Induced Coherence: Mechanisms & Applications

• Induced Coherence is a phenomenon where coherence between indirectly coupled photonic modes is generated via shared interaction channels, such as common vacuum modes or indistinguishable pathways.
• The mechanism is implemented in SPDC setups using two nonlinear crystals where path identity erases which-path information, resulting in high-visibility interference essential for advanced imaging.
• Experimental applications in quantum imaging, tomography, and cavity-QED demonstrate how controlled coherence can enhance measurement sensitivity and nonclassical state investigations.

Induced coherence denotes the generation of coherence between states, modes, or outputs that do not couple directly, but become phase-related through a shared interaction channel, a common vacuum mode, or indistinguishable alternative pathways. In the literature, the term is used most prominently for induced coherence without induced emission in spontaneous parametric down-conversion (SPDC), following the Zou–Wang–Mandel configuration, where two coherently pumped nonlinear crystals produce mutually coherent signal beams when their idler modes are made indistinguishable. Related usages also appear in ultrafast atomic control and cavity QED, where coherence is generated indirectly through a common excited state or a common vacuum-mediated decay channel (Kolobov et al., 2017, Sarma et al., 2011, Vafafard et al., 2017).

1. Canonical induced-coherence architecture

The canonical optical scheme employs two coherently pumped nonlinear crystals. Each crystal can generate a signal–idler pair by SPDC, and the key experimental operation is to align the idler output of the first crystal with the idler mode associated with the second crystal. In the path-identity formulation, this is expressed as

$$\hat a_{i_1}(\vec k_{i_1})=\hat a_{i_2}(\vec k_{i_2})=\hat a_i(\vec k_i).$$

When this condition is met, which-crystal information is erased, and the signal outputs interfere although the signal photons themselves do not directly interact (Shafiee et al., 2023).

A standard single-mode description uses Bogoliubov transformations for each crystal. For the first crystal,

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$

with $|U|^2-|V|^2=1$. If the idler then passes through a lossy element modeling a sample,

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$

and the injected idler becomes indistinguishable from the idler associated with the second crystal, the second signal output carries information about what happened to the idler between the two crystals (Vallés et al., 2017).

This architecture underlies the broader program of imaging with undetected photons, in which the sample acts on the idler path while interference is measured only in the signal path. In one formulation, crystal A emits into signal mode $1'$ and idler mode $3'$; the transmitted idler from A seeds crystal B, which emits into signal mode $2'$; the two signal modes are then recombined on a 50:50 beam splitter and detected. The object or filter in the idler path controls how much which-crystal information remains and therefore controls how much signal coherence is induced (Kolobov et al., 2017).

2. Quantitative description: first-order coherence, visibility, and distinguishability

The induced-coherence observable is usually a first-order coherence between the two signal modes. In induced-coherence tomography, the normalized first-order coherence is

$$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$

and for vacuum inputs and general gain,

$$|g_{s_1,s_2}^{(1)}|=|\mu| \sqrt{\frac{1+|V|^2}{1+|\mu|^2|V|^2}}.$$

In the experimentally relevant low-gain regime, $|V|\ll1$, this simplifies to

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$0

so the sample transmissivity amplitude directly sets the degree of mutual coherence between the signal beams (Vallés et al., 2017).

A closely related formulation for quantum imaging uses

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$1

The corresponding detected photon numbers after the final beam splitter are

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$2

with visibility

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$3

In the low-gain regime,

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$4

whereas in higher-gain operation visibility and coherence separate because the two interfering signal beams can become unequally populated (Kolobov et al., 2017).

A complementary formulation uses a distinguishability parameter $$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$5 that quantifies how well one can tell which nonlinear crystal generated the pair. In the low-gain regime,

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$6

and, more generally,

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$7

This equality was derived for arbitrary parametric gain, establishing that induced-coherence interferometers admit an exact complementarity relation beyond the single-photon regime (Machado et al., 2023).

3. Imaging, tomography, and sensing with undetected photons

The most developed application class is imaging with undetected photons, where the idler probes the object and the signal is detected. In induced coherence tomography, axial sectioning is not obtained by directly measuring a sample-reflected reference beam; instead, the sample changes the first-order coherence between two signal beams generated in a nonlinear interferometer. The axial localization follows from the multimode correlation function

$$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$8

so coherence peaks only when the path delay matches the sample-dependent idler delay, and the axial resolution is set by $$a_{s_1}=U b_{s}+V b_{i}^{\dagger}, \qquad a_{i_1}=U b_{i}+V b_{s}^{\dagger},$$9 (Vallés et al., 2017).

