One-dimensional Harmonic Trap Research Articles

Engineering impurity Bell states through coupling with a quantum bath

We theoretically demonstrate the feasibility of creating Bell states in multicomponent ultracold atomic gases by solely using the ability to control the interparticle interactions via Feshbach resonances. For this we consider two distinguishable impurities immersed in an atomic background cloud of a few bosons, with the entire system being confined in a one-dimensional harmonic trap. By analyzing the numerically obtained ground states we demonstrate that the two impurities can form spatially entangled bipolaron states due to mediated interactions from the bosonic bath. Our analysis is based on calculating the correlations between the two impurities in a two-mode basis, which is experimentally accessible by measuring the particle positions in the left or right sides of the trap. While interspecies interactions are crucial in order to create the strongly entangled impurity states, they can also inhibit correlations depending on the ordering of the impurities and three-body impurity-bath correlations. We show how these drawbacks can be mitigated by manipulating the properties of the bath, namely its size, mass, and intraspecies interactions, allowing one to create impurity Bell states over a wide range of impurity-impurity interactions. Published by the American Physical Society 2024

Characteristic features of strong correlation: lessons from a 3-fermion one-dimensional harmonic trap

Abstract Many quantum phenomena responsible for key applications in material science and quantum chemistry arise in the strongly correlated regime. This is at the same time, a costly regime for computer simulations. In the limit of strong correlation analytic solutions exist, but as we move away from this limit numerical simulation are needed, and accurate quantum solutions do not scale well with the number of interacting particles. In this work we propose to use few-particle harmonic traps in combination with twisted light as a quantum emulator to investigate the transition into a strongly-correlated regime. Using both analytic derivations and numerical simulations we generalize previous findings on 2 Coulomb interacting fermions trapped in a one-dimensional harmonic trap to the case of 3 fermions. The 4 signatures of strong correlation we have identified in the one-dimensional harmonic trap are: (i) the ground state density is highly localized around N equilibrium positions, where N is the number of particles, (ii) the symmetric and antisymmetric ground state wavefunctions become degenerate, (iii) the von Neumann entropy grows, (iv) the energy spectrum is fully characterized by N normal modes or less. Our findings describe the low-energy behavior of electrons in quantum wires and ions in Paul traps. Similar features have also been reported for cold atoms in optical lattices.

Few particles with an impurity in a one-dimensional harmonic trap

We present a comprehensive study of the static properties of a mobile impurity interacting with a bath with a few particles trapped in a one-dimensional harmonic trap. We consider baths with either identical bosons or distinguishable particles and we focus on the limiting case where the bath is non-interacting. We provide numerical results for the energy spectra and density profiles by means of the exact diagonalization of the Hamiltonian, and find that these systems show non-trivial solutions, even in the limit of infinite repulsion. A detailed physical interpretation is provided for the lowest energy states. In particular, we find a seemingly universal transition from the impurity being localized in the center of the trap to being expelled outside the majority cloud. We also develop an analytical ansatz and a mean-field solution to compare them with our numerical results in limiting configurations.

Correlated dynamics of collective droplet excitations in a one-dimensional harmonic trap

We address the existence and dynamics of one-dimensional harmonically confined quantum droplets, appearing in two-component mixtures by deploying a nonperturbative approach. We find that, in symmetric homonuclear settings, beyond Lee-Huang-Yang correlations result in flat-top droplet configurations for either decreasing intercomponent attraction or larger atom number. Asymmetric mixtures feature spatial mixing among the involved components with the more strongly interacting or heavier one exhibiting flat-top structures. Applying quenches on the harmonic trap we trigger the lowest-lying collective droplet excitations. The interaction-dependent breathing frequency, being slightly reduced in the presence of correlations, shows a decreasing trend for stronger attractions. Semi-analytical predictions are also obtained within the Lee-Huang-Yang framework. For relatively large quench amplitudes the droplet progressively delocalizes and higher-lying motional excitations develop in its core. Simultaneously, enhanced intercomponent entanglement and long-range two-body intracomponent correlations arise. In sharp contrast, the dipole motion remains robust irrespectively of the system parameters. Species selective quenches lead to a correlation-induced dephasing of the droplet or to irregular dipole patterns due to intercomponent collisions.

