Download Feshbach Resonances in Ultracold Gases

Feshbach Resonances in Ultracold Gases
Sara L. Campbell
MIT Department of Physics
(Dated: May 15, 2009)
First described by Herman Feshbach in a 1958 paper, Feshbach resonances describe resonant
scattering between two particles when their incoming energies are very close to those of a two-particle
bound state. Feshbach resonances were first applied to tune the interactions between ultracold Fermi
gases in the late 1990’s. Here, we review two simple models of a Feshbach resonance: a spherical
well model and a coupled square well model. coupled square well model of a Feshbach resonance
and explain how it can be implemented on cold alkali atoms experimentally. Then, we review some
recent key developments in atomic physics that would not have been realized without Feshbach
resonances.
1.
INTRODUCTION AND HISTORY
In 1958, Hermann Feshbach developed a new unified
theory of nuclear reactions[1] to treat nucleon scattering. Feshbach described a situation of resonant scattering that occurs whenever the initial scattering is equal
to that of a bound state between the nucleon and the
nucleus.[2] In the late 1990’s, atomic physicists began
to take advantage of the kind of resonant scattering proposed by Feshbach to realize resonant scattering between
ultracold alkali atoms. Previously, laser and evaporative
cooling of atoms had allowed physicists to reach temperatures as low as 1 µK. However, it is much harder to
cool molecules, because their energy levels and structure
are much more complicated than those of a single atom.
Feshbach resonances made it possible to create ultracold
molecules by assembling them from ultracold atoms[2],
thus making phenomena such as molecular Bose-Einstein
condensation and fermionic superfluidity a reality.
We also express the wave function at large r as the solution to Schrödinger’s equation. V (r) is a central potential, so the equation is separable. We use partial wave
analysis to express our wave function as the sum of the
spherical harmonics Yl and solutions to Schrödinger’s radial equation R(r). We assume symmetry about the angle φ, so we ignore the quantum number m. The spherical harmonics then are just proportional to the Legendre
polynomicals Pl .[5]
φlarge r = Y (θ)R(r)
=
∞
X
!
Cl Pl (cos θ)
l=0
(4)
lπ
1
sin kr −
+ δl .
kr
2
(5)
We expand the eikz terms and equate coefficients to
obtain[4],
∞
1X
sin δl Pl (cos θ).
f (θ) =
k
(6)
l=0
2.
ESSENTIAL RESULTS FROM SCATTERING
THEORY
In a general scattering problem, we have an incident
plane wave with wave number k and a radial scattered
wave with anisotropy term f (θ),[3]
φinc = eikz , φsc =
f (θ)e
r
ikr
.
(1)
dσ = |f (θ)|2 dΩ.
(2)
So, we see that |f (θ)|2 relates the amplitude of particles
passing through a particular target area to the number
of particles passing through a solid angle at a particular
angle θ.
Now we consider a particle with mass m, energy
~2 k 2 /2m in a central potential V (r). We can find f (θ)
by considering the wave function at large r, where it is
simply the sum of the incident and scattered waves[4],
φlarge r = φinc + φsc = eikz +
f (θ)e
r
ikr
(3)
The different angular momentum quantum numbers
l correspond to different partial waves.
Recall
Schrödinger’s radial equation,
−
~2 d2 u(r)
+ Veff (r)u(r) = Eu(r),
2m dr2
(7)
where,
~2 l(l + 1)
.
(8)
2mr2
For the low particle energies in ultracold atoms[6], if l > 0
the angular momentum barrier term in Schrödinger’s radial wave equation prevents the particle from reaching
small r. Using the assumption that V (r) is only significant for small r, we find that only the l = 0 or the s-waves
in the expansion interact with the potential. P0 = 1, so
Veff (r) = V (r) +
φlarge r =
C0
sin(kr − δ0 ),
kr
(9)
which implies that,
1
sin δ0
1
1
=
.
f = − eiδ0 sin δ0 =
k
k cos δ0 − i sin δ0
k cot δ0 − ik
(10)
2
Feshbach resonances in ultracold Fermi gases
By definition, the s-wave scattering length a is[7],
a = − lim
k→0
tan δ0 (k)
.
k
(11)
For strong attractive interactions, a is large and negative.
