Broadband entangled-photon omni-resonance in a planar optical cavity

Abstract

Resonant field enhancement in an optical cavity is a promising pathway towards realizing optical nonlinearities at the few-photon level. This quest is hampered by inevitable narrowing of the resonant linewidth as the cavity finesse is increased, which necessitates striking a compromise between the magnitude of the field enhancement and the bandwidth over which it is harnessed. This difficulty is exacerbated for broadband entangled-photon pairs, which are typically frequency-anticorrelated, so that the two photons cannot be simultaneously admitted to a cavity except when the degenerate wavelength coincides with a cavity resonance. Here we show that introducing judicious angular dispersion into single-photon and entangled-photon states before incidence on a planar Fabry-P{é}rot (FP) cavity renders these non-classical fields omni-resonant: the entire spectrum is coupled to a single longitudinal cavity mode. Making use of a planar FP cavity of finesse $\approx100$, resonant linewidth $\approx0.3$~nm, and free spectral range $\approx22$~nm in the near-infrared, we couple single-photon states and frequency-anticorrelated entangled-photon states of 20-nm bandwidth to a broadband achromatic resonance associated with a single underlying narrowband longitudinal FP-cavity mode -- thereby preserving the entangled spectral structure. In general, pre-conditioning the optical field by introducing angular dispersion enables coupling non-classical states of light to a single longitudinal cavity mode, even if the field bandwidth far exceeds the resonant linewidth, or even exceeds the cavity free-spectral-range. These results pave the way to broadband resonant interactions with non-classical states of light in photon-starved applications.

Figures (7)

