Laser Manipulation of Spin-Exchange Interaction Between Alkaline-Earth Atoms in $^1$S$_0$ and $^3$P$_2$ States

Ultracold gases of fermionic alkaline-earth (like) atoms are hopeful candidates for the quantum simulation of many-body physics induced by magnetic impurities (e.g., the Kondo physics), because there are spin-exchange interactions (SEIs) between two atoms in the electronic ground ($^1$S$_0$) and metastable ($^3$P) state, respectively. Nevertheless, this SEI cannot be tuned via magnetic Feshbach resonance. In this work we propose three methods to control the SEI between one atom in the $^1$S$_0$ state and another atom in the $^3$P$_2$ states or $^3$P$_2$-$^3$P$_0$ dressed states, with one or two laser beams.These methods are based on the spin-dependent AC-Stark shifts of the $^3$P$_2$ states, or the $^3$P$_2$-$^3$P$_0$ Raman coupling. We show that due to the structure of alkaline-earth (like) atoms, the heating effects induced by the laser beams of our methods are very weak. For instance, for ultracold Yb atoms, AC-Stark-shift difference of variant spin states of the $^3$P$_2(F=3/2)$ level, or the strength of the $^3$P$_2$-$^3$P$_0$ Raman coupling, could be of the order of $(2\pi)$MHz, while the heating rate (photon scattering rate) is only of the order of Hz. As a result, the Feshbach resonances, with which one can efficiently control the SEI by changing the laser intensity, may be induced by the laser beams with low-enough heating rate, even if the scattering lengths of the bare inter-atomic interaction are so small that being comparable with the length scale associated with the van der Waals interaction.


INTRODUCTION
In recent years, the ultracold gases of fermionic alkaline-earth (like) atoms have attracted much attention [1,2].One important motivation for studying this system is that there are spin-exchange interactions (SEIs) between two fermionic alkaline-earth (like) atoms (e.g., two 173 Yb atoms or two 171 Yb atoms) in the electronic 1 S 0 and 3 P states, respectively, which plays a central role on the quantum simulation of many-body models with magnetic impurities (e.g, the Kondo or Kondo-lattice models) .Explicitly, the atoms in the 3 P state can be individually confined in the sites of a deep optical lattice, and play the role of the local magnetic impurities, so that the two-body loss induced by the collision of two 3 P atoms can be avoided.In addition, the moving 1 S 0 atoms can play the role of the itinerant electrons in the Kondo-type models.To realize this quantum simulation, it is important to develop the techniques to manipulate the SEI [20][21][22][23][24][26][27][28][29].
To avoid the heating loss induced by the spin-exchange process, the difference between the Zeeman energies of the atoms before and after this process should be lower than the temperature of the ultracold gases.Thus, the control of the SEI should be done under zero or lowenough magnetic field, and thus is difficult to be realized via magnetic Feshbach resonance [30].Due to this fact, people studied the manipulation of SE interaction via a confinement-induced resonance (CIR) [31] under zero magnetic field [24,[26][27][28].This technique has been experimentally realized for the control of the nuclear SE interaction between ultracold 173 Yb atoms [21].* pengzhang@ruc.edu.cnNevertheless, mostly the CIR occurs only when the inter-atomic scattering length in the three-dimensional (3D) free space is comparable with the characteristic length of the confinement, which is usually of the order of 1000a 0 .For current experiments of ultracold alkalineearth (like) atoms, this condition is only partly satisfied by 173 Yb atoms, for which one of the two scattering lengths related to the SEI is about 2000a 0 [4][5][6].For other systems, e.g., 171 Yb atoms, the relevant 3D scattering lengths are of the order of 100a 0 [6][7][8]32], i.e., much less than the confinement characteristic length, and thus the control effect of the CIR approach to be weak.On the other hand, the interaction between atoms in 1 S 0 and 3 P 0 states includes not only the SEI but also a spinindependent term.In current experiments of 173 Yb or 171 Yb atoms, this term is very strong, so that the spinexchange effects may be suppressed [11].Therefore, it would be helpful if more control techniques for the SEI between alkaline-earth (like) atoms can be developed.
In this work, we propose three methods for controlling the SEI between two fermionic alkaline-earth (like) atoms with pseudo-spin 1/2 via one or two laser beams.Explicitly, one atom is in the 1 S 0 state and another one in the 3 P 2 state (methods I and II) or a 3 P 2 -3 P 0 dressed state (method III) [33,34].So far the SEI of alkalineearth like atoms has been only observed with atoms being in 1 S 0 and 3 P 0 states.Nevertheless, for our systems with pseudo-spin 1/2 atoms in the 1 S 0 and 3 P 2 states, there also exits SEI (i.e., the exchange of the pseudo-spin states) processes.These processes are induced by a similar mechanism as the one for the 1 S 0 and 3 P 0 atoms, and are permitted by the selection rule of the corresponding inter-atomic interaction potential, as shown below.In this work, we consider the 3 P 2 states because for these states the laser-induced effects which can be used for the manipulation of SEI (e.g., the spin-dependent AC-Stark effects) are much more significant than the ones of 3 P 0 states.
Our approaches are based on the spin-dependent AC-Stark shifts of 3 P 2 states (methods I and II), or the laserinduced Raman coupling between 3 P 2 and 3 P 0 states (method III).Explicitly, for the systems of these methods, there are both open and closed channels of the spinexchange scattering processes, and the energy gap between the open and closed channels just equals to (or has the same order of magnitude with) the AC-Stark shift difference ∆ AC between 3 P 2 states with different magnetic quantum numbers, or the effective Rabi frequency Ω eff of the 3 P 2 -3 P 0 Raman coupling.Therefore, one can control the amplitude of the spin-exchange scattering of the atoms incident from the open channel, or the effective inter-atomic SEI, by tuning ∆ AC or Ω eff via changing the laser intensity.
More importantly, we show that the heating effects induced by the laser beams are quite weak.This is due to the structure of alkaline-earth (like) atoms, and is very different from the situations of the ultracold alkaline atoms under similar laser manipulations (e.g., the Raman coupling between different hyperfine states of electronic ground state), where the lasers mostly induce strong heating.For instance, for a Yb atom the heating rate (photon scattering rate) could be just of the order of Hz when ∆ AC or Ω eff , and thus the energy gap between the open and closed channels, is of the order of (2π)MHz, and is comparable with the van der Waals energy scale E vdW .On the other hand, the potentials of the closed channels are very possible to support s-wave bound states with the binding energies |E b | being comparable with or less than the van der Waals energy E vdW , even in the absence of a s-wave resonance.For instance, for the cases of a single-channel van der Waals interaction we have |E b | < 0.99E vdW for a s > β 6 [35], with β 6 being the length scale associated with this van der Waals interaction [36].Moreover, for the systems of our methods II and III, the s-wave states of the open and closed channels are coupled to each other.Thus, for these systems, no matter if the closed channels are on resonance, it is always very possible that one can make the threshold of the open channel be resonant to a closed-channel bound state by tuning the laser intensity, and thus induce Feshbach resonances for these two atoms, while keeping the heating rate low enough.Using these resonances one can efficiently manipulate the effective SEI.
For the systems of our method I where the s-wave states of the open channels are only coupled to the dwave closed-channel bound states, the above kind of "low-heating" Feshbach resonance occurs when the closed channels are close to a d-wave resonance.Nevertheless, for our system there are four degenerate closed channels which are coupled with each other.Therefore, the probability for the appearance of these resonances is much larger than the one of a single-channel van der Waals potential.
For the systems of all the methods I, II, and III, we can always treat the atoms in the 1 S 0 states and the relevant 3 P states as two distinguishable atoms with pseudo-spin 1/2.The effective Hamiltonian of these two atoms can be expressed as where m is the single-atom mass, p S(P ) is the momentum operator of the atom in the 1 S 0 ( 3 P) state.Here the effective inter-atomic interaction Veff is given by ( = 1): with µ ≡ m/2 and r being reduced mass and the interatomic position, respectively, and σ(S(P )) x,y,z being the Pauli operators for the pseudo-spin of the atom in the 1 S 0 ( 3 P) state.Namely, the pseudo-spin states of the two atoms are degenerate eigen-states of the effective two-atom free Hamiltonian p 2 S /(2m) + p 2 P /(2m).In addition, the effective interaction Veff is described by the parameters A x,y,z,0 .For instance, the strength of the effective SEI is (A x + A y )/2, and the strength of the spin-independent interaction is A 0 .In addition, for the systems of methods I and III we always have A x = A y , while for the system of method II A x and A y may be unequal.Using our methods one can resonantly control the parameters A x,y,z,0 via changing the laser intensity.
Since we lack the detailed parameters for the bare interaction potential between atoms in 1 S 0 and 3 P 2 states, so far we cannot perform quantitatively accurate calculations for experimental systems.Therefore, in this work we qualitatively illustrate the three methods with twobody calculations for a multi-channel square-well interaction model.Our results show that the effective interaction can be tuned to be either "anti-ferromagnetic-like" or "ferromagnetic-like" for many cases, where the lowest eigen state of the pseudo-spin operator in the square bracket of Eq. ( 1) is the singlet state or the pseudo-spinpolarized states, respectively.In addition, the absolute values |A x,y,z,0 | can be controlled in a broad region (e.g., from zero to 1000a 0 ).One can also completely "turn off" the spin-independent interaction (i.e., tune A 0 to be zero) while keeping the SEI strength (A x + A y )/2 to be finite.
The remainder of this paper is organized as follows.In Sec.II, III, and IV we show the principles of methods I, II, and III, respectively, and illustrate the control effects via the square-well model.In Sec.V we provide some discussions, including a comparison of the advantages and disadvantages of these three methods.Some details of our calculations are given in the appendixes.
p 7 5 b n / 5 + 7 t U j V u G U 2 L 9 0 H 5 n / 1 e l a F E 6 w Z W o I q K b E M L o 6 n r t k p i v 6 5 s 6 n q h Q 5 J M R p 3 K a 4 J M y N 8 q P P j t G k p n b d W 2 b i L y Z T s 3 r P 8 9 w M r / q W N G D v + z h / g n q 5 5 K 2 X y n s b x c p 2 P u o J L G E Z q z T P T V S w i y p q 5 H 2 B e z z g 0 T q 0 r q 0 b 6 / Y 9 1 R r K N Y v 4 s q y 7 N / e K l z A = < / l a t e x i t > |c, "i p 7 5 b n / 5 + 7 t U j V u G U 2 L 9 0 H 5 n / 1 e l a F E 6 w Z W o I q K b E M L o 6 n r t k p i v 6 5 s 6 n q h Q 5 J M R p 3 K a 4 J M y N 8 q P P j t G k p n b d W 2 b i L y Z T s 3 r P 8 9 w M r / q W N G D v + z h / g n q 5 5 K 2 X y n s b x c p 2 P u o J L G E Z q z T P T V S w i y p q 5 H 2 B e z z g 0 T q 0 r q 0 b 6 / Y 9 1 R r K N Y v 4 s q y 7 N / e K l z A = < / l a t e x i t > |c, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " R K X f d 8 G + J + j / C r i e F N v s s 8 x S X h L l R j v r s G E 1 q a t e 9 Z S b + a j I 1 q / c 8 z 8 3 w p m 9 J A / a + j / M n O K p W v F q l e r B Z 3 t n N R 1 3 E C l a x T v P c w g 7 2 U U e D v C / w g E c 8 W U 3 r x r q 1 7 j 5 S r U K u W c a X Z d 2 / A 0 H E l h c = < / l a t e x i t > |c,   |g, "i < l a t e x i t s h a 1 _ b a s e 6 4 = " D y p 4 5 p 7 5 b n / 5 + 7 t U j V u G U 2 L 9 0 H 5 n / 1 e l a F E 6 w Z W o I q K b E M L o 6 n r t k p i v 6 5 s 6 n q h Q 5 J M R p 3 K a 4 J M y N 8 q P P j t G k p n b d W 2 b i L y Z T s 3 r P 8 9 w M r / q W N G D v + z h / g n q 5 5 K 2 X y n s b x c p 2 P u o J L G E Z q z T P T V S w i y p q 5 H 2 B e z z g 0 T q 0 r q 0 b 6 / Y 9 1 R r K N Y v 4 s q y 7 N / e K l z A = < / l a t e x i t > |c, #i  < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 0 j u

