Condensate fraction of a Fermi gas in the BCS-BEC crossover
Luca Salasnich and Nicola Manini
Dipartimento di Fisica and INFM, Università di Milano, Via Celoria 16, 20133 Milano, Italy Alberto Parola
Dipartimento di Fisica e Matematica and INFM, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy 共Received 31 May 2005; published 30 August 2005兲
We investigate the Bose-Einstein condensation of fermionic pairs in a uniform two-component Fermi gas, obtaining an explicit formula for the condensate density as a function of the chemical potential and the energy gap. We analyze the condensate fraction in the crossover from the Bardeen-Cooper-Schrieffer共BCS兲 state of weakly interacting Cooper pairs to the Bose-Einstein condensate of molecular dimers. By using the local- density approximation we study confined Fermi vapors of alkali-metal atoms for which there is experimental evidence of condensation also on the BCS side of the Feshbach resonance. Our theoretical results are in agreement with these experimental data and give the behavior of the condensate on both sides of the Feshbach resonance at zero temperature.
DOI:10.1103/PhysRevA.72.023621 PACS number共s兲: 03.75.Hh, 03.75.Ss Several experimental groups have reported the observa-
tion of the crossover from the Bardeen-Cooper-Schrieffer 共BCS兲 state of weakly bound Fermi pairs to the Bose- Einstein condensate 共BEC兲 of molecular dimers with ultra- cold two-hyperfine-component Fermi vapors of 40K atoms 关1–3兴 and 6Li atoms 关4,5兴. The detection of a finite con- densed fraction also in the BCS side of the crossover关2,4兴 stimulated a debate over its interpretation关6–10兴. Extended BCS共EBCS兲 equations 关11–13兴 have been used to reproduce in a satisfactory way density profiles关14兴 and collective os- cillations关15兴 of these Fermi gases. As this EBCS mean-field theory is defined for any value of the coupling, it provides an interpolation between the BCS weak-coupling regime and the BEC strong-coupling limit关16,17兴. Despite well-known limitations 关16兴 the EBCS theory is considered a reliable approximation for studying the whole BCS-BEC crossover at zero temperature, giving a simple and coherent description of the crossover in terms of fermionic variables.
Within the EBCS scheme, we derive an explicit formula for the number of condensed fermionic pairs in the uniform BCS ground state. We use the EBCS equations to study the behavior of this condensate fraction as a function of the in- teratomic scattering length aF in the BCS-BEC crossover:
from the BCS regime 共small negative aF兲 crossing the uni- tarity limit 共infinitely large aF兲 to the BEC regime 共small positive aF兲. In addition, by using the local-density approxi- mation, we calculate the condensate fraction and density pro- files of the Fermi gas in anisotropic harmonic confinement.
With no fitted parameters, we find a remarkable agreement with recent experimental results 关4兴 indicating a relevant fraction of condensed pairs of 6Li atoms also on the BCS side of the Feshbach resonance.
The Hamiltonian density of a dilute interacting two-spin- component Fermi gas in a box of volume V is given by
Hˆ = − ប2 2m
兺
=↑,↓ˆ†ⵜ2ˆ+ gˆ↑†ˆ↓†ˆ↓ˆ↑, 共1兲 whereˆ共r兲 is the field operator that destroys a fermion of spinin the position r, whileˆ†共r兲 creates a fermion of spin
in r. The attractive interatomic interaction is modeled by a contact pseudopotential of strength g共g⬍0兲. The field op- erators satisfy the usual anticommutation rules and can be expanded in Fourier series,ˆ共r兲=V−1/2兺keik·raˆk, in terms of operators aˆk destroying a fermion of spin with linear momentum បk. At zero temperature the BCS variational state关18兴 is given by 兩典=兿k共uk+vkaˆk†↑aˆ−k† ↓兲兩0典, where 兩0典 is the vacuum state, uk andvk are variational amplitudes, and the state兩典 is normalized to unity if uk
2+vk2= 1. The ampli- tudes ukandvkare obtained by imposing the minimization of the thermodynamic potential
⍀ =
冕
d3r具Hˆ共r兲 −nˆ共r兲典, 共2兲where nˆ共r兲=兺=↑,↓ˆ†共r兲ˆ共r兲 is the number density operator andis the chemical potential, determined by the condition N =兰d3r具nˆ共r兲典 that fixes the average number N of fermions. All averages are done over the BCS state 兩典.
