Generalized Mach-Zehnder Interferometer

• GMZI is an interferometric architecture that generalizes the classic Mach–Zehnder design by employing various mode spaces such as frequency, temporal, or qubit systems.
• It utilizes customized beam splitters and generalized internal transformations to enable multiarm phase estimation, weak-value extraction, and enhanced metrological sensitivity.
• Experimental implementations demonstrate high visibilities and innovative adaptations, bridging theoretical models with practical applications in quantum sensing and interferometry.

A generalized Mach–Zehnder interferometer (GMZI) is an interferometric architecture that preserves the Mach–Zehnder logic of two mixing operations separated by a controllable internal transformation, while relaxing the assumptions that the two “paths” are spatial arms and that the internal element is merely a scalar phase shifter. In the literature, this generalization appears in several distinct but structurally related forms: two-mode interferometers acting on frequency, path, time-bin, or hybrid excitations; qubit interferometers with an arbitrary internal unitary; multiport and multiarm devices built from symmetric higher-dimensional couplers; cascaded temporal-mode sorters; and lossy or unbalanced interferometers optimized for metrology and weak-value extraction (Kobayashi et al., 2017, Argentieri et al., 2015, Simon et al., 2022, Markiewicz et al., 2020, Horoshko et al., 2023, Trevillian et al., 23 Feb 2026, Masiello et al., 23 Jun 2026).

1. Conceptual scope and defining features

In its most compact form, the GMZI is the statement that the standard Mach–Zehnder pattern—split, acquire a relative transformation, recombine—can be transplanted from spatial paths to a more general mode space. One line of work formulates this explicitly: a conventional MZI operates on spatial modes, whereas a generalized MZI acts on an arbitrary pair of modes in Hilbert space, such as different spatial modes, different polarizations, different time bins, or different frequency modes (Kobayashi et al., 2017). Another line of work uses the same label for higher-dimensional interferometers in which the two $2\times 2$ beam splitters are replaced by multiports, lifting the device from a two-mode to a four-mode or $d$-mode space (Simon et al., 2022, Markiewicz et al., 2020). A third usage keeps the two-level path structure but generalizes either the central phase element to an arbitrary single-qubit unitary or the first beam splitter to an arbitrary imbalance (Argentieri et al., 2015, Masiello et al., 23 Jun 2026).

This diversity makes one misconception worth dispelling: “generalized” does not refer to a single canonical hardware design. It refers instead to a family resemblance. Across these realizations, the invariant is the interferometric topology; what changes is the space on which the topology acts, the algebra of the internal transformation, or the measurement task.

GMZI variant	Generalized ingredient	Representative source
Frequency-domain GMZI	Paths are frequency modes; beam splitters are quantum frequency converters	(Kobayashi et al., 2017)
Grover-based GMZI	Beam splitters replaced by directionally unbiased 4-port Grover multiports	(Simon et al., 2022)
Generalized qubit interferometer	Central phase shifter replaced by arbitrary single-qubit unitary	(Argentieri et al., 2015)
Weak-value GMZI	Arbitrary first beam splitter, controlled path phase, fixed 50/50 recombiner	(Masiello et al., 23 Jun 2026)
$d$-mode GMZI	Two symmetric multiports with phases in every internal mode	(Markiewicz et al., 2020)
Temporal-mode sorter	Cascade of MZIs with FrFT/time-lens transform in one arm	(Horoshko et al., 2023)
Temporal hybrid GMZI	Beam splitters implemented by pulsed qubit–magnon coupling	(Trevillian et al., 23 Feb 2026)
Three-grating matter-wave GMZI	Diffraction orders serve as momentum-space paths	(Sanz et al., 2014)

2. Unitary structure in two-mode GMZIs

The two-mode GMZI admits a unified operator description. In the frequency-domain formulation, the full interferometer is written as

$$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$

exactly as in a spatial MZI, but acting on the two-dimensional Hilbert space spanned by the frequency states $\lvert \omega_{\mathrm{L}}\rangle$ and $\lvert \omega_{\mathrm{U}}\rangle$ rather than on two spatial ports (Kobayashi et al., 2017). The same abstract structure reappears when the path degree of freedom is treated as a qubit. In the generalized-interferometer duality analysis, the first beam splitter is still a rotation

$B = e^{-i\frac{\pi}{4}\sigma_y},$

but the central element becomes a generic SU(2) unitary

$$U = e^{-i\hat{\mathbf{m}}\cdot\hat{\boldsymbol{\sigma}}\,\phi/2},$$

with the total transformation

$V = B^\dagger U B.$

The interferometer is therefore no longer restricted to relative $z$-axis phase accumulation; it can implement a general Bloch-sphere rotation before the output measurement of $d$0 (Argentieri et al., 2015).

