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Behavior of heat capacity of an attractive Bose-Einstein Condensate approaching

collapse

Sanchari Goswami∗, Tapan Kumar Das† and Anindya Biswas‡

Department of Physics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India

We report calculation of heat capacity of an attractive Bose-Einstein condensate, with the numberN of bosons increasing and eventually approaching the critical number Ncr for collapse, using thecorrelated potential harmonics (CPH) method. Boson pairs interact via the realistic van der Waalspotential. It is found that the transition temperature Tc increases initially slowly, then rapidly as Nbecomes closer to Ncr. The peak value of heat capacity for a fixed N increases slowly with N , forN far away from Ncr. But after reaching a maximum, it starts decreasing when N approaches Ncr.The effective potential calculated by CPH method provides an insight into this strange behavior.

PACS numbers: 03.75.Hh, 03.65.Ge, 03.75.Nt

I. INTRODUCTION

Bose-Einstein condensation (BEC) is the transitionprocess, in which a macroscopic fraction of bosonsgoes into the lowest energy state, as the temperatureis lowered below a certain critical temperature Tc [1].It was predicted by Einstein in 1925, based on Bose’sexplanation of black body radiation. A great deal ofactivity, both theoretical and experimental, has beenseen in this field, since the experimental realization ofBEC in 1995. Although a number of static, dynamicand thermodynamic properties have been studied [2, 3],not much attention has been paid to the heat capacityof attractive condensates. The main motivation of thiswork is to fill this gap.

In laboratory experiments, the condensate is trappedby a confining potential, usually a harmonic oscillatorpotential. An attractive condensate (e.g. 7Li con-densate) has a negative value of the s-wave scatteringlength as and collapses, when the number of particlesN in the condensate exceeds a critical number Ncr.On the other hand a repulsive condensate (e.g. 87Rbcondensate) corresponds to as > 0 and is stable for anyN , since repulsively interacting bosons are containedin the externally applied trap. The situation is quitedifferent for an attractive condensate: attractive bosonstend to come to the center of the trap, which is balancedonly by the kinetic pressure, resulting in a metastablecondensate. The total attraction increases as the numberof pairs N(N − 1)/2, while the kinetic pressure increasesas N . Thus for N larger than a critical value Ncr, thenet attraction dominates and a collapse occurs.

In this communication, we report the calculation ofheat capacity of an attractive condensate containing a

∗e-mail: sg.phys.caluniv@gmail.com†e-mail: tkdphy@caluniv.ac.in‡e-mail: abc.anindya@gmail.com

fixed number of 7Li atoms, interacting via the realisticvan der Waals potential, appropriate for the experi-mental scattering length. The features are markedlydifferent from those of repulsive condensates, only whichhave so far been investigated. In a repulsive condensate,the heat capacity CN (T ) for a fixed number N of bosonsin the trap, as also the critical temperature Tc, smoothlyapproach a constant value as N → ∞ [4, 5]. As afunction of T , the heat capacity for a given N increasesto a maximum (CN )max, then falls rapidly to a satura-tion value 3NkB as T increases (kB is the Boltzmannconstant). These features are qualitatively similar tothose of a trapped non-interacting condensate [3, 5].However for an attractive condensate, there are impor-tant changes in the nature. This is due to the fact thatthe number of available energy levels of the system islimited, especially when N → Ncr, while for any N ,there are infinitely many energy levels for the repulsiveor non-interacting condensate. In the limit of high T ,both the repulsive and the non-interacting condensatebehave as the corresponding trapped Bose-gas, resultingin a saturation in CN (T ). On the other hand, anattractive condensate also shows similar behavior, onlyif it is allowed to absorb energy internally through ro-tational motion involving large orbital angular momenta.

We provide an understanding of this peculiar naturebased on the many-body picture. For the theoreticalcalculation, we adopt the correlated potential harmonic(CPH) method [6, 7] to solve the many-body problemapproximately. This technique is based on the potentialharmonics (PH) expansion method [8]. The laboratoryBEC must be very dilute to preclude three-body colli-sions, which lead to molecule formation and consequentdepletion. Hence only two-body correlations are rele-vant. The PH is a subset [8] of the full hypersphericalharmonics (HH) basis [9], that involves only two-bodycorrelations. Hence the PH basis is a good approxima-tion for expanding the condensate wave function. Itreduces the bulk of the numerical procedure immensely,while retaining the most important basic features of thecondensate. However, the leading members of the PH

