Josephson qubits

Josephson qubits

P. Bertet

SPEC, CEA Saclay (France),Quantronics group

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

11

00

01

switc

hing

pro

babi

lity

swap duration (ns)

10

Introduction : Josephson circuits for quantum physics

M.H. Devoret, J.M. Martinis and J. Clarke, PRL 85, 1908 (1985)

M.H. Devoret, J.M. Martinis and J. Clarke, PRL 85, 1543 (1985)YES THEY CAN

Discrete energy levels

… to genuine quantum information and quantum optics on a chip

CAN MACROSCOPIC « MAN-MADE » ELECTRICAL CIRCUITS BEHAVE QUANTUM-MECHANICALLY ????

From a fundamental question (25 years ago) ….

M. Hofheinz et al., Nature (2009)Q. state engineering and

Tomography M. Neeley et al., Nature (2011)3-qubit entanglement

… ANDMUCH MORE TO COME !!

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2: Qubit readout and circuit quantum electrodynamics

Lecture 3: 2-qubit gates and quantum processor architectures

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2: Qubit readout and circuit quantum electrodynamics

Lecture 3: 2-qubit gates and quantum processor architectures

1) Introduction: Hamiltonian of an electrical circuit

2) The Cooper-pair box3) Decoherence of superconducting qubits

Ene

rgy

(a.u

)

x (a.u.)

Real atoms

Hydrogen atom

2

0

( )4eV rr

2ˆˆ ˆ( )2pH V rm

quantization ˆ ˆ,r p i

|0>

|1>|2>

|3>

« two-level atom »

E01=E1-E0=hn01

I.1) Introduction

Real atoms

Hydrogen atom

2

0

( )4eV rr

2ˆˆ ˆ( )2pH V rm

ˆ ˆ,r p i

Frequency n (a.u)

n01

FWHMG/2

P(1

) (a.

u)

Spectroscopy (weak field)

laser( ) ( )IH t d E t

0( ) cos 2E t E tn

+ spontaneous emission G

OpticalBloch

equations

P(1

)

0

1

Time (a.u)

Rabi oscillations (short pulses, strong field at n=n01)

0R

d Ef

h

|0> |0>

|1>

1 0 12

I.1) Introduction

quantization

X,f

E

0|0>|1>

|2>|3>

|4>

Quantum regime ??

ˆ ˆ,x p i ˆ ˆ, iQ

Electrical harmonic oscillator

k

m

x

2 2

( , )2 2kx pH x p

m

+qf

-q

2 2

( , )2 2

H QQL C

0km

01LC

( ) ( ') 't

t V t dt

( ) ( ') '

tQ t i t dt

I.1) Introduction

LC oscillator in the quantum regime ?

2 conditions :

X,f

E

|0>|1>

|2>|3>

|4>0kT

Typic : L nH C pF 0 5GHzn

At T=30mK : 0 8hkTn

X,f

E

|0>|1>

|2>|3>

|4>1Q

OK if dissipation negligible

Superconductors at T<<Tc

I.1) Introduction

T<<Tc : dissipation negligble at GHz frequencies

Q=4 1010

at 51GHzand at 1K

Microwave superconducting resonators

Tc(Nb)=9.2K

S. Kuhr et al., APL 90, 164101 (2006)

Quantumregime

I.1) Introduction

Necessity of anharmonicity

How to prepare |1> ?

X,f

E

0|0>|1>

|2>|3>

|4>+qf

-q

Need non-linear and non dissipative element : Josephson junction

I.1) Introduction

Basics of the Josephson junction

The building block of superconducting qubits

Josephson AC relation : 2d QV

e dt C

Josephson DC relation : sinCI I

R L= mod(2 )=θ -θ 0,/2e

θ 2

N=Q/2e

RθLθ

( ) ( ') 't

t V t dt

( ) ( ') '

tQ t i t dt

B. Josephson, Phys. Lett. 1, 251 (1962)

P.W. Anderson & J.M. Rowell, Phys. Rev. 10, 230 (1963)

S. Shapiro, Phys. Rev. 11, 80 (1963)

I.1) Introduction

Basics of the Josephson junction

The building block of superconducting qubits

Classical variables ??

