Coherent superpositions of quantum states are very sensitive to coupling to the environment. The fluctuations associated with this coupling tend to wash out quantum phases, so that in practice only quantum mixtures are observed at the macroscopic level. This problem has been the object of recent interest in connection with measurement theory (Schrodinger cat). Theoretical predictions indicate that the decay of quantum coherences in the so-called pointer basis is amplified by a factor proportional to a measure of the distance in Hilbert space between the states. Micromasers offer a possible experimental tool to investigate the decay of such coherences. Theoretical studies of a lossless version of the micromaster have shown the existence of trapping states | n q 〉 of the cavity mode. These number states are such that atoms injected in the upper state into the cavity experience a 2qπ interaction with the field. Cavity states of the form | ψ 〉 = Σ q C q | n q 〉 are easily shown to be dynamic steady states of the system. Number states are not a pointer basis for the micromaser, and our theoretical results will evidence the qualitative difference in the nature of coherence decay between this and previously studied cases. Because the decay times of micromaser cavities are typically of the order of hundreds of milliseconds, our results should be amenable to experimental verification.
Coherent superpositions of quantum states are very sensitive to coupling to the environment. The fluctuations associated with this coupling tend to wash out quantum phases, so that in practice only quantum mixtures are observed at the macroscopic level. This problem has been the object of recent interest in connection with measurement theory (Schrodinger cat). Theoretical predictions indicate that the decay of quantum coherences in the so-called pointer basis is amplified by a factor proportional to a measure of the distance in Hilbert space between the states. Micromasers offer a possible experimental tool to investigate the decay of such coherences. Theoretical studies of a lossless version of the micromaster have shown the existence of trapping states | n q 〉 of the cavity mode. These number states are such that atoms injected in the upper state into the cavity experience a 2qπ interaction with the field. Cavity states of the form | ψ 〉 = Σ q C q | n q 〉 are easily shown to be dynamic steady states of the system. Number states are not a pointer basis for the micromaser, and our theoretical results will evidence the qualitative difference in the nature of coherence decay between this and previously studied cases. Because the decay times of micromaser cavities are typically of the order of hundreds of milliseconds, our results should be amenable to experimental verification.
A single beam of excited two-level atoms couples two micromasers in a series. It is shown that in the absence of dissipation the possible steady states of their fields are superpositions, of two-field trapping number states. Which one is realized depends upon the initial state of the fields and the interaction parameters, g'\ensuremath{\tau}' and g''\ensuremath{\tau}'', of the two cavities. A large number of these trapping states are pure quantum states, some of them showing entanglement of the two nonlocal micromaser fields of the form \ensuremath{\Vert}N,N+M〉\ifmmode\pm\else\textpm\fi{}\ensuremath{\Vert}N+M,N〉. Here, N and N+M are arbitrary trapping numbers that belong to disconnected blocks of the photon number space and M specifies the order of correlation between the fields. The time evolution of the system toward steady states is investigated numerically, mainly concentrating on the production of pure entangled trapping states of the form discussed earlier. We describe a special procedure to amplify a number state, \ensuremath{\Vert}N,N〉, into such states that is based on conditions regarding the interaction parameters. In principle, N and M can be made arbitrarily large resulting in a steady-state nonlocal quantum superposition of distinct macroscopical fields (nonlocal ``Schr\"odinger cat''). We also present a solution of the standard master equation of a damped harmonic oscillator at finite temperature and apply it to study the effect of dissipation on the production of entangled trapping states at regular and Poissonian pump statistics.It is found that although the entanglement does not survive at a steady state it can build up in the short-time transient regime when cavity losses and the number of thermal photons are not too large. In this small-loss (large Nex) regime the quantum correlation between the micromaser fields decays into a steady-state classical superposition at a rate that depends on the cavity lifetime and the difference (or order of correlation), M, between the superposed photon numbers. In the large-loss (small Nex) regime, however, no transient correlation can be produced and the photon statistics spreads out towards a vacuum for increasing losses. The system undergoes a transition from an uncorrelated to a correlated behavior when the pumping parametrized by Nex exceeds the threshold between the large- and small-loss regimes. Thermal photons enhance the decay of the correlation and by coupling the disconnected blocks of the photon number space they populate the trapping number states of adjacent blocks. However, it is shown that the production of transient entanglement is not significantly affected by either thermal radiation or pumping fluctuations. The experimental realization of the entanglement of nonlocal micromaser fields employing these two-field trapping states of small photon numbers is shown to be feasible in the short-time transient regime using the presently available high-Q cavities and low temperatures, and could be possible on the macroscopic scale in the near future.
