A variational model for the delayed collapse of Bose Einstein condensates
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l A variational model for the delayed collapse of Bose Einstein condensates
Stavros Theodorakis and Stavros Athanasiou
Physics Department, University of Cyprus, P.O. Box 20537, Nicosia 1678, Cyprus ∗ (Dated: July 15, 2020)We present an action that can be used to study variationally the collapse of Bose Einstein conden-sates. This action is real, even though it includes dissipative terms. It adopts long range interactionsbetween the atoms, so that there is always a stable minimum of the energy, even if the remainingnumber of atoms is above the number that in the case of local interactions is the critical one. Theproposed action incorporates the time needed for the abrupt and delayed onset of collapse, yieldingin fact its dependence on the scattering length. We show that the evolution of the condensate isequivalent to the motion of a particle in an effective potential. The particle begins its motion farfrom the point of stable equilibrium and it then proceeds to oscillate about that point. We prove thatthe resulting large oscillations in the shape of the wavefunction after the collapse have frequenciesequal to twice the frequencies of the traps. Our results agree with the experimental observations. I. Introduction
Two most intriguing aspects of the experimental re-sults on the collapse of Bose Einstein condensates areits abrupt and delayed onset and then the survival of aremarkably stable remnant condensate with a constantnumber of atoms for more than one second[1]. In theseexperiments the repulsive nature of the interactions be-tween the atoms that were trapped in a magnetic trapwas changed abruptly to attractive, resulting in the col-lapse of the condensate. A proposed explanation for thedelayed onset of this collapse maintained that the con-densate conserves initially the number of its atoms whileshrinking in size. Thus the density gradually increasesat the center of the condensate. When it becomes largeenough, three-body recombination losses set in at thecenter, resulting thus in the expulsion of atoms from thecondensate[2].The theoretical descriptions of the collapse involvedthe usual local Gross-Pitaevskii equation, augmented by ∗ Electronic address: [email protected] a quintic dissipative term due to three body recombi-nation processes[3]. If one also includes the linear termdescribing the atomic feeding of the condensate from thesurrounding nonequilibrium thermal cloud[4], then onewill end up with the generalized Gross-Pitaevskii equa-tion i ¯ h ∂ψ∂t = − ¯ h m ∇ ψ + U ( r ) ψ − g | ψ | ψ + i ¯ hνψ − i ¯ hξ | ψ | ψ, (1)where ν and ξ are real constant parameters. Here U ( r )is the real harmonic potential of the trap, R | ψ | d r = N and g = 4 π ¯ h a/m , N being the number of atoms in thecondensate and a being the absolute value of the negativescattering length. This equation leads to the depletionrate of the condensate: dNdt = 2 νN − ξ Z | ψ | d r. (2)Hence the number of condensed atoms will becomeconstant only if the right hand side becomes zero forlong times. This involves ψ having a steady profile, cor-responding to the compensation for the three-body re-combination losses by a steady influx of thermal non-condensed atoms from the surrounding cloud. In par-ticular, the condensate density at the center of the trapand the width of the wavefunction will have to be con-stant at long times. This disagrees with the experimen-tal results[1], which show that the width of the remnantcondensate keeps oscillating in time, while the number ofcondensed atoms remains constant. Thus the longevityof the remnant condensate for times of the order of onesecond cannot be explained this way.A way out seems to be offered by the realization thatif the attractive interactions between the atoms are long-ranged and nonsingular, then the condensate cannotcollapse[5]. In fact, there are two energy minima belowthe critical point: a large width metastable anisotropiccondensate, which disappears at the critical point, and asmall width isotropic stable remnant, which continues toexist for values of a much higher than the critical one,such as the ones used in the experiments[1].Indeed, the existence of the high density and smallwidth minimum of the energy is absolutely vital for theobservation of a long-lived remnant. It explains, in fact,why the remnant has been often observed for numbersof atoms