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Atom-atom interactions in ultracold gases
Claude Cohen-Tannoudji
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Claude Cohen-Tannoudji. Atom-atom interactions in ultracold gases. DEA. Institut Henri
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Atom-Atom Interactions
in Ultracold Quantum Gases
Claude Cohen-Tannoudji
Lectures on Quantum Gases
Institut Henri Poincaré, Paris, 25 April 2007
Collège de France
1
Lecture 1
(25 April 2007)
Quantum description of elastic collisions
between ultracold atoms
The basic ingredients for a mean-field description of
gaseous Bose Einstein condensates
Lecture 2
(27 April 2007)
Quantum theory of Feshbach resonances
How to manipulate atom-atom interactions in a
ultracold quantum gas
2
A few general references
1 – L.Landau and E.Lifshitz, Quantum Mechanics, Pergamon, Oxford (1977)
2 – A.Messiah, Quantum Mechanics, North Holland, Amsterdam (1961)
3 – C.Cohen-Tannoudji, B.Diu and F.Laloë, Quantum Mechanics, Wiley,
New York (1977)
4 – C.Joachin, Quantum collision theory, North Holland, Amsterdam (1983)
5 – J.Dalibard, in Bose Einstein Condensation in Atomic Gases, edited by
M.Inguscio, S.Stringari and C.Wieman, International School of Physics
Enrico Fermi, IOS Press, Amsterdam, (1999)
6 – Y. Castin, in ’Coherent atomic matter waves’, Lecture Notes of Les
Houches Summer School, edited by R. Kaiser, C. Westbrook,
and F. David, EDP Sciences and Springer-Verlag (2001)
7 – C.Cohen-Tannoudji, Cours au Collège de France, Année 1998-1999
http://www.phys.ens.fr/cours/college-de-france/
8 – C.Cohen-Tannoudji, Compléments de mécanique quantique, Cours de
3ème cycle, Notes de cours rédigées par S.Haroche
http://www.phys.ens.fr/cours/notes-de-cours/cct-dea/index.html/
9 – T.Köhler, K.Goral, P.Julienne, Rev.Mod.Phys. 78, 1311-1361 (2006)
3
1 - Introduction
Outline of lecture 1
2 - Scattering by a potential. A brief reminder
• Integral equation for the wave function
• Asymptotic behavior. Scattering amplitude
• Born approximation
3 - Central potential. Partial wave expansion
• Case of a free particle
• Effect of the potential. Phase shifts
• S-Matrix in the angular momentum representation
4 - Low energy limit
• Scattering length a
• Long range effective interactions and sign of a
5 - Model used for the potential. Pseudo-potential
• Motivation
• Determination of the pseudo-potential
• Scattering and bound states of the pseudo-potential
• Pseudo-potential and Born approximation
4
Interactions between ultracold atoms
At low densities, 2-body interactions are predominant and can be
described in terms of collisions. We will focus here on elastic collisions
(although inelastic collisions and 3-body collisions are also important
because they limit the achievable spatial densities of atoms).
Collisions are essential for reaching thermal equilibrium
At very low temperatures, mean-field descriptions of degenerate
quantum gases depend only on a very small number of collisional
parameters. For example, the shape and the dynamics of Bose
Einstein condensates depend only on the scattering length
Possibility to control atom-atom interactions with Feshbach
resonances. This explains the increasing importance of ultracold
atomic gases as simple models for a better understanding of
quantum many body systems
Purpose of these lectures: Present a brief review of the concepts of
atomic and molecular physics which are needed for a quantitative
description of interactions in ultracold atomic gases.
5
Notation
)
(
Two atoms, with mass m, interacting with a 2-body interaction
G
G
potential V r1 − r2
In lecture 1, we ignore the spins degrees of freedom. They will be
taken into account in lecture 2.
)
(
p12
p22
G
G
(1.1)
H =
+
+ V r1 − r2
2m
2m
Change of variables
G
G
G
G
G
G
RG = ( r1 + r2 ) 2
PG = p1 + p2 Center of mass variables
M = m1 + m2 = 2m Total mass
G
G
G
G
G
G
r = r1 − r2
p = ( p1 − p2 ) 2 Relative variables
μ = m1m2 ( m1 + m2 ) = m 2
Reduced mass
G2
G2
G
H = PG 2 M + p
2μ + V ( r )
(1.2)
Hamiltonian
/
/
/
Hamiltonian of a free
particle with mass M
/ /
/
H CM
(
)
H rel
Hamiltonian of a “fictitious” particle
with mass μ, moving in V(r)
6
Finite range potential
G
Simple case where V ( r ) = 0 for r > b
b is called the range of the potential
One can extend the results obtained in this simple case to
potentials decreasing fast enough with r at large distances.
For example, for the Van der Waals interactions between atoms
decreasing as C6 / r6 for large r, one can define an effective range
bVdW
⎛ 2μ C6 ⎞
= ⎜ 2 ⎟
⎝ = ⎠
/
1
4
(1.3)
See for example Ref. 5
7
Outline of lecture 1
1 - Introduction
2 - Scattering by a potential. A brief reminder
• Integral equation for the wave function
• Asymptotic behavior. Scattering amplitude
• Born approximation
3 - Central potential. Partial wave expansion
• Case of a free particle
• Effect of the potential. Phase shifts
• S-Matrix in the angular momentum representation
4 - Low energy limit
• Scattering length a
• Long range effective interactions and sign of a
5 - Model used for the potential. Pseudo-potential
• Motivation
• Determination of the pseudo-potential
• Scattering and bound states of the pseudo-potential
8
Scattering by a potential. A brief reminder
Shrödinger equation for the relative particle (with E>0)
⎡ =2
=2k 2
G ⎤ G
G
(1.4)
⎢ − 2 μ Δ + V ( r ) ⎥ψ (r ) = Eψ (r ) E = 2 μ
⎣
⎦
G
G
G
2μ
⎡Δ + k 2 ⎤ ψ ( r ) = 2 V ( r ) ψ ( r )
⎣
⎦
=
Green function of Δ + k2
G
G
(1.5)
⎡Δ + k 2 ⎤ G ( r ) = δ ( r )
⎣
⎦
The boundary conditions for G will be chosen later on
(1.6)
:
Integral equation for the solution of Shrödinger equation
G
ϕ0 ( r ) Solution of the equation without the right member
G
⎡ Δ + k 2 ⎤ ϕ0 ( r ) = 0
(1.7)
⎣
⎦
G
G
G G
G
G
(1.8)
ψ ( r ) = ϕ0 ( r ) + ∫ d3 r ′ G ( r − r ′ ) V ( r ′ ) ψ ( r ′ )
9
Choice of boundary conditions
.
.
G
G G
G
ik r
ikκ r
ϕ0 ( r ) = e
= e
G
κ = k
G
/
We choose for ϕ0 a plane
wave with wave vector k
G
k
(1.9)
and we choose, for the Green function G, boundary conditions
corresponding to an outgoing spherical wave (see Ref.2, Chap.XIX)
G G
i k r −r′
G G
1 e
(1.10)
G+ ( r − r ′ ) = −
G G
4π r − r ′
)
(
)
(
.
)
(
We thus get the following solution for Schrödinger
equation
G G
i k r − r′
G G
G
G
1
e
+ G
3
G r′
′
′
(1.11)
ψ k+G r = ei k r −
r
V
r
ψ
d
G
G
k
∫
4π
r − r′
If V has a finite range b, the integral over r’ is restricted to a finite
range and we can write:
G G
G G
G G
′
′
If r b, r − r r − r . n with n = r / r
i kr
G G
G
G
e
+ G
i k .r
ψ kG ( r ) e − f ( k , κ , n)
(1.12)
r
G G
G G
G + G
2μ
3
- i k n.r ′
′
′)ψ kG ( r ′)
f ( k , κ , n) = −
r
V
(
r
d
e
2 ∫
4π =
10
Scattering state with an outgoing spherical wave
)
(
Asymptotic behavior for large r
/
)
(
p
x
e
)
,
,
(
)
(
)
(
)
,
,
(
Scattering amplitude f
(
.
-
G G
G G
G
2μ
3
i k′ r′
+ G
G r′
′
′
f k κ n = −
r
V
r
ψ
d
e
k
2 ∫
G4π = G
G
We have put k ′ = k n = k r r
.
p
x
e
G
The state ψ r is a solution of the Schrodinger equation behaving
GG
for large r as the sum of an incoming plane wave
i k r and of
G G
an outgoing spherical wave f k κ n
ikr r
+
G
k
)
(1.13)
/
)
,
,
(
/
G G
of the outgoing spherical wave in the
f k κ n is the
G amplitude
G
G
direction of k ′ = k n = k rG r. It depends only
G on k and on the
polar angles θ and ϕ of k ′ with respect to k
Differential cross section
)
,
,
(
/
Comparing the fluxes along k and k’, one gets :
G G 2
dσ d Ω = f k κ n
(1.14)
11
Born approximation
In the scattering amplitude, the potential V appears explicitly
)
(
)
(
)
,
,
(
.
∫ d r′ e
3
)
. )
)
( (
( p
x
.
e )
(
)
,
,
(
G G
2μ
f k κ n = −
4π = 2
-
G G
G G
G
2μ
+ G
3
i k′ r′
f k κ n = −
V r ′ ψ kG r ′
d r′ e
(1.15)
2 ∫
4π =
G
To lowest order in V, one can thus replace ψ k+G r ′ by the zeroth
GG
ikr
order solution of the Schrodinger equation
G G G
i k − k′ r′
G
V r′
(1.16)
This is the Born approximation
In this approximation, the scattering amplitude is proportional to
the spatial Fourier transform of the potential
12
Low energy limit
)
(
)
,
,
(
)
(
.
-
The presence of V(r’) in the scattering amplitude
G G
G G
2μ
G
+ G
3
i k ′ r′
G r′
′
′
f k κ n = −
r
V
r
ψ
d
e
k
2 ∫
4π =
restricts the integral over r’ to a finite range r’< b
(1.17)
G
− i k ′.r ′
)
, .
,
(
)
, (
If kb 1 , one can replace e
by 1. The scattering amplitude
G G
G +G G
2μ
3
′
d r V ( r ′)ψ k ( r ′)
f ( k , κ , n) = −
(1.18)
2 ∫
4π =
G
′
then no longer depends on the direction
G of the scattering vector k .
It is spherically symmetric even if V ( r ) is not.
G G
When k → 0
f k κ n → − a
ikr
G G
(1.19)
G
e
a
+
ik r
ψ kG r e
− a
→ 1−
r
r
a is a constant, called “scattering length”, which will be
discussed in more details later on
13
Another interpretation of the outgoing scattering state
/
Another expression for this state (see refs. 4 and 8)
1
(1.20)
V ψ k+G
T = p2 2μ
ε → 0+ E − T + i ε
For ε non zero but very small, ψ k+G appears as the state obtained
ψ k+G = ϕkG + Lim
/
at t = 0 by starting from the free state ϕkG at t = −∞ and by
switching on slowly V on a time interval on the order of = ε
Ingoing scattering state
)
(
G
e
− G
G r′
′
V
r
ψ
G
G
k
r − r′
)
3
d
∫ r′
(
.
