High-intensity two-frequency photoassociation spectroscopy of a weakly bound molecular state: theory and experiment
We investigate two-frequency photoassociation of a weakly bound molecular state, focusing on a regime where the ac Stark shift is comparable to the binding energy. In this “high-intensity” regime, we observe features absent in low-intensity two-frequency photoassociation. We experimentally measure the spectra of Sr atoms coupled to the least bound state of the Sr ground electronic channel through an intermediate electronically excited molecular state. We compare the spectra to a simple three-level model that includes a two-frequency drive on each leg of the transition. With numerical solution of the time-dependent Schrödinger equation, we show that this model accurately captures (1) the existence of experimentally observed satellite peaks that arise from nonlinear processes, (2) the locations of the two-photon peak in the spectrum, including ac Stark shifts, and (3) in some cases, spectral lineshapes. To better understand these numerical results, we develop an approximate treatment of this model, based on Floquet and perturbation theory, that gives simple formulas that accurately capture the binding energies. We expect these expressions to be valuable tools to analyze and guide future two-frequency photoassociation experiments.
The coherent coupling of atoms to bound dimers using light is a fundamental tool in ultracold physics. It provides a sensitive probe of interatomic potentials Lett et al. (1993); Miller et al. (1993), can be used to control the interaction strength with an optical Feshbach resonance Fedichev et al. (1996); Theis et al. (2004); Chin et al. (2010), and is a key step in creating ultracold ground state molecules Jones et al. (2006); Vanhaecke et al. (2004); Kim et al. (2004); Bohn et al. (2017).
Recent experiments Aman et al. (2018) have spectroscopically probed a weakly-bound halo state in ultracold alkaline-earth atoms. Ultracold alkaline-earth atoms are an exciting form of quantum matter Stellmer et al. (2014), offering extremely long-lived electronic clock states Ludlow et al. (2015), a large nuclear spin degeneracy in the fermionic isotopes (up to states for Sr), and an SU() symmetry that can stabilize quantum fluctuations Cazalilla and Rey (2014); Gorshkov et al. (2010).
However, accessing halo states in the alkaline-earth atoms requires different tools than in alkali atoms, as the former’s closed electronic structure precludes the existence of magnetic Feshbach resonances Chin et al. (2010) and the formation of weakly bound molecular states through magnetoassociation Köhler et al. (2006), and it forbids one-photon processes from coupling a well-separated pair of atoms to the halo dimer Chin and Julienne (2005). Consequently, Ref. Aman et al. (2018) used two-photon photoassociation (PA) through an intermediate molecular bound state in an excited electronic channel, following techniques in Refs. Jones et al. (2006); Martinez de Escobar et al. (2008); Abraham et al. (1995), as illustrated in Fig. 1(a). If the frequencies and of the two lasers are sufficiently different, only one laser will be relevant for driving each leg of the two-photon transition, corresponding to the commonly used -model Yoo and Eberly (1985); Bohn and Julienne (1996). When the intermediate state detuning is much larger than the Rabi frequency of each leg of the transition, the -model predicts a single photoassociation peak at , where is the binding energy (including light shift). (Here and throughout, .)
However, this treatment is not always accurate. One such case is when the unperturbed binding energy, , is not large relative to the detuning from the intermediate state. In this case, the PA lasers are similar enough in frequency that both are relevant for each leg of the transition. This two-frequency drive for each leg of the transition leads to additional ac Stark shifts, as we will discuss in Sec. IV.
A second case where treating each leg with a single-frequency drive is inadequate is when the unperturbed binding energy is not much larger than the ac Stark shift. In this regime, we observe that additional peaks appear in the PA spectra at approximately , , , as shown in Fig. 1(c). This occurs at high intensities and/or low intermediate-state detunings, and we will refer to this as the “high-intensity” regime throughout this paper. We will discuss the origin of this effect in detail in Sec. VI. As per Fig. 1(c), we will refer to the most red-detuned line in the spectrum as the main negative peak, the additional peaks as higher order peaks, and the most blue-detuned line as the main positive peak.
In this paper, we measure spectra in the high-intensity regime and develop a theory that accounts for the effects of the two-frequency drive when molecular binding energy is small and excitation laser intensity is high. The model neglects motional degrees of freedom for the initial state of two free atoms and considers only three levels: a dimer in the ground electronic state, the intermediate dimer state in the excited electronic potential, and two well-separated atoms. We numerically solve this model and analyze many of its features with analytic approximations. We also briefly consider extensions of this three-level model that add extra levels to capture some fine details of the ac Stark shift at large detuning.