II. METHOD I
In this and the following two sections, we introduce our methods for the manipulation of SEI in detail.For clearance, we take the system of two 171 Yb(I = 1/2) atoms as an example.The generalization of our methods for atoms of other species is straightforward.
Our method I is based on the strong spin-dependent AC-Stark effect of P 2 states.In the following, we first introduce this effect and then show how to use this effect to control the SEI.

A. AC-Stark Shifts and Heating Effects of 3 P2 States
As shown in Fig. 1(a), in our scheme a π-polarized laser beam is applied at a zero magnetic field, so that the 3 P 2 (F = 3/2) states are far-off resonantly coupled to the excited states (e.g., the 3 S 1 and 3 D 1,2,3 states).Explicitly, all of the detunings of this beam with respect to the transition to the excited states [37] are much larger than the fine splitting of these states.As a result, the energies of the 3 P 2 (F = 3/2) states are shifted via the AC-Stark effect.We denote the 3 P 2 (F = 3/2) states with m F = −1/2(+1/2) and m F = ±3/2 as |c, ↑ (↓) and |c, ±3/2 , respectively.It is clear that AC-Stark shifts ( We further define the difference between the AC-Stark shifts of states |c, ↑ (↓) and |c, ±3/2 as (Fig. 1(a)): Here we emphasize that, the spin-dependence of the AC-Stark effect for the 3 P 2 levels of an alkaline-earth (like) atom is much more significant than the one of the electronic ground states of an ultracold alkali atom.As a result, to induce a given ∆ AC , the heating effect of the laser beams for our system is much lower than the ones for the alkali atoms.
This can be explained as follows.As mentioned before, here we consider the large-detuning cases where the detuning of the laser is much larger than the fine splitting of the electronic excited states.For the electronic ground manifold of an alkali atom, all the spin levels are in the same electronic orbit state, i.e., the S-state, and thus have the same dynamical polarizability.Therefore, the spindependence of the AC-Stark effect is essentially induced by the electronic spin-orbit coupling of the excited states [38,39].Thus, to realize significant spin-dependence AC-Stark shifts in the large-detuning cases one has to apply an extremely strong beam, and thus the heating effect would be quite large.However, for an alkaline-earth (like) atom the electronic orbit states corresponding to the 3 P 2 levels |c, ξ (ξ =↑, ↓, ±3/2) are different from each other.Precisely speaking, there are three electronic orbit P-states that are orthogonal with each other, and the level |c, ξ corresponds to a ξ-dependent probability mixture of these three orbital states.As a result, these levels have different dynamical polarizability.Therefore, even in the large detuning cases, one can still realize very different AC-Stark shifts for these levels with weak laser beams, and thus the corresponding heating effects can be much weaker.
The above discussions yield that for our system one can realize a very large AC-Stark shift difference ∆ AC together with a long lifetime.To illustrate this, we calculate ∆ AC and the photon scattering rate Γ sc which describes the heating effect, for various cases.The details of this calculation are given in Appendix A 1, and the results are shown in Fig. 2. In the calculation we take into account the contributions from the excited states 3 S 1 and 3 D 1,2,3 , which are mostly close to the 3 P 2 levels.In Fig. 2 cases where the detuning ∆ of the laser beams with respect to the 3 P 2 -3 S 1 transition (Fig. 1(a)) takes various values.It is shown that for ∆ = (2π)3.3× 10 14 Hz (corresponding to laser wavelength 5.08µm) we have Γ sc ∼Hz when ∆ AC ∼ (2π)MHz.If we estimate the lifetime of the ultracold gas as 1/Γ sc , then this result yields that the lifetime of our system can be hundreds of milliseconds.In Fig. 2(b) we further show the ratio Γ sc /∆ AC as a function of ∆ or the laser wavelength λ L .Since Γ sc is always positive, the sign of this ratio is the same as the one of ∆ AC .In our scheme, ∆ AC is required to be tuned to be positive.In addition, the divergence of Γ sc /∆ AC for ∆ ≈ (2π)1.85× 10 14 Hz is because that we have ∆ AC = 0 for this particular case.The divergences of Γ sc /∆ AC for the other four special values of ∆ (including ∆ = 0) shown in Fig. 2(b) are due to the resonance between the laser and the transitions from 3 P 2 levels to the 3 S 1 and 3 D 1,2,3 states for these cases.

B. Effective Inter-Atomic Interaction
Our scheme is to control the effective SEI of two atoms in the states |g, ↑ (↓) and |c, ↑ (↓) , respectively (Fig. 1(b)), with |g ↑ (↓) being defined as the 1 S 0 states with m F = −1/2(+1/2).In this subsection, we derive the form of the effective interaction between these two atoms.To this end, we consider the s-wave scattering of these two atoms in the zero-energy limit, and perform discussions in the first-quantization formalism with the two atoms being labeled as 1 and 2. In the scattering process these two atoms are incident from the sub-space spanned by the following four states: Notice that since the atoms are identical fermions, the s-wave scattering occurs only when they are in antisymmetric internal states.Furthermore, during the scattering process the inter-atomic interaction can couple the states in Eq. ( 4) to the states [40] Taking into account these states, we can express Hamiltonian for the two-atom relative motion and internal states as where µ and r are the reduced mass and relative position of the two atoms, respectively, and V (2) (r) is the projection of the interaction potential between one atom in the 1 S 0 state and another atom in the 3 P 2 (F = 3/2) states, respectively, on the subspace spanned by the states in Eqs.(4,5).The explicit form of V (2) (r) is given in Appendix B.Moreover, in Eq. ( 6) the operator Ĥf is the free Hamiltonian of the internal state of the two atoms, which is given by where the free energy of the states |g, σ; c, σ (σ, σ =↑, ↓) is chosen to be zero.Here we ignore the electronic 3 P 0,1 and 3 P 2 (F = 5/2) states, because the energy differences between these states and the ones relevant to our proposal are very large.In summary, there are four open channels (i.e., the channels corresponding to |g, σ; c, σ with σ, σ =↑, ↓) and four closed channels (i.e., the channels corresponding to |g, σ; c, ξ with σ =↑, ↓ and ξ = ±3/2) with the potential of each channel and the coupling between different channels all being determined by the interaction V (2) (r).
Furthermore, as shown in Appendix B, the interaction potential V (2) (r) is anisotropic, and can couple the wave functions with the angular momentum of two-atom relative motion being l and l+2.Nevertheless, the projection of the total angular momentum along the z-axis is conserved in the scattering process, where m F 1,2 is the magnetic quantum number of the atoms 1,2, and m l is the z-component of the angular momentum of two-atom relative motion.Using this fact and other properties of V (2) (r) we find that in the zero-energy limit if the two atoms were incident from one of the following four states, i.e., the polarized states |g, ↑; c, ↑ and |g, ↓; c, ↓ , and the anti-polarized states |± defined by then there is only elastic scattering, i.e., the two-atom internal state cannot be changed by the scattering process.We denote |Ψ σ,σ (r) (σ =↑, ↓) as the zero-energy scattering wave functions corresponding to the polarized incident state |g, σ; c, σ , and |Ψ ± (r) as the ones for the incident state |± defined in Eq. ( 9).The above analysis yields with a ± and a f being the corresponding elastic scattering lengths.Notice that the scattering lengths for the polarized incident states |g, ↑; c, ↑ and |g, ↓; c, ↓ are the same, because our system is invariant under the reflection with respect to the x − y plane.When the atoms are incident from a superposition of the four special states |g, ↑; c, ↑ , |g, ↓; c, ↓ and |± , the scattering state would be the corresponding superposition of the ones in Eqs.(10)(11)(12)(13), and thus the scattering amplitudes can be expressed in terms of a ± and a f .In particular, when the atoms are incident from the antipolarized state |g, ↓; c, ↑ or |g, ↑; c, ↓ , the corresponding scattering wave function and Eqs. (14,15) yield that the spin-exchange scattering process can occur, and the amplitude of spin-exchange is just (a − − a + )/2.According to the above discussion, the low-energy scattering between these two atoms can be described by the pseudo potential where is the projection operator of the polarized states.
In addition, we can treat the electronic states 1 S 0 (g) and 3 P 2 (F = 3/2)(c) as the labels of the two atoms, and treat the g-and c-atoms as two distinguishable particles.Furthermore, both of these two atoms are particles with pseudo-spin 1/2, because in the open channel each atom has two possible magnetic quantum numbers ↑ and ↓.In this treatment, one can express the effective two-atom Hamiltonian in the form mentioned in Sec.I, i.e., Here p S(P ) is the momentum operator of the g-(c-) atom, and p 2 S /(2m) + p 2 P /(2m) is the pseudo-spin independent free Hamiltonian, and is the effective inter-atomic interaction which is equivalent to the one of Eq (17).Here σ(S(P )) x,y,z are the Pauli operators of the pseudo spin of the g-atom (c-atom), and the coefficients A x,y,z,0 are given by Furthermore, as shown before, the order of magnitude of ∆ AC can be as large as (2π)MHz, with the photon scattering rate Γ sc being only of the order of Hz.On the other hand, the characteristic energy corresponding to the length scale of the inter-atomic van der Waals interaction potential, i.e., the van der Waals energy E vdW , is also of this order for Yb atoms [41].Therefore, if the closed channels have d-wave bound states with binding energy |E ↑ | comparable or less than E vdW , by tuning ∆ AC via the laser intensity one can make the open channels to be near-resonant to the closed-channel bound states, i.e., realize Feshbach resonances, while keeping the heating effect being low enough.At each resonance point, one of the three scattering lengths a ±,f diverges.Around the resonances, one can efficiently manipulate a ±,f (or the interaction parameters A x,y,z,0 ).That is the basic principle of this method.

