Abstract

Signal reception of nuclear magnetic resonance (NMR) usually relies on electrical amplification of the electromotive force caused by nuclear induction. Here, we report up-conversion of a radio-frequency NMR signal to an optical regime using a high-stress silicon nitride membrane that interfaces the electrical detection circuit and an optical cavity through the electro-mechanical and the opto-mechanical couplings. This enables optical NMR detection without sacrificing the versatility of the traditional nuclear induction approach. While the signal-to-noise ratio is currently limited by the Brownian motion of the membrane as well as additional technical noise, we find it can exceed that of the conventional electrical schemes by increasing the electro-mechanical coupling strength. The electro-mechano-optical NMR detection presented here opens the possibility of mechanical parametric amplification of NMR signals. Moreover, it can potentially be combined with the laser cooling technique applied to nuclear spins.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Electrical signals can be up-converted from radio frequency (RF) to optical regimes using a high-Q metal-coated silicon nitride membrane, which serves both as a capacitor electrode and a mirror of an optical interferometer [1]. There, the mechanics of the membrane, the electronics of the RF circuit, and the optics of the interferometer interact with one another through the opto-mechanical and the electro-mechanical couplings. Even though the principle of such membrane-based, RF-to-light signal transduction has now been established, its power has yet to be harnessed in various RF-relevant fields. In this work, we report on what we believe is the first RF-to-light up-conversion of nuclear magnetic resonance (NMR) signals.

NMR [24] is a powerful analytical tool, offering access to structure and dynamics in liquid and solid materials of physical/chemical/biological interest. Usually, NMR signal reception relies on nuclear induction [5] causing an electromotive force across the detection coil, followed by electrical amplification of the RF signals [6]. For a given signal strength, which could be significantly enhanced by nuclear hyperpolarization techniques [7,8], the sensitivity is limited by the noises, namely the Johnson noise of the resistive components within the circuit as well as the inevitable noise from the amplifier. While the noise levels in unconventional optical NMR schemes, such as Faraday rotation [9,10], force detection [11], fluorescence [12,13], and atomic magnetometry [14], are much lower than that found in the traditional NMR and can, in principle, be quantum-noise-limited, all existing optical NMR detection schemes lack wide applicability compared to the traditional induction approach, which allows measurements of any bulk samples, including living organisms, placed inside the detection coil.

Here, we put forward a versatile approach to optical NMR readout, applicable straightforwardly to chemical analysis as well as magnetic resonance imaging (MRI) diagnosis, by exploiting the membrane signal transducer system that we designed and fabricated to meet the specific needs for pulsed NMR spectroscopy. In the following sections, we demonstrate the electro-mechano-optical (EMO) NMR detection scheme with proton (H1) spin echoes [15] in water. The signal-to-noise ratio (SNR), albeit currently limited by thermal noise due to the Brownian motion of the membrane as well as additional technical noise, is expected to increase with the electro-mechanical coupling strength. We show that the EMO NMR approach can offer better sensitivity compared to the conventional all-electrical scheme with realistic improvements in the experimental parameters. The EMO approach opens the possibility of mechanically amplifying the NMR signal [16] and even laser cooling nuclear spins [1719] to further enhance the sensitivity of NMR.

2. EXPERIMENT

A. Experimental Setup

We aimed at transducing H1 NMR signals induced in a magnetic field of 1T from the original RF domain (ωs/2π43MHz) to the optical domain (Ωc/2π300THz) for a demonstration of EMO NMR. Figure 1 illustrates the experimental setup. For the opto-mechanical and the electro-mechanical couplings, the mechanically compliant part was a high-stress silicon nitride (Si3N4) membrane (Norcada) with lateral dimensions of 0.5×0.5mm and a thickness of 50 nm. A circular Au layer with a diameter of 0.45 mm and a thickness of 100 nm was deposited on the membrane. The effective mass m of the Au-coated membrane oscillator was 8.6×1011kg. We found the fundamental (1,1)-drum mode oscillation of the Au-coated membrane at ωm/2π180kHz. The Q factor was about 1800 in a vacuum with no air damping. Counter electrodes were patterned on a silica plate, and the membrane capacitor was assembled with a designed gap d0 between the electrodes of 800 nm. The actual gap was estimated to be d01.4μm (see Supplement 1).

 figure: Fig. 1.

Fig. 1. (a) Experimental setup for EMO NMR composed of an orthogonal pair of coils tuned at the NMR frequency, a membrane put inside a vacuum chamber, an optical cavity, and a photodetector. (b) Schematic drawing of the membrane capacitor. The Au layer on the membrane is electrically floating, and coupled capacitively to the Al pattern on the substrate. The two electrodes of the capacitor were electrically connected with the rest of the circuit through a pair of contact probes pushing against the Al pads on the silica substrate. (c) Photograph of the Au-deposited membrane.

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The magnetic field was provided by a nominally 1 T permanent magnet, in which a pair of orthogonal rf coils were embedded for pulsed excitation of nuclear spins and NMR signal reception, respectively. The excitation coil was a two-turn saddle coil, while the detection coil was a 10-turn solenoid coil with a diameter of 3 mm (L=150nH). In addition, a pair of planar coils (not shown) were placed outside the RF coil pairs to vary the static magnetic field with application of dc current around the resonance condition of the proton spins. The membrane capacitor was connected in parallel with the detection coil together with additional trimmer capacitors with capacitances Ct=98pF and Cm=21pF, forming a balanced resonant circuit at ωLC/2πωs/2π43MHz with a Q factor of 26.7. The excitation coil was also impedance-matched at the same frequency. The isolation between these two separate circuits was 22.5 dB at the resonance frequency.

