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

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.

Figure 1

Left: In the attractive regime, the rapidities of the constituents of a bound state share the same real part but are shifted in units of $\left|c\right|$ along the imaginary axis. Right: As $c\to 0^{-}$, the bound states are indistinguishable from unbound particles.

Figure 2

Top: Bound states populations for different choices of $\rho \left(\lambda \right)$ at $c=0^{-}$. The green histogram is obtained using the low-density (incorrect) result ${\rho }_j=\frac{j}{2\pi }{e}^{-\omega j}$ [see Eq. (28) and the relative discussion]. Middle: Evolution of the root densities adiabatically evolving the homogeneous system from $c=4$ to $c=-4$. Note that, in the homogeneous case, the time evolution can be reparametrized in terms of the interaction $c$. The system is initialized in a thermal state with density 0.5 and inverse temperature $\beta =0.1$. Bottom: Energy $E$ and $g_2=\langle {\left({\widehat{\psi }}^{†}\right)}^2{\widehat{\psi }}^2\rangle /{\langle {\widehat{\psi }}^{†}\widehat{\psi }\rangle }^2$ evolution for the same protocol but with different initial temperatures and density 0.5. See Appendix pp1 for the details of the numerical solution of the GHD equations.