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Spin-Light Coherence for Single-Spin Measurement and Control in Diamond

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Science  26 Nov 2010:
Vol. 330, Issue 6008, pp. 1212-1215
DOI: 10.1126/science.1196436

Abstract

The exceptional spin coherence of nitrogen-vacancy centers in diamond motivates their function in emerging quantum technologies. Traditionally, the spin state of individual centers is measured optically and destructively. We demonstrate dispersive, single-spin coupling to light for both nondestructive spin measurement, through the Faraday effect, and coherent spin manipulation, through the optical Stark effect. These interactions can enable the coherent exchange of quantum information between single nitrogen-vacancy spins and light, facilitating coherent measurement, control, and entanglement that is scalable over large distances.

The coherent coupling of light with matter provides a powerful tool for quantum measurement and control. The resulting hybrid states have been employed in individually addressable semiconducting (1), atomic (2), and superconducting (3) quantum systems. Individual diamond nitrogen-vacancy (NV) centers (4) show great promise as solid-state spin qubits with demonstrations of millisecond coherence times (5) and nanosecond manipulation times (6). NV-center spins coherently couple to nearby electronic (7, 8) and nuclear (9, 10) spins, creating few-qubit networks for simple algorithms and quantum memories. NV-center spin states are conventionally read out destructively with the use of spin-dependent photoluminescence (PL). We present nondestructive single-spin measurement via the Faraday effect (FE) and unitary single-spin manipulation via the optical Stark effect (OSE) with the use of a near-resonant laser field coupled to an NV center. With enhanced coupling from an optical cavity, these techniques can be extended to quantum weak and quantum nondemolition measurements of single spins, facilitating applications such as quantum repeaters (11) and spin-photon entanglement (12) on demand. This approach may also assist with the study of newly identified defect-based spin systems (13) with limited single-spin read-out alternatives.

The FE and OSE probe the respective light and spin responses to the near-resonant coupling between an NV center and light, which emerge from spin-dependent optical transitions (14). Each optical transition’s ground (|g〉) and excited (|e〉) states coherently mix with the laser field (Fig. 1A), forming polariton states (|G〉 and |E〉) whose energies repel in the form of a level anticrossing (Fig. 1A, inset) as a function of the energy detuning (Δ) between the light and the optical transition. The groundlike polariton states (|G〉) adiabatically remain occupied as they shift in energy during the interaction. The FE originates from polarization selection rules in which the coupled polarization component of the coherent optical field experiences a spin-dependent phase shift relative to the other. Simultaneously, the OSE energy shifts add quantum phases to the spin states producing spin rotation. Theoretical predictions of the FE and OSE from a modified Jaynes-Cummings model are in quantitative agreement with our measurements (15).

Fig. 1

(A) An optical transition interacting with a laser detuned in energy by Δ relative to the transition energy. Under adiabatic conditions, the spin-light polariton energy levels shift by ε as a result of the interaction, producing measurable effects in both the phase of interacting light and the NV-center spin dynamics. (Inset) ε as a function of Δ. (B) NV-center energy-level diagram depicting ground (3A) and excited (3EX, 3EY) spin triplet levels. From left to right: Diagonal arrows show the intersystem crossing from the excited ms = ±1 states to the ground ms = 0 state through singlet-level (1E, 1A1) metastable decay, which permits spin initialization and measurement. The nonresonant green upward and red downward arrows depict optical excitation and collected PL, respectively. The red double arrow depicts the multiple resonant transitions magnified in (C). (C) Fine structure of the presented NV-center ground- and excited-state energy levels at 1920 G, additionally depicting spin-conserving optical transitions for ms = 0 (black arrows) and ms = –1 (blue arrows) spin states in both excited-state orbitals. The EY, ms = 0 (circled) spin-conserving transition is ~3 GHz larger in energy than the EY, ms = –1 transition for our measurements. With increased strain, 3EX and 3EY orbital energies further separate. An increase in magnetic field shifts the ms = +1 and ms = –1 spin sublevels up and down, respectively, in all orbital branches, having only an indirect effect on changing spin-conserving optical transition energies.

