Photon Gas Research Articles

Dimensionless fluctuations balance applied to statistics and quantum physics

This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi–Dirac, and Bose–Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann’s entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger’s Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics.

Enigmatic factor of 4/3 in electromagnetic momentum of a moving spherical capacitor

The electromagnetic energy–momentum of a moving charged spherical capacitor may be calculated by a 4-vector Lorentz transformation from the energy in the rest frame. However, energy–momentum of the moving system computed directly from electromagnetic fields yields extra terms; in particular a factor of 4/3 in momentum appears, similar to that encountered in the classical electron model, where this enigmatic factor has been a source of confusion for more than a century. There have been many attempts to eliminate this ‘unwanted’ factor, noteworthy among them is a modification in electromagnetic field energy–momentum definition that has entered even standard textbooks. Here it is shown that in a moving charged spherical capacitor, such a factor of 4/3 in the electromagnetic momentum arises naturally from electromagnetic forces in system or equivalently from terms in the Maxwell stress tensor; contributions that do not otherwise show up in 4-vector transformations. A similar factor of 4/3 in the momentum of a perfect fluid comprising a randomly moving ultra-relativistic gas molecules or an isotropic photon gas, filling an uncharged spherical capacitor in motion, appears owing to the contribution of pressure. Thus, genesis of the ‘enigmatic’ factor of 4/3 can be traced to pressure or stress whose presence in the system may even be of non-electromagnetic origin and where the proposed modifications in energy–momentum definition do not even come into picture, implying there is no justification in modifying the standard definition of electromagnetic energy–momentum.

Temperature fluctuations in a relativistic gas: Pressure corrections and possible consequences in the deconfinement transition

In this work, we study the effects of random temperature fluctuations on the partition function of a quantum system by means of the . This picture provides a conceptual model for a quantum nonequilibrium system, depicted as an ensemble of subsystems at different temperatures, randomly distributed with respect to a given mean value. We then assume the temperature displays stochastic fluctuations T=T0+δT with respect to its ensemble average value T0, with zero mean δT¯=0 and standard deviation δT2¯=Δ. By means of the replica method, we obtain the average grand canonical potential, leading to the equation of state and the corresponding excess pressure caused by these fluctuations with respect to the equilibrium system at a uniform temperature. Our findings reveal an increase in pressure as the system’s ensemble average temperature T0 rises, consistently exceeding the pressure observed in an equilibrium state. We applied our general formalism to three paradigmatic physical systems; the relativistic Fermi gas, the ideal gas of photons, and a gas of non-Abelian gauge fields (gluons) in the noninteracting limit. Finally, we explore the implications for the deconfinement transition in the context of the simple bag model, where we show that the critical temperature decreases. Published by the American Physical Society 2024

Bose–Einstein condensation of photons in a vertical-cavity surface-emitting laser

Many bosons can occupy a single quantum state without a limit. It is described by the quantum-mechanical Bose–Einstein statistic, which allows Bose–Einstein condensation at low temperatures and high particle densities. Photons, historically the first considered bosonic gas, were late to show this phenomenon, observed in rhodamine-filled microcavities and doped fibre cavities. These findings have raised the question of whether condensation is also common in other laser systems with potential technological applications. Here we show the Bose–Einstein condensation of photons in a broad-area vertical-cavity surface-emitting laser with a slight cavity-gain spectral detuning. We observed a Bose–Einstein condensate in the fundamental transversal optical mode at a critical phase-space density. The experimental results follow the equation of state for a two-dimensional gas of bosons in thermal equilibrium, although the extracted spectral temperatures were lower than the device’s. This is interpreted as originating from the driven-dissipative nature of the photon gas. In contrast, non-equilibrium lasing action is observed in the higher-order modes in more negatively detuned device. Our work opens the way for the potential exploration of superfluid physics of interacting photons mediated by semiconductor optical nonlinearities. It also shows great promise for enabling single-mode high-power emission from a large-aperture device.

