Energy-participation quantization of Josephson circuits

Zlatko K. Minev1,∗, Zaki Leghtas1,2, Shantanu O. Mundhada1,†,Lysander Christakis1,‡, Ioan M. Pop1,3, Michel H. Devoret1

1Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA2Centre Automatique et Systèmes, Mines-ParisTech,

PSL Research University, 60 Bd Saint Michel, 75006 Paris, France3IQMT, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany

∗Current address: IBM T.J. Watson Research Center, Yorktown Heights,New York 10598, USA; zlatko.minev@aya.yale.edu; www.zlatko-minev.com

†Current address: Quantum Circuit Incorporated (QCI), New Haven, CT 06511, USA and‡Current address: Department of Physics, Princeton University, Princeton, NJ 08540, USA

(Dated: August 17, 2021)

Superconducting microwave circuits incorporating nonlinear devices, such as Josephson junctions,are a leading platform for emerging quantum technologies. Increasing circuit complexity furtherrequires efficient methods for the calculation and optimization of the spectrum, nonlinear inter-actions, and dissipation in multi-mode distributed quantum circuits. Here, we present a methodbased on the energy-participation ratio (EPR) of a dissipative or nonlinear element in an electro-magnetic mode. The EPR, a number between zero and one, quantifies how much of the modeenergy is stored in each element. The EPRs obey universal constraints and are calculated fromone electromagnetic-eigenmode simulation. They lead directly to the system quantum Hamiltonianand dissipative parameters. The method provides an intuitive and simple-to-use tool to quantizemulti-junction circuits. We experimentally tested this method on a variety of Josephson circuits,and demonstrated agreement within several percents for nonlinear couplings and modal Hamiltonianparameters, spanning five-orders of magnitude in energy, across a dozen samples.

I. INTRODUCTION

Quantum information processing based on the control ofmicrowave electromagnetic fields in Josephson circuits isa promising platform for both fundamental physics ex-periments and emerging quantum technologies1–3. Keyto the success of this platform is the ability to quan-titatively model the distributed quantized electromag-netic modes of the system, their nonlinear interactions,and their dissipation (see Fig. 1). This challenge isthe subject of intensifying interest4–26, as experimen-tal architectures27–36 and nonlinear devices37–44 scale inboth complexity and diversity.

In this paper, we introduce a circuit quantizationmethod based on the concept of the energy-participationratio (EPR). We reduce the quantization problem to an-swering the simple question: what fraction of the en-ergy of mode 𝑚 is stored in element 𝑗? This leads toa constrained number between zero and one, the EPR,denoted 𝑝𝑚𝑗

45. This ratio is the key quantity thatbridges classical and quantum circuit analysis; we showit plays the primary role in the construction of the sys-tem many-body Hamiltonian. Furthermore, dissipationin the system is treated on equal footing by calculatingthe EPR 𝑝𝑚𝑙 of lossy element 𝑙 in mode 𝑚.

The EPR method deviates from previous black-box quantization work4,6,7, which uses the impedance-response matrix, denoted 𝑍 𝑗 𝑗′ (𝜔), where 𝑗 and 𝑗 ′ indexports associated with nonlinear elements. For all pairsof ports, the complex function 𝑍 𝑗 𝑗′ (𝜔) is calculated froma finite-element (FE) driven simulation in the vicinityof the eigenfrequency of every mode. Our method re-

places these steps with a more economical FE eigenmodesimulation, from which one extracts the energy partici-pations 𝑝𝑚𝑙 and 𝑝𝑚𝑗 , needed to fully characterize boththe dissipative and Hamiltonian properties of the circuit.

To test the method, we compared EPR calculations ofcircuit parameters to experimentally measured ones for 8superconducting devices designed with the EPR method,comprising a total of 15 qubits, 8 readout and storageresonator modes, and one waveguide system. The re-sults demonstrate agreement for Hamiltonian parametersspanning over five orders of magnitude in energy. Res-onance frequencies were calculated to one percent accu-racy, large nonlinear interactions, such as anharmonici-ties and cross-Kerr frequencies, to five percent, and small,nonlinear interactions to ten percent. This level of accu-racy is sufficient for most current quantum informationexperiments.

II. RESULTS AND DISCUSSION

A. To quantize a simple circuit: qubit coupled to acavity

In this section, we introduce the EPR method of quan-tum circuit design on a modest, yet informative, exam-ple: a transmon qubit coupled to a cavity mode (seeFig. 2). The transmon46 consists of a Josephson junc-tion shunted by a capacitance. It is embedded in thecavity, which we will consider as a black-box distributedstructure. The Hamiltonian of this system 𝐻full can beconceptually separated in two contributions (see Supple-

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Figure 1. Conceptual overview. a Illustration of the physical model of an example quantum device, which comprises twothree-dimensional (3D) cavities (grey enclosures), each housing several qubit chips (green boxes). A close-up view of one of thechips is depicted in the inset. The dotted box in the center of the chip schematically outlines a non-linear inductive sub-circuit,referred to as a Josephson dipole. b Results of a finite-element eigenmode analysis (FEe) of the Josephson circuit linearizedabout its equilibrium. The 𝑚-th mode eigenfrequency and electric and magnetic fields are 𝜔𝑚, ®𝐸𝑚 (®𝑟), and ®𝐻𝑚 (®𝑟), respectively,where 𝑟 denotes spatial position. Center inset:

®𝐸𝑚

profile (red: high; blue: low) for the fundamental mode of one of the 3Dcavities. Additional FE driven simulations (FEd) are unnecessary; i.e., the impedance matrix 𝑍 𝑗 𝑗′ (𝜔) is not calculated. cThe Hamiltonian 𝐻full, which includes nonlinear interactions to arbitrary order (see Results ), is computed directly from theeigenanalysis via the EPRs 𝑝𝑚𝑗 and EPR signs 𝑠𝑚𝑗 = ±1 of the junctions, 𝑗 . Dissipative contributions due to a lossy element 𝑙

are similarly computed from the loss EPRs 𝑝𝑚𝑙 ; for linear dissipation, the EPR signs 𝑠𝑚𝑙 are unnecessary. Direct extractionof Hamiltonian and dissipative parameters from eigensolutions is unique to the EPR method. The geometry of the classicalmodel is modified in an iterative search for the desired dissipative and Hamiltonian parameters (left-pointing arrow).

qa ca

JΦ

a b

Figure 2. Quantizing a simple circuit. a Illustration ofa 3D cavity enclosing a transmon qubit chip. The cross sym-bol marks the location of a Josephson junction. Vertical bluearrows depict the electric field ®𝐸𝑚 (®𝑟) of the fundamental cav-ity mode, TE101. b Equivalent two-mode lumped-element-representation of the distributed circuit. Operators 𝑎𝑞 and 𝑎𝑐denote the qubit and cavity mode operators, respectively.

mentary Section A2),

𝐻full = 𝐻lin + 𝐻nl , (1)

where 𝐻lin consists of all terms associated with the lin-ear response of the junction and the resonator structure,and 𝐻nl consists of terms associated with the nonlinearresponse of the junction. Restricting our attention tothe cavity and qubit modes of the otherwise black-boxstructure, the analytical form of the Hamiltonian followsfrom standard circuit quantization47,48 (see Supplemen-tary Section A4):

𝐻lin = ℏ𝜔𝑐𝑎†𝑐𝑎𝑐 + ℏ𝜔𝑞𝑎

†𝑞𝑎𝑞 , (2)

𝐻nl = −𝐸𝐽

[cos (𝜑𝐽 ) + 𝜑2

𝐽/2], (3)

𝜑𝐽 = 𝜑𝑞

(𝑎𝑞 + 𝑎†𝑞

)+ 𝜑𝑐

(𝑎𝑐 + 𝑎†𝑐

), (4)

where 𝜔𝑐 and 𝜔𝑞 are the angular frequencies of the cav-ity and qubit eigenmodes defined associated with 𝐻lin,respectively, and where 𝑎𝑐 and 𝑎𝑞 are their annihila-tion operators, respectively. The Josephson energy 𝐸𝐽

can be computed from the Ambegaokar-Baratoff for-mula adapted to the measured room-temperature resis-tance of the junction49. The junction reduced general-ized flux 𝜑𝐽 corresponds to the classical variable 𝜑𝐽 (𝑡) B∫ 𝑡

−∞ 𝑣𝐽 (𝜏) d𝜏/𝜙0, where 𝑣𝐽 (𝜏) is the instantaneous volt-age across the junction47,48, and 𝜙0 B ℏ/2𝑒 is the reducedflux quantum. The junction flux operator [Eq. (4)] is alinear, real-valued, and non-negative combination of themode operators (see Supplementary Section A4), and inits expression, 𝜑𝑐 and 𝜑𝑞 are the quantum zero-point fluc-tuations of junction flux in the cavity and qubit mode,respectively. It is worth stating that the linear couplingbetween the cavity and qubit, commonly denoted 𝑔46, isfully factored in our analysis, and is implicitly handled inthe extraction of the operators from the electromagneticsimulation.

Our principal aim is to determine the unknown quanti-ties: 𝜔𝑐 , 𝜔𝑞 , 𝜑𝑞 and 𝜑𝑐. As we will show, we extract andcompute these quantities from an eigenanalysis of theclassical distributed circuit corresponding to 𝐻lin. Thisincludes the qubit-cavity layout, materials, electromag-netic boundary conditions, and a model of the junctionas a lumped-element, linear inductor. The eigensolver re-turns the requested set of eigenmodes and their frequen-cies, quality factors, and field solutions. By running theeigensolver in the frequency range of interest, we obtainthe hybridized cavity and qubit modes, whose eigenfre-

2

quencies 𝜔𝑐 and 𝜔𝑞 fully determine 𝐻lin (see Supplemen-tary Section C for FE methodology).

To determine 𝐻nl, we need the quantum zero-pointfluctuations 𝜑𝑞 and 𝜑𝑐, which are calculated from theparticipation of the junction in the eigenfield solutions.The participation 𝑝𝑚 of the junction in mode 𝑚 ∈ 𝑐, 𝑞is defined to be the fraction of inductive energy stored inthe junction relative to the total inductive energy storedin the entire circuit,

𝑝𝑚 BInductive energy stored in the junction

Total inductive energy stored in mode 𝑚, (5)

evaluated when only mode 𝑚 is excited. Thus, 𝑝𝑚 can becomputed from the electric ®𝐸𝑚 (®𝑟) and magnetic ®𝐻𝑚 (®𝑟)eigenfields as detailed in Supplementary Section C2; ®𝑟denotes spatial position. In the quantum setting, Eq. (5)links 𝑝𝑚, 𝜑𝐽 , and the state of the circuit,

𝑝𝑚 =〈𝜓𝑚 | 12𝐸𝐽𝜑

2𝐽|𝜓𝑚〉

〈𝜓𝑚 | 12𝐻lin |𝜓𝑚〉, (6)

where |𝜓𝑚〉 denotes a coherent state or a Fock excitationof mode 𝑚. Note that normal-ordering must be used inEq. (6); this correct treatment of vacuum fluctuationsis detailed in Supplementary Section A6. SimplifyingEq. (6), one expresses the variance of the quantum zero-point fluctuations 𝜑𝑐 and 𝜑𝑞 as a function of the classicalenergy participations 𝑝𝑚,

𝜑𝑐2 = 𝑝𝑐

ℏ𝜔𝑐

2𝐸𝐽

and 𝜑𝑞2 = 𝑝𝑞

ℏ𝜔𝑞

2𝐸𝐽

, (7)

which completely determines 𝐻nl, and thus completes thedescription of the system Hamiltonian 𝐻full. Here, 𝜑𝑐

and 𝜑𝑞 can be taken as positive numbers. As we will seein the next section, in the presence of multiple junctions,this is not always true.

Designing experiments with the EPR requires one tofurther extract from 𝐻full the transition frequencies andnonlinear couplings between modes. Depending on thecase, this can be done approximately or exactly usingnumerical or analytical techniques20. This task is easilyachieved if 𝐻nl is a perturbation to 𝐻lin

46. In this limit,𝐻full for our qubit-cavity example can be approximatedby the effective, excitation-number-conserving Hamilto-nian, see Supplementary Eq. (B.2),

𝐻eff =(𝜔𝑞 − Δ𝑞

)𝑛𝑞 + (𝜔𝑐 − Δ𝑐) 𝑛𝑐 − 𝜒𝑞𝑐𝑛𝑞𝑛𝑐

− 1

2𝛼𝑞𝑛𝑞

(𝑛𝑞 − 1

)− 1

2𝛼𝑐𝑛𝑐

(𝑛𝑐 − 1

), (8)

where 𝑛𝑞 = 𝑎†𝑞𝑎𝑞 and 𝑛𝑐 = 𝑎

†𝑐𝑎𝑐 denote the qubit and

cavity excitation-number operators, respectively, Δ𝑞 de-notes the ‘Lamb shift’ of the qubit frequency due to thedressing of this nonlinear mode by quantum fluctuationsof the fields, 𝛼𝑞 (𝛼𝑐) is the qubit (cavity) anharmonic-ity, and 𝜒𝑞𝑐 is the qubit-cavity dispersive shift (cross-

Kerr coupling). The Hamiltonian parameters can be cal-culated directly from the EPR, see Supplementary Sec-tion B,

𝛼𝑞 =1

2𝜒𝑞𝑞 = 𝑝2𝑞

ℏ𝜔2𝑞

8𝐸𝐽

, (9)

𝛼𝑐 =1

2𝜒𝑐𝑐 = 𝑝2𝑐

ℏ𝜔2𝑐

8𝐸𝐽

, (10)

𝜒𝑞𝑐 = 𝑝𝑞 𝑝𝑐ℏ𝜔𝑞𝜔𝑐

4𝐸𝐽

. (11)

Experimentally, the qubit Lamb shift can be obtainedas Δ𝑞 = 𝛼𝑞−𝜒𝑞𝑐/2. Since a single EPR 𝑝𝑚 determines thenonlinear interaction for each mode, the parameters 𝜒𝑞𝑐and 𝛼𝑞 are interdependent,

𝜒𝑞𝑐 =√𝜒𝑞𝑞𝜒𝑐𝑐 = 2

√𝛼𝑞𝛼𝑐 . (12)

As shown in Supplementary Section A7, the EPRs 𝑝𝑐and 𝑝𝑞 obey the constraints

0 ≤ 𝑝𝑞 , 𝑝𝑐 ≤ 1 and 𝑝𝑞 + 𝑝𝑐 = 1 . (13)

These relations together with Eqs. (9) and (12) are usefulto budget the dilution of the nonlinearity of the junction(see Supplementary Section B2) and to provide insight onthe limits of accessible parameters (see Methods). Fur-ther, Eq. (13) is used to validate the convergence of theFE simulation.

B. Quantizing the general Josephson system

The simple results obtained in the preceding sectionwill now be generalized to arbitrary nonlinear devices en-closed in a black-box, distributed, electromagnetic struc-ture. While such structures are frequently classifiedas planar27,50–52 (2D), quasi-planar28,29,32,53 (2.5D), orthree-dimensional30,54–56 (3D), we will treat all classeson equal footing. The electromagnetic structure is as-sumed to be linear in the absence of the enclosed nonlin-ear devices. For simplicity of discussion, we can considerthese devices to be inductive and lumped; distributednonlinear devices, such as kinetic-inductance transmis-sion lines57–60, can be thought of as a series of lumpedones.

The simplest nonlinear device comprises a sin-gle element, such as a Josephson tunnel junc-tion [see Fig. 3(a)], an atomic-point contact61,62, ananobridge38,63, a semiconducting nanowire40,41,64–67, oranother hybrid structure68. A multi-element device,such as a SQUID69,70, a SNAIL71 [see Fig. 3(b)], asuperinductance72–74, or a junction array5,75–80 refers toa subcircuit composed of purely inductive lumped ele-ments. This subcircuit can also be subjected to externalcontrols, such as voltage or flux biases.

The general nonlinear device that we now consider,referred to as a Josephson dipole, is any lumped, purely-inductive, nonlinear subcircuit with two terminals. The

3

jΦjextΦ

ca b

)j(ΦjE 3jc

j4Φ

4jc

. . .

j3Φ

mlκ

linH

fullH

nlH

. . .

. . .

= 1j

= 2j

1jΦ

2jΦ

..

....

..

....

geom1−L

geomC

d

Figure 3. Schematic representation of the Josephsoncircuit and its nonlinear elements. a A simple exampleof a Josephson dipole—a Josephson tunnel junction. b Anexample of a composite junction, comprising four Joseph-son junctions in a ring, frustrated by an external magneticflux Φext

𝑗threading the loop. c Conceptual decomposition

of a general Josephson dipole, denoted 𝑗 . For convenience,its potential energy function E 𝑗 (Φ 𝑗 ;Φ

ext𝑗) can be Taylor ex-

panded in a sum of nonlinear inductive contributions of in-creasing order Φ

𝑝

𝑗, with relative amplitude 𝑐 𝑗 𝑝, where 𝑝 de-

notes the index in the series. The energy function can besubjected to external bias parameters Φext

𝑗, such as flux or

voltages. d Schematic diagram of a general Josephson cir-cuit conceptually resolved into a purely dissipative (left, ^𝑙𝑚),linear (middle, 𝐻lin), and nonlinear (right, 𝐻nl) constitutions.

key characteristic of the Josephson dipole is that itpossesses a characteristic energy function, which en-capsulates all details of its constitution. For exam-ple, the two-terminal nonlinear device known as thesymmetric SQUID70 is described by the energy func-tion E 𝑗

(Φ 𝑗 ;Φ

ext𝑗

)= −𝐸 𝑗

(Φext

𝑗

)cos

(Φ 𝑗/𝜙0

), where Φ 𝑗 is

the generalized flux across the device terminals47,48, 𝐸 𝑗

is the effective Josephson energy, Φext𝑗

is the externalflux bias, and the subscript 𝑗 denotes the 𝑗-th Josephsondipole in the circuit. The flux Φ 𝑗 is defined as the de-viation away from the value in equilibrium, as discussedbelow. To ease the notation, parameters such as Φext

𝑗

will be implicit hereafter. Similarly to the example ofthe single-junction transmon , the energy of a Joseph-son dipole can be separated in two parts. One part Elin

𝑗

accounts for the linear response of the dipole, while theother Enl

𝑗accounts for the nonlinear response,

E 𝑗

(Φ 𝑗

)= Elin𝑗

(Φ 𝑗

)+ Enl𝑗

(Φ 𝑗

), (14)

where

Elin𝑗(Φ 𝑗

)B

1

2𝐸 𝑗

(Φ 𝑗

𝜙0

)2, (15)

and where the constant 𝐸 𝑗 sets the scale of the junctionenergy. This energy scale can be represented by the lin-ear inductance 𝐿 𝑗 B 𝜙0/𝐸 𝑗 presented by the Josephsondipole when submitted to a small excitation about itsequilibrium.

Frustrated equilibrium. External biases can setup persistent currents in the circuit. These can alterthe static [direct-current (dc)] equilibrium of the Joseph-son system. For example, frustrating a superconductingring with a magnetic flux sets up a persistent circulatingcurrent in the ring. For a Josephson dipole in such aloop, the definition of the flux Φ 𝑗 will differ in Eqs. (14)and (15) as a function of the equilibrium. Due to this ad-justment, terms linear in Φ 𝑗 are absent from Eq. (15) byconstruction. Supplementary Sections A8 and A9 discussthese equilibrium considerations in detail.

1. Quantum Hamiltonian.

Having conceptually carved out nonlinear contribu-tions from the system Hamiltonian 𝐻full, and collectedthem in the set of Enl

𝑗functions, we define the linearized

Josephson circuit to correspond to everything left overin the system. This linear circuit consists of the elec-tromagnetic circuit external to the Josephson dipoles,combined with their linear inductances 𝐿 𝑗 . We willuse the eigenmodes of the linearized circuit to explic-itly construct 𝐻full. The eigenmode frequencies and fielddistributions are readily obtained using a conventionalfinite-element solver (see Supplementary Section C). TheHamiltonian of the linearized Josephson circuit can thusbe expressed as (see Supplementary Section A5)

𝐻lin =

𝑀∑𝑚=1

ℏ𝜔𝑚𝑎†𝑚𝑎𝑚 , (16)

where 𝑀 is the number of modes addressed by the nu-merical simulation, 𝜔𝑚 is the solution eigenfrequency ofmode 𝑚, and 𝑎𝑚 the corresponding mode amplitude (an-nihilation operator), defined by the mode eigenvector.We emphasize that the frequencies 𝜔𝑚 will be signifi-cantly perturbed by the Lamb shifts Δ𝑚, and should beseen as an intermediate parameter entering in the calcu-lation of the rest of the nonlinear Hamiltonian,

𝐻nl =

𝐽∑𝑗=1

Enl𝑗 =

𝐽∑𝑗=1

𝐸 𝑗

(𝑐 𝑗3𝜑

3𝑗 + 𝑐 𝑗4𝜑

4𝑗 + · · ·

)(17)

=

𝐽∑𝑗=1

𝐸 𝑗

∞∑𝑝=3

𝑐 𝑗 𝑝𝜑𝑝

𝑗, (18)

𝜑 𝑗 =

𝑀∑𝑚=1

𝜑𝑚𝑗

(𝑎†𝑚 + 𝑎𝑚

), (19)

where 𝐽 is the total number of junctions and 𝜑 𝑗 B Φ 𝑗/𝜙0.In Eq. 17, we have introduced a Taylor expansion of Enl

𝑗,

4

where the energy 𝐸 𝑗 and expansion coefficients 𝑐 𝑗 𝑝 areknown from the fabrication of the Josephson circuit, seeFig. 3(c). For example, for a Josephson junction, theconstant 𝐸 𝑗 is just the Josephson energy, while 𝑐 𝑗 𝑝 arethe coefficients of the cosine expansion; i.e., 𝑐 𝑗 𝑝 is 0 forodd 𝑝 and (−1) 𝑝/2+1 /𝑝! for even 𝑝. The expansion ishelpful for analytics but does not need to be used in thenumerical analysis of 𝐻nl, see Supplementary Section A.

The Hamiltonian 𝐻full is specified since the opera-tors 𝜑 𝑗 are expressed in terms of the mode amplitudesas a linear combination (see Supplementary Section A5).Here, 𝜑𝑚𝑗 are the dimensionless, real -valued, quantumzero-point fluctuations of the reduced flux of junction 𝑗

in mode 𝑚. Determination of 𝐻full is now reduced tocomputing 𝜑𝑚𝑗 . We achieve this by employing a gener-alization of the energy-participation ratio.

The energy-participation ratio 𝑝𝑚𝑗 of junction 𝑗 ineigenmode 𝑚 is defined to be the fraction of inductiveenergy stored in the junction when only that mode isexcited,

𝑝𝑚𝑗 BInductive energy stored in junction 𝑗

Inductive energy stored in mode 𝑚

=〈𝜓𝑚 | 12𝐸 𝑗𝜑

2𝑗|𝜓𝑚〉

〈𝜓𝑚 | 12𝐻lin |𝜓𝑚〉, (20)

which is a straightforward extension of Eq. (5), and issimilarly computed using normal ordering (see Supple-mentary Section A6). The EPR 𝑝𝑚𝑗 is computed fromthe eigenfield solutions ®𝐸𝑚 (®𝑟) and ®𝐻𝑚 (®𝑟) as explainedin Supplementary Section C3. It is a bounded, non-negative, real number, 0 ≤ 𝑝𝑚𝑗 ≤ 1. A zero EPR 𝑝𝑚𝑗 = 0means that junction 𝑗 is not excited in mode 𝑚. A unityEPR 𝑝𝑚𝑗 = 1 means that junction 𝑗 is the only inductiveelement excited in the mode.

