Periodic thermodynamics of the parametrically driven harmonic oscillator

Abstract

We determine the quasistationary distribution of Floquet-state occupation probabilities for a parametrically driven harmonic oscillator coupled to a thermal bath. Since the system exhibits detailed balance, and the canonical representatives of its quasienergies are equidistant, these probabilities are given by a geometrical Boltzmann distribution, but its quasitemperature differs from the actual temperature of the bath, being affected by the functional form of the latter's spectral density. We provide two examples of quasithermal engineering, i.e., of deliberate manipulation of the quasistationary distribution by suitable design of the spectral density: We show that the driven system can effectively be made colder than the undriven one, and demonstrate that quasithermal instability can occur even when the system is mechanically stable.

Floquet-state cooling

“…To demonstrate the feasibility of such unexpected Floquet-state cooling we employ the model of a harmonic oscillator with a periodically time-dependent spring function k(t), which often has served as a workhorse in studies of quantum thermodynamics [27][28][29][30]. It is given by the Hamiltonian…”

Environment-controlled Floquet-state paramagnetism

“…We now couple this system weakly to a heat bath consisting of infinitely many, thermally occupied harmonic oscillators, described by a bath Hamiltonian H bath on a corresponding Hilbert space H bath . Let us briefly summarize the required formal steps [22,23,25]: The total Hamiltonian on the composite space H sys ⊗ H bath can be written as [26]…”

“…The analytical calculation of such Floquet-state occupation probabilities is feasible for certain integrable systems only [13,20,22,23,25]; in general, the above framework has to be implemented numerically. Thus, when an ideal paramagnetic material is strongly driven by an oscillating field B 1 , while interacting with a thermal environment and being probed by a static field B 0 directed along the z-axis, the resulting nonequilibrium "quasithermal" magnetization is still given by a relation of the general form (3), but the usual thermal expectation value (4) now has to be replaced by the Floquet-state expectation value [25]…”

“…As a consequence of time-periodic forcing, the driven system possesses a complete set of Floquet states; as a consequence of their interaction with the heat bath, a nonequilibrium steady state establishes itself which is characterized by a quasistationary distribution of Floquet-state occupation probabilities. Quite unlike the universal Boltzmann distribution (2), such Floquet-state distributions depend on the very details of the system-bath interaction [7,[22][23][24]. Therefore, measurements of these distributions, or of observables directly governed by them, provide experimental access to system-bath coupling mechanisms, or to details of the heat bath itself, which remain unobservable when probing equilibrium systems.…”

Periodic thermodynamics of a two spin Rabi model

“…• Similarly, the parametrically driven harmonic oscillator assumes a quasi-stationary state with a quasitemperature that is, however, generally different from the bath temperature, see [26]. • A spin s exposed to both a static magnetic field and an oscillating, circularly polarized magnetic field applied perpendicular to the static one, as in the classic Rabi set-up [27], and coupled to a thermal bath of harmonic oscillators has been shown to approach a quasi Boltzmann distribution, see [28].…”