Quantum measurement and superconducting qubits

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Quantum measurement and superconducting qubits
STMP-09, St. Petersburg 2009, July 3-8
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Outline

• superconducting qubits
• quantum readout
• parametric driving and bifurcation
• stationary states, switching curve

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Why quantum?

• computers
• high computational capacity?
• (R. Feynman, D. Deutsch)
• Shor algorithm factorization
• in polynomial time, 1994

• physics
• fundamental phenomena
• in quantum-mechanical systems
• - devices new challenges

Quantum computers
Quantum bits (qubits) 2-level quantum systems ,
Logic gates unitary evolution Quantum
algorithms sequence of elementary operations
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Physical realizations of qubits
2-level ions
Ion traps Cirac Zoller,
laser
NMR Chuang et al., Cory et al.
Bx(t)
nuclear spins in various environments
Bz
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Requirements to realizations of quantum computers
controlled
controlled
controlled
weak

• 2-level quantum systems N qubits
• controlled dynamics
• initialization - cooling or read-out
• coherence
• read-out quantum measurement

• 1-qubit gates
• 2-qubit gates

enough for error correction
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Electron spins in semiconductors Loss
DiVincenzo, 1998
tj spins gt tj charges exactly 2 states
Kouwenhoven et al. (Delft)
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Josephson quantum bits

• coherence of superconducting state
• advanced control techniques for
• single-charge and SQUID systems

combine
Quantronium (Saclay)
no excitations at low T
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

j
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Single-electron effects
junctions w/ small area 10nm x 10nm typical
capacitance C 10-15 F typical energies EC
e2/2C 1 K
V. Bouchiat et al. (1996)
j
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Josephson charge qubits
N
Fx
tunable
tj 2 ns
Nakamura et al. 99 2 qubits Pashkin et al. 03
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Problems decoherence
noise gt random phases gt dephasing (T2) noise
gt transitions gt energy relaxation (T1)
1/T2 1/(2T1) 1/T2
low-frequency 1/f noise gt strong dephasing
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Quantronium
Charge-phase qubit Vion et al. (Saclay) 2002
operation at a saddle point
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Transmon Koch et al. (Yale) 2007
EJ/EC50
EJ
j (coordinate)
heavy particle (CB) gt flat bands gt
very low sensitivity to charge noise
E(q)
in a periodic potential
still (slightly) anharmonic gt address only
two states
T2 ? 2 ms (close to 2T1 without spin echo)
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Topologically protected qubits
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Quantum measurement
detector
quality of detection
- reliable ? single-shot? - QND? back-action -
fast?
Measurement as entanglement
unitary evolution of qubit detector
Mii --- macroscopically distinct states
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Dynamics of current
probability that m electrons passed
QPC
I(t)
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Qubit dynamics
Reduced density matrix of qubit
tmeas tj
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Mixing (relaxation)
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Quantum readout for Josephson qubits
U - Ic cosj I j
Quantronics Ithier, 2005
j

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

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Quantum readout for Josephson qubits
dispersive readout monitor the eigen-frequency
of LC-oscillator
monitor reflection / transmission amplitude /
phase
Sillanpää et al. 2005
Wallraff et al. 2004
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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 QND higher repetition rate qubit
always close to optimal point
Siddiqi et al. 2003 cf. Ithier, thesis 2005
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w/ A.Zorin
Parametric bifuraction for qubit readout
t wt
I0Ic sin j0
jj0x
Landau, Lifshitz, Mechanics diff. driving Dykman
et al. 98
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Method of slowly-varying amplitudes
A2u2v2
Migulin et al., 1978
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Equations of motion
Hamiltonian
Equations of motion in polar coordinates
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without quartic term
m0
stationary solutions A0 or
A2
x
x-
detuning x
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detuning
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Phase diagram
driving P
bistability
detuning, w-w0
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Switching curves
Pswitching
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contrast
0gt
single-shot readout?
1gt
0
width
detuning
Psw 1 e G t
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Dykman et al. (1970s 2000)
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Velocity profile and stationary oscillating states
0, A stable states A- unstable state
near origin j relaxes fast, A slow degree of
freedom
(at rate q)
2D Focker-Planck eq. gt 1D FPE
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Near bifurcation (x ?x-)
cos 2a (x3/2g P2)/P(b3/2mP2)
ds/dt - dW(s)/ds x(s)
W(s) ? U(s) !
W
gt
s
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Tunneling, switching curves
Focker-Planck equation for P(s)
DW
G exp(-DW/Teff)
W a s2 b s4
width of switching curve x - x- ? T
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For a generic bifurcation (incl. JBA)
controls
width of switching curve
For a parametric bifurcation - additional
symmetry
at origin
u,v -gt -u,-v
shift by period of the drive
should be even !
gt
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Another operation mode
no mirror symmetry gt generic case, stronger
effect of cooling
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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