2

basis do not have the correct short separation behaviorof the interacting Faddeev component. This causesa very slow rate of convergence of the PH expansionbasis. To correct for this, we include a short-rangecorrelation function in the expansion basis. This corre-lation function is obtained as the zero-energy solutionof the two-body Schrodinger equation [7]. It is a correctrepresentation of the short separation behavior and alsoincorporates the s-wave scattering length, as, throughits asymptotic behavior [3]. The technique has beenshown to reproduce known results, both experimentaland theoretical [10]. These include the following: groundstate properties (energy, wave function, condensatesize, one-body density, pair-correlation, etc.) as alsomultipolar moments of both repulsive and attractivecondensates, correct prediction of the critical numberand collapse scenario of attractive condensates, thermo-dynamic properties of repulsive condensates, propertiesof condensates in finite traps, etc.

We can understand the behavior of heat capacityof attractive condensates in terms of the energy levelsof the system produced by the CPH method. Thismethod generates an effective potential in which thecondensate moves. For an attractive condensate, theeffective potential has a metastable region (MSR),separated from a deep well on the inner side by anintermediate finite barrier. A finite number of energylevels are supported by the MSR. As the number Nof atoms increases, the MSR shrinks and the numberof energy levels reduce drastically. As temperature in-creases, particles are distributed in higher energy levels,according to Bose distribution formula. Thus at lowtemperatures the internal energy and CN (T ) increasewith temperature. At higher temperatures, the bosonshave fewer levels to occupy, causing CN (T ) to differ fromthe repulsive case. There is also a dominant effect of thedrastically reducing number of energy levels as N → Ncr.

The paper is organized as follows. For easy readabilityand to introduce our notations, we briefly review thecorrelated potential harmonic method in Section II.Section III provides our numerical procedure. Resultsand discussion are presented in Section IV. Finally wedraw our conclusions in Section V.

II. CORRELATED POTENTIAL HARMONICS(CPH) METHOD

We adopt the correlated potential harmonicsmethod [6, 7] to solve the many-body problem ofthe BEC. We briefly recapitulate the technique in thefollowing. Interested readers can find details in the citedreferences.

For the relative motion of a system of N identical spin-

less bosons, we introduce (N − 1) Jacobi vectors

~ζi =

√

2i

i+ 1

~xi+1 −1

i

i∑

j=1

~xj

, (i = 1, ..., N − 1),

(1)where ~xi is the position vector of the i-th particle. TheSchrodinger equation governing the relative motion of thesystem trapped in a harmonic well, is

[

−~2

m

N∑

i=1

∇2~ζi+ Vtrap(~ζ1, ..., ~ζN ) +

V (~ζ1, ..., ~ζN )− ER

]

ψ(~ζ1, ..., ~ζN ) = 0, (2)

where N = N − 1 and the trapping potential Vtrap andthe interatomic interaction V are expressed in terms ofthe Jacobi vectors. The energy of the relative motion isER. Next, we introduce hyperspherical variables corre-sponding to the set of N Jacobi vectors. First, a hyper-radius is defined as

r =

[

N∑

i=1

ζ2i

]

12

. (3)

The remaining set of (3N − 1) ‘hyperangles’ consists of2N polar angles of N Jacobi vectors and (N − 1) anglesdefining their relative lengths [9]. In the hypersphericalharmonics expansion method (HHEM) ψ is expandedin the complete set of hyperspherical harmonics (HH),which are the eigenfunctions of the grand orbital oper-ator [hyperangular part of the N dimensional Laplaceoperator, given by the sum in the first term of Eq.(2)] [9]. Substitution of this in Eq. (2) and projectionon a particular HH result in a set of coupled differentialequations. Imposition of symmetry of the wave functionand calculation of the matrix elements become increas-ingly difficult and tedious as N increases. In addition,the degeneracy of the HH basis increases very rapidly [9]with the increase in the grand orbital quantum numberK. Hence a convergent calculation using HHEM withthe full HH basis is extremely computer intensive andunmanageable for N > 3. This is the price one pays forkeeping all many-body correlations in ψ.

