Optimized spectral and interferometric techniques for the certification of ETPA

TL;DR

The work tackles the long-standing variability in reported ETPA cross-sections by proposing a spectro-interferometric certification scheme that relies on asymmetrically overlapping the JSI with a molecular two-photon notch filter. Two complementary measurements are proposed: direct spectral JSI spectroscopy to reveal absorption-induced asymmetry and Hong-Ou-Mandel interferometry to detect reduced spectral indistinguishability via a lowered visibility. A quantitative detection-limit framework links the required absorption efficiency $η_E$ and cross-section $σ_E$ to experimental parameters (photon-pair flux, detector noise) via $η_E^{min}= \frac{δR_{det}}{R^{(2)}_{in}}$ and $σ_E^{min}= \frac{η_E^{min}}{C N_A ℓ}$. Preliminary experiments with RhB and Rh6G illustrate feasibility, showing that practical detection requires cross-sections exceeding the computed minima and that the sensitivity depends on notch bandwidth, detuning, and pump configuration.

Abstract

The phenomenon of Entangled Two-Photon Absorption (ETPA) presents a persistent controversy in the literature, evidenced by a wide disparity in the reported values for the $σ_E$ cross-sections. Much of this discrepancy is attributed to the difficulty in discriminating ETPA from various background processes that can mimic its signal, such as linear absorption or scattering. Given this need to certify the presence of ETPA unequivocally, this work introduces a key strategy to isolate the ETPA contribution through its spectral signature. This involves modeling the molecule as a two-photon notch filter and inducing a controlled asymmetric overlap with the joint spectral intensity (JSI) of the incident photons. This asymmetry is used to generate a measurable distortion in the transmitted JSI, and, complementarily, as a reduction in the visibility of the Hong-Ou-Mandel (HOM) dip. To ensure the experimental feasibility of this technique, a comprehensive analysis of the experimental conditions required for its detection is presented, establishing the absorption efficiency limits that must be overcome given the constraints of entangled-photon-pair flux and detector noise. A preliminary experiment addressing the ETPA detection limits using the standard RhB dye is presented.

Figures (8)

• Figure 1: $\abs{\alpha}^2$ function (first column) and the $\abs{\Phi}^2$ function (second column), with the third column showing the JSI $\abs{f(\omega_s,\omega_i)}^2$ and its marginals $f(\omega_s)$, $f(\omega_i)$ for type-0 SPDC generated in a ppKTP crytal (L=10 mm). An inset was added showing the HOM dip and the visibility of the resulting interferogram in the cases of employing: (a) a continuous wave laser ($\sigma_p=0.1$ nm); (b) a pulsed laser ($\sigma_p=5$ nm). [The axes were rescaled using $\Omega_{\mu}=\omega_{\mu}-\omega_{0}$, with $\omega_0=\omega_{p0}/2$ and $\mu$ = $s$ or $i$]
• Figure 2: JSI function for type-II SPDC generated in a ppKTP crytal (L=10 mm) as the product of the $\abs{\alpha}^2$ function (first column) and the $\abs{\Phi}^2$ function (second column). The third column shows the JSI $\abs{f(\omega_s, \omega_i)}^2$ and its marginals $f(\omega_s)$ and $f(\omega_i)$. An inset was added showing the HOM dip and visibility. (a) In the case of using a CW laser ($\sigma_p = 0.1$ nm), the HOM dip reaches $V=94\%$ because, although the $\Phi$ function is highly tilted, the narrow diagonal width of the pump is still cut symmetrically such that the marginal distributions are quite similar. (b) In the case of using a pulsed laser ($\sigma_p = 5$ nm), the JSI loses its symmetry, and it is immediately clear that the marginal distributions are very different. It is observed that visibility is almost completely lost, $V = 8\%$. [The axes were rescaled using $\Omega_{\mu} = \omega_{\mu} - \omega_0$, with $\omega_0 = \omega_{p0}/ 2$ and $\mu$ = $s$ or $i$].
• Figure 3: Effect generated for a type-0 SPDC process on the JSI, $\abs{f(\omega_s,\omega_i)}^2$, and on the HOM visibility interferogram, $R_C(\tau)$, before and after being filtered by a Notch filter. The top row (a) refers to the narrow bandwidth CW case and the bottom row (b) refers to the broad bandwidth case (femtosecond pulsed laser). (i) JSI, $\abs{f(\omega_s,\omega_i)}^2$, marginal $f(\omega_s)$ (red), and the interferogram (white) along with the corresponding visibility. (ii) The Notch filter, $\abs{h_N(\omega_s,\omega_i)}^2$, which forms an anti-diagonal with width, $\sigma_N=1$ nm, and efficiency, $\eta=0.90$. (iii) The JSI modified by the Notch filter, $\abs{f_{sam}}^2=\abs{f}^2 \times \abs{h_N}^2$ [The axes were rescaled using $\Omega_{\mu}=\omega_{\mu}-\omega_{0}$, where $\omega_0=\omega_{p0}/2$] [The vertical marginal was omitted since it is symmetric for this process, $f(\omega_i)=f(\omega_s)$.]
• Figure 4: Effect for a type-II SPDC process on the JSI, $\abs{f(\omega_s,\omega_i)}^2$. The insets show the effect on $R_C(\tau)$, before and after being filtered by a Notch filter. The top column (a) refers to the CW case and the bottom column (b) refers to the pulsed laser case. (i) JSI, $\abs{f(\omega_s,\omega_i)}^2$, marginal $f(\omega_s)$ (red), and the interferogram (white) along with the corresponding visibility. (ii) The Notch filter, $\abs{h_N(\omega_s,\omega_i)}^2$, which forms an anti-diagonal, $\sigma_N=1$ nm, and efficiency, $\eta=0.90$. (iii) JSI modified by the Notch filter, $\abs{f_{sam}}^2=\abs{f}^2 \times \abs{h_N}^2$ [The axes were rescaled using $\Omega_{\mu}=\omega_{\mu}-\omega_{0}$, where $\omega_0=\omega_{p}/2$] [The vertical marginal was omitted since it is symmetric for this process, $f(\omega_i)=f(\omega_s)$.]
• Figure 5: Realistic bandwidths for Notch filters. We selected the best JSI case, generated by a type-II SPDC process with a pump bandwidth of $\sigma_p = 5$ nm, corresponding to Figure \ref{['fig: jsi_notch_comp_type2']} (b). (A) is the JSI before interacting with the sample. (B) is an ideal $\sigma_N=$ 5 nm Notch filter. (D) is a Notch filter with the bandwidth of $\sigma_N=$ 20 nm corresponding to RhB. (F) is a Notch filter with the same bandwidth as RhB, but shifted to the absorption center of this molecule. (C), (E), and (G) show the JSI after passing through filters (B), (D), and (F), respectively.