text: Fig. 14. Overlay of SABOCA contours on a 5 GHz image of the Hii region obtained from ATCA data by Mücke et al. (2002). The blue colored image regions correspond to the largest 5 GHz ﬂuxes, while the yellow color denotes the faintest 5 GHz emission. The contour levels are at 87%, 73%, 60%, 46%, 33%, 19%, and 06% of the maximum. The ATCA beam is 2", the SABOCA beam is 7.8" and is shown in the lower left corner. the mid-infrared. The structure of MM2 as seen in the CS(2–1) line at 3 mm by ATCA is characterized by ﬁlaments at diﬀerent systemic velocities. The CO mapping data do not show conclu- sive evidence for a high velocity outﬂow, certainly not for a very Fig. 15. ATCA CS (2–1) contour overlay on an HST H-α image massive one. However, the presence of SiO emission, with a hint (Brandner et al. 2000). of line wing emission, is indicative of weak outﬂow activity. Acknowledgements. This research has made use of the SIMBAD database, oper- ated at CDS, Strasbourg, France. 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W., & Turner, N. J. 2015, ApJ, 800, 86 Lebouteiller, V., Bernard-Salas, J., Brandl, B., et al. 2008, ApJ, 680, 398 Article number, page 10 of 10 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: Fig.

text: Overlay

text: of

text: SABOCA

text: contours

text: on

text: a

text: GHz

text: image

text: of

text: the

text: Hii

font: SQSUAE+rtxsc

text: region

text: obtained

text: from

text: ATCA

text: data

text: by

text: Mücke

text: et

text: al.

text: The

text: blue

text: colored

text: image

text: regions

text: correspond

text: to

text: the

text: largest

text: GHz

text: ﬂuxes,

text: while

text: the

text: yellow

text: color

text: denotes

text: the

text: faintest

text: GHz

text: emission.

text: The

text: contour

text: levels

text: are

text: at

text: and

text: of

text: the

text: maximum.

text: The

text: ATCA

text: beam

text: is

text: the

text: SABOCA

text: beam

text: is

text: and

text: is

text: shown

text: in

text: the

text: lower

text: left

text: corner.

text: the

text: mid-infrared.

text: The

text: structure

text: of

text: as

text: seen

text: in

text: the

text: line

text: at

text: mm

text: by

text: ATCA

text: is

text: characterized

text: by

text: ﬁlaments

text: at

text: diﬀerent

text: systemic

text: velocities.

text: The

text: CO

text: mapping

text: data

text: do

text: not

text: show

text: conclu-

text: sive

text: evidence

text: for

text: a

text: high

text: velocity

text: outﬂow,

text: certainly

text: not

text: for

text: a

text: very

text: Fig.

text: ATCA

text: CS

text: contour

text: overlay

text: on

text: an

text: HST

text: H-α

text: image

text: massive

text: one.

text: However,

text: the

text: presence

text: of

text: SiO

text: emission,

text: with

text: a

text: hint

text: (Brandner

text: et

text: al.

text: of

text: line

text: wing

text: emission,

text: is

text: indicative

text: of

text: weak

text: outﬂow

text: activity.

text: Acknowledgements.

text: This

text: research

text: has

text: made

text: use

text: of

text: the

text: SIMBAD

text: database,

text: oper-

text: ated

text: at

text: CDS,

text: Strasbourg,

text: France.

text: We

text: thank

text: the

text: anonymous

text: referee

text: for

text: comments

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text: I.,

text: Richardson,

text: K.

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text: 12 Γwφ (0) Γwφ (0.5) ×Γκφ − Within the same approximations used previously [in set 1 13.1 0.56 1 ×10 −6 particular, we assume again Eq. (31) to hold], the total set 2 7.99 1.17 1.5 10 6 Hamiltonian H(cid:88)takes the same form as in Eq. (32): − TABLE II. Estimates (in MHz) for the dephasing rates Γw, N 1 φ Eq. (72), and Γκ, Eq. (73) – see text for details. H = H¯ + H + H + H , (80) φ φ µ int V µ=1 − 1 H¯ = 4E¯φp2 Eb cos (φ + ϕ ) + E φ2 , (81) assumption (valid within our model for typical parameter φ C φ J e 2 L values as in Table I) that the collective modes are much 1 H = 4(cid:88)E p2 + Eaξ2 , (82) more weakly coupled to the cavity than the qubit is, as in µ C,µ µ 2 J µ Fig. 7; if the assumption is not correct, this could result − N 1 in a dephasing time shorter by several orders of magni- H = g p p , (cid:34) (cid:35) (83) int (cid:88)µ µ φ tudes – see also the end of Sec. . µ=1 VII. BROKEN PT-SYMMETRY: AN EXAMPLE − In all the previous sections we have assumed the sys- HV = − Nµ=11 gµp(cid:34)µ eV 4E1Cc − (8δEc)t2 −(cid:88)2oE¯µCφ (cid:0) (cid:1) (8(cid:35)4) tem to be PT-symmetric, which ensures the decoupling − 2 − (δc)2 (δc)2 E c2 of qubit and odd collective modes in the (approximate) E¯φp eV + o C,µ µ . quadratic Lagrangian. In practice it is diﬃcult to fabri- C φ ECc Et 2N µ µ Ega 2 s4µ cate a perfectly symmetric circuit, so it is interesting to investigate what are the main qualitative consequences Despite the formal similarity, there are important dif- of breaking parity symmetry. To this end, we consider a ferences between Eqs. (32) and (80): ﬁrst, all collective simple case in which the symmetry is broken by taking modes appear in H, not just the even ones; in fact, we re- the two coupling capacitors to have diﬀerent values: mind here that EC,µ is given by either Eq. (25) or Eq. (28) depending on the mode parity. Second, due to the asym- − C0 = C + C δc/2 , CN = C C δc/2 (74) metry the qubit charging energy E¯φ is renormalized from c c t c c t C the deﬁnition in Eq. (30): with C deﬁned in Eq. (12). Therefore in this section C t c represent the average of the two coupling capacitors, and 1 1 − (δc)2 = . (85) we have introduced the dimensionless asymmetry param- E¯φ E˜φ 4Et C C eter Third, the coupling constants g are diﬀerent for even − µ C0 CN and odd modes: δc = c c (75) C t √4 E¯φE c − | | ≤ g = C C,µ µ (o o δc) . (86) with δc 1. Inclusion of the asymmetric capacitive µ 2N Ea s2 µ+1 µ g µ coupling in the circuit Lagrangian amounts to the sub- stitutions [38] The structure of Hamiltonian H in Eq. (80) shows that the main consequence of breaking the parity symmetry G → G − 1 (δc)2 , (76) is the introduction of coupling between qubit and odd 00 00 4E modes with coupling strength linear in the asymmetry t | | ∼ → δc √c o parameter δc. Therefore for strong asymmetry, δc 1, G G + µ µ (77) 0µ 0µ 2Ea 2Ns2 the odd modes inﬂuence the qubit in the sam| e |w∼ay as g µ the even ones. Even for moderate asymmetry, δc 0.1, (78) the eﬀect of the lower-energy odd modes may be non- negligible (at least near zero ﬂux, where the qubit fre- in Eqs. (9)-(10), and (cid:88) quency is closer to those of the collective modes): while δc suppresses the coupling of the odd modes to the qubit, L → L (δc)2 − δc √c o − + φ˙eV µ µ ξ˙ eV (79) the odd modes with index 2ρ 1 are closer in frequency V V 8Et 4Ega µ 2Ns2µ µ to the qubit than the even modes with index 2ρ, and the smaller frequency diﬀerence generally increases the dis- in Eq. (17). Note that matrix G is not aﬀected and, persive shift, see Eq. (39); also, the term proportional µν as discussed in Sec. , a rotation is needed to diagonalize to 1/E¯φ in Eq. (84) roughly compensate for the δc sup- C it in the odd sector. This rotation in principle modiﬁes pression of coupling between odd modes and cavity, thus − the new term introduced in Eq. (77); we neglect this giving similar lifetimes for odd (2ρ 1) and even (2ρ) modiﬁcation as it does not introduce any new qualitative modes. On the other hand, small asymmetry at the per- feature – this is also a quantitatively good approximation cent level, as usually present in nominally symmetric de- if the parameter λ, Eq. (13), is suﬃciently small. vices, implies that the odd modes can be safely neglected. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Γwφ

