Inelastic collisions in an exactly solvable two-mode Bose-Einstein Condensate

0.52 MB·English

by  I. Fuentes-Schuller

Inelastic collisions in an exactly solvable two-mode Bose-Einstein Condensate
P. Barberis-Blostein
Instituto de Ciencias F´ısicas, Universidad Nacional Auto´noma de M´exico,
Ap. Postal 48-3, 62251 Cuernavaca, Morelos, M´exico
I. Fuentes-Schuller∗
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, A. Postal 70-543, M´exico D.F. 04510, M´exico
InelasticcollisionsoccurinBose-Einsteincondensates,insomecases,producingparticlelossinthe
system. Nevertheless,theseprocesseshavenotbeenstudiedinthecasewhenparticlesdonotescape
thetrap. Weshowthatsuchinelasticprocessesarerelevantinquantumpropertiesofthesystemsuch
as the evolution of the relative population, the self trapping effect and the probability distribution
of particles. Moreover, including inelastic terms in the model of the two-mode condensate allows
7
foran exact analytical solution. Usingthissolution, weshow thatcollisions favorthegeneration of
0
entanglementbetweenthemodesofthecondensateaslongasthecollision ratedoesnotexceedthe
0
natural frequency of thesystem.
2

n
a I. INTRODUCTION these terms allows for an exact analyticalsolution of the
J
model. Several types of inelastic collisions are known to

1 be relevant in real physical situations [10, 11, 12]. The
2 Bose-Einstein condensates are quantum systems of
inelastic process better known in experiments are those
great interest since they consist of up to 1010 particles
whichresultinparticlelosssuchasbackgroundcollisions,
1 which exhibit collective quantum behavior. Recently,
three-bodyrecombination,spin-exchangeanddipolarre-
v multi-component(ormulti-mode)Bose-Einsteinconden-
8 laxation[11,13]. Theinelasticprocessesthatweconsider
sates have been extensively studied both theoretically
4 herehavenotbeenproperlycharacterizedbyexperiment
[1, 2, 3] and experimentally [4, 5, 6, 7]. The components
1 yet. The main reason for this is that they do not give
of the condensate are either several spatially separated
1 rise to particle loss. Nevertheless, there is good indica-
0 condensates [7] or several internal degrees of freedom of
tion that this process are present in real situations. In
7 the condensed particles [2]. Unfortunately, the lack of
this paper, we present a number of theoretical predic-
0 analytical solutions for these systems limit our under-
tions on the evolution of the relative population of the
/ standing of them. Until now, multi-component conden-
h condensate, the self trapping effect and on the probabil-
p sateshaveonlybeenstudiedthroughnumericalsolutions
ity distribution of particles in the condensates to be cor-
- which are always limited by the growing degrees of free-
t roborated by experiment. We compare our results with
n dom of the system.
the predictions of the canonical two mode Bose-Einstein
a
The simplest multi-component condensate is known condensate. Experimentscouldbeperformedtocompare
u
q as the two-mode Bose-Einstein condensate. The sys- thepredictionsofthemodels. Thegreatadvantageofour
: tem consists of condensed particles in two hyperfine lev- modelisthatithasexactanalyticalsolutionwhichallows
v
els [1]. Transitions between the levels are induced via foracompleteunderstandingofthesystem. Remarkably,
i
X a Josephson-type interaction produced by a laser. Two we find that the groundstate of the system is the coher-
r body collisions are, together with the laser interaction, ent state which was found to provide a good description
a the most relevant physical process in the system. An- of the two-mode Bose-Einstein condensate [14, 15]. We
other incarnation of the two-mode condensate consists takeadvantageofthesimplicityofthemodeltoshowun-
of two spatially separated condensates coupled through derwhatconditionsmaximally entangledstatesbetween
a tunneling barrier corresponding to the Josepshon-type themodescanbe createdandthe roleofcollisionsinthe
interaction [3]. The canonical model used to describe generation of entanglement.
