Hamiltonian of a flux qubit-LC oscillator circuit in the deep-strong-coupling regime

F. Yoshihara,1, ∗ S. Ashhab,2 T. Fuse,1 M. Bamba,3, 4 and K. Semba1

1Advanced ICT Institute, National Institute of Information and Communications Technology,4-2-1, Nukuikitamachi, Koganei, Tokyo 184-8795, Japan

2Qatar Environment and Energy Research Institute,Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar

3Department of Physics, Kyoto University, Kyoto 606-8502, Japan4PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan

(Dated: August 19, 2020)

We derive the Hamiltonian of a superconducting circuit that comprises a single-Josephson-junctionflux qubit and an LC oscillator. If we keep the qubit’s lowest two energy levels, the derived circuitHamiltonian takes the form of the quantum Rabi Hamiltonian, which describes a two-level systemcoupled to a harmonic oscillator, regardless of the coupling strength. To investigate contributionsfrom the qubit’s higher energy levels, we numerically calculate the transition frequencies of thecircuit Hamiltonian. We find that the qubit’s higher energy levels mainly cause an overall shiftof the entire spectrum, but the energy level structure up to the seventh excited states can stillbe fitted well by the quantum Rabi Hamiltonian even in the case where the coupling strength islarger than the frequencies of the qubit and the oscillator, i.e., when the qubit-oscillator circuit isin the deep-strong-coupling regime. We also confirm that some of the paradoxical properties of thequantum Rabi Hamiltonian in the deep-strong-coupling regime, e.g. the non-negligible number ofphotons and the nonzero expectation value of the flux in the oscillator in the ground state, arisefrom the circuit Hamiltonian as well.

Superconducting circuits are one of the most promisingplatforms for realizing large-scale quantum informationprocessing. One of the most important features of su-perconducting circuits is the freedom they allow in theircircuit design. Since the first demonstration of the co-herent control of a Cooper pair box [1], various typesof superconducting circuits have been demonstrated. Inprinciple, the Hamiltonian of a superconducting circuitcan be derived using the standard quantization procedureapplied to the charge and flux variables in the circuit [2].The Hamiltonian of an LC oscillator is well known to bethat of a harmonic oscillator. The Hamiltonians of vari-ous kinds of superconducting qubits have also been wellstudied [3–7] and these Hamiltonians can be numericallydiagonalized to obtain eigenenergies and eigenstates thataccurately reproduce experimental data. On the otherhand, the Hamiltonian of circuits consisting of two ormore components, i.e., qubit-qubit, qubit-oscillator, oroscillator-oscillator systems, are usually treated in sucha way that the Hamiltonian of the individual compo-nents and the coupling among them are separately ob-tained [8–11]. This separate treatment of individual cir-cuit components works reasonably well in most circuits.Even for flux qubit-oscillator circuits in the ultrastrong-coupling [12, 13] regime, where the coupling strength gis around 10% of the oscillator’s frequency ω and thequbit frequency ∆q, or the deep-strong-coupling [14–16]regime, where g is comparable to or larger than ∆q andω, the experimental data can be well fitted by the quan-tum Rabi Hamiltonian [17–19], where a two-level atomand a harmonic oscillator are coupled by a dipole-dipole

∗ fumiki@nict.go.jp

interaction. However, the more rigorous approach basedon the standard quantization procedure has not been ap-plied to such circuits except in a few specific studies [20–22], and the validity of describing a flux qubit-oscillatorcircuit by the quantum Rabi Hamiltonian has not beendemonstrated from this perspective.

