Nonequilibrium heat transport and work with a single artificial atom coupled to awaveguide: emission without external driving

Yong Lu,1, ∗ Neill Lambert,2, † Anton Frisk Kockum,1 Ken Funo,2 Andreas

Bengtsson,1 Simone Gasparinetti,1 Franco Nori,2, 3 and Per Delsing1

1Microtechnology and Nanoscience, MC2, Chalmers University of Technology, SE-412 96 Goteborg, Sweden2Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan

3Department of Physics, The University of Michigan, Ann Arbor, 48109-1040 Michigan, USA(Dated: July 28, 2021)

We observe the continuous emission of photons into a waveguide from a superconducting qubitwithout the application of an external drive. To explain this observation, we build a two-bathmodel where the qubit couples simultaneously to a cold bath (the waveguide) and a hot bath (asecondary environment). Our results show that the thermal-photon occupation of the hot bathis up to 0.14 photons, 35 times larger than the cold waveguide, leading to nonequilibrium heattransport with a power of up to 132 zW, as estimated from the qubit emission spectrum. By addingmore isolation between the sample output and the first cold amplifier in the output line, the heattransport is strongly suppressed. Our interpretation is that the hot bath may arise from activetwo-level systems being excited by noise from the output line. We also apply a coherent drive, anduse the waveguide to measure thermodynamic work and heat, suggesting waveguide spectroscopy isa useful means to study quantum heat engines and refrigerators. Finally, based on the theoreticalmodel, we propose how a similar setup can be used as a noise spectrometer which provides a newsolution for calibrating the background noise of hybrid quantum systems.

I. INTRODUCTION

Over the past 20 years, superconducting qubitcoherence times have increased from less than 1 nsto more than 1 ms [1–4]. Many studies have shownthat such coherence times are limited by two-levelsystems (TLSs) [5–7] and that quasiparticles canalso contribute [8–11]. Moreover, excessive thermalpopulation of a qubit, arising from nonequilibriumquasiparticles, has been observed extensively, witheffective temperatures in the range 30–200 mK [12–14]. More recently, ionizing radiation due to high-energy cosmic rays and radioactive decay has beenshown to reduce qubit coherence [15–17]. Therefore,isolating superconducting qubits from all possible sourcesof noise will be crucial for realizing fault-tolerantsuperconducting quantum computers [16, 18, 19].

In addition, in the field of quantum thermodynamics,alongside the study of quantum heat engines [20–23] andrefrigerators [24, 25], the dynamical control of heat flow,mediated by phonons or photons, has been investigatedin solid-state circuits [26], nanostructures [27, 28], andsuperconducting circuits [29, 30]. More recently, thepotential for significant quantum effects was introducedinto such studies in the form of a superconducting qubitcoupled to cavities [31–35]. In particular, a heat valvefor energy transfer between two artificial heat bathsconstructed from resonators has been realized, wherethe transferred power was derived from the temperaturedifference between the two baths which is measured by

∗ e-mail:kdluyong@outlook.com† e-mail:nwlambert@gmail.com

thermometers [34, 36]. The direct coupling of a qubit to awaveguide, a setting known as quantum electrodynamics(QED) [37, 38] has been suggested as a platform forstudies in thermodynamics [39], but has not yet beenexperimentally explored.

In this work, we first observe the emission from asuperconducting qubit into a cold waveguide due tothermal excitation of the qubit by a hot bath, andinvestigate the potential source of this additional thermalnoise. We then show how the qubit emission spectrumand reflectivity can be used to measure the heat and workrates, in analogy to the heat valves demonstrated withengineered environments [34, 36].

II. RESULTS

Theoretical model. To model our experiment,we introduce a two-bath model where the qubit issimultaneously coupled to a cold (radiative) waveguidebath and a hot auxiliary (nonradiative) environment. Wealso include a coherent drive, at frequency ωp, throughthe waveguide in the system. Thus, we have a qubit andtwo bosonic environments described by the Hamiltonian

H = Hq +Hr +Hn (1)

with

Hq

~= −∆

2σz +

Ω

2σx, (2)

where the qubit Hamiltonian is in the rotating frame with∆ = ωp−ω01 and Ω as the probe strength. The radiative(waveguide) Hamiltonian Hr and the nonradiative bathHamiltonian Hn, including couplings to the qubit (under

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FIG. 1. Sketch of the experimental setup. a, A schematicshowing how the two-level artificial atom (|0〉 and |1〉 are thecorresponding ground and excited states of the atom) absorbsenergy from the hot bath surrounding the atom. Therefore,the excited atom emits photons to the cold waveguide. Wehave Γr and Γn as the emission rates of the atom into thewaveguide and the non-radiative bath, respectively. b, False-color micrograph of the superconducting circuit realizing thesetup in panel a: A transmon qubit consisting of a cross-shapedsuperconducting island shunted by a superconducting quantuminterference device (SQUID) capacitively coupled to a microwavecoplanar waveguide (blue). The inset shows a close-up of theSQUID. [See more details on the measurement setup in theMethods section]

the rotating-wave-approximation), are given by

Hi

~=∑k

ωk,ia†k,iak,i +

∑k

gk,i

(σ−a

†k,i + σ+ak,i

),(3)

where gk,i is the coupling strength to mode k inenvironment i ∈ r ,n at frequency ωk,i, and

ak,i(a†k,i) is the corresponding annihilation (creation)

operator of the mode. Hereafter, we treat bothenvironments under the standard Born-Markov secular(BMS) approximations [40], and use this model to fitthe single-tone spectroscopy and power spectrum. Underthe BMS approximation [Supplementary material S2], wedefine Γr and Γn as the radiative and the nonradiativedecay rates, respectively and nr and nn as the thermaloccupations at temperatures Tr and Tn, respectively, forthe two baths. A corresponding explanatory diagram isshown in Fig. 1(a).

Our device is a frequency-tunable transmon-typeartificial atom [41] coupled to a 1D semi-infinitewaveguide, terminated by an open end, acting as amirror in Fig. 1(b) [5, 42–44, 46]. We operate thequbit at the 10 mK stage of the mixing chamber of thedilution refrigerator at its maximum frequency ω01/2π ≈5.5 GHz, where the qubit is flux-insensitive to first order,thus minimizing the pure dephasing rate.

Qubit spectroscopy. We first characterize ourqubit by single-tone spectroscopy where we send a weakcoherent tone to the qubit (Ω Γr) and then measurethe coherent reflection coefficient r. Sweeping the probefrequency across the resonance of the qubit we observethat the reflection coefficient shows a dip in the amplitude[Fig. 2(a)] and a π phase shift [Fig. 2(b)]. According toinput-output theory with two baths under a weak probe,the reflection coefficient is

r = 1− iΓr(1− 2Γ+/Γ1)

∆ + iΓ2, (4)

where Γ+ = nnΓn +nrΓr, Γ1 = (1+2nn)Γn +(1+2nr)Γr,and Γ2 = Γ1/2 + Γφ. The pure dephasing rate Γφ canalso be affected by the thermal excitation of the thirdlevel of the transmon [Supplementary Material S6]. Byusing Eq. (4), we calculate the theoretical solid curveswith Γr(1−2Γ+/Γ1)/2π = 214 kHz and Γ2/2π = 147 kHzshown in Fig. 2.

Qubit power spectrum density. Without sendingany signal from room temperature to the sample, weobserve a power spectral density (PSD) emitted fromthe qubit into the waveguide, shown as blue dots inFig. 3(a) [more measurement details are presented inthe Methods section]. The Lorentzian shape centeredat the qubit frequency indicates that the environmentsurrounding the qubit is hotter than the waveguide, andis causing heat flow into the waveguide through thequbit. In order to justify this statement, and fit thedata, we use standard input-output theory for the outputof the transmission line bout(t) = bin(t) − i

√Γrσ−(t),

where bin(t) = fin(t) + Ω2√

Γ1includes thermal noise

fin(t) and the coherent drive Ω2√

Γ1, for the cases when

such applied. It is necessary to take into account thefact that the thermal input fin(t) to the waveguide canbecome correlated with the qubit emission operator σ−(t)to evaluate the correct output spectrum [1], given by[Supplementary Material S5]

S(ω) =~ω01

2π

∫ ∞−∞

dte−iωt〈b†out(t)bout(t′)〉(t′→∞) (5)

with

〈b†out(t)bout(t′)〉 = Γr (nr + 1) 〈σ+(t)σ−(t′)〉 (6)

− Γrnr〈σ−(t′)σ+(t)〉+ 〈f†in(t)fin(t′)〉

− iΩ∗

2〈σ−(t′)〉+

iΩ

2〈σ+(t)〉+

|Ω|2

4Γr.

