FLOQUET Theory

The natural quantum analogue of the classical map (1.21) is obtained by considering the evolution of quantum states during one period $T$ of the excitation. Therefore I define, in close analogy with (1.19), the state immediately before the $n$-th kick as

$$\left\vert \psi_n \right> \; := \; \lim_{t\nearrow nT} \left\vert \psi(t) \right>, \quad n\in\mathbb{Z}.$$ (2.6)

The quantum map is then given by the propagator ${\hat{U}}_n$ for a period of time of length $T$, acting on $\left\vert \psi_n \right>$:

$$\left\vert \psi_{n+1} \right> \; = \; {\hat{U}}_n \left\vert \psi_n \right>. % geboxt:$$ (2.7)

${\hat{U}}_n$ is a special case of the general time evolution operator ${\hat{U}}(t',t)$:

$${\hat{U}}_n \; := \; \lim_{t\nearrow nT} \, {\hat{U}}\left( t+T,t \right),$$ (2.8)

where ${\hat{U}}(t',t)$ is defined to take the quantum state from time $t$ to time $t'$,

$${\hat{U}}(t',t): \; \left\vert \psi(t) \right> \longmapsto \left\vert \psi(t') \right>,$$ (2.9)

and satisfies

$${\hat{U}}(t',t) \; = \; {\hat{U}}(t',t'')\, {\hat{U}}(t'',t) \quad \forall\, t''\in\mathbb{R},$$ (2.10)

as usual. By the Hermiticity of the Hamiltonian and the initial condition ${\hat{U}}(t,t)=\mathbbm{1}$, the time evolution operator is unitary,

$$\left( {\hat{U}}(t',t) \right)^{-1} \; = \; {\hat{U}}(t,t') \; = \; \left( {\hat{U}}(t',t) \right)^\dagger.$$ (2.11)

For a system with a time-periodic Hamiltonian

$$\H(t+T) \; = \; \H(t) \quad \forall \, t\in\mathbb{R},$$ (2.12)

as in the present case of equation (2.5), ${\hat{U}}(t',t)$ is also time-periodic in the sense of

$${\hat{U}}(t'+T,t+T) \; = \; {\hat{U}}(t',t).$$ (2.13)

Therefore the propagator (2.8) is the same for all $n$,

$${\hat{U}}\; := \; {\hat{U}}_n \quad \forall\, n,$$ (2.14)

and the quantum map (2.7) simplifies to

$$\left\vert \psi_{n+1} \right> \; = \; {\hat{U}}\left\vert \psi_n \right>$$ (2.15)

for all iterations.

The time-$T$-propagator ${\hat{U}}$ is also known as the FLOQUET operator of the quantum system. This naming convention is due to the fact that, using a time-independent orthonormal basis $\{\left\vert \phi_n \right>\}$ of HILBERT space for expanding $\left\vert \psi(t) \right>$ into $\sum_n a_n(t) \left\vert \phi_n \right>$, the time-dependent SCHRÖDINGER equation (2.1) can be transformed into a system of ordinary linear differential equations, the coefficients of which are $T$-periodic because $H$ exhibits the same periodicity. For a finite basis this is the setting of the FLOQUET theorem which asserts existence and uniqueness of the solutions and explicitly states their functional dependence on $t$ [Flo83,YS75]. In the present case the FLOQUET theorem does not apply as the HILBERT space is infinite-dimensional, but nevertheless the typical form of the FLOQUET solution and several other properties do carry over [Sal74]. Therefore, by analogy, the quantum theory of systems with time-periodic Hamiltonians is often also called FLOQUET theory.