In this paper, we apply the standard quantization pro-cedure to a superconducting circuit in which a single-Josephson-junction flux qubit (an rf-SQUID qubit or anequivalent circuit of a fluxonium) and an LC oscillatorare inductively coupled to each other by sharing an in-ductor [Fig. 1(a)], and derive the Hamiltonian of thecircuit. Note that single-Josephson-junction flux qubitshave been experimentally demonstrated using superin-ductors, realized by high-kinetic-inductance supercon-ductors [23, 24], granular aluminum [25], and Josephson-junction arrays [26]. The derived circuit Hamiltonianconsists of terms associated with the LC oscillator, theflux qubit (and its higher energy levels), and the prod-uct of the two flux operators. Excluding the qubit’s en-ergy levels higher than the first excited state, this circuitHamiltonian takes the form of the quantum Rabi Hamil-tonian. To investigate contributions from the qubit’shigher energy levels, we numerically calculate the tran-sition frequencies of the circuit Hamiltonian. We findthat the qubit’s higher energy levels mainly cause a neg-ative shift of the entire spectrum, and that the calcu-lated transition frequencies are well fitted by the quan-tum Rabi Hamiltonian even when the qubit-oscillator cir-cuit is in the deep-strong-coupling regime. Using thecircuit Hamiltonian, we investigate some of the paradox-ical properties of the quantum Rabi Hamiltonian in thedeep-strong-coupling regime, the non-negligible numberof photons and the nonzero expectation value of the flux

arX

iv:2

008.

0770

8v1

[qu

ant-

ph]

18

Aug

202

0

2

(a)

g

1

2

Φ𝑥𝑥

𝐿𝐿𝑐𝑐

𝐶𝐶𝐿𝐿2 𝐿𝐿1

𝐸𝐸𝐽𝐽 ,𝐶𝐶𝐽𝐽

3

(b)

g

1

𝐿𝐿𝑐𝑐

𝐶𝐶𝐿𝐿1

(c)

2

Φ𝑥𝑥

𝐿𝐿𝑐𝑐

𝐿𝐿2

(d)

g

1

2

Φ𝑥𝑥 𝐿𝐿𝑔𝑔2

𝐶𝐶𝐿𝐿12 𝐿𝐿𝑔𝑔1

𝐸𝐸𝐽𝐽 ,𝐶𝐶𝐽𝐽

g

𝐸𝐸𝐽𝐽 ,𝐶𝐶𝐽𝐽

FIG. 1. Circuit diagrams. (a) A superconducting circuit inwhich a single-Josephson-junction flux qubit and an LC os-cillator are inductively coupled to each other by sharing aninductor. (b) Equivalent circuit of (a) obtained by applyingthe so-called star-delta (Y-∆) transformation to the inductornetwork in circuit (a). (c) The outer loop of circuit (a), whichforms an LC oscillator. (d) The inner loop of circuit (a), whichforms a single-Josephson-junction flux qubit.

in the oscillator in the ground state [27]. Note that al-though the expectation value of the flux in the oscillatoris nonzero, the expectation value of the current is zero,meaning that an unphysical DC current through the in-ductor of an LC oscillator is not predicted.

Following the standard quantization procedure, nodesare assigned to the circuit as shown in Fig. 1(a). Beforederiving the circuit Hamiltonian, the circuit in Fig. 1(a) istransformed to the one shown in Fig. 1(b) by applying theso-called star-delta (Y-∆) transformation to the inductornetwork. Then, node 3 surrounded by the inductors canbe removed. The inductances of the new set of inductorsare given as Lg1 = (LcL1 + LcL2 + L1L2)/L2, Lg2 =(LcL1 + LcL2 + L1L2)/L1, and L12 = (LcL1 + LcL2 +L1L2)/Lc.

The Lagrangian of the circuit can now be obtainedrelatively easily:

Lcirc =C

2˙Φ2

1 +CJ

2˙Φ2

2 + EJ cos

(2π

Φ2 − Φx

Φ0

)

− Φ21

2Lg1− Φ2

2

2Lg2− (Φ2 − Φ1)2

2L12, (1)

where CJ and EJ = IcΦ0/(2π) are the capacitance andthe Josephson energy of the Josephson junction, Ic is thecritical current of the Josephson junction, Φ0 = h/(2e)

is the superconducting flux quantum, and Φk and˙Φk

(k = 1, 2) are the node flux and its time derivative fornode k. The node charges are defined as the conjugate

momenta of the node fluxes as qk = ∂Lcirc/∂˙Φk. Af-

ter the Legendre transformation, the Hamiltonian is ob-

tained as

Hcirc =q212C

+q22

2CJ− EJ cos

(2π

Φ2 − Φx

Φ0

)

+Φ2

1

2LLC+

Φ22

2LFQ− Φ1Φ2

L12(2)

= H1 + H2 + H12,

where

H1 =q212C

+Φ2

1

2LLC, (3)

H2 =q22

2CJ− EJ cos

(2π

Φ2 − Φx

Φ0

)+

Φ22

2LFQ, (4)

H12 = − Φ1Φ2

L12, (5)

L−1LC = L−1g1 + L−112 , and L−1FQ = L−1g2 + L−112 .

