The standard Langevin noise formalism (LNF) is a well-established approach for macroscopic quantum electrodynamics in unbounded lossy media, and accordingly, its application to finite objects requires endowing the surrounding vacuum with a fictitious absorption that is taken to asymptotically vanish. Existing justifications for such a singular limiting procedure have been mainly restricted to phenomena governed by vacuum fluctuations and medium-induced interactions (Casimir-Polder forces, thermal radiation, quantum emitter dynamics, etc.), thereby strictly excluding scenarios that involve the scattering of external quantum radiation. Conversely, the modified Langevin noise formalism (MLNF) has recently been established as the complete canonical quantum theory for arbitrary macroscopic systems and electromagnetic processes, entirely bypassing the need for limiting procedures, treating incident quantum radiation fields as independent degrees of freedom, and exactly reducing to the LNF for unbounded media. In this work, we demonstrate that the LNF limiting procedure is universally valid by showing that it recovers any physical prediction of the MLNF across all regimes, including arbitrary quantum scattering processes. Specifically, we introduce a suitable mapping associating any MLNF state and observable with a corresponding entity within the LNF augmented by fictitious vacuum absorption. We then rigorously prove that, as the fictitious absorption tends to zero, every LNF matrix element constructed via this mapping exactly converges to its MLNF original, so that the limiting procedure establishes the asymptotic convergence of the LNF to the MLNF.
The standard Langevin noise formalism (LNF) is a well-established approach for macroscopic quantum electrodynamics in unbounded lossy media, and accordingly, its application to finite objects requires endowing the surrounding vacuum with a fictitious absorption that is taken to asymptotically vanish. Existing justifications for such a singular limiting procedure have been mainly restricted to phenomena governed by vacuum fluctuations and medium-induced interactions (Casimir-Polder forces, thermal radiation, quantum emitter dynamics, etc.), thereby strictly excluding scenarios that involve the scattering of external quantum radiation. Conversely, the modified Langevin noise formalism (MLNF) has recently been established as the complete canonical quantum theory for arbitrary macroscopic systems and electromagnetic processes, entirely bypassing the need for limiting procedures, treating incident quantum radiation fields as independent degrees of freedom, and exactly reducing to the LNF for unbounded media. In this work, we demonstrate that the LNF limiting procedure is universally valid by showing that it recovers any physical prediction of the MLNF across all regimes, including arbitrary quantum scattering processes. Specifically, we introduce a suitable mapping associating any MLNF state and observable with a corresponding entity within the LNF augmented by fictitious vacuum absorption. We then rigorously prove that, as the fictitious absorption tends to zero, every LNF matrix element constructed via this mapping exactly converges to its MLNF original, so that the limiting procedure establishes the asymptotic convergence of the LNF to the MLNF.
The modified Langevin noise formalism (MLNF) models the interaction of the quantized electromagnetic field with an arbitrary lossy magneto-dielectric object placed in vacuum using three types of non-interacting bosonic polaritons: scattering, electric, and magnetic. These respectively represent free-space photons scattered by the object, and photons radiated by quantized electric and magnetic dipolar sources embedded within its volume. Recently [A. Ciattoni, Phys. Rev. A 110, 013707 (2024)], this formalism was justified from the canonical quantization of macroscopic electromagnetism (CQME) [Philbin, New J. Phys. 12, 123008 (2010)] in the Heisenberg picture. This was achieved by identifying the polariton operators within the formal solution of the macroscopic Maxwell equations, assuming they obey bosonic commutation relations to retrieve the canonical ones, and showing they diagonalize the CQME Hamiltonian. However, the explicit functional dependence of these polaritons on the underlying canonical field operators remained undetermined. In this paper, we derive the exact analytical expressions for the polariton operators in terms of the canonical CQME field operators. Using these mappings, we provide a direct and rigorous derivation of the MLNF from the canonical theory in the Schrödinger picture. Our derivation is structured in three foundational steps: 1) adopting the derived analytical expressions as the constitutive definitions of the polariton operators; 2) mathematically proving that these operators are strictly bosonic as a direct consequence of the canonical commutation relations; and 3) demonstrating that they exactly diagonalize the macroscopic CQME Hamiltonian.
In this paper, we model the interaction of a quantum emitter with a finite-size dispersive dielectric object in an unbounded space within the framework of macroscopic quantum electrodynamics, using the modified Langevin noise formalism, without any restrictions on the emitter level structure or dipole operator. The quantized electromagnetic field consists of two contributions: the medium-assisted field, which accounts for the electromagnetic field generated by the noise polarization currents of the dielectric, and the scattering-assisted field, which takes into account the electromagnetic field incoming from infinity and scattered by the dielectric. We show that the emitter couples to two distinct bosonic baths: a medium-assisted bath and a scattering-assisted bath, each characterized by its own spectral density. We identify the conditions under which the electromagnetic environment composed of these two baths can be effectively replaced by a single bosonic bath, so that the reduced dynamics of the quantum emitter remain unchanged. In particular, when the initial states of the medium- and scattering-assisted baths are thermal states with the same temperature, we find that a single bosonic bath with a spectral density equal to the sum of the medium-assisted and scattering-assisted spectral densities is equivalent to the original electromagnetic environment.
