One-dimensional Bose Gas Research Articles

Gradient corrections to the local-density approximation in the one-dimensional Bose gas

The local density approximation (LDA) is the central technical tool in the modeling of quantum gases in trapping potentials. It consists in treating the gas as an assembly of independent mesoscopic fluid cells at equilibrium with a local chemical potential, and it is justified when the correlation length is larger than the size of the cells. The LDA is often regarded as a crude approximation, particularly in the ground state of the one-dimensional (1D) Bose gas, { where the correlation length is "therefore said to be" infinite (in the sense that correlation functions decay as a power law).} Here we take another look at the LDA. The local density $\rho(x)$ is viewed as a functional of the trapping potential $V(x)$, to which one applies a gradient expansion. The zeroth order in that expansion is the LDA. The first-order correction in the gradient expansion vanishes due to reflection symmetry. At second order, there are two corrections proportional to $d^2V/dx^2$ and $(dV/dx)^2$, and we propose a method to determine the corresponding coefficients by a perturbative calculation in the Lieb-Liniger model. This leads to an expression for the coefficients in terms of matrix elements of the density operator, which can in principle be evaluated numerically for an arbitrary coupling constant; here we show how to efficiently evaluate the coefficient associated to the curvature of the potential $d^2V/dx^2$, which dominates the deviation to LDA near local minima or maxima of the trapping potential. Both coefficients are evaluated analytically in the limits of infinite repulsion (hard-core bosons) and small repulsion (quasi-condensate).} The corrected LDA density profiles are compared to DMRG calculations, with significant improvement compared to zeroth-order LDA.

Bulk Viscosity of Dual Bose and Fermi Gases in One Dimension

One-dimensional Bose and Fermi gases with contact interactions are known to exhibit the weak-strong duality, where the equilibrium thermodynamic properties of one system at weak coupling are identical to those of the other system at strong coupling. Here, we show that such duality extends beyond the thermodynamics to the frequency-dependent complex bulk viscosity, which is provided by the contact-contact response function. In particular, we confirm that the bulk viscosities of the Bose and Fermi gases agree in the high-temperature limit, where the systematic expansion in terms of fugacity is available at arbitrary coupling. We also compute their bulk viscosities perturbatively in the weak-coupling limit at arbitrary temperature, which via the duality serve as those of the Fermi and Bose gases in the strong-coupling limit.

Condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas

The extensive control of spin makes spintronics a promising candidate for future scalable quantum devices. For the generation of spin-superfluid systems, a detailed understanding of the build-up of coherence and relaxation is necessary. However, to determine the relevant parameters for robust coherence properties and faithfully witnessing thermalization, the direct access to space- and time-resolved spin observables is needed. Here, we study the thermalization of an easy-plane ferromagnet employing a homogeneous one-dimensional spinor Bose gas. Building on the pristine control of preparation and readout we demonstrate the dynamic emergence of long-range coherence for the spin field and verify spin-superfluidity by experimentally testing Landau's criterion. We reveal the structure of the emergent quasi-particles: one 'massive'(Higgs) mode, and two 'massless' (Goldstone) modes - a consequence of explicit and spontaneous symmetry breaking, respectively. Our experiments allow for the first time to observe the thermalization of an easy-plane ferromagnetic Bose gas; we find agreement for the relevant momentum-resolved observables with a thermal prediction obtained from an underlying microscopic model within the Bogoliubov approximation. Our methods and results pave the way towards a quantitative understanding of condensation dynamics in large magnetic spin systems and the study of the role of entanglement and topological excitations for its thermalization.

On Bose–Einstein Condensation in One-Dimensional Noninteracting Bose Gases in the Presence of Soft Poisson Obstacles

We study Bose–Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on \(\mathbb R\) with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With a probability arbitrarily close to one when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the rth mean, \(r \ge 1\). Furthermore, in Poisson random potentials of any fixed strength a probability arbitrarily close to one for type-I g-BEC to occur is also obtained, but our upper bound for the number of macroscopically occupied one-particle states may be large in this case.

