Photon-Fluid Model Fundamentals

• Photon-fluid model is a hydrodynamic formulation of light where spatial confinement and diffraction impart an effective mass to photons and enable nonlinear interactions.
• It employs Schrödinger- and Gross-Pitaevskii-type equations to simulate nonlinear optical media, revealing phenomena such as superfluidity, dispersive shocks, and analogue gravity.
• The model applies across diverse platforms—from microcavities and nonlinear fibers to driven-dissipative systems—illustrating the interplay of non-equilibrium dynamics and classical squeezing effects.

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 (Carusotto, 2022, Marino, 2019). 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 (Sadowski et al., 2015).

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 (Carusotto, 2022, Marino, 2019).

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 $m^{(q)}=\hbar k_z^{(q)}/c$ (Carusotto, 2022). 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 (Carusotto, 2022, Marino, 2019).

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 (Carusotto, 2022). 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 (Sadowski et al., 2015). 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 (Marino, 2019, Liu et al., 2024).

For a self-defocusing medium with local Kerr nonlinearity, the resulting equations are

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$0

and

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$1

where the last term is the quantum-pressure contribution inherited from diffraction (Liu et al., 2024). The associated barotropic pressure law is

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$2

and the sound speed on a background density $$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$3 is

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$4

This is the basic compressible-fluid structure underlying superfluidity, sound, shocks, and analogue gravity in propagating photon fluids (Liu et al., 2024).

The same logic underlies fiber-based temporal photon fluids, but with an exchange of the roles of space and time: in the defocusing NLSE,

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$5

the propagation distance $$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$6 becomes the evolution variable and retarded time $$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$7 becomes the effective spatial coordinate. After the Madelung transform,

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$8

one obtains

$$\hbar\omega^{(q)}(k_\parallel)=\hbar c k_z^{(q)}+\frac{\hbar^2 k_\parallel^2}{2m^{(q)}},$$9

and

$m^{(q)}=\hbar k_z^{(q)}/c$0

so optical power acts as density and chirp acts as velocity (Bendahmane et al., 2020, Xu et al., 2017).

Driven-dissipative cavity fluids replace conservative propagation by gain, loss, and coherent injection. A representative mean-field equation is

$m^{(q)}=\hbar k_z^{(q)}/c$1

which makes explicit that many photon fluids are intrinsically non-equilibrium steady states rather than equilibrium condensates (Carusotto, 2022).

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,

$m^{(q)}=\hbar k_z^{(q)}/c$2

the local Kerr part sets the compressibility and sound speed, while the infinite-range thermal response changes the low-energy spectrum qualitatively (Marino, 2019). For a homogeneous background, the Bogoliubov dispersion becomes

$m^{(q)}=\hbar k_z^{(q)}/c$3

with gap

$m^{(q)}=\hbar k_z^{(q)}/c$4

The elementary excitations are then massive phonons, with effective rest mass

$m^{(q)}=\hbar k_z^{(q)}/c$5

and relativistic low-momentum dispersion

$m^{(q)}=\hbar k_z^{(q)}/c$6

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 (Marino, 2019).

Nonlocality also generates qualitatively different phase-space dynamics. In the highly nonlocal regime of the nonlocal photon fluid,

$m^{(q)}=\hbar k_z^{(q)}/c$7

the self-induced nonlocal potential can be Taylor-expanded to quadratic order and the dynamics reduce to the reversed harmonic oscillator

$m^{(q)}=\hbar k_z^{(q)}/c$8

This makes the propagation unitarily equivalent to a squeezing operator with squeezing parameter

$m^{(q)}=\hbar k_z^{(q)}/c$9

and yields quadrature variances

$$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$0

The analysis is explicitly classical: the paper identifies the phenomenon as “classical spontaneous squeezing,” not a full quantum-noise calculation (Braidotti et al., 2016).

