Quantum and classical model<\/p>\n
The theory of Talbot\u2013Lau interference is best formulated in phase space using the Wigner\u2013Weyl representation of quantum mechanics42<\/a>. This framework can account for incoherent particle sources, phase and absorption gratings, and all laser-induced photophysical effects, as well as any relevant decoherence process. It also allows for a direct comparison between the predictions of quantum and classical mechanics within the same formalism and set of assumptions.<\/p>\n For a cluster with mass m and longitudinal velocity vz, the probability of being detected behind the interferometer can be written as a Fourier series in the transverse position x3 of G3:<\/p>\n $$S({x}_{3})=\\mathop{\\sum }\\limits_{{\\ell }=-\\infty }^{\\infty }{S}_{{\\ell }}\\exp \\left({\\rm{i}}\\frac{2{\\rm{\\pi }}{\\ell }}{d}{x}_{3}\\right).$$<\/p>\n \n (1)\n <\/p>\n In a symmetric setup with equal grating separations L and periods d, the Fourier coefficients are<\/p>\n $${S}_{{\\ell }}={B}_{-{\\ell }}^{(1)}(0){B}_{2{\\ell }}^{(2)}\\left({\\ell }\\frac{L}{{L}_{{\\rm{T}}}}\\right){B}_{{\\ell }}^{(3)}(0),$$<\/p>\n \n (2)\n <\/p>\n where the Talbot\u2013Lau coefficients \\({B}_{{\\ell }}^{(j)}\\) of order \u2113 for the jth grating still need to be determined as a function of the Talbot length LT\u2009=\u2009mvzd2\/h.<\/p>\n We assume that every absorbed grating photon results in the ionization of the sodium cluster. The transmission of the particle beam through a standing wave of incident laser power P, wavelength \u03bbL and Gaussian beam waist wy is then characterized by the mean number of ionizing photons absorbed in each grating antinode<\/p>\n $${n}_{0}=\\frac{8{\\sigma }_{{\\rm{ion,266}}}P{\\lambda }_{{\\rm{L}}}}{\\sqrt{2{\\rm{\\pi }}}hc{w}_{y}{v}_{z}},$$<\/p>\n \n (3)\n <\/p>\n as well as by the phase shift induced by the optical dipole potential<\/p>\n $${\\phi }_{0}=\\sqrt{\\frac{8}{{\\rm{\\pi }}}}\\frac{{\\alpha }_{266}P}{\\hbar c{\\varepsilon }_{0}{w}_{y}{v}_{z}}.$$<\/p>\n \n (4)\n <\/p>\n The values of the UV polarizability \u03b1266 and ionization cross-section \u03c3ion,266 are mass-dependent and determined further below. We can then express the Talbot\u2013Lau coefficients as58<\/a><\/p>\n $$\\begin{array}{l}{B}_{n}(\\xi )\\,=\\,{{\\rm{e}}}^{-{n}_{0}\/2}{\\left(\\frac{{\\zeta }_{{\\rm{coh}}}-{\\zeta }_{{\\rm{ion}}}}{{\\zeta }_{{\\rm{coh}}}+{\\zeta }_{{\\rm{ion}}}}\\right)}^{n\/2}\\\\ \\,\\times {J}_{n}({\\rm{sgn}}({\\zeta }_{{\\rm{coh}}}+{\\zeta }_{{\\rm{ion}}})\\sqrt{{\\zeta }_{{\\rm{coh}}}^{2}-{\\zeta }_{{\\rm{ion}}}^{2}}),\\end{array}$$<\/p>\n \n (5)\n <\/p>\n where the coherent phase shift and the ionization depletion are described by<\/p>\n $${\\zeta }_{{\\rm{coh}}}(\\xi )={\\phi }_{0}\\sin ({\\rm{\\pi }}\\xi )$$<\/p>\n \n (6)\n <\/p>\n $${\\zeta }_{{\\rm{ion}}}(\\xi )=\\frac{{n}_{0}}{2}\\cos ({\\rm{\\pi }}\\xi ).$$<\/p>\n \n (7)\n <\/p>\n For short de Broglie wavelengths, as \u03be\u2009\u2261\u2009L\/LT\u2009\u2192\u20090, the latter turn asymptotically into the expressions<\/p>\n $${\\zeta }_{{\\rm{coh}}}^{{\\rm{cl}}}(\\xi )={\\phi }_{0}{\\rm{\\pi }}\\xi $$<\/p>\n \n (8)\n <\/p>\n $${\\zeta }_{{\\rm{ion}}}^{{\\rm{cl}}}={n}_{0}\/2,$$<\/p>\n \n (9)\n <\/p>\n which appear in the classical description. It yields the same expression (equations (2<\/a>)\u2013(5<\/a>)) for the signal, except that equations (6<\/a>) and (7<\/a>) are replaced by equations (8<\/a>) and (9<\/a>).<\/p>\n In our setup, both the quantum and the classical signal are well approximated by a sinusoidal with fringe visibility V\u2009=\u20092|S1|\/S0. We average the predicted signal over the measured velocity and mass distributions, accounting for the mass dependence of both the polarizability and the ionization cross-section.<\/p>\n Macroscopicity assessment<\/p>\n To assess the macroscopicity of the demonstrated quantum superposition, it is necessary to calculate how the predicted interference signal is affected by the class of minimal macrorealist modifications (MMM) of quantum mechanics10<\/a>. These are parameterized by the classicalization time scale \u03c4e, and by the momentum spread \u03c3q and spatial spread \u03c3s of a phase space distribution. The greater the value of \u03c4e, the larger the scales at which the quantum superposition principle still holds.