Berry connection in atom-molecule systems

Abstract

In the mean-field theory of atom-molecule systems, where bosonic atoms combine to form molecules, there is no usual 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.

Quantum phase transition in a three-level atom-molecule system

“…On the other hand, the atom-molecule conversion systems can be well described by a mean-field theory when the particle number is large enough and this treatment will lead to nonlinearity. These features have stimulated much efforts to study the adiabatic evolution [13][14][15][16][17][18], geometric phase [19,20], and phase transition [21,22] of the systems. Instead of the traditional approaches for describing a QTP (i.e., using the concepts of order parameter and symmetry breaking), very recently Santos el al.…”

“…In the three-mode description, each mode |α (α = a, g, and e respectively represent the atomic mode, the ground-state molecular mode, and the excited-state molecular mode) is associated with an annihilation operator β (β = a, b g , and b e ) due to the basic assumption that the spatial wavefunctions for these modes are fixed. By setting the energy of atomic mode as zero, the Hamiltonian of the system takes the following second-quantized form with = 1 [20]:…”

“…To achieve this one can split the laser pulse into two beams and then recombine and focus them on the system. As a result, we can express the complex parameter Ω p as Ω p = ξ 1 + ξ 2 e −iϕ with ξ 1 and ξ 2 being real numbers, where the phase factor ϕ is determined by the difference of optical paths between the two laser beams [20].…”

New method for calculating the Berry connection in atom—molecule systems

“…[12−18] Naturally, there are also efforts to define the Berry connection in this type of systems. [19,20] However, even though the Berry connection is defined for these atom-molecule systems in Ref. [20], it is still difficult to compute it because of the nonlinearity, which is not the usual U (1) symmetry.…”

“…[19,20] However, even though the Berry connection is defined for these atom-molecule systems in Ref. [20], it is still difficult to compute it because of the nonlinearity, which is not the usual U (1) symmetry. Thus it is not trivial to develop a convenient method suitable for computing the Berry connection in these atom-molecule systems.…”