Master Equation Approach
to the Dynamics
of Open Time-Varying
Quantum Systems
M. Scala, B.D. Militello and A. Messina
MIUR and Dipartimento di Scienze Fisiche ed
Astronomiche,
Università di Palermo (Italy)
Summary
• Adiabatic Approximation and Berry Phase
• Environment
• Derivation of the Master Equation
• A Simple Illustrative Application
Adiabatic Approximation
H (t )
•Time-dependent hamiltonian
•Instantaneous eigenstates
{ n(t ) }
•Evolution (non-degenerate spectrum):
n(t = 0) → e
iφ ( t )
n(t )
Berry Phase
H(T ) = H(0)
• Cyclic Evolution
i (φ D (T ) +φG (T ) )
n(0) → e
• Dynamical Phase
n ( 0)
T
φD (T ) = − ∫ En (t ) dt
0
• Berry Phase
T
φG (T ) = i ∫
0
→ 
n R(t ) ∇ R


→
→  d R
n R(t ) ⋅
dt

 dt
Decoherence
• Fluctuations in the classical parameters
• The system is not isolated
H = H S + H env + H int
System-Environment Coupling
Interaction hamiltonian
(A and B hermitian)
Expansion of A (constant eigenvalues of HS(t)):
Properties:
Interaction Picture
H 0 = H S (t ) + H env
Evolution of the system density matrix (Born-Markov approximation):
Master Equation (I)
Our Main Hypothesis:
Rotating Wave Approximation
(RWA)
→
Master Equation (II)
In Schroedinger Picture:
where
and
Prescriptions
1. Write down the Master Equation in the
case of HS independent of time
2. Insert the time dependence of the
parameters directly into the stationary ME
Corrections to the Master Equation
where
Adiabatic Evolution
Lindblad-type Master Equation
• Recent criterion: (Sarandy and Lidar, PRA 71, 012331
(2005))
• Evolution of the Eigenoperators (1-dimensional
Jordan blocks):
Phase factors
where
and
Example: spin ½
Master Equation for the spin
Time-dependent basis
where
Results (coherences) I
Eigenvalue
Right Eigenoperator
Left Eigenoperator
Adiabatic Evolution
Results (coherences) II
Dynamical Phase
Geometric Phase
Comparison with previous results
• Sarandy and Lidar, quant-ph/0507012.
Similar results: no corrections to Berry
phase.
• Whitney et al., PRL 94, 070407 (2005).
Master Eq. in a rotating frame.
Different results: corrections to Berry
phase.