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GROUND STATE ENERGY OF ONE-DIMENSIONAL MACROSCOPIC QUANTUM OSCILLATOR WITH GRAVITATIONAL POTENTIAL

Teguh Budi Prayitno

Published May 1, 2012• 0 citations
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Abstract

We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.

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Current Science•Dec 1, 2015•4 citations
We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.
We have derived some excitation states of the one-dimensional Gross-Pitaevskii equation coupled by the gravitational potential. The methods that we have used here are taken by pursuing the recent work of Kivshar et. al. by considering the equation as a macroscopic quantum oscillator. To obtain the states, we have made the appropriate transformation to reduce the three-dimensional Gross-Pitaevskii equation into the one-dimensional Gross-Pitaevskii equation and applying the time-independent perturbation theory in the general solution of the one-dimensional Gross-Pitaevskii equation as a linear superposition of the normalized eigenfunctions of the Schrödinger equation for the harmonic oscillator potential. Moreover, we also impose the condition by assuming that some terms in the equation should be so small in order to preserve the use of the perturbation method.
Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 We have discussed the partition function of the Bose-Einstein condensation in parabolic trap associated to the one-dimensional Gross-Pitaevskii equation. The partition function itself is constructed by considering all the energy levels of the macroscopic quantum oscillator which is similar to statistical mechanics. The solutions of the energy levels for this case can be derived by pursuing the method proposed by Kivshar et al. that applies the time-independent perturbation theory. In their paper, they demonstrated that the one-dimensional Gross Pitaevskii equation can be treated as the one-dimensional macroscopic quantum oscillator on condition that the nonlinearity is very small. Moreover, they found the analytical expression for the ground state energy by applying the method. However, the higher level states were not explicitly provided. In this research we followed up on their former work to derive explicitly the other states in order to formulate the partition function. However, we did not find the closed form of the partition function since the results of nonlinear term integral could not form the recursion relation. As a consequence, not only should the partition function but also the Helmholtz free energy and entropy should be reevaluated to check their convergences. Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 Keywords: Gross-Pitaevskii equation, partition function, quantum oscillator, thermodinamical properties.
Abstract Considering a set of condensates as N indistinguishable noninteracting quantum oscillators and replacing an available volume with a harmonic volume, we derived an ideal gas model by applying the canonical partition function via the thermodynamic relation. In this case, we take a longitudinal part of the three-dimensional Gross-Pitaevskii equation in which the effect of gravitation of earth works. Since we used the time-independent perturbation theory, to include gravity effect, we must put the higher coefficient in the formulation of each energy state whose value is determined by the separated amplitude formula in each state. It will also be shown that even though the equation of state represents an ideal gas, the appropriate heat capacity approaches Einstein’s solid-like model if the certain conditions are employed.
Asymptotic approximations for the energy and chemical potential of the ground state in Bose-Einstein condensates are presented in the semiclassical regime with several typical trap-ping potentials. As preparatory steps, we begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE), review several typical external trapping potentials, scale the 3D GPE and show how to reduce it to 1D and 2D GPEs in certain limiting trapping frequency regime. For the 1D box potential, we derive asymptotic approximations up to o(1) in term of the scaled interacting pa-rameter d for energy and chemical potential of the ground and all excited states in both weakly interacting regime, i.e. d! 0 and strongly repulsive interacting regime, i.e. d! 1, respec-tively. For the 1D harmonic oscillator, double well and optical lattice potentials, as well as a more general external potential in high dimensions, we get asymptotic approximations up to o(1) in term of the scaled interacting parameter d for the energy and chemical potential of the ground state in semiclassical regime, i.e. d!1. Our extensive numerical results conrm all our asymp-totic approximations, provide convergence rate and suggest several very interesting conclusions.
We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as ≈1020\approx 10^{20}≈1020, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a $1/r^3 $ gravity at short scales.
We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as ≈1020\approx 10^{20}≈1020, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a $1/r^3 $ gravity at short scales.