Creation and manipulation of bound states in continuum with lasers: Applications to cold atoms and molecules
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Creation and manipulation of bound states in continuum with lasers:Applications to cold atoms and molecules
Bimalendu Deb and G. S. Agarwal Department of Materials Science, Raman Center for Atomic, Molecular and Optical Sciences,Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, INDIA. Department of Physics, Oklahoma State University, Stillwater, OK 74078, USA.
Abstract
We show theoretically that it is possible to create and manipulate a pair of bound states in continuum inultracold atoms by two lasers in the presence of a magnetically tunable Feshbach resonance. These boundstates are formed due to coherent superposition of two electronically excited molecular bound states and aquasi-bound state in ground-state potential. These superposition states are decoupled from the continuumof two-atom collisional states. Hence, in the absence of other damping processes they are non-decaying.We analyze in detail the physical conditions that can lead to the formation of such states in cold collisionsbetween atoms, and discuss the possible experimental signatures of such states. An extremely narrow andasymmetric shape with a distinct minimum of photoassociative absorption spectrum or scattering crosssection as a function of collision energy will indicate the occurrence of a bound state in continuum (BIC).We prove that the minimum will occur at an energy at which the BIC is formed. We discuss how a BIC willbe useful for efficient creation of Feshbach molecules and manipulation of cold collisions. Experimentalrealizations of BIC will pave the way for a new kind of bound-bound spectroscopy in ultracold atoms.
PACS numbers: 03.65.Ge,03.65.Nk,32.80.Qk,34.50.Rk . INTRODUCTION First introduced by von Neuman and Wigner more than eighty years ago [1], a bound state incontinuum (BIC) is a counter-intuitive and fundamentally profound concept. The original theoret-ical approach of Neuman and Wigner has undergone extensions and modifications over the years[2–4]. In recent times, it has attracted renewed research interests [5] with prospective applicationsin many areas [6–10]. A BIC refers to a discreet eigenstate with energy eigenvalue above thethreshold of the continuum of a potential. The amplitude of the wave function of this state falls offin space and so the wave function is square-integrable. Normally, the eigenstates of a one-particleor a multi-particle system above the continuum are infinitely extended and sinusoidal at distanceslarger than the range of the potential. Below the threshold, there exists negative-energy spectrumof discrete square-integrable bound states. The idea of von Neuman and Wigner was to assumefirst the existence of a positive-energy square-integrable wave function with its envelop decayingin space, and then to construct an appropriate potential that can support such states. Physically,a BIC occurs due to destructive interference of the outgoing Schr¨odinger waves scattered by thepotential, creating an “unusual” trap [5] for an electron [1]. Hsu et al. [10] have observed trappedlight, namely, a BIC of radiation modes by the destructive interference of outgoing radiationsamplitudes.Nearly forty five years after its discovery [1], Stillinger and Herrick [3] extended the idea ofBIC to two-body interactions, and discussed its applications in atomic and molecular physics. Fortwo interacting particles, a BIC can be identified with a scattering resonance state with zero width.In general, a resonance at finite energies arises due to the existence of a quasi-bound (almostbound) state at positive energy. In the absence of any other source of dissipation, it is the couplingof the quasi-bound state with the continuum of scattering states that results in finite width of theresonance. This means that zero width of the resonance would imply decoupling of the quasi-bound state from the continuum of scattering states. In other words, the resonance state with zerowidth becomes a BIC [2, 4].Here we show that it is possible to create a BIC in cold atom-atom collisions in the presenceof two photoassociation (PA) lasers near a magnetic field-induced Feshbach resonance. Our pro-posed scheme is depicted in Fig.1. The two lasers L and L are tuned near to the resonance oftwo excited molecular (bound) states | b i and | b i , respectively. We consider a magnetic Fes-hbach resonance of two colliding ground-state atoms with two ground-state channels of which2ne is closed and the other open. In the absence of coupling with the open-channel, the closedchannel is assumed to support a bound state | b c i . The two PA lasers couple open-channel con-tinuum of scattering states | E i bare with E being collision energy, and | b c i to both the excitedbound states. Using projector operator techniques, we analyze the resolvent operator ( z − ˆ H ) − of the Hamiltonian operator ˆ H and thereby arrive at an effective complex Hamiltonian ˆ H eff of thethree interacting bound states. ˆ H eff is non-hermitian and its eigenvalues are in general complex.However, as we will demonstrate, under appropriate physical conditions, two of the eigenvaluesof the effective Hamiltonian can be made real. We establish the mathematical relations involvingthe parameters of our model that should hold good for the existence of the real eigenvalues. Theeigenvectors corresponding to the real eigenvalues are non-decaying states and hence representbound states in continuum. Similar effective Hamiltonians and their eigenvalue spectrum werestudied in the context of two-photon dressed atomic continuum or autoionizing states [11–13] in1980s. In passing, we would like to mention that non-hermitian Hamiltonians with real eigenval-ues also arise in other areas such parity-time (PT) symmetric Hamiltonian systems [14, 15] andFriedrichs-Fano-Anderson model [16–18] where similar spectral singularity appears and can beassociated with a non-decaying state in continuum [19, 20].Here we emphasize that it is possible to detect the two predicted bound states in continuum bytwo spectroscopic methods, namely photoassociative absorption and photoassociative ionizationtechniques. Mathematically, a BIC in our model appears as a spectral singularity in the scatteringcross section as a function of energy. The expressions of photoassociation probability of eitherexcited bound states as well as the scattering cross section involve the inverse operator ( E − ˆ H eff ) − . This means that for a real eigenvalue of ˆ H eff the denominator of the expressions goes tozero. This leads to divergence in scattering cross section implying the occurrence of a resonancewith zero width [2, 4], that is, a BIC. However, photoassociative absorption spectrum does notdiverge for a real eigenvalue, because the numerator of the expression for the spectrum also goesto zero for the real eigenvalue canceling out the singularity of ( E − ˆ H eff ) − . Physically, thesingularity in scattering cross section implies that BIC is a non-decaying state and hence decoupledfrom the continuum. However, BIC can make transition to either of the excited bound state viaBIC-bound coupling leading to a finite probability for the absorption of a photon. Practically,spectral singularity can not be observed in an experiment. Instead, the signature of BIC will bemanifested as an ultra narrow in the coherent photoassociative spectrum or scattering cross sectionwhen the collision energy is tuned very close to the energy of the BIC. Usually, photoassociation3s described in terms of atom loss from traps, due to the formation of excited diatomic moleculesdecaying into two hot atoms or to a diatom that can escape from trap. The occurrence of the boundstates in continuum in cold collisions will facilitate to photoassociate two atoms effectively througha bound-bound transition process which can be coherent. We show that a possible signature of aBIC in photoassociative cold collisions appears as a sharp and asymmetric line in photoassociativeabsorption spectrum as a function of collision energy. Close to the sharp spike-like line, there liesa minimum which resembles to well-known Fano minimum [17] and corresponds to the energy ofthe BIC.We further demonstrate that, when the intensities and the detuning parameters of L and L are adjusted appropriately, one of the bound states in continuum can be reduced to a superpositionof | b i and | b i only while the other BIC results from superposition of all three bound states.We call the first one as A-type BIC and the second one as B-type BIC. The existence of A-typeBIC can be probed by a probe laser producing the molecular ion and measuring the ion yield asa function of laser frequency. When the two continuum-bound couplings are much larger thanthe Feshbach resonance linewdith, the superposition coefficient of | b c i state in B-type BIC ismuch larger than those of | b i and | b i . Since the state | b c i has a magnetic moment, B-type BIC can be probed by bound-free or bound-bound radio-frequency spectroscopy. In case ofbound-free spectra, the final state would be two free atoms, and thus B-type BIC can be used forcontrolling collisional properties of cold atoms. Furthermore, Feshbach molecules can be createdby stimulating bound-bound transitions with a radio-frequency pulse at a fixed magnetic fieldstrength. To create Feshbach molecules, the usual method uses a sudden sweep of magnetic fieldfrom large negative to large positive scattering length sides of the Feshbach resonance. However,this sudden sweep of magnetic field leads to substantial atom loss due to increase of kinetic energyand thereby limits the atom-molecule conversion efficiency. In contrast, since a BIC is effectivelydecoupled from the continuum, by creating a B-type BIC, Feshbach molecules can be producedefficiently by inducing stimulated transitions from the BIC to Feshbach molecular states with aradio-frequency pulse.We also show that, when the coupling of the continuum to one the bound states is turned off,one can still find one BIC which can be identified as the familiar “dark state” made of superpositionof the two remaining bound states. Coherent population trapping occurs in this superposition stateresulting in the vanishing of the probability of the continuum of scattering states. When lasercoupling to either of the excited states is turned off, the model reduces to one [21] that describes4eshbach resonance-induced Fano effect in photoassociation. The effective Hamiltonian approachto this model shows that the BIC appears at an energy at which Fano minimum occurs. This can beidentified with the standard result that the population trapping occurs due to the “confluence“ ofcoherences [22] at Fano minimum. When the quasibound state in ground-state potential is absentor the magnetic Feshbach resonance is turned off, the resulting effective Hamiltonian has a realeigenvalue when the corresponding eigenvector is an excited molecular dark state [23].The paper is organized in the following way. In Secs. II and III, we present our model and itssolution, respectively. We analyze in some detail how to realize our model and its application incold atoms and molecules in Sec. IV. Finally, we discuss important conclusions of our study inSec. V. II. THE MODEL
The model is schematically depicted in Fig.1. To begin with, we keep our model most general.Suppose a two-channel model is capable of describing an s -wave Feshbach resonance in ground-state atom-atom cold collision. One of these two channels is open and the other is closed. Theclosed channel is assumed to support a bound state | b c i . The thresholds of these two ground-statechannels and the binding energy of | b c i are tunable with an external magnetic field. Both the barecontinuum of scattering states | E i bare in the open channel with E being the collision energy, and | b c i are coupled to two bound states | b i and | b i in an excited molecular potential by two lasers L and L , respectively. Suppose, the states | b i and | b i have same rotational quantum numbers J = J = 1 , but they have different vibrational quantum numbers if both of them are supportedby the same adiabatic molecular potential. In case they belong to different molecular potentials,their vibrational quantum numbers may be same or different. The energy spacing between | b i and | b i is assumed to be large enough compared to the line widths of the two lasers. Furthermore, | b i and | b i are assumed to be far below the dissociation threshold of excited potential(s) so thatthe transition probability at the single-atom level be negligible.In the rotating wave approximation, the Hamiltonian of our mode can be expressed as ˆ H =ˆ H + ˆ V where ˆ H = X n ( E n − ~ ω L n ) | b n ih b n | + E | b c ih b c | + Z E ′ dE ′ | E ′ i bare bare h E ′ | (1)5 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) } G r ound − s t a t e po t e n ti a l E x c it e d s t a t e po t e n ti a l Interatomic separation
Closed channel bound stateb−1b−2 S+PClosed channel S+SL−1 ContinuumOpen channelL−2
