Resonance inversion in a superconducting cavity coupled to artificial atoms and amicrowave background

Juha Leppakangas,1, 2 Jan David Brehm,1 Ping Yang,1 Lingzhen Guo,3

Michael Marthaler,2, 4, 5 Alexey V. Ustinov,1, 6 and Martin Weides1, 7

1Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany2HQS Quantum Simulations GmbH, 76131 Karlsruhe, Germany

3Max Planck Institute for the Science of Light, 91058 Erlangen, Germany4Institut fur Theorie der Kondensierten Materie,

Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany5 Theoretische Physik, Universitat des Saarlandes, 66123 Saarbrucken, Germany

6Russian Quantum Center, National University of Science and Technology MISIS, 119049 Moscow, Russia7School of Engineering, University of Glasgow, Glasgow G12 8QQ, UK

We demonstrate how heating of an environment can invert the line shape of a driven cavity. Weconsider a superconducting coplanar cavity coupled to multiple artificial atoms. The measured cavitytransmission is characterized by Fano-type resonances with a shape that is continuously tunable bybias current through nearby (magnetic flux) control lines. In particular, the same dispersive shiftof the microwave cavity can be observed as a peak or a dip. We find that this Fano-peak inversionis possible due to a tunable interference between a microwave transmission through a background,with reactive and dissipative properties, and through the cavity, affected by bias-current inducedheating. The background transmission occurs due to crosstalk between the control and transmissionlines. We show how such background can be accounted for by Jaynes-Cummings type models viamodified boundary conditions between the cavity and transmission lines. We find generally thatwhereas resonance positions determine system energy levels, resonance shapes give information onsystem fluctuations and dissipation.

I. INTRODUCTION

A Fano resonance [1] is a fundamental effect of wavepropagation. It appears in a wide range of physical sys-tems, including light propagation in photonic devices [2],light interaction with nano- and microstructures [3–7],charge transport in nanoelectronics [8], and inelastic scat-tering of elementary particles [1]. The Fano resonanceemerges due to an interference effect between two paral-lel paths connecting input and output scattering states:transmission through a continuous-mode or wide back-ground state and transmission through a discrete or nar-row energy-level. The resulting resonance can be bothadmitting or reflecting (anti-resonance) depending on thedetails of the system. As many other interference effects,it has numerous practical applications in metrology andoptical engineering [2, 8].

In microwave devices, wanted or unwanted Fano-typeresonances can easily emerge, for example, from ca-pacitive or inductive background coupling between dif-ferent ports of a resonant circuit [9]. In the follow-ing, we call this type of coupling a microwave back-ground. In state-of-the-art superconducting quantum-information devices [10–13] a large number of qubits withtheir control lines are integrated into a small-sized chip,with possible further size-optimization to reduce deco-herence mechanisms, such as non-equilibrium quasipar-ticle tunneling [14] or field focusing [15]. A quantum-state measurement in such circuits is commonly based onmicrowave transmission through readout resonators [16].When increasing circuit complexity and scaling up qubitnumbers, it is difficult to avoid multiple interference

paths for microwave signals propagating through the cir-cuit. It is then also of high interest to understand indetail how Fano resonances can appear in such circuitsand how to account for them in most commonly usedtheoretical models.

In this article, we investigate microwave transmissionacross a superconducting coplanar resonator coupled tomultiple artificial atoms. The measured transmission ischaracterized by Fano-type resonances with a shape thatis continuously tunable by current bias through nearbymagnetic-flux control lines. In particular, we observethat the very same dispersive shift of the cavity can beseen as either a peak or a dip, depending on the cur-rent bias. The experiment is also characterized by largeoff-resonance transmission.

We investigate the observed effects further by estab-lishing a theoretical model for cavity transmission in thepresence of a microwave background. The backgroundtransmission accounts for a crosstalk between the inputand output transmission lines through control lines by-passing the coplanar microwave cavity. We find that thewell-known Jaynes- and Tavis-Cummings models [17–20]of the cavity-atom interaction are valid also for the con-sidered system. The background transmission can be ac-counted for by modifying boundary conditions betweenthe cavity and transmission-line microwave fields.

Using the established model, we find that dissipationand incoherent transitions can strongly affect the cavityline shape. A transformation from a peak to a dip, whichwe call here a resonance inversion, is possible through achanging interference between a background microwavetransmission, with reactive and dissipative properties,

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and a cavity transmission, with dissipative or incoherentdynamics. A dip appears when the cavity transmission iscomparable with the background transmission and dissi-pation.

Based on the established model, we explain the ob-served experimental features as an interplay betweenbias-induced heating and background transmission: Theapplied DC bias currents cause local heating of the biasleads, which in turn induce incoherent energy-level tran-sitions in the cavity-atom system. Increasing rates ofsuch transitions, by increasing the bias currents, reducesthe cavity transmission and smoothly changes the spec-troscopic response of the device from a peak to a dip, thelatter appearing when the cavity transmission becomescomparable with the background transmission and dissi-pation. This allows for the full tunability of the Fanoresonance shape. We also demonstrate how the localtemperature of the system as well as average qubit T1

and T2 times can be estimated from the line-shape of theresonator.

The article is organized as follows. In Sec. II, we in-troduce our model of microwave transmission across acoplanar resonator coupled to artificial atoms. We showhow microwave propagation beside the cavity can be in-cluded by additional terms in the cavity-line boundaryconditions. In Sec. III, we study theoretically the ef-fect of dissipation and decoherence on the form of Fanoresonances. In particular, we show how the line shapeof a microwave resonator can change from a peak to adip in various situations. In Sec. IV, we present our ex-perimental results and estimate the local temperature ofthe cavity-qubit system under different biasing conditionsfrom the line shape of the resonator. We also discuss howto estimate average qubit T1 and T2 times from the res-onator line shape. Conclusions and discussion are givenin Sec. V.

II. SYSTEM AND MODEL

The system we consider is shown in Fig. 1(a). Mi-crowaves propagate in two semi-infinite transmissionlines (TLs). The TLs are connected to each other througha two-sided cavity and through a background. The cav-ity is described by a resonant element Z and couplingcapacitors Cci. The background coupling between theTLs is described by a parallel impedance Zb. Multiplesuperconducting artificial atoms (transmons) can be em-bedded in the cavity, affecting its resonance frequency, asdescribed below.

We start building a theoretical model for this circuit inSec. II A by introducing a quantized model of microwaveradiation in TLs. In Sec. II B, we account for the couplingbetween the cavity and the TLs and in Sec. II C betweenthe cavity and the artificial atoms. In Sec. II D, we showhow the parallel transmission channel can be included inthe model by additional terms in microwave boundaryconditions. A full solution in the linear limit is derived

FIG. 1: (a) Incoming and outgoing microwave fields propa-gate in two semi-infinite transmission lines with characteristicimpedance Z0. They are connected via capacitors Cc to a mi-crowave resonator modeled by an element Z. The transmis-sion lines are also connected directly to each other throughbackground impedance Zb. (b) A coplanar microwave res-onator can be modeled as an LCR resonator [27]. (c) N su-perconducting transmon artificial atoms [28] (crossed boxes)embedded in the coplanar resonator interact with the cavitymode through coupling capacitors Cgi.

in Sec. II E and a master equation for simulation in amore general situation is given in Sec. II F.

A. Microwave radiation in open transmission lines

1. Radiation states

Microwave radiation in the left TL can be described bya traveling-field solution for the magnetic flux [17, 29–31]

Φ(x < 0, t) = (1)√~Z0

2ω0

[aL

in(t− x/c) + aLout(t+ x/c) + H.c.

].

We associate the position x = 0 as the cavity bound-ary and the space is semi-infinite, −∞ < x < 0. Thecapacitance C ′ and inductance L′ per unit length de-fine the characteristic impedance Z0 =

√L′/C ′ and the

effective speed of light c = 1/√L′C ′. We work in the

narrow-bandwidth approximation [29] and therefore thecavity resonance frequency ω0 (defined more detailed be-

3

low) appears in the equation. The operator aLin(t) anni-

hilates and aLin(t)† creates an incoming photon. These

operators satisfy the commutation relations[aL

in(t), aLin(t′)†

]= δ(t− t′) . (2)

The same relation is valid also for the correspondingoutgoing-field photon operators aL

out(t). We can also de-fine photon operators of specific frequencies

aLin/out(ω) =

1√2π

∫ ∞−∞

dteiωtaLin/out(t) . (3)

We have then

aLin/out(t) =

1√2π

∫ ∞−∞

dωe−iωtaLin/out(ω) . (4)

The commutation relations of these fixed-frequency op-erators have the form[

aLin(ω), aL

in(ω′)†]

= δ(ω − ω′) . (5)

A similar definition is made for the operators decribingfields on the right of the cavity, aR

in and aRout.

2. Microwave transmission and reflection

The microwave properties we study in this paper arethe microwave reflection, s11(ω), and the microwavetransmission, s12(ω). They are defined through the out-put amplitudes when having a coherent input from oneside and no input from the other side. By assuming acoherent input of frequency ω from the left, we define

s11(ω) =〈aL

out(ω)〉〈aL

in(ω)〉(6)

s12(ω) =〈aR

out(ω)〉〈aL

in(ω)〉. (7)

In theoretical modeling, these relations can be deter-mined through solving microwave boundary conditionsand cavity equations of motion, as described below.

B. Open transmission line connected to cavity

We continue building the model by considering first asituation where there are no transmons embedded in thecavity, i.e., the setup of Fig. 1(b). We further assumethe absence of parallel transmission (Zb = ∞) and thatthe cavity has no intrinsic dissipation (R = ∞). Wedescribe the photonic state inside the cavity by a single-mode Hamiltonian

H0 = ~ω0a†a . (8)

Here the operator a(†) is the cavity photon annihilation(creation) operator satisfying [a, a†] = 1. The coupling-normalized resonance frequency is

ω0 =1√

L(C + 2Cc). (9)

Here L (C) is the inductance (capacitance) of the cavityand Cc the coupling capacitance between the cavity anda TL. From here on we assume the case of symmetriccoupling, Cc1 = Cc2 = Cc.

