## Mechanical systems at the quantum level

A number of platforms are being pursued for developing technologies that exploit the enhanced sensing and measurement capabilities of quantum mechanics. Hybrid systems offer the flexibility of combining and optimizing different platforms. Hong *et al.* combined optomechanical control of motion and single-phonon counting techniques to probabilistically generate a single-phonon Fock state within a nanomechanical resonator. Chu *et al.* used electromechanical coupling to address a bulk piezoelectric resonator with a superconducting quantum circuit. Both approaches hold promise for developing hybrid quantum technologies.

## Abstract

Nano- and micromechanical solid-state quantum devices have become a focus of attention. Reliably generating nonclassical states of their motion is of interest both for addressing fundamental questions about macroscopic quantum phenomena and for developing quantum technologies in the domains of sensing and transduction. We used quantum optical control techniques to conditionally generate single-phonon Fock states of a nanomechanical resonator. We performed a Hanbury Brown and Twiss–type experiment that verified the nonclassical nature of the phonon state without requiring full state reconstruction. Our result establishes purely optical quantum control of a mechanical oscillator at the single-phonon level.

Intensity correlations in electromagnetic fields have been pivotal in the development of modern quantum optics. The experiments by Hanbury Brown and Twiss were a particular milestone that connected the temporal and spatial coherence properties of a light source with the second-order intensity autocorrelation function *g*^{(2)}(τ, *x*) (*1*–*3*). In essence, *g*^{(2)} correlates intensities measured at times differing by τ or at locations differing by *x* and hence is a measure of their joint detection probability. At the same time, these correlations allow the quantum nature of the underlying field to be inferred directly. For example, a classical light source of finite coherence time can only exhibit positive correlations at a delay of τ ≈ 0 in the joint intensity detection probability, leading to bunching in the photon arrival time. This result holds true for all bosonic fields. Fermions, on the other hand, exhibit negative correlations and hence antibunching in the detection events (*4*–*6*), which is a manifestation of the Pauli exclusion principle. A bosonic system needs to be in a genuine nonclassical state to exhibit antibunching. The canonical example is a single-photon (Fock) state, for which *g*^{(2)}(τ = 0) = 0 because no joint detection can take place (*7*). For this reason, measuring *g*^{(2)} has become a standard method to characterize the purity of single-photon sources (*8*). In general, *g*^{(2)}(τ) carries a wealth of information on the statistical properties of a bosonic field with no classical analog (*9*, *10*)—specifically, sub-Poissonian counting statistics [*g*^{(2)}(0) < 1] and antibunching [*g*^{(2)}(τ) ≥ *g*^{(2)}(0)]—all of which have been demonstrated successfully with quantum states of light (*11*, *12*).

Over the past decade, motional degrees of freedom (phonons) of solid-state devices have emerged as a quantum resource. Quantum control of phonons was pioneered in the field of trapped ions (*13*), where single excitations of the motion of the ions are manipulated through laser light. These single-phonon states have been used for fundamental studies of decoherence (*14*) and for elementary transduction channels in quantum gates for universal quantum computing (*15*). Cavity optomechanics (*16*) has successfully extended these ideas to optically controlling the collective motion of solid-state mechanical systems. It has allowed for remarkable progress in controlling solid-state phonons at the quantum level, including sideband cooling into the quantum ground state of motion (*17*, *18*), the generation of quantum correlated states between radiation fields and mechanical motion (*19*–*21*), and the generation of squeezed motional states (*22*–*24*).

So far, single-phonon manipulation of micromechanical systems has exclusively been achieved through coupling to superconducting qubits (*25*–*27*), and optical control has been limited to the generation of quantum states of bipartite systems (*20*, *21*, *28*). Here we demonstrate all-optical quantum control of a purely mechanical system, creating phonons at the single-quantum level and unambiguously showing their nonclassical nature. We combined optomechanical control of motion and single-phonon counting techniques (*21*, *29*) to probabilistically generate a single-phonon Fock state from a nanomechanical device. Implementing Hanbury Brown and Twiss interferometry for phonons (*21*, *29*) (Fig. 1) allowed us to probe the quantum mechanical character of single phonons without reconstructing their states. We observed *g*^{(2)}(0) < 1, which is a direct verification of the nonclassicality of the optomechanically generated phonons, highlighting their particle-like behavior.

