Theory of Feshbach molecule formation in a dilute gas during a magnetic field ramp

Starting with coupled atom-molecule Boltzmann equations, we develop a simplified model to understand molecule formation observed in recent experiments. Our theory predicts several key features: (1) the effective adiabatic rate constant is proportional to density; (2) in an adiabatic ramp, the dependence of molecular fraction on magnetic field resembles an error function whose width and centroid are related to the temperature; (3) the molecular production efficiency is a universal function of the initial phase space density, the specific form of which we derive for a classical gas. Our predictions show qualitative agreement with the data from [Hodby et al, Phys. Rev. Lett. {\bf{94}}, 120402 (2005)] without the use of adjustable parameters.

By ramping a magnetic field across a Feshbach resonance, loosely bound diatomic molecules have been created in two-component Fermi gases above [1,2] and below [3,4,5] the superfluid transition temperature. In this process, there appear to be two distinct relaxation timescales: a "two-body" timescale in the range of 10's to 100's of microseconds during which molecules form, and a longer "many-body" timescale in the range of 10's of milliseconds during which a condensate forms [4,6]. A critical unresolved riddle emerging from these experiments is that the short-time molecular formation physics can not be fully accounted for by theories that treat the Feshbach ramp as a purely two-body process [7,8], suggesting a many-body treatment is warranted [9,10,11]. In this letter, we argue that in the above-cited experiments, multiple collisions per atom occur during the ramp. Thus, Feshbach molecule formation in a thermal gas must be viewed as a collisional relaxation process, which we describe using kinetic theory [11,12] and thermodynamics [13,14,15].
Several key experimental quantities in need of a comprehensive theoretical understanding are illustrated qualitatively in Fig. 1. As the magnetic field is tuned across the resonance, the molecular fraction χ ≡ 2N M /N tot increases to a maximum value χḂ, where N tot and N M are the total and molecular populations. This behavior is represented in Fig. 1a by the error function χ erf (B) = χḂ{1 − erf[ √ 2(B − B cen )/δB]}/2, which has been used to fit experimental data [1,4]. Fig. 1b shows how the production of molecules increases and then saturates to a value less than unity as the ramp rate is decreased. An exponential function χḂ = χ 0 [1 − exp(−α/|Ḃ|)] gives a reasonable fit to data [1,2,4], however, a power-law form can not be ruled out [16]. The rate constant α determines whether the sweep is adiabatic and χ 0 is the maximum production efficiency obtained in the adiabatic limit α/|Ḃ| ≫ 1. In this letter, we identify the root cause of the saturation effect and determine the scaling behavior of the measured properties B cen , δB, α and χ 0 .
A useful idealization that is commonly invoked to understand Feshbach molecule formation is to scale the system size down to two isolated trapped atoms initially in their ground state. This description is amenable to a straight-forward application of the Landau-Zener (LZ) level crossing theory [7], for which the effect of the magnetic field ramp is easy to conceptualize: the state of the atom pair tracks the lowest energy state during the ramp, eventually crossing threshold to become a true two-body bound state. In the adiabatic limit, the LZ model predicts that at the end of the sweep the probability of finding the atom pair in a bound molecular state is unity, corresponding to a production efficiency χ 0 = 1. For finite temperature gases, this two-atom LZ model is not directly applicable, and thus one does not necessarily expect that a unit conversion efficiency can be achieved. Nevertheless, is there a way to incorporate the twoatom LZ theory into a model that takes into account the presence of many atoms in a trapped gas? The key to answering this is to realize that in a dilute gas there is an upper bound on the ramp time t ramp set by the average time between collisions τ col : if t ramp ≫ τ col , an individual atom may undergo multiple collisions during the ramp, and this physics is not taken into account by the LZ approach. In the opposite limit t ramp ≪ τ col , each atom will likely encounter at most a single collision partnerits nearest neighbor, so that the LZ theory can be applied to isolated pairs of atoms distributed throughout the gas. Models working in this type of "static gas" approximation (SGA) predict a maximum production efficiency of χ 0 = 0.5 [8,10]. Although this efficiency limit is consistent with the first two Feshbach sweep experiments in a Fermi gas [1,2], the more extensive recent study of Hodby et al. [17] measured values of χ 0 in the range 0.1 − 0.9, in disagreement with the SGA prediction.
We can understand this apparent breakdown of the SGA by estimating τ col for typical experimental parameters using τ −1 col = nσv rel , where σ is the collisional cross section andv rel is the average relative velocity. In the unitarity limit, σ = 4πa 2 /(1 + k 2 a 2 ), which can be approximated by σ = 4π/k 2 F close to the resonance where the scattering length a diverges. Here k F is the Fermi wave vector. Estimating the density as n ≈ (2mk B T F /h 2 ) 3/2 /6π 2 , where m is the atomic mass, and takingv rel = 4 k B T F /πm, the collisional time becomes τ col ≈h/k B T F [6]. In the JILA experiments [17], typical Fermi temperatures are T F = 350 nK, giving τ col = 20 µs. With t ramp > ∼ 50 µs in these experiments, multiple collisions per atom occur during the ramp and an alternative to the SGA is necessary.
