Journal of Low Temperature Physics, Vol. 135, Nos. 3/4, May 2004 (© 2004)
The Schrödinger Cat Family in Attractive Bose Gases
T.-L. Ho and C. V. Ciobanu (1)
Department of Physics, The Ohio State University, Columbus, Ohio 43210
E-mail: ciobanu@engin.brown.edu
(Received December 18, 2003; revised January 21, 2004)
We show that the ground state of an attractive Bose gas in a double well
evolves from a coherent state to a Schrödinger Cat like state as the tunneling
barrier is decreased. The latter exhibits super-fragmentation similar to the
ground state of a spin-1 Bose gas with antiferromagnetic interactions. We
also show that the fragmented condensates of attractive and repulsive Bose
gases in double wells lead to very different interference patterns.
FOR
KEY WORDS : L; L.
G
N
I
AD
1. INTRODUCTION
E
R
F
O
Since the discovery of Bose–Einstein condensation (BEC) in atomic
gases, there have been many studies of the phase coherence properties of
Bose systems. A frequently studied example is a repulsive Bose gas in a
double well, where the ground state changes from a coherent state to a
Fock state (or number state) as the tunnelling is decreased. 1–4 In a coherent
state, the condensates in the two wells are phase-coherent, whereas they
are incoherent in a Fock state. A number of authors 3, 5, 6 have shown that
despite their differences, coherent states and Fock states cannot be distinguished by interference experiments which measure the density profile of
two overlapping condensates that are initially separated. The reason turns
out to be a subtle one: the projective nature of the measurement process
introduces coherence into a Fock state, so much that it gives rise to the
same interference pattern as the coherent state. In contrast, there are few
studies of the phase coherence of Bose gases with attractive interactions.
Such studies are not only important for conceptual reasons, but also relevant for current experiments. 7
PRO
1
LY
N
O
Corresponding author, currently in the Division of Engineering, Brown University, Providence,
Rhode Island 02912.
257
0022-2291/04/0500-0257/0 © 2004 Plenum Publishing Corporation
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 1/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
258
T.-L. Ho and C. V. Ciobanu
We will show that both repulsive and attractive interactions lead to
‘‘fragmented’’ condensates, characterized by more than one macroscopic
eigenvalue in the single-particle density matrix. However, the ground state
of an attractive Bose gas in double wells is in general Schrödinger Cat-like
and is ‘‘superfragmented.’’ 8 The latter means that the internal number
fluctuation DN 2 is enormous, i.e., of order N 2. In contrast, DN 2 ’ 0 for
the Fock state (which is fragmented), and DN 2 ’ N for coherent states.
Because of the huge number fluctuations, the measurement process is
unable to project the system into a single coherent state as in the case
of Bose condensates with repulsive interactions; instead, the system is
projected in a statistical mixture of Fock-like states with unequal number
of particles in each well. Consequently, the interference pattern of attractive and repulsive Bose gases in double wells are very different in the
fragmented regime.
Before proceeding, we shall comment on two recent related developments. There have been increasing efforts in recent years to create a
Schrödinger cat state in condensed matter and atomic systems. The
Schrödinger Cat state (or ‘‘Cat state’’ for short) is a superposition of two
macroscopic quantum states. It is often used to illustrate the peculiarity of
Quantum Theory, which admits such superpositions even though they have
never been observed on the macroscopic scale. The usual explanation is
that Cat states are highly unstable against entanglement with environment. 9 Despite such difficulties, Cat states have recently been created on
the mesoscopic scale. 10 Since the discovery of BEC, there have been
suggestions to produce ‘‘bigger’’ Cat states using atomic Bose condensates.
These include using binary mixtures of Bose condensates, 11 splitting the
microtrap that contains the condensate, 12 as well as performing projections
on the coherent states. 13 The attractive Bose gas in a double well is mathematically similar to the binary mixtures of Bose condensates 11 but is a
much simpler system: we will illustrate that it has Cat-like ground states over
a wide range of parameters, a fact that has not been widely appreciated.
