Many-body treatment of the collisional frequency shift in fermionic atoms
MMany-body treatment of the collisional frequency shift in fermionic atoms
A.M. Rey , A.V. Gorshkov , C. Rubbo JILA, NIST and Department of Physics, University of Colorado Boulder, CO 80309, and Physics Department, Harvard University, Cambridge, MA, 02138 (Dated: November 10, 2018)Recent clock experiments have measured density-dependent frequency shifts in polarized fermionic alkaline-earth atoms using S - P Rabi spectroscopy. Here we provide a first-principles non-equilibrium theoreticaldescription of the interaction frequency shifts starting from the microscopic many-body Hamiltonian. Ourformalism describes the dependence of the frequency shift on excitation inhomogeneity, interactions, and many-body dynamics, provides a fundamental understanding of the effects of the measurement process, and explainsthe observed density shift data. We also propose a method to measure the second of the two S - P scatteringlengths, whose knowledge is essential for quantum information processing and quantum simulation applications. PACS numbers: 03.75.Ss, 06.30.Ft, 06.20.fb, 32.30-r, 34.20.Cf
Experimental efforts in cooling, trapping, and manipulat-ing alkaline-earth-like atoms such as Sr and Yb have led tounprecedented developments in optical clocks based on the S - P transition [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. While theinterrogation of a large number of atoms enhances the sensi-tivity of these clocks, the accompanying interatomic interac-tions degrade clock precision. Thus the understanding of theseinteractions is crucial for precision spectroscopy. Fermionicalkaline-earth atoms have also started to attract considerabletheoretical attention in the context of quantum informationprocessing [11, 12, 13, 14] and quantum simulation [15], ap-plications requiring the knowledge of both S - P scatteringlengths. Here we present a many-body formulation, which,on the one hand, allows us to understand the nature of thecollisional frequency shift (CFS) measured in a recent exper-iment [3], and on the other, allows us to propose a way tomeasure the remaining S - P scattering length, which wasnot probed in Ref. [3]. Our model helps to clarify the roleof excitation inhomogeneities, dynamics, and interactions infermionic clock experiments [3, 16, 17, 18].Clock experiments based on Rabi interrogation start witha nuclear-spin-polarized sample of atoms prepared (for con-sistency with Ref. [3], which we aim to model) in an excitedstate e , which is then transferred to the ground state g by il-luminating the atoms during a time t f with a probe beam de-tuned from the atomic resonance. The CFS δω eg is inferredby recording the final population in g as a function of the de-tuning and looking for changes in the corresponding lineshapedue to interactions. So far most treatments of CFSs in dilutepolarized fermionic gases away from the unitarity limit werebased on a static mean field analysis [16, 17, 19, 20]. Thelatter predicts a frequency shift δω eg = π (cid:126) a − eg M ( ρ g − ρ e ) G (2) ge ,with G (2) ge the two atom correlation function at zero distance,which measures the probability that two particles are simulta-neously detected, a − eg the s-wave scattering length between the g and e atoms with mass M , and ρ g,e the corresponding atomdensities. Here we extend this