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In system’s the [10]. distribu- of ronment on usual form depending very than equilibrium, the universal thermal less characterizing be tions should they that h infiac fteeFoutsae 2 et nthe in rests (2) states Floquet these of significance The nasotpormai oeette Proi Ther- “Periodic entitled note programmatic short a In a led acltdteedsrbtosfrtime- for distributions these calculated already had oaiiisfrabtaiysrn driving. strong arbitrarily for robabilities 2 harmonic n atnHolthaus Martin and , prtr hc sdffrn rmthe from different is which mperature T eulbimsed tt fadriven a of state steady nequilibrium daayial eemn h quasi- the determine analytically nd aaim o h mrigfil of field emerging the for paradigms eim oehls,teresponse the Nonetheless, netism. | proi anr vr solution every manner, -periodic ubr edsoe remarkable a discover We number. ψ 11Odnug Germany Oldenburg, 6111 { gdiig hs eagethat argue we Thus, driving. ng | vrluepce etrs even features, unexpected everal ( c p lfl,Grayand Germany elefeld, t n ) n | i } 2 siltrteeatoshv hw that shown have authors these oscillator = abr¨uck, Germany ept h rsneo time-periodic a of presence the despite , fFoutsaeocpto probabil- occupation Floquet-state of ν X samte fconvention. of matter a is 0 = n c n c n rzto o arbitrary for arization c ontdpn ntm.Hence, time. on depend not do n | u n ol edtrie solely determined be would aig circularly lating, ( t 3 ) i exp( − { p i ε n n } t emphasizing , ) | , ψ ( t ) i othe to (6) et 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 establish a general ⊗ ⊗ formula for the ensuing magnetization by means of an- Stipulating further that the interaction Hamiltonian Hint other systematic use of the lifting technique, and then factorizes according to show that the resulting expression can be interpreted as a derivative of a partition function based on both the Hint = V W, (11) quasitemperature and the system’s quasienergies, in per- ⊗ fect formal analogy to the textbook treatment of para- the golden rule can be applied to joint transitions from magnetism in the absence of time-periodic driving; these Floquet states i to Floquet states f of the system ac- insights are exploited for elucidating the response of an companied by transitions from bath eigenstates n with 3
1 energy En to other bath eigenstates m with energy Em, B. The lift from s = 2 to general s acquiring the form We will make heavy use of a procedure which allows Γmn =2π V (ℓ) 2 W 2 δ(E E + ω(ℓ)) . (12) fi | fi | | mn| m − n fi one to transfer a solution to the Schr¨odinger equation
ℓ∈Z 1 X for a spin with spin quantum number s = 2 in a time- Moreover, following Breuer et al. [11], let us consider a dependent external field to a solution of the correspond- bath consisting of thermally occupied harmonic oscilla- ing Schr¨odinger equation for general s, see also Ref. [30]. tors, and an interaction of the prototypical form This procedure does not appear to be widely known, but has been applied already in 1987 to the coherent evolu- † tion of a laser-driven N-level system possessing an SU(2) W = bωe + bωe , (13) e dynamic symmetry [31], and more recently to the spin-s ω X Landau-Zener problem [32]. Here we briefly sketch this † where bωe (bωe) is the annihilation (creation) operator per- method. taining to a bath oscillator of frequency ω. We now have Let t Ψ(t) SU(2) be a smooth curve such that 7→ ∈ to distinguish two cases: If En Em = ω > 0, so that Ψ(0) = ½, as given by a 2 2-matrix of the form − × the bath is de-excited and transfers energye to the system, the required annihilation-operator matrix element reads z1(t) z2(t) e Ψ(t)= ∗ ∗ (21) z2 (t) z1 (t) Wmn = n(ω) , (14) − 2 p with complex functions z1(t), z2(t) obeying z1(t) + where n(ω) is the occupation number of a bath oscilla- 2 | | tor with frequency ω, and the squaree W 2 entering z2(t) = 1 for all times t. One then has | mn| | | the goldene rule (12) has to be replaced by the thermal d avarage Ψ(t) Ψ(t)−1 iH(t) su(2) , (22) e dt ≡− ∈ 1 N(ω) n(ω) = , (15) ≡ h i exp(βω) 1 where su(2) denotes the Lie algebra of SU(2), i.e., the − space of anti-Hermitean, traceless 2 2-matrices which with β denotinge the inversee bath temperature. Con- is closed under commutation [33]. Hence× the columns versely, if E E = ω < 0 so thate the system loses n − m ψ1(t) , ψ2(t) of Ψ(t) are linearly independent solutions energy to the bath and a bath phonon is created, one of| thei Schr¨odinger| i equation has e d W = n( ω)+1 , (16) i ψj (t) = H(t) ψj (t) , j =1, 2 . (23) mn − dt| i | i giving p e Next we consider the well-known irreducible Lie algebra 1 representation of su(2), N(ω) n( ω) +1= . (17) ≡ h − i 1 exp(βω) − r(s) : su(2) su(2s + 1) , (24) −→ Finally, let Je(ω) be thee spectral density of the bath. Then e the total rate Γfi of bath-induced transitions among the which is parametrized by a spin quantum number s such
