Cavity Quantum Electrodynamics at Arbitrary Light-Matter Coupling Strengths

Yuto Ashida,1, 2, 3, ∗ Ataç İmamoğlu,4 and Eugene Demler51Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan2Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan

3Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan4Institute of Quantum Electronics, ETH Zurich, CH-8093 Zürich, Switzerland

5Department of Physics, Harvard University, Cambridge, MA 02138, USA

Quantum light-matter systems at strong coupling are notoriously challenging to analyze due to the need toinclude states with many excitations in every coupled mode. We propose a nonperturbative approach to analyzelight-matter correlations at all interaction strengths. The key element of our approach is a unitary transformationthat achieves asymptotic decoupling of light and matter degrees of freedom in the limit where light-matter in-teraction becomes the dominant energy scale. In the transformed frame, truncation of the matter/photon Hilbertspace is increasingly well-justified at larger coupling, enabling one to systematically derive low-energy effectivemodels, such as tight-binding Hamiltonians. We demonstrate the versatility of our approach by applying it toconcrete models relevant to electrons in crystal potential and electric dipoles interacting with a cavity mode. Ageneralization to the case of spatially varying electromagnetic modes is also discussed.

Understanding quantum systems with strong light-matterinteraction has become a central problem in both fundamen-tal physics and quantum technologies [1]. Recent experi-mental and theoretical advances in solid-state physics [2–38],quantum optics [39–73], and quantum chemistry [74–93] havemade it possible to achieve strong coupling regimes in a va-riety of setups. In these systems, standard assumptions suchas the rotating wave approximation can no longer be justified,and the inclusion of the diamagnetic Â2 term or multilevelstructure of matter becomes crucial. Thus, quantized light andmatter degrees of freedom must be treated on equal footingwithin the exact quantum electrodynamics (QED) Hamilto-nian. Despite considerable theoretical efforts, a comprehen-sive formulation for analyzing such challenging problems atarbitrary coupling strengths is still lacking.

On another front, strongly correlated many-body systemshave often been tackled by devising a unitary transformationthat disentangles certain degrees of freedom, after which asimplified ansatz can be applied; a highly entangled quan-tum state in the original frame can then be expressed as afactorable state after the transformation. This general ideahas been used in several contexts, such as analyzing quantumimpurity systems [94–97], constructing low-energy effectivemodels [98–101], and solving many-body localization [102]or electron-phonon problems [103].

The aim of this Letter is to extend this nonperturbative ap-proach to strongly correlated light-matter systems, thus de-veloping a consistent and versatile framework to seamlesslyanalyze arbitrary coupling regimes. Specifically, we proposeto use a unitary transformation that asymptotically decoupleslight and matter in the strong-coupling limit. Our approachputs no limitations on the coupling strength and allows usto explore the full range of system parameters, including theregime where light-matter coupling dominates over all otherrelevant energy scales. Importantly, we construct a generalway to systematically derive low-energy effective models byfaithful level truncations, which remain valid at all couplingstrengths. This in particular provides a solution to the long-

standing controversy [104–120] about which frame is bestsuited for studying strong-coupling physics. We demonstratethe versatility of our formalism by applying it to specific mod-els relevant to materials and atomic systems in cavity QED.

Asymptotic decoupling of light-matter interaction.— To il-lustrate the main idea, we first focus on a one-dimensionalmany-body system coupled to a single electromagnetic mode;a generalization to higher-dimensional systems with spatiallyvarying electromagnetic modes will be given later. We start

10-2

10-1

1

10

0

0.2

0.4

-1 101 102

ξ gen

ergy

coupling strength g/ωc

k

0

π/d

ESC

∝1/g

∝1/g1/2

FIG. 1. (Top) Effective parameter ξg characterizing interactionstrength in the asymptotically decoupled frame against the bare light-matter coupling g. (Bottom) Exact spectrum obtained by diagonaliz-ing Eq. (4), or equivalently (5), for an electron in periodic potential.In the extremely strong coupling (ESC) regime, it exhibits equallyspaced flat bands narrowing as ∝ 1/g, corresponding to localizedelectrons with the large renormalized mass (inset). Numerical valuesare shown in the unit ωc = ~ =m = 1 throughout this Letter. Thepotential depth and lattice constant are v=5 and d=4, respectively.

arX

iv:2

010.

0358

3v3

[co

nd-m

at.m

es-h

all]

18

Mar

202

1

2

from the QED Hamiltonian in the Coulomb gauge:

ĤC =

∫dx ψ̂†x

[(−i~∂x−qÂ)2

2m+V (x)

]ψ̂x+~ωcâ†â+Ĥ||,

(1)

where ψ̂x (ψ̂†x) is the annihilation (creation) operator offermions of mass m and charge q at position x, and V (x)is an arbitrary external potential. Equation (1) describesthe coupling between electrons and a cavity electromag-netic mode with frequency ωc and the vector potential op-erator  = A(â + â†), where A is the mode amplitudeand â (â†) is the annihilation (creation) operator of photons.The instantaneous Coulomb interaction is given by Ĥ|| =∫dxdx′ q2ψ̂†xψ̂

†x′ ψ̂x′ ψ̂x/4π�0|x− x′|. We rewrite ĤC as

ĤC =

∫dx ψ̂†x

[−~

2∂2x2m

+ V (x)

]ψ̂x + ~Ωb̂†b̂+ Ĥ||

−gxΩ∫dx ψ̂†x(−i~∂x)ψ̂x (b̂+ b̂†), (2)

where Ω =√ω2c + 2Ng

2 is the dressed photon frequencywith the particle number N [121] and the coupling strengthg = qA

√ωc/m~, and xΩ =

√~/mΩ is a characteristic

length relevant both in weak and strong coupling regimes.Here, the photon part has been diagonalized by a Bogoliubovtransformation: b̂+ b̂† =

√Ω/ωc (â+ â

†) [122].To asymptotically decouple light and matter degrees of

freedom, we propose to use a unitary transformation

Û = exp

[−iξg

∫ ∞−∞

dx ψ̂†x(−i∂x)ψ̂x π̂], (3)

where ξg = gxΩ/Ω is the effective length scale characterizedby the coupling strength g and π̂ = i(b̂†− b̂). The transforma-tion (3) is reminiscent of the Lee-Low-Pines transformationused for polaronic systems [94], and leads to the HamiltonianĤU ≡ Û†ĤCÛ given by

ĤU =

∫dx ψ̂†x

[−~

2∂2x2m

+V (x+ ξgπ̂)

]ψ̂x+~Ωb̂†b̂+Ĥ‖

− ~2g2

mΩ2

[∫dx ψ̂†x(−i∂x)ψ̂x

]2, (4)

where the light-matter interaction is now absorbed by the po-tential term as the shift ξgπ̂ of the electron coordinates. Phys-ically, the unitary operator (3) changes a reference frame insuch a way that quantum particles no longer interact with theelectromagnetic mode through the usual minimal coupling,but through the gauge-field dependent shift of the electron co-ordinates and the associated quantum fluctuations in the ex-ternal potential. Thus, in the transformed frame the effectivestrength of the light-matter interaction is characterized by ξginstead of the original coupling g. Remarkably, as shown inthe top panel of Fig. 1, ξg remains small over the entire re-gion of g and, in particular, vanishes as ξg ∝ g−1/2 in thestrong-coupling limit g→∞. For this reason, we shall call the

present frame as the asymptotically decoupled (AD) frame;the identification of the AD Hamiltonian (4) is the first mainresult of this Letter.

