Abstract
Reminiscent of the bound character of a qubit’s dynamics confined on the Bloch sphere, the observation of a Mollow triplet in the resonantly driven qubit fluorescence spectrum represents one of the founding signatures of quantum electrodynamics. Here we report on its observation in a hybrid spinnanomechanical system, where a nitrogenvacancy spin qubit is magnetically coupled to the vibrations of a silicon carbide nanowire. A resonant microwave field turns the originally parametric hybrid interaction into a resonant process, where acoustic phonons are now able to induce transitions between the dressed qubit states, leading to synchronized spinoscillator dynamics. We further explore the vectorial character of the hybrid coupling to the bidimensional deformations of the nanowire. The demonstrated microwave assisted synchronization of the spinoscillator dynamics opens novel perspectives for the exploration of spindependent forces, the key ingredient for quantum state transfer.
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Introduction
Amechanical oscillator coupled to a twolevel system is a versatile basis to study the interaction between macroscopic and purely quantum objects. This unconventional combination^{1,2} is a promising route towards the generation of nonclassical states of motion of macroscopic objects. Hybridcoupling signatures have now been demonstrated between a mechanical oscillator and Bose–Einstein condensates^{3,4}, superconducting qubits^{5,6,7}, solidstate single spins^{8,9,10,11,12,13,14,15,16,17}, molecules^{18} or quantum dots^{19,20,21,22,23}.
The hybrid interaction coupling phonons and qubits is in profound analogy with quantum electrodynamics (QED) where hallmark experiments revealing the interplay between atoms and photons have permitted exploring the foundations of quantum mechanics. In particular, the appearance of a Mollow triplet in atomic fluorescence spectra^{24}, characterized by the onset of sidebands appearing on each side of the pump frequency with splitting proportional to the laser field strength, is one of the characteristic signatures of the strongly driven Jaynes–Cumming interaction. Along with the Autler–Townes doublet^{25} or vacuum Rabi oscillations, it expresses the dressing of the atom with the optical photon field^{26}. Since then, Mollow triplets were observed in atomic vapours^{27,28}, single molecules^{29,30} single quantum dots^{31,32} or superconducting qubits^{33} coupled to photon fields in the optical or microwave (MW) domain.
Here we report on the observation of a phononic Mollow triplet, where the phonon field of a nanomechanical oscillator dresses a MWdressed single spinqubit immersed in a strong magnetic field gradient. We investigate the dynamics of the spin qubit in presence of large mechanical drive when the spin precession gets locked onto the mechanicaldriving tone. We exploit the bidimensional deformations of the nanowire fundamental eigenmodes to fully investigate the vectorial character of the hybrid coupling. This also represent a novel dynamical regime for hybrid qubitnanomechanical systems: the observed synchronization of the spin precession onto the mechanical oscillation frequency opens new detection strategies for observing spindependent forces.
Results
Hybrid spinnanomechanical interaction
Our hybrid device consists of a single nitrogenvacancy (NV) spinqubit hosted in a diamond nanocrystal attached to the vibrating extremity of a silicon carbide (SiC) nanowire^{10}. A strong magnetic field gradient couples both components through a spatially dependent Zeeman effect (Fig. 1a). Formally, the generic hybrid spinoscillator Hamiltonian can be expressed as , where ω_{0}/2π is the qubit energy, Ω_{m}/2π the oscillator frequency, a^{†}(a) the phonon creation (annihilation) operator, σ_{i} the Pauli matrices of the spinqubit quantized along the z axis and g_{i} the respective coupling constants. As in QED, a distinction between the transverse and parametric regimes can be made. In the first situation, described by the interaction Hamiltonian , the mechanical oscillator and qubit can coherently exchange single excitations if they have similar frequencies. Several hybrid mechanical systems are exploring this regime, either through a direct interaction^{6,34,35} or mediated by a bus cavity^{7,36,37}. In the case of parametric hybridcoupling, the mechanical motion modulates the qubit energy according to the coupling Hamiltonian . Reciprocally, the qubit exerts a state dependent force on the oscillator which represents the key ingredient for quantum state transfer between both components. This configuration enables hybrid coupling between components with totally different excitation energies. Moreover by employing a resonant MW tone driving Rabi precession of the qubit at frequencies Ω_{R}/2π close to the mechanical frequency, it is possible to let both components of the hybrid system evolve on similar time scales^{15,38}. By doing so the parametric interaction with the original qubit is turned into a transverse coupling to the MWdressed qubit. This configuration enables the observation of a phononic Mollow triplet, provided that the phonon field is coherently driven and that the oscillator frequency is larger than the qubit decay rate Γ_{spin}, which corresponds to the socalled resolved sideband regime (Ω_{m}>Γ_{spin}) (ref. 16) of the parametric interaction. In that situation we introduce the dynamical parametric modulation strength δω_{0}, which denotes the classical amplitude of the mechanically driven parametric modulation (Fig. 1b).
