One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment

Abstract

Single electron spins coupled to multiple nuclear spins provide promising multi-qubit registers for quantum sensing and quantum networks. The obtainable level of control is determined by how well the electron spin can be selectively coupled to, and decoupled from, the surrounding nuclear spins. Here we realize a coherence time exceeding a second for a single nitrogen-vacancy electron spin through decoupling sequences tailored to its microscopic nuclear-spin environment. First, we use the electron spin to probe the environment, which is accurately described by seven individual and six pairs of coupled carbon-13 spins. We develop initialization, control and readout of the carbon-13 pairs in order to directly reveal their atomic structure. We then exploit this knowledge to store quantum states in the electron spin for over a second by carefully avoiding unwanted interactions. These results provide a proof-of-principle for quantum sensing of complex multi-spin systems and an opportunity for multi-qubit quantum registers with long coherence times.

Fig. 1

Experimental system and T₁ measurements. a We study a single nitrogen-vacancy (NV) center in diamond surrounded by a bath of ¹³C nuclear spins (1.1% abundance). In this work, we show that the microscopic nuclear-spin environment is accurately described by 7 isolated ¹³C spins, 6 pairs of coupled ¹³C spins and a background bath of ¹³C spins (not depicted). b Longitudinal relaxation of the NV electron spin. The spin is prepared in $m_s=0,-1$, or + 1 and the fidelity with the initial state is measured after time t. The inset shows the microwave (MW) and laser controls for the NV spin and charge states, as well as the pathways for spin relaxation induced by potential background noise from these controls. All error bars are one statistical s.d.

Fig. 2

Quantum sensing of the microscopic spin environment. a Dynamical decoupling spectroscopy revealing a rich nuclear-spin environment consisting of individual ¹³C spins, as well as pairs of coupled ¹³C spins. The electron spin is prepared in a superposition, $\mid x⟩=\left(\mid m_s=0⟩+\mid m_s=-1⟩\right)∕\sqrt{2}$ and a decoupling sequence of N = 32 π-pulses separated by 2τ is applied. Loss of coherence indicates the interaction of the electron spin with nuclear spins in the environment. Blue: data. Purple line: theory (see Methods). The shaded areas mark the signals due to six ¹³C–¹³C pairs labeled 1–6. b Zoom-in showing sharp signals due to coupling to isolated individual ¹³C spins^–. The total signal is well described by seven ¹³C spins (see Supplementary Table 2 for hyperfine parameters) and a bath of 200 randomly generated spins with hyperfine couplings below 10 kHz. c Zoom-in showing a broad signal due to ¹³C–¹³C pair 1^,. Blue: data. The solid orange line is the theoretical signal just due to pair 1, while the purple line includes the seven individual ¹³C spins and the ¹³C spin bath as well

Fig. 3

Direct spectroscopy of nuclear-spin pairs. a Measurement sequences for Ramsey spectroscopy of ¹³C–¹³C pairs, for $Z>>X$ and for $X>>Z$. The controlled ±x (±z) gates are controlled ±π/2 rotations around x (z) with the sign controlled by the electron. The initial states are ${\rho }_0=\mid 0⟩⟨0\mid$ and the mixed state ${\rho }_m$. b, c Nuclear spin Ramsey measurements and obtained precession frequencies for pairs 2, 4, and 6. The electron spin state during the free evolution time t is set to m_s = 0 (b) or $m_s=-1$ (c) and an artificial detuning is applied. Each pair yields a unique set of frequencies, confirming that the pairs are distinct. For pair 2 an additional beating is observed (frequency of 23(3) Hz), indicating a small coupling to one (or more) additional spins. See Supplementary Fig. 3 for the other three pairs and Supplementary Table 4 for fit results. All error bars are one statistical s.d.

Fig. 4

Atomic structure and decoupling signal for the six nuclear-spin pairs. a Structure of the six ¹³C–¹³C pairs within the diamond unit cell (up to symmetries and equivalent orientations). The z-values give the height in fractions of the diamond lattice constant a₀. The magnetic field is oriented along the <111> direction, i.e., along the axis of pair 4. For pair 3 there is an additional possible structure that yields a similar X, Supplementary Table 3. b The calculated signal for the six individual ¹³C–¹³C pairs accurately describes the measured decoupling signal for different number of pulses N. Data are taken for $\tau =m\cdot \frac{2\pi }{{\omega }_{\mathrm{L}}}$ to avoid coupling to single-¹³C spins. See Supplementary Fig. 5 for other values of N

Fig. 5

Protecting quantum states with tailored decoupling sequences. a Normalized signal under dynamical decoupling with the number of pulses varying from N = 4 to N = 10,240. The electron is initialized and readout along x. The thin lines are fits to equation (4), which takes into account the six identified ${}^{13}{\mathrm{C}}^{13}\mathrm{C}$ pairs. We use the extracted amplitudes A to re-normalize the signal. Thick lines are the extracted envelops $\left(0.5+0.5\cdot e^{-{\left(t∕T\right)}^n}\right)$ with T and n obtained from the fits. See Supplementary Fig. 6 for the obtained values n. b Scaling of the obtained coherence time T as function of the number of pulses (error bars are <5%). The solid line is a fit to the power function $T_{N=4}\cdot {\left(N∕4\right)}^{\eta }$, where T_N=4 is the coherence time for N = 4. We find η = 0.799(2). c The average state fidelity obtained for the six cardinal states (Supplementary Fig. 8). Unlike in a, the signal is shown without any renormalization. The number of pulses N is chosen to maximize the obtained signal at the given total evolution time while avoiding interactions with the ¹³C environment. The solid green line is a fit to an exponential decay. The horizontal line at $\frac{2}{3}$ fidelity marks the classical limit for storing quantum states. The two curves cross at t = 1.46 s demonstrating the protection of arbitrary quantum states well beyond a second. All error bars are one statistical s.d.