Holmes, CA

Milburn, GJ

G. J. Milburn

C. A. Holmes

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Physical Review Letters

Dissipative Quantum and Classical Liouville Mechanics of the Anharmonic Oscillator

University of Queensland eSpace

VOLUME 56, NUMBER 21 PH&SICAL REVIEW LETTERS 26 MA.Y 1986
Dissipative Quantum and Classical Liouville Mechanics of the Anharmonic Oscillator
G. J. Milburn
Department ofPhysics and Theoretical Physics, Faculty ofScience, Australian /Vational University,
Canberra, Australian Capita/ Territory 2601, A ustralia
aIld
C. A. Holmes
Department ofMathematics, Emperical College, London SW728Z, United Kingdom
(Received 14 January 1986)
%e present a solution for the dynamics of an anharmonic oscillator coupled to a zero-
temperature heat bath. Comparison of observable properties in a classical and quantum description
uses true joint phase-space probability densities. The time evolution of the density in the quantum
case is rapidly "reduced" to that given in the classical description. The rate of reduction is propor-
tional to the product of the damping rate and the oscillator's initial energy. Quite rapidly, typical
quantum recurrence effects are destroyed and the classical "whorl" structure restored. %e point
out the close similarity with rapid destruction of quantum coherence through dissipation.
PACS numbers: 03.65.-w, 03.65.8z, 05.20.—y, 05.30.—d
A dynamical phase-space picture requires the simul-
taneous specification of position and momentum at
varying time instants. Such measurements made upon
an ensemble of identically prepared systems yield joint
phase-space probability densities. In a quantum
description, it is well known that the possible joint
probability densities which arise are restricted to a sub-
set of those occurring in a classical description. In a
previous paper' one of us investigated the quantum
and classical dynamics of joint phase-space probability
densities describing simultaneous measurements of
position and momentum for a particular anharmonic
oscillator model with the Hamiltonian
H = Hp+ (ix/lt top)Hp,
where Hp is the free Hamiltonian of the simple har-
monic oscillator. The appearance of lr in Eq. (1) is
simply to provide a convenient energy scale; H is in
units ofh top when Hp is in units of @cop.
When the system is described classically, an initial
Gaussian joint density displaced from the origin
develops into a "whorl" 2; contours of the initial densi-
ty undergo a rotational shear and as time proceeds
spiral out from the origin. However, when the same
system is described quantum mechanically the density
undergoes a more complicated evolution; "interfer-
ence" fringes develop, the initial state recurs at a fixed
period, and no whorl develops. This more complicated
behavior is manifested in the density evolution equa-
tion by the appearance of second-order derivatives
with complex coefficients. Similar behavior for a relat-
ed anharmonic model has recently been reported by
Takahashi and Saito.3
In this Letter we consider the effect of dissipation
on the evolution of phase-space densities for a particu-
lar anharmonic oscillator model. We show that quite
apart from the overall contraction of phase space, the
effect of dissipation is to destroy the interference
(3)
terms, prevent a recurrence of the initial state and to
restore the classical whorl structure. The destruction
of interference effects becomes much more rapid as
the average energy of the initial state increases (i.e.,
when the initial density is concentrated at a large ra-
dius). This dependence on the initial energy is similar
to the decay of off-diagonal coherence in the harmonic
oscillator. ~ We are thus lead to interpret the interfer-
ence terms as a manifestation of quantum coherences,
between parts of the density wrapped on neighboring
tori.
We incorporate dissipation in our model by coupling
the anharmonic oscillator to a reservoir of oscillators
which we assume to be at zero temperature. The
reservoir is then eliminated and a Markovian master
equation for the oscillator-density operator in the in-
teraction picture is obtained. A unique phase-space
density is associated with the density operator by
means of a bounded positive map from the state space
of density operators to the classical state space of prob-
ability densities on phase space. This map is then used
to transform the evolution equation for the density
operator to an evolution equation for the density on a
two-dimensional phase space.
The Hamiltonian for the coupled oscillator-reservoir
system is
H = Hii+ (p,jir p) Hp2 + Ht + Hit, (2)
