Main

Intense experimental effort to understand the ES levels of nitrogen–vacancy (NV) centres has focused on spectroscopic investigations both at room temperature7,8,9 and at low temperature9,10,11. The short ES orbital lifetime (10–20 ns), however, presents a challenge for the measurement of ES spin coherence. The recent development of subnanosecond spin manipulation of individual NV centres12 puts these investigations within reach, with the fundamental goal of understanding ES-spin dynamics and decoherence mechanisms. Through these experiments, we establish the presence of room-temperature ES-spin coherence despite rapid phonon-mediated transitions between orbital branches13 that might be expected to destroy the quantum phase of the spin.

NV centres are a naturally occurring lattice defect in diamond. They consist of a substitutional nitrogen atom and an adjacent carbon vacancy. The ground state (GS) is a spin-1 orbital singlet (3A), whereas the ES is a spin-1 orbital doublet (3E). All ES spin levels (spin quantum number ms=0,±1) exhibit spontaneous decay by photon emission. Further radiationless relaxation from the ES ms=±1 levels through an intersystem crossing causes the NV photoluminescence intensity (IPL) to vary linearly with the ms=0 spin population and enables optical pumping into the ms=0 spin state14 (Fig. 1a).

Figure 1: Optical relaxation of NV centres.
figure 1

a, Simplified level diagram of NV centres. b, Photoluminescence as a function of time after the optical pulse. The purple data are for no microwave pulse, and the blue, green and gold data are for a microwave π pulse applied on the ES transition at 0, 7 and 15 ns delays after the optical pulse, respectively. Arrows indicate the microwave pulse timing. Inset: Integrated curves with the same colour labels overlaid with a fit to equations (1) and (2) as described in the text. c, Timing diagram.

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Before probing ES spin dynamics we first characterize the orbital relaxation. Figure 1b shows the decay of IPL as a function of time for a single NV centre at room temperature excited using a pulsed laser (Fig. 1c). The slowest decay rate (purple) occurs when the NV centre is optically polarized into ms=0 with no further spin manipulation. The fastest decay rate (blue) occurs when a 1 ns microwave π pulse on the ES ms=0 to −1 transition is applied simultaneously with optical excitation. Microwave π pulses at later delays (7 and 15 ns) abruptly change the initialized state decay rate to the ms=−1 decay rate (green and gold, respectively) as seen previously8.

The average number of photons emitted by an NV centre during the collection window T evolves as

where p0 is the initial polarization of the NV centre in the ms=0 state, γ0 is the decay rate from the ms=0 state and γ1 is the decay rate from the ms=±1 state including the intersystem relaxation. After a π pulse is applied at time τπ<T, the populations of ms=0 and ms=±1 interchange, and the evolution follows

where we have ignored the π-pulse duration. To determine γ0, γ1 and p0, we integrate the experimental data in Fig. 1d and fit to equations (1) and (2) (Fig. 1d, inset). We found γ0=0.0563±0.0043 ns−1, γ1=0.111±0.014 ns−1 and p0=0.84±0.08. The corresponding lifetimes (τi=1/γi) are τ0=17.8±1.4 ns and τ−1=9.0±1.2 ns. It is necessary to take p0 into account to obtain good simultaneous fits and to obtain correct results for the experiments described below. Details are available in the Supplementary Information.

To probe the ES spin coherence, we measure ES Rabi oscillations by setting the microwave pulse delay to zero and varying the pulse duration. Figure 2a shows ES Rabi measurements at magnetic field B=1,276 G and four values of microwave power. The ES Rabi frequencies are consistent with GS Rabi oscillations measured using the same microwave power at B=261 G, where the Larmor frequency ω0/2π for the ms=0 to −1 transition is also 2.14 GHz. It is noteworthy that the ES Rabi oscillations decay faster than γ=(γ0+γ1)/2=0.084±0.008 ns−1 (lifetime of 11.9±1.1 ns), the rate expected purely from spontaneous decay to the GS.

Figure 2: ES spin dynamics of a single NV centre.
figure 2

a, ES Rabi oscillations for four microwave driving powers at B=1,276 G. The Rabi frequencies from top to bottom are 244±5, 348±7, 477±8 and 634±7 MHz. Inset, left panel: τ* determined from fits to each Rabi curve plotted as a function of driving field H1. The red dashed line is the average of all four points weighted by their uncertainties. For comparison, the orbital lifetimes for each spin state are also shown (grey dashed lines). Inset, right panel: τ* from a fit to the Hahn-echo measurements, plotted on the same scale for comparison with Rabi measurements. b, ES levels as a function of strain at B=0 G and B=1,276 G. For scale, 1 GHz strain corresponds to roughly 1 MPa (ref. 11). For a given value of strain, there are rapid phonon-mediated transitions between the EX and EY orbital branches at rate R. Within each branch, there is a Larmor frequency ω0±/2π. There is also a spin-dependent decay rate γ0, γ1 from each spin sub-level into the orbital GS. The phonon-mediated transitions induce motional narrowing, causing the two separate resonances to appear as one resonance at the average frequency.

