Semiclassical chaos, the uncertainty principle, and quantum dissipation Page: 8,493
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SEMICLASSICAL CHAOS, THE UNCERTAINTY PRINCIPLE,...
where b (Wg ) is the annihilation (creation) operator with
commutation relation [g, ]= 1. The physical system
modeled by (3.1) is that of a spin-} dipole interacting
with a magnetic field directed along the z axis and
modeled by an oscillator of frequency ft with space
coordinate =(b" + b) /V2O and momentum
p= i(gt -)V /2. Note that a, and Oz are spin ma-
trices. Herein we study the dynamics of the spin-boson
system with the initial condition
a I)+)=+I+) , (3.2)
and the boson field is in a state characterized by the aver-
age number of photons (n ).
A system of this kind has recently been studied by Gra-
ham and Hohnerbach  and Fox and Eidson .
They show that a semiclassical approximation of the sys-
tem with Hamiltonian (3.1) exhibits chaotic dynamics for
sufficiently strong coupling g when the system is in reso-
nance. Similar results have been previously found by
Belobrov, Zaslavskii, and Tartakovskii  and Milonni,
Ackerhalt, and Galbraith .
Further insight into the behavior of this system can be
obtained using the Wigner formalism. Let us extend the
arguments used earlier so as to be applicable to the spin-
boson problem. For this system the Wigner density reads
pw(x,q,p; t) fdk exp( -ikx)
S p (217)3
X dadr exp--i(qa+p)F(k,a,7;t) ,
where F(k,a,T; t) is the quantum characteristic function
F(k,a,7;t )=Tr exp[i(kxa, +k2 y +k3z )]
Xexp[i(craq +r-)]p(t)] , (3.4)
where p( t) is the density operator for the complete quan-
tum system. Note that (3.3) and (3.4) extend the usual
treatment of the Wigner distribution to include spin.
The variables x =(x1,x2,x3), p and q are the phase-
space variables associated, via a generalized Weyl rule of
the form (2.1), to the spin-boson operators: aj--*xj,
-+q, p-p. The quantum average values of these opera-
tors on the statistical system described by the density ma-
trix p t) are given by
( j(t)) = fdx dq dp x pw(x,q,p;t) , (3.5)
(q(t))= fdxdq dp q pw(x,q,p;t) . (3.6)
The corresponding equation of evolution for the Wigner
distribution (3.3) is given by
pw(x,q,p;t)= (Lcl+QGD)Pw(x,q,p;t) , (3.7)
where after some substantial algebra we obtain for the
spin-boson system [18,19]
L ~w x -x +2gq X3 xa -X2x3
cl 1 8x 2 2 1 3 x 2 2 3
2 p 1 8+gx
ap a ap
a a 2 a
QGD 1 ax ax1 x2 ax3
The operator c1 is identical to the Liouvillian of a classi-
cal dipole interacting with a classical oscillator, i.e., this
term alone corresponds to the semiclassical set of equa-
tions discussed by various authors [23-26]. We refer to
the calculations based on the study of the single trajecto-
ry solutions of the nonlinear dynamical equations as the
semiclassical predictions. In this regard we point out a
significant feature of the analysis that was apparently
overlooked in previous investigations having to do with
chaotic trajectories [23-26]. These earlier studies fo-
cused on individual trajectories and gave them physical
meaning. However, from (3.5) and (3.6) we see that even
when LQGD is neglected, it is only the ensemble that has
physical significance not the individual trajectories .
The term LQGD in (3.9) has a number of significant im-
plications. It has a diffusionlike structure, but the state
dependence of the "diffusion coefficient" results in its not
being positive definite. It has recently been shown by
Roncaglia et al.  that if the oscillator is coupled to a
heat bath so as to transmit to the spin-1 dipole standard
thermal fluctuations, then this term results in the average
value of the z component of the dipole changing from a
Langevin (classical) function to the hyperbolic tangent
(quantum). In other words, this term was coined quanti-
zation generating diffusion by these authors , precise-
ly because it ensures that the dipole retains its quantum
nature. The operator LQGD acts as an antiduffusional
mechanism, it competes against thermal fluctuations and
constrains the dipole, which otherwise would freely
diffuse over all possible orientations, to vacillate between
two possible orientations. In Sec. V, however, we show
that the influence of the QGD mechanism on the motion
of the oscillator is qualitatively different from that exert-
ed on the spin-2 system. This is in part expected, since
there are no intuitive reasons why the state quantization
of an oscillator should impede diffusion. We shall see via
numerical calculation that both dissipation and the rate
of the growth of quantum fluctuations are enhanced by
the QGD mechanism. The quantitative account of this
effect is left as a subject for a future investigation.
In Ref.  it is shown in detail that if the QGD mech-
anism is neglected, then the solution of (3.7) is equivalent
to determining the trajectories described by the system of
.(t)= C0x2(t) ,
1c2()=-X 1 (t)-2gq(t)x3(t) ,
p(t)= - 2q(t)_gx (t) .
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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/m1/4/: accessed September 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.