these systems considers the Josephson-type interaction
andelastic collisions betweensame anddifferent particle
type [1]. The ground state of the model can be found in II. PARTICLE LOSS DUE TO INELASTIC
the firstorsecondquantizationonlybynumericalmeans COLLISIONS IN BOSE-EINSTEIN
[1, 8]. CONDENSATES
In a recentpublication, we introduceda model for the
two-mode Bose-Einstein condensate which includes in- Inelastic collision in Bose-Eistein condensates occur
elastic collisions [9]. In our model the inelastic collisions with high frequency and have been studied extensively
areinducedbytheJosephsontype-interaction. Including [10,11,13,16]. Theseprocessesarewellknownsincethey
commonly result in particle loss. For this reason there
has been considerable efforts to suppress them in exper-
imental situations [11, 13]. The main inelastic processes
∗PublishedbeforeundermaidennameFuentes-Guridi known are backgroundcollisions, three-body recombina-
2
tion, spin exchange and dipole relaxation. Atoms in a particles in one mode into the other. Ignoring the in-
condensate are commonly knocked out of the trap by elastic terms in Eq.(1), we find that our model coincides
collisions with untrapped, room-temperature molecules with the canonical Josephson Hamiltonian [1] when the
in the background chamber. In the three body recombi- rate of collisions of same particle type is equal. The as-
nationprocessthreeparticlescollideandformamolecule sumption of equal collision rates for the same particle
whichisnolongertrappedbythepotential. Particlesare type is alsomade in [1]in orderto find approximateand
alsolostwhen,duringacollision,theatomsmaketransi- numerical solutions. Our model has the same number of
tions to different energy levels which are not trapped by free parametersas the canonicaltwo-mode Hamiltonian.
the potential. Spin-exchange and dipolar relaxation are The only difference is that the Hamiltonian in Eq.(1) in-
important mechanism of particle loss which are closely cludes inelastic collisions, which are usually present in
related to the process considered in our model. After a real BECs [10].
collisioneitherorbothatomsexitinadifferentspinstate. Including the correct rate of inelastic collisions in the
If the spin re-orientationenergy is larger than the depth Hamiltonian allows for an analytical solution. The solu-
of the trapping potential this mechanism might result in tion is simply found by considering the following Hamil-
the lossof the particlesfrom the trap[16]. Indipolar re- tonian H0 = A1(a†a−b†b)+A2(a†a−b†b)2 with eigen-
laxationoneortwoparticlesinthehigherhyperfinelevel states|N,miwhereN istheeigenvalueofthetotalnum-
decay to the lower level during the collision due to their ber operator Nˆ =n +n =a†a+b†b and m the eigen-
a b
dipole-dipole interaction. The excess of energy is then value ofthe relativepopulationmˆ =a†a−b†b. Since the
transferred to an excess of momentum which produces number of particles in the system N is constant, m is
the particle to escape the trap. In our model the inelas- restricted to values m =−N,...,N. H0 describes a two-
tic process occurs but the excess of energy is not enough modecondensatewithnoJosephson-typeinteraction,i.e,
to help the particle escape from the trap. Theoretically, no laser interaction or tunneling barrier. The parameter
inelastic processesare included in the studies ofthe con- A2 correspondsthe two-bodyelasticscatteringprobabil-
densates by considering classical rate equations for the ity and A1 to the energy difference between the modes.
particle loss [11]. But in the literature there is a lack of It is easy to verify that if the parameters of the model
experimentalandtheoreticalanalysisoninelasticprocess satisfy the following conditions
which do not give rise to particle loss. Our model exem-
plifies the importancethat theseprocessesmighthavein A0 = A2(cos2θN2+sin2θN)/4,
the behavior of the system.
δω = (A1cosθ)/2,
λ = (A1sinθ)/2
III. TWO-MODE BOSE-EINSTIEN U = A2(1−3cos2θ)/4,
CONDENSATE WITH INELASTIC COLLISIONS µ = (A2cosθsinθ)/2,
Λ = (A2sin2θ)/4. (2)
Our model of the two mode Bose-Einstein condensate
consists of the following Hamiltonian, then the Hamiltonian can be written as H2 = UH0U†
where where U is the two-mode displacement operator
U = eξa†b+ξ∗ab† with displacement parameter ξ = θeiφ.
