Resolving the Vacuum Fluctuations of an Optomechanical System Using An

LETTERS PUBLISHED ONLINE: 15 JUNE 2015 | DOI: 10.1038/NPHYS3365

Resolving the vacuum ﬂuctuations of an optomechanical system using an artiﬁcial atom F. Lecocq*, J. D. Teufel, J. Aumentado and R. W. Simmonds

Heisenberg’s uncertainty principle results in one of the Here, we go beyond just cQED by introducing cavity strangest quantum behaviours: a mechanical oscillator can optomechanical interactions. Our unique architecture incorporates never truly be at rest. Even at a temperature of absolute an artificial atom—a superconducting qubit22—into a circuit cavity zero, its position and momentum are still subject to quan- electromechanical system23, on a single chip. Here, a low-frequency, tum fluctuations1,2. However, direct energy detection of the high-quality-factor mechanical oscillator strongly interacts with oscillator in its ground state makes it seem motionless1,3, the microwave cavity photons. The qubit–cavity interaction and in linear position measurements detector noise can realizes a non-classical emitter and detector of photons, thus masquerade as mechanical fluctuations4–7. Thus, how can providing an essential nonlinear resource for the deterministic we resolve quantum fluctuations? Here, we parametrically control of long-lived mechanical quantum states. We demonstrate couple a micromechanical oscillator to a microwave cavity the potential of such an architecture by measuring the quantum to prepare the system in its quantum ground state8,9 and vacuum fluctuations inherently present in the motion of a then amplify the remaining vacuum fluctuations into real macroscopic oscillator. energy quanta10. We monitor the photon/phonon-number A microwave cavity is the central element of this architecture distributions using a superconducting qubit11–13, allowing us to (in blue in Fig. 1a). It is a linear inductor–capacitor (LC) res- resolve the quantum vacuum fluctuations of the macroscopic onator formed by a coil inductor and a mechanically compliant oscillator’s motion. Our results further demonstrate the vacuum-gap capacitor23,24. First, the intra-cavity electromagnetic ability to control a long-lived mechanical oscillator using field is coupled by means of radiation pressure to the vibrational a non-Gaussian resource, directly enabling applications in mode of the compliant capacitor (in red in Fig. 1). Second, the quantum information processing and enhanced detection of microwave cavity is capacitively coupled to a phase qubit (in green displacement and forces. in Fig. 1). A phase qubit is formed from a Josephson junction Cavity optomechanical systems have emerged as an ideal in parallel with an LC oscillator, and it behaves like a nonlin- testbed for exploring the quantum limits of linear measurement of ear resonator at the single-quantum level, that is, an artificial macroscopic motion2, as well as a promising new architecture for atom22. To a good approximation the phase qubit can be oper- ω performing quantum computations. In such systems, a light field ated as a two-level system whose transition frequency qb can reflecting off a mechanical oscillator acquires a position-dependent be widely tuned in situ by applying an external flux, such that ω / phase shift and reciprocally, it applies a force onto the mechanical 9GHz≤ qb 2π≤13.5 GHz. The microwave cavity and the funda- oscillator. This effect is enhanced by embedding the oscillator inside mental flexural mode of the capacitor are two harmonic oscillators ω / a high-quality-factor electromagnetic cavity. Numerous physical with resonance frequencies of respectively c 2π=10.188 GHz and Ω / implementations exist, both in the microwave and optical domains, m 2π=15.9 MHz. and have been used to push the manipulation of macroscopic The qubit and the cavity are both electrical circuits with oscillators into the quantum regime, demonstrating laser cooling quantized energy levels, sharing a voltage through the coupling 8,14 ∆ ω ω to the ground state of motion , coherent transfer of itinerant capacitor. On resonance, qb = qb − c =0, the interaction between light fields into mechanical motion9,15, or their entanglement10. the qubit and the cavity is well described by the Jaynes–Cummings ( σ †σ ) σ σ Thus far, linear position measurements have provided evidence Hamiltonian Hjc = hJ¯ aˆ ˆ+ +a ˆ ˆ− . Here, ˆ+ ( ˆ−) is the raising for the quantization of light fields through radiation pressure shot (lowering) operator for the qubit, aˆ † (aˆ) is the creation (annihilation) noise16,17 and mechanical vacuum fluctuations through motional operator for cavity photons and J is the capacitive coupling strength. 4–7 sideband asymmetries . However, the use of only classical and Hjc describes the exchange of a single quantum between the qubit linear tools has restricted most optomechanical experiments to the and the cavity at a rate 2J. In the strong-coupling regime, when the γ manipulation of Gaussian states. coupling strength J overcomes the decoherence rates of the qubit qb κ >(γ κ) The addition of a strong nonlinearity, such as an atom, has and the cavity , that is, J qb, , the system hybridizes, leading to fostered tremendous progress towards exquisite control over non- the well-known vacuum Rabi splitting, measured spectroscopically Gaussian quantum states of light fields and atomic motion11,12. in Fig. 1e. The qubit–cavity interaction can be effectively turned off First developed in the context of cavity quantum electrodynamics by detuning the qubit. (cQED), these techniques are now widely applied to engineered The position of the mechanical oscillator modulates the cavity systems, such as superconducting quantum bits (qubits) and resonance frequency and thus, the energy stored in the cavity. As microwave resonant circuits13,18–21. In a pioneering cQED-type a result, the microwave photons apply a force on the mechanical experiment, a qubit resonantly coupled to a high-frequency oscillator. This interaction is described by the radiation pressure 3 2 ω / † mechanical oscillator allowed for the control of a single-phonon Hamiltonian Hrp =hG¯ nˆ cxˆ, where G=d c dx, nˆ c =a ˆ aˆ is the cavity (ˆ† ˆ) Fock state. However, short energy lifetimes of the mechanical photon number and xˆ =xzpf b +b is the oscillator’s position. Here, ˆ† ˆ oscillator and the qubit have slowed any further progress. xzpf is the oscillator’s zero-point fluctuation and b (b) is the creation