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The enhanced ES spin decoherence rate motivates a deeper theoretical investigation of the ES spin dynamics. The right-hand panel of Fig. 2b shows the ES level structure9,10,11,14 where ms is a good quantum number at large magnetic field. The ES orbital doublet is split by strain into two branches. The energy splitting for a given spin transition is unique to each branch, with a transition frequency ω0±/2π, where ± denotes the high-/low-energy orbital branch. At room temperature there are stochastic, phonon-mediated transitions between the two orbital branches at rate R (ref. 13). Although these orbital transitions are spin conserving at large magnetic field, they cause motional narrowing: the two sets of ES spin resonances merge into a single set9,11,15. Each transition appears at the average frequency ω0=(ω0++ω0−)/2. In the rotating frame of this average resonance frequency, occupation of the ± orbital branch corresponds to a longitudinal effective magnetic field ±B0=±ℏ(ω0+−ω0−)/(2gesμB), where ℏ is Planck’s constant, μB is the Bohr magneton and ges is the ES g factor. Phonon-mediated orbital transitions cause the effective field to switch between +B0 and −B0, producing spin dephasing at a rate Γ=B02/2R. Depending on the spin transition, the external magnetic field and the strain on the NV centre, B0 may be as large as 5.5 GHz (2 kG). According to recent observations, R is in the range of 1 THz (ref. 13).

Following the approach of refs 16 and 17 we derive the equations of motion for the density matrix of the spin under motional narrowing, intense Rabi driving and spontaneous orbital decay. To compare with the experiment, we calculate the average number of photons emitted within the measurement window as a function of the microwave pulse duration. We fit each Rabi curve in Fig. 2a to the theory (solid lines). Full details of the theory and fits are described in the Supplementary Information. From these fits we extract the total ES spin coherence time τ*, which includes both the spontaneous orbital decay (γ) and spin dephasing (Γ). The values we obtain from fitting the data are plotted in the inset of Fig. 2a along with the spin-dependent orbital lifetimes. As expected, τ* is independent of the microwave driving field H1. The average of τ* from all four fits gives τ*=5.8±0.9 ns (red dashed line in Fig. 2b). The corresponding spin-dephasing time is 1/Γ=10.9±3.3 ns, which is comparable to the lifetime from orbital relaxation 1/γ=11.9±1.1 ns, indicating that both processes are important. This is in contrast to situations where orbital relaxation may be ignored, such as nuclear magnetic resonance measurements of rare-earth-ion ensembles, where the ES orbital lifetimes are much longer than the nuclear-spin coherence18.

Although we analyse the data using the full theory we also present simplified expressions that, although unsuitable for quantitative analysis, clarify the basic physical behaviour. To first order in the small quantity δ=(γ0−γ1)/2, Rabi dynamics are described by IPL(t)=A−Bcos(Ω t+φ)exp[−(γ+Γ/2)t]. Here t is time, is the renormalized Rabi frequency and A, B and φ are constants. The total decay rate is a sum of the orbital decay rate γ and spin dephasing at a rate Γ/2. The factor of 1/2 appears because the dephasing from a longitudinal field affects only the transverse spin component.

To investigate coherent interactions between the ES electronic spin and the 15N nuclear spin, we also carried out a two-pulse Ramsey experiment that is illustrated in Fig. 3a with a Bloch sphere representation. The spin is initialized in the ms=0 state along +Z. Immediately after optical excitation, we apply a microwave π/2 pulse to bring the spin into a superposition state along +Y. This state is sensitive to all longitudinal fields, including the dephasing field B0 and the 61 MHz ES hyperfine interaction7. At the end of a free evolution period τ, we apply a second π/2 pulse to project the spin along −Z for measurement. In the simplified theory, we obtain IPL(t)=A1+B1cos(AHFt)exp[−(Γ+γ)t], where A1 and B1 are constants and AHF is the ES hyperfine coupling. Note that, as the spin has no longitudinal component during free evolution, Γ is not reduced. The results of our measurements are plotted in Fig. 3b along with two predictions from the full theory using τ*=5.8 ns. The first (solid line) includes the hyperfine interaction and the second (dashed line) does not. Even on this short timescale, the ES hyperfine coupling has a strong influence on the coherent spin dynamics.

Figure 3: Ramsey measurement of the ES hyperfine interaction.
figure 3

a, Bloch-sphere representation of the ES Ramsey measurement. b, ES Ramsey data, with an inset describing the pulse sequence used and the simultaneous orbital decay. The solid line is the theoretical prediction that includes the hyperfine interaction and the dashed line does not. Both predictions assume τ*=5.8 ns.

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As the hyperfine interaction is coherent, it may be reversed by inserting a π pulse in the centre of the free-evolution period, as in the three-pulse Hahn-echo sequence outlined in Fig. 4a. The π pulse does not reverse spin dephasing due to orbital telegraphing, which occurs on timescales much shorter than the free evolution time. Thus, the simplified theoretical expression for the evolution is IPL(t)=A2+B2exp[−(Γ+γ)t], and AHF does not appear. The data are plotted in Fig. 4b along with a fit (solid line) to the full theory for this pulse sequence. We obtain τ*=7.1±1.3 ns, which agrees with the Rabi measurements of τ* within the uncertainty (Fig. 2a, inset: right-hand panel).

