Cavity-enhanced solid-state nuclear spin gyroscope

Abstract

Solid-state quantum sensors based on ensembles of nitrogen-vacancy (NV) centers in diamond have emerged as powerful tools for precise sensing applications. Nuclear spin sensors are well-suited for applications benefiting from long coherence times, such as inertial sensing, but suffer from control complexity and limited optical readout efficiency. Cooperative cavity quantum electrodynamic (cQED) coupling enables efficient nuclear spin readout. Unlike cQED methods used to enhance electron spin readout, here we employ two-field interference in the NV hyperfine subspace to directly probe the nuclear spin transitions. The nNV-cQED system can be used as an inertial sensor, indicating a rotation sensitivity improved by three orders of magnitude compared to other solid-state spin demonstrations. Furthermore, the NV electron spin can be simultaneously used as a comagnetometer, with the NVs' four crystallographic axes used for vector resolution in a single nNV-cQED system.

Description

Attorney Docket No. MIT-26298WO01CAVITY-ENHANCED SOLID-STATE NUCLEAR SPIN GYROSCOPECROSS-REFERENCE TO RELATED APPLICATION(S)

[0001] This application claims the priority benefit, under 35 U.S.C. 119(e), of U.S. Application No. 63 / 728,132, filed December 4, 2024, which is incorporated herein by reference in its entirety for all purposes.BACKGROUND

[0002] Nitrogen-vacancy (NV) centers in diamond have emerged as a promising platform for a variety of sensing modalities due to favorable attributes including room-temperature spin polarization and readout, atomic-scale size, and long coherence times. NV sensors generally use magnetic resonance spectroscopy of the electron ground-state spin triplet transition frequencies.SUMMARY

[0003] An inventive gyroscope uses the nuclear spin degree of freedom of an ensemble of nitrogen-vacancy (NV) centers in a diamond host. Although the nuclear spin degree of freedom is not directly accessible for optical polarization and readout and has potentially disadvantageous properties such as reduced resonant frequency and gyromagnetic ratio, it is less sensitive to noisy magnetic fields. This reduced sensitivity to noisy magnetic fields gives the nuclear spin degree of freedom a longer coherence time, which is advantageous for quantum memory and inertial sensing applications.

[0004] An inventive nuclear-spin gyroscope may include an NV ensemble in a diamond host, at least one microwave signal generator, at least one antenna, and a microwave detector. The NV ensemble has electron-nuclear spin states {|1), |2), |e)} that form a A system with a |1) <->|e) spin transition and a |2) <-^|e) spin transition. In operation, the microwave signal generator generates (i) a driving microwave field tuned to the |2) <-^|e) spin transition and (ii) a probe microwave field tuned to the |1)<-»|e) spin transition. The antenna, which is operably coupled to the microwave signal generator and in electromagnetic communication with the NV ensemble, applies the driving and probe microwave fields to the NV ensemble and senses the probe microwave field returned (e.g., reflected or transmitted, depending on the geometry) by the NV ensemble. The microwave detector, which is operably coupled the antenna, senses aAttorney Docket No. MIT-26298WO01change in the returned probe microwave field in response to a shift in the electron-nuclear spin states {| 1), 12)} caused by rotation of the NV ensemble.

[0005] The microwave signal generator can generate the driving microwave field at an amplitude sufficient to cause electromagnetically induced transparency (EIT) of the NV ensemble or to cause masing without inversion (MWI) of the NV ensemble.

[0006] In some cases, the probe microwave field is a first probe microwave field and the microwave signal generator generates a second probe microwave field for measuring a shift in an electron spin resonance of the NV ensemble caused by a change in magnetic field experienced by the NV ensemble.

[0007] The microwave detector can detect a change in amplitude, caused by the rotation of the NV ensemble, of the returned probe microwave field. The microwave detector can also detect common mode and difference-frequency shifts in the driving and returned probe microwave fields. The microwave detector may include a mixer and data acquisition hardware. The mixer, which is operably coupled to the microwave signal generator, mixes the returned probe microwave field with a phase-shifted copy of the probe microwave field. And the data acquisition hardware, which is operably coupled to the mixer, measures / samples an output of the mixer.

[0008] The gyroscope may also include a microwave resonator that is electromagnetically coupled to the|1) spin transition in order to enhance the change in the returned probe microwave field. The gyroscope may also include a light source, such as a laser, that is in optical communication with the nitrogen-vacancy ensemble and that polarizes the nitrogenvacancy ensemble into the electron-nuclear spin state |1). And the gyroscope can also include a magnetic field source, in electromagnetic communication with the NV ensemble, to apply a bias magnetic field to the NV ensemble. If desired, the magnetic field source may be oriented to apply the bias magnetic field to the nitrogen-vacancy ensemble in a direction that breaks a degeneracy among the electron-nuclear spin states {|1), |2), |e)} of NVs in the NV ensemble aligned with different crystallographic axes of the diamond host. In these cases, the driving microwave field may be a first driving microwave field tuned to a |2)spin transition of the nitrogen vacancies aligned with a first crystallographic axis of the diamond host and the microwave signal generator can also generate second, third, and fourth driving microwave fields tuned to |2)spin transitions of the nitrogen vacancies aligned with the diamond host’s second, third, and fourth crystallographic axes, respectively.Attorney Docket No. MIT-26298WO01

[0009] An NV ensemble in a diamond host can be used to measure rotation as follows. A driving microwave field drives a |2) <->|e) spin transition of a A system formed by electron-nuclear spin states {| 1), 12), |e)} of NV ensemble. At the same time, a probe microwave field probes a |1) <->|e) spin transition of the A system. A microwave detector sensor senses a change, caused by rotation of the nitrogen-vacancy ensemble, in the probe microwave field returned (e.g., reflected or transmitted) by the NV ensemble.

