Traps for neutral atoms
Trapping of atoms
• Study of quantum collective behavior (BEC, Fermi gazes)
• Guiding of atoms : towards guided atoms interferometers. Gyroscope ?
• Traps for individual atoms : use as Qbits for quantum computation
Neutral atoms : small interaction with external fields.
→ Use of laser cooling technics to have cold sample : T ' 10 µK
Interaction with electric field
• Static electric field. Induced dipole : V = −αE2 /2. Attraction towards high fields. No trap possible. Used in combination
with other fields.
• Oscillating fields : CO2 laser. Atoms confined in high intensity regions
• Laser light close to an atomic transition. Confinement towards high fields or low fields. Many possible geometries (standing
wave etc ...) Higher confinement. But spontaneous emission.
Interaction with magnetic field
• No population of excited states No spontaneous emission
• Large trapping volume
• Use of Radio-frequency fields : efficient evaporation
Magnetic trapping
Use interaction between a dipole moment and a magnetic field
~ .B
~
W = −M
~
B
~
M
Order of magnitude :
M ' µB = 1.4M Hz/G
To have W = 1K, B ' 1.5 T requiered
Experiment made with H : Superconducting coils, permanent magnets
Trap depth of about 1 K
With laser cooling : temperature of a few 10 µK available
Field of a few G sufficient.
Zeeman effect
Neglecting the diamagnetic term, the magnetic interaction between an atom and a magnetic field writes
W = −M.B,
where M is the magnetic dipole of the atom. M writes
M = MI −
X µB
~
(Li + 2Si )
i
where MI is the magnetic dipole of the nucleous and the sum is performed on each electron.
~|q|
µB = 2m : Bohr magneton
For small magnetic field, W can be computed in each unperturbed energy level.
M is a dipolar operator ⇒ M is proportionnal to F,
where F is the total spin of the atom.
An atom of zero spin such as the the ground state of He or the alkali-earth do not interact with magnetic field.
Zeeman effect large for an alkali.
Zeeman effect : case of alkaline
Spin of the complete shells cancel. Ground state has L = 0.
M = MI − 2
µB
S,
~
where MI is the nuclear spin. The gyromagnetic factor of the nucleus is much smaller than that of the electron (typically 10 3
smaller) as its mass is much larger. In the following we neglect it.
Ground state : hyperfine structure
E
EHF ' GHz
2 cases :µB B EHF
µB B EHF
Fmax = I + 1/2
Fmin = I − 1/2
Zeeman effect : case of alkaline. 2
1. B small
Effect of W in each hyperfine state
B along z. Then Wigner Eckart gives hm| W |mi = µB gF mB
F = Fmax : gFmax = 1/Fmax .
|Fmax , m = Fmax i = |mI = I, mS = 1/2i
Fmax µB gFmax = µB
F = Fmin
: gFmin = −1/Fmax
Seen easily by developing
|Fmax , m = Fmax − 1i and |Fmin , m = Fmax − 1i on
|mI = I, mS = −1/2i and |mI = I − 1, mS = 1/2i.
Case of Rubidum 87
m
=
2
m
1
−
=
B
Zeeman effect : case of alkaline. 3
2. Second order zeeman effect
E = µB gF m + δE (2)
δE (2)
=
(−1)F a
(µB B)2
,
m E
HF
(
a2 = a−2 = 0
a1 = a−1 = 3/4
a0 = 1
2. B large
Good basis : mS .
Zeeman energy : W = ±µB B
For µB B ωHF :
for Rb : B 0.4 T
3. General case
E/~ (MHz)
Case of
12000
8000
4000
0
-4000
-8000
-12000
Field used : e few G
→ frist order zeeman effect sufficient
Rb87
Not sufficient for :
High fields
Spectroscopy
0
2000
B0 (G)
4000
Magnetic trap
Trap for atom requieres a non homogeneous magnetic field. It a priori depends on the projection of the spin on the local direction of
the magnetic field. However, a simplification can be made for large enough magnetic field : the adiabatic following condition
2 time scales : ωL ' µB B
B
ωL
ωrot : rotation of B
If ωL ωrot , then the spin follows adiabatically the direction of B.
The potential seen by the atom reduces to
V = mgµB |B|
We will come back later on this condition.
Wing theorem
To realize a trap : extremum of |B| required.
Wing theorem : a maximum of |B| is impossible in free space
Let us suppose |B| has a maximum in O and let us take B(0) = B0 z0 .
|B|2 = B02 + |δB|2 + 2B0 δBz
where B = B(0) + δB.
Because |δB|2 ≥ 0, if |B|2 is maximum, then δBz should be a minimum. But Maxwell equation gives ∆δBz = 0
and a function f that fulfills ∆f = 0 has no extremum.
Field of a magnetic bottle :
L
O
z
Bz = b0 + b00 (z 2 −
Bρ = −b00 ρz
ρ2
)
2
Trap for atoms : minimum of |B|.
Trapped states : low-field seeker states
Disadvantage : large internal energy ⇒ unstable against spin flip transitions, two body inelastic collisions
Trapping high field seekers : time varying field
Static trap for high field seeker impossible. ac magnetic trap possible
Produces B0
dc
ac
Produce Bz = b0 + b00 (z 2 − ρ2 /2)
Helmoltz : cancel b0
V = −µB |B| = −µB
p
(B0 + b00 (z 2 − ρ2 /2))2 + b002 ρ2 ' −µB B0 − µB b00 (z 2 − ρ2 /2)
Atoms confined radially, expelled longitudinaly
Oscillating b00 : Regimes of stable motion of the atoms (like Paul trap)
Shallow trap (µ K), small confinement (ωosc ' 5 Hz)
Linear traps
Goal : Reach a high confinement. Then linear tarps are a better solution than higher order traps.
R
I
Magnetic field produced by similar coils
Quadratic trap
Linear traps
|B|
0
b '
ll '
µ0 I
R2
kB T R 2
µ0 µB I
|B|
kB T /µB
kB T /µB
r
r
ll
ll
lq
'
q
lq
kB T R
µ0 µB I
'
µ0 I
b00 ' q
R3
kB T µB R 3
lq '
µ0 I
q
ll
R
ll R ⇒ l l l q
Realisation of linear trap
Linear trap : B = 0 at the center. (B has no singularity)
Most simple realsisation : Anti-Helmoltz coils
O
z
I
I
On the z axis , Bkz0
By symmetry : B(O) = 0

 Bz = GBz
Bx = −G/2x

By = −G/2y
x
n
2.7cm
1900At
loaded from a zeeman slower (1985)
First trap for cold atoms
→ depth of 17 mK
Now : ' 100 G/cm, dissipation ' 100 Watt
z
Losses in a quadrupole trap
Losses due to non zero magnetic field at the center
d
v
a
θ
∆θ ' π/2 for d ' a
O
→ θ̇ '
p
~v/GµB .