The design of the optical Fabry–Perot cavity is described in Supplement 1. Here, the metal-coated membrane served as one of the two mirrors of an optical cavity for a laser beam with a wavelength of 780 nm. The other mirror with a reflectance of 97% and a radius of curvature of 75 mm was attached to a ring piezo actuator. The cavity length, which was coarsely adjusted to 17.5 mm, was locked by the feedback on the piezo to the position where the amplitude of the reflected laser beam drops half the dip at cavity resonance, so that the membrane oscillation resulted in amplitude modulation of the laser and thus was imprinted in the optical sideband signal at ωm. Note that the cutoff frequency of the piezo servo system is far below ωm, so that the mechanical response, which would include the RF signal contribution, can be safely transduced to the optical sideband signal at ωm.

B. Electro-Mechano-Optical Signal Transduction

The RF signal developed in the detection LC circuit was parametrically transduced to the membrane oscillation in the presence of the drive signal at either the sum or the difference angular frequency ωD=ωs±ωm, which was applied to bridge the mismatch between the H1 resonance frequency ωs/2π43MHz and the membrane resonance frequency ωm/2π180kHz. The resultant membrane oscillation was then probed by light.

To examine the EMO signal transduction, we applied a continuous-wave RF signal at a frequency ωs/2π+500Hz to port A in Fig. 1, instead of the real EMF signal, together with drive irradiation at various powers. Figure 2 shows the acquired optical sideband spectra, where in each spectrum the mechanical responses of the membrane to the noise (blue) as well as the delta-function-like RF-signal tone (red) are visible. With the increasing power of the drive, the mechanical resonance frequency is shifted downward [1]. In addition to the Johnson noise and the Brownian noise of the mechanical oscillator, we found an increase in the noise floor with the drive power. We ascribed this to the phase noise of the drive, as we will describe in Section 3.D.

 figure: Fig. 2.

Fig. 2. Drive-power dependence of the sideband spectra of the optically-detected membrane oscillation under application of a continuous-wave tone signal with a power of 81dBm. The spectra are plotted with vertical offsets proportional to the drive power. The baselines (horizontal broken lines) indicate the corresponding drive power (right axis) as well as the reference power spectral density of 114.5dBm/Hz. Along with the membrane spectra (blue lines) the peaks corresponding to the tone signals (red lines) appear at 500-Hz off-resonance from the mechanical resonance frequency ωm (black points). The observed downward shifts of the mechanical resonance frequency were fitted with a model discussed in Supplement 1 (orange line).

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C. H1 Spin Echo Experiment

H1 NMR experiments were then carried out at room temperature using a 0.1 mol/L aqueous solution of CuSO4 in a glass test tube (inner diameter 1 mm) with 2.2×1020 H1 spins of water molecules, in which the paramagnetic copper ions accelerate H1 spin relaxation, allowing rapid repetition of signal averaging. The spin-echo measurement [15] was performed by applying RF pulses with a power of +17dBm to the tuned excitation coil through port B in Fig. 1 with the widths of the π/2 and the π pulses of 140 μs and 280 μs, respectively, and a pulse interval of 1.5 ms. The inset of Fig. 3 shows a conventional electrical signal of the H1 spin echo obtained by connecting port A in Fig. 1 to a low-noise amplifier, so that the amplified electrical nuclear induction signal could be sent to the conventional demodulation circuit of the NMR spectrometer. The maximum intensity of the NMR echo signal was 93dBm at the input of the low-noise amplifier. The observed decay with a time constant T2*320μs was dominantly caused by the inhomogeneity of the magnetic field.

 figure: Fig. 3.

Fig. 3. H1 spin echo signals in a 0.1 mol/L aqueous solution of CuSO4 detected by the EMO approach on-resonance (blue line) and +2.5kHz off-resonance (red line). The vertical scale represents the 5000-times average signal intensity in units of the number of photons reaching the photodetector per second. The broken line represents a convolution of the electrically detected spin-echo signal shown in the inset with an exponential function with a time constant 2/γm. The signal-to-noise ratio S/N is about 5.4.

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Next, the low-noise amplifier at port A in Fig. 1 was replaced with a drive source for down conversion of the NMR signal to the mechanical frequency, and the optical output from the Fabry–Perot cavity was measured under the drive power of +15dBm. During the RF pulses, the frequency of the drive was detuned by +400kHz, so as to decouple the electro-mechanical interaction and thereby prevent the membrane from being shaken by the excitation RF pulse leaked to the detection circuit, which, in spite of the 22.5 dB isolation, was still orders of magnitude more intense than the NMR signals induced in the receiving LC circuit (93dBm).

Figure 3 shows the electro-mechano-optically detected spin-echo signal (blue line) accumulated over 5000 times with a repetition interval of 20 ms. For comparison, we performed another measurement with the identical experimental parameters, except for a slight shift in the static magnetic field (0.06mT), to make the H1 spins off-resonant by 2.5 kHz. We verified that the signal disappeared (red line), convincing ourselves that the profile of the optically detected signal (blue line in Fig. 3) does really originate from the nuclear induction signal.