The interaction between the semi-infinite TLs and thecavity is described by boundary conditions at the twosides of the cavity [17]. On the left-hand side we write

aLout(t) =

√γa(t)− aL

in(t) . (10)

The operators are time dependent since the condition isgiven in the Heisenberg picture. Similarly for the right-hand side,

aRout(t) =

√γa(t)− aR

in(t) . (11)

The decay rate is now identical in the two directions andhas the form

γ =

(Cc

C + 2Cc

)2Z0

ZLCω0 , (12)

where the characteristic impedance of the resonator isZLC =

√L/(C + 2Cc). This treatment of the cavity

field is valid for high quality factors, Q = ω0/2γ 1.

The cavity field operator also satisfies the Heisenbergequation of motion [17]

ˆa(t) =i

~

[H0, a(t)

]− γa(t) +

√γ[aL

in(t) + aRin(t)

]. (13)

This is found to be also more generally valid, with properredefinition of Hamiltonian H0, accounting for the pres-ence of artificial atoms, and also in the presence of reac-tive and dissipative parallel coupling.

C. Cavity interacting with artificial atoms

It is straightforward to account for the presence of ar-tificial atoms in the above treatment. We first considerthe case of including them as two-level systems and afterthis generalize the treatment to the case of multi-levelatoms.

1. Interaction with two-level systems

For transverse cavity-atom couplings gi ω0, the sys-tem is decribed in the rotating-wave approximation by

4

the Tavis-Cummings Hamiltonian [17]

H0 = ~ω0a†a+~

n∑i=1

Ωi2σiz+

n∑i=1

~gi(a†σi− + aσi+

). (14)

Here σi+(−) is the spin raising (lowering) operator of two-

level system i. Boundary conditions (10-11) and Heisen-berg equation of motion (13) keep their form.

Here, when gi |∆i|, where ∆i = Ωi − ω0, the sys-tem shows effectively a longitudinal coupling between theresonator and the two-level systems. This is called a dis-persive coupling regime. For a single-atom environment,the resulting Hamiltonian has the form [10, 32]

H ′0 = ~ (ω0 + χσz) a†a+

~2

(Ω + χ) σz . (15)

Here we have defined the dispersive shift

χ =g2

∆. (16)

Corrections to this Hamiltonian are higher orders ing/∆ 1. We see that the effective resonance fre-quency of the cavity then depends on the state of thetwo-level system and can have values ω0 ± χ. This typeof Hamiltonian can be used to describe a driven cav-ity if the neglected energy-level anharmonicity is smallcompared to the energy-level broadening [22]. For lowphoton-number distributions the relevant condition isg2/|∆| × (g/∆)2 γ, which is assumed to be true whenconsidering a dispersive regime in this article. This cou-pling regime is commonly used for readout of supercon-ducting qubits [16].

A generalization of the dispersive Hamiltonian to ac-count for multiple two-level systems is straightforward,

H ′0 = ~

(ω0 +

n∑i=1

χiσiz

)a†a+

~2

n∑i=1

(Ωi + χi) σiz .

Here χi = g2i /∆i, gi |∆i|, and we have neglected

cavity-mediated couplings between the two-level sys-tems [32, 33], i.e., we assume gigj/|∆i/j | (Ωi − Ωj)

2.

2. Interaction with multi-level systems

In our experiment, the cavity interacts with trans-mon artificial atoms. For multi-level artificial atoms withlow anharmonicity like the transmon also higher energylevels need to be accounted for [28]. This can be thecase even though only two lowest (photon-dressed) atomstates would be populated.

A Hamiltonian of a resonator connected to one multi-

level atom can be written generally in the form

H0 = ~ω0a†a+

∑j

Ej |j〉〈j|+∑k,l

~gkl(a+ a†)|k〉〈l| .

(17)

Here Ej is the energy of the atom-state |j〉 and j =0, 1, 2, . . .. The effective Hamiltonian in the dispersiveregime has here the form

H ′0 = HC +∑j

Ej |j〉〈j|+∑j>0

~χj−1,j |j〉〈j| , (18)

where the resonator-atom coupling is described by

HC = ~a†a

ω0 − χ01|0〉〈0|+∑j>0

(χj−1,j − χj,j+1)|j〉〈j|

.

(19)

and

χj−1,j = jg2

Ej/~− Ej−1/~− ω0. (20)

If only two lowest (photon-dressed) transmon levels arepopulated, the correction to the dispersive-shift operatorof Eq. (15) is accounted for by the replacement [28]

g2

∆σza†a←

(g2

∆− g2

∆− EC/~

)σza†a . (21)

Here the charging energy EC corresponds to the anhar-monicity of the artificial atom.

Finally, transmons can be described as parallel LiCicircuits with coupling capacitors Cgi, see Fig. 1(c). Theexplicit form of the cavity-qubit couplings can then be ex-pressed as a function of these linear-circuit elements [28].They have here the form

~gi =~2

√ω0Ωi

Cgi√CCi

. (22)

We assume here small coupling capacitors, Cgi, Cc C,Ci. The lowest energy-level splittings are in the samelimit Ωi = 1/

√Li(Ci + Cgi), the pure cavity frequency

ω0 = 1/√L(C + 2Cc +

∑i Cgi), and the charging ener-

gies EiC = e2/2(Ci + Cgi).The dispersive shifts of a multi-transmon system as

well as of higher excited states of single transmons havebeen studied experimentally in Refs. [34–36].

D. Background

We now extend the above model to also account for aparallel transmission with the cavity, i.e., element Zb inFig. 1(a). We first consider a reactive (capacitive or in-ductive) coupling beside the cavity and after this extend

5

FIG. 2: (a) A background described by an inductor L con-necting the input and output transmission lines. (b) A back-ground with inductive and dissipative properties, the latterdescribed by a resistor Rb.

the analysis to also account for dissipative background.

1. Reactive background

In the case of purely reactive background, as inFig. 2(a), the boundary conditions of Eqs. (10-11) canbe shown to generalize to (Appendix B)

aLout(t) =

√γa(t)− 1

1 + 2iεaL

in(t)− 2iε

1 + 2iεaR

in(t) (23)

aRout(t) =

√γa(t)− 1

1 + 2iεaR

in(t)− 2iε

1 + 2iεaL

in(t) . (24)

Here we have introduced a parameter describing the re-active response of a parallel inductance Lb

ε =Z0

ω0Lb=

Z0

|Zb(ω0)|. (25)

We consider explicitly the case of a parallel inductor,whereas the result for a parallel capacitor is obtained by asign change ε→ −ε (Appendix C). The limit ε→ 0 thengives the previous input-output relations for two-sidedcavity, Eqs. (10-11). Essential is that the Heisenbergequation of motion for the cavity field, Eq. (13), staysthe same, irrespective of the values of γ and ε.

It should be emphasized that since the Heisenbergequation of motion for the cavity field stays the same,models of driven cavities [21, 22, 34, 37] or dispersivequantum-state measurement [16, 28] can be directly gen-eralized to cover also the presence of background trans-mission, just by using the modified boundary condi-tions (23-24) when evaluating properties of the outgoingfields (from the unchanged solution for the cavity field).

2. Dissipative background

A finite resistivity of the background can be introducedby adding a negative imaginary part to parameter ε, i.e.,

by replacing

ε← ε− iεd . (26)

The exact relation to the model shown in Fig. 2 can bederived to be (Appendix A)

ε− iεd = −i Z0

Z∗b(ω0)=Z0(ω0Lb − iRb)

ω20L

2b +R2

b

(27)

Zb(ω) = iωLbω +Rb . (28)

The model is valid for arbitrary strengths of the back-ground transmission, as long as the cavity mode is cou-pled only weakly to the transmission lines and the back-ground changes weakly within cavity linewidth γ. Notethat variables ε and εd are dimensionless.

3. Removing the background from experimental data

We can also remove the Fano resonance from an ex-perimental data to recover the pure cavity spectrum. Forexample, if we have only input from one side of the cavityand we have measured the transmission also when arti-ficial atoms and the resonator are tuned far away, which

result we mark now sbackground12 = 2(ε−iεd)/[i−2(ε−iεd)],

the above results imply that

√γ⟨

aLin

⟩ 〈a〉 =(s12 − sbackground

12

). (29)

Here s12 is the measured transmission in the presence ofresonator and artificial atoms. Since the cavity equationof motion is independent of the background, the left-handside of Eq. (29) is also the transmission in the abence ofthe background. We must have then

sfree12 = s12 − sbackground

12 . (30)

Here sfree12 is the transmission in a hypothetical exper-

iment, where the background is not present. This rela-tion remains to be valid for all stregths of the backgroundtransmission.

The transformation from the measured backgroundtransmission to the cavity boundary parameter ε has theform

ε− iεd =i

2

sbackground12

1 + sbackground12

. (31)

E. Solution for a linear cavity

In the case of a linear cavity, we can solve the out-fieldsas a function of input directly by Fourier transformation.This solution is valid also for a cavity-atom system inthe dispersive limit when no atom transitions occur, or

6

when transitions are slow and can be accounted for bystatistical averaging.

In the case of a dissipationless cavity and backgroundwe have(

aLout(ω)aR

out(ω)

)=

1

(1 + 2iε)(1− 2if)(32)

×(

4εf − 1 −2i(ε+ f)−2i(ε+ f) 4εf − 1

)(aL

in(ω)aR

in(ω)

),

where

f(ω) =γ

2(ω0 − ω). (33)

The possible dispersive shift of the cavity frequency isnow incorporated in ω0. Here we consider explicitly thecase of a non-dissipative background (εd = 0), but theresult in the general case can be obtained by replace-ment (26).

