Feshbach Resonances in Ultracold Gases

Sara L. CampbellMIT Department of Physics

(Dated: May 15, 2009)

First described by Herman Feshbach in a 1958 paper, Feshbach resonances describe resonantscattering between two particles when their incoming energies are very close to those of a two-particlebound state. Feshbach resonances were first applied to tune the interactions between ultracold Fermigases in the late 1990’s. Here, we review two simple models of a Feshbach resonance: a sphericalwell model and a coupled square well model. coupled square well model of a Feshbach resonanceand explain how it can be implemented on cold alkali atoms experimentally. Then, we review somerecent key developments in atomic physics that would not have been realized without Feshbachresonances.

1. INTRODUCTION AND HISTORY

In 1958, Hermann Feshbach developed a new unifiedtheory of nuclear reactions[1] to treat nucleon scatter-ing. Feshbach described a situation of resonant scatter-ing that occurs whenever the initial scattering is equalto that of a bound state between the nucleon and thenucleus.[2] In the late 1990’s, atomic physicists beganto take advantage of the kind of resonant scattering pro-posed by Feshbach to realize resonant scattering betweenultracold alkali atoms. Previously, laser and evaporativecooling of atoms had allowed physicists to reach temper-atures as low as 1 µK. However, it is much harder tocool molecules, because their energy levels and structureare much more complicated than those of a single atom.Feshbach resonances made it possible to create ultracoldmolecules by assembling them from ultracold atoms[2],thus making phenomena such as molecular Bose-Einsteincondensation and fermionic superfluidity a reality.

2. ESSENTIAL RESULTS FROM SCATTERINGTHEORY

In a general scattering problem, we have an incidentplane wave with wave number k and a radial scatteredwave with anisotropy term f(θ),[3]

φinc = eikz, φsc =f(θ)eikr

r. (1)

dσ = |f(θ)|2dΩ. (2)

So, we see that |f(θ)|2 relates the amplitude of particlespassing through a particular target area to the numberof particles passing through a solid angle at a particularangle θ.

Now we consider a particle with mass m, energy~2k2/2m in a central potential V (r). We can find f(θ)by considering the wave function at large r, where it issimply the sum of the incident and scattered waves[4],

φlarge r = φinc + φsc = eikz +f(θ)eikr

r(3)

We also express the wave function at large r as the solu-tion to Schrodinger’s equation. V (r) is a central poten-tial, so the equation is separable. We use partial waveanalysis to express our wave function as the sum of thespherical harmonics Yl and solutions to Schrodinger’s ra-dial equation R(r). We assume symmetry about the an-gle φ, so we ignore the quantum number m. The spheri-cal harmonics then are just proportional to the Legendrepolynomicals Pl.[5]

φlarge r = Y (θ)R(r) (4)

=

( ∞∑l=0

ClPl(cos θ)

)1kr

sin(kr − lπ

2+ δl

).

(5)

We expand the eikz terms and equate coefficients toobtain[4],

f(θ) =1k

∞∑l=0

sin δlPl(cos θ). (6)

The different angular momentum quantum numbersl correspond to different partial waves. RecallSchrodinger’s radial equation,

− ~2

2md2u(r)dr2

+ Veff(r)u(r) = Eu(r), (7)

where,

Veff(r) = V (r) +~2l(l + 1)

2mr2. (8)

For the low particle energies in ultracold atoms[6], if l > 0the angular momentum barrier term in Schrodinger’s ra-dial wave equation prevents the particle from reachingsmall r. Using the assumption that V (r) is only signifi-cant for small r, we find that only the l = 0 or the s-wavesin the expansion interact with the potential. P0 = 1, so

φlarge r =C0

krsin(kr − δ0), (9)

which implies that,

f = −1keiδ0 sin δ0 =

1k

sin δ0cos δ0 − i sin δ0

=1

k cot δ0 − ik .(10)

Feshbach resonances in ultracold Fermi gases 2

By definition, the s-wave scattering length a is[7],

a = − limk→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 k2 gives[6],

k cot(δ0(k)) = −1a

+12reffk

2 + · · · , (12)

where reff is the effective range of the potential. Combin-ing equations (10) and (12) gives the following expressionfor f(k) which will be useful later,

f(k) =1

− 1a + reff

k2

2 − ik. (13)