)
G G
G
1
ik r
−
r = e
4π
(
ψ
−
G
k
G G
−i k r − r′
(1.21)
/
1
−
G
G
V ψ k−G
ψ k = ϕ k + Lim
ε → 0+ E − T − i ε
If one starts from such a state at t = 0 and if one switches
off V slowly on a time interval on the order of = ε , one gets
the free state ϕkG at t = +∞
14
S - Matrix
:
j
S ϕkG
i
= Lim ϕ kG
t1 → −∞
t 2 → +∞
U t 2 t1
j
ϕkG
i
,
S ji = ϕkG
)
,
(
Definition
(1.22)
U evolution operator in interaction representation
One can show that
S ji = ψ k−G ψ k+G
j
(1.23)
i
Qualitative interpretation
,
V is switched on slowly (time scale ħ/ε) between -∞ and 0, and
then switched off slowly (time scale ħ/ε) between 0 and +∞
One starts from ϕi at t = -∞ and one looks for the probability
amplitude to be in ϕj at t = +∞
From t = −∞ to t = 0 the initial free state ϕ i is transformed
into ψ i+ Since the evolution operator is unitary, and since ψ −j
.
,
.
transforms into ϕ j from t = 0 to t = +∞ ψ −j ψ i+ is the
amplitude to find the system in the free state ϕ j at t = +∞ if
one starts from ϕ i at t = −∞
15
Outline of lecture 1
1 - Introduction
2 - Scattering by a potential. A brief reminder
• Integral equation for the wave function
• Asymptotic behavior. Scattering amplitude
• Born approximation
3 - Central potential. Partial wave expansion
• Case of a free particle
• Effect of the potential. Phase shifts
• S-Matrix in the angular momentum representation
4 - Low energy limit
• Scattering length a
• Long range effective interactions and sign of a
5 - Model used for the potential. Pseudo-potential
• Motivation
• Determination of the pseudo-potential
• Scattering and bound states of the pseudo-potential
16
Central potential
V depends only on r
1D radial Schrödinger equation
One looks for solutions of the form
G
G
ϕk l m ( r ) = Rk l ( r )Yl m ( n )
If we put
Rk l ( r ) =
G G
n = r /r
(1.24)
uk l ( r )
r
with the boundary condition
(1.25)
uk l (0) = 0
(1.26)
)
(
)
(
)
(
one gets for ukl the following 1D radial equation
=2 l l + 1
2μ
r2
)
(
⎡ d2
⎤
l l +1
2μ
2
(1.27)
− 2 V r ⎥ uk l r = 0
⎢ 2 + k −
2
=
r
⎣ dr
⎦
1D Schrödinger equation for a particle moving in a potential which
is the sum of V and of the centrifugal barrier
(1.28)
17
Case of a free particle (V=0)
The solutions of the Schrödinger equation are:
)
(
π
)
2k 2
(
)
(
)
(
ϕk0l m
G
r =
G
jl kr Yl m n
(1.29)
)
(
n
i
s
)
(
!
!
) )
(
(1.30)
)
/
(
n
i
s
)
(
)
(
)
(
G
ϕk0l m r
(
)
(
where the jl are the spherical Bessel functions of order l
kr l
1
π
jl kr jl kr kr − l
r → 0 2l + 1
r →∞ kr
2
For large r we thus have
2
G
Yl m n
)
/
)
/
(
)
(
(
kr − l π 2
r →∞
π
r
(1.31)
− i kr − l π 2
i kr − l π 2
⎡
⎤
e
−
e
G ⎣
2
⎦
=
Yl m n
2ir
π
= outgoing spherical wave + ingoing spherical wave
)
(
)
(
)
(
These functions form an orthonormal set (see Appendix)
0
= δ k − k ′ δ ll ′δ mm ′
ϕk0′l ′m ′ ϕklm
(1.32)
18
Expansion of a plane wave in free spherical waves
e
G G
G
G
k′ k = δ k − k′
G G
ik r
)
2
(
−3
.
G G
r k = 2π
/
)
(
Plane wave
(1.33)
The factor (2π) -3/2 is introduced for the orthonormalization
l =0 m = −l
G
i Ylm κ ϕ
l
0
klm
G
r
G
with κ = k
G
The transformation from the orthonormal basis
(1.34)
k
G
k
{ } to t
)
(
*
} is given by the matrix
)
0
ϕklm
(
)
(
ϕ
0
k ′l ′m ′
{
)
(
he orthonormal basis
)
l =0 m = −l
i Ylm
G
G
n Ylm κ jl kr
/
∑∑
∑∑
l
)
(
)
(
)
(
*
m =+l
m =+l
(
)
(
*
4π
∞
)
(
1
=
k
∞
2
)
(
= 2π
−3
)
(
e
G G
ik r
/
)
2
(
−3
.
(
2π
/
)
One can show that:
G
G
1
k = δ k − k′
Yl ′m ′ κ
k
(1.35)
19
Effect of a potential. Phase shifts
We come back to the Schrödinger equation with V≠0.
Consider, for r large, an incoming wave exp[-i(kr-lπ/2)]. Since the
reflection coefficient of V is 1 (conservation of the norm), the
reflected outgoing wave has the same modulus and has just
accumulated a phase shift with respect to the V=0 case. The
superposition of the 2 waves is thus a shifted sinusoid.
)
(
/
n
i
s
)
(
We conclude that there is a set of solutions of the Schrödinger
equation with V≠0 which behave for large r as:
)
(
⎡⎣ kr − l π 2 + δ l k ⎤⎦
ϕklm
Ylm
r →∞
π
r
One can show that these functions are orthonormalized (see
Appendix)
ϕ k ′l ′m ′ ϕklm = δ k − k ′ δ ll ′δ mm ′
G
r
2
G
n
(1.36)
)
(
(1.37)
They don’t form a basis if there are also bound states in the
potential V.
20
Partial wave expansion of the outgoing scattering state
)
(
Consider the linear superposition of the states ϕ klm with the same
coefficients as those appearing in the expansion (1.34) of the plane
0
wave on the ϕklm
, each state being multiplied by the phase factor
iδ l We will show that such a state is nothing but the outgoing
state ψ kG+ (multiplied by 2π −3 2 for having an orthonormalized state)
(
/
)
.
)
(
p
x
e
∑∑
l =0 m = −l
m =+l
∑∑
l =0 m = −l
)
(
*
m =+l
G
i l Ylm κ e iδ l ϕ klm
)
(
=
∞
∞
)
(
ψ kJG+
1
=
k
0
ϕ klm
(1.38)
G iδ
k e l ϕ klm
)
(
Before demonstrating this identity, let us discuss its physical
meaning.The outgoing scattering state is obtained by switching
0
on slowly V onG the free state. Each spherical wave ϕ klm
of the
expansion of k is transformed into ϕklm , but in addition it acquires
a phase factor eiδl which depends on l and which thus varies
from one spherical wave of the expansion to another one
.
21
Demonstration
For large r, the linear superposition introduced in (1.38) behaves as:
i l Ylm κ
2
π
Ylm
2
kr − l π
r
2
(
+
e
(
2 iδ l
)
− 1 e − il π
2i
)
/
l = 0 m = −l
G
⎡
G ⎢
n
⎢
⎣
− i kr − l π
)
/
∑∑
)
(
*
m = +l
)
(
1
k
∞
)
− e (
(1.39)
2ir
2 iδ
i ( kr − l π 2 ) × ⎡1 + e l − 1 ⎤
⎣
⎦
(
n
i
s
)
(
we get:
/
2 + 2δ l ) =
2 + 2δ l )
/
− lπ
Ylm
/
π
Ylm κ
i kr − l π
G e (
n
p
x
e
2
)
(
G
/
l = 0 m = −l
)
(
∑∑ i
i ( kr
l
*
m = +l
)
(
∞
p
x
e
1
k
Using
2
(1.40)
⎤
e ikr ⎥
r ⎥
⎦
(1.41)
)
,
,
(
.
/
)
(
The contribution of the first term of the bracket is nothing but the
asymptotic expansion of the plane wave in spherical waves.
The second term gives an outgoing spherical wave
ikr
G G
⎡
⎤
G
G
e
−3 2
ik r
(1.42)
2π
+ f k κ n
⎢e
⎥
r ⎦
⎣
This demonstrates that the state given in (1.38) is an outgoing
scattering state and gives in addition the expression of the amplitude f
22
Scattering amplitude in terms of the phase shifts
)
(
*
)
(
n
i
s
)
,
,
(
)
s
o
c
(
n
i
s
)
(
G G
G
G
4π ∞ m = + l i δ l
e
f k κ n =
δ l Ylm n Ylm κ
∑
∑
k l =0 m = −l
1 ∞
i δl
l
2
1
e
=
+
δ l Pl
θ
∑
k l =0
where Pl
θ is a Legendre polynomial and where θ
2
G
G
is the angle between n and κ Integrating f over the
G
polar angles of κ gives the scattering cross section
∞
4π
2
σ k = ∑ σl k
σ l k = 2 2l + 1
δl k
k
l =0
(1.43)
)
s
o
c
(
.
)
(
n
i
s
)
(
)
(
)
(
)
(
(1.44)
Scattering of 2 identical particles
)
2π
(
0
2
fk θ → fk θ + ε fk π − θ
ε = +1 −1 for bosons (fermions)
)
(
∫
π
π-θ
n
i
s
σ total =
/
θ
)
)
(
)
(
(
)
(
Quantum interference between
2 different paths
θ dθ f k θ + ε f k π − θ
2
(1.45)
23
Partial wave expansion of the ingoing scattering state
∑∑
l =0 m = −l
G
i Ylm κ e
l
− iδ l
ϕ klm =
∞
m =+l
∑∑
l =0 m = −l
)
(
m =+l
.
)
∞
(
p
x
e
1
=
k
)
(
ψ
−
JG
k
) )
(
*
(
p
x
e
ψ kG− is given by a linear superposition of the states ϕ klm analogous
to the one introduced for ψ kG+ each state being now multiplied by
−iδ l instead of
iδ l
the phase factor
ϕ
0
klm
G − iδ
k e l ϕ klm
The demonstration of this identity is similar to the one given
above for the outgoing scattering state
(1.46)
)
(
If we start from thisG state and if we switch off V slowly, it transforms
0
into the free state k . Each wave ϕ klm is transformed into ϕklm
, but
in addition its phase factor changes from e − iδl to 1 which corresponds
to acquiring a phase factor e + iδl
Finally, when we go from t = −∞ to t = +∞ switching on and then
0
0
switching off V slowly, we start from ϕklm
and we end in ϕklm
acquiring
a global phase factor e + iδl × e + iδl = e +2iδl
,
)
(
)
(
24
S – Matrix in the angular momentum representation
G
G
= k ′ S k = ψ k−G′ ψ k+G
SkG′ kG
(1.47)
)
(
)
(
We use the expansion of ψ kG+ and ψ kJG−′ in spherical waves
∞ m′ = + l ′
∞ m = +l
G − iδ
G iδ
0
−
0
+
l
ψ kJG′ = ∑ ∑ ϕ k ′l ′m′ k ′ e l ϕ k ′l ′m′
ψ kJG = ∑ ∑ ϕklm k e ϕklm
l ′ = 0 m′ = − l ′
l = 0 m = −l
(1.48)
)
(
)
(
This gives a first expression of S kG′ kG
∞ m = + l ∞ m′ = + l ′ G
G
iδ l
+ iδ l
−
+
0
0
G
G
G
G
Sk ′ k = ψ k ′ ψ k = ∑ ∑ ∑ ∑ k ′ ϕ k ′l ′m ′ e
ϕk ′l ′m ′ ϕ klm e ϕklm k
l = 0 m = − l l ′ = 0 m′ = − l ′
G G
)
(
(1.49)
δ k − k ′ δ l l ′δ m m ′
On the other hand, a change of basis gives for S kG′ kG
m′ = + l ′
∑∑∑ ∑
ϕ
0
k ′l ′m ′
S ϕ
0
klm
ϕ
0
klm
G
k
,
l = 0 m = − l l ′ = 0 m′ = − l ′
G
k ′ ϕk0′l ′m ′
)
(
∞
)
(
m =+l
)
(
∞
)
(
SkG′ kG
G
G
= k′ S k =
(1.50)
+2 i δ
0
= e
ϕk0′l ′m ′ S ϕklm
δ k − k ′ δ l l ′ δ m m′
(1.51)
we get
)
l
(
)
(
)
(
Comparing the 2 expressions obtained for S kG′ kG
)
(
p
x
e
which shows that the S -matrix is diagonal in the angular momentum
representation, with diagonal elements
2iδ l clearly showing
the unitarity of S
25
Outline of lecture 1
1 - Introduction
2 - Scattering by a potential. A brief reminder
• Integral equation for the wave function
• Asymptotic behavior. Scattering amplitude
• Born approximation
3 - Central potential. Partial wave expansion
• Case of a free particle
• Effect of the potential. Phase shifts
• S-Matrix in the angular momentum representation
4 - Low energy limit
• Scattering length a
• Long range effective interactions and sign of a
5 - Model used for the potential. Pseudo-potential
• Motivation
• Determination of the pseudo-potential
• Scattering and bound states of the pseudo-potential
26
Central potential. Low energy limit
)
(
rl =
)
r
l l +1
)
2μ
(
k rl =
/
=2k 2
(
l l + 1 =2
=2 k 2
=
2
2μ
2 μ rl
l l + 1 =2
2μ r 2
rl
)
(
Suppose first V=0. The centrifugal barrier in the 1D Schrödinger
equation prevents the particle from approaching near the region r=0
l l + 1 dB
If the range b of the potential is small enough, i.e. if
b dB
(1.52)
.
