗/γ e∗ f

2∆/γ f∗ e

SZn

S+n

S−n

=

SZo

S+o

S−o

(5.4.10)

The fact that the above matrix is not unitary is just a normalisation problem that

arises because the raising and lowering operators are defined as S± = (SX ± iSY ) withouta normalisation constant of 1/

√2.

For completeness, we can write the rotation operators in terms of the operators SZ ,

SX , SY .

2ξF /γ (∆ +∆∗)/γ i(∆−∆∗)/γ(d+d∗)

2(e+f+e∗+f∗)

2 i (e−f−e∗+f∗)

2(d−d∗)

2i(e+f−e∗−f∗)

2i(e−f+e∗−f∗)

2

SZo

SXo

SYo

=

SZn

SXn

SYn

(5.4.11)

This matrix is indeed unitary, as expected.


5.4.2 The Ground State

Our identification of the S operators as spin operators allows us to recognise that the

mean-field Hamiltonian is proportional to an operator that measures the spin in some

direction, n. We can simply rewrite the Hamiltonian (eq. 5.4.3) as:

HMF = 2(εF − µ)SZo −∆S+o −∆∗S−o

= γSZn (5.4.12)

From this, it is obvious that the ground state will be the minimum eigenstate of spin

in the direction n and that the eigenvalue will be −γ l2 . This does not tell us how to write

the ground state in terms of our original operators SZ,+,−o , however. To do this we utilise

the results of section 5.3.2, and note that the minimum spin state in the direction n will be

a coherent state in terms of the operators SZ,+,−o . We still do not know what the direction

n is, and thus do not know the parameter z in equation 5.3.7.

We find the parameter z (and therefore find n) by noting that if a state is the minimal

spin state along the direction n, it will be destroyed by the lowering operator S−n along

that direction i.e. S−n |l, z〉 = 0.

S−n |l, z〉 = 0

=

(SZo +

∆

2ξF + γS+o +

∆∗

2ξF − γS−o

)1√2l

l∑

N=0

(z∗S+o )

N |l, 0〉N !

=l∑

N=0

((N − l

2)(z∗)N

N !+

∆(z∗)N−1

(2ξF + γ)(N − 1)!

+∆∗(N + 1)(l −N)(z∗)N+1

(2ξF − γ)(N + 1)!(S+o )

N

)|l, 0〉 (5.4.13)

If this state is to be equal to zero, the coefficients must be zero for all N . For example,

considering the coefficient of the l = 0 term:

− l2+

∆∗z∗l2ξF − γ

= 0

l

2=

∆∗z∗l2ξF − γ

z∗ =2ξF − γ2∆∗

(5.4.14)


Therefore we have found that the ground state of eq. 5.4.3 is a coherent state (eq.

5.3.7) with z = 2ξF−γ2∆ . Directly calculating the action of SZn on this state confirms it is

indeed the ground state.

HMF |l, z〉 = γSZn |l, z〉

= − l2γ|l, z〉

= − l2

√4(εF − µ)2 + 4∆2|l, z〉

= −l√

(εF − µ)2 +∆2|l, z〉 (5.4.15)

Remembering that we wish to compare the solution to the mean-field spin Hamiltonian,

it may not be immediately obvious how the spin coherent state above relates to the BCS

state (eq. 1.6.6). We can, however, rewrite the coherent state so that the connection is

clear:

|l, z〉 =1√2l

l∑

N=0

(z∗S+)N |l, 0〉N !

=1√

(1 + |z|2)l∏

k

(1 + z∗ c†k↑c†−k↓)|l, 0〉 (5.4.16)

Above we see that the coherent state is equal to the BCS state when all the parameters

uk and vk are equal.

5.4.3 Self Consistency

With the ground state of the mean-field Hamiltonian found, we need to choose the param-

eters ∆ and µ to give self-consistency and also the correct value for the average number

of particles.

We want to choose µ such that the ground state α gives 〈l, z|N |l, z〉 = N . The section

above showed that the state |l, z〉 is the ground state of the Hamiltonian. We calculate

the expectation value of the (pair) number operator, N = SZo + l2 with this state:

N = 〈l, z|N |l, z〉

N =l

2

(1− εF − µ√

(εF − µ)2 − |∆|2

)(5.4.17)


We also have to chose ∆/V to be equal to the expectation value of the S+ operator:

∆∗ = V 〈l, z|S+|l, z〉

= Vl

2

∆∗√(εF − µ)2 − |∆|2

(5.4.18)

In full BCS theory, these equations involve a sum over levels k, and we have two

integral equations. For our simplified spin model, the fact that the εk are all equal means

that we can trivially sum over k, giving two coupled algebraic relations instead of integral

equations. Thus we can evaluate ∆ and µ exactly. We find:

|∆|2 = V 2N(l − N) (5.4.19)

εF − µ = V (l/2− N) (5.4.20)

With these values we have the mean field solution for a given average number N .

Including the constant that we discarded from 5.4.1, we have:

HSMF = 2(εF − µ)SZo −∆S+o −∆∗S−o +

|∆|2V

= γSZn +|∆|2V

(5.4.21)

When we insert 5.4.19 and 5.4.20 into γ and ∆, we obtain:

HSMF |l, z〉 =

(γSZn +

|∆|2V

)|l, z〉

=

(− l2γ +|∆|2V

)|l, z〉

=

(−V l

2

2+ V N(l − N)

)|l, z〉

(5.4.22)

5.4.4 Comparison to the BCS Theory

In order to check the viability of neglecting the differences between electron energy levels

εk, we solve the gap equations for a general BCS case, and then take the limit where


all the levels have the same energy. To take this limit, we consider a top-hat density of

states centred around εF with width w and height l/w. This allows the limit where the

bandwidth w → 0 to be taken whilst keeping a constant number of levels, l. As shown in

appendix B, at zero temperature the integrals can be calculated exactly. We find:

∆2 = (ξFr)2

((x+ 1)2

(x− 1)2− 1

)+(w2

)2((x− 1)2

(x+ 1)2− 1

)(5.4.23)

Nr =lrw

(w

2−√

(ξFr +w

2)2 +∆2 +

√(ξFr −

w

2)2 +∆2

)(5.4.24)

With x = e2wV lr and n = Nr

lr. These two equations can be solved to give ∆ and ξFr.

When the limit V/w →∞ (the strong coupling limit) is taken, we find that we obtain the

same results as given by the spin model, (5.4.19, 5.4.20).

Evidently, the fact that this spin representation reproduces the BCS results, albeit only

in the strong coupling limit, lends credibility to our simplified model, and allows us to con-

sider the situations we wish to describe, namely those of small regions of superconductor,

or ‘Cooper Pair Boxes’, coupled to a superconducting reservoir.

Before we discuss these systems, let us examine the results we obtain for our single

isolated island if we do not make the mean field approximation.

5.5 Solving the Spin Hamiltonian Exactly

Examining the mean field solution of our Hamiltonian confirms that, at least in the mean

field, our spin Hamiltonian behaves regularly, and the results are those that would be

obtained by solving the full (k-dependent) problem and then taking the limit ~ω ¿ V .

However, we do not need to take the mean field approximation to solve the Hamiltonian.

In fact the solution is rather simple.

Hsp = 2εFSZ − V S+S− (5.5.1)

Equation 5.5.1 is the strong coupling limit of the BCS Hamiltonian. It is easy to see

that this is diagonal in |l, N〉, the eigenstates of SZ , as the S− operator reduces by one

the number of Cooper Pairs (N) on the island, and the raising operator increases it by

one. Equations 5.2.12 allow us to find the effect of the operator S+S− on an eigenstate of

5.5 Solving the Spin Hamiltonian Exactly 111

SZ . This is: S+S−|l, N〉 = N(l − N + 1)|l, N〉. With this we can find the action of the

Hamiltonian on its eigenstates:

Hsp|l, N〉 = EN |l, N〉 (5.5.2)

EN = 2(εF − µ)(N −l

2)− V N(l −N + 1) (5.5.3)

We need to find which of these eigenstates is the ground state. We find the lowest

energy solution by differentiating the eigenvalue with respect to N and setting equal to 0:

dENdN

= 2(εF − µ)− V (l + 1) + 2V N = 0

2(εF − µ) = V (l + 1)− 2V N (5.5.4)

This determines the chemical potential for a given N . If we insert this value back into

the Hamiltonian, we have a Hamiltonian which is restricted so that the expectation of the

number operator in its ground state is always equal to a specified value, N .

Hsp = (V (l + 1)− 2N)SZ − V S+S− (5.5.5)

Hsp|l, N〉 = (V (l + 1)− 2N)(N − l

2)− V N(l − N + 1) |l, N〉

=

(−V l

2(l + 1) + V N(l − N)

)|l, N〉

(5.5.6)

If we compare this to the mean-field result:

HSMF |l, z〉 =

(−V l

2

2+ V N(l − N)

)|l, z〉

(5.5.7)

We see that the mean field result has a term l2

2 which in the exact result is replaced

by l2(l + 1). This difference is due to quantum fluctuations, and becomes especially clear

at half-filling (N = l2). The classical energy of a spin J is J2 (=

(l2

)2), but quantum

fluctuations mean the energy is given by S2, i.e. J(J + 1) (= l2

(l2 + 1

)). The quantum

fluctuations are not captured by the mean field theory but are described by the exact

solution.


The mean-field solution gives a non-zero pairing parameter 〈l, z|S−|l, z〉 = ∆V = l

2 . The

exact solution gives 〈l, l2 |S−|l, l2〉 = 0. Thus, the exact solution has zero pairing parameter,

and therefore cannot be said to be superconducting in the usual sense. This is only to be

expected, as we are dealing with a finite system. None the less, the system does possess

pairing fluctuations:

〈l, l2|S+S−|l, l

2〉 − 〈l, l

2|S+|l, l

2〉〈l, l

2|S−|l, l

2〉 =

l

2

(l

2+ 1

)(5.5.8)

5.5.1 Broken Symmetry: Finite size

We have established that the solution to the exact Hamiltonian has zero pairing parameter.

It may be however, that we could find a solution with broken symmetry that has a lower

energy. To find if this is true, we consider a Hamiltonian which includes a symmetry-

breaking field, η (see eg. [99]). We find the ground state of this Hamiltonian, and the

pairing parameter in this ground state. We then let the magnitude of the field go to zero,

and see if we still have a non-zero pairing parameter. This would indicate a spontaneously

broken symmetry.

Take the broken symmetry Hamiltonian:

HBS = 2(εF − µ)SZ − V S+S− − ηS+ − η∗S− (5.5.9)

To investigate this we consider the symmetry-breaking terms η, η∗ as a perturbation

on the exact spin Hamiltonian 5.1.7 (since η → 0 is the limit of interest).

