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

$$\mathrm{\U0001d5c2}\frac{\partial}{\partial t}\psi \left(t\right)=\widehat{H}\left(t\right)\psi \left(t\right).$$(1)

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

$$\widehat{H}\left(t+T\right)=\widehat{H}\left(t\right),$$(2)

where throughout in this paper $T=\frac{2\pi}{\omega}>0$, (1) gives rise to a matrix equation for the evolution operator *U*(*t*, *t*_{0}) that reads

$$\mathrm{\U0001d5c2}\frac{\partial}{\partial t}U(t,{t}_{0})=\widehat{H}\left(t\right)U(t,{t}_{0}),$$(3)

with the initial condition

$$U({t}_{0},{t}_{0})\mathit{\hspace{1em}}=\U0001d7d9.$$(4)

We will assume that *U*(*t*, *t*_{0}) ∈ SU_{2}, 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 su_{2} of anti-Hermitean 2 × 2-matrices with vanishing trace, closed under commutation [ , ]. The relation between (1) and (3) is obvious: If ${\psi}_{0}\in {\u2102}^{2}$ and *U*(*t*, *t*_{0}) is the unique solution of (3) with initial condition (4), then *ψ*(*t*) ≡ *U*(*t*, *t*_{0}) *ψ*_{0} will be the unique solution of (1) with initial condition *ψ*(*t*_{0}) = *ψ*_{0}. Conversely, let *ψ*_{1}(*t*) and *ψ*_{2}(*t*) be the two solutions of (1) with initial conditions ${\psi}_{1}\left({t}_{0}\right)=\left(\genfrac{}{}{0pt}{}{1}{0}\right)$ and ${\psi}_{2}\left({t}_{0}\right)=\left(\genfrac{}{}{0pt}{}{0}{1}\right)$, then $U(t,{t}_{0})\equiv ({\psi}_{1}\left(t\right),{\psi}_{2}\left(t\right))$ will solve (3) and (4).

Further, it follows that any other solution *U*_{1}(*t*, *t*_{0}) of (3) with initial condition *U*_{1}(*t*_{0}, *t*_{0}) = *V*_{0} will be of the form

$${U}_{1}(t,{t}_{0})=U(t,{t}_{0}){V}_{0}.$$(5)

As a special case of (5), we consider

$${U}_{2}(t,{t}_{0})\equiv U(t+T,{t}_{0}),$$(6)

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

$${U}_{2}({t}_{0},{t}_{0})=U({t}_{0}+T,{t}_{0})\equiv \mathcal{F}.$$(7)

Hence (5) implies

$${U}_{2}(t,{t}_{0})=U(t+T,{t}_{0})=U(t,{t}_{0})\mathcal{F}.$$(8)

*ℱ* ∈ SU_{2} is called the “monodromy matrix”. It can be written as

$$\mathcal{F}={e}^{-\mathrm{\U0001d5c2}TF},\mathrm{\U0001d5c2}F\in {\mathrm{su}}_{2}.$$(9)

Now we define

$$\mathcal{P}(t,{t}_{0})\equiv U(t,{t}_{0}){e}^{\mathrm{\U0001d5c2}\left(t-{t}_{0}\right)F}$$(10)

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

$$\mathcal{P}(t+T,{t}_{0})=U(t+T,{t}_{0}){e}^{\mathrm{\U0001d5c2}\left(t+T-{t}_{0}\right)F}$$(11)

$$=U(t+T,{t}_{0}){e}^{\mathrm{\U0001d5c2}TF}{e}^{\mathrm{\U0001d5c2}\left(t-{t}_{0}\right)F}$$(12)

$$\stackrel{(8,9)}{=}U(t,{t}_{0})\mathcal{F}{\mathcal{F}}^{-1}{e}^{\mathrm{\U0001d5c2}\left(t-{t}_{0}\right)F}$$(13)

$$=\mathcal{P}(t,{t}_{0}).$$(14)

Summarising this, we have shown that the evolution operator *U*(*t*, *t*_{0}) can be written as the product of a periodic matrix and an exponential matrix function of time, i.e.

$$U(t,{t}_{0})=\mathcal{P}(t,{t}_{0}){e}^{-\mathrm{\U0001d5c2}\left(t-{t}_{0}\right)F},$$(15)

which is essentially the Floquet theorem for TLSs. Equation (15) is also called the “Floquet normal form” of *U*(*t*, *t*_{0}). For an example where an explicit solution for *U*(*t*, *t*_{0}) is possible for some limit case, see also subsection 6.2.2.