label: table

text: Γwφ

text: ×Γκφ

text: Within

text: the

text: same

text: approximations

text: used

text: previously

text: [in

text: set

text: −6

text: particular,

text: we

text: assume

text: again

text: Eq.

text: to

text: hold],

text: the

text: total

text: set

text: Hamiltonian

text: the

text: same

text: form

text: as

text: in

text: Eq.

text: TABLE

text: II.

text: Estimates

label: caption

text: (in

text: MHz)

text: for

text: the

text: dephasing

text: rates

text: Γw,

text: N

text: φ

text: Eq.

text: and

text: Γκ,

text: Eq.

text: see

text: text

text: for

text: details.

text: H

text: H¯

text: H

text: H

text: H

text: φ

text: φ

text: µ

text: int

text: V

text: H¯

text: Eb

text: cos

text: (φ

text: ϕ

text: E

text: φ2

text: assumption

text: (valid

text: within

text: our

text: model

text: for

text: typical

text: parameter

text: φ

text: C

text: φ

text: J

text: e

text: L

text: values

text: as

text: in

text: Table

text: I)

text: that

text: the

text: collective

text: modes

text: are

text: much

text: H

text: more

text: weakly

text: coupled

text: to

text: the

text: cavity

text: than

text: the

text: qubit

text: is,

text: as

text: in

text: µ

text: C,µ

text: µ

text: J

text: µ

text: Fig.

text: if

text: the

text: assumption

text: is

text: not

text: correct,

text: this

text: could

text: result

text: N

text: in

text: a

text: dephasing

text: time

text: shorter

text: by

text: several

text: orders

text: of

text: magni-

text: H

text: g

text: p

text: p

text: int

text: µ

text: φ

text: tudes

text: see

text: also

text: the

text: end

text: of

text: Sec.

text: VII.

text: BROKEN

text: PT-SYMMETRY:

text: AN

text: EXAMPLE

text: In

text: all

text: the

text: previous

text: sections

text: we

text: have

text: assumed

text: the

text: sys-

text: HV

text: eV

text: tem

text: to

text: be

text: PT-symmetric,

text: which

text: ensures

text: the

text: decoupling

text: E

text: of

text: qubit

text: and

text: odd

text: collective

text: modes

text: in

text: the

text: (approximate)

text: E¯φp

text: eV

text: o

text: C,µ

text: µ

text: quadratic

text: Lagrangian.

text: In

text: practice

text: it

text: is

text: diﬃcult

text: to

text: fabri-

text: C

text: φ

text: ECc

text: Et

text: µ

text: µ

text: Ega

text: cate

text: a

text: perfectly

text: symmetric

text: circuit,

text: so

text: it

text: is

text: interesting

text: to

text: investigate

text: what

text: are

text: the

text: main

text: qualitative

text: consequences

text: Despite

text: the

text: formal

text: similarity,

text: there

text: are

text: important

text: dif-

text: of

text: breaking

text: parity

text: symmetry.

text: To

text: this

text: end,

text: we

text: consider

text: a

text: ferences

text: between

text: Eqs.

text: and

text: ﬁrst,

text: all

text: collective

text: simple

text: case

text: in

text: which

text: the

text: symmetry

text: is

text: broken

text: by

text: taking

text: modes

text: appear

text: in

text: H,

text: not

text: just

text: the

text: even

text: ones;

text: in

text: fact,

text: we

text: re-

text: the

text: two

text: coupling

text: capacitors

text: to

text: have

text: diﬀerent

text: values:

text: mind

text: here

text: that

text: EC,µ

text: is

text: given

text: by

text: either

text: Eq.

text: or

text: Eq.

text: depending

text: on

text: the

text: mode

text: parity.

text: Second,

text: due

text: to

text: the

text: asym-

text: C

text: C

text: CN

text: C

text: C

text: metry

text: the

text: qubit

text: charging

text: energy

text: E¯φ

text: is

text: renormalized

text: from

text: c

text: c

text: t

text: c

text: c

text: t

text: C

text: the

text: deﬁnition

text: in

text: Eq.

text: with

text: C

text: deﬁned

text: in

text: Eq.

text: Therefore

text: in

text: this

text: section

text: C

text: t

text: c

text: represent

text: the

text: average

text: of

text: the

text: two

text: coupling

text: capacitors,

text: and

text: we

text: have

text: introduced

text: the

text: dimensionless

text: asymmetry

text: param-

text: E¯φ

text: E˜φ

text: C

text: C

text: eter

text: Third,

text: the

text: coupling

text: constants

text: g

text: are

text: diﬀerent

text: for

text: even

text: µ

text: CN

text: and

text: odd

text: modes:

text: δc

text: c

text: c

text: C

text: t

text: √4

text: E¯φE

text: c

text: g

text: C

text: C,µ

text: µ

text: (o

text: o

text: δc)

text: with

text: δc

text: Inclusion

text: of

text: the

text: asymmetric

text: capacitive

text: µ

text: Ea

text: µ

text: g

text: µ

text: coupling

text: in

text: the

text: circuit

text: Lagrangian

text: amounts

text: to

text: the

text: sub-

text: stitutions

text: The

text: structure

text: of

text: Hamiltonian

text: H

text: in

text: Eq.

text: shows

text: that

text: the

text: main

text: consequence

text: of

text: breaking

text: the

text: parity

text: symmetry

text: G

text: G

text: is

text: the

text: introduction

text: of

text: coupling

text: between

text: qubit

text: and

text: odd

text: modes

text: with

text: coupling

text: strength

text: linear

text: in

text: the

text: asymmetry

text: t

text: δc

text: √c

text: o

text: parameter

text: δc.

text: Therefore

text: for

text: strong

text: asymmetry,

text: δc

text: G

text: G

text: µ

text: µ

text: 0µ

text: 0µ

text: the

text: odd

text: modes

text: inﬂuence

text: the

text: qubit

text: in

text: the

text: sam|

text: e

text: |w∼ay

text: as

text: g

text: µ

text: the

text: even

text: ones.

text: Even

text: for

text: moderate

text: asymmetry,

text: δc

text: the

text: eﬀect

text: of

text: the

text: lower-energy

text: odd

text: modes

text: may

text: be

text: non-

text: negligible

text: (at

text: least

text: near

text: zero

text: ﬂux,

text: where

text: the

text: qubit

text: fre-

text: in

text: Eqs.

