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Abstract
We present a machine-learning experiment involving evaporative cooling of gaseous 87Rb atoms. The evaporation trajectory was optimized to maximize the number of atoms cooled down to a Bose-Einstein condensate using Bayesian optimization. After 300 trials within 3 hours, Bayesian optimization discovered trajectories that achieved atom numbers comparable with those of manual tuning by a human expert. Analysis of the machine-learned trajectories revealed minimum requirements for successful evaporative cooling. We found that the manually obtained curve and the machine-learned trajectories were quite similar in terms of evaporation efficiency, although the manual and machine-learned evaporation ramps were significantly different.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Machine learning (ML) technology accelerates various fields of experimental science, such as efficient control of experiments [1–4], prediction of physical properties based on statistics termed material informatics [5,6], and reconstruction of low clarity images [7–9]. ML uses available data to estimate (“learn”) a function that maps parameters to experimental results; the constructed function can be used for prediction beyond the data used for learning. This prediction ability is useful for controlling apparatuses or experimental conditions to desired states [10].
We report online optimization of evaporative cooling based on Bayesian optimization (BO), which is an ML technique to efficiently find the best experimental parameters that yield the maximum performance based on statistical prediction with Gaussian processes [11, 12]. Evaporative cooling is a technique to cool down a neutral atomic gas into a degenerate quantum gas such as a degenerate Fermi gas or a Bose-Einstein condensate (BEC) [13]. This technique is widely adopted in various modern quantum studies and applications that employ quantum gasses such as quantum simulation [14,15], quantum sensing and metrology [16–18]. Online optimization means that the statistical prediction is updated as soon as a new observation is available. Online optimization of evaporative cooling is a useful target to test whether the ML technology can be applied for automation of highly-sophisticated machines of quantum technologies. In this work, we reduced the evaporation curve, a time-dependent function, to guide the cooling process into a polygonal line that linearly connects multiple nodes, and the values at the nodes were controlled to maximize the evaporation results by using BO. The search space of this method is larger than that of a previous work that treated the evaporation curve as a closed form function containing several free parameters to be optimized [1]. However, the method can be applicable even if the prediction in closed form is hard to obtain, e.g., for atomic species of which evaporative cooling is difficult due to complicated collision processes. Our goal in this paper is not to improve optimization but to explore possibilities of optimization with larger degrees of freedom.
In this paper, we analyzed the optimal trajectories found by a BO-based online optimization method and found non-standard behavior such as non-monotonic decreases. We used an optical trap with a cross configuration for creating BEC and optimized the powers of the trap beams (Section 2.1). We designed a piecewise linear function for modeling a beam power time sequence, which is to be optimized as an evaporative cooling trajectory (Section 2.2). The performance of BO was evaluated by comparing it with random-search and human performance results. We gained insights into the evaporation by analyzing the correlation of parameters for higher scores (Section 3). The non-standard trajectories were compared with the result in our human experiment, and the trend of the trajectories obtained by the Bayesian optimizer was interpreted from a probabilistic perspective (Section 4).
2. Methods
2.1. Experimental setup
Our experimental setup for laser cooling consisted of a 2D magneto-optical trap (MOT) and a 3D MOT. After loading for 5 s, approximately 108 rubidium-87 (87Rb) atoms were collected in the 3D MOT. The atomic cloud was compressed and then cooled further by polarization gradient cooling (PGC). The cooled atoms were directly transferred into a far off-resonance trap (FORT), which was created by crossing two focused beams at an angle of 90 degrees in the horizontal plane (Fig. 1(a)). The beams were generated from a fiber laser operating at 1064 nm. To avoid interference of the two beams, their frequencies were shifted by acousto-optical modulators (AOMs) driven with different RF frequencies of 80 MHz and 110 MHz; in addition, we used perpendicular linear polarizations for the beams. The power of the beams was independently controlled by the AOMs. The beam waists were approximately 40 μm. The FORT beams were switched on at the maximum power of approximately 4 W for each beam during the MOT stage. After switching off the cooling lasers used for PGC, the atoms were held in the optical trap at a constant trap depth of 500 μK in 300 ms for thermalization. At this stage, 3.8 × 106 atoms at 40 μK were prepared. We note that some atoms were distributed outside of the crossed region. Forced evaporation was applied by decreasing the powers of the FORT beams. We optimized the FORT power ramp in this work. After evaporative cooling and holding time of 500 ms, atoms were suddenly released from the optical trap. The atom distribution was recorded by absorption imaging after free expansion for 25 ms.