NON-LINEAR INDUCTANCE 2( )

2 1 /J

C C

L IeI I I

POTENTIAL ENERGY ( ) cos cos2C

J JIE Ee

Josephson AC relation : 2d QV

e dt C

Josephson DC relation : sinCI I

R L= mod(2 )=θ -θ 0,/2e

θ 2

N=Q/2e

RθLθ

( ) ( ') 't

t V t dt

( ) ( ') '

tQ t i t dt

I.1) Introduction

Hamiltonian of an arbitrary circuit

=

HAMILTONIAN ???

M. H. Devoret, p. 351 in Quantum fluctuations (Les Houches 1995)G. Burkard et al., Phys. Rev. B 69, 064503 (2004)

Correct procedure described in :

G. Wendin and V. Shumeiko, cond-mat/0508729

M.H. Devoret, lectures at Collège de France (2008) accessible online

=


=

HAMILTONIAN ???

=

1) Identify the relevant independent circuit variables

2) Write the circuit Lagrangian

3) Determine the canonical conjugate variables and the Hamiltonian


=

=

1) Choose reference node (ground)

branchnode

Identifying the relevant independent circuit variables


=

=


2) Choose « spanning tree » (no loop)



=

=



3) Define « tree branch fluxes »

a

c d

eb

( ) ( ') 't

i t V t dt



=

=



3) Define « tree branch fluxes »

a

c d

eb

4) Define node fluxes = sum of branch fluxes from ground

4= b+ d

51

3 2

( ) ( ') 't

i t V t dt



=

=

Write Lagrangian ( , ) ( ) ( )i i el i pot iL L L

taking into account constraints imposed by external biases (fluxes or charges)

a

c d

eb

4= b+ d

51

3 2

ext

1 2

21 22 1cos

2ext

pot JL EL


=

=

Conjugate variables : ii

LQ

Classical Hamiltonian ( , )i i i iH Q Q L

Quantum Hamiltonian ˆˆ( , )i iH Q With ˆˆ ,i iQ i

ˆ ˆ( , )i iH n ˆ ˆ,i in i or with

ˆˆ / 2i in Q eˆ ˆ (2 / )i i e

a

c d

eb

4= b+ d

51

3 2

Different types of qubits

Cooper-pair boxes Phase qubits

CJ EE /0.1 1 50

Flux qubits

50 410

Junctions sizes

.01 to 0.04 mm2 .04mm2 100mm2

NISTSanta Barbara

TU DelftMIT

BerkeleyNEC

SaclayChalmers

NECYale

ETH Zurich

ShapeOf the

PotentialEnergy

-1 0 1

Ep (

a.u)

/ (rad)-2 0 2

Ep

(a.u

)

m/

-2 0 2 4

Ep

(a.u

)

(rad)

Charge qubit/Quantronium/Transmon

I.2) Cooper-Pair Box

Different types of qubits

Cooper-pair boxes Phase qubits

CJ EE /0.1 1 50

Flux qubits

50 410

Junctions sizes

.01 to 0.04 mm2 .04mm2 100mm2

NISTSanta Barbara

TU DelftMIT

BerkeleyNEC

SaclayChalmers

NECYale

ETH Zurich

ShapeOf the

PotentialEnergy

-1 0 1

Ep (

a.u)

/ (rad)-2 0 2

Ep

(a.u

)

m/

-2 0 2 4

Ep

(a.u

)

(rad)

Charge qubit/Quantronium/Transmon


Vg

The Cooper-Pair Box

1 degree of freedom1 knob

θ̂,N̂ i

ˆ ˆˆ cos2gC JEH = ( -E N N ) -

JCE ,E

2(2 )2ceEC

« charging energy »

