Matrix Elements of the FLOQUET operator

It is the objective of the next chapters to study the time evolution of initial wave packets $\left\vert \psi_0 \right>$ under the iteration of the quantum map. One way to do this is to expand the initial state into a series of eigenstates of the harmonic oscillator and then use the corresponding matrix representation of ${\hat{U}}$ for the iteration. In this subsection I supply the formulae that are needed for this approach. Similar results may be found in [BRZ91].

The number states $\left\vert m \right>$, i.e. the eigenstates of the unkicked quantum harmonic oscillator given by the Hamiltonian (2.29a), are the solutions of the eigenvalue equation

$$\H_{\mbox{\scriptsize free}}\left\vert m \right> \; = \; \h... ...rac{1}{2} \right)\left\vert m \right>, \quad m\in\mathbb{N}_0.$$ (2.30)

It is well known that in the position representation these eigenstates can be written as

$$\left< x \left\vert m \right> \right. \; = \; \frac{1}{\sqr... ...} } \, {\mbox{H}}_m \! \left( \frac{x}{\sqrt{\hbar}} \right),$$ (2.31)

where the ${\mbox{H}}_m$ are HERMITE polynomials -- see for example [Gra89]. Using $\{\left\vert m \right>\}$ as a basis, any $\left\vert \psi_n \right>$ can be expanded into

$$\left\vert \psi_n \right> \; = \; \sum_{m=0}^\infty a_m^{(n)} \left\vert m \right>$$ (2.32)

(with the expansion coefficients $a_m^{(n)}\in\mathbb{C}$), which evolves, after one iteration of the quantum map, into

$$\left\vert \psi_{n+1} \right> % = \Uop \Ket{\psi_n} \; = \;... ..._{m,m'=0}^\infty U_{mm'} \, a_{m'}^{(n)} \left\vert m \right>,$$ (2.33)

where the

$$U_{mm'} \; := \; % \Matrel{m}{\Uop}{m'}. \big< m\vert{\hat{U}}\vert m' \big>$$ (2.34)

are the matrix elements of the FLOQUET operator. In terms of the coefficients $a_m^{(n)}$ this may also be formulated as

$$a_m^{(n+1)} \; = \; \sum_{m'=0}^\infty U_{mm'} \, a_{m'}^{(n)}.$$ (2.35)

In other words, the matrix $U=\left(U_{mm'}\right)$ is the propagator of the quantum map in the eigenrepresentation of the free harmonic oscillator. It remains to derive an explicit expression for the matrix elements in order to complete the scheme given by equation (2.43) for iterating the quantum map.

In the harmonic oscillator eigenrepresentation the free propagator (2.32b) is diagonal:

$$\big< m \big\vert {\hat{U}}_{\mbox{\scriptsize free}} \big\... ...\textstyle -iT \left( m+\frac{1}{2} \right) } \, \delta_{mm'}.$$ (2.36)

Using this expression and the splitting (2.28), $U_{mm'}$ can be written as

$$U_{mm'} \; = \; e^{ -i T\left( m+\frac{1}{2} \right) } \, K_{mm'},$$ (2.37)

where the kick matrix elements are defined as

Due to the oscillator eigenstates in the position representation (2.39) being real-valued functions, the kick matrix elements are symmetric:

$$K_{mm'} \; = \; K_{m'm}.$$ (2.37)

A FOURIER expansion of the kick propagator $e^{\textstyle -\frac{i}{\hbar}V_0\cos{\hat{x}}}$ (cf. [AS72]) gives

$$K_{mm'} \; = \; \sum_{l=-\infty}^\infty (-i)^l \, {\mbox{J}... ...) \left< m \left\vert e^{il{\hat{x}}} \right\vert m' \right>,$$ (2.38)

with the BESSEL functions ${\mbox{J}}_l$. The matrix element $\left< m \left\vert e^{il{\hat{x}}} \right\vert m' \right>$ on the right hand side can be evaluated by means of the formula (see [GR00]):

$$\int\limits _{-\infty}^\infty e^{-x^2}\, {\mbox{H}}_m(x+y)\... ...^{m}\, \sqrt{\pi}\, z^{m-m'}\, {\mbox{L}}_{m'}^{(m-m')}(-2yz),$$ (2.39)

holding for $m\geq m'$; the ${\mbox{L}}_{m'}^{(m-m')}$ are generalized LAGUERRE polynomials. With equation (2.49) I get

$$\left< m \left\vert e^{il{\hat{x}}} \right\vert m' \right> ... ...right), % \quad \mbox{for} \quad m\geq m', \;\;\; m\geq m',$$ (2.40)

and using the elementary property ${\mbox{J}}_{-l}(x)=(-1)^l\, {\mbox{J}}_l(x)$ of the BESSEL functions, I finally obtain

$$K_{mm'} \; = \; \left\{ \begin{array}{l@{\;}l@{\quad}l} ... ... \displaystyle 0 & & \mbox{$m-m'$\ odd.} \end{array} \right.$$

(2.41)

Note that $U_{mm'}=K_{mm'}=0$ for $m-m'$ odd may be viewed as a selection rule that -- with equation (2.46b) -- follows directly from the fact that the kick potential (1.18) has been chosen as $V(x)=V_0\cos x$; every other kick potential with the same symmetry property -- i.e. being an even function of $x$ -- would lead to the same selection rule. As a result, the dynamics in the harmonic oscillator eigenrepresentation decomposes into two parts, taking place in the dynamically disconnected subspaces of HILBERT space that are given by the states $\left\vert n \right>$ with odd and even indices, respectively.^2.1

With equations (2.45) and (2.51) the matrix elements of the FLOQUET operator are completely specified. For the above derivation no assumption concerning the (resonant or nonresonant) value of $T$ had to be made; also note that $T$ enters $U_{mm'}$ via the elementary exponential $e^{ -i T\left( m+1/2 \right) }$ in equation (2.45) only and does not interfere with the more complicated calculation of $K_{mm'}$.

The rest can be left to a computer with large memory. The way in which formula (2.51) can be evaluated efficiently within a computer program is discussed in chapter 3.

Footnotes

... respectively.^2.1

A similar manifestation of this symmetry of the kick potential arises in subsection 5.3.1 below, where the selection rule (5.88) leads to two discrete SCHRÖDINGER equations (5.94) that are interwoven with but independent of each other.