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arXiv:cond-mat/9810413 · Physical Review A

Markov approximation for the atomic output coupler

Michael W. Jack, M. Naraschewski, Matthew J. Collett, Daniel F. Walls

Published Apr 1, 1999• 29 citations
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Abstract

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.

DOI
10.1103/physreva.59.2962

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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 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.
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.
A linear quantum dynamic theory for output coupler of Bose–Einstein condensed atoms in a trap is considered with the Bogoliubov approximation in the thermodynamical limit based on the recent MIT experiment ( Phys. Rev. Lett. 78 (1997) 582) for atomic laser. In evolution of total system, the solution of the many-body problem shows a factorization of dynamic process, i.e., the wave function initially prepared in a direct product of a vacuum state and a coherent state remains in a direct product of coherent states at any instance. Physically, this factorizable structure predicts that an ideal condensate in the trap will remain in such a condensate state after the radiation frequency interaction while the output-coupler pulse of atoms forms a macroscopic quantum state in a propagating mode, i.e., the atomic laser.
We develop the quantum-mechanical description of output coupling of macroscopic coherent matter waves from a Bose-Einstein condensate (BEC) via a radio-frequency field in the pulsed and cw limits for both strong and weak field coupling. The theory is converted into a Fock state description to point out the stimulated (in Bose particle) nature of the output coupling. A useful analogy with the theory of molecular photodissociation is used to explain the various regimes of output coupling BECs using radiation sources. We present specific calculations of the rates of output coupling as a function of the number of Bose atoms in the condensate and the frequency, detuning, and intensity of the radio-frequency field.
Atom lasers have been demonstrated in both pulsed and semi-continuous modes. The dynamics of these devices is predicted to be dramatically different to the optical laser due to fundamental differences between the two types of boson fields, however to date it is the similarities that have been emphasized. Here we show how the atom laser dynamics are strongly affected by coupling of the output beam back to the condensate. Typically research in the field of Bose-Einstein condensation (BEC) 1 relates to studies of fundamental properties, such as collective modes 2 or to the response of the BEC to external stimuli 3. In contrast, the atom laser is one of the most promising technologies to arise from BEC, being the di-rect atomic equivalent of the optical laser 4. Mewes et al. 5 demonstrated the first atom laser, based on the application of pulsed radio-frequency (RF) fields to induce controlled spin flips from magnetically trapped to un-trapped states of a Bose-Einstein condensate. Later it was shown by Hagley et al. 6 that a pulsed Raman out-coupling could be used to achieve a quasi-continuous multi-state atomic beam. Bloch et al. 7 achieved con-tinuous RF out-coulping for up to 100ms, producing a single state atom laser beam, and showed that this beam could be coherently manipulated in direct analogy to the optical laser 8. These experiments, and others 9,10, have conclusively demonstrated that the outcoupling process can be made coherent. They have also demonstrated a number of other similarities be-tween the optical and atomic lasers such as high flux and low divergence of the output beam. In all these works, the similarities between the two sys-tems have been emphasized, however, ultimately it is the differences such as mass and wavelength that will lead to applications of the atom laser. 1