This expression is easily generalized to include morelevels within the artificial atom. Importantly, if wedefine the Fourier transforms of the first two terms ofEq. (6), s+

q (ω) =∫∞−∞ dt〈σ+(t)σ−(t′)〉e−iωt, s−q (ω) =∫∞

−∞ dt〈σ−(t′)σ+(t)〉e−iωt, then the limited detailedbalance that arises for a qubit coupled to a singleBMS environment, e.g., the waveguide, tells us thats−q (ω) = eβrω01s+

q (ω), where βr = 1/kBTr, and kB is theBoltzmann constant. Hence, in equilibrium situations

3

a b

FIG. 2. Weak-power spectroscopy under a coherent drive. a, Magnitude and b, phase response of the reflection coefficient r asa function of the probe detuning. Blue dots are the experimental data with the red solid curves calculated from Eq. (4).

a b

a b

FIG. 3. Thermal spectrum. a, Power spectral density (PSD) of the output field where the results with and without a strong driveare Son and Sth, respectively. Units are yW/Hz=10−24 W/Hz and S = 2πS(ω). The solid curves are the corresponding fits toEq. (8) and Eq. (7). The Rabi frequency used for Son on the |0〉 ↔ |1〉 transition is Ω/2π ≈ 8.8 MHz. b, Power spectral densityfrom Autler-Townes splitting. The solid curve is calculated from theory. In both a and b, dots are from the experimental data.The Rabi frequency used for the drive on the |1〉 ↔ |2〉 transition is Ω/2π ≈ 1.5 MHz. Note that the PSD in a is subtracted bythe background when tuning the qubit frequency away through the external magnetic flux, whereas in b the PSD is the differencebetween the thermal spectra with and without the drive on the |1〉 ↔ |2〉 transition.

(no drive and no additional environment Hn), the firsttwo terms in Eq. (6) cancel, resulting in S(ω) = 0.

Explicitly evaluating the correlation functions for thequbit using our two-bath master equation [omittingboth delta-function coherent contributions [4] andthe background thermal black-body spectrum of thetransmission line in Eq. (6)], we find the thermal spectraldensity output (Ω = 0) is given by

Sth(ω) = ~ω01Γr

2π

2Γ2Γn∆n/Γ1

δω201 + Γ2

2 , (7)

where ∆n = nn − nr, and δω01 = ω − ω01. In Eq. 7,we find that Sth(ω) = 0 if nn = nr, as expected fromdetailed balance. Without the additional nonradiativebath to thermally excite the qubit, we would not observea thermal response (even in the presence of thermal inputto the transmission line). As shown in Fig. 3(a), Sth(ω) 6=0 in the experiment, which implies that the waveguideand the bosonic bath surrounding the qubit are not inequilibrium. Fitting Eq. (7) to the data in Fig. 3(a)gives ΓrΓn∆n/(2πΓ1) = 5.6 ± 0.2 kHz correspondingto a transported power

∫∞−∞ Sth(ω)dω = 132 ± 5 zW

4

ca b c

FIG. 4. Power loss, work, and heat. a, The lost power Ploss as a function of the Rabi frequency (Ω) of the drive with two and four

isolators. b, The magnitude of the qubit emission as a function of Ω of the drive with two isolators. c, Work (W ) performed and heat

(Q) generated by the drive as a function of Ω. In all panels, the stars are the experimental data and the solid curves are from theory,

where the blue stars in (c) are from the difference between the values of the black stars in c and a according to Qr = Ploss − W .

with the given Γ2 value from the reflection-coefficientmeasurement.

We also send a strong drive on resonance with thequbit (Ω/2π ≈ 8.8 MHz Γ1), where such a large Rabifrequency ensures that there is no overlap between thesidebands and the center of the Mollow-triplet PSD [3,49]. For the middle peak from the elastic scattering, wehave

Son(ω) = ~ω01Γr

2π

Γ2/2

δω201 + Γ2

2 . (8)

By fitting the data with the drive on [black dotsin Fig. 3(a)], we obtain Γr/2π = 227 ± 4 kHz andΓ2/2π = 143 ± 4 kHz. Compared to Eq. (7), the heightof this spectrum is mostly independent of the thermaloccupation difference ∆n, due to the saturation of thefirst excited state.

Heat transport and work. Another straightforwardway to evaluate the heat transport between the twobaths is to measure the lost power Ploss due to thenonradiative decay. Shown as black in Fig. 4(a), wefind that when the coherent drive is weak, the thermalnoise dominates the qubit dynamics. Therefore, thequbit absorbs photons from the hot bath and thendecays into the cold waveguide with a higher probabilitycompared to the overall emission back into the hotbath, resulting in a negative lost power. When weincrease the photon-flux occupation in the waveguideby increasing the drive intensity, the absolute value ofPloss is reduced. At ΩRabi/2π ≈ 95 kHz, we see thatequilibrium is reached, with zero lost power. However,beyond this value, the waveguide occupation becomeslarger than the non-radiative environment. Thus, thequbit is excited by the photons in the waveguide andthen dissipates into both environments via nonradiativedecay. By increasing the drive intensity further, thequbit is saturated, with the population reaching the

maximum value of 0.5 in the steady state. Therefore,Ploss ≈ ~ω01

Γn

2 . Theoretically, the lost power is given by

Ploss = ~ω01(〈b†inbin〉 − 〈b†outbout〉) as

Ploss = ~ω01Γn

2

Ω2 − 2Γ2Γr∆n

Ω2 + Γ2Γ1

, (9)

By fitting Eq. (9) to the data using the values of Γ2 andΓr extracted from the fit of Eq. (8), with Ω calibratedby the Mollow-triplet [black solid curve], we obtainΓn/2π = 55 ± 3 kHz with ∆n = 0.135 photons andΓ1/2π = 299 kHz. The transported power from the hotbath is about 132 zW which is consistent with the integralof the thermal spectrum in Fig. 3(a).

To obtain the temperature of the waveguide and thebath separately, we combine the value of ∆n with theresult from the reflection-coefficient measurement wherewe can obtain the thermal population of the qubitρth

11 = Γ+/Γ1 ≈ 2.86 % from the extracted parameters.Consequently, we obtain nr ≈ 0.004, nq ≈ 0.03 andnn ≈ 0.139 corresponding to Tr ≈ 50 mK for the coldwaveguide, Tq ≈ 78 mK for the qubit and Tn ≈ 131 mKfor the hot bath, respectively. Therefore, we can obtainthe value of Γ1 = (1 + 2nn)Γn + (1 + 2nr)Γr ≈ 2π×299±7 kHz which is very close to 2Γ2, indicating that the puredephasing rate Γφ is negligible at the flux sweet spot.

In order to make sure that the environment is notaffected when changing the external magnetic flux totune the qubit away from its maximum frequency, wesend a strong drive on the transition between thesecond and third levels of the transmon, inducing anAutler-Townes splitting [51]. Then, we obtain the PSDsubtracted from the reference with the drive off, wherewe observe two peaks and one dip [blue in Fig. 3(b)].The two peaks are the thermal spectra from the thermalemission of the dressed states when a strong drive isapplied on the |1〉 ↔ |2〉 transition. The distance betweenthese two peaks are twice of the corresponding Rabi

5

TABLE I. Summary of the system parameters includingdifferent qubit decay rates and temperatures of the qubit,waveguide and the hot bath.

Γr/2π Γn/2π Γ1/2π Γ2/2π Tq Tr Tn

kHz kHz kHz kHz mK mK mK227± 4 55± 3 299± 7 143± 4 78 50 131

frequency : 2Ω ≈ 2π × 3.0 MHz. When the drive isoff, as we discussed, we have a thermal spectrum Sth

at the undressed qubit frequency. Importantly, we havea dip at that frequency when we subtract the data withthe drive off from the data with the drive on. We findthat the numerical result from the master equation [redin Fig. 3(b)] matches well the data [blue in Fig. 3(b)]with parameters Γr/2π = 227 kHz, Γn/2π = 55 kHz and∆n = 0.135. These parameters agree fully with thoseextracted from the power-loss measurements.

When we drive the qubit, the external field is doingwork on this single-atom quantum system. DefiningH0 = ~ω01

2 σz as the bare Hamiltonian which measures

the internal energy of the qubit, and H1 = ~Ω2 σx as the

drive Hamiltonian, the work performed on the qubit bythe coherent drive is defined as [33, 52]

W = Tr−i[H0, H1]ρ

=

~Ω

2〈σy〉 = ~Ω<[i〈σ−〉](10)

Therefore, the work W only depends on the qubitcoherent emission 〈σ−〉 with 〈σ−〉 = i(〈bout〉−〈bin〉)/

√Γr.

