Expansion of a Fermi cloud in the BCS-BEC crossover
G. Diana,1N. Manini,1and L. Salasnich1,21Dipartimento di Fisica and CNR-INFM, Università di Milano, Via Celoria 16, 20133 Milano, Italy 2
CNR-INFM and CNISM, Unità di Milano, Via Celoria 16, 20133 Milano, Italy
共Received 1 March 2006; published 6 June 2006兲
We study the free expansion of a dilute two-component Fermi gas with attractive interspecies interaction in the Bardeen-Cooper-Schrieffer–Bose-Einstein condensate 共BCS-BEC兲 crossover. We apply a time-dependent parameter-free density-functional theory by using two choices of the equation of state: an analytic formula based on Monte Carlo data and the mean-field equation of state resulting from the extended BCS equations. The calculated axial and transverse radii and the aspect ratio of the expanding cloud are compared to experi-mental data on vapors of 6Li atoms. Remarkably, the mean-field theory shows a better agreement with the experiments than the theory based on the Monte Carlo equation of state. Both theories predict a measurable dependence of the aspect ratio on expansion time and on scattering length.
DOI:10.1103/PhysRevA.73.065601 PACS number共s兲: 03.75.Kk
I. INTRODUCTION
Current experiments with cold vapors of 6Li and 40K at-oms can operate in the regime of deep Fermi degeneracy. The available experimental data on two-hyperfine-component Fermi gases are concentrated across a Feshbach resonance, where the s-wave scattering length aF of the in-teratomic potential varies from large negative to large posi-tive values 关1–4兴 and where a crossover from a Bardeen-Cooper-Schrieffer 共BCS兲 superfluid to a Bose-Einstein condensate 共BEC兲 of molecular pairs has been predicted 关5–7兴. In these experiments, the Fermi cloud is dilute be-cause the effective range R0 of the interaction is much smaller than the mean interparticle distance, i.e., kFR0Ⰶ1, where kF=共32n兲1/3 is the Fermi wave vector and n is the gas number density. Even in this dilute regime the s-wave scattering length aF can be made very large: the interaction parameter kFaF diverges and changes sign at a Feshbach resonance, despite kFR0remaining small关1–3,8兴.
Recent experimental and theoretical investigations studied the density profiles 关9–11兴, collective excitations 关9,10,12–17兴, condensate fraction 关18–21兴 and vortices 关22,23兴 of the fermion cloud through the BCS-BEC cross-over. In this Brief Report we analyze the free expansion of the Fermi gas through this crossover by using a parameter-free time-dependent density-functional theory关16,17兴 based on the bulk equation of state of the superfluid, and including a quantum-pressure term. We adopt two possible equations of state: a reliable analytical interpolating formula based on bulk Monte Carlo results关17兴 and the mean-field equation of state based on extended BCS equations关5–7,14兴. Experimen-tally, the free expansion of superfluid 6Li clouds was ob-served by O’Hara et al.关1兴 and by Bourdel et al. 关4兴. The comparison of our theory with these experimental data shows that the effects of interaction could be detected during the expansion if the thermal component was negligible. In addition, by using local scaling equations, we investigate the long-time dynamics of the Fermi gas predicting novel and measurable effects of interaction on the time evolution of the expansion process.