However all these complications can be avoided anda much simpler computational procedure can be formu-lated for the laboratory BEC, which is designed to beextremely dilute (typical number density is ∼ 1015 cm−3)in order to avoid recombination through three-body col-lisions. Thus three-body correlations and three-bodyforces are totally negligible. We can then express ψ asa sum of two-body Faddeev component ψij for the (ij)-interacting pair [8]

ψ =

N∑

i,j>i

ψij(~rij , r). (4)

3

Note that the assuption of two-body correlations onlymakes ψij a function of the pair separation vector andthe hyperradius only. One can then expand ψij in a sub-set of HH, called the potential harmonics (PH) subset,which is sufficient for the expansion of the interactionpotential V (~rij) as a function in the hyperangular spacefor the (ij)-partition. Since the labeling of the particles

is arbitrary, we can choose ~rij = ~ζN . Then the corre-

sponding PH, P lm2K+l(Ω

ijN ) (the argument is the full set

of hyperangles for the (ij)-partition) is independent of

~ζ1, . . . , ~ζN−1 and a simple analytic expression is possi-ble [8]. Expansion of the Faddeev component in the PHbasis reads

ψij(~rij , r) = r−(3N−1)

2

∑

K

P lm2K+l(Ω

ijN )ulK(r), (5)

where P lm2K+l(Ω

ijN ) is a potential harmonic [8]. The r-

dependent factor in front is included to remove the firstderivative with respect to r. Substitution of this expan-sion in the Faddeev equation for the (ij)-partition

(T + Vtrap − ER)ψij = −V (rij)

N∑

k,l>k

ψkl, (6)

[where T = −~2

m

∑Ni=1 ∇

2~ζi] and projection on the PH

corresponding to the (ij)-partition give a set of cou-pled differential equations in r. Note that any realistictwo-body potential, V (~rij) can be used. A realistic in-teratomic potential has a very strong repulsion (arisingfrom the nucleus-nucleus repulsion) at very short separa-tions. Consequently, corresponding ψij must be vanish-ingly small for small values of rij . But the leading PH(corresponding to K = 0) in the expansion in Eq. (5) isa constant and does not have this behavior. Hence con-vergence of the expansion in Eq. (5) will be very slow.To improve the rate of convergence, we include a short-range correlation function η(~rij) in the expansion basis,so that Eq. (5) is replaced by

ψij(~rij , r) = r−(3N−1)

2

∑

K

P lm2K+l(Ω

ijN )ulK(r)η(~rij ). (7)

The short-range correlation function is chosen to have theexpected behavior of ψij(~rij , r) for small rij in the fol-lowing manner. The small rij behavior of ψij will be thatof a zero-energy pair interacting via V (~rij), since the en-ergy of the interacting pair is practically zero. We obtainη(~rij) by solving the zero-energy two-body Schrodingerequation

−~2

m

1

r2ij

d

drij

(

r2ijdη(rij)

drij

)

+ V (rij)η(rij) = 0. (8)

Inclusion of the short-range correlation function η(~rij)enhances the rate of convergence greatly, which has beenchecked in our numerical calculation.

The laboratory BEC is very dilute; hence the averageseparation of the atoms is very large compared withthe range of interatomic interactions. Moreover, theatoms scatter with almost zero energy. Hence theeffective two-body interaction is represented by thes-wave scattering length as. In our calculation, we takeV (~rij) to be the van der Waals potential with a hard

core: V (~rij) = − C6

rij6for rij ≥ rc and = ∞ for rij < rc.

The correlation function obtained by solving Eq. (8)quickly attains its asymptotic form C(1 − as

rij) for large

rij . The asymptotic normalization is chosen to make thewavefunction positive at large rij . The hard core radiusrc is adjusted so that the calculated as is the actualexperimental value of the scattering length [3]. Thisprocedure assures that the realistic two-body interactionappropriate for the condensate has been incorporated.

Substitution of the expansion, Eq. (7) in Eq. (6) andprojection on the PH corresponding to the (ij)-partitionresult in

[

−~2

m

d2

dr2+

~2

mr2L(L+ 1) + 4K(K + α+ β + 1)

+ Vtrap(r) − ER

]

UKl(r)

+∑

K′

fKlVKK′(r)fK′lUK′l(r) = 0, (9)

where UKl(r) = fKlulK(r), L = l+ 3N−6

2 , α = 3N−82 , β =

l+ 12 , l being the orbital angular momentum contributed

by the interacting pair. f2Kl is a constant representing

the overlap of the PH for interacting partition with thefull set of all partitions, which can be found in Ref. [8].The correlated potential matrix element VKK′(r) is givenby [7]

VKK′(r) = (hαβK hαβK′)− 1

2

∫ +1

−1

PαβK (z)V

(

r

√

1 + z

2

)

PαβK′ (z)η

(

r

√

1 + z

2

)

Wl(z)

dz. (10)

Here hαβK and Wl(z) are respectively the norm and

weight function [11] of the Jacobi polynomial PαβK (z).

