Robust dynamical decoupling with concatenated continuous driving

Coherent control of quantum systems has opened a promising route towards novel quantum devices for quantum technologies, such as quantum information processing, quantum metrology and quantum sensing [1–4]. The performance of such quantum devices critically depends on coherence times of their constituent quantum systems which, in turn, are limited by uncontrolled interactions with their surrounding environment. This results in a challenging but fundamentally important task in current quantum experiments, namely how to protect individual quantum states from decoherence by their environment while retaining the ability to control the quantum dynamics of the system, in particular in solid state systems with characteristic complex environments. For slow environmental noise a successful strategy is dynamical decoupling due to rapid sequences of intense pulses of electromagnetic radiation, which has been applied successfully in NMR [5]. Significant progress has been made with various theoretical proposals for dynamical decoupling [6–13], and their experimental demonstration [14–23].

The recently developed dynamical decoupling schemes that require only continuous oscillatory driving fields [7, 8] inherit the advantages of standard dynamical decoupling, namely requiring no encoding overhead, no quantum measurements, and no feedback controls. Moreover, they are easier to realize experimentally and are more naturally combined with other quantum information tasks, such as the implementation of high-fidelity quantum gates [11–14]. In principle, one can apply continuous driving to reduce the noise suffered by a qubit considerably simply by increasing their intensity. However, random and systematic fluctuations which are inevitably present in the driving field itself will ultimately limit the efficiency of dynamical decoupling. The deleterious effect of driving field fluctuations in particular will become significant when employing strong driving fields to achieve ultralong coherence times. This is important in all applications in which environmental noise cannot be avoided (e.g. sensing in biological environments) and thus represents a fundamental obstacle. Overcoming the limitations imposed by driving field fluctuations and thus extending coherence times further represents a key step towards the construction of quantum memory [24], highly sensitive nanoscale magnetometers [1–3] and error-resistant quantum operation [13]. It will also be of particular interest for T₁ limited Rabi-type magnetic resonance imaging, the resolution of which is highly dependent on the stability of microwave driving fields [25, 26].

In this work, we address this challenge by introducing the scheme of concatenated continuous decoupling (CCD), which can significantly extend coherence times by protecting against driving field fluctuations. As a proof-of-principle, we implement a second-order CCD with a single nitrogen–vacancy (NV) center in diamond where a second, weaker driving field, reduces the impact of the amplitude fluctuations of the first-order driving field. We demonstrate experimentally that CCD schemes with only a weak second driving field can already increase coherence times by an order of magnitude as compared to standard schemes based on a single drive.

We start by considering a two-level quantum system with eigenstates $\left \vert \uparrow \right \rangle , \left \vert \downarrow \right \rangle$, see figure 1(a), and Hamiltonian $H_0=\frac {\hbar \omega }{2} \left (\left \vert \uparrow \right \rangle \left \langle \uparrow \right \vert -\left \vert \downarrow \right \rangle \left \langle \downarrow \right \vert \right)$. Environmental noise causes fluctuations to the energies ω_↑, ω_↓ and thus the loss of coherence. To counter such effects, we can apply a driving field on resonance with the energy gap ℏω between $\left \vert \uparrow \right \rangle$ and $\left \vert \downarrow \right \rangle$ as

$$\begin{equation} H_{d_1}= \hbar\Omega_1 \cos{(\omega t)} \sigma_x, \end{equation} \tag{ 1 }$$

where $\sigma _x=\left \vert \uparrow \right \rangle \left \langle \downarrow \right \vert +\left \vert \downarrow \right \rangle \left \langle \uparrow \right \vert$. In the interaction picture with respect to H₀ and with rotating wave approximation, we find $H_{\mathrm {I}}^{(1)}=\frac {\hbar \Omega _1}{2} \sigma _x$, and its eigenstates $\left \vert \uparrow \right \rangle _x=\frac {1}{\sqrt {2}}\left (\left \vert \uparrow \right \rangle +\left \vert \downarrow \right \rangle \right)$ and $\left \vert \downarrow \right \rangle _x=\frac {1}{\sqrt {2}}\left (\left \vert \uparrow \right \rangle -\left \vert \downarrow \right \rangle \right)$ are the dressed states [27]. On this basis, the effect of dephasing noise now induces transitions among these dressed states, which are suppressed by an energy penalty as long as the noise power spectrum at the resonance frequency is negligible [14]. The decoupling efficiency will be limited if the noise has a wide range of frequencies, while it can be very efficient for slow baths, e.g. in diamond [28, 29]. The above analysis is based on the assumption that the amplitude of the driving fields is stable. In realistic experiments, however, the intensity of the driving fields will fluctuate owing to limited stability of microwave sources and amplifiers (the frequency instead can be relatively much more stable), and thus cause fluctuations of the energies of the dressed states. Achievable coherence times of the dressed qubit using a single drive are thus ultimately limited by the stability of the driving fields [25], which appears as the fast decay of Rabi oscillation.

