Coherence de Broglie Wavelength (CBW)

• Coherence de Broglie wavelength (CBW) is an effective reduced interference wavelength generated by deterministic phase control and coherent coupling in Mach–Zehnder interferometers.
• CBW employs various architectures—such as cross-coupled and asymmetrically coupled MZIs—to achieve phase multiplication and superresolved fringes scaling as λ₀/N or λ₀/2.
• Experimental results confirm CBW’s high fringe visibility and robustness against photon loss, underpinning its potential for precision metrology and advanced optical sensing.

Coherence de Broglie wavelength (CBW) denotes, within a specific interferometric literature, an effective reduced interference wavelength generated by coherently coupled Mach–Zehnder interferometers (MZIs) rather than by multiphoton entanglement. It is presented as a coherence-optics analogue or extension of photonic de Broglie waves (PBWs): whereas PBW is conventionally written as $\lambda_B=\lambda_0/N$ for an $N$-photon entangled state, CBW is obtained by deterministic phase control, path indistinguishability, and ordered superposition in coupled interferometers fed by coherent light (Ham, 2020). Later work reformulates the same idea as the $N$-th or $M$-th power of a single-MZI unitary, so that the output depends on $N\phi$ and exhibits superresolved fringes with an effective scale $\lambda_0/N$ (Ham, 23 Feb 2026, Ham, 5 Mar 2026).

1. Conceptual definition and scope

The foundational CBW papers define the phenomenon as a coherence-optics realization of photonic de Broglie-wave behavior using interferometric phase control in Mach–Zehnder-based networks, rather than multiphoton entanglement from SPDC (Ham, 2020). In that formulation, the reduced effective wavelength is attributed to perfect mutual coherence, path indistinguishability, and controlled phase relations between superposed fields. The input is a coherent field $E_0$, and the effect is claimed to arise without entangled input states.

This literature consistently distinguishes CBW from the ordinary de Broglie wavelength $\lambda=h/p$ of a massive particle and from PBW based on N00N-like states. In the CBW setting, “de Broglie wavelength” refers to an effective interferometric wavelength inferred from fringe compression, not to the kinematic wavelength of a free particle. The strongest shared claim is that a coupled interferometer can produce a response equivalent to phase multiplication, yielding fringes narrower than those of a single MZI (Ham, 2020, Ham, 2021).

The same body of work also presents CBW as both a reinterpretation and an extension of PBW. In the 2020–2021 papers, the language alternates between “classical-coherence analogue,” “coherence version of PBW,” and “wave-nature-based analogue of photonic de Broglie waves,” while later 2026 analyses recast the effect in fully unitary SU(2) terms and emphasize first-order intensity readout, loss tolerance, and compatibility with coherence optics (Kim et al., 2021, Ham, 23 Feb 2026).

2. Architectures and order conventions

The basic hardware throughout the CBW literature is a network of 50:50 beam splitters, phase shifters, and coupled MZIs. The earliest formulation uses a cross-coupled double MZI (CCD-MZI), in which the outputs of a first MZI block $D$ feed a second block $D'$ with asymmetric phase placement (Ham, 2020). The first experimental observation uses an asymmetrically coupled double MZI (ACD-MZI), controlled by AOM-induced phase offsets and an intermediate coupling phase $N$0 (Ham, 2020). Later papers generalize the construction to anti-symmetrically coupled MZI chains, where a dummy MZI or a Sagnac-integrated round-trip geometry preserves the path basis required to realize the $N$1-th power of a single-MZI unitary (Ham, 23 Feb 2026, Ham, 5 Mar 2026).

The literature does not use a single normalization for “order,” and the reported scaling laws differ across papers. The difference is architectural rather than rhetorical: some papers count basic CCD-MZI blocks, some count coupled modules, and some count the power of the MZI unitary itself.

