Machine learner optimization of optical nanofiber-based dipole traps Open Access

Optical nanofiber (ONF) waveguides have proven to be a very promising platform for cold atom-based hybrid quantum technologies, with several applications demonstrated in recent years.^1–7 In particular, alkali atoms trapped in a two-color, evanescent field dipole trap around an ONF^8–10 demonstrate long trapping lifetimes and useful collective properties,^11,12 while facilitating direct interrogation of the atoms and integration of the experimental platform into a fiber network. For many applications, trapping alkali atoms without affecting the energies between the relevant atomic levels is desirable, as a deep trap potential can be achieved without broadening the relevant transition beyond usefulness. This can be achieved by tuning the two-color trapping lasers—the red- and blue-detuned lasers—to magic values,^13,14 where the differential ac-Stark shifts cancel. Such a compensated ONF-based dipole trap has been demonstrated for laser-cooled Cs atoms¹⁵ with around 2000 atoms trapped.

For Rb atoms, the situation is somewhat more complicated than for Cs; there are magic wavelengths,^16–19 but these are either too close to resonant transitions, causing heating, or are not convenient to use in nanofiber-based traps, due to absorption by the silica fiber. There has been only one report of a (non-state-compensated) ONF-based dipole trap for Rb atoms.²⁰ The authors estimated that 302 ⁸⁷Rb atoms were trapped, with a lower bound of 123 atoms. This number is relatively low compared to that reported for Cs.

In an uncompensated trap that is deep enough to trap Rb atoms cooled from a magneto-optical trap (MOT), a tens of MHz light shift broadens the absorption profile of a probe beam near-resonant to the cooling transition (⁠$5S1/2→5P3/2$ for ⁸⁷Rb) and creates asymmetries in the observed spectra.²⁰ This makes it difficult to determine the number and temperature of trapped Rb atoms as the saturation absorption method⁹ cannot be used, and dispersive methods¹⁰ are much less effective. Real-time and manual optimization of the trap loading parameters is very time intensive, if not near impossible, in contrast to Cs.¹⁰ Furthermore, optimization of the optical depth (OD)²¹—an important parameter that determines how much of the probe beam is absorbed by the atoms—depends on both the number and temperature of the trapped atoms, rendering OD determination in real-time also very challenging.

With Rb atoms being one of the most widely used atomic species in cold atom quantum technologies, finding techniques to optimize their loading into ONF-based dipole traps is crucial. The aforementioned difficulty in measuring the number of trapped atoms in real-time renders optimization of the trapping sequence during an experiment essentially impossible if one tries to do this manually. Additionally, while free-space dipole traps have been studied systematically,²² this approach becomes significantly harder as the atom trap dynamics become more complex or where non-adiabatic pathways become more effective. A complete quantitative description of atom dynamics for the optimal loading of evanescent field dipole traps from a MOT does not exist, though many of the requirements to efficiently load free-space dipole traps are useful as a starting point.¹³ 

One tractable approach to optimizing a physical system is by using online optimization, which does not require access to the system's quantitative description. Such an approach has previously been used to optimize the loading of atoms into a Bose–Einstein condensate using a Gaussian process learner,^23,24 for example. This technique allows a learner (or agent) to directly interact with the physical system in an in-loop setting, providing it with new parameters to implement while receiving feedback on its response. For larger parameter spaces, it is better to leverage methods, which are computationally more expensive so as to minimize the time spent measuring the physical system. One possibility is the use of deep learning methods, which has been successfully demonstrated for high dimensional problems such as image classification²⁵ and regression.²⁶ 

In this work, we investigated the loading of laser-cooled ⁸⁷Rb atoms into an uncompensated ONF-based dipole trap array using machine learning to optimize the final number of trapped atoms and optical depth of the system. We compared the outcome of the machine learner (ML) optimization with that achieved through manual optimization that took several months, with peak OD increased from 3.2 to 5.3. We discuss how the manual process is unlikely to achieve a comparable outcome due to the non-intuitive optimization process followed by the machine learner, illustrating the advantage of using such techniques for ONF-based dipole trap loading.

