It is well known that any multimode positive definite quadratic Hamiltonian can be transformed into a Hamiltonian of uncoupled harmonic oscillators. Based on this theorem, the multimode thermal squeezed coherent states are constructed in terms of density operators. Decoherence of multimode thermal squeezed coherent states is investigated via the characteristic function and it is shown that the decohered (reduced) states are still thermal squeezed coherent states in general

Yeh, Leehwa

English

Yeh, L.

eScholarship - University of California

Decoherence of Multimode Thermal Squeezed Coherent States

NASA Technical Reports Server

N93-27345
Quantum Harmonic Oscillator In A Thermal Bath
Yuhong Zhang
Biophysics Laboratory
Center for Biologics Evaluation and Research
Food and Drug Administration
8800 Rockville Pike
Bethesda, MD 20982, USA
ABSTRACT
In this talk, we briefly review the influence functional path-integral treatment of quan-
tum Brownian motion. We report on a newly derived exact master equation of a quantum
harmonic oscillator coupled to a general environment at arbitrary temperature. We apply
it to the problem of loss of quantum coherence.
353
https://ntrs.nasa.gov/search.jsp?R=19930018155 2020-03-17T06:06:02+00:00Z
INTRODUCTION
Recently there has been considerable interest in quantum Brownian motion. It was
motivated by possible observation of macroscopic effects in quantum systems. Among
them are quantum tunneling with dissipation [1], loss of quantum coherence due to system-
environment interaction [2], just name a few. The newest application of quantum Brownian
motion is in quantum cosmology, where the issue of quantum-to-classical transition of an
open system is very important [3]. These issues also appear in semiclassical theory of
early universe in which noise, fluctuation and dissipation play important roles in particle
production, back reaction, phase transition, inflation and galaxies formation [4]. In these
problems, the interaction between a system and its environment is quite complicated giving
rise to nonlocal dissipation and colored noise.
The effect of nonlocal dissipation and colored noise in quantum Brownian motion is
an outstanding problem, which has been studied only to a limited extent. In some limiting
cases, the quantum master equation (the time evolution equation) for the reduced density
matrix of the Brownian motion has been derived before by different authors with different
methods. These cases are all in the class of ohmic environment, for which the dissipation
is always local [5]. It corresponds to having a linear damping force proportional to the
velocity of the Brownian particle classically. The noise associated with the dissipation is
colored at low temperature.
Our contribution reported in this talk is the derivation of an exact master equa-
tion for the reduced density matrix of a Brownian harmonic oscillator linearly coupled to
a general environment (with a general thermal bath spectral density) at arbitrary tem-
perature [6]. In our model, the environment is a set of bath harmonic oscillators with
different natural frequencies. The environment is at a thermal equilibrium state. The
system (Brownian particle) is brought to contact with this thermal bath. The derivation is
done from first principles of statistical and quantum physics with Feynman path-integral
354
method and Feynman-Vernon influence functional formalism [7]. This master equation
can accommodate all possible forms of the nonlocal dissipation kernel and nonlocal noise
kernel. It is a linear partial differential equation with time dependent coefficients. The
non-Markovian character resides in these coefficients. In particular we examine the cases
of ohmic, subohmic and superohmic environment and compute these time dependent co-
efficients numerically. We show that all the previous master equations obtained otherwise
are just special examples of our master equation.
INFLUENCE FUNCTIONAL
Let us briefly review the Feynman-Vernon influence functional formalism of quantum
open system. Consider a Brownian particle with mass M = 1 and natural (bare) frequency
_. The environment is modeled by a set of harmonic oscillators with mass mn and natural
frequency wn. The Brownian particle is coupled linearly to each bath oscillator with
strength Cn. The total action of the combined system plus environment is
S[x,q]= S[x]+ SE[q]+ si.,[x, q]
+/0''SZfCnXqnl
 sE{1 1
. 2_q_- 2_q_} (1)
