Laser threshold magnetometry

The electronic ground state manifold of the NV⁻ centre is a spin one triplet, which at zero magnetic field has a spin-0 ground state, and nearly-degenerate spin ±1 states at 2.88 GHz, due to the crystal field splitting [39]. For our purposes we will treat it as a spin half system, assuming that optical pumping into the spin 0 ground state is perfect³ , and the radio-frequency (RF) field that induces spin flips is only resonant with one of the ±1 spin states. We further assume that all of the processes except for the singlet pathway and the RF drive field are spin conserving. Hence we can separate the problem into two manifolds, the spin 0 manifold and the spin 1 manifold. A schematic of the pertinent energy levels of the NV⁻ system is given in figure 1(c). The ³A₂ spin 0 and spin 1 states are denoted by $| 1\rangle$ and $| 4\rangle$ respectively and have an energy difference that changes linearly with external magnetic field. An external RF drive can induce coherent Rabi oscillations at frequency Ω and detuning Δ between these two states, i.e. Δ also changes linearly with external magnetic field. ${{\rm{\Gamma }}}_{14}$ is the ground-state decoherence. A green pump laser drives population to a phonon-added state just above the ³E spin 0 and spin 1 states, denoted by $| 2\rangle$ and $| 5\rangle$ respectively, followed by a rapid decay into $| 2\rangle ,| 5\rangle$. This effectively incoherent driving is spin-conserving: $| 1\rangle \to | 2\rangle$ and $| 4\rangle \to | 5\rangle$. The system decays from $| 2\rangle ,| 5\rangle$ via several decay paths back to states $| 1\rangle$ and $| 4\rangle$: direct decay creating a photon, indirect decay creating a photon of less energy and one or more phonons—the phonon sidebands, and decay via the singlet states ¹A₁ and ¹E, for simplicity represented by only one state $| 7\rangle$. Due to this relatively long-lived singlet pathway a spin polarisation mechanism occurs such that spin 1 population is more likely to be pumped to spin 0. Because the singlet pathway is longer lived than the direct emission, the spin 0 state is slightly brighter than the spin 1 state. This mechanism is used elsewhere for both spin polarisation and readout of an individual NV⁻ centre. We use the difference in fluorescence to shift the laser threshold based on external magnetic fields.

As the branching ratio to the three phonon sideband is the strongest [26, 40, 41], we denote the three phonon added ground state in the respective spin manifold by states $| 3\rangle ,| 6\rangle$ and consider lasing on the $| 2\rangle \leftrightarrow | 3\rangle$ and $| 5\rangle \leftrightarrow | 6\rangle$ transition while all other sidebands are included into the effective direct decays ${L}_{21},{L}_{54}$. The phononic states $| 3\rangle ,| 6\rangle$ are only virtual levels and relax very rapidly via phononic processes to the ground states, $| 1\rangle ,| 4\rangle$. Hence for our purposes we require population inversion between $| 3\rangle$ and $| 2\rangle$ or between $| 6\rangle$ and $| 5\rangle$ respectively. The L_ij represent incoherent decay rates within the NV⁻ centre and the G_ij the coherent (cavity induced) transitions, ${{\rm{\Lambda }}}_{12},{{\rm{\Lambda }}}_{45}$ represent effectively incoherent pumping rates from the green laser pump. Finally n is the number of intracavity photons per NV⁻ centre and κ is the photonic loss rate from the cavity, which determine the red laser output power P_out:

$$\begin{eqnarray}&&{P}_{\mathrm{out}}=2\pi {{nN}}_{\mathrm{at}}\kappa {\hslash }{\nu }_{23},\end{eqnarray} \tag{ 1 }$$

where N_at is the number of NV⁻ centres in the cavity. The laser output power and pumping power both scale linearly with κ in the relevant parameter regime (input via linear dependence on the operating point). This means laser pump power is another tunable parameter although in practice the output will be limited by the input power one can provide to the diamond crystal.