Finally, we quote the Lippmann-Schwinger[8][2] equa-tion, which expresses f in terms of the wavevectorskandk′ of the two particles and the interatomic potentialv,

f(k′,k) = −v(k′ − k)4π

+∫

d3q

(2π)2

v(k′ − q)f(q,k)k2 − q2 + ıη

. (14)

For s-waves, f(q,k) ≈ f(k), so we can take f outside theintegral and also let v0 = v(0. Then,

1f(k)

≈ −4πv0

+4πv0

∫d3q

(2π)3

v(q)k2 − q2 + iη

. (15)

3. APPLICATION TO ULTRACOLD FERMIGASES

Experimentally, we can take advantage of the inter-nal structure of atoms to tune their interaction. Theeffective interaction potential of the two fermionic atomsdepends the angular momentum coupling of their two va-lence electrons, |s1,m1〉 ⊗ |s2,m2〉, where s is the totalspin quantum number and m is the spin projection quan-tum number. If the valance electrons are in the singletstate with the antisymmetric spin wave function,

|0, 0〉 =1√2

(↑↓ − ↓↑), (16)

then they have a symmetric spatial wave function andcan exist close together. If the valance electrons are inthe triplet state and have one of the following symmetricspin wave functions,

|1, 1〉 =↑↑ (17)

|1, 0〉 =1√2

(↑↓ + ↓↑) (18)

|1,−1〉 =↓↓ (19)

then they have an antisymmetric spatial wave functionand cannot exist close together. Therefore, the singlet

potential Vs tends to be much deeper than the triplet po-tential VT . For simplicity, in both models we assume Vsjust deep enough to contain one bound state and that VTcontains no bound states, and 0 < Vhf VT , VS ,∆µB.

In addition to the effective interaction potential, if thenucleus has spin I and the electron has spin S, there is ahyperfine interaction between the electron spin and thenucleus spin[6],

Vhf =α

~2I · S. (20)

If we apply an external magnetic field B, then there willalso be Zeeman shifting ∆µB between the triplet andsinglet states, where ∆µ is the difference in magneticmoments betwee the two states.

In the field of ultracold atoms, the gas temperature Tis very small and the atoms are confined in space usinga magneto-optical trap (MOT) so that the atoms feel aconsiderable magnetic field B. Therefore, ∆µ kBT ,and if we assume sufficiently low gas density, we can ap-ply Maxwell-Botzmann statistics to the system and seethat almost all of the atoms scatter via the | ↑↑〉 stateand very few atoms scatter via the | ↓↓〉 state. Usingthe language of scattering theory, we call | ↑↑〉 an openchannel and | ↓↓〉 a closed channel.[2]

3.1. Spherical Well Model

In this model we only consider one bound state in theclosed channel |m〉 and a continuum of plane waves ofrelative momentum k between the two atoms |k〉 in theopen channel.[10] Without coupling, |m〉 and |k〉 are en-ergy eigenstates of the free Hamiltonian[2],

H0|k〉 = 2εk|k〉, εk =~2k2

2mH0|m〉 = δ|m〉 (21)

Let |φ〉 be the coupled wavefunction, which can bewritten as a superposition of the molecular state and thescattering states,

|φ〉 = α|m〉+∑k

ck|k〉. (22)

Figure 1 shows how the bound states couple to the con-tinuum. We will solve for the coupling explicitly byplugging |φ〉 into Schrodinger’s equation with the Hamil-tonian H = H0 + V , where V includes the coupling〈m|V |k〉 = gk/

√Ω, where Ω is the phase space volume

of the system. Because we are just dealing with s-waves,we make the approximation,

gk =g0, E < ER0, E > ER

, (23)

where ER = ~2/mR2.

Feshbach resonances in ultracold Fermi gases 3

Fig. 32. – Simple model for a Feshbach resonance. The dashed lines show the uncoupled states:The closed channel molecular state |m! and the scattering states |k! of the continuum. Theuncoupled resonance position lies at zero detuning, ! = 0. The solid lines show the coupledstates: The state |"! connects the molecular state |m! at ! " 0 to the lowest state of thecontinuum above resonance. At positive detuning, the molecular state is “dissolved” in thecontinuum, merely causing an upshift of all continuum states as " becomes the new lowestcontinuum state. In this illustration, the continuum is discretized in equidistant energy levels.In the continuum limit (Fig. 33), the dressed molecular energy reaches zero at a finite, shiftedresonance position !0.

