Characterizing Temperature and Strain Variations with Qubit Ensembles for Their Robust Coherence Protection

Solid-state spin defects, especially nuclear spins with potentially achievable long coherence times, are compelling candidates for quantum memories and sensors. However, their current performances are still limited by dephasing due to variations of their intrinsic quadrupole and hyperfine interactions. We propose an unbalanced echo to overcome this challenge by using a second spin to refocus variations of these interactions while preserving the quantum information stored in the nuclear spin free evolution. The unbalanced echo can be used to probe the temperature and strain distribution in materials. We develop first-principles methods to predict variations of these interactions and reveal their correlation over large temperature and strain ranges. Experiments performed in an ensemble of $\sim {10}^{10}$ nuclear spins in diamond demonstrate a 20-fold dephasing time increase, limited by other noise sources. We further numerically show that our method can refocus even stronger noise variations than present in our experiments.

Figure 1

Coherence protection. (a) Energy levels of $\mathrm{N}\text{−}V$ ground state. (b) Coherence protection sequence. A laser pulse polarizes the $\mathrm{N}\text{−}V$ to $|m_S=0,m_I=+1⟩$ state. The nuclear spin state $(|0⟩+|\pm 1⟩)/\sqrt{2}$ is prepared with a $\pi /2$ rf pulse. A microwave $\pi$ pulse is applied to flip the $\mathrm{N}\text{−}V$ electronic spin state from $|0⟩$ to $|\pm 1⟩$. A final $\pi$ pulse transfers the $\mathrm{N}\text{−}V$ state back to $|0⟩$. The $\pi /2$ pulse applied to the nuclear spin associated with a conditional electronic spin $\pi$ pulse performs a readout of the nuclear spin population. (c) Free-evolution (Ramsey) measurements of unprotected and protected nuclear spin states $(|0⟩+|-1⟩)/\sqrt{2}$. At each time $t$, we sweep the phase of the last nuclear spin $\pi /2$ pulse to measure the coherence (inset). The amplitudes of the phase sweep measurements are fit to exponential decay. (d) Sweep of the pulse location $\tau /t$ with $t=1400\mathrm{\mu }\mathrm{s}$.

Figure 2

Inhomogeneity characterization. (a),(b) Sweep of pulse location $\tau /t$ under different free-precession times $t$. The electronic spin flip is between $|m_S=0⟩$ and $|m_S=+1⟩$. The nuclear spin in (a) and (b) is prepared in $(|0⟩+|-1⟩)/\sqrt{2}$, $(|0⟩+|+1⟩)/\sqrt{2}$, respectively. (c),(d) Decay rate $1/T_{2n}^{*}$ as a function of $\tau /t$ for the same nuclear and electron spin states as in (a),(b), respectively.

Figure 3

Theoretical model. (a) Spin-phonon energy levels. (b) Calculated and measured shifts of $\delta Q$, $\delta A_{zz}$ as a function of the temperature. Square and triangle data points are extracted from Soshenko et al. [36] and Jarmola et al. [37]. (c) Calculated shifts of $\delta Q$, $\delta A_{zz}$ as a function of the applied strain (compression). (d) Experimental measurements of frequency shifts as a function of temperature shifts.

Figure 4

Refocusing large temperature inhomogeneities. (a) Calculated ratios ${\alpha }_A/{\alpha }_Q$ and ${\beta }_A/{\beta }_Q$ as a function of temperature. The strain curve is the ratio between the two slopes in Fig. 3. (b) Simulated unbalanced echo ($t=2\mathrm{ms}$) for a qubit ensemble with a Lorentzian-shape temperature inhomogeneity. The nuclear spin state is $(|0⟩+|-1⟩)/\sqrt{2}$ and the electronic spin flips between $|m_S=0⟩$ and $|m_S=+1⟩$. The coherence improvement factor versus inhomogeneity ${\sigma }_T$ is shown in the inset. (c) Simulation of unprotected and protected ($\tau /t=0.172$) Ramsey experiments (here we consider only temperature-induced dephasing and neglect other noise sources, including $\mathrm{N}\text{−}V$ $T_1$.).