The bidimensional vibration properties of our nanowires were described in ref. 39. First signatures of parametric coupling of a single NV spinqubit to the vibrations of a nanowire were reported in ref. 10 in the adiabatic regime through continuous spinqubit electron spin resonance spectroscopy. The mechanism of spin locking on a time varying RF field was first observed in ref. 15 and suggested the possibility to observe a phononic Mollow triplet in our hybrid spinnanomechanical system. Formally, this required entering the resolved sideband regime, developing fast and stable dynamical actuation and readout capacity of the suspended spin qubit and coping with the bidimensional character of the spinqubit trajectories in space. This permitted thereby a full exploration of the intrinsically vectorial nature of the hybrid parametric coupling.
Experimental setup
The experimental setup is sketched in Fig. 1a. The nanomechanical oscillator is a 6μmlong SiC nanowire of 300 nm diameter, suspended from a sharp metallic tip. Its moving extremity is functionalized with a ≃50 nm nanodiamond hosting a single NV defect. The hybrid system is investigated with a confocal microscope apparatus (Supplementary Fig. 1) and a 532nm pump laser. It serves for both measuring the vibrations of the nanowire using either the transmitted or reflected light beams and for optical polarization and readout of the qubit using spinstate dependent fluorescence detection^{40}, see Supplementary Fig. 2.
Measurements of the nanowire Brownian motion permit determining the mechanical properties of the fundamental flexural eigenmodes^{39}. These are aligned along two perpendicular directions e_{1,2}, see Fig. 1f, tilted by ≈10° with respect to the optical axis, at frequencies Ω_{m}/2π (m=1 or 2) of 5.99 and 6.29 MHz respectively, with mechanical damping rates in air, limited by acoustic emission. The measured effective masses of M_{eff}≈10^{−15} kg correspond to a spatial spreading of their Brownian motion over with zeropoint fluctuations of Δx_{q}≈36 fm. Using a resonant force actuation δ F, either piezoelectric or electrostatic, it is possible to drive vibrations of the the nanowire around its rest position r_{0}. Its vectorial deflection δ r(t) can thus be expressed in frequency space as:
using the mechanical susceptibilities and the independent Langevin forces . By adjusting the drive frequency as well as the orientation e_{F} of the force vector with respect to the nanowire eigenmodes it is thus possible to generate different trajectories in the oscillation plane. This will permit exploring the vectorial character of the magnetic hybrid interaction.
