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Physica B Condensed Matter

On the harmonic oscillator wavefunction modified to account for variations in the width of a trapped Bose gas

Asaad R. Sakhel, Roger R. Sakhel, Saleem I. Qashou

Published Jan 14, 2015• 0 citations
Publisher
DOI
10.1016/j.physb.2015.01.009

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Abstract We present an application of the Lagrangian variational method (LVM) to study the small-oscillation dynamics and ground-state properties of an interacting Bose gas in a one-dimensional dimple trap, for the first time. For this purpose, a Gaussian wavefunction ansatz—rich in variational parameters—is used for the analytical solutions of the equations of motion resulting from the LVM. The results of LVM are found to agree well with those due to the numerical Crank–Nicolson method. The strength of this method is shown to reveal details hidden in the wavefunction and its dynamics that are otherwise hard to obtain numerically or even experimentally. In this regard, the important role that the wavefunction-width plays in determining the properties of a trapped Bose gas is manifested. Via this method, the dependence of the breathing-mode frequency on the interactions, width of wavefunction, and dimple trap parameters, is demonstrated. A significant result is that in the ‘width-space’ of the wavefunction, the LVM yields a differential equation that describes the motion of a fictitious particle in an effective potential whose shape is by analogy similar to that of an interatomic interaction potential.
The static fluctuation approximation (SFA) is applied to compute the thermodynamic properties of a trapped two-dimensional (2D) interacting hard-sphere (HS) Bose gas in the weakly and strongly interacting regime. A mean-field approach involving a variational wave function is used to compute the mean-field energy as a function of temperature for each harmonic oscillator (HO) state plugged into the SFA technique. In the variational approach, a parameter $\ensuremath{\alpha}$ is introduced into the harmonic oscillator wave function in order to take into account the changes in the width when the repulsive interactions between the bosons are increased. In the weakly interacting regime, below the critical temperature, the total energy of all HO states (evaluated by our model) matches the noninteracting result very well. However, beyond the critical temperature, we ``fit'' our energies to the classical limit for 2D bosons in a trap by using a suitably proposed weighting function. We compare our results to earlier results of mean-field theory. Further, we evaluate the density matrix arising from correlations between the HO orbitals.
Motivated by a recent experiment (Catani J. et al. , Phys. Rev. A , 85 (2012) 023623 ) we study breathing oscillations in the width of a harmonically trapped impurity interacting with a separately trapped Bose gas. We provide an intuitive physical picture of such dynamics at zero temperature, using a time-dependent variational approach. The amplitudes of breathing oscillations are suppressed by self-trapping, due to interactions with the Bose gas. Further, exciting phonons in the Bose gas leads to damped oscillations and non-Markovian dynamics of the width of the impurity, the degree of which can be engineered through controllable parameters. Our results, supported by simulations, reproduce the main features of the dynamics observed by Catani et al. despite the temperature of that experiment. Moreover, we predict novel effects at lower temperatures due to self-trapping and the inhomogeneity of the trapped Bose gas.