Note that the inclusion of the short-range correlationfunction, η(rij) makes the PH basis non-orthogonal.Numerical solution of Eq. (8) shows that η(rij) dif-fers from a constant value only in a small interval ofsmall rij values. Hence the dependence of the overlap

< P lm2K+l(Ω

(ij)N )|P lm

2K+l(Ω(kl)N )η(rkl) > on the hyperradius

r is quite small. Disregarding derivatives of this overlapwith respect to the hyperradius, we approximately getEq. (9), with VKK′(r) given by Eq. (10). The effectof the overlap being different from unity is representedby the asymptotic constant C of η(rij). The emergingphysical picture is: the effective interaction betweenpairs of atoms at very low energy becomes V (rij)η(rij).

4

This is justified, since at very low kinetic energy, theatoms have a very large de Broglie wave length and donot approach each other close enough to ”see” the actualinteratomic interaction. In the limit of zero energy, thescattering cross section becomes 4π|as|2 and the effectiveinteraction is governed by the s-wave scattering lengthas, through the asymptotic form of η(rij).

Introduction of the PH basis and inclusion of theshort-range correlation function, referred to as thecorrelated potential harmonic (CPH) method, simplifiesthe many-body problem dramatically. A fairly fastcomputer code can solve Eq. (9) with upto 15000particles in the condensate. This technique has beentested against known results, both experimental onesand theoretical ones calculated by other authors, forrepulsive as well as attractive condensates [6, 7, 10].

III. NUMERICAL PROCEDURE

A. Solution of coupled equations

Although Eq. (9) can be solved by an exact numericaltechnique using the Numerov method, we adopt thehyperspherical adiabatic approximation (HAA) [12],which apart from simplifying the computations greatly,provides an effective potential in the hyperradial space,in which the condensate moves. This effective conden-sate potential provides a physical picture for the internalmechanism of the condensate.

In the HAA, one assumes that the hyperangular mo-tion is much faster than the hyperradial motion, sincethe latter corresponds to the breathing mode. There-fore, one can solve the former adiabatically for a fixedvalue of r and obtain the solution as an effective poten-tial for the hyperradial motion, as in Born-Oppenheimerapproximation. The hyperangular motion is solved by di-agonalizing the potential matrix VKK′(r) together withthe hyper-centrifugal potential [second term of Eq. (9)].The lowest eigenvalue ω0(r) [corresponding eigen columnvector being χK0(r)], is the effective potential for thehyperradial motion [12]:

[

−~2

m

d2

dr2+ω0(r)+

∑

K

|χK0(r)

dr|2−ER

]

ζ0(r) = 0. (11)

The third term is an overbinding correction. Eq. (11) issolved by the Runga-Kutta method, subject to appropri-ate boundary conditions to get ER and the hyperradialwave function ζ0(r). The many-body wave function canbe constructed in terms of ζ0(r) and χK0(r) [12]. Totalenergy is obtained by adding the center of mass energy(1.5 o.u.) to ER. Energy levels, Enl, are characterizedby the quantum numbers (n, l), where n represents theexcitation quantum number for a given orbital angular

momentum l. The HAA has been tested for nuclear,atomic and molecular systems and shown to give betterthan 1% accuracy, even for the long-range Coulombpotential [13]. In our case, the van der Waals potentialhas a shorter range and HAA is expected to be better.Moreover, in a BEC, the dominant confining harmonicoscillator potential is smooth and the correspondinghyperradial equation is completely decoupled. Hence ina BEC, the HAA is expected to be far better. We testedthis by solving the CDE, Eq. (9), with the interatomicpotential for the ground state by the renormalizedNumerov method [14, 15], which is an exact numericalalgorithm for solving a set of coupled differential equa-tions. The calculated exact ground state energies are(in o.u.) 948.6420, 1174.5284, 1277.8219, 1460.3706 and1596.2611 respectively for N = 700, 900, 1000, 1200 and1400. These compare very well with the correspondingHAA results: 948.0986, 1173.9809, 1277.2040, 1459.6373and 1595.8163 respectively. The error is less than 0.06%in all cases. Thus we can safely use the HAA, whichreduce the numerical complications to a great extent.

