Scattering and bound states

Transcription

1 Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states [86, 87]. In the first part we consider single-channel scattering and focus on the exaple of the square well. In the second part we consider the situation of two coupled channels, which can give rise to a Feshbach resonance..1 Single-channel scattering: an exaple We consider the situation of two atos of ass that interact via the potential V (r) that vanishes for large distances between the atos. The otion of the atos separates into the trivial center-of-ass otion and the relative otion, described by the wave function ψ(r) where r x 1 x, and x 1 and x are the coordinates of the two atos, respectively. This wave function is deterined by the tie-independent Schrödinger equation [ h ] + V (r) ψ(r) = Eψ(r), (.1) with E the energy of the atos in the center-of-ass syste. Solutions of the Schrödinger equation with negative energy correspond to bound states of the potential, i.e., to olecular states. To describe ato-ato scattering we have to loo for solutions with positive energy E = ɛ, with ɛ h / the inetic energy of a single ato with oentu h. Since any realistic interatoic interaction potential vanishes rapidly as the distance between the atos becoes large, we now that the solution for r of Eq. (.1) is given by a superposition of incoing and outgoing plane waves. More precisely, the scattering wave function is given by an 15

2 16 CHAPTER. SCATTERING AND BOUND STATES θ Figure.1: Scheatic representation of two-ato scattering in the center-of-ass reference frae. The atos are initially in a plane-wave state with relative oentu h, and scatter into the spherical wave with relative oentu h. Due to energy conservation we have that =. The angle between and is denoted by θ. The region where the interaction taes place is indicated by the blac circle. incoing plane wave and an outgoing spherical wave and reads ψ(r) e i r + f (, ) ei r r, (.) where the function f (, ) is nown as the scattering aplitude. The interatoic interaction potential depends only on the distance between the atos and hence the scattering aplitude depends only on the angle θ between and ˆr, and the agnitude. Because of energy conservation we have that =. The situation is shown scheatically in Fig..1. Following the partial-wave ethod we expand the scattering aplitude in Legendre polynoials P l (x) according to f (, ) = l=0 f l ()P l (cos θ). (.3) The wave function is expanded in a siilar anner as ψ(r, θ) = R l (, r)p l (cos θ), (.4) l=0

3 .1. SINGLE-CHANNEL SCATTERING: AN EXAMPLE 17 with R l (, r) = u l (, r)/r the radial wave function and u l (, r) deterined by the radial Schrödinger equation [ d l(l + 1) dr r ] V (r) h + u l (, r) = 0. (.5) By expanding also the incident plane wave in partial waves according to e i r = (l + 1)i l l=0 r ( sin r lπ ) P l (cos θ), (.6) we can show that to obey the boundary condition in Eq. (.), the partial-wave aplitudes f l () have to be of the for f l () = l + 1 i ( e iδ l () 1 ), (.7) where δ l () is the so-called phase shift of the l-th partial wave. For the ultracold alali atos, we are allowed to consider only s-wave (l = 0) scattering, since the colliding atos have too low energies to penetrate the centrifugal barrier in the effective hailtonian in Eq. (.5). Moreover, as we see later on, the lowenergy effective interactions between the atos are fully deterined by the s-wave scattering length, defined by δ 0 () a = li. (.8) 0 Fro Eq. (.7) we find that the s-wave scattering aplitude is given by f 0 () = 1 cot δ 0 () i. (.9) As explained above, we tae only the s-wave contribution into account, which gives for the scattering aplitude at zero-oentu f (0, 0) a. (.10) To illustrate the physical eaning of the s-wave scattering length, we now calculate it explicitly for the siple case that the interaction potential is a square well. We thus tae the interaction potential of the for V (r) = { V0 if r < R; 0 if r > R, (.11)

