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Abstract
The precise calibration of optical lattice depth is an important step in the experiments of ultracold atoms in optical lattices. The Raman-Nath diffraction method, as the most commonly used method of calibrating optical lattice depth, has a limited range of validity and the calibration accuracy is not high enough. Based on multiple pulses Kapitza-Dirac diffraction, we propose and demonstrate a new calibration method by measuring the fully transfer fidelity of the first diffraction order. The high sensitivity of the transfer fidelity to the lattice depth ensures the highly precision calibration of the optical lattice depth. For each lattice depth measured, the calibration uncertainty is further reduced to less than 0.6% by applying the Back-Propagation Neural Network Algorithm. The accuracy of this method is almost one order of magnitude higher than that of the Raman-Nath diffraction method, and it has a wide range of validity applicable to both shallow lattices and deep lattices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
An optical lattice is typically formed by two interfering laser beams, giving rise to a periodic intensity pattern which acts as a periodic potential for cold atoms. By choosing an appropriate set of intensity, angle, phase and polarization of the interfering laser beams, the band structure, geometry [1–3] and topology [4–6] can be directly engineered. Optical lattices have been widely used in atomic physics as a way to cool, trap and manipulate atoms [7–10]. In recent years ultracold atoms in optical lattices have played a key role in quantum simulations [11, 12], atom interferometers [13,14] and optical clocks [15–18]. Storing ultracold quantum gases in optical lattices has created structures far beyond those currently achievable in condensed-matter physics systems, opening innovative manipulation possibilities in a highly controllable and almost pure environment [19].
In the experiments of ultracold atoms in optical lattices, an important issue is how and with what precision the lattice depths can be measured and controlled, especially for experiments in which the agreement with theory depends critically on the exact value of the lattice depth. For example, when the tunneling rate which depends exponentially on the lattice depth is involved, a large error can be made with a calibration method not precise enough. In principle, the lattice depth can be readily calculated if the atomic transition characteristics and the parameters of the lattice laser beam are given. While the atomic transition characteristics are usually well known for the ultracold atoms in optical lattice experiments, the power and waist of the lattice laser beam at the position of the atoms turn out to be difficult to measure owing to the possible attenuation and distortion during the propagation through the windows of the vacuum system, which can result in combined systematic errors on the order of 10 ~ 20% [20].
Measuring the lattice depths based on some effects of ultracold atoms in optical lattices can lead to a more precise calibration [21]. Different methods have been developed consisting of Raman-Nath diffraction [22], parametric heating [23], Rabi oscillation [24] and sudden expansion [25]. Consisting in illuminating a Bose-Einstein condensate (BEC) by the lattice laser for a short time and analyzing the resulting diffraction pattern, Raman-Nath diffraction is the most commonly used method in experiments. Constrained by the short interaction time to ensure the atomic motion could be neglected, the Raman-Nath diffraction method allows to calibrate optical lattice depth accurately in the deep lattice limit. In addition, for the Raman-Nath diffraction method, the relative populations of atoms in each diffraction order are not sensitive enough to the lattice depth, which sometimes leads to a large calibration uncertainty.
In this paper, we demonstrate a high precision method of calibrating optical lattice depth based on multiple pulses Kapitza-Dirac (MPKD) diffraction. Using optimized standing-wave laser pulse sequences with stationary light intensity and different duration time, a BEC can be transferred into the first diffraction order with almost 100% fidelity. This transfer fidelity is extremely sensitive to the lattice depth, promising a high precision method of calibrating optical lattices. The MPKD diffraction method is valid over a wide range of optical lattice depths by taking the atomic motion into account. The high sensitivity of the transfer fidelity to the lattice depth and the Back-Propagation Neural Network (BPNN) algorithm applied to locate the highest fidelity ensure the calibration uncertainty is less than 0.6% for both shallow lattices and deep lattices.