Paper(s)	Architecture / order definition	Reported scaling
(Ham, 2020)	CCD-MZI; $N$2 cascaded CCD-MZI blocks	$N$3
(Ham, 2020)	ACD-MZI; $N$4 asymmetrically coupled MZIs	$N$5
(Kim et al., 2021)	$N$6 CBW modules $N$7 and $N$8	$N$9
(Ham, 23 Feb 2026, Ham, 5 Mar 2026)	$N$0- or $N$1-th power of $N$2	$N$3 or $N$4
(Kim et al., 12 Mar 2026)	Scalable coupled-MZI chain, experimentally $N$5	$N$6

This suggests that CBW is best read as a family of closely related phase-multiplication constructions rather than a single fixed normalization. Across these variants, the invariant content is the same: coherent coupling changes the interferometric response from $N$7 to a higher-harmonic dependence such as $N$8, $N$9, or more generally $M$0.

3. Mathematical formulation and scaling laws

In the initial CCD-MZI treatment, a first MZI fed by $M$1 yields the standard outputs

$M$2

and the second-order correlation

$M$3

After the second, asymmetrically phased MZI block, the outputs become

$M$4

with

$M$5

For a serial chain of $M$6 identical CCD-MZI blocks, the same paper gives

$M$7

and

$M$8

which it interprets as the general CBW scaling law (Ham, 2020).

The 2020 observation paper shifts the emphasis from second-order correlation to first-order interference in a coupled-MZI geometry. Under the CBW condition $M$9 and $N\phi$0, the outputs satisfy

$N\phi$1

so the modulation frequency doubles relative to a single MZI (Ham, 2020). The 2021 attenuated-laser study uses a closely related coupled system and reports

$N\phi$2

under the asymmetric condition $N\phi$3, interpreting this as $N\phi$4 for the demonstrated case (Kim et al., 2021).

The later quantum-mechanical analyses are more compact. They write the single-MZI unitary in SU(2) form and impose an anti-symmetrically coupled architecture such that

$N\phi$5

For input $N\phi$6, the output becomes

$N\phi$7

so the detection probabilities are

$N\phi$8

The 2026 experimental paper states the same result as

$N\phi$9

It also gives a phase basis

$\lambda_0/N$0

which formalizes the superresolved phase structure (Ham, 23 Feb 2026, Kim et al., 12 Mar 2026).

4. Experimental demonstrations

The first reported observation used a coherent laser with $\lambda_0/N$1, four synchronized AOMs, and $\lambda_0/N$2. A single MZI produced a $\lambda_0/N$3 modulation, whereas the ACD-MZI under CBW conditions produced a $\lambda_0/N$4 modulation and complementary outputs $\lambda_0/N$5 and $\lambda_0/N$6, interpreted as fringe-period halving and $\lambda_0/N$7 for the demonstrated case (Ham, 2020).

A subsequent experiment used a $\lambda_0/N$8 CW laser attenuated by ND filters with $\lambda_0/N$9 to reach $E_0$0. In that setup, the single-MZI reference fringes had visibilities $E_0$1 and $E_0$2 in coincidence, while the CBW outputs under $E_0$3 had $E_0$4, $E_0$5, and $E_0$6, again with doubled phase modulation corresponding to $E_0$7 (Kim et al., 2021).

The 2026 scalable demonstration extended the architecture to $E_0$8 and reported near-perfect fringe visibility in both single-photon and CW regimes. In the single-photon case, the quoted visibilities were $E_0$9 and $\lambda=h/p$0 for $\lambda=h/p$1, $\lambda=h/p$2 and $\lambda=h/p$3 for $\lambda=h/p$4, and $\lambda=h/p$5 and $\lambda=h/p$6 for $\lambda=h/p$7. In the CW case, the reported values remained in the $\lambda=h/p$8 to $\lambda=h/p$9 range. The same paper states that the observed CBWs are invariant to photon loss in measurements, although the loss-dependent data are not shown in the supplied extract (Kim et al., 12 Mar 2026).

A parallel 2026 wavemetry paper demonstrated the $D$0 case in a Sagnac-integrated round-trip MZI using a $D$1 He-Ne laser, with a doubled fringe density relative to the reference interferogram. That proof-of-principle experiment validated the $D$2-fold phase-compression design in a compact architecture intended for wavemetry and metrology (Ham, 5 Mar 2026).

