Path-Imbalanced Mach–Zehnder Interferometers

• Path-imbalanced Mach–Zehnder interferometers are optical systems with deliberately unequal arm lengths that create temporal delays and energy-dependent phase shifts.
• They enable key functionalities across fiber optics, integrated photonics, and quantum devices by harnessing imbalance for controlled interference patterns and spectral tuning.
• Design challenges include maintaining high interference visibility and phase sensitivity while compensating for fabrication nonidealities and environmental drifts.

Path-imbalanced Mach–Zehnder interferometers are interferometric systems in which the two interfering trajectories are intentionally or effectively unequal in optical length, propagation delay, enclosed area, or readout balance. Across fiber optics, photonic integration, silicon photonics, quantum Hall electronics, thermoelectric transport, and quantum metrology, this imbalance is not a mere fabrication defect: it is often the operative resource that creates temporal separation, energy-dependent phase accumulation, tunable spectral periodicity, or multi-path interference. In parallel, several closely related literatures use “unbalanced” in a distinct sense to denote non-$50{:}50$ beam splitters rather than unequal arm lengths, so the term must be interpreted with care in context (Xavier et al., 2012, Emadi et al., 27 Mar 2026, Song et al., 10 Aug 2025, Mishra et al., 2022).

1. Core definitions and physical regimes

The basic interference mechanism can be written as

$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$

with

$$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$

In this form, path imbalance enters through the delay $\tau$, which converts an arm-length difference into an energy-dependent phase. The same logic recurs in optical and electronic implementations, although the experimentally relevant control variable may be fiber delay, waveguide-length mismatch, gate-defined loop area, or edge-state geometry (Hofer et al., 2015).

A second important regime is the Franson condition

$\tau_c \ll \Delta t \ll \tau_p,$

where $\tau_c$ is the single-photon coherence time, $\tau_p$ is the pump coherence time, and $\Delta t$ is the interferometer delay. In this regime, single-photon interference is suppressed while two-photon short-short and long-long amplitudes remain indistinguishable, enabling Franson interference in matched unbalanced analyzers (Emadi et al., 27 Mar 2026).

A third usage appears in phase-estimation theory: an interferometer is called unbalanced when at least one beam splitter is not $50{:}50$, namely when

$$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$

This is conceptually distinct from unequal path lengths, although both forms of imbalance alter visibility and phase sensitivity (Mishra et al., 2022).

Platform	Imbalance mechanism	Reported consequence
GV95 fiber MZI	$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$0 m delay, about $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$1 ns	Orthogonal-state QKD over $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$2 km fiber
PIC Franson analyzer	On-chip short and long arms, $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$3 ns	Single-photon interference suppressed, two-photon interference enabled
Silicon SOI MZI	$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$4	FSR $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$5 nm
Graphene quantum Hall MZI	Asymmetric gate potentials shift channel positions	Loop area and oscillation frequency change
QHE edge-state MZI	Magnetic-field-dependent IES rearrangement	Effective paths need not match lithography

These cases show that “path imbalance” spans at least three non-identical but related notions: temporal delay between arms, geometrical inequality of interfering trajectories, and readout asymmetry produced by nonideal couplers or deliberately unbalanced beam splitters.

2. Long fiber-optic path imbalance and orthogonal-state quantum cryptography

A clear optical realization is the long-distance actively stabilized Mach–Zehnder fibre optical interferometer used to implement the GV95 quantum cryptography protocol with orthogonal states. The reported system is built from telecom single-mode fiber and spans $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$6 km of spooled optical fibers in the laboratory. Alice prepares the single-photon input state and selects the input port through an optical switch, while Bob performs detection and active phase stabilization. A $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$7 coupler splits the photon into two spatially separated wavepacket paths denoted $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$8 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$9; the key is encoded in the relative phase between the two paths (Xavier et al., 2012).

In this implementation, path imbalance is a protocol requirement rather than a parasitic effect. GV95 requires temporal separation of the two wavepackets, and the delay must exceed the relevant uncertainty in emission and detection times. The experiment uses a delay of about $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$0 m of fiber in Bob’s station, corresponding to a propagation time of about $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$1 ns, which is stated to be larger than the relevant uncertainty scales. The imbalance is realized with a short spool of $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$2 m fiber, a fiber-optic delay line, and a fibre-piezo stretcher that serves both as adjustable delay and phase actuator. The delay line was first coarsely adjusted with a filtered broadband light source to bring the arm-length mismatch to within about $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$3 mm (Xavier et al., 2012).