Floquet states i and f of the driven system is expressed that 2s Æ, together with the corresponding irreducible e ∈ as a sum of partial rates, group representation (“irrep” for brevity)
Γ = Γ(ℓ) , (18) R(s) : SU(2) SU(2s + 1) . (25) fi fi −→
ℓ∈Z X It follows that where r(s)(is ) = iS , j = x, y, z , (26) Γ(ℓ) =2π V (ℓ) 2 N(ω(ℓ)) J( ω(ℓ) ) . (19) j j fi | fi | fi | fi | where s = 1 σ denote the three s = 1 spin operators These total rates (18) now determine the desired quasi- j 2 j 2 given by the Pauli matrices σ , and the S denote the stationary distribution p as a solution to the equa- j j n corresponding spin operators for general s. Recall the tion [11] { } standard matrices
Γnmpm Γmnpn =0 . (20) − (Sz) = nδmn , (27) m m,n X 1 s(s + 1) n(n 1) : m = n 1 , It deserves to be emphasized again that the very details (S ) = 2 x m,n − ± 0 : else , ± of the system-bath coupling enter here, so that the pre- p cise form of the respective distribution p may depend 1 s(s + 1) n(n 1) : m = n 1 , n (S ) = 2i strongly on such details [10]. { } y m,n ± − ± 0 : else , ± p 4 where m,n = s,s 1,..., s, and The Floquet states (2) brought about by this Hamilto- − − nian (34) are given by [34] S± Sx iSy . (28) ≡ ± e∓iΩt/2 √Ω δ e−iωt/2 It follows from the general theory of representations [33] ψ±(t) = ± ± +iωt/2 , (36) | i √2Ω √Ω δ e that r(s) and R(s) may be applied to Eq. (22) and yield ∓ where d −1 R(s) Ψ(t) R(s) Ψ(t) = r(s) iH(t) . (29) dt − δ = ω0 ω (37) − Since H(t) can always be written in the form of a Zeeman denotes the detuning of the transition frequency ω0 from term with a time-dependent magnetic field b(t), namely, the driving frequency ω, and Ω is the Rabi frequency, 2 2 3 Ω= δ + F . (38) H(t)= b(t) s = b (t) s , (30) · j j The 2 2-matrix Ψ(t) constructedp from these states does j=1 × X not satisfy Ψ(0) = ½. This is of no concern, since Ψ(t) −1 Eq. (26) implies that could be replaced by Ψ(t) Ψ(0) . The distinct advan- tage of these Floquet solutions (36) lies in the fact that 3 they yield a particularly convenient starting point for the r(s) iH(t) = i b(t) S = i b (t) S . (31) · j j lifting procedure outlined in Sec. II B: One has j=1 X 1 √Ω+ δ √Ω δ In (30) we have chosen the + sign convention adapted to Ψ(t) = iωt iωt− − √2Ω e √Ω δ e √Ω+ δ electrons for which the magnetic moment is opposite to − −i(ω+Ω)t/2 their spin s. Hence, the “lifted” matrix e 0 × 0 e−i(ω−Ω)t/2 Ψ(s)(t) R(s) Ψ(t) (32) ≡ −iωt/2 P (t)e exp( iΩtsz) . (39) will be a matrix solution of the lifted Schr¨odinger equa- ≡ − tion This decomposition possesses the general Floquet form d i Ψ(s)(t)= b(t) S Ψ(s)(t) . (33) Ψ(t)= P (t) exp( iGt) , (40) dt · − where the unitary matrix P (t) = P (t + T ) again is Note that the matrix Ψ(s)(t) is unitary, and hence its T -periodic, and the eigenvalues of the “Floquet ma- columns span the general (2s + 1)-dimensional solution trix” G, to be obtained from the matrix logarithm space of the lifted Schr¨odinger equation (33). −1 of Ψ(T ) Ψ(0) = exp( iGT ), provide the system’s quasienergies [35–38]. Since−G already is diagonal in this 1 III. FLOQUET FORMULATION OF THE RABI representation (39), the quasienergies of a spin 2 driven PROBLEM by a circularly polarized field according to the Hamilto- nian (34) can be read off immediately: A. Floquet decomposition for s = 1/2 ω Ω ε± = ± mod ω , (41) 1 2 A spin 2 subjected to both a constant magnetic field applied in the z-direction and an orthogonal, circularly satisfying ε+ + ε− = 0 mod ω. For later application we polarized time-periodic field, as constituting the classic express the periodic part P (t) of the decomposition (39) Rabi problem [24], is described by the Hamiltonian in the following way: −iωt/2 ω F iωt/2 e 0 0 P (t) = e +i 2 H(t)= σz + (σx cos ωt + σy sin ωt) . (34) 0 e ωt/ 2 2 1 √Ω+ δ √Ω δ Here ω0 denotes the transition frequency pertaining to − − ×√2Ω √Ω δ √Ω+ δ the spin states in the static field alone, while F denotes − the frequency associated with the amplitude of the peri- eiωt/2 exp( iωts ) Ξ . (42) odic drive. This is a special form of the Zeeman Hamil- ≡ − z tonian (30) with the particular choices The time-independent matrix Ξ = Ψ(0) introduced here bx(t) = F cos ωt can be written as by(t) = F sin ωt cos(λ/2) sin(λ/2) Ξ = exp( iλsy)= − (43) bz(t) = ω0 . (35) − sin(λ/2) cos(λ/2) 5
