Rubidium II Continuous loading of a non-dissipative trap Christian Roos, Post Doc Philippe Cren, PhD Student David Guéry-Odelin and Jean Dalibard Innsbruck
The main ideas We imagine a beam of particles entering in a potential well. For a collisionless beam: all incoming particles escapes with their incoming energy. For an interacting beam of particles, some particles may be trapped because of sharing of total energy in elastic collisions processes. One first qualitative requirement is that the mean free path is smaller than the size of the well.
A toy model (1) potential well = isotropic and harmonic (w ) Atomic beam with flux Fin and a mean energy U(1+e). Phase space density of the incoming gas
The steady state solution (assumptions) We focus on the steady state solution The steady state energy distribution is assumed to be a truncated Boltzmann distribution: P(E) ~ r (E) e-E/T q (U-E) We assume that atoms are evaporated as soon as their total (kinetic + potential) energy after a collision exceeds the trap depthU (Knudsen assumption ). In other words, the density is not high enough to allow for multiple collisions, or the mean free path is larger than the size of the trapped cloud.
The steady state set of equations To determine T and N, we equalize the incoming and outcoming flux of particles and energy. Fin = Fevap + Fnevap = f Fin + (1-f) Fin  f : fraction of incoming atoms which collide with a trapped atom, Knudsen assumption f < 1. UinFin = UevapFevap + UinFnevap   and  leads to Uin = U(1+e) = Uevap 
The expression for Uevap Uevap has been calculated by Luiten et al. to describe the kinetic of evaporative cooling: his the trap depth over the effective temperature of the cloud. Uevap = U+k(h) T where h = U/T k(h) Evaporated particles have a mean extra energy less thanT.
k(h) k(h) e e = = h h Energy equation Uin = U(1+e) = Uevap = U+k(h) T Uin = U(1+e) = Uevap = U+k(h) T   k(h) k(h) h h h0 solution leads to the steady state temperature T0=U/h0 Threshold condition on the energy of the incoming particle for steady state solution.
Fin N = e e1/e w Number of particles equation f Fin = pgN Fin: incoming flux g : collision rate p: probability that an atom is evaporated after a collision. f: fraction of atoms which collide with trapped atoms p ~ 2h e-h si h >> 1 low collision regime g << w Boltzmann factor The number of atoms can be in principle very large Nmax ~ e U / m w2s Knudsen assumption f < 1
Fin N = e e1/e w Conclusion of the toy model Threshold condition on the energy of the incoming particle for steady state solution. T0=U/h0 can lead to a significant increase of the phase space density
A more elaborated model Anisotropic trap: wz << wT Evaporation can be either longitudinal or transverse. Longitudinal evaporation is a consequence of loading mechanism wz << wT permits to reach HD longitudinal regime g >> wz, all incident atoms undergo collisions with the trapped atoms.
Longitudinal evaporation HD regime: atoms that emerge after a collision with Ez > Uz, undergo other collisions which can bring Ez < Uz. longitudinal evaporation rate is reduced by comparaison with the « collision less» regime. We have calculated by means of a molecular dynamics simulation pz : probability that an atom is longitudinally evaporated Uz+kzT : the average energy carried away by this atom for ; Reducing factor
Transverse evaporation is now a crucial parameter If then With the molecular dynamics simulation, we obtain from the Knudsen regime to the HD one in the range
Steady state Knowing we obtain the equation for the number of atoms and for the energy
An example (1) Typical parameters:87Rb wz = 2p x 10 Hz v = 10 cm/s wx,y = 2p x 1000 Hz Dv = 2 cm/s F = 107 atoms/s Uz ~ 30 mK Average excess of energy eUz ~ 30 mK i.e. e ~ 1
An example (2) Uz ~ 30 mK Average excess of energy eUz ~ 30 mK x 1000 A resonance occurs when the transverse evaporation threshold coincides with the energy of the incident particles Note that the steady state temperature is much lower than eUz, which has to be contrasted with the toy model result.
Around the resonance (1) The resonance is observed when When then and linear variation when longitudinal evaporation can no more be neglected
Around the resonance (2) When the transverse evaporation becomes more and more inefficient This result is reminiscent of the results of the toy model.
Optimum wz = 2p x 10 Hz v = 20 cm/s wx,y = 2p x 1000 Hz Dv = 4 cm/s Uz ~ eUz ~ 130 mK The gain in phase space density saturates when the transverse motion enters the hydrodynamic regime
Numerical simulation: molecular dynamics Flux: 105 atoms/s Phase space density: 0.01 Velocity: 5 cm/s wz = 2p x 10 Hz Uz= 11 mK (4.5 cm/s) e = 0.36 wx,y = 2p x 1000 Hz = 1.5 Uz kT/U N (x 106) 2.4 0.3 1.6 0.2 0.8 0.1 0.0 0 0 50 100 50 100 t=100 seconds: N=2.4 106 atoms @ T=1.4 mKdegenerate gas !
The kinetic aspect The set of kinetic equation is Hint of the behavior: linearizing around the steady state solution
Time constants We find two times: where Dynamics two time scales: The fast one The slow one
Why two time constants ? where depends on the kind of evaporation The steady state is given by and has some exponential behavior, thus
CONCLUSION Continuous loading of a non-dissipative trap can be done with the help of evaporation This process can be very efficient for anisotropic geometry In this case, incoming particle can have an energy eUz of the order of the energy barrier Uz and reach a temperature T << Uz. A gain of phase space density of several orders of magnitude is possible for realistic configuration. N.B. We have neglected losses processes background gas, and inelastic collisions, ... To appear in Europhysics Letters