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An Algorithm Based on the TROTTER Formula

Consider the quantum map (2.37). When discussing states in the position representation, application of the kick propagator as in equation (2.34) is easily accomplished, because ${\hat{U}}_{\mbox{\scriptsize kick}}$ depends on ${\hat{x}}$ only, and not on ${\hat{p}}$. On the other hand, the application of the free propagator,

\left\vert \psi\big((n+1)T-0\big) \right>
\; = \; {\hat{U}}_{\mbox{\scriptsize free}} \left\vert \psi(nT+0) \right> ,
\end{displaymath} (3.1)

poses a greater problem. In the form of equation (2.44) it is straightforward to let ${\hat{U}}_{\mbox{\scriptsize free}}$ act on states expanded in terms of the harmonic oscillator eigenstates only -- I come back to this point later in section 3.3. For now, I want to discuss the application of the free propagator to states in the position representation.

Clearly, the exponential (2.32a) can be evaluated by expanding it into its TAYLOR series,

{\hat{U}}_{\mbox{\scriptsize free}}
\; = \; \sum_{m=0}^\inf...
-\frac{i}{\hbar}\H_{\mbox{\scriptsize free}} T
\right)^m ,
\end{displaymath} (3.2)

allowing to evaluate the mapping (3.1) term by term, but this approach is problematic from a practical point of view. Namely, in practice the infinite sum must be truncated at some finite value of $m$, such that the resulting approximant to the unitary ${\hat{U}}_{\mbox{\scriptsize free}}$ may become nonunitary. Unitarity of the propagator used for the calculation is a natural prerequisite for obtaining an unconditionally stable [DR96] numerical algorithm, because in this way conservation of the norm of the states is automatically built into the method. Unconditional stability being granted implies that the numerical error -- mainly due to discretization with respect to position and time -- that builds up in the course of many iterations of the quantum map remains controllable.

Therefore, from the point of view of unconditional stability, using a plain TAYLOR expansion should be excluded from the considerations, especially when it is intended to study long-time evolution of quantum states. Nevertheless, in [Dro95] a procedure is described that allows to improve on expansions like equation (3.2), but truly long-time dynamics still seems to remain beyond reach even with this refined approach.

The basic idea for constructing an unconditionally stable algorithm is to decompose ${\hat{U}}_{\mbox{\scriptsize free}}$ into a product where all factors are already unitary -- and where each factor can be diagonalized by a suitably simple transformation. In this way the propagation over a full period $T$ gets replaced with successive propagations over smaller time steps, such that a discretization with respect to time is introduced into the method.

Above all, the problems with constructing a unitary approximant to ${\hat{U}}_{\mbox{\scriptsize free}}$ are caused by the fact that $\H_{\mbox{\scriptsize free}}$ is a sum of two noncommuting contributions, namely the kinetic and potential terms ${\hat{p}}^2/2$ and ${\hat{x}}^2/2$. Since $[ {\hat{p}}^2, {\hat{x}}^2 ]\neq 0$, a simple BAKER-CAMPBELL-HAUSDORFF (BCH) decomposition into two unitary factors is not possible:

e^{\textstyle -\frac{i}{\hbar} \H_{\mbox{\scriptsize free}}T...
...{p}}^2 } \;
e^{\textstyle -\frac{iT}{2\hbar} \, {\hat{x}}^2 }
\end{displaymath} (3.3)

(cf. equation (A.4) in appendix A for more on BCH formulae).

An alternative and more suitable approach is based on the TROTTER product formula. For two operators ${\hat{A}}$, ${\hat{B}}$ and $z\in\mathbb{C}$, the exponential of the sum of the operators can be expressed as an infinite ordered product of exponentials of the individual operators:

e^{\textstyle z({\hat{A}}+{\hat{B}}) }
\; = \; \lim_{s\to\...
...\hat{A}}} \,
e^{\textstyle \frac{z}{s}{\hat{B}}}
\right)^s ;
\end{displaymath} (3.4)

for a proof see [Sch81]. Using the TROTTER formula, for sufficiently large $s\in\mathbb{N}$ one may approximate:
e^{\textstyle \frac{z}{s}({\hat{A}}+{\hat{B}}) }
\; \appro...{z}{s} {\hat{A}}} \,
e^{\textstyle \frac{z}{s} {\hat{B}}} .
\end{displaymath} (3.5)

For purely imaginary $z$, the error induced by this approximation, due to $s$ being finite, can be estimated using [DR87]
\bigg\Vert \,
e^{\textstyle \frac{z}{s}({\hat{A}}+{\hat{B}}...
...2s^2} \,
\left\Vert \, [{\hat{A}},{\hat{B}}] \, \right\Vert ,
\end{displaymath} (3.6)

where $\Vert\cdot\Vert$ is a suitable operator norm. Note that, using the estimate (3.6), for commuting ${\hat{A}},\, {\hat{B}}\,$ the simplest BCH result $ e^{{\hat{A}}+{\hat{B}}} = e^{{\hat{A}}} e^{{\hat{B}}} $ is confirmed again.