National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA. *e-mail: ﬂ[email protected]

NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics 635

a b 100 μm d Cavity drive Qubit bias ω c d ωqb ω d p Qubit Pump drive 2 μm

c b c

Circuit Microwave QED optomechanics Phase Microwave Mechanical qubit cavity oscillator J g ω ω Ω T = 25 mK qb c m 2 μm

f e Drive frequency, (GHz) 10.28 2 10.190 Pe |R| (dB) 0.0 0.1 0.2 −1.0 −0.5 0.0

π 10.189

/2 10.22 qb d

ω ω /2π J/π c g/π 10.188 ω d c 10.16 /2 π Ω π − m /2 10.187

Drive frequency, (GHz) 10.10 80 40 0 −40 −80 −17 −16 −15 −14 ω ω ω ω π Qubit detuning, ( qb − c ) /2 π (MHz) Pump detuning, ( p − c )/2 (MHz)

Figure 1 | Device description and strong-coupling regime. a, False-colour optical micrograph of the device. Aluminium is in grey and sapphire in light blue. The phase qubit is in green; the microwave cavity in blue; and the mechanically compliant capacitor in red. b,c, False-colour scanning electron micrograph of the qubit’s Josephson junction and of the mechanical oscillator, respectively. d, Equivalent circuit diagram. e, Qubit–cavity vacuum Rabi splitting. Population of the qubit’s excited state Pe, in green, as a function of the drive frequency and of the qubit detuning with respect to the microwave cavity. The avoided crossing is a clear signature of the single-photon strong-coupling regime. From a fit with theory we extract the qubit/cavity coupling rate J/2π=12.5 MHz. f, Cavity–oscillator normal-mode splitting. Reflected power in blue as a function of the drive and pump frequencies. The drive is applied 5 near the cavity resonance and the pump is applied near ∆p =−Ωm. Here, the pump strength is set to np ≈5×10 . The normal-mode splitting observed in the cavity-driven response corresponds to the hybridization of the cavity and mechanical modes in the driven strong-coupling regime. From a fit with theory we extract κ/2π=163 kHz, Γm/2π=150 Hz and g=2π×242kHz>( n¯mΓm,κ), where n¯m ≈32 at T ≈25 mK.