For strong repulsive interactions, a is large and positive.
Also, Taylor expanding k cot(δ0 (k)) in k 2 gives[6],
1 1
k cot(δ0 (k)) = − + reff k 2 + · · · ,
a 2
(12)
where reff is the effective range of the potential. Combining equations (10) and (12) gives the following expression
for f (k) which will be useful later,
f (k) =
1
− a1
+ reff k2 − ik
2
.
(13)
Finally, we quote the Lippmann-Schwinger[8][2] equation, which expresses f in terms of the wavevectors
kandk0 of the two particles and the interatomic potential
v,
Z
v(k 0 − k)
d3 q v(k0 − q)f (q, k)
f (k0 , k) = −
+
. (14)
4π
(2π)2 k 2 − q 2 + ıη
For s-waves, f (q, k) ≈ f (k), so we can take f outside the
integral and also let v0 = v(0. Then,
Z
1
4π 4π
v(q)
d3 q
≈−
.
(15)
+
f (k)
v0
v0
(2π)3 k 2 − q 2 + iη
3.
APPLICATION TO ULTRACOLD FERMI
GASES
Experimentally, we can take advantage of the internal structure of atoms to tune their interaction. The
effective interaction potential of the two fermionic atoms
depends the angular momentum coupling of their two valence electrons, |s1 , m1 i ⊗ |s2 , m2 i, where s is the total
spin quantum number and m is the spin projection quantum number. If the valance electrons are in the singlet
state with the antisymmetric spin wave function,
1
|0, 0i = √ (↑↓ − ↓↑),
2
(16)
then they have a symmetric spatial wave function and
can exist close together. If the valance electrons are in
the triplet state and have one of the following symmetric
spin wave functions,
|1, 1i =↑↑
1
|1, 0i = √ (↑↓ + ↓↑)
2
|1, −1i =↓↓
potential Vs tends to be much deeper than the triplet potential VT . For simplicity, in both models we assume Vs
just deep enough to contain one bound state and that VT
contains no bound states, and 0 < Vhf VT , VS , ∆µB.
In addition to the effective interaction potential, if the
nucleus has spin I and the electron has spin S, there is a
hyperfine interaction between the electron spin and the
nucleus spin[6],
(17)
(18)
Vhf =
α
I · S.
~2
(20)
If we apply an external magnetic field B, then there will
also be Zeeman shifting ∆µB between the triplet and
singlet states, where ∆µ is the difference in magnetic
moments betwee the two states.
In the field of ultracold atoms, the gas temperature T
is very small and the atoms are confined in space using
a magneto-optical trap (MOT) so that the atoms feel a
considerable magnetic field B. Therefore, ∆µ kB T ,
and if we assume sufficiently low gas density, we can apply Maxwell-Botzmann statistics to the system and see
that almost all of the atoms scatter via the | ↑↑i state
and very few atoms scatter via the | ↓↓i state. Using
the language of scattering theory, we call | ↑↑i an open
channel and | ↓↓i a closed channel.[2]
3.1.
Spherical Well Model
In this model we only consider one bound state in the
closed channel |mi and a continuum of plane waves of
relative momentum k between the two atoms |ki in the
open channel.[10] Without coupling, |mi and |ki are energy eigenstates of the free Hamiltonian[2],
H0 |ki = 2k |ki, k =
~2 k 2
H0 |mi = δ|mi
2m
(21)
Let |φi be the coupled wavefunction, which can be
written as a superposition of the molecular state and the
scattering states,
|φi = α|mi +
X
ck |ki.
