Experimental protection of quantum coherence by using a phase-tunable image drive

The protection of qubit coherence is an essential task in order to build a practical quantum computer able to manipulate, store and read quantum information with a high degree of fidelity. Recently, it has been proposed to increase the operation time of a qubit by means of strong pulses to achieve a dynamical decoupling of the qubit from its environment. We propose and demonstrate a simple and highly efficient alternative pulse protocol based on Floquet modes, which increases the decoherence time in a number of materials with different spin Hamiltonians and environments. We demonstrate the regime $T_2\approx T_1$, thus providing a route for spin qubits and spin ensembles to be used in quantum information processing and storage.

The protection of quantum coherence is essential for building a practical quantum computer able to manipulate, store and read quantum information with a high degree of fidelity. Recently, it has been proposed to increase the operation time of a qubit by means of strong pulses to achieve a dynamical decoupling of the qubit from its environment. We propose and demonstrate a simple and highly efficient alternative route based on Floquet modes, which increases the Rabi decay time (TR) in a number of materials with different spin Hamiltonians and environments. We demonstrate the regime TR ≈ T1, thus providing a route for spin qubits and spin ensembles to be used in quantum information processing and storage. In open quantum systems, coherence of spin qubits is limited by spin-spin interactions, spin diffusion, inhomogeneity of the static and microwave fields 1 as well as charge noise 2 . An increase in coherence time is achieved by dynamically decoupling (DD) qubits from their surroundings using distinct Electron Spin Resonance (ESR) pulses 3,4 . However, such pulses have inherent imperfections and fluctuations, thus requiring their own layer of DD, resulting in a doubly dressed qubit. The technique of concatenated DD 5,6 has been proposed for NV centers up to the second order of dressing 5,[7][8][9] . Here we demonstrate a pulse protocol based on Floquet modes which successfully increases the decoherence time in a number of materials with different spin Hamiltonians and environments, such as low and high spin-orbit coupling for instance. Rather than focusing on decoupling from the bath by strong excitation, we use very weak pulses and alter the dynamics of the entire system. For short spin relaxation times accessible to our measurement setup (at around 40 K) one can do a direct comparison with the coherence time, and we demonstrate the regime T R ≈ T 1 . In magnetic diluted systems T 1 T 2 , e.g. T 1 of several ms in rare earth ions such as Y 2 SiO 5 :Er 3+10 and Y 2 SiO 5 :Yb 3+11 or 28 Si:Bi with a tunable T 1 of thousand of seconds 12 . Our general method can thus lead to very long persistent Rabi oscillations, using a single circularly polarized image pulse.
The use of strong continuous microwave excitation has been proposed as a way to protect qubits 13,14 although the quantum gates would need proper redesigning. In related studies, complex pulse design using an arbitrary waveform generator, proved essential in studying Floquet Raman transitions 15,16 and quantum metric of a two-level system 17 in NV centers. It is worth noting that in the case of concatenated DD, the frequency of the second order (n = 2) excitation has to match the Rabi frequency of the first excitation (n = 1); also, the two excitations are linearly polarized and perpendicular to each other (the method extends to higher orders in n). Experimentally, the protocol quickly becomes complex and demanding in terms of pulse design and frequency stability, above the second order.
Our protocol uses two coherent microwave pulses: a main pulse drives the qubit Rabi precession while a low-power, circularly polarized (image) pulse continuously sustains the spin motion. The image drive has a frequency close to the main drive and its amplitude is 1-2 orders of magnitude smaller. In this way, a quantum gate could be driven by regular pulses, without the image pulse, while the time interval between gates could be filled with an integer number of Rabi nutations that use our protection protocol. Such scheme would protect the coherence of the qubit in-between usual quantum gates. We note that the initial phase difference between the two pulses allows to tune the spin dynamics by enhancing (or diminishing) the Floquet modes 18 of its second dressing. The technical implementation is simple and can be generalized to any type of qubit, such as superconducting circuits or spin systems. In this paper we focus on the experimental implementation and simply observe that numerical simulations based on Bloch model with T 2 = 2T 1 describe the final results very well.

