# 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 [23] and Fox and Eidson [24].

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 [25] and Milonni,

Ackerhalt, and Galbraith [26].

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) ,

(27r)2

(3.3)

where F(k,a,T; t) is the quantum characteristic function

defined by

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

a

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 32 p 1 8+gx

ap a ap(3.8)

and

a a 2 a

QGD 1 ax ax1 x2 ax3

(3.9)

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 [18].

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. [20] 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 [20], 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. [19] 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

equations

.(t)= C0x2(t) ,

1c2()=-X 1 (t)-2gq(t)x3(t) ,(3.10)

~3(t)=2gq(t)x2(t) ,

4l(t)=p(t) ,

p(t)= - 2q(t)_gx (t) .8493

45

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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.