Optical cycling of AlF molecules

The molecules that have been laser-cooled so far (apart from YO) consist of an alkaline earth and alkaline earth-like atom bonded to a fluorine, hydrogen or hydroxyl ligand to form a ²Σ⁺ ground state. These radicals have an unpaired electron which is localized at the metal atom and laser excitation promotes it without changing the bond length of the molecule. This leads to the very diagonal Franck–Condon matrix, which is typical for these molecules, and as a result the number of vibrational repump lasers required is two or three.

The situation is slightly different for AlF, which combines a group III element with fluorine to form a X¹Σ⁺ ground state. The AlF bond is mostly ionic ³ , with the 3p electron of Al being largely transferred to the fluorine atom [33]. Laser excitation promotes one electron from the 7σ² molecular orbital, which has primarily Al 3s character [34], into a 3π orbital which has primarily Al 3p character. This results in a A¹Π or a a³Π state, depending on the orientation of the spin of the electron. We have performed electronic structure calculations to better understand why the Franck–Condon matrix is diagonal, despite the lack of an unpaired electron. Figure 1 shows the calculated electron density and molecular orbitals of the electronic states relevant to this study. The electron density (top of each panel) has been calculated within the multi-reference-configuration-interaction (MRCI) method available in MOLPRO 2019.2 [35]. ⁴ The bottom part of each panel shows the HOMO of the X¹Σ⁺ electronic state, and the HOMO and LUMO for the A¹Π electronic state, respectively. The calculations are carried out by employing the AVQZ [36] basis set for each atom.

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The electron density in the X¹Σ⁺ and A¹Π state is very similar and the electronic excitation resembles that of an atomic transition between an s and p orbital [37]. It involves only a minimal change in the molecular bond which results in a diagonal Franck–Condon matrix and allows optical cycling with only a few vibrational repump lasers.

Spontaneous decay on the spin-forbidden A¹Π → a³Π transition is weakly allowed, due to spin–orbit mixing. This causes a small leak from the optical cycle below the 10⁻⁶ level which we have analyzed in detail previously [26, 38].

Figure 2(a) shows the relevant vibronic energy levels for the optical cycle of AlF, together with the transition wavelengths and calculated branching ratios. The spontaneous decay rate from A¹Π, v' = 0 → X¹Σ⁺, v'' = 0 is proportional to A₀₀, the Einstein A coefficient of the 0–0 band. Molecules that can be laser-cooled typically have A₀₀ ≫ A₀₁ [39]. Without any repump lasers to address the vibrational branching to v'' > 0 the average number of photons that can be scattered is determined by $r={A}_{00}/{\sum }_{{v}^{{\prime\prime}}=0}^{\infty }\enspace {A}_{0{v}^{{\prime\prime}}}={\tau }_{0}{A}_{00}$, where r is the probability for a molecule to decay back to v'' = 0 and τ₀ = 1.9 ns is the lifetime of A¹Π, v' = 0 level. This is often approximated by the Franck–Condon factor which neglects the variation of the transition dipole moment with the internuclear distance.

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The probability distribution for a molecule to scatter exactly n photons, when the number is limited to N, is p(n) = (1 − r)rⁿ⁻¹ for n < N, p(n) = rⁿ⁻¹ for n = N, and p(n) = 0 for n > N. For a molecule that interacts with a laser for a time t_i the maximum number of photons scattered is N = Rt_i, where R is the photon scattering rate. When an ensemble of molecules interacts with a laser beam, the fraction of molecules remaining in the initial state is given by P(N) = r^N [39]. The average number of photons scattered is

$$\begin{equation}\langle {n}_{\text{ph}}\rangle =\sum\limits _{n=0}^{N}\enspace np(n)=\frac{1-{r}^{N}}{1-r}=\frac{1-P(N)}{1-r}.\end{equation} \tag{ 1 }$$

The limiting value for N → ∞ is given by $\langle {n}_{\text{ph}}^{\infty }\rangle =1/(1-r)$, and the standard deviation of the probability distribution is equal to the mean value. This means that there is a wide variation of the number of photons scattered by individual molecules [18, 39]. We can approximate r by the previously calculated ratio of Einstein coefficients r = 1/(1 + A₀₁/A₀₀) = 0.9953 [26] and arrive at a maximum mean number of scattered photons of $\langle {n}_{\text{ph}}^{\infty }\rangle =213$. ⁵ The population in v'' = 0 decreases with N, according to ${P}_{{v}^{{\prime\prime}}=0}(N)={r}^{N}$, which for r ≈ 1 is well approximated by an exponential decay ${P}_{{v}^{{\prime\prime}}=0}(N)\simeq {\mathrm{e}}^{-N(1-r)}$. In the limit of Γt_i ≫ 1 and R ≪ Γ, with Γ = 1/τ₀ being the spontaneous decay rate,

$$\begin{equation}{n}_{\text{ph}}({t}_{\mathrm{i}})=\frac{1-{\mathrm{e}}^{-R(1-r){t}_{\mathrm{i}}}}{1-r}.\end{equation} \tag{ 2 }$$

The limit of $\langle {n}_{\text{ph}}^{\infty }\rangle$ can only be reached if R(1 − r)t_i ≫ 1 and the fluorescence saturates [40].

