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Physical Review A

Rotation-operator approach and spin dynamics in a time-varying magnetic field

Jinyuan Zhou, Chaohui Ye

Published Aug 1, 1994• 10 citations
Publisher

Abstract

The evolution operator for an arbitrary spin system in an arbitrary time-varying magnetic field is disentangled into the rotation operator; consequently, the time-ordering problem is shifted into solving the Euler geometric equations. Some realistic transformations are proposed in order to find their analytic solutions. The probabilities and phases characterizing state transitions are also analyzed.

DOI
10.1103/physreva.50.1903

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•Jul 15, 2003•4 citations
The evolution operator for an arbitrary spin system in an arbitrary time-varying magnetic field is disentangled into the rotation operator; consequently, the time-ordering problem is shifted into solving the Euler geometric equations. Some realistic transformations are proposed in order to find their analytic solutions. The probabilities and phases characterizing state transitions are also analyzed.
The rotation operator approach proposed previously is applied to spin dynamics in a time-varying magnetic field. The evolution of the wave function is described, and that of the density operator is also treated in terms of a spherical tensor operator base. It is shown that this formulation provides a straightforward calculation of accumulated phases and probabilities of spin transitions and coherence evolutions. The technique focuses, not on the rotation matrix, but on the three Euler angles and its characteristic equations are equivalent to the Euler geometric equations long known to describe the motion of a rigid body. The method usually depends on numerical calculations, but analytical solutions exist in some situations. In this paper, as examples, a hyperbolic secant pulse is solved analytically, and a Gaussian-shaped pulse is calculated numerically.
The time evolution of a spin precessing in a large static field with a perpendicular rotating field is a classic problem in magnetic resonance that appears in nearly all quantum textbooks. It can be solved exactly by going to the rotating frame. Often, however, the final transition probabilities are determined only for the spin 1/2 case. In this work, we show how an operator-based approach naturally produces a time-ordered product via a Trotter product formula and then by employing simple operator identities such as the Hadamard lemma, the Baker–Campbell–Hausdorff formula, and the exponential disentangling identity, the problem can be directly solved for arbitrary spin.
A neutral particle with general spin and magnetic moment moving in an arbitrarily varying magnetic field is studied. The time evolution operator for the Schrödinger equation can be obtained if one can find a unit vector that satisfies the equation obeyed by the mean of the spin operator. There exist at least 2 s + 1 cyclic solutions in any time interval. Some particular time interval may exist in which all solutions are cyclic. The nonadiabatic geometric phase for cyclic solutions generally contains extra terms in addition to the familiar one that is proportional to the solid angle subtended by the closed trace of the spin vector.
All the six kinds of rotation approach with the same form to the evolution problem of arbitrarily time-dependent two-level system are investigated in this paper. A time-dependent two-level system can be viewed as a spin-1/2 system in a time-varying magnetic field. It is shown that for each kind of rotation approach, we can always find a rotating frame in which the direction of the effective magnetic field is fixed. This property reduces the problem of finding the time-evolution operator to the solution of a second-order differential equation. Applications are made to the J C model in quantum optics and the L and au–Zener model in resonance physics.
A neutral particle with general spin and magnetic moment moving in an arbitrarily varying magnetic field is studied. The time evolution operator for the Schr\\"odinger equation can be obtained if one can find a unit vector that satisfies the equation obeyed by the mean of the spin operator. There exist at least 2s+12s+12s+1 cyclic solutions in any time interval. Some particular time interval may exist in which all solutions are cyclic. The nonadiabatic geometric phase for cyclic solutions generally contains extra terms in addition to the familiar one that is proportional to the solid angle subtended by the closed trace of the spin vector.