CIRCUIT QED

Circuit QED: A promising advance towardsquantum computing

Himadri Barman

Jawaharlal Nehru Centre for Advanced Scientific ResearchBangalore, India.

QCMJC Talk, July 10, 2012

CIRCUIT QED

Outline

• Basics of quantum computation.

• QED and cavity-QED.

• Superconducting qubits.

• Circuit QED (cQED).

• Quantum Rabi model and its analytical solution.

CIRCUIT QED

Basics of quantum computation

• Qubits: States of a two-level system or their superposition.• Single qubit: a|0〉+b|1〉• Two qubits: a|00〉+b|01〉+ c|10〉+d|11〉

• In general they are entangled: cannot be written as a productof qubits.

• DiVincenzo’s criteria (D. P. DiVincenzo, arXiv:quant-ph/0002077v3)

1 Scalability of physical system with well characterized qubits.2 The ability to initialize the state of the qubits.3 Decoherence (coherence) time >> gate operation time.4 A universal set of quantum gates.5 A qubit specific measurement capability.

• Here we shall mainly talk about the third criterion.

CIRCUIT QED

Basics of quantum computation

Basic anatomy of a quantum algorithm

CIRCUIT QED

Quantum electrodynamics (QED): Semiclassical

• Matters interacting with light.

• Two-state problem with a potential oscillating in time(Chap. 5, J. J. Sakurai, Mod. Q.M.).

H = H0 +V (t)

H0 = E1|1〉〈1|+E2|2〉〈2| (E2 > E1)

V (t) = γeiωt |1〉〈2|+ γe−iωt |2〉〈1|

Probability of finding in state |2〉

|c2(t)|2 =γ2/h2

Ω2 sin2(Ωt)

where Ω≡√

γ2/h2 +(ω−ω21)2/4 and ω21 ≡ (E2−E1)/h.

CIRCUIT QED

QED: Rabi oscillation

• From probability conservation law: |c1(t)|2 = 1−|c2(t)|2

• Resonance condition: ω = ω21, i.e. Ω = γ/h.

• This principle has been applied in nuclear magnetic resonance(NMR) and masers.Nobel prizes: I. I. Rabi (1944); E. M. Purcell and F. Bloch (1952);

C. H. Townes, N. Basov, and A. Prokhorov (1964), N. F. Ramsey

(1989).

CIRCUIT QED

Cavity QED: Atom in a resonator

• A Rydberg atom: hydrogen-like atom in excited state withlarge principal q. no.: En =−Ry/(n−δ )2 ∼−Ry/n2 for largen) placed inside a cavity resonator.

2g = vacuum Rabi fre-quency, κ= cavity decayrate, γ= transverse decayrate, t=transition time.

• Hamiltonian:H = hω(a†a+ 1

2)+ hg(a†σ−+aσ+)+ h∆σ z +Hκ +Hγ where2∆= energy level spacing, ω=resonator’s frequency.

Raimond, Brune, and Haroche, Rev. Mod. Phys. 73, 565 (2001)

CIRCUIT QED

Dephasing and decoherence• Coherence can be tested through interference.• Interference pattern decays under repeated trials.

• T ∗2 =dephasing timescale, T1=equilibration time,T2=decoherence timescale; T1 ≥ T2/2.

Laad et al. , Nature 464, 45 (2010)

CIRCUIT QED

Jaynes-Cummings Hamiltonian (strong coupling limit:g >> γ,κ,1/t)

HJC = hω(a†a+12)+ hg(a†

σ−+aσ

+)+∆σz

One can easily think that this operates only on two possible states:|e, n〉 and |g, n+1〉. |e, n〉= excited state of the atom with nphotons ≡ | ↓, n〉 |g, n+1〉= ground state of the atom with n+1photons ≡ | ↑, n+1〉 Then in matrix form

H = (n+12)hω1+

[δ g

√n+1

g√

n+1 −δ

]where δ ≡ ∆− 1

2 hω. After diagonalizing the off-diagonal term weget the eigen energies

E±n = (n+12)hω±

√δ 2 +g2(n+1)≡ (n+

12)±Ωn

Jaynes and Cummings, Proc. IEEE 51, 89 (1963)

CIRCUIT QED

Vacuum Rabi oscillationWe also can find the probability to find in the ground state |g〉 attime t, for an atom initially at the excited state |e〉.