<\/p>\n For our symmetric Talbot\u2013Lau setup, the impact of an MMM is accounted for by multiplying the Fourier coefficients (equation (2<\/a>)) by<\/p>\n $$\\begin{array}{l}{R}_{{\\ell }}\\,=\\,\\exp \\,[-2\\sqrt{\\frac{2}{{\\rm{\\pi }}}}{\\left(\\frac{3{\\hbar }m}{{R}_{{\\rm{c}}{\\rm{l}}}{{\\sigma }}_{{\\rm{q}}}{m}_{{\\rm{e}}}}\\right)}^{2}\\frac{L}{{v}_{z}{\\tau }_{{\\rm{e}}}}\\\\ \\,\\times \\,{\\int }_{0}^{{\\rm{\\infty }}}{\\rm{d}}z\\,{{\\rm{e}}}^{-{z}^{2}\/2}{j}_{1}^{2}\\left(\\frac{{R}_{{\\rm{c}}{\\rm{l}}}{{\\sigma }}_{{\\rm{q}}}}{{\\hbar }}z\\right)\\,f\\,\\left(\\frac{{\\ell }d{{\\sigma }}_{{\\rm{q}}}L}{{\\hbar }{L}_{{\\rm{T}}}}z\\right)]\\end{array}$$<\/p>\n \n (10)\n <\/p>\n with Rcl the radius of the spherical clusters, me the electron mass, j1 a spherical Bessel function and f(x)\u2009=\u20091\u2009\u2212\u2009Si(x)\/x involving the sine integral10<\/a>. The dependence on \u03c3s can be neglected for this setup. The mean count rate is unaffected by MMM since R0\u2009=\u20091.<\/p>\n The macroscopicity is obtained by using the raw experimental data \\({\\mathcal{C}}\\) (cluster counts at given grating shift x3 and grating powers) for a Bayesian test of the hypothesis that MMM holds with a classicalization time no greater than \u03c4e (ref.\u200931<\/a>). Bayesian updating yields the posterior probability distribution \\(p({\\tau }_{{\\rm{e}}}| {\\mathcal{C}},{{\\sigma }}_{{\\rm{q}}})\\) of the classicalization time \u03c4e, starting from Jeffreys\u2019 prior, by using the likelihoods obtained by incorporating equation (10<\/a>) in the detection probability S(x3) (ref.\u200946<\/a>). The lowest 5% quantile \u03c4m(\u03c3q) of the posterior distribution then determines the macroscopicity as \\(\\mu =\\mathop{\\text{max}}\\limits_{{{\\sigma }}_{{\\rm{q}}}}({\\log }_{10}({\\tau }_{{\\rm{m}}}({{\\sigma }}_{{\\rm{q}}})\/1{\\rm{s}}))\\).<\/p>\n In our case, a total number of 3,895 data points yield a distribution very well approximated by a Gaussian (Kullback\u2013Leibler divergence 1.27\u2009\u00d7\u200910\u22123) whose 5% quantile \u03c4m\u2009=\u20092.84\u2009\u00d7\u20091015\u2009s (maximized at \u0127\/\u03c3q\u2009=\u200910 \u2009nm) remains constant to three decimal places after 3,280 data points. This indicates that sufficient data were recorded and that the distribution is independent of the prior. The resulting macroscopicity is \u03bc\u2009=\u200915.45.<\/p>\n Cluster beam<\/p>\n Large sodium clusters are generated in a custom-built aggregation chamber, inspired by earlier work32<\/a>,59<\/a>. The sodium is evaporated at 650\u2013700\u2009K into a cold mixture of argon and helium at a liquid nitrogen temperature of 77\u2009K and pressure of less than 1\u2009mbar. The resulting distribution covers masses beyond 1\u2009MDa and velocities between 120\u2009m\u2009s\u22121 and 170\u2009m\u2009s\u22121. The clusters exit through a 5-mm aperture and pass three differential pumping stages before they reach the interferometer (Supplementary Information<\/a>).<\/p>\n Two horizontal collimation slits dH1,H2\u2009=\u20090.5\u2009mm spaced by 1.8\u2009m facilitate the alignment of the grating yaw angles perpendicular to the molecular beam with a precision of about 200\u2009\u03bcrad. Two vertical collimation slits dV1\u2009=\u20090.5\u2009mm and dV2\u2009=\u20091\u2009mm, spaced by 2.2\u2009m, confine the beam height and ensure good overlap with the standing light wave. This also reduces the influence of gravitationally induced phase averaging.