Detecting quantum noise of a solid-state spin ensemble with dispersive measurement

TL;DR

This work develops a comprehensive theoretical framework for dispersive readout of a solid-state spin ensemble coupled to a microwave resonator, analyzed via time-integrated homodyne detection. It derives analytic conditions under which the measurement is limited by intrinsic spin-projection noise rather than technical noise, encapsulated in the quality parameter λ, and shows how inhomogeneous broadening and phase noise affect sensitivity. The authors also propose a protocol to detect spin squeezing through fluctuations in the integrated homodyne signal and quantify the required experimental resources, including the number of runs, to certify entanglement-enhanced metrology. The results offer concrete guidelines for achieving quantum-limited magnetometry with solid-state spins and for benchmarking entangled states in realistic, noise-prone solid-state platforms.

Abstract

We theoretically explore protocols for measuring the spin polarization of an ensemble of solid-state spins, with precision at or below the standard quantum limit. Such measurements in the solid-state are challenging, as standard approaches based on optical fluorescence are often limited by poor readout fidelity. Indirect microwave resonator-mediated measurements provide an attractive alternative, though a full analysis of relevant sources of measurement noise is lacking. In this work we study dispersive readout of an inhomogeneously broadened spin ensemble via coupling to a driven resonator measured via homodyne detection. We derive generic analytic conditions for when the homodyne measurement can be limited by the fundamental spin-projection noise, as opposed to microwave-drive shot noise or resonator phase noise. By studying fluctuations of the measurement record in detail, we also propose an experimental protocol for directly detecting spin squeezing, i.e. a reduction of the spin ensemble's intrinsic projection noise from entanglement. Our protocol provides a method for benchmarking entangled states for quantum-enhanced metrology.

Figures (5)

