The manipulation of ultracold atoms by external fields has attracted a lot of interest from researchers in recent years. Among various technologies, magnetic Feshbach resonances have been widely studied and detected in many systems [1–5]. Taking advantage of magnetic Feshbach resonances, researchers can modulate the scattering length of ultracold atoms [6, 7] and prepare the Bose–Einstein condensate (BEC) [8]. Feshbach resonances can also be used to realize the 3D or quasi-2D BEC-BCS crossover with atomic Fermi gases [9–11]. Laser can be used to induce photoassociation resonances and regulate the interaction between ultracold atoms as well [12]. Photoassociation resonances are also widely applied to prepare ultracold molecules [13, 14]. Ultracold atoms can be excited by laser to the excited molecular state during the collision, but this will cause atomic spontaneous emission losses.

In order to control the collision of ultracold atoms effectively, both the magnetic field and laser field are usually applied [15–17]. The frequency and intensity of the laser field are two adjustable parameters, offering more flexibility for researchers in experiments. In the mixture experiments of BECs of two different species [18, 19] or BECs in different internal states of the same isotope [20], several scattering lengths need to be modulated independently. In an optical lattice, the scattering length can be modulated in specific lattice sites [21–25]. The fine spatial modulation of the scattering length can be realized by laser, and this offers experimental feasibility for spatial modulation of the interatomic interaction [26, 27].

A traveling-wave laser beam has been applied to control the magnetic Feshbach resonance [28]. Under the co-action of the magnetic field and laser field, the magnetic Feshbach resonance splits into two resonances, and an Autler–Townes doublet in the particle loss has been observed. Compared with tuning the scattering length with an optically induced Feshbach resonance [29], the loss rate coefficient can be reduced by one order of magnitude. Bauer and co-workers used one specific excited molecular state in this experiment, where the photoassociation coupling between this specific excited molecular state and the continuum state of the incoming atom pair is negligible [28]. Thus, laser only induces the molecular bound-to-bound transition between ground and excited molecular states [30–32]. With the photoassociation coupling not being considered, the resonance width is proportional to | E col − ϵ 3 E col − ϵ 2 + E col − ϵ 3 | , where ϵ ₂ and ϵ ₃ are the energies of the ground and excited molecular states. The energy E _col is the collision energy between two colliding atoms. The probability of being trapped in the excited molecular state is proportional to 1 ( E col − ϵ 3 ) 2 . Therefore, tuning the laser frequency to shift ϵ ₃ far away from E _col suppresses spontaneous emission losses. The “dark-state” optical method is proposed to tune the scattering length and suppress spontaneous emission losses, in which two lasers with different frequencies are applied to couple one excited molecular state to two ground molecular states [33–36]. In this method, the photoassociation coupling was not taken into account.

Friedrich and co-workers demonstrated that when two ultracold atoms are trapped in the bound states of two different closed channels during the collision, the resonance position and width can be altered by changing the external field [37, 38]. However, the particle loss caused by the external field is not considered. The modification of the ground molecular state by laser is not considered either. The external field can induce a resonance with vanishing width. Such a resonance is also called the bound state in the continuum, which has been observed in various systems such as quantum billiard and quantum dot [39, 40]. The bound state in the continuum can be prepared by lasers near a magnetic Feshbach resonance in ultracold atoms but decays fast due to the spontaneous emission loss [41, 42].

