Vacuum Rabi splitting in a plasmonic cavity at the single quantum emitter limit — TEI-tagged view

Kotni SanthoshpersName, Ora BittonpersName, Lev ChuntonovpersName, Gilad HaranpersName (2016date). “Vacuum Rabi splitting in a plasmonic cavity at the single quantum emitter limit.” Nature Communicationstitle·j.

Object of study — silverterm·dom bowtieterm·key plasmonic cavitiesterm·key loaded with one to a few semiconductor quantum dotsterm·dom

Research question — whether strong couplingterm·key can be reached at the limit of a single quantum emitter

Contribution — vacuum Rabi splittingterm·key with a coupling rateterm·key of 120 meV even for a single quantum dotterm·key

Full text — tagged

Vacuum Rabi splittingterm·key in a plasmonic cavity at the single quantum emitter limit Abstract The strong interaction of individual quantum emitters with resonant cavities is of fundamental interest for understanding light–matter interactions. Plasmonic cavitiesterm·key hold the promise of attaining the strong couplingterm·key regime even under ambient conditions and within subdiffraction volumes. Recent experimentsterm·mth revealed strong couplingterm·key between individual plasmonic structures and multiple organic molecules; however, strong couplingterm·key at the limit of a single quantum emitter has not been reported so far. Here we demonstrate vacuum Rabi splittingterm·key, a manifestation of strong couplingterm·key, using silverterm·dom bowtieterm·key plasmonic cavitiesterm·key loaded with semiconductor quantum dotsterm·dom (QDsterm·dom). A transparency dipterm·key is observed in the scattering spectraterm·key of individual bowtiesterm·key with one to a few QDsterm·dom, which are directly counted in their gaps. A coupling rateterm·key as high as 120 meV is registered even with a single QD, placing the bowtie-QD constructs close to the strong couplingterm·key regime. These observations are verified by polarization-dependent experimentsterm·mth and validated by electromagnetic calculations.

The interaction of emitters with an optical cavity falls within the realm of cavity quantum electrodynamics 1 . Approaching the limit of strong couplingterm·key between individual quantum emitters and resonant cavities is important for multiple advanced optical applications, such as quantum information processing 2 , 3 and quantum communication 4 , 5 . Strong couplingterm·key can be observed through the phenomenon of vacuum Rabi splittingterm·key in the optical spectra of the joint systems, as well as in the appearance of non-classical photon correlations. The strength of the interaction depends on the ratio of the quality factorterm·key of the cavity to the mode volumeterm·key, Q/V . In optical cavities made of, for example, photonic crystals 6 or micropillars 7 , the fundamental laws of diffraction set a strict limitation on the cavity size, which cannot be smaller than half the wavelength of the interacting photon. This in turn limits how small V can be and mandates a very high Q , to attain the strong couplingterm·key regime. Obtaining such a high-quality factor requires demanding experimental conditions such as cryogenic temperatures and ultra-narrow frequency light sources.

Cavities made of materials that can sustain surface plasmonterm·dom (SPterm·dom) excitations can beat the diffraction limit by focusing intense electromagnetic fields into volumes much smaller than the wavelength of light 8 . Such cavities should simplify considerably the experimental conditions required for strong light–matter interactions and may allow quantum optical experimentsterm·mth to be conducted under ambient conditions.

Noble metals are most often used for generating structures sustaining SPsterm·dom at visible wavelengths. Although the SPterm·dom relaxation times in such structures are ultrafast, severely limiting Q , their mode volumeterm·key is drastically reduced compared with photonic cavities. In recent years, there has been much interest in studies of coupling between SPsterm·dom, either propagating or localized, and quantum emitters such as molecules 9 . These experimentsterm·mth involved multiple molecules and often multiple or extended plasmonic structures as well. In recent times, strong couplingterm·key has also been probed at the level of a single plasmonic device. Thus, Halas and colleagues 10 used gold dimers and molecular J-aggregates to observe a Rabi splitting of ∼230 meV. Shegai and colleagues 11 observed strong couplingterm·key using either a silverterm·dom nanorod or a silverterm·dom nanoprism 12 , in both cases employing J-aggregates as the quantum emitters. In all of these experimentsterm·mth, the interaction involved hundreds of quantum emitters or more. However, for quantum information operations one needs to approach the limit of a single quantum emitter coupled to the cavity.