• Figure 1: (a) Schematic of a circuit-QED setup relevant to our system, with $N$ spin-1/2 degrees of freedom coupling to a resonator $\hat{a}$ with individual couplings $g_j$. Excitations decay from the resonator to its environment with strength $\kappa$, and from individual spins with rate $\gamma_{-}$. The resonator is driven by an input microwave field $\hat{a}_{\mathrm{in}}$. The quadratures of the light field coming out of the resonator $\hat{a}_{\mathrm{out}}$ are passed through, e.g., an $I$-$Q$ mixer, and measured to obtain a time-integrated homodyne current $M$. (b) Schematic of the system spectrum. The spins have inhomogeneous frequencies $\delta_j$, while the resonator is detuned by $\Delta$ from the spins' center frequency. (c) Schematic of how an initial collective spin polarization $\langle \hat{S}^{z}(0)\rangle$ (either on the equator of the spin Bloch sphere, or inclined by some angle $\theta$) affects the dynamics of the integrated homodyne quadrature measurement outcome $M(T)$ over a collection time $T$. Different trajectories correspond to different runs of an experiment. (d) For a fixed set of $N_{\mathrm{runs}}$ experimental runs, the distribution of measurement outcomes after a collection time $T$ may be binned. The mean of the distribution is the expectation value $\langle \hat{M}(T)\rangle$ dependent on the collective spin polarization. The width of the distribution is set by the variance $(\Delta M(T))^2$, which we seek to make spin-projection-noise limited.
• Figure 2: Experimental setup for spin ensemble detection at and beyond the standard quantum limit (SQL): (a) proposed experimental device and (b) measurement setup. (a) To realize Fig. \ref{['fig:fig1']} (a), a superconducting thin film (grey) is lithographically patterned on top of the diamond substrate (light blue) containing a near-surface defect center spin ensemble (pink). The patterned superconductor defines a lumped-element resonator circuit with a narrow wire forming the inductor (L) carrying current, and planar meandered fingers forming the capacitor (C) carrying charge, with the resonant frequency lying in the microwave regime. A zoomed-in cartoon around the wire is shown as inset. The current-carrying wire produces a microwave magnetic field (gold lines) which couples to the spins located within the microwave resonator mode volume. This coupling is modeled by the Jaynes-Cummings interaction written in Eq. \ref{['eq:1']} and ultimately responsible for the dispersive interaction in Eq. \ref{['eq_SWModel']}. b) Microwave detection circuit of the device shown in (a) and caricatured in Fig. \ref{['fig:fig1']}. The Larmor precession frequency of the spins is tuned by a DC magnetic field produced by a Helmholtz coil (maroon). The measurement chain is kept low-noise by attenuating thermal noise along the input lines and adding low-noise amplification along the output lines. Homodyne microwave detection measures the field quadratures $I$ and $Q$ by interfering the signal exiting the resonator with a known local oscillator (LO) reference. Time-integration of the field quadratures yields the measurement observable $M$.
• Figure 3: (a) Contributions to the measurement variance $(\Delta M(T))^2 = (\Delta M_{\mathrm{spin}}(T))^2+(\Delta M_{\mathrm{shot}}(T))^2$ [defined in Eq. \ref{['eq_NoiseTotal']}] from intrinsic spin-projection noise $(\Delta M_{\mathrm{spin}}(T))^2$ and shot noise $(\Delta M_{\mathrm{shot}}(T))^2 = T$ from homodyne detection. We use parameters from Table \ref{['table_Params']}, but vary the number of resonator photons $\overline{n}$. (b) Total measurement variance $(\Delta M(T))^2$, for different values of the measurement-quality parameter $\lambda$, defined in Eq. \ref{['eq_FinalSmallParameter']}, normalized by the shot noise. We use the same parameters as panel (a), varying $\lambda$ by tuning $\overline{n}=2.5 \times 10^4, 10^5, 10^6$ for $\lambda =1, 10,100$.
• Figure 4: (a) SNR of the measurement relative to the standard quantum limit $\sqrt{N}$ from Eq. \ref{['eq_SNR']}, assuming an infintesimally small angle $\theta \ll 1$ and homogeneous system parameters. (b) Frequency schematic for an inhomogeneous ensemble. We still use the parameters from Table \ref{['table_Params']}, still fix spin-resonator couplings at $g_j/(2\pi) = 50$ Hz, but randomly sample non-uniform spin frequencies $\delta_j$ from a Gaussian distribution with standard deviation $\sigma_{\delta} = 2\pi \times 1$ MHz, leading to non-uniform decay rates $\gamma_j = \gamma_{-} + \kappa |g_j|^2/(\Delta-\delta_j)^2$ and dispersive couplings $\chi_j=|g_j|^2/(\Delta-\delta_j)$. Since this sampling can lead to some spins breaking the dispersive approximation, we discard all spins with sampled frequencies satisfying $|\Delta - \delta_j|\leq g \sqrt{\overline{n}}$ (highlighted red region). (c) SNR for an inhomogeneous ensemble using the above approach. The SNR is calculated for 10 realizations of random spin frequencies $\delta_j$ and averaged together. (d) Difference in signal and spin noise between inhomogeneous and homogeneous parameters from Eq. \ref{['eq_InhomogeneousNoiseComparison']} for a collection time $\gamma_{-} T = 1$.
• Figure 5: (a) Schematic of an entangled initial spin-squeezed state characterized by Wineland parameter $\xi^2$. The subsequent variance in the integrated homodyne current will shrink (grow) for (anti-)squeezed states with $\xi^2 <1$ ($\xi^2 > 1$) for fixed collection time $T$. (b-c) Noise difference (change in variance) of the homodyne measurement from Eq. \ref{['eq_spinNoiseEntangled']}, normalized by the shot noise $(\Delta M_{\mathrm{shot}}(T))^2 = T$. We consider both (b) anti-squeezed and (c) squeezed states. The squeezing is reported in decibels $\xi^2[\text{dB}] = -10 \text{log}_{10}(\xi^2)$. We use homogeneous system parameters from Table \ref{['table_Params']} except for the spin-resonator coupling, which we fix to $g_j = 2\pi \times 63$ Hz, thereby setting the measurement quality parameter to $\lambda = 10$.