Several remarks are in order. First, a specific form ofξg can depend on polarization of an electromagnetic mode.For instance, when matter is coupled to a circularly polar-ized mode, ξg vanishes as ξg ∝ g−1 provided that the cou-pling g is sufficiently large [123]. Second, we note that thetransformation (3) preserves the translational symmetry of the(bare) matter Hamiltonian. This should be compared to, e.g.,the Power-Zienau-Woolley (PZW) frame [124, 125] in whichsuch symmetry is broken due to the additional terms in thetransformed Hamiltonian [112, 123]. Third, in view of ourdefinition of the coupling strength g, the so-called ultrastrong(deep strong) coupling regime should approximately corre-spond to g & 0.3 (g & 3). Below we show that further in-crease of g leads to the new regime, namely, the extremelystrong coupling (ESC) regime. In the latter, truncation of mat-ter/photon levels can no longer be justified in the conventionalframes, but is asymptotically exact in the AD frame as dis-cussed in detail below.

General properties at extremely strong coupling.— Fromnow on, we focus on the single-electron problems and delin-eate universal spectral features in the ESC regime; the role ofelectron interactions will be discussed in a future publication.The AD-frame Hamiltonian is then simplified to

ĤU =p̂2

2meff+ V (x+ ξgπ̂) + ~Ωb̂†b̂, (5)

where renormalization of the effective mass meff = m[1 +2(g/ωc)

2] exactly arises from the last term in Eq. (4); thisrenormalization becomes even more prominent in a many-body case [123]. Note that the last term in Eq. (4) also gen-erates the interaction term ∝ ψ̂†ψ̂†ψ̂ψ̂, which, however, doesnot affect single-electron systems considered below.

One can understand the key spectral features of ĤU in theESC regime as follows. In the limit of large g, the renor-malized photon frequency Ω becomes large, while the effec-tive light-matter coupling, characterized by ξg , eventually de-creases. Thus, in the strong-coupling limit, the lowest-energyeigenstates |ΨU 〉 of ĤU are well approximated by a productstate:

|ΨU 〉 ' |ψU 〉 |0〉Ω, (6)

where |ψU 〉 is an eigenstate of p̂2/2meff+V (x), and |0〉Ω is thedressed-photon vacuum. Now, suppose that potential V haswell-defined local minima, around which it can be expandedas δV ∝ x2. Since the effective mass rapidly increases asmeff ∝ g2, |ψU 〉 is tightly localized around the potential min-ima. The low-lying spectrum of ĤU thus reduces to that of theharmonic oscillator with narrowing level spacing δE ∝ 1/g.

The above argument shows that, in the AD frame, an energyeigenstate can be well approximated by a product of light andmatter states. Nevertheless, they are still strongly entangledin the original frame. To see this, we consider an eigenstate

3

|ΨC〉 = Û |ΨU 〉 of the Coulomb-gauge Hamiltonian ĤC:

|ΨC〉 = Û∫dpψp|p〉|0〉Ω =

∫dpψp|p〉D̂ξgp|0〉Ω, (7)

where∫dpψp|p〉 = |ψU 〉 is the AD-frame eigenstate ex-

pressed in the momentum basis, and D̂β = eβb̂†−β∗b̂ is the

displacement operator. In the ESC regime, |ψU 〉 has vanish-ingly small variance σx ∝ 1/g; accordingly, the momentumdistribution |ψp|2 is very broad with variance σp∝g, showingthat the Coulomb-gauge eigenstate (7) is a highly entangledstate consisting of superposition of coherent states with largephoton occupancy determined by the particle momentum.

Difficulties of level truncations in conventional frames.—The AD frame readily allows us to elucidate the origin of dif-ficulties for level truncations in the Coulomb gauge [112, 117,118, 120]. Namely, if we expand a tightly localized state |ψU 〉in terms of eigenstates of p̂2/2m + V with the bare mass m,we will find substantial contribution from high-energy elec-tron states. This point can be seen from Eq. (7), which con-tains large-momentum eigenstates. Thus, any analysis per-formed in the Coulomb gauge, which uses a fixed UV cut-off for electron states, should become invalid at sufficientlystrong coupling. While the use of the PZW frame can par-tially mitigate the limitations [24, 38, 118] and can be validup to ultrastrong/deep strong coupling regimes [112, 115], itis ultimately constrained by the same restrictions, especiallyin the ESC regime. This holds true even when high-lyingstates appear to be reasonably out of resonance. Moreover,both the mean and fluctuation of the photon number in thePZW frame increases as n, δn∝ g. Thus, the number of pho-ton states required to diagonalize the Hamiltonian diverges atlarge g, making photon-level truncation (that is unavoidablein actual calculations) ill-justified in the deep- or extremely-strong coupling regimes. Altogether, as long as one relies onthe conventional frames, we conclude that effective modelsderived by level truncations, such as tight-binding models orthe quantum Rabi model, must inevitably break down when gbecomes sufficiently large.

In contrast, the AD frame (5) introduced here provides asimple solution to this problem. Specifically, matter-leveltruncation, i.e., tight-binding approximation, is increasinglywell-justified in ĤU at larger g, owing to tighter localizationof the wavefunction |ψU 〉. Similarly, due to the photon dress-ing and asymptotic decoupling, one can always truncate high-lying photon levels; indeed, the mean photon number remainsvery small over the entire region of g and, in particular, van-ishes in the ESC limit. Below we demonstrate such versatilityof the AD frame by applying it to concrete models relevant toquantum electrodynamical materials and atomic dipoles.

