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Physical Review A

Berry connection in atom-molecule systems

Fu-Cheng Cui, Biao Wu

Published Aug 2, 2011• 5 citations
Publisher

Abstract

In the mean-field theory of atom-molecule systems, where bosonic atoms combine to form molecules, there is no usual $\text{U}(1)$ symmetry, presenting an apparent hurdle for defining the Berry phase and Berry curvature for these systems. We define a Berry connection for this system, with which the Berry phase and Berry curvature can be naturally computed. We use a three-level atom-molecule system to illustrate our results. In particular, we have computed the mean-field Berry curvature of this system analytically, and compared it to the Berry curvature computed with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our definition.

DOI
10.1103/physreva.84.024101

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In the mean-field theory of atom-molecule systems, where bosonic atoms combine to form molecules, there is no usual U(1)\text{U}(1)U(1) symmetry, presenting an apparent hurdle for defining the Berry phase and Berry curvature for these systems. We define a Berry connection for this system, with which the Berry phase and Berry curvature can be naturally computed. We use a three-level atom-molecule system to illustrate our results. In particular, we have computed the mean-field Berry curvature of this system analytically, and compared it to the Berry curvature computed with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our definition.
In the mean-field theory of atom-molecule systems,where the bosonic atoms combine to form molecules,there is no usual U(1) symmetry,which presents an apparent hurdle for calculating the Berry connection in these systems.We develop a perturbation expansion method of Hannay’s angle suitable for calculating the Berry curvature in the atom-molecule systems.With this Berry curvature,the Berry connection can be computed naturally.We use a three-level atom-molecule system to illustrate our results.In particular,with this method,we compute the curvature for Hannay’s angle analytically,and compare it to the Berry curvature obtained with the second-quantized model of the same system.An excellent agreement is found,indicating the validity of our method.
In the mean-field theory of atom—molecule systems, where the bosonic atoms combine to form molecules, there is no usual U(1) symmetry, which presents an apparent hurdle for calculating the Berry connection in these systems. We develop a perturbation expansion method of Hannay's angle suitable for calculating the Berry curvature in the atom—molecule systems. With this Berry curvature, the Berry connection can be computed naturally. We use a three-level atom—molecule system to illustrate our results. In particular, with this method, we compute the curvature for Hannay's angle analytically, and compare it to the Berry curvature obtained with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our method.
We discuss how symmetries and conservation laws are affected when Berry’s phase occurs in a quantum system: symmetry transformations of coordinates have to be supplemented by gauge transformations of Berry’s connection, and consequently constants of motion acquire terms beyond the familiar kinematical ones. We show how symmetries of a problem determine Berry’s connection, curvature and, once a specific path is chosen, the phase as well. Moreover, higher order corrections are also fixed. We demonstrate that in some instances Berry’s curvature and phase can be removed by a globally well-defined, time-dependent canonical transformation. Finally, we describe how field theoretic anomalies may be viewed as manifestations of Berry’s phase.
We examine the geometric interpretation of the quantum geometric tensor proposed in [Phys. Rev. B {\bf 110}, 035404 (2024)] for mixed quantum states, focusing on its imaginary part, which is proportional to a weighted sum of the Berry curvatures of the eigenstates of the density operator. While the real part naturally decomposes into Fisher--Rao and weighted Fubini--Study contributions, we show that the imaginary part does not, in general, coincide with the curvature of a connection on a globally defined U(1) line bundle whose holonomy yields a mixed-state geometric phase. Using a two-level system as an explicit example, we demonstrate that its surface integral depends on the choice of surface bounded by the same closed path, with ambiguities that are not integer multiples of 2π2π2π.
We investigate the quantum phase transition in an ultracold atom-molecule conversion system. It is found that the system undergoes a phase transition from a mixed atom-molecule phase to a pure molecule phase when the energy bias exceeds a critical value. By constructing a coherent state as variational state, we get a good approximation of the quantum ground state of the system. Using this variational state, we deduce the critical point analytically. We then discuss the scaling laws characterizing the transition and obtain the corresponding critical exponents. Furthermore, the Berry curvature signature of the transition is studied. In particular, we find that the derivatives of the Berry curvature with respect to total particle number intersect at the critical point. The underlying mechanism of this finding is discussed as well.
Generalizing the concept of the Berry phase, I show that an additional gauge structure associated with a time-component vector potential A0{A}_{0}A0​ is present in the adiabatic evolution of a quantum system. The gauge structure, which generates the usual dynamical phase, is revealed by a formalism in which the time variable is treated on an equal footing with other parameters on which the Hamiltonian depends. The invariant curvature associated with A0{A}_{0}A0​, an electriclike field, determines the phase difference between low- and high-energy paths in a generalized phase-space and time picture. The covariance of the Born-Oppenheimer approximation illustrates the results.