(22)
k
Figure 1 shows how the bound states couple to the continuum. We will solve for the coupling explicitly by
plugging |φi into Schrödinger’s equation with the Hamiltonian H = H√0 + V , where V includes the coupling
hm|V |ki = gk / Ω, where Ω is the phase space volume
of the system. Because we are just dealing with s-waves,
we make the approximation,
(19)
gk =
then they have an antisymmetric spatial wave function
and cannot exist close together. Therefore, the singlet
where ER = ~2 /mR2 .
g0 , E < ER
,
0, E > ER
(23)
3
Feshbach resonances in ultracold Fermi gases
As in figure 1, we see that relative to the “bare” |mi
state, the “dressed” ground state resonance position is
shifted to δ0 .[10]
3.1.2.
Scattering amplitude and scattering length
To find the scattering amplitude and scattering length,
we plug |φi into the time-independent Schrödinger equation H|φi = E|φi for E > 0 and find,
X g0
gk
1
g
√
√q cq
(E − 2)c0 = √ α =
E
−
δ
+
iη
Ω
Ω
Ω
q
X
Fig. 32. – Simple model for a Feshbach resonance. The dashed lines show the uncoupled states:
FIG. 1: Energy as a function of detuning for states in the
=
Veff (k, q)cq .
The closed channel molecular state |m! and the scattering states |k! of the continuum. The
spherical
well model
for fermion
q
uncoupled resonance
position
lies at zero
detuning,collisions.
δ = 0. TheThe
soliduncoupled
lines show the coupled
molecular
state
|mi
and
scattering
states
|ki
as state of the
states: The state |ϕ! connects the molecular state |m! at δ " 0are
to shown
the lowest
Substituting
v0 = Veff Ω into equation ?? we find,
dashed
lines and
coupled
statesthe
aremolecular
shown asstate
solidis lines.
continuum above
resonance.
Atthe
positive
detuning,
“dissolved”
in the
From causing
[2].
continuum, merely
an upshift of all continuum states as ϕ becomes the new lowest
Z
1
4π~2
1
d3 q
continuum state. In this illustration, the continuum is discretized in equidistant energy levels.
≈
(E
−
δ)
+
4π
.
2
In the continuum limit (Fig. 33), the dressed molecular energy reaches zero at a finite, shifted
f0 (k)
mg0
(2π)3 k 2 − q 2 + iη
resonance position δ0 .3.1.1. Bound state energy and resonance shift
Now we substitute in,
Z
Then to find the molecular wavefunction after cou−4π~2 δ0
d3 q 1
.
5 3.1. A pling,
model we
for plug
Feshbach
resonances.
Good insight into
the Feshbach resonance
=
− 4π
|φi into
the time-independent
Schrödinger
(2π)3 q 2
mg02
mechanism can
be gained
by =
considering
equation
H|φi
E|φi for two
E <coupled
0 and spherical
find[2], well potentials, one each
(33)
(34)
(35)
(36)
for the open and closed channel [262]. Other models can be found in [142, 247]. Here,
and also using equation (32) that when δ ≈ δ0 , the energy
g0
we will use an even simpler
model
Feshbach
resonance, in which (24)
there is only one
α
(E −
2)c0of=a √
E is approximately the coupling energy g0 , we find,
bound state of importance |m! in the closed
Ωchannel, the others being too far detuned in
s
energy (see Fig. 32). The continuum of plane
r
1 waves of1relative
g02 α momentum k between the
1
m
2~2 2
1
√
(E
−
δ)α
=
g
c
=
(25)
0 0
two particles in the incoming channel will be denoted
In the absence of coupling, ≈ −
(δ
−
δ)
−
k − ik (37)
ΩasE|k!.
− 2
0
Ω
f0 (k)
2~2 E0
2 mE0
these are eigenstates of the free hamiltonian
2
X
E−δ =
1
Ω
g0
.