As can be seen from Eq. (2), the Hamiltonian Hcirc

can be separated into three parts: the first part H1

consisting of the charge and flux operators of node 1,the second part H2 consisting of node 2 operators, andthe third part H12 containing the product of the twoflux operators. Let us consider a separate treatment ofthe circuit shown in Fig. 1(a), where the circuit is as-sumed to be naively divided into the two coupled com-ponents. The capacitor and the inductors in the outerloop of the circuit in Fig. 1(a) form an LC oscillator[Fig. 1(c)]. The Josephson junction and the inductorsin the inner loop form a single-Josephson-junction fluxqubit [Fig. 1(d)]. The LC oscillator and the single-Josephson-junction flux qubit share the inductor Lc atthe common part of the two loops. It is instructive toinvestigate the relation of the following pairs of Hamil-tonians: H1 and the Hamiltonian of the LC oscillatorshown in Fig. 1(c), H2 and the Hamiltonian of the flux

qubit shown in Fig. 1(d), and H12 and the Hamiltonianof the inductive coupling between the LC oscillator andthe flux qubit MILC Iq = −Lc/[(Lc+L1)(Lc+L2)]Φ1Φ2,where we have used the relations of the oscillator currentILC = Φ1/(Lc+L1), the qubit current Iq = Φ2/(Lc+L2),and the mutual inductance M = Lc. Actually, only theinductances are different in the Hamiltonians of the twopictures: The inductances of the LC oscillator, the fluxqubit, and the coupling Hamiltonians derived from theseparate treatment are respectively Lc + L1, Lc + L2,and −(Lc + L1)(Lc + L2)/Lc, while those in Hcirc areLLC , LFQ, and −L12. Figure 2 shows the inductances

in the separate treatment and in Hcirc as functions of Lc

on condition that the inductance sums are kept constantat Lc + L1 = 800 pH and Lc + L2 = 2050 pH. As Lc

approaches 0, Lc + L1 → LLC , Lc + L2 → LFQ, and(Lc +L1)(Lc +L2)/Lc → L12. In this way, the Hamilto-nian derived from the separate treatment has the same

3

0

500

L LC (p

H)

Lc+ L1=800 pH

(a)

circ separate

0 200 400Lc (pH)

0

1000

2000

L FQ

(pH

)

Lc+ L2=2050 pH

(b)

0 200 400Lc (pH)

0.0

0.2

1/L 1

2 (1

/nH

) (c)

FIG. 2. (a), (b), (c) The inductances of the LC oscillator

LLC , the flux qubit LFQ, and the inverse inductance of H12,1/L12, (solid lines) are plotted as functions of Lc together withtheir counterparts in the separate treatment (dotted lines)on condition that the inductance sums are kept constant atLc + L1 = 800 pH and Lc + L2 = 2050 pH.

form as Hcirc, and the inductances in the separate treat-ment approach those of Hcirc in the case Lc � L1, L2.

Next, we compare the Hamiltonian Hcirc with the gen-eralized quantum Rabi Hamiltonian:

HR/~ = ω

(a†a+

1

2

)− 1

2(εσx + ∆qσz)

+gσx(a+ a†) (6)

=(HLC + HFQ + Hcoup

)/~.

The first part HLC represents the energy of the LC os-cillator, where a† and a are the creation and annihilationoperators, respectively. The second part HFQ representsthe energy of the flux qubit written in the energy eigenba-sis at ε = 0. The operators σx,z are the standard Paulioperators. The parameters ~∆q and ~ε are the tunnelsplitting and the energy bias between the two states withpersistent currents flowing in opposite directions aroundthe qubit loop. The third part Hcoup represents the cou-pling energy.