D. Illustration with Multi-Channel Square-Well Model
We illustrate our approach with a calculation for the scattering lengths a ±,f and the interaction parameters A x,y,z,0 defined above.As shown in Appendix B, the potential V (2) (r) can be expressed as where r = |r| and r = r/r.Here V 1,...,6 (r) are the potential curves corresponding to six different electronic states.
These electronic states are defined in Appendix B, together with the operators D1,...,6 (r).As mentioned before, we do not know the parameters of the potential curves V (2) 1,...,6 (r).Therefore, we can only qualitatively illustrate our proposal with a multi-channel square-well model with V (2)  η (r) = −U (2)  η θ(b − r), (r ≥ 0; η = 1, ..., 6), (24) where θ(x) is the step function satisfying θ(x) = 1 for x ≥ 0 and θ(x) = 0 for x < 0. In our calculation, we choose b = 85a 0 , with a 0 being the Bohr radius and 2b taking a typical value of the length scale β 6 associated with the van der Waals interaction between atoms as heavy as Ytterbium [42].We consider all the involved swave and d-wave channels, and ignore the channels with higher relative angular momentum, for simplicity.We also ignore the centrifugal potential of the d-wave channels in the region r < d because the square-well potentials in this region are very deep.
We display the results for two cases in Fig. 3.In Table I of Appendix C we show the value of the potential depth U (2) η (η = 1, ..., 6) as well as the s-wave scattering length a (2) η corresponding to a single-channel square-well potential V (2) η (r) given in Eq. ( 24).In Fig. 3 we illustrate the behaviors of both the scattering lengths a ±,f and the interaction parameters A x,y,z,0 defined in Eqs.(20)(21)(22) as functions of ∆ AC .It is shown that using the resonances one can tune the intensity A x = A y of the effective SEI as well as the intensities A z,0 of the spinnon-exchanging and spin-independent interaction in a broad region e.g., between −1000a 0 and 1000a 0 through the laser intensity, and may prepare the effective interatomic interaction Veff to be either "anti-ferromagnetic like" (A x = A y > Max[0, −A z ]) with the lowest eigen state being the singlet state |+ defined in Eq. ( 9), or "ferromagnetic like" (A z < 0 and with the lowest eigen states being the polarized ones |g, ↑; c, ↑ and |g, ↓; c, ↓ .It is also possible to "turn off" the spin-independent interaction by tuning A 0 = 0 or prepare the system to other required interaction parameter regions.In Appendix D we show the values of ∆ AC under which we have A 0 = 0 for the two cases in Fig. 3, as well as the corresponding values of A x,y,z and the scattering lengths a ±,f .

III. METHOD II
Now we introduce our second approach for the manipulation of SEI.As above, we also take the system of two 171 Yb atoms as an example.In addition, to avoid using too many different symbols, in this and the next section, we will use some notations which have already been used in Sec.II, for the clear-enough cases (e.g., we still use Γ sc to denote the photon scattering rate of laser beams for the current scheme).The exact definitions of these notations for this section will all be given in the following discussions.

Method II (a) r t k p i v 6 5 s 6 n q h Q 5 J M R p 3 K a 4 J M y N 8 q P P j t G k p n b d W 2 b i L y Z T s 3 r P 8 9 w M r / q W N G D v + z h / g n q 5 5 K 2 X y n s b x c p 2 P u o J L G E Z q z T P T V S w i y p q 5 H 2 B e z z g 0 T q 0 r q 0 b 6 / Y 9 1 R r K N Y v 4 s q y 7 N / e K l z A = < / l a t e x i t > |c, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " h X P I H t 7 u e 9 i 4 k J e V r w J e i y R n P T f g V b d 7 q O P V C y 5 j X 4 S n a h D x s x 7 r h H 7 b 9 5 g i q p l d u + 5 s N     Method II is a generalization of method I.In this approach two laser beams are applied, which are polarized along the z-direction and the the x-direction, respectively (Fig. 4(a)).Both of these two beams are far-off resonant for the transition from the 3 P 2 (F = 3/2) level to the excited states (i.e., the detunings are much larger than the Rabi frequencies of the transitions and the natural linewidth of the excited states), similar as in Sec.II.The frequency difference of these two beams is not required to take any certain value.The only requirement for this frequency difference is that it should be much larger than the AC-Stark shifts induced by each beam so that the two beams induce the second-order effects for the 3 P 2 (F = 3/2) level independently.In this case, the total effect of these two beams is not equivalent to the one of a single beam polarized along some direction between the x-and z-axis.

Method II
As in Sec.II A, the beam polarized along the z-direction can induce spin-dependent AC-Stark shifts for the states in the 3 P 2 (F = 3/2) level.Meanwhile, the beam polarized along the x-direction can be decomposed into two beams with σ + and σ − polarizations, respectively, and thus induces Raman coupling between the 3 P 2 (F = 3/2) state with m F = −3/2(+3/2) and the one with m F = 1/2(−1/2) (Fig. 4(b)).As a result, four dressed states |q(h), ↑ (↓) can be formed (Fig. 4 .The energy gap ∆ hq between these higher and lower dressed states (Fig. 4(c)) is also derived in Appendix A 2, where we find that ∆ hq can be expressed as where ∆ is the AC-Stark shift difference only induced by the laser beams polarized along the z-(x-) direction, as defined in Sec.II.Therefore, ∆ hq has the same order of magnitude with the AC-Stark shift difference ∆ (z(x)) AC , as mentioned in Sec.I. Furthermore, the total photon scattering rate Γ sc of these two beams, which describes the heating effects, can also be calculated directly (Appendix A 2).In Fig. 5(b) we show Γ sc as a function of ∆ hq for a typical case.It is shown that we have Γ sc ∼Hz when ∆ hq ∼ (2π)MHz.Therefore, the laser-induced heating effect is weak, which is similar to the one in Sec.II.
Furthermore, similar as in Sec.II, our method is to control the SEI between two atoms in the 1 S 0 state (g-state) and the lower dressed state (q-state) (Fig. 4(d)).To this end, we consider the scattering processes in the zeroenergy limit, with incident states being in the Hilbert space spanned by the states: Method II In addition, the inter-atomic interaction V (2) (r) couples these open channels corresponding to the above four states to the closed channels corresponding to (σ, σ =↑, ↓; ).(31) The energy gap between the above open and closed channels is just ∆ hp , as shown in Fig. 4(e).In addition, using an analysis based on the properties of the interaction potential V (2) (r), which is similar to the discussion in Sec.II B, we find that in the zeroenergy limit if the two atoms were incident from one of the following four states: then in the zero-energy limit there are only elastic scattering processes in which the two-atom internal state is not changed.We denote the elastic scattering lengths with respect to the incident states |± and |p ± as a ± and a p± , respectively.Notice that for the current system it is possible that a p+ = a p− , i.e., the spin-change processes |g, ↑, q; ↑ ⇔ |g, ↓; q, ↓ are also permitted.Therefore, the low-energy interaction between these two atoms can be described by the pseudo potential Moreover, by treating the atoms in |g, ↑ (↓) and |q, ↑ (↓) as two distinguished particles with pseudo-spin 1/2, one can express the effective two-atom Hamiltonian in the form of Sec.I, i.e., H Here p S(P ) is the momentum operator of the g-(q-) atom and p 2 S /(2m) + p 2 P /(2m) is the pseudo-spin independent free Hamiltonian, and the effective inter-atomic interac- is equivalent to the one of Eq. (34), with σ(S(P )) x,y,z being the Pauli operators of the pseudo spin of the g-(q-) atom.In Eq. (35) the coefficients A x,y,z,0 are related to the scattering lengths a ± and a p± via Notice that in the current system the interaction parameters A x and A y may be unequal.Similar to before, by changing the intensities of the two laser beams, one can tune the energy gap ∆ hp and induce Feshbach resonances.The scattering lengths a ± and a p± or the interaction parameters A x,y,z,0 can be efficiently manipulated via these Feshbach resonances.We illustrate these resonances via the multi-channel squarewell model used in Sec.II D, with width b = 85a 0 and other parameters being given in Table II of Appendix C. The results are shown in Fig. 6.
We emphasis that there is an important difference between the current approach and method I.For the system of method I, the s-wave states of the open channels are coupled only to the d-wave states of the closed channels, as mentioned in the above sections.However, for our current system, the s-wave open-channel states are coupled to both the d-wave and the s-wave states of the closed channels.As discussed in Sec.I, this fact implies that the Feshbach resonances for a ± with a low-enough heating rate are much more possible to appear for realistic systems.That is an important advantage of the current method.