E − 2 !2 k2
(26)
Now our equation looks just like equation (13), so we find
a and reff ,
(195)
H0over
|m! =
δ |m!
withthen
δ ≶ gives[2],
0
s
Integrating
the
continuum
r
2E
1
2~
2~2
0
,
r
=
−
a
=
.
(38)
2 Z Eof
R the “bare” molecular state, is the parameter under
where δ, the bound state energy
eff
ρ()d
g0
m δ0 − δ
mE0
(27)and the |k!’s.
|E| + We
δ =consider
experimental control.
interactions explicitly only between |m!
Ω 0
2 + |E|
If necessary, scattering that occurs
exclusively
can be accounted

s in the incomingschannel
!
Finally, we plug in δ − δ0 = ∆µ(B − B0 ) to find that,
for by including a phase shift
intoR )the scattering
wave functions
|k!
(see
Eq. 213), i.e.
g02 ρ(E
|E|
2E
R 
r

=)/r, where δbg1 −
using ψk (r) ∼ sin(kr + δbg
and abg = arctan
δbg /k are the (so-called
Ω
2Er − limk→0 tan|E|
1
2~E0
.
(39)
a=−
background) phase shift and scattering length, resp., in the open channel. First, let us
m
B
−
B0
(28)
see how the molecular state is modified due to the coupling to the continuum
|k!. For
(
p
This is the scattering length we find without considering
δ0 − 2E0|E|,
124 |E| ER
≈
(29)
background collisions in the open channel. We account
2ER
δ0 3|E| ,
|E| ER
for background scattering by adding a phase shift δbg to
our wavefunction so that δtotal = δ + δbg . To see how this
where,
extra phase shift affects the scattering length, we add δbg
to our definition of scattering length in equation 11,
g2 X 1
4p
δ0 = 0
=
E0 Er ,
(30)
Ω
2k
π
tan(δ0 (k) + δbg )
k
atotal = lim −
(40)
2 k→0
2
k
g0 m 3/2
tan δ0 (k) + tan δbg
E0 =
.
(31)
= lim −
(41)
2π 2~2
k→0
k(1 − tan δ0 tan δbg )
p
(
2
2
tan δ0
tan δbg
−E0 + δ − δ0 + (E
q0 − δ + δ0 ) − (δ − δ0 ) , |E| ER
≈ lim −
+ lim
(42)
E=
2
k→0
k→0
delta
δ
2
k
k
−
+
δ
E
,
|E|
E
0
R
R
2
4
3
= a + abg .
(43)
(32)
H0 |k! = 2!k |k!
kwith !k
=
2m
>0
4
Feshbach resonances in ultracold Fermi gases
So, using this approximation,
atotal = abg + a = abg
2
∆B
1−
B − B0
,
1
(44)
0
a/R
where,
r
∆B =
-1
2~2 E0 1
.
m ∆µabg
(45)
-2
-3
3.2.
Coupled Square Well Model
4.4
4.45
4.5
4.55
4.6
4.65
4.7
! µB/("h2/mR2)
4.75
4.8
4.85
4.9
Fig. 6. Scattering length for two coupled square-well potentials as a function of
Putting all of these effects together into one HamiltoFIG.
2: of
Theoretical
vs.potentials
magneticisfield,
depth
the tripletscattering
and singletlength
channel
VT = nu!2 /mR2 and
nian, the Schrödinger equation for a triplet state |T i ∆µB.
and The
merically
calculated for the coupled square well model. From
2 /mR2 , respectively.
2 /mR2 . The
V
=
10!
The
hyperfine
coupling
is
V
=
0.1!
S
hf
a singlet state |Si with potentials VT (r) and Vs (r) is[4],
[6].
−~2 ∇2
m
! dotted line shows the background scattering length abg .
