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Physical Review A

Schrödinger-cat states at finite temperature: Influence of a finite-temperature heat bath on quantum interferences

M. S. Kim, Vladimír Bužek

Published Oct 1, 1992• 118 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.46.4239

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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.
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.
The observation of quantum phenomena often necessitates sufficiently pure states, a requirement that can be challenging to achieve. In this study, our goal is to prepare a nonclassical state originating from a mixed state, using dynamics that preserve the initial low purity of the state. We generate a quantum superposition of displaced thermal states within a microwave cavity using only unitary interactions with a transmon qubit. We measure the Wigner functions of these "hot" Schrödinger cat states for an initial purity as low as 0.06. This corresponds to a cavity mode temperature of up to 1.8 kelvin, 60 times hotter than the cavity's physical environment. Our realization of highly mixed quantum superposition states could be implemented with other continuous-variable systems, e.g., nanomechanical oscillators, for which ground-state cooling remains challenging.
We study phase properties of quantum mechanical superposition states decaying into phase-sensitive reservoirs. We show that the Wigner phase quasiprobability distribution reflects the quantum interference effects very well while the Pegg–Barnett and the Husimi phase probability distributions are quite insensitive with respect to the character of the quantum interference between component states. We study in detail phase properties of the even coherent state which decays into the thermal and the squeezed reservoirs. In thermal reservoirs the phase of the superposition state becomes randomized under the influence of thermal fluctuations. The higher the number of thermal photons the faster the randomization process is. On the other hand we show that depending on the relative phase between the superposition state and the phase-sensitive reservoir the process of phase randomization can be either enhanced or completely suppressed.
A multiple Schrödinger cat state can be defined as a quantum superposition of an arbitrary number of coherent states. In this paper, we study the process of decoherence of such multiple states in interaction with their environments, using the model of a harmonic oscillator coupled to a thermal bath. The evolution of a superposition of various coherent states into a statistical mixture is described with the Wigner distribution function. The decoherence effect is expressed in analytic form, and it is seen that the damping time is very sensitive to the temperature. Multiple superpositions of states can be conveniently described using crystallographic group techniques; numerical calculations of the Wigner functions for some typical groups are also presented.
The transition from quantum to classical, in the case of a quantum harmonic oscillator, is typically identified with the transition from a quantum superposition of macroscopically distinguishable states, such as the Schr\"odinger-cat state, into the corresponding statistical mixture. This transition is commonly characterized by the asymptotic loss of the interference term in the Wigner representation of the cat state. In this paper we show that the quantum-to-classical transition has different dynamical features depending on the measure for nonclassicality used. Measures based on an operatorial definition have well-defined physical meaning and allow a deeper understanding of the quantum-to-classical transition. Our analysis shows that, for most nonclassicality measures, the Schr\"odinger-cat state becomes classical after a finite time. Moreover, our results challenge the prevailing idea that more macroscopic states are more susceptible to decoherence in the sense that the transition from quantum to classical occurs faster. Since nonclassicality is a prerequisite for entanglement generation our results also bridge the gap between decoherence, which is lost only asymptotically, and entanglement, which may show a ``sudden death.'' In fact, whereas the loss of coherences still remains asymptotic, we emphasize that the transition from quantum to classical can indeed occur at a finite time.
A cat-state is a superposition of two coherent states with amplitudes α0α_{0}α0​ and −α0-α_{0}−α0​. Recent experiments create cat states in a microwave cavity field using superconducting circuits. As with degenerate parametric oscillation (DPO) in an adiabatic and highly nonlinear limit, the states are formed in a signal cavity mode via a two-photon dissipative process induced by the down conversion of a pump field to generate pairs of signal photons. The damping of the signal and the presence of thermal fluctuations rapidly decoheres the state, and the effect on the dynamics is to either destroy the possibility of a cat state, or else to sharply reduce the lifetime and size of the cat-states that can be formed. In this paper, we study the effect on both the DPO and microwave systems of a squeezed reservoir coupled to the cavity. While the threshold nonlinearity is not altered, we show that the use of squeezed states significantly lengthens the lifetime of the cat states. This improves the feasibility of generating cat states of large amplitude and with a greater degree of quantum macroscopic coherence, which is necessary for many quantum technology applications. Using current experimental parameters for the microwave set-up, which requires a modified Hamiltonian, we further demonstrate how squeezed states enhance the quality of the cat states that could be formed in this regime. Squeezing also combats the significant decoherence due to thermal noise, which is relevant for microwave fields at finite temperature. By modeling a thermal squeezed reservoir, we show that the thermal decoherence of the dynamical cat states can be inhibited by a careful control of the squeezing of the reservoir. To signify the quality of the cat state, we consider different signatures including fringes and negativity, and the Cl1C_{l_{1}}Cl1​​ measure of quantum coherence.