where x and qn are the coordinates of the particle and the n-th bath oscillators.
It is well known that the time evolution of the total density matrix of the system plus
environment _(t) is governed by the following quantum Liouville equation
• d^ [H, _(t)]
,h_p(t) = (2)
In the coordinate representation, the solution of the above quantum Liouville equation can
355
be written as
-t-cw +(x) +oo +oo
--00 --00 --00 --00
' ' '-0)x Y(xs, qf, x_,@,tlxi,qi,x_,qi,O ) p(xi,qi;xi,qi,
(3)
where
I i 0J(xl,qs,x),q),t l x,,qi,xi,q,, )
# #
XJ X f qn I qns
:i..iox,.,io,
Xi X( qni tqnlI
exp_{S[x,q]- S[x',q']} (4)
is the propagator of the total density matrix in path-integral form. Here q represents the
full set of bath oscillator coordinates and the subscript i and f denote the initial and final
variables.:
We are only interested in how the dynamics of the system (the Brownian particle)
under the influence of the environment (all bath oscillators). The quantity containing this
information is the reduced density matrix of the system
-t-oo -t-_
--OO --OO
q;x',q')5(q-q') (5)
which is propagated in time by the the evolution operator
--00 --00
I iX's,t l =,,x,,O) pr(=,,x,,o ) (6)
If we assume that at t = 0 the system and the environment are uncorrelated
_;(t= o)= _.(o) × _(o), (7)
then
' 0)=&(zs,x's,t l zi,zi,
l
x! x!
l.xl.,
=i X(
i
356
d'S
exp _ {S[.] - S[x']} F[x, x'] (8)
The functional factor FIx, x'] in (8), called influence functional, is defined as
+oo +oo +oo q! q!
",...',:I'_I'_.I",:I"_So_'
--oo --oo --oo qi q_ (9)
It is first introduced by Peynman and Vernon [7].
For the problem described by (1), the influence functional can be computed exactly.
The result is:
t 81
i ids2[x(s,) x'(s,)]r1(s, s2)[x(s2)+x'(s2)].I.,.,=e-.{-./'.l- -
0 0
l sl
'_f _,I'..[-.,)'.,)_-.,-)I-..).)]}
0 0
(10)
where
is the noise kernel, also
-1-OO
0
d
_(_)= _ _(_)
COS _8 (11)
(12)
and
-+OO
_(_) f d_ I(_)= -- COS 0.,18
u¢
0
is dissipation kernel. Here I(w) is the dissipation spectral density defined as
(13)
2
I(w) = E 5(a_ -wn),) Cn (14)
_?TlnOJ n
n
The kernels r/(s) and v(s) are generally non-local. There exists an important relation
between the noise and dissipation kernels, known as the fluctuation-dissipation relation.
It can be written as
-+00
.(s) = f d_'
--00
K(s - s') 7(s') (15)
357
where the kernel K(s) is
+oo
= -- w coth hw
7r
0
cosws (16)
which is independent of the dissipation spectral density I(w)
EXACT MASTER EQUATION
The detailed derivation of the exact master equation of a quantum harmonic oscillator
with influence functional (10) has been published in Ref.: [6]. Here we just give that
equation below
0
ih_ pr(z, x', t)
[ h2 02
= l--_ (_2
+ _a_(t)(x 2- x,2)pr(x,_',t)
- i_r(t)(z - z')( O O_,)p_(x,x',t)
- ir(t)h(t)(x - x') 2 p_(z,z',t)
, a a t)
+ hF(t)f(t)(z - x )(_x + -_;x') p,(x,x',
_2°2) + _-a_(x2_ _,2)} p,(x,_',t)
(17)
where the time dependent coefficients are
d,(t)
r(t)- 2_1(_)
5_2(t) = d2(t) - 2r(t)_2(t)
f(t) = 2 a12(t) e2(t) - cl(t)
_2(o----_ + 2r(t)_2(o)
h(t) = u2(t) f(t) + 8 all(t) +
el(t)- c2(t)
r(t)
The time dependent functions ci(t), di(s) and el(t) in (18) to (21) are
t t t
0 0 o
+ a:_(_,_,)].(_ - _)_(_)
(18)
(19)
(20)
(21)
(22)
i
i
!
=
i
i
!
!
!
358
t2 / ds rl(t- s)ui(s) (23)di(t)
0
t
el(t) = /ds v(s)ui(s) (24)
P
0
The elementary functions ui(s) satisfy the following boundary value problem
$
d_2 +2 e_'_(_-_')u,(J)+_g_,(8)=o (28)
0
Ul(O)= 1, Ul(_)= o u2(o)= o, _2(t) = 1
and
v,(8) = _2(t - 8) v2(_)= u,(t - 8) (26)
The Green function G12(8,,82) in (22) is
G12(8,,82) = u l ( 81 )'/z2( 82)0(81 - 82) - 112(81)u,(82)0(S 2 - 8,)
_1(82)U2(82) _ U1(32)?_2(82 ) (27)
A similar expression for G21(sz,82) can be written in terms of v,(8). In all the above
equations, the index i runs from 1 to 2.
Let us take a closer look of this master equation. The first line corresponds to the usual
unitary Liouvillian evolution, which is independent of the system-bath interaction. The
second line corresponds to a time-dependent frequency shift (frequency renormalization).
The third line contains a dissipative term with a time-dependent dissipative coefficient
F(t). The last two lines contain two diffusive terms with time-dependent coefficients. All
of these terms depend on the system-bath coupling. Further, one can see that all these time-
dependent coefficients vanish at t = 0, when the initial uncorrelated condition is assumed
valid. The frequency shift and the dissipation coefficient depend only on the dissipation
kernel while the diffusion coefficients depend on the noise kernel. From (11), (12) and (13),
