Ultra-long coherence times amongst room-temperature solid-state spins

Solid-state single spins are promising resources for quantum sensing, quantum-information processing and quantum networks, because they are compatible with scalable quantum-device engineering. However, the extension of their coherence times proves challenging. Although enrichment of the spin-zero 12C and 28Si isotopes drastically reduces spin-bath decoherence in diamond and silicon, the solid-state environment provides deleterious interactions between the electron spin and the remaining spins of its surrounding. Here we demonstrate, contrary to widespread belief, that an impurity-doped (phosphorus) n-type single-crystal diamond realises remarkably long spin-coherence times. Single electron spins show the longest inhomogeneous spin-dephasing time (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2^ \ast \approx 1.5$$\end{document}T2*≈1.5 ms) and Hahn-echo spin-coherence time (T2 ≈ 2.4 ms) ever observed in room-temperature solid-state systems, leading to the best sensitivities. The extension of coherence times in diamond semiconductor may allow for new applications in quantum technology.

Supplementary Note 1: T 2 in phosphorus-doped samples E-H Supplementary Fig. 1 shows T 2 s of NV centres in the samples E-H. Samples E-G have almost the same phosphorus concentrations as sample C in the main text, and sample H has a larger phosphorus concentration. It should be noted that the growth conditions of samples E-H are the same as those of samples A-D, except for the PH 3 /CH 4 gas ratio to change the phosphorus concentration as described in the main text and in the Methods section. In samples E-G, we confirmed that several NV centres in each sample have a T 2 longer than 2.0 ms. In the high-concentration sample H, we found that T 2 was below 2 ms, which indicates again that an optimum exists. Supplementary Note 2: T * 2 measurement T * 2 follows from a free-induction decay measurement (FID) 1,2 . When detuning the frequency of the microwave field from the resonance frequency between the used energy levels of the NV centre, the phase of the electron spin rotates with this detuning frequency (in the rotational frame). The results are fitted to with A and B constants, τ the delay, n the power of the exponent, ω the detuning frequency, and φ the phase (usually −π). Since the points that effect the power n are limited, the fitted n has a large uncertainty. To ensure fair comparison between NV centres, since n and T * 2 depend on each other, it was chosen to fix n at 2. Roughly speaking, there are two different factors limiting the resulting T * 2 . The first are factors inside the sample, such as fluctuations in magnetic field created by nearby spins. The second are factors outside the sample related to the measurement setup, such as change of temperature which shifts the sample location and changes the magnetic field of the static magnet, and hence the magnetic field at the NV centre; and the m s = 0 energy level depends on temperature 3 as well. The former factors limit the actual T * 2 , and this is the T * 2 to measure. The latter factors limit the apparent T * 2 , but only due to external (in principle solvable without changing the sample) issues. For example, this is visible in a daily cycle of the detuning frequency (when keeping the microwave frequency the same), which might be related to the outside temperature.
To investigate the effect of this external additional change in detuning frequency, it was simulated numerically, an example is shown in Supplementary Fig. 2a. Supplementary Fig. 2b shows the effect of a frequency shift of 60 Hz min −1 , a fairly average rate during the day in our measurement setup, on the apparent T * 2 for different measurement times, for a number of actual T * 2 s. As expected, the longer the measurement, the worse the result. For measurements taking longer than 15 min, the actual T * 2 is getting rather obscured by the apparent T * 2 . The effect is stronger for a longer T * 2 . In Supplementary Fig. 2c, the effect of different frequency shifts is simulated for a measurement time of 10 min. For shifts larger than about 80 Hz min −1 , the effect makes it difficult to differentiate between the actual T * 2 s, for longer T * 2 it becomes difficult with slower shifts already. Supplementary Fig. 2b and c are expected to look similar, since both the measurement time and the frequency shift affect the total change in detuning during the complete measurement.
There are several ways to deal with such slow changes. One way is to measure the average shift in the optically detected magnetic resonance (ODMR) spectrum 4 troughs' frequencies over time, and shift the frequency during the measurement accordingly. A second way is limiting the time of the measurement, since the shorter it is, the smaller the total shift. The latter method is shown in the paper, where the measurement has two parts. A relatively high detuning frequency is chosen, which is found by the first part of the pulse sequence, which uses short delays. This part takes relatively little time. The second part of the pulse sequence uses long delays, with enough points to capture the oscillation. Since the points in between are skipped, this limits the measurement time of the total sequence.
To find NV centres with a high potential to have a long actual T * 2 , first, short measurements (∼5 min) with a gap of 0.5 ms are conducted. Since for long T * 2 the decay is rather small in these measurements, these only indicate whether their T * 2 is beyond 0.5 ms, an example is shown in Supplementary Fig. 2d. The most promising ones are measured with a longer gap, as shown in Supplementary Fig. 2e and f, and in the main text in Fig. 2b. However, the measurement time for a 1.0 ms gap is about 10 min, which means that the apparent T * 2 will become shorter (see Supplementary  Fig. 2b). Therefore, although it is easy to find NV centres with results as in Supplementary Fig. 2d, extending the measurement time often reduces the apparent T * 2 to below 1.0 ms, even though the actual T * 2 is probably higher (see Supplementary Fig. 2b).
To conclude, the described effect in our measurement setup is given as example. This effect depends on many factors, such as the design of the experimental room and the season, and thus is different depending on the specific environment. For our setup in summer, the temperature oscillates with an amplitude of 1 K. When merely looking at the shift of the m s = 0 level by -74.2 kHz K −1 3 , this equals a change of ∼150 kHz per ∼12 h, thus a change of ∼2 × 10 2 Hz min −1 . This is larger than the 60 Hz min −1 used in the examples of Supplementary Fig. 2, since the actual shift is lower when closer to the more stable times of the day. Measurements were carried out around these stable times, to limit the effect as much as possible, which enables results as shown in Fig. 2b in the main text.