Hole-induced anomaly in the thermodynamic behavior of a one-dimensional Bose gas

We reveal an intriguing anomaly in the temperature dependence of the specific heat of a one-dimensional Bose gas.~The observed peak holds for arbitrary interaction and remembers a superfluid-to-normal phase transition in higher dimensions, but phase transitions are not allowed in one dimension.~The presence of the anomaly signals a region of unpopulated states which behaves as an energy gap and is located below the hole branch in the excitation spectrum.~The anomaly temperature is found to be of the same order of the energy of the maximum of the hole branch.~We rely on the Bethe Ansatz to obtain the specific heat exactly and provide interpretations of the analytically tractable limits.~The dynamic structure factor is computed with the Path Integral Monte Carlo method for the first time.~We notice that at temperatures similar to the anomaly threshold, the energy of the thermal fluctuations become comparable with the maximal hole energy, leading to a qualitative change in the structure of excitations.~This excitation pattern experiences the breakdown of the quasiparticle description for any value of the interaction strength at the anomaly, similarly to any superfluid phase transition at the critical temperature.~We provide indications for future observations and how the hole anomaly can be employed for in-situ thermometry, identifying different collisional regimes and understanding other anomalies in atomic, solid-state, electronic, spin-chain and ladder systems.

Time-dependent variational Monte Carlo study of the dynamic response of bosons in an optical lattice

We study the dynamics of a one-dimensional Bose gas at unit filling in both shallow and deep optical lattices and obtain the dynamic structure factor \bm{S(k,\omega)}𝐒(𝐤,𝛚) by monitoring the linear response to a weak probe pulse. We introduce a new procedure, based on the time-dependent variational Monte Carlo method (tVMC), which allows to evolve the system in real time, using as a variational model a Jastrow-Feenberg wave function that includes pair correlations. Comparison with exact diagonalization results of \bm{S(k,\omega)}𝐒(𝐤,𝛚) obtained on a lattice in the Bose-Hubbard limit shows good agreement of the dispersion relation for sufficiently deep optical lattices, while for shallow lattices we observe the influence of higher Bloch bands. We also investigate non-linear response to strong pulses. From the power spectrum of the density fluctuations we obtain the excitation spectrum, albeit broadened, by higher harmonic generation after a strong pulse with a single low wave number. As a remarkable feature of our simulations we furthermore demonstrate that the full excitation spectrum can be retrieved from the power spectrum of the density fluctuations due to the stochastic noise inherent in any Monte Carlo method, without applying an actual perturbation.

Inducing spin-order with an impurity: phase diagram of the magnetic Bose polaron

We investigate the formation of magnetic Bose polaron, an impurity atom dressed by spin-wave excitations, in a one-dimensional spinor Bose gas. Within an effective potential model, the impurity is strongly confined by the host excitations which can even overcome the impurity-medium repulsion leading to a self-localized quasi-particle state. The phase diagram of the attractive and self-bound repulsive magnetic polaron, repulsive non-magnetic (Fröhlich-type) polaron and impurity-medium phase-separation regimes is explored with respect to the Rabi-coupling between the spin components, spin–spin interactions and impurity-medium coupling. The residue of such magnetic polarons decreases substantially in both strong attractive and repulsive branches with strong impurity-spin interactions, illustrating significant dressing of the impurity. The impurity can be used to probe and maneuver the spin polarization of the magnetic medium while suppressing ferromagnetic spin–spin correlations. It is shown that mean-field theory fails as the spinor gas approaches immiscibility since the generated spin-wave excitations are prominent. Our findings illustrate that impurities can be utilized to generate controllable spin–spin correlations and magnetic polaron states which can be realized with current cold atom setups.

The dynamics of solitons in one-dimensional Bose gas with strong two-body losses

We study the dynamics of soliton in one-dimensional Bose gas with strong two-body losses which make the system non-conservative. Employing the perturbed variational approximation, we analytically get the dynamics equations which govern the dynamics of soliton in this open system. Our results show that the strong two-body loss rate will dramatically influence the solitons exists, amplitudes and widths, which is contrary different from conservative system without two-body loss.

