Rubidium IIRubidium II
Continuous loading of a non-dissipative trapContinuous loading of a non-dissipative trap
Christian Roos, Post Doc
Philippe Cren, PhD Student
David Guéry-Odelin and Jean Dalibard
Innsbruck
The main ideas
U
U
We imagine a beam of particles entering in a potential well.
For a collisionless beam:all incoming particles escapeswith their incoming energy.
For an interacting beam of particles, some particles may be trappedbecause 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)
U
U
potential well = isotropic and harmonic ()
Atomic beam with flux in and a mean energy U
Phase space density of the incoming gas
The steady state solution (assumptions)
The steady state energy distribution is assumedto be a truncated Boltzmann distribution:
P(E) ~ E e-E/T U-E
We focus on the steady state solution
We assume that atoms are evaporated as soon as their total(kinetic + potential) energy after a collision exceeds the trap depth U (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.
in = evap + nevap = f in + (1-f) in [1]
Uinin = Uevapevap + Uinnevap [2]
f : fraction of incoming atoms which collide with a trapped atom, Knudsen assumption f < 1.
[1] and [2] leads to
Uin = U= Uevap [3]
The expression for Uevap
Uevap has been calculated by Luiten et al. to describe the kineticof evaporative cooling:
Uevap = U+ T where = U/T is the trap depth over the effective temperature of the cloud.
Evaporated particles have a mean extra energyless than T.
Energy equation
Uin = U= Uevap = U+ T [3]
=
Uin = U= Uevap = U+ T [3]
=
Threshold condition on the energy of the incoming particlefor steady state solution.
0 solution leads to the steady statetemperature T0=U/0
Number of particles equation
in: incoming flux
f: fraction of atomswhich collide with trapped atoms
p: probability that an atom is evaporated after a collision.
: collision rate
f in = pN
p ~ 2e si low collision regime <<
N = ein
The number of atoms can be in principle very large
Knudsen assumption f < 1 Nmax ~ U / m 2
Boltzmann factor
Conclusion of the toy model
Threshold condition on the energy of the incoming particle for steady state solution.
T0=U/0 N = ein
can lead to a significant increase of the phase space density
A more elaborated model
z
U(z)
z0
Uz
Incident atoms
Uz
U
Anisotropic trap: z << T
Evaporation can be either longitudinal or transverse.
Longitudinal evaporation is a consequence of loading mechanism
z << T permits to reach HD longitudinal regime >> z,
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 withthe « collision less» regime.
pz : probability that an atom is longitudinally evaporated
Uz+zT : the average energy carried away by this atom
We have calculated by means of a molecular dynamics simulation
; 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
zx 10 Hz
x,yx 1000 Hz
107 atoms/s
v 10 cm/s
v 2 cm/s
Uz ~ 30 K
Average excess of energy Uz ~ 30 K i.e. ~ 1
An example (2)
0 5 0 1 0 0 1 5 00
0 .4
0 .8
1 .2
Phase space density
Transverse evaporation threshold (K)
(c)
0
1.5
1
0.5
N / 108
(b)
0
5
1 0
20
15
T (K)
(a)
0 5 0 1 0 0 1 5 0 0 5 0 1 0 0 1 5 0
x 1000
Uz ~ 30 K Average excess of energy Uz ~ 30 K
A resonance occurs when the transverse evaporation thresholdcoincides with the energy of the incident particles
Note that the steady state temperature is much lower than Uz, whichhas to be contrasted with the toy model result.
Around the resonance (1)
The resonance is observed when
When then and
linear variation
0 5 0 1 0 0 1 5 00
0 .4
0 .8
1 .2
Phase space density
Transverse evaporation threshold (K)
(c)
0
1.5
1
0.5
N / 108
(b)
0
5
1 0
20
15
T (K)
(a)
0 5 0 1 0 0 1 5 0 0 5 0 1 0 0 1 5 0
when
longitudinal evaporation can no more be neglected
Around the resonance (2)
When
the transverse evaporation becomes more and more inefficient
0 5 0 1 0 0 1 5 00
0 .4
0 .8
1 .2
Phase space density
Transverse evaporation threshold (K)
(c)
0
1.5
1
0.5
N / 108
(b)
0
5
1 0
20
15
T (K)
(a)
0 5 0 1 0 0 1 5 0 0 5 0 1 0 0 1 5 0
This result is reminiscent of the resultsof the toy model.
Optimum
zx 10 Hz
x,yx 1000 Hz
v 20 cm/s
v 4 cm/s
Uz ~ Uz ~ 130 K
The gain in phase space density saturates when the transverse motionenters the hydrodynamic regime
Numerical simulation: molecular dynamics
kT/U0.3
0.2
0.10 50 100 0 50 100
N (x 106)2.4
1.6
0.8
0.0
Flux: 105 atoms/s Phase space density: 0.01Velocity: 5 cm/s zx 10 Hz
x,yx 1000 Hz
t=100 seconds: N=2.4 106 atoms @ T=1.4 K degenerate gas !
Uz= 11 K (4.5 cm/s) = 0.36
= 1.5 Uz
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:
00
5
0 .0
T ( K) N
time (s)
10
500 1000
1.6x108
1 .2x108
8.0x107
4.0x107
The fast one
The slow one
Why two time constants ?
where
depends on the kindof evaporation
The steady state is given by and
has some exponential behavior, thus
CONCLUSION
Continuous loading of a non-dissipative trap can be donewith the help of evaporation
This process can be very efficient for anisotropic geometry
In this case, incoming particle can have an energy Uz of theorder of the energy barrier Uz and reach a temperature T << Uz.
A gain of phase space density of several orders of magnitudeis possible for realistic configuration.
N.B. We have neglected losses processes background gas, and inelastic collisions, ...
To appear in Europhysics Letters