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Next: The FLOQUET Operator of Up: The Quantum Map Previous: The Quantum Map   Contents


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}.
\end{displaymath} (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:
\end{displaymath} (2.7)

${\hat{U}}_n$ is a special case of the general time evolution operator ${\hat{U}}(t',t)$:
\; := \; \lim_{t\nearrow nT} \,
{\hat{U}}\left( t+T,t \right),
\end{displaymath} (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>,
\end{displaymath} (2.9)

and satisfies
{\hat{U}}(t',t) \; = \; {\hat{U}}(t',t'')\, {\hat{U}}(t'',t) \quad \forall\, t''\in\mathbb{R},
\end{displaymath} (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.
\end{displaymath} (2.11)

For a system with a time-periodic Hamiltonian

\H(t+T) \; = \; \H(t) \quad \forall \, t\in\mathbb{R},
\end{displaymath} (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).
\end{displaymath} (2.13)

Therefore the propagator (2.8) is the same for all $n$,
{\hat{U}}\; := \; {\hat{U}}_n \quad \forall\, n,
\end{displaymath} (2.14)

and the quantum map (2.7) simplifies to
\left\vert \psi_{n+1} \right> \; = \; {\hat{U}}\left\vert \psi_n \right>
\end{displaymath} (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.
\end{displaymath} (2.16)

the FLOQUET operator can formally be written as [Sch02]
\; = \;
\lim_{t\nearrow nT} \, \hat{{\cal Z}}
...{\hbar}\int\limits _{t}^{t+T} {\mbox{d}}t' \, \H(t')
\end{displaymath} (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>;
\end{displaymath} (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),
\end{displaymath} (2.19)

time-dependent solutions
\left\vert \phi_E(t) \right> \; := \; {\hat{U}}(t,-0) \left\vert \phi_E \right>
\end{displaymath} (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 $\lambda_E$:

$\displaystyle {\hat{U}}(t+T,t) \left\vert \phi_E(t) \right>$ $\textstyle =$ $\displaystyle {\hat{U}}(t+T,t) \, {\hat{U}}(t,-0) \left\vert \phi_E \right>$  
  $\textstyle =$ $\displaystyle {\hat{U}}(t+T,T-0) \, {\hat{U}}\left\vert \phi_E \right>$  
  $\textstyle =$ $\displaystyle \lambda_E \left\vert \phi_E(t) \right>,$ (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}.
\end{displaymath} (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>
\end{displaymath} (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>.
\end{displaymath} (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>,
\end{displaymath} (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.,
\end{displaymath} (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>,
\end{displaymath} (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.

next up previous contents
Next: The FLOQUET Operator of Up: The Quantum Map Previous: The Quantum Map   Contents
Martin Engel 2004-01-01