Principles and techniques of the quantum diamond microscope

2.1 NV ground electronic state in the absence of external fields

Quantum control of NV centers with the QDM driving fields is possible because of the NV electronic- and spin-level structure [12], [13]. An NV center consists of a substitutional nitrogen and an adjacent lattice vacancy defect in a diamond crystal. A negatively charged NV has six electrons, with two electrons from nitrogen, one electron from each of the three carbon atoms, and an additional electron from the lattice. These electrons occupy four sp³ atomic orbitals with electronic spin quantum number S=1. These sp³ orbitals linearly combine to form four molecular orbitals [14], comprising the ground electronic configuration. The lowest energy state of the ground configuration is the orbital singlet, spin triplet state ³A₂, which has fine, Zeeman, and hyperfine structures, as shown in Figure 2. The four molecular orbitals also give rise to electronic excited states: orbital-doublet spin-triplet ³E, and spin-singlet orbital-singlet ¹E, and ¹A₁ shown in Figure 3.

Figure 2:

NV ground-state configurations and ODMR Spectra.

(A) ¹⁴N hyperfine states and (B) ¹⁵N hyperfine states. Schematic optically detected magentic resonance (ODMR) spectra are shown with Zeeman splitting and hyperfine splitting for ¹⁴N and ¹⁵N. The energy levels for ¹⁴N are further shifted by quadrupolar interactions.

Figure 3:

NV radiative and nonradiative state transitions.

Radiative ³E↔³A₂ transition with optical 637-nm zero-phonon line (ZPL), and ¹E↔¹A₁ transition with non-optical 1042-nm ZPL. Phonon sidebands (PSBs) can shift the transition frequencies. Nonradiative intersystem crossing (ISC)-mediated transitions exist between ³E and ¹A₁ and between ¹E and ³A₂.

NV magnetometry uses fluorescence from electronic state transitions to detect changes to the ³A₂ ground state configuration that result from coupling to a sample field. Therefore, focus is placed on the physics of the ³A₂ Hamiltonian. NV centers have C_3v point-group symmetry, and are spatially invariant under the C_3v symmetry transformations (the identity, two 120° rotations about the NV axis, and three vertical reflection planes). NV centers also have a built-in quantization axis along the NV axis (called the NV z-axis, or the crystallographic [111] direction). The ³A₂ electronic ground state is an orbital singlet and spin triplet manifold, with ground state Hamiltonian [15]

where h is Planck’s constant, and S^=(S^x, S^y, S^x) and I^=(I^x, I^y, I^x) are the dimensionless electron and nitrogen nuclear spin operators, respectively. The first term is the fine structure splitting due to the electronic spin-spin interaction, with the fine structure tensor D [16]. The second term is the hyperfine interaction between NV electrons and the nitrogen nucleus, with the hyperfine tensor A. The third term is the nuclear electric quadrupole interaction, with the electric quadrupole tensor Q. Under the C_3v symmetry of the NV center, D, A, and Q are diagonal in the NV coordinate system [17], [18], and H^gs can be written as [12]

D(T) is the fine-structure term called the zero-field splitting (ZFS), A^‖ and A^⊥ are the axial and transverse hyperfine terms, and P is the nuclear electric quadrupole component. Two important features of the ground state are evident from the Hamiltonian. First, the ³A₂m_s=±1 magnetic sublevels and the m_s=0 level have D(T) difference in energy. D(T) is temperature dependent because the spin-spin interaction changes with the lattice constant [19], [20], with D≈2.87 GHz and dD/dT=−74.2 kHz/K at room temperature. Second, the ³A₂ electronic states have an additional hyperfine energy splitting Ags∥S^zI^z due to the nitrogen nucleus. I=1 for a ¹⁴N nucleus, while I=1/2 for a ¹⁵N nucleus. The energy level diagrams for ¹⁴N and ¹⁵N are shown in Figure 2. The hyperfine parameters are AN14∥=−2.14 MHz,AN14⊥=−2.70 MHz,PN14∥=−5.01 MHz,AN15∥=3.03 MHz, and AN15⊥=3.65 MHz [21].

Crystal stress in the diamond also contributes to the ³A₂ Hamiltonian. This is expressed as [22], [23], [24]

Here, M_x, M_y, M_z, N_x, and N_y are stress-dependent amplitudes. The M_z term contributes to the ZFS, while the other terms may be negligible or suppressed depending on the experimental conditions (such as an applied magnetic field [25]). The NV spin sensitivity to this spin-stress-induced interaction can be used to image internal or external diamond stress [26], which is important for diamond material characterization. However, for imaging external magnetic fields, we consider NV stress sensitivity as a potential limitation.

2.2 NV electronic transitions

The first excited electronic configuration of the NV has an orbital-doublet, spin-triplet state, ³E, shown in Figure 3. The two orbital states and three spin states of ³E combine to form six fine-structure states that reduce to three states at room temperature [27], resembling the ³A₂ state. ³E is coupled to the ³A₂ ground state by an optical 637-nm zero-phonon line (ZPL). The ³E↔³A₂ is a radiative transition that generally conserves the electron spin state m_s as a result of weak spin-orbit interaction [28]. The ³E→³A₂ (³A₂→³E) transition works for longer (shorter) wavelengths in fluorescence (absorption) as a result of the phonon sideband (PSB). This behavior is similar to Stokes and anti-Stokes shifted transitions [29]. Figure 3 also shows the radiative spin-conserving ¹E↔¹A₁ transition, which has an infrared 1042-nm ZPL and its own sideband structure.

Nonradiative transitions between states of different spin multiplicity exist between ³E and ¹A₁ and between ¹E and ³A₂. These nonradiative transitions are caused by an electron-phonon-mediated intersystem crossing (ISC) mechanism and do not conserve spin. The probability of the ISC transition occurring for ³E to ¹A₁ is only non-negligible for the m_s=±1 states of ³E and is characterized by the ISC rate of transition [14]. Similarly, the ISC transition probability from ¹E to the m_s=0 state of ³A₂ is approximately 1.1–2 times that of the ISC transition from ¹E to the m_s=±1 states of ³A₂ [30], [31]. These state-selective differences in the ISC transition rate allow for spin polarization of the NV under optical excitation from 532-nm laser illumination.

2.3 Optical pumping and spin polarization

An optical driving field from a pump laser is applied in order to spin polarize the NV electronic state at the start of a QDM measurement. This pump laser is also used at the end of a measurement to read out the final NV spin state through the fluorescence intensity. NV optical pumping takes advantage of the m_s-selective nonradiative ISC decay pathway [30], [31]. An NV that is optically excited from ³A₂ to ³E state by a 532-nm photon, decays along either the optically radiative ³E→³A₂ pathway or the non-optical, ISC-mediated ³E→¹A₁→¹E→³A₂ pathway. The m_s-selectivity of the ISC will preferentially depopulate the m_s=±1 spin projection states. NVs starting in the ³A₂m_s=±1 sublevel are eventually pumped (on average, after a few pump photon absorption cycles) into the ³A₂m_s=0 sublevel. Typically only ∼80% of NVs in an ensemble can be initilallized into the m_s=0 state [32], where they remain in a cycling transition. The ¹E state is metastable with a ~200 ns lifetime at room temperature [33], [34]. The ³E upper state has a t_3E≈13 ns lifetime [30], [31], and the ³A₂→³E absorption cross section at λ=532 nm [35] is σ=3.1×10⁻¹⁷ cm² (although there is disparity in the reported 532-nm absorption cross-section value and saturation intensity [36], [37]). These correspond to (hc)/(λσt_3E)≈0.9 MW/cm² saturation intensity, where c is the speed of light.