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The principal idea of CCD is to provide a concatenated set of continuous driving fields with decreasing intensities (and thus smaller absolute value of fluctuation) such that each new driving field protects against the fluctuations of the driving field at the preceding level. For example, fluctuations in the amplitude of the first-order driving can be suppressed by applying a second-order driving field as follows:

$$\begin{equation} H_{d_2}=2\hbar \Omega_2 \cos \left(\omega t+\frac{\pi}{2}\right) \cos(\Omega_1 t) \sigma_x. \end{equation} \tag{ 2 }$$

This second-order driving field is on resonance with the energy gap of the first-order dressed state, see figure 1(a), which actually describes rotation about the axis $\hat {y}$ in the interaction picture (see the appendix for details) and thus plays the role of decoupling the first-order dressed states from the fluctuation of Ω₁. With these two driving fields as H_d1 and H_d2, we find the effective Hamiltonian in the second-order interaction picture (see the appendix for details) as

$$\begin{equation} H_{\mathrm{I}}^{(2)}=\frac{\hbar\Omega_2}{2} \sigma_y, \end{equation} \tag{ 3 }$$

where $\sigma _y=-i \left \vert \uparrow \right \rangle \left \langle \downarrow \right \vert +i \left \vert \downarrow \right \rangle \left \langle \uparrow \right \vert$ and the second-order dressed states are $\left \vert \uparrow \right \rangle _y=\sqrt {\frac {1}{2}} \left (\left \vert \uparrow \right \rangle +i\left \vert \downarrow \right \rangle \right)$ and $\left \vert \downarrow \right \rangle _y=\sqrt {\frac {1}{2}} \left (\left \vert \uparrow \right \rangle -i\left \vert \downarrow \right \rangle \right)$, which can be used to encode and store quantum information.

In general, we can apply nth order continuous driving fields on the condition that two subsequent drivings describe rotations about orthogonal axes ($\hat {x}$ and $\hat {y}$) in the corresponding interaction picture. The higher-order driving fields can be explicitly written as

$$\begin{eqnarray} H_{d_{2k+1}} &=& 2^{k}\hbar\Omega_{2k+1} \prod_{j=1}^{k}\cos(\Omega_{2j} t) \cos(\omega t) \sigma_x, \end{eqnarray}\noindent \tag{ 4 }$$

$$\begin{eqnarray} H_{d_{2k}} &= &2 ^{k}\hbar\Omega_{2k} \prod_{j=1}^{k}\cos(\Omega_{2j-1} t) \cos\left(\omega t+\frac{\pi}{2}\right)\sigma_x. \end{eqnarray} \tag{ 5 }$$

We would like to stress that the achievable coherence time is limited by the T₁ time, which can usually be achieved using a fourth-order scheme in the context of NV centers in diamond. The decoherence of the nth order dressed qubit stems dominantly from the fluctuation of the nth order driving field. As long as the orthogonality of two consecutive driving field is satisfied, the (n + 1)th driving field protects against the noise of the nth driving field. Hence with a concatenated scheme in which subsequent driving fields have decreasing intensities, the effective dephasing will be sequentially suppressed and coherence times can be extended significantly.

