PHYSICAL REVIEW X 9, 021049 (2019)

Quantum Dynamics of a Few-Photon Parametric Oscillator

Zhaoyou Wang,* Marek Pechal,* E. Alex Wollack, Patricio Arrangoiz-Arriola, † Maodong Gao, Nathan R. Lee, and Amir H. Safavi-Naeini Department of Applied Physics and Ginzton Laboratory, Stanford University 348 Via Pueblo Mall, Stanford, California 94305, USA

(Received 31 January 2019; revised manuscript received 22 March 2019; published 7 June 2019)

Modulating the of a oscillator at nearly twice its natural frequency leads to amplification and self-. Above the oscillation threshold and in the presence of a nonlinearity, the field settles into a coherent oscillating state with a well-defined phase of either 0 or π. We demonstrate a quantum parametric oscillator operating at and drive it into oscillating states containing only a few photons. The small number of photons present in the system and the coherent nature of the nonlinearity prevent the environment from learning the randomly chosen phase of the oscillator. This result allows the system to oscillate briefly in a quantum superposition of both phases at once, effectively generating a nonclassical Schrödinger’s cat state. We characterize the dynamics and states of the system by analyzing the output field emitted by the oscillator and implementing quantum state tomography suited for nonlinear . By demonstrating a quantum parametric oscillator and the requisite techniques for characterizing its quantum state, we set the groundwork for new schemes of quantum and classical information processing and extend the reach of these ubiquitous devices deep into the quantum regime.

DOI: 10.1103/PhysRevX.9.021049 Subject Areas: Nonlinear Dynamics, Quantum Physics Quantum Information

I. INTRODUCTION or photon number. Furthermore, the self-oscillation ampli- tude scales inversely with the magnitude of the non- Parametric and oscillators are quintessential linearity. Nonlinearities have been exceedingly small in devices used to amplify small electromagnetic signals parametric oscillators to date, leading to large oscillation [1–3], convert radiation from one frequency to another amplitudes that result in rapid decay of quantum coherence [4,5], create squeezed light and entangled photons [6,7], and realize new information processing architectures and the appearance of classical dynamics [13]. The [8–12]. They operate by modulating a parameter, the quantum regime of nonlinear parametric oscillators has been extensively studied in theory [14–16] but has received natural frequency of the resonant circuit ωc, at approx- only limited attention in experiments [17]. imately twice its frequency ωp ≈ 2ωc (Fig. 1). The modu- We experimentally realize a quantum Kerr parametric lation amplifies one of the field quadratures at the oscillator (KPO) by implementing an on-chip supercon- half-harmonic frequency ωp=2. A sufficiently large ampli- ducting nonlinear and investigate its quantum fication overtakes the detuning and decay present in the dynamics under parametric driving. In contrast to previ- system and leads to an exponential increase in the half- ously demonstrated optical and microwave parametric harmonic cavity field amplitude. Nonlinearities clamp this oscillators, our device operates in the quantum regime exponential growth and cause the system to enter an oscillating steady state (Fig. 1). These nonlinearities can with a self-oscillating steady state containing only a few be dissipative or dispersive. An example of the latter is the photons. Unlike in the classical KPO, the onset of self- Kerr nonlinearity that induces a change in the cavity oscillation in the quantum system is not a sharp transition. frequency proportional to the intracavity field intensity A reasonable criterion to define self-oscillation in this case is by the narrowing of the KPO’s spectrum to well below the linewidth of the resonator. In this regime, reached for *These two authors contributed equally to this work. a sufficiently strong drive, the KPO also shows other † [email protected] properties typically associated with self-oscillating sys- tems: Relative amplitude and phase fluctuations become Published by the American Physical Society under the terms of small and correlated over a long time. the Creative Commons Attribution 4.0 International license. LC Further distribution of this work must maintain attribution to The resonator is implemented as an circuit with an the author(s) and the published article’s title, journal citation, array of Josephson junctions in place of the [18]. and DOI. The nonlinear inductance of this array induces a Kerr

2160-3308=19=9(2)=021049(13) 021049-1 Published by the American Physical Society ZHAOYOU WANG et al. PHYS. REV. X 9, 021049 (2019)

