The Floquet Theory of the Two-Level System Revisited

Z. Naturforsch. 2018; 73(8)a: 705–731

Heinz-Jürgen Schmidt* The Floquet Theory of the Two-Level System Revisited https://doi.org/10.1515/zna-2018-0211 In the following decades it was realised [3, 4] that the Received April 24, 2018; accepted May 8, 2018; previously underlying mathematical problem is an instance of the published online June 14, 2018 Floquet theory that deals with linear differential matrix equations with periodic coefficients. Accordingly, analyti- Abstract: In this article, we reconsider the periodically cal approximations for its solutions were devised that were driven two-level system especially the Rabi problem with the basis for subsequent research. Especially the seminal linear polarisation. The Floquet theory of this problem can paper of Shirley [4] has been until now cited more than be reduced to its classical limit, i.e. to the investigation 11,000 times. Among the numerous applications of the the- of periodic solutions of the classical Hamiltonian equa- ory of periodically driven TLSs are nuclear magnetic res- tions of motion in the Bloch sphere. The quasienergy is onance [5], ac-driven quantum dots [6], Josephson qubit essentially the action integral over one period and the res- circuits [7] and coherent destruction of tunnelling [8]. On onance condition due to Shirley is shown to be equivalent the theoretical level the methods of solving the RPL and to the vanishing of the time average of a certain compo- related problems were gradually refined to include power nent of the classical solution. This geometrical approach series approximations for Bloch-Siegert shifts [9, 10], per- is applied to obtain analytical approximations to physical turbation theory and/or various limit cases [11–16] and the quantities of the Rabi problem with linear polarisation as hybridised rotating wave approximation [17]. Special ana- well as asymptotic formulas for various limit cases. lytical solutions of the general Floquet problem of TLS can be generated by the inverse method [18–20]. Keywords: Bloch-Siegert Shift; Floquet Theory; Rabi There exists even an analytical solution [21] of the Problem. RPL and its generalisation to the Rabi problem where the angle between the constant field and the periodic field is 1 Introduction arbitrary. This solution bears on a transformation of the Schrödinger equation into a confluent Heun differential Many physical experiments can be described as the inter- equation. A similar approach has been previously applied action of a small quantum system with electromagnetic to the TLS subject to a magnetic pulse [22]. However, radiation. If one tries to theoretically simplify this situa- the analytical solution is achieved by combining together tion as far as possible, one arrives at a two-level system three different solutions and does not yield explicit solu- (TLS) interacting with a classical periodic radiation field. tions for the quasienergy or for the resonance curves. The The special case of a constant magnetic field into, say, z- ongoing research on Heun functions, (see e.g. [23, 24]), direction plus a circularly polarised field in the x-y-plane might facilitate the physical interpretation of these analyt- was solved eight decades ago by Rabi [1] and has found its ical solutions in the future. To summarise, the problem is way into many text books. This case will be referred to as far from being completely solved and it appears still worth- the Rabi problem with circular polarisation (RPC). Shortly while to further investigate the general Floquet problem of after, Bloch and Siegert [2] considered the analogous prob- the TLS and to look for more analytical approximations of lem of a linearly polarised magnetic field, orthogonal to the RPL. the direction of the constant field (hence forward called In this paper we will suggest an approach to the Flo- Rabi problem with linear polarisation, RPL) and suggested quet problem of the TLS via its well-known classical limit, the so-called rotating wave approximation. Moreover, they see e.g. [25]. It turns out that, surely not in general, but investigated the shift of the resonance frequencies due to for this particular problem, the classical limit is already the approximation error of the rotating wave approxima- equivalent to the quantum problem. More precisely, we tion. Since then it was called the “Bloch-Siegert shift”. will show that to each periodic solution of the classical equation of motion there exists a Floquet solution of the original Schrödinger equation that can be explicitly calcu- Universität *Corresponding author: Heinz-Jürgen Schmidt, lated via integrations. Especially, the quasienergy is essen- Osnabrück, Fachbereich Physik, Barbarastr. 7, D – 49069 Osnabrück, Germany, E-mail: [email protected] tially given by the action integral over one period of the 706 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited classical solution. This is reminiscent of the semi-classical function, we show in subsection 4.2 that the resonance Floquet theory developed in [26]. In the special case of condition is equivalent to the vanishing of the time aver- the Rabi problem with elliptical polarisation (RPE), our age of the third component of the classical periodic solu- approach yields the result that Shirley’s resonance condition X(t) (Assertion 2) and that the slope of the function tion is equivalent to the vanishing of the time average of ϵ(ω), where ω denotes the frequency of the periodic mag- the component of the classical periodic solution into the netic field, is equal to the geometric part of the quasienergy direction of the constant magnetic field. When applied to divided by ω (Assertion 3). the RPL, our approach suggests to calculate the truncated The third main part deals with the analytical approx- Fourier series for the classical solution and to obtain from imations to the RPL. If the classical periodic solution X(t) this the quasienergy by the recipe sketched above. For var- is expanded into a Fourier series, the equation of motion ious limit cases of the RPL there also exists a classical ver- can be rewritten as an infinite-dimensional matrix prob- sion that will be analysed and evaluated in order to obtain lem. This is similar to the approach in [3, 14] to the TLS asymptotic formulas for the quasienergy. Schrödinger equation. As the involved matrix A and any The structure of the paper is as follows. We have truncation A(N) of it are tri-diagonal, the determinant of three main parts, Generalities, Resonances and Analytical A(N) and all relevant minors can be determined by recur- Approximations that refer to problems of decreasing gen- rence relations. Thus we obtain, in subsection 5.1, analyt- erality: the general TLS with a periodic Hamiltonian, the ical results for the truncated Fourier series of X(t) that are RPE and the RPL. Moreover, we have four subsections 2.4, arbitrarily close to the exact solution. By means of Asser- 3.2, 4.3 and 3.3, where the explicitly solvable case of the tion 1 these analytical approximations can also be used to RPC and another solvable toy example is used to illustrate calculate the quasienergy in subsection 5.3. As expected, certain results of the first two main parts. we observe different branches and avoid level crossing In subsection 2.1 we start with a short account of at the resonance frequencies. The latter can be approxi- the well-known Floquet theory of TLS that emphasises mately determined, using Assertion 2, via det A(N) = 0, see the group theoretical aspect of the theory. This aspect is subsection 5.2. crucial for the following subsection 2.2 where we show The remainder of the paper, Section 6, is devoted to how to lift Floquet solutions of the TLS to higher spins the investigation of various limit cases that often require s > 1/2. Also this lift procedure has been used before but additional ideas for asymptotic solutions and not simply we present a résumé for the convenience of the readers. the evaluation of the truncated Fourier series. The RPL The next subsection 2.3 is vital for the remainder of the has three parameters, namely the Larmor frequency ω0 of paper in so far as it reduces the Floquet problem for the the constant magnetic field into z-direction, the amplitude TLS to its classical limit. More precisely, the classical equa- F of the periodic field into x-direction and the frequency tion of motion for a spin X in a periodic magnetic field ω of the periodic field. Accordingly, there are the three has, in the generic case, exactly two periodic solutions limit cases where F → 0, see subsection 6.1, ω0 → 0, see ±X(t) and the Floquet solutions u±(t) together with the subsection 6.2, and ω → 0, see subsection 6.3. Moreover, quasienergies ϵ± can be derived from ±X(t). This is the there are also complementary limit cases where ω → ∞, content of Assertion 1. The next Section 3 closely investi- see subsection 6.4 and ω0 → ∞, see subsection 6.5. The gates some geometrical aspects of the problem. The Bloch case F → ∞ is somewhat intricate and will be treated in sphere can either be viewed as the set of one-dimensional Section 6.3 and not in its own subsection. We want to high- projections of the TLS or as the phase space of its classi- light three features among the various limit cases. First, cal limit. The first view leads to a scenario that was anal- by using the resonance condition in the form det A(N) = 0 ysed in [27, 28] in the context of the generalised Berry it is a straight-forward task to calculate a finite number phases. Following this approach, in subsection 3.1, we are of terms of the F-power series for the Bloch-Siegert shifts led to the splitting of the quasienergy into a geometrical that can be compared with known results from the litera- and a dynamical part. The classical mechanics approach ture. Second, for small F it is sensible to expand the Fourier in subsection 3.4 shows that the quasienergy is essentially coefficients of X(t) into power series in F. This leads to the the integral of the Poincaré-Cartan form over one closed so-called Fourier-Taylor series that are defined in-depth in orbit. This result is closely connected to the approach to subsection 6.1.2 and also give rise to analytical approxima- semi-classical Floquet theory in [26]. tions of the quasienergy within their convergence domain. The second main part on resonances essentially bears Finally, the classical RPL equation of motion has an exact on the resonance condition due to Shirley [4]. After a “pendulum” solution for ω0 = 0 that can be extended to short subsection 4.1 on the quasienergy as a homogeneous a solution valid even in linear order w. r. t. ω0. In this H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 707 order it is also possible to obtain a simple expression for As a special case of (5), we consider the quasienergy and to solve the Schrödinger equation, see subsection 6.2.2. Hence this limit case seems to be suited U2(t, t0) ≡ U(t + T, t0), (6) for further studies. We close with a summary and outlook which due to (2), also solves (3) but has the initial in Section 7. condition

U2(t0, t0) = U(t0 + T, t0) ≡ ℱ. (7)

2 Generalities Hence (5) implies

U2(t, t0) = U(t + T, t0) = U(t, t0) ℱ. (8) 2.1 Floquet Theory for SU2

ℱ ∈ SU2 is called the “monodromy matrix”. It can be It appears that the simplest way to explain the general written as ideas of the Floquet theory for TLSs is by again proving −i TF its central claim. In doing so we will emphasise the group- ℱ = e , i F ∈ su2. (9) theoretical aspects of the Floquet theory but otherwise will stick closely to [29]. Now we define The Schrödinger equation for this system is of the form i(t−t0)F 풫(t, t0) ≡ U(t, t0) e (10) ∂ i ψ(t) = H^ (t) ψ(t). (1) and will show that 풫 is T periodic in the first argument: ∂t 풫(t + T, t ) = U(t + T, t ) ei(t+T−t0)F (11) Here we have set ~ = 1 and will assume the Hamilto- 0 0 i TF i(t−t0)F nian H^ (t) to be T-periodic in time, = U(t + T, t0) e e (12) (8,9) = U(t, t ) ℱℱ −1 ei(t−t0)F (13) H^ (t + T) = H^ (t), (2) 0 = 풫(t, t0). (14) where throughout in this paper T = 2π > 0, (1) gives rise ω Summarising this, we have shown that the evolution oper- to a matrix equation for the evolution operator U(t, t0) that ator U(t, t ) can be written as the product of a periodic reads 0 matrix and an exponential matrix function of time, i.e. ∂ i U(t, t ) = H^ (t) U(t, t ), (3) −i(t−t0)F ∂t 0 0 U(t, t0) = 풫(t, t0) e , (15) with the initial condition which is essentially the Floquet theorem for TLSs. Equa- tion (15) is also called the “Floquet normal form” of U(t, U(t0, t0) = . (4) t0). For an example where an explicit solution for U(t, t0) is possible for some limit case, see also subsection 6.2.2. We will assume that U(t, t0) ∈ SU2, the Lie group of unitary The derivation of (15) can be easily generalised from × 2 2-matrices with unit determinant. Consequently, the SU2 to any other finite-dimensional matrix Lie group with Hamiltonian H^ (t) has to be chosen such that i H^ (t) lies in the property that the exponential map from the Lie algebra the corresponding Lie algebra su2 of anti-Hermitean 2 × 2- to the Lie group is surjective, as this has been implicitly matrices with vanishing trace, closed under commutation used in (9). 2 [ , ]. The relation between (1) and (3) is obvious: If ψ0 ∈ C The matrix F is Hermitean and hence has an eigen- and U(t, t0) is the unique solution of (3) with initial condi- basis |n⟩, n = 1, 2 and real eigenvalues ϵn such that tion (4), then ψ(t) ≡ U(t, t0) ψ0 will be the unique solution of (1) with initial condition ψ(t0) = ψ0. Conversely, F = ϵ1 |1⟩⟨1| + ϵ2 |2⟩⟨2|. (16) let ψ1(t) and ψ2(t) be the two solutions of (1) with initial (︀1)︀ (︀0)︀ In this eigenbasis, (15) assumes the form conditions ψ1(t0) = 0 and ψ2(t0) = 1 , then U(t, t0) ≡ −i(t−t0) F (ψ1(t), ψ2(t)) will solve (3) and (4). U(t, t0) |n⟩ = 풫(t, t0) e |n⟩ (17)

Further, it follows that any other solution U1(t, t0) of −i(t−t0)ϵn = 풫(t, t0) e |n⟩ (18) (3) with initial condition U1(t0, t0) = V0 will be of the form −i(t−t0)ϵn = 풫(t, t0) |n⟩ e (19)