Probing many-body effects in harmonic traps with twisted light

We explore the potential of twisted light, a structured beam carrying orbital angular momentum, as a tool to unveil many-body effects in parabolically confined systems. According to the generalized Kohn theorem, the dipole response of such a multiparticle system to a spatially homogeneous probe is indistinguishable from the response of a system of noninteracting particles. Twisted light however can excite internal degrees of freedom, resulting in the appearance of new peaks in the multipole spectrum which are not present when the probe is a plane wave. We also demonstrate the ability of the proposed twisted light probe to capture the transition of interacting fermions into a strongly correlated regime in a one-dimensional harmonic trap. We report that, by suitable choice of the probe's parameters, the transition into a strongly correlated phase manifests itself as an approaching and ultimate superposition of peaks in the second-order quadrupole response. These features are observed in exact calculations for two electrons and well reproduced in adiabatic time-dependent density-functional theory simulations.

Determine the equilibrium distribution by avoiding utilizing the Stirling formula

The derivations of the most probable distribution are not self-consistent in most statistical physics textbooks since the Stirling’s approximation adopted in the derivations is not valid for systems with very few particles. Any improvement by taking more higher terms of the Stirling series is insufficient due to erroneous divergence for small number. In this paper, we derive the equilibrium distribution by extrapolating the factorial function by the gamma function so that the Stirling’s approximation or the Stirling formula is avoided. For the ideal Bose system in a one-dimensional harmonic trap, we find that the equilibrium distribution given by this paper coincide with the rigorous result perfectly even for very small system size.

Sudden quench of harmonically trapped mass-imbalanced fermions

Dynamical properties of two-component mass-imbalanced few-fermion systems confined in a one-dimensional harmonic trap following a sudden quench of interactions are studied. It is assumed that initially the system is prepared in the non-interacting ground state and then, after a sudden quench of interactions, the unitary evolution is governed by interacting many-body Hamiltonian. By careful analysis of the evolution of the Loschmidt echo, density distributions of the components, and entanglement entropy between them, the role of mass imbalance and particle number imbalance on the system’s evolution stability are investigated. All the quantities studied manifest a dramatic dependence on the number of heavy and lighter fermions in each component at a given quench strength. The results may have implications for upcoming experiments on fermionic mixtures with a well-defined and small number of particles.

Two ultracold highly magnetic atoms in a one-dimensional harmonic trap

We theoretically investigate the properties of two interacting ultracold highly magnetic atoms trapped in a one-dimensional harmonic potential. The atoms interact via an anisotropic long-range dipole-dipole interaction, which in one dimension effectively can be modeled by the contact interaction. We investigate the interplay of the external magnetic field, the spin-spin interaction, and the trapping potential and how they affect the magnetization of the system. We show the role of indistinguishability and symmetries in the dynamics by studying the time evolution of the observables that could be measured experimentally. The presented model may depict the on-site interaction of the extended Hubbard models, therefore giving a better understanding of the fundamental building block of the respective many-body quantum simulators.

Unconventional pairing in few-fermion systems at finite temperature

Attractively interacting two-component mixtures of fermionic particles confined in a one-dimensional harmonic trap are investigated. Properties of balanced and imbalanced systems are systematically explored with the exact diagonalization approach, focusing on the finite-temperature effects. Using single- and two-particle density distributions, specific non-classical pairing correlations are analyzed in terms of the noise correlations—quantity directly accessible in state-of-the-art experiments with ultra-cold atoms. It is shown that along with increasing temperature, any imbalanced system hosting Fulde–Ferrel–Larkin–Ovchinnikov pairs crossovers to a standard Bardeen-Cooper-Schrieffer one characterized by zero net momentum of resulting pairs. By performing calculations for systems with different imbalances, the approximate boundary between the two phases on a phase diagram is determined.

