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Physical Review A

Robustness of superposition states evolving under the influence of a thermal reservoir

José S. Sales, N. G. de Almeida

Published Jun 27, 2011• 4 citations
Publisher

Abstract

We study the evolution of superposition states under the influence of a reservoir at zero and finite temperatures in cavity quantum electrodynamics aiming to know how their purity is lost over time. The superpositions studied here are composed of coherent states, orthogonal coherent states, squeezed coherent states, and orthogonal squeezed coherent states, which we introduce to generalize the orthogonal coherent states. For comparison, we also show how the robustness of the superpositions studied here differs from that of a qubit given by a superposition of zero- and one-photon states.

DOI
10.1103/physreva.83.062121

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We study the evolution of superposition states under the influence of a reservoir at zero and finite temperatures in cavity quantum electrodynamics aiming to know how their purity is lost over time. The superpositions studied here are composed of coherent states, orthogonal coherent states, squeezed coherent states, and orthogonal squeezed coherent states, which we introduce to generalize the orthogonal coherent states. For comparison, we also show how the robustness of the superpositions studied here differs from that of a qubit given by a superposition of zero- and one-photon states.
Recently several methods have been proposed for generation of superposition (Schrödinger cat) states in microwave cavities. At microwave frequencies thermal photons can significantly affect statistical properties of superposition states. In the present letter we study the influence of a thermal heatbath on non-classical properties of quantum superposition states. We show that at non-zero temperatures the loss of quantum coherences is much faster than at zero temperature and that the sensitivity of the quantum coherence to the presence of thermal photons can lead to some difficulties in the preparation of Schrödinger cat states in microwave cavities unless the temperature of the microwave cavity is sufficiently low.
Recently several methods have been proposed for generation of superposition (Schr\"odinger-cat) states in microwave cavities. At microwave frequencies, thermal photons can significantly affect statistical properties of superposition states. In the present paper we study the influence of a thermal heat bath on nonclassical properties of quantum superposition states. We show that at nonzero temperature the loss of coherences is much faster than at zero temperature. Using the formalism of quasiprobability distributions and solving the corresponding Fokker-Planck equations, we describe the time evolution of the super-position states in phase space and derive the rate of the decay of quantum coherence. This decay rate depends on the separation between the component states and on the temperature of the heat bath. Moreover, we discuss in detail how the interaction with a nonzero-temperature heat bath leads to a transformation of a nonclassical state to a classical state. We show that the sensitivity of the quantum coherence to the presence of thermal photons can lead to some difficulties in the preparation of Schr\"odinger-cat states in microwave cavities unless the temperature of the microwave cavity is sufficiently low.
We propose an engineered reservoir inducing the relaxation of a cavity field towards non-classical states. It is made up of two-level atoms crossing the cavity one at a time. Each atom-cavity interaction is first dispersive, then resonant, then dispersive again. The reservoir pointer states are those produced by a fictitious Kerr Hamiltonian acting on a coherent field. We thereby stabilize squeezed states and quantum superpositions of multiple coherent components in a cavity having a finite damping time. This robust method could be implemented in state-of-the-art experiments and lead to interesting insights into mesoscopic quantum state superpositions.
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.
In this paper we detail some results advanced in a recent letter [Prado et al., Phys. Rev. Lett. 102, 073008 (2009).] showing how to engineer reservoirs for two-level systems at absolute zero by means of a time-dependent master equation leading to a nonstationary superposition equilibrium state. We also present a general recipe showing how to build nonadiabatic coherent evolutions of a fermionic system interacting with a bosonic mode and investigate the influence of thermal reservoirs at finite temperature on the fidelity of the protected superposition state. Our analytical results are supported by numerical analysis of the full Hamiltonian model.