Using the time ordering operator ${\hat{\cal{Z}}}$,

$${\hat{\cal{Z}}}\left( {\hat{A}}(t)\,{\hat{B}}(t') \right) ... ...] {\hat{B}}(t')\,{\hat{A}}(t ) & & t<t', \end{array} \right.$$ (2.16)

the FLOQUET operator can formally be written as [Sch02]

$${\hat{U}} \; = \; \lim_{t\nearrow nT} \, \hat{{\cal Z}} \... ...{\hbar}\int\limits _{t}^{t+T} {\mbox{d}}t' \, \H(t') \right),$$ (2.17)

but it is difficult to evaluate this expression for general time-dependent Hamiltonians. For autonomous $H$, on the other hand, ${\hat{U}}$ is found in its usual form as an exponential of the Hamiltonian, $e^{-\frac{i}{\hbar}\H T}$; this can be used for the free (i.e. unkicked) propagation part of the Hamiltonian (2.5). The propagator for the explicitly time-dependent (kick) part of the Hamiltonian (2.5) is found in the following subsection by direct integration of the SCHRÖDINGER equation. Before turning to the calculation of ${\hat{U}}$ for the full Hamiltonian (2.5), I now discuss some general properties of the FLOQUET operator that are of importance later on in chapter 5.

Consider the (normalized) eigenstates $\left\vert \phi_E \right>$ of the FLOQUET operator ${\hat{U}}$ with respect to the eigenvalues $\lambda_E$:

$${\hat{U}}\left\vert \phi_E \right> \; = \; \lambda_E \left\vert \phi_E \right>;$$ (2.18)

for the index $E\in\mathbb{R}$ see below. Using

$${\hat{U}}(t,-0) \; := \; \lim_{t_0\nearrow 0} {\hat{U}}(t,t_0),$$ (2.19)

time-dependent solutions

$$\left\vert \phi_E(t) \right> \; := \; {\hat{U}}(t,-0) \left\vert \phi_E \right>$$ (2.20)

of the SCHRÖDINGER equation (2.1) with respect to the initial condition $\left\vert \psi_0 \right>=\left\vert \phi_E \right>$ can be constructed for all times $t$. I now discuss some properties of these solutions for general $t$, although later on mainly solutions for the stroboscopic times $nT-0$ are needed.

Since $\left\vert \phi_E \right>$ is an eigenstate of ${\hat{U}}$ with respect to the eigenvalue $\lambda_E$, $\left\vert \phi_E(t) \right>$ is an eigenstate of ${\hat{U}}(t+T,t)$ with respect to the same eigenvalue $\lambda_E$:

$${\hat{U}}(t+T,t) \left\vert \phi_E(t) \right> = {\hat{U}}(t+T,t) \, {\hat{U}}(t,-0) \left\vert \phi_E \right>$$

$$= {\hat{U}}(t+T,T-0) \, {\hat{U}}\left\vert \phi_E \right>$$

$$= \lambda_E \left\vert \phi_E(t) \right>,$$ (2.21)

where in the last step the periodicity (2.13) of ${\hat{U}}(t',t)$ has been used. Therefore the $\left\vert \phi_E(t) \right>$ can all be labelled by the same index $E$, indicating the same eigenvalue for all $t$.

Because of the unitarity of ${\hat{U}}$ its eigenvalues are of unit modulus and can be written as

$$\lambda_E \; =: \; e^{-\frac{i}{\hbar}ET}.$$ (2.22)

The motivation for this formulation of the parameter dependence of the eigenvalues and for labelling the eigenstates by $E$ becomes clear when investigating the time dependence of the $\left\vert \phi_E(t) \right>$. By the definition

$$\left\vert \phi_E(t) \right> \; =: \; e^{-\frac{i}{\hbar}Et} \left\vert u_E(t) \right>$$ (2.23)

for the reduced states $\left\vert u_E(t) \right>$, the trivial part of the time dependence of the $\left\vert \phi_E(t) \right>$, corresponding to the time dependence of an energy eigenstate with respect to an autonomous system with energy $E$, is effectively separated off, and it remains to discuss the $\left\vert u_E(t) \right>$. Obviously, these states inherit the periodicity (2.12) of the Hamiltonian:

$$\left\vert u_E(t+T) \right> \; = \; \left\vert u_E(t) \right>.$$ (2.24)

This property provides the basis for some of the considerations in chapter 5.

