Suppression of two-body collisional loss in an ultracold gas via the Fano effect
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Suppression of two-body collisional loss in an ultracold gas via the Fano effect
Jianwen Jie, Yawen Zhang, and Peng Zhang
1, 2, 3 Department of Physics, Renmin University of China, Beijing, 100872, China Beijing Computational Science Research Center, Beijing, 100084, China Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices (Renmin University of China)
The Fano effect (U. Fano, Phys. Rev. , 1866 (1961)) shows that an inelastic scattering processcan be suppressed when the output channel (OC) is coupled to an isolated bound state. In thispaper we investigate the application of this effect for the suppression of two-body collisional lossesof ultracold atoms. The Fano effect is originally derived via a first-order perturbation treatment forcoupling between the incident channel (IC) and the OC. We generalize the Fano effect to systemswith arbitrarily strong IC–OC couplings. We analytically prove that, in a system with one IC andone OC, when the inter-atomic interaction potentials are real functions of the inter-atomic distance,the exact s -wave inelastic scattering amplitude can always be suppressed to zero by coupling betweenthe IC or the OC (or both of them) and an extra isolated bound state. We further show that whenthe low-energy inelastic collision between two ultracold atoms is suppressed by this effect, the realpart of the elastic scattering length between the atoms is still possible to be much larger thanthe range of inter-atomic interaction. In addition, when open scattering channels are coupled totwo bound states, with the help of the Fano effect, independent control of the elastic and inelasticscattering amplitudes of two ultracold atoms can be achieved. Possible experimental realizations ofour scheme are also discussed. PACS numbers: 34.50.Cx, 03.65.Nk, 67.85.-d
I. INTRODUCTION
In ultracold gases of neutral atoms prepared in excitedinternal states (e.g., excited hyperfine states correspond-ing to the electronic ground level of alkali atoms or long-lived excited states of alkali-earth (like) atoms), two-bodycollisional losses can be induced by inelastic scatteringprocesses [1–17]. In these processes the atoms can jumpto the lower internal states and gain a large amount ofkinetic energy. Two-body collisional losses can shortenthe lifetime of the atomic gases. For instance, in theBose–Einstein condensate of Rb atoms in the hyper-fine state | F = 2 , m F = 1 i and the ultracold gas Ybatoms in the P states, two-body collisional loss ratesare of the order of 10 − cm / s [2, 3] and 10 − cm / s [4],respectively. This means that for ultracold gases of theseatoms having typical densities of 10 / cm , the lifetimecan be reduced to much less than 1 s or even less than 1ms. In most experiments of optically trapped ultracoldgases, the atoms are prepared in the lowest internal statesso that two-body inelastic scattering can be avoided.Nevertheless, a lot of interesting physics can be studiedwith ultracold gases of atoms prepared in excited internalstates. For instance, the physics of spin-2 Bose Einsteincondensation can be studied with ultracold Rb atomswith F = 2 [18]. Physics related to spin-exchange pro-cesses and the Kondo effect can be studied with a mix-ture of ultracold alkali-earth (like) atoms in the ground S and excited P states [4, 19, 20]. To obtain suchultracold gases with sufficiently long lifetimes, it is im-portant to study how to suppress the two-body inelasticscattering processes between ultracold atoms [12–17].In 1961, Ugo Fano found that inelastic scattering canbe significantly suppressed if the output channel (OC) of that process is coupled to an isolated bound state[21]. The Fano effect can be understood as the resultof destructive interference between the quantum transi-tion from the incident channel (IC) to the OC and thetransition from the isolated bound state to the OC. In theoriginal derivation of the Fano effect, coupling betweenthe IC and OC is treated as a first-order perturbation[21, 22]. This perturbative treatment has also been usedin the previous study for the application of the Fano effecton the suppression of collisional loss in ultracold gases[17].In this paper, we go beyond this first-order perturba-tion approximation and investigate the Fano effect in thetwo-atom scattering problem with arbitrarily strong IC–OC coupling. Then, we study the application of this ef-fect for the suppression of the two-body collisional lossesin ultracold gases. The main results and the structure ofthis paper can be summarized as follows:In Sec. II we study the Fano effect in a two-atomscattering problem in three-dimensional space, with oneIC and one OC. Here each channel corresponds to atwo-atom internal state. With an analytical calculationfor the exact inelastic scattering amplitude, we provethat when the inter-atomic interaction potentials are realfunctions of the distance between these two atoms, the s -wave inelastic scattering amplitude can always be sup-pressed to zero when an isolated bound state is coupledto the IC or the OC, or both of them (Fig. 1). Our resultis applicable for systems with arbitrary IC–OC couplingintensities and incident kinetic energy. In particular, weprove that this suppression effect can occur even whenthe bound state is only coupled to the IC and not di-rectly coupled to the OC, as shown in Fig. 1(c). Weshow that the suppression effect can be understood asresulting from destructive interference between the di- FIG. 1: (color online) Multi-channel models studied in Sec. II(a–c) and Sec. IV (d). Here r is the distance between the twoatoms. The black curves represent the interaction potentialsfor the IC α and OC β of the inelastic scattering processes,while the blue and orange curves represent the interactionpotentials for the closed