Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2301.11993 · Physical Review A

Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves

Yue Jiang, Yefeng Mei, Shengwang Du

Published May 4, 2023• 6 citations
arXivHTMLPDF

Abstract

We provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated electromagnetic fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double-$\mathrm{\ensuremath{\Lambda}}$ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. We find that a complex-valued nonlinear coupling coefficient can lead to noises even without linear gain or loss. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.

DOI
10.1103/physreva.107.053703

Related papers

Theory of high-gain twin-beam generation in waveguides: From Maxwell's equations to efficient simulation

Nicolás Quesada, Gil Triginer, Mihai D. Vidrighin +1 more

We provide an efficient method for the calculation of high-gain, twin-beam generation in waveguides derived from a canonical treatment of Maxwell's equations. Equations of motion are derived that naturally accommodate photon generation via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing and, also, include the effects both of self-phase modulation of the pump and of cross-phase modulation of the twin beams by the pump. The equations we solve involve fields that evolve in space and are labeled by a frequency. We provide a proof that these fields satisfy bona fide commutation relations and that in the distant past and future they reduce to standard time-evolving Heisenberg operators. Having solved for the input-output relations of these Heisenberg operators we also show how to construct the ket describing the quantum state of the twin beams. Finally, we consider the example of high-gain SPDC in a waveguide with a flat nonlinearity profile, for which our approach provides an explicit solution that requires only a single matrix exponentiation.

arXiv:1907.01958•Physical Review A•Sep 21, 2020•41 citations

Microscopic Origin of Superradiant Biphoton Emission in Atomic Ensembles

Zi-Yu Liu, Jiun-Shiuan Shiu, Wei-Lin Chen +1 more

Superradiant biphoton emission from atomic ensembles provides a powerful route to generating highly correlated quantum light, yet its microscopic physical origin has remained incompletely understood. In particular, it is often unclear how collective enhancement, spontaneous emission, and vacuum fluctuations jointly give rise to both paired biphoton generation and unavoidable unpaired background within a single, self-consistent framework. Here we present a fully quantum microscopic theory within a unified Heisenberg--Langevin--Maxwell framework that explicitly incorporates dissipation and quantum noise, thereby revealing the microscopic origin of superradiant biphoton emission in atomic ensembles. The theory provides a consistent description of parametric gain and unpaired noise within the same open-quantum-system framework and applies to both Doppler-free cold atomic ensembles and Doppler-broadened warm vapors. In the high-optical-depth regime, the coupled propagation equations admit analytical solutions, under which the biphoton dynamics rigorously reduce to an effective collective two-level emission process. Within this limit, the biphoton correlation time and spectral properties are shown to obey closed-form scaling relations governed by optical depth and excited-state decoherence. Our results establish a unified microscopic picture of superradiant biphoton generation and clarify the fundamental role of vacuum fluctuations and dissipation in setting the brightness, pairing efficiency, and temporal structure of atomic biphoton sources, with direct relevance to quantum networking and atomic quantum interfaces.

quant-pharXiv:2602.11438•Feb 13, 2026•1 citation

QED in dispersing and absorbing media

L. Knöll, Stefan Scheel, D.‐G. Welsch

After giving an outline of the quantization scheme based on the microscopic Hopfield model of a dielectric bulk material, we show how the classical phenomenological Maxwell equations of the electromagnetic field in the presence of dielectric matter of given space- and frequency-dependent complex permittivity can be transferred to quantum theory. Including in the theory the interaction of the medium-assisted field with atomic systems, we present both the minimal-coupling Hamiltonian and the multipolar-coupling Hamiltonian in the Coulomb gauge. To illustrate the concept, we discuss the input--output relations of radiation and the transformation of radiation-field quantum states at absorbing four-port devices, and the spontaneous decay of an excited atom near the surface of an absorbing body and in a spherical micro-cavity with intrinsic material losses. Finally, we give an extension of the quantization scheme to other media such as amplifying media, magnetic media, and nonlinear media.