From the EPR 𝑝𝑚𝑗 , one directly computes the vari-ance of the quantum zero-point fluctuations (see Ap-pendix A6),

𝜑2𝑚𝑗 = 𝑝𝑚𝑗

ℏ𝜔𝑚

2𝐸 𝑗

. (21)

Equation (21) constitutes the bridge between the classi-cal solution of the linearized Josephson circuit and thequantum Hamiltonian 𝐻full of the full Josephson system,up to the sign of 𝜑𝑚𝑗 .

2. Universal EPR properties

The quantum fluctuations 𝜑𝑚𝑗 are not independent ofeach other, since the EPRs are submitted to three typesof universal constraints—valid regardless of the circuittopology and nature of the Josephson dipoles. Theseare of practical importance, as they are useful guides inevaluating the performance of possible designs and as-sessing their limitations. As shown in Supplementary

Section A7, the EPRs obey one sum rule per junction 𝑗

and one set of inequalities per mode 𝑚,

𝑀∑𝑚=1

𝑝𝑚𝑗 = 1 and 0 ≤𝐽∑𝑗=1

𝑝𝑚𝑗 ≤ 1 . (22)

The total EPR of a Josephson dipole is a quantity thatis independent of the number of modes—it is preciselyunity for all circuits in which the dipole is embedded.It can only be diluted among the modes. On the otherhand, a given mode can accept at most a total EPR ofunity from all the dipoles. In practice, this sum rule canbe fully exploited only if the bound 𝑀 reaches the totalnumber of relevant modes of the system.

The next fundamental property concerns the orthogo-nality of the EPRs. Rewriting Eq. (21) in terms of theamplitude of the zero-point fluctuation we have

𝜑𝑚𝑗 = 𝑠𝑚𝑗

√𝑝𝑚𝑗ℏ𝜔𝑚/2𝐸 𝑗 , (23)

where the EPR sign 𝑠𝑚𝑗 of junction 𝑗 in mode 𝑚 is ei-ther +1 or −1. The EPR sign encodes the relative di-rection of current flowing across the junction. Only therelative value between 𝑠𝑚𝑗 and 𝑠𝑚𝑗′ for 𝑗 ≠ 𝑗 ′ has physi-cal significance (see Supplementary Figure S8). The EPRsign 𝑠𝑚𝑗 is calculated in parallel with the process of cal-culating 𝑝𝑚𝑗 , from the field solution ®𝐻 (®𝑟), see Supple-mentary Section (C.8). We now obtain the EPR orthog-onality relationship

𝑀∑𝑚=1

𝑠𝑚𝑗 𝑠𝑚𝑗′√𝑝𝑚𝑗 𝑝𝑚𝑗′ = 0 , (24)

valid when the sum from 1 to 𝑀 covers all the relevantmodes, see Supplementary Eq. (A.60).

3. Excitation-number-conserving interactions

Thus, as announced, knowledge of the energy-participation ratios completely specifies 𝐻nl, throughEqs. (17), (19), and (21). The Hamiltonian can nowbe analytically or numerically diagonalized using variouscomputational techniques20. In this section, our focuswill be now to explicitly handle the effect of the nonlin-ear interactions 𝐻nl on the eigenmodes. Before treatingthe case of a general nonlinear interaction, we focus onthe leading-order effect of 𝐻nl in the case of the ‘canon-ical’ Josephson system. In this case, the 𝐽 Josephsondipoles are all Josephson tunnel junctions, characterizedby Eq. (A.14b), and the dispersive regime is satisfied forall pairs of modes 𝑘 and 𝑚; i.e., 𝜔𝑘 − 𝜔𝑚 𝐸 𝑗𝑐 𝑗 𝑝

⟨𝜑𝑝

𝑗

⟩for 𝑝 ≥ 3 and in the absence of strong drives. Theleading-order nonlinear terms are the subset of 𝑝 = 4terms that conserve excitation number. After normalordering, see Supplementary Section B1, one finds the

5

effective Hamiltonian

ˆ𝐻4 = −ℏ

𝑀∑𝑚=1

Δ𝑚𝑎†𝑚𝑎𝑚 +

𝛼𝑚

2𝑎†2𝑚 𝑎2𝑚 +

∑𝑛<𝑚

𝜒𝑚𝑛𝑎†𝑚𝑎†𝑛𝑎𝑚𝑎𝑛 ,

(25)which is a generalization of the one found in Eq. (8).In Eq. (25), we have introduced the Lamb shift Δ𝑚 ofmode 𝑚, the anharmonicity 𝛼𝑚 of the mode, and its to-tal dispersive shift 𝜒𝑚𝑛 (so-called cross-Kerr term) witha different mode, labeled 𝑛. Each of these parameters isdirectly calculated from the EPRs. As shown in Supple-mentary Section (B.3a), for arbitrary 𝑚 and 𝑛,

𝜒𝑚𝑛 =

𝐽∑𝑗=1

ℏ𝜔𝑚𝜔𝑛

4𝐸 𝑗

𝑝𝑚𝑗 𝑝𝑛 𝑗 , (26)

while 𝛼𝑚 = 𝜒𝑚𝑚/2 and Δ𝑚 =∑𝑀

𝑛=1 𝜒𝑚𝑛/2. Equation (26)implements, mathematically, the idea that the amplitudeof these nonlinear couplings is the result of a spatial-mode scalar product of the EPRs. Remarkably, fromEq. (26) it is seen that the EPRs are essentially the onlyfree parameters subject to design, when determining thenonlinear couplings, since 𝜔𝑚, 𝜔𝑛, and 𝐸 𝑗 are generallytightly constrained by experimental considerations.

Equation (26) can be cast in matrix form by introduc-ing the EPR matrix

P B©­­«

𝑝11 · · · 𝑝1𝐽...

. . ....

𝑝𝑀1 · · · 𝑝𝑀𝐽

ª®®¬ , (27)

which we have found useful in handling large cir-cuits, especially for those in excess of 100 modes. Wealso introduce the diagonal matrices of eigenfrequen-cies Ω B diag (𝜔1, . . . , 𝜔𝑀 ) and junction energies EJ Bdiag (𝐸1, . . . , 𝐸𝐽 ), which lead to the matrix form ofEq. (26),

Kerr matrix: χ = ℏ4ΩPE

−1J Pᵀ𝛀 ,

Anharmonicity: 𝛼𝑚 = 12 [χ]𝑚𝑚 ,

Lamb shift: Δ𝑚 = 12

∑𝑀𝑚′=1 [χ]𝑚𝑚′ .

(28)

We have defined the symmetric matrix of dispersiveshifts χ, with elements [χ]𝑚𝑚′ = 𝜒𝑚𝑚′ . Further discus-sion of the matrix approach and applications to 𝑝th-ordercorrections is deferred to Supplementary Section B2, andthe amplitude of an arbitrary multi-photon interactionstemming from the full 𝐻nl is calculated in Supplemen-tary Section B3.

4. EPR for dissipation in the circuit

The EPR method treats the calculation of Hamiltonianand dissipation parameters on equal footing. Unlike inthe impedance method4, one can completely character-ize both 𝐻full and the effect of dissipative elements in the

10-1

100

101

102

103

104

10-1 100 101 102 103 104

Theory vs. experiment

R9C1R2C1R7C1R3C1R3C2DT3DTW1WG1

sample

Pre

dic

ted

ener

gy(M

Hz)

Measured energy (MHz)

χmn

m′ωmα

fullHx=y

Figure 4. Comparison between theory and experi-ment over five-orders of magnitude in energy scale of thesystem Hamiltonian 𝐻full for eight distinct, multi-mode de-vice samples, described in detail in the Methods, including3D, flip-chip (2.5D), and 3D waveguide architectures incor-porating readout and storage resonators and qubit modes.For each device, the dominant parameters in 𝐻full, dressedfrequencies 𝜔′𝑚, bare anharmonicities 𝛼𝑚, and cross-Kerr in-teractions 𝜒𝑚𝑛, were measured and calculated using the EPRmethod with our open-source pyEPR package95. Gray lineis of slope one, representing ideal agreement between theoryand experiment.

circuit from the eigenfield solutions, ®𝐸𝑚 (®𝑟) and ®𝐻𝑚 (®𝑟).The list of dissipative elements include bulk and sur-face dielectrics81–83, thin-film metals53,84, surface inter-faces85–89, and metal seams90. The energy-participationratio of a dissipative element 𝑙 in mode 𝑚 will be de-noted 𝑝𝑚𝑙. It is calculated similarly to 𝑝𝑚𝑗 , as summa-rized in Supplementary Section D. The participation 𝑝𝑚𝑙

and the quality factor 𝑄𝑙 of the material of this elementare used to estimate the total quality factor of mode 𝑚 inthe standard way when the fields are not greatly alteredby the dissipation (𝑄𝑚 1)30,91–93,

𝑄−1𝑚 =∑𝑙

𝑝𝑚𝑙𝑄−1𝑙 . (29)

Experimental values of 𝑄𝑙 are found in the literature, andsome are provided in Supplementary Section D. Equa-tion (29) and the dissipative EPR 𝑝𝑚𝑙 provide a dissipa-tion budget for the individual influence of each dissipa-tion mechanism in the system, providing a useful tool tooptimize design layout for quantum coherence94.

6

C. Comparison between theory and experiment

Applying the EPR method, we designed 8 supercon-ducting samples to test the agreement between the EPRtheory and experimental results. We tested several sam-ple configurations, comprising 15 qubits, 8 cavity modes,and one waveguide in three different circuit-QED ar-chitectures. The samples were measured in a standardcQED setup, see Methods, at the 15 mK stage of a dilu-tion unit, over multiple cool downs.

Six of the samples were each composed of 2 qubits andone 3D cavity, one sample was composed of 2 qubits anda waveguide, and one sample was a flip-chip, 2.5D sys-tem28 consisting of a flip-chip qubit embedded in a two-mode whispering gallery mode resonator53 (WGMR).The specifics of each sample are discussed in the Meth-ods.

For each sample, we measured the circuit parametersof interest: dressed mode frequencies 𝜔𝑚 − Δ𝑚, anhar-monicities of qubits and high-Q cavities 𝛼𝑚, cross-Kerrfrequencies 𝜒𝑚𝑛, and input-output quality factors 𝑄𝐶 forany readout modes. Our measurement methodology isdetailed in the Methods.

The measured parameters were compared to those cal-culated using the energy-participation method. The lin-earized Josephson circuit of each sample was modeledin Ansys High-Frequency Electromagnetic-Field Simula-tor (HFSS). Junctions were modeled as lumped induc-tors, whose nominal energy 𝐸 𝑗 was inferred from room-temperature resistance measurements49. To account forthe error bars of the measurement and the drift in resis-tance over time, 𝐸 𝑗 was adjusted by no more than 10%to fit the measured qubit frequency. To minimize thenumber of free parameters, we neglect the small junctionintrinsic capacitance 𝐶𝐽 in our modeling. The tradeoff isa small and estimable systematic offset of the bare simu-lated mode anharmonicities. We estimate this correctionto be on the order of 4% for a 𝐶𝐽 = 4 fF. From theeigenfield solutions, we calculated the EPRs 𝑝𝑚𝑗 and thesign 𝑠𝑚𝑗 to construct 𝐻 and extract its parameters. De-tailed steps of the procedure can be found in Supplemen-tary Section C. The results are presented in Tables I–III.

Figure 4 summarizes the agreement of the measuredand calculated sample parameters, which span five or-ders of magnitude in frequency. Accounting for 𝐶𝐽 , wefind that mode frequencies are calculated to one percentaccuracy, large nonlinear interaction energies (namely,anharmonicity and cross-Kerr frequencies greater than10 MHz) are calculated at the 5% level, and small nonlin-ear interaction energies agree at the 10% level. We high-light that we have used minimal, coarse adjustment toaccount for shifts in 𝐸 𝑗 , and otherwise, by neglecting 𝐶 𝑗 ,the calculation is free from adjustable parameters.

The results of Fig. 4 demonstrate the accuracy andapplicability of the EPR method. For each device, theEPR results are obtained from a single eigenmode sim-ulation, using full automation of the analysis, providedby our open-source package pyEPR95. For current stan-

dard applications, we find the agreement sufficient. Fur-ther improvements in accuracy would require improvedability to estimate the Josephson dipole energy 𝐸 𝑗 andits intrinsic capacitance 𝐶𝐽 . At the same level of accu-racy, improvements in the precision and reproducibilityof the implementation and assembly of the Josephsoncircuit design are needed, such as in chip-clamping tech-niques, precision machining of the device sample holderand input-output couplers.

Conclusion. An intuitive, easy-to-use and efficientmethod is needed to design and analyze Josephson mi-crowave quantum circuits. We have described in thisarticle such a method, based on the distribution of theelectromagnetic energy in the circuit and its participa-tion in nonlinear and dissipative elements. This so-calledEPR method offers physical insight helping the designprocess, and provides a simple link between the clas-sical circuit and its quantum properties. By compar-ing our theory to 8 experimental devices incorporatingJosephson junctions, we have shown that our methodis accurate and applicable to a large range of quan-tum circuit architectures. It is directly applicable toa broader class of nonlinear inductive elements, suchas weak-link nanobridges38,63, nanowires40,41,64,66,67, andkinetic-inductance thin-films43,58,59. While best suitedfor weakly nonlinear systems, the EPR method is de-rived within circuit theory without approximations. Itcan be seen as arising from a change of basis adaptedto nonlinear elements, as detailed in Supplementary Sec-tion A. In practice, the useful reach of the method isset by the numerical ability to include all relevant elec-tromagnetic modes and to compute the spectrum of theextracted Hamiltonian20. We contribute an open-sourcepackage pyEPR95, which automates the EPR method,and was tested in the design of several further experi-ments28,74,96–105.

III. METHODS

A. Methods of the experiment

Device fabrication. Unless otherwise noted, sampleswere fabricated according to the following methodology.Sample patterns, both large and fine features, were de-fined by a 100 kV electron-beam pattern generator (RaithEBPG 5000+) in a single step on a PMAA/MAA (Mi-crochem A-4/Microchem EL-13) resist bilayer coated on a430 `m thick, double-side-polished, c-plane sapphire wafer,grown with the edge-defined film-fed growth (EFG) tech-nique. Using the bridge-free fabrication technique106–108 theAl/AlOx/Al Josephson tunnel junctions were formed by adouble-angle aluminum evaporation under ultra-high vacuumin a multi-chamber Plassys UMS300 UHV. The two depo-sitions were interrupted by a thermal oxidation step, static100 Torr environment of 85% argon and 15% oxygen, to formthe thin AlO𝑥 barrier of the tunnel junction. Prior to the firstdeposition, to reduce junction aging108, the exposed wafersurfaces were exposed to 1 minute oxygen-argon plasma clean-

7

Device Frequency (MHz) Anharmonicity (MHz) Cross-Kerr (MHz) I-O coupling𝜔D/2𝜋 𝜔B/2𝜋 𝜔C/2𝜋 𝛼D/2𝜋 𝛼B/2𝜋 𝜒DB/2𝜋 𝜒BC/2𝜋 𝜒DC/2𝜋 𝑄C

R9C1 4951 5664 9158 138 170 92. 4.7 0.4 5.20 × 1034866 5691 9154 150 185 99. 4.2 0.55 7.40 × 103-1.7% 0.5% -0.04% 8% 8% 7% -12% 27% 29%

R2C1 4823 5567 8947 150 192 64.5 4.8 0.3 4.97 × 1034770 5640 8950 161 211 67.7 5.88 0.46 5.44 × 103-1.1% 1.3% 0.03% 6.8% 9% 4.7% 18% 35% 9 %

R7C1 4726 5475 8999 156 189 67. 4.8 0.34 2.68 × 1034770 5640 8950 161 211 67.7 5.88 0.46 3.07 × 1030.9% 2.9% -0.55% 3.1% 10% 1% 18% 26% 13%

R3C2 4845 5620 8979 152 195 61 5.1 0.3 2.11 × 1034770 5640 8950 161 211 67.7 5.88 0.46 1.78 × 103-1.5% 0.4% -0.3% 5.6% 7.6% 9.9% 13% 35% -19%

R3C1 4688 5300 9003 148 174 85 5. 0.33 2.43 × 1034745 5265 8922 159 198 73 5.1 0.37 5.65 × 1031.2% -0.7% -0.9% 6.9% 12.1% -16% 2% 9% 57%

DT3 6160 7110 9170 130 150 278 3. 2.5 9.17 × 1036100 7141 9155 140 177 312 3.9 3.1 7.33 × 103-1.0% 0.4% -0.15% 7% 15% 11% 23% 19% -25%

Table I. Two-qubit, one-cavity devices. Summary of measured and calculated Hamiltonian and input-output (I-O)coupling parameters for the six devices described in Methods. Indices D, B, C denote the dark, bright, and cavity modesrespectively. The input-output quality factor to the readout cavity is denoted 𝑄C. For each device, the first (second) rowquantifies the measured, 𝑚, (bare calculated, 𝑐) values. The third row quantifies the bare agreement, i.e., (𝑐 − 𝑚) /𝑐. In theanharmonicity column, the bare agreement should be corrected by the systematic shift due to our choice to neglect the junctionintrinsic capacitance in our modeling (see Methods). We evaluate the correction to be of order 4%, estimated by taking anominal junction 𝐶𝐽 = 4 fF; hence, an overall corrected agreement of 4.3% for this column.

Device Frequency (MHz) Anharmonicity (MHz) Cross-Kerr (MHz) I-O coupling𝜔Q/2𝜋 𝜔S/2𝜋 𝜔C/2𝜋 𝛼D/2𝜋 𝜒QS/2𝜋 𝜒QC/2𝜋 𝑄C

WG1 4890 7070 7267 310 0.25 0.30 20 × 1034820 7020 7340 325 0.29 0.33 16 × 103-1.4% -0.7% 1.0% 4.6% 13% 9% -22%

Table II. Flip-chip (2.5D), one-qubit, one-storage-cavity, one-readout-cavity devices. Summary of measured andcalculated Hamiltonian and input-output (I-O) coupling parameters for the device described in Methods. Indices Q, S, Cdenote the qubit, storage, and readout cavity modes respectively. The input-output quality factor to the readout cavity isdenoted 𝑄C. For each device, the first (second) row quantifies the measured, 𝑚, (bare calculated, 𝑐) values. The third rowquantifies the bare agreement, i.e., (𝑐 − 𝑚) /𝑐.

ing, under a pressure of 3 × 10−3 mbar. After wafer dicing(ADT ProVecturs 7100 ) and chip cleaning, the normal-stateresistance 𝑅𝑁 of the Josephson junctions was measured toprovide an estimate of the Josephson energy, 𝐸𝐽 , of the devicejunctions. The junction energy was to first order estimatedby an extrapolation of 𝑅𝑁 from room temperature to the op-erating sample temperature, at approximately 15 mK, usingthe Ambegaokar-Baratoff relation109,

𝐸𝐽 =1

2

ℎΔ

(2𝑒)2𝑅−1𝑁 , (30)

where Δ is the superconducting gap of aluminum, 𝑒 is theelementary charge, and ℎ is Planck’s constant.

Sample holder. Sample holders were machined in aluminumalloy 6061, seams were formed using thin indium gasketsplaced in machined grooves in one of the mating surfaces.Only non-magnetic components were used in proximity tothe samples, molybdenum washers, aircraft-alloy 7075 screws(McMaster/Fastener Express) with less than 1% iron impu-rities, and non-magnetic SubMiniature version A (SMA) con-nectors.

Cryogenic setup. Samples were thermally anchored to the15mK stage of a cryogen-free dilution refrigerator (OxfordTriton 200 ) and were measured using a standard cQED mea-surement setup45,51,96. High-magnetic-permeability, `-metal(Amumetal A4K ) shields together with aluminum supercon-

8

Frequency (MHz) Anharmonicity (MHz) Cross-Kerr (MHz)𝜔D/2𝜋 𝜔B/2𝜋 𝛼D/2𝜋 𝛼B/2𝜋 𝜒DB/2𝜋

6010 8670 85 180 2785824 8878 97 206 281-3.2% 2.3% 12% 13% 1.1%

Table III. Two-qubit, one-waveguide devices. Summary of measured and calculated Hamiltonian parameters for thedevice described in Methods . Indices D and B denote the dark and bright modes, respectively. For each device, the first(second) row summarizes the measured, 𝑚, (bare calculated, 𝑐) values. The third row quantifies the bare agreement, i.e.,(𝑐 − 𝑚) /𝑐.

ducting shields enclosed all samples. Microwave input andoutput lines were filtered with Eccosorb CR-110 infrared-frequency filters55,92, thermally anchored at the 15 mK stage.Output lines were additionally filtered with cryogenic isolators(Quinstar CWJ1019-K414 ) and 12GHz K&L multi-sectionlowpass filters. Output lines leading up to the high-electron-mobility transistor (HEMT) amplifier (Low Noise Factory),anchored at 4 K, were superconducting (CoaxCo Ltd. SC-086/50-NbTi-NbTi PTFE).

Quantum amplifier. The output signal of a sample wasprocessed by a Josephson parametric converter (JPC) an-chored at the 15 mK stage and operated in amplificationmode110,111, before routing to the HEMT. The JPC providea typical gain of 21 dB with a typical noise-visibility ratio of6 dB. See Ref.112 for a review of the parametric amplification.

Frequency and input-output (I-O) coupling measurements.Spectroscopic measurements were used to determine thefrequencies of the resonator modes. Anharmonicitieswere determined in two-tone spectroscopy92,113. Cross-Kerr energies were determined from dressed dephasingmeasurements114,115. In particular, the dressed-dephasingmeasurement sequence consisted of first preparing the qubitin the ground state, then exciting it to the equator by a 𝜋/2pulse. Subsequently, a weak readout tone excited the readoutcavity of the qubit for a fixed duration, 10 times the read-out cavity lifetime ^𝑟 , after which we measure the qubit Xand Y Bloch vectors, after waiting for a time 5/^𝑟 for anyphotons in the cavity to leak out. By varying the amplitudeand frequency of the applied weak-readout tone, we couldcalibrate both the strength of our readout, in steady-statephoton number in the readout cavity, and the value of thecross-Kerr frequency shift between the qubit and readout res-onator. The values could be obtained from fits of the X and Yquadratures. For each sample, the coupling quality factor ofthe readout-cavity mode, denoted 𝑄𝐶 , was extracted fromthe spectroscopic response of the readout cavity at low pho-ton numbers92,113, by measuring the scattering parameters,𝑆21 or 𝑆11.

To test EPR’s robustness to experimental variability andits applicability over wide range of experimental conditions,the presented samples were fabricated in multiple runs andmeasured in different cooldowns. Some devices were subjectedto as many as 6 thermal cycles.

The Hamiltonian parameters and coupling energies for eachsample were also calculated, following the EPR method pre-sented in section on the general approach. In particular, wemodeled the sample geometry and materials in a FE elec-tromagnetic simulation, as explicated in Supplementary Sec-tion C. Our aim in writing this supplementary section hasbeen to provide an easy access point to the practical use of the

EPR method, which we hope will benefit the reader, and al-low them to adopt it easily. Our choice of simulation softwarewas the Ansys High Frequency Electromagnetic Field Simu-lation (HFSS), although we emphasize that the EPR ideastranslate to any standard EM eigenmode simulation package.Further, we modeled the loss due to the input-output couplersin the simulation as 50Ω resistive sheets, see SupplementarySection D4. The eigenmode analysis provided the calculatedI-O quality factors and Purcell limits. All electromagneticand quantum analyses, including the extraction of participa-tions form the eigenfields and the numerical diagonalization ofthe Hamiltonian to extract its quantum spectrum, were per-formed in a fully automated manner using the freely availablepyEPR package95.

The mode quality due to the input-output coupling, 𝑄𝐶 ,was set by the length of the I-O SMA-coupler pin. Its lengthinside the sample-holder box was measured at room temper-ature using calipers. This nominal length was used then usedin the HFSS model to create a 3D model of the pin inside thesample holder. The quality factor 𝑄𝐶 was then obtained fromthe eigenmode eigenvalue. We remark that the measurementof the pin-length is accurate to no more than 20%; further, itcan be affected by various idiosyncrasies, such as bending ofthe thin SMA center pin. Nonetheless, the predictions of thequality factors for low-Q modes were observed to be very rea-sonable estimates, and, similarly, the predicted Purcell limitsfor qubit and high-Q cavity modes were consistent with esti-mates from measurements.

