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Journal of Modern Optics

Berry's Phase in the Coherent Excitation of Atoms

Stephen Mark Barnett, Demosthenes Ellinas, M.‐A. Dupertuis

Published Mar 1, 1988• 13 citations
Publisher

Abstract

The adiabatic variation of a Hamiltonian can cause the wavefunction, governed by the Hamiltonian, to acquire an unexpected phase. The existence of this phase (Berry's phase) is an additional element in the well-known quantum adiabatic theorem. Berry's phase is observable in the interference between two identically prepared systems only one of which is adiabatically varied. We show that a single quantum system prepared in a superposition of the eigenstates of its Hamiltonian leads to observable Berry phase effects at all times. We examine this principle within the context of two-level optical resonance.

DOI
10.1080/09500348814550581

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The adiabatic variation of a Hamiltonian can cause the wavefunction, governed by the Hamiltonian, to acquire an unexpected phase. The existence of this phase (Berry's phase) is an additional element in the well-known quantum adiabatic theorem. Berry's phase is observable in the interference between two identically prepared systems only one of which is adiabatically varied. We show that a single quantum system prepared in a superposition of the eigenstates of its Hamiltonian leads to observable Berry phase effects at all times. We examine this principle within the context of two-level optical resonance.
We examine the role of Berry's geometrical phase [Proc. R. Soc. London, Ser. A 392, 45 (1984)] in the coherent excitation of atoms. The dynamics of a two-level atom prepared in a superposition of its eigenstates will be modified in a nontrivial fashion by the Berry phase. We extend Berry's formalism to treat the Liouville equation for the density matrix. This extension allows us to discuss statistical mixtures and to incorporate Berry's phase within the well-known Bloch-vector description of optical resonance. Furthermore, our density-matrix formalism provides the means of investigating Berry's phase in open, dissipative systems.
The adiabatic theorem states that if we prepare a quantum system in one of the instantaneous eigenstates then the quantum number is an adiabatic invariant and the state at a later time is equivalent to the instantaneous eigenstate at that time apart from phase factors. Recently, Marzlin and Sanders have pointed out that this could lead to apparent violation of unitarity. We resolve the Marzlin-Sanders inconsistency within the quantum adiabatic theorem. Yet, our resolution points to another inconsistency, namely, that the cyclic as well as non-cyclic adiabatic Berry phases may vanish under strict adiabatic condition. We resolve this inconsistency and develop an unitary operator decomposition method to argue for the validity of the adiabatic approximation.
A formalism for calculating Berry phases for nonadiabatic systems containing an additional adiabatic time dependence is developed. This is then applied to a two-level atom in a strong resonant polarized laser, where the laser polarization provides the adiabatic parameter. It is found that, when the laser polarization is kept constant, judicious polarization choices allow one to eliminate either the rotating-wave or non-rotating-wave part of the interaction. Furthermore, the adiabatic Berry phasor obtained when the laser polarization is varied is found to be complex. This can be compared to the case of purely adiabatic time-reversal even systems, which can only have real Berry phasors.
We consider a two-level system coupled to a highly non-Markovian environment when the coupling axis rotates with time. The environment may be quantum (for example a bosonic bath or a spin bath) or classical (such as classical noise). We show that an Anderson orthogonality catastrophe suppresses transitions, so that the system's instantaneous eigenstates (parallel and anti-parallel to the coupling axis) can adiabatically follow the rotation. These states thereby acquire Berry phases; geometric phases given by the area enclosed by the coupling axis. Unlike in earlier proposals for environment-induced Berry phases, here there is little decoherence, so one does not need a decoherence-free subspace. Indeed we show that this Berry phase should be much easier to observe than a conventional one, because it is not masked by either the dynamic phase or the leading non-adiabatic phase. The effects that we discuss should be observable in any qubit device where one can drive three parameters in the Hamiltonian with strong man-made noise.
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.