where Htt is the free Hamiltonian for the reservoir and
Ht is the oscillator-reservoir interaction Hamiltonian.
Fol«wing Agarwal we assume Ht to be of the form of
position-position coupling and we make the rotating-
wave approximation. Thus
Hi= Xtgtabt~+ gt a bt,
where
a = ( gW)'~'q+ t (X,) »2p, - (4)
b, =( t!&)"'0,+t(2' )-'i'P, .
1986 The American Physical Society 2237
VoLUME 56, NUMBER 21 PHYSICAL REVIEW LETTERS 26 MAv 1986
a and b, are the annihilation operators for the oscillator and reservoir, respectively. By use of standard techniques
the following master equation is obtained in the Markov approximation and in the interaction picture:
Bp/8~ = —ip, [(a'a ) ', p] + (y/2) [ap, a'] + (y/2) [a,pa'], (6)
~here 7 = coot.
Classical states are represented by joint probability densities on phase space Q(n, n, 7). [We choose to work
with the complex parameters a defined as
a = (~0/&)'i'q+ i(2ea)0) 'i'p,
where (q,p) is a point in the phase space of the oscillator. ) Quantum states are represented by density operators on
Hilbert space. However, there exists a bounded positive operator on quantum state space which maps every quan-
tum state to a unique element in a subset of classical state space. The map T is defined by
T p(t) =—Tr[p(t) ~a) (a{]= Q(n, n";t), (7)
where {a) is a minimum-uncertainty state. The resulting joint probability density describes the simultaneous mea-
surements of "approximate" canonical variables q and p relative to some apparatus specified by the choice of the
state ~n).7 More generally, {a)(a{d2n is an example of an effect-valued measure. s In fact, the effect ~n) (a~ is a
special example of a more general effect for simultaneous position and momentum measurements. 9 It corresponds
to a situation where the measuring instrument is operated at zero temperature and thus adds no excess noise to the
measurement.
Under the map T the evolution equation in quantum state space becomes an evolution equation in classical
state space. For the evolution equation (6) we have
(n, n';r ) = {[t) (y/2+ i p ( I + 2 {a { )n ) +c c.] + i p, a282 —i p, a'28, + y t) 8 I Q (n, a', ~ ),
where tI =—8/Bn. When y=0 Eq. (8) reduces to the classical Liouville equation ask 0.'
The evolution equation (8) is solved subject to the initial condition
Q(n, n', 0) =exp[ —{n—no{2]. (9)
Jh
With respect to the measurements defined by T discussed above, this density corresponds to the quantum state
p = {ao)(ao{ where ~ao) is a particular coherent minimum-uncertainty oscillator state. The solution is
Q(n, n";~) =exp( —lnl') X (q'.p'. ) '(nno)'(n"no)'f(t)"'""exp{ —lao{'[f(t)++]/(I+ +)],
es -o
(10)
where
5= (p —q)/K,
f(t) =exp[ —Kv —iv(p —q)],
(»)
(12)
(»)
(14)
v = 2p T,
K = (y/2p. ).
The method of solution is similar to that discussed in
(1).
In Figs. 1(a)-1(c) we have plotted Q(a, a";~) for
various values of v, {no{2, and K. When y=0,
Q (a, n';~) exhibits the complicated recurrence
behavior discussed in (1). The initial Gaussian starts
to form a whorl; however, as the leading edge of the
whorl begins to encircle the trailing tail concentrated
on an interior torus, "interference fringes" begin to
develop. This is evident in Fig. 1(a). These interfer-
ence fringes become more evident as the evolution
~ proceeds and eventually result in a complete re-
currence of the initial state at v =2m (up to a phase of
e' ). This is in distinct contrast to the evolution of a
similar density in a classical description, Fig. 1(c); no
"interference" fringes arise, no recurrence occurs,
and the density becomes concentrated on a "thin"
spiral.
However, when ya0, but small, we see a significant
change in the evolution of the density. As shown in
Fig. 1(b) the interference fringes are suppressed and
the density becomes similar to its classical analog, Fig.
1(c). However, it is prevented from becoming too
narrowly concentrated on a phase-space spiral through
the bounded nature of the map T . In fact, for
{ao{)) 1 a very small value of y is sufficient to
suppress the interference features, which decay on a
time scale of ({no{K) ', much shorter than the time
scale of the overall contractive dynamics K
This behavior is reflected in the moments. For ex-
ample, one may show that
(a(t)) =aoeXp[ —iv(1 —iK)/2 —{no{2(l—e '"t' '"i)/(I —lK)]. (IS)
VoLUME 56, NUMBER 21 PHYSICAL REVIEW LETTERS 26 MAv 1986
For vK (( 1 and K (( 1, this may be written
&«r)) =~oe '""exp' —l~ol'(I —e '") }exp' —i~el'~« '"l.
For K = 0& a (r) ) exhibits the typical periodicity expect-
ed; however, for K &0 there is a rapid collapse of the
recurrence on a time scale (Inol2K)
Such a decay rate is quite similar to the decay of