text: and

text: quency

text: is

text: closer

text: to

text: those

text: of

text: the

text: collective

text: modes):

text: while

text: δc

text: suppresses

text: the

text: coupling

text: of

text: the

text: odd

text: modes

text: to

text: the

text: qubit,

text: L

text: L

text: δc

text: √c

text: o

text: φ˙eV

text: µ

text: µ

text: ξ˙

text: eV

text: the

text: odd

text: modes

text: with

text: index

text: 2ρ

text: are

text: closer

text: in

text: frequency

text: V

text: V

text: µ

text: µ

text: to

text: the

text: qubit

text: than

text: the

text: even

text: modes

text: with

text: index

text: and

text: the

text: smaller

text: frequency

text: diﬀerence

text: generally

text: increases

text: the

text: dis-

text: in

text: Eq.

text: Note

text: that

text: matrix

text: G

text: is

text: not

text: aﬀected

text: and,

text: persive

text: shift,

text: see

text: Eq.

text: also,

text: the

text: term

text: proportional

text: µν

text: as

text: discussed

text: in

text: Sec.

text: a

text: rotation

text: is

text: needed

text: to

text: diagonalize

text: to

text: in

text: Eq.

text: roughly

text: compensate

text: for

text: the

text: δc

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text: C

text: it

text: in

text: the

text: odd

text: sector.

text: This

text: rotation

text: in

text: principle

text: modiﬁes

text: pression

text: of

text: coupling

text: between

text: odd

text: modes

text: and

text: cavity,

text: thus

text: the

text: new

text: term

text: introduced

text: in

text: Eq.

text: we

text: neglect

text: this

text: giving

text: similar

text: lifetimes

text: for

text: odd

text: and

text: even

text: modiﬁcation

text: as

text: it

text: does

text: not

text: introduce

text: any

text: new

text: qualitative

text: modes.

text: On

text: the

text: other

text: hand,

text: small

text: asymmetry

text: at

text: the

text: per-

text: feature

text: this

text: is

text: also

text: a

text: quantitatively

text: good

text: approximation

text: cent

text: level,

text: as

text: usually

text: present

text: in

text: nominally

text: symmetric

text: de-

text: if

text: the

text: parameter

text: λ,

text: Eq.

text: is

text: suﬃciently

text: small.

text: vices,

text: implies

text: that

text: the

text: odd

text: modes

text: can

text: be

text: safely

text: neglected.

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text: 13 of 14 resonant frequencies in an array of 200 junctions without shunting capacitor (Cb = 0). We can again J compare our calculated frequencies with the measured ones: setting Ca = 98 aF and optimizing the other pa- g rameters, we ﬁnd again diﬀerences of less than 1% except for the third mode, whose measured frequency is about 7% higher than the calculated one; this larger diﬀerence is likely due to the presence near the frequency of that mode of a spurious resonance [28]. VIII. COUPLING INTO THE SUPERINDUCTANCE So far, both for the PT-symmetric and the broken- symmetry cases, we have taken the coupling capacitors to be connected to the two islands separated by the phase- slip junction. The coupling capacitors can be attached to FIG. 12. (Color online) Comparison between collective modes any island in the circuit, and in fact such a setup has been frequencies measured in Ref. 18 (red circles) with those cal- used in more recent experiments [11, 39]. In general, arbi- culated using the present theory (black crosses). The inset trary placement will immediately break parity symmetry shows the ratio between theoretical and experimental frequen- cies. even if the capacitances are the same for both capacitors. Here we consider brieﬂy the simplest case of equal capac- itors placed symmetrically with respect to the phase-slip junction, so that parity symmetry is preserved and no A. Comparison with experiment qubit-odd mode interaction is allowed. Concretely, we ≥ take the ﬁrst capacitor to be connected to island δ 0, Asymmetrically coupled systems similar to that de- where islands 0 and N are the two island surrounding scribed above have been recently probed experimentally, the phase-slip junction; then the second capacitor is con- − which enable us to test in part our theory. For example, nected to island N δ, and the maximum possible δ is (cid:98) − (cid:99) in Ref. 18 an array of 80 junctions was placed in parallel δ = (N 1)/2 . In this conﬁguration, the coupling M L to a (resonator) capacitor and the nine lowest resonant Lagrangian (cid:18)V [cf. Eq(cid:19). (17)] become(cid:114)s (cid:88) frequencies were measured. The system is described by the Hamiltonian H in Eq. (80) if we set Eb = 0. Then H describes a harmonic oscillator linearly coJupled to N − 1 L = − 1 1 − 2δ φ˙ eV + 1 2 Ne s4δρ η˙ eV . oscillators. The resonant frequencies of the correspond- V 4Ec N 4Ec N s ρ C C ρ=1 2ρ ing N independent oscillators can be easily calculated (87) numerically; to compare with experiments, we note that This formula correctly reduces to Eq. (17) for δ = 0, the lowest mode in the experiment correspond to what while for δ > 0 a new coupling between cavity and even we call the qubit mode φ, and therefore the higher even modes is present. indices correspond to our odd modes and viceversa, odd Changing the position of the coupling capacitors also mode indices in the experiments correspond to our even aﬀects the kinetic energy part T of the Lagrangian [cf. modes. G Eq. (8)], and a general treatment of this modiﬁed term is In calculating the resonant frequencies we use as input quite cumbersome. Here we consider the simple limit in param(cid:39)eters the array junction capacitance CJa = 40.1 fF which we neglect the ground capacitances, Ca, Cb → 0. (Ea 483 MHz) and its Josephson inductance L = g g C (cid:39) J In this case, as we show in Appendix(cid:88), the Hamiltonian 1.94 nH (Ea 84.3 GHz), the capacitance to ground J (cid:39) is Ca = 114 aF (Ea 170 GHz), the two coupling capac- g g (cid:39) itor(cid:39)s Cc0 = 5.5 fF and CcN = 3.16 fF (ECc 2.24 GHz, H = Hˆφ + H2 + Hint + HV + Hµ , (88) δc (cid:39)0.27), and the resonator capacitor CJb = 3.19 fF µ(cid:54)=2 (Eb 6.07 GHz). The frequencies calculated with these C Eˆφ − 1 parameters diﬀer by less than 1% from the measured fre- Hˆ = 4 −C p2 Eb cos (φ + ϕ ) + E φ2 , (89) quencies, see Fig. 12. We note that while the array junc- φ 1 g2 φ J e 2 L tion parameters Ca and L agree with those reported in (cid:113)E 1 J J H = 4 −C2 p2 + Eaξ2 , (90) Ref. 18, the ground capacitance Cga we estimate here is 2 1 g2 2 2 J 2 about three times bigger; since the calculations in the previous sections are based on the original estimate of g EˆCφEC2 ≡ H = − p p g p p , (91) Ref. 18, they could underestimate, e.g., the dispersive int 1 g2 2 φ 2 2 φ shifts by almost one order of magnitude. 1 A very recent experiment [28] reports the measurement Hµ = 4ECa p2µ + 2EJaξµ2 , (92) ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: resonant

text: frequencies

text: in

text: an

text: array

text: of

text: junctions

text: without

text: shunting

text: capacitor

text: (Cb

text: We

text: can

text: again

text: J

text: compare

text: our

text: calculated

text: frequencies

text: with

text: the

text: measured

text: ones:

text: setting

text: Ca

text: aF

text: and

text: optimizing

text: the

text: other

text: pa-

text: g

text: rameters,

text: we

text: ﬁnd

text: again

text: diﬀerences

text: of

text: less

text: than

text: except

text: for

text: the

text: third

text: mode,

text: whose

text: measured

text: frequency

text: is

text: about

text: higher

text: than

text: the

text: calculated

text: one;

text: this

text: larger

text: diﬀerence

text: is

text: likely

text: due

text: to

text: the

text: presence

text: near

text: the

text: frequency

text: of

text: that

text: mode

text: of

text: a

text: spurious

text: resonance

text: VIII.

