Temperature and coupling dependence of the universal contact intensity for an ultracold Fermi gas
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Temperature and coupling dependence of the universal contact intensityfor an ultracold Fermi gas
F. Palestini, A. Perali, P. Pieri, and G. C. Strinati
Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy
Physical properties of an ultracold Fermi gas in the temperature-coupling phase diagram canbe characterized by the contact intensity C, which enters the pair-correlation function at shortdistances and describes how the two-body problem merges into its surrounding. We show that thelocal order established by pairing fluctuations about the critical temperature T c of the superfluidtransition considerably enhances the contact C in a temperature range where pseudogap phenomenaare maximal. Our ab initio results for C in a trap compare well with recently available experimentaldata over a wide coupling range. An analysis is also provided for the effects of trap averaging on C. PACS numbers: 03.75.Ss,03.75.Hh,74.40.-n,74.20.-z
The “contact” C, introduced by Tan [1, 2] to charac-terize the merging of two-body into many-body physicsin systems like ultracold Fermi gases with a short-rangeinter-particle interaction, has attracted much interestlately [3–7]. This is especially relevant in the contextof the BCS-BEC crossover, whereby a smooth evolutionoccurs jointly for the two- and many-body physics, fromthe presence of Cooper pairs with an underlying Fermisurface in the BCS limit, to the formation of molecularbosons with a residual interaction in the BEC limit.Recently, the contact C was measured in an ultracoldgas of trapped fermionic ( K) atoms [8], from the large-momentum tail of the momentum distribution as wellas from the high-frequency tail of the radio-frequencysignal following an earlier suggestion [9]. These mea-surements were done from about unitarity (where thescattering length a F of the Fano-Feshbach resonance di-verges) to deep inside the BCS region (more precisely,in the coupling range − . < ∼ ( k F a F ) − < ∼ +0 . k F = (2 mE F ) / is the Fermi wave vector expressed interms of the Fermi energy E F = ω (3 N ) / , m being theatom mass, N the total number of atoms, and ω theaverage trap frequency [10]), and in a temperature rangeabout the critical temperature T c .Several questions concerning the contact C remainopen. They hinge on the recent experimental results ofRef.[8] (like the temperature and coupling dependence ofC, and the effect that trap averaging has on the values ofC), as well as on more theoretical issues. These includethe identification of the (approximate) spatial boundarybetween short- and medium-range physics that can beassociated with the contact C, the effects that improvedtheoretical approaches have on the values of C, and theinterconnection with the presence of a pseudogap in thesingle-particle excitation spectrum.In this Letter, we address these questions and calcu-late the contact C using a t-matrix approach [11] thatproved successful in comparison with ARPES-like datafor ultracold Fermi atoms [12], and also using a nontriv-ial extension of this theory [13] which takes into account the residual interaction among composite bosons. Thisis to verify to what an extent improvements on the de-scription of the medium-range physics (over and abovethe results of the t-matrix) influence the values of C indifferent coupling and temperature ranges [14].We shall, specifically, be concerned with the temper-ature dependence of C over an extended temperaturerange up to (several times) the Fermi temperature T F ,to determine how the value of C is affected by thepseudogap physics extending above T c in the unitary( − < ∼ ( k F a F ) − < ∼ +1) regime, and to address the re-lated question of how trap averaging influences the valueof C with respect to that of a homogeneous system withthe same nominal temperature and coupling.The contact C was originally introduced to accountfor the large wave-vector behavior of the fermionic dis-tribution n ( k ) (for spin component) of the homogeneoussystem, such that n ( k ) ≈ C h k − . Here, the suffix“h” stands for homogeneous, k = | k | is in units of k F = (3 π n ) / where n is the total particle density, and n ( k ) is normalized such that R d k (2 π ) n ( k ) = 1 /
2. Alterna-tively, C h can be extracted from the high-frequency tailof the radio-frequency (RF) spectrum I RF ( ω ) per unitvolume, so that I RF ( ω ) ≈ (C h / / π ) ω − / . Here, thefrequency ω is in units of E F and the RF spectrum isnormalized such that R + ∞−∞ dω I RF ( ω ) = 1 /
