Decoded Quantum Interferometry (DQI)

• Decoded Quantum Interferometry is a quantum-enhanced interferometric technique that exploits nonlinear interactions and entanglement to convert fragile phase data into robust observables.
• It employs tailored decoding operations in matter-wave and photonic systems to achieve phase sensitivity beyond classical limits, enabling precise quantum metrology.
• The approach integrates many-body physics models and phase-space representations to optimize measurements in quantum sensors, paving the way for advanced computing protocols.

Decoded Quantum Interferometry (DQI) is a class of quantum-enhanced interferometric protocols that leverage system-specific nonlinearities, engineered quantum correlations, and tailored “decoding” operations to extract phase or structural information with sensitivity superior to standard quantum limits. DQI generalizes linear quantum interferometry by using both the generation and controlled measurement of non-classical correlations—typically in many-body or complex quantum systems—to “decode” information stored in fragile phase-like degrees of freedom into accessible observables such as population imbalance, photon number, or spatial interference patterns.

1. Nonlinear Many-Body Dynamics and Quantum Interferometry

At the core of DQI with matter waves is the exploitation of nonlinear interactions arising in systems such as Bose–Einstein condensates (BECs). In these systems, the ultracold atomic ensemble is governed by the Gross–Pitaevskii equation: $$i\hbar\frac{\partial\psi}{\partial t} = \left( -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{ext}} + g|\psi|^2 \right)\psi$$ where $g$ parametrizes the atom-atom s-wave interaction. This term ($g|\psi|^2$) enables a variety of nonlinear phenomena absent in linear interferometers, such as the formation of matter-wave solitons (both bright for $g < 0$ and dark for $g > 0$) and macroscopic quantum self-trapping (MQST) in weakly-linked bosonic Josephson junctions.

In the two-mode approximation for BECs in double-well or internal-state configurations, the Hamiltonian includes nonlinearities via

$$H = \frac{1}{2}\delta(n_2-n_1) + \frac{E_c}{8}(n_2-n_1)^2 - J(\psi_1^*\psi_2 + \psi_2^*\psi_1)$$

yielding coupled equations for population imbalance $z$ and relative phase $\theta$: $$\begin{align*} \dot{z} & = -2J\sqrt{1-z^2}\sin\theta \ \dot{\theta} & = \frac{E_c N}{2}z + \frac{2Jz}{\sqrt{1-z^2}}\cos\theta \end{align*}$$ When interactions dominate ($E_cN\gg J$), the evolution displays MQST, locking population imbalance and leading to nonlinear phase dynamics.

2. Generation and Decoding of Nonclassical Correlations

The atom-atom nonlinear interactions facilitate the dynamical generation of nonclassical correlations, such as spin-squeezed states (with variance suppression in a single quadrature) and highly entangled states (e.g., NOON/GHZ states): $$|\text{NOON}\rangle = \frac{1}{\sqrt{2}}\left( |N,0\rangle + e^{i\theta}|0,N\rangle \right)$$ Such states allow phase sensitivity scaling at the Heisenberg limit, $\Delta\phi \sim 1/N$, contrasting with the Standard Quantum Limit (SQL) of $1/\sqrt{N}$.

DQI protocols employ a controlled “decoding” step to convert fragile, delocalized phase information—imprinted on such nonclassical many-body states during nonlinear evolution—into robust, directly measurable observables. In BEC experiments, this often corresponds to rapid parameter changes in the Hamiltonian (e.g., quenching $J$ or $\delta$) that map accumulated phase differences into population imbalance or parity, optimizing the derivative $\partial\langle O\rangle/\partial\phi$: $$\Delta\phi = \frac{\Delta O}{|\partial\langle O\rangle/\partial\phi|}$$ Direct measurement then achieves sensitivity scaling set by the engineered entanglement, potentially achieving the Heisenberg limit.

