Absorption-Based Diamond Spin Microscopy on a Plasmonic Quantum Metasurface

The principle of NV-based magnetometers lies in the Zeeman energy shift of the NV defect spin sublevels that can be polarized and measured optically. As illustrated in Fig. 1c, the spin sublevels m${}_{\text{s}}$ = 0 and m${}_{\text{s}}$ = $\pm 1$ of the ³A₂ ground triplet state, labelled as $\ket{1}$ and $\ket{2}$, respectively, are separated by a zero-field splitting of D = 2.87 GHz, whose transition can be accessed with a resonant microwave field. Upon spin-conserving off-resonant green laser excitation, a fraction of the population decays non-radiatively into the ¹A₁ metastable singlet state, $\ket{5}$, predominantly from $\ket{4}$ as k₄₅ $\gg$ k₃₅, where k${}_{\text{ij}}$ indicates the decay rate from level i to level j. After a sub-ns decay from $\ket{5}$ to $\ket{6}$, the shelving time at $\ket{6}$ exceeds 200 ns at room temperature Acosta et al. (2010b). Therefore, the population of $\ket{6}$ can be measured by absorption of the singlet state transition resonant at 1042 nm.

Figure 1a shows the proposed PQSM for the IR-absorption-based detection scheme. The sensing surface is pumped with a green laser at 532 nm for NV spin initialization and illuminated with transverse magnetic (TM) polarized probe light at $\lambda_{0}$ = 1042 nm for IR readout. The PQSM-NV layer causes a spin-dependent phase and amplitude of the IR reflection. The spatially well-defined signal beam is separated from the incident probe field in a dark-field excitation geometry (i.e., k-vector filtering). By interfering with a local oscillator, phase-sensitive homodyne detection at the camera enables measurement at the photon shot noise limit. The PQSM doubles as a wire array for NV microwave control Shalaginov et al. (2020); Ibrahim et al. (2020): with a subwavelength spacing, an array of the silver wires produces a homogeneous transverse magnetic field, $\vec{B}$, as shown in Fig. 1b. Local excitation and probing of NVs within a pixel are possible by running a current through an individual wire.

Here we discuss photonic design criteria to maximize the IR signal of spin ensemble sensors. The IR absorption readout has only been successfully implemented with bulk diamond samples due to the intrinsic absorption cross sectional area that is about an order of magnitude smaller than that of the triplet state transition Dumeige et al. (2013); Jensen et al. (2014). This weak light-matter interaction can be enhanced by modifying the electromagnetic environment of quantum emitters. The rate of absorption of a quantum emitter under an oscillating electromagnetic field with frequency, $\omega_{0}$, can be expressed following Fermi’s golden rule.

$$\Gamma_{\text{abs}}=\frac{2\pi}{\hbar}|\braket{5}{\vec{\mu}\cdot\vec{E}}{6}|^{2}\rho(\omega_{0})$$

(1)

