Robust dynamical decoupling with concatenated continuous driving

The loss of coherence is one of the main obstacles for the implementation of quantum information processing. The efficiency of dynamical decoupling schemes, which have been introduced to address this problem, is limited itself by the fluctuations in the driving fields which will themselves introduce noise. We address this challenge by introducing the concept of concatenated continuous dynamical decoupling, which can overcome not only external magnetic noise but also noise due to fluctuations in driving fields. We show theoretically that this approach can achieve relaxation limited coherence times, and demonstrate experimentally that already the most basic implementation of this concept yields an order of magnitude improvement of the decoherence time for the electron spin of nitrogen vacancy centers in diamond. The proposed scheme can be applied to a wide variety of other physical systems including, trapped atoms and ions, quantum dots, and may be combined with other quantum technologies challenges such as quantum sensing and quantum information processing.

Introduction.-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][2][3][4]. The performance of such quantum devices critically depends on coherence times of their constituent quantum systems which, in turn, is 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. Dynamical decoupling originated in liquid state NMR and allows for the resolution of complex spectra [5]. Significant progress has been made with the theoretical proposals of various dynamical decoupling [6][7][8][9][10][11][12][13], and their experimental demonstrations [14][15][16][17][18][19][20][21][22][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][12][13][14]. In principle, one can apply continuous driving to reduce the noise suffered by a qubit considerably simply by increasing their intensity. 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. 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 nano-scale magnetometers [1][2][3] and error-resistant quantum operation [13]. It will also be of particular interest for T 1 limited Rabi-type magnetic resonance imaging, the resolution of which highly depends on the stability of microwave driving fields [25,26].
In this Letter, 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 NV center in diamond where a second, weaker driving field, reduces the impact of the amplitude fluctuations of the first order 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.
Concatenated continuous dynamical decoupling -We start by considering a two-level quantum system with the eigenstates |↑ , |↓ , see Fig.1(a). Its system Hamiltonian is H 0 = ω 2 (|↑ ↑| − |↓ ↓|). 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 ω between |↑ and |↓ as where σ x = |↑ ↓| + |↓ ↑|. In the interaction picture with respect to H 0 and with rotating wave approximation, we find H 2 (|↑ ± |↓ ) are the dressed states. In 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 ef-arXiv:1111.0930v2 [quant-ph] 8 May 2012 Second-order concatenated continuous dynamical decoupling: The first-order driving field with the frequency ω and the amplitude Ω1 creates the firstorder dressed states |+ , |− , which suffer less from the dephasing effect of environment noise, however are subject to the fluctuation in the amplitude Ω1 of the driving field. The subsequent second-order driving field with the amplitude Ω2 (which is generally smaller than Ω1) is off-resonant with the detuning Ω1 and a relative phase φ with respect to the firstorder driving field. With such a second-order driving field, it is possible to overcome the intensity fluctuation of the firstorder driving, and leads to the second-order dressed state |↑ϕ and |↓ϕ that are subject to the much reduced energy fluctuation. b: The diagram for the energy levels of the NV center electron spin. The NV spin triplet electronic ground state is splitted by an applied magnetic field. The effective two-level system used in our experiment is formed by the spin sublevels ms = 0 (labeled as |↓ ) and ms = −1 (labeled as |↑ ). c: The procedure for the Ramsey experiment with the second-order dressed qubit: the NV electron spin is first polarized into the ms = 0 sublevel, which is a coherent superposition of |↑ϕ and |↓ϕ ; the first-and second-order driving fields are simultaneously switched on for time τ followed by the optically readout of the population of ms = 0 via spin-dependent fluorescence.
ficiency 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 [27]. 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. 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 This second-order driving field is on resonance with the the energy gap of the first-order dressed state, see Fig.1 (a), which actually describes rotation about the axiŝ n (ϕ, −π/2) in the interaction picture (see SI for details) and thus plays the role of decoupling the first-order dressed states from the fluctuation of Ω 1 . With these two driving fields as H d1 and H d2 , we find the effective Hamiltonian in the second-order interaction picture as where the second-order dressed states are Therefore, one can encode quantum information in the above second-order dressed states as in Eq. (4)(5). In general, we can apply n-order continuous driving fields on the condition that two subsequent drivings describe rotations about orthogonal axes in the corresponding interaction picture. We remark that the relative phase does not need to take a specific value, once it is fixed the corresponding second-order dressed state can serve as a robust encoded qubit see SI for details.