The difference in the profile of the spin-echo signal obtained by the EMO approach from that in the conventional electrical scheme can be explained by the transient response of the high-Q membrane. In other words, the response b(t) of the membrane to an excitation a(t), the present case of which is the profile of the electrically detected spin echo, is determined by the response function h(t) of the membrane through convolution (i.e., b(t)=th(tτ)*a(τ)dτ). Since the spectrum of the fundamental mode of the membrane was well fitted with a Lorentzian function with a width γm/2π100Hz, we approximated the response function h(t) to be an exponentially decaying function with a time constant 2/γm, and calculated the response b(t), which was found to reproduce the measured profile of the EMO NMR signal (broken line in Fig. 3).

3. THEORY AND DISCUSSION

A. Dynamics of the EMO System

Figure 4 schematically shows the pathway of successive signal transduction through a chain of three harmonic oscillators: the LC circuit, the membrane oscillator, and the optical cavity. Here, q and ϕ are the charge and the flux of the LC circuit, z and p are the displacement and the momentum of the mechanical oscillator, and X and Y are the canonical quadratures of the optical cavity’s field. Gem and Gom are the electro-mechanical and the opto-mechanical coupling strength. γi, γm, and γo are the intrinsic dissipation rates for the LC circuit, the mechanical oscillator, and the optical cavity. Associated with these dissipations, there are rotating-frame thermal fluctuation inputs qin and ϕin for the LC circuit, and a laboratory-frame thermal fluctuation input fin for the mechanical oscillator. The thermal fluctuation inputs for the optical cavity, xin and yin, are negligible and thus omitted. κi and κo are the external coupling rates for the LC circuit and the optical cavity. In addition to the NMR signal input, S, the associated fluctuation inputs are Qin and ϕin for the LC circuit and Xin and Yin for the optical cavity. The total dissipation rates are thus κiT=κi+γi for the LC circuit and κoT=κo+γo for the optical cavity, respectively.

 figure: Fig. 4.

Fig. 4. Schematic diagram of electro-mechano-optical signal transduction of NMR. The three harmonic oscillators—the LC circuit, the membrane, and the optical cavity—are represented with circles, each of which has channels of the input and output, with coupling strengths κi,Gem,Gom, and κo, and dissipation to the bath, with rates γi,γm and γo. The RF signal S generated by nuclear induction at frequency ωsωLC is transduced to the membrane oscillation through the LC circuit with the electro-mechanical coupling under application of the drive signal at ωD=ωLC+ωm. The resultant membrane oscillation is in turn read out optically with the optical cavity through the opto-mechanical coupling.

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Using the input-output formalism in the rotating wave approximation [20], we have the following Heisenberg–Langevin equations of motion,

q˙=ΔiϕκiT2qκi(Qin+S)γiqin,
ϕ˙=ΔiqκiT2ϕκiϕinγiϕinGemz,
z˙=ωmp,
p˙=ωmzγmp2γmfinGemqGomX,
X˙=ΔoYκoT2XκoXin,
Y˙=ΔoXκoT2YκoYinGomz,
where Δi=ωDωLC is the difference between the drive signal frequency and the LC resonance frequency, and Δo=ΩDΩc is the detuning of the optical cavity from the drive laser frequency. We note that the electro-mechanical coupling Gem increases quadratically with decreasing the gap d0 between the electrodes of the capacitor (see Supplement 1).

Now we shall see how the RF NMR signal, S appearing in Eq. (1), is transduced to the optical output Xout, which is given by the input-output relation,

Xout=Xin+κoX.

By taking the time derivative of Eq. (3) and using Eq. (4), we have

z¨ωm=ωmzγmωmz˙2γmfin(Gemq+GomX).
In the frequency domain, the above equation for the displacement can be written as
z(ω)=χm(ω)[2γmfin(ω)(Gemq(ω)+GomX(ω))],
where the mechanical susceptibility χm(ω) is defined by
χm(ω)=(ω2ωmiωγmωm+ωm)1.

In a similar fashion, we obtain the frequency-domain representation for q and X as

q(ω)=χLC(ω)[ΔiGemz+κi(Δiϕin+(iωκiT2)Qin)+γi(Δiϕin+(iωκiT2)qin)],
X(ω)=χc(ω)[ΔoGomz+κo(ΔoYin+(iωκoT2)Xin)],
where the LC susceptibility χLC(ω) and the optical susceptibility χc(ω) are given by
χLC(ω)=[(iω+κiT2)2+Δi2]1,
χc(ω)=[(iω+κoT2)2+Δo2]1.

In our experiment, Δiωm was much smaller than the resonant bandwidth of the LC circuit. Thus, we consider the case of the resonant application of the drive, Δi0. In addition, to detect the membrane displacement through amplitude modulation of the optical output, we detuned the optical cavity approximately by half its bandwidth (i.e., ΔoκoT/2). Further, since the frequency ωm of interest in the optical output signal is much smaller than Δo, we set ω0 in Eqs. (12) and (14). Neglecting Gom21, we obtain, after some algebra,

Xout=κoκoTYin+(1κoκoT)XinGomκoκoT2γmχm(ω)finGomκoκoTGemiωκiT2χm(ω)[κi(Qin+S)+γiqin].