A proof-of-concept implementation used a 532 nm CW pump and two 20-mm-long PPLN crystals, each producing signal photons at about 810 nm and idler photons at about 1550 nm. The measured idler spectrum was centered at 1552.3 nm with FWHM bandwidth of about 1.6 nm. The reported axial resolution was about 500 $|U|^2-|V|^2=1$0m, intentionally not optimized, and the analysis notes that broader source bandwidth would improve sectioning, with submicron axial resolution in principle achievable. The same experiment demonstrated visibility up to about 90% after filtering the signal photons with a fiber Bragg grating around 809.4 nm to improve spectral indistinguishability (Vallés et al., 2017).

A related line of work analyzes how induced coherence can be controlled for quantum imaging. In the two-crystal PDC interferometer, the object transmittance $|U|^2-|V|^2=1$1 in the idler path sets the induced coherence, while gain imbalance can reduce the observed fringe visibility. An explicit optimization chooses

$|U|^2-|V|^2=1$2

which equalizes the signal intensities before interference and yields

$|U|^2-|V|^2=1$3

In the high-gain limit, the optimized visibility approaches unity (Kolobov et al., 2017).

The scheme has also been generalized by polarization-state quantum erasure to interpolate continuously between IC-IUP and NI-IUP operation. In a retro-reflected implementation using a 1064 nm pump, signal at 1550 nm, and idler at 3.4 $|U|^2-|V|^2=1$4, an internal QWP and external HWP control how much which-way information is retained or erased. Fine and coarse scans of the idler mirror reveal two distinct interference regions associated with NI-IUP and IC-IUP, and the measured signal count-rate maps and visibility curves agree well with the low-gain theory (Gemmell et al., 2023).

4. Vacuum fields, complementarity, and nonclassicality

A central interpretive question is whether induced coherence is best understood as an interference effect, a vacuum-field effect, or a genuinely nonclassical resource. In a three-crystal SPDC experiment with BBO1, BBO2, and BBO3, induced coherence appears when BBO1 and BBO2 share the same idler vacuum mode $|U|^2-|V|^2=1$5. Opening an additional photon channel with BBO3 introduces which-path information and adds an incoherent background term to the single-photon count,

$|U|^2-|V|^2=1$6

reducing signal-only visibility from about 80% to about 53% in the experiment. Yet coincidence counting remains nearly perfectly visible because the full biphoton amplitudes still interfere coherently (Heuer et al., 2014).

This behavior is commonly summarized by the complementarity relation

$|U|^2-|V|^2=1$7

or, more generally with partial pump incoherence,

$|U|^2-|V|^2=1$8

In the induced-coherence picture, first-order signal interference is high when idler modes are indistinguishable and which-path knowledge is inaccessible; it is reduced when an additional channel can in principle reveal the origin of the signal photon (Heuer et al., 2015).

The distinction between induced and stimulated coherence sharpens this point. With aligned idler modes but no external seed, the vacuum fields participating in SPDC establish the phase relation. With a He-Ne laser at 632.8 nm seeding the idler, the stimulating field dominates and the observed visibilities are often above 90%. A variable transmission filter of amplitude transmission $|U|^2-|V|^2=1$9 reveals a characteristic difference: $a_{i_1} \Longrightarrow \mu a_{i_1}+f,$0 for induced coherence, but

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$1

for stimulated coherence. In the induced case, the filter changes the effective vacuum-field correlation between the two idler modes rather than merely attenuating a classical field amplitude (Heuer et al., 2015).

Whether such observations alone certify nonclassicality has been disputed. One response is a contextuality-based criterion: in an extended three-crystal ZWM configuration, a KCBS witness,

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$2

takes the low-gain form

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$3

The analysis states that the violation disappears for

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$4

so strong path identity is required for incompatibility with noncontextual hidden-variable models (Shafiee et al., 2023).

A distinct delayed-choice proposal advances a different interpretation: using Poisson-distributed coherent light, linear optics, and gated heterodyne detection, it claims that selective coincidence measurement can induce an effective polarization-path correlation with coincidence fringe

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$5

while the local outputs remain basis-random and fringe-free (Ham, 2023). This suggests that the boundary between induced coherence, measurement-induced selection, and broader nonlocal-correlation language remains an active conceptual issue.