The relation between H1 and HLC is straightforward.The resonance frequency and the operators in HLC canbe analytically described by the variables and operatorsin H1 as ω = 1/

√LLCC and a + a† → Φ1/(LLCIzpf ),

where Izpf =√~ω/(2LLC) is the zero-point-fluctuation

current. To see the relation between H2 and HFQ, we

numerically calculated the eigenenergies of H2 as func-tions of Φx. In the calculation, we use the followingparameters: Lc + L2 = 2050 pH, EJ/h = 165.1 GHz(LJ = 990 pH), and CJ = 4.84 fF (EC/h = 4.0 GHz).As shown in Fig. 3(a) the lowest two energy levels are iso-lated from the higher levels, which are more than 40 GHz

0.4975 0.5000 0.5025Φx/Φ0

120

130

140

150

160

170

180

ωi/(2π

) (G

Hz)

(a)

120

122

124

ωi/(2π

) (G

Hz)

(b)

|g⟩ |e⟩

0.4975 0.5000 0.5025Φx/Φ0

−1

0

12π⟨ Φ

2⟩/Φ

0

(c)

FIG. 3. (a) Numerically calculated energy levels of H2 as

functions of Φx. (b) The lowest two energy levels of H2.

(c) The expectation values of the flux operator⟨g∣∣∣Φ2

∣∣∣ g⟩and

⟨e∣∣∣Φ2

∣∣∣ e⟩. The solid circles are obtained from numerical

calculations of H2, while the lines are obtained from fitting byHFQ. The black and red colors respectively indicate states |g〉and |e〉. In the calculation, we use the following parameters:Lc + L2 = 2050 pH, EJ/h = 165.1 GHz (LJ = 990 pH), andCJ = 4.84 fF (EC/h = 4.0 GHz).

higher in frequency. The lowest two energy levels of H2

are well fitted by HFQ [Fig. 3(b)]. Besides the offset inthe y axis, the fitting parameters are ∆q and the maxi-mum persistent current Ip, which is determined as theproportionality constant between the energy bias andthe flux bias, ε = 2Ip(Φx − 0.5Φ0). We also numeri-cally calculated the expectation values of the flux oper-

ator,⟨g∣∣∣Φ2

∣∣∣ g⟩ and⟨e∣∣∣Φ2

∣∣∣ e⟩, which are well fitted by

〈g |σx| g〉 and 〈e |σx| e〉, respectively [Fig. 3(c)]. Here, |g〉and |e〉 are respectively the ground and excited states ofthe qubit. The only fitting parameter is determined by

the ratio Φ2max = −⟨i∣∣∣Φ2

∣∣∣ i⟩ / 〈i |σx| i〉 (i = g, e). Now,

the Pauli operator σx is identified as being proportionalto the flux operator σx → −Φ2/Φ2max. The relation be-

tween H12 and Hcoup can now be obtained by using therelations for the oscillator and qubit operators identifiedabove. This way we find that the Hamiltonian H12 canbe described as −(LLC/L12)IzpfΦ2maxσx(a+ a†), which

is exactly the same form as Hcoup, with the couplingstrength ~g = −(LLC/L12)IzpfΦ2max. Note that Hcoup

is derived in the flux gauge, which is optimal for oursystem with the single oscillator mode [28–30]. Exclud-ing the qubit’s energy levels higher than the first excitedstates, Hcirc takes the form of HR. In other words, oncethe circuit parameters, i.e. Φx, Lc, L1, L2, C, EJ , andCJ , are given, the corresponding parameters in HR (ω,

ε, ∆q, and g) are obtained. The relation between Hcirc

and HR is summarized in Table I.

