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Physical Review A

Generation of non-Abelian geometric phases in degenerate atomic transitions

Lachezar S. Simeonov, Nikolay V. Vitanov

Published Sep 5, 2017• 6 citations
Publisher

Abstract

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 $J=0$ and $J=1$, 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 $N$ 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 $J$ 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.

DOI
10.1103/physreva.96.032102

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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−2N\ensuremath{-}2N\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↔J\ensuremath−1J\ensuremath{\leftrightarrow}J\ensuremath{-}1J\ensuremath↔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=32\ensuremath↔J=12J=\frac{3}{2}\ensuremath{\leftrightarrow}J=\frac{1}{2}J=23​\ensuremath↔J=21​ and J=2\ensuremath↔J=1J=2\ensuremath{\leftrightarrow}J=1J=2\ensuremath↔J=1 systems. A method for measuring the geometric phase is proposed.
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.
The Morris-Shore (MS) transformation is a powerful tool for decomposition of the dynamics of multistate quantum systems to a set of two-state systems and uncoupled single states. It assumes two sets of states wherein any state in the first set can be coupled to any state in the second set, but the states within each set are not coupled between themselves. Another important condition is the degeneracy of the states in each set, although all couplings between the states from different sets can be detuned from resonance by the same detuning. The degeneracy condition limits the application of the MS transformation in various physically interesting situations, e.g., in the presence of electric and/or magnetic fields or light shifts, which lift the degeneracy in each set of states, e.g., when these sets comprise the magnetic sublevels of levels with nonzero angular momentum. This paper extends the MS transformation to such situations, in which the states in each of the two sets are nondegenerate. To this end, we develop an alternative way for the derivation of Morris-Shore transformation, which can be applied to nondegenerate sets of states. We present an approximated eigenvalue approach, by which, in the limit of small detunings from degeneracy, we are able to generate an effective Hamiltonian that is dynamically equivalent to the nondegenerate Hamiltonian. The effective Hamiltonian can be mapped to the Morris-Shore basis with a two-step similarity transformation. After the derivation of the general framework, we demonstrate the application of this technique to the popular $\mathrm{\ensuremath{\Lambda}}$ three-state system, and the four-state tripod, double-$\mathrm{\ensuremath{\Lambda}}$, and diamond systems. In all of these systems, our formalism allows us to reduce their quantum dynamics to simpler two-state systems even in the presence of various detunings, e.g., generated by external fields of frequency drifts.
Geometric phases encode the holonomy acquired under cyclic control of a quantum Hamiltonian: for nondegenerate eigenstates the effect is Abelian (a scalar phase), while in degenerate manifolds the Wilczek-Zee generalization yields a matrix-valued holonomy that can mix states and depends on path ordering. We present an overview of measuring Abelian and non-Abelian geometric phases in nitrogen-vacancy (NV) centers in diamond and propose a non-rotating measurement protocol enabled by the intrinsic14 N{ }^{14} ~\mathrm{N}14 Nhyperfine manifold. As a first step, we propose to measure the Abelian Berry phase on a single hyperfine-resolved two-level transition using microwaveengineered closed loops with dynamical-phase cancellation by loop reversal. We then propose a non-Abelian protocol that engineers a two-dimensional degenerate dark subspace in an NV-14 N{ }^{14} ~\mathrm{N}14 Nmanifold via RF dressing and three phase-coherent microwave couplings (a tripod Hamiltonian). Two closed control loops implemented by winding independent microwave phases generate non-commuting SU(2) holonomies, which can be verified using final-only population readout and minimal tomography. We also present timing trade-off curves, full unitary Schrödinger evolution of the four-level tripod, and projected dark-subspace diagnostics that quantify both leakage and non-commuting holonomy.
The adiabatic motion of ultra-cold, multi-level atoms in spatially varying laser fields creating a dark state can give rise to effective gauge potentials. We show that these potentials lead to effective magnetic fields if the light fields possess a relative orbital angular momentum. This allows to study quantum-Hall like effects in ultra-cold atomic gases in various geometries. If the atom-light interaction creates several degenerate adiabatic eigenstates, the associated gauge potentials are non-Abelian. A pair of such degenerate dark states emerges e.g. if laser fields couple three internal states of an atom to a fourth common one under pairwise two--photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.
We show that a non-Abelian gauge potential in two nearly degenerated dressed states may be induced by two laser beams interacting with a three-level $\ensuremath{\Lambda}$ atomic system. We demonstrate that the populations of the atomic states at the end of a composed path formed by two closed loops are dependent on the order of those two loops, showing an unambiguous signature of the non-Abelian geometric phase. Through numerical calculations, we show that the non-Abelian feature of the geometric phases can be tested under realistic conditions.
We show that the adiabatic motion of ultracold, multilevel atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges, e.g., if laser fields couple three internal states of an atom to a fourth common one under pairwise two-photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.