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