formulation beyond mean-field and fully account for the many-body dynamics duringRabi interrogation. Our key statements are as follows. (i)Motion-induced excitation inhomogeneity can lead to s-wave CFS even in an initially polarized ensemble of fermions. (ii)CFS is sensitive to the time-averaged population differencebetween g and e atoms, consistently with the mean field ap-proximation, and in particular vanishes when this differencegoes to zero ( π pulse). (iii) For a fixed pulse area, the CFSapproaches zero as t f → , meaning that, in order to experi-ence interactions, atoms should have enough time to feel theexcitation inhomogeneity. (iv) Measurements of δω eg doneby locking the interrogation laser at fixed final ground statefraction are very sensitive to the pulse area and strength of in-teractions. Depending on these parameters, even the sign of δω eg can be reversed.We begin our analysis with the Hamiltonian ˆ H describingcold fermionic alkaline-earth atoms illuminated by a linearlypolarized laser beam with bare Rabi frequency Ω and trappedin an external potential V ( r ) that is the same for g and e (i.e. atthe “magic wavelength” [1]). Assuming that the atoms arepolarized in a state with nuclear spin projection m , we omitthe nuclear spin label and, setting (cid:126) = 1 , obtain [14, 15] ˆ H = (cid:88) α (cid:90) d r ˆΨ † α (cid:18) − M ∇ + V ( r ) (cid:19) ˆΨ α + u − eg (cid:90) d r ˆ ρ e ˆ ρ g + ω (cid:90) d r (ˆ ρ e − ˆ ρ g ) − Ω (cid:90) d r ( ˆΨ † e e − i ( ω L t − k · r ) ˆΨ g + h . c . ) . (1)Here ˆΨ α ( r ) is a fermionic field operator at position r foratoms in electronic state α = g ( S ) or e ( P ), while ˆ ρ α ( r ) = ˆΨ † α ( r ) ˆΨ α ( r ) is the corresponding density opera-tor. Since polarized fermions are in a symmetric nuclear state,their s -wave interactions are characterized by only one scatter-ing length a − eg , with the corresponding interaction parameter u − eg = 4 π (cid:126) a − eg /M , describing collisions between two atomsin the antisymmetric electronic state |−(cid:105) = ( | ge (cid:105) − | eg (cid:105) ) / √ .The laser with frequency ω L and wavevector k is detunedfrom the atom transition frequency ω by δ = ω L − ω .As in the experiment of Ref. [3], we assume that mostatoms are frozen along one (longitudinal) z -direction, leav-ing in the remaining transverse x - y plane an isotropic 2D har-monic oscillator with frequency ω x = ω y . We can then write ˆΨ α ( r ) = φ z ( z ) (cid:80) ν ˆ c α ν φ ν x ( x ) φ ν y ( y ) , where φ zν and φ ν a r X i v : . [ phy s i c s . a t o m - ph ] A ug are, respectively, the longitudinal and the transverse harmonicoscillator eigenmodes and ˆ c † α ν creates a fermion in mode ν = ( ν x , ν y ) and electronic level α . Following Refs. [3, 18],we assume that the probe is slightly misaligned from the z -direction: k = k z ˆ z + k x ˆ x with | k x /k z | (cid:28) . Definingthen Ω ν,ν (cid:48) = Ω e − η z / L ( η z ) (cid:104) φ ν ( x ) | e ik x x | φ ν (cid:48) ( x ) (cid:105) , where η i = k i (cid:113) (cid:126) mω i (cid:28) are the Lamb-Dicke parameters and L n are Laguerre polynomials [21], laser induced sideband transi-tions can be neglected if Ω ν,ν (cid:48) (cid:54) = ν (cid:28) ω x . In this regime, ˆ H can be rewritten in the rotating frame as ˆ H = − δ (cid:88) ν ˆ n e ν + (cid:88) ν ,α E ν ˆ n α ν − (cid:88) ν Ω ν x c † g ν ˆ c e ν + h . c)+ u − eg (cid:88) ν