(annihilation) operator for mechanical phonons. The force applied the cavity lifetime, 1/κ ≈ 1 µs, this device is in the quantum- by a single photon onto the mechanical oscillator is typically weak, coherent regime15. ( Γ κ) with g0 = Gxzpf n¯ m m, , where n¯ m is the equilibrium thermal Measurements in the frequency domain provide an extensive Γ occupancy of the oscillator and m is its intrinsic relaxation rate. characterization of the device’s parameters; however, they probe However, the total force increases significantly with the intensity only the steady state of the system, in equilibrium with the of the intra-cavity field. In the presence of a strong coherent thermal environment. In the next two paragraphs, we will show ω microwave pump of frequency p, the optomechanical interaction that time-domain protocols enable: the preparation of non-classical is linearized and takes two different forms depending on the cavity states and the measurement of the intra-cavity photon- ∆ ω ω 11–13 pump–cavity detuning p = p − c (Supplementary Information). number distribution ; and coherent state transfer by frequency ∆ Ω When p = − m, the annihilation of a mechanical phonon can conversion and entanglement by parametric amplification between up-convert a pump photon into a cavity photon, mediating a the microwave cavity and the mechanical oscillator25,26. ˆ† ˆ † ‘beamsplitter’ interaction, H− =hg¯ (aˆb +baˆ ). This results in the The out-of-equilibrium dynamics between the phase qubit and coherent exchange of the cavity and mechanical states at a rate 2g, the cavity are shown in Fig. 2. First, in Fig. 2a,b we perform the √ 27 where g =g0 np is the enhanced optomechanical coupling and np first basic block of the Law–Eberly protocol . We initialize the is the pump strength expressed in terms of the average number of qubit in the excited state using a resonant microwave pulse, then ∆ Ω intra-cavity photons. When p = + m, pump photons are down- tune the qubit into resonance with the cavity for an interaction τ converted into correlated photon–phonon pairs, mediating a ‘two- time and measure the qubit population Pe (using a destructive † ˆ† ˆ mode squeezer’ interaction, H+ =hg¯ (aˆ b +baˆ). This results in single-shot readout). The coupled system undergoes vacuum Rabi the amplification and entanglement of the cavity field and the oscillations at a single frequency J/π and after half a cycle the cavity mechanical motion10, at a rate 2g. The hallmark for entering is prepared in a single-photon Fock state. Next, in Fig. 2c–f, we >(Γ κ) the strong-coupling regime in our device, g m, , is the exploit the well-known scaling of the Rabi frequency with the cavity hybridization and normal-mode splitting induced by a strong Fock state number to measure the intra-cavity photon-number beamsplitter interaction23, as measured through the cavity-driven distribution13. We initialize the cavity in either a coherent state or a response in Fig. 1f. Finally, with a lifetime of the mechanical thermal state, parametrized by the average photon occupancy hâ†aî. / Γ oscillator’s ground state, 1 n¯ m m ≈ 33 µs, much longer than When the qubit is tuned into resonance, each initial distribution

636 NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics

© 2015 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3365 LETTERS a 〉 〉 〉 〉 a Phase ω |g |e |g |e ±± qb π 〈â â〉 ≈ 0 〈â â〉 = n + |α |2 qubit Microwave ω i ± c c cavity c Parametric τ ω ω Ω 〉 p = c ± m interaction Microwave |0 Qubit ω Mechanical Ω θ = ∫2g(t) dt cavity c m sensor oscillator 〈 〉 i α i 2 〈 〉 State preparation Resonant interaction Qubit measurement b b i = nm + | m| b b ±

e b Beamsplitterc Two-mode squeezer b P 1.0 102 102

Sensor saturation 101 101 0.1 Qubit population, 0 20 40 60 80 100 120 Interaction time, τ (ns) 100 100 c 〉 〉 Phase ω |g |g |e〉 Δ = −Ω Δ = +Ω

Occupancy (quanta) p m p m qubit qb 10−1 Exp. 10−1 Exp. τ 〈 〉 〈 〉 〉 〈 〉 â â â â + Microwave |0 â â − ω f(t) 〈b b〉 〈 〉 cavity c − b b + 10−2 10−2 State preparation Resonant interaction Qubit measurement 0 π 2π 4π 6π 0 π 2π Interaction phase, θ (rad) Interaction phase, θ (rad) deCoherent state f Thermal state −1 0 α i 2 1.0 1.0 d 10 10 | m| (phonons) 〈â â〉 ≈ 10 101 0.1 0.1 θ = π

Two-mode squeezer (phonons)

± 100

G n /

P 〈 〉 ≈ +1 quantum â â 1 ± 〉 Δ Ω

â , p = + m n i e â m Δ Ω 〈 P Beamsplitter , p = − m 10−1 10−15 xzpf 10−14 α i Initial mechanical displacement, | m | (m) 〈â â〉 ≈ 10−1

| Optomechanics with a number-resolving detector.