II. RESULTS AND DISCUSSION
The standard method to induce Rabi oscillations in a two-level system (TLS) is to apply an electro-magnetic pulse of frequency f 0 equal to the TLS level separation (resonance regime, gµ B H 0 = hf 0 ). The pulse will drive the spin population coherently between the two states. Experimentally, the drive is at a frequency f 0 + ∆ (where f 0 is the Larmor frequency and ∆ is a small detuning away from the resonance condition) followed by read out pulses of frequency f 0 to record the state S z (see Fig. 1a). The method introduced here makes use of two coherent microwave pulses (see Fig. 1b,c): the drive pulse at f 0 + ∆ of amplitude h d and length τ Rabi creates quantum Rabi oscillations while the second one sustain them using a very low power image of the drive (h i h d ), operated at f 0 − ∆. In order to probe S z (τ Rabi ) at the end of the Rabi sequence, we wait a time longer than T 2 , such that S x ≈ S y ≈ 0, followed by π/2 − π pulses to create a Hahn echo of intensity proportional to S z (Mn 2+ spins are readout without echo, as detailed in Sect. IIB of 19 ). Fig. 1b shows one way of coherently creating the drive and its image, by means of a mixer multiplying a pulse at frequency f 0 with an intermediate frequency (IF) cosine signal allowing to control the detuning ∆, phase φ and the pulse length, shape and amplitude 19 .
Rabi oscillations of three different types of paramagnetic systems − a rare earth ion (Gd 3+ ), a transition metal ion (Mn 2+ ) and a defect in diamond (P1) − are shown in Fig. 1 (d,e,f), respectively. Their detuned Rabi oscillations induced by the drive pulse only (red curves) are of similar frequency (≈ 20 MHz) and last for a small number of nutations (< 20). The blue curve shows the Rabi oscillation when the image pulse is superimposed. The oscillations remain intense far beyond the decay time of the red curves and their number is dramatically increased. This effect has maximum impact when the frequency difference between the drive and the image pulses 2∆ matches the Rabi frequency induced by the main drive, F R . In addition of the very long coherence time, we observe a slow amplitude modulation, depending on the phase φ and attributed to Floquet modes, as explained below.
We study the new decay time of the Rabi oscillation under image pumping, by tuning the relaxation time via temperature control and by applying the longest drive pulse available to us (Fig. 2) Figure 1. Comparison of Rabi oscillation with or without the image pulse. a,b,c, Schematic of the microwave implementation. a, The drive (f + ∆) and the readout (f0) sources are independent. b, Using the same source as the readout (f0), the drive pulse (f0 − ∆) is generated by a non-linear mixing with a low frequency signal ∆, with f0/∆ ∼ 10 3 , a process that creates a low-amplitude image drive (f0 − ∆) as well. The drive, image and readout pulses are coherent and the phase relationship is tunable. c Pulse sequence (see 19 ): the drive and the image pulses act at the same time on the qubit while the readout is sensitive to the Sz projection by a spin-echo process. d,e,f Rabi oscillation of the CaWO4:Gd 3+ , MgO:Mn 2+ and P1 defects respectively in the presence (blue) or absence (red) of the image pulse at the optimum condition FR = 2∆. The green guideline in f shows the improvement of the coherence time when compared to a CPMG pulse sequence while the blue curve shows not only a long coherence but also beatings which are tunable via phase-tunable Floquet dynamics (see text).
is limited by the pulse power amplifier of the setup, with a maximum pulse length of 15 µs. At 40 K, the relaxation time of the spin system MgO:Mn 2+ (S = 5/2) is also ≈ 15µs. The Rabi oscillation for the transition + 1 2 ↔ − 1 2 is shown in Fig. 2. Guidelines showing the exponential decays due to T 2 (in green) and T 1 (in orange) measured by Hahn echo and inversion recovery respectively, are added as well. While the amplitude of the Floquet mode (slow amplitude modulation) decreases with T 2 , the Rabi oscillations persists with a decay time ≈ T 1 .