The ISC is also responsible for the reduced fluorescence intensities of NVs in the m_s=±1 sublevels, since they emit fewer optical photons when returning to the ³A₂ state through the ISC-mediated pathway. The fractional fluorescence difference between NVs in the m_s=±1 sublevels and NVs in the m_s=0 sublevel is called the fluorescence contrast, which can be as large as ~20% for a single NV [38]. The fluorescence intensity from an optically pumped NV diamond chip therefore indicates the percentage of the NVs in the m_s=0 state or in the m_s=± 1 states. A transition of NVs from the m_s=0 to the m_s=±1 state, e.g. induced by a resonant MW field, drops the fluorescence as more NVs follow the ISC-mediated decay transition. This is the mechanism underlying optical readout for QDM imaging.

2.4 Microwave driving field

A MW driving field resonant with the m_s=0 to +1 or −1 transitions induces Rabi oscillations, transferring the NV population from one sublevel to the other and creating superpositions of m_s states. Either a continuous-wave (CW) or a pulsed MW field can be used. The length of the MW pulse determines its impact on the NV population: π pulses are of sufficient duration to transfer the NV population from m_s=0 to m_s=1 when the NVs are initialized in the m_s=0 state; π/2 pulses are of duration to create an equal superposition of m_s states. The utility of pulsed MW fields for QDM detection of different types of sample fields will be discussed below.

Applying resonant CW MWs simultaneously while optically pumping of the NVs to the m_s=0 sublevel results in MW-induced transfer of the NV population out of the m_s=0 sublevel, spoiling the optical spin polarization and decreasing the emitted fluorescence intensity. Measuring the NV fluorescence intensity as a function of the probing MW frequency is called optically detected magnetic resonance (ODMR) spectroscopy [39]. Simulations of ensemble NV ODMR spectra for NVs with ¹⁴N and ¹⁵N isotopes are shown in Figure 2. The known dependence of the ³A₂ sublevel energy on external fields allows conversion of these ODMR spectra into magnetic field, electric field, temperature, and crystal stress information.

2.5 Static magnetic bias field and Zeeman splitting

A static magnetic field B₀ causes a Zeeman interaction in the ³A₂ state, written as

Here, μ_B=9.27×10⁻²⁴ J/T is the Bohr magneton, g is the electronic g-factor tensor (which is nearly diagonal), g_e≈2.003 is equal to the NV center’s electronic g-factor [12], and γ=g_eμ_B/h is the NV gyromagnetic ratio. The Zeeman interaction lifts the degeneracy between the m_s=±1 sublevels, and for |B₀| along the NV axis, the m_s=±1 sublevel energies split linearly with |B₀| while the m_s=0 sublevel is unaffected. The nuclear Zeeman terms are considered negligible and have been excluded.

A sufficiently large bias magnetic field makes the Zeeman term dominant in the Hamiltonian. Otherwise, terms including stress and electric field would dominate with the Zeeman term acting as a perturbation, reducing magnetic field sensitivity and complicating the data analysis. For magnetic imaging, both the static bias fields, which are part of the QDM, and the sample magnetic fields contribute to the Zeeman interaction.

2.6 Sample fields

QDM experiments create a two-dimensional image of the magnetic fields from a sample containing a distribution of magnetic field sources. It is also possible to image a sample’s temperature distribution and electric fields.

The sample magnetic field is generated by field sources, such as current densities or magnetic dipoles, with either known or unknown distributions. Measurement of the sample magnetic field can be used for the inverse problem of estimating an unknown source distribution under certain conditions [40], [41], [42]. The form of the sample magnetic field in terms of its sources is

where J is the current density of the sample, R=r–r′ is the distance from a magnetic source at r′ to an observation point at r, R=|r–r′|, and R^=R/R. Equation (5) is the Biot-Savart law for static fields and applies to fields in the quasi-static regime for which the characteristic system size is small compared to the electromagnetic wavelength. A sample consisting of small magnetic particles will have a magnetic field composed of single dipole fields of the form

Here, m is the magnetic moment, n=r/r, and the delta function only contributes to the field at the site of the dipole r=0. Other typical sample fields, such as the narrowband magnetic field from the Larmor precession of protons, can also be derived [43]. Figure 4 shows examples of the magnetic fields for a current distribution and a distribution of magnetic dipoles.

Figure 4:

Simulated QDM measurement planes above magnetic samples.

Magnetic field distributions from (A) current distributions and (B) magnetic dipole distributions simulated in COMSOL. The NV layer in the QDM measures the sample magnetic field in the x-y plane at distance z above the sample. Two measurement planes at different values of z are shown for each simulation. A smaller stand-off distance between the measurement plane and the sample gives a magnetic field image with higher spatial resolution.

The time dependence of the sample magnetic field will determine the QDM measurement protocol. Static and quasi-static sample fields will contribute to the NV Hamiltonian by an additional term in Eq. (4):

where B_s is the magnetic field of the sample, which can take the forms given in Eqs. (5) and (6). The magnitude of the sample field along the NV axis is therefore determined by changes in separation in the ODMR resonance features that result from B_s in addition to the effect of the bias field. ODMR measurements, with and without the sample, then allow the determination of the unknown B_s field. Reasonable assumptions can be made to determine B_s without having to take multiple measurements [1]. A time-varying sample magnetic field with frequency components near the 2.87-GHz ZFS will in turn induce NV spin transitions if the B₀ bias field has been tuned to the appropriate Zeeman splitting. Sweeping the B₀ field will then locate the frequency of the sample fields, with the magnitude determined by the ODMR contrast depth and linewidth.

Electric field and temperature distributions from the sample will also change the NV spin states. The external sample electric field E_s=(E_sx, E_sy, E_sz) adds to the internal local electric fields [44] E_loc=(E_loc,x, E_loc,y, E_loc,z) in the diamond, e.g. induced by a high density of P1 (nitrogen) centers, such that E_tot=E_s+E_loc contributes to the Hamiltonian in Eq. (2) [16]:

Here, d_‖ and d_⊥ are the coupling constants related to the NV electric dipole moment, and d_‖=d_⊥ with d_‖ =3.5×10⁻³ Hz/(V/m) and d_⊥ =0.17 Hz/(V/m) [45], [46]. For the typical scale of sample electric fields, coupling to the NVs is small compared to sample magnetic fields of interest. Hence electric fields do not cause noticeable shifts in ODMR resonances for most QDM magnetic imaging experiments. External temperature variations, e.g. from the sample, couple to the NV by the temperature dependence D(T) of the ZFS [19]. Changes in temperature of the diamond due to the sample temperature field will therefore result in a common mode shift of the ODMR resonance, which is distinct from the effect of magnetic fields.