off-diagonal coherence in general quantum systems. 4
For example, a damped simple harmonic oscillator ini-
tially in the superposition state
(a)
p(0) = &(l~) + Ip) )(&~l+ &pl)
evolves to
(17)
y, y a
The off-diagonal elements decay via the factor
&~lp)" ' ""'=«p( ——,'(l~l'+ Ipl' 2~'p—)~K}
for small vK. If n and P lie along a common radius we
put u=xe'a and p= (x+Sx)e'a and the decay factor
becomes
(b)
&~IP)('- "")=exp( -Sx".]2}, (19)
which is small for large 5x. It thus seems reasonable
to suggest that the interference terms discussed above
arise from off-diagonal coherence due to the density
becoming concentrated on adjacent tori. The off-
diagonal coherence and consequently the interference
are suppressed by dissipation at a rate proportional to
the radii of the tori on which the density becomes con-
centrated. The sizes of the radii are determined by the
average energy in the initial state.
There are a number of ways to interpret the dissipa-
tion discussed in this Letter. In the first instance, one
may claim that the inclusion of dissipation models the
effect of connecting an actual measuring instrument to
the system, which generates the observed phase-space
densities. Interpreting a measuring device as essential-
ly a reservoir from the point of view of the measured
system to which it is coupled has been discussed in a
number of recent papers. '0 '2 The subsequent irre-
versible interaction ensures that the state of the sys-
tem becomes diagonal in a unique basis. It thus ap-
pears that the very act of observing the dynamic prop-
erties of a quantum system is sufficient to produce a
rapid approach of the observed behavior to that ex-
pected from a classical model.
Of course, quite apart from whether or not the dissi-
pation is attributed to the act of measurement, one
may claim that no system is truly isolated and some
dissipation, no matter how small, should be included.
For example, the anharmonic oscillator discussed here
FIG. 1. Plot of Q-function probability density on phase
space. The origin is in the center of each figure. (a) Quan-
tum case a0=4i, v =0.6, K =0.0; orientation of position (q)
and momentum (p) axes is shown. (h) Quantum case
~0 = 4i, v = 0.6, ~ = 0.5. (c) Classical case ao = 4i, v =0.6,
K =0.0.
2239
VoLUME 56, NUMBER 21 PHYSICAL REVIEW LETTERS 26 MAY 1986
may be realized as an intracavity field made to interact
with a nonlinear crystal. This mode is unavoidably
coupled to the electromagnetic vacuum through the
cavity mirrors even though for a high-0 cavity this
coupling will be very small. The essential conclusion
of the discussion here is that although the dissipation
may be so small as to be almost unobservable as a gen-
eral contraction of phase-space dynamics, it may be
sufficient to induce a reduction of quantum dynamics
to classical dynamics as the energy of the system ap-
proaches the classical scale. The effectiveness of even
small dissipation in suppressing quantum coherences
has also been emphasized by Caldeira and Leggett'3
and Zurek. '4 We suggest that such a model may pro-
vide a unified description of microscopic and macro-
scopic dynamics. A similar suggestion has recently
been made by Ghirardi, Rimini, and Weber. '
iG. J. Milburn, Phys. Rev. 33, 674 (1985).
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Phys. 122, 26 (1979).
3K. Takahashi and N. Saito, Phys. Rev. Lett. 55, 64S
(1985).
40. F. Walls and G. J. Milburn, Phys. Rev. A 31, 2403
(1985).
sG. S. Agarwal, Phys. Rev. A 4, 739 (1971).
6W. H. Louiseti, Quantum Statistical Properties ofRadiation
(Wiley, New York, 1973).
7E. B. Davies, Quantum Theory of Open System (Academic,
New York, 1979).
A. Barchielli, L. Lanz, and G. M. Prosperi, in Proceedings
of the International Symposium on Foundations of Quantum
Mechanics, Tokyo, I989, edited by S. Kamefuchi et ai (Ph. ys-
ical Society of Japan, Tokyo, 1984), p. 165.
9A. Barchielli, L. Lanz, and G. M. Prosperi, Nuovo
Clmento 728, 79 (1982).
toW. H. Zurek, Phys. Rev. D 24, 1516 (1981).
iiW. H. Zurek, Phys. Rev. D 26, 1862 (1982).
&20. F. Walls, M. J. Collett, and G. J. Milburn, Phys. Rev.
D 32, 3208 (1985).
&3A. 0. Caldeira and A. J. Leggett, Phys. Rev. A 31, 10S9
(1985).
i4W. H. Zurek, in Proceedings of the NATO Summer School
on Statistica/ Mechanics, edited by G. Moore and M. O. Scul-
ly (Plenum, New York, 1986).
i5G. C. Ghirardi, A. Rimini, and T. Weber, in Quantum
Probability and Applications I1, Lecture Notes in Mathematics
Vol. 1136 (Springer-Verlag, Berlin, 1985).


Dissipative Quantum and Classical Liouville Mechanics of the Anharmonic-Oscillator

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