II. THEORY
To describe the dynamics of a zero-temperature Fermi cloud in the external potential U共r,t兲 we use a hydrodynamic model with a von Weizsäcker quantum-pressure term. This theoretical approach is expected to be reliable for studying the collective dynamics of the Fermi gas关16,17兴. The action functional A关兴 of the theory depends on the superfluid order parameter共r,t兲 as follows:
A =
冕
dtd3rL共,t,兲, 共1兲 where the Lagrangian density readsL = i ប* t+
ប2 2m
*2− U兩兩2−E共兩兩2兲兩兩2. 共2兲
E represents the bulk energy per particle of the system,
which is conveniently expressed as a function of the number density n =兩兩2by the following equation:
E共n兲 =3 5 ប2k F 2 2m f共y兲, 共3兲
where f共y兲 is a function of the inverse interaction parameter
y =共kFaF兲−1. In the weakly attractive regime共yⰆ−1兲 one ex-pects a BCS Fermi gas of weakly bound Cooper pairs where the superfluid gap energy ⌬ is exponentially small. In the so-called unitarity limit 共y=0兲 one expects that the energy per particle is proportional to that of a noninteracting Fermi gas with a n-independent coefficient f共0兲=0.42 关24兴. In the weak-coupling BEC regime共yⰇ1兲, a weakly repulsive Bose gas of dimers of mass mB= 2m and density nB= n / 2 is ex-pected. Such Bose-condensed molecules interact with a posi-tive scattering length aB= 0.6aF关25,26兴. The function f共y兲 is modeled by the analytical formula
f共y兲 =␣1−␣2arctan
冉
␣3y1+兩y兩
2+兩y兩
冊
共4兲 recently derived 关17兴 from Monte Carlo 共MC兲 simulations 关26,27兴 and asymptotic expressions. Table 1 of Ref. 关17兴
re-PHYSICAL REVIEW A 73, 065601共2006兲
ports the values of the interpolating␣1,␣2,␣3,1, and2. In the present investigation we take an axially symmetric harmonic potential as a confining trap
U共r,t兲 =m
2关¯共t兲
2共x2+ y2兲 +¯z共t兲2z2兴, 共5兲 where¯j共t兲=j⌰共−t兲, with j=1,2,3=,, z and⌰共t兲 is the step function, so that, after the external trap is switched off at
t⬎0, the Fermi cloud performs a free expansion. The
Euler-Lagrange equation for the field共r,t兲 is obtained by mini-mizing the action functional of Eqs.共1兲 and 共2兲. This leads to a time-dependent nonlinear Schrödinger equation共TDNLSE兲
iបt=
冋
− ប2 2m2+ U +共兩兩2兲
册
. 共6兲 The nonlinear term is the bulk chemical potential of the system. Like the energyE of Eq. 共3兲, also the bulk chemical potential is a function of the number density n. The MC chemical potential is related to the MC energy by the ther-modynamical formula 共n兲 =„nE共n兲… n = ប2k F 2 2m冋
f共y兲 − y 5f⬘
共y兲册
. 共7兲 Instead of the MC equation of state共7兲 based on Eq. 共4兲, in the TDNLSE共6兲 one can plug the mean-field equation of state, obtained from the extended BCS 共EBCS兲 equations 关5–7兴. In this scheme, the chemical potential and the gap energy⌬ of the uniform Fermi gas are found by solving the following EBCS equations:− 1 aF =2共2m兲 1/2 ប3 ⌬ 1/2I 1
冉
⌬冊
, 共8兲 n =共2m兲 3/2 22ប3⌬ 3/2I 2冉
⌬冊
, 共9兲where I1共x兲 and I2共x兲 are two monotonic functions which can be expressed in terms of elliptic integrals关20,28兴. By solving these two EBCS equations we obtain the chemical potential
as a function of n and aF, which can be inserted into the TDNLSE共6兲.
III. EXPANSION OF FERMI GAS AND SCALING EQUATIONS
The free expansion of a droplet of 1.5⫻105 6Li atoms in the unitarity limit共y⯝0兲 was investigated experimentally in Ref. 关1兴. The harmonic potential is anisotropic with =z/= 0.035. The scattering length for the applied mag-netic field B = 910 G is aF= −0.38m = −0.14az关30兴, which corresponds to y = −0.16 at the droplet center. Here az =关ប/共mz兲兴1/2. Figure 1共a兲 compares the observed full width half maximum共FWHM兲 of the transverse and axial size of the expanding cloud as a function of time关1兴 with the ones obtained by the numerical integration of the TDNLSE, based on the MC and EBCS equation of state. The TDNLSE is solved numerically by using a finite-difference algorithm 关31兴 on a real-space grid.