In a separate development, recent studies of spin-1 Bose gases with
antiferromagnetic interactions show that strict spin conservation leads to
fragmented ground states, which can be Fock like (with zero spin fluctuation) or superfragmented (with N 2 spin fluctuation), depending on the
magnetization of the system. 8 Although the superfragmented states of
spin-1 Bose gas and ground states of attractive Bose gas in double wells
assume very different forms, the origin of their formation is the same and
arises from the existence of degenerate minima of the interaction energy,
and the quantum fluctuations between those minima. In contrast, a Fock
state is caused by a single deep minimum in the interaction energy, which
strongly suppresses the number fluctuations. This mechanism can be seen
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 2/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
The Schrödinger Cat Family in Attractive Bose Gases
259
in spin-1 Bose gases, and is best illustrated in the double well example
below.
2. THE TWO-SITE BOSE-HUBBARD MODEL
Consider the Hamiltonian
U
H=−t(a †b+b †a)+ [na (na − 1)+nb (nb − 1)],
2
(1)
with na +nb =N, where a and b destroy bosons at site (well) a and b,
na =a †a, nb =b †b, N is the total number of bosons, t > 0 is the tunneling
matrix element, and U is the interaction between particles which has the
same sign as the s-wave scattering length. 14 Equation (1) is meant to
describe the actual system when reduced to the lowest doublet (a ± b).
Although other (residual) terms appear in the reduction to an effective
Hamiltonian, and t, U in Eq. (1) depend on the condensate density, we
shall not consider such features here because they do not affect the basic
physics of the problem (i.e., the competition between tunneling and
interaction). Instead, we shall focus on the simple model given by Eq. (1),
which deserves to be studied in its own right.
For a system with N bosons, the Hilbert space is
3 |aP=: N2+a, N2 − a 84 ,
|a| [ N/2,
(2)
where |Na , Nb P — a †Na b †Nb |0P/`Na ! Nb ! is the state with Na bosons at site a
and Nb bosons at site b. We shall take N even, a choice of convenience that
has no effects on our results. If |YP=; a Ya |aP is an eigenstate with
energy E, Eq. (1) implies
EYa =−(ta − 1 Ya − 1 +ta Ya+1 )+Ua 2 Ya ,
(3)
where ta =`N2 ( N2+1) − a(a+1). It is clear that ta strongly favors large
amplitudes near a=0. This is reflected in the non-interacting ground state
1
|CP=
(a †+b †) N |0P,
`2 NN!
(4)
which shows that Y Ca is a Gaussian centered at a=0 with a width
2 1/4 −a 2/N
sc =`N/2 since Y Ca % ( pN
) e
for N ± 1. When U > 0, the potential
Ua 2 suppresses particle fluctuations between a and b, narrowing the
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 3/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
260
T.-L. Ho and C. V. Ciobanu
Gaussian toward the delta-function da=0 , (corresponding to the Fock state
|N/2, N/2P). 1, 2 The ground state for general U > 0 (referred to as the
‘‘repulsive family’’) is
Y (+)
a =Gs (a),
2
1
2
1
U
2=
2+
s
N tN
2
1/2
(5)
2
where Gs (a)=e −a /2s /(ps 2) 1/4. Equation (5) 15 follows from the fact that
in the continuum limit Eq. (3) near a=0 is the Schrödinger equation of
a simple harmonic oscillator. For U < 0, the potential Ua 2 favors large
amplitudes at a=± N/2. It competes strongly with tunneling and tends to
split the Gaussian Y Ca at U=0 into two weakly overlapping Gaussians.
These features are found in the numerical solutions of Eq. (3), which also
show that the ground state (referred to as the ‘‘attractive family’’) is, to a
good approximation, given by
L
R
Y (−)
a =Q(Y a +Y a )
(6)
(R)
(L)
where Q is a normalization factor, Y (R)
a =Gs (a − A), Y a =Y −a . Figure 1
shows the amplitudes Ya of the ground state |YP as a function of a, for
different UN/t [ 0. In other words, the ground state is a superposition of
the coherent/Fock-like states |LP and |RP, |Y (−)P=Q(|LP+|RP),
|LP=C Gs (q) |q̃PN+ , N − ,
q
|RP=C Gs (q) |q̃PN − , N+ ,
(7)
q
Fig. 1. Numerical solution of Y (−)
calculated from Eq. (1) for difa
ferent UN/t for a system with N=1000 particles. The results for
U < 0 can be well fitted by the functional form in Eq. (6) with
begins
parameters A and s shown in Fig. 2. For UN/t < − 2, Y (−)
a
to split into two Gaussians. The split-up is complete for UN/t
% − 2.1.