we find all these time dependent coefficients are determined by the dissipation spectral
density (14).
359
A simpler closed formula for the time-dependent coefficients in the master equation
can be found in the weak coupling limit (up to the first order in the coupling constant
between the system and the bath),
t
t
5ft2(t) = 2 / ds rl(s ) cosf_s (28) i
0
t _--
1 / ds r/(s ) sinfts (29)r(t) =
0
t
F(t)f(t) = -_ ds v(s) sinf_s (30) '
0
t
r(t)h(t) = f ds v(s) cosf_s (31) -
0
!
i
I
EXAMPLES
An important class of dissipation spectral density is
2 02 )n--1 '°2
I(w) = _70w(5 e- A-'r (32)
where A is the physical cutoff frequency and & is another frequency scale usually taken to
be A. The environment is classified as ohmic if n = 1, as supra-ohmic if n > 1 or as sub-
ohmic if 0 < n < 1. It is important to introduce the physical cutoff frequency because, on
physical grounds, one expects the spectral density to go to zero for very high frequencies.
It is clear to see that after introducing the physical cutoff frequency, the dissipation kernel
(13) is a non-local kernel even for the ohmic environment.
We have numerically computed the coefficients of the master equation given by (18)
to (21) for three different environments, namely, ohmic (n = 1), subohmic (n = 0.5)
and superohmic (n = 3). The damping constant is % = 0.3 and the cutoff frequency
360
!
J
is A = 2000. The bare frequency fi is determined from the renormalized one, namely,
fi2 = _2 + 5fl2 __ f12 = 1 (see explanation in Ref. [6]). Both high temperature region
(T = 105) and low temperature region (T = 10) have been studied. The numerical plots
of these time dependent coefficients and detail analysis could be found in Ref. [6].
APPLICATION: QUANTUM DECOHERENCE
As a simple application,, we discuss the damping of the interference between two
Gaussian wave packets [8].
Let ¢1,2(x, t) be the wave functions of Gaussian wave packets located initially (t = 0)
at x = +x0 respectively with the same initial spread a
_)l,2(X, 0) = Ne-_ (33)
Let ¢(x, t) be the wave function of a system consisting of the superposition of these two
wave packets,
¢(x,t) = ¢l(x,t) + ¢2(x,t) (34)
The density matrix of the system can be written as the sum of three parts
p(x, x',t) = ¢(x,t)¢t(x',t) = pl(x,x',t) + p2(x, x',t) + Pi,t(x,x',t) (35)
The probability density function
P(x,t) =1 ¢(x,t)12= p(x,x,t) (36)
can also be written as the sum of three parts
Pl(x,t) = Pl(x,t) + P2(x,t) + Pint(x,t) (37)
By using the influence functional (10) and the master equation (17), we get
±
P1,2(x,t) =/_r(t) exp [ 2a2(t) ] (38)L
361
and
p_.,(x,t) = 2x/sl(x,t)_e-s(o cos¢(t) (39)
where
_1(0)
x0(0 = _(0)x° (40)
and
a(t) = [_ + _) + 4a,__,O,/uSI,)J (41)
are respectively the position and spread of the wave packet at time t,
2_0[_(0)z + _(0)z0]¢(t) =
1 + 8a, l(t)c_ 2 + 4_12(0)a t
is the oscillatory angle (which is present even in the absence of the environment) and
(42)
D(t) = 4all(t)x]
1 + 8all(t)a 2 + 4h_(O)a 4 (43)
is the decay factor (which is present only because of the environment). It is this last term
depicting the decay of interference between the two wave packets which is usually regarded
as providing a measure of decoherence.
We have numerically computed the decay factor e -D(t) for all the cases described in
the previous section. The results could be found in Ref. [6].
ACKNOWLEDGEMENT:
Most of the material in this talk is based on research completed in collaborations with
B. L. Hu and Juan Pablo Paz. This work is supported in part by NSF grant PHY-8717155.
362
REFERENCES
[1] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); A. J. Leggett et
al, Rev. Mod. Phys. 59, 1 (1987).
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[3] J. B. Hartle and S. W. Hawking, Phys. Rev. D28, 1960 (1983); A. Vilenkin, Phys.
Lett. l17B, 25 (1985); Phys. Rev. D27, 2848 (1983) D30, 509 (1984).
[4] A. A. Starobinsky, in Field Theory, Quantum Gravity and Strings, ed. H. J. deVega
and N. Sanchez (Springer, Berlin 1986).
[5] W. G. Unruh and W. H. Zurek, Phys. Rev. D40, 1071 (1989).
[6] B. L. Hu, J. P. Paz and Y. Zhang, Phys. Rev. D 45, 2843 (1992).
[7] R. P. Feynman and F. L. Vernon, Ann. Phys. 24, 118 (1963); A. O. Caldeira and A. J.
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=


Decoherence of multimode thermal squeezed coherent states

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Decoherence of multimode thermal squeezed coherent states

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