We have used NV centers in diamond to demonstrate the working principle and efficiency of our concatenated continuous dynamical decoupling scheme. As a promising candidate physical system for modern quantum technologies, NV centers have been used to demonstrate basic quantum information processing protocols [30], as well as ultrasensitive magnetometry and nanoscale imaging at room temperature [31–33]. The electronic ground state of NV centers is a spin triplet with three sublevels with magnetic quantum numbers m_s = 0 and ±1, and the zero field splitting is  ∼ 2.87 GHz [34], see figure 1(b). We apply an additional magnetic field along the axis of the NV center to split the energy levels of m_s = ± 1. The two electronic transition frequencies corresponding to m_s = 0 → m_s = ± 1 are determined to be 2042 and 3696 MHz via an optically detected magnetic resonance (ODMR) measurement. The effective two-level system we use is formed by the sublevel $\left \vert m_{\mathrm { s}}=-1 \right \rangle \equiv \left \vert \uparrow \right \rangle$ and $\left \vert m_{\mathrm { s}}=0 \right \rangle \equiv \left \vert \downarrow \right \rangle$. Interaction with ¹³C nuclear spin bath is the dominant source of decoherence for ultrapure IIa type crystals [29], which were used in our experiments. The magnetic noise can be modelled by a fluctuating field and its spectrum is expected to be Lorentzian [28, 29]. While significant progress has been achieved in material engineering [35], it will not be possible to eliminate all sources of magnetic noise in this way, in particular for implanted NV defects [33]. Thus this relatively slow bath is both practically important and at the same time ideal for decoupling experiments [21] and thus for the demonstration of CCD.

In our experiment, we first polarize the NV center electron spin into the sublevel m_s = 0 with a green laser (532 nm). We note that this is equivalent to the preparation of an initial coherent superposition of the dressed states, namely $\left \vert m_{\mathrm { s}}=0 \right \rangle \equiv \left \vert \downarrow \right \rangle = \sqrt {\frac {1}{2}} (\left \vert \uparrow \right \rangle _x+\left \vert \downarrow \right \rangle _x)$. For comparison, we start by applying a single driving field on resonance with the electronic transition $m_{\mathrm { s}}=0 \leftrightarrow m_{\rm s}=-1$. We measure the oscillation of the state m_s = 0 population. It can been seen from figure 2(a) that the decay of the Rabi oscillation is very fast. Theoretically, we model the microwave fluctuation by an Ornstein–Uhlenbeck process, and thus the amplitude of the driving field is time-dependent with random fluctuation as Ω_i(t) = Ω_i[1 + δ_i(t)]. The system dynamics is described by the following master equation:

$$\begin{equation} \frac{\rm d}{{\rm d}t}\rho =-{\rm i}[H(t),\rho]+ \frac{\Gamma}{2}\sum_{a=\sigma_\pm}(2a^{\dagger} \rho a -\rho a a^{\dagger} -a a^{\dagger}\rho).\end{equation}{\noindent} \tag{ 6 }$$

The relaxation takes the high temperature limit, which is the case for our experiment at room temperature. We choose the relaxation parameter Γ corresponding to T₁ = 1.5 ms, which is close to the value of the diamond used in our experiment. The numerical result agrees well with the experimental data and fits a Gaussian decay envelope $S_1(\tau)=\exp {(-b_1^2 \tau ^2/2 )}$ for slow fluctuating fields [36], see figure 2(a). We thus estimate that the decay rate of Rabi oscillation is b₁ ≈ 98 kHz, and the decay time scale is τ₁ = 2.3 μs defined by S(τ₁) = e⁻¹. We stress that the fast decay is mainly due to intensity fluctuations of the microwave field. To support this observation, we have simulated the decay of coherence for two different Rabi frequencies Ω₁ = 20 and 40 MHz. For comparison, we also perform simulations assuming that there is no microwave fluctuation. Our results are shown in figure 2(a₂), which provides evidence that the fast decay mainly stems from the microwave fluctuation, while the residual effect of magnetic noise from the slow spin bath formed by ¹³C nuclei is comparatively small.

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To demonstrate the working principle of our concatenated continuous dynamical decoupling, we add a second-order driving field with a weaker amplitude than the first driving⁶. We first show that the CCD scheme can sustain Rabi oscillations by applying a second-order driving field, the intensity of which is ten times weaker than the first driving field. In figure 2(b), we observe coherent Rabi oscillation after 300,500,800,1000 μs, and find that it decays significantly more slowly than the one using only a single driving field, see figure 2(a) for comparison. Our experimental data thus demonstrate that the effect of the fluctuation of the first driving field can be significantly suppressed by a second-order driving field; see the appendix for more discussion.

