We examine the possibility of Bose-Einstein condensation in one-dimensional
interacting Bose gas subjected to confining potentials of the form $V_{\rm
ext}(x)=V_0(|x|/a)^\gamma$, in which $\gamma < 2$, by solving the
Gross-Pitaevskii equation within the semi-classical two-fluid model. The
condensate fraction, chemical potential, ground state energy, and specific heat
of the system are calculated for various values of interaction strengths. Our
results show that a significant fraction of the particles is in the lowest
energy state for finite number of particles at low temperature indicating a
phase transition for weakly interacting systems.Comment: LaTeX, 6 pages, 8 figures, uses grafik.sty (included), to be
  published in Phys. Rev. 

A. Minguzzi

A. S. Parkins

A. Smerzi

A. Widom

C. C. Bradley

D. J. Han

E. P. Gross

F. Brosens

F. Dalfovo

G. Baym

G.-L. Ingold

I. F. Silvera

I. Zapata

J. R. Ensher

K. B. Davis

L. C. Ioriatti, Jr.

L. P. Pitaevskii

M. Bayindir

M. H. Anderson

M. Naraschewski

M. Olshanii

M.-O. Mewes

P. C. Hohenberg

R. J. Dodd

S. Giorgini

S. Grossmann

S. Pearson

T. Haugset

V. Bagnato

W. Deng

W. J. Mullin

W. Ketterle

English

arXiv

We examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to confining potentials of the form V ext(x) = V 0(|x|/a) γ, in which γ&lt;2, by solving the Gross-Pitaevskii equation within the semiclassical two-fluid model. The condensate fraction, chemical potential, ground state energy, and specific heat of the system are calculated for various values of interaction strengths. Our results show that a significant fraction of the particles is in the lowest energy state for a finite number of particles at low temperature, indicating a phase transition for weakly interacting systems

Bayındır, Mehmet

Tanatar, Bilal

Gedik, Z.

