In this article we study the formation of the two modes Bose-Einstein
condensate and the correlation between them. We show that beyond the mean field
approximation the dissociation of a molecular condensate due to the chemical
potential curve crossing leads to the formation of two modes condensate. We
also show that these two modes are correlated in a two mode squeezed state.Comment: 10 page

A. Vardi

C. Menotti

C. Zener

D. Gottesman

D. Jaksch

D.J. Heinzen

Franco Dalfovo

K. Goral

L.D. Landau

M. A. Kayali

N. A. Sinitsyn

R. Wynar

S. Brundobler

U.V. Poulsen

V.A. Yurovsky

Yu.N. Demkov

Physical Review A

English

arXiv

Crossref

Formation of a two-component Bose condensate during the chemical-potential curve crossing

Kayali, M. A.

Sinitsyn, N. A.

Texas A&M University

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Formation of Two Component Bose Condensate During The
Chemical Potential Curve Crossing
M.A. Kayali, N.A. Sinitsyn
Department of Physics, Texas A&M University
College Station, Texas 77843-4242, USA
(Dated: June 26, 2018)
Abstract
In this article we study the formation of the two modes Bose-Einstein condensate and the cor-
relation between them. We show that beyond the mean field approximation the dissociation of a
molecular condensate due to the chemical potential curve crossing leads to the formation of two
modes condensate. We also show that these two resulting modes are correlated in a two mode
squeezed state.
PACS numbers: 03.75.Fi, 03.65.Nk, 34.50.Lf, 34.50.RK
1
INTRODUCTION
Recent studies on Bose-Einstein condensate have directed the attention of many theorists
and experimentalists to the multimodes Bose-Einstein condensate. In particular the appear-
ance of multimodes condensate from a molecular one via molecular condensate dissociation
process. In [1], [2] the authors studied the dissociation of a molecular condensate into a
one mode atomic condensate. Moreover, Yurovsky et.al showed that the resulting atomic
condensate is in a squeezed state. They also showed that the number of atoms and
squeezing parameters in the case of time dependent chemical potentials depend on the rate
of curve crossing and the coupling between the molecular and the atomic fields.
The problems of curve crossing can be divided into two different classes. These are the
r-dependent crossings and the t-dependent ones. In the former the potentials are coordi-
nates dependent and the crossing takes place in space, while in the latter the potentials
are time dependent. The standard approach to the crossing problems is the semiclassical
approach which is known as Landau-Zener (LZ) theory. The LZ theory provide a very
good description of the crossing problem for systems with potentials that cross at one point
only. However, if crossing between two potentials occur at two points or more then the
solution of the problem is possible by using a different approach which is out of the LZ theory.
In the production of an atomic condensate from a molecuar one the curve crossing problem
arise from the crossing of the chemical potentials of the condensates. In this article, we
study the creation of multi modes atomic condensate from a molecular one near the curve
crossing of the system’s chemical potentials. We show that within reasonable approximations
the problem can be reduced to the case of two atomic modes and one molecular mode
condensates. We show that exact results for this problem are still possible since the equations
for the operators evolution decouple so that the two level Landau-Zener problem can be
exactly solved. Compared to one atomic mode process, the two modes system provides much
more possibilities for correlated states since the initial conditions depend on two complex
parameters which describe the initial states of the two modes. The aim of this article is to
generalize the results of [1] to the case that includes exited modes of the atomic condensate.
In this article we study the process when the dissociation of atomic collisions in one mode