far greater than the number corresponding tothe critical point. It is not sufficient though, as can beseen from Eq. (2). The nonlocal nature of the attractiveinteractions does alter the cubic term of Eq. (1), but nocubic terms appear in Eq. (2). Therefore the paradoxpersists.This paper will propose a mechanism for resolving thiscontradiction between the experimental results and theformalism of the generalized Gross-Pitaevskii equation.In doing so, it will also give a detailed variational de-scription of the collapse.We shall adopt a phenomenological time-dependent dissipative term that can reproduce the observed evolu-tion of the number of atoms of the condensate, includingthe delayed onset of the collapse. In fact, we shall be ableto describe in detail the collapse of the Bose-Einstein con-densate, finding in addition the dependence of the timeof collapse on the scattering length. We shall achievethis by noting that the terms on the right hand side ofEq. (2) should vanish for long times, irrespective of whatthe width of the wavefunction looks like. This can onlybe achieved if both coefficients of the dissipative termsof Eq. (1) vanish at long times. In fact, both dissipativeterms in Eq. (1) can be modelled by a phenomenologicallinear term with an imaginary time-dependent coefficientthat will effectively encompass both the influx of noncon-densed atoms and the dissipative losses. These will endup balancing each other, resulting thus eventually in thevanishing of this coefficient.A variational description will enable us to examinefully the behavior of the condensate. In order to do thisthough, we shall need to formulate a real action that leadsto the desired generalized Gross-Pitaevskii equation, in-cluding the dissipative terms. We shall write down pre-cisely such an action, enabling us to find the evolution ofthe wavefunction profile. This action will be used in thenext section for determining the evolution of the numberof atoms and comparing it with the observations.In Section III we use a simple trial wavefunction forthe case of an isotropic trap in order to find the effec-tive potential and the critical point, as well as to explainthe persistent oscillations of the remnant condensate. Wealso study the perturbations around the critical point andwe find the dependence of the time of collapse on the scat-tering length. In Section IV we repeat these calculationsusing an anisotropic trial wavefunction and we comparethe results to the experimental observations. Section Vsummarizes our results.II. A real action
We shall adopt the generalized Gross-Pitaevskii equa-tion i ¯ h ∂ Ψ ∂t = − ¯ h m ∇ Ψ + mω ρ ρ mω z z i ¯ hν ( t )Ψ − πa ¯ h m Z V ( r − r ′ ) | Ψ( r ′ , t ) | Ψ( r , t ) d r ′ . (3)The magnetic trap is cylindrically symmetric with fre-quencies ω ρ and ω z . For the long range interaction weassume that R V ( r ) d r = 1, so that in the limit of zerorange it will reduce to a Dirac delta function, turningthen the nonlocal term into the standard cubic local termof Eq. (1). The time dependent coefficient ν ( t ) is com-plex, hence the whole equation is a dissipative one.We can make this equation dimensionless[6], if we mea-sure Ψ in units of p N /d , distances in units of d , timesin units of 1 /ω , V in units of 1 /d and ν ( t ) in units of ω ,where d = p ¯ h/ (2 mω ) and ω = ( ω z ω ρ ) / . Thus Eq. (3)takes the form i ∂ Ψ ∂t = −∇ Ψ + λ − / ρ λ / z iν ( t )Ψ − πk √ Z V ( r − r ′ ) | Ψ( r ′ , t ) | Ψ( r , t ) d r ′ , (4)where λ = ω z /ω ρ , k = N a/ℓ , ℓ = p ¯ h/ ( mω ). Here N is the initial value of the number of atoms N ( t ) in thecondensate, so that R | Ψ | d r = N ( t ) /N = n ( t ).If we multiply Eq. (4) by Ψ ∗ , subtract from the result-ing expression its corresponding complex conjugate andthen integrate over all space, we shall obtain the relation dndt = 2 n ( t ) Reν ( t ) (5)and hence n ( t ) = e R t Reν ( τ ) dτ . (6)Thus dissipation will take place only as long as thereal part of ν ( t ) is nonzero. In this formulation, the de- tails of the nonlocal interactions do not affect directly theevolution of n ( t ).Eq. (4) minimizes the action R L ( t ) dt − i R ν ( t ) | Ψ | d rdt , where L ( t ) = Z d r (cid:16) i ∗ ∂ Ψ ∂t − i ∂ Ψ ∗ ∂t − |∇ Ψ | − λ − / ρ | Ψ | − λ / z | Ψ | (cid:17) +4 πk √ Z d r d r ′ | Ψ( r , t ) | V ( r − r ′ ) | Ψ( r ′ , t ) | (7)The piece R L dt of the action is real, but the term − i R ν ( t ) | Ψ | d rdt is not. Thus we cannot use this actionin a variational