Outline
Theory of atom-molecule
coherence in ultracold gases
Rembert Duine
H.T.C. Stoof
cond-mat/0210544
cond-mat/0211514
cond-mat/0302304
Utrecht University
Atom-atom scattering
Scattering length (I)
Solve the Schrodinger equation in the center-of-mass
system (consider atoms as pointlike).
Look for solutions of the form:
(
internuclear separation
Incoming wave
Outgoing wave. Scattering amplitude
contains only s-wave (l=0) since the
temperatures are low (~nK)
Scattering length (II)
eikr
r →∞

→ r k + f (k )
r
rψ ( r )
)
The s-wave scattering length a
is defined as:
δ (k )
a = − lim
k
k→0
positive a
r
a
Repulsive potential: a positive, effective hard core.
Scattering & bound states
rψ ( r )
Attractive potential:
a can have either sign.
Negative a: attractive
negative a
effective interactions.
Positive a: repulsive
-a
r
effective interactions
Scattering length determined
experimentally
Effective interaction potential: Effective (pseudo) potential yields
exact result for scattering amplitude
4π aℏ 2 in Born approximation (1st order PTB),
δ ( x − x ') since it contains the interaction already
V ( x − x ') =
m
to all orders in PTB.
Two-body T-matrix: T2B
scattering length
r ψ (+)
In terms of the phase shift δ(k) the
scattering amplitude is given by:
1 2iδ ( k )
f (k ) =
e
−1
2ik
 ℏ 2∇ 2
 + V ( r ) ψ ( r ) = Eψ ( r )
−
 m

Two-body physics: scattering & bound states
- single-channel
- multi-channel (Feshbach resonance)
Many-body physics:
- effective theory that incorporates the exact
two-body physics
- Feshbach resonance described by atommolecule model
Applications to experiments
- many-body effects on frequency of coherent
atom-molecule oscillations
Scattering length of
attractive potential
V0
changes sign from −∞ to +∞
as a bound state enters the
potential.
Eb
internuclear separation
R
Near resonance:
Bound state enters the
well. Potential/shape
resonance
V0
Eb = −
ℏ2
ma2
1
Feshbach resonance I
Open channel (down-down)
coupled to closed (up-up)
channel by hyperfine coupling
Closed channel has (bare)
bound state close to two-atom
continuum threshold.
↑ + ↑
detuning
δ
∆µ B
↓ + ↓
internuclear separation
 ℏ2 2

V↑↓ ( r )
 − ∇ + V↓↓ ( r )
  ψ ( r ) 
 ψ (r ) 
 m
  open  = E  open 
2
ψ closed ( r ) 

 ψ closed ( r ) 
ℏ 2

V↑↓ ( r )
− ∇ + ∆µ B + V↑↑ ( r ) 

m


Energy difference between
two atoms and bare molecular state:
δ ( B) = ∆µ ( B − B0 )

∆B 
a(B) = abg 1−

 B − B0 
B0
150
Rb-85
B
125
Also in this case (not obvious):
100
Eb ( B) = −
50
ℏ2
∼ − ( B − B0 ) ∼ −δ 2
2
m [ a( B) ]
2
25
Experimentally
accessible!
156
157
B0
158
159
160
B(G)
What’s next…
Atom-atom interactions are at low energy fully
determined by the s-wave scattering length
If scattering length a large and positive, boundstate with energy ~1/a2.
Feshbach resonance: two-channel problem,
scattering length experimentally adjustable to any
value by tuning magnetic field. Resonance
behaviour due to molecular bound state.
Near Feshbach resonance: a~1/(B-B0),
Molecular binding energy: ~1/a2~(B-B0)2
 ℏ 2∇ 2
† 
Hˆ = ∫ d xψˆ ↑† ( x )  −
+ ∆µ B / 2 + ∫ dx ψ
' ˆ ↑ ( x ')V↑↑ ( x − x ')ψˆ ↑ ( x ') ψˆ ↑ ( x )
 2m

 ℏ2∇ 2
† 
+ ∫ dxψˆ ↓† ( x ) −
+ ∫ dx ψ
' ˆ ↓ ( x ')V↓↓ ( x − x ')ψˆ ↓ ( x ') ψˆ ↓ ( x )
 2m

† † + ∫ dx ∫ dx ψ
' ↓ ( x )ψ ↓ ( x ')V↑↓ ( x − x ')ψˆ ↑ ( x ')ψˆ ↑ ( x ) + h.c
detuning
δ
Derive effective many-body theory (field theory),
describing a Bose gas near a Feshbach resonance,
that incorporates two-atom physics (atom-atom
scattering, molecular binding energy) exactly.
Applications to experiments
Other theories for Feshbach-resonant systems:
1.
2.
3.
4.
E. Timmermans et al., PRL 83, 2691 (1999).
S.J.J.M.F. Kokkelmans and M.J. Holland, PRL 89, 180401 (2002).
M. Mackie, K.-A. Suominen, J. Javanainen, PRL 89, 180403 (2002).
T. Kohler, T. Gasenzer, and K. Burnett, PRA 67, 013601 (2002).
Microscopic atom-molecule
hamiltonian
Microscopic atomic hamiltonian
Take only into account
bound state in closed channel:
Introduce molecular field that
describes this state
(Hubbard-Stratonovich trafo)
During collisions atoms form a
virtual molecule. This leads to
abg
resonant behavior (~1/δ) of the
scattering length in the open channel.
75
So far…
|Eb|(kHz)
Feshbach resonance II
↑ + ↑
∆µ B
↓ + ↓
internuclear separation
 ℏ 2∇ 2
† 
Hˆ = ∫ dxψˆ a† ( x ) −
+ ∫ dx ψ
' ˆ a ( x ')V↓↓ ( x − x ')ψˆ a ( x ') ψˆ a ( x )
 2m