The paper is organized as follows. In Sec. 2, we calculate the optimized standing-wave laser pulse sequences to transfer all atoms into the first diffraction order and address the MPKD diffraction method by measuring the fully transfer fidelity of the first diffraction order. Section 3 describes the experimental procedure of optical lattice depth calibration in detail. We measure the transfer fidelity changes over optical lattice depths under the designed standing-wave laser pulse sequence. The atoms can be transferred into the first diffraction order with the highest fidelity only when the detection voltage exactly corresponds to the lattice depth we expect. We apply the BPNN algorithm to fit the experimental data and obtain the voltage value corresponding to the highest fidelity. The calibration results for different lattice depths are presented in Sec. 4, we also compare the results of the MPKD diffraction method with the conventional Raman-Nath diffraction method. We then conclude the paper with some general comments on our approach.
2. Theory and numerical calculation
For the conventional Raman-Nath diffraction method, the quantitative analysis of the diffraction pattern is simplified by neglecting the atomic motion. This so-called Raman-Nath approximation is valid when the displacement of the atoms during interaction time with the lattice laser is much smaller than the lattice spacing [26, 27], normally taking ten percent of the half-wavelength of the lattice laser. For our system, the lattice laser wavelength is λL = 1064 nm and the pulse duration Δt of the Raman-Nath diffraction method is restricted to be less than 6 µs [28].
Under the Raman-Nath approximation, the periodic potential is directly imprinted on the phase of the atomic wave function and the population of atoms in n th diffraction order yields
Here Jn is the Bessel function of the first kind and VL is the lattice depth. The relation between the lattice depth VL and the shortest laser pulse duration Δt0 to scatter all the atoms into the high diffraction orders is then given by VL = 4.81ħ/Δt0, where ħ is the reduced Planck constant. In the experiment, we measure optical lattice depths in units of the single-photon recoil energy as a natural energy scale, with kL the lattice laser wave number. The Raman-Nath diffraction method allows to calibrate optical lattice depth accurately for relatively deep lattices, typically larger than 20 Er. For smaller lattice depths, longer pulse duration which exceeds the limit is needed and the experimental results will distinctly deviate from the theoretical expectation [29].The main idea of the MPKD diffraction method is to transfer a BEC into the first diffraction order with designed standing-wave laser pulse sequence and calibrate the corresponding lattice depth according to the fully transfer fidelity. In a single pulse Kapitza-Dirac diffraction process, the BEC interacts with a far-detuned standing-wave laser pulse and the initial state |p = 0〉 could be transferred into a superposed state of different momenta |p = 2ℓħkL〉, where ℓ = 0, ±1, ±2 ⋯ is the diffraction order. Using a sequence of standing-wave pulses with different duration time, the amplitude of each diffraction order and the relative phase among them can be adjusted precisely. By adding the kinetic energy term into the Hamiltonian [30, 31], we observe better agreement between the numerical calculations and the experimental results even for pulse duration as long as one hundred microseconds. The MPKD diffraction method eliminates the limit of Raman-Nath approximation and proves to be valid over a wide range of lattice depths.
Assuming that the atoms in a light field can experience a scalar potential whose strength is proportional to the intensity of laser, the single-atom Hamiltonian in optical lattice constructed by the standing-wave laser is then given by:
where m is the atom mass. According to the Bloch theorem, the eigenstates of the Hamiltonian can be expressed as (ħ = 1)We now consider the problem of transferring a BEC with initial wave-function |ψi〉 into the first diffraction order, which means atoms are populated symmetrically in |p = ±2ħkL〉 momentum state with equal and the maximum probability P±2 = 1/2. Suppose that after M steps of standing-wave laser pulses have applied, the final state |ψf〉 is
where is the evolution operator in the jth step, is the Hamiltonian corresponding to the lattice depth Vj and tj is the pulse duration. The final state |ψf〉 is a superposed state of symmetrically separated momenta in momentum space and the population in each momentum state is Pℓ = |〈2ℓħkL|ψf〉|2, where ℓ = 0, ±1, ±2 ⋯ is the diffraction order. The atoms can be transferred between different diffraction orders only when the pulse is on, while the amplitudes of different diffraction orders keep constant and the phases vary linearly in the intervals.To fully transfer the atoms into the first diffraction order, at least two standing-wave laser pulses are needed [32]. A three-stage sequence with two pulses and an interval when the amplitudes of different diffraction orders keep constant and the phases vary linearly is the simplest form, but enough to transfer the atoms into the first diffraction order with high fidelity [33, 34]. In the numerical calculation, we fix the potential depth to Vj = VL and adopt a three-stage standing-wave laser pulse sequence, where [t1, t2, t3] corresponds to the lattice depth [VL, 0, VL]. For the target state |ψt〉, the parameters [t1, t2, t3] can be determined by maximizing the fidelity
