Induced-Coherence Setup in Quantum Optics

• Induced-coherence is an interferometric architecture that uses spatially separated SPDC sources with engineered indistinguishability to create quantum interference.
• It leverages controlled path distinguishability and a complementarity relation (D² + V² = 1) to manipulate fringe visibility and quantify coherence.
• The setup underpins applications in quantum imaging, metrology with undetected photons, and contextuality tests, advancing quantum information processing.

Induced-Coherence (IC) Setup

Induced-coherence (IC) setups form a specialized class of interferometric architectures in quantum optics, fundamentally enabling interference between photons that originate in spatially separated spontaneous parametric down-conversion (SPDC) sources—typically nonlinear χ^2 crystals—whose outputs are rendered quantum-mechanically indistinguishable by the engineering of their undetected photon degrees of freedom. A defining property is the "wave–particle" complementarity relation governing the first-order coherence of the detected photons and the path distinguishability encoded in their elusive quantum origin. This paradigm is central to quantum imaging, metrology with undetected photons, and has implications in contextuality tests and quantum information.

1. Optical Architecture and Quantum State Construction

The canonical IC interferometer employs two identical nonlinear crystals, NLC₁ and NLC₂, each pumped by a mutually coherent continuous-wave laser split with precisely controlled relative phase $\varphi_{p_2}-\varphi_{p_1}$ (Machado et al., 2023). SPDC in either crystal yields a signal photon (s₁ or s₂, λ ≈ 810 nm) and an idler photon (i₁ or i₃, λ ≈ 1550 nm). The idler generated by NLC₁, i₁, is routed through a variable transmissivity element (amplitude transmission $t$, reflection $r$; $|t|^2+|r|^2=1$) and injected into NLC₂, ensuring the two SPDC events are indistinguishable in the idler degree of freedom. Signal outputs s₁ and s₂ are re-combined on a 50:50 beam splitter, and scanning the relative delay yields interference fringes.

In the weak-pump (low-gain) regime, neglecting vacuum, the joint state (Schrödinger picture) reads: $$|\Psi\> = \frac{1}{\sqrt2} \Big[ |1\>_{s_1}|0\>_{s_2}( r|1\>_{i_2}|0\>_{i_3} + t|0\>_{i_2}|1\>_{i_3}) + |0\>_{s_1}|1\>_{s_2}|0\>_{i_2}|1\>_{i_3} \Big]$$ The overlap $\<\Psi_1|\Psi_2\> = t$ quantifies the indistinguishability of the origins.

2. Path Distinguishability, Coherence, and Complementarity

Englert's trace distance formalism yields a path distinguishability metric: $D = \sqrt{1 - |\<\Psi_1|\Psi_2\>|^2} = \sqrt{1 - |t|^2} = |r|$ Maximum indistinguishability ($D = 0$) occurs at $|t|=1$. The first-order coherence between s₁ and s₂, measured via fringe visibility $V$,

$g^{(1)} = \frac{\<\hat a_{s_1}^\dagger \hat a_{s_2}\>}{\sqrt{\<\hat a_{s_1}^\dagger\hat a_{s_1}\> \<\hat a_{s_2}^\dagger\hat a_{s_2}\>}}, \qquad V = |g^{(1)}| = |t|$

enables the central complementarity relation: $D^2 + V^2 = 1$ This exact relation is not only valid in the single-photon regime but, as proven via Bogoliubov transformations in the high-gain (many-photon) regime, holds for arbitrary flux (Machado et al., 2023).

3. Experimental Realization and Measurement Protocols

Implementations use two χ^2 PPLN crystals pumped at 532 nm with dichroics separating s and i channels (Machado et al., 2023). The idler (i₁) passes through a variable transmission filter (controlling $|t|$), with joint detection and coincidence electronics (coincidence window $T_R \approx 2.5$ ns, coherence time $T_c \approx 580$ fs) monitoring the relevant second-order correlations

$g^{(2)}_{1,3}, \quad g^{(2)}_{2,3}$

allowing D to be experimentally obtained—even at high gain—via

$$D = \sqrt{ \frac{g^{(2)}_{2,3} - g^{(2)}_{1,3}}{g^{(2)}_{2,3} - 1} }$$

Signal fringes are fitted to extract V; in low flux, $V_{\max} \approx 0.86$ (modal overlap factor $\gamma \approx 0.86$). Across loss-tuned $|t|$, $D^2 + V^2$ remains unity, empirically verifying the complementarity (Machado et al., 2023).

4. Quantum Imaging, Sensing, and Extended IC Architectures

IC setups underpin quantum imaging with undetected photons ("IUP" imaging), where only the idler interacts with the sample and signal interference fringes reveal sample properties (Gemmell et al., 2023). In IC-IUP, only the idler from the first source seeds the second; in NI-IUP, both signal and idler are reinjected. Polarization quantum erasers with tunable waveplates control which-path marking and erase distinguishability, enabling hybrid sensing configurations. By adjusting beam-splitter angles and polarization, visibility can be maximized even under gain imbalance or loss, extending the tunability (Gemmell et al., 2023).

IC interferometry extends to high-gain regimes, enabling imaging protocols with optimized visibility via balanced gains or signal attenuation. In hybrid IC interferometers combining Mach–Zehnder (visible) and Michelson (infrared), spatial and spectral indistinguishability enable quantum-optical induced-coherence tomography (QICT), with axial and transverse resolution set by spectral bandwidth and focusing optics (Kim et al., 2023).

5. Contextuality and Nonclassicality Signatures

While classical models can mimic visibility in multi-photon regimes, genuine nonclassicality of IC is established through contextuality tests exceeding noncontextual hidden variable model (NCHV) bounds. In a three-crystal setup, the KCBS inequality is violated when single-pair generation and high path-identity ($t > 0.904$) are achieved, confirming quantum nonclassicality: $\kappa_{QM} = 2 + 1/9 > 2$ Only low-gain, high-indistinguishability regimes produce nonclassicality exceeding the KCBS bound (Shafiee et al., 2023).

6. Robustness Against Thermal Noise and Heralded Detection

Thermal photons or background noise degrade singles visibility, especially at low gain. The intrinsic interference contrast is recovered either by optimal attenuation to rebalance arm intensities or by extending the geometry to three-crystal configurations. Heralded detection—conditioning the signal interference on idler clicks—effectively projects out the thermal noise, restoring visibility independent of the mean thermal photon number. These procedures are essential for IC interferometry in thermally noisy bands (IR, THz, microwave) (Theerthagiri et al., 5 Nov 2025).

7. Fundamental Significance and Application Scope

The induced-coherence paradigm explicitly realizes quantum complementarity—coherence versus path distinguishability—in an experimentally tunable system. It enables phase-sensitive quantum metrology with undetected photons, practical quantum imaging across challenging spectral domains, hybrid interferometry, and contextuality investigations. The mechanism fundamentally relies on quantum indistinguishability in the unmeasured idler—vacuum states in SPDC, polarization erasure, or engineered overlap via optical elements.

Applications span undetected-photon interferometric imaging (Gemmell et al., 2023), tomography with sub-wavelength axial resolution (Kim et al., 2023), metrology surpassing shot-noise limits via squeezing and coherent seeding (Miller et al., 2019), contextuality-based nonclassicality verification (Shafiee et al., 2023), and robust phase sensing in high-noise environments (Theerthagiri et al., 5 Nov 2025). The ability to continuously tune between classical visibility and quantum coherence addresses wide-ranging requirements in next-generation quantum optics and information processing.