For convenience, we describe both, the ground and excited state, using Hund's case (a). The angular momenta R, L and J = R + L describe the end-over-end rotation of the nuclei, the total orbital angular momentum of the electrons and the total angular momentum excluding hyperfine interactions, respectively. ⁶ The relevant rotational structure is shown in figure 2(b). To prevent rotational branching it is common to use the strict parity and angular momentum selection rules of electric dipole transitions [41]. In AlF, each Q line of the A¹Π ↔ X¹Σ⁺ transition is rotationally closed. The number of photons a molecule can scatter is in this case limited by losses to higher vibrational levels.

The rotationally open P and R transitions can be used as 'standard candles' to calibrate the laser-induced fluorescence of the cycling transition. The average number of photons scattered by a molecule is limited by optical pumping to dark rotational states that are not coupled to the laser light. Analogous to the vibrational branching discussed above, the rotational branching ratios determine the number of photons a molecule scatters. Contrary to the vibrational case, the branching ratio for the rotational transitions r = r_ΔJ (J) depends on the transition type and J'' and is determined by the Hönl–London factor ${S}_{J}^{{\Delta}J}({J}^{{\prime\prime}})$, which for a ¹Π – ¹Σ⁺ transition is given by

$$\begin{equation}{S}^{\mathrm{P}}({J}^{{\prime\prime}})=\frac{1}{2}{J}^{{\prime\prime}}-\frac{1}{2}\quad \text{for}\enspace {J}^{{\prime\prime}}{\geqslant}2\end{equation} \tag{ 3 }$$

$$\begin{equation}{S}^{\mathrm{Q}}({J}^{{\prime\prime}})={J}^{{\prime\prime}}+\frac{1}{2}\quad \;\;\;\enspace \text{for}\enspace {J}^{{\prime\prime}}{\geqslant}1\end{equation} \tag{ 4 }$$

$$\begin{equation}{S}^{\mathrm{R}}({J}^{{\prime\prime}})=\frac{1}{2}{J}^{{\prime\prime}}+1\quad \enspace \text{for}\enspace {J}^{{\prime\prime}}{\geqslant}0,\end{equation} \tag{ 5 }$$

where ∑_ΔJ S^ΔJ (J'') = 2J'' + 1 is the total degeneracy [42]. The branching ratios can be written as

$$\begin{equation}{r}_{\mathrm{P}}({J}^{{\prime\prime}})=\frac{{S}^{\mathrm{P}}({J}^{{\prime\prime}})}{{S}^{\mathrm{P}}({J}^{{\prime\prime}})+{S}^{\mathrm{R}}({J}^{{\prime\prime}}-2)}\end{equation} \tag{ 6 }$$

$$\begin{equation}{r}_{\mathrm{Q}}({J}^{{\prime\prime}})=1\end{equation} \tag{ 7 }$$

$$\begin{equation}{r}_{\mathrm{R}}({J}^{{\prime\prime}})=\frac{{S}^{\mathrm{R}}({J}^{{\prime\prime}})}{{S}^{\mathrm{R}}({J}^{{\prime\prime}})+{S}^{\mathrm{P}}({J}^{{\prime\prime}}+2)},\end{equation} \tag{ 8 }$$

with a limiting value for the number of photons scattered by a molecule $\langle {n}_{\text{ph,P}}^{\infty }\rangle =2-1/{J}^{{\prime\prime}}$ and $\langle {n}_{\text{ph},\mathrm{R}}^{\infty }\rangle =2+1/({J}^{{\prime\prime}}+1)$ for P and R lines, respectively. There is an additional vibrational loss of about 1/213 per cycle, which lowers the expected number of photons for each line by about 1%. When including the hyperfine structure the limiting values must be corrected to account for additional hyperfine dark states which can lower the expected number of scattered photons by about 10%.

Both the aluminium nucleus and the fluorine nucleus have an unpaired proton. Therefore, both carry a nuclear spin, given by I_Al = 5/2 and I_F = 1/2 and combines to the total angular momentum F = F₁ + I_F, where F₁ = J + I_Al. The hyperfine interaction splits the J = 0, 1 and 2 levels into two, six, and ten hyperfine levels, respectively, and into 12 F-levels for J ⩾ 3.

