Molecule–molecule and atom–molecule collisions with ultracold RbCs molecules

We begin by examining molecule–molecule collisions in an optical trap, where the intensity of the light is modulated as a square wave such that the molecules spend 75% of each modulation cycle in the dark. In doing so, we partially suppress the optical excitation of two-body collision complexes, which is believed to be the dominant mechanism for collisional loss for pairs of molecules in the absolute ground state. When the trap light is off, complexes can form and break apart without the risk of destructive optical excitation. The modulation leads to a reduction in the loss rate, with the maximum fractional reduction in loss simply equal to the duty cycle of the modulation.

We model the rate of change of the densities of 'free' molecules n_m and bimolecular complexes n_c with the rate equations

$$\begin{align}\hfill {\dot{n}}_{\mathrm{m}}& =-{k}_{2}{n}_{\mathrm{m}}^{2}+\frac{2}{{\tau }_{-1}}{n}_{\mathrm{c}},\hfill \\ \hfill {\dot{n}}_{\mathrm{c}}& =+\frac{1}{2}{k}_{2}{n}_{\mathrm{m}}^{2}-\frac{1}{{\tau }_{-1}}{n}_{\mathrm{c}}-\frac{1}{{\tau }_{\text{inel}}}{n}_{\mathrm{c}}-{k}_{\text{laser}}I(t){n}_{\mathrm{c}}.\hfill \end{align} \tag{ 2 }$$

This differs from previous treatments [60, 61, 63] in the inclusion of a term for inelastic loss of complexes. Here k₂ describes the rate of formation of complexes and k_laser is the photon scattering rate of the complexes per unit intensity I(t). The lifetime τ₋₁ is the 1/e time for dissociation of the complexes back to free molecules in the initially prepared state, and τ_inel describes the loss of complexes via a non-optical mechanism, such as dissociation to molecular states other than the one in which the molecules are initially prepared. The lifetime of the complex in the dark τ_c is then given by

$$\begin{equation}1/{\tau }_{\mathrm{c}}=1/{\tau }_{-1}+1/{\tau }_{\text{inel}},\end{equation} \tag{ 3 }$$

and the probability that the molecules return to the initial state is Φ = τ_inel/(τ₋₁ + τ_inel). We have previously measured k₂ by examining the rate of loss of molecules from a CW trap, which is rate-limited by the formation of complexes. The molecules have a temperature of 2 μK and k₂ = 5.4 × 10⁻¹¹ cm³ s⁻¹ [60, 72], which is about a factor of two lower than the thermally averaged universal rate [66]. In addition, we set k_laser = 3 × 10³ W⁻¹ cm² s⁻¹ as measured in our previous experiments [60] on molecules in their absolute ground state (where Φ = 1). The behaviour of our model is however insensitive to this value provided that k_laser I(t) ≫ 1/τ_c when the trap light is on, corresponding to the situation where the loss of complexes is strongly saturated.

In figure 1, we show the effect of modulating the trap intensity as a square wave by solving these rate equations for CW and modulated traps. For the modulated trap, the intensity is modulated at a frequency of 1 kHz and the molecules spend 75% of each cycle in the dark. In figure 1(a), we fix Φ = 1 and τ_c = τ₋₁ = 0.53 ms, as previously measured for molecules in the spin-stretched hyperfine ground state [72]. In this case we expect a slower rate of molecule loss in the modulated trap (dashed line) when compared to the CW trap (solid line). In figure 1(b), we examine the effect of varying τ₋₁ and Φ, for a fixed hold time in the trap of 200 ms. We find that significant suppression of the loss occurs only if τ₋₁ < τ_inel and τ₋₁ < t_dark, where t_dark is the duration the molecules spend in the dark during each cycle of the trap modulation.

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The dark time can be varied in experiments by changing the frequency of the modulation, and we expect any suppression to become greater as the modulation frequency is reduced and the dark time increases correspondingly. Measurements in modulated traps thus allow one to distinguish optical loss from other loss mechanisms only if the lifetime of the complex is significantly shorter than the dark time and the optical loss dominates over any laser-free loss.