† † interaction −ðχ=2Þaˆ aˆ aˆ aˆ, where aˆ is the annihilation peaks are resolvedP (χ ≫ κ), the total power in the nth peak χ=2π 17 3 ∞ p operator of the resonator and ¼ . MHz is the is proportional to k¼n k (see Appendix D). The mea- resonator frequency shift per photon [Fig. 2(a)]. The sured signal is related to the field emitted from the resonator linewidth of the resonator is κ=2π ≈ 1.1 MHz, which by a frequency-dependent gain factor. The calibrated gain means that we are well within the single-photon Kerr allows us to calculate the Fock occupations fpng in the regime [19] with χ=κ ≈ 17. The resonator frequency ωc=2π initial state from the transient PSD. can be tuned down from 8 GHz to below 4 GHz by an on- The quantum parametric oscillator does not have a sharp chip flux line, and most of the measurements are done with self-oscillation threshold. To study the transition to self- the resonator in the 6–8-GHz frequency range. This oscillation, we modulate the flux line with a continuous tunability also enables parametric driving of the form wave at a fixed frequency ωp ¼ 2ðωc − ΔÞ and measure β˜ðtÞðaˆ þ aˆ†Þ2, where β˜ðtÞ is proportional to the voltage the mean photon number [Fig. 2(c)] as well as the steady- VðtÞ applied to the flux line. Additionally, the flux line has state PSD [Fig. 2(d)] for increasing parametric drive a weak capacitive coupling to the resonator, which allows amplitudes β. Somewhat counterintuitively, the parametric us to also use it for coherent displacement of the resonator excitation of a quantum KPO is not the most efficient state by resonant driving. at Δ ¼ 0. Simulations show that detunings close to ˜ χ=2; 3χ=2; 5χ=2; … With parametric driving βðtÞ¼2βðtÞ cos ωpt at fre- result in larger mean photon numbers. quency ωp ¼ 2ðωc − ΔÞ, which is slightly detuned from We therefore choose to study the off-resonant regime. Themeanphotonnumberhnˆi [Fig. 2(c)] is obtained by the parametric 2ωc, the dynamics of the reso- nator in a rotating frame at half the driving frequency is turning the drive off and measuring the transient PSDs as β well described by the Hamiltonian the state relaxes. At large , both the classical and the quantum model predict photon numbers very close to the χ β H=ˆ ℏ Δaˆ†aˆ − aˆ†aˆ†aˆ aˆ β t aˆ2 aˆ†2 ; measured results. For smaller , the measurements show a ¼ 2 þ ð Þð þ Þ ð1Þ gradual transition into self-oscillation, smoothed by quantum fluctuations. This behavior deviates from the where βðtÞ is the slowly varying amplitude of the para- classical prediction but is reproduced well by the quan- metric driving. tum model. The steady-state PSD [Fig. 2(d)] spectrum reduces to 2Δ II. EXPERIMENTAL RESULTS two peaks spaced by in the weak drive limit and can be understood as the result of a two-photon emission A. Parameters of the resonator and its steady state process, with one photon emitted at the cavity frequency under parametric driving ωc ¼ ωp=2 þ Δ and the other, by energy conservation, at We first characterize the energy-level structure of the ωp=2 − Δ [Fig. 2(e)]. For large drive amplitude β,the Kerr nonlinear resonator without parametric driving resonator enters the self-oscillation regime and is phase- (β ¼ 0). To do this, we prepare the resonator in an excited locked to the external parametric drive leading to the state ρˆ0 and let it relax back to the vacuum state while we narrow peak seen in Fig. 2(d). The width of this peak is collect the emitted microwave signal. The power spectral determined by the timescale of the fluctuations in the density (PSD) of this signal, which we call “transient PSD” phase of the resonator state. The main cause of these to distinguish it from the PSD measured at a steady state, fluctuations is switching between the two classically contains multiple peaks evenly spaced by χ [Fig. 2(b)] stable phases 0 and π, which is suppressed exponentially due to the nonlinear energy-level structure. The nth peak with increasing size of the state. captures photons at frequency ωc − ðn − 1Þχ, emitted dur- Characterizing the full quantum state of the parametric ing relaxation from jni to jn − 1i. We measure the transient oscillator is challenging and calls for a distinct approach to PSDs for coherent states ρˆ0 ¼jαihαj, which we prepare by quantum tomography. There are several methods based on driving the system resonantly at ωc with 1-ns pulses of measuring Fock-state populations or parities of displaced different amplitudes. Since the pulses are much shorter than states. These approaches were developed for qubit- 1=χ, they simply displace the state of the resonator from resonator systems [19–22] or qubits with tunable coupling vacuum into a coherent state jαi, with α proportional to the to the environment [23] and require more complicated pulse amplitude. In agreement with theory, the number of devices with auxiliary cavities and control parameters. peaks we observe in the spectrum grows with α as higher Other methods have been developed in quantum optics aˆ Fock states are populated [Fig. 2(b)]. based on studying the statistics of the output field out of – aˆ The transient PSD measurement provides a way to infer linear resonators [24 26], where out is linearly related to the Fock occupations pn ¼hnjρˆ0jni of the initial state ρˆ0. the resonator field aˆðt ¼ 0Þ at some initial time. These The probability that a transition from jni to jn − 1i occurs methods are not suitable for our system since χ ≫ κ leads during free relaxation to j0i is exactly equal to the total to a highly nonlinear relationship between the output field population of all Fock states jni and higher. When the and the intracavity mode.

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B. State tomography We develop a state tomography method suited for the quantum parametric oscillator that does not need any auxiliary systems. Our method is based on measuring the transient PSD using a purely linear detection of the output field. The transient PSD measurement provides us with the diagonal elements of ρˆ. By displacing the state in phase space and detecting the transient PSD, we find the Classical Quantum Dˆ ρˆ Dˆ † diagonal elements of a displaced density matrix that or contain information about the off-diagonal elements of ρˆ. Repeating this for several different displacements, we obtain enough information to estimate the full density matrix by a maximum likelihood method. This aspect of the tomography method is conceptually similar to an existing technique using generalized Q functions [19,27]. We arrive FIG. 1. Schematic of a parametric oscillator above the insta- ρˆ L ρˆ LC at the estimate of by minimizing a loss function ð estÞ, bility threshold. An circuit with a harmonically modulated which quantifies the difference between the measured PSD inductance is an example of a parametric oscillator. Above and the PSD simulated for the state ρˆ (see Appendix E). threshold, the system is oscillating at half the driving frequency est ω =2 This convex optimization problem can be solved efficiently p , which can be described as a parametric down-conversion process. The phase of the oscillator with respect to the external by semidefinite programming. The broadening of the driving can be 0 or π. Classically, this symmetry is broken, and higher Fock-state peaks due to their shorter lifetime limits one of the two cases is realized at random. Quantum mechan- the maximum size of an unknown state that we can ically, the oscillator can be in a superposition of both states. reconstruct. Additionally, we have observed systematic errors that become more severe for larger measured states. We suspect this result to be due to excitation of unwanted nonclassical evolution is important to understand in the off-resonant transitions by strong drive pulses and other context of emerging applications for parametric oscillators nonlinear effects. in quantum information processing [10,11]. To investigate We use this tomography procedure to study the free the transient dynamics, we turn on the parametric drive dynamics of the system, which we visualize by plotting suddenly and measure the time evolution of the mean the Wigner function calculated from the reconstructed photon number for three different drive amplitudes: β=2π 3 5 density matrix. After displacing the resonator from vac- ¼ . , 5.8, and 11.5 MHz [Fig. 4(a)]. The observed uum to a coherent state, we let the system evolve freely for time dependence of the mean photon number is in good a time τ before performing the tomography [Fig. 3(a)]. agreement with a theory fit to all three data sets simulta- Δ κ Here, as well as in the rest of the paper, the displacements neously, using only the detuning , the loss rate , and a β=V used in the tomography procedure are arranged in con- drive conversion factor as free fit parameters. Using centric rings as illustrated in Fig. 3(b). Figure 3(c) shows the previously described tomography procedure, we also β=2π 5 8 an example of a raw tomography data set, consisting of the reconstruct the quantum state for ¼ . MHz at t 20 t 2000 transient PSDs for all displacements. We reconstruct the ¼ ns (transient state) and ¼ ns (steady state) state at different time slices τ and observe the evolution of [Fig. 4(b)]. Comparison between the theoretically predicted the Wigner distributions due to the Kerr nonlinearity and experimentally obtained Wigner functions shows [Fig. 3(d)]. In agreement with simulations, this evolution relatively good agreement with fidelities of 0.93 and involves a “shearing” distortion, which has a classical 0.94 for the transient and steady states, respectively. We counterpart explained by the amplitude dependence of attribute the discrepancies to systematic errors in the the resonator frequency, and also exhibits negative tomography process, which we believe could be mitigated Wigner function values, a signature of quantum mechani- by improving its calibration. The only fit parameters in cal behavior [19]. the theoretical prediction are the overall rotation angle and a short delay (td ¼ 2.5 ns) between the end of the parametric drive pulse and the start of the tomography C. Transient dynamics and adiabatic (see Appendix F). cat-state generation These results demonstrate that the state of the The system can behave nonclassically under a parametric oscillator can be engineered by designing the parametric drive but only before photons leaking out of the oscillator drive βðtÞ. For example, by adiabatically changing βðtÞ, cause the loss of quantum coherence. In our system, we can prepare even-parity energy eigenstates of the quantum dynamics persists in this transient regime for a Hamiltonian for different values of β as long as losses sufficiently long time to allow detailed observation. This are negligible [16]. Intriguingly, for Δ < 0,asβ approaches

