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Physical Review A

Canonical transformation and decay into phase-sensitive reservoirs

ARTUR K. EKERT, P. L. Knight

Published Jul 1, 1990• 51 citations
Publisher

Abstract

We show that the master equation describing the dissipative evolution of a bosonic quantum system coupled to a phase-sensitive reservoir may be reduced to a standard form of dissipation for a thermal reservoir by canonical transformations. The solution to the master equation for complex phase-sensitive interaction induced, for example, by broadband squeezed light, may then be obtained by a transformation from the known solutions for the simpler thermal case. We show that canonical transformations have a particularly useful form when describing the evolution of quantum dissipative systems in phase space using the Wigner function. We illustrate these ideas with two phase-sensitive dissipative problems: that of the correlated-emission laser and of decay induced by a broadband squeezed vacuum field.

DOI
10.1103/physreva.42.487

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We show that the master equation describing the dissipative evolution of a bosonic quantum system coupled to a phase-sensitive reservoir may be reduced to a standard form of dissipation for a thermal reservoir by canonical transformations. The solution to the master equation for complex phase-sensitive interaction induced, for example, by broadband squeezed light, may then be obtained by a transformation from the known solutions for the simpler thermal case. We show that canonical transformations have a particularly useful form when describing the evolution of quantum dissipative systems in phase space using the Wigner function. We illustrate these ideas with two phase-sensitive dissipative problems: that of the correlated-emission laser and of decay induced by a broadband squeezed vacuum field.
We show that atoms interacting with a squeezed radiation field reservoir can exhibit phase-dependent population decay and relax into a highly correlated pure state, in striking difference to decay in a heat bath. We use a Heisenberg operator treatment of radiative transitions in a broadband squeezed light field described as a quantum white-noise field with correlated fluctuations to describe the modified decay of a single atom, and of two atoms cooperatively decaying in a squeezed vacuum. We compare this decay with that generated by a broadband thermal field which lacks the essential field-mode correlations characteristic of squeezing.
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.
A dissipating system consisting of two strongly interacting subsystems, each coupled to its respective thermal reservoir is considered. The usual method for obtaining a master equation for the reduced density operator of such a system is to assume the factorization of this density operator in the derivation of the irreversible term, retaining the coupling in the reversible term alone. This approach has the serious limitation that it fails to preserve detailed balance in general and yields an incorrect canonical form for the stationary solution of the density operator. A general approach, based on the introduction of a new interaction picture, is presented. This method is illustrated by applying it to a coupled boson system and to a two-level atom interacting with a single mode of the electromagnetic field.
Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.
Squeezed thermal reservoirs, characterized by thermal noise with anisotropic fluctuations, have profound implications in quantum thermodynamics and serve as powerful resources for quantum information. However, their experimental realizations remain challenging. Existing schemes typically rely on injected squeezed light, time-dependent modulation, or driven nonlinear interactions, which introduce complexity and limit experimental feasibility. Using only time-independent linear coupling to a lossy mode within a normal thermal environment, we identify a general and experimentally accessible framework for squeezed-reservoir engineering, applicable across platforms such as circuit and cavity quantum electrodynamics as well as coupled cavity systems. We illustrate the framework through two experimentally relevant cases: directional phase coherence extension in two-level systems like qubits or atoms, and dissipative quadrature squeezing in bosonic modes like photons or phonons. By eliminating the need for active control or squeezed input, our passive linear-coupling approach provides a resource-efficient and practical pathway to dissipative squeezing, decoherence suppression, entanglement stabilization, quantum simulation, and the exploration of unconventional quantum thermodynamics and phase transitions.