calculation, since it cannot be minimized.We shall use a modified action instead, similar to the oneused in the problem of a damped harmonic oscillator[7]: S = Z e − R t Reν ( τ ) dτ L ( t ) dt + Z d r dt e − R t Reν ( τ ) dτ Imν ( t ) | Ψ | . (8)We can easily verify that the functional differentiationof S with respect to Ψ ∗ yields Eq. (4). It is the realaction of Eq. (8) the action on which we shall base allour calculations.In the experiment[1] the scattering length is almostzero at time t = 0, the initial condensate wavefunc-tion being the harmonic oscillator ground state. It thenjumps to 36 Bohr radii[8] within 0.1 msec. The attrac-tive interaction is thus switched on suddenly and the con-densate will absorb almost instantaneously any noncon-densed atoms happen to be around it. In the experimentthe condensate contained 97 percent of the total num-ber of atoms. The initial sweeping of the surroundingnoncondensed atoms is the reason for the slight initialpositive slope of n ( t ), seen in Figure 1. Nonetheless, themaximum number of 16000 atoms is reached almost in-stantaneously. This number remains constant while thecondensate is shrinking, till the central density becomeslarge enough to enhance dramatically the expulsion ofatoms due to the three-body recombination losses. Thus n ( t ) starts decreasing, till it reaches eventually an asymp-totic value. The reason the collapse stops is the longrange attraction of each atom by its outlying neighbors,which provides a vital enhancement to the quantum pres-sure and balances the attraction of the trap and of thecentral atoms. This balance is manifested by the ubiqui-tous existence of a stable isotropic small width minimumof the energy, irrespective of the value of n ( t ), and canresult in the appearance of remnants with a number ofatoms far greater than the critical number.We can describe all this behavior by choosing for thephenomenological parameter ν ( t ) the simplest form thatis consistent with the experimental observations: ν ( t ) = − ν r t − t c t e − ( t − t c ) /t (9)Here t c is the collapse time, i.e. the time during whichthere is no change in the total number of condensateatoms. Indeed, if 0 < t < t c , the real part of the ν ( t )given by Eq. (9) is zero. Correspondingly, Eq. (6) indi-cates that the number of atoms remains constant duringthis time and equal to 1. Furthermore, the vanishing of ν ( t ) at long times implies according to Eq. (5) that thenumber of atoms is constant in that limit. This is whatwe mean by remnant condensate.If t > t c , then Eq. (6) gives n ( t ) = Exp h e − ( t − t c ) /t ν √ t √ t − t c −√ πν t erf ( p ( t − t c ) /t ) i (10)We have fitted this expression for n ( t ) to the data ofFigure 4.2 of Ref.[8], as shown in Figure 1. The fit givesthe dimensionless values t c = 0 . t = 0 . ν = 2 . N = 16000,while a rises within 0.1 msec from 0 to a constant value of t n H t L FIG. 1: The number of atoms n ( t ) = N ( t ) /N in the con-densate versus time for a = 36 a and N = 16000. The con-tinuous line is the expression of Eq. (10) when t c = 0 . t = 0 . ν = 2 . a , giving thus k = 9 .
98. The final (asymptotic) valueof n ( t ) is 0.280, corresponding to an asymptotic value of2.79 for n ( t ) k , still much greater than the correspondingcritical value of 0.55[6]. The trap frequencies are ω z =2 π × ω ρ = 2 π × d = 2 . µm .We have also fitted the expression of Eq. (10) to thedata of Figure 1b of Ref.[9], as shown in Figure 2. The fitgives the dimensionless values t c = 0 . t = 0 . ν = 11 . N = 40000, while a rises almost instantaneouslyfrom 0 to a constant value of 20 a , giving thus k = 21 . n ( t ) is 0.166, correspond-ing to an asymptotic value of 3.61 for n ( t ) k , still muchgreater than the critical value of 0.557 for k [6]. Themean trap frequency is ω = 2 π × λ = 2 . d = 1 . µm .Finally, we have fitted the expression of Eq. (10) to thedata of Figure 3 of Ref.[9], as shown in Figure 3. The fitgives the dimensionless values t c = 0 . t = 0 . ν = 8 . N = 40000, while a rises almost instantaneously from t n H t L FIG. 2: The number of atoms n ( t ) = N ( t ) /N in the con-densate versus time for a = 20 a and N = 40000. The con-tinuous line is the expression of Eq. (10) when t c = 0 . t = 0 . ν = 11 . t n H t L FIG. 3: The number of atoms n ( t ) = N ( t ) /N in the conden-sate versus time for a = 8 . a and N = 40000. The con-tinuous line is the expression of Eq. (10) when t c = 0 . t = 0 . ν = 8 . . a , giving thus k = 9 .
