Quantum dark soliton: Nonperturbative diffusion of phase and position

Abstract

The dark soliton solution of the Gross-Pitaevskii equation in one dimension has two parameters that do not change the energy of the solution: the global phase of the condensate wave function and the position of the soliton. These degeneracies appear in the Bogoliubov theory as Bogoliubov modes with zero frequencies and zero norms. These "zero modes" cannot be quantized as the usual Bogoliubov quasiparticle harmonic oscillators. They must be treated in a nonperturbative way. In this paper I develop non-perturbative theory of zero modes. This theory provides non-perturbative description of quantum phase diffusion and quantum diffusion of soliton position. An initially well localized wave packet for soliton position is predicted to disperse beyond the width of the soliton.

Quantum many-body dynamics of dark solitons in optical lattices

“…This is why we are able to make quantitative predictions regarding the accuracy of using meanfield theory to describe soliton dynamics. Related studies regarding the effect of quantum fluctuations on excited condensates in continuous geometries, e.g., dark solitons, using few-mode approximations and BDG-based methods can be found in the works of Dziarmaga, Sacha, and Karkuszewski [14,15,16,41,42], as we have already alluded to. Using related methods, Law studied the dynamical depletion of dark solitons created via phase imprinting [43].…”

“…Second, BDG methods have been used to describe quantum fluctuations of solitons. The main result is that within BDG framework the dark soliton, which manifests as a density notch with a phase jump across it, appears to be filled in: the soliton's position is uncertain, so that in an ensemble average of measurements it appears blurred [14,15,16]. However, these studies predict that in any particular measurement the soliton is indeed localized, and the predictions of mean-field theory for the density, phase, and stability properties of the soliton hold.…”

Interacting multiple zero mode formulation and its application to a system consisting of a dark soliton in a condensate

“…Note that although in Ref. [8], which considered a nonperturbative treatment of the two zero modes, the system is put in an artificial box with a size L and the boundary condition…”

“…In Ref. [8], Eq. (28) is derived from the classical Lagrangian for the collective coordinates as the starting point of the nonperturbative treatment, and the U(1) gauge zero mode sector is restricted to a subspace with a definite total number of atoms, assuming a large phase fluctuation.…”

“…This example was studied in Ref. [8], in which a nonperturbative treatment of the zero modes using an effective Hamiltonian was proposed. However, the resultant Hamiltonian represents the free motion of each zero mode with no interaction, so our approach is essentially different from that in Ref.…”