2.7 NV ground-state Hamiltonian

Detecting the resultant spatial distribution of changes in the ODMR spectra across an NV ensemble due to spatially varying sample fields is the principle underlying QDM high-resolution imaging. The ground-state Hamiltonian necessary to capture the relevant dynamics of single NVs for QDM imaging can be summarized by combining Eqs. (2), (3), (7), and (8):

Equation (9) summarizes the interaction between an NV center and temperature, magnetic field, and electric field. This equation demonstrates that NVs can in principle be used to image all of these quantities. For simplicity, Eq. (9) does not include the nuclear electric quadrupole interaction and the comparatively negligible terms of the crystal strain interaction. The hyperfine splitting terms from Eq. (2), also excluded for simplicity, are important and visibly evident in the hyperfine splitting of the ODMR resonances in Figure 14. Further simplifications can be made to Eq. (9) depending on the magnitude of the parameter of interest and the bias fields.

2.8 Behavior of NV ensembles

QDMs use ensembles of NVs to obtain simultaneous measurements over a wide-field of view. Ensembles yield stronger signal than single NVs because of the larger number of NVs contributing photons to overall fluorescence but introduce ensemble behavior that can worsen contrast compared to single NV performance. Intuition about single NVs also does not simply extend to the ensemble case: the NV spin-ensemble behavior can be substantially different from single-NV behavior. The complex spin bath environment of diamond contributes several mechanisms to NV spin ensemble dephasing and decoherence, which ultimately limit the magnetic field sensitivity of NV ensembles. These ensemble dynamics are characterized by the longitudinal (spin-lattice) relaxation time T₁, the transverse (decoherence) relaxation time T₂, and inhomogeneously broadened transverse (dephasing) relaxation time T2∗ [47]. In particular, understanding and minimizing ensemble NV dephasing is critical for high-performance broadband and static field magnetic imaging with QDMs, informing both diamond material design and quantum control techniques. This topic has been treated extensively in the literature [17], [48], and will be discussed later in this article, after introduction of NV measurement protocols.

3.1 DC magnetometry: static and low-frequency fields

Three established QDM sensing protocols exist for measuring static (DC) and slowly varying magnetic fields: CW ODMR, pulsed ODMR, and Ramsey magnetometry. These protocols have been used to sample time-varying magnetic fields up to 1 MHz in a single-pixel experiment [49].

3.1.1 CW ODMR

CW ODMR is a robust and simple method that can image the vector components of a magnetic field in the QDM modality. Because of its easy implementation, CW ODMR is the most common technique used for QDM applications. Continuous laser pumping, MW driving, and fluorescence readout are performed simultaneously, as shown in Figure 5. The laser is used to both pump the NVs into the m_s=0 spin state and probe the spin states of the population via NV fluorescence. The frequency of the MW drive is swept in time and synchronized with the readout. A decrease in fluorescence occurs when the MW frequency matches the NV resonance due to the spin-state dependence of NV photon emission described in Section 2.2.

Figure 6A shows an example where a change in B₀ shifts the line center of the resonance feature. For an NV ensemble, the resonance lineshape – often modeled as a Lorentzian or Gaussian – is parameterized by the center frequency, linewidth, and fluorescence contrast. The center frequencies of every NV resonance feature are fit to the appropriate Hamiltonian to extract the desired magnetic field, strain, temperature, and electric field. In a magnetic imaging experiment, this analysis yields B₀+B_s, from which the magnetic field of the sample can be determined [11], [49], [50].

Figure 6:

DC magnetometry protocols.

(A) Example CW ODMR lineshape before (blue) and after (red) change in magnetic field. (B) Example pulsed ODMR lineshape before (blue) and after (red) change in magnetic field. (C) Schematic Ramsey free induction decay (FID) to determine the dephasing time (T2∗) and the optimal sensing time (τ_sense). (D) Schematic Ramsey magnetometry curve. Free precession time is fixed to be the point of maximum slope of the FID curve closest to T2∗, indicated by a black circle. Accumulated phase from the sample field results in an oscillatory response of fluorescence with changing amplitude.

Measuring the entire resonance spectrum in CW ODMR limits the sensitivity and the temporal resolution of the measurements because of the significant fraction of experiment time spent interrogating with probe frequencies that yield no signal contrast. Sparse sampling of the resonance spectrum can improve the sensitivity of the measurement by minimizing the dead time. An extreme version of sparse sampling can be achieved using a lock-in modality where the probe frequency is modulated between the points of maximum slope of an ODMR resonance feature [51]. This technique has been extended to monitor multiple ODMR features simultaneously to extract the vector magnetic field by modulating at different frequencies [52]. Frequency-modulated ODMR has been performed with bandwidths up to 2 MHz, but was demonstrated on a small volume and required high laser and MW intensity beyond that typically employed with QDMs [48].

3.2 AC magnetometry: narrowband fields

A QDM can measure narrowband oscillating magnetic fields using AC magnetometry sequences, including Hahn echo and dynamical decoupling. These pulse sequence protocols act as frequency filters and allow the QDM to operate as a sensitive lock-in detector [55] of these AC fields. The frequency range of narrowband signals that are detectable with NV AC magnetometry is ~1 kHz to ~10 MHz, limited at the low end by NV decoherence and at the high end by the amplitude of fast MW pulses that can be realistically applied to an NV ensemble.

3.2.1 Hahn echo

The addition of a π pulse into the middle of a Ramsey sequence mitigates environmental perturbations that are slow compared to the free precession interval between pulses [55]. This pulse sequence is known as the Hahn Echo sequence [54], [56], and results in the refocusing of NV ensemble dephasing such that the limiting measurement timescale becomes the decoherence time T₂ rather than the dephasing time T2∗. The consequence is improved magnetic field sensitivity (discussed in Section 4.1), especially for lower frequency signals, because T₂ typically exceeds T2∗ by over an order of magnitude [57]. Figure 7A demonstrates a decoherence curve when using a Hahn echo pulse protocol. The spacing between MW pulses acts as a narrowband filter in frequency space. The width of this filter is given by the filter response function [58]. Hahn echo uses only one π pulse and therefore has a fairly broad filter, allowing for the sensing of a wide bandwidth of magnetic field frequencies.

Figure 7:

AC magnetometry protocols.

(A) Schematic T₂ decoherence curves for Hahn echo and CPMG-32 sequences. Improved decoupling from interactions with the spin bath environment results in an extended CPMG T₂ decoherence time compared to the Hahn echo T₂. (B) Schematic magnetometry curves. Longer T₂ for CPMG leads to increased magnetic field sensitivity, as indicated by the slope of the CPMG magnetometry curve. Hahn echo is much less sensitive, exhibiting an oscillation similar to CPMG over a much larger magnetic field amplitude range.

To optimally sense external oscillating fields, the spin evolution time is set to be ~T₂; however, the frequency of the sensed magnetic field can lead to operation with a non-optimal τ [55]. For a fixed spin evolution time, a change in magnetic field will lead to a difference in phase accumulation that maps onto the total fluorescence, Figure 7B.

