Quantum Entangled Dark Solitons Formed by Ultracold Atoms in Optical Lattices

Inspired by experiments on Bose-Einstein condensates in optical lattices, we study the quantum evolution of dark soliton initial conditions in the context of the Bose-Hubbard Hamiltonian. An extensive set of quantum measures is utilized in our analysis, including von Neumann and generalized quantum entropies, quantum depletion, and the pair correlation function. We find that quantum effects cause the soliton to fill in. Moreover, soliton-soliton collisions become inelastic, in strong contrast to the predictions of mean-field theory. These features show that the lifetime and collision properties of dark solitons in optical lattices provide clear signals of quantum effects.

Figure 1

Three ways to describe a quantum soliton: (a) particle number density, (b) Penrose-Onsager condensate wave function density, and (c) order parameter density versus position and time during quantum evolution of a dark soliton initial state obtained via method 1. Shown in (d) is the time dependence of the average local impurity, quantum depletion, and order parameter norm.

Figure 2

Quantum soliton lifetimes. The decay time ${\tau }_c$ of the soliton contrast versus interaction strength $\nu U/J$ at four separate filling factors $\nu$. Here we use data corresponding to dark solitons created via method 2 with $M=30$ lattice sites [22]. All values of $\nu U/J$ considered reside within the superfluid region of the BHH ground state phase diagram.

Figure 3

Many-body characterization of a quantum soliton: pair correlations and entanglement entropy. The (a) pair correlation function and (b) local von Neumann entropy for the same simulation presented in Fig. 1.

Figure 4

Quantum-induced inelasticity. Dark soliton collision for (a) the DNLS and the corresponding quantum evolution at filling factors (b) $\nu =1$, (c) $\nu =0.5$, and (d) $\nu =0.1$ at effective interaction strength $\nu U/J=0.35$ for $M=31$ sites. For (b), (c), and (d), the decoherence time ${\tau }_b$ [black dashed line; cf. Fig. 1a] is tuned to occur after, during, and before the collision time, respectively.