From eq. 5.5.2 we have:

E(0)N = 2(εF − µ)(N −

l

2)− V N(l −N + 1) (5.5.10)

Using time-independent perturbation theory, we find the first order perturbation in

the energy:

E(1)N = 〈l, N |H ′|l, N〉

= 〈l, N | − ηS+ − η∗S−|l, N〉

= 0 (5.5.11)

5.5 Solving the Spin Hamiltonian Exactly 113

So there is no first-order correction to the energy. The first-order correction to the

eigenstates is:

|ψ(0)N 〉 = |l, N〉

|ψ(1)N 〉 =

∑

N ′ 6=N

H ′N ′,N

E(0)N − E

(0)N ′

|ψ(0)N 〉 (5.5.12)

where the matrix elements H ′N ′,N of the perturbing Hamiltonian H ′ = −ηS+−η∗S− are:

H ′N ′,N = 〈l, N ′| − ηS+ − η∗S−|l, N〉

= −η√

(N + 1)(l −N)δN ′,N+1

−η∗√N(l −N + 1) δN ′,N−1 (5.5.13)

and the difference in the energy eigenstates EN of the unperturbed Hamiltonian is given

by:

E(0)N − E

(0)N+1 = −2(εF − µ) + V (l + 1)− 2NV − V

= −2(N −N)− V

E(0)N − E

(0)N−1 = 2(εF − µ)− V (l + 1) + 2NV − V

= −2(N − N)− V

(5.5.14)

Substituting 5.5.14 and 5.5.13 into 5.5.12 gives the first order corrections to the eigen-

states:

|ψ(1)N 〉 = −η

√(N + 1)(l −N)

−2(N −N)− V |ψ(0)N+1〉

−η∗√N(l −N + 1)

−2(N − N)− V |ψ(0)N−1〉 (5.5.15)

The second order correction to the energy is:

E(2)N =

∑

N ′ 6=N

|H ′N ′,N |2

E(0)N − E

(0)N ′

(5.5.16)


Inserting N = N into 5.5.13 and 5.5.14 allows us to calculate the correction to the

ground state:

E(2)N = −η

2(N + 1)(l −N)

V− η2(N)(l −N + 1)

V(5.5.17)

By looking at 5.5.15 and 5.5.17 we see that the broken symmetry solution does have

a lower ground state energy than the non-broken symmetry solution, but that when we

take the η → 0 limit, we find that the correction to the states disappears; we do not

have spontaneous symmetry breaking. The symmetry will break, however, if we have an

external term of the form −ηS+ − η∗S−. This happens when we connect the system to a

larger superconductor, as we shall see later.

5.5.2 Broken Symmetry: Infinite Size

Above we have considered a finite spin, and checked to see if there is a broken symmetry

solution. We find that there is not, for any size of the spin. If, however, we allow the size

of the spin to become infinite first, and only then check for a broken symmetry solution,

we may find a different result.

Starting from eq. 5.5.9 we allow the size to go to infinity. By examining the differential

forms of S+, S−, we see that in the l→∞ limit, S+S− becomes l2

4 . This can also be seen

without recourse to the differential forms if we recall that S+S− = S2 − SZ(SZ + 1) and

make the assumption that SZ ¿ l2 in the infinite limit. Thus, 5.5.9 becomes:

HBS = 2(εF − µ)SZ − Vl2

4− ηS+ − η∗S− (5.5.18)

We see then that this Hamiltonian has the same form as the mean-field Hamiltonian.

We can just read off the result, inserting the relevant parameters.

The ground state is then the coherent state |l, z〉, with:

HBS |l, z〉 =

(γSZn − V

l2

4

)|l, z〉

=

(− l2

(√4(εF − µ)2 + 4|η|2

)− V l

2

4

)|l, z〉

(5.5.19)

As η → 0, the energy at half filling becomes simply −V l2

4 , so we have confirmed

that the broken symmetry solution is indeed degenerate with the non-broken symmetry

solution.

5.6 Expanding the Richardson Solution to First Order 115

We need to examine ground states, to see if the broken symmetry ground state reverts

to the non-broken symmetry ground state as η → 0. If it does not, we have a spontaneous

broken symmetry.

z∗ =2(εF − µ)−

√4(εF − µ)2 + 4|η|22η∗

(5.5.20)

If we are not at half filling, then (εF −µ) is either negative or positive. If it is negative,then as η → 0 z∗ goes to infinity. If (εF − µ) is positive, then we use l’Hopital’s rule and

differentiate both the denominator and the numerator. As η → 0 we get:

z∗ =−4η∗√

4(εF − µ)2 + 4|η|2(5.5.21)

which goes to zero. So, in the limit, z∗ goes to either 0 or infinity. A coherent state with

z = 0 is the state |J,m = −J〉, and a coherent state with z =∞ is the state |J,m = +J〉,so the system is only interesting precisely at half-filling.

At half filling,

z∗ =−√

+4|η|22η∗

=|η|η∗

= eiφ (5.5.22)

|l, z〉 =1√2l

l∑

N=0

(eiφS+o )

N

N !|l, 0〉 (5.5.23)

=1√2l

∏

k

(1 + eiφc†k↑c

†−k↓

)|0〉 (5.5.24)

i.e. the ground state is independent of the magnitude of η in the infinite limit, and the

symmetry remains broken for η → 0. Note that the phase of η still enters, and it is this

that gives the phase of the coherent state.

5.6 Expanding the Richardson Solution to First Order

In section 5.4 we compared the mean field solution of our spin model with the strong-

coupling limit of the mean field solution of the BCS Hamiltonian. In this section we shall


derive the exact solution found in section 5.5 as a large-coupling limit of the Richardson

Solution described in section 1.9.2.

− 1

V+

N∑

ν=1

2

EJµ − EJν=

l∑

k=1

1

EJµ − 2εk(5.6.1)

We shall be taking the strong-coupling limit, that is where λ→∞, where λ = V l/ωc

is the dimensionless coupling constant and we write V = λd, with d = ωc/l the mean level

spacing.

In what follows, we reproduce the derivation of Yuzbashyan et al. [90], with a small

but significant difference. In [90], Yusbashyan et al. were concerned with only isolated

dots, and so after taking the limit, where εk → εF , could disregard the kinetic energy part

of the Hamiltonian (essentially setting εF to zero). We will be discussing coupling between

more than one island, and so will need to keep the kinetic energy term.

As λ → ∞, a number of the Richardson parameters EJ diverge (proportional to V ).

The number which diverge is related to the symmetry of the solution. For any given

number of Cooper pairs on the island, the ground state is the symmetrised solution |l, N〉,and the number of diverging parameters for this state is N , i.e. all EJ diverge.

As stated above, we wish to retain the kinetic energy term. We can regard (εk−εF )/Vas zero, because |εk − εF | < ωc, but we cannot discard εF /V as negligible. We expand the

Richardson equations (eq. 5.6.1) in 2(εk − εF )/(EJµ − 2εF ), where we have not made any

assumptions about the size of εF /V .

− 1

V+

N∑

ν=1

2

EJµ − EJν=

l

(EJµ − 2εF )+

l∑

k=1

2(εk − εF )(EJµ − 2εF )2

(5.6.2)

We discard the second term on the right as negligible, multiply by EJµ−2εF , and sum

over the N parameters EJµ

−EJµ − 2εFV

+N∑

ν=1

2EJµEJµ − EJν

−N∑

ν=1

4εFEJµ − EJν

= l

−N∑

µ=1

EJµ − 2εFV

+N(N − 1) + 0 = Nl (5.6.3)

5.6 Expanding the Richardson Solution to First Order 117

where the double sums over µ, ν have either vanished, or gone toN(N−1) due to symmetry.

Finally, we recall that the energy of the Richardson solution is given by a sum over EJ ,

and rewrite 5.6.3 to get the energy of an island containing N Cooper pairs:

EN = N2εF − V N(l −N + 1) (5.6.4)

This should be compared with the energy as calculated directly from the spin Hamil-

tonian:

((SZ +

l

2)2εF − V S+S−

)|l, N〉 = EN |l, N〉

EN =

(N2εF − V N(l −N + 1)

)(5.6.5)

This derivation shows that the spin Hamiltonian can be thought of as the first term in

an expansion in ωc/V . Including all the terms in the expansion allows an exact treatment

of the BCS Hamiltonian in terms of the spin operators SZ , S+, S−.

Chapter 6

The Super Josephson Effect

The spin model for a superconducting system described in chapter 5 allows the easy inves-

tigation of various phenomena. Here we apply the model to an analogue of a well-known

quantum optical effect known as superradiance, in which the intensity of the radiation

emitted by the atoms in a cavity is proportional to the square of the number of atoms.

We shall investigate an analogue of a quantum optical effect known as superradiance.

In superradiance, a group of atoms in a quantised electromagnetic field (i.e. a cavity)

are placed in a particular state. These atoms then emit light with an intensity that is

proportional to the square of the number of atoms [100]. This is an interesting effect, as

its observation relies on the fact that the atoms are placed in a highly entangled state,

but is observable simply by measurement of the intensity of light emitted.

In this chapter, we will describe a similar effect that occurs in an array of supercon-

ducting islands coupled to a superconducting reservoir, in which the current out of the

array is proportional to the square of the number of islands. This work is presented in

[101].

6.1 The Cooper Pair Box Array

The system we wish to describe is an array of Cooper pair boxes, each of which is coupled

individually to a superconducting reservoir. In this chapter, we assume that we can

represent the boxes as two level systems, and we consider the reservoir to be a quantum

spin, the size of which we allow to go to infinity at the end of the calculation.

119

120 The Super Josephson Effect

Figure 6.1: An array of lb superconducting islands, or Cooper Pair Boxes, that are indi-

vidually coupled to a superconducting reservoir, r, through capacitive Josephson tunnel

junctions (with no inter-box coupling).

The Hamiltonian of this system is:

H = Hb +Hr +HT (6.1.1)

where the individual Hamiltonians refer to the array of Cooper pair boxes, the reservoir,

and the tunnelling between the two respectively. The first term is the Hamiltonian of the

boxes:

Hb =

lb∑

i=0

Echi σzi (6.1.2)

There are lb Cooper pair boxes, each of which is described as a two-level system where

the two states are zero or one excess Cooper pair on the box, with Echi the charging energy

of each, as described in chapter 3.

The second term Hr is the well known BCS Hamiltonian [27] (see section 1.6), and the

final term:

HT = −∑

k,i

Ti,k (σ+i c−k↓ck↑ + σ−i c

†k↑c

†−k↓) (6.1.3)

6.1 The Cooper Pair Box Array 121

describes the tunnelling of Cooper pairs from each of the boxes to the reservoir. The

charging energy of each box can effectively be tuned by applying a gate voltage to that

box, and a tuneable tunnelling energy can be achieved by adjusting the flux through a

system of two tunnel junctions in parallel ([68], section 2.3).

In chapter 5 we used a set of finite spin operators to represent a finite superconductor.

We can also use this formalism to describe the array of boxes. If the charging energies Echi

and tunnelling elements Ti,k of each box are all adjusted to be the same, then the charging

energy for all the boxes acts like the z-component of a quantum spin with J = lb/2.

Similarly the sum over all the raising (lowering) operators acts like a large-spin raising

(lowering) operator.

lb∑

i=0

σZi → SZb

lb∑

i=0

σ±i → S±b (6.1.4)

If we consider transitions caused by only these operators, then the symmetrised number

states form a complete set. The eigenstates of SZb represent states where a given number

Nb of the boxes are occupied, with SZb |Nb〉 = (Nb − lb/2)|Nb〉. We now have a description

of the Cooper pair boxes in terms of a single quantum spin:

Hb = Echb S

Zb

HT =∑

k

−Tk(S+b c−k↓ck↑ + S−b c

†k↑c

†−k↓) (6.1.5)

As mentioned above, superradiance is observed when the atoms are placed in a par-

ticular initial state. The initial state is one of the form:

|ψ(0)〉 = |000..111..〉+ |001..110..〉+ · · ·+ |111..000〉 (6.1.6)

where 0 and 1 indicate that an atom (identified by the place of these numbers in the array

which specifies the state |n1, n2, n3 . . . 〉) is in its ground or excited state. Clearly, this

state is an eigenstate of the operator for the total number of excited states (SZb ), but not

of the individual σZi operators. Note that it is the symmetrised state with a total number

of excited states Nb. Evidently, these are exactly the states described by our model.