Figure 1共a兲 shows that the expanding gas accelerates more strongly in the radial direction, where the confinement is tighter than axially. Accordingly, the cloud undergoes a shape transition: from a cigar to disk. This is a consequence of superfluidity and interaction: a noninteracting or a normal Fermi gas would undergo a ballistic expansion, leading even-tually to a spherical shape关8兴. In Fig. 1 the TDNLSE results are plotted as dot-dashed lines and show a fair agreement with the experimental data. It is important to observe that the present theory does not rely on any fitting parameters, while the model curves shown in Ref.关1兴 critically depend on the choice of the initial widths. The initial profile for the TDNLSE is obtained by running the code integrating Eq.共6兲 in imaginary time until the confined ground state is filtered out. Figure 1共b兲 plots the droplet aspect ratio showing that the theory overestimates the experimental data.
From the TDNLSE one can deduce Landau’s hydrody-namic equations of superfluids at zero temperature, by set-ting 共r,t兲=
冑
n共r,t兲eiS共r,t兲 and v共r,t兲=共ប/m兲S共r,t兲, andneglecting the quantum-pressure term 共−ប2ⵜ2
冑
n兲/共2m冑
n兲,that is expected to be comparably small for a large number N of particles关16,17,29兴. These hydrodynamic equations are
tn + · 共nv兲 = 0, 共10兲
mtv +
冋
共n兲 + U共r,t兲 + 1 2mv2
册
= 0. 共11兲 In this approximation, the stationary state in the trap is given by the Thomas-Fermi profile n0共r兲=−1关¯ − U共r,0兲兴. HereFIG. 1. 共Color online兲 Expansion of a cloud of 1.5⫻105 6Li
atoms released from a trap as realized in Ref.关1兴, with anisotropy =z/= 0.035.共a兲 Transverse and axial radii of the6Li atomic cloud close to the unitarity limit: aF= −0.14azwhich corresponds to
y = −0.16.共b兲 Aspect ratio of the cloud as function of the time t.
Circles and squares: experimental data of Ref.关1兴; dot-dashed lines: TDNLSE with the MC equation of state; solid lines: LSE with the MC equation of state; dashed lines: LSE with the EBCS equation of state.
BRIEF REPORTS PHYSICAL REVIEW A 73, 065601共2006兲
¯ , the chemical potential of the inhomogeneous system, is
fixed by the normalization condition N =兰d3rn
0共r兲. We im-pose that the hydrodynamic equations satisfy the scaling so-lutions n共r,t兲=n0(x / b1共t兲,y/b2共t兲,z/b3共t兲)/b¯共t兲 and v共r,t兲 = (xb˙1共t兲/b1共t兲,yb˙2共t兲/b2共t兲,zb˙3共t兲/b3共t兲), where ¯共t兲b =兿k=13 bk共t兲. We obtain three differential equations for the
scaling variables bj共t兲, with j=1,2,3=,, z. These scaling differential equations depend also on the space vector r. Only if the chemical potential satisfies a polytropic power law共n兲=Cn␥then the space dependence drops out关14,16兴. In our problem共n兲 is not a power law but we expect that the dynamics can be well approximated by evaluating the scaling differential equations at the center共r=0兲 of the cloud 关32兴. In this case the variables bj共t兲 satisfy the local scaling equations共LSE兲 b¨j共t兲 +¯j共t兲2bj共t兲 = j 2 b ¯共t兲 共/n兲关n0共0兲/b¯共t兲兴 共/n兲关n0共0兲兴 . 共12兲 The coupled ordinary differential equations 共12兲 are inte-grated accurately and efficiently to arbitrary time by standard algorithms. We check the reliability of the LSE approach by comparing their numerical solutions to the expansion ob-tained by using the full TDNLSE 共6兲, both within the MC equation of state共7兲. Figure 1 reports the LSE results as solid lines, clearly showing that the LSE are extremely accurate: solid lines are practically superimposed to dotted-dashed lines 关relative difference Ⰶ1%, see the inset of Fig. 1共b兲兴. Figure 1共a兲 also reports the transverse and axial radii ob-tained by solving the LSE with the chemical potential共n兲 given by the EBCS equations 共8兲 and 共9兲. Remarkably the mean-field EBCS results are closer to the experimental data than the MC results. The two theories essentially coincide for the aspect ratio.