Application to solid-state systems.— We first consider anelectron in periodic potential and discuss the formation ofelectron-polariton band structures. To be concrete, we assumeV = v[1+cos (2πx/d)] with d and v being the lattice constantand potential depth, respectively. Since the AD frame pre-serves the translational symmetry, Bloch’s theorem remainsvalid and every eigenvalue of ĤU has a well-defined crystal

(a) (b) (e)

(d)(c)

g/ω =10c g/ω =1c

g/ω =10c g/ω =10c

-1

2-2[×

10

]

wavevector k0

0

0

0.2

0.4

0.6

0

2

4

6

8

1

2

3

4

5

0

1

2

3

4

5

π/d-π/dwavevector k

0 π/d-π/dwavevector k

0 π/d-π/d

ener

gyen

ergy

ener

gyen

ergy

0.1

0.2

0.3

0.4

ener

gy

g increase

ExactCoulomb gaugeAD frame

FIG. 2. (a-d) Exact electron-polariton bands obtained by diagonal-izing Eq. (5). Gray dashed curves indicate dispersions at g = 0.(e) Comparisons between the exact results and the tight-bindingmodels at g = 0.1, 1, 2, 10 from top to bottom. Black dashed (reddotted) curves indicate the tight-binding results in the asymptoticallydecoupled frame (Coulomb gauge). We set v=5 and d=4 in (a-e).

wavevector k ∈ [−π/d, π/d). Figures 1 and 2a-d show theobtained exact eigenspectra at different coupling strengths g,in the sense that matter/photon-energy cutoffs are taken to belarge enough such that the results are converged. As g is in-creased, the bands become increasingly flat and form equallyspaced spectra with energy spacing narrowing ∝ 1/g, whichis fully in accord with the universal spectral features discussedearlier. While the signature of band flattening can emerge al-ready at deep strong coupling [118], the drastic level narrow-ing/softening of the whole excitation spectrum is one of thekey distinctive features of the ESC regime [see Fig. 1].

To construct the effective low-energy Hamiltonian, we de-rive the tight-binding model by projecting the continuum sys-tem on the lowest-band Wannier orbitals. Specifically, we firstexpand a matter state in terms of the Wannier basis,

ψ̂x =∑j

wj(x)ĉj , (8)

where wj is the Wannier function at site j for the lowest bandof p̂2/2meff + V with the effective mass, and ĉj is the corre-sponding annihilation operator. We then consider a manifoldspanned by product states of these Wannier orbitals and an ar-bitrary photon state. Projecting ĤU on this manifold and con-sidering the leading contributions, we obtain the tight-bindingHamiltonian in the AD frame as [123]

ĤTBU = (tg + t′g δ̂g)

∑i

(ĉ†i ĉi+1 + h.c.)

+(µg + µ′g δ̂g)

∑i

ĉ†i ĉi + ~Ωb̂†b̂, (9)

where tg =∫dkK �k,ge

ikd is the effective hopping parame-ter with �k,g being the lowest-band energy of p̂2/2meff +Vand K = 2π/d, and µg =

∫dkK �k,g is the effective chemi-

cal potential. The electromagnetically induced fluctuation ofpotential causes the terms with δ̂g = cos (Kξgπ̂)− 1, t′g =v∫dxw∗i cos(Kx)wi+1, and µ

′g=v

∫dxw∗i cos(Kx)wi.

4

Figure 2e shows that this surprisingly simple tight-bindingmodel (black dashed curve) accurately predicts the exact spec-trum (solid curve) at any g, and asymptotically becomes ex-act in the strong-coupling limit as expected. For the sake ofcomparison, we show the tight-binding results in the Coulombgauge (red dotted curve), which are obtained by projectingĤC onto the lowest band of p̂2/2m + V with the bare mass[123]. While this naı̈ve tight-binding model is valid wheng . 0.1, it completely misses key features at larger g, suchas band flattening and level narrowing in the whole excita-tion spectrum. Physically, this drastic failure originates fromill-justified truncation of strongly entangled high-lying light-matter states in the original frame [cf. Eq. (7)].

These results clearly demonstrate that a choice of the frameis essential to construct an accurate tight-binding model instrong-coupling regimes. The AD frame solves this issueby performing the projection after the unitary transformation,which effectively realizes suitable nonlinear truncation in theCoulomb gauge. The most general form of the AD-frametight-binding Hamiltonian is given by

ĤTBU =∑ijνλ

tijνλĉ†iν ĉjλ+

∑lijνλ

t′(l)ijνλĉ

†iν ĉjλπ̂

l+~Ωb̂†b̂, (10)

where ν (λ) labels internal degrees of freedom in each unitcell i (j), and tijνλ =

∫dxw∗iν [p̂

2/2meff +V ]wjλ, t′(l)ijνλ =

ξlgl!

∫dxw∗iνV

(l)wjλ with wiν being the corresponding Wan-nier functions, and V (l) is the l-th derivative of V withl= 1, 2, . . . The renormalized parameters t, t′(l) nonperturba-tively depend on g through the nonlinear truncation. Higher-order terms at larger l contribute less to eigenspectrum owingto smallness of ξg [cf. Fig. 1], which enables a systematicapproximation when necessary. The minimal tight-bindingHamiltonian (10), which should be valid at arbitrary couplingstrengths and even under disorder, provides the material coun-terpart of the quantum Rabi model. Its derivation is the secondmain result of this Letter.

For a general periodic potential, the calculation of matrixelements of the shifted potential V (x+ ξgπ̂) can be sepa-rated into light and matter parts, after which the standardprocedures can be used [123]. Even when a potential is nottranslationally invariant, one can expand it as V (x+ξgπ̂) 'V (x)+

∑lmaxl=1 V

(l)(x)ξlgπ̂l. The truncation order lmax should

scale inversely with g due to decreasing ξg . This expansionshould be valid unless a potential has singular spatial depen-dence and expansion coefficients are not systematically sup-pressed at higher orders.

Application to atomic dipoles.— We next apply the ADframe to the case of a quantum particle in double-well po-tential V =−λx2/2+µx4/4, which is a standard model forthe electrical dipole moment. Blue solid curves in Fig. 3a,bshow the spectra obtained in the AD frame at different poten-tial depths, where the results efficiently converge already ata low photon-number cutoff nc ∼ 5-10 [123]. The spectra atESC exhibit the universal features discussed above, i.e., en-ergies become doubly degenerate corresponding to two wells

(a) (b)

10-2 10-1 101 102coupling strength g/ωc

10-1 101 102coupling strength g/ωc

… …

ESC

……

ESC

AD framePZW frame

AD framePZW frame

1

0.1

0.5

ener

gy

FIG. 3. Low-energy spectra for (a) deep and (b) shallow double-wellpotentials with photon-number cutoff nc = 100. Blue solid curves(red dotted curves) show the results in the AD frame (PZW frame).We choose the parameters (a) λ=50, µ=95 and (b) λ=3, µ=3.85such that the transition frequency of the two lowest matter levels isresonant with ωc in each case.

and are equally spaced with narrowing ∝ 1/g due to tight lo-calization around the minima [cf. insets].