5.3.1. A model for Feshbach resonances. Good insight into the Feshbach resonancemechanism can be gained by considering two coupled spherical well potentials, one eachfor the open and closed channel [262]. Other models can be found in [142, 247]. Here,we will use an even simpler model of a Feshbach resonance, in which there is only onebound state of importance |m! in the closed channel, the others being too far detuned inenergy (see Fig. 32). The continuum of plane waves of relative momentum k between thetwo particles in the incoming channel will be denoted as |k!. In the absence of coupling,these are eigenstates of the free hamiltonian

H0 |k! = 2!k |k! with !k =!2k2

2m> 0

H0 |m! = " |m! with " ! 0(195)

where ", the bound state energy of the “bare” molecular state, is the parameter underexperimental control. We consider interactions explicitly only between |m! and the |k!’s.If necessary, scattering that occurs exclusively in the incoming channel can be accountedfor by including a phase shift into the scattering wave functions |k! (see Eq. 213), i.e.using #k(r) " sin(kr + "bg)/r, where "bg and abg = # limk!0 tan "bg/k are the (so-calledbackground) phase shift and scattering length, resp., in the open channel. First, let ussee how the molecular state is modified due to the coupling to the continuum |k!. For

124

FIG. 1: Energy as a function of detuning for states in thespherical well model for fermion collisions. The uncoupledmolecular state |m〉 and scattering states |k〉 are shown asdashed lines and the coupled states are shown as solid lines.From [2].

3.1.1. Bound state energy and resonance shift

Then to find the molecular wavefunction after cou-pling, we plug |φ〉 into the time-independent Schrodingerequation H|φ〉 = E|φ〉 for E < 0 and find[2],

(E − 2ε)c0 =g0√Ωα (24)

(E − δ)α =1√Ωg0c0 =

1Ω

g20α

E − 2ε(25)

E − δ =1Ω

∑k

g20

E − 2ε. (26)

Integrating over the continuum then gives[2],

|E|+ δ =g2

0

Ω

∫ ER

0

ρ(ε)dε2ε+ |E| (27)

=g2

0ρ(ER)Ω

1−√|E|2Er

arctan

(√2ER|E|

)(28)

≈δ0 −

√2E0|E|, |E| ER

δ0

(2ER

3|E|), |E| ER

(29)

where,

δ0 =g2

0

Ω

∑k

12εk

=4π

√E0Er, (30)

E0 =[g2

0

2π

( m

2~2

)3/2]2

. (31)

E =

( −E0 + δ − δ0 +p

(E0 − δ + δ0)2 − (δ − δ0)2, |E| ERdelta

2−q

δ2

4+ 2

3δ0ER, |E| ER

(32)

As in figure 1, we see that relative to the “bare” |m〉state, the “dressed” ground state resonance position isshifted to δ0.[10]

3.1.2. Scattering amplitude and scattering length

To find the scattering amplitude and scattering length,we plug |φ〉 into the time-independent Schrodinger equa-tion H|φ〉 = E|φ〉 for E > 0 and find,

(E − 2ε)c0 =gk√Ωα =

∑q

g0√Ω

1E − δ + iη

gq√Ωcq (33)

=∑q

Veff(k, q)cq. (34)

Substituting v0 = VeffΩ into equation ?? we find,

1f0(k)

≈ 4π~2

mg20

(E − δ) + 4π∫

d3q

(2π)3

1k2 − q2 + iη

. (35)

Now we substitute in,

− 4π∫

d3q

(2π)3

1q2

=−4π~2δ0mg2

0

(36)

and also using equation (32) that when δ ≈ δ0, the energyE is approximately the coupling energy g0, we find,

1f0(k)

≈ −√

m

2~2E0(δ0 − δ)− 1

2

√2~2

mE0k2 − ik (37)