In this study, we show that one can indeed observe strong couplingterm·key in the limit of a single quantum emitter. In particular, we use silverterm·dom bowtieterm·key plasmonic cavitiesterm·key and couple them to semiconductor quantum dotsterm·dom (QDsterm·dom). Scattering spectraterm·key registered from individual plasmonic cavitiesterm·key containing one to a few QDsterm·dom show vacuum Rabi splittingterm·key, indicating that the strong couplingterm·key regime is approached in these systems. Polarization-dependent experimentsterm·mth verify that the observed Rabi splitting is due to the coupling of the longitudinal plasmon resonance of the bowtiesterm·key with the QDsterm·dom.

Results Construction of plasmonic cavitiesterm·key with QDsterm·dom Inspired by recent computational studies, suggesting that strong couplingterm·key with individual emitters can be achieved in the gaps of plasmonic structures 13 , we turned to silverterm·dom bowtiesterm·key, which can be fabricated with a small gap 14 . Silverterm·dom is preferred as a material for plasmonic devices due to its relatively low SPterm·dom damping and hence the high achievable quality factorsterm·key of such devices compared with other metalsterm·dom 15 . By varying the structural parameters of the bowtiesterm·key, including side length and gap size, the localized SPterm·dom excitations of bowtiesterm·key manufactured by electron-beam lithography can be tuned to resonance with the quantum emitter of choice, while maintaining gaps of the order of ∼20 nm. To achieve strong couplingterm·key, the quantum emitter should possess a large oscillator strength and a narrow linewidth. QDsterm·dom have several advantages over organic emitters in relation to the aforementioned characteristics 16 and (most importantly) are also significantly more photostable. They can also be observed by electron microscopy and therefore can be directly counted. Commercially available CdSe/ZnS QDsterm·dom were employed in the present study. Optical spectroscopy and electron microscopy studies showed that the sample is mildly heterogeneous and the diameter of the QD varies between 6 and 8 nm.

To position QDsterm·dom within the gaps of bowtiesterm·key, we made use of interfacial capillary forces 17 to drive the QDsterm·dom into lithographically patterned holes in the bowtieterm·key gaps. This simple method avoids additional steps such as chemical modification of the QD/nanostructure surfaces. The process, described in Fig. 1 and in the Methods section, led to the trapping in bowtieterm·key gaps of a number of QDsterm·dom that varied from one to several, as could be ascertained by scanning electron microscopy.

Construction of bowtiesterm·key with quantum dotsterm·key in their gaps. ( a ) Schematic illustration of the two-step lithography process for making holes at the centre of bowtie structuresterm·dom and the interfacial capillary force assisted method for driving QDsterm·dom into the holes. ( b ) Scanning electron microscope images of bowtiesterm·key with one, two and multiple QDsterm·dom in their gaps (from left to right). The positions of the QDsterm·dom are marked by red arrows. Scale bars, 20 nm (yellow).

Spectroscopy demonstrates Rabi splitting Dark-field microspectroscopy was used to characterize the plasmonic behaviour of every single bowtieterm·key by measuring its scattering spectrum. As shown in Supplementary Fig. 1 , empty bowtiesterm·key possess two modes, a transverse mode at ≈2.05 eV and a longitudinal mode at ≈1.9 eV, the latter being due to dipolar coupling of the two parts of each bowtieterm·key. The absorption and emission spectra of the QDsterm·dom are shown in Supplementary Fig. 2 and demonstrate that the optical transitions of the QDsterm·dom are in resonance with the plasmon excitations of the bowtiesterm·key. (See also Supplementary Note 1 for further discussion on the role of the absorption and emission spectra in coupling to the plasmonic cavity.) Figure 2a shows bowtiesterm·key with QDsterm·dom in their gaps whose scattering spectraterm·key present transparency dipsterm·key indicative of Rabi splitting. Overall, we collected scattering spectraterm·key from 21 QD-containing bowtiesterm·key, out of which 14 spectra showed a definite signature of splitting. Thus, the top bowtieterm·key in Fig. 2a , with a single QD and a particularly small gap (∼19 nm), presents a clear Rabi splitting feature. Similarly, Rabi splitting is observed in the spectra of the two other bowtiesterm·key shown in Fig. 2a , with two and three QDsterm·dom and gaps of 17 and 30 nm, respectively. The spectra in this figure, as well as other spectra showing Rabi splitting, were fitted using the coupled oscillator model 18 (see Methods). This model represents the plasmon mode and the exciton of the quantum emitter as coupled harmonic oscillators exchanging energy reversibly. The coupling rateterm·key characterizing the interaction is given by g = μ · E , where μ is the transition dipole moment of the exciton and E is the electric field of the plasmon. This coupling leads to the generation of upper ( Ω + ) and lower ( Ω − ) plasmon-exciton hybrid states, with a transparency dipterm·key in between 13 , 19 . The Rabi splitting values, Ω + − Ω − , can be obtained directly from the fitted curves, and for the three examples in Fig. 2a they are 176, 288 and 224 meV, respectively. These values are comparable to recently reported values of Rabi splittings of J-aggregates with individual plasmonic structures 10 , 12 , although hundreds of quantum emitters were involved in the latter. Not all bowtiesterm·key loaded with a single QD showed a clear transparency dipterm·key in the scattering spectrum. Instead, Rabi splitting manifested itself in a significant spectral broadening as compared with that of an empty bowtieterm·key ( Supplementary Fig. 3 ), indicating coupling between plasmons and excitons. Splitting was readily observed in bowtiesterm·key with several QDsterm·dom; see Supplementary Fig. 4 for additional spectra.