A diamond NV center consists of a vacancy adjacent to a substitutional nitrogen atom in the diamond lattice. When negatively charged, the orbital ground state is a spin triplet whose levels function as a qubit (4) and are tunable with an applied magnetic field. Microwave magnetic fields that are resonant and near-resonant (16) with the spin transitions produce unitary rotations of the spin state. NV centers exhibit spin-conserving optical transitions (Fig. 1B) to an orbital-doublet, spin-triplet excited state 1.945 eV above the ground state (17). An NV center may be optically excited into the excited state, either resonantly at 637 nm or with higher-energy light along with phonon creation. Phonons can also be created during spontaneous emission back to the ground state, resulting in red-shifted PL. Optical excitation causes spin polarization from the ms = ±1 spin states into the ms = 0 spin state through an intersystem-crossing decay mechanism. PL is inhibited during the 460-ns cryogenic lifetime (18) of this decay, resulting in a spin-dependent PL intensity that is used for destructive spin-projection measurements. These dynamics allow individual NV centers to be optically addressed.

The NV-center spin coherently couples to light in our measurements because the optical transition energies (Fig. 1C) vary by spin state as a result of spin-spin and spin-orbit interactions (19). These transitions are energetically stable (20) in pure diamond hosts under cryogenic temperatures (21). In our NV centers, the transverse crystal strain is strong enough to energetically separate the excited-state orbital doublet into two branches with approximately linear, orthogonal optical-transition dipoles aligned to the transverse strain: 3EX and 3EY (22). These dipoles couple to orthogonal near-linear polarizations of light, giving rise to the FE in a nonstandard polarization basis. A magnetic field is applied along the NV symmetry axis to produce similar spin eigenstates in both the ground and excited orbitals, providing robust spin-conserving optical transitions. In the range of crystal strains and magnetic fields that we studied, the largest optical transition–energy spin splitting occurs in the lower-energy excited-state orbital branch (3EY). The 3EY, ms = –1 optical transition is roughly 3 GHz lower in energy than the 3EY, ms = 0 optical transition (Fig. 1C, right). The fine structure of the higher-energy excited state orbital branch (3EX) more closely resembles the ground-state fine structure for our experimental conditions, producing spin-conserving optical transitions that are closely spaced in energy. The spin splitting in the 3EY branch is insensitive to changes in strain and magnetic field when these effects dominate spin-orbit interactions, allowing the spin-light interactions investigated here to persist across a broad range of experimental conditions.

We assembled a confocal microscope to study single NV centers at cryogenic temperatures. The NV centers were naturally formed in high-purity diamond grown by chemical vapor deposition specified to contain less than 5 parts per billion nitrogen impurities. A 0.85–numerical aperture microscope objective focused both a red laser, tunable in energy across the NV-center optical transitions at 637 nm, and a green 532-nm laser onto a single NV center for both resonant and nonresonant optical excitation, respectively. PL from the NV center’s phonon sideband was collected with the objective and detected with an avalanche photodiode (APD). The PL provided two important measurements: (i) the spin projection (<SZ>) under green illumination and (ii) resonant optical absorption from photoluminescence excitation (IPLE) under tunable red excitation. We used a lithographically patterned, short-terminated coplanar waveguide (6) to apply microwave-frequency magnetic fields for resonant spin manipulation of the ground-state triplet. The lasers, microwave fields, and measurements of PL were all gated for time-domain measurements. The polarization of the tunable red laser light was prepared relative to the orthogonal optical dipoles of the NV center, ensuring that each dipole interacted with half of the light for polarization-interference measurements. The transmitted red laser light was collected through a back-side window in the cryostat, passed through polarization optics to project the correct polarization, and then analyzed with a near shot-noise–limited balanced photodiode bridge whose signal was measured with a lock-in amplifier (15).

