DECOHERENCE AND DEPHASING IN QUANTUM COMPUTATION

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DECOHERENCE AND DEPHASING IN QUANTUM COMPUTATION
LECTURE III Control Readout

• Environmental impedances
• 2. Effects of the readout device for flux
  qubits
• 3. Driven systems
• Resonant driving
• The frequency of the oscillations is not
  always linear
• in the driving strength!
• High frequency driving

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REALISTIC ENVIRONMENTS
Example 1 Z(w)R
Ohmic form with Drude cut-off
RK 25kW
Example 2 Z(w)iwL
single oscillator
Example 3 Z(w)RiwL
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REALISTIC ENVIRONMENTS II
RC shunt
C schunt
Chiorescu et al., Science 1869, 299 (2003)
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REALISTIC ENVIRONMENTS II
EFFECTIVE ENVIRONMENT
Ieff (w)
Ieff (w)
C schunt
w (a.u.)
w (a.u.)
Iper persistent current
M mutual inductance R resistor from el.
environment
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REALISTIC ENVIRONMENTS III
harmonic bath
g
g
T,
QUBIT
DETECTOR
ELECTROMAGNETIC ENVIRONMENT
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REALISTIC ENVIRONMENTS III
g
QUBIT
DETECTOR
EFFECTIVE BATH
BATH
Garg, Onuchic, Ambegaokar, J. Chem. Phys. 83,
4491 (1985)
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EFFECTS OF READOUT
weak coupling
Gf/D
numerically exact PI simulations two dephasing
rates
Weak coupling theory breaks down
g/g gtgt1 and D W and kBT lthD
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EFFECTS OF READOUT II
BATH
g
g
T,
SYSTEM
Bath is Ohmic
safely apply Bloch-Redfield
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EFFECTS OF READOUT II
g
SYSTEM
Step 1. Diagonalize system Hamiltonian
D W, e 0, RWA

dressed-atom levels
W

three-level approx
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EFFECTS OF READOUT II
Step 2. Matrix elements of the operator coupling
to the bath
forbidden
two frequencies, two dephasing rates
Step 3. Evaluate the elements of the Redfield
tensor
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DRIVEN SYSTEMS
Time-dependent control fields needed for quantum
computation
deterministic HS (t)
time-dependent problems hard to solve

• Thermal equilibrium is not reached at long time
• Decay rates and asymptotic probabilities
  modified by driving
• Extra frequencies and/or frequency
  renormalizations

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DRIVEN SYSTEMS II
Example longitudinal AC-driven spin-boson
system
( used e.g. for
control in JJ qubits)
NO exact reduction to static problem for HS
i)
NOT of Rabi form
ii)
Nonlinear system
linear response
linear susceptibility
thermal equilibrium
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DRIVEN SYSTEMS III
Various methods exist to treat driving exactly
i)
Real-time path integral approach

• weak dissipation

master eqs for the RDM with time-dependent kernels
ii) Projector operator formalism Born-Markov
approximation
Grifoni and Hänggi, Driven Quantum Tunneling,
Phys. Rep. 304, 229 (1998) Hartmann et al. PRE
61, R14005 (2000)
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RESONANT DRIVING
Unitary transformation
yielding
longitudinal
transverse
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RESONANT DRIVING II
longitudinal
transverse
Driving transverse component counter-rotating
term No Rabi
Symmetric TLS e 0

• Degeneracy point longitudinal 0
• Rotating wave approximation for W E/h

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RESONANT DRIVING III
Transform to rotating frame
Static problem

• Rates

• Rabi oscillations with frequency

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HIGH FREQUENCY DRIVING
High frequency regime ? gtgt ?, ?
averaged relaxation rate

• Generalized master equation for RDM with PI
  approach
• Weakdamping kernels linear in the
  coupling to bath
• Time average over one period

infinite set of frequencies !
Hartmann et al. PRE 61, R14005 (2000)
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HIGH FREQUENCY DRIVING II
Special points
only one oscillation
?n 0

symmetric TLS !
Experimental observation Nakamura, Pashkin,
Tsai PRL 87, 246601 (2001)

• linear regime s ltlt ? and

?1 0
Rabi frequency
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HIGH FREQUENCY DRIVING III
high frequency driving
? 10, s 1
(units of D)
dips
e
M.C. Goorden, Diploma Thesis, Delft (2002)
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HIGH FREQUENCY DRIVING IV
First dip

M.C. Goorden, Diploma Thesis, Delft (2002) M.C.
Goorden , F.K. Wilhelm, cond-mat/0305467
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HIGH FREQUENCY DRIVING V
First dip
W 10
(units of D)
s 1
s 10
e
e
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HIGH FREQUENCY DRIVING VI
Control of tunneling
??00, no zero photon process
??10, no one photon process
s 38.317
e
e
s24.042
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HIGH FREQUENCY DRIVING VIII
Relaxation rate
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HIGH FREQUENCY DRIVING VII
Dephasing rates
s 1, W 10
G
e
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m.grifoni_at_tnw.tudelft.nl
under construction
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Peak place
In second order in
Bloch Redfield
High Frequency only qualitatively correct
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Peak width

• First peak

• Bloch with RWA

• Bloch Redfield

High frequency approximation misses saturation
broadening