[0010] The driving microwave field can drive the |2) <-^|e) spin transition is at a rate sufficient to cause electromagnetically induced transparency (EIT) of the NV ensemble or to cause masing without inversion (MWI) of the NV ensemble.

[0011] Another probe microwave field can measure a shift in an electron spin resonance of the NV ensemble caused by a change in magnetic field experienced by the NV ensemble.

[0012] Sensing the change in the returned probe microwave field may include sensing a change in amplitude of the returned probe microwave field. It may also include detecting common mode and difference-frequency shifts in the driving and probe microwave fields returned by the NV ensemble.

[0013] In some cases, a bias magnetic field is applied to the NV ensemble in a direction that breaks a degeneracy among the electron-nuclear spin states {| 1), |2), |e)} of NVs in the NV ensemble aligned with different crystallographic axes of the diamond host. In these case, driving the |2) <-^|e) spin transition comprises driving the |2) <-^|e) spin transitions aligned with the different crystallographic axes with different driving microwave fields.

[0014] All combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are part of the inventive subject matter disclosed herein. The terminology used herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein.BRIEF DESCRIPTIONS OF THE DRAWINGS

[0015] The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in theAttorney Docket No. MIT-26298WO01drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and / or structurally similar elements).

[0016] FIG. 1A shows a cavity-enhanced, solid-state, nuclear spin gyroscope, also called a nuclear spin readout nitrogen vacancy-cavity quantum electrodynamic (nNV-cQED) gyroscope.

[0017] FIG. IB shows a nitrogen vacancy (NV) ensemble in a diamond (top) and diamond with embedded NVs that rotates with rate R = {Rx, Ry, Rz} (bottom).

[0018] FIG. 1C shows the NV energy level structure in the nuclear spin gyroscope of FIG. 1 A. The transition 11) <->|e) is coupled to a cavity mode of the microwave resonator for the cavity-enhanced readout. A driving field fl2is applied between the spin-exchanging transition |2) |e).

[0019] FIG. ID shows a more detailed view of the microwave resonator and NV spin ensemble in the nuclear spin gyroscope of FIG. 1A. A continuous-wave green laser beam continuously polarizes the NV spins to the |ms= 0) state, and a loop antenna measures the reflection signal from the microwave resonator.

[0020] FIG. IE is a plot of |<z0|2as a function of detuning 4 / K within strong coupling regime. An electromagnetically induced transparency (EIT) feature appears around the resonant frequency.

[0021] FIG. IF shows plots of Im(<7le) as a function of fl2(top), with negative (positive) Im(<7le) in the EIT (masing without inversion (MWI)) regime, and of the time dynamics of Im(cr) (bottom).

[0022] FIG. 1G is a plot of a0as a function of P and fl2. The solid line indicates the perfect EIT condition. The boundary of the oscillation regime is marked as a white line.

[0023] FIG. 2A shows the level structure for rotation sensing in the nNV-cQED gyroscope of FIG. 1A. The sensing signal emitted by the nNV-cQED gyroscope can be divided into common-mode signal Acand difference-mode signal AD.

[0024] FIG. 2B is a plot of the signal per unit frequency shift S as a function of Acand ADfor the nNV-cQED gyroscope of FIG. 1A.Attorney Docket No. MIT-26298WO01

[0025] FIG. 2C is a plot of inertial angle random walk (ARW) as a function of probe power P and driving field strength fl2for the nNV-cQED gyroscope of FIG. 1A. The boundary of the oscillatory regime is marked as a solid white line. The nNV-cQED gyroscope can achieve an ARW of 1.5 mdeg / s / VHz.

[0026] FIG. 2D is a plot of the sensitivity as a function of cooperativity for the nNV-cQED gyroscope of FIG. 1A. with the photon- shot-noise limit and SQL plotted for comparison. The best inverse readout fidelity is around an~ 37 at C ~ 20.

[0027] FIG. 3A is a plot of reflection |r|2as a function of A and As(top) and the sensitivity along A = 0 for P2= —40 dBm (bottom). The numerals indicate the frequencies of I: Comagnetometer probe field. II: EIT probe field. Ill: EIT driving field.

[0028] FIG. 3B is a plot of r / in the presence of an additional field with Rabi rate flRand detuning AT. The dots indicate field combinations for co-magnetometry and vector operation.

[0029] FIG. 3C is a plot of the sensing signal S as a function of Ac. S drops significantly beyond the cavity linewidth.

