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arXiv:quant-ph/0612037 · Journal of Physics A Mathematical and Theoretical

Generalized phase-space representation of operators

John R Klauder, Bo-Sture K Skagerstam

Published Feb 14, 2007• 18 citations
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Abstract

Introducing asymmetry into the Weyl representation of operators leads to a variety of phase space representations and new symbols. Specific generalizations of the Husimi and the Glauber-Sudarshan symbols are explicitly derived
DOI
10.1088/1751-8113/40/9/015

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We present \texttt{SymQuPS}, a SymPy-based algebra system in Python mainly aimed toward the phase space representation of quantum mechanics within the Cahill-Glauber formalism (including the Glauber-Sudarshan PPP, Wigner, and Husimi QQQ representations). By extension, the package serves as an algebraic venue for canonical quantization. A key feature is the phase space representation of an arbitrary Lindblad master equation, which gives the phase space equation of motion of the quantum system. We describe the core functionalities of the package, consisting of sss-ordered operators, the star products, and the phase space representation. Some examples of use are given to illustrate the application of the package, and the package's performance in typical use cases is discussed.
The Wigner (W), Husimi-Kano (Q) and Glauber-Sudarshan (P) quasidistributions are generalized to f-deformed quasidistributions which extend the parametric family of s-ordered quasidistributions of Cahill and Glauber. The deformation procedure is obtained via a canonical nonisometric transform of the displacement operators which preserves the form of the standard creation-annihilation commutation relation, hence the Heisenberg-Weyl algebra, but changes the scalar product in the Hilbert space of the oscillator states. A whole class of new resolutions of the identity is introduced. The time evolution equation for the new generalized quasidistributions is derived.
We present a compact, systematic formulation of the dynamics of the Husimi Q- and GlauberSudarshan P-phase space distribution functions expressed in terms of their complementary Hamiltonian symbols: Anti-Wick for Q and Wick for P. The resulting evolution equations have a universal leading structure, the classical Liouvillian drift plus terms with higher-order derivatives of the Hamiltonian. For Hamiltonians no higher than quartic in the moduli of the complex phase space variables αi, the higher-order terms reduce to a second-order Fokker-Planck type term with a traceless diffusion matrix, thereby clarifying and recovering recent results for such Hamiltonians within a simple star-product framework. We further derive a transparent Ehrenfest theorem for Wick/Anti-Wick symbols of the operators representing dynamical observables. Using these results, we show that a previously reported nonclassical contribution to the Q-function drift for the anharmonic oscillator is an artifact of the quantization scheme used. Our paper consolidates the formulation of the dynamics of the phase space distribution functions using complementary symbols and provides an efficient route to compute and interpret quantum phase space evolution.