In particular, when the drive is on resonance with thequbit [see Supplementary Material Eq. (S7)], it leads to

W = i~Ω〈σ−〉 = ~Ω|〈σ−〉|. We obtain 〈σ−〉 from thevalues of 〈bout〉 and 〈bin〉 which are measured with andwithout the qubit by tuning the qubit frequency. Theresults match well with theory [black in Fig. 4(b) and(c)], where the work increases with the drive intensityand then saturates around 3.2 aW due to the saturationof the qubit.

Besides the work, the heat from the waveguide and thehot bath can be derived from Q = Tr

H0L[ρ]

= Qr +

Qn, where L[ρ] is the dissipative part of the Liouvillianfor the interaction with both baths [Supplementary

Material S2], and Qi = ~ω01(Γini〈σ−σ+〉 − Γi(ni +1)〈σ+σ−〉) i ∈ n, r. In particular, we find from Eq. (6)

that Qr = Ploss−W [blue in Fig. 4(c)], where the negative

values of Qr means that the waveguide is heated up dueto the qubit emission. Interestingly, when the drive isoff (W = 0), and in the steady-state (the change of the

internal energy of the qubit U = TrH0∂tρ(t)

= 0),

the first law of thermodynamics Qr + Qn + W = Uimplies that we have Qn = −Qr, indicating the heatexchange between these two baths, consistent with whatwe observed.

Noise origin. In a separate cooldown, we observethat improving the isolation between the sample andthe first amplifier in our chain [see Methods] leads to

a strong reduction in the nonequilibrium heat flow,and a reduced thermal population of the qubit [redstars in Fig. 4(a)]. This observation suggests that thetemperature of the non-radiative environment has beenreduced, with Γn/2π = 53 ± 4 kHz, and that the twobaths are in thermal equilibrium without external drive.Therefore, we have a reduced thermal population of ourqubit: ρth

11 ≈ 0.4 %. It is possible that noise from thehigh-mobility electron transistor amplifier (HEMT) atthe 4K stage induces quasiparticles [53, 54] or activeTLSs [11, 55].

Quasiparticles may in principle be generated fromstray thermal noise propagating to the sample from theoutput line. These “hot” quasiparticles with energyhigher than ∆g +~ω01 could excite the qubit [12, 53, 54],where ∆g is the superconducting gap. However, from thethermal population of our qubit ρth

11 ≈ 2.86 %, we caninfer that the quasiparticle-induced nonradiative decayrate should be Γqp/2π ≈ 84 kHz > Γn (see Methodssection), in conflict with our observations. Moreover,if the excitation is dominated by the quasiparticles, thevalue of Γn should be decreased after we decrease thetemperature of the hot bath which is not the case inour experiment. Conversely, we do not see a substantialincrease either, as might be expected for a purelyTLS bath (see Methods). Thus, we believe our hotbath is primarily from excited two-level systems, butquasiparticles may still play a small role. However,precisely reconstructing the properties and origin of thisthermal noise requires further investigation. We alsopoint out that our model makes it possible to study theenergy exchange between “hot” quasiparticles and a coldwaveguide, if the waveguide is cold enough [See Methods].

III. DISCUSSION

In summary, with the system parameters shownin Table I, we constructed a two-bath model toexplain the heat transport via a superconductingqubit. Our study indicates that the qubit excited-state population is increased up to 2.9 % due tointeraction with a hot environment, probably fromexcited TLSs, with an equivalent temperature of about131 mK. This temperature can be reduced by addingmore isolation between the sample and the outputline, leading to the thermal population of our qubitbeing reduced by almost one order of magnitude. Ourresults point at the necessity of providing sufficientisolation between the sample and the HEMT amplifierto mitigate residual thermal occupation of qubits,which adversely affects state preparation in quantumcomputing applications [18, 19].

Moreover, as presented in the master equation, thenoise will increase the intrinsic qubit relaxation rate byΓ+ = nnΓn + nrΓr. Thus, Γ+ is reduced from about9 kHz with two isolators to 1 kHz with four isolators,which is about 15 % of the nonradiative decay rate. Such

6

a large improvement will be important in circuit QED,where generally the radiative decay Γr is negligible andΓn dominates the qubit decay.

We also showed that, under strong drive, we can usethe waveguide to measure work and heat rates. While ourquantum system is operating in a regime where the driveonly increases the heat rate to both environments, havingdirect access to the work, from measuring the coherentoutput, and the heat, from the power loss, is a powerfultool for the future study of regimes where useful taskslike cooling or work extraction can be performed.

Noise spectrometer. Based on the model wedeveloped here, we propose that our study also enablesthe construction of a practical noise spectrometer. Indetail, we could engineer two waveguide channels coupledto the qubit, similar to the setup in Refs. [56, 57], butwith equally strong coupling on both sides with Γ1 = 2Γr

(Γr Γn). One channel, acting as a probe, could be usedfor inputting noise while the other, acting as a detector,is for measuring the noise PSD. In this setup, the noisecan be very well isolated from the detection channelsince the direct capacitive coupling between the probeand detection channels can be designed to be extremelysmall [56, 57].

In this scenario, the thermal spectrum density inEq. (7) becomes

Sth(ω) = ~ω01Γr

2π

Γ2∆n

δω201 + Γ2

2 , (11)

where ∆n = nth − nr, with nth (nr), the temperature ofthe probe (detection) channel. We notice that Eq. (8)is still valid. Thus, we can evaluate the value of ∆n atthe qubit frequency according to ∆n = Sth(ω)/2Son(ω)if Sth(ω) and Son(ω) are measured. Furthermore, bytaking the integral of the spectra, we have the powersPth = ~ω01Γr∆n/2 and Pon = ~ω01Γr/4 for Sth and Son,respectively. Therefore, we obtain ∆n = Pth/2Pon.

Compared to measuring the whole spectra, the secondmethod will reduce the measurement time by a factorproportional to the number of data points in the spectra.However, by observing the symmetry of the spectra,we can check whether the bandwidth of the noise islarger than Γr or not, which can be possible whena qubit strongly couples to some resonant TLSs [58].Note that it is not necessary to calibrate the systemgain with these two measurements. Moreover, sincethe method is insensitive to Γφ, this insensitivity makesthe method robust against flux noise, which introducesexcess dephasing when the qubit frequency is tuned awayfrom the sweet spot. As a result, the spectrometercan be reliably operated over a broad frequency range.Combined with the reflection coefficient, we will haver = nth + nr, when Γφ Γr. Thus, we can obtainnth and nr, separately.

Compared to thermometers based on tunnelingjunctions [59, 60], our proposal does not requireadditional calibration and enables the possibility to

obtain the noise spectrum by sweeping the qubitfrequency. Circuit-QED radiometers [25, 61] have alsobeen developed, where the bandwidth is limited by thecavity. Very recently, a thermometer based on thereflection coefficient of a qubit in front of a mirror wasdemonstrated [5], where it has a single bosonic bath inthe waveguide, namely, Γr 6= 0 (nr 6= 0) with Γn = 0(nn = 0). Compared to Ref. [5], the thermometrymethod suggested here, based on PSD measurements,can tolerate a higher noise background in the probewaveguide, and, unlike Ref. [5], is insensitive to thepure dephasing rate. Finally, separating the detectionand probe channels can make it easier to connect ourthermometer chip directly to other hybrid quantumsystems [62–65].

IV. METHODS

Measurement of PSD. To obtain the PSD, wemeasure the output signal of the qubit emission into thewaveguide as a voltage in the time domain, normalizedto the system gain based on the Mollow-tripletspectrum [49], and then calculate the PSD accordingto the Welch method [66], where the background noiseis subtracted by the reference found by tuning thequbit frequency away using the external magnetic flux.Experimentally, in order to remove the background andthe gain drift, we first measure the PSD for 1 s with thedrive either on or off, as required. After that, we tunethe qubit away by changing the external flux and repeatthe measurement to obtain the background reference.Finally, we just take the difference between these twomeasurements to obtain either Son or Soff , if the drivewas on or off, respectively.

Quasiparticles. In addition to the hot bath wediscussed, nonequilibrium quasiparticles could also exciteand decay the qubit with rates Γ↑ and Γ↓, respectively.Therefore, we add two additional dissipators (Γ↓D[σ−]ρand Γ↑D[σ+]ρ) into the master equation used for ourmodel [Supplementary Material S2]. By solving themaster equation without external drive, we obtain thelost power, which contains the effects from both the hotbath and quasiparticles as

Ploss

~ω01=

ΓrΓn(nr − nn)− Γr[(nr + 1)Γ↑ − nrΓ↓]

Γ1 + Γ↑ + Γ↓.(12)

When Γn = 0 and nr = 0, the qubit population due

to the nonequilibrium quasiparticles is ρqp11 =

Γ↑Γ↑+Γ↓

.

Experimentally, we can increase the capacitive couplingbetween the waveguide and the qubit in order to make thenonradiative decay negligible (Γr Γn). By suppressingthe noise from the output line, as we did in the main text,we have nr ≈ nn. In these conditions, the lost power ismainly from the quasiparticles. Depending on the valueof the waveguide thermal-photon occupation, we havethree regimes: if nr > Γ↑/(Γ↓−Γ↑), Ploss > 0, the qubit isthermally excited and prefers to be dissipated due to the

7

quasiparticle tunneling through the Josephson junction;if nr < Γ↑/(Γ↓ − Γ↑), Ploss < 0, the quasiparticles excitethe qubit, and then the qubit emits a photon into thewaveguide; if nr = Γ↑/(Γ↓ − Γ↑), Ploss = 0, we canconsider the effects on the qubit from the quasiparticlesand the thermal photons in the waveguide are the same.

The quasiparticle-induced excited-state population canbe approximately written as [53]

ρqp11 ' 2.17

nqp

ncp

(∆g

~ω01

)3.65

, (13)

in which nqp (ncp) is the density of all quasiparticles(Cooper pairs) and ∆g is the superconducting energygap. Combining this with the quasiparticle-induceddecay rate for a transmon qubit [9, 53]

Γ↓ '√

2

RNC

nqp

ncp

(∆g

~ω01

)1.5

, (14)

we have

Γ↓ '√

2

2.17RNC

(∆g

~ω01

)−2.15

· ρqp11 , (15)

where the normal resistance of our SQUID is RN ≈6.3 kΩ, the total capacitance of the qubit is C = 78 fFand ∆g = 170µeV for aluminium. By assuming thatour thermal population is solely from the quasiparticles,i.e., ρqp

11 = ρth11 ≈ 2.86 %, we have Γ↓/2π ≈ 87 kHz.

Therefore, we have Γ↑ ≈ ρqp11Γ↓ ≈ 2π × 3 kHz. Thus,

the quasiparticle-induced nonradiative decay rate wouldbe Γqp = Γ↓ − Γ↑ ≈ 2π × 84 kHz > Γn, see discussion inthe main text.

Finally, as discussed in the main text, we donot observe any considerable decrease in Γn as thetemperature is lowered (by the addition of isolators).Conversely, we also notice that the value of Γn doesnot observably increase. If we assume that Γn issolely from TLSs, then, when we decrease the bathtemperature from 131 mK to 50 mK, the value of Γn

is expected to increase by about 10 kHz according tothe relationship Γn,T = Γn,0 tanh(~ω01/kBT ) [7] (seealso Supplementary Information), where Γn,0 is thenonradiative decay rate due to TLSs at zero temperature.It may be that the high-temperature noise from theoutput may also generate some quasiparticles. Assumingthat the noise temperature is the same as that of the TLSbath, namely, 131 mK, the corresponding quasiparticleinduced decay rate is about 8 kHz, as given by Γqp =ω01

π

√2∆g

~ω01xqp with the normalized quasiparticle density

xqp =

√2π∆gkBT

∆ge−∆g/kBT [9, 67], and said rate would

decrease as the noise temperature is lowered. In total,it is possible that we do not see a change in Γn aftersuppressing the noise because these two effects (TLS rateincreasing, quasiparticle rate decreasing) mitigate eachother.

Measurement setup. Figure 5 shows ourmeasurement setup, where a transmon qubit is

FIG. 5. Experimental setup. Setup inside the dilutionrefrigerator. The signal from the input port is attenuated and fedinto the waveguide through a low-pass filter (LP), Eccosorb, anda -20dB directional coupler. After interacting with the transmon,the signal from the sample goes through the directional coupleragain, passes another LP, two isolators (Iso), and a high-passfilter with amplification from a HEMT amplifier. The qubitfrequency can be changed by sending a current to the magneticflux coil to generate an external magnetic flux Φ through theSQUID.

weakly coupled to a 1D semi-infinite transmissionline. To measure the reflection coefficient, a vectornetwork analyzer (VNA, not shown) generates coherentcontinuous microwaves which are sent into the inputline. This signal is attenuated, pass through a 20 dBdirectional coupler, and then reaches the qubit. Afterthes interaction with the qubit, the electrical field isreflected back, and passes filters, isolators and a HEMTamplifier. Finally, it goes back to the VNA to obtainthe reflection coefficient with some room-temperatureamplifiers.

To measure the power spectral density (PSD) from thethermal noise, turning off the VNA, we use a digitizerto measure the voltage from the output port in thetime domain where the sampling rate is 3 MHz and4 MHz for Fig. 3(a) and (b), respectively. Such ratesare large enough to capture the signal around the qubitin the frequency domain. After that, as discussedabove about the measurement of the PSD, we take theFourier transform to obtain the PSD by using the Welch

8

method [66]. The PSD with strong drive on the qubit isobtained in the same way.

CONTRIBUTIONS

Y.L. planned and performed the measurements.A.B. and Y.L. designed the sample, A.B. fabricated thedevice. N.L. and Y.L. developed the model. N.L., Y.L.,K.F., and A.F.K. performed the theoretical analysis.Y.L. and N.L. wrote the manuscript with help from allauthors. P.D. supervised this work.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with ShahnawazAhmed and Dr. Jonathan Burnett. We also thankDr. Niklas Wadefalk and Dr. Sumedh Mahashabdefor useful discussions about the HEMT amplifiers.

Numerical modelling was performed using the QuTiPlibrary [6, 7]. N.L. acknowledges partial support fromJST PRESTO through Grant No. JPMJPR18GC.F.N. is supported in part by: NTT Research, JapanScience and Technology Agency (JST) (via the Q-LEAPprogram, Moonshot R&D Grant No. JPMJMS2061,and the CREST Grant No. JPMJCR1676), JapanSociety for the Promotion of Science (JSPS) (viathe KAKENHI Grant No. JP20H00134 and theJSPS-RFBR Grant No. JPJSBP120194828), ArmyResearch Office (ARO) (Grant No. W911NF-18-1-0358),Asian Office of Aerospace Research and Development(AOARD) (via Grant No. FA2386-20-1-4069). F.N. andN.L. acknowledge the Foundational QuestionsInstitute Fund (FQXi) via Grant No. FQXi-IAF19-06. We acknowledges the use of the NanofabricationLaboratory (NFL) at Chalmers. Y.L., A.F.K., S.G., andP.D. were supported by the Knut and Alice WallenbergFoundation through the Wallenberg Center for QuantumTechnology (WACQT). Y.L. and P.D. are supported bythe Swedish Research Council [VR Radsprof (5920793)].

[1] X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, andF. Nori, “Microwave photonics with superconductingquantum circuits,” Physics Reports 718, 1 (2017).

[2] M. Kjaergaard, M. E. Schwartz, J. Braumuller, P. Krantz,J. I.-J. Wang, S. Gustavsson, and W. D. Oliver,“Superconducting qubits: Current state of play,” AnnualReview of Condensed Matter Physics 11, 369 (2020).

[3] A. P. Place, L. V. Rodgers, P. Mundada, B. M. Smitham,M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon,S. Sussman, G. Cheng, et al., “New material platformfor superconducting transmon qubits with coherence timesexceeding 0.3 milliseconds,” Nature Communications 12,1779 (2021).

[4] A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. V.Kuzmin, and V. E. Manucharyan, “Millisecond coherencein a superconducting qubit,” (2021), arXiv:2103.08578.

[5] J. Lisenfeld, G. J. Grabovskij, C. Muller, J. H. Cole,G. Weiss, and A. V. Ustinov, “Observation of directlyinteracting coherent two-level systems in an amorphousmaterial,” Nature Communications 6, 6182 (2015).

[6] S. Schlor, J. Lisenfeld, C. Muller, A. Bilmes, A. Schneider,D. P. Pappas, A. V. Ustinov, and M. Weides, “Correlatingdecoherence in transmon qubits: Low frequency noise bysingle fluctuators,” Physical Review Letters 123, 190502(2019).

[7] C. Muller, J. H. Cole, and J. Lisenfeld, “Towardsunderstanding two-level-systems in amorphous solids:insights from quantum circuits,” Reports on Progress inPhysics 82, 124501 (2019).

[8] S. Gustavsson, F. Yan, G. Catelani, J. Bylander,A. Kamal, J. Birenbaum, D. Hover, D. Rosenberg,G. Samach, A. P. Sears, et al., “Suppressing relaxationin superconducting qubits by quasiparticle pumping,”Science 354, 1573 (2016).