In the present work, we investigated the collision property between two ultracold atoms under the co-action of the magnetic field and laser field. The magnetic field is adjusted to the neighborhood of a magnetic Feshbach resonance. Laser can induce the photoassociation process and bound-to-bound transition. With the photoassociation coupling being considered, the resonance width is dependent on the three coupling terms among the ground molecular state, the excited molecular state, and the continuum state of the incoming atom pair. The resonance width can be increased by adjusting the laser intensity. Compared with a narrow resonance, the tunability of the scattering length at a wide resonance is significantly improved and the spontaneous emission loss is reduced. The coupling between the ground and excited molecular states is composed of the direct bound-to-bound coupling and the indirect coupling induced by the photoassociation coupling and Feshbach coupling via the continuum states of the incoming atom pair. The coupling can be almost completely canceled by adjusting the laser intensity. We found that in this case, the spontaneous emission loss at the resonance is significantly suppressed and that the magnetic field position of this resonance is stable against the change in laser frequency. We found that the interference between the bound-to-bound transition and photoassociation transition can be used to prepare the bound state in the continuum, a resonance with a vanishing width. The magnetic field position of the bound state in the continuum can be shifted by changing the laser intensity. At the magnetic field position where the bound state in the continuum occurs, when laser frequency is detuned with respect to the resonant frequency, the scattering length is almost unchanged with laser frequency and the spontaneous emission loss is significantly suppressed.

This paper is organized as follows: in Section 2, we present the solution to the coupled equations for the three-radial channel wave functions in the magnetic field and laser field. The resonance width is given, which depends on the three coupling terms among the ground, excited molecular states, and the continuum state of the incoming atom pair. It is explained why the scattering length is frequency-independent at the magnetic field position where the bound state in the continuum occurs when the frequency is detuned away from the resonance condition. We give the amplitude factors of the ground and excited molecular states in the condition that the open-channel wavefunction remains normalized in energy, which are related to the resonance width. In Section 3, we calculate the magnetic field positions of resonances and the loss rate coefficients at different laser frequencies and intensities. We also investigate the modulation of the real part of the s-wave scattering length at the magnetic field positions near the original magnetic Feshbach resonance. Finally, a conclusion is drawn in Section 4.

The scattering process of two ultracold atoms controlled by the magnetic field and laser field is shown in Figure 1. In the absence of the laser field, channel 1 is only coupled with channel 2. Two colliding atoms are trapped in the bound states of channel 2, and the magnetic field induces a magnetic Feshbach resonance. After laser is applied, two colliding atoms are either directly excited to the bound states of channel 3 by photoassociation coupling or first trapped in the bound states of channel 2 and then excited to the bound states of channel 3 by bound-to-bound coupling. When the two atoms are trapped in the bound states of channel 3, the spontaneous emission may take place and induce atomic losses. In our close-coupling calculations, an open-channel coupled with channel 3 is used to describe atomic losses [43–45]. Under the rotating-wave approximation, the coupled equation of radial channel wavefunctions for the three-channel system is given by the following: − ℏ 2 2 μ d 2 d r 2 + V i + E i u i r + ∑ i ≠ j V i , j u j r = E col u i r , (1) where ℏ is the reduced Planck constant, μ the reduced mass, and r the internuclear separation. E _i and V _i (i = 1,2,3) are the channel energy and interaction potential, respectively. V _i (i = 1,2,3) approaches zero as r → ∞. The threshold energy E ₁ of channel 1 is taken to be zero. Channels 2 and 3 are closed channels with E ₂ and E ₃ > 0. The energy E ₃ of channel 3 is obtained by reducing the energy of one photon ℏω from the original channel energy. E ₃ can be adjusted by changing the laser frequency. V _i,j (i ≠ j) are the coupling potentials between the channels. The coupling potential V _1,2 (V _2,1) between channels 1 and 2 does not vary with the laser frequency and intensity. V _1,3 (V _3,1) and V _2,3 (V _3,2) can be modulated by changing the laser intensity and are independent of the laser frequency. E _col is the collision energy between two ultracold atoms. E _col = 1 μK × k _B in our close-coupling calculations, where k _B is the Boltzmann constant. Here, only the s-wave scattering is considered. The interaction potentials V _i (i = 1,2,3) and coupling potentials V _i,j (i ≠ j) used in our calculation are taken from [46], which change with the magnetic field. In the appendix, V _i (i = 1,2,3) and V _i,j (i ≠ j) are shown in Figure A1 at the magnetic field having the original magnetic Feshbach resonance. Figures A1E, F show the laser-induced coupling potentials V _1,3 and V _2,3 when laser amplitude is set to 10E _s. The minimal laser amplitude in our calculation is taken to be E _s, where the coupling potentials V _1,3 (V _3,1) and V _2,3 (V _3,2) are much weaker than the coupling potential V _1,2 (V _2,1).

