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arXiv:quant-ph/9906125 · Physical Review A

Atom lasers, coherent states, and coherence. I. Physically realizable ensembles of pure states

Howard M. Wiseman, John A. Vaccaro

Published Mar 20, 2002• 8 citations
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Abstract

A laser, be it an optical laser or an atom laser, is an open quantum system that produces a coherent beam of bosons (photons or atoms, respectively). Far above threshold, the stationary state ${\ensuremath{\rho}}_{\mathrm{ss}}$ of the laser mode is a mixture of coherent-field states with random phase, or, equivalently, a Poissonian mixture of number states. This paper answers the question: can descriptions such as these, of ${\ensuremath{\rho}}_{\mathrm{ss}}$ as a stationary ensemble of pure states, be physically realized? Here physical realization is as defined previously by us [H. M. Wiseman and J. A. Vaccaro, Phys. Lett. A 250, 241 (1998)]: an ensemble of pure states for a particular system can be physically realized if, without changing the dynamics of the system, an experimenter can (in principle) know at any time that the system is in one of the pure-state members of the ensemble. Such knowledge can be obtained by monitoring the baths to which the system is coupled, provided that coupling is describable by a Markovian master equation. Using a family of master equations for the (atom) laser, we solve for the physically realizable (PR) ensembles. We find that for any finite self-energy $\ensuremath{\chi}$ of the bosons in the laser mode, the coherent-state ensemble is not PR; the closest one can come to it is an ensemble of squeezed states. This is particularly relevant for atom lasers, where the self-energy arising from elastic collisions is expected to be large. By contrast, the number-state ensemble is always PR. As the self-energy $\ensuremath{\chi}$ increases, the states in the PR ensemble closest to the coherent-state ensemble become increasingly squeezed. Nevertheless, there are values of $\ensuremath{\chi}$ for which states with well-defined coherent amplitudes are PR, even though the atom laser is not coherent (in the sense of having a Bose-degenerate output). We discuss the physical significance of this anomaly in terms of conditional coherence (and hence conditional Bose degeneracy).