2. The aboveasymptotic form of I RF ( ω ) holds provided final-state ef-fects can be neglected [9, 15, 16].Similar asymptotic behaviors can be obtained for thetrapped system. Preserving the above normalization asfor the homogeneous system, we write within a local-density approximation: n ( k ) = Z d r n ( k ; r ) ≈ C t k (1)where C t = 8 π Z d r [3 π n ( r )] / k F C h ( r ) . (2)Here, the suffix “t” and k F refer to the trapped system,the spatial position r is in units of the Thomas-Fermiradius R F = [2 E F / ( mω )] / , and n ( r ) and C h ( r ) arethe density and contact locally in the trap. By a similartoken, the large- ω behavior of the total RF spectrum ofthe trapped system reduces to: I RF ( ω ) = Z d r I RF ( ω ; r ) ≈ C t / π ω / (3)with the overall normalization of the homogeneous case.This local-density analysis shows that, by adding con-tributions of different shells with a weight proportionalto n ( r ) / , the trap modifies the value C h for a homo-geneous system at the same nominal coupling ( k F a F ) − and (relative) temperature T /T F . In this respect, thetemperature dependence of C h provides an important in-formation, because different shells are at different relativetemperature with respect to the local value of T F .We begin our analysis with the homogeneous case at T c and show in Fig. 1(a) the coupling dependence of C h ,when n ( k ) is obtained within the t-matrix approximationof Ref.[11] (full line) and its improved Popov version ofRef.[13] (dashed line). The calculations are done at thevalue of T c of the respective theories. The plot also shows(light-dotted lines) the leading (low-temperature) ap-proximations for C h obtained in the BCS weak-couplingand BEC strong-coupling limits, which are given bythe expressions (4 / k F a F and 4 π/ ( k F a F ), respectively,with the crossover region − < ∼ ( k F a F ) − < ∼ +1 markingthe change between these two limiting behaviors.Note how the difference between the full and dashedlines in Fig. 1(a), which originates from the activation ofthe residual interaction among the composite bosons, isappreciable only close to unitarity where particles cor-relate with each other within the inter-particle spacing k − F . The smallness of this difference resulting fromour calculations confirms the validity of the t-matrix ap-proximation for the contact C h and for the high-energyscale to which C h is associated (see below). Yet, thisdifference is relevant for the physical interpretation ofthe contact C h as characterizing the effects of medium-range (many-body) physics over and above the short-range (two-body) physics. This interpretation is also con-sistent with rewriting C h = (3 π / ∞ /E F ) in termsof the high-energy scale ∆ ∞ introduced in Ref.[9], suchthat ∆ ∞ = 2 π | a F | n/m embodies in weak coupling the ef-fects of surrounding particles through a mean-field shift[17], while ∆ ∞ = 4 πn/ ( m a F ) reflects a standard re-lation in strong coupling [18] between the density andthe gap parameter within BCS theory. For compari-son, Fig. 1(a) also reports the coupling dependence ofC h at zero temperature (dashed-dotted line) within thet-matrix approximation, to which the above approximateexpressions in the BCS and BEC limits converge.Figure 1(b) shows the temperature dependence of C h at unitarity over an extended temperature range withinthe t-matrix approximation. The rather slow decay of C h T/T F (b) F C h (k F a F ) -1 (a) FIG. 1: The contact C h for the homogeneous case: (a) At T c vs the coupling ( k F a F ) − , obtained within the t-matrixapproximation (full line) and its improved Popov version(dashed line). The leading approximations in weak and strongcoupling (light-dotted lines) as well as the T = 0 result withinthe t-matrix approximation (dashed-dotted line) are also re-ported. (b) At unitarity vs the temperature T (in units of theFermi temperature T F ), obtained within the t-matrix (fullline). The high-temperature approximation to this curve isalso reported as a reference (dashed line) and extrapolated to T = 0. [See the text for the meaning of the inset.] C h at high temperature is consistent with the expec-tation that C h is not related to long-range order. Thetemperature behavior of C h steepens up at low tempera-ture when entering the pseudogap region for T /T F < ∼ . h appearsmost evident when the calculation is continued below T c [19], with the result that a cusp appears in C h at T c where the effect of the pseudogap is maximum. To em-phasize this enhancement, we have indicated in Fig. 1(b)the extrapolation of the high-temperature behavior of C h (dashed line) down to T = 0, on top of which the con-tribution associated with the pseudogap region about T c appears evident. Our value (=3.23) for C h at T = 0 com-pares well with that (=3.40) extracted from the Monte-Carlo calculations of Ref.