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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus


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Volume 73, Issue 8

Issues

The Floquet Theory of the Two-Level System Revisited

Heinz-Jürgen Schmidt
Published Online: 2018-06-14 | DOI: https://doi.org/10.1515/zna-2018-0211

Abstract

In this article, we reconsider the periodically driven two-level system especially the Rabi problem with linear polarisation. The Floquet theory of this problem can be reduced to its classical limit, i.e. to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarisation as well as asymptotic formulas for various limit cases.

Keywords: Bloch-Siegert Shift; Floquet Theory; Rabi Problem

1 Introduction

Many physical experiments can be described as the interaction of a small quantum system with electromagnetic radiation. If one tries to theoretically simplify this situation as far as possible, one arrives at a two-level system (TLS) interacting with a classical periodic radiation field. The special case of a constant magnetic field into, say, z-direction plus a circularly polarised field in the x-y-plane was solved eight decades ago by Rabi [1] and has found its way into many text books. This case will be referred to as the Rabi problem with circular polarisation (RPC). Shortly after, Bloch and Siegert [2] considered the analogous problem of a linearly polarised magnetic field, orthogonal to the direction of the constant field (hence forward called Rabi problem with linear polarisation, RPL) and suggested the so-called rotating wave approximation. Moreover, they investigated the shift of the resonance frequencies due to the approximation error of the rotating wave approximation. Since then it was called the “Bloch-Siegert shift”.

In the following decades it was realised [3], [4] that the underlying mathematical problem is an instance of the Floquet theory that deals with linear differential matrix equations with periodic coefficients. Accordingly, analytical approximations for its solutions were devised that were the basis for subsequent research. Especially the seminal paper of Shirley [4] has been until now cited more than 11,000 times. Among the numerous applications of the theory of periodically driven TLSs are nuclear magnetic resonance [5], ac-driven quantum dots [6], Josephson qubit circuits [7] and coherent destruction of tunnelling [8]. On the theoretical level the methods of solving the RPL and related problems were gradually refined to include power series approximations for Bloch-Siegert shifts [9], [10], perturbation theory and/or various limit cases [11], [12], [13], [14], [15], [16] and the hybridised rotating wave approximation [17]. Special analytical solutions of the general Floquet problem of TLS can be generated by the inverse method [18], [19], [20].

There exists even an analytical solution [21] of the RPL and its generalisation to the Rabi problem where the angle between the constant field and the periodic field is arbitrary. This solution bears on a transformation of the Schrödinger equation into a confluent Heun differential equation. A similar approach has been previously applied to the TLS subject to a magnetic pulse [22]. However, the analytical solution is achieved by combining together three different solutions and does not yield explicit solutions for the quasienergy or for the resonance curves. The ongoing research on Heun functions, (see e.g. [23], [24]), might facilitate the physical interpretation of these analytical solutions in the future. To summarise, the problem is far from being completely solved and it appears still worthwhile to further investigate the general Floquet problem of the TLS and to look for more analytical approximations of the RPL.

In this paper we will suggest an approach to the Floquet problem of the TLS via its well-known classical limit, see e.g. [25]. It turns out that, surely not in general, but for this particular problem, the classical limit is already equivalent to the quantum problem. More precisely, we 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 calculated via integrations. Especially, the quasienergy is essentially given by the action integral over one period of the classical solution. This is reminiscent of the semi-classical Floquet theory developed in [26]. In the special case of the Rabi problem with elliptical polarisation (RPE), our approach yields the result that Shirley’s resonance condition is equivalent to the vanishing of the time average of the component of the classical periodic solution into the direction of the constant magnetic field. When applied to the RPL, our approach suggests to calculate the truncated Fourier series for the classical solution and to obtain from this the quasienergy by the recipe sketched above. For various limit cases of the RPL there also exists a classical version that will be analysed and evaluated in order to obtain asymptotic formulas for the quasienergy.

The structure of the paper is as follows. We have three main parts, Generalities, Resonances and Analytical Approximations that refer to problems of decreasing generality: the general TLS with a periodic Hamiltonian, the RPE and the RPL. Moreover, we have four subsections 2.4, 3.2, 4.3 and 3.3, where the explicitly solvable case of the RPC and another solvable toy example is used to illustrate certain results of the first two main parts.

In subsection 2.1 we start with a short account of the well-known Floquet theory of TLS that emphasises the group theoretical aspect of the theory. This aspect is crucial for the following subsection 2.2 where we show how to lift Floquet solutions of the TLS to higher spins s > 1/2. Also this lift procedure has been used before but we present a résumé for the convenience of the readers. The next subsection 2.3 is vital for the remainder of the paper in so far as it reduces the Floquet problem for the TLS to its classical limit. More precisely, the classical equation of motion for a spin X in a periodic magnetic field has, in the generic case, exactly two periodic solutions ± X(t) and the Floquet solutions u±(t) together with the quasienergies ϵ± can be derived from ± X(t). This is the content of Assertion 1. The next Section 3 closely investigates some geometrical aspects of the problem. The Bloch sphere can either be viewed as the set of one-dimensional projections of the TLS or as the phase space of its classical limit. The first view leads to a scenario that was analysed in [27], [28] in the context of the generalised Berry phases. Following this approach, in subsection 3.1, we are led to the splitting of the quasienergy into a geometrical and a dynamical part. The classical mechanics approach in subsection 3.4 shows that the quasienergy is essentially the integral of the Poincaré-Cartan form over one closed orbit. This result is closely connected to the approach to semi-classical Floquet theory in [26].

The second main part on resonances essentially bears on the resonance condition due to Shirley [4]. After a short subsection 4.1 on the quasienergy as a homogeneous function, we show in subsection 4.2 that the resonance condition is equivalent to the vanishing of the time average of the third component of the classical periodic solution X(t) (Assertion 2) and that the slope of the function ϵ(ω), where ω denotes the frequency of the periodic magnetic field, is equal to the geometric part of the quasienergy divided by ω (Assertion 3).

The third main part deals with the analytical approximations to the RPL. If the classical periodic solution X(t) is expanded into a Fourier series, the equation of motion can be rewritten as an infinite-dimensional matrix problem. This is similar to the approach in [3], [14] to the TLS Schrödinger equation. As the involved matrix A and any truncation A(N) of it are tri-diagonal, the determinant of A(N) and all relevant minors can be determined by recurrence relations. Thus we obtain, in subsection 5.1, analytical results for the truncated Fourier series of X(t) that are arbitrarily close to the exact solution. By means of Assertion 1 these analytical approximations can also be used to calculate the quasienergy in subsection 5.3. As expected, we observe different branches and avoid level crossing at the resonance frequencies. The latter can be approximately determined, using Assertion 2, via det A(N) = 0, see subsection 5.2.

The remainder of the paper, Section 6, is devoted to the investigation of various limit cases that often require additional ideas for asymptotic solutions and not simply the evaluation of the truncated Fourier series. The RPL has three parameters, namely the Larmor frequency ω0 of the constant magnetic field into z-direction, the amplitude F of the periodic field into x-direction and the frequency ω of the periodic field. Accordingly, there are the three limit cases where F → 0, see subsection 6.1, ω0 → 0, see subsection 6.2, and ω → 0, see subsection 6.3. Moreover, there are also complementary limit cases where ω → ∞, see subsection 6.4 and ω0 → ∞, see subsection 6.5. The case F → ∞ is somewhat intricate and will be treated in Section 6.3 and not in its own subsection. We want to highlight three features among the various limit cases. First, by using the resonance condition in the form det A(N) = 0 it is a straight-forward task to calculate a finite number of terms of the F-power series for the Bloch-Siegert shifts that can be compared with known results from the literature. Second, for small F it is sensible to expand the Fourier coefficients of X(t) into power series in F. This leads to the so-called Fourier-Taylor series that are defined in-depth in subsection 6.1.2 and also give rise to analytical approximations of the quasienergy within their convergence domain. Finally, the classical RPL equation of motion has an exact “pendulum” solution for ω0 = 0 that can be extended to a solution valid even in linear order w. r. t. ω0. In this order it is also possible to obtain a simple expression for the quasienergy and to solve the Schrödinger equation, see subsection 6.2.2. Hence this limit case seems to be suited for further studies. We close with a summary and outlook in Section 7.

2 Generalities

2.1 Floquet Theory for SU2

It appears that the simplest way to explain the general ideas of the Floquet theory for TLSs is by again proving its central claim. In doing so we will emphasise the group-theoretical aspects of the Floquet theory but otherwise will stick closely to [29].

The Schrödinger equation for this system is of the form

𝗂tψ(t)=H^(t)ψ(t).(1)

Here we have set ℏ = 1 and will assume the Hamiltonian Ĥ(t) to be T-periodic in time,

H^(t+T)=H^(t),(2)

where throughout in this paper T=2πω>0, (1) gives rise to a matrix equation for the evolution operator U(t, t0) that reads

𝗂tU(t,t0)=H^(t)U(t,t0),(3)

with the initial condition

U(t0,t0)=𝟙.(4)

We will assume that U(t, t0) ∈ SU2, the Lie group of unitary 2 × 2-matrices with unit determinant. Consequently, the Hamiltonian Ĥ(t) has to be chosen such that 𝗂 Ĥ(t) lies in the corresponding Lie algebra su2 of anti-Hermitean 2 × 2-matrices with vanishing trace, closed under commutation [ , ]. The relation between (1) and (3) is obvious: If ψ02 and U(t, t0) is the unique solution of (3) with initial condition (4), then ψ(t) ≡ U(t, t0) ψ0 will be the unique solution of (1) with initial condition ψ(t0) = ψ0. Conversely, let ψ1(t) and ψ2(t) be the two solutions of (1) with initial conditions ψ1(t0)=(10) and ψ2(t0)=(01), then U(t,t0)(ψ1(t),ψ2(t)) will solve (3) and (4).

Further, it follows that any other solution U1(t, t0) of (3) with initial condition U1(t0, t0) = V0 will be of the form

U1(t,t0)=U(t,t0)V0.(5)

As a special case of (5), we consider

U2(t,t0)U(t+T,t0),(6)

which due to (2), also solves (3) but has the initial condition

U2(t0,t0)=U(t0+T,t0).(7)

Hence (5) implies

U2(t,t0)=U(t+T,t0)=U(t,t0).(8)

∈ SU2 is called the “monodromy matrix”. It can be written as

=e𝗂TF,𝗂Fsu2.(9)

Now we define

𝒫(t,t0)U(t,t0)e𝗂(tt0)F(10)

and will show that 𝒫 is T periodic in the first argument:

𝒫(t+T,t0)=U(t+T,t0)e𝗂(t+Tt0)F(11)

=U(t+T,t0)e𝗂TFe𝗂(tt0)F(12)

=(8,9)U(t,t0)1e𝗂(tt0)F(13)

=𝒫(t,t0).(14)

Summarising this, we have shown that the evolution operator U(t, t0) can be written as the product of a periodic matrix and an exponential matrix function of time, i.e.

U(t,t0)=𝒫(t,t0)e𝗂(tt0)F,(15)

which is essentially the Floquet theorem for TLSs. Equation (15) is also called the “Floquet normal form” of U(t, t0). For an example where an explicit solution for U(t, t0) is possible for some limit case, see also subsection 6.2.2.