Expansion of a one-dimensional Bose gas: the role of interactions and kinetic-energy driving

We study the expansion of a one-dimensional boson gas by initialising it in a small region of a chain, and then suddenly allowing it to expand into the remainder of the chain. We consider three initial ground-state configurations: the Mott insulator, the conventional superfluid, whose momentum density is sharply peaked at zero momentum, and the cat-like state with momentum peaks at ±π/2, produced by kinetic driving, the latter being a particular case of a flat-band system. In turn, we consider three types of expansion: spectroscopic (with interactions tuned to zero), dynamic (with standard short-range repulsive interactions), and under kinetic driving. The numerical calculations are exact. We compute the momentum and real-space one-particle densities, as well as the two-particle momentum correlations. We find that the spectroscopic time-of-flight experiment reflects the initial momentum distribution except for the larger number of momentum states and at high momenta. For the dynamic expansion starting from an insulator, we recover the non-equilibrium quasi-condensation into momenta ±π/2, provide a physical explanation in terms of interacting bosons that is confirmed by the numerical simulation, and note the existence of nontrivial correlations in the momentum distribution. Under kinetic driving the expansion is comparatively slow, but we conjecture that at high densities it will be much faster. We compare various measures of the two-particle momentum correlations, noting that some of them tend to conceal the possible cat-like structure of a many-body state.

Local spectral density of an interacting one-dimensional Bose gas with an impurity

A weakly interacting Bose gas with a single static impurity is studied in one dimension under the assumption that its phase coherence length is longer than the system size. The local spectral density is evaluated and an analytic expression valid for positive strengths of the impurity coupling to the bosons, at all distances and at all frequencies, is obtained using the Bogoliubov--de Gennes approximation.

Mediated interaction between polarons in a one-dimensional Bose gas

We study a weakly-interacting one-dimensional Bose gas with two impurities coupled locally to the boson density. We derive analytical results for the induced interaction between the impurities at arbitrary coupling and separation $r$. At $r\lesssim \xi$, where $\xi$ denotes the healing length of the Bose gas, the interaction is well described by the mean-field contribution. Its form changes as the coupling is increased, approaching a linear function of $r$ at short distances in the regime of strong coupling. The mean-field contribution decays exponentially at arbitrary coupling for $r\gg\xi$. At such long distances, however, the effect of quantum fluctuations becomes important, giving rise to a long-ranged quantum contribution to the induced interaction. At longest distances it behaves as $1/r^3$, while at strong coupling we find an intermediate distance regime with a slower decay, $1/r$. The quantum contribution in the crossover regime is also calculated. The induced interaction between impurities (i.e., polarons) is attractive and leads to the formation of their bound state, known as bipolaron. We discuss its binding energy.

Dispersion relation of a polaron in the Yang-Gaudin Bose gas

We study a one-dimensional Bose gas with two internal states described by the Yang-Gaudin model and calculate analytically the dispersion relation of a polaron quasiparticle, which is the lowest excitation branch. We study the dispersion in the thermodynamic limit in the regimes of weak and strong interaction without limitations on the momentum. At weak interaction the polaron dispersion is in the vicinity of the dark soliton one; we calculate the leading deviation in the parametric form. At strong interaction we find an ansatz for the explicit form of the polaron dispersion. It has the form of a power series of the sine function of the momentum with interaction-dependent coefficients. By increasing the power of the series, the corresponding coefficients show faster decay and thus one practically needs only a few of them; we give the results for the first three. The coefficients of the series are connected to the Maclaurin series of the polaron dispersion and thus it is sufficient to calculate latter quantity to automatically find the power series result for the dispersion at all momenta. The derived results for the dispersion can be used to obtain explicit expressions for the exponents of the power-law singularities in the response functions at the spectral edge.

The one-dimensional Bose gas with strong two-body losses: the effect of the harmonic confinement

We study the dynamics of a one-dimensional Bose gas in presence of strong two-body losses. In this dissipative quantum Zeno regime, the gas fermionises and its dynamics can be described with a simple set of rate equations. Employing the local density approximation and a Boltzmann-like dynamical equation, the description is extended to take into account an external potential. We show that in the absence of confinement the population is depleted in an anomalous way and that the gas behaves as a low-temperature classical gas. The harmonic confinement accelerates the depopulation of the gas and introduces a novel decay regime, which we thoroughly characterise.

