Phase space representation of quantum dynamics

Journal Article · Sun Aug 15 00:00:00 EDT 2010 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2010.02.006· OSTI ID:21457143

Polkovnikov, Anatoli ^[1]

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1. Department of Physics, Boston University, Boston, MA 02215 (United States)

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose-Hubbard model, Dicke model and others.

• OSTI ID: 21457143
• Journal Information: Annals of Physics (New York), Vol. 325, Issue 8; Other Information: DOI: 10.1016/j.aop.2010.02.006; PII: S0003-4916(10)00038-2; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ATOMS
CORRECTIONS
CORRELATION FUNCTIONS
DEGREES OF FREEDOM
FEYNMAN PATH INTEGRAL
FLUCTUATIONS
HEISENBERG PICTURE
HUBBARD MODEL
INTERACTING BOSON MODEL
NONLINEAR PROBLEMS
PHASE SPACE
QUANTIZATION
SEMICLASSICAL APPROXIMATION
SINE-GORDON EQUATION
SPIN
STOCHASTIC PROCESSES
WAVE EQUATIONS
ANGULAR MOMENTUM
APPROXIMATIONS
CALCULATION METHODS
CRYSTAL MODELS
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD EQUATIONS
FUNCTIONS
INTEGRALS
MATHEMATICAL MODELS
MATHEMATICAL SPACE
NUCLEAR MODELS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE PROPERTIES
PATH INTEGRALS
SHELL MODELS
SPACE
VARIATIONS