Electron shelving of a superconducting artificial atom

Nathanael Cottet,1, 2, ∗ Haonan Xiong,1, ∗ Long Nguyen,1 Yen-Hsiang Lin,1 and Vladimir Manucharyan1

1Physics Department, Joint Quantum Institute, and Quantum Materials Center,University of Maryland, College Park, MD 20742

2Universite Lyon, ENS de Lyon, Universite Claude Bernard Lyon 1,CNRS, Laboratoire de Physique, F-69342 Lyon, France

(Dated: August 7, 2020)

Interfacing stationary qubits with propagatingphotons is a fundamental problem in quantumtechnology1. Cavity quantum electrodynamics2

(CQED) invokes a mediator degree of freedomin the form of a far-detuned cavity mode, theadaptation of which to superconducting circuits(cQED) proved remarkably fruitful3–5. The cav-ity both blocks the qubit emission and it en-ables a dispersive readout of the qubit state.Yet, a more direct (cavityless) interface is pos-sible with atomic clocks, in which an orbitalcycling transition can scatter photons depend-ing on the state of a hyperfine or quadrupolequbit transition6. Originally termed “electronshelving”7, such a conditional fluorescence phe-nomenon is the cornerstone of many quantuminformation platforms, including trapped ions8,solid state defects9,10, and semiconductor quan-tum dots11. Here we apply the shelving idea tocircuit atoms and demonstrate a conditional flu-orescence readout of fluxonium qubit12 placed in-side a matched one-dimensional waveguide13. Cy-cling the non-computational transition betweenground and third excited states produces a mi-crowave photon every 91 ns conditioned on thequbit ground state, while the qubit coherencetime exceeds 50 µs. The readout has a built-in quantum non-demolition property14, allowingover 100 fluorescence cycles in agreement with afour-level optical pumping model. Our result in-troduces a resource-efficient alternative to cQED.It also adds a state-of-the-art quantum memoryto the growing toolbox of waveguide QED15–18.

An important first step towards eliminating the cav-ity was accomplished with the observation of resonantfluorescence by a flux qubit directly coupled to the one-dimensional vacuum of a matched 50 Ω transmissionline13. Resonance fluorescence was also explored withtransmons in relation to several fundamental topics inquantum optics19–22. On one hand, circuit atoms placedinside microwave transmission lines routinely reach anearly unit light-matter coupling efficiency, i.e. the rateof spontaneous emission into the waveguide largely sur-passes the non-radiative decoherence. Achieving suchregime is extremely difficult in conventional quantum op-tics. On the other hand, the lack of diversity in transi-tion frequencies and selection rules prevented a full-blownquantum control of transmons and flux qubits outside the

“qubit-in-a-cavity” paradigm of cQED23.

Our cavityless experiment is made possible by a uniquecombination of strong anharmonicity and long coherencetime of modern fluxonium qubits24,25. Fluxonium con-sists of a weak Josephson junction shunted by an appro-priately high-value inductance and a capacitance, herein the form of bow-tie antennas (Fig. 1a). When cooled-down near 10 mK in a dilution refrigerator the circuitdynamics can be characterized by a single collective de-gree of freedom, the superconducting phase-difference φacross the weak junction. At the half-integer magneticflux through the loop, φext = π, the variable φ be-haves like the coordinate of a particle inside a symmetricdouble-well potential and the spectrum is insensitive toflux noise in the first order (Fig. 1b). Tunneling of φacross the barrier forms a qubit transition between thetwo lowest energy eigenstates |0〉 and |1〉 at frequencyω01 = 2π × 1.152 GHz. The non-computational states|2〉, |3〉, etc., correspond to one-dimensional orbitals ofφ at energies above the barrier. In particular, transition|0〉-|3〉 at frequency ω03 = 2π × 6.544 GHz suits well forfluorescence cycling26. Note that such transition is al-lowed, unlike in transmons, thanks to the absence of aharmonic ladder selection rule in our non-linear circuit.

