--- title: Photon-Fluid Model Fundamentals url: https://www.emergentmind.com/topics/photon-fluid-model type: topic --- # Photon-Fluid Model Fundamentals The photon-fluid model is an effective many-body and hydrodynamic description of light in which photons, or photon-like quasiparticles, acquire a finite effective mass from spatial confinement or diffraction and effective interactions from optical nonlinearity. In this regime, the optical field is governed by Schrödinger- or Gross-Pitaevskii-type dynamics, can be written in density-phase variables, and supports collective excitations, superfluid phenomena, dispersive hydrodynamics, and, in analogue-gravity settings, emergent acoustic spacetimes for fluctuations [2211.10980, 1908.00875]. In current literature the term covers several related but non-identical constructions: paraxial propagation in nonlinear media, nonlocal fluids of light, driven-dissipative cavity and circuit-QED lattices, and, in a distinct astrophysical usage, radiation treated as a Bose-Einstein photon gas with conserved photon number [1508.04980]. ## 1. Foundational concept and physical realizations In propagating optical systems, the basic mechanism is the paraxial reduction of Maxwell optics in a nonlinear medium. Diffraction in the transverse plane generates the analogue of kinetic energy, while the refractive-index shift generated by the medium provides an interaction term. This yields a classical-field model with the same mathematical structure as the nonlinear Schrödinger or Gross-Pitaevskii equation, allowing light to be interpreted as a compressible bosonic fluid [2211.10980, 1908.00875]. Two broad realizations recur. In microcavities, longitudinal confinement quantizes \(k_z\) and gives an in-plane dispersion \[ \hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}}, \] so the cavity mode behaves as a massive two-dimensional boson with \(m^{(q)}=\hbar k_z^{(q)}/c\) [2211.10980]. In propagating media, the paraxial equation \[ i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E \] or, in the nonlinear-optics notation used in analogue-gravity work, \[ \partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n, \] plays the corresponding role, with \(z\) reinterpreted as an effective time [2211.10980, 1908.00875]. The same conceptual framework extends across markedly different platforms. The review literature treats cavity photons, exciton-polaritons, propagating beams in Kerr or thermo-optic media, Rydberg-EIT photons, and superconducting-circuit photons within a common “quantum fluids of light” language [2211.10980]. By contrast, the astrophysical radiation-hydrodynamics literature uses “photon fluid” in a thermodynamic sense, where the radiation field is modeled as a Bose-Einstein fluid with energy density and photon number density evolved separately [1508.04980]. These usages are compatible at the level of fluid language but refer to different effective degrees of freedom. ## 2. Hydrodynamic formulation and governing equations The hydrodynamic mapping is obtained by writing the optical field in Madelung form, \[ E=\rho^{1/2}e^{i\phi}, \] with intensity \(\rho=|E|^2\) interpreted as fluid density and phase gradient interpreted as flow velocity. In the paraxial photon-fluid literature the propagation coordinate is mapped to time via \[ t=\frac{n_0}{c}z, \] and the velocity field is \[ \mathbf v=\frac{c}{k n_0}\nabla\phi . \] This yields continuity and Euler/Bernoulli-type equations for the optical fluid [1908.00875, 2404.04982]. For a self-defocusing medium with local Kerr nonlinearity, the resulting equations are \[ \frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\mathbf v)=0, \] and \[ \frac{\partial \psi}{\partial t}+\frac{1}{2}\mathbf v^2+\frac{c^2 n_2}{n_0^3}\rho-\frac{1}{2}\frac{c^2}{k^2 n_0^2}\frac{\nabla^2 \rho^{1/2}}{\rho^{1/2}}=0, \] where the last term is the quantum-pressure contribution inherited from diffraction [2404.04982]. The associated barotropic pressure