leads to the production of atoms in two different modes.
2
THE PROBLEM
In [7] it was shown that the description of molecular and atomic condensate by two
modes only is generally not realistic, since the exited atomic modes interact with molecular
condensate. Consequently some of them become highly populated and must be taken into
account. Generally speaking the Hamiltonian that includes more than one atomic and
molecular modes can be written as follows[7]:
Hˆ =
∑
k
(µka
†
k
ak + νk
1
2
ψ†
k
ψk) + gk,k′
∑
k,k′
ψ†
k+k′akak′ + ψk+k′a
†
k
a†
k′
+
1
2
gint
∑
k,k′,k′′
a†
k+k′−k′′a
†
k′′
ak′ak (1)
If we assume that the chemical potentials µk depend on time then the Landau-Zener model
for multimode case becomes very complicated to solve even in the case when the molecular
field is treated as a c-number. However, if the number of atoms in the atomic modes is
not very large we can disregard the interaction term (gint = 0). Let us consider a system
that initially has one highly populated mode of the molecular condensate. For example,
molecular mode with total momentum equals zero. Due to momentum conservation, the
atomic mode with momentum ~k would couple via the interaction with the molecular mode
to the atomic mode of momentum −~k only. In such systems the problem is reduced to the
interaction of two atomic modes with molecular condensate. Other molecular modes can
also be considered but in this article we will show that the effective coupling constant in
this case is propotional to the square root of the number of particles in the molecular mode;
therefore, we can disregard the interactions with the initially empty molecular modes.
Let us consider a system of two atomic fields A and B and one molecular field AB.
Suppose aˆ, bˆ and ψˆ are the annihilation operators of A, B and AB field respectively. If we
disregard the interaction between A and B during curve crossing then the Hamiltonian for
molecular dissociation in the process AB → A+B is time dependent and can be written as
follows:
Hˆ = µ1(t)aˆ
+aˆ+ µ2(t)bˆ
+bˆ+ gψ+aˆbˆ+ g∗ψaˆ+bˆ+ (2)
where the molecular field energy is set to zero. This Hamiltonian different from the one
considered in [1] where A = B. Here aˆ and bˆ are anihilation operators of distinct atomic
3
modes so [aˆ+, bˆ] = 0. Only atomic mode with zero momentum can be described by the
process A2 → A + A. In the case of Hamiltonian (2) the atomic modes A and B become
correlated. We assume that there is a macroscopic number of molecules that does not change
considarably during the process so that we can substitute < ψˆ > instead of ψˆ. Substituting
g < ψ >∗= γ into (2) we get the following atomic Hamiltonian:
Hˆ = µ1(t)aˆ
+aˆ + µ2(t)bˆ
+bˆ+ γaˆbˆ+ γ∗aˆ+bˆ+ (3)
Schrodinger’s equation leads to the following operator equations:
i ˙ˆa = µ1(t)aˆ + γ
∗bˆ+
i
˙ˆ
b+ = −µ2(t)bˆ+ − γaˆ (4)
and
i ˙ˆa+ = −µ1(t)aˆ+ − γbˆ
i
˙ˆ
b = µ2(t)bˆ+ γ
∗aˆ+
(5)
If we make the following change of variable:
aˆ→ aˆe−i
∫
t
(µ1(t)/2−µ2(t)/2)dt
bˆ→ bˆei
∫
t
(µ1(t)/2−µ2(t)/2)dt
µ(t) = µ1(t)/2 + µ2(t)/2
(6)
then equations (4),(7) read:
i ˙ˆa = µ(t)aˆ+ γ∗bˆ+
i
˙ˆ
b+ = −µ(t)bˆ+ − γaˆ
(7)
and
i ˙ˆa+ = −µ(t)aˆ+ − γbˆ
i
˙ˆ
b = µ(t)bˆ+ γ∗aˆ+
(8)
In the adiabatic approximation there are no transitions except for the case when adia-
baticity is violated near curve crossing points µ(t) = 0. According to Landau-Zener theory,
to estimate transition amplitude we can approximate µ(t) linearly near the crossing points,
hence the equation that should be considered are:
i ˙ˆa = βtaˆ+ γ∗bˆ+
i
˙ˆ
b+ = −βtbˆ+ − γaˆ
(9)
4
The corresponding second order differential equation for operator aˆ reads:
¨ˆa+ (β2t2 − |γ|2 + iβ2)aˆ = 0 (10)
which is exactly the same operator equation that appears in the case of one mode condensate.
The details of its solution can be found in [1]. The difference from the one mode process is
that the operator aˆ in (9) is coupled to operator bˆ+ rather than aˆ+. The solution in the two
mode case can be written in the form:
aˆ(t) = φc(t)aˆ(t0) + φs(t)bˆ
+(t0) (11)
with initial conditions:
φc(t0) = 1, φs(t0) = 0