The derivation of (15) can be easily generalised from SU2 to any other finite-dimensional matrix Lie group with the property that the exponential map from the Lie algebra to the Lie group is surjective, as this has been implicitly used in (9).

The matrix F is Hermitean and hence has an eigenbasis |n,n=1,2 and real eigenvalues ϵn such that

F=ϵ1|11|+ϵ2|22|.(16)

In this eigenbasis, (15) assumes the form

U(t,t0)|n=𝒫(t,t0)e𝗂(tt0)F|n(17)

=𝒫(t,t0)e𝗂(tt0)ϵn|n(18)

=𝒫(t,t0)|ne𝗂(tt0)ϵn(19)

un(t,t0)e𝗂(tt0)ϵn,(20)

in which the latter functions are called the “Floquet functions” or “Floquet solutions of (1)” and the eigenvalues ϵn of F are called “quasienergies”, see [29]. For the TLS, we have exactly two quasienergies ±ϵ such that ϵ ≥ 0 as Tr F = 0. It follows that any solution ψ(t) of (1) with initial condition ψ(t0)=a1|1+a2|2 can be written in the form

ψ(t)=U(t,t0)ψ(t0)=n=12anun(t,t0)e𝗂(tt0)ϵn,(21)

with the time-independent coefficients an. In this respect un(t, t0), resp. ϵn, generalise the eigenvectors, resp. eigenvalues, of a time-independent Hamiltonian Ĥ. The latter is trivially T-periodic for every T > 0 hence also in this case the Floquet theorem (15) must hold. Indeed it does so with 𝒫(t, t0) = 𝟙 and F = Ĥ.

We remark that the mere analogy between Floquet solutions and eigenvectors can be given a precise meaning by considering the “Floquet Hamiltonian” K defined on the extended Hilbert space L2[0,T]2, see [29] such that the quasienergies are recovered as the eigenvalues of K. This was already anticipated in [3], [4], but we will not go into the details as the extended Hilbert space will not be used in the present paper.

In this account of Floquet theory we have stressed the dependence of the various definitions of the choice of an arbitrary initial time t0. It, hence, remains to investigate the effect of changing from t0 to some other initial time t1. A straightforward calculation using the semi-group property of the evolution operator

U(t,t0)=U(t,t1)U(t1,t0)(22)

gives the result

U(t,t1)=𝒫(t,t0)𝒫(t1,t0)1exp(𝗂(tt1)𝒫(t1,t0)F𝒫(t1,t0)1)(23)

𝒫(t,t1)exp(𝗂(tt1)F~).(24)

It follows that the eigenvalues of F~𝒫(t1,t0)F𝒫(t1,t0)1 and F coincide, hence the change of the initial time will modify the Floquet functions but not the quasienergies. In concrete applications there will often be a natural choice for t0 and the dependence on t0 may be suppressed without danger of confusion.

We will add a few remarks on the uniqueness of the quasienergies ϵn. It is often argued that the quasienergies are only unique up to integer multiples of ω, see, e.g. [29]. It seems at first glance that in our approach uniqueness is guaranteed by the requirement 𝗂 F ∈ su2. For example, the replacement ϵnϵn + ω in (16) would result in FF + ω 𝟙 and violate the condition Tr F = 0. But this uniqueness is achieved by using a complex arg-function with a discontinuous cut. Consider, for example, a smooth 1-parameter family of monodromy matrices (ω) and the corresponding family ϵ1(ω) of quasienergies. It may happen that exp(𝗂Tϵ1(ω)) crosses the cut and hence ϵ1(ω) changes discontinuously. But this discontinuity is not a physical effect and only due to the choice of the arg-function. It could be compensated by, say, passing from ϵ1(ω) to ϵ1(ω) + ω. In this case it would be more appropriate to consider, say, ϵ1(ω) and ϵ1(ω) + ω as physically equivalent quasienergies. Generally speaking, the issue of continuity is an argument in favour of considering the quasienergies modulo ω.

2.2 Lift to Higher Spins

A possible physical realisation of the TLS with Hilbert space 2 is a single spin with spin quantum number s = 12. Even if the TLS is realised in a different way it will be convenient to adopt the language of spin systems. For example, the Lie algebra su2 is spanned by the Pauli matrices σi, i = 1, 2, 3 (times 𝗂) or equivalently, by the three spin operators s^i12σi,i=1,2,3 (times 𝗂). Consequently, the Hamiltonian H^(t) can always be construed as a Zeeman term with a time-dependent dimensionless magnetic field h(t) namely,

H^(t)=h(t)s^i=13hi(t)s^i,(25)

where, following the usual convention, we have omitted a minus sign. We will outline the procedure of lifting a solution of the Schrödinger equation for a spin with s=12 in a time-dependent magnetic field to a solution of the corresponding Schrödinger equation for general s. For this lift the T-periodicity of h(t) is not necessary, but it will later be used to draw conclusions about the Floquet state of the system with general s. The lift procedure is less known but has been already applied in 1987 to the problem of an N-level system in a periodic laser field [30]. Also see [31] for a more recent application of the lift procedure to the problems of Landau-Zener transitions in a noisy environment. For the Bloch-Siegert shift in s = 1 systems see also [32].

Let, as in Section 2.1, tU(t, t0) be a smooth curve in SU2, such that U(t0, t0) = 𝟙 . It follows that

(tU(t,t0))U(t,t0)1𝗂H^(t)su.2(26)

Hence the columns ψ1(t),ψ2(t) of U(t, t0) are linearly independent solutions of the Schrödinger equation (1). Next we consider the well-known irreducible Lie algebra representation (shortly called “irrep”), see, e.g. [33] Chapter 10–4,

r(s): su2su2s+1(27)

of su2 parametrised by the spin quantum number s such that 2s and the corresponding irreducible group representation (also called “irrep”)

R(s):SU2SU.2s+1(28)

It follows that

r(s)(𝗂s^i)=𝗂S^i,i=1,2,3,(29)

where the s^i is defined above as the three s=12 spin operators and the S^i denotes the corresponding spin operators for general s.

It follows from (29) that

r(s)(𝗂H^(t))=𝗂h(t)S^𝗂i=13hi(t)S^i.(30)

Hence

U(t,t0)(s)R(s)U(t,t0)(31)

will be a matrix solution of the lifted time evolution equation

𝗂tU(t,t0)(s)=h(t)S^U(t,t0)(s).(32)

Note that U(t, t0)(s) is a unitary matrix and hence its columns span the general (2s + 1)-dimensional solution space of the lifted Schrödinger equation.

Next we will use that the Hamiltonian is a T-periodic function of time, i.e. that (2) holds. Consequently, we can apply the irrep R(s) to the Floquet normal form (15) of U(t, t0) and obtain

U(t,t0)(s)=R(s)(𝒫(t,t0))R(s)(e𝗂(tt0)F)(33)

𝒫(t,t0)(s)e𝗂(tt0)F(s),(34)

where

𝗂F(s)r(s)(𝗂F).(35)

Recall that the eigenvalues of F are of the form ±ϵ where ϵ ≥ 0 is the quasienergy of the TLS. Moreover, as 𝗂 F ∈ su2, F can be written in the form

F=2ϵi=13fis^i,(36)

such that i=13|fi|2=1. and hence s^=i=13fis^i is the s=12 spin operator into the direction (f1,f2,f3). From this it follows that

F(s)=2ϵi=13fir(s)(s^i)=(29)2ϵi=13fiS^i.(37)

As S^=i=13fiS^i is the general s spin operator into the direction (f1,f2,f3) with eigenvalues m = −s, …, s it further follows that the eigenvalues of F(s) and hence the quasienergies of the lifted Schrödinger equation are of the form

ϵm(s)=2ϵm,m=s,,s.(38)

Also the Floquet functions for the lifted problem can be obtained from those of the TLS, and hence the general solution of the lifted Schrödinger equation can be reduced to the general solution of (1).

2.3 Lift to SO3

We will consider the lift of the two-level problem to the three-level problem with spin s = 1 with more details. Till the end we will not directly use the irrep R(1) but some other well-known representation R that is, however, unitarily equivalent to R(1). It is defined by

us^iu*=j=13R(u)ijs^j,foralli=1,2,3andusu2,(39)

and can be restricted to an irrep R:SU2SO3. The corresponding Lie algebra irrep r:su2so3 maps 𝗂s^i,i=1,2,3 onto three anti-symmetric real matrices that span so3. Let

H(t)(0h3(t)h2(t)h3(t)0h1(t)h2(t)h1(t)0),(40)

then the lifted evolution equation can be written as

tR(t,t0)=H(t)R(t,t0),(41)

where, as usual, R(t, t0) ∈ SO3 and R(t0, t0) = 𝟙 . The underlying “Schrödinger equation” has the form

ddtX(t)=h(t)×X(t),(42)

with X(t)3 and can simultaneously be considered as the classical limit of the lifted Schrödinger equation for s → ∞. A more direct derivation of (42) can be given in the following way.

Let ψ(t) be a solution of the Schrödinger equation (1) and

P(t)=|ψ(t)ψ(t)|(43)

be the corresponding 1-dimensional projector. It can also be written in the form

P(t)=12𝟙+i=1,2,3Xi(t)s^i.(44)

We will write X(t)=(X1(t),X2(t),X3(t)) and note that 0=detP(t)=14(X(t)21). Hence the 1-parameter family P(t) corresponds to a curve X(t) on the “Bloch sphere”, defined by X(t)2=1.

The projector P(t) satisfies the von-Neumann equation

ddtP(t)=𝗂[H^(t),P(t)].(45)

Using the commutation relations of the spin operators ŝi one easily derives from (45) the differential equation (42) for X(t). In the special case where ψ(t)=ψ1(t)=u1(t,t0)e𝗂(tt0)ϵ1 is a Floquet solution of (1) the curve X(t) is a T-periodic solution of (42) and can be visualised as a closed trajectory on the Bloch sphere. The second Floquet solution ψ2(t)=u2(t,t0)e𝗂(tt0)ϵ2 is orthogonal to ψ2(t) for all t and hence must correspond to the “antipode” periodic solution − X(t) of (42).

It will be instructive to check the consistency of our representation by directly applying the Floquet theory to (41). The corresponding monodromy matrix (1)=R(t0+T,t0)=exp(𝗂TF(1)) has as any matrix in SO3, three eigenvalues of the form 1,e𝗂ρ,e𝗂ρ. Consequently, F(1) has the eigenvalues 0,+ρ/T,ρ/T which is in accordance with (38) if 2ϵ = ρ/T. We note in passing that these considerations suggest a simple numerical procedure to determine the quasienergy ϵ: solve (41) numerically over one period T and find the eigenvalues of the corresponding monodromy matrix (1).

As a byproduct of the Floquet theory for SO3 we will prove the existence of periodic classical solutions of (42). Let X0 be the eigenvector of the monodromy matrix (1)=R(t0+T,t0) corresponding to the eigenvalue 1. If we use X0 as the initial value X(t0) of a solution of (42) we conclude

X(t+T)=R(t+T,t0)X0(46)

=(15)𝒫(1)(t+T,t0)e𝗂(t+Tt0)F(1)X0(47)

=(9)𝒫(1)(t,t0)e𝗂(tt0)F(1)(1)X0(48)

=𝒫(1)(t,t0)e𝗂(tt0)F(1)X0(49)

=(15)R(t,t0)X0(50)

=X(t).(51)

This means that this special solution X(t) will be T-periodic. We may hence ask whether it can be obtained as the lift of a Floquet solution of the Schrödinger equation, and, if so, how this Floquet solution can be reconstructed from X(t).