channels η and η ′ , respectively. Thered arrows represent inter-channel coupling. In Sec. II, weconsider cases where a bound state | Φ b i in the closed channel η is coupled to both the IC α and OC β (a), only coupled tothe OC β (b), or only coupled to the IC α (c). In Sec. IV,we consider the system where the open channels α and β arecoupled to two isolated bound states | Φ b i and | Φ b ′ i in the η and η ′ channels (d). rect transition from the IC to the OC, and an indirecttransition from the IC to the isolated bound state andthen back to the IC, and then to the OC.Our results imply that two-body collisional loss in anultracold gas may be completely suppressed when thereis one OC in the inelastic scattering processes. In Sec.III we further show that when this loss is suppressed bythe Fano effect, it is still possible for the elastic scatteringlength between the two ultracold atoms to be either largeor small. That is, it is possible to obtain ultracold gasesof atoms in excited internal states with strong inter-atominteractions and negligible collisional loss rates. To ourknowledge, this result has not been obtained in previousstudies on the control of two-body collisional loss in ul-tracold gases. In this section we also discuss the possibleexperimental realizations of our approach.In Sec. IV we investigate the independent control ofelastic and inelastic collisions between ultracold atoms.We study the system where the IC and OC are coupledto two bound states (Fig. 1(d)). We show that, for thissystem, when the inelastic scattering amplitude is sup-pressed to zero by the Fano effect, the elastic scatteringlength in the IC can be tuned to any value by alteringthe energies of these two bound states.These results are helpful for the study of ultracold gases of atoms prepared in excited internal states. More-over, our generalization of the Fano effect for systemswith strong IC–OC coupling is also useful for the studyof inelastic scattering processes in other physical systems. II. FANO EFFECT IN THE SYSTEM WITHSTRONG IC–OC COUPLING
We consider the three-channel scattering problem oftwo ultracold atoms shown in Fig. 1(a–c). The threechannels α , β , and η correspond to two-atom internalstates | α i I , | β i I , and | η i I , respectively. In natural units ~ = m = 1, with m the single-atom mass, we can expressthe Hamiltonian of our system as H = p + X j = α,β,η E j | j i I h j | + V ( r ) , (1)where p and r are the relative momentum and relativecoordinate of the two atoms, respectively, and r = | r | .The energy E j ( j = α, β, η ) is the threshold energy ofchannel j , with E η > E α > E β . In Eq. (1), V ( r ) is theinteraction potential of the two atoms, and is given by V ( r ) = X l,j = α,β,η V lj ( r ) | l i I h j | . (2)Here V jj ( r ) ( j = α, β, η ) is the potential of channel j ,while V lj ( r ) = V jl ( r ) ( l = j ) is the inter-channel coupling.For the systems shown in Fig. 1(b) and Fig. 1(c), wehave V αη = 0 and V βη = 0, respectively. In this paperwe consider systems where all the components of V area real function of r .We consider the case where the two atoms are incidentfrom channel α , and the incident state is near resonant toan isolated s -wave bound state | Φ b i in channel η . In thiscase, channels α and β are the IC and OC of the inelas-tic scattering process, respectively. The s -wave inelasticscattering amplitude f βα from channel α to β can be ex-pressed as a function of the scattering energy E s and theenergy ǫ b of | Φ b i . In the following we will prove that f βα can always be suppressed to zero by coupling betweenthe bound state in channel η and the channels α and/or β , no matter how strong the IC–OC coupling V αβ . Thatis, for any given value of E s , there always exists a realenergy ˜ ǫ b , which leads to f βα ( E s , ǫ b = ˜ ǫ b ) = 0.In the following subsections we will first derive the an-alytical expression of f βα , and then calculate the non-diagonal element of the K -matrix for our system. Thiselement is proportional to f βα and easier to study. Wewill prove our result by analyzing the character of this K -matrix element. A. Scattering amplitude
In this subsection we calculate the s -wave scatteringamplitude with the method in Ref. [23]. In our systemthe Hilbert space H can be expressed as H = H R ⊗H I , with H R being the Hilbert space for the inter-atomicrelative motion in the spatial space and H I representingthe two-atom internal state. We use |i to denote the statein H , |i R for the state in H R , and |i I for the state in H I .The scattering amplitude from channel l to channel j ( l , j = α, β ) is defined as f jl = − π h Ψ (0) k j ,j | V | Ψ (+) k l ,l i , (3)where | Ψ (+) k l ,l i is the s -wave component of the out-goingscattering state with respect to the incident momentum k l and incident channel l , and the state | Ψ (0) k j ,j i is definedas | Ψ (0) k j ,j i = | ψ (0) k j ,j i R | j i I , with | ψ (0) k j ,j i R the s -wave com-ponent of the eigen-state | k j i R of the relative momentumoperator p . Here we have k l ( j ) = | k l ( j ) | . Notice that the s -wave states | Ψ (+) k l ,l i and | Ψ (0) k j ,j i are independent of thedirections of the momentum k l and k j , respectively. Dueto energy conservation, the momentum k l,j satisfies k l + E l = k j + E j ≡ E s , (4)where E s is defined as the scattering energy.We can obtain the scattering amplitude by solving theLippman–Schwinger equation satisfied by the scatteringstate | Ψ (+) k l ,l i . This equation can be expressed as (Ref.