[0030] FIG. 3D is a plot of the sensing signal S for subensembles Ei as a function of Ac.

[0031] FIG. 4 shows an nNV-cQED vector gyroscope configured for comagnetometry.DETAILED DESCRIPTION

[0032] Nuclear spin gyroscopes offer stable long-term performance with competitive shortterm sensitivity. Devices based on atomic vapors have achieved inertial angle random walk (ARW) comparable with the optical sensors at the mdeg / Vh level, but corresponding performance has not been achieved using nuclei in the solid state. In a nitrogen vacancy (NV)-based solid-state nuclear spin gyroscope, a standard quantum limit (SQL) is described as r / = / NT, where T2* is the nuclear spin dephasing time, and N is the number of spins (NVs). This leads to a quantum-limited ARW of 0.1 mdeg / s / VHz for a spin ensemble of N = 1014and T2= 2 ms, which outperforms similarly sized (millimeter-scale) solid-state systems, such as microelectromechanical systems and fiber optic gyroscopes. However, current state-of-the-art NV-based nuclear spin gyroscopes achieve an ARW of J]exp= 4.7 deg / s / VHz. This ARW remains significantly higher than the theoretical SQL and can be quantified as an inverseAttorney Docket No. MIT-26298WO01readout fidelity an= Texp / ri ~ 104that captures all measurement imperfections. This leaves room for dramatic improvement towards the SQL.

[0033] NV-cavity quantum electrodynamic (cQED) coupling is a promising avenue that has been applied to enhance readout fidelity for electron spin ensembles. However, the manipulation of nuclear spin has not been demonstrated with a cQED scheme, as nuclei are not directly spin-polarizable using optics and have a low gyromagnetic ratio and a resonance frequency that substantially reduces achievable spin-cavity coupling.

[0034] Our cavity-enhanced, solid-state, nuclear spin gyroscope extends our electron spinbased cQED scheme to realize a nuclear spin readout (nNV-cQED) using two-field interference. Like electromagnetically induced transparency (EIT) gyroscopes in atomic vapor systems, nuclear spin readout using two-field interference exhibits a linewidth limited by the nuclear spin coherence, providing an exceptional sensitivity to rotation. The electron spin coherence of the cavity-coupled transition becomes positive, leading to a cavity gain and enhanced sensitivity in the masing without inversion (MWI) regime. Optimizing over performance and additional spin refrigeration effects yields an ARW of 1.5 mdeg / s / VHz with an inverse readout fidelity of an~ 37, surpassing the sensitivity of state-of-the-art NV gyroscopes by three orders of magnitude.

[0035] A 4-EIT scheme enabling vector inertial sensing and electron spin comagnetometry further enhances functionality of nuclear spin gyroscopes. The vector resolution is realized via four NV axes, unlike in atomic gyroscopes, where external magnetic bias primarily determines nuclei orientation. An electron spin comagnetometer is implemented by measuring common mode and difference-frequency shifts of driving fields, improving bias stability by compensating for noise due to electron spin and temperature. Each transition (rotation and magnetic field) in the 4-EIT scheme can be measured independently with minimal crosstalk or performance degradation. The nNV-cQED system enhances inertial sensitivity while providing a pathway to broader applications exploring quantum phenomena in hybrid solid-state systems.

[0036] Our nNV-cQED gyroscope uses nuclear spin readout performed using a strongly coupled microwave cavity, with readout fidelity of an= 37 outperforming current ensemble optical approaches. The extension of nNV-cQED to multiple ensembles enables comagnetometry and vector sensing, which is not currently achieved in atomic inertial sensors or NV electron-spin cQED experiments. A nNV-cQED system operating with high cooperativity enters the counterintuitive MWI regime, potentially into an oscillatory phase.Attorney Docket No. MIT-26298WO01A Nuclear Spin Gyroscope

[0037] FIGS. 1 A and 1C show a cavity-enhanced, solid-state, nuclear spin gyroscope 100, also called an nNV-cQED gyroscope, that uses shifts in nuclear spin resonances of an NV ensemble 110. As explained in greater detail below, the NV ensemble 110 has several electron-nuclear spin states, labeled as {| 1), 12), |e)}, that together form a A system, with the spin transitions 11) <-> |e) and |2) <-> |e) forming the legs of the A system. Rotating the NV ensemble 110 shifts the energy levels of the 11) and |e) electron-nuclear spin states with respect to the energy level of the 12) electron-nuclear spin state. At the same time, changing the magnetic field applied to the NV ensemble 110 shifts the energy level of the |e) electron-nuclear spin state with respect to the energy levels of the |1) and |2) electron-nuclear spin states. Shifts caused by rotation versus magnetic field changes can be disambiguated by measuring changes in the frequencies of the |1) <-> |e) and |2) <-> |e) spin transitions: a magnetic field change produces a common shift in the frequencies of both spin transitions (a common shift), whereas a rotation changes only the frequency of the |1) <-> |e) spin transition (a differential shift).