In the laboratory frame, the linearized rotating-frame signal Xout in Eq. (15) has to be modified to be

X˜out=Xoutcos(Ωt)+Youtsin(Ωt)+κoNDcos(ΩDt),
where the last term comes from the displacement by the optical drive, which oscillates at frequency ΩD. Here, ND is the intracavity photon number (see Supplement 1). Note that Yout, now appearing in X˜out, is given by
Yout=κoκoTXin+(1κoκoT)Yin+GomκoκoT2γmχm(ω)fin+GomκoκoTGemiωκiT2χm(ω)[κi(Qin+S)+γiqin].

In the photo-detected signal |X˜out|2 in the laboratory frame, the components oscillating around ωωm, which are produced by the interference between the term oscillating at ΩD and the ones at ΩD±ωm, are of interest. These components constitute the optical signal output, O(ω), which amounts to the magnitude of the quadrature demodulated signal (see Supplement 1), and can be written as

O(ω)=κoND|Xout|2+|Yout|2.

This indeed contains the RF signal input S along with various noises, which is faithfully transduced from the mechanical response Eq. (9) with the amplification factor proportional to Gom, as seen in Eqs. (15) and (17). The added noise here is just the optical shot noise, which can be quantum-noise-limited. One of the potential advantage of the EMO NMR detection over the conventional NMR is the fact that both the Brownian noise and the optical shot noise can be suppressed by increasing the electro-mechanical coupling Gem as well as the opto-mechanical coupling Gom [1].

B. Noise Spectral Densities

Since the mean value of noise is zero, each noise shall be evaluated in terms of spectral density. For the Brownian noise of the mechanical oscillator, the noise spectral density SFF is defined as SFF=|fin|2. The Johnson noise in the LC circuit can come from the bath as well as from the input channel, and its spectral density, Sqq, is given κiTSqq=κi|Qin|2+γi|qin|2. Assuming that these noise spectra SFF(ω) and Sqq(ω) are white within the bandwidth of the mechanical resonance, we have the Nyquist-type noise spectra,

SFF(ω)=nth(ωm,Teff),
Sqq(ω)=nth(ωLC,T),
with
nth(ω,T)=kBTω.

Here, we assume that the electric bath temperature T is 300 K, while the mechanical bath temperature Teff is not necessarily equal to 300 K but can rather be higher given that the quality factor is good. In this scenario, the ambient noise could easily bring the mechanical oscillator away from the thermal equilibrium. We note that the LC circuit and the mechanical oscillator are both in a high temperature regime where kBTeffωm and kBTωLC. Conversely, we can expect that the noise spectral density SXX|Xin|2 and SYY|Yin|2 for the optical part can be made much smaller.

From Eq. (18), the single-sided spectral density Soo(ω) of the optical signal at frequency ω close to ωm can be written as

Soo(ω)=κoND[((κoκoT)2+(1κoκoT)2)(2SXX(ω)+2SYY(ω))+ComκoκoT2γm2|χm(ω)|24SFF(ω)+ComκoκoTCem(ω)γm2|χm(ω)|2×[4Sqq(ω)+κiκiT4S2δ(ωωm)]].

Here, we introduced the opto-mechanical cooperativity Com and the frequency-dependent electro-mechanical cooperativity Cem(ω) as

Com=Gom2γmκoT,
Cem(ω)=4Gem2γmκiTκiT24ω2+κiT2.

C. Signal-to-Noise Ratio

In the under-coupling limit κoκoT, the SNR S/N in units of a photon number within a narrow frequency range Δωm at around ω=ωm (i.e., from ωmΔ2 to ωm+Δ2), is

S2κiTκi(SXX(ωm)+SYY(ωm)2ComκoκoTCem(ωm)+2SFF(ωm)Cem(ωm)+Sqq(ωm))Δ,
where we used γm2|χm(ωm)|2=1. The form of the SNR consolidates the aforementioned potential advantage of the EMO NMR. All the noise except for the Johnson noise, which is intrinsically inseparable from the RF signal, are suppressed by increasing the electro-mechanical coupling Gem and thus the electro-mechanical cooperativity Cem [1].

Note that yet another figure-of-merit, the signal transfer rate [21], for the current EMO-NMR is given by ComκoκoTCemκiκiT.

D. Comparison to the Experiments

We calibrated the parameters (see Supplement 1) that characterize the EMO signal transduction from the acquired optical sideband spectra shown in Fig. 2. In the presence of a +15dBm drive, the electro-mechanical cooperativity Cem of 0.019 was attained, whereas the opto-mechanical cooperativity Com was 0.32×103. With these values, the signal transfer rate amounts to 1.1×107.

As the drive power is increased to make Cem much larger, however, the phase noise of the drive becomes conspicuous, as mentioned in Section 2.B. In terms of the offset angular frequency ω, the profile of the phase noise can be expressed as

L(ω)=δPω2+δP24,
which is a Lorentzian form with the spectral line width of δP, where 1/f noise and frequency-independent noise are ignored. Then the photon flux associated with the phase noise of the drive at a sideband frequency ω can be given by
L(ω)PDωD,
where PD is the power of the drive. Thus, the spectral density at frequency ω in Eq. (22) must be modified when the phase-noise contribution is appreciable.