4 18 CHAPTER. SCATTERING AND BOUND STATES with R > 0. With this potential, the general solution of Eq. (.5) for l = 0 is given by u < (r) = Ae i<r + Be i<r, for r < R; u > (r) = Ce ir + De ir, for r > R, (.1) with < = V 0 / h. Since the wave function ψ(r) has to obey the Schrödinger equation at the origin we have to deand that the function u < (r) vanishes at this point. This leads to the boundary condition B = A. By coparing the explicit for of the wave function u > (r) with the s-wave coponent of the general scattering wave function for r, we find that e iδ 0() = C D. (.13) Hence, we deterine the phase shift by deanding that the wave functions for r < R and r > R join soothly. This leads to the equations A ( e i< R e i< R ) = e iδ 0() e i R + e i R, A ( < e i< R + < e i< R ) = e iδ 0() e i R e i R, (.14) where we have chosen the noralization such that D = 1. Multiplication of the above equations with e iδ 0() and dividing the result leads to tan( < R) = < tan(δ 0 () + R), (.15) fro which it follows that [ ] δ 0 () = R + tan 1 < tan(< R). (.16) Note that for a repulsive hard-core potential we have that V 0 and therefore, with the use of the definition in Eq. (.8), that the scattering length a = R. This iediately gives a physical picture for a positive s-wave scattering length: at low energy and oenta the details of the potential are uniportant and we are allowed to odel the potential with an effective hard-core potential of radius a. For a fully repulsive potential the scattering length is always positive. For a potential with attractive parts the scattering length can be both negative and positive, corresponding to attractive and repulsive effective interactions, respectively. This is seen by explicitly calculating the scattering length for our exaple in the case that V 0 < 0. As its definition in Eq. (.8) shows, the scattering length is deterined by the linear dependence of the phase shift on the agnitude of the

5 .1. SINGLE-CHANNEL SCATTERING: AN EXAMPLE r eff /R and a/r Figure.: Scattering length (solid line) and effective range (dashed line) for an attractive square well in units of the range of the potential, as a function of the diensionless paraeter γ = R V 0 / h. γ relative oentu h of the scattering atos for sall oentu. Generally, the phase shift can be expanded according to [86 88] cot(δ 0 ()) = 1 a + 1 r eff +, (.17) fro which the scattering length is deterined by ( a = R 1 tan γ ) γ, (.18) with γ = R V 0 / h a diensionless constant. The paraeter r eff is the so-called effective range and is, in our exaple of the square-well potential, given by [ r eff = R tan γ γ (3 + γ ] ) 3γ (γ tan γ ). (.19) In Fig.. the scattering length is shown as a function of γ by the solid line. Clearly, the scattering length can be both negative and positive, and becoes equal to zero at values of γ such that γ = tan γ. In the sae figure, the effective range is shown by the dashed line. Note that the effective range diverges if the scattering length becoes equal to zero. This is because the expansion in Eq. (.17) is ill-defined for a = 0. At values of γ = (n + 1/)π with n a positive integer the scattering length

6 0 CHAPTER. SCATTERING AND BOUND STATES diverges and changes sign. This behaviour is called a potential or shape resonance and in fact occurs each tie the potential is just deep enough to support a new bound state. Therefore, for large and positive scattering length the square well has a bound state with an energy just below the continuu threshold. It turns out that there is an iportant relationship between the energy of this bound state and the scattering length. To find this relation we have to deterine the bound-state energy by solving the Schödinger equation for negative energy V 0 < E < 0. This leads to solutions u < (r) = A ( e i<r e ) i< r, for r < R; u > (r) = Be κr, for r > R, (.0) with < = (E V 0 )/ h and κ = E / h. Deanding again that these solutions join soothly at r = R, we find the equation for the bound-state energy ( ) h E = h (E V 0 ) cot h (E V 0 ). (.1) We can show that for values of γ such that (n 1/)π < γ < (n + 1/)π this equation has n solutions for V 0 < E < 0 [87]. For sall binding energy E V 0 we have fro the equation for the boundstate energy that h E γ cot γ /R 1/a, (.) where we ade use of the fact that γ has to be close to the resonant values (n+1/)π in this case. This leads to the desired relation between the energy of the olecular state and the scattering length given by E = h a. (.3) This result does not depend on the specific details of the potential and it turns out to be quite general. Any potential with a large positive scattering length has a bound state just below the continuu threshold with energy given by Eq. (.3). Moreover, the relation will turn out to hold also in the ultichannel case of a Feshbach resonance as we will see in Section.3. Before discussing this situation, we first turn to soe concepts of scattering theory which are of iportance for the reainder of this thesis.. Single-channel scattering: foral treatent Let us give a ore foral treatent of the scattering theory described above. In a basis-independent forulation the Schrödinger equation we have solved reads, [Ĥ0 + ˆV ] ψ = E ψ, (.4)