Jet formation model from accretion disks of electron-ion-photon gas

The problem of Astrophysical Jet formation from relativistic accretion disks through the establishment of relativistic disk-powerful jet equilibrium structure is studied applying the Beltrami-Bernoulli equilibrium approach of Shatashvili and Yoshida 2011; Arshilava et al. 2019. Accretion disk is weakly magnetized consisting of fully ionized relativistic electron-ion plasma and photon gas strongly coupled to electrons due to Thompson Scattering. Analysis is based on the generalized Shakura-Sunyaev α-turbulent dissipation model for local viscosity (being the main source of accretion), in which the contributions from both the photon and ion gases are taken into account. Ignoring the self-gravitation in the disk we constructed the analytical self-similar solutions for the equilibrium relativistic disk-jet structure characteristic parameters in the field of gravitating central compact object for the force-free condition. It is shown, that the magnetic field energy in the Jet is several orders greater compared to that of accretion disk, while jet-outflow is locally Super-Alfvénic with local Plasma-beta <1 near the jet-axis. The derived solutions can be used to analyze the astrophysical jets observed in binary systems during the star formation process linking the jet properties with the parameters of relativistic accretion disks of electron-ion-photon gas.

Modified Temperature–Redshift Relation and Ultra-high-energy Cosmic Ray Propagation

We reexamine the interactions of ultra-high-energy cosmic rays (UHECRs) with photons from the cosmic microwave background (CMB) under a changed, locally nonlinear temperature–redshift relation T(z). This changed temperature–redshift relation has recently been suggested by the postulate of subjecting thermalized and isotropic photon gases such as the CMB to an SU(2) rather than a U(1) gauge group. This modification of ΛCDM is called SU(2)CMB, and some cosmological parameters obtained by SU(2)CMB seem to be in better agreement with local measurements of the same quantities, in particular H 0 and S8. In this work, we apply the reduced CMB photon density under SU(2)CMB to the propagation of UHECRs. This leads to a higher UHECR flux just below the ankle in the cosmic ray spectrum and slightly more cosmogenic neutrinos under otherwise equal conditions for emission and propagation. Most prominently, the proton flux is significantly increased below the ankle (5 × 1018 eV) for hard injection spectra and without considering the effects of magnetic fields. The reduction in CMB photon density also favors a decreased cosmic ray source evolution than the best fit using ΛCDM. In consequence, it seems that SU(2)CMB favors sources that evolve like the star formation rate, such as starburst galaxies and gamma-ray bursts, over active galactic nuclei as origins of UHECRs. We conclude that the question about the nature of primary sources of UHECRs is directly affected by the assumed temperature–redshift relation of the CMB.

Calorimetry of Photon Gases in Nonlinear Multimode Optical Fibers

Recent studies have shown that light propagating in a nonlinear, highly multimode system can thermalize in a manner totally analogous to that encountered in traditional statistical mechanics. At thermal equilibrium, the system’s entropy is at a maximum, in full accord with the second law of thermodynamics. In such arrangements, the entropy is extremized once the statistical power allocation among modes associated with this photon gas attains a Rayleigh-Jeans distribution that is fully characterized by an optical temperature T and a chemical potential μ. However, it has been theoretically argued that the variables T and μ represent actual thermodynamic forces that control the exchange of the respective conjugate quantities between two subsystems. In this work, we report, for the first time, optical calorimetric measurements in nonlinear multimode fibers, which unambiguously demonstrate that both the temperature T and the chemical potential μ dictate the flow of their associated extensive quantities, i.e., the energy and the optical power. Specifically, we study the process of light thermalization associated with two orthogonally polarized laser beams. Our observations are enabled by recently developed techniques that allow one to judiciously multiplex/demultiplex the optical power within various mode groups. Our results indicate that because of photon-photon collisions, “heat” only flows from a hot to a cold photon gas subsystem—thus providing an unequivocal demonstration of the second law in such all-optical thermodynamic arrangements. In addition to being fundamental, our findings provide a new approach to manipulate laser beams using thermodynamic principles. Published by the American Physical Society 2024

On the Wave-Particle Duality of the Photons and the Matter-Photon Particle Mixture Model