FIG. 1: A schematic diagram for creating BIC in ultracold atoms. Two lasers L and L are used toexcite PA transitions from the magnetic Feshbach-resonant collisional state of two ultracold ground-state( S + S ) atoms to the two bound states b and b , respectively, in the same electronically excited molecularpotential. The magnetic Feshbach resonance is considered as a two-channel model in the electronic ground-state potentials with the lower channel being open and the upper one being closed. In the large separationlimit the ground-state channel potential corresponds to two separated S + S atoms while the excited-statepotential connects to two separated S + P atoms. ˆ V = X n Z dE ′ Λ n ( E ′ ) | b n i bare h E ′ | + Z dE ′ V E ′ | b c i bare h E ′ | + X n ~ Ω n | b n ih b c | +C . c . (2) E n is the binding energy of n -th excited molecular state | b n i , ω L n denotes the frequency ofthe L n laser, E c being the energy of the cosed channel bound state | b c i and | E ′ i bare the barecontinuum of scattering state with energy E ′ . Note that all the energies are measured from the openchannel threshold unless stated otherwise. Here Λ n ( E ) is the dipole matrix element of transition | E i bare →| b n i , V E is the coupling between the closed channel bound state | b c i and the openchannel scattering state | E i bare , and Ω n is the Rabi frequency between | b n i and | b c i . Themagnetic Feshbach resonance linewidth is Γ f = 2 π | V E | .6o study BIC, we analyze the resolvent operator G ( z ) = ( z − ˆ H ) − and introduce the projectionoperators P = | b c ih b c | + X n =1 , | b n ih b n | (3) Q = 1 − P = Z dE | E i bare bare h E | (4)which satisfy the properties P P = P, QQ = Q, P Q = QP = 0 , P + Q = 1 (5)Thus we have G = G + G ˆ V G = 1 E − ˆ H + iǫ + 1 E − ˆ H + iǫ ˆ V G (6)Projecting out the bare continuum states, after some algebra as given in appendix-A, we obtain aneffective Hamiltonian of interacting three bound states. Explicitly, this Hamiltonian is given by H eff = H + X n,n ′ =1 , (cid:20)(cid:18) ~ δ nn ′ − i ~ Γ nn ′ ( E )2 (cid:19) | b n ih b n ′ | + (cid:18) δ c − i ~ Γ f ( E )2 (cid:19) | b c ih b c | + X n ~ Γ nf ( E )2 { q nf − i } | n ih b c | +C . c . (7)where δ nn = ( E n + ∆ shift nn ) / ~ − ω L n is the detuning of the light-shifted n th excited levelfrom the L n laser frequency ω L n , δ nn ′ = ~ − ∆ shift nn ′ ( n = n ′ ), where ∆ shift nn ′ is the real partof the quantity R dE ′ Λ ∗ n ( E ′ )Λ n ′ ( E ′ ) / ( E − E ′ ) between, Γ nn ′ ( E ) = 2 π Λ ∗ n ( E )Λ n ′ ( E ) . Here δ c = ~ − (cid:2) E c ( B ) + ∆ shift f − E th ( B ) (cid:3) r is the detuning of the shifted closed channel bound statelevel from the the threshold E th of the open channel. Note that δ c is a function of the appliedmagnetic field B due to the dependence of E c and E th on B . ∆ shift f = P R dE ′ | V E ′ | / ( E − E ′ ) ,where P stands for Cauchy’s principal value, is the shift due to magnetic coupling V E ′ between | b c i and | E ′ i bare . q nf is the well-known Fano-Feshbach asymmetry parameter defined by q nf = δ shift nf + Ω n Γ nf / . (8)7here Ω n is the Rabi frequency for transition | b c i ↔| b n i and δ shift nf = ~ − P Z dE ′ Λ ∗ n ( E ′ ) V E ′ / ( E − E ′ ) (9)is a frequency-shift of | b c i ↔| b n i transition frequency due to the indirect coupling of the twobound states via the continuum. Here Γ nf = 2 π ~ − Λ ∗ n ( E ) V E . For energy E near the thresholdof the continuum, the region E ′ > E of the above integrand contributes more strongly [24]. Asa result, δ shift nf will be negative at low energy. Since Ω n is positive, the sign of q nf depends on therelative strength between (cid:12)(cid:12) δ shift nf (cid:12)(cid:12) and Ω n . Since the magnetic coupling V E ′ is determined by thehyperfine spin coupling between the closed and the open channel, its value depends on the specificatomic system chosen. In contrast, the laser couplings Ω n and δ shift nf depend on which bound state | b n i is chosen for the laser to be tuned to, in accordance with Franck-Condon principle of molecularspectroscopy. Thus, it is possible to alter the sign and magnitude of Fano-Feshbach asymmetryparameter in our model through the selectivity of the excited molecular bound states. Since inmolecular excited states, there is a host of vibrational levels in different molecular symmetriesthat can be accessed by PA, there is a lot of flexibility in choosing the excited bound states in ourmodel. We will further discuss this point in Sec.IV.In the absence of the lasers fields, the magnetic field-dependent resonant scattering phase shift η r and the s -wave scattering length a s are given by − cot η r = E − ˜ E c ~ Γ f / ≃ ka s + 12 r k (10) ˜ E = E c + ∆ shift f is the shifted energy of | b c i , k is the wave number related to the collision energy E = ~ k / µ with µ being the reduced mass of the two colliding atoms. Here r is the effectiverange of the open-channel ground-state potential. In the limit k → , Γ f / ≃ kG f where G f is aconstant having the dimension L s − . The scattering length a s and the effective range r are relatedto ˜ E c and G f by a s = − ˜ E c ~ G f and r = ~ µG f , respectively. This means the magnetic field dependentdetuning δ c ( B ) = − G f /a s . When ˜ E c > , a s is negative and the | b c i lies above the threshold ofthe open channel and hence | b c i is a quasi-bound state. In contrast, when ˜ E c < , | b c i is a truebound state (Feshbach molecular state) and scattering length is positive. Later, we will show thatby forming a BIC with large scattering length, the BIC can be converted into a Feshbach moleculeby stimulated radio-frequency spectroscopy. 8 II. THE SOLUTION
For simplicity, let us introduce the dimensionless parameters ˜ δ n = δ nn / (Γ f / g n = Γ nn / Γ f , g nn ′ = Γ nn ′ / (Γ f / ( n = n ′ ). We assume that δ shift12 = δ shift21 ≃ , that is, the real parts oflaser-induced couplings between | b i and | b i are negligible. Assuming the two free-boundphotoassociative couplings Λ nE to be real quantities, we have Λ nE V E / | V E | = Λ nE /V E = √ g n Under these conditions, the effective Hamiltonian of Eq.(7) can be written in matrix form H eff = ~ Γ f A + i B ] (11)where A = ˜ δ q f √ g δ q f √ g q f √ g q f √ g − ( ka s ) − (12)and B = − g − g −√ g − g − g −√ g −√ g −√ g − (13)For ( ka s ) − = 0 , these matrices have the same form as the Eq. (2.18) of Ref.