By using the mapped Fourier grid method [47, 48], we calculate the wavefunctions of several stationary s-wave continuum states with the lowest eigenenergies in the three-channel system. We find that the wavefunctions in channels 2 and 3 are both superpositions of bound-state wavefunctions of corresponding closed channels. In the absence of the laser field, the wavefunctions in channel 2 in the neighborhood of the magnetic Feshbach resonance are also superpositions of bound-state wavefunctions. Therefore, when obtaining the solution of Eq. 1 by using the Feshbach theory, we cannot think that the wavefunction in channel 2 or 3 is composed of a single bound-state wavefunction.

The solution of Eq. 1 can be written as follows: U = u 1 r A 2 u 2 0 r A 3 u 3 0 r , (2) where u 2 0 ( r ) and u 3 0 ( r ) are normalized superpositions of bound-state wavefunctions in channels 2 and 3, respectively. The wavefunction u ₁(r) in channel 1 is given by the following: u 1 r = u 1 r e g r + ∫ 0 ∞ G r , r ′ + A 2 V 1,2 r ′ u 2 0 r ′ + A 3 V 1,3 r ′ u 3 0 r ′ d r ′ , (3) where u 1 r e g ( r ) is the solution of the radial equation in channel 1 without the coupling with channel 2 or 3. The asymptotic behavior of u 1 r e g ( r ) is given by the following: u 1 reg r ~ r → ∞ 2 μ π ℏ 2 k sin k r + δ bg , (4) where k is the magnitude of the incoming wave vector and δ _bg the s-wave background phase shift in channel 1. In Eq. 3, G ( r , r ′ ) is the radial Green’s function.

We then obtain two equations about A ₂ and A ₃, A 2 E col − ⟨ u 2 0 | H ̂ 2 | u 2 0 ⟩ − ⟨ u 2 0 | V 2,1 G ̂ V 1,2 | u 2 0 ⟩ = ⟨ u 2 0 | V 2,1 | u 1 reg ⟩ + A 3 ⟨ u 2 0 | V 2,3 | u 3 0 ⟩ + ⟨ u 2 0 | V 2,1 G ̂ V 1,3 | u 3 0 ⟩ , (5) A 3 E col − ⟨ u 3 0 | H ̂ 3 | u 3 0 ⟩ − ⟨ u 3 0 | V 3,1 G ̂ V 1,3 | u 3 0 ⟩ = ⟨ u 3 0 | V 3,1 | u 1 reg ⟩ + A 2 ⟨ u 3 0 | V 3,2 | u 2 0 ⟩ + ⟨ u 3 0 | V 3,1 G ̂ V 1,2 | u 2 0 ⟩ , (6) where H ̂ i = − ℏ 2 2 μ d 2 d r 2 + V i + E i , i = 2,3 . (7)

By using the abbreviations, ϵ i = ⟨ u i 0 | H ̂ i | u i 0 ⟩ + ⟨ u i 0 | V i , 1 G ̂ V 1 , i | u i 0 ⟩ , W i , 1 = ⟨ u i 0 | V i , 1 | u 1 reg ⟩ = W 1 , i * , i = 2,3 (8) and W 2,3 = 〈 u 2 0 | V 2,3 | u 3 0 〉 + 〈 u 2 0 | V 2,1 G ̂ V 1,3 | u 3 0 〉 = W 3,2 * , (9) the solutions of Eqs 5–6 are expressed as A 2 = E col − ϵ 3 W 2,1 + W 2,3 W 3,1 E col − ϵ 2 E col − ϵ 3 − | W 2,3 | 2 , (10) A 3 = E col − ϵ 2 W 3,1 + W 3,2 W 2,1 E col − ϵ 2 E col − ϵ 3 − | W 2,3 | 2 . (11)

With the photoassociation coupling V _1,3 being considered, the coupling W _3,2 between the ground and excited molecular states | u 2 0 〉 , | u 3 0 〉 is induced by the direct bound-to-bound coupling V _2,3 and the indirect coupling induced by the Feshbach coupling V _2,1 and photoassociation coupling V _1,3 via G ̂ in channel 1.