−i(t−t0)ϵn U1(t, t0) = U(t, t0) V0. (5) ≡ un(t, t0) e , (20) 708 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited in which the latter functions are called the “Floquet func- F F + ω and violate the condition Tr F = 0. But this tions” or “Floquet solutions of (1)” and the eigenvalues uniqueness is achieved by using a complex arg-function ϵn of F are called “quasienergies”, see [29]. For the TLS, with a discontinuous cut. Consider, for example, a smooth we have exactly two quasienergies ±ϵ such that ϵ ≥ 0 as 1-parameter family of monodromy matrices ℱ(ω) and the Tr F = 0. It follows that any solution ψ(t) of (1) with initial corresponding family ϵ1(ω) of quasienergies. It may hap- (︀ )︀ condition ψ(t0) = a1 |1⟩+a2 |2⟩ can be written in the form pen that exp −i T ϵ1(ω) crosses the cut and hence ϵ1(ω) changes discontinuously. But this discontinuity is not a 2 ∑︁ −i(t−t0)ϵn physical effect and only due to the choice ofthe arg- ψ(t) = U(t, t0) ψ(t0) = an un(t, t0) e , (21) n=1 function. It could be compensated by, say, passing from ϵ1(ω) to ϵ1(ω) + ω. In this case it would be more appro- with the time-independent coefficients an. In this respect priate to consider, say, ϵ1(ω) and ϵ1(ω) + ω as physically un(t, t0), resp. ϵn, generalise the eigenvectors, resp. eigen- equivalent quasienergies. Generally speaking, the issue ^ values, of a time-independent Hamiltonian H. The latter is of continuity is an argument in favour of considering the trivially T-periodic for every T > 0 hence also in this case quasienergies modulo ω. the Floquet theorem (15) must hold. Indeed it does so with 풫(t, t0) = and F = H^ . We remark that the mere analogy between Floquet 2.2 Lift to Higher Spins solutions and eigenvectors can be given a precise mean- ing by considering the “Floquet Hamiltonian” K defined A possible physical realisation of the TLS with Hilbert 2 2 space 2 is a single spin with spin quantum number on the extended Hilbert space L [0, T] ⊗ C , see [29] such C 1 that the quasienergies are recovered as the eigenvalues of s = 2 . Even if the TLS is realised in a different way it K. This was already anticipated in [3, 4], but we will not go will be convenient to adopt the language of spin systems. into the details as the extended Hilbert space will not be For example, the Lie algebra su2 is spanned by the Pauli used in the present paper. matrices σi, i = 1, 2, 3 (times i) or equivalently, by the 1 In this account of Floquet theory we have stressed the three spin operators ^si ≡ 2 σi , i = 1, 2, 3 (times i). Con- dependence of the various definitions of the choice ofan sequently, the Hamiltonian H^ (t) can always be construed arbitrary initial time t0. It, hence, remains to investigate as a Zeeman term with a time-dependent dimensionless the effect of changing from t0to some other initial time t1. magnetic field h(t) namely, A straightforward calculation using the semi-group prop- 3 ∑︁ erty of the evolution operator H^ (t) = h(t) · ^s ≡ hi(t) ^si , (25) i=1 U(t, t0) = U(t, t1) U(t1, t0) (22) where, following the usual convention, we have omitted a gives the result minus sign. We will outline the procedure of lifting a solu- 1 tion of the Schrödinger equation for a spin with s = 2 in −1 U(t, t1) = 풫(t, t0) 풫(t1, t0) a time-dependent magnetic field to a solution of the corre- (︁ −1)︁ sponding Schrödinger equation for general s. For this lift exp −i(t − t1)풫(t1, t0) F 풫(t1, t0) (23) (︁ )︁ the T-periodicity of h(t) is not necessary, but it will later ≡ 풫(t, t1) exp −i(t − t1)F̃︀ . (24) be used to draw conclusions about the Floquet state of the system with general s. The lift procedure is less known but −1 It follows that the eigenvalues of F̃︀ ≡ 풫(t1, t0) F 풫(t1, t0) has been already applied in 1987 to the problem of an N- and F coincide, hence the change of the initial time will level system in a periodic laser field [30]. Also see [31] for modify the Floquet functions but not the quasienergies. a more recent application of the lift procedure to the prob- In concrete applications there will often be a natural lems of Landau-Zener transitions in a noisy environment. choice for t0 and the dependence on t0 may be suppressed For the Bloch-Siegert shift in s = 1 systems see also [32]. without danger of confusion. Let, as in Section 2.1, t U(t, t0) be a smooth curve We will add a few remarks on the uniqueness of the in SU2, such that U(t0, t0) = . It follows that quasienergies ϵn. It is often argued that the quasiener- (︂ ∂ )︂ gies are only unique up to integer multiples of ω, see, e.g. U(t, t ) U(t, t )−1 ≡ −i H^ (t) ∈ su . (26) ∂t 0 0 2 [29]. It seems at first glance that in our approach uniqueness is guaranteed by the requirement i F ∈ su2. For exam- Hence the columns ψ1(t), ψ2(t) of U(t, t0) are linearly ple, the replacement ϵn ϵn + ω in (16) would result in independent solutions of the Schrödinger equation (1). H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 709

∑︀3 2 ′ ∑︀3 Next we consider the well-known irreducible Lie alge- such that i=1 |fi| = 1. and hence ^s = i=1 fi ^si is the 1 ⊤ bra representation (shortly called “irrep”), see, e.g. [33] s = 2 spin operator into the direction (f1, f2, f3) . From Chapter 10–4, this it follows that

(s) 3 3 r : su2 su2s+1 (27) (s) ∑︁ (s) (29) ∑︁ F = 2 ϵ fi r (^si) = 2 ϵ fi S^i . (37) i=1 i=1 of su2 parametrised by the spin quantum number s such that 2s ∈ N and the corresponding irreducible group rep- ^′ ∑︀3 ^ As S = i=1 fi Si is the general s spin operator into the resentation (also called “irrep”) ⊤ direction (f1, f2, f3) with eigenvalues m = −s, ..., s it (s) (s) further follows that the eigenvalues of F and hence the R : SU2 → SU2s+1. (28) quasienergies of the lifted Schrödinger equation are of the It follows that form

(s) (s) r (i ^si) = i S^i , i = 1, 2, 3, (29) ϵm = 2 ϵ m, m = −s, ... , s. (38)

1 where the ^si is defined above as the three s = 2 spin oper- Also the Floquet functions for the lifted problem can be ators and the S^i denotes the corresponding spin operators obtained from those of the TLS, and hence the general for general s. solution of the lifted Schrödinger equation can be reduced It follows from (29) that to the general solution of (1).

3 (s) (︁ )︁ ∑︁ r −i H^ (t) = −i h(t) · S^ ≡ −i hi(t) S^i . (30) i=1 2.3 Lift to SO3 Hence We will consider the lift of the two-level problem to the (s) (s) three-level problem with spin s = 1 with more details. Till U(t, t0) ≡ R U(t, t0) (31) the end we will not directly use the irrep R(1) but some other will be a matrix solution of the lifted time evolution well-known representation R that is, however, unitarily (1) equation equivalent to R . It is defined by

∂ 3 i U(t, t )(s) = h(t) · S^ U(t, t )(s). (32) * ∑︁ ∂t 0 0 u ^si u = R(u)ij ^sj , for all i = 1, 2, 3 and u ∈ su2, j=1 (s) Note that U(t, t0) is a unitary matrix and hence its (39) columns span the general (2s + 1)-dimensional solution and can be restricted to an irrep R : SU2 → SO3. The cor- space of the lifted Schrödinger equation. responding Lie algebra irrep r : su2 → so3 maps i^si , i = Next we will use that the Hamiltonian is a T-periodic 1, 2, 3 onto three anti-symmetric real matrices that span function of time, i.e. that (2) holds. Consequently, we can so3. Let (s) apply the irrep R to the Floquet normal form (15) of ⎛ ⎞ 0 −h3(t) h2(t) U(t, t0) and obtain H(t) ≡ ⎜ h (t) 0 −h (t) ⎟, (40) (︁ )︁ ⎝ 3 1 ⎠ (s) (s) (︀ )︀ (s) −i(t−t0)F U(t, t0) = R 풫(t, t0) R e (33) −h2(t) h1(t) 0

(s) (s) −i(t−t0)F ≡ 풫(t, t0) e , (34) then the lifted evolution equation can be written as

∂ where R(t, t ) = H(t) R(t, t ), (41) ∂t 0 0 i F(s) ≡ r(s) (i F). (35) where, as usual, R(t, t0) ∈ SO3 and R(t0, t0) = . The Recall that the eigenvalues of F are of the form ±ϵ where underlying “Schrödinger equation” has the form ϵ ≥ 0 is the quasienergy of the TLS. Moreover, as i F ∈ su2, d X(t) = h(t) × X(t), (42) F can be written in the form dt

3 3 ∑︁ with X(t) ∈ R and can simultaneously be considered as F = 2 ϵ fi ^si , (36) i=1 the classical limit of the lifted Schrödinger equation for 710 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

s → ∞. A more direct derivation of (42) can be given in the as the initial value X(t0) of a solution of (42) we conclude following way. Let ψ(t) be a solution of the Schrödinger equation (1) X(t + T) = R(t + T, t0)X0 (46) (1) and (15) (1) −i(t+T−t0)F = 풫 (t + T, t0) e X0 (47) (1) (9) (1) −i(t−t0)F (1) = 풫 (t, t0)e ℱ X0 (48) (1) (1) −i(t−t0)F P(t) = |ψ(t)⟩⟨ψ(t)| (43) = 풫 (t, t0) e X0 (49) (15) = R(t, t0) X0 (50) be the corresponding 1-dimensional projector. It can also = X(t). (51) be written in the form This means that this special solution X(t) will be T- periodic. We may hence ask whether it can be obtained as 1 ∑︁ P(t) = + X (t) ^s . (44) the lift of a Floquet solution of the Schrödinger equation, 2 i i i=1,2,3 and, if so, how this Floquet solution can be reconstructed from X(t). To this end we will start with a given T-periodic solu- ⊤ We will write X(t) = (X1(t), X2(t), X3(t)) and note that tion X(t) of (42) and want to construct a corresponding 1 (︁ 2 )︁ 0 = det P(t) = 4 ‖X(t)‖ − 1 . Hence the 1-parameter Floquet solution of (1). It is not necessary to assume the 2 family P(t) corresponds to a curve X(t) on the “Bloch condition ‖X(t)‖ = 1 from the outset. We may rather use sphere”, defined by ‖X(t)‖2 = 1. the fact that (42) admits the constant of motion The projector P(t) satisfies the von-Neumann equation 2 2 2 2 R = X1(t) + X2(t) + X3(t) , (52)

d [︁ ]︁ P(t) = −i H^ (t), P(t) . (45) and hence the solutions of (42) are trajectories on the Bloch dt sphere of radius R. Then the T-periodic 1-parameter family P(t) of one-dimensional projectors defined by

Using the commutation relations of the spin operators (︃ )︃ 1 R + X3(t) X1(t) − i X2(t) ^si one easily derives from (45) the differential equation P(t) ≡ , (53) 2 R X1(t) + i X2(t) R − X3(t) (42) for X(t). In the special case where ψ(t) = ψ1(t) = −i(t−t0)ϵ1 u1(t, t0) e is a Floquet solution of (1) the curve X(t) is a T-periodic solution of (42) and can be visualised as satisfies the von-Neumann equation (45) that is equiv- a closed trajectory on the Bloch sphere. The second Flo- alent to (42). As P(t) is a projector and hence satisfies 2 −i(t−t0)ϵ2 P(t) = P(t), each non-vanishing column of P(t) will be an quet solution ψ2(t) = u2(t, t0) e is orthogonal to eigenvector of P(t) corresponding to the eigenvalue 1. After ψ2(t) for all t and hence must correspond to the “antipode” periodic solution −X(t) of (42). normalising we thus obtain the T-periodic one-parameter It will be instructive to check the consistency of our family of vectors representation by directly applying the Floquet theory to (︃ )︃ (41). The corresponding monodromy matrix ℱ (1) = R(t + 1 R + X3(t) 0 φ(t) = √︀ , (54) (︁ (1))︁ 2R(R + X (t)) X1(t) + i X2(t) T, t0) = exp −i TF has as any matrix in SO3, three 3 eigenvalues of the form 1, eiρ , e−iρ. Consequently, F(1) such that |φ(t)⟩⟨φ(t)| = P(t). We note in passing that (54) has the eigenvalues 0, +ρ/T, −ρ/T which is in accor- is undefined at the south pole of the Bloch sphere where dance with (38) if 2ϵ = ρ/T. We note in passing that these X3(t) = −R. We cannot expect that φ(t) is already a solu- considerations suggest a simple numerical procedure to tion of the Schrödinger equation (1) but only that φ(t) dif- determine the quasienergy ϵ: solve (41) numerically over fers from a solution ψ(t) of (1) by a time-dependent phase one period T and find the eigenvalues of the corresponding factor ei α(t). After some calculations using (42) we obtain monodromy matrix ℱ (1). As a byproduct of the Floquet theory for SO3 we will (︂ d )︂ H^ (t) − i φ(t) prove the existence of periodic classical solutions of (42). dt Let X be the eigenvector of the monodromy matrix ℱ (1) = (︂ )︂ 0 1 h1(t)X1(t) + h2(t)X2(t) = h3(t) + φ(t) (55) R(t0+T, t0) corresponding to the eigenvalue 1. If we use X0 2 R + X3(t) H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 711

