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Modern Physics Letters B

SPIN-J GEOMETRIC PHASE DRIVEN BY A CLASSICAL FLUCTUATING FIELD

Jibing Liu, Xin-You Lü, Na Liu, Anshou Zheng, Zhiming Zhan

Published Mar 10, 2008• 0 citations
Publisher

Abstract

The authors investigate the geometric phase of a spin-J system interacting with a classical magnetic field which contains a fluctuating component. Assuming an Ornstein–Uhlenbeck process for the fluctuations, we find that the distribution due to the fluctuating component for the geometric phase is a Gaussian. We calculate in detail the variance of geometric and dynamic phases. The results clearly show that the geometric phase can reduce the fluctuations, but the dynamic phase can not.

DOI
10.1142/s0217984908014912

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The authors investigate the geometric phase of a spin-J system interacting with a classical magnetic field which contains a fluctuating component. Assuming an Ornstein–Uhlenbeck process for the fluctuations, we find that the distribution due to the fluctuating component for the geometric phase is a Gaussian. We calculate in detail the variance of geometric and dynamic phases. The results clearly show that the geometric phase can reduce the fluctuations, but the dynamic phase can not.
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.
The geometric phase has been proposed as a candidate for noise resilient coherent manipulation of fragile quantum systems. Since it is determined only by the path of the quantum state, the presence of noise fluctuations affects the geometric phase in a different way than the dynamical phase. We have experimentally tested the robustness of Berry's geometric phase for spin-1/2 particles in a cyclically varying magnetic field. Using trapped polarized ultracold neutrons, it is demonstrated that the geometric phase contributions to dephasing due to adiabatic field fluctuations vanish for long evolution times.
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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.