The backscattered fluorescence of the NV defect is collected by avalanche photodiodes arranged in a Hanbury Brown and Twiss configuration which allows to confirm the presence of a single NV defect through fluorescence intensity autocorrelation functions (Fig. 1d; ref. 41). The qubit can be initialized in its ground state through optical pumping, manipulated with quasiresonant MW fields and readout exploiting its spinstate dependent fluorescence rate^{40}. When immersing the suspended NV defect in a strong magnetic field gradient the spin state becomes position dependent, as described by the Zeeman Hamiltonian −gμ_{B}σ·B(r_{0}+δr). At first order the nanomechanical oscillator’s vectorial deflection δ r and the spin are dynamically coupled through a magnetic field gradient according to the interaction Hamiltonian:
Thus, it is generally possible to tune the interaction from resonant (σ_{x,y}) to parametric (σ_{z}) by adjusting the qubit frequency and the topography of the magnetic field gradient. In the following, we will only investigate the parametric case, where the nanomotion modulates the qubit energy.
A 18μm diameter NdFeB hard magnetic sphere generates the magnetic field gradient needed to generate a large hybridcoupling strength. It is positioned onto a narrow gold stripline antenna, see Fig. 2b, used to deliver the resonant MW field and especially designed to fit between the high numerical aperture objectives without compromising the photon collection efficiency (Supplementary Note 1) and the spatial access to the magnetic gradient source. The latter can be piezo scanned with respect to the suspended NV spin with nanometric precision. The following restrictions dictate the necessary configuration of the experimental setup. Rotating the nanowire while optimizing the collected fluorescence allows to align the intrinsic spin’s quantization direction along the optical z axis^{10}. The remanent magnetization of the NdFeB particle was oriented in a strong magnetic field (≈1.5 T) and aligned with the NV natural quantization axis to preserve the spinselective readout efficiency. The magnetic bead stray field also allows to polarize the ^{14}N nuclear spin by working at the excited state level anticrossing (B≈50 mT e_{z}) (refs 42, 43), which permits to restrict our system to a pure twolevel system through MW frequency selection. Finally, the last fundamental requirement consists in reaching a large parametric coupling strength, orienting ∇ B_{z} with the direction of the eigenmode of interest.
Determination of the vectorial parametric coupling strength
To determine the vectorial parametric coupling strength, the spatial dependence of the spinqubit energy ω_{0}(r) was measured by collecting the NV fluorescence while scanning the position of the micromagnet in presence of a continuous MW tone. Typical fluorescence maps are shown in Fig. 2d for varying MW frequencies. The projections of the qubit isoenergy surfaces on the oscillation plane, see Fig. 2d, appear as dark resonant slices^{44,45,46,47}. In addition, a global fluorescence quenching which indicates regions with a strong offaxis magnetic field^{48}. Reproducing this measurement for varying MW frequencies permits determination of ω_{0}(r) (Fig. 2f). When moving in those strong magnetic field gradients, the suspended spin qubit undergoes a dynamical parametric energy modulation of , which is determined by evaluating the gradient of the isoenergy map in the (e_{1}, e_{2}) oscillating plane. The mapping of the vectorial coupling strength measured in a 10 × 2.5 μm^{2} horizontal area in front of the magnetic bead is reproduced in Fig. 2f. Inherent to the dipolar structure of the microbead magnetic stray field, it strongly varies in magnitude and in orientation which permits a fine adjustment of the vectorial coupling strength with respect to the eigenmodes orientations by properly nanopositioning the micromagnet. Furthermore analysis of the fluorescence quenching in this imaging procedure also permits a direct identification of the locations where the B field is properly aligned with the NV quantization axis (Fig. 2c,d), which is a key requirement to ensure efficient optical spinstate readout. Finally, the triple requirement of avoiding fluorescence quenching, polarizing the nitrogen nuclear spin and operating with a large parametric coupling strength compete in determining the best location in space where to operate the experiment.
