Dimensionality and Finite Number Effect on BCS Transition of Atomic Fermi Gas

Commun. Theor. Phys.(Beijing, China)43(2005)pp. 826-830C International Academic PublishersVol.43,No.5,May15,2005Dimensionality and Finite Number Effect on BCS Transition of Atomic Fermi Gas*CUI Hai-Tao, WANG Lin-Cheng 2 and YI Xue-XiI Physics Department, Dalian University of Techology, Dalian 116024,ChinaInstitute of Theoretical Physics, Northeast Normal University, Changchun 130024, China(Received September 28, 2004; Revised November 16, 2004)Abstract The effect of finite number and dimensionality has been discussed in this paper. The finite number effecthas a negative correction to final temperature for 2D or 3D atomic Fermi gases. The changing of final temperatureobtained by scanning from BEC region to BCS region are 10% or so with N 10 and can be negligible when N>103However, in 1D atomic Fermi gas, the effect gives a positive correction which greatly changes the final temperature inFermi gas. This behavior is completely opposed to the 2D and 3D cases and a proper explanation is still to be foundDimensionality also has a positive correction, in which the more tightly trapping, the higher final temperature one getswith the same particle number. A discussion is also presentedPACS numbers: 03.75.Ss, 05.30. Fk, 36.90+fKey words: finite number effect, dimensionality, atomic BCS transition1 IntroductionBoth types of the bound-weaklypairs can be con-The progress having been made in achieving highly verted from each other adiabatically by tuning Feshbachdegenerate regime of trapped atomic Fermi gas has at- resonance. 1, 2 It provides a straightforward method toracted much attention. It originates from the reason that reach deeply degenerate Fermi gas from a molecules beCBardeen-Cooper-Schriffer(BCS) transition, which hap- By adiabatically changing the scattering length a frompens at very low temperature, compared to Fermi temper- positive to negative, the entropy is held constant, whichture TF, is associated with Bose-Einstein condensation leads to strong decrease at the final temperature 7Itof Cooper-pairs. I Recently a series of experiments 2, 3 is just the main idea of Ref[7]. In this work the en-have demonstrated the possibility of going further into the tropy of atomic Fermi gas is proportional to T/TF.AndBCS-BEC crossover. [1,4 In Regal's experiment trappedn the bec region the same quantity is proportional tatomic gas has been cooled to T/TF =0.07 with w 100 (T/TBEc) with the Bogoliubov-Hartree-Fock approxparticles. While in MIT experiment an even lower tem- mation. Since it is achieved to reach deeply into the tem-perature T/TF=0.05 has been obtained. [5I Since underperature below TBEC, this suggests that extremely lowIch a low temperature the system parameters suchtemperature in atomic Fermi gas could be made. Thistensity, temperature, interaction strength are easy tedetermined, it opens up the opportunity for theoreticalwork predicts theoretically that T/TF N 10-2 could beinvestigation into this novel quantum system. 16.71attained under which it possibly leads to BCS transitionHoInteraction between atomic fermplays a crucialit is worth while to note that the parti-role in explaining the BCS-BEC crossover. When the scle number is very limit in these experiments, for examwave scattering length is positive, it is possible for two ple Nn 107 in MITI5 and N w 10 in JILA (3) Theyfermions to form a bound state called molecular and un- are far below the thermodynamics limit(N-o, V-dergo BEC 15. 8 In the other hand, if a<0, fermionsoo, N/V- constant) and the thermodynamic methodsare also bound weakly together, which are candidates for are not in function. But the work(7 did not take this ef-Cooper-pair, the delocalized pairs in momentum spacefect into account. It is necessary to analyze the effect ofof atomic s-wave scattering length. It opens up the door tems and an analysis is inevitable. In addition to the finiteinto the beBEC crossover region experimentally. In number effect, we also discuss the effect of dimensional-also called BEC region, there exist long-lived ity on the final temperature obtained by adiabatic tuningmolecules, which are weakly bound short-range fermion The method used here is a generalization of Ref [7] withpairs. [ol On the other hand weakly bound fermions pairs consideration of fir中国煤化工 nality.1Thealso exist, which are very different from Cooper pairs be- result shows thatCNMHGes anegativecause of stronger interaction, and the BEC of this type of correction to the final temperature 1% or so when the to-pairs have been observed in JILA. 3tal number is no more than 10 in two-dimensional(2D)The project supported by National Natural Science Foundation of China under Grant No. 10305002No 5Dimensionality and Finite Number Effect on BCS Transition of Atomic Fermi Gas82ase and three-dimensional (3D)case. However, it is sur-Molecules produced from fermions show long lifetimeprising to note that the finite number effect gives a greatly This has been attributed to the Pauli suppression ofpositive correction in one-dimensional(1D)case, which is the vibration quenching process, which couples a verycontrary completely to the 2D, 3D cases. A proper ex- weakly bound molecular state to much more tightly boundplanation is still to be found. At the end of this paper, a lower lying vibrational states. 