Nanoscale Fourier-transform MRI of spin noise

Nanoscale Fourier-transform MRI of spin noise John M. Nichol1, Tyler R. Naibert1, Eric R. Hemesath2, Lincoln J. Lauhon2, and Raffi Budakian1 1 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 2 Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208 ABSTRACT We report a method for Fourier-transform magnetic resonance imaging of statistically polarized nanoscale samples. Periodic magnetic field gradient pulses, which are generated by ultrahigh current densities in a nanoscale metal constriction, spatially encode the spin density in the sample and create temporal correlations in the spin noise. We demonstrate this technique using a silicon nanowire mechanical oscillator as a force sensor to image 1H spins in polystyrene. We obtain a two-dimensional projection of the sample proton density with approximately 10-nm resolution. Fourier encoding offers better sensitivity than point-by-point techniques for high resolution imaging of statistically polarized samples. Magnetic resonance imaging (MRI) is an invaluable tool for three-dimensional biological and materials imaging. The noninvasive tomographic capabilities of MRI together with the techniques of nuclear magnetic resonance spectroscopy offer unique ways to elucidate the physical and chemical structure of objects, usually at millimeter length scales and above. 1 Continued efforts to extend the sensitivity and resolution of MRI to the nanometer scale and below are motivated by the desire to investigate materials on the atomic scale using the same techniques that have proved so powerful in biology and medicine. Promising candidates for nanoscale MRI include force-detected magnetic resonance, which has been used to perform three-dimensional imaging of single tobacco mosaic virus particles with spatial resolution below 10 nm [1], and nitrogen-vacancy-based magnetic resonance, which has been used to detect proton resonance in volumes as small as (5 nm)3 [2, 3]. Despite remarkable progress, nanoscale MRI remains challenging because the relatively weak nuclear magnetic moment makes achieving a high signal-to-noise ratio (SNR) difficult [4]. Therefore, it seems worthwhile to consider spatial encoding techniques that are used in conventional MRI, such as Fourier encoding [5, 6], for nanoscale imaging. Such methods offer better sensitivity [7] than sequential-point imaging by acquiring signal from all voxels in the sample simultaneously. It is difficult, however, to adapt these and most spectroscopic techniques to nanoscale objects for two primary reasons. First, precise spin manipulation pulses require homogeneous magnetic fields, and Fourier encoding requires pulsed gradients. In contrast, most nanoscale MRI techniques use static gradients [1, 8], making both global spin manipulation and Fourier-encoding difficult. For thermally-polarized samples on the micron scale and above, this challenge has been met in part by moving the gradient source to enable encoding [9-11] or by employing uniform-field spin-detection protocols [12]. Second, pulsed magnetic resonance techniques cannot be used per se because statistical fluctuations can equal or exceed the thermal spin polarization in nanoscale objects [13, 14]. To overcome this challenge, back-projection encoding without rf pulses has been used for millimeter-scale imaging of a liquid sample [15], and spectroscopic approaches that correlate the statistical polarization before and after an 2 encoding pulse have been proposed [16, 17] and implemented [13, 18] for micron-scale statistically polarized samples. In the following, we describe a method that overcomes both challenges to enable Fouriertransform imaging of nanometer-size statistically polarized samples. We use a nanoscale metal constriction, which can support current densities greater than 3 × 108 Acm-2 , to generate intense, periodic gradient pulses that create correlations in the statistical polarization, or spin noise, of a solid organic sample. The correlations are recorded for a set of pulse configurations and Fourier transformed to give the spin density. We demonstrate the method using a silicon nanowire force sensor to reconstruct a two-dimensional projection image of the 1H density in a polystyrene sample with roughly 10-nm resolution. We also show that Fourier encoding enhances SNR for high-resolution imaging. Figure 1(a) shows a schematic of the apparatus. A key element of the experiment is an ultrasensitive