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Physical Review A

Generating-function approach to the moment problem for the density matrix of a single mode

Ulrike Herzog

Published Apr 1, 1996• 18 citations
Publisher

Abstract

Equations are derived that express the density matrix of a single mode in terms of the normally ordered moments of the photon annihilation and creation operator and that are convergent for arbitrary radiation fields. The procedure consists of introducing a set of analytic generating functions and applying the method of analytic continuation. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.2889

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Abstract This chapter sets out the second-quantisation treatment for systems of identical bosons or fermions. Key features such as the symmetrisation principle and super-selection rules are presented. Orthogonality and completeness requirements for mode functions are set out, and Fock states are defined in terms of mode occupancy, leading to the definitions of mode annihilation and creation operators and the determination of their commutation (bosons) or anticommutation (fermions) rules. Important two-mode states such as binomial states and relative-phase eigenstates are considered. Field creation and annihilation operators are defined and related to multi-particle position eigenstates. The Hamiltonian is expressed in terms of both mode operators and field operators, state dynamics being treated via Liouville–von Neumann, master or Matsubara equations for the density operator. Normal ordering is introduced and applied to expressions for the vacuum state projector. Multi-particle position probabilities are considered and related to normally ordered quantum correlation functions.

•Nov 13, 2014•0 citations
Equations are derived that express the density matrix of a single mode in terms of the normally ordered moments of the photon annihilation and creation operator and that are convergent for arbitrary radiation fields. The procedure consists of introducing a set of analytic generating functions and applying the method of analytic continuation. \textcopyright{} 1996 The American Physical Society.
A single-mode radiation field with Gaussian Wigner functions (Gaussian state field) is studied. The single-mode Gaussian-field generating function for both the density matrix elements and the moments of the creation and annihilation operators is calculated. It is shown that the matrix elements as well as the moments of different orders can be expressed in terms of Hermitian polynomials of two variables. Furthermore, the cumulants of the Gaussian state field are calculated. Finally, we generalize the Gaussian state field and compute the corresponding moments and density matrix elements.
The transient behavior of the density operator for radiation in a single-mode cavity at a finite temperature is considered. Any initial state will evolve toward thermal equilibrium because of the interaction with the mirrors. This steady state is determined uniquely by the temperature, but the transient state depends on the initial conditions. The equation of motion for the matrix elements of the density operator is solved analytically, given an arbitrary initial state. The factorial moments, the generating function, and the time-dependent spectral distribution are also obtained. The results yield known expressions in the appropriate limits.
The master equation formed from the diagonal elements of the density matrix is solved in general for arbitrary initial photon probability distributions. Several examples are studied including the decay of an initial Poisson distribution (coherent state) to the equilibrium Bose-Einstein distribution. The concept of a mixed Poisson process is introduced and its physical implications examined in the context of the present problem. A general expression for the nonstationary correlation function of the photon field is also obtained.
The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.
Abstract This chapter sets out the second-quantisation treatment for systems of identical bosons or fermions. Key features such as the symmetrisation principle and super-selection rules are presented. Orthogonality and completeness requirements for mode functions are set out, and Fock states are defined in terms of mode occupancy, leading to the definitions of mode annihilation and creation operators and the determination of their commutation (bosons) or anticommutation (fermions) rules. Important two-mode states such as binomial states and relative-phase eigenstates are considered. Field creation and annihilation operators are defined and related to multi-particle position eigenstates. The Hamiltonian is expressed in terms of both mode operators and field operators, state dynamics being treated via Liouville–von Neumann, master or Matsubara equations for the density operator. Normal ordering is introduced and applied to expressions for the vacuum state projector. Multi-particle position probabilities are considered and related to normally ordered quantum correlation functions.