B. Devices

1. Two-qubit, one-cavity devices

Device description. We measured 6 samples that wereeach comprised of two qubits and one cavity. The cavity was astandard, machined aluminum cavity54. It housed either oneor two sapphire chips, which were either patterned with trans-mon qubits or simply blank. Each transmon consisted of twothin-film aluminum pads connected by a Josephson junction.We tested two configurations of chips and patterns. Config-uration A consisted of one chip with two orthogonal qubits,as depicted in Fig. 5(a). Similarly, configuration B consistedof one chip with two parallel qubits, depicted in Fig. 5(b).The two qubits were aligned parallel to each other; however,unlike configuration B, there was no galvanic connection be-tween them. The results of the measurements are presentedin Table I.

9

I-O

cavity

15 mm

E

ca b

Figure 5. Two–qubit, one-cavity devices. a and b Not-to-scale diagram illustrating chip configurations A and B, re-spectively. Vertical blue arrow indicates cavity electric fieldorientation. Crosses mark the location of Josephson tunneljunctions. c Optical photograph of sample R2C1. Bottomhalf of aluminum sample holder is visible; top half is removed.The two-qubit chip (outlined by the dashed green box) ishoused in the middle of the readout cavity (highlighted inblue). Cavity fundamental mode electric field profile ®𝐸 de-picted by arrows. Input-output SMA pin coupler labeled I-O.

Samples R1C9, R2C1, R7C1, and R2C1, R3C2 were fab-ricated in configuration A, sample DT3 was fabricated inconfiguration B. Three of the sample (R2C1, R7C1, R3C1)were fabricated simultaneously on the same sapphire wafer, allwith nominally identical dimensions. Additionally, R2C1 andR7C1 were designed to have nominally the same Josephsonjunctions energy, 𝐸 𝑗 . The rest of the samples (R1C9, DT3,and R3C2) were fabricated at different times and on differentwafers. The dimensions of their transmons and the inductanceof the junctions were designed to be different. Only sampleR3C2 was designed to be very similar to the nominally identi-cal sample R2C1 and R7C1, but with adjusted 𝐸 𝑗 . For sam-ples R2C1, R7C1, R3C1, and R3C2 a second, un-processed,un-patterned, blank sapphire chip was placed in parallel withthe qubit carrying chip [see Fig. 5(c)] to purposefully lowerthe readout cavity frequency, thus bringing it within the JPCamplification band.

Configurations A and B were designed to test the abilityof the EPR method to calculate the mixing between stronglycoupled modes. The strong coupling was achieved in twodistinct ways. First, configuration A used the spatial prox-imity of the two qubits to yielded a strong capacitive cou-pling between them, which resulted in large qubit-qubit mix-ing. Second, instead of spatial proximity, configuration Bused a galvanic connection between the qubits to yield stronghybridization. Our two-qubit designs share some similar-in-spirit characteristics with the promising recent developmentsreported in Refs. 116–121, but our implementation is distinctand is designed to provide several unique advantages.Mode structure and interesting physical insights.Configuration A is characterized by strong capacitive couplingbetween the two transmons, which have different pad sizes, seeFig. 5(c), and hence different normal-mode frequencies. Dueto the strong hybridization, each qubit normal mode consistsof some excitation in the vertical and some in the horizon-tal transmon. With some foresight, we will label the verticalmode bright (B), and the horizontal dark (D). The bright-mode resonance is higher in frequency, and thus is closer tothe resonance of the readout cavity mode (C). This smallerdetuning made it a natural choice for designing stronger cou-

pling between it, (B), and the readout mode (C). This wasimplemented by orienting the transmon design that partici-pates in mode (B) vertical.

To understand this design choice, let us first consider thepopular analogy46,122 between circuit-QED and cavity-QED,often used to discuss mode couplings. In this atomic analogy,the transmon qubit is analogous to a real atom inside the cav-ity. Thus, it can described by an electric dipole moment ®d𝐵.Meanwhile, its coupling, cross-Kerr, etc. to the cavity modeare derived from the electric-dipole coupling interaction. Inparticular, the coupling amplitude is proportional to ®d𝐵 · ®𝐸,where ®𝐸 is the cavity electric field at the transmon junction.From this analogy, one can infer that the coupling is maxi-mized when the two are parallel, ®d𝐵 ‖ ®𝐸, and one could hopeto measure a strong cross-Kerr between the bright qubit andthe cavity. This successful conclusion is true, but a coinci-dence. We will shortly discuss how this popular analogy failsspectacularly for the dark mode in configuration B. Instead,we will argue that a correct way to understand the nonlinearcoupling between the two modes is through the participationratio, which will provide the correct coupling for both config-uration A and B.

Before proceeding to configuration B, we note one furtheruseful features that configuration A exhibits. In particular,while the bright qubit mode can be Purcell limited116,123, thedark mode is simultaneously Purcell protected. Thus, one canpotentially achieve a high ratio in the I-O bath coupling ofthe two qubits.

Configuration B has two qubit modes, which we will alsolabel dark (D) and bright (B). Since both transmons are de-signed with the exact same transmon pad geometry and junc-tion energy 𝐸𝐽 , see Fig. 5, we can expect that no single junc-tion is preferred, due to the symmetry of the sample. Thisis in sharp contrast to the asymmetric energy distributionin configuration A. Returning to configuration B, we can es-timate that in each qubit mode, both junctions participateequally and with near maximal allowed participation,

𝑝D1 = 𝑝D2 = 𝑝B1 = 𝑝B2 ≈1

2. (31)

If the two transmons were well-separated spatially and notconnected, they would be uncoupled. However, the galvanicconnection between the two lower pads, see Fig. 5(a), resultsin a very strong hybridization and splitting between the nomi-nally identical transmons. The result of the strong hybridiza-tion is a symmetric and antisymmetric combination of thetwo bare transmons. In other words, the hybridization re-sults in a common mode, namely (B), where both junctionsoscillate in-phase, and a differential mode namely (D), whereboth junctions oscillate out-of-phase. These phase relation-ships are captured by the signs:

𝑠D1 = 1 , 𝑠D2 = −1 , (32)𝑠B1 = 1 , 𝑠B2 = +1 . (33)

In an attempt to understand how these hybridized qubitmodes will couple to the cavity mode (C), let us first considerthe atomic analogy again. When the two junctions oscillate inphase, in the (B) mode, the net dipole moment of the brightmode, ®𝑑B, must be large, since it is the sum of the two junc-tion dipole contributions. Secondly, ®𝑑B must be oriented inthe vertical direction, parallel to the cavity electric field ®𝐸.Hence, we would conclude that the bright mode coupling is

10

large, ®𝑑B · ®𝐸 0, and there should be a strong cross-Kerrinteraction between the cavity and bright qubit. Continuingthe analogy in the case of the dark mode, we would deducethat the net dipole moment of the dark mode is zero, since thetwo junctions oscillate out of phase, and cancel each other’scontribution, ®𝑑D = 0. Thus, we should not expect any cou-pling between the dark qubit and the cavity mode, ®𝑑D · ®𝐸 = 0.To the contrary of this conclusion, as can be seen in the mea-sured results in Table I, the nonlinear coupling of the darkand bright qubit to the cavity is nearly equal. The atomicanalogy and the dipole argument have failed completely. Wecan understand the origin of this failure and how to arrive atthe correct conclusion by using the energy-participation ratio.As embodied in Eq. (26), in the dispersive regime, the non-linear coupling between two modes, in this case a qubit andcavity, is given by the overlap of the EPR distribution. Inparticular, the cross-Kerr amplitude between the dark qubitand the readout cavity mode is given by

𝜒DC =ℏ𝜔𝐷𝜔𝐶

4𝐸𝐽(𝑝D1𝑝C1 + 𝑝D2𝑝C2) , (34)

where both junctions have the same junction energy 𝐸𝐽 ,and 𝜔𝐵 (resp: 𝜔𝐶) denotes the dark qubit (resp: cavity)mode frequency. The signs, used in the atomic dipole logicdo not factor into the coupling, because the Josephson me-chanics is fundamentally different. To obtain 𝜒BC, one canreplace the label ‘D’ with ‘B’ in Eq. (34). Then, it is easy touse Eq. (31) to show that the ratio of two Kerr couplings isnot zero, but rather of order unity,

𝜒BC/𝜒DC = 𝜔𝐵/𝜔𝐷 . (35)

Failure of the conventional dipole approach. Weshowed that although the heuristic atomic analogy seems se-ductively accurate, it fails completely in some cases to predictthe nonlinear couplings. Instead, one can use the intuitionand calculation method provided by the EPRs.

As an added note, we observe that Eqs. (32) and (32) em-bodies the orthogonality of the participations, see Eq. (24).We also remark that although the atomic analogy fails in thecase of the nonlinear couplings, it can yield some guidancewhen considering the linear mixing of the modes, useful fordiscussing the Purcell effect. To illustrate, let us briefly ex-tend the atomic analogy. The dipole-like coupling between thebright mode and the cavity suggests that the bright mode willinherit some coupling to the environment, mediated by thecavity. Thus, since ®𝑑B · ®𝐸 0, we can expect the bright qubitto potentially be Purcell limited. In contrast, since ®𝑑D · ®𝐸 = 0,we could expect the dark qubit to be Purcell protected. Bothof these qualitative Purcell predictions are valid, but to quan-tify them, we will use the EPR method and FE eigenmodesimulation of the sample, as will be discussed shortly.Experimental results. Table I summarizes the results ofthe agreement between the measured and calculated Hamil-tonian parameters for all two-qubit, one-cavity samples. Thethree modes in each sample are labeled dark (D), bright (B),and cavity (C); the reason for this convention is describedabove. In all samples, the qubits were designed to be in thedispersive regime with respect to the cavity, which was de-tuned by 2–4GHz. However, in a large number of the samples,the two qubits were strongly hybridized, often necessitatinghigher-order nonlinear corrections to be included in the cal-culation. This strong hybridization was used as a test of thetheory in this more challenging and fickle regime.

waveguideWR90

shorttermination

I-Oport

4λ/

Figure 6. Schematic representation (not to scale) ofsuperconducting waveguide device DTW1. Two-qubitchip is placed _/4 from a short termination in the waveg-uide. Guided input-output waves are launched and monitoredthrough the I-O port connecting to a standard SMA adapter(not shown).

In total, for each sample we measured and calculated 8frequency parameters and one dimensionless, coupling qual-ity factor, 𝑄𝐶 , of the readout cavity mode. In particular, inthe low-excitation limit, the nonlinear interactions among themodes were characterized by the effective dispersive Hamilto-nian

ˆ𝐻/ℏ = 𝜔𝐷𝑛𝐷 + 𝜔𝐵𝑛𝐵 + 𝜔𝐶𝑛𝐶

− 1

2𝛼𝐷𝑛𝐷

(𝑛𝐷 − 1

)− 1

2𝛼𝐵𝑛𝐵

(𝑛𝐵 − 1

)− 𝜒𝐷𝐵𝑛𝐷𝑛𝐵 − 𝜒𝐷𝐶𝑛𝐷𝑛𝐶 − 𝜒𝐵𝐶𝑛𝐵𝑛𝐶 , (36)

where 𝑛𝐷 , 𝑛𝐵, and 𝑛𝐶 denote the dark, bright, and cavityphoton-number operator, respectively. The coupling of theresonator mode to the bath is given by the Lindblad super-operator term ^𝐶D[𝑎𝐶 ]𝜌, where ^𝐶 = 𝜔𝐶/𝑄𝐶 , and 𝜌 is thedensity operator.

We remark that all samples in configuration A demon-strated a large asymmetry in the Kerr coupling between thebright-to-cavity and dark-to-cavity coupling, 𝜒𝐵𝐶 𝜒𝐷𝐶 .In contrast, samples in configuration B demonstrated nearequal coupling, 𝜒𝐵𝐶 ≈ 𝜒𝐷𝐶 . In both configurations A andB, the dark mode was Purcell protected, we calculated aPurcell coupling factor of 𝑄𝐷

Purcell 107, using the eigen-mode method described in Supplementary Section D4. Onthe other hand, the bright mode was somewhat Purcell lim-ited, 𝑄𝐵

Purcell ≈ 106. From the relative Rabi amplitudes of thedark and bright qubits, we could verify the order of magnitudescaling calculated for the Purcell effect.

2. Two-qubit, single-waveguide devices

We measured a two qubit sample inside of a waveguide. Fig-ure 6 presents the setup, and depicts the sample, which wasof the configuration B type presented in Fig. 5. The samplechip was positioned inside an aluminum WR90 waveguide.The waveguide was terminated in a short at one side, andattached to an impedance-matched SMA coupler port on theinput-launcher side, which was used to drive and measure thewaveguide. The chip was centered inside the cross-section

11

top chip

bottom chip

2.5D transmon(lower layer)WGMR

a b

Figure 7. Illustration of flip-chip (2.5D) device WG1.a Depiction (not to scale) of chip stack consisting of two chipsseparated by a 100 μm vacuum gap. The inner face of eachchip supports part of the pattern of a multi-layer whisper-ing gallery-mode resonator (WGMR) resonator53. The lowerlayer contains a 2.5D, aperture transmon qubit28 embedded inthe WGMR. b Zoomed-in view of the lower layer of the aper-ture transmon. The cross marks the location of the Josephsontunnel junction device, which connects the lower center traceto an island embedded in the lower aperture. The capacitanceto the top layer significantly participates in determining qubitparameters.

of the waveguide, and placed _/4 away from the termina-tion wall, at the measurement frequency. The rest of theexperimental setup was identical to that described in section‘Methods of the experiment’. The two qubit modes were la-beled dark and bright, similarly to the samples discussed inthe section ‘Two-qubit, one-cavity devices.’.

Table III presents the agreement between the measured andcalculated key Hamiltonian parameters of the sample. Theseconsist of the two mode frequencies, two qubit anharmonic-ities, and the strong cross-Kerr interaction between the twoqubits.

3. Flip-chip (2.5D), one-qubit, one-storage-cavity,one-readout-cavity devices

We also designed a multilayer planar28 (2.5D) circuit-QEDsample, depicted in Fig. 7, with the EPR method. It consistedof high-Q storage mode (S), one low-Q readout cavity (C),and one control transmon qubit (Q). The two cavity modeswere formed in the footprint of a single whispering gallerymode resonator53. The three modes were in the dispersiveregime, and the storage mode was used to encode and decodequantum information, as well as to observe parity revivals.Details of the sample design have been reported in Ref. 28.The agreement between the measured parameters of the sam-ple and those obtained by the EPR calculation methods arepresented in Table II.

Data availability. Data are available from the authorson reasonable request.

Code availability. The source code for theEPR method is open-sourced and can be found at

http://github.com/zlatko-minev/pyEPR.

Acknowledgments. We thank S.M. Girvin,R.J. Schoelkopf, S. Nigg, H. Paik, A. Blais, H.E. Türeci,F. Solgun, V. Sivak, S. Touzard, N. Frattini, S. Shankar,C. Axline, V.V. Albert, K. Chou, A. Petrescu, D. Cody,A. Eickbusch, and E. Flurin for valuable discussions, andthe I. Siddiqi and B. Huard groups for using the earlyversions of pyEPR. This research was supported by theUS Army Research Office (ARO) Grant No. W911NF-18-1-0212. Z.K.M. acknowledges partial support fromthe ARO (W911NF-16-1-0349). M.H.D. acknowledgespartial support from the ARO (W911NF-18-1-0020) andthe Air Force Office of Scientific Research (FA9550-19-1-0399). The view and conclusions contained in thisdocument are those of the authors and should not beinterpreted as representing the official policies, eitherexpressed or implied, of the Army Research Office orthe US Government. The US Government is authorizedto reproduce and distribute reprints for Governmentpurposes notwithstanding any copyright notation herein.

Author contributions. Z.K.M. conceived and devel-oped the method and the theory, performed the exper-iments, and analyzed the data. Z.L. and M.H.D. con-tributed to the theory. I.M.P. and S.O.M. contributedto the measurement and fabrication of the devices L.C.assisted with the simulations. Z.K.M. and M.H.D. wrotethe manuscript. All authors provided suggestions, dis-cussed the results and contributed to the manuscript.

Competing interests. The authors declare no com-peting interests.

12

Supplementary Information:Energy-participation quantization of Josephsoncircuits

Contents

I INTRODUCTION 1

II RESULTS AND DISCUSSION 1A To quantize a simple circuit: qubit coupled to a cavity . . . . . . . . . . . . . . . . . . . . . . . . . 1B Quantizing the general Josephson system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3C Comparison between theory and experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

III METHODS 7A Methods of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7B Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

A Theoretical foundation of the energy-participation method 14A1 Review of electrical circuit theory, and the Josephson junction . . . . . . . . . . . . . . . . . . . . 15A2 The Josephson system and its non-linear Josephson dipoles . . . . . . . . . . . . . . . . . . . . . . 17A3 Energy of the Josephson circuit and its Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A4 Eigenmodes of the linearized Josephson circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20A5 Quantizing the Josephson circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A6 Energy-participation ratio (EPR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22A7 Universal EPR properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24A8 The biased Josephson system: equilibrium state in the presence of persistent currents . . . . . . . 24A9 The biased Josephson system: simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

B Nonlinear interactions, effective Hamiltonians, and the EPR 27B1 Excitation-number-conserving interactions of weakly-nonlinear systems . . . . . . . . . . . . . . . . 27B2 EPR matrices and many-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28B3 General many-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B4 The driven Josephson system: parametrically-activated interactions . . . . . . . . . . . . . . . . . 30

C Finite-element electromagnetic-analysis methodology 31C1 Modeling the Josephson dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31C2 Calculating the EPR 𝑝𝑚 in the case of a single Josephson dipole . . . . . . . . . . . . . . . . . . . 32C3 Calculating the EPR 𝑝𝑚𝑗 in the case of multiple Josephson dipoles . . . . . . . . . . . . . . . . . . 32C4 Remarks on the finite-element eigenmode approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

D Dissipation budget and input-output coupling 33D1 Dissipation budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D2 Capacitive loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34D3 Inductive loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34D4 Radiative loss and input-output coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

References 36

13

A. Theoretical foundation of the energy-participation method

Supplementary Table I: Table of key symbols and relationshipsused in the derivation of the energy-participation-ratio method.

Basic Josephson circuit variables and parameters

Symbol Value Description

Φ0 ℎ/(2𝑒) Superconducting magnetic-flux quantum—the ratio of Planck’s quantum of electromag-netic action ℎ and the charge of a Cooper pair 2𝑒.

𝜙0 Φ0/(2𝜋) Reduced flux quantum.

Φ𝑏 (𝑡)∫ 𝑡

−∞ 𝑣𝑏 (𝜏) d𝜏 Generalized magnetic flux of circuit branch 𝑏, at time 𝑡. The instantaneous voltage 𝑣𝑏 (𝜏)across the terminals of branch 𝑏 at time 𝜏 is equivalently denoted ¤Φ𝑏 (𝜏). In general,Φ𝑏 (𝑡) =

∫ 𝑡

−∞ 𝑣𝑏 (𝜏) d𝜏 + Φ𝑏 (−∞), but we take the initial flux Φ𝑏 (−∞) to be zero, corre-sponding to the circuit in an equilibrium state (see Sec. A8). With this convention, Φ𝑏

is a deviation away from equilibrium. This variable is analogous to the elongation of amechanical spring.

𝑗 , 𝐽 The subscript or index 𝑗 denotes the 𝑗-th Josephson dipole, where 𝑗 ∈ 1, . . . , 𝐽, and 𝐽

denotes the total number of Josephson dipoles.

Φ 𝑗 (𝑡) Generalized magnetic-flux deviation of a non-linear device, the 𝑗-th Josephson dipole.

𝜑 𝑗 (𝑡) Φ 𝑗 (𝑡) /𝜙0 Reduced magnetic-flux variable of Josephson dipole 𝑗 .

E 𝑗

(Φ 𝑗

)Elin𝑗

(Φ 𝑗

)+ Enl

𝑗

(Φ 𝑗

)Energy function of Josephson dipole 𝑗 . Typically split into two component: Elin

𝑗associated

with linear interactions and Enl𝑗

associated with nonlinear ones.

Elin𝑗

(Φ 𝑗

) 12𝐸 𝑗

(Φ 𝑗/𝜙0

)2 Linear component of the energy function of junction 𝑗 , defined by the energy scale 𝐸 𝑗 .Terms linear in Φ 𝑗 are absent since we selected an equilibrium operating point of thecircuit. The energy function Elin

𝑗may never contain non-linear terms.

Enl𝑗

(Φ 𝑗

)E 𝑗

(Φ 𝑗

)− Elin

𝑗

(Φ 𝑗

)The nonlinear component of the energy function of junction 𝑗 . For certain situations, itmay be favorable to select the partition of E 𝑗 such that Enl

𝑗contains linear interactions. If

possible, it is often convenient to expand Enl𝑗

in a Taylor series around the circuit operating

point: Enl𝑗

(Φ 𝑗

)= 𝐸 𝑗

∑∞𝑝=3 𝑐 𝑗 𝑝

(Φ 𝑗

𝜙0

) 𝑝, where 𝑐 𝑗 𝑝 are the dimensionless coefficients of the

expansion.

Φt (𝑡)(Φt1 , Φt2 , . . .

)ᵀ Column vector consisting of the flux deviations of all circuit branches enclosed in theminimum spanning tree. The roman subscript t denotes the tree. The flux of individualtree-branches are denoted Φt1 , Φt2 , . . .

Linearized Josephson eigenmodes

𝑚, 𝑀 The label 𝑚 indexes sets associated with the eigenmodes of the linearized Hamilto-nian Hlin. We consider 𝑚 ∈ 1, . . . , 𝑀, where 𝑀 is the total number of eigenmodesof relevance determined by the context. That is, 𝑀 is either the total number of eigen-modes of Hlin or of the relevant eigenmodes included in the finite-element simulation.

Hfull Hlin + Hnl Hamiltonian function of the Josephson system. It can be partitioned into Hlin, a partcomprised of purely linear terms (quadratic in Φt, or equivalently Φm), and Hnl, a partgenerally comprised of non-linear terms (higher-than quadratic in Φt, or equivalently Φm).

Φm, Q𝑚 Column vectors (of length 𝑀) whose elements are the eigenmode flux Φ𝑚 (consideredas the generalized position) and charge 𝑄𝑚 (considered as the generalized momentum)canonical variables, associated with the Hamiltonian Hlin.

𝜔𝑚 Eigenfrequency of the 𝑚-th mode of Hlin, carries a dimension of circular frequency.

14

Ω Diag (𝜔1, 𝜔2, . . .)ᵀ Diagonal 𝑀 × 𝑀 matrix comprising the eigenfrequencies.

Φt EΦm The spanning-tree branch fluxes Φt are a linear combination of the eigenmode Φm fluxes.The two are related by an affine transformation given by the properly-constructed eigen-vector matrix E, see Eqs. (A.38) and (A.37).

Quantum operators and energy-participation ratios

𝐻lin∑𝑀

𝑚=1 ℏ𝜔𝑚𝑎†𝑚𝑎𝑚 Hamiltonian operator corresponding to the Hamiltonian function Hlin, expressed in sec-

ond quantization with respect to the eigenmodes of Hlin. The eigenmodes (expressed interms of Φm and Q𝑚) define the bosonic amplitude (lowering) operators 𝑎𝑚.

𝜑 𝑗∑𝐽

𝑚=1 𝜑𝑚𝑗𝑎𝑚 +H.c. Operator corresponding to the reduced flux 𝜑 𝑗 of Josephson dipole 𝑗 ; 𝜑 𝑗 admits a lineardecomposition in terms of the mode operators 𝑎𝑚. The coefficients of this expansion arethe quantum zero-point fluctuations 𝜑𝑚𝑗 .