DOI
10.1103/physreva.65.043605

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A laser, be it an optical laser or an atom laser, is an open quantum system that produces a coherent beam of bosons (photons or atoms, respectively). Far above threshold, the stationary state \ensuremathρss{\ensuremath{\rho}}_{\mathrm{ss}}\ensuremathρss​ of the laser mode is a mixture of coherent-field states with random phase, or, equivalently, a Poissonian mixture of number states. This paper answers the question: can descriptions such as these, of \ensuremathρss{\ensuremath{\rho}}_{\mathrm{ss}}\ensuremathρss​ as a stationary ensemble of pure states, be physically realized? Here physical realization is as defined previously by us [H. M. Wiseman and J. A. Vaccaro, Phys. Lett. A 250, 241 (1998)]: an ensemble of pure states for a particular system can be physically realized if, without changing the dynamics of the system, an experimenter can (in principle) know at any time that the system is in one of the pure-state members of the ensemble. Such knowledge can be obtained by monitoring the baths to which the system is coupled, provided that coupling is describable by a Markovian master equation. Using a family of master equations for the (atom) laser, we solve for the physically realizable (PR) ensembles. We find that for any finite self-energy \ensuremathχ\ensuremath{\chi}\ensuremathχ of the bosons in the laser mode, the coherent-state ensemble is not PR; the closest one can come to it is an ensemble of squeezed states. This is particularly relevant for atom lasers, where the self-energy arising from elastic collisions is expected to be large. By contrast, the number-state ensemble is always PR. As the self-energy \ensuremathχ\ensuremath{\chi}\ensuremathχ increases, the states in the PR ensemble closest to the coherent-state ensemble become increasingly squeezed. Nevertheless, there are values of \ensuremathχ\ensuremath{\chi}\ensuremathχ for which states with well-defined coherent amplitudes are PR, even though the atom laser is not coherent (in the sense of having a Bose-degenerate output). We discuss the physical significance of this anomaly in terms of conditional coherence (and hence conditional Bose degeneracy).
As discussed in the preceding paper [Wiseman and Vaccaro, preceding paper, Phys. Rev. A 65, 043605 (2002)], the stationary state of an optical or atom laser far above threshold is a mixture of coherent field states with random phase, or, equivalently, a Poissonian mixture of number states. We are interested in which, if either, of these descriptions of ${\ensuremath{\rho}}_{\mathrm{ss}}$ as a stationary ensemble of pure states, is more natural. In the preceding paper we concentrated upon the question of whether descriptions such as these are physically realizable (PR). In this paper we investigate another relevant aspect of these ensembles, their robustness. A robust ensemble is one for which the pure states that comprise it survive relatively unchanged for a long time under the system evolution. We determine numerically the most robust ensembles as a function of the parameters in the laser model: the self-energy $\ensuremath{\chi}$ of the bosons in the laser mode, and the excess phase noise $\ensuremath{\nu}.$ We find that these most robust ensembles are PR ensembles, or similar to PR ensembles, for all values of these parameters. In the ideal laser limit $(\ensuremath{\nu}=\ensuremath{\chi}=0),$ the most robust states are coherent states. As the phase noise or phase dispersion is increased through $\ensuremath{\nu}$ or the self-interaction of the bosons $\ensuremath{\chi},$ respectively, the most robust states become more and more amplitude squeezed. We find scaling laws for these states, and give analytical derivations for them. As the phase diffusion or dispersion becomes so large that the laser output is no longer quantum coherent, the most robust states become so squeezed that they cease to have a well-defined coherent amplitude. That is, the quantum coherence of the laser output is manifest in the most robust PR ensemble being an ensemble of states with a well-defined coherent amplitude. This lends support to our approach of regarding robust PR ensembles as the most natural description of the state of the laser mode. It also has interesting implications for atom lasers in particular, for which phase dispersion due to self-interactions is expected to be large.
We present a many-body theory of a driven and damped trapped gas of interacting bosons, and demonstrate that one of the trap levels can become coherently populated, thereby leading to a coherent atomic-beam generator, or ``laser for atoms.'' The specific system we consider consists of a sample of bosonic atoms interacting via the near-resonant dipole-dipole interaction. The transverse center-of-mass motion of the atoms is confined by a two-dimensional potential well created by an array of cooling laser beams, while their longitudinal motion is quantized by a Fabry-P\'erot for atoms. Under appropriate conditions, the dipole-dipole selection rules lead to the simplification that only two quantized levels of atomic motion need to be considered explicitly, the other levels being treated as reservoirs. One of the two levels is the ``pump level,'' while the other is the one where atomic coherence builds up (the ``lasing'' level). The master equation describing the dynamics of these levels can be solved numerically, and its solution exhibits a ``threshold behavior'' with a transition from super-Poissonian to Poissonian atom statistics in the ``lasing mode.'' \textcopyright{} 1996 The American Physical Society.
One-atom lasers are important because their governing equations can be solved exactly, even with a quantized field. We present a fully quantum-mechanical treatment of one-atom lasers modeled by quantum-optical master equations. These are solved numerically without any significant approximations. We show that laser action is possible with one atom, and that it might be achievable experimentally. Laser action is characterized by the dominance of stimulated emission over spontaneous emission. We use the one-atom laser model to investigate, without approximation, some interesting generic laser phenomena. Under certain conditions lasers produce intensity squeezed light, and then the laser linewidth increases with the pumping rate, in contrast with standard lasers. We also report ``self-quenching'' behavior: lasers with incoherent pumping out of the lower laser level turn off when the pumping is sufficiently fast because the coherence between the laser levels is destroyed.
Coherent states are normally used to describe the state of a laser field in experiments that generate and detect squeezed states of light. Nevertheless, since the laser field absolute phase is unknown, its quantum state can be described by a statistical mixture of coherent states with random phases, which is equivalent to a statistical mixture of Fock states. Here we describe single-mode squeezed vacuum experiments using this mixed quantum state for the laser field. Representing the laser state in the Fock basis, we predict the usual experimental results without using the squeezing concept in the analysis and concluding that no squeezed state is generated in the experiments. We provide a general physical explanation for the noise reduction in the experiments in terms of a better definition of the relative phase between the signal and local oscillator fields. This explanation is valid in any description of the laser field (in terms of coherent or Fock states), thus providing a deeper understanding of the phenomenon.
It was proposed by two of us (Wiseman H M and Collett M J 1995 Phys. Lett. A 202 246) that a device producing a coherent atomic wave (that is, an atom laser) could be built using dark-state cooling of atoms in a trap. Here we present a more complete analysis of this system, including the effects of atomic interactions through dipole-dipole coupling and hard-core collisions. We show that for reasonable physical parameters the dipole interactions have little contribution to the atomic dynamics. In contrast, the hard-core collisions are likely to dominate the laser linewidth. In fact, it would appear necessary to select (or produce) a species with a hard-core scattering radius of much less than one nm in order for the output of the device to be phase coherent.
The stationary solution ρ of a quantum master equation can be represented as an ensemble of pure states in a continuous infinity of ways. Here we suggest a definition of the the “most natural” ensemble. The first requirement is that the ensemble be physically realizable through monitoring the system’s environment. We call this an ’unraveling’ of the master equation. The second requirement is that it be the most robust of such ensembles. Robustness of an ensemble is defined in terms of the survival probability S(t) which is the average probability for one of its elements to remain unchanged after a time t . The most natural ensemble can thus be said to be the one produced by the maximally robust unraveling. An application of interest is the atom laser. We show that the nature of the maximally robust unraveling is closely related to the question of whether the output beam satisfies the conditions required to be a true laser beam.