Protection of quantum coherence for different initial states
We analized our protocol for different initial states. For initial and final states along +X, +Y and +Z we observe that the protection protocol is effective for times up to the maximum amplifier gate length of 15 µsec.
a. Pulse sequence The protocol proposed here was tested for different initial states and spin systems. In the following, the details of the experiment are given. The ground state of the spin is along +Z axis and is used as initial state. In Fig. 4, this is shown in orange and labeled "preparation". The +Z state is obtained by thermalization. States +X and +Y are obtained from +Z using a hard π 2 pulse around the y or x axes, respectively, able to excite the whole spin ensemble (or spectroscopic line). In MgO:Mn 2+ and P1, this is ensured by their narrow spectroscopic linewidths Γ, while in CaWO 4 :Gd 3+ , Γ is of the same order of magnitude with the maximum excitation bandwidth. Therefore, some Gd 3+ spins of the spin packet might not be in a perfect +X or +Y state. However we didn't notice 0 2 4 6 8 10 12 14 16 τ Rabi (µs)  any particular effect in the final results for Gd. We note that by combining rotations around any of the x, y and z axes, we can prepare the initial state in any position. Once prepared, the spin state is subjected to coherence protected Rabi protocol or to the usual Rabi drive for an integer number of Rabi flops (top and bottom "burst" panels in Fig. 4, respectively). Thus, the final state is along the same direction as the initial one 20 . The echo-based measurement for an initial state +Z is shown in the green panel (with τ wait T 2 ). For initial states +X and +Y (blue panel), one have to wait a time τ f ree in order to let the spin packet defocus before applying a π pulse and detect the subsequent echo signal.
b. P1 defects in diamond In the case of P1 defects in diamond, we present measurements of the spin echo signal when the initial and final states are along +Z, with and without the protection protocol for different lengths of the Rabi pulse τ burst . Without image pulse (Fig. 5 left) the signal is visible after a Rabi burst of 300 ns, but it is rapidly lost for times longer than 1 µs. With the protection protocol in place (Fig. 5 right), the signal is almost entirely conserved for times up to 15 µs. This behavior shows a significant improvement over the CPMG method (see Supplementary  Information, T 2 = 0.69 µs). For this experiment the temperature was set to T =15 K and τ f ree = 300 ns.
c. CaWO 4 :Gd 3+ For Gd 3+ spins, we measured the spin-echo signal after an integer number of Rabi flops for initial/final states along the +X, +Y and +Z axes. The Rabi oscillations are shown in Fig. 6 (a) as the real and imaginary part of the recorded signal (blue and orange lines, respectively). One notes that for the initial state along +X and +Y the Rabi signal starts and ends at zero value, with the end time being indicated by the arrow in each π π/2 π preparation drive + image measurement of Sx and Sy measurment of Sz τ wait τ free τ free τ free τ free τ burst +X +Y +Z π π/2 π preparation drive only measurement of Sx and Sy measurement of Sz τ wait τ free τ free τ free τ free τ burst +X +Y +Z Figure 4. Pulse sequence used for different initial states. First, the spin is prepared in the ground state +Z or, after a ( π 2 )y,x, in the +X or +Y states. A Rabi oscillation is performed during τ burst for an integer number of Rabi flops, with or without the image pulse hi (top and bottom panels respectively). The final state is along the same direction as the initial one, and the readout is done using an usual echo detection (blue and green areas). inset. The bottom panel shows large echo signals after 10 µs, much longer than T 2CP M G = 4 µs (see Supplementary  Information), for each of the initial state preparation. Using a combination of rotations around the x,y and z axes, we can create any initial state for the spin and thus use the image pulse protection for any arbitrary state. The experimental conditions for these measurements are: temperature T =40 K and τ f ree = 200 ns.