In the present context of the kicked harmonic oscillator, I obtain the following approximation for the free propagator,

{\hat{U}}_{\mbox{\scriptsize free}}
\; = \; \left(
e^{ \te...
...}^2 }
+ {\mathcal O}\left( \frac{T^2}{s} \right) ,
\end{displaymath} (3.7)

which can be made as accurate as desired by increasing $s$. Note that the upper bound of the error does not depend explicitly on $\hbar$ any more, such that the quality of the algorithm, as given by the error term ${\mathcal O}(T^2/s)$, does not vary with $\hbar$. Unitarity of the approximating propagator is granted by ${\hat{p}}^2$, ${\hat{x}}^2$ being Hermitian.

As opposed to ${\hat{U}}_{\mbox{\scriptsize free}}$ in the form of equation (3.2), application of its approximant (3.7) is straightforward. For each of the $s$ time steps per period $T$, expressions of the following type have to be evaluated:

$\displaystyle \Big< x \, \Big\vert \, e^{ \textstyle -\frac{iT}{2\hbar s} \, {\...
...tstyle -\frac{iT}{2\hbar s} \, {\hat{x}}^2 } \,
\Big\vert \, \psi \Big>
\; = \;$     (3.8)
    $\displaystyle \hspace*{-2.35cm}
\int\limits _{-\infty}^\infty\!\! {\mbox{d}}p' ...
...rt \, x' \Big>
\left< x' \left\vert \psi \rule{0.0cm}{0.35cm} \right> \right. ,$  

i.e. the ${\hat{x}}$-dependent exponential is multiplied to the initial state in the position representation, the result is transformed into the momentum representation, where the ${\hat{p}}$-dependent exponential is easily applied; finally, the resulting expression is transformed back into the position representation.

In this way, the free harmonic oscillator dynamics of arbitrary states is obtained essentially by successively switching between the position and momentum representations; this can be done quite effectively using the technique of fast FOURIER transformation (FFT) [CT65,EMR93]. Nevertheless, the need for applying the FFT very often in order to obtain the desired accuracy is the limiting bottleneck for this algorithm.3.1

The full kicked harmonic oscillator dynamics is then obtained by alternately applying the simple ${\hat{U}}_{\mbox{\scriptsize kick}}$ of equation (2.34) and the TROTTER-approximated ${\hat{U}}_{\mbox{\scriptsize free}}$ as described above.

Effectively, the TROTTER algorithm implies a discretization of the dynamics with respect to time; the period of time $T$ of the unperturbed harmonic oscillator dynamics is split into $s$ sufficiently small time steps of length

\delta t \; := \; \frac{T}{s}.
\end{displaymath} (3.9)

This may be used to implement an adaptive step size control: each time the approximate propagator (3.7) is applied, $s$ can be tuned to be small enough to make the algorithm not too time-consuming on the one hand, and large enough to achieve the desired accuracy on the other hand; the latter can be checked by applying (3.7) twice each time, for example using $s$ and $2s$, and comparing the two resulting iterated states. (The same applies to the number of nodes used for storing $\left< x \left\vert \psi \right> \right.$ and $\left< p \left\vert \psi \right> \right.$, but in contrast to $s$ this number cannot be changed easily in the course of the calculation, but rather has to be specified before the algorithm is started.)


... algorithm.3.1
In [See95] a stroboscopically kicked (and otherwise free) particle is studied quantum mechanically using a similar algorithm of successive transformations between the position and momentum representations (with modifications as described in [HM87] which regrettably are not applicable to the present case of the cosine kicked harmonic oscillator where the kick potential is not of finite range). Similarly, in subsection 5.1.1 and in [Lan94] the quantum dynamics of the cosine-kicked rotor is discussed. In these systems, only a single pair of FOURIER transformations per kick period is needed, because between the kicks the (free) dynamics in the momentum representation is trivial. The present subsection 3.1.1 illustrates the way in which the situation becomes considerably more intricate when replacing the truly free dynamics between the kicks with the ``free'' harmonic oscillator dynamics.

next up previous contents
Next: The GOLDBERG Algorithm Up: Finite Differences Previous: Finite Differences   Contents
Martin Engel 2004-01-01