Quantum control of proximal spins using nanoscale magnetic resonance imaging
Quantum control of individual spins in condensed matter systems is an emerging field with wide-ranging applications in spintronics Awschalom et al. (2002), quantum computation Nielsen and Chuang (2000), and sensitive magnetometry Chernobrod and Berman (2005). Recent experiments have demonstrated the ability to address and manipulate single electron spins through either optical Jelezko et al. (2004); Xu et al. (2007) or electrical techniques Hanson et al. (2007); Morello et al. (2010); Foletti et al. (2009). However, it is a challenge to extend individual spin control to nanoscale multi-electron systems, as individual spins are often irresolvable with existing methods. Here we demonstrate that coherent individual spin control can be achieved with few-nm resolution for proximal electron spins by performing single-spin magnetic resonance imaging (MRI), which is realized via a scanning magnetic field gradient that is both strong enough to achieve nanometric spatial resolution and sufficiently stable for coherent spin manipulations. We apply this scanning field-gradient MRI technique to electronic spins in nitrogen-vacancy (NV) centers in diamond and achieve nanometric resolution in imaging, characterization, and manipulation of individual spins. For NV centers, our results in individual spin control demonstrate an improvement of nearly two orders of magnitude in spatial resolution compared to conventional optical diffraction-limited techniques. This scanning-field-gradient microscope enables a wide range of applications including materials characterization, spin entanglement, and nanoscale magnetometry.
Magnetic field gradients allow for spatially distinguishing spins in ensembles, as fields locally modify the spins’ resonance frequencies. Spatially separated spins can therefore be addressed selectively, allowing for magnetic resonance imaging (MRI), which has revolutionized the medical and biological sciences by yielding micron-scale imaging of nuclear spins Mansfield (2005); Lee et al. (2001). Performing MRI on single spins with high spatial resolution is attractive both for determining structure on the molecular scale and for achieving individual spin quantum control in ensemble systems. With conventional MRI techniques, however, it is difficult to improve the spatial resolution to the nanoscale due to insufficient readout-sensitivity and inadequate magnetic field gradients Glover and Mansfield (2002). Recently, magnetic field gradients introduced via scanning probe techniques have enabled single spin detection with few-nm resolution Rugar et al. (2004); Balasubramanian et al. (2008); however, control and characterization of individual spins in nanoscale clusters has not been demonstrated thus far.
Here we perform scanning field-gradient MRI on proximal electron spins in nanoscale ensembles and demonstrate a spatial resolution nm under ambient conditions. We show that scanning field-gradient microscopy not only allows for imaging but further provides quantum spin control for characterization and manipulation of individual spins on the nanoscale. By pushing the spatial resolution to few-nm length scales, our results illustrate that quantum control of individual spins can be maintained in dense ensembles of spins, where the mutual coupling between adjacent spins can become very strong Neumann et al. (2010). Thus, scanning-field-gradient MRI will help facilitate the creation of entangled spin-states with applications for quantum information processing and sensitive, nanoscale magnetometry. While our approach is applicable to any spin system where spins can be initialized and read out, we focus here on the electronic spins associated with NV centers in diamond, where spin initialization and readout can be performed optically Gruber et al. (1997). Additionally, NV spins are attractive for performing quantum information processing Neumann et al. (2010); Dutt et al. (2007) and sensitive magnetometry Balasubramanian et al. (2008); Maze et al. (2007); Taylor et al. (2008); Degen (2008). Individual NV spin control in a nanoscale ensemble is a key advance towards the implementation of these applications.