Several fermions strongly interacting with a heavy mobile impurity in a one-dimensional harmonic trap

We propose a numerically exact method for a mixture with a single impurity immersed in several majority fermions, confined in a harmonic potential. We separate one of the degrees of freedom through an appropriately tailored canonical transformation and perform exact diagonalization on the simplified Hamiltonian. This method is especially effective for a heavy impurity, where it outmatches the typical exact diagonalization approach. We used our method to calculate energy and density profiles of the first few eigenstates for the mixture with up to ten majority fermions.

Measurement of one-dimensional harmonic trap frequency through multimode Bose–Einstein condensate interference

Measurement of one-dimensional harmonic trap frequency through multimode Bose–Einstein condensate interference

Breathing dynamics of the few-body Bose polaron in a species-selective harmonic trap

We perform an extensive numerical study on the breathing dynamics of a few-body Bose polaron setup in a one-dimensional species-selective harmonic trap. The dynamics is triggered by a quench of the impurity trap. The excitation of the background majority atoms is mediated via the majority-impurity interaction. The breathing spectrum is obtained for different numbers of majority particles, several values of the majority-component interaction strengths and trap ratios. It is further compared to the breathing spectrum of a particle-balanced few-body Bose-Bose mixture. In particular, for equal post-quench traps the employed protocol allows to couple states of different center-of-mass parity in contrast to species-symmetric trap quenches. Among the participating eigenstates we identify one having odd center-of-mass parity and even global parity. The breathing frequency induced by this state is a monotonically decreasing function of the coupling parameter. Importantly, in order to be observable it requires the entanglement between the species to be taken into account. We demonstrate this by comparing the numerically exact results obtained by means of the Multi-Layer Multi-Configuration Time-Dependent Hartree Method for Mixtures to the ones of a species mean-field ansatz. The entanglement-sensitive breathing frequency persists also for unequal post-quench traps where the center-of-mass cannot be decoupled. Finally, we analyze the impact of parity symmetry on the breathing dynamics by initializing a state of odd global parity. We evidence a striking resemblance to the ground state breathing dynamics.

Connecting Scrambling and Work Statistics for Short-Range Interactions in the Harmonic Oscillator.

We investigate the relationship between information scrambling and work statistics after a quench for the paradigmatic example of short-range interacting particles in a one-dimensional harmonic trap, considering up to five particles numerically. In particular, we find that scrambling requires finite interactions, in the presence of which the long-time average of the squared commutator for the individual canonical operators is directly proportional to the variance of the work probability distribution. In addition to the numerical results, we outline the mathematical structure of the N-body system which leads to this outcome. We thereby establish a connection between the scrambling properties and the induced work fluctuations, with the latter being an experimental observable that is directly accessible in modern cold-atom experiments.

Many-body molecule formation at a domain wall in a one-dimensional strongly interacting ultracold Fermi gas

We analyze how the presence of the bound state on top of strong intercomponent contact repulsion affects the dynamics of a two-component ultracold Fermi gas confined in a one-dimensional harmonic trap. By performing full many-body numerical calculations, we retrieve dynamics of an initially phase-separated state that has been utilized to excite the spin-dipole mode in experimental settings. We observe an appearance of pairing correlations at the domain wall, heralding the onset of a molecular fraction at the interlayer between the components.

