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arXiv:0907.0483 · Physical Review A

Dynamic response of one-dimensional bosons in a trap

Vitaly N. Golovach, Anna Minguzzi, L. I. Glazman

Published Oct 16, 2009• 32 citations
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Abstract

We calculate the dynamic structure factor $S(q,\ensuremath{\omega})$ of a one-dimensional (1D) interacting Bose gas confined in a harmonic trap. The effective interaction depends on the strength of the confinement enforcing the (1D) motion of atoms; interaction may be further enhanced by superimposing an optical lattice on the trap potential. In the compressible state, we find that the smooth variation in the gas density around the trap center leads to softening of the singular behavior of $S(q,\ensuremath{\omega})$ at the first Lieb excitation mode compared to the behavior predicted for homogeneous 1D systems. Nevertheless, the density-averaged response $\overline{S}(q,\ensuremath{\omega})$ remains a nonanalytic function of $q$ and $\ensuremath{\omega}$ at the first Lieb excitation mode in the limit of weak trap confinement. The exponent of the power-law nonanalyticity is modified due to the inhomogeneity in a universal way and thus bears unambiguously the information about the (homogeneous) Lieb-Liniger model. A strong optical lattice causes formation of Mott phases. Deep in the Mott regime, we predict a semicircular peak in $S(q,\ensuremath{\omega})$ centered at the on-site repulsion energy, $\ensuremath{\omega}=U$. Similar peaks of smaller amplitudes exist at multiples of $U$ as well. We explain the suppression of the dynamic response with entering into the Mott regime, observed recently by Cl\'ement et al. [Phys. Rev. Lett. 102, 155301 (2009)], based on an $f$-sum rule for the Bose-Hubbard model.