The s-wave scattering phase shift δ _res caused by the resonance is given by the following [37]: tan δ res = − π E col − ϵ 3 | W 2,1 | 2 + W 1,2 W 2,3 W 3,1 + E col − ϵ 2 | W 3,1 | 2 + W 1,3 W 3,2 W 2,1 E col − ϵ 2 E col − ϵ 3 − | W 2,3 | 2 . (12)

When (E _col − ϵ ₂)(E _col − ϵ ₃) − |W _2,3|² = 0, the resonance takes place under the co-action of the magnetic field and laser field. The resonance width Γ is given by the following: Γ = 2 d δ res d E col − 1 . (13)

When W _i,j (i, j = 1, 2, 3) does not change with E _col, we obtain the following: Γ | D = 0 = 2 1 + N 2 D 2 D 2 D ′ N − N ′ D = 2 N D ′ , (14) where N E col = π E col − ϵ 3 | W 2,1 | 2 + W 1,2 W 2,3 W 3,1 + E col − ϵ 2 | W 3,1 | 2 + W 1,3 W 3,2 W 2,1 , (15) D E col = E col − ϵ 2 E col − ϵ 3 − | W 2,3 | 2 , (16) and N ′ = d N d E col , (17) D ′ = d D d E col . (18)

When (E _col − ϵ ₂)W _3,1 + W _3,2 W _2,1 = 0 at a specific magnetic field position B ₁ and the resonance condition D = 0 is met, we obtain the following: E col − ϵ 3 W 2,1 + W 2,3 W 3,1 = | W 2,3 | 2 E col − ϵ 2 W 2,1 + W 2,3 W 3,1 = W 2,3 E col − ϵ 2 E col − ϵ 2 W 3,1 + W 3,2 W 2,1 = 0 . (19)

From Eqs 14, 15, it is shown that N (E _col) = 0, and hence, the resonance width is zero. Thus, due to the interference between the photoassociation transition induced by W _3,1 and the bound-to-bound transition induced by W _3,2, (E _col − ϵ ₂)W _3,1 + W _3,2 W _2,1 = 0 at B ₁ and the bound state in the continuum occurs when D = 0.

When the energy ϵ ₃ of the excited molecular state is detuned away from the resonance condition by changing the laser frequency, at B ₁ where (E _col − ϵ ₂)W _3,1 + W _3,2 W _2,1 = 0, we obtain A 2 = − W 3,1 W 3,2 = W 2,1 E col − ϵ 2 , A 3 = 0 . (20)

In this case, two colliding atoms will be almost free from being trapped in the bound states of channel 3 during the collision. The spontaneous emission loss is negligible. The phase shift δ _res is entirely dominated by channel 2, that is, tan δ res = − π W 1,2 A 2 = − π | W 1,2 | 2 E col − ϵ 2 . (21)

Tuning the laser frequency does not change the scattering length at B ₁.

In order to let the wavefunction u 1 ′ ( r ) in channel 1 to be normalized in energy, the asymptotic behavior of u 1 ′ ( r ) should be expressed as u 1 ′ r ~ r → ∞ 2 μ π ℏ 2 k sin k r + δ bg + δ res . (22)

To meet this requirement, we multiplied U by cos δ _res, U ′ = u 1 ′ r A 2 ′ u 2 0 r A 3 ′ u 3 0 r = cos δ res U , (23) where A 2 ′ = − sin δ res E col − ϵ 3 W 2,1 + W 2,3 W 3,1 N E col = − 2 ⁡ sin δ res E col − ϵ 3 W 2,1 + W 2,3 W 3,1 D ′ Γ (24) and A 3 ′ = − sin δ res E col − ϵ 2 W 3,1 + W 3,2 W 2,1 N E col = − 2 ⁡ sin δ res E col − ϵ 2 W 3,1 + W 3,2 W 2,1 D ′ Γ . (25)

When the resonance condition D = 0 is fulfilled, we obtain | A 3 ′ | 2 = 8 π Γ | E col − ϵ 2 E col − ϵ 2 + E col − ϵ 3 | . The probability of being trapped in the excited molecular state increases as the resonance width decreases.