Relativistic quantum mechanics and many experimental results and observations show that electron-positron annihilation can produce photons; photon-matter and photon-photon interaction can also create electrons and positrons. The photons have no rest mass, e.g. relative to Lab frame, and always travel at the speed of light in the vacuum. In this paper, a rotational moving electric dipole model with negative and positive charges was proposed for photons. In a vacuum, the electric dipoles are moving in a twisted helical motion around their propagation axis (their center of mass). Photon particles are traveling at the speed of light along the propagation axis. This pair of negative and positive charged particles exerts both electrostatic and magnetic force on each other. Both forces are attractive and act as a centripetal force to keep the electric dipole in a continuous helical motion around the axis of rotation. With this model, the wave-particle duality of photons can be described simultaneously. The space is filled by matter (having rest mass relative to Lab frame) and photons (without rest mass); it is a multicomponent mixture fluid. In "vacuum", though there is no rest matter particle, but there are still photon gas particles, the total energy-momentum tensor should include rest matter particles and photon gas particles.

Thermodynamics of the electron–positron plasma at very high temperatures and sources of annihilation γ-quanta in our Galaxy

Thermodynamic properties and equations of state of the electron–positron plasma (or gas) at high and very high temperatures (T≥170 keV) are derived and investigated. We have derived a number of simple analytical formulas for the Fermi–Dirac distribution functions, which can be applied to various Fermi gases and plasmas in different cases. Almost all these formulas are represented in the form of series expansions. The coefficients in these expansions are relatively simple functions of the μT ratio, where T is the temperature and μ is the chemical potential of this Fermi system. Our approach works well for high-temperature electron–positron plasmas, which are in thermal equilibrium with the photon gas of annihilation γ-quanta, and for the model gas of fermions, where there is no radiation at all. The first case corresponds to the ultra-relativistic limit for high-temperature electron–positron plasma, while the second case represents a model Fermi gas of particles, which has some non-zero chemical potential. By investigating sources of annihilation γ-quanta in our Galaxy, we have arrived to a remarkable conclusion about the high-temperature limit in our regular (photon) optics.

Momentum work and the energetic foundations of physics. III. The unification of mechanics and electrodynamics

Albert Einstein proposed a unification of mechanics and electrodynamics by his special theory of relativity, which describes point-like particles moving through empty space. We show in this paper that mechanics and electrodynamics can be unified in Euclidean space if we relax the idea of point-like particles by describing fermions and bosons as spatially extended wave objects. Based on the process theory of gases presented in Paper II [Kalies et al., AIP Adv. 13(5), 055317 (2023)] of this series, we consider a confined photon gas and show that the momentum work of photons leads to an analogous description of matter and light and to the insight that electromagnetic energy is momentum energy. We present the unified energy equations, which do justice to the wave nature of matter, and it is the momentum work that links physical theories. Without momentum energy, which is larger in real terms than the kinetic energy of mechanics, the theoretical edifice of modern physics, at the fundamental level, is incomplete.

Anisotropic motion of an electric dipole in a photon gas near a flat conducting boundary

The quantum Brownian motion of a single neutral particle with nonzero electric dipole moment placed in a photon gas at fixed temperature and close to a conducting wall is here examined. The interaction of the particle with the photon field leads to quantum dispersions of its linear and angular momenta, whose magnitudes depend on the temperature, distance to the wall, and also on the dipole moment characteristics. It is shown that for typical experimental parameters the amount of energy held by the dipole rotation is expressively larger than the one related to the center of mass translation. Furthermore, the particle kinetic energy in presence of a thermal bath can decrease if the wall is added to the system, representing a novel quantum cooling effect where the work done by the quantum vacuum extracts energy from the particle. Finally, possible observable consequences are discussed.

