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Journal of the Optical Society of America B

Squeezed thermal states: the result of parametric down conversion in lossy cavities

Hossein Seifoory, Sean Doutre, Marc M. Dignam, John E. Sipe

Published Jul 12, 2017• 28 citations
Publisher

Abstract

We theoretically investigate the properties and temporal evolution of squeezed states generated using degenerate parametric down conversion in lossy cavities. We show that the Lindblad master equation, which governs the evolution of this system, has, as its solution, a squeezed thermal state with an effective temperature and squeezing parameter that depends on time. We derive analytical solutions for the time evolution of quadrature noise, thermal photon number, squeezing parameter, and total photon number under different pumping regimes. We also find the steady state limits of the quadrature noises and discuss the g(2) factor of the generated light inside the cavity in the steady state.

DOI
10.1364/josab.34.001587

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We theoretically investigate the properties and temporal evolution of squeezed states generated using degenerate parametric down conversion in lossy cavities. We show that the Lindblad master equation, which governs the evolution of this system, has, as its solution, a squeezed thermal state with an effective temperature and squeezing parameter that depends on time. We derive analytical solutions for the time evolution of quadrature noise, thermal photon number, squeezing parameter, and total photon number under different pumping regimes. We also find the steady state limits of the quadrature noises and discuss the g(2) factor of the generated light inside the cavity in the steady state.
We investigate theoretically the properties of squeezed states generated using degenerate parametric down conversion in lossy cavities. We show that the Lindblad master equation, which governs the evolution of this system, has as its solution a squeezed thermal state with an effective temperature and squeezing parameter that depends on time. We derive analytical solutions for the time-evolution of quadrature noise, thermal photon number, squeezing parameter, and total photon number under different pumping regimes. We also find the steady state limits of the quadrature noises and discuss the $ g^{(2)} $ factor of the generated light inside the cavity in the steady state.
We model squeezed state generation in a lossy optical cavity in the presence of a thermal bath using the Lindblad master equation. We show that the exact solution is a squeezed thermal state, where thermal photons arise both from loss and from the thermal bath. We derive an exact, closed-form solution for the evolution of the quadrature uncertainty arising from pulsed degenerate spontaneous parametric down conversion in the cavity. We apply this solution under different pump conditions and show in detail how the thermal environment reduces quadrature squeezing as well as the second order coherence function.
We model squeezed state generation in a lossy optical cavity in the presence of a thermal bath using the Lindblad master equation. We show that in the undepleted-pump approximation, the exact solution is a squeezed thermal state, where thermal photons arise both from loss and from the thermal bath. We derive a closed-form solution for the evolution of the quadrature uncertainty arising from pulsed degenerate spontaneous parametric down conversion in the cavity under the parametric oscillation threshold. We apply this solution under different pump conditions and show in detail how the thermal environment reduces quadrature squeezing as well as the second-order coherence function.
Continuous-variable (CV) entanglement is a valuable resource in the field of quantum information. One source of CV entanglement is the correlations between the position and momentum of photons in a two-mode squeezed state of light. In this paper, we theoretically study the generation of squeezed states, via spontaneous parametric down conversion, inside a two-mode lossy cavity that is pumped with a classical optical pulse. The dynamics of the density operator in the cavity is modeled using the Lindblad master equation, and we show that the solution to this model is the density operator for a two-mode squeezed thermal state, with a time-dependent squeezing amplitude and average thermal photon number for each mode. We derive an expression for the minimum correlation variance inside the cavity that depends crucially on the difference in the losses between the two modes. We apply our analytic solution to the important example of a microring resonator that is pumped with a Gaussian pulse. The expressions that we derive will help researchers optimize CV entanglement in lossy cavities.
We show that the light generated via spontaneous four-wave mixing or parametric down conversion in multiple, coupled, lossy cavities is a multimode squeezed thermal state. Requiring this state to be the solution of the Lindblad master equation results in a set of coupled first-order differential equations for the time-dependent squeezing parameters and thermal photon numbers of the state. The benefit of this semianalytic approach is that the number of coupled equations scales linearly with the number of modes and is independent of the number of photons generated. With this analytic form of the state, correlation variances are easily expressed as analytic functions of the time-dependent mode parameters. Thus, our solution makes it computationally tractable and relatively straightforward to calculate the generation and evolution of multimode entangled states in multiple coupled, lossy cavities, even when there are a large number of modes and/or photons.
Nonlinear multiphoton conversion processes are theoretically and experimentally challenging systems in quantum optics and related areas. These phenomena bring remarkable quantum properties and can be of fundamental relevance in quantum information and computation applications. Based on recent experimental advances and the possible uses in generalized three-photon squeezing, we investigate degenerate three-photon parametric down-conversion generated inside a cavity fed by a classical pump. In addition, we consider that the process is stimulated by a coherent driving mode and subjected to dissipation by spontaneous emission of the cavity. A treatment using phase-space methods and stochastic differential equations is applied and the stability conditions of its steady states are found. We find that the system does not exhibit multistability, instead it has a limited parameter region leading to a single stable steady branch, alongside multiple unstable branches. Focusing on the stable steady states and linearizing the dynamics around them, we calculate the spectral densities of the quadrature variables. We identify and characterize the squeezing of three-photon down-conversion. Moreover, since the quasi-probability differential equations of the higher-order multiphoton processes have higher-order derivatives, hindering their mapping into stochastic differential equations, we make use of a positive Wigner function approach to calculate two-dimensional spectral densities associated with three-time correlation functions, with the goal of studying non-Gaussian properties of the system.