Quantum Fluctuations of the Center of Mass and Relative Parameters of Nonlinear Schrödinger Breathers

We study quantum fluctuations of macroscopic parameters of a nonlinear Schrödinger breather—a nonlinear superposition of two solitons, which can be created by the application of a fourfold quench of the scattering length to the fundamental soliton in a self-attractive quasi-one-dimensional Bose gas. The fluctuations are analyzed in the framework of the Bogoliubov approach in the limit of a large number of atoms $N$, using two models of the vacuum state: white noise and correlated noise. The latter model, closer to the ab initio setting by construction, leads to a reasonable agreement, within 20% accuracy, with fluctuations of the relative velocity of constituent solitons obtained from the exact Bethe-ansatz results [Phys. Rev. Lett. 119, 220401 (2017)] in the opposite low-$N$ limit (for $N\le 23$). We thus confirm, for macroscopic $N$, the breather dissociation time to be within the limits of current cold-atom experiments. Fluctuations of soliton masses, phases, and positions are also evaluated and may have experimental implications.

Figure 1

A schematic representation of the fundamental “mother soliton”, as the vacuum state including inherent correlated quantum noise (the left panel), transformed into the breather by means of the interaction quench (the right panel).

Figure 2

Variances of fluctuations of the relative parameters of the breather, as a function of time (from top to bottom): the number of atoms $⟨\mathrm{\Delta }{\widehat{n}}^2(t)⟩$, phase $⟨\mathrm{\Delta }{\widehat{\theta }}^2(t)⟩$, velocity $⟨\mathrm{\Delta }{\widehat{v}}^2(t)⟩$, and position $⟨\mathrm{\Delta }{\widehat{b}}^2(t)⟩$, as found for the white-noise vacuum state (red dashed lines) and prequench correlated vacuum state (black solid lines).