Strong couplingterm·key of plasmons and quantum emitters. ( a ) Scattering spectraterm·key of bowtiesterm·key with (from top to bottom) one, two and three QDsterm·dom in the gap, respectively. All spectra show a transparency dipterm·key due to Rabi splitting. The black lines are experimental data and the coloured lines are fits to the coupled oscillator model. Insets show the scanning electron microscope images of the bowtiesterm·key. The positions of the QDsterm·dom are marked by red arrows. Scale bars, 20 nm (yellow). ( b , c ) Coupling ratesterm·key as a function of gap size for bowtiesterm·key with one QD (( b ) red symbols) and two QDsterm·dom (( c ) green symbols). The errors in the coupling rateterm·key values, obtained from the fitted functions, are estimated to be 2–5 meV. The continuous lines represent the numerically calculated coupling ratesterm·key at two configurations along the centre line of the bowtieterm·key: with the QDsterm·dom almost touching one of the prisms (continuous lines) or with the QDsterm·dom at the centre of the bowtieterm·key (dashed-dotted lines). In the experimentsterm·mth, the QDsterm·dom may be positioned away from the centre line so that their coupling ratesterm·key are lower than the calculated lines. ( d ) Polarization series of the middle bowtie structureterm·dom in a . As the polarization of the excitation light is rotated from the direction parallel to the bowtieterm·key long axis to perpendicular to it, the transparency dipterm·key in the spectrum gradually vanishes, indicating that it indeed originates in the coupling of the QD exciton with the longitudinal plasmon resonance of the bowtieterm·key.

Coupling ratesterm·key obtained from the fits of data sets with one and two QDsterm·dom are plotted as a function of gap size in Fig. 2b,c . The figures also contain theoretical estimates of the coupling ratesterm·key, based on the numerical calculations described below. The coupling rateterm·key depends strongly on the gap size of each bowtieterm·key. For instance, for two QDsterm·dom the coupling rateterm·key is more than doubled when the gap size changes from 25 to 17 nm. In general, the coupling increases with the number of quantum emitters ( N ) ( , g 1 being the coupling rateterm·key for one quantum emitter 1 ), although in our experimentsterm·mth this dependence is partially masked by the dependence on gap size and QD position variations. Although the values of the coupling ratesterm·key obtained from the fits are smaller than the values of the plasmon widths (which are ∼400 meV), significant splittings are observed, indicating that the strong couplingterm·key regime has indeed been attained, or is close 9 . Supplementary Fig. 5 shows histograms of the values of all parameters obtained from the coupled oscillator fits to measured spectra.

To obtain an unequivocal proof that the splitting in the spectra is due to strong (or close-to-strong) coupling of the bowtieterm·key longitudinal plasmon mode and the quantum emitter's exciton, we resorted to polarization-dependent experimentsterm·mth in which the polarization of the excitation source was rotated. Figure 2d shows an example of a polarization series measured on the bowtieterm·key with two QDsterm·dom from Fig. 2a . When the excitation is polarized along the longitudinal direction of the bowtieterm·key (that is, along its long axis), the spectrum shows two peaks due to Rabi splitting. As the polarization is rotated to the transverse direction, the double-peaked spectrum is gradually replaced with the single-peaked spectrum of the transverse mode. Similar results with other bowtiesterm·key are shown in Supplementary Fig. 6 . These results clearly indicate that the transparency dipsterm·key in the scattering spectraterm·key are due to a genuine coupling of the plasmon and QD excitations.

