Application of the Lagrangian variational method to a one-dimensional Bose gas in a dimple trap

Abstract

We present an application of the Lagrangian variational method (LVM) to study the smalloscillation 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.

Characterization of the energy level-structure of a trapped dipolar Bose gas via mean-field parametric resonances

“…Other quantities, such as the potential energy, zero-point energy, and the radial size could also have been used here because they reveal the same PRs with the same properties as obtained for E kin . Further, E kin has been used in a number of previous examinations [7,46,[89][90][91][92] that further demonstrate its importance.…”

“…The reasons for the appearance of the PRs at certain values of G 1dd are deeply encrypted in the numerical solutions of the TDGPE and hard to decipher. Therefore, there is no other way to circumvent this problem but to seek qualitative explanations basing on other models and methods, such as LVM [7,[45][46][47][48][49][50][51]97], the WKB approximation, as well as time-dependent and time-independent perturbation theory [98,99]. It should be noted that LVM has been applied earlier to treat the same HO+NGP system as the present one, but without DDI [7].…”

“…In this section, we revisit the frequency of the breathing mode explored earlier in some of our publications [7,46], except that this time the effects of the DDI are added. As before, the breathing-mode frequency can be obtained by a linearization of Eq.…”

“…(i) By a variation of the DDI strength we detect meanfield PRs in the dynamics of a DBEC induced by an NGP whose depth is periodically modulated with time. The PRs obtained are an inherent feature deeply embedded in the physics of the DBEC as they could also be generated by other tools such as the Lagrangian variational method (LVM) [7,[45][46][47][48][49][50][51].…”

Inverse engineering for fast transport and spin control of spin-orbit-coupled Bose-Einstein condensates in moving harmonic traps

“…This Gaussian assumption turns out to be an appropriate choice, particularly for weak interaction g and small number of atoms N [37,38], otherwise the wave function becomes an inverted parabolic shape at strong repulsive interaction, especially N > 600, see [39].…”