In our model, there is a magnetic Feshbach resonance at B = B ₀ in the absence of laser, and B ₀ is given in the appendix. With laser applied, the two colliding atoms are excited to the bound states of channel 3 during the collision. We calculate the real part Re(a) of the scattering length and loss rate coefficient at different magnetic field positions and laser frequencies using the close-coupling method. For the minimal laser amplitude E ₀ = E _s of the electric field E = E ₀ cos (ωt), we find that the bound state in the continuum occurs at the magnetic field position B 1 E 0 = E s = B 0 − 2.49 G. With laser frequency detuned away from the resonance condition, according to Eq. 21, tan δ _res is independent of the energy ϵ ₃ of the excited molecular state at B 1 E 0 = E s so that the tuning laser frequency does not change Re(a) at B 1 E 0 = E s and Re(a) = −130.15 a ₀. As shown in Eq. 20, when the resonance condition is not met, the probability of being trapped in the excited molecular state at B 1 E 0 = E s is significantly suppressed and the loss rate coefficient is lower than 10^–18 cm³s⁻¹.

We then calculate the wavefunction u 2 0 ( r ) | E 0 = E s in channel 2 at B = B 1 E 0 = E s using the mapped Fourier grid method. We find that u 2 0 ( r ) | E 0 = E s is almost invariant with the laser frequency. By comparing u 2 0 ( r ) | E 0 = E s with the wavefunction u 2 0 ( r ) | n o l a s e r in channel 2 at B = B 1 E 0 = E s in the absence of the laser field, we obtain 1 − | ⟨ u 2 | E 0 = E s 0 | u 2 | n o l a s e r 0 ⟩ | 2 < 1 0 − 10 . (26)

It can be seen that the two wavefunctions u 2 0 ( r ) | E 0 = E s and u 2 0 ( r ) | n o l a s e r are almost the same, and hence, W _1,2 and ϵ ₂ are also almost unchanged. Therefore, the scattering length at B 1 E 0 = E s in the absence of the laser field should be very close to the real part Re(a) of the scattering length under the action of laser according to Eq. 20. The calculated scattering length at B 1 E 0 = E s in the absence of the laser field is −130.31 a ₀. The difference between the scattering length without laser and Re(a) under the action of laser is less than 0.16 a ₀. This shows that when the amplitude E ₀ = E _s , the laser-induced bound-to-bound coupling and photoassociation coupling are much weaker than the Feshbach coupling between the ground molecular state and the incoming continuum state so that the wavefunction in channel 2 is slightly changed by the laser field.

When the amplitude E ₀ increases to 5E _s , the bound state in the continuum occurs at the magnetic field position B 1 E 0 = 5 E s = B 0 − 2.49 G. With laser frequency detuned, Re(a) = −126.45 a ₀ at B 1 E 0 = 5 E s and the loss rate coefficient is lower than 10^–16 cm³s⁻¹. Comparing u 2 0 ( r ) | E 0 = 5 E s with u 2 0 ( r ) | n o l a s e r at B 1 E 0 = 5 E s , we obtain 1 − | ⟨ u 2 | E 0 = 5 E s 0 | u 2 | n o l a s e r 0 ⟩ | 2 < 1 0 − 8 . (27)

It can be seen that when E ₀ = 5E _s, the wavefunction u 2 0 ( r ) | E 0 = 5 E s is changed more. Although the magnetic field position where the bound state in the continuum occurs is not shifted, the difference between the scattering length in the absence of laser and Re(a) under the action of laser is more than 3.85 a ₀.

