Non-Markovian Open Quantum Systems

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Non-Markovian Open Quantum Systems

• Sabrina Maniscalco
• Quantum Optics Group, Department of Physics
• University of Turku

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Contents

• Motivation
• Theoretical approaches to the study of open
  systems dynamics
• Markovian approximation and Lindblad Master
  Equation
• Non-Markovian systems
• Examples Quantum Brownian Motion
• and two-level atom

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Open Quantum Systems
Quantum Mechanics
closed systems
unitary dynamics
reversible dynamics
non-unitary and irreversible dynamics
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Motivation
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Approaches to the dynamics of open quantum systems
microscopic approach
phenomenological approach
analytical methods
numerical methods
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Approximations
weak coupling between system and environment
perturbative approaches (expansions in the
coupling constant)
assumes that the reservoir correlation time is
much smaller than the relaxation time of the open
system
coarse-graining in time
changes in the reservoir due to the interaction
with the system do not feed back into the system
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Lindblad master equation 1,2
Weak coupling Markovian approximation RWA or
secular approx.
1 G. Lindblad, Comm. Math. Phys. 48, 119 (1976) 2
V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J.
Math. Phys. 17, 821 (1976)
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Positivity and Complete positivity
Violation of CP incompatible with the assumption
of a total closed system (for factorized initial
condition)
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Importance of CP
CP is not guaranteed and unphysical situations,
showing that the model we are using is not
appropriate, may show up in the dynamics
1 K. Kraus, States, Effects and Operations,
Fundamental Notions of Quantum Theory (Academic,
Berlin, 1983).
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Non-Markovian master equations

• photonic band-gap materials 1
• quantum dots
• atom lasers 2
• non-Markovian quantum information
• processing 3

QUANTUM NANOTECHNOLOGIES 4,5,6,7
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Damped harmonic oscillator
or Quantum Brownian Motion in a harmonic potential
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Two approachesmicroscopic and phenomenologic 1
GOAL To study the dynamics of the system
oscillator, in presence of coupling
with the reservoir beyond the
Markovian approximation
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Exact Master Equation 1,2(time convolutionless
approach)
1 J.P. Paz and W.H. Zurek, Environment-induced
decoherence and the transition from quantum to
classical, Proceedings of the 72nd Les Houches
Summer School on Condensed Matter Waves,
July-August 1999, quant-ph/0010011. 2 F.
Intravaia, S.M., A. Messina, Eur. Phys. J. B 32,
109 (2003).
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Secular approximated master equation (and
applications)
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Exact solution1
NO MARKOVIAN APPROX, NO WEAK COUPLING APPROX, NO
RWA APPROX
1 F. Intravaia, S. M. and A. Messina, Phys. Rev.
A 67, 042108 (2003) 2 S.M.Barnett and P.M.
Radmore, Methods in Theoretical Quantum Optics
(Oxford University Press, Oxford, 1997)
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Example of non-Markovian dynamics
The risks of working with non-Lindblad Master
Equations
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Master equation with memory kernel
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Solution using the QCF
Fourier transform
p 1 P function
p -1 Q function
p 0 Wigner function
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Solution using the QCF
The defining properties are always satisfied
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Density matrix solution
problem of positivity firstly noted by
BarnettStenholm2
1 S.M.Barnett and P.M. Radmore, Methods in
Theoretical Quantum Optics (Oxford University
Press, Oxford, 1997) 2 S.M. Barnett and S.
Stenholm, Phys. Rev. A 64, 033808 (2001).
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Limits in the QCF description
The QCF and the density matrix are not
operatively equivalent descriptions of the
dynamics, in the sense that the QCF may fail in
discriminating when non-physical conditions
(negativity of the density matrix eigenvalues)
show up.
S. M., "Limits in the quantum characteristic
function description of open quantum systems",
Phys. Rev. A 72, 024103 (2005)
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Non-Markovian dynamics of a qubit
Phenomenological model Non-Markovian master
equation with exponential memory
Markovian Liouvillian
memory kernel
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Positivity and Complete Positivity
THE BLOCH SPHERE
Condition for positivity The dynamical map ?
maps a density matrix into another density matrix
if and only if the Bloch vector describing the
initial state is transformed into a vector
contained in the interior of the Bloch sphere,
i.e. the Bloch ball.
THIS RESULT PROVIDES THE EXPLICIT CONDITIONS OF
VALIDITY OF A PARADIGMATIC PHENOMENOLOGICAL MODEL
OF THE THEORY OF OPEN QUANTUM SYSTEMS, NAMELY THE
SPIN-BOSON MODEL WITH EXPONENTIAL MEMORY.
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Summary
Markovian master equation
Non-Markovian master equation
Two approaches microscopic and phenomenological
Paradigmatic model QBM or damped harmonic
oscillator
Memory kernel
TCL (microscopic exact approach)
Generalized master equation
QCF and density matrix solutions
Solution based on algebraic method
Positivity violation
Applications trapped ions, linear amplifier
Limits in the use of the QCF
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References