We studied the FE spin dependence and the corresponding NV-center response to a 1-μs pulse of tunable red laser light as a function of the light’s energy. In the measurement sequence (Fig. 2A), the initial spin state was prepared alternately in either ms = 0 or ms = –1 before the red laser light pulse. The measured spin-dependent FE (ΦF) is the difference in phase shift of the red laser light polarization between the two spin preparations. Additionally, both IPLE and <SZ> were measured for both prepared spin states with PL measured at different times during the sequence. To increase the signal-to-noise ratio, ΦF, IPLE, and <SZ> data were averaged for ~5 × 106 repeated measurements, requiring ~30 s per point. These three independent measurements were correlated as the red laser light energy was tuned across the optical transitions, providing a detailed picture of the full interaction dynamics.

Fig. 2

(A) FE measurement timing sequence showing the gated green laser, tunable red laser, microwave, and multiplexed APD timing, as well as the spin-state evolution and diode bridge signal in time. PL photons are binned separately in time to measure both IPLE (APD bins 1, 3) and <SZ> (APD bins 2,4) for the ms = –1 and ms = 0 prepared spin states. Gray areas indicate possible spin-polarization effects from the red laser. ΦF (denoted by the blue arrow at bottom-left) is the difference in measured red laser polarization response between the prepared ms = 0 and ms = –1 spin states. (B and C) FE data sets scaled identically showing ΦF, as well as IPLE and <SZ>, for both ms = 0 (blue) and ms = –1 (red) prepared spin states of the 3EY orbital-branch optical transitions at 5 and 0.1 μW red laser power, respectively. The ms = 0 optical transition (at 0-GHz laser energy) is more robust against spin polarization than the ms = –1 optical transition (near –3-GHz laser energy). μrad, microradians; kCps, photon kilocounts per second. (D) FE data set at 15 μW scanning across both 3EX and 3EY orbital-branch optical transition energies. The measured FE is substantially reduced for the 3EX orbital transitions (near 16.5 GHz), primarily as a result of the smaller optical transition energy splitting between the spin states. Vertical-pointing double arrows denote expected transition energies, as described in the text.

To illustrate the FE dynamics in our system, simultaneous measurements of ΦF, IPLE, and <SZ> at a temperature of 8 K and an applied magnetic field of 1620 G are shown for the lower-energy EY orbital-branch optical transitions at red laser powers of 5 and 0.1 μW (Fig. 2, B and C, respectively). The EY, ms = 0 optical transition (circled in Fig. 1C) defines the origin of the red laser energy scale, placing the EY, ms = –1 optical transition near –3-GHz laser energy in all presented FE data. As a result of spectral broadening of the optical transitions (21), ΦF is also broadened, resulting in a reduced response near resonances. The ms = ±1 spin states are susceptible to polarization into the ms = 0 spin state when optically excited because of the intersystem-crossing decay, which is measured directly with <SZ>. Therefore, optical spin polarization reduces ΦF and IPLE signals more substantially near resonance with ms = ±1 optical transitions and at higher powers. Photo-ionization (23) may also result from red laser excitation, which reduces ΦF and IPLE signals and skews <SZ>. The NV center is less perturbed at lower red laser powers, producing more symmetric FE data at smaller signal intensities (Fig. 2C). The temperature dependence of the FE that we measured (15) was consistent with previous studies of the excited-state energetic stability with temperature (21). Using our developed model (15) along with our measured FE response, we determined that the average, equal-intensity area of the red laser confocal spot was (0.75 ± 0.1 μm)2, which is consistent with diffraction, drift, and aberration limits.