[0038] The NV ensemble 110 is electromagnetically coupled to a microwave cavity 120, also called a microwave resonator or dielectric resonator, that enhances absorption of the microwave fields that probe the shifts in the spin transition frequencies. The NV ensemble 110 is formed in a diamond 112 held within the microwave cavity 120 by polytetrafluoroethylene (PTFE) mounts 124 and a crystalline silicon carbide substrate 126. Microwave shielding 122 around the microwave cavity 120 protects the NV ensemble 110 from stray microwave radiation.

[0039] Two microwave signal generators 130 and 140 generate the microwave signals that drive and probe the electron-nuclear spin states of the NV ensemble 110. (Alternatively, these microwave signals can be generated with a single microwave signal generator.) Signal generator 130 generates a driving microwave field that is resonant with the |2) <-> |e) spin transition and transmitted to the NV ensemble 110 by a first loop antenna 132 coupled to signal generator 130 and disposed inside the microwave shield 122. Signal generator 140 generates a probe microwave field that is resonant with the 11) <-> |e) spin transition and transmitted to the NV ensemble 110 by a second loop antenna 142 disposed inside the microwave shield 122 and coupled to signal generator 140 via one output of a 3 dB splitter 144 and a circulator 146. The second loop antenna 142 also receives a sensing signal from the NV ensemble 110. In this case, the sensing signal includes both the portion of the probe microwave field that is reflected byAttorney Docket No. MIT-26298WO01the NV ensemble 110 and the portion of the driving microwave field that is transmitted by the NV ensemble 110. (Alternatively, the first loop antenna 132 could be used to receive a sensing signal from the NV ensemble 110 comprising the portion of the probe microwave field that is transmitted by the NV ensemble 110 and the portion of the driving microwave field that is reflected by the NV ensemble 110.) The circulator 146 couples this sensing signal from the second loop antenna 142 to a low-noise amplifier (LNA) 148, which amplifies the sensing signal and couples to the RF port of a mixer 150.

[0040] The other output of the 3 dB splitter 144 is coupled to a phase shifter 156, which in turn is coupled to the local oscillator (LO) port of the mixer 150 via another LNA 158. The 3 dB splitter 144 directs a copy of the probe microwave field to the phase shifter 156, which delays the copy of the probe microwave field. The LNA 158 amplifies the phase-delayed copy of the probe microwave field, which the mixer 150 mixes with the sensing signal to produce a baseband sensing signal at the intermediate frequency (IF) output of the mixer 150-a homodyne measurement of the reflected probe microwave field.

[0041] A data acquisition (DAQ) board 152, which may include an analog-to-digital converter (ADC) and digital processor, measures and records the baseband sensing signal, which may include components representing both the reflected probe microwave field (at DC) and the transmitted driving microwave field (offset from DC by an amount equal to the frequency difference between the probe and driving microwave fields). The DAQ board 150 uses these frequency components to resolve the common mode and / or differential mode frequency shifts caused by rotation and / or changes in ambient magnetic field. For example, rotation causes a change (e.g., a reduction) in complex amplitude of the sensing signal that can be detected by the DAQ board 150.

[0042] A dichroic beam splitter 162 reflects a laser beam 161 from a laser 160 through a hole in the microwave shielding 122 and onto the NV ensemble 110, placing the NV ensemble 110 into the |1) electron-nuclear spin state. A photodetector 164 detects fluorescence 163 emitted by the NV ensemble 110 and transmitted by the dichroic beam splitter 162. This photodetector 164 can be used to make an optically detected magnetic resonance (ODMR) measurement for alignment of the bias magnetic field to the

[0100] crystalline direction.Two-Field Interference

[0043] The nuclear spin gyroscope 100 in FIG. 1 A relies on two-field interference in a hybrid nNV-cQED system, where a probe microwave field’s state is controlled by a driving field viaAttorney Docket No. MIT-26298WO01coherent NV spin interaction. Two-field interference phenomena, such as EIT, have been demonstrated for NVs and atomic systems. However, EIT in highly cooperative cavity-coupled ensembles is rarely reported, and NV ensembles are further complicated by optical polarization and inhomogeneous broadening.

[0044] FIG. IB shows one NV of the NV ensemble 110 in the bulk diamond host 112. The NV ground state comprises an S = 1 electronic triplet state coupled via hyperfine interaction to the 1 = 114N nuclear states. Taking the { 111 } crystallographic direction of the diamond host 112 as the z-axis, the NV Hamiltonian reads:= DS* + YeS • B + QI* - • B + S • • I, (1)where D = 2.87 GHz and Q = —4.945 MHz are the electron and nuclear spin zero-field splittings, respectively, ye= 2TT x 2.8 MHz / G and yn= 2π x 0.308 kHz / G are the electron and nuclear spin gyromagnetic ratios, respectively, and c / Z is the diagonal hyperfine interaction tensor. The NV ground-state level structure based on Eq. (1) is shown in FIG. 1C. A magnetic field B generated by a bias magnet 170 (FIG. 1C) and oriented along the

[0100] crystalline direction of the diamond host 112 mixes the electron and nuclear spin states, enabling forbidden spin-exchange transitions. The nuclear spin gyroscope 100 uses a A system formed by electron-nuclear spin states {|0, — 1), |0,0), |1, — 1)}, which are labeled in FIG. 1C as {|1), |2), |e)}, respectively.