4

HR Hcirc

HLC H1

HFQ H2

Hcoup H12

a+ a† Φ1/(LLCIzpf )

(a− a†)/i q1/(CVzpf )

σx −Φ2/Φ2max

σz —

ω 1/√LLCC

ε 2Ip(Φx − 0.5Φ0)

∆q ∆q

g (LLC/L12)IzpfΦ2max/~

TABLE I. Hamiltonians, operators, and variables in HR

and their counterparts in Hcirc. Izpf =√

~ω/(2LLC) and

Vzpf =√

~ω/(2C) are the zero-point-fluctuation current andvoltage, respectively. The parameters Φ2max, Ip, and ∆q are

obtained by numerically calculating the eigenenergies of H2

as functions of Φx and fitting the lowest two energy levels byHFQ. Note that there is no analytic expression of σz and ∆q.

As previously mentioned, HR considers only the low-est two energy levels of the flux qubit, while Hcirc in-cludes all energy levels. To investigate the effect of thequbit’s higher energy levels, we perform numerical cal-culations and compare the energy levels calculated byHR and Hcirc. The details of the numerical diagonal-ization of Hcirc are given in Appendix A. Since the con-tributions from the qubit’s higher energy levels are ex-pected to become larger as the coupling strength in-creases, the numerical calculations cover a wide rangeof coupling strengths from the weak-coupling to thedeep-strong-coupling regime. In the calculation, we fixLc + L1 = 800 pH, Lc + L2 = 2050 pH, C = 0.87 pF,EJ/h = 165.1 GHz (LJ = 990 pH), and CJ = 4.84 fF(EC/h = 4.0 GHz), and sweep the flux bias Φx aroundΦ0/2 at various values of Lc. Some of our calculationswere performed using the QuTiP simulation package [31].

Transition frequencies of the qubit-oscillator circuit ωij

corresponding to the transition |i〉 → |j〉 numerically cal-

culated from Hcirc around the resonance frequency ofthe oscillator ω are plotted in Fig. 4 for (a) Lc = 20 pHand (b) Lc = 350 pH, where the indices i and j label theenergy eigenstates according to their order in the energy-level ladder, with the index 0 denoting the ground state.In the case Lc = 20 pH, the spectrum indicates that thequbit and the oscillator are strongly coupled. AroundΦx/Φ0 = 0.5, a dispersive shift of the oscillator frequencyis observed. Note that transitions |0〉 → |2〉 and |1〉 → |3〉around Φx/Φ0 = 0.5 respectively correspond to transi-tions |g0〉 → |g1〉 and |e0〉 → |e1〉, where “g” and “e”denote, respectively, the ground and excited states of thequbit, and “0” and “1” the number of photons in the os-cillator’s Fock state. Avoided level crossings between thequbit and oscillator transition signals are also observed

6.00

6.05

6.10

ωij/(2π

) (G

Hz) (a)

|0⟩ → |1⟩|0⟩ → |2⟩

|0⟩ → |3⟩|1⟩ → |2⟩

|1⟩ → |3⟩

5.86.06.26.46.6 (b)

0.495 0.500 0.505Φx/Φ0

6.00

6.05

6.10

ωij/(2π

) (G

Hz) (c)

0.495 0.500 0.505Φx/Φ0

5.8

6.0

6.2

6.4 (d)

FIG. 4. Transition frequencies of the qubit-oscillator cir-cuit for (a), (c) Lc = 20 pH and (b), (d) Lc = 350 pH.(a), (b) Numerically calculated transition frequencies from

Hcirc are plotted as circles, while the transition frequen-cies of HR are plotted as lines. The parameters of HR ob-tained from Hcirc by mapping the lowest two levels of H2 tothe qubit are ω/(2π) = 6.033 GHz, ∆q/(2π) = 1.240 GHz,g/(2π) = 0.424 GHz, and Ip = 281.3 nA for Lc = 20 pHand ω/(2π) = 6.272 GHz, ∆q/(2π) = 2.139 GHz, g/(2π) =7.338 GHz, and Ip = 282.5 nA for Lc = 350 pH. (c),

