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
depends on only, and not on .
On the other hand, the application of the free propagator,

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

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 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 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
are
caused by the fact that
is a sum of
two noncommuting
contributions, namely the kinetic and potential terms
and
.
Since
,
a simple
BAKER-CAMPBELL-HAUSDORFF (BCH) decomposition
into two unitary factors
is not possible:

An alternative and more suitable approach is based on the
*TROTTER product formula*.
For two operators , and
, the exponential
of the sum of the operators can be expressed as an infinite ordered
product of exponentials of the individual operators:

(3.4) |

For purely imaginary , the error induced by this approximation, due to being finite, can be estimated using [DR87]

where is a suitable operator norm. Note that, using the estimate (3.6), for commuting the simplest BCH result is confirmed again.

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

As opposed to
in the form of equation (3.2),
application of its approximant (3.7) is
straightforward.
For each of the time steps per period ,
expressions of the following type have to be evaluated:

(3.8) | |||

i.e. the -dependent exponential is multiplied to the initial state in the position representation, the result is transformed into the momentum representation, where the -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 of equation (2.34) and the TROTTER-approximated as described above.

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

(3.9) |

- ...
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.