[0045] FIG. 1C illustrates the interaction of an ensemble of N NVs with a quantized cavity mode field a coupled to the transition |1) <->|e), driven by an external probe field J. The nNV-cQED Hamiltonian reads:N N0~C = AiJa + ](a^ + d) + A^cigp + A2 ■)cr2p + 9s (™ei +a+fflT);=i 7 = 1In the following analysis, we assume a homogeneous NV ensemble and therefore drop the spin index j. We validate this approach by comparing it with numerical simulations for an inhomogeneous ensemble. Here A = mc— mdis the cavity detuning between the detection frequency mdand the cavity frequency mc, A = ms— mdis the detuning between the spin transition |1) <->|e) frequency a>sand the detection frequency, and A2= m2+ |Q| — mdis the detuning between the driving field m2and the probe field md. The rate gsis the single spinAttorney Docket No. MIT-26298WO01coupling strength, J is the probe field strength, and fl2is the Rabi frequency of the spin-flip driving field. The total coupling strength of the ensemble is g = gsN.

[0046] Analytically and numerically solving the quantum master equation yields the dynamics for nuclear spin manipulation based on the Hamiltonian in Eq. (2), together with loss processes using a Lindbladian open system formalism. The cavity mode has a relaxation rate K = KC+ KC1, where KCis intrinsic and KC1is coupling to a microwave probe line. The NV ensemble includes several polarization and decoherence processes: optical polarization (by the laser beam 161 in FIG. 1A) driving spins to ground states at rate yp; electron spin decoherence at rate f; thermal depolarization at rate yth; and nuclear spin dephasing rate ync].

[0047] The primary parameter of interest for nNV-cQED operation is the cavity field, modulated by the NV ensemble and the driving microwave field. We first consider the case where the spin and cavity are on resonance with an achievable system cooperativity C = 4g2 / KP ~ 20, and the system is driven with a rate fl2= 6 kHz by the driving microwave field, which is on resonance with the spin-exchanging transition, while being probed by the probe microwave field at a power P = — 55 dBm. A solution for the cavity field as a function of input probe detuning is shown in FIG. IE, using an input-normalized cavity field |<z012= | KC1a / J |2. In this strong coupling regime, the hybridization of the cavity mode with the spin ensemble results in a Rabi splitting of the cavity field maxima, characteristic of the collectively coupled system. A narrow EIT feature appears at the two-photon resonance condition where A = 0 due to two-field interference. The EIT linewidth and amplitude are determined by the interplay of driving and probe fields with the nuclear spin decoherence. The minimum linewidth is set by the nuclear-spin coherent dark state’s lifetime, and the EIT contrast compared to the cavity-coupled spin system scales with system cooperativity.

[0048] We now consider the nNV-cQED device under different two-field driving conditions. In the steady-state, the spin ensemble depends on the cavity field via the relation a = (J — igsNale) / (K / 2 where the spin coherence <7ledepends on the spin-exchange driving field,.2, as <7le= (— igsa — i£lcrle) / (F / 2) (valid when most of the population is in the |1) state). The upper plot in FIG. IF shows the relation between fl2and °ie- Low driving fields result in a negative Im(<7le) dominated by the cavity field a, corresponding to imperfect EIT. As the drive fl2increases, the ground state coherence a12is built corresponding to coherent population trapping of a nuclear spin dark state, and the term i.a12compensates the resonant cavity probe field igsa. This increases <7leand reduces the phase shift of the cavity field dueAttorney Docket No. MIT-26298WO01to spins, with <7le= 0 corresponding to complete cancellation (perfect EIT). As fl2continues to increase, < Jlebecomes positive, resulting in gain for the cavity field (MWI). The MWI regime produces the largest two-field interference visibilities and is, therefore, useful for sensor operation. In the limit £12<712» 9sa, the cavity field has a oc gsN.2 / Kr = fl2C / ^s. Increased cooperativity then results in increased MWI gain for fixed driving field.

[0049] FIG. 1G shows the normalized steady-state cavity field a0under varying two-field drive parameters. As the power of the probe microwave field increases, the system exhibits saturation effects. In the low-power regime, increased probe power does not destroy the perfect EIT condition as coherence cr12compensates increased gsa. As the probe power increases to P -> — 55 dBm and beyond, however, the drive fl2should be increased to maintain perfect EIT due to < J12saturation. This combination induces Autler-Townes splitting in the dressed state picture, broadening the resonant peak and decreasing EIT contrast.

