Photonic Counterparts of Cooper Pairs

The microscopic theory of superconductivity raised the disruptive idea that electrons couple through the elusive exchange of virtual phonons, overcoming the strong Coulomb repulsion to form Cooper pairs. Light is also known to interact with atomic vibrations, as, for example, in the Raman effect. We show that photon pairs exchange virtual vibrations in transparent media, leading to an effective photon-photon interaction identical to that for electrons in the BCS theory of superconductivity, in spite of the fact that photons are bosons. In this scenario, photons may exchange energy without matching a quantum of vibration of the medium. As a result, pair correlations for photons scattered away from the Raman resonances are expected to be enhanced. An experimental demonstration of this effect is provided here by time-correlated Raman measurements in different media. The experimental data confirm our theoretical interpretation of a photonic Cooper pairing, without the need for any fitting parameters.

Figure 1

(a) Photons from a monochromatic laser are scattered by a molecule into different angles and energies. Lateral features are found in the spectrum, which correspond to Stokes (redshifted) and anti-Stokes (blueshifted) Raman peaks. (b) Schematically represented Raman scattering processes through phonon-mediated photon-photon interactions, including the usual Stokes (S) and anti-Stokes (AS) scattering, and a two-step correlated real process (real SAS) or an exchange of virtual phonons (virtual SAS).

Figure 2

Stokes–anti-Stokes correlation function $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ for different values of Raman shifts. (a) Experimental values are shown as symbols connected by dashed lines, at two excitation laser powers (see the legend). Each data point was obtained with a pair of bandpass filters equally spaced from the excitation laser line towards the Stokes and anti-Stokes shifts [18]. The green solid line represents the Raman spectrum of water, as obtained with the experimental apparatus [18]. The absolute values for correlation are highly sensitive to instrumental performance; thus, we provide here the percentage $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ with respect to the highest observed value, all obtained with the same instrumental apparatus (except for the Stokes and anti-Stokes bandpass filters). The values were also corrected to account for the dependence of the measured $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ with the band overlap between Stokes and anti-Stokes filters, and the bandpass overlaps are shown as horizontal bars, indicated only in one set of data. For the as-measured $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ values, see [18]. The solid blue line is the correlation calculated within a perturbative BCS approach [Eq. (3)], using as input the experimental Raman spectrum, without fitting parameters. (b),(c) The correlation function $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ is calculated (solid red line) within a perturbative BCS approach for a simplified model with only two vibrational modes of the water molecule [green line in (c)], shown near the real scattering condition ${({\omega }_{\mathbf{k}}-{\omega }_{\mathbf{k}-\mathbf{q}})\approx 1640\mathrm{cm}}^{-1}$. We also present as dashed and dash-dotted lines the numerical solutions to the problem of a single vibrational mode, including two values for the vibrational relaxation time $T_1$.

Figure 3

Stokes–anti-Stokes correlation function $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ for different media. (a) Experimental values for the different media are shown as different symbols (see the legend) for three different Raman shifts. The percentage $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ with respect to the highest observed value is shown, also corrected to account for the dependence of the measured $g_{\mathrm{S},\mathrm{AS}}^{(2)}(0)$ with the band overlap between Stokes and anti-Stokes filters. The horizontal bars indicate the overlap between bandpass filters and are shown for only one set of data. (b) Stokes Raman spectra for the different media, following the same color pattern as in (a), all normalized to the highest-intensity Raman peak. The unusually broad Raman peaks reflect the use of a femtosecond laser to excite the samples.