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arXiv:1806.01820 · New Journal of Physics

Traces of integrability in scattering of one-dimensional dimers on a barrier

Juan Polo, Anna Minguzzi, Maxim Olshanii

Published Feb 11, 2019• 5 citations
arXivHTMLPDF

Abstract

We consider molecules made of two one-dimensional short-range-interacting bosonic atoms. We show that in the process of scattering of these molecules off a narrow barrier, odd incident waves produce no unbound atoms, even when the incident energy exceeds the dissociation threshold . We demonstrate that such a feature originates from the fact that our system shares a part of its spectrum, namely its odd eigenstates, with a Bethe Ansatz integrable model, a δ -interaction Hamiltonian with local and non-local terms, associated with a reflection group C 2 —a full symmetry of a square. This partial integrability induces prohibition of chemical reactions, allowing us to make predictions about scattering processes. We suggest several experimental implementations of the effect. We also propose to use the monomer production as an alternative read-out channel in an atom interferometer: unlike in the standard interferometric schemes, no spatial separation of the output channels will be required. Our approach offers an example of use of integrable problems beyond the realm of fully integrable models.

DOI
10.1088/1367-2630/ab0612

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•May 3, 2018•0 citations
We consider molecules made of two one-dimensional short-range-interacting bosonic atoms. We show that in the process of scattering of these molecules off a narrow barrier, odd incident waves produce no unbound atoms, even when the incident energy exceeds the dissociation threshold . We demonstrate that such a feature originates from the fact that our system shares a part of its spectrum, namely its odd eigenstates, with a Bethe Ansatz integrable model, a δ -interaction Hamiltonian with local and non-local terms, associated with a reflection group C 2 —a full symmetry of a square. This partial integrability induces prohibition of chemical reactions, allowing us to make predictions about scattering processes. We suggest several experimental implementations of the effect. We also propose to use the monomer production as an alternative read-out channel in an atom interferometer: unlike in the standard interferometric schemes, no spatial separation of the output channels will be required. Our approach offers an example of use of integrable problems beyond the realm of fully integrable models.
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Exact solution for two δ\deltaδ-interacting bosons on a ring in the presence of a δ\deltaδ-barrier: Asymmetric Bethe Ansatz for spatially odd states
In this article, we apply the recently proposed Asymmetric Bethe Ansatz method to the problem of two one-dimensional, short-range-interacting bosons on a ring in the presence of a \delta δ -function barrier. Only half of the Hilbert space—namely, the two-body states that are odd under point inversion about the position of the barrier—is accessible to this method. The other half is presumably non-integrable. We consider benchmarking the recently proposed 1/g 1 / g expansion about the hard-core boson point [D. Sen, Int. J. Mod. Phys. A 14, 1789 (1999); A. G. Volosniev et al., Nat. Commun. 5, 5300 (2014)] as one application of our results. Additionally, we find that when the \delta δ -barrier is converted to a \delta δ -well with strength equal to that of the particle-particle interaction, the system exhibits the spectrum of its non-interacting counterpart while its eigenstates display features of a strongly interacting system. We discuss this phenomenon in the “Summary and Future Research” section of our paper.
We consider bound and scattering states of the one-dimensional dimer formed by two coupled non-identical atoms when one of them also interacts with the zero-range potential located at the origin. By calculating the dimer localized and scattering wave functions, we identify properties of the system without the two-body bound-state breakdown. In general, we predict an enhancement of the dimer reflection compared to a single atom, except for a narrow region on the attractive side of the external potential.
Three identical bosons or fermions are considered in the limit of zero-range interactions and finite effective range. By using a two-channel model, we show that these systems are not integrable and that the wave function verifies specific continuity conditions at the contact of three particles. This last feature permits us to solve a contradiction brought by the contact model which can lead to an opposite result concerning the integrability issue. For fermions, the vicinity of integrability is characterized by large deviations with respect to the predictions of the Bethe ansatz.
We analytically study the atom-dimer scattering problem in the near-integrable limit when the oscillator length l0{l}_{0}l0​ of the transverse confinement is smaller than the dimer size, $\ensuremath{\sim}{l}_{0}^{2}/|a|$, where a<0a<0a<0 is the interatomic scattering length. The leading contributions to the atom-diatom reflection and break-up probabilities are proportional to a6{a}^{6}a6 in the bosonic case and to a8{a}^{8}a8 for the $\ensuremath{\uparrow}$-$\ensuremath{\uparrow}\ensuremath{\downarrow}$ scattering in a two-component fermionic mixture. We show that by tuning aaa and l0{l}_{0}l0​ one can control the ``degree of integrability'' in a quasi-1D atom-dimer mixture in an extremely wide range leaving thermodynamic quantities unchanged. We find that the relaxation to deeply bound states in the fermionic (bosonic) case is slower (faster) than transitions between different Bethe ansatz states. We propose a realistic experiment for detailed studies of the crossover from integrable to nonintegrable dynamics.
Scattering a particle on a one-dimensional δ\deltaδ-potential barrier: asymptotic superselection rule
As is stated, the modern mathematical (C∗C^*C∗-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the δ\deltaδ-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit t→∞t\to \inftyt→∞, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.