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Proceedings of the National Academy of Sciences of the United States of America logoLink to Proceedings of the National Academy of Sciences of the United States of America
. 2021 Aug 13;118(33):e2105601118. doi: 10.1073/pnas.2105601118

Multidimensional four-wave mixing signals detected by quantum squeezed light

Konstantin Dorfman a,1,2, Shengshuai Liu a,1, Yanbo Lou a,1, Tianxiang Wei a, Jietai Jing a,b,c,d,2, Frank Schlawin e,f, Shaul Mukamel g,h
PMCID: PMC8379966  PMID: 34389678

Significance

Quantum light and its statistics provide powerful tools for the study of properties of matter that are difficult to retrieve with classical light. Novel spectroscopic and sensing techniques based on quantum light sources can reveal information about complex material systems that is not accessible by varying the frequencies or time delays of classical light pulses. Here, based on a four-wave mixing process, we report an experimental study of the 2D quantum noise spectra of two-beam intensity difference squeezing. External noise erodes the resolution of classical measurements, while quantum signals remain intact. Our results pave the way for exploiting quantum correlations of squeezed light for spectroscopic applications.

Keywords: quantum spectroscopy, multidimensional spectroscopy, squeezed light

Abstract

Four-wave mixing (FWM) of optical fields has been extensively used in quantum information processing, sensing, and memories. It also forms a basis for nonlinear spectroscopies such as transient grating, stimulated Raman, and photon echo where phase matching is used to select desired components of the third-order response of matter. Here we report an experimental study of the two-dimensional quantum noise intensity difference spectra of a pair of squeezed beams generated by FWM in hot Rb vapor. The measurement reveals details of the χ(3) susceptibility dressed by the strong pump field which induces an AC Stark shift, with higher spectral resolution compared to classical measurements of probe and conjugate beam intensities. We demonstrate how quantum correlations of squeezed light can be utilized as a spectroscopic tool which unlike their classical counterparts are robust to external noise.


Quantum light and its statistics (113) provide powerful tools for the study of properties of matter that are hard to retrieve with classical light. Novel spectroscopic and sensing techniques based on quantum light sources (14) can reveal information about complex material systems that is not accessible by simply varying the frequencies or time delays of classical light pulses (15, 16). The state of quantum light provides most valuable control parameters. The matter response imprinted in the quantum light statistics can be retrieved by measuring higher-order correlation functions of the photon number. Spectroscopic measurements with entangled photons provide a unique observation window for the material response by accessing as well as controlling exciton distributions and transport processes (17), and the charge density in diffraction imaging (18, 19). Apart from their novel matter information, quantum light measurements have higher signal-to-noise ratio (20), and allow to shift optical measurements to desired frequency regimes where optical equipment is more readily available (21).

Here we focus on a class of quantum spectroscopy measurements of the multimode correlated squeezed light generated by four-wave mixing (FWM) (4, 2227). Squeezed light can be broadly defined as a state of light whose quadrature amplitudes of the electric and magnetic fields are squeezed, that is, whose quantum uncertainty in one quadrature is smaller than that of a coherent state, typical for lasers. Fig. 1A shows the standard FWM setup used for squeezed light generation, which become a powerful spectroscopy tool. After an FWM process, the probe (blue line) and conjugate (yellow line) beams are multimode squeezed and they can be detected by, for example, a classical intensity measurement. Following the approach outlined in ref. 28, we have calculated the probe and conjugate transmitted intensities,

N^prGN^0,N^c(G1)N^0, [1]

where G(ωpr,ωc;2ωpu)=cosh2[|χ~(3)(ωpr,ωc;2ωpu)|] is the FWM gain governed by a third-order susceptibility χ~(3) dressed by the strong pump field (see Materials and Methods), and N^0=|α|2 is the average photon number of the input probe beam. Rather than detecting classical field intensities, one can measure quantum fluctuations of the relative squeezing spectra defined by