5. Metrological uses and limit claims

From the outset, the CBW papers place the phenomenon in coherence-quantum metrology. The 2020 deterministic-control paper lists high-precision optical spectroscopy, optical clocks, gravitational-wave detection, quantum lithography, quantum sensors, and secure communications or classical key distribution as target applications, with the metrological promise attributed to the shortened effective wavelength $D$3 and its finer fringe structure (Ham, 2020).

A 2021 proposal embeds CBW in a quantum ring gyroscope. There the coupled interferometer is mapped onto a ring cavity with a beam splitter, mirrors, two PZTs, and a Sagnac phase $D$4. Under $D$5, the output fields take the form

$D$6

so the effective wavelength is $D$7. The proposal interprets the narrower resonance structure as enhanced gyroscope phase resolution while preserving reciprocal-noise cancellation characteristic of ring geometries (Ham, 2021).

The 2026 unitary analyses formalize the same metrological intuition through Fisher information. Modeling the fringe as $D$8, they obtain an $D$9 enhancement of the Fisher information with respect to the phase parameter and a Cramér–Rao bound that scales as $D'$0 before full resource accounting (Ham, 23 Feb 2026). The wavemetry paper makes the same point for spatial fringe frequency in a Fizeau-type geometry and argues that multiplied fringe density improves wavelength estimation while remaining compatible with standard coherence optics (Ham, 5 Mar 2026).

At the same time, the strongest “quantum advantage” claims are progressively narrowed in later work. The gyroscope paper says the method can “overcome the Heisenberg photon-phase relation,” but its own discussion already notes that this is better read as transfer-function engineering via $D'$1 rather than as a literal violation of quantum mechanics (Ham, 2021). The 2026 analyses are explicit that CBW does not attain the Heisenberg limit and remains within the shot-noise limit, even while offering superresolution, enhanced practical sensitivity, near-unity visibility, and reduced vulnerability to photon loss relative to high-$D'$2 N00N-state protocols (Ham, 23 Feb 2026, Kim et al., 12 Mar 2026).

6. Interpretation, controversy, and relation to broader de Broglie/coherence usage

CBW belongs to a specialized interferometric usage of “de Broglie wavelength,” and the broader de Broglie literature uses the same vocabulary differently. In the covariant relativistic account of the ordinary de Broglie relation, the fundamental statement is

$D'$3

with the familiar $D'$4 arising as the spatial component and the invariant phase $D'$5 enforcing Lorentz consistency (Soltau, 5 Aug 2025). That is a statement about particle four-momentum and wave four-vector, not about effective fringe compression in coupled optical networks.

Elsewhere, “coherence” linked to de Broglie wavelength refers to different physical questions. In nonrelativistic scattering from a composite target, coherent scattering occurs when the incident de Broglie wavelength satisfies $D'$6, so that the projectile does not resolve internal structure and amplitudes add coherently (Gasbarri et al., 2015). In low-energy electron interferometry, coherence is characterized by transverse fringe contrast and longitudinal coherence length; one measured source had $D'$7 at $D'$8, $D'$9 at $N$00, and a longitudinal coherence length of $N$01, showing explicitly that de Broglie wavelength and coherence length are distinct quantities (Pooch et al., 2017). In de Broglie–Mackinnon packet optics, the relevant scale is an intrinsic localization length $N$02 for $N$03 or $N$04 for $N$05, which the authors do not identify as a standard coherence length (Hall et al., 2023).

These comparisons indicate that CBW refers to a particular interferometric construction: an effective reduced fringe wavelength produced by coherently coupled interferometers. It is not the ordinary $N$06, not a generic coherence length, and not the only meaning of “coherent de Broglie” in the wider matter-wave literature. Within its own line of development, however, CBW has a stable core meaning: deterministic phase multiplication in coupled MZI architectures that yields PBW-like superresolution without requiring high-order entangled input states (Ham, 2020, Kim et al., 12 Mar 2026).