Critical speed of a binary superfuid of light

TL;DR

The paper develops a theory for the critical speed of dissipationless flow in a 2D binary superfluid of light past a polarization-sensitive obstacle. It combines linear-response analysis, where dissipation arises when the flow excites density or spin Bogoliubov modes with speeds $c_d$ and $c_s$, and nonlinear hydraulic/incompressible approaches to treat strong, extended obstacles, yielding a reduced ellipticity condition that determines a velocity-dependent critical speed $V_c$. It finds that optical saturation can invert the ordering of $c_d$ and $c_s$, altering the dominant dissipation channel, and shows that for impenetrable obstacles dissipation sets in via vortex–antivortex nucleation, while for penetrable obstacles it can involve Jones–Roberts soliton–type excitations inside the obstacle. Numerical simulations corroborate the analytical predictions, revealing mode hybridization and distinct pathways to dissipation; the work provides a general framework for 2D binary nonlinear Schrödinger superflows, applicable to Bose–Bose mixtures beyond optics.

Abstract

We theoretically study the critical speed for superfluid flow of a two-dimensional (2D) binary superfluid of light past a polarization-sensitive optical obstacle. This speed corresponds to the maximum mean flow velocity below which dissipation is absent. In the weak-obstacle regime, linear-response theory shows that the critical speed is set by Landau's criterion applied to the density and spin Bogoliubov modes, whose relative ordering can be inverted due to saturation of the optical nonlinearity. For obstacles of arbitrary strength and large spatial extent, we determine the critical speed from the conditions for strong ellipticity of the stationary hydrodynamic equations within the hydraulic and incompressible approximations. Numerical simulations in this regime reveal that the breakdown of superfluidity is initiated by the nucleation of vortex-antivortex pairs for an impenetrable obstacle, and of Jones-Roberts soliton-type structures for a penetrable obstacle. Beyond superfluids of light, our results provide a general framework for the critical speed of 2D binary nonlinear Schrödinger superflows, including Bose-Bose quantum mixtures.

Figures (7)