IV. METHOD III
Now we introduce our third approach for the manipulation of SEI, which is based on the 3 P 2 -3 P 0 Raman coupling, by taking 171 Yb atoms as an example.As before, we will use some notations which have been used in Sec.II, to reduce the number of different symbols.The exact definitions of these notations for this section will be given below.
! " (↓) |d, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " I A N c 3 j w o 6 y a A s k 0 X 0 D U h T t x 6 w + 4 1 R 8 S / 0 D / w j t j B B + I T k h y 5 t x 7 z s y 9 1 4 0 C P 1 a O 8 5 y x x s Y n J q e y 0 7 m Z 2 b n 5 h n 0 2 t i X T t q r d M x 1 9 1 p m L V 3 k l y Y 7 y p W 9 K A i z / H O Q l q p U J x p 1 A 6 3 s 2 X D 5 J R p 7 G O D W z R P P d Q x h E q q J L 3 J R 7 w i C f j 1 L g x b o 2 7 j 1 Q j l W j W 8 G 0 Z 9 + 8 j e Z d C < / l a t e x i t > |u, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " k I b h P a < l a t e x i t s h a 1 _ b a s e 6 4 = " / a u u 0 c v X r F 8 9 x c u u e m K v r n 9 q S p F D i l x G j c p L g n 7 R t n t s 2 0 0 m a l d 9 5 a Z + K v J 1 K z e + 0 V u j j d 9 S x p w 9 f s 4 f 4 L 9 p U p 1 u b K 0 u 1 L e 3 C p G P Y R Z z G G B 5 r m K T e y g h j p 5 X + M B j 3 i y r q w b 6 9 a 6 + 0 i 1 e g r N D L 4 s 6 / 4 d a l a p M Q = = < / l a t e x i t > ( , 0 =", #; cd < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 F a q B 6 K S s q p r N F k u j T R p n t V p 5  A. Low-Heating Raman Coupling between 3 P0 and 3 P2 Levels As shown in Fig. 7(a), in the current method two πpolarized laser beams α and β are applied at a zero magnetic field so that the 3 P 0,2 levels are far-off resonantly coupled to the excited states.These two beams can induce a Raman coupling between the states | 3 P 0 , 1/2, σ and | 3 P 2 , 3/2, σ , with σ = −1/2(↑) or 1/2(↓).Furthermore, the frequency difference of these two beams is tuned to compensate for the difference of the AC-Stark shifts of these states, so that this Raman coupling is resonant.Explicitly, the fluctuation of this frequency difference should be much less than the effective Rabi frequency Ω eff of the Raman coupling, which is of the order of (2π)MHz as shown below.
heating" Feshbach resonances is quite possible for realistic systems, and does not require the scattering lengths of the bare inter-atomic interaction potentials V (0) (r) and V (2) (r) to be very large.For instance, as shown in Tables II and III of Appendix C, in our calculations with multi-channel square-well models for these two methods for 171 Yb atoms, we set the scattering lengths a (0) ± for the potential V (0) (r) to be 232a 0 and 372a 0 , which are reported by the references [6][7][8]32], and set all the other scattering lengths in our interaction model to be less than 200a 0 .As illustrated in Fig. 6 and Fig. 9, Feshbach resonances with low heating rates can appear for these cases.
In the above sections, we take ultracold 171 Yb atoms as an example.Our methods are also possible to be applicable for other types of fermionic alkaline-earth (like) atoms, e.g., 173 Yb [43] or 87 Sr atoms [44].
At the end of this paper, we give the following comments for these methods: (I): Our above analysis as well as the illustrations with the multi-channel square-well models just show it is quite possible to realize Feshbach resonances with the laserinduced heating rate being of the order of Hz or even lower.Nevertheless, for realistic systems there does exist the possibility that the resonances only occur in the regions with the heating rate being larger than 10Hz, since sometimes the binding energy of the shallowest bound state of a van der Waals interaction potential can be larger than E vdW by one order of magnitude.As mentioned above, unfortunately, we cannot make quantitative predictions for realistic systems with specific atoms, due to the lack of detailed parameters of the interaction potential.The positions and widths of the Feshbach resonances for specific atoms, as well as the significance of the laser-induced heating in the resonance region, should be examined via experiments or multi-channel numerical calculations with accurate inter-atomic interaction potentials.(Notice that our system is complicated because there are many degenerate closed channels that are coupled with each other.Thus, for a specific type of atom one cannot predict if resonances can appear for a certain parameter region (e.g., the regions with ∆ AC < (2π)10MHz or Ω eff < (2π)10MHz) even with the analysis based on the theory of single-channel van der Waals potential.)(II): Here we can make a brief comparison for these three methods.Method I is the most simple one because only one laser beam is required.Method II is a little bit complicated because it requires two laser beams.Nevertheless, as shown in Sec.III, the frequency difference of these two beams is not required to be locked to a certain value.Thus, this method is still easier to be realized than the usual Raman schemes.Method III is the most complicated one because the atoms are in a superposition state of 3 P 2 and 3 P 0 levels, and the frequency difference of the two Raman beams should be fixed.Fortunately, since the effective Rabi frequency Ω eff of our system is as large as (2π)MHz, the fluctuation of this frequency difference is required only to be much less than this order.This requirement can be realized in current experiments.
Moreover, in methods II and III the 3 P atoms should be prepared in the lower dressed states |q, σ or |d, σ (σ =↑, ↓).The preparation and detection of the dressed states may induce imperfections and complications for the experiments.More discussions for the detection of the dressed states are given in the following point (III).
On the other hand, by comparing Fig. 2, Fig. 5, and Fig. 8 we find that, for the realization of a fixed energy gap between the open and closed channels via lasers with given one-photon detuning, the photon scattering rate Γ sc of the laser beams of method III is usually lower than the ones of methods I and II.This result implies that method III has the weakest heating effect.This is an important advantage of this method.
Finally, as discussed before, the possibility for the appearance of Feshbach resonances with a low-enough heating rate is higher for the systems with method II and III, due to the coupling between s-wave states of the open and closed channels.In addition, since the potential curves involved in the closed channels of these three methods are quite different, for a specific system it is possible that this kind of resonances cannot be realized via one method, but can be realized via another method.
(III) Here we discuss the measurement of the dressed states of methods II and III.Due to the energy conservation, after the scattering processes the atoms would return to the open channels with lower dressed states |q, ↑ (↓) (method II) or |d, ↑ (↓) (method III).Therefore, in most cases, only these lower dressed states should be detected.According to Eqs. (26,28), |q, ↑ is the superposition of the 3 P 2 (F = 3/2) states with m F = −1/2 and +3/2, while |q, ↓ is the superposition of states with m F = 1/2 and −3/2.Therefore, one can use a Stern-Gerlach experiment to detect the number of atoms for each m F , and then derive the populations of |q, ↑ and |q, ↓ from these atom numbers.Similarly, since the states |d, ↑ (↓) are the 3 P 2 -3 P 0 dressed states with m F = −1/2(+1/2), one can use a Stern-Gerlach experiment to detect the atom numbers for the 3 P 2 (m F = −1/2) and 3 P 2 (m F = +1/2) states, and then derive the populations of |d, ↑ and |d, ↓ from these atom numbers.