[9] G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H.Devoret, and L. I. Glazman, “Quasiparticle relaxation ofsuperconducting qubits in the presence of flux,” Physical

Review Letters 106, 077002 (2011).[10] C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Axline,

T. Brecht, R. W. Heeres, L. Frunzio, M. H. Devoret,G. Catelani, et al., “Measurement and control ofquasiparticle dynamics in a superconducting qubit,”Nature Communications 5, 5836 (2014).

[11] S. de Graaf, L. Faoro, L. Ioffe, S. Mahashabde,J. Burnett, T. Lindstrom, S. Kubatkin, A. Danilov, andA. Y. Tzalenchuk, “Two-level systems in superconductingquantum devices due to trapped quasiparticles,” ScienceAdvances 6, eabc5055 (2020).

[12] X. Jin, A. Kamal, A. Sears, T. Gudmundsen, D. Hover,J. Miloshi, R. Slattery, F. Yan, J. Yoder, T. Orlando,et al., “Thermal and residual excited-state population in a3D transmon qubit,” Physical Review Letters 114, 240501(2015).

[13] J. Wenner, Y. Yin, E. Lucero, R. Barends, Y. Chen,B. Chiaro, J. Kelly, M. Lenander, M. Mariantoni,A. Megrant, C. Neill, P. J. J. O’Malley, D. Sank,A. Vainsencher, H. Wang, T. C. White, A. N. Cleland,and J. M. Martinis, “Excitation of superconducting qubitsfrom hot nonequilibrium quasiparticles,” Physical ReviewLetters 110, 150502 (2013).

[14] D. Riste, C. C. Bultink, K. W. Lehnert, and L. DiCarlo,“Feedback control of a solid-state qubit using high-fidelityprojective measurement,” Physical Review Letters 109,240502 (2012).

[15] L. Cardani, F. Valenti, N. Casali, G. Catelani,T. Charpentier, M. Clemenza, I. Colantoni, A. Cruciani,L. Gironi, L. Grunhaupt, et al., “Reducing the impact ofradioactivity on quantum circuits in a deep-undergroundfacility,” Nature Communications 12, 2733 (2021).

[16] A. P. Vepsalainen, A. H. Karamlou, J. L. Orrell, A. S.Dogra, B. Loer, F. Vasconcelos, D. K. Kim, A. J. Melville,B. M. Niedzielski, J. L. Yoder, et al., “Impact of ionizingradiation on superconducting qubit coherence,” Nature584, 551 (2020).

9

[17] L. Grunhaupt, N. Maleeva, S. T. Skacel, M. Calvo,F. Levy-Bertrand, A. V. Ustinov, H. Rotzinger,A. Monfardini, G. Catelani, and I. M. Pop,“Loss mechanisms and quasiparticle dynamics insuperconducting microwave resonators made of thin-filmgranular aluminum,” Physical Review Letters 121,117001 (2018).

[18] M. McEwen, D. Kafri, Z. Chen, J. Atalaya, K. Satzinger,C. Quintana, P. V. Klimov, D. Sank, C. Gidney,A. Fowler, et al., “Removing leakage-induced correlatederrors in superconducting quantum error correction,”Nature Communications 12, 1761 (2021).

[19] D. Egger, M. Werninghaus, M. Ganzhorn, G. Salis,A. Fuhrer, P. Muller, and S. Filipp, “Pulsed reset protocolfor fixed-frequency superconducting qubits,” PhysicalReview Applied 10, 044030 (2018).

[20] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah,E. Lutz, F. Schmidt-Kaler, and K. Singer, “A single-atomheat engine,” Science 352, 325 (2016).

[21] J. P. Peterson, T. B. Batalhao, M. Herrera, A. M.Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra,“Experimental characterization of a spin quantum heatengine,” Physical Review Letters 123, 240601 (2019).

[22] K. Ono, S. N. Shevchenko, T. Mori, S. Moriyama, andF. Nori, “Analog of a quantum heat engine using a single-spin qubit,” Physical Review Letters 125, 166802 (2020).

[23] H. T. Quan, Y.-x. Liu, C. P. Sun, and F. Nori,“Quantum thermodynamic cycles and quantum heatengines,” Physical Review E 76, 031105 (2007).

[24] G. Maslennikov, S. Ding, R. Hablutzel, J. Gan,A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, andD. Matsukevich, “Quantum absorption refrigerator withtrapped ions,” Nature Communications 10, 202 (2019).

[25] M. Xu, X. Han, C.-L. Zou, W. Fu, Y. Xu, C. Zhong,L. Jiang, and H. X. Tang, “Radiative cooling of asuperconducting resonator,” Physical Review Letters 124,033602 (2020).

[26] W. Schoonveld, J. Wildeman, D. Fichou, P. Bobbert,B. Van Wees, and T. Klapwijk, “Coulomb-blockadetransport in single-crystal organic thin-film transistors,”Nature 404, 977 (2000).

[27] K. Schwab, E. Henriksen, J. Worlock, and M. L. Roukes,“Measurement of the quantum of thermal conductance,”Nature 404, 974 (2000).

[28] C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl,“Solid-state thermal rectifier,” Science 314, 1121 (2006).

[29] M. Partanen, K. Y. Tan, J. Govenius, R. E. Lake, M. K.Makela, T. Tanttu, and M. Mottonen, “Quantum-limitedheat conduction over macroscopic distances,” NaturePhysics 12, 460 (2016).

[30] M. Meschke, W. Guichard, and J. P. Pekola, “Single-mode heat conduction by photons,” Nature 444, 187(2006).

[31] H. T. Quan, Y. D. Wang, Y. X. Liu, C. P. Sun, andF. Nori, “Maxwell’s demon assisted thermodynamic cyclein superconducting quantum circuits,” Physical ReviewLetters 97, 180402 (2006).

[32] M. Naghiloo, D. Tan, P. Harrington, J. Alonso, E. Lutz,A. Romito, and K. Murch, “Heat and work alongindividual trajectories of a quantum bit,” Physical ReviewLetters 124, 110604 (2020).

[33] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffeves, R. Azouit,P. Rouchon, and B. Huard, “Observing a quantum

Maxwell demon at work,” Proceedings of the NationalAcademy of Sciences 114, 7561 (2017).

[34] J. Senior, A. Gubaydullin, B. Karimi, J. T. Peltonen,J. Ankerhold, and J. P. Pekola, “Heat rectificationvia a superconducting artificial atom,” CommunicationsPhysics 3, 40 (2020).

[35] C. Cherubim, F. Brito, and S. Deffner, “Non-thermalquantum engine in transmon qubits,” Entropy 21, 545(2019).

[36] A. Ronzani, B. Karimi, J. Senior, Y.-C. Chang, J. T.Peltonen, C. Chen, and J. P. Pekola, “Tunable photonicheat transport in a quantum heat valve,” Nature Physics14, 991 (2018).

[37] D. Roy, C. M. Wilson, and O. Firstenberg,“Colloquium: Strongly interacting photons in one-dimensional continuum,” Reviews of Modern Physics 89,021001 (2017).

[38] A. F. Kockum and F. Nori, “Quantum bitswith Josephson junctions,” in F. Tafuri (ed.)Fundamentals and Frontiers of the Josephson Effect .(Springer Series in Materials Science, vol 286., 2019) p.703.

[39] J. Monsel, M. Fellous-Asiani, B. Huard, and A. Auffeves,“The energetic cost of work extraction,” Physical ReviewLetters 124, 130601 (2020).

[40] H.-P. Breuer and F. Petruccione, The theory of openquantum systems (Oxford University Press, 2002).

[41] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I.Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin,and R. J. Schoelkopf, “Charge-insensitive qubit designderived from the Cooper pair box,” Physical Review A76, 042319 (2007).

[42] I.-C. Hoi, A. F. Kockum, L. Tornberg, A. Pourkabirian,G. Johansson, P. Delsing, and C. Wilson, “Probing thequantum vacuum with an artificial atom in front of amirror,” Nature Physics 11, 1045 (2015).

[43] P. Y. Wen, A. F. Kockum, H. Ian, J. C. Chen,F. Nori, and I.-C. Hoi, “Reflective amplificationwithout population inversion from a strongly drivensuperconducting qubit,” Physical Review Letters 120,063603 (2018).

[44] Y. Lu, I. Strandberg, F. Quijandrıa, G. Johansson,S. Gasparinetti, and P. Delsing, “Propagating wigner-negative states generated from the steady-state emissionof a superconducting qubit,” Physical Review Letters 126,253602 (2021).

