Analytical pair correlations in ideal quantum gases: Temperature-dependent bunching and antibunching
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Analytical pair correlations in ideal quantum gases:Temperature-dependent bunching and antibunching
J. Bosse , K. N. Pathak , and G. S. Singh , ∗ Institute of Theoretical Physics, Freie Universit¨at, Berlin 14195, Germany Physics Department, Panjab University, Chandigarh 160 014, India Physics Department, Indian Institute of Technology Roorkee, Roorkee 247 667, India(23 October 2011)
Abstract
The fluctuation–dissipation theorem together with the exact density responsespectrum for ideal quantum gases has been utilized to yield a new expression forthe static structure factor, which we use to derive exact analytical expressions forthe temperature–dependent pair distribution function g ( r ) of the ideal gases. Theplots of bosonic and fermionic g ( r ) display “Bose pile” and “Fermi hole” typicallyakin to bunching and antibunching as observed experimentally for ultracold atomicgases. The behavior of spin–scaled pair correlation for fermions is almost featurelessbut bosons show a rich structure including long–range correlations near T c . Thecoherent state at T =0 shows no correlation at all, just like single-mode lasers. Thedepicted decreasing trend in correlation with decrease in temperature for T < T c should be observable in accurate experiments.Key words: Quantum gases, Pair correlation function, Bunching and Antibunching.PACS numbers: 05.30.Fk, 05.30.Jp Introduction : The surge in the study of various properties of ultracold atomic gaseshas prompted search for atomic analog of the Hanbury-Brown–Twiss (HBT) effect [1]by various research groups as reported in Refs. [2, 3, 4] and references therein. Thesuppression of density fluctuations, a signature of the Pauli exclusion principle at work inreal space and thereby antibunching, has been demonstrated in [2, 3]. On the other hand,Jeltes et al. [4] have compared results of the two-particle correlations of a polarized, butnot Bose-condensed, sample of ultracold He* atoms with those of polarized He* atoms.The experimental conditions in [4] were such that the gases could be treated almostideal. Hence bunching for bosons and antibunching for fermions at small interatomicseparations have been attributed to purely quantum effects associated with the exchangesymmetries of wavefunctions of indistinguishable particles. Also, the measurement ofcorrelations has been reported [5] both above and below the Bose-Einstein condensation(BEC) temperature in atomic He*.An elegant form for the correlation function ν ( r ) of the density fluctuations in idealquantum gases (IQG) has been derived by Landau and Lifshitz [6]. Although many qual-itative features and limiting expressions for ν ( r ), which is related to the pair distributionfunction g ( r ), have been discussed by them, quantitative descriptions would require avail-ability of general analytical forms. References [7, 8, 9] have discussed expressions for g BE ( r ) of an ideal Bose gas (IBG) in one or the other range in the temperature domain0 < T < T + c whereas Lee and Long [10] have given g F D ( r ) of an ideal electron gas at T =0.Reference [11] has discussed model analytic expression for the unpolarized homogeneous1lectron gas in solids. However, these expressions cannot be utilized to get comprehensivetheoretical values to compare with the observed HBT effect reported in [4] and [5]. Themain purpose of this Brief Report is to fill up the gap by deriving exact analytical expres-sions for g ( r ) of IQGs. The improvements on some asymptotic results, both for fermionsand bosons, available in the literature are also discussed.A unified approach is presented for evaluation of