Linear Mode Interference: Theory and Applications

• Linear mode interference is the coherent superposition of distinct optical, electronic, or quantum modes via passive, unitary transformations, leading to observable interference fringes and phase-sensitive correlations.
• It employs passive elements like beam splitters and phase shifters to mix modes without nonlinearities, enabling scalable implementations in quantum computing and precision metrology.
• Experimental realizations across frequency, spatial, and time-bin domains demonstrate its role in enhancing interferometric measurements and advancing photonic quantum circuits.

Linear mode interference refers to the coherent superposition of distinguishable optical, electronic, or quantum modes in a system governed by linear unitary transformations, leading to interference fringes, dips, and nontrivial correlations among observables. The defining features are (i) the absence of optical nonlinearities during mode mixing, (ii) the use of passive linear operations (beam splitters, phase shifters, spatial/frequency/time multiplexing), and (iii) observable outputs being sensitive to the relative phases and overlaps between input modes. Linear mode interference is central to quantum optics, photonic quantum computing, condensed matter physics (e.g., mesoscopic superconductivity), and interferometric metrology.

1. Theoretical Formulation of Linear Mode Interference

Let $\{\hat b_n^\dagger\}$ be creation operators for a basis of orthogonal modes (spatial, frequency, time, polarization), satisfying %%%%1%%%%. A general $N$-mode linear interferometer is represented by a unitary $U \in U(N)$, acting as

$$\hat a_k^\dagger = \sum_{n=1}^N U_{k,n}\,\hat b_n^\dagger \,.$$

This unitary preserves bosonic commutation relations and enforces orthogonality among output modes. In the simplest 2-mode case, setting $$U=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \ 1 & 1\end{pmatrix}$$ yields the canonical 50:50 beam splitter transformation in frequency, spatial, or temporal modes.

Linear mode interference arises when photons (or electrons) are prepared in mutually coherent superpositions or arbitrary orthogonal modal functions and then mixed via $U$, with subsequent detection revealing phase-sensitive probabilities and correlations. The observed interference is determined by the combined effect of input modal amplitudes, their overlaps, and mode-dependent phase evolution.

2. Experimental Realizations Across Modal Bases

Frequency Domain Interference

Programmable frequency-domain interference is achieved by generating quantum states in arbitrary orthogonal spectral modes $\{\phi_n(\omega)\}$ using pulse shaping, and employing spectral projectors or measurement bases defined by grating-based wavelength separators. Coincidence probabilities in the frequency Hong–Ou–Mandel (HOM) experiment are

$P_c \propto 1 - |\langle\psi_1|\psi_2\rangle|^2,$

where $\psi_{1,2}(\omega)$ are the spectral wavefunctions, with perfect interference ($P_c=0$) at zero delay for orthogonal modes (Hosaka et al., 2024).

Spatial and Time-Bin Architectures

Time-bin linear mode interference implements multi-mode unitary transformations in a single spatial channel using programmable loop architectures. Input and output relations are constructed using laddered beam splitters and phase shifters (e.g., fiber delay lines, EOM, Sagnac interferometers). The $M$-mode output is given by

$\hat a_{\textrm{out}} = U\,\hat a_{\textrm{in}},$

with $U$ recursively defined by per-cycle transmission/reflection settings and internal phases. n-photon interference is quantitatively modeled via permanents of submatrices of $U$ (Carosini et al., 2023).

Multi-Mode HOM and Multi-Slit Analogies

Several works establish the analogy between multimode HOM interference and classical multi-slit interference. The observed patterns are characterized by two factors: $$P(\tau) = \tfrac{1}{2}\Big[1 - P_0(\tau)\,\frac{\sin(Nx)}{\sin x}\Big],$$ where $P_0(\tau)$ encodes single-mode (slit) envelope, $N$ is mode/slit number, and $x$ is the phase spacing. Both scenarios exhibit analogous comb structures with fringe narrowing $\propto 1/N$ and Fisher information scaling as $FI_{\max}^{1/2}\propto N$ (Guo et al., 2023).