To deduce the expected SNR, one missing element is the bandwidth of the NMR signal. In the echo experiment, the effective bandwidth of the detection is determined by 1/πT2*1kHz, where T2*320μs. Since the bandwidth of the electro-mechano-optical NMR detection is limited by the mechanical response, Δ/2πγm/2π100Hz, the impedance mismatch roughly leads to the factor of γmT2*/2 reduction of the signal strength. The SNR for the echo experiment shown in Fig. 4 is thus expected to be

SN=S2T2*2(γmT2*2)κiTκiSXX(ωm)+SYY(ωm)2ComκoκoTCem(ωm)+κiTκi2SFF(ωm)Cem(ωm)+κiTκiSqq(ωm)+ηpPDωDγm0.12
for the single-shot measurement, where the parameter ηp in Eq. (28) is characterizing the phase noise at around ωm with respect to the carrier at ωD, i.e.,
ηp=ωmΔ2ωm+Δ2dω2πγm2|χm(ω)|2L(ω),
which was evaluated to be ηp1011 (see Supplement 1). The number of total noise quanta [the denominator of Eq. (28)] is estimated to be 2.4×1010, while the signal quanta [the numerator of Eq. (28)] for the echo experiment is on the order of 3.6×108, which are proportional to the noise and the signal voltages squared, respectively. The noise budget of the current EMO NMR detection is shown in Table 1. With 5000-times averaging, the SNR becomes roughly 8, agreeing well with the SNR of the acquired data (S/N5.4) shown in Fig. 3.

Tables Icon

Table 1. Noise Budget of the Current EMO NMR Detection

E. Prospects

Even though the SNR in the present proof-of-principle EMO NMR demonstration is lower than that in the conventional electrical NMR approach, there is plenty of room to improve the sensitivity. In particular, the electro-mechanical cooperativity Cem1/d04 would increase dramatically by reducing the capacitor gap. With a realistic revision that includes the capacitor design, we have a prospect of attaining an effective noise temperature of as low as 6 K at room temperature operation of the transducer with a +30dBm drive (see Supplement 1), which would outperform the conventional NMR approach. If the membrane is put in a cryogenic environment, further improvement is expected.

In addition, the effect of the phase noise of the drive can be made negligibly small by increasing the mechanical oscillation frequency and thereby the difference ωDωs. One way to do this would be to reduce the weight of the metal layer deposited on the membrane. Some filters can also be arranged to prevent the phase noise of the drive from exciting the mechanical oscillator.

Moreover, as increasing the electro-mechanical cooperativity Cem, signal transduction would be accompanied by parametric signal amplification. So far, in NMR and MRI, parametric amplification has been realized using an LC circuit containing a varactor diode, whose capacitance can be varied electrically [22,23]. The present work would lead to electro-mechanical parametric amplification of NMR/MRI signals.

The use of the Fabry–Perot optical cavity in this work opens the possibility of exploiting the effect of radiation-pressure cooling [2426]. If the opto-mechanical and electro-mechanical couplings as well as the laser power are large enough, the membrane’s oscillation modes, and thereby the eigenmode of the LC circuit, can be cooled [17], implying the possibility of cooling nuclear spins through electro-mechanical and opto-mechanical couplings without physically lowering the temperature of the experimental system. If the expected challenges, such as the insufficient Q factor and finite dissipation rates to the bath, have been overcome, laser cooling of the nuclear spins would provide a way to further enhance the NMR sensitivity. It is worth noting that nuclear-spin laser cooling would not require doping of paramagnetic impurities in the sample of interest, in contrast to the current dynamic nuclear polarization schemes [27].

In the coupling between a microwave cavity and an ensemble electron spins [2830], analogous population exchange has been theoretically proposed [18,19] and experimentally reported [31,32]. Its extension to nuclear spins with a cold mechanical nanoresonator is also suggested [33].

With the separate coil used for RF excitation, pulsed-NMR techniques for coherent manipulation of nuclear spin interactions can be applied straightforwardly [34], whereas in the receiving part, the bandwidth is limited by that of the membrane oscillator (100Hz). This can be rather narrow compared to the spectral width of interest in the NMR analysis, where the resonance lines can spread due to the broadening and/or distribution of isotropic shifts. In this context, the EMO approach is compatible with traditional continuous-wave NMR [7] as well as recently reported field-sweep NMR [35,36], where the frequency of interest is fixed throughout measurement and the external magnetic field is varied instead. It is also worth noting that the aforementioned enhancement of the electro-mechanical coupling would cause damping of the membrane’s oscillation, and thereby increase the accessible bandwidth.

4. SUMMARY

RF signals of nuclear induction can be upconverted to light through the membrane oscillator that forms a part of both the LC resonant circuit and the optical cavity. The EMO NMR approach presented here potentially offers better sensitivity than that of a conventional electrical detection scheme.

Funding

SENTAN, Japan Science and Technology Agency (JST) (14537844); Exploratory Research for Advanced Technology (ERATO) (JPMJER1601).

Acknowledgment

We are grateful to Y. Tabuchi, M. Okada, Y. Tominaga, M. Negoro, T. Koshi, K. Yamada, M. Takahashi, A. Saitoh, K. Kusuyama, M. Ataka, H. Fujita, K. Lehnert, E. Zeuthen, A. Sørensen, A. Schliesser, and E. S. Polzik for fruitful discussions and collaborations.

 

See Supplement 1 for supporting content.