label: section

text: COUPLING

text: INTO

text: THE

text: SUPERINDUCTANCE

text: So

text: far,

text: both

text: for

text: the

text: PT-symmetric

text: and

text: the

text: broken-

text: symmetry

text: cases,

text: we

text: have

text: taken

text: the

text: coupling

text: capacitors

text: to

text: be

text: connected

text: to

text: the

text: two

text: islands

text: separated

text: by

text: the

text: phase-

text: slip

text: junction.

text: The

text: coupling

text: capacitors

text: can

text: be

text: attached

text: to

text: FIG.

text: (Color

text: online)

text: Comparison

text: between

text: collective

text: modes

text: any

text: island

text: in

text: the

text: circuit,

text: and

text: in

text: fact

text: such

text: a

text: setup

text: has

text: been

text: frequencies

text: measured

text: in

text: Ref.

text: (red

text: circles)

text: with

text: those

text: cal-

text: used

text: in

text: more

text: recent

text: experiments

text: In

text: general,

text: arbi-

text: culated

text: using

text: the

text: present

text: theory

text: (black

text: crosses).

text: The

text: inset

text: trary

text: placement

text: will

text: immediately

text: break

text: parity

text: symmetry

text: shows

text: the

text: ratio

text: between

text: theoretical

text: and

text: experimental

text: frequen-

text: cies.

text: even

text: if

text: the

text: capacitances

text: are

text: the

text: same

text: for

text: both

text: capacitors.

text: Here

text: we

text: consider

text: brieﬂy

text: the

text: simplest

text: case

text: of

text: equal

text: capac-

text: itors

text: placed

text: symmetrically

text: with

text: respect

text: to

text: the

text: phase-slip

text: junction,

text: so

text: that

text: parity

text: symmetry

text: is

text: preserved

text: and

text: no

text: A.

text: Comparison

text: with

text: experiment

text: qubit-odd

text: mode

text: interaction

text: is

text: allowed.

text: Concretely,

text: we

text: take

text: the

text: ﬁrst

text: capacitor

text: to

text: be

text: connected

text: to

text: island

text: δ

text: Asymmetrically

text: coupled

text: systems

text: similar

text: to

text: that

text: de-

text: where

text: islands

text: and

text: N

text: are

text: the

text: two

text: island

text: surrounding

text: scribed

text: above

text: have

text: been

text: recently

text: probed

text: experimentally,

text: the

text: phase-slip

text: junction;

text: then

text: the

text: second

text: capacitor

text: is

text: con-

text: which

text: enable

text: us

text: to

text: test

text: in

text: part

text: our

text: theory.

text: For

text: example,

text: nected

text: to

text: island

text: N

text: δ,

text: and

text: the

text: maximum

text: possible

text: δ

text: is

text: in

text: Ref.

text: an

text: array

text: of

text: junctions

text: was

text: placed

text: in

text: parallel

text: δ

text: (N

text: In

text: this

text: conﬁguration,

text: the

text: coupling

text: M

text: L

text: to

text: a

text: (resonator)

text: capacitor

text: and

text: the

text: nine

text: lowest

text: resonant

text: Lagrangian

text: [cf.

text: frequencies

text: were

text: measured.

text: The

text: system

text: is

text: described

text: by

text: the

text: Hamiltonian

text: H

text: in

text: Eq.

text: if

text: we

text: set

text: Eb

text: Then

text: H

text: describes

text: a

text: harmonic

text: oscillator

text: linearly

text: coJupled

text: to

text: N

text: L

text: 2δ

text: φ˙

text: eV

text: Ne

text: s4δρ

text: η˙

text: eV

text: oscillators.

text: The

text: resonant

text: frequencies

text: of

text: the

text: correspond-

text: V

text: N

text: N

text: s

text: ρ

text: C

text: C

text: 2ρ

text: ing

text: N

text: independent

text: oscillators

text: can

text: be

text: easily

text: calculated

text: numerically;

text: to

text: compare

text: with

text: experiments,

text: we

text: note

text: that

text: This

text: formula

text: correctly

text: reduces

text: to

text: Eq.

text: for

text: δ

text: the

text: lowest

text: mode

text: in

text: the

text: experiment

text: correspond

text: to

text: what

text: while

text: for

text: δ

text: a

text: new

text: coupling

text: between

text: cavity

text: and

text: even

text: we

text: call

text: the

text: qubit

text: mode

text: φ,

text: and

text: therefore

text: the

text: higher

text: even

text: modes

text: is

text: present.

text: indices

text: correspond

text: to

text: our

text: odd

text: modes

text: and

text: viceversa,

text: odd

text: Changing

text: the

text: position

text: of

text: the

text: coupling

text: capacitors

text: also

text: mode

text: indices

text: in

text: the

text: experiments

text: correspond

text: to

text: our

text: even

text: aﬀects

text: the

text: kinetic

text: energy

text: part

text: T

text: of

text: the

text: Lagrangian

text: [cf.

text: modes.

text: G

text: Eq.

text: and

text: a

text: general

text: treatment

text: of

text: this

text: modiﬁed

text: term

text: is

text: In

text: calculating

text: the

text: resonant

text: frequencies

text: we

text: use

text: as

text: input

text: quite

text: cumbersome.

text: Here

text: we

text: consider

text: the

text: simple

text: limit

text: in

text: the

text: array

text: junction

text: capacitance

text: CJa

text: fF

text: which

text: we

text: neglect

text: the

text: ground

text: capacitances,

text: Ca,

text: Cb

text: (Ea

text: MHz)

text: and

text: its

text: Josephson

text: inductance

text: L

text: g

text: g

text: C

text: J

text: In

text: this

text: case,

text: as

text: we

text: show

text: in

text: the

text: Hamiltonian

text: nH

text: (Ea

text: GHz),

text: the

text: capacitance

text: to

text: ground

text: J

text: is

text: Ca

text: aF

text: (Ea

text: GHz),

text: the

text: two

text: coupling

text: capac-

text: g

text: g

text: fF

text: and

text: CcN

text: fF

text: (ECc

text: GHz,

text: H

text: Hˆφ

text: Hint

text: HV

text: Hµ

text: δc

text: and

text: the

text: resonator

text: capacitor

text: CJb

text: fF

text: (Eb

text: GHz).

text: The

text: frequencies

text: calculated

text: with

text: these

text: C

text: Eˆφ

text: parameters

text: diﬀer

text: by

text: less

text: than

text: from

text: the

text: measured

text: fre-

text: Hˆ

text: −C

text: Eb

text: cos

text: (φ

text: ϕ

text: E

text: φ2

text: quencies,

text: see

text: Fig.