Our scanning-field-gradient MRI system (Figure 1a) is comprised of an atomic force microscope (AFM) outfitted with a magnetic tip and integrated into an optical confocal microscope, all operating under ambient conditions. Small ensembles of shallowly implanted NV centers, nm below the surface of a diamond sample, are placed in the confocal spot (volume cubic micron) where an excitation laser at nm is used to initialize and read out the NV centers’ spin states. Because of the NV centers’ spin-dependent fluorescence (nm), optically detected electron spin resonance (ESR) can be observed by sweeping the frequency of a driving radio-frequency (RF) field through the spin resonance and measuring the corresponding variation in fluorescence (Figure 1b) Gruber et al. (1997), in this case on a single NV center. As has been demonstrated by Balasubramanian et al. Balasubramanian et al. (2008), a magnetic tip in proximity to an NV center shifts the energy of the NV spins, particularly for fields along the NV axis. By selectively detuning the applied RF field and scanning the magnetic tip over a single NV center, a magnetic field map corresponding to the applied detuning can be acquired (Figure 1c). Using the AFM to regulate the magnetic tip’s height allows for precise nanoscale control of the tip’s location in three dimensions: the height of the tip is maintained via a conventional feedback loop on the force from the sample, while several-nm precision in lateral positioning is achieved by using the sample’s topography as a marker.
We first demonstrate how this technique can be used for imaging proximal NV spins. The magnetic response shown in Figure 1c provides a direct means for determining the relative location of proximal NV centers. For NVs with an identical orientation, the indistinguishable nature of their spins leads to an identical but spatially shifted magnetic field map for each NV spin. Thus, magnetic resonance images of single-spins (such as in Figure 1c) can serve as the point spread function for multi-spin imaging. To demonstrate the performance of this technique, we executed scanning-field-gradient imaging on three closely spaced NV centers (Figure 2). As the magnetic tip is scanned laterally across the sample, a magnetic field contour is observed for each center. Here, we selected an ensemble of NVs where all three have a common quantization axis orientation, so that every spin responds in the same way to the presence of the magnetic tip. Thus, the relative distances between NV spins can be obtained by quantifying the spatial shift between the observed resonance rings through a deconvolution procedure. The resulting image (Figure 2a) indicates the relative positions of the NV centers, which we find are spaced by nm and nm with respect to NV II. A spatial resolution of nm can be extracted from the width of the resonance ring along the vector connecting NV centers II and III; a precision of nm can be determined from a two-dimensional Gaussian fit to the deconvolved peak. This estimate of precision only accounts for variance induced by random noise and does not account for systematic deviations caused by effects such as scanner non-linearity or deviations from the peak shape to the Gaussian fit.
Our single-spin MRI technique can be extended to three dimensions by performing magnetic tomography, where the magnetic tip is retracted from the surface in few-nm steps and scanned laterally (Figure 2b). The size of the resonance rings evolves quickly as a function of z-distance as the magnetic field from the tip becomes too weak to bring the spin-transitions into magnetic resonance with the detuned RF driving field. The height differences between the NVs can be seen here as NV I vanishes roughly nm before the other two, which indicates that NV I lies roughly nm further below the surface than both NV II and III. A vertical scan across NVs II and II shows the z-resolution of this measurement to be nm, which is extracted from the width of the resonance line in the z-direction. Since the lateral separation of these two NVs is larger than our spatial resolution, we can determine the relative height of the two NVs with high precision, which we find to be nm.
The NV centers in our demonstration experiments were created through implantation of nitrogen ions forming a layer nm below the diamond surface. Modeling of this implantation procedure Ziegler (2010) predicts a spatial variation of nm; whereas we observe a larger variation in the distribution of NV depths (nm) using our scanning-field-gradient MRI technique. This discrepancy may arise from ion channeling or surface effects. Nanoscale precision in measuring the distance between proximal NV spins also allows determination of the mutual dipole coupling between adjacent spins, a key component for creating entangled spin-states.
Scanning-field-gradient MRI provides not only a method for nanoscale mapping of the spatial locations of proximal spins but also allows for individual spin transitions to be resolved in frequency space with high precision. In the absence of magnetic field gradients, identical spins sharing a quantization axis are indistinguishable making selective control over proximal spins impossible. Performing scanning-field-gradient MRI on proximal spins differentiates their transition frequencies, allowing for coherent manipulation and characterization of individual spins in the ensemble. This can be done simultaneously on several NV spins while preserving independent control of each spin. Such selective control over proximal spins can be maintained as long as the magnetic field gradient separates the transition frequencies of neighboring spins by more than their resonance linewidth, which is the same condition that determines the spatial resolution in MRI. Therefore, spins separated by more than the achieved MRI spatial resolution (nm for the NV spin experiments reported here) can be addressed independently using our technique.