To this end we will start with a given T-periodic solution X(t) of (42) and want to construct a corresponding Floquet solution of (1). It is not necessary to assume the condition X(t)2=1 from the outset. We may rather use the fact that (42) admits the constant of motion

R2=X1(t)2+X2(t)2+X3(t)2,(52)

and hence the solutions of (42) are trajectories on the Bloch sphere of radius R. Then the T-periodic 1-parameter family P(t) of one-dimensional projectors defined by

P(t)12R(R+X3(t)X1(t)𝗂X2(t)X1(t)+𝗂X2(t)RX3(t)),(53)

satisfies the von-Neumann equation (45) that is equivalent to (42). As P(t) is a projector and hence satisfies P(t)2 = P(t), each non-vanishing column of P(t) will be an eigenvector of P(t) corresponding to the eigenvalue 1. After normalising we thus obtain the T-periodic one-parameter family of vectors

φ(t)=12R(R+X3(t))(R+X3(t)X1(t)+𝗂X2(t)),(54)

such that |φ(t)φ(t)|=P(t). We note in passing that (54) is undefined at the south pole of the Bloch sphere where X3(t) = −R. We cannot expect that φ(t) is already a solution of the Schrödinger equation (1) but only that φ(t) differs from a solution ψ(t) of (1) by a time-dependent phase factor e𝗂α(t). After some calculations using (42) we obtain

(H^(t)𝗂ddt)φ(t)=12(h3(t)+h1(t)X1(t)+h2(t)X2(t)R+X3(t))φ(t)(55)

χ(X(t))φ(t)=nane𝗂nωtφ(t),(56)

where the infinite sum in (56) represents the Fourier series of the T-periodic function tχ(X(t)). By integrating this Fourier series over t we obtain

α(t)a0t+n0nane𝗂nωt𝗂nω,(57)

(neglecting an additional integration constant that would only yield a constant phase factor) and further

(H(t)𝗂ddt)ψ(t)=0,(58)

where

ψ(t)exp(𝗂α(t))φ(t)(59)

=exp(𝗂n0nane𝗂nωt𝗂nω)φ(t)exp(𝗂a0t)(60)

u(t)exp(𝗂ϵt).(61)

According to (61) and (58), ψ(t) is indeed a Floquet solution of (1) with quasienergy ϵ = a0 modulo ω since u(t) is T-periodic. The quasienergy ϵ is the time average of χ( X(t)) denoted by an overbar:

ϵ=a0=12(h3(t)+h1(t)X1(t)+h2(t)X2(t)R+X3(t))¯.(62)

Thus we have proven the following:

Assertion 1

There exists a 1:1 correspondence between T-periodic solutions X(t) of (42) such that X(t)2=1 and Floquet solutions ψ(t)=u(t)exp(iϵt) of (1) satisfy the following conditions:

(i) If ψ(t) is a Floquet solution of (1) then

|ψ(t)ψ(t)|=12(1+X3(t)X1(t)𝗂X2(t)X1(t)+𝗂X2(t)1X3(t)),(63)

(ii) If X(t) is a normalised T-periodic solution of (42) then ψ(t)=u(t)exp(𝗂ϵt) will be a Floquet solution of (1) where

u(t)=exp(𝗂n0nane𝗂nωt𝗂nω)×12(1+X3(t))(1+X3(t)X1(t)+𝗂X2(t)),(64)

ϵ=12(h3(t)+h1(t)X1(t)+h2(t)X2(t)1+X3(t))¯,(65)

and the an in (64) are the Fourier coefficients of the T-periodic function tχ(X(t)) defined in (55) and (56).

2.4 The RPC Example I

We will check the results of Assertion 1 for the exactly solvable case of the circularly polarised Rabi problem (RPC) where

h(t)=(FcosωtFsinωtω0).(66)

We obtain T=2πω-periodic solutions X(t) of (42) by the following argument. Obviously,

dhdt=(ωFsinωtωFcosωt0)=(00ω)×(FcosωtFsinωtω0)𝝎×h.(67)

We set X(t)=h(t)𝝎 and conclude

dXdt=dhdt=(67)𝝎×h=h×(𝝎)=h×(h𝝎)=h×X,(68)

and analogously for X(t)=𝝎h(t). Hence one finds two T-periodic solution of (42) of the form

X±(t)=±(FcosωtFsinωtω0ω).(69)

In this case the function χ( X(t)), see (56), turns out to be time-independent which directly yields the quasienergies

ϵ±=12(ω±Ω),(70)

where Ω is the Rabi frequency

ΩR=F2+(ω0ω)2.(71)

Moreover, the corresponding two Floquet solutions ψ±(t) of the Schrödinger equation (1) can be obtained by (60) with the result:

ψ+(t)=(ω+Ω+ω0e12𝗂t(ω+Ω)2ΩFe12𝗂t(ωΩ)2Ω(ω+Ω+ω0)),(72)

ψ(t)=(ω+Ωω0e12𝗂t(ωΩ)2ΩFe12𝗂t(ω+Ω)2Ω(ω+Ωω0)),(73)

in accordance with the well-known result, see, e.g. [34].

3 Geometry of the Two-Level System

The correspondence between T-periodic solutions X(t) of (42) and Floquet solutions ψ(t) of (1) that has been formulated in Assertion 1 can be further analysed w. r. t. two different geometric perspectives: either the map ψ(t) ↦ X(t) can be viewed as the restriction of the map of a Hilbert space onto the corresponding projective Hilbert space P() and the quasienergy as a phase change during a cyclic quantum evolution in the sense of [27]. Or the Bloch sphere can be construed as the phase space of the classical limit of the TLS and the quasienergy can be related to its semi-classical limit in the sense of [26]. We will treat both aspects in the following subsections.

3.1 Geometry of the Fibre Bundle π:C|1|2𝓢2

The map of wave functions to projectors ψ|ψψ| can be viewed as a surjective map π from the unit sphere of |1|2 of 2 to the unit sphere of 3, i.e. as π:|1|2𝒮2. Since |1|2𝒮3 this is essentially the Hopf fibration [35]. The fiber π−1( X) consists of the set of all phase factors eiα and can hence be identified with the one-dimensional unitary group U1. Then the map σ:Xφ(t) according to (54) can be viewed as a local section of the principal fiber bundle π:|1|2𝒮2 with structure group U1. The above remark that σ is undefined at the south pole of the Bloch sphere means that σ cannot be extended to a global section due to topological obstacles. The group SU2 operates in a natural fashion on |1|2 as well as on 𝒮2 via rotations R(u) defined in (39)

Following [27] we may split the function

χ(X(t))=φ(t)|(H^(t)𝗂ddt)|φ(t),(74)

see (56), into a “dynamical” and a “geometrical part”. The dynamical part is defined as

χd(X(t))φ(t)|H^(t)|φ(t)(75)

=12R(h1X+h2Y+h3Z)(76)

and represents the expectation value of the energy. Its time average yields the dynamical part of the quasienergy:

ϵdχd(X(t))¯=12R(h1X+h2Y+h3Z)¯.(77)

The geometrical part of χ( X(t)) is defined as

χg(X(t))φ(t)|𝗂ddtφ(t)(78)

=X(t)Y˙(t)Y(t)X˙(t)2R(R+Z(t)).(79)

Using spherical coordinates θ, ϕ for the Bloch sphere with radius 1, we may write the differential χg dt in the form

χg(X(t))dt=XdYYdX2R(R+Z)(80)

=1cosθ2dϕα.(81)

This yields a differential 1-form α on the Bloch sphere and the time average of χg(X(t)) is, up to the factor 1T=ω2π, the integral of α over the closed curve 𝒞 parametrised by X(t). By applying Stoke’s theorem we obtain

ϵg=χg(X(t))¯=ω2π𝒞α(82)

=ω2π𝒜dα=ω4π|𝒜|,(83)

where 𝒜 denotes the oriented area enclosed by 𝒞 and |𝒜| the correspondingly signed solid angle. Here we have used that the 2-form

dα=12sinθdθdϕ(84)

equals 12 times the surface element on the unit sphere.

Thus we obtain the following interpretation of the quasienergy ϵ = ϵd + ϵg as composed of two parts: the dynamical part ϵd is the time average of the energy (76) and the geometrical part ϵg is ω4π times the solid angle enclosed by the T-periodic solution of (42).

We refer to [27] for the differential-geometric background of this scenario. Obviously, 1-form α is the retract of the canonical connection 1-form γ of the principal fiber bundle π:|1|2𝒮2 w. r. t. the local section σ:𝒮2|1|2 described above. Hence the non-vanishing of ϵg is due to the curvature of this principal fiber bundle and generalises Berry’s phase. Recall further that, by definition, γ has values in the Lie algebra of U1. This Lie algebra is isomorphic to and an analogous identification has been made by considering above α as a usual real-valued 1-form. However, ϵg as the integral over α should properly have values in U1 and not in and this subtle difference is in turn in accordance with the general claim that quasienergies are only defined modulo ω.

3.2 The RPC Example II

We will illustrate the results of the preceding subsection by the explicitly solvable case of RPC. The calculation is essentially identical with that in [27]. It follows from (66) and (69) that the dynamical part of the quasienergy assumes the value (note that the involved functions are constant and taking the time average is superfluous)

ϵd=h1X+h2Y+h3Z2R=F2+ω0(ω0ω)2F2+(ω0ω)2.(85)

On the other hand, the vector X+(t) prescribes a circle on the Bloch sphere with constant Z = ω0ω, see (69). Consider first the case of Z > 0. The corresponding spherical segment has an area of 2πR(RZ) corresponding to a solid angle of |𝒜|=2π(RZ)R, where R=Ω=F2+(ω0ω)2 is the radius of the Bloch sphere. Hence

ϵg=ω4π2π(RZ)R=ω(Rω0+ω)2R,(86)

and

ϵ+=ϵd+ϵg=F2+ω0(ω0ω)2R+ω(Rω0+ω)2R(87)

=12(ω+R),(88)

in accordance with (70). For the second periodic solution X(t) it follows that both terms ϵd and ϵg acquires a minus sign, the latter as the spherical segment encircled by X(t) has a negative orientation. Hence ϵ=ϵ+modω in accordance with (70).

In the case of Z < 0 the dynamical part ϵd remains unchanged whereas the solid angle of the spherical segment encircled by X+(t) assumes the form |𝒜|=2π(R+Z)R. This yields ϵ+=12(ω+R) which differs from (88) only by −ω and is thus equivalent. Analogous arguments hold for ϵ.

