DECOHERENCE AND DEPHASING IN QUANTUM COMPUTATION

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DECOHERENCE AND DEPHASING IN QUANTUM COMPUTATION
LECTURE II Dynamics of single and coupled qubits

• Some examples of qubits
• interactions, control and dissipation
• 2. Coherent dynamics of a single qubit
• Dissipative dynamics of a single qubit (Ohmic
  damping)
• Can we directly apply the results to real
  experiments?
• 4. Coupled qubits

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QUBITS
Many physical realizations of a qubit

• If qubits are isolated, all these realizations
  are equivalent

• Coupling to the environment, among qubits or to
  external
• sources depends on the system

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SPIN QUBITS
Spin degree of freedom e of electron in a quantum
dot
g giromagnetic ratio sx, sy, sz, Pauli
matrices
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CHARGE QUBITS
tunable Josephson energy
charging energy
1 Yu. Makhlin, G. Schön and A. Shnirman, Nature
368, 305 (1999) Rev. Mod. Phys. 73, 357 (2001)
2 Y. Nakamura, Y.A. Pashkin and J.S. Tsai,
Nature 398, 786 (1999) 3 D. Vion et al.
Science 296, 886 (2002)

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CHARGE QUBIT II
EJ 0
Requirements (Nakamura type)
n-2
I)
n0
n-1
n1
DS superconducting gap
-1
1
0
ng

• only Cooper pairs tunnel through
• the junctions
• convenient to choose basis where
• number n of Cooper pairs is well
• defined

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CHARGE QUBIT III
n-1
i) Qubit Hamiltonian
n0
-1
0
1
ng
In the representation
HTLS is non diagonal

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CHARGE QUBITS
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FLUX QUBIT
rf-SQUID
L
charge Q on the leads canonically conjugate to
the flux
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FLUX QUBIT II
Truncation to two-states
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FLUX QUBIT III
less sensitive to external noise than rf-SQUID
Requirements

flux through the loop appropriate degree of
freedom
i) Truncation to the lowest two states
3 Mooij et al., Science 285, 1036 (1999) C. H.
van der Wal et al., Science 290, 773 (2000) 4
I. Chiorescu et al., Science 299, 1869 (2003)
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FLUX QUBITS IV
ii)
AC-control fields
no conventional Rabi oscillations
iii) Inductive qubit-qubit coupling
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L
LT
LT
D diagonal L longitudinal, T transverse
Table updated to 31.07.2003
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COHERENT DYNAMICS
Given the TLS Hamiltonian
we wish to evaluate the occupation probability
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COHERENT DYNAMICS II
Energy representation
TLS Hamiltonian is diagonalized under the
transformation
yielding
where
and
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COHERENT DYNAMICS III
Quantum mechanical calculation of in
energy basis
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COHERENT DYNAMICS IV
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What are the effects of the environment on P(t)
?
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THE REDUCED DENSITY MATRIX II.
with the RDM being in the basis
Here,
represent the occupation probabilitie
s of the right and left state, respectively.
The coherences
describe interference effects between
states.
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REDFIELD EQUATIONS FOR TLS
Redfield Eqs. yield the RDM dynamics in energy
basis
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MATRIX ELEMENTS
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RELAXATION AND DEPHASING
We know r in the energy basis. Then we find
yielding

• dephasing
• relaxation towards equilibrium
• dressing of frequencies

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RELAXATION AND DEPHASING II
Relaxation part, only present for asymmetric TLS
Symmetric TLS, e 0
and
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Rabi oscillations in a resonantly driven qubit
I. Chiorescu et al., Science 299, 1869 (2003)
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ELECTROMAGNETIC ENVIRONMENT
Charge and environmental effects in tunneling
junctions
- RT tunneling resistance, C junction
capacitance - Vext ideal voltage source - Z (w)
environmental impedence (frequency response of
the electromagnetic environment)
The voltage drop U at the junction differs from
the applied voltage Vext by the dc-voltage drop
at the impedance
current-voltage characteristics of the junction
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ELECTROMAGNETIC ENVIRONMENT II
What is the electromagnetic Hamiltonian?
Recall correspondence between electrical and
mechanical quantities
Mechanical
Electrical
mass M momentum p velocity v p/M coordinate
q q, pih/2p spring constant f harmonic
oscillator
capacitance C charge Q voltage V Q/C phase f f
,Qie inverse inductance 1/L LC-circuit
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ELECTROMAGNETIC ENVIRONMENT III
Introduce
Then the fluctuations of charge and phase
are conjugate variables q,fie
The environment junction Hamiltonian
reproducing the same relaxation dynamics as in
the classical limit then reads
with spectral density
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ELECTROMAGNETIC ENVIRONMENT IV
Classical charge relaxation
(i.e. forget tunneling, and only consider the
junction as a capacitor with charge Q CU)
Effects of the environment
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ELECTROMAGNETIC ENVIRONMENT V
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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COUPLED QUBITS
Voltage oscillations in LCt circuit affect the
qubits
L
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COUPLED QUBITS II
Example coupled charge qubits
harmonic bath
Coupled dynamics
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COUPLED QUBITS III
Express H in the singlet/triplet basis
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COUPLED QUBITS IV
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COUPLED QUBITS V
ii) Evaluate the matrix elements of the operator
coupling to the bath
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COUPLED QUBITS VI
iii) Evaluate the Redfield tensor to get the RDM
dynamics in energy basis
dephasing (non degenerate levels)
relaxation
Nlev 4

• At T 0

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COUPLED QUBITS VII
iv) Transform back to energy basis
complicated beating pattern for Pm(t)
M. Governale, M.G., G. Schön, Chem. Phys. 268
(2001)
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COUPLED QUBITS IX
Qubits coupled to separate environments
L
spin-boson
Singlet-triplet basis
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COUPLED QUBITS VIII

• Large g eigenstates approach
  triplet/singlet basis states

• For evaluation of the probabilities follow the
  same steps as before