3. Pairwise Interference, Coherence, and Visibility

For $N$ single-mode fields, the composite intensity and interference visibility follow from first-order coherence: $$G^{(1)}_{ij} = \langle a_i^\dagger a_j\rangle, \quad \mu_{ij} = \frac{G^{(1)}_{ij}}{\sqrt{G^{(1)}_{ii}\,G^{(1)}_{jj}}},$$ with path-indistinguishability $D_{ij} = |\mu_{ij}|$. The fringe visibility for $N$ modes is

$V = \frac{2}{N}\sum_{i<j}|\mu_{ij}|.$

Born's rule dictates that only pairwise (second-order) interference survives in single-photon detection, with no genuine higher-order terms (Das et al., 2020).

4. Modal Distinguishability in Multi-Photon Interference

The interference depth and character are sensitive to polarization and spectro-temporal mismatches. For two independent single photons at a 50:50 beam splitter, the coincidence probability is

$$P_{\textrm{coin}} = \tfrac{1}{2}\Big(1 - \cos^2\Phi\,\cos^2\Theta\Big),$$

where $\cos\Phi$ is the polarization overlap and $\cos\Theta$ is the spectral overlap. For $N$ photons, the success probabilities generalize via sums and permanents over the overlap matrices, with increasing sensitivity to mismatch. Realistic imperfections such as detector efficiency, photon loss, asymmetry in beam splitter reflectivity, and depolarizing channels modulate but do not fundamentally alter the modal overlap dependence (Crum et al., 24 Jan 2025).

5. Applications in Quantum Information, Sensing, and Condensed Matter

Photonic Quantum Computing and Boson Sampling

Linear mode interference enables programmable manipulation of frequency, time-bin, polarization, or spatially multiplexed photons for scalable photonic quantum computation. Arbitrary $N$-mode unitaries—beam splitting, multiport transformations—are implementable by pulse shaping and programmable modal mixing, yielding architectures amenable to large-scale boson sampling, cluster-state generation, and quantum algorithms in a single physical channel (Hosaka et al., 2024, Carosini et al., 2023).

Quantum Metrology

Multimode linear interference schemes (e.g., MM-HOM) provide ultra-high-resolution time-delay/frequency sensing, clock synchronization, and super-resolving phase metrological protocols. The achievable Fisher information increases quadratically with mode count, and coherent programming of modal amplitudes yields interference fringe patterns with tailored envelope and detail structure (Guo et al., 2023, Shabbir et al., 2013).

Condensed Matter Josephson Junctions

Linear mode interference among transverse orbital modes in few-mode nanowire Josephson junctions causes anomalous suppression, revival, and gate-tunable nodes in the critical current under magnetic field. Each mode's phase shift depends on orbital area and Zeeman effect, and their interference produces irregular non-monotonic $I_c(B)$. This mechanism dominates over Zeeman-only and spin-orbit-only scenarios in realistic systems (Zuo et al., 2017).

6. Criteria for Quantum Versus Classical Interference

Neither the shape nor the visibility of $N$-photon interference patterns uniquely certify quantumness. If the modal state can be synthesized by classical coherent states with linear optics, both arbitrary fringe shapes (rectangle, sawtooth, $\lambda/N$ compressed) and near-unity visibility are possible. Quantum advantage in metrology or information processing demands input states that realize phase sensitivity beyond the shot-noise limit (e.g., true NOON states or nonclassical entangled inputs) as well as statistical signatures (Fisher information, success rate in entanglement swapping, or gate fidelity) unattainable classically (Shabbir et al., 2013).

7. Practical Considerations and Scalability

Linear mode interference strategies leveraging frequency multiplexing, time-bin encoding, or waveguide integration enable dramatic reductions in experimental resource overhead—one detector, one programmable modulator, single spatial beam—contrasted with conventional spatial-multiplexed architectures requiring $O(N^2)$ bulk elements. Full unitarity programmability is obtained by controlling spectral or temporal masks, loop settings, or measurement bases, with scalability constrained primarily by source brightness and mode-resolved detection (Hosaka et al., 2024, Carosini et al., 2023). In quantum networks and metrology, modal distinguishability must be optimized to maintain maximal visibility, fidelity, and error rates.

Linear mode interference thus constitutes a universal physical and mathematical framework with demonstrable impact across quantum optics, mesoscopic electronics, and scalable quantum information science.