Bilkent University Institutional Repository

PHYSICAL REVIEW A FEBRUARY 1999VOLUME 59, NUMBER 2Bose-Einstein condensation in a one-dimensional interacting systemdue to power-law trapping potentialsM. Bayindir, B. Tanatar, and Z. GedikDepartment of Physics, Bilkent University, Bilkent, 06533 Ankara, Turkey~Received 31 August 1998!We examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas sub-jected to confining potentials of the formVext(x)5V0(uxu/a)g, in which g,2, by solving the Gross-Pitaevskiiequation within the semiclassical two-fluid model. The condensate fraction, chemical potential, ground stateenergy, and specific heat of the system are calculated for various values of interaction strengths. Our resultsshow that a significant fraction of the particles is in the lowest energy state for a finite number of particles atlow temperature, indicating a phase transition for weakly interacting systems.@S1050-2947~99!00902-6#PACS number~s!: 03.75.Fi, 05.30.Jp, 67.40.Kh, 64.60.2itiot-palseitionabelao1net tthtesthbereonposthillaeCistrn toltsento-on-nedm-tallyran-sI. INTRODUCTIONThe recent observations of Bose-Einstein condensa~BEC! in trapped atomic gases@1–5# have renewed interesin bosonic systems@6,7#. BEC is characterized by a macroscopic occupation of the ground state forT,T0 , whereT0depends on the system parameters. The success of exmental manipulation of externally applied trap potentibrings about the possibility of examining two- or even ondimensional Bose-Einstein condensates. Since the transtemperatureT0 increases with decreasing system dimensiit was suggested that BEC may be achieved more favorin low-dimensional systems@8#. The possibility of BEC inone- ~1D! and two-dimensional~2D! homogeneous Bosgases is ruled out by the Hohenberg theorem@9#. However,due to spatially varying potentials which break the transtional invariance, BEC can occur in low-dimensional inhmogeneous systems. The existence of BEC is shown in anoninteracting Bose gas in the presence of a gravitatiofield @10#, an attractive-d impurity @11#, and power-law trap-ping potentials@12#. Recently, many authors have discussthe possibility of BEC in 1D trapped Bose gases relevanthe magnetically trapped ultracold alkali-metal atoms@13–18#. Pearson and his co-workers@19# studied the interactingBose gas in 1D power-law potentials employing the paintegral Monte Carlo~PIMC! method. They have found thaa macroscopically large number of atoms occupy the lowsingle-particle state in a finite system of hard-core bosonsome critical temperature. It is important to note thatrecent BEC experiments are carried out with a finite numof atoms ~ranging from several thousands to several 106!,therefore the thermodynamic limit argument in some theoical studies@15# does not apply here@8#.The aim of this paper is to study the two-body interactieffects on the BEC in 1D systems under power-law traptentials. For ideal bosons in harmonic oscillator traps trantion to a condensed state is prohibited. It is anticipatedexternal potentials more confining than the harmonic osctor type would be possible experimentally. It was also argu@15# that in the thermodynamic limit there can be no BEphase transition for nonideal bosons in 1D. Since the realsystems are weakly interacting and contain a finite numbePRA 591050-2947/99/59~2!/1468~5!/$15.00neri--on,ly--Daldo-statert--i-at-dicofparticles, we employ the mean-field theory@20,21# as appliedto a two-fluid model. Such an approach has been showcapture the essential physics in 3D systems@21#. The 2Dversion@22# is also in qualitative agreement with the resuof PIMC simulations on hard-core bosons@23#. In the re-maining sections we outline the two-fluid model and presour results for an interacting 1D Bose gas in power-law ptentials.II. THEORYIn this paper we shall investigate the Bose-Einstein cdensation phenomenon for 1D interacting Bose gas confiin a power-law potential:Vext~x!5V0S uxua Dg, ~1!whereV0 anda are some suitable energy and length paraeters defining the external potential, andg controls the con-finement strength. Presumably, they can be experimenadjusted. Using the semiclassical density of states, the tsition temperatureT0 and the fraction of condensed particleN0 /N for the noninteracting system were calculated as@12#kBT05F NkF~g!G~g!G2g/~21g!~2!andN0 /N512S TT0D1/g11/2, ~3!wherek52(2m)1/2a/ghV01/g (m is the mass of bosons andhis Planck’s constant!, andF~g!5E01 x1/g21dxA12x, ~4!andG~g!5E0` x1/g21/2dxex215G~1/g11/2!z~1/g11/2!, ~5!1468 ©1999 The American Physical Societynteet-ewtivmtionnc-atr-ti-s,tice--itoies-slyti-ly.Toialostctse in-ent.elytoPRA 59 1469BOSE-EINSTEIN CONDENSATION IN A ONE- . . .in which G(x) and z(x) are the gamma and the Riemanzeta functions, respectively. The total energy of the noninacting system forT,T0 (m50) is given by^E&NkBT05G~1/g13/2!z~1/g13/2!G~1/g11/2!z~1/g11/2! S TT0D1/g13/2. ~6!Figure 1 shows the variation of the critical temperatureT0 asa function of the exponentg in the trapping potential. Itshould be noted thatT0 vanishes for harmonic potential duto the divergence of the