a particle with l ≠ 0 cannot feel the potential
)
(
)
(
Only l = 0 wave will feel V "s-wave scattering"
⎡ d2
⎤
2μ
2
⎢ 2 + k − 2 V r ⎥ uk 0 r = 0
=
⎣ dr
⎦
(1.53)
27
)
(
n
i
s
Scattering length
.
)
(
n
i
s
⎡⎣ kr + δ 0 k ⎤⎦
For r large enough, uk 0 = uk varies as
⎡⎣ kr + δ 0 k ⎤⎦ extending uk for all r
Let vk be the function
Let P be the intersection point of vk with the r -axis which is the
closest from the origin.By definition, the scattering length a is the
limit of the abscissa of P when k → 0 (see figure)
)
(
s
o
c
)
(
n
i
s
)
(
n
i
s
)
(
Expansion of vk in powers of kr near kr = 0
vk r =
δ 0 k + kr
⎣⎡ kr + δ 0 k ⎤⎦ kr→
→0
1
δ0 k
Abscissa of P : −
k
−
δ0 k
π
π
a =
−
≤ δ0 k ≤ +
k →0
k
2
2
(1.54)
)
(
m
i
l
)
( )
n (
a
t n
a
t
δ0 k
(1.55)
28
Scattering length (continued)
)
(
)
(
Limit k=0
⎡ d2
⎤
2μ
⎢ 2 − 2 V r ⎥ u0 r = 0
=
(1.56)
⎣ dr
⎦
)
(
Far from r=o, the solution of
the S.E. is a straight line and
v0 r ∝ r − a
(1.57)
The abscissa of Q is equal to a
⇒
For identical bosons
σ l = 0 k = 4π
σ l = 0 k = 4π a 2
σ l = 0 k = 8π a 2
2
)
(
)
(
k →0
⇒
) )
( ( )
(
)
(
n
i
s
)
(
)
(
4π
2
σ l k = 2 2l + 1
δl k
k
δ0 k − k a
n
i
s
Scattering cross section
δ0 k
k
2
(1.58)
(1.59)
(1.60)
29
Scattering length for square potentials
Square potential barriers
/
Square barrier of
height V0 = = 2 k02 2 μ
and width b
)
(
V r
For r < b and k = 0
u0′′ r = k02 u0 r
The curvature of u0 is
positive and u0 r = 0 = 0
)
(
r
)
(
b
)
(
a
v0 r ∝ r − a
)
(
0
For r > b and k = 0
u0 r = v0 r ∝ r − a
)
(
)
(
u0 r
)
(
V0
We conclude that the scattering length is always positive and
smaller than the range b of the potential
0 ≤ a ≤ b
When V0 → ∞ (hard sphere potential)
a → b
30
Scattering length for square potentials (continued)
)
(
)
(
For r > b and k = 0
u0 r = v0 r ∝ r − a
)
(
v0 r ∝ r − a
)
(
b
)
(
0
r
)
(
For r < b and k = 0
u0′′ r = − k02 u0 r
The curvature of u0 is
negative and u0 r = 0 = 0
u0 r
a
2μ
)
(
V r
V0 = − = 2 k02
/
Square potential wells
)
(
V0
If V0 is small enough so that there is no bound state in the potential
well, the curvature of u0 for r < b is small and a is negative
.
When V0 increases, the curvature of u0 for r < b increases in
absolute value and a → −∞ Then a switches suddenly to + ∞
and decreases. This divergence of a corresponds to the appearance
of the first bound state in the potential well
31
Square potential wells (continued)
Variations of a with k0
figure taken from Ref.5
/
)
(
When the depth of the potential well increases, divergences
of a occur for all values of V0 such that k0 b = 2n + 1 π 2
corresponding to the appearances of successive bound states
in the potential well.
These divergences of a which goes from − ∞ to + ∞ are
called "zero-energy" resonances
32
Long range effective interactions and sign of a
)
(
The scattering length determines how the long range behavior of the
wave functions is modified by the interactions. To understand how
the sign of a is related to the sign of the effective long range
interactions, it will be useful to consider the particle enclosed in a
spherical box with radius R, so that we have the boundary condition
u0 R = 0
(1.61)
leading to a discrete energy spectrum
.
.
.
,
,
)
/
)
(
)
(
(
n
i
s
In the absence of interactions (V=0), the normalized eigenstates and
the eigenvalues of the 1D Schrödinger equation are:
1
Nπ r R
= 2 N 2π 2
0
EN =
N = 1 2
ψN r =
2
2π R
r
2μ R
(1.62)
Nπ r R
)
/
(
n
i
s
Figure corresponding
to N=3
0
R
r
33
V = 0
R
V ≠ 0
a > 0
a
R
The dotted line is the sinusoid outside the range of the potential
It has a shorter wavelength than for V=0, and thus a larger wave number k.
The kinetic energy in this region, which is also the total energy, is larger
V ≠ 0
a < 0
a
R
The dotted line is the sinusoid outside the range of the potential
It has a longer wavelength than for V=0, and thus a smaller wave number k.
The kinetic energy in this region, which is also the total energy, is smaller
34
Correction to the energy to first order in a
R
a
/
)
(
/
)
δ E N = E N′ − E N
= EN R
2
R − a
)
Finally, we have
k
2
(
) /
→
/
2μ
E N′ = E N k ′2
(
/
/
2
/
EN = = k
2
(
//
For the state ψ N0 , we have R = N λ 2
Because of the interactions, these N half wavelengths occupy
now a length R − a so that
N
→ λ′ = 2 R − a
λ = 2R N
k = 2π λ
→ k ′ = 2π λ ′ = k R R − a
2
⎛
2a ⎞
E N ⎜1 +
⎟
R
⎝
⎠
2a
= 2π 2 N 2
=
EN =
a
3
R
μR
(1.63)
Long range effective interactions are - repulsive if a > 0
- attractive if a < 0
35
Outline of lecture 1
1 - Introduction
2 - Scattering by a potential. A brief reminder
• Integral equation for the wave function
• Asymptotic behavior. Scattering amplitude
• Born approximation
3 - Central potential. Partial wave expansion
• Case of a free particle
• Effect of the potential. Phase shifts
• S-Matrix in the angular momentum representation
4 - Low energy limit
• Scattering length a
• Long range effective interactions and sign of a
5 - Model used for the potential. Pseudo-potential
• Motivation
• Determination of the pseudo-potential
• Scattering and bound states of the pseudo-potential
36
Model used for the potential V(r)
Why not using the exact potential?
The interaction potential is very difficult to calculate exactly.
A small error in V can introduce a very large error on the scattering
length deduced from this potential.
Mean field description of ultracold quantum gases require in general a
first order treatment of the effect of V (Born approximation). But Born
approximation cannot be in general applied to the exact potential
Approach followed here
The motivation here is not to calculate the scattering length. This
parameter is supposed known experimentally. We are interested in
the derivation of the macroscopic properties of the gas from a mean
field description using a single parameter which is a.
The key idea is to replace the exact potential by a “pseudo-potential”
simpler to use than the exact one and obeying 2 conditions:
- It has the same scattering length as the exact potential
- It can be treated with Born approximation so that mean field
descriptions of its effects are possible
37
Determination of the pseudo-potential
Derivation “à la” H.Bethe and R.Peierls (Y.Castin, private communication)
)
(
)
(
)
(
We add to the 3D Schrödinger equation of a free particle (V=0) a
term proportional to a delta function
G
G
G
=2
=2 k 2
(1.64)
−
Δψ r + C δ r =
ψ r
2μ
2μ
To determine the coefficient C, we impose to the solution of this
equation to coincide with the extension to all r of the asymptotic
⎡⎣ kr + δ 0 k ⎤⎦ r of the true wave function u0 r r
behavior
In particular, for k small enough, one should have:
G
(1.65)
ψ r Br −a r
/
)
(
/
)
(
n
i
s
/
)
(
,
)
(
)
/
(
)
)
( (
r →0
Inserting (1.65) into 1.64 and using Δ 1 r = −4π δ r we get
an equation containing a delta function multiplied by a coefficient
4π = 2
C −
aB
2μ
which must vanish. This gives the coefficient C appearing in (1.64)
4π = 2
4π = 2
(1.66)
C = gB
where
g =
a =
a
38
m
2μ
Determination of the pseudo-potential (continued)
)
/
(
It will be more convenient to express C=gB, not in terms of the
coefficient B appearing in the wave function ψ = B(r-a)/r of equation
(1.65), but in terms of the wave function ψ itself. We use for that
⎡d
⎤
B = ⎢ rψ ⎥
(1.67)
⎣ dr
⎦r =0
Equation (1) can be rewritten as:
G
G =2 k 2 G
=2
−
Δψ ( r ) + Vpseudo ψ ( r ) =
ψ (r )
(1.68)
2μ
2μ
G
G d
G
where
Vpseudo ψ ( r ) = g δ ( r ) [ r ψ ( r )]
(1.69)
dr
Vpseudo is called the pseudo-potential. The term ⎡⎣ d dr r ⎤⎦ regularizes
G
the action of δ r when it acts on functions behaving as 1 r near
r = 0 For functions which are regular in r = 0, Vpseudo has the
G
same effect as g δ r
G
G
G
ψ r = u r r with u 0 ≠ 0 ⇒ Vpseudo ψ r = g u′ 0 δ r
G
(1.70)
G
G
G
ψ r regular in r = 0
⇒ Vpseudo ψ r = g ψ 0 δ r
/
)
(
.
