郭西川 國立彰化師範大學物理系 年原子分子及光學物理夏季學校 (Aug. 20~ Aug. 24, 新竹市國立清華大學物理系 )

Transcription

1 超冷原子氣體中的粒子散射及 Feshbach 共振之理論簡介 郭西川 國立彰化師範大學物理系 7 年原子分子及光學物理夏季學校 (Aug. ~ Aug. 4, 新竹市國立清華大學物理系

2 References:. B.R. Holstein, Topics in Advanced Quantum Mechanics, (Addision-Wesley 99.. R.H. Landau, Quantum Mechanics II A Second Course in Quantum Theory, (Wiley 曾謹言, 量子力學,( 凡異出版社 B.H. Bransden and C.J. Joachain, Physics of Atoms and Molecules, (Longman Dalibard, in La condensazione di Bose-Einstein nei gas atomnici, (Societa Italiana Di Fisica Bologna-Italy 6. C.J Pethic and H. Smith, Bose-Einstein Condensation in Dilute Gases, (Cambridge University Press 7. L. Pitaevsii and S. Stringari, Bose-Einstein Condensation, (Oxford University Press

3 . Basic scattering theory Consider a collision process between two particles and with the same mass m, interacting through the potential U ( r r. The Hamiltonian of the system is H p p = + + U( rr (. m m Introducing the center-of-mass variables R,P and the relative variables p, r, which are defined as follows r+ r r = rr R = p p p = P= p+ p (. the Hamiltonian Eq.(. can now be written as H P p = + + U 4m m r (. As it is expected, the center of mass moves as a free particle with a mass m. The interesting collisional dynamics arises from the relative motion corresponding to the scattering of a particle with reduced mass mred = m / by the potential U ( r. The Schrödinger equation in the relative coordinate system is given by where ( + ψ = u ψ r r r (.4 ( r mu u ( r =. In scattering theory, it is generally convenient to rewrite Eq.(.4 as an integral equation, using the outgoing-wave Green s function G ip ( rr ip rr d p e e = = p iδ 4π r r ( r r ( + ( π (.5 where we have added a positive infinitesimal imaginary part δ in the denominator to ensure that the scattered wave has only outgoing terms. The function G (+ satisfies the

4 differential equation ( + ( + ( = δ ( G r r r r (.6 and the Green s theorem then yields the following equation for the scattering wave function ψ ( r representing an incident plane wave with wave vector plus an outgoing scattered wave: ψ ( + ir r = e G rr u( rψ r d r' (.7 Let b be the range of action of U(r. For r >> b, the asymptotic form of the wave function at a large distance from the scattering center taes the form ψ i( r rr / r ir e r e u( r ψ ( r d r' 4 π rrr / r ir ir e irr / r r ψ ( r e e u d r' r ir = e + f (, ir e r (.8 where m = red ir ( ( ψ ( f, e U r r d r' (.9 π is the scattering amplitude and = n, = n with n = r / r. The physical meaning of this collision state Eq.(.8 is rather clear. It is the superposition of an incident plane wave of momentum and of a scattered wave. At a given point r, the scattering amplitude depends on the energy of the particle through, and on the incident direction n and the observation direction n (see Fig.. From Eq.(.9, we see that the scattering amplitude relates the value of wave function far from the scattering region to the values of the same wave function inside the scattering region. The scattering amplitude is a very important quantity in the collision physics, since it determines the differential and total cross-sections of the scattering process: dσ = f (,, σ ( = f (, dω' (. dω 4

5 Fig. Scattering of an incident wave pacet propagating along the direction n by a potential U(r with a finite range b. It is worth noting the following properties of the scattering amplitude:.. The time reversal symmetry ensures that f(, = f(, (.. In the low energy limit f, const. In other words, the scattering process is isotropic for a sufficiently small energy. The energy scale below which this simplification occurs is directly related to the range b of the potential. For / b or, equivalently, E /mredb, the scattering amplitude is independent of the direction n and n.,. In principle, the scattering amplitude as a solution of Eq.(.9 can be evaluated by iteration. The solution of the lowest order is obtained by assuming that the scattered wave is almost unchanged during the scattering, ψ r e ir, such that i.e., mred i( r f (, = e U ( r d r (. π This approximation is nown as the Born approximation. The scattering amplitude in Eq.(. is just the -D Fourier transform of the potential U with respect to the momentum q=. Quite generally, the Born approximation tends to get better at higher energies. The integral of Eq.(.9 can be furthermore rewritten in the convolution relation 5