It is apparent that the value 1 − F describes the difference between the realistic state |ψf〉 and the target state |ψt〉. For the MPKD diffraction method, we set the target state |ψt〉 to be fully transferring the atoms into the first diffraction order. Here, F is equal to the total population of atoms in the first diffraction order and F = 1 means the atoms would be fully prepared in the target state |ψt〉. The calculated standing-wave laser pulse sequences for different lattice depths are shown in Table 1. In fact, the quasi-momentum is conserved in the process of the optical pulse scattering, all the final states can be achieved are symmetrically populated. The target state |ψt〉 is the superposed state with the maximum first diffraction order probability, therefore in the numerical calculation, we just need to maximize the probability of the first diffraction order to optimize the values for [t1, t2, t3]. There is still very few population in other diffraction orders, including the zeroth diffraction order and higher diffraction orders except the first diffraction order. Theoretically achieving 100% transfer fidelity is mainly limited by the finite parameter space. If the parameters [VL, t1, t2, t3] could be adjusted with arbitrary precision, then 100% transfer fidelity should be achieved theoretically.
Table 1. Calculated standing-wave laser pulse sequences
The fidelity of transferring a BEC into the first diffraction order by the MPKD diffraction method as the lattice depth changes is calculated, shown in Fig. 1 with solid line. The transfer fidelity is extremely sensitive to the lattice depth. Because of the small deviations between theoretical calculated transfer fidelities and experimental measurement results, which mainly arises from the elastic scattering between atoms in different momentum components during time-of-flight (TOF) expansion, we calibrate the lattice depth according to the highest transfer fidelity point. Based on the high sensitivity to the lattice depth, we propose the MPKD diffraction method to achieve high precision calibration of optical lattice depth. As a contrast, we plot the corresponding curves for the conventional Raman-Nath diffraction method under the same parameters in dash line. Since the atom population exists in all the high diffraction orders in the Raman-Nath diffraction method, we take the fidelity for the dash line as the total population of atoms in all the diffraction order except the zeroth order. Comparing both theoretical curves for different lattice depths, the change of the transfer fidelity near the highest point of the MPKD diffraction method are clearly more significant than that of the Raman-Nath diffraction method.
Fig. 1 Theoretical calculations reflect the variation of transfer fidelity with lattice depth. The solid lines in A–F correspond to the calculated standing-wave laser pulse sequences (A–F) for different lattice depths in Table 1. The dash lines are the theoretical curves of Raman-Nath method and the fidelity for the dash line means transferring a BEC into the high diffraction orders not just the first order. The change of the transfer fidelity near the highest point of the MPKD diffraction method are clearly more significant than that of the Raman-Nath diffraction method.
3. Experimental measurement and data processing
In our experiment, an atomic BEC of 5 × 104 weakly repulsive 87Rb atoms is produced in the |F = 1, mF = −1〉 state without a discernible thermal fraction. Here, F and mF denote the total angular momentum and the magnetic quantum number of the atom’s hyperfine state, respectively. The BEC is prepared in a crossed-beam optical dipole trap with nearly isotropic trapping frequencies of 2π × 6 Hz [35]. The harmonic trapping frequency is achieved to be small enough to make sure the condensate density is sufficiently low so that mean-field effects that could influence the optical lattice calibration result are minimized.
The optical lattice is created by one retroreflected laser beam with 1/e2 radius around 145 µm. The lattice laser beam is adjusted by imaging it according to the position of the BEC measured on the camera, assuring a good overlap of the incoming and the retroreflected beam by reaching the maximal modulation depth. We further optimize the alignment by observing the transverse oscillation of the atoms in the optical lattice potential and the precision of the alignment is achieved to be less than 1 µm. This precise overlap between the center of the lattice laser beams and the BEC avoids an anharmonic combined potential. The light intensity of the lattice laser is stabilized by a pre-stage feedback loop to reduce the laser power fluctuation to less than 0.05%. A post-stage switch is used to ensure the sharp rising and falling edges of each lattice laser pulse duration are within 100 ns, limited by the acousto-optic modulator. Introduced by this finite switching time, the systematic uncertainties of the target lattice depths that correspond to the peak transfer fidelity are ±0.1%.