As discussed earlier, truncation of high-lying photon statesshould eventually be invalid in conventional frames. Wedemonstrate this by comparing to results obtained in the PZWframe, ĤPZW = Û

†PZWĤCÛPZW with ÛPZW =exp(iqxÂ/~),

at a large cutoff nc = 100 (red dotted curves in Fig. 3). No-tably, the PZW frame dramatically fails in the ESC regime,which has its root in the rapid increase of mean-photon num-ber due to strong light-matter entanglement and sizable prob-ability amplitudes of high photon-number states [123]. Weremark that matter-level cutoff is taken to be sufficiently largesuch that the results converge because the strong light-matterentanglement also invalidates matter-level truncation in theRabi-type descriptions. Since any actual calculation must re-sort to finite cutoffs, these results indicate the fundamentaldifficulties of the conventional frames in the ESC regime.

Beyond the single-mode description.— While the single-mode description can be justified in, e.g., an LC-circuit res-onator [49], it may fail when more than one cavity mode mustbe included depending on the cavity geometry. The unitarytransformation (3) can be generalized to such a case with spa-tially varying electromagnetic modes:

Û = exp

[−i p̂

~·∑α

ξαπ̂α(x)

], (11)

where α labels multiple modes and the electromagnetic fieldsnow depend on position x. At the leading order, the trans-formed Hamiltonian is

ĤU =p̂2

2m−∑α

(p̂ · ζα)2

~Ωα+ V

(x+

∑α

ξαπ̂α (x))

+∑α

~Ωαb̂†α(x)b̂α(x), (12)

where ζα is the effective polarization vector of mode α [123].This simple expression is valid when field variation is small

5

compared to the effective length scale, i.e., k|ξ| � 1 with|∇b̂| ∼ kb̂; this condition is independent of system size andmuch less restrictive than, e.g., the dipole approximation, ow-ing to smallness of ξg . We remark that light-matter decouplingdue to the inhomogeneous diamagnetic term has previouslybeen studied in the case of the quadratic Hamiltonian [66].

Discussions.— In the limit of classical electromagneticfields, our transformation (3) can be compared with theKramers-Henneberger (KH) transformation, which was usedto analyze atoms subject to intense laser fields [126, 127]. Be-sides the full quantum treatment given here, one importantdifference is that the KH transformation does not take into ac-count the diamagnetic A2 term other than its contribution toponderomotive forces appearing in spatially inhomogeneouslaser profiles. In our quantum setting, the asymptotic light-matter decoupling emerges only after the diamagnetic term isconsistently included through the Bogoliubov transformation.

With the advent of new materials and subwavelength cavitydesigns, it is now possible to explore ultra/deep strong cou-pling regimes of light-matter interaction and possibilities forfurther extending the interaction strength. We expect our re-sults to be applicable in the analysis of mono- or (twisted)bilayer-2D materials embedded in high quality-factor lumped-element terahertz cavities [16], where a single mode of theelectromagnetic field is isolated from higher-energy Fabry-Perot-like confined modes, as well as the electromagneticcontinuum. Signatures of the level narrowing/softening inthe ESC regime are already appreciable around g/ωc & 5,which can be realized with current experimental techniques[11, 19, 49].

In summary, we presented a new formulation (4) of stronglycorrelated light-matter systems that is applicable to both quan-tum electrodynamical materials and atomic systems. Sincethis is a nonperturbative approach, it is valid at arbitrary cou-pling strengths and, in particular, allows us to consistently ex-plore the extremely strong coupling regime for the first time.Our formalism elucidates difficulties of level truncations inthe conventional frames from a general perspective, and offersa systematic way to derive the faithful tight-binding Hamilto-nians (10). While the emphasis was placed on the extremelystrong coupling, our formalism is versatile enough to be ap-plied to any coupling regimes, where standard/conventionaldescriptions can be inadequate. It would be interesting toapply the present formulation to identify the correct tight-binding models of more complex light-matter systems. Inparticular, it merits further study to elucidate role of the light-induced band flattening and narrowing in genuine many-bodyregimes.

We are grateful to Jerome Faist, Zongping Gong, and Gi-acomo Scalari for fruitful discussions. Y.A. acknowledgessupport from the Japan Society for the Promotion of Sci-ence through Grant No. JP19K23424. E.D. acknowledgessupport from Harvard-MIT CUA, AFOSR-MURI PhotonicQuantum Matter (award FA95501610323), DARPA DRINQSprogram (award D18AC00014), and the NSF EAGER-QAC-QSA award 2038011 “Quantum Algorithms for Correlated

Electron-Phonon System”.

∗ ashida@phys.s.u-tokyo.ac.jp[1] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Pho-

tons and Atoms (Wiley, New York, 1989).[2] D. N. Basov, M. M. Fogler, and F. J. Garcı́a de Abajo, Science

354 (2016).[3] J. J. Baumberg, J. Aizpurua, M. H. Mikkelsen, and D. R.

Smith, Nat. Mater. 18, 668 (2019).[4] R. J. Holmes and S. R. Forrest, Phys. Rev. Lett. 93, 186404

(2004).[5] S. Kéna-Cohen and S. Forrest, Nat. Photon. 4, 371 (2010).[6] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgen-

frei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, andJ. Hone, Nat. Nanotech. 5, 722 (2010).

[7] G. Constantinescu, A. Kuc, and T. Heine, Phys. Rev. Lett.111, 036104 (2013).

[8] E. Orgiu, J. George, J. Hutchison, E. Devaux, J. Dayen,B. Doudin, F. Stellacci, C. Genet, J. Schachenmayer,C. Genes, G. Pupillo, P. Samori, and T. W. Ebbesen, Nat.Mater. 14, 1123 (2015).

[9] T. Chervy, J. Xu, Y. Duan, C. Wang, L. Mager, M. Frerejean,J. A. Münninghoff, P. Tinnemans, J. A. Hutchison, C. Genet,A. E. Rowan, T. Tasing, and T. W. Ebbesen, Nano Lett. 16,7352 (2016).

[10] C. Jin, J. Kim, J. Suh, Z. Shi, B. Chen, X. Fan, M. Kam,K. Watanabe, T. Taniguchi, S. Tongay, A. Zettl, J. Wu, andF. Wang, Nat. Phys. 13, 127 (2017).

[11] B. Askenazi, A. Vasanelli, Y. Todorov, E. Sakat, J.-J. Greffet,G. Beaudoin, I. Sagnes, and C. Sirtori, ACS photonics 4, 2550(2017).

[12] S. Klembt, T. Harder, O. Egorov, K. Winkler, R. Ge, M. Ban-dres, M. Emmerling, L. Worschech, T. Liew, M. Segev,C. Schneider, and S. Hoefling, Nature 562, 552 (2018).