Sunlight-pumped two-dimensional thermalized photon gas

The Liouville theorem states that the phase-space volume of an ensemble in a closed system remains constant. While gases of material particles can efficiently be cooled by sympathetic or laser cooling techniques, allowing for large phase-space compression upon suitable coupling to the environment, for light both the absence of an internal structure, as well as the nonconservation of photons upon contact with matter imposes fundamental limits for three-dimensional light-harvesting systems, such as fluorescence-based light concentrators. An advantageous physical situation can in principle be expected for dye-solution-filled microcavities with a mirror spacing in the wavelength range, where low-dimensional photon gases with nonvanishing and tunable chemical potential have been experimentally realized. Motivated by the goal to observe phase-space compression of sunlight by cooling the captured radiation to room temperature, in this work we theoretically show that in a lossless, harmonically confined system the phase-space volume scales as ${(\mathrm{\ensuremath{\Delta}}x\mathrm{\ensuremath{\Delta}}p/T)}^{d}=\mathrm{constant}$, where $\mathrm{\ensuremath{\Delta}}x$ and $\mathrm{\ensuremath{\Delta}}p$ denote the position and momentum spread and $d$ the dimensionality of the system ($d=1$ or 2). Experimentally, we realize a corresponding sunlight-pumped dye microcavity and demonstrate thermalization of scattered sunlight to a two-dimensional room temperature photon gas with nonvanishing chemical potential. Prospects of phase-space buildup of light by cooling, as can be feasible in systems with a two- or three-dimensional band gap, range from quantum state preparation in tailored potentials up to technical applications for more efficient collection of diffuse sunlight.

Statistical aspects of the massive photon gases in the presence of a minimal length

Statistical aspects of the massive photon gases in the presence of a minimal length

Ray-tracing Simulations and Spectral Models of X-Ray Radiation in Dusty Media

Dust can play an important role in shaping the X-ray spectra and images of astrophysical sources. In this work we report on the implementation of dust in the ray-tracing platform RefleX. We illustrate the different effects associated with the interaction between X-ray photons and dust grains, such as dust scattering, near-edge X-ray absorption fine structures, and shielding. We show how the cross sections of the photon–gas interaction change depending on the fraction of metals in dust grains (i.e., the dust depletion factor). We compare RefleX simulations to the most widely used absorption model that includes dust and show how X-ray spectra are affected by the presence of dust in the absorbing/reprocessing medium for different geometries. We also show how RefleX can be used to reproduce the dust scattering halos observed in Galactic sources, and we release the first torus X-ray spectral model that considers dust absorption and scattering (RXTorusD), to reproduce the spectra of active galactic nuclei (AGNs). RXTorusD also considers other physical processes that are not included in the most widely used AGN torus models, such as Rayleigh scattering and scattering on molecular gas, which can lead to remarkable differences in the predicted X-ray spectra for the same set of geometrical and physical parameters.

Hartree–Fock analogue theory of thermo-optic interaction

Thermo-optic interaction significantly differs from the usual particle-particle interactions in physics, as it is retarded in time. A prominent platform for realising this kind of interaction are photon Bose–Einstein condensates, which are created in dye-filled microcavities. The dye solution continually absorbs and re-emits these photons, causing the photon gas to thermalize and to form a Bose–Einstein condensate. Because of a non-ideal quantum efficiency, these cycles heat the dye solution, creating a medium that provides an effective thermo-optic photon–photon interaction. So far, only a mean-field description of this process exists. This paper goes beyond by working out a quantum mechanical description of the effective thermo-optic photon–photon interaction. To this end, the self-consistent modelling of the temperature diffusion builds the backbone of the modelling. Furthermore, the manyfold experimental timescales allow for deriving an approximate Hamiltonian. The resulting quantum theory is applied in the perturbative regime to both a harmonic and a box potential for investigating its prospect for precise measurements of the effective photon–photon interaction strength.

Exact diagonalisation of photon Bose–Einstein condensates with thermo-optic interaction

Although photon Bose–Einstein condensates have already been used for studying many interesting effects, the precise role of the photon–photon interaction is not fully clarified up to now. In view of this, it is advantageous that these systems allow measuring both the intensity of the light leaking out of the cavity and its spectrum at the same time. Therefore, the photon–photon interaction strength can be determined once via analysing the condensate broadening and once via examining the interaction-induced modifications of the cavity modes. As the former method depends crucially on the concrete shape of the trapping potential and the spatial resolution of the used camera, interferometric methods promise more precise measurements. To this end, the present paper works out the impact of the photon–photon interaction upon the cavity modes. A quantum mechanical description of the photon–photon interaction, including the thermal cloud, builds the theoretical backbone of the method. An exact diagonalisation approach introduced here exposes how the effective photon–photon interaction modifies both the spectrum and the width of the photon gas. A comparison with a variational approach based on the Gross–Pitaevskii equation quantifies the contribution of the thermal cloud in the respective applications.