To go beyond the SGA, Hodby et al. devised a stochastic phase space sampling (SPSS) model based on the assumption that the probability of two atoms forming a molecule depends on their proximity in phase space [17]. With a single fitting parameter -a cut-off radius in phase space -the phenomenological SPSS model predicts that that χ 0 is a universal function of ρ 0i that agrees remarkably well with their experimental data. In this letter, we provide a theoretical backdrop for understanding this behavior and derive from first principles the universal functional form for χ 0 (ρ 0i ).
Rather than try to adapt the two-atom LZ theory to the case of a finite-temperature gas, we have developed an entirely different many-body approach that is appropriate for the regime t ramp ≫ τ col . Using the Keldysh nonequilibrium Green's function formalism [18], we derived a pair of coupled Boltzmann equations describing the dynamical evolution of the atomic f A (p, r, t) and molecular f M (p, r, t) distribution functions [12]; we assume an equal spin mixture f A ≡ f ↑ = f ↓ . Working within the quasi-particle approximation, we then developed idealized models to study the dynamics [11] and thermodynamics [15] of molecule formation. In this letter, we describe the salient features of this approach and apply our theory to the experiments of Hodby et al. [17].
In [11], we used our kinetic theory to derive rate equations for the atomic (A) and molecular (M ) populations is the density-weighted average rate of molecule formation. The density is defined by n j (r, t) = dpf j (p, r, t)/h 3 . The rates of molecule formation γ f (r, t) and dissociation γ d (t) follow directly from the "in" and "out" terms of the atom-molecule collision integrals in the Boltzmann equations, which describe the relaxation toward chemical equilibrium due to resonant two-body collisions. Within the classical gas approximation (i.e. f i ≪ 1), they are γ d (t) = 4π 2 g 2 h 7 n M (r, t) dp 1 dp 2 dp 3 δ( γ f (r, t) = 4π 2 g 2 h 7 n A (r, t) dp 1 dp 2 dp 3 δ( where the explicit position and time dependence has been suppressed in the integrals. Neglecting self-energy effects, the atom and molecule energies are ǫ A (p, r) ≡ p 2 /2m + U ext (r) and ǫ M (p, r, t) ≡ p 2 /4m + 2U ext (r) +  (2) and (3) can be calculated explicitly when the system is in equilibrium The molecular dissociation rate then simplifies to the position-independent form Here, the unit step function θ(ǫ res ) follows from energy and momentum conservation and signifies that below threshold ǫ res < 0, the pairs become stable and no longer can decay. We note that (4) is consistent with the decay rate derived in Ref. [19]. The formation rate reduces to γ f (r) = 2 3/2 ρ(r)e −ǫres/kBT γ d , where ρ(r) ≡ λ 3 th n A (r) is the phase space density of the atomic component and λ th = h/ √ 2πmk B T is the thermal deBroglie wavelength. The density-weighted average is γ f = γ f (0)/2 3/2 . The formation rate vanishes as ǫ res is tuned below threshold, in the same way that γ d does. This causes the saturation of molecule production during a Feshbach sweep: once ǫ res is tuned below threshold, no more molecules are formed via two-body collisions.
Despite its simple intuitive form, Eq. (1) cannot be solved without knowing the f j (p, r, t) of Eq. (2) where n A0 ≡ n A (r = 0) is the peak atomic density. This result (6) resembles the Landau-Zener rate constant derived for a zero temperature Bose gas in [21]. The experimental data in [17] seem to be consistent with this linear dependence on density.
In the adiabatic regime α/|Ḃ| ≫ 1, the system follows an isentropic path as ǫ res is ramped downward [11]. When the resonance energy ǫ res [B(t)] is ramped to negative values, the two-body rates of molecule formation γ f and dissociation γ d vanish, at which point the process of molecule formation must proceed via three-body collisional relaxation, which may play a role for very slow ramps [22]. When the timescale for three-body relaxation is much longer than t ramp , it can be neglected [23,24]. Thus, molecule formation effectively halts when ǫ res crosses threshold at B = B 0 , giving rise to the observed saturation of molecule production.
In [15] we calculated constant entropy contours for a quantum ideal gas mixture of fermionic atoms and bosonic molecules without accounting for this saturation effect. It is straightforward to incorporate this effect into our earlier thermodynamic calculation. For positive detunings where ǫ res > 0, the results are unchanged: the molecular fraction χ(ǫ res ) is determined according to the conservation of entropy and total atom number with the constraint that the atomic and molecular components are in thermal T M = T A ≡ T and chemical µ M = 2µ A ≡ 2µ equilibrium. For negative detunings, the molecular fraction is held fixed χ(ǫ res < 0) = χ(0) ≡ χ 0 , where χ 0 is the maximum production efficiency. The details of this type of calculation are given in [15].
