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arXiv:1312.7005 · Physical Review A

Breakdown of scale invariance in the vicinity of the Tonks-Girardeau limit

Zhedong Zhang, G. E. Astrakharchik, David C. Aveline, Seok Jin Choi, Hélène Perrin, T. H. Bergeman, Maxim Olshanii

Published Jun 23, 2014• 11 citations
arXivHTMLPDF

Abstract

In this article, we consider the monopole excitations of a harmonically trapped Bose gas in the vicinity of the Tonks-Girardeau limit. Using Girardeau's Fermi-Bose duality and subsequently an effective fermion-fermion odd-wave interaction, we obtain the dominant correction to the scale-invariance-protected value of the excitation frequency, for microscopically small excitation amplitudes. We produce a series of diffusion Monte Carlo results that confirm our analytic prediction for three particles. And less expectedly, our result stands in excellent agreement with the result of a hydrodynamic simulation (with the Lieb-Liniger equation of state as an input) of the microscopically large but macroscopically small excitations. We also show that the frequency we obtain coincides with the upper bound derived by Menotti and Stringari using sum rules. Surprisingly, however, we found that the usually successful hydrodynamic perturbation theory predicts a shift that is $9/4$ higher than its ab initio numerical counterpart. We conjecture that the sharp boundary of the cloud in local density approximation---characterized by an infinite density gradient---renders the perturbation inapplicable. All our results also directly apply to the three-dimensional $p$-wave-interacting waveguide-confined fermions.

DOI
10.1103/physreva.89.063616

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In this article, we consider the monopole excitations of a harmonically trapped Bose gas in the vicinity of the Tonks-Girardeau limit. Using Girardeau's Fermi-Bose duality and subsequently an effective fermion-fermion odd-wave interaction, we obtain the dominant correction to the scale-invariance-protected value of the excitation frequency, for microscopically small excitation amplitudes. We produce a series of diffusion Monte Carlo results that confirm our analytic prediction for three particles. And less expectedly, our result stands in excellent agreement with the result of a hydrodynamic simulation (with the Lieb-Liniger equation of state as an input) of the microscopically large but macroscopically small excitations. We also show that the frequency we obtain coincides with the upper bound derived by Menotti and Stringari using sum rules. Surprisingly, however, we found that the usually successful hydrodynamic perturbation theory predicts a shift that is 9/49/49/4 higher than its ab initio numerical counterpart. We conjecture that the sharp boundary of the cloud in local density approximation---characterized by an infinite density gradient---renders the perturbation inapplicable. All our results also directly apply to the three-dimensional ppp-wave-interacting waveguide-confined fermions.
We consider a homogeneous 1D Bose gas with contact interactions and a large attractive coupling constant. This system can be realized in tight waveguides by exploiting a confinement induced resonance of the effective 1D scattering amplitude. By using the diffusion Monte Carlo method we show that, for small densities, the gaslike state is well described by a gas of hard rods. The critical density for cluster formation is estimated using the variational Monte Carlo method. The behavior of the correlation functions and of the frequency of the lowest breathing mode for harmonically trapped systems shows that the gas is more strongly correlated than in the Tonks-Girardeau regime.
Using a time-dependent modified nonlinear Schrödinger equation (MNLSE)-where the conventional chemical potential proportional to the density is replaced by the one inferred from Lieb-Liniger's exact solution-we study frequencies of the collective monopole excitations of a one-dimensional Bose gas. We find that our method accurately reproduces the results of a recent experimental study [E. Haller et al., Science 325, 1224 (2009)] in the full spectrum of interaction regimes from the ideal gas, through the mean-field regime, through the mean-field Thomas-Fermi regime, all the way to the Tonks-Giradeau gas. While the former two are accessible by the standard time-dependent NLSE and inaccessible by the time-dependent local density approximation, the situation reverses in the latter case. However, the MNLSE is shown to treat all these regimes within a single numerical method.
Abstract 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.
A harmonically trapped, ultracold, one-dimensional (1D) spin-1 Bose gas with strongly repulsive or attractive 1D even-wave interactions induced by a three-dimensional (3D) Feshbach resonance is studied. The exact ground state, a hybrid of Tonks-Girardeau (TG) and ideal Fermi gases, is constructed in the TG limit of infinite even-wave repulsion by a spinor Fermi-Bose mapping to a spinless ideal Fermi gas. It is then shown that in the limit of infinite even-wave attraction this same state remains an exact many-body eigenstate, now highly excited relative to the collapsed generalized McGuire-cluster ground state, showing that the hybrid TG state is completely stable against collapse to this cluster ground state under a sudden switch from infinite repulsion to infinite attraction. It is shown to be the TG limit of a hybrid super-Tonks-Girardeau (STG) state, which is metastable under a sudden switch from finite but very strong repulsion to finite but very strong attraction. It should be possible to create it experimentally by a sudden switch from strongly repulsive to strongly attractive interaction, as in the recent Innsbruck experiment on a spin-polarized bosonic STG gas. In the case of strong attraction, there should also exist another STG state of much lower energy, consisting of strongly bound dimers, a bosonic analog of a recently predicted STG state which is an ultracold gas of strongly bound bosonic dimers of fermionic atoms, but it is shown that this STG state cannot be created by such a switch from strong repulsion to strong attraction.
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 calculate the excitation modes of a one-dimensional (1D) dipolar quantum gas confined in a harmonic trap with frequency ${\ensuremath{\omega}}_{0}$ and predict how the frequency of the breathing n=2n=2n=2 mode characterizes the interaction strength evolving from the Tonks-Girardeau value ${\ensuremath{\omega}}_{2}=2{\ensuremath{\omega}}_{0}$ to the quasi-ordered, superstrongly interacting value ${\ensuremath{\omega}}_{2}=\sqrt{5}{\ensuremath{\omega}}_{0}$. Our predictions are obtained within a hydrodynamic Luttinger-liquid theory after applying the local density approximation to the equation of state for the homogeneous dipolar gas, which are in turn determined from reptation quantum Monte Carlo simulations. They are shown to be in quite accurate agreement with the results of a sum-rule approach. These effects can be observed in current experiments, revealing the Luttinger-liquid nature of 1D dipolar Bose gases.