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arXiv:quant-ph/9801046 · Physical Review A

Born and Markov approximations for atom lasers

G. M. Moy, J. J. Hope, Craig M. Savage

Published Jan 1, 1999• 91 citations
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Abstract

We discuss the use of the Born and Markov approximations in describing the dynamics of an atom laser. In particular, we investigate the applicability of the quantum optical Born-Markov master equation for describing output coupling. We derive conditions based on the atomic reservoir and atom dispersion relations for when the Born-Markov approximations are valid and discuss parameter regimes where these approximations fail in our atom laser model. Differences between the standard optical laser model and the atom laser are due to a combination of factors, including the parameter regimes in which a typical atom laser would operate, the different reservoir state that is appropriate for atoms, and the different dispersion relations between atoms and photons. We present results based on an exact method in the regimes in which the Born-Markov approximation fails. The exact solutions in some experimentally relevant parameter regimes give a nonexponential loss of atoms from a cavity.

DOI
10.1103/physreva.59.667

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We discuss the use of the Born and Markov approximations in describing the dynamics of an atom laser. In particular, we investigate the applicability of the quantum optical Born-Markov master equation for describing output coupling. We derive conditions based on the atomic reservoir and atom dispersion relations for when the Born-Markov approximations are valid and discuss parameter regimes where these approximations fail in our atom laser model. Differences between the standard optical laser model and the atom laser are due to a combination of factors, including the parameter regimes in which a typical atom laser would operate, the different reservoir state that is appropriate for atoms, and the different dispersion relations between atoms and photons. We present results based on an exact method in the regimes in which the Born-Markov approximation fails. The exact solutions in some experimentally relevant parameter regimes give a nonexponential loss of atoms from a cavity.
We discuss the use of the Born and Markov approximations in describing the dynamics of an atom laser. In particular, we investigate the applicability of the quantum optical Born-Markov master equation for describing output coupling. We derive conditions based on the atomic reservoir, and atom dispersion relations for when the Born-Markov approximations are valid and discuss parameter regimes where these approximations fail in our atom laser model. Differences between the standard optical laser model and the atom laser are due to a combination of factors, including the parameter regimes in which a typical atom laser would operate, the different reservoir state which is appropriate for atoms, and the different dispersion relations between atoms and photons. We present results based on an exact method in the regimes in which the Born-Markov approximation fails. The exact solutions in some experimentally relavent parameter regimes give non-exponential loss of atoms from a cavity.
The dynamics of atom lasers with a continuous output coupler based on two-photon Raman transitions is investigated. With the help of the time-convolutionless projection operator technique the quantum master equations for pulsed- and continuous-wave (cw) atom lasers are derived. In the case of the pulsed atom laser the power of the time-convolutionless projection operator technique is demonstrated through comparison with the exact solution. It is shown that in an intermediate coupling regime where the Born-Markov approximation fails, the results of this algorithm agree with the exact solution. To study the dynamics of a continuous-wave atom laser a pump mechanism is included in the model. Whereas the pump mechanism is treated within the Born-Markov approximation, the output coupling leads to non-Markovian effects. The solution of the master equation resulting from the time-convolutionless projection operator technique exhibits strong oscillations in the occupation number of the Bose-Einstein condensate. These oscillations are traced back to a quantum interference which is due to the non-Markovian dynamics and which decays slowly in time as a result of the dispersion relation for massive particles.
The regions of validity of the Markov approximation for the coupling of atoms out of an atomic trap are determined. We consider radio-frequency output coupling in the presence of gravity and collisional repulsion, and Raman output coupling. The Markov approximation is crucial in most theoretical descriptions of an atom laser that assume a continuous process of output coupling from a trapped Bose-Einstein condensate. In this regime many techniques that have proved to be useful for modeling the optical laser, such as master equations, can be used to describe the dynamics of the damping of the condensate mode undergoing output coupling.
A two level atom driven by a strong resonant field and coupled to a structured reservoir is studied. The generalized master equation describing the evolution of the atom is derived under the Born-Markov approximation. To derive the master equation the dressing transformation on the atomic operators is performed first and next the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The modifications introduced by a strong field and/or by the reservoir with non-flat density of modes lead to non-standard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs although the reservoir is not a squeezed vacuum. The optical Bloch equations based on this generalized master equation are obtained and solved for the steady state. The resonance fluorescence spectra of the atom are shortly discussed.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
We present a steady-state analysis of a quantum-mechanical model of an atom laser. A single-mode atomic trap coupled to a continuum of external modes is driven by a saturable pumping mechanism. In the dilute flux regime, where atom-atom interactions are negligible in the output, we find an analytic form for the linewidth and frequency shift of the laser. This result does not make the Born-Markov approximation, but is based on the far less restrictive ``self-consistent Markov approximation.'' The more exact treatment has a different effective damping rate and occupation of the lasing mode, as well as a shifted frequency and linewidth of the output. We examine gravitational damping numerically, finding linewidths and frequency shifts for a range of pumping rates. We treat mean-field damping analytically, finding a memory function for the Thomas-Fermi regime. The occupation and linewidth are found to have a nonlinear scaling behavior which has implications for the stability of atom lasers.
We present a fully quantum-mechanical treatment of a single-mode atomic cavity with a pumping mechanism and an output coupling to a continuum of external modes. This system is a schematic description of an atom laser. In the dilute limit where atom-atom interactions are negligible, we have been able to solve this model without making the Born and Markov approximations. When coupling into free space, it is shown that for reasonable parameters there is a bound state which does not disperse, which means that there is no steady state. This bound state does not exist when gravity is included, and in that case the system reaches a steady state. We develop equations of motion for the two-time correlation in the presence of pumping and gravity in the output modes. We then calculate the energy spectrum of the steady-state output flux from the laser. We present a ``self-consistent Markov approximation'' which allows efficient calculation of this energy spectrum well above threshold.