A related qubit-level parameterization appears in the weak-value GMZI. There the path basis is $d$1, the prepared input state is

$d$2

and the internal arm operation is a projector phase

$d$3

The final recombiner is fixed to 50/50, with post-selected output states

$d$4

This makes clear that a GMZI can remain strictly two-dimensional while still being generalized through amplitude imbalance, projector-based phase control, and nonstandard observables such as weak values (Masiello et al., 23 Jun 2026).

These formulations support a unifying interpretation: a two-mode GMZI is a programmable SU(2)-type interferometer whose “paths” need not be spatial and whose middle element need not be a scalar phase.

3. Frequency-domain GMZI as a canonical nonspatial realization

The frequency-domain realization by Kobayashi and collaborators is the clearest concrete instance of the Hilbert-space generalization. Here the two interferometric arms are the lower-frequency mode $d$5 at 1580 nm and the upper-frequency mode $d$6 at 795 nm, driven by a pump at 1600 nm satisfying

$d$7

The two “beam splitters” are quantum frequency converters in periodically poled lithium niobate (PPLN) waveguides, so the GMZI is implemented entirely in frequency space rather than in path space (Kobayashi et al., 2017).

The effective Hamiltonian of each frequency-domain beam splitter is

$d$8

which yields Heisenberg-picture mode transformations formally identical to those of a two-mode beam splitter. The tunable reflectance and transmittance are

$d$9

so 50:50 operation occurs at $d$0. Experimentally, the two PPLN devices were tuned to approximately 50% conversion efficiency, and the interferometer exhibited visibilities above 0.99 in both frequency outputs. The relevant phase was

$d$1

combining the pump phases of the two converters and the intermediate phase shifter.

The device was also analyzed beyond the idealized lossless model. A lossy GMZI was represented by virtual loss media before and after the ideal frequency-domain beam splitters, with measured transmittances $d$2, $d$3, and $d$4, together with relative-loss parameters $d$5 and $d$6. Even under this realistic model, the measured visibilities remained near 0.99 around 50% conversion.

The quantum-regime analysis is equally significant. Using superconducting single-photon detectors, the background photon rates were characterized as functions of pump power. The 795 nm noise displayed both quadratic and linear power dependence, while the 1580 nm noise was mainly linear, consistent with Raman scattering of the strong pump as the dominant noise source. With average input photon number $d$7 and filters of bandwidth $d$8, the estimated single-photon interference visibility remained above 0.9. The same analysis covered single photons with bandwidth 9.6 MHz or 1.2 GHz, provided suitably narrow filters such as 13 MHz or 1.6 GHz were used. In this sense, the frequency-domain GMZI is not only an analogy to the spatial MZI; it is a practical frequency-qubit interferometer linking telecom and rubidium-memory-compatible wavelengths (Kobayashi et al., 2017).

4. Higher-dimensional GMZIs: multiports, multiarm phase sensing, and simultaneous estimation

A distinct generalization replaces the two-port beam splitter itself. In the Grover-based GMZI of Simon, Schwarze, and Sergienko, each interferometric vertex is a directionally unbiased 4-port Grover multiport with unitary

$d$9

Unlike an ordinary beam splitter, this device allows amplitude to exit any port, including the input port. Two such multiports, connected by four internal lines and three independent phase shifts $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$0, form a four-mode analog of the Mach–Zehnder interferometer. With two-photon input and coincidence detection, the output rates encode the three phases simultaneously, and the interference terms depend on combinations such as $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$1, giving effectively doubled phase sensitivity for $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$2 relative to a standard MZI. The same architecture also interpolates continuously between Hong–Ou–Mandel and anti-HOM behavior, and when its transmission lines are arranged in different planes it functions as a higher-dimensional Sagnac interferometer capable of measuring rotation about three axes with a single device (Simon et al., 2022).