ν ν ν A ν ν ν ν ˆ c † e ν ˆ c e ν ˆ c † g ν ˆ c g ν , (2)where A ν ν ν ν = (cid:82) ( φ z ) dz (cid:82) (cid:81) j φ ν jx dx (cid:82) (cid:81) j φ ν jy dy , ˆ n α ν = ˆ c † α ν ˆ c α ν , Ω ν x = Ω L ν x ( η x ) L ( η z ) e − ( η x + η z ) / , and E ν are single-particle energies.Many interaction terms in Eq. (2) can be ignored providedthat the interaction is weaker than ω x and that the 2D oscilla-tor is slightly anharmonic as in the experiment of Ref. [3].Furthermore, we note that unless ν = ν and ν = ν , | (cid:82) φ ν φ ν | (cid:29) | (cid:82) (cid:81) j φ ν j | . As a result, the interaction isdominated by the terms where ( ν , ν ) is equal to ( ν , ν ) , ( ν x , ν y , ν x , ν y ) , ( ν x , ν y , ν x , ν y ) , or ( ν , ν ) , whichdescribe the exchange of modes along neither direction, along x , along y , and along both directions, respectively. Postponingthe study of the terms exchanging modes along one directiononly, the remaining terms conserve the number of particles permode ν . Assuming there is only one atom in each of N non-empty modes (cid:126) ν = { ν , . . . , ν N } , as is the case if all atomsare initially in the same internal state, ˆ H can then be reducedto a spin- / model describing these modes: ˆ H S = − δ ˆ S z − (cid:88) ν Ω ν x ˆ S ν x − (cid:88) ν (cid:54) = ν (cid:48) U νν (cid:48) ( (cid:126) ˆ S ν · (cid:126) ˆ S ν (cid:48) − / . (3)Here U νν (cid:48) = u − eg A ννν (cid:48) ν (cid:48) , (cid:126) ˆ S ν = (cid:80) α,α (cid:48) ˆ c † α ν (cid:126)σ αα (cid:48) ˆ c α (cid:48) ν ,where (cid:126)σ are Pauli matrices in the { e, g } basis, ˆ S i = x,y,z = (cid:80) ν ˆ S ν i , and constant terms were dropped. ˆ H S is reminiscentof solid-state spin Hamiltonians, which can also feature long-range interactions and rich nonequilibrium dynamics [22].The rotational invariance of the interaction term in ˆ H S ( ∝ U νν (cid:48) ) is key to understanding some of the basic features ofthe model. The interaction term is diagonal in the collectiveangular momentum basis | S, M, q (cid:105) , satisfying ˆ S | S, M, q (cid:105) = S ( S + 1) | S, M, q (cid:105) and ˆ S z | S, M, q (cid:105) = M | S, M, q (cid:105) , with S = 0 , . . . N/ and − S ≤ M ≤ S . Here the extra label q is required to uniquely specify each state. The fully symmet-ric (Dicke) S = N/ states do not interact. They are uniqueand the label q can be omitted for them.For a homogeneous excitation, Ω ν = ¯Ω , the term (cid:80) ν Ω ν x ˆ S ν x commutes with ˆ S , the interaction energy is con- (cid:45) (cid:45) (cid:45) (cid:45) averaged ground state fraction Sh i ft (cid:64) ∆ Ω e g (cid:144) Π (cid:68) (cid:72) H z (cid:76) a eg (cid:45) (cid:61) a a eg (cid:45) (cid:61) a a eg (cid:45) (cid:61) a Ν (cid:217) (cid:200) (cid:198) Υ (cid:198) Υ (cid:200) (cid:226) x Ν (cid:61) Ν (cid:61) Ν (cid:61) FIG. 1: (color online) CFS, δω eg / (2 π ) , at resonant transfer, for dif-ferent a − eg (in Bohr radii, a ). The time-averaged ground state frac-tion was varied by changing t f . The solid lines were calculated bythermally averaging N g computed from Eq. (3), and the dots bya single realization of (cid:126) ν randomly chosen out of those satisfying ∆Ω( (cid:126) ν ) = (cid:104) ∆Ω (cid:105) T , ¯Ω( (cid:126) ν ) = (cid:104) ¯Ω (cid:105) T , and ¯ U ( (cid:126) ν ) = (cid:104) U (cid:105) T . The dashedlines show Eq. (5) evaluated at thermally averaged parameters. Here