Theory of strong down-conversion in multi-mode cavity and circuit QED Nitish Mehta1Cristiano Ciuti2Roman Kuzmin1and Vladimir E. Manucharyan1 3 1Department of Physics University of Maryland College Park MD 20742 USA

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Theory of strong down-conversion in multi-mode cavity and circuit QED

Nitish Mehta,1Cristiano Ciuti,2Roman Kuzmin,1and Vladimir E. Manucharyan1, 3

1Department of Physics, University of Maryland, College Park, MD 20742, USA

2Universit´e Paris Cit´e, CNRS, Mat´eriaux et Ph´enom`enes Quantiques, F-75013 Paris, France

3Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1005 Lausanne, Switzerland

(Dated: October 27, 2022)

We revisit the superstrong coupling regime of multi-mode cavity quantum electrodynamics (QED),

deﬁned to occur when the frequency of vacuum Rabi oscillations between the qubit and the nearest

cavity mode exceeds the cavity’s free spectral range. A novel prediction is made that the cavity’s

linear spectrum, measured in the vanishing power limit, can acquire an intricate ﬁne structure

associated with the qubit-induced cascades of coherent single-photon down-conversion processes.

This many-body eﬀect is hard to capture by a brute-force numerics and it is sensitive to the light-

matter coupling parameters both in the infra-red and the ultra-violet limits. We focused at the

example case of a superconducting ﬂuxonium qubit coupled to a long transmission line section.

The conversion rate in such a circuit QED setup can readily exceed a few MHz, which is plenty

to overcome the usual decoherence processes. Analytical calculations were made possible by an

unconventional gauge choice, in which the qubit circuit interacts with radiation via the ﬂux/charge

variable in the low-/high-frequency limits, respectively. Our prediction of the ﬁne spectral structure

lays the foundation for the “strong down-conversion” regime in quantum optics, in which a single

photon excited in a non-linear medium spontaneously down-converts faster than it is absorbed.

I. INTRODUCTION

Quantum electrodynamics (QED) is a branch of

physics describing a remarkable list of fundamental phe-

nomena produced by the quantum nature of electromag-

netic ﬁelds [1–3]. In cavity QED, the electromagnetic

modes are spatially conﬁned and the corresponding vac-

uum ﬁelds can be dramatically enhanced [4]. A cele-

brated manifestation of cavity QED is the strong cou-

pling regime, in which an atom (or a qubit) and a cav-

ity mode coherently exchange a single excitation – un-

dergoing the vacuum Rabi oscillations – faster than the

decoherence rate in either system. This regime has en-

abled one to control qubits with radiation and to control

radiation with qubits, and among other directions it in-

ﬂuenced the development of circuit QED and supercon-

ducting quantum computing [5,6].

More recently, two new kinds of strong coupling

regimes of cavity QED have been explored. The ﬁrst

one is the ultrastrong coupling regime that is achieved

when the non-rotating-wave terms of light-matter inter-

action become relevant, resulting in the non-conservation

of the total number of excitations in the system [7–10].

In the simplest case of a single mode cavity, ultrastrong

coupling physics kicks in when the vacuum Rabi fre-

quency becomes comparable to the atom/cavity tran-

sition frequencies, as demonstrated in the semiconduc-

tor [11,12] and the circuit QED [13,14] platforms. An-

other kind of strong coupling regime, termed superstrong

coupling [15], can be obtained in massively multi-mode

cavities, when the vacuum Rabi frequency exceeds the

free spectral range of the resonator, that is the frequency

spacing between the modes. In this case, the qubit ex-

changes an excitation with the cavity faster than light

can traverse the cavity length, and hence the spatial pro-

ﬁle of the cavity modes becomes dependent on the qubit

state. This regime was approached in circuit QED, using

a meter-long on-chip superconducting transmission line

resonator [16] and in a cold atom setup using a 30 m long

optical resonator [17].

The superstrong coupling has an intuitive spectro-

scopic manifestation in the single-particle approximation.