[20]. For completeness, theinset of Fig. 1(b) reports the temperature dependence ∆ pg / E F , ∆ ∞ / E F (k F a F ) -1 FIG. 2: The high-energy scale ∆ ∞ (full line) and the low-energy scale ∆ pg (dashed line) are reported (in units of E F )vs the coupling ( k F a F ) − at T c . of C h for ( k F a F ) − = − . k F a F ) − = +1 . T /T F ≃ . h is here reportedfor the first time and deserves further comments. Onphysical grounds, the enhancement of C h when enter-ing the fluctuative region from above T c is due to thestrengthening of local pairing correlations, still in theabsence of long-range order. In the present approach,pairing correlations are embodied by the pair-fluctuationpropagator in the particle-particle channel, whose wave-vector and frequency structures give rise to a characteris-tic low-energy scale ∆ pg in the single-particle excitations[11], which is referred to as the pseudogap. The con-tact C h , through its alternative definition in terms of thehigh-energy scale ∆ ∞ that was previously mentioned, isalso related to a wave-vector and frequency averaging ofthe very same structures in the pair-fluctuation propa-gator [9]. The inter-dependence between the two energyscales ∆ ∞ and ∆ pg can then be explicitly appreciatedin Fig. 2, where they are shown at T c vs the coupling( k F a F ) − . The two energy scales do not fully relate toeach other in weak coupling where the contact is dom-inated by the mean-field interaction, but become veryclose in value at unitarity where strong local-pairing cor-relations dominate both thermodynamic (∆ ∞ ) and dy-namic (∆ pg ) quantities.Although the values of C h are extracted from theasymptotic (scale-free) power law n ( k ) ≈ C h k − , a char-acteristic value k C can nevertheless be identified at which n ( k ) has reached C h k − within, say, a few percent accu-racy. The length scale corresponding to k − is approxi-mately the spatial range at which two-body physics inter-faces with medium-range physics. This range is expectedto be about the size of the composite bosons in strong k C / k F (k F a F ) -1 ω C / E F (k F a F ) -1 FIG. 3: The characteristic values of k C (in units of k F ), atwhich the asymptotic power-law behavior n ( k ) ≈ C h k − isreached within 2 .
5% (dashed line), 5% (dashed-dotted line),and 10% (full line) accuracy, are shown at T c vs ( k F a F ) − .The inset shows, correspondingly, the characteristic value ω C (in units of E F ), at which the asymptotic power-law behavior I RF ( ω ) ≈ (C h / / π ) ω − / of the RF spectrum is reachedwithin 10% accuracy. coupling and the inter-particle spacing in weak coupling.Figure 3 shows the values of k C extracted in this wayat T c vs ( k F a F ) − within the t-matrix approximation,for the three distinct values (2 . , , k C reaches a pronounced min-imum for the coupling at which the chemical potentialvanishes, away from which k C increases most markedlyon the strong-coupling side as expected from the increas-ing spatial localization of composite bosons.In addition, the inset of Fig. 3 shows the value of ω C at T c vs ( k F a F ) − , extracted within a 10% accuracy fromthe large- ω behavior of the RF spectrum. (Recall thatthe ω − / tail of the RF spectrum originates from theshort-range behavior of the 2-body wave function [22, 23]in the absence of final-state effects [24].) Comparisonwith k C with the same accuracy yields the relation 2 ω C = k C that holds approximately for all couplings.The values of the contact C t obtained from Eq.(1),by adding the asymptotic contributions from all shellsin the trap within the t-matrix approximation, are re-ported in Fig. 4 at T c vs ( k F a F ) − (full line). Theyare compared with the values (filled and empty circles,stars) obtained experimentally in Ref.[8] through alter-native procedures. The theoretical value obtained atunitarity for T = 0 is also reported for comparison(empty square). Due to difficulties in extracting theasymptotic behavior of n ( k ) from experimental data, thefigure also shows the theoretical values of C t (dashed-dotted line) obtained upon averaging k n ( k ) over theinterval k min ≤ k ≤ k max . This follows the procedure C t (k F a F ) -1 C t T/T F FIG. 4: The contact C t obtained within the t-matrix ap-proximation for the trapped case (full line) is shown at T c vsthe coupling ( k F a F ) − , and compared with the experimentalvalues of Ref.[8] (filled and empty circles, stars). The insetshows, correspondingly, C t at unitarity vs T /T F . used to extract the experimental values of C t , for which k min = 1 .