Fig. 1 Optimization of evaporation in a crossed FORT. (a) Schematic figure of a crossed FORT and evaporative cooling. (b) Optimization of the evaporation trajectory. FORT power intermediate control voltage (red filled circles) is adjusted within the range indicated by the black arrows, while the initial and the final values are fixed to the power in the plain evaporation (PE) and the hold time (HT) after evaporation, respectively. (c) Absorption image of a 87Rb BEC. The BEC obtained with the manually optimized evaporation trajectory is shown. The black dashed circle, with a diameter of 100 μm, indicates the accumulation area for counting the number of atoms as the score of optimization.
2.2. Optimization
When we decrease the power of the FORT beams, atoms with higher energy escape from the optical trap, and the remaining atoms become colder after rethermalization (Fig. 1(a)). The efficiency of cooling depends on how the trap beam power is reduced. Often, an exponential ramp is used for efficient evaporation [19,20].
In this work, we split the cooling process into five stages with equal time durations and changed the intermediate power of the FORT beams. More precisely, we used the control voltage input into the two AOMs to adjust the power of the beams as optimization parameters. At each stage of evaporation, the voltage was linearly ramped. To avoid the complexity of optimized parameters and focus on the physics of optimized evaporation trajectories, we fixed the total evaporation time at 6350 ms as well as the initial and final power of the beams. The fixed values were based on the evaporation sequence we manually optimized to obtain an almost pure BEC with a large number of atoms. We also restricted the search region of the control voltage to under the line linearly connecting the initial and final voltages. The shaded convex region in Fig. 1(b) shows the area spanned by the evaporation curves drawn by our optimization scheme. The four intermediate powers of the two FORT beams were independently controlled; thus, we optimized 8 parameters in total.
The phase space density (PSD) of the samples obtained shows the quality of cooling [19,20]. In the present study, the temperature or energy of the samples was determined by the fixed final trap depth; thus, we could simply use the number of atoms for the score of optimization. The atom number can be extracted from an absorption image, and often, an adequate fit, e.g. a Gaussian fit for a thermal cloud and a parabolic fit for a pure condensate, is used to remove noise from the images. However, such a fit sometimes fails if there are too few atoms. For robust evaluation of the score, we summed the atoms in a certain region in the image. The region was inside the circle with a diameter of 100 μm shown in Fig. 1(c). The diameter roughly corresponded to the full width at half the maximum of the optical density of the typical BEC images. It is notable that the atom number evaluated in a small area around the cloud center after a sufficiently long time-of-flight is related to the PSD of the samples, assuming that the interatomic interaction effect is negligible. Therefore, the score evaluation method used here can be applied to more general cases, where the final trap depths are also varied. To reduce shot-to-shot atom number fluctuation, we ran a sequence twice for each trial and averaged the atom numbers. In each shot, there was about 4.5 s for the experimental sequence preparation by the controller, which did not include the calculation for BO. The total time for a single trial with two shots was approximately 35 s.
Bayesian optimization [21] was utilized for automatic online optimization of the cooling. We employed the COMBO library [22] to implement BO in our setup. The parameter optimization with BO is based on prediction from past data. The expected number of atoms and its probabilistic uncertainty are predicted for parameters that compose evaporation trajectories. BO determines the parameters used for the next trial by automatically balancing exploitation (taking high expectation) and exploration (reducing uncertainty). For taking a good balance, several techniques are available such as Thompson sampling, probability of improvement, and expected improvement [12]. COMBO employs Thompson sampling, which randomly draws a parameter set with the probability that the parameter set is optimal; that is, even parameters that are expected to yield low scores can be selected for the next trial with certain probability. The first 20 trials were conducted with a random search, and a prediction model was constructed. The parameters of the next trial were selected based on the previous results, and the prediction model was updated as a new measurement was found. Our code is available at https://github.com/atsu-kan/quml under the Apache 2.0 License.
3. Result
The optimization results for performance are represented in Fig. 2. We plotted the maximum score among all the searched results up to that trial, as a function of the number of trials (Fig. 2(a)). Because the optimization result depended on the random process of the first 20 trials as well as fluctuations and drift of the atom number, we repeated the optimization search 16 times, starting with different initial searches. The result of BO is described by the blue solid line representing the average score of 16 optimization sets and the blue shaded area showing the standard deviation. The Bayesian optimizer performed better than random search (the red dot-dashed line and shaded area). The score reached to the values comparable to the manually optimized result (the green dashed line) within 300 trials, which required approximately 3 hours. We conducted optimization searches up to 400 trials. After all the optimization experiments, we checked the reproducibility and stability of the atom number with the optimized parameters (Table 1). It is notable that the stability of the scores is better for machine-learned trajectories than for the manual curve.