For a cavity with a free spectral range ∆, the qubit res-

onance at frequency feg =ωeg/2πsimultaneously hy-

bridizes with about Γ/∆ standing-wave modes of the cav-

ity that are nearby in frequency (Fig. 1b). The quantity

Γ is, in fact, the rate of spontaneous emission of the qubit

in the limit of an inﬁnitely long cavity, corresponding to

∆→0, and the superstrong coupling condition formally

reads ∆ Γfeg. Because there are many cavity

modes and only one qubit, one can think that each cavity

mode from the Γ-vicinity of the qubit resonance becomes

weakly dressed by the qubit. The dressed modes acquire

a small frequency shift, of the order ∆2/Γ, which rapidly

vanishes outside the hybridization window. This spectral

property was indeed veriﬁed in recent cQED experiments,

where the ratio Γ/∆ was increased further compared to

the Ref. [16] using compact high-impedance/slow light

transmission lines [18,19].

Here we reveal a surprisingly important role of the cav-

ity’s lowest energy modes in the superstrong coupling

dynamics (Fig 1a,b). Even though these far-detuned

modes are negligibly dressed by the qubit, one can use

them to construct a large number of multi-particle ex-

citations with energies near the qubit resonance. For

example, in addition to a number around Γ/∆ of single-

particle states hybridizing with the qubit, there is a much

larger number of two-particle states, scaling as (Γ/∆)2,

in the same energy window. These states consist of one

“high-frequency” photon at a frequency near feg and one

“low-frequency” photon at a frequency of about Γ or less.

There is an even larger number of three-particle states,

arXiv:2210.14681v1 [quant-ph] 26 Oct 2022

FIG. 1. (a) Schematic of modes in an ideal Fabry-Perot cav-

ity resonator coupled to a single atom (qubit) at frequency

ωeg The bare mode frequencies are equally spaced by fre-

quency ∆ and are labeled ω1,ω2, ..., ω100, ... (b, left side)

Single-particle states of the uncoupled system, separated into

one photon in the low-frequency modes (red), one photon in

the high-frequency modes (blue), as well as one qubit excita-

tion, labeled ωeg (black). In the superstrong coupling regime,

a number of high-frequency modes in the Γ-vicinity of the

qubit transition hybridize with the qubit. (b, right) Exam-

ples of two-particle excitations in the same energy window,

involving one photon in the cavity’s lowest-frequency modes.

For concreteness, we set ωeg =ω100.

involving two low-frequency photons in the cavity, and so

forth. The presence of low-frequency modes is crucial for

such a massive multi-particle degeneracy: if we ignore the

cavity’s spectrum below half the qubit frequency, there

would be room for only one two-particle excitation and no

three-particle ones. How strong can the coupling between

single-particle and multi-particle manifolds be in the su-

perstrong coupling regime, and how would this many-

body interaction eﬀect manifest in the simplest linear

(vanishing probe power) spectroscopy experiment? Our

work comprehensively answers this question.

To understand the cavity’s multi-particle dynamics,

one has to ﬁrst face the notorious problem of the gauge

choice in quantum electrodynamics (see e.g. the ref-

erences [20–32]). Indeed, a coherent hybridization be-

tween a single-particle and, say, a two-particle excitation

in Fig 1b clearly violates the conservation of the total

number of excitations, which is the key attribute of the

ultrastrong coupling regime. However, it is not obvious

which Hamiltonian parameters control the emergence of

ultrastrong coupling physics in a massively multi-mode

system. To make matters worse, the light-matter cou-

pling Hamiltonians are generally gauge-dependent [22],

which in the past has led to paradoxical predictions for

multi-mode problems, such as a divergent Lamb shift or

spontaneous emission rate [33–35]. Furthermore, simula-

tions of multi-mode QED systems are impossible without

truncating the system’s Hilbert space, and the accuracy

of such a truncation is also gauge-dependent, as it was

ﬁrst pointed out for the problem of a two-photon ab-

sorption in hydrogen [20]. To our knowledge, no general

recipe exists for choosing the most computationally eﬃ-

cient gauge for a multi-mode cavity QED problem [26].

