Interacting Bose Gas Research Articles

Generalized Hydrodynamics with Space-Time Inhomogeneous Interactions.

We provide a new hydrodynamic framework to describe out-of-equilibrium integrable systems with space-time inhomogeneous interactions. Our result builds up on the recently introduced generalized hydrodynamics (GHD). The method allows us to analytically describe the dynamics during generic space-time-dependent smooth modulations of the interactions. As a proof of concept, we study experimentally motivated interaction quenches in the trapped interacting Bose gas, which cannot be treated with current analytical or numerical methods. We also benchmark our results in the XXZ spin chain and in the classical sinh-Gordon model.

Quantum work of an optical lattice

A classic example of a quantum quench concerns the release of a interacting Bose gas from an optical lattice. The local properties of quenches such as this have been extensively studied however the global properties of these non-equilibrium quantum systems have received far less attention. Here we study several aspects of global non-equilibrium behavior by calculating the amount of work done by the quench as measured through the work distribution function. Using Bethe Ansatz techniques we determine the Loschmidt amplitude and work distribution function of the Lieb-Liniger gas after it is released from an optical lattice. We find the average work and its universal edge exponents from which we determine the long time decay of the Loshcmidt echo and highlight striking differences caused by the the interactions as well as changes in the geometry of the system. We extend our calculation to the attractive regime of the model and show that the system exhibits properties similar to the super Tonks-Girardaeu gas. Finally we examine the prominent role played by bound states in the work distribution and show that, with low probability, they allow for work to be extracted from the quench.

Bogoliubov theory for dilute Bose gases: The Gross-Pitaevskii regime

In 1947, Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely, to test the validity of Bogoliubov’s predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes, we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in Boccato et al. [Commun. Math. Phys. (to be published); preprint arXiv:1812.03086 and Acta Math. 222, 219–335 (2019); e-print arXiv:1801.01389], which extends a previous result obtained in Boccato et al. [Commun. Math. Phys. 359, 975 (2018)], removing the assumption of a small interaction potential.

Thermodynamics of rotating weakly interacting Bose gases in a harmonic trap

The analytical expressions of particle number, critical temperature, condensate fraction, total energy, specific heat and spatial distribution of particle current for rotating weakly interacting Bose gases in a harmonic trap are derived by using the local density approximation (LDA). The effects of weakly repulsive interaction on the above thermodynamics are discussed by introducing a positive scattering length. Our results completely coincide with those for the case of rotating ideal Bose gases when the weak interaction disappears.

Hydrodynamics of the interacting Bose gas in the Quantum Newton Cradle setup

Describing and understanding the motion of quantum gases out of equilibrium is one of the most important modern challenges for theorists. In the groundbreaking Quantum Newton Cradle experiment [Kinoshita, Wenger and Weiss, Nature 440, 900 (2006)], quasi-one-dimensional cold atom gases were observed with unprecedented accuracy, providing impetus for many developments on the effects of low dimensionality in out-of-equilibrium physics. But it is only recently that the theory of generalized hydrodynamics has provided the adequate tools for a numerically efficient description. Using it, we give a complete numerical study of the time evolution of an ultracold atomic gas in this setup, in an interacting parameter regime close to that of the original experiment. We evaluate the full evolving phase-space distribution of particles. We simulate oscillations due to the harmonic trap, the collision of clouds without thermalization, and observe a small elongation of the actual oscillation period and cloud deformations due to many-body dephasing. We also analyze the effects of weak anharmonicity. In the experiment, measurements are made after release from the one-dimensional trap. We evaluate the gas density curves after such a release, characterizing the actual time necessary for reaching the asymptotic state where the integrable quasi-particle momentum distribution function emerges.

Quantum dark solitons in the one-dimensional Bose gas

Dark and grey soliton-like states are shown to emerge from numerically constructed superpositions of translationally-invariant eigenstates of the interacting Bose gas in a toroidal trap. The exact quantum many-body dynamics reveals a density depression with superdiffusive spreading that is absent in the mean-field treatment of solitons. A simple theory based on finite-size bound states of holes with quantum-mechanical center-of-mass motion quantitatively explains the time-evolution of the superposition states and predicts quantum effects that could be observed in ultra-cold gas experiments. The soliton phase step is shown to be a key ingredient of an accurate finite size approximation, which enables us to compare the theory with numerical simulations. The fundamental soliton width, an invariant property of the quantum dark soliton, is shown to deviate from the Gross-Pitaevskii predictions in the interacting regime and vanishes in the Tonks-Girardeau limit.