7 .. SINGLE-CHANNEL SCATTERING: FORMAL TREATMENT 1 with Ĥ 0 = ˆp / the inetic energy operator for the atos. To describe scattering, we have to loo for solutions which asyptotically represent an incoing plane wave, and an outgoing spherical wave. In the absence of the potential ˆV there is no scattering, and hence we deand that the solution of Eq. (.4) reduces to a plane wave in the liit of vanishing potential. The foral solution that obeys this condition is given by ψ (+) = + 1 E + Ĥ 0 ˆV ψ (+), (.5) where represents the incoing plane wave and we recall that E = ɛ is the inetic energy of the atos. This energy is ade slightly coplex by the usual liiting procedure E + li η 0 E + iη. Moreover, we have for the scattering aplitude that f (, ) = 4π h ˆV ψ (+). (.6) To deterine the scattering aplitude directly, we introduce the two-body T(ransition) atrix by eans of ˆV ψ (+) = ˆT B (E + ). (.7) Multiplying the foral solution in Eq. (.5) by ˆV we have that ˆT B (E + 1 ) = ˆV + ˆV ˆT B (E + ). (.8) E + Ĥ 0 Since this equation holds for an arbitrary plane wave and because these plane waves for a coplete set of states we have the following operator equation for the two-body T-atrix ˆT B 1 (z) = ˆV + ˆV ˆT B (z). (.9) z Ĥ 0 This equation is called the Lippann-Schwinger equation and fro its solution we are able to deterine the scattering properties of the potential ˆV. To see this we first note that fro the definition of the T-atrix in Eq. (.7), together with Eq. (.6), it follows iediately that f (, ) = 4π h ˆT B (ɛ + ). (.30) Therefore, we indeed see that the two-body T-atrix copletely deterines the scattering aplitude. The Lippann-Schwinger equation for the two-body T-atrix can be solved in perturbation theory in the potential. This results in the so-called Born series given by ˆT B (z) = ˆV + ˆV Ĝ 0 (z) ˆV + ˆV Ĝ 0 (z) ˆV Ĝ 0 (z) ˆV +, (.31)

8 CHAPTER. SCATTERING AND BOUND STATES where Ĝ 0 (z) = 1 z Ĥ 0, (.3) is the noninteracting propagator of the atos. By using, instead of the true interatoic interaction potential, a pseudopotential of the for V (x x ) = 4πa h δ(x x ), (.33) the first ter in the Born series iediately yields the correct result for the scattering aplitude at low energies and oenta, given in Eq. (.10). Such a pseudopotential should therefore not be used to calculate higher-order ters in the Born series, but should be used only in first-order perturbation theory. The poles of the T-atrix in the coplex-energy plane correspond to bound states of the potential. To see this we note that the foral solution of the Lippann- Schwinger equation is given by ˆT B 1 (z) = ˆV + ˆV z Ĥ ˆV. (.34) After insertion of the coplete set of eigenstates ψ α of Ĥ = Ĥ 0 + ˆV we have ˆT B (z) = ˆV + α ˆV ψ α ψ α z ɛ α ˆV, (.35) where the suation over α is discrete for the bound-state energies ɛ α < 0, and represents an integration for positive energies that correspond to scattering solutions of the Schrödinger equation, so explicitly we have that ˆT B (z) = ˆV + κ ˆV ψ κ ψ κ ˆV + z ɛ κ (+) d ψ ψ ˆV (+) ˆV. (.36) (π) 3 z ɛ Fro this equation we clearly see that the two-body T-atrix has poles in the coplexenergy plane, corresponding to the bound states of the potential. In addition, the T-atrix contains a branch cut on the positive real axis due to the continuu of scattering states. As an exaple, we note that for s-wave scattering the T-atrix T B (ɛ + ) ˆT B (ɛ + ) is independent of the angle between and. Fro the relation between the T-atrix and the scattering aplitude, and the expression for the latter