Figure 2(c) presents the relevant energy level structure for J' = 1 and J'' = 1. In the electronic ground state, the hyperfine structure is dominated by the interaction of the electric quadrupole moment of the Al nucleus with the local electric field gradient. As a result, the intermediate angular momentum F₁ = J + I_Al is a good quantum number. The total span of the hyperfine structure in the X¹Σ⁺ state lies well within the natural linewidth Γ/(2π) = 84 MHz of the A − X transition, where ${\Gamma}={\sum }_{{v}^{{\prime\prime}}}\enspace {A}_{0{v}^{{\prime\prime}}}=1/{\tau }_{0}$ is the spontaneous decay rate. This means that radiofrequency sidebands or additional lasers to cover the spin-rotation or hyperfine splittings in the ground state are not required.

In the excited state, each J'-level comprises of a closely-spaced pair of opposite parity levels, a Λ-doublet, with Λ being the projection of L onto the internuclear axis. Each Λ-doublet component is split by the hyperfine interaction. The interaction arising from the Al and F nuclei is comparable and F₁ is not a good quantum number. The total span of the hyperfine intervals in the lowest J' level is about 50 times larger (≈6Γ/(2π)) than in the ground state and reduces to ≈2Γ/(2π) for J' = 4. The decreasing span of the hyperfine structure in the A state can be seen in the Q-branch spectrum of the A¹Π, v' = 0 ← X¹Σ⁺, v'' = 0 band provided in figure 3(a). The hyperfine structure in the A¹Π, v' = 0 level is partially resolved for the Q(1) line only. Panel (b) gives a more detailed representation of the hyperfine levels of the A¹Π, v' = 0 level for J' = 1 to J' = 4 with respect to their gravity centre by using the parameters from our previous spectroscopic study of AlF [26].

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According to [43], a J' = J'' ↔ J'' transition, such as the cycling Q lines discussed here, has one dark state for any choice of polarisation in zero magnetic field. Assuming linearly polarised light and the quantisation axis along the polarisation axis, the molecules are pumped into the dark $\left\vert {J}^{{\prime\prime}},{M}_{{J}^{{\prime\prime}}}=0\right\rangle$ state after scattering only a few photons. When a molecule is pumped into such a dark state it ceases to fluoresce because it does not couple to the excitation light anymore. This is detrimental for Doppler cooling. ⁷

In general, a dark state can either be an angular momentum eigenstate or a coherent superposition of these eigenstates. The former type is stationary, i.e. it does not evolve in time, and molecules accumulate and remain there indefinitely. The latter type of dark state can be non-stationary, i.e. it precesses between a dark and a bright state. We determine the dark state composition by using equation (23) in [45] and find that for linear polarisation they are superpositions of states with different values of F₁. The smallest F₁ splitting is ∼Γ/30 (see figure 2(c)) and for a fixed linear polarisation it is this separation that limits the scattering rate (see below).

A common method to increase the scattering rate is to lift the degeneracy of the ground states by inducing a Zeeman splitting of the order of Γ [43]. To achieve this for AlF a magnetic field of a few T is needed, due to the small magnetic g-factor in the ¹Σ⁺ ground state. A second method is to rapidly switch the polarisation of the light. Each polarisation is associated with a set of dark states, and switching the polarisation recovers a high scattering rate, provided the number of ground states is less than three times the number of excited states. For the Q-lines of AlF, the number of ground states is equal to the number of excited states, and all are accessible with a single laser frequency. As a result, polarisation modulation can be applied to increase the scattering rate above the limit set by the ground state hyperfine structure (see below).

To simulate the optical cycling rate of the Q(1) line of AlF, we solve the optical Bloch equations for the 72-level system, following the notation presented in [46, 47]. The solid red lines in figure 4(a) represent the excited-state population, ρ_ee for the Q(1) line as a function of the laser intensity, normalized to the two-level saturation intensity I_sat = πhcΓ/(3λ³) = 0.93 W cm⁻². The laser frequency is set to the center of the Q(1) line. This is an idealized case, assuming no losses to v'' > 0 to demonstrate that a high scattering rate can be achieved despite the large number of levels involved and despite the unresolved hyperfine structure in the ground state. As expected, with a linearly polarised laser and no polarisation modulation (δ_PM = 0), the scattering rate peaks at about Γ/30, limited by the precession of dark superposition states in the ground state. At high intensity, the laser interaction dominates over the hyperfine interaction, and this begins to stabilise the dark states. The excited state fraction falls off, and the system increasingly resembles the fine structure picture described above. The simulations also show that if the polarisation axis rotates about the k-vector of the light at a rate δ_PM = Γ/2, the dark states are effectively destabilized, and a high scattering rate is restored. The time-evolution of the population in the ground and excited states is plotted for I = 8I_sat and δ_PM = 0 in figure 4(b), showing that the steady state is reached at Γt = 1000.