[13] S. Ravets, P. Knüppel, S. Faelt, O. Cotlet, M. Kroner,W. Wegscheider, and A. Imamoglu, Phys. Rev. Lett. 120,057401 (2018).

[14] A. J. Giles, S. Dai, I. Vurgaftman, T. Hoffman, S. Liu, L. Lind-say, C. T. Ellis, N. Assefa, I. Chatzakis, T. L. Reinecke, J. G.Tischler, M. M. Fogler, J. H. Edgar, D. N. Basov, and J. D.Caldwell, Nat. Mater. 17, 134 (2018).

[15] G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Val-morra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn,K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, Nat. Phys. 15,186 (2019).

[16] J. Keller, G. Scalari, F. Appugliese, S. Rajabali, M. Beck,J. Haase, C. A. Lehner, W. Wegscheider, M. Failla, M. My-ronov, D. R. Leadley, J. Lloyd-Hughes, P. Nataf, and J. Faist,Phys. Rev. B 101, 075301 (2020).

[17] T. Chervy, P. Knüppel, H. Abbaspour, M. Lupatini, S. Fält,W. Wegscheider, M. Kroner, and A. Imamoǧlu, Phys. Rev. X10, 011040 (2020).

[18] E. Cortese, N.-L. Tran, J.-M. Manceau, A. Bousseksou,I. Carusotto, G. Biasiol, R. Colombelli, and S. De Liberato,Nat. Phys. , 1 (2020).

[19] N. S. Mueller, Y. Okamura, B. G. Vieira, S. Juergensen,H. Lange, E. B. Barros, F. Schulz, and S. Reich, Nature 583,780 (2020).

[20] A. Thomas, E. Devaux, K. Nagarajan, T. Chervy, M. Sei-del, D. Hagenmüller, S. Schütz, J. Schachenmayer, C. Genet,

mailto:ashida@phys.s.u-tokyo.ac.jphttps://science.sciencemag.org/content/354/6309/aag1992https://science.sciencemag.org/content/354/6309/aag1992https://www.nature.com/articles/s41563-019-0290-yhttp://dx.doi.org/10.1103/PhysRevLett.93.186404http://dx.doi.org/10.1103/PhysRevLett.93.186404https://www.nature.com/articles/nphoton.2010.86https://www.nature.com/articles/nnano.2010.172http://dx.doi.org/10.1103/PhysRevLett.111.036104http://dx.doi.org/10.1103/PhysRevLett.111.036104https://www.nature.com/articles/nmat4392https://www.nature.com/articles/nmat4392https://pubs.acs.org/doi/abs/10.1021/acs.nanolett.6b02567https://pubs.acs.org/doi/abs/10.1021/acs.nanolett.6b02567https://www.nature.com/articles/nphys3928https://pubs.acs.org/doi/abs/10.1021/acsphotonics.7b00838https://pubs.acs.org/doi/abs/10.1021/acsphotonics.7b00838https://www.nature.com/articles/s41586-018-0601-5http://dx.doi.org/ 10.1103/PhysRevLett.120.057401http://dx.doi.org/ 10.1103/PhysRevLett.120.057401https://www.nature.com/articles/nmat5047https://www.nature.com/articles/s41567-018-0346-yhttps://www.nature.com/articles/s41567-018-0346-yhttp://dx.doi.org/ 10.1103/PhysRevB.101.075301http://dx.doi.org/10.1103/PhysRevX.10.011040http://dx.doi.org/10.1103/PhysRevX.10.011040https://www.nature.com/articles/s41567-020-0994-6https://www.nature.com/articles/s41586-020-2508-1https://www.nature.com/articles/s41586-020-2508-1

6

G. Pupillo, and T. W. Ebbesen, arXiv:1911.01459 (2019).[21] M. Ruggenthaler, J. Flick, C. Pellegrini, H. Appel, I. V.

Tokatly, and A. Rubio, Phys. Rev. A 90, 012508 (2014).[22] J. Schachenmayer, C. Genes, E. Tignone, and G. Pupillo,

Phys. Rev. Lett. 114, 196403 (2015).[23] J. Feist and F. J. Garcia-Vidal, Phys. Rev. Lett. 114, 196402

(2015).[24] A. Cottet, T. Kontos, and B. Douçot, Phys. Rev. B 91, 205417

(2015).[25] D. Hagenmüller, J. Schachenmayer, S. Schütz, C. Genes, and

G. Pupillo, Phys. Rev. Lett. 119, 223601 (2017).[26] D. Hagenmüller, S. Schütz, J. Schachenmayer, C. Genes, and

G. Pupillo, Phys. Rev. B 97, 205303 (2018).[27] M. A. Sentef, M. Ruggenthaler, and A. Rubio, Sci. Adv. 4

(2018).[28] F. Schlawin, A. Cavalleri, and D. Jaksch, Phys. Rev. Lett. 122,

133602 (2019).[29] J. B. Curtis, Z. M. Raines, A. A. Allocca, M. Hafezi, and

V. M. Galitski, Phys. Rev. Lett. 122, 167002 (2019).[30] D. M. Juraschek, T. Neuman, J. Flick, and P. Narang,

arXiv:1912.00122 (2019).[31] V. Rokaj, M. Penz, M. A. Sentef, M. Ruggenthaler, and A. Ru-

bio, Phys. Rev. Lett. 123, 047202 (2019).[32] G. Mazza and A. Georges, Phys. Rev. Lett. 122, 017401

(2019).[33] M. Kiffner, J. Coulthard, F. Schlawin, A. Ardavan, and

D. Jaksch, New J. Phys. 21, 073066 (2019).[34] X. Wang, E. Ronca, and M. A. Sentef, Phys. Rev. B 99,

235156 (2019).[35] Y. Ashida, A. Imamoglu, J. Faist, D. Jaksch, A. Cavalleri, and

E. Demler, Phys. Rev. X 10, 041027 (2020).[36] K. Lenk and M. Eckstein, arXiv:2002.12241 (2020).[37] A. Chiocchetta, D. Kiese, F. Piazza, and S. Diehl,

arXiv:2009.11856 (2020).[38] O. Dmytruk and M. Schiró, arXiv:2009.11088 (2020).[39] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys.

73, 565 (2001).[40] P. Forn-Dı́az, L. Lamata, E. Rico, J. Kono, and E. Solano,

Rev. Mod. Phys. 91, 025005 (2019).[41] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang,

J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature431, 162 (2004).

[42] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J.Schoelkopf, Phys. Rev. A 69, 062320 (2004).

[43] P. Forn-Dı́az, J. Lisenfeld, D. Marcos, J. J. Garcı́a-Ripoll,E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev.Lett. 105, 237001 (2010).