The definition (2.23) is tailored to make the description of the time dependence of the $\left\vert \phi_E(t) \right>$ as similar to the dynamics of an autonomous system as possible. What is more, for stroboscopic times these two types of dynamics coincide,

$$\left\vert \phi_E(t+T) \right> \; = \; e^{-\frac{i}{\hbar}ET} \left\vert \phi_E(t) \right>,$$ (2.25)

which can be seen by combining equations (2.23) and (2.24); as required, equation (2.25) reproduces the result (2.21).

Since the parameter $E$ plays a similar role as the energy eigenvalue of a time-independent system, $E$ is called a quasienergy of the time-periodic Hamiltonian, and the FLOQUET states $\left\vert \phi_E(t) \right>$ are referred to as its quasienergy states [Zel67]. For brevity, often the states $\left\vert u_E(t) \right>$ are called (reduced) quasienergy states, too. The quasienergy is defined modulo $2\pi\hbar/T$ only, since it originates from the exponential in equation (2.22). Due to this nonuniqueness of the quasienergy it cannot be identified with any physical observable in a straightforward way, but note that, for an unscaled system, the quasienergy has the dimension of an energy. A discussion of the problems potentially arising from identifying the quasienergy with the conventional energy -- and thereby linking the quasienergy spectrum directly with the resonance (emission/absorption) spectrum of the respective system -- may be found in [DM98]. Normally, one restricts $E$ to the interval $\left[ 0,2\pi\hbar/T \right)$. By equation (2.21), the case of $E=0$, i.e. $\lambda_E=1$, corresponds to the quantum map's stationary states, for which not only the reduced $\left\vert u_0(t) \right>$, but also the full FLOQUET states $\left\vert \phi_0(t) \right>$ are periodic with period $T$.

The quasienergy states are characterized by several useful properties. It is easy to show that for $E_1\neq E_2$ the quasienergy states $\left\vert \phi_{E_1}(t) \right>$, $\left\vert \phi_{E_2}(t) \right>$ are orthogonal: their scalar product is

$$\left< \phi_{E_1}(t) \left\vert \phi_{E_2}(t) \right> \righ... ..._2)t} \left< u_{E_1}(t) \left\vert u_{E_2}(t) \right> \right.,$$ (2.26)

where the exponential is periodic with a period larger than $T$, whereas the period of $\left< u_{E_1}(t) \left\vert u_{E_2}(t) \right> \right.$ is smaller than or equal to $T$; this means that $\left< \phi_{E_1}(t) \left\vert \phi_{E_2}(t) \right> \right.$ must be zero, as every pair of solutions of the SCHRÖDINGER equation yields a constant scalar product. Furthermore, $\left\{ \, \left\vert \phi_{E}(t) \right> \, \vert \, 0\leq E< 2\pi\hbar/T \, \right\}$ is a complete set [Zel67,Per93]. The last two properties combined imply that the set $\{\left\vert \phi_{E}(t) \right>\}$ can be used in the conventional way as a basis for expanding arbitrary states of the system,

$$\left\vert \psi(t) \right> \; = \; \sum_E A_E \left\vert \phi_E(t) \right>,$$ (2.27)

with constant (i.e. time-independent) expansion coefficients $A_E\in\mathbb{C}$ [KW96].

Summarizing, with respect to a time-periodic Hamiltonian the quasienergies and the quasienergy states play much the same role as the energy eigenvalues and the stationary energy eigenstates do with respect to a time-independent Hamiltonian [Sam73]. This analogy includes the observation that in the same way as any solution of the time-independent SCHRÖDINGER equation can be expanded in terms of energy eigenstates with constant coefficients, the same can be accomplished using quasienergy states in the FLOQUET case.