φ˙c(t0) = βt0, φ˙s(t0) = −iγ∗
(12)
The average numbers of the atoms in the A-mode are:
< aˆ+(t)aˆ(t) >= nst + nsp (13)
where
nsp = |φs(t)|2 (14)
corresponds to the spontaneous transitions into the A−atomic vacuum state. This term
does not appear if the problem is treated in the mean field approximation and is the result
of quantum effects [1], [13], [14]
The quantity nst corresponds to the stimulated transitions in the case when atomic states
were initially populated:
nst = |φc|2 < aˆ+(t0)aˆ(t0) > +|φs|2 < bˆ+(t0)bˆ(t0) > +2Re(φ∗sφc < aˆ(t0)bˆ(t0) >) (15)
The functions φc(t) and φs(t) are c-functions that are independent solutions of the equa-
tion (10). This equation with the initial conditions (12) appears also in the one mode case
which was investigated in [1], (see appendix). Using the results of [1] and substitute it into
(11) we find assimptotics for the self-consistent solution of our initial valued problem is as
follows:
5
φs(t→∞) = |γ|γ
√
exp(2πλ)− 1e−i3pi4 −iS(|τ0|)−iS(τ)−i arg Γ(iλ),
φc(t→∞) = epiλ+iS(|τ0|)−iS(τ)
(16)
where τ =
√
2βt, S(τ) = τ 2/4− λ ln τ and λ = |γ|2/(2β) Using this one can derive
|φs(t→∞)|2 = e2piλ − 1
|φc(t→∞)|2 = nsp = e2piλ
(17)
so if initially we have coherent atomic states |α > |β > where aˆ|α >= α|α > and bˆ|β >=
β|β > then
nst = |α|2e2piλ + |β|2(e2piλ − 1) + 2
√
e2piλ − 1epiλ|α||β| ×
cos(
3π
4
+ 2 S(|τ0|) + arg(Γ(iλ)) + arg(γ) + arg(α) + arg(β)) (18)
so, depending on the initial arguments of α and β number of produced particles due to the
stimulated transitions can be in the range n− < nst < n+ where
n± = (|α|epiλ ± |β|
√
e2piλ − 1)2 (19)
Now, let us proceed to the problem of correlations between the two modes. We will
investigate the case when initially there were no particles in atomic condensates. For this
purpose we consider the quadrature phase amplitudes:
Xˆθ(t) = (aˆ(t) + bˆ(t))e
iθ + (aˆ+(t) + bˆ+(t))e−iθ (20)
If initially the atomic condensates are in vacuum states then:
< Xˆθ
2
(t) >= |φc(t)eiθ + φ∗s(t)e−iθ|2 (21)
and long time after curve crossing event we obtain:
< Xˆθ
2
(t→∞) >= |
√
e2piλ − 1 + epiλ− ı3pi4 −ı arg γ−2ıS(τ)−ı arg Γ(iλ)+2ıθ|2 (22)
Choosing two orthogonal phase angles θ+ = 1/2(
3pi
4
+ arg γ + 2S(τ) + arg Γ(iλ)) and
θ− = 1/2(
−pi
4
+ arg γ + 2S(τ) + arg Γ(iλ)) we find:
< X2θ± >= |
√
e2piλ − 1± epiλ|2 (23)
6
which is what one gets in the one mode case [1] but with different interpretation. The
result (23) means that dissociation forms atoms in a two modes squeezed state. On the
applications of two mode squeezed states in atomic condensate see for example [6], [15], [16].
Finally we would like to discuss another application of our results. The same Hamiltonian
(3) would appear in three components condensate (that differ for example by spin projection)
if the following reaction during atomic collisions is possible:
C + C −→ A +B (24)
This process leads to a gψˆψˆaˆ+bˆ+ term in the Hamiltonian which in the case of large number
of C atoms can be substituted by g < ψˆψˆ > aˆ+bˆ+.
The time dependent chemical potential crossing can be achived by changing adiabatically
the shape of the trap. This would produce the change of condensate concentration and hence
of the chemichal potentials.
If components A and B are distinct by some internal degree of freedom that is not
conserved during the reaction then the Hamiltonian (2) should be generalized so that reaction
go in all possible channels:
Hˆ = µ1(t) = aˆ
+aˆ+ µ2(t)bˆ
+bˆ+ γaˆaˆ+ γ∗aˆ+bˆ+ + γaaˆaˆ+ γ
∗
aa
+a+ + γbbb+ γ
∗
b b
+b+ (25)
In this case even if we treat the field of initial condensate as a c-number, the resulting
Landau-Zener problem in not analytically solvable, since all four operator equations become
coupled. However, if additional symmetry exists, exact asymptotics can be found. For
example, if the chemical potentials are equal due to some symmetry:
µ1(t) = µ2(t) (26)
then the equivalent Landau-Zener problem can be solved. For simplicity let us also assume
that γb = γa = γ1. Then the operators evolution is govenered by the following equations:
i ˙ˆa = taˆ+ γb+ − 2γ∗1 aˆ+
i
˙ˆ