In a micromaser where a classical field strongly drives the atoms while they cross the cavity, remarkable atom–atom correlations show up at steady state, which vanish much faster than dissipative decay. Hence we consider atom pair correlation measurements in which the detection of the first probe atom prepares a mesoscopic superposition state of the cavity field, that entangles with a second probe atom. The conditional probabilities for the latter atomic detection provide a description of the decoherence of the superposition state, occurring in an open system in the presence of pumping, driving, dissipative, and thermal effects. The decoherence rate scales as the squared interaction time, that sets the separation in phase space between the superposition components, whereas the quantum coherence is unaffected by the atomic pumping. Hence we further investigate the system when the cavity is not pumped. Starting the correlation measurements from a thermal state, we can describe the effect of temperature on decoherence. Starting from a vacuum state, the superposition states are maximally separated Schrodinger cat states, whose decoherence can thus be monitored.
A distinctive signature of the granulated nature of the electromagnetic field is the revivals known to occur in the dynamics of a two-level atom interacting with a single-field mode (Jaynes-Cummings model). The development of high-Q single-mode cavities combined with progress in Rydberg atom spectroscopy has led to a possibility of verifying these predictions. Revival experiments using micromasers have been performed by Walther et al. Here the cavity mode is first allowed to reach a steady state, and then probe atoms are injected into the resonator. By measuring their state as they exit the cavity, one can infer the state of the field. We show that the micromaser experiences Jaynes-Cummingslike revivals when the pump parameter θ is varied. However, in contrast to the conventional Jaynes-Cummings situation, which considers initially coherent fields, the micromaser photon statistics are usually not Poissonian and depend sensitively on θ. The micromaser revivals cannot be explained by a semiclassical theory or linearized quantum fluctuations about the semiclassical solution. Rather, an exact quantum treatment, considering fully the granularity of the cavity mode, is required.
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.
This dissertation considers both a lossless and a dissipative micromaser in which a monoenergetic beam of two-level atoms in a coherent superposition of their upper and lower states is injected inside a single mode, high-Q cavity. In the lossless case, we find that under appropriate conditions a field initially in a mixed state will evolve to previously unknown pure states, which we call the tangent and cotangent states. In various limits, these states exhibit interesting properties such as sub-Poissonian photon statistics and squeezing, and most importantly they acquire the characteristics of "macroscopic" quantum superpositions. When dissipation of the cavity mode is incorporated into the model, we find that although the field no longer evolves to a pure state, the mixed steady-state field may still retain the properties of a macroscopic superposition under experimentally realizable damping rates. We then evaluate the experimental conditions necessary for the preparation and detection of such macroscopic superpositions.
We show that a coherently driven dissipative micromaser may display a new kind of bistability of the cavity field and/or the photon number. The bistability is a direct consequence of the presence of trapping states and the robustness of the related cotangent states against the perturbations caused by the cavity losses. The novel features of the bistability stem from the fact that even in the presence of cavity losses, the stationary state of the maser is close to a macroscopic superposition of number states (Schrödinger's cat) in the bistable regime. With increasing injection rate, the bistable regime of the micromaser is reached by means of a route that resembles a first-order phase transition. Hysteresis and other related irreversible effects are discussed.