[5] M. Scigliuzzo, A. Bengtsson, J.-C. Besse, A. Wallraff,P. Delsing, and S. Gasparinetti, “Primary thermometry ofpropagating microwaves in the quantum regime,” PhysicalReview X 10, 041054 (2020).

[46] W.-J. Lin, Y. Lu, P. Wen, Y.-T. Cheng, C.-P. Lee, K.-T.Lin, K.-H. Chiang, M. Hsieh, J. Chen, C.-S. Chuu, et al.,“Deterministic loading and phase shaping of microwavesonto a single artificial atom,” arXiv:2012.15084 (2020).

[1] C. Gardiner and P. Zoller, Quantum noise: a handbookof Markovian and non-Markovian quantum stochasticmethods with applications to quantum optics, Vol. 56(Springer Science, 2004).

[4] K. Koshino and Y. Nakamura, “Control of the radiativelevel shift and linewidth of a superconducting artificialatom through a variable boundary condition,” NewJournal of Physics 14, 043005 (2012).

[49] O. Astafiev, A. M. Zagoskin, A. Abdumalikov, Y. A.Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and

10

J. S. Tsai, “Resonance fluorescence of a single artificialatom,” Science 327, 840 (2010).

[3] Y. Lu, A. Bengtsson, J. J. Burnett, E. Wiegand,B. Suri, P. Krantz, A. F. Roudsari, A. F. Kockum,S. Gasparinetti, G. Johansson, and P. Delsing,“Characterizing decoherence rates of a superconductingqubit by direct microwave scattering,” npj QuantumInformation 7, 35 (2021).

[51] S. H. Autler and C. H. Townes, “Stark effect in rapidlyvarying fields,” Physical Review 100, 703 (1955).

[52] F. W. J. Hekking and J. P. Pekola, “Quantum jumpapproach for work and dissipation in a two-level system,”Physical Review Letters 111, 093602 (2013).

[53] J. Wenner, Y. Yin, E. Lucero, R. Barends, Y. Chen,B. Chiaro, J. Kelly, M. Lenander, M. Mariantoni,A. Megrant, et al., “Excitation of superconducting qubitsfrom hot nonequilibrium quasiparticles,” Physical ReviewLetters 110, 150502 (2013).

[54] K. Serniak, M. Hays, G. De Lange, S. Diamond,S. Shankar, L. Burkhart, L. Frunzio, M. Houzet,and M. Devoret, “Hot nonequilibrium quasiparticles intransmon qubits,” Physical Review Letters 121, 157701(2018).

[55] W. D. Oliver and P. B. Welander, “Materials insuperconducting quantum bits,” MRS Bulletin 38, 816(2013).

[56] Z. H. Peng, S. E. De Graaf, J. S. Tsai, and O. V.Astafiev, “Tuneable on-demand single-photon source inthe microwave range,” Nature Communications 7, 12588(2016).

[57] Y. Zhou, Z. H. Peng, Y. Horiuchi, O. V. Astafiev, andJ. S. Tsai, “Tunable microwave single-photon source basedon transmon qubit with high efficiency,” Physical ReviewApplied 13, 034007 (2020).

[58] X. You, A. A. Clerk, and J. Koch, “Positive-andnegative-frequency noise from an ensemble of two-levelfluctuators,” Physical Review Research 3, 013045 (2021).

[59] L. Spietz, K. Lehnert, I. Siddiqi, and R. Schoelkopf,“Primary electronic thermometry using the shot noise ofa tunnel junction,” Science 300, 1929 (2003).

[60] L. Wang, O.-P. Saira, and J. Pekola, “Fast thermometrywith a proximity Josephson junction,” Applied PhysicsLetters 112, 013105 (2018).

[61] Z. Wang, M. Xu, X. Han, W. Fu, S. Puri, S. M. Girvin,H. X. Tang, S. Shankar, and M. H. Devoret, “Quantummicrowave radiometry with a superconducting qubit,”Physical Review Letters 126, 180501 (2021).

[62] B. Albanese, S. Probst, V. Ranjan, C. W. Zollitsch,M. Pechal, A. Wallraff, J. J. Morton, D. Vion, D. Esteve,E. Flurin, et al., “Radiative cooling of a spin ensemble,”Nature Physics 16, 751 (2020).

[63] M. Mirhosseini, A. Sipahigil, M. Kalaee, and O. Painter,“Superconducting qubit to optical photon transduction,”Nature 588, 599 (2020).

[64] W. Hease, A. Rueda, R. Sahu, M. Wulf, G. Arnold, H. G.Schwefel, and J. M. Fink, “Bidirectional electro-opticwavelength conversion in the quantum ground state,”PRX Quantum 1, 020315 (2020).

[65] X. Han, W. Fu, C. Zhong, C.-L. Zou, Y. Xu, A. Al Sayem,M. Xu, S. Wang, R. Cheng, L. Jiang, et al., “Cavity piezo-mechanics for superconducting-nanophotonic quantuminterface,” Nature Communications 11, 3237 (2020).

[66] P. Welch, “The use of fast Fourier transform for theestimation of power spectra: a method based on timeaveraging over short, modified periodograms,” IEEETransactions on Audio and Electroacoustics 15, 70 (1967).

[67] R. Barends, J. Wenner, M. Lenander, Y. Chen, R. C.Bialczak, J. Kelly, E. Lucero, P. OMalley, M. Mariantoni,D. Sank, et al., “Minimizing quasiparticle generation fromstray infrared light in superconducting quantum circuits,”Applied Physics Letters 99, 113507 (2011).

[6] J. Johansson, P. Nation, and F. Nori, “Qutip: Anopen-source python framework for the dynamics of openquantum systems,” Computer Physics Communications183, 1760 (2012).

[7] J. Johansson, P. Nation, and F. Nori, “QuTiP 2: Apython framework for the dynamics of open quantumsystems,” Computer Physics Communications 184, 1234(2013).

11

Supplementary Material for Nonequilibrium heat transport and work with a singleartificial atom coupled to a waveguide: emission without external driving

Yong Lun1,∗, Neill Lambertn2,∗, Anton Frisk Kockum1, Ken Funo2, Andreas Bengtsson1, Simone Gasparinetti1,Franco Norin2,3, Per Delsing1

1Microtechnology and Nanoscience, MC2, Chalmers University of Technology, SE-412 96 Goteborg, Sweden2Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan

3Department of Physics, The University of Michigan, Ann Arbor, 48109-1040 Michigan, USA

S1. MASTER EQUATION

The dynamics and steady state of the qubit in contact with two bosonic heat baths can be found, in the weak-coupling and Markovian approximations, by solving the equation of motion

∂

∂tρS(t) = − i

~[Hq, ρ(t)] + L[ρ(t)], (S1)

where

L[ρ(t)] =Γr

2(nr + 1)D[σ−]ρ+

Γr

2nrD[σ+]ρ+

Γn

2(nn + 1)D[σ−]ρ+

Γn

2nnD[σ+]ρ+

Γφ4D[σz]ρ, (S2)

where the Lindblad operator is D[σi]ρ(t) = 2σiρ(t)σ†i −σ†iσi, ρ(t), and the thermal occupation of the bosonic baths

are given by ni = [exp(~ω01/kBTi)− 1]−1, where kB is the Boltzmann constant.The qubit Hamiltonian is

Hq

~= −∆

2σz +

Ω

2σx, (S3)

where the detuning ∆ = ωp − ω01 is the energy difference between the drive at frequency ωp and the bare qubitfrequency ω01.

In the above, we explicitly assume that the qubit only has two levels, and that the radiative (transmission line) andnon-radiative baths are bosonic, and obey the Born-Markov secular (BMS) approximation. The influence of a thirdlevel is described below.

The second assumption, the bosonicity of the non-radiative bath, can be replaced by assuming a TLS bath. Thissimply changes the temperature dependence of the Lindblad operators for that bath, such that

Γn

2(nn + 1)D[σ−]→ Γn

2

(1 + nn

1 + 2nn

)D[σ−],

Γn

2(nn)D[σ+]→ Γn

2

(nn

1 + 2nn

)D[σ+]. (S4)

Ultimately, this implies some ambiguity in the temperature ascertained from the height of the thermal peak in thespectrum calculation. For simplicity we use that obtained from the bosonic assumption, but note that a TLS bathrequires a larger temperature to produce the equivalent thermal peak in the spectrum, since kBTn = ~ω01/[ln(1 ±nn)− ln(nn)], where ± = + for a bosonic bath, and ± = − for a TLS bath. For example, for the ∆n = 0.135 observedin the main text we would find, assuming nr = 0, kBTn/~ω01 = 0.47 for a bosonic bath and 0.54 for a TLS one.