Because a kilometer-scale fiber MZI is highly sensitive to environmental phase drift, the experiment uses active stabilization via a classical control channel wavelength-multiplexed with the quantum channel using a DWDM. The control laser is an external-cavity tunable laser at $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$4 with coherence length greater than $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$5 m, while the quantum channel is centered at $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$6. At Bob, the classical channel is detected by a p-i-n photodetector, and an FPGA drives the fiber stretcher to cancel phase drifts. The DWDM has about $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$7 dB insertion loss, and together with fiber Bragg gratings and additional FBG/circulator filtering it suppresses crosstalk before single-photon detection (Xavier et al., 2012).

The reported visibility is defined as

$$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$8

where $$\varphi=\varphi_a-\varphi_b=\phi+\frac{E\tau}{\hbar},\qquad \tau=\frac{L_a-L_b}{v_D}.$$9 and $\tau$0 are the count rates at detectors $\tau$1 and $\tau$2. Raw visibilities of $\tau$3 and $\tau$4 were reported, improving after dark-count correction to $\tau$5 and $\tau$6. The raw QBER values were $\tau$7 and $\tau$8 for the two states, and after correcting for the higher dark count rate of detector $\tau$9, the average total QBER was $\tau_c \ll \Delta t \ll \tau_p,$0, summarized as $\tau_c \ll \Delta t \ll \tau_p,$1. In this setting, a balanced MZI would not serve GV95 in the intended way, because the deliberate delay is what simultaneously enforces non-simultaneous wavepacket transmission and preserves the interferometric basis needed to distinguish the orthogonal states (Xavier et al., 2012).

3. Integrated photonic implementations: passive analyzers and narrow-FSR silicon devices

A distinct integrated-photonics realization uses two matched unbalanced Mach–Zehnder interferometers on a photonic integrated circuit fabricated on a thermally stable borosilicate glass platform. Each uMZI contains an input coupler, short and long arms, and an output combiner; the path imbalance is encoded directly in the physical waveguide lengths and corresponds to a temporal delay of approximately $\tau_c \ll \Delta t \ll \tau_p,$2 ns. This value is intentionally chosen so that $\tau_c \ll \Delta t \ll \tau_p,$3, thereby suppressing single-photon interference while maintaining indistinguishability of the two-photon $\tau_c \ll \Delta t \ll \tau_p,$4 and $\tau_c \ll \Delta t \ll \tau_p,$5 amplitudes that interfere in the central Franson coincidence peak (Emadi et al., 27 Mar 2026).

These analyzers are described as fully passive because they contain no on-chip phase shifters, no heaters, no electro-optic tuners, and no active stabilization loop. Instead, the relative phase is scanned by uniform thermal tuning of the entire chip through the thermo-optic effect. The monolithic glass PIC is reported to provide good passive stability, with common-mode perturbations strongly reduced and only minimal phase drift over hours once the operating point is set. In the reported experiment, a narrow-linewidth CW laser at $\tau_c \ll \Delta t \ll \tau_p,$6 nm pumps a cascaded PPLN source, DWDM separates the photons into CH22 at about $\tau_c \ll \Delta t \ll \tau_p,$7 nm and CH20 at about $\tau_c \ll \Delta t \ll \tau_p,$8 nm, and the two photons are sent into the matched passive uMZIs before coincidence analysis (Emadi et al., 27 Mar 2026).

The coincidence fringe is fitted with

$\tau_c \ll \Delta t \ll \tau_p,$9

At the optimum operating point, the measured raw visibility is $\tau_c$0, the background-corrected visibility is $\tau_c$1, and the fit-derived visibility is $\tau_c$2. The same experiment reports a maximum system heralding efficiency of about $\tau_c$3, a typical range of $\tau_c$4, and a coincidence-to-accidental ratio exceeding $\tau_c$5 at only $\tau_c$6 mW of pump power. The reported delay of about $\tau_c$7 ns is much larger than the approximately $\tau_c$8 ps single-photon coherence time after DWDM filtering, which is the stated basis for suppression of single-photon interference (Emadi et al., 27 Mar 2026).

A different integrated purpose for path imbalance appears in silicon-on-insulator strip-waveguide MZIs designed for low free-spectral range. In the “narrow FSR” device, the waveguide cross section is $\tau_c$9 nm by $\tau_p$0 nm, the interferometer supports single-mode TE polarization, standard $\tau_p$1 bend-radius bends are used, and the longer arm includes long horizontal and vertical straight sections between bends. The reported arm lengths are $\tau_p$2, $\tau_p$3, so that

$\tau_p$4

This large imbalance yields a measured free-spectral range of $\tau_p$5 nm at $\tau_p$6 nm, with fitted group index $\tau_p$7, dispersion $\tau_p$8, and $\tau_p$9 (Warner, 1 Jul 2025).