In the circuit QED platform, particularly suited for ex-

ploring quantum eﬀects beyond the strong coupling, the

majority of multi-mode experiments involved a transmon

type qubit, which is in fact a weakly anharmonic oscil-

lator circuit rather than a two-level system [36]. In this

case, the black box circuit quantization (BBQ) theory

provides an eﬃcient solution to all challenges related to

the gauge choice [37,38]. Within the BBQ theory, the

qubit’s non-linearity is treated as a perturbation to the

normal modes of the coupled system, the parameters of

which are given entirely in terms of the circuit impedance

or admittance functions. However, the conversion pro-

cesses are suppressed, at least at the single-photon level,

because the interaction felt by the low-frequency modes

is controlled entirely by the degree of their hybridization

with the qubit, a vanishing quantity in the superstrong

coupling regime. Here we consider the case of a highly

anharmonic ﬂuxonium qubit [39,40], to which the BBQ

theory cannot be directly applied.

Our new result can be summarized as follows. The

single-particle excitations of the superstrongly coupled

qubit-cavity system can be formally viewed as polari-

tons, although we prefer to call them “dressed photons”,

because the qubit degree of freedom is also of the elec-

tromagnetic nature. These dressed photons experience a

non-linearity, the dominant eﬀect of which is to convert a

single dressed photon into a lower energy dressed photon

and a number of low-frequency photons (the dressing of

which is negligible). We focused at the most eﬃcient con-

version process, involving just one low-frequency photon

production. Such a parity-ﬂipping process is allowed by

symmetry when operating the qubit circuit away from

a half-integer ﬂux bias. In this case, the single-photon

conversion rate can readily reach a few MHz, which is

well above the decoherence ﬂoor in superconducting cir-

cuits, and which makes the conversion process reversible

and coherent. Consequently, the cavity’s linear spectrum

of standing-wave modes would acquire a ﬁne structure of

multi-particle resonances, consisting of hybridized single-

particle and two-particle excitations. Higher-order con-

version processes can lead to even ﬁner spectral features,

involving three-particle states and so forth.

Our prediction has important connections with two

other topics. The ﬁrst connection is the quantum impu-

rity dynamics of the Ohmic bath spin-boson model [41,

42]. The hybridization of single-particle and multi-

particle excitations would lead, in the inﬁnite cavity size

limit, to a non-zero probability of the inelastic scattering

of a single photon by the qubit [43–46]. The second con-

nection is to the phenomenon of spontaneous parametric

down-conversion (SPDC) [47,48]. To our knowledge,

there hardly exist a non-linear optical material where

the SPDC rate would be larger or at least comparable

to the absorption rate [49]. Consequently, the probabil-

ity of observing a reversible SPDC process is practically

zero. Our synthetic system implements the opposite sce-

nario: a single photon can coherently down-convert into

a photon pair, or even to a superposition of multiple pho-

ton pairs, and then up-convert back into the original sin-

gle photon, and so on, at a sub-microsecond time scale.

By analogy with the vacuum Rabi oscillations in cavity

QED, we refer to this previously unavailable regime of

non-linear optics as a regime of strong down-conversion.

The article is organized as follows. In Sec. II, we

present the proposed circuit implementation of multi-

mode cavity QED and describe its Hamiltonians in three

gauges, namely the charge gauge, the ﬂux gauge and a

newly introduced mixed gauge. The mixed gauge is our

key innovation as it provides a shortcut to calculating

the two-particle amplitudes. In Sec. III, we describe an

eﬀective photon-photon interaction Hamiltonian respon-

sible for coherent conversion processes. An example of

the resulting ﬁne-structured cavity spectrum is given in

Section IV. In Section Vwe draw our conclusions and

perspectives. Technical details are organized in several

Appendices. In particular, in Appendix Dwe present

a successful numerical benchmark of the proposed eﬀec-

tive Hamiltonian with a direct numerical diagonalization,

which was made possible by considering a much shorter-

length transmission line and a weaker qubit coupling.

II. CIRCUIT MODEL AND GAUGE CHOICE

In our system, the multi-mode cavity is implemented

as a superconducting transmission line of length `, char-

acterized by the wave impedance Z∞and the speed of

light v, as shown in Fig. 2(a). This transmission line

is terminated on one end by a superconducting loop of

inductance L, interrupted by a single weak Josephson

junction, characterized by the Josephson energy EJand

the charging energy EC. With the proper choice of pa-

rameters, such a loop implements a circuit atom known

as ﬂuxonium. Fluxonium interacts with the transmis-

sion line modes by sharing a fraction x(0 < x ≤1)

of its loop inductance with the line. Fluxonium’s spec-

tral properties can be tuned by varying an external mag-

netic ﬂux (~/2e)ϕext pierced through the superconduct-

ing loop. The hybridization parameter Γ grows with x

and it also depends on Z∞and the ﬂuxonium circuit pa-

rameters. All numerical calculations in the main text are

done using example device parameters listed in Table I.

FIG. 2. (a) Schematic of a telegraph transmission line sec-

tion terminated by a superconducting ﬂuxonium qubit. The

transmission line is characterized by its wave impedance Z∞,

speed of light v, and length `. The ﬂuxonium circuit is de-

ﬁned by the Josephson energy EJ, the charging energy EC

of the small junction and the loop inductance L. A fraction

xof the ﬂuxonium loop inductance is shared with the trans-

mission line. The qubit transition frequency can be tuned by

changing the external ﬂux bias (~/2e)ϕext threading the su-

perconducting loop. (b), (c) and (d): Exact lumped-element

circuit model of the device obtained using Foster’s decompo-

sition. The values Ciand Liare derived in the Appendix A.

The ﬁlled red dots indicate the choice of generalized coordi-

nate variables ϕicorresponding to (b) ﬂux gauge, (c) charge

gauge, and (d) mixed gauge. Note that the variable ϕMis

dependent on ϕiand ϕJvia current conservation.

The atom-cavity coupling Hamiltonian for the system

shown in Fig. 2(a) takes the usual form:

H=ˆ

Hatom +ˆ

Hmodes +ˆ

Hint,(1)

where the ﬁrst two terms describe respectively an atom

and a set of non-interacting bosonic modes hosted by the

transmission line, while the third term describes their

interaction. In the limit x→0+, the ﬂuxonium atom

decouples from the transmission line modes. The bare

atom Hamiltonian is then given in terms of the phase-

diﬀerence operator ˆϕJacross the junction and the con-

jugate Cooper pair number operator ˆnJ, which obey the

commutation relation [ ˆϕJ,ˆnJ] = i. Such Hamiltonian

reads:

Hatom = 4ECˆn2

J−EJcos( ˆϕJ−ϕext) + ELˆϕ2

J/2,(2)

where EJis the Josephson junction’s energy, EC=

e2/2CJis the charging energy, EL= (~/2e)2/L is the

inductive energy of the loop inductance, and ϕext is the

external phase-bias, created by piercing the ﬂuxonium

loop with an externally applied magnetic ﬁeld.

In order to obtain the Hamiltonian (1) for x > 0, as

the ﬁrst step, we replace the continuous electromagnetic

structure of the transmission line by its lumped circuit

element representation. The lumped element model in-

volves Nseries-LC circuits connected in parallel. The

values of the individual capacitors Ciand inductors Li

(i= 1,2, ..., N) can be obtained using Foster’s theorem

(see Appendix A), which states that for N→ ∞ the

lumped element representation becomes mathematically

exact. To simplify the discussion, we consider a non-

dispersive, open-ended transmission line, characterized

by only two parameters: the wave impedance Z∞, and

the free spectral range ∆ = v/2`, where vis the speed of

light and `is the transmission line length.

In the second step, we must choose an electromagnetic

gauge, that is a set of independent generalized coordi-

nates of the circuit. Depending on the gauge, the circuit

atom can be coupled to the modes via the ﬂux variable,

charge variable, or both. We will show that both the

pure ﬂux and pure charge gauge present challenges for

truncating either the cavity modes or the atomic levels,

and introduce a special mixed gauge which resolves the

truncation problem.