We use the model discussed in chapter 5 to describe the superconducting reservoir.

This model allows us to take into account the discrete nature of the Cooper pairs, whilst

being simple enough to allow ease of calculation.

In the superconducting analogue we consider three cases. The first is when the system

is in an eigenstate of both the number operator of the large superconductor, and the

operator representing the total number of Cooper pairs on the boxes. This is an artificial

case if we think of the boxes coupled to a truly bulk superconductor, but instead would

represent many boxes coupled to an island with only a few levels. It is, however, the

closest to superradiance, where the atoms are in a symmetrised number eigenstate, and

the field contains a given number of photons. The second case is when the boxes are in a

number eigenstate and the reservoir in a coherent state, and finally we consider when the

boxes are in a coherent state as well as the reservoir.

6.2 Superradiant Tunnelling Between Number Eigenstates

The simplest case is when both the boxes and the reservoir are in eigenstates of the

number operators SZb , SZr respectively. We consider the unperturbed Hamiltonian as the

sum of the Hamiltonians that act individually on the array and on the reservoir. As we are

discussing the case of superradiance between number eigenstates, we take the Hamiltonian

of the reservoir to be diagonal in SZr , i.e. , we do not break the symmetry with a pairing

parameter.

H = Echb S

Zb + 2ξFrS

Zr (6.2.1)

As the Hamiltonian is diagonal in SZr and SZb , the eigenstates are product states of

the number states of the reservoir, |Nr〉 and of the array, |Nb〉.This Hamiltonian is perturbed by the tunnelling Hamiltonian:

HT = −T (S+b S

−r + S−b S

+r ) (6.2.2)

and the current, i.e. the rate of change of charge on the islands, is given by the commutator

of the number operator with the Hamiltonian:

6.2 Superradiant Tunnelling Between Number Eigenstates 123

dNb

dt=dSZbdt

=1

i~[SZb , H]

I = T (S+b S

−r − S−b S+

r ) (6.2.3)

The expectation value of the current operator with the ground state is equal to zero,

so we go to time dependent perturbation theory:

|ψ(t)〉 = U(t)|ψ(0)〉

|ψ(t)〉 =

1 +

1

i~

t∫

0

e−iHt′/~HT e

−iHt′/~dt′ + · · ·

|ψ(0)〉 (6.2.4)

The expectation value of any operator at a given time is, to first order in HT :

〈ψ(t)|I|ψ(t)〉 = 〈ψ(0)|I|ψ(0)〉

+2<〈ψ(0)|I 1

i~

t∫

0

eiHt′/~HT e

−iHt′/~dt′|ψ(0)〉 (6.2.5)

The symbol < indicates we are taking the real part of the expression. The tunnelling

Hamiltonian and current consist of a sum (or difference) of raising and lowering operators.

When calculating the time-dependent current given a number eigenstate as the initial

state, the only non-zero terms are those that raise the number eigenstate and then lower

it, or lower it then raise it, i.e. only the terms S−b S+r S

+b S

−r and S+

b S−r S

−b S

+r survive.

Thus the expectation value of the current when the system is initialised in a number

eigenstate is a simple sum over two other number eigenstates, with a time dependent factor

from the exponentials.


〈ψ(t)|I|ψ(t)〉 = 2<〈Nr, Nb|TS+b S

−r

1

(i~)2

t∫

0

e−iHt′/~(−T )S−b S+

r eiHt′/~dt′|Nr, Nb〉

−2<〈Nr, Nb|TS−b S+r

1

(i~)2

t∫

0

e−iHt′/~(−T )S+

b S−r e

iHt′/~dt′|Nr, Nb〉

= 2< −T 2〈Nb|S+b S

−b |Nb〉〈Nr|S−r S+

r |Nr〉

1

(i~)2

t∫

0

e−i(EbNb−1−E

bNb

)−i(ErNr+1−ErNr ) t′/~dt′

+2< T 2〈Nb|S−b S+b |Nb〉〈Nr|S+

r S−r |Nr〉

1

(i~)2

t∫

0

e−i(EbNb+1−EbNb )−i(E

rNr−1

−ErNr ) t′/~dt′

(6.2.6)

Taking the real part and noting that we have even level spacing (i.e. EbNb

and ErNr

are

equal for all Nb, Nr), we find the current is:

〈ψ(t)|I|ψ(t)〉 =−2T 2

(i~)2

t∫

0

cos((Ech

b − 2ξFr) t′/~)dt′

×(〈Nb|S+

b S−b |Nb〉〈Nr|S−r S+

r |Nr〉 − 〈Nb|S−b S+b |Nb〉〈Nr|S+

r S−r |Nr〉

)

=−2T 2

(i~)2

t∫

0

cos((Ech

b − 2(εF − µ)) t′/~)dt′

× (Nr(lr −Nr)(lb − 2Nb)−Nb(lb −Nb)(lr − 2Nr) +Nrlb − lrNb)

The time dependence is determined by the energy difference between adjacent number

eigenstates, i.e. by Echb and 2ξFr. If we evaluate the expectation values, we find terms like

Nb(lb −Nb)(lr − 2Nr) which is quadratic in lb and linear in lr (and vice versa, i.e. terms

quadratic in lr and linear in lb). In particular, to have the largest effect, we set Nb = lb/2

i.e. half the boxes are occupied.

〈ψ(t)|I|ψ(t)〉 =−2T 2

(i~)2sin((Ech

b − 2ξFrt/~)

((Ech

b − 2ξFr/~)

(lb2

(lb2+ 1

)(lr − 2Nr)

)

(6.2.7)

6.3 Superradiant Tunnelling: Number State to Coherent State 125

We see that the current is proportional to l2b , the square of the number of Cooper Pair

boxes. This is in contrast to the situation where the initial state of the boxes is a product

state of lb independent wavefunctions, in which any current can be at most linear in the

number of boxes. The current is also proportional to (nr − 1/2)lr, where nr = Nr/lr, i.e.

how far away the large superconductor is from half filling.

6.3 Superradiant Tunnelling Between a Number Eigenstate

and a Coherent State

The second case we consider is when the Cooper Pair boxes are in an eigenstate of the

number operator, and the large superconductor is in a coherent, that is to say BCS-like,

state. We ensure the coherent state of the large superconductor by introducing a pairing

parameter into the unperturbed Hamiltonian:

H = Echb S

Zb + 2ξFrS

Zr −∆rS

+r −∆∗rS

−r (6.3.1)

The ground state of the large superconductor is a coherent state and ∆r = 〈S−r 〉 cannow be found self-consistently (section 5.4).

We make a basis transform (section 5.4.1) and write the Hamiltonian in terms of this

new basis, n:

H = Echb S

Zb + 2EFrS

Zr,n (6.3.2)

This basis transform makes the unperturbed Hamiltonian simpler, but at the expense

of making the perturbation more complicated in the new basis:

HT = −T (S+b S

−r + S−b S

+r )

= −TS+b (

2∆r

EFrSZr,n + f∗r S

+r,n + erS

−r,n)

−TS−b (2∆∗rEFr

SZr,n + e∗rS+r,n + frS

−r,n) (6.3.3)

Again, the tunnelling current is zero to first order in T, as the boxes are in a number

eigenstate. With this basis transform, we can easily calculate the second-order current.

Following the same procedure as before, we get:


〈ψ(t)|I|ψ(t)〉 =−2T 2

(i~)2|∆r|2E2Fr

〈0r,n|SZr,nSZr,n|0r,n〉t∫

0

dt′ cosEchb t′/~

×(〈Nb|S+b S

−b |Nb〉 − 〈Nb|S−b S+

b |Nb〉)

− 2T 2

(i~)2〈0r,n|S−r,nS+

r,n|0r,n〉 ×

|e|2t∫

0

dt′ cos (2EFr − Echb ) t′/~〈Nb|S+

b S−b |Nb〉

−|f |2t∫

0

dt′ cos (2EFr + Echb ) t′/~〈Nb|S−b S+

b |Nb〉 (6.3.4)

The terms e and f are as defined in 5.4.8.

The time dependence is again given by the level spacing, but for the large supercon-

ductor this is now given by EFr =√ξ2Fr − |∆|2 rather than (εFr − µ).

〈ψ(t)|I|ψ(t)〉 =−2T 2

(i~)2sinEch

b t/~Echb /~

|∆|2E2Fr

(lr2

)2

(lb − 2Nb)

+2T 2

(i~)2sin (Ech

b + γ2ξF )t/~(Ech

b + γ2ξF )/~lr|f |2〈Nb|S−b S+

b |NB〉

− 2T 2

(i~)2sin (Ech

b − γ2ξF )t/~(Ech

b − γ2ξF )/~lr|e|2〈Nb|S+

b S−b |NB〉 (6.3.5)

If we assume the levels of the large superconductor are much more finely spaced than

those of the boxes, i.e. EFr ¿ Echb , then we obtain the expression below. If we do not

make this assumption, we find a similar result (current proportional to l2b ), but with a

more complicated time dependence.

〈ψ(t)|I|ψ(t)〉 (6.3.6)

=−2T 2

(i~)2sinEch

b t/~Echb /~

|∆|2E2Fr

(lr2

)2

(lb − 2Nb)

− 2T 2

(i~)2sinEch

b t/~Echb /~

lr

ξFrEFr

Nb(lb −Nb) + |e|2Nb − |f |2(lb −Nb)

Remembering that lrξFr/EFr = lr/2 − Nr, (where Nr is the expectation value of

the reservoir number operator, see section 5.4.17) and setting Nb = lb/2, we find this is

proportional to:

6.4 Superradiant Tunnelling Between Coherent States 127

lr(1− 2nr)lb2

(lb2+ 1

)(6.3.7)

This is very similar to the case described in (6.2.7) where both the boxes and the

reservoir are in number eigenstates. In both cases, the effect is largest when the boxes are

at half-filling, and the large superconductor is far from half-filling. The difference is that

here it is the average number Nr that is relevant, not the number eigenvalue.

6.4 Superradiant Tunnelling Between Coherent States

The third case is where both the boxes and the large superconductors are in the coherent

state. Pairing parameters are introduced for each:

H = 2ξFbSZb −∆bS

+b −∆∗bS

−b

+2ξFrSZr −∆rS

+r −∆∗rS

−r (6.4.1)

The ground state of this Hamiltonian is for both superconductors to be in a coherent

state, which again is determined self consistently. As before we make a change of basis,

this time for both the boxes and the large superconductor, which makes the unperturbed

Hamiltonian:

H = 2EFbSZb,n + 2EFrS

Zr,n (6.4.2)

The tunnelling Hamiltonian becomes:

HT = −T (S+b S

−r + S−b S

+r ) (6.4.3)

= −T ( ∆∗b

EFbSZb,n + e∗bS

+b,n + fbS

−b,n)(

∆r

EFrSZr,n + f∗r S

+r,n + erS

−r,n)

−T ( ∆b

EFbSZb,n + f∗b S

+b,n + ebS

−b,n)(

∆∗rEFr


−r,n)

If we insert this into (6.2.5) we find that we do have a non-zero term linear in T , i.e.

the expectation value of I with |ψ(0)〉. To second order in T we have terms proportional

to lr and quadratic in lb and vice versa, and terms linear in both, i.e.