Fig. 2 Optimization of evaporative cooling performance. (a) Optimization result of a single set. The red filled circles show scores obtained with 400 trials, which consist of 20 random search and 380 BO trials. The blue solid line represents the maximum score among all the searched results up to that trial. The score of the manually optimized evaporation trajectory is indicated by the green dashed line, as a guide. (b) Performance averaged over 16 different optimization sets. The blue solid line and shaded area show the average and standard deviation of the 16 sets, respectively. The width of the shaded area represents the scatter of optimization results with different random searches. The BO experiment shared the result of the initial search of 20 measurements with the random search experiment, whose average and standard deviation are described by the red dot-dashed line and shaded area, respectively.
Table 1. Stability of Evaporation Trajectoriesa
We now discuss the evaporation curves searched for in the optimization process. Figure 3 shows the evaporation trajectories obtained from the measurements for BO and random search. It was surprising that some trajectories with unexpected features such as non-monotonic decreases exhibited a score comparable to the manually found curve. Correlation plots of the two FORT powers enabled us to investigate requirements that ensured evaporation success. Figure 4 displays the two-dimensional dot plots concerning the powers of the two FORT beams for 4 intermediate stages of evaporation. The dot color represents the score obtained by the evaporation trajectory through the parameters. Plots in the upper panels (Fig. 4(a)) contain all the 12,480 data points obtained by the optimization experiments in this work. In the former (i.e., 1st and 2nd) stages of the evaporation, high-score parameters were widely distributed. In contrast, in the latter stages, high-score parameters were limited to a small area. This trend reflects a trivial requirement that precise control of the FORT powers is necessary in the latter part of the evaporation, where the trap potential is shallow and the trapped gas is more fragile because of lower temperatures. Interestingly, the correlation in the third stage was different from the rest. The distribution was elongated along the balance line (dashed line), which indicates that the balance between the powers of the two FORT beams is more important than the powers themselves in the stage before the final part of the evaporation. The nonintuitive shapes found in some of the obtained evaporation curves originated from less restriction in the former stages and the requirement for balance only in the third stage.
Fig. 3 Optimized evaporation trajectory. (a) The evaporation trajectories with higher scores (top 5,000 of 12,480 trials) are shown for both FORT beams. The color of the lines represents the score obtained with a corresponding experiment. The gray dashed line shows the evaporation trajectory we manually optimized. (b) The best evaporation trajectories for 16 different optimization sets are exhibited. Each different color corresponds to different optimization sets.
Fig. 4 Correlation plots between powers of two FORT beams. (a) Full plot of the 12,480 trials. Horizontal and vertical axes show the actual beam powers conversed from the control voltages. The color of the dots represents the score obtained by the evaporation trajectory through that data point. The dashed gray line shows a “balance line” where two FORT powers are equal. (b) Same plot only using the Bayesian optimizer. (c) Same plot with the random search, for comparison. Plot areas in (b) and (c) are rescaled for clarity.
In Figs. 4(b) and 4(c), the same correlation plots as Fig. 4(a) are depicted with data points obtained only with the BO and the random search, respectively. From these plots, it is clearly seen that almost all the high-score parameters were found by the BO experiment. In contrast, as shown in Table 1, the score achieved after 400 BO trials was limited to only 1.3 times higher than that by the random search. The BO method showed a clear advantage in the quality of the entire sets of the searched trajectories. The distribution of high-score parameters in the extensive parameter space was efficiently revealed by utilizing the Bayesian optimizer as a sampler.
4. Discussion
An important question is whether there is an essential difference between the manually obtained curve and the machine-learned trajectories. We refer the reader to Figure 3. We can confirm that the manual evaporation curve (gray dashed line) satisfies the requirements of balance and power in the latter part of the evaporation, which we discovered from the correlation plots. In the former part of the evaporation, however, the FORT powers were more rapidly decreased for the manual curve than for the machine-learned trajectories. This dissociation can also be seen in changes of PSDs in the evaporation sequence. Figure 5(a) demonstrates a plot of PSDs for 18 representative evaporation trajectories, including the manual curve, the machine-learned trajectories from 16 sets of the BO experiment, and the averaged curve of the 16 machine-learned curves. Indeed, the PSD of the manual curve grew faster than that of the other trajectories. This result is consistent with the procedure to optimize the evaporation curve manually, where the FORT powers are ramped down as fast as possible without significant decreases in the number of atoms remaining in the trap.
Fig. 5 Change in PSD during the evaporation process. PSD is plotted as a function of evaporation time (a) and the number of remaining atoms (b). Evaluation of PSD in the final stages is not exact because PSD was calculated from the “temperature” extracted with a Gaussian fit to condensed atoms.
