Hard-wall Trap Research Articles

Two mass-imbalanced atoms in a hard-wall trap: Deep learning integrability of many-body systems.

The study of integrable systems has led to significant advancements in our understanding of many-body physics. We design a series of numerical experiments to analyze the integrability of a mass-imbalanced two-body system through energy-level statistics and deep learning of wave functions. The level spacing distributions are fitted by a Brody distribution and the fitting parameter ω is found to separate the integrable and nonintegrable mass ratios by a critical line ω=0. The convolutional neural network built from the probability density images could identify the transition points between integrable and nonintegrable systems with high accuracy, yet in a much shorter computation time. A brilliant example of the network's ability is to identify a new integrable mass ratio 1/3 by learning from the known integrable case of equal mass, with a remarkable network confidence of 98.22%. The robustness of our neural networks is further enhanced by adversarial learning, where samples are generated by standard and quantum perturbations mixed in the probability density images and the wave functions, respectively.

Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D6 Symmetry.

SummaryWe show that a system consisting of two interacting particles with mass ratio 3 or 1/3 in a hard-wall box can be exactly solved by using Bethe-type ansatz. The ansatz is based on a finite superposition of plane waves associated with a dihedral group D6, which enforces the momentums after a series of scattering and reflection processes to fulfill the D6 symmetry. Starting from a two-body elastic collision model in a hard-wall box, we demonstrate how a finite momentum distribution is related to the D2n symmetry for permitted mass ratios. For a quantum system with mass ratio 3, we obtain exact eigenenergies and eigenstates by solving Bethe-type-ansatz equations for arbitrary interaction strength. A many-body excited state of the system is found to be independent of the interaction strength, i.e., the wave function looks exactly the same for non-interacting two particles or in the hard-core limit.

Equilibrium and nonequilibrium entanglement properties of two- and three-dimensional Fermi gases

We investigate the entanglement properties of the equilibrium and nonequilibrium quantum dynamics of two-dimensional (2D) and three-dimensional (3D) Fermi gases, by computing entanglement entropies of extended space regions, which generally show multiplicative logarithmic corrections to the leading power-law behaviors, corresponding to the logarithmic corrections to the area law. We consider 2D and 3D Fermi gases of $N$ particles constrained within a limited space region, for example, by a hard-wall trap, at equilibrium at $T=0$, i.e., in their ground state, and compute the first few terms of the asymptotic large-$N$ behaviors of entanglement entropies and particle fluctuations of subsystems with some convenient geometries, which allow us to significantly extend their computation. Then we consider their nonequilibrium dynamics after instantaneously dropping the hard-wall trap, which allows the gas to expand freely. We compute the time dependence of the von Neumann entanglement entropy of space regions around the original trap. We show that at small time it is characterized by the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ with the particle variance, and multiplicative logarithmic corrections to the leading power law, i.e., $S\ensuremath{\sim}{t}^{1\ensuremath{-}d}\mathrm{ln}(1/t)$.

Comparative study of quantum dynamics of a few bosons in a one-dimensional split hard-wall trap: exact results versus Bose–Hubbard-model approximations

We study the dynamical properties of a few bosons confined in a one-dimensional split hard-wall trap with the interaction strength varying from the weakly to strongly repulsive regime. The system is initially prepared on 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 the 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 diagonalization verify the enhancement of correlated tunnelling 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 the quantitative discrepancy, we validate that the form of correlated tunnelling gives an important contribution to tunnelling 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.

Lowest scattering state of one-dimensional Bose gases with attractive interactions

We investigate the lowest scattering state of a one-dimensional Bose gas with attractive interactions trapped in a hard-wall trap. By solving the Bethe ansatz equation numerically, we determine the full energy spectrum and the exact wave function for different attractive interaction parameters. The resultant density distribution, momentum distribution, reduced one-body density matrix, and two-body correlation show that a decreased attractive interaction induces rich density profiles and specific correlation properties in the weakly attractive Bose gas.

Hard-core Bose-Fermi mixture in one-dimensional split traps

We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a hard-wall trap or a harmonic oscillator trap with a tunable $\ensuremath{\delta}$-function barrier at the trap center. The mixture consists of a 1D Bose gas with repulsive interactions and of a 1D noninteracting spin-aligned Fermi gas, with both species interacting through hard-core interactions. Using a generalized Bose-Fermi mapping, we calculate the reduced single-particle density matrix and the momentum distribution of the gas as a function of barrier strength and the parity of particle number. The secondary peaks in the momentum distribution show remarkable correlation between particles on the two sides of the split.

Momentum distribution of a freely expanding Lieb-Liniger gas

We numerically study free expansion of a few Lieb-Liniger bosons, which are initially in the ground state of an infinitely deep hard-wall trap. Numerical calculation is carried out by employing a standard Fourier transform, as follows from the Fermi-Bose transformation for a time-dependent Lieb-Liniger gas. We study the evolution of the momentum distribution, the real-space single-particle density, and the occupancies of natural orbitals. Our numerical calculation allows us to explore the behavior of these observables in the transient regime of the expansion, where they are nontrivially affected by the particle interactions. We derive analytically (by using the stationary phase approximation) the formula which connects the asymptotic shape of the momentum distribution and the initial state. For sufficiently large times, the momentum distribution coincides (up to a simple scaling transformation) with the shape of the real-space single-particle density (the expansion is asymptotically ballistic). Our analytical and numerical results are in good agreement.