Figure 4: ES Hahn-echo measurement of spin dephasing.
figure 4

a, Bloch-sphere representation of the ES Hahn-echo experiment. b, ES Hahn-echo data, with an inset showing the pulse sequence and orbital decay. The solid line is a fit to the full theory. c, B0 plotted as a function of strain. d, Schematic of an ES swap between the NV centre electronic spin and the nuclear spin of nitrogen as described in the text.

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The strong ES hyperfine interaction may be a promising route to an optically controlled swap gate between the NV electronic spin and its intrinsic nitrogen nuclear spin. The nitrogen nuclear spin is attractive as a quantum memory because they are present in every NV centre with the same coupling strength and because nuclear spins have long coherence times3. The ES hyperfine coupling has been used to polarize the 15N nuclear spin of the NV centre and to increase the spin-dependent contrast of IPL (refs 7, 19, 20, 21). Our studies demonstrate that controlled coherent interactions between the nuclear spin and the ES electronic spin are also possible. One scheme is outlined in Fig. 4d. Working near the avoided crossing of the motionally narrowed ES levels (B∼500 G) where the eigenstates of the coupled electron–nuclear spin states become hybridized, the NV centre can be optically pumped into the |0,+1/2〉 state in the |ms,mI〉 basis19,22. Next, a GS microwave π pulse rotates the NV electronic state into |−1,+1/2〉, followed by a green optical pulse that stimulates excitation and ‘turns on’ coherent precession between |−1,+1/2〉 and |0,−1/2〉 in the ES. After a 16–18 ns precessional half-cycle, a red laser pulse stimulates emission to the ground state, swapping the state of the electronic spin into the nuclear spin.

To achieve the ES nuclear swap operation, both spin dephasing and orbital relaxation need to be controlled. Spin dephasing can be reduced by engineering higher strain to reduce B0 (Fig. 4c) and by increasing the temperature to increase R. Although orbital relaxation is more difficult to control, advances in fabrication may provide solutions. If the collection efficiency can be increased to near unity then we may use postselection, where the operation is repeated whenever a photon is detected before the precessional half-cycle is complete23. A number of efforts are already underway to boost collection efficiency, including the fabrication of diamond nanowires with embedded NV centres24 and the development of hybrid photonic structures and waveguides25. Controlling the density of optical modes can also suppress spontaneous emission, which will minimize the number of unsuccessful attempts. Such suppression is observed in diamond nanocrystals26, and further suppression may be achieved using photonic bandgap structures tuned to overlap the energy range of the NV centre emission27. Although significant challenges remain, there is a clear pathway towards realizing an ES nuclear swap.

In semiconductor spintronics, spin decoherence determined by both orbital relaxation and spin dephasing is a common situation. These experiments demonstrate that fast, coherent control over an ES electronic spin can be used to differentiate between these effects and to study coherent electron–nuclear interactions. Techniques presented here may prove useful in studies of other spin systems where orbital and spin relaxation act on similar timescales. Moreover, these experimental and theoretical results provide a means to engineer the NV centre ES as a valuable quantum resource for solid-state quantum information processing at room temperature.

Methods

Samples consist of single-crystal diamond grown by chemical vapour deposition (Element-6 electronic grade). The sample described here was implanted with 109 15N ions cm−2 at 20 keV and subsequently annealed at 850 C for 10 min. We examine single NV centres located near the short termination of lithographically fabricated waveguides12. Photon antibunching measurements ensure that each NV centre is a single quantum emitter. We studied a total of five NV centres, which all showed consistent behaviour.

To study spin coherence during the short orbital lifetime requires on-resonance Rabi frequencies greater than 200 MHz. Such large driving fields will also stimulate transitions on nearby states within a frequency range comparable to the Rabi frequency owing to the Lorentzian pre-factor in Rabi’s formula15. To constrain our manipulation to two-level dynamics and to ensure that our driving will produce spin rotation only in the excited state, we work at B=1,276 G. At this field the ES ms=0 to ms=−1 transition frequency is 2.14 GHz. All other GS- and ES-spin transitions are strongly detuned compared with H1. We optimized our microwave pulses at this frequency to limit the pulse rise and fall times to approximately 200 ps.

There are two sources of illumination in our experiment. The first is a 532 nm continuous-wave laser used for initialization and then turned off with an acousto-optic modulator. Co-aligned with this laser is fibre-coupled 554 nm light from an optical parametric oscillator pumped with a titanium:sapphire pulsed laser used for timed excitation. We estimate that the pulses are roughly 6 ps in duration after passing through the fibre. They are pulse-picked to a 66 ns pulse period. Synchronized electronics generate microwave pulses at a fixed delay with respect to illumination.