SNVar(N^prN^c)N^pr+N^c=12G1, [2]

which can be reduced to below the shot noise limit (SNL), providing a notable quantum advantage in weakly absorbing materials. To observe quantum squeezing of the probe and conjugate beams, we measure their intensity difference noise power spectrum and compare it with its corresponding SNL. The extent of quantum squeezing is given by the degree to which the intensity difference noise power is lower than SNL. As shown in Fig. 1A, the output probe and conjugate beams are sent to two silicon photodetectors (D1 and D2, respectively). We can then obtain the direct current (dc) and radio frequency (rf) components of the photocurrents. The rf components from the two ports are subtracted by using an rf subtractor (S) and then analyzed with a spectrum analyzer (SA). The subtracted result constitutes their intensity difference noise power spectrum. To obtain the corresponding SNL, we use a coherent beam with a power equal to the total power of the output beams. We then divide it into two beams with a 50:50 beam splitter and send the obtained beams into the two previously used photodetectors to get the noise power of the differential photocurrent, which gives the corresponding SNL. By scanning the pump and probe frequencies across several hundred megahertz, one can obtain a two-dimensional (2D) spectrum containing valuable matter information. While, in the standard FWM squeezed light generation scheme, the squeezing occurs between single modes at fixed frequencies, here the pump and probe frequencies scanned over the broad range ensure a multimode squeezing. While this measurement is not novel, but the spectroscopic advantage is certainly unique. By scanning the pump and probe pulse frequencies across several hundred megahertz, one obtains a 2D spectrum containing valuable matter information. In the standard FWM squeezed light generation scheme, the squeezing occurs between single modes at fixed frequencies; here, in contrast, the pump and probe frequencies scanned over the broad range ensure a multimode squeezing. A perturbative theoretical analysis provides a simple account of the squeezing measurements in SiV color centers in diamond (29) (see SI Appendix, section S1). Note that, in contrast with techniques where the quantum light sources are directed at the material to probe its response (14), here the generation of quantum light combined with quantum detection of squeezing serves as a probe of the nonlinear response of matter.

Fig. 1.

Fig. 1.

(A) The experimental layout. A strong classical pump beam (red line) and a weak probe (yellow line) (in a coherent state with |α|600 which corresponds to 10-μW power) are crossed in an 85Rb vapor cell at a small angle. The probe beam is amplified, and a new conjugate beam (blue line) is generated on the other side of the vapor cell. The phase matching is given by 2kpu = kpr + kc, where subscripts indicate pump, probe, and conjugate beams. The probe and conjugate beam intensities can be either measured separately (classical measurement) or combined on an S to reveal the noise spectra of the intensity difference (Eq. 2) (quantum measurement). HWP, half-wave plate; PBS, polarization beam splitter; BS, beam splitter; AOM, acousto-optic modulator; PZT, piezoelectric transducer; GL, Glan−Laser polarizer; GT, Glan−Thompson polarizer; D1 (D2), photodetector; OS, oscilloscope; yellow lines, signal beam; blue lines, idler beam; red lines, pump beam. (B) Level scheme for the FWM process. Since the pump beam drives both electronic transitions ge and se, while the probe (conjugate) interacts only with es (eg), i indicates the case where initial state of the vapor is ground state g, while ii corresponds to the case when s is initially populated. (C) The 85Rb-level scheme in the presence of the strong pump.

Results

Experimental Scheme.

We consider an FWM process based on the double-Λ level scheme shown in Fig. 1B. Two lower hyperfine states g (F=2) and s (F=3) are separated by 3.035 GHz, while the upper states e1 (F=3) and e2 (F=2) are separated by 361.58 MHz. At vapor temperature 113 °C, both g and s states are almost equally populated. The FWM may take place starting from either the g or s states as shown by level schemes i and ii, since the strong pump beam interacts with both ge and se transitions. In diagram i, the pump first drives the ge transition, while the probe drives the es transition. Another pump photon promotes the system via the se transition, while the conjugate beam generated by the FWM brings the system back to its ground state via eg transition. The bookkeeping of the field−matter interactions is shown in diagram ia in SI Appendix, Fig. S1. One can describe diagram ii similarly by exchanging states gs and probing with conjugate beams. Due to the strong pump (180 mW), all four transitions, ge1,2 and se1,2, show an AC Stark splitting which results in the transition frequency detunings ωejm(±)=δejm/2±Ωejm, where Ωejm=δejm2/4+Ωejm2, δejm=ωpuωejm is the detuning and Ωejm=μejmεpu, j=1,2, m=g,s is a Rabi frequency corresponding to the pump field amplitude εpu for a given transition dipole moment μejm (Fig. 1C). The following two points should be noted. First, the pump should be strong enough to induce an AC Stark splitting, thus doubling the number of measured resonances. Peaks missed by classical intensity measurements clearly show up in the squeezing detection, thanks to the higher signal-to-noise ratio. Second, in the absence of losses, both intensity and squeezing measurements carry identical information associated with the FWM gain. This is no longer the case when optical losses exist. The squeezing measurement is robust to external noise added to the pump and therefore to all the output fields of the FWM process. However, classical measurements become unstable and hard to interpret under these conditions.

