The Density Dependence of the Transition Temperature in A

arXiv:cond-mat/9910013v1 1 Oct 1999 aoaoyo tmcadSldSaePyisadTeCornell The and Physics State Solid and Atomic of Laboratory ayyas nsieo oghsoyo theoretical tem- transition elementary of the history 1950s, in shift the long perature, possible the to a for as back such of interest questions dating of spite [1] topic In investigation a con- been years. Bose-Einstein has density many systems low of (BEC) properties densed the of tion a o mre osnu 2 3 4 that [4] there [3] [2] gas, consensus Bose a of dilute case emerged the the now in In has interactions past. repulsive recent the the until unsettled remained nicesn ucino h neato aaee,na parameter, interaction the of function increasing an eaueo uefli atcenme o the tem- for number transition particle the superfluid of on dependence perature the examined have we of consequence a transi- as the reduced that interaction. interparticle be found would had temperature number [5] tion calculations a Moreover, earlier the density. of number and mass transi- particle the below well temperature, temperature tion a at occurs transition est.Ti a emasrrsn eut ic tis it liquid since of result, case particle the the surprising in n a that and seem known diameter well may sphere This hard the density. is a where ciglmtt h togyitrcigrgm currently regime interacting strongly the to limit inter- can weakly acting low-density, parameter the interaction from continuously the varied gases, case, be atomic Vycor trapped the in of because systems BEC the over vantages h oetdniyahee nteeeprmnswson 2 was of gas. experiments order these Bose the in “dilute” achieved re- or density experimental demon- lowest interacting an The we weakly [6], the system 1983 of in this time alization with first work the for early strated, our In system. rvd eino vra,i em fteinteraction na the of of values terms the in overlap, with of parameter, region a provide civdwt oecnesdN,R rHe. or be Rb can Na, that Bose-condensed values the with below achieved na magnitude of values of to orders system density,several this particle s- restrict the recombination, small on by the set limits [9] and the hydrogen length In atomic scattering wave condensed traps. Bose optical of or case magnetic within confined atoms neato aaee,the parameter, interaction oteBCssesraie with realized systems BEC the to h oeo nepril neatosi h determina- the in interactions interparticle of role The oiae ytercn hoeia oki hsarea, this in work theoretical recent the by Motivated o usin uha h ffc on effect the as such questions For h est eedneo h rniintmeauei ho a in temperature transition the of dependence density The T c ihdniyaditrcinsrnt have strength interaction and density with , oprdwt eettertclcluain o DBs c Bose 3D for calculations theoretical recent with compared iueBs a eiew n,i gemn ihter,apos a theory, ∆ with agreement form in the find, of we ture regime gas Bose dilute o au fteitrcinprmtr na parameter, interaction the of value a for ASnmes 37.i 53.p 67.40.-w 05.30.Jp, 03.75.Fi, numbers: PACS calculations. rniintmeauedt banda ucino parti of function a as obtained data temperature Transition × ..Rpy ..Crooker B.C. Reppy, J.D. 10 T 18 0 o nielBs a ihtesame the with gas Bose ideal an for , e cm per 4 3 eVcrsse ffr ad- offers system He-Vycor hsi ucetylwto low sufficiently is This . T/T 0 3 urnl accessible currently , = 87 T γ b[]or [7] Rb 4 c ( etesuperfluid the He na ficesn the increasing of 3 a ) .Hebral B. , 1 / 3 4 thge este aiu sfudi h ai of ratio the in found is maximum a densities higher At . T He-Vycor c taa Y14853-2501 NY Ithaca, 23 Nvme ,2018) 5, (November ilbe will a[8] Na 3 hti nareetwt ahitga ot Carlo Monte path-integral with agreement in is that , b 3 3 ..Cri,J e n ..Zassenhaus G.M. and He, J. Corwin, A.D. , , 1 etrfrMtrasRsac,CakHl,CrelUnivers Cornell Hall, Clark Research, Materials for Center xetta o o este T we