:
)
(
)
(
)
(
)
(
)
(
) )
( (
)
(
/
)
(
) )
( (
39
Scattering states of the pseudo-potential
,
We are looking for solutions of equation (1.68) with E > 0
,
)
(
,
For l ≠ 0 the centrifugal barrier prevents the particle
from
G
approaching r = 0 and one can show that ψ 0 = 0 so that,
G
according to (1.70), Vpseudoψ r = 0 Vpseudo gives only s-scattering
G
and one can write ψ (r ) = u0 r r where u0 0 can be ≠ 0
2
⎡ u0 0
u0 r
u0 r − u0 0 ⎤
1 d u0
Δ
= Δ⎢
+
(1.71)
⎥ = −4π u0 0 δ r +
2
r
r
r dr
⎣ r
⎦
The Schrödinger equation for u0 becomes:
u0′′ r ⎤
G
G ′
=2 ⎡
= 2 k 2 u0 r
(1.72)
⎢ −4π u0 0 δ r +
⎥ + gδ r u0 0 =
−
2μ ⎢
2μ
r ⎥
r
⎣
⎦
G
Cancelling the term proportional to δ r and the term independant
G
of δ r we get 2 equations:
.
)
(
.
)
/ (
) )
(
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
μ
= −a
2
2π =
)
(
u0′ 0
= −g
)
(
, ) )
) ( (
(
u0 0
u0′′ r + k 2 u0 r = 0
(1.73)
40
Scattering states of the pseudo-potential (continued)
Inserting this solution into the first equation gives:
δ0 = − k a
(1.75)
n
i
s
(1.74)
)
(
The solution of the second equation can be written:
u0 r =
( k r + δ0 )
n
a
t
n
i
s
)
(
On the other hand, the s-wave scattering amplitude is equal to:
1 i δ0
(1.76)
f0 k =
e
δ0
k
Using equation (1.75) giving tanδ0 finally gives after simple algebra:
a
f0 k = −
(1.77)
1+ ika
Vpseudo is proportional to a A first order treatment of Vpseudo
thus gives the correct result for the scattering amplitude in
the zero energy limit (k a = 0 This shows that Born
approximation can be used with Vpseudo for ultracold atoms
)
(
.
.
)
The 2 conditions imposed above on Vpseudo are thus fulfilled
41
The unitary limit
)
(
From the expression of the scattering amplitude obtained above,
we deduce the scattering amplitude for identical bosons
8π a 2
σ k =
(1.78)
1 + k 2a 2
which is valid for all k.
)
(
The low energy limit ka 1 gives the well known result:
σ (k ) 8π a 2
ka 1
(1.79)
)
(
There is another interesting limit, corresponding to high energy,
or strong interaction ka 1 leading to result independent of a:
8π
σ (k ) 2
ka 1 k
This is the so called “unitary limit”
(1.80)
42
Bound state of the pseudo-potential
)
(
u0′ 0
)
(
) )
( (
u0 0
/
/
The calculation is the same as for the scattering states, except that
we replace the positive energy = 2 k 2 2 μ by a negative one -= 2κ 2 2 μ
The 2 equations derived from the Schrodinger equation are now:
u0′′ r − κ 2 u0 r = 0
= −a
)
(
The solution of the second equation (finite for r→∞) is:
u0 r = e −κ r
which inserted into the first equation gives:
(1.81)
(1.82)
/
κ = 1 a
(1.83)
p
x
e
The pseudo-potential thus has a bound state with an energy
=2
E = −
(1.84)
2μ a 2
and a wave function:
⎛ r⎞
(1.85)
⎜− a ⎟
⎝
⎠
43
Energy shifts produced by the pseudo-potential
)
(
)
(
We come back to the problem of a particle in a box of radius R.
We have calculated above the energy shifts of the discrete energy
levels of this particle produced by a potential characterized by a
scattering length a. To first order in a, we found:
= 2π 2 N 2
(1.86)
δ EN =
a
3
μR
This result was deduced directly from the modification induced by
the interaction on the asymptotic behavior of the wave functions
and not from a perturbative treatment of V. We show now that:
δ E N = ψ N0 Vpseudo ψ N0
to first order in Vpseudo
(1.87)
which is another evidence for the fact that the effect of the pseudo
potential can be calculated perturbatively, which is not the case for
the real potential. For example, a hard core potential (V=∞ for r< a)
cannot obviously be treated perturbatively, but its scattering length
is a , and using a pseudo-potential with scattering length a allows
perturbative calculations.
44
Demonstration
)
/
)
(
)
(
(
n
i
s
The unperturbed normalized eigenfunctions of the particle in the
spherical box are:
1
Nπ r R
0
(1.47)
ψN r =
2π R
r
ψ N0 r is regular in r = 0 and
)
(
)
(
0 =
2π
Nπ
1
R
/
1
3
(1.88)
2
)
(
)
(
)
(
)
(
)
(
so that
)
(
)
(
ψ
0
N
G
V pseudoψ N0 r = gδ r ψ N0 0
)
(
δ EN = g ψ
)
(
We deduce that
0
N
0
2
= 2π 2 N 2
N 2π 2
= g
=
a
3
3
2π R
μR
(1.89)
(1.90)
which coincides with the result obtained above in (1.63).
45
Conclusion
Elastic collisions between ultracold atoms are entirely
characterized by a single number, the scattering length
Effective long distance interactions are attractive if a<0 and
repulsive if a>0
Giving the same scattering length as the real potential, the
pseudo-potential gives the good asymptotic behavior for the
wave function describing the relative motion of 2 atoms, and
thus correctly describes their long distance interactions
In a dilute gas, atoms are far apart. The pseudo-potential is
proportional to a and can be treated perturbatively. A first order
treatment of the pseudo-potential is the basis of mean field
description of Bose Einstein condensates where each atom
moves in the mean field produced by all other atoms.
Next step: can one change the scattering length?
46
Atom-Atom Interactions
in Ultracold Quantum Gases
Claude Cohen-Tannoudji
Lectures on Quantum Gases
Institut Henri Poincaré, Paris, 27 April 2007
Collège de France
1
Lecture 1
Quantum description of elastic collisions
between ultracold atoms
The basic ingredients for a mean-field description of
gaseous Bose Einstein condensates
Lecture 2
Quantum theory of Feshbach resonances
How to manipulate atom-atom interactions in a
quantum ultracold gas
2
A few general references
1 – L.Landau and E.Lifshitz, Quantum Mechanics, Pergamon, Oxford (1977)
2 – A.Messiah, Quantum Mechanics, North Holland, Amsterdam (1961)
3 – C.Cohen-Tannoudji, B.Diu and F.Laloë, Quantum Mechanics, Wiley,
New York (1977)
4 – C.Joachain, Quantum collision theory, North Holland, Amsterdam (1983)
5 – J.Dalibard, in Bose Einstein Condensation in Atomic Gases, edited by
M.Inguscio, S.Stringari and C.Wieman, International School of Physics
Enrico Fermi, IOS Press, Amsterdam, (1999)
6 – Y. Castin, in ’Coherent atomic matter waves’, Lecture Notes of Les
Houches Summer School, edited by R. Kaiser, C. Westbrook,
and F. David, EDP Sciences and Springer-Verlag (2001)
7 – C.Cohen-Tannoudji, Cours au Collège de France, Année 1998-1999
http://www.phys.ens.fr/cours/college-de-france/
8 – C.Cohen-Tannoudji, Compléments de mécanique quantique, Cours de
3ème cycle, Notes de cours rédigées par S.Haroche
http://www.phys.ens.fr/cours/notes-de-cours/cct-dea/index.html/
9 – T.Köhler, K.Goral, P.Julienne, Rev.Mod.Phys. 78, 1311-1361 (2006)
3
Outline of lecture 2
1 - Introduction
2 - Collision channels
• Spin degrees of freedom.
• Coupled channel equations
• Strong couplings and weak couplings between channels
3 - Qualitative interpretation of Feshbach resonances
4 - Two-channel model
• Two-channel Hamiltonian
• What we want to calculate
5 - Scattering states of the 2-channel Hamiltonian
• Calculation of the outgoing scattering states
• Asymptotic behavior. Scattering length
• Feshbach resonance
5 - Bound states of the 2-channel Hamiltonian
• Calculation of the energy of the bound state
• Calculation of the wave function
4
Feshbach Resonances
Importance of Feshbach resonances
Give the possibility to manipulate the interactions between ultracold
atoms, just by sweeping a static magnetic field
- Possibility to change from a repulsive gas to an attractive one and
vice versa
- Possibility to turn off the interactions → perfect gas
- Possibility to study a regime of strong interactions and correlations
- Possibility to associate pairs of ultracold atoms into molecules and
vice versa
Example of a recent breakthrough using Feshbach resonances (MIT)
Investigation of the BEC-BCS crossover
Ultracold atoms with interactions manipulated by Feshbach
resonances become a very attractive system for getting a better
understanding of quantum many body systems
5
Purpose of this lecture
- Provide a physical interpretation of Feshbach resonances in terms
of a resonant coupling of the state of a colliding pair of atoms to a
metastable bound state belonging to another collision channel
- Present a simple two-channel model allowing one to get analytical
predictions for the scattering states and the bound states of the
two colliding atoms near a Feshbach resonance
• How does the scattering length behave near a resonance?
• When can we expect broad resonances or narrow resonances?
• Are there bound states near the resonances? What are their binding
energies and wave functions?
- In addition to their interest for ultracold atoms, Feshbach resonances
are a very interesting example of resonant effect in collision processes
deserving to be studied for themselves
This lecture will closely follow the presentation of Ref.9:
T.Köhler, K.Goral, P.Julienne, Rev.Mod.Phys. 78, 1311-1361 (2006)
See also the references therein
6
Microscopic atom-atom interactions
,
Case of two identical alkali atoms
,
G G
Unpaired electrons for each atom with spins S1 S 2
G G
Nuclear spins I1 I 2
Hyperfine states f1 m f 1 f 2 m f 2
,
;
,
Born Oppenheimer potentials (2 atoms fixed at a distance r)
2 potential curves:
VT(r) for the triplet state S=1
VS(r) for the singlet state S=0
VT(r)
:
)
(
)
(
)
(
S quantum
G number
G
Gfor the total spin
S = S1 + S 2
V r = VS r PS + VT r PT
VS(r)
r
(2.1)
PS :Projector on S = 0 states
PT :Projector on S = 1 states
7
Microscopic atom-atom interactions (continued)
)
(
)
(
)
(
Electronic interactions
.
)
(
)
(
)
(
)
(
Vel r = VS r PS + VT r PT
G G
1
3
1
(2.2)
⎡
⎤
= VS r + VT r +
−
V
r
V
r
S
S
T
S
⎦ 1 2
4
4
2= 2 ⎣
This interaction depends on the electronic spins because of Pauli
principle (electrostatic interaction between antisymmetrized states).
It is called also “exchange interaction”
Does not depend on the orientation in space of the molecular axis
(line joining the nuclei of the 2 atoms)
Magnetic spin-spin interactions Vss
Dipole-dipole interactions between the 2 electronic spin magnetic
moments. Depends on the orientation in space of the molecular axis
el
+
s
Vel is much larger than Vss
=
Vs
int
V
V
Interaction Hamiltonian
(2.3)
8
Outline of lecture 2
1 - Introduction
2 - Collision channels
• Spin degrees of freedom.