+ VT (r) − E
Vhf
(46)
2 2
Vhf
− ~ m∇ + ∆µB + VS (r) − E
„
«
ψT (r)
×
= 0. (47)
ψS (r)




>
 u↑↑ (R) 
<
 u↑↑ (R) 
−1 >
Q−1 (θ< ) 
 = Q (θ ) 
 ;
<
u↓↓the
(R) s-wave scattering
u>
↓↓ (R)
Now we find
length. Similar to
'
'


' scattering, for r > >
traditional
s-wave
R we''assume the
<
'
u
(r)
u
(r)
∂
∂ solutions
↑↑
↑↑

'

'


spherical
'
= Schrödinger’s
,
(54)
Q−1 (θ< ) to the 
Q−1 (θ> )  equation,
'
'
'
<
>
∂r
As before, we use square well models (for simplicity we ∂r
u↓↓
u↓↓ (r) '
(r) 'r=R ikr
ikr
r=R
use the same radius R) for the two interaction potentials,
Ce + De
u↑↑ (r)
0
=
,
(55)
<
u
(r)
F e−k rfour equations determine the
where tan θ = 2Vhf /(VS↓↓− VT − ∆µB). These
−VT,S , r < R
coefficients A, B, C, Dpand F up to a normalization factor, and therefore also
(48)
VT,S (r) =
0
0,
r>R
where
= them(E
)/~2 −Although
k 2 . And itforis rpossible
< R weto find an
the phase
shift kand
scattering
length.
+ − E−
assume
the solutions,
analytical
expression
for the scattering length as a function of the magnetic
The kinetic energy operator is diagonal because our
field, the resulting expression is rather
formidable andis omitted here. The
basis kets are two different internal states of the atoms,
1r
A(eikin1 r Fig.
− e−ik
) VS = 10!2 /mR2 , VT =
v↑↑length
(r)
result for the scattering
is
shown
6, for
0
=
,
(56)
so we need to diagonalize the Hamiltonian,
ik1 r
−ik10 r
2
2
2
2
v
B(e
−
e
! /mR and Vhf = 0.1!↓↓(r)
/mR , as a function of ∆µB. )The resonant behaviour
is due to the bound state of the singlet potential VS (r). Indeed, solving the
0 Vhf
where,
Hint =
,
(49)
equation
for the binding energy in Eq. (21) with V0 = −VS we find that
Vhf ∆µB
p 2 , which is approximately the position of the resonance in
2
|Em | " 4.62!
2
2
k1 /mR
= m(E
(57)
− − E1,− )/~ + k
Fig. 6. The difference
is
due
to
the
fact
that
the hyperfine interaction leads to
p
and the energy eigenvalues are,
0
2
k1 position
= m(Eof
− Eresonance
(58)
− the
1,+ )/~ + k2
a shift in the
with respect to Em .
p
∆µB
1p
∆µB
−
V
−
V
1
T
s
E1,± =
∓
(Vs − VT − ∆µB)2 + (2Vhf )2 .
E± =
(∆µB)2 + (2Vhf )2
(50)
±
The magnetic-field
dependence
of the
2
2 scattering length near a Feshbach reso2
2
(59)
nance is characterized experimentally by a width ∆B and position
B0 accordLet, Q(θ) be the rotation matrix that diagonalizes ing
theto
Finally, u↑↑ and v↑↑ and their derivatives must match at
Hamiltonian H and
)
r = R. (The same
∆B applies to u↓↓ and v↓↓ . The analytical
a(B)solution
= abg 1is−complicated,
. but a numerical solution of the
(55)
| ↑↑i
|T i
B − B0
=Q
.
(51)
| ↓↓i
|Si
scattering length a for VS = 10~2 /mR2 , VT = ~2 /mR2
and Vhf = .1~2 /mR2 was calculated as a function of ∆µB
This
explicitly
shows in
that
the scattering
length, andlooks
therefore
magnitude
Also, define V↑↑ (r), V↑↓ (r) and V↓↓ (r) by the change of
and plotted
figure
2, and qualitatively
verythe
simbasis formula,
ilar to the spherical well solution in equation (44). Both
models show the essential 20
physics - that the scattering
V↑↑ (r) V↑↓ (r)
VT (r)
0
length is a widely-tunable function of the applied mag= Q(θ)
Q−1 (θ).
