The realization of Bose-Einstein condensation in dilute atomic gases opens an exciting way to quantum mechanics and begins a new area of quantum simulation. As a macroscopic quantum object and a many-body bosonic system, the Bose-Einstein condensates can show numerous exotic quantum effects and have naturally attracted great attention. One of the simplest quantum many-body systems to be realized experimentally and studied theoretically is ultra-cold atoms in a double-well potential. This system can exhibit a great variety of quantum interference phenomena such as tunneling oscillation, self-trapping and the entanglement of macroscopic superpositions. Specifically, the double-well potentials built by optical or magnetic fields are easy to change and the many-body interaction between ultra-cold atoms can be changed by the method of Feshbach resonance, enabling the precise quantum control of the double-well dynamics of the condensates. In the present work, we study the dynamics of a condensate in a trapping potential consisting of an unalterable double-well trap and an additional moving optical lattice. If the lattice space is much smaller than the size of the double-well trap, the system can be simplified into a double-well trapped condensate with a tunable effective mass. Using the mean-field factorization assumption, together with a two-mode approximation, we obtain the analytic expressions for the dependence of the tunneling rate and the self-collision strength on the effective mass. The tunneling rate decays and the collision strength grows up with the increase of the effective mass. As a consequence of their different changes, we conclude that the adjustment of the effective mass of the ultra-cold atoms, rather than the changing of the trap barrier or adjusting of the atomic scattering length, is an alternative approach to controlling the double-well dynamics of the condensate. Via numerical simulations of the mean-field dynamical equations with some realistic parameters, we show that a transition between the quantum coherent tunneling and the self-trapping behaviors is experimentally realizable with the mass-control approach. Specifically, we show that the approach is still valid for the case of negative mass. Moreover, we find that the negative-mass case can be used even to stimulate the double-well dynamics of the condensate with a negative atomic scattering length.
The realization of Bose-Einstein condensation in dilute atomic gases opens an exciting way to quantum mechanics and begins a new area of quantum simulation. As a macroscopic quantum object and a many-body bosonic system, the Bose-Einstein condensates can show numerous exotic quantum effects and have naturally attracted great attention. One of the simplest quantum many-body systems to be realized experimentally and studied theoretically is ultra-cold atoms in a double-well potential. This system can exhibit a great variety of quantum interference phenomena such as tunneling oscillation, self-trapping and the entanglement of macroscopic superpositions. Specifically, the double-well potentials built by optical or magnetic fields are easy to change and the many-body interaction between ultra-cold atoms can be changed by the method of Feshbach resonance, enabling the precise quantum control of the double-well dynamics of the condensates. In the present work, we study the dynamics of a condensate in a trapping potential consisting of an unalterable double-well trap and an additional moving optical lattice. If the lattice space is much smaller than the size of the double-well trap, the system can be simplified into a double-well trapped condensate with a tunable effective mass. Using the mean-field factorization assumption, together with a two-mode approximation, we obtain the analytic expressions for the dependence of the tunneling rate and the self-collision strength on the effective mass. The tunneling rate decays and the collision strength grows up with the increase of the effective mass. As a consequence of their different changes, we conclude that the adjustment of the effective mass of the ultra-cold atoms, rather than the changing of the trap barrier or adjusting of the atomic scattering length, is an alternative approach to controlling the double-well dynamics of the condensate. Via numerical simulations of the mean-field dynamical equations with some realistic parameters, we show that a transition between the quantum coherent tunneling and the self-trapping behaviors is experimentally realizable with the mass-control approach. Specifically, we show that the approach is still valid for the case of negative mass. Moreover, we find that the negative-mass case can be used even to stimulate the double-well dynamics of the condensate with a negative atomic scattering length.
The overlap of localized wave functions in a two-mode approximation leads to interaction (cross-collision) between ultra-cold atoms trapped in distinct sites of a double-well potential. We show that this interaction can significantly change the atom tunneling rate for special trap configurations resulting in an effective linear Rabi regime of population oscillation between the trap wells. In this sense, we demonstrate that cross-collisional effects can significantly extend the validity of the two-mode model approach allowing it to be alternatively employed to explain the recently observed increase of tunneling rates due to nonlinear interactions. Moreover, we investigate the extension for ultra-cold atoms trapped in an optical lattice. Control over the cross-collisional terms, obtained through manipulation of the optical trapping potential, can be used as an engineering tool to study many-body physics.