(IV): In this work we perform the calculations in the zero-energy limit.For realistic systems, one may require to consider the effect induced by the finite incident momentum k.When k is finite there may be new spin-change scattering processes, e.g., the process |g, ↑; ξ, ↑ ⇔ (|g, ↑; ξ, ↓ + |g, ↓; ξ, ↑ )/ √ 2, with ξ = c for method I and ξ = d for method III.These processes are induced by the fact that the interaction potential V (2) (r) is anisotropic.However, the scattering amplitudes of these processes are proportional to k 2 and thus can be ignored when k is low enough, because the dwave motional states are involved.More importantly, direct analysis based on the symmetry of the interaction potential shows that the interaction potential can never couple the singlet state to the triplet states of the twoatom pseudo-spin we defined before, even in the finite-k cases.Therefore total pseudo-spin of the two atoms is always conserved, and the singlet state is always a nondegenerate eigen state of the effective inter-atomic interaction.
Appendix B, for our system the inter-atomic interaction potential is anisotropic, and couples the wave functions with relative angular momentum l and l+2.On the other hand, the z-component of the total angular momentum, i.e., M ≡ mF 1 + mF 2 + m l , is conserved, with mF j (j = 1, 2) being the magnetic quantum number of the atom j and m l being the z-component of the angular momentum of two-atom relative motion.The internal-state channels |g, σ; c, σ and |g, σ; c, ξ (σ, σ , σ =↑, ↓, ξ = ±3/2) are coupled according to this selection rule.For instance, the s-wave state of the channel |g, ↑; c, ↓ is coupled to the d-wave state with m l = −2 of the channel |g, ↓; c, 3/2 ..
[41] For two 171 Yb atoms, the length scale associated with the van der Waals interaction is β6 ≈ 155a0 when both of them are in 1 S0 state, and β6 ≈ 168a0 when the two atoms are in 1 S0 and 3 P0 states, respectively [9,36].Therefore, we have E vdW ≈ (2π)2.5MHzand E vdW ≈ (2π)3.0MHzfor these two cases, respectively.For the interaction between two 171 Yb atoms in 1 S0 and 3 P2 states, we do not know the precise value of β6 and E vdW .Nevertheless, according to the above two facts it is reasonable to estimate that for this case E vdW is also of the order of (2π)MHz.
[42] Following the previous studies for the magnetic Feshbach resonances of alkaline atoms with square-well models, we choose the width of the square well as b = ā, where ā ≡ 2π/Γ[1/4] 2 β6 is the "van der Waals length scale" [48].
For the 1 S0-3 P0 interaction of two 171 Yb atoms, it was found that β6 ≈ 168a0 [9] and thus ā ≈ 81a0.However, for the 1 S0-3 P2 interaction of two 171 Yb atoms, we do not know the value of β6 or ā.We estimate ā for this case to be 85a0, and thus choose b = 85a0 for methods I and II.In addition, in the system of method III, one atom is in a 3 P0-3 P2 dressed state and the other one is in the 1 S0.The inter-atomic interaction potential is different with both the 1 S0-3 P0 potential and the 1 S0-3 P2 potential, and the value of ā is unknown.In the corresponding calculations, for simplicity, we just choose b = 81a0.Here we emphasis that the qualitative results (i.e., the order of magnitude of the laser intensities with which the resonances can occur ) of our calculations are the same when b is around the values used in our manuscript..
t e x i t s h a 1 _ b a s e 6 4 = " z 9 D M v v h N e K 7 H T s X + Y T C 7 O d T o r F Q = " >A A A C y X i c j V H L S s N A F D 2 N r 1 p f V Z d u g k V w V Z I q 6 L K o C 8 F N B f u A t k i S T u v Y y c N k I t b i y h 9 w q z 8 m / o H + h X f G C G o R n Z D k z L n 3 n J l 7 r x s J n k j L e s k Z U 9 M z s 3 P 5 + c L C 4 t L y S n F 1 r Z G E a e y x u h e K M G 6 5 T s I E D 1 h d c i l Y K 4 q Z 4 7 u C N d 3 h o Y o 3 r 1 m c 8 D A 4 k 6 O I d X 1 n E P A + 9 x x J V K N z x I R 0 z o s l q 2 z p Z U 4 C O w M l Z K s W F p / R Q Q 8 h P K T w w R B A E h Z w k N D T h g 0 L E X F d j I m L C X E d Z 7 h D g b Q p Z T H K c I g d 0 n d A u 3 b G B rR X n o l W e 3 S K o D c m p Y k t 0 o S U F x N W p 5 k 6 n m p n x f 7 m P d a e 6 m 4 j + r u Z l 0 + s x A W x f + k + M / + r U 7 V I 9 L G v a + B U U 6 Q Z V Z 2 X u a S 6 K + r m 5 p e q J D l E x C n c o 3 h M 2 N P K z z 6 b W p P o 2 l V v H R 1 / 1 Z m K V X s v y 0 3 x p m 5 J A 7 Z / j n M S N C p l e 6 d c O d 0 t V Q + y U e e x g U 1 s 0 z z 3 U M U x a q i T 9 y U e 8 I g n 4 8 S 4 M m 6 M 2 4 9 U I 5 d p 1 P 2 l L 5 d 2 r U y N q K 9 8 U y s m t M p I b 2 K l C 5 W S C M p T x E 2 p 7 k 2 n l p n w / 7 m P b C e 5 m 7 n 9 P c z r x 6 x G i f E / q X 7 y P y v z t S i c Y w t W 0 N A N c W W M d X x z C W 1 X T E 3 d z 9 V p c k h J s 7 g D s U V Y W 6 V H 3 1 2 r S a x t Z v e M h t / s Z m G N X u e 5 a Z 4 N b e k A Z e / j / M n q F d K 5 f V S Z X + j u L 2 T j T q P J S x j l e a 5 i W 3 s o Y o a e V / g H g 9 4 d F r O t X P j 3 L 6 n O r l M s 4 A v y 7 l 7 A 0 e Z m B c = < / l a t e x i t > |c, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " R K X f d 8 G + J + j / C r i e F N v s s 8 X I / o M = " > A A A C 2 X i c j V H L S s N A F D 2 N r / q u j 5 2 b Y B F c S E m r o E v R j c s K 9 g G t y G Q 6 1 m C a C Z O J R a s L d + L W H 3 C r P y T + g f 6 F d 8 Y I P h C d k O T M u f e c m X u v H 4 d B o j 3 v O e c M D Y + M j u X H J y a n p m d m C 3 P z 9 U S m i o s a l 6 F U T Z 8 l I g w P 2 l L 5 d 2 r U y N q K 9 8 U y s m t M p I b 2 K l C 5 W S C M p T x E 2 p 7 k 2 n l p n w / 7 m P b C e 5 m 7 n 9 P c z r x 6 x G i f E / q X 7 y P y v z t S i c Y w t W 0 N A N c W W M d X x z C W 1 X T E 3 d z 9 V p c k h J s 7 g D s U V Y W 6 V H 3 1 2 r S a x t Z v e M h t / s Z m G N X u e 5 a Z 4 N b e k A Z e / j / M n q F d K 5 f V S Z X + j u L 2 T j T q P J S x j l e a 5 i W 3 s o Y o a e V / g H g 9 4 d F r O t X P j 3 L 6 n O r l M s 4 A v y 7 l 7 A 0 e Z m B c = < / l a t e x i t > |c, 3 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " o j G 7 V O 5 I Y + / d f c 7 N I D e X l 5 u 0 + L M = " > A A A C 1 3 i c j V H L S s N A F D 3 G V 3 3 H u n Q T L I I L L W k r 6 L L o x m U F W 5 V W Z D J O a z B N w m Q i l i r u x K 0 / 4 F b / S P w D / Q v v j C m o R X R C k j P n 3 n N m 7 r 1 e H P i J c t 3 X E W t 0 b H x i M j c 1 P T M 7 N 7 9 g L + Y b S Z R K L u o 8 C i J 5 5 L F E B H 4 o 6 s p X g T i K p W B d L x C H 3 s W u j h 9 e C p n 4 U X i g e r E 4 6 b J O 6 L d 9 z h R R p 3 b + m q 9 v t N q S 8 U q 5 J V n Y C c S p X X C L r l n O M C h l o I B s 1 S L 7 B S 2 c I Q J H i i 4 E Q i j C A R g S e p o o w U V M 3 A n 6 x E l C v o k L 3 G C a t C l l C c p g x F 7 Q t 0 O 7 Z s a G t N e e i V F z O i W g V 5 L S w S p p I s q T h P V p j o m n x l m z v 3 n 3 j a e + W 4 / + X u b V J V b h n N i / d I P M / + p 0 L Q p t b J s a f K o p N o y u j m c u q e m K v r n z p S p F D j F x G p 9 R X B L m R j n o s 2 M 0 i a l d 9 5 a Z + J v J 1 K z e 8 y w 3 x b u + J Q 2 4 9 H O c w 6 B R L p Y q x f L + Z q G 6 k 4 0 6 h 2 W s Y I 3 m u Y U q 9 l B D n b y v 8 I g n P F v H 1 q 1 1 Z 9 1 / p l o j m W Y J 3 5 b 1 8 A H W J J Z O < / l a t e x i t > |c, i < l a t e x i t s h a 1 _ b a s e 6 4 = " o j G 7 V O 5 I Y + / d f c 7 N I D e X l 5 u 0 + L M = " > A A A C 1 3 i c j V H L S s N A F D 3 G V 3 3 H u n Q T L I I L L W k r 6 L L o x m U F W 5 V W Z D J O a z B N w m Q i l i r u x K 0 / 4 F b / S P w D / Q v v j C m o R X R C k j P n 3 n N m 7 r 1 e H P i J c t 3 X E W t 0 b H x i M j c 1 P T M 7 N 7 9 g L + Y b S Z R K L u o 8 C i J 5 5 L F E B H 4 o 6 s p X g T i K p W B d L x C H 3 s W u j h 9 e C p n 4 U X i g e r E 4 6 b J O 6 L d 9 z h R R p 3 b + m q 9 v t N q S 8 U q 5 J V n Y C c S p X X C L r l n O M C h l o I B s 1 S L 7 B S 2 c I Q J H i i 4 E Q i j C A R g S e p o o w U V M 3 A n 6 x E l C v o k L 3 G C a t C l l C c p g x F 7 Q t 0 O 7 Z s a G t N e e i V F z O i W g V 5 L S w S p p I s q T h P V p j o m n x l m z v 3 n 3 j a e + W 4 / + X u b V J V b h n N i / d I P M / + p 0 L Q p t b J s a f K o p N o y u j m c u q e m K v r n z p S p F D j F x G p 9 R X B L m R j n o s 2 M 0 i a l d 9 5 a Z + J v J 1 K z e 8 y w 3 x b u + J Q 2 4 9 H O c w 6 B R L p Y q x f L + Z q G 6 k 4 0 6 h 2 W s Y I 3 m u Y U q 9 l B D n b y v 8 I g n P F v H 1 q 1 1 Z 9 1 / p l o j m W Y J 3 5 b 1 8 A H W J J Z O < / l a t e x i t > |c, 3 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " E d X h P