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The results from the optical Bloch equations can be compared to the solutions of a rate equation model, which neglects all coherences (dashed, red line in figure 4(a)). The higher Q lines involve many more levels for which solving the Bloch equations becomes impractical. However, we solve the rate equations for the Q(2) and Q(3) lines shown as dashed green and dashed blue curves, respectively. The figure demonstrates that, for low laser intensities (s ≪ 1), the rate-equation model describes the complex multi-level system well. However, for high intensities there are significant differences. The rate equations predict that the scattering rate increases for higher Q lines because the span of the hyperfine structure reduces with increasing J'. The number of coherent dark states decreases from 1/3 to 1/9 of the total number of hyperfine transitions for the Q(1) to Q(4) lines. It is therefore expected that the peak scattering rate also increases for higher Q lines.

It is interesting to contrast AlF to the case of TlF, where recently it was shown that unresolved hyperfine structure in the ground state significantly limits the optical cycling rate [48]. Here, the hyperfine structure in the excited state is about 30 times larger than in AlF and the natural linewidth of the optical cycling transition is about 50 times narrower. The number of excited states which are accessible with a single laser is less than a third of the number of ground states, and polarisation modulation alone is insufficient to recover the molecules pumped into dark states. Instead, laser polarisation modulation must be combined with resonant microwave radiation to recover the molecules from the dark states.

A straightforward way to determine the number of optical cycles is to compare the fluorescence yield from a cycling transition to a standard candle. For the cycling Q lines, the average number of photons that can be scattered is determined by the vibrational branching ratio to $\langle {n}_{\text{ph},\mathrm{Q}}^{\infty }\rangle =213(30)$ (see section 2.2). All other potential loss channels can be neglected at this level and are summarized in our previous study [26]. To restrict off-axis excitation, the molecular beam is collimated with a round, 2 mm diameter aperture placed 4 cm before the first LIF detector. For the data presented here, the laser beam is collimated to a e⁻² diameter of w_z = 3.7 mm and w_y = 2.9 mm. This ensures sufficient intensity is available to saturate the transition, that the transverse extent of the laser exceeds that of the molecular beam, and that the interaction volume can be reliably imaged onto the PMT.

First, we perform a consistency check to make sure that the standard candles are indeed a good reference to calibrate the fluorescence of the cycling transitions. Figure 7(a) shows the laser induced fluorescence for the R(1) and P(3) line, time-integrated over the duration of the molecular pulse, as a function of the laser intensity. The ratio of the two limiting values is fixed by the ratio of the Hönl–London factors. A deviation from this ratio would lead to a systematic error in the calibration. The solid curves are the result of a theoretical simulation of the light–molecule interaction using rate equations, the measured velocity distribution and spatial profile of the molecular and laser beam. The P line saturates at 1.5 photons and not at $\langle {n}_{\text{ph},\mathrm{P}}^{\infty }\rangle =5/3$ as predicted in section 2.3. This is due to optical pumping of molecules into the dark ${M}_{{F}^{{\prime\prime}}}={\pm}6$ Zeeman sublevels lowering the average number of photons scattered per molecule slightly. A Q-branch spectrum allows us to normalize the LIF signals to the respective rotational state population in J'' = 1 and J'' = 3. We display the saturation of the R(1) and P(3) line because they both reach the same excited state. This is important, because the excited state hyperfine structure can influence the detected fluorescence yield in subtle ways (see below). In saturation, the two LIF signals indeed reflect the predicted ratio by the Hönl–London factors.

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The excited state hyperfine structure can lead to subtle interference effects between decays from the different excited states. The relative weight of σ⁺, σ⁻ and π polarised radiation emitted by the molecules depends on the angle of the polarisation of the excitation light with respect to the detector. The fluorescence intensity of π polarised light is I_π ∝ sin² Θ and for σ polarised light is I_σ ∝ (1 + cos² Θ)/2, where Θ is the angle between the k-vector of the fluorescence light and the polarisation vector of the excitation light. The position of the fluorescence detector is fixed at a right angle to the k vector of the excitation light. By averaging over all possible decay channels we expect a periodic modulation of the detected fluorescence with an amplitude of 0.18 for the R(1) and −0.03 for the P(3) line. Figure 7(b) summarizes this effect. The points with errorbars are the measured fluorescence normalized to their value for Θ = 0 and the solid curves are the predicted modulation by calculating the weighted sum over all possible decay channels. Therefore, it is important to choose Θ = 0, where the ratio of the two fluorescence signals in figure 7(a) corresponds to the ratio of the Hönl–London factors. The same is true for the Q(1) line, whose periodic modulation has an amplitude of 0.11, slightly less compared to the R(1) line.

