Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,498
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BONCI, RONCAGLIA, WEST, AND GRIGOLINI
0 5 10 15 20 25 30
t (arb. units)
FIG. 6. The quantum uncertainty U(t) as a function of time. In all curves g= 10, wo= -Z = 271, and the starting point is condition
(i) of Sec. IV, namely the oscillator in the coherent state, with (n )= 10. Curves (a) and (b) refer to the RWA approximation. (a) is
the result of a calculation without the QGD mechanism and (b) is the result of a fully quantum mechanical RWA treatment. Curve
(e) is a guide for the eye and indicates the effective increase of quantum uncertainty, if we disregard the Eiselt-Risken effect .
Curves (c) and (d) refer to a calculation made with no RWA approximation. Curve (c) is the result of a calculation neglecting the
QGD mechanism, whereas (d) is the result of a calculation with no approximation.
chaos: these are irreversibility and unexpected growth of
quantum uncertainty. Surprisingly enough these quan-
tum manifestations of semiclassical chaos are accounted
for by using classical arguments, namely, the classical
contribution to the Wigner equation of motion, Lc1l of
(3.8), and the sensitive dependence on initial conditions of
classical chaos. Quantum mechanics, though, is subtlety
involved, since the QUP sets a constraint on the initial
Wigner density so that it cannot be a mere Dirac 8 func-
tion. Thus adopting classical arguments and the QUP,
one would be led to conclude with Fox  that "to
properly describe classical-mechanical chaos, one must
do quantum mechanics." In conclusion, on the latter
property we substantially agree with the point of view of
Fox . However, according to the theoretical analysis
of Fox, based on the adoption of arguments similar to the
expansion of the master equation in terms of the macro-
scopic volume (1/11 is the macroscopic parameter of his
analysis), the growth of quantum uncertainty is proved to
depend only on what we call Lcl1. In the spin-boson mod-
el here under study (it must be noticed that the Fox
analysis was applied in Ref.  to the multidimensional
version of the nonlinear oscillator studied for illustrative
purposes in Sec. IV, and not to the spin-boson model of
the present paper) we see, on the contrary, that also the
QGD mechanism contributes to the growth of quantum
uncertainty and makes it grow significantly faster than it
does with this mechanism absent. In a future publication
we will give theoretical arguments to account for the
speeding up role of the QGD mechanism on the growth
of the quantum uncertainty of the oscillator. In the case
where the semiclassical equations of motion have chaotic
solutions, the quantum fluctuations that in Ref.  are
expected to become macroscopic, within the spin-boson
model of this paper are shown to quickly fill the phase
space which would be explored by the single semiclassical
and chaotic trajectory. This means an extended cloud of
quantum uncertainty the size of which is the available
phase space. The consequences of this property might be
quite remarkable, since distinct regions of the same quan-
tum cloud should bear precise phase relations (with the
capability of producing interference effects of a single
particle with itself), and this appears to conflict with the
fact that within the semiclassical approximation the same
phase-space region is the domain of randomness.
The former consequence of semiclassical chaos, quan-
tum irreversibility, is intimately connected to the latter,
but surprisingly it has been overlooked by the investiga-
tions carried out so far in the field of quantum chaos.
Quantum irreversibility as a manifestation of semiclassi-
cal chaos is distinct from the irreversibility generated by
the interaction of the system with a bath with many de-
grees of freedom. The assumption that the radiation
field, the oscillator of our model, is in a coherent state,
produces irreversibility features  which are reminis-
cent of those observed in this paper (see Fig. 2). It must
be noticed indeed that Phoenix and Knight study the
RWA approximation of the spin-boson model and thus in
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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/m1/9/: accessed January 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.