b = tbˆ+ γ∗a+ − 2γ∗1 bˆ+
(27)
this is accomplished by the hermitian conjugate equations.
If we add the equations in (27) and denote cˆ = aˆ+ bˆ we get
7
i ˙ˆc = tcˆ+ (γ∗ − 2γ∗1)cˆ+ (28)
so that for cˆ and cˆ+ we have a solvable two state Landau-Zener system. Accordingly, we
have similar equation for operators dˆ and dˆ+, where dˆ = aˆ− bˆ:
i
˙ˆ
d = tdˆ− (γ∗ + 2γ∗1)dˆ+ (29)
At the end we can switch back to operators aˆ and bˆ. Such solution would be very unstable
in respect of the terms that break the symmetry appear in the Hamiltonian given by (26)
since there are states that have the same energy participating in the curve crossing. This
instability was investigated in a set of articles [17], [18], [19].
In conclusion, we considered a time dependent problem of coherent dissociation of molec-
ular condensate into distinct modes beyond the mean field approximation. We showed that it
can be reduced to the case of two distinct atomic modes and molecular mode. We discussed
in particular the case when the molecular condensate has zero momentum because in this
case the interaction term with the two resulting modes is equal hence the coupling constant
g between the molecular and the atomic field becomes 2g. This fact is very important since
it leads to an enhancement in the number of the atomic condensate atoms that exist in the
squeezed state. In the case of two modes condensate the corresponding multistate Landau-
Zener problem can be solved exactly since the operators evolution equations decouple. The
produced condensates are entangled in a two modes squeezed state. A more general case of
the two mode condensate production during a curve crossing could also be exactly solvable
if additional symmetry is imposed. However such system could be very unstable if terms
that break the symmetry participate in the Hamiltonian of the system.
APPENDIX
The solutions of eq.(9) that satisfy the initial conditions (11) are given by
φc(t) = −iγ∗φ
∗
1(t0)φ1(t)− φ∗2(t0)φ2(t)
W (φ1, φ2)
(30)
φs(t) = 2iγ
∗φ2(t0)φ1(t)− φ1(t0)φ2(t)
W (φ1, φ2)
(31)
where W (φ1, φ2) = i(2β/γ)exp(−πλ/2) and
8
λ = |γ|2/(2β) (32)
is Landau-Zener parameter. The Functions φ1 and φ2 are two standard solutions of parabolic
cylinder equation with asymptotics at t0 −→ −∞:
φ1(t0) ∼ 1
τ0
exp(−π
4
λ+ i
π
4
+ iS(|τ0|)) (33)
φ2(t0) ∼ 1
γ
√
2β exp (−π
4
λ− iπ
4
− iS(|τ0|)), (34)
and at t −→ +∞:
φ1(t) ∼ 2|γ|
√
β sinh (πλ) exp (
π
4
λ− iπ
2
− iS(τ)− iargΓ(iλ)) (35)
φ2(t) ∼ 1
γ
√
2β exp (
3π
4
λ− iπ
4
− iS(|τ0|)) (36)
Acknowledgments. This work was supported by NSF under the grant DMR 0072115 and
by DOE under the grant DE-FG03-96ER45598. It is our pleasure to thank V.L. Pokrovsky
and L.P. Pitaevskii for the fruitful discussions and encouragement.
[1] V.A. Yurovsky, a. Ben-Reuven, P.S. Julienne cond-mat 0108372
[2] A. Vardi, V. A. Yurovsky, J. R. Anglin, cond-mat/0105439
[3] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, Sandro Stringari, Rev. Mod. Phys. V71,
Issue 3 pp. 463-512 (1999)
[4] L.D.Landau, Physik Z. Sowjetunion 2, 46 (1932)
[5] C.Zener, Proc. Roy. Soc. Lond. A 137, 696 (1932)
[6] D. Jaksch, J.I. Cirac P. Zoller cond-matt/0110494
[7] K. Goral, M. Gajda, K. Rzaz˙ewski Phys.Rev.Lett V86, 1397 (2001), cond-mat/0006192
[8] C. Menotti, J.R. Anglin, J.I. Cirac, P. Zoller, Phys. Rev. A, V63, 023601
[9] D. J. Heinzen, R. H. Wynar cond-mat/0110565
[10] S. Brundobler, V. Elser, J. Phys. A: Math.Gen. 26 (1993) 1211-1227
[11] Yu. N. Demkov, V. I. Osherov, Zh. Exp. Teor. Fiz. 53 (1967) 1589 (Engl. transl. 1968 Sov.
Phys.-JETP 26, 916)
9
[12] L.D.Landau, E.M. Lifshitz, Quantum Mechanics Pergamon, Oxford
[13] A. Vardi, J.R. Anglin, Phys. Rev. Lett, V86, No4, 568 (2001)
[14] V. A. Yurovsky cond-mat/0109463
[15] Uffe V. Polsen, K. Molmer, Phys. Rev. A, V63, 023604
[16] D. Gottesman, J. Preskill, Phys.Rev. A, V63, 022309
[17] V.A. Yurovsky, A. Ben-Reuven, Phys. Rev. A. V63, 043404 (2001)
[18] V. A. Yurovsky, A. Ben-Reuven, P.S. Julienne, Y.B. Band, J. Phys. B 32, 1845, (1999)
[19] V.A. Yurovsky, A. Ben-Reuven, Phys. Rev. A 60, 4561 (1999)
10


