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Coherent Dynamics of Coupled Electron and Nuclear Spin Qubits in Diamond

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Science  13 Oct 2006:
Vol. 314, Issue 5797, pp. 281-285
DOI: 10.1126/science.1131871

Abstract

Understanding and controlling the complex environment of solid-state quantum bits is a central challenge in spintronics and quantum information science. Coherent manipulation of an individual electron spin associated with a nitrogen-vacancy center in diamond was used to gain insight into its local environment. We show that this environment is effectively separated into a set of individual proximal 13C nuclear spins, which are coupled coherently to the electron spin, and the remainder of the 13C nuclear spins, which cause the loss of coherence. The proximal nuclear spins can be addressed and coupled individually because of quantum back-action from the electron, which modifies their energy levels and magnetic moments, effectively distinguishing them from the rest of the nuclei. These results open the door to coherent manipulation of individual isolated nuclear spins in a solid-state environment even at room temperature.

The controlled, coherent manipulation of quantum-mechanical systems is an important challenge in modern science and engineering (1). Solid-state quantum systems such as electronic spins (210), nuclear spins (11, 12), and superconducting islands (13) are among the most promising candidates for realization of qubits. However, in contrast to isolated atomic systems (14), these solid-state qubits couple to a complex environment, which often leads to rapid loss of coherence and, in general, is difficult to understand (1519).

We used spin-echo spectroscopy on a single-electron solid-state qubit to gain insight into its local environment. We investigated a single nitrogen-vacancy (NV) center in a high-purity diamond sample and showed that its electron spin coherence properties are determined by 13C nuclear spins. Most importantly, we demonstrated that the electron spin couples coherently to individual proximal 13C spins. By selecting an NV center with a desired nearby 13C nucleus and adjusting the external magnetic field, we could effectively control the coupled electron-nuclear spin system. Our results show that it is possible to coherently address individual isolated nuclei in the solid state and manipulate them via a nearby electron spin. Because of the long coherence times of isolated nuclear spins (20), this is an important element of many solid-state quantum information approaches from quantum computing (11, 12) to quantum repeaters (21, 22).

Spin echo is widely used in bulk electron spin resonance (ESR) experiments to study interactions and to determine the structure of complex molecules (23). Recently, local contact interactions were observed between single-NV electronic spins and the nuclear spins associated with the host nitrogen and the nearest-neighbor carbon atoms (3, 24). In the latter case, coherent dynamics of electron and nuclear spins were observed (3). We show that coherent coupling extends to separated isolated nuclei, which nominally constitute the electron environment and couple weakly to the electron spin.

The NV center stands out among solid-state systems because its electronic spin can be efficiently prepared, manipulated, and measured with optical and microwave excitation (2). The electronic ground state of the NV center is a spin triplet that exhibits a 2.87-GHz zero-field splitting, defining the axis of the electron spin (Fig. 1A). Application of a small magnetic field splits the magnetic sublevels ms = ±1, allowing selective microwave excitation of a single spin transition.

Fig. 1.

(A) The energy level structure of the NV center. (B) Scanning confocal image showing locations of single NV centers A to F. (Inset) A representative measured auto-correlation function g(2)(τ) from a single NV center, where g(2)(0) ≪ ½ indicates that we are exciting a single quantum emitter. (C) Experimental procedure. (D) Driven spin oscillations (Rabi nutations). The percent change in fluorescence between the signal and reference is observed as a function of resonant microwave (MW) pulse duration (inset) for NV center B. (E) Electron-spin free precession. The data were taken with a microwave detuning of 8 MHz as a function of delay between the two π/2 pulses (inset). The Ramsey signal was fitted (red) to Embedded Image, where fi values correspond to the level shifts from the host 14N nuclear spin, obtaining T2* = 1.7 ± 0.2 μs.

Our observations can be understood by considering how the NV electron spin interacts with a proximal spin-½ nucleus in the diamond lattice. If the electron spin is in the state with zero magnetic moment (ms = 0), it does not interact with the nuclear spin, which is thereby free to precess under the influence of a small magnetic field applied externally. However, if the electron is in either of the ms = ±1 states, then it generates a large local magnetic field that inhibits the free precession of nearby nuclei (25, 26). Hence, the nuclear precession is conditional on the state of the electron. In particular, if the electron spin is prepared in a superposition state, then it becomes entangled with the nuclear spins at a rate determined by the external magnetic field, i.e., the Larmor frequency. In practice, the diamond lattice contains a large number of randomly placed nuclear spins. The electron becomes entangled with all of them and thus decoheres on the time scale of the Larmor period. Coherent coupling to individual proximal nuclear spins is nevertheless possible, because the electron spin effectively enhances their magnetic susceptibilities and hence their precession frequency.

