Induced Coherence Interferometer

• Induced Coherence Interferometer is a quantum optical device that uses indistinguishable SPDC paths to create high-visibility interference without photon overlap.
• The method involves precise alignment of nonlinear crystals and polarization quantum erasure to control interference and compensate for gain imbalances and losses.
• Experimental implementations enable interaction-free imaging and enhanced phase sensing by transferring information via undetected idler photons.

An induced coherence interferometer is a quantum optical device that exploits the indistinguishability of pathways in spontaneous parametric down-conversion (SPDC) to generate high-visibility interference using photons that have never themselves overlapped. Such interferometers enable imaging, phase sensing, and metrological applications with the detected photons carrying information imprinted via undetected modes. The operational principle leverages "induced coherence without induced emission," in which single-photon interference arises due to indistinguishability in shared quantum vacuum modes, while the system remains in the quantum (low-gain) regime (Gemmell et al., 2023, Hochrainer et al., 2016).

1. Fundamental Principle and Canonical Architectures

The canonical induced coherence interferometer employs two sequential or parallel nonlinear crystals, each pumped coherently. Each SPDC event produces a pair of photons: a signal (s) and an idler (i). The idler from the first crystal is spatially, spectrally, and temporally aligned to seed the second crystal, rendering the idler pathways indistinguishable. The two signal modes, which have never physically interacted, can then interfere at a beam splitter. Critically, the idler is not detected; instead, interference in the signal photons encodes changes in the idler channel.

Two principal operational modes exist:

• Induced Coherence Interferometry with Undetected Photons (IC-IUP): Only the idler from the first source is routed through the second, so only indirect coupling occurs.
• Nonlinear Interferometry with Undetected Photons (NI-IUP): Both signal and idler from the first SPDC process are directed through the second, allowing for direct and indirect pathway superpositions.

The overall state, in first-order perturbation theory, is

$$|\psi\rangle \approx |0\rangle + \xi_A e^{i\varphi_A}|1_{s}^A, 1_{i}^A\rangle + \xi_B e^{i\varphi_B}|1_{s}^B, 1_{i}^B\rangle$$

where $\xi_{A,B}$ are small SPDC gains. Signal interference arises solely when the indistinguishable idler pathways erase which-way information (Gemmell et al., 2023).

2. Quantum Optical Theory and Signal Interference

The generated state combines contributions from both SPDC passes, with the signal detection rate following

$$R_s(\phi, \theta) = R_0(\theta_2) + A(\theta_1,\theta_2)\cos[\phi_i + \delta(\theta_1,\theta_2)]$$

where $A(\theta_1, \theta_2)$ is the interference amplitude and $R_0$ collects background terms.

Signal visibility in the IC and NI modes is given by

$$V_\mathrm{IC}(\theta_2) = \frac{|\sin\theta_2\cos\theta_2|}{1/2+\sin^2\theta_2} \qquad V_\mathrm{NI}(\theta_2) = \frac{\cos^2\theta_2}{1/2+\cos^2\theta_2}$$

for ideal balanced gain ($\xi_A=\xi_B$) and no loss, which yields $V\to1$ in the respective maximized polarizations (Gemmell et al., 2023).

A practical induced coherence interferometer must address residual which-path information (via loss or imperfect indistinguishability), which directly reduces observed visibility. The complementarity between path distinguishability $\mathcal{D}$ and interference visibility $\mathcal{V}$ can be captured at all gain regimes via

$\mathcal{D}^2 + [g_{12}^{(1)}]^2 = 1$

where $g_{12}^{(1)}$ is the normalized first-order coherence between the two signal modes (Machado et al., 2023).

3. Control and Optimization: Quantum Erasure and Balancing

Experimental realization involves both state-preparation and mode-matching optimizations.

