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arXiv:1104.2675 · Physical Review A

Loschmidt echo in one-dimensional interacting Bose gases

Karlo Lelas, T. Ševa, Hrvoje Buljan

Published Dec 2, 2011• 11 citations
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Abstract

We explore Loschmidt echo in two regimes of one-dimensional interacting Bose gases: the strongly interacting Tonks-Girardeau (TG) regime, and the weakly interacting mean-field regime. We find that the Loschmidt echo of a TG gas decays as a Gaussian when small (random and time independent) perturbations are added to the Hamiltonian. The exponent is proportional to the number of particles and the magnitude of a small perturbation squared. In the mean-field regime the Loschmidt echo shows richer behavior: it decays faster for larger nonlinearity, and the decay becomes more abrupt as the nonlinearity increases; it can be very sensitive to the particular realization of the noise potential, especially for relatively small nonlinearities.

DOI
10.1103/physreva.84.063601

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We explore Loschmidt echo in two regimes of one-dimensional interacting Bose gases: the strongly interacting Tonks-Girardeau (TG) regime, and the weakly interacting mean-field regime. We find that the Loschmidt echo of a TG gas decays as a Gaussian when small (random and time independent) perturbations are added to the Hamiltonian. The exponent is proportional to the number of particles and the magnitude of a small perturbation squared. In the mean-field regime the Loschmidt echo shows richer behavior: it decays faster for larger nonlinearity, and the decay becomes more abrupt as the nonlinearity increases; it can be very sensitive to the particular realization of the noise potential, especially for relatively small nonlinearities.
We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation V(q,t)V(\mathbf{q},t)V(q,t) with typical strength $\ensuremath{\hbar}∕{\ensuremath{\tau}}_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length ${\ensuremath{\xi}}_{0}$ and a correlation time ${\ensuremath{\tau}}_{0}$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi ``golden rule,'' $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}∕{\ensuremath{\tau}}_{V}^{2}$, where ${\ensuremath{\tau}}_{c}\ensuremath{\sim}\mathrm{min}[{\ensuremath{\tau}}_{0},{\ensuremath{\xi}}_{0}∕v]$ and vvv is the typical particle velocity. Whenever the rate $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence one can protect the system against decoherence.
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 address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
We study the quantum Loschmidt echo (LE) in a Bose-Einstein condensate (BEC) in a double-well potential. The BEC may undergo a dynamical phase transition between two phases: a tunneling phase and a self-trapping phase. For sufficiently weak perturbation, the LE has Gaussian decay in both phases. While, for relatively strong perturbation, the LE has a Gaussian decay in the self-trapping phase and has a stretched exponential decay in the tunneling phase. This qualitative difference in the decaying law of the LE in the two phases provides a characterization of the dynamical phase transition of the BEC. The semiclassical theory is used to explain the numerically observed behaviors of the LE decay.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.