Laser cooling with adiabatic passage for type-II transitions

Abstract

We extend the idea of laser cooling with adiabatic passage to multi-level type-II transitions. We find the cooling force can be significantly enhanced when a proper magnetic field is applied. That is because the magnetic field decomposes the multi-level system into several two-level sub-systems, hence the stimulated absorption and stimulated emission can occur in order, allowing for the multiple photon momentum transfer. We show that this scheme also works on the laser-coolable molecules with a better cooling effect compared to the conventional Doppler cooling. A reduced dependence on spontaneous emission based on our scheme is observed as well. Our results suggest this scheme is very feasible for laser cooling of polar molecules.

Graphical abstract

Keywords

laser cooling of polar molecule / adiabatic passage / type-II transition / cold molecule / cold atom

1 Introduction

Allowing fast producing and precise manipulation of cold atomic samples [1–3], laser cooling technique has revolutionized the field of atomic, molecular, optical and quantum physics during last several decades [4–8]. For conventional Doppler cooling on atoms such as alkali atoms, the main cooling laser is typically red-detuned with only single frequency (without consideration on the repump laser). It is good enough for most laser-coolable atomic species. But for some special cases, multi-frequency cooling schemes [9], including adiabatic rapid passage [10–12] and bichromatic force [13–16], have been proposed and predicted to have a better performance, but with the price of increasing the complexity of the system. However, some features, for example, stronger cooling forces and a weak dependence on spontaneous emission [9], make them potentially be applied to direct laser cooling of molecules where the cooling transition is quasi-closed [17–20]. Besides the leakage channels, the energy levels for molecules are complicated and the type-II transitions (in which the total angular momentum quantum number in the ground state J is no less than that in the excited state J^{\prime }) are dominant [21, 22], making the cooling process much more complex and the Doppler cooling force much weaker than those in atoms. To achieve a better cooling efficiency, it might be worthwhile to sacrifice the simplicity of the cooling scheme by introducing a light field with multiple frequencies [23].

The mechanism of laser cooling can be explained with an atom-light interaction approach in either semi-classical or quantum picture [24–26]. Briefly speaking, in Doppler cooling process, the momentum from photons is transferred to atoms by scattering the photons. The Doppler effect breaks the symmetry in the momentum space, making the atoms focus to the zero momentum [24]. The strength of the cooling force depends on how fast the momentum exchange is repeated, that is, the decay rate from the excited state. Once successive scattering is achieved in a closed cycling transition, the force would only be limited by the spontaneous decay rate Γ. For a two-level system, the maximum force is F_{\text{max}}=\hslash k\mathrm{\Gamma }/2 with \hslash k the photon momentum. The cooling velocity range and the temperature limit also depend on the decay rate Γ [24].

One way to go beyond the limit of the spontaneous emission is using the stimulated emission[27]. Recently, an adiabatic-passage cooling scheme by rapidly sweeping the laser frequency in a sawtooth wave shape (Sawtooth-wave adiabatic passage, or SWAP, for short) has been theoretically and experimentally demonstrated in narrow-line transitions [28–31]and Raman transitions [32]. Such a scheme uses the stimulated emission to transfer the excited atoms back to the ground state, therefore strong forces can be achieved with a high sweeping rate even for a narrow transition. Let us consider a moving particle with a velocity of v in a light field, which is formed by two counter-propagating laser beams with the same time-dependent frequency \omega (t); see Fig. 1(a). Due to the Doppler effect, resonances of the particle with the two beams happen at different time. For a frequency ramping scheme shown in Fig. 1(b), an adiabatic excitation is first induced by the counter-propagating beam, followed by another adiabatic de-excitation process induced by the co-propagating beam. Ideally, the particle loses twice the photon momentum during one sweeping period, resulting in a maximum average force of F_{\text{SWAP}}=2\hslash k{T}^{-1} with T the sweep time in one period. To guarantee the adiabatic transfer, the Landau–Zener condition {\mathrm{\Omega }}^2\gg \alpha should be fulfilled [33], here Ω is the Rabi frequency and \alpha \equiv \frac{\text{d}\omega }{\text{d}t} is the frequency ramp speed.

Now an interesting question is whether the SWAP scheme can be extended to multi-level type-II transitions and further complex molecules. The type-II transitions generally show a weak Doppler cooling force [34, 35] due to the remixing process of the dark states via either rapidly switching the polarization of the light [18, 36] or introducing a magnetic field that has an angle to the laser beams to re-define the quantum axis [17, 37]. Here we consider the latter case. Different from the ideal two-level system, the problem of the multi-level systems lies in that the excitation and stimulated de-excitation pairs might happen out of order. The three polarization components of the two beams take part in the interaction, and therefore lead to the lost of the resonance order. This will degrade the efficiency of the momentum transfer.

Lots of molecules that have been directly laser cooled, such as ⁸⁸SrF, ⁴⁰CaF, have a 4+12 hyperfine structure (under weak magnetic field), i.e., there are 4 sublevels in the A^2{\mathrm{\Pi }}_{1/2} state and 12 sublevels in the X^2{\mathrm{\Sigma }}_{1/2} state, as illustrated in Fig. 1(c). It makes the momentum transfer much more complicated. In this work, we apply a time-dependent master equation approach to deal with the interaction between a multi-level system and a frequency-chirped light field for the one-dimensional case. Then, we investigate the SWAP cooling effect for type-II transitions in Section 3, where the motion of the particles is classically treated. Note that our work is different from the discussion on the Raman transition in Ref. [32] where the frequency ramp is much narrower than the ground-state splitting, and so they require an explicit optical pumping step to reinitialize the state matrix before every frequency ramp. Here we use the frequency ramp that covers the entire hyperfine manifold and addresses all sublevels in each ramp. Next we implement the SWAP scheme on the molecule to analyze its cooling effect in Section 4. Finally, in Section 5, we consider a three-level model with one dark state to check the resistance of the SWAP scheme on the unwanted loss channels.

2 Time-dependent master equation approach

2.1 Semi-classical treatment

Without loss of generality, we consider the interaction between a multi-level particle and a multi-frequency laser field. An arbitrary multi-frequency light field can be decomposed as

\begin{array}{}\text{(1)}& \mathit{E}={\displaystyle \sum _{n\mathrm{,}q}}\frac{1}{2}{E}_n{M}_{nq}{\widehat{ϵ}}_q{\text{e}}^{\text{i}{\mathit{k}}_n\cdot \mathit{z}}{\text{e}}^{-\text{i}{\beta }_n(t)}+c\mathrm{.}c\mathrm{.,}\end{array}

where n is the laser beam index with frequency {\omega }_n and wave vector (propagating direction) {\mathit{k}}_n, and E_n is the amplitude of the n-th beam. The wave number k_n here can be approximately treated as constant for simplicity. q=\mathrm{0,}\pm 1 correspond to π, {\sigma }^{\pm } polarizations respectively. The polarization vectors under the Cartesian coordinate axis are: {\widehat{ϵ}}_0={\widehat{e}}_z, {\widehat{ϵ}}_{\pm }=\mp ({\widehat{e}}_x\pm \text{i}{\widehat{e}}_y)/\sqrt{2}. Here the Cartesian coordinate is assumed to have a specific orientation to the laser beam. |M_{nq}{|}^2 gives the intensity fraction of the light field for each polarization component in the n-th beam. {\beta }_n(t)\equiv {\displaystyle {\int }_{t_0}^t}{\omega }_n(t^{\prime })\text{d}{t}^{\prime } is the accumulated phase of the n-th laser beam from the initial time t₀.

Using Eq. (1), the Hamiltonian in the interaction picture is

\begin{array}{}\text{(2)}& H=\frac{\hslash }{2}{\displaystyle \sum _{u\mathrm{,}l}}{\displaystyle \sum _{n\mathrm{,}q}}{\mathrm{\Omega }}_{ul}^{nq}|u\rangle \langle l|+h\mathrm{.}c\mathrm{.,}\end{array}

with |l\rangle and |u\rangle represent the ground and excited states respectively, and

\begin{array}{}\text{(3)}& {\mathrm{\Omega }}_{ul}^{nq}={\text{e}}^{\text{i}{\mathit{k}}_n\cdot \mathit{z}}{\text{e}}^{-\text{i}{\text{Θ}}_{nq}^{ul}(t)}{M}_{nq}{A}_{ul}^{(q)}{\mathrm{\Omega }}_n\mathrm{.}\end{array}

Here {\mathrm{\Omega }}_n is the total Rabi frequency of the n-th beam. {\mathrm{\Theta }}_{nq}^{ul}(t)\equiv {\displaystyle {\int }_{t_0}^t}[{\omega }_n(t^{\prime })-{\omega }_{ul}+{\delta }_{ul}(B)]\text{d}{t}^{\prime } is the time-dependent phase of the laser detuning. {\omega }_n(t) has a sawtooth shape versus t as shown in Fig. 1(b), and {\omega }_{ul} is the resonant frequency for the |l\rangle \to |u\rangle transition. {\delta }_{ul}(B)={\delta }_l(B)-{\delta }_u(B), where {\delta }_{l(u)}(B) is the energy shift of the sublevel |l\rangle (|u\rangle ) under a magnetic field with strength of B. A_{ul}^{(q)} are the relative transition strengths for all electric dipole allowed transitions.

The force can be derived from the gradient of the total energy of the system, that is,

\begin{array}{}\text{(4)}& \widehat{F}=-\nabla H=-\frac{\hslash }{2}{\displaystyle \sum _{u\mathrm{,}l}}{\displaystyle \sum _{n\mathrm{,}q}}(\nabla {\mathrm{\Omega }}_{ul}^{nq})|u\rangle \langle l|+h\mathrm{.}c\mathrm{.,}\end{array}

then the expectation value of the force for a system represented by a density matrix ρ is \langle F\rangle =\text{Tr}(\rho \widehat{F}). The motion of the particles is treated classically. By letting z=vt in Eq. (3), the Hamiltonian (2) and thus the force operator (4) become velocity dependent.

The time evolution of the density matrix ρ is determined by the master equation

\begin{array}{}\text{(5)}& \frac{\partial \rho }{\partial t}=\frac{1}{\text{i}\hslash }[H\mathrm{,}\rho ]+\widehat{\mathcal{L}}(\rho )\mathrm{,}\end{array}

where the Linblad operator

\begin{array}{}\text{(6)}& \widehat{\mathcal{L}}(\rho )=\frac{1}{2}{\displaystyle \sum _{ul}}(2C_{ul}\rho C_{ul}^{†}-C_{ul}^{†}{C}_{ul}\rho -\rho C_{ul}^{†}{C}_{ul})\end{array}

with the collapse operator C_{ul}={\displaystyle {\sum }_q}{A}_{ul}^q\sqrt{\mathrm{\Gamma }}|l\rangle \langle u|. We start with an initial state and evolve the system in time until the density matrix finally changes periodically with the frequency sweeping. Then we calculate the average force over one SWAP ramp period T.

2.2 Remixing of the Zeeman dark state

For laser cooling with the type-II transitions and molecules, a magnetic field B is introduced to eliminate the dark states. The direction of the magnetic field re-defines the local quantum axis. We assume the field is on the lab coordinate yz plane and has an angle of θ to the z axis. So the Cartesian components of the polarization vector {\widehat{ϵ}}_{\mathrm{0,}\pm } is rotated by transforming from \left\{{\widehat{e}}_x\mathrm{,}{\widehat{e}}_y\mathrm{,}{\widehat{e}}_z\right\} of the old frame to \left\{{{\widehat{e}}^{\prime }}_x\mathrm{,}{{\widehat{e}}^{\prime }}_y\mathrm{,}{{\widehat{e}}^{\prime }}_z\right\} of the re-defined frame. The new M vector for the three polarizations in each laser beam can be obtained from the old one, i.e.,

\begin{array}{}\text{(7)}& \left(\text{}\begin{array}{l}{M}_{+1}^{\prime }\hfill \\ M_0^{\prime }\hfill \\ M_{-1}^{\prime }\hfill \end{array}\text{}\right)\text{}\text{}\text{}=\text{}\text{}\frac{1}{2}\text{}\text{}\left(\text{}\text{}\begin{array}{ccc}1+\cos \theta & \sqrt{2}\sin \theta & 1-\cos \theta \\ -\sqrt{2}\sin \theta & 2\cos \theta & \sqrt{2}\sin \theta \\ 1-\cos \theta & -\sqrt{2}\sin \theta & 1+\cos \theta \end{array}\right)\text{}\text{}\text{}\text{}\left(\text{}\text{}\begin{array}{l}{M}_{+1}\hfill \\ M_0\hfill \\ M_{-1}\hfill \end{array}\text{}\text{}\right)\text{}\text{}\mathrm{.}\end{array}

In our calculations below, we consider the one-dimensional case that the particle moves along \widehat{z}-axis for simplicity. A light field from two counter-propagating beams in {\sigma }^{+}–{\sigma }^{-} configuration [the polarization here is defined by the lab \widehat{z}-axis before the transformation with Eq. (7)] is applied, and the two beams have an equal Rabi frequency Ω. When the magnetic field is carefully chosen (\theta \ne 0), they contain all three polarization components and thus can effectively eliminate all of the Zeeman dark states. We first investigate the possible type-II transitions in Fig. 1(c) under a weak magnetic field, and then turn our attention to strong magnetic field regime.

3 SWAP cooling for type-II transitions

A typical 4+12 energy structure for the diatomic molecule is shown in Fig. 1(c), there are three different multi-level type-II transitions: (i) F=1\to F^{\prime }=0, (ii) F=1\to F^{\prime }=1 and (iii) F=2\to F^{\prime }=1. We first focus on the case (i) and use the parameters for the F=1^{(-)}\to F^{\prime }=0 transitions for the typical laser coolable diatomic molecule (in our case, it is BaF, but the conclusions are similar with other molecules). The Landé g-factor for the ground state is g_{F=1}=-0.51, and a magnetic field of B=5\hspace{0.33em}\text{G} (\theta =\pi /2) leads to a Zeeman splitting of {\mathrm{\Delta }}_z=\left|{\mu }_B{g}_{F}B\right|/\hslash \sim 1.25\mathrm{\Gamma } for \mathrm{\Delta }{m}_F=1 (Here {\mu }_B/\hslash \approx 1.4\text{MHz}/\text{G}).

To check the effect of the magnetic field on the SWAP cooling scheme, we first consider a low laser intensity case, i.e., a small Rabi frequency {\mathrm{\Omega }}_p=\mathrm{\Omega }=10\mathrm{\Gamma } for each beam. To fulfill the adiabatic condition |M_{pq}{A}_{ij}^{(q)}{\mathrm{\Omega }}_p{|}^2\gg \alpha, we choose the frequency ramp speed \alpha =4{\mathrm{\Gamma }}^2 and the sweeping period T=5{\mathrm{\Gamma }}^{-1}. The ramp frequency should cover the resonant positions for all possible transitions[33], which gives the requirement for the sweeping range \alpha T,

\begin{array}{}\text{(8)}& \alpha T>2kv+2{\mathrm{\Delta }}_z\mathrm{.}\end{array}

The condition is well fulfilled for velocities shown in Fig. 2.

The cooling force has been enhanced in the small velocity range, as shown in Fig. 2. We label the critical velocity that characters this regime as v_b. To determine v_b, let us analyze the different cooling mechanisms for large and small velocities in the SWAP scheme for the F=1\to F^{\prime }=0 transition. For a relatively small velocity, the Doppler shift {\mathrm{\Delta }}_{\text{D}}=kv is smaller than the Zeeman shift {\mathrm{\Delta }}_z, as shown in Fig. 3(a), the time sequence of the stimulated excitation and de-excitation of the three transitions is still in order. At time t₁ in Fig. 3(a), the particles in Zeeman sublevel m_F=-1 get excited by the {\sigma }^{+} component in the counter-propagating beam, then the stimulated de-excitation back to m_F=-1 happens at time t₂ with the co-propagating beam. This is similar to the two-level case. As time goes on, this process happens for the other two Zeeman states. However, when t₃ is close to t₂ [see Fig. 3(a)], the de-excitation to the m_F=0 sublevel also happens. Competition between the de-excitation to m_F=-1 via the co-propagating beam and to m_F=0 via the counter-propagating beam certainly suppresses the net momentum transfer in one sweeping period. The critical velocity v_b is defined when t_2=t_3. Taking both the Doppler shift and the Zeeman shift into consideration, we have {\omega }_0-{\mathrm{\Delta }}_z+{\mathrm{\Delta }}_{\text{D}}={\omega }_0-{\mathrm{\Delta }}_{\text{D}}, resulting in

\begin{array}{}\text{(9)}& v_b={\mathrm{\Delta }}_z/(2k)\mathrm{.}\end{array}

In Fig. 2, a rapid decrease of the SWAP cooling force for velocities larger than v_b is consistent with the above discussion.

Moreover, there is an enhanced cooling effect in the small velocity region. The maximum achievable force holds about \sim 0.6\hslash k\mathrm{\Gamma }, which surpasses the limit that the above mechanism allowed. Such a strong force located near zero velocity should partly attribute to the nontrivial Bragg oscillations, which can induce more than 2\hslash k of momentum transfer during one sweeping period. As discussed in Ref. [33], the Bragg effect induced momentum change does not contribute to the cooling process, one should trace this effect out under the dressed state picture. However, these oscillations only play an important role at small velocity regime () and cause the force displaying an oscillatory behaviour that may restrict the cooling limit. For more details of the cooling behavior in the near-zero velocity regime, we discuss that in Appendix 1.

When , we can depict the physical picture by going to another limit {\mathrm{\Delta }}_{\text{D}}\gg {\mathrm{\Delta }}_z, as shown in Fig. 3(b). At time t₁, the {\sigma }^{+} component in the counter-propagating beam becomes on resonance for the m_F=-1\to m_F^{\prime }=0 transition. Since the Doppler shift is large, before the co-propagating beam reaches the resonant point, the de-excitations back to the other two ground sublevels already happen. The π and {\sigma }^{-} components in the counter-propagating beam first become near resonance and drive the stimulated emission processes from m_F^{\prime }=0 to m_F=0 and m_F=+1 respectively. Therefore, the exchanged momenta from excitation and de-excitation are in opposite directions, making the net momentum transfer smaller than \hslash k.

Another issue we should pay attention to is the role of the spontaneous emission in the sweeping process. The time order of momentum transfer will be messed up if the stimulated emission is replaced by the spontaneous emission, which would reduce the cooling force. Therefore, an effective SWAP cooling requires a low probability of spontaneous dacay, which means the time interval that a particle spends in the excited state should be less than the lifetime {\mathrm{\Gamma }}^{-1}. This effect also set an effective cooling velocity range . For the case in Fig. 3(a) where {\mathrm{\Delta }}_z\gg {\mathrm{\Delta }}_{\text{D}}, the time interval {\tau }_e=t_2-t_1=2{\mathrm{\Delta }}_{\text{D}}/\alpha. On the other hand, when {\mathrm{\Delta }}_{\text{D}}\gg {\mathrm{\Delta }}_z, they are {\tau }_{e1}=t_3-t_1={\mathrm{\Delta }}_z/\alpha and {\tau }_{e2}=t_2-t_4=2({\mathrm{\Delta }}_{\text{D}}-{\mathrm{\Delta }}_z)/\alpha. To guarantee all of the time intervals to fulfill the condition that , the velocity must be smaller than the upper limit v_r, where

\begin{array}{}\text{(10)}& v_r=\alpha /(2k\mathrm{\Gamma })\mathrm{.}\end{array}

Although we only discuss the SWAP cooling on the F=1\to F^{\prime }=0 transition, the results for other type-II transitions are similar. The complex multi-level system can always be decomposed into several two-level sub-systems if the magnetic field is large enough. And the particle will experience similar dynamics with absorption and stimulated emission in one sub-system. So we conclude that, in a multi-level transition, an excitation by the counter-propagating beam must be followed by an emission stimulated by the co-propagating beam to ensure a large momentum transfer. Anything that breaks the order will degrade the cooling effect. In order to achieve a better performance of the SWAP cooling, a relatively strong magnetic field is preferred to meet the condition that {\mathrm{\Delta }}_z\gg {\mathrm{\Delta }}_{\text{D}}. This also gives an ideal velocity range that \left|v\right|\ll v_b\le v_r, where the cooling effect is significantly enhanced. Of course, larger Zeeman shifts require larger sweeping range as to higher power of cooling lasers to ensure the adiabatic condition.

4 SWAP cooling for molecules

The results in the type-II transition in Section 3 indicate that application of the SWAP scheme on molecules might be possible. Just as an example, and for checking the conclusions derived above, we perform a numerical calculation for a real molecule (BaF). Without the external magnetic field, the energy gap {\mathrm{\Delta }}_{\text{hf}} between the F=1^{(-)} state and the F=2 state is about \sim 56\mathrm{\Gamma } for the ¹³⁸BaF molecule [20]. To avoid the population accumulating into sublevels that are dark, we have to sweep over all lower manifolds. It is almost impossible to apply a scheme that sweep over part of the manifolds and repump from the other, as the hyperfine splitting of the X^2{\mathrm{\Sigma }}_{1/2}\mathrm{,}N=1 for ¹³⁸BaF is not that large enough to guarantee that the molecule get cooling through SWAP without interacting with the repump laser. Therefore, a large frequency sweeping range \alpha T and a large total Rabi frequency Ω are required. Again, we first consider the case with a weak magnetic field B=10\text{ G}. The Zeeman shift {\mathrm{\Delta }}_z={\mu }_{B}B{m}_F{g}_F is much smaller than the hyperfine splitting {\mathrm{\Delta }}_{hf}. As shown in Fig. 4, even with \mathrm{\Omega }=100\mathrm{\Gamma }, the maximum SWAP force is about 0.1\hslash k\mathrm{\Gamma }, at the same magnitude with that for the conventional Doppler cooling. Although the enhancement does not appear, a wider coolable velocity region is still observed. It can be estimated from Eq. (10) that v_r\sim 50\mathrm{\Gamma }/k with \alpha =100{\mathrm{\Gamma }}^2, which indicates the effective cooling range.

The behaviours of the SWAP force under a large magnetic field are different; see Fig. 4. Compared with low field condition, the maximum achievable cooling force is 4× larger, that is, \sim 0.4\hslash k\mathrm{\Gamma }. The enhanced cooling velocity limit v_b is determined by the energy gaps between every two neighbouring sublevels, as discussed in Section 3. Note that in real molecules, the behaviours of the transition strength and the Zeeman splitting under a large magnetic field different from that under a weak magnetic field. One must take a different quantum label system and recalculate the Zeeman splitting; please see the Appendix 2 for more details. The energy gaps between the neighbouring Zeeman sublevels maintain around \sim 15\mathrm{\Gamma } at B=200\hspace{0.33em}\text{G}, leading to v_b\sim 7.5\mathrm{\Gamma }/k, which is consistent with the result in Fig. 4. On the other hand, the SWAP force has a weak velocity sensitivity, and the cooling forces for large velocities are similar to those with B=10\hspace{0.33em}\text{G}, approximate \sim 0.1\hslash k\mathrm{\Gamma }.

5 Resistance to leakage channels

For molecules, the closed cycling transitions generally can not be perfectly realized due to the existence of the leakage channels, such as the higher vibrational states and the intermediate Δ state [20, 38]. Since the SWAP cooling scheme employs the stimulated emission to transfer the excited particles back to the ground state, the fraction of spontaneous decay gets partially suppressed. Hence, compared to the conventional Doppler cooling, a smaller loss to the possible leakage channels might be expected. In order to understand such an effect, we analyse a simple three-level system, i.e., a two-level transition driven by the frequency-chirped cooling lasers with an additional loss channel (the leakage branch ratio is assumed to be \gamma =0.05) with our semi-classical model in Section 2; see Fig. 5(a). In the conventional Doppler cooling, \gamma =0.05 allows a maximum scattering photon number of {\gamma }^{-1}\sim 20 before the particles populate the leakage state. As shown in Fig. 5(b), with an evolution time of 50\mathrm{\Gamma }/k, the loss fraction reaches \sim 0.7 after scattering \sim 14 photons. However, under the SWAP scheme, with an equivalent evolution time, i.e., five sweeping periods, the loss {\mathcal{l}}_{\text{SWAP}} is several times smaller.

To quantitatively evaluate the resistance of the SWAP cooling to the leakage, we introduce a loss ratio defined by

\begin{array}{}\text{(11)}& R=r_{\text{SWAP}}/r_{\text{Conv}}\mathrm{.}\end{array}

Here r_{\text{SWAP}} indicates the population loss in the SWAP cooling once the moving particle changing its momentum by -\hslash k, resulting a definition as r_{\text{SWAP}}={\mathcal{l}}_{\text{SWAP}}\hslash k/\delta p_{\text{SWAP}} with \delta p_{\text{SWAP}}(\tau )=|{\displaystyle {\int }_0^{\tau }}\langle F(t)\rangle \text{d}t|. In the conventional Doppler cooling, r_{\text{Conv}}\sim \gamma. Figure 5(c) shows the loss ratios R for two different velocities with various Ω under the SWAP scheme. For a large velocity, for example, v=20\mathrm{\Gamma }/k, the ratios are always larger than one, which means that the cooling effect of the SWAP scheme is less significant than the conventional Doppler case as less momenta (\sim 10 photons for five periods) are transferred but the loss lies in a similar or larger level. This can be easily understood by introducing the interval time {\tau }_e\equiv 2kv/\alpha that the particle spends in the excited state. For the parameters we use here, it is estimated to be {\tau }_e\sim 1/\mathrm{\Gamma }. Considering the lifetime of the particle {\tau }_n=1/\mathrm{\Gamma }, the effect of swap scheme would be similar to that of the Doppler cooling, which explains the loss ratio R\sim 1.

For a small velocity, v=2\mathrm{\Gamma }/k, the loss ratio is typically around \sim 0.1. The SWAP scheme is much better compared with the conventional scheme. The reason lies in that, for a small velocity, {\tau }_e is small enough to suppress the spontaneous decay and make the stimulated emission dominate. For v=2\mathrm{\Gamma }/k, {\tau }_e happens to be 1/(10\mathrm{\Gamma }). Considering the ideal case that no spontaneous decay happens and \delta p_{\text{SWAP}}\sim 2\hslash k, r_{\text{SWAP}} is about \gamma /20, leading to the loss ratio R\sim 0.05. However, the effect is weakened due to the non-zero Γ. We can also find that with a better fulfillment of the adiabatic condition {\mathrm{\Omega }}^2\gg \alpha by increasing Ω, the ratio R decreases but always larger than the limit R=0.05, as shown Fig. 5(c). To get a suppression of the spontaneous decay and consequently achieve an effective SWAP cooling, the condition v\ll \alpha {\mathrm{\Gamma }}^{-1}/(2k) should be fulfilled. We claim here that, compared to the conventional Doppler cooling, a better resistance to the leakage channels in the SWAP cooling occurs in the small velocity region, i.e., v\ll \alpha {\mathrm{\Gamma }}^{-1}/(2k).

In the real molecular cooling experiment, scattering of thousands of photons is required, indicating that the vibrational repumping is still needed in the SWAP scheme. For the system in Fig. 5, a loss ratio of R\sim 0.1 generally means that only about a hundred of photons are scattered before the particle all populates the |g_1\rangle state. This tells us that the SWAP scheme still leads to non-negligible losses to the v=1 vibrational state, i.e., one should work with at least one extra v=1 vibrational repumping laser, but might get rid of the v=3 and even v=2 repumpers.

6 Conclusion

In summary, we have analyzed the SWAP cooling force for the multi-level type-II transitions and further diatomic molecules. The magnetic field that has an angle θ to the axis of laser beams cannot only remix the Zeeman dark states in the type-II transitions, but also introduce an enhancement to the SWAP cooling force at the small velocity region. When the Zeeman energy splitting is larger than the Doppler shift, the multi-level system can be decomposed into several two-level sub-systems in time ordering. Ideally, a momentum transfer of twice the photon momentum during one sweeping period is allowed and results in an enhanced cooling force for velocities below v_b.

The SWAP scheme does not require a sideband modulation, and is less sensitive to the long-term stabilization of the laser frequency. Unlike the bichromatic cooling, the frequencies of co- and counter-propagating beams of the SWAP scheme have no difference in the laboratory frame and can be done with a retro-reflected mirror. Such properties indicate an easier experimental realization, and the applications in laser slowing of a molecular beam might be possible as well if one can get access to both a high laser intensity and a rapid frequency ramp speed. For SWAP, the cooling force relies on stimulated emissions and the spontaneous emission rate is reduced, causing the resistance of the leakage channels. This will be useful to reduce the number of the vibrational repump lasers needed for an ordinary molecule that lacks a closed-cycling transition. Also, due to the independence on the spontaneous emission of the SWAP cooling, it will be beneficial for the narrow linewidth transition, which lacks large number of scattered photons, to provide an effective cooling force [29, 33].

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