# Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,491

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SEMICLASSICAL CHAOS, THE UNCERTAINTY PRINCIPLE,...

tioned above. This number drops precipitously with in-

creasing frequency of the external field. Koch et al. [15]

explain that one may use a severely truncated

"quasiresonant" state basis [14] in which the effective

density of states is much smaller than one would estimate

at first, so that quantum effects become significant. Final-

ly, it has been demonstrated by Mackey and Meiss [16]

that classical chaos is suppressed when the phase-space

area escaping through classical cantori (holes broken in

Kolmogorov-Arnold-Moser surfaces) in each cycle of the

electric field, is small compared to Planck's constant.

Such cantori are opaque to quantum transport, but do

not significantly impede classical transport until the driv-

ing field is raised to sufficiently high levels [15].

Herein we introduce a fifth explanation in that we use

the Wigner pseudoprobability density [17] and a generali-

zation of the attendant formalism to establish that the dy-

namics of a microscopic system can be explicitly separat-

ed into two pieces [18,19]. The Liouvillian describing the

evolution of the Wigner density is 4cl + LQGD; the first

operator cl gives the exact semiclassical description of

the evolution of the system, i.e., the evolution in the h-)0

limit; the second operator LQGD is the quantum generat-

ing diffusion (QGD) mechanism which has been recently

shown to constrain the classical dynamics in such a way

that the quantization prescriptions of quantum mechan-

ics [20] are satisfied. If it were possible to ignore the ac-

tion of the QGD mechanism, then we would make the

following two predictions regarding the quantum effects

of semiclassical chaos.

(i) The quantum fluctuations would undergo a massive

growth, unexpected on the basis of standard quantum

mechanics [21].

(ii) The time evolutions of the quantum expectation

values would become irreversible and dissipative [18,19].

Both property (i), which was recently pointed out by

Fox [21], and property (ii), more recently discussed by

Bonci et al. [18] and Roncaglia et al. [19], are immedi-

ately derived from the Wigner formalism. This is so be-

cause the adoption of the Wigner formalism supplement-

ed by the neglect of the QGD mechanism is equivalent to

the evaluation of the classical trajectories with a distribu-

tion of initial conditions. The initial distribution cannot

be a 8 function in the phase space variables in order to

satisfy the quantum uncertainty principle. In the case

where the classical approximation to the equation of

motion yields chaotic trajectories this initial uncertainty

is expected to grow exceptionally large, if not macroscop-

ic as it did in the Fox paper [21]. On the other hand,

since, according to the Wigner formalism, the time evolu-

tion of the expectation value of a quantum observable is

formally obtained as an average over the corresponding

classical trajectories, and these averages quickly lose their

correlations when the trajectories are chaotic, the result

is expected to be a relaxation. In other words, the time

evolution of a quantum expectation value would be

roughly equivalent to the correlation function of a ran-

dom process [18,19]. We use numerical techniques to

verify these theoretical expectations and to assess the role

of the QGD mechanism on both quantum irreversibility

and the growth of quantum fluctuations. We study theseproperties with the help of a spin-boson model and exam-

ine the dynamics of the oscillator (boson field) as well as

that of the spin-{ system.

The outline of the paper is as follows. Section II is de-

voted to a brief review of the Wigner method. In Sec. III

we extend the Wigner formalism to include spin and to

initiate the study of the spin-boson Hamiltonian. The

quantum irreversibility triggered by semiclassical chaos

and its competition with the restraining role of the QGD

mechanism is illustrated in Sec. IV for the spin-} system.

In Sec. V we show that the sudden growth of quantum

fluctuations is a manifestation of semiclassical chaos.

Another, less marked manifestation, is the increased rate

of the process of regression to equilibrium of the oscilla-

tor. It is also shown that the QGD mechanism markedly

enhances the growth of quantum fluctuations. Section VI

is devoted to concluding remarks.

II. THE WIGNER DISTRIBUTION

The Wigner distribution allows one to express

quantum-mechanical averages in the same form one

writes for classical averages. In this formalism every

operator in Hilbert space, corresponding to a physical ob-

servable, is associated, via the Weyl transformation [22],

to a function of suitable variables [17]. Consider a parti-

cle in one dimension with position operator 4 and

momentum operator p such that the operator A (4,f) has

the associated function A w(q,p):

Aw(q,p)=J+ f dz exp iPz q--- A q+ z

00 hz 2 2

(2.1)

Equation (2.1) can be used to put the expectation value of

the operator A into the form

( A(t)) =Tr[ Alt)]= fdq dp Aw(q,p)pw(q,p;t) ,

(2.2)

where pw(q,p; t) is the Wigner pseudoprobability func-

tion associated with the density matrix p,

pW(q,p;t)= f-dz(q -zpl q +z )exp [2iz .

(2.3)

Note the similarity between (2.2) and the classical

definition of an average. For convenience, we shall refer

to p, as the Wigner distribution or the Wigner density.

The most remarkable property of the Wigner distribu-

tion is its dynamical evolution, which also closely resem-

bles the classical situation. Let us suppose that the quan-

tum system is described by the following Hamiltonian:2= + V(2)= a- q2 V() .

2m 2m aq2(2.4)

The evolution of pw(q,p ; t) is then determined by [17]

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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/m1/2/: accessed September 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.