Qubit dynamics
The qubit dynamics in the absence of a bath, is described by the spin Hamiltonian in the laboratory frame 19 : where f 0 is the Larmor frequency caused by the static field, h d and h i are the microwave drive and image field, respectively, Fig. 1) and θ is a small additional phase, possibly created by imperfections of the setup (as discussed in 19 ). Variables f 0 , h d,i and ∆ are expressed in units of MHz. After a transformation in a frame rotating with ω + , the Hamiltonian (1) becomes : When the image field h i is absent, the equation (2) has no explicit time dependence and the Rabi frequency is simply When h i is present, the dynamics of S z can be solved numerically, as it is shown in Fig. 3 for the case of CaWO 4 :Gd 3+ .
For a fixed power h d of the drive pulse, Rabi oscillations are measured as a function of the detuning ∆. As shown in the contour plot of Fig. 3a, S z (t) vanishes after few oscillations except when the condition F R ∼ 2∆ is met. At this Floquet resonance, S z keeps oscillating for a very long time (> 15µs). Its Fast Fourier Transform (FFT) is presented in Fig. 3b. The free (unprotected) Rabi oscillations mode F R is rather weak and broad showing the large damping caused by the environment. However, when the mode crosses the frequency of the image pulse, indicated by the vertical white dashed line, the peak becomes intense and narrower, as the qubit protection from environment is activated. The condition gives the most efficient protection of the Rabi oscillation (see 19 ). The general condition is F R = n∆, n = 2k, k ∈ N showing a comensurate motion of the qubit and h i on the Bloch sphere.
We can compare the experimental result to the model described by the Hamiltonian (2) which can be rewritten as in Eq. S14 in 19 : /2) ]. When the "image" pulse is not applied, the Hamiltonian is time independent and the propagator is simply the matrix exponential of the Hamiltonian: U p (t) = exp(−i2πH RF t). When the image pulse is present (h i > 0), the Hamiltonian becomes explicitly time-dependent. Although a second canonical transformation RWA could remove the time dependence if ∆ h i , it is importat to leave ∆ as a free parameter since the methods works at resonance as well (∆ = 0). Thus, for the sake of generality, we solved numerically the explicit time-dependent differential equations using QuTIP 21 . The parameters used in the simulation have been measured independently: the microwave drive field h d has been calibrated using the frequency of Rabi oscillations at no detuning (∆ = 0), the image drive h i was measured by a spectrum analyzer directly connected to the output of the AWG (h i /h d ≈ 0.12), relaxation (T 1 ) and decoherence (T 2 ) times were measured by inversion recovery and Carr-Purcell-Meiboom-Gill (CPMG) protocol, respectively (see 19 ). We used QuTiP implementation of Lindblad's master equation with S − as collapse operator which is equivalent to the phenomenological Bloch model for the case T 2 = 2T 1 . Figure 3c MHz as a function of φ (amplitude is normalized to the highest peak). For φ = 2kπ/4, k ∈ N , Rabi oscillations have a single mode FR while for φ = (2k + 1)π/4 a splitting of twice the Floquet mode frequency is observed. While its frequency is fixed, the intensity of the Floquet mode changes gradually, with a period of π/2.
(2) describes very well the protection of the coherence by means of the image pulse. Note the existence of a Floquet mode at ∆=7.5 MHz of frequency ∼ 1 MHz, visible in both the experimental and theoretical contour plots of Fig. 3.
The Floquet mode appears as beatings of the Rabi frequency and is φ-tunable. Similarly to the case of Gd 3+ , the qubit protection and Floquet mode dynamics is obtained for the S = 5/2 spin of MgO:Mn 2+ , here measured under the experimental conditions of Fig. 2. Rabi oscillations and corresponding FFT spectra are shown in Fig. 7 for two values of the initial phase, φ = 0 • (green) and φ = 45 • (gold), while simulations are shown in black. The decay times are much larger than T 2 for both values (here T 1 ≈ 15 µs, see Fig. 2); however, the dynamics is strikingly different. When the drive and image pulses have the same initial phase (one can consider the initial time in Eq. 2 as − θ 4π∆ without loss of generality), the Rabi oscillations have maximum visibility, with almost no beatings (see 19 ). At φ = 45 • , the spin torques generated by the h d and h i fields induce strong beatings or a Floquet mode creating two additional modes of the Rabi frequency. The left panel shows Rabi splittings equal to the Floquet frequency for φ = 45 • and a single Rabi oscillation for φ = 0 • . Experimentally, we can continuously vary the value of φ and analyze the frequency and intensity of the Floquet mode. As an example, a comparison between theory and experiment is shown in Fig. 8 for the case of CaWO 4 :Gd 3+ for ∆ = h d / √ 3 = 34 MHz. For even and odd multiples of π/4, a single and a splitted Rabi mode is observed, respectively. The Rabi splitting is the Floquet mode and is constant as a function of φ but its intensity oscillates with a period of π/2. The effect is evident in simulations as well, since the terms in h d,i of H RF are along the same direction or orthogonal, for φ = 0 • and 45 • respectively.
In regards to decoherence sources, it is safe to assume that the main contribution comes from the spin bath made by nuclear spins surrounding the central spin, as well as other electronic spins located in its closed vicinity. Such scenario is the typical situation in spin systems operated at low enough temperatures to reduce the role of the phonon bath on T 2 . The details of the entangled qubit-bath dynamics are outside the scope of the current study. We do observe that the final results are well described when only dissipation is the source of decoherence, leading to T 2 = 2T 1 . This may indicate that the image pulse h i is able to control the dynamics and thus the decoherence of the spin bath.
The qubit rotation is thus tunable by using a pre-selected value of φ, allowing to create complex rotations. With a decoherence time approaching spin lifetime T 1 , the value of φ can be changed while qubit control is still ongoing. Our study demonstrates a sustained quantum coherence using a general protocol that can be readily implemented to any type of qubit. Our approach can be used in other detection schemes, such as sensitive spin detection using on-chip resonance techniques [22][23][24] .