For our experiments, we create a sample of molecules [29, 82–85] starting from an ultracold atomic mixture of magnetically levitated Rb and Cs atoms confined to a crossed optical dipole trap (with a wavelength, λ = 1550 nm) [82]. We first form weakly bound molecules from the atomic mixture using magnetoassociation on an interspecies Feshbach resonance at 197 G [83]. Following this, we remove the remaining atoms from the trap using the Stern–Gerlach effect [83]. We then transfer the molecules into an optical trap where the intensities of the beams are modulated as a square wave. This is achieved by ramping up the intensity of the modulated trap and switching off the 1550 nm CW trap and magnetic levitation gradient.

The modulated trap is formed from a single beam (λ = 1064 nm) in a bow-tie configuration. The trap frequencies experienced by molecules in this trap are [ω_x , ω_y , ω_z ] = [96(2), 160(3), 185(3)] × π rad s⁻¹. The square-wave intensity modulation is achieved by blocking the light using an optical chopper wheel. Using this method, we can modulate the trap intensity with cycle frequencies of up to 5 kHz. We find significant trap loss for modulation frequencies below 1 kHz, consistent with loss due to parametric heating resonances, as occurs in time-dependent potentials that are not fully in the time-averaged trap regime [86–88].

We transfer the molecules in the modulated trap to the ¹Σ rovibrational ground state using STImulated Raman Adiabatic Passage (STIRAP) [29, 84, 85]. The STIRAP is performed at a magnetic field of B = 181.6 G during the trap dark time. Immediately after STIRAP, the sample typically consists of up to 5000 molecules with a mean density of ∼10¹¹ cm⁻³. STIRAP prepares the molecules in a single hyperfine level of the rovibrational ground state.

For RbCs at B = 181.6 G the rovibrational ground state consists of 32 hyperfine states spread across 1.3 MHz with separations between neighbouring states typically ranging from 10 to 100 kHz as shown in figure 2(a). We label these hyperfine states by ${(n,{m}_{f,\mathrm{R}\mathrm{b}\mathrm{C}\mathrm{s}})}_{k}$ where n is the quantum number for rotational angular momentum, m_f,RbCs = m_Rb + m_Cs + m_n is the sum of the angular momentum projections for the nuclear spins m_Rb, m_Cs and the rotation of the molecule m_n , and k is an index that counts up the states in order of increasing energy for a given value of n and m_f,RbCs. For magnetic fields above 90 G the spin-stretched state (0, 5)₀ state is the hyperfine ground state of RbCs. With the STIRAP, we are able to populate either the (0, 5)₀ or (0, 4)₁ hyperfine states directly. In this work, we also prepare molecules in (0, 4)₀, which we achieve using a pair of one-photon π-pulses to drive transitions in the molecule coherently between n = 0 and n = 1 [89]. The states used in this work are highlighted in figure 2(a).

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Our understanding of the formation and optical excitation of molecule–molecule collision complexes is far from complete. This is evidenced by observations in ²³Na³⁹K, ²³Na⁴⁰K, and ²³Na⁸⁷Rb [61, 63] where no suppression of loss in modulated traps was seen, despite RRKM predictions of lifetimes for complexes [69] that are much shorter than the dark time in the trap. We have previously observed a suppression of loss for RbCs molecules in the spin-stretched hyperfine ground state, so here we explore collisions of molecules prepared in different hyperfine states, where other loss channels and mechanisms could affect the lifetime of the complex.

To measure the effect of the dark time in the modulated trap, we measure the number of molecules remaining after a 200 ms hold in the modulated trap, both with (N_mod+CW) and without (N_mod) an additional CW source of 1550 nm light. The CW light is derived from the 1550 nm trap used to prepare the Feshbach molecules, and has total peak intensity of ∼3 × 10² W cm⁻². This intensity is sufficiently low that it does not significantly affect the trap frequencies or the trap depth experienced by the molecules, but high enough to remove complexes continuously from the trap [72]. By performing a comparative measurement, we remove any sensitivity to the effects of residual heating and associated evaporative loss that may arise due to the modulation of the trap intensity. To measure the molecule number, we reverse the STIRAP and association process, to break the molecules back apart into their constituent atoms for detection by absorption imaging. For each measurement of loss, we perform 50 interleaved measurements of N_mod and N_mod+CW and extract a mean and standard error from the resulting distributions.

We characterise the suppression of loss due to the dark time by calculating the fractional difference in the molecule number N_mod/N_mod+CW − 1. This is shown as a function of modulation frequency in figure 2(b) for molecules prepared in several different hyperfine states. For molecules in (0, 5)₀ we observe a suppression of loss characterised by N_mod/N_mod+CW − 1 > 0. The suppression is greatest at the lowest modulation frequencies we are able to reach where t_dark/τ_-1 is greatest, see figure 1(b). We fit the results using the rate equation model (equation (2)), making the assumption that the molecules remain at a fixed temperature. We fix Φ = 1 because the (0, 5)₀ state is both spin-stretched and is the lowest-energy hyperfine state; the molecules do not have sufficient kinetic energy to leave the complex in any other state. As such, the lifetime of the collision complex in the dark is the only free parameter in the fitting. We find an optimal value for the lifetime of the (RbCs)₂ collision complex is 0.8(3) ms in the dark, where the number in brackets is the 1σ uncertainty. This is consistent with our previously measured value of 0.53(6) ms [72].

Figure 2(b) also shows similar measurements for molecules prepared in the higher-energy hyperfine states (0, 4)₀ and (0, 4)₁. In these cases, we generally observe a lower suppression of loss, although the measurements on the highest and lowest hyperfine states deviate by only 1σ at modulation frequencies of 1 kHz and 1.5 kHz. We now present analysis of our results to identify possible changes in model parameters that could explain such a hyperfine state dependence.

We first analyse results with the assumption τ_inel → ∞, Φ = 1. In this limit, our results would suggest that the lifetime of the complex in the dark depends on the hyperfine state, which might be associated with an increase in the effective density of states. This change might be caused by the increased number of nuclear spin arrangements that have the same value of the total spin projection M_F = m_f,1 + m_f,2 as the incoming pair. For the (0, 4)₀ state our results are consistent with τ_c = τ₋₁ = 2.1(1.3) ms. In contrast, for the (0, 4)₁ state our results are consistent with no suppression of loss and we can place only a lower limit on the lifetime of the complex in the dark, τ_c = τ₋₁ > 3.3 ms (at the 68% confidence level).

An alternative interpretation of the reduced suppression for the states with m_f,RbCs = 4 is the presence of one or more additional loss channels. Collisions between molecules in either state have at least one energetically accessible inelastic channel that conserves M_F. Decay of complexes to form molecules in different states would be observed as loss in our experiments, because we detect molecules only in the specific hyperfine state in which they are prepared. To test this interpretation, we restrict τ₋₁ to the range of values in the 68% confidence interval found for the (0, 5)₀ state, τ₋₁ = 0.8(3) ms, and fit the results for (0, 4)₀ and (0, 4)₁ with τ_inel as a free parameter.

For molecules prepared in (0, 4)₀ there is one energetically available channel for spin exchange, with the molecules exiting the complex in (0, 5)₀ + (0, 3)₀. This combination of states is higher in energy than the prepared state by only k_B × 0.16 μK, which is much smaller than the temperature of the molecules T = 2 μK. For molecules in (0, 4)₀, our results find an optimum value ${\tau }_{\text{inel}}=0.{2}_{-0.1}^{+0.5}$ ms, leading to ${\Phi}=0.{2}_{-0.1}^{+0.4}$. For molecules in (0, 4)₁, several spin-exchange channels are available. In addition, there is the possibility of exchange between nuclear spins in the same molecule such that it exits in the lower-energy (0, 4)₀ state. For the (0, 4)₁ state, we find τ_inel < 0.13 ms and Φ < 0.08 at the 68% confidence level.

This analysis demonstrates that our experimental results for excited molecular states can be explained either by a longer lifetime for the complex in the dark τ₋₁ or by the presence of laser-free inelastic decay. We cannot at present distinguish between these explanations.

