Hybrid continuous dynamical decoupling: a photon-phonon doubly dressed spin

We study the parametric interaction between a single nitrogen-vacancy electronic spin and a diamond mechanical resonator in which the spin is embedded. Coupling between spin and oscillator is achieved by crystal strain, which is generated upon actuation of the oscillator and which parametrically modulates the spins’ energy splitting. Under coherent microwave driving of the spin, this parametric drive leads to a locking of the spin Rabi frequency to the oscillator mode in the megahertz range. Both the Rabi oscillation decay time and the inhomogeneous spin dephasing time increase by two orders of magnitude under this spin-locking condition. We present routes to prolong the dephasing times even further, potentially to the relaxation time limit. The remarkable coherence protection that our hybrid spin-oscillator system offers is reminiscent of recently proposed concatenated continuous dynamical decoupling schemes and results from our robust, drift-free strain-coupling mechanism and the narrow linewidth of the high-quality diamond mechanical oscillator employed. Our findings suggest feasible applications in quantum information processing and sensing.

(Some figures may appear in colour only in the online journal) Solid-state spins rank amongst the most promising sources for quantum information processing and sensing, due to their ease of use and the in-principle scalability they offer. Exploiting their quantum nature for computation or sensing requires quantum coherence to be preserved for a time long compared to the speed of their coherent manipulation. The fundamental limit of these coherence times is set by the spin relaxation rate T 1 [1,2]-in some cases exceeding seconds [3] -while spin manipulation rates in the Gigahertz range have recently been demonstrated [4]. Despite these interesting prospects, reaching relaxation-limited coherence times for solid state spins remains highly challenging due to additional noise sources in the spins' environment and driving fields, which significantly reduce the number of coherent spinmanipulation steps that can be experimentally achieved.
Various approaches to enhance spin coherence times towards the relaxation time limit have been put forward. Most notably, dynamical decoupling-a method of filtering out environmental noise through either pulsed [5,6] or continuous [7][8][9][10] driving of the spin-can isolate the spins from the low-frequency environmental fluctuations responsible for dephasing. In this regard, pulsed schemes have proven especially effective [11], robust to pulse errors [12] and even allow for decoherence protected quantum gates [13]. However, they come at the cost of increased experimental complexity and potentially harmful, high intensity driving pulses. Decoupling schemes relying on continuous driving, on the other hand, are experimentally simpler [7,8], do not suffer from pulsing errors, and can be readily combined with quantum gate operations [14,15]. Unfortunately, the effectiveness of these schemes is limited by the spin's high sensitivity to the ubiquitous low-frequency fluctuations of the driving field. As recently shown [16,17], applying higher order drivings fields, with each additional field decoupling the spin from the driving field fluctuations of the preceding driving field, would protect the spin even further. In principle, this procedure can be iterated ad infinitum and may then yield relaxation limited coherence times [17]. The application of many consecutive decoupling fields, however, exposes the spin to significant driving field powers and sets intrinsic constraints to the speed at which the final, decoherence protected spin states can be coherently manipulated [17]. New approaches to continuous dynamical decoupling are therefore required to yield fully robust spin systems which are of practical use to quantum information processing and sensing.
In this work, we experimentally demonstrate a novel and efficient approach to continuous dynamical decoupling, through the parametric interaction of a single electronic spin with a mechanical resonator. We employ a coherent microwave drive for first order decoupling and use the spin-oscillator interaction to decouple the spin from amplitude fluctuations in the microwave field. This concatenated, hybrid continuous dynamical decoupling (HCDD) builds on two key advances over past approaches [17]: (1) second order decoupling is achieved by a parametric drive along the quantisation axis of the undriven spin. The second order driving field is thus orthogonal to the first order drive, irrespective to the phase between these two fields, in contrast to the conventional dynamical decoupling by concatenated driving, where phase-locking between the driving field is required to ensure the necessary orthogonality. (2) The second order decoupling field is transduced to the spin through a mechanical oscillator, whose resonant behaviour effectively low-pass filters amplitude noise and yields a highly stable, second order driving field amplitude. A concatenation of only two driving fields thereby yields a coherence time nearly two orders of magnitude longer than that of an undriven spin, while also maintaining a final dressed state splitting close to one MHz. By using mechanical oscillators with even higher quality-factors, our scheme should allow us to prolong coherence times even further and ultimately reach the limit imposed by energy relaxation.
Our experiments were performed on individual, negatively charged nitrogen-vacancy (NV) defect centres embedded in singly clamped cantilever diamond mechanical oscillators ( figure 1(b)). The cantilevers exhibited typical fundamental mode-frequencies w m in the MHz range (w p =2 5.81MHz m for the cantilever studied here) and were fabricated along the [ ] 110 crystal axis of ultra-pure, single-crystal, synthetic diamond using top-down nanofabrication described elsewhere [18,19]. NV centres were created at densities <1 μm −2 by 14 N ion implantation and subsequent high-temperature annealing [20]. We used a homebuilt confocal microscope to study spin-dynamics of a NV centre located at the base of the cantilever, where strain-fields for parametric driving are maximised [21,22]. The cantilevers were mechanically actuated using a piezoelectric transducer, which was placed nearby the sample. All our experiments were performed under ambient conditions.
