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arXiv:1104.5299

Berry Phase for Systems with Angular Momenta in Electric and Magnetic Fields

Kumar Ghosh, De Munshi, D., Binayak Dutta-Roy

Published Apr 28, 2011• 0 citations
arXivHTMLPDF

Abstract

The Berry phase for a variety of systems comprising of two angular momenta is discussed. These include the electron and proton in the ground state of the hydrogen atom (taking into account the hyperfine interaction), the positronium atom, the spin and orbital angular momenta of a single electron, muon capture by the deuteron in an external magnetic field, etc. Though the time scales involved, the underlying intrinsic Hamiltonians are quite different, as also the possible experimental probes, the geometric nature of the results for the Berry phase due to a time varying externally imposed magnetic field is found to be quite robust. Some indications are also put forward as to the possible interesting studies with time varying electric fields. The objective of this work is an attempt to broaden the scope of such studies in both the experimental and theoretical directions.

DOI
10.48550/arxiv.1104.5299

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The Berry phase for a variety of systems comprising of two angular momenta is discussed. These include the electron and proton in the ground state of the hydrogen atom (taking into account the hyperfine interaction), the positronium atom, the spin and orbital angular momenta of a single electron, muon capture by the deuteron in an external magnetic field, etc. Though the time scales involved, the underlying intrinsic Hamiltonians are quite different, as also the possible experimental probes, the geometric nature of the results for the Berry phase due to a time varying externally imposed magnetic field is found to be quite robust. Some indications are also put forward as to the possible interesting studies with time varying electric fields. The objective of this work is an attempt to broaden the scope of such studies in both the experimental and theoretical directions.
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.
We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].
We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].
We derive the general formula giving Berry's phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the 'effective' electric and magnetic fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to useful selection rules. Furthermore, the Hamiltonian parameter space coincides with the density matrix space for a spin S = 1. This implies a mathematical identity between Berry's phase and the Aharonov–Anandan phase, which is lost for S > 1. We have found that new physical features of Berry's phases emerge for integer spins ⩾2. We provide explicit numerical results of Berry's phases for S = 2, 3, 4. For any spin, one easily finds well-defined regions of the parameter space where instantaneous eigenstates remain widely separated. We have decided to stay within them and not deal with non-Abelian Berry's phases. We present a thorough and precise analysis of the non-adiabatic corrections with separate treatment for periodic and non-periodic Hamiltonian parameters. In both cases, the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept below the 0.1% level. For a class of quantum cycles, involving as a sole periodic parameter the precession angle of the electric field around the magnetic field, the corrections odd upon the reversal of the associated rotation velocity can be canceled exactly if the quadrupole to dipole coupling is chosen appropriately ('magic values'). The even ones are eliminated by taking the difference of the Berry phases generated by two 'mirror' cycles. We end by indicating a possible application of the theoretical tools developed in this paper. We propose a way to perform a holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian given as a nonlinear function of the total spin operator S , defined as the sum of the N individual spin operators. The basic idea behind this proposal is the mathematical fact that any non-correlated states can be expanded into eigenstates of S 2 and S z . The same eigenvalues appear several times in the decomposition when N > 2 but all these states differ by their symmetry properties under the N -spin permutations. The case N = 4 and S z = 1 is treated explicitly and a maximum entanglement is achieved.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects