Bichromatic slowing of MgF molecules in multilevel systems

In the bichromatic field, assuming that a pair of monochromatic beams is symmetrically detuned from a carrier frequency ω by δ and both frequency components have the same amplitude E₀, the electric field from the counter propagating beams aligned along the z-axis can be given by

$$\begin{eqnarray}&&\begin{array}{c}E(z,t)={E}_{0}\mathrm{Re}\{{{\rm{e}}}^{{\rm{i}}((k+{\rm{\Delta }}k)z-(\omega +\delta )t)}+{{\rm{e}}}^{{\rm{i}}((k-{\rm{\Delta }}k)z-(\omega -\delta )t)}\\ \,\,\,\,+{{\rm{e}}}^{{\rm{i}}(-(k+{\rm{\Delta }}k)z-(\omega +\delta )t)}+{{\rm{e}}}^{{\rm{i}}(-(k-{\rm{\Delta }}k)z-(\omega -\delta )t)}\}.\end{array}\end{eqnarray} \tag{ 6 }$$

Where k = 2π/λ = ω/c and Δk = δ/c. Grouping terms, we can rewrite equation (6) as

$$\begin{eqnarray}E(z,t) & = & 4{E}_{0}\left\{\displaystyle \frac{1}{2}({\rm{\cos }}(kz){\rm{\cos }}(\delta t){\rm{\cos }}(\chi /2)\right.\\ & & +{\rm{i}}\,{\rm{\sin }}(kz){\rm{\sin }}(\delta t){\rm{\sin }}(\chi /2)){{\rm{e}}}^{-{\rm{i}}\omega t}\\ & & +\displaystyle \frac{1}{2}({\rm{\cos }}(kz){\rm{\cos }}(\delta t){\rm{\cos }}(\chi /2)\\ & & -{\rm{i}}\,{\rm{\sin }}(kz){\rm{\sin }}(\delta t){\rm{\sin }}(\chi /2)){{\rm{e}}}^{{\rm{i}}\omega t}\phantom{\displaystyle \frac{1}{2}}\,\}.\end{eqnarray} \tag{ 7 }$$

Where χ = 2Δkz = 2δz/c is the phase difference between the electric fields of the counterpropagating beat notes. If the slowing length satisfies the condition of $z\ll c/\delta ,$ the χ remains approximately constant. Each transition shown in table 1 is associated with a Rabi frequency defined by

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{ij}^{R}(z,t)\equiv \displaystyle \frac{{d}_{ij}E(z,t)}{\hslash }=\displaystyle \frac{{r}_{ij}dE(z,t)}{\hslash }.\end{eqnarray} \tag{ 8 }$$

Where d_ij is the electric-dipole transition matrix element between states $| i\rangle$ and $| j\rangle ,$ and d is the total A²Π → X²Σ⁺ (or B²Σ⁺ → X²Σ⁺) dipole transition moment, $d\ =\langle {\rm{\Lambda }}^{\prime} ,S,{\rm{\Sigma }}^{\prime} | {T}_{q}^{(1)}(d)| {\rm{\Lambda }},S,{\rm{\Sigma }}\rangle .$ We can find that the same optical field can simultaneously drive multiple transitions at different Rabi frequencies with reference to table 1. The Rabi frequency ${{\rm{\Omega }}}_{ij}^{R}$ can be written into two terms:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{ij}^{R}(z,t)=\displaystyle \frac{1}{2}({{\rm{\Omega }}}_{ij}{{\rm{e}}}^{-{\rm{i}}\omega t}+{{\rm{\Omega }}}_{ij}^{* }{{\rm{e}}}^{{\rm{i}}\omega t}).\end{eqnarray} \tag{ 9 }$$

Combining equations (7) and (8) we can find that

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{ij}(z,t)=\displaystyle \frac{4{r}_{ij}d{E}_{0}}{\hslash }(\cos (kz)\cos (\delta t)\cos (\chi /2)\\ &&\,\,\,\,+i\,\sin (kz)\sin (\delta t)\sin (\chi /2)).\end{eqnarray} \tag{ 10 }$$

In which we define Rabi frequency amplitude ${{\rm{\Omega }}}_{ij}^{0}$ as

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{ij}^{0}\equiv \displaystyle \frac{{r}_{ij}d{E}_{0}}{\hslash }.\end{eqnarray} \tag{ 11 }$$

The Hamiltonian for the multilevel system in the bichromatic field consists of the zero-field energies and the interaction energies between the system and the light field. This Hamiltonian is given by

$$\begin{eqnarray}&&\displaystyle \frac{H}{\hslash }=\left(\displaystyle \sum _{i}{\omega }_{{e}_{i}}| {e}_{i}\rangle \langle {e}_{i}| +\displaystyle \sum _{l}{\omega }_{{g}_{l}}| {g}_{l}\rangle \langle {g}_{l}| \right)\\ &&\,\,-\left(\displaystyle \sum _{i,l}{{\rm{\Omega }}}_{{e}_{i}{g}_{l}}^{R}| {e}_{i}\rangle \langle {g}_{l}| +{\rm{c.c}}\right).\end{eqnarray} \tag{ 12 }$$

Here $| {g}_{l}\rangle$ is the ground state and $| {e}_{i}\rangle$ is the excited state.

The equations of motion of the density matrix, including the contribution from spontaneous emission, are given by

$$\begin{eqnarray}&&\dot{\rho }=\displaystyle \frac{1}{{\rm{i}}\hslash }[H,\rho ]+{\left(\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}t}\right)}_{sp}.\end{eqnarray} \tag{ 13 }$$

In the rotating-wave approximation, the full set of equations of motion of density matrix of a multilevel system with an upper-state manifold of N_e levels and a lower-state manifold of N_g levels are given by

$$\begin{eqnarray}&&\begin{array}{c}{\dot{\rho }}_{{e}_{i}{e}_{i}}=\displaystyle \sum _{m}\text{Im}[{{\rm{\Omega }}}_{{e}_{i}{g}_{m}}^{\ast }{\tilde{\rho }}_{{e}_{i}{g}_{m}}]-{{\rm{\Gamma }}}_{i}{\rho }_{{e}_{i}{e}_{i}}\\ {\dot{\rho }}_{{e}_{i}{e}_{j}}=\displaystyle \frac{{\rm{i}}}{2}\left[2({\omega }_{{e}_{j}}-{\omega }_{{e}_{i}}){\rho }_{{e}_{i}{e}_{j}}-\displaystyle \sum _{m}({{\rm{\Omega }}}_{{e}_{j}{g}_{m}}^{\ast }{\tilde{\rho }}_{{e}_{i}{g}_{m}}-{{\rm{\Omega }}}_{{e}_{i}{g}_{m}}{\tilde{\rho }}_{{e}_{j}{g}_{m}}^{\ast })\right]\\ \,\,\,\,-\displaystyle \frac{{{\rm{\Gamma }}}_{i}+{{\rm{\Gamma }}}_{j}}{2}{\rho }_{{e}_{i}{e}_{j}}\\ {\dot{\tilde{\rho }}}_{{e}_{i}{g}_{l}}=\displaystyle \frac{{\rm{i}}}{2}\phantom{\rule[-0000]{}{3.5ex}}[2(\omega +{\omega }_{{g}_{l}}-{\omega }_{{e}_{i}}){\tilde{\rho }}_{{e}_{i}{g}_{l}}-{{\rm{\Omega }}}_{{e}_{i}{g}_{l}}({\rho }_{{e}_{i}{e}_{i}}-{\rho }_{{g}_{l}{g}_{l}})\\ \,\,\,\,+\left.\displaystyle \sum _{m\ne l}{{\rm{\Omega }}}_{{e}_{i}{g}_{m}}{\rho }_{{g}_{m}{g}_{l}}-\displaystyle \sum _{k\ne i}{{\rm{\Omega }}}_{{e}_{k}{g}_{l}}{\rho }_{{e}_{i}{e}_{k}}\right]-\displaystyle \frac{{{\rm{\Gamma }}}_{i}}{2}{\tilde{\rho }}_{{e}_{i}{g}_{l}}\\ {\dot{\rho }}_{{g}_{l}{g}_{l}}=\displaystyle \sum _{k}(-\text{Im}[{{\rm{\Omega }}}_{{e}_{k}{g}_{l}}^{\ast }{\tilde{\rho }}_{{e}_{k}{g}_{l}}]+{{\rm{\Gamma }}}_{kl}{\rho }_{{e}_{k}{e}_{k}})\\ {\dot{\rho }}_{{g}_{l}{g}_{n}}=\displaystyle \frac{{\rm{i}}}{2}\left[2({\omega }_{{g}_{n}}-{\omega }_{{g}_{l}}){\rho }_{{g}_{l}{g}_{n}}-\displaystyle \sum _{k}({{\rm{\Omega }}}_{{e}_{k}{g}_{n}}{\tilde{\rho }}_{{e}_{k}{g}_{l}}^{\ast }-{{\rm{\Omega }}}_{{e}_{k}{g}_{l}}^{\ast }{\tilde{\rho }}_{{e}_{k}{g}_{n}})\right].\end{array}\end{eqnarray} \tag{ 14 }$$

Where ${\tilde{\rho }}_{{e}_{i}{g}_{l}}\equiv {\rho }_{{e}_{i}{g}_{l}}{{\rm{e}}}^{{\rm{i}}\omega t},$ ${{\rm{\Gamma }}}_{i}$ is the total decay rate of the excited state $| {e}_{i}\rangle$ and ${{\rm{\Gamma }}}_{kl}$ is a specific channel decay rate. Here we should know the sums over an excited state index (i, j, k) from 1 to N_e and a ground state index (l, m, n) from 1 to N_g. Numerical solutions to this set of equations are computed using the built-in numerical differential equation solver in MatLab. The average force on the multilevel system is given by

$$\begin{eqnarray}&&F(r,t)=-Tr(\rho {\rm{\nabla }}H)=\hslash \displaystyle \sum _{i,l}\mathrm{Re}({\tilde{\rho }}_{{e}_{i}{g}_{l}}{\rm{\nabla }}{{\rm{\Omega }}}_{{e}_{i}{g}_{l}}^{\ast }).\end{eqnarray} \tag{ 15 }$$

In the two-level systems, the optimal Rabi frequency amplitude is ${\rm{\Omega }}=\sqrt{\mathrm{3/2}}\delta$ and the corresponding laser irradiance for each traveling wave is given by

$$\begin{eqnarray}&&I=2{I}_{s}{\left(\displaystyle \frac{{\rm{\Omega }}}{{\rm{\Gamma }}}\right)}^{2}=3{I}_{s}{\left(\displaystyle \frac{\delta }{{\rm{\Gamma }}}\right)}^{2}.\end{eqnarray} \tag{ 16 }$$

Where I_s = πhcΓ/(3λ³) is the saturation irradiance and Γ is the upper-level decay rate. Whereas in the multilevel systems each of the lower states can have a different electric dipole coupling to a given upper level as shown in table 1. So the same optical field can simultaneously drive multiple transitions at different Rabi frequencies. For the given bichromatic detunings δ, to find the optimal laser irradiance, we need to utilize a total Rabi frequency amplitude Ω^tot to describe our system. The Ω^tot is defined as the quadrature sum of the each Rabi amplitude,

$$\begin{eqnarray}&&{{\rm{\Omega }}}^{{\rm{t}}{\rm{o}}{\rm{t}}}=\sqrt{{({{\rm{\Omega }}}_{{e}_{j}{g}_{1}}^{0})}^{2}+{({{\rm{\Omega }}}_{{e}_{j}{g}_{2}}^{0})}^{2}+\cdots }.\end{eqnarray} \tag{ 17 }$$

From equations (11) and (17) we can express each Rabi amplitude as

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{e}_{j}{g}_{m}}^{0}=\displaystyle \frac{{r}_{{e}_{j}{g}_{m}}}{\sqrt{\displaystyle \sum _{m}{({r}_{{e}_{j}{g}_{m}})}^{2}}}{{\rm{\Omega }}}^{{\rm{t}}{\rm{o}}{\rm{t}}}.\end{eqnarray} \tag{ 18 }$$

Here, the index m denotes all possible Rabi oscillations for the excited state $| {e}_{j}\rangle$ for π-polarized light in our work.

Using the relations $I=(\mathrm{1/2}){\varepsilon }_{0}c{E}_{0}^{2}$ and

$$\begin{eqnarray}&&{\rm{\Gamma }}=\displaystyle \frac{1}{3\pi {\varepsilon }_{0}\hslash {c}^{3}}\displaystyle \sum _{p}{\omega }_{jp}^{3}{d}_{jp}^{2},\end{eqnarray} \tag{ 19 }$$

we can obtain

$$\begin{eqnarray}&&I=\displaystyle \frac{\pi hc{\rm{\Gamma }}}{3{\lambda }^{3}}\displaystyle \frac{1}{{f}_{00}\displaystyle \sum _{m}{r}_{jm}^{2}}\displaystyle \frac{2{({{\rm{\Omega }}}^{{\rm{t}}{\rm{o}}{\rm{t}}})}^{2}}{{{\rm{\Gamma }}}^{2}}.\end{eqnarray} \tag{ 20 }$$

Here, Γ is the spontaneous decay rate of the upper level, ω_jp is the resonance frequency between the excited state $| {e}_{j}\rangle$ and the ground state $| {g}_{p}\rangle ,$ the index p denotes all possible final states in the decay of the excited state. We have made the approximation that ω_jp = 2πc/λ for all final states. In the numerical simulations we can find the optimal Ω^tot to determine the required laser irradiance I.

From figure 1 and table 1 we can find that the dark states exist in the ground state X²${{\rm{\Sigma }}}_{\mathrm{1/2}}^{+}$ (v = 0, N = 1), in which F = 2, m = ±2 these two states are dark states for π-polarized light. In the absence of a remixing mechanism, population will accumulate in these dark states and cycling will cease, which causes the BF to rapidly diminish to zeros. Here, we apply a magnetic field with magnitude B₀ at an angle θ_B relative to the laser polarization to remix dark states.

In the presence of an external magnetic field B, the Zeeman Hamiltonian is well approximated by, in the space-fixed axis system,

$$\begin{eqnarray}&&{H}_{z}={g}_{s}{\mu }_{{\rm{B}}}{\bf{S}}\cdot {\bf{B}}+{g}_{{\rm{L}}}{\mu }_{{\rm{B}}}{\bf{L}}\cdot {\bf{B}}-{g}_{{\rm{I}}}{\mu }_{{\rm{N}}}{\bf{I}}\cdot {\bf{B}}.\end{eqnarray} \tag{ 21 }$$

Here the electron g-factor is ${g}_{s}\approx 2.002,$ the electron orbital g-factor is ${g}_{{\rm{L}}}\approx 1,$ and the nuclear g-factor is ${g}_{{\rm{I}}}\approx 5.585;$ ${\mu }_{{\rm{B}}}$ is the Bohr magneton, and ${\mu }_{{\rm{N}}}$ is the nuclear magneton. Since ${\mu }_{{\rm{N}}}/{\mu }_{{\rm{B}}}\approx \mathrm{1/1836},$ the last term can be neglected and the Zeeman Hamiltonian can be expressed as

$$\begin{eqnarray}&&\begin{array}{c}{H}_{z}=\displaystyle \sum _{m,n}\langle {g}_{m}| {g}_{s}{\mu }_{{\rm{B}}}{\bf{S}}\cdot {\bf{B}}+{g}_{{\rm{L}}}{\mu }_{{\rm{B}}}{\bf{L}}\cdot {\bf{B}}| {g}_{n}\rangle \\ \,\,\,+\displaystyle \sum _{i,j}\langle {e}_{i}| {g}_{s}{\mu }_{{\rm{B}}}{\bf{S}}\cdot {\bf{B}}+{g}_{{\rm{L}}}{\mu }_{{\rm{B}}}{\bf{L}}\cdot {\bf{B}}| {e}_{j}\rangle .\end{array}\end{eqnarray} \tag{ 22 }$$

Here magnetic coupling between ground states and excited states is neglected because this coupling is small compared with optical coupling. For the A²Π → X²Σ⁺ transition, mixing caused by magnetic fields in the excited state is negligible since the A²Π_1/2 state lacks a magnetic moment. The magnetic Hamiltonian in equation (22) can be evaluated using equations (A4) and (A5) in the appendix. We should note that the eigenstates of F = N = 1 states in X²${{\rm{\Sigma }}}_{\mathrm{1/2}}^{+}$ are of mixed and not pure J. The applied skewed magnetic field will cause Larmor precession of the ground-state Zeeman sublevels. This precession will act to mix the bright states with the dark states so that the dark state destabilization is realized.

