Far-field spatial coherence driven by lossy objects: first-principles approach unifying scattering of quantum light and thermal emission

Abstract: Far-field spatial coherence dictates the interference properties of scattered light and thermal emission. Traditionally, these phenomena are treated through disjointed paradigms: classical scattering descriptions assume cold objects lacking quantum fluctuations, idealized quantum scattering schemes ignore dissipation, and semiclassical fluctuational electrodynamics relies on phenomenological noise currents, precluding the consistent treatment of incident quantum states. Here, we develop a first-principles framework based on the modified Langevin noise formalism to unify the scattering of quantum light and the intrinsic thermal emission of finite dissipative objects. We demonstrate that the outgoing far-field spatial coherence separates into an algebraic superposition of two geometry-driven mechanisms, coupled by the global unitarity of the radiation-matter dynamics. The first mechanism, elastic scattering, acts as a non-unitary spatial filter, mode-selectively attenuating and reshaping incident quantum correlations. The second mechanism, thermal emission, originates from localized material dissipation and projects the object's absorption profile into the far field, providing a quantum-vectorial derivation of the macroscopic van Cittert-Zernike theorem. Applying this framework across optical regimes, we determine operational bounds for lossy quantum photonics. Under chaotic thermal illumination, we analytically demonstrate thermal cloaking at equilibrium and show that a passive sink casts a structured thermal shadow geometrically identical to a primary emitter. Under coherent illumination, we derive a thermodynamic phase diagram bounding macroscopic phase correlations, demonstrating that subwavelength nanostructures undergo substantial coherence degradation compared to bulk objects. Finally, under spatially entangled illumination...

• The paper introduces a unified theory that separates elastic scattering and thermal emission, offering an exact operator-level treatment of far-field coherence in lossy systems.
• It employs the Modified Langevin Noise Formalism to derive rigorous input-output relations, ensuring energy conservation and Kirchhoff’s law at the quantum level.
• Numerical and analytic results reveal that thermal fluctuations can both suppress and purify coherence, guiding the design of quantum optical systems resilient to loss.

Unified Quantum Theory of Far-Field Spatial Coherence from Lossy Objects

Introduction and Motivation

The spatial coherence of light in the far field fundamentally controls interference phenomena, with implications spanning classical imaging and stellar interferometry to advanced quantum communication and continuous-variable entanglement distribution. Conventional treatments decompose spatial coherence effects into distinct paradigms:

• Classical Scattering: Describes correlation redistribution by cold, lossless objects, neglecting intrinsic quantum fluctuations and thermal emission.
• Quantum Scattering Theory (QST): Considers nonclassical states but typically omits dissipative processes and thermal emission, treating media as ideal dielectrics.
• Fluctuational Electrodynamics: Captures thermal emission via phenomenological noise currents, precluding consistent accounting of incident quantum states and unable to describe their concurrent interplay with dissipation-induced emission.

The absence of a rigorous, unified approach prevents accurate modeling of realistic open quantum optical systems, especially where geometric shape, absorption, and quantum statistics all crucially interact. This work develops a first-principles theory unifying quantum scattering and thermal emission from arbitrary lossy objects, leveraging the modified Langevin noise formalism (MLNF) and recent advances in canonical quantization for macroscopic electromagnetism.

Theoretical Framework

Modified Langevin Noise Formalism and Quantum Scattering Structure

The MLNF provides a canonical quantization prescription for lossy, dispersive magneto-dielectric objects of arbitrary shape. The total field operator decomposes into contributions from:

• Scattering Polariton Operators ($\hat{\mathbf{g}}_{\omega s}$): External, propagating vacuum modes incident on the object.
• Localized Electric and Magnetic Polariton Operators ($\hat{\mathbf{f}}_{\omega e,m}$): Internal, dissipative polaritonic degrees of freedom, proportional to $\text{Im}[\varepsilon_\omega(\mathbf{r})]$ and $\text{Im}[\mu_\omega(\mathbf{r})]$, vanishing in the lossless limit.

This yields an explicit separation of elastic scattering and emission processes at the operator level, enabling the exact tracing of quantum coherence and dissipative effects throughout interaction. The outgoing far-field correlation is captured via transmission and emission dyadics derived from the object's classical electromagnetic response.

Rigorous Input-Output Relations and Unitarity

The scattering process is governed by algebraic input-output (IO) relations connecting incoming and outgoing polariton operators. Energy conservation and global unitarity enforce constraints:

• The emission dyadics are strictly linked to absorption dyadics by microscopic symmetry, enforcing Kirchhoff's law as a direct corollary of the IO formalism.
• The total correlation in the outgoing field is an algebraic superposition:

$$\text{Outgoing Coherence} = \text{Transmission-filtered Incident Coherence} + \text{Intrinsic Thermal Emission}$$

with the unitarity defect of the scattered field exactly balanced by the thermal emission channels.

Far-Field Spatial Coherence: Mechanism and General Results

Separation of Elastic Scattering and Thermal Emission

For arbitrary quantum illumination and object temperature, the outgoing spatial correlation dyadic is:

• First Term (Scattering): Transmission dyadic non-unitarily filters the incident quantum state's spatial correlations.
• Second Term (Thermal Emission): Internal dissipation stochastically generates far-field coherence, formally providing a quantum-vectorial generalization of the van Cittert-Zernike theorem.

The two processes differ fundamentally in the transparency limit: elastic scattering persists, thermal emission strictly vanishes. The emission term precisely compensates for the transmission's non-unitarity as required by macroscopic conservation.