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 4/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
The Schrödinger Cat Family in Attractive Bose Gases
261
where |q̃PN+ , N − — |N++q, N− − qP, N± =N2 ± A, and N+ is the average
number of a (or b) particle in |LP (or |RP). We also found numerically that
the quantities A g — 2A/N and s g — s/sc (sc =`N/2) depend only on the
combination UN/t, 16 with a dividing behavior at UN/t=−2 as shown in
Fig. 2. As UN/t decreases below − 2, we have A Q N/2 and s Q 0. The
system is driven towards the ‘‘extreme’’ Cat state
|N, 0P+|0, NP
|Cat gP=
.
`2
(8)
In fact, the overlap between |LP and |RP vanishes rapidly when A > s, or
A g/s g > `2/N. For a system with N=1000 particles, we have found that
|LP and |RP cease to overlap when UN/t < − 2.1; thus, for these values of
UN/t, the system is essentially a superposition to two non-overlapping
mesoscopic condensates.
3. FRAGMENTATION AND SUPER-FRAGMENTATION
That the interaction will lead to fragmentation can be seen from the
single-particle density matrices of both |Y (+)P and |Y (−)P, which can be
written as
r̂ —
R
Oa †aP Oa †bP
Ob aP
†
S R S
N 1 x
=
,
2 x 1
Ob bP
†
2
where x=e −1/(4s ) for the repulsive case (i.e., |Y (+)P with s given in Eq. (5)),
2
2
2
2
2
) 2+e −(A+1/2) /s )/(1+e −A /s ) for the attractive
and x=(e −1/(4s ) `1 − ( 2A
N
case (i.e., |Y (−)P with s given in Eq. (6) and Fig. 2). The eigenvalues of r̂
are l ± =(N/2)(1 ± x). When U Q 0, we have x=1 for both signs of U,
and r̂ has only one macroscopic eigenvalue (l+=N and l − =0). This is
expected since the ground state reduces to the coherent state |CP. As |U|
increases, x Q 0 for both signs of U. The condensate becomes fragmented
since both eigenvalues of r̂ become macroscopic, l+, l − Q N/2.
However, when examining the internal number fluctuations DN 2a —
O(Na − ONa P) 2P, one finds DN 2a =s 2/2 for |Y (+)P, which vanishes as U/tN
2
2
increases. In contrast, for |Y (−)P DN 2a =s4 + −A2A 2/s 2 is of order N 2 since
1+e
A ’ N and s ’ `N for large |U| N/t. The values of DN 2a obtained from
the numerical solution of Eq. (3) are shown in Fig. 2. We have plotted
ln(4 DN 2a )/ln N instead of DN 2a because the former assumes the simple
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 5/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
262
T.-L. Ho and C. V. Ciobanu
Fig. 2. With the solutions of Eq. (1) for U < 0 well fitted by Eq. (6),
we find that for different values of N, U, t, each of the parameters
A g — A/(N/2), s g — s/sc , and ln(4 DN 2a )/ln N falls onto a single
curve when plotted against UN/t. Note that ln(4 DN 2a )/ln N=2
for the Cat state (|N, 0P+|0, NP)/`2, and that ln(4 DN 2a )/ln N=1
for the coherent state |CP, Eq. (4). At UN/t=−2, ln(4 DN 2a )/
ln N=1.32.
values of 2 and 1 for the ‘‘extreme’’ Cat state |Cat gP and the coherent state
|CP, respectively. Figure 2 shows that DN 2a reaches a substantial fraction of
its maximum value of N 2/4 over the interval − 3 < UN/t − 2. Thus, from
the viewpoint of achieving a superfragmented structure, it is not necessary
to go deeply into the Cat regime characterized by U < 0 and |UN/t| ± 1.