5. Gain engineering, bright operation, and noisy environments

Induced-coherence interferometers are not restricted to spontaneous, low-flux operation. In a bright, coherently seeded Mandel-type interferometer, three modes are used: signal mode $a_{i_1} \Longrightarrow \mu a_{i_1}+f,$6, which passes through the sample; idler mode $a_{i_1} \Longrightarrow \mu a_{i_1}+f,$7 from the first nonlinear interaction; and idler mode $a_{i_1} \Longrightarrow \mu a_{i_1}+f,$8 from the second. The total transformation is written as

$a_{i_1} \Longrightarrow \mu a_{i_1}+f,$9

and the phase sensitivity is evaluated through

$1'$0

The analysis finds below shot noise scaling in the fully squeezed, coherent-seeded regime, including when seeding the undetected mode, and emphasizes that the second gain is often optimal at a relatively low value rather than being maximized (Miller et al., 2019).

For coherent seeding in mode $1'$1 and intensity-difference detection between $1'$2 and $1'$3, the reported expression is

$1'$4

The same work stresses that subtracted intensity measurements offer a practical advantage because common-mode noise cancels, and that the super-sensitive effect does not appear within the spontaneous-parametric-down-conversion approximation (Miller et al., 2019).

Noise in the undetected idler arm can strongly degrade the observable interference. In a noisy ZWM interferometer with thermal background $1'$5, the low-gain singles intensities become

$1'$6

so the term

$1'$7

acts as an incoherent thermal pedestal. The corresponding visibility is reduced by this background contribution (Theerthagiri et al., 5 Nov 2025).

Two remedies are analyzed. The first is optimal attenuation, which balances the arms and recovers the coherence-bound visibility. The second is a three-SPDC configuration, which balances the intensities intrinsically. The most decisive remedy is heralded detection, for which the heralded signal intensity is

$1'$8

The thermal pedestal is absent, and the heralded visibility and SNR become independent of $1'$9. The analysis interprets heralding as a projection onto the correlated pair sector rather than as the creation of new coherence (Theerthagiri et al., 5 Nov 2025).

6. Atomic and cavity-QED realizations

Outside SPDC interferometry, induced coherence also denotes coherence generation in multilevel quantum systems through a common intermediate state or common vacuum-mediated decay channel. In a three-level $3'$0-type atomic configuration, with lower states $3'$1, $3'$2 and excited state $3'$3, two nonlinearly chirped few-cycle pulses

$3'$4

drive the dipole-allowed transitions $3'$5 and $3'$6. Solving the density-matrix equations without invoking the rotating wave approximation, the scheme generates the lower-state coherence $3'$7; at optimum parameters,

$3'$8

while $3'$9 and $2'$0 and $2'$1 become equal, corresponding to a maximally coherent ground-state superposition (Sarma et al., 2011).

The same $2'$2-scheme is interpreted through bright and dark states. Population transfer from the bright state to the dark state gives coherent population trapping, and the depletion of the excited state together with the buildup of $2'$3 is described as electromagnetically induced transparency for the resonant $2'$4 transition. By changing the pulse parameters to

$2'$5

the system instead ends nearly entirely in the initially empty ground state $2'$6, demonstrating electromagnetically induced population transfer. A contour scan further shows near-maximal coherence over a broad range of nonlinear chirp parameter and peak Rabi frequency, and the authors state that the effect is fairly insensitive to changes in pulse duration and $2'$7 as well (Sarma et al., 2011).

A related but distinct mechanism appears in vacuum-induced coherence in cavity QED. There, a three-level V-type system with excited states $2'$8, $2'$9 and ground state $$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$0 is strongly coupled to a leaky cavity, so both excited states decay through the same cavity vacuum mode. In the symmetric/antisymmetric basis,

$$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$1

the state $$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$2 is dark to the cavity field. For $$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$3, the long-time populations satisfy

$$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$4

demonstrating trapped population and steady-state coherence created by interference between two vacuum-mediated decay channels (Vafafard et al., 2017).

Taken together, these atomic and cavity-QED examples show that induced coherence is not confined to photon-pair interferometry. The common structural element is indirect coherence generation through a shared mediating channel—an excited state in the $$g_{s_1,s_2}^{(1)}=\frac{\langle a_{s_1}^{\dagger} a_{s_2}\rangle}{\sqrt{\langle a_{s_1}^{\dagger} a_{s_1} \rangle}\sqrt{\langle a_{s_2}^{\dagger} a_{s_2} \rangle}},$$5 system, or a common cavity vacuum mode in the V-type system—together with the emergence of bright/dark-state structure, population trapping, and controllable steady-state coherence (Sarma et al., 2011, Vafafard et al., 2017).