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