It is convenient to use the Heisenberg equations of motion generated by Eq. (S1) for the operators s1(t) = 〈σ−(t)〉,s∗1(t) = 〈σ+(t)〉, and s2(t) = 〈σ+(t)σ−(t)〉:

d

dt

s1

s∗1s2

= M

s1

s∗1s2

+B, (S5)

where

M =

i∆− Γ2 0 iΩ0 −i∆− Γ2 −iΩ∗

iΩ∗/2 −iΩ/2 −Γ1

(S6)

12

and

B =

−iΩ/2iΩ∗/2Γ+

(S7)

and where, as in the main text, Γ+ = nnΓn + nrΓr, Γ1 = (1 + 2nn)Γn + (1 + 2nr)Γr, and Γ2 = Γφ + Γ1/2. For thesteady state t→∞ this gives

〈σ−(t→∞)〉 =Ω(Γ1 − 2Γ+) (∆− iΓ2)

2Ω2Γ2 + 2(∆2 + Γ22)Γ1

(S8)

〈σ+σ−(t→∞)〉 =|Ω|2Γ2 + 2Γ+(∆2 + Γ2

2)

2|Ω|2Γ2 + 2(∆2 + Γ22)Γ1

(S9)

〈σ−σ+(t→∞)〉 =|Ω|2Γ2 + 2Γ−(∆2 + Γ2

2)

2|Ω|2Γ2 + 2(∆2 + Γ22)Γ1

, (S10)

where in the last equation Γ− = (nn + 1)Γn + (nr + 1)Γr.

S2. REFLECTIVITY

To calculate the reflectivity, we use Eq. (S8) with

r =bout

bin= 1− i2Γr

Ω〈σ−(t→∞)〉, (S11)

where assuming weak drive Ω Γ1 gives

r = 1− iΓr(1− 2Γ+/Γ1)

∆ + iΓ2. (S12)

S3. POWER OUTPUT

Recalling from the main text, we employ input-output theory for the output of the transmission line bout(t) =bin(t)− i

√Γrσ−(t), where bin(t) = fin(t) + Ω

2√

Γ1includes the thermal noise fin(t) and coherent drive Ω

2√

Γ1. To obtain

the correct expression for the output power (and the power spectral density) we need to calculate correlations betweenthe input thermal field in the waveguide and the qubit of the form 〈σ+(t)fin(t′)〉. We can can evaluate these using theresult from Ref. [1], where, assuming the effect of the waveguide on the dynamics of the system obeys the standardBMS approximation, it is possible to show that for t < t′, the input field has not yet interacted with the qubit, so〈σ+(t)fin(t′)〉 = 0. For t = t′ and t > t′, the thermal input can be correlated with the qubit, and the following holds:

〈σ+(t)fin(t′)〉 = −i√

ΓrnrΘ(t− t′)〈[σ+(t), σ−(t′)

]〉 (S13)

and

〈f†in(t)σ−(t′)〉 = i√

ΓrnrΘ(t′ − t)〈[σ+(t), σ−(t′)

]〉. (S14)

Splitting the input field into thermal terms and coherent terms, one can evaluate the output field intensity correlatoras

〈b†out(t)bout(t′)〉 = Γr (nr + 1) 〈σ+(t)σ−(t′)〉 − Γrnr〈σ−(t′)σ+(t)〉+ 〈f†in(t)fin(t′)〉

− iΩ∗

2〈σ−(t′)〉+

iΩ

2〈σ+(t)〉+

|Ω|2

4Γr. (S15)

With this we can also evaluate the equal-time terms for the power output, but it is more instructive to show theevaluation more explicitly, following the steps outlined in Ref. [2] modified to accommodate a qubit instead of cavity.

13

As described in the main text, we start with the full system Hamiltonian for the qubit and the two environments,

Hsys = Hq +Hr +Hn. (S16)

The radiative (Hr) and non-radiative bath Hamiltonians (Hn) are

Hi

~=∑k

ωk,ia†k,iak,i +

∑k

gk,i

(σ−a

†k,i + σ+ak,i

), (S17)

and include the interaction with the qubit.First we absorb the non-radiative bath into the qubit Hamiltonian,

H ′q = Hq +Hn. (S18)

This gives us Langevin equations for the system operators coupled to the waveguide (setting ~ = 1 in the followingsteps for simplicity),

σ− = −i[σ−, Hsys] = −i[σ−, H ′q]− i[σ−, σ+]∑

k

gk,rak,r. (S19)

We then define the spectral density for the transmission line as J(ω)r = π∑k |gk,r|2δ(ω−ωk,r), insert the definition

of the integral of the transmission-line modes into Eq. (S19), and make the standard Markovian approximationJ(ω)r = Γr. Evaluating the result gives

σ− = −i[σ−, H ′sys]−Γr

2σ− + iσz

√Γrbin(t). (S20)

To obtain correlation functions between the thermal input and system operators, we omit the coherent contributionto bin(t) = fin(t) + Ω

2√

Γ1and multiply the equation of motion from the left first with σz and then with the required

system operator for the correlation function we wish to evaluate, say σ+. Rearranging gives the form

i√

Γr〈σ+fin〉 = −〈σ+σ−〉 −Γr

2〈σ+σ−〉 − i〈σ+[σ−, H

′q]〉 (S21)

To evaluate the term 〈σ+σ−〉 we use the formula

2〈O1O2〉 =d

dt〈O1O2〉+ Tr

O2Lr[ρO1]−O1Lr[O2ρ]

, (S22)

where ρ is the system density matrix after tracing out the waveguide, and Lr is the Liouvillian describing the evolutionof the system in contact with the waveguide,

Lr[ρ] = −i[H ′sys, ρ] +Γr

2(nr + 1)D[σ−]ρ+

Γr

2nrD[σ+]ρ. (S23)

Evaluating this gives

〈σ+σ−〉 = −i〈σ+[σ−, H′sys]〉 −

Γr

2〈σ+σ−〉 −

Γrnr

2〈[σ+, σ−]〉 (S24)

Combining this with Eq. (S21) we find

i√

Γr〈σ+fin〉 =Γrnr

2〈[σ+, σ−]〉. (S25)

Performing the same steps for the other system-thermal noise correlator, and inserting into the output-power formula,we find, as expected

〈b†outbout〉 = Γr (nr + 1) 〈σ+σ−〉 − Γrnr〈σ−σ+〉+ 〈f†infin〉

− iΩ∗

2〈σ−〉+

iΩ

2〈σ+〉+

|Ω|2

4Γr. (S26)

One can use this to show that the power loss Ploss = ~ω01

(〈b†inbin〉 − 〈b

†outbout〉

)under zero drive (Ω = 0) is given

by

Ploss =~ω01ΓrΓn(nr − nn)

Γ1. (S27)

14

S4. POWER SPECTRAL DENSITY

As described in the main text, to evaluate the power spectral density

S(ω) =~ω01

2π

∫ ∞−∞

dte−iωt〈b†out(t)bout(t′)〉(t′→∞) (S28)

we need to evaluate the correlation functions contained in Eq. (S15) for the system operators. In some simple cases,this can be done by hand. For example, in the zero-drive limit, one can easily show that

〈σ+(t)σ−(t′)〉t′→∞ = e−Γ2|t|e−i∆t〈σ+σ−(t′)〉

and

〈σ−(t′)σ+(t)〉t′→∞ = e−Γ2|t|e−i∆t〈σ−σ+(t′)〉,

where 〈σ+σ−(t′)〉t′→∞ = Γ+

Γ1and 〈σ−σ+(t′)〉t′→∞ = Γ−

Γ1. Combining these results with Eq. (6) in the main text and

Eq. (S28) gives Eq. (7) in the main text.A general solution can be conveniently found by combining Eq. (S5) with the quantum regression theorem to define

the two-time correlation functions, and the Fourier transform can be performed following the approach used in [3, 4].The result is cumbersome, but taking the strong-driving limit we obtain the result in Eq. (8) in the main text.

S5. THREE-LEVEL MASTER EQUATION

To evaluate the influence of higher levels in the transmon on our results, we consider a three-level master-equationmodel. We describe the energy difference between levels 1 and 2 with the parameter ω12, such that the anharmonicityis given by δ/2π = (ω12 − ω01)/2π = −250 MHz.

To describe the Autler-Townes splitting in the main text we apply two drives at two different frequencies, and inthe rotating frame of the two-drive terms we find the system Hamiltonian to be

HS

~= (ω01 − ω(1)

p )|1〉〈1|+ (ω12 + ω01 − ω(1)p − ω(2)

p )|2〉〈2|+ Ω1

2σ(01)x +

Ω2

2

√2σ(12)

x , (S29)

where ω(i)p is the drive frequency of input drive i, and we define σ

(ij)x = |i〉〈j|+ |j〉〈i|, σ(ij)

− = |i〉〈j|, and σ(ij)+ = |j〉〈i|.

