Heralded Induced-Coherence Interferometry in a Noisy Environment

TL;DR

This work analyzes induced-coherence interferometry in the Zou–Wang–Mandel geometry under thermal seeding. It derives closed-form expressions for singles intensities, first-order coherence, visibility, and SNR in both low- and high-gain regimes, revealing a thermal pedestal proportional to $(1-T)N_B$ that degrades contrast. Visibility can be recovered passively by attenuation or by a three-SPDC configuration, and is made robust against thermal noise through heralded detection, which projects measurements onto the two-photon subspace and eliminates uncorrelated background. The heralded scheme yields a visibility independent of $N_B$ in the ideal limit and offers a noise-resilient route to induced-coherence sensing in thermally bright environments, enabling imaging and sensing in mid-IR, THz, and microwave bands. Collectively, the results extend induced-coherence techniques to noisy settings and underscore heralding as a practical tool for robust quantum interferometry.

Abstract

Induced-coherence interferometry, first introduced in the Zou-Wang-Mandel (ZWM) setup, enables retrieval of object information from the interference pattern of light that never interacted with the object. This scheme relies on two identically correlated photon pairs and the absence of "which-way" information about the photons illuminating the object to induce coherence in their companions. In previous studies, the effect of thermal background on the ZWM interferometer was considered; here we explicitly include background noise and analyze the interference visibility in both low- and high-gain regimes, revealing how thermal photons introduce an incoherent offset that lowers the observed interference contrast. We show that the visibility can be restored either by optimal attenuation or by extending the geometry to a three-SPDC configuration. Furthermore, we demonstrate that introducing heralded detection removes the detrimental effect of thermal background noise, restoring high-contrast interference fringes.

Figures (5)

• Figure 1: Experimental setup for observing induced coherence. Two coherently pumped nonlinear crystals (A, B) generate signal ($\hat{a}_{1}^{1},\hat{a}_{2}^{1}$)-idler pairs ($\hat{a}_{3}^{1},\hat{a}_{3}^{3}$). The idler modes are aligned into a common path that includes a filter $S$ with a variable transmittance $T$ (modeled as a beam splitter) that both attenuates the idler and injects a thermal mode with a mean photon number $N_B$. The signal modes are combined at a beam splitter $S_1$ and detected at outputs F and G. The induced coherence in crystal B is thus controlled by attenuation and thermal injection in the shared idler channel.
• Figure 2: Singles visibility $\mathcal{V}$ versus idler transmittance $T$ for three thermal backgrounds $N_B\in\{0,10,100\}$ (blue, orange, green), computed from Eq. \ref{['eq:V_general_short']}. Each panel corresponds to the crystal gains: Low gain: (a) $V_A=V_B=0.1$, High gain: (b) $V_A=V_B=1$, (c) $V_A=V_B=10$, (d) $V_A=V_B=100$. For low gain the visibility increases monotonically with $T$ and approaches the vacuum-limit curve as $N_B\!\to\!0$. At high gain the $N_B=0$ curve exhibits a maximum at finite $T$ due to competition between the $\sqrt{T}$ scaling of the numerator and the $T V_A V_B$ term in the denominator of Eq. \ref{['eq:V_general_short']}; thermal backgrounds suppress visibility except near $T\!\approx\!1$, where the pedestal $(1-T)N_B V_B$ vanishes.
• Figure 3: (Color online) Singles visibility $\mathcal{V}$ versus idler transmittance $T$ for three configurations: two crystals (2-SPDC, solid blue), three crystals with an added source in the $A$-signal arm (3-SPDC, solid red), and 2-SPDC with an optimized attenuator in the $B$-signal arm (green dashed), computed from the analytic expressions in Sec. II and App. \ref{['apa']}. Panels show the effect of thermal background $N_B$ and High gain: (a) $V_A=V_B=V_C=10$; $N_B=0$,(b) $V_A=V_B=V_C=100$, $N_B=0$; (c) $V_A=V_B=V_C=10$, $N_B=10$; (d) $V_A=V_B=V_C=10$, $N_B=100$. Thermal injection suppresses the 2-SPDC visibility except near $T\!\to\!1$ where the pedestal $(1-T)N_BV_B$ vanishes. Rebalancing the signal powers—either by adding a third SPDC (3-SPDC) or by optimally attenuating the stronger arm—drives the singles visibility toward the first-order coherence bound $|g^{(1)}_{12}|=\sqrt{T(1+V_A)/(1+T V_A+(1-T)N_B)}$, restoring high contrast at large $T$ and in bright backgrounds.
• Figure 4: (Color online) Visibility $\mathcal{V}$ versus idler transmittance $T$ for three detection schemes: heralded (solid blue, Eq. \ref{['eq:Vherald-final']}), two-SPDC singles (solid red, Eq. \ref{['eq:V_general_short']}), and two-SPDC with optimal attenuation (green dashed, $|g^{(1)}_{12}|$). Low gain: (a) $V_A=V_B=0.1$, $N_B=10$: the thermal pedestal $(1-T)N_BV_B$ suppresses singles visibility over most of $T$, whereas heralding remains high and monotonic with $T$. High gain: (b) $V_A=V_B=10$, $N_B=10$: heralded visibility peaks at small $T$ and decreases slightly as stimulated imbalance ($\propto T V_A V_B$) grows.
• Figure 5: Log--log scaling of the photon–number–difference SNR versus idler transmittance $T$ in the low gain (a) and high gain (b). Blue: heralded SNR, independent of thermal background $N_B$ [Eq. \ref{['eq:SNR-herald-LB']}]. Colored curves: unconditional two-SPDC SNR for $N_B\!\in\!\{0,1,10,100\}$ [Eq. \ref{['eq:SNR_2SPDC_max']}]. For small $T$, all curves scale linearly with $T$; as $N_B$ increases, the unconditional SNR decreases while the heralded SNR remains constant. At high $T$, both converge as the thermal pedestal $(1-T)N_BV_B\!\to\!0$.