Feshbach Resonances in Ultracold Gases Sara L. Capbell MIT Departent of Physics Dated: May 5, 9) First described by Heran Feshbach in a 958 paper, Feshbach resonances describe resonant scattering between two particles when their incoing energies are very close to those of a two-particle bound state. Feshbach resonances were first applied to tune the interactions between ultracold Feri gases in the late 99 s. Here, we review two siple odels of a Feshbach resonance: a spherical well odel and a coupled square well odel. coupled square well odel of a Feshbach resonance and explain how it can be ipleented on cold alali atos experientally. Then, we review soe recent ey developents in atoic physics that would not have been realized without Feshbach resonances.. INTRODUCTION AND HISTORY In 958, Herann Feshbach developed a new unified theory of nuclear reactions[] 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.[] In the late 99 s, atoic physicists began to tae advantage of the ind of resonant scattering proposed by Feshbach to realize resonant scattering between ultracold alali atos. Previously, laser and evaporative cooling of atos had allowed physicists to reach teperatures as low as µk. However, it is uch harder to cool olecules, because their energy levels and structure are uch ore coplicated than those of a single ato. Feshbach resonances ade it possible to create ultracold olecules by assebling the fro ultracold atos[], thus aing phenoena such as olecular Bose-Einstein condensation and ferionic superfluidity a reality.. ESSENTIAL RESULTS FROM SCATTERING THEORY In a general scattering proble, we have an incident plane wave with wave nuber and a radial scattered wave with anisotropy ter fθ),[3] φ inc = e iz, φ sc = fθ)eir. ) r dσ = fθ) dω. ) So, we see that fθ) relates the aplitude of particles passing through a particular target area to the nuber of particles passing through a solid angle at a particular angle θ. Now we consider a particle with ass, energy / in a central potential V r). We can find fθ) by considering the wave function at large r, where it is siply the su of the incident and scattered waves[4], φ large r = φ inc + φ sc = e iz + fθ)eir r 3) 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 su of the spherical haronics Y l and solutions to Schrödinger s radial equation Rr). We assue syetry about the angle φ, so we ignore the quantu nuber. The spherical haronics then are just proportional to the Legendre polynoicals P l.[5] φ large r = Y θ)rr) 4) ) = C l P l cos θ) r sin r lπ ) + δ l. l= 5) We expand the e iz ters and equate coefficients to obtain[4], fθ) = sin δ l P l cos θ). 6) l= The different angular oentu quantu nubers l correspond to different partial waves. Recall Schrödinger s radial equation, d ur) dr + V eff r)ur) = Eur), 7) V eff r) = V r) + ll + ) r. 8) For the low particle energies in ultracold atos[6], if l > the angular oentu barrier ter in Schrödinger s radial wave equation prevents the particle fro reaching sall r. Using the assuption that V r) is only significant for sall r, we find that only the l = or the s-waves in the expansion interact with the potential. P =, so φ large r = C r sinr δ ), 9) which iplies that, f = eiδ sin δ = sin δ = cos δ i sin δ cot δ i. )
Feshbach resonances in ultracold Feri gases By definition, the s-wave scattering length a is[7], tan δ ) a = li. ) For strong attractive interactions, a is large and negative. For strong repulsive interactions, a is large and positive. Also, Taylor expanding cotδ )) in gives[6], cotδ )) = a + r eff +, ) where r eff is the effective range of the potential. Cobining equations ) and ) gives the following expression for f) which will be useful later, f) = a + r eff i. 3) Finally, we quote the Lippann-Schwinger[8][] equation, which expresses f in ters of the wavevectors and of the two particles and the interatoic potential v, f, ) = v ) d 3 q v q)fq, ) + 4π π) q. 4) + ıη For s-waves, fq, ) f), so we can tae f outside the integral and also let v = v. Then, f) 4π + 4π d 3 q