Abstract Bose-Einstein Condensate (BEC) is a paradigmatic representative of macroscopic quantum states. 39 K atoms serve as an ideal system for realizing a nearly ideal BEC owing to their ultra-small background scattering length. This study further investigates the nonlinear dynamics of 39 K BEC confined in a double-well potential under attractive interactions. In this work, a Gaussian barrier is integrated with a double-well potential, and we innovatively adopt a transient dynamics approach to construct a highly non-equilibrium initial state where all atoms are confined to the left well. We track the full evolutionary process of the system from non-equilibrium excitation to stable self-trapping, elucidating wave function contraction and energy conversion. Under high barrier height conditions, tunneling is substantially suppressed: atoms localize in a single well, the longitudinal position expectation value remains constant, and the phase difference is locked, leading to the formation of a macroscopic quantum self-trapping state. Radial oscillations exhibit reduced amplitude and a slower evolution rate owing to density homogenization. A smooth, fluctuation-free momentum density profile corresponds to the static localized regime of self-trapping. Coherence transitions from dynamic oscillation to static locking, and weak attractive interactions in 39 K atoms further prolong the coherence time. This study elucidates key microscopic mechanisms, including the nonlinear coupling mechanism of attractive interactions, the dynamic-static transition rules of coherence, and momentum density acting as a microscopic probe for tunneling and soliton dynamics, thereby providing a quantitative basis for the preparation of tunable quantum states.
In this paper, we study the interaction-modulated tunneling dynamics of a Bose-Fermi superfluid mixture, where a Bose-Einstein condensate (BEC) with weak repulsive interaction is confined in a symmetric deep double-well potential and an equally populated two-component Fermi gas in a harmonic potential symmetrically is positioned in the center of the double-well potential. The tunneling between the two wells is modulated by fermions trapped in a harmonic potential. When the temperature is adequately low and the bosonic particle number is adequately large, we can employ the mean-field theory to describe the evolution of the BEC in the double-well potential through the time-dependent Gross-Pitaevskii equation. For the Fermi gas in the harmonic potential trap, we consider the case where the inter-fermion interaction is tuned on the deep Bose-Einstein condensate of the inter-fermion Feshbach resonance, where two fermions of spin-up and spin-down form a two-body bound state. Within the regime, the Fermi gas is well described by a condensate of these fermionic dimers, and hence can be simulated as well by a Gross-Pitaevskii equation of dimers. The inter-species interactions couple the dynamics of the two species, which results in interesting features in the tunneling oscillations. The dynamic equations of the BEC in the double-well potential is described by a two-mode approximation. Coupling it with time-dependent Gross-Pitaevskii equation of the harmonically potential trapped molecular BEC, we numerically investigate the dynamical evolution of the Boson-Fermi hybrid system under different initial conditions. It is found that the interaction among fermions in a harmonic potential leads to strong non-linearity in the oscillations of the bosons in the double-well potential and enriches the tunneling dynamics of the bosons. Especially, it strengthens macroscopic quantum self-trapping. And the macroscopic quantum self-trapping can be expressed in three forms: the phase tends to be negative and monotonically decreases with time, the phase evolves with time, and the phase tends to be positive and increases monotonically with time. This means that it is possible the tunneling dynamics of the BEC in double-well potential is adjustable. Our results can be verified experimentally in a Bose-Fermi superfluid mixture by varying different interaction parameters via Feshbach resonance and confinement-induced resonance.
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.
We study dynamical behaviors of the weakly interacting Bose-Einstein condensate in the one-dimensional optical lattice with an overall double-well potential by solving the time-dependent Gross-Pitaevskii equation. It is observed that the double-well potential dominates the dynamics of such a system even if the lattice depth is several times larger than the height of the double-well potential. This result suggests that the condensate flows without resistance in the periodic lattice just like the case of a single particle moving in periodic potentials. Nevertheless, the effective mass of atoms is increased, which can be experimentally verified since it is connected to the Josephson oscillation frequency. Moreover, the periodic lattice enhances the nonlinearity of the double-well condensate, making the condensate more “self-trapped” in the π -mode self-trapping regime.
Dynamics of the repulsive Bose–Einstein condensate (BEC) in a double-well trap is explored within the 3D time-dependent Gross–Pitaevskii equation. The model avoids numerous common approximations (two-mode treatment, time-space factorization, fixed values of the chemical potential and barrier penetrability, etc) and thus may provide a realistic description of BEC dynamics, including both weak-coupling (sub-barrier) and strong-coupling (above-barrier) regimes and their crossover. The strong coupling regime is achieved by increasing the number N of BEC atoms and thus the chemical potential. The evolution with N of Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST) is examined and the crucial impact of the BEC interaction is demonstrated. At weak coupling, the calculations well reproduce the JO/MQST experimental data. At strong coupling with a significant overlap of the left and right BECs, we observe a remarkable persistence of the Josephson-like dynamics: the JO and MQST converge to a high-frequency JO-like mode where both population imbalance and phase difference oscillate around the zero averages.