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arXiv:cond-mat/0402319 · Physical Review B

Spin precession in an adiabatically rotating electric field

Yuri A. Serebrennikov

Published Aug 30, 2004• 9 citations
arXivHTMLPDF

Abstract

Due to spin-orbit coupling, the adiabatic perturbation of an electron's orbital motion induced by a revolving external electric field lead to the electron spin-precession. The obtained results describe both transverse and longitudinal dynamics of an effective spin, for conical and figure-8 trajectories of a driving electric field, and elucidate the link between the quantum-mechanical, geometrical, and classical pictures of spin rotation. Within the limit of adiabatic approximation the derived formulas are valid regardless of whether eigenvectors of the total Hamiltonian of the problem are explicitly available or not, and are convenient for approximate calculations. The main expression for the time evolution of the Bloch vector has pure geometric character and is independent of the physical context.

DOI
10.1103/physrevb.70.064422

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Due to spin-orbit coupling, the adiabatic perturbation of an electron's orbital motion induced by a revolving external electric field lead to the electron spin-precession. The obtained results describe both transverse and longitudinal dynamics of an effective spin, for conical and figure-8 trajectories of a driving electric field, and elucidate the link between the quantum-mechanical, geometrical, and classical pictures of spin rotation. Within the limit of adiabatic approximation the derived formulas are valid regardless of whether eigenvectors of the total Hamiltonian of the problem are explicitly available or not, and are convenient for approximate calculations. The main expression for the time evolution of the Bloch vector has pure geometric character and is independent of the physical context.
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This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.
We calculate Berry’s phase when the driving field, to which a spin-[Formula: see text] is coupled adiabatically, rather than the familiar classical magnetic field, is a quantum vector operator, of noncommuting, in general, components, e.g. the angular momentum of another particle, or another spin. The geometric phase of the entire system, spin plus “quantum driving field”, is first computed, and is then subdivided into the two subsystems, using the Schmidt decomposition of the total wave function — the resulting expression shows a marked, purely quantum effect, involving the commutator of the field components. We also compute the corresponding mean “classical” phase, involving a precessing magnetic field in the presence of noise, up to terms quadratic in the noise amplitude — the results are shown to be in excellent agreement with numerical simulations in the literature. Subtleties in the relation between the quantum and classical case are pointed out, while three concrete examples illustrate the scope and internal consistency of our treatment.
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We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation with the time-dependent Hamiltonian can be reduced to a Schr\"odinger-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Schr\"odinger equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explicitly available. For the alkaline atomic electron and a strong magnetic field, the eigenvalue problem of the effective Hamiltonian is completely solved, and the geometric phase turns out to be a linear combination of two solid angles. For a weak magnetic field, the same problem is solved partly. For a general charged particle, the problem is solved approximately in a slowly rotating magnetic field, and the geometric phases are also calculated.
We consider a semi-classical model to describe the origin of the spin-orbit interaction in a simple system such as the hydrogen atom. The interaction energy U is calculated in the rest-frame of the nucleus, around which an electron, having linear velocity v and magnetic dipole-moment μ, travels in a circular orbit. The interaction energy U is due to the coupling of the induced electric dipole p=(v/c)×μ with the electric field En of the nucleus. According to quantum mechanics, the radius of the electron’s orbit remains constant during a spin-flip transition. Under such circumstances, our model predicts that the energy of the system changes by ΔE=1/2U, the factor 1/2 emerging naturally as a consequence of equilibrium and the change of the kinetic energy of the electron. The correct 1/2 factor for the spin-orbit coupling energy is thus derived without the need to introduce the well-known Thomas precession in the rest-frame of the electron.