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
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