3.3 Another Solvable Example

The idea of a “reverse engineering of the control” h(t), see [18], [19], [20], can also be applied to the classical Floquet problem. Given a normalised T-periodic function X(t) we may choose h(t)X(t)×X˙(t) such that (42) is satisfied. This special choice of h entails hX=0 and hence the dynamical part χd of χ vanishes according to (76). The geometrical part χg = χ will be calculated according to (79) for the following example:

X(t)=(cos(ωt)sin(fsin2(ωt2))sin(ωt)sin(fsin2(ωt2))cos(fsin2(ωt2))),(89)

f being a real parameter, which leads to

h(t)=(12ω(fsin2(ωt)+cos(ωt)sin(ffcos(ωt)))12ωsin(ωt)(fcos(ωt)sin(ffcos(tω)))ωsin2(fsin2(ωt2))),(90)

and

χ(t)=ωsin2(12fsin2(ωt2)),(91)

see Figure 1 visualising the example f = 1, ω = 1. The Fourier series of χ(t) can be explicitly calculated:

χ(t)=ϵ+n=1bncosnωt,(92)

where

ϵ=ω2(1cos(f2)J0(f2)),(93)

bn=ωJn(f2){(1)n+12sin(f2):n odd,(1)n+22cos(f2):n even,.(94)

Here the Jn(…) denote the Bessel functions of the first kind and integer order. The constant term of (92) is the quasienergy ϵ according to (93) that only depends linearly on ω. The Fourier series of χ(t) could be utilised to explicitly determine the Floquet solutions u±(t)e𝗂ϵt of the corresponding Schrödinger equation according to Assertion 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 trajectory on the Bloch sphere (blue curve). The corresponding magnetic field h(t) according to (90) is visualised by the red curve. At the time t = π the three vectors X(π)\(\textbf{X}(\pi)\), h(π)\(\textbf{h}(\pi)\) and X˙(π)=h(π)×X(π)\(\dot{\textbf{X}}(\pi)=\textbf{h}(\pi)\times\textbf{X}(\pi)\) are indicated by the coloured arrows.
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 trajectory on the Bloch sphere (blue curve). The corresponding magnetic 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) are indicated by the coloured arrows.

Note further that Assertion 3 of Section 4.2 can be sharpened to ϵω=ϵω as ϵd=0 in this example.

3.4 Classical Mechanics of the Two-Level System

First we will introduce some concepts of classical mechanics suited for the present case. Let z and φ be coordinates of the unit Bloch sphere defined by

X/R=1z2cosφ,(95)

Y/R=1z2sinφ,(96)

Z/R=z.(97)

Further we define the classical Hamiltonian

H=1R(h1X+h2Y+h3Z)(98)

=1z2(h1cosφ+h2sinφ)+h3z,(99)

and rewrite the differential equation (42) in terms of the two functions z(t) and φ(t):

z˙=1z2(h1sinφh2cosφ)=Hφ,(100)

φ˙=h3z1z2(h1cosφ+h2sinφ)=Hz.(101)

Note that due to (76), H is twice the expectation value of the Hamiltonian Ĥ. Obviously, (100)–(101) can be viewed as Hamiltonian equations of motions in a two-dimensional phase space isomorphic to 𝒮2 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 space 𝒫=𝒮2× with coordinates p, q, t and the Poincaré-Cartan form

𝜶pdqHdt(103)

defined on 𝒫 (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 on 𝒫 that is given by the unique null direction field of the exterior derivative of the Poincaré-Cartan form

𝝎d𝜶=dpdqdHdt,(104)

see Section 44 of [36] for the details.

Periodic solutions of (100), (101) correspond to curves γ in 𝒫 that are not closed as after one period T the coordinate t has changed from t = 0 to t = T. This can be repaired by defining another extended phase space 𝒫 via identifying points with t-coordinates that differ by an integer multiple of T, or, more formally,

𝒫𝒮2×/T.(105)

Of course, /T is isomorphic to 𝒮1. Locally the manifolds 𝒫 and 𝒫 are isomorphic, but globally they are not. The differential forms 𝜶 and 𝝎 can be transferred to 𝒫 as all functions involved in 𝜶 and 𝝎 are T-periodic. Now periodic solutions of (100), (101) correspond to closed curves γ in 𝒫.

Next we will rewrite the expression (62) for the quasienergy ϵ. By (76) and (81) we obtain

χddt=12Hdt,(106)

χgdt=12(1z)dφ=12(1p)dq.(107)

Hence

ϵ=χg+χd¯=ω4πγ(pdqHdt)+n2ω,(108)

where the n in the last term denotes the winding number of γ around the z-axis. This result is in close analogy to equation (2.35) of [26] that represents the semi-classical limit of the quasienergy for integrable Floquet systems. It thus seems that for the TLS, similar as in the case of the driven harmonic oscillator, the semi-classical limit of the quasienergy and the exact quantum-theoretical expression coincide. However, it has not yet been shown that the quantisation procedure adopted in [26] yields the quantum TLS when starting from its classical limit.

4 Resonances

In this and the following sections we will restrict the Hamiltonian (25) to the following special case

h1=Fcosωt,(109)

h2=Gsinωt,(110)

h3=ω0,(111)

that will be referred to as the RPE. It includes the two limit cases G → 0, the RPL, and GF, the RPC. Hence the 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 have positive values.

The corresponding classical Hamiltonian (99) reads

H(z,φ)=1z2(Fcosωtcosφ+Gsinωtsinφ)+ω0z.(112)

If the TLS is coupled to a second weak electromagnetic field there may occur transitions between the two different Floquet states, analogously as in the case of two energy levels for a time-independent Hamiltonian. Shirley has computed the time-averaged probability P¯ of such a transition, see (26) in [4], with the remarkably simple result

P¯=12(14(ϵω0)2).(113)

Although Shirley’s derivation of (113) refers to the RPL case, see (1) in [4], one can easily check that it also holds in the more general RPE case. It implies that the transition probability assumes its maximal value P¯max=12 for

ϵω0=0.(114)

Hence the condition (114) will be called the “resonance condition”. It will be further analysed in the following subsections.

4.1 Homogeneity of the Quasienergy

For the RPE the classical equation of motion (42) reduces to

X˙=Gsin(ωt)Zω0Y,(115)

Y˙=ω0XFcos(ωt)Z,(116)

Z˙=Fcos(ωt)YGsin(ωt)X.(117)

It is invariant under the transformation

ω0λω0,(118)

FλF,(119)

GλG,(120)

ωλω,(121)

t1λt,(122)

for all λ > 0. Under this transformation the quasienergy (62) scales with λ and hence is a positively homogeneous function of degree 1:

ϵ(λω0,λF,λG,λω)=λϵ(ω0,F,G,ω)(123)

for all λ > 0. This could be used to eliminate the variable ω0 (by choosing λ=ω01 which transforms ω0 into 1) but for some purposes, e.g. the investigation of the limit ω0 → 0 or the analysis of the resonance condition (114), the elimination of ω0 is not appropriate.

By the Euler theorem the positive homogeneity of ϵ implies

ϵ(ω0,F,G,ω)=ω0ϵω0+FϵF+GϵG+ωϵω.(124)

4.2 Calculation of ϵ

We first consider ϵω0. For the sake of simplicity we assume that the classical equations of motion (115)–(117) for the RPE has two unique normalised T-periodic solutions of the form ± X(t). The latter assumption is, for example, violated in the case ω0 = 0, see Section 6.2, where we have a 1-parameter family of periodic solutions and a quasienergy ϵ = 0. Under these assumptions we will prove the following:

Assertion 2

ϵω0=Z(t)2R¯.(125)

Hence the resonance condition ϵω0=0 is equivalent to Z(t)¯=0.

For the proof of this assertion we will adopt the language of classical mechanics introduced in 3.4. In order to calculate ϵω0 we have to vary the parameter ω0. To this end we define still another extension of the phase space by

𝒫′′𝒮2×/T×>0,(126)

with coordinates p, q, t and ω0 > 0. Again the differential forms 𝜶 and 𝝎 can be transferred to 𝒫′′ but instead of d𝜶=𝝎 we now have

d𝜶=𝝎Hω0dω0dt=(112)𝝎zdω0dt.(127)

The closed curves γ corresponding to periodic solutions of (100), (101) smoothly depend on ω0 and hence will be denoted by γ(ω0). Geometrically, this defines a tube τ in 𝒫′′ parametrised by γ(ω0, t), see Figure 2. We will consider values of ω0 running through some closed interval ω0[ω0(1),ω0(2)] and restrict the tube τ to these values. Hence the boundary ∂τ of the tube can be identified with γ2γ1, where γiγ(ω0(i)),i=1,2 and the minus sign in γ2γ1 accounts for the correct orientation. We consider the difference between the quasienergies

ϵ2ϵ1ϵ(ω0(2))ϵ(ω0(1))=(103),(108)ω4π(γ2𝜶γ1𝜶)+δn2ω,(128)

where δn denotes the difference of the winding numbers. In this paper we will make the general assumption that the quasienergy can be chosen as a smooth function ϵ(ω0) of ω0 taking into account that it is only defined up to integer multiples of ω. Hence a discontinuous change of the winding number by an even number can be compensated by the choice of the right branch of ϵ(ω0). In the RPL example there are only even changes of the winding number due to the symmetry of the periodic solution but, according to our general assumption, this must hold generally. Hence we can neglect the term δn2ω in (128).

The tube τ in extended phase space 𝒫′′\(\mathcal{P}^{\prime\prime}\) generated by the family of closed curves γ(ω0)\(\gamma({\omega_{0}})\), where ω0(1)≤ω0≤ω0(2)\(\omega_{0}^{(1)}\leq{\omega_{0}}\leq\omega_{0}^{(2)}\). The two curves −γ1\(-{\gamma_{1}}\) and γ2 that form the boundary of τ are displayed together with their orientations.
Figure 2:

The tube τ in extended phase space 𝒫′′ generated by the family of closed curves γ(ω0), where ω0(1)ω0ω0(2). The two curves γ1 and γ2 that form the boundary of τ are displayed together with their orientations.

Then

ϵ2ϵ1=ω4π(γ2𝜶γ1𝜶)(129)

=ω4π(γ2γ1𝜶)(130)

=ω4π(τ𝜶)(131)

=ω4π(τd𝜶)(132)

=(127)ω4π(τ𝝎τ(zdω0dt)),(133)

invoking Stokes theorem in (132). Now we use the fact that τ𝝎=0 as the tangent plane at any point xτ contains a null vector of 𝝎, namely the vector tangent to the curve γ passing through xτ. Here we have employed the above-mentioned results of analytical mechanics according to [36]. It follows that

ϵ2ϵ1=ω4πτzdω0dt(134)

=12ω0(1)ω0(2)z(ω0,t)¯dω0,(135)

where z(ω0,t)¯ denotes the time average w. r. t. the curve γ(ω0) and hence depends on ω0. Choosing ω0(1)=ω0δ and ω0(2)=ω0+δ we obtain the limit δ → 0:

ϵω0=12z(ω0,t)¯,(136)

which concludes the proof of Assertion 2.

The calculation of ϵF and ϵG can be performed analogously. For example, the analogue of (135) reads

ϵ2ϵ1=ω4πτHFdFdt(137)

=12F(1)F(2)1z2cosωtcosφ¯dF(138)

=12F(1)F(2)xccos2ωt¯dF(139)

(F(2)F(1))xc4,(140)

and hence

ϵF=xc4,(141)

where xc is the coefficient of the term cosωt in the Fourier series of x(t). xc depends on F and hence (140) only holds asymptotically for F(2)F(1)0.

Similarly,

ϵG=ys4,(142)

where ys is the coefficient of the term sinωt in the Fourier series of y(t).