arXiv:quant-ph/0006121•Jun 27, 2000•49 citations

First order Maxwell operator formalism for macroscopic quantum electrodynamics

Ishita Agarwal, Ankit Kundu, Christian M. Lange +1 more

Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both $\mathbf{E}$ and $\mathbf{H}$ and keeps those boundary terms, naturally leading to a quantum input-output formalism. We recast Maxwell's equations as an operator equation for the dual field $\mathit{E}$=$[\mathbf{E},\mathbf{H}]^T$, whose first-order Green operator $g$ propagates the electromagnetic state between surfaces. Symmetries of the Maxwell operator under energy and reciprocal inner products yield the propagation formula, Lorentz reciprocity, and a generalized optical theorem, with minimal vector calculus. Quantizing via a Heisenberg-Langevin approach for absorptive, dispersive media yields two independent quantum noise sources: bulk Langevin operators from material absorption and input-output field operators at the boundary. Expressing the interior field in terms of these operators and the Green propagator yields an exact closed commutation relation $[{\mathit{E}},{\mathit{E}}^\dagger]\propto \mathrm{Im}\,g$, consistent with the fluctuation-dissipation theorem. This identity holds even when dielectrics extend to the boundary, as in waveguide input-output problems, and enables quantum input-output descriptions of complex photonic structures where the Green's function is obtained numerically, extending the framework beyond cavities and waveguides.

quant-pharXiv:2603.27475•Mar 31, 2026•0 citations

Photon-pair generation in a lossy microring resonator. II. Entanglement in the output mixed Gaussian squeezed state

Paul M. Alsing, Edwin E. Hach

In this work we examine the entanglement of the output signal-idler squeezed vacuum state in the Heisenberg picture as a function of the coupling and internal propagation loss parameters of a microring resonator. Using the log negativity as a measure of entanglement for a mixed Gaussian state, we examine the competitive effects of the transfer matrix that encodes the classical phenomenological loss, as well as the matrix that that incorporates the coupling and internal propagation losses due to the quantum Langevin noise fields required to preserve the unitarity of the composite system (signal-idler) and environment (noise) structure.

arXiv:1708.01338•Physical Review A•Sep 27, 2017•6 citations

Quantization of the electromagnetic field in dielectrics

B. Hüttner, Stephen Mark Barnett

We present a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on a microscopic model, in which the medium is represented by a collection of interacting matter fields. We calculate the exact eigenoperators for the coupled system and express the electromagnetic field operators in terms of them. The dielectric constant of the medium is explicitly derived and is shown to satisfy the Kramers-Kronig relations. We apply these results to treat the propagation of light in dielectrics and obtain simple expressions for the electromagnetic field in the medium in terms of space-dependent creation and annihilation operators. These operators satisfy a set of equal-space commutation relations and obey spatial Langevin equations of evolution. This justifies the use of such operators in phenomenological models in quantum optics. We also obtain two interesting relationships between the group and the phase velocity in dielectrics.