silicon nanowire force transducer [19], which vibrates in response to the force exerted by nuclear spins on resonance in the presence of a magnetic field gradient. The nanowire used in this study was grown epitaxially on a Si[111] substrate using a controlled-diameter vapor-liquid-solid approach with silane as a precursor at 600 °C [20]. The nanowire was roughly 15 µm long, with a tip diameter of 50 nm. The fundamental flexural mode had a spring constant k = 150 µN/m, a resonance frequency ω0/2π = 333 kHz, and an intrinsic quality factor Q = 1.8 × 104 at a temperature of approximately 6 K. The displacement of the nanowire was measured using a polarized fiber-optic interferometer [19, 21]. The sample consisted of a thin polystyrene coating on the tip of the nanowire (Figure 1b). A second important element of the experiment is a constriction in an Ag current-carrying wire (Figure 1c). The constriction focuses current passing through the wire to densities 3 exceeding 3×108 Acm-2. Such locally intense current densities are used to generate large timedependent magnetic field gradients that couple spins in the sample to the resonant displacement of the nanowire, rf magnetic fields to excite magnetic resonance, and pulsed gradients for imaging [21]. The constriction was patterned by electron beam lithography on an MgO (100) substrate, and the Ag film was deposited in an ultra-high vacuum electron beam evaporator (see Supporting Information). The constriction used in this study was 240 nm wide and 100 nm thick. Both the nanowire substrate and constriction were cooled to 4.2 K in high vacuum, and the sample was positioned directly above the center of the constriction. A small solenoid provided a uniform static field B0 = 181 mT along the z direction. We used the MAGGIC spin detection protocol [21] to measure the longitudinal component of the 1H spins in the sample near the constriction. Fourier-transform imaging generally involves measuring the transverse component of the sample magnetization as it evolves in the presence of an external field gradient. To accomplish this using the MAGGIC protocol, we used an encoding pulse sequence, which creates a transverse component of the magnetization and projects its evolution onto the longitudinal axis. We inserted this sequence every τp (Figure 2a) in the MAGGIC protocol. Provided τp<< τm, where τm is the statistical spin correlation time, the encoding sequence creates measurable correlations in the spin noise. To verify this procedure, we measured the free precession of the sample magnetization in a uniform field. The pulse sequence consists of an adiabatic halfpassage (AHP) [22], an evolution period te, and a time-reversed AHP. The first AHP rotates the spins away from the z axis onto the xy plane. During the period te, the spins precess about B0 . The second AHP, which is phase-shifted by φ ( te ) = −γ B0te relative to the first AHP, projects the magnetization back onto the z axis. The time-dependent phase shift creates a longitudinal 4 projection that oscillates at the Larmor frequency as te varies. This free-precession sequence is related to a Ramsey-fringe measurement [23]. Since the statistical polarization fluctuates randomly, the encoding sequence has no impact on the mean or variance of the polarization. However, we may still infer the effect of the pulses by measuring correlations in the force signal. Specifically, we measure the (nonnormalized) autocorrelation R ff (τ p , te ) of the force signal f ( t ) at lag τ p : R ff (τ p , te ) = T 1 dt f ( t ) f ( t − τ p ) . T ∫0 (1) Here, T is the measurement time, and the force signal implicitly depends on te . In the Supporting Information, we show that the time-averaged value of the autocorrelation is R ff (τ p , te ) = e−τ p τm µ 2 D2 2 ∫ drρ ( r )G 2 ( r ) M pulse (τ p , te , r ) . (2) Here, µ is the spin magnetic moment, D is the duty cycle of the MAGGIC gradient modulation [21], ρ ( r ) is the spin density, G ( r ) = dBz ( r ) dx is the gradient modulation strength, and pulse pulse M pulse (τ p , te , r ) = 1 − 2 Pflip ( te , r ) describes the effect of the encoding pulses. Pflip ( te , r ) is the probability for a spin located at r to have reversed its orientation after a single encoding sequence. The exponential factor in eq. (2) expresses the requirement that τ p should be much less than τ m to prevent statistical fluctuations from destroying the desired correlations. For the free-precession sequence, M pulse (τ p , te , r ) = Ev ( te ) cos ( γ B0te ) , where γ 2π = 42.6 MHz/T is the proton gyromagnetic ratio. The free precession will decay with envelope Ev ( te ) due to the fluctuating local fields experienced by the spins. Hence, 5 R ff (τ p , te ) = e −τ p τm µ 2 D2 2 Ev ( te ) cos ( γ B0te ) ∫ drρ ( r )G 2 ( r ) . (3) By sweeping te , we measured the Larmor precession of the statistical polarization (Figure 2b). We found the decay envelope to be well described by a Gaussian: Ev ( te ) = e ( − te T2* ) 2 with T2* = 14 µs (Figure 2c). The non-exponential envelope is characteristic of free precession in solids [24], and the decay time is consistent with previous measurements in polystyrene [25]. By cosine-transforming the free-precession data, we obtain the nuclear magnetic resonance spectrum of our statistically polarized sample (Figure 2c). The essential feature of the free-precession encoding is the use of repeated, identical pulse sequences to induce correlations in the spin noise. Such a paradigm permits the use of established magnetic resonance techniques not only for spectroscopy, but also for imaging of statistically polarized samples. The most common MRI technique, Fourier encoding, uses a pulsed gradient during the free precession to encode the location of a spin in either the phase or frequency of its Larmor precession. With the use of the constriction, which enables the generation of intense, pulsed gradients, this technique can be easily adapted for nanoscale imaging. DC current through the constriction produces a strong gradient in the x direction of the total field Btot ( r ) and a spatially varying Larmor frequency ωLarmor ( r ) = γ Btot ( r ) . Additionally, rf current through the constriction at frequency γ B0 produces a field in the rotating frame B1 ( r ) = Bx ( r ) 2 that excites magnetic resonance. Directly above the center of the constriction, B1 ( r ) , and hence the Rabi frequency ωRabi ( r ) = γ B1 ( r ) , vary strongly with distance above the constriction. With these two independent gradients, the constriction enables spatial encoding in 6 two dimensions. To simplify notation, we introduce the generalized coordinates u ( r ) ≡ ωRabi ( r ) and v ( r ) ≡ ωLarmor ( r ) . Figure 1a shows constant u and v contours above the constriction. These contours form the coordinate system in which the spin density will be encoded. Because dv dy << dv dx and du dy << du dz over the dimensions of the sample for fixed x and z, we make the reasonable assumptions that u ( r ) = u ( x, z ) and v ( r ) = v ( x, z ) for the purposes of imaging. The imaging sequence (Figure 3a) is composed of two parts. The first is similar to the free-precession sequence except that a static gradient pulse of length tv is applied during the evolution period to encode the spin density along v contours. The second part consists of an rf pulse of length tu with center frequency γ B0 , which nutates the spins by an angle γ B1 ( r ) tu and encodes them along u contours. For this sequence, M pulse (τ p , tu , tv , r ) = Eu ( tu ) Ev ( tv ) cos ( u ( r ) tu ) cos ( v ( r ) tv ) , where Eu ( tu ) describes the transverse spin relaxation in the rotating frame [26], Hence, R ff (τ p , tu , tv ) = e−τ p τm E u ( tu ) Ev ( tv ) µ 2 D 2 2 ∫ dudv p ( u, v ) cos ( utu ) cos ( vtv ) , where p ( u , v ) = G 2 ( u , v ) J ( u , v ) ∫ dy ρ ( y, u , v ) is the projected signal density in the (4) ( u, v ) coordinate system, and J ( u, v ) is the Jacobian of the ( x, z ) → ( u, v ) coordinate transformation. We have also assumed that G ( r ) = dBz ( x, z ) dx , since the readout gradient does not vary significantly with y for fixed x and z in the sample. To record an image, the sample was positioned 40 nm above the surface of the constriction. The pulse interval was τ p = 11 ms, and the spin correlation time was τ m ≈ 400 ms . 