where $\vec{\mu}=e\cdot\vec{r}$ is the transition dipole moment operator, $\vec{E}$ is the electric fields, and $\rho(\omega)=\frac{1}{\pi\hbar}\frac{\frac{1}{2}\gamma^{*}}{(\omega-\omega_{0})^{2}+(\frac{1}{2}\gamma^{*})^{2}}$ is the electronic density of states, which is modeled as a continuum of final states with a Lorentzian distribution centered at $\omega_{0}$ with linewidth $\gamma^{*}$. The equation shows that the rate can be enhanced by increasing the electric field at the emitter position. Plasmonic structures can focus light intensity at subwavelength scales, and thus, they have been used to increase spontaneous emission rates of single emitters or ensembles of emitters confined in a nanometer-scale volume Choy et al. (2013); Akselrod et al. (2014); Hoang et al. (2015); Karamlou et al. (2018); Bogdanov et al. (2017); De Leon et al. (2012); Hausmann et al. (2013); Jun et al. (2010); Russell et al. (2012). However, this field concentration comes with the trade-off of reducing the number of NV centers, $N_{\text{NV}}$, that are coupled to the optical field. Balancing this trade-off depends on the use case: sensing applications with a spatial resolution below the grating period benefit from highly localized SPP-like modes, while applications with larger lateral resolution benefit from more RWA-like modes that average over a larger $N_{\text{NV}}$. To guide the PQSM optimization, we adopt a figure of merit addressing the latter of $\braket{}{E/E_{0}}{{}^{2}}V_{\text{pixel}}n_{\text{NV}}$, where $\braket{}{E/E_{0}}{{}^{2}}=\int_{\text{pixel}}|E/E_{0}|^{2}dV/\int_{\text{pixel}}dV$ is the spatially averaged optical field enhancement over the single-pass field without plasmonic structure, $E_{0}$, $d_{\text{NV}}$ is the thickness of the diamond sensing layer with NV density of $n_{\text{NV}}$ , and $V_{\text{pixel}}=L^{2}d_{\text{NV}}$ is the sensing volume.

The SPP-RWA resonance delocalizes the plasmonic modes, creating a large field enhancement within a few-micron-thick surface layer of diamond, as shown in the corresponding spatially-resolved electric field intensity profile (Fig. 1b). The structure dimensions can be chosen to find the desired balance between SPP modes and RWA modes: stronger SPP localization resulting in more localized sensing can be traded against better sensitivity, and vice versa. SPPs do not couple with free-space light without satisfying the momentum matching conditions. An incoming far-field optical excitation can excite SPPs modes via a grating structure with period $p$ given by $G=2m/p$ where $m$ is an integer. To form a metallodielectric grating, the plasmonic nanostructures are arranged periodically with a period of $\lambda_{0}/n$, where $\lambda_{0}$ is 1042 nm and $n$ is the refractive index of diamond. As evident in the dispersion relation of RWA (Eq. 2), this period satisfies the condition for first-diffraction-order RWA mode under normal incidence (i.e., $k_{x}=0$).

$$\frac{\omega}{c}n=k_{x}+m\frac{2\pi}{p}$$

(2)

where $c$ is the speed of light in vacuum, $k_{x}=k_{0}\sin(\theta_{i})$ is the momentum component of free-space light in the direction of grating period, and $m$ denotes the diffraction order Gao et al. (2009); Steele et al. (2003). When the RWA mode is excited, the incident electromagnetic wave diffracts parallel to the grating surface and creates a field profile that extends vertically away from the grating surface Wood (1935); Rayleigh (1907). Thus, the SPP-RWA hybrid mode shows the electric field intensity profile extending a few microns from the grating surface while maintaining a large field concentration near the Ag-diamond interface as shown in Fig. 1b. While the RWA mode alone does not depend on the properties of the plasmonic material (Eq. 2), the SPP-RWA hybrid mode is highly dependent on the dispersive permittivity of the plasmonic material. When silver is replaced with palladium (Pd), a weak plasmonic material, the electric field enhancement is heavily suppressed (refer to SI Section 1). As the SPP-RWA mode delocalizes the field away from the lossy material, this PQSM exhibits a quality factor of 935 at 1042 nm, an exceptionally high value for a plasmonic coupled mode.

To determine the signal from a pixel of the PQSM containing an ensemble of emitters, the rate of absorption is averaged over all four orientations of NV emitters. For a given angle, $\theta$, between the emitter’s transition dipole orientation and the electric field created by the PQSM in the diamond layer, Eq. 1 can be expressed as Eq. 3 in terms of the spontaneous emission rate of the singlet state transition, $\gamma=\frac{\omega_{0}^{3}|\braket{5}{\vec{\mu}}{6}|^{2}}{3\pi\epsilon\hbar c^{3}}$.