OAKTrust Digital Repository (Texas A&M Univ)

arXiv:cond-mat/0112365v1  [cond-mat.stat-mech]  19 Dec 2001Formation of Two Component Bose Condensate During TheChemical Potential Curve CrossingM.A. Kayali, N.A. SinitsynDepartment of Physics, Texas A&M UniversityCollege Station, Texas 77843-4242, USA(Dated: June 26, 2018)AbstractIn this article we study the formation of the two modes Bose-Einstein condensate and the cor-relation between them. We show that beyond the mean field approximation the dissociation of amolecular condensate due to the chemical potential curve crossing leads to the formation of twomodes condensate. We also show that these two resulting modes are correlated in a two modesqueezed state.PACS numbers: 03.75.Fi, 03.65.Nk, 34.50.Lf, 34.50.RK1INTRODUCTIONRecent studies on Bose-Einstein condensate have directed the attention of many theoristsand experimentalists to the multimodes Bose-Einstein condensate. In particular the appear-ance of multimodes condensate from a molecular one via molecular condensate dissociationprocess. In [1], [2] the authors studied the dissociation of a molecular condensate into aone mode atomic condensate. Moreover, Yurovsky et.al showed that the resulting atomiccondensate is in a squeezed state. They also showed that the number of atoms andsqueezing parameters in the case of time dependent chemical potentials depend on the rateof curve crossing and the coupling between the molecular and the atomic fields.The problems of curve crossing can be divided into two different classes. These are ther-dependent crossings and the t-dependent ones. In the former the potentials are coordi-nates dependent and the crossing takes place in space, while in the latter the potentialsare time dependent. The standard approach to the crossing problems is the semiclassicalapproach which is known as Landau-Zener (LZ) theory. The LZ theory provide a verygood description of the crossing problem for systems with potentials that cross at one pointonly. However, if crossing between two potentials occur at two points or more then thesolution of the problem is possible by using a different approach which is out of the LZ theory.In the production of an atomic condensate from a molecuar one the curve crossing problemarise from the crossing of the chemical potentials of the condensates. In this article, westudy the creation of multi modes atomic condensate from a molecular one near the curvecrossing of the system’s chemical potentials. We show that within reasonable approximationsthe problem can be reduced to the case of two atomic modes and one molecular modecondensates. We show that exact results for this problem are still possible since the equationsfor the operators evolution decouple so that the two level Landau-Zener problem can beexactly solved. Compared to one atomic mode process, the two modes system provides muchmore possibilities for correlated states since the initial conditions depend on two complexparameters which describe the initial states of the two modes. The aim of this article is togeneralize the results of [1] to the case that includes exited modes of the atomic condensate.In this article we study the process when the dissociation of atomic collisions in one modeleads to the production of atoms in two different modes.2THE PROBLEMIn [7] it was shown that the description of molecular and atomic condensate by twomodes only is generally not realistic, since the exited atomic modes interact