The atom is lost if a <
The loss rate is the flux of atoms going inside
the sphere of radius a surrounding O
~v 2
dN/dt ' 4πa nv ' n
GµB
2
Using mv 2 ' kB T and
n ' N (µB G/kB T )3 we obtain
dN/dt =
Adiabatic condition : θ̇ ωl
(GµB )2
−N ~ m(k T )2
B
~ 1
= −N m
l2
v
a
Larmor frequency : ωl ' GµB a
Solution : TOP trap
To remedy the Majorana losses problem, two solutions were proposed : the Time Orbital Potential and the addition of an optical plug.
• Time Orbital Potential Trap (TOP trap)
The idea is to displace the zero of the trap using an external field bf B0 . If the direction of B0 is turned, the zero of the field moves
around a circle. If this rotation is fast enough, the atoms do not have time to follow and experience a time average potential.
Instantaneous potential seen by the atoms
B0
U 'G
p
4z 2 + (x − r0 cos(ωt))2 + (y − r0 sin(ωt))2
' µB Gr0 (1 − x/r0 cos(ωt) − y/r0 sin(ωt) +
√
Here we used 1 + = 1 + /2 − 1/82 .
After average over a period
UT OP = µB Gr0 +
.
Condition of validity :
Adiabatic following of the spins : ω µB Gr
p0 = µB B0
µB G/2mr0
Atom motion do not follow : ω ωvib =
FIRST BEC obtained in such a trap in 1995. ω = 7.5kHz, ωvib = 25 Hz
µB G 2
(x + y 2 + 8z 2 )
4r0
x2 +y 2
4r 2
0
+ 2z 2 /r02 )
Quadrupole trap used for transportation
Quadrupole trap used for transportation of “hot” clouds.
• Moving coils
Motorized translation
• Transfer from coil to coil
Static traps of non vanishing magnetic field : IOFFE
configuration
Traps with non vanishing B at the center : B(O) = B0 z0
|B|2 = B02 + 2B0 δBz + |δB|2
|δB|2 at most of second order because δB at most of first order.
|B| minimum ⇒ ∇|B|2 = 0 ⇒ ∇δBz = 0. Thus δBz is at least of second order.
First order expansion of B
∇B = 0 ⇒ ∂Bx /∂x = −∂By /∂y
∇ ∧ B = 0 ⇒ ∂Bx /∂y = ∂By /∂x
Up to a rotation in the (xy) plane,
⇒ B⊥
G
=
G0
G0
x
−G y
Bz = B 0
B x = b0 x
By = −b0 y
2D quadrupole : responsible for transverse confinement.
Longitudinal confinement
2nd order term in δBz : b00 z 2 . Maxwell equations requieres that it does not come alone !
Assuming invariance by rotation along z,
Bz(2) = b00 (z 2 −
Bρ(2) = −b00 ρz
ρ2
2
)
y
B0
x
IOFFE configuration
Magnetic field modulus developed to second order
b02
|B| = B0 + b z + r (
− b00 /2).
2B0
00 2
Potential V = µ|B| =
2 2
1
2 M ωz z
00
02
B0 chosen small so that b
⇒ cigar shape trap
+
2 2
1
2 M ωρ ρ ,
b /B0
ωρ =
p
2
µ(b02 /B
0
−
b00 /2)/M,
V
ωz =
p
2µb00 /M
r
z
Validity of harmonic expansion : r B0 /b0 .
At r⊥ large, linear trapping domain.
r
Realization
4 current carrying bars → 2 dimensional quadrupole field.
Longitudinal confinement realized with two coils running parallel current. (more separated than in the Helmholtz configuration)
Two coils in Helmholtz configuration : decrease B0 in order to increase the transverse confinement.
I
I
I
I
BEC obtained in such a trap.
I ' 6 × 575A. b0 = 275 G/cm.
Dipole coils :
I = 6 × 455A , b00 = 365 G/cm2 .
Oscillation frequencies
ω⊥ = 2π × 280Hz, ωz = 2π × 24Hz.
Value of B0
In order to achieve a high transverse confinement, one want to decrease the minmum field B 0 . But if B0 is reduced too much, then
spin flip losses may occur.
Naive approach
As the atoms move in the transverse direction, the magnetic field direction rotates. The rotation frequency is smaller than ω vib . Then
if ωvib ωl = µB B0 /~, adiabaticity is fulfilled. How small is the spin flip rate ?
A quantum approach
Let us study here a very simplified model where we consider spin 1/2.
Assumptions :
•Magnetic field direction depends only on x, y
B
• Size of cloud l b00 . angles given to first order in x, y
p
Case of ground state wave function : l =
~/2M ωvib ⇒ µB0 ~ωvib , B03 Case of temperature T : kB T µB0
V
Adiabatic states :
|1i
|1i (x) = e−i(θx Sy +θy Sx ) |↑i
' |↑i + (b0 x/2B0 − ib0 y/2B0 ) |↓i
x
|2i (x) = e−i(θx Sy +θy Sx ) |↓i
' |↓i + (−b0 x/2B0 − ib0 y/2B0 ) |↑i
b02 ~2
Mµ
θx = b0 /B0 x
B
|2i
Kinetic energy operator :
P2
2m
P2
=
2m
2
~
= − 2m
(
∂2
∂x2
+
∂2
∂y 2
+
∂2
∂z 2
)
2
~
∆(|1i h1| + |2i h2|)
− 2m
2
~ 0
∂
∂
∂
~2 0
∂
−
b /B0 (−
−i
) |2i h1| −
b /B0 (
+i
) |1i h2|
2m
∂x
∂y
2m
∂x
∂y
|
{z
W
}
Spin flip rate
W couple |1i to untrapped states |2i.
Initial state : ground state of |1i = |1, ϕ0 i = |1, nx = 0, ny = 0, nz = 0i.
Final state : continuum
W acts only on x and y. → |f i = |2, kf , nz = 0i.
kf = kfx x0 + kfy y0 (We will neglect the potential in the state |2i.)
Γ=
Fermi Golden Rule :
2
2π
~ ρ(Ef )|h2, kf |W |1, ϕ0 i| .
~ 2 b0
~ 2 b0
h2, kf |W |1, ϕ0 i =
hkf |∂x + i∂y |ϕ0 i = i
(kx + iky )hkf |ϕ0 i
2m B0
2m B0
|h2, kf |W |1, ϕ0 i|
2
density of states n 2D
We use :
b02
B0
02
4
= ~ 2 b 2 kf2 12
4m B
L
0
2
: ρ2D = mL 2
2π~
~
2πmω
~k2
f
e− mω
ϕ0 =
,
p
= mω 2 /2µB and ~2 kf2 /(2m) ' µB B0
Γ=
ω −
2π e
2µB B0
~ω
Thermal state
Similar calculation. Calculation of the probability to occupy the momentum kf .
