Spectroscopic readout of chiral photonic topology in a single-cavity spin-orbit-coupled Bose-Einstein condensate

Abstract

Topological photonic phases are typically identified through band reconstruction, steady-state transmission, or real-space imaging of edge modes. In this work, we present a framework for spectroscopic readout of chiral photonic topology in a single driven optical cavity containing a spin-orbit-coupled Bose-Einstein condensate. We demonstrate that the cavity transmission power spectral density provides a direct and measurable proxy for a momentum- and frequency-resolved photonic Chern marker, enabling topological characteristics to be inferred from spectral data without the need for bulk-band tomography. In the loss-dominated regime, where cavity decay exceeds atomic dissipation, the power spectral density exhibits Dirac-like gapped hybrid modes with a vanishing Chern marker, indicating a trivial phase. When the dissipation imbalance is reversed, a bright, gap-spanning spectral ridge emerges, co-localized with peaks in both the Chern marker and Berry curvature. The complex spectrum reveals parity-time symmetric coalescences and gain-loss bifurcations, marking exceptional points and enabling chiral, gap-traversing transport. By linking noise spectroscopy to geometric and non-Hermitian topology in a minimal cavity-QED architecture, this work provides a framework for spectroscopic detection of topological order in driven quantum systems. This approach offers a pathway to compact, tunable topological photonics across a broad range of light-matter platforms, providing a method for the study and control of topological phases in hybrid quantum systems.

Figures (4)

• Figure 1: Influence of SOC and interspecies interactions on the transmission PSD. (a)--(c) Transmitted-field PSD $S_{\rm out}(k,\omega)$ for increasing SOC strength $\alpha=1.5\,\Omega$ (a), $2.5\,\Omega$ (b), and $3.5\,\Omega$ (c), at fixed Raman and dissipation parameters (as in the main text). Larger $\alpha$ enhances spin--momentum locking, splits an almost degenerate response into two SOC-hybridized polaritonic branches, and strengthens the $\Omega_z$-controlled gap while increasing the $k$-asymmetry. (d)--(f) PSD at fixed $\alpha$ for increasing interspecies interaction ratio $\epsilon\equiv U_{\uparrow\downarrow}/U$: $\epsilon=0$ (d), $\epsilon=1$ (e), and $\epsilon=2$ (f). Increasing $\epsilon$ renormalizes the atomic dispersion and dispersive light shift, shifting ridge frequencies and reducing gap contrast as the system approaches the SU(2)-symmetric point $\epsilon=1$. Across all panels, bright ridges trace poles of the linearized response, while linewidths reflect the net damping set by $\kappa$ and $\gamma$.
• Figure 2: Emergence of gap-spanning edge states and PSD-derived topology in the gain-dominated regime ($\gamma>\kappa$).a, Power spectral density $S_{\mathrm{out}}(k,\omega)$ at Raman coupling $\Omega_{z}=3\,\Omega$ for $\gamma>\kappa$. A bright, continuous branch traverses the bulk gap, revealing an edge mode that connects the upper and lower polaritonic bands across wavevector $k$. b, Magnified view of the gap region in a. The inset arrows (blue/red) indicate opposite group velocities along the traversing branch, consistent with two-way chiral transport supported by non-Hermitian gain–loss imbalance. c, Chern-marker distribution reconstructed from the PSD in a. Two well-resolved positive peaks align with the gap-crossing branch and encode opposite transport directions of the chiral edge modes (signalled by the local spectral slope). The vanishing background away from the branch indicates that topology is concentrated at the gap-spanning trajectory. d, Chern marker as a function of input pump power $P$ for selected wavevectors $k_x$. The black curve ($k_x=0$) corresponds to the bulk region where no edge mode exists, giving a vanishing marker at all powers. The blue ($k_x=11\times2\pi$) and red ($k_x=31\times2\pi$) curves track the momenta of the first and second edge modes, respectively, both showing a pronounced increase of the marker with power as the chiral edge channels emerge and intensify with stronger light–matter coupling.
• Figure 3: Berry-curvature cartography and non-Hermitian band topology.a, Berry-curvature density $\Omega_{B}(k,\omega)$ reconstructed from the PSD for $\Omega_{z}=3\,\Omega$, shown as a 3D surface with a base-plane density projection. Curvature concentrates along the gap-edge trajectories identified in transmission. b, Same as a for $\Omega_{z}=5\,\Omega$, revealing enhanced, more sharply localized curvature lobes and a redistribution of weight as the Raman-induced mass increases. c, Real parts of the eigenvalues $\mathrm{Re}\,\lambda(k)$ at $\Omega_{z}=3\,\Omega$, displaying coalescence points characteristic of $\mathcal{PT}$-symmetric band crossings. d, Imaginary parts $\mathrm{Im}\,\lambda(k)$ at the same parameters, showing gain–loss bifurcation at the coalescence points in c, thus identifying the exceptional points (EPs) that delimit the transition between unbroken and broken $\mathcal{PT}$ phases.
• Figure 4: Phase–detuning control of edge transport and Chern-marker localization. Upper row: a–c, Transmission PSD $S_{\mathrm{out}}(k_x,\omega)$ for Raman detuning $\delta/\Omega=-1,\,0,\,+1$, respectively, at fixed $\Omega_z=5\,\Omega$ in the gain-dominated regime ($\gamma>\kappa$). A finite detuning breaks the $k_x\!\to\!-k_x$ symmetry of the SOC–BEC phase and re-positions the gap-spanning edge branch within the bulk gap: for $\delta=-\Omega$ it is biased to negative $k_x$ (a), for $\delta=0$ it is centered (b), and for $\delta=+\Omega$ it shifts to positive $k_x$ (c). Lower row: d–f, Corresponding Chern-marker profiles along $k_x$ (frequency-integrated around the gap) showing a rigid displacement of the topological weight that tracks the edge-branch drift: left-shift for $\delta=-\Omega$ (d), symmetric/centered for $\delta=0$ (e), and right-shift for $\delta=+\Omega$ (f). The marker magnitude remains comparable, indicating that detuning redistributes rather than quenches topology, enabling phase-bias control of chiral transport.