with molecularcondensate. Consequently some of them become highly populated and must be taken intoaccount. Generally speaking the Hamiltonian that includes more than one atomic andmolecular modes can be written as follows[7]:Hˆ =∑k(µka†kak + νk12ψ†kψk) + gk,k′∑k,k′ψ†k+k′akak′ + ψk+k′a†ka†k′+12gint∑k,k′,k′′a†k+k′−k′′a†k′′ak′ak (1)If we assume that the chemical potentials µk depend on time then the Landau-Zener modelfor multimode case becomes very complicated to solve even in the case when the molecularfield is treated as a c-number. However, if the number of atoms in the atomic modes isnot very large we can disregard the interaction term (gint = 0). Let us consider a systemthat initially has one highly populated mode of the molecular condensate. For example,molecular mode with total momentum equals zero. Due to momentum conservation, theatomic mode with momentum ~k would couple via the interaction with the molecular modeto the atomic mode of momentum −~k only. In such systems the problem is reduced to theinteraction of two atomic modes with molecular condensate. Other molecular modes canalso be considered but in this article we will show that the effective coupling constant inthis case is propotional to the square root of the number of particles in the molecular mode;therefore, we can disregard the interactions with the initially empty molecular modes.Let us consider a system of two atomic fields A and B and one molecular field AB.Suppose aˆ, bˆ and ψˆ are the annihilation operators of A, B and AB field respectively. If wedisregard the interaction between A and B during curve crossing then the Hamiltonian formolecular dissociation in the process AB → A+B is time dependent and can be written asfollows:Hˆ = µ1(t)aˆ+aˆ+ µ2(t)bˆ+bˆ+ gψ+aˆbˆ+ g∗ψaˆ+bˆ+ (2)where the molecular field energy is set to zero. This Hamiltonian different from the oneconsidered in [1] where A = B. Here aˆ and bˆ are anihilation operators of distinct atomic3modes so [aˆ+, bˆ] = 0. Only atomic mode with zero momentum can be described by theprocess A2 → A + A. In the case of Hamiltonian (2) the atomic modes A and B becomecorrelated. We assume that there is a macroscopic number of molecules that does not changeconsidarably during the process so that we can substitute < ψˆ > instead of ψˆ. Substitutingg < ψ >∗= γ into (2) we get the following atomic Hamiltonian:Hˆ = µ1(t)aˆ+aˆ + µ2(t)bˆ+bˆ+ γaˆbˆ+ γ∗aˆ+bˆ+ (3)Schrodinger’s equation leads to the following operator equations:i ˙ˆa = µ1(t)aˆ + γ∗bˆ+i˙ˆb+ = −µ2(t)bˆ+ − γaˆ (4)andi ˙ˆa+ = −µ1(t)aˆ+ − γbˆi˙ˆb = µ2(t)bˆ+ γ∗aˆ+(5)If we make the following change of variable:aˆ→ aˆe−i∫t(µ1(t)/2−µ2(t)/2)dtbˆ→ bˆei∫t(µ1(t)/2−µ2(t)/2)dtµ(t) = µ1(t)/2 + µ2(t)/2(6)then equations (4),(7) read:i ˙ˆa = µ(t)aˆ+ γ∗bˆ+i˙ˆb+ = −µ(t)bˆ+ − γaˆ(7)andi ˙ˆa+ = −µ(t)aˆ+ − γbˆi˙ˆb = µ(t)bˆ+ γ∗aˆ+(8)In the adiabatic approximation there are no transitions except for the case when adia-baticity is violated near curve crossing points µ(t) = 0. According to Landau-Zener theory,to estimate transition amplitude we can approximate µ(t) linearly near the crossing points,hence the equation that should be considered are:i ˙ˆa = βtaˆ+ γ∗bˆ+i˙ˆb+ = −βtbˆ+ − γaˆ(9)4The corresponding second order differential equation for operator aˆ reads:¨ˆa+ (β2t2 − |γ|2 + iβ2)aˆ = 0 (10)which is exactly the same operator equation that appears in the case of one mode condensate.The details of its solution can be found in [1]. The difference from