Accordingly, the standard BCS equation for the number of particles is
N = 2
兺
k
vk2, 共3兲
and for the BCS gap is
− 1/g =共1/V兲
兺
k 共1/2Ek兲 共4兲关18,19兴. Here
Ek=关共⑀k−兲2+⌬2兴1/2 共5兲 and
vk2=12关1 − 共⑀k−兲/Ek兴, 共6兲 with the noninteracting fermion kinetic energy ⑀k
=ប2k2/共2m兲. The chemical potentialand the gap energy⌬ are obtained by solving Eqs. 共3兲 and 共4兲. Unfortunately, in the continuum limit, due to the choice of a contact potential, the gap equation diverges in the ultraviolet. A suitable regu- larization is obtained by introducing the scattering length aF
via the equation PHYSICAL REVIEW A 72, 023621共2005兲
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m/4ប2aF=共1/g兲 + 共1/V兲
兺
k 共1/2⑀k兲, 共7兲and then subtracting this equation from the gap equation 关11–13兴. In this way one obtains the regularized gap equation
− m/4ប2aF=共1/V兲
兺
k
共1/2Ek− 1/2⑀k兲. 共8兲 The EBCS equations 共3兲 and 共8兲 can be used to study the evolution from BCS superfluidity of Cooper pairs 共aF⬍0兲 to the Bose-Einstein condensation of molecular dimers 共aF⬎0兲 关11–13,16兴. This transition is conveniently studied as a function of the dimensionless inverse interaction parameter y =共kFaF兲−1, where kF=共32n兲1/3 is the Fermi wave vector of noninteracting fermions of the same density 关20–22兴.
As previously stressed, several properties of ultracold Fermi gases have been investigated in the last few years by using the EBCS equations 关14,15兴. Here we analyze the condensate fraction of fermionic pairs that is strictly re- lated to the off-diagonal long-range order共ODLRO兲 关23兴 of the system. As shown by Yang关24兴, the BCS state 兩典 guar- antees the ODLRO of the Fermi gas, namely, that, in the limit wherein both unprimed coordinates approach an infinite distance from the primed coordinates, the two-body density matrix factorizes as follows:
具ˆ↑†共r1
⬘
兲ˆ↓†共r2⬘
兲ˆ↓共r1兲ˆ↑共r2兲典 = 具ˆ↑†共r1⬘
兲ˆ↓†共r2⬘
兲典具ˆ↓共r1兲ˆ↑共r2兲典.共9兲 The largest eigenvalue N0of the two-body density matrix共9兲 gives the number of Fermi pairs in the lowest state, i.e., the condensate number of Fermi pairs 关24,25兴. This number is given by
N0=
冕
d3r1d3r2兩具ˆ↓共r1兲ˆ↑共r2兲典兩2, 共10兲and it is straightforward to show关25兴 that
N0=
兺
k uk2vk2. 共11兲It is quite remarkable that such a formula can be expressed in a simple form as a function of the chemical potential and the energy gap ⌬. In fact, in the continuum limit 兺k→V/共2兲3兰d3k→V/共22兲兰k2dk, taking into account the functional dependence共6兲 of the amplitudes ukandvkon
and⌬, we find
n0= N0/V =共m3/2/8ប3兲⌬3/2冑/⌬ +冑1 +2/⌬2. 共12兲 By the same techniques, also the two EBCS equations can be written in a more compact form as
− 1/aF=关2共2m兲1/2/ប3兴⌬1/2I1共/⌬兲, 共13兲 n = N/V =关共2m兲3/2/22ប3兴⌬3/2I2共/⌬兲, 共14兲 where I1共x兲 and I2共x兲 are two monotonic functions which can be expressed in terms of elliptic integrals, as shown by Marini, Pistolesi, and Strinati关26兴.
Simple analytical results from Eqs.共12兲–共14兲 can be ob- tained in three limiting cases: 共i兲 the BCS regime, where
yⰆ−1,/⌬Ⰷ1, and the size of weakly bound Cooper pairs exceeds the typical interparticle spacing kF−1;共ii兲 the unitarity limit, with y = 0共thus aF= ±⬁兲; 共iii兲 the BEC regime, where yⰇ1,/⌬Ⰶ−1 and the fermions condense as a gas of tight dimers.
(i) BCS regime. approaches the noninteracting Fermi energy ⑀F=ប2kF2/共2m兲. One finds that I1共x兲⯝冑x关ln共8x兲−2兴 and I2共x兲⯝2x3/2/ 3. It follows that the condensate density is given by
n0⯝ 共mkF/8ប2兲⌬ = 共3/2e2兲n exp关共/2兲y兴, 共15兲 with an exponentially small energy gap ⌬
= 8e−2⑀Fexp共y/ 2兲.