9 .3. EXAMPLE OF A FESHBACH RESONANCE 3 in ters of the phase shift, we have for low positive energies T B (E + ) = 4π h 4πa h E h ( cot δ 1 ( )) E h i E h 1, (.37) 1 + ia E ar effe h h where we ade use of the expansion in Eq. (.17). Fro this result we deduce by analytic continuation that T B (z) 4πa h 1 1 a. (.38) z ar effz h h Clearly, for large and positive scattering length the T-atrix has a pole at negative energy E = h /a, in coplete agreeent with our previous discussions. Suarizing, we have found that the scattering length of an attractive potential well can have any value and depends strongly on the energy of the wealiest bound state in the potential. In principle therefore, if we have experiental access to the energy difference of this bound state and the continuu threshold we are able to experientally alter the scattering length and thereby the effective interactions of the atos. In the single-channel case this is basically ipossible to achieve. In a ultichannel syste, however, the energy difference is experientally accessible, which aes the low-energy effective interactions between the atos tunable. In the next section we discuss this situation..3 Exaple of a Feshbach resonance We consider now the situation of ato-ato scattering where the atos have two internal states [89]. These states correspond, roughly speaing, to the eigenstates of the spin operator S of the valence electron of the alali atos. The effective interaction potential between the atos depends on the state of the valence electrons of the colliding atos. If these for a singlet the electrons are in principle allowed to be on top of each other. For a triplet this is forbidden. Hence, the singlet potential is generally uch deeper than the triplet potential. Of course, in reality the ato also has a nucleus with spin I which interacts with the spin of the electron via the hyperfine interaction V hf = a hf I S, (.39) h

10 4 CHAPTER. SCATTERING AND BOUND STATES with a hf the hyperfine constant. The hyperfine interaction couples the singlet and triplet states. Moreover, in the presence of a agnetic field the different internal states of the atos have a different Zeean shift. In an experient with agneticallytrapped gases, the energy difference between these states is therefore experientally accessible. Putting these results together, we can write down the Schödinger equation that odels the above physics ( h + V T(r) E V hf V hf h + µb + V S(r) E ) ( ) ψt (r) = 0. (.40) ψ S (r) Here, V T (r) and V S (r) are the interaction potentials of atos with internal state T and S, respectively, and µb is their difference in Zeean energy due to the interaction with the agnetic field B, with µ the difference in agnetic oent. In agreeent with the above rears, T is referred to as the triplet channel, whereas S is referred to as the singlet channel. The potentials V T (r) and V S (r) are the triplet and singlet interaction potentials, respectively. As a specific exaple, we use for both interaction potentials again square well potentials, { VT,S if r < R V T,S (r) = 0 if r > R, (.41) where V T,S > 0. For convenience we have taen the range the sae for both potentials. Furtherore, we assue that the potentials are such that V T < V S and that V S is just deep enough such that it contains exactly one bound state. Finally, we assue that 0 < V hf V T, V S, µb. The potentials are shown in Fig..3. To discuss the scattering properties of the atos, we have to diagonalize the hailtonian for r > R, in order to deterine the incoing channels, which are superpositions of the triplet and singlet states T and S. Since the inetic energy operator is diagonal in the internal space of the atos, we have to find the eigenvalues of the hailtonian ( ) 0 H > Vhf =. (.4) µb These are given by V hf ɛ > ± = µb ± 1 ( µb) + (V hf ). (.43) The hailtonian H > is diagonalized by the atrix ( cos θ sin θ Q(θ) = sin θ cos θ ), (.44)

