Dynamics of Trapped Fermion Clouds

Laser Physics, Vol. 14, No. 2, 2004, pp. 302 306. Original Text Copyright 2004 by Astro, Ltd. Copyright 2004 by MAIK Nauka /Interperiodica (Russia). PHYSICS OF COLD TRAPPED ATOMS Dynamics of Trapped Fermion Clouds F. Toschi 1, 2, P. Vignolo 3, P. Capuzzi 3, S. Succi 1, 3, and M. P. Tosi 3 1 Istituto per le Applicazioni del Calcolo, CNR, Viale del Policlinico 137, Roma, 00161 Italy 2 INFM, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma, 00133 Italy 3 NEST-INFM and Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, 56126 Italy e-mail: vignolo@sns.it Received July 31, 2003 Abstract Experiments at JILA on the collisions between two oscillating spin-polarized components of a Fermi gas of 40 K atoms have shown that this setup is an important tool for studying the dynamics of quantum gases. We have developed advanced numerical methods to solve the Vlasov Landau equations for a binary mixture of atomic Fermi gases inside a harmonic trap at very low temperatures. In particular, a locally adaptive importance-sampling technique allows us to handle collisional interactions several orders of magnitude more efficiently than in standard Monte Carlo calculations. These methods have been applied to study the dynamical and transport properties of a fermion mixture in the presence of mean-field interactions and of collisions between its components for system parameters ranging from the collisionless to the hydrodynamic regime. 1. INTRODUCTION Since the achievement of Bose Einstein condensation [1], a main goal of experimentalists has been to cool a gas of fermionic atoms down to an expected superfluid regime. The cooling process is an evaporative process where at each step the most energetic atoms are expelled from the trap and the others rethermalize through collisions. However, at very low temperatures, the main collisional events should occur in the s-wave channel but are inhibited by the Pauli principle in a spin-polarized Fermi gas. A way to circumvent this problem is to add a second fermionic component, which is polarized in another isospin state inside the magnetic trap. Experiments at JILA [2] have studied the collisions between two oscillating spin-polarized components of a Fermi gas of 40 K atoms and have given evidence for a transition from collisionless (zerosound) to hydrodynamic (first-sound) behavior through measuring the damping time τ. This becomes very long at both small and large values of the estimated collision rate. Following earlier numerical studies of the dynamics of various types of atomic clouds [3 5], Geist and Kennedy [6] described the evaporative cooling process in a two-component Fermi gas from semiclassical Boltzmann equations as functions only of the energy. A more comprehensive numerical study of the dynamics of such a system in coordinate and momentum space has been realized by Toschi et al. [7] by solving the Vlasov Landau equations (VLE) for the fermionic Wigner distributions. We have examined the collisional properties of the two-component Fermi gas as functions of both temperature T and rate Γ q of quantum collisions, which are counted step by step in the simulation. In this paper, we present in detail the strategy used in the numerical solution of the VLE and report some illustrative results. We treat the quantum fluid using a particle-dynamics approach [8] accounting for both mean-field interactions and collisions between the two components. Upon inclusion of collisional events at very low temperature, the calculation would grind to a halt, basically through a saturation of phase space resulting in a vanishing efficiency of the Monte Carlo sampling. We have developed a locally adaptive importance-sampling technique, which allows us to handle collisional interactions several orders of magnitude faster than in standard Monte Carlo techniques. Moreover, since we focus on the JILA setup [2], where axial symmetry is maintained during the experiment, we can use a code in which the angular degree of freedom is taken into account via an effective weight. We find that, even though most collisions become forbidden both classically and by the Pauli principle as temperature is lowered, the few collisions that can occur involve particles increasingly clustered around the Fermi level and suffice to drive the gas from the collisionless to the hydrodynamic regime. 2. THE MODEL AND ITS NUMERICAL SOLUTION The purpose of our numerical code is to solve the VLE for two interacting Fermi gases in the presence of collisions at finite temperature T down to T ~ 0.2T F, where T F is the Fermi degeneracy temperature. 2.1. VLE The two fermionic components in external potentials (r) are described by distribution functions V ext 302