[44] C. Maissen, G. Scalari, F. Valmorra, M. Beck, J. Faist,S. Cibella, R. Leoni, C. Reichl, C. Charpentier, andW. Wegscheider, Phys. Rev. B 90, 205309 (2014).

[45] C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, Phys. Rev.Lett. 118, 133604 (2017).

[46] A. Bayer, M. Pozimski, S. Schambeck, D. Schuh, R. Huber,D. Bougeard, and C. Lange, Nano Lett. 17, 6340 (2017).

[47] P. Forn-Dı́az, J. J. Garcı́a-Ripoll, B. Peropadre, J.-L. Orgiazzi,M. Yurtalan, R. Belyansky, C. M. Wilson, and A. Lupascu,Nat. Phys. 13, 39 (2017).

[48] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito,and K. Semba, Phys. Rev. A 95, 053824 (2017).

[49] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito,and K. Semba, Nat. Phys. 13, 44 (2017).

[50] X. Li, M. Bamba, N. Yuan, Q. Zhang, Y. Zhao, M. Xiang,K. Xu, Z. Jin, W. Ren, G. Ma, S. Cao, D. Turchinovich, andJ. Kono, Science 361, 794 (2018).

[51] S. K. Ruddell, K. E. Webb, M. Takahata, S. Kato, and T. Aoki,Opt. Lett. 45, 4875 (2020).

[52] K. Wang, R. Dahan, M. Shentcis, Y. Kauffmann, S. Tsesses,and I. Kaminer, Nature 582, 50 (2020).

[53] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs,G. Rupper, C. Ell, O. Shchekin, and D. Deppe, Nature 432,200 (2004).

[54] G. Khitrova, H. Gibbs, M. Kira, S. W. Koch, and A. Scherer,Nat. Phys. 2, 81 (2006).

[55] L. Greuter, S. Starosielec, A. V. Kuhlmann, and R. J. Warbur-ton, Phys. Rev. B 92, 045302 (2015).

[56] R. Albrecht, A. Bommer, C. Deutsch, J. Reichel, andC. Becher, Phys. Rev. Lett. 110, 243602 (2013).

[57] D. Riedel, I. Söllner, B. J. Shields, S. Starosielec, P. Appel,E. Neu, P. Maletinsky, and R. J. Warburton, Phys. Rev. X 7,031040 (2017).

[58] K. Le Hur, L. Henriet, A. Petrescu, K. Plekhanov, G. Roux,and M. Schiro, C. R. Phys. 17, 808 (2016).

[59] A. F. Kockum, A. Miranowicz, S. De Liberato, S. Savasta, andF. Nori, Nat. Rev. Phys. 1, 19 (2019).

[60] A. Le Boité, Adv. Quantum Technol. 3, 1900140 (2020).[61] A. Stokes and A. Nazir, arXiv:2009.10662 (2020).[62] C. Ciuti, G. Bastard, and I. Carusotto, Phys. Rev. B 72,

115303 (2005).[63] S. D. Liberato, C. Ciuti, and I. Carusotto, Phys. Rev. Lett. 98,

103602 (2007).[64] J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev,

Y. Nakamura, and A. Blais, Phys. Rev. A 80, 032109 (2009).[65] J. Casanova, G. Romero, I. Lizuain, J. J. Garcı́a-Ripoll, and

E. Solano, Phys. Rev. Lett. 105, 263603 (2010).[66] S. De Liberato, Phys. Rev. Lett. 112, 016401 (2014).[67] J. Pedernales, I. Lizuain, S. Felicetti, G. Romero, L. Lamata,

and E. Solano, Sci. Rep. 5, 1 (2015).[68] S. Ashhab and K. Semba, Phys. Rev. A 95, 053833 (2017).[69] T. Jaako, Z.-L. Xiang, J. J. Garcia-Ripoll, and P. Rabl, Phys.

Rev. A 94, 033850 (2016).[70] D. De Bernardis, T. Jaako, and P. Rabl, Phys. Rev. A 97,

043820 (2018).[71] S. Felicetti and A. Le Boité, Phys. Rev. Lett. 124, 040404

(2020).[72] P. Pilar, D. De Bernardis, and P. Rabl, arXiv:2003.11556

(2020).[73] M. Bamba, X. Li, N. Marquez Peraca, and J. Kono,

arXiv:2007.13263 .[74] T. W. Ebbesen, Acc. Chem. Res. 49, 2403 (2016).[75] J. Feist, J. Galego, and F. J. Garcia-Vidal, ACS Photonics 5,

205 (2017).[76] J. R. Tischler, M. S. Bradley, V. Bulovic, J. H. Song, and

A. Nurmikko, Phys. Rev. Lett. 95, 036401 (2005).[77] T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbesen,

Phys. Rev. Lett. 106, 196405 (2011).[78] J. A. Hutchison, T. Schwartz, C. Genet, E. Devaux, and T. W.

Ebbesen, Angew. Chem. Int. Ed. 51, 1592 (2012).[79] D. M. Coles, Y. Yang, Y. Wang, R. T. Grant, R. A. Taylor,

S. K. Saikin, A. Aspuru-Guzik, D. G. Lidzey, J. K.-H. Tang,and J. M. Smith, Nat. Commun. 5, 5561 (2014).

[80] A. Thomas, J. George, A. Shalabney, M. Dryzhakov, S. J.Varma, J. Moran, T. Chervy, X. Zhong, E. Devaux, C. Genet,J. A. Hutchison, and T. W. Ebbesen, Angew. Chem. Int. Ed.55, 11462 (2016).

[81] R. Chikkaraddy, B. De Nijs, F. Benz, S. J. Barrow, O. A.Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, andJ. J. Baumberg, Nature 535, 127 (2016).