H2 = A0+δω(a†a−b†b) (1) The solution of the Hamiltonian (1) is simply U†|N,mi
+ λ(eiφa†b+e−iφab†)+Ua†b†ab with energy Em =A1m+A2m2.
Note that by fixing the free parameters δω, λ and U
+ Λ(e2iφa†a†bb+h.c.)
theinelasticcollisionconstants,Λandµ,aredetermined.
+ µ((a†a†ab−b†a†bb)eiφ+h.c.). This is because in our model, the inelastic collisions are
produced by the effect of the Josephson-type interaction
Themodesa†,aandb†,bwithfrequencydifferenceδω, oncollidingparticles. Thisphysicalrelationshipismath-
correspond to either atoms with two different hyperfine ematically expressedby the relationshipbetween the co-
levels [5] or alternatively, two spatially separated con- efficients. The relationship between the parameters im-
densates [4]. The Josephson-type interactions is induced ply that, for fixed θ, the scattering probabilities for in-
by applying a laser [5] or a magnetic field gradient [4]. elastic collisions are higher if the scattering for elastic
In our Hamiltonian the Josephson-type term, in which collisions are more probable. If sin(θ) = 0, the Joseph-
one particle is annihilated in one mode and created in son coupling is zero and there is no inelastic processes
the other, has coupling constant λ and phase φ. The occurringin the system. There are only elastic collisions
terms with four bosonic operators describe two-particle between particles of the same type. We also see that
elasticandinelasticcollisions. Theelasticcollisionshave for a resonant laser or a symmetric well δω = 0, which
interaction strength U. The inelastic collisions have in- corresponds to cosθ = 0, there are elastic and inelas-
teractionstrengthµwhentwoparticlesinthesamemode tic collisionsbut the inelastic interactionalwaysinvolves
collideandoneofthemistransformedintotheotherand the exchange of two particles simultaneously. We would
interactionstrength Λ when the collision transformstwo like to emphasize that if for a specific physical system,
3
the ratesofinelastic to elastic collisionsdoes notholdor is a function of the tunneling probability and the trap-
cannotbearrangedbyexternalmanipulation,ananalyt- ping potential. The probability of inelastic events is also
ical solution cannot be found using our method for such a function of the probability of elastic events which de-
a condensate. Fortunately, in the laboratory the rate of pendonthe scatteringlengthsofparticlesinthesystem.
elastic and inelastic collisions can be manipulated, for It seams logical and natural to consider such inelastic
example, by applying a magnetic potential [17] making processes. Note that there are other theoretical motiva-
possible to meet the experimental values for the produc- tions to support the inclusion of such terms [18]. The
tion ratios of those terms. The advantage of having a specific relationship between the parameters for which
region in the parameter space of the system where we the model has an exact analytical solution corresponds
can find an exact solution is that it is then possible to to a special type of potential.
find solutions for other parameters using perturbation
theory.
Due to the simplicity of our solution the ground state
V. INELASTIC COLLISIONS BETWEEN
U†|N,m0i of H2 is trivially found by minimizing the PARTICLES IN THE PRESENCE OF A LASER
energy Em2 = A1m + A2m2 with respect to m. For FIELD
A2 > 0, m0 is the nearest integer to −A1/(2A2) or
m0 =−A1N/|A1| when |−A1/(2A2)|>N. For A2 <0,
In the physicalrealizationof the model where the two
the minimum corresponds to m0 = N if A1 < 0 or
modescorrespondtotwohyperfinelevels,theJosephson-
m0 =−N otherwise. ThesolutionforA2 <0andA1 >0
type interaction correspondsto the interactionof a laser
is the spin coherent state which has been argued to de-
field which induces transitions between the atoms hy-
scribethe Bose-Einsteincondensatewellenough[14,15].
perfine levels. In these picture, same and different spice
particlecancollide. Theinelasticcollisionscorrespondto
the collision of particles in the presence of a laser field.
