Phase space representation of quantum dynamics

By A. Polkovnikov. 2009

544 citations recorded by the source corpus.

Abstract 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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/81f4b0d0beead9a04fd89c03e2696f66a2cd3836

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Citations in the corpus

Many-body energy localization transition in periodically driven systems (2012, unknown): “Let us briefly review the formalism [33].”
Many-Body Quantum Spin Dynamics with Monte Carlo Trajectories on a Discrete Phase Space (2014, methodology): “Another direction is to try to combine higher-order corrections to the TWA [18, 22] with the idea of discrete phase spaces; this could lead to an even more powerful method capable of capturing quantum many-body dynamics for longer time scales.”
Colloquium: Nonequilibrium dynamics of closed interacting quantum systems (2010, methodology): “For particular noninteracting models, the scaling of the diagonal entropy was found to be the same as that of the density of quasi-particles (4) (De Grandi et al., 2010a,b; Mukherjee et al., 2008).”
Scrambling Dynamics and Out-of-Time-Ordered Correlators in Quantum Many-Body Systems (2022, methodology): “There are many other wholly or partially numerical approaches to calculating OTOCs, such as the truncated Wigner approximation in the semi-classical limit [79, 80], but these are among the most general purpose.”