Our optomechanical crystal (*18*) consists of a microfabricated silicon nanobeam patterned so that it simultaneously acts as a photonic and phononic resonator (Fig. 2). The resulting optical and mechanical modes couple through radiation pressure and the photoelastic effect so that a displacement equivalent to the zero-point fluctuation of the mechanical mode leads to a frequency shift of the optical mode by *g*_{0}/2π = 869 kHz (*g*_{0}, optomechanical coupling rate). The optical resonance has a wavelength λ = 1554.35 nm and a critically coupled total quality factor *Q*_{o} = 2.28 × 10^{5} (cavity energy decay rate κ/2π = 846 MHz), whereas the mechanical resonance has a frequency of ω_{m}/2π = 5.25 GHz and a quality factor of *Q*_{m} = 3.8 × 10^{5}. The device is placed in a dilution refrigerator with a base temperature of *T* = 35 mK. When the device is thermalized, its high frequency guarantees that the mechanical mode is initialized deep in its quantum ground state (*21*).

We utilized two types of linearized optomechanical interactions—the parametric down-conversion and the state swap—which can be realized by driving the system with detuned laser beams in the limit of weak coupling (, where *n*_{c} is the intracavity photon number) and resolved sidebands (κ ≪ ω_{m}) (*16*). The parametric down-conversion interaction has the form , where is the reduced Planck constant; and are the phononic creation and annihilation operators, respectively; and and are the respective photonic operators. This interaction is selectively turned on by detuning the laser frequency ω_{L} to the blue side of the cavity resonance ω_{c} (ω_{L} = ω_{c }+ ω_{m}). *H*_{dc} drives the joint optical and mechanical state, initially in the ground state, into the state . For low excitation probabilities *p*_{b} ≪ 1, higher-order terms can be neglected so that the system can be approximated as emitting a pair consisting of a resonant signal photon and an idler phonon with a probability *p*_{b} (*30*). Detection of the signal photon emanating from the device heralds a single excitation of the mechanical oscillator , in close analogy to heralded single photons from spontaneous parametric down-conversion. To read out the phonon state, we send in another laser pulse that is now red-detuned from the cavity resonance by ω_{m} (ω_{L} = ω_{c }– ω_{m}). This realizes a state-swap interaction , which transfers the mechanical state to the optical mode with efficiency *p*_{r}. We can therefore use the scattered light field from this “read” operation to directly measure the second-order intensity correlation function *g*^{(2)} of the mechanical oscillator mode, which is defined as(1)where τ is the time between the first and the second detection event. Like for any other bosonic system, *g*^{(2)}(0) > 1 means that the phonons exhibit super-Poissonian (classical) behavior, whereas *g*^{(2)}(0) < 1 is direct evidence of the quantum mechanical nature of the state and implies sub-Poissonian phonon statistics (*10*).

We implemented the experimental approach (Fig. 1) by repeatedly sending a pair of optical pulses, the first one blue-detuned [pump pulse, full width at half maximum (FWHM) ≈ 32 ns] and the second one red-detuned (read pulse, FWHM ≈ 32 ns), with a fixed repetition period *T*_{r} = 50 μs. Photons generated through the optomechanical interactions were reflected back from the device and analyzed by a Hanbury Brown and Twiss interferometer using two superconducting nanowire single-photon detectors (SNSPDs). We set the mean pump pulse energy to 27 fJ so that *p*_{b} = 1.2% (*31*). Detection of resonant (signal) photons created by this pulse heralds the preparation of the mechanical oscillator in a single-phonon Fock state, in principle with a probability of 98.8%. Owing to a small amount of initial thermal phonons and residual absorption heating, a fraction of unwanted phonons were incoherently added to the quantum states that we prepared (*21*). After each pump pulse, a red-detuned read pulse was sent to the device with a programmable delay *t*_{d}, reading out phonons stored in the device by converting them into photons on resonance with the cavity. The mean read pulse energy was set to 924 fJ, corresponding to a state-swap efficiency *p*_{r} ≈ 32.5%. Taking into account subsequent optical scattering losses, this yields an absolute quantum efficiency for the detection of phonons of 0.9% (*31*). Last, the pulse repetition period of *T*_{r} = 50 μs, which is long compared with the mechanical damping time of 11 μs, provides ample time for dissipating any excitation or unwanted heating generated by optical absorption. This ensured that each experimental cycle started with the mechanical mode well in the quantum ground state. The pulse sequence was repeated more than 7 × 10^{9} times to acquire enough statistics. Conditioned on heralding events from detector D1 by the blue-detuned pulses, we analyzed the coincidence detection probability of photons at D1 and D2 that are transferred from phonons by the swap operation.

