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Physical Review A

Theory of detection in the micromaser

Rebecca R. McGowan, William C. Schieve

Published Jan 1, 1999• 6 citations
Publisher

Abstract

The field in the micromaser cavity, the atoms which both pump and probe the micromaser, and the detectors that determine the outgoing states of the atoms are treated as a single system in order to derive a nonlinear master equation for the field in the micromaser cavity which takes the effect of undetected atoms into account. Analytic solutions for this master equation are found for both phase-insensitive and phase-sensitive detection. These solutions can be used to explicitly show the back-action effects that measurement has on the spectrum of the micromaser. In addition, these solutions are used to find analytic expressions for the mean number of successive detector clicks of the same kind as well as atom-atom and photon-photon correlation functions. In this way, the analytic solutions are linked to experimentally accessible data.

DOI
10.1103/physreva.59.778

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We reconsider a variant of the one-atom-maser setup in which the emerging atoms are probed for coherent superpositions of the two Rydberg states of the maser transition. The statistical properties of the corresponding detector clicks contain information about the temporal evolution of the phase of the quantized electromagnetic field inside the resonator. We derive analytical expressions for certain statistical quantities, such as the mean number of successive clicks of the same kind, or the atom-atom and the photon-photon correlation functions. The latter are closely related if the photon-number distribution in steady state is dominated by a single narrow peak. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Jun 1, 1996•30 citations
The field in the micromaser cavity, the atoms which both pump and probe the micromaser, and the detectors that determine the outgoing states of the atoms are treated as a single system in order to derive a nonlinear master equation for the field in the micromaser cavity which takes the effect of undetected atoms into account. Analytic solutions for this master equation are found for both phase-insensitive and phase-sensitive detection. These solutions can be used to explicitly show the back-action effects that measurement has on the spectrum of the micromaser. In addition, these solutions are used to find analytic expressions for the mean number of successive detector clicks of the same kind as well as atom-atom and photon-photon correlation functions. In this way, the analytic solutions are linked to experimentally accessible data.
We present a general method for the derivation of various statistical quantities describing the detection of a beam of atoms emerging from a micromaser. The use of non-normalized conditioned density operators and a linear master equation for the dynamics between detection events is discussed as are the counting statistics, sequence statistics, and waiting time statistics. In particular, we derive expressions for the mean number of successive detections of atoms in one of any two orthogonal states of the two-level atom. We also derive expressions for the mean waiting times between detections. We show that the mean waiting times between detections of atoms in like states are equivalent to the mean waiting times calculated from the uncorrelated steady state detection rates, although like atoms are indeed correlated. The mean waiting times between detections of atoms in unlike states exhibit correlations. We evaluate the expressions for various detector efficiencies using numerical integration, reporting results for the standard micromaser arrangement in which the cavity is pumped by excited atoms and the excitation levels of the emerging atoms are measured. In addition, the atomic inversion and the Fano-Mandel function for the detection of deexcited atoms are calculated for comparison to the experimental results of Weidinger et al. [Phys. Rev. Lett. 82, 3795 (1999)], who report the observation of trapping states.
We present a general theory for the calculation of various characteristic properties of the beam of atoms emerging from a resonator in one-atom-maser experiments. The beam is described in terms of the statistics of the detector clicks. The evolution of the state of the maser photons between clicks is governed by a nonlinear master equation. The nonlinearity originates in the necessity to account for the atoms that escape detection. Despite the permanent reductions of the photon state, resulting from the detections, the steady state of the conventional linear master equation determines the statistics of the detector clicks. The whole process is ergodic, in the sense that a single run of the experiment contains all reproducible statistical data, provided the duration of the run is much longer than all relevant correlation times. The nonlinear master equation is used to calculate the distribution of waiting times between detector clicks. Other statistical properties of the clicks that are derived include correlation functions and variances of the counting statistics. The formalism is applied to standard one-atom-maser experiments and to parity measurements on both unpumped and pumped cavities. We find that, for the standard one-atom-maser operation, none of the said statistical properties is a simple immediate indicator for a sub-Poissonian variance of the photon number. For example, the detector clicks may be antibunched although the photon distribution has a super-Poissonian variance.
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.
We investigate signals of trapping states in the micromaser system in terms of the average number ⟨n⟩⟨n⟩⟨n⟩ of cavity photons as well as a suitably defined correlation length of atoms leaving the cavity. In the description of collective two-atom effects we allow the mean number, ϵϵϵ, of pump atoms inside the cavity during the characteristic atomic cavity transit time to be as large as of order one. The master equation we consider, which describes the micromaser including collective two-atom effects, still exhibits trapping states for $ϵ\ensuremath{\ne}0$, even for a mean number of atoms inside the cavity close to one. We, however, argue more importantly that the trapping states are more pronounced in terms of the correlation length as compared to ⟨n⟩⟨n⟩⟨n⟩, i.e., we suggest that trapping states can be more clearly revealed experimentally in terms of the atom correlation length. For axion detection in the micromaser this observable may, therefore, be an essential ingredient.
In this paper results are reported for the application of the quantum trajectory method to analysing the dynamics of the micromaser. The master equation for the cavity field is derived by a method which highlights the close relationship between the quantum trajectory method and the usual model for the one-atom micromaser. Possible quantum trajectory unravellings of the micromaser dynamics are discussed, and a simple number-state unravelling is described and used to generate simulations from which the cavity-field intensity correlation function is calculated. Attention is principally confined to the case of a Poissonian atomic beam, though a method of dealing with the non-Markovian master equation obtained for non-Poissonian beams is also examined. A formal theory of atomic detection analogous to the Glauber - Kelly - Kleiner theory of photodetection is proposed, based on a quantum-field description of the atomic beam. This theory is used to establish the connection between the atomic detection record obtained by continuous monitoring of the atoms leaving the cavity and the quantum trajectories followed by the cavity field. Results obtained by other treatments of atomic detection are regained. A classical correlation function obtained from the detection record of ground-state atoms emerging from a micromaser cavity is then shown, by use of a quantum trajectory analysis, to be identical to the cavity field intensity correlation function, apart from shot-noise-type contributions, provided the cavity reservoir is at zero temperature.
We reconsider a variant of the one-atom-maser setup in which the emerging atoms are probed for coherent superpositions of the two Rydberg states of the maser transition. The statistical properties of the corresponding detector clicks contain information about the temporal evolution of the phase of the quantized electromagnetic field inside the resonator. We derive analytical expressions for certain statistical quantities, such as the mean number of successive clicks of the same kind, or the atom-atom and the photon-photon correlation functions. The latter are closely related if the photon-number distribution in steady state is dominated by a single narrow peak. \textcopyright{} 1996 The American Physical Society.