15. Thefinal (asymptotic) value of n ( t ) is 0.3737, correspond-ing to an asymptotic value of 3.42 for n ( t ) k , still muchgreater than the critical value of 0.557 for k [6]. The meantrap frequency is ω = 2 π × d = 1 . µm .In all three of these cases the final value of n ( t ) k is much greater than the critical value of 0.55. This partic-ular value corresponds to the collapse in the local case,where the condensate collapses to a singularity when k becomes large enough. The long range interactions gen-erate however a stable isotropic minimum of the effectivepotential, the remnant condensate, so the concept of col-lapsing to a point singularity becomes irrelevant, alongwith the value k crit =0.55.III. The isotropic trap
We shall use Eq. (8) in order to study variationallythe case of an isotropic trap with λ = 1 and ω z = ω ρ .We shall be interested in values of k much greater than k crit , thus we expect the density to be acutely peakedat the center. We shall adopt thus an exponential trialwavefunction of r = | r | , rather than a gaussian:Ψ( r , t ) = p n ( t ) √ πs ( t ) / e − r/s ( t )+ ic ( t ) r + iw ( t ) , (11)where s ( t ), c ( t ) and w ( t ) are real. This wavefunctionsatisfies the relation R | Ψ | d r = n ( t ).The initial wavefunction at t = 0 is (2 π ) − / e − r / ,the ground state of the harmonic oscillator, in which themean value of r is 3. In contrast, the mean value of r forour exponential trial wavefunction is 3 s (0) at time t = 0.We shall require our trial wavefunction to have initiallythe same width as the initial wavefunction. Hence s (0) =1. Furthermore, c (0) = 0 since the initial wavefunctionhas no ir term in the exponent.We shall assume a particular form now for the longrange interaction: V ( r ) = e − r/ℓ πℓ . (12)When we insert our trial wavefunction of Eq. (11) intoEq. (8), we obtain the action Z dt (cid:16) − s ( t ) c ′ ( t ) − s ( t ) − c ( t ) s ( t ) s ( t ) + n ( t ) k (32 ℓ + 10 ℓs ( t ) + s ( t ) ) √ ℓ + s ( t )) − w ′ ( t ) + Imν ( t ) (cid:17) (13)The last two terms do not contribute to the dynamics.The functional derivative of this action with respect to c ( t ) gives c ( t ) = s ′ ( t ) / (4 s ( t )). We insert this expressionfor c ( t ) into Eq. (13), obtaining finally the effective action S eff = Z dt (cid:16) s ′ ( t ) − U eff (cid:17) (14)and the effective energy H eff = 34 s ′ ( t ) + U eff , (15)where U eff = 1 s ( t ) + 3 s ( t ) − k eff ( t )(32 ℓ + 10 ℓs ( t ) + s ( t ) ) √ ℓ + s ( t )) , (16)and k eff ( t ) = n ( t ) k . We see thus that the dynamics isdetermined by the instantaneous value k eff ( t ).There is always at least one minimum of U eff . Forexample, if k eff ( t ) = 9 .
98 and ℓ = 0 .
05 the wavefunc-tion width is very small and the corresponding singleminimum very deep (see Figure 4). In fact, minimiz-ing U eff for large k eff ( t ) and small s ( t ) yields the width s ( t ) = 1 . ℓ / /k eff ( t ) / .If however we take the example k eff ( t ) = 0 .
33 and ℓ = 0 .
05, there are two minima (see Figure 5), a minimumwith a large width and low density and the high densityremnant with a small width. The remnant exists due tothe long range interactions.In the local case ( ℓ = 0) the width of the remnantbecomes zero, hence the remnant becomes a singularity. In that case the collapse is associated with the loss ofstability of the unique minimum, the one with the largewidth. We can find the critical value k crit of k eff ( t ) inthis local case by requiring that both the first and secondderivatives of the effective potential U eff with respect to s ( t ) vanish there. This happens when s ( t ) = 0 . k crit = 0 . ℓ = 0 .
05. The first andsecond derivatives of U eff with respect to s ( t ) now be-come simultaneously zero for the values ( s ( t ), k eff ( t ))=(0.1486, 0.297) and (0.7013, 0.575). If we start with azero value of k eff ( t ) and then increase it, we shall haveinitially a large width minimum, then at k eff ( t ) = 0 . k eff ( t ) = 0 .
