Abstract
The motion of micrometresized mechanical resonators can now be controlled and measured at the fundamental limits imposed by quantum mechanics. These resonators have been prepared in their motional ground state^{1,2,3} or in squeezed states^{4,5,6}, measured with quantumlimited precision^{7}, and even entangled with microwave fields^{8}. Such advances make it possible to process quantum information using the motion of a macroscopic object. In particular, recent experiments have combined mechanical resonators with superconducting quantum circuits to frequencyconvert, store and amplify propagating microwave fields^{9,10,11,12}. But these systems have not been used to manipulate states that encode quantum bits (qubits), which are required for quantum communication and modular quantum computation^{13,14}. Here we demonstrate the conversion of propagating qubits encoded as superpositions of zero and one photons to the motion of a micromechanical resonator with a fidelity in excess of the classical bound. This ability is necessary for mechanical resonators to convert quantum information between the microwave and optical domains^{15,16,17} or to act as storage elements in a modular quantum information processor^{12,13,18}. Additionally, these results are an important step towards testing speculative notions that quantum theory may not be valid for sufficiently massive systems^{19}.
Main
Quantum communication networks that use superconducting qubits and modular quantum computing architectures require the ability to store, amplify or frequencyshift propagating microwave fields. A single electromechanical device provides all of these functions by rapidly varying the parametric coupling between mechanical motion and microwave fields. For example, the ability to suddenly turn off the interaction between a microwave field and mechanical motion allows the state of a field propagating through a transmission line to be converted to, and trapped in, the motional state of the resonator^{12}. To use this capture process in a general quantum information processor, one must work with states that have nonGaussian statistics, such as qubits encoded as superpositions of zero and one photons. In contrast, any process using only Gaussian states can be simulated efficiently on a classical computer^{20}. But in the regime that most of these devices operate, the equations that describe the coupling are linear, ensuring that a Gaussian state of the microwave field or mechanical resonator will never evolve into a nonGaussian state.
For electromechanical devices that manipulate propagating microwave fields, accessing nonGaussian mechanical states requires either a source of nonGaussian microwave fields or a nonlinear detector such as a singlephoton counter. Forgoing the control of propagating fields, nonGaussian mechanical states have been accessed by resonantly coupling a mechanical system to a qubit^{1}. More recent work has shown that parametrically coupling a mechanical resonator to a qubit via an intermediate cavity bus may enable access to nonGaussian states^{21}. But in this integrated device, the predicted transfer fidelity is low primarily because of the intrinsic loss in the cavity. So far, the demands of fabricating such hybrid devices have reduced the coherence of the mechanical resonator, qubit or cavity far below the state of the art.
In this work, we convert nonGaussian states from propagating microwave fields to the motion of a micrometresized mechanical resonator. We use an electromechanical device to capture, store and amplify single photons generated by a superconducting qubit and then determine the density matrix of the mechanical resonator using quantum state tomography. We find that the quantum state can be stored on a timescale exceeding 100 μs, an improvement of over four orders of magnitude compared with previous work that demonstrated the storage of a nonGaussian state in an electromechanical device^{1}. To characterize how the capture process affects arbitrary propagating qubit states, we capture superpositions of zero and one photons. The degree to which this process preserves quantum information is quantified by the average fidelity^{22}, which we find to be F_{avg} = 0.83_{−0.06}^{+0.03} where the limits are the 90% confidence interval. This level of performance exceeds the fidelity achievable using only classical resources, indicating that our electromechanical device is suitable for the transduction of quantum information.
The electromechanical device consists of an inductor–capacitor (LC) circuit that is tunable and coupled to a mechanical resonator (Fig. 1a). The tunability and coupling arise from the upper plate of the capacitor, which is a 100nmthick suspended and tensioned aluminium membrane that is free to vibrate. The fundamental drumheadlike vibrational mode of this membrane forms the mechanical resonance at ω_{m}/2π ≈ 9.3 MHz. If displaced by the resonator’s zeropoint motion of 6.4 fm, the circuit’s resonant frequency shifts by g_{0}/2π ≈ 280 Hz. The circuit also couples inductively to propagating microwave fields in a nearby transmission line at a rate of κ_{LC}/2π ≈ 3 MHz. We tune the LC circuit into precise resonance with a narrowband and fixedfrequency photon source by using a third electrode, biased at V_{d.c.} relative to the membrane, to control the static separation between the membrane and the microwave electrode^{9}.
