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arXiv:0810.2912 · Physical Review A

Entanglement, Berry phases, and level crossings for the atomic Breit-Rabi Hamiltonian

Sangchul Oh, Zhen Huang, Uri Peskin, Sabre Kais

Published Dec 9, 2008• 31 citations
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Abstract

The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided crossing points.

DOI
10.1103/physreva.78.062106

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The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided crossing points.
A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methods to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair $^{23}\mathrm{Na}\text{\ensuremath{-}}^{85}\mathrm{Rb}$ which are of current interest.
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.
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.
Impurity spins in semiconductors are potential quantum bits. Entanglement and topological phases are key resources in quantum computation. We prove that the coupled electron spins carried by a diamond nitrogen-vacancy color center (NV${}^{\ensuremath{-}}$) and a single substitutional nitrogen impurity (P1 center) are entangled in the immediate vicinity of the level anticrossing that appears in the Zeeman energy diagram at about 500 G. We also determine the Aharonov-Anandan, Berry, and marginal geometric phases that can be accumulated by the state vectors of this spin system when it is magnetically transported around a closed path. At the resonance where the gap between two energy levels is minimum, the geometric phases undergo discontinuities, and the entanglement of the two electron spins is maximal.
The Berry phase, acquired by an anisotropic spin system with spin J ≥1 when it is adiabatically rotated in a closed circuit, is considered to be associated with a non-quantized Dirac monopole, and has a geometrical as well as a topological component owing to the Dirac string. Here, it is argued that the Berry phase of a spin state with J ≥1 can be associated with a quantized Dirac monopole when the corresponding spin is taken to be an entangled state of a composite system of 1/2 spins. Evidently, this avoids the contribution of the Dirac string, and the Berry phase is purely geometrical in nature.