the one mode process isthat the operator aˆ in (9) is coupled to operator bˆ+ rather than aˆ+. The solution in the twomode case can be written in the form:aˆ(t) = φc(t)aˆ(t0) + φs(t)bˆ+(t0) (11)with initial conditions:φc(t0) = 1, φs(t0) = 0φ˙c(t0) = βt0, φ˙s(t0) = −iγ∗(12)The average numbers of the atoms in the A-mode are:< aˆ+(t)aˆ(t) >= nst + nsp (13)wherensp = |φs(t)|2 (14)corresponds to the spontaneous transitions into the A−atomic vacuum state. This termdoes not appear if the problem is treated in the mean field approximation and is the resultof quantum effects [1], [13], [14]The quantity nst corresponds to the stimulated transitions in the case when atomic stateswere initially populated:nst = |φc|2 < aˆ+(t0)aˆ(t0) > +|φs|2 < bˆ+(t0)bˆ(t0) > +2Re(φ∗sφc < aˆ(t0)bˆ(t0) >) (15)The functions φc(t) and φs(t) are c-functions that are independent solutions of the equa-tion (10). This equation with the initial conditions (12) appears also in the one mode casewhich was investigated in [1], (see appendix). Using the results of [1] and substitute it into(11) we find assimptotics for the self-consistent solution of our initial valued problem is asfollows:5φs(t→∞) = |γ|γ√exp(2πλ)− 1e−i3pi4 −iS(|τ0|)−iS(τ)−i arg Γ(iλ),φc(t→∞) = epiλ+iS(|τ0|)−iS(τ)(16)where τ =√2βt, S(τ) = τ 2/4− λ ln τ and λ = |γ|2/(2β) Using this one can derive|φs(t→∞)|2 = e2piλ − 1|φc(t→∞)|2 = nsp = e2piλ(17)so if initially we have coherent atomic states |α > |β > where aˆ|α >= α|α > and bˆ|β >=β|β > thennst = |α|2e2piλ + |β|2(e2piλ − 1) + 2√e2piλ − 1epiλ|α||β| ×cos(3π4+ 2 S(|τ0|) + arg(Γ(iλ)) + arg(γ) + arg(α) + arg(β)) (18)so, depending on the initial arguments of α and β number of produced particles due to thestimulated transitions can be in the range n− < nst < n+ wheren± = (|α|epiλ ± |β|√e2piλ − 1)2 (19)Now, let us proceed to the problem of correlations between the two modes. We willinvestigate the case when initially there were no particles in atomic condensates. For thispurpose we consider the quadrature phase amplitudes:Xˆθ(t) = (aˆ(t) + bˆ(t))eiθ + (aˆ+(t) + bˆ+(t))e−iθ (20)If initially the atomic condensates are in vacuum states then:< Xˆθ2(t) >= |φc(t)eiθ + φ∗s(t)e−iθ|2 (21)and long time after curve crossing event we obtain:< Xˆθ2(t→∞) >= |√e2piλ − 1 + epiλ− ı3pi4 −ı arg γ−2ıS(τ)−ı arg Γ(iλ)+2ıθ|2 (22)Choosing two orthogonal phase angles θ+ = 1/2(3pi4+ arg γ + 2S(τ) + arg Γ(iλ)) andθ− = 1/2(−pi4+ arg γ + 2S(τ) + arg Γ(iλ)) we find:< X2θ± >= |√e2piλ − 1± epiλ|2 (23)6which is what one gets in the one mode case [1] but with different interpretation. Theresult (23) means that dissociation forms atoms in a two modes squeezed state. On theapplications of two mode squeezed states in atomic condensate see for example [6], [15], [16].Finally we would like to discuss another application of our results. The same Hamiltonian(3) would appear in three components condensate (that differ for example by spin projection)if the following reaction during atomic collisions is possible:C + C −→ A +B (24)This process leads to a gψˆψˆaˆ+bˆ+ term in the Hamiltonian which in the case of large numberof C atoms can be substituted by g < ψˆψˆ > aˆ+bˆ+.The time dependent chemical potential crossing can be achived by changing adiabaticallythe shape of the trap. This would produce the change of condensate concentration and henceof the chemichal potentials.If components A and B are distinct by some internal degree of freedom that is notconserved during the reaction then