Pe(t) = ∑n

p(n)sin2(g√

n+1t)

where p(n) is the photon number distribution. ⇒ In ν space,maxima should occur at hν ,

√2hν ,

√3hν , · · · .

Also Tφ ∼ 10 ns.Brune et al. Phys. Rev. Lett. 76, 1800 (1996)

CIRCUIT QED

Quantum harmonic oscillator (QHO) on an electricalcircuit

LC circuit:

• In quantum version H = φ 2

2L +q2

2C .

• Analogous to a QHO: H = 12m p2 +

mω20

2 x2.

• So H = hω0(a†a+ 12); ω = 1/

√LC.

• However, energy levels are equally spaced ∆E = hω0.

• We need transition to be restricted only between two levels.

• So we need non-linearity: anharmonic quantum oscillator.

CIRCUIT QED

Josephson junction: a non-linear inductor

• I = Ic sinφ where Ic = (2π/Φ0)EJ; Φ = h/(2e); andφ = φL−φR.

• Also we have dφ/dt = (2π/Φ0)V• Using the relation V = LJ dI/dt we get

LJ = Φ0/(2πIc cosφ)

• So a Josephson junction is equivalent to a non-linear inductor(which we want).

CIRCUIT QED

Superconducting qubits: Cooper pair box (CPB)

• HCPB = EC(N−Ng)2−EJ cos φ where

EC = (2e)2/(2(CJ +Cg))=charging energy; Ng =CgVg/(2e).• Just projecting on the space formed by the two states| ↑〉 ≡ |n〉 and | ↓〉 ≡ |n+1〉, we can rewrite in the matrix form

HCPB =CgVg

2(C+2CJ)σ

z +EJ

2σ

x

• In absence of tunneling, ∆E = EC(1−2Ng), degeneracy atNg = 1/2, but lifted in presence of tunneling.

• This is a desirable region for qubit operation.Shnirman, Schon, and Hermon, Phys. Rev. Lett. 79, 2371 (1997); Bouchiat,

Vion, Joyez, Esteve, and Devoret, Phys. Scr. T76, 165 (1998)

CIRCUIT QED

Superconducting qubits: Cooper pair box (CPB)

Bouchiat et al. Phys. Scr. T76, 165 (1998) Vion et al. Science 296, 886 (2002)

• Degeneracy points (Ng = 1/2-integer) zero slope ⇒ leastaffected by noise.

• Shows optimal coherence (“sweet spots”).

• Tφ ∼ 0.5 µs.

CIRCUIT QED

Circuit QED: On-chip realization of cavity QED

CIRCUIT QED

Circuit QED: On-chip realization of cavity QED

• Cavity replaced by a 1D superconducting transmission line.

• A transmission line is a chain of LC oscillators.

• This creates coplanar waves that get reflected in the gap.

• Use superconducting charge qubits with EJ/EC >> 1:transmon.

• The oscillators generate a microwave photon coupled to thetransmon.

Blais et al. , Phys. Rev. A 69, 062320 (2004)

CIRCUIT QED

Circuit QED: On-chip realization of cavity QED

Everywhere are the “sweets spots” for large EJ/EC.Koch et al. PRA 76, 042319 (2007)

Tφ = 5.5±0.2 µsSchreier et al. PRB 77, 180502 (R) (2008)

CIRCUIT QED

Quantum Rabi model

• a†σ++aσ− terms added to the Jaynes-CummingHamiltonian:

HR = hωa†a+ hgσx(a† +a)+∆σ

z

• Why HJC is solvable, HR seems not?• Charge C = a†a+ 1

2(σz +1) is conserved (good q. #),

[C,HJC] = 0.• Braak (2011) pointed out that HR has a Z2 (parity)

symmetry can be decomposed into two subspaces.

CIRCUIT QED

Comparison with other candidates

Source: http://universe-review.ca/R13-11-QuantumComputing.htm

CIRCUIT QED

Some comments

Advantages

• Can study open quantum systems in contrast to ultra-coldatoms.

• More practical for implementation since easy to fabricate on achip.

• Can be tuned easily by suitably designing the circuits.

Disdvantages

• Coherence time is still smaller.

• Can work at low temperature only.

• Presence of disorder is still a persisting problem.

CIRCUIT QED

Thanks for your kind attention !