Abstract
Quantum integrable models display a rich variety of non-thermal excited states with unusual properties. The most common way to probe them is by performing a quantum quench, i.e., by letting a many-body initial state unitarily evolve with an integrable Hamiltonian. At late times these systems are locally described by a generalized Gibbs ensemble with as many effective temperatures as their local conserved quantities. The experimental measurement of this macroscopic number of temperatures remains elusive. Here we show that they can be obtained for the Bose gas in one spatial dimension by probing the dynamical structure factor of the system after the quench and by employing a generalized fluctuation-dissipation theorem that we provide. Our procedure allows us to completely reconstruct the stationary state of a quantum integrable system from state-of-the-art experimental observations.
Highlights
The experimental measurement of this macroscopic number of temperatures remains elusive. We show that they can be obtained for the Bose gas in one spatial dimension by probing the dynamical structure factor of the system after the quench and by employing a generalized fluctuation-dissipation theorem that we provide
The left and right panels correspond to quenches to c = 2 and c → ∞, respectively. In the latter case, the curves clearly show that Ξ(k, ω) vanishes, for fixed k, as |ω| → ∞, as it was analytically shown in Ref. [64] and the same behavior is expected for any quench to c > 0 despite the fact that numerical artifacts (due to the truncated diagonal ensemble used in order to numerically compute the dynamical structure factor as in (39)) hide it
In this work we have presented a method which allows one to completely determine the stationary state of a quantum integrable model from the knowledge of a dynamic structure factor
Summary
Motivated by the impressive experimental progress in engineering and manipulating cold atomic gases, the dynamics of isolated quantum many-body systems has recently been the subject of very intense theoretical [1,2,3] and experimental [4,5,6,7,8,9] investigations. Our approach can be extended to any integrable model with one single type of quasi-particle in its spectrum It was recently shown [49, 50] that given the DSF of an operator which is globally conserved, as for example the density ρ(x) in the one-dimensional Bose gas (as the total density n = 〈 ρ(x)d x〉 is conserved by the Lieb-Liniger Hamiltonian (6)), relation (5) holds generically for any model as a consequence of the generalized hydrodynamics theory [51,52]. S(k, ω) (see Eq (3)) at late times after the quench, the right hand side in Eq (33) or (34) can be computed once the total density ρt (λ), the sound velocity v(λ), and the mode energy ω(λ) are known These functions are given by integral equations depending on θ(λ) = (1 + eβ(λ)(ω(λ)−μ))−1 (see Eq (9) for ρt and Eqs. Note that the equivalence in Eqs. (33) and (34) holds only in the limit k → 0, the finite k corrections are of (k2) as in Eq (29), which makes it possible to compute θ(λ) by using the DSF at small but finite values of k
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