The out-of-equilibrium quantum dynamics of a Bose gas trapped in an asymmetric double well and interacting with a finite-range interaction has been studied in real space by solving the time-dependent many-body Schrödinger equation numerically accurately using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We have focused on the weakly interacting limit where the system is essentially condensed. We have examined the impact of the range of the interaction on the dynamics of the system, both at the mean-field and many-body levels. Explicitly, we have studied the maximal and the minimal values of the many-body position variance in each cycle of oscillation, and the overall pace of its growth. We find that the range of the interaction affects the dynamics of the system differently for the right well and the left well. We have also examined the infinite-particle limit and find that even there, the impact of the range of the interaction can only be described by a many-body theory such as MCTDHB.
The out-of-equilibrium quantum dynamics of a Bose gas trapped in an asymmetric double well and interacting with a finite-range interaction has been studied in real space by solving the time-dependent many-body Schrödinger equation numerically accurately using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We have focused on the weakly interacting limit where the system is essentially condensed. We have examined the impact of the range of the interaction on the dynamics of the system, both at the mean-field and many-body levels. Explicitly, we have studied the maximal and the minimal values of the many-body position variance in each cycle of oscillation, and the overall pace of its growth. We find that the range of the interaction affects the dynamics of the system differently for the right well and the left well. We have also examined the infinite-particle limit and find that even there, the impact of the range of the interaction can only be described by a many-body theory such as MCTDHB.
Abstract The out-of-equilibrium quantum dynamics of an interacting Bose gas trapped in a one-dimensional asymmetric double-well potential is studied by solving the many-body Schrödinger equation numerically accurately. We examine how the gradual loss of symmetry of the confining trap affects the macroscopic quantum tunneling dynamics of the system between the two wells. In an asymmetric double well, the two wells are not equivalent anymore, say, the left well is deeper than the right one. Accordingly, we analyze the dynamics by initially preparing the condensate in both the left and the right wells. The dynamics of the system is characterized by the time evolution of a few physical quantities of increasing many-body complexity, namely, the survival probability, depletion and fragmentation, and the many-particle position and momentum variances. In particular, we have examined the frequencies and amplitudes of the oscillations of the survival probabilities, the time scale for the development of fragmentation and its degree, and the growth and oscillatory behavior of the many-particle position and momentum variances. There is an overall suppression of the oscillations of the survival probabilities in an asymmetric double well, except for resonant values of asymmetry for which the one-body ground state energy in the right well matches with one of the one-body excited states in the left well, thereby resulting in resonantly enhanced tunneling from the right well ground state. Overall, depending on whether the condensate is initially prepared in the left or right well, the repulsive inter-atomic interactions affect the survival probabilities differently. For a sufficiently strong repulsive interaction, the system is found to become fragmented. The degree of fragmentation depends both on the asymmetry of the trap and the initial well in which the condensate is prepared in a non-trivial manner. Furthermore, we show that the phenomenon of resonantly enhanced tunneling can be accompanied by a large degree of fragmentation (depletion) for the strong (weak) interaction. The many-particle position and momentum variances follow the density oscillations of the system in the asymmetric double well and bears prominent signatures of the degree of depletion or fragmentation, depending on the strength of the interactions. These quantities further exhibit a fine structure signifying a breathing-mode oscillation. Finally, a universality of fragmentation for systems made of different numbers of particles but the same interaction parameter is also found and its dependence on the asymmetry is investigated. The phenomenon is robust despite the asymmetry of the junction and admits a macroscopically-large fragmented condensate characterized by a diverging many-particle position variance. This is as far as one can get from the dynamics of the density in the junction.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of N interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M), where M specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number M of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
Tunneling in a many-body system appears as one of the novel implications of quantum physics, in which particles move in space under an otherwise classically-forbidden potential barrier. Here, we theoretically describe the quantum dynamics of the tunneling phenomenon of a few intricate bosonic clouds in a closed system of a two-dimensional symmetric double-well potential. We examine how the inclusion of the transverse direction, orthogonal to the junction of the double-well, can intervene in the tunneling dynamics of bosonic clouds. We use a well-known many-body numerical method, called the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method. MCTDHB allows one to obtain accurately the time-dependent many-particle wavefunction of the bosons which in principle entails all the information of interest about the system under investigation. We analyze the tunneling dynamics by preparing the initial state of the bosonic clouds in the left well of the double-well either as the ground, longitudinally or transversely excited, or a vortex state. We unravel the detailed mechanism of the tunneling process by analyzing the evolution in time of the survival probability, depletion and fragmentation, and the many-particle position, momentum, and angular-momentum expectation values and their variances. As a general rule, all objects lose coherence while tunneling through the barrier and the states which include transverse excitations do so faster. In particular for the later states, we show that even when the transverse direction is seemingly frozen, prominent many-body dynamics in a two-dimensional bosonic Josephson junction occurs. Implications are briefly discussed.