Discussion The experimental results described above show that the positioning of one or a few QDsterm·dom within a silverterm·dom bowtieterm·key can bring this plasmonic cavity very close to the strong couplingterm·key regime. To support the experimental results and shed further light on them, we performed electromagnetic calculations, using the boundary element method 20 . The simulations revealed some interesting facts. The top spectrum in Fig. 3a is the calculated scattering spectrum of an empty bowtieterm·key. The spectrum for a bowtieterm·key with a single QD at the centre of the gap is shown below, demonstrating a small change compared with the empty bowtieterm·key. However, when the QD is positioned closer to one of the prisms constituting the bowtieterm·key, or the number of QDsterm·dom in the hotspot is increased from one to two, a more significant transparency dipterm·key appears. The Rabi splitting is observed in both calculated scattering and extinction spectra (the latter not shown). Importantly, no such splitting is found when the simulation is repeated with a single metallic prismterm·dom rather than a bowtie structureterm·dom.

Electromagnetic simulations of strongly coupled plasmonic structures. ( a ) Simulated scattering spectraterm·key of bowtiesterm·key with one and two QDsterm·dom organized as indicated in the inset structures. ( b ) Distribution of the coupling rateterm·key (in eV) of a quantum emitter with an oscillator strength of 0.6 in a bowtie structureterm·dom. The white bar represents 10 nm.

The simulation allows us to directly calculate the coupling rateterm·key for a quantum emitter with an oscillator strength similar to a QD (∼0.6) 21 , 22 , 23 . The spatial distribution of the coupling rateterm·key across the gap is shown in Fig. 3b , clearly indicating that the electromagnetic field in the hotspot is not uniform and is concentrated at the edges rather than at the centre. Indeed, in many of the bowtiesterm·key exhibiting strong couplingterm·key ( Fig. 2 and Supplementary Figs 3 and 4 ) the QDsterm·dom are close to one of the edges, in agreement with the simulation results. We estimate from the simulation that at the centre of a bowtie structureterm·dom g is ∼100 meV, whereas g is as high as 200 meV very close to one of the prisms.

In conclusion, we have successfully integrated lithographically fabricated silverterm·dom bowtiesterm·key with one to a few QDsterm·dom that reside exactly within the plasmonic cavity. The very small plasmon mode volumesterm·key of the bowtiesterm·key allowed us to demonstrate vacuum Rabi splittingterm·key in the limit of an individual quantum emitter. A key advantage of using QDsterm·dom is that we can directly count them under the electron microscope and verify the relation between coupling and the number of quantum emitters within each cavity. An important question to ask is: what value of the coupling may deem it ‘strong'? An often-used rule of thumb for the strong couplingterm·key regime is , where γ 0 and γ pl are the QD and plasmonic cavity linewidths, respectively 9 , 24 . The maximal coupling ratesterm·key obtained in this work for two and three QDsterm·dom (∼200 meV; Fig. 2b,c and Supplementary Fig. 5d ) fulfil the above criterion, whereas the maximum value obtained for a single QD (120 meV) is somewhat smaller. This shows that the constructs of QDsterm·dom with plasmonic cavitiesterm·key studied here are definitely close to the strong couplingterm·key regime. In future experimentsterm·mth we plan to significantly increase the coupling between individual QDsterm·dom and bowtiesterm·key. To this end, we will reduce the mode volumeterm·key of the bowtiesterm·key, which can be achieved by decreasing their gap size and improving control over their shape in the fabrication process. Further improvement in QD positioning methodology will allow us to locate a single QD reproducibly at the position of the highest field within a bowtieterm·key cavity. Such improvements will enable systematic variation of experimental parameters and therefore provide additional proofs for strong couplingterm·key. Importantly, our results also pave the way to more sophisticated quantum nonlinear optics experimentsterm·mth 25 . In particular, by probing time-resolved photoluminescence of the coupled QD-bowtie system we hope to directly demonstrate quantum correlations 2 , 26 .