〈ψ(t)|I|ψ(t)〉 = 〈ψ(0)|I|ψ(0)〉+ 〈ψ(t)|I|ψ(t)〉l2blr

+〈ψ(t)|I|ψ(t)〉lbl2r + 〈ψ(t)|I|ψ(t)〉lblr (6.4.4)

The first term is just the expectation value of I with the ground state:

〈ψ(0)|I|ψ(0)〉 =T

i~|∆b|2EFb

|∆r|2EFr

lb lr 2 sin (φb − φr)

=T

i~|∆b|Vb

|∆r|Vr

2 sin (φb − φr) (6.4.5)

where φb, φr are the phases of ∆b,∆r. This is the usual Josephson effect.

The term quadratic in lb is:

〈ψ(t)|I|ψ(t)〉l2blr

=−T 2

(i~)2l2b lr

sin 2EFrt/~2EFr/~

|∆b|22E2

Fb

ξFrEFr

− T 2

(i~)2l2b lr

cos 2EFrt/~2EFr/~

|∆b|24E2

Fb

|∆r|24E2

Fr

2 sin 2(φb − φr) (6.4.6)

The first term is proportional to |∆b|2/E2Fb and ξFr/EFr, and so is largest when the

boxes are half occupied and the large superconductor is far from half occupied. This term

is also phase-independent and is like the effects seen in sections 6.2 and 6.3

The second term is largest when both are half occupied, and also depends on the phase

difference of the two pairing parameters with a frequency twice that of the usual Josephson

effect. As such, it can be considered a higher harmonic in the Josephson current. The

term quadratic in lr is identical apart from an anti-symmetric exchange of the labels, b, r.

Finally, the term that is linear in both contains a phase - independent term that is

largest when both the boxes and the reservoir are far from half-filling, and a term that is

phase dependent with twice the Josephson frequency that is largest at half filling.

〈ψ(t)|I|ψ(t)〉lblr=−T 2

(i~)2lblr

sin 2(EFb + EFr)t/~2(EFb + EFr)/~

(ξ2Fb4E2

Fb

ξFrEFr

− ξ2Fr4E2

Fr

ξFbEFb

)

+T 2

(i~)2lb lr

cos 2(EFb + EFr)t/~2(EFb + EFr)/~

( |∆b|24E2

Fb

|∆r|24E2

Fr

2 sin 2(φb − φr))

(6.4.7)

6.4 Superradiant Tunnelling Between Coherent States 129

In summary we conclude that tunnelling into or out of a Cooper pair reservoir from a

coherent ensemble of Cooper Pair Boxes can lead to a tunnelling current proportional to

the square of the number of ‘boxes.’ Interestingly, an observation of such scaling, in turn,

could be taken as a demonstration of coherence and entanglement of the ‘boxes’ as such

states are necessary to produce the phenomena.

Chapter 7

Revival

A further quantum optical effect that can be observed in an analogous superconducting

system is the effect of revival, where the state of a superconducting island connected to a

reservoir, initially in a pure state, decays to a mixed state but returns to a pure state at

a later time. The fact that the island is in a pure state is indicated by oscillations in the

probability of occupation of the charge states. The effect can be extended to the case of

multiple islands connected to a reservoir

7.1 The Superconducting Analogue of the Jaynes-Cummings

Model

In the previous chapter we investigated an analogue of superradiance. Another well-known

quantum optical phenomena is that of quantum revival (e.g. [102]), which occurs when

an atom is coupled to a single mode of the electromagnetic field. The initial pure state of

the atom decays into a mixed state, but reappears at a later time. This apparent decay

and subsequent revival occur because the environment of the atom is a discrete quantum

field. Here we examine a Josephson system analogue.

As in the last chapter, we consider an array of Cooper pair boxes individually connected

to a superconducting reservoir (see fig. 6.1). Again, we treat each of the boxes as a

simple two-level system and treat the reservoir as a superconductor represented by a large

quantum spin (see chapters 5, 6). We consider a Hamiltonian (eq. 6.3.1) discussed in

section 6.3.

131

132 Revival

H = Echb S

Zb + 2ξFrS

Zr −∆S+

r −∆∗S−r

HT = −T (S+b S

−r + S−b S

+r ) (7.1.1)

As in section 6.3, we make a basis transform so that the reservoir is written as a

quantum spin in another direction.

H = Echb S

Zb + 2EFrS

Zr,n

HT = −TS+b (

2∆r

EFrSZr,n + f∗r S

+r,n + erS

−r,n)

−TS−b (2∆∗rEFr


−r,n) (7.1.2)

We can write a general state of the system in terms of the number eigenstates:

|ψ(t)〉 =lr,lb∑

Nr,Nb=0

aNr,Nb(t)|Nr〉|Nb〉 (7.1.3)

For a single two level system, lb = 1, but we can also do the calculations for a general

number of symmetrised boxes. We can calculate the time evolution of the system using

the Schrodinger equations:

i~∑

Nr,Nb

d aNr,Nb(t)

dt|Nr〉|Nb〉 =

∑

Nr,Nb

aNr,Nb(t)

(Echb (Nb −

lb2) + 2ξFr(Nr −

lr2)

)|Nr〉|Nb〉

−∑

Nr,Nb

aNr,Nb(t)TS+b (

2∆r

EFrSZr,n + f∗r S

+r,n + erS

−r,n)|Nr〉|Nb〉

−∑

Nr,Nb

aNr,Nb(t)TS−b (

2∆∗rEFr


−r,n)|Nr〉|Nb〉

(7.1.4)

More explicitly, we have a set of coupled differential equations for the coefficients

aNr,Nb(t):

7.1 The Superconducting Analogue of the Jaynes-Cummings Model 133

i~d aNr,Nb(t)

dt= aNr,Nb(t)

(Echb (Nb −

lb2) + 2ξFr(Nr −

lr2)

)

−aNr,Nb+1(t)T√

(Nb + 1)(lb −Nb)2∆∗rEFr

(Nr − lr/2)

−aNr−1,Nb+1(t)T√

(Nb + 1)(lb −Nb)e∗r

√Nr(lr −Nr + 1)

−aNr+1,Nb+1(t)T√

(Nb + 1)(lb −Nb)fr√

(Nr + 1)(lr −Nr)

−aNr,Nb−1(t)T√Nb(lb −Nb + 1)

2∆r

EFr(Nr − lr/2)

−aNr−1,Nb−1(t)T√Nb(lb −Nb + 1)f∗r

√Nr(lr −Nr + 1)

−aNr+1,Nb−1(t)T√Nb(lb −Nb + 1)er

√(Nr + 1)(lr −Nr)

(7.1.5)

We consider an approximation that allows a simple analytical model.

The ∆ terms in eq. 7.1.1 mean that the total particle number Nr+Nb is not conserved.

We note that whilst the commutator of the total number operator [Nr + Nb, H] = ∆S+r −

∆∗S−r is not zero, its expectation value in the coherent state is. With this in mind, we

approximate and assume no fluctuations in the total number (although we of course retain

fluctuations in Nr −Nb) and discard any terms in eq. 7.1.5 that change the total number.

With these terms discarded, we have a set of coupled differential equations in which

each coefficient aNr,Nb(t) only couples to aNr−1,Nb+1(t) and aNr+1,Nb−1(t).

i~d aNr,Nb(t)

dt= aNr,Nb(t)

(Echb (Nb −

lb2) + EFr(Nr −

lr2)

)

−aNr+1,Nb−1(t)T√Nb(lb −Nb + 1)

√(Nr + 1)(lr −Nr)

−aNr−1,Nb+1(t)T√

(Nb + 1)(lb −Nb)√Nr(lr −Nr + 1)

(7.1.6)

These equations can be restated in terms of an eigenvalue problem. Each level Nr has

a set of eigenvectors for the boxes. If the boxes and the reservoir are initially in a product

state, the probability to be in a given value of Nb (regardless of Nr) is:

PNb =

lb∑

Nb=0

|aNr(0)|2∣∣∣∣∣∑

i

〈Nb|νi,Nr〉e−iEi,Nr t/~〈νi,Nr |ψNb(0)〉∣∣∣∣∣

2

(7.1.7)

134 Revival

where |νi,Nr〉 (Ei,Nr) are the eigenvectors (values) of the Nb-level system associated with

the reservoir level |Nr〉, aNr(0) are the initial amplitudes of |Nr〉, and |ψNb(0)〉 is the initialstate of the boxes.

This may not be exactly solvable in general, but can be solved for few-level systems,

lb. For example, if we have a single two level system and specify the initial state to be a

product state of the box state |0b〉 with some state∑aNr(0)|Nr〉 of the reservoir, we have

the probabilities that the box and reservoir are in a given state:

P1,Nr−1(t) = |aNr(0)|2T ′Nr

2

ΩNr2 sin

2(ΩNr t/~)

P0,Nr(t) = |aNr(0)|2 (1− P1,Nr−1(t)) (7.1.8)

with T ′Nr = T√Nr(lr −Nr + 1) and ΩNr =

√(Ech

b − EFr)2/4 + T ′2. This is the same

result as in quantum optics, except that the photon creation/ annihilation operators give

T ′Nr = T√Nr .

7.1.1 Quantum Revival of the Initial State

In quantum optics, the phenomenon of revival is seen when the electromagnetic field is

placed in a coherent state (i.e. |aN |2 = exp(−N)NN/N !, with the sum over N running to

infinity). We place the reservoir in the spin coherent state, i.e.

|aNr(0)|2 = |α|2Nrlr!

(lr −Nr)!Nr!(7.1.9)

where α is determined by the average number |α| = Nr/√

(lr − Nr)Nr.

The probability of the box being in the state |0b〉 at a given time is:

P0(t) =

lr∑

Nr

|aNr(0)|2P0,Nr(t) (7.1.10)

For simplicity we consider the case when Echb = EFr, i.e. ΩNr = T ′Nr . This is shown

in Figure 7.1, for the values Nr = 10, lr = 50. The initial state dies away, to be revived at

a later time. We have chosen the above values for Nr and lr in order to show the revival

clearly, and also to show the difference between the spin and quantum optics cases. If the

value of lr is too small (compared to Nr) the oscillations never fully decay. On the other

hand, if lr À Nr, the system behaves identically to the quantum optics case. The effects


0

0.2

0.4

0.6

0.8

1

P0(t)

1 2 3 4 5 6time

Figure 7.1: Spin Coherent State Revival for Nr = 10, lr = 50 (solid line). The faint line

indicates the usual quantum optical coherent state revival with Nr = 10, scaled along the

t axis by√lr and the vertical dash-dot line indicates the calculated revival time, where

the time is in units of 2π/EJ .

of revival are much more generic than this, and also can be observed in systems with, for

example, non-identical parameters for the Cooper pair boxes.