575 the large width minimum disappears,leaving only the remnant with the narrow width as a pos-sible state. We can see in fact the regions of existenceof one or two possible minima in the ( k eff ( t ), ℓ ) graphof Figure 6. Within the triangular region shown in the k eff ( t )- ℓ space of that figure, there are two possible min-ima of the effective potential. The borders of this regionrender both the first and the second derivative of U eff with respect to s ( t ) equal to zero. Outside this region,there is only one minimum. The minimum on the right ofthe curved line corresponds to the remnant condensate.We can explore further the dynamics of the conden-sate close to the point of collapse. Let us assume thatthe effective potential has a minimum at s ( t ) = s for agiven set of ℓ and k eff ( t ). We can expand the effectivepotential around s , obtaining U eff ( s ( t )) ≈ U eff ( s ) + B ( k eff ( t ))( s ( t ) − s ) / . (17)For a given value of ℓ there is a value k crit which is on s H t L - - - - - U eff FIG. 4: The effective potential U eff of Eq. (16) when k eff ( t ) = 9 .
98 and ℓ = 0 . s H t L - U eff FIG. 5: The effective potential U eff of Eq. (16) when k eff ( t ) = 0 .
33 and ℓ = 0 . the curved border of the triangular region of Figure 6. Atthis value the first and second derivatives of U eff becomezero and the large width minimum ceases to exist. Hence B ( k crit ) = 0. For values of k eff ( t ) above k crit B ( k eff ( t ))will be negative. We may expand it around k crit thenand obtain B ( k eff ( t )) ≈ − ( k eff ( t ) − k crit )Ω (18)Thus the action becomes Z dt (cid:16) s ′ ( t ) − U eff ( s ) k eff H t L { FIG. 6: Within the triangular region shown above in the k eff ( t )- ℓ space, there are two possible minima of the effec-tive potential. The borders of this region make both the firstand the second derivative of U eff with respect to s ( t ) equalto zero. Outside this region, there is only one minimum. Theminimum on the right of the curved line corresponds to theremnant condensate. +( k eff ( t ) − k crit )Ω ( s ( t ) − s ) / (cid:17) (19)The corresponding equation of motion is32 s ′′ ( t ) = ( k eff ( t ) − k crit )Ω ( s ( t ) − s ) (20)For times before t c we expect k eff ( t ) = k . The solutionsof this equation involve then exponentials of the form e t/τ , where the quantity τ = q k − k crit )Ω indicates thetime needed for the manifestation of the instability. Itis, in other words, essentially the time of collapse. Weexpect therefore the time of collapse t c to be proportionalto ( k − k crit ) − / , a conclusion that can also be reached byalternative arguments[10]. This collapse time determinesthe beginning of the collapse towards the small widthminimum.We can test this prediction by comparing it with theexperimental data. The experiments used an anisotropictrap, but t c varies like ( k − k crit ) − / in that case too, aswe shall see in the next section. In Figure 7 we show the k t c H ms L FIG. 7: The collapse time (in msec) as a function of k , for N = 6000 and d = 2 . µm . The experimental points aretaken from Figure 4.3 of Ref.[8] and fit quite well to the func-tion f / √ k − k c , where f = 3 . k c = 0 . collapse time t c (in msec) from Figure 4.3 of Ref.[8], forwhich N = 6000 and d = 2 . µm . This data is fitted toa function of the form f / √ k − k c , where f = 3 . k c = 0 . k = 0 . U eff to disappear at this value of k eff ( t ),then we would need the minimum to occur at s ( t ) = 0 . ℓ would need to be equal to 0.047.In Figure 8 we show the collapse time t c (in msec)from Figure 2 of Ref.[9], for which N = 40000 and d =1 . µm . This data is fitted to a function of the form f / √ k − k c , where f = 2 . k c = 0 . k = 0 . U eff to disappear at this value of k eff ( t ), then we would needthe minimum to occur at s ( t ) = 0 .