• Figure 1: By slightly tilting uniform, equally bright left- (${+}$) and right- (${-}$) circularly polarized beams with same angle with respect to the ${z}$ axis (blue arrows), we create, in the transverse ${x{-}y}$ plane, a fully balanced binary superflow of light with total density ${\rho_{0}}$ and mean velocity ${V_{0}}$, here along the positive-${x}$ direction (blue spot in the zoom). This 2D flow impinges on a birefringent optical obstacle of radius ${w}$, characterized by total and relative potential amplitudes ${U}$ and ${u}$ (red spot in the zoom). The obstacle can be created using an auxiliary beam along the ${z}$ axis (red arrow). The medium (brown box) through which the lasers propagate provides ${\pm}$/${\pm}$ and ${+}$/${-}$ interactions (${\alpha}$) which saturate (${\beta}$) at high laser intensity.
• Figure 2: Density speed of sound ${c_{\mathrm{d}}}$ [first equation in \ref{['Eq:cdcs']}; solid curve] and spin speed of sound ${c_{\mathrm{s}}}$ (second equation; dashed curve) as functions of the saturation parameter ${\beta}$ for the intercomponent interaction constant ${\alpha\simeq0.41}$ of Ref. Piekarski2025. The coordinates of the intersection point are ${\beta=2\alpha/(1-\alpha)\simeq1.39}$ and ${c_{\mathrm{d},\mathrm{s}}=(1-\alpha)/[2(1+\alpha)]^{1/2}\simeq0.35}$. At low ${\beta}$, in the Kerr regime, ${c_{\mathrm{d}}\simeq[(1+\alpha)/2]^{1/2}\simeq0.84}$ and ${c_{\mathrm{s}}\simeq[(1-\alpha)/2]^{1/2}\simeq0.54}$. The shaded area represents the parameter regime where condition \ref{['Eq:VcLandau']} holds and the two-component flow of mean asymptotic velocity ${V_{0}}$ is superfluid, within perturbation theory. The arrows indicate the opening of the density ("${\mathrm{d}}$") and spin ("${\mathrm{s}}$") dissipation channels as ${V_{0}}$ is increased from the superfluid phase at fixed ${\beta}$'s below (${\beta=0.5}$; ${V_{\mathrm{c}}=c_{\mathrm{s}}\simeq0.44<c_{\mathrm{d}}\simeq0.56}$) and above (${\beta=2.5}$; ${V_{\mathrm{c}}=c_{\mathrm{d}}\simeq0.24<c_{\mathrm{s}}\simeq0.29}$) the curve inversion at ${\beta\simeq1.39}$.
• Figure 3: Normalized drag force ${F/(\pi w^{2}U^{2})}$ [Eqs. \ref{['Eq:FPert_bis']} and \ref{['Eq:Ids']}] experienced by the obstacle of radius ${w}$ and total and relative potential amplitudes ${U}$ and ${u}$ as a function of the mean asymptotic velocity ${V_{0}}$ of the flow. The two graphs are plotted for the intercomponent interaction constant ${\alpha\simeq0.41}$ of Ref. Piekarski2025, ${w=10}$, and ${|u|=U}$. In panel (a), ${\beta=0.5}$ (left arrow in Fig. \ref{['Fig:Landau']}) and ${c_{\mathrm{s}}\simeq0.44<c_{\mathrm{d}}\simeq0.56}$; in panel (b), ${\beta=2.5}$ (right arrow) and ${c_{\mathrm{d}}\simeq0.24<c_{\mathrm{s}}\simeq0.29}$. The shaded region corresponds to the dragless, superfluid regime, which breaks down as soon as ${V_{0}}$ exceeds the critical speed (a) ${V_{\mathrm{c}}=c_{\mathrm{s}}}$ or (b) ${V_{\mathrm{c}}=c_{\mathrm{d}}}$.
• Figure 4: Threshold speeds ${V_{1}}$ [Eq. \ref{['Eq:VdImpene']}; dark solid curve] and ${V_{2}}$ [Eq. \ref{['Eq:VsImpene']}; dark dashed curve] for an impenetrable obstacle potential (${U/\mu_{0}>1}$). While ${V_{1}}$ and ${V_{2}}$ are formally derived from the density- and spin-sector local constraints \ref{['Eq:LocalLandau']}, respectively, the resulting excitations above ${V_{\mathrm{c}}=\min\{V_{1},V_{2}\}}$ are hybridized (see Sec. \ref{['SubSec:Num']}). The curves are plotted as functions of the saturation parameter ${\beta}$ for the intercomponent interaction constant ${\alpha\simeq0.41}$ of Ref. Piekarski2025. The light curves correspond to the linear-response, density and spin speeds of sound ${c_{\mathrm{d}}}$ and ${c_{\mathrm{s}}}$, originally shown in Fig. \ref{['Fig:Landau']}. The shaded area highlights the parameter regime where inequality \ref{['Eq:VcImpene']} holds, and the flow of mean asymptotic velocity ${V_{0}}$ is superfluid. Below (above) the intersection point ${\beta=\alpha(11+5\alpha)/(4-3\alpha-\alpha^{2})\simeq2.06}$, the critical speed for superfluidity is ${V_{\mathrm{c}}=V_{2}}$ (${V_{\mathrm{c}}=V_{1}}$). In the Kerr regime (${\beta\ll1}$), ${V_{\mathrm{c}}=V_{2}\simeq0.25}$ while ${V_{1}\simeq0.36}$.
• Figure 5: Surface plot of the critical speed ${V_{\mathrm{c}}(U,|u|,\alpha,\beta)}$ as a function of the normalized obstacle amplitudes ${U/\mu_{0}}$ and ${|u|/\mu_{0}}$ in the penetrable regime \ref{['Eq:UCond']}. The orange region in the ${U/\mu_{0}{-}|u|/\mu_{0}}$ plane corresponds to the range of applicability \ref{['Eq:uCond']} of our approach. In this plot, the interaction constant is set to the value ${\alpha\simeq0.41}$ used in the experiment of Ref. Piekarski2025. The blue, shaded sheets show ${V_{\mathrm{c}}}$ for (a) ${\beta=0}$ (Kerr regime), (b) ${\beta=1}$, and (c) ${\beta=3}$. For ${U/\mu_{0},|u|/\mu_{0}\ll1}$, ${V_{\mathrm{c}}}$ converges to the linear-response predictions: (a) ${c_{\mathrm{s}}\simeq0.54}$, (b) ${c_{\mathrm{s}}\simeq0.38}$, and (c) ${c_{\mathrm{d}}\simeq0.21}$ (see Fig. \ref{['Fig:Landau']}).