We can calculate the linear entropy of the system over time (fig. 7.2). This is defined as

2(1−Tr ρ2) and is a measure of how mixed the state is, with 0 indicating a pure state, and

1 indicating a maximally mixed qubit state. As expected, the entropy [9] increases (purity

decreases) as the oscillations decay, and then the entropy is reduced (purity increased) as

the oscillations revive.

The revival occurs when the terms in the sum are in phase. We can make an estimate

of this by requiring the terms close to Nr to be in phase:

2π = 2T√

(Nr + 1)(lr − Nr) trev − 2T√Nr(lr − Nr + 1) trev

trev =2π

T

n1/2r (1− nr)1/2(1− 2nr)

(7.1.11)

where, as before nr = Nr/lr. Note that the revival time remains finite in the lr → ∞limit, as long as Nr/lr is finite.

136 Revival

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

Figure 7.2: Decay and revival of a single Cooper pair island state, showing the degree

of linear entropy, which increases at first, and is then reduced. The oscillations are the

probability that the island is in state |0〉 (solid line) or state |1〉 (dashed line) against time

in units of 2π/EJ .


0

0.2

0.4

0.6

0.8

1

P0(t)

0.2 0.4 0.6 0.8 1 1.2 1.4time

Figure 7.3: Analytic asymptotic form for decay oscillations in the probability against time

in units of 2π/EJ(solid line) compared with the numeric evolution (dashed line) Nr = 10,

lr = 100

We can find an analytic form for the initial decay, and also demonstrate the equivalence

of this ‘spin revival’ to the well known quantum optics revival, in the limit lr → ∞,

lr/Nr → ∞. Note that this is an unrealistic limit for our case, where we would keep

the filling factor Nr/lr constant, but it allows us to make a connection with the quantum

optics case.

The coefficients, aNr(0) for the coherent and spin-coherent states are the Poisson and

binomial distributions respectively, and both can be approximated by a Gaussian distri-

bution.

exp(−Nr)NNrr /Nr! ' (2πNr)

−1/2 exp[−(Nr − Nr)

2/2Nr

]

|α|2Nr lr!

(lr −Nr)!Nr!' (2πNr)

−1/2 exp[−(Nr − Nr)

2/2Nr

](7.1.12)

For the coherent state we have taken the limit Nr À 1. For the spin-coherent case we

have also assumed lr À Nr. The Gaussian factor suppresses any terms not around the

138 Revival

average Nr, so we can expand Nr ' Nr + (Nr − Nr).

In the limit, the spin coherent state gives:

P0(t) =1

2+

1

2(2πN)1/2

∫dN exp

(−(Nr − Nr)

2

2Nr

)

× cos

[2T (lrNr)

1/2t

(1 +

(Nr − Nr)2

2Nr

)](7.1.13)

This is the same as for the coherent state apart from the factor of l1/2r in the frequency.

Doing the integral gives:

P0(t) =1

2+

1

2cos(2T (lrNr)

1/2t) exp(−(T lrt)2

2) (7.1.14)

This is plotted in Figure 7.3.

Thus the phenomenon of quantum revival has a direct analogy in our ‘spin supercon-

ductor’ model. In both cases, the revival is due to constructive interference between the

terms in the sum over the reservoir (field) number states.

7.2 Revival of Entanglement

Revival is usually considered for single two-level systems, but the formula in (7.1.7) gives

the state for a general system. In particular, the dynamics of a pair of two-level systems

can be easily calculated, either numerically, or analytically, remembering that the singlet

state is uncoupled due to radiation trapping. The Hamiltonian, 7.1.1, does not cause

transitions between the triplet states and the singlet state, so if we start the system in

any of the triplet states, we essentially have only a three level system to solve.

Starting in the state |00〉, we find a very similar effect to the single-island case. Figure

7.4 shows the oscillations of one Cooper pair box other traced over, compared to the

oscillations of a single box. Note that whilst the decays in the single and double box

systems are in phase, the first revival of the double box is out of phase with that of the

single box, and the second revival is in phase.

If we do not trace over one island, but consider the revival of the state |00〉, we find

oscillations in the probability of returning to this state earlier than in the case of a single

island.

7.2 Revival of Entanglement 139

1 2 3 4 5

0.2

0.4

0.6

0.8

Figure 7.4: Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. The

solid line represents the probability that, when the system is initialised in the state |00〉,one of the Cooper pair boxes is in the state |0〉 at time t (in units of 2π/EJ), after tracing

over the other box. The dashed line indicates the revival of a single isolated Cooper Pair

box connected to a reservoir. Note that the revivals occur at the same time, but out of

phase.

To investigate the revival of entanglement, we start the system in the state (|01〉 +|10〉)/

√2. As before we note that the oscillations in the probability of being in the state

(|01〉+ |10〉)/√2 decay and then revive. It is interesting to note, however, that in this case

the initial state is an entangled one. One measure of entanglement is the negativity [103],

i.e. the sum of the negative eigenvalues of the partially transposed density matrix. If we

plot the negativity over time, we see that it dies away at first, as do the oscillations. As

the oscillations revive, we also see an revival in the negativity of the two islands (Figure

7.5).

This revival of entanglement, an initially counterintuitive phenomenon, is of course due

to the fact that we have considered the entanglement between the two islands only, just

as when we considered a single island, we calculated the purity of the island as an isolated

system and not the entropy of the whole system. When the entanglement ‘disappears,’

it is only the entanglement between the islands that has disappeared. If we were to

consider the whole system, including the reservoir, the evolution would be unitary and the

entanglement would remain constant.

It is worth noting that this ‘entanglement revival’ is not a unique feature of our spin

model, and would be expected whenever two-level systems are coupled to a field with (a

140 Revival

0.2

0.4

0.6

0.8

1

1 2 3 4

Figure 7.5: Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. The

dashed line represents the probability that the Cooper Pair boxes have returned to their

initial state (|01〉 + |10〉)/√2 at time t,in units of 2π/EJ . The dashed line indicates the

negativity of the boxes as a function of time.

7.2 Revival of Entanglement 141

possibly infinite number of) discrete energy levels.

Chapter 8

An Electronic Derivation of The

Qubit Hamiltonian

In this chapter we examine in greater detail the derivation of the tunnelling term of the

qubit Hamiltonian for superconducting charge qubits. We consider a rigorous derivation

of the qubit Hamiltonian from single-electron creation and annihilation operators using the

spin model described in chapter 5.

8.1 Introduction

In chapter 3 we derived a Hamiltonian for the qubit circuit from the Josephson relations.

These equations give the rate of change of phase and the current across each junction. To

attempt to capture the behaviour of junctions with small capacitances, we quantise these

relations. A Lagrangian is found from which the Euler-Lagrange equations reproduce the

Josephson equations. This leads to a Hamiltonian for the system, and tells us that the

phase and charge are conjugate variables.

This procedure is somewhat empirical, in the sense that the equations of motion for

the phase and charge are treated as if they were the equations of motion for any classical

system. These equations are, of course, derived from a microscopic theory of the electrons

in the superconductor. In the derivation, it is assume that the superconductors are bulk so

that BCS theory is the correct description. In particular, taking the infinite limit means

that the phase becomes a well defined classical variable.

In the course of this derivation, then, we have taken a microscopic quantum description

143

144 An Electronic Derivation of The Qubit Hamiltonian

(superconductivity theory), taken its classical limit (the Josephson relations) and then

‘re-quantised’ it (the qubit Hamiltonian) in order to attempt to capture the finite size

behaviour. If nothing else, this procedure is inelegant, and it may be that important

features have been neglected. Furthermore, quantising the Josephson relations in this way

implies the existence of a ‘phase operator’ analogous to the position operator X, which is

problematic. We would like to go directly from an electronic description of the system to

the qubit Hamiltonian. The model described in chapter 5 will allow us to do this.

8.2 The Phase Representation

Quantum mechanics is usually first taught to undergraduates in terms of wavefunctions

(the co-ordinate representation). The position wavefunction ψ(x) gives the amplitude that

a particle will be found between x and x+ dx. The whole of the function ψ(x) defines the

state of the particle, and we may have a set of such states ψn(x) that are of interest to

us, for example that are energy eigenstates. Operators OC are represented by differential

operators that act on these states.

An alternative representation is ket notation. In this, each state is denoted by an

abstract ket, |n〉, and the operators OK are defined by their actions on these states.

We can easily translate between these two equivalent representations. The position

wavefunction is given in terms of ket notation by ψn(x) = 〈x|n〉, where |x〉 is an eigenket

of the position operator X with eigenvalue x (see eg. [39]). The operators are represented

by the differential operators in a way that is consistent with this.

OK |ψa〉 = |ψb〉

OC〈x|ψa〉 = 〈x|ψb〉 (8.2.1)

If we have a complete set of eigenstates |n〉 of the operator OK with eigenvalue λn, we

can fully define the operators in terms of their actions on these states:

OK |ψn〉 = λn|ψn〉

OC〈x|ψn〉 = λn〈x|ψn〉 (8.2.2)

In going from an electronic description of the qubit to one in terms of phase, we have

to go from ket to coordinate representation.

8.2 The Phase Representation 145

8.2.1 Phase Representation of Spins

The model set up in chapter 5 describes a small region of superconductor in terms of a

large but finite quantum spin. Before returning to our superconducting system, we shall

examine how to describe quantum spins in terms of phase.

A quantum spin of size l2 has a complete set of l states, |l, N〉. Operations on this

system can be written in terms of the diagonal z-operator, SZ , and the raising and lowering

operators, S±. The action of these operators on the states l, N〉 is:

SZ |l, N〉 = (N − l

2)|l, N〉

S+|l, N〉 =√

(N + 1)(l −N)|l, N + 1〉

S−|l, N〉 =√N(l − (N − 1))|l, N − 1〉 (8.2.3)

We wish to write both the states and the operator in some coordinate representation,

specifically a phase coordinate representation.

We define a state |φ〉:

|φ〉 =∑

eiφN |l, N〉 (8.2.4)

We now follow the procedure described above. An SZ eigenstate |l, N〉 is now described

by a wavefunction ψN (φ) given by ψN (φ) = 〈φ|l, N〉 = e−iφN . The wavefunction of any

general state |a〉 can be expressed by a linear combination of these eigenfunctions:

|a〉 =∑

N

aN |l, N〉

ψa(φ) =∑

N

aNψN (φ)

=∑

N

aNe−iφN (8.2.5)

We now need the coordinate representation of the operators. Our task is to find

differential operators which are equivalent to our spin operators (eq. 8.2.3) in the sense

defined by eq. 8.2.2.

The action of SZ is reproduced by id/dφ− l/2:


SZ〈φ|l, N〉 =

(id

dφ− l

2

)e−iφN

=

(N − l

2

)e−iφN

SZ〈φ|l, N〉 =

(N − l

2

)〈φ|l, N〉 (8.2.6)

So our coordinate representation of SZ is id/dφ − l/2. We now want to do the same

thing for the raising and lowering operators. Examining the action of the raising and low-

ering operators on the state |l, N〉 (eq. 8.2.3) tell that the that the appropriate differentialoperator form is:

S+ = e−iφ

√(id

dφ+ 1

)(l − i d

dφ

)

S− = eiφ

√id

dφ

(l − i d

dφ+ 1

)(8.2.7)

We write these in a form that will be more convenient. For example, it is not imme-

diately obvious from the above form that S+ and S− are Hermitian conjugates of each

other.