We show that the three-level model captures key experimentally observed features despite its simplicity. Numerical simulations show that it accurately predicts the appearance of additional spectral lines in the high-intensity regime, and that these correspond to higher order non-linear processes. It accurately captures the location of the spectral peaks, including ac Stark shifts from both lasers when detuning is comparable to . Furthermore, when the intensity of the lasers is not too small, the spectral lineshapes of individual lines are reasonably well-captured. This indicates that effects missing from this theory – the motion of the atoms Bohn and Julienne (1996), interaction shifts due to molecules and atoms scattering off of other molecules and atoms Wynar et al. (2000), and the varying density within the trap Martinez de Escobar et al. (2008) – are less relevant in the regime of interest here.
We also develop a simplified analytic theory based on a Floquet treatment of this model, similar to Ref. Ho and Chu (1985); Wang et al. (1985), and we show that it captures the binding energy of the spectral peaks, including their dependence on laser frequency and intensity. The simple expression for binding energy is given in Eq. (11). This will allow future experiments to easily predict and analyze these features.
In Sec. II, we describe the PA experiment and the key features observed from the spectra. We introduce and develop the three-level model in Sec. III. In Sec. IV, we evaluate the model’s ability to replicate the binding energy obtained in experiment and introduce an analytic treatment of the model using Floquet and perturbation theory. Sec. V examines the effectiveness of our three-level model in predicting spectral linshapes. Sec. VI explores higher order processes that become possible when both lasers driving each leg of the transition are relevant, and describes additional peaks observed in experiment.
We perform two-frequency PA spectroscopy of Sr atoms, where the atoms are coupled to the ground electronic state of the Sr halo molecule through an intermediate molecular bound state. Two PA lasers with frequencies and , intensities and , and the same linear polarization are applied to ultracold Sr in an optical dipole trap, as illustrated in Fig. 1(a). We use equal PA laser intensities in the experiment, . By decreasing the frequency of laser B, we obtain a spectrum of atom loss as a function of the frequency difference of PA lasers, i.e. , in the regime . When the two-photon process is near resonant with the unperturbed binding energy, , there is significant atom loss through inelastic processes acting on the bound halo state and intermediate excited molecular state, and we expect a peak to be present in the atom loss spectrum. From the spectral peaks, we extract the binding energy by fitting the lineshape to a convolution of the asymmetric loss function described in Ref. Aman et al. (2018) with a Lorentzian, which provides good fit to the lineshape for all intensities. We measure spectra for different intermediate-state detunings, , and PA laser intensities to determine the dependence of light shift on detuning and PA laser intensity. More details of the experimental setup can be found in Ref. Aman et al. (2018), where low-intensity spectra are reported.
In high-intensity measurements reported in the present paper, we observe features absent in the -model where only a single laser drives each leg of the PA process. Most notable are additional peaks near for integer , as shown in the spectra in Fig. 1(c-e). In Sec. VI, we will understand the origins of these peaks in terms of non-linear processes illustrated in Fig. 1(b) that become possible with the two-frequency drive. These additional peaks are more apparent at higher PA laser intensity and lower detuning.
The peaks we observe in the experiment also experience strong light shifts, and understanding these is complicated by both lasers driving each leg of the transition. We can gain insight into the challenges that the two-frequency drive presents for predicting the ac Stark shift by considering two regimes. The first limit is when and are far apart, corresponding to large binding energies. The two lasers oscillate at very different frequencies and thus we do not expect the interference term to contribute. In this case, the ac Stark effects from each laser can be independently added, giving a contribution proportional to for , a formula that has been suggested and used in Refs. Kitagawa et al. (2008); Wynar et al. (2000). The second limit is when and are nearly equal and the electric fields and add constructively, where the total electric field amplitude is . In this case, one obtains a Stark shift proportional to . The experiments reported in this paper were performed at an intermediate regime to these limits. Section IV.1 will present results from numerical simulations that describe the shifts in this intermediate regime, and Section IV.2 derives analytic expressions capable of capturing the behavior in relevant limits.
