Evaluation of the particle numbers via the two root mean square radii in a 2-species Bose-Einstein condensate
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Evaluation of the particle numbers via the two root mean square radii in a 2-speciesBose-Einstein condensate
Y.Z.He , Y.M.Liu , and C.G.Bao , ∗ State Key Laboratory of Optoelectronic Materials and Technologies,School of Physics, Sun Yat-Sen University, Guangzhou, 510275, P. R. China Department of Physics, Shaoguan University, Shaoguan, 512005, P. R. China and State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing, 100190, China
The coupled Gross-Pitaevskii equations for two-species BEC have been solved analytically underthe Thomas-Fermi approximation (TFA). Based on the analytical solution, two formulae are derivedto relate the particle numbers N A and N B with the root mean square radii of the two kinds of atoms.Only the case that both kinds of atoms have nonzero distribution at the center of an isotropic trapis considered. In this case the TFA has been found to work nicely. Thus, the two formulae areapplicable and are useful for the evaluation of N A and N B . PACS numbers: 03.75.Mn,03.75.KkKeywords: Bose-Einstein condensation,2-species BEC, root mean square radius, determination of particlenumbers
Since the pioneer theoretical study by Ho andShenoy[1] in 1996, the interest in two-species Bose-Einstein condensate (2-BEC) is increasing in recentyears. There are many theoretical studies [1–19]. Ex-perimentally, this system was first achieved by Myatt, etal.[20] in 1997. Making use of a magnetic trap, an opticaltrap, or a combined magneto-optical trap, various typesof 2-BEC can be created [21–24] (also refer to the refer-ences listed in [24]). In related experiments most param-eters can be known quite accurately (say, the strengthsof interaction can be precisely determined via the photo-association spectroscopy), but the particle numbers N A and N B can not. With this background we propose anapproach which can be used for the evaluation of the par-ticle numbers. In details, the followings are performed.(i) For the condensate with the A- and B-atoms, wehave derived two formulae to relate the two root meansquare radii r uRMS and r vRMS , respectively, to the param-eters involved in the experiments. Since the root meansquare radii are observable, these two formulae are usefulfor the determination or refinement of the parameters.(ii) We have find out the border separating thewhole parameter-space into two subspace for miscibleand immiscible phases, respectively. The determinationof the border provides a base for plotting the phase-diagrams,[25] and therefore helps to understand intu-itively the inherent physics.(iii) Since we have introduced the Thomas-Fermi ap-proximation (TFA) in the derivation (in which the ki-netic energy has been neglected), we have performed anumerical calculation to evaluate the error caused by theTFA. In this way the applicability of the two formulae is ∗ Corresponding author: [email protected], Tel:020-84037356, Guangzhou, Xingangxi Road, 135, Zhongshan Daxue,612-601 (code:510275) clarified.Let the masses of the A- and B-atoms be m A and m B .These cold atoms are subjected to the isotropic parabolicpotentials m S ω S r ( S = A or B ). We introduce a mass m and a frequency ω . ~ ω and λ ≡ p ~ / ( mω ) are usedas units for energy and length in this paper. Then, theintra-species interaction V S = c S P i v/r | r =0 > r tendsto zero, both u and v should tend to zero as fast as r ), andare distributed compactly (i.e., not distributed in discon-nected regions), then it is in miscible phase. Otherwise,in immiscible phase. For the miscible states, under theTFA, the analytical expression for u/r and v/r have beengiven previously [19] but in a rather complicated form.In this paper, by introducing the W-strengths definedahead eq.(2), we obtain a much simpler expression asgiven in the Appendix. Where the kind of atoms hav-ing a narrower distribution is named as the A-atom anddescribed by u , while the other kind by v . The borderin the parameter-space that separates the two phases isalso given in the Appendix.With this very simple analytical expression of u/r and v/r , it is straight forward to obtain the root mean squareradii from the definitions r u RMS ≡ ( R u r d r ) / and r v RMS ≡ ( R v r d r ) / . Thus we have( r u RMS ) = 37 [ 15( α α − β β ) α − β ] / (4) β ( r u RMS ) + α ( r v RMS ) = 15 /