densities [4], low [3] for [2] torsional that calulations our expect recent of tempera- the sensitivity Following zero mass to oscillator. calibrated signal the an and superfluid from ture the estimated is of density extrapolation number The transition the corresponding temperature. are the theory and density with The number comparison particle regime. a gas for Bose required dilute the quantities for data the consider first hroer ehd r hs tnadi h td of study and the in techniques standard superfluid those cryogenic are The methods thermometry methods. state. experimental superfluid BEC the of observation for time ai rprissmlrt hs fte“re rideal or m “free” mass, the thermody- effective an of limit, with those density gas to Bose low then similar the We properties the in atoms. namic of observe, superfluid modification to mobile small expect the a of in ge- mass reflected pore effective and anticipate, be substrate We may the of ometry sample. constrained influence su- Vycor the gas the the that however, Bose of model volume homogeneous we the at a within Therefore pores as many phase link [11]. perfluid will density state particle superfluid characteriz- pore low or cycles BEC the exchange Feynman the than ing the larger that be the and can of size wavelength particles thermal appreciate superfluid the to mobile important temperatures is low It at pores. that the sur- by 3D-connected der provided network. complex van face the by 3D over move constrained interconnected to are forces highly 4 Waals atoms a from helium form superfluid diameter glass and The in Vycor nm range porous 8 measurements the to these of for channels temper- used interior transition and superfluid The the superfluid determine ature. the to in as participating well as particles den- the number the for estimate sity Vycor to us our allows within propor- then density which signal particle sample, superfluid a a the obtain used to have to tional we technique experiments, oscillator present torsional the For details. for ie,o h re cm a order the on sizes, ncesbewt h laivprsses ute,work- the Further, with ing systems. vapor alkali the with inaccessible ntedsuso foreprmna eut,w shall we results, experimental our of discussion the In nti etrw ilgv nyabifsmayo our of summary brief a only give will we letter this In l est nthe in density cle nesdsses ntelwdensity low the In systems. ondensed tv hf ntetasto tempera- transition the in shift itive 3 e n h edri eerdeswee[10] elsewhere referred is reader the and He, 4 eVcrsse losmc agrsample larger much allows system He-Vycor 4 3 eVcrsse are system He-Vycor iha setal unlimited essentially an with , oeosBs fluid Bose mogenous c ilb ie by given be will ∗ . T c c /T 0 ity, 3 1/3 Tc = T0[1 + γ(na ) ], (1) and result in a range of overlap in terms of the interaction parameter between all three systems. This is illustrated 2 ∗ 2/3 2/3 where T0 = [2π¯h /m kB ζ(3/2) ]n is the transition in Figure 1, where we indicate the range of the quan- ∗ temperature for an ideal Bose gas with particle mass m tity n1/3a for the 4He, experiments 23Na and 87Rb ex- and density n. The coefficient for γ is positive as required periments. Since the 4He-Vycor system clearly exhibits 3 for Tc to rise above T0 as the interaction parameter, na superfluid properties, one may also expect superfluidity increases. Although there is agreement on the form of to exists in the Bose-condensed alkali systems at compa- Eq. 1, the theoretical estimates for γ range over more rable values of the interaction parameter and sufficiently than an order of magnitude, from an estimate of 0.34 low temperature. reported by Grüter, Ceperley, and Laloë (GCL) [3], 2.9 by Baym et al. [4], to a value of 4.66 given by Stoof [2]. 0.05 Stoof For a convenient comparison to experimental results Best Fit 1/3 we cast Eq. 1 in a linear form with the variable, n a, 0.04 Baym et al. as 2 0.03 Ideal Gas Tc 2π¯h 1/3 )

