Quantum suppression of chaos in the spin-boson model
G. A. Finney
*
and J. Gea-Banacloche
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701
Received 6 November 1995; revised manuscript received 8 May 1996
We identify a transition to chaos in the semiclassical spin-boson model that occurs for relatively large boson
fields, as one of two periodic orbits becomes unstable. We have studied the quantum dynamics in the vicinity
of this transition, as characterized by i phase-space trajectories followed by quantum expectation values, ii
spectra of such trajectories, iii subsystem entropies for the spin and boson systems, and iv growth of
operator variances for the boson system. We find that the transition to chaos in the classical system has no
apparent effect on any of these variables, in the spin-
1
2
case. This is in disagreement with some claims made in
earlier studies of this system. S1063-651X9610106-9
PACS numbers: 05.45.+b, 03.65.Sq, 42.65.Sf
I. INTRODUCTION
The spin-boson Hamiltonian
H =
1
2
0
z
+ a
†
a + g
x
a +a
†
1
where
i
are Pauli spin matrices and a
†
, a are boson cre-
ation and annihilation operators may describe a number of
physical systems, including a two-level atom coupled to a
single mode of the quantized radiation field 1 or to its own
center-of-mass motion in an atomic trap 2. Taking expec-
tation values in the Heisenberg equations and making a fac-
torization assumption i.e.,
z
a
z
a , etc. yields the
semiclassical equations
a ˙
1
= a
2
, 2a
a ˙
2
=- a
1
-gx , 2b
x ˙ =-
0
y , 2c
y ˙ =
0
x -4 ga
1
z , 2d
z ˙ =4 ga
1
y , 2e
where a
1
= a +a
†
/2, a
2
= a -a
†
/2 i , x =
x
, y =
y
,
and z =
z
. The system 2 has long been known to exhibit
chaos for certain values of the parameters 3. A continuing
question has been whether or not there is any signature of
this semiclassical chaos in the solutions, especially the dy-
namics, of the full quantum problem 14–8. This is the
subject of the present paper.
The system 1 is not of the ‘‘standard’’ form of most
quantum chaos problems, which typically involve particles
in externally prescribed potentials, but this very difference
makes it interesting in the context of the hitherto relatively
little studied ‘‘dynamically driven’’ systems, where new
phenomena may arise due to, for instance, the nonunitary
nature of each subsystem’s evolution 9. Moreover, the sys-
tem 1 is the restriction to the j =
1
2
subspace of the more
general Hamiltonian
H =
0
J
z
+ a
†
a +2 gJ
x
a +a
†
, 3
which describes, in general, a quantum rotor coupled to a
quantum harmonic oscillator Eq. 3 may also describe a
collection of N =2 j two-level atoms interacting with the ra-
diation field. In some limits if the reaction of the rotor on
the oscillator and/or the quantum nature of the latter are neg-
ligible this may be like a periodically driven rotor, which is
an archetypal model for quantum chaos 10.
It has been shown by Graham and Ho
¨
hnerbach see 4b
for details that a classical Hamiltonian for this problem may
be written as
H
c
=
0
I
1
+ I
2
+4 g I
2
J
2
-I
1
2
cos
1
cos
2
, 4
where I
i
and
i
are canonical action-angle variables and J
2
is a constant, corresponding to the total angular momentum
for correspondence with the quantum problem,
J
2
= j ( j +1) or, in terms of N equivalent two-level atoms,
J
2
=( N /2)( N /2 +1) . It can be verified immediately that
the canonical equations of motion
˙
1
= H
c
/ I
i
and
I
˙
i
=- H
c
/
i
are identical to the equations 2 obtained
from the factorization assumption in the quantum Heisenberg
equations of motion, with the correspondence
a
1
= I
2
cos
2
,
a
2
=- I
2
sin
2
,
5
x = J
2
-I
1
2
cos
1
,
y = J
2
-I
1
2
sin
1
,
z =I
1
.
In the classical problem 4, J is an arbitrary constant that
can, in fact, be scaled away by redefining the coupling con-
stant g and the boson field amplitude, as in I
1
' =I
1
/ J ,
I
2
' =I
2
/ J , and g ' =g J . For the quantum Hamiltonian 3,
*
Present address: USAFA/DFP, U.S. Air Force Academy, Colo-
rado Springs, CO 80840.
PHYSICAL REVIEW E AUGUST 1996 VOLUME 54, NUMBER 2
54 1063-651X/96/542/14498/$10.00 1449 © 1996 The American Physical Society