The scattering amplitude can be shown to be here

s12 =2ε

i− 2ε+

γ

γ + i(ω0 − ω)(34)

=γ + 2ε(ω0 − ω)

(1 + 2iε) [γ + i(ω0 − ω)].

The special cases ε = 0 or γ = 0 give the transmis-sion amplitudes when the parallel transmission does notcontribute or the cavity does not contribute, correspond-ingly. The reflection amplitude has the form

s11 = − 1

1 + 2iε+

γ

γ + i(ω0 − ω)(35)

=2εγ + ω − ω0

(−i+ 2ε) [γ + i(ω0 − ω)].

For a non-dissipative system the transmission and reflec-tion powers sum to 1,

|s11|2 + |s12|2 = 1 . (36)

Here, the solution also satisfies the commutation rela-tions [

aLout(t), a

Lout(t

′)†]

= δ(t− t′) (37)[aR

out(t), aRout(t

′)†]

= δ(t− t′) , (38)

which is obtained only by assuming that this is true forthe input fields, Eq. (2), demonstrating consistency ofthe theory.

In the linear solution, the effect of intrinsic dissipationof the cavity field can be accounted for by adding animaginary part to the resonance frequency, i.e., replacing

ω0 ← ω0 − iκ

2. (39)

This corresponds in the parallel LCR-circuit of Fig. 1

to [30]

κ =1

RC=

ω0

Qint, (40)

where in the second form we have defined an internalquality factorQint = ω0RC. The equivalent energy decayrate is then

1

RC= κ . (41)

This approach assumes implicitly κ ω0.

At finite temperatures, a dissipative system performsfluctuations. In our analysis, based on expectation valuesof Eqs. (6-7), temperature plays a role only in the case ofa non-linear system, since otherwise thermal fluctuationsaverage out. This is since for a linear system, thermalfluctuations result in an additional width of the Gaussianprobability distribution of the field quadratures aroundthe classical mean [17]. Similarly, thermal radiation emit-ted by the background is not expected to contribute toaverage transmission and reflection.

F. General system: Master equation simulation

We can simulate a more general cavity-atom systemusing a Lindblad master equation [17]

ˆρ =i

~[ρ, H] + LL[ρ] + LR[ρ] +

n∑i=1

Liq[ρ] . (42)

Here ρ is the reduced density matrix of the resonatorand artificial atoms. The Hamiltonian H now accountsfor a coherent drive, i.e., operator aL

in, in a way shownbelow. Lindblad operators LL/R then describe decay to

(and excitations from) TLs and Lindblad operators Liqdecoherence (decay and dephasing) of artificial atoms.

1. Lindblad operators

As previously, we assume that cavity-qubit couplingsare small, gi ω0,Ωi, and that possible differences be-tween decay rates related to different shifted values of thecavity frequency can be neglected. The Lindblad super-operator LL then describes cavity photon transitions dueto interaction with the left TL,

LL[ρ] =γ−

2

(2aρa† − a†aρ− ρa†a

)+

γ+

2

(2a†ρa− aa†ρ− ρaa†

). (43)

7

The decay rate to the left TL satisfies in thermal equi-librium

γ− = γ

1 +1

exp(

~ω0

kBT

)− 1

, (44)

and correspondingly for the thermal excitation rate

γ+ = γ

1

exp(

~ω0

kBT

)− 1

. (45)

We have then γ− = γ−(T = 0) + γ+. Similarly for theinteraction with right TL described by super-operatorLR.

Intrinsic dissipation and fluctuations of artificial atomscan also be added by Lindblad super-operators. In thecase of two-lvel systems we have

Liq[ρ] =κ−i2

(2σi−ρσ

i+ − σi+σi−ρ− ρσi+σi−

)+

κ+i

2

(2σi+ρσ

i− − σi−σi+ρ− ρσi−σi+

), (46)

where κ±i are the corresponding transition rates of two-level system i. These rates are affected by the electro-magnetic environment as seen by the transmons [26]. Thetemperature dependence of κ±i is equivalent to Eqs. (44-45).

Additionally, qubit pure dephasing [38] can be ac-counted for within an operator

Lφ[ρ] =∑i

κiφ2

(σizρσ

iz − ρ

). (47)

Similarly, it is possible to account for cavity dephasing byreplacement σz ← σa = 2a†a−1. In the experiment anal-ysis, we find that resonator dephasing due to coupling tomultiple atoms can be modeled rather well within suchmodel. Furthermore, in the linear case (Sec. II E), suchcavity dephasing can also be accounted for as cavity loss(κ > 0), discussed more detailed below in Sec. III D 1.

2. Coherent drive in the Hamiltonian

We consider the case of an incoming coherent radiationfrom the left-hand side TL. The presence of such coherentdrive is accounted for by an effective Hamiltonian [17]

H = H0 + Hd , (48)

where the drive appears as a term

Hd = i~√γA(t)a† + H.c. . (49)

This form is derived by assuming⟨aL

in(t)⟩

= A(t) , (50)

and⟨aR

in(t)⟩

= 0. For example, in the dispersive regime,the corresponding total Hamiltonian of a driven cavitycoupled to single two-level system can be written in theform

H ′0 = ~ (ω0 − ω + χσz) a†a+

~2

(Ω0 + χ) σz

+α

2

(a+ a†

). (51)

Here we have gone into the rotating frame with respect todrive frequency ω and assumed that α = 2i~√γA(t)eiωt

is a real number (so that A(t) ∝ ie−iωt). The general-ization to multi-level atoms and multi-atom systems isstraightforward.

3. Solution for the out field

For obtaining the average output fields, 〈aout(t)〉 and

〈bout(t)〉, we first determine the steady-state solution forthe cavity field, 〈a(t)〉, obtained from solving the mas-ter equation (42) with Hamiltonian (48). After this thesolution for the out-field is obtained from the boundaryconditions (23-24) with inserting the assumed form of theinput field

⟨aL

in(t)⟩

= A(t).

III. FANO RESONANCES AND THE EFFECTOF DECOHERENCE

In this section, we investigate microwave transmissionacross a two-sided cavity in the presence of backgroundtransmission and decoherence (dissipation and fluctua-tions). We start in Sec. III A by giving a short sum-mary of the form of the conventional Fano-resonance. InSec. III B we solve the Fano resonance in the case of alinear cavity with no internal or background dissipation.In Sec. III C, we study the influence of dissipation in thecase of a linear cavity. In Sec. III D, we study a cav-ity connected to single or multiple dissipative two-levelsystems subjected to heating and fluctuations. Finally,in Sec. III E, we estimate the size of contribution fromhigher energy levels of transmons.

A. Conventional Fano function

In a Fano resonance [1, 2, 8], the spectral response of aresonant system is asymmetric around the resonance fre-quency due to an interference effect between two scatter-ing amplitudes: scattering through a background with aconstant (or wide) state density and scattering through adiscrete (or narrow) energy-level. The conventional formof the Fano interference is characterized by only single

8

variable: Fano parameter q. Here, the total scatteringamplitude |s|, or spectral density |s|2, are of the form(neglecting normalization factors)

|s| ∝ |q + η|√1 + η2

, |s|2 ∝ (q + η)2

1 + η2, (52)

where η is a broadening-normalized drive frequency withrespect to the resonance frequency, η = (ω0 − ω)/(Γ/2),with Γ being a parameter describing the resonant-statebroadening. Two central limits of this function are q →∞, giving a Lorentzian shaped peak, and q = 0, giving aa Lorentzian shaped dip. This description then catchesresonant enhancement as well as resonant suppression astwo limits of one formalism.

B. Decoherence-free linear oscillator

Consider first the case of dissipation-free linear cavity.Here one can use the analytical solutions of Eqs. (34-35),which give

|s12|2 =1

1 + q2

(q + η)2

1 + η2. (53)

Here we have identified

q =1

2ε(54)

η =ω0 − ωγ

. (55)

From Eq. (25) we obtain that for |Zb(ω0)| → ∞, q →∞,and transmission probability |s12|2 is a Lorentzian peak.For any finite parallel coupling (finite q) interference oc-curs, which is perfectly destructive when q = −η, mean-ing

ω = ω0 +γ

2ε. (56)

The response near the resonance frequency can also bea dip, q = 0, when |Zb(ω0)| → 0. Such form then needsa very strong parallel transmission. In the following, weshow that when the system is dissipative, a dip in thetransmission can appear also for a weak parallel trans-mission.

C. Lossy linear oscillator

Consider now including dissipation in the cavity whenhaving a weak parallel transmission, ε 1. A finite in-trinsic quality factor of the cavity can be accounted forby adding an imaginary part ω0 ← ω0 − iκ/2. The solu-tion of Eq. (34) is valid also here. Transmission |s12(ω)|for several values of loss rate κ (or equivalently dephasingrate κ/2) is shown in Fig. 3 (εd = 0). We find that the

Frequency ω (units of ω0)

Transm

ission|s

12|

εd = 0 (Rb = 0) εd = 0.05 (Rb > 0)

Increasing cavity loss

?6

?

?

FIG. 3: The effect of intrinsic cavity loss on transmis-sion |s12(ω)| in the presence of weak background transmissionε = 0.05. We increase the dephasing rate κ/2 as indicated bythe plot legends in units of 10−2ω0. When the rate is increasedthe curve minimum and maximum change as visualized by thearrows. In particular, without dissipation in the background(εd = 0), increase of cavity decoherence lowers the maximumand increases the minimum of |s12(ω)|. With a dissipativebackground (εd = 0.05) both minimum and maximum de-crease with increasing cavity decoherence in the consideredrange κ/2 ∈ [0, 0.01]ω0. For higher κ the minimum starts torise. We consider a linear cavity with resonance frequency ω0

and coupling to each transmission line γ = 2× 10−3ω0.

intrinsic loss reduces transmission and ’straightens’ theinterference structure. We note that simultaneously thereflection dip also gets less deep (not plotted).