Figure 8(a) shows two time of flight (TOF) profiles of the molecular pulse with the laser tuned to the Q(1) (blue) and R(1) (black) lines. Both traces are taken with the same laser intensity to demonstrate the signal enhancement due to optical cycling. The time-integrated fluorescence signal as a function of the laser frequency for low and high laser intensity is shown in figure 8(b). The low intensity spectrum is scaled by a factor of 20. Each Q-line shows a similar signal enhancement at high laser power, demonstrating that they can all cycle effectively.

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The saturation of the Q(1) fluorescence for two forward velocities v_z = 150 m s⁻¹ (blue) and v_z = 300 m s⁻¹ (red), is shown in figure 8(c). The solid curves are simulations based on the optical Bloch equations and the dashed lines are the predictions from the rate equation model. We use the measured laser beam profile and forward velocity to predict N = ∫R(t)dt for each theoretical model and equation (1) to predict ⟨n_ph⟩. It is clear that a high optical cycling rate can be obtained despite the large number of hyperfine components involved in the cycling scheme. The rate equations predict that the fluorescence saturates at a lower intensity because they do not account for the effects of coherent dark states. Figure 8(d) shows the number of photons scattered as a function of the interaction time, defined by t_i = w_z /v_z , where w_z is the e⁻² diameter of the laser beam. The solid curve is a fit to the data using equation (1) with R as fit parameter and r = 0.9953 the theoretical branching ratio. The fit gives R = 17.2(2) × 10⁶ s⁻¹ for the effective scattering rate for a saturation parameter s = 6. We estimate the systematic uncertainty in determining the number of photons scattered to be 10%. This is mainly due to uncertainties in the calibration procedure that arise from a non-linearity in the PMT gain, combined with variations in the source flux and velocity distribution.

The second method to determine the photon-scattering rate is to measure the rate at which the molecules are pumped out of X¹Σ⁺, v'' = 0 and into the dark X¹Σ⁺, v'' = 1 level. For this we use a high-intensity laser beam (≈20 W cm⁻², w_z = 1.3 mm, w_y = 1.1 mm) tuned to the A¹Π, v' = 0 ← X¹Σ⁺, v'' = 0 transition. To probe the molecules remaining in X¹Σ⁺, v'' = 0 we split off a small fraction of the pump laser beam and intersect it with the molecular beam a second time in LIF-2. Figure 9(a) shows a Q-branch spectrum recorded in LIF-2 with the pump laser present (blocked) in LIF-1 in blue (red). The fraction of the v'' = 0 population that is pumped out by the high-intensity laser is 29%, 37%, 55% and 67% for the Q(1) to Q(4) lines, respectively. The data indicates that higher Q lines scatter faster at the same laser intensity and therefore pump out more molecules for the same interaction time. This is consistent with the predictions of the rate equation model presented in figure 4.

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The number of photons scattered by the molecules is related to the population remaining in v'' = 0 by $\langle {n}_{\text{ph}}\rangle =-\mathrm{ln}\enspace ({P}_{{v}^{{\prime\prime}}=0})/(1-r)=73(11)$, 98(15), 170(25) and 236(35), for the Q(1) through Q(4) lines, respectively (see section 2.2). The pumped-out fraction for higher rotational lines is slightly overestimated because the molecules experience a radiation pressure force which leads to a Doppler shift in LIF-2 of about 17 MHz per 100 photons scattered (see next section). Additionally, the molecules are displaced from the center of the detection region which decreases the detection efficiency slightly. The number of photons scattered on the Q(1) line corresponds to an effective scattering rate of R = 16(2) × 10⁶ s⁻¹ which increases to R = 42(7) × 10⁶ s⁻¹ for the Q(4) line. This is in good agreement with the scattering rate determined in the previous section. We have not performed optical Bloch equation simulations for the high Q-lines, but we expect that the peak scattering rate also increases for higher Q lines. In addition, the higher Q lines reach the peak scattering rate at a lower intensity or shorter interaction time which leads to an effectively higher scattering rate for the same laser-beam diameter.