In Ref. 关4兴 the free expansion of 7⫻104 cold 6Li atoms was studied for different values of y =共kFaF兲−1 around the Feshbach resonance共y=0兲. Unfortunately, in this experiment the thermal component is not negligible and thus the com-parison with the present T = 0 theory is not fully satisfactory. Figure 2 compares the experimental data of Ref.关4兴 with the LSE based on both the MC and EBCS equations of state. Figure 2 shows that the aspect ratio predicted by the two T = 0 theories exceeds the finite-temperature experimental re-sults. This is not surprising because the thermal component tends to suppress the hydrodynamic expansion of the super-fluid. On the other hand, the released energy of the atomic gas is well described by the two T = 0 theories, and again the mean-field theory seems more accurate, also probably due to the thermal component. For completeness, Fig. 2共c兲 reports the actual released energy兰d3rn
0共r兲E(n0共r兲).
In the two experiments of Refs.关1,4兴 the time evolution is sufficiently short for a full TDNLSE simulation. It would be computationally impractical to integrate the TDNLSE for times much longer than H−1, where H=共2z兲1/3. As the LSE are very reliable at small and intermediate times, we use them to investigate the time evolution of the Fermi cloud for longer times. Figure 3 shows the aspect ratio of the expand-ing cloud as a function of the inverse interaction parameter y
at subsequent time intervals. At t = 0 the aspect ratio equals the trap anisotropy=0.34. According to these calculations, during the cloud expansion the aspect ratio in the BCS re-gime 共yⰆ−1兲 is measurably different from the one of the BEC regime共yⰇ1兲. Thus the free expansion enables one to
FIG. 2. 共Color online兲 Properties of a 6Li cloud after 1.4 ms expansion from the trap realized in Ref. 关4兴, of anisotropy =z/= 0.34.共a兲 Aspect ratio of the6Li atomic cloud as a function of the inverse interaction parameter y =共kFaF兲−1. 共b兲 Released en-ergy of the same cloud defined as in Ref. 关4兴 based on the rms widths of the cloud.共c兲 Actual released energy of the atomic cloud. Squares report the experimental data of Ref.关4兴; solid lines: LSE with the MC equation of state; dashed lines: LSE with the EBCS equation of state.
FIG. 3. 共Color online兲 Successive frames of the aspect ratio of the Fermi gas as a function of the inverse interaction parameter y =共kFaF兲−1in the experimental conditions of Ref.关4兴. At t=0 the
Fermi cloud is cigar-shaped with a constant aspect ratio equal to the initial trap anisotropy =z/= 0.34. Solid lines: LSE with the MC equation of state; dashed lines: LSE with the EBCS equation of state.
BRIEF REPORTS PHYSICAL REVIEW A 73, 065601共2006兲
recognize the regime involved. Figure 3 predicts a novel in-teresting effect: initially 共tHⱗ3兲 the cloud aspect ratio evolves faster in the BCS region, but then at some interme-diate time共here tH⯝4兲 the BEC side reaches and eventu-ally overtakes the BCS side at larger times共here tHⲏ5兲. Of course, the detailed sequence of deformations depends on the experimental conditions and in particular on the initial aniso-tropy, but the qualitative trend of an initially faster reversal on the BCS side, later surpassed by the BEC gas, is predicted for the expansion of any initially cigar-shaped interacting fermionic cloud. Similarly, starting from a disk-shaped cloud 共⬎1兲, the aspect ratio reduces more quickly initially on the BCS side, and later on the BEC side.
IV. DISCUSSION
Comparison of the EBCS 共dashed lines兲 and MC 共solid lines兲 data shows that beyond mean-field effects do not alter
qualitatively the general trend, but they affect the aspect ratio quantitatively, to an extent which could be appreciated by very accurate experiments carried out at extremely low tem-perature. In particular, the mean-field curves flatten to the asymptotic values共for 兩y兩 Ⰷ1兲 closer to the unitary limit than the MC ones. Present-day experimental data, including mea-surements of collective oscillation frequencies关9,10,12–17兴, are equally well compatible with the EBCS mean field and the MC-based analysis accounting for beyond mean-field ef-fects. New experiments could shed light on these correlation effects and verify the predictions of the present calculations.
ACKNOWLEDGMENT
This work was funded in part by the EU’s 6th Framework Programme through the NANOQUANTA Network of Excel-lence共Grant No. NMP4-CT-2004-500198兲.
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