• Coupled channel equations
• Strong couplings and weak couplings between channels
3 - Qualitative interpretation of Feshbach resonances
4 - Two-channel model
• Two-channel Hamiltonian
• What we want to calculate
5 - Scattering states of the 2-channel Hamiltonian
• Calculation of the outgoing scattering states
• Asymptotic behavior. Scattering length
• Feshbach resonance
5 - Bound states of the 2-channel Hamiltonian
• Calculation of the energy of the bound state
• Calculation of the wave function
9
Channels
,
,
,
{f
,
α
:
Two atoms entering a collision in a s-wave (A = 0) and in well defined
hyperfine and Zeeman states. This defines the “entrance channel” α
defined by the set of quantum numbers:
}
m f 1 f2 m f 2 A = 0
1
ψ
=
∑
α
)
(
The eigenstates of the total Hamiltonian with eigenvalues E can be
written:
G
α ψα r
(2.4)
)
,
(
where ψα(r) is the wave function in channel α whose radial part is of
the form:
Fα r E
r
Because the interaction has off diagonal elements between different
channels, the Fα do not evolve independently from each other
10
Coupled channel equations
The coupled equations of motion of the Fα are of the form:
)
,
(
)
,
(
∂2
2μ
Fα r E + 2 ∑ ⎡⎣ E δαβ − Vαβ ⎤⎦ Fβ r E = 0
2
= β
∂r
⎡
A A + 1 =2 ⎤
int
Vαβ = ⎢ E f m + E f m +
V
r
δ
+
⎥ αβ
αβ
2
i
f1
2
f2
2μ r
⎣
⎦
(2.5)
)
(
)
(
,
,
(2.6)
Solving numerically these coupled differential equations gives the
asymptotic behavior of Fα for large r from which one can determine
the phase shift δ0 and the scattering length in channel α.
Importance of symmetry considerations
The symmetries of Vel(r) and Vss determine if 2 channels can be
coupled by the interaction. In particular, if 2 channels can be
coupled by Vel, the Feshbach resonance which can appear due to
this coupling will be broad because Vel is large. If the symmetries
are such that only Vss can couple the 2 channels, the Feshbach
resonance will be narrow.
11
Examples of symmetry considerations
If the magnetic field B0 is the only external field, the projection M of
the total angular momentum along the z-axis of B0 is conserved.
M = m f 1 + m f 2 + mA
Only states with the same value of m f 1 + m f 2 + mA can be coupled
by the interaction Hamiltonian
The s-wave entrance channel can be coupled to A ≠ 0 channels only
by Vss because Vel , which depends only on the distance r between the
G
2 atoms, commutes with the molecule orbital angular momentum L
,
,
.
Consider the various states M = m f 1 + m f 2 + mA with a fixed value of
G2
M They can be also classified by the eigenvalues of F Fz where
G
G
G
F = F1 + F2 This gives the states { f1 f 2 F M F mA } with M F + mA = M
G G
G
G
G
G
G
G
Since S1 S 2 and thus Vel commutes with F = S1 + S2 + I1 + I 2 and L
Vel can couple only states with the same value of F and A
,
,
,
,
.
,
,
,
.
Examples of application of these symmetry considerations to the
identification of broad Feshbach resonances will be give later on
12
Outline of lecture 2
1 - Introduction
2 - Collision channels
• Spin degrees of freedom.
• Coupled channel equations
• Strong couplings and weak couplings between channels
3 - Qualitative interpretation of Feshbach resonances
4 - Two-channel model
• Two-channel Hamiltonian
• What we want to calculate
5 - Scattering states of the 2-channel Hamiltonian
• Calculation of the outgoing scattering states
• Asymptotic behavior. Scattering length
• Feshbach resonance
5 - Bound states of the 2-channel Hamiltonian
• Calculation of the energy of the bound state
• Calculation of the wave function
13
Open channel and closed channel
The 2 atoms collide with a very
small positive energy E in an
channel which is called “open”
V
The energy of the dissociation
threshold of the open channel is
taken as the zero of energy
Closed
channel
Eres
E
0
r
Open
channel
There is another channel above
the open channel where
scattering states with energy E
cannot exist because E is below
the dissociation threshold of this
channel which is called “closed”
There is a bound state in the
closed channel whose energy
Eres is close to the collision
energy E in the open channel
14
Physical mechanism of the Feshbach resonance
The incoming state with energy E of the 2 colliding atoms in the
open channel is coupled by the interaction to the bound state ϕres in
the closed channel.
The pair of colliding atoms can make a virtual transition to the
bound state and come back to the colliding state. The duration of
this virtual transition scales as ħ / I Eres-E I, i.e. as the inverse of the
detuning between the collision energy E and the energy Eres of the
bound state.
When E is close to Eres, the virtual transition can last a very long
time and this enhances the scattering amplitude
Analogy with resonant light scattering when an impinging photon of
energy hν can be absorbed by an atom which is brought to an
excited discrete state with an energy hν0 above the initial atomic
state and then reemitted. There is a resonance in the scattering
amplitude when ν is close to ν0
15
Sweeping the Feshbach resonance
The total magnetic moment of the atoms are not the same in the 2
channels (different spin configurations). The energy difference between
the 2 channels can thus be varied by sweeping a magnetic field
V
Closed
channel
E
0
r
Open
channel
16
Shape resonances
Metastable
state
0
/
)
(
)
(
V r + A A + 1 =2
2μ r 2
Can appear in a A≠0 channel
where the sum of the potential
and the centrifugal barrier gives
rise to a potential well
Incoming
state
r
The 2 colliding atoms arrive in
a state with positive energy
In the potential well, there are quasi-bound states with positive energy
which can decay by tunnel effect through the potential barrier due to
the centrifugal potential. This is why they are metastable
If the energy of the incoming state is close to the energy of the
metastable state, there is a resonance in the scattering amplitude
These resonances are different from the zero-energy resonances
studied in this lecture. They explain how scattering in A≠0 waves can
become as important as s-wave scattering at low temperatures
17
Outline of lecture 2
1 - Introduction
2 - Collision channels
• Spin degrees of freedom.
• Coupled channel equations
• Strong couplings and weak couplings between channels
3 - Qualitative interpretation of Feshbach resonances
4 - Two-channel model
• Two-channel Hamiltonian
• What we want to calculate
5 - Scattering states of the 2-channel Hamiltonian
• Calculation of the outgoing scattering states
• Asymptotic behavior. Scattering length
• Feshbach resonance
5 - Bound states of the 2-channel Hamiltonian
• Calculation of the energy of the bound state
• Calculation of the wave function
18
Two-channel model
Only two channels are considered, one open and one closed
)
(
op ϕop
)
(
State of the atomic system
G
G
r + cl ϕcl r
(2.7)
The wave function has two components, one in each channel
W r ⎞
⎟ (2.8)
H cl ⎟⎠
)
(
H 2-channel
⎛ H op
= ⎜
⎜W r
⎝
)
(
Hamiltonian
H op
H cl
=2
= −
Δ + Vop
2μ
=2
= −
Δ + Vcl
2μ
(2.9)
)
(
)
(
Resonant bound state in the closed channel
H cl ϕres r = Eres ϕres r
Eres = = Δ
(2.10)
The energy Eres of this state, denoted also =Δ, is close to the energy
E 0 of the colliding atoms in the open channel
19
What we want to calculate
)
(
We want to calculate the eigenstates and eigenvalues of H2-channel
⎛ ϕop ⎞
W r ⎞ ⎛ ϕop ⎞
⎟⎜
⎟ = E⎜
⎟
⎟
⎜
⎟
⎜
⎟
H cl ⎠ ⎝ ϕcl ⎠
⎝ ϕcl ⎠
G
G
G
H op ϕop r + W r ϕcl r = E ϕop r
G
G
G
W r ϕop r + H cl ϕcl r = E ϕcl r
)
(
⎛ H op
⎜
⎜W r
⎝
(2.11)
) )
( (
) )
( (
)
(
) )
( (
)
(
(2.12)
Eigenstates with positive eigenvalues E>0
They describe the scattering states of the 2 atoms in the presence
of the coupling W. In particular, we are interested in the behavior of
the scattering length when Eres is swept around 0
The 2 components
stateG corresponding to an
G
G of the scattering
k
incoming wave k are denoted ϕop
and ϕclk
Eigenstates with negative eigenvalues Eb<0
They describe the bound states of the 2 atoms in the presence of W
b
Their 2 components are denoted ϕop
and ϕclb
20
Single resonance approximation
We will neglect all eigenstates of Hcl other than ϕres
Near the resonance we want to study (Eres close to 0), they are too
far from E=0 and their contribution is negligible
We will use the following expression for the Hamiltonian of the
closed channel
H cl = Eres ϕres
ϕres
(2.13)
The resolvent operator (or Green function) of H cl will be thus
given by:
)
(
ϕres ϕres
1
=
Gcl z =
z − H cl
z − Eres
(2.14)
21
Outline of lecture 2
1 - Introduction
2 - Collision channels
• Spin degrees of freedom.
• Coupled channel equations
• Strong couplings and weak couplings between channels
3 - Qualitative interpretation of Feshbach resonances
4 - Two-channel model
• Two-channel Hamiltonian
• What we want to calculate
5 - Scattering states of the 2-channel Hamiltonian
• Calculation of the outgoing scattering states
• Asymptotic behavior. Scattering length
• Feshbach resonance
5 - Bound states of the 2-channel Hamiltonian
• Calculation of the energy of the bound state
• Calculation of the wave function
22
Scattering states of the
two-channel Hamiltonian H2-channel
Open channel component of the scattering state of H2-channel
)
(
)
)
(
(
)
(
The first equation (2.12) can be written
G
G
k G
k G
E − H op ϕop r = W r ϕcl r
(2.15)
)
(
)
(
Its solution is the sum of a solution of the equation without the rightside member and a solution of the full equation with the right-side
member considered as a source term.
G
G
1
k
+
+
k
+
ϕop = ϕkG + Gop E W ϕcl
Gop
E =
(2.16)
E − H op + i ε
In (2.16), G+op(E) is a Green function of Hop. The term +iε, where ε is
a positive number tending to 0, insures that the second term of (2.16)
has the asymptotic behavior of an outgoing scattered state for r→∝.
)
(
.
(
/
)
)
(
The first term of (2.16), involving only Hop, is chosen as an outgoing
scattering state of Hop, in order to get the good behavior for r→∝.
G2
G G
⎡ ikr
G
1
1
p
+ G ⎤
(2.17)
G r
V
T
ϕk+G r =
e
ϕ
+
=
op
⎥
k
3 2 ⎢
E − T + iε
2μ
2π
⎣
⎦
23
Scattering states of the
two-channel Hamiltonian H2-channel (continued)
Closed channel component of the scattering state of H2-channel
)
(
)
(
)
(
The second equation (2.12) can be written:
G
G
k G
k G
( E − H cl ) ϕcl r = W r ϕop r
(2.18)
= Gcl E W ϕ
)
(
ϕ
G
k
cl
)
(
Its solution can be written in terms of the Green function of Hcl:
G
k
op
Gcl E = ( E − H cl )
−1
(2.19)
Using the single resonance approximation (2.14), we get:
ϕres W ϕ
)
(
)
(
G
G
ϕ r = ϕres r
G
k
cl
G
k
op
E − Eres
The closed channel component ϕ is thus proportional to ϕres
(2.20)
G
k
cl
Dressed states and bare states
G
k
op
G
k
cl
.