To verify that the molecules are indeed pumped to X¹Σ⁺, v'' = 1, we probe the v'' = 1 population in LIF-2. Figure 9(b) shows Q-branch spectra of the A¹Π, v' = 0 ← X¹Σ⁺, v'' = 1 band using a second UV laser tuned to 231.7 nm with an output power of up to 150 mW. The top panel shows the 0–1 spectrum with (without) the 0–0 pump laser present in LIF-1 in blue (red). The top panel shows the 0–1 spectrum with the pump laser tuned to the Q(1) line. The bottom panel is the same spectrum, but with the pump laser tuned to the Q(2) line. The total number of v'' = 1 molecules that are produced in the source relative to the number of molecules in v'' = 0 is typically 9(2)%. The increase in v'' = 1 population (blue curves) corresponds to a fraction of molecules that is pumped out of v'' = 0 of 36(8)% and 31(7)% for the Q(1) and Q(2) lines, respectively. This is consistent with the pumped-out fraction shown in the inset of figure 9(a). The uncertainty is dominated by fluctuations in the v'' = 1 population relative to v'' = 0.

After establishing an optical cycling scheme, the next step towards laser slowing and magneto-optical trapping is to demonstrate a measurable radiation pressure on the molecular beam. This method is also a very direct way to determine the number of photons scattered by the molecules. Robert Frisch, motivated by Otto Stern's proposal [55], has demonstrated the deflection of an atomic beam using resonance radiation already in 1933 [56]. This was followed by many more experiments using lasers in combination with a variety of atomic and molecular beams [18, 57–60].

Here, we use a high-intensity (10 W cm⁻², w_z = 0.9 mm and w_y = 2 mm) laser tuned to the Q(2) rotational line to exert radiation pressure onto the molecular beam and measure the transverse deflection by imaging the molecular fluorescence onto an sCMOS camera. A round 2 mm aperture located in front of the pump region in LIF-1 collimates the molecular beam transversely to 1 m s⁻¹ (FWHM). This guarantees that the entire molecular beam can be imaged onto the camera. The imaging system is calibrated by translating the collimating aperture in LIF-1 along x and measuring the average displacement of the molecular fluorescence on the camera. A clean-up beam (100 mW in w_z = w_y = 2 mm) intersects the molecular beam a few cm downstream from the pumping region (see figure 5) and recovers >95% of the molecules lost to v'' = 1. The v'' = 0 molecules are detected in LIF-2 by splitting off a fraction of the 0–0 pump light and directing it through LIF-2.

The short excited state lifetime and small Franck–Condon factor of the 0–1 transition requires a high laser intensity to saturate the transition. For future laser-slowing and cooling experiments it is important to reach a repumping rate that exceeds the rate at which molecules are pumped into v'' = 1. We measure a repumping rate of ≈10⁶ s⁻¹, fast enough to guarantee efficient repumping. The clean-up beam pumps also the small population of v'' = 1 molecules produced in the source to v'' = 0 which is then detected in LIF-2 by the probe beam. We measure this fractional increase in v'' = 0 population by turning off the pump laser and determining the difference in LIF signal with and without the clean-up beam. The difference is subtracted from the image of the deflected beam because it originates from molecules that are not deflected by the radiation pressure force in LIF-1.

By absorbing a photon, the molecule acquires a linear momentum p = ℏ k in the direction of this photon. The resulting deflection of the molecular beam, measured in LIF-2, is d = ⟨n_ph⟩v_r L/v_z , where v_r = ℏk/m = 3.8 cm s⁻¹ is the recoil velocity per photon, L = 28 cm is the distance between the pump and probe regions and v_z = 270(20) m s⁻¹ is the forward velocity of the molecules.

Figure 10 summarizes the results of this experiment. Panel (a) shows a binned camera image with the pump laser turned off (top) and turned on (bottom). The image is integrated and fitted with a Gaussian to determine the average displacement of d = 2.15 mm (panel (b)), indicating that the molecules scatter ⟨n_ph⟩ = 55(4) photons. This is consistent with the value derived from optical pumping, given the shorter interaction time due to the smaller waist diameter w_z . The recoil velocity is comparable to the width of the transverse velocity distribution of the molecular beam which leads to a broadening of the molecular beam along x. This is a result of the random direction of the spontaneously emitted photons which leads to velocity diffusion and therefore a symmetric broadening of the molecular beam. The diffusion coefficient can be approximated by $D\approx {\left(\hslash k/m\right)}^{2}R$, with R being the photon scattering rate. This momentum diffusion gives rise to a characteristic velocity change of the molecules of R/k = 0.54 m s⁻¹ [61], and therefore broadens the transverse velocity distribution by about 54%. A more accurate approximation is given in [58]. The upper bound for the transverse velocity after scattering ⟨n_ph⟩ = 55 photons is ${v}_{\mathrm{t}}=2({v}_{z}{\Delta}\theta +\sqrt{\langle {n}_{\text{ph}}\rangle /3}h/(m\lambda ))=1.33$ m s⁻¹, adding 0.33 m s⁻¹ to the 1 m s⁻¹ of the original beam. Here Δθ = 1.9 mrad is the half-angular divergence of the molecular beam, before the interaction with the laser. We measure a 33% increase in the width of the molecular beam, consistent with the simple model presented above. The symmetric broadening in figure 10(b) indicates that the scattering rate along y is homogeneous. This is surprising since the laser beam w_y = 2 mm has the same diameter as the aperture that is used to collimate the molecular beam. However, the inset of figure 8(d) shows that for high laser intensities (s ⩾ 10), the Gaussian wings contribute significantly to the overall scattering rate.