640 and ℓ would needto be equal to 0.105.We can find the time evolution of the condensate bysolving the Euler-Lagrange equation for the action ofEq. (14). We shall do so in fact for the data of Fig- k t c H ms L FIG. 8: The collapse time (in msec) as a function of k , for N = 40000 and d = 1 . µm . The experimental points aretaken from Figure 2 of Ref.[9] and fit quite well to the function f / √ k − k c , where f = 2 . k c = 0 . ure 4.2 of Ref.[8], shown in Figure 1. We shall adoptthe values k = 9 . ℓ = 0 . t c = 0 . t = 0 . ν = 2 . k eff ( t ) is 9.98, since the value of a is shifted almostinstantaneously from 0 to 36 a . The initial values s (0)and s ′ (0) are then 1 and 0 respectively. In the interval(0, t c ) we have n ( t ) = 1, but for later times it is given byEq. (10). The resulting numerical solution of the Euler-Lagrange equation for the action of Eq. (14) is shown inFigure 9.We see that the corresponding oscillations are huge andpersistent. The reason for this can be understood if welook at Figure 4. The action of Eq. (14) is the action of aparticle moving in the effective potential U eff . The parti-cle starts at rest at the point s [0] = 1, at the right edge ofthe deep potential well of this figure. It then acceleratestowards the minimum and passes it, overshooting till itreaches a stopping point at a value of s ( t ) = 0 . s ( t ) in these os-cillations. For large values of s ( t ) the Euler-Lagrange t s H t L FIG. 9: The oscillations of s ( t ) as a function of time. Here n ( t ) is equal to 1 if 0 < t < t c , but it is given by Eq. (10)for later times. We adopt the values k = 9 . ℓ = 0 . t c = 0 . t = 0 . ν = 2 . equation for Eq. (14) becomes3( s ( t ) + s ′′ ( t ))2 = 2 s ( t ) (21)The solution of this differential equation is s ( t ) = sr
43 + δ + δ cos(2 t − t ) , (22)where δ and t are integration constants. For the os-cillation of Figure 9, in which a maximum occurs at t = 1 . s ( t ) = 6 . δ = 22 . t = 1 . ω inan isotropic trap, irrespective of the value of k eff ( t ).IV. The anisotropic trap
We shall now study variationally the case of ananisotropic trap. Here we shall use a gaussian wavefunc-tion of the cylindrical coordinates ρ and z , since an ex-ponential wavefunction would involve very complicatedintegrals: Ψ( ρ, z, t ) = p n ( t ) π / s t ) p s t ) × e − ρ s t )2 − z s t )2 + ic ( t ) ρ + ic ( t ) z + iw ( t ) , (23)where s ( t ), s ( t ), c ( t ), c ( t ) and w ( t ) are real. Thiswavefunction satisfies the relation R | Ψ | d r = n ( t ).The initial wavefunction is(2 π ) − / e − ρ / (4 λ / ) − λ / z / , the ground state of theanisotropic harmonic oscillator. Hence s √ λ / and s (0) = √ λ − / . Furthermore, c (0) = c (0) = 0since the initial wavefunction has no iρ and iz termsin the exponent.We shall assume a gaussian form now for the long rangeinteraction: V ( ρ, z ) = e − ρ /ℓ − z /ℓ π / ℓ . (24)When we insert our trial wavefunction of Eq. (23) intoEq. (8), we obtain the action Z dt (cid:16) − s ( t ) − s ( t ) − s ( t ) (cid:0) λ − / c ( t ) + c ′ ( t ) (cid:1) − s ( t ) (cid:0) λ / + 16 c ( t ) + 4 c ′ ( t ) (cid:1) + 4 n ( t ) k p /π ( ℓ + 2 s ( t ) ) p ℓ + 2 s ( t ) − w ′ ( t ) + Imν ( t ) (cid:17) (25)The last two terms do not contribute to the dynamics.The functional derivatives of this action with respect to c ( t ) and c ( t ) give c ( t ) = s ′ ( t ) / (4 s ( t )) and c ( t ) = s ′ ( t ) / (4 s ( t )). We insert these expressions for c ( t ) and c ( t ) into Eq. (25), obtaining finally the effective action S eff = Z dt (cid:16) s ′ ( t ) + 18 s ′ ( t ) − U eff (cid:17) (26)0and the effective energy H eff = 14 s ′ ( t ) + 18 s ′ ( t ) + U eff , (27)where U eff = 1 s ( t ) + s ( t ) λ / + 12 s ( t ) + λ / s ( t ) − k eff ( t ) p /π ( ℓ + 2 s ( t ) ) p ℓ + 2 s ( t ) (28)and k eff ( t ) = n ( t ) k . We see thus that the dynamics isdetermined by the instantaneous value k eff ( t ).There is always at least one minimum of U eff . Forlarge k eff ( t ) the wavefunction widths are very small andthe corresponding single minimum very deep. In fact,minimizing U eff for large k eff ( t ) and small s ( t ) and s ( t ) yields the widths of the remnant that is stabilisedby the long range interactions: s ( t ) ≈ √ ℓ / λ / ( ℓ + 32 λ / p /πk eff ( t )) / (29)and s ( t ) ≈ √ ℓ / ( ℓ λ / + 32 p /πk eff ( t )) / . (30)For large k eff ( t ) these widths are essentially equal, re-sulting in a spherical remnant.In the local case ( ℓ = 0) the width of the remnantbecomes zero, hence the remnant becomes a singularity.In that case the collapse is associated with the loss ofstability of the unique minimum, the one with the largewidth. We can find the critical value k crit of k eff ( t ) inthis local case, for the case of λ = 6 . / . U eff . This happens when s ( t ) = 0 . s ( t ) = 0 . k crit = 0 . ℓ = 0 .
