Broadband microwave detection using electron spins in a hybrid diamond-magnet sensor chip

Quantum sensing has developed into a main branch of quantum science and technology. It aims at measuring physical quantities with high resolution, sensitivity, and dynamic range. Electron spins in diamond are powerful magnetic field sensors, but their sensitivity in the microwave regime is limited to a narrow band around their resonance frequency. Here, we realize broadband microwave detection using spins in diamond interfaced with a thin-film magnet. A pump field locally converts target microwave signals to the sensor-spin frequency via the non-linear spin-wave dynamics of the magnet. Two complementary conversion protocols enable sensing and high-fidelity spin control over a gigahertz bandwidth, allowing characterization of the spin-wave band at multiple gigahertz above the sensor-spin frequency. The pump-tunable, hybrid diamond-magnet sensor chip opens the way for spin-based sensing in the 100-gigahertz regime at small magnetic bias fields.

Electron spins associated with nitrogen-vacancy (NV) defects in diamond are magnetic field sensors that provide high spatial resolution and sensitivity at room temperature [1,2]. They have been used to study nuclear magnetic resonance at the nanoscale [3,4], bio- [5], paleo- [6], and solid-state magnetism [7], and electric currents in quantum materials [8,9]. Most of these applications focus on detecting magnetic fields in the 0-100 megahertz (MHz) frequency range, in which a toolbox of spin-control techniques enables high sensitivity and a tunable detection frequency without requiring a specific electron spin resonance (ESR) frequency [1]. In contrast, NV-based sensing in the microwave regime [1-100 gigahertz (GHz)] currently relies on tuning the ESR to the frequency of interest using a magnetic bias field [10]. This bias field changes the properties of e.g. magnetic or superconducting samples under study [11,12], for instance by altering their excitation spectrum, which limits its application in materials science. Furthermore, the field must be on the Tesla scale for operation in the 10-100 GHz range [13], making the required magnets large and slow to adjust, precluding the small sensor packaging desired for technological applications.
Here, we enable broadband spin-based microwave sensing by interfacing a diamond chip containing a layer of NV sensor spins with a thin-film magnet. The central concept is that the non-linear dynamics of spin waves -the collective spin excitations of the magnetic film [14] -locally convert a target signal to the NV ESR frequency under the application of a pump field (Fig. 1A-B). We realize a ∼1-GHz detection bandwidth at fixed magnetic bias field via four-spin-wave mixing, and microwave detection at multiple GHz above the ESR frequency via difference-frequency generation.
The pump-tunable detection frequency enables characterizing the spin-wave band structure despite a multi-GHz detuning and provides insight into the non-linear spin-wave dynamics limiting the conversion process. Furthermore, the converted microwaves are highly coherent, enabling high-fidelity control of the sensor spins via off-resonant drive fields.
Our hybrid diamond-magnet sensor platform consists of an ensemble of near-surface NV spins in a diamond membrane positioned onto a thin film of yttrium iron garnet (YIG) -a magnetic insulator with low spin-wave damping [14] (Fig. 1B). A stripline delivers the "two-color" signal and pump microwave fields to the YIG film, in which they excite spin waves at the signal and pump frequencies, f s and f p , respectively. The frequency-converted microwaves at the ESR frequency f NV are detected by measuring the spin-dependent NV photoluminescence under green laser excitation (Methods and Fig. 1C). The ESR frequency is fixed by an external magnetic bias field B NV (Fig. 1D).
Our first detection protocol harnesses degenerate four-spin-wave mixing [15-17] -the magnetic analogue of optical four-wave mixing ( Fig. 2A). In the quasiparticle picture, this process corresponds to the scattering of two "pump" magnons into a "signal" magnon and an "idler" magnon at frequency f i = 2f p − f s . This conversion enables the detection of a microwave signal that is detuned from the ESR frequency, which would be otherwise invisible in the optical response of the NV centers (Fig.   2B). By tuning the frequency of the pump, we enable the detection of signals of specific microwave A "spin-wave mixer" uses a pump to convert a microwave signal at frequency f s to an output frequency f NV that is detectable by nitrogenvacancy (NV) sensor spins in diamond. (B) Sketch of the setup. A diamond with NV centers implanted ∼ 10-20 nm below its surface is placed onto a film of yttrium iron garnet (YIG, thickness: 235 nm). A microstrip delivers the signal and pump microwaves, which excite spin waves in the YIG. Spin-wave mixing enables detection of the signal field by converting its frequency to the NV electron spin resonance (ESR) frequency. Inset: Atomic structure of an NV center in the diamond carbon (C) lattice. (C) Initialization and readout of the NV spins is achieved through excitation by a green laser and detection of the red photoluminescence (PL). The PL is stronger in the m s = |0 state than in the m s = |±1 states. (D) NV spin energy levels in the electronic ground state. A magnetic field B NV along the NV axis splits the m s = |±1 states via the Zeeman interaction. From the four possible configurations in the diamond lattice, we use the NV orientation with in-plane projection parallel to the stripline. f NV denotes the |0 ↔ |−1 ESR transition frequency. frequencies (Fig. 2C).