The derivation of (15) can be easily generalised from SU_{2} 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 $|n\u27e9,n=1,2$ and real eigenvalues *ϵ*_{n} such that

$$F={\u03f5}_{1}|1\u27e9\u27e81\left|+{\u03f5}_{2}\right|2\u27e9\u27e82|.$$(16)

In this eigenbasis, (15) assumes the form

$$U(t,{t}_{0})|n\u27e9=\mathcal{P}(t,{t}_{0}){e}^{-\mathrm{\U0001d5c2}\left(t-{t}_{0}\right)F}|n\u27e9$$(17)

$$=\mathcal{P}(t,{t}_{0}){e}^{-\mathrm{\U0001d5c2}\left(t-{t}_{0}\right){\u03f5}_{n}}|n\u27e9$$(18)

$$=\mathcal{P}(t,{t}_{0})|n\u27e9{e}^{-\mathrm{\U0001d5c2}\left(t-{t}_{0}\right){\u03f5}_{n}}$$(19)

$$\equiv {u}_{n}(t,{t}_{0}){e}^{-\mathrm{\U0001d5c2}\left(t-{t}_{0}\right){\u03f5}_{n}},$$(20)

in which the latter functions are called the “Floquet functions” or “Floquet solutions of (1)” and the eigenvalues *ϵ*_{n} 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 $\psi \left({t}_{0}\right)={a}_{1}|1\u27e9+{a}_{2}|2\u27e9$ can be written in the form

$$\psi \left(t\right)=U(t,{t}_{0})\psi \left({t}_{0}\right)=\sum _{n=1}^{2}{a}_{n}{u}_{n}(t,{t}_{0}){e}^{-\mathrm{\U0001d5c2}\left(t-{t}_{0}\right){\u03f5}_{n}},$$(21)

with the time-independent coefficients *a*_{n}. In this respect *u*_{n}(*t*, *t*_{0}), resp. *ϵ*_{n}, 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*, *t*_{0}) = 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 ${L}^{2}[0,T]\otimes {\u2102}^{2}$, 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 *t*_{0}. It, hence, remains to investigate the effect of changing from *t*_{0} to some other initial time *t*_{1}. A straightforward calculation using the semi-group property of the evolution operator

$$U(t,{t}_{0})=U(t,{t}_{1})U({t}_{1},{t}_{0})$$(22)

gives the result

$$\begin{array}{ccccc}U(t,{t}_{1})\hfill & =\mathcal{P}(t,{t}_{0})\mathcal{P}{({t}_{1},{t}_{0})}^{-1}\hfill & & & \\ & \mathrm{exp}\left(-\mathrm{\U0001d5c2}\left(t-{t}_{1}\right)\mathcal{P}({t}_{1},{t}_{0})F\mathcal{P}{({t}_{1},{t}_{0})}^{-1}\right)\hfill & & & \end{array}$$(23)

$$\equiv \mathcal{P}(t,{t}_{1})\mathrm{exp}\left(-\mathrm{\U0001d5c2}\left(t-{t}_{1}\right)\stackrel{~}{F}\right).$$(24)

It follows that the eigenvalues of $\stackrel{~}{F}\equiv \mathcal{P}({t}_{1},{t}_{0})F\mathcal{P}{({t}_{1},{t}_{0})}^{-1}$ 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 *t*_{0} and the dependence on *t*_{0} may be suppressed without danger of confusion.

We will add a few remarks on the uniqueness of the quasienergies *ϵ*_{n}. 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* ∈ su_{2}. For example, the replacement *ϵ*_{n} ↦ *ϵ*_{n} + *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 *ϵ*_{1}(*ω*) of quasienergies. It may happen that $\mathrm{exp}\left(-\mathrm{\U0001d5c2}T{\u03f5}_{1}\left(\omega \right)\right)$ crosses the cut and hence *ϵ*_{1}(*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 *ϵ*_{1}(*ω*) to *ϵ*_{1}(*ω*) + *ω*. In this case it would be more appropriate to consider, say, *ϵ*_{1}(*ω*) and *ϵ*_{1}(*ω*) + *ω* as physically equivalent quasienergies. Generally speaking, the issue of continuity is an argument in favour of considering the quasienergies modulo *ω*.

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