Polaron-Depleton Transition in the Yrast Excitations of a One-Dimensional Bose Gas with a Mobile Impurity

We present exact numerical data for the lowest-energy momentum eigenstates (yrast states) of a repulsive spin impurity in a one-dimensional Bose gas using full configuration interaction quantum Monte Carlo (FCIQMC). As a stochastic extension of exact diagonalization, it is well suited for the study of yrast states of a lattice-renormalized model for a quantum gas. Yrast states carry valuable information about the dynamic properties of slow-moving mobile impurities immersed in a many-body system. Based on the energies and the first and second-order correlation functions of yrast states, we identify different dynamical regimes and the transitions between them: The polaron regime, where the impurity’s motion is affected by the Bose gas through a renormalized effective mass; a regime of a gray soliton that is weakly correlated with a stationary impurity, and the depleton regime, where the impurity occupies a dark or gray soliton. Extracting the depleton effective mass reveals a super heavy regime where the magnitude of the (negative) depleton mass exceeds the mass of the finite Bose gas.

Beyond Gross-Pitaevskii equation for 1D gas: quasiparticles and solitons

Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.

Generalized hydrodynamics in the one-dimensional Bose gas: theory and experiments

We review the recent theoretical and experimental progress regarding the generalized hydrodynamics (GHD) behavior of the one-dimensional (1D) Bose gas with contact repulsive interactions, also known as the Lieb–Liniger gas. In the first section, we review the theory of the Lieb–Liniger gas, introducing the key notions of the rapidities and of the rapidity distribution. The latter characterizes the Lieb–Liniger gas after relaxation and is at the heart of GHD. We also present the asymptotic regimes of the Lieb–Liniger gas with their dedicated approximate descriptions. In the second section we enter the core of the subject and review the theoretical results of GHD in 1D Bose gases. The third and fourth sections are dedicated to experimental results obtained in cold atom experiments: the experimental realization of the Lieb–Liniger model is presented in section 3, with a selection of key results for systems at equilibrium, and section 4 presents the experimental tests of the GHD theory. In section 5 we review the effects of atom losses, which, assuming slow loss processes, can be described within the GHD framework. We conclude with a few open questions.

Ideal Bose gas in steep one-dimensional traps

We study thermodynamic properties of a one-dimensional ideal Bose gas trapped by a steep potential of an exponential type U(q)=U0[e(2q/a)b−1]. Fugacity, energy, and heat capacity of such a system are calculated for various combinations of the potential parameters as well for several values of the number of particles N. Both the thermodynamic limit and finite N are considered. Estimations for the single-particle spectrum asymptotics are obtained in the analytical form involving the Lambert W function. In the thermodynamic limit, the Bose–Einstein condensation is predicted for 0 < b < 2. We associate such behavior with an effective temperature-dependent space dimensionality arising due to the influence of the external potential of the analyzed type.

Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons

We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.

Quantum generalized hydrodynamics of the Tonks–Girardeau gas: density fluctuations and entanglement entropy

We apply the theory of quantum generalized hydrodynamics (QGHD) introduced in (2020 Phys. Rev. Lett. 124 140603) to derive asymptotically exact results for the density fluctuations and the entanglement entropy of a one-dimensional trapped Bose gas in the Tonks–Girardeau (TG) or hard-core limit, after a trap quench from a double well to a single well. On the analytical side, the quadratic nature of the theory of QGHD is complemented with the emerging conformal invariance at the TG point to fix the universal part of those quantities. Moreover, the well-known mapping of hard-core bosons to free fermions, allows to use a generalized form of the Fisher–Hartwig conjecture to fix the non-trivial spacetime dependence of the ultraviolet cutoff in the entanglement entropy. The free nature of the TG gas also allows for more accurate results on the numerical side, where a higher number of particles as compared to the interacting case can be simulated. The agreement between analytical and numerical predictions is extremely good. For the density fluctuations, however, one has to average out large Friedel oscillations present in the numerics to recover such agreement.

Effective scaling approach to frictionless quantum quenches in trapped Bose gases

We work out the effective scaling approach to frictionless quantum quenches in a one-dimensional Bose gas trapped in a harmonic trap. The effective scaling approach produces an auxiliary equation for the scaling parameter interpolating between the noninteracting and the Thomas-Fermi limits. This allows us to implement a frictionless quench by engineering inversely the smooth trap frequency, as compared to the two-jump trajectory. Our result is beneficial to design the shortcut-to-adiabaticity expansion of trapped Bose gases for arbitrary values of interaction, and can be directly extended to the three-dimensional case.