6 ( π ( π ( π ir ( ( r ψ ( r 6 4 π f, = e u d r = dre dqe u d pe ψ ( q ψ ( q ( q ψ ( p ir iqr ipr = dq d puq p δ qp = dqu ( ( (. where ψ ipr ψ, ipr = dre u = dre u p r p r (.4 are the corresponding Fourier component of ψ ( r, u ( r in the momentum space Also in momentum space, the scattered wave can be expressed as ψ ipr ( p = dre ψ ( r ipr ir ( + = dre e G ( rr u( r ψ ( dr r iqr ( dq e u r' ψ r = ( π δ ( p dr π δ ( pq π q iδ ( ipr ' = ( π δ ( p dre ur ψ r p i ( dq = ( π δ ( p ψ u q p q p iδ π = ( π δ ( ( p f ( p, δ p iδ (.5 where dq f( p, = 4 π f ( p, = u( q ψ ( pq (.6 is the modified scattering amplitude. ( π Multiplying the term u ( q p to the last line in Eq.(.5 and subsequently taing the integral over the momentum p, we get 6

7 d p ( π u ( q p ψ ( p d p ( = ( π δ ( p u ( qp ( π d p f ( p, u( q p = u ( q π p iδ ( f ( p, u q p p iδ (.7 Maing exchange of variables, q p, yields dq ( π u ( p = u ( p q ψ ( q dq ( π ( f ( q, u p q q iδ (.8 Let p q= w, it follows that + d ( w ( π u ψ = p w w f( p, (.9 Thus we arrive at the integral equation of the modified scattering amplitude ( π ( dq u p q f( q, f( p, = u( p (. q iδ In the formal scattering theory, the scattering amplitude f ( p, and the matrix element of the T matrix is related by and thus Eq.(. is rewritten as or ( π m f( p, = T( p, (. dq q T( p, = U( p + U ( + iδ T(, p q m m q (. 7

8 q T( p, = U( p + U( pq + iδ T( q, (. V m m q which is nown as the Lippmann-Schwinger equation. In the operator notations, Eq.(. can be expanded as T = U + UG T = U + UG U + UG T = U+ UGU+ UGUGU+ UGUGUGU+ (.4 where G = E H + iδ Hamiltonian. is the free propagator, and H is the unperturbed The formal solutions of the Lippmann-Schwinger equation Eq.(.4 can be obtained in the following way: ( T = U + UG T UG T = U ( ( T = UG U = UG UG G = UG UG G using UG, UG = ( [ ] = U G UG G = U G U = U GU (.5 or, equivalently, T = U = U + GU + GUGU + GU ( ( = U + UG U + UG UG U + = + UG + UG UG + U = U UG (.6 8

9 . Partial wave expansion In considering the scattering in a spherically symmetric potential, one often examine how states with definite angular momenta are affected by the scatters. Such considerations lead to the method of partial waves. For spherically symmetric potentials, U(r U(r, the scattering amplitude depends only on the angel between n and n, that is, f (, = f (, θ ( = cos θ nn. Since the angular momentum is conserved in a spherically symmetric potential, it is convenient to expand the incident and scattered wave functions on a basis set of eigenfunctions of ˆL and L ˆz. To be specific, we shall assume that the particle is incident in the z-direction. Accordingly, the wave function has axial symmetry with respect to z-axis, so that it can be expanded in terms of Legendre polynomials, AP( cosθ R ( r ψ r = l l l (. l= The radial wave function R l r satisfies the equation d dr ( + l l + + U( r Rl ( r = rdr r d mred (. For r, the radial function is given in terms of the phase shifts δ l according to the equation π Rl ( r sin r l+ δl r iδ ilπ / l e e ir iδ l = e e + ( e ir l+ ir (. In order to derive the form of the partial wave scattering amplitude, we begin iz with a plane wave e for which we write iz ir cosθ l l l= ( ( cosθ e = e = l+ i j r P (.4 l where 9

10 π jl r Jl+ / r r = (.5 are the spherical Bessel functions. Specifically, the first few lowest order forms of the spherical Bessel functions are given below sin x sin x cos x j( x =, j( x =, x x x j ( x = sin cos x x x x x etc. (.6 For large values of r, we have so that lπ jl ( r sin r r (.7 iz l ( π / ( π / ( ir l ir l e l+ i e e P cos l ir l= ( ir irlπ = l+ e e P cos l ir l= ( θ ( θ for r (.8 We observe that this representation is in terms of an incoming outgoing ir ir ( r e and ( r e spherical wave with the phase of the latter being shifted by an angle lπ with respect to the former. In the presence of an interaction potential, the scattering wave function must be slightly different. The relative normalization between incident and outgoing spherical waves should remain the same (since there should be identical probability flow into and out of the scattering region by unitarity and the form of the incoming wave function should be unchanged since it has not yet interacted with the scattering potential. The only modification allowed then is in the phase of the outgoing spherical wave, and the scattering wave function must assume the form