text: We

text: note

text: that

text: while

text: the

text: array

text: junc-

text: φ

text: φ

text: J

text: e

text: L

text: tion

text: parameters

text: Ca

text: and

text: L

text: agree

text: with

text: those

text: reported

text: in

text: J

text: J

text: H

text: Ref.

text: the

text: ground

text: capacitance

text: Cga

text: we

text: estimate

text: here

text: is

text: J

text: about

text: three

text: times

text: bigger;

text: since

text: the

text: calculations

text: in

text: the

text: previous

text: sections

text: are

text: based

text: on

text: the

text: original

text: estimate

text: of

text: g

text: H

text: p

text: p

text: g

text: p

text: p

text: Ref.

text: they

text: could

text: underestimate,

text: e.g.,

text: the

text: dispersive

text: int

text: φ

text: φ

text: shifts

text: by

text: almost

text: one

text: order

text: of

text: magnitude.

text: A

text: very

text: recent

text: experiment

text: reports

text: the

text: measurement

text: Hµ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 17 (cid:80) where the inductive energy E should be replaced by E˜ where for example the elements T of matrix T are de- L L λ,ρσ λ [Eq. (53)]. For the experiments reported in Refs. 9 and ﬁned via T = T ζ˙ ζ˙ . We can similarly choose λ ρ,σ λ,ρσ ρ σ 11, with N = 43 and N = 95 array junctions respec- S to eliminate the oﬀ-diagonal terms at order λ2. The 2 tively, this procedure leads to the parameters reported remaining diagonal elements are then, up to order λ2, in Table III. We also give there our rough estimate of the ratio Cb/Ca which is based on the geometry of the − λ2 g g T + λT [S , T ] . (C3) devices in the two experiments. d,ρρ λ,ρρ 2 1 λ ρρ The phase-slip junction Josephson energy Eb in Ta- (cid:88) J ble I is taken directly from the experimental estimates The last term is more explicitly written as in Table III. For the coupling capacitor energy Ec , we C T T use for set 1 the value of coupling capacitance given in [S , T ] = 2 λ,ρσ−λ,σρ . (C4) Ref. 9, while for set 2 we take as an example the value re- 1 λ ρρ (cid:54) Td,σσ Td,ρρ σ=ρ ported in Ref. 18. The latter experiment was performed with junctions fabricated with the same procedure used By substituting the matrix elements into Eq. (C3), we (cid:0) (cid:1) to fabricate the ﬂuxonium junctions of Ref. 11; for this arrive at Eq. (25). reason, we also use the value of ground capacitance in Note that the rotated modes ζ˜ are related to the orig- ρ Ref. 18 to estimate Ea for set 2. The value of Ea for inal modes ζ via ζ˜ = eS ζ . For use in Sec. , we g g σ ρ ρσ σ set 1 is smaller than that of set 2 by a factor of 0.4 be- deﬁne the matrixΛ(cid:0)µν (cid:1)that performs the rotation in the cause, according to the supplementary to Ref. 18, the odd sector while leaving the even sector unchanged: diﬀerent fabrication processes lead to such a diﬀerence in capacitances to ground. Values for Egb then follows  eS 2µ−1,2ν−1 µ, ν odd from the geometrically estimated ratio Cgb/Cga. To fur- Λµν = δµν µ, ν even (C5) ther constraint the parameters, since the only diﬀerence 0 otherwise between phase slip and array junctions is in their area, we assume that their plasma frequency is the same, so Appendix D: Even sector Hamiltonian that EaEa = EbEb . We then choose the parameters J C J C L Ea, Ea , and Eb as to obtain, using Eqs. (30) and (53), The Lagrangian in Eq. (27) has a quadratic kinetic J C C e the values of E˜Cφ and E˜L reported in Table III. energy part, so that transforming t×o the Hamiltonian Appendix C: Rotation in the odd sector amounts to inverting the (Ne + 1) (Ne + 1) matrix K = K + K whose diagonal part K has entries d i d The diagonalization of the kinetic energy matrix To 1 K = , (D1) [Eq. (20)] of the odd sector can be seen as a rotation d,00 8E˜φ among the odd-sector coordinates ζ . Such a rotation C ρ 1 can be constructed perturbatively order by order in the K = , ρ = 1, . . . , N , (D2) parameter λ, with a procedure analogous to that used to d,ρρ 8Ee e (cid:80) C,ρ perform a Schrieﬀer-Wolﬀ transformation to an eﬀective and the interaction matrix K has elements Hamiltonian: we want to obtain an antisymmetric matrix i − S = n=1 Snλn such that the the product e SToeS is K = K = − √1 1 c2ρ (D3) diagonal. Expanding this product up to second order we i,0ρ i,ρ0 32 2N Ea s2 have g 2ρ (cid:18) (cid:19) − − and all other elements are zero. Indeed, after performing e ST eS = T + λ (T [S , T ]) o d λ 1 d a standard Legendre transformation the kinetic part H K 1 − − (C1) of the Hamiltonian has the form + λ2 [S , [S , T ]] [S , T ] [S , T ] + . . . 2 1 1 d 2 d 1 λ 1 − − − H = nK 1n nK 1v , (D4) K 2 To eliminate the oﬀ-diagonal terms at order λ, we take (cid:54) S1 to have elements S1,ρρ = 0 and for ρ = σ where T n = (n , n , . . . , n ) , (D5) S = λ−,ρσ , (C2) −φ 1 Ne 1,ρσ T T v = ( eV/4Ec , 0, . . . , 0) (D6) d,σσ d,ρρ C are two N + 1-dimensional vectors. We can formally e N Eb E˜φ E˜ Cb/Ca write the inverse matrix as J C L g g set 1 43 8.93 2.39 0.52 32 − − 1 set 2 95 10.2 3.60 0.46 0.6 K 1 = K 1 − − − d 1 K K 1K K 1 i d i d (D7) TABLE III. Additional ﬂuxonium parameters estimated as − − − 1 K 1K K 1 − − − explained in Appendix . Energies are given in GHz. d i d 1 K K 1K K 1 i d i d ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: the

text: inductive

text: energy

text: E

text: should

text: be

text: replaced

text: by

text: E˜

text: where

text: for

text: example

text: the

text: elements

text: T

text: of

text: matrix

text: T

text: are

text: de-

text: L

text: L

text: λ,ρσ

text: λ

text: [Eq.

text: For

text: the

text: experiments

text: reported

text: in

text: Refs.

text: and

text: ﬁned

text: via

text: T

text: T

text: ζ˙

text: ζ˙

text: We

text: can

text: similarly

text: choose

text: λ

text: ρ,σ

text: λ,ρσ

text: ρ

text: σ

text: with

text: N

text: and

text: N

text: array

text: junctions

text: respec-

text: S

text: to

text: eliminate

text: the

text: oﬀ-diagonal

text: terms

text: at

text: order

text: The

text: tively,

text: this

text: procedure

text: leads

text: to

text: the

text: parameters

text: reported

text: remaining

text: diagonal

text: elements

text: are

text: then,

text: up

text: to

text: order

text: in

text: Table

text: III.

text: We

text: also

text: give

text: there

text: our

text: rough

text: estimate

text: of

text: the

text: ratio

text: Cb/Ca

text: which

text: is

text: based

text: on

text: the

text: geometry

text: of

text: the

text: λ2

text: g

text: g

text: T

text: λT

text: [S

text: T

text: devices

text: in

text: the

text: two

text: experiments.

