Mapping of the Quantum Kicked Harmonic
Oscillator onto an ANDERSON-like Model

Although classically the kicked rotor (5.1) and the kicked harmonic oscillator (1.17) are fundamentally different, for example in that the kicked rotor is characterized by one frequency only (namely the frequency of the kick) whereas the kicked oscillator has two frequencies (the second being the frequency of the unperturbed harmonic dynamics), the quantum localization phenomena in both models can be explained by the same mechanism. This mechanism is described by the theory of ANDERSON localization as outlined in section 5.1. I now show that this theory -- with some alterations due to the differing eigenstates of the two model systems -- can also be applied to the quantum kicked harmonic oscillator.

As in equation (5.43) I begin with the quantum map (2.37) for the kicked harmonic oscillator,

$$\left\vert \psi_{n+1} \right> \; = \; e^{\textstyle -iT\le... ...style -\frac{i}{\hbar}V({\hat{x}})} \left\vert \psi_n \right>,$$ (5.52)

with the kick potential $V(x)$ not yet specified. This equation of motion is to be reformulated in terms of the (reduced) quasienergy states

which are similarly defined as their rotor counterparts in equations (5.45) and (5.47). Using the quasienergy states $\left\vert u_E^\mp \right>$ before and after the kick, the quantum map (5.72) becomes

With the operator

$$W({\hat{x}}) \; := \; -\tan \frac{V({\hat{x}})}{2\hbar}$$ (5.51)

I transform the kick equation (5.74a) into

which in terms of the averaged quasienergy state $\big\vert \overline{u}_E \big>$ gives

$$\Big(1-iW({\hat{x}})\Big) \big\vert \overline{u}_E \big> \; ... ...ht) } \Big(1+iW({\hat{x}})\Big) \big\vert \overline{u}_E \big>$$ (5.51)

by combination with the equation (5.74b) describing the free part of the dynamics. As in the case of the kicked rotor the original dynamical system given by equation (5.72) is thus mapped onto a static problem by exploiting the FLOQUET character of the system.

Projecting equation (5.77) onto the eigenstates $\left\vert m \right>$ of the free harmonic oscillator (cf. equation (2.38)), I get

$$\overline{u}_{E,m} - i \big< m \big\vert \hat{W} \big\vert... ...< m \big\vert \hat{W} \big\vert \overline{u}_E \big> \right),$$ (5.52)

where $\overline{u}_{E,m}$ is defined as in equation (5.52), but here with respect to oscillator eigenstates rather than rotor eigenstates:

$$\overline{u}_{E,m} \; := \; \left< m \left\vert \overline{u... ...right> \right. % \quad \mbox{with} , \quad m\in\mathbb{N}_0.$$ (5.53)

Collecting terms I then arrive at the equation

$$\epsilon_m \overline{u}_{E,m} + \big< m \big\vert \hat{W} \big\vert \overline{u}_E \big> \; = \; 0$$ (5.54)

with the diagonal energies

$$\epsilon_m(E,\hbar,T) \; := \; \tan \left( \frac{T}{2\hbar} \left( E-\hbar\left( m+\frac{1}{2} \right) \right) \right).$$ (5.55)

Completeness of $\{\left\vert m \right>\}$ implies that $\big< m \big\vert \hat{W} \big\vert \overline{u}_E \big>$ can be expanded according to

$$\big< m \big\vert \hat{W} \big\vert \overline{u}_E \big> \; = \; \sum_{m'=0}^\infty W_{m,m'} \; \overline{u}_{E,m'} \, ,$$ (5.56)

where the hopping matrix elements $W_{m,m'}\in\mathbb{R}$,

$$W_{m,m'} \; := \; \big< m \big\vert \hat{W} \big\vert m' \big> % \quad \mbox{with} \, , \quad m,m'\in\mathbb{N}_0,$$ (5.57)

have been introduced. Summarizing, the discrete SCHRÖDINGER equation that describes the quantum dynamics of the kicked harmonic oscillator has been derived:

$$\epsilon_m \overline{u}_{E,m} + \sum_{m'=0}^\infty \, W_{m,m'} \, \overline{u}_{E,m'} \; = \; 0.$$ (5.58)

The argument list $(E,\hbar,T)$ of the $\epsilon_m$ is dropped for convenience, as usual. But note that here -- in contrast to the diagonal energies $\epsilon_m(E,\hbar)$ of the rotor (5.58) -- the $\epsilon_m$ depend on the kick period $T$ as well. In the context of localization, this is a very important observation, as will be seen shortly when the issue of resonances is discussed. While the above derivation at first sight seems to be entirely analogous to the situation in subsection 5.1.3, there are several important differences that I discuss in the following.

