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arXiv:physics/0108052 · Physical Review A

Collective dynamics of internal states in a Bose-Einstein gas

Mehmet Özgür Oktel, Leonid S Levitov

Published May 30, 2002• 14 citations
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Abstract

A theory for the Rabi and internal Josephson effects in an interacting, homogeneous Bose-Einstein gas in the cold collision regime is presented. By using the microscopic transport equation for the density matrix the problem is mapped onto a problem of precession of two coupled classical spins. In the absence of an external excitation field our results agree with the theory of density induced frequency shifts in atomic clocks. In the presence of the external field, the internal Josephson effect occurs in a condensed Bose-Einstein gas as well as in a noncondensed gas. The crossover from Rabi oscillations to Josephson oscillations as a function of interaction strength is studied in detail.

DOI
10.1103/physreva.65.063604

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A theory for the Rabi and internal Josephson effects in an interacting, homogeneous Bose-Einstein gas in the cold collision regime is presented. By using the microscopic transport equation for the density matrix the problem is mapped onto a problem of precession of two coupled classical spins. In the absence of an external excitation field our results agree with the theory of density induced frequency shifts in atomic clocks. In the presence of the external field, the internal Josephson effect occurs in a condensed Bose-Einstein gas as well as in a noncondensed gas. The crossover from Rabi oscillations to Josephson oscillations as a function of interaction strength is studied in detail.
We consider the Josephson effect between two spatially separated Bose-Einstein condensates of atoms each of which can be in two hyperfine states. We derive simple equations of motion for this system closely analogous to the Bloch equations. We also map the dynamics of the system onto those of a classical particle in a well. We find density and spin modes of oscillation and stable equilibrium points of the motion that are unstable in the spinless case. Finally we analyze the oscillation modes in the spin-1 (F=1)(F=1)(F=1) case.
The study of physical properties of Bose condensed gases has recently become a very active topic due to the experimental realization of Bose-Einstein condensation in the magnetically trapped gases of alkali atoms. In this paper we theoretically study the Rabi oscillation of a weakly interacting Bose gas in the regime of microwave electronic spin resonance. For a Bose gas above the critical temperature for Bose-Einstein condensation, the microwave radiation induced by the driving microwave field exhibits the ordinary Rabi oscillation damped by the inhomogeneous Doppler dephasing. For a Bose gas with a condensate, the microwave radiation is composed of two components. One is the strong coherent Rabi oscillation from the condensate. The other is a modulated Rabi oscillation due to the noncondensed part of the gas. We show that the modulation of Rabi oscillation of the noncondensed gas is directly relative to the elementary collective excitation of atoms in the gas.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
We calculate the frequencies and damping rates of the low-lying collective modes of a Bose-Einstein condensed gas at nonzero temperature. We use a complex nonlinear Schr\"odinger equation to determine the dynamics of the condensate atoms, and couple it to a Boltzmann equation for the noncondensate atoms. In this manner we take into account both collisions between noncondensate-noncondensate and condensate-noncondensate atoms. We solve the linear response of these equations, using a time-dependent Gaussian trial function for the condensate wave function and a truncated power expansion for the deviation function of the thermal cloud. As a result, our calculation turns out to be characterized by two dimensionless parameters proportional to the noncondensate-noncondensate and condensate-noncondensate mean collision times. We find in general quite good agreement with experiment, both for the frequencies and damping of the collective modes.
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We theoretically study the internal Josephson effect, which is driven by spin-exchange interactions and magnetic dipole-dipole interactions, in a three-level system for spin-1 Bose--Einstein condensates, obtaining novel spin dynamics. We introduce single spatial mode approximations into the Gross--Pitaevskii equations and derive the Josephson-type equations, which are analogous to tunneling currents through three junctions between three superconductors. From an analogy with two interacting nonrigid pendulums, we identify unique varied oscillational modes, called the $0--\ensuremath{\pi}$, 0−−running0--\mathit{running}0−−running, running−−running\mathit{running}--\mathit{running}running−−running, $2n\ensuremath{\pi} & \mathit{running}--2\ensuremath{\pi}$, singlenonrigidpendulum\mathit{single nonrigid pendulum}singlenonrigidpendulum, and tworigidpendulums\mathit{two rigid pendulums}tworigidpendulums phase modes. These Josephson modes in the three states are expected to be found in real atomic Bose gas systems.