a r X i v :c o n d -m a t /0308259v 1  [c o n d -m a t .s o f t ]  13 A u g  2003
Eigenstates and excitations of the simple atom-molecule Bose-Einstein condensate
Marijan Koˇs trun and Juha Javanainen
Department of Physics,University of Connecticut,Storrs,Connecticut 06269-3046
∗
(Dated:February 2,2008)
We analyze the mean-field eigenstates of the atom-molecule Bose-Einstein condensate (AMBEC)under the assumption that the background (elastic)scattering length of the atoms can be ignored.It is shown that the relevant eigenstates are localized in the space of the condensate parameters:The eigenstate has a different character in different regions of the parameter space,and at the interface of two local eigenstates the properties of the system may change nonanalytically.Using both analytical and numerical techniques,we find the approximate boundaries of the local eigenstates and identify the types of parametric excitations that occur when an eigenstate is forced outside of its region of validity by a parameter sweep.We contrast the properties
of the mean-field parametric excitations found in AMBEC with the experimentally observed excitations of the BEC.
PACS numbers:03.75.F,05.30.J,34.50
I.INTRODUCTION
The dynamics of the dilute,trapped Bose-Einstein condensates (BEC)of a single atomic species has been successfully modeled using the Gross-Pitaevskii equation (GPE)[1,2][toss these reference],
i
∂Φ(r ,t )
2m ∇2
+V 0(r ,t )
Φ(r ,t )
+
4π 2a N
∗Electronic
address:kostrun@phys.uconn.edu ;URL:
krampus.phys.uconn.edu
or with electromagnetic radiation (photoassociation)[15–19].[add Stwalley’s suggestion].These schemes may be modeled with a quantum field theory of coupled atomic and molecular fields.A formally identical field theory can be devised for elastic resonant scattering [20],with the difference that the scattering resonances takes the role of the molecular dimer.All of these approaches lead to the same mean-field theory of the atom-molecule BEC (AM-BEC)in which there are three parameters:K ,coupling strength between pairs of atoms and molecules;δ,detun-ing of the molecular state from the edge of the two-atom continuum;and a bg ,background atom-atom scattering length.
In this paper we examine the eigenstates of the mean-field AMBEC under the simplifying assumption that a bg =0.The focus is on two eigenstates:the thermody-namically stable ground state,and a “twin”state that,we believe,is often more relevant than the ground state in experiments starting with a
n atomic condensate.Us-ing numerical and analytical techniques we show that the {δ,K }parameter space is divided into regions,between which the nature of the eigenstate abruptly changes.We associate the existence of such “local”eigenstates within a “global”eigenstate with the non-linearity of the mean-field theory of the coupling between the atoms and the molecules.
When the system is created in a local eigenstate and is subsequently pushed by a time-dependent variation of the parameters into a region in which the initial local eigenstate no longer exists,parametric excitations ensue.We will characterize and classify the excitation analyti-cally and numerically.
Finally,we will point out intriguing similarities with experiments.Caution is due in such comparisons,as the motion of the atoms and the molecules in the trap is significant in our calculations and not necessarily so in all experiments.Conversely,though,in genuine trap ex-periments heretofore unrecognized possibilities open up.In particular,with a proper variation of the parameters the collapse of a condensate with a negative (effective)scattering length may be achieved continuously and in a
2 controllable fashion.
II.MEAN FIELD MODEL OF SIMPLE AMBEC
We use the meanfieldsϕandψto describe atomic and
molecular condensates,respectively.The interaction en-
ergy produced by the coupling between the condensates
reads
E I=−
1
∂τ
=H aϕ−Kϕ∗ψ,(3a)
i
∂ψ
2∇2+
1
4∇2+ iω2m,i x2i.(4b)
For simplicity,we assume that the trap is isotropic with ω=1.Lastly,we scale thefields so that the normaliza-tion reads
ϕ|ϕ + ψ|ψ =1.(5) The expression for the conserved total energy of the sys-tem is then
E= ϕ|H a|ϕ +1
6
+
√
6
,for¯δ≤2,
1,for¯δ>2,
(8) and the frequency is
µGS(¯δ)= −¯δ¯δ2+12
2
,for¯δ>2.
