Abstract
We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers.
Highlights
Entropy 2020, 22, 382 e.g., by the theory of open quantum systems [19,20,21,22], modern semiclassics [23], or random matrix theory [24,25]—there is an intermediate range of system sizes where efficient numerical methods can (a) be gauged against each other, to benchmark their quantitative reliability, without any a priori restriction on the explored portion of Hilbert space, and (b) contribute to gauge effective theories against exact solutions [26,27,28], at spectral densities where quantum granular effects induce possibly sizeable deviations [29] from effective theory predictions
The most striking feature is the opening of an energy gap, clearly observed at large Amax : At the ground-state level, the three-fold degenerate states for λ = 0 split into a unique ground state which remains unperturbed by the interaction, plus two degenerate excited states which are affected by the non-vanishing interaction strength λ 6= 0
The Fourier Grid Hamiltonian method was employed to extract the full spectral information for two interacting bosons, whereas a Bose–Hubbard representation of the continuous double-well potential was found to be more efficient to describe the spectral structure of the three-particle case
Summary
The detailed microscopic understanding of interacting many-particle quantum dynamics in state-of-the-art experiments with ultracold atoms [1,2,3,4,5,6,7,8,9,10] in well-characterized potential landscapes remains a challenging task for theory: While a large arsenal of advanced numerical techniques has been developed over the past two decades to efficiently simulate interacting many-particle dynamics [11,12,13,14,15], all of them must surrender when confronted with truly complex dynamics, i.e., under conditions where a generic initial state fully explores, on sufficiently long time scales, an exponentially large Hilbert space in the number of particles and/or degrees of freedom. By the very meaning of complexity, even the most efficient numerical methods can only be expected to yield reliable results when the dynamics can be restricted to finite sub-spaces of the exponentially large Hilbert spaces—either by reducing the time window over which the evolution is followed, or by choosing physical situations which a priori confine the many-particle state This has been long understood in the light–matter interaction of atoms and molecules [16], as well as in quantum chaos [17], and meets revived interest given the experimental progress in the control of cold matter [18].
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