Quantum many-body dynamics of dark solitons in optical lattices

Abstract

We present a fully quantum many-body treatment of dark solitons formed by ultracold bosonic atoms in one-dimensional optical lattices. Using time-evolving block decimation to simulate the single-band Bose-Hubbard Hamiltonian, we consider the quantum dynamics of density and phase engineered dark solitons as well as the quantum evolution of mean-field dark solitons injected into the quantum model. The former approach directly models how one may create quantum entangled dark solitons in experiment. While we have already presented results regarding the latter approach elsewhere [Phys. Rev. Lett. 103, 140403 (2009)], we expand upon those results in this work. In both cases, quantum fluctuations cause the dark soliton to fill in and may induce an inelasticity in soliton-soliton collisions. Comparisons are made to the Bogoliubov theory which predicts depletion into an anomalous mode that fills in the soliton. Our many-body treatment allows us to go beyond the Bogoliubov approximation and calculate explicitly the dynamics of the system's natural orbitals.

Tensor network states and algorithms in the presence of a global U(1) symmetry

“…2,5,14,[54][55][56][57][58][59][60][61][62] Symmetries have also been used in more recent proposals to simulate time evolution with MPSs. [10][11][12][13][14][63][64][65][66][67][68] When considering symmetries, it is important to notice that an MPS is a trivalent tensor network. That is, in an MPS each tensor has at most three indices.…”

Many-Body Matter-Wave Dark Soliton

“…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.…”

“…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.…”

“…Following Refs. [19,20,34] we choose as initial state the ground state of the system in the presence of a Gaussian barrier V (x) = V 0 e −x 2 /σ 2 0 located at the center of the box. For V 0 = 18 and σ 0 = 0.2 the one-body density profile is nearly identical to the density profile of the dark soliton described by eq.…”

“…The corresponding parameter for the Lieb-Liniger model [24] is γ = 0.04, indicating the weakly interacting regime. Space is discretized on a grid with ∆x = 0.2, so that the many-body Hamiltonian in the second-quantization formalism is a Bose-Hubbard Hamiltonian with tunneling amplitude 1/(2∆x 2 ) and interaction energy g 0 /∆x [19,20,29,30] [37]. An N -body state of the system can be represented by a matrix product state (MPS) α l =1,2... are rapidly decaying numbers and a cutoff χ can be introduced in the sum over Greek indices.…”