temperature–dependent g ( r ) for idealBose–Einstein (BE), Fermi–Dirac (FD) and Maxwell–Boltzmann (MB) gases wherein theunification has been achieved using polylogarithm [12, 13, 14, 15, 16, 17]. Our methodstarts from the expression for the density response function derived in [17] using themethod of second quantization, applies the fluctuation–dissipation theorem to get a gen-eral new expression for the static structure factor, introduces a function which is relatedto the one-particle density matrix, and ultimately gets the general form for g ( r ) validfor all quantum gases at all temperatures. The computed values of g BE ( r ) and g F D ( r )using our analytical expressions are depicted graphically as a function of r at varioustemperatures showing at small– r appearance of bump and dip which we term as “Bosepile” and “Fermi hole”. The plots are further compared with the experimental results[4, 5] for ultracold atomic gases. Basic Expressions : The pair–distribution function g ( r ) of a uniform one–component fluidconsisting of N particles in volume V is defined by the thermal average of an operatorthat counts pairs of particles located distance r apart, divided by the square of numberdensity. It is related to the static structure factor S ( q ) of the fluid by the spatial Fouriertransform: n [ g ( r ) −
1] = 1 V X q e i q · r [ S ( q ) −
1] (1)with n = N/V denoting the number density. Taking due account of the fact that theoperator of the total particle–number, ˆ N ≡ N q = , is a constant of motion, the fluctuation–dissipation theorem for a uniform system, χ ′′ ( q, ω ) = ( nπ/ ¯ h )(1 − e − β ¯ hω ) S ( q, ω ), canbe solved for the van Hove function S ( q, ω ) = (¯ h/nπ ) (1 − δ q , ) χ ′′ ( q, ω ) / (cid:16) − e − β ¯ hω (cid:17) + D ( δ ˆ N ) /N E δ q , δ ( ω ), where h ... i represents averaging in the grand canonical ensemble(GCE). The expression S ( q ) = R ∞−∞ d ω S ( q, ω ) then yields S ( q ) = (1 − δ q , ) ¯ hnπ Z ∞ d ω coth β ¯ hω ! χ ′′ ( q, ω ) + δ q , D ( δ ˆ N ) E N . (2)Upon inserting Eq. (2) in Eq. (1), the q –sum on the right–hand side separates into twoparts: the first part contains the summation with the restriction q = while the secondresults in an additive constant, h n D ( δ ˆ N /N ) E − /V i , which vanishes in the thermo-dynamic limit for all IQGs except for an IBG at T ≤ T c . The pathological aspect ofGCE for the condensate fluctuations has been ameliorated by replacing D [ δ ( a † a )] E GCE by D [ δ ( a † a )] E CE , with a and a † being the ground–state annihilation and creation op-erators, as suggested in Ref. [18] based on results in [19], and which has been utilizedby others, see, e.g., [20]. The “law of large numbers” considered by us in order to makethe constant to vanish in the Bose-condensed phase has the form rD ( δ ˆ N /N ) E ∝ N − / ,see, e.g., [21, Eqs. (3.55) & (3.57)]. 2e substitute χ ′′ ( q, ω ) = nπ P k C k [ δ (¯ hω − ∆ k ( q )) − δ (¯ hω + ∆ k ( q ))] from [17] for theIQG with ∆ k ( q ) = ε | k + q | − ε k , and obtain for q = , S ( q ) = X k C k coth (cid:20) β (cid:16) ε | k + q | − ε k (cid:17)(cid:21) ; ( q > , (3)where C k denotes the thermal–average fraction of particles having momentum ¯ h k , C k = g s N β ( ε k − µ ) − η ≡ g s N η ζ (cid:16) ηλ e − βε k (cid:17) , X k C k = 1 . (4)Here η = +1 , − , g s =2 s + 1 is the spin-degeneracy factor for spin s , λ = e βµ is the fugacity, and the function ζ ν ( x ) denotes thepolylogarithm [12, 13] of order ν . The solution of the equation µ ≡ µ η ( n, T ) =0 gives thecharacteristic temperature which can be expressed as T ( η )0 = ε u /k B (cid:16) √ π ζ / ( η ) /η (cid:17) − / with ε u = ¯ h k / (2 m ) and k u = 2 (6 π n/g s ) / serving as units of energy and wave number,respectively.For further analytical discussions, Eq. (3) will now be recast into an appropriate form by(i) substituting k → k + q and to get 2 S ( q ), (ii) expressing exponentials in coth–functionsin accordance with the first equation in (4), and (iii) using P k (cid:16) C k + C | k + q | (cid:17) = 2. Theprocedure finally yields a new form: S ( q ) − ηNg s X k C k C | k + q | , ( q >