𝜑𝑚𝑗 𝑠𝑚𝑗

√ℏ𝜔𝑚

𝐸 𝑗𝑝𝑚𝑗 The quantum zero-point fluctuation (ZPF) of the reduced magnetic flux 𝜙 𝑗 of Josephson

dipole 𝑗 due to mode 𝑚; i.e., for a given mode, the ZPF magnitude gives the non-zero standard deviation of the magnetic flux in the ground state. The amplitude of theZPF 𝜑𝑚𝑗 is determined by the physical structure of mode 𝑚, and can be understood interms of the energy-participation ratio 𝑝𝑚𝑗 and its sign 𝑠𝑚𝑗 .

𝑝𝑚𝑗 , 𝑝𝑚𝑙 Energy-participation ratios (EPR) of Josephson dipole 𝑗 and lossy element 𝑙 in mode 𝑚,comprised between zero and unity, 0 ≤ 𝑝𝑚𝑗 , 𝑝𝑚𝑙 ≤ 1.

𝑠𝑚𝑗 , 𝑠𝑚𝑙 Sign 𝑠𝑚𝑗 (resp., 𝑠𝑚𝑙) of the energy-participation ratio 𝑝𝑚𝑗 (resp., lossy EPR 𝑝𝑚𝑙). Thevalue of the sign is either +1 or −1. See Supplementary Figure S8.

In this section, we derive the EPR method from firstprinciples. We first review quantum electromagnetic cir-cuit theory (Sec. A1), then use it to find the quantumeigenmodes of the circuit (Secs. A2–A5). In Sec. A6,we define the EPR of a Josephson dipole in a quantummode, and use it to find the quantum zero-point fluctu-ations (ZPF). The universal properties and sum rules ofthe EPRs are detailed in Sec. A7.

In Section A8, we detail use of the EPR method forbiased systems—those that incorporate active elements(such as current and voltage sources) or external biasconditions that result in persistent currents (such as afrustration by an external magnetic flux).

The EPR derivation consists of a series of exact trans-formation. No approximation are made in deriving theEPR or in using it to find the quantum Hamiltonian ofa general Josephson system. In this sense, the resultsare universal. Approximation are made in practice whenusing numerical methods.

A1. Review of electrical circuit theory, and theJosephson junction

We briefly recount the basic formulation of classicalelectrical theory. This formulation will be used as a step-ping stone in the following derivation. We focus on thelumped-element regime. An electrical element is said tobe in the lumped-element regime when its physical di-mensions are negligible with respect to the electromag-netic wavelengths considered in the analysis. In otherwords, self-resonances and parasitic internal dynamical

degrees of freedom of the element are neglectable.Electrical theory describes the physical laws governing

four basic manifestations of electricity. For each element,these are captured in the variables for voltage 𝑣 (𝑡), cur-rent 𝑖 (𝑡), charge 𝑄 (𝑡), and magnetic flux Φ (𝑡), as func-tions of time 𝑡. Maxwell’s equations and conservation ofcharge lead to the six universal relationships presentedin Eqs. (A.1)–(A.3)124,125.

The time-instantaneous voltage 𝑣 (𝑡) and general-ized magnetic flux across an element are relatedby the lumped-element version of Faraday’s law ofinduction47,48,124,125,

𝑣 (𝑡) = d

d𝑡Φ (𝑡) and Φ (𝑡) =

∫ 𝑡

−∞𝑣 (𝜏) d𝜏 +Φ (−∞) .

(A.1)By convention, the reference orientation of voltage is op-posite to that of current. This is not the conventiontypically used in Lenz’s law; in electromagnetism, thecurrent-density vector ®𝐽 and the electric-field vector ®𝐸are usually projected on the same orientation.

From charge conservation, it follows that the time-instantaneous current 𝑖 (𝑡) and the charge 𝑄 (𝑡) havingpassed through the element, obey a relation similar tothat of the voltage and flux,

𝑖 (𝑡) = d

d𝑡𝑄 (𝑡) and 𝑄 (𝑡) =

∫ 𝑡

−∞𝑖 (𝜏) d𝜏 +𝑄 (−∞) .

(A.2)All four variables have their support on a non-compactset; i.e., 𝑖, 𝑣, 𝑄, Φ ∈ [−∞,∞].

Reference state and initial conditions. In Eqs. (A.1)and (A.2), we now assume zero-valued initial condi-

15

tions, Φ (−∞) = 𝑄 (−∞) = 0. In the case of a circuit frus-trated by sources such that the equilibrium system stateis non-zero, we can define the variables 𝑖, 𝑣, 𝑄, and Φ todenote deviations away from the global equilibrium of thecircuit, as discussed in more detail in Sec. A8. By wayof analogy, imagine a mechanical spring. The spring isstretched from its rest position to a new equilibrium by asecond stretched spring. The deviations of the spring aremeasured from the equilibrium position of the combinedsystem (this is what we mean by global equilibrium), notthe spring in isolation.

Power and energy. The instantaneous power 𝑝 (𝑡) de-livered to an element and the total energy absorbed bythe element E (𝑡) are

d

d𝑡E (𝑡) = 𝑝 (𝑡) = 𝑣 (𝑡) 𝑖 (𝑡) and E (𝑡) =

∫ 𝑡

−∞𝑝 (𝑡) d𝜏 .

(A.3)Given our convention for the orientation of 𝑖 and 𝑣, powerdelivered to the element is positive if 𝑝 (𝑡) is positive.Passive elements (i.e., non-source elements) obey E (𝑡) −E (−∞) ≥ 0 for all 𝑡, and lossless elements convert all oftheir energy into stored electric or magnetic energy.

Capacitive and inductive elements. A capacitive(resp., inductive) element is described by an algebraicrelationship between charge and voltage (resp., flux andcurrent). Let us introduce the simplest linear, passive,time-invariant elements. The simplest capacitor is de-fined by the constitutive relationship 𝑄 (𝑡) = 𝐶𝑣 (𝑡),where 𝐶 is its capacitance—a positive, real constant. Thedual relationship Φ (𝑡) = 𝐿𝑖 (𝑡) defines the simplest in-ductor, where 𝐿 is its inductance, also a positive, realconstant. In terms of the flux across the element, theenergies of the simple capacitor and inductor are

Ecap( ¤Φ)

=1

2𝐶 ¤Φ2 and Eind (Φ) =

1

2𝐿Φ2 , (A.4)

respectively (and assuming Φ (−∞) = 𝑄 (−∞) = 0).Since 𝐶, 𝐿 ≥ 0 and Φ (𝑡) ∈ R for all 𝑡, we can verifythat the total energy gained by these elements is alwayspositive, as required for passive elements.

Josephson tunnel junction. The chief non-linear el-ement used in circuit quantum electrodynamics isthe Josephson tunnel junction37,48,126, characterizedby the flux-controlled inductive relationship 𝑖 (𝑡) =

𝐼0 sin (Φ (𝑡) /𝜙0), where 𝜙0 B ℏ/2𝑒 is the reduced mag-netic flux quantum and 𝐼0 is the junction critical current.The energy function of the junction in terms of flux is

E𝐽 (Φ) = 𝐸𝐽 (1 − cos (Φ/𝜙0)) , (A.5)

where 𝐸𝐽 B 𝐼0𝜙0 denotes the Josephson energy. Forsmall deviations about Φ = 0, ignoring the constant en-ergy offset,

E𝐽 (Φ) ≈1

2𝐸𝐽 (Φ/𝜙0)2 −

1

4!𝐸𝐽 (Φ/𝜙0)4 + O

(Φ6) . (A.6)

To lowest order, the junction responds as a linear in-ductor with inductance 𝐿𝐽 B 𝐸𝐽/𝜙2

0 [compare this to

Eq. (A.4)] It is useful to introduce the reduced magneticflux 𝜑 B Φ/𝜙0 of the junction.

Single- vs. multi-valued energy functions. TheJosephson junction exhibits a fundamental asymmetry.Inverting the current-flux relationship leads to a multi-valued function with an infinite number of branches; i.e.,𝜑 = sin−1 (𝑖 (𝑡) /𝐼0) + 2𝜋𝑘 or 𝜑 = 𝜋 − sin−1 (𝑖 (𝑡) /𝐼0) + 2𝜋𝑘,where 𝑘 is some integer. For flux-controlled inductors,the multi-valued situation can be avoided by favoring adescription in terms of flux, rather than charge.

Junction in a frustrated circuit. Embedding the junc-tion in a frustrated circuit can lead to a non-zero equilib-rium value for 𝜑. For deviations 𝜑 − 𝜑eq away from theequilibrium value 𝜑eq, see also Eq. (A.64),

E𝐽 (𝜑) ≈ 𝐸𝐽

[sin

(𝜑eq

) (𝜑 − 𝜑eq

)+ 12cos

(𝜑eq

) (𝜑 − 𝜑eq

)2− 1

6sin

(𝜑eq

) (𝜑 − 𝜑eq

)3 ] + O (𝜑4) . (A.7)

We can identify the differential inductance of the junctionat 𝜑 = 𝜑eq to be 𝐿𝐽

(𝜑eq

)= 𝐿𝐽/cos

(𝜑eq

). In Sec. A8, we

discuss how 𝜑 can be taken as a deviation away from theequilibrium value 𝜑eq, in effect mapping 𝜑 − 𝜑eq ↦→ 𝜑,see Eq. (A.63). Also in Sec. A8, we discuss sources termssuch as the one presented by 𝐸𝐽 sin

(𝜑eq

) (𝜑 − 𝜑eq

). Such

energy terms linear in 𝜑 turn the junction into an activecomponent, capable of supplying current; i.e., we observethat the current relationship of the junction to lowestorder 𝐼 (𝑡) = 𝜕E𝐽

𝜕Φ= 𝐼0 sin

(𝜑eq

)+ 𝐿−1

𝐽

(𝜑eq

)Φ (𝑡) + O

(Φ2

)contains the constant term 𝐼0 sin

(𝜑eq

), which acts like a

current source.Relationship between the gauge-invariant supercon-

ducting phase difference of a tunnel junction and the re-duced magnetic flux 𝜑 (compact vs. non-compact vari-ables). In superconductivity, the Josephson energy cou-pling two small superconducting islands has a cosine de-pendence on the gauge-invariant phase difference \ 𝑗 ofthe superconducting phases of the two islands70,127. Thismacroscopic variable is a phase angle—a compact vari-able in the half-open interval \ ∈ [0, 2𝜋[; in contrast withthe non-compact variable 𝜑 ∈ [−∞, +∞], which must beused in circuits where a superconducting wire connectsthe two sides of the junction.

In our treatment of the Josephson circuits so far, wehave completely ignored this subtlety. Rather, we havebased our discussion on the non-compact variable 𝜑. Therelationship between these two collective, macroscopicvariables is \ = Φ/𝜙0 mod 2𝜋. Although the variable 𝜑

is non-compact, the associated wavefunction 𝜓 (𝜑) is sub-mitted in practice to constraints (like confinement in oneor a few potential wells) such that it is decomposed ontoa basis set of wavefunctions that are indexed by a singlediscrete (rather than continuous) index; for example, theFock basis 𝜓𝑛 (𝜑), with 𝑛 ∈ N. Quantum-mechanically,the representation of 𝜓 (𝜑) and 𝜓 (\) is therefore not verydifferent since 𝜓 (\) is represented by the discrete rotorbasis, 𝜓𝑘 (\), with 𝑘 ∈ Z.

16

A broken symmetry. The compact support of \ cor-responds to a symmetry that is usually broken in mostcircuits—that of the impossibility to distinguish betweendifferent values of 𝜑 differing by 2𝜋. Losses associatedwith the junction, or coupling to other elements, suchas in the RF-SQUID or fluxonium qubit128, render 2𝜋turns of 𝜑 macroscopically distinguishable—hence de-manding a description in terms of non-compact supportcorresponding to a point on an open-ended line ratherthan a circle. For certain special cases, however, it maybe advantageous to retain the compact support version.But even in such cases, one can start with the non-compact version of 𝜑 and recover the compact versionby a limit procedure46. Thus, in this article, we useonly gauge-invariant phases with non-compact support;i.e., 𝜑 ∈ [−∞, +∞].

Flux-controlled inductor. To generalize from theJosephson junction and introduce more general non-linear elements, consider the flux-controlled inductor. Itis defined by an algebraic relationship 𝑖 (𝑡) = ℎ (Φ (𝑡) , 𝑡),where ℎ is a single-valued function. If ℎ is time-invariant,such as for the case of the Josephson tunnel junction, theenergy function [see Eq. (A.3)] is

Eind (Φ) =∫ Φ(𝑡)

Φ(−∞)ℎ (Φ′) dΦ′ . (A.8)

Similar results can be obtained for current controlled in-ductors and for generalized capacitors.

A network of circuit elements. An electrical circuitis an interconnected collection of circuit elements. Theconnectivity of the elements can be described by an ori-ented graph. Each branch in the graph corresponds toone element. The 𝑏-th element is associated with the in-stantaneous voltage 𝑣𝑏 (𝑡), current 𝑖𝑏 (𝑡), charge 𝑄𝑏 (𝑡),flux Φ𝑏 (𝑡), and reduced flux 𝜑𝑏 (𝑡). The universal re-lationships Eqs. (A.1)–(A.3) link the variables at eachbranch. Variables across different branches are linked byKirchhoff’s laws.

Kirchhoff’s two universal circuit laws. The followingtwo laws are universal and topological in nature. Theydescribe relationship among the branch variables, inde-pendent of the constitution of the branch elements. Inother words, they apply to nonlinear, time-dependent,and even hysteric elements.

Kirchhoff’s voltage law (KVL) is the lumped-elementmanifestation of the Maxwell-Faraday equation, ∇ × ®𝐸 =

− 𝜕 ®𝐵𝜕𝑡

. By applying Stokes’ theorem to the Maxwell-Faraday equation along an oriented loop of lumped el-ements (and a surface associated with the closed loop),one finds a relationship valid for any closed circuit loop:the oriented sum of the fluxes along the 𝑙-th loop is equalto the external applied flux Φext

𝑙(𝑡) threading the loop,∑

𝑏∈loop𝑙

±Φ𝑏 (𝑡) = Φext𝑙 (𝑡) , (A.9)

where the sum runs over all branches 𝑏 that form the 𝑙-thloop. For a given branch 𝑏, the positive (resp., negative)

sign in Eq. (A.9) is selected if its flux reference directionaligns (resp., is opposite to) the loop orientation. Alge-braically, KVL leads to a set of constraints among thenetwork variables. Thus, in the context of a Lagrangiandescription of the circuit in which the generalized positionvariables are taken to be fluxes Φ𝑏, the KVL conditionsexpress a set of holonomic constraints that need to beeliminated in order to obtain a Lagrangian of the secondkind129.

Kirchhoff’s current law (KCL) is a statement of theconservation of charge: at every node in the circuit,the algebraic sum of all the current leaving or enteringthe node is equal to zero. Recast another way, for allbranches 𝑏 connected to node 𝑛,∑

𝑏∈node𝑛±𝑖𝑏 (𝑡) = 0 . (A.10)

The negative sign is chosen for branches whose currentreference direction points toward the 𝑛-th node. In theflux-based Lagrangian description of the circuit, the KCLalgebraic conditions become the Lagrangian equations ofmotion.

Eliminating the KVL constraints using the method ofthe minimum spanning-tree. The set of KVL algebraicequations defined in Eq. (A.9) reduce the number of inde-pendent branch fluxes Φ𝑏. We can systematically choosea minimal set of independent branch fluxes using the min-imum spanning-tree graph method48,126,130,131. For ourderivation, it is not necessary to explicitly construct thetree. A set of branches from the graph can be selectedto form a complete and minimum spanning tree. In gen-eral, there are many satisfactory tree sets of branches.Different trees are related by a simple algebraic trans-formation; similar to a basis change. The branches thatbelong to the spanning tree can be labeled t1, t2, . . . Theflux of the 𝑘-th spanning-tree branch is denoted Φt𝑘 (𝑡).In subscripts, roman (resp., italic) symbols denote labels(resp., variables). The spanning-tree can be organized ina column vector

Φt (𝑡) B©­­«Φ𝑡1 (𝑡)Φ𝑡2 (𝑡)

...

ª®®¬ , (A.11)

which serves the purposes of a basis for the description ofthe circuit. The branch-fluxes not in the spanning tree(links or chords of the graph) are obtained by a lineartransformation of Φt. We define the energy functions andLagrangian of the system in terms of Φt in the Sec. A3.

A2. The Josephson system and its non-linearJosephson dipoles

In the main text, under Quantizing the general Joseph-son system, we introduced the notion of a Josephsonsystem—a general electromagnetic environment that in-corporates nonlinear devices, referred to as Josephson

17

dipoles. The Josephson system is treated as a distributedblack-box structure.

Discretization. We aim to model the Josephson sys-tem as realistically as possible. To account in detailfor its physical layout, materials, boundary conditions,and dipole structures, we aim to leverage conventionalelectromagnetic analysis techniques, such as the finite-element (FE) method. The FE method subdivides thephysical layout of the system. Using a set of basis func-tions, the electromagnetic circuit is discretized132,133.The discretized circuit can be represented by a lumped-element model. In principle, we can take the limit ofinfinite subdivision. The Josephson dipoles are assumedto be the only non-linear elements in the circuit. Allother elements are linear, as representative of the linearnature of Maxwell’s equations.

Dissipation. As the object of interest is the controlof quantum information in the system, in this section,we focus on systems with low dissipation. This conditionrequires that the quality factor of all modes of relevanceis high, 𝑄𝑚 1, where 𝑚 is the mode index. In Sec. D,we treat dissipation as a perturbation to the lossless so-lutions.

Josephson dipole. The Josephson dipole was intro-duced in the main text, under Quantizing the gen-eral Josephson system. For simplicity of discussion,here, we treat the dipole as a lumped, two-terminal,flux-controlled element. The 𝑗-th Josepson dipolein the system is fully specified by its energy func-tion E 𝑗

(Φ 𝑗 ;Φ 𝑗 ,ext

), see Eq. (A.8). The energy depends

only on the magnetic flux Φ 𝑗 (𝑡) across the terminals ofthe dipole and on any external parameters Φ 𝑗 ,ext thatcontrol the energy landscape; these can include a volt-age bias, a current bias, and an external magnetic fieldbias. The index 𝑗 runs from unit to the total number 𝐽

of Josephson dipoles in the circuit.

This formulation is rather general. It encapsulate thewide span of Josephson dipoles discussed in the maintext. These include simple devices, such as the Joseph-son tunnel junction or a nanowire, and also compositedevices, such as SQUIDs, SNAILs, or more general sub-circuits. The underlying physical phenomenon giving riseto the low-loss, non-linearity of the dipole is immaterial.

Partition of the Josephson dipole energy-function. Itis always possible to partition E 𝑗

(Φ 𝑗 ;Φ 𝑗 ,ext

)into a linear

and non-linear part,

E 𝑗

(Φ 𝑗 ;Φ 𝑗 ,ext

)= Elin𝑗

(Φ 𝑗 ;Φ 𝑗 ,ext

)+ Enl𝑗

(Φ 𝑗 ;Φ 𝑗 ,ext

).

(A.12)This division is purely a conceptual one—the Josephsondipole cannot be physically divided into a linear and non-linear part. By selecting an equilibrium point of thecircuit and defining the branch fluxes Φ 𝑗 as deviationsaway from the equilibrium [discussed in more detail in

Sec. (A8)], the partitions take the concrete form

Elin𝑗(Φ 𝑗

)B

1

2𝐸 𝑗

(Φ 𝑗/𝜙0

)2, (A.13a)

Enl𝑗(Φ 𝑗

)B E 𝑗

(Φ 𝑗

)− Elin𝑗

(Φ 𝑗

)(A.13b)

= 𝐸 𝑗

∞∑𝑝=3

𝑐 𝑗 𝑝

(Φ 𝑗/𝜙0

) 𝑝, (A.13c)

where 𝐸 𝑗 is an overall scaling factor of the energy func-tion, defined in Eq. (A.13a); we refer to it as the Joseph-son dipole energy scale. In Eq. (A.13c), we have intro-duced the Taylor series expansion of Enl

𝑗around the equi-

librium state of the circuit and have introduced the di-mensionless expansion coefficients 𝑐 𝑗 𝑝. We stress thatthe expansion is not needed for the EPR method. It ismerely a convenient tool for working analytically withweakly non-linear circuits, such as the transmon qubit.For notation simplicity, in Eq. (A.13) the dependanceof Elin

𝑗, Enl

𝑗, 𝐸 𝑗 , and 𝑐 𝑗 𝑝 on the external bias parame-

ter Φ 𝑗 ,ext is made implicit, and we will continue to doso henceforth; in other words, keep in mind that 𝐸 𝑗 B𝐸 𝑗

(Φ 𝑗 ,ext

)and 𝑐 𝑗 𝑝 B 𝑐 𝑗 𝑝

(Φ 𝑗 ,ext

).

Example of a Josephson dipole: the Josephson junc-tion. We illustrate the partitioning construction definedin Eq. (A.13) using the example of the Josephson tunneljunction. For an un-frustrated junction, it follows fromEq. (A.5) that

Elin𝑗(Φ 𝑗

)B

1

2𝐸 𝑗

(Φ 𝑗/𝜙0

)2, (A.14a)

Enl𝑗(Φ 𝑗

)B −𝐸 𝑗

[cos

(Φ 𝑗/𝜙0

)+ 1

2

(Φ 𝑗/𝜙0

)2],(A.14b)

where 𝐸 𝑗 is the Josephson energy. The energy func-tion Elin

𝑗is associated with the linear response of the

junction. It presents the inductance 𝐿 𝑗 = 𝜙20/𝐸 𝑗 . The

energy Enl𝑗

is associated with the response of non-linearinductor. The expansion coefficient of Enl

𝑗as defined in

Eq. (A.13c) are

𝑐 𝑗 𝑝 =

(−1) 𝑝/2+1

𝑝! for even 𝑝 ,

0 for odd 𝑝 .(A.15)

In partitioning E 𝑗 , we included all of the linear responseof the junction in Elin

𝑗, leaving none for Enl

𝑗; i.e., Enl

𝑗lacks

quadratic terms in Φ 𝑗 . However, this is not required.There are certain cases for which retaining some part ofthe linear response in Enl

𝑗is advantageous.

For notational ease, we now introduce the reducedflux 𝜑 𝑗 (𝑡) B Φ (𝑡) /𝜙0.

Example of a Josephson dipole in a frustrated circuit.Imagine a Josephson tunnel junction incorporated in aclosed loop of several circuit elements. Suppose the loopsupports the flow of a direct current. A current sourcein the path of the loop (or perhaps an external magneticflux threading the loop) establishes a persistent current.

18

The equilibrium flux 𝜑 𝑗 of the junction shifts to a non-zero equilibrium value 𝜑eq, 𝑗 , as determined by the circuitequilibrium considerations (see Sec. A8). Applying thepartition defined in Eq. (A.14) to Eq. (A.7), we find thatin terms of the out-of-equilibrium flux deviation 𝜑 𝑗 ,

E 𝑗

(𝜑 𝑗 ; 𝜑eq, 𝑗

)=1

2𝐸 𝑗

(𝜑eq, 𝑗

)𝜑2𝑗

+ 𝐸 𝑗

(𝜑eq, 𝑗

)𝑐 𝑗3

(𝜑eq, 𝑗

)𝜑3𝑗 + · · · , (A.16)

where 𝐸 𝑗

(𝜑eq, 𝑗

)= 𝐸𝐽/cos

(𝜑eq, 𝑗

), 𝐸𝐽 is the 𝑗-th junc-

tion Josephson energy, and 𝑐 𝑗3

(𝜑eq, 𝑗

)= − 1

6 sin(𝜑eq, 𝑗

).