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(a)(b) (c) (e) 10 0 3 21 -20 2 32 1 -40 43 0

-60 Mean photon number 0 51015 (d) -80 Theory Experiment 20 -100 PSD (MHz-1) (MHz) 10 0.2 0 Detuning (MHz) -120 PSD (MHz-1) Detuning -10 1 -140 10 0 -20 -6 -160 10 123 51525 515 25 Coherent state amplitude

FIG. 2. (a) Energy-level diagram for the Kerr nonlinear resonator with each additional photon reducing the transition frequency by χ. (b) Transient PSD measurements for coherent states with different amplitudes α. The curve plotted on the right is the PSD corresponding to α indicated by the white dashed line. (c) Mean photon number hnˆi at a steady state versus the amplitude of the parametric driving at Δ=2π ¼ 25.3 MHz. The quantum prediction (solid line) agrees well with the experimental data. The classical model predicts two thresholds and hysteretic behavior (dashed line), corresponding to transitions between the monostable, tristable, and bistable regimes of the KPO [15]. (d) Steady-state PSD (logarithmic scale) for different driving amplitudes at Δ=2π ¼ 11.2 MHz. The spectrum goes from a double-peaked shape at small β to a single narrow peak in the self-oscillation regime, passing through a multipeaked spectrum at intermediate values of β. The white dot indicates the approximate onset of self-oscillation. The spacing between the two peaks in the weak drive limit is 2Δ, as explained by the two-photon emission process shown in diagram (e).

(a) (b) (d) Preparation Tomography 3 Amplitude Phase 0

Transient PSD -3 measurement Experiment (c) 3 0.2

0

Frequency -3 Theory -0.2 -3 0 3 -30 3 -30 3 -3 0 3 Displacements

FIG. 3. Schematic of the tomography and time evolution of a coherent state under Kerr nonlinearity. The tomography pulses are (a) delayed by τ from the state preparation pulse and (b) swept in phase space with two different amplitudes, which corresponds to displacements with 16 different phases and two different amplitudes of about 0.5 and 1.0. (c) An example tomography data set for a coherent state (τ ¼ 0). (d) Time evolution of the system’s Wigner function WðαÞ at four different time slices τ, starting from a coherent state at τ ¼ 0. The distortion in phase space is caused by the Kerr nonlinearity and is reproduced by simulations. The fidelities of the reconstructed states compared with simulated ones are 0.93, 0.92, 0.90, and 0.89, respectively. and exceeds χ, the energy eigenstate adiabatically con- begin with the resonator in the vacuum state with β ¼ 0. nected to the vacuum state begins to closely approximate We increase β slowly so the system follows the eigenstate the even-parity Schrödinger’s cat state. We set the pump of the Hamiltonian. Numerical simulations suggest that, detuning in the experiment to Δ=2π ¼ −6.7 MHz and given our system’s parameters,

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πt β t β sin2 ; 0 ≤ t ≤ t ; 2 approximation, the generated states after the short free ð Þ¼ max 2t max ð Þ ˆ † † † max evolution generated by H0 ¼ Δaˆ aˆ − χaˆ aˆ aˆ a=ˆ 2 are with t 22 ns [Fig. 4(c)] is a reasonable parametric −iHˆ t max ¼ jψðαÞi ∝ e 0 d ðjαiþj− αiÞ; ð3Þ driving profile for preparing a cat state. The length of this signal is much shorter than the cavity decay time 1=κ ≈ 150 ns but long enough to ensure approximately with α ¼ 0.64, 0.88, 1.08, and 1.2 for the data shown in adiabatic evolution. We perform the experiment ramping to Fig. 4(d). The largest of these corresponds to a 4jαj2 ¼ 5.8 β ’ β ≫ χ different values of max and verify the result with the state photon Schrödinger scatstate[22,28].Inthe max tomography procedure. The results are compared to the regime, the relevant eigenstate exponentially approaches Wigner functions found theoretically [Fig. 4(d)]. In the a cat state of size 8β=χ due to the double-well shape simulations, the rotation angle and the drive conversion of the system’s effective potential [Fig. 4(c)](see factor β=V are the only fit parameters and are common to Appendix C for more details). For very large cat states, all four data sets. As described above, we again assume a imperfections in the tomography process that grow with short (td ¼ 2.5 ns) period of free evolution between the end the number of photons in the analyzed state prevent its of state preparation and the start of our tomography, which faithful reconstruction. In comparison to other schemes causes a small distortion (due to χ) of the reconstructed for cat-state generation [22,28–31], our method is sig- state with respect to the eigenstate of the driven system. In nificantly more hardware efficient as it requires only a β ∼ χ our experiment, we ramp the drive up to max , which is resonator with one input and one output line for both state necessary to see the emergence of the cat state. To a good generation and read-out.