Iii Three Level Model
We model the PA experiment using a three level system with representing two well-separated Sr atoms in the S state, representing the second least-bound vibrational state of the molecular potential, and representing the halo state, as shown in Fig. 1(a). Two lasers with frequencies and , corresponding to the PA lasers, are included to drive the two-frequency PA process. Unlike the -model where each laser will only drive one transition, our model includes additional couplings where laser A drives the transition and laser B drives the transition . We include decay terms to describe atom loss in the system, with representing atom loss from the intermediate bound state due to natural decay and representing loss from the halo state due to collisions with background atoms. The Hamiltonian for our model is thus
where is the coupling from laser driving transition . Using the rotating wave approximation and a suitable basis change, we obtain
where . In the experiment, both laser intensities are equal, thus we expect and . However, the couplings for the two transitions are different since the corresponding matrix elements are different, and they can be characterised by the parameter
From the results of Ref. Aman et al. (2018), we know that is large () in our experiment. Since the Rabi frequencies depend on the laser electric field as , we have
for some constant that we will determine through fits with experimental data.
Our model neglects motional effects. However, photoassociation is a scattering process, involving an initial state of two free atoms, and thus the collision amplitude will depend on energy. As we will show in Sec. V, this limits our ability to accurately predict lineshapes in the low intensity regime in which spectra are broadened by the distribution of relative kinetic energies for initial scattering states.
Nevertheless, numerically simulated spectra using the three-level model do reproduce many features observed in the experiments, such as additional peaks, ac Stark shifts for a wide range of intensities and detuning, and high-intensity lineshapes, as shown in Fig. 1(c-f).
Iv Binding Energy
iv.1 Numerical solution of three-level model
To determine the model parameters , , , , and , we numerically solve the time-dependent Schrödinger equation with the Hamiltonian in Eq. (2) to obtain total atom loss as a function of after a given evolution time with initial state . We find that , , and have little impact on the location of the main negative peak [labeled n1 in Fig. 1(c)]. Therefore, we fix 40, and for our analysis, which can replicate roughly the main peak’s lineshape for the and spectrum in Fig. 1(c). More accurate ways of determining the values of , , and are examined in Sec. V.
At the range of intensities and detunings shown in Fig. 2(a), we observe that the three-level model generally yields a linear relation between binding energy and intensity. Hence, we obtain a value for the unperturbed binding energy of by linear extrapolation of the low-intensity detuning data shown in Fig. 2(a). This value is consistent with the more accurate determination given in Ref. Aman et al. (2018), which additionally takes into account effects of atomic densities, trapping lasers and initial scattering state. A global value of is then determined by minimizing the least square distance between the model’s predicted binding energy and the same detuning data.
With just two effective fitting parameters, and (, , and have negligible effect on peak location), we find that the three-level model reproduces the experimental binding energies’ dependence on intensity for seven values. Within the range of intensities and detuning explored in Fig. 2(a), the difference between binding energies obtained in model simulation and experiment is below .
However, deviations between experiment and model binding energies are observed at higher intensity values for small detuning (when ac Stark shift is large), as shown in Fig. 2(b). Our model tends to underestimate the binding energy, with the deviation getting larger with increasing intensity. One possible reason for this is the presence of an additional, high-lying level, , that can couple with state 1 and alter the intermediate state energy. This modifies the ac Stark shift for state 0 and state 2. We find through numerical simulations (not shown) with added to the three-level model (Eq. (2)), suitable values of coupling, and , that the additional state can account for this deviation. Although this should not be construed as strong evidence for an additional level at this energy, it does suggest that additional levels at high energies are a candidate to be responsible for this effect.
A second discrepancy occurs at larger intermediate state detunings (). Here, there are also significant differences between binding energy predicted by the three-level model and experimental data. Considering only low-intensity data where the light shift varies linearly with intensity, Fig. 3 shows the susceptibility, , defined by , against detuning. This figure includes higher detuning data not included in Fig. 2(a). These high-detuning data show a large discrepancy, especially for data with , likely due to the presence of another intermediate state close to state 1 that couples to state 0. Further analysis is provided in Sec. IV.3.
iv.2 Floquet Treatment
To obtain analytic insight into our model, we analyze the Hamiltonian with Floquet theory. Floquet theory states that for a periodic Hamiltonian with period , there exist Floquet state solutions of form
where is periodic with period Shirley (1965); Grifoni and HÃ¤nggi (1998). Substituting the Floquet state solutions into the Schrödinger equation and performing a Fourier expansion of the Hamiltonian and , we arrive at the eigenvalue equation
where , , and is an orthonormal basis. is termed the Floquet Hamiltonian and has the form where is the n-th Fourier coefficient of the Hamiltonian . Hence, we have transformed the time-dependent periodic Hamiltonian into a time-independent one, where familiar techniques can be used to obtain solutions to the dynamics of the original Hamiltonian.