We emphasize that 𝜑 𝑗 denotes deviations away from theequilibrium; compare Eq. (A.16) to Eq. (A.7).

A3. Energy of the Josephson circuit and itsLagrangian

Capacitive energy. The total capacitive energy Ecapof the Josephson system is simply the algebraic sum ofthe total energy of all its capacitive elements. UsingEq. (A.4), and summing over all capacitive branches,Ecap B

∑𝑏∈cap.

12𝐶𝑏¤Φ2𝑏, where 𝐶𝑏 is the capacitance of

branch 𝑏. Each of these fluxes can be expressed in termsof the linearly-independent spanning-tree fluxes Φt. Theenergy function is quadratic in Φt,

Ecap( ¤Φt

)=1

2¤ΦTt C ¤Φt , (A.17)

where C is the capacitance matrix of the circuit47,48,126.It follows from KVL and the constitutive relationships ofthe capacitors that C is a positive-definite, real, symmet-ric (PDRS) matrix. In the continuous limit of space, thetotal capacitive energy Ecap can be found using Eq. (C.3).

Inductive energy. The total inductive energy in thecircuit Eind is similarly the algebraic sum of the totalenergy of all circuit inductive branches; i.e., Eind B∑

𝑏∈ind. E𝑏 (Φ𝑏). In the Josephson system, inductivebranches come in two distinct flavors: linear and non-linear. Physically, linear inductive branches are asso-ciated with the geometry and magnetic fields. We de-note their total energy Emag. The non-linear induc-tive branches (Josephson dipoles) are generally associ-ated with the kinetic inductance of electrons. We denotetheir total energy Ekin. Hence,

Eind (Φt) = Emag (Φt) + Ekin (Φt) . (A.18)

The energy of the linear branches Emag is the dual ofEq. (A.17). It is also a quadratic form,

Emag (Φt) =1

2ΦT

t L−1magΦt , (A.19)

where the inductance matrix L−1mag completely de-scribes all linear, magnetic-in-origin inductances in thecircuit47,48,126. Due to its nature, L−1mag is PDRS. In the

continuous limit of space, the total magnetic inductiveenergy Emag can be found using Eq. (C.4).

The total inductive kinetic energy of the circuit, asso-ciated with the non-linear dipoles, is

Ekin =

𝐽∑𝑗=1

E 𝑗

(Φ 𝑗

)=

𝐽∑𝑗=1

Elin𝑗(Φ 𝑗

)+

𝐽∑𝑗=1

Enl𝑗(Φ 𝑗

).

(A.20)

This energy is not stored in the magnetic fields. How-ever, we can group the magnetic energy and the linearpart of Ekin together to express the inductive energy asa partition of linear and non-linear contributions

Eind (Φt) =1

2ΦT

t L−1Φt + Ekin (Φt) , (A.21a)

L−1 B L−1mag +1

2

𝐽∑𝑗=1

𝐸 𝑗

(Φ 𝑗/𝜙0

)2, (A.21b)

Enl (Φt) B𝐽∑𝑗=1

Enl𝑗(Φ 𝑗

), (A.21c)

where we have introduced the total inductance matrix ofthe circuit L−1 and the total non-linear energy functionof the circuit Enl. For later use, we can obtain the se-ries expansion of Enl in terms of that of Enl

𝑗, defined in

Eq. (A.13b),

Enl (Φt) =𝐽∑𝑗=1

∞∑𝑝=3

𝐸 𝑗𝑐 𝑗 𝑝

(Φ 𝑗/𝜙0

) 𝑝. (A.22)

Generalized coordinates for the Lagrangian. We haveexpressed Ecap and Eind in terms of the independent set ofspanning-tree fluxes Φt. We could equivalently have ex-pressed Ecap and Eind in terms of the charge variables 𝑄𝑏.However, as discussed in Sec. A1, the flux-controlledJosephson dipoles present an asymmetry which favorstreatment in the flux basis. We hence employ Φt asthe generalized position coordinates in the Lagrangiandescription of the circuit, and ¤Φt as the generalized ve-locity.

Lagrangian of the Josephson circuit. The Lagrangianfunction of the Josephson circuit follows from KCL. En-ergy functions with Φt as their argument (resp., ¤Φt) playthe role of potential (resp., kinetic) energies. The sys-tem Lagrangian is the difference of the total kinetic andpotential energy functions,

Lfull

(Φt, ¤Φt

)B Ecap

( ¤Φt

)− Eind (Φt) (A.23a)

= Llin

(Φt, ¤Φt

)+ Lnl (Φt) , (A.23b)

where we have partitioned the Lagrangian into a lin-ear Llin and nonlinear Lnl part. Substituting inEqs. (A.17) and (A.21),

Llin

(Φt, ¤Φt

)=1

2¤ΦTt C ¤Φt −

1

2ΦT

t L−1Φt , (A.24a)

Lnl (Φt) = −Enl (Φt) = −𝐽∑𝑗=1

Enl𝑗(Φ 𝑗

).(A.24b)

19

Equation (A.24) explicitly constructs the Josephsonsystem Lagrangian. It partitions it into a linear and non-linear part. In Sec. A4, we diagonalize Llin to find theeigenmodes of the linearized system. In Sec. B, we treatthe effect of Lnl.

A4. Eigenmodes of the linearized Josephson circuit

In this sub-sectino, we diagonalize Llin and find itseigenmodes, eigenfrequencies 𝜔𝑚 and eigenvectors (i.e.,spatial-mode profiles). These intermediate result pro-vide a key stepping stone on our path to quantizing theJosephson system and treating Lnl. The process con-ceptually parallels that taken by the finite-element (FE)electromagnetic (EM) solver in an eigenanalysis of thelinearized Josephson system.

The Lagrangian Llin is the sum of two quadratic forms,see Eq. (A.24). We use the standard method for theirsimultaneous diagonalization129, based on a series ofprinciple-axis transforms. We then transform the La-grangian into a diagonalized Hamiltonian.

Diagonalizing the inductance matrix. Since the in-verse inductance matrix L−1 is a PDRS matrix, we candiagonalize it with a real orthogonal matrix OL, obey-ing OLO

ᵀL= O

ᵀLOL = I, where I is the identity matrix,

OTL L−1OL = Λ−1L I−1L , (A.25)

where Λ−1L is a diagonal matrix comprising the (dimen-sionless) eigenvalue magnitudes, and IL is the identitymatrix with physical dimensions of inductance. Theeigenvectors of L−1 form the columns of OL.

Employing Eq. (A.25) with Eq. (A.24), the system La-grangian takes the suggestive form

Lfull

(Φt, ¤Φt

)=1

2¤ΦTt C ¤Φt + Lnl (Φt)

− 1

2

(ΦT

t OLΛ−1/2L

)I−1L

(Λ−1/2L

OTLΦt

), (A.26)

which motivates the principle-axis transformation of themagnetic flux defined by

Φ B Λ−1/2L

OTLΦt , (A.27)

where Φ is a rotated and then scaled version of Φt. Underthis transformation, the transformed Lagrangian func-tion becomes

Lfull

(Φ,¤Φ)B

1

2¤ΦTC

¤Φ − 1

2ΦTI−1L Φ + Lnl

(Φ), (A.28)

where Lnl

(Φ)B Lnl

(Φ(Φ))

and

C B(Λ1/2L

OTL

)C(OLΛ

1/2L

). (A.29)

Since the capacitance matrix C is PDRS and it is trans-formed by a rotation and then a dilation, it follows that Cis also PDRS. More generally, the eigenvalues of a ma-trix are invariant under a similarity transform, such asthe one employed in Eq. (A.29).

Diagonalizing the capacitance matrix. Since C isPDRS, we diagonalize it with a real, orthogonal trans-formation OC, such that

OT

CCOC = ΛCIC , (A.30)

where ΛC is the dimensionless, diagonal matrix con-structed from the eigenvalues of C and IC is the identitymatrix with physical dimensions of capacitance.

Employing the orthogonality transformation OT

Cin a

manner similar to the one used with OL, we rotate (but donot scale) the coordinates for a second time. Under thissecond principle-axis transform, we define the eigenmodemagnetic flux variable

Φm B OT

CΦ , (A.31)

in terms of which the Lagrangian Llin is diagonal,

Lfull

(Φm, ¤Φm

)B

1

2¤ΦTmΛCIC

¤Φm −1

2ΦT

mI−1L Φm

+ Lnl (Φm) , (A.32)

where the nonlinear part under the transformation is

Lnl (Φm) B Lnl (Φt (Φm)) . (A.33)

Equations of motion. The Lagrangian equations ofmotion, 𝜕L

𝜕Φm− d

d𝑡𝜕L𝜕 ¤Φm

= 0, yield the harmonic eigenvalue

equation d2

d𝑡2Φm + Ω2Φm = 0, where 0 is the column vec-

tor of all zero elements and Ω2 B Λ−1CI2𝜔. The identity

matrix I𝜔 has physical dimensions of circular frequency.The generalized momentum canonical to Φm is the vectorof charge variables Qm B

𝜕L𝜕 ¤Φm

= ΛCIC¤Φm.

Diagonalized Hamiltonian of the Josephson system.The system Hamiltonian follows from the Legendre trans-form on Lfull, Hfull (Φm,Qm) =

( ¤Φm (Qm))T

Qm − Lfull,which we can partition into

Hfull (Φm,Qm) = Hlin (Φm,Qm)+Hnl (Φm,Qm) , (A.34)

where the linear and nonlinear parts of the Hamiltonianexpressed in the eigenmode coordinates are

Hlin (Φm,Qm) B1

2QT

mΩ2ILQm +

1

2ΦT

mI−1L Φm (A.35a)

=

𝑀∑𝑚=1

1𝐿2𝜔2𝑚𝑄

2𝑚 +

1

21𝐿Φ

2𝑚 , (A.35b)

Hnl (Φm,Qm) B −Lnl (Φm) , (A.35c)

where the diagonal eigenfrequency matrix Ω of the Hamil-tonian Hlin is

Ω B Λ−1/2C

I𝜔 =©­­«𝜔1

. . .

𝜔𝑀

ª®®¬ , (A.36)

and 1𝐿 is unity carrying physical dimensions of induc-tance. The entries of Ω are the eigenmode frequencies 𝜔𝑚

of the linearized circuit. These correspond to the eigen-frequencies solved for by the FE eigenanalysis.

20

Eigenvectors of the Josephson system. The eigenvec-tor matrix E relates the spanning-tree fluxes Φt to theeigenmode ones Φm,

Φt = EΦm . (A.37)

It is found by concatenating the principle-axis transfor-mations defined in Eqs. (A.27) and (A.31),

E B OLΛ1/2L

OC . (A.38)

The eigenvector matrix E is real and positive-definite,since it is the produce real, positive-definite transforms.It is dimensionless and, in general, non-symmetric. Itis related to the square root of the inductance ma-trix, EEᵀ = LI−1H and

(E−1

)ᵀE−1 = L−1IH. The eigen-

vector matrix E represents the eigenfield solutions foundin the FE analysis. It is key in determining the quantumzero-point fluctuations of the mode and dipole fluxes, asshown in the following section.

A5. Quantizing the Josephson circuit

We quantize Hfull using Dirac’s canonical approach134.Before taking passage from classical to quantum, we in-troduce the complex mode amplitude operator 𝛼𝑚—theclassical analog of the bosonic amplitude operator 𝑎𝑚.This provides a direct path to second quantization in theeigenmode basis of Hlin.

Complex action-angle variables. We define the vectorof action-angle variables α = (𝛼1, . . . , 𝛼𝑀 )T by the non-canonical, complex transformation

α (𝑡) B 1√2ℏΩ

(Φm (𝑡) 1−1/2H + 𝑖ΩQm (𝑡) 11/2H

), (A.39)

where 1H is unity with dimensions of inductance. Thenormalization 1/

√2ℏΩ is chosen so that the Poisson

bracket of the action-angles is𝛼𝑚, 𝛼

∗𝑚′P= 1/(𝑖ℏ)𝛿𝑚𝑚′ .

In terms of the action-angles, the Hamiltonian remaindiagonal,

Hlin =ℏ

2

(α𝑇Ωα∗ +α∗𝑇Ωα

). (A.40)

We have symmetrized Hlin to avoid operator order am-

biguity. The flux is Φm =

√ℏΩ1H

2 (α∗ +α).Quantizing the action-angle variables. Following

Dirac’s prescription134, we supplant Poisson brackets bycommutators and promote observables to operators. Theaction-angles 𝛼𝑚 and 𝛼∗𝑚 promote into the ladder annihi-lation 𝑎𝑚 and creation 𝑎

†𝑚 operators. Their commutator

follows from the Poisson bracket47,48,135, 𝑖ℏ×𝛼𝑚, 𝛼∗𝑚 ↦→[

𝑎𝑚, 𝑎†𝑚

], [

𝑎𝑚, 𝑎†𝑘

]= 𝛿𝑚𝑘 𝐼 , (A.41)

where[𝐴, 𝐵

]B 𝐴𝐵 − 𝐵𝐴 is the commutator, 𝛿𝑚𝑘 is the

Kronecker delta function, and 𝐼 is the identity opera-tor. Particles of the electromagnetic field (photons) aredistinguishable bosons48; symmetrization of the many-body wavefunction is not required. Operators are in theSchrödinger picture, unless otherwise indicated by an ex-plicit time argument.

Hamiltonian. The quantized form of Hlin, seeEq. (A.40), is

𝐻lin =

𝑀∑𝑚=1

ℏ𝜔𝑚𝑎†𝑚𝑎𝑚 . (A.42)

This is Eq. (13) of the main text. The quantizedform of Hnl follows from combining Eqs. (A.21), (A.22),(A.24b) and (A.35c),

𝐻nl =

𝐽∑𝑗=1

Enl𝑗(𝜑 𝑗

)=

𝐽∑𝑗=1

∞∑𝑝=3

𝐸 𝑗𝑐 𝑗 𝑝𝜑𝑝

𝑗. (A.43)

where 𝜑 𝑗 B Φ 𝑗/𝜙0 is the reduced magnetic-flux operatorfor Josephson dipole 𝑗 . This is Eq. (14) of the main text.In the following, we decompose 𝜑 𝑗 in terms of 𝑎𝑚; i.e.,in second quantization with respect to the eigenmodesofHlin. In writing Eq. (A.42), we have omitted the 1

2ℏ𝜔𝑚

ground-state energy of every mode. In the limit of infinitediscretization, the sum of these ground energies tends toinfinity, a standard conceptual difficulty in quantum fieldtheory. Since physical experiments observe changes inthe energy of the field, the vacuum energy can be ne-glected (except for special cases; e.g., Casimir effect). Toproceed, we introduce helpful notation.

Notation for vectors of operators. A bold symboltypeset in roman, such as Φt, denotes a vector or ma-trix, whose elements are constants or variables. Sincevariable, such as Φt1 are promoted to quantum opera-tors, Φt1 , we can accommodate vectors of operators inour notation with a hat symbol; e.g.,

Φt =©­­«Φt1

Φt2...

ª®®¬ and Φm =©­­«Φm1

...

Φm𝑀

ª®®¬ . (A.44)

The spanning-tree flux operator Φt𝑘 corresponding tothe 𝑘-th spanning-tree-branch flux variable Φt𝑘 . Simi-larly, the eigenmode flux operator Φm𝑘

corresponding tothe 𝑘-th eigenflux variable Φm𝑘

.Zero-point fluctuations of the eigenoperators. Invert-

ing Eq. (A.39), one finds the vectors of eigenflux andeigencharge operators,

Φm B ΦZPFm

(a† + a

), (A.45a)

Qm B 𝑖QZPFm

(a† − a

), (A.45b)

21

respectively, where the diagonal matrices of the quantumZPF of the operators are

ΦZPFm B

√ℏ

2Ω1/2IH1/2 (A.46a)

=

©­­­­«√

ℏ𝜔1

2 1H. . . √

ℏ𝜔1

2 1H

ª®®®®¬,

QZPFm B

√ℏ

2Ω−1/2IH−1/2 . (A.46b)

We recall that the operator Φ𝑚 = ΦZPF𝑚

(𝑎†𝑚 + 𝑎𝑚

)has

a zero-mean, gaussian-distributed distribution in theground state; i.e., its mean is

⟨0Φ𝑚

0⟩ = 0 and its vari-

ance is⟨0

(Φ𝑚

)20⟩ =(ΦZPF

𝑚

)2. The non-zero variance is

representative of the ground state energy and the quan-tum zero-point fluctuations of the flux. The ZPF saturatethe Heisenberg uncertainty bound, ΦZPF

m QZPFm = ℏ

2 I. Allmode ZPFs are positive, ΦZPF

𝑚 , 𝑄ZPF𝑚 ∈ R>0.

Interpretation of the eigenmode ZPFs and effec-tive mode inductances, and impedances. The modeimpedance 𝑍𝑚 = ΦZPF

𝑚 /𝑄ZPF𝑚 = 𝜔𝑚1H and ZPF ΦZPF

𝑚 =√ℏ𝜔𝑚/2 × 1H depend only on 𝜔𝑚. These quantities are

in general removed from direct physical meaning. Physi-cal meaning is extracted by using them as computationaltools to calculate the quantum ZPF of the branch fluxesand charges. The EPR method negates the need to ex-plicitly compute them; rather, we only directly work withphysically measure quantities, such as the eigenfields ofthe modes and the Josephson dipole EPRs, as discussedin the next sub-section.

From abstract to physical ZPFs. The zero-pointfluctuations of the spanning-tree fluxes follow fromEqs. (A.38) and (A.46a),

ΦZPFt = EΦZPF

m . (A.47)

The (𝑘, 𝑚) element of ΦZPFt is the ZPF of the 𝑘-th

spanning-tree branch due to mode 𝑚 and is either pos-itive or negative. The overall sign is arbitrary. Hover,the relative sign between two branches 𝑘 and 𝑘 ′ in thesame mode determines if they are excited by the modein-phase or out-of phase with each other. The eigenvaluematrix E is constructed canonically so as to correctlyrelate the fluctuations. The eigenvector matrix of thestandard product C−1L−1, obtained in the Lagrangianequations of motion, is not canonical and cannot be usedin Eq. (A.47).

Second quantization of the dipole flux operator.Since E is real, the reduced-magnetic-flux ZPF ampli-tudes 𝜑𝑚𝑗 can be chosen to be real-valued numbers;Eq. 14(c) of the main text follows,

𝜑 𝑗 B Φ 𝑗/𝜙0 =

𝑀∑𝑚=1

𝜑𝑚𝑗

(𝑎𝑚 + 𝑎†𝑚

). (A.48)

A6. Energy-participation ratio (EPR)

We define energy-participation ratio in the context ofquantum circuits. We motivate the omission of vacuumenergy contributions and use the EPR to find the quan-tum zero–point fluctuations 𝜑𝑚𝑗 .

Definition of EPR in plain english. The EPR 𝑝𝑚𝑗 ofJosephson dipole 𝑗 in eigenmode 𝑚 is the fraction of in-ductive energy allocated to the Josephson dipole whenmode 𝑚 is excited.

Interpretation of the energy-participation ratio. TheEPR quantifies how much of the inductive energy of amode is allocated to a Josephson dipole. The lowest pos-sible participation is zero—the Josephson dipole inductordoes not participate in the mode. When the mode is ex-cited, none of the excitation energy flows to the dipole.The largest possible participation is unity. When themode is excited, all of its excitation flows to the Joseph-son dipole; none of it is distributed to any other inductor.As we show in the following, a larger participation leadsto larger quantum vacuum fluctuations.

Transmon-qubit example. Consider the transmonqubit coupled to a readout cavity mode (see Methods).The Josephson junction has an EPR of near unity in thequbit eigenmode, since nearly all of the inductance of thetransmon is due to the kinetic inductance of the tunneljunction. On the other hand, the EPR of the junctionin the cavity mode is on the order of 10−2. The junctionkinetic inductance contributes only on the order of onepercent to the total inductance of the cavity, associatedwith the large cavity box. This leads to the large ZPF ofthe junction flux in the transmon mode, 𝜑ZPF

𝑞 ∼ 1, andthe therefore much smaller fluctuations of the junction inthe cavity mode 𝜑ZPF

𝑞 ∼ 10−1 (see also Sec. A7).Energy offset. The EPR is defined in terms of the

energy excitation of a mode, rather than in terms of itsabsolute energy.

Vacuum energy. Mathematically, the vacuum energyis a consequence of the non-commutativity of 𝑎 and 𝑎†,which in effect adds a constant offset to the energy ofeach mode. In calculating the EPR, we take the referenceenergy of an element relative to its vacuum energy.

Equipartition theorem. Classically, for a linear circuit,the energy of an eigenmode oscillates in time between in-ductive and capacitive energy. At periodic intervals givenby 𝜋/𝜔𝑚, the inductive energy of the mode is exactlyzero. For this reason, we use the time-average energy ofthe Josephson dipole and the eigenmode. We recall thatthe time-averaged energy is half of the peak energy. Thiscan be exploited in expressing the total inductive energyas half the total mode energy,⟨

Eind⟩=1

2

⟨𝐻lin

⟩=1

2

∑𝑚

ℏ𝜔𝑚

⟨𝑎†𝑚𝑎𝑚

⟩. (A.49)

We will discuss what states are used in calculating theexpectation values below. The ground state energy istaken to be zero.

22

Calculating the EPR. These ideas lead us to the fol-lowing definition of the EPR in the quantum setting,

𝑝𝑚𝑗 B⟨E 𝑗 ,lin

⟩/⟨Eind

⟩, (A.50)

where the over-line denotes a time average and the ex-pectation value is taken over a state with an excitationonly in mode 𝑚. As discussed above, all energies arereferenced to their ground-state expectation values, todisregard vacuum-energy contributions (for special no-tation to accommodate this, see remark on Wick or-dering at the end of this section). Using Eqs. (A.49)

and (A.42), we simplify the denominator,⟨Eind

⟩=∑

𝑚12𝑎†𝑚𝑎𝑚. The operator for the junction energy follows

from Eq. (A.13a), E 𝑗 ,lin = 12𝐸 𝑗

(𝜑2𝑗−⟨𝜑2𝑗

⟩0

), where 〈 〉0

indicates an expectation value over the ground state.Thus,

𝑝𝑚𝑗 =

⟨12𝐸 𝑗𝜑

2𝑗

⟩−⟨12𝐸 𝑗𝜑

2𝑗

⟩0

12

∑𝑚 ℏ𝜔𝑚

⟨𝑎†𝑚𝑎𝑚

⟩ . (A.51)

Using Eq. (A.48),

𝜑2𝑗 −

⟨𝜑2𝑗

⟩0=

𝑀∑𝑚=1

𝜑2𝑚𝑗

(2𝑎†𝑚𝑎𝑚 + 𝑎2𝑚 + 𝑎†2𝑚

)+

𝑀∑𝑚′≠𝑚

𝜑𝑚𝑗𝜑𝑚′ 𝑗

(𝑎𝑚𝑎𝑚′ + 𝑎𝑚𝑎†𝑚′ + 𝑎

†𝑚𝑎𝑚′ + 𝑎†𝑚𝑎†𝑚′

).

(A.52)

EPR for a Fock state excitation. We can take expec-tation value in Eq. (A.51) for a Fock excitation of 𝑛

photons in mode 𝑚, denoted |𝑛𝑚〉. We recall that themany-body vacuum state is |0〉 B |vac〉 B ∏𝑀

𝑘=1 ⊗ |0〉𝑘 ,where |0〉𝑘 denotes the single-particle, zero Fock state ofmode 𝑘; thus, |𝑛𝑚〉 = 1√

𝑛𝑚!