We find
Γ=
~ω 2
4πkB T
e
−
µB B0
kB T
2
− ~k
~
mω
2πmω e
Different coils configurations
Many different coils configurations can lead to a IOFFE trap. We give here some examples.
• Cloverleaf configuration
Advantage : Good optical access.
typical parameters :
b0 = 80 G/cm, b00 = 25 G/cm2
Heat dissipation : ' kW, water cooling with pressurised water (15 bar)
• Baseball configuration
Advantage : Less heat dissipation
b0 = 300 G/cm, b00 = 30 G/cm2
For Rb87 in |F = 2, m = 2i : ωperp = 400 Hz, ωz = 10 Hz
Different coils configurations
• 3 coils trap
3 coils of same radius with same current.
Advantage : small heat dissipation, simpler than baseball configuration.
For a few hundred W dissipation :
b0 = 100 G/cm
b00 = 60 G/cm2
• Not symmetric 3 coils trap
QUIC trap
For same current in the three coils, smaller bias field.
No need of extra compensation coils.
y
T. Esslinger et al. : b0 = 220 G/cm, b00 = 260 G/cm2 .
For Rb87 , B0 = 2 G : ω⊥ = 2π × 200 Hz, ωz = 2π Hz
z
Loading : merging of two quadrupoles
trap position
Bz
BzIO
−BzQ
z
Use of permanent magnet
Some experiments use permanent magnet to create the magnetic field.
Advantage : no heat dissipation, high gradiant
N
S
S
N
N
S
S
N
Cornell experiment : b0 = 450 G/cm
Inconvenient : No possibility to switch off
Higher order fields
Transverse field : grows in ρn , n > 1.
n = 2 : octopole field
|B| = b00 ρ2
Bx = b00 (x2 − y 2 )
By = −2b00 xy
Steeper potential.
Towards smaller structures
Gradiant and curvature proportionnal to 1/R2 and 1/R3 respectively. Thus small coils are preferred. But coils with their water
cooling system takes a lot of room.
Size of the system : compromize between optical access and magnetic confinement.
To overcome this difficulty, one solution is to use magnetic material.
Use of magnetic material
B0
H=
B
µ0
−M
Magnetic material : H =
B
,
µr µ0
µr ' 104 for iron
Ampere law : ∇ ∧ H = jext
NI =
e
R
C
H.dl =
R
B
dl
C µr µ0
Neglecting the contribution of the part inside the material gives
B0 ' µ0 N I/e
I
e
N turns
Magnetic field similar to that produced by coils distant by e.
Coils can be put far away from the interesting region. Can be larger. Less heat dissipation.
Heat dissipation : ' 200 W
Confinement :
ωperp = 750 Hz .
ωz = 5 Hz
N I/2 N I/2
Use of micro-structures
2-D quadrupole field produced with wires of radius
R = 260 µm
Gradiant : b0 = 5000 G/cm
Guiding atoms using a single wire and an homogeneous magnetic field
Single current carrying wire + homogeneous magnetic field
h=
0
b =
µ0 I
2πBbais
25
Btotal
µ0 I
2πh2
ex e
z
Bbiais
ey
h
Realised with 50 µm wire.
h = 300 µm
b0 = 1000 G/cm
Bfil
I
(a)
20
B [G]
2 D quadrupole field :
15
10
5
0
0
h
1
x [mm]
(b)
2
Chip mounted micro-structures
Wires mounted on a chip :
→ Good heat dissipation
→ Mechanical stability
Fabrication technologies:
Substrate : Good heat conduction. Si, GaAS, Sapphire (glass also used)
Pattern design :
•UV lithography, electroplating for wires of size ' 10µm
Example of fabrication process
UV light
mask
200 nm gold
planarizing
resine
Photoresist
(6 µm)
Ti/Au
14 µm
Si
SiO2 (200 nm)
UV photolithography
Electrodeposition
Realisation of a miror
•Electron beam lithography, metal evaporation for size < 1µm
1 cm
Maximum confinement
µ0 I
2πh2
µ0 I
b0max '
2πW 2
For infinilty small wire : b0 =
Infinitly thin wire
Wire of size W=20 µm
B (G)
For a wire of size W :
200
150
100
How large can be I ?
Limited by heat dissipation process
50
0
10 20 30 40 50 60 70 80 90 100
h(µm)
Heat dissipation
Assumptions : Substrate at temperature T0 . Insulating layer : thermal conductivity K.
Energy conservation for a unit length along the wire :
• Energy released by Joules effect in a second : W =
ρ
2
We I
T
• Energy flux from the wire to the substrate : J = KW (T − T0 )
T = T0
• Energy stored in the wire : E = cW eT
dT
K
ρI 2
⇒
−
=
(T − T0 ).
dt
c(W e)2
ce
Equilibrium temperature : Teq = T0 +I 2
ρ
KW 2 e
2
Growing time : τ =
500 nm SiO2 on Si : K ' 106 W/Km .
τ ' 0.7 µs for e = 1µm.
For Tmax = T0 + 30K, W = 20µm, e = 5µm, Imax = 0.6A
Scaling law for squarre wire (e = W )
T − T0 = ∆Tmax ⇒ Imax ∝ W 3/2 ⇒ b0max ∝
√1
W
W
ce
K
Insulating layer
J
Heat transfer
Limit of the model
• Temperature dependence of ρ : ρ = ρ0 (1 + α(T − T0 ))
ρ ↑ when T ↑. → instability if I > Icrit =
For W = 20µm, e = 5µm, Ic ' 2A.
q
KeW 2
αρ0
• Heating of the substrate
Slow increase of temperature on time scales of seconds.
Importance of the substrate mount.
.
α = 3.5 × 10−3 K−1 for gold
Other guide geometries
• A three wire guide
Field perpendicular to the wire produced by side wires.
Possibility to realize curved guide.
• A 2 wires guide
Other geometries
• A 2 wire guide with opposite current
Atoms have been guided along a spiral
(group of J.Schmiedmayer)
Experiment of M.Printiss : S = 200 µm, ∇B ' 800 G/cm
Towards atom interferometry ?
2 wires and a magnetic field
By
z
Bbais
Coalescence point
Bbais
z
y
Possibility to split a wave packet in 2. → interferometer
Tunnel coupling between the potential minima near the coalescence point ?
