Clock transition by continuous dynamical decoupling of a three-level system

We present a novel continuous dynamical decoupling scheme for the construction of a robust qubit in a three-level system. By means of a clock transition adjustment, we first show how robustness to environmental noise is achieved, while eliminating drive-noise, to first-order. We demonstrate this scheme with the spin sub-levels of the NV-centre’s electronic ground state. By applying drive fields with moderate Rabi frequencies, the drive noise is eliminated and an improvement of 2 orders of magnitude in the coherence time is obtained compared to the pure dephasing time. We then show how the clock transition adjustment can be tuned to eliminate also the second-order effect of the environmental noise with moderate drive fields. A further detailed theoretical investigation suggests an additional improvement of more than 1 order of magnitude in the coherence time which is supported by simulations. Hence, our scheme predicts that the coherence time may be prolonged towards the lifetime-limit using a relatively simple experimental setup.

We used a standard home-build confocal microscope to address individual NV centres in a diamond sample. The diamond sample was a 98.9 % 12 C electronic grade substrate with a 40 µm thick and 99.8 % 12 C layer grown on top by chemical vapour deposition. The growth layer has a low concentration nitrogen doping forming individual NV centers which can be optically resolved in our confocal microscope. The top side of the diamond was attached to a glass slide with a lithographically deposited microwave structure to enable close vicinity to the NV center and hence efficient microwave driving. The glass structure with the diamond was attached to a printed circuit board (PCB) sample holder with SMA connectors for microwave field delivery. The PCB board was mounted on a xyz micrometer stage to position the sample relative to the microscope piezo scanner holding the Olympus UPlanFL N 100X/1.30 objective. Our xyz piezo scanner is a P-527.3CD from Physik Instrumente with an E-725.3CD Digital Piezo Controller. The excitation light is provided by a Coherent Verdi G2-SLM OPS 532 nm laser pulsed with an AOM from Brimrose. In our setup, we separate the red fluorescence from the green illumination using a dichroic mirror FF552-Di02-25x36 from Semrock and detect the signal with an avalanche photon diode (APD), SPCM-AQRH from Excelitas Technologies. The electrical signal is either recorded by a custom configured XEM6310 LX45 FPGA from Opal Kelly, with a 950 MHz detection bandwidth (for fast counting), or with a PCI-6232 X-series card from National Instruments (for slow counting). The hardware triggered confocal scanning operation as well as the ODMR measurements are controlled with the NI PCI-6232 card. The creation of the microwave field is either realised with a Rohde&Schwarz SMR20 microwave signal generator (for ODMR measurements) or with an arbitrary waveform generator AWG70002A from Tektronix (borrowed for a short period). In both cases, the microwave field was amplified by a Mini-Circuits ZHL-16W-43 S+ high power amplifier. All the instruments are remote controlled by a PC, which handles the sequence control and data acquisition. The powerful software suite Qudi [1] serves as the operational tool to establish the control over all hardware components and performs the measurements. The NV selected for this experiment was about 1 µm deep in the diamond and about 10 µm away from the microwave structure. We performed this experiment at a magnetic field strengh of 358 G in the vicinity of the excited state level anti-crossing [2][3][4], in which case the nitrogen nuclear spin becomes polarised within a few optical pumping cycles. The static magnetic field was created by two neodymium magnets in a quadrupole configuartion and aligned along the quantization axis. The magnets were attached (with a 3D printed holder) to a three axes NRT150/M 150 mm motorised linear translation stage with a BDC103 three channel controller from Thorlabs. In addition, a Thorlabs motorised rotation state with a TST001 driver controled the angular movement of the magnet. For an excitation power of about 250 µW, measured before the objective, we obtain a fluorescence signal of around 200 kcounts/s at zero magnetic bias fields.

A. Pulsed ODMR measurements
We performed high resolution pulsed ODMR measurements on both resonances, ν 1 and ν 2 , to identify potential coupling to adjacent 13 C nuclear spins. Within the full width half maximum of a Lorentzian fit, no additional couplings to 13 C nuclear spins are visible (cf. Fig. S1). The average of both transition frequencies yields (ν 1 +ν 2 )/2 = (2870.7005 ± 0.008) GHz, which corresponds to the zero-field splitting parameter D = 2.87 GHz and hence indicats a well aligned bias magnetic field.