FE measurements showing both EX and EY orbital-branch optical transitions at a temperature of 8 K, an applied magnetic field of 1920 G, and a higher red laser power of 15 μW are shown in Fig. 2D. The ms = 0 and ms = –1 optical transitions in the EX orbital branch (near 16.5-GHz laser energy) are closer together in energy than their spectrally broadened absorption and spin polarization widths, making them difficult to differentiate and reducing ΦF for this orbital branch. Optical coupling to the EX orbital branch is 40% weaker relative to the EY orbital branch due to the NV center’s orientation, further reducing ΦF for the EX orbital branch. The EX orbital branch picks up a stronger ms = 0 optical transition response in ΦF, probably because the ms = –1 spin decays more easily through the intersystem crossing, which inhibits the FE during its lifetime. Data from the EY orbital-branch optical transitions in Fig. 2D resemble those of Fig. 2, B and C, but at a higher power, showing an increased asymmetry between the ms = 0 and ms = –1 optical transitions. Optical transitions that do not conserve the spin state can also be observed at this red laser power. Black arrows between IPLE and <SZ> graphs denote the four spin-conserving transition energies, with the EX transitions being nearly degenerate. Blue and red arrows denote the expected positions of spin-nonconserving transitions coupled to the ms = 0 and ms = –1 ground spin states, respectively. Sharp features in both IPLE and <SZ> are consistent with these spin-nonconserving optical transition energies but do not line up exactly with their expected positions because the hysteretic and nonlinear response of the red laser energy tuning was not actively corrected. FE measurements on a different NV center gave nearly identical results for both EX and EY orbital branches, although these orbital transitions were further separated in energy as a result of higher crystal strain.

Complementary to the light’s FE response, the NV-center spin also coherently reacts to its dispersive interactions with light in the form of the OSE. The OSE manifests as relative energy shifts of the ground spin eigenstates, which alter the spin’s Larmor precession rate during interaction with the red laser light pulse. This process is equivalent to a σZ spin rotation of the spin qubit in the Bloch sphere rotating frame representation. We measured the OSE with the use of a modified Hahn echo microwave pulse sequence (Fig. 3A) to mitigate slow environmental dephasing. The red laser interaction was timed between the two final Hahn microwave pulses such that the OSE spin evolution is encoded in the final spin state that was measured with PL intensity (IPL). Through phase control of the final Hahn echo pulse, we measured multiple spin projections to give a more detailed picture of the OSE spin dynamics. Fig. 3B shows the OSE spin rotation as a function of red laser pulse duration. All OSE spin projection data were averaged over 2.5 × 105 repeated measurements, taking 2.6 s for each data point and ~15 min for typical scans. An exponentially decaying sinusoidal fit to these data determines the Larmor frequency shift (ΣS) induced by the OSE for fixed red laser energy and power. When the laser detuning is much larger than the resonant optical Rabi frequency, ΣS is nearly linear with laser power (Fig. 3C). We determined the ground-state spin coherence to be T2 = 480 μs with Gaussian decay using a Hahn echo sequence without applying the OSE, revealing that pure spin decoherence is a negligible effect in the OSE data.

Fig. 3

(A) OSE measurement timing sequence. The red laser produces a σZ spin rotation proportional to its 0- to 4-μs pulse duration. <SZ> follows ±sin(σZ) as a result of a ±90° microwave phase shift of the final Hahn echo pulse. (B) IPL of sequences that have these two microwave phases. kCts, photon kilocounts. (C) ΣS as a function of laser power at ~2-GHz laser energy. The error is represented by the data-point size and is dominated by the 15% uncertainty in the laser power calibration. The red line is a linear fit of ΣS with 5.74 MHz per microwatt slope. (D) Comparison of ΣS and ΦF showing their complimentary response as a function of laser energy and also showing IPLE and <SZ> taken with ΦF for both prepared spin states. OSE and FE data sets were taken under the same experimental conditions at 0.66-μW laser power. The ΣS data point denoted by the blue arrow is the frequency fit of the data presented in Fig. 3B. (E) OSE spin coherence measured in number (ND) of σZ rotations to the 1/e decay point. Black circles represent the measurements with 0.66-μW red laser power (Fig. 3D), and green triangles represent measurements with 1.59-μW red laser power. The blue arrow denotes the ND data point fit from in Fig. 3B. The red line is a fit to the data using a model incorporating dephasing from spectral broadening and laser intensity fluctuations. Error bars indicate standard errors of OSE data fits.