<\/p>\n Photophysics<\/p>\n The optical polarizability \u03b1266, absorption cross-section \u03c3abs,266 and ionization potential Ei depend on the size, mass and purity of the cluster. They determine transmission, the maximal matter-wave phase shift \u03d50 and the mean number of absorbed photons n0 in the antinodes of the grating. Photophysics60<\/a> and thermodynamics61<\/a> of small sodium clusters have been extensively studied, and the preparation of particles up to 1\u2009MDa has been demonstrated before59<\/a>. However, the mass-selected UV polarizability has not been known. Here, we use the high-contrast fringe patterns of clusters between 0.4\u2009MDa and\u00a01\u2009MDa to determine it in a mass range for which the classical and quantum models predict the same visibilities. We derive a value of \u03b1266\/atom\u2009=\u2009\u22124\u03c0\u03b50\u2009\u00d7\u2009(4.5\u2009\u00b1\u20090.5)\u2009\u00c53 (Supplementary Information<\/a>), which is consistent with the experiments and the quantum model for m\u2009=\u2009100\u2013200\u2009kDa.<\/p>\n The photo-ionization cross-section \u03c3ion,266 is a product of the absorption cross-section \u03c3abs,266 and the ionization yield. It determines the total transmission through the interferometer and influences the highest possible interference contrast. By measuring the mass-selected transmission of the interferometer for different grating powers, we determine an effective cross-section of \u03c3ion,266\u2009=\u2009(0.537\u2009\u00d7\u2009m [kDa]\u2009\u2212\u20091.5)\u2009\u00d7\u200910\u221220\u2009m2 for our clusters.<\/p>\n Mass selection and detection<\/p>\n After passing all gratings, the cluster beam is photo-ionized using 425\u2009nm light and the cations are filtered by their m\/z ratio using a quadrupole mass spectrometer. The mass filter includes guiding ion optics (Extrel) and 300\u2009mm long quadrupole rods (Oxford Applied Research) with a diameter of 25.4\u2009mm. The mass filter is operated at a resolution of \u0394m\/m\u2009=\u20090.32. Interference scans centred on mass m, therefore, involve clusters within a mass range of \u00a0\u00b1\u0394m\/2, where the transmission function is close to rectangular shape and taken into account in our models. The mass filter was centred at 170\u2009kDa. The underlying mass distribution, convoluted with the trapezoidal transmission, shifts the effective mass centre towards 172\u2009kDa.<\/p>\n The selected cluster ions are counted by a channel electron multiplier with a conversion dynode at 10\u2009kV. Electronic dark counts range from 15\u00a0to\u00a0100\u2009 counts\u2009s\u22121.<\/p>\n We must also account for the mixing of multiply charged ions with identical m\/z ratios. Based on the measured work function of W\u2009=\u2009(2.4\u2009\u00b1\u20090.1)\u2009eV (Supplementary Information<\/a>), neutral clusters with a diameter of dCl\u2009~\u20098 \u2009nm exhibit an ionization threshold of Ei\u2009=\u20092.53\u2009eV, followed by Ei,+1\u2009=\u20092.88\u2009eV and Ei,+2\u2009=\u20093.23\u2009eV for subsequent ionization processes. The detection laser has a photon energy of Eph\u2009=\u20092.92\u2009eV and can generate doubly charged ions, whereas triply charged ions remain energetically out of reach.<\/p>\n We have selected doubly charged clusters in the detector and verified the correct cluster mass by analysing mass spectra at both low and high detection laser powers (Supplementary Information<\/a>). In the antinodes of the gratings, the 266\u2009nm light can also lead to multiply charged ions. However, this does not affect the interference pattern, because every ion is removed from the cluster beam by electrostatic deflection, independent of its charge state. Only clusters that remain neutral while passing through all gratings contribute to the final interference pattern.<\/p>\n Velocity distribution<\/p>\n The cluster velocity distribution is determined from a time-of-flight measurement, in which we imprint a start time signal onto the cluster beam by UV photodepletion close to G1, and we measure the cluster arrival time behind the ionizing mass spectrometer. The time-of-flight data are corrected for the drift time inside the quadrupole, where it is slightly accelerated by the entrance voltage U to \\(v{\\prime} =v+\\sqrt{2eU\\,\/\\,m}\\). A convolution of a Gaussian drift time distribution and a rectangular chopper opening function is then fitted to the corrected unsmoothed data. The results are converted to a velocity distribution. We determine the average velocity and the width of the distribution from the standard deviation of the Gaussian fit.<\/p>\n