In Fig. 2 we plot the molecular fraction along an isentropic path (solid blue lines) for the specific case of an initial temperature T i /T F = 0.5, taking values for 40 K: B 0 = 202 × 10 −4 T, ∆B = 7.4 × 10 −4 T, and a bg = 174a 0 [3]. The inset shows χ(ǫ res ) versus ǫ res compared to the case where the gas remains in chemical equilibrium for negative detunings (dashed blue line). To plot χ versus magnetic field, the small detuning limit was used for ǫ res (B). The resulting molecular formation curve χ(B) is fit remarkably well by the error function χ erf (B) (dashed red line). The fitted width and centroid for four different initial temperatures T /T F = {0.25, 0.5, 0.75, 1.0} are δB = {0.14, 0.19, 0.21, 0.24} × 10 −4 T and B cen − B 0 = {0.37, 0.34, 0.35, 0.36}×10 −4 T. These widths δB are consistent with the value 0.2 × 10 −4 T measured in experiments [1,4], which is much smaller than the Feshbach resonance itself (∆B = 7.4×10 −4 T). A crucial point arising from our analysis is that the centroid B cen of the error function does not coincide with the resonance position B 0 . The centroid occurs roughly where ǫ res (B) ≈ k B T F , so that B cen − B 0 ≈ ∆Ba bg √ mk B T F /h. Our saturation mechanism is quite general and occurs in both Bose and Fermi gases; examples of both systems have been studied by Hodby et al. [17]. Within the classical gas approximation it is straightforward to relate χ 0 to the initial peak atomic phase space density ρ 0i . The maximum molecular fraction is given by χ 0 ≡ 2N M (T f , ρ 0f , ǫ res = 0)/N tot , where T f and ρ 0f are the temperature and peak atomic phase space density at ǫ res = 0. The total atomic population is N tot = ηN A + 2N M , where for generality we allow for  7), for η = 2. The solid black circles are experimental data for 40 K from Hodby et al. [17]. The two earliest experiments are those of JILA (red square) [1] and Rice (red diamond) [2]. The inset compares the η = 1 case of Eq. (7) (dashed green line) to data for 85 Rb taken in the classical regime of low phase space density [17].
the number of atomic spin components to be η ∈ {1, 2}. The atomic and molecular populations can be written as N A (T, ρ 0 ) = (k B T /hω) 3 ρ 0 and N M (T, ρ 0 , ǫ res ) = (k B T /hω) 3 ρ 2 0 exp(−ǫ res /k B T ), where the mean trapping frequency isω = (ω x ω y ω z ) 1/3 . Using these expressions, the molecular fraction takes the form χ 0 = ρ 0f /(η/2 + ρ 0f ). We can relate ρ 0f to ρ 0i using entropy conservation. The atomic and molecular entropies can be written as S A (T, ρ 0 ) = k B N A (4 − ln ρ 0 ) and S M (T, ρ 0 , ǫ res ) = k B N M (4 + ǫ res /k B T − 2 ln ρ 0 ). Substituting these into the entropy conservation equation ηS A (T i , ρ 0i ) = ηS A (T f , ρ 0f )+S M (T f , ρ 0f , ǫ res = 0) yields the relation ln ρ A0i = ln ρ A0f + 2χ 0 . From these observations, the following key result can be derived The solution χ 0 of this transcendental equation is a universal function of the the initial peak atomic phase space density ρ 0i and the number of spin components η. A similar derivation for an ideal quantum gas results in a much more complicated expression, in which χ 0 also depends on the quantum statistics of the different components. In Fig. 3 we compare our theoretical prediction of χ 0 , both with (solid blue) and without (dashed green) quantum statistical effects, to experimental data and find qualitative agreement with the JILA data [1,17] but not with the Rice data point (red diamond) [2].
In summary, we have treated Feshbach molecule formation in a dilute gas during a magnetic field ramp as a relaxation process described by kinetic theory [11,12] and thermodynamics [15]. Such a treatment is appropriate to the conditions of present experiments [1,2,17].
Our key findings are: (i) the rate factor α is proportional to density, (ii) the width δB and centroid B cen of the formation curve are related to the temperature T of the gas, and (iii) the molecule production efficiency χ 0 is a universal function of the initial peak atomic phase space density ρ 0i , which in the classical regime is approximated by the ideal gas result Eq. (7). Our predictions are consistent with most of the existing experimental data.
We thank E. Hodby and C. Regal for providing the experimental data used in Fig. 3. We appreciate useful discussions with T. Nikuni, P. Julienne and E. Tiesinga.