• Polarization Quantum Erasure: By introducing quarter- and half-wave plates at controlled angles $(\theta_1, \theta_2)$, the overlap of polarization modes in the signal arm can be tuned, providing continuous variation between pure IC and pure NI operation. The optimal erasure angle for maximizing visibility is given analytically as $\theta_2 = \arctan(\xi_B/\xi_A)$, compensating gain imbalance between passes (Gemmell et al., 2023).
• Attenuation and Gain Engineering: Direct attenuation (variable filter in the idler arm) or optimization of SPDC gains ($\xi_A$, $\xi_B$) can balance the interference amplitudes, maximizing coherent signal even in the presence of inherent loss or modal mismatch (Gemmell et al., 2023, Theerthagiri et al., 5 Nov 2025).

The quantum eraser modality uniquely enables dynamic optimization not available through linear interferometry, especially in the presence of non-unit transmission $T$ or unbalanced gain.

4. Experimental Implementations and Metrological Implications

Realizations of the induced coherence interferometer span retro-reflected two-crystal setups, double-pass SPDC loops, and polarization-based geometries. Experimentally, the induced coherence effect manifests as interference in the detected signal photons—whose visibility is a direct probe of the idler path integrity and indistinguishability.

Key experimental attributes include:

• Fringe Visibility: Maximal IC-IUP visibility of $V_\mathrm{IC}(45^\circ)=0.80\pm0.02$ (Gemmell et al., 2023).
• Coherence Envelope: Measured Fourier width matches the predicted coherence length $L_\mathrm{coh}=0.20\pm0.01\,\mathrm{mm}$.
• Loss Impact: Idler path attenuation $T$ observed at $0.25\pm0.01$ significantly limits visibility but can be compensated by erasure/balancing.
• Spatial and Spectral Control: Alignment of idler with spatial and spectral precision (via 4f systems and dichroics) is crucial for achieving high visibility (Hochrainer et al., 2016).

Interference fringes can be controlled entirely by phase or alignment changes in the undetected idler, enabling interaction-free imaging and metrology using visible photons while probing with IR or mid-IR undetected modes.

5. Distinction from Stimulated Coherence and Classical Analogues

Induced coherence is distinct from stimulated coherence produced by external seeding. In the spontaneous (vacuum-induced) regime, the phase relation between signal and idler fields is random; first-order coherence emerges solely due to vacuum indistinguishability. In the presence of an external seed, near-unity visibility can be achieved via phase-locked classical fields, but this is not true induced coherence (Heuer et al., 2015).

A defining feature is the potential for high-visibility single-photon interference without any coincidence counting, a capability not accessible in classical division-of-amplitude interferometers, which require direct detection and cannot transfer image or phase information between disjoint spectral regimes (Hochrainer et al., 2016).

6. Applications and Extensions

Induced coherence interferometry underpins a variety of quantum-enhanced sensing and imaging modalities:

• Quantum Imaging with Undetected Photons (IUP): Enables imaging and phase retrieval using signal detection at a wavelength where detectors are optimal, while the object is probed at an entirely different, potentially undetectable frequency (Gemmell et al., 2023).
• Interaction-Free Sensing: Absorptive or phase-shifting samples placed in the idler path modulate the signal interference pattern, facilitating interaction-free spectroscopy and imaging, including at mid-IR or THz frequencies (Hochrainer et al., 2016).
• Quantum Optical Tomography: Hybrid interferometers incorporating Mach-Zehnder (signal) and Michelson (idler) architectures facilitate time-domain and frequency-domain tomographic readouts of multi-layer samples with en face and depth profiling capabilities (Kim et al., 2023).
• Quantum-Enhanced Metrology: Multi-photon variants (e.g., two-photon induced coherence) have demonstrated phase amplification beyond the single-photon regime, opening the route to Heisenberg-limited precision in phase estimation using undetected photons (Im et al., 10 Oct 2025).

Heralded induced coherence schemes can be leveraged to suppress thermal noise and background, maintaining high-visibility fringes in the targeted signal channel even in challenging environments (Theerthagiri et al., 5 Nov 2025).

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References

• (Gemmell et al., 2023, Hochrainer et al., 2016, Machado et al., 2023, Theerthagiri et al., 5 Nov 2025, Heuer et al., 2015, Im et al., 10 Oct 2025, Kim et al., 2023)