N = 7 , T = 1 µ K, (cid:104) ∆Ω (cid:105) T / (cid:104) ¯Ω (cid:105) T = 0 . and ¯Ω = 6 π/ (80 ms ) . Theinset shows R φ ν ( x ) φ ν ( x ) dx in arbitrary units. served, and no CFS will be observed provided the initial stateis an eigenstate of the interaction or a classical mixture ofthem, consistent with Ref. [17]. If the system is preparedin the Dicke manifold, states with S < N/ are never pop-ulated and the ground state population evolves collectivelyas N (0) g ( t, δ ) = N ¯Ω ¯Ω + δ sin (cid:18) t √ ¯Ω + δ (cid:19) , where the super-script (0) indicates a homogeneous excitation.In the presence of excitation inhomogeneity, there aretwo simple limiting cases where CFSs are absent: the non-interacting regime and the strongly interacting regime whereinteractions dominate over Rabi frequency inhomogeneity.The suppression of CFSs in the latter case is a consequenceof the large energy gap between states with different S , whichbrings S -changing transitions out of resonance [23]. In theintermediate interaction regime, on the contrary, the excita-tion inhomogeneity cannot be ignored. It will transfer atomsbetween states with different S generating a net CFS even atzero temperature.In most experiments, e.g. Ref. [3], δω eg is measured byfirst locking the spectroscopy laser at two points, δ , , ofequal height in the transition lineshape (equal final groundstate fraction under the initial condition of all atoms instate e ) and then determining the change in the mean fre-quency as the interaction parameters or density are varied, δω eg = ( δ + δ ) / . Note that for a homogeneous excitation N (0) g ( t, δ ) = N (0) g ( t, − δ ) and therefore δω eg = 0 . Defining ¯Ω( (cid:126) ν ) to be the mean Rabi frequency over modes (cid:126) ν and treat-ing (cid:80) ν (Ω ν x − ¯Ω( (cid:126) ν )) ˆ S ν x as a perturbation, we write N g ( t f , δ ) as N (0) g ( t f , δ ) + N (2) g ( t f , δ ) (the first order term vanishes),Taylor expand it around ± δ (0)1 , and obtain δω eg ≈ N (2) g ( t f , − δ (0)1 ) − N (2) g ( t f , δ (0)1 )2 ∂N (0) g ( t f ,δ ) ∂δ | δ (0)1 . (4)To proceed further, we note that U νν (cid:48) is a slowly varyingfunction of | ν i − ν (cid:48) i | ( i = x, y ) [see Fig. 1 (inset)], exceptwithin a narrow range near ν i = ν (cid:48) i . Provided k B T (cid:38) N ω z (which was satisfied in Ref. [3]), the occupied modes (cid:126) ν aresufficiently sparse for the behavior of U ν i ν j to be dominatedby its slowly varying part. Therefore, we can approximate U ν i ν j → ¯ U ( (cid:126) ν ) . Under this approximation, the states with S = N/ − , so called spin-wave states, are separated inenergy from the Dicke states by ¯ U ( (cid:126) ν ) N and are the onlystates exited to first order by the vector perturbation opera-tor (cid:80) ν (Ω ν x − ¯Ω( (cid:126) ν )) ˆ S ν x . This allows us to obtain an analyticexpression for δω eg that depends on Ω ν ix only through ¯Ω( (cid:126) ν ) and ∆Ω( (cid:126) ν ) , the root-mean-square Rabi frequency. This ex-pression is particularly simple and illuminating when evalu-ated at resonant population transfer ( δ (0)1 → ): δω eg ˛˛˛˛ δ (0)1 → = − ∆Ω N ¯ U ¯Ω sin( A ) A sin[( N ¯ U ) t f ]( N ¯ U ) t f f ( ¯Ω , t f , N ¯ U ) . (5) Here A ( (cid:126) ν ) = ¯Ω( (cid:126) ν ) t f is the pulse area and f ( ¯Ω , t f , N ¯ U ) = − ¯Ω / ( N ¯ U ) tan( t f ( N ¯ U ) /