3.3 Resonant coupling to external GHz fields

Applications that require measurement of GHz-scale oscillations can leverage interactions between the NV and magnetic signals near the NV resonance as a probe [62], [63], [64]. CW ODMR constitutes a simple protocol that can be used in this manner. Measurements of the contrast and linewidth enable the determination of the optical power and MW power that broaden the lines, Figure 8A, in addition to other mechanisms that contribute to the inhomogeneous dephasing of the ensembles [38]. However, this method is not very sensitive and difficult to quantify because of the various ways the contrast and linewidth can vary over a field of view [1]. Alternative methods to CW ODMR include Rabi driving and T₁ relaxometry.

Figure 8:

GHz magnetometry protocols.

(A) CW ODMR broadening: increased MW power will increase ODMR fluorescence contrast and linewidth (red). (B) Rabi oscillation: increased amplitude of the MW field will increase the Rabi frequency (red). (C) T₁ relaxometry: phonon-limited T₁ decay rate (blue) is increased (red) by high-frequency magnetic noise near the NV resonance frequency.

4.1 Magnetic field sensitivity

The minimum detectable field difference is defined as the change in magnetic field magnitude δB for which the resulting change in a given measurement of the field equals the standard deviation of a series of identical measurements. However, characterizing the minimum detectable field difference must consider the total measurement duration as well as the total number of NVs that contribute to the measurement for meaningful determination of sensor performance. The magnetic field sensitivity scales as the square root of the number of detected photons. The number of photons collected over unit time from a unit volume of NVs increases proportionally with time and volume. To account for measurement time, sensitivity is represented as η=δBtmeas with units of T Hz^−1/2, where t_meas is the total measurement time. To account for the number of NV spins required to reach a given sensitivity, a volume-normalized sensitivity is defined as ηvol=ηV with units T μm^3/2 Hz^−1/2, where V is the volume for a fixed density of NVs [67], [68].

CW ODMR magnetometry is the most widely used QDM measurement technique because of its simplicity. The sensitivity of a CW ODMR magnetometry sequence is characterized by the slope of the resonance line ∂I/∂ν₀, with fluorescence intensity I and frequency ν₀, and the rate R of photon detection from a cubic micrometer of NVs. The CW ODMR shot-noise-limited sensitivity is

where C is the contrast and Δν is the linewidth of the ODMR resonance. The resonance line shape is typically fit by a Lorentzian, giving the 4/(33) factor for the maximum slope. The relationship between the ODMR linewidth and the previously defined dephasing time T2∗ is approximated by T2∗=(πΔν)−1 [17], [48], [57], [69]. When performing ensemble measurements, many mechanisms can contribute to the linewidth, as demonstrated in Figure 9.

Figure 9:

Mechanisms for ensemble broadening.

Different colors represent resonance features from different NV sub-ensembles that contribute to the measured resonance linewidth from the entire NV ensemble. (A) Differences in linewidth center frequency due to variations in local environment, e.g. magnetic field and/or strain gradients, and (B) differences in contrast and linewidth due to dephasing from variations in T2∗ and/or power broadening. From Ref. [48].

The sensitivity of CW ODMR magnetometry is limited by laser and MW ODMR lineshape power broadening. Solving the Bloch equation for a simplified two-level model yields the contrast, linewidth, and volume-nomralized magnetic sensitivity, shown in Figure 10. The calculations are based on [38] for CW ODMR using parameters from Table 1, Figure 10 displays a broad range of laser and MW powers to indicate how these affect the sensitivity. The tradeoff between laser and MW power limits the achievable volume-normalized sensitivity of CW ODMR, precluding simultaneous optimal contrast and narrow linewidth. Applications that require higher temporal and spatial resolution must use techniques more sensitive than CW ODMR.

Figure 10:

Ensemble CW ODMR sensitivity analysis.

Simulations of (A, D) CW ODMR contrast, (B, E) linewidth, and (C, F) volume-normalized magnetic field sensitivity (η_vol) as a function of laser intensity and MW Rabi (which scales as the square root of the input MW power) with parameters from Table 1 for a diamond with 1 ppm of nitrogen (top row), and a diamond with 20 ppm nitrogen (bottom row). Laser intensity scale assumes saturation intensity of 0.9 MW/cm².

Ramsey magnetometry achieves the best DC magnetic field sensitivity of the QDM protocols because of its pulse scheme: the NV spins interrogate the sample fields during an interval without simultaneous interaction of the optical and MW driving fields. The shot-noise-limited sensitivity for DC magnetic fields using a Ramsey pulse is [48]

where N is the number of noninteracting NVs contributing to the measurement S=1/2 spins, and Δm_s is the generalization to >1 spin state difference used for measurement (e.g. Δm_s=2 for the NV m_s=−1 to m_s=1 transition when operating with a double-quantum coherence [17]), C is the resonance contrast, n is the average number of photons collected per NV per measurement, τ is the spin interrogation time, and t₁ and t_R are the optical spin-state initialization and readout times, respectively (t_meas=t_I+τ+t_R). The spin-projection-noise-limited sensitivity is given by the first two terms of Eq. (11). It is evident that longer interrogation time τ and larger number of spins N allow for better sensitivity to small magnetic fields. However, several factors cause Ramsey magnetometry to fall short of this limit: a decrease in sensitivity due to spin dephasing with characteristic time T2∗ is accounted for in the exponential term with parameter p depending on the origin of dephasing; imperfect readout contributes the first square root term; and the reduced fraction of total measurement time allocated for spin interrogation due to the overhead time from t₁ and t_R is accounted for in the last term. Optimal DC sensitivity is achieved for τ~T2∗ [48]. Figure 11 compares the sensitivity of Ramsey magnetometry as a function of the frequency of the field being measured for the two diamonds in Table 1.

Figure 11:

Frequency dependence of QDM volume-normalized sensitivity.

Achievable volume-normalized magnetic field sensitivity as a function of the sample field frequency for DC broadband and AC narrowband QDM protocols. Calculations use the parameters from Table 1. Ramsey is broadband and is sensitive to magnetic fields of differing frequencies without requiring changes in the pulse sequence. CPMG is narrowband and requires a change in the pulse sequence based on the field frequency being sensed in order to maintain optimal sensitivity. The gray region indicates high sample frequencies where experimental requirements on MW pulses and power become technically challenging. Dotted lines are for a single pulse, which achieves the same sensitivity as a Hahn echo sequence. Solid lines are for CPMG-k protocols limited up to 1024 pulses.

The sensitivity for measurement of AC magnetic fields using the Hahn echo protocol is

Hahn echo magnetometry builds on the Ramsey protocol as discussed in Section 3.2.1, resulting in similar physics underlying the AC magnetic field sensitivity to that of DC fields. The additional MW π pulse in the Hahn echo sequence refocuses the dephasing NV ensemble such that the sensing duration τ approaches the spin decoherence time T₂. Because T₂ is at least an order of magnitude longer than the spin dephasing time T2∗ [57], the sensing duration increases translating to an improvement in sensitivity. AC sensing protocols are thus limited by T₂, whereas DC sensing protocols are limited by T2∗; because T2≫T2∗, the AC protocols can generally achieve better sensitivity than DC protocols. However, the benefit of being T₂-limited can be degraded by coherent interactions between the NV spin ensemble and other spin impurities, which decrease the T₂ coherence time. The optimal spin interrogation time τ for Hahn echo magnetometry is τ~T₂; additionally, τ should match the period of the AC magnetic field T_AC. Consequently, maximum sensitivity is achieved for AC magnetic fields with T_AC~T₂ of the diamond.