(d) Numerically calculated transition frequencies from Hcirc

are plotted as circles, while the results of the fitting by HR areplotted as lines. The parameters of HR obtained by fitting thespectra of Hcirc to HR are ω/(2π) = 6.033 GHz, ∆q/(2π) =1.240 GHz, g/(2π) = 0.430 GHz, and Ip = 281.3 nA forLc = 20 pH and ω/(2π) = 6.064 GHz, ∆q/(2π) = 2.388 GHz,g/(2π) = 7.822 GHz, and Ip = 282.9 nA for Lc = 350 pH.Black, gray, orange, magenta, and red colors indicate transi-tions |0〉 → |1〉, |0〉 → |2〉, |0〉 → |3〉, |1〉 → |2〉, and |1〉 → |3〉,respectively.

approximately at Φx/Φ0 = 0.497 and 0.503. In the caseLc = 350 pH, the characteristic spectrum indicates thatthe qubit-oscillator circuit is in the deep-strong-couplingregime [15]. Transition frequencies of HR are also plot-ted in Figs. 4(a) and (b). It should be mentioned that

the parameters of HR are obtained in two different ways.Here, the parameters of HR are obtained from Hcirc bymapping the lowest two levels of H2 to the qubit. Theoverall shapes of the spectra of HR and Hcirc look simi-lar. On the other hand, the shift of the entire spectrumbecomes as large as more than 200 MHz for Lc = 350 pH.To quantify the difference of the spectra between HR andHcirc, transition frequencies up to the third excited statenumerically calculated from Hcirc are fitted by HR. Fig-ures 4(c) and (d) show that the same spectra of Hcirc

in Figs. 4(a) and (b) are well fitted by HR, but with adifferent parameter set.

Figure 5 shows parameters of HR obtained from Hcirc

by mapping the lowest two energy levels of H2 to thequbit and by fitting the spectra of Hcirc up to the third

5

0 200 400

6.2

6.4

ω/(2

π) (G

Hz) (a)

lowest two levels all levels

0 200 4000

5

10

Δ q/(2

π) (G

Hz) (b)

0 200 4000

5

10

g/(2π)

(GH

z) (c)

0 200 400Lc (pH)

278280282284286

I p (n

A)

(d)

0 200 400Lc (pH)

0

10

20

[δωij/(2π

)]2 (M

Hz2

)

(e)

FIG. 5. Parameters of HR obtained from Hcirc by mappingthe lowest two energy levels of H2 to the qubit (red solid lines)

and by fitting the spectra of Hcirc up to the third excited stateto HR (black dashed lines) as a function of Lc. Panels (a)-(d)respectively correspond to the oscillator frequency ω, qubitfrequency ∆q, coupling strength g, and persistent current Ipin HR. (e) The average of the squares of the residuals in the

least-squares method of obtaining the parameters of HR byfitting, [δω0i/(2π)]2 (i = 1, 2, and 3).

excited state to HR as a function of Lc. The parametersof HR obtained in these two different ways become quitedifferent from each other at large values of Lc due tocontributions of the qubit’s higher energy levels. On theother hand, the average of the squares of the residuals inthe least-squares method of obtaining the parameters ofHR by fitting, [δω0i/(2π)]2 (i = 1, 2, and 3), remains atthe moderate level, at most 25 MHz2, which is consistentwith the good fitting shown in Figs. 4(c) and (d). In

fact, the energy level structure of Hcirc up to the seventhexcited state can still be fitted well by the quantum RabiHamiltonian as shown in Appendix B.

One of the most paradoxical features of HR in thedeep-strong-coupling regime is the non-negligible expec-tation number of photons in the oscillator in the groundstate. In the case of the circuit shown in Fig. 1(a), theexpectation number of photons can be investigated usingthe operator H1/(~ω) − 0.5 in Hcirc, which corresponds

to a†a in HR. As shown in Fig. 6(a), a non-negligible ex-pectation number of photons in the oscillator is obtained.Another paradoxical feature of HR in the deep-strong-coupling regime is a non-negligible expectation value ofthe field operator

⟨a+ a†

⟩when the qubit flux bias is

Φx 6= 0.5Φ0. The expectation value of the field operatorcan also be investigated using the operator Φ1/(LLCIzpf )

0

1

2

⟨pho

ton

num

ber⟩ (a)

−1

0

1

2π⟨ Φ

k⟩/Φ

0 (b)

1 2

0 200 400Lc (pH)

