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arXiv:2109.01500 · Physical Review Letters

Self-Pinning Transition of a Tonks-Girardeau Gas in a Bose-Einstein Condensate

Tim Keller, Thomás Fogarty, Thomas Busch

Published Jan 31, 2022• 14 citations
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Abstract

We show that a Tonks-Girardeau (TG) gas that is immersed in a Bose-Einstein condensate can undergo a transition to a crystal-like Mott state with regular spacing between the atoms without any externally imposed lattice potential. We characterize this phase transition as a function of the interspecies interaction and temperature of the TG gas, and show how it can be measured via accessible observables in cold atom experiments. We also develop an effective model that accurately describes the system in the pinned insulator state and which allows us to derive the critical temperature of the transition.

DOI
10.1103/physrevlett.128.053401

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We show that a Tonks-Girardeau (TG) gas that is immersed in a Bose-Einstein condensate can undergo a transition to a crystal-like Mott state with regular spacing between the atoms without any externally imposed lattice potential. We characterize this phase transition as a function of the interspecies interaction and temperature of the TG gas, and show how it can be measured via accessible observables in cold atom experiments. We also develop an effective model that accurately describes the system in the pinned insulator state and which allows us to derive the critical temperature of the transition.
We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87 atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution for the ground state of a 1D Bose gas.
We study the pinning quantum phase transition in a Tonks-Girardeau gas, both in equilibrium and out of equilibrium, using the ground-state fidelity and the Loschmidt echo as diagnostic tools. The ground-state fidelity will have a dramatic decrease when the atomic density approaches the commensurate density of one particle per lattice well. This decrease is a signature of the pinning transition from the Tonks to the Mott insulating phase. We study the applicability of the fidelity for diagnosing the pinning transition in experimentally realistic scenarios. We find that fidelity can predict the particle number(s) at which the pinning occurs. In addition, we explore the out-of-equilibrium dynamics of the gas following a sudden quench with a lattice potential. We find that all properties of the ground-state fidelity are reflected in the Loschmidt echo dynamics, i.e., in the nonequilibrium dynamics of the Tonks-Girardeau gas initiated by a sudden quench of the lattice potential.
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half of the maximum density and the optical lattice is modeled by a periodic potential of strength $\ensuremath{\lambda}$. For small $\ensuremath{\lambda}$ and temperature, BEC is proved to occur, while at large $\ensuremath{\lambda}$ or temperature there is no BEC. At large $\ensuremath{\lambda}$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the p=0\mathbf{p}=\mathbf{0}p=0 mode in this model, although the density itself has the periodicity of the imposed potential.
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice.We present a model of this phenomenon which we are able to analyze rigorously.The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength λ.For small λ and temperature, BEC is proved to occur, while at large λ or temperature there is no BEC.At large λ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase.The interparticle interaction is essential for this transition, which occurs even in the ground state.Surprisingly, the condensation is always into the p = 0 mode in this model, although the density itself has the periodicity of the imposed potential.
In this paper, we investigate the ground-state properties of a bosonic Tonks-Girardeau (TG) gas confined in a one-dimensional periodic potential. The single-particle reduced density matrix is computed numerically for systems up to N=265N=265N=265 bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein condensation (BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, $\ensuremath{-}0.5$ and 1, respectively, while in the incommensurate case, they are 0.5, $\ensuremath{-}0.5$, and 1.5, respectively. These exponents are related to each other in a universal relation.
We investigate the properties of a Tonks–Girardeau gas in the presence of a one-dimensional lattice potential. Such a system is known to exhibit a pinning transition when the lattice is commensurate with the particle density, leading to the formation of an insulating state even at infinitesimally small lattice depths. Here we examine the properties of the gas at all lattices depths and, in addition to the static properties, also consider the non-adiabatic dynamics induced by the sudden motion of the lattice potential with a constant speed. Our work provides a continuum counterpart to the work done in discrete lattice models.