Abstract
We discuss the hydrodynamic approach to the study of the time evolution—induced by a quench—of local excitations in one dimension. We focus on interaction quenches: the considered protocol consists of creating a stable localized excitation propagating through the system, and then operating a sudden change of the interaction between the particles. To highlight the effect of the quench, we take the initial excitation to be a soliton. The quench splits the excitation into two packets moving in opposite directions, whose characteristics for short times can be expressed in a universal way. Our treatment allows for the description of the internal dynamics of these two packets in terms of the different velocities of their components. We confirm our analytical predictions through numerical simulations performed with the Gross–Pitaevskii equation and with the Calogero model (as an example of long range interactions and solvable with a parabolic confinement). Through the Calogero model we also discuss the effect of an external trapping on the protocol. The hydrodynamic approach shows that there is a difference between the bulk velocities of the propagating packets and the velocities of their peaks: it is possible to discriminate the two quantities, as we show through the comparison between numerical simulations and analytical estimates. We show that our analytical results capture with remarkable precision the findings of the numerical simulations also for intermediate times and we provide predictions for the time at which the two packets becomes distinguishable. In the realizations of the discussed quench protocol in a cold atom experiment, these different velocities are accessible through different measurement procedures.
Highlights
One of the experimentally most relevant protocol to study out-of-equilibrium dynamics and the issue of thermalization is the quench protocol, in which, typically, a system described by a Hamiltonian H is prepared in its ground state and at a given moment of time, let evolved using a different Hamiltonian H [37]
The hydrodynamic approach shows that there is a difference between the bulk velocities of the propagating packets and the velocities of their peaks: it is possible to discriminate the two quantities, as we show through the comparison between numerical simulations and analytical estimates
In this work our main goal is to investigate in detail the hydrodynamic approach for the study of the dynamics of solitonic excitations in one-dimensional systems, clarifying the hypothesis behind the derivations based on the hydrodynamic approach and to discuss on the possible experimental realization of the quench protocol in cold atom systems
Summary
All quantum systems at sufficiently low temperature acquire a collective behavior In many such cases, the expectation value of the particle density operator ρ(x) = j δ(x − xj) becomes a smooth function and the quantum fluctuations around it are negligible. For larger deviations one has to study the quantum dynamics using directly the Lieb-Liniger model (which might be at present rather challenging from the computational point of view) or resort to 1D mean-field effective equations [69,70,71,72] from which hydrodynamic equations may be derived as above producing (time-dependent) non-linear Schrödinger equations with suitable general non-linear terms (ρ), the GP equation corresponding to (ρ) ∝ ρ. The ultimate fate of a soliton configuration due to classical to quantum crossover has been studied in [74]
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