IV. INELASTIC COLLISIONS IN A DOUBLE
WELL Energyisabsorbedandliberatedintothefieldduringcol-
lisioninwhichtheparticleschangetheirhyperfinelevels.
Although a detailed microscopic calculation of this pro-
Inthe physicalrealizationofthemodel,wherethetwo
cess has not been yet calculated, it has been observedin
modesa,a†andb,b†ofthecondensatecorrespondtopar-
experiments that the rate of inelastic collisions between
ticles in two spatially separated condensates, A and B,
atoms is increased when there is an interaction with a
the Josephson-type interaction corresponds to the tun-
laser field [12]. We provide in this paper severaltheoret-
nelingofparticlesthroughabarrierwithprobabilitypro-
icalpredictionsto be experimentally testedofthe effects
portional to λ.
of inelastic collusion which do not give rise to particle
The canonical model, as well as our model, considers
loss in the condensate.
elastic collisions between the particles in each well given
by the terms proportional to a†aa†a and b†bb†b. Two
particles in one well are annihilated and two particles in
the same well are created. Note that in our model these VI. EFFECTS OF INELASTIC COLLISIONS IN
termsarepartoftheconstanttermN2 whichisincluded THE SYSTEM
in the parameter A0. Both models also consider colli-
sionsbetweenparticlesindifferentwellsgivenbya†ab†b. In this section, we analyze the effects of the inelastic
One particle in well A and one particle in well B are an- collisiontermsaddedtothe Hamiltonianinthe behavior
nihilated and one particle in well A and one particle in ofthetwo-modeBose-Einsteincondensate. Westudythe
well B are created. These collisions occur in the region particle distribution of the ground state and the evolu-
where the spatialwavefunctions of eachwelloverlap. In tion of relative population in each mode. We will show
all these elastic interactions the particle number in each that the presence of inelastic collisions have important
well is conserved. The new feature in our model is that consequences in the self trapping effect.
weconsidercollisionsoccurringintheoverlappingregion
inwhichparticles ina wellendupin the other wellafter
the collision. We consider terms in which two particles
A. Particle distribution of the ground state
in one well collide (close to the tunneling barrier) and
one or two of them are simultaneously tunneled to the
second well. We also include situations in which parti- Let us first consider the analytical solution to our
cles from different wells collide in the overlapping region model. The particle distribution P = |hN,m|ψ0i|2 of
andboth ofthemend upin the same wellafter the colli- the ground state |ψ0i = U†|N,m0i is equal to the mod-
sion. The probabilityofsuchinelastic eventsdepends on ulus square of the Clebsch-Gordon coefficients (Wigner)
the overlapping region between the condensates, which P =|dN |2 where,
m,m0
4
dN = (−1)k−m0+m (N +m0)!(N −m0)!(N +m)!(N −m)! (cos(θ/2))2N−2k+m0−m(sin(θ/2))2k−m0+m.
m,m0 k (Np+m0−k)!k!(N −k−m)!(k−m0+m)!
X
(3)
Thesummustbedoneoverkwhenevernoneoftheargu- distribution with one maxima to a superposition distri-
ments offactorialsin the denominatorare negative. Dif- bution is no longer abrupt. We find the abrupt transi-
ferent ground states parameterized by m0 are obtained tion only when the parameters change along the values
bychangingtherateA1/A2sincem0istheclosestinteger corresponding to those for which the system has exact
to −A1/2A2 (for |−A1/2A2|< N otherwise, m0 = N). analytical solution. In the canonical model, which does
When m0 =N, the particle distribution is not include inelastic collisions, the change from a sin-
gle peaked distribution to a double superposition is also
2N! smoothasshownin[1]. Themaindifferenceobservedbe-
|dN |2 = cos(θ/2)N+msin(θ/2)N−m
N,m s(N +m)!(N −m) tween the canonical model and our model is that in the
canonicalmodelonlysingleanddoublesuperpositiondis-
(4)
tributions are observed. In our model, it is possible to
which corresponds to a distribution with one maximum.