0) (5)from which we read1 V X q = e i q · r [ S ( q ) −
1] = nηg s C X q = C q e i q · r + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X q = C q e i q · r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6)with C = N ( T ) /N denoting the fraction of particles which occupy the zero–momentumstate. Introducing the thermal de Broglie wavelength Λ= q π ¯ h β/m and the dimension-less function F ( r ) = X k e i k · r C k = C ( T ) + 2 g s √ π n Λ η Z ∞ d κ ζ (cid:16) ηλ e − κ (cid:17) cos √ π r Λ κ ! (7)which is related to the one–body density matrix [21] by n (1) ( r , r ′ )=( n/g s ) F ( | r − r ′ | ) , thecondensed fraction C ( T ) can be extracted from the normalization condition F (0) = 1.The evaluation of the κ –integration thus leads to C ( T ) = δ η, Θ( T c − T ) h − ( T /T c ) / i ,with Θ( x ) the Heaviside unit step and δ i,j the Kronecker delta, in conformity with [17]and the condensed–IBG result [21, Chap. 3.2].Equations (5) and (6) constitute to be our basic results which are valid at all temper-atures and for all ideal gases. Inserting Eq. (6) into Eq. (1), we find g ( r ) = 1 + ηg s h F ( r ) − C ( T ) i (8)which, in conjunction with Eq. (7), yields an expression in agreement with that discussedin Problem 4 of [6, § T < T c . Thus Eq. (8), which is valid at all3emperatures for all IQGs, generalizes [6, Eq. (117.8)] whose validity is for an FD gas atall T but for a BE gas at T > T c only.It seems pertinent to mention that for bosons g ( r ) is not simply the sum of condensateand non–condensate (or thermal) contributions as it is for the one–body density matrix,Eq.(7), or the density–response function [17, 21]. The presence of the Fourier convolutionin Eq.(5) has resulted into Eq. (8) wherein the thermal contribution [ F ( r ) − C ( T )]appears as a factor in the second term on the right hand side. This factor vanishes in thelimit T → g ( r ) →
1, observed at T ≪ T c .There is another significant aspect regarding the derivation of our results in context ofthe GCE used here. On the dynamic route leading to Eq. (3), and thereby Eq. (5), weneither needed to nor did we use the Bogoliubov prescription which replaces a and a † by c –numbers. For a consistency check, we took recourse to the static route (not elabo-rated here) starting from S ( q )= D δN q δN † q E /N with N q = P k ,σ a † k ,σ a k + q ,σ . We obtain fora uniform fluid the counterpart of Eq. (5) wherein the right hand side contains additionalterms representing correlations of number fluctuations, D δ (cid:16) a † k ,σ a k ′ ,σ ′ (cid:17) δ (cid:16) a † k ′ + q ,σ ′ a k + q ,σ (cid:17) E .However, for any q = 0, these extra terms vanish for ideal gases, irrespective of populationof any single–particle state. Hence it is comforting to note that static and dynamic routeslead to exactly the same result. Analytic Expressions for F ( r ): For a BE or an FD gas, the integral in Eq. (7) can becarried out analytically in the region −∞ < µ ≤
0, i.e. for 0 < λ ≤
1. This region covers thecomplete domain of the IBG while it describes only the high–T domain ( T ( − ≤ T < ∞ )of the ideal Fermi gas (IFG). On series expansion of ζ ( z ) and subsequent term–by–termintegration, we get F ( r ) = C ( T ) + g s n Λ η ∞ X ℓ =1 ( ηλ ) ℓ ℓ / exp − πr Λ ℓ ! . (9)An alternative form, equivalent to Eq. (9) and most suitable for evaluation at r ≪ Λ, is F ( r ) = C ( T ) + 1 − C ( T ) ζ / ( ηλ ) ∞ X ℓ =0 ( − ℓ ℓ ! πr Λ ! ℓ ζ ℓ +3 / ( ηλ ) , (10)which results from series expansion of the exponential function in Eq. (9) and subsequentinterchange of summations. Also, we have used the relation obtained on implementing F (0)=1 in Eq. (9).For analytical evaluation of the integral in Eq.(7) for η = − µ > λ >