In standard BJJs, particles tunnel between two single-well potentials linked by a finite barrier. The dynamics of standard BJJs have been extensively studied, both at the many-body and mean-field levels of theory. In the present work, we introduce the concept of a composite BJJ. In a composite BJJ, particles tunnel between two double-well potentials linked by a finite potential barrier between them. We focus on many-body facets of quantum dynamics and investigate how the complex structure of the junction influences tunnelling. Employing the multiconfigurational time-dependent Hartree for bosons method, highly-accurate many-boson wave-functions are obtained from which properties are computed. We analyze the dynamics using the survival probability, the degree of fragmentation, and the fluctuations of observables, and discuss how many-boson tunnelling behaves, and how it may be controlled. We also provide strong evidence that a universal degree of fragmentation into multiple time-dependent modes takes place. Further applications are briefly discussed.
In order to solve the many-boson Schr\"odinger equation we utilize the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). To be able to attack larger systems and/or to propagate the solution for longer times, we implement a parallel version of the MCTDHB method, thereby realizing the recently proposed [Streltsov et al., Phys. Rev. A 81, 022124 (2010)] idea on how to construct efficiently the result of the action of the Hamiltonian on a bosonic state vector. As an illustrative example of its own interest, we study the real-space dynamics of repulsive bosonic systems made of N=12, 51, and 3003 bosons in triple-well periodic potentials. The ground state of this system is threefold fragmented. By suddenly strongly distorting the trap potential, the system performs complex many-body quantum dynamics. At long times it reveals a tendency to an oscillatory behavior around a threefold fragmented state. These oscillations are strongly suppressed and damped by quantum depletions. In spite of the richness of the observed dynamics, the three time-adaptive orbitals of MCTDHB(M=3) are capable of describing the many-boson quantum dynamics of the system for short and intermediate times. For longer times, however, more self-consistent time-adaptive orbitals are needed to correctly describe the nonequilibrium many-body physics. The convergence of the MCTDHB(M) method with the number M of self-consistent time-dependent orbitals used is demonstrated.
Understanding how coherent quantum dynamics give way to correlation-dominated behavior in low-dimensional systems remains a central challenge in quantum many-body physics. Here, we investigate a one-dimensional Bose-Josephson junction confined in a box trap using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). By varying the interaction strength and initial population imbalance, we identify distinct dynamical regimes governed by the competition between coherence and correlation-induced fragmentation. Weak interactions support coherent Josephson oscillations, whereas increasing imbalance leads to damping. At intermediate interaction strength, varying only the initial imbalance induces a crossover from nearly pure coherent oscillations to many-body dephasing with collapse-and-revival dynamics, and ultimately to equilibration accompanied by strong fragmentation and the saturation of many-body observables. In the strongly interacting regime, the system enters a dynamical freezing regime characterized by pronounced fragmentation, well-separated particle-resolved density peaks, and strongly suppressed tunneling. A systematic comparison with the Bose-Hubbard model reveals excellent agreement in the weakly interacting regime, while progressively larger deviations emerge as higher-orbital occupations beyond the two-mode approximation become significant. These results provide a unified picture of the emergence and competition of coherence, many-body dephasing, equilibration, and dynamical freezing, while delineating the regime of validity of the Bose-Hubbard description.