Ground-state properties of interacting two-component Bose gases in a hard-wall trap

We investigate ground-state properties of interacting two-component Bose gases in a hard-wall trap using both the Bethe ansatz and exact numerical diagonalization method. For equal intra-atomic and interatomic interaction, the system is exactly solvable. Thus the exact ground-state wave-function and density distributions for the whole interacting regime can be obtained from the Bethe ansatz solutions. Since the ground state is a degenerate state with total spin $S=N/2$, the total density distributions are the same for each degenerate state. The total density distribution evolves from a Gauss-like Bose distribution to a Fermi-like one as the repulsive interaction increases. The distribution of each component is ${N}_{\ensuremath{\alpha}}/N$ of the total density distribution. This is approximately true even in the experimental situation. In addition the numerical results show that with the increase in interspecies interaction the distributions of two Tonks-Girardeau gases exhibit composite-fermionization crossover with each component developing $N$ peaks in the strongly interacting regime.

Bright and dark solitary waves in a one-dimensional spin-polarized gas of fermionic atoms withp-wave interactions in a hard-wall trap

We elucidate the physics underlying the fact that both bright and dark solitary waves can arise in a one-dimensional spin-polarized gas of fermionic atoms with attractive three-dimensional $p$-wave interactions in a hard-wall trap. This is possible since the one-dimensional fermion system can be mapped to a system of bosons described by the Lieb-Linger model with either repulsive or attractive $\ensuremath{\delta}$-function interactions which can support solitary waves.

One-dimensional fermionic gases with attractivep-wave interaction in a hard-wall trap

We investigate the ground state of the one-dimensional fermionic system enclosed in a hard-wall trap with attractive contact $p$-wave interactions. Based on the Bethe ansatz method, the explicit wave function is derived by numerically solving the Bethe ansatz equations for the full physical regimes $(\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{c}_{F}\ensuremath{\le}0)$. With the exact wave function some quantities which are important in many-body physics are obtained, including the one-body density matrix and the momentum distribution of the ground state for finite system. It is shown that the shell structure of the density profiles disappears with the increase of the interaction and in the fermionic Tonks-Girardeau limit the density distribution shows the same behavior as that of an ideal Bose gas. However, the one-body density matrix and the momentum distribution exhibit completely different structures compared with their bosonic counterparts.

Two-dimensional binary clusters in a hard-wall trap: Structural and spectral properties

Within the Monte Carlo formalism supplemented by the modified Newton-Raphson optimization technique, we investigated structural and dynamical properties of two-dimensional binary clusters confined in an external hard-wall potential. Two species of differently charged classical particles, interacting through the repulsive Coulomb force are confined in the cluster. Subtle changes in the energy landscape and the stable cluster configurations are investigated as a function of the total number of particles and the relative number of each of the two particle species. The excitation spectrum and the normal modes corresponding to the ground-state configuration of the system are discussed, and the lowest nonzero eigenfrequency as a measure of the stability of the cluster is analyzed. The influence of the particle mass on the eigenfrequencies and eigenmodes are studied, i.e., we study a binary system of particles with different charge and different mass. Several unique features distinct from a monodisperse system are obtained.

Ground-state properties of one-dimensional ultracold Bose gases in a hard-wall trap

We investigate the ground state of the system of $N$ bosons enclosed in a hard-wall trap interacting via a repulsive or attractive $\ensuremath{\delta}$-function potential. Based on the Bethe ansatz method, the explicit ground state wave function is derived and the corresponding Bethe ansatz equations are solved numerically for the full physical regime from the Tonks limit to the strongly attractive limit. It is shown that the solution takes a different form in different regime. We also evaluate the one body density matrix and second-order correlation function of the ground state for finite systems. In the Tonks limit the density profiles display the Fermi-like behavior, while in the strongly attractive limit the Bosons form a bound state of $N$ atoms corresponding to the $N$-string solution. The density profiles show the continuous crossover behavior in the entire regime. Further, the correlation function indicates that the Bose atoms bunch closer as the interaction constant decreases.

Matter-wave diffraction in time with a linear potential

Diffraction in time of matter waves incident on a shutter which is removed at time $t=0$ is studied in the presence of a linear potential. The solution is also discussed in phase space in terms of the Wigner function. An alternative configuration relevant to current experiments where particles are released from a hard wall trap is also analyzed for single-particle states and for a Tonks-Girardeau gas.

Interference of a Bose-Einstein condensate in a hard-wall trap: From the nonlinear Talbot effect to the formation of vorticity

We theoretically study the coherent expansion of a Bose-Einstein condensate in the presence of a confining impenetrable hard-wall potential. The nonlinear dynamics of the macroscopically coherent matter field results in rich and complex spatio-temporal interference patterns demonstrating the formation of vorticity and solitonlike structures, and the fragmentation of the condensate into coherently coupled pieces.