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arXiv:2003.05868 · Physical Review A

Full counting statistics of the momentum occupation numbers of the Tonks-Girardeau gas

P. Devillard, Denis Chevallier, Patrizia Vignolo, Mathias Albert

Published Jun 1, 2020• 10 citations
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Abstract

We compute the fluctuations of the number of bosons with a given momentum for the Tonks-Girardeau gas at zero temperature. We show that correlations between opposite momentum states, which are an important identifying characteristic of long-range order in weakly interacting Bose systems, are suppressed and that the full distribution of the number of bosons with nonzero momentum is exponential. The distribution of the quasicondensate is however quasi-Gaussian. The experimental relevance of our findings for recent cold-atom experiments is discussed.

DOI
10.1103/physreva.101.063604

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We compute the fluctuations of the number of bosons with a given momentum for the Tonks-Girardeau gas at zero temperature. We show that correlations between opposite momentum states, which are an important identifying characteristic of long-range order in weakly interacting Bose systems, are suppressed and that the full distribution of the number of bosons with nonzero momentum is exponential. The distribution of the quasicondensate is however quasi-Gaussian. The experimental relevance of our findings for recent cold-atom experiments is discussed.
We compute the fluctuations of the number of bosons with a given momentum for the Tonks-Girardeau gas at zero and finite temperature in a harmonic trap. We show that correlations between opposite momentum states ppp, which is an important fingerprint of long-range order in weakly interacting Bose systems, are suppressed. Nontrivial correlations, including negative correlations, are observed for momenta smaller than or of the order of the inverse radius of the gas. The full distribution of the number of bosons with momentum ppp exhibits an interesting crossover from a nontrivial distribution at zero momentum to an exponential distribution. The distribution of the quasicondensate occupation is also studied. The experimental relevance of our findings for recent cold-atom experiments is discussed.
We consider the single-particle correlations and momentum distributions in a gas of strongly interacting, spinless 1D fermions with zero-range interactions. This system represents a fermionic version of the Tonks-Girardeau gas of impenetrable bosons as it can be mapped to a system of noninteracting 1D bosons. We use this duality to show that the T = 0, single-particle correlations exhibit an exponential decay with distance. This strongly interacting system is experimentally accessible using ultracold atoms and has a Lorentzian momentum distribution at large momenta whose width is given by the linear density.
Inspired by recent advances in cold atomic systems and nonequilibrium physics, we introduce a characterization scheme, the time-of-flight full counting statistics. We benchmark this method on an interacting one-dimensional Bose gas and show that there the time-of-flight image displays several universal regimes. Finite momentum fluctuations are observed at larger distances, where a crossover from exponential to Gamma distribution occurs upon decreasing momentum resolution. Zero-momentum particles, on the other hand, obey a Gumbel distribution in the weakly interacting limit, characterizing the quantum fluctuations of the former quasicondensate. Time-of-flight full counting statistics is demonstrated to capture (pre-)thermalization processes after a quantum quench and can be useful for characterizing exotic quantum states such as many-body localized systems or models of holography.
We analyze the two-body momentum correlation function for a uniform, weakly interacting, one-dimensional Bose gas. We show that the strong positive correlation between opposite momenta, expected in a Bose-Einstein condensate with a true long-range order, almost vanishes in a phase-fluctuating quasicondensate where the long-range order is destroyed. Using the Luttinger liquid approach, we derive an analytic expression for the momentum correlation function in the quasicondensate regime, showing (i) the reduction and broadening of the opposite-momentum correlations (compared to the singular behavior in a true condensate) and (ii) an emergence of anticorrelations at small momenta. We also numerically investigate the momentum correlations in the crossover between the quasicondensate and the ideal Bose-gas regimes using a classical field approach and show how the anticorrelations gradually disappear in the ideal-gas limit.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.