[0050] In this high-cooperativity device, an oscillatory regime exists where igsNaleexceeds the probe microwave field / , providing sustaining gain that overcomes cavity losses. Compared with the EIT and MWI regimes, where the output frequency locks with the probe field, the frequency in the oscillating regime is determined by spin-cavity parameters and the driving field. The lower plot in FIG. IE shows Im(cr) as a function of time for the detuning A&= 2TT X 80 Hz in the oscillating regime, showing a beat note with frequency Abcompared to the probe field.Gyroscope Performance

[0051] Having shown the features of the nNV-cQED system under two-field drive, we now consider performance in an inertial sensing application. The nuclear spin behavior under a rotation R, shown in FIGS. IB and 1C, is described by the Hamiltonian J-CR= R • I. FIG. 2A shows the sensing signal divided into a common mode (Ac) and a difference mode (AD). The common mode describes the sensing signal shifting the state |e) while keeping nuclear spin energies constant. The difference mode describes the sensing signal shifting |2) while changing nuclear energy difference from |Q| to |Q| + AD. A rotation R results in both common- and difference-mode shifts equal to the rotation rate, but the resonant linewidth for the difference mode is limited by nuclear spin linewidth (~ 80 Hz), which is three orders of magnitude smaller than the electron spin linewidth-limited common signal (~ 0.33 MHz).

[0052] The device’s sensitivity is determined by the signal and noise when probing the system: g = LfeS. The signal per unit frequency shift is defined as S = [7 dIm(r)] / d4D, where V =Attorney Docket No. MIT-26298WO01J lh(i)dR / Kclis the probe field input voltage (R is the resistance) and r = — 1 + a0is the probe reflection coefficient. FIG. 2B is a plot of S as a function of Acand ADwith the same spincavity parameters as above. The optimal S is 18 V / Hz with a linear dynamical range of 20 Hz.

[0053] The noise performance is typically limited by Johnson-Nyquist noise at room temperature £JN. However, the NV ensemble can act as a spin refrigerator, performing below the Johnson-Nyquist limit even with active microwave probes. The noise performance can then be expressed as L = ^CjN, with the refrigeration prefactor f = 0.45.

[0054] FIG. 2C is a plot of the sensitivity T as a function of 122and P An optimal sensitivity of 1.5 mdeg / s / VHz is achievable in the MWI regime, while in the EIT regime the best sensitivity is 3.5 mdeg / s / VHz. In the oscillation regime, the field frequency is determined by spin-cavity parameters and the driving field. The sensitivity of such a frequency modulation system depends on oscillator phase noise, which is outside the scope of our discussion. The rotation standard quantum limit (SQL) of this system is 0.04 mdeg / s / VHz, and the optimum sensitivity corresponds to an inverse readout fidelity of an= 37, three orders better than photon-shot-noise-limited sensitivity.

[0055] Finally, we analyze the ARW dependence on nNV-cQED system cooperativity. FIG.2D is a plot of the ARW dependence on the cooperativity C, given the same single-spin coupling strength and loaded quality factor. Here, the cooperativity is tuned by changing the ensemble size (number of NVs). In the low-C regime (C < 10), the ARW decreases linearly with C. The cooperativity is then decreased super-linearly in the regime of 10 < C < 24, attributed to MWI. After C > 24, the maximum-sensitivity operating point enters the oscillating regime and degrades performance. The best readout fidelity for our system is an= 37 with C = 20 and an ensemble size of 2.4 x 1015. The sensitivity is independent of the Q^ / K ratio for fixed cooperativity.Comagnetometry

[0056] One challenge for spin-based gyroscopes in ambient conditions is the magnetic field environment. The magnetic field generates a varying nuclear phase via Yn^z- indistinguishable from the rotation signal in the Hamiltonian, setting the noise floor r / r. A comagnetometer uses the electron spin to measure the magnetic field, which is then used to compensate for nuclear spin phase accumulation induced by the magnetic environment.Attorney Docket No. MIT-26298WO01

[0057] Here, we apply a second probe microwave field to measure the electron spin frequency. The upper plot in FIG. 3A shows the cavity reflection |r |2as a function of A and As. FIG. 3A shows three avoid-crossing features, corresponding to the three14A hyperfine transitions, assuming each subensemble has an equal population with C = 10. The lower plot in FIG. 3A shows the magnetic sensitivity-limited ARW floor produced by the second probe tone as a function of its detuning A. With the two-field interference applied in the |1), |2) subspace, however, the coupling strengths become imbalanced. The driving microwave field polarizes the nuclear spin transition to state 11) from |2), reducing the coupling strength for the 2+transition compared to 1+and producing an asymmetry in the sensitivity spectrum. The comagnetometer sensitivity peaks at a probe detuning of A = —4.7 MHz (point I in FIG. 3A), where the noise floor scales with comagnetometer probe power as r / r= 1 mdeg / s / VHz with power P2= —40 dBm. This resonates with the 3+transition, effectively using the third nuclear spin hyperfine subensemble as a comagnetometer.

[0058] FIG. 3B shows the sensitivity degradation induced by the second probe microwave field, detuned from the EIT probe by a frequency detuning ATand Rabi frequency flR. The ideal co-magnetometry operating point is shown by the dot I in FIG. 3B, with a Rabi frequency flR= 4.4 kHz producing a r]r= 1 mdeg / s / VHz. At this point, the EIT rotational sensitivity is not affected by the comagnetometer probe, with Ar / / ri < 1%. The compensation for the magnetic field drift also provides potential for long-term stability for the nNV-cQED system.Vector Gyroscope

[0059] We extend the nNV-cQED system to a vector gyroscope with a multifield EIT scheme. The external magnetic bias is adjusted such that the electron spin transitions are near the cavity resonance yet resolvable (Et— Ej > ), with multiple fields applied for each spin subensemble. Each subensemble, with the same cooperativity as above, measures the rotation projection on its axis, as the external bias is small compared to the electron zero-field splitting. Subensembles should be densely packed within the cavity linewidth since the rotation signal S drops significantly beyond the cavity linewidth, as shown in FIG. 3C.

