QUBITS AND ENVIRONMENT

1
QUBITS AND ENVIRONMENT
There are many two-level systems in nature that
can be used as qubits

• Spin ½ systems
• e.g. electrons in quantum dots

0?
1?

• Effective TLSs
• e.g. current states in superconducting rings,
• charge states in Cooper-pair box

0?
1?
. however, they are not isolated from their
surroundings !!!
2
DECOHERENCE AND DEPHASING IN QUANTUM COMPUTATION
LECTURE I. Open quantum systems

• System reservoir model
• The quantum Langevin equation
• The influence functional
• Master Eqs. for weak system-bath coupling
• Bloch-Redfield Eqs.

Milena Grifoni, Wittenberg 29-30 Juli, 2003
3
OPEN QUANTUM SYSTEMS

• What are the effects of the environment ?

• How can we quantitatively determine them ?

• How can we minimize those effects ?
• What are the predictions for real physical
  systems ?

4
SYSTEM-PLUS-RESERVOIR APPROACH
0?
BATH
T, J(w)
1?

• Global system is conservative, and as such obeys
  standard rules
• of quantization
• Friction comes about by the energy transfer from
  small to large system
• Influence of the bath fully characterized by a
  stochastic force z(t)

5
THE REDUCED DENSITY MATRIX
We wish to include the bath degrees of freedom,
and evaluate expectation values of an observable
O of the relevant system
where
is the total density matrix of the
system at time t, and U is the time-evolution
operator.
Upon performing a trace over the reservoir one
obtains
Here is called
the reduced density matrix (RDM)
6

• single and coupled qubits
• with/without driving
• realistic environments, e.g. in presence of a
  detector

7
Step 1. We need a microscopic, tractable model
for the environment
8
HARMONIC OSCILLATOR BATH MODEL
Environment only weakly perturbed by coupling to
the small system
The bath is considered as linear, and therefore
it can be described by a set of harmonic
oscillators
References 1 Weiss Quantum dissipative
systems, World Scientific Singapore, 2nd ed.
(1999). 2 Zwanzig, J. Stat.
Phys. 9, 215 (1973). 3
Caldeira and Leggett, Ann. Phys. 149, 374 (1983).
9
SYSTEM OSCILLATOR BATH MODEL
The most general Hamiltonian reads
10
SPECTRAL DENSITIES
It is convenient to introduce the spectral
density of the environment
and regard J(w) a smooth function of w ,

11
THE QUANTUM LANGEVIN EQUATION
Harmonic bath system QLE for
systems dynamics
12
THE QLE II
Fourier transform of g (t)
13
PROPERTIES OF THE STOCHASTIC FORCE
In general z(t) is a coloured, Gaussian
fluctuating force

• Classically

dissipation
fluctuation

• Quantum mechanically

Z partition function
14
EXAMPLE CHARGE QUBIT
15
Step 2. Exact elimination of the bath degrees of
freedom using the real time
path-integral approach
Exact expression for
16
SHORT ABOUT THE PATH INTEGRAL APPROACH

• The path integral approach
• Is a way of formulating a quantum theory
  withouth operators
• It smoothly links quantum with classical
  mechanics
• The trace over the degrees of freedom of a
  harmonic reservoir
• is performed exactly.

propagator
17
SHORT ABOUT THE PATH INTEGRAL APPROACH II
For one
has the original Feynman formula
with S the classical action
18
THE INFLUENCE FUNCTIONAL
To evaluate the RDM r, we start from the full
density matrix
In the position representation it reads
19
THE INFLUENCE FUNCTIONAL II
According to Feynmann formula, the propagator has
the PI form
with the total action
given in terms of the Lagrangian L LS LB
LS-B
20
THE INFLUENCE FUNCTIONAL III
After tracing out the bath degrees of freedom one
obtains 4
where the propagator J reads
influence of the bath
probability amplitudes of the isolated system
4 H. Grabert, P. Schramm and G.-L. Ingold,
Phys. Rep. 168, 115 (1998)
21
THE INFLUENCE FUNCTIONAL IV
The influence phase has the explicit form
where we set x
q-q, h (qq)/2 and
22
THE INFLUENCE FUNCTIONAL V
h quasi-classical path (it obeys the Langevin
eq. in classical limit)
x describes excursion to off-diagonal states of
the RDM it relates to the quantum
fluctuations
damping
decoherence
23
Step 3. Weak dissipation perturbation theory
in the system-bath coupling
24
WEAK SYSTEM-BATH COUPLING
tB thermal time d typical energy difference for
the conservative system
25
MASTER EQS. FOR THE RDM
The master Eq is Markovian and reads 5
free propagator
5 R.P. Feynman and A.R. Hibbs, Quantum
Mechanics and Path Integrals (McGraw-Hill, New
York, 1965)
26
MASTER EQS. FOR THE RDM II
Note the obtained results are equivalent to
those of a projection operator approach 6,7
yielding the operator Eq.
We assumed Y(t) 0 for t gttB and extended
integration to infinity
Where A,BABBA, and the Heisenberg position
and momentum operators qH and pH are defined
according to
6 C.W. Gardiner, Quantum Noise (Springer,
Berlin 1991) 7 P.N. Argyres, P.L. Kelley, Phys.
Rev. 134, A98 (1964)
27
MASTER EQS. FOR THE RDM III
Ohmic dissipation, where g (t) 2g d(t),
Here
Note The positivity of the density matrix might
not be guaranteed for preparations far from
equilibrium. In this case however, the
conditions for the validity of the master eq. are
violated.
and
28
Step 4. Weak dissipation it is convenient to
work out the master Eq. in the energy basis
29
REDFIELD TENSOR
For weak dissipation, it is convenient to work
out the master equation in the energy basis, such
that
30
REDFIELD TENSOR I
Note
The master equation can be generalized to other
kind of couplings
If the bath is coupled to the operator O
in Eqs. of motion for the RDM
31
REDFIELD TENSOR II
Introducing the Redfield tensor
one gets the so-termed Bloch-Redfield Eqs.
32
REDFIELD TENSOR III
Secular approximation neglect all but
33
REDFIELD TENSOR IV
Solution of the Redfield Eqs.
energy shift
dephasing rate ()
equilibrium value
relaxation rate ()
() relative phase between eigenstates becomes
random on the time scale (Gab)-1
() occupation probabilities of eigenstates
change on the time scale G-1
34
REDFIELD TENSOR V
Dephasing rates
absorbtion or emission of a single phonon of the
bath