7 R ff (τ p , tu , tv ) was measured for 305 different ( tu , tv ) configurations (Figure 3b). The u encoding pulse was stepped in increments of 0.26 µs up to Tu = 5.2 µs, and the v encoding pulse was stepped in increments of 0.625 µs up to Tv = 10.6 µs. The data were zero-padded by a factor of 4 in each dimension. A discrete cosine transform (see Supporting Information) was used to recover the signal density p ( u, v ) (Figure 3c). To suppress noise, all negative values of p ( u, v ) were set to zero. To obtain the ( x, z ) -space representation of the image (Figure 3d), p ( u, v ) was divided by G2 ( u, v ) J ( u, v ) , and the coordinates were transformed from ( u , v ) to ( x, z ) . The magnetic field distribution from the constriction was calculated using COMSOL Multiphysics (COMSOL, Inc.) and was used to obtain G ( u, v ) , J ( u, v ) , and the ( u , v ) ↔ ( x, z ) coordinate transformation. The reconstructed spin density strongly resembles the expected shape of the polystyrene coating (Figure 1b). The nanowire and the gold catalyst particle are clearly visible through the polystyrene in the reconstructed image as a reduction in the spin density. Figures 3e and 3f show a simulated signal density and reconstructed image, respectively. To generate the simulations, R ff (τ p , tu , tv ) was calculated for each ( tu , tv ) point using the calculated magnetic field distribution and the profile of a nanowire tip and polystyrene coating (Figure 1b) prepared in the same fashion as the tip and sample used in this study. The actual tip and sample used here were not imaged in a scanning electron microscope to avoid electron beam damage. The simulated data were inverted and transformed in the same manner as the experimental data. Both simulations appear qualitatively similar to the actual data and image. In order to determine the spatial resolution, we simulated the image from a point source located near the tip of the sample and find the resolution in the x and z directions to be 8 approximately 10 nm and 15 nm, respectively. The voxel size increases with increasing distance from the constriction because the magnetic field gradients are strongest near the constriction. We estimate the peak imaging gradients in this study to be u z ,max γ ≈ 2.0 × 105 Tm-1 for 56 mA of current through the constriction, and vx ,max γ ≈ 1.4 × 105 Tm-1 for 20 mA of current [27]. Here, vx,max = dv dx max and u z ,max = du dz max are the maximum gradients experienced by the sample. Such gradients are more than 104 times stronger than the highest gradients used in inductively-detected MRI [28]. During the readout, the peak gradient was approximately 5.0 × 105 Tm-1 for 71 mA of current through the constriction, corresponding to a current density of 3.0 × 108 Acm-2. The most notable feature of the constriction is its ability to sustain ultrahigh current densities comparable to the best values obtained with carbon nanotubes [29], metal-silicide nanowires [30], and other metal nanostructures [31]. Such high current densities permit the generation of intense, pulsed gradients, which are necessary for nanoscale solid-state Fouriertransform imaging. Both the cryogenic operating temperature and ac nature of the current in the constriction likely contribute to its ability to survive such high current densities [32]. For thermally polarized samples, Fourier encoding offers better sensitivity by a factor of N , where N is the total number of points in the image, over sequential-point methods by acquiring signal from the entire sample at all times [7]. For such samples, the dominant noise source is voltage noise in the receiver circuitry for inductively-detected MRI or oscillator force noise in the case of force-detected MRI. When the sample is statistically polarized, however, the spin noise [33] also contributes to the total noise, and a new analysis is required. In the following, we show that Fourier-transform imaging of statistically polarized samples improves SNR for high resolution imaging when force noise becomes the limiting factor. 9 In the Supporting Information, we show that, in d dimensions, the average SNR of an image acquired via Fourier encoding is SNRFourier ≈ e −τ p τ m  2d NAτ p 2d NS 2f 2d NS f + +  4 2 T 4Tτ pσ spin T σ spin     −1 2 . (5) 2 is the variance of the spin-component of the force signal from the entire sample, S is Here, σ spin f the oscillator force noise power spectral density, T is the averaging time per point, and A , which is approximately 2 for the present experiment, characterizes the average error in the autocorrelation integrated over the sample. Spin relaxation effects have been neglected in the above estimate. For comparison, the average SNR of an image in which each voxel is measured sequentially is SNR point  2τ m N 2 S 2f 2 NS f ≈ + + 4 2  T Tτ mσ spin T σ spin     −1 2 . (6) Equation (6) assumes that the signal in each voxel is the same (see Supporting Information). While the resonant-slice imaging techniques used in previous force-detected MRI experiments [1, 34] are not, strictly speaking, point-by-point methods, they do involve