(︀ )︀ ∑︁ i n ω t ≡ χ X(t) φ(t) = an e φ(t), (56) and the an in (64) are the Fourier coefficients of the (︀ )︀ n∈Z T-periodic function t χ X(t) defined in (55) and (56). where the infinite sum in (56) represents the Fourier series of the T-periodic function t χ (︀X(t))︀. By integrating this Fourier series over t we obtain 2.4 The RPC Example I ∑︁ ei n ω t α(t) ≡ a0 t + an , (57) i n ω We will check the results of Assertion 1 for the exactly solv- n∈Z n=/ 0 able case of the circularly polarised Rabi problem (RPC) (neglecting an additional integration constant that would where only yield a constant phase factor) and further ⎛ ⎞ F cos ω t (︂ )︂ d h(t) = ⎜ F sin ω t ⎟. (66) H(t) − i ψ(t) = 0, (58) ⎝ ⎠ dt ω0 where 2π We obtain T = ω -periodic solutions X(t) of (42) by the ψ(t) ≡ exp (︀−i α(t))︀ φ(t) (59) following argument. Obviously, ⎛ ⎞ i n ω t ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∑︁ e −ωF sin ω t 0 F cos ω t = exp ⎜−i an ⎟ φ(t) exp (−ia0 t) (60) dh ⎝ i n ω ⎠ = ⎜ ω F cos ω t ⎟ = ⎜0⎟ × ⎜F sin ω t⎟ ≡ ω × h. n∈Z ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ n=/ 0 dt 0 ω ω0 ≡ u(t) exp (−i ϵ t) . (61) (67) We set X(t) = h(t) − ω and conclude According to (61) and (58), ψ(t) is indeed a Floquet solution of (1) with quasienergy ϵ = a0 modulo ω since u(t) is dX dh (67) = = ω × h = h × (−ω) = h × (h − ω) = h × X, T-periodic. The quasienergy ϵ is the time average of χ(X(t)) dt dt denoted by an overbar: (68) and analogously for X(t) = ω − h(t). Hence one finds two (︂ )︂ 1 h1(t)X1(t) + h2(t)X2(t) T-periodic solution of (42) of the form ϵ = a0 = h3(t) + . (62) 2 R + X3(t) ⎛ ⎞ F cos ω t Thus we have proven the following: ⎜ ⎟ X±(t) = ± ⎝ F sin ω t ⎠. (69) Assertion 1. There exists a 1:1 correspondence between T- ω0 − ω periodic solutions X(t) of (42) such that ‖X(t)‖2 = 1 and Floquet solutions ψ(t) = u(t) exp(−i ϵ t) of (1) satisfy the In this case the function χ(X(t)), see (56), turns out to be following conditions: time-independent which directly yields the quasienergies (i) If ψ(t) is a Floquet solution of (1) then 1 (︃ )︃ ϵ± = (ω ± Ω), (70) 1 1 + X (t) X (t) − i X (t) 2 |ψ(t)⟩⟨ψ(t)| = 3 1 2 , 2 X (t) + i X (t) 1 − X (t) 1 2 3 where Ω is the Rabi frequency (63) (ii) If X(t) is a normalised T-periodic solution of (42) then √︁ 2 2 ψ(t) = u(t) exp (−i ϵ t) will be a Floquet solution of (1) Ω ≡ R = F + (ω0 − ω) . (71) where Moreover, the corresponding two Floquet solutions ψ (t) ⎛ ⎞ ± of the Schrödinger equation (1) can be obtained by (60) ∑︁ ei n ω t u(t) = exp ⎜−i a ⎟ × ⎝ n i n ω ⎠ with the result: n∈Z n=/ 0 ⎛ √ − 1 it(ω+Ω) ⎞ (︃ )︃ −ω + Ω + ω0 e 2 1 1 + X (t) √ √ 3 , (64) ⎜ 2 Ω ⎟ √︀ ⎜ 1 ⎟ 2(1 + X (t)) X1(t) + i X2(t) ⎜ 2 it(ω−Ω) ⎟ 3 ψ+(t) = ⎜ Fe ⎟ , (72) (︂ )︂ ⎜ √ √︀ ⎟ 1 h1(t)X1(t) + h2(t)X2(t) ⎝ 2 Ω(−ω + Ω + ω0) ⎠ ϵ = h3(t) + , (65) 2 1 + X3(t) 712 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

⎛ √ − 1 it(ω−Ω) ⎞ ω + Ω − ω e 2 see (56), into a “dynamical” and a “geometrical part”. The √ √0 ⎜ 2 Ω ⎟ dynamical part is defined as ⎜ 1 it(ω+Ω) ⎟ ψ−(t) = ⎜ Fe 2 ⎟ , (73) ⎜ −√ ⎟ ⃒ ⃒ ⎜ √︀ ⎟ χ (︀X(t))︀ ≡ ⟨φ(t) ⃒H^ (t)⃒ φ(t)⟩ (75) ⎝ 2 Ω(ω + Ω − ω0) ⎠ d ⃒ ⃒ 1 = (h X + h Y + h Z) (76) 2R 1 2 3 in accordance with the well-known result, see, e.g. [34]. and represents the expectation value of the energy. Its time average yields the dynamical part of the quasienergy:

1 ϵ ≡ χ (︀X(t))︀ = (h X + h Y + h Z). (77) d d 2R 1 2 3

3 Geometry of the Two-Level The geometrical part of χ(X(t)) is defined as System ⟨ ⃒ ⟩ (︀ )︀ ⃒ d χg X(t) ≡ φ(t) ⃒−i φ(t) (78) The correspondence between T-periodic solutions X(t) of ⃒ dt X(t)Y˙ (t) − Y(t)X˙ (t) (42) and Floquet solutions ψ(t) of (1) that has been for- = . (79) mulated in Assertion 1 can be further analysed w. r. t. 2R(R + Z(t)) two different geometric perspectives: either the map Using spherical coordinates θ, ϕ for the Bloch sphere with ψ(t) X(t) can be viewed as the restriction of the map radius 1, we may write the differential χg dt in the form of a Hilbert space ℋ onto the corresponding projective Hilbert space P(ℋ) and the quasienergy as a phase change X dY − Y dX χ (︀X(t))︀ dt = (80) during a cyclic quantum evolution in the sense of [27]. Or g 2R(R + Z) the Bloch sphere can be construed as the phase space of 1 − cos θ = dϕ ≡ α. (81) the classical limit of the TLS and the quasienergy can be 2 related to its semi-classical limit in the sense of [26]. We This yields a differential 1-form α on the Bloch sphere and will treat both aspects in the following subsections. (︀ )︀ 1 ω the time average of χg X(t) is, up to the factor T = 2π , the integral of α over the closed curve 풞 parametrised by X(t). By applying Stoke’s theorem we obtain ∮︁ 3.1 Geometry of the Fibre Bundle (︀ )︀ ω ϵg = χg X(t) = α (82) π 2 풮 2 2π : C|1| 풞 ω ∫︁ ω = dα = |풜| , (83) The map of wave functions to projectors ψ |ψ⟩⟨ψ| can 2π 4π be viewed as a surjective map π from the unit sphere of 풜 2 of 2 to the unit sphere of 3, i.e. as π : 2 풮2. C|1| C R C|1| where 풜 denotes the oriented area enclosed by 풞 and |풜| Since 2 =∼ 풮3 this is essentially the Hopf fibration [35]. C|1| the correspondingly signed solid angle. Here we have used −1 iα The fiber π (X) consists of the set of all phase factors e that the 2-form and can hence be identified with the one-dimensional uni- 1 tary group U1. Then the map σ : X φ(t) according to dα = sin θ dθ ∧ dϕ (84) 2 (54) can be viewed as a local section of the principal fiber 2 2 bundle π : 풮 with structure group U1. The above 1 C|1| equals 2 times the surface element on the unit sphere. remark that σ is undefined at the south pole of the Bloch Thus we obtain the following interpretation of the sphere means that σ cannot be extended to a global section quasienergy ϵ = ϵd + ϵg as composed of two parts: the due to topological obstacles. The group SU operates in a 2 dynamical part ϵd is the time average of the energy (76) and natural fashion on 2 as well as on 풮2 via rotations R(u) ω C|1| the geometrical part ϵg is 4π times the solid angle enclosed defined in (39) by the T-periodic solution of (42). Following [27] we may split the function We refer to [27] for the differential-geometric back- ground of this scenario. Obviously, 1-form α is the retract ⃒(︂ )︂⃒ of the canonical connection 1-form γ of the principal (︀ )︀ ⃒ d ⃒ χ X(t) = ⟨φ(t) ⃒ H^ (t) − i ⃒ φ(t)⟩, (74) 2 2 ⃒ dt ⃒ fiber bundle π : C|1| 풮 w. r. t. the local section σ : H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 713

2 2 풮 C|1| described above. Hence the non-vanishing of 3.3 Another Solvable Example ϵg is due to the curvature of this principal fiber bundle and generalises Berry’s phase. Recall further that, by defini- The idea of a “reverse engineering of the control” h(t), see tion, γ has values in the Lie algebra of U1. This Lie alge- [18–20], can also be applied to the classical Floquet prob- bra is isomorphic to R and an analogous identification lem. Given a normalised T-periodic function X(t) we may has been made by considering above α as a usual real- choose h(t) ≡ X(t) × X˙ (t) such that (42) is satisfied. This valued 1-form. However, ϵg as the integral over α should special choice of h entails h · X = 0 and hence the dynam- properly have values in U1 and not in R and this subtle dif- ical part χd of χ vanishes according to (76). The geometri- ference is in turn in accordance with the general claim that cal part χg = χ will be calculated according to (79) for the quasienergies are only defined modulo ω. following example:

⎛ (︂ (︂ ωt )︂)︂ ⎞ cos(ωt) sin f sin2 ⎜ 2 ⎟ 3.2 The RPC Example II ⎜ ⎟ ⎜ (︂ (︂ ωt )︂)︂ ⎟ ⎜ sin(ωt) sin f sin2 ⎟ ⎜ ⎟ We will illustrate the results of the preceding subsection X(t) = ⎜ 2 ⎟ , (89) ⎜ (︂ (︂ )︂)︂ ⎟ by the explicitly solvable case of RPC. The calculation ⎜ 2 ωt ⎟ ⎜ cos f sin ⎟ is essentially identical with that in [27]. It follows from ⎝ 2 ⎠ (66) and (69) that the dynamical part of the quasienergy assumes the value (note that the involved functions are f being a real parameter, which leads to constant and taking the time average is superfluous) ⎛ ⎞ 2 1 (︁ 2 )︁ h1X + h2Y + h3Z F + ω0(ω0 − ω) − ω f sin (ωt) + cos(ωt) sin(f − f cos(ωt)) ϵd = = √︁ . (85) ⎜ 2 ⎟ 2R 2 ⎜ ⎟ 2 F2 + (ω − ω) ⎜ ⎟ 0 ⎜ 1 ⎟ ⎜ ω sin(ωt)(f cos(ωt) − sin(f − f cos(tω))) ⎟ h(t) = ⎜ 2 ⎟, On the other hand, the vector X (t) prescribes a cir- ⎜ ⎟ + ⎜ (︂ (︂ )︂)︂ ⎟ ⎜ 2 2 ωt ⎟ cle on the Bloch sphere with constant Z = ω0 − ω, see ⎜ ω sin f sin ⎟ (69). Consider first the case of Z > 0. The correspond- ⎝ 2 ⎠ ing spherical segment has an area of 2πR(R − Z) corre- (90) sponding to a solid angle of |풜| = 2π(R−Z) , where R = √︁ R and 2 2 Ω = F + (ω0 − ω) is the radius of the Bloch sphere. (︂1 (︂ ωt )︂)︂ Hence χ(t) = ω sin2 f sin2 , (91) 2 2 ω 2π(R − Z) ω (R − ω0 + ω) ϵg = = , (86) 4π R 2 R see Figure 1 visualising the example f = 1, ω = 1. The Fourier series of χ(t) can be explicitly calculated: and