0.0

0.2

0.4

2π⟨ Φ

1⟩/Φ

0 (c)

0.499 0.500 0.501Φx/Φ0

−200

0

200

⟨ I k⟩

(nA)

(d)

FIG. 6. (a) Expectation numbers of photons in the os-cillator in the ground state as a function of Lc. The solid

curve corresponds to⟨

0∣∣∣H1

∣∣∣ 0⟩ /(~ω)−0.5, while the dashed

line indicates the simple expression (g/ω)2, which is true inthe limit ∆q � ω. (b) [(d)] Expectation values of fluxes

2π⟨

Φk

⟩/Φ0 [Expectation values of currents

⟨Ik⟩

] as func-

tions of the flux bias Φx in the case Lc = 350 pH. Black andred colors respectively indicate k = 1 and 2, and solid anddashed lines indicate the ground and the excited states, re-

spectively. (c) Expectation value of flux 2π⟨

Φ1

⟩/Φ0 in the

ground state as a function of Lc at Φx/Φ0 = 0.498.

in Hcirc, which corresponds to a + a† in HR. Fig. 6(b)

shows the expectation value of fluxes,⟨

Φ1

⟩and

⟨Φ2

⟩, as

functions of the flux bias Φx for the ground and the firstexcited states in the case Lc = 350 pH. At Φx/Φ0 6= 0.5,

a nonzero expectation value⟨

Φ1

⟩is demonstrated. One

may suspect that a nonzero⟨

Φ1

⟩is unphysical because

it might result in a nonzero DC current through the in-ductor of an LC oscillator in the energy eigenstate. Therelation between the flux and current operators is explic-itly given in the circuit model:(

Lc + L1 Lc

Lc Lc + L2

)(I1I2

)=

(Φ1

Φ2

), (7)

where Ik (k = 1, 2) is defined as the operator of thecurrent flowing from the ground node to node k. Notethat in the case Lc = 0, I1 = Φ1/L1 and I2 = Φ2/L2. Asshown in Fig. 6(c), at Φx/Φ0 = 0.498, the expectation

value 2π⟨

Φ1

⟩/Φ0 in the ground state linearly increases

as Lc, and is zero when Lc = 0. Fig. 6(d) shows the

expectation values of currents⟨I1

⟩and

⟨I2

⟩as functions

of Φx in the case Lc = 350 pH. The expectation values⟨I2

⟩are nonzero at Φx/Φ0 6= 0.5, while

⟨I1

⟩are exactly

zero at all values of Φx. In this way, although⟨

Φ1

⟩are

6

nonzero in the case Lc 6= 0, an unphysical DC currentthrough the inductor of an LC oscillator is not predicted.

In conclusion, we have derived the Hamiltonian of a su-perconducting circuit that comprises a single-Josephson-junction flux qubit and an LC oscillator using the stan-dard quantization procedure. Excluding the qubit’shigher energy levels, the derived circuit Hamiltoniantakes the form of the quantum Rabi Hamiltonian. Thequbit’s higher energy levels mainly cause a negative shiftof the entire spectrum, but the energy level structure canstill be fitted well by the quantum Rabi Hamiltonian evenwhen the qubit-oscillator circuit is in the deep-strong-coupling regime. We found that some of the paradoxicalproperties of the quantum Rabi Hamiltonian, the non-negligible expectation number of photons and expecta-tion value of the flux operator in the oscillator in theground state, also arise from the circuit Hamiltonian.

ACKNOWLEDGMENTS

We are grateful to M. Devoret for valuable discussions.This work was supported by Japan Society for the Pro-motion of Science (JSPS) Grants-in-Aid for Scientific Re-search (KAKENHI) (No. JP19H01831), Japan Scienceand Technology Agency (JST) Precursory Research forEmbryonic Science and Technology (PRESTO) (GrantNo. JPMJPR1767), and JST Core Research for Evo-lutionary Science and Technology (CREST) (Grant No.JPMJCR1775).