functionG(g52). It appears thatthe maximumT0 is attained forg'0.5, and for a constantrap potential@i.e., Vext(x)5V0# the BEC disappears, consistent with the Hohenberg theorem.We are interested in how the short-range interactionfects modify the picture presented above. To this end,employ the mean-field formalism and describe the collecdynamics of a Bose condensate by its macroscopic tidependent wave functionY(x,t)5C(x)exp(2imt), wheremis the chemical potential. The condensate wave funcC(x) satisfies the Gross-Pitaevskii~GP! equation@24,25#F2 \22m d2dx21Vext~x!12gn1~x!1gC2~x!GC~x!5mC~x!,~7!whereg is the repulsive, short-range interaction strength, an1(x) is the average noncondensed particle distribution fution. We treat the interaction strengthg as a phenomenological parameter without going into the details of actually reling it to any microscopic description@26#. In thesemiclassical two-fluid model@27,28# the noncondensed paticles can be treated as bosons in an effective poten@21,29#Veff~x!5Vext~x!12gn1~x!12gC2~x!. ~8!The density distribution function is given byn1~x!5E dp2p\ 1exp$@p2/2m1Veff~x!2m#/kBT%21 ,~9!FIG. 1. The variation of the critical temperatureT0 with theexternal potential exponentg.r-f-eee-nd--aland the total number of particlesN fixes the chemical potential through the relationN5N01E r~E!dEexp@~E2m!/kBT#21 , ~10!whereN05*C2(x)dx is the number of condensed particleand the semiclassical density of states is determined byr~E!5A2mh EVeff~x!,EdxAE2Veff~x!. ~11!The GP equation yields a simple solution when the kineenergy term is neglected~the Thomas-Fermi approximation!,C2~x!5m2Vext~x!22gn1~x!gu@m2Vext~x!22gn1~x!#,~12!where u@x# is the unit step function. More precisely, thThomas-Fermi approximation@7,20,30# would be valid whenthe interaction energy;gN0 /L far exceeds the kinetic energy \2/2mL2, whereL is the spatial extent of the condensate cloud. For a linear trap potential~i.e., g51), a varia-tional estimate for L is given by L5@\2/2m(p/2)1/22a/V0#1/3. We note that the Thomas-Fermapproximation would break down for temperatures closeT0 whereN0 is expected to become very small.The above set of equations@Eqs.~9!–~12!# must be solvedself-consistently to obtain the various physical quantitsuch as the chemical potentialm(N,T), the condensate fraction N0 /N, and the effective potentialVeff . In a 3D system,Minguzzi et al. @21# solved a similar system of equationnumerically and also introduced an approximate semianacal solution by treating the interaction effects perturbativeMotivated by the success@21,22# of the perturbative ap-proach we consider a weakly interacting system in 1D.zero order ingn1(r ), the effective potential becomesVeff~x!5H Vext~x! if m,Vext~x!2m2Vext~x! if m.Vext~x!. ~13!Figure 2 displays the typical form of the effective potentwithin our semianalytic approximation scheme. The mnoteworthy aspect is that the effective potential that affethe bosons acquires a double-well shape because of thteractions. We can explain this result by a simple argumLet the number of particles in the left and right wells beNLandNR , respectively, so thatN5NL1NR . The nonlinear orinteraction term in the GP equation may be approximatregarded asV5NL21NR2 . Therefore the problem reducesthe minimization of the interaction potentialV, which isachieved forNL5NR .The number of condensed atoms is calculated to beN052ga~11g!gV01/g m1/g11. ~14!The density of states is given byumth tedn.foresteari-nhe1470 PRA 59M. BAYINDIR, B. TANATAR, AND Z. GEDIKr~E!5kH H~g,m,E!~2m2E!1/g21/2 if m,E,2mF~g!E1/g21/2 if E.2m,~15!whereH~g,m,E!5E1E/~2m2E! x1/g21dxAx21.Using the above density of states, conservation of total nber of particles gives us a transcendental equation forchemical potentialN5N01k~kBT!1/g11/2I ~g,m,T!, ~16!whereI ~g,m,T!5F~g!E2m/kBT` x1/g21/2dxzex211Em/kBT2m/kBTH~g,m,xkBT!3~2m/kBT2x!1/g21/2dxzex21,in which z5e2m/kBT. The chemical potentialm(N,T) is de-termined from the solution of Eq.~16!. Finally, the totalenergy of the interacting system can be written as^E&5@^E&nc~N2N0!/21^E&c#/N, ~17!where^E&nc is the energy of the noncondensed particles^E&nc5E Er~E!dEexp@~E2m!/kBT#215k~kBT!1/g11/2J~g,m,T!, ~18!whereFIG. 2. Effective potentialVeff(x) in the presence of interactio@x05(m/V0)1/ga#. Thick dotted line represents the lower part of texternal potentialVext(x).-eJ~g,m,T!5E2m/kBT` x1/g11/2dxzex211Em/kBT2m/kBTH~g,m,x!~2m/kBT2x!1/g11/2dxzex21and ^E&c is the energy of the particles in the condensate,^E&c5g2 E C4~x!dx5 2ag2m211/g~11g!~2g11!gV01/g . ~19!The kinetic energy of the condensed particles is neglecwithin our Thomas-Fermi approximation to the GP equatioIII. RESULTS AND DISCUSSIONUp to now we have based our formulation on arbitraryg,but in the rest of this work we shall present our resultsg51. Our calculations show that the results for other valuof g are qualitatively similar. In Figs. 3 and 4 we calculathe condensate fraction as a function of temperature for vous values of the interaction strengthh5g/V0a ~at constantFIG. 