We further demonstrate a significant improvement of the dephasing time. In our experiment, the concatenated driving fields are generated using arbitrary waveform generator. With a second-order driving field about 10³ times less intense than the first-order driving, we observed a prolongation of the coherence time by an order of magnitude from T₂ = 2.3 μs to T₂≅21 μs (see footnote ⁶) as estimated by numerical fitting the decay of beat oscillation, see figure 3(a). We stress that such a weak additional driving significantly improves the coherence time, mainly by decoupling the amplitude fluctuation of the first driving field (and incidentally further suppressing the residual effect of magnetic noise). By increasing the amplitude of the second-order driving one can suppress the effect of the fluctuation in the first driving field more effectively; however, this will result in larger fluctuations of the second-order driving itself. The compromise between these two effects leads to the optimal choice of second-order driving. This observation has also been confirmed by our experiment data, from which we estimate the coherence times for various strength of the second-order driving field, see figure 3(c) (red curve with error bars). In the experiment we used the simplified second-order field $H_{d_2}=\hbar \Omega _2 \cos {\left [ \left (\omega +\Omega _1 \right)(t)+\varphi \right ]}\sigma _x$ for which some counter-rotating terms persist that limit the achievable coherence time. This limitation can be overcome by using the refined second-order field given in equation (2), as confirmed by our numerical simulation shown in figures 3(b) and (c) (red squares). These simulations use the parameters estimated from the experiment and exhibit the same qualitative features as the measured data shown in figure 3(c) (red curve with error bars). To further extend T₂, one would need to apply higher-orders driving fields. Our calculations show that (see figure 3(d)) the coherence time can approach the T₁ limit (T₁ = 1.5 ms in the present example) with the fourth-order scheme (i.e. adding the driving fields as in equations (4)–(5)) for the current experiment parameters. The general ability of high-order schemes to suppress decoherence from the environment and the driving fields is valid on condition that the microwave fluctuation is slow (i.e. its bandwidth is smaller 1/T₁), which is usually the case in experiments. This implies that the width of the microwave fluctuation spectrum (which is proportional to the inverse of the correlation time of fluctuation) is relatively small. Thus, high-order driving fields can still effectively suppress the effect of the proceeding field fluctuation as long as their amplitude is much larger than the width of the microwave noise spectrum.

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Dressed qubits can be coherently manipulated by external driving fields and coupled with each other via electron spin dipole interaction. For single qubit rotation, we can use the following radio-frequency field:

$$\begin{equation} H_{\mathrm{sg}}=\hbar\Omega \left[\prod_{k=1,\ldots, n} \cos(\Omega_k t+\varphi_k) \right] \sigma_z. \end{equation}\noindent \tag{ 7 }$$

To demonstrate explicitly how this works, we consider the second-order dressed qubit as an example and apply the field H_sg = ℏΩ cos (Ω₁t) cos (Ω₂t + φ₂)σ_z. In the first-order interaction picture with respect to the original Hamiltonian $H_0^{(1)}=\frac {\hbar \omega }{2} \sigma _z$, the coupling can be written as

$$\begin{equation} H_{\mathrm{c}}^{(1)}= \frac{\hbar\Omega_1 }{2} \sigma_x + \hbar\Omega_2 \cos(\Omega_1 t) \sigma_y+\hbar\Omega \cos(\Omega_1 t)\cos(\Omega_2 t+ \varphi_2)\sigma_z. \end{equation} \tag{ 8 }$$

In the second-order interaction picture with respect to $H_0^{(2)}=\frac {\hbar \Omega _1}{2} \sigma _x$, and using the rotating wave approximation, it becomes

$$\begin{equation} H_{\mathrm{c}}^{(2)}=\frac{\hbar\Omega_2 }{2} \sigma_y + \frac{\hbar\Omega}{2} \cos{\left(\Omega_2 t + \varphi_2 \right)} \sigma_z. \end{equation} \tag{ 9 }$$