(a) (d) Experiment Theory

3 3 0 2 -3

1 3 Mean photon number 0 0 0100 200 300 400 2000 Driving time (ns) -3 (b) 0.2 0.2 3 3 3 0 0 0 -3 -3 -0.2 -0.2 -30 3 -30 3 -30 3 -3 0 3 -3 (c) 3 0.2

0

-3 Drive -0.2 -3 0 3 -3 0 3 Time

FIG. 4. Transient-state dynamics of the parametrically driven KPO. (a) Time evolution of the mean photon number up to 400 ns at Δ=2π ¼ 24.6 MHz for β=2π ¼ 3.5 MHz (red line), β=2π ¼ 5.8 MHz (yellow line), and β=2π ¼ 11.5 MHz (blue line). (b) Both state reconstructions at t ¼ 20 ns (transient state) and t ¼ 2000 ns (steady state), showing good agreement with theory (left is experiment; right is theory) with fidelities of 0.93 and 0.94, respectively. (c) Pulse amplitude profile βðtÞ for cat-state generation. A pulse length of 22 ns is chosen from simulation. Ramping β up from 0 to βmax drives the resonator from the vacuum state into an oscillating cat state. W α β Δ=2π −6 7 (d) Wigner functions ð Þ of generated cat states, reconstructed from tomography, for different max at ¼ . MHz. These measurement results match closely (fidelities 0.98, 0.95, 0.92, 0.89) with simulations.

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III. CONCLUSIONS AND OUTLOOK (a) (b) (c) The parametric oscillator is one of the paradigmatic systems in quantum optics and has found an enormous range of applications over the years. We have experi- mentally demonstrated the few-photon quantum dynam- ics of a parametric oscillator by introducing a large Kerr nonlinearity in the microwave frequency regime. We show that the system can adiabatically generate cat states of five to six photons and have developed a tomography 50 µm 5 µm method suited for the characterization of its state and dynamics. FIG. 5. (a) False colored micrograph of the Kerr parametric Our work demonstrates that nontrivial quantum states oscillator. The coupling capacitor and flux bias line are shown in can be engineered and characterized with nearly minimal blue and red, respectively. The capacitor shunting the SQUID hardware complexity. The quantum coherence and hard- array is shaped in an unconventional way in order to accom- ware efficiency of the system bode well for prospective modate an array of nanomechanical resonators introduced in later applications of these devices in emerging quantum infor- devices [36]. (b) SEM image of the SQUID array. (c) Circuit mation processing and optimization architectures [8–12]. diagram of the Kerr parametric oscillator, showing the SQUID array and shunting capacitor (black), coupling capacitor (blue) For instance, parametric preparation and manipulation of and transmission line (grey), and flux bias line (red). cat states may be a convenient alternative to qubit-based approaches [32] in continuous variable encodings of quantum information. In the context of quantum optimi- 1. Device parameters zation, networks of coupled KPOs could provide a useful platform to study quantum counterparts of classical Ising Table I gives the device parameters for the Kerr para- machines [8] and explore the possibility of speed-up in metric oscillator. Here, the maximum resonator frequency ω quantum annealing algorithms [33]. c;max is determinedpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from a fit to the flux-bias tuning curve ω Φ ω πΦ =Φ Φ cð eÞ¼ c;max j cosð e 0Þj, where e is the exter- Φ ACKNOWLEDGMENTS nally applied magnetic flux and 0 is the flux quantum. The maximum Josephson energy of each of the N ¼ 10 This work was supported by the U.S. Department of E SQUIDs, denoted by J;max is determined from normal- Energy through Grant No. DE-SC0019174. Part of this E ω state resistance measurements. Together, J;max and c;max work was performed at the Stanford Nano Shared Facilities are used to extract the resonator charging energy EC, (SNSF), supported by the National Science Foundation which closely matches predictions from finite-element under Grant No. ECCS-1542152, and the Stanford capacitance simulations. The resonator intrinsic and extrin- Nanofabrication Facility (SNF). sic decay rates are denoted as κi and κe, respectively. Here, χ is the resonator frequency shift per photon, APPENDIX A: DEVICE FABRICATION determined from the peak-to-peak splitting in transient PSD measurements of a coherent state. The maximum The device, shown in Fig. 5(a), was fabricated using a parametric drive amplitude β=2π used in the experiment is five-mask lithography process on a 500-μm high-resistivity about 20 MHz. Larger β are possible but lead to large states (>10 kΩ · cm) Si substrate. First, the aluminum ground that cannot be faithfully reconstructed by the employed planes and feed lines are defined in photolithography tomography procedure due to the nonlinearity of the using a liftoff process. Next, palladium marks are added system. in preparation for aligning subsequent electron-beam (e-beam) lithography masks. The SQUID array shown in Fig. 5(b) is fabricated using a Dolan-bridge double-angle technique for growing Al=AlOx=Al junctions via in situ TABLE I. Device parameters for the Kerr parametric oscillator. oxidation [34,35]. After junction growth, the resonator capacitor is defined using e-beam lithography; narrow Parameter Value wires and an unconventional capacitor design were chosen N 10 ω =2π to accommodate an array of nanomechanical resonators c;max 8.35 GHz E =h introduced in later devices [36] and are not essential to the C 1.053 GHz E =h J;max 82.79 GHz design. Finally, a superconducting connection between the κ =2π SQUID array and capacitor leads is formed using a bandage i 200 kHz κe=2π 900 kHz process [37]. An equivalent circuit diagram of the device is χ=2π 17.3 MHz shown in Fig. 5(c).

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APPENDIX B: EXPERIMENTAL SETUP we can use instead of sending the displacement pulses through the resonator input/output line. This way, we avoid 1. Up-conversion board saturation of the measurement setup by the reflection of the We use two separate channels of a Tektronix series 5200 strong resonant pulse. The input/output line is not used for arbitrary waveform generator (AWG) to synthesize the driving the system, except in initial characterization mea- temporal profile of the pulses at an intermediate frequency surements of the resonator frequency using a vector net- around 4 GHz and then further up-convert them to the work analyzer. desired frequencies close to the first (for the displacement) and second (for the parametric drive) harmonic of the 2. Phase locking resonator frequency using single sideband mixers (Fig. 6). Both pulses are then combined together and sent to the For a state prepared by parametric driving, the orienta- sample through the flux line. Thanks to a weak but nonzero tion of the reconstructed quasiprobability distribution in direct coupling of the flux line to the resonator, it can phase space is determined by the difference between the effectively double as a weakly coupled charge line, which phase ϕd of the tomography pulses and the phase ϕp=2 of

AWG AWG ADC ADC channel 1 LO channel 2 channel B channel A LO

4.4 GHz 4 GHz 225 MHz 222255 MHz

I I d r

L L a +50 dB +2+200 ddBB o R R b ion s r

12-18 GHz e 4-8 GHz 225 MHz 225 MHz onv Phase locking c

Up-conversion board +14 dB IIown- I Down-conversion board L D L

RRR

+12 dB +40 ddBB +24 dB

44-8-8 GGHzHz -20 dB

Room temperature

HEMT Cryostat -33 dB

TWPA Flux Output Sample Line Line

Low-pass Band-pass Attenuator Splitter Mixer dc block Directional filter filter coupler

FIG. 6. Schematic diagram of the experimental setup.

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ω =2 ˆ the p subharmonic of the parametric driving. These EJ þ δEJ cos ωpt. After Taylor expanding cosðϕ=NÞ to phases in turn depend on the absolute phases of the up- fourth order, we get conversion local oscillators. Slow changes of this relative       phase due to phase drifts of the signal generators would 1 ϕˆ 2 1 ϕˆ 4 H=ˆ ℏ 4E nˆ2 − NE 1 − lead to gradually accumulating errors over long measure- ¼ C J 2 N þ 24 N þ ments. To mitigate these errors, we monitor the phase over     1 ϕˆ 2 the course of the measurement using a second down- − NδEJ 1 − þ cos ωpt: ðC2Þ conversion board. Here, we somewhat unconventionally 2 N feed both the parametric driving pulse at ωp and the The quadratic time-independent part of the Hamiltonian displacement pulse at ωd into the RF port of the mixer can be diagonalized by defining while the LO port is 50-ohm terminated (Fig. 6), relying on intermodulation to produce a signal at ωp − 2ωd.By † nˆ ¼ −in0ðaˆ − aˆ Þ; measuring this signal, we can determine the phase differ- ϕˆ ϕ aˆ aˆ† ; ence ϕ ¼ ϕp − 2ϕd, which needs to be constant to ensure ¼ 0ð þ Þ ðC3Þ correct performance of the tomography measurements even pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 E =32NE ϕ2 2NE =E with very long acquisition times. We observe a slow drift of where 0 ¼ J C and 0 ¼ C J are the ϕ of about 1 radian per hour, which we then eliminate by zero-point fluctuations. We also drop c-valued terms in the measuring ϕ in approximately 1-minute intervals and expression above and get appropriately adjusting the phase of the up-conversion LO. E H=ˆ ℏ ω ð0Þaˆ†aˆ − C aˆ aˆ† 4 ¼ c 12N2 ð þ Þ 3. Down-conversion and digitization of the signal δE ω ð0Þ J c aˆ aˆ† 2 ω t; The signal emitted by the resonator at about 7 GHz is þ 4E ð þ Þ cos p ðC4Þ first amplified by a traveling-wave parametric amplifier J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (TWPA) [2] and a low- high-electron-mobility tran- ð0Þ where ωc ¼ 8ECEJ=N. We then transform the sistor (HEMT) amplifier inside the cryostat. At room Hamiltonian into a rotating frame at the frequency ωp=2 temperature, it is further amplified and then converted to an intermediate frequency of 125 MHz by a single- and perform a rotating-wave approximation, assuming that ω ð0Þ − ω =2 E =12N2 δE ω ð0Þ=4E sideband mixer (Fig. 6). The resulting signal is then j c p j, C , and J c J are all much recorded by a digitizer card (AlazarTech ATS9350) with smaller than 2ωp. After normal ordering the resulting a 12-bit resolution and a sampling rate of 500 MS=s. The expression, we obtain acquired data are first saved in the on-board memory buffer χ and then transferred to GPU for real-time data processing. H=ˆ ℏ Δaˆ†aˆ − aˆ†aˆ†aˆ aˆ β aˆ2 aˆ†2 ; ¼ 2 þ ð þ Þ ðC5Þ In the transient PSD measurements, 1-μs waveforms are collected and Fourier transformed, and their squared 2 where we have defined the Kerr nonlinearity χ ¼ EC=N , absolute values are averaged. The waveform length is β ω ð0ÞδE =8E chosen to be approximately 6=κ, long enough to make the parametric drive strength ¼ c J J, the ω ω ð0Þ − χ sure the resonator has relaxed to its ground state with high dressed resonator frequency c ¼ c , and the detun- Δ ω − ω =2 probability. ing ¼ c p .