obtain the superposition of N distributions. Therefore,
We plot an example for N =1000 particles and θ =1 in
the presence of more than two components in the prob-
Fig. 1(a). As we change the rate A1/2A2 from 1000 to
ability distribution is a clear footprint of the presence of
977, the distribution changes abruptly from a distribu-
inelastic collisions in the system.
tion with a single maximum to a distribution with sev-
eralpeaks correspondingto multiple cat states as shown
in Fig. 1. These discontinuous changes occur when the
systemundergoesfirstordertransitionscorrespondingto
levelcrossings. Thestructureofthegroundstatechanges
abruptly and this is reflected in the probability distribu-
tion. As m0 approaches zero, the number of maxima in
the distributions grows. Controllingthe inelastic param-
eters allows us to prepare the condensate in such super-
position states.
FIG.2: Groundstaterelativepopulationdistributionfor1000
atoms. δω=109J, λ=487J and U =0.214027J.
FIG.1: Groundstaterelativepopulationdistributionfor1000
atoms and θ=1. Different m0 correspond to a different rate
between the parameters A1 and A2. Quantum superposition B. Inelastic collisions in the evolution of the
appears when m0<1000. relative population and the self trapping effect
Now consider the Hamiltonian (1) with general pa- We now calculate the evolutionof the averagerelative
rameters. In this general case it is not possible to find populationha†a−b†bi,fortheinitialcondition|ψ(t=0)i.
an analytical solution. Nevertheless we find, by numeri- In the case where the system has an analytical solution,
calmeans,the probabilitydistributionofparticlesinthe theevolutionoftherelativepopulationforagiveninitial
groundstate. InFig. 2weshowtheprobabilitydistribu- state |ψ(t = 0)i = N C U†|N,mi, with coeffi-
m=−N m
tion for N = 1000. We leave δω = 109J, λ = 487J and
cients C , is given by
m
U = 0.214027J fixed and vary µ and Λ as shown in the P
figure. By varying these parameters we show that the N
transition from different probability distributions of the ha†a − b†bi=cosθ m|C |2 (5)
m
ground state can be done smoothly. The change from a
−N
X
5
N
− sinθ CmCm−1(N(N +1)−m(m−1))1/2Lm
−N+1
X
Lm = cos(φ+(Em−1−Em)t) (6)
In the general case, the evolution of the relative pop-
ulation is calculated numerically. In Fig. (3) we plot
the evolution of the relative population for δω = 109J, (a)
λ = 487J and U = 0.214027J and vary µ and Λ as
shown in the figure. Fig3(a) correspond to the evolu-
tionofthe canonicalmodel(without inelasticcollisions).
In Figs.(3(b)-3(c)) we plot the evolution of the general
modelwhichincludesinelasticterms. Weconsideredsev-
eral cases with different amount of inelastic collisions.
Until here all the results where numerical. Finally, the
exact analytical solution given by Eq. (5) is plotted in
Fig.3(d). In the canonical model the collapse and re- (b)
vivals are not uniform. Each revival becomes wider and
at some point one revival gets mixed up with the next
one. We can see that adding inelastic collisions changes
drastically the behavior of the system. As the rate of
elastic and inelastic collision approaches the values for
which we have analytical solutions, we observe that the
revivals start becoming more uniform. We can go from
the solution of the canonical model to the analytical so-
lution ofourmodel continuouslyby varyingthe inelastic (c)
collision parameters. Finally, for the analytical solution
of the model, obtained when the rate ofelastic to inelas-
tic collisions is equal to 1/2, the evolution of the rela-
tivepopulationisperiodicandallrevivalshavethe same
width. One can see in Fig.3(d) that the structure of the
revivals repeats after two collapses.