text: d,ρρ

text: λ,ρρ

text: λ

text: ρρ

text: The

text: phase-slip

text: junction

text: Josephson

text: energy

text: Eb

text: in

text: Ta-

text: J

text: ble

text: I

text: is

text: taken

text: directly

text: from

text: the

text: experimental

text: estimates

text: The

text: last

text: term

text: is

text: more

text: explicitly

text: written

text: as

text: in

text: Table

text: III.

text: For

text: the

text: coupling

text: capacitor

text: energy

text: Ec

text: we

text: C

text: T

text: T

text: use

text: for

text: set

text: the

text: value

text: of

text: coupling

text: capacitance

text: given

text: in

text: [S

text: T

text: λ,ρσ−λ,σρ

text: Ref.

text: while

text: for

text: set

text: we

text: take

text: as

text: an

text: example

text: the

text: value

text: re-

text: λ

text: ρρ

text: Td,σσ

text: Td,ρρ

text: σ=ρ

text: ported

text: in

text: Ref.

text: The

text: latter

text: experiment

text: was

text: performed

text: with

text: junctions

text: fabricated

text: with

text: the

text: same

text: procedure

text: used

text: By

text: substituting

text: the

text: matrix

text: elements

text: into

text: Eq.

text: we

text: to

text: fabricate

text: the

text: ﬂuxonium

text: junctions

text: of

text: Ref.

text: for

text: this

text: arrive

text: at

text: Eq.

text: reason,

text: we

text: also

text: use

text: the

text: value

text: of

text: ground

text: capacitance

text: in

text: Note

text: that

text: the

text: rotated

text: modes

text: ζ˜

text: are

text: related

text: to

text: the

text: orig-

text: ρ

text: Ref.

text: to

text: estimate

text: Ea

text: for

text: set

text: The

text: value

text: of

text: Ea

text: for

text: inal

text: modes

text: ζ

text: via

text: ζ˜

text: eS

text: ζ

text: For

text: use

text: in

text: Sec.

text: we

text: g

text: g

text: σ

text: ρ

text: ρσ

text: σ

text: set

text: is

text: smaller

text: than

text: that

text: of

text: set

text: by

text: a

text: factor

text: of

text: be-

text: deﬁne

text: the

text: performs

text: the

text: rotation

text: in

text: the

text: cause,

text: according

text: to

text: the

text: supplementary

text: to

text: Ref.

text: the

text: odd

text: sector

text: while

text: leaving

text: the

text: even

text: sector

text: unchanged:

text: diﬀerent

text: fabrication

text: processes

text: lead

text: to

text: such

text: a

text: diﬀerence

text: in

text: capacitances

text: to

text: ground.

text: Values

text: for

text: Egb

text: then

text: follows

text: eS

text: µ,

text: ν

text: odd

text: from

text: the

text: geometrically

text: estimated

text: ratio

text: Cgb/Cga.

text: To

text: fur-

text: Λµν

text: δµν

text: µ,

text: ν

text: even

text: ther

text: constraint

text: the

text: parameters,

text: since

text: the

text: only

text: diﬀerence

text: otherwise

text: between

text: phase

text: slip

text: and

text: array

text: junctions

text: is

text: in

text: their

text: area,

text: we

text: assume

text: that

text: their

text: plasma

text: frequency

text: is

text: the

text: same,

text: so

text: Appendix

text: D:

text: Even

text: sector

text: Hamiltonian

text: that

text: EaEa

text: EbEb

text: We

text: then

text: choose

text: the

text: parameters

text: J

text: C

text: J

text: C

text: L

text: Ea,

text: Ea

text: and

text: Eb

text: as

text: to

text: obtain,

text: using

text: Eqs.

text: and

text: The

text: Lagrangian

text: in

text: Eq.

text: has

text: a

text: quadratic

text: kinetic

text: J

text: C

text: C

text: e

text: the

text: values

text: of

text: E˜Cφ

text: and

text: E˜L

text: reported

text: in

text: Table

text: III.

text: energy

text: part,

text: so

text: that

text: transforming

text: t×o

text: the

text: Hamiltonian

text: Appendix

text: C:

text: Rotation

text: in

text: the

text: odd

text: sector

text: amounts

text: to

text: inverting

text: the

text: (Ne

text: (Ne

text: matrix

text: K

text: K

text: K

text: whose

text: diagonal

text: part

text: K

text: has

text: entries

text: d

text: i

text: d

text: The

text: diagonalization

text: of

text: the

text: kinetic

text: energy

text: matrix

text: To

text: K

text: [Eq.

text: of

text: the

text: odd

text: sector

text: can

text: be

text: seen

text: as

text: a

text: rotation

text: 8E˜φ

text: among

text: the

text: odd-sector

text: coordinates

text: ζ

text: Such

text: a

text: rotation

text: C

text: ρ

text: can

text: be

text: constructed

text: perturbatively

text: order

text: by

text: order

text: in

text: the

text: K

text: ρ

text: N

text: parameter

text: λ,

text: with

text: a

text: procedure

text: analogous

text: to

text: that

text: used

text: to

text: d,ρρ

text: e

text: C,ρ

text: perform

text: a

text: Schrieﬀer-Wolﬀ

text: transformation

text: to

text: an

text: eﬀective

text: and

text: the

text: interaction

text: matrix

text: K

text: has

text: elements

text: Hamiltonian:

text: we

text: want

text: to

text: obtain

text: an

text: antisymmetric

text: matrix

text: i

text: S

text: Snλn

text: such

text: that

text: the

text: the

text: product

text: e

text: SToeS

text: is

text: K

text: K

text: √1

text: diagonal.

text: Expanding

text: this

text: product

text: up

text: to

text: second

text: order

text: we

text: Ea

text: have

text: g

text: 2ρ

text: and

text: all

text: other

text: elements

text: are

text: zero.

text: Indeed,

text: after

text: performing

text: e

text: ST

text: eS

text: T

text: λ

text: (T

text: [S

text: T

text: o

text: d

text: λ

text: d

text: a

text: standard

text: Legendre

text: transformation

text: the

text: kinetic

text: part

text: H

text: K

text: of

text: the

text: Hamiltonian

text: has

text: the

text: form

text: λ2

text: [S

text: [S

text: T

text: [S

text: T

text: [S

text: T

text: d

text: d

text: λ

text: H

text: nK

text: nK

text: K

text: To

text: eliminate

text: the

text: oﬀ-diagonal

text: terms

text: at

text: order

text: λ,

text: we

text: take

text: to

text: have

text: elements

text: and

text: for

text: ρ

text: σ

text: where

text: T

text: n

text: (n

text: n

text: n

text: S

text: λ−,ρσ

text: −φ

text: Ne

text: 1,ρσ

text: T

text: T

text: v

text: d,σσ

text: d,ρρ

text: C

text: are

text: two

text: N

text: vectors.

text: We

text: can

text: formally

text: e

text: N

text: Eb

text: E˜φ

text: E˜

text: Cb/Ca

text: write

text: the

text: inverse

text: matrix

text: as

text: J

text: C

text: L

text: g

text: g

text: set

text: set

text: K

text: K

text: d

text: K

text: K

text: K

text: i

text: d

text: i

text: d

text: TABLE

text: III.

text: Additional

text: ﬂuxonium

text: parameters

text: estimated

text: as

text: K

text: K

text: explained

text: in

text: Appendix

text: Energies

text: are

text: given

text: in

text: GHz.