We characterize the bandwidth of the four-wave-mixing detection scheme by measuring the NV photoluminescence contrast as a function of the microwave signal frequency and magnetic bias field. Fig. 2B, when the pump field is switched off, we only detect signals resonant with f NV (Fig.   2D). In contrast, when the pump is switched on, a broad band of frequencies becomes detectable  ( Fig. 2E). The bandwidth ∆f of ∼ 1 GHz is limited from below by the ferromagnetic resonance (FMR), the spatially homogenous spin-wave mode below which spin waves cannot be excited in our measurement geometry, and from above by the limited efficiency of our 5-micron-wide stripline to excite high-momentum spin waves. As such, the bandwidth can be extended by using narrower striplines or magnetic coplanar waveguides [19].

As in
At 14 dBm signal and pump power, consecutive mixing processes generate higher-order idler modes at discrete and equally spaced frequencies (Fig. 2F). Motivated by the success of their optical coun-terparts in high-precision spectrometry [20], such "spin-wave frequency combs" are of great interest because of potential applications in microwave metrology [21][22][23]. We use the spin-wave comb to realize sensitivity to multiple microwave frequencies by detecting the n-th order idler frequency, when it is resonant with the ESR frequency (Fig. 2F, upper inset). An increasing number of idler modes appears with increasing drive power (Fig. S3), such that at large powers we resolve up to the n = 10th idler order (Fig. 2F, bottom inset). The shift of the idler frequency is amplified by the integer n over the shift of the signal frequency (Eq. 1), leading to a 1/n decrease in the linewidth of the NV ESR response [22] (Fig. 2F) and a correspondingly enhanced ability to resolve closely spaced signal frequencies.
In addition to enabling off-resonant quantum sensing, the idlers also provide a resource for off- We only observe ESR contrast when both f s and f p are above the FMR (Fig. 4D), confirming that the conversion is mediated by spin waves. As such, the efficiency of the conversion process can be used to characterize high-frequency magnetic band structures. Similar to Fig. 2E, the conversion is limited by the spin-wave excitation efficiency, which explains the observation of the largest ESR contrast for long-wavelength spin waves (i.e. just above the FMR).
We demonstrated magnon-mediated, spin-based sensing of microwave magnetic fields over a gigahertz bandwidth at a fixed magnetic bias field. The frequency of the pump determines the detection frequency, and the detection range is set by the frequencies at which spin waves can be excited efficiently.
The range could be extended to the 10-100 GHz scale using materials with a larger magnetization that increases the spin-wave group velocity or crystal anisotropies that increase the spin-wave gap NV microwave magnetometry. The four NV-center families are sensitive to microwave magnetic fields at their electron spin resonance (ESR) frequencies, which are determined by the magnetic bias field B NV via the NV spin Hamiltonian H = DS 2 z + γB NV · S, with D = 2.87 GHz the zero-field splitting, γ = 28 GHz/T the electron gyromagnetic ratio and S i∈{x,y,z} the ith spin-1 Pauli matrix.
In this work, we align the field along one of the NV orientations, such that this "on-axis" family has |0 ↔ |±1 ESR frequencies given by D ± γB NV (with B NV = |B NV |). For the other three "off-axis" families, the bias field is equally misaligned by ∼ 71 • due to crystal symmetry, leading to the ESR frequency plotted in Fig. 4B (labeled "Off-axis"). The photoluminescence dips were recorded using continuous-wave microwaves and non-resonant optical excitation at 515 nm. For the Rabi oscillations we first initialize the NV spin in the |0 -state via a ∼ 1-µs green laser pulse, then we drive the spin using an idler pulse and finally we read out the NV photons in the first 300-400 ns of a second laser pulse.