(ii) Unitarity limit. I1共x兲=0 and I2共x兲⯝1.16 with x
=/⌬⯝0.85. In addition one finds ⯝0.59⑀F and ⌬
⯝0.69⑀F, and the condensate fraction is
2N0/N⯝ 0.6994. 共16兲
(iii) BEC regime. The condensate fraction approaches the ideal Bose gas value 2N0/ N⯝1, with all pairs 共molecular dimers兲 moving into the condensate. In addition one finds that I1共x兲⯝−冑兩x兩/2 and I2共x兲⯝/共8冑兩x兩兲. From these ex- pressions we obtain
n0⯝ n/2 ⯝ 共m2aF/8ប4兲⌬2, 共17兲 which is precisely the condensate density deduced by Pieri and Strinati in this BEC regime using a Green-function for- malism关27兴. In this limit the chemical potential approaches half the dimer binding energy ⯝−ប2/共2maF2兲=−⑀Fy2 and the gap⌬⯝4共3兲−1/2⑀Fy1/2.
The behavior of the condensate fraction 2N0/ N through the BCS-BEC crossover is shown in Fig. 1共solid line兲 as a function of the Fermi-gas parameter y. The figure shows that a large condensate fraction builds up in the BCS side already before the unitarity limit, and that on the BEC side it rapidly converges to the ideal boson gas value. The finding of a finite FIG. 1. 共Color online兲 Condensate fraction N0/共N/2兲 of Fermi pairs in the uniform two-component dilute Fermi gas as a function of y =共kFaF兲−1 共solid line兲. The same quantity computed in the local-density approximation for a droplet of N = 6⫻106fermions in an elongated harmonic trap with⬜/z= 47, as in the experiment of Ref.关4兴, plotted against the value of y at the center of the trap 共joined diamonds兲. Open circles with error bars: experimentally de- termined condensed fraction关4兴.
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nontrivial 2N0/ N⫽1 around the unitarity limit contrasts with the suggestion 关6兴 that the condensate fraction could reach unity already at y = 0.
The local-density approximation 共LDA兲 can be imple- mented to calculate the properties of a nonuniform Fermi gas of sufficiently slowly varying density, as in the case of a large number N of particles in a smooth trapping potential U共r兲. In the spirit of the Thomas-Fermi method, we replace the chemical potentialwith the quantity− U共r兲 into Eqs.
共12兲–共14兲. In this way the energy gap ⌬共r兲, the total density n共r兲, and the condensate density n0共r兲 become local scalar fields. In order to compare with the experiment of Ref.关4兴 done with 6Li atoms, we solve the LDA equations for N = 6⫻106 fermions trapped in an external anisotropic har- monic potential well defined by a transverse frequency ⬜
and an axial frequencyz:
U共r兲 = 共m/2兲关⬜2共x2+ y2兲 +z2z2兴. 共18兲 We take an anisotropy ratio ⬜/z= 47. For given aF, and for a given initial guess of , the local chemical potential
− U共r兲 is computed and Eq. 共13兲 for ⌬ is solved numeri- cally, and this calculation is repeated for each r on a spatial grid. The total number N of fermions in the trap is then computed by integrating Eq.共14兲 over the position r. The parameter is then adjusted iteratively in order to obtain the required total number N of fermions within the droplet. The final density, gap, and n0functions are determined according to Eqs.共12兲–共14兲.
We find that at the border of the droplet, all three of these functions vanish together, for both positive and negative aF. Figure 2 reports the computed axial density
nz共z兲 =
冕
dx dy n共x,y,z兲 共19兲 and condensed density 2n0z共z兲 共defined analogously兲 for three values of the scattering length aF. The fermionic cloud is rather diffuse in the BCS region, and gets more and more compact as the BEC regime is approached. The smoothly vanishing behavior of nz共z兲 near the cloud border is due to the integration over transverse coordinates: n共r兲 instead van- ishes with a finite jump in its gradient component across the surface. The same applies for the condensed quantities. Of course, this sharp vanishing behavior is a consequence of the LDA, while quantum delocalization effects共here neglected兲 should smooth the density profile at the surface of the actual droplet.As shown in Fig. 2, the condensed fraction共dotted curve兲 vanishes as one moves from center to border of the BCS cloud, as pairing acts less efficiently as the density is low. On the contrary, on the BEC side, the condensed fraction in- creases to unity toward the cloud border, as the bosonic gas suffers less from quantum depletion wherever its density is smaller. In the unitarity limit, the condensed fraction is in fact constant, and equals the uniform value of Eq.共16兲: this reflects the scale invariance of this special limit of infinitely large scattering length.
