The micromaser is an archetype experimental setting where a beam of excited two-level atoms is injected into a high-finesse cavity. It has played a pivotal role as a testbed for predictions of quantum optics. We consider a generalized micromaser setting consisting of high-quality cavity pumped by a beam of three-level atoms. The atoms are assumed to be prepared to carry quantum coherence between their excited state doublet. Our objective is to produce quantum entanglement between the right-handed circular (RHC) and left-handed circular (LHC) polarized photons in the cavity, exploiting the quantum coherence in the pump atoms. For that aim, we derive the generalized micromaser master equation for our system. We find that the dynamics of the micromaser field driven by the pump beam is equivalent to two non-interacting RHC and LHC photonis systems sharing a common non-equilibrium environment. The effect of the shared bath is to mediate an incoherent interaction between the otherwise non-interacting cavity photons, which emerges only if the atoms carry quantum coherence. We take into account cavity losses as a source of quantum decoherence and characterize the quantum entanglement between the LHC and RHC polarized photons in terms of logarithmic negativity, calculated using the dynamical solution of the master equation. Our reseults reveal that while there is no steady-state entanglement, LHC and RHC polarzied photons can be entangled in the transient regime.
We describe the dynamics of a micromaser pumped by atoms in superposition states and driven by a resonant coherent field. By a quantum jump approach to the master equation solution, we show how the coherences excited in the cavity mode affect the nonclassical steady-state behaviour of the system, in a way that depends on the relative phase between the polarized atoms and the driving field. Both the mean photon number and the phase of the cavity field can be controlled via the atomic transit time and pump rate, or the driving field intensity. The predicted coherence and phase competition effects can show up in the statistics of the populations of outgoing atoms.
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.
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 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.
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 general theory of the micromaser is described. The model is based on treating the input atomic beam as a two-component quantum field so that the two-level atoms in the beam are ``quanta'' of this field. This approach makes it possible to formulate a general quantum Langevin description of the dynamics with the input atomic field as a source of quantum noise. The passage to a master-equation description is then effected by use of an adjoint-operator method, and by introducing a general class of statistical states for the atomic beam known as generalized shot noise. The result is a (non-Markovian) master equation for the field inside the cavity which is valid for a broad range of statistical properties of the input atomic beam. The approximate steady-state solutions to this master equation for the photon statistics of the cavity field for sub-Poissonian (antibunched) atomic beams found by other researchers are regained. The theory is then extended to treat super-Poissonian (bunched) atomic beams. An exact result is found in the limit of a strongly bunched beam in which the cavity-field state is shown to converge to a mixture of a thermal-field state and the state produced by a random beam of twice the intensity. A physical explanation of this result is also presented.