sequentially scanning the sample with respect to a magnetic tip. Furthermore, achieving high spatial resolution with resonant-slice techniques necessitates acquiring signal from small volumes in the sample, as opposed to spatial encoding techniques, which acquire signal from the entire sample at all times. Because SNR in resonant-slice imaging is difficult to calculate, however, we shall compare sequential-point imaging to Fourier encoding in order to illustrate its benefits. In both equations (5) and (6), the first term in the parentheses represents the spin noise, the second term represents the oscillator force noise, and the last term is the covariance of the 10 force noise and the spin noise. The main difference between (5) and (6) is how the different sources of noise scale with the number of points N . For sequential-point imaging, the spin noise power is independent of N , but the force noise power scales as N 2 . For Fourier encoding, the spin noise scales less favorably with N because spin noise from all voxels contributes to every data point. However, the force noise contribution scales more favorably with N because all voxels in the image are measured N times, compared with only once in the sequential-point case. When N is large enough that the force noise significantly exceeds the spin noise per 2 >> 1 , then SNR voxel, i.e., when NS f 2τ mσ spin point ∝ 1 N . Because SNRFourier ∝ 1 N, Fourier encoding can be expected to offer better sensitivity than sequential point imaging in this 2 2 ≈ 300 aN2, S ≈10 aN2Hz-1, and NS case. In the present experiment, σ spin f f 2τ mσ spin ~ 13. For example, consider a hypothetical (20 nm)3 biological sample with a spin density of 4.9 ×1028 m-3 experiencing a uniform gradient of 6 × 106 Tm-1 [35], with S f = 4 aN2Hz-1 [21], τ m = 20 ms [1], and τ p = 10 ms. Under these conditions, the two imaging methods yield approximately the same SNR for a voxel size of (3 nm)3. However, for a (1 nm)3 voxel size, SNRFourier SNR point ≈ 5, and for a (.5 nm)3 voxel size, SNRFourier SNR point ≈ 15. The sensitivity enhancement for small voxel sizes suggests that Fourier encoding is advantageous when force noise makes highresolution imaging difficult. In this work we have demonstrated nanoscale Fourier-transform MRI by creating correlations in the spin noise of a nanometer-scale sample via pulsed gradients generated by ultrahigh current densities in a metal constriction. We have further shown that Fourier encoding can enhance sensitivity for high-resolution imaging. This technique could be readily extended to enable full three-dimensional encoding with constrictions capable of producing two orthogonal 11 static gradients. A small coil could also be used to generate a uniform B1 in the sample, which would enable the use of solid-state line-narrowing pulses for high-resolution spectroscopy as well as longer encoding times and enhanced spatial resolution. More generally, our approach serves as a model for leveraging these and other sophisticated magnetic resonance tools to aid nanoscale MRI in its progress toward atomic-scale imaging. ACKNOWLEDGEMENTS This work was supported by the Army Research Office through grant No. W911NF-12-1-0341 and by the Department of Physics and the Frederick Seitz Materials Research Laboratory Central Facilities at the University of Illinois. Work at Northwestern University was supported by the National Science Foundation Grant Nos. DMI-0507053 (E.R.H) and DMR-1006069 (L.J.L.). 12 REFERENCES [1] C. L. Degen, M. 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During imaging, the spin density was encoded along contours of constant Larmor and Rabi frequency, which are illustrated as blue and green lines, respectively. (b) Scanning electron micrograph of a representative nanowire and polystyrene coating prepared in the same manner as the nanowire and sample used in this study. The dashed lines indicate the outer diameter of the nanowire. (c) Scanning electron micrograph of the constriction used in this study. The constriction is 100 nm thick and 240 nm wide. 15 Figure 2. Measurement of free precession in a statistically polarized sample. (a) Periodic encoding pulses were inserted in the MAGICC protocol every τp. To measure the free precession, the encoding sequence consisted of two adiabatic half passages (AHP) separated by an evolution period te . The first AHP rotates the spins onto the xy plane, where they precess for a time te , and the second AHP projects the magnetization back along the z axis. (b) Autocorrelation of the force signal R ff (τ p , te ) showing free precession of the statistical spin polarization and fit to a cosine. (c) Amplitude of the free precession and fit to a Gaussian. From the fit, we infer that T2* = 14 µs. Inset: nuclear magnetic resonance spectrum of the statistically polarized sample. 