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arXiv:quant-ph/0101072 · Physical Review A

Rapid and slow decoherence in conjunction with dissipation in a system of two-level atoms

Péter Földi, Attila Czirják, M. G. Benedict

Published Feb 9, 2001• 28 citations
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Abstract

We investigate the time evolution of the superposition of macroscopically distinct quantum states in a system of two-level atoms interacting with a thermal environment of photon modes. We show that the atomic coherent states are robust against decoherence; therefore, we call their superpositions atomic Schr\"odinger cat states. The initial fast regime of the time evolution is associated with the process of decoherence and it is directed toward the statistical mixture of the constituent coherent states of the original state for most of the initial conditions. However, certain superpositions, called symmetric, exhibit exceptionally slow decoherence. By introducing a new measure, we generalize the usual decoherence scheme regarding the evolution of the state to account also for the symmetric case. To stress the fact that the environment preserves symmetric superpositions much longer than the other ones, we present Wigner function images of the decoherence of a suitably oriented four-component cat state.

DOI
10.1103/physreva.63.033807

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We investigate the time evolution of the superposition of macroscopically distinct quantum states in a system of two-level atoms interacting with a thermal environment of photon modes. We show that the atomic coherent states are robust against decoherence; therefore, we call their superpositions atomic Schr"odinger cat states. The initial fast regime of the time evolution is associated with the process of decoherence and it is directed toward the statistical mixture of the constituent coherent states of the original state for most of the initial conditions. However, certain superpositions, called symmetric, exhibit exceptionally slow decoherence. By introducing a new measure, we generalize the usual decoherence scheme regarding the evolution of the state to account also for the symmetric case. To stress the fact that the environment preserves symmetric superpositions much longer than the other ones, we present Wigner function images of the decoherence of a suitably oriented four-component cat state.
We consider a class of states in an ensemble of two-level atoms: a mesoscopic superposition of two distinct atomic coherent states, which can be regarded as atomic analogs of the states usually called Schr\"odinger-cat states in quantum optics. According to the relation of the constituents, we define polar and nonpolar cat states. The properties of these are investigated by the aid of the spherical Wigner function. We show that nonpolar cat states generally exhibit squeezing, the measure of which depends on the separation of the components of the cat, and also on the number of the constituent atoms. By solving the master equation for the polar cat state embedded in an external environment, we determine the characteristic times of decoherence and dissipation, and also the characteristic time of a new parameter, the nonclassicality of the state. This latter one is introduced by the help of the Wigner function, which is used also to visualize the process. The dependence of the characteristic times on the number of atoms of the cat and on the temperature of the environment shows that the decoherence of polar cat states is surprisingly slow.
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 dynamics of quantum decoherence have been experimentally studied for the first time. A single circular Rydberg atom prepares in a high-quality superconducting cavity a ‚Schrödinger cat‘ state of the radiation field. This highly non-classical state is a quantum superposition of two coherent components with different classical phases. A second atom probes the cavity state after a tunable delay. The decay of the quantum correlations between the two atoms reveals the evolution of the initial quantum superposition into a mere statistical mixture. The time scale of this process decreases when the separation beween the two components increases. A simple theoretical model of the experiment is presented. The excellent agreement with the experimental signals confirms in a striking way the decoherence approach.
A mesoscopic superposition of quantum states involving radiation fields with classically distinct phases was created and its progressive decoherence observed. The experiment involved Rydberg atoms interacting one at a time with a few photon coherent field trapped in a high QQQ microwave cavity. The mesoscopic superposition was the equivalent of an `` atom+measuringapparatus\mathrm{atom}+\mathrm{measuring}\mathrm{apparatus}atom+measuringapparatus'' system in which the ``meter'' was pointing simultaneously towards two different directions---a ``Schr\"odinger cat.'' The decoherence phenomenon transforming this superposition into a statistical mixture was observed while it unfolded, providing a direct insight into a process at the heart of quantum measurement.
In the quantum trajectory approach an open quantum system is represented by a stochastic pure-state wave function. The mixed-state density operator that satisfies the usual master equation is recovered from the quantum trajectories by performing either an ensemble average or, for stationary systems, an average over time. This unraveling of the master equation dynamics into pure-state trajectories provides new insight into the decay of quantum coherence in systems that are open to the environment. Under some conditions macroscopic superposition states are preserved as macroscopic superposition states along individual trajectories, but are reduced to mixtures by the trajectory average. Under other conditions one of the states in a macroscopic superposition becomes dominant (in amplitude) over the other along each quantum trajectory. We discuss the mechanisms that produce these dynamics and assess the implications for the observation of Schrodinger cat states in optics experiments. We present results for some specific examples of macroscopic superposition states generated by a cavity QED system. The system involves a small collection of atoms in an optical cavity driven by a coherent laser field. Under strong-coupling conditions this system produces a variety of Schrodinger cat states whose precise form depends on the number of atoms, the method of excitation (through a cavity mirror or from the side), the initial state of the atoms, and the position of the atoms in the cavity.