Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers

Abstract

We consider a spin s subjected to both a static and an orthogonally applied oscillating, circularly polarized magnetic field while being coupled to a heat bath, and analytically determine the quasistationary distribution of its Floquet-state occupation probabilities for arbitrarily strong driving. This distribution is shown to be Boltzmannian with a quasitemperature which is different from the temperature of the bath, and independent of the spin quantum number. We discover a remarkable formal analogy between the quasithermal magnetism of the nonequilibrium steady state of a driven ideal paramagnetic material, and the usual thermal paramagnetism. Nonetheless, the response of such a material to the combined fields is predicted to show several unexpected features, even allowing one to turn a paramagnet into a diamagnet under strong driving. Thus, we argue that experimental measurements of this response may provide key paradigms for the emerging field of periodic thermodynamics.

Periodic thermodynamics of the parametrically driven harmonic oscillator

“…This definition of the quasitemperature formally yields negative τ when r > 1. While such negative quasitemperatures are quite natural and physically meaningful in systems with a finite-dimensional Hilbert space, such as periodically driven spin systems [9], here they signal quasithermal instability, implying Γ n+1,n > Γ n,n+1 , so that the particle tends to climb the oscillator ladder to infinite height.…”

“…Now let the parametrically driven "system" (1) with T -periodic spring function k(t) be weakly coupled to a "bath" consisting of infinitely many harmonic oscillators with a prescribed temperature, causing transitions among the system's Floquet states; our goal is to find the corresponding quasistationary distribution [5,[8][9][10]. Following the general theory of open quantum systems [32] we then require, besides the Hilbert space H system that the driven part H 0 (t) is acting on, the Hilbert space H bath pertaining to the bath Hamiltonian H bath , and construct the composite space H system ⊗ H bath .…”

“…Thus, the master equation (71) simplifies considerably, reducing to (Γ n,n−1 p n−1 − Γ n−1,n p n ) + (Γ n,n+1 p n+1 − Γ n+1,n p n ) = 0 (74) for n ≥ 1; for n = 0 the first bracket disappears. This tridiagonal form implies detailed balance [9], meaning that both brackets vanish individually for n ≥ 1: Setting the second bracket to zero gives the forward relation…”

“…As in the case of a spin driven by a circularly polarized field [9], the existence of such a geometric distribution (77), combined with equidistantly spaced canonical representatives (51) of the system's quasienergies, now allows one to introduce a quasitemperature τ for the periodically driven nonequilibrium system: Setting…”

“…Remarkably, a linearly driven harmonic oscillator interacting with a harmonic-oscillator heat bath retains the Boltzmann distribution of the bath, that is, the Floquet states of the linearly driven harmonic oscillator are occupied according to a Boltzmann distribution with the bath temperature [5,8]. On the other hand, the Floquet substates of a spin s exposed to a circularly polarized driving force while being coupled to a heat bath develop a Boltzmann distribution with an effective quasitemperature which differs from the actual temperature of the bath [9]. In the case of strongly driven anharmonic oscillators the quasistationary Floquet-state distributions can be significantly influenced by phase-space structures of the corresponding classical system [5,10].…”