By choosing appropriate phase φ₂, one can implement general rotations of the second-order dressed qubit (encoded in the eigenstates of σ_y: $\left \vert \uparrow \right \rangle _y$ and $\left \vert \downarrow \right \rangle _y$), e.g. see figure 4(a). Similar results can be obtained for general higher-order dressed qubits. In fact, this also implies the possibility to measure the amplitude of the radio-frequency field Ω by Rabi-type spectroscopy with the robust dressed qubit, e.g. see equation (9). The sensitivity is determined by the coherence time of the dressed qubit [25, 37]. Using concatenated continuous dynamical decoupling, the coherence time of the dressed qubit is possible to be prolonged to the relaxation time, and one could thus in principle construct a magnetometer with the sensitivity scaling with $1/\sqrt {T_1}$. We would also like to point out that CCD can also be combined with the sensing protocols in biological environments proposed in [26], where the noisy biological environment requires strong driving. With the robust energy gap provided by the CCD scheme, the Hartmann–Hahn condition [38] will be more stably matched, and thus the measurement of the coupling between NV center and the target nuclear spin will be more efficient and precise.

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Regarding the coupling between dressed qubits, we again take the second-order decoupling scheme as an example. The dipole–dipole interaction, e.g. between NV electron spins, is described by

$$\begin{equation} H_{\mathrm{d-d}}=J(3S_z^a S_z^b-\vec{S}_a\cdot \vec{S}_b), \end{equation} \tag{ 10 }$$

where S is the spin-1 operator of NV centers. In the first-order interaction picture, the electron–electron interaction with the driving fields becomes [12]

$$\begin{equation} H_{\mathrm{d-d}}^{(1)}=2J S_z^a S_z^a + \hbar \Omega_1^a \sigma_x^a+ \hbar \Omega_1^b \sigma_x^b+\hbar\Omega_2^a \cos(\Omega_1^a t) \sigma_y^a+\hbar\Omega_2^b \cos(\Omega_1^b t) \sigma_y^b. \end{equation} \tag{ 11 }$$

We use the two sublevels $\left \vert m_{\mathrm { s}}=-1 \right \rangle \equiv \left \vert \uparrow \right \rangle$ and $\left \vert m_{\mathrm { s}}=0 \right \rangle \equiv \left \vert \downarrow \right \rangle$ of NV electron spin as a qubit. The above interaction in the qubit Hilbert subspace can be written as

$$\begin{equation} \fl H_{\mathrm{d-d}}^{(1)}=\frac{J}{2} \left(\sigma_z^a \sigma_z^b+\sigma_z^a +\sigma_z^b \right)+ \frac{\hbar \Omega_1^a}{2} \sigma_x^a+ \frac{\hbar \Omega_1^b}{2} \sigma_x^b+\hbar\Omega_2^a \cos(\Omega_1^a t) \sigma_y^a+\hbar\Omega_2^b \cos(\Omega_1^b t) \sigma_y^b, \end{equation} \tag{ 12 }$$

where σ_x, σ_y and σ_z are Pauli operators in the spin up and down basis. The effective Hamiltonian in the second-order interaction picture with respect to H^(d)₀ = ℏΩ^a₁σ^a_x + ℏΩ^b₁σ^b_x is

$$\begin{equation} H_{\mathrm{d-d}}^{(2)}=\frac{J}{4}\left(\sigma^a_y \sigma^b_y + \sigma^a_z \sigma^b_z \right) +\frac{\hbar\Omega_2 ^a}{2} \sigma_y^a+\frac{\hbar\Omega_2 ^b}{2} \sigma_y^b. \end{equation} \tag{ 13 }$$

The effective entangling coupling between dressed qubits is thus feasible (as in equation (13)), and can be exploited to construct two-qubit gates [39]. We use the following quantity to characterize the fidelity [40] between the coherent coupling of two dressed qubits and the evolution (U_ideal) from the ideal (noiseless) effective coupling Hamiltonian

$$\begin{equation} \fl F\left(U_{\rm{ideal}},\mathcal{M} \right)=\frac{1}{16}\left[ 4+\frac{1}{5}\sum_{\mu,\nu}{\rm tr}\left[ \left(U_{\rm{ideal}}\left(\sigma_{\mu}\otimes \sigma_{\nu} \right) U^{\dagger}_{\rm{ideal}} \right) \mathcal{M}\left(\sigma_{\mu}\otimes \sigma_{\nu} \right) \right] \right], \end{equation} \tag{ 14 }$$

where σ_μ,σ_ν = I,σ_x,σ_y,σ_z and $\left (\sigma _{\mu }\otimes \sigma _{\nu } \right)\neq \mathbf {I}\otimes \mathbf {I}$. We numerically simulate the dynamics $\mathcal {M}$ by solving the master equation in equation (A.9), including both the external magnetic noise and microwave field fluctuations. The result is shown in figure 4(b), which demonstrates that the CCD scheme can also be helpful in the protection of two-qubit coupling.

