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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