V↑↓ (r) V↓↓ (r)
0
VS (s)
netic field.
(52)
Plugging these new basis states into the Schrödinger
equation gives,
4. EXPERIMENTAL OBSERVATION OF
!
2 2
− ~ m∇ + V↑↑ (r) − E
V↑↓ (r)
2 2
V↑↓ (r)
− ~ m∇ + E+ − E− + V↓↓ (r) − E
(53)
`
´
× φ↑↑ (r)φ↓↓ (r) = 0.
(54)
FESHBACH RESONANCES
4.1.
Setup
First predicted for ultracold gases in 1995[11], Feshbach resonances in a Bose-Einstein condensate were
responsible for the acceleration of the atoms. After ballistic expansion of the condensate (either 12 or 20 ms), the atoms were optically
pumped into the jF ¼ 2! ground state and probed using resonant
light driving
the cycling
disk-like
expansion of the
Feshbach
resonances
intransition.
ultracoldThe
Fermi
gases
the observed onset of trap loss at least a factor of ten sharper than for
the first. As the upper resonance was only reached by passing
through the lower one, some losses of atoms were unavoidable;
for example, when the lower resonance was crossed at 2 G ms!1,
5
Figure
Observation
the Feshbach
at 907 Gnumber
using phase-contrast
4 we1 also
see aofsharp
dropresonance
in the atom
and mean
a
imaging in an optical trap. A rapid sequence (100 Hz) of non-destructive, in situ
density at the resonance.
phase-contrast images of a trapped cloud (which appears black) is shown. As the
b
magnetic field was increased, the cloud suddenly disappeared for atoms in the
j mF ¼ þ1! state (see images in c), whereas nothing happened for a cloud in the
(images
in d). The height
of the
images is 140 "m.
A diagram of
j mF ¼ ! 1! state
4.3.
Measuring
the
scattering
length
the optical trap is shown in a. It consisted of one red-detuned laser beam
200 µm
c
providing radial confinement, and two blue-detuned laser beams acting as end-
interaction
energyof E
of the field
condensate
is proporcapsThe
(shown
as ovals). The minimum
theImagnetic
was slightly offset
from
m =+1
F
tional
the
scattering
the
centre to
of the
optical
trap. As a length,[12][14]
result, the condensate (shaded area) was
pushed by the magnetic field curvature towards one of the end-caps. The axial
profile of the total potential is shown
E in b.2π~2
I
N
908 G
905 G
d
m =-1
F
905 G
(0 ms)
Magnetic field
(Time)
=
EK
1
2
= mvrms
.
N
2
(70 ms)
Nature © Macmillan Publishers Ltd 1998
From [14] we know that,
Finding the resonances
To realize a Feshbach resonance, a large magnetic field
was applied to the condensate and swept in magnitude.
On one side of a Feshbach resonance, the scattering
length is very large and negative, interactions between
the atoms are very strong and attractive, and the condensate collapses. On the other side of a Feshbach resonance, the scattering length is very large and positive,
interactions between the atoms are very strong and repulsive, and the condensate is quickly lost. The exact locations of the Feshbach resonances in sodium were found
by sweeping the magnetic field, finding the magnetic field
values with condensate loss, and then sweeping in smaller
and smaller intervals about those intervals.[12]. Figure
3c shows condensate loss in the |F = 1, mF = +1i state
as the magnetic field is swept from 905 to 908 G. In figure
(61)
(62)
Combining, we find,
2
vrms
.
N
(63)
2
and N from time of flight
We can determine both vrms
images at a particular magnetic field. This is how we
obtain data points for a scattering length vs. magnetic
field curve shown in figure 4. The results are very similar
to the theoretical model (44)
5.