Now our equation looks just like equation (13), so we finda and reff,

a =

√2~2E0

m

1δ0 − δ , reff = −

√2~2

mE0. (38)

Finally, we plug in δ − δ0 = ∆µ(B −B0) to find that,

a = −√

2~E0

m

1B −B0

. (39)

This is the scattering length we find without consideringbackground collisions in the open channel. We accountfor background scattering by adding a phase shift δbg toour wavefunction so that δtotal = δ+δbg. To see how thisextra phase shift affects the scattering length, we add δbg

to our definition of scattering length in equation 11,

atotal = limk→0− tan(δ0(k) + δbg)

k(40)

= limk→0− tan δ0(k) + tan δbgk(1− tan δ0 tan δbg)

(41)

≈ limk→0− tan δ0

k+ limk→0

tan δbgk

(42)

= a+ abg. (43)

Feshbach resonances in ultracold Fermi gases 4

So, using this approximation,

atotal = abg + a = abg

(1− ∆B

B −B0

), (44)

where,

∆B =

√2~2E0

m

1∆µabg

. (45)

3.2. Coupled Square Well Model

Putting all of these effects together into one Hamilto-nian, the Schrodinger equation for a triplet state |T 〉 anda singlet state |S〉 with potentials VT (r) and Vs(r) is[4],

−~2∇2

m+ VT (r)− E Vhf

Vhf −~2∇2

m+ ∆µB + VS(r)− E

!(46)

ׄψT(r)ψS(r)

«= 0. (47)

As before, we use square well models (for simplicity weuse the same radius R) for the two interaction potentials,

VT,S(r) = −VT,S , r < R

0, r > R(48)

The kinetic energy operator is diagonal because ourbasis kets are two different internal states of the atoms,so we need to diagonalize the Hamiltonian,

Hint =(

0 Vhf

Vhf ∆µB

), (49)

and the energy eigenvalues are,

E± =∆µB

2± 1

2

√(∆µB)2 + (2Vhf)2 (50)

Let, Q(θ) be the rotation matrix that diagonalizes theHamiltonian H and( | ↑↑〉

| ↓↓〉)

= Q

( |T 〉|S〉

). (51)

Also, define V↑↑(r), V↑↓(r) and V↓↓(r) by the change ofbasis formula,(

V↑↑(r) V↑↓(r)V↑↓(r) V↓↓(r)

)= Q(θ)

(VT (r) 0

0 VS(s)

)Q−1(θ).

(52)Plugging these new basis states into the Schrodingerequation gives, −~2∇2

m+ V↑↑(r)− E V↑↓(r)

V↑↓(r) −~2∇2

m+ E+ − E− + V↓↓(r)− E

!(53)

× ` φ↑↑(r)φ↓↓(r)´

= 0.

(54)

-3

-2

-1

0

1

2

4.4 4.45 4.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9

a/R

! µB/("h2/mR

2)

Fig. 6. Scattering length for two coupled square-well potentials as a function of!µB. The depth of the triplet and singlet channel potentials is VT = !2/mR2 andVS = 10!2/mR2, respectively. The hyperfine coupling is Vhf = 0.1!2/mR2. Thedotted line shows the background scattering length abg.

Q!1(!<)

!"# u<""(R)

u<##(R)

$%& = Q!1(!>)

!"# u>""(R)

u>##(R)

$%& ;

"

"rQ!1(!<)

!"# u<""(r)

u<##(r)

$%&'''''''r=R

="

"rQ!1(!>)

!"# u>""(r)

u>##(r)

$%&'''''''r=R

, (54)

where tan !< = 2Vhf/(VS ! VT ! !µB). These four equations determine thecoe"cients A, B, C, D and F up to a normalization factor, and therefore alsothe phase shift and the scattering length. Although it is possible to find ananalytical expression for the scattering length as a function of the magneticfield, the resulting expression is rather formidable and is omitted here. Theresult for the scattering length is shown in Fig. 6, for VS = 10!2/mR2, VT =!2/mR2 and Vhf = 0.1!2/mR2, as a function of !µB. The resonant behaviouris due to the bound state of the singlet potential VS(r). Indeed, solving theequation for the binding energy in Eq. (21) with V0 = !VS we find that|Em| " 4.62!2/mR2, which is approximately the position of the resonance inFig. 6. The di#erence is due to the fact that the hyperfine interaction leads toa shift in the position of the resonance with respect to Em.