law is \[ P=\frac{c^2 n_2 \rho^2}{2 n_0^3}, \] and the sound speed on a background density \(\rho_0\) is \[ c_s^2=\frac{c^2 n_2 \rho_0}{n_0^3}. \] This is the basic compressible-fluid structure underlying superfluidity, sound, shocks, and analogue gravity in propagating photon fluids [2404.04982]. The same logic underlies fiber-based temporal photon fluids, but with an exchange of the roles of space and time: in the defocusing NLSE, \[ i\frac{\partial E}{\partial Z}-\frac{k''}{2}\frac{\partial^2E}{\partial T^2}+\gamma|E|^2E=0, \] the propagation distance \(Z\) becomes the evolution variable and retarded time \(T\) becomes the effective spatial coordinate. After the Madelung transform, \[ E(T,Z)=\sqrt{P_0}\sqrt{\rho(t,z)}\exp\!\left(-i\int_{-\infty}^{t}u(t',z)\,dt'\right), \] one obtains \[ \rho_z+(\rho u)_t=0, \] and \[ u_z+\left(\frac{u^2}{2}+\rho\right)_t=\text{dispersive term}, \] so optical power acts as density and chirp acts as velocity [2007.16060, 1703.09019]. Driven-dissipative cavity fluids replace conservative propagation by gain, loss, and coherent injection. A representative mean-field equation is \[ i\frac{\partial \psi}{\partial t}=\left[\omega_0-\frac{\hbar\nabla^2}{2m^*}\right]\psi+g_{\rm nl}|\psi|^2\psi+\frac{i}{2}\left[\frac{P}{1+|\psi|^2/n_s}-\gamma\right]\psi+E_{\rm inc}(\mathbf r,t), \] which makes explicit that many photon fluids are intrinsically non-equilibrium steady states rather than equilibrium condensates [2211.10980]. ## 3. Interaction structure, nonlocality, and collective excitations The minimal interaction structure is a local repulsive nonlinearity, but several important photon-fluid models include nonlocal terms. In the thermo-optic model with both local and nonlocal third-order nonlinearities, \[ \Delta n(|E|^2,\mathbf r,z)=n_2|E|^2+\hat n_{\rm nl}|E|^2, \] the local Kerr part sets the compressibility and sound speed, while the infinite-range thermal response changes the low-energy spectrum qualitatively [1908.00875]. For a homogeneous background, the Bogoliubov dispersion becomes \[ \Omega^2=\Omega_0^2+c_s^2K^2\left(1+\frac{\xi^2K^2}{4\pi^2}\right), \] with gap \[ \Omega_0=c_s\sqrt{\frac{\alpha|\beta|}{\kappa n_2}}=\frac{c_s}{\lambda_C}. \] The elementary excitations are then massive phonons, with effective rest mass \[ m=\frac{\hbar\Omega_0}{c_s^2}, \] and relativistic low-momentum dispersion \[ E^2=m^2c_s^4+c_s^2p^2\left(1+\frac{p^2}{p_c^2}\right). \] In inhomogeneous flows and in the long-wavelength regime, density perturbations satisfy the massive Klein-Gordon equation on the acoustic metric, making the photon fluid an explicit emergent-gravity model [1908.00875]. Nonlocality also generates qualitatively different phase-space dynamics. In the highly nonlocal regime of the nonlocal photon fluid, \[ i\partial_z \psi+\frac{1}{2}\partial_x^2 \psi-PK(x)\ast|\psi|^2\psi=0, \] the self-induced nonlocal potential can be Taylor-expanded to quadratic order and the dynamics reduce to the reversed harmonic oscillator \[ \hat H_{RHO}=\frac{\hat p^2}{2}-\frac{\gamma^2\hat x^2}{2}. \] This makes the propagation unitarily equivalent to a squeezing operator with squeezing parameter \[ r=\gamma z,\qquad \theta=0, \] and yields quadrature variances \[ \Delta u^2=\frac{e^{-2r}}{4}\left(\frac{1+\gamma^2}{\gamma}\right),\qquad \Delta v^2=\frac{e^{2r}}{4}\left(\frac{1+\gamma^2}{\gamma}\right). \] The analysis is explicitly classical: the paper identifies the phenomenon as “classical spontaneous squeezing,” not a full quantum-noise calculation [1611.01793]. At the many-body lattice level, strongly interacting photon fluids admit a different excitation taxonomy. In a driven array of nonlinear cavities with non-Markovian incoherent pumping, a Gutzwiller treatment of the non-equilibrium Mott/superfluid transition yields particle/hole excitations in the insulating phase and a diffusive Goldstone mode in the superfluid phase, with the order parameter \[ \psi_0(t)=|\psi_0|e^{-i\omega_0 t} \] emerging