We take advantage of having an analytical solution to
calculate (using Eq. (5)) the time at which the oscilla-
tions of the relative population collapse and the period (d)
atwhich the collapseand revivalsoccur. In orderto cal-
culate the time of collapse we find the condition when FIG. 3: Evolution of the relative population hmi,δω =109J,
the different terms in Eq. (5) cancel each other. This λ=487J andU =0.214027J andvaryµandΛareshown in
occurs when the phase difference between the cosines in thefigure.
L is (2n+1)π, where n is an integer. Therefore, we
m
obtain the condition (Em−1−Em)t−(Em −Em+1)t =
π(2n + 1) which implies that the time of collapse is write Eq. (7) as,
tr = (2n+1)π/(2A2). We will assume that t = 0 cor-
responds to exactly half of the revival (as in Fig. 3). (−A1−A2(2m−1))t1 =2πnm . (8)
Therefore,t (n=0)correspondstohalfthe distancebe-
r
tween the revivals. If A1/A2 is not a rational number Eq. (8) is never sat-
To calculate the period of the evolutionof the relative isfied and the function is not periodic. We will have
populationwesupposethatthe initialtime ist0 =0 and collapse and revivals, but the particular form of every
consideralatertimet1 >t0 atwhichtheaveragerelative revival would be different. When A1/A2 is a ratio-
population is the same ha†a−b†bi(t0)=ha†a−a†ai(t1). nal number, the time t1 corresponds to the period, and
If for every integer m exists an integer nm where the pr = t1/(2tr(n = 0)) is the period of the relative pop-
following condition is satisfied, ulation in units of number of revivals. Remember that
t (n = 0) corresponds to half the distances between re-
r
(Em−1−Em)t1 =2πnm, (7) vivals. Inotherwords,pr =kmeansthatafterkrevivals
the particular structure of the first revival repeats itself.
fromEq. (5)wecanseethattheevolutionofalltheterms Considering A1 = cA2 with c = p/q and p,q integers,
Lm is identical from t1 as it is from t0. If we consider one can se that if −q +p is even then pr = q, and if
t1 to be the smallest number which satisfies Eq. (7) the −q +p is odd then pr = 2q. An example is shown in
function ha†a−b†bi is periodic with period t1. We can Fig.4, where the evolution of the relative population is
6
plotted for the following values θ = 1.35, A2 = 1 and sate. This means that when the collisions rate is much
A1 = 49,50,101/3,59/2. The values are chosen so that smallerthanthe frequencydifference betweenthe modes
pr =1,2,3,4. in the absence of Josephson-type interaction A2 ≪ A1,
In the canonical model (µ = 0,Λ = 0), there are no the entanglementgeneratedafter turning onthe interac-
Rabi oscillations if U ≫ λ. The condensate remains tion, will be maximum if the double well is symmetric
trappedinonemodeifthescatteringprobabilityismuch witha hightunneling probability. Ina twospeciesBose-
larger than the Josephson-type interaction. This corre- Einstein condensate, maximum degrees of entanglement
sponds to the well-known self trapping effect [19]. Now will be generated, in this case, by turning on an intense
consider our model, when the parameters satisfy condi- resonant laser. Note that when m0 = N the probability
tions (2) and the evolution of the relative population is distribution has a single component.
givenbyEq. (5). IfA2 andA1 arefixed,theratioU/λis We plot in Fig.(5a) the entanglement between the
maximumforθ =0. Underthesecircumstances,itcanbe modes for different ground states m0 as a function of
trivially seen from Eq. (5) that there are no oscillations θ for N =1000particles. We find that the entanglement
and thus, we observe self trapping. This case is trivial isgenerallyhigherwhenm0issmall. Thiscorrespondsto
since there is no Josephson-typeinteraction. However,if a regime where the probability distribution of particles
we consider θ fixed, we find that the inelastic collisions has many maximumpeaks. This occurs when the elastic
can prevent self trapping from occurring even if U ≫ λ. collision rate becomes comparable to the frequency dif-
For instance, consider θ = π/2, there is a collapse at ference between the modes in the absence of Josephson-
time tr (defined above). In this case, although A2 ≫A1 type interaction, A1 ≈ A2. But note that by turning on
(implying that U ≫λ) there is never self trapping. the Josephson interaction with θ close to π/2, there is
a local minimum at m0 = 0. In Fig. (6b) we plot en-
tanglement at θ = π/2 and observe this local minimum.