At first sight, such an interval may seem physically irrelevant because
for any finite interval D(UN/t), the corresponding range in U/t vanishes as
1/N in the thermodynamic limit, and the system is either deep in the Cat
regime or is a coherent state depending on whether U < 0 or U=0. This,
however, is not true. In general, the tunneling matrix element t depends on
an energy barrier Vo in an exponential fashion, ln t 3 − Vo . 17 Thus, the
range DVo corresponding to the interval D(UN/t) is only proportional to
ln N, making the system highly tunable from one regime to another. Two
other considerations also make this transition region relevant. Firstly,
quantum gases are mesoscopic instead of macroscopic systems, with
N < 10 5 instead of N ’ 10 23. Secondly, recent experiments have shown that
the scattering length of Rb 85 can be tuned through zero by varying the
external magnetic field. 18 Therefore, it is possible to have a system with a
small UN even for N as high as 10 5. The fact that the ground state changes
continuously from a coherent to a Cat-like structure over a wide range of
parameter allows us to explore the phase coherence of the system while the
crossover takes place.
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 6/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
The Schrödinger Cat Family in Attractive Bose Gases
263
4. INTERFERENCE OF THE ATTRACTIVE BOSE GAS
IN SUPERFRAGMENTED REGIME
As mentioned, many authors have pointed out that there are no distinctions between the interference patterns of a coherent state and a Fock
state. Numerical evidence of this effect was given Javanainen and Yoo
(JY). 3 Later, using analogies with quantum optics, Castin and Dalibard
(CD) simulated the particle collection process by the ‘‘beam splitter’’
operators a ± b and showed explicitly how the measurement process
changes a Fock state into a coherent state. 6 The exact spatial pattern,
however, has not been derived. In the following, we shall modify the calculation of CD to obtain the spatial interference pattern of the attractive
family described by Eq. (6). We shall see that the operators for particle
collection are different from the ‘‘beam splitter’’ operators. Furthermore,
our calculation furnishes another derivation of JY’s result 3 when applied to
Fock states.
To illustrate the key features, it is sufficient to consider the onedimensional case. At time t=0, the trap is turned off and the condensates
in wells a and b begin to expand and to overlap. For simplicity, we shall
assume the atoms expand as non-interacting particles for t > 0, which
2
m
implies k̂(x, t)=`i2p(t
> ds e iM(x − s) /2(t k̂(s). If a and b are Wannier states
localized at ± xo , we then have k(s) % c(e iz(x)/2a+e −iz(x)/2b), z(x)=
2
2
m
e iM(x +x o )/2(t. If the bosons were photons, the
(2xo M/(t) x, and c=`i 2p(t
operators (a+b) k+ (a − b) k− represent detecting different number of photons
in different beam splitters. However, in an interference experiment, particle
detections are products of k̂(x, t), which are specific combinations of a and
b rather than products of (a+b) k+ (a − b) k−.
Next we consider a sequence of D particle detectors located at xi ,
i=1, 2, . . ., D. The joint probability of detecting a total of k particles
(k ° N) with ki particles in the detector at xi can be written as (see
Appendix):
(N − k)!
k!
||Ô |YPN || 2,
P({ki })=
N! < Di=1 ki !
(9)
where Ô=< i k̂ k i (xi ) removes ki particles at xi , |YPN is a normalized state
with N particles, and ; Di=1 ki =k. The measured density at xi is given by
the most probable set {k̄i } which optimizes P({ki }), i.e., n(xj )=k̄j . In case
there are many such sets, the measured density will change from experiment to experiment, as each experiment samples a different optimal set.
For Cat like states (Eq. (6)), we have P({ki })=Q 2[PL ({ki })+PR ({ki })
+PLR ({ki })], where PL and PR are Eq. (9) evaluated at |YP=|LP and |RP,
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 7/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
264
T.-L. Ho and C. V. Ciobanu
and PLR is Eq. (9) with the norm replaced by OL| Ô †Ô |RP+OR| Ô †Ô |LP.
