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arXiv:cond-mat/9911390 · Physical Review A

Density-matrix approach to a strongly coupled two-component Bose-Einstein condensate

Andal Narayanan, Hema Ramachandran

Published Oct 12, 2000• 2 citations
arXivHTMLPDF

Abstract

The time evolution equations for average values of population and relative phase of a strongly coupled two-component Bose-Einstein condensate (BEC) are derived analytically. The two components are two hyperfine states, which are coupled by an external laser that drives fast Rabi oscillations between these states. Specifically, this derivation incorporates the two-mode model proposed in J. Williams et al., e-print cond-mat 9904399 for the strongly coupled hyperfine states $|1,\ensuremath{-}1〉$ and $|2,1〉$ of ${}^{87}\mathrm{Rb}.$ The fast Rabi cycles are averaged out and the rate equations so derived represent the slow dynamics of the system. These include the collapse and revival of Rabi oscillations and their dependence on detuning and trap displacement as reported in experiments of J. Williams. A procedure for stabilizing vortices is also suggested.

DOI
10.1103/physreva.62.055602

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The time evolution equations for average values of population and relative phase of a strongly coupled two-component Bose-Einstein condensate (BEC) are derived analytically. The two components are two hyperfine states, which are coupled by an external laser that drives fast Rabi oscillations between these states. Specifically, this derivation incorporates the two-mode model proposed in J. Williams et al., e-print cond-mat 9904399 for the strongly coupled hyperfine states ∣1,\ensuremath−1〉|1,\ensuremath{-}1〉∣1,\ensuremath−1〉 and ∣2,1〉|2,1〉∣2,1〉 of 87Rb.{}^{87}\mathrm{Rb}.87Rb. The fast Rabi cycles are averaged out and the rate equations so derived represent the slow dynamics of the system. These include the collapse and revival of Rabi oscillations and their dependence on detuning and trap displacement as reported in experiments of J. Williams. A procedure for stabilizing vortices is also suggested.
The equilibrium and stability properties of a coupled two-component BEC is studied using a variational method and the one-dimensional model of Williams and collaborators. The variational parameters are the population fraction, translation and scaling transformation of the condensate densities, assumed to have a Gaussian shape. We study the equilibrium and stability properties as a function of the strength of the laser field and the traps displacement. We find many branches of equilibrium configurations, with a host of critical points. In all the cases, the signature of the onset of criticality is the collapse of a normal mode which is a linear combination of the out-of-phase translation and an in-phase breathing oscillation of the condensate densities. Our calculations also indicate that we have symmetry breaking effects when the traps are not displaced.
In terms of exact solutions of the time-dependent Schr\"odinger equation for an effective giant spin modeled from a coupled two-mode Bose-Einstein condensate (BEC) with adiabatic and cyclic time-varying Raman coupling between two hyperfine states of the BEC, we obtain analytic time-evolution formulas of the population imbalance and relative phase between two components with various initial states, especially the SU(2) coherent state. We find the Berry phase depending on the number parity of atoms, and particle number dependence of the collapse revival of population-imbalance oscillation. It is shown that self-trapping and phase locking can be achieved from initial SU(2) coherent states with proper parameters.
Summary form only given. Recent elegant experimental observations of Rabi-type oscillations in a two-component Bose-Einstein condensate (BEC) of 87Rb brings up an issue of collapse and revival in BEC which has resisted any conventional theoretical analysis. Though a heuristic argument using "twists" of an SU(2) order parameter was given in, the fundamental dynamics of the observed oscillations has not been understood. In this talk, we present a resolution to this problem in terms a new model of BEC dynamics. We go beyond revival analysis to present a new quantum mechanical picture of the multicomponent BEC physics based on a strong confinement model, which explains experimentally observed features quite well. We show that the coupled Gross-Pitaevski (GP) equations for the two-component Bose-Einstein condensate can be decoupled by a time-dependent dressing transformation.
Two internal hyperfine states of a Bose-Einstein condensate in a dilute magnetically trapped gas of 87Rb{}^{87}\mathrm{Rb}87Rb atoms are strongly coupled by an external field that drives Rabi oscillations between the internal states. Due to their different magnetic moments and the force of gravity, the trapping potentials for the two states are offset along the vertical axis, so that the dynamics of the internal and external degrees of freedom are inseparable. The rapid cycling between internal atomic states in the displaced traps results in an adiabatic transfer of population from the condensate ground state to its first antisymmetric topological state. This has a pronounced effect on the internal Rabi oscillations, modulating the fringe visibility in a manner reminiscent of collapses and revivals. We present a detailed theoretical description based on zero-temperature mean-field theory.
Using the numerical calculation,we have investigated the long-time evolution of atomicity density of two-component BEC,and have found that the evolution of condensate density in long time,which presents a typical quantumn collapse and revival with a complex period,is rather different from that in short time,which displays a regular cosine oscillation.Futhermore,we have also discussed the influence of the Rabi frequency of coupling drive field,various values for the relative phase,the strength of the coupling drive,the strength of the interatomic interaction in each condensate and the strength of the interspecies interaction on the evolution properties of atomicity density.
Bose-Einstein condensates present to us the opportunity to probe into the atomic interac- tions that govern a macroscopic quantum mechanical system. The degenerate hyperfine manifold in the bosonic atoms splits in the presence of an external B-field; radio-frequency induced coupling releases experimentally-attainable knowledge about the Zeeman mani- fold in 87Rb. The purpose of this study is to investigate quantum dynamics in two and three-component Bose-Einstein condensate (BEC) systems. We start with a theoretical analysis of Josephson tunneling dynamics between identical BECs trapped in a double potential well, paying particular attention to the non-linear self-trapping effect observed as a consequence of the intra-well interaction. We present a model for the non-equilibrium dynamics in a two-level system and introduce the Rabi oscillations. This is followed by a numerical and experimental investigation of Rabi oscillations in a three-level 87Rb BEC between the F = 1 hyperfine level spin states mF = {1, 0, 1}. The relation between the observed total Rabi frequency and the detuning, along with its effect on the BEC population dynamics is explored. Finally, we explore the possible suspects for the shift of the resonance at strong Rabi frequencies .