Neglecting the decay terms in our three level model, Eq. (2), which we numerically find affect the binding energy negligibly, our model Hamiltonian is periodic with period . Applying Floquet theory, we obtain the Floquet Hamiltonian
Using the Floquet Hamiltonian, we find the binding energy following the argument of Shirley Shirley (1965). Shirley has shown that the probability of a system initially in at time to be in state at time is
When , corresponding to where refers to the energy of state , the near resonance between and implies strong transition between the two states. Therefore, the term with in Eq. (8) would dominate and we neglect all other terms in the summation. Hence, we focus only on these two states and the near resonance condition allows us to apply second order perturbation theory in and , with conditions for integer , with , and . The coupling of and to states , and generates the second order perturbation terms, allowing us to rewrite the Floquet Hamiltonian into a 22 effective Hamiltonian
This analysis where gives the main positive peak [peak p1 in Fig. 1(c)]. If we use the resonance condition for the main negative peak [peak n1 in Fig. 1(c)], , similar expressions for the light shifts and off-diagonal terms are obtained. Substituting the effective Hamiltonian into Shirley’s equation to obtain an expression for transition probability, we find that the transition probability amplitude at long times is maximized when , which corresponds to the binding energy where there is maximum atom loss.
Given that the laser coupling to the transition is small in our experiment (), the binding energy can be obtained by solving the quadratic equation
where the negative sign corresponds to the peak associated with [process p1 in Fig. 1(b)] and the positive sign to the case of [process n1 in Fig. 1(b)]. Furthermore, since the two PA lasers have the same intensity, we define and the result simplifies to
Fig. 4 shows that binding energy predicted by Floquet analysis closely reproduces the results from the three-level model simulation. They closely agree even outside the approximation’s strict regime of validity, since even points in Fig. 4 with Stark shifts of are close to the three-level model simulation results. Thus, Eq. (11) provides a simple analytic formula to analyze similar PA experiments in the future without the need to do the full three-level simulation. Eq. (11) also matches the light shift observations in past experiments Wynar et al. (2000); Kitagawa et al. (2008). The forms of these light shift cannot be captured by the -model where only one laser drives each leg of the PA process. Upon inspection, one notices that light shift in Eq. (11) is the sum of ac Stark shifts when the two PA lasers act independently on the transition, corresponding to the limit of and being well-separated. This suggests that for light shifts observed for a bichromatic drive, one can safely treat the effect of ac Stark shift from each laser independently at low intensities where the binding energy is not too large compared to the ac Stark shift. However, at higher intensities, the laser interference is no longer negligible and our Floquet approximation would deviate from the numerically calculated binding energy from Eq. (2).
iv.3 Four Level Model
Considering only low-intensity data where the binding energy varies linearly with intensity, Fig. 3 shows that there are deviations between susceptibility predicted by the three-level model and experimental results when the detuning is large. This may be due to the presence of another level . To test this hypothesis, we extend our model by introducing the additional level X, which couples to state 0 and is of higher energy than state 1, as shown in Fig. 5. One could also assume a coupling between state 2 and X, but we will see that it is not necessary to reproduce the observations. The new Hamiltonian has three additional terms,
Applying the rotating wave approximation and following the Floquet treatment, we can predict the binding energy at low intensity by solving
where . Additionally, we introduce a parameter to relate the Rabi frequency of the transition and laser intensity,
There is close agreement between the four-level model and experimental light shifts even for large detuning values shown in Fig. 3, where we performed a global fit of parameters , and to susceptibility as a function of detuning. This supports the hypothesis that an additional level approximately higher in energy than state 1 causes light shift deviation which are unaccounted for in the three-level model, Eq. (2). If one assumes instead that state X is coupled only to state 2, the sign of the ac Stark shift due to the additional level would be negative at red-detuning, which is inconsistent with experimental observation in Fig. 3, where the susceptibility is positive when red-detuned from state X. Hence, the dominant coupling is between state X and state 0. Two candidates for state X are unbound atoms, with one atom in the excited atomic state, P+S, and the least bound state in the excited channel. However, for any molecular level, it is very unlikely that coupling to state 0 (free atoms) is stronger than coupling to state 2 (molecule). Therefore, we conclude that the dominant effect observed is the ac Stark shift of the atomic levels due to coupling to the P+S excited atomic state, which is approximately higher in energy than the intermediate bound state Borkowski et al. (2014); Aman et al. (2018).