2) cot( A / − ( N ¯ U ) / ¯Ω )[4 sin ( A / / A − sin( A ) / A ] . The dependence of ¯ U , ¯Ω , and ∆Ω on (cid:126) ν is implied. We now make a few impor-tant remarks: (a) In the limit t f → ( N e → ), Eq. (5)reproduces the mean-field expression δω eg → − (∆Ω) N ¯ U ¯Ω ∝− N ¯ U G (2) ge [ t f → , δ → [3, 18]. (b) δω eg depends onthe time-averaged population difference, (cid:104) N g − N e (cid:105) t f = t f (cid:82) t f ( N g ( τ ) − N e ( τ )) dτ | δ =0 , ∆Ω=0 = − N sin( A ) A and ex-actly vanishes at A = π when (cid:104) N g (cid:105) t f = (cid:104) N e (cid:105) t f . (c) For afixed pulse area A , as t f → , the frequency shift vanishes as δω eg → − (∆Ω t f ) N ¯ U cot[ A / / (2 A ) , implying that in or-der to experience a CFS atoms need time to feel the excitationinhomogeneity. (d) At finite times t f N ¯ U (cid:38) and N ¯ U (cid:29) ¯Ω , δω eg ∝ sin[ t f ( N ¯ U )]( N ¯ U ) reproducing the expected suppression inthe strongly interacting limit.So far we have assumed a fixed set of populated modes, (cid:126) ν . At finite temperature, expectation values need to be calcu-lated by averaging over all possible combinations of modes { (cid:126) ν } weighted according to their Boltzmann factor. How-ever, since the quantities ∆Ω( (cid:126) ν ) , ¯Ω( (cid:126) ν ) , and ¯ U ( (cid:126) ν ) are sharplypeaked around their thermal averages, to a good approxima-tion, (cid:104) N g (cid:105) T and thus (cid:104) δω eg (cid:105) T can be calculated by replac-ing ∆Ω( (cid:126) ν ) → (cid:104) ∆Ω (cid:105) T , ¯Ω( (cid:126) ν ) → (cid:104) ¯Ω (cid:105) T , and ¯ U ( (cid:126) ν ) → (cid:104) ¯ U (cid:105) T .Here (cid:104)O(cid:105) T = P (cid:126) ν O ( (cid:126) ν ) e − E ( (cid:126) ν ) / ( kBT ) P (cid:126) ν e − E ( (cid:126) ν ) / ( kBT ) . The validity of this ap-proximation is demonstrated in Fig. 1, which also shows thatEq. (5) is in fair agreement with Eq. (3).Experimentally it is hard to measure CFSs close to resonantpopulation transfer due to the small signal-to-noise ratio, andinstead the probe laser is generally locked at a finite detuning[3]. Away from δ (0)1 = 0 , we have to consider the more gen-eral expression given by Eq. (4), and an intuitive interpretationis not straightforward. The interaction induced asymmetry inthe lineshape not only can change the sign of the frequencyshift, but, in general, makes CFS a very sensitive function of (cid:104)A(cid:105) T , δ (0)1 , and (cid:104) N ¯ U (cid:105) T . In particular, the solid lines in Fig. 2 (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:45) (cid:45) (cid:64) N g (cid:72) t f (cid:76)(cid:144) N (cid:68) Sh i ft (cid:64) ∆ Ω e g (cid:144) Π (cid:68) (cid:72) H z (cid:76) (cid:65) (cid:144) Π(cid:61) (cid:65) (cid:144)
Π(cid:61)
FIG. 2: (color online) CFS for different pulse areas, A . The finalground state fraction, N g ( t f ) /N , was varied by changing the de-tuning. Here four spatial modes at the corners of a square in the ν x - ν y plane, ν = (30 , , ν = (101 , , ν = (30 , , and ν = (101 , , were assumed with N = 2 atoms occupying ν and ν at t = 0 . The solid blue (red) lines, for ¯ UN/ ¯Ω = 1 (3 . and t f = π/ Ω = 7 ms, were obtained using Eq. (2), which allows topopulate ν and ν . The empty squares (dots) were obtained using ˆ H S with an effective Ω eff = Ω / and t efff = π/ Ω eff . show the CFS as a function of the ground state fraction at t f ,computed using Eq. (2) for two atoms and four spatial modes.For