B. Coherence times of the Rabi drives
To ensure that both transitions, ν 1 and ν 2 , experience the same Rabi frequency, Ω, we had to adjust the drive strength on the transitions individually. The inset of Fig. S2 shows the dependency of the Rabi frequency, Ω, on the amplitude of the signal generator. It becomes apparent that the lower transition, ν 1 requires a larger drive amplitude to yield a similar Rabi frequency as on ν 2 . The reasons for the imbalances can be manifold and are discussed in Sec. S4.
The coherence time of the individual transitions is rather different. In order to record the coherence times, we first determined the correct Rabi frequency for a given drive amplitude. Then we measured the coherence time by deliberately undersampling the obtained Rabi frequency in order to substantially reduce the total measurement time (for more details see Sec. S3).
The most relevant characteristics can be extracted from the coherence time measurements presented in Fig. S2. Each driving field, Ω i , posses a noise component δΩ i . The amplitude of the driving noise depends on the strength of the drive and its scaling with amplitude depends on the measurement configuration (the frequency dependent amplification and attenuation properties of the components). The coherence time curves of both transitions coincide for small Rabi frequencies, as the external magnetic noise, δB, constitutes to be the main dephasing contribution in this regime. By increasing the drive a larger dressed state energy gap can be created protecting the system from magnetic noise, δB. At the same time drive fluctuations become more prominent and dominate from a certain drive value on the noise contribution. The maximum coherence time indicates a balanced situation, where the dressed states of the drive experience similar noise contributions from the drive and the environment. Increasing the drive further leads to a non-linear increase of the drive noise and hence the coherence time decreases.

S4
The drastic difference in the appearance of the coherence time curves indicates a frequency dependent noise of the drive fields, respectively for ν 1 and ν 2 . In contrast to this, the magnetic noise appear to have an equal impact on both transitions, which will be further addressed in the following sections.

C. Ramsey measurements
We performed a Ramsey measurement on both transitions to identify that external and internal magnetic noise have the same impact on the m s = −1 and m s = +1 spin states. The measurement results presented in Fig. S3 demonstrate that within the error bars both transitions have an identical pure dephasing time, and the dephasing time averaged over both transitions is T * 2 = (1.78 ± 0.24) µs.

D. Hahn Echo measurements
We also carried out a coherence time measurement with a Hahn Echo scheme (cf. Fig. S4). Within the error bar, both transitions yield the same coherence time, T 2 . An averaged coherence time of T 2 = (215 ± 31) µs can be stated for the system and hence we can consider that the magnetic noise acting on both the m s = −1 and m s = +1 states is identical.

S3. MEASUREMENT PROCEDURE
In this section, we want to explain the technique applied to measure the coherence times, which are shown in the main text. We would like to emphasise that a compromise was needed between the measurement time, the number of data points and the resulting error bars, which are responsible for the larger error bars in Fig. 4 of the main text.
Almost all measurement curves in this experiment can be represented by where ν Rabi = Ω B /2π is the Rabi frequency and T Ω B 2 is the coherence time depending on the detuning, ∆, in the double drive (see Fig. 2 in the main text). Consequently, we could simply measure the whole trace with a resolution of about 4 points per period. This would require a sampling frequency of ν samp = 4 · ν Rabi ≈ 13 MHz, which corresponds to a time step of ∼ 77 ns. In Fig. S5 the Rabi period is sampled with roughly 10 points, resulting in a good estimate of the frequency (relative error of about ∼ 0.15%). Hence, to sample a full decay with a length up to 250 µs would require to record (at 4 points per period) about 3250 points. Then a signal to noise of about 10 requires more than 1 day of measurement for one detuning value. Since in this case it becomes impractical to carry out the experiment, we significantly reduced the measurement time by acquiring the data in a different way as explained in the following. The simplest idea is to sample the oscillation at every period (ν samp = ν Rabi ) to extract directly the exponential decay from the data. However, this would result in larger fit errors since the exponential fit requires a much smaller noise component on the data to yield a good coherence times estimate. Moreover, if the Rabi frequency, ν Rabi was measured with 0.15% accuracy (this corresponds to an absolute error of ±5 kHz), it would mean that there might be oscillations in the measured time trace with frequencies ν osci resulting from an incorrect estimation of the Rabi frequency. At these time scales the frequency component of a decaying signal cannot be extracted accurately since only a part of the full period could be sampled. These reasons required to refine the measurement technique. In this work we choose to deliberately undersample the known Rabi frequency by a specific amount to measure effectively a slowly oscillating decaying image frequency. Such time traces show a better robustness in the fitting algorithm and are eventually a compromise in measurement time and the resulting error bars in the fit.  was averaged for about 1 hour yielding a measurement result displayed in Fig. S6.