Benchmarking the multiconfigurational Hartree method by the exact wavefunction of two harmonically trapped bosons with contact interaction

We consider two bosons in a one-dimensional harmonic trap, interacting by a contact potential, and compare the exact solution of this problem to a self-consistent numerical solution by using the multiconfigurational time-dependent Hartree (MCTDH) method. We thereby benchmark the predictions of the MCTDH method with a few-body problem that has an analytical solution for the most commonly experimentally realized interaction potential in ultracold quantum gases. It is found that exact ground state energy and first order correlations are accurately reproduced by MCTDH up to the intermediate dimensionless coupling strengths corresponding to typical background scattering lengths of magnetically trapped ultracold dilute Bose gases. For larger couplings, established for example by (a combination of) Feshbach resonances and optical trapping, the MCTDH approach overestimates the depth of the trap-induced correlation dip of first order correlations in position space, as well as overestimates the fragmentation, defined as the average relative occupation of orbitals other than the energetically lowest one. We anticipate that qualitatively similar features in the correlation function may arise for larger particle numbers, paving the way for a quantitative assessment of the accuracy of MCTDH by experiments with ultracold atoms.

Variational ansatz for p-wave fermions confined in a one-dimensional harmonic trap

We propose a very accurate and efficient variational scheme for the ground state of the system of p-wave attractively interacting fermions confined in a one-dimensional harmonic trap. By the construction, the method takes the non-analytical part of interactions exactly into account and thus it approximates the true ground-state wave function in a whole range of interactions very accurately. Within the method, we determine different properties of the system for a different number of particles and different interactions. In this way, we explore how the system and its features transit from the ideal non-interacting Fermi gas to the system of infinitely strong attractions. Additionally, we demonstrate that the ansatz may also be used on a repulsive branch of interactions where other numerical methods break down. The presented method of including zero-range interactions is very universal and may be easily generalized to other one-dimensional confinements.

Microcanonical condensate fluctuations in one-dimensional weakly-interacting Bose gases

Weakly interacting Bose gases confined in a one-dimensional harmonic trap are studied using microcanonical ensemble approaches. Combining number theory methods, I present a new approach to calculate the particle number counting statistics of the ground state occupation. The results show that the repulsive interatomic interactions increase the ground state fraction and suppresses the fluctuation of ground state at low temperature.

Comment on the paper “Microcanonical analysis of Boltzmann and Gibbs entropies in trapped cold atomic gases, K.J. Higginbotham, D.E. Sheehy, Physica A 536(2019) 122547”

In the commented paper, the authors study a gas of noninteracting fermions confined in a one-dimensional harmonic trap, and calculate the Boltzmann temperature and the Gibbs temperature of this system. They find that the Gibbs temperature is closer to the corresponding temperature given by the grand-canonical approach than the Boltzmann temperature. In this comment we point out their conclusion is not always true.

Breakdown of the mean field for dark solitons of dipolar bosons in a one-dimensional harmonic trap

We directly compare the mean-field and the many-body approach in a one-dimensional Bose system in a harmonic trap. Both contact and dipolar interactions are considered. We propose a multi-atom version of the phase imprinting method to generate dark solitons in the system. We begin with a general analysis of system dynamics and observe the emergence of a dark soliton and a shock wave. Center of mass and soliton motion become decoupled because the shock wave oscillates with the trap frequency and soliton does not. A detailed investigation of frequencies reveals significant differences between results obtained in the mean-field and the many-body pictures.

Unconventional pairing in one-dimensional systems of a few mass-imbalanced ultracold fermions

We study the ground-state properties of a two-component fermionic mixture effectively confined in a one-dimensional harmonic trap. We consider scenarios when numbers of particles in components are the same but particles have different masses. We examine whether it is possible to detect signatures of an unconventional pairing between opposite-spin fermions in the presence of attractive interactions. For this purpose, we perform the exact diagonalization of the many-body Hamiltonian and study the two-particle reduced density matrix. In agreement with expectations, we confirm that the many-body ground state is dominated by conventional pairs with a negligible total momentum for a small mass imbalance. Furthermore, we show that for sufficiently large mass ratios the domination of fundamentally different pairs is established and the Fulde-Ferrell-Larkin-Ovchinnikov phase is supported. Finally, we argue that the two mechanisms can coexist in the regime of moderate mass ratios. Due to the current experimental progress in obtaining ultra-cold fermionic systems in a few-body regime, our predictions may have some importance for the upcoming experiments.