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The area under both curves in figure 10(b) is identical, showing that the total number of molecules is conserved and verifying the effect of the repump laser. The Doppler shifts induced by the deflection are negligible because of the large transition linewidth. Figure 10(c) shows the deflection and the inferred average number of scattered photons for a slightly faster molecular beam (v_z = 330 m s⁻¹) as a function of the laser intensity. The red solid curve is the result of a Monte Carlo trajectory simulation using the rate equation model to predict the scattering rate, with the measured laser beam profile, velocity distribution and spatial distribution of the molecular beam as input parameters. The simulation results fit the data well for low laser intensities, but, as expected, the rate equations slightly overestimate the number of scattered photons for high laser intensities.

The mean scattering rate of the molecules, averaged over the interaction volume is R = 23(4) × 10⁶ s⁻¹, which corresponds to an acceleration of a = v_r R = 8.7(1.5) × 10⁵ m s⁻².

To slow the molecular beam to rest using radiation pressure, the leak to v'' = 1 must be closed with a repump laser, which allows scattering about 10⁴ photons. This typically lowers the scattering rate by at least a factor of two due to the additional ground-state levels that couple to the same excited state. However, the molecules also reach a higher peak scattering rate as they are not pumped to v'' = 1 by the low intensity Gaussian wings of the laser beam. We estimate the stopping distance for the SF₆ and NF₃ beam presented in figure 6, conservatively to 6 and 2 cm, respectively. By modulating the polarisation of the 0–0 light in a slowing configuration the peak scattering rate could be quadrupled, reducing the stopping distance to below 1 cm. Conveniently, the large hyperfine structure of the Q lines covers the Doppler width of the molecular beam.

A small electric field can mix the closely spaced, opposite parity Λ-doublet levels in the A¹Π state, and therefore break the parity selection rule of an electric dipole transition. After scattering only a few photons, the molecules are pumped into dark rotational states in the ground state and are lost from the optical cycle.

The strongest mixing in the A¹Π, J' = 1 level occurs between the opposite parity M_F' = 4 sublevels of F' = 4, whose energy-level spacing is only a few MHz (see figure 3(b)). For very low electric fields, i.e. a small Stark shift compared to the energy-level spacing, the fraction of the opposite parity that is mixed into a given parity level increases quadratically with the electric field, transitions to a linear increase for higher fields and converges to an equal mixture at high electric fields. Given the electric dipole moment of 1.45 Debye, an electric field of a few V cm⁻¹ is sufficient to open a significant loss-channel to the optical cycle. In the X¹Σ⁺ state the opposite parity levels are separated by at least twice the rotational constant, and parity mixing can be neglected.

When exciting on the Q(1) line, the parity mixing in the A¹Π, J' = 1 level leads to rotational branching to the dark J'' = 0 and J'' = 2 rotational levels in the electronic ground state. This effect reduces the average number of photons that can be scattered from $\langle {n}_{\text{ph},\mathrm{Q}}^{\infty }\rangle =1/(1-{\tau }_{0}{A}_{00})$ to a limiting value in high electric fields of only two.

To quantify this potential loss channel, we install electrodes in the optical pumping region (LIF-1), apply a uniform electric field and measure the fraction of molecules that is pumped into X¹Σ⁺, v'' = 1. The frequency of the pump laser is locked to the center of the Q(1) line and the population in v'' = 1 is probed in LIF-2, similar to the experiment described in section 5.2. Figure 11(a) shows that the fraction of molecules pumped into the v'' = 1 level decreases rapidly with increasing electric field. Following the model described in section 2.2 the population in the v'' = 1 level after N optical cycles is given by

$$\begin{equation}{P}_{{v}^{{\prime\prime}}=1}(N,E)={\tau }_{0}{A}_{01}\sum\limits _{n=0}^{N-1}{\left(r-\gamma (1-{\tau }_{0}{A}_{00}){E}^{2}\right)}^{n}\simeq \frac{1}{1+\gamma {E}^{2}}\left(1-{\mathrm{e}}^{-N(1-{\tau }_{0}{A}_{00})(1+\gamma {E}^{2})}\right),\end{equation} \tag{ 9 }$$

where τ₀ A₀₁ ≈ (1 − τ₀ A₀₀). We express the losses from the optical cycle induced by parity mixing as γ(1 − τ₀ A₀₀)E² and assume a quadratic dependence with the electric field E. Assuming N = 55 ± 15 we fit the model using γ as a fit parameter. The model fits the data well for γ = 0.0034(6)cm² V⁻². The losses due to vibrational branching and due to parity mixing become equal when γE² = 1, which occurs at 17(2)V cm⁻¹. Only if the parity and vibrational losses are equal and N → ∞ will ${P}_{{v}^{{\prime\prime}}=1}=0.5$. The loss-channel induced by parity mixing becomes negligible for stray electric fields below 1 V cm⁻¹.