Statistical fluctuations and quasi phase transition of mesoscopic Bose–Einstein condensation in anharmonic trap

The possibility of quasi phase transition of interacting Bose gas in mesoscopic regime and in the anharmonic trap is studied by a correlated two-body basis function approach. Our methodology accurately describes all measures of statistical fluctuations in anharmonic trap with different anharmonicity. The results addressed here go beyond the mean-field theory, correctly describe finite-sized effect and one unique formalism is valid for entire temperature region. Our main discovery : the lambda-structure in all fluctuation measures near the transition temperature remains same qualitatively in all traps. We also define three characteristic temperatures calculated from the point of inflexion of the lambda-structure of standard deviation, third and fourth central moment. All the three characteristic temperatures smoothly increase with increase in anharmonicity which jointly confirm that the achievement of condensation is more favourable in anharmonic trap. The possibility of true phase transition is ruled out for all traps for the mesoscopic Bose–Einstein condensate. However the additional calculation of transition exponent for the specific heat capacity nicely exhibits that quasi phase transition is more prominent in harmonic trap in comparison with anharmonic traps.

Bunching, clustering, and the buildup of few-body correlations in a quenched unitary Bose gas

We study the growth of two- and three-body correlations in an ultracold Bose gas quenched to unitarity. This is encoded in the dynamics of the two- and three-body contacts analyzed in this work. Via a set of relations connecting many-body correlations dynamics with few-body models, signatures of the Efimov effect are mapped out as a function of evolution time at unitarity over a range of atomic densities $n$. For the thermal resonantly interacting Bose gas, we find that atom-bunching leads to an enhanced growth of few-body correlations. These atom-bunching effects also highlight the interplay between few-body correlations that occurs before genuine many-body effects enter on Fermi timescales.

Self-consistent determination of the many-body state of ultracold bosonic atoms in a one-dimensional harmonic trap

We study zero-temperature quantum fluctuations in harmonically trapped one-dimensional interacting Bose gases,using the self-consistent multiconfigurational time-dependent Hartree method. We define phase fluctuations from the full single-particle density matrix by the spatial decay exponent of off-diagonal long-range order. In a regime of mesoscopic particle numbers and moderate contact couplings, we derive the spatial dependence of the amplitude of phase fluctuations, determined from the self-consistently derived shape of the field operator orbitals and Fock space orbital occupation amplitudes. It is shown that the phase fluctuations display a peak, which in turn corresponds to a dip of the first-order correlations in position space, akin to what has previously been obtained in the Tonks–Girardeau limit of very large interactions and low densities.

Low-Lying Energy Levels of a One-Dimensional Weakly Interacting Bose Gas under Zero Boundary Conditions

We diagonalize the second-quantized Hamiltonian of a one-dimensional Bose gas with a non-point repulsive interatomic potential and zero boundary conditions. At a weak coupling, the solutions for the ground-state energy E0 and the dispersion law E(k) coincide with the Bogoliubov solutions for a periodic system. In this case, the single-particle density matrix F1(x, x′) at T = 0 is close to the solution for a periodic system and, at T > 0, is significantly different from it. We also obtain that the wave function ⟨w(x, t)⟩ of the effective condensate is close to a constant √︀N0/L inside the system and vanishes on the boundaries (here, N0 is the number of atoms in the effective condensate, and L is the size of the system). We find the criterion of applicability of the method, according to which the method works for a finite system at very low temperature and with a weak coupling (a weak interaction or a large concentration).

Intertwined superfluidity and density wave order in a p -orbital Bose condensate

We study a continuum model of the weakly interacting Bose gas in the presence of an external field with minima forming a triangular lattice. The second lowest band of the single-particle spectrum ($p$-band) has three minima at non-zero momenta. We consider a metastable Bose condensate at these momenta and find that, in the presence of interactions that vary slowly over the lattice spacing, the order parameter space is isomorphic to $S^{5}$. We show that the enlarged symmetry leads to the loss of topologically stable vortices, as well as two extra gapless modes with quadratic dispersion. The former feature implies that this non-Abelian condensate is a "failed superfluid" that does not undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition. Order-by-disorder splitting appears suppressed, implying that signatures of the $S^5$ manifold ought to be observable at low temperatures.

