Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers Heinz-J¨urgen Schmidt1, J¨urgen Schnack2, and Martin Holthaus3 1Universit¨at Osnabr¨uck, Fachbereich Physik, D-49069 Osnabr¨uck, Germany 2Universit¨at Bielefeld, Fakult¨at f¨ur Physik, D-33501 Bielefeld, Germany and 3Carl von Ossietzky Universit¨at, Institut f¨ur Physik, D-26111 Oldenburg, Germany 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 quasi- stationary 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. I. INTRODUCTION where the choice of the “canonical representative” distin- guished by setting ν = 0 is a matter of convention. A quantum system governed by an explicitly time- The significance of these Floquet states (2) rests in the dependent Hamiltonian H(t) which depends periodically fact that, as long as the Hamiltonian depends on time in on time t, such that a strictly T -periodic manner, every solution ψ(t) to the time-dependent Schr¨odinger equation can be| expandedi H(t)= H(t + T ) , (1) with respect to the Floquet basis, possesses a complete set of Floquet states, that is, of solu- ψ(t) = c u (t) exp( iε t) , (6) tions to the time-dependent Schr¨odinger equation having | i n | n i − n n the particular form X where the coefficients cn do not depend on time. Hence, ψ (t) = u (t) exp( iε t) . (2) | n i | n i − n the Floquet states propagate with constant occupation probabilities c 2, despite the presence of a time-periodic The Floquet functions un(t) share the T -periodic time | n| dependence of their Hamiltonian,| i drive. Under conditions of perfectly coherent time evo- lution these coefficients cn would be determined solely u (t) = u (t + T ) ; (3) | n i | n i by the system’s state at the moment the periodic drive is turned on. However, if the periodically driven sys- the quantities εn, which accompany their time evolu- tion in the same manner as energy eigenvalues accom- tem is interacting with an environment, as it happens in pany the evolution of unperturbed energy eigenstates, many cases of experimental interest [4–9], that environ- are known as quasienergies [1–3]. Here we assume that ment may continuously induce transitions among the sys- the quasienergies constitute a pure point spectrum, asso- tem’s Floquet states, to the effect that a quasistationary ciated with square-integrable Floquet states in the sys- distribution pn of Floquet-state occupation probabil- ities establishes{ } itself which contains no memory of the tem’s Hilbert space S; we also adopt a system of units H initial state, and the question emerges how to quantify arXiv:1902.05814v1 [quant-ph] 15 Feb 2019 such that both the Planck constant ~ and the Boltzmann this distribution. constant kB are set to one. Evidently the factorization of a Floquet state (2) into In a short programmatic note entitled “Periodic Ther- a Floquet function and an exponential of a phase which modynamics”, Kohn has drawn attention to such quasi- stationary Floquet-state distributions p , emphasizing grows linearly in time is not unique: Defining ω =2π/T , { n} and taking an arbitrary, positive or negative integer ν, that they should be less universal than usual distribu- one has tions characterizing thermal equilibrium, depending on the very form of the system’s interaction with its envi- iνωt un(t) exp( iεnt)= un(t)e exp i[εn+νω]t , (4) ronment [10]. In an earlier pioneering study, Breuer et | i − | i − al. had already calculated these distributions for time- where u (t)eiνωt again is a T -periodic Floquet function, n periodically forced oscillators coupled to a thermal os- representing| thei same Floquet state as u (t) . There- n cillator bath [11]. For the particular case of a linearly fore, a quasienergy is not to be regarded| as justi a sin- forced harmonic oscillator these authors have shown that gle number equipped with the dimension of energy, but the Floquet-state distribution remains a Boltzmann dis- rather as an infinite set of equivalent representatives, tribution with the temperature of the heat bath, whereas ε ε + νω ν Z , (5) it becomes rather more complicated in the case of forced n ≡{ n | ∈ } 2 anharmonic oscillators. These investigations have been ideal paramagnet to a circularly polarized driving field. extended later by Ketzmerick and Wustmann, who have In Sec. VI we consider the rate of energy dissipated by demonstrated that