text: d

text: i

text: d

text: K

text: K

text: K

text: i

text: d

text: i

text: d

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text: 5 Here p is the momentum conjugate to η . The above ρ ρ expression for the Hamiltonian is one of the main re- sults of this paper: it contains the leading interaction terms between the qubit mode and the collective modes of the junctions forming the superinductance due to ca- pacitance to ground in the array. The coupling constant g is proportional to the array capacitance to ground and ρ monotonically decreases with ρ, see Fig. 2; therefore the higher collective modes couple more weakly to the qubit. Similarly, since g determines also the coupling of the ρ collective modes to the ac voltage V , the higher modes are more weakly coupled to V than the lower ones; more- over, the low-energy ones are more weakly coupled to V than the qubit mode. As we will see below, this implies a lower decay rate for the collective modes compared to the qubit. FIG. 3. (Color online) Main panel: matrix elements squared |(cid:104) | | (cid:105)| |(cid:104) | | (cid:105)| A. Dispersive shifts 0 p l 2 (ﬁlled circles) and 1 p l 2 (empty circles) at f = φ ≤ φ 0.35 for l 20 calculated using parameter set 1 [see Table I]. To further study the eﬀect on the qubit of the interac- Inset: same as main panel but for parameter set 2. We stress that the overall decrease of the matrix elements with l is valid tion with the collective modes, we perform a Schrieﬀer- for any ﬂux f, not just for the particular value used here as Wolﬀ transformation and project the Hamiltonian H , e an example. Eq. (32), onto the qubit subspace to ﬁnd the eﬀective Hamiltonian [29] (cid:88) ω (f) Ne † this is not possible, so we resort to numerical estimates. H = 10 σz + [ω + χ (f)σz] a a , (37) eﬀ 2 ρ ρ ρ ρ Fast convergence of the sums in the above equation is ρ=1 ensured by the rapid decrease of the matrix elements (cid:113) (cid:104) | | (cid:105) (cid:104) | | (cid:105) 0 p l and 1 p l as l increases, see Fig. 3; further aid- where σz is a Pauli matrix in the qubit subspace, φ φ ing the convergence is the approximately linear increase ∼ of the energy of the states with slope (8E˜φE )1/2. The ωe = 8Ee Ea (38) C L ρ C,ρ J reason for the decay of the matrix elements is the follow- ing: the low-lying states are localized near φ , while the is the harmonic oscillator frequency of the even collec- e † high-energy states are to a good approximation the eigen- tive mode ρ, and a , a the creation and annihilation ρ ρ states of the harmonic oscillator obtained by neglecting operators. As indicated by the presence of the parame- the Josephson term in H (this also explain the linear ter f = ϕ /2π, the qubit frequency ω and the ac Stark φ e 10 increase in their energy). Since these high-energy states shifts χ depend on ﬂux through the loop, and we ne- ρ display oscillations of small magnitude at the center of glect for the moment the coupling to external bias given the potential, the overlap of their derivative with the low- by H . Here with the frequency ω we indicate the en- V 10 lying states is small, and the increase of the number of ergy diﬀerence between the two lowest eigenstates of the oscillations with l causes the decrease of the matrix ele- Hamiltonian H , Eq. (33); below we will discuss a small φ ments. renormalization δω of the qubit frequency due to the 10 interaction with the collective modes. The dispersive shifts calculated via Eq. (39) diverge as The dispersive shifts χρ depend on matrix elemen|ts(cid:105) the energy diﬀerences ωl0 or ωl1 approach one of the col- of charge ope(cid:115)rator pφ th(cid:20)at involve all the eigenstates l lective modes frequencies ωρ. This divergence, however, with energy (cid:15) (l = 0, 1, 2, . . .) of Hamiltonian H [29]: only signals the breakdown of the perturbative calcula- l φ tion near such resonant conditions: the actual dispersive χρ = 21 8EEeJa g(cid:88)ρ2 |(cid:104)0|pφ|1(cid:105)|2 ω22ω−10ω2 sghρ,iftsoisvalilmidiitteyd|oifn|thm(cid:28)eapgnerittuudrbeatbiyvethaeppcroouapclhinigs gcoivnesntabnyt C,ρ 10 ρ the condition χ g [31]. Despite this limitation, ρ ρ + (cid:88) |(cid:104)0|p |l(cid:105)|2 ω−l0 (cid:21) Eq. (39) correctly estimates the dispersive shifts at most l≥2 φ ωl20 ωρ2 (39) ﬂux values. We show in Figs. 4 and 5 the ﬂux depen- dence of dispersive shifts χ and χ , respectively. We see − |(cid:104) | | (cid:105)| ω 1 2 1 p l 2 −l1 that away from resonances the shifts for set 2 are usually l≥2 φ ωl21 ωρ2 larger, as expected due to the larger coupling strengths [cf. Fig. 2]. Moreover, due to the higher energy of the − with ω = (cid:15) (cid:15) . While in the case of the transmon [30] collective mode involved, the shifts χ displays a richer lj l j 2 selection rules and low anharmonicity enable the analyt- resonance structure. We will comment on the eﬀect of ical calculation of the dispersive shift, for the ﬂuxonium these shifts on qubit coherence in Sec. . Next, we con- ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Here

text: p

text: is

text: the

text: momentum

text: conjugate

text: to

text: η

text: The

text: above

text: ρ

text: ρ

text: expression

text: for

text: the

text: Hamiltonian

text: is

text: one

text: of

text: the

text: main

text: re-

text: sults

text: of

text: this

text: paper:

text: it

text: contains

text: the

text: leading

text: interaction

text: terms

text: between

text: the

text: qubit

text: mode

text: and

text: the

text: collective

text: modes

text: of

text: the

text: junctions

text: forming

text: the

text: superinductance

text: due

text: to

text: ca-

text: pacitance

text: to

text: ground

text: in

text: the

text: array.

text: The

text: coupling

text: constant

text: g

text: is

text: proportional

text: to

text: the

text: array

text: capacitance

text: to

text: ground

text: and

text: ρ

text: monotonically

text: decreases

text: with

text: ρ,

text: see

text: Fig.

text: therefore

text: the

text: higher

text: collective

text: modes

text: couple

text: more

text: weakly

text: to

text: the

text: qubit.

text: Similarly,

text: since

text: g

text: determines

text: also

text: the

text: coupling

text: of

text: the

text: ρ

text: collective

text: modes

text: to

text: the

text: ac

text: voltage

text: V

text: the

text: higher

text: modes

text: are

text: more

text: weakly

text: coupled

text: to

text: V

text: than

text: the

text: lower

text: ones;

text: more-

text: over,

text: the

text: low-energy

text: ones

text: are

text: more

text: weakly

text: coupled

text: to

text: V

text: than

text: the

text: qubit

text: mode.

text: As

text: we

text: will

text: see

text: below,

text: this

text: implies

text: a

text: lower

text: decay

text: rate

text: for

text: the

text: collective

text: modes

text: compared

text: to

text: the

text: qubit.

text: FIG.

text: (Color

text: online)

text: Main

text: panel:

text: matrix

text: elements

text: squared

text: A.

