The Breakdown of the Born and Markov approximations for atom lasers

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 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.

arXiv:quant-ph/9801046Physical Review AJan 1, 199991 citations

Non-Markovian dynamics in pulsed- and continuous-wave atom lasers

Heinz‐Peter Breuer, Daniel Faller, Bernd Kappler +1 more

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 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.

arXiv:quant-ph/9906119Physical Review AOct 1, 199928 citations

Markov approximation for the atomic output coupler

Michael W. Jack, M. Naraschewski, Matthew J. Collett +1 more

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.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.

arXiv:cond-mat/9810413Physical Review AApr 1, 199929 citations

Exact steady-state solution of the quantum-mechanical single-mode laser model

V. Dohm

The fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function $p(n,m)$ to find $n$ photons in the light field and $m$ atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.The fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function $p(n,m)$ to find $n$ photons in the light field and $m$ atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.

Physical Review AJul 1, 19769 citations

Steady-state quantum statistics of a non-Markovian atom laser. II

Ashton S. Bradley, J. J. Hope, Matthew J. Collett

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 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.

arXiv:cond-mat/0210688Physical Review ADec 22, 20037 citations

Two-level atom in a structured reservoir

A. Kowalewska-Kudłaszyk, R. Tanaś

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.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.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEMar 7, 20010 citations

p(n,m)

to find

n

photons in the light field and

m

atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.