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arXiv:2301.03082 · Few-Body Systems

On the Exponential Decay of Strongly Interacting Cold Atoms from a Double-Well Potential

Przemysław Kościk

Published Feb 14, 2023• 2 citations
arXivHTMLPDF

Abstract

Abstract In this article, we study an exponential decay for the gas of bosons with strong repulsive delta interactions from a double-well potential. We consider an exactly solvable model comprising an infinite wall and two Dirac delta barriers. We explore its features both within the exact method and with the resonance expansion approach. The study reveals the effect of the splitting barrier on the decay rate in dependence on the number of particles. Among other things, we find that the effect of the splitting barrier on the decay rate is most pronounced in systems with odd particle numbers. During exponential decay, the spatial correlations in an internal region are well captured by the “radiating state”.

DOI
10.1007/s00601-023-01792-2

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We study the decay properties of Tonks-Girardeau gases escaping from a double-well potential. Initially, the gases are constrained between two infinite $\ensuremath{\delta}$ barriers with an on-center $\ensuremath{\delta}$-split potential. The strength of one of the obstacles is suddenly reduced, and the particles start to tunnel to the open space. Using the resonance expansion method, we derive the single-term approximate expression for the NNN-particle survival probability and demonstrate its effectiveness in both the exponential and nonexponential regimes. We also predict a parity effect and provide physical insights into its nature at different stages of the time evolution. We conclude that only the initial phase of the decay of the many-particle state may comply with an exponential law. The decay properties are dramatically affected by the presence of the split barrier. Our results reveal the overall decay mechanism of unstable Tonks-Girardeau gases from single and double quantum wells.
We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.
We study the dynamical properties of a few bosons confined in an one-dimensional split hard wall trap with the interaction strength varying from the weakly to strongly repulsive regime. The system is initially prepared in one side of the double well by setting the barrier strength of the split trap to be infinity and then the barrier strength is suddenly changed to a finite value. Both exact diagonalization method and Bose-Hubbard model (BHM) approximation are used to study the dynamical evolution of the initial system. The exact results based on exact diagonaliztion verify the enhancement of correlated tunneling in the strongly interacting regime. Comparing results obtained by two different methods, we conclude that one-band BHM approximation can well describe the dynamics in the weakly interacting regime, but is not efficient to give quantitatively consistent results in the strongly interacting regime. Despite of the quantitative discrepancy, we validate that the form of correlated tunneling gives an important contribution to tunneling in the large interaction regime. To get a quantitative description for the dynamics of bosons in the strongly interacting regime, we find that a multi-band BHM approximation is necessary.
We study the time dependence of a quantum particle initially localized'behind two delta-function barriers of strengths v 1 and v 2 . We determine the conditions under which the decay is exponential by varying the ratio v 1 /v 2 and the separation between the delta-barriers. We find that exponential decay occurs even when v 1 /v 2 is of order one. We also find cases of non-exponential decay. An interesting example of non-exponential decay is when the two barriers are very close. We show that the exponential decay is due to poles in the complex k-plane, where k denotes the wave numbers of the energy eigenstates, and we find that the decay rates are fully understood in many cases as being due to a single pole.