To demonstrate such selective nanoscale characterization and control of proximal spins we examined a pair of proximal NV centers (NV IV and NV V, separated by nm, which share the same NV axis orientation. In the absence of the magnetic tip, we performed a continuous-wave ESR measurement on both NV spins simultaneously (Figure 3a, left panel), finding no difference in their spectra, as expected. After tuning the RF frequency to the pair of NV spins, we drove Rabi oscillations and observed the free induction decay of the two NV spins via a Ramsey sequence (Fig 3a, right panels). The measured Ramsey fringes show a pronounced beating pattern due to the hyperfine structure of the NV spin transitions and are damped due to inhomogeneous dephasing of the two NV centers. However, information distinguishing the individual coherence-properties of the proximal NV centers cannot be extracted from these measurements. In contrast, when the magnetic tip is brought in close proximity to the NV spins, it splits their resonance frequencies due to the differing magnetic fields applied to each NV, thereby allowing each NV spin to be addressed and characterized individually and coherently (Figure 3b) without modifying the spin of the neighboring NV.
Such coherent individual spin control was realized by tuning the RF frequency to the ESR resonance of the target NV center. Pulsing the RF with a variable duration induced coherent Rabi oscillations of either NV IV or NV V, dependent on the tuning of the RF frequency. Similarly, characterization of individual spin coherence properties was achieved by measuring the spin’s free-induction decay via a Ramsey sequence using the appropriate RF frequency for the target NV spin. We observed that NV IV has a faster free-induction decay rate than NV V, indicating that the two spins have different inhomogeneous dephasing times, . The measured free-induction decay rate for NV V with and without the magnetic tip are comparable, indicating that additional decoherence induced by the tip is small compared to that spin’s ambient dephasing . In order to prevent tip-induced spin-decoherence, we employed a three-dimensional spatial feedback scheme, which ensured that variations in the applied tip-field were smaller than the intrinsic NV inhomogeneous dephasing rates. Thus, the applied magnetic field gradient can be used to characterize individual spin coherence properties, a direct consequence of achieving single-spin control.
Under the influence of the tip’s magnetic field gradient, manipulating one NV spin does not perturb the state of a neighboring NV spin. To experimentally verify this selective, independent spin control, we simultaneously drove Rabi oscillations on NV IV while measuring the ESR spectrum of NV V (Figure 4a) by applying two separate RF fields (RF 1 and RF 2), each used to drive one NV spin-resonance. To illustrate the independence of the two measurements, we performed a two-dimensional sweep over the RF 1 pulse duration () and the frequency of RF 2 () (Figure 4b). We then subtracted out the mean of each row or column to show that the two measurements are independent. Rows or columns were summed to reconstruct the resulting Rabi oscillations of NV IV or the ESR curve of NV V, respectively. The only deviations from simultaneous, independent NV spin measurements are observed when is swept close to the resonance of NV IV, yielding a slight damping in the Rabi oscillations. These deviations do not persist through the ESR transition of NV V, which shows that both NVs can be manipulated independently when they are driven on resonance.
Scanning-field-gradient MRI of spins yields precise determination of their relative locations with nanometric spatial resolution, which will be crucial for optimizing the performance of spin-based magnetometers and the functionality of spin-based quantum bit ensembles. Once suitable spin ensembles are identified and spatially mapped, individual spin control both allows for the determination of individual spin properties and - when combined with magnetic dipole coupling between adjacent spins - provides a method for achieving complete control of the quantum state of spin ensembles. The spatial resolution of our scanning technique is extendable to the atomic scale by using stronger magnetic field gradients Mamin et al. (2007) and narrower ESR linewidths Balasubramanian et al. (2009). For NV spins, the current experimental barrier to improving this spatial resolution is overcoming the reduction in ESR contrast due to a strong off-axis magnetic field created by the tip Epstein et al. (2005); however, this can be prevented by either applying a strong bias-field along the NV axis or by tailoring the domain structure of the magnetic tip to produce high gradients but only moderate total fields.