In our experiments, single NV centers were isolated and addressed at room temperature by using optical scanning confocal microscopy (Fig. 1B) with excitation at 532 nm and fluorescence detection over the range from 650 to 800 nm. Each circled spot is a single NV center, which was verified by photon correlation measurements (inset). We investigated over 20 individual centers in detail, and where relevant we indicate which center we observed. The 532-nm excitation polarizes the spin triplet into the ms = 0 state on the time scale of a few microseconds. After microwave manipulation of the spin, we detected the remaining population in the ms = 0 state by again applying the excitation laser. Just after the 532-nm light is applied, the ms = 0 state fluoresces more strongly than the ms = ±1 states, allowing measurement of the spin (Fig. 1C) (27). Oscillations in fluorescence occur as a function of the duration of a microwave pulse resonant with the ms = 0 to ms = 1 transition (2) (Fig. 1D). These Rabi nutations should correspond to complete population transfer within the two-state system. Fluorescence data were thus normalized in units of ms = 0 probability, p, where p = 1and p = 0 correspond to the maximum and the minimum fluorescence, respectively, in a fit to Rabi oscillations observed under the same conditions.

To probe coherence properties of single electron spins, we make use of Ramsey spectroscopy and spin echo techniques (28). The free electron spin precession [Ramsey signal (28)] dephases on a fast time scale, T2* = 1.7 ± 0.2 μs (Fig. 1E). Moreover, the signal exhibits a complex oscillation pattern caused by level shifts from the host 14N nucleus and other nearby spins (27). These frequency shifts can be eliminated by using a spin-echo (or Hahn echo) technique (29). It consists of the sequence π/2 – τ – π – τ′ – π/2, where π represents a microwave pulse of sufficient duration to flip the electron spin from ms = 0 to ms = 1 and τ and τ′ are durations of free precession intervals. When the two wait times are equal, τ = τ′, this sequence decouples the spin from an environment that changes slowly compared with τ (Fig. 2A). Decay of a typical Hahn echo signal (Fig. 2B) yielded a much longer coherence time, τC ≈ 13 ± 0.5 μs δ T2*/2, thus indicating a long correlation (memory) time associated with the electron spin environment.

Fig. 2.

(A) Spin echo. The spin-echo pulse sequence (left) is shown along with a representative time-resolved spin echo (right) from NV center B. A single spin echo is observed by holding τ fixed and varying τ′. (B) Spin echo decay for NV center B in a small magnetic field (B ∼5 G). Individual echo peaks are mapped out by scanning τ′ for several values of τ (blue curves). The envelope for the spin echoes (black squares), which we refer to as the spin-echo signal, maps out the peaks of the spin echoes. It is obtained by varying τ and τ′ simultaneously so that τ = τ′ for each data point. The spin-echo signal is fitted to exp[–(τ/τC)4] (red curve) to obtain the estimated coherence time τC = 13 ± 0.5 μs. (C) Collapse and revival of the spin-echo signal from NV center B in a moderate magnetic field (53 G). The decay of the revivals (blue curve) is found by fitting the height of each revival to exp[–(2τ/T2)3], as would be expected from 13C dipole-dipole induced dephasing (24, 31), with T2 ≈ 242 ± 16 μs. (D) Simulation of collapse and revival for an NV center in 53 G applied magnetic field, surrounded by a random distribution of 1.1% 13C spins (27). Additional structure in the simulation arises from coherent interactions with the nearest 13C in the lattice, via the same mechanism shown in Fig. 4. The phenomenological decay is added to the simulation for comparison with experimental data.