DOI
10.1103/physreva.80.043611

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We calculate the dynamic structure factor S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) of a one-dimensional (1D) interacting Bose gas confined in a harmonic trap. The effective interaction depends on the strength of the confinement enforcing the (1D) motion of atoms; interaction may be further enhanced by superimposing an optical lattice on the trap potential. In the compressible state, we find that the smooth variation in the gas density around the trap center leads to softening of the singular behavior of S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) at the first Lieb excitation mode compared to the behavior predicted for homogeneous 1D systems. Nevertheless, the density-averaged response S‾(q,\ensuremathω)\overline{S}(q,\ensuremath{\omega})S(q,\ensuremathω) remains a nonanalytic function of qqq and \ensuremathω\ensuremath{\omega}\ensuremathω at the first Lieb excitation mode in the limit of weak trap confinement. The exponent of the power-law nonanalyticity is modified due to the inhomogeneity in a universal way and thus bears unambiguously the information about the (homogeneous) Lieb-Liniger model. A strong optical lattice causes formation of Mott phases. Deep in the Mott regime, we predict a semicircular peak in S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) centered at the on-site repulsion energy, \ensuremathω=U\ensuremath{\omega}=U\ensuremathω=U. Similar peaks of smaller amplitudes exist at multiples of UUU as well. We explain the suppression of the dynamic response with entering into the Mott regime, observed recently by Cl'ement et al. [Phys. Rev. Lett. 102, 155301 (2009)], based on an fff-sum rule for the Bose-Hubbard model.
We study the quantum (zero-temperature) critical behaviors of confined particle systems described by the one-dimensional (1D) Bose-Hubbard model in the presence of a confining potential, at the Mott insulator to superfluid transitions, and within the gapless superfluid phase. Specifically, we consider the hard-core limit of the model, which allows us to study the effects of the confining potential by exact and very accurate numerical results. We analyze the quantum critical behaviors in the large trap-size limit within the framework of the trap-size scaling (TSS) theory, which introduces a new trap exponent $\ensuremath{\theta}$ to describe the dependence on the trap size. This study is relevant for experiments of confined quasi-1D cold atom systems in optical lattices. At the low-density Mott transition, TSS can be shown analytically within the spinless fermion representation of the hard-core limit. The trap-size dependence turns out to be more subtle in the other critical regions, when the corresponding homogeneous system has a nonzero filling fff, showing an infinite number of level crossings of the lowest states when increasing the trap size. At the n=1n=1n=1 Mott transition, this gives rise to a modulated TSS: The TSS is still controlled by the trap-size exponent $\ensuremath{\theta}$, but it gets modulated by periodic functions of the trap size. Modulations of the asymptotic power-law behavior is also found in the gapless superfluid region, with additional multiscaling behaviors.
While the one-dimensional (1D) Bose gas appears to exhibit superfluid response under certain conditions, it fails the Landau criterion according to the elementary excitation spectrum calculated by Lieb. The apparent riddle is solved by calculating the dynamic structure factor of the Lieb-Liniger 1D Bose gas. In a dual representation, strongly interacting bosons are mapped into weakly interacting fermions, which are treated in the Hartree-Fock and generalized random-phase approximations. The dynamic structure factor is calculated analytically in this approximation, which is valid to first order in $1∕\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is the dimensionless interaction strength of the model. The results clearly indicate a suppression of superfluidity-breaking umklapp excitations near the Tonks-gas limit, which should be observable by Bragg scattering in current experiments.
Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model and can exist in two different phases, Mott insulator or superfluid, depending on the strength of the system parameters. In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent Mott lobes. Because of this, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition in a point of the phase diagram where no transition occurs. Motivated by this, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Moreover, we study the one-dimensional repulsively interacting Bose gas under harmonic confinement, with special attention to the asymptotic behavior of the momentum distribution, which is a universal k−4k^{-4}k−4 decay characterized by the Tan's contact. The latter constitutes a direct signature of the short-range correlations in such an interacting system and provides valuable insights about the role of the interparticle interactions. We investigate the system constituted of NNN interacting particles in the strongly interacting limit. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas. Because of the difficulty in analytically solving the system for NNN particles at finite interaction, the Tonks-Girardeau regime provides a favorable scenario to probe the contact. Therefore, we are able to provide an analytical formula for the Tan's contact. Furthermore, we analyze the scaling properties of the Tan's contact in terms of NNN in the high-temperature regime as well as in the strongly interacting regime. Finally, we compare our analytical calculations of the Tan's contact to quantum Monte Carlo simulations and discuss some fundamental differences between the canonical and the grand-canonical ensembles.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.
We investigate a one-dimensional Bose--Hubbard model in a parametrically driven global harmonic trap. The delicate interplay of both the local interaction of the atoms in the lattice and the driving of the global trap allows us to control the dynamical stability of the trapped quantum many-body state. The impact of the atomic interaction on the dynamical stability of the driven quantum many-body state is revealed in the regime of weak interaction by analyzing a discretized Gross--Pitaevskii equation within a Gaussian variational ansatz, yielding a Mathieu equation for the condensate width. The parametric resonance condition is shown to be modified by the atom interaction strength. In particular, the effective eigenfrequency is reduced for growing interaction in the mean-field regime. For a stronger interaction, the impact of the global parametric drive is determined by the numerically exact time-evolving block decimation scheme. When the trapped bosons in the lattice are in a Mott insulating state, the absorption of energy from the driving field is suppressed due to the strongly reduced local compressibility of the quantum many-body state. In particular, we find that the width of the local Mott region shows a breathing dynamics. Finally, we observe that the global modulation also induces an effective time-independent inhomogeneous hopping strength for the atoms.
We analyze the macroscopic dynamics of a Bose gas in a harmonic trap with a superimposed two-dimensional optical lattice, assuming a weak coupling between different lattice sites. We consider the situation in which the local chemical potential at each lattice site can be considered as that provided by the Lieb-Liniger solution. Because of the weak coupling between sites and the form of the chemical potential, the three-dimensional ground-state density profile and the excitation spectrum acquire remarkable properties different from both 1D and 3D gases. We call this system a quasi-Tonks gas. We discuss the range of applicability of this regime, as well as realistic experimental situations where it can be observed.