Qubit dynamics in presence of coherent mechanical motion
Having thus fully characterized the static properties of the system, we now investigate the qubit dynamics in presence of coherent mechanical motion, generated by a modulated piezoelectricdriving force. We first restrict our analysis to one single mechanical mode (m=2) by positioning the gradient source at a location allowing a large parametric coupling strength along the e_{2} orientation and tuning the external drive frequency Ω_{d}/2π to the resonance of the second eigenmode (6.29 MHz). The qubit is initialized in its ground state with laser illumination while the MW power is adjusted so that Ω_{R}≈Ω_{d}. The subsequent Rabi evolution of the population σ_{z}(t) is shown in Fig. 3a, in absence (top) and in presence (bottom) of the coherent mechanical drive. The corresponding Fourier transforms are shown in Fig. 3b. While first a single decaying oscillation is observed, a longerlasting beating signal can be observed when the qubit is coherently oscillating in space, presenting a characteristic triplet spectrum, whose central component coincides with Ω_{d}, revealing the synchronization of the spin on the oscillator dynamics. Increasing the oscillation amplitude δr[Ω_{d}] from 0 to 9 nm results in a linear increase of the triplet separation, see Fig. 3c, up to δω_{0}≈2π × 5 MHz, corresponding to a 0.5 MHz nm^{−1} parametric coupling strength (20,000 T m^{−1} equivalent magnetic gradient or g_{z}/2π=20 Hz), in good agreement with the measured coupling strength at the target position, see Fig. 2f. Scanning the MW power permits to illustrate the dependence of the triplet structure in the detuning between Ω_{R} and Ω_{d}, and underlines the synchronization of the qubit precession on the driven nanomotion^{15,38}.
A doubly dressed spin qubit
These observations can be explained by a double dressing of the spin qubit with MW photon and acoustic phonon fields as follows. The resonant interaction of the MW pump field with the qubit can be described in the dressed states basis , see Fig. 3e, parameterized by the number of excitations N shared between the qubit and the MW pump field^{49}. Under intense coherent excitation, the dynamics of the spinMW (polaritonlike) subsystem can be formally described by a pseudo qubit (), see Supplementary Note 4, quantized along the MW polarization axis with a characteristic energy splitting of Ω_{R}/2π (ref. 49). As a consequence of this rotation of perspective in the Bloch sphere, the respective roles of the σ_{y}, σ_{z} operators are consequently exchanged. Therefore, the phonon field parametrically coupled to the spin qubit (∝(a+a^{†})σ_{z}) is now able to resonantly drive the pseudo qubit, if the resonance condition Ω_{R}≈Ω_{d} is met. This second interaction can similarly be described by a second dressing of the pseudo qubit by the phonon field. This gives rise to a ladder of phonon dressed states, see Fig. 3e, with eigenstates parameterized by the number M of phononic and dressed qubit excitations. The energy splitting within multiplicities can be expressed as (Supplementary Note 4) , which simplifies to when the phononic dressing field frequency Ω_{d}/2π is resonant with the dressed qubit energy splitting Ω_{R}/2π. As a consequence the spectrum of Rabi oscillations is peaked at frequencies corresponding to the allowed transitions for the σ_{z} operator (Fig. 3e and Supplementary Fig. 3).
Measuring the temporal evolution of the spinqubit population indeed permits to record the temporal evolution of the dipole of the MWdressed qubit () (ref. 49), see Supplementary Note 4. This dipole governs the dressed qubit emission (in analogy with the atomic case), whose spectrum (Fig. 3b) reflects the cascade among phononic dressed states. This situation is precisely the one permitting the observation of a Mollow triplet in QED when the atomic fluorescence spectrum under intense illumination was measured^{24}. An important distinction is that here the time resolved evolution of the ‘atomic’ dipole (the dressed qubit) is accessible.
