A three-dimensional multimode lumped-element resonator for collective spin manipulation and dispersive readout

Abstract

We report a three-dimensional lumped-element multimode microwave resonator that enables homogeneous collective manipulation and dispersive readout of a macroscopic spin ensemble. By exploiting geometric symmetry, two antisymmetric modes with strongly suppressed cross-talk are engineered to spatially overlap and couple to the same ensemble at distinct frequencies. Using negatively charged nitrogen-vacancy centers in diamond at 28 mK, we observe collective strong coupling with a coupling strength of 5.0 MHz and demonstrate non-destructive dispersive readout via a detuned mode. The compact design, tunable coupling, and high field homogeneity make this resonator a versatile device for hybrid spin-photon systems and multimode solid-state quantum technologies.

Figures (4)

• Figure 1: (a) Photographs of the resonator base and enclosure. Both parts are fabricated from oxygen-free copper and plated with silver; the surfaces are polished to mirror-level roughness, and four microwave connectors are integrated into the enclosure. (b) Top-view schematics of the resonator modes classified by the symmetry of the electric potential. The blue regions indicate locations where spin ensembles can be installed. The plus and minus signs denote the electric potential. (c) Measured transmission spectra of the multimode resonator. The all-symmetric mode $S(x,y)$ produces a magnetic field distributed away from the center. The anti-symmetric modes $A(x)$ and $A(y)$ generate magnetic fields concentrated in the central slots, where the spin ensembles are located. For the hybrid mode $A(S(x),S(y))$, the magnetic field is distributed in both regions. The mode $A(y)$ has a higher resonance frequency because the wings along the $y$ axis are shorter than those along $x$. (d) Side views of the current flow for the symmetric and anti-symmetric modes. The modes are formed by a lumped capacitance between the wings and the top lid in series with a geometric inductance defined by the current path.
• Figure 2: Simulated magnetic field distribution of the anti-symmetric mode $A(x)$. Top: side view, where the cross section is taken parallel to the $z$ axis and oriented at $45^\circ$ with respect to the $x$ axis. Bottom: top view, taken parallel to the $x$--$y$ plane. All contour lines correspond to a 5% deviation in magnetic field strength relative to the field magnitude at the center of each slot.
• Figure 3: Collective strong coupling between multiple cavity modes and an $\text{NV}^-$ spin ensemble, involving the $m_s = 0 \rightarrow m_s = +1$ transition, whose frequency is tuned by an external static magnetic field. The left heatmap shows the measured transmission from port 1 to port 2 as a function of magnetic field, while the middle heatmap shows the transmission from port 3 to port 4. The right panel displays line cuts of the two transmission spectra when modes $A(x)$ and $A(y)$ are tuned into resonance with the spin ensemble at approximately 14.2 mT and 17.0 mT, respectively. The resonance frequency of mode $A(x)$ is 3.1598 GHz with a half width at half maximum (HWHM) of 0.995 MHz. The resonance frequency of mode $A(y)$ is 3.2275 GHz with an HWHM of 0.911 MHz. The white dotted lines indicate the calculated transition frequency of the $\text{NV}^-$$m_s = +1$ state under a magnetic field applied along the $[100]$ crystallographic direction. All transmission amplitudes are normalized. The extracted collective coupling strength is 5 MHz for both modes $A(x)$ and $A(y)$.
• Figure 4: Measured transmission amplitude and phase of the readout mode $A(y)$ from port 4 to port 3 for different spin transition frequencies. Red traces correspond to zero external magnetic field, where the $\text{NV}^-$ spin transition frequency is 2.87 GHz. Black traces correspond to the case where the spins are tuned into resonance with mode $A(x)$ at 3.1598 GHz. The detuning between mode $A(y)$ and the spin ensemble is 67.7 MHz, which is much larger than the collective coupling strength. A clear dispersive frequency shift of $\chi = 0.27$ MHz is observed for mode $A(y)$.