In practice, the optical lattice depth is controlled by the detection signal of the lattice laser beam, which is the detection voltage captured by a sampling photodetector in our experiment. Using the standing-wave laser pulse sequences in Table 1, we calibrate different lattice depths independently to obtain the conversion relation between the optical lattice depth and detection signal. For each of the numerically optimized pulse sequences above, measurements of atom population in the first diffraction order as a function of the detection voltage are recorded, with multiple repeats for each set of parameters. The detection voltage output by the sampling photodetector is obtained by a data acquisition card (sampling rate 125 MHz, resolution 16 Bits). Since generally, the theoretical curves that is based on numerical calculation have no analytical form, we adopt BPNN algorithm which is approximately the steepest descent method of minimum mean square error to fit the data.
The structure of the BPNN algorithm is shown in Fig. 2. Generally, the BPNN algorithm could approach any kind of continuous nonlinear functions. The basic principle of the BPNN algorithm is to obtain an output (predicted transfer efficiency) from an input (detection voltage) through a series of transformations of the hidden layer, the forward propagated input information and the back propagated output error constitute the information loop of the BP network. The hidden layer is the middle layer of the neural network between the input layer and the output layer, extracting the pattern characteristics of the input and passing it to the output layer. The BPNN algorithm modifies the weight of each connection based on the output error to further make the output error within the preset range. For the selection of specific parameters such as the number of layers and units, it should ensure that the error of the fitting curve is small enough and no overfitting due to the random errors. The fitting model cannot give an analytical formula but a mapping relationship based on the sample. From the fitting result, we can locate the corresponding detection signal value of the highest transfer fidelity point by differentiating directly.
Figure 3 provides an example of such a data fitting process for the lattice depth of 9.97 Er, in which the fidelity variation of transferring a BEC into the first diffraction order can be clearly seen. In actual operation, we set one hidden layer of six hidden units. Training a neural network model essentially means selecting one model from the set of allowed models that minimizes the cost. By choosing Levenberg-Marquardt training algorithms, we take eighty percent of data as training data (blue data point in Fig. 3), ten percent as validation data (green data point in Fig. 3) and remaining ten percent data as test data (red data point in Fig. 3). A reasonable training precision is set according to the data sample so as to raise the accuracy in the fitting process. We calibrate the lattice depth according to the fully transfer fidelity, the highest transfer fidelity indicates the lattice depth we suppose to calibrate.
Fig. 3 Data fitting process for the lattice depth of 9.97 Er using BPNN algorithm. The blue, green and red dots are the experimental data used for training, validation and test in turn. The black line is the final fit to the data. The errors of the fitting are also shown. The three insets a–c are the corresponding integrated density profiles for the selected data to reveal the transfer fidelity variation of the MPKD diffraction method. The slight asymmetry of ±2ħkL peaks is mainly due to the imperfect overlap of the retroreflected lattice laser beam. The lattice depth values (up horizontal axis) put here as a reference come from the final conversion of lattice depth to detection voltage in Eq. (6).
To further minimize the calibration uncertainty, we select 10, 000 reasonable fittings during each lattice depth calibration and find the corresponding voltage value of the peak fidelity point from each fitting curve by differentiating directly. The selection here means we abandon the fittings with coefficient of determination less than 0.95. These 10, 000 voltage values are calibration samples for the given lattice depth and their probability distribution can be seen in Fig. 4. We take the expectation value of the distribution as the calibrated voltage value for the lattice depth. As for the calibration uncertainty, we use the 95% confidence bounds of the distribution to reflect the accuracy of calibration. The inset of Fig. 4 shows how the uncertainty scales with number of fittings. From our result of each numerically optimized pulse sequence, 3,000 fittings which requires about one hour is sufficient in actual application. Plus one additional hour of data measurement, the total time cost of each lattice depth calibration is about two hours.