VII. ENTANGLEMENT This means that for a large Josephson-type interaction
strength, there is a local minimum when A2 >A1.
Entanglement is a property of multipartite quantum We have learned something interesting about the role
statesthatarisesfromthetensorproductstructureofthe of two-body collisions in the generation of entanglement
Hilbertspaceandthesuperpositionprinciple. Itplaysan in a Bose-Einstein condensate. Higher degrees of entan-
importantroleintheunderstandingofmanybodyquan- glement are generated by switching the Josephson-type
tum systems. Moreover,it is consideredto be a resource interaction when the rate of collisions is comparable to
for quantum informationtasks [20] such as teleportation the frequency difference between the modes (in the ab-
and has applications in quantum control and quantum sence of the Josepson-type interaction) A1 ≈ A2. How-
simulations [21]. It is therefore of interest to find the ever, if A2 > A1, lower degrees of entanglement are
necessary conditions to prepare states with high degrees generated. In the case of a double well condensate, if
entanglement. the collision probability in the absence of tunneling is
Entanglementbetween the modes of a two component comparable (but lower)to the energy difference between
Bose-Einsteincondensatehavebeenstudiedinthecanon- the wells, then, by letting the atoms tunnel lowering the
ical model [15, 22]. In this section, we calculate entan- barriermaximally,entanglementwill be maximum if the
glementbetweenthe modes using the analyticalsolution wellsaresymmetric. However,ifthecollisionprobability
to our model and show the conditions to generate max- exceedstheenergydifferencebetweenthewellsintheab-
imal entanglement in the system. We analyze in detail senceoftunneling,maximumdegreesofentanglementare
the role of collisions in the generation of entanglement. achievedwhenloweringmaximallythebarrierifthewells
Quantifying entanglement for pure bipartite systems are slightly asymmetric. This will favor the generation
is simple. Entanglement between the systems is given of entanglement. In the case of a Bose-Einstein conden-
by the von-Neumann entropy S(ρ ) = −tr(ρ log (ρ )), sate consisting of two internal degrees of freedom, if col-
a a 2 a
where ρ is the reduced density matrix. The entangle- lisions occur with probability comparable (but lower) to
a
ment between the modes for the ground state of our the energy difference between the modes, entanglement
system U†|N,m0i is given simply by S(θ,N,m0) = will become maximum by coupling the internal degrees
− m|dN (θ)|2log |dN (θ)|2 where dN (θ) are with a strong resonant laser. But if the collision proba-
m0,m 2 m0,m m0,m
the Clebsch-Gordon coefficients. The entanglement is a bility exceeds the energy difference between the internal
funPction of the angle θ, the number of particles N and levels (in the absence ofthe laser),it becomes more con-
m0. The ground state is determined by the parameter venient for the generation of entanglement, to turn on
m0 which depends on the rate A1/2A2. This parameter a slightly detuned laser coupling. The same conclusions
regulates the size of the elastic collision probability in can we drawn from the analysis of Fig. (6a) where we
comparison to the difference of frequency of the modes plot the entanglement at m0 = 0, i.e. A2 > A1. We ob-
in the absence of Josephson-type interaction. servelocalminimum atθ =π/2. Therefore,weconclude
It is easy to see from the formula that the entangle- that in this case, it is more convenient to have θ sightly
ment of the state m0 = N is maximum when θ = π/2 different to π/2.
independently of the number of particles in the conden- In Fig. (5b) we plot the entanglementas a function of
7
(a)
(b)
(c)
(d)
FIG. 4: The evolution of the relative population hmi is plotted for the following values theta = 1.35, A2 = 1 and A1 =
49,50,101/3,59/2. The values are chosen so pr =1,2,3,4
thenumberofparticlesN anddifferentgroundstatesm0 frequency difference between the modes in the absence
forθ =π/2. Wecanseethattheentanglementisstronger of the Josephson-type interaction. One then turns on a
as we have more particles in the condensate. We plotted strong near resonant laser (or lower maximally the tun-
theentanglementasafunctionofthenumberofparticles neling barrier in an almost symmetric well).
for N as big as 105 and found the same behavior.