Momentum distribution and coherence of a weakly interacting Bose gas after a quench

We consider a weakly interacting uniform atomic Bose gas with a time-dependent nonlinear coupling constant. By developing a suitable Bogoliubov treatment we investigate the time evolution of several observables, including the momentum distribution, the degree of coherence in the system, and their dependence on dimensionality and temperature. We rigorously prove that the low-momentum Bogoliubov modes remain frozen during the whole evolution, while the high-momentum ones adiabatically follow the change in time of the interaction strength. At intermediate momenta we point out the occurrence of oscillations, which are analogous to Sakharov oscillations. We identify two wide classes of time-dependent behaviors of the coupling for which an exact solution of the problem can be found, allowing for an analytic computation of all the relevant observables. A special emphasis is put on the study of the coherence property of the system in one spatial dimension. We show that the system exhibits a smooth "light-cone effect," with typically no prethermalization.

Dimensional crossover for the beyond-mean-field correction in Bose gases

We present a detailed beyond-mean-field analysis of a weakly interacting Bose gas in the crossover from three to low dimensions. We find an analytical solution for the energy and provide a clear qualitative picture of the crossover in the case of a box potential with periodic boundary conditions. We show that the leading contribution of the confinement-induced resonance is of beyond-mean-field order and calculate the leading corrections in the three- and low-dimensional limits. We also characterize the crossover for harmonic potentials in a model system with particularly chosen short- and long-range interactions and show the limitations of the local-density approximation. Our analysis is applicable to Bose-Bose mixtures and gives a starting point for developing the beyond-mean-field theory in inhomogeneous systems with long-range interactions such as dipolar particles or Rydberg-dressed atoms.

Prethermalization to thermalization crossover in a dilute Bose gas following an interaction ramp

The dynamics of a weakly interacting Bose gas at low temperatures is close to integrable due to the approximate quadratic nature of the many-body Hamiltonian. While the short-time physics after an abrupt ramp of the interaction constant is dominated by the integrable dynamics, integrability is broken at longer times by higher-order interaction terms in the Bogoliubov Hamiltonian, in particular by Beliaev-Landau scatterings involving three quasiparticles. The two-stage relaxation process is highlighted in the evolution of local observables such as the density-density correlation function: a dephasing mechanism leads the system to a prethermal stage, followed by true thermalization conveyed by quasiparticle collisions. Our results bring the crossover from prethermalization to thermalization within reach of current experiments with ultracold atomic gases.

Exponents of Spectral Functions in the One-Dimensional Bose Gas

The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe’s ansatz by Lieb and Liniger (Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State). The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge c = 1 . For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The “impurity” term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole Green’s function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.

Proof of Bose condensation for weakly interacting lattice bosons

A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence of the standard or zero-mode Bose condensation at sufficiently low temperature. This result is valid for sufficiently small interaction potential and small values of chemical potential. Our method exploits infra-red bound for the suitable two-point Bogolyubov’s inner product. We do not use the reflection positivity or some expansion methods.

Application of the functional renormalization group to Bose gases: From linear to hydrodynamic fluctuations

We study weakly interacting Bose gases using the functional renormalization group with a hydrodynamic effective action. We use a scale-dependent parametrization of the boson fields that interpolates between a Cartesian representation at high momenta and an amplitude-phase one for low momenta. We apply this to Bose gases in two and three dimensions near the superfluid phase transition where they can be described by statistical O(2) models. We are able to give consistent physical descriptions of the infrared regime in both two and three dimensions. In particular, and in contrast to previous studies using the functional renormalization group, we find a stable superfluid phase at finite temperatures in two dimensions. We compare our results for the superfluid and boson densities with Monte-Carlo simulations, and we find they are in reasonable agreement.

Re-examining the quadratic approximation in theory of a weakly interacting Bose gas with condensate: the role of nonlocal interaction potentials

We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its c-number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.

INFLUENCE OF THE FINITE SIZE EFFECT ON PROPERTIES OF A WEAKLY INTERACTING BOSE GAS IN IMPROVED HATREE-FOCK APPROXIMATION

The finite size effect causes many interesting behaviors in properties of a weakly interacting Bose gas. These behaviors were considered in one-loop approximation of quantum field theory. In this paper the influence is investigated in improved Hatree-Fock approximation, which gives more accurate results.

Finite-Temperature Disordered Bosons in Two Dimensions.

We study phase transitions in a two dimensional weakly interacting Bose gas in a random potential at finite temperatures. We identify superfluid, normal fluid, and insulator phases and construct the phase diagram. At T=0 one has a tricritical point where the three phases coexist. The truncation of the energy distribution at the trap barrier, which is a generic phenomenon in cold atom systems, limits the growth of the localization length and in contrast to the thermodynamic limit the insulator phase is present at any temperature.