We applied a machine learner (ML) optimization protocol to an experimental control sequence for cooling the atoms in a MOT and subsequently loading them into an ONF-based dipole trap array. Figure 1 illustrates how the ML controlled the experiment and explored the experimental parameter landscape that we made available to it. The ML optimization protocol was based on earlier work²⁷ in which a predictive agent, in this case a stochastic artificial neural network (SANN), explored a parameter space by predicting new optima based on the results of its previous predictions. The SANN is an ensemble of neural networks acting as surrogate models that constitute the mapping from parameter space to a physical cost, which represents the experimental output to be optimized. In our case, we aimed to maximize the number of atoms trapped in the dipole trap by minimizing the transmission of a probe beam through the ONF; hence, the probe beam transmission was our cost and the optimization process minimized this value. “Stochasticity” arose from the independent random initialization of the neural networks in the ensemble, which generated multiple unique representations of the experimental response landscape. This provided multiple regions of parameter space to explore and allowed further optimization or dismissal of those regions as the neural network model was refined, allowing us to balance the exploration versus exploitation trade-off. Unlike other surrogate methods, the SANN was given direct control of the experiment optimization as part of an in-loop feedback. See supplementary material appendix for further technical details on the SANN.

In our experiments, a cloud of cold ⁸⁷Rb atoms was formed using a MOT based on a three-beam retro-reflected configuration with 15-mm-diameter cooling beams of intensity 8 mW/cm² and a maximum magnetic field gradient of 15 G/cm. This configuration was used in previous experiments investigating atom–light interactions in the ONF evanescent field by placing at the nanofiber waist a magneto-optically trapped cloud of $∼5×106$ atoms at a density $∼109$ cm³, or a sub-Doppler-cooled cloud at temperatures $∼40$ μK. An optical nanofiber with an exponentially shaped taper profile was installed in the ultrahigh vacuum (UHV) chamber so that it was positioned at the intersection of the laser-cooling beams. The ONF was fabricated from commercial single-mode optical fiber (Fibercore SM800-5.6-125)²⁸ and had a ∼4-mm-long waist region and a ∼400-nm-diameter waist region. The basics of ONF fabrication are described in some of our earlier work.²⁹ 

The ONF-based two-color optical dipole trap array was created using a combination of red- and blue-detuned beams relative to the cooling transition, $5S1/2→5P3/2$⁠, in ⁸⁷Rb. A schematic of the experimental setup is shown in Fig. 2. We sent 1064-nm (red-detuned) light through the ONF in a counterpropagating configuration with 1.8 mW in one direction and 2.1 mW in the opposite direction to provide (i) an attractive force for the atoms toward the ONF surface and (ii) the one-dimensional (1D) optical lattice along the fiber. The difference in the powers is due to different losses in each of the taper regions. Additionally, 1.23 mW of 762 nm (blue-detuned) light was sent through the ONF along one direction to provide a repulsive force from the fiber surface, moving the dipole trap potential minima away from the surface of the nanofiber. A weak probe beam (5 pW) resonant with the $5S1/2(F=2)→5P3/2(F′=3)$ transition at 780 nm was sent through the ONF counterpropagating to the 762-nm light and was used to measure the absorption profiles using an in-fiber analogous technique to free-space absorption spectroscopy. The stated powers of each beam were those measured at the fiber outputs after passing through the ONF.

All the fiber-guided beams were quasi-linearly polarized along the x axis, that is, perpendicular to the fiber axis, using the method described in Ref. 30. For parallel-polarized red- and blue-detuned beams, we would expect the vector light shifts to be large, but this configuration reduced the amount of power needed in the blue-detuned beam.¹⁰ Note that we manually adjusted the polarization of each beam slightly to maximize the probe beam absorption (in other words, the number of atoms within the evanescent field region) prior to conducting any experiments. The combined evanescent fields of the fiber-guided light beams formed a dipole trap array along the waist located at $∼200$ nm from the ONF surface.

