Optically Coherent Nitrogen-Vacancy Defect Centers in Diamond Nanostructures

Optically active solid-state spin defects have the potential to become a versatile resource for quantum information processing applications. Nitrogen-vacancy defect centers (NV) in diamond act as quantum memories and can be interfaced with coherent photons as demonstrated in entanglement protocols. However, particularly in diamond nanostructures, the effect of spectral diffusion leads to optical decoherence hindering entanglement generation. In this work, we present strategies to significantly reduce the electric noise in diamond nanostructures. We demonstrate single NVs in nanopillars exhibiting a lifetime-limited linewidth on a timescale of one second and long-term spectral stability with an inhomogeneous linewidth as low as 150 MHz over three minutes. Excitation power and energy-dependent measurements in combination with nanoscopic Monte Carlo simulations contribute to a better understanding of the impact of bulk and surface defects on the NV’s spectral properties. Finally, we propose an entanglement protocol for nanostructure-coupled NVs providing entanglement generation rates up to hundreds of kHz.

Popular Summary

Diamond is often associated with expensive jewelry or industrial abrasives; however, it is also a promising platform for the next era in quantum computing and communication. Light-sensitive atomic-scale defects in diamond can act as excellent quantum bits, or qubits, and can be used as nodes in prospective quantum networks. One well-studied defect, the nitrogen-vacancy (NV) defect, has proven itself as a quantum memory, but the efficient use of NVs for quantum information processing tasks remains a major challenge. As a step toward overcoming this challenge, we present a detailed analysis and methods for improvement of the spectral properties of NV defect centers in diamond nanostructures.

NVs are extremely sensitive to electronic noise induced by fluctuations in the local charge environment, resulting in spectral shifts over time known as spectral diffusion. This effect disrupts the NV’s most important task: the creation of spin-photon entanglement. In our work, we use extensive experimental investigations and theoretical modeling to arrive at a detailed understanding of an NV’s spectral properties. We then develop strategies to significantly reduce the electronic noise by carefully choosing material properties, fabrication recipes, and experimental control schemes. Marking a milestone in NV research, we demonstrate lifetime-limited transition linewidths for NVs in nanostructures under experimental conditions suitable for entanglement generation.

Mitigation of spectral diffusion makes NVs in nanostructures compatible with coherent optical control and lays the foundation for efficient entanglement generation with rates orders of magnitude higher than presently achieved in bulklike diamond structures.

Figure 1

Nanostructure and single-scan linewidth. (a) Schematic sketch of a NV incorporated in a nanopillar. Green laser irradiation causes ionization of present surface and bulk defects; i.e., charges are released and move within the lattice, for example, to other defects that act as charge traps. This dynamic induces fluctuations of the electrostatic environment of the NV and leads to spectral diffusion of the NV ZPL. Insets: scanning electron microscopy images of a single nanopillar with a radius of 125 nm and a height of $1.6\mathrm{\mu }\mathrm{m}$ (top) and an array of identical nanopillars (bottom). The scale bar corresponds to a dimension of $2\mathrm{\mu }\mathrm{m}$. (b) Single PLE scan described by a Voigt profile. The exemplarily selected spectrum shows a linewidth close to the lifetime limit maintained for about one second. (c) Histogram presenting the occurrence of single-scan linewidth obtained from 3031 PLE scans. The measured data are simulated by a Monte Carlo approach. From the simulations, a power-broadened linewidth of 28 MHz is determined. The gray dashed line indicates the lifetime-limited linewidth of 14 MHz. (d) Single-scan linewidth as a function of excitation power at a scan speed of $6\mathrm{GHz}/\mathrm{s}$. The single-scan linewidths are evaluated using three methods. The data can be fitted by the function $\gamma ={\gamma }_0\sqrt{1+P/P_{\mathrm{sat}}}$ describing laser-induced power broadening of the linewidth. For the Monte Carlo simulation, the saturation power and natural linewidth are determined as $P_{\mathrm{sat},\mathrm{MC}}=(5.4\pm 2.2)$ nW and ${\gamma }_{0,\mathrm{MC}}=(16.2\pm 1.8)\mathrm{MHz}$, and for the mean value approach as $P_{\mathrm{sat},\mathrm{MV}}=(8.1\pm 2.9)\mathrm{nW}$ and ${\gamma }_{0,\mathrm{MV}}=(24.4\pm 2.2)\mathrm{MHz}$. (e) Plot summarizing the single-scan linewidths of eight single NVs in different nanopillars from different sample regions (blue circles and red squares). The NVs are labeled with the number of PLE scans that were taken into account. The excitation power is 19 nW, and the scan speed is $50\mathrm{GHz}/\mathrm{s}$. In contrast to panel (d), we apply a 525 nm initialization pulse before every line scan. For the linewidth analysis, the Monte Carlo simulation (filled circles and squares) and inverse-variance weighting (empty diamonds) methods are used. The uncertainties of the single-scan linewidths determined by the Monte Carlo simulation are given by the 99% confidence interval. For this fast scan speed, the inverse-variance method is not valid anymore.