Methods Materials Glass/ITO (10 × 10 × 1.1 mm) substrates were purchased from Xin Yan Technology Ltd., China. The ITO layer is 180 nm thick and is characterized by 83% transparency to visible light and a sheet resistance of 10 Ω −2 . Poly(methyl methacrylate) 950 K A2 (PMMA) electron beam resist, used for the fabrication of bowtiesterm·key, was procured from MicroChem. Chromium (used as an adhesive layer) and silverterm·dom were obtained from Kurt J. Lesker. Water-soluble mercapto-undecanoic acid capped CdSe/ZnS core/shell nanocrystals (QDsterm·dom) were acquired from MK Impex Corp.

Electron beam lithography and fabrication of silverterm·dom bowtiesterm·key Before fabrication, glass/ITO substrates were cleaned with acetone followed by isopropanol for 3 min in each step. The cleaned glasses were dried using a nitrogen flow and then PMMA was spin-coated at a speed of 6,000 r.p.m. for 50 s to achieve a thickness of 60 nm, followed by baking at 180 °C for 90 s. The PMMA-coated glasses were loaded into a Raith E_line Plus electron beam lithography system and the PMMA was exposed to define the shape of bowtiesterm·key. The accelerating voltage used for the exposure was 30 kV and the beam current was 30 pA. The design consisted of matrices of bowtiesterm·key, with each matrix hosting 64 bowtiesterm·key. Each bowtieterm·key was separated by 10 μm from its neighbouring partner to guarantee no interaction. To remove the exposed PMMA, the substrates were developed using 1:3 methyl isobutyl ketone:isopropyl alcohol for 30 s, followed by immersion in a stopper (isopropyl alcohol) for 30 s and drying with a nitrogen flow.

An electron beam evaporator (Odem) was used for the metal deposition on the patterned substrates. Initially, chromium was evaporated to deposit a 2-nm adhesion layer, followed by silverterm·dom. Different prism lengths from 75 to 200 nm and thicknesses from 20 to 35 nm were examined and finally the dimensions were fixed to a length of ≈85 nm and a thickness of ≈30 nm. These bowtiesterm·key showed a plasmon resonance that overlapped the QD exciton frequency. After metallization, a lift-off step was performed in acetone for 5 min to obtain silverterm·dom bowtiesterm·key on the glass/ITO substrate.

A second stage of E-beam lithography was performed, to expose holes in the gap regions of the bowtiesterm·key. A 60-nm layer of PMMA was spin-coated on the substrate and then the gap regions of the bowtiesterm·key were exposed to create 10–30 nm holes, into which the QDsterm·dom were driven. (See below and in Fig. 1 of the main text.) To reach an overlay accuracy as high as a few nanometres in the second exposure, we made use of alignment marks that were fabricated in the first E-beam exposure step.

Incorporation of QDsterm·dom into bowtie structuresterm·dom To incorporate QDsterm·dom into the bowtieterm·key gaps we adopted a method originally developed by Alivisatos and colleagues 17 . Substrates with bowtiesterm·key and holes in the PMMA layer prepared as described above were placed vertically in an aqueous solution of QDsterm·dom. Controlled solvent evaporation was then used to exert a capillary force along the receding line of contact of the QD solution and drive QDsterm·dom successfully into the holes. By varying the QD concentration in solution (from 10 to 100 nM), bowtiesterm·key with one to a few QDsterm·dom in the gap were successfully obtained ( Fig. 1 ). Although this method is limited in controlling the exact position of the QDsterm·dom within the gap, it was found to be facile and relatively reproducible.

Bulk absorption and emission measurements of QDsterm·dom Steady-state absorption and emission spectral measurements ( Supplementary Fig. 2 ) were carried out on a Cary-100 ultraviolet–visible spectrophotometer (Varian) and a Fluorolog-3 spectrofluorimeter (Horiba Jobin Yvon), respectively. Extinction coefficients of the QDsterm·dom were estimated by employing the method developed by Peng and colleagues 23 .