Figure 5(b) displays a log-log plot of the PSDs for the 18 curves as a function of the remaining atoms in the trap. In these plots, the evaporation efficiency is evaluated from the slope. The plot concerning the evaporation efficiency infers that there is no significant difference between the manual and the machine-learned curves. The manual and the machine-learned curves are almost identical in terms of the atoms remaining in the trap and evaporation progress. The optimization with polygonal lines utilizing BO discovered various evaporation trajectories with cooling efficiency equal to or greater than that for the manually obtained curve.
A remaining question is why the Bayesian optimizer tended to find evaporation trajectories with slow growing of PSD in the early stages. We interpret this from a probabilistic perspective. For successful evaporation, possible parameter regimes decrease with cooling progress, as seen in Fig. 4. Evaporation with rapidly growing PSD in the early stages, as for the manually obtained curve, requires precise control of the parameters in the following stages. In contrast, the slow growing strategy, learned by the Bayesian optimizer, makes the parameters in the early stages insensitive to the evaporation result, and the number of sensitive parameters can be reduced. Among the solutions that give similar scores, trajectories with insensitive parameters can be discovered easily because the Bayesian optimizer refers to the result of the initial random search. The stability of the scores shown in Table 1, which is better for machine-learned trajectories than for the manual curve, also implies a difference a sensitivity. Evaporation trajectories with less sensitive parameters are robust against errors such as apparatus fluctuations, which is an additional advantage of machine-learned trajectories.
We should note the fact that the manually obtained curve cannot be reproduced by the piecewise linear function we used in this optimization. When the evaporation curve was optimized manually, nodes were implemented in the early part of the evaporation to achieve a smooth curve with a rapid ramp-down and gentle slope for the final trap potential. In the current optimization, the nodes were implemented with equal separation in time to investigate the properties of each stage of the evaporation. The number of nodes could not be added to obtain an optimization result within a realistic experimental time. Due to these limitations, the machine-learned curves cannot achieve the rapid ramp in the early part of the manual evaporation. Despite the concern of the boundary limitation of the searching, the correlation plots shown in Fig. 4 demonstrate less importance of the parameters in the early stages of the evaporation. The machine-learned curves achieved evaporation efficiency comparable with the manual curve without the rapid falling (Fig. 5(b)). Furthermore, the difference between the manual and machine-learned evaporation ramp can be interpreted in terms of the probabilistic preference of the Bayesian optimizer. Drawing a conclusion from the above discussion, the search with polygonal lines performed in this work involves enough degrees of freedom to shed light on the global texture of our evaporation problem and to compare the manual curve with machine-learned trajectories.
5. Conclusion
We demonstrated an automatic evaporative cooling optimization for gaseous 87Rb atoms based on BO. The optimization was carried out by reducing an evaporation curve into a piecewise linear function. The Bayesian optimizer performed optimization faster than the random search. In spite of a large degree of freedom given to the present optimization based on the piecewise linear function, 300 trials by the Bayesian optimizer within 3 hours provided parameters for achieving a BEC as large as that obtained by human experiments. We found that the stability of the atom number is better for machine-learned trajectories than for the manual curve. The large number of trajectories automatically obtained by the optimization experiments reveals the minimum requirements for successful evaporation. These findings also enable one to compare the cooling strategy learned by BO with the manually obtained evaporation curve.
What is occurring in evaporative cooling with the non-monotonically decreasing ramps discovered by BO is an interesting question because there is no evaporation when the trap depth is increased. A possible explanation is that the density of atomic gases increases (i.e., compression occurs), and the higher density speeds up rethermalization process, which leads to an efficient evaporation in the later part. This work may stimulate a study on evaporation with non-standard ramps.
The Bayesian optimizer showed advantages not only for mere optimization of physics experiments but also for efficient sampling from an extensive parameter space. ML technology will pave the way for a new experimental paradigm, where a machine-learner executes the presampling of successful experimental parameters and a human experimentalist gains insights into the experiment from the obtained results. It is, of course, difficult to transform machine-learned results to a comprehensible form. This difficulty has already been demonstrated in optimization of the cold atom experiments with deep learning [23]. In our study, searching evaporation trajectories with the piecewise linear function helped us directly relate conformation of multiple parameters to the physical substance of the experimental sequence at the expense of its optimization performance. Finally, we note that no special modification was applied to the optimization kernel of COMBO in our study. BO can be a versatile solver of various optimization problems in experiments.
Funding
ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan).
Acknowledgments
We acknowledge the stimulating discussion in the meeting of the Cooperative Research Project of the Research Institute of Electrical Communication, Tohoku University.
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