05 with λ =6 . / .
5. There are saddle points at ( s ( t ), s ( t ), k eff ( t ))=(0.0707, 0.0707, 0.1095) and (0.876, 0.993, 0.6395). Ifwe start with a zero value of k eff ( t ) and then increase it,we shall have initially an anisotropic minimum, then at k eff ( t ) = 0 . k eff ( t ) = 0 . s ( t ) = s and s ( t ) = s for a given set of ℓ , λ and k eff ( t ). Wecan expand the effective potential around this minimum,obtaining U eff ≈ U + A ( k eff ( t ))( s ( t ) − s ) / A ( k eff ( t ))( s ( t ) − s ) / A ( k eff ( t ))( s ( t ) − s )( s ( t ) − s ) . (31)For a given value of ℓ there is a value k crit at which theanisotropic minimum becomes a saddle point. At thisvalue A and A are positive, while A = A A . Forvalues of k eff ( t ) just above k crit the quantities s ( t ) and s ( t ) will increase exponentially as e W t . If we solve theresulting linear equations of motion, we shall find W asa function of k eff ( t ): W = − A − A q A + 8 A − A A + 4 A . (32)At the critical point, where A = A A , this W be-comes zero. Hence, if we expand W just above k crit ,we shall obtain W ≈ W ( k eff ( t ) − k crit ). There-fore the time 1 /W , the time that characterizes themanifestation of the instability, will be proportional to1 / p k eff ( t ) − k crit . Consequently, the collapse time,which is of the same order of magnitude as 1 /W , willalso vary like 1 / p k eff ( t ) − k crit , as was already shownto be the case for the isotropic trap and as the experi-mental observations in Figure 7 and Figure 8 indicate aswell.We can find the time evolution of the condensate bysolving the Euler-Lagrange equation for the action ofEq. (26). We shall do so in fact for the data of Figure 4.2of Ref.[8], shown in Figure 1. We shall adopt the values λ = 6 . / . k = 9 . ℓ = 0 . t c = 0 . t = 0 . ν = 2 . k eff ( t ) is 9.98, since the value of a is shifted almostinstantaneously from 0 to 36 a . The initial values s (0), s (0), s ′ (0) and s ′ (0) are then 1.208, 1.938, 0 and 0 re-spectively. In the interval (0, t c ) we have n ( t ) = 1, butfor later times it is given by Eq. (10). The resulting nu-merical solutions of the Euler-Lagrange equation for theaction of Eq. (26) are shown in Figure 10 and Figure 11.We see that the corresponding oscillations are againhuge and persistent, just as in the isotropic case. Theaction of Eq. (26) is the action of a particle moving inthe effective potential U eff . The particle starts at restat the edge of the deep potential well. It then acceleratestowards the spherical narrow width minimum and passesit, overshooting till it reaches a stopping point well be-yond the minimum. Finally, it moves in the oppositedirection, completing thus a full oscillation.We can find the approximate form of the widths inthese oscillations. For large values of s ( t ) and s ( t ) the t s H t L FIG. 10: The oscillations of the radial width s ( t ) as a func-tion of time. Here n ( t ) is equal to 1 if 0 < t < t c , but itis given by Eq. (10) for later times. We adopt the values λ = 6 . / . k = 9 . ℓ = 0 . t c = 0 . t = 0 . ν = 2 . t s H t L FIG. 11: The oscillations of the axial width s ( t ) as a functionof time. Here n ( t ) is equal to 1 if 0 < t < t c , but it is given byEq. (10) for later times. We adopt the values λ = 6 . / . k = 9 . ℓ = 0 . t c = 0 . t = 0 . ν = 2 . Euler-Lagrange equations for Eq. (26) become s ( t ) λ / + s ′′ ( t ) = 4 s ( t ) (33)and λ / s ( t ) + s ′′ ( t ) = 4 s ( t ) (34)The solutions of these differential equations are2 s ( t ) = rq λ / + δ + δ cos(2 λ − / ( t − t )) , (35)and s ( t ) = rq λ − / + δ + δ cos(2 λ / ( t − t )) , (36)where δ , δ , t and t are integration constants. Forthe oscillations of Figure 10 and Figure 11, in which amaximum for s ( t ) occurs at t = 1 . s ( t ) =19 . δ = 187 .