Since PLR depends on the overlap of |LP and |RP, it is non-vanishing only
within the range of UN/t for which A < s. For A > s, P is dominated by
PL and PR , and the interference pattern n(xj ) is proportional to k̄ Lj +k̄ Rj ,
where {k̄ Li } and {k̄ Ri } are the optimal set of PL and PR , respectively.
To calculate {k̄ Lj }, we consider the coherent state
1
|a, bP N =
(ua †+vb †) N |0P
`N!
(10)
where u — e −ia/2 cos b2 , v=e ia/2 sin b2 , cos 2 b2 — N+/N. It follows from Eq. (10)
that Oa †aP=N+, Ob †bP=N− , N± — N/2 ± A. Equation (10) has the
expansion
−i(A+q) a
|a, bPN =C Y (o)
|q̃P N
q e
+, N −
q
(11)
2
where Y (o)
q =Gso (q) with s o =2N+ N− /N up to 1/N+ , 1/N− corrections.
Inverting Eq. (11), we can express |q̃P N , N and hence |LP [Eq. (7)] in
+
−
terms of |a, bPN . We then have
Ô |LP=C C f(q) F
q
p
−p
da ia(A+q) D
e
D [Wx i (a)] k i |a, bPN − k
2p
i=1
(12)
where f(q)=Gs (q)/Gso (q), Wx (a)=e iz(x)/2u+e −iz(x)/2v, and C=c k `(NN!
.
− k)!
Since |a, bPN − k is close to the Fock state e iAa |N+, N− P, and since the range
of q is restricted to 1/s, the combination e ia(A+q) |a, bPN − k varies slowly
with a. Since ki ± 1, one can use the method of steepest descents
to determine the phase angle a g that maximizes the magnitude
< Di=1 |Wx i (a)| k i , and we have Ô |LP 3 |a gP. This shows that the measurement process (specified by the set {ki }) projects the state |LP into the
coherent state |a g, bP.
To calculate PL , we note that OaŒ, b | a, bP % exp[− ( N 8− k sin 2 b)(aŒ − a) 2
A
+i N (N − k)(aŒ − a)] for N − k ± 1. We then have
PL ({ki })=gk! F
D |W (a)| 2k i
da
xi
|f̃(a)| 2 D
2p
ki !
i=1
2
(13)
2
−2
where f̃(a)=; q f(q) e iaq=`so /s ; q e −q /2s eff e iaq, where s eff
=s o−2 +s −2,
8p
and g=`(N − k) sin 2 b . As the ground state becomes more Cat-like, s Q 0,
f̃(a) has a weak a dependence. To find the optimal set {k̄ Li }, we rewrite the
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 8/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
The Schrödinger Cat Family in Attractive Bose Gases
265
ki ! in Eq. (13) using Stirling’s formula, and optimize the product in Eq. (13)
subject to the constraint ; Di=1 ki =k. One then obtains k̄ Li 3 |Wx i (a g)| 2, or
5
k̄ Li =l 1+sin b cos
1 2Mx(t x − a 26 ,
o
g
(14)
where l is a constant. The stationary condition for a g can be derived in
a straightforward manner and will not be presented here. Repeating the
calculation for PR we find k̄ Ri =k̄ Li . The measured density n(xj )=k̄ Lj +k̄ Rj
therefore consists of a uniform background and a sinusoidal oscillation
with wavevector ( −1M(2xo /t). The amplitude of oscillation sinb vanishes
as the systems becomes more Cat-like, since b Q 0 as A Q N/2. This is in
contrast to the repulsive case, where the interference pattern is independent
of the barrier height which causes the system to fragment into a Fock state.
APPENDIX
Equation (9) was derived in Ref. 6 using continuous quantum measurement theory. It can also be derived from the following elementary
considerations. For a complete set of states (labelled by p) with creation
operators {A †p }, the probability of detecting a particle in state i in an
N-particle system |YP is:
OA † A P
||A |YP|| 2
.
P (1)= i i Y = i
NO1PY
N |||YP|| 2
(15)
The probability of detecting a second particle in state i after the first particle is detected (provided that the state evolves very little between detections) is
||A i (A i |YP)|| 2
P (2)=
.