Difficulty : Very sensitive to magnetic field fluctuations
b)
c)
d
LVIS mirror
d)
Electromagnet
for transverse
bias field
Wires on
glass substrate
16
14
12
10
8
6
4
2
0
-2
a)
b)
0.8
0.7
0.6
0.5
Thermal
atoms. No demonstration
guide 1
0.4
guide 2
0.3
total (1+2)
0.2
0.1
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Iguide2/Iguide1
Guided Atom Flux (arb. units)
a)
Bbias
Guided Atom Flux (103/s)
Demonstration of an atomic beam splitter
Iguide2/Iguide1
of coherence
Realization of a longitudinal confinement
Longitudinal magnetic field presenting a minimum
• Use of coils
• Use of wires : the Z shape trap
Field produced by the bars for h L
B0
I0
I0
z
I
z
Bz
1 cm
z
L
Z-shape wire
For h L
∂ 2 Bz /∂z 2 =
hµ0 I
L2
0I
∂ 2 Bz /∂ 2 z ∝ hµ
L4
Bz '
48µ0 I h
π
L4
Advantage : less electrical connections
Addition of an extra longitudinal field to prevent spin flip
L = 2.8 cm, I = 300 mA
B0 = 14 G, Bbias = 1 G
h = 170 µm
ωperp = 3.5 kHz
ωz = 16 Hz
Other geometries
• A micro quadrupole trap
P
z
Larger domain of quadrupole field for a ruban shape
Bz = 0 in the plane P
Bz linear in z
For L = 2.5mm :
b0x = 85 G/cm
b0y = −74G/cm
b0z = −11 G/cm
I
ideal
quadrupole
Used for MOT
• Ioffe trap with only two wires
B0
i
z
I
B0
i
z
I
Bz
|Bz |
wire
ruban
Realization of a longitudinal confinement
• T trap
2 wires trap simplified in T trap.
Advantage : less electrical connections
I
B0
i
• Conveyor belt for atoms
i = i0
t=0
I
i0 = 0
i=0
t=T/4
I
I
I
Bz
z
i0 = 0
i=0
t=3T/4
Bz
z
i0 = i 0
i = −i0
t=T/2
Bz
z
i0 = −i0
Bz
z
Use of magnetic material
Equivalence between magnetization and a fictitious current distribution
Case of a layer of magnetic material with magnetization perpendicular to the surface.
Geometry equivalent to a Z-shape trap
jef f
equivalent current distribution :
jef f = ∇ × M
Zone of unifrom magnetisation M :
equivalent current going around : I = eM
Possibility to write the magnetic pattern with laser (coercitive field decreases with temperature)
Size of pattern : limited to about 1 µm.
Large magnetic field gradients : 106 G/cm
Not tested on atoms yet.
Advantages of magnetic structures
Absence of conductors nearby the atoms → no thermally excited currents.
Stable in time
Disadvantages of magnetic structures
No possibility to turn on and off the magnetic field.
20 µm
Micro-magnetic traps produced by video tapes.
z
Periodic magnetisation
Equivalent currents
x
λ
Magnetic field produced by a periodic current distribution :
P i2nπx/λ −2nπz/λ
~ = ∇φ, ∆φ = 0
e
e
∇ × B = ∇.B = 0 → B
Periodicity and ∆φ = 0: → φ =
n
~ = B1 e−kz (− cos(kx)x0 + sin(kx)z0 ).
For z > λ, terms n = ±1 dominate : φ = φ1 e2iπx/λ e−2πz/λ + c.c. ⇒ B
z
Addition of a uniform field B0 in the plane xz
|B|
λ
Exponential decay of |B|
Used as a miror for atoms
Array of micro-traps
BEC achieved in such microtraps
Limitation of micro-traps
Advantages of microtraps realized by microstructures
• Very high confinement.
• Very compact apparatus.
• Large variety of trapping and guiding configurations
Limitations of micro-traps
• Roughness of the magnetic potential
• Interaction with nearby materials at high temperature
Coupling with thermally excited currents
• Van-der-Waals attraction force to the surface
Effect of atoms adsorbed on the surface.
Roughness of the longitudinal potentiel in an atom guide
Cloud at small temperature get fragmented when brought close to wire
50µm from wire
500µm from wire
V
V
δV ' µK
z
z
δV ∝ I
Due to distortion of current flow in wire
Origin of the roughness
Roughness of the border edge
Induces deviation of the current flow inside the wire
→ rough longitudinal magnetic field → potential roughness
∆B/B0 ' 10−4 : → very sensitive to wire border fluctuations
Fluctuations of Bx produce a small displacement of the guide height.
Origin of the roughness : experimental evidence
Potential roughness measured with cold atomic cloud at thermal equilibrium . n(z) ∝ e −V (z)/kB T
Confining potential substracted.
Wire border roughness measured with an SEM.
x
Expected potential roughness computed.
z
y
Potentials
5µ m
40
176µm
107µm
80µm
69µm
V /I(µ K/A)
35
30
25
20
Potential calculated from
15
54µm
10
5
46µm
0
33µm
-5
Potential measured with the atoms
-800 -600 -400 -200 0
200 400 600 800
z(µm)
the measurement of the wire edge roughness
Scaling laws of the potential roughness
fl
x
h
z
z
W/2
y
y
fr
We consider only h W → Bz = Bz [Iy (z)], Iy =
R R
x
y
• k component fl/r = fk cos(kz). h W → k 1/W .
Only symmetric part fl + fr contribute to Iy .
→ Bk = µ0 Ifk g(k, h) =
µ I
fk 02
h
g̃(kh)
dxdyjy
Iyk = −Ifk k sin(kx)
NB : If k W , then current distortion localised near the borders
Case of white noise rougness of f
0.01
|g̃|2
Fluctuations avergades
to zero
|fk |2 = Jf = J0
hBz2 i = 0.044J0
0 1 2
jy small
3
4
5
kh
(µ0 I)2
h5
Fluctuations of Bz peacked at k ' 1/h
Effect of Casimir Polder
Interaction between an atom and a dielectric wall :
h
C3
For h λat : Van der Waals force VV dW = − R
3
C4
For h λat : Casimir-Polder potential V (h) = − R
4
C4 ' 6 × 10−4 µKµm4
C4 = K()α α :polarisability
: dielectric constant for Rb and = 4
1.0
0.8
with CP potential
0.6
6
0.4
E/kB [µK]
Experimental study
Effect of Casimir-Polder force :
smaller depth of the potential
→ losses of atoms by evaporation
Measured loss rate compared with calculation based on 1D evaporation model
fraction χ
λat atomic transition wavelength
0.2
4
2
0
0
0
0
Without CP potential
Limit the distance of approach to the surface
0.5
1.0
1.5
2.0
2.5
d [µm]
3.0
distance from the surface d [µm]
3
Effect of adsorption of atoms on the surface
Atoms adsorbed on some surfaces get polarized → electric field E ∝
2
⇒ interaction energy with atoms : U = − α
2E
⇒ Attraction towards the surface.
a)
b)
+
E
~105-107 adatoms
E field
V
+
Typical potential for 107 Rb atoms adsorbed on 4µm×150µm area
4
Energy (kHz)
3
2
1
0
0
2
4
6
8
Distance from surface (µm)
10
d
r3
Stability of a trapped cloud
• Background gas
• Fluctuations of the trap
• Spin flip transitions induced by magnetic field fluctuations
• interaction with environment at temperature T = 320 K
Effect of the background gas
• Losses produced by collisions
Atoms of background gas at T ' 320 K depth of the trap.
