Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,497
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SEMICLASSICAL CHAOS, THE UNCERTAINTY PRINCIPLE,.
calculation, upon increasing the interaction strength g.
The Liapunov coefficients are not a monotonic function
of g . However, for the five values of g chosen, larger
g's involve larger Liapunov coefficients. Curve (e) corre-
spond to a fully chaotic regime for the equation of
motion. We thus see that classical chaos is consistent
with the quantum-mechanical uncertainty quickly reach-
ing a fixed asymptotic value. This signals that quantum
fluctuations grow so fast as to quickly invade the whole
region of the available phase space.
Figure 5 illustrates the relaxationlike nature of the
time evolution of (q(t)). This character is more distinct
when the QGD mechanism is included. The comparison
with the RWA condition points out an element of ambi-
guity of "irreversibility" as a proper indicator of semi-
classical chaos. As mentioned earlier in this paper (see
the remarks in Sec. IV on the entropy of Ref.  con-
cerning the RWA Hamiltonian), the initial condition
with the oscillator in the coherent state supplemented by
the RWA can generate some degree of irreversibility.
This is due to the fact that the dynamics of the system de-
0 5 10 15 20 25 30
t (arb. units)
FIG. 5. Time evolution of (q(t)). In all curves g=10,
Co= f=- 2r, and the starting point is condition (i) of Sec. IV,
namely the oscillator in the coherent state, with (n ) = 10. The
full line is the result of the "exact" numerical calculation. The
dashed line denotes the result of averaging on the semiclassical
trajectories over the Wigner density corresponding to the above
condition (i). (a) refers to the complete Hamiltonian, with no
RWA and (b) to the RWA.
pends on an infinite number of eigenstates of the oscilla-
tor with incommensurate frequencies , and this is
reflected in a overall increase of the entropy of the oscil-
lator system. The results illustrated in Fig. 5(b) on the ir-
reversibility of (q (t)) under the RWA reflects this prop-
erty. We see indeed that, whereas the averaging of the
RWA trajectories over the condition (i) of Sec. IV results
in a nondissipative motion, the full quantum-mechanical
RWA calculation exhibits some elements of dissipation,
albeit less marked than in case without RWA. We be-
lieve that semiclassical chaos is manifest in a relaxation
behavior of the oscillator, but this is mixed in this case
with the irreversibility stemming from the conventional
source of a heat bath played by the coherent state.
Figure 6 shows that the QGD mechanism increases the
rate of growth of quantum fluctuations. A quite surpris-
ing result illustrated in Fig. 6 is that of the quantum
RWA. The initial increase of U of this case is the most
violent. We see that it grows very quickly, much faster
than any other process; it reaches a maximum, then it un-
dergoes a fast decrease until it reaches a first minimum.
For simplicity of description, we are not taking into ac-
count the ultrafast oscillations of this curve. The process
is repeated over and over again with minima of ever in-
creasing value. The envelope of the maxima of this curve
would correspond to a process of growth for the uncer-
tainty U(t) comparable to that triggered by semiclassical
chaos. However, if we focus our attention on the en-
velope of the minima, we see that this turns out to be, as
must be, an increase faster than the semiclassical RWA
case, but slower than the predictions of both quantum-
mechanical calculations without RWA.
The reason why the actual increase of quantum uncer-
tainty is indicated by the envelope of the minima [curve
(e) of Fig. 6] is given by an interesting quantum-
mechanical property recently discovered by Eiselt and
Risken . The Eiselt-Risken effect is a quantum
phenomenon associated with the well-known quantum
effect of collapses and revivals . The collapse process
is shown  to correspond to a splitting of the quantum
cloud into two distinct clouds. These two clouds quickly
move far apart, thereby provoking a fast increase of U.
The minima are easily explained by the fact that periodi-
cally, and precisely in the correspondence of the revivals,
the two distinct clouds merge again into a single cloud.
In conclusion, in this case U is not a reliable indicator of
the quantum uncertainty, since the splitting of the cloud
into two clouds results in a larger U, whereas the total
volume of the quantum cloud (the sum of the volumes of
the two single clouds) does not change appreciably com-
pared to the volume of the cloud before splitting. If we
replace the function U(t) with the envelope of its minima
[curve (e)], then we see that our theoretical expectations
[that for a case with semiclassical chaos the rate of in-
crease U(t) evaluated with the RWA is slower than
without RWA] are fulfilled.
VI. CONCLUDING REMARKS
Adopting the Wigner formalism we have established
two important quantum manifestations of semiclassical
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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/m1/8/: accessed October 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.