Abstract
Heisenberg’s uncertainty principle results in one of the strangest quantum behaviours: a mechanical oscillator can never truly be at rest. Even at a temperature of absolute zero, its position and momentum are still subject to quantum fluctuations^{1,2}. However, direct energy detection of the oscillator in its ground state makes it seem motionless^{1,3}, and in linear position measurements detector noise can masquerade as mechanical fluctuations^{4,5,6,7}. Thus, how can we resolve quantum fluctuations? Here, we parametrically couple a micromechanical oscillator to a microwave cavity to prepare the system in its quantum ground state^{8,9} and then amplify the remaining vacuum fluctuations into real energy quanta^{10}. We monitor the photon/phononnumber distributions using a superconducting qubit^{11,12,13}, allowing us to resolve the quantum vacuum fluctuations of the macroscopic oscillator’s motion. Our results further demonstrate the ability to control a longlived mechanical oscillator using a nonGaussian resource, directly enabling applications in quantum information processing and enhanced detection of displacement and forces.
Main
Cavity optomechanical systems have emerged as an ideal testbed for exploring the quantum limits of linear measurement of macroscopic motion^{2}, as well as a promising new architecture for performing quantum computations. In such systems, a light field reflecting off a mechanical oscillator acquires a positiondependent phase shift and reciprocally, it applies a force onto the mechanical oscillator. This effect is enhanced by embedding the oscillator inside a highqualityfactor electromagnetic cavity. Numerous physical implementations exist, both in the microwave and optical domains, and have been used to push the manipulation of macroscopic oscillators into the quantum regime, demonstrating laser cooling to the ground state of motion^{8,14}, coherent transfer of itinerant light fields into mechanical motion^{9,15}, or their entanglement^{10}. Thus far, linear position measurements have provided evidence for the quantization of light fields through radiation pressure shot noise^{16,17} and mechanical vacuum fluctuations through motional sideband asymmetries^{4,5,6,7}. However, the use of only classical and linear tools has restricted most optomechanical experiments to the manipulation of Gaussian states.
The addition of a strong nonlinearity, such as an atom, has fostered tremendous progress towards exquisite control over nonGaussian quantum states of light fields and atomic motion^{11,12}. First developed in the context of cavity quantum electrodynamics (cQED), these techniques are now widely applied to engineered systems, such as superconducting quantum bits (qubits) and microwave resonant circuits^{13,18,19,20,21}. In a pioneering cQEDtype experiment, a qubit resonantly coupled to a highfrequency mechanical oscillator^{3} allowed for the control of a singlephonon Fock state. However, short energy lifetimes of the mechanical oscillator and the qubit have slowed any further progress.
Here, we go beyond just cQED by introducing cavity optomechanical interactions. Our unique architecture incorporates an artificial atom—a superconducting qubit^{22}—into a circuit cavity electromechanical system^{23}, on a single chip. Here, a lowfrequency, highqualityfactor mechanical oscillator strongly interacts with the microwave cavity photons. The qubit–cavity interaction realizes a nonclassical emitter and detector of photons, thus providing an essential nonlinear resource for the deterministic control of longlived mechanical quantum states. We demonstrate the potential of such an architecture by measuring the quantum vacuum fluctuations inherently present in the motion of a macroscopic oscillator.
A microwave cavity is the central element of this architecture (in blue in Fig. 1a). It is a linear inductor–capacitor (LC) resonator formed by a coil inductor and a mechanically compliant vacuumgap capacitor^{23,24}. First, the intracavity electromagnetic field is coupled by means of radiation pressure to the vibrational mode of the compliant capacitor (in red in Fig. 1). Second, the microwave cavity is capacitively coupled to a phase qubit (in green in Fig. 1). A phase qubit is formed from a Josephson junction in parallel with an LC oscillator, and it behaves like a nonlinear resonator at the singlequantum level, that is, an artificial atom^{22}. To a good approximation the phase qubit can be operated as a twolevel system whose transition frequency ω_{qb} can be widely tuned in situ by applying an external flux, such that 9 GHz ≤ ω_{qb}/2π ≤ 13.5 GHz. The microwave cavity and the fundamental flexural mode of the capacitor are two harmonic oscillators with resonance frequencies of respectively ω_{c}/2π = 10.188 GHz and Ω_{m}/2π = 15.9 MHz.