For both manual and ML optimization, the sequence for cooling atoms in the MOT and subsequently loading them into the dipole trap array was computer-controlled via LabVIEW. Timed analog and transistor-transistor logic (TTL) voltages were used to control acousto-optic modulator (AOM) frequencies and amplitudes (thereby controlling laser frequencies and powers), and to open/close coil circuits to switch the MOT's magnetic field on and off. A general purpose interface bus (GPIB) controller set the programmable current supplies for the coils. The probe beam transmission through the ONF was measured using a single-photon counting module (Excelitas SPCM-AQRH-FC). A volume Bragg grating (VBG) and ruled grating (RG) were placed in the probe beam path to filter the trapping fields.

Manual optimization of the dipole trap loading took us several months of constant fine tuning of the experimental parameters. During this process, we adjusted essentially the same set of experimental parameters and followed the same timing sequence as was later used for the ML optimization, described below. However, manually, we could also adjust the cooling and repump beam alignments. A typical manual optimization consisted of first maximizing the geometric overlap of the cold atom cloud with the ONF during the optical molasses phase by adjusting the magnetic field amplitude and zero position, then slightly realigning the cooling and repump beams. Finally, absorption of the probe through the ONF by the dipole-trapped atoms was maximized by adjusting the red- and blue-detuned beam powers and their relative polarization.

For the ML-optimized experiments, the above manual optimization process was followed by iterative adjustment of n = 19 parameters, corresponding to timings, magnetic field amplitudes, and optical field powers and detunings, via the ML algorithm, see Fig. 1. During an overall 2-s-long experimental cycle, the MOT was first operated with constant optical and magnetic fields set by the ML for 1620 ms. The ML controlled the cooling laser detuning (12 to 20 MHz) and intensity (4 to 8 mW/cm²), repump intensity, the magnetic quadrupole field gradient (13 to 15 G/cm), the zero magnetic field position along the x axis (±5 mm), and the magnetic field generated by the compensation coils (up to 1 G). Numbers in parentheses indicate the range of values the ML was allowed to use when exploring the parameter space. Next, the cooling laser detuning was swept for 180 ms (to 12 to 36 MHz) before the magnetic field Helmholtz coils were switched off. The cooling laser detuning/intensity and the repump beam intensity were ramped for 5 ms (divided into two variable-length subperiods) to form a 40-μK cloud of rubidium atoms. The red-detuned nanofiber dipole trap light intensity was ramped down during these 5 ms, but was otherwise kept constant during the experiment in order to prevent thermal stresses from causing movement of the ONF. During this final cooling process, cold atoms from the MOT were loaded into the dipole trap array.

Once dipole trap loading was achieved, the cooling and repump lasers were switched off, the probe was detuned +10 MHz from the $5S1/2(F=2)→5P3/2(F′=3)$ transition to give a high signal-to-noise ratio (SNR) and sent through the ONF for 65 ms, and absorption of the probe by the dipole-trapped ⁸⁷Rb atoms was measured for the first 10 ms. A dummy cycle was run each time the parameters were changed to equilibrate recapture of atoms in the MOT. The transmission of the probe was integrated over the 10-ms window, background subtracted, and normalized relative to a probe sent later in the cycle once atoms were cleared from the traps, and averaged over three experimental cycles.

The aim of the ML optimization was to minimize the averaged probe transmission through the ONF and, correspondingly, increase the probe's absorption by increasing the number of atoms in the dipole trap array, see Fig. 1(b). Therefore, the cost function was set as the averaged probe transmission, $C(x)=Tprobe$⁠, and it was used to train five neural networks, see Fig. 1(c). Predictions from the SANN were used for online optimization, according to the method of Ref. 27 with some modifications. The cost function was derived from the probe transmission because direct measurement of the atom number after each cycle was not possible. Existing methods would have been too time-consuming and would have significantly impacted the usefulness of the ML optimization process itself.