The magnetic-field dependence of the scattering length near a Feshbach reso-nance is characterized experimentally by a width !B and position B0 accord-ing to

a(B) = abg

(1! !B

B ! B0

). (55)

This explicitly shows that the scattering length, and therefore the magnitude

20

FIG. 2: Theoretical scattering length vs. magnetic field, nu-merically calculated for the coupled square well model. From[6].

Now we find the s-wave scattering length. Similar totraditional s-wave scattering, for r > R we assume thesolutions to the spherical Schrodinger’s equation,(

u↑↑(r)u↓↓(r)

)=(Ceikr +Deikr

Fe−k′r

), (55)

where k′ =√m(E+ − E−)/~2 − k2. And for r < R we

assume the solutions,(v↑↑(r)v↓↓(r)

)=(A(eik1r − e−ik1r)B(eik

′1r − e−ik′

1r)

), (56)

where,

k1 =pm(E− − E1,−)/~2 + k2 (57)

k′1 =pm(E− − E1,+)/~2 + k2 (58)

E1,± =∆µB − VT − Vs

2∓ 1

2

p(Vs − VT −∆µB)2 + (2Vhf)2.

(59)

Finally, u↑↑ and v↑↑ and their derivatives must match atr = R. The same applies to u↓↓ and v↓↓. The analyticalsolution is complicated, but a numerical solution of thescattering length a for VS = 10~2/mR2, VT = ~2/mR2

and Vhf = .1~2/mR2 was calculated as a function of ∆µBand plotted in figure 2, and qualitatively looks very sim-ilar to the spherical well solution in equation (44). Bothmodels show the essential physics - that the scatteringlength is a widely-tunable function of the applied mag-netic field.

4. EXPERIMENTAL OBSERVATION OFFESHBACH RESONANCES

4.1. Setup

First predicted for ultracold gases in 1995[11], Fes-hbach resonances in a Bose-Einstein condensate were

Feshbach resonances in ultracold Fermi gases 5

Nature © Macmillan Publishers Ltd 1998

8

Experimental set-upBose–Einstein condensates in the jF ! 1; mF ! ! 1! state wereproduced as in our previous work by laser cooling, followed byevaporative cooling in a magnetic trap21. The condensates weretransferred into an optical dipole trap formed at the focus of aninfrared laser beam19. Atoms were then spin-flipped with nearly100% efficiency to the jF ! 1; mF ! "1! state with an adiabatic r.f.sweep while applying a 1 G bias field. Without large modifications ofour magnetic trapping coils, we could provide bias fields of up to!1,200 G, but only by using coils producing axial curvature21,which for high-field-seeking states generated a repulsive axialpotential. At the highest magnetic fields, this repulsion was strongerthan the axial confinement provided by the optical trap. To preventthe atoms from escaping, two ‘end-caps’ of far-off-resonant blue-detuned laser light were placed at the ends of the condensate,creating a repulsive potential, and confining the atoms axially(Fig. 1a). For this, green light at 514 nm from an argon-ion laserwas focused into two sheets about 200 "m apart. The focus of theoptical trap was placed near the minimum of the bias field in orderto minimize the effect of the destabilizing magnetic field curvature.The axial trapping potential at high fields was approximately ‘‘W’’-shaped (Fig. 1b), and had a minimum near one of the end-caps asobserved by phase-contrast imaging22 (Fig. 1c, d).

The calibration factor between the current (up to !400 A) in thecoils and the magnetic bias field was determined with an accuracy of2% by inducing r.f. transitions within the jF ! 1! ground-statehyperfine manifold at about 40 G. Additionally, an optical reso-nance was found around 1,000 G, where the Zeeman shift equalledthe probe light detuning of about 1.7 GHz and led to a sign-reversalof the phase-contrast signal. These two calibrations agreed withintheir uncertainties.