Vg

20ˆ cosˆ

cosˆˆ ˆ

J2

gCH = ( -N ) -2

E E2L

N

1θ̂ , ˆ 1N i2 d° of freedom

JCE ,E1N

2N

2θ̂ , ˆ 2N i

θ1

θ2L

inductance

2 knobs

small

0

2

J

LE

2 11 2

ˆ ˆθ -θ ˆ ˆ ˆ2

,θ̂= =N N -N

ior

1 21 2

ˆ ˆˆˆ ˆ ˆ=θ +θ =2

, N +NK

i

θ

The split CPB

Vg

ˆ cos cosˆ ˆ2JgCH = (E N E-N ) -

2

θ̂,N̂ i1 d° of freedom

JCE ,E

N

2 knobs

0

δ=

tunable JE

The split CPB

Energy levels of the CPB

ˆ ( , ) ( )cosˆˆ C J2

g gH N = ( - EN ) -E N

Solve either in charge basis |N>

( , ), ( , )k g k gE N N

N ( ) ,k k NN

c N

ˆg

N

J2C

EE N N N N+1 NH = ( -N ) -2

N N+1

... ......

...

...

...

...

2

C Jg

2

C g

2

g

J J

CJ

N=-1 N=0 N=1

... ... ...

0 ...

... ...

... 0

... ... ..

...

E (-1- N )

E (0 - N )

-E /2

-E /2 -

E (1- N

E /2

-E

.

/2 )

Diagonalize


Energy levels of the CPB

ˆ ( , ) ( )cosˆˆ C J2

g gH N = ( - EN ) -E N

… or in phase basis |>

( , ), ( , )k g k gE N N

0,2 ( )2

0

( )k kd

ˆ ( , ) ( ) cos

2g gC JH N = ( -N ) -E1E

i

Solve Mathieu equation

( )cos

Jk kgC k k2 ( ) (( -N ) - =E)1 E

i(E )


Two simple limits : (1) ( )J CE E

0 0.5 1 1.5 2 2.5 3

0

1

2

Ng

Ene

rgy

0 0.5 1 1.5 2 2.5 3

0

1

2E jE c 0

|N=0>

|N=1>

|N=2>

EJ/Ec=0

N

0 ( )c N

N

1( )c N3 2 1 0 1 2 3

1

3 2 1 0 1 2 3

1 0 0

0.5

0 0

0.5

20 ( )

21( )

(charge regime)


0 0.5 1 1.5 2 2.5 3

0

1

2

Ng

Ene

rgy

EJ/Ec=0.1

E0(Ng)E1(Ng)

E2(Ng)

3 2 1 0 1 2 3

1

3 2 1 0 1 2 3

1

0 ( )c N

1( )c N

0 0

0.52

0 ( )

21( )

0 0

0.5

Ng=0.01

(charge regime)

QUBIT

0 0

0.5


0 0.5 1 1.5 2 2.5 3

0

1

2

Ng

Ene

rgy

EJ/Ec=0.1

E0(Ng)

E1(Ng)

E2(Ng)

0 ( )c N

1( )c N

20 ( )

21( )

Ng=0.5

3 2 1 0 1 2 3

1

3 2 1 0 1 2 3

1

0 0

0.5

(charge regime)

QUBIT

0.0 0.2 0.4 0.6 0.8 1.0 1.0 0.5

0.00.51.01.52.0

From ( )J CE E to ( )J CE E E

nerg

y

Ng

EJ/Ec=0.5

0.0 0.2 0.4 0.6 0.8 1.0 2

1

0

1

2

3

Ng

EJ/Ec=2

0.0 0.2 0.4 0.6 0.8 1.0

4

2

0

2

Ene

rgy

Ng

EJ/Ec=5

0.0 0.2 0.4 0.6 0.8 1.0 8

6

4

2

0

2

Ng

EJ/Ec=10

STILL A QUBIT !



