# 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 September 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.