4.2.
(60)
NATURE | VOL 392 | 12 MARCH 1998
hni ∝ N (N a)−3/5 .
a∝
first observed by the Ketterle group in 1998[12][13]. A
schematic of the experimental setup for the 1998 experiment is shown in figure 3. Condensates were prepared
in a MOT in the |F = 1, mF = −1i state, moved to an
optical dipole trap (ODT), adiabatically spin-flipped by
an RF field to the |F = 1, mF = +1i state. The condensate was observed using both phase-contrast imaging
and time-of-flight absorption imaging[12].
ahni,
where N is the total number of atoms, m is the mass of
an atom, and hni is the average density of the condensate. Because the kinetic energy of atoms in a trapped
condensate is insignificant compared to the interaction
energy, the kinetic energy of a freely expanding condensate defined by the root mean square velocity vrms equals
the the interaction energy of the gas while it was trapped,
908 G
152
FIG.
3: Illustration of experimental method of phase-contrast
imaging in an optical trap. a. Shows the optical trap, made
of a red-detuned laser beam for radial confinement and two
blue-detuned beams for axial confinement (“endcaps”). b.
Shows the confining potential in the axial direction. c and d.
show images of the condensate as a function of magnetic field
for the mF = ±1 substates. From [12]
m
CONCLUSIONS
We have discussed the theory of Feshbach resonances in
ultracold gases for two simple models and have shown the
major physical result - the scattering length is a widely
tunable function of an applied magnetic field. The scattering length dependence on the magnetic field has been
observed experimentally and agrees with our derived results. Feshbach resonances have been crucial to the field
of ultracold Fermi gases. Because of the Pauli exclusion
principle, fermions tend to stay away from each other
in space, but by tuning a magnetic field near a Feshbach resonance, we can cause them to interact more
strongly than they would otherwise. Important recent
developments like fermionic superfluidity and molecular
Bose-Einstein condensates (where two fermions make the
molecule) were only realized by using a magnetic field to
tune the fermion-fermion interaction. Experimetnal control over the scattering length is a very powerful tool.
m vrms
.
(3)
4πh̄2 a
The peak density n0 is given by n0 = (7/4)"n# in a parabolic
Feshbach
resonances in ultracold Fermi gases
potential.
"n# =
mF = +1
6
Number of Atoms N
mF = -1
5
a)
10
10
d)
4
10
5
10
0.13 G/ms
0.31 G/ms
-0.06 G/ms
3
10
0.28 G/ms
4
10
900
905
910
915
1170
b)
1180
1190
1200
e)
Magnetic Field [G]
1
1
0.1
( )
895
-3
Mean Density <n> [10 cm ] Scattering Length a/a0
10
895
900
905
910
c)
10
8
8
6
6
4
4
2
2
1180
1190
Magnetic Field [G]
14
10
1170
915
( )
1200
f)
0
0
895
900
905
910
Magnetic Field [G]
915
1170
1180
1190
1200
Magnetic Field [G]
FIG.
4: Number
of atoms,
normalized
scattering length
and
Figure
1: Number
of atoms
N , normalized
scattering
mean
density
a function
of magnetic
field around
the 907
length
a/a0asand
mean density
"n# versus
the magnetic
G field
(left)near
and the
1195907
G (right)
resonancesresonances.
in a BoseG and Feshbach
1195 G Feshbach
Einstein
condensate
of
Na.
From
[13]
The different symbols for the data near the 907 G resonance correspond to different ramp speeds of the magnetic
field. All data were extracted from time–of–flight images.
The errors due to background noise of the images, thermal atoms, and loading fluctuations of the optical trap
areH.indicated
byAnnals
single error
bars in
each curve.
[1]
Feshbach,
of Physics
(1958).