In order to calculate ϵω it is advisable first to simplify the ω-dependence of ϵ by the coordinate transformation tτωt together with H𝖧Hω. The latter transformation insures that the Hamiltonian equations of motion retain their canonical form

dφdτ=𝖧z,(143)

dzdτ=𝖧φ.(144)

After the coordinate transformation the Poincaré-Cartan form reads

𝜶=pdq𝖧dτ.(145)

Recall that according to (108)

ϵ(ω)=ω4πγ(pdq𝖧dτ)+n2ω.(146)

Together with an analogous calculation as in the proof of Assertion 2 this implies

ϵω=ϵω+ω4π02π𝖧ωdτ(147)

=ϵω14πω02πHdτ(148)

=ϵω12ωH¯(149)

=(77)ϵgω,(150)

using 𝖧ω=1ω2H in (148). An alternative derivation of (149) would consist in the evaluation of the Euler relation (124) using (125), (141) and (142). Hence the calculation of the partial derivatives of ϵ does not lead to a simplified formula for ϵ itself. However, the relation (150) can be used for a geometrical interpretation of the splitting ϵ=ϵg+ϵd, see Figure 3. We will formulate this result separately and stress the fact that the above proof is independent the particular form (109)–(111) of H.

The quasienergy ϵ as a function of ω for fixed ω0,F\({\omega_{0}},\ F\) and G is considered as the sum of the geometrical part ϵg and the dynamical part ϵd satisfying (150). Hence the values of ϵg and ϵd can be geometrically determined by the intersection of the tangent to the graph of ϵ(ω)\(\epsilon(\omega)\) with the ϵ axis.
Figure 3:

The quasienergy ϵ as a function of ω for fixed ω0,F and G is considered as the sum of the geometrical part ϵg and the dynamical part ϵd satisfying (150). Hence the values of ϵg and ϵd can be geometrically determined by the intersection of the tangent to the graph of ϵ(ω) with the ϵ axis.

Assertion 3

Under the assumptions of Sections 2 and 3 the following holds:

ϵω=ϵgω.(151)

Hence ϵω equals the solid angle |A| encircled by the corresponding periodic solution of the classical RPE equation divided by 4π.

4.3 The RPC Example III

The quasienergies

ϵ±=(70,71)12(ω±F2+(ω0ω)2)(152)

are obviously positively homogeneous functions of ω0, F, ω.

Further, the normalised third component of X(t) has the constant value

z=ω0ωF2+(ω0ω)2=z(t)¯,(153)

see (69). On the other hand, by (152),

ϵ+ω0=12Ωω0=12ω0ωF2+(ω0ω)2,(154)

which confirms Assertion 2. The resonance condition ϵ+ω0=0 is equivalent to ωres = ω0.

Moreover, Assertion 3 is confirmed by the following calculation:

ϵ+ω=(152)12(1+ωω0F2+(ωω0)2)(155)

=12(1+ωω0R)(156)

=(86)ϵgω.(157)

5 Analytical Approximations

5.1 Truncated Fourier Series Solution

In this and the following sections we specialise in the RPL. Thus the classical equation of motion (42) reduces to

X˙=ω0Y,(158)

Y˙=ω0XFcos(ωt)Z,(159)

Z˙=Fcos(ωt)Y.(160)

According to our general approach we are looking for T-periodic solutions of these equations. The space of real T-periodic functions is spanned by the four subspaces defined by even/odd sin/cos-series. Assume, for example, that X(t) is given by an odd cos-series. Then it follows that X˙(t) is an odd sin-series and, by (158), also Y(t) will be an odd sin-series. By (160) we further conclude that Z˙(t) is an even sin-series and hence Z(t) will be an even cos-series. One easily derives that also both sides of (159) are odd cos-series, and hence probably there exists a solution of (158)–(160) that fulfills the above requirements concerning the subspaces in which X, Y and Z lie.

On the basis of these considerations and numerical investigations we obtain the following ansatz of a (not necessarily normalised) Fourier series solution of (158)–(160):

X(t)=n=0ω0x2n+1cos(2n+1)ωt,(161)

Y(t)=n=0x2n+1(2n+1)ωsin(2n+1)ωt,(162)

Z(t)=z0+n=1x2ncos2nωt.(163)

The form of (162) is already uniquely determined by the differential equation X˙=ω0Y. The Fourier coefficients x2n of Z(t) in (163) are written in such a way that the vector of unknown Fourier coefficients assumes the form x=(x1,x2,). The validity of the ansatz (161)–(163) will not be rigorously proven but appears highly plausible due to the investigation of the analytical approximations to these solutions in what follows.

If we insert the ansatz (161)–(163) into (158)–(160) we obtain an infinite system of linear equations of the form A x = f, where

f=(Fz00).(164)

The matrix A is tri-diagonal due to the simple form of the h1=Fcosωt 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 series has rapidly decreasing coefficients and hence represents a good analytic approximation to the infinite Fourier series. The matrix elements of A are given by

Anm={n2ω2ω02:n=modd,nω:n=meven,F2:n=m±1odd,mFω2:n=m±1even,0:else.(165)

For example, the truncated matrix A(6) has the form

A(6)=(ω2ω02F20000Fω22ω3Fω20000F29ω2ω02F200003Fω24ω5Fω20000F225ω2ω02F200005Fω26ω).(166)

The truncated system of linear equations of the form A(N)x=f has the formal solution x=Fz0(A(N))11, where (A(N))11 denotes the first column of the inverse matrix of A(N). Fortunately, there exists a recurrence formula for the inverse of tri-diagonal matrices in terms of leading principal minors and co-leading principal minors, see [37]. Recall that a leading minor of A(N) of order n is the determinant of the sub-matrix of matrix elements in rows and columns from 1 to n. Similarly, we will denote by the “co-leading principal minor ϕn of co-order n” the determinant of the sub-matrix of matrix elements of A(N) in rows and columns from n to N. As we do not need the whole matrix (A(N))1 but only its first column it turns out that only co-leading principal minors are involved. It is well-known that the determinant of a tri-diagonal matrix satisfies a three-term recurrence relation. For our problem this implies the following system of recurrence relations for the ϕn,n=N,N1,,1.

ϕN=AN,N={Nω:N even,(Nω)2ω02:N odd,(167)

ϕN1=AN1,N1AN,NAN,N1AN1,N(168)

={14F2ω(N1)ωN(ω2(N1)2ω02):N even,14F2ωNω(N1)(ω2N2ω02):N odd,(169)

ϕn={nωϕn+1+n+14F2ωϕn+2:n even,((nω)2ω02)ϕn+1+n4F2ωϕn+2:n odd,(170)

Especially, ϕ1=detA(N). Then the first column of (A(N))1 can be expressed in terms of the co-leading principal minors and certain products of the lower secondary diagonal elements of A, see the theorem in [37] for the special case of j = 1. We write down the corresponding result for the Fourier coefficients xn:

x1=Fz0ϕ2/ϕ1,(171)

xn={z0(1)n2(F2)n(ω)n/2(n1)!!ϕn+1ϕ1:n even,z0(1)n2(F2)n(ω)n12(n2)!!ϕn+1ϕ1:n odd,(172)

where (172) holds for n=2,3,,N. z0 is a free parameter that necessarily occurs due to the fact that the Fourier series solution is not yet normalised. It could be chosen as z0 = 1 or, alternatively, as z0=ϕ1. Depending on the context both choices will be adopted in what follows. The latter choice has the following advantage: if ϕ1=0 the above solution (171)–(172) is no longer defined, but choosing z0=ϕ1 and cancelling the fraction z0/ϕ1 to 1 we obtain a solution that is always defined. Upon the choice z0=ϕ1 the vanishing of ϕ1=detA(N) is equivalent to the vanishing of the time average Z(t)¯=z0. We recall the fact that this in turn characterises the occurrence of resonances, see Assertion 2.

In any case, from these recursion relations it is clear that each xn is a rational function ρ(n,N,F,ω,ω0) in the variables F,ω and ω0. It can hence be viewed as a kind of Padé approximation for xn that becomes more and more exact for increasing N. For the choice z0=ϕ1 the rational function ρ(n,N,F,ω,ω0) becomes a polynomial in the variables F, ω and ω0.

In order to give an impression of the structure of ρ(n,N,F,ω,ω0) we give the results for the N = 4 truncation with z0=ϕ1 although this will not yet be good approximations of the exact RPL solutions:

X(t)=12Fω2ω0(9F2+16(ω029ω2))cos(ωt)F3ω2ω0cos(3ωt),(173)

Y(t)=12Fω3(9F2+16(ω029ω2))sin(ωt)3F3ω3sin(3ωt),(174)

Z(t)=δ18F2ω2(3F2+16(ω029ω2))cos(2ωt)+3F4ω28cos(4ωt),(175)

δ=116ω2(3F48F2(27ω211ω02)+128(ω2ω02)(9ω2ω02))(176)

In Figure 4 we show solutions of the classical RPL for different F values at the resonance frequency ωres(1)(F) that will be calculated in the next subsection. These solutions are based on the truncated Fourier series (161)–(163) with N = 20 and the choice z0=ϕ1=0. They can be either calculated by directly evaluating the Fourier series or by numerically solving the equation of motion with the initial values obtained by the Fourier series. The differences between both methods are negligible. For small F the 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 approaches the limit solution for F, ω → ∞, see subsection 6.2.

Various periodic solutions of the classical RPL for ω=ωres(1)(F)\(\omega=\omega_{res}^{(1)}(F)\) and ω0=1\({\omega_{0}}=1\) visualised as closed trajectories on the Bloch sphere. The coloured curves correspond to different values of F, namely F = 1/10 (blue), F = 1/2 (red), F = 1 (green), F = 2 (magenta), F = 5 (cyan), F = 10 (brown), and F = 100 (black).
Figure 4:

Various periodic solutions of the classical RPL for ω=ωres(1)(F) and ω0=1 visualised as closed trajectories on the Bloch sphere. The coloured curves correspond to different values of F, namely F = 1/10 (blue), F = 1/2 (red), F = 1 (green), F = 2 (magenta), F = 5 (cyan), F = 10 (brown), and F = 100 (black).

5.2 Calculation of the Resonance Frequencies

We have shown in Section 4.1 that ϵ is a positively homogeneous function and the same holds for its restriction to the variables ω0, ω, F in the limit G → 0. The domain of these variables can be restricted to a two-dimensional domain without loss of information. Instead of eliminating one of the three variables, which is inappropriate in some cases, one could introduce the scaled variables

ω0~ω0ω+ω0+F,ω~ωω+ω0+F,F~Fω+ω0+F,(177)

that satisfy 0<ω0~,ω~,F~<1 and ω0~+ω~+F~=1. The domain of the scaled variables is an open equilateral triangle Δ, see Figure 5. If the values of a positively homogeneous function like the quasienergy ϵ are known for arguments in Δ the function can be uniquely extended to the whole positive octant. If it is clear that we mean the scaled variables the tilde will be omitted.

The domain of the three variables ω0, ω, F of the homogeneous function ϵ can be realised by an equilateral triangle Δ. Each 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 positive coefficients that add to unity. The three open edges represent limit cases, e.g. the edge opposite to the vertex labelled “ω” represents the limit case ω → 0 such that ω0 and F are kept finite. The three vertices themselves represent the limit cases where two of the three variables approach 0 and the third one approaches 1.
Figure 5:

The domain of the three variables ω0, ω, F of the homogeneous function ϵ can be realised by an equilateral triangle Δ. Each 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 positive coefficients that add to unity. The three open edges represent limit cases, e.g. the edge opposite to the vertex labelled “ω” represents the limit case ω → 0 such that ω0 and F are kept finite. The three vertices themselves represent the limit cases where two of the three variables approach 0 and the third one approaches 1.