The condensate was observed in the trap directly using phase-contrast imaging22 or by using time-of-flight absorptionimaging1,2,21. In the latter case, the optical trap was suddenlyswitched off, and the magnetic bias field was shut off 1–2 ms laterto ensure that the high-field value of the scattering length wasresponsible for the acceleration of the atoms. After ballistic expan-sion of the condensate (either 12 or 20 ms), the atoms were opticallypumped into the jF ! 2! ground state and probed using resonantlight driving the cycling transition. The disk-like expansion of the

cloud and the radial parabolic density profile were clear evidence forthe presence of a Bose condensed cloud.

Locating the resonancesWhen the magnetic field is swept across a Feshbach resonance onewould expect to lose a condensate due to an enhanced rate ofinelastic collisions (caused either by the collapse in the region ofnegative scattering length or by an enhanced rate coefficient forinelastic collisions). This allowed us to implement a simple pro-cedure to locate the resonances: we first extended the field rampuntil the atoms were lost and then used successively narrower fieldintervals to localize the loss. This procedure converged much fasterthan a point-by-point search. As we could take many non-destructivephase-contrast images during the magnetic field ramp, the sharponset of trap loss at the resonance was easily monitored (Fig. 1c, d).

The most robust performance was obtained by operating theoptical dipole trap at 10 mW laser power focused to a beam waist of6 "m, resulting in tight confinement of the condensate and there-fore rather short lifetimes owing to three-body recombination19.This required that the magnetic field be ramped up in two stages: afast ramp at a rate of !100 G ms!1 to a value slightly below thatexpected for a Feshbach resonance, followed by a slow ramp at a ratebetween 0.05 and 0.3 G ms!1 to allow for detailed observation. Near907 G, we observed a dramatic loss of atoms, as shown in Figs 1c and2a. This field value was reproducible to better than 0.5 G and had acalibration uncertainty of #20 G.

To distinguish between an actual resonance and a threshold fortrap loss, we also approached the resonance from above. Fieldsabove the Feshbach resonance were reached by ramping at a fast rateof 200 G ms!1, thus minimizing the time spent near the resonanceand the accompanying losses. The number of atoms above theresonance was typically three times smaller than below. Approach-ing the resonance from above, a similarly sharp loss phenomenonwas observed about 1 G higher in field than from below (Fig. 2a),which roughly agrees with the predicted width of the resonance. Asecond resonance was observed 54 # 1 G below the first one, withthe observed onset of trap loss at least a factor of ten sharper than forthe first. As the upper resonance was only reached by passingthrough the lower one, some losses of atoms were unavoidable;for example, when the lower resonance was crossed at 2 G ms!1,

articles

152 NATURE | VOL 392 | 12 MARCH 1998

905 G 908 GMagnetic field

mF

= - 1

mF

= + 1

908 G

a

b

c

d

905 G

(Time)(0 ms) (70 ms)

200 µm

Figure 1 Observation of the Feshbach resonance at 907G using phase-contrast

imaging in an optical trap. A rapid sequence (100 Hz) of non-destructive, in situ

phase-contrast images of a trapped cloud (which appears black) is shown. As the

magnetic field was increased, the cloud suddenly disappeared for atoms in the

j mF ! "1! state (see images in c), whereas nothing happened for a cloud in the

j mF ! ! 1! state (images in d). The height of the images is 140 "m. A diagram of

the optical trap is shown in a. It consisted of one red-detuned laser beam

providing radial confinement, and two blue-detuned laser beams acting as end-

caps (shownasovals). The minimumof the magnetic field was slightly offset from

the centre of the optical trap. As a 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 in b.

FIG. 3: Illustration of experimental method of phase-contrastimaging in an optical trap. a. Shows the optical trap, madeof a red-detuned laser beam for radial confinement and twoblue-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 fieldfor the mF = ±1 substates. From [12]

first observed by the Ketterle group in 1998[12][13]. Aschematic of the experimental setup for the 1998 exper-iment is shown in figure 3. Condensates were preparedin a MOT in the |F = 1,mF = −1〉 state, moved to anoptical dipole trap (ODT), adiabatically spin-flipped byan RF field to the |F = 1,mF = +1〉 state. The con-densate was observed using both phase-contrast imagingand time-of-flight absorption imaging[12].