Data processing. The data presented in Figs. 2F and 4D is normalized by the median of each row and column (Fig. S2).     S1. Derivation of the spin-wave dispersion for bias fields along the NV axis Here we derive the spin-wave dispersion for a magnetic film in the xy-plane with perpendicular magnetic anisotropy (PMA) and a magnetic bias field B B in an arbitrary direction. The dispersion is given by the poles of the transverse magnetic susceptibility [1, 2] that relates the transverse magnetization to a drive field B AC . We derive the magnetic susceptibility from the Landau-Lifshitz-Gilbert (LLG) equation that describes the dynamics of the unit magnetization vector ṁ

References
where α G is the Gilbert damping and the "overdot" denotes the time derivative. We solve this equation in the (x ,y ,z ) magnet frame that is tilted with respect to the (x,y,z) lab frame by an angle θ 0 , such that the equilibrium magnetization points in theẑ direction and theŷ ( ) axes overlap.
B = B eff +B AC , with B eff the effective magnetic field as derivative of the magnetic free energy density where M s is the saturation magnetization and α ∈ {x , y , z } indicates the vector components in the magnet frame. The free energy density includes the Zeeman energy, the demagnetizing field B d , the PMA energy F A , and the exchange interaction In the magnet frame such that the x and z components of the anisotropy effective field are The contributions of the Zeeman-, demagnetizing-and exchange energy to B eff have been derived in Refs. [1,2].
In linear response with m z ≈ 1, the LLG equation describes the transverse magnetization dynamics.
In the frequency domain it reads where ω is the angular frequency. Substituting the components of the effective magnetic field and rewriting the equations in matrix form, where and ω B = γB B , ω M = γµ 0 M s , ω D = γD Ms and ω K = γK Ms . µ 0 is the vacuum permeability, k = |k| is the modulus of the wavevector along an angle φ with respect to the in-plane projection of the magnetization, θ B is the angle of the magnetic bias field with respect to the plane normal (z axis), and f = 1 − 1−e −kt kt depends on the film thickness t. By inverting the matrix in Eq. (S7), we obtain the transverse magnetic susceptibility, which is singular when Assuming α G 1, the real part of the solutions of this quadratic equation gives the spin-wave dispersion as a function of k The theoretical lines in Figs. 2 and 4 in the main text are based on Eq. (S10). We assume that the field is applied parallel to the NV axis, such that θ B = 54.7 • , with in-plane projection along the stripline. We consider only spin waves with φ = π/2, since these are most efficiently excited by our 150-micron-long stripline. θ 0 minimizes the free energy density and is found by numerically solving ∂F ∂θ 0 = 0. The ferromagnetic resonance (FMR) frequency corresponds to k = 0. Table S1 states the values of the saturation magnetization, exchange and uniaxial anisotropy constants for different magnetic materials used for calculating the spin-wave dispersions in Fig. 4E of the main text.
S2. Dependence of the detection bandwidth on the microwave drive field For efficient frequency conversion, the microwaves should excite propagating spin waves with a significant amplitude. The spin-wave excitation efficiency depends on the microwave power and the spatial mode overlap between the drive field and the spin waves [3]. In our experiment, a 5-micronwide stripline creates an inhomogeneous microwave drive field with a sinc-like amplitude in k-space (Fig. S1A). The efficiency drops with decreasing wavelength with nodes at λ = w/n, where w is the stripline width and n is an integer.
To characterize the dependence on the drive field, we measure the bandwidth induced by four-wave mixing as a function of the pump power (Fig. S1B). As expected, the bandwidth increases with microwave power. The photoluminescence contrast is suppressed at spin-wave frequencies that correspond to the nodes of the drive field in Fig. S1A (colored dashed lines). The frequencies of these modes agree with the spin-wave dispersion derived in the previous section [Eq. (S10)]. The spin-wave excitation antenna is therefore an important design parameter for hybrid diamond-magnet microwave sensors.

Stark shift
A strong microwave field detuned by δf from the NV ESR frequency (f NV ), causes the latter to shift, an effect known as the AC (or dynamical) Stark shift [4]. The Stark shift increases with drive power and is inversely proportional to δf , which allows detecting the presence of an off-resonant microwave signal. We show here that the idler-driven Rabi frequency resulting from four-spin-wave mixing is about an order magnitude larger than the Stark shift at the same off-resonant drive power.
We measure the Stark shift via pump-probe microwave spectroscopy. The high-power pump is detuned from f NV by 10-1000 MHz, while a low-power probe measures the ESR frequency. We determine the Stark shift for every detuning by measuring the ESR frequency with and without pump (blue data in Fig. S4A).