The transformation between the three scaled variables ω0~,ω~,F~ and the two Cartesian coordinates x, y defining the points of Δ has the following form:

x=12(ω~ω0~),(178)

y=32F~,(179)

ω0~=12xy3(180)

ω~=12+xy3(181)

F~=2y3.(182)

The resonance frequencies ωres(n),n=1,2, can be represented by “resonance curves” n in the triangular domain Δ, see Figure 6. These curves have been numerically calculated by setting z0=ϕ1=detA(50)=0, see subsection 5.1 and Assertion 2. Later we will show that their intersections with the two edges F = 0 and ω0 = 0 can be analytically determined, see Figure 6. For large n the resonance curves approach the straight line segments determined by these intersections.

The first 10 resonance curves ℛn,n=1,…,10\({\mathcal{R}_{n}},\ n=1,\ldots,10\) in the triangular domain of variables ω0,ω,F\({\omega_{0}},\omega,F\). They intersect the line F = 0 at ωω0=12n−1\(\frac{\omega}{{{\omega_{0}}}}=\frac{1}{{2n-1}}\) and the line ω0 = 0 at Fω=j0,n\(\frac{F}{\omega}={j_{0,n}}\) (the latter denoting the zeroes of the Bessel function J0), as will be shown below, see (183) and (225).
Figure 6:

The first 10 resonance curves n,n=1,,10 in the triangular domain of variables ω0,ω,F. They intersect the line F = 0 at ωω0=12n1 and the line ω0 = 0 at Fω=j0,n (the latter denoting the zeroes of the Bessel function J0), as will be shown below, see (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 ϵ=12(ω0+(Fcos(ωt)X(t)R+Z(t))¯) by a numerical integration. Alternatively, we can determine the integral over t[0,2πω] analytically in the following way: We choose F, ω0 and ω as rational numbers. Substituting cos(nωt)=12(zn+1zn) converts Fcos(ωt)X(t)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 theorem. The poles of the rational function to be integrated are obtained by using computer algebra software packets as “root objects” (this is the reason to choose rational 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 that the quasienergy determined by these methods jumps into the “wrong branch” and has to be corrected by adding an integer multiple of ω, see the corresponding discussion in subsection 2.1. In this way we obtain representations of the branches ϵi(F,ω) without any restriction to the values of F and ω0.

We have drawn a couple of branches of the function ϵ(ω) for fixed F and ω0, see Figures 7 and 8. At the resonance frequencies ωres(n),n=1,,5 that were calculated according to the method described in subsection 5.2 we observe avoided level crossings analogous to those obtained in the literature, see, e.g. [3], Figure 1 or [4], Figure 1.

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 neighbourhood of the resonance frequencies ωres(n)\(\omega_{res}^{(n)}\) 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 impression of crossing levels due to the low resolution of the figure but, e.g. the comparison with Figure 8 for n = 2 shows that the levels actually do not cross. The dotted horizontal line indicates the value of limω→0ϵ(ω)≈0.52992\(\mathop{\lim}\nolimits_{\omega\to 0}\epsilon(\omega)\ \approx 0.52992\), see (2.4).
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 neighbourhood of the resonance frequencies ωres(n) 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 impression of crossing levels due to the low resolution of the figure but, e.g. the comparison with Figure 8 for n = 2 shows that the levels actually do not cross. The dotted horizontal line indicates the value of limω0ϵ(ω)0.52992, see (2.4).

The quasienergies ϵi,i=1,2\({\epsilon_{i}},\;i=1,2\) for F = 1/2 and ω0 = 1 as functions of the frequency ω in the neighbourhood of the resonance frequency ωres(2)≈0.355776\(\omega_{res}^{(2)}\approx 0.355776\). ϵ1 (green curve) has been calculated analytically and numerically as described in the text and ϵ2 (red curve) was chosen as ϵ2=3ω−ϵ1\({\epsilon_{2}}=3\ \omega-{\epsilon_{1}}\).
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 frequency ωres(2)0.355776. ϵ1 (green curve) has been calculated analytically and numerically as described in the text and ϵ2 (red curve) was chosen as ϵ2=3ωϵ1.

6 Special Limit Cases

The introduction of Δ in subsection 5.2 as the natural domain of the arguments of ϵ also clarifies the consideration of the various limit cases. We have three limit cases where one of the scaled variables approaches 0 but the other two variables remain finite. These three cases correspond to the three open edges of Δ and will be considered in the corresponding following subsections. First, the limit case F → 0 is covered by a Fourier-Taylor series solution for X(t) and ϵ, see subsection 6.1. The second case of ω0 → 0 is considered in subsection 6.2 where we have calculated the asymptotic solution X(t) and the quasienergy ϵ up to linear terms in ω0. It is very difficult to extend these results to higher orders of ω0 and hence we will contend ourselves with numerical approximations. Finally, in the limit case ω → 0 we have recursively determined the terms of an ω-power series for X(t) and explicitly calculated the first two terms of ϵasy=ϵ0+ϵ2ω2+O(ω4), see subsection 6.3.

There are three further “limit cases of the limit cases” where two of the three scaled variables approach 0 and the third one necessarily approaches 1. They correspond to the three vertices of Δ and are not automatically included in the previous limit cases where we assumed that one scaled variable approaches 0 but the other two remain finite. Consider first the case where the unscaled variable ω approaches ∞ and the other two unscaled variables F, ω0 remain finite. Then, by (177), the scaled variables F~ and ω0~ will approach 0 whereas ω~ → 1. In this case we have calculated an FT series for X(t) in powers of 𝒯1ω and the corresponding power series of the quasienergy, see Section 6.4.

The next case of F~,ω~0, or, equivalently ω0 can be treated either by considering the lowest order of F in (2.3) and (246) or the lowest order of ω in (195) and (196). It follows that both cases yield the same result, see Section 6.4.

The last case ω~,ω0~0 or, equivalently F → ∞, is somewhat subtle as the two limits cannot be interchanged, see the discussion in Section 6.3. It will not be treated in a separate subsection.

6.1 Limit Case F → 0

6.1.1 Resonance Frequencies

A glimpse of (166) shows that for F = 0 the determinant of A(N) vanishes for ω=ω02n+1,n=1,2,. Hence the resonance condition z0=ϕ1=detA(N)=0 for N arbitrarily large, see Assertion 2, leads to

ωres(n)=ω02n1,forn=1,2,andF=0.(183)

This explains the intersections of the resonance curves n with the edge F = 0 of Δ, see Figure 6.

By an analogous reasoning we may also calculate the first terms of the power series w. r. t. F of ωres(n) for small n or for small m. The power series has the form:

ωres(n)=ω02n1+m=1σ2m(n)ω02m1F2m.(184)

Recall that the differences ωres(n)ω02n1 are traditionally called “Bloch-Siegert shifts”. The coefficients σ2m(n) of (184) can be determined as follows: we insert the power series (184) into the expression of detA(N) (for a suitable large N) and set the first few coefficients of the resulting power series w. r. t. F to zero. This yields recursive equations from which the σ2m(n) may be determined, independent of N. The corresponding results for n = 1, 2, 3 are contained in the Tables 13. They are in accordance with the three coefficients for n = 1 published in [4] and with the results of [9], [10]. Table 4 contains the first non-vanishing coefficients σ2m(n) for n = 1, …, 10 and m = 1, 2, 3 that were determined in the same way. A closed formula for the σ2m(n) is not known, but the coefficients σ2(n) that we have calculated satisfy the recursion relation

σ2(n+1)=(n1)2n+1σ2(n)+716(n+1)18.(185)

Table 1:

Coefficients of the power series (184) for the resonance frequencies ωres(1).

Table 2:

Coefficients of the power series (184) for the resonance frequencies ωres(2).

Table 3:

Coefficients of the power series (184) for the resonance frequencies ωres(3).

Table 4:

Coefficients σ2m(n) of the power series (184) for the resonance frequencies ωres(n) for m = 1, 2, 3 and n = 1, … 10.

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

Various resonance frequencies ωres(n)\(\omega_{res}^{(n)}\) (blue curves) for ω0 = 1 as functions of F for n = 1, 2, …, numerically determined by the resonance condition detA(20)=0\(\det{A^{(20)}}=0\). The thin continuous 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 dashed curves are the asymptotic four terms limits of ωres(n)\(\omega_{res}^{(n)}\) for ω, F → ∞ according to (226)–(228).
Figure 9:

Various resonance frequencies ωres(n) (blue curves) for ω0 = 1 as functions of F for n = 1, 2, …, numerically determined by the resonance condition detA(20)=0. The thin continuous 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 dashed curves are the asymptotic four terms limits of ωres(n) for ω, F → ∞ according to (226)–(228).

6.1.2 Fourier-Taylor Series

In the Section 6.1.3 we will present a solution of the classical RPL in terms of so-called Fourier-Taylor (FT) series. A few explanations will be in order. An FT series is a Taylor series of a (vector) quantity A(F, t), periodic in t, w. r. t. the parameter F such that each coefficient of Fn is a finite Fourier series w. r. t. the time variable t of maximal order n:

A(F,t)=n=0Fnm=nnAnme𝗂mωt.(186)

Put differently, the T=2πω-periodic function A(F, t) is expanded into a Fourier series such that each Fourier coefficient of order m is a Taylor series w. r. t. F that starts with the lowest order of n = m. Fourier series with components that are in turn Laurent series of a suitable parameter are known as “Poisson series” in celestial mechanics, see, e.g. [38]. FT series are special Poisson series characterised by the restriction m=nn in (186) and have been applied in [39] to a couple of physical problems by utilising computer algebraic means. It is possible to consider the more general case where the frequency ω is not given but also calculated iteratively in terms of a Taylor series, but this generalisation is not needed in the present context of RPL.

It is obvious that sums and products of FT series are again FT series. More generally, the power series of an FT series is again an FT series, at least in the sense of formal power series. For practical applications the size of the convergence radius becomes important.

6.1.3 FT Series for X(t) and ϵ

In the case F = 0 there are only two normalised solutions of the classical RPL that are T-periodic for all T > 0, namely X(t)=±(0,0,1). Hence for infinitesimal F we expect that we still have Z(t)=±1+O(F2) but (X(t), Y(t)) will describe an infinitesimal ellipse, i.e. X(t)=FAcosωt+O(F3) and Y(t)=FBsinωt+O(F3). These considerations and numerical investigations suggest the following FT series ansatz:

X(t)=n=0F2n+1m=0nRn,m(ω,ω0)cos(2m+1)ωt,(187)

Y(t)=1ω0n=0F2n+1×m=0n(2m+1)ωRn,m(ω,ω0)sin(2m+1)ωt,(188)

Z(t)=n=0F2nm=0nSn,m(ω,ω0)cos2mωt.(189)

In the ansatz (188) for Y(t) we have already used that Y(t) is completely determined via Y(t)=1ω0ddtX(t) and hence need not be further considered. The other two differential equations (159) and (160) yield recursion relations for the functions Rn,m(ω,ω0) and Sn,m(ω,ω0). As initial conditions we impose the following choices:

S0,0(ω,ω0)=1,(190)

R0,0(ω,ω0)=ω0(ωω0)(ω+ω0),(191)

Sn,0(ω,ω0)=0, for n=1,2,.(192)

For n > 0 the FT coefficients Rn,m and Sn,m can be recursively determined by means of the following relations:

Sn,m=14mω0((2m+1)Rn1,m+(2m1)Rn1,m1) for 1mn,(193)

Rn,m=ω02((2m+1)2ω2ω02)(Sn,m+Sn,m+1) for 0mn,(194)

where, of course, we have to set Sn,n+1=0 in (194). It follows that Rn,m(ω,ω0) and Sn,m(ω,ω0) are rational functions of ω and ω0.