DYNAMICS OF TRAPPED FERMION CLOUDS 303 f (j) (r, p, t) with j = 1 or 2. These obey the kinetic equations t f + p m --- r f r U p f = C 12 [ f ], (1) where the Hartree Fock (HF) effective potential is U (j) (r, t) V ext (r) + gn (r, t), with j denoting the species different from j. Here, we have set = 1, g = 2πa/m r, with a being the s-wave scattering length between two atoms and m r being their reduced mass, and n (j) (r, t) is the spatial density given by integration of f (j) (r, p, t) over momentum degrees of freedom. Collisions between atoms of the same spin can be neglected at low temperature, so that in Eq. (1) the term C 12 involves only collisions between particles with different polarizations: C 12 [ f ] = 22π ( ) 4 g 2 /V 3 Δ p Δ ε f ( j f 2 f ) 3 f 4 f ( j f ) [ 2 f 3 f 4 ], p 2, p 3, p 4 f with f (j) f (j) (r, p, t), 1 f (j), f (j) (r, p i, t), and 1. V is the volume occupied by the gas, and the factors Δ p and Δ ε are the usual delta functions accounting for conservation of momentum and energy, with the energies given by p 2 + U (j) i /2m j (r, t). 2.2. Numerical Solution The numerical procedure by which the VLE are solved in time consists of three basic steps: (i) initialization of the fermionic distributions, (ii) propagation in phase space driven by external and mean-field forces, and (iii) propagation in phase space through collisional events. 2.2.1. Initializing the VLE. The initial phase-space distributions at equilibrium inside harmonic traps are generated from the HF expression f eq ( rp, ) = β -------- p2 + U ( r) μ 1 exp + 1 2m j (2) where β = 1/k B T and μ (j) is the chemical potential of species j [9]. The particle densities entering U (j) (r) are determined self-consistently together with the chemical potentials. This is done by means of a standard rootfinding subroutine. We exploit the axial symmetry of the system by introducing the angularly integrated particle densities (j) (r, z) = 2πrn (j) (r, z) defined on a {r, z} grid in cylindrical coordinates. To evaluate them numerically, we, move to a particle-in-cell description by locating a number (j) (r, z)δrδz of fermions inside each cell of volume ΔrΔz [5]. Low statistical noise is achieved by representing each fermion by means of a number N q of computational particles ( quarks ). To generate the initial momentum distribution, the ith quark is located at point {p ir, p iθ, p iz } in momentum space by using a Monte Carlo sampling and making sure that each cell of volume h 3 is occupied by no more than N q quarks. This control in 3D phase space is transferred to 2D by imposing a maximum number wn q of quarks in the 2D cell {Δr, Δz, Δp r, Δp z } of volume h 2. Here, the weight w = int(2p F r/ ) takes into account that the number of available cells in 2D depends on the radial position and on the number of particles through the Fermi momentum p F = 2mk B T F. The use of N q quarks to represent a single atom is the analog of the splitting techniques widely used in kinetic Monte Carlo studies [10]. In our work on quantum gases, it has a further beneficial effect on the acceptance rate of the Monte Carlo procedure used to implement the Pauli exclusion principle. 2.2.2. Propagating the VLE. Taking our first experiment as a specific example, in the propagation step, the two clouds are rigidly displaced from the center of their traps along the z direction and start evolving in time in the {r, z} plane by performing oscillations at their respective frequencies. The VLE are advanced in time by a standard particle-in-cell (PIC) method [11] in a modified Verlet timemarching scheme on a grid with mesh spacing dx > vdt, where v is a typical particle velocity and dt is the time step. This inequality is dictated by the accuracy and stability of the propagation step [5]. We obtain the following set of discrete algebraic equations: r i ( t + dt) = r i () t + v ri ()dt t + z i ( t + dt) = z i () t + v zi ()dt t + v ri ( t + dt) = v ri () t + a ri ()dt t v zi ( t + dt) = v zi () t + a zi ()dt t a ri ()dt t 2 /2 a zi ()dt t 2 /2 (3) for i = 1 to N. Here, a r, a z and v r, v z are the two components of the acceleration and of the velocity along the {r, z} coordinates. The algorithm is standard except for (i) the decoupling of the variable p θ from the equations of motion, exploiting the fact that, in terms of symmetry, the average value of p θ at each value of r is zero and does not change in time, and (ii) the specification of the self-consistent coupling arising from the forces due to the density gradients. The grid-forces which cause the accelerations are evaluated at the discrete particle locations by means of a bilinear cloud-in-cell interpolator [5]. With the force/acceleration fields transferred to the par-