 ℏ 2∇ 2
+ ∫ dxψˆ m† ( x )  −
+ δ ( B)  ψˆ m ( x )
 2(2m)

 x + x' ˆ ˆ + ∫ dx ∫ dx 'g *B ( x − x ')ψˆ m† 
ψ a ( x )ψ a ( x ') + h.c.
 2 
V (x)
Microscopic (bare)
g ( x ) = ↑↓
χm ( x)
atom-molecule coupling. B
2
Atom-atom interactions and atom-molecule coupling are strong: must be
calculated to all orders in perturbation theory. Because we are dealing with
a dilute gas we only have to take into account two-atom collisions. This means
we have to sum all ladder (Feynman) diagrams.
2
+
+
Geometric series
+...
Series is summed by
Lippmann-Schwinger equation
for T-matrix
T
=
T
+
Second quantization: atomic field operator coupled to molecular
field operator
∂ψˆ ( x , t )  ℏ 2∇ 2

iℏ a
= −
+ Tbg2 Bψˆ a† ( x , t )ψˆ a ( x , t ) ψˆ a ( x , t ) + 2 gψˆ a† ( x , t )ψˆ m ( x , t )
∂t
 2m

m3
∂ψˆ ( x, t )  ℏ2∇2
∂ ℏ2∇2 
2 iℏ m
i iℏ +
= −
+ δ ( B(t )) − g 2
ψˆ m ( x, t ) + gψˆ a ( x, t )
3
∂t
2π ℏ
∂t 2(2m) 
 2(2m)
ℏΣm (ℏω) =
Correct result!
g
Tbg2 Bψˆ a†ψˆ aψˆ a + 2 gψˆ a†ψˆ m →
Correct Feshbachresonant scattering
amplitude
≈ −g2
m3
i ℏω
2π ℏ3
4π a ( B ) ℏ 2
m
ψˆ a†ψˆ aψˆ a
abg

∆B 
a(B) = abg 1−

 B − B0 
B0
Load the magnetic trap with atoms
Create BEC of atoms
m3
i ℏω
2π ℏ3
magnetic field
B(G)
Ebound (MHz)
156
gB
This leads to effective atom-atom interaction [δ∼(Β−Β0)]
This propagator has for negative detuning a pole at:
ℏ2
Ebound (B) = − 2
ma
g
B
Experiments
To find the molecular bound-state ( + ) ℏ
Gm (0, ω ) =
energy we have to find the poles
ℏω + − δ ( B) + g 2
of the propagator:
Close to resonance:
≡
ψˆ m = − ψˆ a
δ
Effective theory contains all ladder diagrams: ladders
must not be summed again.
Does the effective theory contain the Feshbach-resonant
scattering properties and the molecular binding energy?

g 4m3 
16π 2ℏ6
Ebound (B) = δ (B) + 2 6  1− 4 3 δ ( B) −1
8π ℏ 
gm

gB
Adiabatic elimination of (“integrating out the”) molecular
g 2
field:
Bound-state energy
+
gB
Usual square-root behavior for decay into continuum
+…=
Scattering properties
Effective atom-molecule model
gB
+
T
We have g B ( x − x ') → gδ ( x − x ') where g is taken from
experiment.
Proportional to g2, due to optical theorem
Molecular self-energy:
gB
V(x-x’)
For the low-energy effective theory this means
that the atom-atom interaction renormalizes to:
4π abg ℏ 2 V ( x − x ') →
δ ( x − x ') Scattering length abg taken from
m
experiment. May also be calculated
+
gB
atom
atom
V↓↓ ( x − x ')
Renormalization of atom-molecule coupling:
Renormalization of atom-atom interaction:
atom
Ladder summations (II)
atom
Ladder summations (I)
157
158
159
160
161
162
Measure atom number.
Molecules cannot be
detected directly
wait for time tevolve
time
-0.1
B0
-0.2
-0.3
-0.4
-0.5
Rb-85
pulse (∼µs) towards resonance
to create superposition
Donley et al., Nature 417, 531 (2002)
3
N.R. Claussen et al., cond-mat/0302195
Experiments: Results I
Experiments: Results II
Bevolve=159,7 G
Image taken after opening the magnetic trap:
Deviation from two-body
result due many-body
effects
52 µm
remnant number
366 µm
Bevolve=157,6 G
•Density of atoms consists of
condensate + high-energetic part (“burst”)
tevolve (µs)
•
•
Mean-field theory
Mean-field approximation: oscillations are understood as
Josephson-like oscillations between atomic and molecular
condensate:
2
∂φ
iℏ a = Tbg2 B φa φ a + 2 gφ a*φm
∂t
∂φm 
m3
∂
iℏ
= δ ( B(t )) − g 2
i iℏ  φm + gφa2
∂t 
2π ℏ3
∂t 
Oscillation frequency is found by linearization: this takes
the fractional derivative into account exactly!
Oscillations in remnant number
Damping!
Oscillation corresponds to
two-body binding energy
Oscillation frequency: results
Josephson frequency:
ℏω J =
( coupling ) + ( Eb ( B ) )
2
2
proportional to condensate
density na
Excellent agreement with
experiment.
For large detuning reduces to
two-body result
Conclusions & Outlook
Coherent atom-molecule oscillations understood
in terms of effective atom-molecule model.
Excellent agreement on deviation from two-body
result, due to many-body effects.
Further simulations with time-dependent magnetic
field (Niels de Keijzer and Bart Vlaar).
Applications to/with Fermions (Gianmaria Falco).
BEC/BCS crossover (See also: Y. Ohashi and A. Griffin, PRL
89, 140402 (2002)).
Collective modes (inhomogeneous)
Normal state
4