55 when ( k F a F ) − < − . k min = 1 . − . < ( k F a F ) − , while k max = 2 . t at T = 0 was obtained across the unitary regime byinterpolating known results in the BCS and BEC limits.That interpolation has to be contrasted with the com-pletely ab initio theoretical calculation at T c reported bythe full line in Fig. 4, which spans a wide coupling rangejust across the unitary regime.Finally, the inset of Fig. 4 displays the temperaturedependence of C t for the trapped case at unitarity withinthe t-matrix approximation (full line), with the verticalarrow indicating the corresponding value of T c . Contraryto the result for the homogeneous case of Fig. 1(b), nocusp now appears in C t at T c for the trapped case. Anexpanded discussion about the effects that trap averaginghas on the values of C is provided in [25].In conclusion, we have presented a detailed analy-sis of the contact intensity C for a Fermi gas over thetemperature-coupling phase diagram. The values of Chave been obtained from the large- k behavior of the mo-mentum distribution n ( k ) as well as from the large- ω tailof the RF spectrum I RF ( ω ), both for the homogeneousand trapped case. For the latter case, good agreement isobtained with recent experimental data. The effects ofpairing fluctuations on C have been determined by thet-matrix approximation and its improved Popov versionthat takes into account the interaction among compositebosons. We have found that the values of C are stronglyaffected by the emergence of a pseudogap in the single- particle excitations about T c .Discussions with E. Cornell, D. Jin, and J. T. Stew-art are gratefully acknowledged. This work was partiallysupported by the Italian MIUR under Contract PRIN-2007 “Ultracold Atoms and Novel Quantum Phases”. [1] S. Tan, Ann. Phys. , 2952 (2008).[2] S. Tan, Ann. Phys. , 2971 (2008).[3] E. Braaten and L. Platter, Phys. Rev. Lett. , 205301(2008).[4] F. Werner, L. Tarruell, and Y. Castin, Eur. Phys. J. B , 401 (2009).[5] S. Zhang and A. J. Leggett, Phys. Rev. A , 023601(2009).[6] R. Combescot, F. Alzetto, and X. Leyronas, Phys. Rev.A , 053640 (2009).[7] F. Werner and Y. Castin, arXiv:1001.0774v1.[8] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin,Phys. Rev. Lett. , 235301 (2010).[9] P. Pieri, A. Perali, and G. C. Strinati, Nature Phys. ,736 (2009), arXiv:0811.0770.[10] We set ¯ h = 1 and k B = 1 throughout.[11] A. Perali, P. Pieri, G. C. Strinati, and C. Castellani,Phys. Rev. B , 024510 (2002).[12] J. P. Gaebler, J. T. Stewart, T. E. Drake, D. S. Jin,A. Perali, P. Pieri, and G. C. Strinati, Nature Phys. (inpress), and arXiv:1003.1147.[13] P. Pieri and G. C. Strinati, Phys. Rev. B , 094520(2005).[14] Z. Yu, G. M. Bruun, and G. Baym, Phys. Rev. , 023615(2009).[15] W. Schneider and M. Randeria, Phys. Rev. A , 021601(2010).[16] E. Braaten, D. Kang, and L. Platter, Phys. Rev. Lett. , 223004 (2010).[17] In weak coupling k F | a F | ≪
1, the zero-temperaturemean-field gap ∆ /E F = (8 /e ) exp {− π/ (2 k F | a F | ) } asso-ciated with long-range order is sub-leading with respectto ∆ ∞ associated with medium-range order.[18] P. Pieri and G. C. Strinati, Phys. Rev. Lett. , 030401(2003).[19] The t-matrix approximation is extended to the superfluidcase below T c following P. Pieri, L. Pisani, and G. C.Strinati, Phys. Rev. B , 094508 (2004).[20] R. Combescot, S. Giorgini, and S. Stringari, Eur. Lett. , 695 (2006).[21] V. A. Belyakov, Sov. Phys. JETP , 850 (1961).