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

$$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$1

emerging spontaneously at the transition (Caleffi et al., 2022). 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 $$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$2-dimensional acoustic spacetime with horizon, ergoregion, and frame dragging (Ciszak et al., 2021, Liu et al., 2024). 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 (Marino, 2019, Ciszak et al., 2021).

For rotating photon-fluid black holes, stationary bound states arise at the synchronization condition

$$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$3

the direct analogue of Kerr-cloud resonance. The radial eigenfunctions are regular at the horizon, decay exponentially at infinity, and exist in the regime

$$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$4

In the rapid-rotation regime, the stationary cloud spectrum is discrete, and the maximal allowed ratio of mass to synchronization frequency is

$$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$5

which analytically explains the previously observed numerical upper limit (Hod, 2021).

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 $$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$6 of the peak of the non-monotonic radial eigenfunction, the theorem establishes the parameter-independent bound

$$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$7

where $$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$8 is the acoustic horizon radius and $$i\frac{\partial E}{\partial z}=-\frac{1}{2\beta_0}\nabla_\perp^2E+V(\mathbf r)E+G_{\rm nl}|E|^2E$$9 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 (Hod, 2022).

The same background supports quasinormal ringing. In the $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$0-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 $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$1 show $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$2 increasing strongly with $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$3, while counter-rotating modes with $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$4 show $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$5 decreasing and $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$6 approaching zero from below, suggesting long-lived quasi-resonant behavior (Liu et al., 2024).

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

$$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$7

which yields

$$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$8

In a graphene/methanol thermal optical solution, the measured radial flow and sound profiles exhibit two sonic crossings interpreted as inner and outer horizons (Wu et al., 25 Apr 2025). Exact spectral analyses of the same $$\partial_z E=\frac{i}{2k}\nabla^2E-i\frac{k}{n_0}E\,\Delta n,$$9-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

$z$0

and negative greybody factors for co-rotating modes in the superradiant regime (Senjaya et al., 3 Sep 2025).

5. Dispersive hydrodynamics: shocks, cavitation, and soliton ensembles

Photon-fluid models have also become a laboratory for dispersive hydrodynamics. In the temporal fiber setting, a step-like dam-break initial condition in the defocusing NLSE produces a right-going rarefaction wave and a left-going dispersive shock wave rather than a viscous shock. The plateau density in the corresponding shallow-water reduction is

$z$1

and Whitham modulation theory gives the DSW edge speeds. A central result is the observation of self-cavitation at the universal threshold

$z$2

where the DSW develops a vacuum point and the edge soliton becomes black (Xu et al., 2017).

The virtual piston problem generalizes this to a symmetric velocity jump imposed by cross-phase modulation. For the pure piston problem, $z$3 produces two DSWs separated by a constant plateau, while $z$4 produces two rarefaction waves. The dispersive phase transition occurs at

$z$5

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 $z$6, the transition threshold becomes

$z$7

These regimes are described quantitatively by Whitham modulation theory and verified experimentally in fiber optics (Bendahmane et al., 2020).

A genuinely two-dimensional extension has now been demonstrated in a photorefractive crystal with saturable focusing response. A box-shaped optical field evolves under

$z$8

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,

$z$9

which the authors interpret as evidence for a gaseous phase of solitons in a nonintegrable $E=\rho^{1/2}e^{i\phi},$0-dimensional photon fluid (Dieli et al., 2024).

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 (Carusotto, 2022). 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 (Caleffi et al., 2022).

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 (Braidotti et al., 2016). 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 $E=\rho^{1/2}e^{i\phi},$1 need not obey Kerr bounds (Liu et al., 2024, Ciszak et al., 2021).

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

$E=\rho^{1/2}e^{i\phi},$2

with an approximate inversion

$E=\rho^{1/2}e^{i\phi},$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 $E=\rho^{1/2}e^{i\phi},$4 in the funnel region of supercritical accretion flows (Sadowski et al., 2015). 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 (Carusotto, 2022, Caleffi et al., 2022, Sadowski et al., 2015).