Next, we measure Rabi oscillations using the four-spin-wave mixing technique. We extract the Rabi frequency for signal spin waves detuned from 10 to 710 MHz (red data in Fig. S4A). We attribute the small oscillations in the Rabi frequency and Stark shift to frequency-dependent (cable) resonances in the microwave transmission of the stripline. Fig. S4B shows that the Rabi frequencies are larger than the Stark shift by about an order of magnitude over the measurement range.

S4. Eight-modes model
Here we describe the details of the model for the spin-wave dynamics under a two-tone drive used to calculate the idler amplitude as a function of pump and signal power, as plotted in Fig. 3E in the main text. Fig. S5 shows the spin-wave dispersion of a YIG film for θ k = 0 (blue line) and θ k = π/2 (black line), where θ k is the angle between the in-plane spin-wave wavevector k and the static magnetization for the parameters in Table S1. Since the out-of-plane component of the applied bias field B NV is small compared to the demagnetizing field of ∼ 178 mT in YIG, we assume that the static magnetization lies in-plane alongẑ (x is the out-of-plane axis), parallel to Bẑ, the in-plane component of B NV . The long stripline alongẑ excites signal and pump spin waves with θ k = π/2. Conservation of momentum dictates that the two created idler spin waves also lie on the θ k = π/2 branch with wavevectors k i = 2k p − k s and k i = 2k s − k p (Fig. S5).
When the pump mode is strongly driven beyond a certain threshold, the four magnon scattering term in the spin-wave Hamiltonian c † kp c † kp c k c k leads to a Suhl instability. Here c ( †) k is the annihilation (creation) operator for a magnon with wavevector k, which is normalized by √ S, where S = V s n /V n is the total number of spins, V is the volume, s n is the number of spins per unit cell, and V n is the unit cell volume. A specific pair of magnons wins the "instability competition", k = k p,1 and k = k p,2 = 2k p − k p,1 , which we call the "efficient Suhl pair" of the pump mode. The efficient Suhl pair for the signal k s,1 and k s,2 should also be considered when its mode amplitude is sufficiently large. We disregard cascades that lead to the weak higher-order idlers in Fig. 2F of the main text, as well as the Suhl pairs of the idlers that are safely below their instability threshold at the presently applied powers. A minimal model should therefore include the eight modes indicated in Fig. S5.
The efficient pump and signal pairs can be identified from the threshold amplitude of the pump (signal) mode x = |α p,s | 2 above which the Suhl instability leads to {k , k } pairs, which solve [5] (D Suhl p(s);k ;k where ∆ = ω p(s) − (ω k + ω k )/2, with ω an angular frequency, and ξ is a dissipation rate chosen here to be 10 MHz for all modes. D Suhl p(s);k ;k (D CK p(s);k ) is the matrix element for the scattering process [6]. First, we numerically calculate the threshold amplitude |α p,s | 2 as a function of |k | and θ k as in Fig. S6A. We identify the minimum threshold amplitude in the (|k |,θ k ) plane of Fig. S6A When driving two spin waves at frequencies ω s and ω p , and amplitudes m s and m p , the squared modulus    Fig. S3: Emergence of a spin-wave comb generated by two microwave drives RF 1 and RF 2 . Normalized NV photoluminescence at B NV = 28 mT as a function of RF 1 frequency (RF 2 is kept at 2.2 GHz, red dashed line), and RF 2 power (RF 1 is kept at 4 dBm). An increasing number of higher-order idlers appear at increased drive power. RF 1 and RF 2 function either as pump or signal field depending on which frequency is closer to the ESR frequency f NV = 2.086 GHz, as is indicated by the labels f s and f p with matching colors.   Fig. S5: Eight-modes model for the calculation of the idler amplitude. In our model we consider the signal (blue dot), pump (red dot), idlers (green and black dots), "efficient signal Suhl instability pair" (blue stars) and "efficient pump Suhl instability pair" (red stars) spin waves. The lines are branches of the spin-wave dispersion corresponding to different angles θ k of the wavevector with respect to the static magnetization (see legend). The dispersion is symmetric upon rotations of θ k by π. Calculations like those presented in Fig. S6 lead to the wavevectors of the efficient pump and signal pairs. Here Bẑ = 23 mT.  3.76 · 10 5 9.5 · 10 −13 1.46 · 10 6 Ref. [8] Table S1: Values of the saturation magnetization (M s ), exchange constant (D) and uniaxial anisotropy constant (K) used to calculate the spin-wave dispersions in Fig. 4F of the main text.