Supplementary Note 3: Magnetic field calibration
When applying a magnetic field along the symmetry axis of the NV centre (here the z-axis), the m s = ±1 energy levels are Zeeman split 5 , with Zeeman energy with µ z the magnetic moment along the z-axis, B z the magnetic field along the z-axis, g the g-factor of the electron spin, e the electron charge, m e the mass of an electron, S z the angular momentum along the z-axis, reduced Planck's constant, and µ B the Bohr magneton. Since ∆E = h∆ f , with h Planck's constant and ∆ f the shift in resonance frequency, the change in applied magnetic field amplitude follows from the locations of the troughs in the opticallydetected magnetic resonance (ODMR) spectrum 4 as To calibrate the magnetic field induced by the coil near the sample, a series of DC voltages was applied to the coil, and the magnetic field amplitude was derived from the shift of the troughs in the ODMR spectrum. The result, plotted in Supplementary Fig. 3, is fitted to a first-order polynomial, from which the calibration factor follows (|δB| = 12.10 ± 0.05 µT V −1 ).

Supplementary Note 4: Explanation of δB min
To derive the minimum detectable magnetic field amplitude δB min , first the magnetic field measurement itself needs to be defined. To measure the magnetic field amplitude, a Hahn-echo measurement is performed (see Fig. 3a in the main text). Then, the resulting intensity is converted into a magnetic field amplitude via their relationship (see Fig. 3b in the main text). This means, since this relation is sinusoidal, a certain working region around a working point needs to be chosen. The sensor would be most sensitive when the gradient (grad) in this point is largest. Thus, the initial system would be in one of these working points, then the magnetic field to measure should be applied, and after measuring the Hahn-echo intensity, the magnetic field can be computed. For the AC magnetic field measurement, δB min is the essential measure for its quality (the smaller the better). It is related to the uncertainty in the detected magnetic field amplitude. Given the uncertainty σ 1 of the measured variable (the intensity of a Hahn-echo measurement, as described above), the uncertainty in the magnetic field amplitude σ B follows from the gradient grad, as illustrated in Supplementary Fig. 4a. This uncertainty is defined as the minimum detectable magnetic field amplitude, thus giving Please note that since this is merely a single standard deviation, for Gaussian variables, this only encompasses 38% of the results (so it has a 38% confidence interval), as shown in Supplementary Fig. 4b. Although this is not convincing as certainty (an 87% confidence interval, equivalent to δB min = 3σ B , would be more appropriate), as long as the same definition is used (in others' experiments), the results can be compared.
Supplementary Fig. 4. Origin of δB min . a The uncertainty in magnetic field amplitude σ B relates directly to the uncertainty in the Hahn-echo intensity σ 1 via the gradient grad at the working point. b Gaussian distribution with several standard deviations indicated: a single standard deviation σ includes 38% of the results (inner magenta dashed lines), two standard deviations 2σ include 68% of the results (middle cyan dashed lines), and three standard deviations 3σ include 87% of the results (outer orange dashed lines).

Supplementary Note 5: Optimum frequency for AC magnetic field measurement
Below, the optimum time period for the AC magnetic field measurement, and hence the frequency, is derived. The derivation starts with describing the sensitivity. Such descriptions exist, and are encouraged to read for comparison 6 . However, please note two important differences. The first, here, the focus is on calculating the optimum frequency. Secondly, the approach taken here is more experimentally oriented, while the formulae in for example 6 are more theoretically oriented. The frequency of the magnetic field f B = 1/t that yields the lowest sensitivity is found as follows (definitions in Supplementary Fig. 5). The sensitivity η is defined by where δB min follows from Supplementary Note 4, and T meas is the time it takes to measure the Hahn-echo intensity used to derive the magnetic field amplitude. In the shot-noise limit, the uncertainty of a single measurement of the Hahn-echo intensity σ 1 is (essentially a Poisson distribution divided by its mean) with N ph the average photon count per sequence, and N the number of iterations of the sequence. The latter is simply N = T meas t sequence with t sequence = t + t overhead the time length of a single sequence, as shown in Supplementary Fig. 5.