Polariton dynamics in one-dimensional arrays of atoms coupled to waveguides

Photons strongly coupled to material systems constitute a novel system for realizing non-linear optics at the level of individual photons and studying the dynamics of non-equilibrium quantum many-body system. We give a simple physical polariton-picture of the dynamics of photons coupled to a one-dimensional array of two-level atoms. This picture allows a fully analytical description of the dynamics in terms of polariton scattering inside the medium and reflections of the polaritons from the edge of the array. We show that inelastic collisions, previously identified in small systems, also occur in infinite systems and are related to the existence of multiple bands in the dispersion relation. The developed theory constitutes an effective field theory for the dynamics, which can be used for studies of non-linear optics and many-body dynamics. As a specific example we map the system to the Lieb–Liniger model and show that a so-called Tonks–Girardeau gas of photons is a stable eigenstate of the system in the limit of many emitters.

Resetting photons

Starting from a frequency diffusion process for a tagged photon which simulates relaxation to the Planck law, we introduce a resetting where photons lower their frequency at random times. We consider two versions, one where the resetting to low frequency is independent of the existing frequency and a second case where the reduction in frequency scales with the original frequency. The result is a nonlinear Markov process where the stationary distribution modifies the Planck law by abundance of low-frequency occupation. The physical relevance of such photon resetting processes can be found in explorations of nonequilibrium effects, e.g., via random expansions of a confined plasma or photon gas or via strongly inelastic scattering with matter.

3D stochastic interferometer detects picometer deformations and minute dielectric fluctuations of its optical volume

AbstractSpeckle metrology harnesses the interferometric properties of disordered light to achieve remarkable sensitivities. Often relying on time-domain analysis, it is rate-limited by the acquisition of speckle images. In the present work instead, we use a frequency-domain approach which spans 8 to 10 frequency decades up to 100 MHz, and reveals minute changes of speckle decorrelation spectra. We built a 3D stochastic interferometer using a centimeter-sized quartz-powder cavity with arbitrary shape and high Lambertian reflectivity. Filled with a coherent monochromatic photon gas, it creates statistically isotropic and homogeneous 3D interference patterns, whose variations arise from cavity deformations or fluctuations of the dielectric tensor field inside. Speckle decorrelation depends neither on where the perturbation sits nor on where it is measured. With an average 62 m photon transit path and a finesse of 10500, cavity deformations are detected with a power noise floor of 4 × 10−3 pm2, i.e., 2.7 pm at 1 kHz. We also demonstrate a 100-fold sensitivity gain compared to conventional light scattering techniques when probing thermal motions of single and multiply scattering colloids.

Revisiting the Boltzmann Derivation of the Stefan Law

The Stefan–Boltzmann (SB) law relates the radiant emittance of an ideal black-body cavity at thermal equilibrium to the fourth power of the absolute temperature [Formula: see text] as [Formula: see text], with [Formula: see text][Formula: see text]W[Formula: see text]m[Formula: see text][Formula: see text]K[Formula: see text] the SB constant, first estimated by Stefan to within [Formula: see text] of the present theoretical value. The law is an important achievement of modern physics since, following Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563], its microscopic derivation implies the quantization of the energy related to the electromagnetic field spectrum. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By contrast, the Boltzmann derivation implies two assumptions not justified within a classical approach, namely: (i) the zero value of the chemical potential and (ii) the internal energy of the black body with a finite value and dependent from both temperature and volume. By using Planck [Ueber das Gesetz der Energieverteilung im Normalspectrum [On the law of distribution of energy in the normal spectrum], Ann. Phys. 4 (1901) 553–563] quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation free from the above assumptions that also provides the value of the SB constant on the basis of a set of universal constants including the quantum action constant [Formula: see text]. However, the successive consideration by Planck [Uber die Begründung des Gesetzes der schwarzen Strahlung [On the grounds of the law of black body radiation], Ann. Phys. 6 (1912) 642–656] concerning the zero-point energy contribution was found to be responsible of another divergence of the internal energy for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh–Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we would expect the divergence of the SB law (here called second Boltzmann paradox). Here, both the Boltzmann paradoxes are revised by accounting for both the quantum-relativistic photon gas properties, and the Casimir force.