2 ∞ F + ω0(ω0 − ω) ω (R − ω0 + ω) ∑︁ ϵ+ = ϵd + ϵg = + (87) χ(t) = ϵ + bn cos n ω t, (92) 2 R 2 R n=1 1 = (ω + R), (88) 2 where in accordance with (70). For the second periodic solution ω (︂ (︂ f )︂ (︂ f )︂)︂ ϵ = 1 − cos J , (93) X−(t) it follows that both terms ϵd and ϵg acquires a minus 2 2 0 2 sign, the latter as the spherical segment encircled by X−(t) ⎧ n+1 (︁ f )︁ (︂ )︂ ⎨ (−1) 2 sin : n odd, has a negative orientation. Hence ϵ− = −ϵ+ mod ω in f 2 bn = ω Jn n+2 (︁ )︁ . (94) 2 2 f accordance with (70). ⎩ (−1) cos 2 : n even, In the case of Z < 0 the dynamical part ϵd remains unchanged whereas the solid angle of the spherical Here the Jn(...) denote the Bessel functions of the first segment encircled by X+(t) assumes the form |풜| = kind and integer order. The constant term of (92) is the 2π(R+Z) 1 − R . This yields ϵ+ = 2 (−ω + R) which differs from quasienergy ϵ according to (93) that only depends lin- (88) only by −ω and is thus equivalent. Analogous early on ω. The Fourier series of χ(t) could be utilised ∓i ϵ t arguments hold for ϵ−. to explicitly determine the Floquet solutions u±(t)e 714 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

as Hamiltonian equations of motions in a two-dimensional phase space isomorphic to 풮2 with canonical coordinates

p = z, q = φ. (102)

We will hence forward often use p and q instead of z and φ. Following [36] we consider the extended phase 2 space 풫 = 풮 × R with coordinates p, q, t and the Poincaré-Cartan form

α ≡ p dq − H dt (103)

defined on 풫 (not to be confounded with the 1-form α defined in Section 3.4). The Hamiltonian equations (100), (101) can be geometrically construed as the direction field Figure 1: A periodic solution X(t) of the classical equation of motion (42) according to (89) with f = 1, ω = 1 represented by a closed on 풫 that is given by the unique null direction field of the trajectory on the Bloch sphere (blue curve). The corresponding mag- exterior derivative of the Poincaré-Cartan form netic field h(t) according to (90) is visualised by the red curve. At the time t = π the three vectors X(π), h(π) and X˙ (π) = h(π) × X(π) ω ≡ dα = dp ∧ dq − dH ∧ dt, (104) are indicated by the coloured arrows. see Section 44 of [36] for the details. of the corresponding Schrödinger equation according to Periodic solutions of (100), (101) correspond to curves Assertion 1. γ in 풫 that are not closed as after one period T the coor- Note further that Assertion 3 of Section 4.2 can be dinate t has changed from t = 0 to t = T. This can be ∂ϵ ϵ ′ sharpened to ∂ω = ω as ϵd = 0 in this example. repaired by defining another extended phase space 풫 via identifying points with t-coordinates that differ by an integer multiple of T, or, more formally, 3.4 Classical Mechanics of the Two-Level ′ 2 System 풫 ≡ 풮 × R/TZ. (105)

1 First we will introduce some concepts of classical mechan- Of course, R/TZ is isomorphic to 풮 . Locally the manifolds ′ ics suited for the present case. Let z and φ be coordinates 풫 and 풫 are isomorphic, but globally they are not. The ′ of the unit Bloch sphere defined by differential forms α and ω can be transferred to 풫 as all functions involved in α and ω are T-periodic. Now peri- √︀ X/R = 1 − z2 cos φ, (95) odic solutions of (100), (101) correspond to closed curves γ √︀ ′ Y/R = 1 − z2 sin φ, (96) in 풫 . Z/R = z. (97) Next we will rewrite the expression (62) for the quasienergy ϵ. By (76) and (81) we obtain Further we define the classical Hamiltonian 1 χ dt = H dt, (106) 1 d 2 H = (h X + h Y + h Z) (98) R 1 2 3 1 1 χg dt = (1 − z) dφ = (1 − p) dq. (107) √︀ 2 = 1 − z (h1 cos φ + h2 sin φ) + h3 z, (99) 2 2 Hence and rewrite the differential equation (42) in terms of the two functions z(t) and φ(t): ω ∮︁ n ϵ = χg + χ = − (p dq − H dt) + ω, (108) d 4π 2 √︀ ∂H γ z˙ = 1 − z2 (h sin φ − h cos φ) = − , (100) 1 2 ∂φ where the n in the last term denotes the winding number z ∂H φ˙ = h3 − √ (h1 cos φ + h2 sin φ) = . (101) of γ around the z-axis. This result is in close analogy to 1 − z2 ∂z equation (2.35) of [26] that represents the semi-classical Note that due to (76), H is twice the expectation value of limit of the quasienergy for integrable Floquet systems. It the Hamiltonian H^ . Obviously, (100)–(101) can be viewed thus seems that for the TLS, similar as in the case of the H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 715 driven harmonic oscillator, the semi-classical limit of the 4.1 Homogeneity of the Quasienergy quasienergy and the exact quantum-theoretical expression coincide. However, it has not yet been shown that the For the RPE the classical equation of motion (42) reduces quantisation procedure adopted in [26] yields the quan- to tum TLS when starting from its classical limit. X˙ = G sin(ωt) Z − ω0 Y, (115)

Y˙ = ω0 X − F cos(ωt) Z, (116) Z˙ = F cos(ωt) Y − G sin(ωt) X. (117) 4 Resonances It is invariant under the transformation In this and the following sections we will restrict the ω λ ω , (118) Hamiltonian (25) to the following special case 0 0 F λ F, (119) G λ G, (120) h1 = F cos ωt, (109) ω λ ω, (121) h2 = G sin ωt, (110) 1 t t, (122) h3 = ω0, (111) λ for all λ > 0. Under this transformation the quasienergy that will be referred to as the RPE. It includes the two limit (62) scales with λ and hence is a positively homogeneous cases G → 0, the RPL, and G → F, the RPC. Hence the function of degree 1: quasienergy ϵ can be written as a function ϵ(ω0, F, G, ω) of the four parameters ω0, F, G, ω that will be assumed to ϵ(λ ω0, λ F, λ G, λ ω) = λ ϵ(ω0, F, G, ω) (123) have positive values. The corresponding classical Hamiltonian (99) reads for all λ > 0. This could be used to eliminate the vari- −1 able ω0 (by choosing λ = ω0 which transforms ω0 into √︀ 1) but for some purposes, e.g. the investigation of the limit H(z, φ) = 1 − z2(F cos ωt cos φ ω0 → 0 or the analysis of the resonance condition (114), + G sin ωt sin φ) + ω0 z. (112) the elimination of ω0 is not appropriate. By the Euler theorem the positive homogeneity of ϵ If the TLS is coupled to a second weak electromagnetic implies field there may occur transitions between the two different ∂ϵ ∂ϵ ∂ϵ ∂ϵ Floquet states, analogously as in the case of two energy ϵ(ω , F, G, ω) = ω + F + G + ω . (124) 0 0 ∂ω ∂F ∂G ∂ω levels for a time-independent Hamiltonian. Shirley has 0 computed the time-averaged probability P of such a transition, see (26) in [4], with the remarkably simple result 4.2 Calculation of ∇ϵ

(︃ (︂ )︂2)︃ We first consider ∂ϵ . For the sake of simplicity we assume 1 ∂ϵ ∂ω0 P = 1 − 4 . (113) that the classical equations of motion (115)–(117) for the 2 ∂ω0 RPE has two unique normalised T-periodic solutions of the form ±X(t). The latter assumption is, for example, violated Although Shirley’s derivation of (113) refers to the RPL in the case ω0 = 0, see Section 6.2, where we have a 1- case, see (1) in [4], one can easily check that it also holds parameter family of periodic solutions and a quasienergy in the more general RPE case. It implies that the transition ϵ = 0. Under these assumptions we will prove the follow- 1 probability assumes its maximal value Pmax = 2 for ing:

∂ϵ Assertion 2. = 0. (114) ∂ω0 ∂ϵ Z(t) = . (125) ∂ω0 2 R Hence the condition (114) will be called the “resonance Hence the resonance condition ∂ϵ = 0 is equivalent to condition”. It will be further analysed in the following ∂ω0 subsections. Z(t) = 0. 716 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

For the proof of this assertion we will adopt the language γ2 of classical mechanics introduced in 3.4. In order to calculate ∂ϵ we have to vary the parameter ω . To this end we ∂ω0 0 define still another extension of the phase space by τ

′′ 2 풫 ≡ 풮 × R/TZ × R>0, (126)

ω0 –γ1 with coordinates p, q, t and ω0 > 0. Again the differential forms α and ω can be transferred to 풫′′ but instead of dα = ω we now have Figure 2: The tube τ in extended phase space 풫′′ generated by (1) (2) the family of closed curves γ(ω0), where ω0 ≤ ω0 ≤ ω0 . The two curves −γ1 and γ2 that form the boundary of τ are displayed ∂H (112) together with their orientations. dα = ω − dω0 ∧ dt = ω − z dω0 ∧ dt. (127) ∂ω0 Then ⎛ ⎞ ω ∮︁ ∮︁ The closed curves γ corresponding to periodic solu- ϵ2 − ϵ1 = − ⎝ α − α⎠ (129) 4π tions of (100), (101) smoothly depend on ω0 and hence γ2 γ1 ⎛ ⎞ will be denoted by γ(ω0). Geometrically, this defines a tube ∮︁ ′′ ω τ in 풫 parametrised by γ(ω0, t), see Figure 2. We will = − ⎝ α⎠ (130) 4π consider values of ω0 running through some closed inter- γ2−γ1 (1) (2) val ω0 ∈ [ω0 , ω0 ] and restrict the tube τ to these val- ⎛ ⎞ ω ∮︁ ues. Hence the boundary ∂τ of the tube can be identified = − ⎝ α⎠ (131) (i) 4π with γ2 − γ1, where γi ≡ γ(ω0 ), i = 1, 2 and the minus ∂τ sign in γ2 − γ1 accounts for the correct orientation. We ⎛ ⎞ ω ∫︁ consider the difference between the quasienergies = − ⎝ dα⎠ (132) 4π τ ⎛ ⎞ (127) ω ∫︁ ∫︁ = − ω − (z dω ∧ dt) , (133) (2) (1) ⎝ 0 ⎠ ϵ2 − ϵ1 ≡ ϵ(ω ) − ϵ(ω ) 4π 0 0 τ τ ⎛ ⎞ (103),(108) ω ∮︁ ∮︁ δn = − ⎝ α − α⎠ + ω, (128) invoking Stokes theorem in (132). Now we use the fact that 4π 2 ∫︀ γ2 γ1 τ ω = 0 as the tangent plane at any point x ∈ τ contains a null vector of ω, namely the vector tangent to the curve γ passing through x ∈ τ. Here we have employed the above- mentioned results of analytical mechanics according to where δn denotes the difference of the winding numbers. [36]. It follows that In this paper we will make the general assumption that the ∫︁ quasienergy can be chosen as a smooth function ϵ(ω0) of ω ϵ2 − ϵ1 = z dω0 ∧ dt (134) ω0 taking into account that it is only defined up to inte- 4π τ ger multiples of ω. Hence a discontinuous change of the (2) ω0 winding number by an even number can be compensated 1 ∫︁ = z(ω , t) dω , (135) by the choice of the right branch of ϵ(ω0). In the RPL exam- 2 0 0 (1) ple there are only even changes of the winding number due ω0 to the symmetry of the periodic solution but, according to our general assumption, this must hold generally. Hence where z(ω0, t) denotes the time average w. r. t. the curve δn (1) we can neglect the term 2 ω in (128). γ(ω0) and hence depends on ω0. Choosing ω0 = ω0 − δ H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 717