A more algebraic multiarm generalization appears in the three- and four-mode GMZI of simultaneous multiphase estimation. There the interferometer is

$$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$3

where $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$4 is a symmetric multiport and $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$5 places one phase in each of the $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$6 internal modes. For $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$7 and $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$8, the generators are written in a Heisenberg–Weyl operator basis, and the full device is a direct $$\hat{U}_{\mathrm{MZI}} \equiv \hat{U}_{\mathrm{BS2}}\hat{U}_{\mathrm{PS}}\hat{U}_{\mathrm{BS1}},$$9-mode extension of the MZI topology. Because only relative phases are observable, the full quantum Fisher information matrix is singular: all $\lvert \omega_{\mathrm{L}}\rangle$0 phases cannot be estimated simultaneously. Nevertheless, any subset of $\lvert \omega_{\mathrm{L}}\rangle$1 phases can be jointly estimated with Heisenberg-like scaling in a completely fixed setup. For the optimized three-mode case,

$\lvert \omega_{\mathrm{L}}\rangle$2

and for the optimized four-mode case,

$\lvert \omega_{\mathrm{L}}\rangle$3

This establishes a precise higher-dimensional sense in which a GMZI is a multiparameter interferometer built from generalized beam splitters rather than from passive two-port elements (Markiewicz et al., 2020).

Taken together, these papers show that “generalized” can mean either higher local scattering dimensionality, a larger interferometric mode space, or both.

5. Temporal, hybrid, and momentum-space GMZIs

Another major branch of the literature relocates the interferometric arms into time, hybrid excitations, or momentum space. In the temporal Hermite–Gauss sorter, the device consists of $\lvert \omega_{\mathrm{L}}\rangle$4 cascaded Mach–Zehnder interferometers, each containing in one arm a temporal fractional Fourier transform implemented by a time lens and dispersive sections. Because temporal Hermite–Gauss modes are eigenfunctions of the FrFT, the transform imposes an order-dependent temporal Gouy phase

$\lvert \omega_{\mathrm{L}}\rangle$5

which the interferometer converts into constructive or destructive output routing. By choosing $\lvert \omega_{\mathrm{L}}\rangle$6 in stage $\lvert \omega_{\mathrm{L}}\rangle$7, the network sorts the first $\lvert \omega_{\mathrm{L}}\rangle$8 temporal Hermite–Gauss modes. For a two-interferometer sorter applied to SPDC Schmidt modes, the derived theoretical lower bound on cross-talk probability is 5.5% (Horoshko et al., 2023).

In the temporal magnon–qubit interferometer, the two paths are not optical modes at all but the states “excitation in the qubit” and “excitation in the magnon.” Short magnetic-field pulses act as temporal beam splitters by tuning the qubit–magnon detuning into resonance. In the single-excitation sector, a pulse scatters

$\lvert \omega_{\mathrm{L}}\rangle$9

with transition probability

$\lvert \omega_{\mathrm{U}}\rangle$0

Two such pulses, separated by free evolution at large detuning, form a temporal MZI whose output is the final qubit excitation probability. Within a Lindblad description, amplitude noise and phase noise in the magnon channel affect, respectively, the average output level and the fringe visibility, enabling separate extraction of $\lvert \omega_{\mathrm{U}}\rangle$1 and $\lvert \omega_{\mathrm{U}}\rangle$2 from $\lvert \omega_{\mathrm{U}}\rangle$3 and $\lvert \omega_{\mathrm{U}}\rangle$4 (Trevillian et al., 23 Feb 2026).

Matter-wave interferometry supplies a further generalization. In the atomic three-grating Mach–Zehnder analysis, the first grating acts as an input beam splitter by producing multiple diffraction orders, the second grating redirects selected orders as the effective mirrors, and the third grating samples the recombined interference pattern. In this formulation, the interferometric paths are diffraction orders with transverse momenta

$\lvert \omega_{\mathrm{U}}\rangle$5

and the local transverse momentum

$\lvert \omega_{\mathrm{U}}\rangle$6

provides a path-resolved description of the probability flow. The transmitted flux through the final grating varies periodically with the lateral shift $\lvert \omega_{\mathrm{U}}\rangle$7, with period equal to the grating period $\lvert \omega_{\mathrm{U}}\rangle$8, recovering the standard interferometric cosine modulation in a momentum-path setting (Sanz et al., 2014).

