Time evolution of the Husimi and Glauber-Sudarshan functions in terms of complementary Hamiltonian symbols
Mritunjay Tyagi, Simon Friederich
TL;DR
The paper develops a unified, complementary-symbol framework for the time evolution of the Husimi Q-function and Glauber-Sudarshan P-function using Anti-Wick and Wick Hamiltonian symbols, respectively, within a star-product formalism. It reveals a universal drift-diffusion structure where the leading term is classical Liouville flow and higher-derivative corrections appear, with quartic-or-less Hamiltonians reducing to a zero-trace diffusion (Fokker-Planck) form. An Ehrenfest theorem is established in this complementary setting, and the work resolves a known discrepancy by Milburn by showing that the nonclassical drift arises from using a non-Anti-Wick quantized Hamiltonian; quantizing the classical Hamiltonian via Anti-Wick recovers the classical drift in the Q-function dynamics. For the anharmonic oscillator, the results demonstrate that the appropriate pairing with Anti-Wick quantization yields drift terms consistent with classical dynamics, while higher-order corrections and multi-mode generalizations are systematically addressed. Overall, the framework provides a concise route to compute and interpret quantum phase-space evolution across Wick, Anti-Wick, and Weyl representations.
Abstract
We present a compact, systematic formulation of the dynamics of the Husimi Q- and Glauber-Sudarshan P-phase space distribution functions expressed in terms of their \emph{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 $α_i$, the higher-order terms reduce to a second-order Fokker-Planck type term with a \emph{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.