The experiment was run $2n+1$ times (where n is the number of parameters being adjusted) using random sets of parameters within the allowed space and measuring the cost each time. This parameter-cost data were used to train the neural networks. The choice of $2n+1$ runs was to ensure that random training grew along with the number of parameters being controlled and could be viewed as the minimum number of runs needed while ensuring the process remained time efficient. The first neural network was then used to predict a set of optimal parameters, and the experiment was run with these parameters, generating a new parameter-cost pair to add to the data. The entire data set was used to train the next neural network, which was then used to predict a new set of optimal parameters. This process was continually iterated and, if the ML detected convergence (predicted optima continually lying within a small parameter range), local cost minima outside this region were explored to improve the model around any other possible global minima in the parameter space.

On completion of a preset number (usually 300) of iterations by the ML, the experimental parameters that yielded the maximum measured absorption signal (due to the minimized cost) were tested in a second series of experiments to determine the number of trapped atoms and their temperature. Here, the transmission of the probe beam was measured over the entire absorption window of the $5S1/2(F=2)→5P3/2(F′=3)$ transition, from −35 to +85 MHz detuning, immediately after the molasses stage, and was typically averaged over 25 experimental cycles, see Fig. 3(a).

To determine the number of atoms loaded into the dipole trap array, the absorption of a fiber-guided probe by the trapped atoms was modeled for various trap powers and temperatures. We model the absorption of the fiber-guided probe by the trapped atoms according to the following equation, which is modified from Ref. 20:

The atoms were assumed to have a thermal ensemble density, $ρ$⁠, with an average temperature, T, due to an approximate van der Waals potential, V_vdW,³¹ and an optical potential, V_opt induced by light shifts^32,33 of the $(5S1/2,F=2,mF)$ states by the fiber-guided trapping fields.³⁴ The optical depth per atom for each transition is given by $OD(mF,eprobe)$⁠. We assumed only Lorentz broadening due to the natural linewidth, $l$⁠, and a local frequency shift due to the light shifts of the upper and lower levels, $δ(eprobe(r),mF))$⁠. Integrating over the trap volume produces the observed broadening.

The OD was calculated for the local probe intensity and polarization, with transition strengths given by the Clebsch–Gordan (C–G) coefficients and the transition cross section. The atoms were assumed to have an m_F population distribution due to optical pumping by the probe field, with the m_F distribution determined by the steady state of the population transfer due to the probe polarization at the trapping potential minimum, with C–G coefficients for the various transitions to the $5P3/2,F′=3,mF′$ levels and instantaneous spontaneous decay back to $5S1/2F=2$ (ignoring stimulated emission and other excitation processes).

In Fig. 3(a), we plot the probe beam transmission as a function of detuning for both the manually optimized (black curve) and ML-optimized (red curve) dipole traps. The theoretical model was fitted using 300 trapped atoms for the manually optimized trap and 450 atoms for the ML-optimized trap. The atoms loaded into the ML-optimized trap were also at a slightly lower temperature of 140 μK compared to 150 μK for the manual trap. Figure 3(b) plots the same data as optical depth of the probe absorption, showing more clearly the fitting of the model at intermediate detunings. The powers of the dipole trap light fields used to fit the spectra were based on the fitting of the modeled spectra to the probe transmission data and are significantly different to experimentally measured dipole trap transmissions at the output of the nanofiber. Losses at the waist due to fiber degradation, particularly for the 1064-nm light, mean the actual powers at the waist were likely higher than those measured. For example, the measured 1064-nm output power was 2.1 and 1.8 mW at either end of the ONF, whereas the modeled absorption spectrum fits a power of 3.25 mW in each direction. For 762-nm light, the measured output power was 1.23 mW, whereas 1.3 mW was fitted by the model. Note that during the manual optimization, we adjusted the polarization of each beam slightly to maximize the probe beam absorption.