The control and manipulation of individual spins using magnetic field gradients is independent of the method used for spin readout. For optically addressable spins, such as NV spins, integrating far-field, sub-diffraction schemes - such as stimulated emission depletion (STED) Rittweger et al. (2009) and reversible saturable optical linear fluorescence (spin-RESOLFT) Maurer et al. (2010) - with a scanning magnetic field gradient would allow for both robust individual spin control and readout with nanometric resolutions. Additionally, selective optical control of such systems is possible via the incorporation of an electric field gradient to the scanning tip Hettich et al. (2002), which would allow both spin and electronic degrees of freedom to be both addressed individually. Alternatively, using demonstrated single-shot electrical readout of individual spins Morello et al. (2010); Barthel et al. (2009) would allow for MRI to be performed rapidly and efficiently, as acquisition times would not be limited by the readout integration time. Individual control of spins via the present technique is also extendable to nuclear spins, provided methods for reading out nuclear spins reach single-spin sensitivity Mamin et al. (2007). Nanoscale or atomic spatial resolution of nuclear spins is feasible as their long spin coherence times help to compensate for nuclear spins’ small dipole moment. For any spin system, providing individual control of spins in dense ensembles, where mutual coupling is strong, allows for the creation of arbitrary entangled states. Such states have intriguing potential applications ranging from sensitive nanoscale magnetometers to scalable quantum information processors Yao et al. (2010).
We gratefully acknowledge G. Balasubramanian and P. Hemmer for fruitful technical discussions, as well as B. Hausmann and M. Loncar for instruction in the fabrication of NV center containing nanostructures. M.S.G. is supported through fellowships from the Department of Defense (NDSEG program) and the NSF. P.M. acknowledges support from the Swiss National Science Foundation, and S.H. thanks the Kwanjeong Scholarship Foundation for fellowship funding. This work was supported by NIST and DARPA.
Correspondence and requests for materials should be addressed to A.Y. (firstname.lastname@example.org)
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NV center samples:
NV centers were created through implantation of 15N ions Rabeau et al. (2006) into ultrapure diamond (Element Six, electronic grade diamond, ppb nitrogen). The implantation was done at keV to give a nominal nitrogen depth of nm Ziegler (2010). To form NV centers, the sample was annealed in vacuum at C, where existing vacancies are mobile and can pair with the implanted nitrogen atoms. The resulting density of NV centers corresponds to one center every 50-100 nanometers, forming a layer roughly nm from the surface. To isolate small NV clusters (i.e., a few proximal NV centers), we selectively etched Lee et al. (2008) away the majority of the shallow diamond surface layer, leaving proximal NV-containing nanostructures. This was done using electron beam lithography to define an etch mask from a flowable oxide (Dow Corning, XR-1541) Hausmann et al. (2010). A reactive-ion etch then removed any exposed diamond surfaces, resulting in shallow diamond nanostructures (nm across) containing ensembles of proximal NV spins.
Magnetic tips were created by evaporating a magnetic layer onto quartz tips of roughly 80 nm in diameter, which were fabricated using a commercial laser-pulling system (Sutter Instrument Co., P-2000). Using a thermal evaporator, a nm layer of cobalt-iron was deposited on the side of the pulled quartz tip. A nm chrome layer was then evaporated, which serves as a capping layer to prevent oxidation of the magnetic material. These tips result in magnetic field gradients of roughly 1G/nm at distances of roughly nm.
Magnetic tip positioning with an AFM:
To achieve high spatial resolution in NV imaging and manipulation, it is necessary to control precisely the relative distance between the magnetic tip and the addressed NV centers. Such nanoscale control can be challenging under ambient conditions (standard temperature and pressure); e.g. temperature drifts of a small fraction of a degree can induce few nm drifts between the magnetic tip and the diamond sample, which would inhibit local spin-control.
To overcome this problem, we used an AFM to position the magnetic tip in three dimensions with precisions of a fraction of a nanometer in z and a few nanometers in both x and y. Height control was achieved through normal AFM operation in which a feedback loop modulates the height of the tip to keep the sample-tip interaction constant. Lateral positioning was achieved through intermittently locating topographic features on the sample to ensure that the tip’s relative distance to the sample is fixed during any performed measurements.