= [1 + γ(n a)]. (2) 2/3 ∗ 2/3 k (

n m kBζ(3/2) c

T 0.02 2/3 In Figure 1, we then plot Tc/n against the parameter n1/3a, taking a value of 0.22 nm [12] for the helium hard 0.01 core diameter. As expected from theory, a linear fit gives a good representation for our data in the low density 0.00 regime. The zero density intercept of the fit serves to 0 10 20 30 40 50 ∗ determine an effective mass ratio of m /m = 1.34, where n (1018 atoms/cc) m is the 4He mass, and the slope yields a value of 5.1 for

γ, which is within a factor of 1.09 of the value given by FIG. 2. The observed transition temperature, Tc, is plot- Stoof. ted as a function of the particle density, n. The top curve

through the data is the function given by Eq. 1 with the pa- ∗ 5 rameters for the eﬀective mass, m and γ determined as from 3 / 2

) the linear ﬁt shown in Fig. 1. The two other curves are for γ 4He cc = 4.66 as given by Stoof and γ = o, the noninteracting case. /

m 4 o t 1H(x10) In Figure 2 we show a more conventional plot of the a (

low density data. Here we have plotted T as a function

/ c

K of the particle density, n. The curve through the data is 15 - 3 23 the theoretical expression given in Eq. 1 with our best Na ∗ 10

( ﬁt values for γ and m as determined previously. Two 87 3 / Rb 2 curves based on the γ estimates of Stoof and Baym et al. n

/ are also shown. The lowest curve is for the case of the

c 2

T 0.00 0.02 0.04 0.06 0.08 noninteracting Bose gas case (i.e., γ = 0 in Eq. 1). The

1/3 relatively close agreement with the estimate of Stoof is n a clear in this plot. In contrast to the other theoretical treatments of the 2/3 FIG. 1. The quantity Tc/n is plotted as a function of BEC problem for the interacting Bose gas [2] [4], the 1 3 the parameter n / a. The straight line is our ﬁt to these calculations of GCL are not restricted to the low den- data. The arrows indicate the range of interaction parameter sity limit. The path-integral Monte Carlo technique em- 1 4 87 23 covered in the H, He, Rb, and Na experiments. Note 0 1 ployed by GCL allows a prediction for the ratio of Tc/T that in the case of H the scale has been enlarged by a factor over the entire range of na3, from the dilute gas limit of 10. to densities approaching the freezing point of liquid helium. In the top panel of Figure 3 we show the results of It is of interest to compare the range of the interac- the GCL calculation taken from Ref. [3]. At low values tion parameter explored in the various BEC experiments. of the interaction parameter, GCL also ﬁnd that the ra- 14 1 3 The highest particle densities achieved to date are 4x10 0 / 3 87 15 3 23 tio, Tc/T , increases in proportion to n a in agreement atoms/cm in the case of Rb and 3x10 cm for Na, with the calculation of Stoof and of Baym et al. An in- however, for both the alkali atoms the s-wave scattering teresting feature of the GCL calculation is the maximum lengths are much larger than for the respective value for 0 4 in Tc/T found near a value of the interaction parameter the He atom. The larger scattering lengths for the alka- na3 = 0.01. lies more than compensate for the lower particle densities In the lower panel of Figure 3, we show the data ob-

2 tained in a number of different 4He-Vycor experiments at ACKNOWLEDGMENTS Cornell, including a recent measurement designed specif- ically to map out the region of the peak found by GCL. The authors wish to acknowledge stimulating conver- Although the higher density data show more scatter than sations with Gordon Baym, David Ceperley, Michael the data of Figs. 1 and 2, the trends are clear, and there Fisher, Geoffrey Chester, Veit Elser, Wolfgang Ketterle, is a pleasing agreement in the qualitative form of the and many others. JDR would like to thank the Aspen experimental data as compared to the GCL calculation. Center for Physics for its hospitality during the time the In particular, there is an excellent match in the position present letter was conceived. GMZ and ADC have been of the peak value for the ratio Tc/T0 as a function of supported by the US Dept. of Ed. through the GAANN the interaction parameter. A remaining problem, how- Program-Physics Grants P200A80709 and P200A70615- ever, is the as yet unexplained disagreement in the mag- 98. This work is supported by the National Science Foun- nitude of the effect estimated by GCL for low densities dation through Grants No. DMR-9971124 and DMR- as compared to our experimental data or the calculations 9705295, and by the Cornell Center for Materials Re- of Stoof [2] and Baym et al. [4]. search, through Grant No. DMR-9632275, CCMR Re- port No. 8405.