Experimental Results.

We start with a classical measurement of the transmission intensities of the probe and conjugate beams given by Eq. 1. In standard treatments of FWM-generated squeezed light fields, all transitions are kept off-resonant with respect to matter, and χ(3) can be replaced by a frequency-independent prefactor. Here, in contrast, we are interested in resonant properties of the nonlinear response which can be measured through the probe or conjugate intensities. Fig. 2A demonstrates that, for a weak pump (100 mW), the 2D gain spectra (Eq. 1) displayed vs. one-photon detuning δ1=ωpuωpu(0) with the reference frequency arbitrarily fixed by the experimental setup is ωpu(0)=377109.2 GHz and two-photon detuning δ2=ωpuωprωsg shows a total of four peaks which can be described by ωpuωpr=ωeqg(μ)ωeps(ν), μ,ν=±, p,q=1,2. Two central peaks denoted 2 and 3 correspond to the two-photon resonances corresponding to μ=, ν=+ with p=q=1 and p=q=2, respectively. Two weaker side peaks denoted as 1 and 4 correspond to μ=ν=, p=q=1 and μ=ν=+, p=q=1, respectively. A similar pattern is observed for the conjugate beam in Fig. 2B. The quantum squeezing signal SN (Eq. 2) represents the relative noise corresponding to the degree of squeezing between the probe and conjugate fields. It is defined as a ratio of the relative intensity noise to the sum of the individual beams shot noise figures. SN is depicted on a log scale 10log(2G1) in Fig. 2C. The use of a log scale for quantum measurement (1) is natural since the noise spectra are normalized to unity for classical fields. Therefore, the noise of the quantum fields must be below one, which can be better visualized in a log scale. While the number and positions of peaks remain similar to the classical measurement, their shapes and relative intensities are different. For instance, peaks 1 and 4, which are barely visible in the gain spectra, are well pronounced in the squeezing signal. Note that the noise spectra in Fig. 2C are not identical to the classical signals of Fig. 2 A and B. To make a fair comparison, we used a logarithmic scale for the SN measurement using classical gain from Fig. 2D, which is shown in SI Appendix, Fig. S3. It contains the same number of peaks as a nonlogarithmic classical probe gain in Fig. 2D, highlighting the difference between squeezed measurement (Eq. 2) and the classical gain measurements (Eq. 1), providing a different observation window onto the susceptibility χ~(3) composed of the terms given by Eq. 5 and derived in SI Appendix, section S1.

Fig. 2.

Fig. 2.

Experimental 2D spectra of the classical and quantum signals. Eq. 3 displayed vs. the one-photon δ1=ωpuωpu(0) where ωpu(0) = 377,109.2 GHz and two-photon δ2=ωpuωprωsg detunings (A) probe and (B) conjugate photon numbers, and (C) squeezing Eq. 4 , for the weak 100-mW pump. (DF) Same as AC but for a strong 180-mW pump. Color lines indicate positions of the AC Stark-shifted resonances calculated using Eq. 5.

As the pump intensity is further increased to 180 mW, the AC Stark shift grows, and the four peaks described above are shifted accordingly, as seen in Fig. 2 D and E. However, the squeezing spectra undergo more dramatic changes. In addition to the original four peaks 1 through 4, Fig. 2F contains four additional peaks (Table 1). This additional information is accessible only by a strong field and quantum squeezing detection, and is missed by classical detection. This arises since the quantum squeezing measurement is higher order in field−matter interactions and thus is not polluted by linear processes which may preclude the detection of weaker resonances. When optical losses are included, the signal-to-noise ratio of such higher-order correlation measurements is significantly increased.

Table 1.