Studying the change in molecule loss rate as a function of atom density can yield insight into the kinetics of the underlying loss mechanism. For reactive RbCs + Rb collisions, the energetically allowed atom-exchange reaction is likely to be the dominant loss mechanism, although it is possible that laser-induced loss may also be important [77]. As this is a two-body (atom + molecule) process, we expect that the rate of loss of molecules will depend linearly upon the density of atoms. For nonreactive RbCs + Cs collisions however, the expectation is less clear. Possible mechanisms for loss in the nonreactive mixture include (but may not be limited to) optical excitation of two-body (atom–molecule) complexes and three-body (atom–atom–molecule) collisions. We expect the loss rate associated with these processes to depend on the atom density, or the square of the atom density, respectively.

To vary the atom density we use resonant light to remove a fraction of the chosen atomic species from the trap. This is performed in parallel with the removal of the unwanted atomic species after STIRAP. The additional light is tuned to be resonant with the 5²S_1/2(f_Rb = 1, m_f,Rb = 1) → 5²P_3/2(2, 2) electronic transition for Rb and the 6²S_1/2(f_Cs = 3, m_f,Cs = 3) → 6²P_3/2(4, 4) transition for Cs. The light is horizontally polarised and delivered in a collimated beam of radius ∼1 mm that propagates orthogonal (designated x axis) to the 21.3 G magnetic field (along the z axis). Throughout the resonant light pulse, we also switch on cooling light from the magneto-optical trap which removes any atoms that spontaneously decay into the f_Rb = 2 state of Rb and the f_Cs = 4 state of Cs. We observe loss of atoms with an exponential time constant of 88(2) μs for Rb and 6.4(2) μs for Cs. The difference in removal rate is primarily due to the different laser intensities used for the different species. Using this method, we do not observe any significant build-up of population in other atomic hyperfine states, such that the atoms that survive the resonant light pulse remain in the target (1, 1) state for Rb and (3, 3) state for Cs. However, we do observe that the temperature of the atoms increases with the duration of the light pulse, with approximately a factor of 2 greater increase in the direction of the laser propagation. This change in temperature is taken into account when calculating the atomic density.

We present measurements of molecule loss rate as a function of mean atom density in figure 4. In figure 4(a), we examine the reactive combination RbCs + Rb. Specifically, we measure collisions between molecules prepared in (0, 4)₁ with ground-state Rb atoms. As expected, we observe a linear dependence of the molecule loss rate on atom density. A linear fit to the results, constrained to pass through the origin, yields a gradient of 3.3(2) × 10¹¹ cm³ s⁻¹. In figure 4(b), we show a measurement using the nonreactive combination of RbCs + Cs. In this case the molecules are prepared in (0, 5)₀ so that both the Cs atoms and RbCs molecules occupy their respective hyperfine ground states. For the nonreactive combination we also find a linear dependence of the molecule loss rate on the atom density, this time with a gradient of 2.05(7) × 10¹¹ cm³ s⁻¹. This indicates that the loss mechanisms for both reactive and nonreactive collisions have rate-limiting steps that depends on a two-body RbCs + atom collision.

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As both reactive and nonreactive atom–molecule collisions appear to be rate-limited by second-order kinetics, we can quantify the loss rate using a second-order rate coefficient ${k}_{2}^{\mathrm{a},\mathrm{m}}$. To extract a second-order rate coefficient from a measurement of molecule loss, we model the loss of molecules due to atom–molecule collisions with the rate equation

$$\begin{equation}{\dot{n}}_{\mathrm{m}}(\mathbf{r},t)=-{k}_{2}^{\mathrm{a},\mathrm{m}}{n}_{\mathrm{a}}(\mathbf{r},t){n}_{\mathrm{m}}(\mathbf{r},t),\end{equation} \tag{ 5 }$$

where n_a, n_m represent the densities of atoms and molecules, respectively. The rate of change of the number of molecules can be obtained by integrating equation (5), giving

$$\begin{equation}{\dot{N}}_{\mathrm{m}}(t)=-{k}_{2}^{\mathrm{a},\mathrm{m}}\int {n}_{\mathrm{a}}(\mathbf{r},t){n}_{\mathrm{m}}(\mathbf{r},t){\mathrm{d}}^{3}\mathbf{r}.\end{equation} \tag{ 6 }$$