The Generalized Degree of Spatial Coherence

The outgoing degree of spatial coherence, $\mu^2$, is defined through the Frobenius norm of the correlation dyadic, quantifying the impact of geometry and dissipation on observable interference. Crucially, the outgoing coherence structure can be entirely captured via the classical transmission dyadics, making the approach operationally accessible for numerical or analytic computation for arbitrary geometries.

Numerical and Analytical Results Across Key Regimes

Thermal Illumination (Chaotic Bath)

Under isotropic blackbody illumination, the framework exactly recovers and extends macroscopic thermodynamic limits:

• Transparent Objects: No spatial correlations are generated; outgoing coherence is identical to the chaotic incident background ($\mu^2=0$ for $\mathbf{n} \neq \mathbf{n}'$).
• Global Equilibrium: Scattering and emission perfectly cancel, resulting in perfect thermal cloaking; the object cannot be detected via spatial coherence.
• Cold Scatterer Limit ($T_\mathrm{em}=0$): Passive absorbers in a hot bath create structured "thermal shadows," exhibiting spatial coherence profiles identical to those of thermal emitters in vacuum—a prediction that rigorously links absorption and emission at the quantum correlation level.
• Analytic Results for Rayleigh Particles: For a subwavelength spherical scatterer, the spatial coherence degree is explicitly derived, revealing that transverse vectorial effects impose a non-uniform, dipole-like spatial coherence even for purely incoherent sources.

Coherent Illumination (Deterministic Phase)

With monochromatic laser fields:

• Thermal emission acts as a non-unitary decoherence channel: Spatial coherence is always suppressed below unity except for zero temperature or perfectly transparent objects.
• Thermodynamic phase diagram: There is an exact analytic threshold temperature $T_\mathrm{th}$ above which thermal fluctuations overwhelm coherent phase. For nanostructures, this threshold is severely reduced due to the divergent absorption-to-scattering ratio: $\hat{\mathbf{f}}_{\omega e,m}$0 for $\hat{\mathbf{f}}_{\omega e,m}$1 (Rayleigh regime). For bulk objects, $\hat{\mathbf{f}}_{\omega e,m}$2 saturates, set by the incident photon energy and cross-sectional ratio.
• Exact analytic formulas: For a Rayleigh scatterer, the transition from fully coherent to geometry-driven partially coherent emission is given, with limiting behaviors directly governed by the operational signal-to-noise ratio between elastic scattering and thermal emission channels.

Nonclassical Illumination (Spatially Entangled Biphotons)

Illumination by continuous-variable entangled biphoton states yields:

• Direct tradeoff between non-local quantum entanglement and spatial coherence: Higher Schmidt rank (entanglement) directly translates to reduced observable spatial coherence, as established by explicit analytic expressions.
• Geometric purification via lossy filtering: In regimes where thermal emission is negligible, dissipative objects can enhance outgoing spatial coherence (increase $\hat{\mathbf{f}}_{\omega e,m}$3) by mode-selectively filtering and absorbing higher Schmidt modes, effectively reducing the rank of the mixed state and enhancing the degree of coherence. This process, however, strictly reduces the total transmitted biphoton flux, representing a direct tradeoff between coherence and transmission.
• Scaling with geometry and thermodynamics: For subwavelength objects, the quantum signature is dominated by thermal emission unless $\hat{\mathbf{f}}_{\omega e,m}$4 is very small; in bulk or resonance-engineered systems, quantum and thermal effects can compete at accessible photon energies.

Implications and Perspective

Theoretical and Practical Advances

This approach rigorously unifies the quantum optical theory of scattering and thermal emission, providing:

• First-principles derivation (without phenomenological assumptions or ad hoc noise sources) of spatial coherence in open, dissipative quantum systems.
• Explicit algebraic bounds on coherence degradation, spatial shadowing, and the crossover between classical and quantum regimes, applicable to any geometry or material.
• Operational protocols for engineering quantum photonic components and structured-light networks robust to loss, and for optimizing entanglement transfer through dissipative channels.

Experimental Pathways and Future Directions

Recent progress in mid-infrared single-photon detection and intensity interferometry directly enables the experimental verification of predicted phenomena such as:

• Observation of thermal shadows cast by passive absorbers in chaotic backgrounds.
• Geometry-induced purification of incident entangled states via engineered scattering structures.
• Testing the derived thermodynamic phase diagrams for nanostructures at different frequencies and temperatures.

Natural extensions include:

• Characterization of near-field spatial coherence where evanescent, dissipative quantum fluctuations dominate.
• Design of quantum thermal metamaterials with tailored coherence properties.
• Integration with quantum information processing platforms for robust entanglement transport and detection in the presence of realistic environmental coupling.

Conclusion

This work establishes a unified, canonical framework for the far-field spatial coherence generated by lossy objects, accommodating any quantum or classical illumination and arbitrary dissipative geometry. By analytically separating and linking the roles of mode-selective filtering (scattering) and stochastic quantum emission (dissipation), the theory rigorously quantifies the interplay between geometry, loss, and quantum statistics from first principles. It provides a robust foundation for the development and analysis of practical, loss-resilient quantum optical systems and sets the stage for the exploration and engineering of coherence and entanglement in realistic, open electromagnetic environments.

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Reference:

A. Ciattoni, "Far-field spatial coherence driven by lossy objects: first-principles approach unifying scattering of quantum light and thermal emission" (2607.00653)