We firstly use the simplified modeling without considering the high vibrational levels but the effects of small internal splittings and degeneracies are considered, including fine structure and hyperfine structure as well as the magnetic quantum numbers, as shown in figure 1. In the modeling, a full 16-level simulation of the system is carried out (for X²${{\rm{\Sigma }}}_{\mathrm{1/2}}^{+}$ (v = 0, N = 1), 12 magnetic sublevels; for A²Π_1/2 (v' = 0, J' = 1/2), four magnetic sublevels). To obtain the optimum BF, we first should investigate the influences of the main parameters on force.

From table 1 and figure 2, we can find that for each of the four sublevels in the excited state, the factor $\displaystyle \sum _{m}{r}_{jm}^{2}$ in equation (20) is 1/3 for π-polarized light, so the same laser irradiance can drive all transitions with the same Ω^tot. Using the optimal χ from the two-level systems and the optimal Ω^tot referred to [15], which has been verified in our work, the remaining parameters, including the magnitude and angle of the magnetic field and carrier frequency detuning from the transition center of mass of the system (CM), are surveyed within several velocities near zeros, in order to find optimal parameters. Our simulation results show that the optimal magnitude and angle of the magnetic field are not unique. Using those optimal parameter values at each of δ = 15Γ, 25Γ, 50Γ, simulations across the full velocity range are carried out as shown in figure 3, in which the Γ/k = 7.9 m s⁻¹ for the A → X transition in MgF. The velocity capture range and the velocity-averaged force near zero are summarized in table 3 in the columns labeled as method (a).

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In a real molecular transition, the high vibrational levels should be taken into account because in the actual optical cycles the molecule will be lost into some inaccessible high vibrational levels. For the A → X transition, the Franck–Condon factor f₀₀ of 0.9978 means that the molecules will be lost to out-of-system in hundreds of spontaneous decay cycles without the repumping lasers. Therefore, a 16-level simulation adding a high vibrational level should be carried out and this model evaluating the BF on molecule is more accurate [15]. Clearly the available interaction time is constrained by out-of-system radiative decays, hence a continuous repumping laser field should be used to transfer population from the dark state to the same excited state used by the cycling BF transition. Using the optimal parameter values from the simplified model at each bichromatic detuning, the BF as a function of repumping laser Rabi frequency is surveyed. For each of the several detunings δ in the figure 4, there is an optimum value for the repumping laser irradiance and the optimum repumping Rabi frequency is about $0.2\sqrt{\delta {\rm{\Gamma }}}.$ Here the integral time of computation of the time-dependence of the density matrix is about 10 μs, with longer integral time the amplitude of the BF will be to zero without repumping laser but it will be almost unchanged under the repumping laser irradiance. Taking into account that it will take too much time to complete the computation for longer integral time, the integral time is set to a suitable value of 10 μs. Then, using those parameters, simulations across the full velocity range are carried out and the calculated results are summarized in table 3 in the columns labeled as method (b).

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In this paper, another more simplified modeling method to evaluate the BF on the MgF is carried out. Using this modeling method, the computation of the time-dependence of the density matrix requires much less time compared with the other two modelings without giving up a great deal of accuracy. As shown in figure 2, no ground-state sublevel is laser-coupled to more than one excited-state sublevel, so that the entire system can be viewed as two three-level and two four-level lambda-type subsystems [15]. In this modeling, we think these four subsystems are independent by neglecting the radiative coupling between them and proportionally increase the decay rates within each subsystem to compensate for the neglected decays. Using the optimal Ω^tot and χ at each of δ = 10Γ, 15Γ, 25Γ and 50Γ, each of the four subsystems is independently simulated for a range of BF carrier frequencies. Then the results from each subsystem are combined together with a weighting given by the fraction of the ground state sublevels included in each subsystem. Figure 5 shows the calculated averaged force with near-zero velocities as a function of the BF carrier frequencies and indicates there is an optimum in the carrier frequencies for each of the detunings δ. As shown in figure 5, the averaged force peaks at a carrier frequency detuned from the hyperfine center of mass, at a location near the F = 0. This carrier frequency is compromise between two three-level subsystems and two four-level subsystems. Because transitions from F = 1⁻ are the strongest transitions in the three-level subsystems but in the four-level subsystems transitions from F = 1⁺ are the strongest transitions. Using the optimal carrier frequencies, simulations across the full velocity range are carried out and the calculated results are summarized in table 3 in the columns labeled as method (c).

