Commuting Heisenberg operators as the quantum response problem: Time-normal averages in the truncated Wigner representation

B. Berg, L. I. Plimak, A. Polkovnikov, M. K. Olsen, M. Fleischhauer, and W. P. Schleich
Phys. Rev. A 80, 033624 – Published 29 September 2009

Abstract

The applicability of the so-called truncated Wigner approximation (W) is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the W approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (in principle, exact) path-integral approach in phase space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The W equations emerge as an approximation within these path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages, Kubo’s renowned formula relating the linear-response function to two-time commutators suffices. The W is directly generalized to the response properties of the system allowing one to calculate approximate time normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.

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  • Received 20 May 2009

DOI:https://doi.org/10.1103/PhysRevA.80.033624

©2009 American Physical Society

Authors & Affiliations

B. Berg1,*, L. I. Plimak1,2, A. Polkovnikov3, M. K. Olsen1,2, M. Fleischhauer4, and W. P. Schleich1

  • 1Institut für Quantenphysik, Universität Ulm, D-89069 Ulm, Germany
  • 2ARC Centre of Excellence for Quantum-Atom Optics, School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australia
  • 3Department of Physics, Boston University, Boston, Massachusetts 02215, USA
  • 4Fachbereich Physik, Technische Universität Kaiserslautern, D-67633 Kaiserslautern, Germany

  • *bettina.berg@uni-ulm.de

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Vol. 80, Iss. 3 — September 2009

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