Figure 3(c) shows the trapping potential in the x–y plane for the modeled trap. The quasilinear polarization of the trapping fields in the model was adjusted to fit the experimental spectrum. Contour lines emphasize that this resulted in an asymmetric trap. This configuration was previously shown to increase the OD per atom by moving the trapping sites closer to the ONF.²⁰ Figure 3(d) shows the corresponding light shifts of a probe transition $mF=0→mF′=0$⁠, with the potential minima marked by dots.

The number of atoms trapped in the dipole array was determined from the fit of the modeled spectra, which were calculated for a range of combinations of dipole trap laser powers, polarizations, atom temperatures, and fiber diameters, then fit with a skew Gaussian model. This yielded the central frequency, width, and asymmetry parameters. The experimental data were fit with the same function, and the model with the closest matching parameters was chosen. The Gaussian fit width increases with temperature, the central frequency increases with overall trap power, and the asymmetry increases as the trap minima move closer to the fiber. These trends make us confident in the matching of the modeled spectra to the experimental data, clearly indicating an increase in the number of trapped atoms for the ML-optimized system. The uncertainty in the atom number is obtained by comparing modeled spectra that fit the experimental spectra but give a different peak OD per atom. See supplementary material appendix for further details of how the model was used to analyze the experimental data.

Figure 4 shows the dependence of the learned cost on a selection of the experimental parameters, centered on the best observed. These data are not entirely accurate as the slices represent mostly unsampled areas of parameter space. However, these plots are somewhat useful as they indicate which of the parameters played an important role and which had little role in the ML optimization and can, therefore, be neglected. They also allow us to determine whether the parameter range should be changed for the ML to explore a different landscape. For example, the repump laser intensity (i), the cooling laser detuning (iv), and the gradient magnetic field offset along the y axis (v) have the strongest effect over their allowed ranges, showing the sensitivity of the cost to these parameters. The numbers in parentheses correspond to the labels in Fig. 4. Due to this sensitivity, we set the magnetic quadrupole parameters as a gradient and position offset to allow the ML to more quickly explore this space (as compared to setting parameters for individual currents through the two quadrupole coils). Learned parameters were stable over time, and reusing the parameters from a single optimization produced a reasonably constant number of trapped atoms over a period of about one month, with no more than $±10%$ change observed.

By instigating ML optimization, we observed a clear increase in the peak OD from 3.2 ± 0.2 to 5.3 ± 0.3, representing a 66% increase in the optical depth of the system. From the OD, we deduce that the number of trapped atoms in the ONF-based dipole traps increased from 300 ± 60 to 450 ± 90. Our manually adjusted dipole trap array held a very similar number of atoms as that reported in Ref. 20. The increase in atom number and decrease in temperature from 150 to 140 μK in the ML-optimized experiments are likely due to improved cooling and increased atomic density in the vicinity of the ONF. Notably, the ML chose non-intuitive experimental parameters in achieving the optimization. For example, the ML was inclined to position the cloud to one side of the fiber, in stark contrast to our manual optimization where we centered the cloud on the ONF. This positioning of the cloud is likely due to irregularities in the ONF (such as stresses and Rb depositions after 5 years of experiments) that lead to optimum light coupling at specific locations. While the achieved differences may appear unremarkable, a 66% increase in the optical depth of the trapped atoms has been observed. Given the sensitivity of the ML to the magnetic fields, we hypothesize that optimal positioning of the MOT as well as the zeroing of the magnetic field for polarization-gradient cooling are the largest improvements given by the ML. By increasing the number of parameters available to the ML beyond 19, we would expect further improvements. In particular, experimental improvements including a time-varying magnetic field to compress the atomic cloud and increase density before loading, and the ability to change the dipole trapping powers more during loading without causing movement of the fiber, should allow more atoms to be loaded.

It was not possible for us to make a direct measurement of atom number by measuring the total absorbed power of a strong probe beam saturating the trapped atoms.⁹ The force due to the light shift by a strong probe pushed atoms from the dipole trap array before enough photons could be detected, preventing us from making a reliable measurement. A heterodyne setup should make this type of measurement possible as it would enable more sensitive detection of coherent light. This would also allow for a more direct measurement of the trap properties using the method described by Solano et al.³⁵ This will be implemented in future experiments.