4.2. Finding the resonances

To realize a Feshbach resonance, a large magnetic fieldwas applied to the condensate and swept in magnitude.On one side of a Feshbach resonance, the scatteringlength is very large and negative, interactions betweenthe atoms are very strong and attractive, and the con-densate collapses. On the other side of a Feshbach res-onance, the scattering length is very large and positive,interactions between the atoms are very strong and re-pulsive, and the condensate is quickly lost. The exact lo-cations of the Feshbach resonances in sodium were foundby sweeping the magnetic field, finding the magnetic fieldvalues with condensate loss, and then sweeping in smallerand smaller intervals about those intervals.[12]. Figure3c shows condensate loss in the |F = 1,mF = +1〉 stateas the magnetic field is swept from 905 to 908 G. In figure

4 we also see a sharp drop in the atom number and meandensity at the resonance.

4.3. Measuring the scattering length

The interaction energy EI of the condensate is propor-tional to the scattering length,[12][14]

EIN

=2π~2

ma〈n〉, (60)

where N is the total number of atoms, m is the mass ofan atom, and 〈n〉 is the average density of the conden-sate. Because the kinetic energy of atoms in a trappedcondensate is insignificant compared to the interactionenergy, the kinetic energy of a freely expanding conden-sate defined by the root mean square velocity vrms equalsthe the interaction energy of the gas while it was trapped,

EKN

=12mv2

rms. (61)

From [14] we know that,

〈n〉 ∝ N(Na)−3/5. (62)

Combining, we find,

a ∝ v2rms

N. (63)

We can determine both v2rms and N from time of flight

images at a particular magnetic field. This is how weobtain data points for a scattering length vs. magneticfield curve shown in figure 4. The results are very similarto the theoretical model (44)

5. CONCLUSIONS

We have discussed the theory of Feshbach resonances inultracold gases for two simple models and have shown themajor physical result - the scattering length is a widelytunable function of an applied magnetic field. The scat-tering length dependence on the magnetic field has beenobserved experimentally and agrees with our derived re-sults. Feshbach resonances have been crucial to the fieldof ultracold Fermi gases. Because of the Pauli exclusionprinciple, fermions tend to stay away from each otherin space, but by tuning a magnetic field near a Fes-hbach resonance, we can cause them to interact morestrongly than they would otherwise. Important recentdevelopments like fermionic superfluidity and molecularBose-Einstein condensates (where two fermions make themolecule) were only realized by using a magnetic field totune the fermion-fermion interaction. Experimetnal con-trol over the scattering length is a very powerful tool.

Feshbach resonances in ultracold Fermi gases 6

a ! v5rms

N. (2)

Normalized to unity far away from the resonance, v5rms/N

is equal to a/a0. The proportionality factor in eqn.(2)involves the mean trapping frequency which was not ac-curately measured. However, we can obtain absolute val-ues by multiplying the normalized scattering length withthe theoretical value of a0. The o!–resonance scatteringlength a0 = 2.75 nm of sodium at zero field increases tothe triplet scattering length a0 = aT = 3.3 nm [8] at thehigh fields of the resonances. Using this value, the meandensity is obtained from the root–mean–square velocity,

"n# =m2v2

rms

4!h2a. (3)

The peak density n0 is given by n0 = (7/4)"n# in a parabolicpotential.

104

105

106

Nu

mb

er

of

Ato

ms N

915910905900895

0.13 G/ms

0.31 G/ms

-0.06 G/ms

a)mF = +1

1

10

Sca

tte

rin

g L

en

gth

a/a

0

915910905900895

b)

103

104

105

1200119011801170

Magnetic Field [G]

0.28 G/ms

d)mF = -1

0.1

1

1200119011801170

Magnetic Field [G]

e)

( )

10

8

6

4

2

0

1200119011801170

Magnetic Field [G]

f)( )

10

8

6

4

2

0

Me

an

De

nsity

<n

> [

10

14cm

-3]

915910905900895

Magnetic Field [G]

c)

Figure 1: Number of atoms N , normalized scatteringlength a/a0 and mean density "n# versus the magneticfield near the 907 G and 1195 G Feshbach resonances.The di!erent symbols for the data near the 907 G reso-nance correspond to di!erent ramp speeds of the magneticfield. All data were extracted from time–of–flight images.The errors due to background noise of the images, ther-mal atoms, and loading fluctuations of the optical trapare indicated by single error bars in each curve.

