Geometric Quantum Noise of Spin

Abstract

The presence of geometric phases is known to affect the dynamics of the systems involved. Here, we consider a quantum degree of freedom, moving in a dissipative environment, whose dynamics is described by a Langevin equation with quantum noise. We show that geometric phases enter the stochastic noise terms. Specifically, we consider small ferromagnetic particles (nanomagnets) or quantum dots close to Stoner instability, and investigate the dynamics of the total magnetization in the presence of tunneling coupling to the metallic leads. We generalize the Ambegaokar-Eckern-Schön effective action and the corresponding semiclassical equations of motion from the U(1) case of the charge degree of freedom to the SU(2) case of the magnetization. The Langevin forces (torques) in these equations are strongly influenced by the geometric phase. As a first but nontrivial application, we predict low temperature quantum diffusion of the magnetization on the Bloch sphere, which is governed by the geometric phase. We propose a protocol for experimental observation of this phenomenon.

Strong nonequilibrium effects in spin-torque systems

“…Here, extending Ref. [12], we derive an effective action of the AmbegaokarEckern-Schön (AES) type [13,14], which governs the dynamics of both the charge (U (1)) and the spin (SU (2)) degrees of freedom. This allows us to obtain equations of motion describing simultaneously the induction of spin precession by current and the generation of current by rotating magnetization.…”

U(1) and SU(2) quantum dissipative systems: the Caldeira–Leggett Versus Ambegaokar–Eckern–Schön approaches

“…The central result of the current paper is the AES-like 9,10 effective action, which we obtain by expanding (37) to the first order inR…”

“…The idea now is to expand the action (37) in bothQ (which is small due to the slowness of n(t)) andR †ΣR (which is small due to the smallness of the tunneling amplitudes). A straightforward analysis reveals that a naive expansion to the lowest order in both violates the gauge invariance with respect to the choice of χ(t).…”

“…Our approach 37 can be viewed as a generalization of the Landau-Lifschitz-Gilbert (LLG)-Langevin equation 38,39 , central to the field of spintronics 40 , to a regime where quantum dynamics dominates. Stochastic LLG equations have been derived in numerous publications for both a localized spin in an electronic environment (a situation of the Caldeira-Leggett type) 41,42 and for a magnetization formed by itinerant electrons 43,44 .…”

“…Thus, if the spin could be held for a long time on a constant θ c = θ 0 trajectory (one possible way to do so was proposed in Ref. 37 ), the diffusion would be determined by the quantum noise at frequencies ω c and ω s , which are governed by the geometric phase. More precisely, the spread of θ c and φ c (in the rotating frame) will be given by (∆θ) 2 = sin 2 θ 0 (∆φ) 2 = Dt, where…”