For effective ML optimization, it is crucial that the cost function, which dictates how the experiment is measured and controlled, is carefully chosen. In our experiment, the goal was to increase the number of atoms trapped in an ONF-based dipole trap. Since a direct measurement of the atom number was impossible, the reduced transmission of a probe due to absorption by the atoms was the next obvious choice. Averaging over several experimental runs, applying background and baseline subtraction, and detuning the probe from the absorption maximum were all necessary to provide a sufficiently high SNR. Probing at a single frequency meant the measurement was also sensitive to factors such as residual magnetic field and atom temperature. These could affect the cost by changing the frequency of the absorption maximum.

The chosen cost function was deemed suitable as most of the improvement in cost corresponded to a related increase in atom number. This indicates that other influences on the cost were small. Similar cost functions—for example, derived from a probe with a different detuning or sent at a different time—performed similarly or worse, likely due to a lower SNR. However, it is possible that a cost function derived in a different way, such as by incorporating the more direct atom number measurement of Ref. 35 could yield a larger increase in atom number when optimizing over the same parameters.

The optimizer did increase the overall optical depth by a larger percentage (66%) than the number of atoms (50%), by also decreasing the temperature. The optical depth per atom increased as the atoms spent more time closer to the fiber, increasing the probe absorption and reducing the cost. The compromise on the ML made between reducing temperature and increasing atom number was sensitive to the cost function, as the absorption spectrum changes shape with temperature, and has slightly different time dependence as colder atoms take a little longer to be heated out of the trap. Additionally, the parameters that affect these, such as magnetic field bias, also change the position of the MOT. These factors, which provide a challenging endeavor for even experienced operators to balance, are implicit in the cost function, thereby allowing the ML to consider them simultaneously. This highlights the difference between the manual and ML optimizations. A human operator has limited ability to simultaneously change a large number of parameters by large amounts and make sensible inferences about the results. The ML has more power to do this and, while it often makes incorrect inferences due to necessarily having limited information about the entire parameter space, it can correct its mistakes and make better informed guesses. These guesses quickly add up to an improved result.

The ML could also be used to produce specific m_F state populations in the trap, but this would require changes to the experiment so that a specific bias magnetic field could be set after loading the dipole traps. The probe detection would need to be sensitive enough to measure the absorption prior to optical pumping caused by the probe itself. It could be possible to achieve even further improvements by giving additional parameter controls to the ML.

We used a machine learner optimizer to increase the optical depth of optical nanofiber-based evanescent field dipole traps for ⁸⁷Rb atoms by 66% from 3.2 ± 0.2 to 5.3 ± 0.3. We derived a microscopic theoretical model that fits the experimental probe transmission spectra and enabled us to determine the number and average temperature of the trapped atoms. According to this model, ML optimization increased the number of trapped atoms from 300 ± 60 to 450 ± 90 and reduced the temperature from 150 to 140 μK. We expect this to be further improved by (i) increasing the number of parameters controlled by the machine learner and (ii) using a new ONF in the experimental setup. Additional investigations of the capabilities of this setup are planned, including optimizing the loading of atoms for a collective atom–light interaction such as four-wave mixing, and increasing the number of nearest-neighbor interactions in a 1D lattice of Rydberg atoms. We envision that this will be feasible with the obtained number of atoms. The techniques developed herein can also be extended to any atomic species where it is desirable to increase the atom number.

See the supplementary material for more information about the artificial neural network and numerical modeling.

This work was supported by OIST Graduate University and Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research Grant Nos. 19K05316 and 20K03795. S.N.C. acknowledges support by Investments for the Future from Laboratoire d'excellence Physique Atomes Lumière Matière (No. ANR-10-LABX-0039-PALM). A.T. acknowledges support from the Australian Research Council (Grant No. CE170100012) R.H acknowledges support from the German Academic Exchange Service (Award/Contract No. 57380758).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.