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Physical Review A

Theory of input and output of atoms from an atomic trap

J. J. Hope

Published Apr 1, 1997• 72 citations
Publisher

Abstract

We consider the interaction between a single-mode atomic trap and a set of external modes that are decribed by a continuum of eigenstates. This theory is designed to model output coupling from current and proposed Bose-Einstein condensation experiments and proposed models for atom lasers. This work generalizes the optical input-output formalism [M. J. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984)].

DOI
10.1103/physreva.55.r2531

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•Jan 1, 1999•0 citations
We consider the interaction between a single-mode atomic trap and a set of external modes that are decribed by a continuum of eigenstates. This theory is designed to model output coupling from current and proposed Bose-Einstein condensation experiments and proposed models for atom lasers. This work generalizes the optical input-output formalism [M. J. Collett and C. Gardiner, Phys. Rev. A 30, 1386 (1984)].
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 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 a general theory of adiabatic output coupling from trapped, weakly interacting, atomic Bose-Einstein condensates at finite temperatures. For weak coupling, the output rate from the condensate and the excited levels in the trap settles in a time proportional to the inverse of the spectral width of the coupling to the output modes. We discuss the properties of the output atoms in the quasi-steady-state, where the population inside the trap is not appreciably depleted. We show how the composition of the output beam, containing the condensate and the thermal component, may be controlled by changing the frequency of the output coupling lasers. This composition determines the first- and second-order coherence of the output beam. We discuss the changes in the composition of the Bose gas left in the trap, and show how nonresonant output coupling can stimulate either the evaporation of the thermal excitations in the trap or the growth of the nonthermal excitations, when pairs of correlated atoms leave the condensate.
Summary form only given. Hope, Savage and Moy have presented a model Hamiltonian for an atom mirror. In conjunction with a Bose-Einstein condensate such a device can model the output coupler for an atom laser. The basic structure of the Hamiltonian is identical to the optical output coupler. A cavity mode at a particular wavevector is coupled to a set of output modes over a range of wavevectors. The difference is that the parameter in the Hamiltonian which would be the frequency in the optical case is the square of the wavevector for atoms. This has the consequence that the cavity mode is only coupled to one of the two superposition (sum and difference) modes of the output. We present a Fano diagonalisation of this output coupler model. The Hamiltonian can be expressed in terms of the diagonal operators together with operators representing a bound mode. This mode appears when the coupling is strong compared to the energy of the cavity mode. Any states of the bound mode represent excitations which will not propagate out of the cavity. We give the criterion for the existence of the mode in the cases where the interaction strengths are real and complex.
We generalize the quantum optical input-output theory developed for optical cavities to ultracold fermionic atoms confined in a trapping potential, which forms an ``atom cavity.'' In order to account for the Pauli exclusion principle, quantum Langevin equations for all cavity modes are derived. The dissipative part of these multimode Langevin equations includes a coupling between cavity modes. We also derive a set of boundary conditions for the Fermi field that relate the output fields to the input fields and the field radiated by the cavity. Starting from a constant uniform current of fermions incident on one side of the cavity, we use the boundary conditions to calculate the occupation numbers and current density for the fermions that are reflected and transmitted by the cavity.
In this thesis a number of aspects on possible manipulation of Bose-Einstein condensate in trapped atomic gases is investigated. First, a model for atom optical experiments involving Bose condensates is proposed and numerical simulations are presented to illustrate its characteristics. We demonstrate ways of focusing and splitting the condensate by modifying experimentally adjustable parameters. We show that there are at least two ways of implementing atom optical elements: one may modulate the interatomic scattering length in space, or alternatively, use a sinusoidal, externally applied potential. The temporal evolution of quasiparticle excitations is studied via the Gross-Pitaevskii Equation. Nonlinear mode mixing of quasiparticles is introduced, and is observed using a quasiparticle projection method. This is used as a basis of time-dependent finite temperature simulations, which we argue to be valid under regimes of high occupation number. An illustration via a closely related evaporative cooling simulation is provided. A phenomenological damping formalism for superfluidity near the λ point is adopted to describe the damping of excitations in a Bose-Einstein condensate. An estimate for the damping parameter is found. The damping formalism as a numerical tool to calculate the ground eigenstate of the condensate is explored. A novel, experimentally realisable interferometry for Bose-Einstein condensates using near-field diffraction is proposed. The scheme is based on the phenomenon of intermode traces or quantum carpets; we demonstrate the structured spatio-temporal pattern for the dilute, atomic Bose-Einstein condensate. The pattern is found to change with temperature, which allows us to perform interferometric temperature measurements. Finally, an output coupler for Bose-Einstein Condensates based on stimulated Raman transition is investigated. The spectrum and coherence are calculated for an atomic beam slowly coupled out of a trap containing a partially condensed Bose gas at finite temperatures. A number conserving Hartree-Fock-Bogliubov formalism has been used to incorporate finite temperature effects. Various different processes are found to become dominant for a suitable choice of the coupling parameters.