METHODS
Spectrometer setup. The measurements have been performed on a conventional pulse ESR spectrometer Bruker E680 equipped with an incoherent electron double resonance (ELDOR) bridge and a coherent arbitrary waveform generator (AWG) bridge. In the ELDOR bridge (Fig. 1a), the drive and the read out pulses come from two independent sources while with the AWG bridge (Fig.1b) all the pulses are generated using the same microwave source and thus they are all phase coherent. The drive frequency is generated by mixing the source f 0 (used as a local oscillator) with a low frequency and phase controllable signal IF (∆, φ) through an in-phase quadrature (IQ) mixer. Ideally, the output of the mixer is monochromatic with the frequency f 0 + ∆. In reality, the output consists of a principal frequency f 0 + ∆ (the drive) and of lower amplitude images f 0 + n∆ (see Supplementary Information for more information). Since the effect of the image is the central part of this paper, we have characterized the AWG bridge using a spectrum analyzer, right before the power amplification stage. An example of spectrum is presented in Fig. S6 of the Supplementary Information. The power of the image f 0 − ∆ is lower by ≈ −18 dB than f 0 + ∆. Consequently, an amplitude ratio of the MW magnetic fields h i /h d around ∼0.12 is used in simulations.
Pulse sequence. First, the system is set to be in resonance condition gµ B H 0 = hf 0 . The drive pulse of amplitude h D , frequency f 0 + ∆ and length τ Rabi induces Rabi oscillation in detuning regime. At the same moment the image pulse (generated through the IQ mixer) of amplitude h I , frequency f 0 − ∆ and the same length τ Rabi also irradiate the spins. In order to probe S z (τ Rabi ) at the end of the Rabi sequence, we wait a time longer than T 2 , such that S x ≈ S y ≈ 0, followed by π/2 − π pulses to create a Hahn echo of intensity proportional to S z .
Spin systems. The methodology presented here is demonstrated on different spin systems: the nitrogen substitution in diamond P1 defect (S = 1/2) (concentration :100 ppm), Mn 2+ impurities in MgO (S = 5/2) with a concentration of 10 ppm and Gd 3+ impurities in CaWO 4 (S = 7/2) with a concentration of 50 ppm. Despite the large Hilbert space of the Mn 2+ and Gd 3+ spin Hamiltonians, the orientation of the magnetic field and the frequency and power of the microwave excitation are chosen to avoid multiple level transitions and thus select only one resonance 25 . Therefore, the spin systems can be considered as effective two-level systems undergoing coherent Rabi rotations. The spin Hamiltonians, operating parameters (fields and frequencies) as well as characteristic T 1,2 times for these materials are given in the Supplementary Information.
Simulation explanation. In the rotating frame, the Hamiltonian describing the dynamics of the electron spin is given by Eq. S14 in the Supplementary Information: H /2) ]. When the "image" pulse is not applied, the Hamiltonian is time independent and the propagator is simply the matrix exponential of the Hamiltonian: U p (t) = exp(−i2πH RF t). When the image pulse is present (h i > 0), the Hamiltonian becomes explicitly time-dependent. Although a second canonical transformation RWA could remove the time dependence if ∆ h i , it is importat to leave ∆ as a free parameter since the methods works at resonance as well (∆ = 0). Thus, for the sake of generality, we solved numerically the explicit time-dependent differential equations using the quantum toolbox QuTIP 21 . The parameters used in the simulation have been measured independently: the microwave drive field h d has been calibrated using the frequency of Rabi oscillations at no detuning (∆ = 0), the image drive h i was measured by a spectrum analyzer directly connected to the output of the AWG, relaxation (T 1 ) and decoherence (T 2 ) times were measured by inversion recovery and Carr-Purcell-Meiboom-Gill (CPMG) protocol, respectively (see