Optical charge state conversion can occur in NV⁻ centres [42–48], and the steady state NV⁻/NV⁰ ratio as a function of intensity has been observed in nanodiamond and single crystal samples [49, 50]. Because we are considering a large ensemble of NV⁻ centres with effectively instantaneous charge conversion events, the steady state NV⁻ population is the correct number of NV⁻ centres to use in the laser threshold calculations.

For simplicity and due to lack of separate experimental data on each spin manifold, we will assume the cavity-induced transitions ${G}_{23}={G}_{56}=G$ and all corresponding rates in the two spin manifolds to be the same ${L}_{23}={L}_{56},{L}_{31}={L}_{64}$ and ${L}_{21}={L}_{54}$. The only exceptions are ${L}_{74}=(462\;{ns})$ ${}^{-1}$, ${L}_{71}={L}_{74}/2$ and ${L}_{57}=(24.9\;{ns})$ ${}^{-1}$, based on [26], which define the non-spin-conserving singlet decay path. The cavity-induced transitions G describe the stimulated emission and absorption rates and are dependent on the cavity volume V_c, and the number of NV⁻ centres N_at

$$\begin{eqnarray}&&G=3{\nu }_{23}{L}_{23}{\lambda }^{3}{N}_{\mathrm{at}}/(4{\pi }^{2}{\rm{\Delta }}{\nu }_{23}{V}_{{\rm{c}}})\end{eqnarray} \tag{ 2 }$$

with the transition frequency ${\nu }_{23}=c/709$ nm (1.75 eV photon energy, see figure 11 in [26]), the corresponding wavelength $\lambda =709$ nm $/2.4$, where 2.4 is the refractive index of diamond, the peak width of the three-phonon sideband⁴ ${\rm{\Delta }}{\nu }_{23}\approx 24$ THz. Assuming a density of NV⁻ centres of 1 per (100 nm)${}^{3}$, which corresponds to 5.7 ppb, a laser medium volume of 1 mm³ and an external cavity volume of V_c = 2 mm³ we find a cavity-induced transition rate of ${G}_{23}={G}_{56}=G=308$ MHz. The spontaneous decay rate to the three-phonon sideband is ${L}_{23}={L}_{56}=18$ MHz and the direct decay rate ${L}_{21}={L}_{54}=68.2$ MHz [52, 53]. The phononic decay is much faster and set to ${L}_{31}={L}_{64}=1$ THz, since the exact value is unknown, but unimportant as it is fast compared to the other decay rates, and small compared with the transition frequencies. Since our scheme does not perform spin-echo of any kind the dephasing is set by the inhomogeneous coherence time ${T}_{2}^{*}$. For the assumed low density of 5.7 ppb NV⁻ this time is determined by the ¹³C concentration, which can be reduced from the natural abundance of ∼1% by isotopic purification [54]. We set the dephasing rate to ${{\rm{\Gamma }}}_{14}={(1\mu {\rm{s}})}^{-1}$ which is above already experimentally achieved values [26, 54]. The cavity loss rate is set to $\kappa =3$ MHz, which corresponds to a cavity $Q=2\pi {\nu }_{23}/\kappa =8.9\times {10}^{8}$, neglecting all other losses. These values are easily achievable with an external cavity. This cavity design is large compared to those used in semiconductor laser diodes, but compatible with the dimensions of readily available synthetic diamond crystals.

The NV⁻ centres are aligned along four directions within the single crystal diamond. Our calculations assume the magnetic field to be oriented along one of them, and take into account the background fluorescence produced by the other three orientations. We note that even better sensitivities than our estimates here might be achieved by preferentially aligned NV⁻ centres [55] and polarisation-selective excitation[56].

