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arXiv:cond-mat/0410311 · Physical Review A

Dynamic structure factor of the one-dimensional Bose gas near the Tonks-Girardeau limit

Joachim Brand, Alexander Yu. Cherny

Published Sep 28, 2005• 44 citations
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Abstract

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.

DOI
10.1103/physreva.72.033619

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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∕\ensuremathγ1∕\ensuremath{\gamma}1∕\ensuremathγ, where \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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.
Correlation functions related to the dynamic density response of the one-dimensional Bose gas in the model of Lieb and Liniger are calculated. An exact Bose-Fermi mapping is used to work in a fermionic representation with a pseudopotential Hamiltonian. The Hartree-Fock and generalized random phase approximations are derived and the dynamic polarizability is calculated. The results are valid to first order in $1∕\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is Lieb-Liniger coupling parameter. Approximations for the dynamic and static structure factor at finite temperature are presented. The results preclude superfluidity at any finite temperature in the large-$\ensuremath{\gamma}$ regime due to the Landau criterion. Due to the exact Bose-Fermi duality, the results apply for spinless fermions with weak ppp-wave interactions as well as for strongly interacting bosons.
We calculate the dynamic polarizability of a one-dimensional Bose gas confined into a parabolic trap near the Tonks–Girardeau limit. We use the duality between the one-dimensional Bose gas with δ-interaction and the one-dimensional Fermi gas with δ''-interaction to map the strong interaction regime of the first into the weak one of the latter. The quantities which depend only on the modulus of the wavefunction can be calculated with either of the two theories; therefore, we have used the random-phase approximation in the weak coupling regime of the fermionic system to access dynamic properties in the strong coupling regime of the bosonic one.
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 qqq 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 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.
Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate-like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In this work, we trace these elusive excitations via their effects on the dynamical structure factor of 1D strongly-interacting Bose gases at low temperature. An array of 1D Bose gases is obtained by loading a 87^{87}87Rb condensate in a 2D lattice potential. The dynamical structure factor of the system is probed by energy deposition through low-momentum Bragg excitations. The experimental signals are compared to recent theoretical predictions for the dynamical structure factor of the Lieb-Liniger model at T>0T > 0T>0. Our results demonstrate that the main contribution to the spectral widths stems from the dynamics of the interaction-induced excitations in the gas, which cannot be described by the Luttinger liquid theory.
Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In this work, we trace these elusive excitations via their effects on the dynamical structure factor of 1D strongly interacting Bose gases at low temperature. An array of 1D Bose gases is obtained by loading a 87Rb^{87}\mathrm{Rb}87Rb condensate in a two-dimensional lattice potential. The dynamical structure factor of the system is probed by energy deposition through low-momentum Bragg excitations. The experimental signals are compared to recent theoretical predictions for the dynamical structure factor of the Lieb-Liniger model at T>0T>0T>0. Our results demonstrate that the main contribution to the spectral widths stems from the dynamics of the interaction-induced excitations in the gas, which cannot be described by the Luttinger liquid theory.