[13]. The secularequation for B matrix is x + ( g + g + 1) x = 0 and it has two roots equal to zero and thethird one equal to − ( g + g + 1) . When the two eigenvectors of B with zero eigenvalues becomesimultaneous eigenvectors of A with real eigenvalues, we have two real roots of the effectiveHamiltonian. In addition, when A and B commute, both these matrices are diagonalizable withinsimultaneous eigenspace, with H eff having two real eigenvalues. The commutative condition canbe easily found to be q f + ˜ δ = q f + ˜ δ = q f g + q f g − ( ka s ) − (14)To evaluate the two real eigenvalues of the complex Hamiltonian H eff , we proceed in thefollowing way. We first get an eigenvector of the matrix A with an unknown eigenvalue λ in theform X = C x x (15)9here C is normalization constant and x and x are the two elements of the vector. All these threequantities C , x and x are the functions of λ . Assuming that X is also an eigenvector of B withzero eigenvalue, the eigenvalue equation B X = 0 leads to a quadratic equation for λ , the solutionsof which are the desired eigenvalues. For ( ka s ) − = 0 , that is, for a s → ∞ or for the magneticfield tuned on the Feshbach resonance, the two real eigenvalues of H eff are E ± = λ ± ~ Γ f / where λ ± = 12 (˜ δ − q f ) ± (cid:20)(cid:16) ˜ δ + q f (cid:17) − g q f ( q f − q f ) (cid:21) / (16)provided (cid:16) ˜ δ + q f (cid:17) ≥ g q f ( q f − q f ) . λ + and λ − are the two eigenvalues of A , the cor-responding eigenvectors are also the eigenvectors of B with both eigenvalues being zeros. Thetwo eigenstates corresponding to these two real eigenvalues are the coherent superpositions of thethree bound states, and represent two bound states in continuum. Note that the two real eigen-values are expressed in terms of g , ˜ δ and the two Fano-Feshbach asymmetry parameters. But,both the remaining parameters ˜ δ and g can not be arbitrary when H eff has the real eigenvalues.By expressing x , x and C in terms of the set of the parameters g , ˜ δ , q f and q f , from theeigenvalue equation A X = λX one obtains g = (cid:16) ˜ δ − λ (cid:17) (cid:16) ˜ δ − λ − g q f (cid:17) q f (cid:16) ˜ δ − λ (cid:17) (17)with λ = ˜ δ . Now, replacing λ by a real eigenvalue of Eq. (16), one can use the above equation toset the appropriate parameter space of g and ˜ δ for which λ remains real and fixed for a fixed setof other parameters.Let us now consider the special case of both excited bound states belonging to the same excitedmolecular potential with closely lying vibrational quantum numbers ≤ | v − v | ≤ . Hence,the bound-bound Franck-Condon (FC) factors for transitions | b c i ↔| b i and | b c i ↔| b i will benearly equal. Similarly, the free-bound FC factors for transitions | E i bare ↔| b i and | E i bare ↔| b i will also be almost equal. Since Fano asymmetry parameters q f and q f are independent oflaser intensities, but are dependent of these FC factors, we expect q f ≃ q f . The BIC condition ofEq.(14) then implies ˜ δ = ˜ δ . Now, putting q f = q f = q f , the commutativity condition implies ˜ δ = ˜ δ = q f ( g + g − . Under these conditions, two real roots of the effective Hamiltonian are E A = ~ Γ f λ + = ~ Γ f q f ( g + g − (18) E B = ~ Γ f λ − = − ~ Γ f q f (19)10he BIC state corresponding to E A is | A i BIC = 1 √ g + g [ √ g | b i − √ g | b i ] (20)Note that this state does not mix with the closed channel bound state | b c i , and so this eigenvectoris immune to magnetic field tuning of the Feshbach resonance. Nevertheless, | b i and | b i remaincoupled with | b c i and | E i bare due to the lasers. It is easy to see that the photoassociative transitionmatrix element for the interaction Hamiltonian ˆ V P A = √ g | b i bare h E | + √ g | b i bare h E | +C . c . between | E i bare and | A i BIC is zero. This means that this is an excited molecular dark statethat is predicted to play an important role in suppression of photoassociative atom loss [23]. Wecall this dark state as A-type BIC. The eigenstate corresponding to the eigenvalue E B is given by | B i BIC = (cid:20) g + g )( g + g + 1) (cid:21) [ √ g | b i + √ g | b i + ( g + g ) | b c i ] (21)This is a superposition of all three bound states. The involvement of | b c i makes this BIC depen-dent on the magnetic field B . We call this state as B-type BIC. Near unitarity regime, | ka s | is largeand consequently | ( ka s ) − | << , and hence the effect of finite ( ka s ) − can be taken into accountperturbatively. The perturbation part of the Hamiltonian is then V = − ~ Γ f ( ka s ) − | b c ih b c | , andthe first order correction to the energy E B is given by ∆ E B = BIC h B | V | B i BIC = − ~ Γ f ka s ) − g + g g + g + 1 (22)The signature of this BIC can be detected in a number of coherent spectroscopic methods aswill be discussed in the next section. For example, a BIC may be manifested as a strong andnarrow photoassociative absorption line. IV. APPLICATIONS: RESULTS AND DISCUSSIONS
Before we discuss some specific applications, it is worthwhile to make some general observa-tions on the dependence of the two real eigenvalues E A and E B on the magnetic field tuning ofFeshbach resonance. In the zero energy limit ( E → ) and near the vicinity of Feshbach reso-nance, the applied magnetic field B and the scattering length a s are related by a − s = − a − bg B − B ∆ (23)where B is the resonance magnetic field at which a s → ∞ and a bg is the background scatteringlength. Since a bg ∆ > , a s < for B > B and a s > for B < B . In case of Fermionic11toms, B > B ( B < B ) region is commonly known as BCS (BEC) side of the resonance. Theparameter range − . ≤ ( ka s ) − ≤ . is usually referred to as ‘unitarity’ regime.Though E A changes with the change of ( ka s ) − , the corresponding eigenstate of Eq. (20)remains intact and so coherent population trapping occurs (CPT) in A-type BIC that remains pro-tected against the tuning of magnetic field or the scattering length. In contrast, both eigenvalueand eigenstate of B-type BIC depends on B or a s . For a s → ∞ and g + g > , as q f → ± , E A → ± and E B → ∓ as can be inferred from the expressions (18) and (19). Note that theeigenvalues of both A- and B-type BIC depend inversely on ka s . A. Detection of BIC via photoassociation
Modifications of photoassociation probability as a result of the formation of BIC can be ascer-tained by making use of isometric and invertible Møller operators Ω ± of scattering theory. Sincebefore turning on the lasers and the magnetic field, the atoms are in a resonant collisional state,incoming state of the problem can be taken to be the bare continuum | E i bare . The dressed contin-uum state | E + i is given by | E + i = Ω + | E i bare (24)where Ω + = 1 + G ( z + iǫ ) V (25)The probability of photoassociative transition | E i →| b n i is given by P n = Z dE |h b n | E + i| (26)The quantity S n ( E ) = |h b n | E + i| (27)is the photoassociation probability per unit collision energy. Now, we have h b n | E + i = h b n | ( P + Q ) G ( z + iǫ )( P + Q ) V | E i = h b n | P G ( z + iǫ )( P + Q ) V | E i bare (28)12sing Eq. (A.3) we have P GQ = P ( Q + GP V Q ) 1 E − H − V Q + iǫ = P GP V Q E − H − V Q + iǫ (29) h b n | Ω + | E i bare = h b n | P G ( E + iǫ ) P R ( E + iǫ ) | E i bare (30)where R ( E + iǫ ) is given in Eq. (A5). Now, P G ( E + iǫ ) P = ( E − H eff + iǫ ) − = 2 ~ Γ f A Det[( ˜ E − ˜ H eff ] (31)where ˜ E = 2 E/ ~ Γ f , ˜ H eff = A + i B and A is the transpose of the co-factor matrix of ( ˜ E − ˜ H eff ) .Thus we have h b n | Ω + | E i = s π ~ Γ f E − ˜ H eff ] " X m =1 , A nm √ g m + A n (32)The quantity within the third bracket in the numerator of the above equation, for n = 1 and ( ka s ) − = 0 , can be expressed as − √ g h − ˜ E (˜ δ − ˜ E ) − g q f − i (˜ δ − ˜ E ) + ig q f + ig q f i + √ g h − q f q f √ g g + i √ g g ( q f + q f ) + i √ g g ˜ E i − h − q f √ g (˜ δ − ˜ E ) − ig √ g q f + ig √ g q f + i √ g (˜ δ − ˜ E ) i (33)the zeros of which are the roots of the quadratic equation ˜ E (˜ δ − ˜ E ) + g q f − q f q f g + q f (˜ δ − ˜ E ) = 0 (34)Thus, the numerator has two zeros at ˜ E ± = ˜ δ + 12 (cid:20) − (˜ δ + q f ) ± q (˜ δ + q f ) + 4 q f g ( q f − q f ) (cid:21) (35)It is important to note that the two zeros of the numerator are the same as the two real eigenvaluesof H eff as given in Eq. (16). This means that, although the denominator Det h ˜ E − ˜ H eff i = X i =1 (cid:16) ˜ E − ˜ E i (cid:17) (36)13 .5 1 1.5 20200040006000800010000 S ( E ) E f E/E f FIG. 2: Dimensionless spectrum S ( E ) E f as a function of dimensionless energy E/E f (where E f = ~ Γ f / ) for g = 0 . (solid), g = 0 . (dashed) and g = 0 . (dashed-dotted) with q f = − . , q f = − . , g = 2 . , ˜ δ = − . and ˜ δ = 1 . . For the solid curve the three complex eigenvalues of ˜ H eff are ˜ E = 1 . − . i , ˜ E = 1 . − . i , and ˜ E = − . − . i ; for dashed curve ˜ E = 1 . − . i , ˜ E = 1 . − . i , and ˜ E = − . − . i ; for dashed-dotted curve ˜ E = 0 . − . i , ˜ E = 1 . − . i , and ˜ E = − . − . i . When g = 0 . , ˜ E becomes real and is equal to 1.1180 which correspond to the minimum of the spectral lines. The spikeheight of the solid curve is . × . where ˜ E i represents an eigenvalue of ˜ H eff , may become zero for a real eigenvalue of ˜ H eff , thespectrum remains finite in the limit ˜ E → E i for a real eigenvalue ˜ E i ≡ λ ± . When the real part of acomplex eigenvalue is nearly equal to a real eigenvalue, and the imaginary part is extremely small( << ~ Γ f ), the spectrum S ( E ) as a function of collision energy E will exhibit Fano-like minimumand a highly prominent maximum lying close to the minimum. Experimentally, searching for sucha spectral structure is possible by choosing the parameters ˜ δ , g , q f and q f for a zero of thenumerator, but g and ˜ δ are chosen such that the real part of one of the eigenvalues of H eff isnearly equal to a zero of the numerator with the imaginary part being small. A Fano-like spectralstructure with a nearby narrow spectral spike will indicate the existence of BIC [13].Figures 2 and 3 display photoassociative absorption spectra S ( E ) (which is a measure ofprobability of transition to | b i ) as a function function of E for different values of g keeping allother parameters fixed. We scale all energy quantities by E f = ~ Γ f / . The fixed parameters arechosen such that they fulfill the Eq. (35). The different values of g are chosen close to a value14 .6 0.65 0.70246810 x 10 S ( E ) E f E/E f FIG. 3: Same as in Fig.2 but for g = 5 . (solid), g = 5 . (dashed) and g = 4 . (dashed-dotted) withfixed parameters q f = 0 . , q f = 1 . , g = 5 . , ˜ δ = 1 . and ˜ δ = − . . For solid curve the eigenvaluesare ˜ E = 0 . − . × − i , ˜ E = − . − . i , and ˜ E = 1 . − . i ; for dashed curve ˜ E = 0 . − . i , ˜ E = − . − . i , and ˜ E = 1 . − . i ; for dashed-dottedcurve ˜ E = 0 . − . i , ˜ E = − . − . i , and ˜ E = 1 . − . i . For g = 5 . , ˜ E is real and equal to 0.6583. The spike height of the solid curve is . × . that is given by the condition (17) for the occurrence of a real eigenvalue λ = ˜ E ± for the given ˜ δ . The results displayed in Figs. 2 and 3 and the data mentioned in the figure captions clearlysupport our analytical results described above. The spikes in solid curves occur due to the BICwith real eigenvalue close to the real part of ˜ E . In Fig. 2, the bumps near E = 1 . E f occursdue to the complex eigenvalue ˜ E . Note that, as can be seen from the Eq.(8), the Fano-Feshbachasymmetry parameter will be positive (negative) when the Rabi frequency Ω is greater (smaller)than the magnitude δ shift nf which is negative. B. Possible realizations of the model
We next discuss the possibility of experimental realization of our model in ultracold atomicgases that are of current experimental interests. The discussed BIC can be realized in ultracoldatoms with currently available experimental techniques of magnetic Feshabch resonances [25]and photoassociation [26, 27]. In particular, our theoretical proposal can be implemented in caseof experimentally observed narrow Feshbach resonances in cold alkali atoms like Na [28–33],15 Rb [34–36], Li [37, 38], Li [39, 40], etc.. Narrow or close-channel dominated Feshbach reso-nances [25] are preferable since the life-time of the quasi-bound state in such resonances will beappreciable for the lasers to excite bound-bound transitions. For | χ i ↔| b n i bound-bound lasercoupling Ω n to be significant , one needs to choose the excited bound state | b n i such that its outerturning point lies within an intermediate separation r ( ≤ r ≤ ), since the wavefunction of | χ i is usually peaked in this range. It is possible to find such excited bound states of alkali dimershaving outer turning points at such intermediate separations, and such bound state are accessibleby PA transitions as demonstrated in a number of experiments.As an example, let us consider narrow Feshbach resonance in ultracold Rb atomic gas nearmagnetic field strength 1007.4 G [35, 36] in order to realize BIC in cold collisions. This resonanceis characterized by the parameters: zero crossing width ∆ = 0 . G, background scattering length a bg = 100 . a , the difference between magnetic moments of the closed-channel bound state | b c and the two free atoms is δµ = 2 . µ B , where µ B is Bohr magnetron. This means the Feshbachresonance linewidth Γ f = ka bg ∆ δµ where k is the collision wave number related to the collisionenergy E = ~ k / (2 µ ) with µ being the reduced mass of the two atoms. For E = 50 nK, Γ f ∼ kHz. The values of parameters g and g used in Figs. 2 and 3 would correspond to stimulatedline widths of the order of 10 or 100 kHz. From the positions of the minimum in Figs. 2 and 3, itmay be noted that BIC will occur at sub- µ K energy, requiring Bose-Einstein condensate of Rbatoms in order to realize a BIC near this particular Feshbach resonance. However, the theoreticalresults depicted in Figs. 2 and 3 can fit into several other alkali atoms for which a condensate isnot essential. Moreover, different parameter regimes can be used for different alkali systems. Inshort, our model provides a vast range of parameter space with well defined relationship amongthe various parameters for searching for BIC in cold collisions.