[22] C. Chin and P. S. Julienne, Phys. Rev. A , 012713(2005).[23] A. Perali, P. Pieri, and G. C. Strinati, Phys. Rev. Lett. , 010402 (2008).[24] The asymptotic behavior of transition matrix elementsfor large energy transfer depends on the short-range be-havior of the two-body wave function [cf. A. R. P. Rauand U. Fano, Phys. Rev. , 68 (1967)]. When final-state effects were taken into account, the RF signal wouldthus eventually acquire an ω − / asymptotic tail [22, 23].[25] See the Supplemental Material below reported for moredetails. SUPPLEMENTAL MATERIAL:“TEMPERATURE AND COUPLINGDEPENDENCE OF THE UNIVERSALCONTACT INTENSITY FOR ANULTRACOLD FERMI GAS”
We discuss the connection between the values of thecontact intensity for the whole trap and for a homoge-neous system.
It is interesting to compare the values of C t obtainedfor the whole trap with its approximation obtained fromEq.(2) of the main text, whereby one identifies theshell at r max corresponding to the maximum of the ra-dial weight function (32 /π ) r [3 π n ( r )] / /k F , and thentakes C h ( r max ) therein outside the integral. The result ofthis procedure at T c is reported vs the coupling ( k F a F ) − in Fig. 5(a), where the inset shows an example of theshape of the radial weight function (full line/right scale)and of C h ( r ) (dashed line/left scale) at unitarity. Thegood agreement, which results between the calculationfor the trap (full line) and the approximation that se-lects the contribution of the most important shell (dashedline), shows to what an extent the results of C t for thewhole trap can be used, together with knowledge of thedensity profiles, to extract the values of C h for the ho-mogeneous case.The same procedure can be applied to interpret thetemperature dependence of C t , reported in the inset ofFig. 4 of the main text and reproduced here in Fig. 5(b)for convenience (full line). In particular, it is interestingto understand how the trap averaging washes out thepeak about T c obtained for the homogeneous case (seeFig. 1(b) of the main text).To appreciate this effect, we have reported in the insetof Fig. 5(b) the temperature dependence at unitarity ofthe integral of the radial weight function (full line/rightscale) and of C h ( r max ) (dashed line/left scale) from aboveto below T c . While C h ( r max ) retains the characteris-tic cusp feature of the homogeneous case (with a maxi-mum at T = 0 . T c ), the steady increase of the integratedweight for decreasing temperature more than compen-sates for the decrease of C h ( r max ) when T < . T c , thusmasking eventually the cusp feature in the integratedquantity.The same approximate procedure, that resulted in thedashed line of Fig. 5(a), can be applied to reproduce thetemperature dependence of C t at unitarity above T c , be-cause in this case the two functions in the integral ofEq.(2) of the main text have a smooth behavior similarto that shown in the inset of Fig. 5(a). At given T below T c , however, the cusp present in C h ( r ) requires us to split it as the sum of a smooth background C (b)h ( r ) and of apeaked contribution C (p)h ( r ), yielding approximately:C t ≃ C (b)h ( r max ) 32 π Z ∞ dr r [3 π n ( r )] / k F + 32 π Z ∞ dr r [3 π n ( r )] / k F C (p)h ( r ) . (4)These approximate results, from above to below T c , areshown by the dashed line in Fig. 5(b) (to which the sec-ond term on the right-hand side of Eq.(4) below T c givesat most a 15% contribution).