spontaneously at the transition [2211.07246]. This places photon-fluid modeling directly in the domain of open quantum many-body theory. ## 4. Emergent spacetime and rotating acoustic black holes One of the most developed branches of photon-fluid theory is analogue gravity. In rotating draining-vortex backgrounds, long-wavelength density fluctuations propagate on an effective \(2+1\)-dimensional acoustic spacetime with horizon, ergoregion, and frame dragging [2101.07508, 2404.04982]. In the thermo-optic model with a mass gap, these fluctuations obey a massive Klein-Gordon equation, so the analogue field theory is not limited to massless phonons [1908.00875, 2101.07508]. For rotating photon-fluid black holes, stationary bound states arise at the synchronization condition \[ \Omega=m\Omega_H, \] the direct analogue of Kerr-cloud resonance. The radial eigenfunctions are regular at the horizon, decay exponentially at infinity, and exist in the regime \[ 1<\frac{m^2\Omega_H^2}{\Omega_0^2}<\frac{32}{27}. \] In the rapid-rotation regime, the stationary cloud spectrum is discrete, and the maximal allowed ratio of mass to synchronization frequency is \[ \alpha_{\max}=\sqrt{\frac{32}{27}}, \] which analytically explains the previously observed numerical upper limit [2102.02215]. The paper “No-short scalar hair theorem for spinning acoustic black holes in a photon-fluid model” proves an analogue no-short-hair theorem for these stationary co-rotating acoustic clouds. Defining the effective cloud length by the location \(r_{\max}\) of the peak of the non-monotonic radial eigenfunction, the theorem establishes the parameter-independent bound \[ r_{\max}>\frac{1+\sqrt{5}}{2}\,r_H>r_{\mathrm{null}}, \] where \(r_H\) is the acoustic horizon radius and \(r_{\mathrm{null}}\) is the co-rotating null circular geodesic radius. The proof is analytic and uses the exact radial potential, the horizon regularity condition, the bound-state decay at infinity, and the sign of the potential at the outer extremum [2202.00688]. The same background supports quasinormal ringing. In the \(2\)-dimensional rotating analog black hole studied numerically, quasinormal frequencies were computed with the asymptotic iteration method, Leaver’s continued-fraction method, and sixth-order WKB. Co-rotating modes with \(m>0\) show \(\omega_R\) increasing strongly with \(\Omega_H\), while counter-rotating modes with \(m<0\) show \(\omega_R\) decreasing and \(\omega_I\) approaching zero from below, suggesting long-lived quasi-resonant behavior [2404.04982]. More recent work extends the analogue geometry to a rotating BTZ-like construction with both outer and inner horizons by using an improved vortex phase profile \[ \varphi(r,\theta)=m\theta-2\pi(Ar-B\ln r-Cr^2), \] which yields \[ v_r=\frac{-2\pi c}{k n_0}\left(A-\frac{B}{r}-2Cr\right),\qquad v_\theta=\frac{cm}{k n_0 r}. \] In a graphene/methanol thermal optical solution, the measured radial flow and sound profiles exhibit two sonic crossings interpreted as inner and outer horizons [2504.18120]. Exact spectral analyses of the same \(2+1\)-dimensional rotating analogue then treat quasibound states, scalar clouds, Hawking radiation, superradiance, and greybody factors for relativistic massive phonons, with superradiance occurring in the frequency window \[ \varpi<\omega0\) produces two DSWs separated by a constant plateau, while \(u_0<0\) produces two rarefaction waves. The dispersive phase transition occurs at \[ u_0^{th}=2, \] where the DSW minima reach zero and the constant plateau disappears, giving way to a DSW-per-DSW state joined by an unmodulated nonlinear periodic wave. In the mixed density-velocity problem with \(\rho_0<1\), the transition threshold becomes \[ u_0^{th}=1+\sqrt{\rho_0}. \] These regimes are described quantitatively by Whitham modulation theory and verified experimentally in fiber optics [2007.16060]. A genuinely two-dimensional extension has now been demonstrated in a photorefractive crystal