Qubit population, Figure 3 a, Sequence diagram. b, Measured cavity occupancy (blue circles) as a function of the

Cavity Fock state probability, probability, state Fock Cavity 5 interaction duration in reduced unit θ, for ∆p =−Ωm (np =3.8×10 and g=2π×198 kHz). c, The same as in b for ∆ =+Ω . In both b and c the 〈â â〉 ≈ 10−2 p m predictions of the Heisenberg–Langevin equations of motion are shown in solid blue lines (Supplementary Information). The mechanical occupancy is shown as solid red lines. d, Measurement of the vacuum fluctuations of the mechanical oscillator. The cavity occupancy is measured as a function of 0481216 0204002040 the initial mechanical displacement for θ =π and for ∆p =Ωm. The gain τ τ Photon number n Time, (ns) Time, (ns) at each pump frequency, G, is the ratio of final cavity displacement to =|α|2/|αi |2 initial mechanical displacement G c m , which is constant for all αi h † i / h † i / Figure 2 | Cavity state preparation and readout. a, Sequence diagram for m. We show â â − G− in red and â â + G+ in blue. The amplification of the preparation of a single-photon Fock state in the cavity. The qubit is the mechanical vacuum fluctuations appears as one additional quantum prepared in the first excited state |ei, with a 75% eciency, and then when ∆p =+Ωm. The error bars are the measurement uncertainty of the interacts resonantly with the cavity for a time τ before the qubit state is photon-number distribution (Supplementary Information). measured. b, The population of the qubit Pe plotted as a function of the interaction time τ. The black line is a fit to a master equation prediction the thermal, noise-like component of the cavity state from the (Supplementary Information). c, Sequence diagram for the readout of the coherent component. cavity state. The cavity is prepared in a coherent (or thermal) state by We will now exploit this measurement technique to explore the driving it with a coherent tone (or with white noise). d, The corresponding out-of-equilibrium optomechanical dynamics (Fig. 3). To acquire intra-cavity photon distributions for four drive amplitudes, parametrized by some physical intuition one can solve the lossless equations of ¯ the average cavity occupancy nc. The distributions for thermal and motion describing the time evolution of the microwave and coherent states are respectively in bright and dark blue. e,f, For each drive mechanical field amplitudes (Supplementary Information). The † amplitude, the evolution of the qubit population Pe is plotted in green for average photon occupancy after a beamsplitter interaction, hâ aî−, † the coherent states (e), and for the thermal states (f). The solid lines are a or a two-mode squeezer interaction, hâ aî+, follows: fit to a master equation prediction. † † 2(θ/ ) ˆ† ˆ 2(θ/ ) hâ aî− =hâ aîi cos 2 +hb bii sin 2 (1) (τ) (Fig. 2d) produces a distinct time evolution of Pe (Fig. 2e,f), † † 2(θ/ ) ˆ ˆ† 2(θ/ ) in good agreement with simulations that include all sources of hâ aî+ =hâ aîi cosh 2 +hbb ii sinh 2 (2) decoherence and where the average photon number n¯ c is the only † ˆ† ˆ free parameter (Supplementary Information). We resolve the cavity where hâ aîi and hb bii are respectively the initial cavity and occupancy down to hâ†aî≈0.02 and have the ability to distinguish mechanical occupancy, and θ = R 2g(t) dt is the accumulated

NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics 637

© 2015 Macmillan Publishers Limited. All rights reserved LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3365 interaction phase. The periodic functions in equation (1) describe distinguish classical noise from quantum noise, as only classical the state exchange induced by the beamsplitter interaction (see noise can lead to real cavity quanta that can excite the qubit out Fig. 3b) and the hyperbolic functions in equation (2) describe of its ground state. In addition, our technique is not sensitive to the amplification induced by the two-mode squeezer interaction the correlations between the electromagnetic noise and mechanical (see Fig. 3c). Experimentally, we start by actively preparing the noise, which would induce ‘squashing’ of the output field7. hˆ† î = ¯ i +|αi |2 mechanical state in a nearly pure coherent state, b b i nm m , Looking forward, with more complex protocols, we could: |αi |2 = where m 23 is the coherent component (displacement) and exploit the qubit as a deterministic single-phonon source to ¯ i = 20 nm 0.25 represents the residual thermal (incoherent) phonon generate arbitrary quantum states of motion ; perform full-state occupancy (Supplementary Information). We then pulse either tomography of the mechanical oscillator29. The ability to encode ∆ Ω optomechanical interaction using a microwave pump at p = m, complex quantum states in these long-lived mechanical systems has followed by tuning the qubit into resonance with the cavity to important implications for quantum information and for the study measure the subsequent photon-number distribution (as described of the fundamental quantum behaviour of massive objects30. previously in Fig. 2) for each pump duration. As expected this distribution corresponds to a displaced thermal state, characterized Received 30 January 2015; accepted 11 May 2015; ¯ α by an incoherent component nc and a coherent component c , for a published online 15 June 2015 hˆ † î = ¯ +|α|2 total average photon occupancy of a a nc c . In Fig. 3b,c, † we show hâ aî as a function of the interaction phase θ. The data in Fig. 3b (Fig. 3c) qualitatively agree with equation (1) (equation (2)), References and quantitatively agree with full numerical simulations (solid blue 1. Clerk, A. A. et al. Introduction to quantum noise, measurement, and lines) that include the finite linewidth and bath temperature of amplification. Rev. Mod. Phys. 82, 1155–1208 (2010). 2. Aspelmeyer, M., Kippenberg, T. & Marquardt, F. Cavity optomechanics. Rev. each mode (Supplementary Information). The expected evolution Mod. Phys. 86, 1391–1452 (2014). ˆ† ˆ of hb bi follows the solid red line. The only free parameter is the 3. O’Connell, A. D. et al. Quantum ground state and single-phonon control of a ˆ† ˆ initial mechanical occupancy hb bii. We emphasize our ability to mechanical resonator. Nature 464, 697–703 (2010). resolve, with a sensitivity well below the single-quantum level, the 4. Jayich, A. M. et al. Cryogenic optomechanics with a Si3N4 membrane and coherent exchange of mechanical phonons and cavity photons or classical laser noise. New J. Phys. 14, 115018 (2012). the amplification of the two localized modes, with both processes 5. Safavi-Naeini, A. H. et al. Observation of quantum motion of a occurring at a rate faster than decoherence. nanomechanical resonator. Phys. Rev. Lett. 108, 033602 (2012). A striking signature of the quantum nature of the oscillator’s 6. Safavi-Naeini, A. H. et al. Laser noise in cavity-optomechanical cooling and ˆ ˆ† = ˆ† ˆ + thermometry. New J. Phys. 15, 035007 (2013). motion resides in the commutation relation bb b b 1. 7. Weinstein, A. J. et al. Observation and interpretation of motional sideband Together with equation (2), we can see that even with both asymmetry in a quantum electromechanical device. Phys. Rev. X 4, † ˆ† ˆ modes initially in their ground state, hâ aîi = hb bii = 0, the 041003 (2014). zero-point motion of the oscillator alone feeds the parametric 8. Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantum amplification process, with a gain sinh2(θ/2), leading to a finite ground state. Nature 475, 359–363 (2011). † cavity occupancy hâ aî+. In contrast, from the same initial 9. Palomaki, T. A. et al. Coherent state transfer between itinerant microwave conditions, equation (1) shows no interesting dynamics for the fields and a mechanical oscillator. Nature 495, 210–214 (2013). † 10. Palomaki, T. A. et al. Entangling mechanical motion with microwave fields. beamsplitter interaction, with hâ aî− = 0. Thus, to quantitatively extract the ‘+1’ contribution during amplification, we compare Science 342, 710–713 (2013). 11. Brune, M. et al. Quantum Rabi oscillation: A direct test of field quantization in the two processes, as shown in Fig. 3d. With the cavity initially in a cavity. Phys. Rev. Lett. 76, 1800–1803 (1996). † its ground state, hâ aîi =0, equation (1) and equation (2) relate 12. Meekhof, D. M. et al. Generation of nonclassical motional states of a trapped † the final average cavity occupancy hâ aî to the initial average atom. Phys. Rev. Lett. 76, 1796–1799 (1996). ˆ† ˆ mechanical occupancy hb bii through the gain of the parametric 13. Hofheinz, M. et al. Generation of Fock states in a superconducting quantum interactions, sin2(θ/2) for the beamsplitter or sinh2(θ/2) for the circuit. Nature 454, 310–314 (2008). two-mode squeezer. First, we set the interaction phase to θ = π 14. Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum and measure the final photon distribution as a function of the ground state. Nature 478, 89–92 (2011). initial mechanical displacement αi , for ∆ =+Ω and ∆ =−Ω . 15. Verhagen, E. et al. Quantum-coherent coupling of a mechanical oscillator to an m p m p m optical cavity mode. Nature 482, 63–67 (2012). In the presence of loss, both gains are less than their maximum 16. Purdy, T. P. et al. Observation of radiation pressure shot noise on a macroscopic < 2( / ) < 2( / ) values, G− sin π 2 and G+ sinh π 2 . By taking the ratio object. Science 339, 801–804 (2013). of final cavity displacement to initial mechanical displacement, 17. Suh, J. et al. Mechanically detecting and avoiding the quantum fluctuations of a =|α|2/|αi |2 = = G c m , we measure G− 0.25 and G+ 0.88, at all microwave field. Science 344, 1262–1265 (2014). coherent drive amplitudes (Supplementary Information). We show 18. Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit † / ˆ† ˆ † / ˆ† ˆ the results for hâ aî− G− =hb bii in red and hâ aî+ G+ =hb bii +1 using circuit quantum electrodynamics. Nature 431, 162–167 (2004). in blue. The difference between these two optomechanical 19. Sillanpaa, M. A. et al. Coherent quantum state storage and transfer between interactions is clear in Fig. 3d, showing the extra ‘+1’ contribution two phase qubits via a resonant cavity. Nature 449, 438–442 (2007). sourced directly from the commutator between the position and 20. Hofheinz, M. et al. Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459, 546–549 (2009). momentum of the macroscopic mechanical oscillator due to the 21. Pirkkalainen, J. M. et al. Hybrid circuit cavity quantum electrodynamics with a quantum vacuum fluctuations. micromechanical resonator. Nature 494, 211–215 (2013). This signature has been discussed in terms of asymmetry 22. Martinis, J. M. Superconducting phase qubits. Quantum Inf. Process. 8, between the rates of phonon absorption and emission5,7, or in 81–103 (2009). terms of added noise in the context of three-wave mixing1,28. Our 23. Teufel, J. D. et al. Circuit cavity electromechanics in the strong-coupling architecture is however uniquely suited to explore quantitatively regime. Nature 471, 359–363 (2011). this phenomenon in optomechanics, because we have the ability 24. Cicak, K. et al. Low-loss superconducting resonant circuits using to measure directly the mechanically scattered photons, localized vacuum-gap-based microwave components. Appl. Phys. Lett. 96, 093502 (2010). in the cavity. By combining the measurement of the intra-cavity 25. Zakka-Bajjani, E. et al. Quantum superposition of a single microwave photon photon-number distribution with the optomechanical interactions, in two different ‘colour’ states. Nature Phys. 7, 599 (2011). we have realized a phonon-number distribution measurement. The 26. Hofer, S. G. et al. Quantum entanglement and teleportation in pulsed cavity nonlinearity of the qubit–cavity interaction allows us to clearly optomechanics. Phys. Rev. A 84, 052327 (2011).