The qubit and the cavity are both electrical circuits with quantized energy levels, sharing a voltage through the coupling capacitor. On resonance, Δ_{qb} = ω_{qb} − ω_{c} = 0, the interaction between the qubit and the cavity is well described by the Jaynes–Cummings Hamiltonian . Here, () is the raising (lowering) operator for the qubit, () is the creation (annihilation) operator for cavity photons and J is the capacitive coupling strength. describes the exchange of a single quantum between the qubit and the cavity at a rate 2J. In the strongcoupling regime, when the coupling strength J overcomes the decoherence rates of the qubit γ_{qb} and the cavity κ, that is, J > (γ_{qb}, κ), the system hybridizes, leading to the wellknown vacuum Rabi splitting, measured spectroscopically in Fig. 1e. The qubit–cavity interaction can be effectively turned off by detuning the qubit.
The position of the mechanical oscillator modulates the cavity resonance frequency and thus, the energy stored in the cavity. As a result, the microwave photons apply a force on the mechanical oscillator. This interaction is described by the radiation pressure Hamiltonian^{2} , where G = dω_{c}/dx, is the cavity photon number and is the oscillator’s position. Here, x_{zpf} is the oscillator’s zeropoint fluctuation and () is the creation (annihilation) operator for mechanical phonons. The force applied by a single photon onto the mechanical oscillator is typically weak, with , where is the equilibrium thermal occupancy of the oscillator and Γ_{m} is its intrinsic relaxation rate. However, the total force increases significantly with the intensity of the intracavity field. In the presence of a strong coherent microwave pump of frequency ω_{p}, the optomechanical interaction is linearized and takes two different forms depending on the pump–cavity detuning Δ_{p} = ω_{p} − ω_{c} (Supplementary Information). When Δ_{p} = − Ω_{m}, the annihilation of a mechanical phonon can upconvert a pump photon into a cavity photon, mediating a ‘beamsplitter’ interaction, . This results in the coherent exchange of the cavity and mechanical states at a rate 2g, where is the enhanced optomechanical coupling and n_{p} is the pump strength expressed in terms of the average number of intracavity photons. When Δ_{p} = + Ω_{m}, pump photons are downconverted into correlated photon–phonon pairs, mediating a ‘twomode squeezer’ interaction, . This results in the amplification and entanglement of the cavity field and the mechanical motion^{10}, at a rate 2g. The hallmark for entering the strongcoupling regime in our device, g > (Γ_{m}, κ), is the hybridization and normalmode splitting induced by a strong beamsplitter interaction^{23}, as measured through the cavitydriven response in Fig. 1f. Finally, with a lifetime of the mechanical oscillator’s ground state, μs, much longer than the cavity lifetime, 1/κ ≈ 1 μs, this device is in the quantumcoherent regime^{15}.
Measurements in the frequency domain provide an extensive characterization of the device’s parameters; however, they probe only the steady state of the system, in equilibrium with the thermal environment. In the next two paragraphs, we will show that timedomain protocols enable: the preparation of nonclassical cavity states and the measurement of the intracavity photonnumber distribution^{11,12,13}; and coherent state transfer by frequency conversion and entanglement by parametric amplification between the microwave cavity and the mechanical oscillator^{25,26}.
The outofequilibrium dynamics between the phase qubit and the cavity are shown in Fig. 2. First, in Fig. 2a, b we perform the first basic block of the Law–Eberly protocol^{27}. We initialize the qubit in the excited state using a resonant microwave pulse, then tune the qubit into resonance with the cavity for an interaction time τ and measure the qubit population P_{e} (using a destructive singleshot readout). The coupled system undergoes vacuum Rabi oscillations at a single frequency J/π and after half a cycle the cavity is prepared in a singlephoton Fock state. Next, in Fig. 2c–f, we exploit the wellknown scaling of the Rabi frequency with the cavity Fock state number to measure the intracavity photonnumber distribution^{13}. We initialize the cavity in either a coherent state or a thermal state, parametrized by the average photon occupancy . When the qubit is tuned into resonance, each initial distribution (Fig. 2d) produces a distinct time evolution of P_{e}(τ) (Fig. 2e, f), in good agreement with simulations that include all sources of decoherence and where the average photon number is the only free parameter (Supplementary Information). We resolve the cavity occupancy down to and have the ability to distinguish the thermal, noiselike component of the cavity state from the coherent component.