We connect the electromechanical device to an ondemand source of single photons using the network depicted in Fig. 1a. To efficiently generate single photons compatible with the narrow bandwidth requirements^{12} of the electromechanical device, we use a circuit quantum electrodynamics system^{23}. It consists of a transmon qubit with a transition frequency ω_{q}/2π = 5.652 GHz in a microwave cavity, whose resonance frequency is ω_{c}^{g}/2π = 7.290 GHz when the qubit is in the ground state, g〉, and ω_{c}^{e}/2π = (ω_{c}^{g} − 2χ)/2π = 7.283 GHz when it is in the excited state, e〉, where χ is the dispersive shift. We use a control pulse^{24} to drive the transition g〉 0〉→ e〉 1〉, where 0〉 and 1〉 correspond to zero and one cavity photons, respectively. The cavity state then evolves into a field propagating through the transmission line with the centre frequency ω_{c}^{e} and narrow bandwidth κ_{c}/2π = 60 kHz.
The propagating microwave field is parametrically coupled to the membrane’s motion by applying pumps to the LC circuit. To capture the state of the propagating field^{12}, we use a pump that is detuned below (reddetuned) the LC resonance with detuning Δ_{r} = −ω_{m}. This pump creates an interaction that can exchange the states of the input microwave field and the mechanical resonator. For a given temporal envelope of the input field, the coupling, Γ_{r}(t) = 4g_{0}^{2}n_{r}(t)/κ_{LC}, must be modulated for optimal capture efficiency^{9}, where n_{r}(t) is the number of photons induced in the LC circuit by the pump. If instead we apply a bluedetuned pump at Δ_{b} = +ω_{m}, a twomode squeezer interaction is created that amplifies both the motion of the resonator and the incident microwave field^{8}. During amplification, the LC circuit emits a propagating field with a temporal envelope that rises exponentially at a rate of Γ_{b}/2, where Γ_{b}(t) = 4g_{0}^{2}n_{b}(t)/κ_{LC} is set by the average number of photons, n_{b}(t), induced in the circuit by the bluedetuned pump. Crucially, the state of the emitted field depends on both the states of the resonator and the input field before amplification^{25}.
We exploit the parametric interactions in two protocols that are used to characterize the capture process. Because the process maps states at the input of the electromechanical device to the resonator, we must determine the input state and compare it with the captured state. To this end, we have developed ‘calibration’ and ‘capture’ protocols that enable us to determine the input and captured states, respectively (Fig. 2a). We initially test the two protocols with coherent signals whose frequency and bandwidth are chosen to match those created by the circuit quantum electrodynamics system (Fig. 2b).
In the calibration protocol, the input field is amplified directly and then measured. We implement this protocol by applying the bluedetuned pump coincident with the input field. In this case, the electromechanical device functions as a linear phasepreserving amplifier whose input and output are the incident and reflected microwave fields, respectively. These pulsed fields have different envelopes; nevertheless, with an appropriate filter (Supplementary Information) they are related by an energy gain of cosh^{2}(r/2), where r = Γ_{b}τ_{b} and τ_{b} is the pump’s duration. If we regard the input of the amplifier as the incident microwave field, the fluctuations of the resonator’s motion are the source of the amplifier’s added noise, reaching the quantum limit^{25} if the resonator is in its ground state^{2}.
After obtaining the input state, we use the capture protocol to determine the resonator state. We first apply the reddetuned pump coincident with the input field. Once it is captured, we then apply the bluedetuned pump to amplify the resonator’s state. In contrast to the calibration protocol, we now regard the amplifier’s input to be the state of the resonator. The output is still the reflected field, but the added noise is due to the vacuum fluctuations of the incident field. When interpreted this way, we realize a linear phaseconjugating amplifier with an energy gain of sinh^{2}(r/2).
Operating the electromechanical device as a lownoise amplifier enables us to perform state tomography on both the input microwave field and on the motion of the resonator. For each repetition of the two protocols depicted in Fig. 3, we record a voltage signal, V (t), at the detector during amplification. For each voltage record, we extract a pair of quadrature amplitudes, X and Y, for the state of either the resonator or input field (Supplementary Information). By making repeated measurements of V (t), we obtain a set of quadrature amplitudes and use this information to extract a density matrix ρ via a method of maximum likelihood state tomography^{26} (Supplementary Information). We refer to the states of the input microwave field and of the mechanical resonator as ρ_{e} and ρ_{m}, respectively.
