Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,495
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45 SEMICLASSICAL CHAOS, THE
consistent with the conclusion drawn in Ref. [20] that the
QGD mechanism counterbalances the action of diffusion.
It must be stressed that these remarks apply to the orien-
tational diffusion of the spin-{ dipole. Quantization rules
imply that only two directions of this dipole are admit-
ted, and free diffusion away from these two admitted
orientations produced by cl must be inhibited in some
way.
The calculation of the entropy confirms the generation
of irreversibility triggered by semiclassical chaos [see
Figs. 1(b) and 2(b)]. It must be pointed out that this is
distinct from the behavior of the entropy of the same sys-
tem, studied under the rotating-wave approximation
(RWA) by Phoenix and Knight [29]. In their case [29]
the entropy is shown to exhibit a sort of overall increase,
though not as marked as in the case of Fig. 2. However,
this latter increase has to do with the fact that a coherent
state can mimic a bath with an infinite number of degrees
of freedom [18]. The collapses occur because of the many
different frequencies contributing to the time evolution of
the mean values of interest, one for each eigenstate In ) of
the unperturbed Hamiltonian of the oscillator
(n =0, 1,...) necessary to build a coherent state. The re-A
0.0
v0.0 05 1.0 1.5 2.0 2.5 3.0
t (arb. units)
FIG. 2. Resonance dynamics of the spin-} system. In all
curves g= 20, coo= 27-r, and the initial condition (i) of Sec.
IV, namely the oscillator in the coherent state, with (n)= 10.
(a) Time evolution of (az(t)). The solid curve is the "exact"
numerical result. The dashed curve is obtained using the
Wigner equation without the QGD mechanism. (b) Time evolu-
tion of the entropy S(t) of spin-' system. The solid curve is
"exact" numerical result. The dashed line is the theoretical
maximum value of the entropy.UNCERTAINTY PRINCIPLE,...
vivals are not perfect because the frequency involved go
as n 1/2, not as an integer multiple of n [29]. As remarked
in Ref. [18], to make it clear that we are dealing with
another kind of irreversibility, it is convenient to reduce
the boson field to a single phonon as given by condition
(ii). In this case the RWA leads to reversible behavior,
whereas our exact calculations, with no RWA approxi-
mation, show that the entropy of the system exhibits a
behavior more and more distinctly irreversible upon in-
crease of g (see Fig. 3).
Note that the behavior illustrated by the time evolu-
tion of the entropy S is the result of an exact calculation,
thereby including the effect of the QGD mechanism. We
see that the QGD mechanism does not completely des-
troy dissipation. The ghosts of reversible behavior are
manifest by dips at integer multiple of 1, which have,
however, a decreasing intensity as time increases, thereby
implying a global irreversibility for sufficiently large cou-
pling coefficient g.
V. NUMERICAL RESULTS ON THE MOTION
OF THE OSCILLATOR
In this section we illustrate the results of calculations
on the dynamics of the oscillator utilizing the initial con-
dition (i) of Sec. IV. We also monitor the time evolution
of the corresponding uncertainty U [cf. Eq. (2.8)]. An ob-
servation of the same kind, on the oscillator rather than
on the spin-i dipole system, was recently made by Miiller
et al. [30]. These authors used the Husimi [31] rather
than the Wigner density to build up the quantum Poin-
care maps for the oscillator. As correctly stated by these
authors, the Wigner density, due to violent oscillations
between positive and negative values, cannot be used for
this purpose. However, we want to stress that their study
of the quantum Poincare map did not lead them to real-
ize that the growth of the quantum uncertainty due to
semiclassical chaos persists until the maximum possible
value of U admitted by the finite size of the available
phase space is obtained.
It must be pointed out that since the region of phase
space available to the oscillator has a finite size, the
growth of U must be limited from above. Thus we expect
that the growth of U(t) is characterized by the attain-
ment of an asymptotic value for t going to infinity. If our
arguments on the growth of quantum fluctuations as a re-
sult of chaos are correct, we expect that the phenomenon
of the growth of fluctuations to be negligible when the
RWA is made since this approximation is known to be
incompatible with chaos [23-26].
These theoretical expectations are checked using both
the solution without the QGD mechanism and the exact
numerical solution of the quantum-mechanical Liouville
equation (i.e., the solution of the Wigner equation of
motion, also including the QGD mechanism).
Figure 4 shows that our theoretical expectation on the
semiclassical chaos as a source of fast growth of quantum
fluctuations is correct. In this figure we show the time
evolution of U(t), as it results from an "exact" numerical8495
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Bonci, Luca; Roncaglia, Roberto; West, Bruce J. & Grigolini, Paolo. Semiclassical chaos, the uncertainty principle, and quantum dissipation, article, June 15, 1992; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc268887/m1/6/: accessed April 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.