[23], Appendix A) | Ψ (+) k l ,l i = | Ψ ( αβ +) k l ,l i + G ( αβ ) ( E s ) W | Ψ (+) k l ,l i , (5)where the operator W is defined as W = V ηα ( r ) | η i I h α | + V ηβ ( r ) | η i I h β | + h.c., (6)and describes the coupling between channels α, β , and η .Here | Ψ ( αβ +) k l ,l i is the s -wave component of the out-goingscattering state for the case with W = 0, with respect tothe incident channel l and incident momentum k l , and G ( αβ ) ( E ) is the Green’s operator for this case. It is givenby G ( αβ ) ( E ) = 1 E + i + − ( H − W ) . (7)As shown above, we consider the case where E s isnear resonant to an isolated s -wave bound state | Φ b i ≡| φ b i R | η i I in channel η . Here | φ b i I satisfies the eigen-equation H η | φ b i R ≡ (cid:2) p + V ηη ( r ) + E η (cid:3) | φ b i R = ǫ b | φ b i R (8)of the self-Hamiltonian H η of channel η , and “near reso-nant” means that E s is close to ǫ b . In this case, we canneglect the contribution from other eigen-states of H η .Under this single-resonance approximation, the Green’soperator G ( αβ ) ( E ) can be re-expressed as G ( αβ ) ( E ) = 1 E + i + − h + | Φ b ih Φ b | E − ǫ b , (9) where h = p + X j = α,β E j | j i I h j | + X l,j = α,β V lj ( r ) | l i I h j | (10)is the “self-Hamiltonian” of channels α and β . With Eq.(9), we can analytically solve the Lippman–Schwingerequation (5) for the scattering state | Ψ (+) k l ,l i , and thus ob-tain the s -wave scattering amplitude f jl ( E s ) defined inEq. (3) (Ref. [23], Appendix B): f jl ( E s , ǫ b ) = f ( αβ ) jl ( E s ) − π A j ( E s ) A l ( E s ) B ( E s ) − ǫ b , (11)where f ( αβ ) jl ( E s ) is the scattering amplitude for the casewith W = 0, and the functions A l ( j ) ( E s ) and B ( E s ) aredefined as A l ( j ) ( E s ) = h Φ b | W | Ψ ( αβ +) k l ( j ) ,l ( j ) i ; (12) B ( E s ) = E s − h Φ b | W G ( αβ ) ( E s ) W | Φ b i . (13) B. S -matrix and K -matrix In this subsection we introduce the S -matrix and K -matrix related to the s -wave scattering in our system. Inthe s -wave subspace the S -matrix is a 2 × S ( E s , ǫ b ) = (cid:20) S αα ( E s , ǫ b ) , S αβ ( E s , ǫ b ) S βα ( E s , ǫ b ) , S ββ ( E s , ǫ b ) (cid:21) . (14)Here the matrix element S jl ( E s , ǫ b ) ( l, j = α, β ) is relatedto the scattering amplitude via the relation f jl ( E s , ǫ b ) = S jl ( E s , ǫ b ) − δ jl i p k l k j . (15)In Appendix C we show the relation between this S -matrix and the S -operator of our system [24], and provethat this S -matrix is a unitary matrix [24].In our system the K -matrix is defined as [25] K ( E s , ǫ b ) = i − S ( E s , ǫ b )1 + S ( E s , ǫ b ) (16) ≡ (cid:20) K αα ( E s , ǫ b ) , K αβ ( E s , ǫ b ) K βα ( E s , ǫ b ) , K ββ ( E s , ǫ b ) (cid:21) . According to this definition, the non-diagonal elementsof the K -matrix and S -matrix satisfy the relation K βα = − iS βα S ] + S αα + S ββ . (17)With direct calculation based on Eqs. (11, 15) and (17),we can obtain the expression of K βα ( E s , ǫ b ). Since theterms S jl ( j, l = α, β ) and Det[ S ] in Eq. (17) are lin-ear and quadratic functions of the scattering amplitude f ij given by Eq. (11), respectively, K βα ( E s , ǫ b ) can beexpressed as K βα ( E s , ǫ b ) = F ( E s ) ǫ b + C ( E s ) ǫ b + D ( E s ) F ( E s ) ǫ b + C ( E s ) ǫ b + D ( E s ) , (18)and we can obtain the coefficients C , , D , and F , viasubstituting Eqs. (11, 15) into Eq. (17). With directcalculation, we are surprised to find that the coefficients F ( E s ) and F ( E s ) of the ǫ b -terms in Eq. (18) are exactlyzero, i.e., F ( E s ) = F ( E s ) = 0. As a result, K βα ( E s , ǫ b )has a simple expression K βα ( E s , ǫ b ) = C ( E s ) ǫ b + D ( E s ) C ( E s ) ǫ b + D ( E s ) , (19)where the coefficients C , ( E s ) and D , ( E s ) are given by C ( E s ) = 2 is βα ( E s ); (20) C ( E s ) = − − s αα ( E s ) s ββ ( E s ) + s αβ ( E s ) s βα ( E s ) − s αα ( E s ) − s ββ ( E s ); (21) D ( E s ) = − i [ s αβ ( E s ) B ( E s ) + A αβ ( E s )] ; (22) D ( E s ) = − C ( E s ) B ( E s ) + s αα ( E s ) A ββ ( E s ) + s ββ ( E s ) A αα ( E s ) − A ( E s ) [ s βα ( E s ) + s αβ ( E s )]+ A αα ( E s ) + A ββ ( E s ) , (23)with A lj ( E s ) = − π i p k j k l A l ( E s ) A j ( E s ) ( l, j = α, β ).Here, the functions A l ( j ) ( E s ) and B ( E s ) are defined inEqs. (12) and (13), and s lj ( E s ) is the element of the S -matrix for the case with W = 0. C. Suppression of inelastic scattering
Based on our above results, now we prove the centralresult of this section.Because the interaction potential in our system is real,the S -matrix S ( E s , ǫ b ) is a symmetric unitary matrix(Ref. [23], appendix C), and thus can be formally ex-pressed as S ( E s , ǫ b ) = (cid:18) ζe iξ p − ζ e iξ ′ p − ζ e iξ ′ − ζe − iξ e iξ ′ (cid:19) , (24)where ζ, ξ , and ξ ′ are real numbers and 0 < ζ ≤ K -matrix element K βα ( E s , ǫ b ) can be re-expressed as K βα ( E s , ǫ b ) = p − ζ − sin ξ ′ + ζ sin( ξ − ξ ′ ) , (25)and thus must be real for all values of E s and ǫ b . Usingthis result and the expression (19) for K βα ( E s , ǫ b ), it canbe proved that (Appendix D) the ratio D ( E s ) /C ( E s ) isalways real. Thus, according to Eq. (19), non-diagonalelement K βα ( E s , ǫ b ) of the K -matrix becomes zero whenthe energy ǫ b of the bound state in channel η takes thevalue ǫ b = − D ( E s ) C ( E s ) . (26) Furthermore, according to Eqs. (17) and (15), we have K βα ( E s , ǫ b ) ∝ S βα ( E s , ǫ b ) ∝ f βα ( E s , ǫ b ) . (27)Therefore, under the condition in Eq. (26), we have f βα = 0 , (28)i.e., the inelastic scattering from the IC α to the OC β iscompletely suppressed by coupling W between these twochannels and the bound state | Φ b i in the closed channel η. Because we do not treat the IC–OC coupling V αβ asa perturbation in our proof, our result is applicable tosystems with arbitrarily strong IC–OC coupling.Our proof shows that the inelastic scattering amplitudecan be suppressed as long as the inter-channel coupling W defined in Eq. (6) is nonzero. This is regardless ofwhether the coupling V βη between the bound state | Φ b i and the OC β is zero or nonzero. When V βη = 0, thesuppression effect can be understood as a result of inter-ference between the quantum transition from channel α to channel β and the one from | Φ b i to channel β . Nev-ertheless, in systems