(2) and ω0 = ω0 + δ we obtain the limit δ → 0: Together with an analogous calculation as in the proof of Assertion 2 this implies ∂ϵ 1 = z(ω0, t), (136) ∂ω0 2 2π ∂ϵ ϵ ω ∫︁ ∂H = + dτ (147) which concludes the proof of Assertion 2. ∂ω ω 4π ∂ω ∂ϵ ∂ϵ 0 The calculation of ∂F and ∂G can be performed analogously. For example, the analogue of (135) reads 2π ϵ 1 ∫︁ = − Hdτ (148) ω ∫︁ ∂H ω 4πω ϵ2 − ϵ1 = dF ∧ dt (137) 0 4π ∂F ϵ 1 τ = − H (149) F(2) ω 2ω 1 ∫︁ √︀ (77) ϵg = 1 − z2 cos ωt cos φ dF (138) = , (150) 2 ω F(1) F(2) using ∂H = − 1 H in (148). An alternative derivation of 1 ∫︁ ∂ω ω2 = x cos2 ωt dF (139) (149) would consist in the evaluation of the Euler relation 2 c F(1) (124) using (125), (141) and (142). Hence the calculation of (F(2) − F(1)) x the partial derivatives of ϵ does not lead to a simplified for- → c , (140) 4 mula for ϵ itself. However, the relation (150) can be used for a geometrical interpretation of the splitting ϵ = ϵg + ϵd, and hence see Figure 3. We will formulate this result separately and ∂ϵ x stress the fact that the above proof is independent the = c , (141) ∂F 4 particular form (109)–(111) of H. where xc is the coefficient of the term cosωt in the Fourier Assertion 3. Under the assumptions of Sections 2 and 3 the series of x(t). xc depends on F and hence (140) only holds following holds: asymptotically for F(2) − F(1) → 0. ∂ϵ ϵ Similarly, = g . (151) ∂ω ω ∂ϵ y = s , (142) ∂G 4 ∂ϵ Hence ∂ω equals the solid angle |풜| encircled by the cor- where ys is the coefficient of the term sinωt in the Fourier responding periodic solution of the classical RPE equation series of y(t). divided by 4π. ∂ϵ In order to calculate ∂ω it is advisable first to simplify the ω-dependence of ϵ by the coordinate transfor- H mation t τ ≡ ω t together with H H ≡ ω . The latter transformation insures that the Hamiltonian equations of motion retain their canonical form d φ ∂H = , (143) d τ ∂z d z ∂H = − . (144) d τ ∂φ

After the coordinate transformation the Poincaré-Cartan form reads

α = p dq − H dτ. (145)

Figure 3: The quasienergy ϵ as a function of ω for fixed ω0, F and G Recall that according to (108) is considered as the sum of the geometrical part ϵg and the dynam-

ical part ϵd satisfying (150). Hence the values of ϵg and ϵd can be ω ∮︁ n ϵ(ω) = − (p dq − H dτ) + ω. (146) geometrically determined by the intersection of the tangent to the 4π 2 graph of ϵ(ω) with the ϵ axis. γ 718 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

4.3 The RPC Example III One easily derives that also both sides of (159) are odd cos- series, and hence probably there exists a solution of (158)– The quasienergies (160) that fulfills the above requirements concerning the subspaces in which X, Y and Z lie. (70,71) (︂ √︁ )︂ 1 2 2 ϵ± = ω ± F + (ω0 − ω) (152) On the basis of these considerations and numerical 2 investigations we obtain the following ansatz of a (not necessarily normalised) Fourier series solution of (158)–(160): are obviously positively homogeneous functions of ω0, F, ω. Further, the normalised third component of X(t) has ∞ ∑︁ the constant value X(t) = ω0 x2n+1 cos(2n + 1)ωt, (161) n=0 ω0 − ω ∞ z = √︁ = z(t), (153) ∑︁ 2 2 Y(t) = x2n+1 (2n + 1) ω sin(2n + 1)ωt, (162) F + (ω0 − ω) n=0 ∞ ∑︁ see (69). On the other hand, by (152), Z(t) = z0 + x2n cos 2nωt. (163) n=1 ∂ϵ 1 ∂Ω 1 ω − ω + = = 0 , (154) ∂ω 2 ∂ω 2 √︁ 0 0 F2 + (ω − ω)2 0 The form of (162) is already uniquely determined by the ˙ = − which confirms Assertion 2. The resonance condition differential equation X ω0 Y. The Fourier coefficients ∂ϵ+ x2n of Z(t) in (163) are written in such a way that the vec- = 0 is equivalent to ωres = ω0. ∂ω0 x = Moreover, Assertion 3 is confirmed by the following tor of unknown Fourier coefficients assumes the form calculation: (x1, x2, ...). The validity of the ansatz (161)–(163) will not be rigorously proven but appears highly plausible due to (︃ )︃ ∂ϵ (152) 1 ω − ω the investigation of the analytical approximations to these + = 1 + 0 (155) √︀ 2 2 solutions in what follows. ∂ω 2 F + (ω − ω0) 1 (︁ ω − ω )︁ If we insert the ansatz (161)–(163) into (158)–(160) we = 1 + 0 (156) 2 R obtain an infinite system of linear equations of the form (86) ϵ A x = f, where = g . (157) ω ⎛ ⎞ −F z0 ⎜ ⎟ f = ⎜ 0 ⎟. (164) 5 Analytical Approximations ⎝ . ⎠ . 5.1 Truncated Fourier Series Solution

In this and the following sections we specialise in the RPL. The matrix A is tri-diagonal due to the simple form of the Thus the classical equation of motion (42) reduces to h1 = F cos ωt which couples only neighbouring modes. Although A is unbounded it may be sensible to truncate it to some N × N-matrix A(N) if the resulting finite Fourier X˙ = −ω0 Y, (158) series has rapidly decreasing coefficients and hence repre- Y˙ = ω0 X − F cos(ωt) Z, (159) sents a good analytic approximation to the infinite Fourier ˙ Z = F cos(ωt) Y. (160) series. The matrix elements of A are given by

According to our general approach we are looking for ⎧ 2 2 2 T-periodic solutions of these equations. The space of real n ω − ω0 : n = m odd, ⎪ T-periodic functions is spanned by the four subspaces ⎪ −nω : n = m even, ⎪ defined by even/odd sin/cos-series. Assume, for example, ⎪ F ⎨⎪ : n = m ± 1 odd, that X(t) is given by an odd cos-series. Then it follows that 2 Anm = mFω (165) ˙ ⎪ − : n = m ± 1 even, X(t) is an odd sin-series and, by (158), also Y(t) will be an ⎪ ⎪ 2 odd sin-series. By (160) we further conclude that Z˙ (t) is an ⎪ 0 : else. ⎪ even sin-series and hence Z(t) will be an even cos-series. ⎩ H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 719

(6) For example, the truncated matrix A has the form result for the Fourier coefficients xn:

⎛ 2 2 F ⎞ x1 = −F z0 ϕ2/ϕ1, (171) ω −ω0 2 0 0 0 0

⎜ Fω 3Fω ⎟ − −2ω − 0 0 0 ⎧ n F n n/2 ⎜ 2 2 ⎟ (−1) 2( 2 ) (−ω) (n−1)!! ϕn+1 ⎜ ⎟ z0 : n even, ⎪ ϕ1 ⎜ 0 F 9ω2−ω2 F 0 0 ⎟ ⎨ (6) ⎜ 2 0 2 ⎟ xn = A = . n n−1 ⎜ ⎟ (−1)n2 F (−ω) 2 (n−2)!! ϕ ⎜ 0 0 − 3Fω −4ω − 5Fω 0 ⎟ ⎩⎪ ( 2 ) n+1 ⎜ 2 2 ⎟ z0 : n odd, (172) ⎜ ⎟ ϕ1 ⎜ 0 0 0 F 25ω2−ω2 F ⎟ ⎝ 2 0 2 ⎠ 5Fω where (172) holds for n = 2, 3, ... , N. z0 is a free para- 0 0 0 0 − 2 −6ω (166) meter that necessarily occurs due to the fact that the The truncated system of linear equations of the form Fourier series solution is not yet normalised. It could be (N) (N) −1 chosen as z0 = 1 or, alternatively, as z0 = ϕ1. Depend- A x = f has the formal solution x = −F z0 (A )1 , (N) −1 ing on the context both choices will be adopted in what where (A )1 denotes the first column of the inverse matrix of A(N). Fortunately, there exists a recurrence for- follows. The latter choice has the following advantage: if mula for the inverse of tri-diagonal matrices in terms of ϕ1 = 0 the above solution (171)–(172) is no longer defined, leading principal minors and co-leading principal minors, but choosing z0 = ϕ1 and cancelling the fraction z0/ϕ1 see [37]. Recall that a leading minor of A(N) of order n is to 1 we obtain a solution that is always defined. Upon the (N) the determinant of the sub-matrix of matrix elements in choice z0 = ϕ1 the vanishing of ϕ1 = det A is equiv- rows and columns from 1 to n. Similarly, we will denote alent to the vanishing of the time average Z(t) = z0. We recall the fact that this in turn characterises the occurrence by the “co-leading principal minor ϕn of co-order n” the determinant of the sub-matrix of matrix elements of A(N) of resonances, see Assertion 2. in rows and columns from n to N. As we do not need the In any case, from these recursion relations it is clear (︁ )︁−1 that each x is a rational function ρ(n, N, F, ω, ω ) in whole matrix A(N) but only its first column it turns n 0 the variables F, ω and ω . It can hence be viewed as a out that only co-leading principal minors are involved. It is 0 kind of Padé approximation for xn that becomes more and well-known that the determinant of a tri-diagonal matrix more exact for increasing N. For the choice z = ϕ the satisfies a three-term recurrence relation. For our problem 0 1 rational function ρ(n, N, F, ω, ω ) becomes a polynomial this implies the following system of recurrence relations 0 in the variables F, ω and ω . for the ϕ , n = N, N − 1, ... , 1. 0 n In order to give an impression of the structure of {︃ ρ(n, N, F, ω, ω0) we give the results for the N = 4 trun- −N ω : N even, cation with z0 = ϕ1 although this will not yet be good ϕN = AN,N = 2 2 (167) (N ω) − ω0 : N odd, approximations of the exact RPL solutions:

1 (︁ (︁ )︁)︁ X(t) = Fω2ω 9F2 + 16 ω2 − 9ω2 2 0 0 ϕN−1 = AN−1,N−1 AN,N − AN,N−1 AN−1,N (168) 3 2 cos(ωt) − F ω ω0 cos(3ωt), (173) 1 (︁ (︁ )︁)︁ Y(t) = Fω3 9F2 + 16 ω2 − 9ω2 2 0 ⎧ 1 2 (︀ 2 2 2)︀ sin(ωt) − 3F3ω3 sin(3ωt), (174) ⎨ 4 F ω(N − 1) − ωN ω (N − 1) − ω0 : N even, = 1 2 2 (︁ 2 (︁ 2 2)︁)︁ ⎩ 1 2 (︀ 2 2 2)︀ Z(t) = δ − F ω 3F + 16 ω0 − 9ω 4 F ωN − ω(N − 1) ω N − ω0 : N odd, 8 (169) 3F4ω2 cos(2ωt) + cos(4ωt), (175) 8 ⎧ n+1 2 (︁ (︁ )︁ −n ω ϕn+1 + F ω ϕn+2 : n even, 1 2 4 2 2 2 ⎨ 4 δ = ω 3F − 8F 27ω − 11ω0 ϕn = 16 (︀ 2 2)︀ n 2 (︁ )︁ (︁ )︁)︁ ⎩ (n ω) − ω0 ϕn+1 + F ω ϕn+2 : n odd, 2 2 2 2 4 + 128 ω − ω0 9ω − ω0 (176) (170) (N) Especially, ϕ1 = det A . Then the first column of −1 In Figure 4 we show solutions of the classical RPL for dif- (︁ (N))︁ (1) A can be expressed in terms of the co-leading ferent F values at the resonance frequency ωres(F) that principal minors and certain products of the lower sec- will be calculated in the next subsection. These solutions ondary diagonal elements of A, see the theorem in [37] for are based on the truncated Fourier series (161)–(163) with the special case of j = 1. We write down the corresponding N = 20 and the choice z0 = ϕ1 = 0. They can be either 720 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

z F

Figure 4: Various periodic solutions of the classical RPL for (1) ω = ωres(F) and ω0 = 1 visualised as closed trajectories on the Bloch sphere. The coloured curves correspond to different values Figure 5: The domain of the three variables ω0, ω, F of the homoge- of F, namely F = 1/10 (blue), F = 1/2 (red), F = 1 (green), F = 2 neous function ϵ can be realised by an equilateral triangle ∆. Each (magenta), F = 5 (cyan), F = 10 (brown), and F = 100 (black). point inside the triangle can be uniquely written as a convex sum of the vectors pointing to the three vertices, i.e. as a sum with pos- calculated by directly evaluating the Fourier series or by itive coeflcients that add to unity. The three open edges represent numerically solving the equation of motion with the ini- limit cases, e.g. the edge opposite to the vertex labelled “ω” repre- tial values obtained by the Fourier series. The differences sents the limit case ω → 0 such that ω0 and F are kept finite. The between both methods are negligible. For small F the three vertices themselves represent the limit cases where two of the three variables approach 0 and the third one approaches 1. solution approximately corresponds to a uniform rotation in the x-y-plane, whereas for large F the solution curve folds into a sort of pendulum motion in the y-z-plane that The transformation between the three scaled variables approaches the limit solution for F, ω → ∞, see subsec- ω0, ω, F̃︀ and the two Cartesian coordinates x, y defining tion 6.2. ̃︁ ̃︀ the points of ∆ has the following form:

5.2 Calculation of the Resonance 1 (︀ )︀ Frequencies x = ω̃︀ − ω̃︁0 , (178) √2 3 We have shown in Section 4.1 that ϵ is a positively homoge- y = F̃︀, (179) 2 neous function and the same holds for its restriction to the 1 y variables ω , ω, F in the limit G → 0. The domain of these ω0 = − x − √ (180) 0 ̃︁ 2 3 variables can be restricted to a two-dimensional domain 1 y without loss of information. Instead of eliminating one of ω = + x − √ (181) ̃︀ 2 3 the three variables, which is inappropriate in some cases, 2 y F̃︀ = √ . (182) one could introduce the scaled variables 3 ω0 ω ω̃︁0 ≡ , ω̃︀ ≡ , ω + ω0 + F ω + ω0 + F F (n) F̃︀ ≡ , (177) The resonance frequencies ωres , n = 1, 2, ... can be rep- ω + ω0 + F resented by “resonance curves” ℛn in the triangular that satisfy 0 < ω̃︁0, ω̃︀ , F̃︀ < 1 and ω̃︁0 + ω̃︀ + F̃︀ = 1. The domain ∆, see Figure 6. These curves have been numeri- (50) domain of the scaled variables is an open equilateral tri- cally calculated by setting z0 = ϕ1 = det A = 0, see angle ∆, see Figure 5. If the values of a positively homoge- subsection 5.1 and Assertion 2. Later we will show that neous function like the quasienergy ϵ are known for argu- their intersections with the two edges F = 0 and ω0 = 0 ments in ∆ the function can be uniquely extended to the can be analytically determined, see Figure 6. For large n whole positive octant. If it is clear that we mean the scaled the resonance curves approach the straight line segments variables the tilde will be omitted. determined by these intersections. H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 721

Figure 7: The various branches of the quasienergy ϵ for F = 1/2 and ω0 = 1 as functions of the frequency ω calculated by an N = 20 truncated Fourier series solution of the classical RPL. In the neigh- (n) bourhood of the resonance frequencies ωres we have an avoided level crossing visualised by choosing the same colour for the continuous branches of the quasienergy. For n > 1 one has the Figure 6: The first 10 resonance curves ℛn , n = 1, ... , 10 in the tri- impression of crossing levels due to the low resolution of the figure angular domain of variables ω0, ω, F. They intersect the line F = 0 but, e.g. the comparison with Figure 8 for n = 2 shows that the lev- ω 1 F at = and the line ω0 = 0 at = j0,n (the latter denoting els actually do not cross. The dotted horizontal line indicates the ω0 2n−1 ω the zeroes of the Bessel function J0), as will be shown below, see value of limω→0 ϵ(ω) ≈ 0.52992, see (254). (183) and (225).

5.3 Calculation of the Quasienergy

After approximating X(t), Y(t), Z(t) by a truncated Fourier series it is possible to calculate the quasienergy (︂ )︂ 1 (︁ F cos(ωt) X(t) )︁ ϵ = 2 ω0 + R+Z(t) by a numerical integration. Alternatively, we can determine the integral over 2π t ∈ [0, ω ] analytically in the following way: We choose F, ω0 and ω as rational numbers. Substituting cos(n ωt) = 1 (︀ n 1 )︀ F cos(ωt) X(t) 2 z + zn converts R+Z(t) into a rational function of the complex variable z and transforms the t-integral into a contour integral over the unit circle in the complex plane that can be evaluated by means of the residue the- Figure 8: The quasienergies ϵi , i = 1, 2 for F = 1/2 and ω0 = 1 as functions of the frequency ω in the neighbourhood of the resonance orem. The poles of the rational function to be integrated (2) frequency ωres ≈ 0.355776. ϵ1 (green curve) has been calculated are obtained by using computer algebra software pack- analytically and numerically as described in the text and ϵ2 (red ets as “root objects” (this is the reason to choose rational curve) was chosen as ϵ2 = 3 ω − ϵ1. numbers for F, ω0 and ω). The corresponding residues are exactly evaluated, mostly again in the form of root objects. Both methods agree within the working precision but the numerical results are obtained faster. It may happen We have drawn a couple of branches of the function that the quasienergy determined by these methods jumps ϵ(ω) for fixed F and ω0, see Figures 7 and 8. At the res- (n) into the “wrong branch” and has to be corrected by adding onance frequencies ωres , n = 1, ... , 5 that were calcu- an integer multiple of ω, see the corresponding discussion lated according to the method described in subsection in subsection 2.1. In this way we obtain representations of 5.2 we observe avoided level crossings analogous to those obtained in the literature, see, e.g. [3], Figure 1 or [4], the branches ϵi(F, ω) without any restriction to the values Figure 1. of F and ω0. 722 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

6 Special Limit Cases large, see Assertion 2, leads to ω ω(n) = 0 , for n = 1, 2, ... and F = 0. (183) The introduction of ∆ in subsection 5.2 as the natural res 2n − 1 domain of the arguments of ϵ also clarifies the consider- ation of the various limit cases. We have three limit cases This explains the intersections of the resonance curves ℛn where one of the scaled variables approaches 0 but the with the edge F = 0 of ∆, see Figure 6. other two variables remain finite. These three cases cor- By an analogous reasoning we may also calculate the (n) respond to the three open edges of ∆ and will be consid- first terms of the power series w. r.t. F of ωres for small n ered in the corresponding following subsections. First, the or for small m. The power series has the form: limit case F → 0 is covered by a Fourier-Taylor series solu- ∞ tion for X(t) and ϵ, see subsection 6.1. The second case of ω0 ∑︁ − ω(n) = + σ(n) ω2m 1 F2m . (184) res 2n − 1 2m 0 ω0 → 0 is considered in subsection 6.2 where we have cal- m=1 culated the asymptotic solution X(t) and the quasienergy (n) ϵ up to linear terms in ω0. It is very difficult to extend ω0 Recall that the differences ωres − 2n−1 are traditionally these results to higher orders of ω0 and hence we will con- (n) called “Bloch-Siegert shifts”. The coefficients σ2m of (184) tend ourselves with numerical approximations. Finally, in can be determined as follows: we insert the power series the limit case ω → 0 we have recursively determined the (184) into the expression of det A(N) (for a suitable large terms of an ω-power series for X(t) and explicitly calcu- N) and set the first few coefficients of the resulting power 2 4 lated the first two terms of ϵasy = ϵ0 + ϵ2 ω + O(ω ), see series w. r. t. F to zero. This yields recursive equations from subsection 6.3. (n) which the σ2m may be determined, independent of N. The There are three further “limit cases of the limit cases” corresponding results for n = 1, 2, 3 are contained in the where two of the three scaled variables approach 0 and Tables 1–3. They are in accordance with the three coeffi- the third one necessarily approaches 1. They correspond to cients for n = 1 published in [4] and with the results of the three vertices of ∆ and are not automatically included [9, 10]. Table 4 contains the first non-vanishing coefficients in the previous limit cases where we assumed that one (n) σ2m for n = 1, ..., 10 and m = 1, 2, 3 that were deter- scaled variable approaches 0 but the other two remain (n) mined in the same way. A closed formula for the σ2m is not finite. Consider first the case where the unscaled variable (n) known, but the coefficients σ2 that we have calculated ω approaches ∞ and the other two unscaled variables F, satisfy the recursion relation ω0 remain finite. Then, by (177), the scaled variables F̃︀ and ω0 will approach 0 whereas ω → 1. In this case we 2 ̃︁ ̃︀ (n+1) (n − 1) (n) 7 1 1 σ2 = σ2 + − . (185) have calculated an FT series for X(t) in powers of 풯 ≡ ω n + 1 16(n + 1) 8 and the corresponding power series of the quasienergy, see Section 6.4. The next case of F̃︀, ω̃︀ → 0, or, equivalently ω0 → ∞ Table 1: Coeflcients of the power series (184) for the resonance (1) can be treated either by considering the lowest order of F frequencies ωres. in (243) and (246) or the lowest order of ω in (195) and (196). m σ(1) It follows that both cases yield the same result, see Section 2 2m 1 6.4. 2 16 The last case ω, ω0 → 0 or, equivalently F → ∞, is ̃︀ ̃︁ 1 somewhat subtle as the two limits cannot be interchanged, 4 1024 see the discussion in Section 6.3. It will not be treated in a 35 6 − separate subsection. 131072 103 8 8388608 1873 6.1 Limit Case F → 0 10 805306368 1577257 12 − 6.1.1 Resonance Frequencies 3710851743744 67429531 14 A glimpse of (166) shows that for F = 0 the determinant of 17099604835172352 A(N) vanishes for ω = ω0 , n = 1, 2, .... Hence the res- 304008125947 2n+1 16 (N) 39397489540237099008 onance condition z0 = ϕ1 = det A = 0 for N arbitrarily H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 723

(n) Table 2: Coeflcients of the power series (184) for the resonance Table 4: Coeflcients σ2m of the power series (184) for the resonance (2) (n) frequencies ωres. frequencies ωres for m = 1, 2, 3 and n = 1, ... 10.

m σ(2) n σ(n) σ(n) σ(n) 2 2m 2 4 6 3 1 1 35 2 1 − 32 16 1024 131072 135 3 135 2133 4 − 2 − 8192 32 8192 1048576 2133 5 2125 1146875 6 3 − 1048576 96 221184 254803968 588789 7 12005 120892751 8 4 − 536870912 192 1769472 40768634880 98579025 9 43011 235598949 10 − 5 − 68719476736 320 8192000 104857600000 19157942853 11 118459 10123182707 12 6 − 17592186044416 480 27648000 5573836800000 13 274625 32687521841 7 − Table 3: Coeflcients of the power series (184) for the resonance 672 75866112 21412451450880 frequencies ω(3) . 15 563625 23778534375 res 8 − 896 179830784 18046378835968 m σ(3) 17 1056295 2573069114971 2 2m 9 − 5 1152 382205952 2219118333788160 2 19 1845071 2204002956989 96 10 − 2125 1440 746496000 2128409395200000 4 − 221184 1146875 ω 6 254803968 2.0 3244765625 8 − 1174136684544 n = 1 2045715078125 10 1352605460594688 1.5 558332576171875 12 − 1038800993736720384

(n) Figure 9 shows a couple of resonance curves ωres 1.0 as functions of F together with their F-expansions. The numerical agreement between both curves is excellent but n = 2 limited to bounded values of F. This clearly indicates a finite radius of convergence for the power series (184). 0.5 n = 3

6.1.2 Fourier-Taylor Series

In the Section 6.1.3 we will present a solution of the classi- F cal RPL in terms of so-called Fourier-Taylor (FT) series. A 1 2 3 4 5 few explanations will be in order. An FT series is a Taylor Various resonance frequencies ω(n) (blue curves) for series of a (vector) quantity A(F, t), periodic in t, w. r. t. Figure 9: res ω = 1 as functions of F for n = 1, 2, ..., numerically determined n 0 the parameter F such that each coefficient of F is a finite by the resonance condition det A(20) = 0. The thin continuous Fourier series w. r. t. the time variable t of maximal order n: red/green/cyan curves correspond to the power series (184) according to the results of Tables 1–3 for n = 1, 2, 3. The corresponding ∞ n (n) ∑︁ ∑︁ dashed curves are the asymptotic four terms limits of ωres for A(F, t) = Fn A ei m ω t . (186) nm ω, F → ∞ according to (226)–(228). n=0 m=−n

2π Put differently, the T = ω -periodic function A(F, t) is the lowest order of n = m. Fourier series with components expanded into a Fourier series such that each Fourier coef- that are in turn Laurent series of a suitable parameter are ficient of order m is a Taylor series w. r. t. F that starts with known as “Poisson series” in celestial mechanics, see, e.g. 724 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

ω [38]. FT series are special Poisson series characterised by R = − 0 S + S n,m (︀ 2 2 2)︀ ( n,m n,m+1) ∑︀n 2 (2m + 1) ω − ω0 the restriction m=−n in (186) and have been applied in [39] to a couple of physical problems by utilising com- for 0 ≤ m ≤ n, (194) puter algebraic means. It is possible to consider the more general case where the frequency ω is not given but also where, of course, we have to set Sn,n+1 = 0 in (194). It fol- calculated iteratively in terms of a Taylor series, but this lows that Rn,m(ω, ω0) and Sn,m(ω, ω0) are rational func- generalisation is not needed in the present context of RPL. tions of ω and ω0. It is obvious that sums and products of FT series are We recall that under the transformation (118)– again FT series. More generally, the power series of an FT (122), X, Y and Z remain invariant which entails −2n−1 −2n series is again an FT series, at least in the sense of for- Rnm λ Rnm and Snm λ Snm. Then it eas- mal power series. For practical applications the size of the ily follows that both sides of the recursion relations (193) convergence radius becomes important. and (194) transform in the same way, which can be viewed as a consistency check of the ansatz (187)–(189). We will show the first few terms of the FT series for X(t) t ϵ 6.1.3 FT Series for X( ) and and Z(t):