• Figure 1: Conventional narrowband resonance versus broadband omni-resonance for single-photon and entangled-photon states. (a) A broadband collimated single-photon state $\widetilde{G}^{(1)}(\omega)$ incident on a planar FP cavity is filtered by the resonant spectral transfer function $T(\omega)$ corresponding to a single conventional FP resonance of linewidth $\delta\omega$ at frequency $\omega_{m}$, $\widetilde{G}^{(1)}_{\mathrm{FP}}(\omega)$. (b) The biphoton spectrum $\widetilde{G}^{(2)}(\omega_{\mathrm{s}},\omega_{\mathrm{i}})$ for broadband frequency-anticorrelated entangled photons is also filtered by the FP cavity with two-photon resonant spectral transfer function $T_{2}(\omega_{\mathrm{s}},\omega_{\mathrm{i}})=T(\omega_{\mathrm{s}})T(\omega_{\mathrm{i}})$, to yield $\widetilde{G}_{\mathrm{FP}}^{(2)}(\omega_{\mathrm{s}},\omega_{\mathrm{i}})=T_{2}(\omega_{\mathrm{s}},\omega_{\mathrm{i}})\widetilde{G}^{(2)}(\omega_{\mathrm{s}},\omega_{\mathrm{i}})$. Both photons couple to the cavity only if the degenerate condition $\omega_{\mathrm{s}}=\omega_{\mathrm{i}}=\omega_{\mathrm{o}}$ coincides with a cavity resonance $\omega_{\mathrm{o}}=\omega_{m}$. Otherwise only one photon is coupled to the FP cavity while the other is rejected. (c,d) In the omni-resonant condition, pre-conditioning the incident light by introducing angular dispersion (AD) enables coupling the photon to a broadband achromatic resonance of the same FP cavity in (a,b). The omni-resonant transfer function now has a broad bandwidth $\Delta\omega\gg\delta\omega$, and may even exceed its FSR. (c) The single-photon and (d) the biphoton spectra are coupled in their entirety to the FP cavity.
• Figure 2: Transforming a conventional narrowband resonance into a broadband achromatic resonance in the omni-resonant configuration. (a-c) Three broadband field configurations incident on a planar FP cavity: (a) collimated light at normal incidence; (b) obliquely incident collimated light at an angle $\varphi$ with the cavity normal, whereupon the resonant wavelength blue-shifts; and (c) angularly dispersed light with wavelength-dependent incident angle $\varphi(\lambda)$. The cases (a, b) correspond to the conventional resonant condition, whereas (c) corresponds to the broadband omni-resonant condition. The first row is a schematic of the incident-field configuration. The second row depicts the wave vectors in the cavity, the axial components of which must match certain values to resonate. The third row is the single-photon spectral transfer function $T(\lambda)$ of width $\delta\lambda$ overlaid with the input spectrum of width $\Delta\lambda$ (dashed spectrum). In (c) the bandwidth of the achromatic resonance is broadened to coincide with $\Delta\lambda$. The fourth row is the two-photon spectral transfer function $T_{2}(\lambda_{\mathrm{s}},\lambda_{\mathrm{i}})=T(\lambda_{\mathrm{s}})T(\lambda_{\mathrm{i}})$, which is the portion of the biphoton spectrum that resonates with the cavity. The resonant wavelength $\lambda_{m}$ differs from the degenerate wavelength $\lambda_{\mathrm{o}}$ in (a), but the two frequencies coincide in (b). Overlaid with the two-photon spectral transmission is a typical biphoton frequency-anticorrelated spectrum for reference (the faint anti-diagonal linear feature).
• Figure 3: Realizing an achromatic resonance. (a) The angular dependence of the spectral transfer function $T(\lambda,\varphi)$ of the planar FP cavity used in our experiments, where $\varphi$ is the external incident angle with the cavity normal of a collimated field; see Fig. \ref{['Fig:OmniResConcept']}a,b. We plot $T(\lambda,\varphi)$ with a reduced finesse of $\mathcal{F}=10$ for clarity. (b) A single conventional resonance from (a) corresponding to $m=29$ is isolated, and we highlight the spectral range utilized for omni-resonance. (c) The optical system to introduce AD into the incident field. G: Diffraction grating; L$_{\mathrm{a}}$, L$_{\mathrm{b}}$: lenses of focal lengths 200 mm and 55 mm, respectively; C: planar FP cavity. (d) The spectral transfer function $T(\lambda,\psi)$ of the achromatic resonance after introducing AD into the conventional resonance in (b). Here $\psi$ (the cavity tilt angle) is the angle made by $\lambda=810$ nm with the cavity normal, corresponding to the dashed line in (c). The direction of this wavelength defines the optical axis. Other wavelengths travel at an angle $\Delta\varphi(\lambda)$ with the respect to this optical axis, thereby making an angle $\psi+\Delta\varphi(\lambda)$ with the cavity normal.
• Figure 4: Configuration for single-photon omni-resonance. (a) Schematic of the angular emission pattern from SPDC illustrating the signal and idler photons emitted on opposite sides of the emission cone; NLC: nonlinear crystal. (b) The spatiotemporal spectrum $\widetilde{G}^{(1)}(q_{\mathrm{s}},\lambda_{\mathrm{s}})$ for the signal photon. (c) The biphoton spectrum $\widetilde{G}^{(2)}(\lambda_{\mathrm{s}},\lambda_{\mathrm{i}})$ showing the frequency-anticorrelation structure, calculated for a 3-mm-long BBO crystal. A spectral uncertainty of $\delta\lambda\approx2.4$ nm is introduced into the frequency anticorrelation as expected from the finite spectral resolution of the measurement system; see Fig. \ref{['Fig:EntangledPhotonSetup']}c. (d) Schematic of the heralded measurements. (e) Setup for omni-resonance measurement configuration. L$_{1}$-L$_{5}$, L$_{\mathrm{a}}$, L$_{\mathrm{b}}$: Lenses; BPF: bandpass filter; BS: beam splitter; SMF: single-mode fiber; MMF: multimode fiber; D$_{1}$, D$_{2}$: single-photon detectors. (f) The measured single-photon spectrum $\widetilde{G}^{(1)}(\lambda_{\mathrm{s}})$ after removing the cavity from the setup.
• Figure 5: Demonstration of broadband single-photon omni-resonance in a planar FP cavity. (a) Measured singles $\widetilde{G}_{\mathrm{FP}}^{(1)}(\lambda_{\mathrm{s}},\varphi)$ and (b) heralded $\widetilde{G}_{\mathrm{FP}}^{(2)}(\lambda_{\mathrm{s}},\varphi)$ rates for the signal photon after traversing the FP cavity in the conventional resonant configuration. The FP cavity is placed before the pre-conditioning system in Fig. \ref{['Fig:heralded_setup']}e. A collimated single-photon state is incident on the FP cavity. The bottom panels are sections through the measurements at $\varphi=0^{\circ}$, $10^{\circ}$, and $58^{\circ}$. The dashed curves are measurements using a strong classical source acquired with an optical spectrum analyzer and not the measurement setup in Fig. \ref{['Fig:heralded_setup']}e, and are provided as a reference for comparison. (c) Measured singles $\widetilde{G}_{\mathrm{FP}}^{(1)}(\lambda_{\mathrm{s}},\psi)$ and (d) heralded $\widetilde{G}_{\mathrm{FP}}^{(2)}(\lambda_{\mathrm{s}},\psi)$ rates for the signal photon in the omni-resonant configuration. The FP cavity is placed after the pre-conditioning system as shown in Fig. \ref{['Fig:heralded_setup']}e, so that the single-photon state incidence on the FP cavity incorporates AD. The bottom panels are sections through the measurements at $\psi=0^{\circ}$, $10^{\circ}$, and $59^{\circ}$, the latter of which is the target omni-resonant condition.