The CCD scheme can be combined with various quantum information processing tasks, making them robust not only against noise, but also the fluctuations of decoupling fields. The coherent manipulation of the nth order dressed qubit can be implemented with a radio frequency field, see section 4. Such a qubit encoded with the sequential dressed states has an ultralong coherence time limited only by its T₁ time, and thereby can be exploited to construct a single-spin magnetometer [37] to probe a weak oscillating magnetic field with an improved sensitivity. The scheme can also be beneficial for the construction of a precise noise spectrometer [41–43]. We note that the inhomogeneity over an ensemble of quantum systems (namely spatial fluctuation) leads to dephasing of the ensemble collective state. However, if the amplitudes of the first order driving field exceeds the disorder, then the effect of spatial fluctuations is suppressed (see e.g. [44] for a detailed discussion). This again points to the usefulness of CCD as it allows us to increase the driving field amplitudes without suffering from the concomitant noise due to amplitude fluctuations.

We have introduced the concept of CCD and implemented it experimentally. Using an NV center in diamond we demonstrated the superior performance of concatenated continuous dynamical decoupling compared to single driving fields in extending coherence times of a dressed qubit. A qubit encoded in the concatenated dressed states is robust against both environmental dephasing noise and intensity fluctuation of driving fields. Our schemes can be applied to a wide variety of quantum systems where they may find applications in the construction of nanoscale magnetometry and imaging, e.g. with the NV center in diamond, and in the construction of fault-tolerant quantum gates that are protected against noise and control errors.

The work was supported by the Alexander von Humboldt Foundation, the EU Integrating Project Q-ESSENCE, the EU STREP PICC and DIAMANT, the BMBF Verbundprojekt QuOReP, DFG (FOR 1482, FOR 1493, SFB/TR 21) and DARPA. J-MC was supported also by a Marie-Curie Intra-European Fellowship within the 7th European Community Framework Program.

A.1. Driving fields for general concatenated continuous dynamical decoupling

In our concatenated scheme, in general we can apply nth order continuous driving fields on the condition that two subsequent drivings describe rotations about orthogonal axes in the interaction picture (see the example for second order driving fields for illustration). The original system Hamiltonian is

$$\begin{equation} H_0=\frac{\hbar\omega}{2} \sigma_z + \frac{\hbar\delta_b(t)}{2} \sigma_z, \end{equation} \tag{ A.1 }$$

where $\sigma _z=\left (\left \vert \uparrow \right \rangle \left \langle \uparrow \right \vert -\left \vert \downarrow \right \rangle \left \langle \downarrow \right \vert \right)$ and we have explicitly written the magnetic noise term δ_b(t). The first-order driving field (in the lab frame) is

$$\begin{equation} H_{d_1}= \hbar\Omega_1 [1+\delta_1(t)]\cos{(\omega t)} \sigma_x, \end{equation} \tag{ A.2 }$$

where δ₁(t) represents the amplitude fluctuation of the first order driving field. In the interaction picture with respect to $H_0^{(1)}=\frac {\hbar \omega }{2} \sigma _z$, we have the following effective Hamiltonian as

$$\begin{equation} H_{\mathrm{I}}^{(1)}=\frac{\hbar\Omega_1 }{2} [1+\delta_1(t)]\sigma_x + \frac{\hbar\delta_b(t)}{2} \sigma_z. \end{equation} \tag{ A.3 }$$