vq) v v π) 3 q + iη. 5) 3. APPLICATION TO ULTRACOLD FERMI GASES Experientally, we can tae advantage of the internal structure of atos to tune their interaction. The effective interaction potential of the two ferionic atos depends the angular oentu coupling of their two valence electrons, s, s,, where s is the total spin quantu nuber and is the spin projection quantu nuber. If the valance electrons are in the singlet state with the antisyetric spin wave function,, = ), 6) then they have a syetric spatial wave function and can exist close together. If the valance electrons are in the triplet state and have one of the following syetric spin wave functions,, = 7), = + ) 8), = 9) then they have an antisyetric spatial wave function and cannot exist close together. Therefore, the singlet potential V s tends to be uch deeper than the triplet potential V T. For siplicity, in both odels we assue V s just deep enough to contain one bound state and that V T contains no bound states, and < V hf V T, V S, 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], V hf = α I S. ) If we apply an external agnetic field B, then there will also be Zeean shifting µb between the triplet and singlet states, where µ is the difference in agnetic oents betwee the two states. In the field of ultracold atos, the gas teperature T is very sall and the atos are confined in space using a agneto-optical trap MOT) so that the atos feel a considerable agnetic field B. Therefore, µ B T, and if we assue sufficiently low gas density, we can apply Maxwell-Botzann statistics to the syste and see that alost all of the atos scatter via the state and very few atos scatter via the state. Using the language of scattering theory, we call an open channel and a closed channel.[] 3.. Spherical Well Model In this odel we only consider one bound state in the closed channel and a continuu of plane waves of relative oentu between the two atos in the open channel.[] Without coupling, and are energy eigenstates of the free Hailtonian[], H = ɛ, ɛ = H = δ ) Let φ be the coupled wavefunction, which can be written as a superposition of the olecular state and the scattering states, φ = α + c. ) Figure shows how the bound states couple to the continuu. We will solve for the coupling explicitly by plugging φ into Schrödinger s equation with the Hailtonian H = H + V, where V includes the coupling V = g / Ω, where Ω is the phase space volue of the syste. Because we are just dealing with s-waves, we ae the approxiation, g = where E R = /R. { g, E < E R, E > E R, 3)
Feshbach resonances in ultracold Feri gases 3 As in figure, we see that relative to the bare state, the dressed ground state resonance position is shifted to δ.[] 3... Scattering aplitude and scattering length To find the scattering aplitude and scattering length, we plug φ into the tie-independent Schrödinger equation H φ = E φ for E > and find, E ɛ)c = g Ω α = q g g q c q 33) Ω E δ + iη Ω FIG. : Energy as a function of detuning for states in the spherical well odel for ferion collisions. The uncoupled olecular state and scattering states are shown as dashed lines and the coupled states are shown as solid lines. Fro []. 3... Bound state energy and resonance shift Then to find the olecular wavefunction after coupling, we plug φ into the tie-independent Schrödinger equation H φ = E φ for E < and find[], E ɛ)c = g Ω α 4) E δ)α = g c = gα Ω Ω E ɛ 5) E δ = g Ω E ɛ. 6) Integrating over the continuu then gives[], E + δ = g ER Ω E = = g ρe R ) Ω ρɛ)dɛ ɛ + E E E r arctan E R E ) 7) 8) { δ E ) E, E E R δ ER 9) 3 E, E E R δ = g = 4 E E r, 3) Ω ɛ π [ g E = ) ] 3/ π. 3) E + δ δ + p E q δ + δ ) δ δ ), E E R delta δ + δer, 4 3 E ER 3) = q V eff, q)c q. 34) Substituting v = V eff Ω into equation?? we find, f ) 4π d 3 q E δ) + 4π π) 3 q + iη. 35) g Now we substitute in, d 3 q 4π π) 3 q = 4π δ g 36) and also using equation 3) that when δ δ, the energy E is approxiately the coupling energy g, we find, f ) δ δ) i 37) E E Now our equation loos just lie equation 3), so we find a and r eff, E a = δ δ, r eff =. 38) E Finally, we plug in δ δ = µb B ) to find that, E a =. 39) B B This is the scattering length we find without considering bacground collisions in the open channel. We account for bacground scattering by adding a phase shift δ bg to our wavefunction so that δ total = δ +δ bg. To see how this extra phase shift affects the scattering length, we add δ bg to our definition of scattering length in equation, a total = li tanδ ) + δ bg ) = li tan δ ) + tan δ bg tan δ tan δ bg ) 4) 4) li tan δ tan δ bg + li 4) = a + a bg. 43)