B. Calculation of specific heat

At a temperature T > 0, bosons are distributed inavailable energy levels Enl according to Bose distributionfunction

f(Enl) =1

eβ(Enl−µ) − 1(12)

where β = 1/kBT and µ is the chemical potential. Thelatter is determined from the constraint that the totalnumber of particles is N . Clearly, µ has a temperaturedependence. The total number of bosons in the trap isfixed and at any temperature it can be written as

N =

∞∑

n=0

∞∑

l=0

(2l + 1)f(Enl) (13)

At a particular temperature T , µ is determined from theconstraint Eq. (13). The total energy of the system at Tis given by

E(N, T ) =∞∑

n=0

∞∑

l=0

(2l + 1)f(Enl)Enl (14)

The specific heat at fixed particle number N is calculatedusing the relation

CN (T ) =∂E(N, T )

∂T

∣

∣

∣

N(15)

5

Using (12), (14), (15) one can obtain the heat capacityas

CN (T ) = β

∞∑

n=0

∞∑

ln=0

(2ln + 1)Enln exp (β(Enln − µ))

(exp (β(Enln − µ))− 1)2

×[Enln − µ

T+∂µ

∂T

]

(16)

where∂µ

∂T=

−

∑∞m=0

∑∞lm=0(2lm + 1)(Emlm

− µ) exp (β(Emlm− µ))(f(Emlm

))2

T∑∞

p=0∑∞

lp=0(2lp + 1) exp (β(Eplp

− µ))(f(Eplp))2

(17)

For an ideal non-interacting bosonic gas containing Nbosons in a three-dimensional isotropic harmonic well,the critical temperature T 0

c is well defined [2]. µ remainsequal to the energy of the single particle ground statefor T < T 0

c and start decreasing rapidly for T > T 0c . In

the standard text book treatment [1] the sums in eqs.(13-14) are replaced in the semi-classical approximationby integrals over energy, assuming a continuous energyspectrum. In a harmonic trap the energy spectrum isdiscrete and this assumption is not valid, particularly forsmall N at low energies. A correct treatment [5] showsthat µ decreases slowly from its maximum value (equalto the ground state energy) as T increases from zero,the rate of decrease becoming suddenly rapid at sometemperature close to the reference temperature T 0

c corre-sponding to the same value of N [4]. Thus, in this casethe critical temperature is not well defined. In the cor-rect treatment, the heat capacity also becomes a smoothfunction of T , attaining a maximum at a temperature, atwhich µ suddenly becomes a rapidly decreasing functionof T . In the limit of large N , the behaviors of µ(T ) andCN (T ) curves approach those of the text book treatment.The transition temperature Tc for a finite interacting sys-tem is defined as the temperature at which CN (T ) is amaximum [5]

∂CN (T )

∂T

∣

∣

∣

Tc

= 0 (18)

For the numerical calculation for a chosen particlenumber N , the CPH equations are solved for a largenumber of energy levels – typically n running from 0 to300 and l running from 0 to 200, subject to an upperenergy cutoff value, EUL (see below), so that Enl < EUL.Using previously calculated values of Enl, Eq. (13) issolved for µ, at a chosen temperature T , by a modifiedbisection method. Next EUL is increased and the processrepeated, until convergence in µ is achieved. Using thisupper energy cutoff, CN (T ) is determined using Eq. (16).

IV. RESULTS AND DISCUSSIONS

We consider the attractive condensate of 7Li atoms inthe trap used in the experiment at Rice University [16].

The harmonic trap used was axially symmetric withνx = νy = 163 Hz and νz = 117 Hz. For simplicity,

we consider an isotropic trap with ν = (νxνyνz)13 . The

experimental value of as is −27.3 o.u. We use oscillator

unit (o.u.) of length (√

~

mω) and energy (~ω). As

mentioned earlier, we choose van der Waals (vdW)potential for the interatomic interaction, with knownvalue [3] of C6 = 1.715 × 10−12 o.u. The value of rc isobtained by the procedure discussed following Eq. (8),so that calculated as has the experimental value [3].Its numerical value is 5.338 × 10−4 o.u. The calculatedeffective potential, ω0(r) is plotted as a function of r inFig. 1 for N = 1300. In the r → 0 limit, ω0(r) becomes

1500

1550

1600

1650

1700

30 40 50 60

ω0(r

)

r

-4×10100×100

4×1010

0 0.5 1

ω0(r

)

r

FIG. 1: (Color online) Calculated effective potential ω0(r)against r in o.u. for the attractive 7Li condensate with N =1300. The narrow and deep well near origin is shown in theinset (note that different scales are used).