The decoherence of such n-th order dressed qubit stems dominantly from the fluctuation of the nth-order driving field and the residual effect of the fluctuations of the preceding driving fields. As long as the orthogonality of two consecutive driving field is satisfied, the next consecutive field reduces the noise of the previous. Hence coherence times of the nth-order dressed qubit will mainly be limited by the fluctuations of the nth-order driving field. With a concatenated scheme in which subsequent driving fields have decreasing intensities, the effective dephasing will be sequentially suppressed and coherence times of the nth-order dressed qubit can be extended significantly.
Experimental demonstration of CCD.-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 [28], as well as ultrasensitive magnetometry and nano-scale imaging at room temperature [29][30][31]. The electronic ground state of NV center is a spin triplet with three sublevels with magnetic quantum numbers m s = 0 and m s = ±1, and the zero field splitting is ∼ 2.87GHz [32], see Fig.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 2042MHz and 3696MHz via an ODMR measurement. The effective two-level system we use is formed by the sublevel m s = −1 ≡ |↑ and m s = 0 ≡ |↓ . Interaction with 13 C nuclear spin bath is the dominating source of decoherence for ultrapure IIa type crystals which were used in our experiments. This relatively slow bath is ideal for decoupling experiments [21]. Significant progress was achieved in material engineering [33] but not all sources of magnetic noise can be eliminated in this way, in particular for implanted NV defects [31]. 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 for the relative phase ϕ = nπ, namely |m s = 0 ≡ |↓ ∝ cos ϕ 2 |↑ ϕ − sin ϕ 2 |↓ ϕ . For comparison, we start by applying a single driving field on resonance with the electronic transition m s = 0 ↔ m s = −1. We measure the oscillation of the population of state m s = 0. It can been seen from Fig.2(a) that the Rabi oscillation decays in a few microseconds. By fitting the experimental data with a Gaussian decay envelope S 1 (τ ) = exp −b 2 1 τ 2 /2 (see SI for details), we estimate that the decay rate of Rabi oscillation is b 1 ≈ 88kHz. We stress that the fast decay is mainly due to intensity fluctuations of the microwave field, as the Rabi frequency is 40MHz, which is strong enough to suppress the effect of magnetic noise almost completely (the nuclear spin bath formed by 13 C nuclei is slow compared to driving frequency). This statement will be supported and verified by the experimental data that we present here.
To demonstrate the working principle of our concatenated continuous dynamical decoupling, we add a secondorder driving field with a weaker amplitude than the first driving. The measured population of the sublevel m s = 0 can be written as (see SI for a derivation) We remark that it is not necessary to fix the relative phase ϕ between two driving fields for the estimation of coherence times. The measured signal averaging over random phase ϕ leads toP 0 (τ ) = 1 2 + 1 4 cos (Ω 1 τ ) + 1 4 cos (Ω 1 τ ) cos Ω2 2 τ . The oscillatory components and their decay features remain the same as a fixed phase.
The second term in Eq. (6) represents Rabi oscillations from the first-order driving. Its decay is mainly due to the fluctuation of the first-order driving field, the effect of which now in the interaction picture is causing the relaxation of the dressed qubit and will be suppressed by the energy gap induced by the second-order driving. To demonstrate this, we first show that 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 Fig.2 (b), we observe coherent Rabi oscillation after 300, 500, 800, 1000µs, and find that it decays significantly slower than the one using only a single driving field (see Fig. 2a for comparison). Our experimental data thus demonstrates that the effect of the fluctuation of the first driving field can be significantly suppressed by a second-order driving field. The measured state population of the sublevel ms = 0 as a function of the evolution time τ . The blue curve is the experimental data, and the purple curve is numerical fitting. The Rabi frequency of the first driving is Ω1 = 40MHz, and the intensity of the second driving is three orders of magnitude weaker. b: The estimated coherence time of the second-order dressed qubit as a function of the amplitude of the second-order driving. The Rabi frequency of the first driving is Ω1 = 40MHz. We fit the experimental data with a Gaussian decay envelop for the Rabi oscillation and the beatings from the second-order driving respectively.
