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arXiv:2403.10078 · Physical Review A

Impact of dark states on the stationary properties of quantum particles with off-centered interactions in one dimension

G. Bougas, Nathan L. Harshman, Peter Schmelcher

Published Aug 27, 2024• 0 citations
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Abstract

We present a generalization of the two-body contact interaction for nonrelativistic particles trapped in one dimension. The particles interact only when they are a distance $c$ apart. The competition of the interaction length scale with the oscillator length leads to three regimes identified from the energy spectra. When $c$ is less than the oscillator length, particles avoid each other, whereas in the opposite case bunching occurs. In the intermediate region where the oscillator length is comparable to $c$, both exclusion and bunching are manifested. All of these regions are separated by dark states, i.e., bosonic or fermionic states which are not affected by the interactions.

DOI
10.1103/physreva.110.023327

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We present a generalization of the two-body contact interaction for nonrelativistic particles trapped in one dimension. The particles interact only when they are a distance ccc apart. The competition of the interaction length scale with the oscillator length leads to three regimes identified from the energy spectra. When ccc is less than the oscillator length, particles avoid each other, whereas in the opposite case bunching occurs. In the intermediate region where the oscillator length is comparable to ccc, both exclusion and bunching are manifested. All of these regions are separated by dark states, i.e., bosonic or fermionic states which are not affected by the interactions.
We consider a generalization of the two-body contact interaction for nonrelativistic particles confined to a one-dimensional box, in which the interaction is decentered, i.e., the particles interact only when they are separated by a distance c. In contrast to the harmonically trapped system, this model is nonintegrable. Despite this, we demonstrate that the system exhibits partial solvability due to the presence of dark states, i.e., bosonic or fermionic states unaffected by the interaction. These states form exactly solvable subspaces embedded within an interacting spectrum. We characterize the stationary properties of the system, identify the conditions for the appearance of dark states, and show how they structure the spectrum and delineate interacting and noninteracting sectors.
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We investigate finite-number effects in collisions between two states of an initially well-known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2N/2N/2 (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of N=4N=4N=4. In the noninteracting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition of contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. Additionally, we study the time evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create behavior qualitatively different from that of repulsive interactions, due to the presence of bound states (quantum solitons), and explain the processes behind this.
We show that a system of three species of one-dimensional fermions, with an attractive three-body contact interaction, features a scale anomaly directly related to the anomaly of two-dimensional fermions with two-body contact forces. We show, furthermore, that those two cases (and their multispecies generalizations) are the only nonrelativistic systems with contact interactions that display a scale anomaly. While the two-dimensional case is well known and has been under study both experimentally and theoretically for years, the one-dimensional case presented here has remained unexplored. For the latter, we calculate the impact of the anomaly on the equation of state, which appears through the generalization of Tan's contact for three-body forces, and determine the pressure at finite temperature. In addition, we show that the third-order virial coefficient is proportional to the second-order coefficient of the two-dimensional two-body case.
We unravel the stationary properties and the interaction quench dynamics of two bosons, confined in a two-dimensional anisotropic harmonic trap. A transcendental equation is derived giving access to the energy spectrum and revealing the dependence of the energy gaps on the anisotropy parameter. The relation between the two- and one-dimensional scattering lengths as well as the Tan contacts is established. The contact, capturing the two-body short-range correlations, shows an increasing tendency for a larger anisotropy. Subsequently, the interaction quench dynamics from attractive to repulsive values and vice versa is investigated for various anisotropies. A closed analytical form of the expansion coefficients of the two-body wave function, during the time evolution is constructed. The response of the system is studied by means of the time-averaged fidelity, the spectra of the spatial extent of the cloud in each direction, and the one-body density. It is found that as the anisotropy increases, the system becomes less perturbed independently of the interactions, while for fixed anisotropy quenches toward the noninteracting regime perturb the system in the most efficient manner. Furthermore, we identify that in the tightly confined direction more frequencies are involved in the dynamics stemming from higher lying excited states.
Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.