2 7 9
A b f 6 S e I f 6 F 9 4 Z 0 z B B 6 I T k p w 5 9 5 4 z c + / 1 k z B I l e u + F K y x 8 Y n J q e J 0 a W Z 2 b n 7 B X l w 6 S u N

2 7 9
A b f 6 S e I f 6 F 9 4 Z 0 z B B 6 I T k p w 5 9 5 4 z c + / 1 k z B I l e u + F K y x 8 Y n J q e J 0 a W Z 2 b n 7 B X l w 6 S u N 9 Z S b + a j I 1 q / c 8 z 8 3 w p m 9 J A / a + j / M n O K p W v F q l e r B Z 3 t n N R 1 3 E C l a x T v P c w g 7 2 U U e D v C / w g E c 8 W U 3 r x r q 1 7 j 5 S r U K u W c a X Z d 2 / A 0 H E l h c = < / l a t e x i t > |g, "i < l a t e x i t s h a 1 _ b a s e 6 4 = " hV W Z 4 Y B G 1 N z J R A f e V V e F g D 8 w m 0 U = " > A A A C 1 3 i c j V H L S s N A F D 2 N r 1 p f t S 7 d B I v g Q k p a B V0 W 3 b i s Y B / S l j J J p z U 0 T c J k o p Y q 7 s S t P + B W / 0 j 8 A / 0 L 7 4 w R 1 C I 6 I c m Z c + 8 5 M / d e O / T c S F r W S 8 q Y m p 6 Z n U v P Z x Y W l 5 Z X s q u 5 W h T E w u F V J / A C 0 b B Z x D 3 X 5 1 X p S o 8 3 Q s H Z 0 P Z 4 3 R 4 c q n j 9 n I v I D f w T O Q p 5 e 8 j 6 v t t z H S a J 6 m R z V / 3 t V h w y I Y K L l m B + 3 + O d b N 4 q W H q Z k 6 C Y g D y S V Q m y z 2 i h i w A O Y g z B 4 U M S 9 s A Q 0 d N E E R Z C 4 t o Y E y c I u T r O c Y 0 M a W P K 4 p T B i B 3 Q t 0 + 7 Z s L 6 t F e e k V Y 7 d I p H r y C l i U 3 S B J Q n C K v T T B 2 P t b N i f / M e a 0 9 1 t x H 9 7 c R r S K z E G b F / 6 T 4 z / 6 t T t U j 0 s K 9 r c K m m U D O q H y S j T m M d G 9 i i e e 6 h j C N U U C X v S z z g E U / G q X F j 3 B p 3 H 6 l G K t G s 4 d s y 7 t 8 B A V m X N A = = < / l a t e x i t > |g, #i< l a t e x i t s h a 1 _ b a s e 6 4 = " F / Y / u / M d C F f J 8 + Q x B / Z V f w S 7 q R 4 = " > A A A C 2 X i c j V H L S s N A F D 3 G d 3 3 V x 8 5 N s A g u p K R V 0 G X R j U s F + 4 B W y i S d 1 t A 0 E y Y T S 3 0 s 3 I l b f 8 C t / p D 4 B / o X 3 h l T U I v o h C R n z r 3 n z N x 7 3 S j w Y + U 4 r 2 P W + M T k 1 P T M b G Z u f m F x K b u 8 U o l F I j 1 e 9 k Q g Z M 1 l M Q / 8 k J e Vr w J e i y R n P T f g V b d 7 q O P V C y 5 j X 4 S n a h D x s x 7 r h H 7 b 9 5 g i q p l d u + 5 s N 1 q i H z I p R b 8 h W d g J e D O b c / K O W f Y o K K Q g h 3 Q d i + w L G m h B w E O C H j h C K M I B G G J 6 6 i j A Q U T c G a 6 I k 4 R 8 E + e 4 Q Y a 0 C W V x y m D E d u n b o V 0 9 Z U P a a 8 / Y q D 0 6 J a B X k t L G J m k E 5 U n C + j T b x B P j r N n f v K + M p 7 7 b g P 5 u 6 t U j V u G c 2 L 9 0 w 8 z / 6 n Q t C m 3 s m x p 8 q i k y j K 7 O S 1 0 S 0 x V 9 c / t L V Y o c I u I 0 b l F c E v a M c t h n 2 2 h i U 7 v u L T P x N 5 O p W b 3 3 0 t w E 7 / q W N O D C z 3 G O g k o x X 9 j J F 0 9 2 c 6 W D d N Q z W M c G t m i e e y j h C M c o k / c l H v G E Z 6 t u 3 V p 3 1 v 1 n q j W W a l b x b V k P H 1 F h m B s = < / l a t e x i t > (b) < l a t e x i t s h a 1 _ b a s e 6 4 = " R K X f d 8 G + J + j / C r i e F N v s s 8X I / o M = " > A A A C 2 X i c j V H L S s N A F D 2 N r / q u j 5 2 b Y B F c S E m r o E v R j c s K 9 g G t y G Q 6 1 m C a C Z O J R a s L d + L W H 3 C r P y T + g f 6 F d 8 Y I P h C d k O T M u f e c m X u v H 4 d B o j 3 v O e c M D Y + M j u X H J y a n p m d m C 3 P z 9 U S m i o s a l 6 F U T Z 8 l I g w i U d O B D k U z V o L 1 / F A 0 / N N d E 2 + c C Z U E M j r Q 5 7 E 4 7 L F u F B w H n G m i j g q L l 3 y t 3 Z H 9 i C k l + 2 3 F o m 4 o j g p F r + T Z 5 f 4 E 5 Q w U k a 2 q L D y h j Q 4 k O F L 0 I B B B E w 7 B k N D T Q h k e Y u I O M S B O E Q p s X O A K E 6 R N K U t Q B iP 2 l L 5 d 2 r U y N q K 9 8 U y s m t M p I b 2 K l C 5 W S C M p T x E 2 p 7 k 2 n l p n w / 7 m P b C e 5 m 7 n 9 P c z r x 6 x G i f E / q X 7 y P y v z t S i c Y w t W 0 N A N c W W M d X x z C W 1 X T E 3 d z 9 V p c k h J s 7 g D s U V Y W 6 V H 3 1 2 r S a x t Z v e M h t / s Z m G N X u e 5 a Z 4 N b e k A Z e / j / M n q F d K 5 f V S Z X + j u L 2 T j T q P J S x j l e a 5 i W 3 s o Y o a e V / g H g 9 4 d F r O t X P j 3 L 6 n O r l M s 4 A v y 7 l 7 A 0 e Z m B c = < / l a t e x i t > |g, ; c, 0 i < l a t e x i t s h a 1 _ b a s e 6 4 = " y i O O T i v 4 o Z G + + c c 9 6 q 3 E 3 x p 0 6 s c 9 9 m 1 m t z W b n r L b P z O Z h r W 7 H m Z W + D e 3 J I G 3 H w 5 z t e g u 9 p o r j V W d 9 b r r e 1 y 1 B U s 4 S u W a Z 7 f 0 c J P t N E h 7 w t c 4 y / + O e f O p X P l / H 5 M d S Z K z S K e L e f P A z 4 K q R 8 = < / l a t e x i t > |g, ; c, ±3/2i

FIG. 1 .
FIG. 1. (color online) Schematic illustration of method I. (a): Single-atom energy levels and the π-polarized laser beam (blue lines with arrows).We also show the 3 P2 levels shifted by the beam.(b): A Two-atom spin-exchange process.The black (red) filled and dashed circles represent the g(c)−atom before and after a collision, respectively.Both this process and the inverse one are studied in this work.(c): The interatomic scattering channels.Here the solid curves represent the potentials of each channel.The coupling potentials between different channels are not shown in the figure.
FIG. 3. (color online) The scattering lengths a ±,f ((a), (c)) and interaction parameters Ax,y,z,0 ((b), (d)) of the system of method I.Here we show the results for two cases, which are given by the multi-channel square-well model in Sec.II D, with width b = 85a0 and other parameters being given in Table I of Appendix C.
H J m X P v O T P 3 3 i A T U a 4 8 7 7 n P 6 R 8 Y H B o e G R 0 b n 5 i c m i 7 N z O 7 m a S F D 3 g h T k c p m w H I u o o Q 3 V K Q E b 2 a S s z g Q f C / o b e r 4 3 g m X e Z Q m O + o 8 4 4 c x 6 y 8 e M u I O c U G c J B S Z O M c V x k h b U B a n D E Z s j 7 4 d 2 h 1 Y N q G 9 9 s y N O q R T B L 2 S l C 4 W S Z N S n i S s T 3 N N v D D O m v 3 N + 8 J 4 6 r u d 0 z + w X j G x C l 1 i / 9 J 9 Z P 5 X p 2 t R O M a 6 q S G i m j L D 6 O p C 6 1 K Y r u i b u 5 + q U u S Q E a f x E c U l 4 d A o P / r s G k 1 u a t e 9 Z S b + Y j I 1 q / e h z S 3 w q m 9 J A / a / j / M n 2 K 1 W / J V K d X u 1 X N u w o x 7 B P B a w R P N c Q w 1 b q K N B 3 m e 4 x w M e n X 3 n 2 r l x b t 9 T n T 6 r m c O X 5 d y 9 A Q P J l z U = < / l a t e x i t > |h, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " C l H 1 O 2 s i 5 g T l t / f n f 8 J j c I z m y PE = " > A A A C 2 X i c j V H L S s N A F D 2 N r / q O j 5 2 b Y B F c S E m r o E v R j c s K V o W 2 l E k 6 b U P T T J h M F K 0 u 3 I l b f 8 C t / p D 4 B / o X 3 h l T 8 I H o h C R n z r 3 n z N x 7 v T g M E u W 6 L z l r Z H R s f C I / O T U 9 M z s 3 b y 8 s H i c i l T 6 v + i I U 8 t R j C Q + D i F d V o E J + G k v O + l 7 I T 7 z e v o 6 f n H G Z B C I 6 U h c x b / R Z J w r a g c 8 U U U 1 7 + a q 7 U W + J 8 4 h J K c 7 r k k W d k D f t g l t 0 z X J + g l I G C s h W R d j P q K M F A R 8 p + u C I o A i H Y E j o q a E E F z F x D Q y I k 4 Q C E + e 4 x h R p U 8 r i l M G I 7 d G 3 Q 7 t a x k a 0 1 5 6 J U f t 0 S k i v J K W D N d I I y p O E 9 W m O i a f G W b O / e Q + Mp 7 7 b B f 2 9 z K t P r E K X 2 L 9 0 w 8 z / 6 n Q t C m 3 s m B o C q i k 2 j K 7 O z 1 x S 0 x V 9 c + d T V Y o c Y u I 0 b l F c E v a N c t h n x 2 g S U 7 v u L T P x V 5 O p W b 3 3 s 9 w U b / q W N O D S 9 3 H + B M f l Y m m z W D 7 c K u z u Z a P O Y w W r W K d 5 b m M X B 6 i g S t 6 X e M A j n q y a d W P d W n c f q V Y u 0 y z h y 7 L u 3 w F T 0 5 g c < / l a t e x i t > |q, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " W 3 5 t C v O G t R 7 l X B M e G U s 5 P Y j e T v 8 = " j l e a 5 i W 3 s o Y o a e V / g H g 9 4 d F r O t X P j 3 L 6 n O r l M s 4 A v y 7 l 7 A 0 e Z m B c = < / l a t e x i t > |c, 3 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = " Y O P 9 r p H y n e 9 S 1 p w N 7 P c Q 6 D a q n o l Y u l o 6 3 C 7 l 4 + 6 i m s Y g 0 b N M 9 t 7 O I A h 6 i Q 9 y U e 8 Y R n 6 8 S 6 t e 6 s + 8 9 U a y T X r O D b s h 4 + A L f m l k E = < / l a t e x i t > |g, "i < l a t e x i t s h a 1 _ b a s e 6 4 = "

hq < l a t e x i t s h a 1 _
b a s e 6 4 = " C b G / 3 k + 8 q 3 f H P p f J n 2 F d I 4 K H z G U = " > A A A C z n i c j V H L T s J A F D 3 U F + I L d e m m k Z i 4 I i 2 a 6 J K o C 5 e Y y C M B Q t o y w I S h r e 2 U h B D i 1 h 9 w q 5 9 l / A P 9C + + M J V G J 0 W n a n j n 3 n D t z 7 3 V D w W N p W a 8 Z Y 2 l 5 Z X U t u 5 7 b 2 N z a 3 s n v 7 t X i I I k 8 V v U C E U Q N 1 4 m Z 4 D 6 r S i 4 F a 4 Q R c 0 a u Y H V 3 e K n i 9 T G L Y h 7 4 t 3 I S s v b I 6 f u 8 x z 1 H E t V s X T E h n c 5 0 c D f r 5 A t W 0 d L L X A R 2 C g p I V y X I v 6 C F L g J 4 S D A C g w 9 J W M B B T E 8 T N i y E x L U x J S 4 i x H W c Y Y Y c e R N S M V I 4 x A 7 p 2 6 d d M 2 V 9 2 q u c s X Z 7 d I q g N y K n i S P y B K S L C K v T T B 1 P d G b F / p Z 7 q n O q u 0 3 o 7 6 a 5 R s R K D I j 9 y z d X / t e n a p H o 4 V z X w K m m U D O q O i / N k u i u q J u b X 6 q S l C E k T u E u x S P C n n b O + 2 x q T 6 x r V 7 1 1 d P x N K x W r 9 l 6 q T f C u b k k D t n + O c x H U S k X 7 p F i 6 O S 2 U L 9 J R Z 3 G A Q x z T P M 9 Q x j U q q Oq O P + I J z 0 b F G B s z 4 / 5 T a m R S z z 6 + L e P h A 3 c M k + M = < / l a t e x i t > z < l a t e x i t s h a 1 _ b a s e 6 4 = " q 4 R 3 P