[2] W. Ketterle and M. Zwierlein, in Proceedings of the International
School results
of Physics
”Enrico
Fermi”,
The experimental
for the
907 G and
1195 Course
G resCLXIVare(2006).
onances
shown in Fig. 1. The number of atoms, the
[3]
R. L. Liboff,
Introductory
normalized
scattering
lengthQuantum
and the Mechanics
density are(Addison
plotted
Wesley,
San
Francisco,
2003),
4th
ed.
versus the magnetic field. The resonances can be identi[4]
Griffiths,
Introduction
to Quantum
Mechanics
(PrenfiedD.by
enhanced
losses. The
907 G resonance
could
be
tice
Hall,
Upper
Saddle
River,
1987),
2nd
ed.
approached from higher field values by crossing it first
[5] D. Bohm, Quantum Theory (Prentice Hall, N.J., 1951).
with very high ramp speed. This was not possible for the
[6] R. A. Duine and H. T. C. Stoof (2008).
[7] J. J. Sakurai, Modern Quantum Mechanics (AddisonWesley, Reading, 1994).
2
[8] B. A. Lippmann and J. Schwinger, Physical Review 79
(1950).
[9] W. D. Phillips, in Proceedings of the International School
of Physics ”Enrico Fermi” (1992).
[10] M. W. Zwierlein, Ph.D. thesis, MIT (2006).
[11] A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Physical
Review A (1995).
[12] S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner,
D. M. Stamper-Kurn, and W. Ketterle, Nature 392
(1998).
[13] J. Stenger, S. Inouye, M. Andrews, H. Miesner,
D. Stamper-Kurn, and W. Ketterle, Physical Review Letters (1999).
[14] G. Baym and C. J. Pethick, Physical Review Letters 76
(1996).
!
Ṅ
Ki "ni−1 #,
=−
N
i
(4)
6
where Ki denotes an i–body loss coefficient, and "n# the
spatially averaged density. In general, the density n depends on the numberAcknowledgments
of atoms N in the trap, and the loss
curve is non–exponential. One–body losses, e.g. due to
background gas collisions or spontaneous light scattering,
are negligible under our experimental conditions. An increase of the dipolar relaxation rate (two–body collisions)
near Feshbach resonances has been predicted [1]. However, for sodium in the lowest energy hyperfine state F =
1, mF = +1 binary inelastic collisions are not possible.
Collisions involving more than three atoms are not expected to contribute. Thus the experimental study of the
loss processes focuses on the three–body losses around the
F = 1, mF = +1 resonances, while both two– and three–
body losses must be considered for the F = 1, mF = −1
resonance.
Figs. 1 (c) and (f) show a decreasing or nearly constant
density when the resonances were approached. Thus the
enhanced trap losses can only be explained with increasing coefficients for the inelastic processes. The quantity
Ṅ /N "n2 # is plotted versus the magnetic field in Fig. 2 for
both the 907 G and the 1195 G resonances. Assuming that
mainly three–body collisions cause the trap losses, these
Thank
you to Martin
me907
to helpplots
correspond
to the Zwierlein
coefficient for
K3pointing
. For the
G
ful
references,
and
to
everyone
else
in
the
Center
(1195 G) resonance, the off–resonant value of Ṅ /N "n2for
# isUltracold
about
20Atoms.
(60) times larger than the value for K3 measured
at low fields [7]. Close to the resonances, the loss coefficient strongly increases both when tuning the scattering
length larger or smaller than the off–resonant value. Since
the density is nearly constant near the 1195 G resonance,
the data can also be interpreted by a two–body coefficient K2 = Ṅ /N "n# with an off–resonant value of about
30 × 10−15 cm3 /s, increasing by a factor of more than 50
near the resonance. The contribution of the two processes
cannot be distinguished by our data. However, dipolar losses are much better understood than three–body
losses. The off–resonant value of K2 is expected to be
about 10−15 cm3 /s [9] in the magnetic field range around