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European Journal of Physics

Smooth and oscillatory geometric phase corrections for driven spins

Michael Berry

Published Sep 8, 2023• 2 citations
Publisher

Abstract

Abstract For a quantum spin driven cyclically by a slowly-rotated magnetic field, geometric phases are well understood. If the cycle takes a long time T , the leading-order (dynamical) phase is proportional to T and the geometric phase is the contribution independent of T . The dynamical and geometric phases are the first two terms of a series in slowness 1 / T . Here it is shown with an exactly solvable example that the corrections are of two types: smooth, proportional to powers of slowness, and oscillatory: essential singularities in 1 / T , in the form of trigonometric functions of T divided by powers of T . The calculations are elementary and therefore suitable for presentation in graduate quantum theory courses.

DOI
10.1088/1361-6404/acf81e

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Abstract For a quantum spin driven cyclically by a slowly-rotated magnetic field, geometric phases are well understood. If the cycle takes a long time T , the leading-order (dynamical) phase is proportional to T and the geometric phase is the contribution independent of T . The dynamical and geometric phases are the first two terms of a series in slowness 1 / T . Here it is shown with an exactly solvable example that the corrections are of two types: smooth, proportional to powers of slowness, and oscillatory: essential singularities in 1 / T , in the form of trigonometric functions of T divided by powers of T . The calculations are elementary and therefore suitable for presentation in graduate quantum theory courses.
It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geometric phase and the solid angle are both calculated explicitly. It turns out that the nonadiabatic geometric phase contains an extra term in addition to the one proportional to the solid angle. The extra term vanishes automatically for spin 1/2. For higher spin, however, it depends on the initial condition. We also consider the valence electron of an alkaline atom. For cyclic solutions with special initial conditions in an arbitrary strong magnetic field, we prove that the nonadiabatic geometric phase is a linear combination of the two solid angles subtended by the traces of the orbit and spin angular momenta. For more general cyclic solutions in a strong rotating magnetic field, the nonadiabatic geometric phase also contains extra terms in addition to the linear combination.
With the help of the time-dependent gauge transformation technique, we have studied the geometric phase of a spin-half particle in a rotating magnetic field. We have found that the slow but finite frequency of the rotating magnetic field will make the difference between the adiabatic geometric phase and the exact geometric phase. When the frequency is much smaller than the energy space and the adiabatic condition is perfectly guaranteed, the adiabatic approximation geometric phase is exactly consistent with the adiabatic geometric phase. A simple relation for the accuracy of the adiabatic approximation is given in terms of the changing rate of the frequency of the rotating magnetic field and the energy level space.
We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V(\mathbit{\ensuremath{\lambda}}\left(t\right))$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (noncommuting) geometric phases appearing when the slowly varying parameter $\mathbit{\ensuremath{\lambda}}=\mathbit{\ensuremath{\lambda}}\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.
Quantum states which evolve cyclically in their projective Hilbert space give rise to a geometric (or Aharonov-Anandan) phase. An aspect of primary interest is stable cyclic behaviour as realized under a periodic Hamiltonian. The problem has been handled by use of time-dependent transformations treated along the lines of Floquet's theory as well as in terms of exponential operators with a goal to examine the variety of initial states exhibiting cyclic behaviour. A particular case of special cyclic initial states is described which is shown to be important for nuclear magnetic resonance experiments aimed at the study of the effects of the geometric phase. An example of arbitrary spin j in a precessing magnetic field and spin j=1 subject to both axially symmetric quadrupolar interaction and a precessing magnetic field are presented. The invariant (Kobe's) geometric phase is calculated for special cyclic states.
Quantum invariant theory is an effective method to study geometric phase problems. By making use of the Lewis Riesenfeld invariant theory, a particle with spin j in an oscillating magnetic field is studied and the exact solutions for two different cyclic periods are found. It is then used to obtain two different geometric phases corresponding to two different cyclic periods. It is quite clear from our derivation that the geometric phases corresponding to two different cyclic periods are closely associated with the initial states of the system.
We consider a quantum limit-cycle oscillator implemented in a spin system whose quantization axis is slowly rotated. Using a kinematic approach to define geometric phases in nonunitary evolution, we show that the quantum limit-cycle oscillator attains a geometric phase when the rotation is sufficiently slow. In the presence of an external signal, the geometric phase as a function of the signal strength and the detuning between the signal and the natural frequency of oscillation shows a structure that is strikingly similar to the Arnold tongue of synchronization. Surprisingly, this structure vanishes together with the Arnold tongue when the system is in a parameter regime of synchronization blockade. We derive an analytic expression for the geometric phase of this system, valid in the limit of slow rotation of the quantization axis and weak external signal strength, and we provide an intuitive interpretation for this surprising effect.