structures found in the phase space of the driven spins into the bath, thus generalizing results 1 classical forced anharmonic oscillators leave their distinct derived previously for s = 2 in Ref. [29]. In Sec. VII we traces in the quasistationary Floquet-state distributions summarize and discuss our main findings, emphasizing of their quantum counterparts [12]. To date, a great va- the possible knowledge gain to be derived from future riety of different individual aspects of the “periodic ther- measurements of paramagnetic response to strong time- modynamics” envisioned by Kohn has been discussed in periodic forcing, carried out along the lines drawn in the the literature [13–23], but a coherent overall picture is present work. still lacking. In this situation it seems advisable to resort to mod- els which are sufficiently simple to admit analytical so- II. TECHNICAL TOOLS lutions and thus to unravel salient features on the one hand, and which actually open up meaningful perspec- A. Golden-rule approach to open driven systems tives for new laboratory experiments on the other. To this end, in the present work we consider a spin s ex- Let us consider a quantum system evolving according posed to both a static magnetic field and an oscillating, to a T -periodic Hamiltonian H(t) on a Hilbert space S circularly polarized magnetic field applied perpendicu- which is perturbed by a time-independent operatorHV . lar to the static one, as in the classic Rabi set-up [24], Then the transition matrix element connecting an initial and coupled to a thermal bath of harmonic oscillators. Floquet function ui(t) to a final Floquet function uf (t) The experimental measurement of the thermal paramag- can be expanded| into ai Fourier series, | i netism resulting from magnetic moments subjected to a static field alone has a long and successful history [25, 26], (ℓ) uf (t) V ui(t) = Vfi exp(iℓωt) , (7) having become a standard topic in textbooks on Statis- h | | i ℓ∈Z tical Physics [27, 28]. We argue that a future generation X of such experiments, including both a static and a strong and consequently the “golden rule” for the rate of tran- oscillating field, may set further cornerstones towards the sitions Γfi from a Floquet state labeled i to a Floquet development of full-fledged periodic thermodynamics. state f is written as [29] We proceed as follows: In Sec. II we collect the nec- Γ =2π V (ℓ) 2 δ(ω(ℓ)) , (8) essary technical tools, starting with a brief summary of fi | fi | fi ∈Z the golden-rule approach to time-periodically driven open Xℓ quantum systems in the form developed by Breuer et al. [11], thereby establishing our notation. We also sketch where a technique which enables one to “lift” a solution to the (ℓ) 1 ωfi = εf εi + ℓω . (9) Schr¨odinger equation for a spin s = 2 in a time-varying − magnetic field to general s. In Sec. III we discuss the Thus, a transition among Floquet states is not associated Floquet states for spins in a circularly polarized driving with only one single frequency, but rather with a set of field, obtaining the states for general s from those for frequencies spaced by integer multiples of the driving fre- s = 1 with the help of the lifting procedure. In Sec. IV 2 quency ω, reflecting the ladder-like nature of the system’s we compute the quasistationary Floquet-state distribu- quasienergies (5); this is one of the sources of the pecu- tion for driven spins under the assumption that the spec- liarities which distinguish periodic thermodynamics from tral density of the heat bath be constant, and show that usual equilibrium thermodynamics [10, 11]. this distribution is Boltzmannian with a quasitempera- Let us now assume that, instead of merely being per- ture which is different from the actual bath temperature; turbed by V , the periodically driven system is coupled the dependence of this quasitemperature on the system to a heat bath, described by a Hamiltonian H acting parameters is discussed in some detail. In Sec. V we bath on a Hilbert space B, so that the total Hamiltonian on determine the magnetization of a spin system which is the composite HilbertH space takes the form subjected to both a static and an orthogonally applied, HS ⊗ HB circularly polarized magnetic field while being coupled ½ Htotal(t)= H(t) ½ + Hbath + Hint . (10) to a heat bath. To this end, we first