(9) The twin state is specified by
y T W(¯δ)= −1,for¯δ<−2,
¯δ¯δ2+12
2
,for¯δ<−2,
−¯δ¯δ2+12
3
all-molecule state(y≡−1).However,as was the case with the ground state,the transition to an all-molecule state is continuous but not smooth.
We next examine the stability of the stationary solu-tions of Eq.(7)byfinding their frequencies of small oscil-lations.The most convenient variables are the molecular amplitude y and the phase difference between the square of the atomic amplitude and the molecular amplitude,θ=2arg(ϕ)−arg(ψ).In these new variables the origi-nal system of ODE’s(7)reads
˙y=(1−y2)sinθ,(12a)
˙θ=¯δ+ 1
y2
+3),(13)
which always gives a real number.
For the all-molecule solution the analysis is somewhat more complicated.For|¯δ|<2the all-molecule state is unstable[25],while for|¯δ|>2it is stable.To get a feel for how the stably the all-molecule state b
ehaves,let us consider the large¯δ=δ/K limit and write the molecular amplitude as y=1−δy.In that limit the approximate solution for phaseθisθ≈¯δt.Solving(12a)forδy(t) yields
δy(t)≈δy(0)·exp −√¯δ(1−cos(¯δt)) .(14)
That is,for|¯δ|>2the all-molecule solution is stable, but not an attractor:the oscillations of amplitudes in the vicinity of the all-molecule state are persistent and do not decay with time.
To conclude,the zero-dimensional model(7)has three stationary states:the ground state,the twin state and the all-molecule state.The ground state(twin state)has a non-analytic point at¯δ=+2(−2)where it merges with the all-molecule state.A detuning sweep across either of the non-analytic points creates parametric excitations of the amplitudes,the details of which depend on the numerical details(sweep rate,initial conditions).Com-monly,these excitations manifest themselves in small, persistent,non-linear oscillations near the all-molecule state.
IV.GROUND STATE AND TWIN STATE IN SIMPLE AMBEC-NUMERICAL STUDY
The simple AMBEC(3)inherits much of the behav-ior of the zero-dimensional version,but also adds some unexpected twists.Before going into the details,we ex-pand on some terminology we have already brought up in passing.
The zero-dimensional model has stationary states,or eigenstates.A notable one is the ground state,the eigen-state with the lowest chemical potential.There are no general criteria for the existence of eigenstates for non-linear differential operators,but it appears from our nu-merical calculations that the simple AMBEC,too,has a unique ground state for all parametersδand K.As the ground state can be identified,seemingly unambigu-ously,in the entire parameter space,we call it a global eigenstate.
In the zero-dimensional model the nature of the ground state is different depending on the parameters.For in-stance,if¯δ>2,the ground state is all molecules.A similar behavior is found in the ground state of the AM-BEC.Locally,in different regions of the parameter space, the ground state may be(nearly)all molecules,(nearly) all atoms,or a mixture of atoms and molecules.As in the zero-dimensional case,the nature of the ground state may change nonanalytically when the AMBEC parameters are varied.We say that at such a point of nonanalyticity the global ground state of the AMBEC switches from one local(in the{δ,K}space)eigenstate to another.We characterize local eigenstates as all-atom,all-molecule, or mixed atom-molecule.On occasion,this distinction is qualitative only.For instance,the all-atom state may not be all atoms,but the fraction of atoms is much larger than in the adjacent mixed atom-molecule state. However,the experiments do not always deal
with the ground state.In a Feshbach resonant system,what seem-ingly is an atomic BEC may be prepared with parameters such that the ground state of the AMBEC would be all molecules.We introduce the twin state analogous of the twin state in the zero-dimensional case to model this type of a situation.We construct the twin state numerically by starting with an all-atom ground state of the nonin-teracting system(K=0),and then vary K adiabatically. This procedure appears to produce an eigenstate of the AMBEC system for allδ>0and K.It appears that for certain K>0andδ<0the twin state is unstable,and in practice it is not possible to construct it numerically by integrating the AMBEC system in time.Nevertheless, we surmise that we have here another global eigenstate of the AMBEC.Just like the ground state,we expect the twin state to be split into three local eigenstates,:all-atom,all-molecule,and mixed atom-molecule eigenstate. Except for stationary states,we will also investigate what happens when the parametersδand K are varied in time.The general observation is that,where the system switches from one local eigenstate to another,various type parametric excitations set in.
