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Journal of Physics A Mathematical and General

Quantum systems at negative temperatures: a holomorphic approach based on coherent states

Apostol Vourdas, R. F. Bishop

Published Oct 23, 1998• 12 citations
Publisher

Abstract

A quantum statistical formalism which accommodates both positive and negative temperatures is studied. The Dirac contour representation is used to extend the harmonic oscillator Hubert space into a larger space which is suitable for the description of quantum systems at both positive and negative temperatures. The analytic continuation of various physical quantities into the negative temperature region is examined in detail. The formalism can be useful for the description of systems which are excited to higher states, and which decay into the lower states.

DOI
10.1088/0305-4470/31/42/015

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A quantum statistical formalism which accommodates both positive and negative temperatures is studied. The Dirac contour representation is used to extend the harmonic oscillator Hubert space into a larger space which is suitable for the description of quantum systems at both positive and negative temperatures. The analytic continuation of various physical quantities into the negative temperature region is examined in detail. The formalism can be useful for the description of systems which are excited to higher states, and which decay into the lower states.
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.
The Dirac contour representation functions fk(z) and fb(z) are employed to represent theket states |f ⟩ and bra states ⟨f |, respectively, in quantum systems with a finite-dimensional Hilbertspace H_2j+1. The scalar product within these quantum systems is defined using a contour integral.Moreover, a numerical approach is utilized to examine the time evolution of both periodic and non-periodic systems, utilizing several Hamiltonian matrices. Furthermore, the stability of periodic systemsis investigated. In addition to these aspects, we study the most significant application of the Dirac con-tour representation, which is its capability to handle an extended Hilbert space, suitable for describingquantum physics at negative temperatures.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.
This research explores analytic and Dirac contour representation systems with finite d-dimensional Hilbert space, where the dual variables, position x, and momentum p, take values in Z(d)(the integer modulo d). It introduces the SU(2) formalism, defining angular momentum states and operators, and investigates the SU(2) formalism in the harmonic oscillator context and SU(2) coherent states and their resolution of identity. Analytical representations based on SU(2) coherent states are developed and numerically implemented in the extended complex plane. The investigation explores time evolution in periodic and non-periodic systems, examining special cases like joining two paths and the behavior of zeros of the analytic representation function approaching zero and ±∞. Furthermore, the research explores Dirac contour representation in a finite dimensional Hilbert space in the extended complex plane. It defines the scalar product of states and proves the relationship between Dirac ket and Dirac bra representations. Time evolution in periodic and non-periodic systems is investigated with numerical implementations and visualizations using MATLAB. Additionally, this study applies Dirac contour representation to finite-dimensional Hilbert spaces at positive and negative temperatures, denoted as H(2j+1)+ and H(2j+1)−, respectively. It defines operators within these spaces and introduces the scalar product of two states in each system. The investigation explores L operators capable of mapping states from H(2j+1)+ to H(2j+1)− and vice versa, examining their Dirac contour representation and obtaining intriguing numerical results visualized using MATLAB.