a 6 9
A f c 6 n + J f 6 B / 4 Z 0 x B b W I T k h y 5 t x z 7 s y 9 1 4 l 9 L 5 G W 9 Z o z 5 u Y X F p f y y 4 W V 1 b X 1 j e L m V i O J U u H y u h v 5 k W g 5 L O G + F / K 6 9 K T P W 7 H g L H B 8 3 n S G p y r e v O U i 8 a L w S o 5 i 3 g n Y I P T 6 n s s k U a 3 2 G f c l 6 4 6 7 x ZJ V t v Q y Z 4 G d g R K y V Y u K L 2 i j h w g u U g T g C C E J + 2 B I 6 L m G D Q s x c R 1 M i B O E P B 3 n u E e B v C m p O C k Y s U P 6 D m h 3 n b E h 7 V X O R L t d O s W n V 5 D T x B 5 5 I t I J w u o 0 U 8 d T n V m x v + W e 6 J z q b i P 6 O 1 m u g F i J G 2 L / 8 k 2 V / / W p W i T 6 O N Y 1 e F R T r B l V n Z t l S X V X 1 M 3 N L 1 V J y h A T p 3 C P 4 o K w q 5 3 T P p v a k + j a V W + Z j r 9 p p W L V 3 s 2 0 K d 7 V L W n A 9 s 9 x z o J G p W w f l C u X h 6 X q S T b q P H a w i 3 2 a 5 x G q O E c N d T 3 H R z z h 2 b g w E m N s 3 H 1 K j V zm 2 c a 3 Z T x 8 A M w f k m 4 = < / l a t e x i t > x < l a t e x i t s h a 1 _ b a s e 6 4 = " O x A N b y 8 N m g l p E J y Q 5 c + 4 5 d + b e 6 8 Y + T 4 V l v e a M m d m 5 + Y X 8 Y m F p e W V 1 r b i + U U + j L P F Y z Y v 8 K G m 6 T s p 8 H r K a 4 M J n z T h h T u D 6 r O E O T 2 S 8 c c 2 S l E f h p R j F r B M 4 g 5 D 3 u e c I o p r t U + Y L p 3 v T L Z a s s q W W O Q 1 s D U r Q q x o V X 9 B G D x E 8 Z A j A E E I Q 9 u E g p a c F G x Z i 4 j o Y E 5 c Q 4 i r O c I 8 C e T N S M V I 4 x A 7 p O 6 B d S 7 M h 7W X O V L k 9 O s W n N y G n i R 3 y R K R L C M v T T B X P V G b J / p Z 7 r H L K u 4 3 o 7 + p c A b E C V 8 T + 5 Z s o / + u T t Q j 0 c a R q 4 F R T r B hZ n a e z Z K o r 8 u b m l 6 o E Z Y i J k 7 h H 8 Y S w p 5 y T P p v K k 6 r a Z W 8 d F X 9 T S s n K v a e 1 G d 7 l L W n A 9 s 9 x T o P 6 X t n e L + 9 d H J Q q x 3 r U e W x h G 7 s 0 z 0 N U c I Y q a m q O j 3 j C s 3 F u p M a t c f c p N X L a s 4 l v y 3 j 4 A M d f k m w = < / l a t e x i t > (d) |g, "i < l a t e x i t s h a 1 _ b a s e 6 4 = " h V W Z 4 Y B G 1 N z J R A f e V V e F g D 8 w m 0 U = " > A A A C 1 3 i c j V H L S s N A F D 2 N r 1 p f t S 7 d B I v g Q k p a B V 0 W 3 b i s Y B / S l j J J p z U 0 T c J k o p Y q 7 s S t P + B W / 0 j 8 A / 0 L 7 4 w R 1 C I 6 I c m Z c + 8 5 M / d e O / T c S F r W S 8 q Ym p 6 Z n U v P Z x Y W l 5 Z X s q u 5 W h T E w u F V J / A C 0 b B Z x D 3 X 5 1 X p S o 8 3 Q s H Z 0 P Z 4 3 R 4 c q n j 9 n I v I D f w T O Q p 5 e 8 j 6 v t t z H S a J 6 m R z V / 3 t V h w y I Y K L l m B + 3 + O d b N 4 q W H q Z k 6 C Y g D y S V Q m y z 2 i h i w A O Y g z B 4 U M S 9 s A Q 0 d N E E R Z C 4 t o Y E y c I u T r O c Y 0 M a W P K 4 p T B i B 3 Q t 0 + 7 Z s L 6 t F e e k V Y 7 d I p H r y C l i U 3 S B J Q n C K v T T B 2 P t b N i f / M e a 0 9 1 t x H 9 7 c R r S K z E G b F / 6 T 4 z / 6 t T t U j 0 s K 9 r c K m m U D O q O i d x i X V X 1 M 3 N L 1 V J c g i J U 7 h L c U H Y 0 c r P P p t a E + n a V W + Z j r / q T M W q v Z P k x n h T t 6 Q B F 3 + O c x L U S o X i T q F 0 v J s v H y S j T m M d G 9 i i e e 6 h j C N U U C X v S z z g E U / G q X F j 3 B p 3 H 6 l G K t G s 4 d s y 7 t 8 B A V m X N A = = < / l a t e x i t > |g, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " F / Y / u / M d C F f J 8 + Q x B / Z V f w S 7 q R 4 = " > A A A C 2 X i c j V H L S s N A F D 3 G d 3 3 V x 8 5 N s A g u p K R V 0 G X R j U s F + 4 B W y i Sd 1 t A 0 E y Y T S 3 0 s 3 I l b f 8 C t / p D 4 B / o X 3 h l T U I v o h C R n z r 3 n z N x 7 3 S j w Y + U 4 r 2 P W + M T k 1 P T M b G Z u f m F x K b u 8 U o l F I j 1 e 9 k Q g Z M 1 l M Q / 8 k J e V r w J e i y R n P T f g V b d 7 q O P V C y 5 j X 4 S n a h D x s x 7 r h H 7 b 9 5 g i q p l d u + 5 s N 1 q i H z I p R b 8 h W d g J e D O b c / K O W f Y o K K Q g h 3 Q d i + w L G m h B w E O C H j h C K M I B G G J 6 6 i j A Q U T c G a 6 I k 4 R 8 E + e 4 Q Y a 0 C W V x y m D E d u n b o V 0 9 Z U P a a 8 / Y q D 0 6 J a B X k t L G J m k E 5 U n C + j T b x B P j r N n f v K + M p 7 7 b g P 5 u 6 t U j V u G c 2 L 9 0 w 8 z / 6 n Q t C m 3 s m x p 8 q i k y j K 7 O S 1 0 S 0 x V 9 c / t L V Y o c I u I 0 b l F c E v a M c t h n 2 2 h i U 7 v u L T P x N 5 O p W b 3 3 0 t w E 7 / q W N O D C z 3 G O g k o x X 9 j J F 0 9 2 c 6 W D d N Q z W M c G t m i e e y j h C M c o k / c l H v G E Z 6 t u 3 V p 3 1 v 1 n q j W W a l b x b V k P H 1 F h m B s = < / l a t e x i t > (e) Interatomic distance  Energy ( , 0 =", #) < l a t e x i t s h a 1 _ b a s e 6 4 = " D y p 4 5 Z V P u 1 E m W 6 / Q R l K j F / O Q 1 o I = " > A A A C 7 3 i c j V H L S s Q w F D 3 W 9 3 v U p Z v i I K j I 0 F F B N 8 K g G 5 c K j g p W J e 3 E M U z b 1 D R V Z P A f 3 L k T t / 6 A W / 0 L 8 Q / 0 L 7 y J F X w g m t L k 5 N x 7 T n J z g z Q S m f a 8 5 w 6 n s 6 u 7 p 7 e v f 2 B w a H h k t D Q 2 v p P J X I W 8 H s p I q r 2 A Z T w S C a 9 r o S O + l y r O 4 i D i u 0 F r 3 c R 3 z 7 j K h E y 2 9 U X K D 2 L W T M S x C J k m 6 q g 0 N + N n o h m z + f f l 0 E + V i P m q n 6 d M K X k + 7 z f k e W L h 7 F G p 7 F U 8 O 9 y f o F q A M o q x K U t P 8 N G A R I g c M T g S a M I R G D L 6 9 l G F h 5 S 4 A 7 S J U 4 S E j X N c Y o C 0 O W V x y m D E t m h u 0 m 6 / Y B P a G 8 / M q k M 6 J a J f k d L F N G k k 5 S n C 5 j T X x n P r b N j f v N v W 0 9 z t g t a g 8 I q J 1 T g h 9 i / d R + Z / d a Y W j W O s 2 B o E 1 Z R a x l Q X F i 6 5 f R V z c / d T V Z o c U u I M b l B c E Q 6 t 8 u O d X a v J b O 3 m b Z m N v 9 h M w 5 p 9 W O T m e D W 3 p A Z X v 7 f z J 9 h Z q F Q X K w t b S + X a W t H q P k x i C j P U z 2 X U s I F N 1 M n 7 C v d 4 w K N z 6 l w 7 N 8 7 t e 6 r T U W g m 8 G U 4 d 2 / d 0 q E W < / l a t e x i t > |q, "i < l a t e x i t s h a 1 _ b a s e 6 4 = " M c U I v 2 6 w A i c I h c w c 0 j z j T z n d 0 y z g y 7 L u 3 g B p 1 Z g l < / l a t e x i t > |g, ; q, 0 i < l a t e x i t s h a 1 _ b a s e 6 4 = " / y V d X G p + A a 6 i H c t e V i p E w H g 3 o y s = "> A A A C / 3 i c j V H L L g R B F D 3 a e 7 w G S 5 u O i c R C J j 1 I C B t h Y 0 l i h k Q j 1 T 2 l V f R L d b V E x i z 8 i Z 2 d 2 P o B W 3 b i D / g L t 0 p P 4 h G h O t 1 9 6 t x 7 T t W 9 1 0 t D k S n H e e m y u n t 6 + / o H B k t D w y O j Y + X x i U a W 5 N L n d T 8 J E 7 n n s Y y H I u Z 1 J V T I 9 1 L J W e S F f N c 7 3 d D x 3 X M u M 5 H E O + o i 5 Q c R C 2 J x L H y m i D o q r 1 y 2 X O P S k r z Z D t p z b i a C i K 1 + Z s 8 6 7 K G b S h F x V 7 I 4 C P l R u e J U H b P s n 6 B W g A q K t Z W U n + G i i Q Q + c k T g i K E I h2 D I 6 N l H D Q 5 S 4 g 7 Q I k 4 S E i b O 0 U a J t D l l c c p g x J 7 S N 6 D d f s H G t N e e m V H 7 d E p I r y S l j R n S J J Q n C e v T b B P P j b N m f / N u G U 9 9 t w v 6 e 4 V X R K z C C b F / 6 T q Z / 9 X p W h S O s W x q E F R T a h h d n V + 4 5 K Y r + u b 2 p 6 o U O a T E a d y k u C T s G 2 W n z 7 b R Z K Z 2 3 V t m 4 q 8 m U 7 N 6 7 x e 5 O d 7 0 L W n A t e / j / A k a 8 9 X a Q n V + e 7 G y t l 6 M e g B T m M Y s z X M J a 9 j E F u r k f Y 0 H P O L J u r J u r F v r 7 i P V 6 i o 0 k / i y r P t 3 Y H 6 p L Q = = < / l a t e x i t > |g, ; h, 0 i < l a t e x i t s h a 1 _ b a s e 6 4 = " l + I u G W a e i 9 5 m 5 o 8 z S q 6 FIG. 4. (color online) Schematic illustration for method II.(a): Two laser beams polarized along the z-direction and the x-direction, respectively, are applied to the atoms.(b): 1atom bare levels as well as the influences of the laser beams.Here the beam polarized along the x-direction is decomposed into two beams with σ+ and σ− polarizations, respectively.(c): 1-atom dressed levels in the rotating frame.(d): A 2atom spin-exchange process.The black (red) filled and dashed circles represent the g(q)−atom before and after a collision, respectively.Both this process and the inverse one are studied in this work.(e):The inter-atomic scattering channels.Here the solid curves represent the potentials of each channel.The coupling potentials between different channels are not shown in the figure.