The governing trend is summarized by

$\Delta t$0

with the reported shorter-imbalance devices showing FSRs from about $\Delta t$1 nm to $\Delta t$2 nm. Equally important, the paper distinguishes the total path-length difference from the geometric distribution of that extra length. The stated result is that longer straight waveguide sections between bends reduce dispersion and improve agreement between measured and simulated spectra, whereas shorter straight segments lead to higher dispersion. This suggests that path imbalance in integrated silicon devices is a two-parameter design problem: $\Delta t$3 sets the spectral period, while inter-bend straight length controls dispersive distortion (Warner, 1 Jul 2025).

4. Electronic and graphene Mach–Zehnder interferometers in the quantum Hall regime

In electronic Mach–Zehnder interferometers based on the quantum Hall effect, the effective paths are determined by incompressible edge states rather than by lithography alone. A self-consistent Hartree/Thomas–Fermi analysis computes the gate-induced confinement, screened external potential, electron density, and Hartree potential, yielding the spatial density profile $\Delta t$4 and the location and width of incompressible strips. The central conclusion is that the edge-state pattern depends strongly on screening and magnetic field, so the actual interferometer arms may run as separate channels, approach each other near quantum point contacts, or merge at the constrictions (0707.1125).

In this framework, path imbalance originates from magnetic-field-dependent rearrangement of the incompressible edge states. As $\Delta t$5 varies, the Landau-level filling changes, the widths and positions of incompressible strips shift, the edge reconstruction near the QPCs changes, and the two arms may acquire different effective lengths and separations. The reported conclusion is that interference is most likely when the two incompressible edge states merge or come very close near the QPCs. Equally explicitly, being on a quantized Hall plateau does not guarantee observable interference, because a plateau does not ensure that the two MZI arms are well defined and coherently connected at the constrictions (0707.1125).

Graphene quantum Hall MZIs introduce a related but not identical notion of path imbalance. In the asymmetric-gate configuration, the p and n regions are controlled by different electrostatic potentials,

$\Delta t$6

so the junction is not mirror-symmetric across the interface. The paper states that this shifts the interface channels on one side relative to the other, with the result that the two interfering chiral edge trajectories no longer enclose identical areas. This is explicitly identified as a path-imbalanced MZI, because the two interfering trajectories do not have the same geometry or enclosed flux (Song et al., 10 Aug 2025).

The interferometric phase is governed by the magnetic flux through the loop,

$\Delta t$7

For $\Delta t$8, the loop width is written as

$\Delta t$9

leading to a first-harmonic frequency estimate

$50{:}50$0

At higher filling factors such as $50{:}50$1, multiple Fermi-level crossings $50{:}50$2 create two distinct loop areas, denoted $50{:}50$3 and $50{:}50$4, as well as a larger composite loop enclosing $50{:}50$5. The resulting conductance traces show beat patterns and multiple Fourier peaks. To resolve these, the paper uses a machine-learning-based Fourier transform implemented as a single-hidden-layer neural network with sinusoidal activation functions, $50{:}50$6 hidden nodes, a linear output layer, Adam optimizer, mean-squared-error loss, cosine-annealing learning-rate schedule, and weight clipping on the input-to-hidden weights. The reported design rule is that visibility is enhanced under symmetric gate conditions and reduced by asymmetry, even though asymmetry generates richer multi-frequency structure (Song et al., 10 Aug 2025).

5. Visibility, decoherence, and correction of imbalance

Path imbalance does not by itself determine visibility. Several of the cited works emphasize that visibility can be limited, or even extinguished, by mechanisms that encode which-path information in additional degrees of freedom. In the fractional quantum Hall case with upstream neutral modes, a tunneling event at one QPC creates not only a charge packet but also upstream neutral wavepackets. The resulting state at the drain is written schematically as a coherent superposition of the two path amplitudes tensor-producted with different neutral excitations, and interference survives only if the corresponding neutral states have nonzero overlap. The stated physical conclusion is that upstream neutral modes act as a built-in which-path detector, so tracing them out suppresses the Aharonov–Bohm interference term (Goldstein et al., 2016).

This suppression persists even in a geometrically symmetric interferometer. Without neutral modes, a symmetric device with $50{:}50$7 can retain visibility if the charge wavepackets overlap. With neutral modes, however, the relevant timescale becomes

$50{:}50$8

for $50{:}50$9, or more generally $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$0. The paper states that even a perfectly symmetric interferometer loses coherence once either $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$1 or $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$2 exceeds $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$3, because the neutral sector stores which-path information and $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$4 is typically small. A plausible implication is that geometric balancing is insufficient whenever additional propagating modes become entangled with the interfering degree of freedom (Goldstein et al., 2016).