(s) 2s+1 C with where ½ denotes the unit matrix in . Therefore, the quasienergies now read δ +Ω λ/2 = arccos . (44) ω 2Ω εm = + m Ω mod ω (52) r ! 2 Hence, one has the identities for half-integer m = s,...,s. − Denoting the eigenstates of Sz by m , in both cases (i) 2 2 | i † δ √Ω δ and (ii) the Floquet states can now be written as Ξ sxΞ = sx + − sz Ω Ω (s) (s) Ψ (t) m = P (t) m exp( iεmt) † | i | i − Ξ syΞ = sy , u (t) exp( iε t) , (53) ≡ | m i − m 2 2 † δ √Ω δ thereby introducing the T -periodic Floquet functions Ξ szΞ = sz − sx , (45) Ω − Ω (s) um(t) = P (t) m . (54) which may be easily verified. | i | i Since Ω ω0 ω when F 0, the quasienergies (50) and (52)→ properly | − connect| to the→ respective eigenvalues of B. Floquet decomposition for general s ω0Sz for vanishing driving amplitude. There is, however, an important further distinction to be observed at this 1 σj point: In either case (i) and (ii) one finds Replacing the spin- 2 operators sj = 2 in the Hamil- tonian (34) by their conterparts Sj for general spin quan- εm mω0 mod ω if ω<ω0 , (55) tum number s, one obtains → whereas H(s)(t) = b(t) S · εm mω0 mod ω if ω>ω0 . (56) = ω0 Sz + F (Sx cos ωt + Sy sin ωt) . (46) →− That is, if the driving frequency ω is detuned to the blue According to Sec. II B the general matrix solution to the side of the transition frequency ω0, the labeling of the corresponding Schr¨odinger equation (33) now is obtained quasienergy representatives (50) and (52) differs from as the lift (32) of the 2 2-matrix (39). Invoking Eqs. (42) that of the eigenvalues of S by a minus sign, effectively × z and (43), and applying the irrep R(s) to this decomposi- reversing their order. This feature needs to be kept in tion yields mind for correctly assessing the following results. We also note that in the adiabatic limit, when the spin Ψ(s)(t) = R(s) exp( iωts )exp( iλs ) exp( iΩts ) − z − y − z is exposed to an arbitrarily slowly varying magnetic field enabling adiabatic following to the instantaneous energy = exp( iωtSz)exp( iλSy) exp( iΩtSz) . (47) eigenstates, the quasienergies should be given by the one- − − − cyle averages of the instantaneous energy eigenvalues. In- In order to bring this factorization into the standard Flo- deed, in this limit the Rabi frequency (38) reduces to 2 2 quet form analogous to Eq. (40), Ω = ω0 + F , while the time-independent instanta- neous energy levels are Em = m Ω, yielding the expected (s) (s) (s) Ψ (t)= P (t)exp iG t (48) identity.p − with a T -periodic matrix P (s)(t)= P (s)(t + T ), we have to distinguish two cases: IV. THE QUASISTATIONARY DISTRIBUTION (i) For integer s we may set
(s) Now we stipulate that the periodically driven spin be P (t) = exp( iωtSz) exp( iλSy) , − − coupled to a thermal bath of harmonic oscillators, as (s) G = Ω Sz , (49) sketched in Sec. II A, taking the coupling operator to be obtaining the quasienergies of the natural form [39] V = γSx . (57) εm = m Ω mod ω (50) In order to calculate the Fourier decompositions (7) of with integer m = s,...,s. − (s) the Floquet matrix elements of V , and referring to the (ii) For half-integer s the requirement that P (t) above representation (54) of the Floquet functions, we be T -periodic demands insertion of additional factors thus need to consider the operator e±iωt/2, in analogy to the representation (42) for s = 1 . 2 (s)† (s) This gives P (t) Sx P (t)
(s) iωt/2 = exp (iλSy) exp (iωtSz) Sx exp( iωtSz)exp( iλSy) P (t) = e exp( iωtSz) exp( iλSy) , − − − − (ℓ) (s) ω (s) V exp(iℓωt) ; (58) G = ½ +Ω Sz , (51) ≡
∈Z 2 Xℓ 6 note that the additional phase factor eiωt/2 contained in for all n = s + 1, ... , s 1, since the term with m = n the expression (51) for P (s)(t) with half-integer s cancels in Eq. (20)− cancels. In the− border cases n = s or n = s here. Using the su(2) commutation relations and their this identity still holds, but only one bracket− survives. counterparts for general s, we deduce Upon setting the first bracket in this Eq. (64) to zero, one obtains exp (iωtSz) Sx exp( iωtSz) − pn Γn,n−1 1 iωt −iωt = (65) = e S +e S− . (59) 2 + pn−1 Γn−1,n Hence, as in the case s = 1 studied in Ref. [29], the only for n = s+1, ... , s 1. Together with the normalization 2 − − non-vanishing Fourier components V (ℓ) occur for ℓ = 1: requirement, this relation already determines the entire ± distribution pm . In particular, it entails (±1) γ { } V = exp (iλSy) S± exp( iλSy) . (60) 2 − pn+1 Γn+1,n = , (66) Applying R(s) to Eqs. (45), this yields pn Γn,n+1
2 2 thus ensuring that also the second bracket in Eq. (64) (±1) γ δ √Ω δ V = Sx + − Sz i Sy . (61) vanishes, confirming detailed balance. 2 Ω Ω ± ! A further factor of substantial importance is the spec-
(±1) tral density J(ω), which may allow one to manipulate the Thus, V is a tridiagonal matrix. For computing the quasistationary distribution to a considerable extent. For partial transition rates (19) we therefore have to con- the sake of simplicity and transparent discussion, here we (±1) e sider only frequencies ωmn of pseudotransitions [29], for assume that J(ω) J0 is constant. The distinction be- which m = n, and of transitions between neighboring tween the physically≡ different positive and negative tran- Floquet states, m = n 1. In view of the definition (9), sition frequencies, which lead to the two different expres- ± e and taking the quasienergies given by Eqs. (50) and (52) sions (15) and (17) entering the transition rates (19), now without “mod ω” as the canonical representatives of their prompts us to treat the cases 0 <ω< Ω and 0 < Ω <ω respective quasienergy