Shelving requirements were met by combining the par-ity selection rules at φext = π with a properly engineeredmicrowave environment. We mount the device chip insidea rectangular enclosure with a specially designed singleinput/output port (Fig. 1c). The port can be viewed asa broadband coaxial-to-waveguide adapter27. Above thecut-off frequency, given by the enclosure dimensions, theport coupling is near unity in a few-GHz band; below thecut-off, the port coupling drops rapidly with frequency(Fig. 1d, solid line). Therefore, the radiative lifetime ofstate |1〉 can be orders of magnitude longer than that ofstate |3〉, even though transitions |0〉-|1〉 and |0〉-|3〉 havecomparable dipoles – matrix elements of the Cooper pairnumber operator −i∂φ (Fig. 1d). Moreover, radiative de-cays |3〉 → |1〉 and |3〉 → |2〉 → |1〉 are inhibited by theparity selection rule and the weak port coupling, respec-tively. As a result, resonant driving of |0〉-|3〉 transitionwould continuously generate fluorescence if and only ifthe atom (and hence the qubit) is initially in state |0〉.Despite a periodic transit of population through the non-computational state |3〉, the atom rapidly relaxes back tostate |0〉 on switching off the drive. Likewise, the atomstarting in state |1〉 remains unperturbed by the far off-resonant readout drive and hence our conditional fluores-

arX

iv:2

008.

0242

3v1

[co

nd-m

at.m

es-h

all]

6 A

ug 2

020

2

EJECEL

ext =

IF

|0i|1i

|2i

|3i

qubit

readout

|0+i|1i

|2+i

ext

(b) (c)

(d)

Frequency (GHz)

ext

|0i-|3

i|1i-|2

i|2i

-|3i

|0i-|2

i

|1i-|3

i

iff

qubit

Port coupling (dB)

50 µm

50 µm

5 µm

1

2

1

2

50 µm1 ! 2

|1i ! |2i|1i $ |2i

|0i|1i

|2i

|3i

|0i(a)

|0i-|

1i

Transition dipole

Tran

sition

dipo

le|hi

|@|ji

|

20

40

0

10 2 3 4 5 6 7 8

0.2

0.4

0.6 0

*

**

* **

0.0

FIG. 1. Cavityless circuit quantum electrodynamics. (a) Scanning electron microscope image of the fluxonium’s super-conducting loop, made of one weak Josephson junction and an array of stronger junctions. Top inset: fluxonium’s effectivethree elements circuit model. Bottom inset: the weak junction is connected to a capacitive bow-tie antenna (optical image).(b) Effective potential seen by the phase-difference φ across the weak junction (black), energy levels (horizontal black lines) andwavefunctions (gray) at the half-integer external flux through the loop, φext = π. The qubit levels are |0〉 and |1〉, the readouttransition is |0〉 − |3〉, and even transitions are dipole-forbidden. (c) Coaxial-to-waveguide adapter for broadband coupling ofthe on-chip circuit to a 50 Ω-cable. (d) Measured port coupling (solid black) vs. frequency and the calculated transition dipoles(colored stars). Note the absence of resonant modes in the adapter in the entire 0− 8 GHz frequency range.

cence readout has a built-in QND property.Following previous work13,15, we detect fluorescence

using a phase-sensitive microwave reflectometry setup.For the time being, let us neglect the possibility of tran-sitions outside the cycling manifold |0〉, |3〉, and assumethe environment to be at zero temperature. According tothe input/output theory, the complex amplitude of theoutgoing reflected wave αout is given by the sum of theincoming wave amplitude αin and the fluorescence fieldemitted by the atom28. For a drive near the readoutfrequency ω03, this relation takes the simple expressionαout = αin −

√Γ〈|0〉〈3|〉, where Γ is the rate of direct

radiative decay |3〉 → |0〉, and the expectation value istaken over the dissipative driven steady state. The flu-orescence contribution is zero if the atom is not initiallyin the ground state, which allows linking the complexreflection amplitude r = αout/αin to the ground statepopulation p0:

r = 1− 2p0 ×Γ2/2− i(ω − ω03)Γ

Γ2/2 + Ω2 + 2(ω − ω03)2(1)

with ω the drive frequency. Here the Rabi rateΩ = 2

√Γαin conveniently represents the readout drive

strength in units of [Hz]. The information about the stateof the atom is carried by the fluorescence signal, henceis proportional to Ω|1 − r(ω,Ω)|. It is maximal when

the drive amplitude and frequency obey Ω = Γ/√

2 andω = ω03, which reduces Eq. (1) to r = 1− p0. Note thatphase-sensitive fluorescence detection relies on the drive-induced coherence between states |0〉 and |3〉, rather thanon the population of state |3〉 in the more common caseof optical photodetection.