with V βη = 0 and V αη = 0, i.e., thesystem shown in Fig. 1(c), this effect is not attributableto direct interference of these quantum transitions.To understand the suppression effect in this spe-cial case, we consider a system where the IC–OCcoupling V αβ is very weak and can be treated as afirst-order perturbation. For this system the inelas-tic scattering amplitude f βα can be approximated as f βα ≈ − π ´ drr ψ ∗ β ( r ) V αβ ( r ) ψ α ( r ), where ψ α ( β ) ( r ) isthe component of the s -wave scattering wave function inchannel α ( β ) for the case with V αβ = 0. In the presenceof the coupling between the IC α and the bound state | Φ b i in channel η , the wave function ψ α ( r ) can be for-mally expressed as ψ α ( r ) = ψ (bg) α ( r ) + δψ α ( ǫ b , r ). Here ψ (bg) α ( r ) is the s -wave scattering wave function in channel -16 -6 40.00.81.6 -10 -8 -60.00.51.0 -11.0 -8.5 -6.00.000.150.30-20 -8 40.00.61.2 -14 -10 -60.00.61.2 -12 -9 -60.00.20.4-20 -8 40.00.61.2 -12 -9 -60.00.40.8 -12 -9 -60.000.150.30U =1/b | f | / b (a) k=0.0 k=0.3 k=1.0 U =3/b (b) U =10/b (c) U =1/b (d) | f | / b U =3/b (e) U =10/b (f) U =1/b (g) U b | f | / b U =3/b (h) U b U =10/b (i) U b FIG. 2: (color online) The absolute value | f βα | of the inelastic scattering amplitude in the square-well model, as a function ofthe potential energy U ηη of channel η . In (a–c), we show results for cases where U αη = 2 /b , U βη = 3 /b , i.e., the cases wherethe closed channel η is coupled to both IC α and OC β (the cases in Fig. 1(a)). In (d–f), we show results for cases where U αη = 0, U βη = 3 /b , i.e., the cases where the closed channel η is only coupled to OC β (the cases in Fig. 1(b)). In (g–i), weshow results for cases where U αη = 3 /b , U βη = 0, i.e., the cases where the closed channel η is only coupled to the IC α (thecases in Fig. 1(c)). Here we consider systems with potential energies U αα = − /b , U ββ = − . /b ; threshold energies E α = 0, E β = − . /b ; inter-channel coupling U αβ = 1 /b (a,d,g), 3 /b (b,e,h) and 10 /b (c,f,i); and incident momentum k α = 0 (solidblack line), 0 . /b (dashed blue line), and 1 /b (dash-dotted red line). α for the case with V αη = V αβ = 0, and the ǫ b -dependentwave function δψ α ( ǫ b , r ) is the modification induced by V αη . This term describes the change of the atomic wavefunction in channel α , which is induced by the second-order process where the atoms transit from channel α to | Φ b i and then return to α . Accordingly, the inelasticscattering amplitude f βα can be expressed as f βα ( E s , ǫ b ) ≈ − π ˆ drr ψ ∗ β ( r ) V αβ ( r ) ψ (bg) α ( r ) − π ˆ drr ψ ∗ β ( r ) V αβ ( r ) δψ α ( ǫ b , r ) . (29)Eq. (29) clearly shows that the inelastic scattering ampli-tude includes contributions from the transition processes from the states ψ (bg) α ( r ) and δψ α ( ǫ b , r ) to the state ψ β ( r ).When the interference of these two transition processesis destructive, the inelastic scattering can be suppressed.This analysis shows that, in a system where the boundstate | Φ b i is only coupled to IC α and not coupled to OC β , the suppression of the inelastic scattering can be un-derstood as a result of destructive interference betweenthe direct transition from channel α to β and the indirecttransition process along the path α → | Φ b i → α → β . D. Illustration
Now we illustrate our results with a simple multi-channel square-well model. In this model the potential V lj ( r ) ( l, j = α, β, η ) defined in Eq. (2) is given by V lj ( r ) = (cid:26) U lj , for r < b , for r > b , (30)with b the range of these potentials. We further choosethe threshold energies E α,β,η to satisfy E η = ∞ , E α = 0 , E β < . (31)We calculate the inelastic scattering amplitude f βα fromthe higher channel α to the lower channel β , for caseswhere the incident state is near resonant to the lowestbound state in channel η . Figure 2 shows | f βα | as afunction of the potential energy U ηη of channel η . Here,we consider cases where the closed channel η is coupledto both the IC α and OC β (Fig. 2(a-c)), and caseswhere channel η is only coupled to the IC α (Fig. 2(d-f))or OC β (Fig. 2(g-i)). It is clearly shown that in all ofthese cases, for the system with any incident momentum k α and IC–OC coupling U αβ , the inelastic amplitude f βα can always be suppressed to zero. III. SUPPRESSION OF INELASTICSCATTERING PROCESSES IN ULTRACOLDGASES
In this and the next section, we study the applicationof our results in ultracold gases. In this system, if thereis only one possible two-atom inelastic scattering process(e.g., the scattering from channel α to channel β ) thenthe two-body collisional loss rate is determined by theinelastic scattering amplitude f βα for E s = E α , i.e., theamplitude of the threshold inelastic scattering. As wasshown in Sec. II, when the open channels α and β arecoupled to an isolated bound state | Φ b i with energy ǫ b ,this scattering amplitude can be suppressed to zero, pro-vided that the condition (26) with E s = E α is satisfied.With straightforward calculation, we find that this con-dition can be re-expressed as ǫ b = ǫ ∗ b ≡ E α − h Φ b | W G ( αβ ) ( E α ) W | Φ b i− π h Φ b | W | Ψ ( αβ +)0 ,α ih Φ b | W | Ψ ( αβ +) √ E α − E β ,β i f ( αβ ) βα ( E α ) . (32)When Eq. (32) is satisfied, the two-body collisional lossis completely suppressed.On the other hand, the interaction between two ultra-cold atoms in state | α i I can be described by the real partof the scattering length a ( ǫ b ), which is defined as a ( ǫ b ) ≡ − f αα ( E α , ǫ b ) . (33)Substituting Eq. (32) into Eq. (11) and using the opticaltheorem, we find that under the conditions of Eq. (32) we haveRe [ a ( ǫ ∗ b )] = a (bg) + f ( αβ ) βα ( E α ) h Φ b | W | Ψ ( αβ +)0 ,α ih Φ b | W | Ψ ( αβ +) √ E α − E β ,β i ;(34)Im [ a ( ǫ ∗ b )] = 0 , (35)where a (bg) = − f ( αβ ) αα ( E α ) is the scattering length in thesystem with W = 0. Eq. (34) implies