Temperature-dependent coherence properties of NV ensemble in diamond up to 600 K

The nitrogen-vacancy (NV) center in diamond is an ideal candidate for quantum sensing because of its excellent spin coherence property as well as the possibility for optical initialization and readout. Previous studies, on the other hand, have typically been conducted at low or room temperature. The inability to fully understand the coherence properties of the NV center at high temperatures limits NV's further applications. We systematically investigate the coherence properties of NV center ensemble at temperatures ranging from 300 K to 600 K in this paper. Coherence time $T_2$ decreases rapidly from $184\mu \text{s}$ at 300 K to $30\mu \text{s}$ at 600 K due to the interaction with paramagnetic impurities. At all experiment temperatures, both single-quantum and double-quantum transitions exhibit a $T^5$ relaxation rate, which is attributed to the two-phonon Raman process. Nonetheless, the inhomogeneous dephasing time $T_2^{*}$ and thermal-echo decoherence time $T_{\text{TE}}$ remain almost unchanged at temperature up to 600 K. A thermal-echo-based thermometer is demonstrated to have a sensitivity of $41\text{mK}/\sqrt{\mathrm{Hz}}$ at 450 K. These findings will pave the way for NV-based high-temperature sensing and provide a more comprehensive understanding of solid-state qubit decoherence.

Figure 1

(a) Left: The electronic energy level diagram for the NV center; the degenerated sublevel $|m_s=\pm 1\rangle$ is split with the magnetic field applied. The straight lines represent optical transitions, and the snaked lines represent nonradiative decay. Right: Ground state energy level of the NV center. The blue arrows indicate the single-quantum channel, and the corresponding relaxation rate is $\mathrm{\Omega }$ (here we set ${\mathrm{\Omega }}_{0,+1}={\mathrm{\Omega }}_{0,-1}=\mathrm{\Omega }$, as verified experimentally in Ref. [19]). The red arrow indicates the double-quantum channel, and the corresponding relaxation rate is $\gamma$. (b) Schematic showing the experimental arrangement of the sample and the heating part. The diamond sample is adhered to a resistance heating plate with the temperature detected by a resistive temperature detector (RTD) and controlled with a temperature controller. A ring-shaped antenna is set on the sample to deliver the MW field. (c) Zero field CW ODMR spectra at three temperatures, 300 K, 450 K, 600 K (from the bottom up). (d) CW ODMR at three temperatures with a 50 G magnetic field applied.

Figure 2

Ramsey and thermal echo measurement. (a) Pulse sequence for Ramsey and thermal echo, respectively. Upper panel: Ramsey; lower panel: thermal echo. Minus ($-$) denotes the transition $|0\rangle \leftrightarrow |-1\rangle$, and plus ($+$) denotes the transition $|0\rangle \leftrightarrow |+1\rangle$. (b) Ramsey fringes at 300 K, 450 K, and 600 K (from the bottom up). The solid line represent the fitting curves. The fitting $R$ squares are $R^2=0.92,0.95,0.83$ from the bottom up. (c) Inhomogeneous dephasing time $T_2^{*}$ from 300 K to 600 K. (d) Thermal echo curves at 300 K, 450 K, and 600 K (from the bottom up). The solid lines represent the fitting curves ($R^2=0.94,0.90,0.84$, from the bottom up). (e) Thermal echo coherence time $T_{\text{TE}}$ from 300 K to 600 K. Error bars in (c) and (e) represent one standard deviation found from the fitting.

Figure 3

DQ and SQ relaxation experiment. (a),(b) Measurement sequences to extract the relaxation rate $\mathrm{\Omega }$ and $\gamma$. Minus (-) denotes the transition $0\leftrightarrow -1$, and plus (+) denotes the transition $0\leftrightarrow +1$. (c) Relaxation curves acquired by using the sequence in (a) at 300 K, 450 K, 600 K, respectively. The data are fitted with function $\text{exp}(-3\mathrm{\Omega }\tau )$, with the fitting $R$ square being 0.96 (300 K), 0.98 (450 K), and 0.92 (600 K). Since spin-lattice relaxation $T_1$ is the time it takes for the curves to decay to $1/e$ of its initial value, we will have $T_1=1/3\mathrm{\Omega }$. (d) Relaxation curves acquired by using the sequence in (b) at 300 K, 450 K, and 600 K, respectively. The data are fitted by function $\text{exp}[-(\mathrm{\Omega }+2\gamma \left)\tau \right]$, with the fitting $R$ square being 0.88 (300 K), 0.96 (450 K), and 0.88 (600 K). (e) Temperature dependence of single-quantum relaxation rate $\mathrm{\Omega }$. Red points are experiment data, and the blue solid line is the fitting curve [$A_{\mathrm{\Omega }}{T}^5+B_{\mathrm{\Omega }}$, with $A_{\mathrm{\Omega }}=1.46\left(7\right)\times {10}^{-11}{\text{K}}^{-5}{\text{s}}^{-1}$ and $B_{\mathrm{\Omega }}=79\left(7\right){\text{s}}^{-1}$]. The fitting $R$ square is $R^2=0.98$. (f) Temperature dependence of double-quantum relaxation rate $\gamma$. Red points are experimental data, and the solid line is the fitting curve [$A_{\gamma }{T}^5+B_{\gamma }$, with $A_{\gamma }=0.85\left(9\right)\times {10}^{-11}{\text{K}}^{-5}{\text{s}}^{-1}$ and $B_{\gamma }=215\left(20\right){\text{s}}^{-1}$]. The fitting $R$ square is $R^2=0.93$. The error bars in (e) and (f) represent one standard deviation found from the fitting.

Figure 4

Spin-echo measurement. (a) Pulse sequences for spin-echo measurement. (b) Spin-echo curves at four temperatures: 300 K, 400 K, 500 K, 600 K (from the bottom up). The experiment data are fitted by function $\text{exp}[-{(\tau /T_2)}^p]\times {\sum }_i\text{exp}[-{(\tau -i\times T_R)}^2/T_w^2]$, with $p, T_R, T_w$ being the fit parameters. The fitting $R$ square is $R^2=0.91,0.91,0.90,0.85$ (from the bottom up). (c) The temperature dependence of coherence time $T_2$ (decoherence rate $1/T_2$). (d) The temperature dependence of pure dephasing rate, ${\mathrm{\Gamma }}_d=1/T_2-(3\mathrm{\Omega }+\gamma )/2$. (e) The temperature dependence of decoherence effect from relaxation (blue dots), P1 center (red solid line), ${}^{13}C$ nuclear spin (black dashed line). The contribution from P1 center and ${}^{13}C$ nuclear spin are estimated from Eq. (2) and Eq. (3), while the contribution from the relaxation is determined from the experiment. Error bars represent one standard deviation found from fitting.

Figure 5

Thermal echo based thermometry. (a) Thermal echo curves at three temperatures (456.9 K, 454.3 K, and 450.6 K, from the bottom up), with the applied microwave frequencies fixed. Solid lines are fitting curves. (b) Temperature dependence of thermal echo curve oscillation frequency. The red solid line is a linear fitting ($R^2=0.98$), with a slope of $132\pm 6\text{kHz/K}$. The red points indicate the data extract from (a). (c) The thermal sensitivity obtained from Eq. (5), with all parameters determined from the experiment.