By comparing the FE and OSE responses together as a function of red laser energy, we analyzed both light and spin components of the polariton dynamics, which gives deeper insight into the full NV-center spin interactions with coherent light. ΦF and ΣS data are superimposed as a function of laser energy at a temperature of 8 K, an applied magnetic field of 1920 G, and a red laser power of 0.66 μW (Fig. 3D, top). By fitting ΣS as a function of laser energy, the resonant optical Rabi frequency was indirectly determined to be 2π × 70 MHz at this laser power. The relation between ΦF and ΣS gives information regarding the mean photon number of the interaction (15), which can be used to calibrate the response to stronger light coupling, down to the single-photon level.

The decay of the OSE oscillations present in Fig. 3B can be analyzed in terms of dephasing between the repeated interrogations of the OSE-shifted spin. The number of OSE oscillations to 1/e exponential decay (ND) in Fig. 3E is plotted as a function of laser energy for two laser powers. The smallest number of coherent oscillations occurs near the optical resonances where absorption, spin polarization (Fig. 3D, bottom), and spectral diffusion become most prominent. Although there are probably several sources of fundamental decoherence and experimental dephasing, these data qualitatively agree with the red line in Fig. 3E, which is a fit using a model that includes only two contributions (15): first, dephasing from spectral hopping giving a distribution of transition energies, and second, dephasing from the red laser intensity fluctuations. The fit yields a spectral hopping half width at half maximum of 63 ± 13 MHz and laser power fluctuations of 3.5 ± 0.8%. These values, along with the 71-MHz power-broadened natural linewidth, are consistent with the 150-MHz IPLE-measured linewidth of the ms = 0 optical transition shown in Fig. 3D. This suggests that spectral diffusion and local laser intensity fluctuations play primary roles in the decay of the OSE oscillations, consistent with recent studies of optical Rabi oscillations (24). The cumulative fidelity for a π spin rotation in Fig. 3B is 89 ± 1% with resistance to back-action spin flips of 98.9 ± 0.3% (15). These two fidelities set limits on the expected fidelity of either single-spin/single-photon entanglement generation or quantum nondemolition measurements, given an appropriately designed cavity to enhance the per-photon spin-light interaction under otherwise similar conditions. Improvements in the experimental design and material quality may mitigate the external sources of dephasing and lead to even higher-fidelity spin-light coupling for future applications of NV centers in diamond.

The coherent coupling between an NV-center spin and light is a critical advancement for the use of NV centers in quantum information science. For example, by mixing the spin eigenstates in either the ground or excited orbitals to form a V (25) or Λ (26, 27) configuration, respectively, all-optical spin manipulation and measurement could be performed in other spin bases to span the full Bloch sphere. With the increased coupling offered by photonic cavities (28), these techniques enable optical quantum nondemolition measurements (29) of single electron spins for fundamental studies of light-matter interactions, potentially leading to solid-state implementations of quantum key distribution (30). Furthermore, these hybrid light-spin states could controllably couple distant NV-center spins through photonic networks (31) for increasingly complex quantum information processing operations.

Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1196436/DC1

Materials and Methods

SOM Text

Figs. S1 to S5

References

References and Notes

  1. Materials and methods and theoretical analysis are available as supporting material on Science Online.
  2. We sincerely thank D. M. Toyli, C. G. Yale, F. J. Heremans, and C. Santori for thoughtful comments. We gratefully acknowledge support from the Air Force Office of Scientific Research, the Army Research Office, and Defense Advanced Research Projects Agency.
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