When the amplitude E ₀ increases to 10E _s , the bound state in the continuum occurs at B 1 E 0 = 10 E s = B 0 − 2.47 G. With laser frequency detuned, Re(a) = −119.52 a ₀ at B 1 E 0 = 10 E s and the loss rate coefficient is lower than 10^–15 cm³s⁻¹. Comparing u 2 0 ( r ) | E 0 = 10 E s with u 2 0 ( r ) | n o l a s e r at B 1 E 0 = 10 E s , we obtain 1 − | ⟨ u 2 | E 0 = 10 E s 0 | u 2 | n o l a s e r 0 ⟩ | 2 < 1 0 − 6 . (28)

It can be seen from the aforementioned results that the wavefunction in channel 2 will be more significantly changed as laser intensity increases. As a result, the magnetic field position B ₁ where the bound state in the continuum occurs is shifted and the real part Re(a) of the scattering length is changed.

According to Eq. 12, the s-wave resonance takes place under the co-action of the magnetic field and laser field when (E _col − ϵ ₂)(E _col − ϵ ₃) − |W _2,3|² = 0. The energy ϵ ₃ can be modulated by changing laser frequency, and hence, the magnetic field position of the resonance shifts with laser frequency. Figure 2 shows the magnetic field positions of resonances at different laser frequencies ω when the amplitude E ₀ = 10E _s. When ω = ω 0 E 0 = 10 E s , the two resonances are located at the magnetic field positions B ₀ ± 1.08 G. It can be seen that as ω is changed, one of the resonances is obviously shifted, while the other resonance is just located near B ₀. For the resonance located far away from B ₀, |E _col − ϵ ₃|≪|E _col − ϵ ₂|. For the resonance located close to B ₀, |E _col − ϵ ₃|≫|E _col − ϵ ₂|. When ω is tuned to ω 0 E 0 = 10 E s , ϵ ₂ and ϵ ₃ are close to each other, and hence, the magnetic field positions of the two resonances deviate from B ₀, and the deviation is about 1 G.

Figure 3 shows the loss rate coefficients as a function of the magnetic field for different laser frequencies around ω 0 E 0 = 10 E s . The loss rate coefficient behaves like the Autler–Townes doublet, which was observed in [28]. With laser frequency detuned away from ω 0 E 0 = 10 E s , one of the two loss rate coefficient peaks increases and the other decreases. As laser frequency decreases from ω 0 E 0 = 10 E s , the loss rate coefficient peak on the left side decreases and the peak on the right side increases. As laser frequency increases from ω 0 E 0 = 10 E s , the loss rate coefficient peak on the left side increases and the peak on the right side decreases. Different from the case observed in [28], it can be seen from Figure 3 that the loss rate coefficient peak on the left side increases much faster than that on the right side. We then calculate the loss rate coefficients at the two resonances versus laser frequency for E ₀ = 10E _s, as shown in Figure 4. As laser frequency increases from ω 0 E 0 = 10 E s to ω 0 E 0 = 10 E s + 80.5 MHz, the resonance on the left side is shifted to the magnetic field position B 1 E 0 = 10 E s , where the bound state in the continuum occurs and the resonance width decreases. The loss rate coefficient at the left side resonance increases rapidly and reaches its maximum when the laser frequency ω is tuned close to ω 0 E 0 = 10 E s + 80.5 MHz. According to | A 3 ′ | 2 = 8 π Γ | E col − ϵ 2 E col − ϵ 2 + E col − ϵ 3 | , the probability of being trapped in the excited molecular state increases as the resonance width decreases, so the loss rate coefficient at the left side resonance increases rapidly.