The 2 components ϕ and ϕ of the scattering states of H 2 − channel
are called dressed states because they include the effect of W
The eigenstates ϕ k+G and ϕres of H op and H cl are called bare states
24
Open channel components of the scattering
states of H2-channel in terms of bare states
ϕ
G
k
op
= ϕ
+
G
k
+
op
+G
)
(
Inserting (2.20) into (2.16), we get:
E W ϕres
ϕres W ϕ
G
k
op
(2.21)
E − Eres
G
k
op
,
In order to eliminate ϕ in the right side, we multiply both sides of
(2.21) by ϕres W which gives:
E − Eres
=
ϕres W ϕk+G
E − Eres − ϕres W G
+
op
)
(
ϕres W ϕ
G
k
op
(2.22)
E W ϕres
Inserting (2.22) into (2.21), we finally get:
+
op
+G
E
W ϕres
ϕres W
E − Eres − ϕres W G
+
op
)
(
= ϕ
+
G
k
)
(
ϕ
G
k
op
E W ϕres
ϕ k+G
(2.23)
Only the bare states appear in the right side of (2.23).
25
Connection with two-potential scattering
Equation (2.23) can be rewritten in a more suggestive way. Il we
introduce the effective coupling Veff defined by:
E − Eres − ϕres W G
+
op
)
(
Veff = W
ϕres ϕres
E W ϕres
W
(2.24)
we get, by inserting (2.24) into (2.23):
G
1
+
k
G
(2.25)
ϕop = ϕ k +
Veff ϕk+G
E − H op + i ε
Veff acts only, like Vop, inside the open channel space. It describes the
effect of virtual transitions to the closed channel subspace. The twochannel scattering problem can thus be reformulated in terms of a
single-channel scattering problem (in the open channel), but with a
new potential Vtot in this channel, which is the sum of 2 potentials
Vtot = Vop + Veff
(2.26)
Equation (2.25) then appears as the scattering produced by Veff on
waves “distorted” by Vop. (Generalized Lippmann-Schwinger equation)
(see for example ref.4, Chapter 17)
26
Asymptotic behavior
of the scattering states of H2-channel
/
)
,
(
.
/
)
(
)
(
Let us come back to (2.23). Only the asymptotic behavior of the
open channel component is interesting because the closed channel
component, proportional to ϕres vanishes Gfor large r.
k
We expect the asymptotic behavior of ϕop
to be of the form:
ikr
G
G G
⎡
⎤
G
G
G
G
1
e
k
ik r
(2.27)
e
+
=
ϕop r f
k
n
n
r
r
⎢
⎥
r →∞ 2π 3 2
r ⎦
⎣
In the limit k→0, the scattering amplitude becomes spherically
symmetric and gives the scattering length we want to calculate
G
f k n → − a
(2.28)
)
,
(
k →0
The asymptotic behavior of the first term of (2.23) describes the
scattering in the open channel without coupling to the closed channel.
It gives the scattering length aop in the open channel alone (W = 0).
This scattering length is often called the background scattering length.
aop = abg
(2.29)
27
Position of the resonance
)
(
The second term of (2.23) is the most interesting since it gives the
effects due to the coupling W.
The scattering amplitude given by its asymptotic behavior becomes
large if the denominator of the second term of (2.23) vanishes, i.e. if:
+
E = Eres + ϕres W Gop
E W ϕres
(2.30)
When E is close to 0, the last term of (2.30) is equal to:
)
(
ϕres W G
+
op
0 W ϕres
=
∑
G
k
ϕres W ϕ
+
G
k
− EkG + i ε
2
= =Δ 0
(2.31)
Its interpretation is clear. It gives the shift ħΔ0 of ϕres due to the second
order coupling induced by W between ϕres and the continuum of Hop
We thus predict that the scattering amplitude, and then the scattering
length, will be maximum (in absolute value), not when Eres is close to
0, but when the shifted energy of ϕres
(2.32)
E res = Eres + = Δ 0
is close to the energy E 0 of the incoming state
28
+
op
Strictly speaking, the Green function G
appearing in (2.30) is equal to:
)
(
Remark
E = (E − E + i ε )
G
k
⎛
⎞
1
1
= P ⎜
− i π δ ( E − EkG )
⎟
⎜
⎟
G
E − EkG + i ε
⎝ E − Ek ⎠
where P means principal part.
−1
(2.33)
,
Because of the last term of (2.33), equation (2.31) should also
contain an imaginary term describing the damping of ϕres due to its
coupling induced by W with the continuum of Hop.
But we are considering here the limit of ultracold collisions E → 0
and the density of states of the continuum of H op vanishes near
EkG = 0 which means that the damping of ϕres can be ignored in
the limit E → 0
For large values of Eres, the imaginary term of (2.33) can no longer
be ignored, and it can be shown that it gives rise to an imaginary
term in the scattering amplitude, proportional to k.
.
29
Variations of Eres and Eres with B
The spin configurations of the two channels have different magnetic
moments. The energies of the states in these channels vary
differently when a static magnetic field B is applied and scanned. If
ξ is the difference of magnetic moments in the 2 channels, the
difference between the energies of 2 states belonging to the
channels varies linearly with B with a slope ξ.
If we take the energy of the dissociation threshold of the open
channel as the zero of energy, the energy Eres of ϕres is equal to:
Eres = ξ ( B − Bres )
(2.34)
Eres is degenerate with the energy of the ultracold collision state
when B=Bres
In fact, the position of the Feshbach resonance is given, not by
the zero of Eres , but by the zero of Eres
(2.35)
E res = Eres + =Δ 0 = ξ ( B − B0 )
This equation gives the correct value, B0, at which we expect a
divergence of the scattering length.
30
E
= Δ0
Eres
Bres
B0
B
E res
We suppose here ξ < 0
Since Δ0 is also negative according to (2.31), B0 is smaller than Bres.
31
Contribution of the inter channel coupling W
to the scattering length
Asymptotic behavior of the W-dependent term of ϕ
G
k
op
Using (2.30) and (2.32), we can rewrite (when E0) equation (2.23):
= ϕ
+
G
k
+
op
+G
W ϕres
)
(
ϕ
G
k
op
ϕres W
ϕk+G
(2.36)
E − Eres
To find the contribution of W to the scattering length, we have to find
the asymptotic behavior for r large of the wave function of the last term
E
)
(
W ϕres ϕres W
G +
+
G
r Gop E
ϕ
k
E − E res
=
(2.37)
)
(
G G W ϕres ϕres W
+
G
r′ r′
ϕ
k
E − E res
We need for that to know the asymptotic behavior for r large of the
Green function of Hop
G G
G
G
1
+
′
′
Gop
E
r
r
r
r
=
(2.38)
(
)
E − H op + i ε
G +
′
∫ d r r Gop E
3
,
,
32
Contribution of the inter channel coupling W
to the scattering length (continued)
/
*
,
,
One can show (see Appendix) that:
G G
G
G
G
ei k r 2μ π
+
−
⎡
⎤
′
′
G
−
=
ϕ
Gop ( E r r ) r
n
r r (2.39)
k n ( )⎦
2
⎣
r →∞
r =
2
*
G
G
G
Using ⎣⎡ϕk−nG ( r ′ ) ⎦⎤ = ϕ k−nG r ′ and the closure relation for r ′, we get
for the asymptotic behavior of (2.37 ):
ϕ k− nG W ϕres ϕres W ϕk+G
e
,
,
2μ
2
(2.40)
2
π
r =2
E − E res
In the limit k → 0 E → 0 ϕ k+G → ϕ0G+ and ϕk− nG → ϕ0G− = ϕ0G+ since
e ± i kr r → 1 r so that (2.40 ) can be also written, using (2.35 ):
−
i k r
/
/
ϕ W ϕres
1 2μ
2
2π
−
2
r =
0 − E res
+
G
0
2
ϕ W ϕres
1 2μ
2
2π
= +
2
r =
ξ ( B − B0 )
+
G
0
2
(2.41)
The coefficient of -1/r in (2.41) gives the contribution of the interchannel coupling to the scattering length
33
Scattering length
The asymptotic behavior of the first term of (2.23) gives the background
scattering length. Adding the contribution of the second term we have
just calculated, we get for the total scattering length:
a = abg
where:
ϕ W ϕres
2μ
2
− 2 2π
−ξ ( B − B0 )
=
+
G
0
ϕ
2μ
2
ΔB = 2 2π
=
+
G
0
2
= abg
W ϕres
⎡
ΔB ⎤
⎢1 −
⎥
−
B
B
0 ⎦
⎣
(2.42)
2
ξ abg
(2.43)
This is the main result of this lecture.
- The scattering length diverges when B = B0
- It changes sign when B is scanned around B0
- It vanishes for B – B0 = ΔB
The variations of the scattering length with the static field are
represented in the next figure
34
Scattering length versus magnetic field
a
0
Bres
B0
B
abg
ΔB
Figure corresponding to two colliding Rb85 atoms each in the state
f = 2, m f = -2 in a s-wave (A = 0).
In this case, we have abg < 0 and ξ < 0
35
Examples of broad and narrow Feshbach resonances
- Entrance channel : ee
f1 = 2 , m f 1 = −2 , f 2 = 2 , m f 2 = −2, A = mA = 0
M = m f 1 + m f 2 + m A = −4
- Other channels with the same
M = −4 A = mA = 0
gg, fh
eg,df
They are open because they are
above the entrance channel.
They have the same negative slope ξ
Zeeman and hyperfine levels of Rb85
with respect to ee when B is varied
(Figure taken from Ref.9)
G G G
Classification by other quantum numbers ( f1 , f 2 ) F , M , A = mA = 0 ( F = f1 + f 2 )
If f1 = f 2 = 2, F = 0, 2, 4 (Odd values of F are forbidden for identical bosons)
Only F = 4 can give M = −4 ⇒ Channel ee corresponds to ( 22), F = 4, M = −4
If f1 = f 2 = 3, F = 0, 2, 4, 6 (Odd values of F are forbidden for identical bosons)
Only F = 4, 6 can give M = −4 ⇒ Channel gg and fh give rise to 2 types of
states
(33), F = 4, M = −4 and (33), F = 6, M = −4
36
Feshbach resonances associated with gg and fh
In the potential wells of the channels
(33) F = 6 or 4, M = - 4, there are
vibrational levels v = -1,-2,-3,…
staring from the highest one v = -1
(Figure taken from Ref.9 )
The energy level
(33) F = 4, M= - 4,v = -3
crosses the energy (~0) of the
entrance channel around B=155 G
The energy level
(33) F = 6, M= - 4,v = -3
crosses E~0) around B=250 G
(Lower part of the figure)
The 2 levels which cross at B=155 G correspond to the same value of F
and can thus be coupled by the strong interaction Vel . This is why the
corresponding Feshbach resonance is broad
The 2 levels which cross at B=250 G correspond to different values of F
and can thus be coupled only by the weak interaction Vss . This is why the
corresponding Feshbach resonance is narrow
(Upper part of the figure)
37
Outline of lecture 2
1 - Introduction
2 - Collision channels
• Spin degrees of freedom.
• Coupled channel equations
• Strong couplings and weak couplings between channels
3 - Qualitative interpretation of Feshbach resonances
4 - Two-channel model
• Two-channel Hamiltonian
• What we want to calculate
5 - Scattering states of the 2-channel Hamiltonian
• Calculation of the outgoing scattering states
• Asymptotic behavior. Scattering length
• Feshbach resonance
5 - Bound states of the 2-channel Hamiltonian
• Calculation of the energy of the bound state
• Calculation of the wave function
38
Bound states of the two-channel
Hamiltonian H2-channel
Are there bound states for H2-channel for B close to B0?
How are they related to the bound state ϕres of Hcl?
How do their energy Eb and wave function vary with B?