14! Hence we can cool suchdiscussion is givenmolecules for bec. 5. 8 Since we are interested in the tem-2 One-Dimensional Caseperatures below TBEC, the Bose-type molecule system canbe treated as weakly interacting Boson system. withConsider atomic Fermi gas trapped by ID isotropicmation. Bogoliubharmonic oscillator potential and described by the grand Hartree-Fock approximation (or mean-field approxima-canonical ensemble with two equally populated spinstates. The grand potential &, without the mean-field tion)is available. [15]The grand potential iscorrection isQ2=kBT/dep(e)In(1-n(e))Q=kET/dep(e)In(1-e-Bewhere B denotes the Bose temperature. The state den-where p(a)is the state density andsity for a lD interacting Bose system may be calcun()(2)lated directly from the Bogoliubov Hamiltonian p(a)Tr[8(E-HBog)]. Under the semi-class limit approxima-With respect to the finite number effect, the effective tion, the state densty can be expressed as approxima-state density for a non-interacting two-component gas under the Thomas-Fermi approximation can be expressed asp(=)drdp(10)(hp(e)=心here the Bogoliubov energy spectrum of elementary ex(3)citation is[1s)Substitute it into Eq. (1)and one gets the grand potentialr|≤R,+-mmopmol>R,where we have expanded the Fermi integral fn(t) in high-regenerate region aswhere mmol= 2m. no(r) is determined by the Gross-Pitaevskii equation below TBEc and equalsfn(t)f1+n(n-1)t-2r(n+1)no(r)=molR+n(n-1)(n-2)(nwith/2in which t= Bu. Then the entropy may be expressed asSk2(6) the radius of BEC cloud (15] The coupling constant nowThe chemical potential u is determined by the total par-equalsticle number w asgmola direct calculation yieldsAnd the final expression for entropy is thenNk0=(222The mean-field correction has little contribution to en-+2 arccos)tropy for weakly bound case and therefore is negligible in√1+zour calculation [7where a= E/umol. It can be simplified under the Thomas-Now consider the analogous calculation for Bose con- Fermi limit as中国煤化工densate of weakly bound molecules at a>0. It meanskJamoll I or nmlamoll< 1, where nrCNMHG(14)molecules density and amol is the molecules s-wave scat- Then the entropy is obtainedtering length, which is related to the atomic scatteringlength a by amol=0.6a. [1412k2(1+√2)(2)828CUI Hai-Tao, WANG Lin-Cheng, and YI Xue-XiVol. 43The critical temperature TBEc is determined by the correction, which will have an extra contribution to thefollowing equationsystem quantities. Then the entropy becomes092)2m2k(pk TBeckB TBec(16)+-)Twhere Eo=w/2 in 1D case. Equation(16) can be con- The chemical potential u can be determined by the totalverted into two independent functions y=In(0.5/a) and particle number Ny=-Nmol/a, where x= kgt/u. The solution of Eq(16)is just the point of intersection of the two functions(k7)2+2It is natural for the next step to set Eqs. (8)and(15) Hence, by solving the above equation, the chemical poten-equal to each other. Afterward one obtains the final tem-peraturetial isTu= hwv2NG()2 In(2)\TBEC/initialThus the final expression for entropy becomesThe diagram demonstration is provided in Fi1S=30A(+2)7-(7)(23)on the final temperature, which is opposed completely to where kBtEN and the factor 1/2+1/v2N apthe 2D case and 3D case below. The changing is very pears because of the finite number effectobvious and has great influence on the final temperatureThe stability of the molecules in 2D-Bose condensatBut a proper explanation about this abnormal behavior is is guaranteed by the same reason stated in Sec.2.Thestill to be found. A further discussion is given in Sec. 5. Bogoliuboy-Hartree-Fock approximation is also in function in 2D Bose condensate. which has the same excitaticpectrum as in the 1D, 3D cases. Hence the state densitycan be calculated directly based onP(a)g(,7),and one can show0.11+√1+p(e)pmol+In0.10where z=Eumol. It can be simplified under the Thomas-1.5BECThen entropy is obtainedFig. 