The first remark concerns the diagonal energies. As their rotor counterparts, the $\epsilon_m(E,\hbar,T)$ can be viewed as being generated by a (quasi-) random number generator that follows a Lorentzian distribution (5.70), provided a condition of nonresonance is satisfied. Because of the different expressions (5.58) and (5.81) defining the respective diagonal energies, the (non-) resonances in question differ, too. For the quantum kicked oscillator, the quantum resonances inhibiting quasi-randomness of the $\epsilon_m$ are given by

$$T_{\rm res} \; = \; \frac{P}{Q} \, \pi, \quad P,Q\in\mathbb{N},$$ (5.59)

as can be concluded from equation (5.81), thus coinciding with the classical resonance condition (1.22) on the kick period. Figure 5.17 verifies this observation:

for the nonresonant value $T=1.0$ in figure 5.17a, a Lorentzian distribution of the values of $\epsilon_m$ is obtained, while the resonant values $T=7\pi/22$ and $T=\pi /2$ in figures 5.17b and 5.17c yield no continuous distribution for $\epsilon_m$ at all, although $T=7\pi/22\approx 0.9996$ is very close to the nonresonant value $T=1.0$ of figure 5.17a. Thus for nonresonant values of $T$ in the sense of equation (5.85) one can expect localization as in the ANDERSON-LLOYD model, as long as the other conditions discussed below are met. In this sense, quantum localization is complementary to the quantum webs discussed in chapter 4, which develop in the resonance case as defined by equation (5.85).

The fact that the resonance condition (1.22/4.21a) -- and thereby the important special case (1.23/4.22) as well --, which is essential for the formation of classical and quantum stochastic webs, naturally arises as equation (5.85) in this quantum theory and characterizes those situations where no quantum localization can be expected, is a strong argument in favour of this formulation of the theory of localization in the kicked harmonic oscillator: in this way proper quantum-classical correspondence is automatically guaranteed. What is more, in this way also the quantum theories for the cases of resonance and nonresonance nicely fit together in a complementary fashion. The resonances with respect to $T$ are true quantum resonances in the sense that they are obtained here as a consequence of a genuinely quantum mechanical theory. On the other hand, as discussed above, these resonances play an important role classically as well, such that the term ``quantum resonance'' might be questioned. This situation is to be compared with the quantum resonances (5.67) of the kicked rotor and (4.7, 4.44, 4.49) of the resonant kicked harmonic oscillator which have no classical counterpart.

The second important difference to the case of the rotor is that for the oscillator the evaluation of the hopping matrix elements is not as simple as in subsection 5.1.3, since the oscillator eigenfunctions $\left< x \left\vert m \right> \right.$ (2.39) are algebraically much more complicated than the simple exponentials of the rotor eigenfunctions (5.12). Explicitly, the matrix elements (5.83) for the cosine kick potential (1.18) are given by^5.10

$$W_{m,m'} \; = \; - \frac{1}{\sqrt{\pi 2^{m+m'}m!m'!}} \, \... ...\hbar}x}{2\hbar} e^{-x^2} {\mbox{H}}_m(x) {\mbox{H}}_{m'}(x).$$ (5.60)

The indices $m$, $m'$ run over the non-negative integers only, as opposed to all the integers in the case of the rotor. This leads to an asymmetric situation, since the lattice on which the discrete SCHRÖDINGER equation is defined is infinite on the right hand side only, rather than being bi-infinite. More asymmetry is added by the fact that the interaction with the left and right neighbours is not the same in general:

$$W_{m,m-m'} \; \neq \; W_{m,m+m'} \, ,$$ (5.61)

while for the rotor $W_{m'}=W_{-m'}$ holds.

It is obvious that, because of the dependence on the HERMITE polynomials, the formula (5.86) cannot be simplified to yield an expression that depends on a single index only as the corresponding rotor matrix element (5.54): the matrix elements $W_{m,m'}$ have to be viewed as functions of both the position $m$ and the difference $m-m'$. This means that the interaction between the sites of the ANDERSON lattice for the oscillator is not translation invariant; in other words, the strength of the interaction between two sites does not depend on their distance on the lattice alone. For the proof of ANDERSON localization this is not an obstacle, as long as for all $m$ the absolute values $\vert W_{m,m'}\vert$ of the matrix elements decay rapidly enough with the distance $\vert m-m'\vert$ from the site $m$, in much the same way as the absolute values of the rotor matrix elements do.