(N − 1) ||A i |YP|| 2
(16)
The joint probability of detecting ki particles in state i is
(N − k)! ||A ki i |YP|| 2
(k − 1)
(1)
P (k)
P
·
·
·
P
=
.
i
i
i
N!
|||YP|| 2
(17)
Equation (9) is obtained by generalizing to the detection of more than one
state, and with A i replaced by k̂(xi ). The combinatoric factors in ki
account for the arbitrary ordering in the detection of different i states.
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 9/10 - Op: SD - Time: 08:28 - Date: 25:03:2004
266
T.-L. Ho and C. V. Ciobanu
ACKNOWLEDGMENTS
Tin-Lun Ho would like to thank Mark Kasevich for pointing out the
similarity between the attractive Bose gas in a double well potential and the
systems described in Ref. 11, and for discussions of his experiments during
the Aspen Workshop (1999). We gratefully acknowledge support from
NASA (Grant NAG8-1441) and NSF (Grants DMR-9705295 and DMR9807284). C. V. Ciobanu was supported through a Presidential Fellowship
from The Ohio State University.
REFERENCES
1. The general formulation of Fock to coherent transition was given in D. Rokhsar, condmat/9812260.
2. R. W. Spekkens and J. E. Sipe, Phys. Rev. A 59, 3868 (1999).
3. J. Javanainen and S. M. Yoo, Phys. Rev. Lett. 76, 161 (1996).
4. See also A. J. Leggett and F. Sols, Found. Physics 21, 353 (1991); A. J. Leggett, in
Bose–Einstein Condensation, A. Griffin, D. W. Snoke, and S. Stringari (eds.), Cambridge
University Press (1995), pp. 452–462.
5. J. I. Cirac, C. W. Gardiner, M. Naraschewski, and P. Zoller, Phys. Rev. A 54, R3714
(1996).
6. Y. Castin and J. Dalibard, Phys. Rev. A 55, 4330, (1997).
7. M. Kasevich, private communication.
8. The notion of superfragmentation was introduced in T. L. Ho and S. K. Yip, Phys. Rev.
Lett. 84, 4031 (2000).
9. See for example, W. Zurek, Physics Today 44 (10), 36, (1991).
10. C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, Science 272, 1131 (1996).
11. J. I. Cirac, M. Lewenstein, K. Mølmer, and P. Zoller, Phys. Rev. A 57, 1208 (1998).
12. A. M. Dudarev, R. B. Diener, B. Wu, M. G. Raizen, and Q. Niu, Phys. Rev. Lett. 91,
010402 (2003).
13. D. Gordon and C. M. Savage, Phys. Rev. A 59, 4623 (1999); J. Ruostekoski, in Lecture
Notes in Physics: Directions in Quantum Optics, H. J. Carmichael et al. (eds.), Springer,
Berlin (2001); J. Ruostekoski, M. J. Collett, R. Graham, and D. F. Walls, Phys. Rev. A
57, 511 (1998); D. A. R. Dalvit, J. Dziarmaga, and W. H. Zurek, Phys. Rev. A 62, 013607
(2000).
14. For attractive Bose gases we shall consider a density range in which the condensate
remains stable.
15. The result represented by Eq. (5) has been reported previously in A. Imamoglu,
M. Lewenstein, and L. You, Phys. Rev. Lett. 78, 2511 (1997).
16. For U < 0, the change in potential energy and tunneling energy by removing a particle
from a to b is ’ UN and t respectively, hence the scaling variable |U| N/t. For U > 0, the
corresponding quantities are U and Nt, and hence a different scaling parameter U/tN as
seen in Eq. (5).
17. The tunnelling barrier Vo is proportional to the intensity of the laser that produces the
barrier, so it can be controlled by tuning the laser intensity.
18. S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Phys.
Rev. Lett. 85, 1795 (2000).
File: KAPP/901-jolt/135_3-4 485758(10p) - Page : 10/10 - Op: SD - Time: 08:28 - Date: 25:03:2004