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arXiv:2401.16860 · Journal of Physics A Mathematical and Theoretical

Quasicondensation and off-diagonal long-range order of hard-core bosons during a free expansion

Attila Takács, Stefano Scopa, Pasquale Calabrese, Lev Vidmar, Jérôme Dubail

Published Sep 25, 2024• 2 citations
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Abstract

Abstract Quasicondensation in one dimension is known to occur for equilibrium systems of hard-core bosons (HCBs) at zero temperature. This phenomenon arises due to the off-diagonal long-range order in the ground state, characterized by a power-law decay of the one-particle density matrix g 1 ( x , y ) ∼ | x − y | − 1 / 2 —a well-known outcome of Luttinger liquid theory. Remarkably, HCBs, when allowed to freely expand from an initial product state (i.e. characterized by initial zero correlation), exhibit quasicondensation and demonstrate the emergence of off-diagonal long-range order during nonequilibrium dynamics. This phenomenon has been substantiated by numerical and experimental investigations in the early 2000s. In this work, we revisit the dynamical quasicondensation of HCBs, providing a fully analytical treatment of the issue. In particular, we derive an exact asymptotic formula for the equal-time one-particle density matrix by borrowing ideas from the framework of quantum Generalized Hydrodynamics. Our findings elucidate the phenomenology of quasicondensation and of dynamical fermionization occurring at different stages of the time evolution, as well as the crossover between the two.

DOI
10.1088/1751-8121/ad7fa8

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Abstract Quasicondensation in one dimension is known to occur for equilibrium systems of hard-core bosons (HCBs) at zero temperature. This phenomenon arises due to the off-diagonal long-range order in the ground state, characterized by a power-law decay of the one-particle density matrix g 1 ( x , y ) ∼ | x − y | − 1 / 2 —a well-known outcome of Luttinger liquid theory. Remarkably, HCBs, when allowed to freely expand from an initial product state (i.e. characterized by initial zero correlation), exhibit quasicondensation and demonstrate the emergence of off-diagonal long-range order during nonequilibrium dynamics. This phenomenon has been substantiated by numerical and experimental investigations in the early 2000s. In this work, we revisit the dynamical quasicondensation of HCBs, providing a fully analytical treatment of the issue. In particular, we derive an exact asymptotic formula for the equal-time one-particle density matrix by borrowing ideas from the framework of quantum Generalized Hydrodynamics. Our findings elucidate the phenomenology of quasicondensation and of dynamical fermionization occurring at different stages of the time evolution, as well as the crossover between the two.
Long-range order in quantum many-body systems is usually associated with equilibrium situations. Here, we experimentally investigate the quasicondensation of strongly interacting bosons at finite momenta in a far-from-equilibrium case. We prepare an inhomogeneous initial state consisting of one-dimensional Mott insulators in the center of otherwise empty one-dimensional chains in an optical lattice with a lattice constant d. After suddenly quenching the trapping potential to zero, we observe the onset of coherence in spontaneously forming quasicondensates in the lattice. Remarkably, the emerging phase order differs from the ground-state order and is characterized by peaks at finite momenta ±(π/2)(ℏ/d) in the momentum distribution function.
We develop an exact approach to study the quench dynamics of hard-core bosons initially in thermal equilibrium in one-dimensional lattices. This approach is used to study the sudden expansion of thermal states after confining potentials are switched off. We find that a dynamical fermionization of the momentum distribution occurs at all temperatures. This phenomenon is studied for low initial site occupations, for which the expansion of the cloud is self-similar. In this regime, the occupation of the natural orbitals allows one to distinguish hard-core bosons from noninteracting fermions. We also study the free expansion of initial Mott insulating domains at finite temperature and show that the emergence of off-diagonal one-body correlations is suppressed gradually with increasing temperature. Surprisingly, the melting of the Mott domain is accompanied by an effective cooling of the system. We explain this phenomenon analytically using an equilibrium description based on an emergent local Hamiltonian.
Abstract We investigate the nature of density-density correlations in a 1D gas of hard-core particles initially prepared at equilibrium (either at zero or finite temperature) on a semi-infinite line and subsequently let to expand into the other (initially empty) half of the system. Using a combination of analytical techniques based on exact methods and asymptotic hydrodynamic approaches, we discuss the behavior of the gas as its initial temperature varies, and back up our derivations with numerical exact diagonalization of the model. Our findings reveal that, irrespective of the initial temperature, the non-equilibrium behavior of density-density correlations at sufficiently large times is characterized by algebraic decay. Furthermore, we provide analytical results based on quantum generalized hydrodynamics that match with the numerical data both at zero and finite temperature.
Bosons in one dimension display a phenomenon called quasicondensation where correlations decay in a power-law fashion. We study the fate of this quasicondensed state in an open-system setup where two macroscopic leads, held at different chemical potentials, drive the system far away from equilibrium. For generic leads, we find that a finite bias destroys the quasicondensed state and the system exhibits exponential-decay correlations and subleading power-law corrections that are bias dependent. Near the equilibrium, we found a diverging correlation length and determine the critical properties, including the critical scaling form of the correlations. Also, we contrast our findings against the fate of quasicondensation in thermal equilibrium and discuss the special case of reflectionless coupling where the quasicondensation survives out of equilibrium. This exactly solvable interacting nonequilibrium system has the remarkable property that the near-equilibrium state cannot be obtained within linear response. These results aid in unraveling the enigmatic properties spawned by strong interactions once liberated from equilibrium constraints.
Long-range order in quantum many-body systems is usually associated with equilibrium situations.Here, we experimentally investigate the quasicondensation of strongly interacting bosons at finite momenta in a far-from-equilibrium case.We prepare an inhomogeneous initial state consisting of one-dimensional Mott insulators in the center of otherwise empty one-dimensional chains in an optical lattice with a lattice constant d.After suddenly quenching the trapping potential to zero, we observe the onset of coherence in spontaneously forming quasicondensates in the lattice.Remarkably, the emerging phase order differs from the ground-state order and is characterized by peaks at finite momenta Æðπ=2Þðℏ=dÞ in the momentum distribution function.
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.