Thermal coherent states in the Bargmann representation

The thermal coherent states considered previously by the present authors represent an alternative mixed-state generalization of the usual pure-state coherent states. They describe displaced harmonic oscillators in thermodynamic equilibrium with a heat bath at nonzero temperature. We show how they provide a ''random'' (or ''thermal'' or ''noisy'') basis on a quantum-mechanical Hilbert space scrH. Their usefulness rests on the fact that the corresponding statistical density operator provides a probability operator measure on scrH. We thereby show how the thermal coherent states permit a generalization to nonzero temperatures of the well-known P and Q representations of operators in scrH. Particular emphasis here is placed on imbedding the formulation in the Bargmann or holomorphic representation of scrH. We examine the corresponding Bargmann representations of both state vectors and operators, and show how the former relate to the usual position and momentum representations and the latter to the usual P, Q, and W (or Weyl) representations. A particularly important and unexpected result is that the present temperature-dependent generalized P and Q representations are the analytic continuations to negative temperatures of each other. The usual Q and P representations thus represent the limits as the temperature approaches zero along the positive and negative real axes, respectively, of the enlarged generalized Q representation, suitably analytically continued to negative temperatures. We discuss the possible physical applications of the present thermal coherent states to both quantum optics situations involving coherent signals in the presence of thermal noise and to signal and image processing. © 1994 The American Physical Society.