Abstract
The Bloch sphere is a generic picture describing the coherent dynamics of coupled classical or quantummechanical twolevel systems under the control of electromagnetic fields^{1,2}. It is commonly applied to systems such as spin ensembles^{3}, atoms^{4}, quantum dots^{5} and superconducting circuits^{6}. The underlying Bloch equations^{7} describe the state evolution of the twolevel system and allow the characterization of both energy and phase relaxation processes^{3,8,9}. Here we realize a classical nanomechanical twolevel system^{2} driven by radiofrequency signals. It is based on the two orthogonal fundamental flexural modes of a highqualityfactor nanostring resonator that are strongly coupled by dielectric gradient fields^{10}. Full Bloch sphere control is demonstrated by means of Rabi^{11}, Ramsey^{12} and Hahn echo^{13} experiments. Furthermore, we determine the energy relaxation time T_{1}and phase relaxation times T_{2} and T_{2}*, and find them all to be equal. Thus decoherence is dominated by energy relaxation, implying that not only T_{1} but also T_{2} can be increased by engineering larger mechanical quality factors.
Main
Whereas the dynamics of semiclassical twolevel systems under the influence of a pulsed external electromagnetic field was observed decades ago in manyspin NMR experiments, a completely classical analogue remained elusive for a long time. Such a classical twomode system can for example be created using two optical cavity modes^{14} or mechanical resonators. Only recently, several approaches were employed to achieve purely mechanical resonant coupling either between separate resonators^{15,16,17} or different modes of the same resonator^{10,18} in the classical regime. So far, the pulsed coherent control of the system was prevented by weak coupling, low quality factors or the lack of a sufficiently strong and fast tuning mechanism.
We present the successful implementation of a purely mechanical, classical twolevel system, consisting of the two coupled fundamental flexural modes of a nanomechanical resonator with coherent timedomain control (see also the experiments independently performed using parametric coupling^{19} instead of the linear coupling employed here). To this end, we use a 250 nm wide and 100 nm thick, strongly stressed^{20} silicon nitride string resonator with a length of 50 μm dielectrically coupled to a pair of electrodes used for detection^{21} as well as actuation and tuning^{22}. The two fundamental flexural modes of the mechanical resonator oscillating in the outofplane and inplane direction (Fig. 1a) are linearly coupled by crossderivatives of the strong inhomogeneous electric field generated between the electrodes (see the Supplementary Information of ref. 10 for a theoretical analysis). A constant d.c. voltage of −15 V is employed to dielectrically tune the system close to the resulting avoided crossing, while the signals generated by an arbitrary waveform generator (AWG) enable timeresolved control in the vicinity of the avoided crossing (Fig. 1b,c). Both voltages are added, combined with the radiofrequency actuation at a biastee, and applied to one electrode. The other electrode is connected to a 3.6 GHz microstrip cavity, enabling heterodyne detection of the string deflection^{21} after addition of a microwave bypass capacitor at the first electrode^{22}. These components, as well as the mechanical resonator, are placed in a vacuum of ≤10^{−4} mbar and the system is cooled to 10.00±0.02 K to improve the temperature stability as well as the cavity quality factor. The microwave cavity is interfaced to the readout with a single coaxial cable and a circulator.
When the system is driven by an external whitenoise source and the AWG output voltage is swept, the avoided crossing of the two modes shown in Fig. 1c can be mapped out, exhibiting a frequency splitting Ω = 24,249±4 Hz. With a quality factor Q = f/Δf≈2×10^{5} and a linewidth of Δf≈40 Hz at the resonance frequency f, the system is clearly in the strong coupling regime of Δf≪Ω. For all measurements discussed in the following, a radiofrequency drive of −59 dBm at 7.539 MHz, resonantly actuating the string at an AWG voltage of 0 V, is applied, which initializes the system in its inplane mode (Fig. 1c, black circle). A 1 ms long, adiabatic voltage ramp up to 2.82 V brings the state to the point of minimal frequency splitting Ω between the coupled modes. Here, the system dynamics is described by two normal modes, hybrid states formed by the inphase and outofphase combinations of the fundamental flexural modes. The adiabatic ramp thus transforms all the energy of the inplane mode into the lower hybrid state, such that the classical twolevel system^{2}, consisting of the two hybrid modes, is prepared in its lower state. As the drive frequency remains constant (Fig. 1c, dashed line), the string is no longer actuated and its energy is slowly decaying.
Now, the AWG is used to apply a continuous pump tone with frequency Ω to the drive electrodes which starts Rabi oscillations^{3,11} between the lower and upper state, as shown in Fig. 2. They can be measured directly by monitoring the time evolution of the output power spectrum at the frequency of one of the hybrid modes, here shown for the upper state at 7.6028 MHz, and measured with a bandwidth of 10 kHz. All timeresolved measurements are averaged over 20 (Rabi oscillations and T1 measurement) or 10 pulse sequences (Ramsey fringes and Hahn echo). For a drive amplitude of 100 mV (half peaktopeak) we find a Rabi frequency of 8.3 kHz (Supplementary Section SIIA). These strong Rabi oscillations demonstrate that the transition between the two hybrid modes forms a classical twolevel system, in contrast to the modes themselves, which can be modelled as highly populated harmonic oscillators.
In principle, the decay of these oscillations is governed by both energy relaxation, characterized by a rate 1/T_{1}, and phase decoherence, characterized by 1/T_{2} or 1/T_{2}*, where T_{2}*≤T_{2} includes reversible processes caused by slow fluctuations or spatial inhomogeneity of the coupling. For clarity, we use these wellknown phenomenological constants in the same way as, for example, in spin systems^{3}, as discussed in detail in Supplementary Section SI.