1, which describes the low–T domain (0 ≤ T < T ( − ) of an IFG, we split theintegral into a sum of two integrals over intervals (0 , √ βµ ) and ( √ βµ, ∞ ), respectively.In the latter integral, the polylogarithm ζ ( z ) with z = − e βµ − κ can be expanded into apower series in z , since 0 ≤ | z | <
1. In the former integral, where | z | >
1, we apply theidentity ζ ( z ) = ζ (1 /z ) − ln( − z ) valid for z / ∈ (0 ,
1) and subsequently expand ζ (1 /z ) intoa power series in 1 /z , since | /z | = 1 / | z | <
1. Term–by–term integration of the resultinginfinite sum finally yields for low– T IFG ( µ >
0) : F FD ( r ) = ˜ k k j (˜ k F r )˜ k F r − π (cid:16) ˜ k F Λ (cid:17) ∞ X ℓ =1 ( − ℓ ℓ / ℑ erfc (cid:16) √ π r Λ 1 √ ℓ − i ˜ k F Λ2 √ π √ ℓ (cid:17) exp (cid:18)(cid:16) ˜ k F Λ2 √ π (cid:17) ℓ − (cid:16) √ π r Λ (cid:17) ℓ (cid:19) + ℜ erfc (cid:16) ˜ k F Λ2 √ π √ ℓ + i √ π r Λ 1 √ ℓ (cid:17) exp (cid:18)(cid:16) √ π r Λ (cid:17) ℓ − (cid:16) ˜ k F Λ2 √ π (cid:17) ℓ (cid:19) (11)where j ( x ) is the spherical Bessel function of the first kind and order 1, and erfc( x ) isthe complementary error function. Also, ˜ k F = ¯ h − q mµ − ( n, T ) denotes a generalizedFermi wave number with lim T → ˜ k F = k F = (6 π n/g s ) / and is a measure of the chemicalpotential. In the low–temperature limit, i.e. for T → → ∞ , one easily retrievesthe following from Eq. (11): F FD ( r ) T → −→ j ( k F r ) / ( k F r ), the expression given, see e.g.,in Ref. [10]. Pair Distribution Functions : From Eq. (8), one readily finds for a dilute quantum gas, g MB ( r ) = 1 which coincides with the classical ideal–gas result. One also deduces the results g (0) = 1 + ( η/g s )[1 − C ( T )] and g ( ∞ ) = 1 leading to the following bounds: g F D (0) ≡ − /g s ≤ g F D ( r ) ≤ g F D ( ∞ ) ≡ g BE ( ∞ ) ≡ ≤ g BE ( r ) ≤ g BE (0) with g BE (0) = ( /g s if T > T c /g s ) h T /T c ) / − ( T /T c ) i if T ≤ T c . (12)The small– and large– r asymptotic behaviors for T < T c are obtained as g BE ( r →
0) = g BE (0) − (2 πr /g s Λ ) ( T /T c ) / [ ζ (5 / /ζ (3 / O (( r/ Λ) ) and g BE ( r → ∞ ) = 1 +2 C ( T ) / ( n Λ r )+ g s / ( n Λ r ). We find that the latter asymptote improves the expressiongiven in [9, Eq.(21)], and is in agreement with [6, p.359]; the expression given in [9] wouldbe valid only at T ≪ T c whereas the validity of ours is in the entire range 0 ≤ T ≤ T c . Also,for T →
0, one gets g F D ( r →
0) = ( g s − /g s + ( k F r / g s ) [1 − k F r /
35 + O ( r )] whereinthe first two terms on the right–hand side give the result as obtained by Lee and Long[10] while discussing the static structure for an ideal electron gas at T =0. And for largedistance r , we get g F D ( r → ∞ ) = 1 − ( k F r ) − sin(2 k F r ) / ( k F r )] / ( g s k r ) + O ( r − )approaching unity as r − by damped oscillations, which improves on a result given in [6, § T asymptote of F ( r ) deduced from Eq.(9), theGaussian form, g ( r ) T ≫ T ( η )0 −→ ηg s exp − π r Λ ! (13)is obtained. In fact, the asymptote (13) generalizes to the spinor gases the earlier resultsderived by others for zero-spin particles, see e.g., [22]. Although the experimental resultsfor a thermal bosonic gas were fitted by Schellekens et al. [5] using an expression like Eq.(13), a detailed analysis together with the theoretical plots showing contrasting behaviorfor spinor bosons and fermions is lacking.The computed values of (2 s + 1) [ g ( r ) −
1] = η [ F ( r ) − C ( T )] are depicted in Fig.1 for a set of six reduced temperatures T ∗ ≡ k B T /ε u =0 . , . , . , . , . .