classes (5), one has separately, while the resonant case ω = Ω can be dealt with by means of limit procedures. ω : m = n , (±1) ± ωmn = Ω ω : m = n +1 , (62) Ω ± ω : m = n 1 . − ± − A. Low-frequency case 0 <ω< Ω According to the Pauli master equation (20), the quasi- stationary distribution pm m=s,...,−s which establishes In order to utilize the above Eq. (66) for determining itself under the combined{ influence} of time-periodic driv- the distribution pm recursively we only need to eval- { } (±1) (±1) ing and the thermal oscillator bath is the eigenvector of uate the partial rates Γm,m+1 and Γm+1,m according to a tridiagonal matrix Γ corresponding to the eigenvalue 0, the general prescription (19). In the low-frequency case where Γ is obtained from Γ Γ(1) +Γ(−1) by subtracting 0 <ω< Ω this leads to the expressions ≡ from the diagonal elementse the respective column sums, 2 ± s(s + 1) m(m + 1) Ω δ i.e., e Γ( 1) = − ∓ m,m+1 16 1 e−β(Ω∓ω) Ω s − 2 Γmn =Γmn δmn Γkn . (63) s(s + 1) m(m + 1) Ω δ − (±1) − Γm+1,m = − ± (67) kX= s 16 eβ(Ω±ω) 1 Ω e − Since Γ is tridiagonal with non-vanishing secondary di- which have been scaled by Γ = 2πγ 2J , and have thus agonal elements, this eigenvector is unique up to nor- 0 0 been made dimensionless. Evidently, these representa- malization.e Moreover, it is evident that we only need tions imply that the desired ratio the matrix elements of Γ in the secondary diagonals for calculating the quasistationary distribution, whereas the Γm+1,m qL (68) diagonal elements will be required for computing the dis- Γm,m+1 ≡ sipation rate [29]. The very fact that V (±1), and hence Γ, merely is a is independent of both s and m; a tedious but straight- tridiagonal matrix has a conceptually important conse- forward calculation readily yields quence: It enforces detailed balance, meaning that each 2δΩ sinh(βω)+ δ2 +Ω2 e−βΩ cosh(βω) term of the sum (20) vanishes individually. With Γ being − qL = 2 2 βΩ . tridiagonal, this sum reduces to 2δΩ sinh(βω) (δ +Ω) (e cosh(βω)) − − (69) (Γ − p − Γ − p ) n,n 1 n 1 − n 1,n n Therefore, not only does one find detailed balance here, + (Γ p Γ p )=0 (64) but the occupation probabilities even generate a finite n,n+1 n+1 − n+1,n n 7 geometric sequence: By construction, the geometric se- Boltzmann distribution with the inverse bath temper- quence ature β, thereby indicating thermal equilibrium of the spin system; the unfamiliar “plus”-sign appearing in this 2s 2s−1 qL (q , q ,...,qL, 1) (70) limit (76) in the case of blue detuning merely reflects the ≡ L L reversed labeling (56). is an eigenvector of the matrix Γ introduced in Eq. (63) More importantly, in both cases 0 < ω < Ω and with eigenvalue 0, and hence yields the desired quasista- 0 < Ω <ω the occupation probabilities p of the Flo- { m} tionary Floquet-state occupatione probabilites pm after quet states constitute a geometric sequence even under normalization. { } arbitrarily strong driving, when the system is far remote from usual thermal equilibrium. Since the quasiener- gies (50) and (52) are equidistant, the quasistationary B. High-frequency case 0 < Ω <ω distribution can therefore still be regarded as a Boltz- mann distribution, but now being parametrized by an Analogously, in the high-frequency case 0 < Ω <ω we effective quasitemperature: require the dimensionless partial rates 1 1 1 p = qs+m = e−ϑεm = e−ϑΩm (77) 2 m ± s(s + 1) m(m + 1) Ω δ Z1 Z2 Zq Γ( 1) = − ∓ m,m+1 ± 16 eβ(±ω−Ω) 1 Ω − with m = s, . . . , s for both integer and half-integer s, 2 − s(s + 1) m(m + 1) Ω δ where the factors Z1, Z2, and Zq are adjusted to ensure (±1) s Γm+1,m = − ± (71) the normalization p = 1, and ± 16 eβ(Ω±ω) 1 Ω m=−s m − P ln q for constructing the matrix Γ = Γ(1) +Γ (−1). Once more, ϑ (78) the ratio ≡− Ω is the inverse quasitemperature of the periodically driven Γm+1,m qH (72) spin system coupled to an oscillator bath with inverse Γm,m+1 ≡ regular temperature β. When defining the quasitemper- does depend neither on s nor on m, so that the occupa- ature in this analytically convenient manner (78) we do tion probabilities again form a geometric sequence; after not distinguish between the cases (76) of red and blue some juggling, one finds detuning, so that our quasitemperature will be negative under weak driving with blue detuning. In accordance δ2 +Ω2 sinh(βω)+2δΩ e−βΩ cosh(βω) with the distinction (75) we will also designate the inverse qH = − . quasitemperature as ϑ and ϑ , respectively. Note that (δ2 +Ω2) sinh(βω) 2δΩ (eβΩ cosh(βω)) L H the parametrization of the geometric distribution (77) − − (73) in terms of the system’s quasienergies and a quasitem- This quantity determines the solution vector perature, in formal analogy to the canonical distribu- 2s 2s−1 tion of equilibrium thermodynamics [28], requires that qH (q , q ,...,qH , 1) (74) ≡ H H the “modulo ω-indeterminacy” of the quasienergies has been resolved in some way, so that the quasienergies en- to the equation Γ q = 0 in the high-frequency regime, H tering this distribution refer to particular, well defined providing the corresponding quasistationary Floquet dis- representatives of their classes (5); here we have selected tribution upon normalization. e the representatives given by Eqs. (50) and (52) without “mod ω”. This implies that the definition of the quasi- C. The quasitemperature temperature still contains a certain degree of arbitrari- ness. Notwithstanding this remark, the very distribution pm itself, governing the observable physics, is defined For ease of notation, let us define uniquely.