S+ = e−iφ

√(l

2+

(id

dφ− l

2

)+ 1

)√(l

2−(id

dφ− l

2

))

(8.2.8)

We now want to commute the e−iφ term past the first square root. To do this we

expand the square root in(i ddφ − l

2

).

√(l

2+

(id

dφ− l

2

)+ 1

)=

√l

2

∞∑

p=0

1

p !

(id/dφ− l/2 + 1

l/2

)p(8.2.9)

Note that this is an infinite expansion, and so it is valid regardless of the size of the

term we are expanding in. We have to commute e−iφ past terms like (SZ + 1)p, where

we have just used SZ as a shorthand for id/dφ − l/2. Using the commutation relation

[e−iφ, i ddφ ] = −e−iφ, it is easy to show that e−iφ(SZ + 1)p = (SZ)pe−iφ. We find that we

can write the coordinate representation of the raising and lowering operators as:


S+ =

√(l

2+

(id

dφ− l

2

))e−iφ

√(l

2−(id

dφ− l

2

))

S− =

√(l

2−(id

dφ− l

2

))eiφ

√(l

2+

(id

dφ− l

2

))

(8.2.10)

This form is much more convenient; the two operators are obviously Hermitian conju-

gates of each other, and the action of S+S− is much clearer. As an example of how these

operators act on a phase wavefunction, we can see how the raising operator acts on the

wavefunction ψl(φ) for the highest spin state, where N = l:

S+ψl(φ) =

√(l

2+

(id

dφ− l

2

))e−iφ

√(l

2−(id

dφ− l

2

))e−il

=

√(l

2+

(l + 1− l

2

))√(l

2−(l − l

2

))e−i(l+1)

=√

(l + 1)√

(l − l) e−i(l+1)

= 0 (8.2.11)

The raising operator destroys the N = l wavefunction, as we would expect. A similar

calculation confirms that the lowering operator destroys the N = 0 wavefunction, and that

these forms act on all the N wavefunctions in the manner required by eq. 8.2.3.

We also require that the coordinate forms for the operators have the commutation

relations given in eq. 5.2.2:

[SZ , S+] =

[SZ ,

√l

2− SZe−iφ

√l

2+ SZ

]

= SZ

(√l

2− SZe−iφ

√l

2+ SZ

)−(√

l

2− SZe−iφ

√l

2+ SZ

)SZ

= SZ

(√l

2− SZe−iφ

√l

2+ SZ

)−√l

2− SZ

(SZe−iφ − e−iφ

)√ l

2+ SZ

=

√l

2− SZe−iφ

√l

2+ SZ

[SZ , S+] = S+ (8.2.12)


So the above commutation relation matches the form we desired. A similar calculation

confirms that the other two commutation relations match also.

8.2.2 The Limit of Large Size

We would like to find the form of these operators in the limit where the size of the spin (or

the size of the superconductor) becomes infinite, l2 →∞. We have written our operators

in a form that will make this easy.

The limit we will be taking in order to make the expansion is the limit SZ ¿ l2 . In

a spin, this refers to the z-value of the spin always being close to zero, i.e. the spin lies

close to the x − y plane. In a superconducting island, this corresponds to the number of

Cooper pairs being close to l2 . Specifically, we assume:

|〈l, N |ψ〉| ∼ 0 for |N − l

2| &

(l

2

) 12

(8.2.13)

For the states |ψ〉 that the system occupies. As a particular example of a state that

obeys this condition, we consider a coherent state:

|〈l, N |l, z〉|2 = 1

(1 + |z|2)ll!

(l −N)! N !|z|2 (8.2.14)

We recognise that this expression for |〈l, N |ψ〉|2 is just the binomial coefficient,

C lNpN (1 − p)l−N , with p = |Z|2. Recalling the well known large-l limit for the bino-

mial coefficient (see eg. [104]):

|〈l, N |l, z〉|2 ' 1√2πlp(1− p)

e−(N−lp)2/2lp(1−p) (8.2.15)

If the average number N is set to l2 , this is a Gaussian centred around l

2 with width(l2

)1/2, and satisfies our condition for expansion. We shall take the full expansion later,

but for now we only take the leading term. In this case, Our three operators have the

simple form:


SZ → (id

dφ− l

2)

S+ →(l

2

)e−iφ

S− →(l

2

)eiφ (8.2.16)

The commutation relation between the raising and lowering operators is now zero:

[S+, S−] = 0

We can use this representation to show how the qubit Hamiltonian described in chapter

3 can be derived from electronic theory, in the limit where the size of both the island

and reservoir become large. If they are not both infinite, we shall need to calculate the

corrections to this.

We take an expansion of the differential forms of the raising and lowering operators.

For the case of an infinitely large spin, we took the limit described in eq. 8.2.13. Here we

find the next order terms by expanding in SZ

l/2 .

S+ =

(l

2+ SZ

) 12

e−iφ(l

2− SZ

) 12

= l/2

(1 +

SZ

l− (SZ)2

2l2+ . . .

)e−iφ

(1− SZ

l− (SZ)2

4l2+ . . .

)

= l/2

(1 +

SZ

le−iφ − e−iφS

Z

l− (SZ)2

4l2e−iφ − (SZ)2

4l2− e−iφ

−(SZe−iφSZ)l2

. . .

)

(8.2.17)

To combine these terms we must commute the exponential past the SZ operators.

S+ = l/2

(e−iφ(1 +

1

l− (SZe−iφSZ)

l2/2+e−iφ

4l2+ . . .)

)(8.2.18)

Our original expansion was an expansion in SZ

l/2 . Commuting the exponential has given

terms like 1l/2 , so we need to consider the limit. The assumption in eq. 8.2.13 means that

the only states that contribute are those such that |N − l2 | ≤

(l2

) l2 . We therefore consider

N − l2 to be of order O((l/2)

12 ) and discard terms of order O((l/2)−2).


S+ ' (l

2+

1

2)e−iφ − (SZe−iφSZ)

l+O(

2

l)

S− ' (l

2+

1

2)eiφ − (SZeiφSZ)

l+O(

2

l) (8.2.19)

8.2.3 Phase Representation of BCS States

Instead of treating the superconductors as spins, we consider the BCS state. As has been

discussed in chapter 5, the ω ¿ V limit of the BCS state is simply the spin coherent state

|l, z〉 (eq. 5.3.7). Instead of considering the superconductor to be in a state in the Hilbert

space spanned by |l, N〉 (or equivalently, by |l, z〉), we consider the space spanned by the

BCS states.

The BCS states, like the spin-coherent states, are parameterised by a phase φ, and can

be written:

|BCS, φ〉 =∏

k

(uk + vkeiφ)|l, 0〉 (8.2.20)

By integrating over φ we can extract a number eigenstate from the BCS states:

|BCS,N〉 =

2π∫

0

|BCS, φ〉e−iφNdφ

=

(∏

k

uk

)(S+BCS)

N |l, 0〉 (8.2.21)

where S+BCS is an operator that raises the number of Cooper pairs by one but does not

distribute the extra Cooper pair over all levels k equally.

S+BCS =

∑

k

vkukc†k↑c

†−k↓ (8.2.22)

The number states described above are eigenstates of the operator SZ , and the operator

S+BCS raises the number of the state. As in the case of spin states, we have a phase and

number state, and it might be desirable to describe the superconductor in terms of these

states rather than the spin states, which do not take into account the distribution functions

uk, vk.


In order to investigate whether this would be a sensible description, we first need to

calculate the commutation relations of these operators.

[SZ , S+BCS ] =

′∑

k, k′

[1

2(c†k′↑ck′↑ + c†−k′↓c−k′↓ − 1),

vkukc†k↑c

†−k↓]

=∑

k

vkukc†k↑c

†−k↓

= S+BCS (8.2.23)

We find that two of the commutation relations are as desired: [SZ , S±BCS ] = ±S±BCS .However, when we calculate the third, we find:

[S+BCS , S

−BCS ] =

′∑

k, k′

[vkukc†k↑c

†−k↓,

v∗ku∗kc−k′↓ck′↑]

=∑

k

|vk|2|uk|2

(c†k′↑ck′↑ + c†−k′↓c−k′↓ − 1)

6= 2SZ (8.2.24)

The operators do not form a closed basis, and so we cannot form an exact representation

as we have done for the spin operators. We can, however, attempt to find an approximate

representation.

We wish to produce a phase coordinate representation for these states, as described

in sections 8.2 and 8.2.1. Following the procedure we used before, we examine the matrix

elements of the operators SZ , S+, S− with BCS states with different φ.

〈BCS, φ|SZ |BCS, φ′〉 =

(id

dφ− l

2

)〈BCS, φ|BCS, φ′〉

〈BCS, φ|S+|BCS, φ′〉 =∑

k

ukvke−iφ〈BCS, φ|BCS, φ′〉+D

〈BCS, φ|S−|BCS, φ′〉 =∑

k

ukvkeiφ〈BCS, φ|BCS, φ′〉+D

D =

∑k

v4k

(∑k

ukvk)2(8.2.25)

This shows that, as before, we can make the connection:


SZ ⇒ (id

dφ− l

2) (8.2.26)

In addition, we can make the approximate connection:

S+ ∼(∑

k

ukvk

)e−iφ

S− ∼(∑

k

ukvk

)eiφ (8.2.27)

where D is a measure of the error incurred by making this approximation.

We can estimate the size of this error term. We use the definition for |vk|2, and make

the usual approximation of the sum by an integral:

∑

k

v4k =∑

k

1

4

(1− 2ξk

Ek+ξ2kEk

)

= N (0)V ol

~ωc∫

−~ωC

dξ1

4

(1− 2ξ

(ξ2 +∆2)1/2)+

ξ2

(ξ2 +∆2)1/2)

)

= N (0)V ol

(~ωc −

∆

2tan−1

~ωc∆

)(8.2.28)

with V ol being the volume of the system. From the gap equation, we know∑ukvk = ∆/V .

In the weak coupling limit, we find:

D = V 2N (0)V ol~ωc∆2

=d

∆λ2~ωc∆

(8.2.29)

where in the last line we have written D in terms of the dimensionless coupling constant

λ = V olN (0)V and the level spacing d = 1/N (0)V ol.

This term goes to zero as the superconductor becomes larger (as 1/V olume), so in the

bulk limit the above forms for the operators are exact.

If we consider D in the strong coupling limit, we expand tan−1 in ~ωc/V , and find:

D =V 2N (0)V ol

2

~ωc∆2

=1

l(8.2.30)

8.3 Island - Reservoir Tunnelling 153

where we have use ~ωc/d = l/2 to regain the spin result.

So we see that the term indicating the size of the error in expressing the operators

S+, S− as differential operators acting on a phase, D, goes to zero when we take the size

of the system to infinity, even in the weak coupling limit (as it also does in the strong

coupling limit). We are therefore justified in saying that we can approximate the operators

S+, S− as proportional to e−iφ, eiφ, and that this approximation gets better as the size of

the superconductor becomes larger. In particular, we note that this representation is not

a vagary of the spin representation or strong coupling limit, but a general feature. The

large size limit of the operators in the spin case (eq. 8.2.16) and the BCS case (eq. 8.2.26

8.2.27 are identical up to a constant.