We begin our experiment by spectroscopically prob-

ing the readout transition (Fig. 2a,b). The measuredreflection amplitude r as a function of drive frequency ωand amplitude Ω fits well to the Eq. (1) model. Onlyfour adjustable parameters are used in the global fit:ω03 = 2π × 6.544 GHz, Γ = 2π × 1.74 MHz, the thermalequilibrium ground state population pth

0 = 0.78, and thescaling factor converting the value of Ω to the RF-sourceamplitude. We integrate the reflection signal during atime τm = 0.5 µs after a delay τw = 0.5 µs. The waittime is chosen so that it is much longer than the charac-teristic emission time 1/Γ = 91 ns but much shorter thanthe time scale of non-radiative processes in the system.As a result the atom is in the driven steady state whilefluorescence is being integrated. At low power, Ω Γ,the reflection amplitude r as a function of frequency be-comes a circle in the parametric IQ-plot, bounded by thevalues Re[r] = 1 for |ω−ω03| Γ and Re[r] = 1−2pth

0 atω = ω03. As stronger driving saturates the |0〉-|3〉 transi-tion, the circle deforms into an ellipse and progressivelyshrinks into a point at r = 1 for Ω Γ. This good agree-ment between spectroscopy data and the Eq. (1) modelindicates that transition |0〉-|3〉 is coupled to itinerant ra-diation with a nearly unit efficiency, similarly to previousworks on artificial two-level atoms.

The characterization presented in Fig. 2a,b providesabsolute calibration of the measurement records in termsof the ground state population p0. This is in regard withstandard superconducting qubit readout based on thedispersive interaction with a far-detuned cavity, wherethe measurement records are only known up to a scalingfactor to the qubit population and have to be indepen-dently calibrated. In order to account for small thermalfluctuations over the course of the experiment, we re-

3

6.540 6.550

(a) (b)

6.545-1 -0.5 0 0.5 1- 1

-0.5

0

0.5

1

- 1

-0.5

0

0.5

1

0 50 100 1500 0.2 0.4 0.6

0.2

0.4

0.6

0.8 0.2

0.4

0.6

0.8

200

1.0

0

0

1.0

1

2

0

*

(c)

FIG. 2. Conditional fluorescence readout. (a) Complex plane representation and (b) real part of the reflection coefficientr as a function of readout frequency ω at various powers represented by the |0〉-|3〉 transition Rabi frequency Ω (inset). Theexperimental data (colored dots) are fitted by the theory of Eq. (1) (lines) with Γ = 2π × 1.74 MHz and p0 = 0.78 due tothermal occupation of excited states. In the limit Ω → 0, the data in Fig. 2a becomes a circle with a radius p0 (gray circle,p0 = 1 and dashed circle, p0 = 0.78). The optimal readout settings are highlighted by the black star and arrow (see text). (c)Rabi oscillations (left panel), energy relaxation (right panel, yellow circles), and spin-echo coherence decay (green triangles)measured using conditional fluorescence readout. Fitting to exponential decay yields (coincidentally) T1 = T2 = 52 µs.

peated the previous calibration over 5 days. We find anaverage thermal equilibrium value of pth

0 = 0.76 ± 0.03,which corresponds to the atom being at thermal equi-librium with the effective temperature T = 45 ± 5 mK,which is indeed commonly encountered in superconduct-ing qubits. In what follows, we operate the readout at theoptimal power/frequency settings (Fig. 2a – black star)where the reflection amplitude becomes a real positivenumber between zero and unity, with p0 = 1− r.

