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arXiv:cond-mat/9912217 · Physical Review A

Time-dependent variational analysis of Josephson oscillations in a two-component Bose-Einstein condensate

Chi-Yong Lin, E. J. V. de Passos, Da-Shin Lee

Published Oct 13, 2000• 9 citations
arXivHTMLPDF

Abstract

The dynamics of Josephson-like oscillations between two coupled Bose-Einstein condensates is studied using the time-dependent variational method. We suppose that the quantum state of the condensates is a Gaussian wave-packet which can translate and perform breathing shape oscillations. Under this hypothesis we study the influence of these degrees of freedom on the tunneling dynamics by comparing the full-model with one where these degrees of freedom are ``frozen'' at its equilibrium values. The result of our calculation shows that when the traps are not displaced the two models agree, whereas when they are, the models differ considerably, the former being now closer to its linear approximation.

DOI
10.1103/physreva.62.055603

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The dynamics of Josephson-like oscillations between two coupled Bose-Einstein condensates is studied using the time-dependent variational method. We suppose that the quantum state of the condensates is a Gaussian wave-packet which can translate and perform breathing shape oscillations. Under this hypothesis we study the influence of these degrees of freedom on the tunneling dynamics by comparing the full-model with one where these degrees of freedom are ``frozen'' at its equilibrium values. The result of our calculation shows that when the traps are not displaced the two models agree, whereas when they are, the models differ considerably, the former being now closer to its linear approximation.
We investigate the dynamics of two-component Bose–Einstein condensates, composed of atoms in two distinct hyperfine states, which are linearly coupled by two-photon Raman transitions. The condensate is loaded into a double-well potential. A variety of dynamical behaviour, ranging from regular Josephson oscillations to mixed Rabi–Josephson oscillations and to regimes featuring increasing complexity are described in terms of a reduced Hamiltonian system with four degrees of freedoms, which are the numbers of atoms in each component in the left and right potential wells, whose canonically conjugate variables are phases of the corresponding wavefunctions. Using the system with four degrees of freedom, we study the dynamics of fractional imbalances of the two bosonic components and compare the results to direct simulations of the Gross–Pitaevskii equations describing the bosonic mixture. We perform this analysis when the fractional imbalance oscillates around a zero-time averaged value and in the self-trapping regime as well.
In this work, we investigate the quantum dynamics of a model for two single-mode Bose–Einstein condensates which are coupled via Josephson tunnelling. Using direct numerical diagonalization of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalized and self-trapping phases. We show that these behaviours are dependent on both the initial state of the system and regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail.
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.
We present particular examples of tunneling and transport dynamics of Bose–Einstein condensate (BEC) in a two-well trap: Josephson oscillations (JO), macroscopic quantum self-trapping (MQST) and complete adiabatic transport. Being quite general, these dynamical regimes have close analogies with Josephson junctions in superconductors and transport problems in nanosystems. The calculations are performed within the time-dependent mean-field prescription within (transport) and beyond (JO and MQST) two-mode approximation. For the transport a universal adiabatic population transfer scheme is proposed, allowing a robust and complete transport even under strong nonlinear effects.
The wave packet dynamics in atomic systems and in Bose-Einstein condensates is investigated by means of a time-dependent variational principle. The wave packets are assumed to be parametrized by a set of time-dependent parameters. The time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations, as obtained from the time-dependent variational principle. Unfortunately, the set of equations is ill-behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. A general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well-behaved is presented. The power of the method is demonstrated for a non-integrable system with two degrees of freedom. The Gaussian wave packet (GWP) method is applied to the three-dimensional hydrogen atom. The regularization based on the introduction of Kustaanheimo-Stiefel coordinates and a fictitious time is performed. The regularization implies a transition from the three-dimensional physical position space to a four-dimensional parameter space, spanned by the Kustaanheimo-Stiefel coordinates. The regularization is accompanied by a restriction on physically allowed wave functions. The Coulomb potential is transformed to a harmonic potential and GWPs are the exact solutions, provided they fulfill the restriction. The effect of the restriction on the four-dimensional GWP is discussed and it is shown that the GWPs can satisfy the restriction if the Gaussian parameter space is reduced in a certain way. The exact analytic evolution of the restricted GWP is presented, and the expansion of a localized initial wave function in the restricted Gaussian basis set and its analytic propagation in the fictitious time are shown. Symmetry subspaces with conserved magnetic quantum number m and with conserved angular momentum quantum numbers l,m are treated separately. The method is also applied to the non-integrable H atom in in a homogeneous magnetic field and in perpendicular external electric and magnetic fields. The evolution of the wave packets is determined by the constrained time-dependent variational principle. The numerical results are compared to numerically exact values and show excellent agreement. Another class of systems where the variational Gaussian wave packet method yields good results are cold gases. In this thesis Bose-Einstein condensates are investigated, where in addition to the common short-range contact interaction two different long-range interactions, viz. a laser induced gravity-like 1/r-interaction or a magnetic dipole-dipole interaction are present. The dynamics as resulting from the time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r-interaction is investigated with both the GWP method and numerically exact calculations. It is shown that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses. The dynamics of condensates with inter-atomic magnetic dipole-dipole interaction is investigated in the mean field limit using a Gaussian trial function. The anisotropy of the magnetic dipole-dipole interaction breaks the spherical symmetry and for ordered dipoles an effectively two-dimensional system with cylindrical symmetry is obtained. Special attention is payed to the regularity of the dynamics and it is shown that a transition from regular to chaotic motion takes place with increasing energy where regions of regular and chaotic motion coexist. It is shown that stable modes exist at energies high above the saddle point energy, i.e. at energies where a collapse of the condensate is expected.
We study the dynamics of the relative phase between two atomic Bose-Einstein condensates coupled via collisions and via a Josephson-like coupling. We derive the equations of the motion for the relative phase and the relative number operators from the second-quantized Hamiltonian of the system using a quantum field theoretical approach. We distinguish the cases in which the two condensates are in the same trap or in two different traps and study the influence of this difference on the first-order correlation function of atomic fields. In identical traps this function does not undergo dephasing. We calculate the dephasing time for the case of different traps.