Consider then adding small dissipation also in the(weak) parallel transmission. A finite resistivity in theparallel channel can be accounted for by a replacementε← ε− iεd. We first note that without a reactive part inthe parallel channel (finite ε), we never get an asymmetricresonance curve (tilt). Again, resistivity in the parallelchannel reduces transmission on resonance, but the twodissipative effects do not simply sum up as an effectiveincreased cavity dissipation rate. Instead, if we assumea fixed background dissipation and increase the cavitydissipation (or dephasing), an interesting effect appears:the minimum value of |s12| decreases with increasing cav-ity decoherence and reaches zero, see Fig. 3 (εd = 0.05).Using Eq. (34) one can derive that the minimum trans-mission is exactly zero when

κ = γεd

ε2 + ε2d. (57)

This zero transmission occurs for

ω = ω0 +γ

2

ε

ε2 + ε2d. (58)

For larger cavity decoherence rates the minimum in-creases again (plotted later in Fig. 8). The reflection|s11|, Eq. (35), is also here always a dip. We then findthat an inverted line shape can occur also in the case of

9

Frequency ω (units of ω0)

Transm

ission|s

12|

εd = 0 (Rb = 0) εd = 0.05 (Rb > 0)

Increasing two-level system hopping (fast decay)

?

6

?

?

FIG. 4: The effect of incoherent transitions in a dispersivelycoupled two-level system on cavity transmission |s12(ω)| inthe presence of weak background transmission ε = 0.05. Thetransition rates κ− and κ+ are increased as indicated by theplot legends in units of 10−2ω0. When the transition ratesare increased the curve minimum and maximum change asvisualized by the arrows. We consider the case of a largedispersive shift, g2/∆ = −15γ, and fast decay to ground state,κ−(T = 0) = 4γ. The resonance frequency ω0 accounts forthe dispersive shift of the ground state and coupling to eachtransmission line γ = 2× 10−3ω0.

weak parallel transmission, when both the cavity and thebackground dissipate radiation.

D. Coupling to two-level systems

We consider now the case of a cavity with no inter-nal losses (κ = 0) but which is coupled to a two-levelsystem exhibiting incoherent transitions between its twostates. In particular, we demonstarte two other ways toobtain peak inversion: (1) fast-incoherent-transitions in-duced cavity dephasing and (2) reduced resonant-statepopulation. Furthermore, a simple fitting formula com-bining these two effects is useful in the analysis of exper-imental data, as discussed in Sec. IV.

1. Cavity dephasing due to cavity frequency switching

Stochastic switchings of the two-level systems inducedephasing of the signal propagating through the cav-ity. In the dispersive coupling regime, this noise corre-sponds to stochastic jumping of the frequency betweentwo values, i.e., to telegraphic noise. An example ofthis effect (not mixed with the reduced population effect,Sec. III D 2) is when one has a high asymmetry betweenexcitation and decay rates, κ+ κ−. Here, the two-level system is mostly in its ground state with stochasticshort-time visits in its excited state.

Numerical results for the effect of incoherent transi-tions in a dispersively coupled two-level system on trans-

mission |s12(ω)| are shown in Fig. 4. We simultaneouslyincrease the excitation and relaxation rate of the two-level system with the same amount, i.e., keeping detailedbalance (thermal equilibrium statistics), Eqs. (44-45).The total dispersive shift is assumed to be much largerthan the cavity broadening, |2g2/∆| γ, and we con-sider a weak drive power. We find that here incoherenthopping can create a very similar effect to s12 as cavitydissipation, see Fig. 3. We however note that for ground-state populations p < 1, the above dephasing mechanismcan be differentiated from pure cavity loss by seeking forweak additional peaks (or dips) corresponding dispersiveshifts from the excited states of the artificial atoms.

The similarity to internal decay can be understood asthat in this limit a single jump of the two-level systemis enough to dephase the system, and that for superpo-sitions of photon numbers the effect of dephasing anddecay is qualitatively the same. More detailed, intrinsiccavity loss with rate κ and pure dephasing of the cavitywith κφ affect the average transmission amplitude s12

equivalently when

κ

2= κφ . (59)

In the presence of multiple two-level systems, the dy-namics also have few simple limits. For relaxation ratesmuch larger than the excitation rates, κ−i κ+

i , withstrong dispersive shifts, the effective dephasing rate isroughly the sum of individual two-level system excita-tions rates,

κφ ≈∑i

κ+i . (60)

Furthermore, the noise spectral density of an ensembleof dispersively-coupled two-level systems is in thermalequilibrium [40]

S(ω) =

∫dteiωt

⟨X(t)X(0)

⟩−⟨X2⟩

(61)

=∑i

χ2i

1

cosh2(

~Ωi

2kBT

) 2κiω2 + κ2

i

,

where X =∑i χiσ

iz and κi is the decay rate of two-level

system i at zero temperature. This describes randomtelegraphic noise of thermally excited two-level systems.For low temperatures we have

S(ω) ≈∑i

χ2i

4

1 + exp(

~Ωi

kBT

) 2κiω2 + κ2

i

, (62)

= 8∑i

(χiκi

)2

κ+i

1

1 +(ωκi

)2 ,

In the limit of fast decay, κi χi, the dephasing can

10

Frequency ω (units of ω0)

Transm

ission|s

12|

εd = 0 (Rb = 0) εd = 0.05 (Rb > 0)

Reducing resonant-state population p

?

6

?

?

FIG. 5: Transmission through a cavity with strong dispersivecoupling (χ γ) to a two-level system that is with proba-bility p in its ground state giving the effective resonance fre-quency ω0 (including the dispersive shift). We assume a weakbackground transmission ε = 0.05 and decrease populationp as indicated by the plot legends. When the population isdecreased the curve minimum and maximum change as visual-ized by the arrows. In the case of dissipation-free background(εd = 0) the reduction in population decreases the maximumand increases the minimum transmission. In the case of dis-sipative background (εd = 0.05) the reduction in populationdecreases the maximum and the minimum transmission inthe shown range of p. The coupling to each transmission lineγ = 2× 10−3ω0.

be treated perturbatively within the Lindblad formalism,Sec. II F 1. Here one gets the dephasing rate

κφ ≈ 8∑i

κ+i

(χi

κ−i

)2

. (63)

We note that in this limit, the contribution from individ-ual two-level systems is then weaker than (maximally) inthe strong-coupling case, κφ

∑i κ

+i .

2. Reduced resonant-state population

In the considered system, resonance inversion canemerge also without the presence of cavity loss or dephas-ing, if the population of the probed state goes essentiallybelow 1. A clean example is a regime, where the qubitswitching is much slower than the cavity decay to trans-mission lines, κ+/− γ, so that the effect of hoppingis to just modify qubit state populations. In particularfor multi-qubit environments under heating, it can thenoccur that the population of the probed state is not 1,but well below it.

Applying Eq. (34) and assuming that the transmissionthrough the cavity in the non-resonant state is negligible,meaning here a background transmission 2(ε− iεd)/(i−2ε+ 2iεd) with probability 1− p, we get for the average

transmission

s12 =2(ε− iεd)

i− 2(ε− iεd)+ p

γ

γ + i(ω0 − ω). (64)

The effect of reducing probability p is visualized in Fig. 5.For the dissipation-free background, Fig. 5 (εd = 0), theinterference structure ’straightens’, but for the dissipa-tive background, Fig. 5 (εd = 0.05), the transmission be-comes increasingly skewed and again touches zero. Thecurve touches zero with population

p = 1− εdεd + 2ε2d + 2ε2

(65)

and the frequency where this occurs is

ω = ω0 +εγ

εd + 2ε2d + 2ε2. (66)

3. Combined effect

In a setup with multiple two-level systems, the twoabove described effects (stochastic artificial-atom swith-ings and reduced resonant-state population) can coexist.An approximative model then includes both the finiteadditional broadening, κ > 0, and the reduced resonant-state population, p < 1. Here we have the transmission

s12 =2(ε− iεd)

i− 2(ε− iεd)+ p

γ

γ + κ/2 + i(ω0 − ω)(67)

=2(ε− iεd)

i− 2(ε− iεd)+ p′

γ′

γ′ + i(ω0 − ω). (68)

In the second form, we have defined effective populationp′ and coupling γ′,

p′ = pγ

γ + κ/2(69)

γ′ = γ +κ

2. (70)

Within these new parameters also Eqs. (65-66) are valid.Furthermore, the population p′ is now the effective spec-troscopic signal strength of resonance at frequency ω0.

4. Motional averaging and a limitation of the dephasingrate by dispersive coupling

As an interesting example, showing the diversity ofthe problem as well as a limitation of the effective cav-ity dephasing rate, we consider a cavity frequency thatswitches with identical hopping rates in the two direc-tions, i.e., κ+ = κ− = κ. Such symmetry appears in thehigh-temperature limit.