A. Flux gauge

One common option is to choose the generalized coor-

dinate variables ϕisuch that they represent the phase-

diﬀerence across the capacitors Ci(see Fig. 2(b), red

nodes). Note that the phase ϕMis linked to ϕiand ϕJ

via the current conservation at the node M(see Fig. 2b,

gray node). Following the standard quantization pro-

cedure detailed in Appendix B, we obtain the following

Hamiltonian operators:

Hmodes =

i=1

~ω(f)

iˆ

b†

iˆ

bi,(3)

Hint =−ˆϕJ

i=1

hg(f)

i(ˆ

bi+ˆ

b†

i).(4)

Here ˆ

bi(ˆ

b†

i) is the photon annihilation (creation) operator

for the photon in the mode iof the transmission line,

and the mode frequencies ω(f)

i, calculated through the

procedure described in Appendix B, are numerically close

to the values (LiCi)−1/2. The expression for the coupling

constants g(f)

iis given by Eq. (B35). Since the coupling

is achieved via the ﬂux variable ϕJ, we call the present

variable choice the “ﬂux” gauge. Importantly, in the ﬂux

gauge, the atomic part ˆ

Hatom is modiﬁed from the bare

form of Eq. (2) by the replacement (see Appendix B):

EL→˜

EL,(5)

where the renormalized inductive energy ˜

ELis given by

Eq. (B9). Note that for x= 0 we have ˜

EL=EL, but for

x > 0 the two quantities can diﬀer signiﬁcantly.

Unfortunately, in the ﬂux gauge the separation of

Hamiltonian (1) into the three terms has a number of

disturbing properties. To start, the value of the cou-

pling constant g(f)

igrows with iapproximately as i1/2

(see Fig. 3(a)). Therefore, truncating the model to a

ﬁnite number of low-energy modes cannot be readily jus-

tiﬁed. In addition, the renormalized inductive energy ˜

also depends on the total number of modes used in the

Foster’s decomposition of the transmission line (N) and

can become much larger than ELfor 1 −x1 (see

Fig. 6). In fact, for x→1−, the parameter ˜

ELdiverges

as ˜

EL→EL/(1 −x) for N→+∞. Such a behavior

of ˜

ELis problematic for interpreting the dynamics of the

renormalized atom’s Hamiltonian, because the two lowest

energy eigenstates of ˆ

Hatom loose their spin-like nature

for EL> EJ(the Josephson potential looses its degener-

ate local minima at ϕext =π). For the reasons above, it

is challenging to produce even qualitative predictions for

the outcome of a simple spectroscopy experiment in the

single-photon excitation regime.

B. Charge gauge

We now show that the problem of divergent coupling

constants in the ultraviolet limit along with the accom-

panying N-dependence of the model parameters can be

eliminated simply by switching to the charge gauge.

Speciﬁcally, we consider a dual choice of circuit vari-

ables shown in Fig. 2(c), where ϕiis the phase-diﬀerence

across the inductance Li. The junction variable ϕJre-

mains the same as in the ﬂux gauge, and the dependent

variable ϕMis still given by the current conservation at

the node M. With the new choice of variables, the atom

couples to the bosonic annihilation (creation) operators

bi(ˆ

b†

i) of the transmission line modes via the Cooper pair

number operator ˆnJ. The transmission line and interac-

tion terms of the Hamiltonian read:

Hmodes =

i=1

~ω(c)

iˆ

b†

iˆ

bi,(6)

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Theoryofstrongdown-conversioninmulti-modecavityandcircuitQEDNitishMehta,1CristianoCiuti,2RomanKuzmin,1andVladimirE.Manucharyan1,31DepartmentofPhysics,UniversityofMaryland,CollegePark,MD20742,USA2UniversiteParisCite,CNRS,MateriauxetPhenomenesQuantiques,F-75013Paris,France3EcolePolytechniqueFed...

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Theory of strong down-conversion in multi-mode cavity and circuit QED Nitish Mehta1Cristiano Ciuti2Roman Kuzmin1and Vladimir E. Manucharyan1 3 1Department of Physics University of Maryland College Park MD 20742 USA

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