The exponential decay of a state’s energy defines T_{1}. The corresponding measurement is shown in Fig. 3 for both the lower and upper state: the system is once again prepared in the lower hybrid state. To reach the upper state, a subsequent πpulse is applied, thus performing one half of a Rabi cycle, which transfers the system to the upper state (Supplementary Section SII for details on the frequency and amplitude calibration of the applied pulses). The exponential decay is then measured directly with a spectrum analyser at a bandwidth of 3 kHz, exhibiting different relaxation times T_{1,l} = 4.83±0.1 ms and T_{1,u} = 4.02±0.1 ms for the lower and upper mode, respectively. They correspond to the spectrally measured quality factors. Previously, it has been shown that, at maximum coupling, the two hybrid modes should have the same quality factor and thus T_{1} time^{10}. However, both modes are affected by dielectric damping^{22} (Supplementary Section SIB), leading to the observed difference.
To measure the T_{2}* time, a π/2pulse is used after the preparation in the lower state to bring the system into a superposition state between the lower and upper hybrid modes. The frequency of the pulse is detuned to Ω+500 Hz, leading to a slow precession of the state vector around the z axis of the Bloch sphere^{3,12}. As a result, a second π/2pulse after time τ does not always bring the system into the upper state, but a slow oscillation, the socalled Ramsey fringes, is observed when the delay τ between the two pulses is varied and the zprojection of the state vector is measured after the second pulse, as shown in Fig. 4. The decay constant of this oscillation is T_{2}*, whereas the decay of the mean value can be interpreted as an effective T_{1} of both modes. The fit in Fig. 4b results in T_{2}* = 4.44±0.1 ms and T_{1} = 4.31±0.1 ms. The energy relaxation time of the superposition state T_{1} is identical to the reciprocal rate average of the two hybrid modes
as the mechanical energy oscillates between the two modes with frequency Ω (Supplementary Movie).
By including an additional πpulse at τ/2 into the Ramsey pulse scheme and replacing the final π/2pulse by a 3π/2pulse to once again rotate to the upper state (Fig. 5), the T_{2} time can be measured in a Hahn echo experiment^{3,13}. The 180°rotation flips the state vector in the x yplane of the Bloch sphere, thus reversing the effects of a fluctuating or inhomogeneous coupling strength Ω in the second delay interval of τ/2 and thereby cancelling their contribution. The frequency of the pulses is once again exactly Ω, as all three pulses need to be applied exactly around the same axis. The resulting decay curve represents T_{2}, for which a value of T_{2} = 4.35±0.1 ms can be extracted from the fit in Fig. 5b.
The good agreement between T_{2} and T_{2}* clearly shows that reversible elastic dephasing, for example caused by temporal and spatial enviromental fluctuations or spatial inhomogeneities, does not noticeably increase decoherence. The experiment is performed with the two hybrid modes occupied by billions of phonons, making it analogous to a manyspin NMR measurement, only that the macroscopic magnetization is replaced by the mode polarization and a single spin flip corresponds to the transfer of a single phonon between the two modes. But in contrast to the NMR system, here all quasiparticles reside in the same collective mechanical mode and thus experience an identical environment (Supplementary Section SIII).
It is more surprising that the phase coherence time T_{2} is equal to the average energy relaxation time T_{1}. This indicates the absence of measurable elastic phase relaxation processes in the nanomechanical system, such that the observed loss of coherence is essentially caused by the energy decay of the mechanical oscillation (Supplementary Section SIII). Earlier research^{20} suggests that the dominant relaxation mechanism in silicon nitride strings is mediated by localized defect states of the amorphous resonator material, described as twolevel systems at low temperature. They facilitate energy relaxation by providing the momentum required to transform a resonator phonon into a bulk phonon. For this process to lead to elastic phase relaxation, an excited defect state would have to reemit the phonon back into the resonator mode, which is extremely unlikely owing to the weak coupling between the two.
In conclusion, we demonstrate coherent electrical control of a strongly coupled (Ω≫f/Q) classical nanomechanical twolevel system, employing the pulse techniques wellknown from coherent spin dynamics in the field of nanomechanics. Each superposition state of the two hybrid modes on the Bloch sphere can be addressed by a sequence of the described pulses. The presented system stands out by the finding that the elastic phase relaxation rate Γ_{φ} is negligible compared with the energy decay rate 2πf/Q, leaving room for improvement of the coherence by means of increased quality factors.
The coherent manipulation schemes presented here allow the simulation of quantum systems using a classical twolevel system^{2}. Furthermore, in light of the recent breakthrough in groundstate cooling of nanomechanical resonators^{23,24,25,26}, they can be directly transferred to quantum nanomechanical systems, where they open up new applications in quantum information processing. Not only can they be used as efficient interfaces for quantum state transfers in hybrid quantum systems^{27,28}, but by creating coupled, quantized resonators^{29} quantum computations can be carried out directly using nanoelectromechanical twolevel systems^{30}.
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Acknowledgements
Financial support by the Deutsche Forschungsgemeinschaft through Project No. Ko 416/18, the German Excellence Initiative through the Nanosystems Initiative Munich (NIM) and LMUexcellent, as well as the European Commission under the FETOpen project QNEMS (233992) is gratefully acknowledged. We thank G. Burkard for his comments on decoherence in a threelevel system and H. Okamoto, I. Mahboob and H. Yamaguchi for critically reading the manuscript.
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J.R. and M.J.S. designed and fabricated the sample, T.F. conducted the measurements and analysed the data. T.F., J.P.K. and E.M.W. wrote the paper with input from the other authors. The results were discussed by all the authors.
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Faust, T., Rieger, J., Seitner, M. et al. Coherent control of a classical nanomechanical twolevel system. Nature Phys 9, 485–488 (2013). https://doi.org/10.1038/nphys2666
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