A canonical approach to quantum fluctuations

TL;DR

This paper develops a canonical formalism to analytically compute quantum fluctuations in integrable PDE systems, applied to soliton solutions of the nonlinear Schrödinger equation post-quench.

Contribution

It introduces a novel analytic method for quantum fluctuation calculations in classical-field approximations of many-body quantum systems, including particle-number-conserving modes.

Findings

Analytic solutions for quantum fluctuations of solitons after a quench.

Comparison of white-noise and correlated-noise models for fluctuation vacuum.

Particle-number-conserving modes often do not alter the fluctuation results.

Abstract

We present a canonical formalism for computing quantum fluctuations of certain discrete degrees of freedom in systems governed by integrable partial differential equations with known Hamiltonian structure, provided these models are classical-field approximations of underlying many-body quantum systems. We then apply the formalism to both the 2-soliton and 3-soliton breather solutions of the nonlinear Schr\"odinger equation, assuming the breathers are created from an initial elementary soliton by quenching the coupling constant. In particular, we compute the immediate post-quench quantum fluctuations in the positions, velocities, norms, and phases of the constituent solitons. For each case, we consider both the white-noise and correlated-noise models for the fluctuation vacuum state. Unlike previous treatments of the problem, our method allows for analytic solutions. Additionally, in the correlated-noise case, we consider the particle-number-conserving (also called $U(1)$-symmetry-conserving) Bogoliubov modes, i.e., modes with the proper correction to preserve the total particle number. We find that in most (but not all) cases, these corrections do not change the final result.