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arXiv:2506.21709 · Physical Review A

Canonical approach to quantum fluctuations

Joanna Ruhl, Vanja Dunjko, Maxim Olshanii

Published Apr 10, 2026• 0 citations
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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 Schrödinger 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.

DOI
10.1103/4zpr-5hh6

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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 Schrödinger 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)U(1)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.
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)PRLTAO0031-900710.1103/PhysRevLett.119.220401] in the opposite low-N limit (for N≤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.
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We investigate the noise in a fundamental soliton state constructed from the exact energy eigenstates of the quantized nonlinear Schr\"odinger equation. It is shown that such a state has statistical properties that are substantially different from those of a coherent state with the same mean field. We can construct, for example, a fundamental soliton state that has initially minimum fluctuations for the four linearized soliton operators (particle number, phase, momentum, and position). We also show that the linearized noise analysis is not valid for nonlinear phase shifts larger than n01/4{\mathit{n}}_{0}^{1/4}n01/4​, where n0{\mathit{n}}_{0}n0​ is the average number of particles in the fundamental soliton. \textcopyright{} 1996 The American Physical Society.
The dynamics of interacting many-particle systems on quantum mechanical scale is a broad subject of research in modern physics. The understanding of the quantum correlations, or entanglement, between the particles in an isolated many-body quantum system may however be challenging, as the amount of interactions may be large. To be able to describe the system in a simpler way, mean field theory is often applied; the interactions of all individual particles are substituted by an averaged effect. First and second order mean field approximations are applied to the equations of motion of the amplitudes <ân> for each mode of the many-body system. In first order mean field, under the assumption that each mode has the same magnitude of the (constant) amplitude, the Kuramoto equation follows for the phases of the complex amplitudes. To study this first order approximation and to refine the mean field solution, the equations of motion have been extended to second order mean field. After that, numerical integration is used to solve the system of coupled differential equations for two modes. The solutions seem unstable, which is verified by defining quantum fluctuations beyond mean field. These are shown not to be negligible for a two mode system.It was discovered that the mean field equations of motion show divergent fluctuations for a system of two modes, which opens a whole set of questions on the validity of the first order mean field approximation and therefore the use of the Kuramoto model for quantum many-body dynamics. The extension to the second order mean field solution might be promising as this does include correlations between the operators.