)
(
)
(
We denote such a bound state
G
b G
op ϕop
r + cl ϕclb r
(2.44)
b
ϕop
and ϕclb are the components of the bound state in the open
channel and the closed channel, respectively, obeying the
normalization condition:
b
b
ϕop
ϕop
+ ϕclb ϕclb = 1
(2.45)
) )
( (
) )
( (
)
(
)
) (
(
)
(
Expressing that the state (2.44) is an eigenstate of the Hamiltonian
(2.8) with eigenvalue Eb, we get the following 2 equations:
b G
b G
b G
H op ϕop r + W r ϕcl r = Eb ϕop r
(2.46)
b G
b G
b G
W r ϕop r + H cl ϕcl r = Eb ϕcl r
39
Bound states of the two-channel
Hamiltonian H2-channel (continued)
) )
( (
To solve equation (2.46), we can use the Green functions of Hop and Hcl
without the iε term because Eb is negative (below the threshold of Vop)
b
ϕop
= Gop Eb W ϕclb
ϕ
b
cl
= Gcl Eb W ϕ
b
op
(2.47)
)
(
As above, we can use the single resonance approximation for Gcl:
Gcl Eb =
ϕres ϕres
(2.48)
,
Eb − Eres
Inserting (2.48) into the second equation (2.47) shows that ϕclb is
proportional to ϕres so that we can write:
)
(
(2.49)
)
Nb =
(
b
⎛ ϕop
⎞
1 ⎛ Gop Eb W ϕres ⎞
⎜ b⎟ =
⎜
⎟
⎜
⎟
⎜ϕ ⎟
Nb ⎝
ϕres
⎠
⎝ cl ⎠
where Nb is a normalization factor
2
1 + ϕres W Gop
Eb W ϕres
(2.50)
40
Implicit equation for the energy Eb
)
(
Inserting (2.48) into the second equation (2.47) gives:
1
b
b
ϕcl =
ϕres ϕres W ϕop
(2.51)
Eb − Eres
which, inserted into the first equation (2.47) leads to:
1
b
b
ϕop
=
Gop Eb W ϕres ϕres W ϕop
(2.52)
Eb − Eres
b
As for equation (2.21), we can eliminate the dressed state ϕop
by
multiplying both sides of this equation at left by ϕres W This gives:
.
)
(
Now, using the identity
1
1
1
1
= −
+ Eb
Gop Eb =
Eb − H op
H op
H op Eb − H op
we can rewrite (2.53) as:
(2.54)
)
(2.53)
(
Eb − Eres = ϕres W Gop Eb W ϕres
)
(
)
(
)
(
Eb = Eres + ϕres WGop 0 W ϕres − Eb ϕres WGop 0 Gop Eb W ϕres)
(2.55)
41
.
Implicit equation for the energy Eb (continued)
,
,
The second term of the right side of (2.55) is the shift =Δ 0 of ϕres
Adding it to Eres we get E res so that (2.55) can be rewritten:
E = E − E ϕ
W G 0 G E W ϕ
(2.56)
res
b
op
op
)
res
(
)
(
b
res
b
To go further, we introduce the spectral decomposition of Gop(z)
)
(
3
d
∫ k
ϕ k+G
/
)
(
Gop z =
ϕk+G
b
+ Gop
z
(2.57)
z − =k
2μ
The last term of (2.57) gives the contribution of the bound states of
H op Se suppose here that their energy if far below E = 0 so that we
can ignore this term. Using (2.57), we can then write (2.56) as:
2
2
,
.
Eb = E res − ( 2 μ ) Eb ∫ d k
2
ϕres W ϕ
3
=k
2
2
(= k
2
2
+
G
k
2
+ 2 μ Eb
)
(2.58)
This is an implicit equation for Eb that we will try now to solve
42
Calculation of the energy Eb
To calculate the integral of (2.58), we introduce the new variable:
=k
u =
(2.59)
2 μ Eb
which allows one to rewrite, after angular integration, the integral of
(2.58) as:
1
=3
4π
Let k0 be the width of
0
ϕres W ϕ
du
(u
ϕres W ϕ
This defines a value u0 of u
u0 =
+
G
k
= k0
2
2
)
+1
2
(2.60)
.
2 μ Eb
∫
∞
+
G
k
considered as a function of k
(2.61)
?
2 μ Eb
characterizing the width in u of the numerator of the integral of (2.60).
Two different limits can then be considered: u0 1 and u0 1
43
First limit
u0 1
⇔
Eb = 2 k02
/
Calculation of the energy Eb (continued)
2μ
/
The denominator of the integral of (2.60) varies more rapidly
G with
G u
than the numerator which can be replaces by its value for k = 0
Equation (2.60) can then be approximated by:
2
∞
1
4π
du
+
(2.62)
G
ϕ
ϕ
W
res
0
∫0 u2 + 1
=3 2 μ E
b
=π
2
/
Replacing the integral of (2.58) by (2.62) then leads to:
Eb = E res +
Eb
2π 2 ( 2 μ )
3
=
W ϕ0G+
2
ϕres W ϕ
3
2
+
G
0
2
(2.63)
)
(
One can then reexpress ϕres
in terms of ΔB thanks to
(2.43) and E res in terms of ξ B − B0 thanks to (2.35) and finally use
(2.43) to show that the solution of (2.6 ) is, to a good approximation:
=2
Eb = −
(2.64)
2
2μ a
44
Second limit
u0 1
⇔
Eb = 2 k02
/
Calculation of the energy Eb (continued)
2μ
The numerator of the integral of (2.60) varies more rapidly with u
than the denominator, so that we can neglect the term u 2 in the
denominator.
In fact, this approximation amounts to neglecting = 2 k 2 com pared to
2 μ Eb in the denominator of the integral of (2.58)
This approximation allows one to transform (2.58) into:
Eb = E res + ( 2 μ )
= E res
∫dk
3
∫dk
3
ϕres W ϕ
2
2μ =2 k 2
ϕres W ϕ
/
= E res +
2
+
G
k
+
G
k
2
(2.65)
=2 k 2 2μ
− =Δ 0 = Eres = ξ ( B − Bres )
We have used the expression (2.31) of ħΔ0 and equation (2.35)
45
Eb
= Δ0
Eres
Bres
B0
B
Eres
Eb
Asymptote
with a slope ξ
- The bound state of H2-channel appears for B > B0, in the region a>0.
- Eb first decreases quadratically with B-B0 and then tends to the unperturbed
energy Eres of the bound state ϕres of the closed channel
- If B0 is swept through the Feshbach resonance from the region a<0 to the
region a>0, a pair of ultracold atoms can be transformed into a molecule
46
Wave function of the bound state
Weight of the closed channel component of the bound state
(
)
2
Eres
N
(2.68)
)
(
Eb =
(2.67)
)
(
we can rewrite the second equation (2.66) as:
∂
N b2 = 1 −
ϕres W Gop Eb W ϕres
∂Eb
The last term of (2.68) can be transformed using (2.53)
2
= −Gop
Eb
)
∂
1
Gop Eb = −
∂Eb
Eb − H op
(
⇒
)
1
Eb − H op
(
)
(
Gop Eb =
)
(
According to (2.49) and (2.50), the relative weight of ϕclb in the
(normalized) wave function of H 2-channel is given by:
1
b
b
2
ϕcl ϕcl = 2
N b2 = 1 + ϕres W Gop
Eb W ϕres
(2.66)
Nb
Using
+ ϕres W Gop Eb W ϕres
(2.69)
= ξ ( B − Bres )
47
Wave function of the bound state (continued)
(2.70)
=1 − N b2
/
This finally gives:
)
(
Taking the derivative of (2.69) with respect to B, we get:
∂Eb
∂Eb
∂
ϕres W Gop Eb W ϕres
= ξ +
∂B
∂Eb
∂B
∂Eb ∂B
1
(2.71)
=
ξ
N b2
The weight of the closed channel component in the wave function of
the bound state, for a given value of B, is thus equal to the slope of
the curve giving Eb(B) versus B, divided by the slope ξ of the
asymptote of the curve giving Eb(B) versus B (see Figure page 46)
Conclusion
When the bound state of the 2-channel Hamiltonian appears near
B=B0 in the region a > 0, the slope of the curve Eb(B) is equal to 0
and the weight of the closed channel component in its wave function
is negligible. For larger values of B, near the asymptote of Eb(B),
this weight tends to 1
48
Wave function of the bound state (continued)
Expression of the wave function of the bound state
The previous conclusion means that, near the Feshbach resonance,
the coupling with the closed channel can be neglected for calculating
the wave function of the bound state and that we can thus look for
the eigenfunction of Hop with an eigenvalue –ħ2/2μa2.
2
= 2 d u0 r
−
2μ d r 2
)
(
)
(
The asymptotic behavior of this wave function (at distances larger
than the range of Vop) can be obtained by solving the 1D radial
Schrödinger equation for u0(r) with Vop=0.
=2
u0 r
= −
2
2μ a
(2.72)
−r
r
)
/
(
p
x
e
The 3D wave function of the bound state thus behaves
asymptotically as
a
(2.73)
49
Comparison with quantitative calculations
Note the logarithmic scale
of the r-axis
When one gets closer to
the Feshbach resonance,
the extension of the wave
function becomes bigger
and the weight of the
closed channel component
smaller:
4.7 % at B=160 G
0.1 % at B=155.5 G
Figure taken from Ref. 9
50
Conclusion
The coupling between the collision state of 2 ultracold atoms and a
bound state of these 2 atoms in another closed collision channel gives
rise to resonant variations of the scattering length a when the energy
of the bound state is varied around the threshold of the closed
channel by sweeping a static magnetic field B.
The scattering length a diverges for the value B0 of B for which the
energy of the bound state in the closed channel, perturbed by its
coupling with the continuum of collision states in the open channel,
coincides with the threshold of the open channel.
The scattering length can thus take positive or negative values, very
large values. It vanishes for a certain value of B depending on the
background scattering length in the open channel.
By choosing the value of B, one can thus obtain an attractive gas, a
repulsive one, a perfect gas without interactions (a=0), a gas with very
strong interactions (a very large, corresponding to the unitary limit).
51
Conclusion (continued)
The width of the resonance, given by the distance between the value
of B for which a diverges and the value of B for which it vanishes,
depends on the strength of the coupling between the 2 channels. The
resonance is broad if the 2 channels are coupled by the spin
exchange interaction, narrow if they can be coupled only by the
magnetic dipole-dipole spin interactions.
Near B=B0, in the region a>0, the two-atom system has a bound state,
with a very weak binding energy, equal to ħ2/2μa2. The wave function
of this bound state has a very large spatial extent of the order of a. Its
closed channel component is negligible compared to the open
channel component.
By sweeping B near B0, one can transform a pair of colliding atoms
into a molecule or vice versa.
A few problems not considered here:
- Influence of the speed at which B is scanned.
- Stability of the “Feshbach molecules”. How do inelastic and 3-body
collisions limit their lifetime. Bosonic versus fermionic molecules.