strongly repulsive, due to the repulsive core of vdW po-tential and the hypercentrifugal repulsion of Eq. (9). Asr increases, there is a deep narrow well (DNW) arisingfrom the strong interatomic attraction at small values ofr. This attraction is proportional to the number of pairsand hence increases rapidly as N increases. For stilllarger r, the effects of the kinetic pressure (including thecentrifugal repulsion), interatomic attraction and theharmonic confinement together produce a metastableregion (MSR). An intermediate barrier (IB) appearsbetween the DNW and MSR. The DNW is very deepand narrow, hence it is shown as an inset in Fig. 1 (notelarge changes in scale for both horizontal and verticalaxes). As N increases, the DNW becomes deeper, IBshallower and the minimum of the MSR higher. At thecritical value Ncr, the maximum of IB and the minimumof MSR merge to form a point of inflexion and the MSRdisappears. At this point, the condensate falls into theDNW, resulting in a collapse of the condensate andformation of clusters within the DNW. Our calculatedvalue of Ncr is 1430. In panels (a) – (f) of Fig. 2, wedemonstrate how the MSR shrinks, with N approaching

6

700

1000

1300

20 40 60

ω0(r

)

r

1520

1580

1640

35 50 65

ω0(r

)

r

(a) (b)

1590

1605

1620

40 50 60

ω0(r

)

r

1590

1605

1620

40 45 50ω

0(r

)

r

(c) (d)

1600

1606

1612

40 45 50

ω0(r

)

r

1600

1606

1612

41 45 49

ω0(r

)

r

(e) (f)

FIG. 2: (Color online) Plot of effective potential ω0(r) againstr (both in appropriate o.u.) for the attractive 7Li condensatewith N = 500, 1300, 1400, 1410, 1420 and 1426 in panels (a)– (f) respectively, showing how the MSR shrinks in depth andwidth. Note that different scales have been used in differentpanels, to bring out the features of the MSR as N → Ncr.

Ncr, for N = 500, 1300, 1400, 1410, 1420, and 1426 re-spectively. From Fig. 2, one notices that both the depthand width of MSR decrease as N increases towards Ncr.Hence the number of bound energy levels supported bythe MSR decreases rapidly with N (see also Fig. 7).

However, the effective potentials shown in Fig. 1 andFig. 2 are obtained for l = 0. For higher l, the effectivepotential has a higher IB, arising from the l-dependentterms of the hyper-centrifugal repulsion [see Eq. (9)].Thus the position of the MSR rises higher in energy as lincreases, as can be seen in Fig. 3 for l = 0, 1, 2, 3, 4 for acondensate containing 1420 atoms. Hence particles withl > 0 can attain higher energy levels. Inclusion of theselevels will have a profound effect on the heat capacity.If such energy states were ignored (i.e., only the energylevels supported by the l = 0 MSR considered), theheat capacity would reduce drastically and Tc wouldincrease indefinitely as N → Ncr, since all the atomswould be forced into the few remaining energy levelsavailable for internal excitation. In our calculation, wehave retained all energy levels supported by a given l. Aquestion arises as to whether the metastable condensatecan have large l values. Intuition indicates that, withincrease of temperature, the system can absorb energyonly by increasing its rotational kinetic energy, thereby

1580

1585

1590

1595

1600

1605

1610

1615

1620

1625

36 38 40 42 44 46 48 50

ωl (r

)

r

l=0l=1l=2l=3l=4

FIG. 3: (Color online) Plot of effective potential ωl(r) againstr (both in appropriate o.u.) for the attractive 7Li condensatewith N = 1420 atoms, corresponding to l = 0, 1, 2, 3, 4. Thecurves show how the IB increases as l increases.

increasing the stability of the metastable system withenhanced centrifugal repulsion. Increase of kineticenergy due to faster linear motion alone would causethe system to fall in the DNW near the center of thecondensate. Compared with the non-interacting orrepulsive condensates, the attractive condensate has aclear distinction, viz. while the number of hyperradialexcitations for a given l in the former is not limited, it isdrastically limited in the latter. Thus for an attractivecondensate, there are fewer energy states, in which thesystem can reside. This causes (CN )max to increaseinitially for N ≪ Ncr (when energy levels are not greatlyrestricted), but as N → Ncr, it starts decreasing, afterattaining a maximum. Fig. 4 shows how CN (T ) dependson T , for selected values of N . It is seen that thetransition temperature Tc increases gradually with N ,but (CN )max increases up to N = 1300 and for largerN , it starts decreasing. When N ≪ Ncr, the nature issimilar to that of a repulsive condensate [4], since in thiscase, the number of available energy levels are still largeenough (the top most energy level – including l 6= 0 –in the MSR has an energy much greater than kBT ), sothat the top most levels are still practically unoccupiedand there is scope for further internal excitation as T in-creases. Consequently Tc increases gradually with N , asin the repulsive case. As N approaches Ncr, the numberof energy levels supported by the MSR decreases rapidlyand there is less scope for absorbing energy internally asT increases. Hence (CN )max decreases and Tc increasesfaster, as N increases towards Ncr. At higher temper-atures, higher l states are excited, which push atomsfurther outwards, increasing the average interatomicseparation. Consequently, the system behaves ultimatelyas a non-interacting Bose gas. Thus the asymptoticvalue of CN (T ) becomes 3NkB. In Fig. 4, we plotthe dimensionless quantity CN (T )/(NkB) against T (in