11 ψ ( δ ( π ir+ l irl + cos l e e P l ir r ( r l= ( θ ( ( ir+ δl ir ir irlπ = + + cos l e e e e P l ir l= iz ( + δ ( ir l ir = e + + cos l e e P l ir ir l= iz e = e + + r l= ( l iδ l e Pl i ( cos θ ( θ ( θ (.9 where δ l is the scattering phase shift in the l-th partial wave. Comparing with the asymptotic form iz e ψ ( r e + f (, ' r (. + δ = + cosθ l ir i l ir + l Pl e e e ir l= ir We identify iδ e f = l+ Pl i l (, θ ( ( cosθ l= (. and the total scattering cross-section in terms of the phase shifts is given by ( = d = f (, dω = d( cos f (, σ σ θ π θ θ 4π = ( l + sin δl l= (. It is interesting to chec the imaginary part of the scattering amplitude in the forward direction θ =, and we find l= l= iδ l e Im f (, = Im ( l+ Pl( = ( l+ sin δl i (. From Eq.(., we conclude that

12 Im f, = σ (.4 4π which is nown as the optical theorem in the scattering theory. Optical theorem is a consequence of conservation of probability flux. So far, we have only considered the scattering between distinguishable particles. For identical particles, we must tae into account the proper symmetrization of the two-particle wave functions. In other words, the wave function must be symmetric under exchange of the coordinates of the two particle if they are bosons, and antisymmetric if they are fermions. Exchange of the two particle coordinates corresponds to changing the sign of the relative coordinate, that is r r or, equivalently, r r, θ π θ and φ π + φ, where φ is the azimuthal angle. Accordingly, the symmetrized wave function is given by ir iz iz e (, θ (, π θ e ± e + f ± f r (.5 Here the + sign applies to bosons and the sign to fermions. The differential cross-section is dσ = f ± dω (, θ f (, π θ (.6 The physical content of Eq.(.6 is that the amplitude for an identical particle to be scattered into some direction is the sum or difference of the amplitude for of the particles to scattered through an angle θ and the amplitude for the other particle to be scattered through an angle π θ ( see Fig.. Fig. Two scattering processes leading to the same final state of indistinguishable particles.

13 Furthermore, substituting Eq.(.6 into Eq.(. and using the parity ( l of the spherical harmonic functions, we find σ σ 8π ( = ( l+ even l 8π ( = ( l+ odd l sin δ for bosons l sin δ for fermions l (.7

14 . Scattering in low-energy regime For the partial wave l =, the potential entering the -D radial equation Eq.(. is simply the inter-atomic potential (see Fig.a. For other partial waves, this potential is superimposed with the centrifugal barrier + / red l l m r (see Fig.b. In the latter case, the relative particle with an energy E much lower than the height of the resulting barrier will not feel the potential U(r and it will simply be reflected by the centrifugal barrier. We therefore expect, qualitatively, that at sufficiently low energy, the scattering due to U(r does not occur for all partial waves except l =. Fig. Potential entering the -D Schrödinger equation. (a s-wave scattering (b scattering with l >. The argument given above can be further verified by examining the behavior of the phase shift in the low energy limit. For a finite-range potential the phase shifts l+ n vary as for small. For a potential varying as r at large distance, this result is true if l < ( n /, but δ n l for higher partial waves. To be specific, we shall consider the case of a finite-range potential, in which l+ δ l modulo π when (. As a result, the cross-section for the l-th partial wave ( l is 8π 4l σl ( = ( l+ sin δl when (. while the cross-section for the s-wave ( l = is 4

15 lim σ l = 8 π a for bosons = (. for fermions Here the s-wave scattering length a is defined as e a= f =lim E iδ ( sin δ ( (.4 Thus again, we arrive at the same conclusion in Chapter that the low-energy scattering processes are isotropic. An important consequence of Eq.(. is that polarized fermions (fermions belonging to the same spin state do not see each other at low temperatures. This is due to the fact that for fermions, the scattering occurs only through partial waves of l =,,5, whose cross-section tends to zero at low temperatures. This property maes the evaporative cooling of a polarized fermionic gas quite difficult. 5