CPMG pulse sequences improve the sensitivity by extending T₂ even further [48]:

where k is the number of pulses, τ=k/(2f₀) is the full spin evolution time, and f₀ is the frequency of the sample field. The optimal number of pulses for a given frequency is given by kopt=(12p(1−s)(2T2f0)p)1/(p(1−s)), with s~2/3 and p~3/2, and is set mostly by the spin bath dynamics [59]. The measurement time increases linearly with increased number of pulses, whereas the decoherence time T₂ increases sublinearly; the optimal number of pulses balances these effects [48]. Extensions of Eq. (13) exist to take into account multipulse dynamical decoupling protocols.

4.2 Temporal resolution and frequency bandwidth

QDM’s temporal resolution is defined as the time required between subsequent measurements of the sample field. The physical limitation determining the fastest temporal resolution is set by the time it takes for the NVs to react to a change in the sample field. The temporal resolution can never be faster than ~5 MHz (the maximum optical pumping rate), which is limited by the ¹E metastable state lifetime of 200 ns, discussed in Section 2.3. The same is true for pulsed measurements, because NVs are optically reinitialized to the m_s=0 state before each measurement. For a measurement with continuous laser illumination and MW field, the NV temporal resolution is set by the optical pumping rate and the MW Rabi frequency. There is also a practical limit to the temporal resolution, set by signal-to-noise ratio (SNR) tolerance: faster temporal resolution gives worse SNR per measurement.

The NV sensor frequency bandwidth is the range of sample frequencies that can be interrogated with the same experimental protocol. A DC magnetometry experiment has a frequency bandwidth spanning DC to the NV temporal resolution cutoff. A dynamical decoupling AC magnetometry experiment has an approximate frequency bandwidth which is roughly equal to 1/T_tot (the Fourier limit), where T_tot is the duration of the dynamical decoupling pulse sequence. An AC magnetometry measurement based on driving the spin population between m_s sublevels (Rabi) or spoiling of the initialized spin state (T₁ relaxometry) has a frequency bandwidth corresponding to the NV resonance linewidth, i.e. the frequency span over which the NVs are in resonance with the MW field, which is >1/(πT2∗).

AC magnetometry sequences that are based on pulse control of the NV spin state have a frequency bandwidth dictated by the filter function S(f) for the specific pulse sequence being used. The center frequency and bandwidth are defined by the number of pulses, k, and the spacing between the π pulses, τ [58]. The center frequency of the filter is given by f₀=1/2τ. For a sequence of k pulses, with total measurement time T=kτ, the width of the filter function is given by Δf=1/T=1/kτ. The filter function S(f) depends upon a protocol-specific response F(fT):

where an example response function for the CPMG protocol is [58]

Figure 12A demonstrates the need to change the number of pulses in order to operate at the sensitivity limit shown in the CPMG curves in Figure 11. Figure 12B gives the filter functions for the most sensitive points along the curves presented in Figure 11.

Figure 12:

CPMG protocol bandwidth.

(A) The optimal number of pulses for the CPMG protocol changes with the sample field frequency. (B) Example filter functions S(f) at the most sensitive sample frequencies for each of the CPMG curves in Figure 11. Dotted lines represent the response for a one-pulse CPMG. Solid lines are for the most sensitive center sample field frequencies for CPMG limited to 1024 pulses. The solid lines are ~1000 times narrower than the dotted lines due to having ~1000 times more pulses.

It is tempting to conflate temporal bandwidth and frequency bandwidth, but they in fact represent different characteristics. For example, an NV T₁ measurement can detect magnetic noise across a few MHz frequency bandwidth around a central frequency ranging from near zero to many GHz (depending on the applied B₀), but measurements may only be repeatable at <1 kHz (temporal resolution). Only in the case of DC magnetometry protocols do the temporal and frequency bandwidth correspond to the same sensor property.

4.3 Spatial resolution and field of view

QDM magnetic imaging seeks to resolve magnetic fields with high spatial variation over a wide-field of view and to successfully invert the magnetic field measurements to generate a map of closely separated magnetic sources. Both goals have fundamental and sensor-specific limitations. It is ideal to operate at the limit of magnetic field inversion and not to be limited by the sensor properties such as resolution and field of view.

The magnetic inversion problem does not generally have a unique solution. Only if the current distribution is limited to two dimensions (2D) can the inverse problem be solved uniquely from a planar measurement of the magnetic field. A magnetometer must sample the field at discrete points in a 2D plane with a sufficient sampling density to recover the continuous magnetic field created by the sample sources. The spatial resolution that can be obtained from this 2D map of the field is then limited by the offset distance between the measurement plane and the sources, as well as by the noise in the data [40], [70]. In general, the offset distance should be as small as, or smaller than, the characteristic length scale of the magnetic field sources, as shown in Figure 4, for reliable inversion of the magnetic image to the source distribution. In analogy to the Nyquist sampling theorem, the pixel size sets the maximum spatial (k-space) frequency. The size of the field of view sets the spatial frequency resolution (again by a Fourier transform argument). Both of these effects impact the ability to perform magnetic field inversions and map the underlying sources [40].

The in plane pixel size is made too small, the noise level could preclude detection of the magnetic fields of interest. This is similar to the negative impact to δB that can result from pushing the temporal resolution, discussed in the previous section. On the other hand, if the pixel size is too large, then small-length-scale signals of interest will be blurred out and the fidelity of the magnetic field amplitude will be degraded. Figure 13 illustrates an example of this tradeoff for magnetic fields simulated in Figure 4.

Figure 13:

QDM spatial resolution and SNR tradeoff.

(A) Magnetic field from the current distribution in Figure 4A for different planar binning sizes. No additional noise is applied. Scale bar is 50 μm. (B) Binning with fractional noise leading to SNR of 1 for a bin size of 1. (C) Magnetic field from magnetic dipole sources in Figure 4B. No additional noise is applied. Scale bar is 10 μm (B) for different planar binning sizes. (D) Binning with fractional noise leading to SNR of 1 for a bin size of 1.

The QDM’s spatial resolution is set by the following:

1. NV-sample standoff distance. As the standoff distance Δz increases, the 2D magnetic map is convolved with a Lorentzian of width Δz, reducing the ability to resolve closely separated magnetic sources [71]. Reducing the standoff distance improves the field strength and sometimes the spatial resolution.

2. NV layer thickness. A thick NV layer has a layer-sample separation Δz corresponding to somewhere between the NVs in the layer nearest to and farthest from the sample. An NV layer that is thin compared to the sample could have better sensitivity than with a thicker NV layer; the far-standoff NVs will measure a B_s comparable to that of the near-standoff NVs, and the photon shot noise improves with a thicker layer. Conversely, an NV layer that is thick compared to the sample will have far-standoff NVs that measure almost no field but add background fluorescence and can cause deleterious artifacts [72].

3. Optical diffraction limit, set by the numerical aperture (NA) of the microscope objective λ/(2NA) for a typical fluorescence wavelength of λ≈700 nm. This assumes that the camera pixel size is small compared to the diffraction-limited spot size in the image plane. The spherical aberration from the diamond chip or other optics can also degrade the resolution.