Resonant structure of the susceptibility in Eq. 5 depicted in Fig. 2 highlighting the sign of the AC Stark shifts

Peak no. μg νg λg μs νs p q Diagram
1 1 1 iia, iib
2 + + + 2 2 iia, iib
3 + + 1 1 ia, ib
4 + + + + + 2 2 ia, ib
5 + + 2 2 iia, iib
6 + + + 1 1 ia
7 + + + 1 1 iia, iib
8 + + 2 2 iia

To rationalize the experimental observations of Fig. 2, we developed a microscopic theoretical model for the χ~(3) susceptibilities, including the AC Stark shifts due to the strong pump. Here the matter response is governed by a χ~(3) susceptibility dressed by a strong pump field, which is different from the standard weak field susceptibility χ(3). Details are presented in SI Appendix, section S2. We maintain a nonperturbative treatment of the strong classical pump, while retaining the lowest-order perturbation expansion in the probe and conjugate beams. Eq. 5 reveals the resonant pattern of the susceptibility. We include optical losses during the propagating through the material cell after the FWM process described by Eqs. 6 and 7. These losses occur when the strong pump field−driven transitions sej (gej), j=1,2 undergo a spontaneous or stimulated emission with frequency matching the probe (conjugate) field. We thus obtain, for the classical measurement (28) (see SI Appendix, section S3),

N^prηprGN^0,N^cηc(G1)N^0. [3]

The corresponding noise figure is given by

SN=1+2(G1)(G(ηprηc)2ηc2)ηprG+ηc(G1). [4]

Fig. 3A shows the 2D spectra of the simulated probe gain for the strong 180-mW pump. All four peaks shown in Fig. 2D are reproduced with good agreement with experiment. The 1D segments of the spectra vs. single-photon detuning δ1 for a given two-photon detuning δ2=30, 0, and 50 MHz depicted by dashed white lines are displayed separately in Fig. 3 B, C, and D, respectively, and show good agreement between theory and experiment (30). The corresponding squeezing measurement is shown in two dimensions in Fig. 3E together with 1D cross-sections depicted in Fig. 3 FH. All eight peaks are well reproduced by the theory. In addition, the quantum regime (negative noise spectra) shown in Fig. 3G indicates the correct magnitude of the noise figure in both quantum (squeezing) and classical regimes. To demonstrate the merits of quantum over classical detection, we added a random time modulation of the input probe beam intensity by utilizing a Mach−Zehnder interferometer as shown in Fig. 4. The red line shows the squeezing, while blue and yellow correspond to classical separate intensity measurements of the conjugate and probe fields, respectively. While the output fields intensity is proportional to the input intensities, the variance of the intensity difference is governed by the sum of variances of the individual classical fields, which is then governed by the sum of the probe and conjugate field intensities. Thus, the overall noise contribution from classical fields will add up. On the other hand, the variance of the photon number difference of the quantum fields is different due to nonzero covariance due to quantum correlations shared between the fields, which reduces the photon number difference below classical noise levels (23). The corresponding squeezing measurement is then governed only by the gain of the FWM. Therefore, the same noise reduction achieved by quantum measurement may not be reached by using classical measurements. This conclusion holds when squeezing is below the shot noise as shown in Fig. 4A (corresponding to the dark blue area in Fig. 2C) as well as in the opposite limit, when noise is above the shot noise level as seen in Fig. 4B. A similar effect was observed in entangled photon spectroscopy, where the spectroscopic information has been obtained in the presence of an external noise such as background thermal radiation, added at the detection level (20). In our scheme, the noise is added to the input pump field prior to the FWM, and thus the noise is present throughout the FWM process in all four fields involved, and yet the quantum measurement’s resolution is stable. Note that the improvement of the SNR due to quantum correlations is not universal and is only applicable for certain parameter regimes. In particular, Fig. 2F shows regions of the negative signal (highlighted by the dark blue color), where quantum correlations yield squeezing SN<0. As has been shown in the two-photon detuning dependence of SN demonstrated in earlier works (31), the region of quantum correlations results in a higher SNR for squeezing measurement, compared to regions of classical correlations SN>0. Fig. 2F shows a more general dependence with respect to both one- and two-photon detunings. Nevertheless, the correlated measurement is robust against the noise in both quantum (Fig. 4A) and classical (Fig. 4B) regions of parameters.