D.Petrov, C.Salomon, G.Shlyapnikov, Phys.Rev.Lett. 93, 090404 (2004)
52
APPENDIX
For the 2 lectures of Claude Cohen-Tannoudji
on “Atom-Atom Interactions
in Ultracold Quantum Gases”
1
Purpose of this Appendix
)
(
1 – Demonstrate the orthonormalization relation
ϕ k ′l ′m ′ ϕklm = δ k − k ′ δ ll ′δ mm ′
(A.1)
- The wave function
)
(
)
(
)
(
)
(
)
(
2 ukl ( r )
Yl m (θ , ϕ )
ϕk l m ( r ) =
(A.2)
π r
describes, in the angular momentum representation, a particle of
mass μ, with energy E=ħ2k2/2μ, in a central potential V(r)
- The radial wave function ukl(r) is a regular solution of
2
⎡ d2
⎤
2μ
+ 1 (A.3)
2
+
k
−
V
r
u
r
=
V
r
=
V
r
+
0
⎢ 2
⎥ kl
tot
tot
2
2μ
r
r2
d
⎣
⎦
)
(
ukl 0 = 0
)
(
/
n
i
s
)
(
which behaves, for r→∞, as:
⎡⎣ kr − l π
ukl r
(A.4)
2 + δ l k ⎤⎦
r →∞
(A.5)
2 ) ⎤⎦ = ∓ i
l
± ikr
)
(
p
x
e
)
(
r →∞
⎡±i ( k r − lπ
⎣
/
ukl± r
p
x
e
)
(
- There are other (non regular) solutions behaving, for r→∞, as:
(A.6)
2
)
(
/
2
2μ + V r
2 – Calculate the Green function of: H = p
with outgoing and ingoing asymptotic behavior
)
(
/
)
(
)
(
)
,
(
)
,
(
*
)
(
p
x
e
,
)
(
− H ) G ± ( r r ′) = δ ( r − r ′)
E = 2 k 2 2μ
(A.7)
- Show that:
2μ 1
±
′
′
±
G ± ( r r ′) = − 2
i
δ
Y
θ
ϕ
Y
θ
ϕ
u
r
u
r
∑
l
lm
lm
kl <
kl >
krr ′ lm
(A.8)
where r> (r<) is the largest (smallest) of r and r’
,
(E
(A.9)
)
,
(
)
,
(
*
)
(
p
x
e
,
)
(
- Introducing the Heaviside function:
θ ( r − r ′ ) = +1 if r > r ′
= 0 if r < r ′
(A.8) can also be written:
2μ 1
±
′
G (r r ) = − 2
± i δ l Ylm θ ϕ Ylm θ ′ ϕ ′ ×
∑
krr ′ lm
× ⎡⎣θ ( r − r ′ ) ukl r ′ ukl± r + θ ( r ′ − r ) ukl r ukl± r ′ ⎤⎦
)
(
)
(
)
(
)
(
(A.10)
3 – Calculate the asymptotic behavior of these Green functions
and demonstrate Equation (2.39) of Lecture 2
3
Wronskian Theorem
The calculations presented in this Appendix use the Wronskian
theorem (see demonstration in Ref.2 Chapter III-8)
)
(
)
(
)
(
- Consider the 1D second order differential equation:
(A.11)
)
(
)
(
y ′′ r + F r y r = 0
Equation (A.4) is of this type with:
2μ
F r = k 2 − 2 Vtot r
(A.12)
)
(
)
(
,
)
(
)
(
- Let y1 r and y2 r be 2 solutions of this equation corresponding to
2 different functions F1 r and F2 r respectively.
The wronskian of y1 and y 2 is by definition:
)
(
)
(
)
(
)
(
)
,
(
W y1 y2 = y1 r y2′ r − y2 r y1′ r
= ⎡⎣W y1 y2 ⎤⎦ r = b − ⎡⎣W y1 y2 ⎤⎦ r = a
=
∫
b
a
)
(
a
) )
(
, )
( (
b
)
)
, (
(
W y1 y2
)
,
(
- One can show that:
⎡⎣ F1 r − F2 r ⎤⎦ y1 r y2 r dr
(A.13)
(A.14)
4
Demonstration of (A.1)
)
(
)
(
We consider 2 different values k1 and k2 of k. According to (A.12):
F1 r − F2 r = k12 − k22
(A.15)
)
,
(
)
(
)
(
(A.14) then gives the scalar product of y1 = uk l and y2 = uk l
1
2
b
b
1
y
r
y
r
d
r
=
W
y
y
1
2 a
∫a 1 2
k12 − k22
If we take a = 0 ⎡⎣W y1 y2 ⎤⎦ r = a = 0 because of (A.4)
(A.16)
)
,
(
,
,
)
(
)
(
,
)
(
)
(
)
(
2
1
r dr = −
2
⎡( k1 + k2 ) R − π + δ1 + δ 2 ⎤
⎣
⎦ +
k1 + k2
⎡( k1 − k2 ) R + δ1 − δ 2 ⎤
⎣
⎦
n
i
s
uk l r uk l
1
n
i
s
)
(
0
)
(
∫
R
)
(
,
)
(
)
(
)
(
If we take b = R very large compared to the range of V r we can
use the asymptotic behavior (A.5) of uk l and uk l
1
2
R
1
∫0 uk1l r uk2l r dr = k 2 − k 2 ⎡⎣uk1l r uk′2l r − uk2l r uk′1l r ⎤⎦ r = R (A.17)
1
2
Using (A.15) and putting δ l k1 = δ1 δ l k2 = δ 2 we get:
1
+
2
k1 − k2
(A.18)
5
- When R→∞, the first term of the right side of (A.18) vanishes as a
distribution, because it is a rapidly oscillating function of k1+k2
(k1 and k2 being both positive k1+k2 cannot vanish)
- The second term becomes important when k1-k2 is close to zero
(we have then δ1-δ2=0)
1
R→∞
1
2
π
2
)
uk l r uk l r dr =
(A.19)
(
0
π
)
(
∫
∞
Rx
= δ x
x
)
(
we get:
)
(
n
i
s
m
i
l
- Using:
δ k1 − k2
(A.20)
- We then have, according to (A.2):
)
(
r uk ′l r dr
kl
π
2
δ k − k′
)
=
(
)
(
= δ k − k ′ δ ll ′δ mm ′
)
= δ ll ′δ mm′
∫u
(
θ ϕ Ylm θ ϕ
)
,
(
l ′m ′
)
π
∫ dΩ Y
,
)
(
)
(
2
(
*
*
3
d
∫ r ϕk ′l ′m′ r ϕklm r =
(A.21)
which demonstrates (A.1).
6
Demonstration of (A.8)
)
(
)
(
Let us apply E-H to the right side of (A.8). Using (A.10) and:
2
2
⎡ 1 ∂2
⎤
L2
2μ
(A.22)
Δ +V r = −
− 2 2 − 2 V r ⎥
H = −
⎢
2
2μ
2μ ⎣ r ∂ r
r
⎦
we get, using (A.12):
1
±
′
± i δ l Ylm θ ϕ Ylm θ ′ ϕ ′ ×
( E − H ) G ( r r ) = − krr ′ ∑
lm
⎧⎪⎛
⎫⎪
∂2 ⎞
±
±
× ⎨⎜ F r + 2 ⎟ ⎡⎣θ ( r − r ′ ) ukl r ′ ukl r + θ ( r ′ − r ) ukl r ukl r ′ ⎤⎦ ⎬
∂r ⎠
⎩⎪⎝
⎭⎪
(A.23)
To calculate the second line of (A.23), we use:
∂
∂
θ ( r1 − r2 ) = −
θ ( r2 − r1 ) = δ ( r1 − r2 )
)
,
(
)
,
(
*
)
(
p
x
e
,
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
∂r1
∂r1
⎡ ∂
⎤
⎡ ∂
⎤
δ ( r1 − r2 ) ⎥ f r1 = − f ′ r2 δ ( r1 − r2 ) + f r2 ⎢ δ ( r1 − r2 ) ⎥
⎢
⎣ ∂r1
⎦
⎣ ∂r1
⎦
(A.24)
)
(
)
(
)
(
The second order derivative of the second line of (A.23) gives 3 types
of terms: proportional to θ r − r ′ and θ r ′ − r , to δ r − r ′ and
to ∂δ r − r ′ ∂r
/
)
(
7
)
)
(
/
)
(
)
(
(
)
(
(
(
)
)
/
( )
/
)
(
(
)
- The terms ∝ θ r − r ′ are multiplied by ⎡ F r + ∂ 2 ∂r 2 ⎤ ukl± r
⎣
⎦
which vanishes because ukl± r is a solution of (A.3).
The same argument applies for the terms ∝ θ r ′ − r which are
multiplied by ⎡ F r + ∂ 2 ∂r 2 ⎤ ukl r = 0
⎣
⎦
- The terms proportional to ∂δ r − r ′ ∂r cancel out
∂r ′ ) ⎤ δ r − r ′
⎦
)
∂r ′ − ukl± r ′ ( ∂ ukl r ′
(
)
/
)
(
/
)
(
)
(
(
⎡ u r ′ ∂ u± r ′
kl
⎣ kl
, )
)
(
(
- The only terms surviving in the second line of (A.23) are
those proportional to δ r − r ′ which gives for this line:
(A.25)
- We recognize in the bracket of (A.25) the Wronskian of ukl and ukl±
.
We can thus use (A.14) with F1 = F2 since ukl and ukl± correspond
to the same value of k
.
- Equation (A.14) shows that the Wronskian is independant of r when
F1 = F2 We can thus calculate it for very large values of r where we
know the asymptotic behavior (A.5) and (A.6) of ukl and ukl±
8
∓ iδ l
)
= −k
(
p
x
e
W ukl ukl+
)
,
(
- The calculation of the Wronskian appearing in (A.25) is
straightforward using (A.5) and (A.6) and gives:
(A.26)
- Inserting (A.26) into (A.25) and then in (A.23) gives:
)
,
(
)
,
(
*
)
(
,
)
(
1
( E − H ) G ( r r ′) = r 2 δ r − r ′ ∑ Ylm θ ϕ Ylm θ ′ ϕ ′ (A.27)
lm
- We can then use the closure relation for the spherical harmonics
(see Ref. 3, Complement AVI):
±
θ ′ δ ϕ − ϕ′
)
(
)
θ −
s
o
c
Ylm θ ϕ Ylm θ ′ ϕ ′ = δ
s
o
c
(
)
,
(
)
,
(
*
∑
(A.28)
lm
θ ′ δ ϕ − ϕ′
)
(
)
θ −
s
o
c
)
(
,
1
( E − H ) G ( r r ′) = r 2 δ r − r ′ δ
= δ r − r′
which demonstrates (A.8).
±
s
o
c
(
)
(
to obtain:
)
(
(A.29)
9
Asymptotic behavior of G+
)
(
)
(
)
,
(
)
,
(
*
,
)
(
For r very large, only the first term of the bracket of (A.10) is non zero
and we get:
2μ 1
iδ l
+
+
′
′
′
− 2
G ( r r ′)
e
Y
θ
ϕ
Y
θ
ϕ
u
r
u
r
∑
lm
lm
kl
kl
r →∞
krr ′ lm
(A.30)
e
)
(
)
,
(
)
,
(
*
)
(
,
)
(
According to (A.6), we have
2μ 1
l iδ l
− 2
−
G + ( r r ′)
i
e Ylm θ ϕ Ylm θ ′ ϕ ′ ukl r ′
∑
r →∞
k r ′ lm
ikr
r
(A.31)
i
− i δ l Ylm n Ylm
)
(
∑
π
l
)
(
2
)
1
′
r =
k
(
p
x
e
)
(
)
(
ϕ
−
kn
)
(
*
On the other hand, from Eq. (1.46) of lecture 1 and (A.2), we have:
n′
uk l r ′
r′
lm
( r r ′)
r →∞
−
2
π
⎡ϕ
2 ⎣
−
kn
r′
r′
e
*
)
(
,
)
(
G
2μ
n′ =
(A.32)
Using (A.32), we can rewrite (A.31) as:
+
r
n =
r
r ′ ⎤⎦
ikr
(A.33)
r
which demonstrates Eq. (2.39) of lecture 2.
10