weak repulsive interactions < (cid:104) N ¯ U (cid:105) T (cid:46) (cid:104) ¯Ω (cid:105) T (bluelines) and pulse areas greater than π , δω eg > for any detun-ing and approaches zero only as δ (0)1 → and A → π (con-sistently with remark b above). For A < π , δω eg changes signas a function of N g ( t f ) , and the zero crossing point moves to-wards smaller N g ( t f ) with decreasing pulse area. For strongerinteractions (cid:104) N ¯ U (cid:105) T > (cid:104) ¯Ω (cid:105) T > (red lines), while at large N g ( t f ) the sign of δω eg also depends on whether (cid:104)A(cid:105) T islarger or smaller than π , at small N g ( t f ) the magnitude ofthe CFS becomes less sensitive to pulse area variations andrecovers the expected negative sign for repulsive interactions(since at t = 0 all atoms are in e ). All these conclusions areconsistent with the measurements reported in Ref. [3] since ∼ variations in A over the course of a day could not beexcluded.We now discuss the effect of the terms whose omission inthe derivation of Eq. (3) was not justified: the terms exchang-ing modes along one direction only. As shown in Fig. 2 for thecase of two atoms occupying four modes, the spin model ˆ H S (dots) reproduces the result of Eq. (2) (lines) very well pro-vided that we reduce Ω in ˆ H S by a factor of 2: Ω eff = Ω / .This approximate result can be derived analytically by notingthat the ν - ν singlet ( |−(cid:105) ) from the spin model is replacedin Eq. (2) by the symmetric linear combination of the ν - ν and the ν - ν singlets. We have checked that for N = 3 and with N modes, ˆ H S with Ω eff = α Ω ( / < α < ) alsoreproduces well the results of Eq. (2).Conjecturing the validity of ˆ H S with Ω eff at larger N aswell, we used ˆ H S to calculate the CFS for the parameters ofRef. [3]. The results are shown in Fig. 3. Our model givesreasonable agreement within the experimental data error barsfor a range of scattering lengths a − eg = (100 − a with (cid:232) (cid:232)(cid:232) (cid:232)(cid:232) (cid:232)(cid:232) (cid:242)(cid:242) (cid:242) (cid:242) (cid:242)(cid:242) (cid:242) (cid:64) N g (cid:72) t f (cid:76)(cid:144) N (cid:68) S h i f t (cid:64) ∆ Ω e g (cid:144) Π (cid:68) (cid:72) H z (cid:76) (cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232)(cid:232) (cid:232)(cid:232) (cid:232) (cid:232) (cid:232)(cid:232)(cid:232)(cid:232) (cid:232)(cid:232)(cid:232) (cid:232)(cid:232) (cid:232) (cid:232) (cid:232) E x c it . fr ac ti on t (cid:72) ms (cid:76) FIG. 3: (color online) CFS predicted by our spin model for T =1(3) µ K and (cid:104) ∆Ω (cid:105) T / (cid:104) ¯Ω (cid:105) T = 0 . ( . ), determined from theRabi flopping curves [see inset]. The blue (pink) shaded area showsthe uncertainty in CFS for T = 1(3) µ K, assuming a variation inpulse area of (cid:104)A(cid:105) T /π = 1 ± . . We used Ω eff = 4 π/ , t f = π/ Ω eff , N = 15 [i.e. ρ ∼ cm − ], and a − eg = 200 a .The circles (triangles) show the T = 1(3) µ K experimental data ofRef. [3] at t f = 80 ms and ρ ∼ cm − . Inset: The shaded areashows Rabi flopping curves calculated at (cid:104) ∆Ω (cid:105) T / (cid:104) ¯Ω (cid:105) T = 0 . for a − eg = 0 − a . The dashed black and solid blue lines correspondto two specific scattering lengths, a − eg = 0 and a , respectively.The dotted purple line is for (cid:104) ∆Ω (cid:105) T / (cid:104) ¯Ω (cid:105) T = 0 . and a − eg = 0 ,while the