III. CARR-PURCELL-MEIBOOM-GILL (CPMG) MEASUREMENTS
In the absence of inhomogeneous broadening, the dephasing time or transverse relaxation time T * 2 is directly determined by the ESR linewidth. In solids, and in particular for single crystals, the anisotropic interactions as well as their distribution throughout the crystal induce an inhomogeneity of the line. To measure the dephasing time, the most simple pulse sequence is the Hahn primary echo: a π/2 pulse rotates the spins in a plane transverse to the static field; due to field inhomogeneity, spin will defocus and spread within the transverse field. A subsequent π pulse reverses spin motion and creates the observed echo signal when refocusing is achieved. By characterizing the exponential decay of the echo signal as a function of the time between the first and the second pulse, one obtains the dephasing time T 2 .
However, because of diffusion mechanisms, imperfection of pulses or microwave field inhomogeneity, the refocusing is not complete and the intrinsic dephasing time measured by this sequence is under-evaluated. To reduce these unwanted effects, we used the dynamical decoupling offered by the Carr-Purcell-Meiboom-Gill (CPMG) sequence: after a ( π 2 ) x pulse applied along x, a train of π pulses is applied along y with alternate orientations (π y , π −y , . . . ). Many primary echoes are thus generated, with an intensity decreasing exponentially as a function of time and characterized by the intrinsic dephasing time T 2 . The measurements of T 2 using CPMG sequence for the diamond and CaWO 4 :Gd 3+ are given in Fig. S1, panels (a) and (b) respectively. Measurements are done at the same temperatures as for the data in the main article Fig. 1 and lead to a T 2 of 0.69 µs and 4 µs respectively. A similar result is obtained if the train of π pulses has alternate orientations along x and y axes ((π x , π y , π −x , π −y , . . . ).
In the case of MgO:Mn 2+ , the inhomogenous absorption linewidth is only 0.05 mT and the corresponding spin-echo measurement gives a decoherence time of 3 µs (Fig. 2). Since the absorption linewidth is very narrow, the spin-echo signal of MgO:Mn 2+ has a small amplitude and thus subsequent CPMG pulses are not helpful in this case.

IV. SPIN SYSTEMS CHARACTERISTICS
The methodology presented here is applied to different spin systems: the nitrogen substitution in diamond P1 defect (S = 1/2) (concentration :100 ppm), Mn 2+ impurities in MgO (S = 5/2) with a concentration of 10 ppm and Gd 3+ impurities in CaWO 4 (S = 7/2) with a concentration of 50 ppm. Despite the large Hilbert space of the Mn 2+ and Gd 3+ spin Hamiltonians, the orientation of the magnetic field and the frequency and power of the microwave excitation are chosen to avoid multiple level transitions and thus select only one resonance 25 . Therefore, the spin systems can be considered as effective two-level systems undergoing coherent Rabi rotations. The spin Hamiltonians, operating parameters (fields and frequencies) as well as characteristic T 1,2 times for these materials are given below.

A. P1 defects in diamond
The substituting nitrogen in diamond has covalent bonds to three surrounding carbons and leaves an unpaired electron on the fourth one, giving rise to a spin S = 1/2 : this is the P1 defect. Its spin Hamiltonian is given by 26 : where g P 1 = 2.0024 is the g-factor, I = 1 is the nuclear spin of 14 N and [A N ] is its hyperfine tensor with A ⊥ = 81 MHz and A = 114 MHz. A measurement of echo signal as a function of field is shown in Fig. S2. We studied the central line, corresponding to m I = 0, which is not affected by the orientation of the crystal and has the strongest signal. In our experiments, the temperature was set at 15 K, the external field at B 0 = 343.62 mT and the microwave frequency at f 0 = 9.645 GHz. The concentration of nitrogen is about 100 ppm, which leads to values of the linewidth 2Γ=4 G and dephasing time T 2 = 0.69µs, similar to values presented in Ref. 27 .  Figure S2. Echo field sweep of P1. Echo signal recorded as a function of static field at T=15 K. The most intense line (mI = 0, shown with a star symbol) is not affected by cristal orientation and was selected for the study presented here.

B. MgO:Mn 2+
MgO:Mn 2+ is the second studied system. The non-magnetic matrix of MgO contains a very low concentration (∼ 10 ppm) of 55 Mn 2+ spins with S = I = 5/2. The spin Hamiltonian is given by 28 : where g M n = 2.0014 is the g-factor, A = 244 MHz is the hyperfine constant and H CF is a crystal field term resulting from the cubic symmetry F m3m of MgO: In previous studies 25,28-30 we detailed the effect of H CF on the spin eigenvalues, and showed that the static field orientation can tune in-situ their values to be perfect equidistant or non-harmonic. In the present study we operate the static field under an alignment generating sufficient non-harmonicity such that one can resonantly select two of the six levels and consider the Mn 2+ a two-level system. Because of the cubic symmetry and the absence of nuclear spin in the host matrix, the line is very narrow with 2Γ = 0.5 G. In this system the detection of S z is not done by echo, but by integrating the area under the Free Induction Decay (FID) signal, measured immediately after τ wait and one π/2 readout pulse.
The measurements were performed at a temperature of 40 K, with the external field at B 0 = 353.7 mT and the microwave frequency f 0 = 9.734 GHz.

