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  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  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 ,  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  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 , ac-driven quantum dots , Josephson qubit circuits  and coherent destruction of tunnelling . 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 , , perturbation theory and/or various limit cases , , , , ,  and the hybridised rotating wave approximation . Special analytical solutions of the general Floquet problem of TLS can be generated by the inverse method , , .
There exists even an analytical solution  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 . 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. , ), 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. . 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 . 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 ,  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 .
The second main part on resonances essentially bears on the resonance condition due to Shirley . 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 ,  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.
2.1 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 .
The Schrödinger equation for this system is of the form
Here we have set ℏ = 1 and will assume the Hamiltonian Ĥ(t) to be T-periodic in time,
where throughout in this paper
with the initial condition
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
Further, it follows that any other solution U1(t, t0) of (3) with initial condition U1(t0, t0) = V0 will be of the form
As a special case of (5), we consider
Hence (5) implies
ℱ ∈ SU2 is called the “monodromy matrix”. It can be written as
Now we define
and will show that 𝒫 is T periodic in the first argument:
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.
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
In this eigenbasis, (15) assumes the form
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 . 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
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
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
gives the result
It follows that the eigenvalues of
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. . It seems at first glance that in our approach uniqueness is guaranteed by the requirement 𝗂 F ∈ su2. For example, the replacement ϵn ↦ ϵn + ω in (16) would result in F ↦ F + 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 ℱ(ω) and the corresponding family ϵ1(ω) of quasienergies. It may happen that
2.2 Lift to Higher Spins
A possible physical realisation of the TLS with Hilbert space ℂ2 is a single spin with spin quantum number s =
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
Let, as in Section 2.1, t ↦ U(t, t0) be a smooth curve in SU2, such that U(t0, t0) = 𝟙 . It follows that
Hence the columns
of su2 parametrised by the spin quantum number s such that
It follows that
It follows from (29) that
will be a matrix solution of the lifted time evolution equation
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.
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
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
and can be restricted to an irrep
then the lifted evolution equation can be written as
where, as usual, R(t, t0) ∈ SO3 and R(t0, t0) = 𝟙 . The underlying “Schrödinger equation” has the form
Let ψ(t) be a solution of the Schrödinger equation (1) and
be the corresponding 1-dimensional projector. It can also be written in the form
We will write
The projector P(t) satisfies the von-Neumann equation
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
It will be instructive to check the consistency of our representation by directly applying the Floquet theory to (41). The corresponding monodromy matrix
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
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
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
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
where the infinite sum in (56) represents the Fourier series of the T-periodic function
(neglecting an additional integration constant that would only yield a constant phase factor) and further
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:
Thus we have proven the following:
(i) Ifψ(t) is a Floquet solution of (1) then
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
and analogously for
In this case the function χ( X(t)), see (56), turns out to be time-independent which directly yields the quasienergies
where Ω is the Rabi frequency
in accordance with the well-known result, see, e.g. .
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 ℋ onto the corresponding projective Hilbert space P(ℋ) and the quasienergy as a phase change during a cyclic quantum evolution in the sense of . 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 . We will treat both aspects in the following subsections.
3.1 Geometry of the Fibre Bundle
The map of wave functions to projectors
Following  we may split the function
see (56), into a “dynamical” and a “geometrical part”. The dynamical part is defined as
and represents the expectation value of the energy. Its time average yields the dynamical part of the quasienergy:
The geometrical part of χ( X(t)) is defined as
Using spherical coordinates θ, ϕ for the Bloch sphere with radius 1, we may write the differential χg dt in the form
This yields a differential 1-form α on the Bloch sphere and the time average of
where 𝒜 denotes the oriented area enclosed by 𝒞 and
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
We refer to  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
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 . 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)
On the other hand, the vector X+(t) prescribes a circle on the Bloch sphere with constant Z = ω0 − ω, see (69). Consider first the case of Z > 0. The corresponding spherical segment has an area of 2πR(R − Z) corresponding to a solid angle of
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
In the case of Z < 0 the dynamical part ϵd remains unchanged whereas the solid angle of the spherical segment encircled by
3.3 Another Solvable Example
The idea of a “reverse engineering of the control” h(t), see , , , can also be applied to the classical Floquet problem. Given a normalised T-periodic function X(t) we may choose
f being a real parameter, which leads to
see Figure 1 visualising the example f = 1, ω = 1. The Fourier series of χ(t) can be explicitly calculated:
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
Note further that Assertion 3 of Section 4.2 can be sharpened to
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
Further we define the classical Hamiltonian
and rewrite the differential equation (42) in terms of the two functions z(t) and φ(t):
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
We will hence forward often use p and q instead of z and φ. Following  we consider the extended phase space
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
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,
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  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  yields the quantum TLS when starting from its classical limit.
In this and the following sections we will restrict the Hamiltonian (25) to the following special case
that will be referred to as the RPE. It includes the two limit cases G → 0, the RPL, and G → F, 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
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
Although Shirley’s derivation of (113) refers to the RPL case, see (1) in , one can easily check that it also holds in the more general RPE case. It implies that the transition probability assumes its maximal value
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
It is invariant under the transformation
for all λ > 0. Under this transformation the quasienergy (62) scales with λ and hence is a positively homogeneous function of degree 1:
for all λ > 0. This could be used to eliminate the variable ω0 (by choosing
By the Euler theorem the positive homogeneity of ϵ implies
4.2 Calculation of
We first consider
Hence the resonance condition
For the proof of this assertion we will adopt the language of classical mechanics introduced in 3.4. In order to calculate
with coordinates p, q, t and ω0 > 0. Again the differential forms 𝜶 and 𝝎 can be transferred to
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
invoking Stokes theorem in (132). Now we use the fact that
which concludes the proof of Assertion 2.