blue circles are experimental data points from Ref. [3]. Ω eff = 4Ω . The net effect of Ω eff is to rescale the shiftby a factor of four [28], which is justified by the exchanging-modes corrections discussed above and by the large (up toa factor of 5 [24]) uncertainty in the experimental determi-nation of the density. We also note that in Ref. [3] (cid:104) ∆Ω (cid:105) T was inferred by fitting Rabi oscillations with a non-interactingmodel. However, the inset of Fig. 3 shows that interactionsmodify Rabi oscillations and that the non-interacting modelcan underestimate (cid:104) ∆Ω (cid:105) T by a factor as large as three. Allthese issues combined with the experimental uncertainty inpulse area complicate the determination of the magnitude of a − eg from the current data. Nevertheless, our model at leastsuggests it to be positive, a − eg > .In addition to a − eg , there is another scattering length a + eg ,which characterizes collisions between a g and an e atom. a + eg collisions require a symmetric electronic state and con-sequently an anti-symmetric nuclear spin configuration. Thusthe interaction Hamiltonian describing g - e collisions beyondthe polarized regime is [14, 15] ˆ H int = u seg (cid:90) d r ˆ ρ e ˆ ρ g + u aeg (cid:88) mm (cid:48) (cid:90) d r ˆΨ † gm ˆΨ † em (cid:48) ˆΨ gm (cid:48) ˆΨ em . (6)Here u s,aeg = ( u + eg ± u − eg ) / , s ( a ) stands for symmetric (anti-symmetric), m, m (cid:48) = − I, . . . , I label the nuclear Zeeman lev-els, u + eg = 4 π (cid:126) a + eg /M , and ˆ ρ α ( r ) = (cid:80) m ˆΨ † αm ( r ) ˆΨ αm ( r ) .We now propose how to estimate a + eg experimentally. Themethod is identical to that of Ref. [3] except that probe lightpolarization should be circular instead of linear. The idea isto Rabi interrogate an ensemble of | e, m (cid:105) atoms with a cir- cularly polarized probe driving the | e, m (cid:105) − | g, m + 1 (cid:105) tran-sition. If a + eg = a − eg , then u aeg = 0 and the dynamics of thesystem are identical to the ones described above, except thatone needs to identify {| e, m (cid:105) , | g, m + 1 (cid:105)} as the spin basis.However, if a + eg (cid:54) = a − eg , the term proportional to u aeg will pop-ulate | e, m + 1 (cid:105) and | g, m (cid:105) . This issue can be overcome byapplying an external magnetic field. If the applied magneticfield satisfies Bµ N ∆ g (cid:29) (cid:104) ¯ U a (cid:105) T , with µ N the nuclear mag-neton, ∆ g the differential g -factor between e and g [25, 26],and (cid:104) ¯ U a (cid:105) T the thermally averaged antisymmetric interaction,then the processes populating | e, m + 1 (cid:105) and | g, m (cid:105) will beenergetically suppressed, and the dynamics will be identicalto the ones in Ref. [3] with CFS proportional to u seg . By com-paring the CFS between the linearly and circularly polarizedcases, one can in principle infer a + eg .In summary, fermionic clocks are sensitive to the CFS in-duced by excitation inhomogeneities. The CFS is sensitiveto pulse area, detuning, and interaction strengths, but if mea-sured close to resonant transfer, it is in qualitative agreementwith the expected mean field expression. To improve the ex-perimental resolution of the CFS, better control over pulsearea variations is required. Rabi interrogation schemes canalso be used to estimate a + eg . Note added in proof:
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