C. Detection of BIC via photoassociative ionization spectroscopy
BIC can also be detected by photoassociative ionization spectroscopy. In BIC scheme of Fig.1,a third laser L can be applied to excite molecular auto-ionization transitions | A i BIC →| b i ,where | b i represents a bound state in an excited potential that asymptotically corresponds to twoatoms in P + P electronic states. Since molecular state | b i is made of two doubly excited atoms,it can autoionize to produce molecular ion. Since | b i and | b can be chosen to be energeti-cally close, L can couple both of them to | b i . Therefore, we can construct a photoassociative16onization (PAI) interaction operator ˆ V P AI ( t ) = Ω e − i ( ω L − ω ) t | b ih b | +Ω e − i ( ω L − ω ) t | b ih b | +C . c (37)where Ω and Ω are the Rabi frequencies for the transitions | b i →| b i and | b i →| b i ,respectively; ω L is the frequency of L laser, and ω n is the transition frequency for the transition | b i ↔| b i . The PAI spectrum is given by S P AI ( ω L ) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ dτ Z dEe − iEτ/ ~ − γτ/ h b | ˆ V P AI ( τ ) | E + i (cid:12)(cid:12)(cid:12)(cid:12) (38)where γ is the non-radiative autoionizing line width of | b i . Using the eigenstates | λ i i ( i =1 , , ) of H eff , one can employ the identity operator P i | λ i ih λ i | to express h b n | E + i in termsof | λ i i basis h b n | E + i = X i =1 h b n | λ i ih λ i | V | E i bare E − λ i ~ Γ f / iǫ (39)When BIC conditions for q f = q f = q f are fulfilled, two of the h λ i are bound states in continuum,of which one is A-type and the other is B-type. For g >> and g >> , the probabilityamplitudes of | b i and and | b i in B-type BIC will be very small. Now, since the operator ˆ V P AI couples | b i and and | b i only to | b i , it is expected that, under the conditions g >> and g >> , the laser L will predominantly couple A-type BIC to | b i . Thus PAI spectrum can beapproximated as S P AI ≃ |
BIC h A | V | E i bare | × (cid:12)(cid:12)(cid:12)(cid:12) Ω √ g ( ω − ω A BIC − ω L ) + iγ/ − Ω √ g ( ω − ω A BIC − ω L ) + iγ/ (cid:12)(cid:12)(cid:12)(cid:12) (40)where ω A BIC = E A / ~ is the eigenfrequency of A-type BIC. Clearly, the spectrum will show a shiftequal to ω A BIC . The spectral intensity will be suppressed (enhanced) depending on whether thequantity Re (cid:20) Ω Ω ∗ √ g g ( ω − ω A BIC − ω L + iγ/ ω − ω A BIC − ω L ) − iγ/ (cid:21) (41)is positive (negative). Thus one can detect A-type BIC by PAI with a probe laser ( L ) in thepresence of two PA lasers and a magnetic field under BIC conditions.17 . Controlling cold collisions with BIC When BIC conditions as discussed in the model and solution sections are fulfilled, the eigen-state with real eigenvalue (i.e., BIC) effectively becomes decoupled from the bare continuum whilethe optical and the magnetic transitions between the continuum and the bound states remain ac-tive. As the system parameters are being tuned very close to the BIC conditions, the complexeigenvalue will tend to become real. The complex eigenvalue with small imaginary part impliesthe leakage of the probability amplitude of the BIC into the continuum. This will give rise to aresonant structure with extremely narrow width [2] in the variation of the scattering cross sectionas a function of energy. To calculate the scattering T -matrix, we follow the standard method ofscattering theory based on Møller operators Ω ± . The dressed continuum | E + i describes out-going scattering waves that are influenced by laser light and the magnetic field. The part of thescattering T -matrix element that is modified by the two laser fields and the magnetic field is T field ( E ) = bare h E | ˆ V | E + i = bare h E | ˆ V Ω + | E i bare can be written as T field ( E ) = bare h E | V ( P + Q ) G ( z + iǫ )( P + Q ) V | E i bare . (42)Since bare h E | V Q = 0 and QV | E i bare = 0 , we have T field ( E ) = bare h E | V P GP V | E i bare .Now, using the relation P G ( E + iǫ ) P = ( E − H eff + iǫ ) − we can express T field ( E ) = 1Det[( ˜ E − ˜ H eff ] X n =1 V ∗ n ( E ) X m =1 A nm V m ( E ) (43)where V n ( E ) = Λ n ( E ) , for n = 1 , and V ( E ) = V E are the free-bound coupling constants.Assuming these coupling constants to be real, we have Λ n ( E ) = p Γ f /π √ g n and V E = p Γ f /π .The form of the term N n = P m =1 A nm V m ( E ) for each n = 1 , is equivalent to that of thenumerator of S n ( E ) described in subsection A. We have proved earlier that the numerator of S n ( E ) has a zero for a real eigenvalue of H eff . For n = 3 we have the term N = − Γ f π h ( ˜ E − ˜ δ ) + ( ˜ E − ˜ δ ) n ˜ δ − ˜ δ + q f g + q f g o + q f g (˜ δ − ˜ δ ) i (44)This expression shows that while the term N will not, in general, vanish for a real eigenvalue of ˆ H eff while the others two terms N and N will do. N will vanish for a real eigenvalue when thecommutative condition is fulfilled. This means that for A-type or B-type BIC as discussed earlier,all three terms N n ( n = 1 , , ) in the numerator will lead to Fano-like structures with a minimumand spike-like maximum, and as the energy will approach towards the minimum the spike willbecome narrower as in PA absorption spectrum discussed earlier.18deally speaking, exactly at BIC there will be no outgoing scattered waves. The reason isobvious - a bound state with infinite lifetime can not give rise to any outgoing wave. So, todetect a signature of BIC via scattering resonances, the BIC should have a small but finite widthmeaning the eigenvalue should have small imaginary part. The above analysis implies that, whenthe system parameters are tuned closed to a BIC, the first two terms N and N that describethe contributions from the two excited bound states will cause Fano-like structure in the T -matrixelement. Further, when the commutativity condition is fulfilled or nearly fulfilled, A-type or B-type BIC will show up as prominent Fano-like resonances since all three terms in the numeratorof Eq. (43) will contribute to the resonance structures. Thus, BIC in cold atoms can be utilized fornarrowing magnetic or optical Feshbach resonances or enhancing the lifetime of the resonances.Thus, creating a BIC in cold atoms with lasers, it is possible to manipulate resonant interactionsbetween the atoms. E. Efficient production of Feshbach molecules using BIC