Spin-echo spectroscopy provides a useful tool for understanding this environment: By observing the spin-echo signal under varying conditions, we can indirectly determine the response of the environment and, from this, glean details about the environment itself. In particular, we observe that the echo signal depends on the magnetic field. As the magnetic field is increased, the initial decay of the spin echo signal occurs faster and faster. However, the signal revives at longer times, when τ equals τR (30). Figure 2C shows a typical spin-echo signal (center B) in moderate magnetic field as a function of time (τ = τ′). The initial collapse of the signal is followed by periodic revivals extending out to 2τ ∼240 μs. We find that the revival rate, 1/τR, precisely matches the Larmor precession frequency for 13C nuclear spins of 1.071 kHz/G (Fig. 3A). This result indicates that the dominant environment of the NV electron spin is a nuclear spin bath formed by the spin-½ 13C isotope, which exists in 1.1% abundance in the otherwise spinless 12C diamond lattice (Fig. 3B). The 13C precession induced periodic decorrelation and rephasing of the nuclear spin bath, which led to collapses and revivals of the electron spin-echo signal (Fig. 2C).

Fig. 3.

(A) Spin-echo revival frequency as a function of magnetic field amplitude. Data from three representative centers—NV B, NV F, and NV G (not shown in Fig. 1B)—exhibit revivals that occur with the 13C Larmor precession frequency (red). The data points for NV centers B and F were taken with B, whereas the data for NV center G were taken in a variety of magnetic field orientations. (B) Illustration of the 13C environment surrounding the NV center. (C) Physical model for spin-echo modulation.

Every NV center studied exhibited spin-echo collapse and revival on long time scales, but many also showed more complicated evolution on short time scales. As an example, the spinecho signal from NV center E (Fig. 4A) showed oscillations with slow and fast components at ∼0.6 MHz and ∼9 MHz, respectively. The fast component (referred to as the modulation frequency) was relatively insensitive to the magnetic field (Fig. 4B), but the slow component (envelope frequency) varied dramatically with the magnetic field amplitude and orientation (Fig. 4, C and D). These observations indicate that the electron spin gets periodically entangled and disentangled with an isolated system until the spin echo finally collapses from interactions with the precessing bulk spin bath. Although the data are not shown, some NV centers, for example NV C, exhibited several envelope and modulation frequencies, indicating that the electron spin interacts coherently with multiple 13C nuclei. Other centers, for example NV F, showed no evidence of proximal 13C spins.

Fig. 4.

(A) Spin-echo modulation as observed for NV E with B = 42 ± 6 G ∥ – + . The red curve represents a theoretical fit with the expected functional form exp[–(τ/τC)4][ab sin(ω0 τ/2)2sin(ω1 τ/2)2], yielding the modulation frequency ω0 ∼2π 9 MHz and envelope frequency ω1 ∼2π 0.6 MHz. (B) Modulation frequency for NV B to E as a function of magnetic field B ∥ – + . (C) Envelope frequency [same conditions as (B)]. The envelope frequencies are different for each center, but they all exceed the bare 13C Larmor precession frequency (dashed line). (D) Effective gyromagnetic ratio (envelope frequency/magnetic field) versus magnetic field orientation for NV centers D and E. The amplitude of the magnetic field is fixed at 40 ± 4 G. The magnetic field is varied in the xz plane for NV center D (red boxes) and NV center E (green triangles) and yz plane for NV center D (dark red stars) and NV center E (dark green diamonds). Six free parameters that describe the interaction with the nearest 13C spin were fit to the envelope and modulation frequency data (27), yielding the solid curves shown in (B) to (D). Error bars indicate a 95% confidence interval for fits to the data (y axis) and estimated error in magnetic field measurement (x axis) obtained from the discrepancy between Hall sensor measurements and the observed Zeeman splitting of the NV center.

To provide a quantitative explanation for these results, we first considered the spin-echo signal arising from a single 13C nucleus I(j) located a distance rj in the direction nj from the NV spin. This 13C spin couples to the electron spin via the hyperfine interaction (23, 31): Math(1) Math where μe and μn are the electron and nuclear magnetic moments, respectively, |ψe(rj)|2 is the electron spin density at the site of the nuclear spin, and angle brackets denote an average over the electron wavefunction, ψe(r). The essence of this Hamiltonian, which can be represented as V(j) = B(j)ms·I(j), is that the nuclear spin experiences an effective magnetic field, B(j)ms, that depends on the electron spin state ms. This electron spin state–dependent magnetic field leads to conditional evolution of the nuclear spin, thereby entangling the two spins. Because of the spatial dependence of the hyperfine interaction, these effects decrease rapidly with distance from the NV center, making proximal nuclei stand out from the remainder of the spin bath.