16 4. The scattering length for some simple potentials In the previous section, we have shown that for low enough energy, the collisions are essentially occurring in the s-wave regime. Thus the solution of the scattering problem at ultra-low energies amounts to determine the scattering length a for a given potential. Such a determination is equivalent to solve the -D radial eigenvalue equation (. subject to the boundary condition Eq.(.. To solve the Eq.(., it is convenient to mae the change of variable, χ = l such that Eq.(. for the s-wave ( l = can be rewritten as rr l, d dr + U r = mred χ ( (4. and the boundary condition Eq.(. becomes iδ e δ χ lim ir i ir = e e e i lim ( + ( ( ir ia ir ra (4. In what follows, we shall demonstrate the determination of scattering length for some simple potentials. Example : The square potential barrier. Consider the case of the square spherical barrier as shown in Fig.4a, where U > r b = U r (4. otherwise The solution of Eq.(4. is then χ ( r C ( r a r > b = Csinh( r r b (4.4 where red = m U / and C and C are normalization coefficients. The continuity conditions for the wave function at r = b leads to tanh a= b ( b (4.5 6

17 Fig.4 (a Square potential barrier. (b Scattering length as a function of b. The numerical solution is plotted in Fig.4b. In this case, the scattering length is always positive. For b, the hard-sphere scattering problem is resumed; the scattering length in this case is just equal to the radius b of the hard sphere. Example : The square potential well. Consider the case of the square spherical well as shown in Fig.5a, where U < r b = U r (4.7 otherwise The solution of Eq.(4.7 is then χ ( r C ( r a r > b = Csin( r r b (4.8 where again = mredu / and C and C are normalization coefficients. The continuity conditions for the wave function at r = b leads to tan a= b ( b (4.9 7

18 Fig.5 (a Square potential well. (b Scattering length as a function of b The solution of Eq.(4.8 is plotted in Fig.5b. Note that in this case, the scattering length can be positive or negative and it diverges for whenever a new bound state appears in the well. In the following, some remars for the results in Fig.5b are listed: If U is too small to have a bound state in the potential well, i.e., b < π /, the scattering length a is negative. The scattering length a becomes divergent when = ( + b n / π, where n is an integer. Each of these discrete values of U corresponds to the appearance of a new bound state in the potential well. These resonances leading to a divergence of the scattering length are called zero-energy resonance, which is a general result of the Levinson theorem. When U is slight lower than the threshold for the appearance of a new bound state, the scattering length a is large and negative; if U is slightly larger than this threshold, a is large and positive. This is also a general result. We have demonstrated the procedures for the determination of the scattering length by considering the cases of square potential barrier and square potential well. For other more complicated model potentials, these procedures apply equally well in principle. The crucial point is that the scattering length is determined by the intercept of the asymptotic wave function χ r with the coordinate axis. The schematic behavior of the solutions of Eq.(4. is shown in Fig.6 for positive and negative 8

19 values of the scattering length, respectively. Fig.6 Schematic behaviour of the solution of Eq.(4. for the scattering problems (a positive scattering length (b negative scattering length. In the following example, we show that in the low-energy regime in which only the s-wave scattering is considered, the phase shift δ in the presence of a finite-ranged potential can be expanded in terms of the energy. This relation is nown as the effective range expansions. 9

20 Example : Effective range expansions Consider an arbitrary potential U(r which vanishes beyond some radius b, i.e. Fr r b U( r = otherwise (4. Accordingly, the wave function can be expressed in the form l= ( r χ ψ = l r AP l l ( cosθ r (4. The s-wave radial function χ satisfies the differential equation as Eq.(4. d dr m + F( r χ ( r = red (4. Denote χ lim χ r r (4. which satisfies the equation d dr m F( r χ ( r = red (4.4 Furthermore, we require that χ = χ = at r =. Since U(r vanishes in the region r > b, the solutions of Eqs.(4. and (4.4 in this region are given by χ r = C( ra χ r = Asin( r+ δ (4.5 We now attempt to find out the relation between the phase shift δ and the energy E. Multiplying, respectively, χ to Eq.(4. and χ the latter from the former, we get to Eq.(4.4, and then subtracting d χ d χ χ χ = χ χ dr dr (4.6 or equivalently, d d χ dχ χ χ = χχ dr dr dr (4.7

21 Since χ = χ = at r =, the integration upon (4.7 gives d χ r dχ r = χ r dr χ r dr χ χ r χχdr (4.8 The upper limit of the integration in Eq.(4.8 is arbitrary. However, if we choose r = R> b, then the wave functions on the left-hand-side of Eq.(4.8 are replaced by those in Eq.(4.5, i.e., R cot ( R+ δ = χ ( r χ ( r dr R a R R χ χ (4.9 Let cot R+ δ = Q, it follows then cot ( R δ ( R δ cotδ tan ( R Q + ( R δ ( R ( R tan tanδ + = = tan + tan + tanδ = = tan cot (4. As a consequence, we find Q tan R Q R + + cotδ = Qtan R QR Q R (4. Hence, to the second order, Eq.(4.9 can be approximated by Q= D R a (4. where = χ ( χ R D r dr R (4. and hence a QR = + RD R a (4.4