Here we discuss how BIC can help in efficient production of Feshbach molecules [41] bystimulated radio-frequency spectroscopy. Note that, for g >> and g >> , the amplitudecoefficient of | b c i in B-type BIC is much greater than those of the two excited bound states. Thismeans that when the two stimulated linewidths Γ and Γ are much greater than the Feshbachresonance linewidth Γ f , by tuning the two detuning parameters to fulfill the BIC condition δ = δ = q (Γ +Γ − Γ f ) , one can prepare a B-type BIC with large probability amplitude for the closedchannel bound state, which then can be converted into a Feshbach molecule by stimulated radio-frequency spectroscopy. The efficiency of the commonly used method of magnetic field sweep forconversion of pairs of bosonic atoms into Feshbach molecules can not usually go beyond 30%. Incontrast, the efficiency of BIC-assisted Feshabch molecule formation can be close to unity. Forexperimental realization of BIC-assisted Feshbach molecule formation, one can use ultracold Naatoms in the parameter regime of the experiment by Inouye et al. [28] and Xu et al. [32].It is thus possible to suppress the atom loss in magnetic Feshbach resonance (MFR) in a Bose-Einstein condensates by creating BIC with two lasers. This loss occurs primarily due to the disinte-gration of quasi-bound states into non-condensate atoms that can escape from the trap. To accountfor the loss, van Abeelen and Verhaar [30] have introduced a “local” lifetime of quasibound state | χ i due to its exchange coupling to the incoming open channel at an intermediate separation. For19odium condensate, this coupling occurs at r ≤ and the local lifetime τ = γ = 1 . µ s [30],where γ is the width due to the coupling. By choosing the bound states | b i and | b i havingouter turning points near 24 a and making the bound-bound laser couplings Ω and Ω greaterthan γ , one can expect to suppress the atom loss to some extent. But, substantial suppressionof atom loss will result when the collision energy is tuned closed to the energy of a BIC. As wehave analyzed earlier, scattering T -matrix element shows a minimum when energy becomes equalto the energy of one of the two bound states in continuum. Since the width γ is given by theenergy derivative of the scattering phase shift at the energy of quasi-bound state [42], in the con-text of our model γ will correspond to the energy derivative of the phase shift at the minimumpoint. Thus, our model provides γ ≃ and so atom loss in magnetic Feshbach resonances ofBose-Einstein condensates can be largely suppressed. Experimental realization of the effect ofthe suppression of atoms loss in MFR in sodium BEC or in ultracold sodium gas is possible. Be-cause, photoaasociation of sodium atoms into relatively shorter-ranged (outer turning points near r ∼ a ) bound states in g potential have been experimentally demonstrated [43–45] and usedto create light force in PA [46] and to manipulate higher partial-wave interactions [47] via opticaloptical Feshbach resonance (OFR) [48].MFR induced Atom loss in BEC is more severe partly due to bosonic stimulation unlike that indegenerate Fermi gases. Feshbach molecular dimers formed of fermionic atoms are found to bemore stable [49, 50] due to Pauli blocking. We therefore predict that the formation of fermionicFeshbach molecules by stimulated radio-frequency spectroscopy using BIC will be quite efficient. F. Two bound states coupled to the continuum
In our model we have so far considered three bound states coupled to the continuum, with onebeing quasibound state embedded in the ground-state continuum and the two others being excitedmolecular states. Naturally, question arises as to what happens to the BIC if coupling to one of thebound states is turned off. Let us first consider that one of the lasers, say L is absent, that is g = 0 .Then the effective Hamiltonian reduces to a × matrix with second row and second column ofthe matrix being removed. Writing the resulting × effective Hamiltonian in the form A + i B ,the matrix B has one eigenvalue equal to zero and the other one equal to − ( ~ Γ f / g + 1) .Taking E c ≃ , the condition for the existence of a real eigenvalue is ˜ δ = q f g − q which isalso the condition for the commutativity between A and B . The real eigenvalue is - q and the20orresponding eigenvector is | ψ i BIC = 1 p Γ + Γ f hp Γ f | b i − p Γ | χ i i (45)Now, for E c = 0 , by measuring the energy from E c , we recover the standard result for conditionof the occurrence of Fano minimum E − E c ~ Γ f / − q f (46)at which population trapping occurs in the state | ψ i BIC . This should be manifested as a prominentminimum in the scattering cross section or PA rate versus energy plot [21] as in the case of 3 boundstates coupled to continuum as discussed above. In fact, a few years back, two experiments [51, 52]have demonstrated minimum in PA loss rate near the resonant value B of the magnetic field thatinduces a Feshbach resonance. Though, spectroscopy of photoassociative atom loss or PA lossis an incoherent method, the spectral minimum observed in such incoherent spectrum might berelated to a state closely related to | ψ i BIC . It is expected that in coherent PA spectroscopy or inthe measurement of scattering cross sections near B under the above-mentioned BIC condition,one would be able to observe the discussed minimum and an ultra-narrow resonant structure asa clear signature of the occurrence of BIC. In the experiment of Junker et al. [51], the quasi-bound state | χ i is probably weakly coupled ( Ω being small) to the excited bound state, sincethe bound-state chosen was relatively long-ranged ensuring stronger free-bound Franck-Condonoverlap rather than bound-bound coupling. This means that q f should be negative [21] and sothe minimum was expected to occur on the positive side of the scattering length, and indeed thatwas the case in Ref.