[82] X. Zhong, T. Chervy, L. Zhang, A. Thomas, J. George,

https://arxiv.org/abs/1911.01459http://dx.doi.org/ 10.1103/PhysRevA.90.012508http://dx.doi.org/10.1103/PhysRevLett.114.196403http://dx.doi.org/10.1103/PhysRevLett.114.196402http://dx.doi.org/10.1103/PhysRevLett.114.196402http://dx.doi.org/ 10.1103/PhysRevB.91.205417http://dx.doi.org/ 10.1103/PhysRevB.91.205417http://dx.doi.org/10.1103/PhysRevLett.119.223601http://dx.doi.org/10.1103/PhysRevB.97.205303https://advances.sciencemag.org/content/4/11/eaau6969https://advances.sciencemag.org/content/4/11/eaau6969http://dx.doi.org/10.1103/PhysRevLett.122.133602http://dx.doi.org/10.1103/PhysRevLett.122.133602http://dx.doi.org/ 10.1103/PhysRevLett.122.167002https://arxiv.org/abs/1912.00122http://dx.doi.org/ 10.1103/PhysRevLett.123.047202http://dx.doi.org/10.1103/PhysRevLett.122.017401http://dx.doi.org/10.1103/PhysRevLett.122.017401http://dx.doi.org/ 10.1088/1367-2630/ab31c7http://dx.doi.org/10.1103/PhysRevB.99.235156http://dx.doi.org/10.1103/PhysRevB.99.235156http://dx.doi.org/ 10.1103/PhysRevX.10.041027https://arxiv.org/abs/2002.12241https://arxiv.org/abs/2009.11856https://arxiv.org/abs/2009.11088http://dx.doi.org/10.1103/RevModPhys.73.565http://dx.doi.org/10.1103/RevModPhys.73.565http://dx.doi.org/ 10.1103/RevModPhys.91.025005https://www.nature.com/articles/nature02851/https://www.nature.com/articles/nature02851/http://dx.doi.org/10.1103/PhysRevA.69.062320http://dx.doi.org/10.1103/PhysRevLett.105.237001http://dx.doi.org/10.1103/PhysRevLett.105.237001http://dx.doi.org/ 10.1103/PhysRevB.90.205309http://dx.doi.org/ 10.1103/PhysRevLett.118.133604http://dx.doi.org/ 10.1103/PhysRevLett.118.133604https://pubs.acs.org/doi/10.1021/acs.nanolett.7b03103https://www.nature.com/articles/nphys3905http://dx.doi.org/ 10.1103/PhysRevA.95.053824https://www.nature.com/articles/nphys3906http://dx.doi.org/10.1126/science.aat5162http://dx.doi.org/ 10.1364/OL.396725https://www.nature.com/articles/s41586-020-2321-xhttps://www.nature.com/articles/nature03119https://www.nature.com/articles/nature03119https://www.nature.com/articles/nphys227http://dx.doi.org/10.1103/PhysRevB.92.045302http://dx.doi.org/ 10.1103/PhysRevLett.110.243602http://dx.doi.org/10.1103/PhysRevX.7.031040http://dx.doi.org/10.1103/PhysRevX.7.031040http://dx.doi.org/ https://doi.org/10.1016/j.crhy.2016.05.003https://www.nature.com/articles/s42254-018-0006-2http://dx.doi.org/https://doi.org/10.1002/qute.201900140https://arxiv.org/abs/2009.10662http://dx.doi.org/10.1103/PhysRevB.72.115303http://dx.doi.org/10.1103/PhysRevB.72.115303http://dx.doi.org/10.1103/PhysRevLett.98.103602http://dx.doi.org/10.1103/PhysRevLett.98.103602http://dx.doi.org/10.1103/PhysRevA.80.032109http://dx.doi.org/10.1103/PhysRevLett.105.263603http://dx.doi.org/10.1103/PhysRevLett.112.016401https://www.nature.com/articles/srep15472http://dx.doi.org/10.1103/PhysRevA.95.053833http://dx.doi.org/10.1103/PhysRevA.94.033850http://dx.doi.org/10.1103/PhysRevA.94.033850http://dx.doi.org/10.1103/PhysRevA.97.043820http://dx.doi.org/10.1103/PhysRevA.97.043820http://dx.doi.org/10.1103/PhysRevLett.124.040404http://dx.doi.org/10.1103/PhysRevLett.124.040404https://arxiv.org/abs/2003.11556https://arxiv.org/abs/2003.11556https://arxiv.org/abs/2007.13263https://pubs.acs.org/doi/10.1021/acs.accounts.6b00295https://pubs.acs.org/doi/abs/10.1021/acsphotonics.7b00680https://pubs.acs.org/doi/abs/10.1021/acsphotonics.7b00680http://dx.doi.org/10.1103/PhysRevLett.95.036401http://dx.doi.org/10.1103/PhysRevLett.106.196405http://dx.doi.org/ 10.1002/anie.201107033https://www.nature.com/articles/ncomms6561http://dx.doi.org/10.1002/anie.201605504http://dx.doi.org/10.1002/anie.201605504https://www.nature.com/articles/nature17974

7

C. Genet, J. A. Hutchison, and T. W. Ebbesen, Angew. Chem.Int. Ed. 56, 9034 (2017).

[83] K. Stranius, M. Hertzog, and K. Börjesson, Nat. Commun. 9,2273 (2018).

[84] L. A. Martinez-Martinez, E. Eizner, S. Kena-Cohen, andJ. Yuen-Zhou, J. Chem. Phys. 151, 054106 (2019).

[85] E. Eizner, L. A. Martı́nez-Martı́nez, J. Yuen-Zhou, andS. Kéna-Cohen, Sci. Adv. 5 (2019).

[86] D. Polak, R. Jayaprakash, T. P. Lyons, L. A. Martı́nez-Martı́nez, A. Leventis, K. J. Fallon, H. Coulthard, D. G.Bossanyi, K. Georgiou, A. J. Petty, II, J. Anthony, H. Bron-stein, J. Yuen-Zhou, A. I. Tartakovskii, J. Clark, and A. J.Musser, Chem. Sci. 11, 343 (2020).

[87] B. Xiang, R. F. Ribeiro, M. Du, L. Chen, Z. Yang, J. Wang,J. Yuen-Zhou, and W. Xiong, Science 368, 665 (2020).

[88] L. A. Martı́nez-Martı́nez, M. Du, R. F. Ribeiro, S. Kéna-Cohen, and J. Yuen-Zhou, J. Phys. Chem. Lett 9, 1951 (2018).

[89] A. Thomas, L. Lethuillier-Karl, K. Nagarajan, R. M. A.Vergauwe, J. George, T. Chervy, A. Shalabney, E. Devaux,C. Genet, J. Moran, and T. W. Ebbesen, Science 363, 615(2019).

[90] M. Ruggenthaler, N. Tancogne-Dejean, J. Flick, H. Appel,and A. Rubio, Nat. Rev. Chem. 2, 1 (2018).

[91] J. Galego, F. J. Garcia-Vidal, and J. Feist, Phys. Rev. X 5,041022 (2015).

[92] J. Flick, M. Ruggenthaler, H. Appel, and A. Rubio, Proc. Natl.Acad. Sci. U.S.A. 112, 15285 (2015).

[93] J. Flick, M. Ruggenthaler, H. Appel, and A. Rubio, Proc. Natl.Acad. Sci. U.S.A. 114, 3026 (2017).

[94] T. D. Lee, F. E. Low, and D. Pines, Phys. Rev. 90, 297 (1953).[95] H. Fröhlich, Proc. R. Soc. A 215, 291 (1952).[96] R. Silbey and R. A. Harris, J. Chem. Phys. 80, 2615 (1984).[97] Y. Ashida, T. Shi, M. C. Bañuls, J. I. Cirac, and E. Demler,

Phys. Rev. Lett. 121, 026805 (2018); Phys. Rev. B 98, 024103(2018).