APPENDIX C: HAMILTONIAN OF A 1. Effective potential PARAMETRIC OSCILLATOR An intuitive way to understand some aspects of the The Hamiltonian of the SQUID array resonator includ- dynamics of this system is to define the following effective ing the parametric driving is potential [16] in phase space: χ ϕˆ V α α Hˆ α Δ α 2 − α 4 β α2 α2 H=ˆ ℏ 4E nˆ2 − NE (Φ t ) ; ð Þ¼h j j i¼ j j 2 j j þ ð þ Þ ¼ C J ð Þ cos N ðC1Þ   χ ¼jαj2 Δ − jαj2 þ 2β cos 2θ ; ðC6Þ where nˆ is the number of Cooper pairs and ϕˆ the overall 2 phase across the junction array. Here, EC is the resonator’s charging energy, N is the number of SQUIDs in the array, where α ¼jαjeiθ. In the regime 2β > jΔj, the effective and EJ is the Josephson energy for a single SQUID in the potential has two symmetric local maxima (corresponding array, which depends on the external flux ΦðtÞ. The flux to classical stationary points of the system) at jαj > 0 and is harmonically modulated around its mean value with a θ ∈ f0; πg. Therefore, in this case, the effective potential small amplitude such that EJ(ΦðtÞ) can be approximated as has the shape of an inverted double well (Fig. 7). When its

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(a) (b)

Eigenenergy Drive

FIG. 7. Illustration of the effective potential VðαÞ plot. (a) Po- β 0 tential at ¼ (below threshold) where there is only one global FIG. 8. Eigenenergies of the Kerr parametric oscillator as a β ≫ χ maximum. (b) The regime (far above threshold) where the function of the driving strength. The plot is made in the classical potential has two symmetrically placed local maxima. limit χ ≪ Δ, where the oscillation threshold is at β ¼jΔj=2. The undriven eigenenergies are decreasing due to our use of a rotating frame at ωp=2 > ωc. The pairs of neighboring energy levels merge together and become nearly degenerate at large β where maxima α are sufficiently well separated, i.e., β is large, the degenerate subspaces are spanned by displaced Fock states. the potential can be approximated by a quadratic function When we adiabatically increase β from 0 to above threshold, the α near . Therefore, the eigenstates of the system are close vacuum state will evolve into an even cat state since parametric to Fock states displaced by α, as described in more driving preserves parity. detail below. Note that V is not the energy of the resonator in the lab frame but rather an effective potential describing the system subspaces is close to a . In other Dˆ † α Hˆ Dˆ α ≈ Dˆ † −α Hˆ Dˆ −α ≈−2χα2aˆ†aˆ in a rotating frame. In particular, stable equilibrium states of words, ð Þ ð Þ ð Þ ð Þ in α ≫ 1 α ≈ the classical dynamics are still given by stationary points of ptheffiffiffiffiffiffiffiffiffiffi limit of . Here, we have used the relation V but do not need to be local minima. 2β=χ and only kept the highest-order terms in α. This method confirms that displaced Fock states indeed ˆ 2. Eigenstates of the Hamiltonian approximate the pairwise degenerate eigenstates of H in the large α limit. Understanding the structure of the eigenvalues and It follows from the adiabatic theorem that if we prepare eigenstates of the Hamiltonian at different β helpsussee the system in the vacuum state and adiabatically increase β, how adiabatic ramping up of the drive amplitude can lead the system will follow the eigenstate and end up within the to Schrödinger’s-cat state generation [16].Atβ ¼ 0,the corresponding degenerate subspace, as determined by eigenstates of the Hamiltonian are simply all Fock states the order of eigenenergies of the Fock states at β ¼ 0, fjnig. As we increase β, two adjacent energy levels get which in turn depends on Δ and χ.ForΔ < 0, j0i has the closer and eventually merge together at very large β highest eigenenergy at β 0 and evolves into Dˆ α 0 (Fig. 8). In the limit where β=χ ≫ 1, as suggested by the ¼ f ð Þj ig at large β. Since parametric driving preserves the parity of effective potential argument above, the eigenstates of the the state, the final state is an even cat state jαiþj− αi. The Hamiltonian form many two-dimensional nearly degen- case Δ > 0 is more complicated since j0i may not have the erate subspaces spanned by Dˆ α n for each n, pffiffiffiffiffiffiffiffiffiffi f ð Þj ig highest eigenenergy at β ¼ 0 and can thus evolve into some α 2β=χ where ¼ are the locations of the local ½Dˆ ðαÞþDˆ ð−αÞjni, where n ≠ 0. Therefore, for cat-state maxima. To show this result, we first note that the ˆ generation, having a negative detuning is helpful since that coupling between states of the form DðþαÞjni and gives a larger energy gap between j0i and all other higher ˆ ˆ Dð−αÞjmi under the Hamiltonian H decreases exponen- Fock states and thus allows a faster adiabatic tuning of β. tially with jαj2, i.e., Driving with an appropriately chosen positive detuning or nonadiabatic drive variations could, on the other hand, be † −2α2 useful for generating displaced Fock states or their super- hmjDˆ ð−αÞHˆ Dˆ ðαÞjni ∼ e ∀ m; n: ðC7Þ positions [16].

Intuitively, this follows from the large separation of the APPENDIX D: DISPLACEMENT two potential wells. The Hilbert space therefore effec- PULSE CALIBRATION tively decomposes into two nearly decoupled subspaces consisting of states localized around þα and −α.Next, The displacement pulses used in the experiments are we observe that the Hamiltonian within each of these 1-ns short pulses created by an arbitrary waveform

021049-9 ZHAOYOU WANG et al. PHYS. REV. X 9, 021049 (2019) generator. To calibrate them, we apply pulses with (a) different amplitudes fVig to the vacuum state of the resonator and measure the transient PSDs for the gen- erated states. In order to simplify the analysis of the PSD ˜ measurements, we preprocess the raw data fSðω; ViÞg by integrating over frequency bins centered around each of the individual transition peaks, thus effectively reducing the dimensionality of the analyzed data to n × m,where n is the number of bins and m the number of different (b) pulse amplitudes,