A different limitation arises in integrated photonics from coupler asymmetry. In a cascaded silicon-photonic MZI architecture with two variable beam splitters $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$5 and $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$6, fabricated MMI couplers that deviate from $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$7 produce amplitude imbalance between the interferometer arms and therefore incomplete destructive interference. The normalized output power at port 3 is written in terms of the reflectivity offsets $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$8, showing explicitly that deviations from ideal splitting ratios reduce interference contrast. Experimentally, a single fixed-splitter MZI on the same chip achieved only $$T \neq \frac{1}{\sqrt{2}} \quad\text{and/or}\quad T' \neq \frac{1}{\sqrt{2}}.$$9 dB extinction, with MMIs corresponding to a $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$00 splitting ratio, whereas the self-optimized device achieved $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$01 dB extinction (Wilkes et al., 2016).

The correction strategy is an automated progressive optimization algorithm with no pre-calibration. It scans the $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$02 voltage range of the central heater to find output extrema, then adjusts the outer heaters in coordinated directions to minimize or maximize the optical power at one output port until the settings stop changing significantly. The stated target is $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$03, so that the effective transfer function of the cascaded device reproduces that of an ideal MZI. This establishes an important counterpoint to the decoherence results above: some imbalances encode unavoidable path information, whereas others are engineering nonidealities that can be compensated in situ (Wilkes et al., 2016).

6. Functional consequences: thermoelectricity and phase metrology

In quantum Hall thermoelectric devices, path imbalance is the source of energy dependence and therefore the source of thermoelectric response itself. The stated mechanism is

$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$04

so that a length difference makes the scattering amplitudes energy dependent. The paper explicitly describes this as the sole origin of thermoelectricity in its noninteracting scattering-theory model. For the experimentally standard three-terminal setup, the optimal point is reported near $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$05 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$06, yielding

$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$07

with

$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$08

In the four-terminal double-MZI geometry, the optimum is reported near $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$09 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$10, with

$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$11

The same work estimates experimentally realistic operation around $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$12 for $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$13 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$14 (Hofer et al., 2015).

In phase metrology, by contrast, “unbalanced” refers primarily to beam-splitter transmission coefficients rather than unequal arm lengths. The first beam splitter $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$15 with transmission coefficient $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$16 is optimized through quantum Fisher information, while the second beam splitter $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$17 with transmission coefficient $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$18 is optimized for the actual detection scheme. The stated conceptual result is that $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$19 is fixed by the ultimate statistical limit, whereas $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$20 is detection-scheme dependent. Difference-intensity detection and single-mode intensity detection usually prefer a balanced second beam splitter,

$$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$21

whereas balanced homodyne detection often benefits from an unbalanced $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$22 (Mishra et al., 2022).

The reported examples show that an unbalanced MZI can outperform a balanced one in some state-and-detector combinations. For squeezed-coherent plus squeezed-vacuum input, the paper reports $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$23 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$24, yielding about a $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$25 improvement at peak phase sensitivity. For a coherent plus Fock input with $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$26 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$27, the reported optimum is $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$28 and $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$29. In a balanced-homodyne example for squeezed-coherent plus squeezed-coherent input, the optimized unbalanced device yields $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$30, compared with $$\left|A_a e^{i\varphi_a}+A_b e^{i\varphi_b}\right|^2 = A_a^2+A_b^2+2A_aA_b\cos\varphi,$$31 for the balanced counterpart. This suggests that, in metrology, interferometer balance is not a universal optimum but a detector-contingent design choice (Mishra et al., 2022).

Taken together, these results delimit the modern meaning of path-imbalanced Mach–Zehnder interferometry. In some settings the imbalance is a deliberate temporal resource, as in GV95 and Franson interferometry. In others it is a spectral design variable, as in low-FSR silicon photonics; a field- and gate-dependent geometric effect, as in quantum Hall and graphene devices; a thermoelectric resource, as in electronic heat engines; or a detection-matched readout parameter, as in phase-sensitive metrology. A recurring misconception is that symmetry, or operation in a nominally favorable transport regime, automatically guarantees maximal interference. The cited literature instead shows that visibility depends on the detailed physical origin of imbalance, on auxiliary modes that may encode which-path information, and on whether the relevant nonidealities can be stabilized or actively corrected (0707.1125, Goldstein et al., 2016).