Since we are in a regime where the anharmonicity of the qubit is larger than the decoherence rates [5], our masterequation is now

∂

∂tρS(t) = − i

~[HS , ρ(t)]

+Γr

2

(n(01)

r + 1)D[σ

(01)− ]ρ(t) +

Γr

2n(01)

r D[σ(01)+ ]ρ(t)

+Γn

2

(n(01)

n + 1)D[σ

(01)− ]ρ(t) +

Γn

2n(01)

n D[σ(01)+ ]ρ(t)

+ Γr

(n(12)

r + 1)D[σ

(12)− ]ρ(t) + Γrn

(12)r D[σ

(12)+ ]ρ(t)

+ Γn

(n(12)

n + 1)D[σ

(12)− ]ρ(t) + Γnn

(12)n D[σ

(12)+ ]ρ(t)

+Γφ2

∑i

D[|i〉 〈i|]ρ(t). (S30)

The output power can be extended to consider two independent coupling operators associated with σ(01)− and σ

(12)− .

Importantly, the |1〉 ↔ |2〉 transition has a dipole moment which is√

2 times larger than the |0〉 ↔ |1〉 transition, andwhich enhances the coupling to the transmission line, increasing both the Rabi drive term and the dissipation rates.

For weak drives on the |0〉 ↔ |1〉 transition alone we can observe that the main influence of the third level isto induce an additional dephasing on the qubit proportional to the thermal excitation rate from the second to thethird level. To evaluate the reflectivity and power spectrum in this case, it is convenient to again work with theHeisenberg equations of motion, where in the limit of Ω2 = 0 we obtain a closed set of equations for the operators

w1(t) = 〈σ(01)− (t)〉, w∗1(t) = 〈σ(01)

+ (t)〉, w2(t) = 〈σ(01)+ (t)σ

(01)− (t)〉 and w3(t) = 〈σ(01)

− (t)σ(01)+ (t)〉, using the normalization

condition 〈σ(01)+ (t)σ

(01)− (t)〉+ 〈σ(01)

− (t)σ(01)+ (t)〉+ 〈σ(12)

+ (t)σ(12)− (t)〉 = 1:

15

d

dt

w1

w∗1w2

w3

= M2

w1

w∗1w2

w3

+B2, (S31)

where

M2 =

i∆−ΓT

2 0 iΩ1

2−iΩ1

2

0 −i∆−ΓT2

−iΩ1

2iΩ1

2iΩ1

2−iΩ1

2 −Γ (01)− −Γ (12)

1 Γ(01)+ −Γ (12)

−−iΩ1

2iΩ1

2 Γ(01)− −Γ (01)

+

and

B2 =

00

Γ(12)−0

(S32)

and where we use ΓT2 =Γ

(01)2 +Γ

(12)+ , indicating the added dephasing from thermalization of the |1〉 ↔ |2〉 transition,

and where Γ(01)2 = Γφ+Γ

(01)1 /2, Γ

(ij)1 = Γ

(ij)+ +Γ

(ij)− , Γ

(ij)+ = Γrn

(ij)r + Γnn

(ij)n , Γ

(ij)− = Γr(n

(ij)r + 1) + Γn(n

(ij)n + 1), and

∆01 = (ω01 − ω(1)p ). Note that here we have accounted for the larger rates in the |1〉 ↔ |2〉 transitions in Eq. (S30),

hence parameters like Γ(12)+ are just for convenience, and differ from the actual thermalization rate of the |1〉 ↔ |2〉

transition by a factor of 2.Reflection coefficient and PSD. Evaluating the reflectivity of the |0〉 ↔ |1〉 transition with the above equations,

in the limit of weak drive Ω1, gives

r = 1−iΓr(1− 2Γ

(01)+ /Γ

(01)1 )

∆ + i(Γ(01)2 + Γ

(12)+ )

G, (S33)

where

G =Γ

(12)− Γ

(01)1

(Γ(01)+ Γ

(12)+ + Γ

(12)− Γ

(01)1 )

≈ 1. (S34)

Similarly, for the output power spectrum, we can obtain the strong-drive (large Ω1) result by generalizing the stepsused in the two-level case. For the central peak, around the |0〉 ↔ |1〉 qubit transition frequency, we find

S(ω) =~ω01Γr

2π

ΓT2

(ω − ω01)2

+ (ΓT2 )2F , (S35)

where

F =Γ

(12)−

2Γ(12)− + Γ

(12)+

= [〈σ(01)+ σ

(01)− 〉(t→∞) = 〈σ(01)

− σ(01)+ 〉(t→∞)]Ω1→∞.

Intuitively, we see here that under a strong drive Ω1 we still observe the added dephasing (ΓT2 = Γ

(01)2 +Γ

(12)+ ), as

well as a small change in the steady-state population, on top of the expected result from the two-level model understrong drive, which captures the thermalization with the third level.

For no drive, Ω1 → 0, we find the thermal spectrum as

S(ω) =~ω01Γr

2π

2ΓT2 Γn∆n

(ω − ω01)2

+ (ΓT2 )2Y, (S36)

where

Y =Γ

(12)−

Γ(12)− Γ

(01)1 + Γ

(12)+ Γ

(01)+

(S37)

16

and ∆n = n(01)n − n(01)

r .The side-peak contributions to the Autler-Townes data presented in the main text are based on applying a drive

on the |1〉 ↔ |2〉 while monitoring the emission from the |0〉 ↔ |1〉 transition. We can also obtain these side-peaksanalytically, in a similar fashion as the above calculations. The Heisenberg equations of motion in this case, for Ω1 = 0

and Ω2 6= 0, form a closed set of equations for the operators z1(t) = 〈σ(12)− (t)〉, z∗1(t) = 〈σ(12)

+ (t)〉, z2(t)〈σ(12)+ (t)σ

(12)− (t)〉,

z3(t) = 〈σ(12)− (t)σ

(12)+ (t)〉, z4 = 〈σ(01)

− (t)〉, z4∗ = 〈σ(01)+ (t)〉, z5 = 〈σ(02)

− (t)〉, and z5∗ = 〈σ(02)+ (t)〉. The full equations

of motion and the full result for the output spectrum are cumbersome, but evaluating the side-peaks for the limit oflarge drive Ω2 we find

S(ω)± =~ω01Γr

2π

2(ΓT

2 + Γ(02)2

)(Γ

(01)+ + n

(01)r

[Γ

(01)1 − 2Γ

(12)−

])(

4[ω − ω01 ± Ω2√

2

]2+[ΓT

2 + Γ(02)2

]2)(2[Γ

(12)− + Γ

(01)+

]− Γ

(01)−

) . (S38)

Here we introduced a new parameter which describes the dephasing rate of z5, Γ(02)2 = Γφ + Γ

(01)+ /2 + Γ

(12)− .

If we assume n(01)n = n

(12)n and n

(01)r = n

(12)r , the second term in the numerator simplifies to(

Γ(01)+ + n(01)

r

[Γ

(01)1 − 2Γ

(12)−

])= 2Γn∆n. (S39)

The various analytical results obtained in this supplementary material were checked against numerical simulationsusing QuTiP [6, 7].

[1] Crispin Gardiner and Peter Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochasticmethods with applications to quantum optics, Vol. 56 (Springer Science, 2004).

[2] H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer-Verlag, Berlin, Heidelberg, 1999).[3] Yong Lu, Andreas Bengtsson, Jonathan J Burnett, Emely Wiegand, Baladitya Suri, Philip Krantz, Anita Fadavi Roudsari,

Anton Frisk Kockum, Simone Gasparinetti, Goran Johansson, and Per Delsing, “Characterizing decoherence rates of asuperconducting qubit by direct microwave scattering,” npj Quantum Information 7, 35 (2021).

[4] K. Koshino and Y. Nakamura, “Control of the radiative level shift and linewidth of a superconducting artificial atom througha variable boundary condition,” New Journal of Physics 14, 043005 (2012).

[5] Marco Scigliuzzo, Andreas Bengtsson, Jean-Claude Besse, Andreas Wallraff, Per Delsing, and Simone Gasparinetti,“Primary thermometry of propagating microwaves in the quantum regime,” Physical Review X 10, 041054 (2020).

[6] J.R. Johansson, P.D. Nation, and F. Nori, “Qutip: An open-source python framework for the dynamics of open quantumsystems,” Computer Physics Communications 183, 1760 (2012).

[7] J.R. Johansson, P.D. Nation, and F. Nori, “QuTiP 2: A python framework for the dynamics of open quantum systems,”Computer Physics Communications 184, 1234 (2013).