Multimode phononic Mollow triplet
To fully explore the vectorial character of the parametric interaction, we now sweep the drive tone across both mechanical eigenfrequencies. This permits moving the qubit in both directions in the (e_{1}, e_{2}) oscillating plane. For each drive frequency, the MW power is adjusted to reach the resonant condition Ω_{R}≈Ω_{d}. The measured Mollow triplet spectra are acquired and shown in Fig. 4a. The central component of the triplets is locked onto the drive frequency Ω_{d}, while the splitting of the Mollow triplet presents two maxima, corresponding to the response of each eigenmode. To properly understand the observed signature, it is necessary to precisely determine the spatial trajectories followed by the moving spinqubit. To do so, an optical measurement similar to the one shown in Fig. 1c permits establishing the local orientation e_{F} and magnitude δF of the electrostaticdriving force field^{39} and determining the driven trajectories using equation (1). The slight spectral overlap between the eigenmodes leads to elliptical trajectories (Fig. 4b) which explore the oscillation plane and the magnetic field gradient over nanometric distances. Finally, the Mollow triplet's motional splitting can be adjusted with, see Supplementary Note 2:
both the deduced magnitude () and the orientation ( reported in Fig. 4b) of the vectorial coupling constant are in good agreement with the static measurements described above, at the position marked in Fig. 2f. Geometrically, the magnitude of the parametric coupling strength corresponds to the length of the projection of the ellipses on the axis. Pursuing this geometrical approach, it is possible to introduce a characteristic length, , here (reported in Fig. 4b), which represents the minimum oscillation amplitude along necessary to resolve the phononic Mollow triplet. It is interesting to point out that this quantity is comparable to the spatial spreading of the nanowire Brownian motion of ≈52 pm, responsible for an equivalent incoherent parametric modulation of , which could alter the Mollow triplet structure^{50} in larger magnetic field gradients. Understanding the coherence properties of the dressed qubit and the contribution of Brownian motion will be the subject of future investigation.
Discussion
We have demonstrated the observation of a phononic Mollow triplet in a spinnanomechanical hybrid system, reproducing with phonons and a spin qubit one of the founding signatures of QED based on photons and atoms. The observed signatures also demonstrate the synchronization of the spin precession onto the mechanical oscillation frequency. This opens the road towards new detection strategies for observing dynamical spindependent forces which can be expressed as . In a 10^{6}T˙m^{−1} gradient, its differential amplitude amounts to ≈20 aN, which favourably compares to the demonstrated force sensitivities falling in the aN Hz^{1/2} with our nanowires at room temperature, at the condition to make the spinqubit precess at mechanical frequency. The required stability on the Rabi precession is however impossible to reach in a standard experimental setting but this is precisely what is enabled by the spinlocking mechanism. This opens new detection strategies for the observation of spindependent forces, which are the key ingredient for quantum state transfer. Also, the phononic Mollow triplet structure should also be imprinted on the oscillator dynamics through the spectrum of σ_{z}[Ω], as mechanical sidebands, leading to a rich and novel oscillating dynamics for the oscillator. Pushing further the analogy with QED (ref. 49) where signatures of photon cascade were observed experimentally^{29}, this should also provide a direct mechanism for creating single phonon sources through cascaded phonon emission within the dressed state ladder.
Additional information
How to cite this article: Pigeau, B. et al. Observation of a phononic Mollow triplet in a multimode hybrid spinnanomechanical system. Nat. Commun. 6:8603 doi: 10.1038/ncomms9603 (2015).
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Acknowledgements
We thank C. Fabre, G. Nogues, O. Buisson, J.F. Roch, J.P. Poizat, P. Vincent, P. Poncharal, A. Auffeves, A. Kuhn, P. Verlot, E. DupontFerrier, C. Hoarau, D. Lepoittevin and E. Eyraud for theoretical, experimental and technical assistance. This work was supported by the Agence Nationale de la Recherche (RPDoc2010, FOCUS 2013), Lanef (CryOptics) and the European Research Council (ERCStG2012, HQNOM). S.R. acknowledges funding from the Nanoscience Foundation.
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Pigeau, B., Rohr, S., Mercier de Lépinay, L. et al. Observation of a phononic Mollow triplet in a multimode hybrid spinnanomechanical system. Nat Commun 6, 8603 (2015). https://doi.org/10.1038/ncomms9603
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DOI: https://doi.org/10.1038/ncomms9603
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