S4. ESTIMATED NOISE CHARACTERISTICS
In this section, we try to identify the components contributing to drive noise of the fields. Based on the Ramsey and the Hahn Echo measurements, we can assume identical external magnetic noise at both transition frequencies, ν 1 and ν 2 . In continuous dynamical decoupling the NV is permanently subjected to strong drive fields and therefore, drive noise will be the strongest noise contribution our scheme. We did not have access to a fast oscilloscope and because of that could not measure the fast noise components directly. However, based on conservative estimates we would like to state the various noise contributions in our system.
The drivings field in our work are created by the Tektronix AWG70002A. The maximal output amplitude for this device is 0.5 Vpp (or 1 Vpp by including the inverted channel) with a DAC accuracy of 8bit, yielding a maximum precision of 0.5 Vpp/2 8 ≈ 2 mVpp or 1 mV for the amplitude of an oscillating signal. Consequently, a sinusoidal field with an amplitude of 0.25 V (0.5 Vpp) has an accuracy of (250 ± 1) mV, which yields a relative error of 0.78 %. The vendor specifies the amplitude accuracy to ±10 mVpp [5], which increases the relative error to 2 %, indicating that besides the DAC inaccuracy, other noise sources exist. Therefore, the application of 4 sinusoidal fields at different frequencies can have at maximum 1/4 of the total amplitude noise, if the noise is equally distributed on all channels (which is not necessarily the case). Thus, one frequency component is accurate up to (62.5 ± 1.3) mV, yielding a relative amplitude error of 2 %.
A field of 62.5 mV (= 125 mVpp) would create (see the inset of Fig.1 in the main text) a Rabi drive of the order of 10 MHz. Increasing the drive field above that value caused heating problems, as a lot of power is dissipated in the microwave structure causing a drastic decrease in ODMR contrast and fluorescence signal. To prevent that from happening we lowered the amplitude. We found a balance between the thermal effects and the maximum obtainable Rabi frequency for fields of the order of 40 mVpp, if 4 fields are used simultaneously, resulting in a total amplitude of 160 mVpp. Given these limitation on the drive fields, the relative noise for each frequency component is on average 6.25 %.
Moreover, the amplitude noise of the AWG is correlated to its base clock, which introduces correlated (amplitude) noise on a µs timescale and may very well be responsible for frequency dependent (fast) noise imbalances, translated to the dressed states of the considered three-level system (see main text Fig. 4). The origin of this noise character can be grasped by investigating how data becomes sampled in an AWG.
Modern, very fast AWGs (with sample rates exceeding 1GS/s) do not process each sample directly as the rate the data is read exceeds the speed of common semiconductor memories [6]. To circumvent this problem, data is processed in parallel and then multiplexed inside the high-speed digital logic that drives the DAC. However, the multiplexing operation is directly bound to the internal clock of the AWG, which is usually of the order of 10MHz and thus creates correlations on that time base. Any errors in the timing of the internal clock edge translates directly into timing errors in the analogue output, which reflects phase noise and is immediately detectable by amplitude fluctuations. This is a noise source, which adds to the DAC inaccuracy and creates (among other effects) the total noise floor of the device. Eventually, a high-speed oscilloscope would be the preferential choice to identify fast noise on the individual frequency components. The clock stability of the device is denoted by 1 ppm for 10 MHz [5], i.e. the total frequency variations are on the order of (10 MHz/10 6 ) = ±10 Hz.
On top of this, we also assume a frequency dependent noise, since the output of the amplifier is not flat over a range of ν 2 − ν 1 ≈ 2 GHz. Moreover, the microwave structure experiences frequency dependent attenuation of the applied drive fields leading to (among other contributions) different Rabi frequencies for ν 1 and ν 2 at the same amplitude output (cf. the inset of Fig.1 in the main text).
All these various noise contributions have a strong impact on the drive of the NV system as they cause the dressed states to fluctuate. Therefore it becomes a crucial task in CCD to reduce the impact of noise, introduced by multifrequency drive fields, by creating robust states, which suffer less from drive noise contributions. By demonstrating robustness to all these noise sources underlines the importance of the present work.