The protocol for time-domain qubit manipulationsmatches that of traditional circuit quantum electrody-namics. For example, applying a drive at the qubitfrequency ω01 = 2π × 1.15 GHz prior to the readoutpulse results in Rabi oscillations of p0 as a function ofdrive duration (Fig. 2c - left panel). Operating the cir-cuit slightly off the sweet-spot at φext/2π = 0.507, i.e.weakly breaking the parity selection rule, we could pro-duce Rabi oscillations between states |0〉 and |2〉 by com-pensating the vanishing transition dipole with a strongdrive at frequency ω02 = 2π × 3.88 GHz26. Impor-tantly, the non-zero decay rates of even transitions re-sulting from breaking the parity selection rule remainseveral orders of magnitude smaller than other decoher-ence processes26. The reflection coefficient measured dur-

ing Rabi oscillations between |0〉 and |1〉 or |2〉 yieldsequilibrium ratios pth

1 /pth0 = 0.329 and pth

2 /pth0 = 0.007

independently of the absolute calibration of pth0 . Assum-

ing pth0 + pth

1 + pth2 = 1, we obtain the equilibrium value

of pth0 = 0.75, in close agreement with the spectroscopic

calibration. We checked that the qubit can be initializedwith p0 ≈ 0.9 by strongly driving the |1〉-|3〉 transition.During such a drive, entropy is removed from the on-chipcircuit via spontaneous emission of photons |3〉 → |0〉 intothe measurement line26. However, the value pth

0 turns outto be sufficiently high for the purpose of our time-domainexperiments and hence we performed them all startingfrom thermal equilibrium.

The energy relaxation signal following the 0−1 swap isexponential with a characteristic decay time T1 = 52 µs,which is nearly three orders of magnitude longer thanthe radiative decay time of the 3-state (Fig. 2c - rightpanel). Yet, given the highly suppressed port couplingat the qubit frequency, such value of T1 is still too shortto be accounted for by radiative decay. In fact, thequbit decay is likely due to the dielectric loss in the cir-cuit’s antenna capacitance with an effective quality factorQdiel = 5× 105. The spin-echo signal decay is also expo-nential with a characteristic coherence time T2 = 52 µs.

4

X02

X01

X02

I

@!03X02

X01

I(a) (b)

population

0 10 20 30 40 500.

0.2

0.4

0.6

0.8

1.

0p1p2p

20 1p pp+ +population

0 50 100 150 200 2500.

0.2

0.4

0.6

0.8

1.

p0

p1

p2

p0

p1

p2

FIG. 3. Transient dynamics of fluorescence. (a) Time-domain evolution of populations p0 (down blue triangles), p1(up red triangles), and p2 (yellow squares) induced by cycling of the readout transition. Note that p0 decays approximatelyexponentially in time with a characteristic time scale τcyc = 9.6 µs, after which the population is transfered from state |0〉 tostates |1〉 and |2〉. The black circles indicate p0 + p1 + p2 and the error bars represent the standard deviation coming from therepetition of the measurements over two days. (b) Free evolution of populations starting predominantly from state |2〉. Thedata in both (a) and (b) is adequately explained by an optical pumping model (solid lines) (see text) where the dominant errormechanism is a non-radiative decay due to dielectric loss.

The additional dephasing Γϕ = 1/T2 − 1/2T1 is likelydue to measurement induced dephasing caused by ther-mal photons at readout frequency. In fact, the rate ofthermal jumps |0〉 → |3〉 can be linked to the effectivephoton temperature T at the microwave port via the de-tailed balance equation Γϕ = Γ exp(−~ω03/kBT ). Plug-ging in the same value T = 45± 5 mK corresponding tothe equilibrium ground state population leads to a closeagreement with the measured dephasing rate. Such anerror process could be exponentially suppressed in thefuture by a moderate improvement of circuit thermaliza-tion and increasing the readout transition frequency.

