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arXiv:2011.13250 · Physical Review Letters

Breakdown of Tan’s Relation in Lossy One-Dimensional Bose Gases

Isabelle Bouchoule, Jérôme Dubail

Published Apr 23, 2021• 47 citations
arXivHTMLPDF

Abstract

In quantum gases with contact repulsion, the distribution of momenta of the atoms typically decays as ∼1/|p|^{4} at large momentum p. Tan's relation connects the amplitude of that 1/|p|^{4} tail to the adiabatic derivative of the energy with respect to the coupling constant or scattering length of the gas. Here it is shown that the relation breaks down in the one-dimensional Bose gas with contact repulsion, for a peculiar class of stationary states. These states exist thanks to the infinite number of conserved quantities in the system, and they are characterized by a rapidity distribution that itself decreases as 1/|p|^{4}. In the momentum distribution, that rapidity tail adds to the usual Tan contact term. Remarkably, atom losses, which are ubiquitous in experiments, do produce such peculiar states. The development of the tail of the rapidity distribution originates from the ghost singularity of the wave function immediately after each loss event. This phenomenon is discussed for arbitrary interaction strengths, and it is supported by exact calculations in the two asymptotic regimes of infinite and weak repulsion.

DOI
10.1103/physrevlett.126.160603

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Abstract This chapter is devoted to the derivation of various thermodynamic relations, known as Tan relations, involving the so-called contact parameter. This physical quantity, which determines the behaviour of the momentum distribution of an interacting gas at large momenta, plays a crucial role in characterizing various thermodynamic quantities, like the dependence of the energy on the scattering length (sweep theorem), the relation between the energy and the momentum distribution, the behaviour of the static structure factor at large wave vectors, and the virial theorem in the presence of harmonic trapping.

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In quantum gases with contact repulsion, the distribution of momenta of the atoms typically decays as ∼1/|p|^{4} at large momentum p. Tan's relation connects the amplitude of that 1/|p|^{4} tail to the adiabatic derivative of the energy with respect to the coupling constant or scattering length of the gas. Here it is shown that the relation breaks down in the one-dimensional Bose gas with contact repulsion, for a peculiar class of stationary states. These states exist thanks to the infinite number of conserved quantities in the system, and they are characterized by a rapidity distribution that itself decreases as 1/|p|^{4}. In the momentum distribution, that rapidity tail adds to the usual Tan contact term. Remarkably, atom losses, which are ubiquitous in experiments, do produce such peculiar states. The development of the tail of the rapidity distribution originates from the ghost singularity of the wave function immediately after each loss event. This phenomenon is discussed for arbitrary interaction strengths, and it is supported by exact calculations in the two asymptotic regimes of infinite and weak repulsion.
We study Tan’s contact, i.e. the coefficient of the high-momentum tails of the momentum distribution at leading order, for an interacting one-dimensional Bose gas subjected to a harmonic confinement. Using a strong-coupling systematic expansion of the ground-state energy of the homogeneous system stemming from the Bethe-Ansatz solution, together with the local-density approximation, we obtain the strong-coupling expansion for Tan’s contact of the harmonically trapped gas. Also, we use a very accurate conjecture for the ground-state energy of the homogeneous system to obtain an approximate expression for Tan’s contact for arbitrary interaction strength, thus estimating the accuracy of the strong-coupling expansion. Our results are relevant for ongoing experiments with ultracold atomic gases.
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We study the expansion of an interacting atomic system at zero temperature, following its release from an isotropic three-dimensional harmonic trap and calculate the time dependence of its density and momentum distribution, with special focus on the behavior of the contact parameter. We consider different quantum systems, including the unitary Fermi gas of infinite scattering length, the weakly interacting Bose gas, and two interacting particles with highly asymmetric mass imbalance. In all cases analytic results can be obtained, which show that the initial value of the contact, fixing the 1/k41/{k}^{4}1/k4 tail of the momentum distribution, disappears for large expansion times. Our results raise the problem of understanding the recent experiment of R. Chang et al. [Phys. Rev. Lett. 117, 235303 (2016)] carried out on a weakly interacting Bose gas of metastable 4He^{4}\mathrm{He}4He atoms, where a 1/r41/{r}^{4}1/r4 tail in the density distribution was observed after a large expansion time, implying the existence of the 1/k41/{k}^{4}1/k4 tail in the asymptotic momentum distribution.
One-dimensional Bose gases with contact repulsive interactions are characterized by the presence of infinite-lifetime quasiparticles whose momenta are called the "rapidities." Here, we develop a probe of the local rapidity distribution, based on the fact that rapidities are the asymptotic momenta of the particles after a long one-dimensional expansion. This is done by performing an expansion of a selected slice of the gas. We first apply this idea to a cloud in the quasicondensate regime at equilibrium in a trap. We obtain an experimental picture of the position-dependent rapidity distribution which is in fair agreement with the theory prediction. The asymptotic regime is barely reached, but we show that finite expansion time can be taken into account using the generalized hydrodynamics theory. We then apply this local probe to an out-of-equilibrium situation where the local rapidity distribution is expected to be doubly peaked-a hallmark of a nonthermal state-even though the global rapidity distribution would possess no such distinctive feature. We observe the doubly peaked local rapidity distribution.
The absence of a characteristic momentum scale in the pseudopotential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan's contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit. In the sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to the equilibrium value for the final value of the scattering length.
Abstract This chapter is devoted to the derivation of various thermodynamic relations, known as Tan relations, involving the so-called contact parameter. This physical quantity, which determines the behaviour of the momentum distribution of an interacting gas at large momenta, plays a crucial role in characterizing various thermodynamic quantities, like the dependence of the energy on the scattering length (sweep theorem), the relation between the energy and the momentum distribution, the behaviour of the static structure factor at large wave vectors, and the virial theorem in the presence of harmonic trapping.