Laser
Polarisation Readout Supplementary Fig. 5. Definitions. Times used in the derivation of the optimum magnetic field frequency.
The maximum gradient grad max follows from the sine-shaped fit (A sin(ωB AC + φ)+S ) to the intensity vs magnetic field amplitude plot (see main text Fig. 3b), and is grad max = Aω.
The amplitude A depends on the maximum contrast possible (for example measurable via Rabi oscillations) and the fraction of this maximum available, which depends on the t sequence and T 2 , since at t sequence T 2 , the maximum contrast is reached, while at t sequence T 2 , there is no contrast at all. Hence, the amplitude is given via the fit to the T 2 Hahn-echo data (Fig. 2e in the main text) as with D a constant which is half the maximum contrast, and both T 2 and n follow from fitting the Hahn-echo data. Finally, the frequency ω (in magnetic field, so units are T −1 ) follows from the accumulated phase ∆φ due to the magnetic field. The latter stems from detuning due to the applied AC magnetic field, since the field shifts the resonance frequency (see Supplementary Note 3). The spin (in the rotational frame) rotates with the shift in detuning frequency ∆ f , hence the phase follows from integrating this frequency shift over the period between the π/2-pulses, taking into account the phase shift at the π-pulse halfway: where τ is the delay time in the pulse sequence (between the two π/2-pulses). The detuning follows from Eq. (3).
As can be seen from the integral, the effect of a DC magnetic field is cancelled due to the π-pulse. The effect of a sinusoidal AC magnetic field is with g the g-factor of the electron spin, µ B the Bohr magneton, B AC the amplitude of the AC magnetic field, h Planck's constant, ν the frequency of the magnetic field (so units are s −1 ), and φ AC the phase of the magnetic field at the first π/2-pulse. With some trigonometry this can be reduced to In the measurement, the magnetic field is synchronised (φ AC = 0), the delay time τ is given as t (see Supplementary  Fig. 5), and the period of the AC magnetic field is chosen as the delay time (thus ν = 1/τ = 1/t). A full period in magnetic field happens every −2π of accumulated phase (it is effectively moving anticlockwise), therefore the period in magnetic field follows Combining all above equations, the sensitivity is For comparison, with t overhead = 0, = h/ (2π), choosing t = αT 2 , and absorbing D, α, N ph and the now-constant exponent into a constant C, the expression often used for the sensitivity given shot-noise 6,7 follows where for example with 30% Rabi contrast, N ph ≈ 0.03 7 and α = 1, C ≈ 0.3/2 × √ 0.03 × e −1 ≈ 0.01. To find the minimum sensitivity, the maximum of its inverse 1/η is determined by differentiation to t, and finding the zeros of the result. Performing the quotient and chain rules, left as exercise for the reader, the differentiation gives Equating this to zero, realising the exponent cannot be zero and the denominator is positive definite, the result for time t that gives the optimum time t optimum is (and f B, optimum = 1/t optimum ) When using the same waiting time in the magnetic field pulse sequence as in the Hahn-echo sequence, effectively t wait = 0 (since the fitting function for the Hahn-echo measurement used in Eq. (8) should include a −t wait as well, which is negligible for T 2 measurements). Finally, to get an idea of the result, without loss of generality, t overhead = αt optimum is chosen. Now, the solution to the above equation is For negligible overhead (α = 0) and for extreme overhead (α = ∞), the solutions are for t overhead t optimum These solutions are plotted in Supplementary Fig. 6 as the fraction of T 2 vs the exponent of the Hahn-echo data n.
For quantum-projection noise limited measurements, the uncertainty σ 1 of a single measurement of the Hahn-echo intensity is with N the number of iterations of the sequence, p the chance to measure the |0 -state (1/2 at the maximum gradient), and Y a constant which depends on the contrast between the |0 and |±1 states. This has exactly the same shape as for the shot noise, hence the final result for the optimum frequency is the same. Analogue to Eq. (14), the sensitivity can be given as since Y from Eq. (19) and A from Eq. (8) cancel except for a factor of 2 (Y = 2A). Here, compared with shot noise (Eq. (14)), the sensitivity for projection noise is two orders of magnitude better. For completeness, for the sensitivity of DC magnetic field measurements (using a FID measurement), the same procedure can be followed. Now, the phase is (equivalent to Eq. (9)) The frequency in magnetic field with measurement time τ = t (equivalent to Eq. (12)) is Combining this as in Eq. (13), with T * 2 instead of T 2 , the sensitivity for DC fields is Therefore, the optimum measurement time follows the same formula as the optimum time period for AC magnetic field measurements, and is shown in Supplementary Fig. 6. Finally, with t overhead = 0, choosing t = αT * 2 , and absorbing D, α, N ph and the now-constant exponent into a constant C, the expression often used for the sensitivity given shot-noise 6 follows where for example with 30% Rabi contrast, N ph ≈ 0.03 and α = 1, C ≈ 0.3/2 × √ 0.03 × e −1 ≈ 0.01.

Supplementary Note 7: Additional T 1 measurements
Additional T 1 data are given in Supplementary Table 1. Please note that, just like the data in the main text, these data are measured with common-mode noise rejection. However, less iterations are used, hence the results have larger uncertainties, nonetheless a similar trend for the rates seems rather likely (3Ω > γ).