[0060] Two kinds of crosstalk may affect the nNV-cQED vector device. First, since the EIT condition is met simultaneously for the subensembles detuned in common-mode frequency, the signal measured from a particular EIT probe should be associated with shifts of one ensemble. To this end, consider the signal response S from an EIT probe at frequency Acassociated with an energy shift of each subensemble, given that the ensembles are separated byAttorney Docket No. MIT-26298WO010.8 MHz. As shown in FIG. 3D, an EIT probe on resonance with the central ensemble under these conditions would have a maximum relative sensitivity of 10% to the nearest off-resonant ensembles (typically 0.5%). A linear crosstalk elimination process can be performed for further correction.

[0061] Second, the comagnetometer and multiple closely packed EIT probes may induce additional interference effects that degrade performance, as discussed above with respect to the comagnetometer session. The sensitivity degradation is also within 1% indicated in FIG. 3B (dots labeled Ei).

[0062] FIG. 4 shows a nuclear spin vector gyroscope 400. Like the nuclear spin gyroscope 100 in FIG. 1A, the nuclear spin vector gyroscope 400 includes an NV ensemble 110 in a host diamond 112. In the nuclear spin vector gyroscope 400, however, a bias magnet 470 applies a bias magnetic field at angle oriented anywhere from the

[0100] crystalline direction to the

[0111] crystalline direction of the diamond 112. This breaks the degeneracy among the {| 1), 12), |e)} electron-nuclear spin states so that the |1) <-> |e) and |2) <-> |e) spin transitions of the NVs oriented along the four different crystallographic axes of the diamond host are resonant at different frequencies. In addition, the driving microwave field is replaced by four different driving microwave field, each resonant with a different |2) <-> |e) spin transition (different crystallographic axis NV alignment). These four different driving microwave fields can be generated simultaneously with four different signal generators 430a-430d, all of which are coupled to the same loop antenna 432, which transmits the four different driving microwave fields simultaneously toward the NV ensemble 110. Alternatively, all four different driving microwave fields can be generated simultaneously by the same signal generator (e.g., an arbitrary waveform generator).

[0063] The nuclear spin vector gyroscope 400 is also configured as a comagnetometer — its probe signal generator 440 generates both a first probe microwave field for probing the 11) <-> |e) spin transition and the second probe microwave field for probing the electron spin frequency. The loop antenna 142 applies both the first and second probe microwave fields to the NV ensemble 110 at the same time. (Alternatively, the first and second probe microwave fields can be generated with separate RF synthesizers and either combined and used to drive the antenna or applied with separate antennas.) The mixer 150 mixes the reflected probe microwave fields and transmitted driving microwave fields down to baseband for detection and processing by the DAQ 152 board as described above.Attorney Docket No. MIT-26298WO01Conclusion

[0064] While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and / or structures for performing the function and / or obtaining the results and / or one or more of the advantages described herein, and each of such variations and / or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and / or configurations will depend upon the specific application or applications for which the inventive teachings is / are used. Those skilled in the art will recognize or be able to ascertain, using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and / or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and / or methods, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.

[0065] Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0066] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0067] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

[0068] The phrase “and / or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elementsAttorney Docket No. MIT-26298WO01listed with “and / or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and / or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and / or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0069] As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and / or” as defined above. For example, when separating items in a list, “or” or “and / or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of’ or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.

[0070] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and / or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including moreAttorney Docket No. MIT-26298WO01than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

[0071] In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of’ and “consisting essentially of’ shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.

Claims

Attorney Docket No. MIT-26298WO01CLAIMS1. A gyroscope comprising:a nitrogen-vacancy ensemble in a diamond host, the nitrogen-vacancy ensemble having electron-nuclear spin states {| 1), 12), |e)} that form a A system with a |1)spin transition and a |2)spin transition;at least one microwave signal generator to generate (i) a driving microwave field tuned to the |2)spin transition and (ii) a probe microwave field tuned to the |1)spin transition;at least one antenna, operably coupled to the at least one microwave signal generator and in electromagnetic communication with the nitrogen-vacancy ensemble, to apply the driving microwave field and the probe microwave field to the nitrogen-vacancy ensemble and to sense the probe microwave field returned by the nitrogen-vacancy ensemble; anda microwave detector, operably coupled to the at least one antenna, to sense a change in the probe microwave field returned by the nitrogen-vacancy ensemble in response to a shift in the electron-nuclear spin states {| 1), 12)} caused by rotation of the nitrogen-vacancy ensemble.

2. The gyroscope of claim 1, wherein the at least one microwave signal generator is configured to generate the driving microwave field at an amplitude sufficient to cause electromagnetically induced transparency (EIT) of the nitrogen-vacancy ensemble.