{ } The dimensionless inverse quasitemperature ω0ϑ ulti- qL : 0 <ω< Ω q (75) mately depends on the scaled driving amplitude F/ω0, ≡ qH : 0 < Ω < ω . the scaled driving frequency ω/ω0, and the dimension- less inverse actual temperature ω β of the heat bath, but Recalling Ω ω ω in the limit F 0 of vanishing 0 0 not on the spin quantum number s. In contrast, the par- driving amplitude,→ | − one| then finds → tition function Zq depends on s: −βω0 e : ω<ω0 q +βω0 (76) 2s +1 → e : ω>ω0 . s sinh ϑΩ 2 Zq = exp( ϑΩm)= . (79) Hence, in the absence of the time-periodic driving field − ϑΩ m=−s sinh the above solutions (70) and (74) correctly lead to a X 2 8
where the latter equation defines the boundary ω = Ωbe- tween the low- and the high-frequency regime, see Fig. 1. At this boundary there is a continuous change from ϑL to ϑH , since
lim ϑL = lim ϑH =0 . (81) ω↑ωc ω↓ωc
However, the two functions ϑL and ϑH do not join smoothly at ω = ωc, since their derivatives with respect to ω adopt limits with opposite signs:
3 4 4 ∂ϑL 4βω0 F + ω0 lim = 2 , FIG. 1: Dimensionless inverse quasitemperature ω0ϑ of the ω↑ωc ∂ω − 4 2 2 F (F + ω0) spin system under the influence of a circularly polarized 3 4 4 monochromatic driving force with amplitude F and fre- ∂ϑH 4βω0 F + ω0 lim = 2 . (82) quency ω, being coupled to a harmonic-oscillator bath of in- ω↓ωc ∂ω 4 2 2 F (F + ω0) verse actual temperature ω0β = 1. The blue part of the graph corresponds to the high-frequency regime 0 < Ω <ω, the red part to the low-frequency regime 0 <ω< Ω. Along the line We will now investigate the behavior of the function segment ω = ω0 with F <ω0 and along the parabola ω = ωc given by Eq. (80), both marked by blue dashes, the inverse ω0ϑ(ω0β,ω/ω0,F/ω0) in the limits corresponding to the quasitemperature vanishes. A few functions ϑ(ω) for con- four sideways faces of the box bounding the plot dis- stant F are highlighted, showing a kink at ω = ωc according played in Fig. 1: to Eqs. (82). The limit ω0ϑ = ω0β for ω/ω0 → 0 is indicated (i) As already noted at the end of Sec. IIIB, in the low- by the green line. frequency limit ω/ω0 0 the canonical representatives of the quasienergies (50)→ and (52) approach the actual 2 2 energies Em = m ω0 + F (with m = s, . . . , s) of a The inverse quasitemperature ϑ vanishes — meaning that spin exposed to a slowly varying drive. Hence,− in this the periodically driven system effectively becomes in- limit the “periodicp thermodynamics” investigated here finitely hot, so that all its Floquet states are populated must reduce to the usual thermodynamics described by equally — regardless of the bath temperature, if either a canonical ensemble; in particular, the inverse quasi- ω = ω0 while 0 2ω0β(ω/ω0) ω0ϑL = ω0β + 2 (F/ω0) +2 4 ω0β 2 2 (F/ω0) ω0β coth (F/ω0) +1 ω0β(ω/ω0) 2 2 3 + 8 4(F/ω0) + O(ω/ω0) . (83) 2 2 2 2 (F/ω0) +2 − − (F/ω0) p+1 p (ii) Next we consider the ultrahigh-frequency limit quencies all Floquet states are populated almost equally, ω/ω , keeping both ω β and F/ω fixed. Inspecting independent of both the driving amplitude and the bath 0 → ∞ 0 0 qH as defined by Eq. (73), and observing that asymptot- temperature. ically Ω = (ω ω)2 + F 2 ω for ω/ω , one (iii) The static limit of vanishing driving amplitude, 0 − ∼ 0 → ∞ finds q 1, and hence F/ω0 0, yields H → p → ω0β lim ω0ϑH =0; (84) lim ω0ϑL = lim ω0ϑH = . (85) ω/ω0→∞ F/ω0→0 F/ω0→0 1 ω/ω − 0 as shown by Fig. 1, this limit is approached with negative Once again, this apparently strange expression, exhibit- quasitemperatures. This means that for high driving fre- ing a pole at ω = ω0 which is prominently visible in Fig. 1, 9 ω0 ϑ ω0 ϑ β 3 ω0 β 2 F2 ω0 β 0 ω/ω F2 F2 F2 0 2 5 10 2 2 2 ω0 ω0 ω0 3 - ω0 β 2 F2 F F ω ω0 2 FIG. 3: Inverse dimensionless quasitemperature ω0ϑ for ω0 ω0 ω0β = 1 and F/ω0 = 100 as a function of ω/ω0 in the 2 regime ω/ω0 > (F/ω0) /2. We show the exact values of ω0ϑL FIG. 2: Inverse dimensionless quasitemperature ω0ϑ for (red line) and ω0ϑH (blue line), together with the asymptotic ω0β = 1 and F/ω0 = 100 as a function of ω/ω0 in the regime form (87) (green dashes). 