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The A¹Π, v' = 0 state of AlF is located 5.45 eV above the X¹Σ⁺, v'' = 0 state, more than half way up to the ionisation potential at 9.73 eV. Therefore, after excitation on the A¹Π, v' = 0 ← X¹Σ⁺, v'' = 0 band near 227.5 nm, a second photon from the same (cw) laser can ionize the molecule, creating an AlF cation in the ²Σ⁺ electronic ground state and a free electron.

It is quite uncommon to observe such a single color (1 + 1) resonance enhanced multiphoton ionisation ((1 + 1)-REMPI) process for molecules using a cw laser. The cross section for the excitation into the ionisation continuum is many orders of magnitude smaller than the cross section of the first, resonant excitation step. The total ionisation yield scales linearly with the time that the molecules spend in the electronically excited state. This makes ionisation with a cw laser difficult to observe, in particular if the intermediate state lives less than 2 ns. However, AlF can be excited on a quasi-cycling transition to increase the time the molecules spend in the excited state to n_ph τ, which can be much larger than τ.

Ionisation leads to a loss from the optical cycle, which can be expressed as

$$\begin{equation}{\gamma }_{\mathrm{i}}(I)=\frac{{{\Gamma}}_{\text{ion}}}{R}=\frac{{\sigma }_{\mathrm{i}}I\tau }{\hslash \omega },\end{equation} \tag{ 10 }$$

where Γ_ion = σ_i ρ_ee(I, δ)I/ℏω, ρ_ee(I, δ) is the fraction of AlF molecules in the excited state, which depends on the intensity I and the detuning from resonance δ, σ_i is the photoionisation cross section from the A¹Π state and R = ρ_ee(I, δ)Γ is the photon scattering rate. The ionisation cross section for the A¹Π state is independent of the rotational quantum number J'', as is the lifetime τ. Figure 11(b) shows the total number of AlF cations produced as a function of the laser frequency while scanning a high power cw laser over the Q-branch of the A¹Π, v' = 0 ← X¹Σ⁺, v'' = 0. The spectrum resembles a power-broadened version of figure 3(a), with a slightly shifted Q(1) peak. The weak electric field mixes the opposite parity states in A¹Π which increases the number of accessible levels and therefore the ionisation probability from the F' = 3 and F' = 4 states, whose Λ-doublet components are particularly close. This causes the Q(1) line to appear shifted by about 100 MHz. Moreover, the electric field used to extract the molecular ions from the ionisation region must be low to prevent opening a rotational loss channel (see previous subsection). The ion-yield from (1 + 1) REMPI decreases with increasing electric field.

The number of AlF molecules passing through the ionisation volume per pulse is determined by absorption spectroscopy to N_mol = 1.2 × 10⁸, with a peak density of 5.5 × 10⁷ cm⁻³. The number of ions detected from the same interaction region is measured using a compact, linear, Wiley–McLaren time-of-flight setup to N_ion = 259/ = 650 per pulse. The ion detection efficiency, ≃ 0.4, is mainly determined by the open area ratio of the microchannel plates [62]. The number of detected ions is likely to be underestimated, due to the low electric field (5 V cm⁻¹) used to extract the ions. With the laser frequency locked to the center of the Q(1) line, the fraction of the molecules that is ionised by the 17 W cm⁻² laser beam (peak intensity) N_ion/N_mol = 5.4 × 10⁻⁶ with Γ_ion = N_ion/(t_i N_mol) = 1.6 s⁻¹, where t_i = 3.3 × 10⁻⁶ s is the interaction time of the molecules with the laser beam. During this time the molecules scatter on average ⟨n_ph⟩ = 60 photons (see figure 7(c)). With R = 18 × 10⁶ s⁻¹, the fractional loss from the optical cycle, due to ionisation, becomes γ_i = 4.7 × 10⁻⁹ $\frac{I}{{I}_{\mathrm{sat}}}$, with σ_i = 2 × 10⁻¹⁸ cm². The results presented here should be used as an order-of-magnitude estimate for the loss rate and the ionisation cross section. The low extraction voltage prevents determining the ionisation volume precisely.