Dark-field microspectroscopy Our dark-field microspectrometer is described schematically in Supplementary Fig. 8 . Briefly, scattering spectraterm·key of single bowtiesterm·key, either with or without QDsterm·dom, were measured with an inverted microscope equipped with a dark-field condenser, a 75 W Xenon lamp (Olympus) and a × 100 oil-immersion objective with a numerical aperture of 0.6. A SpectraPro-150 spectrograph with a 1,200 g mm −1 grating (Acton) and a Newton spectroscopy charge-coupled device camera (Andor Technology) were used to disperse scattered photons and register spectra. Raw spectra were smoothed using a Fourier low-pass filter. Before the spectral measurements, the camera was calibrated in reference to the spectrum of fluorescent dyes. Polarization of the excitation light was controlled by a combination of a polarizer and a selecting sector, such that essentially only s -polarized light reached the sample.

Data fitting to the coupled oscillator model Scattering spectraterm·key were fitted to the following equation, which represents the scattering of two coupled harmonic oscillators at frequency ω 18 : In this expression ω pl and ω 0 are the plasmon and QD resonance frequencies, respectively, whereas γ pl and γ 0 are the corresponding linewidths. g is the coupling rateterm·key and A is a scaling parameter. Parameters obtained from fits to all our data sets are shown in Supplementary Fig. 5 . We fixed the value of γ 0 to 130 meV, which is the measured linewidth of the fluorescence spectra of individual QDsterm·dom ( Supplementary Fig. 2 ). In the case of some of the data sets with a single QD in the gap, where broadening of the spectrum was observed, although no clear splitting, we also fixed the value of γ pl to 385 meV, which is the average value of this parameter obtained from multiple spectra of empty bowtiesterm·key.

Electromagnetic simulations Numerical simulations of the silverterm·dom bowtiesterm·key were performed using the boundary-element method as implemented in MNPBEM—a toolbox developed by Hohenester and Trügler 20 . A numerical model of a bowtieterm·key was constructed from two prisms with equilateral triangular bases of 80 nm side length, 30 nm heights, radii of curvature at the vertices of 7 nm and a gap size of 17 nm. The complex refractive index of silverterm·dom from Johnson and Christy 27 was used and a refractive index of 1.33 was assumed for the ambient medium. The bowtieterm·key was excited by a plane wave at normal incidence and polarized along the longitudinal direction and the plasmon scattering spectraterm·key were calculated at the far field. From the spectrum of an unloaded cavity, we calculated γ pl =0.25 eV, which is smaller than the typical experimental linewidth. However, the cavity quality factorterm·key Q = ω pl / γ pl =7.3 was only slightly higher than the measured quality factorterm·key. Simulations of scattering spectraterm·key for a bowtieterm·key loaded with QDsterm·dom were performed with dots modelled as spheres of 8 nm located at various positions within the gap. The complex dielectric function of these dots was approximated by a Lorentz model with a high-frequency dielectric constant =6.1, ω 0 =1.8 eV, γ 0 =0.08 eV and an oscillator strength f =0.6 (refs 21 , 22 , 23 ). As the computed plasmon linewidth of the idealized bowtie structureterm·dom was consistently smaller than the measured linewidth of a real bowtieterm·key, we scaled the linewidth of the QD accordingly.

The position-dependent coupling rateterm·key of a QD strongly coupled to the bowtieterm·key g ( r ) was evaluated using E ( r ), the electric field numerically calculated at the plasmon resonance energy ω pl =1.83 eV. To this end, the mode volumeterm·key of the antenna was calculated using the approach of Koenderink 28 and was used to normalize the energy density within the cavity, assuming it is occupied with a single photon of energy 1.83 eV. A map of g ( r ) plotted on a horizontal plane that intersects the bowtieterm·key at its half-height is shown in Fig. 3b . Supplementary Fig. 7 shows a similar calculation for a single prism (panel a), and compares the coupling rateterm·key of a bowtieterm·key and a single prism (panel b).

Data availability All relevant data are available from the authors on request.

Additional information How to cite this article: SanthoshpersName, K. et al. Vacuum Rabi splittingterm·key in a plasmonic cavity at the single quantum emitter limit. Nat. Commun. 7:11823 doi: 10.1038/ncomms11823 (2016).

Supplementary Material Supplementary Figures 1-8 and Supplementary Note 1.

Acknowledgments G.H. is the incumbent of the Hilda Pomeraniec professorial chair. We thank Dr Barak Dayan for useful comments on the manuscript.

Footnotes References Associated Data Supplementary Materials Supplementary Figures 1-8 and Supplementary Note 1.

Data Availability Statement All relevant data are available from the authors on request.