The present Section IV reports on numerical studies about the breakdown of the ground state and the twin state into local eigenstates,and describes the parametric excitations that ensue when a parameter is swept across a border between local eigenstates.
4
A.Ground State
Here we calculate the ground state numerically.While the work in three dimensions is feasible in principle,the computations are restricted to spherically symmetric sit-uations for simplicity.We employ the DS-method[26]. The iteration that is repeated until convergence consists of integration in complex time followed by the renormal-ization,see Appendix A.For a point(δ,K)in parameter space wefind the solutionsϕandψfor the amplitudes. Next we determine the atomic fraction,which is the same as the norm of atomic amplitude,N a= ϕ|ϕ ,and the half-size R1/2,Eq.(A4),of the atomic distribution.The calculation of the ground state amplitudes is done over an integer meshδ=−30...30,and K=0.30; where we use K=0.1as an approximation of the limit K→0.
The fraction of the atoms and the half-size in the ground state are shown in Figs.1and2,respectively, for different points in the{δ,K}parameter space.We observe that the ground state consists of three local eigen-states:all-atom state,mixed or atom-molecule state,and all-molecule state.We refer to the boundaries of partic-ular eigenstate as a fractures,and label them as follows. The all-atom state is bounded by the fracture f1GS from the mixed or atom-molecule state and similarly,the all-molecule eigenstate is bounded by the fracture f2GS from the mixed or atom-molecule state.
We next demonstrate how a parameter sweep of the eigenstate across the fracture creates a parametric ex-citation.In thefirst example we perform an intensity sweep,with 18andδ=−30.The behavior of the atomic fraction and the half-size is shown in Fig.3. As surmised,at the fracture f1GS the local all-atom eigen-state disappears causing a parametric excitation of both amplitudes.The extent of the excitation is determined by the relative position of the mixed or atom-molecule eigenstate(which becomes a new ground state)and the numerical details of the sweep(rate,initial conditions). In the second example we perform a positive detuning sweep,withδ=−30...30and K=10.Fig.4shows the behavior of the norm and the half-size of the atomic distribution.Forδ<0wefind no evidence for para-metric excitations in the amplitudes,meaning that the all-atom state may continuously evolve into the mixed or atom-molecule state in some regions of parameter space. Forδ>0,close to the fracture f2GS,we observe another parametric excitation.
The parametric excitations at f2GS and f1GS differ.At f2GS the oscillations of the atomic quantities are erratic, whereas at f1GS the oscillations are periodic.We return to these differences in Sec.V,when we analyze the solutions for the amplitudes analytically.
B.Twin State
In the zero-dimensional model,the twin state and the ground state are related via the transformationδ→
−δ,K→−K.Applying this transformation to the ground state,we surmise that the twin state contains
the all-atom state forδ>0,K→0,and may include the all-molecule state.
We test our assumptions using numerical methods.For
parameters K>0andδ>0,the twin state is prepared from the all-atom state,ground state of the atomic sys-tem in the given trap,by slowly varying the matrix ele-ment K from zero to the desired value at thefixed de-tuningδ.Following such an adiabatic preparation,the twin state is monitored in the standard fashion while one of the parameters,either the detuningδor the matrix element K,is varied.