$$\Gamma_{\text{abs}}=\frac{3}{\pi^{2}\hbar}\frac{\gamma}{\gamma^{*}}(\frac{\lambda_{0}}{n})^{3}\cdot\frac{1}{2}\epsilon_{0}\epsilon|\vec{E}|^{2}\cos^{2}(\theta)$$

(3)

where $\epsilon$ is the relative permittivity of diamond. For a [100] diamond plane, all four orientations of NVs are expected to have equal contributions for the given SPP-RWA-induced field profile. The signal-to-noise ratio (SNR) of the pixelated plasmonic imaging surface is given by Eq. 4 under the assumption of the shot noise limit.

$$\begin{split}\text{SNR}&=\frac{|N_{0}-N_{1}|}{\sqrt{N_{0}+N_{1}}}\\ &=\sqrt{\frac{\Delta t_{\text{mea}}L^{2}}{\hbar\omega_{0}}}\frac{I_{\text{out}}(I_{t},0)-I_{\text{out}}(I_{t},\Omega_{R})}{\sqrt{I_{\text{out}}(I_{t},0)+I_{\text{out}}(I_{t},\Omega_{R})}}\end{split}$$

(4)

where $N_{0}$ and $N_{1}$ are the average numbers of photons detected from the m${}_{\text{s}}$ = 0 and m${}_{\text{s}}$ = $\pm 1$ states, respectively, per measurement, $\Delta t_{\text{mea}}$ is the total readout time, and $I_{\text{out}}(I_{t},\Omega_{R})$ is the reflected intensity under green laser intensity, $I_{t}$, and an applied microwave Rabi field, $\Omega_{R}$. In the limit of low contrast, the SNR can be re-written as follows:

$$\begin{split}\text{SNR}&=\sqrt{\frac{I_{\text{out}}(0,0)\Delta t_{\text{mea}}L^{2}}{2\hbar\omega_{0}}}(I_{\text{NV}}(I_{t},\Omega_{R})-I_{\text{NV}}(I_{t},0))\\ &\propto\braket{}{E/E_{0}}{{}^{2}}V_{\text{pixel}}n_{\text{NV}}\end{split}$$

(5)

where $I_{\text{NV}}(I_{t},\Omega_{R})=\frac{I_{\text{out}}(0,0)-I_{\text{out}}(I_{t},\Omega_{R})}{I_{\text{out}}(0,0)}$ is the fractional change in IR intensity due to NV absorption, and $I_{\text{NV}}(I_{t},\Omega_{R})-I_{\text{NV}}(I_{t},0)\propto\braket{}{E/E_{0}}{{}^{2}}V_{\text{pixel}}n_{\text{NV}}$ as described in detail in SI Section 3. Thus, the SNR scales with both spatially averaged electric field intensity enhancement factor and sensing volume for a given NV density. The PQSM combines a large field enhancement of the SPP mode and delocalization of the RWA mode, making the SPP-RWA hybrid mode well suited for ensemble-based sensing.

The spin-dependent IR absorption depends on the population of the ground singlet state, and it can be obtained by calculating the local density of each sublevel, $n_{\ket{i}}$, based on the coupled rate equations described in SI Section 2. As shown in Fig. 1c, we use an eight-level rate equation model that accounts for photoionization. Under continuous wave (CW)-ODMR, Fig. 2a shows the calculated $n_{\ket{6}}$, indicating that the population of the singlet states weakly depends on the IR probe intensity, $I_{s}$, until the absorption rate becomes comparable to the excited state decay rate. The SPP-RWA-induced field enhancement modifies the absorption rate as well as the radiative decay rate by $\gamma_{\text{rad}}\rightarrow F_{p}\gamma_{\text{rad}}+\gamma_{\text{quenching}}$, where $F_{p}$ is the Purcell factor. However, due to the low intrinsic quantum efficiency of the singlet state transition ($\frac{\gamma_{\text{rad}}}{\Gamma}\approx$0.1% Ulbricht and Loh (2018)), the plasmonic structures have minimal effect on the overall excited state decay rate, $\Gamma$. Because the lifetime of the ground state is approximately two orders of magnitude longer than that of the excited state Acosta et al. (2010b), the singlet state transition has an exceptionally high saturation intensity, enabling each NV to absorb multiple photons per cycle. With resonant structures, the system can be brought to its saturation level at a lower incident intensity due to the transition rate enhanced by a factor of $\sim\braket{}{E/E_{0}}{{}^{2}}$. Similarly, the current PQSM structure can induce a grating mode resonant at 532 nm with an off-normal back illumination (refer to SI Section 1).

