Anomalous diffusion and environment-induced quantum decoherence
Luca Bonci,
1
Paolo Grigolini,
1,2,3
Adam Laux,
1,4
and Roberto Roncaglia
1
1
Dipartimento di Fisica dell’Universita ` di Pisa, Piazza Torricelli 2, 56127 Pisa, Italy
2
Istituto di Biofisica del Consiglio Nazionale delle Ricerche, Via San Lorenzo 26, 56127 Pisa, Italy
3
Department of Physics of the University of North Texas, P.O. Box 5368, Denton, Texas 76203
4
Department of Solid State Physics, Lora ´nd Eo ¨tvo ¨s University of Sciences, Muzeum krt. 6-8, Budapest 1088, Hungary
Received 21 November 1995
We study the anomalous diffusion resulting from the standard map in the so-called accelerating state, and we
observe that it is determined by unusually large times of sojourn of the classical trajectories in the fractal
region at the border between the chaotic sea and the acceleration island. The quantum-mechanical breakdown
of this property implies a coherence among so slightly different values of momentum as to become much more
robust against environment fluctuations than the quantum localization corresponding to normal diffusion.
S1050-29479601107-9
PACS numbers: 03.65.Bz, 05.40.+j, 05.45.+b
I. INTRODUCTION
In quantum mechanics the superposition of two indepen-
dent states, both referring to a physically plausible solution
of the Schro ¨dinger equation, is still a plausible solution of
the Schro ¨dinger equation. If the two solutions have a classi-
cal meaning, for instance, they are two distinct trajectories, it
is not clear how to interpret their superposition, which, yet,
according to quantum mechanics should be regarded as a
valid picture of reality. A widely accepted solution of this
problem is afforded by the decoherence theory DT recently
made quite popular by Zurek 1 but corresponding to a
viewpoint shared by many other authors 2. It rests on the
assumption that there are no such things in nature as isolated
systems. Thus if a macroscopic oscillator, with mass M and
friction , is prepared in a superposition, c
+
| + +c
-
| - ,
of two states, | + and | - , corresponding to two distinct
positions separated by the distance Q , as an effect of the
environmental fluctuations the coherence between these two
states is lost within a time defined by
t
dec
=
2
2 M Q
2
, 1
with = k
B
T . Notice that this decoherence time is propor-
tional to the square of the Planck constant. Thus it turns out
to be virtually instantaneous, if the parameters of the oscil-
lator have values corresponding to a macroscopic body
yielding for the denominator of 1 an extremely large value.
This is the key physical interpretation behind all the ap-
proaches resting on the essential role of the environment to
recover classical from quantum physics 1–3. A careful ex-
amination of this theoretical approach to the transition from
quantum to classical mechanics 4 reveals that it rests on the
assumption that the system of interest, as well as the envi-
ronment, is driven by ordinary equilibrium statistical phys-
ics. The main purpose of this paper is to study the conse-
quence of rejecting the assumption that the system undergoes
an ordinary process of statistical mechanics, while retaining
the plausible assumption that the environment still obeys this
condition.
II. QUANTUM-CLASSICAL CORRESPONDENCE
IN ORDINARY STATISTICAL MECHANICS
Let us now focus our attention on the kicked rotator 5.
In the classical case, this system is described by the standard
map
p
t +1
= p
t
+K sin
t
, 2
t +1
=
t
+ p
t +1
mod 2 .
This is an area-preserving map driving the discrete evolution
of a classical rotator kicked at regular intervals of time by an
impulsive torque with a strength proportional to K sin. In
the case of strong chaos ( K 1) the variable K sin, as well
as , can be perceived as a quickly fluctuating variable, re-
sulting in a diffusional process for the momentum p . The
fluctuating variable K sin is characterized by a finite-time
scale, and in the long-time limit this diffusion process turns
out to be Gaussian on the basis of the central limit theorem
and the second moment p
2
( t ) becomes a linear function of
time. We shall refer to this process as normal diffusion and
as an example of ordinary statistical mechanics.
To discuss the corresponding quantum-mechanical de-
scription let us observe the time evolution of the Wigner
distribution subsequent to an initial condition with the
Wigner distribution virtually identical to the Liouville distri-
bution. This implies the adoption of a Planck constant
much smaller than the volume of the classical phase space
explored by the system during the process of observation.
Note that, as usual in the field of quantum chaos, the
‘‘Planck constant’’ we adopt is actually a parameter express-
ing the ratio of the characteristic classical action to the real
Planck constant. Zurek and Paz 6 show that as a result of
the process of fragmentation of the Liouville distribution, the
quantum corrections to the classical Poisson brackets clas-
sical term become important at the time
t
=
1
ln
, 3
where is the Lyapunov coefficient of the classical and
chaotic trajectories and is a scale parameter proportional to
PHYSICAL REVIEW A JULY 1996 VOLUME 54, NUMBER 1
54 1050-2947/96/541/1127/$10.00 112 © 1996 The American Physical Society