The experiment of Ref.关4兴 addresses the problem of mea- suring the total condensate fraction of trapped interacting fermions near the unitarity limit. Even though it would be hard to demonstrate that the measured “condensed fraction”
coincides with the ODLRO 2N0/ N defined in Eq.共10兲, we attempt a comparison of the computed total condensed frac- tion, reported in Fig. 1, with the experimental data at the lowest temperature kBT = 0.05⑀F. In the experiment the scat- tering length is tuned by means of an external magnetic tuned across a Feshbach resonance. Following Ref.关28兴, the scattering length aFas a function of the magnetic field B near the Feshbach resonance is given by
FIG. 2.共Color online兲 Axial total density 关nz共z兲, solid兴 and con- densed density 关2n0z共z兲, dashed兴 of a droplet composed of N=6
⫻106fermions in an elongated harmonic trap with⬜/z= 47, as in the experiment of Ref.关4兴. With6Li atoms, the harmonic oscil- lator length scale az=ប1/2共mz兲−1/2= 12m in the setup of Ref. 关4兴.
Dotted curves: the axial local condensate fraction 2n0z共z兲/nz共z兲. The values of y at the center of the droplet are −1.2共aF= −0.01az兲,
−10−4共aF= −100az兲, and 0.87 共aF= 0.01az兲, representative of the BCS, unitarity, and BEC regimes, respectively.
CONDENSATE FRACTION OF A FERMI GAS IN THE… PHYSICAL REVIEW A 72, 023621共2005兲
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aF= ab关1 +␣共B − B0兲兴关1 + Br/共B − B0兲兴, 共20兲 where B0= 83.4149 mT, ab= −1405a0, Br= 30.0 mT, and
␣= 0.0040共mT兲−1. The measured condensed fraction at the lowest temperature is reported in Fig. 1 as open circles with error bars. The general increasing trend and the substantially large value of the condensed fraction on the BCS side and close to the unitarity limit agree quite well with the EBCS LDA calculation 共joined diamonds兲. On the BEC side the rapid drop of experimental data is due to inelastic losses and thus these data are not reliable关4兴. Note also that the con- densate fraction measured near a Feshbach resonance of40K atoms关2兴 is much smaller than the values reported in Ref.
关4兴. The origin of this discrepancy has not been clarified yet 关4,6–9兴.
As suggested in the Introduction, the zero-temperature EBCS theory has some limitations. In particular, it overesti- mates the bosonic scattering length aB between molecular dimers in the BEC regime: the theory predicts aB= 2aF, where aF is the two-fermion scattering length, but Monte Carlo results关20兴 confirm the four-body scattering analysis 关29兴 which gives aB= 0.6aF. To overcome such difficulties, the EBCS theory can be further extended by including be- yond mean-field corrections which improve the determina- tion of the chemical potential, and reduce the bosonic scat- tering length to aB= 0.75aF 关30兴.
A finite-temperature formulation of the EBCS mean-field theory is not difficult关19,31兴: for the condensed number of Fermi pairs we find the formula N0=兺kuk2vk2tanh2共Ek/ 2兲, where = 1 /共kBT兲. It is then straightforward to express this relation as an explicit function of , ⌬, and , pretty much like Eqs. 共12兲 at T=0. On the other hand, it is well known 关13,14,16,17兴 that in the BEC regime the finite- temperature mean-field theory overestimates the critical Tc.
This is related to the dimer breaking energy rather than the loss of coherence of the bosonic gas as it should. As a con- sequence, the condensed fraction is overestimated near the unitarity limit and in the BEC region: to get reasonable re- sults beyond mean-field corrections are needed 关13,14,16,17,32兴.
In this paper we have shown that the zero-temperature EBCS mean-field theory gives a simple and nice formula for the condensed fraction of Fermi pairs in the full BCS-BEC crossover. On the BCS side of the Feshbach resonance our theoretical results are in agreement with the low-temperature measurements of Zwierlein et al.关4兴. In fact, the quantitative agreement of the measured and computed condensate frac- tion共based on the off-diagonal long-range-order parameter兲 is no demonstration that these two quantities can be identi- fied. A quantitative microscopic simulation of the experimen- tal procedure could in fact rigorously prove or disprove the identification we are putting forward in the present work.
Such a simulation should be based on a microscopic theory which contains an essentially correct internal dynamics of the pairs across the crossover: this goes likely beyond the capability of a mean-field method such as EBCS. Also alter- native experimental techniques could be developed to pro- vide an independent measurement of the condensed fraction, possibly in closer contact with the off-diagonal long-range order. Further work, both experimental and theoretical, could help to elucidate this interesting issue.
On the BEC side of the Feshbach resonance the experi- mental data of Ref.关4兴 are not reliable due to inelastic losses.
In this region our predictions may be a guide for further experimental investigations.
The authors acknowledge D. Pini and L. Reatto for en- lightening discussions, and C. E. Campbell for useful email suggestions.
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