Adiabatic formation of bound states in the one-dimensional Bose gas

Abstract

We consider the one-dimensional interacting Bose gas in the presence of time-dependent and spatially inhomogeneous contact interactions. Within its attractive phase, the gas allows for bound states of an arbitrary number of particles, which are eventually populated if the system is dynamically driven from the repulsive to the attractive regime. Building on the framework of generalized hydrodynamics, we analytically determine the formation of bound states in the limit of adiabatic changes in the interactions. Our results are valid for arbitrary initial thermal states and, more generally, generalized Gibbs ensembles.

Generalized hydrodynamics of the attractive non-linear Schrӧdinger equation

“…Here, the evolution with the repulsive GHD stops and continues from c = 0 − within the attractive regime with the initial conditions being determined by the endpoint of the repulsive evolution. This boundary condition is not contained within the GHD equations presented so far, which must be supplemented by further considerations: in reference [62], the three of us proposed an analytical ansatz to accomplish this task (see also [42,94]). Our reasoning is based on entropic arguments and on the fact that at c = 0 bound states have purely real rapidities and are indistinguishable from unbound particles with the same rapidity.…”

“…Following the advent of GHD, the hydrodynamics of classical integrable models has been successfully derived from their quantum relatives [67,70], in particular the hydrodynamics of the repulsive NLS has been proven to be extremely fruitful in benchmarking GHD predictions against ab initio numerical simulations. Here, we will follow a similar route and tackle the NLS in its attractive phase κ < 0, which becomes accessible in view of our previous results in the attractive quantum Bose gas [62]. The semiclassical limit is most conveniently accessed by introducing a small parameter h which plays the role of Planck's constant.…”

“…In the quantum case, its expectation value can be computed on arbitrary GGEs by means of a generalization of the Hellmann-Feynmann theorem (see e.g. [62]). From this point onwards, the semiclassical limit is readily taken leading to the classical expressions.…”

“…Also in this case, one can envisage ∼T −1 max corrections in view of the forthcoming argument. Following the quantum case discussed in reference [62], immediately after crossing κ = 0 − , bound state recombination is allowed as long as the typical size of the bound state ∝ |κ| −1 exceeds the typical correlation length ζ corr . In dimensionless units, this sets a recombination time t BS ∝ T max ζ −1 corr .…”

“…Fortunately, GHD comes in handy when facing this challenge: in a recent work [62], the three of us proposed to initialize the gas in the attractive regime by means of a slow interaction tuning starting from the repulsive phase. Within each phase, GHD describes the interaction changes [46] and must be supplemented by a description of what happens crossing the noninteracting point c = 0.…”