Master equation and initial factorization

1
Master equation and initial factorization

• Paolo Facchi
• Università di Bari, Italy
• in collaboration with
• S. Pascazio, K. Yuasa (Bari)
• H. Nakazato, I. Ohba, S. Tasaki (Tokyo)
• G. Kimura (Sendai)

2
Introduction

• Closed quantum system
• unitary evolution (reversible)
• Schrödinger equation
• small systems / discrete spectra

• Open Quantum Systems
• dissipation/decoherence (irreversible)
• system S reservoir B
• derivation of master equation
• for the reduced density ops.

infinitely extended systems
System S

• (Thermal) Quantum Field Theory

Reservoir B
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Master equation
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Derivation of master equation

• System S Reservoir B
• B is infinitely large
• Weak coupling
• Trace over B
• Nakajima-Zwanzigs Projection Method
• Weak-Coupling Markov Approximation
• van Hoves Limit
• No Initial Correlation between S and B

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Assumptions of Factorization
factorized initial state
factorization at later times
D.F. Walls and G.J. Milburn, Quantum Optics
(1994) M.O. Scully and M.S. Zubairy, Quantum
Optics (1997) C. Cohen-Tannoudji, J. Dupont-Roc,
and G. Grynberg, Atom-Photon Interactions
(1998) H.J. Carmichael, Statistical Methods in
Quantum Optics 1 (1999) H.-P. Breuer and F.
Petruccione, The Theory of Open Quantum Systems
(2002) C.W. Gardiner and P. Zoller, Quantum
Noise, 3rd ed. (2004)
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Reference State

• For a factorized initial state

• For a correlated initial state

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Notation

• Liouvillians
• Projections
• Properties

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Projection Method
initial correlation
non-Markovian

• F. Haake, in Quantum Statistics in Optics and
  Solid-State Physics, Vol. 66 of Springer Tracts
  in Modern Physics, edited by G. Höhler (Springer,
  Berlin, 1973), pp. 98-168
• R. Kubo, M. Toda, and N. Hashitsume, Statistical
  Physics II, 2nd ed. (Springer, Berlin, 1995)
• L. Mandel and E. Wolf, Optical Coherence and
  Quantum Optics (1995)

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Requirements

• 0 point spectrum of
• (absolutely) continuous spectrum
• is a bounded perturbation of
• Projection

Tasaki et al., Ann. Phys. 322, 631-656 (2007)
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Mixing reservoir

• is mixing (with respect to ).
• e.g.)

for any bounded ops.
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Mixing reservoir
is mixing (with respect to ).
for any bounded ops.

• For the total system SB

mixing
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Open system in a mixing reservoir

• Perturbations/correlation propagate away to
  infinity.
• There remains the mixing state and the system is
  factorized.
• Correlation through interaction.
• ? Mixing quickly in t.

perturbations
System S
free evolution in t
correlation

SB looks factorized in
Reservoir B