3. The condensate fractionN0 /N versus temperatureT/T0for N5105 and for various interaction strengths.FIG. 4. The condensed fractionN0 /N versus temperatureT/T0for h50.001 and for different numbers of particlesN.tiobcualvcforgnaioeiv-nsofes.talhofchis-thetofruectedcalpntnt1DPRA 59 1471BOSE-EINSTEIN CONDENSATION IN A ONE- . . .N5105) and different numbers of particles~at constanth50.001), respectively. We observe that as the interacstrengthh is increased, the depletion of the condensatecomes more appreciable~Fig. 3!. As shown in the corre-sponding figures, a significant fraction of the particles ocpies the ground state of the system forT,T0 .The temperature dependence of the chemical potentiplotted in Figs. 5 and 6 for various interaction strengths~con-stant N5105) and different numbers of particles~constanth50.001), respectively. Effects of interactions onm(N,T)are seen as large deviations from the noninteracting behafor T,T0 .In Fig. 7 we show the ground state energy of an interaing 1D system of bosons as a function of temperaturedifferent interaction strengths. For smallh, andT,T0 , ^E&is similar to that in a noninteracting system. Ash increases,some differences start to become noticeable, and forh'1we observe a small bump developing in^E&. This may indi-cate the breakdown of our approximate scheme for laenough interaction strengths, as we can find no fundamereason for such behavior. It is also possible that the ThomFermi approximation employed is violated as the transitFIG. 5. The temperature dependence of the chemical potem(N,T) for various interaction strengths and forN5105 particles.FIG. 6. The temperature dependence of the chemical potem(N,T) for different numbers of particlesN and forh50.001.ne--isiort-retals-nto a condensed state is approached. Although it is concable to imagine the full solution of the mean-field equatio@Eqs.~9!–~12!# may remedy the situation for larger valuesh, the PIMC simulations@19# also seem to indicate that thcondensation is inhibited for strongly interacting systemThe results for the specific heat calculated from the toenergy curves, i.e.,CV5d^E&/dT, are depicted in Fig. 8.The sharp peak atT5T0 tends to be smoothed out witincreasing interaction strength. It is known that the effectsa finite number of particles are also responsible for subehavior@20#. In our treatment these two effects are not dentangled. It was pointed out by Ingold and Lambrecht@14#that the identification of the BEC should also be based onbehavior ofCV aroundT'T0 . Our calculations indicate thathe peak structure ofCV remains even in the presenceweak interactions, thus we are led to conclude that a ttransition to a Bose-Einstein condensed state is prediwithin the present approach.IV. CONCLUDING REMARKSIn this work we have applied the mean-field, semiclassitwo-fluid model to interacting bosons in 1D power-law traialialFIG. 7. The temperature dependence of the total energy ofBose gas for various interaction strengthsand the Maxwell-Boltzmann distribution forN5105 particles.FIG. 8. The temperature dependence of the specific heatCV forvarious interaction strengthsh and the Maxwell-Boltzmann distri-bution for N5105 particles.iorthcutoreaebnodn-flucopoourle-sheentsuldo-hof-calerel.1472 PRA 59M. BAYINDIR, B. TANATAR, AND Z. GEDIKpotentials. We have found that for a range of interactstrengths the behavior of the thermodynamic quantitiessembles that of noninteracting bosons. Thus BEC insense of macroscopic occupation of the ground state ocwhen the short-range interparticle interactions are notstrong. Our results are in qualitative agreement with thecent PIMC simulations@19# of similar systems. Both 2D and1D simulation results@19,23# indicate a phase transition forfinite number system, in contrast to the situation in the thmodynamic limit. Since systems of much larger size canstudied within the present approach, our work complemethe PIMC calculations.The possibility of studying the tunneling phenomenoncondensed bosons in spatially different regions separatea barrier has recently attracted some attention@31–34#. Inparticular, Dalfovoet al. @32# have shown that a Josephsotype tunneling current may exist for bosons under the inence of a double-well trap potential. Zapataet al. @34# haveestimated the Josephson coupling energy in terms of thedensate density. It is interesting to speculate on such aann,tt.Eetnri,llssne-erso-r-etsfby-n-s-sibility in the present case, since the effective potential indescription is of the form of a double-well potential~cf. Fig.2!. In our treatment, the interaction effects modify the singwell trap potential into one which exhibits two minima. 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Physical Review A

Bose-Einstein condensation in a one-dimensional interacting system due to power-law trapping potentials

B. Tanatar

Z. Gedik

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Bose-Einstein condensation in a one-dimensional interacting system due
  to power-law trapping potentials

Bose-Einstein condensation in a one-dimensional interacting system due to power-law trapping potentials

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