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19. C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016). [CrossRef]  

20. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010). [CrossRef]  

21. E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

22. C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012). [CrossRef]  

23. C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013). [CrossRef]  

24. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006). [CrossRef]  

25. O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006). [CrossRef]  

26. A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006). [CrossRef]  

27. L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993). [CrossRef]  

28. D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010). [CrossRef]  

29. Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010). [CrossRef]  

30. E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011). [CrossRef]  

31. A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016). [CrossRef]  

32. C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017). [CrossRef]  

33. M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011). [CrossRef]  

34. M. Mehring, Principles of High-Resolution NMR in Solids, 2nd ed. (Springer, 1983).

35. K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012). [CrossRef]  

36. K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015). [CrossRef]  

References

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  1. T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
    [Crossref]
  2. F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
    [Crossref]
  3. E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
    [Crossref]
  4. E. Purcell, “Nuclear magnetism in relation to problems of the liquid and solid states,” Science 107, 433–440 (1948).
    [Crossref]
  5. F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
    [Crossref]
  6. D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
    [Crossref]
  7. A. Abragam, Principles of Nuclear Magnetism (Oxford University, 1961).
  8. C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
    [Crossref]
  9. J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
    [Crossref]
  10. I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
    [Crossref]
  11. M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
    [Crossref]
  12. H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
    [Crossref]
  13. T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
    [Crossref]
  14. I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
    [Crossref]
  15. E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
    [Crossref]
  16. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
    [Crossref]
  17. J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
    [Crossref]
  18. C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
    [Crossref]
  19. C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
    [Crossref]
  20. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
    [Crossref]
  21. E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.
  22. C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
    [Crossref]
  23. C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
    [Crossref]
  24. S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
    [Crossref]
  25. O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
    [Crossref]
  26. A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
    [Crossref]
  27. L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
    [Crossref]
  28. D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
    [Crossref]
  29. Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
    [Crossref]
  30. E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
    [Crossref]
  31. A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
    [Crossref]
  32. C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
    [Crossref]
  33. M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
    [Crossref]
  34. M. Mehring, Principles of High-Resolution NMR in Solids, 2nd ed. (Springer, 1983).
  35. K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
    [Crossref]
  36. K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
    [Crossref]

2017 (1)

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

2016 (2)

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

2015 (1)

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

2014 (3)

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

2013 (3)

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

2012 (2)

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

2011 (4)

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

2010 (4)

M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

2006 (4)

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[Crossref]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

2005 (1)

I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
[Crossref]

2000 (1)

J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
[Crossref]

1997 (1)

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

1993 (1)

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

1950 (1)

E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
[Crossref]

1948 (1)

E. Purcell, “Nuclear magnetism in relation to problems of the liquid and solid states,” Science 107, 433–440 (1948).
[Crossref]

1946 (3)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Abe, E.

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

Abragam, A.

A. Abragam, Principles of Nuclear Magnetism (Oxford University, 1961).

Appel, J.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Arcizet, O.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[Crossref]

Ardavan, A.

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

Aspelmeyer, M.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

Auffeves, A.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Awschalom, D. D.

H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

J. M. Kikkawa and D. D. Awschalom, “All-optical magnetic resonance in semiconductors,” Science 287, 473–476 (2000).
[Crossref]

Bagci, T.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Barthe, M. F.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bäuerle, D.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

Becerra, L. R.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Bergonzo, P.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bertet, P.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Bhakar, B.

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

Bienfait, A.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Blaser, F.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

Bloch, F.

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

F. Bloch, W. Hansen, and M. Packard, “The nuclear induction experiment,” Phys. Rev. 70, 474–485 (1946).
[Crossref]

Böhm, H. R.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

Borneman, T. W.

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Briant, T.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[Crossref]

Briggs, G. A. D.

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Buckley, B. B.

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F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

Simonsen, A.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Singel, D. J.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Slichter, C. P.

C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
[Crossref]

Sørensen, A.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Sørensen, A. S.

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Staudacher, T.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Stern, M.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Takahashi, M.

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Takeda, K.

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

Takegoshi, K.

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

Taylor, J. M.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Temkin, R. J.

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

Torrey, H.

E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Usami, K.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Vahala, K. J.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

Villanueva, L. G.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

Vion, D.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Weis, C. D.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Weitekamp, D. P.

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

Wood, C. J.

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Wrachtrup, J.

T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Wu, H.

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Yamada, K.

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Zabow, G.

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

Zeilinger, A.

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
[Crossref]

Zeuthen, E.

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

Zheng, D.

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

Zhou, X.