text: Dispersive

text: shifts

text: p

text: l

text: (ﬁlled

text: circles)

text: and

text: p

text: l

text: (empty

text: circles)

text: at

text: f

text: φ

text: φ

text: for

text: l

text: calculated

text: using

text: parameter

text: set

text: [see

text: Table

text: I].

text: To

text: further

text: study

text: the

text: eﬀect

text: on

text: the

text: qubit

text: of

text: the

text: interac-

text: Inset:

text: same

text: as

text: main

text: panel

text: but

text: for

text: parameter

text: set

text: We

text: stress

text: that

text: the

text: overall

text: decrease

text: of

text: the

text: matrix

text: elements

text: with

text: l

text: is

text: valid

text: tion

text: with

text: the

text: collective

text: modes,

text: we

text: perform

text: a

text: Schrieﬀer-

text: for

text: any

text: ﬂux

text: f,

text: not

text: just

text: for

text: the

text: particular

text: value

text: used

text: here

text: as

text: Wolﬀ

text: transformation

text: and

text: project

text: the

text: Hamiltonian

text: H

text: e

text: an

text: example.

text: Eq.

text: onto

text: the

text: qubit

text: subspace

text: to

text: ﬁnd

text: the

text: eﬀective

text: Hamiltonian

text: ω

text: (f)

text: Ne

text: this

text: is

text: not

text: possible,

text: so

text: we

text: resort

text: to

text: numerical

text: estimates.

text: H

text: σz

text: [ω

text: χ

text: (f)σz]

text: a

text: a

text: eﬀ

text: ρ

text: ρ

text: ρ

text: ρ

text: Fast

text: convergence

text: of

text: the

text: sums

text: in

text: the

text: above

text: equation

text: is

text: ensured

text: by

text: the

text: rapid

text: decrease

text: of

text: the

text: matrix

text: elements

text: p

text: l

text: and

text: p

text: l

text: as

text: l

text: increases,

text: see

text: Fig.

text: further

text: aid-

text: where

text: σz

text: is

text: a

text: Pauli

text: matrix

text: in

text: the

text: qubit

text: subspace,

text: φ

text: φ

text: ing

text: the

text: convergence

text: is

text: the

text: approximately

text: linear

text: increase

text: of

text: the

text: energy

text: of

text: the

text: states

text: with

text: slope

text: The

text: ωe

text: Ea

text: C

text: L

text: ρ

text: C,ρ

text: J

text: reason

text: for

text: the

text: decay

text: of

text: the

text: matrix

text: elements

text: is

text: the

text: follow-

text: ing:

text: the

text: low-lying

text: states

text: are

text: localized

text: near

text: φ

text: while

text: the

text: is

text: the

text: harmonic

text: oscillator

text: frequency

text: of

text: the

text: even

text: collec-

text: e

text: high-energy

text: states

text: are

text: to

text: a

text: good

text: approximation

text: the

text: eigen-

text: tive

text: mode

text: ρ,

text: and

text: a

text: a

text: the

text: creation

text: and

text: annihilation

text: ρ

text: ρ

text: states

text: of

text: the

text: harmonic

text: oscillator

text: obtained

text: by

text: neglecting

text: operators.

text: As

text: indicated

text: by

text: the

text: presence

text: of

text: the

text: parame-

text: the

text: Josephson

text: term

text: in

text: H

text: (this

text: also

text: explain

text: the

text: linear

text: ter

text: f

text: ϕ

text: the

text: qubit

text: frequency

text: ω

text: and

text: the

text: ac

text: Stark

text: φ

text: e

text: increase

text: in

text: their

text: energy).

text: Since

text: these

text: high-energy

text: states

text: shifts

text: χ

text: depend

text: on

text: ﬂux

text: through

text: the

text: loop,

text: and

text: we

text: ne-

text: ρ

text: display

text: oscillations

text: of

text: small

text: magnitude

text: at

text: the

text: center

text: of

text: glect

text: for

text: the

text: moment

text: the

text: coupling

text: to

text: external

text: bias

text: given

text: the

text: potential,

text: the

text: overlap

text: of

text: their

text: derivative

text: with

text: the

text: low-

text: by

text: H

text: Here

text: with

text: the

text: frequency

text: ω

text: we

text: indicate

text: the

text: en-

text: V

text: lying

text: states

text: is

text: small,

text: and

text: the

text: increase

text: of

text: the

text: number

text: of

text: ergy

text: diﬀerence

text: between

text: the

text: two

text: lowest

text: eigenstates

text: of

text: the

text: oscillations

text: with

text: l

text: causes

text: the

text: decrease

text: of

text: the

text: matrix

text: ele-

text: Hamiltonian

text: H

text: Eq.

text: below

text: we

text: will

text: discuss

text: a

text: small

text: φ

text: ments.

text: renormalization

text: δω

text: of

text: the

text: qubit

text: frequency

text: due

text: to

text: the

text: interaction

text: with

text: the

text: collective

text: modes.

text: The

text: dispersive

text: shifts

text: calculated

text: via

text: Eq.

text: diverge

text: as

text: The

text: dispersive

text: shifts

text: χρ

text: depend

text: on

text: matrix

text: the

text: energy

text: diﬀerences

text: or

text: approach

text: one

text: of

text: the

text: col-

text: of

text: charge

text: pφ

text: involve

text: all

text: the

text: eigenstates

text: l

text: lective

text: modes

text: frequencies

text: ωρ.

text: This

text: divergence,

text: however,

text: with

text: energy

text: (l

text: of

text: Hamiltonian

text: H

text: only

text: signals

text: the

text: breakdown

text: of

text: the

text: perturbative

text: calcula-

text: l

text: φ

text: tion

text: near

text: such

text: resonant

text: conditions:

text: the

text: actual

text: dispersive

text: χρ

text: gcoivnesntabnyt

text: C,ρ

text: ρ

text: the

text: condition

text: χ

text: g

text: Despite

text: this

text: limitation,

text: ρ

text: ρ

text: ω−l0

text: Eq.

text: correctly

text: estimates

text: the

text: dispersive

text: shifts

text: at

text: most

text: φ

text: ωρ2

text: ﬂux

text: values.

text: We

text: show

text: in

text: Figs.

text: and

text: the

text: ﬂux

text: depen-

text: dence

text: of

text: dispersive

text: shifts

text: χ

text: and

text: χ

text: respectively.

text: We

text: see

text: ω

text: p

text: l

text: that

text: away

text: from

text: resonances

text: the

text: shifts

text: for

text: set

text: are

text: usually

text: φ

text: ωρ2

text: larger,

text: as

text: expected

text: due

text: to

text: the

text: larger

text: coupling

text: strengths

text: [cf.

text: Fig.

text: Moreover,

text: due

text: to

text: the

text: higher

text: energy

text: of

text: the

text: with

text: ω

text: While

text: in

text: the

text: case

text: of

text: the

text: transmon

text: collective

text: mode

text: involved,

text: the

text: shifts

text: χ

text: displays

text: a

text: richer

text: lj

text: l

text: j

text: selection

text: rules

text: and

text: low

text: anharmonicity

text: enable

text: the

text: analyt-

text: resonance

text: structure.

text: We

text: will

text: comment

text: on

text: the

text: eﬀect

text: of

text: ical

text: calculation

text: of

text: the

text: dispersive

text: shift,

text: for

text: the

text: ﬂuxonium

text: these

text: shifts

text: on

text: qubit

text: coherence

text: in

text: Sec.

text: Next,

text: we

text: con-

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