Feshbach resonances in ultracold Feri gases 4 So, using this approxiation, a total = a bg + a = a bg B ), 44) B B B = E µa bg. 45) a/r - - 3.. Coupled Square Well Model Putting all of these effects together into one Hailtonian, the Schrödinger equation for a triplet state T and a singlet state S with potentials V T r) and V s r) is[4], + VT r) E V hf V hf 46) + µb + VSr) E «ψtr) =. 47) ψ Sr) As before, we use square well odels for siplicity we use the sae radius R) for the two interaction potentials, { VT,S, r < R V T,S r) = 48), r > R The inetic energy operator is diagonal because our basis ets are two different internal states of the atos, so we need to diagonalize the Hailtonian, ) Vhf H int =, 49) µb V hf and the energy eigenvalues are, E ± = µb ± µb) + V hf ) 5) Let, Qθ) be the rotation atrix that diagonalizes the Hailtonian H and ) ) T = Q. 5) S Also, define V r), V r) and V r) by the change of basis forula, ) ) V r) V r) VT r) = Qθ) Q V r) V r) V S s) θ). 5) Plugging these new basis states into the Schrödinger equation gives,! + V r) E V r) V r) + E+ E + V r) E 53)! ` φ r)φ r) =. 54) -3 4.4 4.45 4.5 4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9! µb/ " h /R ) FIG. : Theoretical scattering length vs. agnetic field, nuerically calculated for the coupled square well odel. Fro [6]. Now we find the s-wave scattering length. Siilar to traditional s-wave scattering, for r > R we assue the solutions to the spherical Schrödinger s equation, u r) u r) ) Ce ir + De ir = F e r ), 55) where = E + E )/. And for r < R we assue the solutions, ) ) v r) Ae i r e ir ) = v r) Be i r e i r, 56) ) = p E E, )/ + 57) = p E E,+)/ + 58) E,± = µb VT Vs p Vs V T µb) + V hf ). 59) Finally, u and v and their derivatives ust atch at r = R. The sae applies to u and v. The analytical solution is coplicated, but a nuerical solution of the scattering length a for V S = /R, V T = /R and V hf =. /R was calculated as a function of µb and plotted in figure, and qualitatively loos very siilar to the spherical well solution in equation 44). Both odels show the essential physics - that the scattering length is a widely-tunable function of the applied agnetic field. 4. EXPERIMENTAL OBSERVATION OF FESHBACH RESONANCES 4.. Setup First predicted for ultracold gases in 995[], Feshbach resonances in a Bose-Einstein condensate were
Feshbach resonances in ultracold Feri gases 5 a b 4 we also see a sharp drop in the ato nuber and ean density at the resonance. 4.3. Measuring the scattering length c µ F = + The interaction energy E I of the condensate is proportional to the scattering length,[][4] E I N = π a n, 6) d 95 G 98 G F = - where N is the total nuber of atos, is the ass of an ato, and n is the average density of the condensate. Because the inetic energy of atos in a trapped condensate is insignificant copared to the interaction energy, the inetic energy of a freely expanding condensate defined by the root ean square velocity v rs equals the the interaction energy of the gas while it was trapped, 95 G 98 G Magnetic field s) Tie) 7 s) FIG. 3: Illustration of experiental ethod of phase-contrast iaging in an optical trap. a. Shows the optical trap, ade of a red-detuned laser bea for radial confineent and two blue-detuned beas for axial confineent endcaps ). b. Shows the confining potential in the axial direction. c and d. show iages of the condensate as a function of agnetic field for the F = ± substates. Fro [] first observed by the Ketterle group in 998[][3]. A scheatic of the experiental setup for the 998 experient is shown in figure 3. Condensates were prepared in a MOT in the F =, F = state, oved to an optical dipole trap ODT), adiabatically spin-flipped by an RF field to the F =, F = + state. The condensate was observed using both phase-contrast iaging and tie-of-flight absorption iaging[]. 