7

0

3

6

9

20 60 100 140

CN

(T)/

(Nk

B)

T

N=500N=1000N=1300N=1350N=1400

FIG. 4: (Color online) Plot of heat capacity CN (T )/(NkB)(dimensionless) against T (in nK), for indicated number of7Li atoms in the metastable condensate.

nK) for 7Li condensate with N = 500, 1000, 1300, 1350and 1400. The features discussed above are clearlyvisible. One notices that the behavior for N < 1300is similar to that of a repulsive condensate, but asN exceeds 1300, the curves become flatter near their

7.2

7.4

7.6

7.8

8

8.2

8.4

500 800 1100 1400

(CN

) max/(

Nk

B)

N

FIG. 5: (Color online) Plot of the peak heat capacity,(CN)max/(NkB) (dimensionless), against the number N ofbosons in the attractive 7Li condensate.

maxima and the peak value of CN (T )/(NkB), namely,(CN )max/(NkB), decreases fairly rapidly, as N → Ncr.All the curves appear to converge to the Bose gaslimit. But a closer scrutiny shows that the curves forN = 1350 and 1400 show a slight downward trend.This is due to a limitation in the higher energy cut-offused in our calculation. In Fig. 5, we plot calculated(CN )max/(NkB) as a function of N . It is seen that thisquantity increases gradually up to N = 1300. Beyondthis value, (CN )max/(NkB) decreases fairly rapidly asN → Ncr. A plot of transition temperature Tc (in nK) asa function of N is shown in Fig. 6. Initially Tc increases

50

60

70

80

90

100

110

120

500 800 1100 1400

Tc

N

FIG. 6: (Color online) Plot of transition temperature (in nK)versus N for the attractive 7Li condensate.

linearly for N < 1300. As discussed above, this behavioris expected for small N , as in the case of a repulsivecondensate. But for N > 1300, Tc increases rapidly.Both the decrease of (CN )max and faster increase of Tcare due to reduction in the number of available energy

0

20

40

60

80

100

120

140

160

1200 1250 1300 1350 1400

Num

ber

of e

nerg

y le

vels

N

FIG. 7: (Color online) Plot of total number of available l = 0energy levels versus N , close to Ncr for the attractive 7Licondensate.

levels as N approaches Ncr. We demonstrate this in Fig.7 for the l = 0 energy levels as N increases from 1200to Ncr. The decrease in the number of available energylevels forces a larger fraction of the bosons to be in theground state as T increases [see Eq. (13)]. This causesa decrease of (CN )max and an increase of Tc, asN → Ncr.

A possible scenario of the attractive condensate as itstemperature is gradually increased is the following. Thestandard definition of critical number Ncr referred to inthe literature, corresponds to the l = 0 condensate atzero temperature. As T is gradually raised, the system

8

absorbs energy by occupying higher available energy lev-els upto the top of the MSR. However, there is a finitelife time of atoms in higher energy levels due to tunnel-ing through the IB into the DNW. Thus there will be adecrease in the number of atoms in the MSR. The rate ofloss of atoms will increase with the energy of the level, asalso with N (increase of N will lower the IB). The usualdefinition of heat capacity, CN (T ), is the rate of changeof internal energy with respect to T [Eq. (15)], for a

fixed number N of atoms in the condensate. This defi-nition is unambiguous for a repulsive or non-interactingcondensate, since in these cases there is no loss. However,since the loss is appreciable for an attractive condensatefor N close to Ncr, or if T is such that kBT is compa-rable with the highest excitation energy allowed by theIB, this definition demands that atoms be pumped intothe condensate at the same rate as the loss rate fromthe condensate. Such an experimental procedure has notbeen adopted yet. However, for N ≪ Ncr and T ≪ Tc,the highest appreciably occupied levels will have negli-gible tunneling probability into the DNW. Under theseconditions, the metastable attractive condensate is fairlylong lived and the standard definition of CN (T ) is ac-ceptable. Hence standard experimental techniques canbe adopted. Thus our results presented in Figs. 4 – 6 areexperimentally verifiable in the small N , small T limit.We have presented, for theoretical completeness, resultsfor N close to Ncr and for T beyond Tc. The questionof how the system can absorb energy internally for suchvalues of N and T was already discussed above.