We find that the three-level model accurately reproduces the lineshape at high laser intensities. For low-intensity spectra, the peak width and area are qualitatively accurate, but the detailed lineshape is less well reproduced, as shown in Fig. 6. Our model gives a symmetric lineshape, which can be adjusted to match the lineshape at high intensity where the experimental peak is symmetric. However, an asymmetry that our model is unable to capture is present in low-intensity spectral peaks.
The asymmetry in experimental spectral peaks at low intensity results from the distribution of possible collision energies in a finite-temperature gas Aman et al. (2018). For an initial state consisting of two colliding Sr atoms, the energy of the state is given by the relative kinetic energy, , instead of zero, where zero is the energy of two atoms at rest at large separation. This results in a larger energy gap between the initial state and the bound state and thus requires a larger laser frequency difference to be on resonance. The initial collisional energies should follow a Boltzmann distribution which would cause asymmetric broadening of the lineshape towards larger frequency differences Bohn and Julienne (1996). Since our model does not take into account collisional effects, it is unable to capture the asymmetry in these lineshapes accurately. At higher intensities, collisional broadening becomes less dominant compared to power broadening. Hence, we observe more symmetric lineshapes, which the model captures better.
Experimental data show a general increase of line widths, as measured by the full-width half-maximum (FWHM), and area with intensity for the main negative peaks of . As shown in Fig. 7(a), FWHM increases approximately linearly with intensity, suggesting that power broadening is causing the increasing linewidth. The peak area shows an initial rise in value at small intensities before leveling off at a higher intensity, as shown in Fig. 7(b).
The three-level model can qualitatively capture the general increase in FWHM and peak area as a function of intensity observed in experiment. Fixing (from Sec. IV.1) and 37.5 (using area of main negative peak, which is largely insensitive to and ), we test different values of and , and find that the FWHM of the main negative peaks can be reproduced when and . The plot of FWHM as a function of intensity using these parameters (red triangles in Fig. 7) shows a general rising trend similar to experimental data, but there is a downward curvature as intensity increases. Our model can also qualitatively replicate the general trend of a rise in area at low intensity, but shows a decrease at higher intensities. The value of obtained here is much higher than its expected value as discussed in Ref. Aman et al. (2018), whereas the value of is very small, consistent with its expected value. corresponds to the particle decay rate of state 1, the second least bound state of the molecular potential, which is expected to be , twice the decay rate of the P excited atomic state. corresponds to the decay from the molecular ground state via collisions with background atoms, which is expected to be small, of the order of Aman et al. (2018).
Vi Higher Order Peaks
Additional peaks, most prominent for spectra at high intensity and low detuning, are observed at approximately for integer in our experimental spectra in Fig. 1(c-f), and these can be accounted for by our model. With the inclusion of additional laser couplings, and , absent in the -model, the additional peaks can be understood as resulting from higher order processes. For instance, the second order resonance peaks present in Fig. 1(c) result from a four-photon process [p2 and n2 in Fig. 1(b)], with transitions through either 01012 or 01212, and detuning from states 0 or 2 in the intermediate step. These processes require either laser B driving transition or laser A driving transition , which are not present in typical three-level models Bohn and Julienne (1996); Yoo and Eberly (1985).
These peaks, which are absent in conventional two-frequency PA experiments, become prominent due to the unusually low binding energy provided by the halo state of Sr. When the binding energy is large, the detuning from states 0 and 2 in the intermediate step of the four-photon process is large, resulting in a small effective coupling between the two states relative to the coupling in the two-photon process. This results in a significantly lower transition probability through this pathway.