22 Substituting Eq.(4.4 into Eq.(4. and eep all terms up to, we obtain cot δ + r + O a (4.5 where the effective range r > is defined by R a + a R a r = a a D (4.6 When the energy is sufficiently low, we may neglect all terms in the right-hand-side of Eq.(4.5 except a and obtain cot or tanδ δ = a = a (4.7 This approximation, nown as the shape-independent approximation says that in the low-energy limit, the potential acts as if it were a hard-sphere potential of diameter a. As we have illustrated in Example, the scattering length diverges whenever the potential well is about to support a new bound state. It is interesting to see that when the two colliding atoms form a bound state at an energy just below the threshold of dissociation. In this case, the wave vector becomes imaginary, namely, mred E m E iκ = i = i (4.8 where E < is the energy of the bound state and κ = mred E / = m E /. Accordingly, the radial wave function of the bound state has the asymptotic form κr = ( χ r Ae r b (4.9 If the condition κb is satisfied, the wave function Eq.(4.9 solves the corresponding Schrödinger equation in a wide interval of values of r b, where the potential U(r can be safely ignored. With this consideration, it is feasible to assume that Eq.(4.9 can be expanded in the form of Eq.(4., i.e, κr e - κr ra (4. which immediately leads to the identification, is given by a = κ, and hence the binding energy

23 E b = ma (4. In general, if the value of κ is small, the energy dependence of the scattering amplitude is important even at small. From the expression for the s-wave scattering amplitude we see that f iδ e sinδ sinδ = = = = e ( cotδ iδ i cotδ + i (4. In the shape-independent approximation, the scattering amplitude is f = a + i (4. For a bound state, a ±, the scattering amplitude Eq.(4. gives rise to a divergent cross-section in the low-energy limit 8 σ = lim π (4.4 The leading order of correction beyond the shape-independent approximation is obtained by substituting Eq.(4.5 into Eq.(4. and this gives f = r a + i (4.5 Since E, in some situations, the scattering amplitude Eq.(4.5 can be parameterized to have the form f E ε +Γ i / (4.6 which exhibits the strong resonant character at E = ε, if the width of the resoannce Γ<< ε. It is important to point out that the constant κ can be very sensitive to the actual value of the inter-atomic potential, as well as to the presence of the external fields. A particularly important situation is given by the occurrence of the so-called Feshbach resonance which will be discussed in the next section.

24 5. Feshbach Resonance So far, in our studies of atom-atom collisions, we have neglected the effects due to the internal degrees of freedom of the atom. However, the internal states of the atoms indeed play an important role in the ultracold-atom experiments based on the laser trapping and cooling techniques as the electronic ground state of an alali atom is degenerate which is consisted of several different hyperfine states. In a scattering process, the internal states of the particles in the initial or final states are described by a set of quantum numbers, such as those for the spin, the atomic species, and their state of excitation. We shall refer to a possible choice of these quantum numbers as a channel. Due to the existence of several hyperfine states for a single atom, the scattering of cold alali atoms is a multi-channel problem. One important consequence of the multi-channel configuration is that the inter-atomic interactions give rise to transition between these states and such inelastic processes are a major mechanism for loss of trapped atoms. Besides inelastic scattering that leads to trap loss, another important effect in the multi-channel problem is that the coupling between channels could give rise to Feshbach resonance, in which a low-energy bound state in one channel strongly modifies scattering in another channel. Feshbach resonance is a powerful tool for experimental investigating, which enables the experimentalists to tune both the magnitude and the sign of the effective interaction. In this section, we shall theoretically loo into the basic principle of the Feshbach resonance. To be specific in the following context, we consider only the hyperfine spins of the atom as the internal degrees of freedom. These single-particle hyperfine states are denoted as α, β,. Thus two atoms initially in the states αβ may be scattered by atom-atom interactions to the state α β, and as a consequence, scattering becomes a multi-channel problem. According to our previous results, the relative part of the Hamiltonian of two atoms interacting through the potential U ( r can be generalized as where H = H + U r (5. = p H + H ( + H ( spin spin mred (5. 4