Finally, we explore the quantum non-demolition(QND) character of conditional fluorescence measure-ment. An elementary QND test consists of measuringthe atom population dynamics during readout. To do sowe drive the |0〉-|3〉 transition for a duration τ 1/Γ,pause for a brief period for state |3〉 to relax, swap thestate of interest with state |0〉, and perform the readout(Fig. 3a - top panel). As a result, we acquire the popu-lation transients p0(t), p1(t), and p2(t) of states |0〉, |1〉,and |3〉, respectively, during the readout. The fidelity ofthis population measurement is limited by the swap gatefidelities, which is above 99% for |0〉− |1〉 and above 90%for |0〉−|2〉26. The p0(t)-data indicates a gradual leakageof population outside the cycling manifold |0〉, |3〉 ona characteristic time scale τcyc = 9.6 µs. Importantly,the leakage involves only states |1〉 and |2〉 as the data

satisfies p0 + p1 + p2 = 1 within the measurement uncer-tainty (Fig. 3a - bottom panel). The average number offluorescence cycles Ncyc = Γ × τcyc ≈ 105 quantifies thedeviation of our readout from ideal QND behavior.

The finite fluorescence lifetime is expected in presenceof non-radiative transitions outside the cycling manifold.Additional information about energy relaxation in ourcircuit was obtained by measuring populations of states|0〉, |1〉, and |2〉 as a function of time following the |0〉-|2〉 swap (Fig. 3b). Combining the two data sets rep-resented on Fig. 3a and b, we constructed a relaxationmodel based on two additional decay mechanisms, di-electric loss and quasiparticle tunneling across the weakjunction. Indeed, such model adequately fits the six ex-perimental curves with only three adjustable parameters:the dielectric loss quality factor Qdiel = 5 × 105, pre-viously introduced to explain the |1〉 → |0〉 decay, thedimensionless quasiparticle density xqp = 10−6, and afudge-factor of 2.5 in front of the |3〉 → |2〉 decay rate.The presence of additional decay for this precise transi-tion could be due to a nearly resonant two-level system29.From this model we conclude that fluorescence lifetimeoriginates from two error processes. A smaller contribu-tion comes from the direct decay |3〉 → |1〉 with a charac-teristic time 36 µs due to quasiparticle tunneling. Suchprocess may break the parity selection rule for φext 6= 0,a prediction that has not been tested so far. The domi-nant error though comes from the decay |3〉 → |2〉 with a

5

characteristic time of 3.6 µs due to dielectric loss in thefluxonium capacitance26. Therefore, improving our fab-rication procedures will likely increase the fluorescencelifetime and hence the QND-fidelity. Alternatively, witha modified choice of circuit parameters, one can increasethe frequency of |1〉-|2〉 transition and use it for cycling,in which case the fluorescence lifetime would be limitedby the qubit T1.

In summary, we showed that a single macroscopicdegree of freedom – the superconducting phase-difference– can combine a highly coherent qubit and its QNDreadout via conditional fluorescence, in full analogy withDemhelt’s electron shelving scheme. The present experi-mental setup can be supplemented with quantum-limitedlinear amplifiers30 or recently developed microwave pho-ton counters31–33 for a single-shot operation. Besidesnovel quantum optics applications, eliminating the cav-ity from cQED can save hardware resources in scalingup quantum processors and constructing networks formodular quantum computing.

Methods

Derivation of Eq. (1)

A drive at angular frequency ω near readout frequencyinduces damped Rabi oscillations of the readout transi-tion with the Hamiltonian

H/~ =(ω − ω03)

2(|3〉〈3|−|0〉〈0|)−iΩ

2(|3〉〈0|−|0〉〈3|) (2)

and Ω = 2√

Γαin. In all generality one should consider allsources of loss and dephasing such as spontaneous emis-sion in the line, non-radiative decay and extra-dephasingdue to flux noise, including transitions from the read-out subspace to other states of the atom. However, thereadout transition has been designed so that its emissionrate in the line overcomes by many orders of magnitudethe other decay and dephasing processes. Assuming anenvironment at zero temperature, the density matrix ofthe atom ρ evolves according to the following Lindbladequation

ρ = − i~

[H, ρ] + ΓD[|0〉〈3|](ρ) (3)

where D is the Lindblad superoperator. The steady-stateof this equation yields to the expectation value

〈|0〉〈3|〉 =Ω

Γ

Γ2/2− i(ω − ω03)Γ

Γ2/2 + Ω2 + 2(ω − ω03)2, (4)

from which we derive Eq. (1). The good agreement be-tween the theoretical model and the experimental datavalidates the assumptions made in the previous deriva-tion.