QDM magnetic field imaging is best used for applications that need high spatial resolution over a wide-field of view and can afford small NV-sample separation. The intuitive rule of thumb is to have NV layer thickness, standoff distance, and sample thickness of comparable sizes.

5.1 Diamond

Properties of the diamond chip that impact QDM performance, include NV layer thickness, NV concentration, isotope and impurity concentration, and diamond cut. These properties are controlled during the diamond fabrication process. Single-crystal diamond substrates used as the platform for QDM imaging are grown in one of two ways. One technique, high-pressure high-temperature (HPHT) growth, resembles natural diamond formation, requires an anvil press at ~1700 K and 5 GPa, and produces diamond samples with ~100 ppm nitrogen density. The second technique, chemical vapor deposition (CVD), grows diamond substrates layer by layer from a plasma, and yields diamond samples with low ppb nitrogen concentration.

Imaging a thin two-dimensional magnetic sample is optimal when the NV layer thickness is comparable to the magnetic sample thickness as discussed in Section 4.3.

The typical NV layer thickness for QDMs ranges from ~10 nm to ~100 μm. There are several methods available to make NV layers of varying thickness.

1. N⁺ or N2+ is implanted in a type IIa diamond with ppb impurity density to create a ~10–100 nm shallow layer. Annealing the diamond improves the NV yield and NV density [73].

2. A ppm-density nitrogen-rich layer is grown on top of a type IIa diamond substrate using CVD. After growth, electron irradiation of the diamonds introduces vacancies, and annealing improves the NV yield by converting substitutional nitrogen atoms (P1 centers) into NVs with a ~10% conversion rate [74]. The nitrogen-rich layer can be from several micrometers down to several nanometers in thickness [75].

3. Instead of irradiating in #2, naturally formed NVs are used and can be preferentially oriented along one of the crystallographic directions (instead of equal NV fractions along all four orientations). Removing three of the NV orientations can improve the magnetic field sensitivity by ~2×, but can come at the expense of reduced NV density and fluorescence [76].

4. Similar to #2, nitrogen is temporarily introduced during CVD diamond growth to create a nitrogen-rich layer of a few nanometers, followed by a nitrogen-free diamond cap layer. NV centers are then created by electron irradiation and annealing. This technique is called delta doping [77]. The cap layer adds to the standoff distance, so the surface layer version in #2 is often preferred, or the cap layer is etched away [78].

5. An HPHT diamond with uniform NV volume density can be cut into a ~35-μm-thick slice. Alternatively, an HPHT diamond can be implanted with helium ions to form a shallow NV layer [79], [80], [81].

The NV density in the NV layer is optimized to achieve a desired magentic field sensitivity. High NV density yields more fluorescence intensity and good photon shot noise. However, the greater density of P1 paramagnetic impurities – required for high NV yield – contributes to magnetic inhomogeneity, thereby broadening ODMR resonances and spoiling magnetic field sensitivity. Optimal sensitivity therefore requires balancing the ODMR linewidth and contrast with the NV density in Eq. (10). Conditions for a favorable ratio of the two NV charge states, NV⁻/ NV⁰, are also needed to ameliorate the NV⁰ contribution to background fluorescence, which spoils the NV⁻ contrast used for imaging [82].

The performance of diamonds with different C and N isotopes is an important consideration. The ¹⁵NV (spin-1/2 nucleus) is more favorable for QDM imaging because it gives greater ODMR contrast and requires a narrower range of MW probe frequencies than the more common ¹⁴NV (spin-1 nucleus). However, because ¹⁵N is the less abundant isotope, diamonds fabricated without special procedures for isotopic control will typically be dominated by ¹⁴N.

Magnetic inhomogeneity from ¹³C (spin-1/2) and paramagentic P1 defect centers limits the NV T2∗; thus, isotopically purified ¹²C (I=0) diamonds are ideal [17], [54]. For diamonds with a 1.1% natural abundance of ¹³C present, it is advantageous to increase the P1 density, resulting in larger NV density without contributing too much to the P1-limited T2∗ [81]. An NV layer fabricated in an isotopically enriched ¹²C layer can reduce the ODMR linewidth. However, this may be irrelevant for NVs shallower than ~10 nm because of magnetic inhomogeneity introduced by electrons on the diamond surface [83].

Synthetic diamond chips used in QDMs are available in several cuts. The most common are diamonds with the top face along the [100] plane and the sides along the [100] or [110] planes (Figure 14A). The NVs in these diamonds point roughly 35 ° out of the plane. Less common diamond cuts include [110] and [111] top faces. The former has two NV orientations in the plane, while the latter has one NV orientation pointing normal to the face. Other more exotic diamond cuts exist: for instance, Ref. [63] used a diamond with a [113] NV layer. The choice of diamond cut does not usually impact the QDM performance. However, different cuts of diamond have different availability and pricing because of the challenge of producing crystals that are not grown along diamond’s preferential growth axis. Surface termination effects can be of impact [84].

Figure 14:

Experimental ODMR spectra for different bias magnetic field magnitudes and orientations.

(A) Example of four possible NV orientations in the diamond lattice, and the crystallographic directions. (B) Example ensemble NV ODMR spectrum with |B₀|=0. The resonance is centered at ~2.87 GHz, but splits into two peaks around this resonance frequency because of the ¹⁵N hyperfine coupling. Strain and electric field also contribute to the ODMR lineshape and broadening, and can cause a variety of lineshapes at |B₀|=0 for different samples. (C) Ensemble NV ODMR spectrum with |B₀| pointing along one axis. The frequency separation between the outer resonance peaks is proportional to the applied field. The inner peaks are from the three other NV orientations overlapping with each other due to equal Zeeman interactions for each. The ¹⁵N hyperfine interaction again splits each resonance into a doublet. (D) Ensemble NV ODMR spectrum with |B₀| orientation such that each axis has a different projection of bias field. (E) Ensemble NV ODMR spectrum with |B₀| along the [001] direction, such that each NV orientation has the same Zeeman interaction. The peak separation is proportional to the |B₀| field projection along the NV axes.

The impact of diamond characteristics on specific QDM techniques is summarized as follows:

1. For CW ODMR imaging, the laser and MW linewidth broadening should match the diamond T2∗ (Eq. (10)).

2. For Ramsey imaging, the diamond T2∗ limits the phase accumulation time.

3. For dynamical decoupling imaging, the diamond T₂ limits the phase accumulation time (depending on the magnetic noise spectrum and pulse sequence).

4. For Rabi and T₁ imaging, the diamond T2∗ sets the spectral filter function. The intrinsic NV T₁ depends on the NV density and depth.

5.2 Laser

A QDM typically uses a 532-nm solid-state laser for optical pumping because of its availability and performance. The green pump laser intensity is weak, typically ~10–1000 W/cm², when illuminating a field of view of a few millimeters, which can be a limitation for pulsed readout techniques. The NV ³A₂→³E optical transition spans hundreds of nanometers because of the phonon sideband as discussed in Section 2.2, which allows for laser excitation wavelengths ranging from 637 to ~470 nm [85]. Past experiments have pumped NVs with 532-nm frequency-doubled Nd:YAG and Nd:YVO₄ lasers, 637-nm and 520-nm diode lasers and LEDs, 594-nm HeNe lasers, argon-ion laser lines (457, 476, 488, 496, and 514 nm), and supercontinuum lasers with an acousto-optic tunable filter [86], [87], [88]. There have been attempts to find the illumination wavelength with the most favorable cross-section and NV⁻/NV⁰ charge-state ratio [89]. Since the NV readout measures a fluorescence intensity, fluorescence intensity instability from the laser or the optics must be minimized for the QDM magnetic sensitivity to reach the photon shot noise limit.