25 Here the first term in H is the inetic energy for the relative motion, and H spin is the Zeeman term for each single particle. The eigenstates of H are denoted by αβ, αβ, where αβ is the relative momentum. If the eigenvalues of the spin Hamiltonian are given by H α = ε α (5. spin α the energies of H are E αβ αβ ( αβ = + εα + ε β mred (5.4 The scattering amplitude is now introduced by generalizing Eqs.(.8 and (.9 to include the internal states. The corresponding asymptotic form the wave function is ψ iαβ r iαβ r e = e + fα β ; αβ αβ, α β αβ αβ (5.5 r where αβ is the relative momentum in the incoming state, which is referred to as the entrance channel. The scattered wave has components in the different internal states α β which are referred to as the exit channel. Since energy must be conserved during the scattering process, the energy of the entrance channel must be equal to that of the exit channel, and this gives αβ αβ = + ε + ε ε ε m m red red α β α' β' (5.6 If α' β', the channel is said to be closed, since the energy is insufficient for the pair of atoms to be at rest far from each other, and the corresponding term should not be included in the sum in Eq.(5.5. Another way of stating a closed channel α β is that the energy of the relative motion given by Eq.(5.4 is less than the threshold energy E = + (5.7 threshold α, β εα ' εβ ' For many purposes, it is convenient to wor in terms of the T matrix rather than the scattering amplitude. In this consideration, the scattering amplitude is defined by 5

26 f m αβ (5.8 π (, = red α β T(, αβ ; αβ αβ αβ αβ αβ which satisfies the Lippmann-Schwinger equation Eq.(.4 Conceptually, the multi-channel scattering problems might be understood with the following picture. In fact, the term channel here is used as an analogy to the flow of a fluid through a channel. The incident channel is the elastic one and is always open. Each excited state of either the projectile or target is considered a separate channel, and the channel is closed unless there is enough energy to reach that excited state (open the gate. When a channel opens, additional flux from the incident channel flows into the open channel. When a virtual process occurs in quantum mechanics, it is analogous to fluid leaing through a flood-gate but with no net current flowing into the inelastic channel. Liewise, as the energy increases, we envision the gates on some energy-forbidden channels as gradually lifting open and permitting current to flow into them. Fig.7 Flux flowing from the entrance elastic channel into the exit of elastic channel and into an inelastic channel. The gate in front of the inelastic channel indicates that it may be closed for low energies. Feshbach resonances appear when the total energy in an open channel matches the energy of a bound state in a closed channel. We now describe the general formalism for the derivation of the Feshbach resonances. At the beginning, we note that the Hilbert space describing the external and internal degrees of freedom can be divided into two subspaces, P, which contains the open channels, and Q, contains the closed channels. We can write the state 6

27 vector Ψ as the sum of its projections onto the two subspaces, Ψ = Ψ P + ΨQ (5.9 where Ψ = ˆ P P Ψ and Ψ = ˆ Q Q Ψ. Here Pˆ and ˆQ are projection operators for the two subspaces, which satisfy the conditions: Pˆ+ Qˆ = I, PQ ˆ ˆ = ˆ ˆ ˆ P = P, Q = Qˆ (5. By multiplying the Schrödinger equation H Ψ = E Ψ, we obtain the two coupled equations for the projections of the state vector onto the two subspaces, namely PH ˆ Ψ = PE ˆ Ψ = E Ψ PH ˆ Ψ + PH ˆ Ψ = E Ψ (using Ψ = Ψ + Ψ PHP ˆ ˆ Ψ + PHQ ˆ ˆ Ψ = E Ψ ( E-H P ˆˆ ˆ ˆ P PHPP ˆ ˆ ˆ Ψ + PHQQ ˆ Ψ = E Ψ (using Pˆ = Pˆ, Q = Q PHP ˆ ˆ Ψ + PHQ ˆ ˆ Ψ = E Ψ H Ψ + H Ψ = E Ψ Ψ = H Ψ P P Q P P Q P Q P PP P PQ Q P PP P PQ Q (5. and similarly ( E-HQQ Ψ Q = H QP ΨP (5. where H = PHP ˆ ˆ, H = QHQ ˆ ˆ, PP QQ H = PHQ ˆ ˆ, H = QHP ˆ ˆ PQ QP (5. The operator and H PQ and HQP solution of Eq.(5. is H PP is the Hamiltonian in the P subspace, HQQ that in the Q space, represent the coupling between the two subspaces. The formal Ψ = H Ψ E-H + iδ Q QP P QQ (5.4 where we have added a positive infinitesimal imaginary part δ in the denominator to 7