Reflection coefficient calibration

The measured reflection coefficient is only known upto a scaling factor and has to be calibrated. In fact, thereadout drive undergoes several stages of attenuation be-fore reaching the artificial atom, while the fluorescencesignal is amplified before digitization. Calibration is per-formed as follows. The circuit is flux biased at φext = 0and is probed in reflection between 6.3 and 6.6 GHz. Theexternal flux is chosen so that no circuit transition existin the frequency band of interest. The acquired signalscal(ω) therefore accounts for the temperature-dependentattenuation and filtering of the lines as well as the wholeacquisition setup amplification chain. Once the atom isbiased at the wanted flux, the reflection coefficient r isdeduced from the experimentally measured signal sexp(ω)by

r =sexp(ω)

scal(ω)10(Pcal−Pexp)/20 (5)

with Pcal/exp the room-temperature power of the read-out drive during the calibration/experiment, expressedin dBm.

Relaxation model of transient dynamics

The relaxation model used to reproduce the data rep-resented on Fig. 3 considers three distinct decay mech-anisms. A jump from a state |i〉 to |j〉 can occur dueto radiative decay in the line, dielectric loss, or quasi-particle tunneling in the weak junction, with the ratesΓradij , Γdiel

ij , Γqpij , respectively. All rates are computed as-

suming an equal bath temperature T = 50 mK for alldecay mechanisms26.

We simulate the dynamics of the atom using thePython library QuTiP34. The spectrum of the fluxoniumhamiltonian Hf = 4EC(−i∂φ)2+ELφ

2−EJ cos(φ−φext)is first computed in a Hilbert space of dimension 100 ×100, to ensure accuracy of the computed eigenfrequenciesωi and eigenstates |i〉, i ≤ 99. We then solve numericallythe Lindblad master equation in the Fock state basis for asmaller Hilbert space containing the first 10 eigenstates.This takes into account the fact that high-energy statesdo not contribute to the atom dynamics and allows fora faster computing time. The master equation contains

all the jump operators √

Γradij + Γdiel

ij + Γqpij |j〉〈i|i,j≤9

computed from the diagonalization of Hf . The initialstate of the atom is the thermal equilibrium state ρth forFig. 3a and a perfect swap between |0〉 and |2〉 on ρth forFig. 3b.

The Hamiltonian for Fig. 3a is H = −iΩ2 (|3〉〈0|−|0〉〈3|)

and represents the readout drive, while it is 0 for Fig. 3b.They correspond to computing the evolution in the framerotating at all the eigenfrequencies, i.e. applying theunitary U(t) = exp(−iHf t/~) = exp(−i∑j ωj |j〉〈j|t)to the driven and undriven hamiltonians, respectively.

6

We find that the transient dynamics of fluorescence isbest reproduced for Ω = 0.95Γ/

√2. This deviation from

the ideal value can be due to small drifts between themeasurements of Fig. 2 and 3.

Data Availability. The data that support the findingsof this study are available from the corresponding authorupon reasonable request.Acknowledgement. We thank Ray Mencia for provid-ing samples at the initial stages of this work and NathanLangford, Shimon Kolkowitz, and Benjamin Huard foruseful discussions. We acknowledge funding from Sloan

Foundation, NSF-PFC at JQI, and ARO-MURI.

Author contributions. H.X. fabricated the deviceand along with Y.H.L acquired and analyzed the data.Y.H.L. designed the coaxial-to-waveguide adaptor andbuilt the low-temperature measurement setup. L.B.N.contributed to circuit design, room-temperature instru-mentation, and to identifying decoherence mechanisms.N.C. performed initial conditional fluorescence experi-ments and developed procedures for analyzing and mod-eling the data. V.E.M. managed the project. The au-thors declare no financial conflict of interest.

∗ These two authors contributed equally1 H Jeff Kimble, “The quantum internet,” Nature 453,

1023–1030 (2008).2 Serge Haroche and Jean-Michel Raimond, Exploring the

quantum: atoms, cavities, and photons (Oxford universitypress, 2006).

3 Andreas Wallraff, David I Schuster, Alexandre Blais,L Frunzio, R-S Huang, J Majer, S Kumar, Steven MGirvin, and Robert J Schoelkopf, “Strong coupling ofa single photon to a superconducting qubit using circuitquantum electrodynamics,” Nature 431, 162–167 (2004).

