Quantum measurement and superconducting qubits

Yuriy Makhlin (Landau Institute)

STMP-09, St. Petersburg 2009, July 3-8

• superconducting qubits

• quantum readout

• parametric driving and bifurcation

• stationary states, switching curve

Outline

Quantum bits (qubits): 2-level quantum systems ,

Logic gates: unitary evolution

Quantum algorithms: sequence of elementary operations

Quantum computers

Why quantum?

computers:

- high computational capacity?(R. Feynman, D. Deutsch)

Shor algorithm: factorizationin polynomial time, 1994

physics:

- fundamental phenomenain quantum-mechanical systems

- devices: new challenges

NMR[Chuang et al., Cory et al.] nuclear spins in

various environmentsBx(t)

Bz

Physical realizations of qubits

Ion traps [Cirac & Zoller, …]

laser

2-level ionsg

e

0 pheE E

• 2-level quantum systems — N qubits• controlled dynamics

• initialization - cooling or read-out

• coherence:

• read-out: quantum measurement

controlled weak

enough forerror correction

Requirements to realizations of quantum computers

controlled controlled

• 1-qubit gates

• 2-qubit gates

Josephson quantum bits

quantum degree of freedom: charge or phase (magn. flux)

• macroscopic quantum physics (unlike in ion traps, NMR, optical resonators)

• artificial atoms:flexibility in fabrication

• scalability (many qubits)

• easy to integrate in el. structures

combine:

• coherence of superconducting state

• advanced control techniques for single-charge and SQUID systems

Quantronium

(Saclay)no excitations at low T

V. Bouchiat et al. (1996)

Single-electron effects

junctions w/ small area 10nm x 10nm

typical capacitance C ≈ 10-15 F

typical energies EC =e2/2C ≈ 1 K

Josephson charge qubits

θ̂,N̂

gV

Vg

x

N x

θ, N

tunable JE

0

g xJ

gC 2e

EC

H cosN (π ) c θV

E os

2 ( )

2 states only, e.g. for EC » EJ

z xh xJgc1

2

1

2σE )V σ(H E) (

Nakamura et al. ’992 qubits: Pashkin et al. ‘03

~ 2 ns

Problems: decoherence

noise => random phases => dephasing (T2)

noise => transitions => energy relaxation (T1)

low-frequency 1/f noise => strong dephasing

1/T2 = 1/(2T1) + 1/T2*

Quantronium

operation at a saddle point

Charge-phase qubitVion et al. (Saclay) 2002

1 1

22( ) ( )z xxg JchEH V E

“Transmon”Koch et al. (Yale) 2007

(“coordinate”)

EJ

heavy particle (CB) => flat bands => very low sensitivity to charge noise

still (slightly) anharmonic => address only two states

T2 2 s (close to 2T1 without spin echo)

in a periodic potential

e-EJ/EC

EJ/EC~50

E(q)

Quantum measurementdetector

result 0 1state |0> |1>

probability |a|2 |b|2

Measurement as entanglement

unitary evolution of qubit + detector

|Mii --- macroscopically distinct states

- reliable ? single-shot?- QND? back-action

- fast?

quality of detection:

probability that melectrons passed

Dynamics of current

I(t)

QPC

Qubit dynamics

Reduced density matrix of qubit:

meas≥

Mixing (relaxation)

Quantum readout for Josephson qubits

switching readout – monitor the critical current• no signal at optimal point• strong back-action by voltage pulse (no QND), quasiparticles• threshold readout

QuantronicsIthier, 2005

U= - Ic cos – I

Quantum readout for Josephson qubits

dispersive readout – monitor the eigen-frequency of LC-oscillator

Sillanpää et al. 2005

Wallraff et al. 2004

monitor reflection / transmissionamplitude / phase

Quantum readout for Josephson qubits

Josephson bifurcation amplifier (Siddiqi et al.) – dynamical switching

exploits nonlinearity of JJ

switching between two oscillating states(different amplitudes and phases)

advantages:no dc voltage generated, close to QNDhigher repetition ratequbit always close to optimal point

Siddiqi et al. 2003

cf. Ithier, thesis 2005

Parametric bifuraction for qubit readout

I0=Ic sin 0 =0+x = t

Landau, Lifshitz, Mechanicsdiff. driving: Dykman et al. ‘98

w/ A.Zorin

Method of slowly-varying amplitudes

A2=u2+v2

Migulin et al., 1978

Equations of motion in polar coordinates:

Equations of motion

Hamiltonian

without quartic term

stationary solutions:

A=0 or

A2

detuning

=0

detuning

detuning, -0

driving P

bistability

“Phase diagram”

width detuning

Pswitching

0

1

Psw = 1 – e –

|0>

|1>

Switching curves

single-shot readout?

contrast

Dykman et al. (1970’s – 2000’)

0, A+ – stable states A- – unstable state

Velocity profile and stationary oscillating states

2D Focker-Planck eq. => 1D FPE

near origin: relaxes fast, A – slow degree of freedom

(at rate )

Near bifurcation -

cos 2 = (+3/2P2)/P(+3/2P2)

ds/dt = - dW(s)/ds + (s)

=>

W

s

W(s) ≠ U(s) !

Tunneling, switching curves

Focker-Planck equation for P(s)

W

~ exp(-W/Teff)

W = a s2 – b s4

width of switching curve - ~ T

For a generic bifurcation (incl. JBA)

controls

width of switching curve

For a parametric bifurcation - additional symmetry

u,v -> -u,-v shift by period of the drive

=>

should be even !

at origin

Another operation mode

no mirror symmetry => generic case, stronger effect of cooling

Conclusions

Period-doubling bifurcation readout:• towards quantum-limited detection?• low back-action• rich stability diagram• various regimes of operation• tuning amplitude or frequency

results• bifurcations, tunnel rates,• switching curves,• stationary states for various parameters,• temperature and driving dependence

of the response with double period