Fig. 3.

Fig. 3.

Calculated 2D probe gain spectra Eq. 3 (A) with 1D slices (red line, theory; black dots, experiment) displayed vs. δ1 evaluated at (B) δ2=30 MHz, (C) δ2=0 MHz, and (D) δ2=50 MHz. (EH) Same as AD but for the noise figure in Eq. 4. Rb gas parameters used in simulations are taken from ref. 28. The values of the coefficient of determination (R2) defined in SI Appendix, Eq. S22 in Fig. 3 BE, G, and H are 0.41, 0.92, 0.72, 0.85, 0.58, and 0.51, respectively (see SI Appendix, section S4).

Fig. 4.

Fig. 4.

(A) Experimental time evolution of the signals due to random time modulation of the input probe intensity. Output probe intensity Eq. 3 (yellow), conjugate (blue), and squeezing Eq. 4 (red) evaluated at δ1 = 0.9 GHz and δ2 = 6 MHz. (B) Same as A but at δ1 = 0.5 GHz and δ2 = −10 MHz. The red arrow indicates that the red dot lines correspond to the axis of “Noise Power”; the yellow arrow indicates that the blue and yellow traces correspond to the axis of “Intensity.”

Discussion

We have carried out multidimensional FWM spectroscopy with squeezed light detection in hot Rb vapor. We find that quantum squeezing measurements provide additional valuable information compared to classical intensity measurements, through higher-order matter correlations. When optical losses are included, the spectra show different resonance patterns and provide a most valuable probe of the third-order response. A theoretical model provides an adequate microscopic account of the experiments. Our simulations allow to extract the actual model parameters from the AC Stark shift between the peaks corresponding to μ,ν=±. For instance, one can obtain the relative strengths of the dipole moments |μe2g|/|μe1g||μe1s|/|μe2s|8. This is consistent with the D1 line 52S1/252P1/2 π-transitions. Here the dipole moments expressed in multiples of J=1/2|er|J=1/22.99ea0, where a0 is Bohr radius, are given by μe1g=μe2s=1/27, μe2g=μe1s=8/27. We can further obtain the dephasing rate characterizing the linewidth given by γeγs10Γ57.5MHz, γgΓ (where Γ is the natural linewidth of the D1 transition). Squeezed light quantum spectroscopy is robust against external noise and yields sub-shot noise signals. Quantum light generated by the FWM process serves as a useful source for quantum spectroscopy and magnetic field sensors (32), complementing spontaneous parametric down-conversion sources. Our results suggest quantum sensing applications with multiphoton correlated light sources with an unprecedented level of microscopic detail beyond classical measurements.

Materials and Methods

Details of the Experiment.

A cavity-stabilized Ti:sapphire laser is used. A polarization beam splitter is used to divide the laser into two beams. One beam serves as the pump beam with frequency ωpu which is vertically polarized. The other beam passes through an acousto-optic modulator to get the probe beam at frequency ωpr. The horizontally polarized probe beam is weak (about 20 μW) and is equally divided into two by a 50/50 beam splitter. These two beams are used to construct a Mach−Zehnder interferometer, which is used to introduce intensity noise to the FWM process. A piezoelectric transducer is placed in the Mach−Zehnder interferometer to introduce intensity noise. The 85Rb vapor cell is 12 mm long and the temperature of the 85Rb vapor cell is stabilized at 113○C. At the center of the vapor cell, the waist of pump beam is about 620 μm, and the waist of probe beam is about 330 μm. Combined by a Glan−Laser polarizer, the pump and the probe beams are crossed in the center of the 85Rb vapor cell. The angle between the signal and pump beams is about 7 mrad. The residual pump beam after the FWM process is eliminated by a Glan−Thompson polarizer. The output probe and conjugate beam with frequency ωc=2ωpuωpr are sent to two silicon photodetectors (D1 and D2, respectively). The detector’s transimpedance gain is 105 V/A, and quantum efficiency is 96%. After the output beams are received by the detectors, we can obtain the dc and rf components of the photocurrents. The dc components from the two ports are sent to an oscilloscope to measure the intensity gain of the system. The rf components from the two ports are subtracted from each other by using an rf S and then analyzed with an SA. The SA is set to a 30-kHz-resolution bandwidth and a 300-Hz video bandwidth.