The experimental results for the 907 G and 1195 G res-onances are shown in Fig. 1. The number of atoms, thenormalized scattering length and the density are plottedversus the magnetic field. The resonances can be identi-fied by enhanced losses. The 907 G resonance could beapproached from higher field values by crossing it firstwith very high ramp speed. This was not possible for the

1195 G resonance due to strong losses, and also not forthe 853 G resonance due to its proximity to the 907 Gresonance and the technical di"culty of suddenly revers-ing the magnetic field ramp. For the 907 G resonance thedispersive change in a can clearly be identified. The solidlines correspond to the predicted shapes with width pa-rameters # = 1 G in Fig. (b), and # = $ 4 G in Fig. (e).The negative width parameter for the 1195 G resonancereflects the decreasing scattering length when the reso-nance is approached from the low field side. The uncer-tainties of the positions are mainly due to uncertaintiesof the magnetic field calibration.

The time dependent loss of atoms from the condensatecan be parameterized as

N

N= $

!i

Ki "ni!1#, (4)

where Ki denotes an i–body loss coe"cient, and "n# thespatially averaged density. In general, the density n de-pends on the number of atoms N in the trap, and the losscurve is non–exponential. One–body losses, e.g. due tobackground gas collisions or spontaneous light scattering,are negligible under our experimental conditions. An in-crease of the dipolar relaxation rate (two–body collisions)near Feshbach resonances has been predicted [1]. How-ever, 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 ex-pected to contribute. Thus the experimental study of theloss processes focuses on the three–body losses around theF = 1, mF = +1 resonances, while both two– and three–body losses must be considered for the F = 1, mF = $1resonance.

Figs. 1 (c) and (f) show a decreasing or nearly constantdensity when the resonances were approached. Thus theenhanced trap losses can only be explained with increas-ing coe"cients for the inelastic processes. The quantityN/N"n2# is plotted versus the magnetic field in Fig. 2 forboth the 907 G and the 1195 G resonances. Assuming thatmainly three–body collisions cause the trap losses, theseplots correspond to the coe"cient K3. For the 907 G(1195 G) resonance, the o!–resonant value of N/N"n2# isabout 20 (60) times larger than the value for K3 measuredat low fields [7]. Close to the resonances, the loss coe"-cient strongly increases both when tuning the scatteringlength larger or smaller than the o!–resonant value. Sincethe density is nearly constant near the 1195 G resonance,the data can also be interpreted by a two–body coe"-cient K2 = N/N"n# with an o!–resonant value of about30% 10!15 cm3/s, increasing by a factor of more than 50near the resonance. The contribution of the two processescannot be distinguished by our data. However, dipo-lar losses are much better understood than three–bodylosses. The o!–resonant value of K2 is expected to beabout 10!15 cm3/s [9] in the magnetic field range around

2

FIG. 4: Number of atoms, normalized scattering length andmean density as a function of magnetic field around the 907G (left) and 1195 G (right) Feshbach resonances in a Bose-Einstein condensate of Na. From [13]

Acknowledgments

Thank you to Martin Zwierlein for pointing me to help-ful references, and to everyone else in the Center for Ul-tracold Atoms.

[1] H. Feshbach, Annals of Physics (1958).[2] W. Ketterle and M. Zwierlein, in Proceedings of the In-

ternational School of Physics ”Enrico Fermi”, CourseCLXIV (2006).

[3] R. L. Liboff, Introductory Quantum Mechanics (AddisonWesley, San Francisco, 2003), 4th ed.

[4] D. Griffiths, Introduction to Quantum Mechanics (Pren-tice Hall, Upper Saddle River, 1987), 2nd ed.

[5] D. Bohm, Quantum Theory (Prentice Hall, N.J., 1951).[6] R. A. Duine and H. T. C. Stoof (2008).[7] J. J. Sakurai, Modern Quantum Mechanics (Addison-

Wesley, Reading, 1994).[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 Let-ters (1999).

[14] G. Baym and C. J. Pethick, Physical Review Letters 76(1996).