The resulting transmission |s12(ω)| as a function of thecommon hopping rate κ and in the absence of the back-ground is visualized in Fig. 6 (left panel). Three regimes

11

Switching rate κ± (units of g2/∆)

Increasing two-level system hopping (high T )

|s12|

Fre

quen

cyω−ω0

(g2/∆

)

FIG. 6: Transmission across a linear oscillator with a fre-quency switching between two values separated by 2g2/∆.We consider a situation without (ε = εd = 0, left panel)and with (ε = εd = 0.05, right panel) a background trans-mission. The switching rate κ± is the same for both direc-tions and the dispersive shift is much larger than the cav-ity linewidth, g2/∆ = 13γ, and γ = 10−3ω0. Without abackground transmission (left panel) and with weak switchingrates, κ± < γ, we observe a statistical average of transmissionfor two equally-populated cavity-frequency states. When in-creasing the switching rate, the two transmission peaks mergeinto a motional-averaged single resonance peak [41]. In thepresence of a weak dissipative background transmission (rightpanel), the two Fano-type peaks evolve to two dips (whenκ± & γ) and finally from a single motional-averaged dip to asingle tilted peak.

can be identified. Firstly, if the excitation and decay ratesare small compared to the system dynamics, κ γ, thesolution for the transmission is a statistical average overtwo results, corresponding to the two frequencies of thedispersively-shifted cavity. Secondly, when the hoppingrate exceeds γ, maximum transport reduces significantlydue to effective dephasing of cavity resonance frequency.These are the two regimes considered in the precedingsubsections. Thirdly, when the excitation and decay ratesdominate the dispersive shift, 2g2/∆, motional averagingemerges [41], where switching is so fast that only the av-erage value of the cavity frequency is observed.

When a weak parallel transmission with dissipation isthen included, Fig. 6 (right panel), the low-hopping ratepeaks are changed to skewed Fano-type peaks. When thehopping rates are increased, the skewed peaks evolve intodips. A motionally-averaged common dip emerges in thelimit of high hopping rates. Finally, a skewed Fano-typepeak is recovered in the limit of very high hopping rates.Our important obervation here (case γ χ) is then thatthe effective dephasing rate due to stochastic switching ofdispersively-coupled two-level systems is limited by thedispersive coupling, i.e., κφ < χ.

We note that a similar line-shape transformation canalso occur without the presence of dispersive-shift hop-ping, but, through other phenomena reducing the cavity

transmission, such as photon blockade when increasingdrive power [21–23]. Such higher drive amplitudes canbe useful for quantum sensors [24] or quantum simula-tion [25, 26].

E. Higher levels of artificial atoms

Higher excited states of artificial atoms can influencethe observed peak transformation. A Hamiltonian thatcan be used to account for higher excited states is givenin Sec. II C 2.

For transmon artificial atoms, a simple estimate forthe population of excited states at a given temperaturecan be made. Here, the anharmonicity EC is usually 5-10 percent of the lowest energy-level spitting ~Ω. Thismeans that for moderate temperatures the populationscan be assumed to be the ones of a harmonic oscillator,

Pn ≈1

1 +N

(N

N + 1

)2

(71)

N =1

expβΩ− 1. (72)

This formula can then be used to estimate how signifi-cantly the higher energy levels contribute to cavity de-phasing. In our experiment, the local temperature variesin a range that implies P1 . 0.2, giving P2 . 0.05.This means that higher excited states are not signifi-cantly populated and their contribution to the analyzedeffects stays negligible. For simplicity, we then neglecttheir contribution from the theoretical model. It shouldhowever be noted that for transmons, the second excitedenergy-levels strongly modify the dispersive shifts of thefirst excited states, see Sec. II C 2.

IV. EXPERIMENT

Our experiment includes 8 transmon artificial atomsembedded in a driven coplanar microwave resonator [23].The equivalent circuit model is shown in Fig. 2(a) withthe cavity-atom coupling scheme of Fig. 2(c). The en-ergy levels of the transmons are individually tunable bylocal magnetic fluxes applied across Josephson inductorsrealized in a SQUID geometry. The magnetic fluxes arecreated by DC bias currents through 8 nearby coils. Theenergy levels of the multi-atom environment affect theeffective cavity resonance frequency, which is probed bymeasuring the microwave transmission s12 through thecavity. Further technical details of the experiment aregiven in Appendix E and in Ref. [23].

12

Coil current I (mA)

Frequen

cyω/2π(G

Hz)

Transmission |s12|

FIG. 7: Measured microwave transmission |s12| througha coplanar cavity coupled to eight transmons as a functionof coil current I, used to tune one transmon, and drive fre-quency ω. The resonance position is periodic as a functionof the current (transmon inductance), whereas the line shapetransforms from a peak to a dip when current |I| is increased.Transmission curves at pointed current bias points are shownin Fig. 8.

A. Uncalibrated system: Resonance inversion byheating

An example of measured microwave transmission s12 isshown in Fig. 7. Here we probe the system in the neigh-borhood of the dispersively-shifted cavity n = 1 modeat base temperature T ≈ 20 mK. This mode has purefrequency ω0/2π = 6.674 GHz. We are in the weak-drivelimit and the data is normalized according to maximaltransmission at high powers. We sweep single coil cur-rent I to tune one transmon. All other coil currents arezero.

In Fig. 7, we observe an avoided energy-level cross-ing between the cavity and the tuned transmon. Theobserved resonance-frequency variation is periodic as afunction of current I, as expected from SQUID flux pe-riodicity. An additional random offset flux threads theSQUID loops leading to an offset from the symmetry-point I = 0. An unexpected feature is the reduction ofon-resonance transmission with increasing |I|, with a lineshape changing from a peak to a dip. This is a generalfeature of the experiment in different biasing conditionsand cool downs.

The experimental data at several current biases is com-

a) b)

c) d)

e) f)

Frequency ω/2π (GHz)

Transm

ission|s

12|

Transm

ission|s

12|

Disp.shift−χ(M

Hz)

Transm

ission|s

12|

FIG. 8: a) Experimental data of Fig. 7 for indicated values ofcoil current I, showing a reduction of maximum transmissionand transition from a peak to a dip when current is increased.b) The change of the line shape as given by master-equation(ME) simulation of cavity coupled dispersively to 8 two-levelsystems. When temperature is increased the transmissionminimum and maximum reduce. c-d) A comparison for widerrange of applied currents and temperatures. e) The transmontwo-level system energies and equivalent dispersive shifts usedin the ME simulation. f) A ME simulation with T = 190 mKand varying common qubit decay κi. The curve keys giveequivalent decay rates at T = 0 in units of the resonatorcoupling γ = 2π × 0.7 MHz. The curve minimum decreaseswith increasing κi. Fitting such curve to experimental data(at known temperatures) can be used to estimate the averagedecay rate of qubits.

pared to master-equation simulations in Fig. 8(a-d). Atthese bias points the cavity-transmon coupling is disper-sive. We simulate the system as a linear resonator cou-pled to eight two-level systems. An important featureof the experiment is the large background transmission,here |s12| . 0.2. This background transmission corre-sponds to model parameter ε − iεd = 0.062 − i0.06 andwas measured separately when the cavity resonance fre-quency was tuned far away (by tuning qubits) and byusing the connection (31). The background transmis-sion originates most probably in a crosstalk between the

13

transmission lines, flux-bias lines, and the sample-boxbonding. For more details see Appendix E.

In master-equation simulations, we use the hypothe-sis that DC current causes heating of the cavity-atomsystem. As a result, stochastic excitations (and decays)of the heated two-level systems make the resonator fre-quency fluctuate. The assumption of elevated tempera-ture is supported by additional experiments on the base-temperature dependence, which imply that for large coilcurrents T ∼ 0.2 K. Furthermore, a spectroscopy ofhigher exciation manifolds described in Ref. [23] impliedT ∼ 0.15 K. At such temperatures, thermal populationsof higher excited-states of transmons are however small:it is consistent to decribe transmons as two-level systems.The base temperature itself was experimentally found todepend on the coil currents only weakly. The origin of theobserved local heating is most probably resistive heatingin the Copper-PCB leads in combination with a weakthermalization of the chip to the sample box. We alsonote that a qualitatively similar dip in microwave trans-mission can be caused by a direct reflection from two-levelsystems [39]. However, in this case the effect of a tem-perature would be opposite, i.e., zero temperature wouldgive the (deepest) dip. This process can also be ruled outby tuning the qubits, since such mechanism would workonly at qubit frequencies, whereas the observation is thatthe dip appears at the cavity frequency.

In master-equation simulations, we use measuredcavity-transmon couplings gi/2π ≈ 110 MHz and an-harmonicities EC ≈ 410 MHz [23]. The coupling ofthe cavity to transmission lines as given by high-powertransmission is γ = 2π × 0.7 MHz, in accordance withthe value obtained from microwave circuit simulations.The used qubit energies are distributed around the reso-nance frequency, with a restriction that total dispersiveshift 20 MHz is reproduced. Therefore here only an es-timation of the qubit frequencies can be made, followingfrom that the experimental control was not calibrated(a calibrated situation is studied below). In our system,only this uncalibrated situation however allows for ob-serving the heating induced inversion of the line shape,since after the calibration the system is always at an ele-vated temperature. The distribution of frequencies (andthereby dispersive shifts) used in the simulation is shownin Fig. 8(e). It should also be noted that in reality thequbit frequencies also shift slightly when magnetic fluxacross the target qubit is being changed, which is notaccounted for by the simulation. An exception is thelast transmission curve (T = 250 mK), where the reso-nance curve is shifted 10 MHz upwards to account forthe changed resonator frequency-shift due to tuning ofthe target qubit.

In Fig. 8(f), we visualize how the peak form can alsobe used to estimate the T1 times of the qubits. Here, wefix the system temperature to T = 190 mK. The higherthe decay rate, the higher the dephasing rate becomes.Simultaneously, the fine structure smoothens out. Theobserved peak depths and forms, together with assuming

Frequency ω/2π (GHz) ME Temperature (mK)

Transm

ission|s

12|

Rate

(unitsofγ)

a) b)

FIG. 9: a) Transmission across a linear cavity with internaldephasing κφ (solid lines) compared to the master equation(ME) simulation of 8 heated two-level systems (dashed lines).In the linear cavity model, the effect of heated multi-qubitenvironment is accounted for by the introduced phenomeno-logical dephasing rate κφ, given in the units of cavity cou-pling γ = 2π × 0.7 MHz. The parenthesis value correspondsto the sum of excitation rates of all qubits in the ME model,κΣ =

∑8i=1 κ

+i . The curve minima decrease with increasing

κ. b) A comparison between the phenomenological dephasingrate κφ and the corresponding master-equation rate κΣ. Herethe temperature (x-axis) is defined by the master-equationsimulation.

temperatures obtained from other observations, lets usto conclude that an average qubit decay rate is between2− 3γ, which means T1 ∼ 50− 80 ns. It should be notedthat the T1 time measured for a single-qubit sample wasessentially longer, T1 ∼ 500 ns. Such direct measurementof T1 in the 8 qubit sample was however not possible. Therelatively short theoretical value is further supported bysimulations when qubits are tuned on resonance with thecavity, see Sec. IV B.

