Multiband dispersion and warped vortices of strongly-interacting photons

Abstract

We present a theoretical study of quantum correlations between interacting photons realized through co-propagating Rydberg polaritons. We show that the spatial evolution of the $n$-photon wavefunction is governed by a multiband dispersion featuring one massive mode and multiple massless modes with degenerate Dirac points and $n$-fold rotational symmetry. The resulting band structure is warped, departing from the single-band, parabolic approximation commonly assumed for interacting polaritons. Our analytical results are supported by rigorous numerical modeling that fully accounts for photon propagation inside the finite atomic medium. These findings advance the understanding of multi-photon interactions and support the development of future multi-photon control tools.

Figures (6)

• Figure 1: Analytical momentum-space dispersion. (a) Two photons: Center-of-mass dispersion $K(k)$ for two non-interacting polaritons. The dashed line represents the single-band Schrödinger approximation of the massive polariton branch, while the solid lines show the full two-band Dirac-like dispersion. (b) Three photons: Center-of-mass dispersion $K(k_{\upeta},k_{\upzeta})$ for three polaritons, highlighting the emergence of trigonal warping. The black surface represents the single-band Schrödinger approximation, which fails to capture the warping. $K(k)$ and $K(k_{\upeta},k_{\upzeta})$ are found from Eqs. \ref{['eq:K2b']} and \ref{['eq:K3']}, and all units in $\rho g^2/(c\Delta)$.
• Figure 2: Analytical multiband model, real-space propagation. (a) Two-photon coordinate space: Schematic showing the propagation center-of-mass coordinate $R$ and relative coordinate $r$. Two-photon interaction occurs only inside the medium ($0<x_{1,2}<L$, gray area). The entry points $x_1=0$ and $x_2=0$ (green lines) serve as light-cone boundaries, below which two-photon correlations cannot develop. (b) Three-photon coordinate space: Interaction occurs already when photon pairs are inside the medium (light blue sections), while full three-photon interaction requires all $0<x_{1,2,3}<L$. Regions above the green (red) lines correspond to a closely spaced photon pair propagating behind (ahead of) a single photon. (c,d) Two-photon phase profiles: The phase of the two-photon wavefunction under the single-band model (c) shows a vortex-antivortex pair at finite propagation distance $R$ (gray line) but also unphysical correlations outside the light-cone boundaries ($x_{1,2}<0$). The full two-band dispersion (d) corrects the unphysical behavior, confining correlations within the light cone and introducing a propagation delay $\Delta R$ in vortex formation. The calculation has been performed for $R=0\ldots 20c/\nu$ for $\nu r_{\rm b}/c=0.43$. Thin vertical lines show the scale of the interaction potential, $|x_1-x_2|=\pm r_{\rm b}$. (e-g) Three-photon phase profiles: The phase of the three-photon wavefunction under the single-band model (e) retains a 6-fold rotational symmetry and fails to capture the warping. The multiband model (f) reduces the rotational symmetry to 3-fold, with the delay $\Delta R$ manifesting here as a delay of the vortex formation along the pair-ahead (red) lines. In (g), the isosurfaces of $|\nabla\mathop\mathrm{arg}(\psi)|$ reveal a vortex ring with trigonal warping, a hallmark of the three-photon interaction captured by the multiband model. Calculations in (e,f,g) follow Eqs. \ref{['eq:K3c']}--\ref{['eq:K3d']}, with $\nu r_{\rm b}/c=1$. Panels (e,f) correspond to $R=7.6c/\nu$.
• Figure 3: Numerical simulations of three-photon vortices with three-fold rotational symmetry $C_{3v}$. A Gaussian-shaped atomic cloud is centered at $x=100~\mu$m ($1\sigma=30~\mu$m, $\mathrm{OD}=78-115$). (a) Phase of the stationary two-photon wavefunction $\psi(x_1,x_2)$, showing a symmetric vortex-antivortex pair. (b,c,e,f) Phase of the stationary three-photon wavefunction $\psi(x_1,x_2,x_3)$, displaying different vortex tube and ring configurations for (b,e) intermediate and (c,f) strong interactions. (e,f) show isosurfaces of $|\nabla\mathop\mathrm{arg}(\psi)|$, while (b,c) present cross-sections of $\mathop\mathrm{arg}(\psi)$ along the shaded planes in (e,f). For an intermediate interaction strength (b,e), three-photon vortex tubes form only in the single-ahead configurations (e.g., $x_1\approx x_2< x_3$), consistent with faster pair propagation. For longer and stronger interactions (c,f), vortex tubes also form in the pair-ahead configurations (e.g., $x_3<x_1\approx x_2$), and the six tubes merge into a warped vortex ring with a reduced $C_{3v}$ symmetry. (d) Phase diagram of three-photon vortex formation. The shaded regions mark the parameter regimes for (gray) single-ahead and (green) pair-ahead vortices as a function of interaction strength ($\lambda$) and duration ($\varphi$). The red circle and square correspond to the conditions of (b,e) and (c,f), respectively. Calculations here follow Eqs. (\ref{['eq:3_photon_equations']}), with the following parameters: $\sqrt{2\pi}\sigma=75$$\mu \rm m$, $r_\mathrm{b}=15.3$$\mu \rm m$, $\Omega=9.5$ MHz, $\Gamma= 3.03$ MHz, $\gamma=0.07$ MHz, $\Delta=28.5$ MHz, $\delta=1.03$ MHz (note that decay rates and Rabi frequencies are given in the half-width convention throughout this paper). The optical depths are: OD= 90 ($\varphi=2.21\pi$, $\lambda=2.03$) for Fig. \ref{['fig:Chiral']}a, OD= 78 ($\varphi=1.92\pi$, $\lambda=0.42$) for Figs. \ref{['fig:Chiral']}b,e, and OD= 115 ($\varphi=2.82\pi$, $\lambda=0.47$) for Figs. \ref{['fig:Chiral']}c,f.
• Figure 4: Analytical multiband model for $n=4$ photons. The surfaces show momentum-space dispersion calculated following Eq. \ref{['eq:K4']} for $\kappa_3=0$. Shaded paraboloid shows the Schrödinger approximation result, Eq. \ref{['eq:K4b']}. All units are $\nu$.
• Figure 5: Two-photon dispersion law $\omega(k_1,k_2)=0$ under the stationary conditions. Dashed lines correspond to the dual-band model based on Eq. (\ref{['eq:two']}), solid lines correspond to the 9-band model. Calculation has been performed for $\sqrt{\rho}g/\Delta=300$, $\Omega/\Delta=1/3$.