Therefore, to create high degrees of entanglement be-
tween the modes, it is convenient to have a condensate
with large number of particles where the two-body elas-
tic collision rate is comparable (but not larger) to the
8
effectsofinelasticcollisionsinthesystemwhentheparti-
cles do not escape the trap. We show that such inelastic
collisionsproduceimportantqualitativeandquantitative
effects. Moreover, our model has the great advantage of
havingexactanalyticalsolutions for a setof parameters.
Previously,thestudyofthetwo-modeBose-Einsteincon-
densate was restricted to numerical treatment.
Inelastic collisions which do not lead to particle loss
(a) have not been yet a topic of experimental research. The
mainintensionofourworkistopresentanumberofpre-
dictionsinthe evolutionofthe relativepopulationofthe
condensateandintheprobabilitydistributionofparticles
in the system which would allow to detect such inelastic
process in the laboratory. The presence of these effects
can be easily detected by comparing the results to the
canonical two-mode hamiltonian which does not include
inelasticcollisions. We findthatthe presence ofinelastic
collisions produce multicomponent superposition states,
(b)
ratherthantwocomponentcatstatesaspredictedbythe
canonical model. We show how inelastic processes have
FIG.5: Entanglementa)asafuntionofθandm0forN =100
qualitative effects in evolution of the relative population
and b) as a funtion of N and m0 =0 for θ=π/2.
andarecapableofpreventingtheselftrappingeffectfrom
occurring,evenwhen the collisionrate is largerthan the
Josephson-type coupling. Taking advantage of the ana-
lytical solution of the system, we include in our work a
studyonmodeentanglementanddiscussunderwhatcon-
ditions high degrees of entanglement can be generated.
We find that higher degreesof entanglementcan be gen-
erated when the condensate consists of large number of
(a) particles and the two-body collision probability is com-
parable(butnotlarger)totheenergydifferencebetween
the modes in absence of Josephson-type interaction. In
suchcondensates,maximallydegreesofentanglementare
generated by a strong resonant laser coupling (or small
symmetric tunneling barrier in the case of a double well
condensate). Theanalyticalsolutiontoourmodelhelped
us learn details about the role of two-body collisions in
(b) the generationof entanglementin the system. We found
that collisions favor entanglement as long as the colli-
FIG. 6: Entanglement for N=100 as a function of a) θ for sionprobabilitydoes notexceedthe frequency difference
m0 =0 and b) m0 for θ=π/2. betweenthe modes(inthe absenceofJosephson-typein-
teraction). If such is the case, higher degrees of entan-
glement can achieved by allowing maximal tunneling in
VIII. DISCUSSION AND OUTLOOK a slightly asymmetric well (or turning on a strong but
slightly detuned laser coupling).
We have presented several predictions of a new model To complete ourstudy of inelastic spin-exchangecolli-
describing a two-mode Bose-Einstein condensate which sionsinthetwo-modeBose-Einsteincondensate,weplan
includes spin-exchange inelastic collisions. These colli- tointroducethelossofparticlesduetoinelasticcollisions
sions have been observed in experiments when they lead asadecoherenceprocessusingamasterequation. Weare
to particle loss. Nevertheless, there are good reasons to currently working in the generalization of our model to
believethattheseprocessesdonotalwaysleadtoparticle spin-1 (or three well) condensates. We are also studding
loss. The particlesescape the trapafter aninelastic pro- the effects of many-body collisions in the system.
cessonlywhenthereisanexcessofenergylargerthanthe We thank C. G. Hern´andez-Salinasfor his help verify-
trapping potential. Our work is the first to analyze the ing some of our results.
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