The hyperfine interaction between the electron spin and a single nuclear spin has a dramatic effect on the spin-echo signal. After the initial π/2 pulse in the spin-echo sequence, the electron spin state Math becomes entangled with the nuclear spin state at a rate determined by B(j)0 and B(j). As the electron spin becomes entangled with the nuclear spin, the spin-echo signal diminishes; when it gets disentangled, the signal revives. The resulting spin-echo signal thus exhibits periodic reductions in amplitude, with modulation frequencies ωj,ms associated with each spin-dependent field B(j)ms. By considering the unitary evolution associated with the dipole Hamiltonian [see, e.g., (26) for derivation], we obtained a simple expression for the spin echo signal, pj = (Sj + 1)/2, with pseudospin Sj given by Math(2) Math

Because the electron spin dipole field is stronger for ms = 1, we associated ωj,1 with the fast modulation frequency and ωj,0 with the slower envelope frequency. Furthermore, we included multiple 13C nuclei in our description by taking a sum over the dipole interactions, V = ΣjV(j); the corresponding unitary evolution yields the echo signal p = (S + 1)/2 with S = Πj Sj.

We began with a simple treatment, which neglected the terms proportional to Sx and Sy because they are suppressed by the large electron-spin splitting Δ ≈ 2.87 GHz [the so-called secular approximation (23)]. In this model (Fig. 3C), the ms = 1 nuclear-spin states have a fixed hyperfine splitting, ωj,1 ∼μeμn[<1/rj3> + 8π|Ψe(rj)|2/3], whereas the degenerate ms = 0 nuclear-spin states can precess in a small applied magnetic field at the bare 13C Larmor frequency ωj,0 = ω0. When we included many nuclear spins in the echo signal, the fast echo modulations ωj,1 interfered with each other, causing initial decay of the signal as exp–(τ/τC)4. However, when τ = τ′ = 2π/ω0, Sj equaled 1 for all j, and the spin-echo signal revived. Simulations based on Eqs. 1 and 2 (Fig. 2D) are in good agreement with the observed collapses and revivals.

Such a simple picture cannot explain the observed echo modulation, however, because it predicts that the spin-echo signal should collapse before coherent interactions with individual 13C spins become visible. In fact, the nonsecular terms in the Zeeman and dipole interactions slightly mix the electronspin levels, introducing some electronic character to the nearby nuclear-spin levels and thus augmenting their magnetic moment by ∼μej,1/Δ). Because μe δ μn, this greatly enhances the nuclear Larmor precession rate for nearby spins. Furthermore, the enhancement is anisotropic: It is strongest when the external field is oriented perpendicular to the NV axis, corresponding to the largest degree of mixing. For a properly oriented magnetic field, proximal nuclei can thereby entangle and disentangle with the NV spin on time scales much faster than the bare 13C Larmor period.

These theoretical predictions are in good agreement with our observations (Fig. 4). In addition, by fitting the envelope frequency as a function of magnetic field, we were able to estimate the six coupling parameters that characterize the hyperfine interaction (27). In principle, these parameters should also allow for precise determination of absolute nuclear position in the diamond lattice. However, direct comparison to the microscopic model depends sensitively on the details of the electronic wave function because of both the isotropic contact contribution and the averaged dipolar term in Eq. 1. Our results indicate that both terms can be important. For example, although the point dipole approximation yields results that are qualitatively similar to experimental data, it underestimates the coherent coupling strength as well as the echo collapse rate. At the same time, fits for NV centers D and E (Fig. 4, B to D, solid lines) indicate that some amount of anisotropic dipolar contribution is present (27). In fact, these fits yield an estimate of the electron-spin density at the positions of the proximal nuclei (27); by analyzing such data from a sufficiently large number of individual NV centers, it may be possible to determine the electronic wavefunction. This intriguing problem warrants future investigation.

Beyond providing a detailed insight into the mesoscopic environment of the spin qubit, our observations demonstrate a previously unknown mechanism for selective addressing and manipulation of single, isolated nuclear spins, even at room temperature. For example, such nuclear spins could be used as a resource for long-term storage of quantum information. They can be effectively manipulated via nearby electronic spins and potentially coupled together to explore a variety of proposed quantum information systems (11, 21).

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