Appendix A: Numerical calculation method

As a simple example, let us consider H1 in the maintext:

H1 =q212C

+Φ2

1

2LLC

= 4EC n2 +

1

2ELφ

2, (A1)

where, EC = e2/(2C), EL = [Φ0/(2π)]2/LLC , and φand n are operators of dimensionless magnetic flux and

charge, respectively, and satisfy [φ, n] = i. Using the

relation φ = −(1/i)∂/∂n, the Hamiltonian is rewrittenas

H1 = 4EC n2 − 1

2EL

∂2

∂n2. (A2)

Let us calculate wavefunctions ψ(n) and their eigenener-gies E of this Hamiltonian. We expand the wavefunctionwith the plane waves as

ψ(n) =∑k

ψkeikn√2nmax

. (A3)

Here, 2nmax is the length of the n-space. Consideringthe periodic boundary condition ψ(−nmax) = ψ(nmax),the wave number k is given by

k =2π

2nmaxη, (η = 0,±1,±2, . . . ). (A4)

Then, the equation for determining ψk and E is obtainedas

H1ψ(n) = Eψ(n)∑k′

[4ECn

2 +1

2ELk

′2]ψk′

eik′n

2nmax= E

∑k′

ψk′eik

′n

2nmax∫ nmax

−nmax

dne−ikn√2nmax

∑k′

[4ECn

2 +1

2ELk

′2]ψk′

eik′n

2nmax=

∫ nmax

−nmax

dne−ikn√2nmax

E∑k′

ψk′eik

′n

2nmax∑k′

[4ECfk−k′(n2) + δk,k′

1

2ELk

2

]ψk′ = Eψk, (A5)

where

fk−k′(n2) ≡ 1

2nmax

∫ nmax

−nmax

dne−i(k−k′)nn2. (A6)

The set of wavefunctions and eigenenergies are obtained

by solving Eq. (A5). Numerically, we can get fk−k′(n2)by the fast Fourier transform (FFT) for discretized n-space. After solving the eigenvalue equation in Eq. (A5),the wavefunctions ψ(n) are also obtained from ψk by the

FFT. For the calculation of H2 in the main text, we usethe following equation instead of Eq. (A5),

∑k′

{4ECJfk−k′(n2) + δk,k′

[−EJ cos (k − kx) +

1

2ELFQk

2

]}ψk′ = Eψk, (A7)

7

0.495 0.500 0.505Φx/Φ0

0

5

10

15

20

ω0i/(2

π) (G

Hz)

(a)

15

26

37

4

0.495 0.500 0.505Φx/Φ0

5.8

6.0

6.2

6.4 (d)

11.8

12.0

12.2

12.4 (c)

17.8

18.0

18.2

18.4 (b)

FIG. 7. (a) Transition frequencies of the qubit-oscillator cir-cuit up to the seventh excited state for Lc = 350 pH. Numer-ically calculated transition frequencies from Hcirc are plottedas circles, while the results of the fitting by HR are plotted aslines. The parameters of HR obtained by fitting the spectra ofHcirc to HR are ω/(2π) = 6.054 GHz, ∆q/(2π) = 2.133 GHz,g/(2π) = 7.562 GHz, and Ip = 282.2 nA. Red, green, blue,cyan, magenta, yellow, and black colors indicate transitionfrequencies ω01, ω02, ω03, ω04, ω05, ω06, and ω07, respectively.(b), (c), (d) The same spectra with smaller range of frequencyaround 3ω, 2ω, and ω.

where ECJ = e2/(2CJ), ELFQ = [Φ0/(2π)]2/LFQ, andkx = 2πΦx/Φ0.

Appendix B: Fitting of Hcirc up to the seventhexcited state

Transition frequencies of the qubit-oscillator circuit nu-merically calculated from Hcirc up to the seventh excitedstate are plotted in Figure 7 for Lc = 350 pH. The calcu-lated spectra can still be fitted well by HR. The averageof the squares of the residuals in the least-squares methodof obtaining the parameters ofHR by fitting, [δω0i/(2π)]2

(i = 1,2,3,4,5,6, and 7), is 152 MHz2. This value is al-most 10 times larger than the case of the fitting up tothe third excited state, but still within a moderate levelconsidering that the largest transition frequency is morethan 20 GHz.

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