that when theinelastic collision is completely suppressed, the scatter-ing length a ( ǫ ∗ b ) still depends on details of the two-atominteraction potential V ( r ) via the factors h Φ b | W | Ψ ( αβ +)0 ,α i and h Φ b | W | Ψ ( αβ +) √ E α − E β ,β i . In principle, it is possible forthe value of Re [ a ( ǫ ∗ b )] to be much larger than the range r ∗ of V ( r ) (e.g., the van der Waals length), or comparableto r ∗ , or much smaller than r ∗ . When Re [ a ( ǫ ∗ b )] is muchlarger than the range r ∗ , the two atoms in channel α have a large probability to be close to each other. Nev-ertheless, because of quantum interference between the α → β and η → β transitions, the atoms do not decay tochannel β . In this case, the interaction between the twoatoms in channel α is still strong, while the collisionalloss is completely suppressed.We illustrate our result with the square-well model inSec. II. D. Figure 3(a–c) shows the absolute value of theinelastic scattering amplitude f βα ( E s = E α ) and the realpart of the scattering length a ( ǫ ∗ b ) as functions of thepotential energy U ηη of the closed channel η , for threetypical cases with the potential energy U αα of channel α taking the values U αα = − . /b , − . /b , and − . /b . It is shown that in these three cases, when | f βα | is suppressed to zero, the scattering length could beeither comparable or much larger than the range b of theinteraction potential. Figure 3(d) shows the scatteringlength a ( ǫ ∗ b ) as a function of U αα . It is clearly shownthat for systems with different interaction potentials, thevalue of a ranges from −∞ to + ∞ .Now we investigate possible experimental realizationsof our scheme. In ultracold gases of alkali atoms, thestates | β i I , | α i I , and | η i I can be chosen as the lowest,second lowest, and higher two-atom hyperfine states withthe same total magnetic quantum number m (1) F + m (2) F .Here m (1(2)) F is the magnetic quantum number of atom1(2). For instance, for ultracold Li atoms, one canchoose | α i I = 1 √ (cid:20) | , − i | , i − | , i | , − i (cid:21) ;(36) | β i I = 1 √ (cid:20) | , i | , − i − | , − i | , i (cid:21) ;(37) | η i I = 1 √ (cid:20) | , − i | , i − | , − i | , i (cid:21) , (38)where | c, d i i is the hyperfine state of the i -th atom with F = c and m F = d . -21.80 -21.78 -21.76-150001500-93 -87 -811.83.55.2 Re[a] |f |
U b R e [ a ]/ b | f | / b (a) -68.8 -68.6 -68.4015003000 Re[a] |f |
U b R e [ a ]/ b | f | / b -68.8 -68.3 -67.8-2000-10000 Re[a] |f |
U b R e [ a ]/ b | f | / b R e [ a ]/ b U b FIG. 3: (color online) (a-c): The absolute value of the inelastic scattering amplitude f βα ( E s = E α ) and the real part of thescattering length as functions of the potential energy U ηη of the closed channel η , for the square-well model in Sec. II. D. Thepotential energy U αα of channel α has the values U αα = − . /b (a), − . /b (b), and − . /b (c). In these three cases,when | f βα | is suppressed to zero we have Re[ a ( ǫ ∗ b )] = 3 . b (a), 1541 b (b), and − b (c). (d): The scattering length a ( ǫ ∗ b ) as afunction of U αα . Our calculation is done with the parameters E α = 0, E β = − . /b , U ββ = U αβ = U βη − /b , andU αη = 2 /b . Since the total magnetic quantum numbers of threehyperfine channels α , β , and η are the same, these threechannels are coupled to each other via the hyperfine spin-exchange interaction. Therefore, when we prepare theatoms in channel | α i I (e.g., prepare the ultracold Liatoms in the hyperfine states | , − i and | , i ) and thethreshold energy E α of channel α is near resonant to abound state in channel η , a system shown in Fig. 1(a)can be realized.In our system, the threshold energies E α,β of channels α , β , and the energy ǫ b of the bound state in channel η can be controlled by a static magnetic field via theZeeman effect. Therefore, the collisional loss of atoms inchannel α can be suppressed by tuning the magnetic fieldsuch that the condition (32) is satisfied. When collisionalloss is suppressed, the elastic scattering length betweentwo atoms is determined by the details of the inter-atomicinteractions, and can be either large or small.One can also couple the open channel α or β and thebound state | Φ b i in a closed hyperfine channel using amicrowave field. In this way it is possible to effectivelycontrol the bound-state energy ǫ b by changing the fre-quency of that microwave field [26]. In this case, thetotal magnetic quantum number of state | η i I would dif-fer from that of states | α i I and | β i I [26]. In addition,in ultracold gases of alkali atoms or alkali-earth (like)atoms, a laser beam can be used to couple the open scat-tering channels and a bound state where one atom is inthe electronic ground state and the other atom is in the electronic excited state [27, 28]. However, in this systemthe spontaneous emission of the excited atom can alsoinduce atomic losses. As a result, the two-body loss ratecan no longer be suppressed to zero. IV. INDEPENDENT CONTROL OF ELASTICAND INELASTIC COLLISIONS BETWEEN TWOULTRACOLD ATOMS