In this paper, we model the interaction of a quantum emitter with a finite-size dispersive dielectric object in unbounded space within the framework of macroscopic quantum electrodynamics, using the modified Langevin noise formalism. The quantized electromagnetic field consists of two contributions: the medium-assisted field, which accounts for the electromagnetic field generated by the noise polarization currents of the dielectric, and the scattering-assisted field, which takes into account the electromagnetic field incoming from infinity and scattered by the dielectric. We show that the emitter couples to two distinct bosonic reservoirs: a medium-assisted reservoir and a scattering-assisted reservoir, each characterized by its own spectral density. We then use emitter-centered modes to reduce the degrees of freedom of both reservoirs. Eventually, we identify the conditions under which the electromagnetic environment composed of these two reservoirs can be effectively replaced by a single bosonic reservoir so that the reduced time evolution of the quantum emitter remains unchanged. In particular, when the initial states of the medium- and scattering-assisted reservoirs are thermal quantum states at the same temperature, we find that a single bosonic bath with a spectral density equal to the sum of the medium- and scattering-assisted spectral densities is equivalent to the original electromagnetic environment.
We investigate the overall optomechanical force experienced by a macroscopic lossy object in free space under external quantum illumination. To this end, utilizing the Modified Langevin Noise Formalism (MLNF), we derive the time-averaged expectation value of the Maxwell stress tensor for a non-equilibrium scenario in which the incoming scattering field is prepared in an arbitrary mixed quantum state, while the medium-assisted field is maintained in local thermal equilibrium. In the limit of full radiation-matter thermal equilibrium, our expression exactly recovers the well-known fluctuation-dissipation relation governing the Casimir effect, and, under coherent illumination, it yields the standard classical radiation pressure. We demonstrate that by driving the scattering field with an anisotropic, multimode squeezed vacuum state, the spatial profile of the electromagnetic quantum fluctuations can be engineered to exhibit broken rotational symmetry, thereby inducing a purely quantum mechanical force acting on the object. Such mechanical interaction is generated in the strict absence of a mean field, ⟨E^⟩=0, and its non-classical nature is evidenced by its reliance on second-order field correlations ⟨E^2⟩, unlike classical optical radiation pressure governed by the squared mean field ⟨E^⟩2. Applying this exact formulation to a homogeneous lossy sphere, we demonstrate the experimental feasibility of the effect using realistic material parameters and optical estimations. Ultimately, we establish a general formalism for macroscopic quantum optomechanics that operates beyond the constraints of thermal equilibrium, enabling the prediction of regimes where the purely quantum force circumvents classical mean fields and shot noise while preserving the object's macroscopic quantum coherence.
The modified Langevin noise formalism [A. Drezet, Phys. Rev. A 95, 023831 (2017); O. D. Stefano, S. Savasta, and R. Girlanda, J. Mod. Opt. 48, 67 (2001)] has been proposed for the correct characterization of quantum electromagnetic fields in the presence of finite-size lossy dielectric objects in free space. The main modification to the original one [T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (1996); H. T. Dung, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998)] (also known as the Green's function approach for a bulk inhomogeneous lossy dielectric medium) was to introduce another fluctuating source in reaction to the radiation loss. Consequently, the resulting electric field operator is now determined by (i) boundary-assisted and (ii) medium-assisted fields on an equal footing, which originate from radiation and medium losses, respectively. However, due to the lengthy mathematical manipulation and complicated concepts, the validity of the modified Langevin noise formalism has not been clearly confirmed yet. In this work, we propose and develop a numerical framework for the modified Langevin noise formalism by exploiting computational electromagnetic methods. Specifically, we present utilization of the finite-element method to numerically solve plane-wave-scattering and point-source-radiation problems whose solutions are boundary-assisted and medium-assisted fields, respectively. We numerically validated the modified Langevin noise model calculating the Purcell factor of a two-level atom inside or outside a lossy dielectric slab. The proposed numerical framework is particularly useful for analyzing the dynamics of multilevel atoms near plasmonic structures or metasurfaces in the open space.
Macroscopic quantum electrodynamics (MQED) provides a unified framework for describing quantum electromagnetic fields in arbitrary macroscopic environments. Central to this theory is the field correlation, which governs both radiative (e.g., Lamb shifts and the Purcell effect) and mechanical phenomena, such as van der Waals and Casimir forces. In this tutorial, we provide an overview of MQED and its extension to active media, highlighting fluctuation-induced forces as manifestations of gain-modified field correlations.