We recall that under the transformation (118)–(122), X, Y and Z remain invariant which entails Rnmλ2n1Rnm and Snmλ2nSnm. Then it easily follows that both sides of the recursion relations (193) 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) and Z(t):

X(t)=Fω0ω2ω02cosωt+F3(ω08(ω2ω02)2cosωtω08(9ω2ω02)(ω2ω02)cos3ωt)+O(F5),(195)

Z(t)=1+F214(ω2ω02)cos2ωt+F4(3ω2ω028(ω2ω02)(9ω2ω02)2cos2ωt+364(ω2ω02)(9ω2ω02)cos4ωt)+O(F6).(196)

We note that the coefficients contain denominators of the form ω2ω02 and 9ω2ω02 due to the denominator (2m+1)2ω2ω02 in the recurrence relation (194). Hence the FT series breaks down at the resonance frequencies ωres(m)=ω02m1. This is the more plausible since according to the above ansatz z0 = 1 which is not compatible with the resonance condition z0 = 0, see Assertion 2.

Using the FT series solution (187)–(189) it is a straightforward task to calculate the quasienergy ϵ=a0 as the time-independent part of the FT series of (62) which in the present case of RPL assumes the form

12(ω0+Fcos(ωt)X(t)R+Z(t))=a0+n0nane𝗂nωt.(197)

The first few terms of the result are given by

ϵ=ω02F2ω08(ω2ω02)+F4ω0(ω2+3ω02)128(ω2ω02)3F6ω0(5ω06+35ω2ω04+33ω4ω02+ω6)512(ω2ω02)5(9ω2ω02)+O(F8).(198)

This is in agreement with [4], (29), except for the first term which is probably a typo.

It will be instructive to check the first two terms of (198) by using the decomposition of the quasienergy into a dynamical and a geometrical part in Section 3.1. In lowest order in F the classical RPL solution is a motion on an ellipse with semi axes a=Fω0|ω2ω02| and b=Fω|ω2ω02|. Hence the geometrical part of the quasienergy reads

ϵg=ω4ππab+O(F4)=F2ω2ω04(ω2ω02)2+O(F4).(199)

The dynamical part is obtained as

ϵd=ω0Z+XFcosωt2R¯=ω02+ω0(ω023ω2)8(ω2ω02)2F2+O(F4).(200)

The sum of both parts together correctly yields

ϵ=ϵd+ϵg=ω02F2ω08(ω2ω02)+O(F4).(201)

Moreover,

ϵω=ωω04(ω2ω02)2F2+O(F4)=ϵgω,(202)

in accordance with Assertion 3.

However, it is plausible from (198) that the FT series for the quasienergy has poles at the values ω=ωres(m)=12m1,m=1,2, and hence the present FT series ansatz is not suited to investigate the Bloch-Siegert shift for small F. We have also found a modified FT series that is valid in the neighbourhood of ωres(1) but will not dwell upon this.

6.2 Limit Case ω0 → 0

6.2.1 The Classical Equation of Motion

We reconsider the classical RPL equations of motion (158)–(160) and look for solutions that are at most linear in ω0, neglecting higher order terms. For ω0 = 0 we have the exact “pendulum solution”

X(t)=0,(203)

Y(t)=sin(fsinωt)=2n=1,3,Jn(f)sinnωt,(204)

Z(t)=cos(fsinωt)=J0(f)+2n=2,4,Jn(f)cosnωt.(205)

Here Jn(…) denotes the Bessel functions of first kind and integer order and we have set fFω as this combination permanently occurs in what follows. Equations (204) and (205) follow from the Jacobi-Anger expansion. Moreover, we will assume

0<f<π,(206)

in order to avoid problems with the following integrations. We note that if a constant x-component X(t) = x0 would be added to the above solution it would still solve (158)–(160) for ω0 = 0. But only the choice X(t) = x0 = 0 is suited as a starting point for higher orders of ω0.

The next linear order of the solution of (158)–(160) is obtained by replacing (2.3) by

X(t)=ω0X1(t), where(207)

X˙1(t)=Y(t)=sin(fsinωt).(208)

A periodic solution of (208) is given by the Fourier series

X1(t)=2ωn=1,3,1nJn(f)cosnωt.(209)

The radius of the Bloch sphere for this solution is still R=1+O(ω02).

We want to determine the quasienergy ϵ in linear order in ω0 which according to (62) reads

ϵ=12(ω0+(h1X1+Z)¯)(210)

=(207)ω02(1+(h1X11+Z)¯),(211)

where h1Fcosωt. For the time average we make the substitution τ = ωt and perform the τ-integration over the interval [0, 2π]. This yields a factor 1/ω for each time integral which is partially compensated by the factor ω2π due to the time average. Then the time average integral in (211) can be transformed by partial integration into

02πX1h11+Zdτ02πudvdτdτ=[uv]02π02πdudτvdτ.(212)

By (208) we have

dudτ=1ωdudt=1ωX˙1=1ωsin(fsinτ).(213)

In order to calculate v we consider the integral

v=h11+Zdτ(214)

=Fcosτ1+cos(fsinτ)dτ.(215)

Substituting x = fsinτ, hence dx=fcosτdτ, we obtain

v=ω11+cosxdx(216)

=ωtanx2=ωtan(f2sinτ),(217)

suppressing irrelevant integration constants. As u and v are 2π-periodic functions the term [uv]02π in (212) vanishes. By (213) and (217) the remaining integral reads

02πdudτvdτ=02πsin(fsinτ)tan(f2sinτ)dτ(218)

=02π(1cos(fsinτ))dτ(219)

==2π(1J0(f)),(220)

using (205) in the last step. After dividing by 2π due to the τ-average we obtain for (211):

ϵ=ω02J0(Fω).(221)

The decomposition into dynamical and geometrical part of the quasienergy according to Section 3.1 reads

ϵg=ω02(FωJ1(Fω)),(222)

ϵd=ω02(J0(Fω)FωJ1(Fω)).(223)

Note further that

ϵω=Fω02ω2J1(Fω)=ϵgω(224)

in accordance with Assertion 3.

Moreover, it is clear from (2.1) that the resonance condition ϵω0=0, cp. (114), is equivalent to

ω=ωres(n)=Fjn,0,(225)

where jn,0 denotes the n-th zero of the Bessel function J0. This yields the intersections of the resonance curves n with the line ω0 = 0, (see Figure 6). Note further that, by (205), J0(Fω) is the Fourier coefficient of Z(t) corresponding to the constant term and hence Z(t)¯=J0(Fω) vanishes exactly in the resonance case, in accordance with Assertion 2.

Unfortunately, the integrals occurring in the next, quadratic and cubic orders in ω0 cannot be solved in closed form and we cannot extend our analysis to this case in a straightforward way. As a way out we return to the Fourier series solution (161)–(163) and the approximate determination of the resonance frequencies by the solution of detA(50)=0. From this the asymptotic form of ωres(n)(F) can be obtained by inserting ω=m=03amF12m into detA(50) and setting the four highest even orders of F to zero. This yields

ωres(1)(F)0.415831F+0.87256F+0.404226F33.83313F5+O(F7),(226)

ωres(2)(F)0.181157F+0.496818F+1.03437F312.9166F5+O(F7),(227)

ωres(3)(F)0.115557F+0.356526F+1.32633F325.278F5+O(F7).(228)

The first terms proportional to F are the numerical approximations of the known exact value Fj0,n, but the next terms could only be determined numerically. For an alternative approach see [9], [10]. Figure 9 shows the numerically determined resonance curves n together with the approximations (226)–(228) for n = 1, 2, 3 that are valid for large F and ω. Recall that according to (177) the limit of the unscaled quantities F, ω → ∞ is equivalent to ω0~0 for the scaled quantity ω0~.

6.2.2 The Schrödinger Equation

For the sake of completeness we will show that the limit ω0 → 0 can also be considered directly for the Schrödinger equation and yields an equivalent result for the linear term of the quasienergy series w. r. t. ω0.

It is convenient to consider the Hamiltonian

H^=12(Fcosωtω0ω0Fcosωt),(229)

that is unitarily equivalent to the RPL Hamiltonian hitherto considered. We make the following series ansatz for the solution of the corresponding Schrödinger equation:

ψ1=n=0ψ1(2n)ω02n,(230)

ψ2=n=0ψ2(2n+1)ω02n+1,(231)

and obtain the following system of (in)homogeneous linear differential equations:

𝗂ddtψ1(0)=F2cosωtψ1(0),(232)

𝗂ddtψ2(2n+1)=12ψ1(2n)F2cosωtψ2(2n+1),(233)

𝗂ddtψ1(2n+2)=12ψ2(2n+1)+F2cosωtψ1(2n+2),(234)

for n = 0, 1, …. The two lowest terms of the series (230) and (232) can be obtained in a straightforward manner:

ψ1(0)=exp(𝗂F2ωsinωt),(235)

ψ2(1)=𝗂2(0texp(𝗂Fωsinωt)dt)×exp(𝗂F2ωsinωt).(236)

We could not calculate the integral in (236) in closed form but only in form of a series using again the Jacobi-Anger expansion and setting fFω:

0texp(𝗂fsinωt)dt=J0(f)t+2𝗂n=0J2n+1(f)cos((2n+1)ωt)1(2n+1)ω+2n=1J2n(f)sin(2nωt)2nω.(237)

For t = 0 we have ψ1(0)=1 and ψ2(1)=0. A second solution can be obtained that is orthogonal to the first one such that the resulting unitary evolution operator

U(t)=(ψ1(0)ψ2(1)¯ω0ψ2(1)ω0ψ1(0)¯)+O(ω02)(238)

satisfies (3) with initial condition (4). The corresponding monodromy matrix reads

=U(T)=(1𝗂ω02J0(f)T𝗂ω02J0(f)T1)+O(ω02),(239)

and has the eigenvalues 1±𝗂ω02J0(f)T. This yields the quasienergies ϵ=±ω02J0(f)+O(ω03) in accordance with (2.1). One may show that ϵ can be expanded into an odd series w. r. t. ω0, whence the above term O(ω03) for the next order.

6.3 Limit case ω → 0

It is plausible that for ω → 0 the classical spin vector X(t) follows the magnetic field, i.e. X(t)=h(t)h(t). We will confirm this by calculating the Taylor series expansion of X(t) w. r. t. ω:

X(t)=n=0Xn(t)ωn.(240)

Note that X(t)2 has the series expansion (suppressing the t-dependence):

XX=X0X0+2ωX0X1+ω2(2X0X2+X1X1)+2ω3(X0X3+X1X2)+ω4(2X0X4+2X1X3+X2X2)+(241)

As the normalisation condition XX=1 must hold in each order of ω it follows that X0X0=1, but X0X1=0 and all other terms in the brackets of (2.1) have to vanish.