Figure 2

Inhomogeneous linewidth and ionization time (regime 1). (a) More than 250 PLE scans performed without reinitialization of the NV with green laser pulses. The excitation power of the resonant laser is 2 nW, and the scan speed is $50\mathrm{GHz}/\mathrm{s}$ (bottom). The inhomogeneous linewidth is given by the intensity sum of all scanned lines recorded over a time of more than three minutes and is fitted by a Gaussian function (top). The determined FWHM of $(150\pm 3)\mathrm{MHz}$ is indicated by dashed lines (bottom). (b) Cumulative inhomogeneous linewidth evolution in time. Data of the single trajectory corresponding to the PLE scans in (a) ($n=1$, blue circles), average of postselected trajectories ($n=14$, red triangles), and all trajectories ($n=42$, dark blue squares) are shown. The simulated data (gray line) are obtained from modeling a normal diffusion process. The experimental data are fitted by a power law ${\gamma }_{\mathrm{Inh}}=b{P}^a$ with exponents $a=0.22$ (blue line), $a=0.41$ (red line), and $a=0.42$ (dark blue line). A scaling exponent $a<1$ is characteristic for subdiffusive processes. (c) Average ionization time for different excitation powers extracted from many tens of PLE data sets recorded at a scan speed of $12\mathrm{GHz}/\mathrm{s}$. The uncertainty is given by the standard deviation of the individual ionization times. The ionization time decays exponentially for increased excitation power with a power constant of $(8.7\pm 1.4)\mathrm{nW}$. The NV transition is driven at a duty cycle of about 0.2%. Inset: ionization rate $\mathrm{\Gamma }$ as a function of excitation power.

Figure 3

Spectral diffusion dynamics (regime 2). (a) Hundreds of PLE scans performed with only resonant excitation laser irradiation (bottom) and with a mixture of resonant and green (525 nm) laser irradiation, with an equal ratio of a total power of 4 nW (top). While red laser irradiation induces a small drift of the ZPL resonance frequency, additional green laser light causes large spectral jumps. (b) Spectral diffusion (SD) rate measured for different excitation wavelengths. A two-color PLE scheme, involving resonant and off-resonant laser irradiation, is applied with a total power of 4 nW. The gray dashed line highlights the wavelength of 564 nm, which corresponds to the ionization energy of substitutional nitrogen defects. As a guide for the eye, the supposed spectral diffusion rate evolution is drawn, assuming bulk diamond containing only nitrogen defects (orange dashed line) and additional surface defects (red dashed line). Inset: relative occurrence of spectral shifts described by a normal distribution function. Here, the data excited with a mixture of resonant and 575 nm light are shown. (c) Spectral diffusion rate as a function of the total excitation power, recorded for resonant excitation and two-color excitation with 525 nm light (equal power ratio of on- and off-resonant excitation laser). The spectral diffusion rate scales like a square-root power law with respect to the excitation power: ${\mathrm{\Gamma }}_{\mathrm{SDR}}=b{P}^a$, with exponents $a=0.49$ for red only and $a=0.53$ for two-color excitation. The filled markers correspond to data recorded after ten weeks at cryogenic temperatures, and the empty markers to data recorded directly after a warm-up and a second cooldown. (d) Monte Carlo approach with the spectral diffusion of the NV ZPL simulated by placing a certain number of bulk and surface charges at random positions and evaluating the resulting line shift, repeated over many iterations. The number of surface charges is increased from 2 to 452 in the direction of the arrow. Different charge configurations correspond to the fluctuating charge environment during repeated PLE line scans. The increase in spectral diffusion follows an approximate square-root power law by increasing the number of charges.

Figure 4

NV in the dark (regime 3). (a) Shutter experiment in which we alternate between PLE scanning for 20 s and blocking the radiation for 60 s. When PLE scans are performed, the center frequency of the ZPL resonance is extracted from Voigt fits (gray dots). Here, a data set is exemplarily presented. (b) Occurrence of spectral shifts obtained from many data sets. The extracted spectral diffusion value for “Laser on” corresponds to the spanned frequency range recorded in a period of 20 s. The spectral diffusion for “Laser off” is extracted from the spectral difference of the last PLE scan before and the first scan after blocking the laser, as illustrated in panel (a).