16 Figure 3. Two-dimensional MRI of the polystyrene sample. (a) Image acquisition encoding sequence. Free precession in the presence of a gradient for a time tv permits encoding along v . In the u -encoding step, an rf pulse lasting a time tu nutates the spins about the effective field in the rotating frame. (b) Raw data. Cross-sections corresponding to R ff (τ p ,0, tv ) and R ff (τ p , tu ,0) are shown. (c) Signal density in the ( u, v ) coordinate system obtained by cosine-transforming the raw data. The arrow indicates the position of v = γ B0 . (d) Real-space reconstruction of the projected spin density. The nanowire and gold catalyst are clearly visible through the polystyrene in the image as a reduction in the spin density. The cross-sections above and to the right of the image are taken along the lines indicated by the arrows. (e) Simulated signal density in ( u, v ) space calculated for the sample and nanowire geometry shown in Figure 1b. (f) Real-space reconstruction of the simulation in (e). 17 Supporting Information Nanoscale Fourier-transform MRI of spin noise John M. Nichol1, Tyler R. Naibert1, Eric R. Hemesath2, Lincoln J. Lauhon2, and Raffi Budakian1 1 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 2 Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208 1. FABRICATION OF THE CONSTRICTION The constriction was fabricated on a single-crystal MgO (100) substrate using a liftoff process with electron beam lithography and an MMA/PMMA resist bilayer. AquaSAVE (Mitsubishi Rayon Co., Ltd.) was used as a conductive layer on top of the resist. Ag was deposited via ultra-high vacuum electron beam evaporation. The MgO surface, because it is hygroscopic, was cleaned briefly by argon ion milling in situ before deposition. Conventional photoresist delaminated during aqueous development, so the contact pads and leads were defined using deep ultraviolet optical lithography and an MMA/PMMA resist bilayer. The same deposition and liftoff procedure used for the constriction was used for the pads and leads. Thin copper wires were gap welded to the pads for good electrical contact. The substrate was diced and polished to ensure that the constriction was within 30 µm of the substrate edge to allow an unobstructed optical path between the nanowire and the optical fiber. 1 2. DERIVATION OF THE TIME-AVERAGED AUTOCORRELATION We derive equation (2) in the main text. Consider a set of N spins nuclear spins constituting the sample, which evolve in time under the influence of statistical fluctuations and encoding pulses. The output of our lock-in amplifier is the (time-dependent) rms amplitude of the force exerted on the nanowire by the spins: f (t ) = D 2 N spins ∑ i =1 µGi mi ( t ) . (S1) Here, µ is the spin magnetic moment, Gi is the peak gradient experienced by the ith spin, D is the duty cycle of the MAGGIC readout [1], and mi ( t ) is a variable taking on the values ±1 describing the orientation of the ith spin as it evolves in time. The time-averaged autocorrelation of the force signal at lag τ p is R ff (τ p ) = limT →∞ T 1 dt f ( t ) f ( t − τ p ) T ∫0 T = limT →∞ = limT →∞ = limT →∞ D2 dt 2T ∫0 N spins N spins ∑ ∑ i =1 D2 2 N spins N spins D2 2 N spins N spins ∑ ∑ i =1 µ 2Gi G j mi ( t ) m j ( t − τ p ) µ 2Gi G j j =1 ∑ ∑ i =1 j =1 T 1 dt mi ( t ) m j ( t − τ p ) T ∫0 µ 2Gi G jδ i , j j =1 D2 = 2 Nspins ≈ µ2 D2 drρ ( r )G 2 ( r ) Rmm (τ p , r ) 2 ∫ ∑ i =1 T 1 dt mi ( t ) mi ( t − τ p ) T ∫0 µ 2Gi2 Rmm,i (τ p ) . (S2) 2 The 4th equality follows because statistical flips between different spins are independent. In the last equality, we have passed into the continuum limit, where ρ ( r ) is the spin number density, and G ( r ) is the peak value of the readout gradient. Note that T 1 Rmm (τ p , r ) = limT →∞ ∫ dt m ( r, t ) m ( r, t − τ p ) T0 = m ( r, t ) m ( r, t − τ p ) , (S3) = P+ ( r ) − P− ( r ) where P+ is the probability that m ( r, t ) = m ( r, t + τ p ) , and P− is the probability that m ( r , t ) = − m ( r, t + τ p ) . Note also pulse pulse pulse P+ ( r ) − P− ( r ) = Peven Pnopulse flip ( r ) + Podd Pflip ( r ) − Podd Pno flip ( r ) − Peven Pflip ( r ) , (S4) where Peven is the probability for an even number of statistical flips during τ p , Podd is the pulse probability for an odd number of statistical flips, Pflip ( r ) is the probability for the spin to have reversed its orientation after a pulse, and Pnopulse flip ( r ) is the probability for no change in pulse orientation after the pulse. Since Pnopulse flip ( r ) = 1 − Pflip ( r ) , ( ) pulse P+ ( r ) − P− ( r ) = ( Peven − Podd ) 1 − 2 Pflip (r ) . Assuming that the statistical fluctuations obey Poisson statistics [2], Peven = (1 + e−τ p Podd = (1 − e−τ p τm ) (S5) τm ) 2 , and 2 . Hence, Rmm (τ p , r ) = e−τ p τm M pulse (τ p , r ) , (S6) 3 pulse where M pulse (τ p , r ) = 1 − 2 Pflip ( r ) describes the effect of a single encoding pulse on the spin at pulse location r . M pulse (τ p , r ) depends on any parameters that affect Pflip ( r ) , such as the pulse length, for example. 3. IMAGE RECONSTRUCTION Here we describe the mathematical image reconstruction procedure for a twodimensional data set. The generalization to other dimensions is straightforward. Recall from the main text that R ff (τ p , tu , tv ) = ∫ dudv p ( u , v ) cos ( utu ) cos ( vtv ) . (We have here neglected the effects of relaxation, which will be considered in a later section.) By incrementing the pulse lengths tu = ku ∆tu with ku = 0,1,⋯, Nu − 1 , and tv = kv ∆tv with kv = 0,1,⋯, Nv − 1 , we may record a series Rk = R ff (τ p , ku ∆tu , kv ∆tv ) , where k = (ku , kv ) . A discrete cosine transformation (DCT) may be applied to the data to recover the spin density. Setting umax = π vmax = π ∆tv ∆tu and , the appropriate transformation is the DCT-I [3]: 2 2 Rɶn = umax vmax Nu −1 Nv −1 πn k  πn k  ∑ ∑ Rk w ( k ) cos  Nuu−u1  cos  Nv v−v1  , (S7) ku =0 kv =0 where n = ( nu , nv ) , and the weighting function is 1 2 ku = 0 or ku = N u − 1 w (k ) =  1 otherwise   1 2 kv = 0 or kv = N v − 1 ×  1 otherwise    .  (S8) n  n  As Nu , Nv → ∞ , Rɶn → p  u umax , v vmax  . Having obtained p ( u , v ) by this method, Nv − 1  Nu − 1  the spin density may be recovered as discussed in the main text. 4 4. ERROR IN MEASUREMENT OF THE AUTOCORRELATION In practice, the measured value of the autocorrelation is the expected value plus random noise: Rk = Rk + rk . Noise in the raw data produces noise in the image: Rɶn = Rɶn + rɶn . Here we will compute the value of rk , and in the next section we will compute the average value of rɶn . We assume that the force signal f ( t ) is digitized much more rapidly than τ p and that all samples between consecutive pulses are averaged together. The signal is thus filtered synchronously with the encoding using a convolution filter [4] with time constant τ p . Hence, a continuous record of the measured signal sk ( t ) = fk ( t ) + n ( t ) containing the desired force signal fk ( t ) and random instrumentation noise n ( t ) spanning a time T becomes a discrete set of points sk ,i = fk ,i + ni , where i = 0,1,⋯ , N pts − 1 , and N pts = T τ p . We have retained the index k to indicate the encoding pulse configuration, and the index i is associated with a point in time. The autocorrelation is: 1 Rk = N pts N pts −1 ∑ i =0 1 sk ,i sk ,i +1 = N pts 1 Only the first term N pts N pts −1 ∑ fk ,i fk ,i +1 + ni ni +1 + fk ,i ni +1 + ni fk ,i +1 . (S9) i =0 N pts −1 ∑ fk ,i fk ,i +1 has a non-vanishing average value. However, all i =0 of the terms have a non-zero variance. It can be shown, using the results of Degen et al. [4], and Bartlett [5] [6], that 5  1 var   N pts  N pts −1 ∑ i =0  var ( ni ) var ( f k ,i ) ni fk ,i +1  =  N pts  = =  1 var   N pts  N pts −1 ∑ i =0 (S f 2 2τ p ) σ spin T τp , (S10) 2 S f σ spin 2T  var ( ni )2 ni ni +1  =  N pts  (S f = 2τ p ) 2 T τp , (S11) S 2f = 4Tτ p and  1 var   N pts  N pts −1 ∑ i =0  var ( fk ,i )2 τ p fk ,i fk ,i +1  =  T  2 (σ spin ) = 2 τp T = Ak ∞ ∑ j =−∞ ∞ ∑ j =−∞ ρk2, j + ρk , j −1ρk , j +1 ρk2, j + ρk , j −1ρk , j +1 , (S12) 4 τ σ spin p T where var (⋯) indicates the variance of the quantity in parentheses, and vmax 2 = ρk , j = R ff ( jτ p , ku ∆tu , kv ∆tv ) σ spin ∫ 0 umax dv ∫ 0 πk u  πk v  du p ( u , v ) cos j  u  cos j  v  e− jτ p  umax   vmax  vmax ∫ 0 umax dv ∫ τm is du p ( u , v ) 0 the time-averaged normalized autocorrelation of the force signal at lag jτ p . The quantity 6 ∞ Ak = ∑ j =−∞ ρk2, j + ρk , j −1ρk , j +1 characterizes the average error in the non-normalized autocorrelation [6]. Putting it all together: rk2 5. = Ak 4 τ σ spin p T + 2 S f σ spin T + S 2f 4Tτ p . (S13) IMAGE SIGNAL TO NOISE The orthogonality relation for the DCT-I is Nu −1 Nv −1 πk n   π k′ n  πk n   π k′n  ∑ ∑ a 2 ( n ) cos  Nuu−u1  cos  Nuu−u1  cos  Nvv−v1  cos  Nvv−v1  nu =0 nv =0 2 ku = 0 or N u − 1 2 kv = 0 or N v − 1 N − 1 Nv − 1 δ k ,k ′ ×  = u ×   2 2  1 otherwise   1 otherwise  , (S14) where 1 a (n ) =   2 nu = 0 or nu = N u − 1  1 ×  1 otherwise   2 nv = 0 or nv = N v − 1  . 