304 TOSCHI et al. ticle locations, only the collisions need setting in order to march the VLE in time. 2.2.3. The Collision step. Collisions are tracked on a grid of mesh spacing λ B (the de Broglie wavelength) both along r and z. λ B is smaller than the particle mean free path l and larger than the spacing dl of the propagation mesh (l > λ B > dl). The first inequality enhances the statistical accuracy of the collision step, whereas coarse graining with respect to the propagation step avoids the need for Pauli-principle constraints in the Lagrangian evolution, at least down to 0.2T F. In this step, again we exploit axial symmetry: we make the approximation that the angular momentum of each quark is left unchanged by collisional events and take into account the third dimension in the exclusion principle by suppressing collisions whose final states are occupied by wn q quarks. The number of classically probable collisions between all possible pairs of quarks belonging to the two species in each cell of volume dv = 2πrdrdz is evaluated at each computational step as dn coll dt = ------ v dv ij σ ij. (4) Here, v ij is the magnitude of the relative speed of quark i of species 1 and quark j of species 2 and σ ij is the corresponding cross section σ ij = σ/n q, with σ being the atom atom cross section σ = πa 2. If dn coll < 1, the collision probability is accumulated over the subsequent time steps until an integer number di coll int(dn coll ) of collisions occurs. The collision probability becomes smaller than the classical one after multiplication by the quantum suppression factor 1 N(r, z, p r, p z )/(wn q ) due to the occupancy of the final state, and the efficiency of the Monte Carlo sampling drops with decreasing temperature. To enhance the acceptance rate by two to three orders of magnitude at the lowest temperatures, we proceed as follows. First, within each spatial cell, the pairs of particles are ordered according to the value of their classical transition probability and pairs with classical probability below a given threshold are filtered out. The threshold is dynamically adjusted cell-by-cell in order to guarantee a correct supply of di coll collisions at each time step. Each particle is then allowed to collide only with the partner which maximizes the product v ij σ ij, so that the original pool of N 1 N 2 collisions is cut down to N 1 collisions. This strategy has allowed us to circumvent the numerical attrition problems due to Pauli blocking down to 0.2T F. 3. RESULTS AND CONCLUDING REMARKS As a first application of the numerical method, we have considered a system of 200 magnetically trapped ij 40 K atoms represented by a total number of 4 10 3 quarks [7]. As in the JILA experiments [2], the atoms are equally shared among two hyperfine states (m f = 9/2 and m f = 7/2) inside harmonic traps with somewhat different longitudinal frequencies (ω 9/2 = 2π 19.8 s 1 and ω 7/2 = 2π 17.46 s 1 ). The two clouds after initialization are rigidly displaced from the trap centers and in the absence onteractions would keep oscillating at their respective trap frequencies without damping. Mean-field interactions would only modify these bare frequencies. This picture is profoundly changed when collisions are allowed to occur as the two clouds meet in their trajectories. We vary the collision rate Γ q at each given temperature by changing the scattering length a, thus mimicking the exploitation of a Feshbach resonance [12]. As in the JILA experiments [2], with increasing Γ q there is an increasing transfer of relative momentum from one cloud to the other, up to the point where the relative motion of the two clouds is suppressed and they start oscillating together at the same frequency. This transition from collisionless-ballistic motions (for Knudsen number Kn ) to a collisional-hydrodynamic regime (for Kn 0), as driven by varying the scattering length at each temperature, is displayed both in the plot of the dipole-mode frequencies of the two clouds in Fig. 1a and in the plot of the damping rate γ = 1/τ of the relative motion of the centers of mass z cm (t) in Fig. 1b. It should be noticed that, since the number of atoms used in the simulation (N = 200) is much smaller than in the experiments (N exp 10 6 ), we have upscaled the scattering cross section σ by a factor of N exp /N ~ 10 4 in order to keep the same Knudsen numbers. In fact, the effects of statistical errors due to the limited number of particles in the simulation are best controlled by letting the simulated and physical system share the same Knudsen numbers, namely, the ratio of the scattering length to the macroscopic size of the experimental sample. We can argue that physical results are not affected by such rescaling as long as the diluteness condition, nσ 3/2 1, remains fulfilled. In Fig. 1a, the oscillation frequencies ω j of the two clouds at two values of T/T F, with k B T F = ω 9/2 (6N 9/2 ) 1/3, have been obtained by fitting z cm (t) with the functions cos(ω j t)e γt. At very low Γ q, the dipole-mode frequencies are given by the corresponding trap frequencies with a small shift due to mean-field interactions. At intermediate values of Γ q, the data points show large fluctuations due to the fact that in this region just a few collisions can drastically alter the motion of the clouds. The trend towards a locking of the two dipole modes at large Γ q is very clear, and the location of the locking is identified with reasonable accuracy.

