Polariton-polariton coherent coupling in a molecular spin-superconductor chip

TL;DR

This work tackles scalable, coherent interfacing of distant polaritons in a modular circuit QED platform. By integrating seven detuned LER pairs on a common readout line and depositing model $S=1/2$ spin ensembles onto the inductors, the authors realize tunable local spin–photon interactions and enable circuit-mediated polariton–polariton coupling. They demonstrate remote spin–photon coupling to an empty resonator, an avoided crossing between upper polariton branches indicating direct polariton–polariton interaction, and pump–probe readout that maps inter-polariton dynamics, all supported by input–output theory and a Schrieffer–Wolff-type analysis. The results establish controllable spin–spin and photon–photon correlations and entanglement between distant polariton nodes, offering a scalable path toward modular quantum networks and simulations using molecular spin ensembles coupled to superconducting circuits.

Abstract

The ability to establish coherent communication channels is key for scaling up quantum devices. Here, we engineer interactions between distant polaritons, hybrid spin-photon excitations formed at different lumped-element superconducting resonators within a chip. The chip consists of several resonator pairs, slightly detuned in frequency to make them addressable, capacitively coupled within each pair and inductively coupled to a common readout line. They interact locally with samples of PTMr and Tripak$^{-}$ organic free radicals, deposited onto their inductors, which provide model $S = 1/2$, $g \simeq 2$ spin ensembles. Frequency-dependent microwave transmission experiments, performed at very low temperatures, measure polariton frequencies as a function of magnetic field in different scenarios. When only one resonator within a pair hosts a molecular sample, the results evidence that spins couple remotely to the empty LER as well as to the local cavity mode. If both resonators interact with a spin ensemble, the magnetic field tunes the polariton frequencies relative to each other, on account of the different spin-photon interactions at each LER. When polaritons are brought into mutual resonance, an avoided level crossing emerges that gives direct spectroscopic evidence for a coherent polariton-polariton interaction mediated by the circuit. Pump-probe experiments reveal that the excitation of a polariton within a connected pair is felt, thus it can be read out, by the other one. These observations, backed by model calculations, illustrate the control and detection of distant photon-photon and spin-spin correlations and entanglement in a scalable modular chip.

Figures (28)

• Figure 1: a) Image of a chip hosting seven superconducting lumped element resonator (LER) pairs coupled to a common transmission line. b) Optical microscopy image of one of these capacitively coupled LER pairs. c) Resonance frequencies of a LER pair calculated as a function of the difference between their finger capacitor lengths. The coupling introduces a mode splitting $\Delta \omega_{\rm r} = 2 \kappa_{i,j}$ when the two LERs become identical. d) Single-photon current density simulations for the two resonant modes of a detuned LER pair (vertical line in panel c). Further details on the design, simulation, fabrication and characterization of the chips are given in Appendices \ref{['ssec:design']} and \ref{['ssec:microwave']}.
• Figure 2: Molecular structures of the organic free radical molecules used in this study: a) Tripak$^{-}$Pakulski2024; b) PTMr Schafter2023. Here, carbon atoms are represented in grey, chlorine atoms in green, nitrogen atoms in blue, sulfur atoms in yellow and oxygen atoms in red. c) and d) Broadband microwave absorption spectra of Tripak$^{-}$ and PTMr, respectively, measured at $T = 11$ mK by coupling them to a superconducting transmission line. The insets show normalized transmission data measured at the magnetic fields indicated by vertical dashed lines.
• Figure 3: a) Optical microscopy image of a $N\sim 5 \times 10^{12}$ PTMr free radical deposit on the inductor of LER-10, forming a coupled pair with LER-9, which remains empty. b) Colour plot of the microwave transmission amplitude $\vert S_{21} \vert$ measured, at $T=11$ mK, near the resonance frequencies of the two LERs. The data show a detectable coupling of the spins to both resonant modes, being $G_{10}/2 \pi = 5.4$ MHz the coupling strength to the 'local' LER-10 and $G_{9,10}/2 \pi = 2.5$ MHz the remote interaction with LER-9. c) Theoretical calculation of the microwave transmission obtained from Eq. \ref{['eq:hamiltonian']} by using input-output theory (Appendices \ref{['ssec:input-output']} and \ref{['ssec:application']}), for $G_{10}/2 \pi=5.4$ MHz and $\kappa_{9,10}/2 \pi = 6.5$ MHz.
• Figure 4: Experimental setup for polariton-polariton coupling. Two LERs belonging to either a coupled pair (LER-1 and LER-2) or to two different and distant pairs (LER-5 and LER-7) host different molecular spin ensembles (blue: Tripak$^{-}$, orange: PTMr). Panels b), d), f) and c), e), g) provide experimental data and simulations obtained for the coupled and the uncoupled LERs, respectively: colour plots of the microwave transmission measured at $T=11$ mK near the resonances of each spin ensemble with its local LER (b, c); experimental (dots) and theoretical (lines) frequency differences between the two upper polaritons as a function of magnetic field (d, e); experimental (dots) and theoretical (lines) visibilities of these modes as a function of magnetic field (f, g). The fits are based on the model outlined in the text [Eq. \ref{['eq:hamiltonian']}] and detailed in Appendix \ref{['sssec:2-spin-theory']}, with parameters $G_{1}/2 \pi = 19.5$ MHz, $G_{2}/2 \pi = 8.5$ MHz and $\kappa_{1,2}/2 \pi = 1.06$ MHz for the coupled LER pair and $G_{7}/2 \pi = 22$ MHz, $G_{5}/2 \pi = 9.7$ MHz and $\kappa_{5,7}/2 \pi = 0$ for the uncoupled LERs.
• Figure 5: a) Sketch of the polariton-polariton coupling model showing the main interactions present in the system \ref{['eq:hamiltonian']}. Details of the eigenstates and of the microwave transmission calculations are described in Appendix \ref{['sec:theory']}. b) and c) Simulated $2D$ microwave transmission plots for two uncoupled ($\kappa_{i,j}=0$) and two coupled ($\kappa_{i,j} = 1.06$ MHz) LERs, respectively. The insets show the probabilities of the single spin and single photon excitation states in each polariton. Striped sectors in the inset of panel c) signal negative probability amplitudes.