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Journal of Modern Optics

Density matrix for photons in a phase-sensitive reservoir

Ho Trung Dung, L. Knöll

Published Apr 1, 1999• 3 citations
Publisher

Abstract

The transient behaviour of the density matrix for a radiation field in a single-mode cavity coupled to a phase-sensitive reservoir is considered using the Fokker-Planck equation approach. The matrix elements of the density operator in the number state basis and the antinormally ordered correlation functions are derived in terms of the initial density matrix elements. Limiting cases of the reservoir state and the cavity-field initial state are investigated and numerical examples are given.

DOI
10.1080/09500349908231309

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The transient behaviour of the density matrix for a radiation field in a single-mode cavity coupled to a phase-sensitive reservoir is considered using the Fokker-Planck equation approach. The matrix elements of the density operator in the number state basis and the antinormally ordered correlation functions are derived in terms of the initial density matrix elements. Limiting cases of the reservoir state and the cavity-field initial state are investigated and numerical examples are given.
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.
We consider the dynamics of a single-mode cavity field damped via its coupling to a broad-band squeezed heat bath. The density operator equation of motion is transformed to a Fokker-Planck equation (FPE) for the s-parametrized quasiprobability distributions. We show that the field amplitude, intensity, spectrum and quadrature variances can be evaluated easily using the differential equations derived directly from the FPE. The FPE is then solved subject to different initial conditions, and the solution is used to investigate squeezing and field state evolution. The time behaviour of the purity Tr ([pcirc] 2) is treated in detail both analytically and numerically. The general steady-state solution is found in the operator form. We also show that, when certain conditions are met, dissipation without change in fluctuation properties is possible. A comparison is made with the cases of single-atom damping and off-resonance field damping.
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 authors present a method of deriving a reduced density operator equation for a quantized field interacting with an arbitrary number of atoms that does not neglect atom-atom correlations. As a particular example they analyse a three-level laser operating well below saturation and solve the density matrix equations in steady state. For weak pumping the system is equivalent to the Scully-Lamb model (1966) and the corresponding equations are recovered. Using the saturation photon number as a (large) scaling parameter one can derive a Fokker-Planck equation for the Glauber-P-distribution. The stationary solution of the reduced density matrix equation is compared with the solution of the Fokker-Planck equation and-for the case of a single-atom laser-with a numerical solution of the total density matrix equation.
The transient behavior of a laser is discussed using the quantum theory as did Scully and Lamb. The formal solution of the density-matrix equation is expressed in terms of exponentially decaying eigenmodes. Some of the lower decay constants are obtained numerically. The equations for the moments of the density matrix are then derived and solved by a truncation method. The equations of motion are integrated numerically for the case where the average number of photons in a laser cavity has the realistically large value 1.3 \ifmmode\times\else\texttimes\fi{} 105{10}^{5}105. An alternative Fokker-Planck-equation approach is discussed.
Using the Wigner function formalism in phase space, we analyze the decay of quantum coherences in phase-sensitive reservoirs. We show that the decay rate of quantum coherences in phase-sensitive reservoirs can be significantly modified compared to the decay rate in ordinary (phase-insensitive) thermal reservoirs. Depending on the phases of the quantum system (field mode) and the squeezed reservoir, the decay rate of the quantum coherence can be either enhanced or significantly suppressed, which is in agreement with the results obtained recently by other methods [T. A. B. Kennedy and D. F. Walls, Phys. Rev. A 37, 152 (1988)]. We show that in an ideally squeezed reservoir with a high degree of squeezing, the decay rate of the quantum coherence (i.e., the decay rate of off-diagonal terms of the density matrix in the coherent-state basis) can be equal to the decay rate of the energy of the system (i.e., the decay rate of diagonal terms of the density matrix). Suppression of the decay rate of the quantum coherence leads to preservation of nonclassical effects such as the oscillations in the photon number distribution. Moreover, we find that some initial superposition states of light exhibiting super-Poissonian photon statistics can be transformed into intermediate sub-Poissonian states under the influence of phase-sensitive reservoirs.