1.08 a. Present address: Physics Dept., Fordham University, Bronx, NY 10458-5198. 1.06 b. Present address: CNRS-CRTBT, BP 166, Grenoble

O Cedex 38042 France. T

1.04

/ c. Present address: McKinsey & Company, Three Land-

T mark Square, 1st Floor, Stamford, CT 06901. 1.02

1.00 3.5 3.0

O 2.5

T 112

/ [1] T.D. Lee and C.N. Yang, Phys. Rev. , 1419 (1957);

c 2.0 Studies in Statistical Mechanics T also K. Huang, in , Vol. 1.5 Stoof II, J. de Boer and G.E. Uhlenbeck, eds. (North Holland 1.0 Publishing, Amsterdam, 1964), 1. -5 -4 -3 -2 -1 [2] H.T.C. Stoof, Phys Rev. A 45, 8398 (1992); and M. Bi- 10 10 10 10 10 jsma and H.T.C. Stoof, Phys. Rev. A 54, 5085 (1996). na3 [3] P. Grüter, D. Ceperley, and F. Laloë, Phys. Rev. Lett. 79, 3549 (1997). FIG. 3. The top panel shows the values of the ratio Tc/T0 calculated by GCL as a function of the interaction parameter [4] G. Baym, J.-P. Blaizot, M. Holtzmann, F. Laloë, na3. The curve through the lower density points is given by and D. Vautherin, submitted to Phys. Rev. Lett., 3 1/3 cond/mat/9905430. T /T0 =1+0.34(na ) . The curve through the higher den- c 5 sity data including the region of the peak is merely a guide [5] M. Giraradeau, The Physics of Fluids, , 1458 (1962); A. Fetter and J.D. Walecka, Quantum Theory of Many- for the eye. In the lower panel we plot the values of Tc/T0, obtained from a number of different Vycor experiments (dis- Particle Systems, (McGraw-Hill, New York, 1971), Sec. 3 141 tinguished by different symbols), as a function of na . The 28, 261; and T. Toyoda, Ann. Phys. (NY) , 154 dashed curve is the theoretical estimate of Stoof [2]. (1982). [6] B.C. Crooker, B. Hebral, E.N. Smith, Y. Takano, and 51 In conclusion, we wish to emphasis that studies with J.D. Reppy, Phys. Rev. Lett. , 666 (1983); for an ex- tensive discussion of these measurements in terms cross- low density Bose condensed helium systems can provide over in critical phenomena between the dilute Bose gas a useful and illuminating contrast to the trapped gas sys- regime and the strongly interacting regime see: M. Ra- tems for the study of properties of the weakly interacting 4 solt, M.J. Stephen, M.E. Fisher, and P.B. Weichman, BEC fluids. Particularly important aspects of the He- Phys. Rev. Lett. 53, 798 (1984); and P.B. Weichman, M. Vycor system are the ability to study long term phenom- Rasolt, M.E. Fisher, and M.J. Stephen, Phys. Rev.B 33, ena, such as the question of robustness or stability of flow 4632 (1986). states in a Bose-condensed superfluid, and to extend ob- [7] M.H. Anderson, J.R. Ensher, J.R. Matthews, C.E. servations of BEC to the realm of larger interaction pa- Weiman, and E.A. Cornell, Science 269, 198 (1995). rameters than are accessible to the trapped gas systems. [8] K.B. Davis, M.-O. Mewes, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995); J. Stenger, D.M. Stamper-Kurn, M.R. Andrews,

3 A.P. Chikkatur, S. Inouye, H.-J. Miesner, W. Ketterle, J. Low Temp. Phys. 113, 167-188 (1998). [9] D.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, S.C. Moss, D. Keppner, and T.J. Greytak, Phys. Rev. Lett. 81, 3811 (1998). [10] Experimental Techniques in Condensed Matter Physics at Low Temperatures, Frontiers in Physics Series, R.C. Richardson and E.N. Smith eds. (Addison Wesley Pub- lishing Company, Menlo Park, CA, 1988). [11] For a discussion of superﬂuidity in terms of exchange cycles and path-integrals see D.M. Ceperley, Rev. Mod. Phys. 67, 1601 (1995); E.L. Pollock and D.M. Ceperley, Phys. Rev. B 36, 8343 (1987); and R.P. Feynman, Phys. Rev. 90, 1116 (1953) and 91, 1291 (1953). [12] K.J. Runge and G.V. Chester, Phys. Rev. B 38, 135 (1988).