→ After a collision with a background atom, the trapped atom is lost almost certainly.
Γ = nbg v̄T σ
where nbg ∝ P is the density of background gas at pressure P and v̄T =
Γ ∝ P.
For Rb atoms, for P = 10−9 mbar, Γ ' 0.1 s−1 .
p
πkB T /2m is the thermal velocity.
• Heating produced by large scattering impact collisions
Production of a hot atoms cloud → heating rate of the cold cloud.
cold cloud
Production of a cloud of hot atoms : the Oort cloud
Very difficult to quantify. Role of multiple scattering : effect more important for dense
cloud
Solution : Trap depth small enough. Use of an RF shield.
A typical value : dT /dt ' 1 µK/s
Heating induced by trap fluctuations
Technical fluctuation of current in coils ⇒ fluctuation of trap parameter
Examples :
Transverse magnetic field B → position fluctuation δx = B/b0
Fluctuation in current in the dipole coils → fluctuation of ωz .
Produces heating
• trap position fluctuations
Fluctuating force : F (t) = mω 2 δx.
Equation : ẍ = −ω 2 x + F (t)/m
Solution x = x0 cos(ωt) +
Calculation of hx2 i (statistical average over noise) :
2
2
2
hx i(t) = x0 cos (ωt) +
Z tZ
0
t
dτ1 dτ2
0
0
F (τ )
mω
sin(ω(t − τ ))dτ
hF (τ1 )F (τ2 )i 1
(cos(ω(τ1 − τ2 )) − cos(2ωt))
m2 ω 2
2
For t tcor the correlation time of F , using JF (ω) =
have
2
R∞
2
R∞
∞
dte−iωt hF (0)F (t)i and averaging over a period, we
hx i(t) = x0 /2 + t
JF (ω)
2m2 ω 2
Thus, the potential energy Ep = mω 2 x2 /2 average over a period fulfills
oscillating period, Ep = Ec = E/2 so that
dE
JF
=
dt
2m
dEp
dt
Heating linear in time. Driven oscillator. Excitation of the dipole mode
m2 ω 4
Heating rate for a fluctuating homogeneous magnetic field : dE
=
02 JB (ω)
dt
b
= t
JF (ω)
4m
After average over an
oscillation frequencies fluctuations
ω 2 = ω02 (1 + (t))
Calculation easier using quantum mechanics
p2
1
2
2
H=
+ mω0 (1 + )x
2m
2
Using x =
p
~
2mω (a
+ a+ ),
|ni coupled to |n − 2i , |n + 2i
H = ~ω0 (1 + /2)a+ a +
2
1
~
mω02 (a2 + a+ )
2
2mω
Transition rates induced by a fluctuating coupling
Sates |ii and |f i coupled by v(t)Aif , v fluctuates.
Initial state : ψ = |ii.
Perturbation theory :
At time t, |ψi = e−iωi t |ii + cf e−iωf t |f i,
1
ci =
i~
Z
t
Aif v(τ )e
i(ωi −ωf )τ
dτ.
0
Population in f (averaged over random realization of v),
1
h|cf | i = 2
~
2
Z tZ
0
t
0
dτ1 dτ2 |Aif |2 hv(τ1 )v(τ2 )ie−iωif (τ1 −τ2 )
For t tcor the correlation time of v,
|Aif |
2
h|cf | i =
2~2
Thus the transition rate is
Γif
2
Z
2t
Jv (ωif )dτ
0
|Aif |2
=
Jv (ωif )
~2
Link with
Golden rule :
P Fermi
−iωi t
v =
vi e
. In a dressed state approach, transition |ii → |f i due to absorption of a quanta in a mode of v. Time
i
independent coupling |i, ni i → |f, ni − 1i. Fermi Golden rule gives transition rate.
Heating rate induced by oscillation frequencies fluctuations
2
2
+
Here V = ~ω
).
4 (t)(a + a
Initial state |ni :
Γn→n+2
ω2
J (2ω)(n + 1)(n + 2)
=
16
Γn→n−2
ω2
J (2ω)n(n − 1)
=
16
Heating rate (thermal average over initil state)
dE
=
dt
X
ω2
pn 2~ω
((n + 1)(n + 2) − n(n − 1))J (2ω)
16
n
dE
ω2
=
J (2ω)hEi.
dt
2
Exponential heating. Parametric driving
Higher order : heating for ωmod = 2ω/p, p integer.
1 D calculation. For a three dimensionnal harmonic trap, Ex = E/3 and heating rate decreased by a factor 3.
Heating produced by current fluctuations in a wire guide
• Heating due to position fluctuation
Using F = mω 2 δx =
B0
JF =
Heating rate,
I
mω 2 µ0
2πB0
h=
I,
mω 2 µ0
2πB0
2
JI .
dE
h2 mω 4 1
JI
=
dt
2
I02
µ0 I
2πB0
Case I0 = 1 A, h = 10 µm, ω = 2π × 10 kHz, JI = 1(µA2 )/Hz : dE/dt = 8 µK/s.
Very good power supply : JI ' 10(nA2 )/Hz
Do not happen if B0 and I have correlated fluctuations : ∆I/I = ∆B0 /B0 (circuit in series)
• Heating due to fluctuations of oscillation frequency
ω ∝ 1/I. Thus
dE
ω2
= E 2 JI .
dt
I
Ratio of heating rates :
dE/dt)dip
mω 2 h2
h2
=
= 2 1,
dE/dt)param
2E
2l
for cloud of size l h .
Spin flip transitions induced by fluctuating magnetic field
V
B along z in the trap
bx and by couple trap state other Zeeman level
~ωl
r
Transition rate : Γ '
µ2
B
~2
→ transition to untrapped states : losses
JBx (ωl )
Case of current fluctuations in a micro-trap : Γ '
µB µ0
2πh
2
JI .
For h = 1 µm, JI = 0.3 (nA)2 /Hz , Γ ' 1.3 s−1
Interaction with environement at temperature T ' 320 K
• blackbody radiation
µ2
B
~2
Field at ω ' ωl = µB Bb : induce transitions to an untrapped state Γ '
Black body field fluctuations :
√
Mode n of the electromagnetic field. Bx = Bxn0 (a + a+ )/ 2.