The NV centre orbital ground state is a spin-triplet, witĥ S z eigenstates -ñ ñ + ñ | | | 1 , 0 , 1 , whereŜ z is the angular momentum operator along the NV binding axis z. The magnetic sublevels  ñ | 1 are split from ñ |0 by a zero-field splitting = D 2.87 GHz 0 and can be further split in energy by a magnetic field B z along z [23]. Optical spin-readout and initialisation into ñ |0 is readily achieved by green optical illumination and detection of red NV fluorescence, while the spin can be coherently driven by applying resonant microwave magnetic fields,B AC , transverse to z [24,25]. In this work, we consider the dynamics of the effective two-level system formed by , while the state ñ |1 is split off in energy by a static magnetic field = B 10.7 G z and ignored in the following. In addition to nearresonant microwave driving with transverse magnetic fields, we employ parametric driving by time-varying (AC) strain fields along z generated by the cantilever (figure 1(a)). The effective Hamiltonian for the two-level system spanned by then reads the microwave driving field with frequency w MW and amplitude (Rabi frequency) W MW . Parametric driving of the spin is achieved by on-axis, AC strain w P = P  couples to the NV centre with strength d 5.5 GHz/strain [21,22] 1 transition energy are resonant [9]. To suppress this coupling, we used the bias field B z to set the transition energy g » B 2 60 z NV MHz far off-resonance from the cantilever frequency. The effect of transverse strain can therefore be neglected for the present work. Note that while data for only one NV centre are presented here, we found consistent results for all investigated NV centres, which were oriented along [ ] 111 or [¯] 111 , i.e. for 50% of all NVs present. This yield could be increased to close to 100% using recently developed growth methods for creation of fully aligned NV centres [26], together with an appropriate cantilever fabrication procedure.
To provide a baseline for our subsequent measurements, we first determined the relevant NV spin relaxation times in the absence of mechanical driving. The NV population decay time T 1 was determined using the experimental pulse sequence illustrated in figure 1(c). Following initialisation in ). We obtain DP directly from the transient NV fluorescence photons c 1 and c 0 , as defined in figure  1 . Therefore, DP yields a measure for the spin population decay, from which we determine =  T 5.1 0.8 1 ms through an exponential fit ( figure 1(c)). Similarly, we determined the decay-time T R of the spin's Rabi oscillations by pulsed, coherent driving of the  ñ «  ñ | | transition with a resonant microwave magnetic field of variable duration τ ( figure 1(d)). The observed Rabi oscillations show a pronounced beating pattern that results from the ∼2.18 MHz hyperfine-splitting between the NV electronic spin and the nitrogen's 14 N nuclear spin ( figure 1(d , which is three orders of magnitude shorter than the relaxation-limit set by T 1 [1]. The Gaussian decay-envelope of our Rabi oscillations suggests that slowly fluctuating noise sources are responsible for the excess dephasing we observed [28]. While both W MW and d j may fluctuate, in our experiment, where  d W 2 j MW , only the former contributes to first order to dephasing. Indeed, we were able to quantitatively reproduce the observed decay envelope (orange line in figure 1(d)) by numerically averaging over Rabi oscillations with random microwave amplitude noise of relative amplitude´-6 10 3 -a typical value for the commercial microwave sources we employ.
In the presence of continuous, resonant microwave driving, the eigenstates of the driven spin system are the 17,29], with energy difference  W MW and N the number of microwave photons dressing the spin (i.e. the mean photon number in the coherent microwave field which drives the spin). These new basis states form a potential resource for quantum information processing [14] or sensing [17]. The Rabi decay time T R , which can be interpreted as the dressed state relaxation time [30], is then a key figure of merit for such applications. To further prolong T R , we decouple  ñ | N from fluctuations in W MW by applying an additional driving field, which near-resonantly and coherently drives the figure 2(b)) and consequently lead to higher-order dressed states-the principle underlying dynamical decoupling by concatenated continuous driving [17].
We achieve second order dressing and the associated dynamical decoupling by driving the + ñ « -ñ | | N N transition using the time-varying, longitudinal strain field generated by the diamond cantilever in which the NV spin is embedded. Such driving is enabled by the coupling term s P   d z 0 (see equation (1) figure 2(a)). The weak, additional spectral features visible in this regime stem from the two additional, hyperfine-split NV spin transitions, which are weakly, offresonantly driven. For w p W » =2 5.81MHz MW m , however, we observe a spectrum that shares striking similarities with the well-known Mollow-triplet in quantum electrodynamics [31,32]: the measured coherent spin oscillations peak at a single frequency w p 2 m , irrespective of the exact value of W MW , with only two weak side bands, which appear at w p  W ( ) 2 2 m m and thereby allow us to quantitatively determine W m [31]. Such spin-oscillator frequency-locking, induced by parametric strain-driving of a bulk NV centre in a mechanical oscillator, was previously observed for NV centres in diamond nanocrystals parametrically driven by the magnetic fields from a nearby antenna [31] or by the mechanical motion of a spin in a strong magnetic field gradient [32].
This phenomenon of frequency-locking is at the heart of our HCDD scheme and indeed efficiently decouples the NV spin from environmental fluctuations. The parametric drive couples the microwave dressed states  ñ | N and thereby yields new eigenstates  ñ | N M , , now doubly dressed by N photons and M cantilever phonons [31]. For resonant strain-driving ( w W = spins for applications in quantum information processing and quantum sensing.