Theoretical Methods.

The third-order susceptibility that enters the FWM gain is derived by second-order perturbation theory with respect to probe and conjugate fields, while the pump field is treated nonperturbatively. The corresponding diagrams and details of the derivations are shown in SI Appendix, section S1. The third-order susceptibility has four terms, χ~(3)=kμg,νg,λg,μs,νsAkχ~kμgνgλgμsνs(3), where Ak are normalization functions that depend on the propagation length inside the sample and other experimental parameters, k runs over the diagrams k=ia,ib,iia,iib shown in SI Appendix, Fig. S1, and μm, νm, and λm indicate the AC Stark-shifted branches of the transition resonances μg,μs,νg,νs,λg=±. The first two terms can be written as

χ~ia(3)=i,j=12μgeiμeis*μsejμejg(ωeig(λg)iγe)(Ωejg2+γe2)(Δijμgμs+iγs)(Δijνgνs+iγe), [5]
χ~ib(3)=i,j=12μgeiμeis*μsejμejg(ωeig(λg)iγe)(Ωejg2+γe2)(Δijμgμsiγs)(Δijνgνsiγe),

where Δijμgμs=ωpuωpr+ωeig(μg)ωejs(μs), and γl, l=s,e are the dephasing rates of the atomic levels. The corresponding expressions for diagrams iia and iib can be obtained by replacing ωprωc and gs. A summary of the resonant structure of the susceptibility is shown in Table 1.

Note that the discrepancy in the linewidth broadening in simulations in Fig. 3 compared to experimental results of Fig. 2 is caused by a simplified model which does not take into account inhomogeneous broadening. The input/output relations for the field operator in the presence of the optical losses accumulated during the propagation after FWM (28) are given by

âprηprâpr+1ηprx^pr, [6]
âcηcâc+1ηcx^c,

where ηr=cos(|χ~r(1)|)2, r=pr,c and the noise operators x^ satisfy the standard boson commutation rules [x^r,x^r]=1. The corresponding susceptibility of the losses due to the spontaneous/stimulated emission is given by

χ~r(1)=Brj=1,2μmejμejmδejm2Ωejm2, [7]

where m=s for r=pr and m=g for r=c and Br is a normalization function that depends on the propagation length inside the vapor cell and other experimental parameters. In simulations shown in Fig. 3, both Ak and Br are used as fitting parameters. Spectra of the noise are shown in SI Appendix, Fig. S4.

Supplementary Material

Supplementary File
pnas.2105601118.sapp.pdf (684.2KB, pdf)

Acknowledgments

K.D. gratefully acknowledges support from the National Science Foundation of China (Award 11934011), the Zijiang Endowed Young Scholar Fund, East China Normal University, and the Overseas Expertise Introduction Project for Discipline Innovation (111 Project, B12024). J.J. gratefully acknowledges support from Innovation Program of Shanghai Municipal Education Commission (Grant 2021-01-07-00-08-E00100), the National Natural Science Foundation of China (Awards 11874155, 91436211, and 11374104), Basic Research Project of Shanghai Science and Technology Commission (Award 20JC1416100), Natural Science Foundation of Shanghai (Award 17ZR1442900), Minhang Leading Talents (Award 201971), Program of Scientific and Technological Innovation of Shanghai (Award 17JC1400401), National Basic Research Program of China (Award 2016YFA0302103), Shanghai Municipal Science and Technology Major Project (2019SHZDZX01), and the 111 Project (B12024). F.S. acknowledges support from the Cluster of Excellence “Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, Project 390715994. S.M. gratefully acknowledges the support of the NSF through Grant CHE-1953045.

Footnotes

The authors declare no competing interest.

This article is a PNAS Direct Submission. G.S.E. is a guest editor invited by the Editorial Board.

See online for related content such as Commentaries.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2105601118/-/DCSupplemental.

Data Availability

The data that support the plots within this paper are available at Open Science Framework: https://osf.io/5kt96/.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supplementary File
pnas.2105601118.sapp.pdf (684.2KB, pdf)

Data Availability Statement

The data that support the plots within this paper are available at Open Science Framework: https://osf.io/5kt96/.


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