Average over thermal state (no coherence between different modes)
hBx (t)Bx (0)i =
X Bx2n
2
0
h(e
−iωn t
iωt +
+
an +e
an )(an +an )i
=
n
Spetcral density : JBx =
R
hBx (t)Bx (0)ie
~ωe−~ω/kB T ω 2
2 3
1−e−~ω/kB T π c
X
2
Bxn (npn
0
e−iωt
2
cos(ωn t)+
| {z }
n
−iωt
dt =
Repartition of energy between E and B and isotropie :
where dE =
JB
3
µ0
P
n
vaccum contribution
2
Bx
npn πδ(ω − ωn )
n
JBx = π
0
P
~ω
npn δ(ω
n L3
− ωn ) = π
L3 dω. is the enrgy in the range ω → ω + dω.
JBx
µ0 ~ω 3 e−~ω/kB T
kB T ω 2 µ 0
=
'
3πc3 1 − e−~ω/kB T
3πc3
Induced loss rate : Γ = 7 × 10−18 s−1 for ωl = 2π × 1 MHz.
Negligible
dE
,
L3 dω
).
Interaction with nearby conductors
Atom chip : atoms close to a surface at temperature T : h ' 1µm..100µm
Effect of thermally excited current in the wire or substrate ?
We need to know J B
Fluctuating magnetic field close to a conductor
z M
p2
θ
2I
2
Small oscillating dipole. θ 1 : H =
+ K
2 θ ,
ω 2 = K/I.
Interaction with the metal : V = −B.M ' θM Bz
θ
• damping of oscillations
• surface at temperature T → fluctuating currents → fluctuating field → heating of
the dipole
• Heating due to the fluctuating field : Using previous results : dE/dt)f luctu =
1
2I
JB (ω)
• Energy dissipation due to damping
Linear response : B = θa M (Re (α) cos(ωt) + Im (α) sin(ωt)) for θ = θa cos(ωt).
Energy decrease :
dE
dt
Average over a period :
=
∂V
∂t
diss
dE
dt diss
2
= θa
M 2 ω cos(ωt)(−Re (α) sin(ωt) + Im (α) cos(ωt))
2
= θa
M 2 ωIm (α)/2
At thermal equilibrium dE/dt)diss = dE/dt)f luctu
⇒
JB = 2
kB T
ω
Im (α)
2
we used θa
= 2hEi/K ' 2kB T /K for kB T ~ω
Theorem fluctuation/dissipation
Scaling low for the fluctuation of magnetic field near conductors
k
T
JB = 2 B
ω Im α(ω)
Im α(ω) : component in quadrature of the magnetic field produced in r0 by the current induced by an oscillating dipole in r0 .
Case of low frequencies :
Oscillating dipole → oscillating magnetic field Bd ∝ 1/|r − r0 |3
→ induced electric field E ∝ ω/|r − r0 |2 (∇vectE = −∂B∂t)
→ induced currents j ∝ ω/(|r − r0 |2 ρ) (j = E/ρ) : Foucault currents
→ induced magnetic field in r
Z
ω
1
3
Bind ∝ Im α ∝
d r
ρ
|r − r0 |4
r0
fluctuations produced by a wire
h
JB ∝
a
insulating substrate
fluctuations produced by a half space
r0
h
ρ
JB ∝
k B T a2
ρh3
kB T
ρh
Case of high frequencies
Electromagnetic field attenuated in the metal after a distance δ(ω). δ =
If h > δ, then previous scaling laws wrong.
For ω = 1.4 MHz (1G/µB ), in gold, δ = 60 µm.
p
2ρ/µ0 ω :skin depth
Correct calculation of α : Maxwell equations for evanescent waves.
Calculations for a half metallic space :
• h δ:
JBi
µ20 kB T 3δ(ω)3
=
si
16πρ
z4
• hδ:
JBi
si = 1/2 for i=x,y and sz = 1, (z normal to surface)
µ20 kB T 1
=
si
16πρ
δ
Losses induced by thermally excited currents
Magnetic field fluctuation at ω = ωl produces losses.
(a) : ωl = 1.8 MHz . Copper, titanium and silicon
(b) : ωl = 1.8 MHz and ωl = 6.24 MHz. Copper
Stability of a magnetically trapped cloud with respect to
collisions
Magnetically trapped cloud : metastable state
• Atoms are not in the lower energy state
E
Exothermic 2 body collisions to an untrapped state.
B
V
• Molecules (and ultimately a solid) should form
r
exothermic 3 body collisions
molecule ' 40 K for Rb
' 0.1 nm
A little of collision theory
2 body elastic collision by a central potential.
r +r
ψ(r1 , r2 ) = φ( 1 2 2 , r1 − r2 ) = φ(R, r)
Stationnary state : φ = eiKR ϕkdif f (r),
~k2
m
2
ϕkdif f = − ~m ∆ϕkdif f + V (r)ϕkdif f
ikr
Asymptotic behavior : ϕkdif f → eikz + f (θ, φ) e r
f (θ) : scattering amplitude
θ
Flux in the solid angle dΩ :
∗
∗
~
im (ϕ ∇ϕ − ϕ∇ϕ ) .dS '
Γout
φin
2
2k
m |f (θ, φ)| dΩ
RR
Collisionnal cross section :
=
|f (θ, φ)|2 dΩ = σ
Low energy behavior
f (θ, φ) = f (k)
k → 0:
⇒ σ → σ0 independant of k
f (k) → f0
Identical atoms
Symmetrisation of theR wave
R function.2
For Bosons : σ = 2
|f (θ, φ)| dΩ
For fermions : σ → 0 as k → 0
Phase shift and scattering length
Central potential : angular momentum is concerved.
Low energy : f isotrop ⇒ only the wave function L = 0 is scattered.
Only
: phase shift of the scattered wave.
n one parameter
1
f ' 2ik
(e2ik − 1)
a : scattering length.
δ ' ka
e−ikr
r
eikr i(π+2δ)
e
r
Collisions rates
Elastic collision rate
3
3
Element of volume d3 r. Number of atoms of velocity
R 3 0between v 0and v 0+ d v : f (v)d v.
Collision rate for an atom of velocity v1 γ = d v σ|v1 − v |f (v ).
√
Collision rate per atom : Γcoll ' nσ 4kB T m.
2 body inelastic collision.
|ai , k
|bi , k 0
Output flux : Γout =
2k0
m
|f |2 dΩ
Low energy behavior : f independant on incident energy.
Rate of inelsatic collision per atom :
⇒ Γinel = Gn independant on incident enrgy.
Ratio between elastic to inelastic collisions
Γel
Γinel
∝
√
T.
Interaction between two alkali
First approximation : V = Vel electrostatic interaction
• Central potenial
• Depends on the spin via the symmetrisation of wave function
2 alkaline atoms : 2 spin 1/2.