In the preceding sections, we studied the suppressionof two-body collisional losses of ultracold atoms via theFano effect. We show that when the collisional loss iscompletely suppressed, it is still possible for the two-atomscattering length, i.e., the threshold elastic scattering am-plitude, to be either large or small. Nevertheless, in thatsystem there is only one control parameter, i.e., the en-ergy ǫ b of the isolated bound state. As a result, when thecollisional loss is suppressed by tuning this bound-stateenergy to some particular value, the scattering length ofthese two atoms would also be entirely fixed, and cannotbe altered.In this section, we study the independent controlof elastic and inelastic collisions between two ultracoldatoms. To this end, we first consider the four-channelmodel shown in Fig. 1(d), where the IC and OC of theinelastic scattering process are coupled to two isolatedbound states, rather than a single bound state. We showthat, in this “ideal” model, when the collisional loss oftwo atoms in the IC is suppressed by the Fano effect, thescattering length can still be tuned over a very broad re-gion by changing the energies of the two bound states.At the end of this section we will discuss a possible ex-perimental realization of this model.In the model shown in Fig. 1(d), there are two boundstates, | Φ b i and | Φ b ′ i , with energies ǫ b and ǫ b ′ , whichare located in the closed channels η and η ′ , respectively.Each bound state is coupled to the IC α or the OC β ,or both of these two open channels. It is clear that,in this system, the two-atom scattering amplitude f jl ( l, j = α, β ) from channel l to channel j depends onboth of the bound-state energies ǫ b and ǫ b ′ , i.e., we have f jl = f jl [ E s , ǫ b , ǫ b ′ ]. This scattering amplitude can becalculated with the method in Sec. II. Notice that in thecalculation we should replace the channel α in Sec. IIwith both the channel α and the bound state | Φ b ′ i in ourcurrent system. This straightforward calculation showsthat because of the Fano effect, for any given value ofthe energy ǫ b ′ of the bound state | Φ b ′ i , the threshold in-elastic collision can always be completely suppressed ifthe energy ǫ b of the bound state | Φ b i takes a particular ǫ b ′ -dependent special value χ ( ǫ b ′ ), i.e., we have f βα [ E α , ǫ b = χ ( ǫ b ′ ) , ǫ b ′ ] = 0 . (39)Furthermore, when ǫ b = χ ( ǫ b ′ ), the scattering length a between the two atoms becomes real, and can be ex-pressed as a function ¯ a ( ǫ b ′ ) ≡ − f αα [ E α , ǫ b = χ ( ǫ b ′ ) , ǫ b ′ ]of the energy ǫ b ′ . According to the direct calculationshown in Appendix E, we have¯ a ( ǫ b ′ ) = a ( αβη ′ ) + A ′ E α − ǫ b ′ − B ′ , (40)where a ( αβη ′ ) is the scattering length in the system with V βη = V αη = 0. The expressions of the parameters A ′ and B ′ are given in Appendix E. In this appendix wealso prove that B ′ is a ǫ b ′ -independent real parameter.Because of this and considering Eq. (40), ¯ a ( ǫ b ′ ) can becontrolled in a very broad region by tuning the bound-state energy ǫ b ′ in the region around E α − B ′ .Here we illustrate this control effect using calcu-lations with a square-well potential. In our model,the total Hamiltonian is p + P j = α,β,η,η ′ E j | j i I h j | + P l,j = α,β,η,η ′ V lj ( r ) | l i I h j | , where V lj ( r ) = U lj for r < b , V lj ( r ) = 0 for r > b , E η = E η ′ = ∞ , E α = 0, and E β <
0. In Fig. 4 we illustrate the scattering length¯ a ( ǫ b ′ ) as a function of the energy ǫ b ′ of the lowest boundstate in channel η ′ . It is clearly shown that this scat-tering length can be resonantly controlled by ǫ b ′ or thepotential energy U η ′ η ′ of channel η ′ .Here, we propose one possible experimental realizationof the model discussed in this section. In an ultracoldgas of alkali atoms, the states | l i I ( l = α, β, η, η ′ ) can bechosen as a two-atom hyperfine state | l i I , which satisfies( F (1) z + F (2) z ) | l i I = M l | l i I . Under the condition, M α = M β = M η , the channels α , β , and η are coupled viahyperfine interactions. Aided by the Zeeman effect, the -4.60 -4.53 -4.46-180001800 a ( b ’ ) / b U ’ ’ b FIG. 4: The elastic scattering length ¯ a ( ǫ b ′ ) ≡ − f αα [ E α , ǫ b = χ ( ǫ b ′ ) , ǫ b ′ ] for the square-well model in Sec. IV. Here ǫ b ′ is the lowest bound state in channel η ′ . The calculationis executed with the parameters E α = 0, E β = − . /b , U αα = − . /b , U ββ = − /b , U αβ = U βη = U αη ′ =3 /b , andU αη = U ηη ′ = U βη ′ = 0. energy ǫ b of the bound state | Φ b i in channel η can becontrolled by a static magnetic field. In addition, with amicrowave field one can further couple the open channels α and β with the bound state | Φ b ′ i in channel η ′ , andeffectively control the energy ǫ b ′ by altering the frequencyof that microwave field. V. SUMMARY
In this paper we generalize the Fano effect to systemswith arbitrary IC–OC coupling strengths. We prove thatin systems with one IC and one OC, when the inter-atomic interaction potential is real, the s -wave inelasticscattering amplitude can always be suppressed to zeroby the coupling between these open channels and an iso-lated bound state. Using our result, we further show thatwhen the two-body collisional loss of an ultracold gas issuppressed via the Fano effect, it is possible for the two-atom elastic scattering length to be either much larger,comparable to or smaller than the van der Waals length.We also show that when the open channels are coupledto two bound states, the elastic scattering length of theatoms in the higher open channel can be resonantly con-trolled, while the inelastic scattering is completely sup-pressed. Our results show that the Fano effect may be avery powerful technique for the suppression of collisionallosses in ultracold gases. Furthermore, the generalizedFano effect we derived in Sec. II may also be useful forthe study of the inelastic scattering processes in othersystems.It is pointed out that in this paper we consider sys-tems with spherically symmetrical interaction potentials.Nevertheless, the Fano effect can also be used to suppressthe collisional losses induced by anisotropic interactions,e.g., dipolar losses caused by dipole-dipole interactions[12, 13]. In these cases, although the collisional lossescannot be suppressed to zero, they can also be signifi-cantly decreased (e.g., decreased by more than one orderof magnitude [12, 13]) when one or several open channelsare coupled to an isolated bound state. Acknowledgments
This work has been supported by the National NaturalScience Foundation of China under Grant Nos. 11222430and 11434011, and by NKBRSF of China under GrantNo. 2012CB922104. Peng Zhang also thanks Hui Zhaiand T. L. Ho for helpful discussions.