To test the conversion of nonGaussian states, we inject single photons into the electromechanical device. We also operate the calibration and capture protocols (Fig. 3a, d) without generating single photons. In this case, we inject a vacuum state to determine the gain of the detector which we use to scale X and Y in units of (quanta)^{1/2} (Supplementary Information). Prior to the execution of each protocol, we cool the resonator close to its quantum ground state^{2} with an occupancy of approximately 0.1 quanta (Supplementary Information). For both protocols, the tomography yields density matrix estimates containing significant elements only on the diagonals (Fig. 3c,f). In particular, we find that the probability of detecting a single photon is [ρ_{e}]_{11} = 0.33_{−0.01}^{+0.02} (Supplementary Information). After capture, the probability of a single phonon occupying the mechanical mode is [ρ_{m}]_{11} = 0.26_{−0.02}^{+0.01}. To distinguish the captured state from a thermal or coherent state, we calculate the degree of secondorder coherence g_{m}^{(2)} = 0.89_{−0.17}^{+0.05} (Supplementary Information). For comparison, a thermal or coherent state of motion yields g_{m}^{(2)} ≥ 1. After capturing single photons, we vary the storage time τ_{s} and test the ability to mechanically store a nonGaussian state (Fig. 3g). We use a master equation formalism to model the evolution of ρ_{m} with the characteristic storage time τ_{m} as the only free parameter (Supplementary Information). We extract τ_{m} = 137 ± 6 μs, which is about ten times longer than the time used to capture the input photon state.
Having demonstrated the ability to capture single photons, we then characterize how the capture process affects arbitrary qubit states encoded as superpositions of zero and one photons. This process is described by a map between incident and captured states whose quality is characterized by the average fidelity^{22}which measures how indistinguishable the output of the process is from the input, averaged over all pure input states Ψ〉. To determine F_{avg}, it is sufficient to capture a set of states that includes a singlephoton state and superpositions of zero and one photons. We can create superposition states by first preparing the transmon qubit in the superposition , with varying phase φ, as shown in Fig. 4a, b. By driving the transition g〉 0〉→ e〉 1〉, we transfer the superposition state from the transmon to the cavity and then let the cavity state evolve into the propagating field. Operating the capture protocol on this set of states shows that the phase of the qubit state is converted to the motion of the mechanical resonator (Fig. 4c). More quantitatively, we follow the procedure illustrated in Fig. 3, determining both ρ_{e} and ρ_{m} for this set of states (Fig. 4d). From the input and output density matrices, we calculate for arbitrary qubit states F_{avg} = 0.83_{−0.06}^{+0.03}, which is consistent with a cascaded beamsplitter model of the capture process (Supplementary Information). Crucially, the average fidelity exceeds 2/3, the highest possible fidelity for transferring qubits using only classical resources (Supplementary Information).
Converting microwave qubit states to mechanical motion opens up new possibilities to process quantum information using micrometresized mechanical resonators. To communicate quantum information between remote modules in a network, such resonators may be the key element in the transduction of microwave quantum signals to telecommunications light^{15,16,17}. For quantum computation protocols that require the feedforward of information, such as teleportation^{27} and error correction schemes^{28}, mechanical resonators can act as ondemand memories for quantum states. As microfabrication advances continue to reduce mechanical dissipation, it could become possible to store a quantum state in the motion of a macroscopic object for about one minute^{29,30}.
Data availability. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge advice from C. Axline, M. CastellanosBeltran, L. Frunzio, S. Glancy, W. F. Kindel and F. Lecocq as well as technical assistance from R. Delaney and H. Greene. We thank P. Blanchard for assistance in taking the micrograph shown in Fig. 1b. We acknowledge funding from the National Science Foundation (NSF) under grant number 1125844, AFOSR MURI under grant number FA95501510015, and the Gordon and Betty Moore Foundation. A.P.R. acknowledges support from the NSF Graduate Research Fellowship under grant number DGE 1144083. L.D.B. acknowledges the support of the ARO QuaCGR Fellowship.
Author information
Affiliations
JILA, Boulder, Colorado 803090440, USA
 A. P. Reed
 , L. Sletten
 , X. Ma
 & K. W. Lehnert
Department of Physics, University of Colorado, Boulder, Colorado 803090390, USA
 A. P. Reed
 , K. H. Mayer
 , L. Sletten
 , X. Ma
 & K. W. Lehnert
National Institute of Standards and Technology (NIST), Boulder, Colorado 80305, USA
 K. H. Mayer
 , J. D. Teufel
 , E. Knill
 & K. W. Lehnert
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA
 L. D. Burkhart
 , W. Pfaff
 , M. Reagor
 & R. J. Schoelkopf
Rigetti Computing, 775 Heinz Avenue, Berkeley, California 94710, USA
 M. Reagor
Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
 E. Knill
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Contributions
A.P.R. and K.W.L. designed the experiment. A.P.R. performed the measurements. K.H.M. designed the tomographic analysis. A.P.R., K.H.M., J.D.T., E.K. and K.W.L. analysed the results. A.P.R., K.H.M., J.D.T. and K.W.L. wrote the manuscript. L.D.B. and A.P.R. fabricated the devices. A.P.R., L.D.B., W.P., M.R., L.S., X.M. and R.J.S. designed and constructed the photon source.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to A. P. Reed.
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