As the series coefficients Xn(t) are T-periodic functions of t and can be written as Fourier series each differentiation of Xn(t) w. r. t. t produces a factor ω and both sides of the equation of motion

ddtX(t)=h(t)×X(t)(242)

are Taylor series in ω. This yields a recursive procedure to determine Xn(t).

The ω0-terms of (2.2) yields =h(t)×X0(t). Together with the normalisation condition this implies (up to a sign)

X0(t)=1ω02+F2cos2ωt(Fcosωt0ω0),(243)

which confirms the above assertion that the classical spin vector, up to normalisation, follows the magnetic field.

The next order, linear in ω, yields

ddtX0(t)=ωh(t)×X1(t).(244)

This is an inhomogeneous linear equation with the general solution

ωX1(t)=ddtX0(t)×h(t)h(t)2+ωλ1(t)h(t).(245)

The normalisation condition implies X0X1=0 and hence λ1(t) = 0. It follows that

X1(t)=(0,Fω0sin(ωt)(F2cos2(ωt)+ω02)3/2,0)T.(246)

In the next quadratic order of ω we analogously have

ddtX1(t)=ωh(t)×X2(t),(247)

with the general solution

ωX2(t)=ddtX1(t)×h(t)h(t)2+ωλ2(t)h(t).(248)

This time the normalisation condition up to the quadratic order of ω gives 2X0X2+X1X1= and hence

λ2(t)=12hX1X1=F2ω02sin2(ωt)2(F2cos2(ωt)+ω02)7/2.(249)

The corresponding X2(t) will not be displayed here.

In this way we may recursively determine an arbitrary number of the term Xn(t). It can be shown by induction over n that for odd n the Xn(t) have only a non-vanishing y-component and for even n the y-component vanishes. Hence Xn(t)Xm(t)=0 if n and m have different parity. This implies that the pre-factors of ωn in (2.1) vanish and hence λn(t) = 0 for all odd n.

We note that the Taylor expansion (240) breaks down for ω0 → 0. This follows already from the observation that the velocity ddtX0(t) would assume arbitrary large values for ω0 → 0 if (240) would be a correct description of the solution X(t). However, the velocity is bounded by h and hence (240) cannot longer hold. For example, we may consider a small fixed value of ω and a finite value of ω0 such that X0(t) is a good approximation of the exact solution X(t). If we lower ω0 to smaller and smaller values we would obtain a sudden switch to the behaviour described in Section 6.2 for the limit ω0 → 0. In this sense the two limits ω → 0 and ω0 → 0 cannot be interchanged.

Finally, we will consider the quasienergy for the lowest orders of ω. In the limit ω → 0 the geometrical part of the quasienergy vanishes as the solution X0(t) is confined to the x-z-plane. The dynamical part (77) has the value

ϵd=ϵ=12R(FcosωtX+ω0Z)¯(250)

=((ω02+F2cos2ωt)2ω02+F2cos2ωt)¯(251)

=ω4π02π/ωω02+F2cos2ωtdt(252)

=ω0πE(F2ω02),(253)

where E(…) denotes the complete elliptic integral of the 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 have

ϵϵ0E(14)π0.52992(254)

for ω → 0.

The next corrections to ϵ are in the form ϵ2ω2+ϵ4ω4. We obtain

ϵ2=(2F2+ω02)E(F2F2+ω02)ω02K(F2F2+ω02)6πω02F2+ω02,(255)

where K(…) denotes the complete elliptic integral of the first kind. ϵ4 is too complicated and will only be calculated for the special values F=1/2,ω0=1. It is plausible that the asymptotic limit ϵasy of ϵ for ω → 0 does not approximate a single branch of the quasienergy but rather represents a kind of envelope of the various branches, see Figure 10. In the special case F = 1/2 and ω0 = 1 that is portrayed in Figure 10 we have

ϵ2=3E(15)2K(15)65π0.0272334,(256)

and

ϵ4=203E(15)24K(15)15005π0.0249063.(257)

A detail of Figure 7 showing various branches of the quasienergy (continuous coloured curves), the corresponding resonance frequencies ωres(n)\(\omega_{res}^{(n)}\) (vertical dashed lines) and the asymptotic limit ϵasy=ϵ0+ϵ2ω2+ϵ4ω4\({\epsilon_{asy}}={\epsilon_{0}}+{\epsilon_{2}}{\omega^{2}}+{\epsilon_{4}}{\omega^{4}}\) (blue dashed curve) for ω → 0 according to (2.4), (256) and (257).
Figure 10:

A detail of Figure 7 showing various branches of the quasienergy (continuous coloured curves), the corresponding resonance frequencies ωres(n) (vertical dashed lines) and the asymptotic limit ϵasy=ϵ0+ϵ2ω2+ϵ4ω4 (blue dashed curve) for ω → 0 according to (2.4), (256) and (257).

As ϵasy represents the envelope of the branches of ϵ the asymptotic form of the resonance frequencies cannot be determined by the present method. However, the inspection of Figure 6 suggests that for ω → 0 the resonance frequencies are given by an interpolation between the limits for F → 0 and ω0 → 0, namely

ωres(n)Fj0,n+ω02n1.(258)

This approximation is of reasonable quality for small F or small ω0 but of poor quality for Fω0 as there the small curvature of the resonance curves n in the triangular domain Δ, see Figure 6, should be taken into account.

6.4 Limit Case ω → ∞

To investigate the limit ω → ∞ we set 𝒯1ω and make the following ansatz of an FT series:

X(t)=n=2,4,𝒯nm=1,3,n1xn,mcos(mωt),(259)

Y(t)=n=1,3,𝒯nm=1,3,nyn,msin(mωt),(260)

Z(t)=1+n=2,4,𝒯nm=2,4,nzn,mcos(mωt).(261)

This ansatz is inserted into the classical equations of motion (158)–(160) in such a way that each factor ω resulting from the differentiation ddt is replaced by 1/𝒯. 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 𝒯 = 1/ω):

X(t)=Fω0ω2cos(ωt)+O(ω4)=X(t)R,(262)

Y(t)=Fωsin(ωt)+O(ω3)=Y(t)R,(263)

Z(t)=1+F24ω2cos(2ωt)+O(ω4),(264)

Z(t)R=1F2(1cos(2ωt))4ω2+O(ω4).(265)

We will compare this result with the first terms of the 1/ω-Taylor expansion of the normalised classical RPC solution X(t) according to (69):

X(t)=(Fω+Fω0ω2)cos(ωt)+O(ω3),(266)

Y(t)=(Fω+Fω0ω2)sin(ωt)+O(ω3),(267)

Z(t)=1F22ω2+O(ω3).(268)

Despite some similarities we come to the conclusion that both solutions are different, even in the lowest non-vanishing order w. r. t. 1/ω. This is in contrast to the view that the rotating wave approximation is an analytical approximation to the RPL solution that is asymptotically valid in the limit of large ω.

According to the FT solution the quasienergy ϵ(ω0, F, ω) can be calculated as a power series in 1/ω the first terms of which are:

ϵ(ω0,F,ω)=ω02F2ω08ω2+F2ω0(F216ω02)128ω4+O(ω6).(269)

This is in accordance with the series expansion of (2.1)

ω02J0(Fω)=ω02F2ω08ω2+F4ω0128ω4+O(ω6),(270)

keeping in mind that (2.1) holds only in first order in ω0.

6.5 Limit Case ω0 → ∞

As remarked above, due to (177) this limit is equivalent to the limit F~0,ω~0 and ω0~1 of the scaled quantities. First we will compare the limit of X(t) for ω → 0 according to (2.3) and (246) with FT series expansion (195), (196) of X(t) that holds for F → 0. Note that for the comparison the latter one has to be normalised. We obtain the result that both limits coincide if we ignore terms of the order O(F3) and O(ω2):

X(t)=Fcos(ωt)ω0+O(F3,ω2),(271)

Y(t)=Fωsin(ωt)ω02+O(F3,ω2),(272)

Z(t)=1F2(1+cos2ωt)4ω02+O(F3,ω2).(273)

In deriving this result we used, of course, a restricted series expansion w. r. t. ω that leaves the terms cosnωt and sinmωt of the Fourier series intact.

Analogously, we will compare the asymptotic forms of the quasienergy for ω → 0 according to (2.3) and for F → 0 according to (198). Again, we find that both limits are compatible and yield the common result:

ϵ=ω02+F28ω0+O(F3,ω2).(274)

7 Summary and Outlook

We have revisited the Floquet theory of TLSs and suggested a kind of geometrical approach based on periodic solutions of the classical equation of motion that can be visualised by closed trajectories on the Bloch sphere. From these solutions one can reconstruct the Floquet solutions of the underlying Schrödinger equation including the quasienergy ϵ by calculating the coefficients of a Fourier series. The relation of ϵ to the classical action integral and the splitting of the quasienergy into a geometrical and a dynamical part, ϵ=ϵg+ϵd, fits well into this geometrical setting. In the case of the RPE the partial derivatives of ϵ w. r. t. the system’s parameters ω0,F and ω can be calculated by methods of analytical mechanics. Thereby the resonance condition of Shirley receives a geometrical/dynamical interpretation and the noteworthy relation ϵω=ϵgω is derived that holds generally for TLSs.

The mentioned results are proven not with strict mathematical rigor, but according to the usual standards of theoretical physics. This means that there are, besides the technical subtleties, still minor logical gaps. For example, it would be desirable to clarify the validity of the assumptions of Assertion 2 on the existence of exactly two normalised periodic solutions of the classical equation of motion. Another interesting open problem is the proof of the continuity or even analyticity of the quasienergy as a function of one of the parameters ω0, F and ω. As briefly mentioned in Section 2.1 the quasienergy can be viewed as an eigenvalue of the Floquet Hamiltonian defined on an extended Hilbert space. Hence one might invoke the corresponding theory of analytical perturbations, e.g. Rellich’s theorem [40] or similar tools, but it is not clear whether the Floquet Hamiltonian satisfies the pertaining conditions.

We have checked our results for simple solvable examples, but the main intended application is the RPL case. Here our approach leads to certain analytical approximations that can be conveniently handled by computer-algebraic aids. It is also possible to perform the geometrical approach for the various limit cases of the RPL. We have compared these results with those known from the literature only in a few cases, as a thorough comparison would need too much space, but such a comparison is nevertheless desirable.

Another future task would be the attempt to utilise the geometrical approach to obtain examples of the theory of periodic thermodynamics that describe periodically driven TLSs coupled to a heat-bath. For recent approaches to this problem, see e.g. [17], [34], [41], [42], [43], [44].

Acknowledgement

I am indebted to the members of the DFG research group FOR 2692 for continuous support and encouragement, especially to Martin Holthaus and Jürgen Schnack. Moreover, I gratefully acknowledge discussions with Thomas Bröcker on the subject of this paper. I dedicate this work to my friend and mentor Marshall Luban on the occasion of his upcoming 82nd birthday.

References

About the article

Received: 2018-04-24

Accepted: 2018-05-08

Published Online: 2018-06-14

Published in Print: 2018-08-28


Citation Information: Zeitschrift für Naturforschung A, Volume 73, Issue 8, Pages 705–731, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2018-0211.

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