We further estimate the dephasing time of the secondorder dressed qubit by numerically fitting the decay of beatings (i.e. the third term in Eq.(6)), which reflects the coherence of the second-order dressed qubit. The decay stems from the fluctuations of both the first and second driving fields. Our experimental data, in combination with numerical fitting, as plotted in Fig.3 (b), shows that there is an optimal amplitude for the second-order driving field in order to achieve the best coherence time. By increasing the amplitude of the second-order driving one can suppress the effect of the fluctuation in the first driving field, 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 a second-order driving. In our experiment, with a secondorder driving field about 1000 times less intense than the first-order driving, one can prolong the coherence by an order of magnitude from T 2 = 2.5µs to T 2 ∼ = 25µs, see Fig.3 (a-b). Such a weak additional driving would not have apparent effect in suppressing environmental magnetic noise, nevertheless it significantly improves the dynamical decoupling efficiency. This fact supports our claim that the fast decay of Rabi oscillation in Fig.2(a) is mainly due to the fluctuation in the driving field, which is shown to be a severe problem in conventional continuous dynamical decoupling with a single drive.
We also demonstrate another important ingredient for concatenated continuous dynamical decoupling, namely fixing the relative phase between consecutive driving fields, in order to generate a long living dressed qubit. We achieved this goal by using arbitrary waveform generator (AWG) to generate the first-and second-order driving fields. In Fig.4, we show an example measurement with an estimated fixed phase ϕ = 2π/5.
Potential applications of CCD.-The CCD scheme can be combined with various quantum information process- ing tasks and make them robust against not only noise but also the fluctuations of decoupling fields. The initialization, coherent manipulation and readout for the second-order dressed qubit can be easily achieved with a continuous driving field at the frequency ω c = ω + Ω 2 /2 and with the same phase as the first-order driving. Such a qubit encoded with the sequential dressed states has an ultralong coherence time, and thereby can be exploited to construct a single-spin magnetometer to probe a weak oscillating magnetic field b(t) = b AC cos[(Ω 1 + Ω/2)t]. If b AC is much larger than the fluctuation of the second driving field, the sensitivity will be ∼ 1/ √ T 1 with T 1 the coherence time demonstrated in Fig.2(a); otherwise it scales as 1/ √ T 2 with T 2 the prolonged dephasing time. The scheme can also be beneficial for the construction of a precise noise spectrometer [26,35]. We note that the inhomogeneity over an ensemble of quantum systems (namely spatial fluctuation) also leads to dephasing of the ensemble collective state. Our schemes can suppress this kind of spatial fluctuation and in the mean time can be robust against the inhomogeneity of the driving field itself, which may be useful in the ensemble-based magnetometer and quantum memory [36].
Summary.-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 nano-scale 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.  5. (Color online) a: Numerical simulation of coherent driven oscillation of NV center with a single microwave field (red curve). The blue curve is the experimental data. The Rabi frequency is Ω1 = 40MHz, and its relative amplitude fluctuation is estimated to be δ1/Ω1 ≈ 2 × 10 −3 . b: Numerical simulation of persistent Rabi oscillation by adding a second-order driving field, the intensity of which is 10 times weaker than the first driving (Ω1 = 20MHz). The plot shows that Rabi oscillation can sustain for much longer time than one single driving field. The four panels show the very slow decay of the Rabi oscillation beyond 300µs. The decay is mainly due to the relaxation of NV center itself, we use the value of the relaxation time T1 = 1.5ms in the numerical simulations. The relative phase is fixed as ϕ = 0.