The calculation of
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
where ys is the coefficient of the term sinωt in the Fourier series of y(t).
In order to calculate
After the coordinate transformation the Poincaré-Cartan form reads
Recall that according to (108)
Together with an analogous calculation as in the proof of Assertion 2 this implies
Under the assumptions of Sections 2 and 3 the following holds:
4.3 The RPC Example III
are obviously positively homogeneous functions of ω0, F, ω.
Further, the normalised third component of X(t) has the constant value
which confirms Assertion 2. The resonance condition
Moreover, Assertion 3 is confirmed by the following calculation:
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
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
The form of (162) is already uniquely determined by the differential equation
The matrix A is tri-diagonal due to the simple form of the
For example, the truncated matrix A(6) has the form
The truncated system of linear equations of the form
where (172) holds for
In any case, from these recursion relations it is clear that each xn is a rational function
In order to give an impression of the structure of
In Figure 4 we show solutions of the classical RPL for different F values at the resonance frequency
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
The transformation between the three scaled variables
The resonance frequencies
5.3 Calculation of the Quasienergy
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
We have drawn a couple of branches of the function
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
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
The next case of
The last case
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
This explains the intersections of the resonance curves
By an analogous reasoning we may also calculate the first terms of the power series w. r. t. F of
Recall that the differences
Coefficients of the power series (184) for the resonance frequencies
Coefficients of the power series (184) for the resonance frequencies
Coefficients of the power series (184) for the resonance frequencies
Figure 9 shows a couple of resonance curves
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:
Put differently, the
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
In the ansatz (188) for Y(t) we have already used that Y(t) is completely determined via
For n > 0 the FT coefficients Rn,m and Sn,m can be recursively determined by means of the following relations:
where, of course, we have to set
We recall that under the transformation (118)–(122), X, Y and Z remain invariant which entails
We will show the first few terms of the FT series for X(t) and Z(t):
We note that the coefficients contain denominators of the form
Using the FT series solution (187)–(189) it is a straightforward task to calculate the quasienergy
The first few terms of the result are given by
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
The dynamical part is obtained as
The sum of both parts together correctly yields
in accordance with Assertion 3.
However, it is plausible from (198) that the FT series for the quasienergy has poles at the values
6.2 Limit Case ω0 → 0
6.2.1 The Classical Equation of Motion
Here Jn(…) denotes the Bessel functions of first kind and integer order and we have set
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.
A periodic solution of (208) is given by the Fourier series
The radius of the Bloch sphere for this solution is still
We want to determine the quasienergy ϵ in linear order in ω0 which according to (62) reads
By (208) we have
In order to calculate v we consider the integral
Substituting x = fsinτ, hence
The decomposition into dynamical and geometrical part of the quasienergy according to Section 3.1 reads
Note further that
in accordance with Assertion 3.
where jn,0 denotes the n-th zero of the Bessel function J0. This yields the intersections of the resonance curves
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
The first terms proportional to F are the numerical approximations of the known exact value
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
that is unitarily equivalent to the RPL Hamiltonian hitherto considered. We make the following series ansatz for the solution of the corresponding Schrödinger equation:
and obtain the following system of (in)homogeneous linear differential equations:
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
For t = 0 we have
and has the eigenvalues
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.
As the normalisation condition
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
are Taylor series in ω. This yields a recursive procedure to determine Xn(t).
The ω0-terms of (2.2) yields
which confirms the above assertion that the classical spin vector, up to normalisation, follows the magnetic field.
The next order, linear in ω, yields
This is an inhomogeneous linear equation with the general solution
The normalisation condition implies
In the next quadratic order of ω we analogously have
with the general solution
This time the normalisation condition up to the quadratic order of ω gives
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
We note that the Taylor expansion (240) breaks down for ω0 → 0. This follows already from the observation that the velocity
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
for ω → 0.
The next corrections to ϵ are in the form
where K(…) denotes the complete elliptic integral of the first kind. ϵ4 is too complicated and will only be calculated for the special values
This approximation is of reasonable quality for small F or small ω0 but of poor quality for
6.4 Limit Case ω → ∞
To investigate the limit ω → ∞ we set
This ansatz is inserted into the classical equations of motion (158)–(160) in such a way that each factor ω resulting from the differentiation
We will compare this result with the first terms of the 1/ω-Taylor expansion of the normalised classical RPC solution
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:
This is in accordance with the series expansion of (2.1)
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
In deriving this result we used, of course, a restricted series expansion w. r. t. ω that leaves the terms
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:
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,
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  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. , , , , , .
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.
A. Gangopadhyay, M. Dzero, and V. Galitski, Phys. Rev. B 82, 024303 (2010).
L. J. El-Jaick and B. D. B. Figueiredo, J. Math. Phys. 49, 083508 (2013).
H. P. Breuer and M. Holthaus, Ann. Phys. 211, 2499291 (1991).
M. Hamermesh, Group Theory and its Application to Physical Problems, 2nd Ed., Addison–Wesley, Reading, MA, 1962.
M. Langemeyer and M. Holthaus, Phys. Rev. E 89, 012101 (2014).
V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd Ed., Springer, New York 1989.
R. A. Usmani, Linear Algebra Appl. 212–213, 413 (1994).
H.-J. Schmidt and T. Bröcker, arXiv:1509.01827v1[physics.class-ph], 2015.
R. Alicki, D. Gelbwaser-Klimovsky, G. Kurizki, arXiv:1205.4552v1[quant-ph], 2012.
T. Shirai, T. Mori, and S. Miyashit, arXiv:1801.02838v1[cond-mat.stat-mech], 2018.