C. CaWO4:Gd 3+
CaWO 4 :Gd 3+ is the third studied system. Gd 3+ spins S = 7/2 are diluted in non-magnetic matrix of CaWO 4 and described by the spin Hamiltonian 22,31 : where g Gd = 1.991 and B 0 2 = −916, B 0 4 = −1.14, B 4 4 = −7.02, B 0 6 = −5.94 × 10 −4 B 4 6 = 4.77 are in MHz units. In our experiments, the temperature was set at 40 K, the external field at B 0 = 377.0 mT along to the crystallographic a-axis (tetragonal symmetry I4/a) and the microwave frequency at f 0 = 9.633 GHz . The full linewidth is 2Γ =6 G A precise analysis of the resonance fields shows a misalignment of about 2.7 • . The crystal field anisotropy ensures that the Zeeman levels are not equally spaced, similarly to the case of MgO:Mn 2+ discussed above. The eigenvalues of H Gd are presented in Fig. S3 (top). The bottom panel shows a typical intensity absorption spectrum at constant frequency as a function of B 0 . Peaks appear at resonance fields, indicated by the red segments in the Zeeman diagram (top) which also show the two levels selected for Rabi oscillations. Thus, it is evident that only two levels are involved in the spin dynamics, making the system an effective TLS. Similar type of spectra are measured for the other two samples (diamond and Mn) in order to select the value of the resonance field. The drive and imaging methodology presented here are independent on which resonance is selected . Their intensity depends on the probability to have a transition between the two levels connected by the red segment.

A. Generation of coherent pulses using a Hartley mixer
Coherent image and Rabi drives can be obtained with a Hartley mixer having slightly unbalanced RF ports, similar to the one used in our setup. An example of such mixer is shown in Fig. S4 and we will discuss how a circularly polarized image pulse can be constructed. The input radio-frequency (RF) signal is LO = 2 sin(ω LO t) and therefore each branch will see LO 1 = sin(ω LO t) and LO 2 = cos(ω LO t). Similarly, IF = 2A sin(ω IF t + φ IF ). We introduce amplitude and phase mismatches for the IF ports in the following manner: The outputs on the two RF ports are:  Figure S4. Hartley mixer. A schema of a mixer allowing the creation of the Rabi drive h d and its coherent image hi, as explained in the text. The main RF drive is at the LO port and it is mixed with a low frequency signal sent into the IF port. The resulting signal exits through the RF port.
and thus We get with the second term being neglible in first order. The first term represents the Rabi drive pulse with an amplitude A d = A cos δφ while the last two show the image pulse which can be rewritten as: with A i = δA cos δφ cos θ and tan θ = A sin δφ δA cos δφ = tan δφ δA/A .

B. Effect of the unperfect Hartley mixer on the spin dynamics
To study the qubit dynamics, we write its Hamiltonian first in the laboratory frame and then in the rotating frame approximation. A static magnetic field B z gives a Zeeman splitting in resonance withhω LO , while the microwave excitation is ∝ RF (t)S x with RF (t) given by Eq. S6. With notations ∆ = ω IF 2π , f 0 = ω LO 2π and ω +,− = ω LO ± ω IF , we have (in units of h): This leads to: The spin systems studied here behave as two-level systems and therefore one assumes S = 1/2 in the matrix above: with ∆ and h d of comparable magnitudes and much larger than h i . In the limit h i → 0, we can replace the 2 × 2 diagonal blocks with a diagonal [ . . ] and zero-ed out subdiagonals, with: The condition for the image pulse h i to sustain the coherence between even-and odd-numbered quasi-energies is therefore −F R /2 = F R /2 − n∆ or F R = n∆ and ∆ 2 + h 2 d = n∆ with n = 2k, k ∈ N (S20) In our study, n = 2 and the coupling is done between the second and third element of the diagonal, via off-diagonal terms containing both h d and h i . The splitted eigenvalues of the Shirley-Floquet Hamiltonian are φ-independent. Experimentally, the off-diagonal coupling is larger than the linewidth of the Rabi modes, such that a splitting is observed.
An analytical estimation of H SF eigenvalues can be done using the second order perturbation theory for the 6x6 block shown in Eq. S18. The determinant det[H SF − λI] = 0 with I the identity matrix and λ representing the eigenvalues, is expanded with Mathematica in powers of h i and equated to zero up to h 4 i . For the levels involved in the n = 2 resonance, the eigenvalues are given by: The splitting can thus be estimated as E + − E − = − 3hi 4 .