Z ˜ ˜ SjðViÞ¼ Sðω; ViÞdω: ðD1Þ bin j

The number of bins that can be usefully analyzed is limited by the increasing overlaps between the peaks FIG. 9. Schematic of pulse calibration and tomography. corresponding to higher transitions with larger linewidths. (a) Pulse calibration is done by minimizing the loss function Lðk; c⃗Þ, where k is the scaling factor that converts the voltage of When multiple transitions contribute to the same bin, the the pulse into the amplitude of displacement in phase space and c⃗ assumptions we use in our model to arrive at Eq. (D9) fail, are the coefficients for the linear transformation that maps a and a more complex model needs to be used. In most of our quantum state into its transient PSD. (b) The loss function for measurements, we use n ¼ 4 to 5 bins. state tomography contains the contributions from the distance The calibration is done under the following assumptions: between measured and predicted transient PSDs for an unknown (i) The state generated by a single pulse with voltage Vi state ρˆ after each displacement. is a coherent state jαii. (ii) αi depends linearly on the voltage Vi, i.e., αi ¼ kVi, ZZ∞ where k is a single fit parameter common to all 0 † 0 −iωðt0−tÞ Sðω; ρˆ0Þ¼ dtdt κhaˆ ðt ÞaˆðtÞie pulses. ˜ 0 (iii) The bin powers fSjðViÞg calculated from the mea- sured PSDs are related to the theoretical predictions ZZ∞ † −iωτ fSjðjαiiÞg by a gain factor cj, which may, in ¼ 2Re dtdτκhaˆ ðt þ τÞaˆðtÞie ; ðD3Þ principle, be different for each bin j. 0 The calibration parameters k and c⃗¼ðc1; …;cnÞ are obtained by minimizing the loss function [Fig. 9(a)] which is normalized so that Z∞ 1 Xm Xn dωS ω; ρˆ aˆ† 0 aˆ 0 : ˜ 2 ð 0Þ¼h ð Þ ð Þi ðD4Þ Lðk; c⃗Þ¼ kSjðViÞ − cjSjðjkViiÞk : ðD2Þ 2π −∞ i¼1 j¼1 From the quantum regression theorem, we can rewrite the two-time correlation function as For a given k, finding optimal c⃗reduces to a simple linear k fitting problem, and the value of is then calculated by † † Lˆτ Lˆt haˆ ðt þ τÞaˆðtÞi ¼ Trfaˆ e aeˆ ρˆ0g; ðD5Þ minimizing LðkÞ¼minc⃗Lðk; c⃗Þ. To evaluate the loss function above, we need to calculate Lˆ the theoretically expected total power Sjðρˆ0Þ in each where is the Liouvillian for the nonlinear resonator with the Hamiltonian transient PSD peak for a given state ρˆ0. The result represented by Eq. (D9) is outlined in the main text, and χ Hˆ − aˆ†aˆ†aˆ aˆ its full derivation is given below. ¼ 2 ðD6Þ and energy decay rate κ. 1. Analytical formula for the transient PSD By solving the master equation and Fourier transforming The transient PSD for an initial state ρˆ0 is given the two-time correlation function, the analytical result for formally by the transient PSD is

021049-10 QUANTUM DYNAMICS OF A FEW-PHOTON … PHYS. REV. X 9, 021049 (2019)     X∞ Xn−1 1 nX−j−1 n − j − 1 uk Sðω; ρˆ0Þ¼2Re pn ; ðD7Þ 1 u n−j−1 k i ω kχ 2 k 1 κ=2 n¼0 j¼0 ð þ Þ k¼0 ð þ Þþ ð þ Þ

where u ¼ iχ=κ and pn ¼hnjρˆ0jni. Notably, Sðω; ρˆ0Þ matrix ρˆ can be estimated by minimizing the difference depends only on the diagonal part of ρˆ0, and this depend- between the predicted transient PSDs and the measured ence is linear. ones, which is expressed by the loss function [Fig. 9(b)] Our system satisfies juj¼χ=κ ≫ 1. In this regime, the transient PSD takes the form of a sum of relatively well- Xm Xn L ρˆ S˜ V − c S Dˆ kV ρˆDˆ † kV separated peaks. The last equation can be approximated as ð Þ¼ k jð iÞ j jð ð iÞ ð iÞÞk ðE1Þ i¼1 j¼1 X∞ Xn−1 ð2j þ 1Þκ ρˆ 1 Sðω; ρˆ0Þ ≈ pn ; ðD8Þ under the linear constraint Trð Þ¼ and under the con- ω jχ 2 2j 1 κ=2 2 ρˆ n¼0 j¼0 ð þ Þ þ½ð þ Þ dition that is positive semidefinite. Notice that both the displacement and the map from a density matrix to a where the transient PSD is decomposed into a sum of transient PSD are linear transformations, and therefore this Lorentzians with different linewidths and different center minimization problem is convex and can be efficiently frequencies corresponding to the different peaks we mea- solved by the MATLAB package CVX [38,39]. sured. Therefore, the total power in the jth peak is expected to be 1. Parity constraint X∞ Xj−1 The form of the master equation of the parametrically Sjðρˆ0Þ¼ hnjρˆ0jni¼1 − hnjρˆ0jni: ðD9Þ driven system implies that all states that can be generated n¼j n¼0 from a vacuum state are mixtures of states with even and odd parity. To see this, we only need to observe that the Hamiltonian conserves parity and the collapse operator pffiffiffi κaˆ flips the parity of a state. APPENDIX E: STATE TOMOGRAPHY When reconstructing states prepared by parametric To reconstruct an unknown quantum state ρˆ, we displace driving, we use this condition as an additional constraint it in phase space [Figs. 3(a)–3(b)] using short calibrated on the unknown density matrix ρˆ, requiring that pulses with complex voltages fVig (including phase) and ˆ ˆ† then measure the transient PSDs and calculate the corre- P ρˆ P ¼ ρˆ; ðE2Þ ˜ sponding integrated powers SjðViÞ for each of the bins j ¼ ˆ iπaˆ†aˆ 1; …;n after each displacement i ¼ 1; …;m. The density where P ¼ e is the parity operator. We justify this assumption by verifying the corresponding symmetry of the transient PSDs measured in the tomography process 4 under a rotation of the applied displacement by π. For a state ρˆ satisfying Eq. (E2), we expect that S˜ðω; þαÞ¼ S˜ ω; −α , and we check this by plotting the difference 3 PSD (arb. units) ð Þ S˜ðω; þαÞ − S˜ðω; −αÞ and observing that it is negligible (a) (b) when compared with the PSDs S˜ ω; α themselves 2 ð Þ (Fig. 10). Frequency 1 APPENDIX F: DATA FITTING 0 χ (c) (d) Some parameters of the system such as its nonlinearity and linewidth κ do not change significantly among different Displacements experiments, and we therefore assume constant values, which are obtained from initial characterization measure- FIG. 10. Justification of the parity constraint for tomography S˜ ω; α ments. Other parameters like the detuning Δ and the of parametrically prepared states. (a) Raw data f ð Þg of β [Fig. 4(b), left], including transient PSDs for all displacements α. parametric driving amplitude vary between experiments, (b) The difference of the raw data under parity transformation and therefore their values are determined separately in each fS˜ðω; þαÞ − S˜ðω; −αÞg. Panels (c) and (d) are the same as panels instance, either directly from the settings of the experiment (a) and (b) but for the tomography data set in [Fig. 4(b), right]. or by fitting.