0 < ω/ω0 < 2F/ω0. We show the exact values of ω0ϑL (red line), as well as the asymptotic form (86) (green dashes). V. APPLICATION: PERIODICALLY DRIVEN PARAMAGNETS is fully in agreement with the expectation that the “peri- odic thermodynamics” should reduce to ordinary thermo- A. Calculation of quasithermal expectation values dynamics when the time-periodic driving force vanishes. Namely, for F/ω0 0 the system possesses the energy Starting from the proposition that the Floquet states of → eigenvalues Em = mω0, whereas the quasienergies (50) a periodically driven spin system be populated according or (52) reduce to εm = m ω0 ω or εm = ω/2+m ω0 ω to the distribution (77) we introduce the quasithermal | − | | − | with m = s, . . . , s, while the Floquet states approach average of the spin component in the direction of the − the energy eigenstates. Since the parametrization of their static field, occupation probabilities in terms of either the distribu- s 1 tion (77) or the standard canonical distribution then S u (t) S u (t) exp( ϑε ) . (88) must lead to identical values, one obtains the require- h ziq ≡ Z h m | z| m i − m 2 m=−s ment βω0 = ϑ(ω0 ω) for 0 <ω<ω0, which immediatly X furnishes the above− limit. In the case of blue detuning, In order to evaluate this expression we utilize the repre- for 0 < ω0 < ω, the formal inversion of the quasienergy sentation (54) of the Floquet functions, giving levels expressed by Eq. (56) yields an an additional “mi- s nus” sign, again leading to the limit (85). Z S = m P (s)† S P (s) m exp( ϑε ) 2h ziq h | z | i − m (iv) In the converse strong-driving limit F/ω we m=−s 0 → ∞ X first focus on the regime ω F where ϑ = ϑL > 0. After (s)† (s) (s) = Tr P Sz P exp ϑG . (89) some transformations we obtain∼ the asymptotic form − Next, we resort once more to the lifting technique: For 1 ωβ s = 2 , the decomposition (42) readily yields ω0ϑL ω0β (86) ∼ − 2 2 † † (F/ω0) + (ω/ω0) P szP = Ξ szΞ p (45) δ √Ω2 δ2 = s − s . (90) in this regime, so that the inverse quasitemperature de- Ω z − Ω x creases monotonically with increasing driving amplitude, (s) as exemplified by Fig. 2. Applying the irrep r , we deduce (s)† (s) (s) † In contrast, for ω>ωc as given by Eq. (80) we have P Sz P = r P szP ϑ = ϑH < 0. Asymptotically, here we find δ √Ω2 δ2 = S − S . (91) Ω z − Ω x ω β ω β 0 0 2 Inserting this into the above identity (89), and calcu- ω0ϑH + 2 (F/ω0) . (87) ∼−ω/ω0 2(ω/ω0) lating the trace in the eigenbasis of Sz, we obtain the important result This asymptotic function exhibits a pronounced mini- m 2 3 2 1 ω0 ω −ϑΩ m mum at ω/ω0 = (F/ω0) of depth ω0ϑH ω0β/(2F ), S = − m e , (92) ∼− h ziq Z Ω depicted in Fig. 3. q m=−s X 10 valid for both integer and half-integer s. Although logarithm, namely, the unusual-looking prefactor (ω ω)/Ω indeed implies 0 − N N that the z-component of the magnetization vanishes for M = µ = g µ m h i V h i − V J Bh i ω = ω0, it does not imply that the magnetization re- verses its direction when ω is varied across ω , since the N ∂ 0 = kBT ln Z0 . (96) reversal of the prefector’s sign can be compensated by V ∂B0 a simultaneous change of the sign of the quasitempera- Here N denotes the density of contributing atoms. ture, as it happens for low driving amplitudes according V Working out this prescription, one finds the magnetiza- to Eq. (85). tion [27, 28] M = M0 BJ (y0) , (97) B. Response of paramagnetic materials to h i circularly polarized driving fields where N M = g µ J (98) As an experimentally accessible example of the above 0 V J B considerations, and thus as a possible laboratory appli- denotes the saturation magnetization, and cation of periodic thermodynamics, we consider the mag- netization of an ideal paramagnetic substance under the 2J +1 2J +1 1 y B (y) coth y coth (99) influence of both a static magnetic field applied in the z- J ≡ 2J 2J − 2J 2J direction, and a circularly polarized oscillating magnetic field applied in the x-y-plane. In order to facilitate com- is the so-called Brillouin function of order J [25]; this parison with the literature, here we re-install the Planck theoretical prediction (97) has been beautifully confirmed constant ~ and the Boltzmann constant kB. in low-temperature experiments with paramagnetic ions We assume that the magnetic atoms of the substance by Henry [26] already in 1952. In the weak-field limit µ B k T one may use to approximation have an electron shell with total angular momentum J, B 0 ≪ B resulting from the coupling of orbital angular momentum J +1 y and spin, giving the magnetic moment µ = g µ J. BJ (y) for 0 〈M〉/〈M〉 where τ =1/(kBϑ) is the quasitemperature, and sat 1.0 J ~Ω Z = exp m (105) 0.8 q − k τ − B mX= J 0.6 is the corresponding partition function (79). Quite re- markably, this expression (104) is a perfect formal analog 0.4 of the previous Eq. (96), since we have 0.2 N ∂ M q = kBτ ln Zq , (106) h i V ∂B0 ω /ω 0.5 1.0 1.5 2.0 2.5 3.0 0 taking into account the nonlinear dependence of the Rabi frequency (102) on the static field strength B0. FIG. 4: Magnetization hMi divided by the saturation mag- Hence, the resulting quasithermal magnetization can be netization (98) as a function of ω0/ω. The