Figure 1(b) shows the laser output power as a function of the laser pumping rate ${{\rm{\Lambda }}}_{12}={{\rm{\Lambda }}}_{45}={\rm{\Lambda }}$ with resonant (${\rm{\Delta }}=0$) and off-resonant (${\rm{\Delta }}=100$ MHz) RF drive. At off-resonant driving, the singlet pathway L₇₁ guarantees population to only be in the spin 0 manifold. At resonance, the RF drive mixes the populations of $| 1\rangle$ and $| 4\rangle$ and works against the singlet pathway, which has a long life-time relative to the triplet emission rates. At resonant driving, the singlet pathway behaves like an additional loss channel of the laser medium, which increases the lasing threshold. In other words, on resonance the emission is reduced due to some population being in the relatively long-lived, non-radiative state $| 7\rangle$.

To operate as a high precision magnetometer we choose an operating point such that, on resonance (with population shared amongst the spin 0 and spin 1 manifold) the population is just below threshold, whereas off-resonance (when the system has been optically pumped into the spin 0 ground state) the system is above threshold. This operating range is around ${\rm{\Lambda }}=1.06$ MHz in figures 1(b) and 2(a).

Figure 2(a) shows the laser ouput power P_out in mW as a function of Δ and Λ. Once the pumping rate Λ is set to the desired operating point (purple line), nonzero detuning pushes the system over the lasing threshold. The RF drive sets the initial value for Δ, e.g. zero; any further changes to Δ are then caused by changes in the outer magnetic field, i.e. any non-zero external magnetic field turns the laser on, with an intensity indicative of the magnetic field strength. With no detuning bias, the external magnetic field is given by

$$\begin{eqnarray}&&B={\rm{\Delta }}\displaystyle \frac{{\hslash }}{{g}_{e}{\mu }_{{\rm{B}}}}\end{eqnarray} \tag{ 3 }$$

i.e. $B/{\rm{\Delta }}=5.68\;\mu {\rm{T}}$ MHz⁻¹, where g_e is the Landé g-factor and ${\mu }_{{\rm{B}}}$ the Bohr magneton. In this way the device is a highly sensitive magnetic field sensor, or magnetometer, with very high contrast.

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Figure 2(b) shows how P_out changes with Δ at the operating point ${\rm{\Lambda }}=1.06$ MHz (solid line). Selecting different values of the Rabi-frequency Ω and the pumping rate Λ provides either increased dynamical measurement range (dotted) or increased measurement precision (dashed) as it increases the laser output gradient for small B-fields. As both the RF drive and laser pump are externally controlled parameters, these adjustments do not require changes of fabrication parameters but simply represent different operating modes. The precision could furthermore be increased by scanning across the entire dip via a variation of the detuning to obtain one high-precision measurement of the magnetic field.

The device has a response time t_r to sudden changes of the magnetic field which is essentially set by the slowest process out of RF driving Ω, laser pumping rate Λ, cavity decay κ, and effective non-spin conserving transition rate. For the parameters of the purple solid line in figure 2, the laser pumping rate is the slowest process and sets ${t}_{r}\approx 1/{\rm{\Lambda }}=0.94\;\mu {\rm{s}}$ (examples in Supplementary material). This can be shortened by increasing $\kappa ,{\rm{\Lambda }},{\rm{\Omega }}$ such that the non-spin conserving transition rate sets ${t}_{r}\approx 1/{L}_{57}+1/{L}_{71}=0.5\;\mu {\rm{s}}$. Since this non-spin conserving transition rate is not precisely known experimentally, the device can also serve as a probe for this parameter.

The shot-noise-limited sensitivity of the device is defined as $\eta \equiv {B}_{\mathrm{min}}\sqrt{T}$ where B_min is the minimum field that can still be detected in the total measurement time T. Since longer measurements in the presence of shot-noise decrease the error as $1/\sqrt{T}$ this factor must be multiplied with B_min to achieve a T-independent value. The sensitivity is:

$$\begin{eqnarray}&&{\eta }_{{\rm{d}}c}=\displaystyle \frac{{\rm{d}}B}{{\rm{d}}n}\sqrt{\displaystyle \frac{n}{{N}_{\mathrm{at}}\kappa }}.\end{eqnarray} \tag{ 4 }$$

The sensitivities corresponding to the different tuning parameters of figure 2(b) are plotted with corresponding colours and linestyle in figure 3(a) as a function of the magnetic field strength being measured.