FIG. 5 .
FIG. 5. (color online) (a): The parameters C and C in Eqs.(25-28) as a function of Ix/Iz, with I z(x) being the intensity of the laser beam polarized along the z-(x-)direction.Here we show the results for the cases with ∆z = (2π)3.3× 10 14 Hz and ∆x = (2π)3.4× 10 14 Hz, where ∆ z(x) is the detuning of the z(x)-polarized laser beam with respect to the 3 P2-3 S1 transition (Fig. 4(a)).In this case the wave length of the z(x)-polarized laser beam is 5.08µm (6.12µm).(b): The total photon scattering rate Γsc as a function of the energy gap ∆ hq between the higher dressed states |h, ↑ (↓) and the lower dressed states |q, ↑ (↓) .Here we show the results for the cases with C = C = 1/ √ 2, and the frequencies of the laser beams polarized along the z-(x-) direction being the same as (a).

FIG. 6 .
FIG. 6. (color online) The scattering lengths a± and p± ((a), (c)) and interaction parameters Ax,y,z,0 ((b), (d)) of the system of method II.Here we show the results for two cases, which are given by the multi-channel square-well model in Sec.II D, with width b = 85a0 and other parameters being given in Table II of Appendix C. The values of ∆ hq under which we have A0 = 0, as well as the corresponding values of Ax,y,z and a±,p±, are shown in Appendix D.
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8 6 9 5 8 z c e 9 3
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d l 3 b 8 D
k 1 K Y L w = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " O 7 C 7 + 1 g e 2 1 O P z 4 / 4 k i 6 4 P V O F n c Y = " > A A A C 1 X i c j V H L S s N A F D 3 G V 6 2 v q E s 3 w S K 4 K k k V d F l 0 4 8 4 K 9 g F t K U k 6 q a F 5 M Z k U S u l O3 P o D b v W X x D / Q v / D O m I J a R C c k O X P u P W f m 3 u s k g Z 8 K 0 3 x d 0 B a X l l d W C 2 v F 9 Y 3 N r W 1 9 Z 7 e R x h l 3 W d 2 N g 5 i 3 H D t l g R + x u v B F w F o J Z 3 b o B K z p D C 9 k v D l i P P X j 6 E a M E 9 Y N 7 U H k e 7 5 r C 6 J 6 u t 6 5 C t n A 7 k 0 6 P D S Y 5 0 1 7 e s k s m 2 o Z 8 8 D K Q Q n 5 q s X 6 C z r o I 4 a L D C E Y I g j C A W y k 9 L R h w U R C X B c T 4 j g h X 8 U Z p i i S N q M s R h k 2 s U P 6 D m j X z t m I 9 t I z V W q X T g n o 5 a Q 0 c E i a m P I 4 Y X m a o e K Z c p b s b 9 4 T 5 S n v N q a / k 3 u F x A r c E v u X b p b 5 X 5 2 s R c D D m a r B p 5 o S x c j q 3 N w l U 1 2 R N z e + V C X I I S F O 4 j 7 F O W F X K W d 9 N p Q m V b X L 3 t o q / q Y y J S v 3 b p 6 b 4 V 3 e k g Z s / R z n P G h U y t Z x u X J 9 U q q e 5 6 M u Y B 8 H O K J 5 n q K K S 9 R Q J + 8 R H v G E Z 6 2 p T b U 7 7 f 4 z V V v I N X v 4 t r S H D + Z 3 l f g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " z 9 D M v v h N e K 7 H T s X + Y T C 7 O d T o r F Q = " > A A A C y X i c j V H L S s N A F D 2 N r 1 p f V Z d u g k V w V Z I q 6 L K o C 8 F N B f u A t k i S T u v Y y c N k I t b i y h 9 w q z 8 m / o H + h X f G C G o R n Z D k z L n 3 n J l 7 r x s J n k j L e s k Z U 9 M z s 3 P 5 + c L C 4 t L y S n F 1 r Z G E a e y x u h e K M G 6 5 T s I E D 1 h d c i l Y K 4 q Z 4 7 u C N d 3 h o Y o 3 r 1 m c 8 D A 4 k 6 O I d X 1 n E P A + 9 x x J V K N z x I R 0 z o s l q 2 z p Z U 4 C O w M l Z K s W F p / R Q Q 8 h P K T w w R B A E h Z w k N D T h g 0 L E X F d j I m L C X E d Z 7 h D g b Q p Z T H K c I g d 0 n d A u 3 b G B r R X n o l W e 3 S K o D c m p Y k t 0 o S U F x N W p 5 k 6 n m p n x f 7 m P d a e 6 m 4 j + r u Z l 0 + s x A W x f + k + M / + r U 7 V I 9 L G v a + B U U 6 Q Z V Z 2 X u a S 6 K + r m 5 p e q J D l E x C n c o 3 h M 2 N P K z z 6 b W p P o 2 l V v H R 1 / 1 Z m K V Xs v y 0 3 x p m 5 J A 7 Z / j n M S N C p l e 6 d c O d 0 t V Q + y U e e x g U 1 s 0 z z 3 U M U x a q i T 9 y U e 8 I g n 4 8 S 4 M m 6 M 2 4 9 U I 5 d p 1 v F t G f f v e C C R g Q = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " 3 F a q B 6 K S s q p r N F k u j T R p n t V p 5+ 4 = " > A A A C z n i c j V H L S s N A F D 2 N r 1 p f V Z d u g k V w V Z I q 6 L L o x m U F 2 w p t K Z P J t I b m R T I p l F L c + g N u 9 b P E P 9 C / 8 M 6 Y g l p E J y Q 5 c + 4 5 d + b e 6 8 S + l 0 r L e i 0 Y S 8 s r q 2 v F 9 d L G 5 t b 2 T n l 3 r 5 V G W c J F k 0 d + l N w 6 L B W + F 4 q m 9 K Q v b u N E s M D x R d s Z X a p 4 e y y S 1 I v C G z m J R S 9 g w 9 A b e J x J o j p d V / i S 9 a f c n f X L F a t q 6 W U u A j s H F e S r E Z V f 0 I W L C B w Z A g i E k I R 9 M K T 0 d G D D Q k x c D 1 P i E k K e j g v M U C J v R i p B C k b s i L 5 D 2 n V y N q S 9 y p l q N 6 d T f H o T c p o 4 I k 9 E u o S w O s 3 U 8 U x n V u x v u a c 6 p 7 r b h P 5 O n i s g V u K O 2 L 9 8 c + V / f a o W i Q H O d Q 0 e 1 R R r R l X H 8 y y Z 7 o q 6 u f m l K k k Z Y u I U d i m e E O b a O e + z q T 2 p r l 3 1 l u n 4 m 1 Y q V u 1 5 r s 3 w r m 5 J A 7 Z / j n M R t G p V + 6 R a u z 6 t 1 C / y U R d x g E M c 0 z z P U M c V G m j q j j / i C c 9 G w x g b M + P + U 2 o U c s 8 + v i 3 j 4 Q O Z V Z P x < / l a t e x i t > (c) |g, "i< l a t e x i t s h a 1 _ b a s e 6 4 = " h V W Z 4 Y B G 1 N z J R A f e V V e F g D 8 w m 0 U = " > A A A C 1 3 i c j V H L S s N A F D 2 N r 1 p f t S 7 d B I v g Q k p a B V 0 W 3 b i s Y B / S l j J J p z U 0 T c J k o p Y q 7 s S t P + B W / 0 j 8 A / 0 L 7 4 w R 1 C I 6 I c m Z c + 8 5 M / d e O / T c S F r W S 8 q Y l a t e x i t > |d, #i < l a t e x i t s h a 1 _ b a s e 6 4 = " I A N c 3 j w o 6 y a A s k 0 X 0 D UV P I x F k c E = " > A A A C 2 X i c j V H L S s N A F D 2 N r 1 p f 9 b F z E y y C C y l p F X R Z d O O y g n 1 A K 2 W S j j W Y Z s J k Y q n Vh T t x 6 w + 4 1 R 8 S / 0 D / w j t j B B + I T k h y 5 t x 7 z s y 9 1 4 0 C P 1 a O 8 5 y x x s Y n J q e y 0 7 m Z 2 b n 5 h f 2 w p t K Z P J t I b m R T I p l F L c + g N u 9 b P E P 9 C / 8 M 6 Y g l p E J y Q 5 c + 4 5 d + b e 6 8 S + l 0 r L e i 0 Y S 8 s r q 2 v F 9 d L G 5 t b 2 T n l 3 r 5 FIG. 7. (color online) Schematic illustration of method III.(a): 1-atom bare levels and π-polarized Raman beams (blue and red lines).Here ∆ is the single-photon detuning.(b): 1-atom dressed levels in the rotating frame.(c): A 2-atom spin-exchange process.The black (red) filled and dashed circles represent the g(d)−atom before and after a collision, respectively.Both this process and the inverse one are studied in this work.(d):The inter-atomic scattering channels.Here the solid curves represent the potentials of each channel.The coupling potentials between different channels are not shown in the figure.