8.3 Island - Reservoir Tunnelling

With the spin operators written in terms of differential operators, we are now ready to

write the Hamiltonian for our superconducting charge qubit in a differential operator form.

Initially we consider tunnelling between one finite and one bulk superconductor. We write

the Hamiltonian in four parts:

H = H1 +Hr +HC +HT (8.3.1)

where H1 is the standard BCS Hamiltonian for the island,

H1 =∑

k,σ

(εk − µ)c†k,σck,σ − V1kF+ωc∑

k,k′=kF−ωcc†k′↑c

†−k′↓c−k↓ck↑ (8.3.2)

The term Hr is the standard BCS Hamiltonian for the reservoir. The Charging Hamil-

tonian measures the electrostatic energy due to charges on the island, as discussed in

chapter 3.

HC =4e2

2C(N1 − ng)2 (8.3.3)

where N1 represents the number of pairs on the island, and ng allows us to incorporate a

gate voltage.

The final term is the pair - tunnelling term derived in section 1.8:


HT = −T∑

k,q

c†kc†−kc−qcq + c†qc

†−qc−kck

(8.3.4)

where the subscripts k and q refer to the island and reservoir superconductors respectively.

If we represent both superconductors by spins, the Hamiltonian can be written in terms

of two sets of spin operators.

H1 = 2(εF1 − µ1)SZ1 − V1S+1 S

−1

Hr = 2(εFr − µr)SZr − VrS+r S

−r

HC =4e2

2C(SZ1 − ng)2

HT = −T(S+1 S

−r + S−1 S

+r

)(8.3.5)

We would like to consider the case where we have one small superconductor (the island,

1) and one bulk superconductor (the reservoir, r). We can treat the large superconductor in

two ways; as the large - spin limit of our spin representation, or as a BCS superconductor.

As described in section 8.2, we get the same result up to a multiplicative constant. When

the reservoir is considered infinitely large, we can replace the raising and lowering operators

S+r , S

−r by the differential forms (eq. 8.2.16). The tunnelling operator then becomes:

HT = −T lr2(S+

1 eiφr + S−1 e

−iφr) (8.3.6)

The effect of considering the reservoir as a BCS superconductor is just to replace the

term lr2 with

∑k

ukvk. Combining the tunnelling Hamiltonian with the charging energy

found in section 3.1.2, we find the Hamiltonian for a single qubit is:

H = E′C(N1 − n′g)2

−T lr2(S+

1 eiφr + S−1 e

−iφr) (8.3.7)

where E′C and n′g are the renormalised charging energy and gate charge which take into

account the terms in H1 linear and quadratic in SZ1 , and N1 = SZ1 − l12 .

8.3 Island - Reservoir Tunnelling 155

We see that this has the same form as the broken-symmetry Hamiltonian discussed

in 5.5.1; the presence of the bulk superconductor has broken the symmetry of the small

one. The eigenstates are no longer the |l1, N1〉 states, and there will be a non-zero pairing

parameter in the ground state. In the limit where the size of the island as well as the

reservoir becomes large, the raising and lowering operators for the island also take the

form 8.2.16, and eq. 8.3.7 becomes:

H = E′C(N1 − n′g)2

−T lrl14

(ei(φ1−φr) + e−i(φ1−φr))

= E′C(N1 − n′g)2 − Tlrl12

cos(φ1 − φr) (8.3.8)

We have regained the Hamiltonian (eq. 2.2.12) obtained by quantising the Josephson

relations. Thus we see that eq. 2.2.12 is the semiclassical limit of 8.3.7, which becomes

valid as the island becomes large as well as the reservoir.

The corrections to this can be found by using the form for S+1 , S

−1 found in section

8.2.2:

H = E′C(N1 − n′g)2 − Tlr2

(l1 + 1) cos(φ1 − φr)

−2(N1 − l12 )

2

l1cos(φ1 − φr)−

2(N1 − l12 )

l1isin(φ1 − φr)

(8.3.9)

We have written the correction terms above as cosine and sine terms with velocity (SZ)

dependent pre-factors. Note that in this form it is not so obvious that the Hamiltonian

is Hermitian (it is) but this form shows the sine and cosine dependence on phase of the

correction terms explicitly.

These corrections destroy the periodicity with ng in the energy levels (figure 8.1). It

then becomes relevant what the absolute value of ng is (rather than just its value modulo

1). We can consider the extra terms in the expansion to be corrections to the existing

cosine term. In this case, using the analogy of a particle in one dimension, the extra terms

give rise to a change in the cosine potential which depends on SZ = N1 − l12 , i.e. , these

terms are a velocity dependent potential. The tunnelling rate is then dependent on the

value of SZ , i.e. the rate of tunnelling depends on the number of Cooper pairs on the


–2

0

2

4

6

8

10

2 4 6 8 10 12 14

Figure 8.1: Energy levels of a charge qubit as a function of ng including the finite size

correction to the tunnelling energy. The dashed lines indicate the uncorrected energy

levels. The parameters are EC = 10 EJ = 1 and l = 100

8.4 Inter - Island Tunnelling 157

island. This can be seen in figure 8.1, where the energy gap between the ground and first

excited states depends on ng.

Alternatively, we can consider the terms to be corrections to the charging energy -

they represent a position dependent mass (a phase - dependent capacitance). Although

this interpretation is perhaps less natural, as the correction terms are proportional to the

tunnelling matrix element T , it might be useful when EJ À EC .

The variation of the level splitting with ng, as well as being of interest from a funda-

mental point of view, offers the possibility of control of the effective Josephson tunnelling

with ng, and as such may be of interest in quatum computing applications.

8.4 Inter - Island Tunnelling

The next case we consider is that of two small superconductors with tunnelling between

them. We treat both as finite quantum spins. The tunnelling Hamiltonian contains terms

like:

S+1 S

−2 =

√l12+ SZ1 e

−iφ1√l12− SZ1

√l22− SZ2 e+iφ2

√l22+ SZ2 (8.4.1)

We re-arrange this:

S+1 S

−2 =

l1l24

√1 +

2

l1SZ1 −

2

l2SZ2 −

4

l1l2SZ1 S

Z2

e−iφ1e+iφ2

l1l24

√1− 2

l1SZ1 +

2

l2SZ2 −

4

l1l2SZ1 S

Z2 (8.4.2)

As before, we expand and combine the terms, discarding those of order O( 1l2) and

higher (assuming l1 and l2 are of the same order).

S+1 S

−2 =

l1l24

e−iφ + (

1

l1+

1

l2)e−iφ

− 2

l21(SZ1 e

−iφSZ1 )−2

l22(SZ2 e

−iφSZ2 )

(8.4.3)

where we have written φ = φ1 − φ2. The tunnelling Hamiltonian is then given by:


HT = −T1

2(l1l2 + l1 + l2) cosφ

−( l2l1(SZ1 )

2 +l1l2(SZ2 )

2)cosφ−

( l2l1iSZ1 +

l1l2iSZ2)sinφ

(8.4.4)

As before, the tunnelling energy is no longer constant with respect to SZ1 and SZ2 , but

varies as the number of Cooper pairs on each island changes.

Chapter 9

Conclusion

In this thesis we have discussed some of the advantages of quantum computing and dis-

cussed some of the problems that the construction of a quantum computer might face.

We have described how constructing qubits from superconductors might help with these

problems.

We have detailed theories of superconductivity, both phenomenological and micro-

scopic and shown how these lead to the Josephson effect, where the current across a

superconducting tunnel junction is proportional to the sine of the difference between the

phases of the superconducting condensates on either side of the junction. We have also

described how for small superconducting grains, finite size effects can lead to the parity

effect, where the condensation energy and pairing parameter are dependent on whether

there is an odd or even number of electrons on the grain.

Experimental progress on superconducting qubits has been reviewed, and the basic

circuit layout of a charge qubit described. A Hamiltonian for such a circuit was derived,

by treating the Josephson equations as classical equations of motion, and then quantising

them in the usual manner by constructing a classical Hamiltonian that leads to those

equations, and then elevating the conjugate variables to operators. We explained how if the

charging energy is much larger than the tunnelling energy, the circuit can function as a two-

level system, and investigated the dynamics of this two level system. We briefly discussed

the effect of noise on a charge qubit, and presented a general method for evaluating the

effect of discrete noise in the Josephson energy.

We examined the charging energy of various relevant circuits, in particular focussing

on the effect of connecting a reservoir to small islands. We found a general computational

159

160 Conclusion

method to establish the correct form for the charging energy by taking the limit where the

capacitance between the islands and the outside world goes to zero and the capacitance

between the reservoir and the outside world goes to zero. These detailed calculations al-

lowed us to construct a quantum Hamiltonian from the charging energy and the tunnelling

energy across each junction. We have shown that this naive method indeed produces the

correct equations of motion for each junction, and confirmed that the gate voltages applied

to each island act as independent tunings to the ‘zero position’ of the individual charging

energies, as is generally assumed.

We described the energy levels of a coupled two-qubit system and described the opera-

tions necessary to test a Bell inequality in this system. We showed how passing a magnetic

flux through the circuit lead to oscillations in both time and flux. We showed that these

oscillations can be considered as being caused by interference between virtual tunnelling

paths between states and showed how diagrams could be a useful visual aid in ‘keeping

track’ of perturbative expansions of the occupation probabilities of each state.

We described the strong coupling limit of the BCS Hamiltonian, where the sums over

electron creation and annihilation operators can be replaced by operators representing

large quantum spins. We then solved this Hamiltonian in both the exact and mean-

field limits, where we found the ground state was the spin coherent state. As part of

this solution, we investigated the rotation of finite spin operators. We investigated the

relationship between the solutions of the spin Hamiltonian and the solutions of the full

Hamiltonian, exact and mean-field. We increased our confidence in the model by showing

that the spin solutions are equal to the full solutions in the limit where the interaction

energy is much larger than the cutoff energy.

We showed how this simple model could be used to easily investigate various phenom-

ena. We described an analogue of the quantum optical effects of superradiance where the

current out of an array of Cooper pair boxes is proportional to the square of the number

of boxes. We showed that this effect remains present when the reservoir is in the spin co-

herent state. This effect illustrates clearly the quantum mechanical nature of the Cooper

pair boxes, and does so using only current measurements, rather than the experimentally

more difficult single-shot measurements.

We also described an analogue of quantum revival, where the coherent oscillations of a

Cooper pair box coupled to a reservoir decay, only to revive again later. The revival is due

to the fact that the Cooper pairs are discrete, and so the phase information is transferred

Conclusion 161

to the levels of the reservoir. We described a similar two-qubit effect, where the oscillations

of two entangled Cooper pair boxes decay and revive. If the entanglement between the

boxes is calculated, we find that the entanglement also decays and then revives. This

effect not only demonstrates the quantum nature of the Cooper pair boxes, but also of the

reservoir, and it is possible that this behaviour could be used as a probe of the reservoir.

We described how quantum spins can be represented using a wavefunction that is

a function of phase. Rather than the standard derivation which treats the Josephson

relations as (classical) equations of motion to be quantised, we derived these equations

directly from a microscopic electronic Hamiltonian, using the spin-Hamiltonian described

in chapter 5. The form for the spin operators in the phase representation were found.