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
[Crossref]

Appl. Phys. Lett. (1)

E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton, “Electron spin ensemble strongly coupled to a three-dimensional microwave cavity,” Appl. Phys. Lett. 98, 251108 (2011).
[Crossref]

Chem. Phys. Lett. (1)

K. Yamada, K. Kitagawa, and M. Takahashi, “Field-swept 33S NMR study of elemental sulfur,” Chem. Phys. Lett. 618, 20–23 (2015).
[Crossref]

Concepts Magn. Reson. (1)

D. I. Hoult and B. Bhakar, “NMR signal reception: virtual photons and coherent spontaneous emission,” Concepts Magn. Reson. 9, 277–297 (1997).
[Crossref]

J. Magn. Reson. (2)

K. Takeda, N. Ichijo, Y. Noda, and K. Takegoshi, “Elemental analysis by NMR,” J. Magn. Reson. 224, 48–52 (2012).
[Crossref]

C. Qian, G. Zabow, and A. Koretsky, “Engineering novel detectors and sensors for MRI,” J. Magn. Reson. 229, 67–74 (2013).
[Crossref]

Magn. Reson. Med. (1)

C. Qian, J. Murphy-Boesch, S. Dodd, and A. Koretsky, “Sensitivity enhancement of remotely coupled NMR detectors using wirelessly powered parametric amplification,” Magn. Reson. Med. 68, 989–996 (2012).
[Crossref]

Nanotechnology (1)

M. Poggio and C. L. Degen, “Force-detected nuclear magnetic resonance: recent advances and future challenges,” Nanotechnology 21, 342001 (2010).
[Crossref]

Nature (6)

I. M. Savukov, S.-K. Lee, and M. V. Romalis, “Optical detection of liquid-state NMR,” Nature 442, 1021–1024 (2006).
[Crossref]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature 480, 351–354 (2011).
[Crossref]

T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
[Crossref]

S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006).
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O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation pressure cooling and optomechanical instability of a micromirror,” Nature 444, 71–74 (2006).
[Crossref]

A. Bienfait, J. J. Pla, Y. Kubo, X. Zhou, M. Stern, C. C. Lo, C. D. Weis, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, and P. Bertet, “Controlling spin relaxation with a cavity,” Nature 531, 74–77 (2016).
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E. Purcell, H. Torrey, and R. Pound, “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37–38 (1946).
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E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 580–594 (1950).
[Crossref]

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
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Phys. Rev. A (2)

C. J. Wood and D. G. Cory, “Cavity cooling to the ground state of an ensemble quantum system,” Phys. Rev. A 93, 023414 (2016).
[Crossref]

M. C. Butler and D. P. Weitekamp, “Polarization of nuclear spins by a cold nanoscale resonator,” Phys. Rev. A 84, 063407 (2011).
[Crossref]

Phys. Rev. Lett. (8)

C. Eichler, A. J. Sigillito, S. A. Lyon, and J. R. Petta, “Electron spin resonance at the level of 104 spins using low impedance superconducting resonators,” Phys. Rev. Lett. 118, 037701 (2017).
[Crossref]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97, 243905 (2006).
[Crossref]

L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
[Crossref]

D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010).
[Crossref]

Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J.-F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
[Crossref]

I. M. Savukov and M. V. Romalis, “NMR detection with an atomic magnetometory,” Phys. Rev. Lett. 94, 123001 (2005).
[Crossref]

J. M. Taylor, A. S. Sørensen, C. M. Marcus, and E. S. Polzik, “Laser cooling and optical detection of excitations in a LC electrical circuit,” Phys. Rev. Lett. 107, 273601 (2011).
[Crossref]

C. J. Wood, T. W. Borneman, and D. G. Cory, “Cavity cooling of an ensemble spin system,” Phys. Rev. Lett. 112, 050501 (2014).
[Crossref]

Rep. Prog. Phys. (1)

C. P. Slichter, “The discovery and renaissance of dynamic nuclear polarization,” Rep. Prog. Phys. 77, 072501 (2014).
[Crossref]

Rev. Mod. Phys. (1)

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
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H. J. Mamin, M. Kim, M. H. Sherwood, C. T. Rettner, K. Ohno, D. D. Awschalom, and D. Rugar, “Nanoscale nuclear magnetic resonance with a nitrogen-vacancy spin sensor,” Science 339, 557–560 (2013).
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T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup, “Nuclear magnetic resonance spectroscopy on a (5-nanometer)3 sample volume,” Science 339, 561–563 (2013).
[Crossref]

Other (3)

A. Abragam, Principles of Nuclear Magnetism (Oxford University, 1961).

E. Zeuthen, A. Schliesser, A. S. Sørensen, and J. M. Taylor, “Figures of merit for quantum transducers,” arXiv: 1610.01099.

M. Mehring, Principles of High-Resolution NMR in Solids, 2nd ed. (Springer, 1983).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplementary Material

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Figures (4)

Fig. 1.
Fig. 1. (a) Experimental setup for EMO NMR composed of an orthogonal pair of coils tuned at the NMR frequency, a membrane put inside a vacuum chamber, an optical cavity, and a photodetector. (b) Schematic drawing of the membrane capacitor. The Au layer on the membrane is electrically floating, and coupled capacitively to the Al pattern on the substrate. The two electrodes of the capacitor were electrically connected with the rest of the circuit through a pair of contact probes pushing against the Al pads on the silica substrate. (c) Photograph of the Au-deposited membrane.
Fig. 2.
Fig. 2. Drive-power dependence of the sideband spectra of the optically-detected membrane oscillation under application of a continuous-wave tone signal with a power of 81 dBm . The spectra are plotted with vertical offsets proportional to the drive power. The baselines (horizontal broken lines) indicate the corresponding drive power (right axis) as well as the reference power spectral density of 114.5 dBm / Hz . Along with the membrane spectra (blue lines) the peaks corresponding to the tone signals (red lines) appear at 500-Hz off-resonance from the mechanical resonance frequency ω m (black points). The observed downward shifts of the mechanical resonance frequency were fitted with a model discussed in Supplement 1 (orange line).
Fig. 3.
Fig. 3. H 1 spin echo signals in a 0.1 mol/L aqueous solution of CuSO 4 detected by the EMO approach on-resonance (blue line) and + 2.5 kHz off-resonance (red line). The vertical scale represents the 5000-times average signal intensity in units of the number of photons reaching the photodetector per second. The broken line represents a convolution of the electrically detected spin-echo signal shown in the inset with an exponential function with a time constant 2 / γ m . The signal-to-noise ratio S / N is about 5.4.
Fig. 4.
Fig. 4. Schematic diagram of electro-mechano-optical signal transduction of NMR. The three harmonic oscillators—the LC circuit, the membrane, and the optical cavity—are represented with circles, each of which has channels of the input and output, with coupling strengths κ i , G em , G om , and κ o , and dissipation to the bath, with rates γ i , γ m and γ o . The RF signal S generated by nuclear induction at frequency ω s ω LC is transduced to the membrane oscillation through the LC circuit with the electro-mechanical coupling under application of the drive signal at ω D = ω LC + ω m . The resultant membrane oscillation is in turn read out optically with the optical cavity through the opto-mechanical coupling.