Totla spin :
S = 1 : Symmetric by exchange of spins (|S = 1, m = 1i = |1/2, 1/2i)
S = 0 : Antisymmetric by exchange of spins
Singulet potential
r
c6 /r 6
Singlet state : Wave function symmetric by exchange of electrons
Two electrons in the same lowest energy wave function
Triplet state : Wave function antisymmetric by exchange of electrons
An electron in the first excited state.
Triplet potential
V
Hyperfine interaction.
At large distance, the hyperfine interaction in each atom dominates.
State discribed by the hyperfine level f1 , f2 of each atom.
r
Selection rule and stable states
~ = I 1 + I2 + S 1 + S 2
Total spin : F
Potential invariant by a rotation of all the spins.
⇒ F, mF concerved
⇒ mf1 + mf2 is conserved.
Stable states
• |Fmax , mmax i : Maximum value of mF . Pure triplet state
• |Fmin , m = Fmin i : exchange collisions energetically forbidden
Unstable states
Any collision that preserved mf1 + mf2 is possible.
In Na : |2, 1i + |1, −1i → |1, 0i + |1, 0i
Typical rates : G ' 10−11 cm3 /s
⇒ Mixture of states unstable
⇒ Any state different from |Fmax , mmax i and |Fmin , m = Fmin i unstable
A special case : Rubidum 87
For Rb : exchange collisions rate very small : G(2,2)+(1,−1) = 2.3 × 10−14 cm3 s−1 .
Very naive explanation
Neglect hyperfine energy during scattering
Initial state :
P
|f1 , f2 , m1 , m2 i
m
,
m
,
m
,
m
s1
s2
i1
i2
i1 ,mi2
mi ,mi
1
2
P
mi1 , mi2 c0,mi ,mi |S = 0i + c0,mi
=
cm
=
mi ,mi
1
2
1
Scattering of (c0 |S = 0i + c1 |S = 1, mi)
ikr
ψscatt = c0 (eikr + fs e r ) |S = 0i + c1 (eikr + fT
For Rubidium, aT ' aS ⇒ fT ' fT = f
ikr
2
eikr
r
→ scattered component : f e r (c0 |S = 0i + c1 |S = 1, mi)
No change of state during scattering
Scattering amplitude is f : independant of internal state.
) |S = 1, mi
1
,mi
2
S = 1, m1 + m2 − mi1 − mi2
Spin changing collisions
Interactions between magnetic moment of the electrons
s1
r
V =
s2
V is not conserved by a rotation of the spins only.
⇒ F is not conserved.
µ0 µB
(s1 .s2 − 3(s1 .r)(s1 .r))
4πr 3
Transfer of angular momentum from spin degree of
freedom to orbital momentum of the relative motion.
Usually very small. For example, in Na, G ' 10−15 cm3 s−1 .
Expected to decrease for low magnetic field in the lower hyperfine state. The energy realized becomes small
Case of Cesium
Important inelastic rate. Both for higher hyperfine state and lower hyperfine state. G ' 10 −12 cm3 s− 1.
Prevent the realization of BEC in magnetic trap for Cs.
The spin-orbit coupling interaction has for Cs a similar role.
3 body losses
Formation of a molecule : 3 atoms required
V
r
molecule ' 40 K for Rb
Exothermic collision :
At + At + At → At2 + At
Collision rate per atom: Γ3b = Ln2 .
Measured rate for Rb : L = 1.8 × 10−29 cm6 s−1
Limit the lifetime of BEC.
Ratio with elastic collision rate :
' 0.1 nm
Γ3b
n
∝ √
Γel
T
In a harmonic trap,
Γ3b
N ω3
∝
Γel
T2
Adiabatic decompression : T ∝ ω
⇒
Γ3b
∝ Nω
Γel
Experimental observations
Evolution of atom numbre :
dN
= −ΓN − Gn̄N − Ln¯2 N
dt
Inelastic process → non exponential decay.
• 2 body spin relaxation losses in magntically trapped Cs
fit : Assume thermal distribution.
• 3 body decay in a Rubidium BEC
Very low rate → high density requiered → BEC used
n̄ depends on N in a BEC (mean field interaction)
Use of radio-frequency magnetic fields to decrease
potential depth
Spin 1/2
Magnetic field : Bx cos(ωt)
V
Can make a transition between |↑i and |↓i
z
Transition resonnant at ωl = ω
~ω
atoms lost when they reach the equipotential V = ~ω/2
RF magnetic field : current loop
Inside the vacuum chamber if metallic !
How strong should be the RF field ?
What happens for an atom of spin F>1/2 ?
Stationary states at a given position
Stationary magnetic field : Bz z0 . ωl = 2µB Bz
RF field : B(t) = BRF cos(ωt)x0
H=
ωl
(|↑i h↑| − |↓i h↓|) + µB BRF cos(ωt)(|↑i h↓| + |↓i h↑|)
2
Rotating wave approximation
|ψi = c↑ e−iωt/2 |↑i + c↓ eiωt/2 |↓i
Evolution :
n
∆ = ωl − ω,
ω
µ
B
d
B RF c
i dt
c↑ = ( 2l − ω
↓
2 )c↑ +
2
ωl
µB BRF
d
ω
i dt c↓ = (− 2 + 2 )c↓ +
c↑
2
iωS t
ψ̃ = e z |ψi evolves with
H̃ =
∆
2
µB BRF
2
µB BRF
2
∆
−2
E
z
Eigenstates : |↑iu ,|↓iu
Energies :
u
∆/µB
Bef f
x
p
∆2 + (µB BRF )2
Case ∆ = 0 : |↑iu =
y
BRF /2
± 12
1
√
(|↑i
2
+ |↓i
µB BRF /2
µB BRF
∆
Dressed states picture
N : number of photons in the RF field
µB BRF /2
|N, ↑i
Energies :
E
N ~ω + ωl /2
neglecting the state |N − 1, ↓iequivalent to
rotating wave approximation
|N + 1, ↓i
(N + 1)~ω − ωl /2
BRF 6= 0
µB BRF
BRF = 0
z
ωl
y
Bef f
θ
x
Dressed state potential
BRF /2
Adiabatic following :
V
θ̇ ωl = µB BRF /2~
r
b0 v
2
µB BRF
~
(
b0 = 100G/cm
v = 15cm/s
BRF > 20mG
Another point of view : ∆T 1/ωl
Case of an atom of spin F>1/2
Case F=2
√
Bx ∝ Fx = (F− + F+ )/ 2
|N, m = 2i
Em
|N + 1, m = 1i
|N + 2, m = 0i
Ed
m=2
|N + 3, m = −1i
|N + 4, m = −2i
V
E
m=1
~ω
Bz
Bz
Case of high magnetic fields : non linear Zeeman effect
m=−1
Em
m=−2
E
BRF
Atom stays tarpped
Happens when
m=0
∆Bz > BRF
m=1
Bz
∆Bz
→ Does not work at high fields
z
Bz
No problem if
m=2
Bz
BRF Bz µEB Bz
HF
A trap for dressed states
Dressed states energy
Vm
E
Egg shell potential for dressed states
Ω
µ B B0
r
Confinement ωosc =
b0
r
p
2µB /m~BRF
losses : same calculation as for normal trap → Γ ∝ e
µ BRF
− B
ω
osc
Loading : start from ωRF < µB B0
Increase ωRF
Effect of gravity : atoms accumlate at the bottom
Evaporative cooling
Principle Thanks to elastic collisions, an atom can gain an
energy larger than the depth of the trap U . He takes
away an energy
∆E > U > hE/N i.