Analytically not much can be said in general about the values $W_{m,m'}$ takes on. The only straightforward property to notice is that every second of them vanishes,

$$W_{m,m'} \; = \; 0 \quad \mbox{for $m+m'$\ odd},$$ (5.62)

since the HERMITE polynomials ${\mbox{H}}_m$ in equation (5.86) are odd if and only if $m$ is odd, and the kick potential $V(x)=V_0\cos x$ is an even function of $x$. (A similar selection rule, but with respect to the matrix elements (2.51) of the FLOQUET operator, has been discussed in subsection 2.1.3; in both cases the selection rules stem from the same symmetry of the kick potential.) Therefore, without loss of generality, it suffices to consider $W_{m,m+2m''}$ with $m''\in\mathbb{Z}$ (and $m''\geq -m/2$). In the special case of small values of $V_0/\hbar$, an estimate for $\vert W_{m,m+2m''}\vert$ can be obtained: using just the lowest order term of the TAYLOR series of the tangent I can write

$$W_{m,m+2m''} \approx \frac{-V_0}{2^{m+m''}\hbar\sqrt{\pi m!... ...r}x \; e^{-x^2} \, {\mbox{H}}_m(x) \, {\mbox{H}}_{m+2m''}(x).$$

(5.63)

According to [GR00] the integral can be evaluated in the following way: (For notational convenience, the following formulae up to (5.93) are meant for positive values of $m''$ only; similar formulae hold for negative $m''$ and large enough $m$.)

$$\int\limits _0^\infty \! {\mbox{d}}x \, \cos\sqrt{\hbar}x \... ...{\mbox{L}}_m^{(2m'')} \!\! \left( \frac{\hbar}{2} \right) \! ,$$

(5.64)

and the generalized LAGUERRE polynomials ${\mbox{L}}_m^{(\alpha)}$ satisfy the inequality [AS72]

$$\left\vert {\mbox{L}}_m^{(\alpha)}(x) \right\vert \; \leq ... ...alpha+1)}{m!\:\Gamma(\alpha+1)} \; e^{\textstyle \frac{x}{2}},$$ (5.65)

such that I get the estimate

$$\vert W_{m,m+2m''}\vert \; { {\protect\begin{array}{c} <\... ...2} \right)^{m''-1} \sqrt{\frac{{m+2m'' \choose m}}{(2m'')!}}.$$ (5.66)

By STIRLING's formula [AS72], in the limit of large $m''$ this expression becomes

$$\vert W_{m,m+2m''}\vert \; { {\protect\begin{array}{c} <\... ...\, \left( \frac{27e^2\hbar^2}{64m''} \right)^\frac{m''}{2},$$ (5.67)

which shows that the absolute values of the matrix elements decay more than exponentially fast with the distance $2m''$; in addition, as in the case of the rotor the amplitudes of all $\vert W_{m,m+2m''}\vert$ are controlled by the factor $V_0/\hbar$ and can thus be made as small as desired by taking the limit $V_0/\hbar\to 0$. Summarizing, the inequality (5.93) shows that for weak perturbations and for not too large values of $\hbar$ the discrete SCHRÖDINGER equation (5.84) with the cosine kick potential (1.18) indeed approaches a tight binding equation.

The convergence $\vert W_{m,m+2m''}\vert \to 0$ for larger values of $\vert m''\vert$ is checked in figures 5.18 and 5.19; since the integral in equation (5.86) cannot be solved analytically in general, the matrix elements are evaluated numerically and plotted for several parameter combinations. Due to the oscillatory integrand, it is difficult to obtain exact results by numerical computation for some of these parameter combinations. This difficulty has an impact especially on the results for larger values of $V_0$ and becomes worse in the case of small values of $\hbar$, as can be seen in the figures: the data for $V_0=1$ and $V_0=0.1$ in figures 5.18c and 5.19c are spoiled by numerical round-off errors, and the same appears to be true for the points with $\vert m''\vert {\protect\begin{array}{c} >\protect\\ [-0.3cm]\sim \protect\end{array}} 10$ and $V_0=1$ or $V_0=0.1$ in figure 5.19a. In the opposite case of small $V_0$ and large $\hbar$ the numerical results nicely agree with the convergence to zero as predicted by the relation (5.93).

Apart from the numerical problems, the figures give a clear indication of the typical behaviour of the matrix elements. The $\vert W_{m,m+2m''}\vert$ are peaked around $m''=0$ and decay more or less exponentially on both sides. Figure 5.19 also illustrates that in general for any site $m$ the speed of this decay is not the same for $m''<0$ and $m''>0$, in agreement with the inequality (5.87). By taking suitable limits (with respect to the parameters), the $\vert W_{m,m+2m''}\vert$ decay fast enough to allow equation (5.84) to approach a tight binding equation. Some aspects of the case of those parameter combinations which do not give rise to tight binding are addressed in subsection 5.3.4 below.