In the case F = 0 there are only two normalised solu- ω X(t) = −F 0 cos ωt + tions of the classical RPL that are T-periodic for all T > 0, 2 2 ω − ω0 ⊤ namely X(t) = ±(0, 0, 1) . Hence for infinitesimal F we (︃ 2 ω expect that we still have Z(t) = ±1 + O(F ) but (X(t), F3 − 0 cos ωt (︀ 2 2)︀2 Y(t)) will describe an infinitesimal ellipse, i.e. X(t) = 8 ω − ω0 3 3 FA cos ωt + O(F ) and Y(t) = FB sin ωt + O(F ). These )︃ ω considerations and numerical investigations suggest the − 0 cos 3ωt + O(F5), (︀ 2 2)︀ (︀ 2 2)︀ following FT series ansatz: 8 9ω − ω0 ω − ω0

∞ n ∑︁ 2n+1 ∑︁ X(t) = F Rn,m(ω, ω0) cos(2m + 1)ωt, (187) (195) n=0 m=0 ∞ 1 1 ∑︁ 2n+1 Z(t) = 1 + F2 cos 2ωt + Y(t) = F × 2 2 ω0 4(ω − ω0) n=0 n (︃ 2 2 ∑︁ 4 3ω − ω (2m + 1)ω R (ω, ω ) sin(2m + 1)ωt, (188) F 0 cos 2ωt n,m 0 8 (︀ω2 − ω2)︀ 2 (︀9ω2 − ω2)︀ m=0 0 0 ∞ n )︃ ∑︁ 2n ∑︁ 3 Z(t) = F Sn,m(ω, ω0) cos 2mωt. (189) + cos 4ωt + O(F6). (︀ 2 2)︀ (︀ 2 2)︀ n=0 m=0 64 ω − ω0 9ω − ω0 In the ansatz (188) for Y(t) we have already used that Y(t) is completely determined via Y(t) = − 1 d X(t) and (196) ω0 dt hence need not be further considered. The other two differ- We note that the coefficients contain denominators of the 2 2 2 2 ential equations (159) and (160) yield recursion relations form ω − ω0 and 9ω − ω0 due to the denominator 2 2 2 for the functions Rn,m(ω, ω0) and Sn,m(ω, ω0). As initial (2m + 1) ω − ω0 in the recurrence relation (194). Hence conditions we impose the following choices: the FT series breaks down at the resonance frequencies (m) ω0 ωres = 2m−1 . This is the more plausible since according S0,0(ω, ω0) = 1, (190) to the above ansatz z0 = 1 which is not compatible with ω0 R0,0(ω, ω0) = − , (191) the resonance condition z0 = 0, see Assertion 2. (ω − ω0)(ω + ω0) Using the FT series solution (187)–(189) it is a straight- Sn,0(ω, ω0) = 0, for n = 1, 2, ... . (192) forward task to calculate the quasienergy ϵ = a0 as the For n > 0 the FT coefficients Rn,m and Sn,m can be recur- time-independent part of the FT series of (62) which in the sively determined by means of the following relations: present case of RPL assumes the form 1 (︀ Sn,m = − (2m + 1)R (︂ )︂ n−1,m 1 F cos(ωt)X(t) ∑︁ i n ω t 4m ω0 ω + = a + an e . (197) )︀ 2 0 R + Z(t) 0 + (2m − 1)Rn−1,m−1 for 1 ≤ m ≤ n, (193) n∈Z n=/ 0 H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 725

The first few terms of the result are given by neglecting higher order terms. For ω0 = 0 we have the exact “pendulum solution” 2 4 (︀ 2 2)︀ ω F ω F ω0 ω + 3ω ϵ = 0 − 0 + 0 2 8 (︀ω2 − ω2)︀ (︀ 2 2)︀3 0 128 ω − ω0 X(t) = 0, (203) 6 (︁ 6 2 4 4 2 6)︁ F ω0 −5ω0 + 35ω ω0 + 33ω ω0 + ω Y(t) = − sin (f sin ωt) − 5 ∑︁ (︀ 2 2)︀ (︀ 2 2)︀ = −2 Jn(f ) sin nωt, (204) 512 ω − ω0 9ω − ω0 n=1,3,... + O(F8). (198) Z(t) = cos (f sin ωt) = J0(f ) ∑︁ This is in agreement with [4], (29), except for the first term + 2 Jn(f ) cos nωt. (205) which is probably a typo. n=2,4,... It will be instructive to check the first two terms of (198) by using the decomposition of the quasienergy into Here Jn(...) denotes the Bessel functions of first kind and F a dynamical and a geometrical part in Section 3.1. In low- integer order and we have set f ≡ ω as this combination est order in F the classical RPL solution is a motion on permanently occurs in what follows. Equations (204) and F ω0 F ω (205) follow from the Jacobi-Anger expansion. Moreover, an ellipse with semi axes a = 2 2 and b = 2 2 . |ω −ω0| |ω −ω0| Hence the geometrical part of the quasienergy reads we will assume

2 2 ω (︁ 4)︁ F ω ω0 (︁ 4)︁ ϵg = π a b + O F = + O F . 0 < f < π, (206) 4π (︀ 2 2)︀2 4 ω − ω0 (199) in order to avoid problems with the following integrations. The dynamical part is obtained as We note that if a constant x-component X(t) = x0 would be (︀ 2 2)︀ added to the above solution it would still solve (158)–(160) ω0 Z + XF cos ωt ω0 ω0 ω0 − 3ω 2 ϵd = = + F for ω = 0. But only the choice X(t) = x = 0 is suited as 2R 2 (︀ 2 2)︀2 0 0 8 ω − ω0 a starting point for higher orders of ω0. (︁ 4)︁ + O F . The next linear order of the solution of (158)–(160) is (200) obtained by replacing (203) by The sum of both parts together correctly yields X(t) = ω0 X1(t), where (207) 2 ω0 F ω0 (︁ 4)︁ ϵ = ϵ + ϵ = − + O F . (201) X˙ 1(t) = −Y(t) = sin (f sin ωt) . (208) d g 2 2 2 8(ω − ω0) Moreover, A periodic solution of (208) is given by the Fourier series ∂ϵ ω ω ϵ = 0 F2 + O(F4) = g , (202) 2 ∑︁ 1 ∂ω (︀ 2 2)︀2 ω X1(t) = − Jn(f) cos nωt. (209) 4 ω − ω0 ω n n=1,3,... in accordance with Assertion 3. However, it is plausible from (198) that the FT series The radius of the Bloch sphere for this solution is still (m) (︀ 2)︀ for the quasienergy has poles at the values ω = ωres = R = 1 + O ω0 . 1 We want to determine the quasienergy ϵ in linear order 2m−1 , m = 1, 2, ... and hence the present FT series ansatz is not suited to investigate the Bloch-Siegert shift in ω0 which according to (62) reads for small F. We have also found a modified FT series that is (︃ )︃ valid in the neighbourhood of ω(1) but will not dwell upon 1 (︂ h X )︂ res ϵ = ω + 1 (210) this. 2 0 1 + Z (︃ (︂ )︂)︃ (207) ω h X = 0 1 + 1 1 , (211) 2 1 + Z 6.2 Limit Case ω0 → 0

6.2.1 The Classical Equation of Motion where h1 ≡ F cos ωt. For the time average we make the substitution τ = ωt and perform the τ-integration over We reconsider the classical RPL equations of motion (158)– the interval [0, 2π]. This yields a factor 1/ω for each time ω (160) and look for solutions that are at most linear in ω0, integral which is partially compensated by the factor 2π 726 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited due to the time average. Then the time average integral in in accordance with Assertion 3. (211) can be transformed by partial integration into Moreover, it is clear from (221) that the resonance ∂ϵ condition ∂ω = 0, cp. (114), is equivalent to 2π 2π 2π 0 ∫︁ ∫︁ ∫︁ h1 dv 2π du X1 dτ ≡ u dτ = [u v]0 − v dτ. (n) F 1 + Z dτ dτ ω = ωres = , (225) 0 0 0 jn,0 (212) By (208) we have where jn,0 denotes the n-th zero of the Bessel function J0. This yields the intersections of the resonance curves ℛn du 1 du 1 1 = = X˙ 1 = sin (f sin τ) . (213) with the line ω0 = 0, (see Figure 6). Note further that, dτ ω dt ω ω (︁ F )︁ by (205), J0 ω is the Fourier coefficient of Z(t) corre- In order to calculate v we consider the integral (︁ F )︁ sponding to the constant term and hence Z(t) = J0 ω ∫︁ h vanishes exactly in the resonance case, in accordance with v = 1 dτ (214) 1 + Z Assertion 2. ∫︁ cos τ Unfortunately, the integrals occurring in the next, = F dτ. (215) 1 + cos(f sin τ) quadratic and cubic orders in ω0 cannot be solved in closed form and we cannot extend our analysis to this Substituting x = fsinτ, hence dx = f cos τ dτ, we obtain case in a straightforward way. As a way out we return to the Fourier series solution (161)–(163) and the approxi- ∫︁ 1 v = ω dx (216) mate determination of the resonance frequencies by the 1 + cos x solution of det A(50) = 0. From this the asymptotic form of x (︂ f )︂ = ω tan = ω tan sin τ , (217) ω(n) (F) can be obtained by inserting ω = ∑︀3 a F1−2m 2 2 res m=0 m into det A(50) and setting the four highest even orders of F suppressing irrelevant integration constants. As u and v to zero. This yields are 2π-periodic functions the term [u v]2π in (212) vanishes. 0 0.87256 0.404226 By (213) and (217) the remaining integral reads ω(1) (F) ≈ 0.415831 F + + res F F3 3.83313 2π 2π − + O(F−7), (226) ∫︁ du ∫︁ (︂ f )︂ F5 − = − v dτ sin (f sin τ) tan sin τ dτ (218) 0.496818 1.03437 dτ 2 ω(2) (F) ≈ 0.181157F + + 0 0 res F F3 2π 12.9166 ∫︁ − + O(F−7), (227) = − (1 − cos (f sin τ)) dτ (219) F5 0 (3) 0.356526 1.32633 ωres(F) ≈ 0.115557F + + 3 = = −2π (︀1 − J (f ))︀, (220) F F 0 25.278 − + O(F−7). (228) F5 using (205) in the last step. After dividing by 2π due to the τ-average we obtain for (211): The first terms proportional to F are the numerical approximations of the known exact value F , but the next terms ω (︂ F )︂ j0,n ϵ = 0 J . (221) could only be determined numerically. For an alternative 2 0 ω approach see [9, 10]. Figure 9 shows the numerically deter- The decomposition into dynamical and geometrical part of mined resonance curves ℛn together with the approxima- the quasienergy according to Section 3.1 reads tions (226)–(228) for n = 1, 2, 3 that are valid for large F and ω. Recall that according to (177) the limit of the (︂ (︂ )︂)︂ ω0 F F unscaled quantities F, ω → ∞ is equivalent to ω0 → 0 for ϵg = J1 , (222) ̃︁ 2 ω ω the scaled quantity ω̃︁0. ω (︂ (︂ F )︂ F (︂ F )︂)︂ ϵ = 0 J − J . (223) d 2 0 ω ω 1 ω 6.2.2 The Schrödinger Equation Note further that (︂ )︂ For the sake of completeness we will show that the ∂ϵ Fω0 F ϵg = J1 = (224) ∂ω 2ω2 ω ω limit ω0 → 0 can also be considered directly for the H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 727

(0) (1) Schrödinger equation and yields an equivalent result for For t = 0 we have ψ1 = 1 and ψ2 = 0. A second solu- the linear term of the quasienergy series w. r. t. ω0. tion can be obtained that is orthogonal to the first one such It is convenient to consider the Hamiltonian that the resulting unitary evolution operator

(︃ )︃ (︃ (0) (1) )︃ ψ −ψ ω0 2 ^ 1 F cos ωt ω0 U(t) = 1 2 + O(ω ) (238) H = , (229) (1) (0) 0 2 ω0 −F cos ωt ψ2 ω0 ψ1

satisfies (3) with initial condition (4). The corresponding that is unitarily equivalent to the RPL Hamiltonian hith- monodromy matrix reads erto considered. We make the following series ansatz for the solution of the corresponding Schrödinger equation: ⎛ iω ⎞ 1 − 0 J (f)T 2 0 2 ℱ = U(T) = ⎜ ⎟ + O(ω0), ∞ ⎝ iω0 ⎠ ∑︁ − J0(f)T 1 ψ = ψ(2n)ω2n , (230) 2 1 1 0 (239) n=0 ω0 ∞ and has the eigenvalues 1 ± i 2 J0(f ) T. This yields the ∑︁ (2n+1) 2n+1 ω0 3 ψ2 = ψ2 ω0 , (231) quasienergies ϵ = ± 2 J0(f) + O(ω0) in accordance with n=0 (221). One may show that ϵ can be expanded into an odd 3 series w. r. t. ω0, whence the above term O(ω0) for the next and obtain the following system of (in)homogeneous order. linear differential equations:

d F ω → i ψ(0) = cos ωt ψ(0), (232) 6.3 Limit case 0 dt 1 2 1 d 1 F It is plausible that for ω → 0 the classical spin vector X(t) i ψ(2n+1) = ψ(2n) − cos ωt ψ(2n+1), (233) dt 2 2 1 2 2 h(t) follows the magnetic field, i.e. X(t) = ‖h(t)‖ . We will con- d (2n+2) 1 (2n+1) F (2n+2) firm this by calculating the Taylor series expansion of X(t) i ψ1 = ψ2 + cos ωt ψ1 , (234) dt 2 2 w. r. t. ω:

∞ for n = 0, 1, .... The two lowest terms of the series (230) ∑︁ X(t) = X (t) ωn . (240) and (232) can be obtained in a straightforward manner: n n=0

(︂ F )︂ Note that || X(t)||2 has the series expansion (suppressing ψ(0) = exp −i sin ωt , (235) 1 2ω the t-dependence): ⎛ t ⎞ ∫︁ (︂ )︂ (1) i F ′ ′ X · X = X0 · X0 + 2ω X0 · X1 ψ2 = − ⎝ exp −i sin ωt dt ⎠ × 2 ω 2 0 + ω (2X0 · X2 + X1 · X1) (︂ F )︂ 3 exp i sin ωt . (236) + 2ω (X0 · X3 + X1 · X2) 2ω 4 + ω (2X0 · X4 + 2X1 · X3 + X2 · X2)

We could not calculate the integral in (236) in closed form + ... (241) but only in form of a series using again the Jacobi-Anger As the normalisation condition X·X = 1 must hold in each expansion and setting f ≡ F : ω order of ω it follows that X0 · X0 = 1, but X0 · X1 = 0 and all other terms in the brackets of (241) have to vanish. t ∫︁ As the series coefficients X (t) are T-periodic functions (︁ ′)︁ ′ n exp −i f sin ωt dt of t and can be written as Fourier series each differentia- 0 tion of Xn(t) w. r. t. t produces a factor ω and both sides of =J0(f) t the equation of motion ∞ ∑︁ cos((2n + 1)ωt) − 1 + 2 i J (f) d 2n+1 (2n + 1) ω X(t) = h(t) × X(t) (242) n=0 dt ∞ ∑︁ sin(2nωt) are Taylor series in ω. This yields a recursive procedure to + 2 J2n(f ) . (237) 2n ω X n=1 determine n(t). 728 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

0 The ω -terms of (242) yields 0 = h(t)×X0(t). Together the solution X(t). However, the velocity is bounded by ‖h‖ with the normalisation condition this implies (up to a sign) and hence (240) cannot longer hold. For example, we may consider a small fixed value of ω and a finite value of ω ⎛ ⎞ 0 F cos ω t 1 such that X0(t) is a good approximation of the exact solu- X (t) = ⎜ 0 ⎟, (243) 0 √︁ ⎝ ⎠ tion X(t). If we lower ω0 to smaller and smaller values we ω2 + F2 cos2 ω t 0 ω0 would obtain a sudden switch to the behaviour described in Section 6.2 for the limit ω0 → 0. In this sense the two which confirms the above assertion that the classical spin limits ω → 0 and ω0 → 0 cannot be interchanged. vector, up to normalisation, follows the magnetic field. Finally, we will consider the quasienergy for the low- The next order, linear in ω, yields est orders of ω. In the limit ω → 0 the geometrical part of d the quasienergy vanishes as the solution X0(t) is confined X (t) = ω h(t) × X (t). (244) dt 0 1 to the x-z-plane. The dynamical part (77) has the value

This is an inhomogeneous linear equation with the general 1 ϵ = ϵ = (F cos ωt X + ω Z) (250) solution d 2R 0 d h(t) ⎛ ⎞ (︀ 2 2 2 )︀ ω X1(t) = X0(t) × 2 + ω λ1(t) h(t). (245) ω0 + F cos ωt dt ‖h(t)‖ = ⎝ √︁ ⎠ (251) 2 2 2 2 ω0 + F cos ωt The normalisation condition implies X0·X1 = 0 and hence 2π/ω λ1(t) = 0. It follows that ∫︁ √︁ ω 2 2 2 = ω0 + F cos ωt dt (252) (︃ )︃T 4π Fω0 sin(ω t) 0 X1(t) = 0, , 0 . (246) (︀ 2 2 2)︀3/2 (︂ 2 )︂ F cos (ω t) + ω0 ω0 F = E − 2 , (253) π ω0 In the next quadratic order of ω we analogously have

d where E(...) denotes the complete elliptic integral of the X (t) = ω h(t) × X (t), (247) dt 1 2 second kind. Also see [4] for a similar result. In the special case F = 1/2 and ω0 = 1 that is portrayed in Figure 7, we with the general solution have

d h(t) (︀ 1 )︀ ω X (t) = X (t) × + ω λ (t) h(t). (248) E − 2 dt 1 2 2 ϵ → ϵ ≡ 4 ≈ 0.52992 (254) ‖h(t)‖ 0 π

This time the normalisation condition up to the for ω → 0. quadratic order of ω gives 2X0 · X2 + X1 · X1 = 0 and 2 4 The next corrections to ϵ are in the form ϵ2 ω +ϵ4 ω . hence We obtain 2 2 2 1 F ω0 sin (ω t) (︁ 2 )︁ (︁ 2 )︁ λ2(t) = − X1 · X1 = − . (︀ 2 2)︀ F 2 F 2 ‖h‖ (︀ 2 2 2)︀7/2 2F + ω0 E 2 2 − ω0K 2 2 2 F cos (ω t) + ω F +ω0 F +ω0 0 ϵ2 = √︁ , (255) (249) 2 2 2 6πω0 F + ω0 The corresponding X2(t) will not be displayed here. In this way we may recursively determine an arbitrary where K(...) denotes the complete elliptic integral of the number of the term Xn(t). It can be shown by induction first kind. ϵ4 is too complicated and will only be calcu- over n that for odd n the Xn(t) have only a non-vanishing lated for the special values F = 1/2, ω0 = 1. It is plausi- y-component and for even n the y-component vanishes. ble that the asymptotic limit ϵasy of ϵ for ω → 0 does not Hence Xn(t) · Xm(t) = 0 if n and m have different parity. approximate a single branch of the quasienergy but rather n This implies that the pre-factors of ω in (241) vanish and represents a kind of envelope of the various branches, see hence λn(t) = 0 for all odd n. Figure 10. In the special case F = 1/2 and ω0 = 1 that is We note that the Taylor expansion (240) breaks down portrayed in Figure 10 we have for ω0 → 0. This follows already from the observation that ⃦ d ⃦ the velocity ⃦ X0(t)⃦ would assume arbitrary large val- (︀ 1 )︀ (︀ 1 )︀ ⃦ dt ⃦ 3E 5 − 2 K 5 ϵ2 = √ ≈ 0.0272334, (256) ues for ω0 → 0 if (240) would be a correct description of 6 5π H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited | 729

d resulting from the differentiation dt is replaced by 1/풯 . As usual, the condition that the resulting FT series has vanishing coefficients yields linear equations that determine the xn,m , yn,m and zn,m and hence X(t), Y(t) and Z(t) up to any finite order. In lowest non-trivial order the asymptotic form of the solution reads (re-substituting 풯 = 1/ω):

Fω X(t) X(t) = − 0 cos(ω t) + O(ω−4) = , (262) ω2 R F Y(t) Y(t) = − sin(ω t) + O(ω−3) = , (263) ω R F2 Z(t) = 1 + cos(2 ω t) + O(ω−4), (264) Figure 10: A detail of Figure 7 showing various branches of the 4 ω2 quasienergy (continuous coloured curves), the corresponding res- 2 Z(t) F (1 − cos(2ωt)) −4 (n) = 1 − + O(ω ). (265) onance frequencies ωres (vertical dashed lines) and the asymptotic 2 2 4 R 4 ω limit ϵasy = ϵ0 + ϵ2ω + ϵ4ω (blue dashed curve) for ω → 0 according to (254), (256) and (257). We will compare this result with the first terms of the 1/ω-Taylor expansion of the normalised classical RPC and solution X−(t) according to (69): (︀ 1 )︀ (︀ 1 )︀ 203E 5 − 24K 5 (︂ F Fω )︂ ϵ4 = √ ≈ 0.0249063. (257) X(t) = − + 0 cos(ω t) + O(ω−3), (266) 1500 5π ω ω2 (︂ )︂ F Fω0 −3 As ϵasy represents the envelope of the branches of Y(t) = − + 2 sin(ω t) + O(ω ), (267) ϵ the asymptotic form of the resonance frequencies can- ω ω F2 not be determined by the present method. However, the Z(t) = 1 − + O(ω−3). (268) inspection of Figure 6 suggests that for ω → 0 the reso- 2 ω2 nance frequencies are given by an interpolation between Despite some similarities we come to the conclusion the limits for F → 0 and ω0 → 0, namely that both solutions are different, even in the lowest non- vanishing order w. r. t. 1/ω. This is in contrast to the (n) F ω0 ωres ∼ + . (258) view that the rotating wave approximation is an analytical j0,n 2n − 1 approximation to the RPL solution that is asymptotically This approximation is of reasonable quality for small F or valid in the limit of large ω. small ω0 but of poor quality for F ∼ ω0 as there the small According to the FT solution the quasienergy ϵ(ω0, F, curvature of the resonance curves ℛn in the triangular ω) can be calculated as a power series in 1/ω the first terms domain ∆, see Figure 6, should be taken into account. of which are:

2 2 (︀ 2 2)︀ ω F ω F ω0 F − 16ω ϵ(ω , F, ω) = 0 − 0 + 0 +O(ω−6). ω → 0 2 8ω2 128ω4 6.4 Limit Case ∞ (269) 1 This is in accordance with the series expansion of (221) To investigate the limit ω → ∞ we set 풯 ≡ ω and make the following ansatz of an FT series: (︂ )︂ 2 4 ω0 F ω0 F ω0 F ω0 −6 J0 = − 2 + 4 + O(ω ), (270) ∞ n−1 2 ω 2 8ω 128ω ∑︁ n ∑︁ X(t) = 풯 xn,m cos(mωt), (259) n=2,4,... m=1,3,... keeping in mind that (221) holds only in first order in ω0. ∞ n ∑︁ n ∑︁ Y(t) = 풯 yn,m sin(mωt), (260) n=1,3,... m=1,3,... 6.5 Limit Case ω0 → ∞ ∞ n ∑︁ ∑︁ Z(t) = 1 + 풯 n z cos(mωt). (261) n,m As remarked above, due to (177) this limit is equivalent to n=2,4,... m=2,4,... the limit F̃︀ → 0, ω̃︀ → 0 and ω̃︁0 → 1 of the scaled quan- This ansatz is inserted into the classical equations tities. First we will compare the limit of X(t) for ω → 0 of motion (158)–(160) in such a way that each factor ω according to (243) and (246) with FT series expansion (195), 730 | H.-J. Schmidt: The Floquet Theory of the Two-Level System Revisited

(196) of X(t) that holds for F → 0. Note that for the com- the continuity or even analyticity of the quasienergy as a parison the latter one has to be normalised. We obtain the function of one of the parameters ω0, F and ω. As briefly result that both limits coincide if we ignore terms of the mentioned in Section 2.1 the quasienergy can be viewed order O(F3) and O(ω2): as an eigenvalue of the Floquet Hamiltonian defined on an extended Hilbert space. Hence one might invoke the corre- F cos(ωt) X(t) = + O(F3, ω2), (271) sponding theory of analytical perturbations, e.g. Rellich’s ω0 theorem [40] or similar tools, but it is not clear whether the Fω sin(ωt) 3 2 Floquet Hamiltonian satisfies the pertaining conditions. Y(t) = 2 + O(F , ω ), (272) ω0 We have checked our results for simple solvable exam- 2 F (1 + cos 2ωt) ples, but the main intended application is the RPL case. Z(t) = 1 − 2 4ω0 Here our approach leads to certain analytical approxi- +O(F3, ω2). (273) mations that can be conveniently handled by computer- algebraic aids. It is also possible to perform the geomet- In deriving this result we used, of course, a restricted series rical approach for the various limit cases of the RPL. We expansion w. r. t. ω that leaves the terms cos nωt and have compared these results with those known from the sin mωt of the Fourier series intact. literature only in a few cases, as a thorough comparison Analogously, we will compare the asymptotic forms would need too much space, but such a comparison is of the quasienergy for ω → 0 according to (253) and for nevertheless desirable. F → 0 according to (198). Again, we find that both limits Another future task would be the attempt to utilise are compatible and yield the common result: the geometrical approach to obtain examples of the theory of periodic thermodynamics that describe periodically 2 ω0 F driven TLSs coupled to a heat-bath. For recent approaches ϵ = + + O(F3, ω2). (274) 2 8ω0 to this problem, see e.g. [17, 34, 41–44].

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