These examples broaden the ontology of an interferometric “arm.” In GMZIs, a path can be a temporal mode, a quasiparticle channel, or a diffraction order just as naturally as a spatial beam.

6. Complementarity, weak values, loss, and nonclassical readout

GMZIs have also been used to probe foundational and metrological questions that are either absent or less transparent in the standard MZI. In the generalized-interferometer duality study, replacing the central phase shifter by an arbitrary single-qubit unitary changes the usual predictability–visibility tradeoff. Instead of

$\lvert \omega_{\mathrm{U}}\rangle$9

the generalized bound becomes

$B = e^{-i\frac{\pi}{4}\sigma_y},$0

For suitable choices of the internal unitary $B = e^{-i\frac{\pi}{4}\sigma_y},$1, full predictability and full visibility are not mutually exclusive. This does not abolish complementarity; it shows that the familiar bound is architecture-dependent rather than universal across all two-way interferometers (Argentieri et al., 2015).

The weak-value GMZI pushes this further by using an unbalanced first beam splitter, a controllable path phase $B = e^{-i\frac{\pi}{4}\sigma_y},$2, and a fixed 50/50 recombiner to extract path weak values directly from output intensities. The interferogram is

$B = e^{-i\frac{\pi}{4}\sigma_y},$3

By combining intensities at selected phase points with single-path measurements, the full complex weak values of the path projectors can be reconstructed without meter states and without weak interactions. In the unbalanced configuration, the real parts can become anomalous, lying outside the eigenvalue interval $B = e^{-i\frac{\pi}{4}\sigma_y},$4 of the projector; the same anomalies are identified with negative quasiprobability distributions of the Kirkwood–Dirac type (Masiello et al., 23 Jun 2026).

Lossy interferometry provides a more operational use of the GMZI concept. In the generalized lossy MZI, both beam splitter reflectivities $B = e^{-i\frac{\pi}{4}\sigma_y},$5 and both internal losses $B = e^{-i\frac{\pi}{4}\sigma_y},$6 are treated as free parameters. For coherent input, both single-intensity and difference-intensity detection can reach the standard interferometric limit

$B = e^{-i\frac{\pi}{4}\sigma_y},$7

when the interferometer is optimized. In particular,

$B = e^{-i\frac{\pi}{4}\sigma_y},$8

while for difference detection one sets $B = e^{-i\frac{\pi}{4}\sigma_y},$9 and $$U = e^{-i\hat{\mathbf{m}}\cdot\hat{\boldsymbol{\sigma}}\,\phi/2},$$0. Experimentally, sensitivity improvements were demonstrated for losses from 0.4 to 0.998; at loss 0.998 in one arm, optimizing the reflectivity produced a 2.5 dB sensitivity improvement in difference-intensity detection, corresponding to 5.5 dB relative to single-intensity detection (Huang et al., 2023).

A further nonclassical variant appears when one arm contains a quantum optomechanical scatterer rather than a passive mirror. In that GMZI, a vibrating micromirror couples to travelling light through radiation pressure and is modeled with the Hudson–Parthasarathy equation. The reflected field acquires the operator phase

$$U = e^{-i\hat{\mathbf{m}}\cdot\hat{\boldsymbol{\sigma}}\,\phi/2},$$1

so the interferometer becomes a probe of continuous-time optomechanical dynamics. The output can display a negative Mandel $$U = e^{-i\hat{\mathbf{m}}\cdot\hat{\boldsymbol{\sigma}}\,\phi/2},$$2-parameter under direct detection, while the difference-current spectrum at the two output ports exhibits nearly complete cancellation of shot noise; in that regime, the Mach–Zehnder plus difference detection is equivalent to homodyne readout of squeezing generated by pure scattering, without a cavity (Barchielli et al., 2021).

A broader interpretive usage, clearly marked as such in the literature, extends the GMZI label to MZIs acting on multi-time-bin, multi-photon Hilbert spaces. In that reading, temporally delayed two-photon inputs can support simultaneous HOM-like and N00N-like interference within a single interferometric geometry, making the “generalized” attribute refer to the enlarged temporal and photon-number mode space rather than to a modified hardware topology (Kim et al., 2016).

Across these developments, the GMZI emerges not merely as a modified interferometer but as a general interferometric template for mode engineering, multiparameter sensing, weak-value reconstruction, decoherence spectroscopy, and tests of nonclassicality.