Many-Body Matter-Wave Dark Soliton

Abstract

The Gross-Pitaevskii equation -which describes interacting bosons in the mean-field approximation -possesses solitonic solutions in dimension one. For repulsively interacting particles, the stationary soliton is dark, i.e. is represented by a local density minimum. Many-body effects may lead to filling of the dark soliton. Using quasi-exact many-body simulations, we show that, in single realizations, the soliton appears totally dark although the single particle density tends to be uniform. [4][5][6][7][8][9]. Within the mean field approach and at zero temperature, an atomic Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation (GPE). In one-dimensional (1D) space, this non-linear wave equation possesses a dark (bright) soliton solution if particle interactions are effectively repulsive (attractive). The GPE assumes all particles in the same single particle state and neglects that the interactions can populate other modes.There is a debate in the literature concerning manybody effects in a dark soliton [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The GPE predicts a stable dark soliton state in 1D. The Bogoliubov corrections show that signatures of the soliton tend to disappear in the single particle density because particles depleted from the condensate fill the soliton notch [12][13][14][15]. However, the single particle density refers to results averaged over many experimental realizations and is not able to predict a single experimental outcome. Within the Bogoliubov approach, it is possible to simulate single experiments: in a single photo, the soliton is completely dark but its position varies randomly from realization to realization [14,15]. This is a signature of the quantum character of the soliton, i.e. the soliton position is not described by a classical variable but by a probability density. The Bogoliubov analysis is limited to quantum fluctuations of the soliton position on a scale smaller than the healing length and cannot predict what happens when many-body effects become stronger. The latter situation has been considered in Refs. [19,20] within many-body numerical calculations. Analysis of the second order correlation function leads the authors to the conclusion that a dark soliton cannot be observed when the single particle density becomes uniform. However, full simulations of single experiments involve much higher order correlation functions [21,22]. It is the goal of the present work to perform many-body numerical simulations of the entire experiment starting from soliton preparation till the destructive measurement of all particle positions. We show that a fully dark soliton can be observed when manybody effects are strong, although its measured position is random and varies among experimental realizations. In the following we assume the zero temperature limit and do not consider thermodynamical instability of the dark soliton [26,27].The stationary solution of the 1D GPE for particles of mass m and interaction coefficient g 0 , associated with a da...

Many-body processes in black and gray matter-wave solitons

“…Moreover, at times when the reduced one-body density, i.e. the average over many single shot measurements, has already relaxed to a flat distribution, the histograms of simulated destructive N -atom single shot measurements have revealed a soliton-like density minimum at random positions [34,36,[41][42][43]. These findings suggest the existence of highly non-trivial correlations being unravelled in single shot measurements.…”

Two-body correlations and natural-orbital tomography in ultracold bosonic systems of definite parity

“…1(b) and [9]). Observing numerically that |g 2 (0, 0) − 1| ≪ 1, we may now understand these non-local correlations as a generic property of parity-symmetric systems with essentially two oc-cupied NOs: Denoting the position of the soliton moving to the right/left with x R/L , our above results imply Second, in order to realize a situation with significant deviations of g 2 (0, 0) from unity, we additionally imprint a relative phase of π between the two half-spaces at t = 0 [6][7][8][9]. Thereby, a black soliton is initialized at x = 0.…”

“…e.g. [6][7][8][9]. Due to the increasing population of this orbital, the depth of the characteristic minimum in the reduced one-body density is reduced, i.e.…”

“…Examples for such phenomena are the Mott-insulating phase, where e.g. an intra-well Tonks-Girardeau transition for a filling factor of two can be understood in the NO framework [4,5], the quantum-fluctuations induced decay of dark solitons [6][7][8][9], where a NO of particular shape is dominantly responsible for the soliton contrast reduction in the reduced one-body density, and fragmented condensates [10][11][12][13], which are defined as many-body states with two or more macroscopically occupied NOs. While there are proposals for the detection of fragmentation and its degree [14,15], the one-body density ρ 1 (r) has, to the best of our knowledge, not yet been unraveled into the * skroenke@physnet.uni-hamburg.de † pschmelc@physnet.uni-hamburg.de contributions |φ i (r)| 2 of the individual NOs by means of a measurement protocol.…”

Variational Monte Carlo study of soliton excitations in hard-sphere Bose gases

“…Using time-dependent Bogoliubov theory [18][19][20] as well as more sophisticated numerical techniques [21][22][23][24][25] to treat the many-body dynamics of one-dimensional bosons at zero temperature, it has been shown that beyond-mean-field effects tend to deplete the condensate and to fill the soliton notch making dark solitons unstable. Here, however, we study theoretically the role of many-body correlations in determining relevant properties of gray solitons in a three-dimensional (3D) Bose gas, such as their excitation energy, density profile, and effective mass.…”