1 otherwise  (S15) Making use of Parseval’s Theorem for the DCT-I, 1 Nu −1 Nv −1 ∑ ( Nu − 1)( N v − 1) n∑ u =0 nv =0 a 2 ( n ) rɶn2 = 2 2 2 2 umax vmax Nu −1 Nv −1 ∑ ∑ w ( k ) rk2 ku =0 kv =0 4 τ 2 S 2f S f σ spin 2 ( N u − 1) 2 ( N v − 1)  σ spin p A = + +  2 2 4Tτ p umax vmax T T  Here A =    .(S16) 1 1 Nu −1 Nv −1 ∑ ∑ w ( k )Ak . Note that w ( k ) appears only to the first power to N u − 1 N v − 1 ku =0 kv =0 account for the normalization factor in eq. (S14). For the current experiment, we estimate that A ≈ 2. 7 A useful quantity to calculate is the average SNR in the image, which we define as the average signal divided by the rms noise. The average variance in the image is given by eq. (S16), and the average signal is R0 = e−τ p SNRFourier = e −τ p τ m ≈ e −τ p τm τ mσ 2 ( Nu − 1) spin . −1 2 Hence, the average SNR in d dimensions is: ( N v − 1) −1 2  2d S 2f 2d Aτ p 2d S f + +  2 4 T σ spin 4Tτ pσ spin  T  2d NAτ p 2d NS f + +  2 σ T T spin  NS 2f 4 4Tτ pσ spin 2d    −1 2    −1 2 , (S17) provided that N u >> 1 and N v >> 1 , and where N = N u N v . For comparison, the SNR of a sequential-point image may be calculated using equation 2 (4) of Degen et al [4]. Assuming that the signal per voxel is σ spin N , where N is the total 4 2  2S 2f 2 S f σ spin  2τ   σ spin number of points, the noise energy in each voxel is  m   + 2 +  . Hence, 2τ m N   T   N 2 4τ m SNR point ≈ −1 2 2 4 σ spin  2τ m   σ spin  =   6.   N  T   2  N S 2f N 2 2τ m + 4 T Tτ mσ spin + 2  2 S f σ spin + 2 +  4τ m 2τ m N  2 S 2f 2S f N   2 T σ spin  −1 2 −1 2 . (S18) SPIN RELAXATION We now discuss the effects of spin relaxation on spatial resolution and SNR. Expanding eq. (S7): 8 2 2 Rɶn = umax vmax = vmax ∫ dv ∫ ∫ 0 umax vmax dv πn k  πn k  ∑ ∑ Rk w ( k ) cos  Nuu−u1  cos  Nv v−v1  ku =0 kv =0 umax 0 = Nu −1 Nv −1 ∫ 4 du p ( u , v ) umax vmax Nu −1 Nv −1 πn k  πk u  πn k  πk v  ∑ ∑ w ( k ) cos  Nuu−u1  cos  umaxu  cos  Nvv−v1  cos  vmaxv  ku =0 kv =0 du p ( u , v )g ( u, v, u′, v′ ) 0 0 ≈ p ( u′, v′ ) (S19) where g ( u, v, u′, v′ ) = u′ = Nu −1 Nv −1  π k u′   π ku u   π kv v′   π kv v   cos  u  cos  v  cos  v ,   max   max   max  u ∑ ∑ w ( k ) cos  umax ku =0 kv =0 nu umax nv , and v′ = v max . Note that g ( u , v, u ′, v′ ) → δ ( u − u ′, v − v′ ) as Nu , Nv → ∞ . Note also Nu − 1 Nv −1 vmax that 4 umax vmax ∫ umax dv′ 0 ∫ du ′ g ( u , v, u ′, v′ ) = 1 . The kernel g ( u, v, u′, v′ ) is strongly peaked about u = u ′ and 0 v = v′ and defines the impulse response of the image transformation and the resulting spatial resolution. With regard to spatial resolution, the effect of spin relaxation is to modify the kernel: g ( u , v, u ′, v′ ) → 4 umax vmax Nu −1 Nv −1  π k u′   π ku u   π kv v′   π kv v   cos  u  cos  v  cos  v ,   max   max   max  u ∑ ∑ w ( k ) E ( k ) cos  umax ku = 0 kv =0 where E ( k ) describes the effect of spin relaxation. In the present experiment E ( k ) is expected to be of the form E ( k ) = e ( − ku ∆tu T2*ρ ) e−( kv∆tv T2* )2 , where T2*ρ is the transverse spin relaxation time in 2 the rotating frame. Provided that E ( 0 ) = 1 , as is usually the case, the average value of the signal density is preserved. The form of E ( k ) affects the shape of g ( u, v, u ′, v′ ) and hence the spatial 9 resolution. In general, E ( k ) → 0 as k → ∞ . The more rapidly E ( k ) decays, the broader g ( u, v, u ′, v′ ) becomes. Spin relaxation affects SNR via Ak → Ak E 2 ( k ) . Although the spatial resolution in the image degrades the more rapidly E ( k ) decays to zero, the SNR can be expected to improve slightly because of the reduced spin noise. REFERENCES [1] J. M. Nichol, E. R. Hemesath, L. J. Lauhon, and R. Budakian, Phys. Rev. B 85, 054414 (2012). [2] W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill Book Company, Inc., New York, 1958). [3] Z. Wang, and B. R. Hunt, in IEEE ICASSPBoston, MA, 1983), pp. 1256. [4] C. L. Degen, M. Poggio, H. J. Mamin, and D. Rugar, Phys. Rev. Lett. 99, 250601 (2007). [5] M. S. Bartlett, J. R. Stat. Soc. 98, 536 (1935). [6] M. S. Bartlett, J. R. Stat. Soc. Ser. B-Stat. Methodol. 8, 27 (1946). 10