Displacement-noise-free gravitational-wave detector using quantum tomography of photon orbital angular momentum states

Gravitational waves (GWs) induce quantum transitions between photon orbital-angular-momentum (OAM) states with amplitudes ${\alpha }_{\pm }$. Exploiting this effect, we present a displacement-noise-free detection framework that extracts GW information from the OAM Hilbert space. Parametrized quantum state tomography of the GW-induced OAM sidebands enables a single V-shaped detector to resolve all seven wave parameters: the frequency, propagation direction, two polarization amplitudes, and initial phase of each polarization. Direct detection of the sidebands yields a weak signal of approximately ${10}^{-15}\mathrm{W}$. We therefore introduce a stronger reference state $r\mathrm{\mid }0{>}_{\mathrm{P}}$, with $r=0.01$, to enhance the detected signal. For an arm length of $L=4\mathrm{m}$, $N=160$ round trips, a paraxial parameter of $\gamma =0.1$, an incident light power of $P_{\mathrm{B}\mathrm{S}}=10\mathrm{k}\mathrm{W}$, a wavelength of ${\lambda }_{\mathrm{o}\mathrm{p}\mathrm{t}}=1064\mathrm{n}\mathrm{m}$, and GW amplitudes of $A={10}^{-18}$, we estimate sideband transition amplitudes of $\mathrm{\mid }{\alpha }_{\pm }\mathrm{\mid }\sim {10}^{-10}$ and signal powers of order $1\mathrm{n}\mathrm{W}$, scaling as $P_{\mathrm{B}\mathrm{S}}{\alpha }_{\pm }\propto P_{\mathrm{B}\mathrm{S}}ANL\gamma \mathrm{/}{\lambda }_{\mathrm{o}\mathrm{p}\mathrm{t}}$. The shot-noise-limited sensitivity floor is $S_n^{1\mathrm{/}2}=1.6\times {10}^{-20}{\mathrm{H}\mathrm{z}}^{-1\mathrm{/}2}$. Because the scheme measures OAM transition amplitudes rather than differential arm lengths, it is intrinsically free from displacement noise and provides a broadband probe of GWs, specifically bridging the $0.1$–$10\mathrm{H}\mathrm{z}$ mid-frequency gap.