In Fig. 9(a), we additionally consider fitting the peakinversion by the model of a linear resonator with increas-ing internal dephasing rate κφ (or equivalently internalloss rate κ/2). Such a simplified model accounts for tran-sitions in the heated two-level system ensemble as an ef-fective cavity dephasing rate κφ. We obtain that an in-crease of dephasing rate gives a qualitatively similar line-shape transformation as the full master-equation modelwith heating (the latter plotted here as dashed lines).Furthermore, the dephasing rates are close to the sumsof the excitation rates of two-level systems, see Fig. 9(b)[see also Eq. (60)].

B. Calibrated system: Transmission in differentbackgrounds

In this section, we study transmission while tuning theeffective resonance frequency of the cavity. Since thebackground transmission is in our system frequency de-pendent, see Appendix E, the change of the cavity fre-quency corresponds to changing the background.

We consider here the case of a fully calibrated sys-tem, allowing for the control of all eight transmons and

14

Frequency ω/2π (GHz)

Transm

ission|s

12|

Transm

ission|s

12|

Transm

ission|s

12|

Disp.shift−χ(M

Hz)

εtot = 0.12− 0.22i εtot = −0.16− 0.14i

εtot = −0.15− 0.04i εtot = −0.16− 0.05i

εtot = −0.15

FIG. 10: Measured transmission when different number ofqubits (from N = 0 to N = 4) are on resonance with thecavity. The background parameter, εtot = ε − iεd, is differ-ent for each N . The master-equation (ME) simulation treatstemperature T as a free fitting parameter. A fit based on themodel of a linear resonator with resonant-state population pand dephasing rate κφ, Eq. (67), is plotted as dashed lines.(Bottom right) Two-level system frequencies and dispersiveshifts before the on-resonance tuning, i.e., case N = 0. Thetwo-level systems tuned on resonance are marked by numbers1→ 4.

thereby the dressed-cavity frequency. In comparison tothe uncalibrated system studied in Sec. IV A, here a sig-nificant total current flows always in the flux-bias lines,leading to an elevated temperature of the system. Wethen cannot reproduce the resonance inversion effect inthis operation scheme (transformation from a peak to adip by heating).

In Fig. 10, we consider transmission when differentnumber of qubits (from N = 0 to N = 4) are on res-onance with the cavity mode. We study transmissionaround the lower resonance peak of the splitted cavity-qubit peak [23]. The equivalent background parameters,εtot = ε − iεd, are also shown in Fig. 10 (from N = 0to N = 4) and were determined from off-resonance data.

The experimental data are compared to master-equationsimulations with calibration-determined qubit frequen-cies and dispersive shifts, shown in Fig. 10 (bottomright). We have fixed the qubit decays to κi = 3γ, in ac-cordance with the dispersive-regime analysis, Sec. IV A.We use temperature as a free fitting parameter. We finda good agreement between the theory and experimentby varying simulation temperature between 130-175 mK.This range of temperatures is consistent with the analysisof Sec. IV A. We have then obtained a reliable estimatefor the local temperature at each bias point.

In Fig. 10, we additionally fit the observed line shapeswith the model of a linear resonator with (resonant-state)population p and phenomenological dephasing rate κφ,see Sec. III D. The fit is similar to the one obtained fromthe master-equation simulation. We find that the popu-lation p reduces when the first qubit is brought on reso-nance. This is since on-resonance transition frequenciesbetween nearby photon numbers are not anymore ap-proximately constant: heating of the resonator takes thesystem away from the subspace providing the transmis-sion resonance. On the other hand, bringing dispersively-coupled qubits on resonance decreases broadening, sincetheir stochastic-hopping induced dephasing is removed.This leads to tendency of reducing κφ when N increases.It also leads to reduction of p and finally to disappearanceof the resonance behind the noise when N > 5 [23].

We also note that the lineshapes are generally affectedby pure dephasing of artificial atoms. In the consideredsetup, however, the width given by average qubit life-times already reproduces the observed linewidths well.This means that here dephasing of qubits is dominatedby the fast qubit decay, T1 ∼ 50 − 80 ns. Such line-width fitting, together with the dispersive-regime line-shape analysis, demonstrates how T1 and T2 times can,in principle, be estimated independently, without resort-ing to time-domain measurements.

V. CONCLUSIONS AND DISCUSSION

In this work we have investigated Fano resonances inmicrowave transmission across a two-sided cavity cou-pled to multiple artificial atoms and in the presence of amicrowave background. The background was effectivelyformed by the artificial-atom control circuitry. Presentand future microwave quantum-information applicationswrap together high number of qubits and control linesin a finite sized chip and a sample box, where Fano res-onances can easily occur. We have then studied in de-tail how to account for such resonances in most commonquantum-microwave models and particularly how the lineshapes of the energy levels can connect to dissipation andfluctuations in such systems, helping to better under-stand possibly complex spectroscopic data of this type ofdevices.

An important theoretical result was that the back-ground does not necessarily affect the equation of mo-

15

tion of the cavity and artificial atoms. Instead, it canbe included to theoretical results obtained, for exam-ple, from the well-known Jaynes- and Tavis-Cummingsmodels afterwards by applying modified linear boundaryconditions. Its effect can also be substracted out (andundone) straightforwardly from measurement data [23].This property remains to be valid also for time-dependentfields (measurement pulses), higher drive powers, and be-yond the two-level system and dispersive-regime approx-imations. Furthermore, we showed how temperature ofthe multi-qubit environment can be estimated from theline shape of the resonator, as well as how average T1

and T2 times of qubits can be determined without doingdirect time-domain measurements (or independently ofthem).

It should also be noted that all dephasing mechanismsof superconducting microwave resonators due to couplingto spurious two-level systems [42–45] are not yet fully un-derstood. The results obtained here for the behavior ofFano-type resonances also apply to studies of such sys-tems generally. Systems as described here can also be

used as quantum simulators to experimentally study theinvolved physical phenomena.

Acknowledgments

This work was supported by the European ResearchCouncil (ERC) under the Grant Agreement 648011,Deutsche Forschungsgemeinschaft (DFG) within ProjectNo. WE4359/7-1, the Initiative and Networking Fund ofthe Helmholtz Association, the China Scholarship Coun-cil (CSC), and Studienstiftung des deutschen Volkes. Wealso acknowledge support provided by the Initiative andNetworking Fund of the Helmholtz Association, withinthe Helmholtz Future Project Scalable solid state quan-tum computing. This work was also partially supportedby the Ministry of Education and Science of the RussianFederation in the framework of the Program to IncreaseCompetitiveness of the NUST MISIS, contract no. K2-2017-081.

[1] U. Fano, Phys. Rev. 124, 1866 (1961), URL https://

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Appendix A: Microwave scattering in a linear circuit

In this appendix, we apply a classical circuit modelto study microwave transmission and reflection in oursystem. This model assumes that the (whole) systemcan be described as a set of lumped circuit elements.The model accounts for superconducting artificial atomsas LC resonators.

In this approach, we derive scattering properties by ap-plying Kirchhoff rules at the cavity boundaries. For this,we first identify the total voltage and total current dueto forward (in) and backward (out) propagating fields atthe two sides of the cavity. On the left-hand side theseare [30]

V L = V Lin + V L

out (A1)

IL =V L

in

Z0− V L

out

Z0(A2)

and on the right-hand side

V R = V Rin + V R

out (A3)

IR = −VRin

Z0+V R

out

Z0. (A4)

These variables are Fourier components of the total prop-agating field, e.g., V L = V L(ω). The reflection s11 and

17

the transmission s12 amplitudes are here

s∗11 =V L

out

V Lin

(A5)

s∗12 =V R

out

V Lin

. (A6)

We assume here V Rin = 0. The complex conjugation

is needed here in comparison to Eqs. (6-7) since theimpedance treatment assumes implicitly a time depen-dence ∼ V (ω)eiωt, which is opposite to the time depen-dence of annihilation operators ∼ a(ω)e−iωt.

We first consider the case of parallel impedance (Cc =0). This gives two boundary conditions, which state cur-rent conservation and voltage drop across the impedanceZb(ω),

V Lin

Z0− V L

out

Z0= −V

Rin

Z0+V R

out

Z0(A7)

Zb

(V L

in

Z0− V L

out

Z0

)= V L

in + V Lout −

(V R

in + V Rout

). (A8)

Using V Rin = 0 the solution is

s∗11 =1

1 + 2Z0

Zb

(A9)

s∗12 =

2Z0

Zb

1 + 2Z0

Zb

. (A10)

We can now study more detailed the effect of dissipa-tion in the parallel channel. In the main text this wasdone by introducing the imaginary part, εtot = ε − iεd.In the impedance approach the equivalent parameter isiεtot = Z0/Z

∗b(ω) = Z0Zb/|Z∗b(ω)|2. Dissipation is in-

cluded, for example, by changing the impedance fromZb(ω) = iωLbω to Zb(ω) = iωLbω + Rb, with a seriesresistance Rb > 0. This approach also shows that a dissi-pative part renormalizes the reactive term ε. This is stud-ied further below. We note that in a direct comparisonof Eqs. (A9-A10) to s11 and s12 given in the main text,Eqs. (34-35), an overall minus-sign difference appears dueto different definition of scattering-state phases, for moredetails see Appendix B.