4 S Haroche, M Brune, and JM Raimond, “From cavity tocircuit quantum electrodynamics,” Nature Physics , 1–4(2020).

5 Alexandre Blais, Steven M Girvin, and William D Oliver,“Quantum information processing and quantum opticswith circuit quantum electrodynamics,” Nature Physics ,1–10 (2020).

6 Andrew D Ludlow, Martin M Boyd, Jun Ye, EkkehardPeik, and Piet O Schmidt, “Optical atomic clocks,” Rev.Mod. Phys. 87, 637 (2015).

7 Warren Nagourney, Jon Sandberg, and Hans Dehmelt,“Shelved optical electron amplifier: Observation of quan-tum jumps,” Physical Review Letters 56, 2797 (1986).

8 Colin D Bruzewicz, John Chiaverini, Robert McConnell,and Jeremy M Sage, “Trapped-ion quantum computing:Progress and challenges,” Applied Physics Reviews 6,021314 (2019).

9 Lucio Robledo, Lilian Childress, Hannes Bernien, BasHensen, Paul FA Alkemade, and Ronald Hanson, “High-fidelity projective read-out of a solid-state spin quantumregister,” Nature 477, 574–578 (2011).

10 Denis D Sukachev, Alp Sipahigil, Christian T Nguyen,Mihir K Bhaskar, Ruffin E Evans, Fedor Jelezko, andMikhail D Lukin, “Silicon-vacancy spin qubit in diamond:a quantum memory exceeding 10 ms with single-shot statereadout,” Physical review letters 119, 223602 (2017).

11 A Nick Vamivakas, Yong Zhao, Chao-Yang Lu, and MeteAtature, “Spin-resolved quantum-dot resonance fluores-cence,” Nature Physics 5, 198–202 (2009).

12 Vladimir E. Manucharyan, Jens Koch, Leonid I. Glazman,and Michel H. Devoret, “Fluxonium: Single cooper-paircircuit free of charge offsets,” Science 326, 113–116 (2009).

13 O Astafiev, Alexandre M Zagoskin, AA Abdumalikov,Yu A Pashkin, T Yamamoto, K Inomata, Y Nakamura,and Jaw Shen Tsai, “Resonance fluorescence of a single

artificial atom,” Science 327, 840–843 (2010).14 Vladimir B Braginsky and F Ya Khalili, “Quantum non-

demolition measurements: the route from toys to tools,”Reviews of Modern Physics 68, 1 (1996).

15 Io-Chun Hoi, C. M. Wilson, Goran Johansson, Tauno Palo-maki, Borja Peropadre, and Per Delsing, “Demonstrationof a single-photon router in the microwave regime,” Phys-ical Review Letters 107, 073601 (2011).

16 Arjan F Van Loo, Arkady Fedorov, Kevin Lalumiere,Barry C Sanders, Alexandre Blais, and Andreas Wall-raff, “Photon-mediated interactions between distant artifi-cial atoms,” Science 342, 1494–1496 (2013).

17 Mohammad Mirhosseini, Eunjong Kim, Xueyue Zhang,Alp Sipahigil, Paul B. Dieterle, Andrew J. Keller, AnaAsenjo-Garcia, Darrick E. Chang, and Oskar Painter,“Cavity quantum electrodynamics with atom-like mir-rors,” Nature (2019), 10.1038/s41586-019-1196-1.

18 Bharath Kannan, Daniel Campbell, Francisca Vasconce-los, Roni Winik, David Kim, Morten Kjaergaard, PhilipKrantz, Alexander Melville, Bethany M. Niedzielski, Joni-lyn Yoder, Terry P. Orlando, Simon Gustavsson, andWilliam D. Oliver, “Generating Spatially Entangled Itiner-ant Photons with Waveguide Quantum Electrodynamics,”(2020), arXiv:2003.07300.

19 AA Houck, DI Schuster, JM Gambetta, JA Schreier,BR Johnson, JM Chow, L Frunzio, J Majer, MH Devoret,SM Girvin, et al., “Generating single microwave photonsin a circuit,” Nature 449, 328–331 (2007).