021049-11 ZHAOYOU WANG et al. PHYS. REV. X 9, 021049 (2019)

The nonlinearity χ=2π ¼ 17.3 MHz was calculated from possible small time interval td of free evolution between the transient PSD measurements as the mean spacing between preparation pulse and the tomography pulses. This param- adjacent peaks. To fit the time evolution of the photon eter is to account for a potential delay between the two number [Fig. 4(a)], we introduce the detuning Δ, the pulses that are generated by different channels of the AWG linewidth κ, and the drive conversion factor β=V that and processed by separate up-conversion boards. relates the voltage amplitude V applied to the flux line For the tomography measurements of the steady state to the parametric driving amplitude β as fit parameters; we under parametric driving [Fig. 4(b), right], both the detun- obtain Δ=2π ¼ 24.6 MHz and κ=2π ¼ 1.1 MHz by min- ing Δ and the parametric driving amplitude β have been imizing the L2 distance between the simulation results and fixed through fitting to the time evolution of the photon the measured data. The value of Δ=2π found by fitting is number [Fig. 4(a)]. Therefore, the only remaining fit very close to the value 24 MHz set in the experiment, and parameters are the phase of the state and the delay time the small difference is likely due to slow magnetic flux td. By maximizing the fidelity between the reconstructed noise, which causes variations in the resonator frequency. density matrix and the simulation result, we get td ¼ The linewidth is also consistent with direct VNA measure- 2.5 ns, which is then kept fixed for all other tomography ments, which give values around 1 MHz, slightly depend- measurements with parametric driving. Consequently, for ing on the resonator frequency. Since none of the results the transient state at 20 ns [Fig. 4(b), left], the only fit in this work is very sensitive to the exact value of the parameter is its phase θ. linewidth, we fix κ=2π to be 1.1 MHz in all subsequent For the tomography measurements of the adiabatically theory fits. prepared cat states, the phase θ is again unknown but The process of fitting the steady-state mean photon should be the same for all four states. Therefore, the phase θ number n¯ [Fig. 2(c)] is similar to the case of the time- and the drive conversion factor β=V are the only fit dependent n¯ measurement described above [Fig. 4(a)] parameters for all four data sets. We again assume that except that κ is fixed and the detuning resulting in the the final state is distorted by a short period of free evolution best fit is Δ=2π ¼ 25.3 MHz. For fitting the PSDs at a whose length we fix at td ¼ 2.5 ns based on previous steady state [Fig. 2(d)], we fix Δ=2π to the value 11.2 MHz measurements. set in the experiment. The only fit parameter is the drive conversion factor β=V. The tomography measurement of the freely evolving state [Fig. 3(d)] has two fit parameters: the size and phase [1] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. of the coherent state at τ ¼ 0. In principle, the observed θ Vale, and K. W. Lehnert, Amplification and Squeezing of phase of the state should be easily predictable since it Quantum Noise with a Tunable Josephson Metamaterial, only depends on the relative phase between the preparation Nat. Phys. 4, 929 (2008). pulse and the tomography pulse, both of which are [2] C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. generated by the AWG and proceed along the same path Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, A Near- through the up-conversion chain. A calculation based on Quantum-Limited Josephson Traveling-Wave Parametric the experimental settings used gives a predicted phase of Amplifier, Science 350, 307 (2015). θ ≈ 1.19π. The size of the state α can also be estimated [3] P. Krantz, A. Bengtsson, M. Simoen, S. Gustavsson, V. from the pulse calibration parameter k, which gives α ≈ 1.5. Shumeiko, W. D. Oliver, C. M. Wilson, P. Delsing, and J. Treating both θ and α as unknown fit parameters, we get Bylander, Single-Shot Read-out of a Superconducting Qubit Using a Josephson Parametric Oscillator, Nat. Commun. 7, θ ≈ 1.24π and α ≈ 1.0. In this fitting, to achieve a simulta- 11417 (2016). neous match to the different states at each of the different [4] S. Harris, Tunable Optical Parametric Oscillators, Proc. τ evolution times , we choose the objective function to be IEEE 57, 2096 (1969). the geometric mean of the fidelities between each measured [5] L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. state and the corresponding theoretical prediction. Bosenberg, and J. W. Pierce, Quasi-Phase-Matched Optical For states prepared by parametric driving, their phase θ Parametric Oscillators in Bulk Periodically Poled LiNbO3, depends on the absolute phase of the signal generator. J. Opt. Soc. Am. B 12, 2102 (1995). Through the feedback loop described in Appendix B2,we [6] B. Yurke, P. G. Kaminsky, R. E. Miller, E. A. Whittaker, stabilize the phase over the measurement time at a fixed A. D. Smith, A. H. Silver, and R. W. Simon, Observation value. This value, in principle, depends on the frequency of 4.2-K Equilibrium-Noise Squeezing via a Josephson- Parametric Amplifier, Phys. Rev. Lett. 60, 764 (1988). of the signal in a complex way due to the variation of the ’ [7] R. Schnabel, Squeezed States of Light and Their Applica- system s S-parameters with frequency. Since the different tions in Interferometers, Phys. Rep. 684, 1 (2017). tomography measurements are mostly performed at differ- [8] P. L. Mcmahon, P. L. Mcmahon, A. Marandi, Y. Haribara, ent resonator frequencies, we treat the phase of the state θ at R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. each of these frequencies as a fit parameter. Another fit Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, parameter we introduce for these measurements is the H. Mabuchi, and Y. Yamamoto, A Fully-Programmable

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