Similarly, we can construct boundary conditions forarbitrary cavity couplings Zc1 = 1/iωCc1 and Zc2 =1/iωCc2. Here we allow for different coupling capaci-tance of the cavity to the left-hand side (Cc1) and theright-hand side (Cc2) TLs. We then consider Kirchhoffequations for the input and output fields as well for thevoltage on the island between capacitances Cc1/2, whichwe mark now V . The resulting equations have the form

1Z0

+ 1Zc1(ω) + 1

Zb(ω) − 1Zc1(ω) − 1

Zb(ω)

− 1Zc1(ω)

1Z(ω) + 1

Zc1(ω) + 1Zc2(ω) − 1

Zc2(ω)1

Zb(ω)1

Zc2(ω) − 1Z0− 1

Zc2(ω) −1

Zb(ω)

V L

out

VV R

out

=

1Z0− 1

Zc1(ω) −1

Zb(ω)1

Zc1(ω)

− 1Zb(ω)

V Lin ,

The answer for the output fields and for the island volt-age as a function of the input V L

in can then be found eas-ily by a matrix inversion. For the considered symmetriccoupling, Cc1 = Cc1 = Cc, the analytical solution is

s∗11 =2Z0

[Z2

c + Z(2Zc + Zb)]

(2Z + Z0 + Zc) [Z0(2Zc + Zb) + ZcZb](A11)

s∗12 =ZcZb(2Z + Zc)− Z2

0 (2Zc + Zb)

(2Z + Z0 + Zc) [Z0(2Zc + Zb) + ZcZb]. (A12)

A comparison between the single-mode treatment ofthe main text and the impedance approach consideredhere is shown in Fig. 11. We consider dissipationlessbackground transmission. Here, the impedance model isexact whereas the single-cavity-mode model used in themain text is an approximation. We find that if Cc C,the single-cavity-mode model works well for all strengthsof parallel transmission, i.e., for all values of ε.

A comparison in the case of dissipative background isshown in Fig. 12. We determine εtot = ε− iεd from iden-tification iεtot = Z0/Z

∗b(ω) = Z0Zb/|Z∗b(ω)|2 by using

Zb(ω) = iωLbω + Rb and insert this parameter into thesingle-mode model. We find again that for Cc C theexact linear solution and the approximative single-cavity-mode model are practically the same.

Appendix B: Deriving boundary conditions andHeisenberg equations of motion

The boundary conditions and Heisenberg equations ofmotion can be derived by starting from a Lagrangian ap-proach for an open transmission line interacting with acavity. Here, we discretize the transmission line to ele-ments of length δx with capacitance to ground δxC ′ andinductance δxL′ in between. We apply this approach tothe case of free cavity (no transmons), whereas general-

18

Frequency ω (units of ω0)

Transm

ission|s

12| a) b)

FIG. 11: Microwave transmission amplitude |s12| across a lin-ear two-sided cavity. We do comparison between the single-mode treatment (solid lines) and impedance approach (dashedlines). The cavity is described by an LC oscillator. (a) Trans-mission peak without the presence of background transmis-sion with Cc/C =0.05, 0.2, 0.4. In the single-mode approxi-mation this corresponds to γ = (0.5, 8, 32)×10−3ω0. The res-onance peak widens with increasing Cc or γ. The single-modetreatment is here a good approximation and becomes exact inthe limit Cc/C → 0. (b) Transmission for Cc/C = 0.1 with in-creasing dissipationless background transmission correspond-ing to ε = 0.05, 0.2, 1.0. Here, the single-mode treatmentis a good approximation for all strengths of the backgroundtransmission. The off-resonance transmission increases whenincreasing ε.

Frequency ω (units of ω0)

Transm

ission|s

12| a) b)

FIG. 12: Microwave transmission amplitude |s12| across a lin-ear two-sided cavity for varying background inductance Lb

and fixed background resistivity Rb. We do comparison be-tween the single-mode treatment (solid lines) and impedanceapproach (dashed lines) of the cavity. (a) Transmission forCc/C = 0.1 with reducing inductance with fixed resistiv-ity. The equivalent ε-parameters are ε = 0.027, 0.10, 0.18 anddissipative part εd = 0.0015, 0.023, 0.08. The off-resonancetransmission increases when increasing ε (reducing induc-tance). (b) Transmission for Cc/C = 0.1 with reducinginductance further with fixed resistivity. The equivalent ε-parameters are ε = 0.21, 0.32, 0.50 and dissipative part εd =0.046, 0.12, 0.55.

ization to the case of cavity embedding multiple trans-mons is straightforward.

In this approach, the total Hamiltonian of the systemcan be derived to be

Htot = HR +HL +Hres +Hint . (B1)

The Hamiltonian that describes the right-hand opentransmission line is

HR =

∞∑r=2

Q2r

2δxC ′+

∞∑r≥2

(Φr − Φr−1)2

2L′δx(B2)

+QQ1

C+Q2

1

2Cs.

Indices r ∈ [1,∞] refer to nodes of the discretized trans-mission line, value r = 1 corresponds to the node nextto the cavity. Variables Φr and Qr correspond to a mag-netic flux and charge at node r and an effective seriescapacitance is defined as 1/Cs = 1/Cc + 1/C. The flux(charge) variable of the resonator is Φ (Q). Similarly forthe left-hand side Hamiltonian,

HL =

−2∑l=−∞

Q2l

2δxC ′+∑l≤−1

(Φl − Φl−1)2

2L′δx(B3)

+QQ−1

C+Q2−1

2Cs.

A Hamiltonian that describes the in-line resonator is

Hres =Q2

2C+

Φ2

2L. (B4)

The inductance L and capacitance C are defined inFig. 1(b). Finally, the direct interaction between the twotransmission lines (background) is described by

Hint =

(Φ1 − Φ−1

)2

2Lb+Q−1Q1

C, (B5)

where Lb is the assumed inductive coupling through thebackground. Also a direct-coupling-type term throughthe resonator appears.

The Heisenberg equations of motion for the transmis-sion lines in the limit δx → 0 result in a wave equationwhose solution can be written as in Eq. (1).

The Heisenberg equations at the node r = 1 are

ˆΦ1(t) =Q

C+Q−1

C+Q1

Cs(B6)

ˆQ1(t) =1

L′∂Φ(x = 0+, t)

∂x+

Φ−1 − Φ1

Lb. (B7)

Similarly for the left-hand side (l = −1)

ˆΦ−1(t) =Q

C+Q1

C+Q−1

Cs(B8)

ˆQ−1(t) = − 1

L′∂Φ(x = 0−, t)

∂x+

Φ1 − Φ−1

Lb. (B9)

19

Eq. (B7) is satisfied by

Q1(t) =

√~

2ω0Z0

[−aR

in(t) + aRout(t)

](B10)

+i

ω0Lp

√~Z0

2ω0

[aL

in(t) + aLout(t)

]− i

ω0Lp

√~Z0

2ω0

[aR

in(t) + aRout(t)

]+ H.c. .

Similarly

Q−1(t) =

√~

2ω0Z0

[−aL

in(t) + aLout(t)

](B11)

+i

ω0Lp

√~Z0

2ω0

[aR

in(t) + aRout(t)

]− i

ω0Lp

√~Z0

2ω0

[aL

in(t) + aLout(t)

]+ H.c. .

Our approach to find an approximative solution forthis problem is the following. We first assume thatωc ≡ 1/CsZ0 ω0 and can thereby neglect the timederivatives in Eqs. (B6) and (B8). Within this approx-imation we can directly establish boundary conditionsbetween the cavity and TL fields to be used later. Thisapproximation can be shown to correspond to neglectingterms γ in final equation of motion for the cavity. Thissimplification leads to boundary conditions

Q−1 = −Q Cc

C + 2Cc(B12)

Q1 = −Q Cc

C + 2Cc. (B13)

Using Eqs. (B10-B12) and inserting Q = i√

~2ZLC

[a†(t)−a(t)] we get

αa(t) = −aLin(t) + aL

out(t) + ε[aL

in(t) + aLout(t)

]+ ε∗

[aR

in(t) + aRout(t)

](B14)

ε = iZ0

ω0Lb(B15)

α = −i Cc

C + 2Cc

√Z0

ZLC

√ω0 . (B16)

Here we choose ZLC =√L/(C + 2Cc) (and ω0 =

1/√L(C + 2Cc)) since this choice removes mixing of a

and a† in the following cavity equations of motion, i.e.,diagonalizes an equivalent cavity Hamiltonian. We havealso implicitly assumed that there is no mixing betweenannihilation and creation operators between the systemand the environment, following from a rotating-wave ap-proximation.

