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Journal of Modern Optics

Geometric phases of two- and three-level atomic systems related to group theory

Y. Ben-Aryeh

Published Mar 10, 2007• 0 citations
Publisher

Abstract

The Hamiltonians of two- and three-level atomic systems interacting with nearly resonant electro-magnetic fields are described. The geometric phase related to the σ z operator of a two-level system is analysed and it is shown that this phase is well defined also for an open circuit. For a closed circuit this phase becomes equivalent to twice the Berry phase. For three-level atomic system the eigenenergies of the Hamiltonian are obtained by solving a cubic equation. Simplified forms for these eigenenergies and a condition for degeneracy are obtained. For non-degenerate three-level system geometric phases related to Cartan subalgebra are analysed under the adiabatic approximation and their relations with the three-level Berry phases are clarified. We show how to use the general theory of non-Abelian connections to the Hamiltonian of a three-level atomic system which includes a degeneracy.

DOI
10.1080/09500340601004951

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The Hamiltonians of two- and three-level atomic systems interacting with nearly resonant electro-magnetic fields are described. The geometric phase related to the σ z operator of a two-level system is analysed and it is shown that this phase is well defined also for an open circuit. For a closed circuit this phase becomes equivalent to twice the Berry phase. For three-level atomic system the eigenenergies of the Hamiltonian are obtained by solving a cubic equation. Simplified forms for these eigenenergies and a condition for degeneracy are obtained. For non-degenerate three-level system geometric phases related to Cartan subalgebra are analysed under the adiabatic approximation and their relations with the three-level Berry phases are clarified. We show how to use the general theory of non-Abelian connections to the Hamiltonian of a three-level atomic system which includes a degeneracy.
A theoretical analysis of Berry's phases, which is based on the Cartan subalgebra, is given for three-level atomic systems. The analysis is developed under the adiabatic approximation and is related to topological fibre bundle theories. The topological Berry's phases derived in the present study, are related to phase shifts and can be applied, also, for partial cycles. Some possible applications of the present theory are discussed, for three-level and two-level atomic systems, interacting with nearly resonant monochromatic fields, under the semiclassical and rotating-wave approximations.
The full effect of adiabatically varying Hamiltonians on systems prepared in eigenstates of the stationary Hamiltonian often includes geometric Berry's phases. For degenerate systems these effects may result in transitions between distinct degenerate eigenstates that can be described in terms of a non-Abelian gauge potential. I demonstrate the explicit dynamic origin of such transitions between degenerate angular-momentum sublevels in an atom that is subject to collinear electric and magnetic fields that rotate adiabatically. The origin of the effects becomes clear when eigenstates of the total Hamiltonian, which includes the rotating fields, are calculated. The total Hamiltonian removes the degeneracy, and the eigenstates are in some cases linear combinations of the angular-momentum sublevels. Thus a system prepared in a specific sublevel may not remain in that sublevel, and the transition probability is exactly that given by an analysis of the geometric phase.
Group-theoretical methods are developed for treating Berry phase effects, which are related to Cartan subalgebra. The theory is applied to two-level and three-level atoms interacting with perturbations that are described by the SU(2) or SU(3) algebra. By using fiber-bundle theories, it is found that a time development operator that depends on Cartan group generators can represent a fiber while a time development operator that depends on other generators of the group represents the base of the quantum manifold. The total time development operator is obtained by multiplication of these two parts and the fiber-bundle theory is applied for calculating Berry phase effects. Explicit expressions for Berry phases are obtained under the adiabatic approximation.
A popular quantum system, in which the Pancharatnam-Berry non-Abelian geometric phase has been generated and exploited, is the atomic tripod system. It is conveniently created by linking a single atomic state with three other states by three electromagnetic fields. Such a linkage pattern naturally emerges between the magnetic sublevels of two atomic levels with angular momenta J=0J=0J=0 and J=1J=1J=1, although tripod implementations between other suitable sublevels are also used. Here we go beyond the limitation of a tripod system and show that it is possible to generate the non-Abelian geometric phase in a quantum system composed of NNN lower and $N\ensuremath{-}2$ upper sublevels. The theoretical instrument is the Morris-Shore transformation which reveals the existence of two uncoupled (dark) states composed of the lower sublevels only. A possible physical implementation is the atomic transition $J\ensuremath{\leftrightarrow}J\ensuremath{-}1$, with JJJ arbitrary, which is driven, as in the case of tripod system, by three electromagnetic fields of different polarizations. This generalization considerably broadens the range of systems that can be used to generate a geometric phase, with the the same experimental complexity as in the tripod system. Specific calculations of the non-Abelian geometric phase are presented for $J=\frac{3}{2}\ensuremath{\leftrightarrow}J=\frac{1}{2}$ and $J=2\ensuremath{\leftrightarrow}J=1$ systems. A method for measuring the geometric phase is proposed.
Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕21∕21∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.