1 Final temperature in Fermi gas versus theinitial temperature in 1D molecules condensate. TheS=3队Nm()(8+2)(27solid, dashed, dotted lines are corresponding to N=100,1000,104 respectively with a=15.1,929,659.02where the limit kBT n umol has been used 7 and one hadefined the critical temperature3 Two-Dimensional CasekBTBEc=hw(rioThe effective state density of atomic Fermi gastrapped by 2D isotropic harmonic oscillator potential, for 2D non-interacting Bose gas. Setting the entropy ofwith consideration of finite number under Thomas-Fermi the molecules condensate equal to the same quantity ofapproximation [131Fermi gas and ignoring the second term in Eq.(23),oneobtains the final expression for the temperature of Fermip()==5+(18)Substitute it into Eq(1)and one gets the grand potential, (T唱YH中国煤化工CNMHG⊥⊥T(finalinitial(kBT)2where the second factor comes from the correction of finiteThe first term is just the result under the thermodynamic number. The relation between Tfinal and Tinitial have beenlimit and the second term comes from the finite number shown in fig. 2. We note that the finite number effect hasNo 5Dimensionality and Finite Number Effect on BCS Transition of Atomic Fermi Gas8a negative correction, which is 10% or so with N 103. where Ao=w(3N)1/3. The final equation for entropy isIt can be negligible when N>103N+(3N)2/3123(N)1/31.T4(3N)2/3,m4(3N)1/37rN15Setting the above equation equal to Eq(18) in Ref. 70.03one gets the final temperature under corrections of thefinite number as0.02G(Bumo(1-=x(3N.01hw[Nmol/S(3)].T/Tjust coming from the finite number correctionc/ initialThe result has been shown in Fig 3. One notes thatFig. 2 Final temperature in Fermi gas versus the initialhe finite number has also a negative correction, which isone in 2D molecules condensate. The dotted, dashed10% or so with n 1000. All lines are overlapped whenolid lines are corresponding to N= 100, 1000, 10 re-N>10 and the effect is negligiblespectively.4 Three-Dimensional caseFermi gas trapped by 3D isotropic harmonic oscillatepotential has been discussed in Ref. [7, but without consideration of the finite number. In this section, a detailed 0.03The effective state density is>>nstrate this effectcalculation has been made to demThe first term is the result of the thermodynamic limit3.5The other terms come from the correction of finite numhas an extra effect on the final temperatureobtained. The grand potential then can be expressed inFig. 3 Final temperature in Fermi gas versus the ini-tialal one in 3D molecules condensate. The dottedhigh-degeneracy region asdashed, solid lines are corresponding to N= 100, 10000 respectively. The daes are over12a36u(kBT)(kBT)kbT)+(30) 5 DiscussionThe effect of dimensionad finiteand then the entropy becomesbeen discussed. The finite number effect gives a negativeSDI2+JkET. (31)3D atomic Fermi gas. The corrections are 10% or so withN< 10 and they are negligible when N>103. In 1DThe chemical potential can be determined by the total atomic Fermi gas the finite number gives a great positivenumber ncorrection, opposed completely to 2D and 3D cases. Thusone has toYHS中国煤化工 ge as possibleN(hBTe) to obtain lower tenCNMHhavior maybeIt is expressed according to the Thomas-Fermi limit as But a more proper explanation is still to be found.Thisargument is just a crude ideal and the essence of this be-=P030(TF8) havior is unveiledCUI Hai-Tao, WANG Lin-Cheng, and YI Xue-XiVol. 430.10final temperature obtained. The result is demonstratedin Fig. 4 with N= 10, which happened in Ref. 3.Onenotes that with more tightly trap, one gets higher tem-perature when TBEC/Tinitial > 2. It may be the reason0.06that a much more tightly trapped Bose system has highercritical temperaturel16 and consequently affects the finaltemperature when the system is tuned from BEC regionto bCS region over Feshbach resonance. According to thisdiagram, lD Fermi systems can more easily be cooled tethe temperature below TF than higher-dimensional sys-tems. Since superfluidity and superconductivity are com-monly regarded as being connected with some type beC ofFig. 4 Final temperature in Fermi gas versus the ini-Cooper-pair, the tightly trapped system can also raise thetial one in molecules condensate with consideration of critical temperature of this transition at the same timedimensionality. The solid, dashed, dotted lines are corre-sponding to 3D, 2D, 1D respectively with N=106 where Acknowledgments44 526. 18 for 1D systemOne of the authors(H.T. Cui) acknowledges helpfulDimensionality has also made an obvious effect on the discussions with Dr L.D. Carr.References[10 C.A. Regal, C. Ticknor, J. L Bohn, and D.S. Jin, Physose-Einstein Condensation. eds. ARev. Lett. 92(2004)083201; C.A. Regal, M. Greiner, andGriffin, D. Snoke, and S. Stringari, Cambridge Univer-D.S. Jin, Nature 424 (2003)47; J. Cubizolles, T. Bourdel, S.J.J. M.F. Kokkelmans, G.V. Shlyapnikov, and C.sity Press, Cambridge(1995) pp. 355-392Salomon, Phys. Rev. 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