D. Torque considerations
The qubit dynamics imposed by Hamiltonian H RF can be simulated using QuTIP 21 as discussed in the main article. Here, we supplement the understanding of qubit dynamics from the point of view of spin's S torque in two situations, φ = 0 and π/4 (see Fig. S5 a and b, respectively) in absence of decoherence. Starting from ground state and for a Rabi frequency of 20 MHz, we compute torque magnitudes for the drive and image pulses as a function of time, | S × h δ |(t) and | S × h i |(t) respectively, with h δ = h d + ∆ẑ . For simplicity and without loss of generality, the phase θ, an angular shift of the initial phase of h i , is assumed equal to zero. For comparison, the resonance case at ∆ = h i = 0 is shown as an horizontal dashed line with maximum torque value since the spin and the microwave field are orthogonal during the Rabi nutation. All other values are normalized to this maximum value. The zero torque dashed line represents the case of "spin locking" when S||h δ . The zero-torque concept of spin-locking applies independently on detuning ∆ but it is valid only if h i ≡ 0; otherwise, the image pulse pulls the spin off the axis of h δ .
In contrast to the spin locking case, both torques are non-zero for our protection protocol. The drive torque at resonance h d = ∆ √ 3 (in red) is performing the Rabi nutation needed for gate operations while the image torque (in blue) acts as a small perturbation; in this example h i is 5% of h d or -26dB in power. The effect of the initial phase φ of h i is essential to qubit dynamics although h i remains a small perturbation during the gate operation. When the initial torques are parallel or antiparallel (φ = 0 or π/2 respectively) the Rabi rotation is closer to full amplitude and the Floquet mode is less visible. This is shown by a small modulation in drive torque (in red, panel (a)) with the period of the Floquet mode. In contrast, when torques are initially perpendicular to each other (φ = π/4, panel (b)), the Rabi nutation and drive torque has strong beatings, while retaining the same amount of coherence protection. The Floquet mode is clearly visible in this case, as discussed in Fig. 4 of the main article.

A. Spectrometer setup
The measurements have been performed on a conventional pulse ESR spectrometer Bruker E680 equipped with an incoherent electron double resonance (ELDOR) bridge and a coherent arbitrary waveform generator (AWG) bridge. In the ELDOR bridge (Fig. 1a), the drive and the read out pulses come from two independent sources while with the AWG bridge (Fig.1b) all the pulses are generated using the same microwave source and thus they are all phase coherent. The drive frequency is generated by mixing the source f 0 (used as a local oscillator) with a low frequency and phase controllable signal IF (∆, φ) through an in-phase quadrature (IQ) mixer. Ideally, the output of the mixer is monochromatic with the frequency f 0 + ∆. In reality, the output consists of a principal frequency f 0 + ∆ (the drive) and of lower amplitude images f 0 + n∆. Since the effect of the image drive is the central part of this paper, we have characterized the AWG bridge using a spectrum analyzer, right before the power amplification stage. An example of spectrum is presented in Fig. S6. The power of the image drive f 0 − ∆ is lower by ≈ −18 dB than f 0 + ∆. Consequently, an amplitude ratio of the MW magnetic fields h i /h d around ∼0.12 is used in simulations.

B. Calibration of the microwave fields
The method presented in this article is highly dependent on the value of the microwave fields h d and h i . The drive intensity h d is easy to calibrate by measuring the Rabi frequency at resonance while knowing its expected value from the spin Hamiltonian in the rotating frame (∆ = 0).
On the contrary, h i is more difficult to calibrate. As seen in Section V A, h i comes from the inherent unbalance of a real mixer and therefore it can be device depended. To calibrate h i , we have used a spectrum analyzer Agilent Technologie, PXA Signal Analyzer N9030A. We have measured the Fourier transform of the microwave coming from the AWG bridge before the power amplification stage as shown in Fig. S6: f 0 is the carrier frequency, f 0 + ∆ is the Rabi drive frequency (h d ) and f 0 − ∆ is the image frequency. The image pulse is about 100 times (∼ 18 dB) weaker than the Rabi drive and thus it can be used to sustain the motion rather than driving it. The signal at f 0 , about 4-6 dB smaller than the image drive, is not in a Floquet resonance with the Rabi drive (see Eq. S20) and thus not contributing to spin dynamics.  Fig. 1 of the main article. The Rabi drive pulse is located at f0 + ∆, ∼ 18 dB stronger than the image pulse used for qubit protection, located at f0 − ∆.