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arXiv:2409.16929 · Physical Review A

One-body correlations and momentum distributions of trapped one-dimensional Bose gases at finite temperature

Attila Takács, Yicheng Zhang, Pasquale Calabrese, Jérôme Dubail, Marcos Rigol, Stefano Scopa

Published Mar 20, 2025• 1 citation
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Abstract

We introduce a general approximate method for calculating the one-body correlations and the momentum distributions of one-dimensional Bose gases at finite interaction strengths and temperatures trapped in smooth confining potentials. Our method combines asymptotic techniques for the long-distance behavior of the gas (similar to Luttinger liquid theory) with known short-distance expansions. We derive analytical results for the limiting cases of strong and weak interactions and provide a general procedure for calculating one-body correlations at any interaction strength. A step-by-step explanation of the numerical method used to compute Green's functions (needed as input to our theory) is included. We benchmark our method against exact numerical calculations and compare its predictions to recent experimental results.

DOI
10.1103/physreva.111.033317

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Abstract We present a comprehensive study shedding light on how thermal fluctuations affect correlations in a Bose gas with contact repulsive interactions in one spatial dimension. The pair correlation function, the static structure factor, and the one-body density matrix are calculated as a function of the interaction strength and temperature with the exact ab-initio Path Integral Monte Carlo method. We explore all possible gas regimes from weak to strong interactions and from low to high temperatures. We provide a detailed comparison with a number of theories, such as perturbative (Bogoliubov and decoherent classical), effective (Luttinger liquid) and exact (ground-state and thermal Bethe Ansatz) ones. Our Monte Carlo results exhibit an excellent agreement with the tractable limits and provide a fundamental benchmark for future observations which can be achieved in atomic gases, cavity quantum-electrodynamic and superconducting-circuit platforms.
Describing the collective behavior of many body interacting systems is a central challenge in coldatom physics. Systems confined in one dimension are particularly interesting due to the fact thatin many cases they are integrable. In this work, we focus on such systems—specifically, Bose gasessolvable via the Bethe Ansatz. We begin the thesis by preparing the background, the tools and themotivation needed in the following, inlcuding a review of the Lieb-Liniger gas, it's Bethe Ansatz andthe recent advancements in the experimental realization of it. We include a brief introduction of thetheory of Generalized Hydrodynamics as well as effective field theoretical methods, such as theLuttinger liquid theory, used to compute correlation functions in ground states of low dimensionalsystems. We then turn to the study of lattice hard core bosons out- of euqilibrium. By computing analyticalresults for the one particle density matrix of a freely expanding system of hard-core bosons from aninitial double domain wall potential we provide a fully analytical treatment of the phenomenon ofdynamical quasicondensation and discuss the emergence of off-diagonal long range order. Next, we investigate the dynamics of hard-core bosons released from a domain wall initial statein the presence of two conformal defects. The observed density profile differs from the Euler-scalehydrodynamic description due to quantum interference effects. By analyzing boson propagators, weshow that repeated scattering on the defects leads to interference patterns that cannot be capturedby the hydrodynamic approach. We derive an analytic expression for the density profile during freeexpansion and demonstrate that a modified semi-classical scattering framework is required for thedescription of systems with such conformal defects. The final part of thesis deals with the calculation of the one particle density matrix of Lieb-Lingergas in smooth trapping potentials at finite temperature and interaction strengths. We introduce amethod to obtain such correlation functions by combining long-distance asymptotics with knownshort distance expansions. For the limits of weak and strong interactions we derive analytical re-sults, as well as a general protocol for computing one-particle correlations at any arbitrary repulsiveinteraction strengths. We include a detailed explanation of the numerical method used to computeGreen's functions of field of the effective theory needed as an input to our procedure. We benchmarkour results by comparing them to exact numerical calculations and to recent experimental findings.
We derive exact closed-form expressions for the first few terms of the short-distance Taylor expansion of the one-body correlation function of the Lieb-Liniger gas. As an intermediate result, we obtain the high-p asymptotics of the momentum distribution of both free and harmonically trapped atoms and show that it obeys a universal 1/p(4) law for all values of the interaction strength. We discuss the ways to observe the predicted momentum distributions experimentally, regarding them as a sensitive identifier for the Tonks-Girardeau regime of strong correlations.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.
We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.