20 Philippe Campagne-Ibarcq, Landry Bretheau, EmmanuelFlurin, Alexia Auffeves, Francois Mallet, and BenjaminHuard, “Observing interferences between past and futurequantum states in resonance fluorescence,” Physical Re-view Letters 112, 180402 (2014).

21 DM Toyli, AW Eddins, S Boutin, S Puri, D Hover,V Bolkhovsky, WD Oliver, A Blais, and I Siddiqi, “Reso-nance fluorescence from an artificial atom in squeezed vac-uum,” Physical Review X 6, 031004 (2016).

22 Nathanael Cottet, Sebastien Jezouin, Landry Bretheau,Philippe Campagne-Ibarcq, Quentin Ficheux, Janet An-ders, Alexia Auffeves, Remi Azouit, Pierre Rouchon, andBenjamin Huard, “Observing a quantum Maxwell demonat work,” Proceedings of the National Academy of Sciences114, 7561–7564 (2017), arXiv:1702.05161.

23 Barbara Gabriele Ursula Englert, G Mangano,M Mariantoni, R Gross, J Siewert, and E Solano,“Mesoscopic shelving readout of superconducting qubits

7

in circuit quantum electrodynamics,” Physical Review B81, 134514 (2010).

24 Long B Nguyen, Yen-Hsiang Lin, Aaron Somoroff, Ray-mond Mencia, Nicholas Grabon, and Vladimir EManucharyan, “High-coherence fluxonium qubit,” Physi-cal Review X 9, 041041 (2019).

25 Helin Zhang, Srivatsan Chakram, Tanay Roy, NathanEarnest, Yao Lu, Ziwen Huang, Daniel Weiss, JensKoch, and David I Schuster, “Universal fast flux con-trol of a coherent, low-frequency qubit,” arXiv preprintarXiv:2002.10653 (2020).

26 See Supplementary Material .27 A Kou, W. C. Smith, U Vool, I. M. Pop, K. M. Sliwa,

M Hatridge, L Frunzio, and M. H. Devoret, “Simultaneousmonitoring of fluxonium qubits in a waveguide,” PhysicalReview Applied 9, 064022 (2018).

28 C. W. Gardiner and M. J. Collett, “Input and output indamped quantum systems: Quantum stochastic differen-tial equations and the master equation,” Physical ReviewA 31, 3761–3774 (1985).

29 PV Klimov, J Kelly, Z Chen, M Neeley, A Megrant, B Bur-kett, R Barends, K Arya, B Chiaro, Yu Chen, et al.,“Fluctuations of energy-relaxation times in superconduct-ing qubits,” Physical review letters 121, 090502 (2018).

30 N Bergeal, F Schackert, M Metcalfe, R Vijay, V. E.Manucharyan, L Frunzio, D. E. Prober, R. J. Schoelkopf,S. M. Girvin, and M. H. Devoret, “Phase-preserving am-plification near the quantum limit with a josephson ringmodulator,” Nature 465, 64–68 (2010).

31 A Opremcak, I. V. Pechenezhskiy, C. Howington, B. G.Christensen, M. A. Beck, E Leonard, J Suttle, C Wilen,K. N. Nesterov, G. J. Ribeill, et al., “Measurement of asuperconducting qubit with a microwave photon counter,”Science 361, 1239–1242 (2018).

32 Shingo Kono, Kazuki Koshino, Yutaka Tabuchi, AtsushiNoguchi, and Yasunobu Nakamura, “Quantum non-demolition detection of an itinerant microwave photon,”Nature Physics 14, 546–549 (2018).

33 Jean-Claude Besse, Simone Gasparinetti, Michele C Col-lodo, Theo Walter, Philipp Kurpiers, Marek Pechal,Christopher Eichler, and Andreas Wallraff, “Single-shotquantum nondemolition detection of individual itinerantmicrowave photons,” Physical Review X 8, 021003 (2018).

34 J. R. Johansson, P. D. Nation, and Franco Nori, “QuTiP2: A Python framework for the dynamics of open quantumsystems,” Computer Physics Communications 184, 1234–1240 (2013), arXiv:1211.6518.