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arXiv:1005.1061 · Physical Review A

Nonlinearity from quantum mechanics: Dynamically unstable Bose-Einstein condensate in a double-well trap

Juha Javanainen

Published May 6, 2010• 9 citations
arXivHTMLPDF

Abstract

We study theoretically an atomic Bose-Einstein condensate in a double-well trap, both quantum-mechanically and classically, under conditions such that in the classical model an unstable equilibrium dissolves into large-scale oscillations of the atoms between the potential wells. Quantum mechanics alone does not exhibit such nonlinear dynamics, but measurements of the atom numbers in the potential wells may nevertheless cause the condensate to behave essentially classically.

DOI
10.1103/physreva.81.051602

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We study theoretically an atomic Bose-Einstein condensate in a double-well trap, both quantum-mechanically and classically, under conditions such that in the classical model an unstable equilibrium dissolves into large-scale oscillations of the atoms between the potential wells. Quantum mechanics alone does not exhibit such nonlinear dynamics, but measurements of the atom numbers in the potential wells may nevertheless cause the condensate to behave essentially classically.
We examine the nonlinear dynamical behavior of a spinor Bose-Einstein condensate in a double-well potential. Considering a condensate with large number of atoms, such that it can be described using the mean field theory, we separate the spinor dynamics from the spatial dynamics under the single-mode approximation. We limit ourselves to certain initial conditions under which the spatial mode is frozen so that we can focus on the spinor dynamics only. Identifying collective spin variables of our system, we derive the corresponding nonlinear equations of motion for them. Employing standard stability analysis, we find and characterize fixed points of the system. For a wide range of physical parameters such as tunneling strength and non-linear interactions, as well as for various initial preparations of the system, we identify qualitatively different dynamical regimes possible in the system. In particular, complete and incomplete oscillations of spin variables between quantum wells are found. We also show that by bringing some fixed points close to each other in the phase space of the system, it is possible to induce amplitude modulation to those otherwise regular tunneling oscillations.
The quantum nonlinear dynamics of an atomic Bose-Einstein condensate (BEC) in a double-well potential is studied. Using a double-well single-particle potential model system and a two-mode approximation, results showed that the coherent oscillations resulting from tunneling are suppressed when the number of atoms is greater than a critical value. The spectrum if nonresonant light scattered by the condensate to include an asymmetric double peak structure with peak separation and relative heights determined by the number of atoms in the condensate and the interaction strength for hard sphere collisions.
Josephson effect of Bose-Einstein condensate in double-well potential is an obvious manifestation of macroscopic quantum coherence. Most of researches focus on the symmetric double-well potential. In this work, we investigate the dynamic of Bose-Einstein condensates in an asymmetric double-well potential by using two-mode theory and computer simulation. In the absence of the interaction between atoms, the dynamic equation of condensate can be solved analytically. The amplitude as a function of energy difference of two wells is obtained. We can find that the change of energy difference will lead to the different dynamic behaviors of condensate. If the energy difference is relatively large, the condensate will primarily occupy the well that is occupied more than the other well at the beginning time. It is interesting that such a trap phenomenon is not dependent on the position of the high energy potential well nor the position of low energy potential well. If the energy difference becomes small, the tunneling and oscillation of condensate will be enhanced. When the interaction between atoms is present, our numerical calculations show that as the nonlinear interaction increases, the dynamic behavior of condensate exhibits different characteristics, such as trapping in a well, enhancing the tunneling and oscillation between two wells, and enhancing the trapping in a well at large nonlinear interaction, which is similar to the dynamic change caused by the energy difference in the case of ideal condensate. That is to say, on the one hand, the nonlinear interaction can lead to the trap of condensate as well as the dynamic trap to happen in symmetric double-well potential. On the other hand, the nonlinearity can promote the tunneling of condensate, counteracting the effect of the asymmetry of potential. And, this counteracting effect is related to the difference in energy between asymmetric potential wells. To understand the underlying mechanism, the full dynamic behavior of two-mode model is illustrated and the dynamic transition can be seen clearly. Combining the results obtained with and without interaction, regarding nonlinear interaction as effective potential provides a clear way to explain dynamic transition of condensate in an asymmetric double-well potential. In addition, we also perform the numerical simulations of the Gross-Pitaevskii equation, and the results are consistent with the conclusions obtained by using the two-mode theory.
Bose Einstein condensates (BECs) in double well potentials have drawn a lot of attention for the possibilities they offer to study fundamental quantum mechanical effects at the macroscopic level as well as for potential applications like interferometry, high precision measurements or thermometry. In contrast to Josephson junctions realized in superconductors and superfluids, in BEC the interatomic interactions play a crucial role leading to anharmonic Josephson oscillations, if the initial population imbalance of the two wells is below a critical value, and to macroscopic quantum self-trapping i.e, inhibition of large amplitude Josephson oscillations above the threshold for the population imbalance. In this paper, a novel technique to adiabatically control of BEC in a double-well potential is proposed. The adiabatic dynamics of a BEC in a double well potential can be described in terms of a dark variable resulting from a combination of the population imbalance and the spatial atomic coherence.
Bose-Einstein condensates in a double-well potential contain the essential ingredients to study many-body systems within a rich classical phase-space that includes an unstable point and a separatrix. Employing a selfconsistent finite difference method, we study some of their quantum properties and their dependency on the strength of the boson-boson interaction. We observe a deviation in the critical parameters associated with a behavior change in both the energy distribution and the eigenstates of the system. We also examine the trends of the nonclassicality via the Wigner function, the tunneling transmission coefficient, and the nonorthogonality of eigenstates associated with the nonlinearity aspects of the Gross-Pitaevskii equation.