The first-order dressed states are the eigenstates of σ_x: $\left \vert \uparrow \right \rangle _x=\frac {1}{\sqrt {2}}\left (\left \vert \uparrow \right \rangle +\left \vert \downarrow \right \rangle \right)$ and $\left \vert \downarrow \right \rangle _x=\frac {1}{\sqrt {2}}\left (\left \vert \uparrow \right \rangle -\left \vert \downarrow \right \rangle \right)$. The effect of the magnetic noise term $\frac {\hbar \delta _b(t)}{2} \sigma _z$ will induce transitions between the dressed states $\left \vert \uparrow \right \rangle _x$ and $\left \vert \downarrow \right \rangle _x$, whose rate can be estimated by the noise spectrum of δ_b(t) at the transition frequency ℏΩ₁. The microwave fluctuation δ₁(t) causes the dephasing of the dressed states. To suppress the effect of fluctuations in the amplitude of the first-order driving, we apply a second-order driving field (in the labe frame) as

$$\begin{equation} H_{d_2}=2\hbar \Omega_2[1+\delta_2(t)] \cos\left(\omega t+\frac{\pi}{2}\right) \cos(\Omega_1 t) \sigma_x. \end{equation} \tag{ A.4 }$$

With such an additional driving field, the effective Hamiltonian with respect to $H_0^{(1)}=\frac {\hbar \omega }{2} \sigma _z$ (as in equation (A.2)) becomes

$$\begin{equation} H_{\mathrm{I}}^{(1)}=\frac{\hbar\Omega_1 }{2}[1+\delta_1(t)] \sigma_x + \hbar\Omega_2 [1+\delta_2(t)] \cos(\Omega_1 t) \sigma_y+\frac{\hbar\delta_b(t)}{2} \sigma_z. \end{equation} \tag{ A.5 }$$

In the second-order interaction picture with respect to $H_0^{(2)}=\frac {\hbar \Omega _1 }{2} \sigma _x$ we have

$$\begin{equation} H_{\mathrm{I}}^{(2)}=\frac{\hbar\Omega_2}{2} [1+\delta_2(t)] \sigma_y + \frac{\hbar\Omega_1 }{2} \delta_1(t)\sigma_x, \end{equation}\noindent \tag{ A.6 }$$

where $\sigma _y=-i \left \vert \uparrow \right \rangle \left \langle \downarrow \right \vert +i \left \vert \downarrow \right \rangle \left \langle \uparrow \right \vert$ and the second-order dressed states are $\left \vert \uparrow \right \rangle _y=\sqrt {\frac {1}{2}} \left (\left \vert \uparrow \right \rangle +i\left \vert \downarrow \right \rangle \right)$ and $\left \vert \downarrow \right \rangle _y=\sqrt {\frac {1}{2}} \left (\left \vert \uparrow \right \rangle -i\left \vert \downarrow \right \rangle \right)$. For simplicity, here we only explicitly write the noise terms of δ₁(t) and δ₂(t) in equation (A.6), nevertheless we stress that we do take it into account in our numerical simulations. Similarly, the fluctuation of the first driving field $\frac {\hbar \Omega _1 }{2} \delta _1(t)\sigma _x$ induces the transitions between the second-order dressed states $\left \vert \uparrow \right \rangle _y$ and $\left \vert \downarrow \right \rangle _y$. Its effect can be characterized by the power spectrum of δ₁(t) at the transition frequency ℏΩ₂. In the above derivations, we have adopted the rotating wave approximations which hold when Ω₂ ≪ Ω₁ ≪ ω, which are fulfilled in the present context. In a similar way, one can find the required driving fields for higher-order decoupling. For example, the third- and fourth-order driving fields can be provided by

$$\begin{eqnarray} &&\fl H_{d_3} = 2\hbar\Omega_3 \cos(\omega t) \cos(\Omega_2 t) \sigma_x\quad {\rm and} \quad H_{d_4} = 4\hbar\Omega_4 \cos(\Omega_1 t)\cos(\Omega_3 t) \,\cos\left(\omega t+\frac{\pi}{2}\right)\sigma_x.\nonumber\\ \end{eqnarray} \tag{ A.7 }$$

The general higher-order driving fields can be written in a similar way as in equations (4) and (5) of the main text.