the Hamiltonian (2) should be generalized so that reactiongo in all possible channels:Hˆ = µ1(t) = aˆ+aˆ+ µ2(t)bˆ+bˆ+ γaˆaˆ+ γ∗aˆ+bˆ+ + γaaˆaˆ+ γ∗aa+a+ + γbbb+ γ∗b b+b+ (25)In this case even if we treat the field of initial condensate as a c-number, the resultingLandau-Zener problem in not analytically solvable, since all four operator equations becomecoupled. However, if additional symmetry exists, exact asymptotics can be found. Forexample, if the chemical potentials are equal due to some symmetry:µ1(t) = µ2(t) (26)then the equivalent Landau-Zener problem can be solved. For simplicity let us also assumethat γb = γa = γ1. Then the operators evolution is govenered by the following equations:i ˙ˆa = taˆ+ γb+ − 2γ∗1 aˆ+i˙ˆb = tbˆ+ γ∗a+ − 2γ∗1 bˆ+(27)this is accomplished by the hermitian conjugate equations.If we add the equations in (27) and denote cˆ = aˆ+ bˆ we get7i ˙ˆc = tcˆ+ (γ∗ − 2γ∗1)cˆ+ (28)so that for cˆ and cˆ+ we have a solvable two state Landau-Zener system. Accordingly, wehave similar equation for operators dˆ and dˆ+, where dˆ = aˆ− bˆ:i˙ˆd = tdˆ− (γ∗ + 2γ∗1)dˆ+ (29)At the end we can switch back to operators aˆ and bˆ. Such solution would be very unstablein respect of the terms that break the symmetry appear in the Hamiltonian given by (26)since there are states that have the same energy participating in the curve crossing. Thisinstability was investigated in a set of articles [17], [18], [19].In conclusion, we considered a time dependent problem of coherent dissociation of molec-ular condensate into distinct modes beyond the mean field approximation. We showed that itcan be reduced to the case of two distinct atomic modes and molecular mode. We discussedin particular the case when the molecular condensate has zero momentum because in thiscase the interaction term with the two resulting modes is equal hence the coupling constantg between the molecular and the atomic field becomes 2g. This fact is very important sinceit leads to an enhancement in the number of the atomic condensate atoms that exist in thesqueezed state. In the case of two modes condensate the corresponding multistate Landau-Zener problem can be solved exactly since the operators evolution equations decouple. Theproduced condensates are entangled in a two modes squeezed state. A more general case ofthe two mode condensate production during a curve crossing could also be exactly solvableif additional symmetry is imposed. However such system could be very unstable if termsthat break the symmetry participate in the Hamiltonian of the system.APPENDIXThe solutions of eq.(9) that satisfy the initial conditions (11) are given byφc(t) = −iγ∗φ∗1(t0)φ1(t)− φ∗2(t0)φ2(t)W (φ1, φ2)(30)φs(t) = 2iγ∗φ2(t0)φ1(t)− φ1(t0)φ2(t)W (φ1, φ2)(31)where W (φ1, φ2) = i(2β/γ)exp(−πλ/2) and8λ = |γ|2/(2β) (32)is Landau-Zener parameter. The Functions φ1 and φ2 are two standard solutions of paraboliccylinder equation with asymptotics at t0 −→ −∞:φ1(t0) ∼ 1τ0exp(−π4λ+ iπ4+ iS(|τ0|)) (33)φ2(t0) ∼ 1γ√2β exp (−π4λ− iπ4− iS(|τ0|)), (34)and at t −→ +∞:φ1(t) ∼ 2|γ|√β sinh (πλ) exp (π4λ− iπ2− iS(τ)− iargΓ(iλ)) (35)φ2(t) ∼ 1γ√2β exp (3π4λ− iπ4− iS(|τ0|)) (36)Acknowledgments. This work was supported by NSF under the grant DMR 0072115 andby DOE under the grant DE-FG03-96ER45598. It is our pleasure to thank V.L. Pokrovskyand L.P. Pitaevskii for the fruitful discussions and encouragement.[1] V.A. Yurovsky, a. Ben-Reuven, P.S. 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