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arXiv:0909.2061 · Physical Review A

Berry-like phases in structured atoms and molecules

Edmund Meyer, A. E. Leanhardt, Eric Cornell, John L. Bohn

Published Dec 15, 2009• 19 citations
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Abstract

Quantum mechanical phases arising from a periodically varying Hamiltonian are considered. These phases are derived from the eigenvalues of a stationary, ``dressed'' Hamiltonian that is able to treat internal atomic or molecular structure in addition to the time variation. In the limit of an adiabatic time variation, the usual Berry phase is recovered. For more rapid variation, nonadiabatic corrections to the Berry phase are recovered in perturbation theory, and their explicit dependence on internal structure emerges. Simple demonstrations of this formalism are given, to particles containing interacting spins, and to molecules in electric fields.

DOI
10.1103/physreva.80.062110

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Quantum mechanical phases arising from a periodically varying Hamiltonian are considered. These phases are derived from the eigenvalues of a stationary, ``dressed'' Hamiltonian that is able to treat internal atomic or molecular structure in addition to the time variation. In the limit of an adiabatic time variation, the usual Berry phase is recovered. For more rapid variation, nonadiabatic corrections to the Berry phase are recovered in perturbation theory, and their explicit dependence on internal structure emerges. Simple demonstrations of this formalism are given, to particles containing interacting spins, and to molecules in electric fields.
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.
The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.
Berry's work on quantum systems described by adiabatically varying Hamiltonians is introduced. Some simple systems exhibiting the "Berry phase" phenomenon are discussed, including: a spin-½ particle in a slowly varying magnetic field, linearly polarised light propagating down a helically wound optical fibre, and dynamic Jahn-Teller E ⊗ systems. The way in which the Berry phase concept illuminates the origin of anomalies in chiral gauge field theories, and the fermionic behaviour of solitons in certain scalar field theories (skyrmions), is also indicated.
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.
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.