V. CONCLUSIONS

In this work, we report a detailed calculation of theheat capacity CN (T ) of an attractive Bose-Einsteincondensate containing N atoms of 7Li. The correlatedpotential harmonics method, which is appropriate forthe dilute BEC, has been used. The effective potential,in general, supports a large number of energy levels.At T = 0, the lowest energy level accommodates allthe bosons. As temperature increases, particles aredistributed in higher energy levels, according to Bosedistribution formula. Thus the internal energy of thesystem increases. Heat capacity CN for a fixed numberN of particles in the condensate is defined as thetemperature derivative of the total internal energy. Fora repulsive condensate trapped by an ideal harmonicoscillator, the effective potential has no upper cutoff. Hence the energy levels are not limited in energy.Consequently, total internal energy and CN increaseas temperature increases upto Tc. For T > Tc, theground state occupation becomes suddenly microscopic(negligible) and the system behaves like a harmonicallytrapped Bose gas. Hence CN decreases rapidly aboveTc, reaching its asymptotic value 3NkB. Thus CN firstincreases, reaches a maximum value (CN )max and thendecreases rapidly to its asymptotic value, as T increases

from zero.

For N < Ncr bosons with mutual attraction, ametastable condensate is formed in the metastableregion (MSR) of the effective potential. On the left ofthe MSR, an intermediate barrier (IB), followed by adeep narrow well and finally a strongly repulsive coreappear, as one approaches the center of the condensate.For N ≪ Ncr, the IB is very high and the minimumof the MSR is very low, so that the metastable wellis sufficiently deep compared with thermal excitationenergy kBTc at Tc, and a large number of energylevels are supported. Hence for T ≤ Tc, even the mostthermally excited particles do not feel the effect of theIB and CN increases gradually, as in the repulsive case.

With increase of temperature, the system with a fixedN absorbs energy internally by increasing the occupa-tion probability of higher energy levels supported by themetastable region of the effective potential. Atoms inenergy levels close to the top of the intermediate barrierhave appreciable probability to tunnel into the deep nar-row well, causing the condensate to loose atoms. Butsuch levels are not occupied with any appreciable proba-bility if N ≪ Ncr and T ≪ Tc. Hence such a condensateis quasi-stable and CN (T ) calculated for a fixed N is ap-propriate. When the rate of atom loss from the conden-sate is appreciable, standard definition of heat capacityat constant N requires feeding the attractive condensatewith additional atoms at a rate such as to compensate forthe loss rate. Although this is not the usual experimen-tal technique, we investigate such cases for a completetheoretical study. In such a situation, there are dras-tic changes. The peak value of CN (T ) (the temperatureat which this occurs is the transition temperature Tc)initially increases gradually with N , then after reachinga maximum, decreases fairly rapidly near Ncr. On theother hand, for small N , Tc increases almost linearly upto N ∼ 1300. For larger N , the transition tempera-ture increases rapidly with N . We provide an explana-tion of this behavior, based on the microscopic mecha-nism of absorption of internal energy, as T increases. AsN increases towards Ncr, the depth and width of themetastable well decrease rapidly. As a result, the num-ber of energy levels supported by the metastable welldecreases rapidly. This tends to increase Tc, since fewerenergy levels are available for absorption of internal en-ergy, and bosons are forced to be in lower energy levels asT increases. Rapid reduction of available energy levels asN → Ncr, causes quicker saturation of internal energy ofthe condensate. Consequently, the maximum of CN (T )decreases rapidly as N → Ncr.

VI. ACKNOWLEDGEMENT

We would like to thank Dr. Parongama Sen for usefuldiscussions. SG acknowledges CSIR (India) for a Ju-

9

nior Research Fellowship [Grant no. 09/028(0762)/2010-EMR-I], TKD acknowledges UGC (India) for the Emer-itus Fellowship [Grant no. F.6-51(SC)/2009(SA-II)] and

AB acknowledges CSIR (India) for a Senior Research Fel-lowship[Grant no. 09/028(0773)/2010-EMR-I].

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