The term in the integral contains the overlap between the distributions of atoms and molecules, and we can use this integral to define a mean interspecies density

$$\begin{equation}{\bar{n}}_{\mathrm{a},\mathrm{m}}=\frac{1}{{N}_{\mathrm{m}}(t)}\int {n}_{\mathrm{a}}(\mathbf{r},t){n}_{\mathrm{m}}(\mathbf{r},t){\mathrm{d}}^{3}\mathbf{r}.\end{equation} \tag{ 7 }$$

If we assume the atoms and molecules are thermally distributed in the harmonic region of the trap with temperatures T_a and T_m, respectively, then,

$$\begin{equation}{\bar{n}}_{\mathrm{a},\mathrm{m}}={N}_{\mathrm{a}}{F}_{z}({{\Delta}}_{z}){\left[\frac{{m}_{\mathrm{m}}{\omega }_{\mathrm{m}}^{2}{m}_{\mathrm{a}}{\omega }_{\mathrm{a}}^{2}}{2\pi {k}_{\mathrm{B}}({m}_{\mathrm{m}}{\omega }_{\mathrm{m}}^{2}{T}_{\mathrm{a}}+{m}_{\mathrm{a}}{\omega }_{\mathrm{a}}^{2}{T}_{\mathrm{m}})}\right]}^{3/2},\end{equation} \tag{ 8 }$$

where N_a is the number of atoms (which remains constant over the duration of the measurement), k_B is the Boltzmann constant, m_a, m_m are the masses and ω_a, ω_m are the geometrically-averaged trap frequencies experienced by the atoms and molecules, respectively. F_z (Δ_z ) describes the reduction in overlap due to the difference in gravitational sag Δ_z between the two clouds [90],

$$\begin{equation}{F}_{z}({{\Delta}}_{z})=\mathrm{exp}\left[-\frac{{m}_{\mathrm{m}}{\omega }_{z,\mathrm{m}}^{2}{m}_{\mathrm{a}}{\omega }_{z,\mathrm{a}}^{2}{{\Delta}}_{z}^{2}}{2{k}_{\mathrm{B}}({m}_{\mathrm{a}}{T}_{\mathrm{m}}{\omega }_{z,\mathrm{a}}^{2}+{m}_{\mathrm{m}}{T}_{\mathrm{a}}{\omega }_{z,\mathrm{m}}^{2})}\right],\end{equation} \tag{ 9 }$$

where ω_z,a, ω_z,m are the vertical trap frequencies of the atoms and molecules, respectively. The largest difference in gravitational sag in our experiments, Δ_z = 2.7 μm, is between Rb and RbCs. In this case, we find F_z (2.7 μm) = 0.98; the difference in gravitational sag changes the interspecies density by only 2%, which is much less than the typical uncertainty in either the atom or the molecule density alone (each typically ∼10%). We therefore neglect the F_z (Δ_z ) term in equation (8). Returning to equation (6), we have

$$\begin{equation}{\dot{N}}_{\mathrm{m}}(t)=-{k}_{2}^{\mathrm{a},\mathrm{m}}{\bar{n}}_{\mathrm{a},\mathrm{m}}{N}_{\mathrm{m}}.\end{equation} \tag{ 10 }$$

We have confirmed experimentally that the atom number and temperature do not change appreciably over the course of the measurements presented. If we additionally assume that the molecule temperature does not change significantly, then ${\bar{n}}_{\mathrm{a},\mathrm{m}}$ is a constant. This produces pseudo-first-order kinetics with the solution

$$\begin{equation}{N}_{\mathrm{m}}(t)={N}_{0}\enspace \mathrm{exp}\left(-{k}_{2}^{\mathrm{a},\mathrm{m}}{\bar{n}}_{\mathrm{a},\mathrm{m}}t\right),\end{equation} \tag{ 11 }$$

where N₀ is the initial number of molecules. This allows us to extract two-body rate coefficients from the measured exponential time constants.