37 ( α + β ) / (5)Obviously, when all the parameters are known except N A and N B , and when r u RMS and r v RMS have been measured, N A and N B can be known from eqs.(4,5).Since the TFA has been adopted, we have to evaluatethe deviation caused by the TFA. For this aim, we go tothe one-species BEC. When ~ ω S and λ S ≡ p ~ / ( m S ω S )are used as units, the dimensionless Gross-Pitaevskiiequation is ( − d r + 12 r + α u r ) u = εu (6)where α = N S | c S | / (4 π ). Under the TFA, u/r = q r (1 − r r ) / , where r = (15 α ) / . The root meansquare radius R TFA = p / α ) / . Let the radius ob-tained from the exact solution of eq.(6) be denoted as R exac , and we define α ′ = / / R , where α ′ /α measures the deviation in α caused by TFA. For Rb,the dimensionless strength | c S | = 0 . √ ω · sec. When ω = 1000 /sec as an example, α = 0 . N S , where N S is assumed to be very large. For a general evaluation, α = 10, 100, 1000, and 10000 are adopted. The wavefunction u/r obtained under TFA and from exact calcu-lation are plotted in Fig.1, R TFA , R exac , and α ′ /α arelisted in Table I.The above results demonstrate that, when the wavefunctions obtained under TFA and from exact calculationoverlap nicely (say, when α ≥ R TFA is close to R exac and α ′ is close to α . It turns out that, for 2-BECand for the case that both u/r and v/r are nonzero at r = 0, the overlap of the wave functions from TFA andbeyond TFA overlap nicely (refer to Fig.1a and 1b of[19]). Therefore, N A and N B obtained via eqs.(4,5) isreliable. r u / r (a) = 10 (b) = 100 (c) = 1000 (d) = 10000 FIG. 1: (color on line) The wave function u/r under TFA(solid line) and obtained from exact calculation (dash line).TABLE I: The root mean square radii and α ′ /α . α R TFA R exac α ′ /α .
783 1 .
883 1 . .
826 2 .
859 1 . .
480 4 .
490 1 . .
100 7 .
103 1 . In conclusion, we have proposed an approach helpfulto the determination of the particle numbers, at leastin the qualitative aspect. This approach is limited tothe case that the numbers of both kinds of atoms arehuge and they have nonzero distribution at the center.Incidentally, if the parameters other than N A and N B are tuned to ensure r u RMS = r v RMS , then eqs.(4,5) togetherwill lead to the equation given as eq.(6) in the preprint[25] which can be used to determine the ratio of the twoparticle numbers.
Acknowledgments
Supported by the National Natural Science Founda-tion of China under Grants No.11372122, 11274393,11574404, and 11275279; the Open Project Program ofState Key Laboratory of Theoretical Physics, Instituteof Theoretical Physics, Chinese Academy of Sciences,China; and the National Basic Research Program ofChina (2013CB933601); and Guangdong Natural ScienceFoundation (2016A030313313).
Appendix: Analytical solutions of the CGP underTFA for the case related to this paper
Let Y = ( α − β ) / ( α α − β β ) and Y = ( α − β ) / ( α α − β β ). For the case that both u/r and v/r are nonzero at r = 0 and u/r has a narrower distribution, u is distributed in the domain (0 ≤ r ≤ ( Y ) / ≡ r a )and appears as u /r = X − Y r (7)where X = (15 / / Y / . v is distributed in the do-main (0 ≤ r ≤ √ ε ), where ε = [15( α + β )] / .When r ≤ r a , v /r = X − Y r (8)where X = ( ε − β X ) /α .When r a < r ≤ √ ε u /r = 0 (9) v /r = 1 α ( ε − r /
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