We limit the exposition of our results to two charac-
teristic examples.In thefirst example,shown in Fig.5, wefix the detuning atδ=100and vary K in the 36.As can be seen,the fraction of the atoms stays close to1at all times,while the half-size of the atomic condensate increases slightly.We observe that there are no parametric excitations,and conclude that the all-atom state exists forδ>0and K>0.This reaffirms adiabatic intensity sweep as a m
ethod for preparing the twin state for positive detunings,and demonstrates the absence of the analogue of the ground state fracture f1GS in the twin state.
In the second example we take the twin state prepared
atδ=50and K=10,and perform a negative detun-ing sweepδ=50...−50.The behavior of the norm and the half-size of the atomic distribution are shown in Fig.6.Forδ>0the all-atom state has no parametric excitations,in accord with thefirst twin-state example. Aroundδ≈0the fraction of the molecules starts to in-crease and the all-atom eigenstate becomes a mixed or atom-molecule eigenstate with an increasing fraction of molecules.At some negativeδwefind familiar signatures of a strong parametric excitation in the atomic quanti-ties,which suddenly start to oscillate chaotically.We refer to the set of points in the{δ,K}parameter space where the parametric excitations set in as the fracture f T W.
V.ANALYTIC BOUNDARIES OF LOCAL
EIGENSTATES
Both the twin state and the ground state have as a lo-cal eigenstate either an all-molecule state(zero-fraction of atoms),or an all-atom state(negligible fraction of molecules).The boundaries of t
hose two special eigen-states in the parameter space may be found using an-alytic methods.Our next task thus is tofind analytic expressions for the boundaries of the local eigenstates. We also compare the results with direct numerical com-putations.
5
3020
10
10
20
30
Detuning 0
5
10
15
20
25
Matrix element K
0.10.20.30.40.50.60.70.80.91N(atoms)FIG.1:Fraction of atoms (N a )in the ground state as a function of the detuning δand the atom-molecule coupling K .
30
20
10
10
20
30
Detuning
5
10
15
20
25
30
Matrix element K
0.20.40.60.811.2R(1/2) atoms FIG.2:Half-size R 1/2,Eq.(A4),of the atomic distribution in the ground state as a function of the detuning δand the atom-molecule coupling K .
A.
All-atom state;fracture f 1
GS
In the limit K →0and −δ≫K the fraction of molecules becomes negligible,so it is possible to elimi-nate the molecules from the theory.Performed in the standard way,this procedure leads to a single-condensate GPE with a negative scattering length.Its solution is ei-ther a single collapsed state or a pair consisting of the col-lapsed state and the “metastable”(non-collapsed)state.The collapse boundary in this approach is the boundary of the metastable state in the parameter space.In the variational approach [3]the collapse boundary is given by
a c ≈−0.67,which in terms of the AMBEC parameters
reads K
2
a1/2
the atomic distribution for the intensity sweep K=0 (18)
withfixedδ=−30.Integration is performed in50,000steps
of
FIG.4:(Ground State)Norm N a and the half-size R1/2of
the atomic distribution for the detuning sweepδ=−30 (30)
withfixed K=10,done in50,000steps of size(A2).Fol-
lowing f2GS the half-size of the atomic condensate begins to
a1/2
atomic distribution for the intensity sweep 36with
fixedδ=100,done in150,000steps of size(A2).We see no
evidence of parametric excitations.
FIG.6:(Twin State)Norm N a and the half-size R1/2of
the atomic condensate for the negative detuning sweepδ=
<−20withfixed K=10,done in100,000steps of size
(A2).The time is reversed(goes from right to left)so that the
comparisonsdetuning axis has the same orientation as in Fig.4.Following
f T W the molecules decay rapidly into atoms and both atomic
and molecular distributions begin to oscillate erratically in
half-size and fraction.
can be formally solved for thefieldψ[27],yielding
ψ=−K i∂δ ∞j=0 1∂t−H m  jϕ2.(15)
In the limit N m= ψ|ψ ≪1and large|δ|,the term
i∂ψ
δ
ψcan be neglected in(15)when com-
pared to H mψ∼3
δ|ϕ|2ϕ+
K2
2δ ϕ2|ϕ2 +K2

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