Single-shot simulations of dynamics of quantum dark solitons

Abstract

Eigenstates of Bose particles with repulsive contact interactions in one-dimensional space with periodic boundary conditions can be found with the help of the Bethe ansatz. The type~II excitation spectrum identified by E. H. Lieb, reproduces the dispersion relation of dark solitons in the mean-field approach. The corresponding eigenstates possess translational symmetry which can be broken in measurements of positions of particles. We analyze emergence of single and double solitons in the course of the measurements and investigate dynamics of the system. In the weak interaction limit, the system follows the mean-field prediction for a short period of time. Long time evolution reveals many-body effects that are related to an increasing uncertainty of soliton positions. In the strong interaction regime particles behave like impenetrable bosons. Then, the probability densities in the configuration space become identical to the probabilities of non-interacting fermions but the wave-functions themselves remember the original Bose statistics. Especially, the phase flips that are key signatures of the solitons in the weak interaction limit, can be observed in the time evolution of the strongly interacting bosons.Comment: 11 pages, 9 figure

Quantum dark solitons in a Bose gas confined in a hard-wall box

“…Although the method is very efficient, only a few body systems are numerically attainable. In the studies of the Lieb-Liniger model with periodic boundary conditions we were able to investigate 8-particle system [49,50]. There, number of terms in the Bethe ansatz solutions increases like N !…”

“…As in Refs. [50,61] we perform Markovian walk in the configuration space. That is, by sampling N -dimensional probability distribution |Ψ(x 1 , .…”

“…When quantum many-body effects are taken into account, one may expect that the position of the dark soliton starts fluctuating in different realizations of the particle positions measurements. The fluctuations are the more significant the stronger interactions are present [49,50]. Hence, we expect that the averaged particle density ρ(x), i.e.…”

“…Smearing of the density notch with increasing γ is clearly visible at the center of the box. It is caused by the fact that the soliton position varies randomly from realization to realization and the range of the fluctuations is the larger the stronger interactions are [49,50]. In a single realization of the measurement process one obtains a set of particle positions {x 1 , .…”

“…Especially, it has been shown that dark soliton signatures are visible in the reduced single-particle density if the system is prepared in a proper superposition of the eigenstates coming from the second branch of the excitation spectrum [45][46][47]. Ultimately, it was demonstrated that dark soliton signatures emerge in the course of measurement of particle positions if the system is prepared initially in a type II eigenstate [49,50].…”