(𝑎†𝑚

)𝑛𝑚|vac〉. To take the ex-

pectation value of the Josephson dipole flux⟨𝜑2𝑗

⟩, we

recall that Fock states have trivial time dynamic, anduse Eq. (A.52). We observe that the expectation valueof any product of two amplitude operators from differentmodes is zero. The only non-zero term in Eq. (A.52) isthe 𝑎

†𝑚𝑎𝑚 term. Thus, the EPR of the Josephson dipole

for the Fock state |𝑛𝑚〉 is

𝑝𝑚𝑗 =𝐸 𝑗𝜑

2𝑚𝑗

12ℏ𝜔𝑚

. (A.53)

The EPR is independent of the excitation amplitude 𝑛𝑚.EPR for a coherent state excitation. A coherent state

excitation of mode 𝑚 is denoted |𝛽𝑚〉 B (𝛽𝑚, 𝑎𝑚) |vac〉,where is the displacement operator of the 𝑚th mode, (𝛽𝑚, 𝑎𝑚) B exp

(𝛽𝑚𝑎

†𝑚 − 𝛽∗𝑚𝑎𝑚

), and 𝛽 is a non-zero

complex number. The coherent state time-evolution isa rotation, |𝛽𝑚 (𝑡)〉 =

𝛽𝑚 (0) 𝑒−𝑖𝜔𝑚𝑡⟩. Using Eq. (A.51),

we find the EPR to again be excitation independent andexactly equal to that given in Eq. (A.53).

Quantum fluctuations in terms of EPR. The EPR fora Fock or coherent state excitation is given by Eq. (A.53).Inverting this expression, we find the ZPF in terms of theEPR,

𝜑2𝑚𝑗 = 𝑝𝑚𝑗

ℏ𝜔𝑚

2𝐸 𝑗

. (A.54)

In the case of a single Josephson dipole in the circuit,Eq. (A.54) reduces to Eq. (6) of the main text. The left-hand side of Eq. (A.54) gives the root-mean-square de-viation of the quantum fluctuations of the reduced mag-netic flux of Josephson dipole 𝑗 , referenced to its equilib-rium value. The right-hand side of Eq. (A.54) comprisesthree classically-known parameters: the eigenmode fre-quency 𝜔𝑚, the Josephson dipole energy scale 𝐸 𝑗 , andthe EPR 𝑝𝑚𝑗 .

Sign of the EPR. Solving Eq. (A.54) explicitly,

𝜑𝑚𝑗 = 𝑠𝑚𝑗

√𝑝𝑚𝑗

ℏ𝜔𝑚

2𝐸 𝑗

, (A.55)

where 𝑠𝑚𝑗 is the EPR sign, 𝑠𝑚𝑗 ∈ −1, +1. In prac-tice, the sign is calculated using Eq. (C.8). Algebraically,the sign can be found from the eigenvector matrix E.If the 𝑗th row of the eigenvector matrix correspondsto the 𝑗th spanning-tree branch flux and the 𝑚th col-umn corresponds to the 𝑚th eigenmode, then 𝑠𝑚𝑗 =

sign([E]𝑚𝑗

), where [E]𝑚𝑗 is the (𝑚, 𝑗)th entry of the

matrix.EPR sign: sign freedom. The value of an individual

sign 𝑠𝑚𝑗 is completely arbitrary. It does not have mea-surable consequences in the same way that the amplitudeof a standing mode can be taken to be positive or neg-ative, indicating a 𝜋-phase shift. The sign 𝑠𝑚𝑗 derivesits physical meaning in relationship to other signs 𝑠𝑚𝑗′ ofthe same mode but different Josephson dipoles (see Sup-plementary Figure S8). If the signs of two dipoles arethe same (resp., different), then the two dipoles oscillatein-phase (resp., out-of-phase). The sign of the EPR canflipped by either flipping the reference direction of thejunction or the overall phase of the mode.

EPR for circuit design. The EPR 𝑝𝑚𝑗 in Eq. (A.54)is essentially the only free parameter in engineering thequantum ZPF and the amplitudes of the non-linear cou-plings. It is readily calculated from the classical eigen-mode FE simulation of the distributed physical layout ofthe circuit, as detailed in Sec. C. In this way, the EPRserves as a bridge between the linearized classical circuitstreated with FE solvers and the Josephson circuit in thequantum domain.

Remark: Equivalent formulation using Wick nor-mal-ordered expectation value for energies. Mathemat-ically enforcing the energy to be referenced to that of

23

the vacuum state can be equivalently accomplished bydisregarding the commutation relationships in expecta-tion values of the energy. The compact notation thataccomplishes this was introduced in quantum optics andis also used in quantum field theory: for an operator 𝑂,one takes the (Wick) normal-ordered form of the oper-ator136, denoted by a colon on either side of the opera-tor, :𝑂 :. Thus, :𝑎𝑎† : = 𝑎†𝑎 and :

(𝑎†𝑎

)2: = 𝑎†2𝑎2. The

normal-ordered form of the operators should not be con-fused with the normal-ordering transformation 𝒩 appliedto the operator, which takes the commutation relationsinto account; e.g., 𝒩

[𝑎𝑎†

]= 𝑎†𝑎 + 𝐼 and 𝒩

[ (𝑎†𝑎

)2]=

𝑎†2𝑎2 + 𝑎†𝑎. In the presence of non-linear interactions,the vacuum energy leads to observable effects, such asthe Lamb shift, introduced in Eq. (7) of the main text.The definition given in Eq. (A.50) can be equivalentlystated using Wick notation:

𝑝𝑚𝑗 B⟨:E 𝑗 ,lin :

⟩/⟨:Eind :

⟩. (A.56)

A7. Universal EPR properties

The energy-participation ratios obey four universalproperties. These properties are valid regardless of thecircuit topology and nature of the Josephson dipoles.They follow directly from the normal-mode structure ofthe eigenmodes of 𝐻lin, obtained in Sec. A4 and are linkedto the ZPF, as discussed in Sec. A6. In the following, 𝑀denotes the total number of modes.

The EPR is bounded. The EPR is an energy fraction.It follows from its definition in Eq. (A.50) that it is areal number comprised between zero and one—since theJosephson dipole energy is always positive and equal toor smaller than the total inductive energy of the mode,

0 ≤ 𝑝𝑚𝑗 ≤ 1 . (A.57)

The total EPR of a dipole follows a sum rule—it is con-served while being diluted among the modes. The totalEPR of a dipole is conserved—it is exactly unity acrossall modes,

𝑀∑𝑚=1

𝑝𝑚𝑗 = 1 for 𝑗 ∈ 1, . . . , 𝐽 , (A.58)

which follows from Eqs. (A.38) and (A.51). Increasingthe EPR of a Josephson dipole in one mode will propor-tionally reduce its participation across other modes. TheEPR is neither created, nor destroyed. It is distributedamong the modes. Adding additional modes to the cir-cuit or removing modes from the circuit does not increaseor decrease the total EPR of a dipole.

The total EPR of a mode is at most unity. A singlemode 𝑚 can at most have a total EPR of unity, and no

(a)

extΦ

Ej

extΦ

L Ej

sv−+

si

(b)

Supplementary Figure S1. Example of a Josephsondipole embedded in frustrated loop. (a) Illustration ofa Josephson tunnel junction (with Josephson energy 𝐸 𝑗) em-bedded in a distributed, superconducting ring subjected to anexternal magnetic-flux bias Φext. (b) Lumped-element modela junction in an inductive ring (with inductance 𝐿) frustratedby an external flux bias Φext, a voltage source 𝑣𝑠, and a cur-rent source 𝑖𝑠.

less than zero,

0 ≤𝐽∑𝑗=1

𝑝𝑚𝑗 ≤ 1 for 𝑚 ∈ 1, . . . , 𝑀 . (A.59)

The upper bound is saturated when there are no linearinductors excited in the mode.

The vector EPRs of two dipoles are orthogonal. Foreach dipole, we can define a vector EPR by the compo-nents

𝑠𝑚𝑗√𝑝𝑚𝑗 |𝑚 = 1, . . . , 𝑀

. The vector EPRs of two

dipoles are orthogonal in the following sense

𝑀∑𝑚=1

𝑠𝑚𝑗 𝑠𝑚𝑗′√𝑝𝑚𝑗 𝑝𝑚𝑗′ = 0 for 𝑗 ≠ 𝑗 ′ , (A.60)

where 𝑠𝑚𝑗 is the EPR sign.Remark: Use of these four properties in quantum cir-

cuit design. These universal EPR constraints are usefulin the design of quantum circuits, especially for designsthat require weak and strong nonlinear interactions si-multaneously. For example, see Methods, where we dis-cuss the impossibility of a design we targeted due to theEPR constraints. We subsequently use the constraintsto obtain a best approximation of the design, and togain insight into the range of possible non-linear cou-plings. In our experience, the EPR has allowed us tothus circumvent the need to run a prohibitively expen-sive finite-element sweep to explore the many possibledesign-parameter regimes.

A8. The biased Josephson system: equilibriumstate in the presence of persistent currents

The magnetic flux across a Josephson dipole Φ 𝑗 , in-troduced in Sec. A1, is defined as a deviation away fromthe equilibrium configuration of the Josephson system. Asystem that incorporates an active element—one that canact as a voltage or current source—or a system subjectedto a frustrated constraint—such as found in a conductingloop frustrated by an external magnetic flux—can have

24

a non-zero equilibrium state. Referencing the Joseph-son dipole and spanning-tree fluxes Φt from equilibriumguarantees that the total inductive energy Eind (Φt) is ata minimum for Φt = 0.

Static-equilibrium conditions. In static equilibrium,the net generalized forces (currents) at each node in thesystem vanish and, consequently, the generalized accel-eration vanishes, d2Φt

d𝑡2= 0. The corresponding mag-

netic flux of the spanning-tree branches in static equilib-rium Φt,eq can be found by extremizing the Lagrangianwith respect to the generalized position variables129,

− 𝜕L𝜕Φt

(Φt,eq.

)=

𝜕Eind𝜕Φt

(Φt,eq

)= 0 . (A.61)

Solving these conditions for the equilibrium flux Φt,eq

amounts to solving for the direct-current (dc) operat-ing point of the circuit—a standard, classical problem;although one that is non-local in nature and, in thepresence of Josephson dipoles, one that involves solu-tions to non-linear equations. In general, classical nu-merical methods can be used to find the equilibrium ofthe circuit, especially when the equilibrium equations aretranscendental. However, for many practical situations,Eq. (A.61) simplifies or need not be evaluated at all, asdiscussed in Sec. A9.

Equilibrium flux of a Josephson Dipole in isolationvs. in a circuit. In general, a Josephson dipole has adifferent equilibrium flux across its terminals when in iso-lation vs. when embedded in a system. When in isolation,the native equilibrium flux of the Josephson dipole isfound from the simple local condition 𝜕E 𝑗

𝜕Φ 𝑗= 0, where E 𝑗

is the energy function of the Josephson dipole and Φ 𝑗 isthe magnetic flux across the dipole. When embedded ina system, the equilibrium flux of the Josephson dipole,found from Eq. (A.61), is in general not a local prop-erty of the dipole anymore—but one of the system, asillustrated by the following example.

Example: Josephson junction in a frustrated ring.Consider a Josephson tunnel junction in isolation—notconnected to any other elements. It’s energy function,see Eq. (A.5), is E𝑖 (Φ𝑖) = −𝐸𝐽 cos (Φ𝑖/𝜙0), where Φ𝑖 isthe total magnetic flux across the junction in isolation.(Here, the subscript 𝑖 denotes the junction in isolation.)The energy is minimized at the native equilibrium-fluxvalue Φnat

eq = 0. Now, consider embedding this junc-tion in a superconducting ring frustrated by an externalmagnetic flux Φext, as depicted in Supplementary Fig-ure S1(a). Suppose the geometric ring inductance is 𝐿,see Supplementary Figure S1(b). The equilibrium condi-tion of the circuit, given by Eq. (A.61), reduces to Ke-pler’s transcendental equation

𝐿−1Φ𝑖 + 𝜙−10 𝐸𝐽 sin (Φ𝑖/𝜙0) = 𝑖eff𝑠 , (A.62)

where the effective dc current sourced to the loop is 𝑖eff𝑠 =

𝐿−1Φext. In the additional presence of a voltage source 𝑣𝑠and current source 𝑖𝑠 as depicted in see SupplementaryFigure S1(b), the net effective current is 𝑖eff𝑠 = 𝐿−1Φext+𝑖𝑠.

eq′′Leq

′L

(a) (b)

x

extL

k

x

′k ′′k

eq0L

Supplementary Figure S2. Mechanical analogy of aJosephson dipole in isolation vs. in a frustrated sys-tem. (a) Depiction of a mechanical spring in isolation. With-out applied forces on the spring, the native equilibrium lengthof the spring is 𝐿0eq. Stretching or compressing the springfrom equilibrium is measured by the deviation 𝑥 away fromthe native equilibrium. The spring constant is 𝑘. (b) De-piction of two connected springs, with spring constant 𝑘 ′

and 𝑘 ′′, constrained between two walls separated by a fixeddistance 𝐿ext. The equilibrium lengths of the two springs inthe system are 𝐿′eq and 𝐿′′eq. These differ from their equilib-rium lengths in isolation; i.e., 𝐿eq ≠ 𝐿′eq. Deviations from thesystem equilibrium are denoted by 𝑥.

A solution Φeq (Φext,Φ𝑠 , 𝑖𝑠) of Eq. (A.62) can be ob-tained numerically or using Lagrange inversion. In theEPR method, the canonical flux deviation away from theselected equilibrium Φeq is then

Φ 𝑗 B Φ𝑖 −Φeq . (A.63)

The energy function of the junction now in terms of thedeviation Φ 𝑗 is

E 𝑗

(Φ 𝑗

)B −𝐸𝐽 cos

( (Φeq +Φ 𝑗

)/𝜙0

). (A.64)

Using the series expansion of Eq. (A.64), given inEq. (A.7), the Josephson dipole energy of the junc-tion, as defined in Eq. (A.13), is 𝐸𝐽 (Φext,Φ𝑠 , 𝑖𝑠) B𝐸𝐽 cos

(Φeq/𝜙0

)and the leading-order non-linear coeffi-

cient is 𝑐 𝑗3 (Φext,Φ𝑠 , 𝑖𝑠) = − 16 tan

(Φeq/𝜙0

).

Mechanical analogy: spring in isolation. By way ofanalogy, a Josephson dipole is like a mechanical spring.Imagine a spring in isolation, see Supplementary Fig-ure S2(a). No external forces act on it. Its native equi-librium length (when the spring is in isolation) is 𝐿0

eq.Stretching the spring to length 𝐿 results in the force 𝐹 =

−𝑘 (𝑥𝑖) on its ends, where 𝑥𝑖 B 𝐿 − 𝐿0eq denotes a devi-

ation away from equilibrium. In general, 𝑘 (𝑥𝑖) is a non-linear function; however, in equilibrium, 𝑘 (0) = 0. Fora linear spring, the energy, E𝑖 (𝑥𝑖) = 1

2 𝑘𝑥2𝑖, is minimized

for 𝑥 = 0.Mechanical analogy: spring in a frustrated system.

Imagine embedding the spring in a simple system as de-picted in Supplementary Figure S2(b). The system im-poses the KVL-like constraint condition 𝐿 ′ + 𝐿 ′′ = 𝐿ext,where 𝐿 ′ is the length of the second spring, and 𝐿ext is thetotal distance between the two walls. For linear springs,the length of the spring in the equilibrium of the sys-tem is 𝐿eq =

[𝑘𝐿 ′eq + 𝑘 ′

(𝐿ext − 𝐿 ′′eq

)]/(𝑘 + 𝑘 ′), where 𝑘 ′

and 𝐿 ′′eq are the spring constant and native equilibrium

25

length of the second spring, respectively. The energy ofthe spring is E (𝑥) = 1

2 𝑘𝑥2, where we omit constant en-

ergy terms and use 𝑥 B 𝐿−𝐿eq to denote deviations awayfrom the equilibrium length of the spring in the system(not the spring in isolation).

Remark: voltage source and equilibrium. The voltagebias 𝑣𝑠 does not appear in the effective current bias 𝑖𝑠of the junction. The capacitor isolates it in dc from therest of the circuit and prevents it from establishing adc current in any loop. In Sec. A9, we discuss furthersituation in which a bias has no effect on the equilibrium.

Remark: voltage source and dc loops. A dc voltagesource is absent from the dc-conducting superconductingloop formed by the junction and inductor, since its mag-netic flux grows linearly in time, Φ𝑠 = 𝑣𝑠𝑡, and it wouldhence supply a current to the loop that grows infinitelylarge. At some time, this would break the abstraction ofthe ideal voltage source or quench the superconductivity.

A9. The biased Josephson system: simplifications

In Sec. A8, we reviewed the equilibrium conditions ofthe Josephson system. Here, we develop several commonsituations in which the analysis of these conditions sim-plifies from a global to a local one or one that need notbe done at all.

A frustrated loop not incorporating Josephson dipoles.If an external bias sets up a persistent current involv-ing only linear parts of the circuit [no Josephson dipoles;see Supplementary Figure S3(a)], the equilibrium of thecircuit does not need to be calculated, since the dc cur-rent will not affect the eigenmodes of the circuit. Putanother way, the alternating-current (ac) response of alinear sub-circuit is independent of its dc configuration.

An open loop incorporating a Josephson dipole withoutinternal loops. If a Josephson dipole is only a member ofloops in the circuit that cannot not support a dc current[see Supplementary Figure S3(d)], any external frustra-tion or bias of the loop is impotent. Thus, the equilibriumflux of the Josephson dipole in isolation Φnat

eq can be takenas the equilibrium flux of the Josephson dipole in the cir-cuit. In other words, we can treat the dipole locally inthe steady-state analysis—i.e., we can forget about therest of the distributed circuit in which it is embedded.This is one of the most common cases of practical inter-est, and is the one of the transmon and SQUID-basedqubit.

An open loop incorporating a Josephson dipole with in-ternal loops. A Josephson dipole may have frustratedloops internal to its structure [see Supplementary Fig-ure S3(e)]. For example, a lumped-element SQUID, RF-SQUID, or SNAIL is subjected to an external magneticflux Φ′ext. However, since a Josephson dipole is purely in-ductive and if it is not part of a global conducting loop,the equilibrium conditions of its internal nodes are solvedlocally.

extΦ

(a)

(d)

extΦ

(e)

(b)

extΦ

extΦ

ext′Φ

si

(c)

extΦ

Supplementary Figure S3. Schematic of five typesof elementary flux-biased Josephson circuits. (a) Ex-ample of a fully linear circuit (no Josephson dipoles). Asuperconducting ring is frustrated by an external magneticflux Φext and, through a mutual inductive coupling, a currentsource 𝑖𝑠. (b) Example of system incorporating a Josephsondipole (here, a Josephson tunnel junction) in a dc-conductingdistributed loop frustrated by Φext and 𝑖𝑠. (c) Same aspanel (b), but with a Superconducting Nonlinear AsymmetricInductive eLement (SNAIL) instead of a junction in the loop.The SNAIL, a composite Josephson dipole, contains internalnodes and loops. Its internal loop is frustrated by the ex-ternal magnetic flux Φ′ext. (d) Example of Josephson tunneljunction embedded in a distributed non-dc-conducting loop.In the region of the gap, we define the segment of the line-contour for the magnetic flux Φext to be the a minimal-lengthone across the gap—i.e., in terms of electrical schematics, wetake the flux across the effective capacitance associated withthe gap in the loop. (e) Example similar to (d) but with aSNAIL element instead of tunnel junction.

A dc-conducting loop incorporating a Josephson dipolewith no internal loops. If a simple Josephson dipole withno loops internal to it (such as the Josephson tunneljunction; not a SQUID or SNAIL), is embedded in adc-conducting distributed and frustrated loop, see Sup-plementary Figure S3(b), then the flux of the dipole inthe equilibrium of the total circuit cannot be determinedlocally. Rather, calculating the equilibrium requires in-corporating information about the geometric-inductanceresponse of the distributed loop at dc. Classical simula-tion methods can be used to obtain an equilibrium point.

A dc-conducting loop incorporating a Josephson dipolewith internal loops. In the most general case, a Joseph-son dipole is part of a dc-conducting loop; see Supplemen-tary Figure S3(c). If the loop is not frustrated, then theequilibrium of flux of the dipole can be taken as the na-tive equilibrium flux of the dipole in isolation. However,in practice, even if no frustration is intended, there may

26

be spurious magnetic flux that will frustrate the loop.If the loop is frustrated by sources (which can includeother Josephson dipoles able to source current, such asa biased SNAIL), the equilibrium condition of the globalloop should be calculated accounting for the geometricinductances.

Voltage offset. A voltage offset will have also have afrustrating effect on the eigenspectrum of the Hamilto-nian. However, for the purposes of calculating the poten-tial energy minima for use in the EPR linearization, it issufficient to consider only the frustration due to currents.

Multiple equilibrium states. The equilibrium condi-tions can yield multiple solutions that locally minimizethe energy function Eind. It is preferable to choose alowest-energy, stable equilibrium state for the operatingpoint of the circuit. In principle, any of the minima canbe used in the EPR method as the operational point; thephysics of all other minima can be reconstructed fromthe operating point. However, we note that the presenceof multiple minima can lead to phase slips72,77,137–140.

B. Nonlinear interactions, effective Hamiltonians,and the EPR

In this section, we find the amplitude of a general non-linear interaction due to 𝐻nl explicitly. In Sec. B1, wefind the effective interaction Hamiltonian of weakly non-linear systems in the dispersive regime to leading-order.In Sec. B2, to more systematically handle large systems,we introduce the EPR matrix P and use it find the Kerrmatrix and the vectors of anharmonicities and Lambshifts. In Sec. B3, we find the general, normal-orderedform of 𝐻nl and an expression to calculate the ampli-tude of any many-body interaction contained within. InSec. B4, we use these results to describe the paramet-ric activation of nonlinearities in a pumped Josephsoncircuit.

B1. Excitation-number-conserving interactions ofweakly-nonlinear systems

Josephson circuits that are weakly non-linear and inthe dispersive regime, such as the transmon-cavity andtransmon-transmon circuits, have, in the absence ofdrives, dominant non-linear interactions that conserveexcitation numbers.

Perturbative and dispersive. For such weakly nonlin-ear systems with small zero-point fluctuations 𝜑𝑚𝑗 1,for all modes 𝑚 and junctions j, the Hamiltonian 𝐻nl

exerts only a perturbative effect on the eigenspectrumof 𝐻lin. Hence, we treat its effect order-by-order us-ing the expansion 𝐻nl =

∑𝐽𝑗=1

∑∞𝑝=3 𝐸 𝑗𝑐 𝑗 𝑝𝜑

𝑝

𝑗, obtained

in Eq. (A.43). For this approach, we focus on the regimein which the energy difference between two modes ismuch larger than the phase excitation of the Josephson

dipoles, ℏ (𝜔𝑘 − 𝜔𝑚) 𝐸 𝑗𝑐 𝑗 𝑝

𝜑𝑝

𝑗

for all 𝑘, 𝑚, 𝑗 , and 𝑝.Example: transmon coupled to a readout cavity mode.

Recall the simple example circuit quantized at the begin-ning of the main text. A qubit (𝑞) is coupled to a cavity(𝑐). To leading order in 𝑝, the circuit Hamiltonian 𝐻nl

is, using Eq. (A.43),

𝐻nl = −𝐸𝐽

24

(𝜑𝑐𝑎𝑐 + 𝜑𝑞𝑎𝑞 +H.c.

)4 + O (𝜑6𝐽

), (B.1)

where 𝐸𝐽 and 𝜑𝐽 are the Josephson energy and flux oper-ator of the tunnel junction, and 𝜑𝑞 and 𝜑𝑐 are the qubitand cavity mode ZPF, respectively.

Expanding the multinomial. Expanding the 𝑝 = 4multinomial term in Eq. (B.1), we find a weighted sum ofall possible four-body interactions between the qubit and

cavity; example terms include 𝑎𝑞

(𝑎†𝑞

)3and 𝑎𝑐𝑎

†𝑐𝑎𝑞𝑎

†𝑞.