with saturable focusing response. A box-shaped optical field evolves under \[ 2ik\,\partial_z A+\partial_x^2A+\partial_y^2A+2k^2\frac{\delta n(I,t)}{n_0}A=0, \qquad \delta n(I,t)=\delta n_0\frac{f(t)}{1+I/I_s}, \] and exhibits a transition from effectively one-dimensional breaking to fully two-dimensional wave breaking controlled by the box aspect ratio. The collision of dispersive shocks propagating in orthogonal directions produces a random two-dimensional ensemble of localized modes whose width-intensity relation matches the soliton existence curve. For sufficiently large boxes, the soliton number fluctuates weakly around a plateau and the intensity probability density is well fitted by a single exponential law, \[ P(I/\langle I\rangle)\propto e^{-I/\langle I\rangle}, \] which the authors interpret as evidence for a gaseous phase of solitons in a nonintegrable \(2\)-dimensional photon fluid [2409.18738]. ## 6. Non-equilibrium structure, strong correlations, and scope of the term A persistent theme across photon-fluid research is that many realizations are intrinsically non-equilibrium. Cavity fluids require pumping to balance losses, so their steady states are non-equilibrium steady states rather than thermal equilibria; propagating fluids are often conservative over the sample length but evolve from imposed initial or boundary data rather than by relaxation to equilibrium [2211.10980]. This distinction matters for collective modes: open condensates can display diffusive Goldstone modes, complex excitation spectra, and even negative density of states in linear response [2211.07246]. The term “photon fluid” therefore does not imply a single ontology. In the nonlinear-optical and analogue-gravity literature it usually denotes an effective classical or semiclassical hydrodynamic description of a coherent optical field. In the squeezing literature, for example, the nonlocal photon fluid is a classical field theory whose exact reversed-harmonic-oscillator reduction reproduces the mathematics of squeezing without invoking a full operator-valued quantum-optical noise treatment [1611.01793]. In the analogue-gravity literature, the emergent metric is kinematic: it is determined by the background optical flow and does not satisfy Einstein equations, so quantities such as \(\Omega_H\) need not obey Kerr bounds [2404.04982, 2101.07508]. A second, distinct usage appears in general relativistic radiation magnetohydrodynamics. “Photon-conserving Comptonization” treats radiation as a gray Bose-Einstein fluid and evolves both radiation temperature and photon number density, rather than assuming a local blackbody. The closure is based on \[ \widehat{E}=C(kT_r)^4 e^{-\xi}I_E(\xi),\qquad \widehat{n}=C(kT_r)^3 e^{-\xi}I_N(\xi), \] with an approximate inversion \[ kT_r=\frac{\widehat E/\widehat n}{3-2.449724(\widehat n^4/C\widehat E^3)}. \] Implemented in KORAL, this model predicts gas and radiation temperatures larger by up to a factor of two than blackbody Comptonization and spectral hardening factors as large as \(5\) in the funnel region of supercritical accretion flows [1508.04980]. A plausible implication is that “photon fluid” has become an umbrella term for several effective fluidizations of light, ranging from coherent nonlinear-wave hydrodynamics to radiation thermodynamics. Taken together, these developments define the photon-fluid model as a family of effective theories in which optical or radiative fields are reorganized into fluid variables and collective degrees of freedom. The unifying structures are finite effective mass, interaction-induced nonlinearity, hydrodynamic variables, and emergent collective phenomena; the differences lie in whether the underlying system is conservative or driven-dissipative, local or nonlocal, weakly or strongly correlated, and kinematic or fully thermodynamic in its use of the fluid analogy [2211.10980, 2211.07246, 1508.04980]. Source: https://www.emergentmind.com/topics/photon-fluid-model