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Optics and Spectroscopy

A solvable open quantum system: The strongly driven micromaser

Federico Casagrande, Berthold‐Georg Englert, Pavel Lougovski, ALFREDO LULLI, E. Solano, H. Walther

Published Aug 1, 2005• 3 citations
Publisher

Abstract

A micromaser, where an additional classical field strongly drives the atoms crossing the cavity, provides an example of a solvable open quantum system with pumping and dissipation. The coherent time evolution consists in cavity field displacements correlated with atomic internal states, a different entanglement than the usual Jaynes-Cummings interaction. We provide analytical results and phase space descriptions via the quantum characteristic function. We illustrate bunching and antibunching effects at steady state in atom pair correlation measurements, where the control parameter is simply the atomic velocity.

DOI
10.1134/1.2034619

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A micromaser, where an additional classical field strongly drives the atoms crossing the cavity, provides an example of a solvable open quantum system with pumping and dissipation. The coherent time evolution consists in cavity field displacements correlated with atomic internal states, a different entanglement than the usual Jaynes-Cummings interaction. We provide analytical results and phase space descriptions via the quantum characteristic function. We illustrate bunching and antibunching effects at steady state in atom pair correlation measurements, where the control parameter is simply the atomic velocity.
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.
We study the dynamics of a micromaser where the pumping atoms are strongly driven by a resonant classical field during their transit through the cavity mode. We derive a master equation for this strongly driven micromaser, involving the contributions of the unitary atom-field interactions and the dissipative effects of a thermal bath. We find analytical solutions for the temporal evolution and the steady state of this system by means of phase-space techniques, providing an unusual solvable model of an open quantum system, including pumping and decoherence. We derive closed expressions for all relevant expectation values, describing the statistics of the cavity field and the detected atomic levels. The transient regime shows the buildup of mixtures of mesoscopic fields evolving towards a super-Poissonian steady-state field that, nevertheless, yields atomic correlations that exhibit stronger nonclassical features than the conventional micromaser.
We consider the resonant interaction of a cavity mode with a two-level atom that is driven by a coherent field while it crosses the cavity. Starting from the cavity field in a coherent state, we show that the state of the system can reach the maximum entanglement after a unitary evolution for long enough interaction times. Also we illustrate how the generation of cavity field superposition states can allow, in the open system dynamics, the observation of their decoherence in atomic correlation measurements, for any initial cavity field state, and even under the combined effects of dissipation, thermal noise, and atomic pumping.
A detailed theoretical and experimental study of the atom-field interaction starting from first principles was made possible by the realization of the single-atom micromaser. The situation is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a superconducting cavity. In spite of the considerable amount of work devoted to the study of the micromaser, it still remains to be established how the system behaves under coherent pumping. We present a comprehensive study of the coherently pumped micromaser; we develop an analytic method to obtain the steady-state solution of the master equation governing the time evolution of the cavity field. We illustrate this method with a simple example and present results for the photon distribution and purity of the steady state of the coherently pumped micromaser.
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.
We investigate the dynamics of a micromaser continuously driven by a resonant coherent field and pumped by atoms injected in superposition states. We formulate a master equation for the system, and by a Monte Carlo wave-function approach we describe the steady-state behavior as a function of atomic-transit time or driving-field intensity. The combined action of polarized atoms and driving field can lead to enhancement or suppression of coherence in system dynamics, including nonclassical effects such as quadrature squeezing, sub-Poissonian photon statistics, and revival of Rabi oscillations, as well as to shifts of the induced cavity-field phase. These results show the possibility of manipulating both the photon number and the phase of the cavity field.