11 .3. EXAMPLE OF A FESHBACH RESONANCE 5 according to ( ɛ Q(θ > )H > Q 1 (θ > > ) = 0 0 ɛ + > ), (.45) which deterines tan θ > = V hf / µb. We define now the hyperfine states and according to ( ) ( ) T = Q(θ > ), (.46) S which asyptotically represent the scattering channels. In this basis the Schrödinger equation for all r reads ( ) h + V (r) E V (r) V (r) h + ɛ> + ɛ> + V (r) E ( ) ψ (r) = 0, (.47) ψ (r) where the energy E is easured with respect to ɛ > and we have defined the potentials according to ( ) ( ) V (r) V (r) = Q(θ > VT (r) 0 ) Q 1 (θ > ). (.48) V (r) V (r) 0 V S (r) Since all these potentials vanish for r > R we can study scattering of atos in the states and. Because the hyperfine interaction V hf is sall we have that ɛ + > µb and ɛ> 0. Moreover, for the experients with agnetically-trapped gases we always have that µb B T where B is Boltzann s constant and is T the teperature. This eans that in a realistic atoic gas, in which the states and are available, there are in equilibriu alost no atos that scatter via the latter state. Because of this, the effects of the interactions of the atos will be deterined by the scattering aplitude in the state. If two atos scatter in this channel with energy E B T µb they cannot coe out in the other channel because of energy conservation. Therefore, the index refers to an open channel, whereas is associated with a closed channel. Note that, since we describe a collision of two atos in the center-of-ass frae, the open and closed channel are two-ato states. The situation is further clarified in Fig..3. To calculate the s-wave scattering length in the open channel we have to solve the Schrödinger equation. In the region r > R the solution is of the fro ( ) ( ) u > (r) Ce u > (r) = ir + De ir, (.49) Fe κr

12 6 CHAPTER. SCATTERING AND BOUND STATES V (r) + ε + > Bound state µβ V (r) + µβ S V (r) T V (r) + ε > R Figure.3: Feshbach resonance in a two-channel syste with square-well interaction potentials. The triplet potential V T (r) is indicated by the thic dashed line. The singlet potential that contains the bound state responsible for the Feshbach resonance is indicated by the thin dashed line. Due to the Zeean interaction with the agnetic field, the energy difference between the singlet and triplet is equal to µb. The interactions in the open and closed hyperfine channels are indicated by V (r) and V, respectively. where κ = (ɛ + > ɛ )/ h > and, because we have used the sae notation as in Eq. (.1), the s-wave phase shift is again deterined by Eq. (.13). In the region r < R the solutions are of the for ( ( ) u < (r) u < (r) = A e i< r e i< r) B (e i< r e i< r), (.50) where and ɛ < ± = µb V T V S < = (ɛ > ɛ < )/ h + ; < = (ɛ > ɛ < +)/ h +, (.51) 1 (VS V T µb) + (V hf ). (.5) are the eigenvalues of the atrix ( ) H < VT V = hf. (.53) V hf µb V S In order to deterine the phase shift we have to join the solution for r < R and r > R soothly. This is done ost easily by transforing to the singlet-triplet