Fig. 4 The probability distribution of corresponding voltage values of the peak fidelity points from 10, 000 reasonable fittings for the lattice depth of 9.97 Er. The expectation value of the distribution is taken as the calibrated voltage value for the lattice depth. The corresponding calibration uncertainty is determined by the 95% confidence bounds of the distribution. The inset shows how the 95% confidence interval scales with the number of fittings.
4. Result and discussion
A set of optimized standing-wave laser pulse sequences for different lattice depths in Table 1 is experimentally investigated. For all the lattice depths calibrated by the MPKD diffraction method, the largest calibration uncertainty is 0.6%, mainly limited by the resolution of the standing-wave laser pulse duration and the first diffraction order population. Figure 5(a) exhibits the calibration results of the MPKD diffraction method with a linear fit to the data. We compare our results with Raman-Nath diffraction method around 20 Er and 50 Er, as shown in Fig. 5(b) and Fig. 5(c). Under the same data measurement and fitting process, the Raman-Nath diffraction method (blue square with error bar) for our system has a typical calibration uncertainty around 3%, which is consistent with the result in latest work [36, 37] and five times larger than the calibration uncertainty of the MPKD diffraction method.
Fig. 5 Lattice depth calibration results. The error bars reflect the 95% confidence bounds of each calibrated value. a, Calibration results of the MPKD diffraction method for different lattice depths. The black solid line is a linear fit to the calibration results. b and c, Comparison of the Raman-Nath diffraction method (blue square with error bar) and the MPKD diffraction method (green circle with error bar) around 20 Er and 50 Er respectively. The corresponding color shaded areas indicate the uncertainty of the linear fitting of the calibration results. The calibration uncertainty of the MPKD diffraction method is much smaller than that of the Raman-Nath diffraction method.
Our method yields an averaged lattice depth sampled by the area of the whole atom cloud, there is an energy offset of 0.022 Er from the center to the outermost lattice site. This energy offset is small owing to the small harmonic trapping frequency and the atom number in our experiment. With the calibration results, we can obtain the conversion of lattice depth VL (in units of Er) to detection voltage UD (in units of mV) from the linear fitting
Here, the uncertainty of the slope mainly comes from the calibration uncertainty of each lattice depth. The precise measurement and control of the optical lattice depth can be achieved by the high precision conversion of these two quantities over a wide range with small uncertainty. By such an optical lattice depth gauge, the depth of the optical lattice as well as its uncertainty can be well determined. For comparison, the linear fit to the results of the Raman-Nath diffraction method is UD = (26.12 ± 0.97) × VL − (26.58 ± 36.89), of which the uncertainty of the slope and the intercept is almost one order of magnitude larger than that of the MPKD diffraction method.Through our approach, the conversion of lattice depth to detection voltage maintains high linearity over a wide range with the determination coefficient of the linear fitting in Fig. 5(a) equal to 1. Considering the largest deviation from the linear fitting at 9.97 Er which is 3.81 mV, the non-linearity error of the conversion of lattice depth to detection voltage is 3.81 mV/(UD(60.02Er) − UD(9.97Er)) = 3.0‰.
5. Conclusion
In conclusion, we have demonstrated a high precision approach to calibrating optical lattice depth over a wide range. The key merit of the method is the high sensitivity of the transfer fidelity to the optical lattice depths under the MPKD diffraction process. We further minimized the measure uncertainty to less than 0.6% by applying the BPNN algorithm to find the fidelity maximum and the corresponding detection signal. The accuracy of the MPKD diffraction method is almost one order of magnitude higher than that of the Raman-Nath diffraction method. The final obtained conversion relation between the optical lattice depths and detection signals shows a high linearity, with the nonlinear error as small as 3.0‰. The high precision applicable for a wide range of optical lattice depths gives our approach further potentials in many optical lattice related experiments.
Funding
National Key Research and Development Program of China (Grant No. 2016YFA0301501); National Natural Science Foundation of China (Grant No. 91736208, No. 11504328, No. 61703025, No. 61475007, No. 11334001).
Acknowledgments
We would like to thank Jörg Schmiedmayer for stimulating discussions. We thank Shengjie Jin for the advice and support in the numerical calculation.
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