The experimentally observed peaks deviate slightly from the naive expectation that higher order peaks will occur at , but these deviations are captured by our numerical simulations of the three-level model, Eq. (2), as shown in Fig. 6 where the model peak location matches the experimental peaks better than the naively expected value. The Floquet treatment provides an approximate equation predicting the binding energy of the -th order peak in the experiment,
The light shift varies across the full spectrum of peaks because is varying, giving rise to a changing ac Stark shift and deviations from the scaling of the peak location. For the second order perturbation theory to be valid, (low-intensity limit), where the higher order peaks are small relative to the main negative peak. However, the location of higher order peaks relative to the main negative peak appears to be well captured even when ac Stark shift is not low compared to binding energy. Using Eq. (15), we fit for each intensity value such that the predicted main negative peak binding energy matches the experimental main peak, and use the same to generate predictions of binding energies of higher order peaks. We perform a similar calculation to obtain predictions from the numerical solution of the three-level model, Eq. (2). The predicted values from both the numerical solution and Floquet analysis shown in Table 1 generally match the experimental values better than the naive expectation. When , as in our experiment, the higher order peaks have a higher binding energy than naively expected. In contrast, when , the higher order peaks have a lower binding energy than naively expected.
|Intensity||Peak 1||Peak 2 ||Peak 3 |
Despite reproducing the binding energy and qualitatively explaining the behavior of satellite peak areas, the three-level model does not fully capture the number of atoms lost for higher order processes, as observed in Fig. 6 where the higher order peaks in the three-level model simulation have a smaller area than experimentally observed. This could be linked to the model’s limitation in capturing the collisional aspects of the process, so as a result it fails to account for the change in excitation rate with atom density. During the experiment, the atomic density decreases over time as atoms are lost. This decrease in density leads to lower atomic collision rates and thus rate of atom loss, which scales as for a two-body scattering process Aman et al. (2018). In contrast, the loss rate in our model is simply proportional to the number of molecules, and thus scales as . Therefore, our model overestimates the number of atoms lost when decreases and this discrepancy is most significant when the experimentally measured atom loss number is high. Since we normalise our model to the main negative peak, where atom loss is the highest in the spectrum, we observe that the model underestimates the atom loss in the higher order peaks, as shown in Fig. 6.
We have shown that for bichromatic driving of a PA process with unperturbed binding energy that is not small compared to detuning or ac Stark shifts, it is important to consider that each laser can couple to both transitions. Including this effect results in strong light shifts arising from both PA lasers and higher order peaks arising from multi-photon processes, which we observe in ultracold Sr PA experiments.
Using a simple three level model, we explain the origin of higher order peaks as alternate resonance processes that only appear with the consideration of extra laser couplings and . The model also predicts the binding energies at various detuning and intensities with just two fitting parameters, and . Although binding energies in principle also depend on , , and , the dependence on these parameters is negligible. We derive simple analytic forms for the binding energy by combining Floquet and perturbation theory, confirming earlier conjectures that the light shift is a sum of independent ac Stark shifts from each of the lasers Kitagawa et al. (2008); Wynar et al. (2000). Although the analytic results rely on certain limits, the numerical calculations and comparison with experiment indicate that these expressions are quite accurate for most intensities and detunings studied in this work. This provides a foundation for future work, circumventing the need for simulations when analyzing experimental PA spectra under many experimentally-relevant conditions.
We also consider the lineshapes, where our model is able to capture the symmetric lineshapes for high-intensity spectral peaks, and also replicate qualitatively the general trends in linewidth and area for detuning data. However, the model is unable to reproduce the asymmetric broadening in low-intensity spectral peaks, likely due to neglecting collisional and scattering effects. Therefore, to have a complete theoretical model of the PA spectral lineshapes, it is essential to include collisional effects of two-atom scattering.
The presence of a weakly bound halo state of Sr at approximately , determined in Ref. Aman et al. (2018), and the strong light shift experienced by this state, observed in this work, opens up the possibility of using coupling to the intermediate state, , to tune the halo state into collisional resonance, and thus tune the atom-atom scattering length. This could serve as a new mechanism for optically controlling a Feshbach resonance in alkaline-earth atoms, where a magnetic Feshbach resonance is not possible due to the lack of hyperfine structure. The ability to easily tune the scattering length broadens the explorable range of few and many-body physics. Our results here, both theory and experimental, provide an foundation to implement this technique.
Acknowledgements.This work was supported by the Welch Foundation (C-1844 and C-1872) and the National Science Foundation (PHY-1607665).
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