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Comparing the results for three different methods estimating the BF in the 16-level A²Π → X²Σ⁺ Q₁₂(0.5)/P₁₁(1.5) system in MgF, as shown in table 3, we can find that the calculated results using methods (a) and (b) are almost consistent in the velocity range and force as well as average excited state fraction. However, the calculations using method (a) are not only simpler, but the computational time needed in method (a) is much less than that in method (b). It is also clear that the subsystem approach consistently underestimates the force and velocity range as well as average excited state fraction.

Without the repumping laser, the molecules will be lost to out-of-system in hundreds of spontaneous decay cycles in MgF. The characteristic out-of-system decay time can be evaluated by

$$\begin{eqnarray}&&T=\displaystyle \frac{{\tau }_{A}}{(1-{f}_{00}){P}_{e}}.\end{eqnarray} \tag{ 23 }$$

Where τ_A is the radiative decay lifetime of the A state, P_e is the average excited state fraction. The P_e can be attained by averaging the calculated excited state fraction over the velocity range in which the force is large. The result is P_e ≈ 1/8 and we can estimate an out-of-system decay time of about 25 μs. Actually, in our calculations we find the out-of-system decay time is about 100 μs. Because when the integral time is set to 100 μs, at last, the BF will be to zero without a repumping laser.

The source of MgF in our simulations is a cryogenic buffer gas cell [28], the final output diameter of molecular beam is 1.5 mm and the transverse velocity is ±1 m s⁻¹ after two times collimation [29]. Five cm after the collimating aperture, the beam enters the bichromatic-force slowing and cooling region. We assume that the MgF in the beam has a forward velocity of v_∣∣ = 120 m s⁻¹ with a spread of Δv_∣∣ ≈ 40 m s⁻¹. In the longitudinal slowing of MgF molecular beam, the bichromatic laser beams are tightly focused to waists with a top-hat diameter of 1 mm to provide the required irradiance. Using the BF with a detuning of δ = 15Γ, a cryogenic buffer-gas-cooled beam of MgF can be decelerated nearly to 4 m s⁻¹ within a longitudinal distance of about 0.5 cm and the efficiency is about 6.23% with respect to the source of MgF, taking into account that the molecular beam flies free for a distance of 5 cm before it is illuminated by bichromatic lasers. Under this condition using equation (20), the laser irradiance for each traveling wave is 127 W cm⁻² and the corresponding laser power is 0.994 W. The time to slow is about 1.08 ms but the out-of-system decay time is about 100 μs, so the repumping laser should be added. Using these three different methods at each of δ = 15Γ, 25Γ and 50Γ, estimated values for the key properties relevant to longitudinal slowing of MgF molecular beam are summarized in table 3. When the laser irradiances for each traveling wave are 351 W cm⁻² and 1405 W cm⁻², respectively, the corresponding laser powers is 2.757 W and 11.035 W. With the limitation of the laser technique, nowadays the required powers for the δ of 25Γ and 50Γ cannot be realized. But these results indicate that as long as the required laser irradiance is available, the BF can provide larger force and much better slowing results.

We also compare the results of the A → X system and the B → X system in MgF at detuning δ of 10Γ, as shown in table 4. For each of the systems, Γ is the spontaneous decay rate of the upper level. The saturation irradiance of B → X system is almost twice as much as that of A → X system, so laser irradiance for each traveling wave of B → X system is almost twice that of the A → X system. In addition, we find that the velocity range in A → X system is larger than that in B → X system because the velocity range is proportional to δ/k. However, in the A → X system Γ/k = 7.9 m s⁻¹, but in the B → X system Γ/k = 4.9 m s⁻¹. The deceleration efficiency in the A → X system is also twice that of the A → X system when the MgF molecular beam is slowed by 60 m s⁻¹. From these results, we can find that the A → X transition in MgF is more suitable for the BF cooling and slowing.