Under an applied microwave field, an increase in the population of the m${}_{\text{s}}=\pm 1$ states incurs an increase in absorption at 1042 nm (i.e., $I_{\text{NV}}(I_{t},\Omega_{R})>I_{\text{NV}}(I_{t},0)$). This state-selective NV infrared absorption produces spin-dependent amplitude and phase changes in the IR probe field reflected from the PQSM. The input-intensity-normalized NV absorption, $A_{\text{pixel}}$, (equivalently, $I_{\text{NV}}(I_{t},\Omega_{R})\times\frac{I_{\text{out}}(0,0)}{I_{s}}$) with varying sensing depth is shown in Fig. 2b. To account for stimulated emission, a net population (i.e., $n_{\ket{6}}$-$n_{\ket{5}}$) is used to calculate the net spin-dependent NV absorption. Based on the calculated $A_{\text{pixel}}$, the NV-induced corresponding phase change, $\Delta\phi_{\text{NV}}$, is obtained from the complex reflection coefficients simulated with FDTD method (Fig. 2b).

The spin-dependent phase and amplitude changes of the signal allow for a phase-sensitive measurement. Here, we implement a phase-sensitive coherent homodyne detection, where a local oscillator interferes with the spin-dependent signal from the PQSM. A combination of $R$ and $\Delta\phi_{\text{LO}}$ is chosen to maximize the SNR (refer to SI Section 4) and the readout fidelity. The incident-intensity-normalized interfered intensity detected by the camera is given by Eq. 6.

$$\begin{split}&\frac{I_{\text{out}}(I_{t},\Omega_{R},R,\Delta\phi_{\text{LO}})}{I_{s}}=(1-R)+R|r(I_{t},\Omega_{R})|^{2}\\ &+2\sqrt{(1-R)R}|r(I_{t},\Omega_{R})|\cos(\Delta\phi_{\text{LO}}+\Delta\phi_{\text{NV}}(I_{t},\Omega_{R}))\end{split}$$

(6)

where $R$ is the power splitting ratio of the beam splitter, $r(I_{t},\Omega_{R})$ is the complex reflection coefficient of the PQSM, $\Delta\phi_{\text{LO}}$ is the relative phase difference between the LO and the reflected light of the PQSM when $I_{t}$ = 0 and $\Omega_{R}$ = 0, and $\Delta\phi_{\text{NV}}(I_{t},\Omega_{R})$ is an additional phase change incurred by the NV absorption. Figure 2c shows SNR that is normalized by the measurement time and pixel area of $L^{2}$. Under the conditions considered in this work, the photon shot noise dominates, and a better SNR is achieved by biasing the interferometric readout with a controlled phase difference. Homodyne detection is particularly advantageous for fast imaging on focal plane arrays.

The shot-noise-limited sensitivity of a CW-ODMR-based magnetometer per root area based on IR absorption measurement is given by Eq. 7.

$$\eta_{\text{CW}}^{A}=\frac{\hbar\Gamma_{\text{MW}}}{g\mu_{\text{B}}}\frac{\sqrt{\Delta t_{\text{mea}}L^{2}}}{\text{SNR}}$$

(7)

where $g\approx 2.003$ is the electronic g-factor of the NV center, $\mu_{\text{B}}$ is the Bohr magneton, and $\Gamma_{\text{MW}}$ is the magnetic-resonance linewidth which can be approximated as $\Gamma_{\text{MW}}=2/T_{2}^{*}$, assuming no power broadening from pump or microwaves. For a given NV density of $\sim$16 ppm, the dephasing time is limited by paramagnetic impurities with the conversion efficiency conservatively approximately as 16% Ulbricht and Loh (2018). An alternative magnetometry method to CW-ODMR, such as pulsed ODMR or Ramsey sequences, can be exploited to achieve $T_{2}^{*}$-limited performance.

