Lieb-Liniger model describes bosons with contact interactions in one-dimensional space. In the limit of weak repulsive particle interactions, there are two types of low lying excitation spectrum. The first is reproduced by the Bogoliubov dispersion relation, the other is believed to correspond to dark soliton excitations. While there are various evidences that the type II spectrum is related to dark solitons, it has not been shown that measurements of positions of particles reveal dark soliton density profiles. Here, we employ the Bethe ansatz approach and show that dark solitons emerge in the measurement process if the system is prepared in an eigenstate corresponding to the type II spectrum. We analyze single and double dark solitons as well as weak and strong interaction regime.

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

In analogy to spontaneous breaking of continuous space translation symmetry in the process of space crystal formation, it was proposed that spontaneous breaking of continuous time translation symmetry could lead to time crystal formation. In other words, a time-independent system prepared in the energy ground state is expected to reveal periodic motion under infinitely weak perturbation. In the case of the system proposed originally by Frank Wilczek, spontaneous breaking of time translation symmetry can not be observed if one starts with the ground state. We point out that the symmetry breaking can take place if the system is prepared in an excited eigenstate. The latter can be realized experimentally in ultra-cold atomic gases. We simulate the process of the spontaneous symmetry breaking due to measurements of particle positions and analyze the lifetime of the resulting symmetry broken state. 03.75.Lm, Hamiltonians of condensed matter systems are invariant under translation of all particles by the same vector in space and so are the eigenstates. Consequently probability density for detection of a single particle must be uniform in space if a system is prepared in the ground state or any other eigenstate. Space crystals emerge due to spontaneous symmetry breaking that can be induced by an external perturbation or by, e.g., measurements of particle positions. If the single particle probability density is uniform but the density-density correlation function reveals periodic behavior, measurement of a position of a particle breaks the continuous space translation symmetry and the probability density for the detection of a next particle shows crystalline structure [1, 2]. In the thermodynamic limit, the lifetime of the symmetry broken state goes to infinity and the stable space crystal is formed.Similar phenomenon was postulated to exist in the time domain [3]. A spontaneous breaking of the continuous time translation symmetry in the ground state of the model system was suggested to lead to a periodic motion of a nonuniform density. Soon the other experiment involving trapped ions in a ring [4] was proposed. However, at the same time the original proposition has been put in question [5,6]. While the discussion interestingly evolved [7][8][9][10][11][12][13] strong arguments have been presented [14][15][16] against the existence of time crystals. The proposals, nevertheless, became inspiring and triggered a new field of research. It turns out that, by analogy to condensed matter physics, where space periodic potentials allow for modeling of space crystals, periodically driven systems can model crystalline behavior in time [17]. Anderson localization and Mott insulator phase in the time domain can be realized [18][19][20] and spontaneous breaking of discrete time translation symmetry to another discrete symmetry can be investigated [17]. The latter phenomenon, termed discrete time crystal, was recently observed in two experiments [21,22] following independent theoretical suggestions [23][24][25][26][27][28][29]. Thos...

The two-component Fermi gas with contact attractive interactions between different spin components can be described by the Yang-Gaudin model. Applying the Bethe ansatz approach, one finds analytical formulae for the system eigenstates that are uniquely parametrized by the solutions of the corresponding Bethe equations. Recent numerical studies of the so-called yrast eigenstates, i.e. lowest energy eigenstates at a given non-zero total momentum, in the Yang-Gaudin model show that their spectrum resembles yrast dispersion relation of the Lieb-Liniger model which in turn matches the dark soliton dispersion relation obtained within the nonlinear Schrödinger equation. It was shown that such conjecture in the case of the Lieb-Liniger model was not accidental and that dark soliton features emerged in the course of measurement of positions of particles, when the system was initially prepared in an yrast eigenstate. Here, we demonstrate that, starting with yrast eigenstates in the Yang-Gaudin model, the key soliton signatures are revealed by the measurement of pairs of fermions. We study soliton signatures in a wide range of the interaction strength.

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