Figure 5

Towards entanglement generation. (a) Entanglement protocol consisting of periods of system preparation (red) and entanglement generation (blue), with different periods assigned to the three excitation regimes discussed in this work. Here, we assume a threefold Purcell enhancement of the ZPL emission, and the protocol is bounded to the condition that only entanglement attempts (including optical spin-repump pulses) can be applied until the inhomogeneous linewidth is broadened by 1%. Then, a spectral stabilization protocol needs to be applied to tune the resonance back to the target frequency (5 ms). When the NV ionizes, the system is initialized by applying a green laser pulse. The protocol sequence for NV preparation involves charge-state initialization as well as spectral realignment. Inset: illustration of entanglement generation of two distant NV spin qubits embedded in nanostructures by applying coherent laser pulses (red) and microwave pulses (wavy line). (b) Entanglement attempts (number of $\pi$ pulses) that can be applied until the total linewidth broadens by $p$ percent due to spectral diffusion at a rate of $640\mathrm{MHz}/\mathrm{s}$ as a function of Purcell enhancement, which can be engineered in different types of nanostructures. The estimated natural linewidth of ${\gamma }_0=14.2\pm 2.0\mathrm{MHz}$ considered in the model is increased by the corresponding Purcell factor. We model the inhomogeneous broadening of the linewidth with a Wiener process. Here, an inhomogeneous linewidth broadening of 100% corresponds to a broadening of twice the homogeneous linewidth. (c) Entanglement attempt rate using the protocol depicted in panel (a) as a function of the Purcell factor. For large Purcell factors, the entanglement attempt rate approaches a value of 370 kHz, which is given by the $\pi$-pulse separation of $2\mathrm{\mu }\mathrm{s}$, the estimated ionization time, and the initialization time.

Figure 6

Fabrication steps. Dry-etching fabrication process of diamond nanostructures.

Figure 7

Saturation measurement. (a) Mean peak count rate as a function of the resonant excitation power (scan speed $6\mathrm{GHz}/\mathrm{s}$) extracted from the excitation power-dependent PLE measurements. The error bars are given by the standard deviations. The fit function is from Eq. (c6). The background at the corresponding excitation power was already subtracted. (b) Background count rate increasing linearly with excitation power. The intercept at zero excitation power is at 128 cts/s.

Figure 8

Modified version of Fig. 5. We add to the proposal of entanglement generation (solid lines) the results of a modified protocol without considering resonant optical spin-repump pulses at 637 nm (20 nW, $1\mathrm{\mu }\mathrm{s}$) before every entanglement attempt (dashed lines).

Figure 9

Spectral diffusion rate distribution. Probability distribution of ${\mathrm{\Gamma }}_{\mathrm{SDR}}$ for red and green excitation light at 5 nW. The distribution agrees well with the theoretical prediction (red) derived from the Wiener process at $\eta ({\omega }_I,\tau )=3.41\times {10}^{24}/\mathrm{Joule}/{\mathrm{s}}^2$ for a spectral diffusion rate of $75.35\mathrm{MHz}/\mathrm{s}$ at 5 nW and $\tau =2.3s$, which was used for modeling the diffusion process.

Figure 10

Inhomogeneous broadening. (a) Inhomogeneous broadening caused by random distribution of charges with a density of 1 ppm (approx. 13800) in a cylindrical volume with a height of 1600 nm and a radius of 125 nm. The blue triangles are calculated using the electric field, including the contribution produced by the polarization surface charge [$\tilde{\mathbf{E}}$ in Eq. (g9)]. The red squares show the result of only including the direct field. The black line represents a fit with the power law $f(x)=b(x-x_0{)}^a$. (b) Blue triangles (100 charges) and squares (1000 charges) mark the FWHM of the inhomogeneous line for two different charge densities depending on the number of realizations of distributing the charges among the charge traps. The red squares and triangles show the root-mean-square error (RMSE) of the fit and the spectrum. The error drops for a higher count of realizations and appears to be saturating, indicating that the resulting spectrum reaches its equilibrium value. We chose 1000 realizations for the Monte Carlo simulation in panel (a).

Figure 11

Effect of surface charges on inhomogeneous linewidth. Inhomogeneous broadening due to surface charges for different pillar radii. The lines represent a fit with the power law $f(x)=b(x-x_0{)}^a$. The power law noticeably differs from the power law found for bulk charges [Fig. 10].

Figure 12

Joint effect of surface and bulk charges. (a) Inhomogeneous broadening for a range of surface and bulk charges. (b) SDR calculated by fixing the time step per reordered charge configuration at 2000 bulk charges, which corresponds to a broadening of approximately 5 GHz (without screening).

Figure 13

Spectral diffusion rate scaling behavior. Shown are the SDR for no bulk charges (red triangles) and no surface charges (green squares) as a function of the number of charges contributing to the diffusion process. The parameters are the same as for Fig. 12. The solid lines correspond to a power-law fit with ${\mathrm{\Gamma }}_{\mathrm{SDR}}=b(n-n_0{)}^a$. The power laws show a surprising similarity to the power laws in Fig. 3.

Figure 14

Effect of trap densities. (a) Inhomogeneous linewidth and (b) spectral diffusion rate as a function of the trap density $\rho /\mathrm{ppm}$ for a range of charges and for 10 000 different configurations. Each point is averaged over ten trap densities equally spaced within 1-ppm increments. The error bars represent the standard deviation for both the linewidths and diffusion rates. The trap density appears to have little impact on the line broadening and the spectral diffusion range. The time step for the spectral diffusion rate is fixed to some arbitrary value, as it does not impact the qualitative behavior of the spectral diffusion rate.