3. The gyroscope of claim 1, wherein the at least one microwave signal generator is configured to generate the driving microwave field at an amplitude sufficient to cause masing without inversion (MWI) of the nitrogen-vacancy ensemble.

4. The gyroscope of claim 1, wherein the probe microwave field is a first probe microwave field and the at least one microwave signal generator is configured to generate a second probe microwave field for measuring a shift in an electron spin resonance of the nitrogen-vacancy ensemble caused by a change in magnetic field experienced by the nitrogen-vacancy ensemble.

5. The gyroscope of claim 1, wherein the microwave detector is configured to detect a change in amplitude, caused by the rotation of the nitrogen-vacancy ensemble, of the probe microwave field returned by the nitrogen-vacancy ensemble.Attorney Docket No. MIT-26298WO016. The gyroscope of claim 1, wherein the microwave detector is configured to detect common mode and difference-frequency shifts in the driving microwave field and the probe microwave field returned by the nitrogen-vacancy ensemble.

7. The gyroscope of claim 1, wherein the microwave detector comprises:a mixer, operably coupled to the at least one microwave signal generator, to mix the probe microwave field returned by the nitrogen-vacancy ensemble with a phase-shifted copy of the probe microwave field; anddata acquisition hardware, operably coupled to the mixer, to measure an output of the mixer.

8. The gyroscope of claim 1, further comprising:a microwave resonator, electromagnetically coupled to the |1)spin transition, to enhance the change in the probe microwave field returned by the nitrogen-vacancy ensemble.

9. The gyroscope of claim 1, further comprising:a light source, in optical communication with the nitrogen-vacancy ensemble, to polarize the nitrogen-vacancy ensemble into the electron-nuclear spin state |1).

10. The gyroscope of claim 1, further comprising:a magnetic field source, in electromagnetic communication with the nitrogen-vacancy ensemble, to apply a bias magnetic field to the nitrogen-vacancy ensemble.

11. The gyroscope of claim 10, wherein the magnetic field source is oriented to apply the bias magnetic field to the nitrogen-vacancy ensemble in a direction that breaks a degeneracy among the electron-nuclear spin states {| 1), 12), |e)} of nitrogen vacancies in the nitrogenvacancy ensemble aligned with different crystallographic axes of the diamond host.

12. The gyroscope of claim 11, wherein the driving microwave field is a first driving microwave field tuned to a |2)spin transition of the nitrogen vacancies aligned with a first crystallographic axis of the diamond host and the at least one microwave signal generator is further configured to generate:a second driving microwave field tuned to a |2)spin transition of the nitrogen vacancies aligned with a second crystallographic axis of the diamond host;a third driving microwave field tuned to a |2)spin transition of the nitrogen vacancies aligned with a third crystallographic axis of the diamond host; andAttorney Docket No. MIT-26298WO01a fourth driving microwave field tuned to a |2) spin transition of the nitrogen vacancies aligned with a fourth crystallographic axis of the diamond host.

13. A method of measuring rotation of a nitrogen-vacancy ensemble in a diamond host, the nitrogen-vacancy ensemble having electron-nuclear spin states {| 1), 12 ), |e)} that form a A system with a |1)spin transition and a |2)spin transition, the method comprising:driving the |2)spin transition with a driving microwave field;while driving the |2)spin transition, probing the |1)spin transition with a probe microwave field; andsensing a change, caused by rotation of the nitrogen-vacancy ensemble, in the probe microwave field returned by the nitrogen-vacancy ensemble.

14. The method of claim 13, wherein the nitrogen-vacancy ensemble is electromagnetically coupled to a microwave cavity resonant with the 11)spin transition.

15. The method of claim 13, wherein driving the |2)spin transition is at a rate sufficient to cause electromagnetically induced transparency (EIT) of the nitrogen-vacancy ensemble.

16. The method of claim 13, wherein driving the |2)spin transition is at a rate sufficient to cause masing without inversion (MWI) of the nitrogen-vacancy ensemble.

17. The method of claim 13, further comprising:measuring a shift in an electron spin resonance of the nitrogen-vacancy ensemble caused by a change in magnetic field experienced by the nitrogen-vacancy ensemble.

18. The method of claim 13, wherein sensing the change in the probe microwave field returned by the nitrogen-vacancy ensemble comprises sensing a change in amplitude of the probe microwave field returned by the nitrogen-vacancy ensemble.

19. The method of claim 13, wherein sensing the change in the probe microwave field returned by the nitrogen-vacancy ensemble comprises detecting common mode and difference-frequency shifts in the driving microwave field and the probe microwave field returned by the nitrogen-vacancy ensemble.

20. The method of claim 13, further comprising:Attorney Docket No. MIT-26298WO01applying a bias magnetic field to the nitrogen-vacancy ensemble in a direction that breaks a degeneracy among the electron-nuclear spin states {| 1), 12), |e)} of nitrogen vacancies in the nitrogen-vacancy ensemble aligned with different crystallographic axes of the diamond host,wherein driving the|2) spin transition comprises driving the |2)spin transitions aligned with the different crystallographic axes with different driving microwave fields.

Images