3. Quantum Interferometry Beyond Linear Devices: Three-Dimensional Integrated Photonics

DQI also encompasses photonic architectures where quantum interference in multi-arm, multi-port geometries is exploited. Devices such as the “tritter” (3-port splitter) and “quarter” (4-port splitter), implemented in three-dimensional integrated photonic circuits, generalize the beam splitter unitary operations: $b_i^\dagger = \sum_j \mathcal{U}_{ij} a_j^\dagger$ Injecting Fock states into such geometries yields output distributions with interference patterns characterized by nonclassical visibilities exceeding classical bounds. Notably, sub-Rayleigh fringes (harmonics $\cos(N\phi)$) arise, and quantum Fisher Information (QFI) exceeds that accessible with all classical probe states. Simultaneous multiparameter phase estimation becomes practical by exploiting the full structure of these multi-mode entangled photonic states, provided that measurements saturate the quantum Cramér–Rao bound.

4. Matter-Wave Interferometry and "Decoding" in Composite Systems

DQI protocols have been extended to high-mass and composite quantum objects in near-field matter-wave interferometry, using Talbot–Lau (TLI), Kapitza–Dirac–Talbot–Lau (KDTLI), and pulsed optical grating (OTIMA) arrangements. Here, the full Wigner phase-space formalism is used to express the quantum evolution and measurement readout: $$w(x,p) = \frac{1}{2\pi\hbar}\int ds~ e^{ips/\hbar}\rho(x-s/2, x+s/2)$$ Interference patterns, which encode both environmental decoherence and the internal degrees of freedom (e.g., polarizability, mass, vibrational state), act as a "code" that is decoded via detailed inversion of the theoretical model—allowing extraction of molecular or nanoparticle parameters with high precision. Experimentally observed fringe shifts and contrast modulations as functions of external perturbations are quantitatively mapped back to microscopic particle properties or signatures of quantum macrorealism modifications.

5. Interferometric Computation and General Theories

DQI also encompasses quantum and post-quantum computation protocols in many-armed interferometers. These frameworks formalize the concept of distributed phase or oracle encoding by decomposing the state space with branch-localized operations, and systematically analyze under which logical theories (quantum, quaternionic, classical with epistemic restrictions) nontrivial phase interferometry can perform computational tasks (e.g., Deutsch–Jozsa, Grover search) with quantum or post-quantum efficiency. The operational concept is based on the existence and structure of a 'phase group'—a set of localized transformations that can be coherently recombined to yield globally meaningful interference.

6. Quantum Metrology and Measurement Enhancement

DQI protocols deliver practical quantum advantage in various metrological regimes. In quantum-enhanced white-light interferometry, use of energy–time entangled photon pairs (e.g., N00N states), with the interference phase

$$\phi_{\text{N00N}} = \phi(\lambda_1) + \phi(\lambda_2)$$

enables phase sensitivity and systematic error reduction superior to classical techniques, thanks to correlations that cancel higher-order dispersion terms and reduce fitting parameters. The precision per photon is dramatically increased (sometimes by orders of magnitude) relative to classical or shot-noise limited schemes, as explicitly demonstrated by two-photon N00N experiments in fiber dispersion metrology.

Similarly, DQI-inspired protocols using squeezed states, SU(1,1) nonlinear interferometers, or twin beam correlation schemes enable phase sensitivity approaching Heisenberg scaling and robust noise cancellation—now realized in environments such as gravitational-wave observatories. These advances rely on tailored quantum states, nonlinear interactions, and measurements that optimally decode the encoded phase information.

7. Fundamental and Applied Implications

Decoded Quantum Interferometry provides a unifying conceptual and operational framework for harnessing nonlinear and non-classical physics—whether in ultracold atomic gases, photonic circuits, or complex matter-wave systems—toward optimal extraction of phase, parameter, or computational information. The key signature is the translation of quantum information stored in delicate, typically phase-like, or delocalized degrees of freedom (often protected or enhanced by engineered entanglement) into robust, accessible observables via precisely timed and system-specific decoding operations.

This approach supports advanced quantum metrology, new computational protocols, and foundational probes of quantum coherence and decoherence at increasingly macroscopic scales, and underpins the design and implementation of next-generation quantum sensors and quantum-enhanced computing platforms. The theoretical machinery, ranging from the Gross–Pitaevskii and two-mode Bose–Hubbard models to Wigner/phase-space representation, quantum Fisher information, and generalized probabilistic theory, provides the rigorous foundation for analyzing, predicting, and optimizing DQI protocols for a wide class of quantum hardware and applications.