Increasing the illumination intensity improves the NV fluorescence intensity, the photon shot noise, and sometimes the ODMR lineshape. The ³A₂→³E optical transition is dipole-allowed when illuminating with light polarized in the xy-plane of the NV coordinate system defined in Section 2.1 [34]. Thus, in a projection magnetic microscopy experiment (Figure 14C), a laser polarization is chosen that favors the optical absorption selection rules for the selected NV orientation. If all NV orientations are interrogated, a laser polarization is selected that addresses all NV orientations with comparable strength. Increasing the laser illumination power increases diamond and sample heating on approximately linear scaling, while the photon shot noise limit only increases as the square root of the laser power. Furthermore, as the fractional photon shot noise improves, the analog-to-digital conversion bit depth must also improve to avoid being limited by quantization noise.

The available laser intensity affects the various QDM techniques in the following ways:

1. For CW ODMR imaging, varying the laser intensity affects the ODMR linewidth (Figure 10).

2. For pulsed imaging experiments, ideally the laser intensity should be close to optical saturation. Weaker laser intensity, longer t_I, and longer t_R will worsen the experiment time resolution.

5.3 Microwave source

The simplest way to apply a MW field to the NVs is with a piece of wire connected to a coaxial cable. The QDM MW field is ideally uniform across the NV layer’s field of view, and there are a variety of alternative engineered MW antennas that aim to optimize the MW field homogeneity, efficiency, or bandwidth [90], [91], [92], [93], [94], [95], [96]. By the transition selection rules, the transitions between the ³A₂ sublevels require left-circularly or right-circularly polarized MW [97]. One QDM MW antenna option is a MW loop as shown in Figure 1; another option is a pair of crossed MW stripline resonators [1]. The striplines are excited in phase (or 90° out of phase) to produce a linearly (or circularly) polarized MW field as needed for a given sensing protocol.

Increasing the MW power improves the contrast in a CW ODMR measurement but also increases the linewidth between ³A₂ resonances. Figure 10 demonstrates that, for simulated CW ODMR measurements, optimizing QDM magnetic sensitivity implies tradeoffs of laser and MW power [38]. Choosing an optical pumping rate much greater than the MW transition rate results in weak contrast, since the laser quickly repumps any NV population fraction removed by resonant MWs. Increasing the MW field amplitude improves the fluorescence contrast but also increases the ODMR linewidth. The MW intensity noise also affects the QDM sensitivity by influencing the ODMR contrast and linewidth in a manner similar to fluorescence intensity noise.

Selecting an appropriate MW frequency sweep rate is critical in CW ODMR measurements, where the probe MW frequency is swept across the NV resonance. The NV reaction time depends on the NV optical pumping rate and MW transition rate, and a sufficient response time is needed for the NVs to re-equilibrate to the updated conditions after updating in the MW probe frequency. This also applies to experiments using lock-in detection to combat fluorescence intensity noise: the MW modulation rate must be slower than the NV reaction time, typically set by the optically pumping rate [49].

When deciding how to apply the microwave field, some of the options affect the various QDM modalities differently:

1. For CW ODMR imaging, increasing the MW power broadens the ODMR line but also improves the contrast.

2. For Ramsey, Hahn echo, and dynamical decoupling imaging, spatial MW inhomogeneity and pulse errors can reduce the NV contrast and worsen the sensitivity.

5.4 Static magnetic field

The QDM bias magnetic field B₀ can be provided by electromagnets (Helmholtz coil sets, solenoids, and C-frame/H-frame electromagnets) or permanent magnets [1], [43], [98] as shown in Figure 1. Electromagnets allow the selection of any arbitrary |B₀| up to a few tesla. However, they require a stable current supply, may need water cooling for the magnet, and can add to sample and system heating. Permanent magnets allow higher B₀ in a more compact instrument, though the applied B₀ can drift with temperature.

The choice of the bias field amplitude |B₀| depends on the samples being measured. Soft magnetic samples that might have their magnetization changed by an applied magnetic field require |B₀| to be minimized. This has the added benefit that small |B₀| typically implies a small |B₀| gradient across the field of view. A large |B₀| can be beneficial when imaging paramagnetic minerals, since the magnetization from paramagnetic particles scales with |B₀| until saturation [99]. For Rabi imaging or T₁ magnetometry, |B₀| is chosen such that the NV spin transition frequency matches the AC sample frequency being interrogated [63], [98]. Due to nitrogen nuclear polarization, |B₀| ~30–50 mT improves the NV fluorescence contrast [100], [101]. Finally, |B₀|=0 is an intuitive choice for NV thermometry or electrometry experiments (Figure 14B).

The direction of B₀ also factors into the specific QDM application [1]. Alignment of B₀ along the NV axis ([111] crystallographic direction), allow for interrogating the NVs along this direction, ignoring the other three NV directions, Figure 14C. This approach allows the optimization of the other measurement parameters, e.g. the optical polarization, to maximize the fluorescence and contrast from the selected NV orientation. Alternatively, the B₀ magnitude and direction are chosen such that each NV orientation has different resonance frequencies and nonoverlapping spectra, Figure 14D. This approach allows the reconstruction of vector magnetic field information from the eight NV resonance frequencies. B₀ can also be aligned with the crystallographic [100] or [110] directions, such that the resonance frequencies for different NV orientations are degenerate, leading to improved contrast. If B₀ is aligned along the [100] direction with a diamond cut along [100], the magnetic field projection direction is normal to the chip, though the Zeeman shift is 3 times weaker than for B₀ along the [111] direction (Figure 14E). Finally, there may be some experiments where the choice of B₀ is forced by the sample being tested. This could cause the NV ODMR lines to overlap, making it difficult to resolve the resonance frequencies and extract vector magnetic field information. This difficulty can be ameliorated using the ³A₂→³E optical polarization selection rules to distinguish the light contributions from each NV orientation [102].

The B₀ field should ideally be as uniform as possible. B₀ inhomogeneity can cause the following problems:

1. In pulsed NV experiments, B₀ inhomogeneity will cause spatially dependent pulse errors, which limit the NV contrast and sensitivity.

2. For all experiments, a B₀ gradient on top of a the desired B_s is something that should be subtracted out. A uniform B₀, allows for subtraction of a constant offset.