Using the commutation relations [Eq. (A.41)], we nor-mal order the polynomial. For example, the qubit-onlyoperators, this yields

𝑎4𝑞 + 4𝑎†𝑞𝑎3𝑞 + 6𝑎†2𝑞 𝑎2𝑞 + 4𝑎†3𝑞 𝑎𝑞 + 𝑎†4𝑞+ 6𝑎2𝑞 + 12𝑎†𝑞𝑎𝑞 + 6𝑎†2𝑞 + 3𝐼 .

Higher-order nonlinearity yields lower-order coupling.Due to the non-commutativity of the operators, thenormal-ordering procedure results in terms that havelower polynomial order than those in original unorderedexpression; these new terms include 𝑎

†𝑞𝑎𝑞 and 3𝐼. Such

quadratic terms (e.g., 𝑎†𝑞𝑎𝑞, 𝑎

†𝑐𝑎𝑐, and 𝑎

†𝑐𝑎𝑞) dress the

modes of 𝐻lin in a linear manner—both renormalizingtheir frequencies and hybridizing them. These new linearcouplings cannot be a-priori straightforwardly includedin 𝐻lin, since they occur only as a non-classical conse-quence of the ZPF in the non-linearity; they do not ap-pear in a classical treatment of the Josephson system,which lacks the non-commuting operator aspect. Hence,their amplitudes are determined by the quantum ZPF ofthe eigenmodes. Remark: A self-consistent approach toinclude these linear couplings in 𝐻lin is possible undercertain assumptions17.

Rotating-wave approximation in the dispersive regime.To leading-order, under the rotating-wave approximation(RWA), only excitation-number conserving interactionsin 𝐻nl (those with an equal number of raising and lower-ing operators in each mode) contribute. Thus, Eq. (B.1)reduces to the effective, dispersive qubit-cavity Hamilto-nian

ˆ𝐻 𝑝=4 =

∑𝑚

−ℏΔ𝑚𝑎†𝑚𝑎𝑚 −

1

2ℏ𝛼𝑚𝑎

†2𝑚 𝑎2𝑚

−∑𝑛≠𝑚

1

2ℏ𝜒𝑚𝑛𝑎

†𝑚𝑎𝑚𝑎

†𝑛𝑎𝑛 ,

(B.2)

where 𝑚 ∈ 𝑞, 𝑐 is the mode label, Δ𝑚 is the effectiveLamb shift, 𝛼𝑚 is the anharmonicity, and 𝜒𝑚𝑛 is the to-tal cross-Kerr frequency shift induced between modes 𝑚

27

and 𝑛. The bar over 𝐻 denotes the RWA; the sub-script 𝑝 = 4 denotes the highest the power of the Taylorexpansion included in the effective Hamiltonian. The ex-citation spectrum of this Hamiltonian is illustrated inSupplementary Figure S4. Remark: Using first-ordertime-independent perturbation theory in place of theRWA yields the same result as the RWA.

Hamiltonian parameters. The Hamiltonian parame-ters are obtained from the normal ordering and usingEq. (A.54),

𝜒𝑚𝑛 =

𝐽∑𝑗=1

ℏ−1𝐸 𝑗𝜑2𝑚𝑗𝜑

2𝑛 𝑗 =

𝐽∑𝑗=1

ℏ𝜔𝑚𝜔𝑛

4𝐸 𝑗

𝑝𝑚𝑗 𝑝𝑛 𝑗 , (B.3a)

Δ𝑚 =1

2

𝑀∑𝑛=1

𝜒𝑚𝑛 , (B.3b)

𝛼𝑚 =1

2𝜒𝑚𝑚 . (B.3c)

Constraints in the Hamiltonian. First, the structureof the non-linearity imposes certain constraints. For ex-ample, 𝜒𝑚𝑛 ≤ 2

√𝛼𝑚𝛼𝑛, where the equality holds only for

the one-junction case, 𝐽 = 1. Second, the universal EPRproperties impose a strict limit on physically realizabledesigns, subject to Eqs. (A.57)–(A.60).

Degeneracies. So far, we implicitly assumed the sys-tem does not have accidental degeneracies, such as thosethat occur when the frequency of a mode is an integermultiple of that of another, 𝜔𝑘 ≈ 𝑧 × 𝜔𝑚, where 𝑧 ∈ Z.In the case of such a degeneracy, additional terms sur-vive the RWA; for example, if 𝜔𝑐 = 3𝜔𝑞, then the non-excitation conserving term 𝑎3𝑞𝑎

†𝑐 survives the RWA.

EPR sign & example. Due to the even-power natureof the nonlinear interactions, the EPR sign drops outaltogether from Eq. (B.3). The interaction strength be-tween the qubit and cavity modes only depends on theoverlap int the EPRs of the two modes alone. The sig-nificance of this is curiously showcased by Device DT3,presented in Methods. The qubit eigenmodes of DT3 arethe equally-hybridized symmetric (bright, B) and anti-symmetric (dark, D) combinations of the two bare trans-mon qubit modes. Naively, using the analogy of atoms ina cavity, one reasons that the symmetric mode couples tothe cavity (C), |𝜒BC | 0, and the antisymmetric modesdoes not, |𝜒DC | = 0, due to the out-of-phase oscillation ofthe junction current dipole—a cancelation effect. How-ever, from the EPR expression, it is seen that, to thecontrary, the signs are irrelevant—no such cancelationis possible. Rather, according to the EPR method, sinceboth modes have equal participation in the two junctions(𝑝𝐷1 = 𝑝𝐷2 = 𝑝𝐵1 = 𝑝𝐵2), the couplings to the cavity areessentially equal, 𝜒DC ≈ 𝜒BC; indeed, this is observed inthe experiment (see the corresponding data table in theMethods section).

Generalizing to many modes. Equations (B.2)and (B.3) were derived for the example of a qubit-cavitycircuit; however, they generalize straightforwardly to 𝑀

modes under the simple extension 𝑚 ∈ 1, . . . , 𝑀.

(a)

c′ω

qαqα

q′ω

(b) (c)

c′ω

qcχqcχ

q′ω

...

...

ω

Supplementary Figure S4. Excitation spectrum (not-to-scale) of a transmon-cavity circuit, corresponding to theeffective dispersive Hamiltonian 𝐻lin+ ˆ𝐻 𝑝=4, see Eq. (B.2). (a)The cavity and qubit frequencies in the ground state are 𝜔′𝑐and 𝜔′𝑞 , respectively. The qubit anharmonicity is 𝛼𝑞 , and thequbit-cavity dispersive cross-Kerr shift is 𝜒𝑞𝑐 . (b) Zoom-inon the qubit spectrum in the vicinity of 𝜔′𝑞 . Each photonin the resonator loads the qubit frequency down by 𝜒𝑞𝑐 . (c)Zoom-in on the cavity spectrum in the vicinity of 𝜔′𝑐 .

B2. EPR matrices and many-body interactions

To more easily and systematically handle large-scalecircuits and higher-order non-linearities, we introduce theEPR matrix P, comprising the 𝑀 × 𝐽 EPRs 𝑝𝑚𝑗 as itsentries,

P B©­­«

𝑝11 · · · 𝑝1𝐽...

. . ....

𝑝𝑀1 · · · 𝑝𝑀𝐽

ª®®¬ . (B.4)

The cross-Kerr interaction amplitudes 𝜒(𝑝)𝑚𝑛 due to the 𝑝-

th order terms of 𝐻nl [for 𝑝 = 4, see Eq. (B.3a)] cansimilarly be organized in an 𝑀 × 𝑀 matrix,

χ𝑝 B©­­­«𝜒(𝑝)11 · · · 𝜒

(𝑝)1𝑀

.... . .

...

𝜒(𝑝)𝑀1 · · · 𝜒

(𝑝)𝑀𝑀

ª®®®¬ . (B.5)

It follows from Eq. (B.3a) that the leading-order Kerrmatrix is

χ4 =ℏ

4(ΩP) E−1𝑗 (ΩP)T , (B.6)

where Ω is the diagonal eigenfrequency matrix and

E−1𝑗 B©­­«𝐸−11

. . .

𝐸−1𝐽

ª®®¬ (B.7)

is the diagonal matrix of inverse Josephson energies.To leading order, 𝑝 = 4, the vector of anharmonicities

is the diagonal of χ4,

α4 B(𝛼(4)1 , . . . , 𝛼

(4)𝑀

)T=1

2diagχ4 , (B.8)

28

and the Lamb shift of mode 𝑚 is the 𝑚-th row sum ofthe Kerr matrix.

∆4 B(Δ(4)1 , . . . ,Δ

(4)𝑀

)T=1

2χ41𝑀 , (B.9)

where 1𝑀 denotes a column vector of length 𝑀 with allentries equal to unity.

Dilution of the nonlinearity. The dilution of the non-linearity of the Josephson dipole elements among theeigenmodes is neatly expressed in Eq. (B.6). The Joseph-son dipole energies EJ

−1 are diluted by ΩP through acongruence transform to the Kerr coefficients. Whilethe eigenfrequencies Ω weighs the contribution to eachmode, the participation matrix P dictates the dilution ofthe junction energies and their nonlinearity among themodes.

Higher-order nonlinear corrections and dilution. Us-ing the results of Sec. B3, the 𝑝-th other correction tothe Kerr matrix is

χ𝑝 = ℏ𝑐𝑝 (ΩP) E−1𝑗 ϕ𝑝−4tot (ΩP)

T , (B.10)

where ϕtot = Diag(𝜑1,tot, . . . , 𝜑𝐽 ,tot

)is the diagonal ma-

trix of the total ZPF fluctuation of the Josephson dipolereduced fluxes, defined in Eq. (B.18). The Kerr matrixincorporating corrections due to all orders of the nonlin-earity, see Eq. (B.22), is χ B

∑∞𝑝=3 χ𝑝. The congruence-

transformation form of Eq. (B.10) is identical to that ofEq. (B.6); it governs the dilution of the nonlinearity inthe same manner, subject to ΩP.

Similarly to the results of Eqs. (B.8) and (B.9), the 𝑝-th order corrections to the anharmonicity and Lamb-shiftvectors are

α𝑝 =1

2diagχ𝑝 and ∆𝑝 = 𝑐𝑝 (𝛀P)ϕ𝑝−2

tot 1𝑀 . (B.11)

In general, the Lamb shift correction depends on the totalZPF frustration of the junctions.

B3. General many-body interactions

So far, we explicitly calculated the leading-order cor-rection on the spectrum of 𝐻lin due to 𝐻nl. We ex-pressed the eigenmode interactions using normal-orderedmany-body terms, such as 𝜒𝑎

†𝑞𝑎†𝑐𝑎𝑞𝑎𝑐. Here, we extend

the analysis and compute the amplitude of any generalnormal-ordered term in 𝐻nl.

The form a general many-body interaction. A generalnormal-ordered many-body interaction has the form

𝐶𝑝

α,β𝑎†𝛽1

1 · · · 𝑎†𝛽𝑀

𝑀𝑎𝛼1

1 · · · 𝑎𝛼𝑀

𝑀C 𝐶

𝑝

α,βa†𝜷a𝜶 , (B.12)

where 𝐶𝑝

α,βis an energy-dimensioned amplitude, de-

termined by the 𝑝-th-order of 𝐻nl [see Eqs. (A.43)and (A.48)], and the multi-index 𝑀-tuples

α B (𝛼1, . . . , 𝛼𝑀 ) and β B (𝛽1, . . . , 𝛽𝑀 ) (B.13)

describe the distribution of annihilation and creation op-erators involved in the interaction among the 𝑀 modes,respectively. The entires of the multi-index tuples αand β are non-negative integers, 𝛼𝑚, 𝛽𝑚 ∈ Z≥0. Theright-hand side of Eq. (B.12) introduces the multi-indexshorthand notation for the interaction terms.

Multi-index shorthand and operator powers. The to-tal number of lowering and raising operators in the ex-pression of Eq. (B.12) is equal to the 1-norm of α and β,

|α| B𝑀∑𝑚=1

𝛼𝑚 and |β | B𝑀∑𝑚=1

𝛽𝑚 , (B.14)

respectively. For a given power 𝑝 of 𝐻nl, the total numberof resulting operators is bounded, |α| + |β | ≤ 𝑝.

Expanding the 𝐻nl multinomials. To arrive atEq. (B.12) from Eq. (A.43), we first group the sum ofthe annihilation operators for a Josephson dipole, seeEq. (A.48), and define

𝐴 𝑗 B𝑀∑𝑚=1

𝜑𝑚𝑗𝑎𝑚 ; (B.15)

𝐻nl =

∞∑𝑝=3

𝐽∑𝑗=1

𝐸 𝑗𝑐 𝑗 𝑝

(𝐴 𝑗 + 𝐴†𝑗

) 𝑝. (B.16)

Importantly, the commutator[𝐴 𝑗 , 𝐴

†𝑗

]=

∑𝑀𝑚=1

𝜑𝑚𝑗

2 isscalar-valued, which allows us to use the normal-orderingnon-commutative binomial theorem to expand the 𝑝-thpower term of 𝐻nl

141–143. Using the non-commuting ex-pansion,(

𝐴 𝑗 + 𝐴†𝑗) 𝑝

=

floor𝑝

2∑𝑘=0

𝑝−2𝑘∑𝑖=0

𝑝!

2𝑘 𝑘!𝑖!(𝑝 − 2𝑘 − 𝑖)!×(

𝜑2𝑗 ,tot

) 𝑘 (𝐴†𝑗

) 𝑖 (𝐴 𝑗

) 𝑝−2𝑘−𝑖,

(B.17)

where floor 𝑝

2 gives the greatest integer that is less than orequal to 𝑝

2 and the total variance of the ZPF of the 𝑗-thdipole is

𝜑2𝑗 ,tot B

[𝐴, 𝐴†

]=ℏ

2𝐸−1𝑗

𝑀∑𝑚=1

𝑝𝑚𝑗𝜔𝑚 . (B.18)

Since 𝐴 𝑗 is the sum of operators that commute, seeEq. (B.15), we can now expand the powers of 𝐴 𝑗 usingthe classical multinomial theorem,(

𝐴 𝑗

) 𝑖=

∑|α |=𝑖

(|α|α

)𝜑α𝑚𝑗 a

α , (B.19)

where the multi-index shorthand(𝜑𝑚𝑗

)αB

∏𝑀𝑚=1 𝜑

𝛼𝑚

𝑚𝑗,

the multinomial coefficient is(|α|α

)B

|α|!𝛼1! · · · 𝛼𝑀 !

, (B.20)

and the sum condition |α| = 𝑖 means that the sum in-cludes terms with all possible tuples α such that their1-norm has the value 𝑖.

29

The general normal-ordered form. CombiningEqs. (B.16)–(B.19), we find the general normal-orderedmany-body form of 𝐻nl to all orders in 𝑝 and withoutapproximations,

𝐻nl =

∞∑𝑝=3

floor𝑝

2∑𝑘=0

𝑝−2𝑘∑𝑖=0

∑|β |=𝑖,

|α |=𝑝−2𝑘−𝑖

𝐶𝑝

α,βa†βaα . (B.21)

The energy-dimensioned amplitude of an interaction dueto the 𝑝-th power of 𝐻nl for a general many body mode-interaction term is the sum of the individual junctioncontributions,

𝐶𝑝

α,βB

𝑝!

α!β!𝑘!2𝑘

𝐽∑𝑗=1

𝐸 𝑗𝑐 𝑗 𝑝𝜑α𝑚𝑗𝜑

β𝑚𝑗

𝜑2𝑘𝑗,tot , (B.22)

where 𝑘 B 12 (𝑝 − |α| − |β |).

Equation (B.22) provides the amplitude for any modeinteraction. Its exact value is calculated using the EPR toobtain the ZPF 𝜑𝑚𝑗 , using Eq. (A.55). Thus, we have an-alytically fully constructed 𝐻nl, and, individually, everyterm contained within, from the FE simulations throughthe EPR.

Use cases. Equation (B.22) can be used to explicitlycalculate higher-order corrections to an effective Hamil-tonian; for example, see Eqs. (B.10) and (B.11). Us-ing Eq. (B.22), we can calculate mode parameters evenwhen the numerical diagonalization of 𝐻nl becomes in-tractable—an issue that occurs at even moderate num-ber of modes. Moreover, Eq. (B.22) can be used to en-gineer pumped multi-photon drive processes and to acti-vate non-RWA interactions144, as discussed in Sec. B4.

B4. The driven Josephson system:parametrically-activated interactions

The Josephson system can be subjected to strong ex-ternal drives used to parametrically activate or enhancenonlinear couplings. We illustrate the use of Equa-tion (B.22) in this context by calculating the amplitudeof a excitation-swapping beam-splitter interaction.

Example: parametrically-activated beam-splitter inter-action. Motivated by the setup of device WG1, we aimto parametrically activate a beam-splitter (BS) interac-tion between two non-resonant modes of a Josephson sys-tem; for example, such an interaction can be used as aQ-switch96,145. The system has a high-quality, storage-cavity mode (𝑐) and another low-quality, readout-cavitymode (𝑟). The two modes are far detuned and obey theconditions outlined at the start of Sec. B1. From the sys-tem 𝐻nl, we aim to obtain the effective BS Hamiltonian

𝐻eff = ℏ𝑔𝑎†𝑐𝑎𝑟 + ℏ𝑔∗𝑎†𝑟𝑎𝑐 , (B.23)

where 𝑔 is the rate excitation exchange.

cω rω

(a) (b)

1pω

2pω

cω rω

1pω

2pω

2∗ξ1ξr

†aca 2ξ1ξr†aca

(c)

cω rω

1pω

2pω

(d)

cω rω

1pω

2pω

2ξ1ξr†ac

†a2∗ξ1ξr

†ac†a

Supplementary Figure S5. Depiction of four-wavemixing in a Josephson junction used to paramet-rically activate a bilinear interaction between twomodes. Off-resonant pumping of the Josephson circuitat 𝜔𝑝1 and 𝜔𝑝2 with effective straights b1and b2, respec-tively, can activate a specific four-wave-mixing process presentin 𝐻nl. The pump frequencies 𝜔𝑝1 and 𝜔𝑝2 could be degen-erate. The activation condition is determined by the storage-and readout-mode frequencies 𝜔𝑐 and 𝜔𝑟 , respectively; theircorresponding mode operators are 𝑎𝑐 and 𝑎𝑞 . (a), (b) The ac-tivated interaction is a beam-splitter-like conversion process,containing exactly one mode annihilation and one creationoperator. The resonance condition is determined by 𝜔𝑐 −𝜔𝑟 .(c), (d) Two-mode squeezing interaction, contains exactly twocreation operators. The resonance condition is set by 𝜔𝑐 +𝜔𝑟 .In the first column, panels (a) and (c), the resonance condi-tion is set by 𝜔𝑝1−𝜔𝑝2, whereas in the second column, panels(b) and (d), it is set by 𝜔𝑝1 +𝜔𝑝2. Note, for each of the fourdiagram, the conjugate process (all arrow directions flipped)is also activated by the pumps.

Interaction in the rotating frame. In the rotatingframe with respect to 𝐻lin, defined by the trans-form 𝑈 (𝑡) B exp

[𝑖𝑡

(𝜔𝑐𝑎

†𝑐𝑎𝑐 + 𝜔𝑟𝑎

†𝑟𝑎𝑟

)], the mode op-

erators 𝑎𝑚 acquire a harmonic time dependence, 𝑎𝑚 ↦→𝑎𝑚 (𝑡) B 𝑈† (𝑡) 𝑎𝑚𝑈 (𝑡) = 𝑎𝑚𝑒

−𝑖𝜔𝑚𝑡 , where 𝑚 ∈ 𝑐, 𝑟.While a term in 𝐻nl of the form 𝑎

†𝑐𝑎𝑟 exists, this terms

is non-stationary, 𝐶 𝑝=4

(1,0) , (0,1)𝑒−𝑖𝑡 (𝜔𝑟−𝜔𝑐)𝑎†𝑐𝑎𝑟 +H.c. and is

eliminated in the RWA in deriving the effective Hamilto-nian.

Parametric activation and interaction rate. A thirdmode of the system, indexed by 𝑝, is off-resonantly drivenat frequencies 𝜔𝑃 and 𝜔′

𝑃with amplitudes 𝜖1 and 𝜖2,

respectively; within the RWA, the drive Hamiltonian is

30

𝐻𝑝 B 𝜖𝑝𝑒−𝑖𝜔𝑃 𝑡𝑎𝑝 + 𝜖∗𝑝𝑒+𝑖𝜔𝑃 𝑡𝑎†𝑝 . (B.24)

Consider the following term contained in 𝐻nl, seeEq. (B.21),

𝐶𝑝=4

(0,1,2) , (1,0,0)𝑎†𝑟 (𝑡) 𝑎𝑐 (𝑡) 𝑎2𝑝 , (B.25)

where the mode label order is (𝑟, 𝑐, 𝑝). Mov-ing into a displaced and rotating frame definedby

(b𝑝

)B exp

(b𝑝𝑎

†𝑝 − b∗𝑝𝑎𝑝

)exp

(−𝑖ℏ𝜔𝑃𝑡𝑎

†𝑝𝑎𝑝

),

where b𝑃 B 𝜖/(𝜔𝑃 − 𝜔𝑝

), the 𝑝 mode operator is

shifted and rotated, 𝑎𝑝 ↦→ 𝑎𝑝 (𝑡) B (b𝑝

)−1𝑎𝑝

(b𝑝

)=(

𝑎𝑝 + b𝑝)𝑒−𝑖𝜔𝑃 𝑡96,146. In this frame, expanding

Eq. (B.25) yields the suggestive interaction term

𝐶𝑝=4

(0,1,2) , (1,0,0)𝑎†𝑟𝑎𝑐b

2𝑝𝑒−𝑖𝑡 (𝜔𝑟−𝜔𝑐−2𝜔𝑃) , (B.26)

which is resonantly activated when 𝜔𝑟 − 𝜔𝑐 − 2𝜔𝑃 = 0,see Supplementary Figure S5. Under this condition,this term survives the RWA when deriving the effectiveHamiltonian. We find the effective BS rate by castingEq. (B.26) in the canonical BS form, Eq. (B.23),

ℏ𝑔 = 𝐶𝑝=4

(0,1,2) , (1,0,0)b2𝑝 . (B.27)

General parametrically-activated interaction. Theprocedure illustrated with the BS example general-izes straightforwardly to the parametric activationof most other interactions and to finding their ratesusing Eq. (B.22). The procedure is useful for morecomplex and even cascaded processes97,144 used indissipation engineering147. It is reported that the limitof procedure is typically reached when b𝑃 approachesunity, and other activated nonlinear processes becomenon-negligible148–151.

C. Finite-element electromagnetic-analysismethodology

We detail the EPR methodology for the finite-element(FE) analysis of the Josephson circuit. In Sec. C1, wemodel a Josephson dipole in the FE simulation as a rect-angular sheet with an inductive lumped-element bound-ary condition with inductance 𝐿 𝑗 . In Secs. C2 and C3, weextract the EPR 𝑝𝑚𝑗 and EPR signs 𝑠𝑚𝑗 from the resultof an eigenanalysis simulation of the linearized Joseph-son circuit, corresponding to 𝐻lin. This step completesthe classical analysis part of the EPR method; from here,the Hamiltonian is fully specified, as described in Secs. Aand B. The eigenresult also provides complete informa-tion on the dissipation and input-output coupling of thecircuit, as described in Sec. D.

These steps, and the calculations detailed in thistext, are automated by the open-source software pack-age pyEPR152, which we offer to the community.

C1. Modeling the Josephson dipole

In the FE model of the Josephson circuit, we modela Josephson dipole as a simple rectangular sheet witha lumped-element boundary condition4, see Supplemen-tary Figure S6. The sheet abstracts away the physicallayout of the Josephson dipole and its wiring leads.