DYNAMICS OF TRAPPED FERMION CLOUDS 305 ω, s 1 126 (a) γ, s 1 10 (b) 122 T/T F = 0.3 T/T F = 1.0 T/T F = 1.0 T/T F = 0.2 118 114 110 0 20 40 60 80 1 0 10 100 Fig. 1. (a) The oscillation frequencies ω (in units of s 1 ) and (b) the damping coefficient γ (in units of s 1 ) as functions of the quantum collision rate Γ q (in units of s 1 ) for the two components of the gas at various temperatures. The horizontal dashed lines in (a) show the bare trap frequencies. The damping rate γ is shown in Fig. 1b at two values of T/T F and has been obtained from the correlation functions φ (j) (t) = z cm ( t' ) z cm ( t' + t) between the magnitude of the turning points z cm, which decay exponentially as exp( γt) at long times. The rate is essentially the same for the two components and increases linearly with Γ q in the collisionless regime, 1 while in the collisional one it scales as Γ q. It is also seen from Fig. 1 that the locking transition shifts to lower Γ q as temperature is lowered (see Fig. 2). This effect is a consequence of Fermi statistics: at lower temperature, the collisions involve particles in a narrower region around the Fermi level and a smaller number of collisions is needed to produce locking of the two species. The transition to hydrodynamic behavior can thus be driven by purely thermal means, taking advantage of the increasing importance of a decreasing set of 100 80 60 40 20 0.2 0.3 0.4 0.5 0.6 0.8 1.0 T/T F Fig. 2. Log log plot of the collision rate Γ q at mode locking (in units of s 1 ) as a function of temperature T (in units of T F ). The fitting curve (dashed line) is (T/T F ) 0.40 ± 0.07. strategic collisions between particles in states clustered around the Fermi level. To quantify this effect, we have performed a fit to our data and obtained a power law behavior of the form Γ q (T/T F ) 0.40 ± 0.07. This is shown in log log scale in Fig. 2. The role of trap anisotropy in the transition from the collisionless to the hydrodynamic regime continues to be explored both experimentally and numerically. Moreover, the effects of collisions during expansion provide decisive information about the nature of the trapped state. In view of these facts, we are at present developing a fully 3D concurrent code to solve the Vlasov Landau equations. This code is allowing us to explore regimes with complete anisotropy of the trapping potential and thus to remove all symmetry constraints during, e.g., the collisional step. In addition, preliminary tests suggest that larger numbers of particles (N = 10 5 ) can be handled [13]. These further studies should allow us to investigate strategies by which one may circumvent Pauli blocking in reaching down to lower temperatures under general trapping situations. ACKNOWLEDGMENTS This work was supported by INFM through the PRA-Photonmatter Initiative. M.P.T. acknowledges the hospitality of the Abdus Salam International Center for Theoretical Physics during the preparation of the manuscript. REFERENCES 1. M. H. Anderson, J. R. Ensher, M. R. Matthews, et al., Science 269, 198 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, et al., Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995).

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