4.. Finding the resonances To realize a Feshbach resonance, a large agnetic field was applied to the condensate and swept in agnitude. On one side of a Feshbach resonance, the scattering length is very large and negative, interactions between the atos 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 atos are very strong and repulsive, and the condensate is quicly lost. The exact locations of the Feshbach resonances in sodiu were found by sweeping the agnetic field, finding the agnetic field values with condensate loss, and then sweeping in saller and saller intervals about those intervals.[]. Figure 3c shows condensate loss in the F =, F = + state as the agnetic field is swept fro 95 to 98 G. In figure Fro [4] we now that, Cobining, we find, E K N = v rs. 6) n NNa) 3/5. 6) a v rs N. 63) We can deterine both vrs and N fro tie of flight iages at a particular agnetic field. This is how we obtain data points for a scattering length vs. agnetic field curve shown in figure 4. The results are very siilar to the theoretical odel 44) 5. CONCLUSIONS We have discussed the theory of Feshbach resonances in ultracold gases for two siple odels and have shown the ajor physical result - the scattering length is a widely tunable function of an applied agnetic field. The scattering length dependence on the agnetic field has been observed experientally and agrees with our derived results. Feshbach resonances have been crucial to the field of ultracold Feri gases. Because of the Pauli exclusion principle, ferions tend to stay away fro each other in space, but by tuning a agnetic field near a Feshbach resonance, we can cause the to interact ore strongly than they would otherwise. Iportant recent developents lie ferionic superfluidity and olecular Bose-Einstein condensates where two ferions ae the olecule) were only realized by using a agnetic field to tune the ferion-ferion interaction. Experietnal control over the scattering length is a very powerful tool.
Feshbach resonances in ultracold Feri gases 6 Scattering Length a/a 5 4 Nuber of Atos N 6 895 F = +.3 G/s.3 G/s -.6 G/s 9 95 9 a) 95 b) 5 4 3. F = - d).8 G/s 7 8 9 Magnetic Field [G] e) ) Acnowledgents Mean Density <n> [ 4 c -3 ] 8 6 4 895 895 9 9 95 95 9 9 95 c) 95 8 6 4 7 7 8 9 ) Magnetic Field [G] 8 9 f) Magnetic Field [G] Magnetic Field [G] FIG. 4: Nuber of atos, noralized scattering length and ean density as a function of agnetic field around the 97 G left) and 95 G right) Feshbach resonances in a Bose- Einstein condensate of Na. Fro [3] Than you to Martin Zwierlein for pointing e to helpful references, and to everyone else in the Center for Ultracold Atos. [] H. Feshbach, Annals of Physics 958). [] W. Ketterle and M. Zwierlein, in Proceedings of the International School of Physics Enrico Feri, Course CLXIV 6). [3] R. L. Liboff, Introductory Quantu Mechanics Addison Wesley, San Francisco, 3), 4th ed. [4] D. Griffiths, Introduction to Quantu Mechanics Prentice Hall, Upper Saddle River, 987), nd ed. [5] D. Boh, Quantu Theory Prentice Hall, N.J., 95). [6] R. A. Duine and H. T. C. Stoof 8). [7] J. J. Saurai, Modern Quantu Mechanics Addison- Wesley, Reading, 994). [8] B. A. Lippann and J. Schwinger, Physical Review 79 95). [9] W. D. Phillips, in Proceedings of the International School of Physics Enrico Feri 99). [] M. W. Zwierlein, Ph.D. thesis, MIT 6). [] A. J. Moerdij, B. J. Verhaar, and A. Axelsson, Physical Review A 995). [] S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Staper-Kurn, and W. Ketterle, Nature 39 998). [3] J. Stenger, S. Inouye, M. Andrews, H. Miesner, D. Staper-Kurn, and W. Ketterle, Physical Review Letters 999). [4] G. Bay and C. J. Pethic, Physical Review Letters 76 996).