We will now exploit this measurement technique to explore the outofequilibrium optomechanical dynamics (Fig. 3). To acquire some physical intuition one can solve the lossless equations of motion describing the time evolution of the microwave and mechanical field amplitudes (Supplementary Information). The average photon occupancy after a beamsplitter interaction, , or a twomode squeezer interaction, , follows:
where and are respectively the initial cavity and mechanical occupancy, and θ = ∫ 2g(t) dt is the accumulated interaction phase. The periodic functions in equation (1) describe the state exchange induced by the beamsplitter interaction (see Fig. 3b) and the hyperbolic functions in equation (2) describe the amplification induced by the twomode squeezer interaction (see Fig. 3c). Experimentally, we start by actively preparing the mechanical state in a nearly pure coherent state, , where α_{m}^{i} ^{2} = 23 is the coherent component (displacement) and represents the residual thermal (incoherent) phonon occupancy (Supplementary Information). We then pulse either optomechanical interaction using a microwave pump at Δ_{p} = ±Ω_{m}, followed by tuning the qubit into resonance with the cavity to measure the subsequent photonnumber distribution (as described previously in Fig. 2) for each pump duration. As expected this distribution corresponds to a displaced thermal state, characterized by an incoherent component and a coherent component α_{c}^{±}, for a total average photon occupancy of . In Fig. 3b, c, we show as a function of the interaction phase θ. The data in Fig. 3b (Fig. 3c) qualitatively agree with equation (1) (equation (2)), and quantitatively agree with full numerical simulations (solid blue lines) that include the finite linewidth and bath temperature of each mode (Supplementary Information). The expected evolution of follows the solid red line. The only free parameter is the initial mechanical occupancy . We emphasize our ability to resolve, with a sensitivity well below the singlequantum level, the coherent exchange of mechanical phonons and cavity photons or the amplification of the two localized modes, with both processes occurring at a rate faster than decoherence.
A striking signature of the quantum nature of the oscillator’s motion resides in the commutation relation . Together with equation (2), we can see that even with both modes initially in their ground state, , the zeropoint motion of the oscillator alone feeds the parametric amplification process, with a gain sinh^{2}(θ/2), leading to a finite cavity occupancy . In contrast, from the same initial conditions, equation (1) shows no interesting dynamics for the beamsplitter interaction, with . Thus, to quantitatively extract the ‘+1’ contribution during amplification, we compare the two processes, as shown in Fig. 3d. With the cavity initially in its ground state, , equation (1) and equation (2) relate the final average cavity occupancy to the initial average mechanical occupancy through the gain of the parametric interactions, sin^{2}(θ/2) for the beamsplitter or sinh^{2}(θ/2) for the twomode squeezer. First, we set the interaction phase to θ = π and measure the final photon distribution as a function of the initial mechanical displacement α_{m}^{i}, for Δ_{p} = +Ω_{m} and Δ_{p} = −Ω_{m}. In the presence of loss, both gains are less than their maximum values, G_{−} < sin^{2}(π/2) and G_{+} < sinh^{2}(π/2). By taking the ratio of final cavity displacement to initial mechanical displacement, G_{±} = α_{c}^{±} ^{2}/ α_{m}^{i} ^{2}, we measure G_{−} = 0.25 and G_{+} = 0.88, at all coherent drive amplitudes (Supplementary Information). We show the results for in red and in blue. The difference between these two optomechanical interactions is clear in Fig. 3d, showing the extra ‘+1’ contribution sourced directly from the commutator between the position and momentum of the macroscopic mechanical oscillator due to the quantum vacuum fluctuations.
This signature has been discussed in terms of asymmetry between the rates of phonon absorption and emission^{5,7}, or in terms of added noise in the context of threewave mixing^{1,28}. Our architecture is however uniquely suited to explore quantitatively this phenomenon in optomechanics, because we have the ability to measure directly the mechanically scattered photons, localized in the cavity. By combining the measurement of the intracavity photonnumber distribution with the optomechanical interactions, we have realized a phononnumber distribution measurement. The nonlinearity of the qubit–cavity interaction allows us to clearly distinguish classical noise from quantum noise, as only classical noise can lead to real cavity quanta that can excite the qubit out of its ground state. In addition, our technique is not sensitive to the correlations between the electromagnetic noise and mechanical noise, which would induce ‘squashing’ of the output field^{7}.
Looking forward, with more complex protocols, we could: exploit the qubit as a deterministic singlephonon source to generate arbitrary quantum states of motion^{20}; perform fullstate tomography of the mechanical oscillator^{29}. The ability to encode complex quantum states in these longlived mechanical systems has important implications for quantum information and for the study of the fundamental quantum behaviour of massive objects^{30}.
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Acknowledgements
We thank A. W. Sanders for taking the micrographs in Fig. 1b, c. This work was supported by the NIST Quantum Information Program.
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F.L., J.D.T. and R.W.S. conceived and designed the experiment. F.L. fabricated the device and performed the measurement. F.L. and J.D.T. analysed the data. F.L., J.D.T. and R.W.S. wrote the manuscript. J.A. contributed to the fabrication process and provided experimental support.
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Lecocq, F., Teufel, J., Aumentado, J. et al. Resolving the vacuum fluctuations of an optomechanical system using an artificial atom. Nature Phys 11, 635–639 (2015). https://doi.org/10.1038/nphys3365
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DOI: https://doi.org/10.1038/nphys3365
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