638 NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics

27. Law, C. K. & Eberly, J. H. Arbitrary control of a quantum electromagnetic field. Author contributions Phys. Rev. Lett. 76, 1055–1058 (1996). F.L., J.D.T. and R.W.S. conceived and designed the experiment. F.L. fabricated the device 28. Bergeal, N. et al. Analog information processing at the quantum limit with a and performed the measurement. F.L. and J.D.T. analysed the data. F.L., J.D.T. and R.W.S. Josephson ring modulator. Nature Phys. 6, 296–302 (2010). wrote the manuscript. J.A. contributed to the fabrication process and provided experimental support. 29. Leibfried, D. et al. Experimental determination of the motional quantum state of a trapped atom. Phys. Rev. Lett. 77, 4281–4285 (1996). Additional information 30. Pikovski, I. et al. Probing Planck-scale physics with quantum optics. Nature Supplementary information is available in the online version of the paper. Reprints and Phys. 8, 393–397 (2012). permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to F.L. Acknowledgements We thank A. W. Sanders for taking the micrographs in Fig. 1b,c. This work was supported Competing ﬁnancial interests by the NIST Quantum Information Program. The authors declare no competing financial interests.

NATURE PHYSICS | VOL 11 | AUGUST 2015 | www.nature.com/naturephysics 639