804 and t = 1 . s ( t ) occurs at t =3 . s ( t ) = 17 . δ = 149 .
619 and t = 3 . s ( t ) and s ( t ) oscillate atthe frequencies 2 λ − / ω = 2 ω ρ and 2 λ / ω = 2 ω z in ananisotropic trap, irrespective of the value of k eff ( t ). Thisis in fact what the experiments showed[1].The single anisotropic gaussian trial function ofEq. (23) is not very accurate. We can obtain a muchmore accurate trial wavefunction if we use the sum oftwo such anisotropic gaussians[11]. However, the result-ing expressions are too lengthy and complicated. For thelocal case ℓ = 0 and for λ = 6 . / .
5, the anisotropic min-imum of the effective potential becomes a saddle pointand the collapse occurs when k eff ( t ) = 0 . λ = 6 . / . k eff ( t ) = 0 . ℓ = 0 . λ = 2 .
175 and k eff ( t ) = 0 . ℓ = 0 . ℓ of the nonlocal interactions seems to beassociated to the already existing lengths d and ℓ B , where ℓ B = p ¯ h/ ( eB ) is the magnetic length. The scatteringlength is too small to be of the order of the interactionrange for the nonlocal interactions.The magnetic length is equal to 2 . µm/ √ B if B ismeasured in Gauss. The scattering length a near a Fes-hbach resonance is given by the expression a ( B ) = a bg (cid:0) − ∆ B − B p (cid:1) , (37)where a bg is the background scattering length, ∆ isthe resonance width and B p is the resonance centre,these quantities having the values[12] a bg = − a ,∆ = 10 . G and B p = 155 . G for the condensatesused in [1]. For the case of Figure 7 we have λ = 6 . / . d = 2 . µm and k eff ( t ) = 0 . ℓ B = 0 . µm . Since the interaction range we found is0 . d = 0 . µm , after making the various lengths di-mensionful again we notice that the dimensionless ratio ℓd/ℓ B is approximately equal to 15.For the case of Figure 8 we have λ = 2 . d =1 . µm and k eff ( t ) = 0 . ℓ B = 0 . µm . Since the interaction range we foundis 0 . d = 0 . µm , after making the various lengthsdimensionful again we notice that the dimensionless ratio ℓd/ℓ B is approximately equal to 15.It seems then that the dimensionful length ℓ is propor-tional to ℓ B /d .IV. Conclusions
The remnant condensate observed after the collapse ofattractive Bose Einstein condensates has been a puzzlefor some time, because the conventional Gross Pitaevskiiformalism cannot readily account for its existence andits longevity. There have also been difficulties in its the-oretical description because of its dissipative origin andof the abrupt and delayed onset of the collapse. Thecomplex terms that had to be included in the extendedGross Pitaevskii equation seemed furthermore to makeimpossible the variational study of this equation.In our paper we have addressed all these issues. Byincluding nonlocal interaction terms in the action, wemake the existence of the remnant inevitable and un-avoidable. The collapse is understood now simply as the3disappearance of the large width anisotropic condensateand its evolution to a narrow width spherical conden-sate. This evolution necessarily reduces the number ofatoms. This reduction necessitates however the inclusionof complex dissipative terms in the action. We presenteda real action that results in a field equation incorporat-ing the desirable complex terms. These terms containexplicitly the delayed onset of the collapse, and they aretime-dependent, so that after the elapse of enough timethey vanish, leading to a constant again number of atomsfor the remnant. The reality of the action enables us to perform various variational calculations. These demon-strate that even though the remnant has eventually aconstant number of atoms, it performs persistent andhuge oscillations at frequencies 2 ω ρ and 2 ω z .The proposed action can be used to perform variationalcalculations on any aspect of the behavior of the remnantcondensate. Acknowledgement
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