Physical Review A•Oct 1, 1992•631 citations
We provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated electromagnetic fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double-\ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. We find that a complex-valued nonlinear coupling coefficient can lead to noises even without linear gain or loss. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.
We provide an efficient method for the calculation of high-gain, twin-beam generation in waveguides derived from a canonical treatment of Maxwell's equations. Equations of motion are derived that naturally accommodate photon generation via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing and, also, include the effects both of self-phase modulation of the pump and of cross-phase modulation of the twin beams by the pump. The equations we solve involve fields that evolve in space and are labeled by a frequency. We provide a proof that these fields satisfy bona fide commutation relations and that in the distant past and future they reduce to standard time-evolving Heisenberg operators. Having solved for the input-output relations of these Heisenberg operators we also show how to construct the ket describing the quantum state of the twin beams. Finally, we consider the example of high-gain SPDC in a waveguide with a flat nonlinearity profile, for which our approach provides an explicit solution that requires only a single matrix exponentiation.
Superradiant biphoton emission from atomic ensembles provides a powerful route to generating highly correlated quantum light, yet its microscopic physical origin has remained incompletely understood. In particular, it is often unclear how collective enhancement, spontaneous emission, and vacuum fluctuations jointly give rise to both paired biphoton generation and unavoidable unpaired background within a single, self-consistent framework. Here we present a fully quantum microscopic theory within a unified Heisenberg--Langevin--Maxwell framework that explicitly incorporates dissipation and quantum noise, thereby revealing the microscopic origin of superradiant biphoton emission in atomic ensembles. The theory provides a consistent description of parametric gain and unpaired noise within the same open-quantum-system framework and applies to both Doppler-free cold atomic ensembles and Doppler-broadened warm vapors. In the high-optical-depth regime, the coupled propagation equations admit analytical solutions, under which the biphoton dynamics rigorously reduce to an effective collective two-level emission process. Within this limit, the biphoton correlation time and spectral properties are shown to obey closed-form scaling relations governed by optical depth and excited-state decoherence. Our results establish a unified microscopic picture of superradiant biphoton generation and clarify the fundamental role of vacuum fluctuations and dissipation in setting the brightness, pairing efficiency, and temporal structure of atomic biphoton sources, with direct relevance to quantum networking and atomic quantum interfaces.
After giving an outline of the quantization scheme based on the microscopic Hopfield model of a dielectric bulk material, we show how the classical phenomenological Maxwell equations of the electromagnetic field in the presence of dielectric matter of given space- and frequency-dependent complex permittivity can be transferred to quantum theory. Including in the theory the interaction of the medium-assisted field with atomic systems, we present both the minimal-coupling Hamiltonian and the multipolar-coupling Hamiltonian in the Coulomb gauge. To illustrate the concept, we discuss the input--output relations of radiation and the transformation of radiation-field quantum states at absorbing four-port devices, and the spontaneous decay of an excited atom near the surface of an absorbing body and in a spherical micro-cavity with intrinsic material losses. Finally, we give an extension of the quantization scheme to other media such as amplifying media, magnetic media, and nonlinear media.
Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both E\mathbf{E}E and H\mathbf{H}H and keeps those boundary terms, naturally leading to a quantum input-output formalism. We recast Maxwell's equations as an operator equation for the dual field E\mathit{E}E=[E,H]T[\mathbf{E},\mathbf{H}]^T[E,H]T, whose first-order Green operator ggg propagates the electromagnetic state between surfaces. Symmetries of the Maxwell operator under energy and reciprocal inner products yield the propagation formula, Lorentz reciprocity, and a generalized optical theorem, with minimal vector calculus. Quantizing via a Heisenberg-Langevin approach for absorptive, dispersive media yields two independent quantum noise sources: bulk Langevin operators from material absorption and input-output field operators at the boundary. Expressing the interior field in terms of these operators and the Green propagator yields an exact closed commutation relation [E,E†]∝Im g[{\mathit{E}},{\mathit{E}}^\dagger]\propto \mathrm{Im}\,g[E,E†]∝Img, consistent with the fluctuation-dissipation theorem. This identity holds even when dielectrics extend to the boundary, as in waveguide input-output problems, and enables quantum input-output descriptions of complex photonic structures where the Green's function is obtained numerically, extending the framework beyond cavities and waveguides.
In this work we examine the entanglement of the output signal-idler squeezed vacuum state in the Heisenberg picture as a function of the coupling and internal propagation loss parameters of a microring resonator. Using the log negativity as a measure of entanglement for a mixed Gaussian state, we examine the competitive effects of the transfer matrix that encodes the classical phenomenological loss, as well as the matrix that that incorporates the coupling and internal propagation losses due to the quantum Langevin noise fields required to preserve the unitarity of the composite system (signal-idler) and environment (noise) structure.
We present a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on a microscopic model, in which the medium is represented by a collection of interacting matter fields. We calculate the exact eigenoperators for the coupled system and express the electromagnetic field operators in terms of them. The dielectric constant of the medium is explicitly derived and is shown to satisfy the Kramers-Kronig relations. We apply these results to treat the propagation of light in dielectrics and obtain simple expressions for the electromagnetic field in the medium in terms of space-dependent creation and annihilation operators. These operators satisfy a set of equal-space commutation relations and obey spatial Langevin equations of evolution. This justifies the use of such operators in phenomenological models in quantum optics. We also obtain two interesting relationships between the group and the phase velocity in dielectrics.