[51]. In contrast, the experiment by Bauer et al. [52] used a relatively shorterranged excited bound state and the minimum (though not very prominent) occurred very close tothe resonant magnetic field where a s → ∞ . The minimum position shows slight shift towardsnegative side of a s as the laser is blue-detuned by about 3 MHz (the subplots of Fig.3 of Ref. [52]should be compared). Assuming q f to be positive, the BIC condition provides δ = ω b − ω L = − q f (Γ − Γ f ) . Since in experiment of Ref. [52], a narrow Feshbach resonance is used, andrelatively strong PA laser is used, (Γ − Γ f ) > . With blue detuning ( ω L > ω b , the BIC conditionwill only be fulfilled if q f is positive ensuring significant bound-bound coupling. Autler-Townesdouble-peaked spectral shape will arise when the the real parts of the two eigenvalues are not veryfar apart. If BIC condition is maintained more precisely, it is expected that one of the peaks wouldbe very narrow and sharp and would correspond to BIC while the other would be relatively broaddue to the fact that the other eigenvalue being essentially complex.21e next discuss the situation when the state | χ i or the magnetic field is is absent. Writing theresulting × matrix in the form A + i B , one finds that B has a zero eigenvalue and the othereigenvalue is equal to − (Γ + Γ ) . The effective Hamiltonian has a real eigenvalue when δ = δ .Note that δ and δ refer to the detuing from the light-shifted bound states. The eigenvalue is δ = δ = δ and the corresponding eigenvector is | φ i BIC = 1 √ Γ + Γ hp Γ | b i − p Γ | b i i (47)which is an excited molecular dark state which has been found to be useful to make an opticalFeshbach resonance (OFR) more efficient [23]. V. CONCLUSIONS
In conclusions, we have demonstrated theoretically that it is possible to create and manipulatebound states in continuum in atom-atom cold collisions by lasers and a magnetic field, employ-ing currently available techniques of photoaasociation and magnetic Feshbach resonance. Ourmodel is composed of 3 bound states interacting with the continuum of scattering states betweenground-state cold atoms. Within the framework of effective Hamiltonian methods, we eliminatethe continuum and obtain an effective Hamiltonian. The eigenvectors of this effective Hamilto-nian with real eigenvalues represent the bound states in continuum. We have provided specificconditions for the occurrence of a BIC in the form of analytical expressions of the relationshipsbetween parameters of our model. We have derived photoassociative absorption spectrum andscattering cross sections that can exhibit signatures of a BIC as an ultra-narrow asymmetric peaknear a prominent minimum. The minimum occurs exactly at the energy at which BIC occurs. Wehave analyzed in some detail the possible applications of BIC in controlling cold collisions andefficient production of Feshbach molecules.The original proposal of von Neuman and Wigner to create a BIC of a particle was through de-structive quantum interference of Schr¨odinger’s waves scattered by a specially designed potentialso that there exists no outgoing waves resulting in the trapping of the particle in the continuum.The key mechanism for creating a BIC is the quantum interference which happens not only in scat-tering of waves but also in different transition pathways in atomic and molecular physics. Boundstates in continuum in our model result from the quantum interference in three possible free-boundtransition pathways. In case of 2 bound states interacting with the continuum, the effective Hamil-22onian yields one BIC that occurs at an energy at which Fano minimum takes place. This indicatesthat BIC in our model does occur due to quantum interference in possible transition pathways.In recent times, utilization and manipulation of quantum interference effects have been essentialin demonstrating a number of coherent phenomena, paving the way for emerging quantum tech-nologies. Of late, quantum interference are being considered for manipulating ultracold collisions[53, 54]. The realization of our proposed bound states in continuum in cold collisions will open anew perspective in quantum interference phenomena with cold atoms and molecules.
Appendix A: Derivation of effective Hamiltonian ˆ V G = ˆ V ( P + Q ) G = ˆ V P G + ˆ
V QG (A1) QG = QG + QE − H + iǫ ( ˆ V P G + ˆ
V QG ) (A2)which leads to QG = 1 E − H − Q ˆ V + iǫ (cid:16) Q + Q ˆ V P G (cid:17) (A3)Substituting (A3) in (A1) and Eq. (6), after some algebra, we get
P GP = 1 E − H − P ˆ V P − P ˆ V Q E − H − Q ˆ V Q + iǫ Q ˆ V P which suggests that the effective Hamiltonian is H eff = H + P RP (A4)where R = ˆ V + ˆ V Q E − H − Q ˆ V Q + iǫ Q ˆ V (A5)Now, in the subspace of 3 bound states, we need to diagonalize H eff . Using Eqs. (2) and (4), wehave Q ˆ V Q = 0 (A6)and 23
V Q E − H − Q ˆ V Q + iǫ Q ˆ V = X n,n ′ (cid:20) ∆ shift nn ′ ( E ) − i Γ nn ′ ( E )2 (cid:21) | b n ih b n ′ | + (cid:20) ∆ shift f ( E ) − i Γ f ( E )2 (cid:21) | b c ih b c | + "X n (cid:26) ∆ shift nf ( E ) − i Γ nf ( E )2 (cid:27) | n ih b c | +C . c . (A7)where ∆ shift nn ′ ( E ) = P Z dE ′ Λ n ( E ′ )Λ ∗ n ′ ( E ′ ) E − E ′ (A8) ~ Γ nn ′ ( E ) = 2 π Λ n ( E )Λ ∗ n ′ ( E ) (A9) ∆ shift f ( E ) = P Z dE ′ | V E ′ | E − E ′ (A10) ~ Γ f ( E ) = 2 π | V E ′ | (A11) ∆ shift nf ( E ) = P Z dE ′ Λ n ( E ′ ) V ∗ E ′ E − E ′ (A12) ~ Γ nf ( E ) = 2 π Λ n ( E ) V ∗ E (A13) Γ f is the Feshbach resonance line width. Using (A6) in (A4) and (A3), one can obtain the effectiveHamiltonian of Eq. (7) the matrix elements of which are given by h b n | H eff | b n ′ i = ( E n − ~ ω L n ) δ nn ′ + ∆ shift nn ′ ( E ) − i ~ Γ nn ′ ( E )2 (A14) h b c | H eff | b c i = E + ∆ shift f ( E ) − i ~ Γ f ( E )2 (A15) h n | H eff | b c i = ∆ shift nf ( E ) + ~ Ω n − i ~ Γ nf ( E )2 (A16) [1] J. von Neuman and E. Wigner, Phys. Z. , 465 (1929).
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