Thus
hE/N i ↓→ T ↓ .
2 conflicting conditions :
• Strong decrease of T per lost atom : U large required
• Time of evaporation smaller than the life time of the cloud : U not too large
Forced evaporative cooling
While T decreases, U is decreased to maintain a high cooling time.
In the following we give a very simple model describing evaporative cooling.
U
Evaporative cooling : a simple model
We assume :
• Any atom of energy larger than U leaves the trap immediately. This will be a good assumption for 3 dimensional
evaporation and if γcol ωosc .
• There is a quasi-thermal equilibrium of the cloud in the trap. Energy distribution is given by the Boltzmann law.
• The trap is harmonic
We note η = kU T. We assume η 1 .
B
Effect of evaporation only Number of atom decrease :
Ṅ = −Γev N.
Each lost atom carry an energy E > U . In average, the energy is hi = kB T (η + κ). κ is of the order of 1. Thus,
Ė = −N Γev (η + κ)kB T.
kB T = E/3N . Thus, if no other losses,
Ṫ
Ṅ
=
T
N
Scaling lows : Γel ∝ nv ∝ N 1−α
η+κ
−1
3
=α
Ṅ
→ T ∝ Nα
N
n
D ∝ 3/2
∝ N 1−3α
T
If α > 1, Γel increases in time. Runaway evaporation
Role of atom losses due to background collisions ?
Equations in presence of a loss rate
Losses due to background gas :
Ṅbg = −ΓN.
Not an energy dependent loss rate. Thus
Ėbg = −ΓE.
We thus have
(
Ṅ = −N Γev − ΓN
Ė = −Γev E η+κ
− ΓE
3
We will consider the quantities n and v which are the peak density and the thermal velocity v = 2
n ∝ NT
We obtain (using E = 3N kB T ),
(
−3/2
and v ∝
η+κ−5
Γev
2
ṅ
n
=
v̇
v
= −Γev η+κ−3
6
√
T.
−Γ
p
2kB T /πm.
Calculation of Γev
py
local loss rate.
Small region of volume δV .
Let us suppose the trap has an infinite depth.
√
Number of atoms of energy larger than ηkB T : δNexc ' nδV √2π ηe−η .
√ √
√ p
Velocity of atoms of energy ηkB T : η 2kB T /m = π ηv/2.
√ √
Collision rate for an atom of energy ηkB T : Γ = nσv π η/2.
ΓδNexc = n2 δV σvηe−η ' loss rate from the high energy tail.
Thermal equilibrium :
2
n δV σvηe
−η
p
px
After a collision, the probability that an
atom stays in the high energy region is
very small
= δΓevap .
Sum on the volume of the trap
At a position r, U = U (r).
Evaporation parameter : η(r) = η − U (r).
Density of atoms : n(r) = n0 e−U (r) .
R
δΓevap ' nn0 e−U (r)/kB T δV ηe−(η−U (r)) σv̄.
δΓevap d3 r = N Γev ⇒
Γev ' n0 σvηe−η .
Runnaway evaporation
n
Solution :
r=
ṅ
n
v̇
v
−η
= −Γ + η−4
2 nσvηe
η
= − η−2
6 nσvηe
e−Γt
nv = n(t = 0)v(t = 0)
1 + rg(e−Γt − 1)
√
√ .
2n(t = 0)v(t = 0)σ/Γ = Γel (t = 0)/Γ and g = ηe−η η−5
3
2
If r > 1/g, divergence : Runnaway evaporation
1/g
1400
1200
1000
800
600
400
200
0
5
6
7
8
9
10
50
40
30
γel /γel0
20
10
0
0
η
50 100 150 200 250 300
γel0 t
Γel (t = 0)/Γ > 300
Limit of the model
Limit of the model
• Beyond the approximation η 1 for the calculation of Γevap . Use kinetic theory. (Walravan 1996)
• Spilling. Losses of atoms due to decrease of U , even in the absence of collisions.
• Computation of κ. Beyond the approximation κ = 1.
Presence of 2 body losses
Dipolar spin changing collisions. Case for Hydrogen and Cesium.
Γdip ∝ n → Γdip /Γel ∝ √1 increases in time. no runaway regime
T
Losses more important at the center of the trap where U minimum → evaporative heating
For Rubidium : Γdip = nG2 = Γel for Tc =
πmG2
16σ 2
' 5 pK
Three body losses
Actual limit for evaporation with alkali.
Γ3b ∝ n2 . Are more and more important
→ decompression of the trap at the end of evaporation.
Hydrodynamic regime
√
σ
1
Model : Γel ω , ie n
= l L = kB T ω m
.
→ Spatial evaporation equivalent to a,n energy cut off.
Case Γel ω ?
Inhomogeneity of temperature. Diffusion equations for heat transport and hydrodynamic equations.
Role of the dimensionnality
Evaporation only in 1D : Case for the first experiment with H
Evaporation efficiency decreased :
Γevap1D = Γevap3D /(4η)
√
mkB T
py
√
θ ' 1/ η
z
p
px
p=
At high temperature, ergodicity → evaporation 3D
Case of RF evaporation
p
ηm2kB T
z
Evaporation surface
x
mgZ
constant potential energy
Z=2
p
2kB T /m/ω
Evaporation becomes 1D if mgZ > kB T ⇒ kB T <
Case of Rubidium, with ω = 50 Hz, T1D ' 20 µK.
Evaporation 1D at the end of cooling
2ηmg 2
ω2
Experimental strategy and results
Strategy :
Optimize the phase space density as a function of time.
Results :
Typically, D ∝ N1γ , γ ' 2, .., 3.
10-17
Con
den
s
-18
Piège Z (15 G)
n
dé
co
mp
re
ss
io
n
10
atio
n
10-20
po
ra
tio
10-21
10-22
év
a
ΛdB3 [m-3]
10-19
Piège Z (8 G)
comp
ressio
10-23
10-24
n
Piège Z (40 G)
PMOexterne PMOU
1017
1018
1019 -3 1020
n [atomes.m ]
1021