This leads to another observation which marks a noteworthy difference to the rotor. Above, I have discussed which conditions have to be met for equation (5.84) to become a tight binding equation. Strictly speaking, equation (5.84) with the cosine kick potential cannot become a tight binding equation under any circumstances, because equation (5.88) indicates that no site is coupled to its nearest neighbour, and the closest interacting sites are characterized by indices $m$, $m'$ with $\vert m-m'\vert=2$. Even more, the dynamics on the sites with even indices and on the sites with odd indices are completely decoupled. This means that the cosine-kicked oscillator must be described not by a single, but by a pair of discrete SCHRÖDINGER equations which are interwoven with but independent of each other:

In each of these two subsystems tight binding is possible within the approximations discussed above. Of course, if one has tight binding in both subsystems at the same time, then the entire system is tightly bound and prone to localization.

On the other hand, by choosing a different kick potential as a replacement for the cosine potential (1.18),

$$\tilde{V}(x) \; := \; -2\hbar \arctan \left( \sqrt{\frac{2}{\hbar}} \, \tilde{\lambda} \, x - \, \tilde{\eta} \right)$$ (5.67)

with arbitrary nonzero real constants $\tilde{\eta}$ and $\tilde{\lambda}$, a single true tight binding model for the kicked harmonic oscillator can be obtained. Again, as in the case of the alternate kick potential (5.64) of the quantum kicked rotor, the diagonal energies (5.81) remain unchanged by the introduction of this alternate potential. The new hopping matrix elements

$$\tilde{W}_{m,m'} \; := \; - \bigg< m \bigg\vert \tan \frac{\... ...2\hbar} \bigg\vert m' \bigg> \, , \quad m,m'\in\mathbb{N}_0,$$ (5.68)

corresponding to $\tilde{V}(x)$ (cf. equation (5.75)) are given by

$$\tilde{W}_{m,m'} \; = \; \frac{1}{\sqrt{\pi \, 2^{m+m'}\, ... ...ilde{\eta} \right) \, {\mbox{H}}_m(x) \, {\mbox{H}}_{m'}(x),$$ (5.69)

where the integration can be accomplished by employing the recurrence relation (3.42) and the orthonormality relation

$$\int\limits _{-\infty}^\infty {\mbox{d}}x \, e^{-x^2} {\mbo... ...{H}}_{m'}(x) \; = \; \sqrt{\pi} \, 2^m \, m! \: \delta_{m,m'}$$ (5.70)

of the HERMITE polynomials [AS72]. It yields

$$\tilde{W}_{m,m'} \; = \; \left\{ \begin{array}{ccl} \sqr... ...'=m+1 \\ [0.2cm] 0 & \: \mbox{else} \, . \end{array} \right.$$ (5.71)

The result is the tight binding equation

$$\epsilon_m \overline{u}_{E,m} + \tilde{\lambda} \Big( \sq... ...u}_{E,m+1} \Big) \; = \; \tilde{\eta} \, \overline{u}_{E,m},$$ (5.72)

coupling each site to its nearest neighbour sites. Note that for the kicked harmonic oscillator no alternate potential could be constructed that yields a tight binding system coupling each site to its respective second-nearest neighbours only, as motivated by equations (5.94).

The alternate kick potential (5.95) is intended here for nothing more than demonstrating how a tight binding system describing the quantum kicked harmonic oscillator can be constructed; in particular I do not discuss the classical dynamics of the harmonic oscillator with this kick potential here. However, motivated by the results described in [Jun95], one might speculate about the existence of classical -- and perhaps quantum mechanical? -- stochastic webs even for this aperiodic kick potential (in cases of resonance with respect to $T$). In the following I do not discuss the alternate system specified by the potential (5.95) any further.

In this subsection I have shown that the quantum kicked harmonic oscillator can be modelled by a discrete SCHRÖDINGER equation which is similar to the model used in ANDERSON's theory. Provided certain conditions are met, it is also possible to obtain an approximative tight binding model as in the case of the rotor. In the following two subsections I discuss how these findings can be used to prove ANDERSON localization in the oscillator.

Footnotes

... by^5.10

For $V_0/\hbar\geq\pi$ the integral in (5.86) is not RIEMANN integrable, because in this case the argument of the tangent can take on values equal to half-integer multiples of $\pi$, making the integrand singular. However, with probability one (with respect to the values of $V_0$ and $\hbar$), the $W_{m,m'}$ still take on well-defined values if the integral is evaluated as the corresponding CAUCHY principal value of (5.86). Therefore this issue does not challenge the proper physical interpretation of the discrete SCHRÖDINGER equation (5.84).