As shown in Figure 2, the magnetic field position of the resonances can be shifted by changing the laser frequency. However, the resonance widths are narrow in the neighborhood of the magnetic field position B 1 E 0 = 10 E s , and the loss rate coefficients are large. Figure 5 shows the real part Re(a) of the scattering length and the loss rate coefficient at three resonances, when the laser frequency ω = ω 0 E 0 = 10 E s + 142.3 , ω 0 E 0 = 10 E s + 86.6 , and ω 0 E 0 = 10 E s − 167.1 MHz. These three resonances are located at the magnetic field positions B = B ₀ − 3.8 G, B ₀ − 2.6 G, and B ₀ + 4.6 G, respectively. The real part Re(a) of the scattering length can be tuned by changing the magnetic field B or laser frequency. With the loss rate coefficient being limited below 10^–11 cm³s⁻¹, Re(a) can be tuned in the range from −430.27 to 572.10 a ₀ for the resonance located at B = B ₀ − 3.8 G. For the resonance located at B = B ₀ + 4.6 G, Re(a) can be tuned in the range from −808.71 to 2719.43 a ₀. However, for the very narrow resonance located at B = B ₀ − 2.6 G, Re(a) can only be tuned in the range from −130.19 to −66.04 a ₀. The tunability of Re(a) at narrow resonances is severely limited.

Therefore, when laser amplitude E ₀ = 10E _s, Re(a) cannot be tuned effectively in the neighborhood of the magnetic field position B 1 E 0 = 10 E s . As mentioned previously, when stronger laser is applied, the wavefunction in channel 2 is more significantly changed by laser. When laser amplitude E ₀ increases to 50E _s, compared with the cases of E ₀ = 1 ∼ 10E _s, we find the magnetic field position where the bound state in the continuum occurs has been significantly shifted. The bound state in the continuum occurs at B 1 E 0 = 50 E s = B 0 + 2.75 G. With laser frequency being detuned, Re(a) = 280.05 a ₀ at B 1 E 0 = 50 E s , and the loss rate coefficient is lower than 10^–14 cm³s⁻¹. Comparing u 2 0 ( r ) | E 0 = 50 E s with u 2 0 ( r ) | n o l a s e r at B 1 E 0 = 50 E s , we obtain 1 − | ⟨ u 2 | E 0 = 50 E s 0 | u 2 | n o l a s e r 0 ⟩ | 2 < 2.5 × 1 0 − 3 . (29)

By increasing laser amplitude from 10E _s to 50E _s, the bound state in the continuum is shifted from B 1 E 0 = 10 E s to B 1 E 0 = 50 E s . Therefore, when laser amplitude E ₀ = 50E _s, a wide resonance occurs at B 1 E 0 = 10 E s . With the loss rate coefficient being limited below 10^–11 cm³s⁻¹, Re(a) can be tuned in the range from −461.90 to 478.00 a ₀ at this wide resonance. The tunability of Re(a) in the neighborhood of B 1 E 0 = 10 E s is significantly improved.

Changing laser amplitude not only shifts the magnetic field position where the bound state in the continuum occurs but also adjusts the coupling W _2,3 between the ground and excited molecular states. A special case is that the coupling W _2,3 approaches zero due to the interference between the direct bound-to-bound coupling and the indirect coupling. Figure 6 shows the magnetic field positions of resonances at different laser frequencies when laser amplitude E ₀ = 60E _s. One of the two resonances is linearly shifted, while the other resonance is unmoved. Due to the small |W _2,3|, the minimum difference between the magnetic field positions of the two resonances is only 0.21 G.

We calculate the real part Re(a) of the scattering length versus the magnetic field B for different laser frequencies, as shown in Figure 7A. With |W _2,3| approaching zero, at the linearly shifted resonance |E _col − ϵ ₂|≫|W _2,3|≫|E _col − ϵ ₃| and the resonance width Γ ≈ 2π|W _3,1|². At the unmoved resonance, |E _col − ϵ ₃|≫|W _2,3|≫|E _col − ϵ ₂| and the resonance width Γ ≈ 2π|W _2,1|². We also calculate the loss rate coefficient at the two resonances for different laser frequencies. The maximum loss rate coefficients at the linearly shifted resonances for three frequencies are almost the same, which are 2.44 × 10⁻¹¹, 2.49 × 10⁻¹¹, and 2.59 × 10⁻¹¹ cm³s⁻¹, respectively. At the linearly shifted resonance A 3 ′ ≈ − 2 ⁡ sin δ res 1 W 1,3 according to Eq. 25. The probability of being trapped in the excited molecular state is independent of the energy ϵ ₂ of the ground molecular state and only dependent on the photoassociation coupling W _1,3. Thus, the maximum loss rate coefficient at the linearly shifted resonance changes little as the magnetic field position of this resonance is shifted. However, the loss rate coefficient peak does not occur at the unmoved resonance, as shown in Figure 7B. At the unmoved resonance A 3 ′ ≈ − 2 ⁡ sin δ res ( E col − ϵ 2 ) W 3,1 + W 3,2 W 2,1 ( E col − ϵ 3 ) | W 2,1 | 2 . The photoassociation transition and bound-to-bound transition are both suppressed because (E _col − ϵ ₂) → 0 and |W _3,2| → 0. The real part Re(a) of the scattering length can be tuned over a large range without rapid losses.