011 corresponding to
T /T c =4 . , . , . , . , . , .
1, respectively. The value g MB ( r )=1 of an ideal dilute quantum gas coincides with the abscissa at all T and, itis to be noted, that the pair–correlation properties of an IQG differ qualitatively fromthose of the corresponding dilute gas, even at high temperatures. As displayed in Fig.1, g BE ( r ) ≥ g FD ( r ) ≤ T . We term the appearance of a bump (dip) as the“Bose pile” (“Fermi hole”) which reflects the statistical “effective” attraction (repulsion)5 a bb cc def k u r - - H + L@ g H r L - D Figure 1: (Color online). Pair correlation functions of ideal BE (above ab-scissa), FD (below abscissa), and MB (coinciding with the abscissa for all T ) gases at reduced temperatures T ∗ = 0.491 (a, red), 0.164 (b, green), 0.114(c, blue), 0.104 (d, short–dashed), 0.055 (e, dot–dashed), and 0.011 (f, long–dashed). T =0 results: bosons (coinciding with abscissa), fermions (lowermostcurve, black, shadowing the long–dashed line). in ideal gases, an interaction which weakens with increasing spin. For s ≫
1, one wouldrecover the result g ( r )=1 true for a classical ideal gas.It is found in low– r region that g F D ( r ) monotonically decreases (increases in magnitude)with increase in r as T decreases. However, for bosons in that domain, g BE ( r ) increasesas T decreases for T > T c and the trend reverses with decrease in T below T c . Let us lookfurther into the behavior of fermionic curves at all T and bosonic at T > T c , i.e. thosemarked (a) to (c) above abscissa. It can be seen that the pile (hole) being generated byrotating the Gaussian curves about the ordinate merely narrows down retaining its originalspin–dependent height (depth) as T is increased. They ultimately acquire congruentbell-shaped forms (cf. Eq.(13)) in the high– T regime having width (at half maximum)= q ln 2 / π Λ ≈ .
332 Λ. Also, it can be seen that g BE ( r ) and g F D ( r ) show differentbehaviors at T = 4 . T c as opposed to studies wherein it was found that response functions[17], dynamical structure factors [23], and momentum distribution functions [17] of BE,FD, and MB gases are essentially independent of statistics at this temperature. However,in the limit T → ∞ , we get g ( r )=1 independent of statistics, as expected.The bosonic curve marked (c) clearly demonstrates that g BE ( r ) becomes long–rangedas T reaches in the close vicinity of T c from high– T side. However, unlike this, g F D ( r ) isof much shorter range at all T . It is estimated that the correlation length ξ FD ( T ) ≤ k − which presents a measure of the largest distance at which fermion pairs are still correlated.If T is raised, ξ FD ( T ) further decreases and at high T it is of the order of Λ (cf. Eq.(13)).It is tempting to compare both the curves marked (a) in Fig. 1 with Fig. 2 of Ref.[4] wherein the HBT effect showing bunching and antibunching for ultracold atoms of He ∗ and He ∗ have been depicted. The striking resemblance between the theoreticaland experimental plots is remarkable. The direct comparison of our curves with theexperimental ones will have to await availability of data free from the uncertainty andsystematic errors in measurements, mentioned by Jeltes et al. [4]. However, the interesting6hysics is unfolded in the BEC phase represented by the curves (d) to (f). The height ofthe Bose pile goes on decreasing and the curve gets increasingly flattened as T → T = 0 . T c is almost flat and ultimately the plot for an IBGat T = 0 coincides with the abscissa analogous to the Bose-condensed phase experimentalresult [5, 24] revealing that the system is completely coherent. In fact, it is just like thesituation in a single-mode laser in which the photons are not bunched [25]. Furthermore,our studies depict temperature-dependent aspects of bunching and antibunching. Acknowledgments : The work is partially supported by the Indo–German (DST–DFG)collaborative research program. JB and KNP gratefully acknowledge financial supportfrom the Alexander von Humboldt Foundation.*Address for correspondence. Email: [email protected]
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