In our first experiment, we set *t*_{d} = 115 ns and measured *g*^{(2)}(0) of the heralded phonons. One of our SNSPDs, D2, exhibited a longer dead time than *t*_{d} (*31*), and we therefore only used photon counts from D1 for heralding the phonon states. From these measurements, we obtained a *g*^{(2)}(0) of (Fig. 3C), demonstrating a nonclassical character of the mechanical state.

The observed *g*^{(2)}(0) of 0.65 is considerably higher than what we expect in the ideal case, (*31*). We attribute this to heating induced by the absorption of the pump and read pulses. Although a detailed physical mechanism for the absorption and subsequent heat transfer into the mechanical mode is still a subject of study (*21*), the influx of thermal phonons caused by the absorption of drive laser pulses can be experimentally deduced from the (unconditional) photon count rates generated by the read pulses (*31*). Including an estimation of the initial thermal phonon number *n*_{init}, which is likewise inferred from the unconditional photon counts associated with the pump and read pulses, we constructed a theoretical model that predicts *g*^{(2)}(0) as a function of *p*_{b}, *n*_{init}, and . Given the measured *n*_{init} ≈ 0.20 and (*31*) within the read pulse, our model predicts *g*^{(2)}(0) ≈ 0.76, which is consistent with the experimental value.

To further probe the effect of thermal phonons, we performed a set of experiments with reduced repetition periods *T*_{r}, while keeping the other settings for the pump pulses the same. This effectively increases *n*_{init}, because the absorbed heat does not have enough time to dissipate before the next pair of pulses arrives. As expected, as *T*_{r} was reduced, we observed an increase in *g*^{(2)}(0). With the measured *n*_{init} and from the same data set, we can plot the predicted *g*^{(2)}(0) values. The experimental values and theoretical bounds on *g*^{(2)}(0) are in good agreement (Fig. 3).

We also measured *g*^{(2)}(0) for *t*_{d} = 350 ns and found that it increased to . This increase is consistent with previously observed delayed heating effects of the absorption (*21*) and is in good agreement with the theoretical prediction of 0.84. Even for these longer delays, the value is still below 1, demonstrating the potential of our device as a single-phonon quantum memory on the time scale of several hundred nanoseconds.

We experimentally demonstrated the quantum nature of heralded single phonons in a nanomechanical oscillator by measuring their intensity correlation function *g*^{(2)}(0) < 1. The deviation from a perfect single-phonon state can be modeled by a finite initial thermal occupation and additional heating from our optical cavity fields. We achieved conversion efficiencies between phonons and telecom photons of more than 30%, only limited by our available laser power and residual absorption. Full state reconstruction of the single-phonon state, as demonstrated with phononic states of trapped ions (*14*), should be realizable with slightly improved readout efficiency and through homodyne tomography. The demonstrated fully optical quantum control of a nanomechanical mode, preparing sub-Poissonian phonons, shows that optomechanical cavities are a useful resource for future integrated quantum phononic devices, as both single-phonon sources and detectors. They are also an ideal candidate for storage of quantum information in mechanical excitations and constitute a fundamental building block for quantum information processing involving phonons. Some of the potential applications include quantum noise–limited, coherent microwave-to-optics conversion, as well as studying the quantum behavior of individual phonons of a massive system.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

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**Acknowledgments:**We thank V. Anant, J. Hofer, C. Loeschnauer, R. Patel, and A. Safavi-Naeini for experimental support and K. Hammerer for helpful discussions. We acknowledge assistance from the Kavli Nanolab Delft, in particular from M. Zuiddam and C. de Boer. This project was supported by the European Commission under the Marie Curie Horizon 2020 initial training programme OMT (grant 722923); the Foundation for Fundamental Research on Matter (FOM) Projectruimte (grants 15PR3210 and 16PR1054;) the Vienna Science and Technology Fund (WWTF; grant ICT12-049); the European Research Council (Consolidator Grant QLev4G and Starting Grant Strong-Q); the Austrian Science Fund (FWF) under projects F40 (SFB FOQUS) and P28172; and the Netherlands Organisation for Scientific Research (NWO/OCW) as part of the Frontiers of Nanoscience program, as well as through a Vidi grant (016.159.369). R.R. is supported by the FWF under project W1210 (CoQuS) and is a recipient of a DOC fellowship of the Austrian Academy of Sciences at the University of Vienna. All relevant data to understand and assess the conclusions of this research are available in the main text and supplementary materials.