Figure 8 is a schematic illustration of the variation of A 3 ′ with the magnetic field B at the unmoved resonance, where |W _2,3| is much smaller than |W _1,3| and |W _1,2|, and |E _col − ϵ ₃|≫|E _col − ϵ ₂|. The two energies ϵ ₂ and ϵ ₃ decrease as B increases, and the energy interval between ϵ ₂ and ϵ ₃ at the resonance is altered by changing ϵ ₃. At the resonance position, the position and width of the peak of | sin δ _res|² are almost unchanged when changing ϵ ₃. As ϵ ₃ gradually approaches ϵ ₂, the value |E _col − ϵ ₃| decreases at the resonance, and hence, |N (E _col)| decreases. The slope of [ ( E c o l − ϵ 2 ) W 3,1 + W 3,2 W 2,1 ] / N ( E c o l ) at the resonance increases gradually. As a result, | A 3 ′ | increases faster on both sides of the resonance as ϵ ₃ approaches ϵ ₂. However, | A 3 ′ | at the resonance is still a small value because | ( E c o l − ϵ 2 ) W 3,1 + W 3,2 W 2,1 | → 0 . This explains the suppressed loss rate coefficient at the nearly immovable resonance in Figure 7 and the increase in the loss rate coefficient on both sides of this resonance as the two resonances are close to each other.

In this paper, we investigate the s-wave scattering of ultracold atoms controlled by the magnetic field and laser field in the neighborhood of the original magnetic Feshbach resonance. We find that the bound state in the continuum occurs at the magnetic field position B ₁ near the original magnetic Feshbach resonance due to the interference between the photoassociation and bound-to-bound transitions. Changing the laser frequency can shift the magnetic field positions of resonances, and the widths of resonances in the neighborhood of B ₁ become narrow. Because the probability of being trapped in the excited molecular state increases as the resonance width decreases, the loss rate coefficients at narrow resonances are large. The tunability of the real part Re(a) of the scattering length is severely limited at narrow resonances. The wavefunction of the ground molecular state is more significantly changed as laser intensity increases. Therefore, changing the laser intensity can shift the magnetic field position B ₁ to induce wide resonances at desired magnetic field positions. This paves the way to tune the scattering length at a wide range of magnetic fields near the original magnetic Feshbach resonance. Changing the laser intensity also adjusts the coupling between the ground and excited molecular states. With the coupling canceled, a resonance is induced at which the loss rate coefficient is significantly suppressed. The scattering length can be tuned over a large range without causing rapid atomic losses. At the magnetic field position where the bound state in the continuum occurs, when the laser frequency is detuned away from the resonance condition, the scattering length does not change with the laser frequency and the spontaneous emission losses are significantly suppressed. Therefore, the laser frequency can be used as the control parameter to manipulate ultracold systems, for example, when other scattering lengths in this system need to be tuned. In this work, the s-wave scattering is manipulated by the magnetic field and one laser. In the future work, we would consider adding another laser to couple the excited molecular state with a deeply bound ground molecular state. In this way, more control parameters will be used to manipulate ultracold systems. Moreover, ultracold atoms are trapped in the deeply bound ground molecular state during the collision, which may be helpful in the preparation of ultracold molecules.