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Ultrahigh precision can be achieved by increasing our conservative estimates for the NV⁻ density. However, for high NV⁻ concentrations the ${T}_{2}^{*}$ time is reduced by the large number of N impurities and NV⁻ to NV⁻ interactions, which has negative effects on the sensitivity; diamond samples with high conversion efficiency from N to NV⁻ and low numbers of other impurities are therefore ideal in maximising both NV⁻ concentration and ${T}_{2}^{*}$ time. For the rest of the paper we use the combination of 16 ppm NV⁻ concentration and ${T}_{2}^{*}=0.181\;\mu {\rm{s}}$, which has been measured in [28]. These adaptations enable an optimal sensitivity of $\eta =1.86$ fT $/\sqrt{{\rm{Hz}}}$, i.e. a precision of 1.86 fT can be achieved in a 1 s-long measurement. This sensitivity is reached at the centre of the dip and behaves rather smoothly, see blue line in figure 3(a). This is about 6 orders of magnitude better than several previous NV⁻ magnetometry experiments [31, 57–59], 2–3 orders of magnitude better than the best recent ensemble NV⁻ experiments [32, 33] and reaches the sensitivity of cryogenically operated SQUIDS. LTMs could therefore become a room-temperature operated alternative to SQUIDs. However compared to single NV⁻ centre magnetometry we have sacrificed the nanometre spatial resolution since our device is about 1 mm³.

So far we have only considered the measurement of constant (dc) magnetic fields. If the sensor is placed in an oscillating (ac) magnetic field signal $B(t)={B}_{S}\mathrm{cos}(\omega t)$ then the cavity photon number n and the laser output will vary in phase with the signal. To achieve good response even for very small B-field amplitudes we measure in the presence of an effective dc field bias to offset the laser output and maximise the response ${\rm{d}}{P}_{\mathrm{out}}/$ dB to external ac fields. In the parameters of figure 3(a) this offset is approximately 164 μT for the blue line. The ac sensitivity ${\eta }_{\mathrm{ac}}$ for a laser signal which is caused by an oscillating cavity photon number $n(t)={n}_{S}\mathrm{cos}(\omega t)+{n}_{o}$ is calculated as:

$$\begin{eqnarray}&&{\eta }_{\mathrm{ac}}=\displaystyle \frac{{\rm{d}}{B}_{S}}{{\rm{d}}{n}_{S}}\sqrt{\displaystyle \frac{{n}_{o}I}{{N}_{\mathrm{at}}\kappa }},\end{eqnarray} \tag{ 5 }$$

where the factor I = 2.43 stems from the signal's contribution in each oscillation period. The sensitivity as a function of frequency is shown in figure 3(b). Sensitivities down to 3.97 fT/$\sqrt{{\rm{Hz}}}$ are achieved for low frequencies and small signal amplitudes (here: B_S = 1 nT). Sensitivities to frequencies above 0.1 MHz deteriorate with increasing frequency due to the finite response time ${t}_{r}\approx 0.5\;\mu {\rm{s}}$ of the lasing process, which we identified above.

We take as our reduced model a seven state system, with the intracavity photon field, shown in figure 1(a). We also assume that the only non-negligible coherence ${\rho }_{14}$ is at the ground state transition. The equations of motion for this simplified structure are:

$$\begin{eqnarray}{\dot{\rho }}_{11} & = & -2{\rm{\Omega }}\ \mathrm{Im}({\rho }_{14})-{{\rm{\Lambda }}}_{12}{\rho }_{11}+{L}_{21}{\rho }_{22}+{L}_{31}{\rho }_{33}+{L}_{71}{\rho }_{77},\\ {\dot{\rho }}_{14} & = & ({\rm{i}}{\rm{\Delta }}-{{\rm{\Gamma }}}_{14}-{{\rm{\Lambda }}}_{12}/2-{{\rm{\Lambda }}}_{45}/2){\rho }_{14}-{\rm{i}}{\rm{\Omega }}({\rho }_{44}-{\rho }_{11}),\\ {\dot{\rho }}_{22} & = & {{\rm{\Lambda }}}_{12}{\rho }_{11}-({L}_{21}+{L}_{23}){\rho }_{22}-{G}_{23}({\rho }_{22}-{\rho }_{33})n,\\ {\dot{\rho }}_{33} & = & {L}_{23}{\rho }_{22}-{L}_{31}{\rho }_{33}-{G}_{23}({\rho }_{33}-{\rho }_{22})n,\\ {\dot{\rho }}_{44} & = & 2{\rm{\Omega }}\ \mathrm{Im}({\rho }_{14})-{{\rm{\Lambda }}}_{45}{\rho }_{44}+{L}_{54}{\rho }_{55}+{L}_{64}{\rho }_{66}+{L}_{74}{\rho }_{77},\\ {\dot{\rho }}_{55} & = & {{\rm{\Lambda }}}_{45}{\rho }_{44}-({L}_{54}+{L}_{56}+{L}_{57}){\rho }_{55}-{G}_{56}({\rho }_{55}-{\rho }_{66})n,\\ {\dot{\rho }}_{66} & = & {L}_{56}{\rho }_{55}-{L}_{64}{\rho }_{66}-{G}_{56}({\rho }_{66}-{\rho }_{55})n,\\ {\dot{\rho }}_{77} & = & {L}_{57}{\rho }_{55}-({L}_{71}+{L}_{74}){\rho }_{77},\\ \dot{n} & = & {G}_{23}({\rho }_{22}-{\rho }_{33})n+{G}_{56}({\rho }_{55}-{\rho }_{66})n-\kappa n,\end{eqnarray} \tag{ 6 }$$

where the ${{\rm{\Lambda }}}_{{ij}}$ represent effectively incoherent pumping rates from the green laser pump, which model the coherent excitation to a phonon-added state just above $| 2\rangle$, followed by a rapid decay into $| 2\rangle$. The L_ij represent incoherent decay rates within the NV⁻ centre and the G_ij the coherent (cavity induced) transitions. The strength of the RF drive is given by Ω—the Rabi frequency, which has detuning Δ, whilst ${{\rm{\Gamma }}}_{14}$ is the ground-state decoherence. The ρ represent the density matrix elements for the NV⁻ centres (fraction of the population in each state for the on-diagonal elements). Finally n is the number of intracavity photons per NV⁻ centre, κ is the photonic loss rate from the cavity and conservation of population implies ${\displaystyle \sum }_{i}{\rho }_{{ii}}=1$.

The steady state solution of the equations of motion gives insight into the lasing threshold and laser output of the system and their dependence on the tunable parameters, such as laser pumping rate ${{\rm{\Lambda }}}_{12},{{\rm{\Lambda }}}_{56}$, Rabi frequency Ω and detuning Δ. We obtain the steady state by diagonalisation of the superoperator for the density matrix and subsequent solution of equation (6), see supplementary material. We have a full analytical solution and inserted the realistic parameters given in the text.

In all calculations we ignored the possibility of a smaller transition rate in the other spin direction, from $| 2\rangle$ to $| 7\rangle$. Such a non-radiative rate has been found to be negligible, relative to the radiative decay [26], and 'radiative non spin-conserving transitions ... are less than 1% of the spin-conserving radiative decay rate' [26]. An inclusion of such a rate in the calculations, even at a higher non-negligible value of ${L}_{27}=0.1{L}_{57}$, yielded only minor changes in the results, particularly the changes in the sensitivity values stay below 10%, see figure 4.

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Similar to other lasing schemes, our LTM requires a stable laser pump rate Λ since fluctuations would lead to corresponding changes in the laser output, which look like signal changes. A stable source and/or laser amplitude stabilisers can help to minimise fluctuations. Furthermore the remaining input fluctuations can be monitored, by insertion of a beam splitter into the laser pump beam, and removed from the output signal.