3. In an extreme case, B₀ inhomogeneity can contribute to NV linewidth broadening within each pixel.

5.5 Optics

QDMs employ various ways to illuminate the NV layer with the pump laser light, depending on other experimental constraints. Side illumination of the diamond chip [1] is a good method for QDMs using a low-magnification (long working distance) objective with a large field of view since the beam will have enough space to avoid clipping the objective and also illuminate a large area. Another approach is to illuminate through the objective by focusing the pump laser at the back aperture to get parallel rays out of the objective [103]. This method works better for QDMs operating with high-magnification microscopes. The laser polarization is easier to control, but focusing the laser at the objective back aperture can lead to burns. Techniques to avoid illuminating the sample as well as the NVs include illumination via total internal reflection in the diamond, shaping the pump laser beam into a light sheet using cylindrical lenses, or coating the NV surface with a reflective layer to reduce the optical intensity through the diamond chip [11], [51], [63].

Optimal photon collection efficiency requires the largest achievable NA for the microscope objective. In practice, the NA for a given magnification is limited, and high-NA objectives are often also high-magnification objectives with a short working distance (sometimes shorter than the diamond thickness). Imaging NV fluorescence through the diamond chip may cause optical aberrations that can spoil the image quality, though the authors are unaware of any QDM experiment that corrects for this. As with other optical microscopes, a QDM images a broadband NV fluorescence (~637–800 nm), so chromatic aberration in the microscope optics is also important to mitigate. Pulsed NV experiments commonly use an acousto-optic modulator (AOM) as an optical switch. For the AOM, the rise time, extinction ratio, and efficiency are the parameters to consider for a given application.

5.6 Camera

QDM camera selection for a targeted application requires consideration of the expected photon collection rate from the NV layer, camera read noise and dark current noise, well depth, global/rolling shutter capability, software and external triggering, frame rate, data transfer rate, pixel size, and quantum efficiency [104]. For experiments with a high photon count rate, the camera must handle enough photoelectrons per second without saturating. Here, the pixel well depth, number of pixels, quantum efficiency, and frame rate are the important quantities to consider, because they determine the maximum photon count rate for fluorescence detection.

The camera frame rate can limit the experimentally realizable temporal resolution. Increasing the camera frame rate is possible by using only a fraction of the sensor. However, the resulting product between the frame rate and the number of pixels usually decreases, indicating that use of the full camera sensor is better for maximizing the number of photoelectrons per second. Alternatively, if the photon count rate is low, parameters such as the read noise and dark current noise should be minimized, while the quantum efficiency is maximized. For pulsed experiments, a low camera frame rate can throttle the experiment repetition rate and sensitivity.

The camera sensor size determines the microscope magnification for a desired field of view. The microscope spatial resolution can be set by the camera pixel size (rather than the optical diffraction limit) if the camera pixels are too widely spaced for the microscope magnification. The diffraction-limited spatial resolution should be oversampled by at least 2× to avoid having the pixel size spoil the diffraction-limited spatial resolution. A given choice of microscope optics has a finite effective image area, and the camera image can have darkened corners (vignetting) if the camera sensor area is too large.

As previously mentioned, the optical readout time t_R must be balanced with the minimum camera exposure time and the maximum camera frame rate for pulsed QDM experiments. Specifically:

1. Sensitivity is lost for experiments with a measurement time t_meas faster than the camera frame rate because the camera is too slow to acquire a new frame at the rate it takes to do each experiment.

2. Experiments for which the minimum camera exposure time is longer than t_R require the readout laser to be off for the duration of the time difference.

5.7 Diamond mounting and configuration

There are two primary ways to prepare the diamond sensor chip and the sample in the QDM. The first method is to fix the diamond chip in the optical microscope setup and move the sample independently with kinematic stages. This way, the diamond chip position (and all other optics) are permanent, keeping the relative positions of the optics, diamond location, and orientation, MW field, and magnetic field constant for all measurements to improve reproducibility and enable faster setup time for new samples. The second method is to mount the diamond chip directly on the sample, and then move the diamond and the sample together within the microscope field of view. This integrated diamond/sample approach offers more certainty that the NV-sample separation is minimized. Generally, sample mounting and manipulation in a QDM is easier with an upright microscope setup rather than with an inverted microscope.

5.8 General design considerations

Table 2 summarizes equipment parameters that optimize QDM operation. While some of the above specifications are technique- or application-specific, this table describes general design choices that affect all QDM instruments.

6.1 Broadband imaging of 0–1 MHz magnetic fields

CW ODMR imaging experiments of static magnetic fields is among the most successful QDM imaging applications to date. The relatively low MW and laser power requirements and simplicity of the experimental control allow for imaging of static magnetic fields up to a 4×4 mm region (limited by the size of a diamond substrate.) Most experiments up to this point have chosen to focus on large quasi-static magnetic fields because of the relatively loose requirements for performance of the QDM. Figure 15 shows several examples.

Figure 15:

Examples of QDM DC magnetic imaging.

(A) Imaging the vector magnetic field from a wire on the diamond. (Reprinted with permission from Ref. [105]). (B) Example of magnetic field and reconstructed current from current flow in graphene [106]. (C) Image showing parabolic profile of hydrodynamic flow in graphene at the Dirac point [107]. (D) Magnetic field image of magnetite intrinsic to MTB bacteria. (Reprinted with permission from Ref. [11]). (E) Imaging static magnetic field profile associated with magnetic memory [108]. (F) Measurement of remnant magnetization from geological sample. (Reprinted with permission from Ref. [1]). (G) Imaging magnetic field from iron mineralization in chiton teeth [109]. (H) Visualization of trafficking of magnetite particles in biological tissue [110].

6.2 Narrowband imaging of ~1 kHz–20 MHz magnetic fields

Narrowband magnetic imaging in an intermediate frequency range is mostly applicable for imaging magnetic fields originating from current distributions and the magnetic field from precessing spins in nuclear magnetic resonance (NMR) applications (see Figure 16). Much of the early work in NV magnetic imaging pushed the state of the art in these regimes, but more development is needed to explore the full range of applications.

Figure 16:

Examples of low-frequency QDM AC magnetic imaging.

(A) Imaging magnetic field from current oscillating at 109.5 kHz [120]. (B) Imaging magnetic field from a wider field of view of current oscillating at 4.75 kHz [120]. (C) Imaging the presence of ¹⁹F on diamond surface through NMR signal. (Reprinted with permission from Ref. [121]). (D) High spatial resolution imaging of patterned ¹⁹F on diamond surface [122].

6.3 Narrowband imaging of 10 MHz–100 GHz magnetic fields

6.3.1 Microwave imaging

QDM imaging of the microwave field from wires, resonators, and structures is possible by measuring the Rabi frequency in a pulsed experiment [63], or by using the fluorescence contrast in a CW ODMR experiment [62] (see Figure 17A and B). An initial step is to compare the NV measurement to a predicted magnetic field map from a finite element method (FEM) calculation. One goal is to use NV microwave imaging as a tool to validate that the FEM or the fabrication is what is expected for more nontrivial devices such as atom chips.

Figure 17:

Examples of GHz-frequency QDM AC magnetic imaging.

(A) Imaging the presence of MW field through the contrast and linewidth of the ODMR. (Reprinted with permission from Ref. [62]). (B) High spatial and temporal resolution Rabi imaging. (Reprinted with permission from Ref. [63]). (C) T₁-weighted imaging of patterned Gd³⁺ and a Gd³⁺ labeled cell membrane. (Reprinted with permission from Ref. [64]). (D) Demonstration of T₁ imaging of patterned Cu²⁺ ions and sensitivity to the bias magnetic field [134].