Analogously we establish a solution using Eq. (B13),

αa(t) = −aRin(t) + aR

out(t) + ε[aR

in(t) + aRout(t)

]+ ε∗

[aL

in(t) + aLout(t)

]. (B17)

The boundary conditions given in the main text fol-low a convention used in Ref. [17] and correspond toredefinition of the phase of the incoming field opera-tor as aL

in ← −iaLin and the outgoing-field operator as

aLout ← iaL

out. Similarly aRin ← −iaR

in and aRout ← iaR

out.It should be noted that in comparison to the impedanceapproach, we have now effectively changed the signs ofout field amplitudes, changing signs of functions s11 ands12. The previously-derived boundary conditions are inthis notation

αa(t) = aLin(t) + aL

out(t) + iε[−aL

in(t) + aLout(t)

]− iε

[−aR

in(t) + aRout(t)

](B18)

αa(t) = aRin(t) + aR

out(t) + iε[−aR

in(t) + aRout(t)

]− iε

[−aL

in(t) + aLout(t)

](B19)

ε =Z0

ω0Lb(B20)

α =Cc

C + 2Cc

√Z0

ZLC

√ω0 =

√γ . (B21)

We can also express the out-fields as a function of in-fields

aLout(t) = αa(t)− 1

1 + 2iεaL

in(t)− 2iε

1 + 2iεaR

in(t) (B22)

aRout(t) = αa(t)− 1

1 + 2iεaR

in(t)− 2iε

1 + 2iεaL

in(t) . (B23)

The Heisenberg equations of motion for the cavity are(under a rotating-wave approximation and using the newnotation)

ˆa(t) = −iω0a(t) +α

2

[aL

in − aLout

]+

α

2

[aR

in − aRout

]. (B24)

Inserting the solutions of Eqs. (B22-B23) in the cavityequation of motion we get

ˆa(t) = −iω0a(t)− α2a(t) + α[aL

in(t) + aRin(t)

]. (B25)

We obtain that (in the presence of the background) theequation of motion for the cavity remains unchanged, i.e.,is independent of ε. The interference between propaga-tion through the cavity and parallel inductor is describedby Eqs. (B22-B23). These important results can be un-derstood qualitatively as a consequence of that the twotransmission channels, through cavity and through thebackground, are in parallel and therefore their commontransmittance is the sum of the individual ones. Theonly limitation of this model is therefore the validity ofthe single-mode treatment of the cavity (weak coupling

20

of the cavity to the transmission lines).

Appendix C: Fano curve tilt direction

An important detail of a Fano interference is the tiltdirection of the Fano curve, i.e., on which side of the res-onance frequency is the minimum. The direction of thetilt is in our examples always rightwards. It is determinedby two properties: (i) The change of the phase betweenthe resonator input and output and (ii) the nature of theparallel coupling (capacitive or inductive).

In the case of a λ/2-resonator the output and in-put fields are related at resonance as bout(ω0)/ain(ω0) =(−1)n+1 for modes n = 1, 2, . . .. The first mode (n = 1)then keeps the sign and the second mode (n = 2) in-verts the sign. The in-line resonator of Fig. 1(b) in-verts the sign at the resonance and is then equivalentto n = 2, 4, . . .. In other words, the in-line resonatorand coplanar resonator are equivalent for full-wavelengthmodes. However, we find that the model based on anin-line resonator can effectively describe also odd modes(n = 1, 3, . . .), if one makes a switch between an inductiveand capacitive parallel coupling, i.e., changes the sign ofε. This is also the reason why our theoretical model con-siders explicitly a parallel inductor, but analysis of theexperiment interprets the value of ε to originate from aparallel capacitor.

Appendix D: Transmission in the I-Q plane

In this paper we mainly concentrate on describing theeffect of decoherence by looking at the amplitude of trans-mission, |s12|. The amplitude shows a peak (or a dip) atresonance, where the phase shifts rapidly. In particular,at a perfect Fano dip the amplitude touches zero and thephase jumps between two values.

In this appendix, we visualize the behavior of the am-plitude and the phase at the same time by representingthe previously studied line shape transformation in theI-Q plane, see Fig. 13. Here, the I-axis corresponds to thein-phase component, i.e., to the real part of s12. The theQ-axis corresponds to the quadrature component, i.e., tothe imaginary part of s12.

For completeness, we first plot (on the left-hand side ofFig. 13) the result for the case of background-free system,ε = εd = 0, where the amplitude is a Lorentzian. Thebackground-free transmission forms a circle in the I-Qplane, approaching zero for |ω − ω0| → ∞. For a lossycavity, κ > 0, the radius of the circle decreases below1. In the case of reducing probability p, the behavior isanalogous. A difference is that off-resonance transmissionapproaches faster zero for p→ 0.

When adding weak non-dissipative background trans-mission, 0 < ε 1 with εd = 0, the point (0,0) (zerotransmission) can be can be touched from bottom right,but only if κ = 0 and p = 1. Otherwise this point

is not touched. For ε = εd > 0, however, loss in thecavity transport can be balanced by a loss in the back-ground transmission, so that zero transmission is againreached. As found before, this occurs for the highestplotted κ and lowest plotted p. Here, unlike in the caseof a weak non-dissipative background transmission, thecircles touch zero almost exactly from below, resultingin that the transmission dip is rather symmetric aroundthis point. Similarly, in the case of strong parallel trans-port ε →= ∞, εd = 0, and p = 1, corresponding to thecase of a conventional Fano dip as described in Sec. III B(q = 0), we obtain a transmission circle which has a cen-ter at (-0.5,0), touching zero exactly from below, at theresonance ω = ω0.

Appendix E: Experiment details

In this appendix, we give additional information tothe experimental setup, the control-line layout, the back-ground transmission, and the used lumped-element back-ground model. More details of the experiment are givenin Ref. [23].

In Fig. 14(a), we show an optical micrograph of thesample. The experimental realization includes eighttransmon artificial atoms embedded in a driven copla-nar microwave resonator. The transmon artificial atomsare tuned by applying a magnetic flux across the super-conducting loops of two parallel Josephson junctions, seeFig. 14(b). The equivalent circuit model is given in Fig. 1.The bonding in the sample box is shown in Fig. 14(c).

The measured transmission |s12| is characterized bylarge background transmission. In Fig. 15 we show wide-bandwidth transmission, between 6.5 and 6.8 GHz, in thesetup studied in Figs. 7 and 8. Such background trans-mission appears in different cool-downs and is constantas a function of drive power. This means that it canbe modeled as a linear circuit element in transmission.The background transmission (as well as the reduction ofthe cavity-transmission when increasing control current)was negligibly small in an additional experiment that in-cluded only a single transmon in the cavity. These resultsimply that the Fano resonance occurs due to a crosstalkbetween the transmission lines and the multiple controllines.

The background transmission is not a constant as afunction of frequency. In most parts the variation is how-ever small within the bandwidth of the resonator (withinfew MHz), and can be treated as a constant in the model.The large-scale variations then correspond to differentparameters in the equivalent model, studied in Fig. 10.Notable is that the maximal background transmission at6.55 GHz is of similar magnitude as maximal transmis-sion on-resonance with the cavity. At high powers, how-ever, the maximal (cavity-peak) transmission |s12| wasroughly two times higher.

The background impedance can be both inductive orcapacitive. In particular, it is inductive on the left-hand

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Quadrature

(Im[s

12])

ε = εd = 0 ε = 0.05, εd = 0 ε = εd = 0.05

In phase (Re[s12])

κφ

p

?

?

FIG. 13: In-phase (Re[s12]) and quadrature (Im[s12]) components of the transmitted field of Figs. 3 and 5. The frequencyis sweeped from 0.95ω0 to 1.05ω0. Dots mark frequency differences of 1.25γ. The three top figures represent changes whenincreasing cavity loss rate κ/2 = κφ (as given by the curve keys) in three different backgrounds (titles). The three bottomfigures represent changes when decreasing resonant-state population p in the same backgrounds. The radius of the transmissioncircles reduce when increasing κ or decreasing p, as visualized by the arrows. For ε = εd = 0, zero transmission [point (0,0)]appears only in the asymptotic limits |ω − ω0| → ∞ or p→ 0. For ε = εd > 0 the transmission circles can overtake and touchzero transmission [point (0,0)] from below, corresponding to a dip and zero in the transmission.

(b)

(c)

FIG. 14: (a) Optical micrograph of the chip including 8 transmon artificial atoms embedded in a driven coplanar microwaveresonator. (b) The transmon qubits are tuned by local current lines connected to DC bias lines outside the chip. (c) The chipinside the sample box.

side and capacitive on the right-hand side of the localmaximum near 6.56 GHz (see also the discussion of theFano tilt direction in Appendix C). This change reflectsthe appearance of a wide resonance in the backgroundtransmission around this region. At each frequency thebackground transmission can however be modeled withinthe simple lumped-element model, having resistor Rb inseries with capacitor or inductor, see Fig 15. The param-eters at the specifically studied point in Fig. 10 are listed

in Table I.

In Fig. 10, to remove small structure originating in thechanging background transmission, the measured data|s12| is modified by first removing the constant (but fre-quency dependent) off-resonance transmission from theon-resonance data, and then replacing it by a backgroundtransmission given by the parameter εtot, which was cho-sen to be the background transmission at the exact res-onance frequency (given in Table I). Such substitution is

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FIG. 15: Measured transmission |s12| in a broader bandwidth, drive frequency ω/2π varying between 6.5 GHz and 6.8 GHz, forthree values of the coil current used to tune one transmon. The data is normalized according to high power cavity transmission,assumed to reach the theoretical maximum ≈ 0.9 for ε− iεd = 0.062− 0.06i. The maximal transmission on-resonance reduceswhen increasing the coil current.

ω/2π (GHz) Rb (Ω) 1/ωCb (Ω) ωLb (Ω) ε εd6.405 0 0 330 −0.15 06.415 10 0 310 −0.16 0.056.46 78 0 300 −0.15 0.046.51 160 0 170 −0.16 0.146.56 160 0 0 0 0.316.585 170 100 0 0.12 0.226.66 400 420 0 0.062 0.06

TABLE I: Summary of model parameters at different drive frequencies.

allowed as long as changes within the cavity bandwidthγ stay small. This substitution helps to better identifythe transmission changes due to the cavity transmissiononly.

The observed capacitive coupling is supported by mi-crowave simulations of the experimental layout, whichgive a capacitive coupling ∼ 50 fF across the multipleDC flux bias lines beside the cavity (impedance ∼ 400 Ω).However, the coupling from the input conductor to the

bias leads does not occur through a direct coupling be-tween the cavity and nearby conducting strips, for whichwe simulate a still a relatively high value . 3 fF (occuringdue to narrow ground conductors in the cavity). It canhowever be mediated via sample box walls. In the probedfrequency range, the equivalent resistance Rb varies fromzero up to the free space impedance (377 Ω), suggestingradiation loss to nearby conductors and/or to free space.