A.2. Numerical simulations of persistent Rabi oscillation

In our experiment, we apply a simplified second-order field as $H_{d_2}=\hbar \Omega _2 \cos {\left [ \left (\omega +\Omega _1 \right)t+\varphi \right ]}\sigma _x$, which leads to the second-order effective Hamiltonian as

$$\begin{equation} H_{\mathrm{I}}^{(2)}=\frac{\hbar\Omega_2}{4} \left(\sin{\varphi}\sigma_y-\cos{\varphi}\sigma_z \right).\end{equation}\noindent \tag{ A.8 }$$

We initially prepare NV spin in the state $\left \vert \downarrow \right \rangle \equiv \left \vert m_{\mathrm { s}}=0 \right \rangle$, which is one of the second-order dressed states if the relative phase φ = 0. As we point out in the main text, the fluctuation of the first driving field would induce transitions of second-order dressed states, which can be suppressed due to the energy penalty induced by the second-order driving field. This is the working principle of CCD scheme. In figure 2(a) of the main text, our experimental data demonstrates this principle by the observation of the persistent oscillation of the state $\left \vert \downarrow \right \rangle \equiv \left \vert m_{\mathrm { s}}=0 \right \rangle$ population. It represents the long lifetime of the second-order dressed states, which is dependent on the relaxation time T₁ of the bare spin states and the residual effect of the first driving field fluctuation. Our experimental data thus demonstrates that the effect of the first driving field fluctuation can be significantly suppressed by a second-order driving field. We have performed numerical simulation, as shown in figure A.1, with a second-order driving field using the estimated parameters. We note that in the numerical simulation we fixed the relative phase, thus the contrast of Rabi oscillation is twice of figure 2(b) in the main text. Nevertheless, the essential feature of the results from our numerical simulation, namely the slow decay of Rabi oscillation, agrees well with the experimental data.

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A.3. Numerical simulation of system dynamics with concatenated driving

We model the magnetic noise and the microwave fluctuation by Ornstein–Uhlenbeck processes [28]. The system dynamics is described by the following quantum master equation:

$$\begin{equation} \frac{\rm d}{{\rm d}t}\rho =-{\rm i}[H(t),\rho]+ \frac{\Gamma}{2}\sum_{a=\sigma_\pm}(2a^{\dagger} \rho a -\rho a a^{\dagger} -a a^{\dagger}\rho), \end{equation}\noindent \tag{ A.9 }$$

where H(t) is the total Hamiltonian including both the magnetic noise and the driving fields with fluctuations. The Lindblad operators represent the relaxation process with the rate Γ corresponding to T₁ = 1.5 ms which is close to the value of the diamond used in our experiment. The relaxation in the above master equation takes the high temperature, which is valid for our experiment at room temperature. We generate 2000 realizations of Ornstein–Uhlenbeck processes with the exact simulation algorithm [45] for the magnetic noise δ_b(t) and the microwave fluctuations δ_i(t). We choose the correlation time for the spin bath as τ_c = 25 μs. The fluctuation of microwave filed amplitude is usually much slower and we use the value of correlation time τ_m = 1 ms. We first simulated numerically the experimental data in figure A.2(a) for the case of Rabi oscillation with one single driving field. The numerical result agrees well with the experimental data and the Gaussian fit of the decay envelope (arising from slow noise [36]) $S_1(\tau)=\exp {(-b_1^2\tau ^2/2)}$ (see figure 2(a) of the main text). This supports that an Ornstein–Uhlenbeck process serves as a good model for the driving field fluctuation. We estimate that the relative amplitude fluctuation of the driving field is about 2.4 × 10⁻³. We use these estimated parameters for the numerical simulations of higher-order driving schemes by solving the master equation in equation (A.9). In figure A.2, we plot the coherence decay of the dressed qubit quantified by

$$\begin{equation} f(\tau)=\left \vert \left\langle \psi_{\mathrm{I}}(0) \vert \psi_{\mathrm{I}}(\tau) \right \rangle\right \vert, \end{equation} \tag{ A.10 }$$

with $\left \vert \psi _{\mathrm {I}}(0) \right \rangle =\left \vert \downarrow \right \rangle$, which is a coherent superposition of the dressed qubit ($\left \vert \uparrow \right \rangle _{x/y}$ and $\left \vert \downarrow \right \rangle _{x/y}$), and $\left \vert \psi _{\mathrm {I}}(\tau) \right \rangle$ is the state of the dressed qubit after time τ. The results in figure A.2 (see also figure 3(d) in the main text) shows the scaling of the coherence time with high-order driving and the possibility to approach T₁. We remark that for the first- and second-order drivings, the decay envelop is more like Gaussian, while for third- and fourth-order drivings the decay envelope becomes exponential (arising from the relaxation) as the effect of microwave fluctuations is increasingly suppressed.

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