28 ensure that the scattered wave has only outgoing terms. Inserting Eq.(5.4 into Eq.(5., we get E-H Ψ = H Ψ = H H Ψ E-H + iδ PP P PQ Q PQ QP P QQ ' ( E-H H Ψ = PP PP P (5.5 Here ' PP = PQ QP E-HQQ + iδ H H H (5.6 is the term that describes Feshbach resonances. It represents an effective interaction in the P subspaces due to transitions from that subspace to the Q subspace and bac again to the P subspace. Equation (5.6 has a form similar to the energy shift in second order perturbation theory, and corresponds to a non-local potential in the open channels. Due to the energy dependence of the interaction, it is also retarded in time. It is convenient to divide the diagonal parts H + H of the Hamiltonian into PP QQ a term H independent of the separation of the two atoms, and an interaction contribution. Here H is the sum of the inetic energy and the Zeeman terms. We write H = H + U (5.7 PP where U is the interaction term for the P subspace. Equation (5.5 may be rewritten as E-H U (5.8 Ψ P = where the total effective atom-atom interaction in the subspace of open channels is given by with U = U+ U (5.9 U = H PP (5. To solve the problem more generally, we shall calculate the T matrix corresponding 8

29 to the interaction given in Eq.(5.5. Recall the formal solution of T given in Eq.(.5 T = U = U UG G U (5. Inserting G = E H + iδ into the first equality in the last expression, we obtain As a consequence, we find and ( ( ( ( T = UG U = G G UG U = G U G U = G G U U = ( E H + iδ U E H + iδ U ( UG = = ( E H + iδ UG E H + iδ U = + GU E H + iδ U ( E H iδ (5. (5. (5.4 Next, using the operator identity for the two operators A and B, = + B + B B + A B A A A A A A = + B + B + B B + A A A A A A A = + B A A B (5.5 and putting A= E H + iδ U, B= U (5.6 we find = + U E H + iδ U E H + iδ U E H + iδ U (5.7 Inserting Eq.(5.7 into Eq.(5., yields 9

30 T = ( E H + iδ + U U E H + iδ U E H + iδ U = + + ( E H iδ ( U U E H + iδ U ( E H iδ U ( U U E H + iδ U E H + iδ U = T+ ( E H + iδ U + U E H + iδ U E H + iδ U = T+ U G U + G U U ( (5.8 where T = E H + i U ( δ E H + iδ U (5.9 which satisfies the equation T = U+ UGT. Physically, T is the T matrix in the P subspace if transitions to the Q subspace are neglected. We note that the term + G U U can be further reduced as + G U U = + G U G G U ( ( ( ( GU ( GU GU ( GU ( = + G G U G G U = + GU GG GU= + GU GU = GU GU + GU GU = + = (5. As a result, Eq.(5.8 becomes ( [ ] T = T + U G U G U (5. To gain more insight into the result Eq.(5., let us consider the matrix elements between plane-wave states with relative momenta and ( [ ] T = T + U G U G U (5.

31 To evaluate the matrix element Eq.(5., we first note that GU eigenstate of H + U (5. This inference can be easily verified. Let ( GU ( GU GUGU G ( UG U ψ = = = = + G U UG (5.4 By multiplying G from left in both sides of Eq.(5.4, we are led to E H + iδ ψ = G ψ = G + G G U UG E H i U UG ( δ = + + = + U = ( + UG + UGUG + U UG = U ψ ( = U + G U + G UG U + (5.5 Accordingly, we obtain H + U ψ = E ψ (5.6 which verifies the inference of Eq.(5. that the state ( GU generates an eigenstate of the total Hamiltonian. indeed At large distance, the state ( GU consists of a plane wave and an outgoing spherical wave because of the infinitesimal imaginary part iδ in G. This state is denoted by ; U, +, where the + sign indicates that the state has outgoing spherical waves. On the other hand, since the operators U H Hermitian and thus for real E, we find are both UG = GU = U EH iδ (5.7 Note that there is a negative infinitesimal imaginary part in G. Consequently, we

32 have EH iδ UG = U ; U, (5.8 Here the sign indicates an incoming spherical wave at large distance. Hence the scattering amplitude Eq.(5. can be written as ' T = ' T + '; U, U ; U, + (5.9 This is the general expression for the scattering amplitude in the P subspace. We now consider the contribution of the leading order of U. This is equivalent to let U U in Eq.(5.9, which then gives ( [ ] + T T U G U G U (5.4 As a result, the matrix elements between plane-wave states are given by T = T + ; U, U ; U, + (5.4 Note that the interaction in the P subspace gives a contribution T to the T matrix, but the contribution due to U is to be evaluated using scattering states that tae into account the potential U rather than the plane waves. Let us now neglect coupling between the open channels. The scattering between those states having zero relative momenta, i.e.,,, is then described by the scattering length, which is related to the T matrix by: 4π 4π T a T a P, (, m m (5.4 where ap is the scattering length when coupling between open and closed channels is neglected. Next, it suffices to assume that in the low-energy limit we can neglect the difference between the scattering states with incoming and outgoing spherical waves, that is, lim ; U, + Ψ, lim ; U, ' Ψ (5.4