We first compare the loss rates measured with molecules prepared in the hyperfine ground state (0, 5)₀ at 181.6 G. Example molecule loss measurements are presented in figure 3. For the reactive combination RbCs + Rb, we find ${k}_{2}^{\mathrm{a},\mathrm{m}}=2.0(4)\times 1{0}^{-11}$ cm³ s⁻¹, while for the nonreactive combination RbCs + Cs we find ${k}_{2}^{\mathrm{a},\mathrm{m}}=1.8(3)\times 1{0}^{-11}$ cm³ s⁻¹. Additional loss rates measured at a magnetic field of 21.3 G and for molecules prepared in (0, 4)₁ are presented in figure 5. We observe no significant variation with the large change in magnetic field, and only marginally higher loss rates for atom–molecule collisions involving molecules prepared in (0, 4)₁.

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We compare the loss rates measured in experiments to a single-channel model based on quantum defect theory (QDT) [65, 66]. The model and its underlying theory have been described at length elsewhere [66, 91, 92], so we omit further description here; we have previously applied it to RbCs + RbCs and Rb + CaF collisions [60, 75, 76]. The long-range interactions are approximated by their leading term −C₆ R⁻⁶ and the short-range interactions are modelled by an absorbing boundary condition. We use values of C₆ from Żuchowski et al [93]. The boundary condition is parameterized by the loss parameter 0 ⩽ y ⩽ 1 of Idziaszek and Julienne [65] and a short-range phase shift δ^s, which is related to the scattering length and controls interference effects including resonances. In the limit of y = 1 all flux that is transmitted past the long-range potential is lost; this is termed the universal limit. The thermally averaged universal loss rates at the current experimental temperatures are 6.0 × 10⁻¹¹ cm³ s⁻¹ for RbCs + Rb and 5.0 × 10⁻¹¹ cm³ s⁻¹ for RbCs + Cs; these are indicated by horizontal dashed lines in figure 5.

The results of the QDT model as a function of y and δ^s, for RbCs + Rb and RbCs + Cs, are shown in figure 6, with the experimental results and their uncertainties. The present results limit the loss parameter to less than 0.35, with the most likely range being 0.2 < y < 0.3 for both species. It is notable that the ranges are very similar for the two systems, even though one is potentially reactive and the other is non-reactive. The ranges also overlap with the result for RbCs + RbCs, y = 0.26(3) [60]. All three results are consistent with a recent prediction [94] that rapid loss from collision complexes would result in an effective loss rate described by y = 0.25.

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Finally, motivated by the observation of Feshbach resonances in ²³Na⁴⁰K + ⁴⁰K collisions [34], we check for the possibility of resonant behaviour close to the magnetic fields at which we perform our experiments. To do this, we hold the atom–molecule mixture in the trap for 10 ms (6 ms for RbCs + Rb around 181.6 G) and vary the magnetic field by ±5 G in steps of ∼0.1 G, as shown in figure 7 for molecules prepared in (0, 5)₀. To determine if there is any variation from background we make a histogram of the results and extract a mean and standard deviation. For all four measurements we observe that only ∼5% of the points are outside the interval defined by the mean ±2σ, as would be expected for normally distributed noise. Additionally, we see no obvious large dips in the molecule number that would indicate the presence of resonances.

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It is possible that the dominant loss process for atom + molecule collisions is the same fast optical excitation of two-body collision complexes that we have observed for molecule + molecule collisions [60, 72]. In the RRKM limit, the laser-free lifetime of the RbCs + Cs collision complex in the dark is expected to be a factor of 2 × 10⁴ shorter [69, 95] than the lifetime of ∼0.5 ms for the (RbCs)₂ complex. If the lifetime of the atom–molecule complex is so short, we may expect that there is not enough time for significant laser excitation before the complex dissociates, or that at least this process may not be saturated. However, recent experiments [77] studying nonreactive collisions of KRb with Rb found a photon-free lifetime for the complex of 0.39(6) ms, which is 5 orders of magnitude longer than the RRKM prediction, along with evidence for loss associated with optical excitation of the complexes. The reason for such long-lived complexes in KRb + Rb and whether optical excitation of complexes can contribute significantly to loss in collisions between other combinations of atoms and molecules are open questions.