13 .3. EXAMPLE OF A FESHBACH RESONANCE a/r µb/( h /R ) Figure.4: Scattering length for two coupled square-well potentials as a function of µb. The depth of the triplet and singlet channel potentials is V T = h / R and V S = 10 h / R, respectively. The hyperfine coupling is V hf = 0.1 h / R. The dotted line shows the bacground scattering length a bg. basis { T, S } since this basis is independent of r. Deanding the solution to be continuously differentiable leads to the equations ( u Q 1 (θ < < ) (R) u < ( (R) ) u r Q 1 (θ < < ) (r) u < (r) ) r=r ( u = Q 1 (θ > > ) (R) u > (R) = r Q 1 (θ > ) ) ( u > (r) u > (r) ; ) r=r, (.54) where tan θ < = V hf /(V S V T µb). These four equations deterine the coefficients A, B, C, D and F up to a noralization factor, and therefore also the phase shift and the scattering length. Although it is possible to find an analytical expression for the scattering length as a function of the agnetic field, the resulting expression is rather foridable and is oitted here. The result for the scattering length is shown in Fig..4, for V S = 10 h / R, V T = h / R and V hf = 0.1 h / R, as a function of µb. The resonant behaviour is due to the bound state of the singlet potential V S (r). Indeed, solving the equation for the binding energy in Eq. (.1) with V 0 = V S we find that E 4.6 h / R, which is approxiately the position of the resonance in Fig..4. The difference is due to the fact that the hyperfine interaction leads to a shift in the position of the resonance with respect to E. The agnetic-field dependence of the scattering length near a Feshbach reso-

14 8 CHAPTER. SCATTERING AND BOUND STATES 0 ε /( h /R ) e-05 -e-05-3e e-05-5e-05-6e µb/( h /R ) Figure.5: Bound-state energy of the olecular state near a Feshbach resonance for two coupled square-well interaction potentials. The solid line and the inset show the result for V hf = 0.1 h / R. The dashed line corresponds to V hf = 0. The other paraeters are the sae as in Fig..4. nance is characterized experientally by a width B and position B 0 according to ( a(b) = a bg 1 B ). (.55) B B 0 This explicitly shows that the scattering length, and therefore the agnitude of the effective interatoic interaction, ay be altered to any value by tuning the agnetic field. The off-resonant bacground scattering length is denoted by a bg and is, in our exaple, approxiately equal to the scattering length of the triplet potential V T (r). Using the expression for the scattering length of a square well in Eq. (.18) for γ = 1, we find that a bg 0.56R. Furtherore, we have for our exaple that the position of the resonance is given by B h / µr and that the width is equal to B 0.05 h / µr. Next, we calculate the energy of the olecular state for the coupled-channel case which is found by solving Eq. (.47) for negative energy. In particular, we are interested in its dependence on the agnetic field. In the absence of the hyperfine coupling between the open and closed channel we siply have that ɛ (B) = E + µb. Here, E is the energy of the bound state responsible for the Feshbach resonance, that is deterined by solving the single-channel Schödinger equation for the singlet potential. This bound-state energy as a function of the agnetic field is shown in Fig..5 by the dashed line. A nonzero hyperfine coupling drastically changes this result. For our exaple the bound-state energy is easily calculated. The result is shown by the solid

15 .3. EXAMPLE OF A FESHBACH RESONANCE 9 line in Fig..5 for the sae paraeters as before. Clearly, close to the resonance the dependence of the bound-state energy on the agnetic field is no longer linear, as the inset of Fig.5 shows. Instead, it turns out to be quadratic. Moreover, the agnetic field B 0 where the bound-state energy is equal to zero is shifted with respected to the case where V hf = 0. It is at this shifted agnetic field that the resonance is observed experientally. Moreover, for agnetic fields larger than B 0 there no longer exists a bound state and the olecule now decays into two free atos due to the hyperfine coupling, because its energy is above the continuu threshold. Close to resonance the energy of the olecular state turns out to be related to the scattering length by h ɛ (B) = [a(b)], (.56) as in the single-channel case. As we will see in the next chapters, the reason for this is that close to resonance the effective two-body T-atrix again has a pole at the energy in Eq. (.56). This iportant result will be proven analytically in Chapter 4. First, we derive a description of the Feshbach resonance in ters of coupled atoic and olecular quantu fields.