Idealization of the deep-sub-wavelength features. Thisidealization of the Josephson dipole is justified in that itsphysical size is in the deep-sub-wavelength regime withrespect to the eigenmodes of interest. For example, forthe devices described in the Methods section, the sepa-ration between the Josephson dipole size and the modewavelengths of interest is approximately five orders ofmagnitude. We hence treat a Josephson dipole in the FEmodel as lumped-element inductor with inductance 𝐿 𝑗 ,given by Eq. (A.13a); the linearization is taken with re-spect to the circuit equilibrium, see Sec. A8.

Electromagnetic model of the lumped inductance.The 𝑗-th Josephson dipole is modeled as a two-dimensional sheet 𝑆 𝑗 , see Supplementary Figure S6. Thesheet is assigned a surface-impedance boundary condi-tion, which imposes ®𝐸 ‖ = 𝑍𝑠 (𝑛 × ®𝐻 ‖) across the sheet,where ®𝐸 ‖ and ®𝐻 ‖ are the tangential electric and mag-netic fields of the sheet, respectively, 𝑛 is the sheet normalvector, and 𝑍𝑠 is the complex-valued surface impedancecorresponding to a total sheet inductance of 𝐿 𝑗 . The hatsymbol over 𝑛 denotes a unit vector in the context ofelectromagnetic fields; it is not to be confused with thehat notation used for quantum operators.

Reducing the model complexity: ignoring leads. Ifthe geometric inductance of the wire leads connectinga Josephson dipole to larger distributed surfaces (such asthe pads of a transmon qubit) is negligible in compari-son to 𝐿 𝑗 and the wire leads are deeply sub-wavelengthin size, then the wire leads can be ignored, as depictedin Supplementary Figure S6. If the wire leads have non-negligible inductance, then they can either be included inthe simulation or their linear inductance can be includedin the definition of the Josephson dipole. In practice, itis preferable to be able to abstract away the wire leadsas much as possible, since their feature sizes are typicallyorders of magnitude smaller than other all other designfeatures. The inclusion of such fine detail in the modelcan result in very large geometric aspect ratios, whichin turn result in increased computational costs. How-ever, while this finer detail is more representative of thephysical design, we have found that it does not lead tonoticeable corrections for typical cQED devices.

Performance tip: mesh operations. The sheet 𝑆 𝑗 isgenerally one of the smallest features of the FE model.Due to this small size but critical role, one can gently seeda higher level of mesh on 𝑆 𝑗 to speed up the eigenanal-ysis. However, caution should be used to avoid seedingtoo heavy of an initial mesh, which can instead lead topoor convergence of the simulation. Convergence can beverified by plotting 𝑝𝑚𝑗 as a function of the simulationadaptive pass number and by verifying the conditions de-

31

Sj

Lj

Supplementary Figure S6. Illustration (not-to-scale)of the representation of a Josephson dipole in a finite-element (FE) simulation model. The two large grey rect-angles on the edges of the illustration depict metal pads of atransmon qubit. The light-grey sheet 𝑆 𝑗 in the center depictsthe sheet used to model the Josephson dipole (and poten-tially its leads) in the FE model. The sheet is assigned alumped-element inductive boundary condition, with induc-tance 𝐿 𝑗 (sheet inductance symbolized by floating inductor-element symbol), corresponding to the Josephson dipole in-ductance with respect to the operating equilibrium point, seeSec. A9. The structural details of the Josephson dipole (loca-tion marked by brown cross) and its lead wires (brown wiresconnected to cross) can be abstracted away.

tailed in Sec. A7.

C2. Calculating the EPR 𝑝𝑚 in the case of a singleJosephson dipole

If a Josephson circuit incorporates exactly one Joseph-son dipole, then we use the global electric and magneticeigenmode energies to directly calculate the EPR 𝑝𝑚 ofthe dipole in the mode.

Energy balance. The time-averaged electromagneticenergy in a resonantly excited mode is equally split intoan inductive Eind and capacitive Ecap contribution125.This detailed balance, Eind = Ecap, holds even in thepresence of dissipation and defines the eigenmode condi-tion. In the presence of a Josephson dipole, the inductiveenergy is split into a magnetic Emag and a kinetic Ekincontribution; Eind = Emag + Ekin. The magnetic contri-bution is associated with magnetic fields and geometricinductance. The kinetic contribution is associated withthe Josephson dipole kinetic inductance and the flow ofelectrons, and their inertia. From the point of view of theFE analysis, the magnetic energy is stored in the mag-netic eigenfields ®𝐻𝑚 and the kinetic energy is stored in thelumped-element boundary condition on 𝑆 𝑗 . If lumped-element capacitive boundary conditions are absent fromthe model, then the capacitive eigenmode energy is storedentirely in electric eigenfields ®𝐸𝑚, Ecap = Eelec; hence,

Eelec = Ecap = Eind = Emag + Ekin . (C.1)

Calculating EPR from the energy balance. UsingEq. (C.1) and Eq. (5) of the main text, 𝑝𝑚 = Ekin/Eind,the EPR is calculated from the ratio of global energyquantities,

𝑝𝑚 =Eelec − Emag

Eelec, (C.2)

where the total magnetic- and electric-field energies arecomputed from the eigenfields phasors,

Eelec =1

4Re

∫𝑉

®𝐸∗max←→𝜖 ®𝐸max d𝑣 , (C.3)

Emag =1

4Re

∫𝑉

®𝐻∗max←→ ®𝐻max d𝑣 , (C.4)

where ®𝐸max (𝑥, 𝑦, 𝑧) [resp., ®𝐻max (𝑥, 𝑦, 𝑧)] is the eigen-mode electric (resp., magnetic) phasor, and ←→𝜖(resp., ←→) denotes the electric-permittivity (resp.,magnetic-permeability) tensor. The spatial integrals areperformed over total volume 𝑉 of the device. The electriceigenfield is related to the phasor by

®𝐸 (𝑥, 𝑦, 𝑧, 𝑡) = Re[®𝐸max (𝑥, 𝑦, 𝑧) 𝑒𝑖𝜔𝑚𝑡

]. (C.5)

Js

lj

Sj

∣∣∣

sJ∣∣∣

0 max

Supplementary Figure S7. Model of a transmon qubitoverlaid with the surface-current eigendensity ®𝐽𝑠 (®𝑟) of thequbit eigenmode, obtained using a FE simulation. Trans-mon pads depicted by the two large gray rectangles, sepa-rated by distance 𝑙 𝑗 . Small center rectangle represents thesheet model 𝑆 𝑗 of the Josephson junction.

C3. Calculating the EPR 𝑝𝑚𝑗 in the case ofmultiple Josephson dipoles

EPR 𝑝𝑚𝑗 of the 𝑗-th Josephson dipole. In the case ofmultiple Josephson dipoles, the total kinetic energy Ekin

32

is itself split among the 𝐽 dipoles, see Eq. (A.20). Wecalculate the EPR 𝑝𝑚𝑗 of junction 𝑗 in mode 𝑚 using theEPR definition, Eq. (A.50),

𝑝𝑚𝑗 =1

2𝐿 𝑗 𝐼𝑚𝑗

2/Eind , (C.6)

where 𝐼𝑚𝑗 is the peak value of the Josephson dipole cur-rent in mode 𝑚. The current 𝐼𝑚𝑗 is calculated from theintegral of the mode surface-current density ®𝐽𝑠,𝑚 (𝑥, 𝑦, 𝑧)over the dipole sheet 𝑆 𝑗 ,𝐼𝑚𝑗

= 𝑙−1𝑗

∫𝑆 𝑗

®𝐽𝑠,𝑚 d𝑠 , (C.7)

where 𝑙 𝑗 is the length of the sheet, see SupplementaryFigure S7.

EPR sign. The EPR sign is calculated from the orien-tation of the current 𝐼𝑚𝑗 . The absolute orientation is rel-ative, as described in the main text, and is determined bydefining a directed line DL 𝑗 across 𝑆 𝑗 . If the phasor ®𝐽𝑠,𝑚is aligned with DL 𝑗 then 𝑠𝑚𝑗 = +1; otherwise, 𝑠𝑚𝑗 = −1.Hence, we extract the sign using

𝑠𝑚𝑗 = sign

∫DL 𝑗

®𝐽𝑠,𝑚 · d®𝑙 . (C.8)

The direction of the line merely establishes a conventionfor a positive EPR sign.

Enforcing energy balance. The convergence of 𝑝𝑚𝑗 , aquantity extracted from the local eigenfield solutions, canbe enhanced by re-normalizing the set of mode EPR toensure energy balance, see Eq. (C.1). If lumped-elementcapacitive boundary conditions are absent, then the EPRcan be renormalized to ensure that their total sum isequal to the ratio

∑𝐽𝑗=1 𝑝𝑚𝑗 = Ekin/Eind, which is a glob-

ally calculated quantity, and is therefore expected to con-verge quicker.

The above calculations are automated by pyEPR95.

C4. Remarks on the finite-element eigenmodeapproach

Finding the eigenfrequencies. Rather than searchingfor the location of an unknown pole in an impedance re-sponse of the circuit and then honing in on it to performfiner sweeps, the eigenmode analysis returns the lowest 𝑀modes above a minimum frequency of interest. This typ-ically lifts the requirement for a prior knowledge of themode frequencies.

Single simulation. The FE eigenmode performs a sin-gle simulation from which complete information of theJosephson circuit is extracted. This is in contrast to thetypical process flow used in an impedance analysis, whichrequires that mode frequencies be first identified so thata series of individual narrow-frequency-range impedance-response sweeps (one for each mode and junction) can beperformed.

Closed-circuit optimization. We have found that theabove two feature (see remarks) can speed up the itera-tive refinement of a quantum design and can circumventsdifficulties associated with finding and fitting unknown,narrow-line poles in an impedance analysis.

D. Dissipation budget and input-output coupling

In this section, we summarize the methodology usedto fully characterize dissipation and input-output cou-pling in the Josephson system. Loss of energy re-sults from material losses and radiative boundaries whichguide energy away from the system. Additionally, con-trol of the system is achieved by means of the radia-tive boundaries, such as input-output (I-O) coupling.The dissipation budget, comprising the individual loss-contribution bound of each lossy and radiative element,is extracted from the same eigensolution used to calcu-late the EPR 𝑝𝑚𝑗 , in essentially the same way—by cal-culating the fraction of the 𝑚-th mode energy stored inthe 𝑙-th lossy element—the lossy EPR 𝑝𝑚𝑙. While lossyEPR signs 𝑠𝑚𝑙 can also be calculated for each element,these are not needed for linear dissipation; however, theirrole in I-O coupling is detailed in Sec. D4.

D1. Dissipation budget

The dissipation budget comprises the estimatedenergy-loss-rate contributions due to each loss mech-anism and each lossy object in the Josephsoncircuit85,88,91,92,113,125,153–156. We classily losses as ca-pacitive, inductive, or radiative, and summarize theircalculation here. Each loss mechanism is described bya corresponding lossy EPR 𝑝𝑚𝑙 and an intrinsic qualityfactor 𝑄. Assuming the dissipation is linear and the modeof interest is underdamped, the loss rates of each mecha-nism simply add; so, the total quality factor of the 𝑚-thmode is92,113,154

1

𝑄total=

1

𝑄cap+ 1

𝑄ind+ 1

𝑄rad, (D.1)

where 𝑄cap and 𝑄ind are the total mode quality factorsdue to capacitive and inductive losses (i.e., those propor-

tional to the intensity of the electric ®𝐸 2 and magnetic

field ®𝐻2, respectively, see Secs. D2 and D3), and 𝑄rad is

the total radiative mode quality factor, see Sec. D4. Inthis section, the mode index 𝑚 is implicit.

Remark. Beyond these intrinsic mechanisms, extrin-sic factors, such as ionizing radiation, can play a sig-nificant role in understanding loss mechanisms, such asthose due to quasiparticle in superconducting quantumcircuits157,158.

33

D2. Capacitive loss

The total capacitive mode quality 𝑄cap is the weightedsum of intrinsic quality factors 𝑄

cap𝑙

(i.e., the inverse ofthe dielectric loss tangent) of all lossy dielectrics 𝑙 in theJosephson circuit92,154,

1

𝑄cap=∑𝑙

𝑝cap𝑚𝑙

𝑄cap𝑙

, (D.2)

where 𝑝cap𝑚𝑙

is the lossy energy-participation ratio of the 𝑙-th dielectric in the mode—i.e., 𝑝cap

𝑚𝑙is the fraction of ca-

pacitive energy stored in the dielectric element 𝑙. Weclassify lossy capacitive elements as either bulk or sur-face. For example, the volume of three-dimension dielec-tric object, such as a chip substrate, is associated withbulk capacitive loss83,94,159. On the other hand, the sur-face of the substrate, which may be a surface-dielectriclayer is classified as a surface-loss element88,94. The lossyEPR for bulk capacitive loss is calculated from the eigen-field solutions,

𝑝cap𝑚𝑙

=1

Eelec1

4Re

∫𝑉𝑙

®𝐸∗max←→𝜖 ®𝐸max d𝑣 , (D.3)

where the integral is carried over the volume 𝑉𝑙 of the 𝑙-th bulk dielectric object; the total electric energy Eelecis defined in Eq. (C.3). The lossy EPR for a surfacedielectric is approximated by

𝑝cap,surf𝑚𝑙

=1

Eelec𝑡𝑙𝜖𝑙

4Re

∫surf𝑙

®𝐸max

2 d𝑠 , (D.4)

where the surface-dielectric layer thickness is 𝑡𝑙 and itspermittivity is 𝜖𝑙.

D3. Inductive loss

Physically, inductive losses originate from the dissipa-tive flow of electrical current in metals or through metal-metal seams. Supercurrent loss due to quasiparticles andvortices can be accounted for in an effective quality factorof the conducting surface. We denote the intrinsic induc-tive quality factor of a lossy object 𝑄ind

𝑙. The bound on

the total inductive-loss quality factor 𝑄ind of mode 𝑚 isa weighted sum of 𝑄ind

𝑙,

1

𝑄ind=∑𝑙

𝑝ind𝑚𝑙

𝑄ind𝑙

, (D.5)

where 𝑝ind𝑚𝑙

is the lossy EPR for inductive element 𝑙. Weclassify inductive losses as either surface, bulk, or seam.

Surface conductive loss (in the skin-depth). The frac-tion of eigenmode energy stored in the skin depth _0 ofmetal surface 𝑙, denoted surf 𝑙, if the lossy EPR of thesurface, and is obtained from the eigenfield solutions,

𝑝ind,surf𝑚𝑙

=1

Emag

_0`𝑙

4Re

∫surf𝑙

®𝐻max, ‖

2 d𝑠 , (D.6)

where `𝑙 is the magnetic permeability of the surface; typ-ically, `𝑙 = `0, and ®𝐻max, ‖ is the magnetic field phasorparallel to the surface. For superconductors, 𝑝

ind,surf𝑚𝑙

is the kinetic inductance fraction91,154, commonly de-noted 𝛼. The total magnetic energy Emag is defined inEq. (C.4).

Surface conductive loss: intrinsic quality. In thenormal state, a metal, such as copper or aluminum,has an intrinsic inductive quality factor of orderunity125, 𝑄

ind,surf𝑙

≈ 1. However, in the supercon-ducting state, the lower bound on the quality fac-tor is typically found to be in the range of severalthousand160, 𝑄ind,surf

𝑙& 103. The lower bound for thin-

film superconducting aluminum has been measured toexceed 𝑄

ind,surf𝑙

> 10553.Bulk magnetic loss. The mode magnetic field can cou-

ple to bulk magnetic impurities present in the volume 𝑉𝑙of lossy object 𝑙. The lossy EPR for the bulk-inductive-loss mechanism of volume 𝑉𝑙 is

𝑝ind,bulk𝑚𝑙

=1

Emag

1

4Re

∫𝑉𝑙

®𝐻∗max←→ ®𝐻max d𝑣 . (D.7)

This coupling is typically negligible in current supercon-ducting quantum circuits.

Seam loss. The seam formed by pressing two met-als together provides an electrical bridge for the cur-rent to flow across but also introduces a key dissipationmechanism90. A common example of a seam is the oneformed by the two halves of the metal sample holdersused to house a cQED chip. In the FE analysis, we modelthe seam by a line path seam𝑙 tracing out the locationof the seam at the surface of the two mating walls. Theinductive lossy EPR participation for the seam is

𝑝ind,seam𝑚𝑙

=1

Emag

_0𝑡𝑙`𝑙

4Re

∫seam𝑙

®𝐻max,⊥

2 d𝑙 , (D.8)

where the seam thickness is denoted 𝑡𝑙, its magnetic per-meability `𝑙, and its the penetration depth _0. It is con-venient to rewrite the total seam loss due to seam 𝑙 interms of an effective seam admittance 𝑔seam, defined inRef. 90,

𝑝ind,seam𝑚𝑙

𝑄seam=

1

𝑔seam

∫seam𝑙

®𝐽𝑠 × ®𝑙2 d𝑙

𝜔`0∫all |𝐻max |2 d𝑉

. (D.9)

D4. Radiative loss and input-output coupling

The Josephson circuit incorporates radiative bound-aries, typically purposefully introduced to serve as input-output ports of the circuit125, thus providing a means toperform system measurement and control. For cQED de-vices, the port exposes the circuit to an external trans-mission line conduit, such as a coaxial cable or a co-planar waveguide. The total radiative quality factor 𝑄rad

34

a

jΦ

ain,1 ain,2

aout,2aout,1

(a)

(b)

a

jΦ

Supplementary Figure S8. Schematic of a transmon-qubit circuit (mode operator 𝑎), comprising a Josephsontunnel junction (flux operator Φ 𝑗) and a capacitor, coupledto two input-output ports. The input and output fields ofthe left (resp., right) transmission line are 𝑎in,1 and 𝑎out,1(resp., 𝑎in,2 and 𝑎out,2). (a) Ports 1 and 2 both coupledto the top qubit node; hence, both port EPR signs 𝑠𝑚𝑝 areequal, 𝑠𝑞1 = 𝑠𝑞2. (b) Port 1 couples to the top node, butport 2 couples to the bottom qubit node. The port EPRsigns 𝑠𝑚𝑝 have opposite signs, 𝑠𝑞1 = −𝑠𝑞2.

of mode 𝑚 is the sum of the individual port contribu-tions 𝑄−1rad =

∑𝑃𝑝=1𝑄

−1𝑚𝑝, where 𝑃 is the total number of

ports and 𝑄𝑚𝑝 is the quality factor due to port 𝑝. Below,we describe how port 𝑝 is modeled in the FE simulationto extract 𝑄𝑚𝑝 from the eigensolutions.

Radiative energy loss. Energy stored in mode 𝑚 canleak at a rate ^𝑚𝑝 through port 𝑝 and be guided away bythe transmission line. While for certain modes, such asreadout ones, this coupling is desired, for other modes,such as a qubit one, this coupling is often consideredspurious. In the case of a qubit mode, the energy lossto a readout port is seen as a manifestation of the Pur-cell effect123; while, for the readout mode of the samestructure, the energy loss to the readout port sets therate of information gain161. The rate ^𝑚𝑝 is calculatedfrom port EPR 𝑝𝑚𝑝, as detailed in the following for bothwanted and spurious terms. The port EPR sign 𝑠𝑚𝑝 iscalculated concurrently and is important for the systemdrive configuration.

FE model of the port. In the presence of a port, theboundary of the Josephson circuit is somewhat ambigu-ous—we can include more or less of the port and conduitstructure in the model. We choose to include a minimalbut sufficiently large portion of these to faithfully modelthe disturbing effect of the boundary condition on theeigenmodes. For example, in the case of the qubit-cavitystructure of Figure 1(a) of the main text, we include ashort stub of the I-O coaxial cable in the FE model. Thelength of the stub can be determined from the effect ofa sweep of its length on the target parameters. We havefound that an alternative heuristic measure is to use the

decay of the eigenfields inside the port structure and tomake sure that the end of the port structure is at leastseveral exponential decay lengths long (the field in theport structure decays exponentially since the eigenmodesare generally below its cutoff). The port structure ter-mination surface 𝑆𝑝 is treated as a resistive sheet with tomodel the effect of the transmission line guiding wavesaway from the Josephson circuit. The sheet effective re-sistance is 𝑅𝑝. In the case of a 50 Ω port line, 𝑅𝑝 = 50Ω.While a resistive boundary condition can be assigned tothe sheet, in the case of high-quality modes, it is possibleto use a perturbative approach and to perform a losslesssimulation of the FE model, from which the loss can beextracted (see also remark at end of this section).

Calculating the input-output (I-O) coupling rate ^𝑚𝑝

from the port EPR 𝑝𝑚𝑝. From the lossless eigensolu-tions of mode 𝑚, the energy lost to the effective re-sistor 𝑅𝑝 of port 𝑝 during one mode oscillation pe-riod 𝑇𝑚 = 2𝜋/𝜔𝑚 is

𝑃𝑚𝑝 =1

2𝑅𝑝 𝐼

2𝑚𝑝𝑇𝑚 , (D.10)

where 𝐼𝑚𝑝 is the peak current across the port due to theexcitation of mode 𝑚; see Eq. (C.7). The total modeenergy E𝑚 (𝑡) at time 𝑡 decays at rate

𝑑

𝑑𝑡E𝑚 = −

∑𝑝

^𝑚𝑝E𝑚 . (D.11)

Hence, assuming a high quality mode, the energy lossduring one oscillation period, between times 𝑡 = 0 and 𝑇𝑚,is

E𝑚 (𝑇𝑚) = E𝑚 (0) −∑𝑝

𝑇𝑚^𝑚𝑝E𝑚 (0) . (D.12)

This shows that the loss to port 𝑝 during one periodis 𝑃𝑚𝑝 = 𝑇𝑚^𝑚𝑝E𝑚 (0), which we equate to the expressionof Eq. (D.10) to find the I-O coupling rate in terms ofquantities calculated from the eigenfields,

^𝑚𝑝 =

12𝑅𝐼

2𝑚𝑝

E𝑚 (0). (D.13)

Thus, the mode coupling quality factor is

𝑄𝑚𝑝 B 𝜔𝑚/^𝑚𝑝 =𝜔𝑚E𝑚 (0)12𝑅𝐼

2𝑚𝑝

. (D.14)

Sign of the I-O participation. If there are multipleports, the port EPR sign 𝑠𝑚𝑝, calculated using Eq. (C.8),is important in a manner similar to that explained for thecase of the EPR sign 𝑠𝑚𝑗 in the case of multiple Josephsondipoles; see the main text. To illustrate, consider a sim-ple transmon-qubit circuit coupled to two transmissionlines in two different ways as depicted in the two panelsof Supplementary Figure S8. While in both configura-tion the eigenmode frequency and quality factor is iden-tical, the configurations are inequivalent. Consider driv-ing both transmission lines with the same amplitude andin-phase, then the circuit of Supplementary Figure S8(a)is excited but that of Supplementary Figure S8(b) is not.

35

Remark on modeling the port termination as resistivevs. lossless. The port sheet 𝑆𝑝 can be treated as eitherresistive or lossless. In the former, the sheet is assigneda lumped-element boundary condition with impedancematching the port input impedance as seen from thesystem; typically designed to be 𝑅𝑝 = 50Ω10,162. Theeigenresults fully account for the effect of the dissipa-tion on the mode profile. These effects are negligible inthe case of high-quality modes, but become significantas 𝑄𝑚 approaches unity. For lossless treatment of 𝑆𝑝,suitable when 𝑄𝑚 1, the termination simply assigned aperfectly conducting boundary condition. In both treat-ment, the eigenmode fields can be used to calculate the

loss due to 𝑅𝑝.Reactive ports. In addition to being resistive, ports

can have a reactive component, typically occurring in thepresence of non-idealities. For example, a port coupledto transmission line suffering from a down-line reflectionwill have some of the energy it leaks out to the line comeback to it163. The reactive part of the port structure thushouses energy in its internal modes. There can be treatedby modifying the boundary condition on 𝑆𝑝 or for a moregeneral treatment can be accounted for by including theport, line, and scatterer structure in the FE model. Thislarger model will account for hybridization between theline modes and those of the system100.

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