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arXiv:2505.12347 · Physical Review A

Nonlinear-optical realization of nonintegrable phases accompanying quantum phase transitions

Chon‐Fai Kam

Published Sep 15, 2025• 1 citation
arXivHTMLPDF

Abstract

In this work, we propose an experimentally feasible nonlinear optical platform to realize nonintegrable geometric phases that emerge in interacting quantum systems undergoing quantum phase transitions. Specifically, we demonstrate that an exotic nonlinear term in the dynamical equations governs the polarization evolution of the optical field propagating along an anisotropic low-birefringence fiber with tetragonal symmetry. Intriguingly, by adiabatically tuning the nonlinear susceptibilities along the fiber, the Stokes vector on the Poincar\'e sphere accumulates a nonintegrable phase known as the Hannay angle---the classical counterpart to the Berry phase---which exhibits the same geometric gauge structure, including conic singularities in parameter space, as those associated with quantum phase transitions in systems like Bose-Einstein condensates. We further discuss a practical experimental implementation through the adiabatic deposition of nanocrystals along the fiber to control these susceptibilities, highlighting assumptions such as negligible losses and low birefringence for realistic fiber lengths.

DOI
10.1103/7xy5-cx8l

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In this work, we propose an experimentally feasible nonlinear optical platform to realize nonintegrable geometric phases that emerge in interacting quantum systems undergoing quantum phase transitions. Specifically, we demonstrate that an exotic nonlinear term in the dynamical equations governs the polarization evolution of the optical field propagating along an anisotropic low-birefringence fiber with tetragonal symmetry. Intriguingly, by adiabatically tuning the nonlinear susceptibilities along the fiber, the Stokes vector on the Poincar'e sphere accumulates a nonintegrable phase known as the Hannay angle---the classical counterpart to the Berry phase---which exhibits the same geometric gauge structure, including conic singularities in parameter space, as those associated with quantum phase transitions in systems like Bose-Einstein condensates. We further discuss a practical experimental implementation through the adiabatic deposition of nanocrystals along the fiber to control these susceptibilities, highlighting assumptions such as negligible losses and low birefringence for realistic fiber lengths.
We investigate the geometric phase or Berry phase of adiabatic quantum evolution in the Bose-Einstein condensate (BEC) systems governed by nonlinear Gross-Pitaevskii(GP) equations. We study how this phase is modified by the nonlinearity and find that the Bogoliubov fluctuations around the eigenstates are accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode BEC model is used to illustrate our theory. Our theory is applicable to other nonlinear systems such as paraxial wave equation for nonlinear optics and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
The Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution is investigated thoroughly. The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase, while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode model is used to illustrate our theory. Our theory is applicable to Bose-Einstein condensate, nonlinear light propagation, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
Abstract The geometric phase for classical electromagnetic light beams, in its original formulation as introduced by Pancharatnam, concerns fields experiencing cyclic, discrete in-phase polarization-state changes. A similar phase was later recognized by Berry to govern the behavior of adiabatic quantum systems, with consequent extensions to nonadiabatic and noncyclic evolutions of the quantum state. However, no optical counterpart for the noncyclic, adiabatic (continuous) evolution has been demonstrated. Here we employ a modified Young’s two-pinhole setup with controlled pinhole polarizations and intensities to generate on interference an arbitrary continuous spatial evolution of the polarization state, an optical analogue to the adiabatic case. The customized arrangement allows separating at any point the accumulated dynamical and geometric phases from the total phase, enabling a detailed study of the noncyclic Pancharatnam–Berry phase in a continuous transformation. Our theoretical and experimental results are in excellent agreement and consistent with the geodesic rule for noncyclic evolutions.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
Geometric Phase in Quantum Mechanics is generally formulated entirely in terms of geometric structure of the Complex Hilbert Space. We will exploit this fact in case of mixed states for three level open systems undergoing depolarization using the eight di- mensional Poincare sphere and non unit vector rays in H3 within the limit of pure state approach may be found to be in agreement with the Pancharatnam Phase, Berry Phase and Aharonov-Anandan Phase. We will consider the Depolarization of the Three di- mensional fields by introducing the stokes parameters and by redefining the definition of Degree of Polarization using the SU(3) algebra to describe the quantum light depolariza- tion of Non-paraxial beams by minimal reservoir coupling models modeled by Lindblad Master equations and connect the solution of that with the geometric phase for systems undergoing depolarization.