blue curve rep- expressed in a manner analogous to Eq. (97), namely, resents the ordinary thermal magnetization (97), the red one the quasithermal magnetization (107) for weak driving, M q = M1 BJ (y1) , (107) gJ µBB1/(~ω) = 0.01. Here we have set kBT = ~ω and h i J = 7/2. with modified saturation magnetization ω ω M = 0 − M , (108) 1 Ω 0 and the argument of the Brillouin function now depend- ing on the quasitemperature, ~Ω y1 = J. (109) kBτ For consistency, this prediction (107) must reduce to the usual weak-field magnetiziation (101) when both B1 0 and µ B k T . This is ensured by the limit (85):→ For B 0 ≪ B sufficiently small B1, the quasitemperature τ is related to the actual bath temperature T through 1 ω 1 0 . (110) k τ ≈ ω ω k T B 0 − B FIG. 5: Ratio hMiq /hMi of the quasithermal magnetiza- Inserting this into Eq. (107), one can employ the approx- tion (107) and the customary magnetization (97) as func- imation (100) for frequencies ω not too close to ω0. In tion of ω/ω0 and F/ω0, with F = gJ µBB1/~. Parameters this way one recovers the expected expression (101) un- chosen here are kBT = ~ω0 and J = 1. Along the green less ω ω0, in which case one has M q 0. line ω = ω0 and along the green parabola ω = ωc given by As is≈ evident from the above discussion,h i ≈ under typical Eq. (80) the quasithermal magnetization vanishes, so that ESR conditions with weak driving amplitudes, such that hMiq/hMi becomes negative for strong driving with frequen- −2 B1/B0 is on the order of 10 or less [40], the difference cies ω0 <ω<ωc. between the quasithermal magnetization M q and the customary thermal magnetization M ish morei or less negligible, except for driving frequenciesh i close to reso- for frequencies ω0 <ω<ωc, implying that the param- agnetic material effectively becomes a diamagnetic one. nance. This is illustrated in Fig. 4 for gJ µBB1/(~ω) = 0.01, where we have chosen the bath temperature accord- The possibility of turning a paramagnet into a diamagnet through the application of strong time-periodic forcing is ing to the fixed driving frequency, kBT = ~ω. The van- a “hard” prediction of periodic thermodynamics which ishing of the magnetization at ω0 = ω is due to the pref- now awaits its experimental verification. actor ω0 ω in (108). The second sharp dip at ω0 2ω can be explained− by the vanishing of the inverse≈ qua- sitemperature ϑ, see (80). Other measurable effects have to be expected in the strong-driving regime. VI. DISSIPATION A particularly striking observation can be made in Fig. 5, where kBT = ~ω0 and J = 1: Under strong driv- Since a bath-induced transition from a Floquet state n ing, there the ratio M / M actually becomes negative to a Floquet state m is accompanied by all frequencies h iq h i 12 (ℓ) ωmn as introduced in Eq. (9), the rate of energy dissi- pated in the quasistationary state is given by [29] R = ~ω(ℓ) Γ(ℓ) p . (111) − mn mn n mnℓX This expression can now be evaluated for all spin quan- tum numbers s. In addition to the partial transition rates (19) for neigboring Floquet states m = n 1 listed in Secs. IVA and IVB, Eq. (111) also requires± the rates for pseudotransitions with m = n. Again dividing by 2 Γ0 =2πγ J0, we obtain the dimensionless diagonal tran- sition rates 2 2 (±1) (2m) F FIG. 6: The dimensionless dissipation rate R for s = 1 and Γmm = (112) ±16(e±βω 1)Ω2 ω0β = 1 as a function of ω/ω0 and F/ω0. The blue part of the − graph corresponds to the high-frequency regime 0 < Ω < ω, for m = s, . . . , s, valid for both cases 0 <ω< Ω and the red one to the low-frequency regime 0 <ω< Ω. Along − 0 < Ω < ω. In order to represent R in a condensed the line ω = ω0 with 0 eβ(±ω+Ω)q 1 (δ Ω)2(ω Ω) A±(q) − ± ± . (114) ≡ eβ(±ω+Ω) 1 − Dividing by ω0Γ0, one obtains a dimensionless dissipation rate which can now be written in the form 2 + − Ps(q) ω F + Qs(q) A (q) A (q) R = 2 ∓ , (115) 8 zs(q)Ω FIG. 7: As Fig. 6, but for s = 10. Along the line ω = where one has to insert either qL or qH for q, in accor- ω0 with 0 R adopting the value s (s + 1) 1 2 rm = 1+ (F/ω0) +1 , (119) rm 8 p as depicted in Figs. 8 and 9. 1 6 VII. DISCUSSION A simple harmonic oscillator which is permanently driven by an external time-periodic force while kept in contact with a thermal oscillator bath represents a 0 ω/ω nonequilibrium system, but nonetheless adopts a steady ωc ωm 0 0 1 2 ω0 ω0 state and develops a quasistationary distribution of Floquet-state occupation probabilities which equals the 2 1 FIG. 8: The scaled dissipation rate r = R/(s + s) for s = 2 , Boltzmann distribution of the equilibrium model ob- 5, 10, 20, 50, 200, with bath temperature ω0β = 1 and driving tained in the absence of the driving force, being char- amplitude F/ω0 = 1/4, as a function of ω/ω0. The six curves acterized by precisely the same temperature as that of increase with s for ω>ωc. They all meet at the two points the bath it is coupled to [11, 29]. with coordinates (ωc/ω0, 1/6) and (1, 1/6). The asymptotic The system considered in the present work, a spin with envelope (117) is given by the red curve, with its maximum arbitrary spin quantum number s exposed to a circu- (ωm/ω0,rm) being determined by Eqs. (118) and (119). larly polarized driving field while interacting with a bath of thermally occupied harmonic oscillators, may be re- garded as the next basic model in a hierarchy of analyti- cally solvable models on Periodic Thermodynamics. Ex- actly as in the case of the linearly forced harmonic oscil- lator, the system-bath interaction here induces nearest- neighbor coupling among the Floquet states of the time- periodically driven system, so that the model’s transi- tion matrix (18) is tridiagonal, thus enforcing detailed balance. Again, the resulting quasistationary Floquet distribution turns out to be Boltzmannian, but now with a quasitemperature which differs from the physical tem- perature of the bath. Already the mere fact that a time- periodically driven quantum system in its steady state may exhibit a quasitemperature which is different from the actual temperature of its environment, and which can be actively controlled by adjusting, e.g., the ampli- tude or frequency of the driving force, in itself constitutes a noteworthy observation, suggesting that periodic ther- modynamics generally may be far more subtle than usual equilibium thermodynamics based on some effective Flo- FIG. 9: The asymptotic limit r∞(ω/ω0,F/ω0) of the scaled 2 quet Hamiltonian. dissipation rate R/(s + s) for s →∞ according to Eq. (117). Importantly, our model system not only is of basic the- The particular curve with F/ω0 = 1/4 considered in Fig. 8 is shown in red color. The maximum values of the func- oretical interest, but also leads to novel predictions con- tions r∞(ω/ω0,F/ω0) with constant F according to Eqs. (118) cerning future experiments with paramagnetic materials and (119) are indicated by a black curve. in strong circularly polarized fields. The very existence of a quasistationary Floquet distribution which is different from the distribution characterizing thermal equilibrium implies that the magnetic response of such a periodically which is independent of the heat-bath temperature. As driven material can be quite different from that of the un- illustrated by Fig. 8 the convergence proceeds pointwise driven one; as we have demonstrated in Sec. V B, a strong except for ω = ωc and ω = ω0 when F <ω0, where circularly polarized driving field effectively may turn a r = 1/6 for all s. For fixed F the asymptotic function paramagnetic material into a diamagnetic one. While we r∞(ω/ω0,F/ω0) has a global maximum at are not in a position to ascertain whether the correspond- ing parameter regime can be reached with already exist- ing experimental set-ups [41], it might be worthwhile to 2 ωm/ω0 = (F/ω0) +1 , (118) design specifically targeted measurements for confirming p 14 this particularly striking prediction of periodic thermo- ation and annihilation operators (13) on the side of the dynamics. bath, combined with the assumption of a constant spec- Yet, there is still more at stake here. When Bril- tral density of the bath, but there are other possibilities. louin published his now-famous treatise [25] on thermal Measurements of magnetism under strong driving will paramagnetism in 1927, this was essentially a blueprint be sensitive to such issues; two materials which exhibit for an experimental demonstration of the quantization precisely the same paramagnetic response in the absence of angular momentum, whereas the further thermody- of time-periodic forcing may react differently to a static namical input into the theory was not to be questioned, magnetic field once an additional time-periodic field has being backed by the overwhelming generality of equilib- been added. Thus, despite the formal similarity of our rium thermodynamics [27, 28]. At the advent of peri- key results (106) and (107) to their historical anteces- odic thermodynamics more than 90 years later, one faces sors (96) and (97), these former equations may have the an inverted situation: With the quantization of angular potential to open up an altogether new line of research. momentum being firmly established, it is nonequilibrium physics in the guise of periodic thermodynamics which is to be examined in measurements of paramagnetism under time-periodic driving. As has been stressed al- Acknowledgments ready by Kohn [10] and clarified by Breuer et al. [11], quasistationary Floquet distributions are not universal, This work has been supported by the Deutsche For- depending on the very form of the system-bath interac- schungsgemeinschaft (DFG, German Research Foun- tion. Here we have assumed an interaction of the natural- dation) through Projects 355031190, 397122187 and appearing type (11) with coupling (57) proportional to 397300368. 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