33 Thus the matrix element of U is given by ; U, U ; U, + Ψ H Ψ PP = Ψ H ψ ψ H Ψ = n PQ n n QP E-HQQ + iδ n ψ n H QP threshold Ψ E -E n (5.44 where the states ψ n are the energy eigenstates of the Hamiltonian H QQ which form a complete in the Q subspace. As a consequence, Eq.(5.9 becomes 4 4 ψ n HQP a = ap + red red n threshold π π Ψ m m E -E n (5.45 If the threshold energy E threshold is close to the resonance energy E res of one particular bound state, the contributions from the all other states will vary slowly with energy and the last expression can be written as 4 4 ψ res H π π Ψ a = a + QP m m E -E red red threshold res (5.46 where a is the effective scattering length due to other non-resonant states. Since the energies of the internal atomic states depend on the external parameters, this implies that the scattering length and hence the inter-atomic interactions can be tuned by varying the external parameters such as the strengths of magnetic and electric fields. To be specific, let us consider the case of an external magnetic field. Assume that the energy denominator in Eq.(5.46 vanishes for a particular value of magnetic field B = B. Expanding the energy denominator about this value of the magnetic field we find threshold res ( µ res µ threshold ( ( µ res µ α µ β ( B B E -E B B = (5.47 where ε ε α β E µ α =, µ β =, µ res = B B B res (5.48

34 are the magnetic moments of the two atoms in the open channel αβ, and that of the molecular bound state, respectively. Substituting Eq.(5.47 into Eq.(5.46, we obtain the expression of the scattering length in the Feshbach resonance, B a = a + B B (5.49 where the width parameter B is given by B = m ψ red res H QP Ψ 4π a µ res µ α µ β (5.5 Fig.8 The Feshbach resonance can be realized by tuning the relative position between the two potential curves with an external magnetic field. In (a and (b the bound state is, respectively, just below and just above the threshold energy (the dotted line. 4

35 As we have stated earlier, the Feshbach resonances tae place when the energy associated with the scattering event between two particles is close to the energy of a bound state relative to a closed channel. This situation is illustrated in Fig.8. The lower curve describes the scattering potential between two atoms in a given spin state while the upper curve represents the scattering potential in a different spin state. To be specific, we may refer to the lower curve as the scattering channel and the upper one as the closed channel. If the magnetic moments of the atoms considered in the two channels are different, the relative position between the two curves can be tuned continuously by changing the external magnetic field. Consequently, one can go from a situation where a bound state is just below (Fig.8a to a situation where the same bound state is just above (Fig.8b the threshold energy. The transition occurs at the critical value B = B of the magnetic field, which is characterized by a divergent scattering length. Because of the dependence of ( B B, large changes in the scattering length can be produced by small changes in the field. It is particularly significant that the sign of the interaction can be changed by a small change in the field. Experimental realizations of the Feshbach resonances have been reported recently for several inds of alali atoms, in which the experimental data highly agree with the theoretical predictions (See Fig.9. In calculating the matrix element Eq.(5.4, we have only considered the contribution of lowest order in U. The high order contribution in U may included by adding an effective interaction to the denominator in Eq.(5.6, i.e., where ' HQQ U = H = H H E-H + iδ H PQ PP PQ QP QQ H E-H H + iδ QQ QQ QP (5.5 H = H E-H + iδ H QQ QP PQ PP (5.5 Hence, the exact solution of Eq.(5. may be expressed as T T U G H H G U ( ( + PQ E-H + δ QP QQ HQQ i (5.5 By comparing this result with Eq.(5.4, we see that the only effect of high-order terms in U is to introduce an extra contribution HQQ in the Hamiltonian acting on 5

36 Q subspace. This effective interaction in the Q subspace is resulted due to the coupling to the open channels. Physically, it has two effects. One is to shift the energies of the bound states, and the second is to give them a non-zero width Γ res if decay into open channel is possible. The width leads to the Breit-Wigner form for the scattering amplitude just as that in Eq.(4.6, i.e., f E E + iγ threshold res res / (5.54 In the regime far from resonance, Ethreshold E res Γres, the scattering amplitude E E behaviour predicted by the simpler calculations. In exhibits the threshold res the strong resonance regime, Ethreshold E res Γres, the divergence is cut off and the scattering amplitude increases smoothly. However, the width of resonant state close to the threshold energy in the open channel is generally very small because of the small density of final states, and Feshbach resonances for cold atoms are consequently very sharp. 6

37 Fig.9 Observation of Feshbach resonance (a in Na (Inouye et al., 998; Nature, 9, 5 (b in 85 Rb (Roberts et al., 998 7