It is useful to compare the photon-shot-noise-limited sensitivity with the spin-projection-noise-limited sensitivity of an ensemble magnetometer consisting of non-interacting spins.

$$\eta_{\text{sp}}^{A,\text{ensemble}}=\frac{\hbar}{g\mu_{\text{B}}\sqrt{n_{\text{NV}}d_{\text{NV}}\tau}}$$

(8)

where $\tau$ is the free precession time per measurement. Figure 3a shows that the PQSM can achieve sub-nT Hz${}^{-\frac{1}{2}}$ sensitivity per µm² sensing surface area for a given an NV layer thickness of $d_{\text{NV}}$ = 5 µm and remaining experimental parameters listed in Table 1. As expected, sensitivity improves with increasing green laser intensity until two-photon-mediated photo-ionization processes start to become considerable. Furthermore, as shown in Fig. 3b, increasing the sensing depth from 500 nm to 10 µm improves the sensitivity by a factor of up to $\sim$ 9. There exists a trade-off between achievable sensitivity and spatial resolution.

Sources of the NV spin dephasing can be largely eliminated with coherent control techniques such as the Hahn echo sequence. With an added $\pi$-pulse halfway through the interrogation time, a net phase accumulated due to a static or slowly varying magnetic field cancels out, and the interrogation time can be extended to a value of $\sim$ $T_{2}$. Thus, the AC sensitivity can improve by a factor of approximately $\sqrt{T_{2}^{*}/T_{2}}$ at the cost of a reduced bandwidth and insensitivity to magnetic field with an oscillating period longer than $T_{2}$. The sensitivity per root area for an ensemble-based AC magnetometer is given by Eq. 9 Barry et al. (2020).

$$\eta_{\text{a.c.}}^{A}=\frac{\hbar\sigma_{\text{R}}e^{\tau/T_{2}}}{g\mu_{\text{B}}\sqrt{n_{\text{NV}}d_{\text{NV}}\tau}}\sqrt{1+\frac{t_{\text{I}}+t_{\text{R}}}{\tau}}$$

(9)

where $T_{2}$ is the characteristic dephasing time, $t_{\text{I}}$ is the initialization time, $t_{\text{R}}$ is the readout time, and $\sigma_{\text{R}}$ is the readout fidelity. For a given NV density of $\sim$ 16 ppm, $T_{2}$ is about an order of magnitude longer than $T_{2}^{*}$ Barry et al. (2020). Rate equations are solved as a function of time to obtain time-dependent population evolution as shown in Fig. 4a. The system loses spin polarization after a few microseconds of readout time. For given $I_{t}$ and $I_{s}$, an optimal readout time that maximizes the time-integrated signal is calculated and is shown to be near 500 ns (refer to SI Section 5). An additional shot noise introduced by the optical readout is quantified with the parameter $\sigma_{\text{R}}$ (Eq. 10), which is equivalent to an inverse of readout fidelity Barry et al. (2020).

$$\sigma_{\text{R}}=\sqrt{1+\frac{2(a+b)}{(a-b)^{2}}}$$

(10)

where $a$ and $b$ are the average numbers of photons detected from the m${}_{\text{s}}$ = 0 and m${}_{\text{s}}$ = $\pm 1$ states per spin per measurement, respectively. The PQSM achieves spin readout fidelity near 0.5 per shot. Figure 4c shows the AC sensitivity down to 10 pT Hz${}^{-\frac{1}{2}}$ per 1-µm² sensing surface area.