It was found that in the limit of large size, the Hamiltonian found by quantising the

Josephson equations was regained. Corrections to this phase Hamiltonian were described.

In this thesis we have developed a simple model of superconductors that can be easily

applied to the study of various interesting aspects of superconducting systems. We have

used this model to derive the standard macroscopic qubit Hamiltonian directly from a

quantum description of the electrons. This effectively justifies the use of the ‘quantised

Josephson junction Hamiltonian’ that is commonly used to study these systems. The

microscopic derivation allowed us to calculate the corrections to this Hamiltonian, and

show how these corrections affect the energy levels of the system. We have also investigated

some interesting new phenomena (superradiance and revival) in superconducting systems,

which in principle could be observed by measuring the occupation probabilities of the

Cooper pair boxes rather than requiring single - shot measurements. Given that coherent

oscillations between two qubits have been observed, it is hoped that the effects described

above could be observed and illuminate the quantum behaviour of these systems.

Superconductivity remains a fascinating and broad subject nearly a century after its

first discovery. In particular the macroscopic quantum coherent nature of the electron

condensate offers an intriguing glimpse of the connections between the microscopic world

of quantum mechanics and the large-scale everyday macroscopic world. This is clearly

seen in superconducting qubits, where superpositions of macroscopically differing states

can be prepared.

For these reasons it is clear that mesoscopic superconducting systems such as charge

qubits will continue to be of great interest for quantum computing as well as condensed

matter physics, and also for demonstrating and investigating basic aspects of quantum

162 Conclusion

mechanics.

Appendix A

Eigenstates of the Two Qubit

Hamiltonian

In this Appendix we examine the eigenstates and eigenvalues of a coupled two-qubit system.

In particular we find these exactly for the case when all the tunnelling matrix elements are

real and use perturbation theory to generate approximate expressions when the tunnelling

matrix elements are complex, up to third order in T/EC12.

A.1 Exact Eigenstates with Real T12

We are interested in the Hamiltonian of the two - qubit system when the gate charges

are set to the degeneracy point. We can solve this analytically when all the tunnelling

matrix elements are real, i.e. when there is an integer or half integer number of flux quanta

through the circuit (see section 2.3).

H =

0 −T1 −T2 0

−T1 −EC12/2 −T12 −T2−T2 −T12 −EC12/2 −T10 −T2 −T1 0

(A.1.1)

The eigenvalues |vn〉 are given below.

163

164 Appendix A. Eigenstates of the Two Qubit Hamiltonian

e1,4 = −1

4EC12 ∓

1

2T12 +

1

4

√(EC12 ± 2T12)2 + 16(T1 ± T2)2

e2,3 = −1

4EC12 ∓

1

2T12 −

1

4

√(EC12 ± 2T12)2 + 16(T1 ± T2)2 (A.1.2)

In the limit of small tunnelling energies,the eigenvectors |v1〉 and |v4〉 are the symmetric

and anti-symmetric combinations of |00〉 and |11〉, and |v2〉 and |v3〉 are the symmetric

and anti-symmetric combinations of |01〉 and |10〉. When the tunnelling energies go to

zero,e1,4 go to zero, and e2,3 go to −EC12/2 as would be expected.

|v1〉 =

1

−e1/(T1 + T2)

−e1/(T1 + T2)

1

1√2 + 2(e1)2

(T1+T2)2

|v2〉 =

1

−e2/(T1 + T2)

−e2/(T1 + T2)

1

1√2+

2(e2)2

(T1+T2)2

|v3〉 =

1

−e3/(T1 − T2)e3/(T1 − T2)

−1

1√2 + 2(e3)2

(T1−T2)2|v4〉 =

1

−e4/(T1 − T2)e4/(T1 − T2)

−1

1√2+

2(e4)2

(T1−T2)2

(A.1.3)

These expressions for the eigenvectors contain terms like e2/(T1 + T2), which go to

infinity as the tunnelling energies go to zero. When doing calculations, it is helpful to note

that each of the eigenvectors can be rewritten in a different form:

|v1〉 =

e2/(T1 + T2)

1

1

e2/(T1 + T2)

1√2 + 2(e2)2

(T1+T2)2

|v2〉 =

e1/(T1 + T2)

1

1

e1/(T1 + T2)

1√2+

2(e1)2

(T1+T2)2

|v3〉 =

e4/(T1 − T2)1)

−1−e4/(T1 − T2)

1√2 + 2(e4)2

(T1−T2)2|v4〉 =

e3/(T1 − T2)1

−1−e3/(T1 − T2)

1√2+

2(e3)2

(T1−T2)2

(A.1.4)

A.2 Perturbation in T12 165

These two forms for the eigenvalues are equal, and the most convenient expression can

be chosen for each calculation.

A.2 Perturbation in T12

In section 4.4 we considered the Hamiltonian with no inter-qubit tunnelling as our unper-

turbed Hamiltonian, and then did perturbation theory in a complex T12.

H = H0 +H ′

=

0 −T1 −T2 0

−T1 −EC12/2 0 −T2−T2 0 −EC12/2 −T10 −T2 −T1 0

+

0 0 0 0

0 0 −T ∗12 0

0 −T12 0 0

0 0 0 0

(A.2.1)

The unperturbed eigenvalues (e0n) and eigenvectors (|v0n〉) can be trivially found from

the above expressions by setting T12 = 0. The first order correction e1n = 〈v0n|H ′|v0n〉to the energies can then be calculated.

e11 = 〈v01|H ′|v01〉 (A.2.2)

= (T12 + T ∗12)(e01)

2

(T1 + T2)21√

2 +2(e01)

2

(T1+T2)2

(A.2.3)

This expression is correct to all orders in T1 and T2, but if we want to do an expansion

(we do!) we find that:

e01,4 = −1

4EC12 +

1

4

√(EC12)2 + 16(T1 ± T2)2

= −1

4EC12 +

EC124

(1 + 16

(T1 ± T2)2(EC12)2

)1/2

≈ 2(T1 ± T2)2EC12

(A.2.4)

Similarly,

166 Appendix A. Eigenstates of the Two Qubit Hamiltonian

e02,3 = −1

4EC12 −

1

4

√(EC12)2 + 16(T1 ± T2)2

= −1

4EC12 −

EC124

(1 + 16

(T1 ± T2)2(EC12)2

)1/2

≈ −EC122− 2(T1 ± T2)2

EC12(A.2.5)

We can now calculate the correction to the unperturbed energies:

e11 = 〈v01|H ′|v01〉

≈ (T12 + T ∗12)2(T1 + T2)

2

(EC12)2

(1 +

4(T1 + T2)2

(EC12)2

)−1

≈ (T12 + T ∗12)2(T1 + T2)

2

(EC12)2(A.2.6)

When calculating the correction to e02 it is more convenient to use the second form (eq.

A.1.4) for the unperturbed eigenvectors. To second order in T1, T2:

e12 = 〈v02|H ′|v02〉

= (T12 + T ∗12)1√

2 +2(e02)

2

(T1+T2)2

≈ (T12 + T ∗12)2

(1 +

4(T1 + T2)2

(EC12)2

)−1

≈ (T12 + T ∗12)2

− (T12 + T ∗12)2(T1 + T2)

2

(EC12)2(A.2.7)

We calculate the corrections to the remaining two energies in the same way;

e13 ≈ −(T12 + T ∗12)2

+ (T12 + T ∗12)2(T1 + T2)

2

(EC12)2

e14 ≈ −(T12 + T ∗12)2(T1 + T2)

2

(EC12)2(A.2.8)

The energy differences, and therefore the frequencies, can be easily calculated from

this.

Appendix B

Limit of BCS Self Consistency

Equations

In this Appendix we solve the BCS gap equations exactly at zero temperature, using a

top hat density of states. This then allows us to take the strong coupling limit where the

pairing potential is much larger than the cutoff energy. We show that solving the BCS

gap equations with the energy levels and then taking the strong coupling limit yields the

same results as assuming all the energy levels to be equal initially, and then finding the

mean-field solution

As confirmation of the validity of our spin model, we wish to compare it with the

standard BCS theory. In particular, we wish to check that solving the gap equations from

the spin model gives the same result (5.4.19, 5.4.20) as if we solve the usual BCS gap

equations (1.6.8,1.6.9) first, and then take the limit. We wish to take the limit where the

single-electron energy levels, εk are equal. To do this we consider a density of states which

is a top-hat function centred around εF with width ω and height lr/ω. Taking the limit

ω → 0 ensures that the interaction occurs in a narrow region around εF whilst ensuring

the integral is over a constant number of levels, lr. We have two integral equations that

must be solved self-consistently. The first is an equation for the pairing parameter, ∆:

∆

V=

1

2

εF+ω∫

εF−ω

dε∆

((ε− µ)2 + |∆|2)) 12(B.0.1)

167

168 Appendix B. Limit of BCS Self Consistency Equations

and the second is an equation for the average number of Cooper Pairs, N :

N =1

2

εF+ω∫

εF−ω

dε

(1− ε− µ

((ε− µ)2 + |∆|2)) 12

)(B.0.2)

As we are at zero temperature, both these integrals can be done exactly:

1

V=

l

2ωln

((ξF + ω/2) +

√(ξF + ω/2)2 + |∆|2

(ξF − ω/2) +√

(ξF − ω/2)2 + |∆|2

)(B.0.3)

N =l

ω

(ω2−√

(ξF + ω/2)2 + |∆|2 +√

(ξF − ω/2)2 + |∆|2)

(B.0.4)

Again, ξF is shorthand for εF − µ. These can be rearranged to give two equations for

|∆|2 (B.0.5, B.0.6), or two equations for ξ2F (B.0.8, B.0.9). The equations for |∆|2 are:

|∆|2 = (ξF )2

((x+ 1)2

(x− 1)2− 1

)+(w2

)2((x− 1)2

(x+ 1)2− 1

)(B.0.5)

|∆|2 =N(l − N)

l2(l − 2N)2(4ξF

2l2 − ω2(l − 2N)2)

(B.0.6)

Equating the two and rearranging gives us an expression for ξF2:

ξF2 =

(ω2

)2((x+ 1)2

(x− 1)2− 1 + 4(1− 2n)2

)/(4n(1− n)(1− 2n)2

− (x− 1)2

(x+ 1)2+ 1

)(B.0.7)

where x = e2wV l and n = N

l . If we rearrange eqns. (B.0.3, B.0.4) for ξF we get:

ξF2 =

(ω2

)2 (x+ 1)2

(x− 1)2− |∆|2 (x+ 1)2

4x(B.0.8)

ξF2 =

(ω2

)2(1− 2n)2 + |∆|2 (1− 2n)2

4n(1− n) (B.0.9)

Which yield:

|∆|2 =(ω2

)2((x+ 1)2

(x− 1)2− (1− 2n)2

)/((1− 2n)2

4n(1− n) +(x+ 1)2

4x

)(B.0.10)

From eqns. (B.0.7, B.0.10), it is clear that the correct limit to take is ω/V → 0. Taking

this limit of eqns. (B.0.7, B.0.10), we get:

Appendix B. Limit of BCS Self Consistency Equations 169

ξF2 =

V 2l2

4(1− 2n)2 (B.0.11)

|∆|2 = V 2l2n(1− n) (B.0.12)

That is, we have regained the forms for ∆ and ξ given by the spin model (5.4.19,

5.4.20).

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