(phase regime)

0.0 0.2 0.4 0.6 0.8 1.0 8

6

4

2

0

2

Ng

EJ/Ec=10

4 2 0 2 4 1.0 0.5

0.00.51.0

4 2 0 2 4 1.0 0.5

0 .00 .51 .0

0 ( )c N

1( )c N

0 0

0.5

0 0

0.5

20 ( )

21( )

J. Koch et al., PRA (2008)

Experimental spectrum of a transmon

J. Schreier et al., PRB (2008)

Ng(t)=DNgcost

One-qubit gates

TRANSMON QUBIT

ˆ ˆ ˆ( ) cost 2

C g JH = E N-N -E

ˆ ˆ ˆˆcos 2 gN D2C J CH = E N -E - E cos tN

transmon drive

01( )2 z

cosR xt Two-level

approximation

2 0 1R C gE N N DI.2) Cooper-Pair Box


One-qubit gates

TRANSMON QUBIT

f0

Rotation : X

|0>

|1>

X

Z

Y

=01


One-qubit gates

TRANSMON QUBIT

f0f0f

Rotation : Xf

|0>

|1>

X

Z

Y fXf

X

=01


One-qubit gates

TRANSMON QUBIT

f0f0f

Rotation : Xf

|0>

|1>

X

Z

Y fXf

X

Z rotation

=01

All rotations on Bloch sphere

Fidelity ? 99%J.M. Chow et al., PRL 102, 090502 (2009)

Decoherence

d(t)

Ng+dNg(t)

Noise in Hamiltonian parameters

DECOHERENCE

MAJOR OBSTACLE TO QUANTUM COMPUTING

I.3) Decoherence

1

0

01

Relaxation(Spontaneous emission)

Decoherence in superconducting qubits(Ithier et al., PRB 72, 134519, 2005)

1 21 1 , 01( )

2T D S

G

environmental density ofmodes at qubit frequency

Pure dephasing

( )i te 0 1

1

0

01( )i

t

i(t)

1 12 2 2T

GG G

2, (0)zD S G

2( )( ) ti tf t e e

G

Low-frequencynoise

01,zD

,1 0 1HD

I.3) Decoherence

Decoherence

d(t)

Ng+dNg(t)


DECOHERENCE

Origin of the noise ???

I.3) Decoherence

Decoherence

d(t)

Ng+dNg(t)


DECOHERENCE


R

R

1) ELECTROMAGNETIC

Low-frequency : Johnson-Nyquist due to thermal noise

High-frequency : spontaneous emission (quantum noise)

Under control

I.3) Decoherence

Decoherence

d(t)

Ng+dNg(t)


DECOHERENCE


R

R

2) MICROSCOPIC

e- Spin flips

Low-frequency noise well studied :

Charge noise3 2(10 )( )NgS

Flux noise6 2

010( )( )S

I.3) Decoherence

High-frequency (GHz)microscopic noise TOTALLY UNKNOWN !!

I.3) Decoherence

Decoherence

d(t)

Ng+dNg(t)


DECOHERENCE


R

R

2) MICROSCOPIC

e- Spin flips

Low-frequency noise well studied :

Charge noise3 2(10 )( )NgS

Flux noise6 2

010( )( )S

High-frequency (GHz)microscopic noise TOTALLY UNKNOWN !!

CPB in charge regime 2 10 100T ns 2 1 100T sm

Transmon 2 1 10T ms 2 1 100T sm

State-of-the-art coherence times

T1=1-2ms

Schreier et al., PRB 77, 180502 (2008)

T2=1-3ms

I.3) Decoherence

Very recent breakthrough: transmon in 3D cavity

T1=60msT2=15ms

H. Paik et al., quant-ph (2011)

ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET

I.3) Decoherence

I.3) Decoherence

SiO2

PMMA-MAAPMMA

e-

O2Al/Al2O3/Al junctions

1) e-beam patterning2) development3) first evaporation4) oxidation5) second evap.6) lift-off7) electrical test

small junctions Multi angle shadow evaporation

Fabrication techniquessmall junctions e-beam lithography

I.3) Decoherence

gate 160 x160 nm

QUANTRONIUM (Saclay group)

FLUX-QUBIT (Delft group)

I.3) Decoherence

I.3) Decoherence

40mm

2mm

TRANSMON QUBIT (Saclay group)

END OF FIRST LECTURE

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Josephson qubits