To investigate this possibility, we have measured the lifetime of RbCs molecules, in the ground state (0, 5)₀, in the presence of Cs atoms, with and without 1 kHz square-wave intensity modulation, in the λ = 1064 nm optical trap described in section 1. As for the experiments on RbCs alone, the square-wave modulation is set such that the atom–molecule mixture spends 75% of each cycle in the dark. The peak intensity in the modulated trap is 4 times that of the CW trap, so that the trap frequencies and trap depths are the same for both trap configurations.

A comparison of the molecule lifetimes for the CW and modulated traps is shown in figure 8. We find two-body rate coefficients of k_mod = 2.1(6) × 10⁻¹¹ cm³ s⁻¹ for the modulated trap, and k_CW = 2.6(6) × 10⁻¹¹ cm³ s⁻¹ for the CW trap. As such, there is no statistically significant difference in the loss rates.

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To examine our results in the context of optical excitation of atom + molecule complexes, we construct the rate-equation model

$$\begin{align}\hfill {\dot{n}}_{\mathrm{m}}& =-{k}_{2}^{\mathrm{a},\mathrm{m}}{n}_{\mathrm{a}}{n}_{\mathrm{m}}+\frac{1}{{\tau }_{-1}}{n}_{\mathrm{c}},\hfill \\ \hfill {\dot{n}}_{\mathrm{c}}& =+{k}_{2}^{\mathrm{a},\mathrm{m}}{n}_{\mathrm{a}}{n}_{\mathrm{m}}-\frac{1}{{\tau }_{-1}}{n}_{\mathrm{c}}-{k}_{\text{laser}}I(t){n}_{\mathrm{c}}.\hfill \end{align} \tag{ 12 }$$

This describes the loss of molecule density due to the atom–molecule collisions and the dissociation and optical excitation of the associated atom–molecule collision complexes. We neglect the formation (and loss) of complexes resulting from molecule–molecule collisions owing to the much longer associated timescale. Additionally, there are no inelastic decay channels for the state combination used in the measurement, such that the lifetime of the complex τ_c = τ₋₁. The predictions of this model are shown in figure 9 and lead to two interpretations of the experimental observations.

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In figure 9(a) we show the predicted ratio (k_CW/k_mod − 1) as a function of k_laser I_avg τ₋₁ under the assumption that the lifetime of the complex is much shorter than the trap dark time (τ₋₁ ≪ t_dark) so that the suppression of loss in the modulated trap is independent of the trap modulation frequency. Here I_avg is the average trap intensity experienced by the molecules in either trap. For modulation where the trap light is off for 75% of each cycle, we expect k_CW/4 < k_mod < k_CW such that the ratio (k_CW/k_mod − 1) can take values between 0 and 3. The ratio extracted from our experiments is (k_CW/k_mod − 1) = 0.2(5). This is indicated by the horizontal solid and dashed lines in figure 9. The grey shaded region in figure 9(a) indicates the range k_laser I_avg τ₋₁ < 0.3 consistent with our experimental results at the 68% confidence level. This corresponds to our results being consistent with optical excitation that is not saturated. This could be due to the predicted short lifetime of the complex, but also depends on the unknown laser scattering rate for the atom–molecule complexes.

In figure 9(b) we examine an alternative interpretation of the experimental results, namely that the lack of suppression in the loss results from the formation of complexes with long photon-free lifetimes. In this case, we assume the optical excitation of the complexes is saturated k_laser I_avg ≫ 1/τ₋₁. Little or no suppression can then occur if the dark time is too short for a significant number of complexes to decay before the next bright trap pulse. Figure 9(b) shows the prediction of our model in this limit as a function of τ₋₁. Our results are consistent with a photon-free lifetime of the complex τ₋₁ > 0.4 ms, again at the 68% confidence level. This would be similar to the recently measured lifetime for complexes in KRb + Rb collisions [77], but is 5 orders of magnitude longer than the RRKM prediction.