Appendix A: Proof of Eq. (5)
In this appendix we prove Eq. (5). According to theformal scattering theory, the scattering state | Ψ (+) k l ,l i sat-isfies the equation [29] | Ψ (+) k l ,l i = lim λ → + iλE s + iλ − H | Ψ (0) k l ,l i , (A1)where E s and H are defined in Eq. (C8) and Eq. (1),respectively, and the state | Ψ (0) k l ,l i is defined in Sec. II. A.Similarly, the state | Ψ ( αβ +) k l ,l i , which is the s -wave com-ponent of the out-going scattering state for the case with W = 0, satisfies the equation | Ψ ( αβ +) k l ,l i = lim λ → + iλE s + iλ − ( H − W ) | Ψ (0) k l ,l i . (A2)Substituting the relation1 E s + iλ − H = 1 E s + iλ − ( H − W )+ 1 E s + iλ − ( H − W ) W E s + iλ − H (A3)into Eq. (A1), and using Eq. (A2) and Eq. (7), we canobtain Eq. (5). Appendix B: Proof of Eq. (11)
In this appendix we prove Eq. (11). To this end, wesubstitute Eq. (9) into Eq. (5). Then we find that thesolution of Eq. (5) can be expressed | Ψ (+) k l ,l i = | Γ k l ,l i + κ | Φ b i , (B1)where the state | Γ k l ,l i is in the subspace spanned by | α i I and | β i I , and κ is a c-number. Furthermore, using Eq. (9) we can rewrite Eq. (5) as the equations of | Γ k l ,l i and κ : | Γ k l ,l i = | Ψ ( αβ +) k l ,l i + κG ( αβ ) ( E s ) W | Φ b i , (B2) κ = h Φ b | W | Γ k l ,l i E s − ǫ b . (B3)Substituting Eq. (B2) into Eq. (B3), we obtain the equa-tion κ = h Φ b | W | Ψ ( αβ +) k l ,l i E s − ǫ b + κ h Φ b | W G ( αβ ) ( E s ) W | Φ b i E s − ǫ b , (B4)which gives κ = h Φ b | W | Ψ ( αβ +) k l ,l i E s − ǫ b − h Φ b | W G ( αβ ) ( E s ) W | Φ b i . (B5)Substituting this result into Eq. (B2), we can furtherderive the state | Γ k l ,l i .Using these results, we can calculate the scattering am-plitude f jl ( E s , ǫ b ). Substituting Eq. (B1) into Eq. (3),we obtain f jl ( E s , ǫ b ) = − π h Ψ (0) k j ,j | ( V − W ) | Γ k l ,l i− π κ h Ψ (0) k j ,j | W | Φ b i . (B6)Substituting Eqs. (B5, B2) into Eq. (B6), and using therelation f ( αβ ) jl ( E s ) = − π h Ψ (0) k j ,j | ( V − W ) | Ψ ( αβ +) k l ,l i (B7)satisfied by the scattering amplitude f ( αβ ) jl ( E s ) for thecase with W = 0, we obtain f jl ( E s , ǫ b ) = f ( αβ ) jl ( E s ) − π h Ψ ( αβ − ) k j ,j | W | Φ b ih Φ b | W | Ψ ( αβ +) k l ,l i E s − ǫ b − h Φ b | W G ( αβ ) ( E s ) W | Φ b i . (B8)Here | Ψ ( αβ − ) k j ,j i is the s -wave component of the incomingscattering state for the case with W = 0, with respectto incident channel j and incident momentum k j . Itsatisfies the Lippman–Schwinger equation | Ψ ( αβ − ) k j ,j i = | Ψ (0) k j ,j i + G ( αβ ) ( E s ) † ( V − W ) | Ψ (0) k j ,j i (B9)and the relation I h l | R h r | Ψ ( αβ − ) k j ,j i = I h l | R h r | Ψ ( αβ +) k j ,j i ∗ (B10)for l = α, β . Here | r i R is the eigen-state of the relativeposition of the two atoms. Because of the relation (B10),we have h Ψ ( αβ − ) k j ,j | W | Φ b i = h Φ b | W | Ψ ( αβ +) k j ,j i . (B11)Substituting Eq. (B11) into Eq. (B8), we can obtain Eq.(11).0 Appendix C: S -matrix in the s -wave subspace In this appendix we prove some properties of the S -matrix related to the s -wave scattering in our system,which is introduced in Sec. II. B.We first study the relation between this S -matrix andthe S -operator in our system. To this end, we introducea state | Φ (0) k,l i ( l = α, β ), which is defined as | Φ (0) k,l i = √ πk | Ψ (0) k,l i . Here | Ψ (0) k,l i is defined in Sec. II. A. It iseasy to prove that R h r | Φ (0) k,l i = sin( kr )2 π √ kr | l i I . (C1)This relation yields [24] h Φ (0) k ′ ,l ′ | Φ (0) k,l i = δ l,l ′ δ ( E k,l − E k ′ ,l ′ ) (C2)and X l ˆ dE | Φ (0) √ E − E l ,l ih Φ (0) √ E − E l ,l | = 1 , (C3)where the energy E k,l ( l = α, β ) is defined as E k,l = k + E l . Now let us consider the factor h Φ (0) k ′ ,l ′ | ˆ S | Φ (0) k,l i ( l, l ′ = α, β ), where ˆ S is the S -operator of our system. It isdefined as ˆ S = Ω †− Ω + , where Ω ± are the M φ ller operators[24]. According to the formal scattering theory [24], wehave h Φ (0) k ′ ,l ′ | ˆ S | Φ (0) k,l i = h Φ ( − ) k ′ ,l ′ | Φ (+) k,l i , (C4)with | Φ ( ± ) k,l i = √ πk | Ψ ( ± ) k,l i (C5)( l = α, β ). Here | Ψ (+ / − ) k,l i is the s -wave component ofthe incoming/out-going scattering state with scatteringenergy E k,l and incident channel l , as defined in Sec. II.A. They satisfy the Lippman–Schwinger equation | Ψ ( ± ) k,l i = | Ψ (0) k,l i + lim λ → + E k,l ± iλ − H V | Ψ (0) k,l i , (C6)the Schr¨odinger equation H | Ψ ( ± ) k,l i = E k,l | Ψ ( ± ) k,l i , and thenormalization condition h Ψ (+) k ′ ,l ′ | Ψ (+) k,l i = h Ψ ( − ) k ′ ,l ′ | Ψ ( − ) k,l i = h Ψ (0) k ′ ,l ′ | Ψ (0) k,l i . These facts yield | Φ ( − ) k ′ ,l ′ i = | Φ (+) k ′ ,l ′ i + (C7)lim λ → + (cid:18) E k ′ ,l ′ − iλ − H − E k ′ ,l ′ + iλ − H (cid:19) V | Φ (0) k ′ ,l ′ i ,H | Φ ( ± ) k,l i = E k,l | Φ ( ± ) k,l i , (C8) h Φ (+) k ′ ,l ′ | Φ (+) k,l i = h Φ ( − ) k ′ ,l ′ | Φ ( − ) k,l i = δ ( E k ′ ,l ′ − E k,l ) δ l,l ′ . (C9) Substituting Eq. (C7) into Eq. (C4), and using Eq. (C9)and (C8), we obtain h Φ (0) k ′ ,l ′ | ˆ S | Φ (0) k,l i = δ ( E k ′ ,l ′ − E k,l ) δ l,l ′ + (cid:18) x + i + − x − i + (cid:19) h Φ (0) k ′ ,l ′ | V | Φ (+) k,l i , (C10)where x = E k ′ ,l ′ − E k,l . With the help of the relation1 x + i + − x + i − = − πiδ ( x ) , (C11)and Eqs. (3) and (C5), we can further rewrite Eq. (C10)as h Φ (0) k ′ ,l ′ | ˆ S | Φ (0) k,l i = δ ( E k ′ ,l ′ − E k,l ) S l ′ ,l ( E k,l ) , (C12)where S l ′ ,l ( E k,l ) is defined in Eq. (15). This is the rela-tion between the S -matrix and the S -operator ˆ S in oursystem.Now we prove the S -matrix is a unitary matrix. Sincethe S -operator ˆ S is a unitary operator [24], it satisfiesˆ S † ˆ S = 1 . (C13)Using this result and Eqs. (C2) and (C3), we obtain X l ′′ ˆ dE ′′ h Φ (0) √ E ′′ − E l ′′ ,l ′′ | ˆ S | Φ (0) k ′ ,l ′ i ∗ h Φ (0) √ E ′′ − E l ′′ ,l ′′ | ˆ S | Φ (0) k,l i = δ l,l ′ δ ( E k ′ ,l ′ − E k,l ) . (C14)Substituting Eq. (C12) into Eq. (C14), we find that the2 × S l ′ ,l ( E ), i.e., the S -matrix weintroduced in Eq. (14), is a unitary matrix.Now we consider the S -matrix in the system with realinteraction potential. In such a system, the S -operatorˆ S satisfies [24] h Φ ′ | ˆ S | Φ i = h e Φ | ˆ S | e Φ ′ i , (C15)where the state | e Φ i is defined as | e Φ i = T | Φ i , with T thetime-reversal operator for the spatial motion. The state | e Φ i satisfies the relation [24] R h r | e Φ i = R h r | Φ i ∗ . (C16)From Eqs. (C1) and (C16), we know that | e Φ (0) k,l i = | Φ (0) k,l i for l = α, β . Therefore, we have h Φ (0) k ′ ,l ′ | ˆ S | Φ (0) k,l i = h Φ (0) k,l | ˆ S | Φ (0) k ′ ,l ′ i . (C17)This result and the relation (C12) indicates that the S -matrix defined in Eq. (14) is a symmetric matrix for thesystem with real potentials.1 Appendix D: The ratio D ( E s ) /C ( E s ) In this appendix we prove that the ratio D ( E s ) /C ( E s ) appearing in Sec. II. C is real.To this end, we will first prove that all the ratios C ( E s ) /C ( E s ), D ( E s ) /D ( E s ), and D ( E s ) /C ( E s )are real.According to Eqs. (20) and (21), the ratio C ( E s ) /C ( E s ) is just the non-diagonal element of the K -matrix for the case with W = 0. As shown in Sec.II. C, in our system this matrix element is real. Thus, C ( E s ) /C ( E s ) is real.Moreover, according to Eq. (19), we have D ( E s ) /D ( E s ) = K βα ( E s , K βα ( E s , ǫ b ) is realfor any ǫ b , the ratio D ( E s ) /D ( E s ) is also real.Now we prove that D ( E s ) /C ( E s ) is also real. Wecan prove this result by contradiction. To this end, were-express Eq. (19) as (cid:18) C ( E s ) C ( E s ) (cid:19) K βα ( E s , ǫ b ) = ǫ b + h D ( E s ) C ( E s ) i ǫ b + h D ( E s ) C ( E s ) i . (D1) Since both K βα ( E s , ǫ b ) and C ( E s ) /C ( E s ) are real, theright-hand side of Eq. (D1) is real for any ǫ b . Thus,if D ( E s ) /C ( E s ) is not real, we must have D ( E s ) C ( E s ) = D ( E s ) C ( E s ) . Using Eq. (19), we find that this result yieldsthat K βα ( E s , ǫ b ) = C ( E s ) /C ( E s ), i.e., K βα ( E s , ǫ b ) is independent of ǫ b . Furthermore, with Eqs. (17, 11, 15)we can express the diagonal elements of the K -matrix as K αα ( E s , ǫ b ) = C ′ ( E s ) ǫ b + D ′ ( E s ) C ( E s ) ǫ b + D ( E s ) ; (D2) K ββ ( E s , ǫ b ) = C ′′ ( E s ) ǫ b + D ′′ ( E s ) C ( E s ) ǫ b + D ( E s ) , (D3)where the factors C ′ ( E s ), D ′ ( E s ), C ′′ ( E s ), and D ′′ ( E s )are given by C ′ ( E s ) = − i [1 − s αα ( E s ) s ββ ( E s ) + s αβ ( E s ) s βα ( E s ) − s αα ( E s ) + s ββ ( E s )] ; (D4) D ′ ( E s ) = − C ′ ( E s ) B ( E s ) − i { s αα ( E s ) A ββ ( E s ) + s ββ ( E s ) A αα ( E s ) − A αβ ( E s ) [ s βα ( E s ) + s αβ ( E s )] }− i A αα ( E s ) + i A ββ ( E s ); (D5) C ′′ ( E s ) = − i [1 − s αα ( E s ) s ββ ( E s ) + s αβ ( E s ) s βα ( E s ) + s αα ( E s ) + s ββ ( E s )] ; (D6) D ′′ ( E s ) = − C ′′ ( E s ) B ( E s ) − i { s αα ( E s ) A ββ ( E s ) + s ββ ( E s ) A αα ( E s ) − A αβ ( E s ) [ s βα ( E s ) + s αβ ( E s )] } i A αα ( E s ) − i A ββ ( E s ) . (D7)Since the K -matrix is a Hermitian matrix, K αα ( E s , ǫ b )and K ββ ( E s , ǫ b ) are real for any ǫ b . Therefore, witha similar method to that used above, we find thatif D ( E s ) /C ( E s ) is not real, both K αα ( E s , ǫ b ) and K ββ ( E s , ǫ b ) are independent of ǫ b . Therefore, if D ( E s ) /C ( E s ) is not real, all the K -matrix elements areindependent of ǫ b . According to Eqs. (16) and (15), thisresult indicates that all scattering amplitudes f jl ( E s , ǫ b )( j, l = α, β ) are independent of ǫ b . However, accordingto Eq. (11), f jl ( E s , ǫ b ) takes different values for different ǫ b . Therefore, in our system the ratio D ( E s ) /C ( E s ) isreal.So far we have shown that the ratios C ( E s ) /C ( E s ), D ( E s ) /D ( E s ), and D ( E s ) /C ( E s ) are all real. It fol-lows that the ratio D ( E s ) /C ( E s ) is also real. Appendix E: Eq. (40) and The Parameter B ′ In this appendix we will prove Eq. (40) in Sec. IV,and prove that the parameter B ′ is real in this equation. Appendix E.A Proof of Eq. (40)