Tables (1)

Tables Icon

Table 1. Noise Budget of the Current EMO NMR Detection

Equations (29)

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q ˙ = Δ i ϕ κ iT 2 q κ i ( Q in + S ) γ i q in ,
ϕ ˙ = Δ i q κ iT 2 ϕ κ i ϕ in γ i ϕ in G em z ,
z ˙ = ω m p ,
p ˙ = ω m z γ m p 2 γ m f in G em q G om X ,
X ˙ = Δ o Y κ oT 2 X κ o X in ,
Y ˙ = Δ o X κ oT 2 Y κ o Y in G om z ,
X out = X in + κ o X .
z ¨ ω m = ω m z γ m ω m z ˙ 2 γ m f in ( G em q + G om X ) .
z ( ω ) = χ m ( ω ) [ 2 γ m f in ( ω ) ( G em q ( ω ) + G om X ( ω ) ) ] ,
χ m ( ω ) = ( ω 2 ω m i ω γ m ω m + ω m ) 1 .
q ( ω ) = χ LC ( ω ) [ Δ i G em z + κ i ( Δ i ϕ in + ( i ω κ iT 2 ) Q in ) + γ i ( Δ i ϕ in + ( i ω κ iT 2 ) q in ) ] ,
X ( ω ) = χ c ( ω ) [ Δ o G om z + κ o ( Δ o Y in + ( i ω κ oT 2 ) X in ) ] ,
χ LC ( ω ) = [ ( i ω + κ iT 2 ) 2 + Δ i 2 ] 1 ,
χ c ( ω ) = [ ( i ω + κ oT 2 ) 2 + Δ o 2 ] 1 .
X out = κ o κ oT Y in + ( 1 κ o κ oT ) X in G om κ o κ oT 2 γ m χ m ( ω ) f in G om κ o κ oT G em i ω κ iT 2 χ m ( ω ) [ κ i ( Q in + S ) + γ i q in ] .
X ˜ out = X out cos ( Ω t ) + Y out sin ( Ω t ) + κ o N D cos ( Ω D t ) ,
Y out = κ o κ oT X in + ( 1 κ o κ oT ) Y in + G om κ o κ oT 2 γ m χ m ( ω ) f in + G om κ o κ oT G em i ω κ iT 2 χ m ( ω ) [ κ i ( Q in + S ) + γ i q in ] .
O ( ω ) = κ o N D | X out | 2 + | Y out | 2 .
S F F ( ω ) = n th ( ω m , T eff ) ,
S q q ( ω ) = n th ( ω LC , T ) ,
n th ( ω , T ) = k B T ω .
S oo ( ω ) = κ o N D [ ( ( κ o κ oT ) 2 + ( 1 κ o κ oT ) 2 ) ( 2 S X X ( ω ) + 2 S Y Y ( ω ) ) + C om κ o κ oT 2 γ m 2 | χ m ( ω ) | 2 4 S F F ( ω ) + C om κ o κ oT C em ( ω ) γ m 2 | χ m ( ω ) | 2 × [ 4 S q q ( ω ) + κ i κ iT 4 S 2 δ ( ω ω m ) ] ] .
C om = G om 2 γ m κ oT ,
C em ( ω ) = 4 G em 2 γ m κ iT κ iT 2 4 ω 2 + κ iT 2 .
S 2 κ iT κ i ( S X X ( ω m ) + S Y Y ( ω m ) 2 C om κ o κ oT C em ( ω m ) + 2 S F F ( ω m ) C em ( ω m ) + S q q ( ω m ) ) Δ ,
L ( ω ) = δ P ω 2 + δ P 2 4 ,
L ( ω ) P D ω D ,
S N = S 2 T 2 * 2 ( γ m T 2 * 2 ) κ iT κ i S X X ( ω m ) + S Y Y ( ω m ) 2 C om κ o κ oT C em ( ω m ) + κ iT κ i 2 S F F ( ω m ) C em ( ω m ) + κ iT κ i S q q ( ω m ) + η p P D ω D γ m 0.12
η p = ω m Δ 2 ω m + Δ 2 d ω 2 π γ m 2 | χ m ( ω ) | 2 L ( ω ) ,

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