[98] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966).[99] S. D. Głazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993).

[100] F. Wegner, Ann. Phys. 506, 77 (1994).[101] S. Bravyi, D. P. DiVincenzo, and D. Loss, Ann. Phys. 326,

2793 (2011).[102] J. Z. Imbrie, J. Stat. Phys. 163, 998 (2016).[103] T. Shi, E. Demler, and J. I. Cirac, arXiv:1912.11907 (2019).[104] W. E. Lamb, Phys. Rev. 85, 259 (1952).

[105] K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phys. Rev.Lett. 35, 432 (1975).

[106] J. Keeling, J. Phys. Cond. Matt. 19, 295213 (2007).[107] P. Nataf and C. Ciuti, Nat. Commun. 1, 72 (2010).[108] L. Chirolli, M. Polini, V. Giovannetti, and A. H. MacDonald,

Phys. Rev. Lett. 109, 267404 (2012).[109] A. Vukics, T. Grießer, and P. Domokos, Phys. Rev. Lett. 112,

073601 (2014).[110] M. F. Gely, A. Parra-Rodriguez, D. Bothner, Y. M. Blanter,

S. J. Bosman, E. Solano, and G. A. Steele, Phys. Rev. B 95,245115 (2017).

[111] S. J. Bosman, M. F. Gely, V. Singh, A. Bruno, D. Bothner, andG. A. Steele, npj Quant. Info. 3, 1 (2017).

[112] D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, andP. Rabl, Phys. Rev. A 98, 053819 (2018).

[113] G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H.MacDonald, and M. Polini, Phys. Rev. B 100, 121109 (2019).

[114] G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H.MacDonald, and M. Polini, arXiv:2005.09088 (2020).

[115] A. Stokes and A. Nazir, Nat. Commun. 10, 1 (2019); A. Stokesand A. Nazir, arXiv:1902.05160 (2019).

[116] A. Stokes and A. Nazir, arXiv:1905.10697 (2019).[117] O. Di Stefano, A. Settineri, V. Macrı̀, L. Garziano, R. Stassi,

S. Savasta, and F. Nori, Nat. Phys. 15, 803 (2019).[118] J. Li, D. Golez, G. Mazza, A. J. Millis, A. Georges, and

M. Eckstein, Phys. Rev. B 101, 205140 (2020).[119] C. Schafer, M. Ruggenthaler, V. Rokaj, and A. Rubio, ACS

photonics 7, 975 (2020).[120] M. A. D. Taylor, A. Mandal, W. Zhou, and P. Huo, Phys. Rev.

Lett. 125, 123602 (2020).[121] We note that, in solid-state systems, the electron density N/V

with V being the volume should be a natural quantity in thedressed frequency, where the volume factor arises from themode amplitude A ∝ 1/

√V .

[122] The coefficients in the Bogoliubov transformation need tobe real-valued in order to diagonalize the quadratic photonHamiltonian.

[123] See Supplemental Materials for further details on the state-ments and the derivations.

[124] E. A. Power and S. Zienau, Phil. R. Soc. A 251, 427 (1959).[125] R. G. Woolley, Proc. R. Soc. A 321, 557 (1971).[126] H. A. Kramers, Collected Scientific Papers (North-Holland,

Amsterdam, 1956) p. 866.[127] W. C. Henneberger, Phys. Rev. Lett. 21, 838 (1968).

http://dx.doi.org/10.1002/anie.201703539http://dx.doi.org/10.1002/anie.201703539https://www.nature.com/articles/s41467-018-04736-1https://www.nature.com/articles/s41467-018-04736-1http://dx.doi.org/10.1063/1.5100192https://advances.sciencemag.org/content/5/12/eaax4482http://dx.doi.org/10.1039/C9SC04950Ahttps://science.sciencemag.org/content/368/6491/665https://pubs.acs.org/doi/10.1021/acs.jpclett.8b00008http://dx.doi.org/10.1126/science.aau7742http://dx.doi.org/10.1126/science.aau7742https://www.nature.com/articles/s41570-018-0118http://dx.doi.org/10.1103/PhysRevX.5.041022http://dx.doi.org/10.1103/PhysRevX.5.041022http://dx.doi.org/ 10.1073/pnas.1518224112http://dx.doi.org/ 10.1073/pnas.1518224112http://dx.doi.org/ 10.1073/pnas.1615509114http://dx.doi.org/ 10.1073/pnas.1615509114http://dx.doi.org/10.1103/PhysRev.90.297http://dx.doi.org/10.1098/rspa.1952.0212http://dx.doi.org/10.1063/1.447055http://dx.doi.org/ 10.1103/PhysRevLett.121.026805http://dx.doi.org/ 10.1103/PhysRevB.98.024103http://dx.doi.org/ 10.1103/PhysRevB.98.024103http://dx.doi.org/10.1103/PhysRev.149.491http://dx.doi.org/10.1103/PhysRevD.48.5863http://dx.doi.org/10.1002/andp.19945060203http://dx.doi.org/https://doi.org/10.1016/j.aop.2011.06.004http://dx.doi.org/https://doi.org/10.1016/j.aop.2011.06.004https://link.springer.com/article/10.1007%2Fs10955-016-1508-x#citeashttps://arxiv.org/abs/1912.11907http://dx.doi.org/10.1103/PhysRev.85.259http://dx.doi.org/10.1103/PhysRevLett.35.432http://dx.doi.org/10.1103/PhysRevLett.35.432http://dx.doi.org/10.1088/0953-8984/19/29/295213https://www.nature.com/articles/ncomms1069http://dx.doi.org/10.1103/PhysRevLett.109.267404http://dx.doi.org/10.1103/PhysRevLett.112.073601http://dx.doi.org/10.1103/PhysRevLett.112.073601http://dx.doi.org/ 10.1103/PhysRevB.95.245115http://dx.doi.org/ 10.1103/PhysRevB.95.245115https://www.nature.com/articles/s41534-017-0046-yhttp://dx.doi.org/ 10.1103/PhysRevA.98.053819http://dx.doi.org/10.1103/PhysRevB.100.121109https://arxiv.org/abs/2005.09088https://www.nature.com/articles/s41467-018-08101-0https://arxiv.org/abs/1902.05160https://arxiv.org/abs/1905.10697https://www.nature.com/articles/s41567-019-0534-4http://dx.doi.org/ 10.1103/PhysRevB.101.205140https://pubs.acs.org/doi/10.1021/acsphotonics.9b01649https://pubs.acs.org/doi/10.1021/acsphotonics.9b01649http://dx.doi.org/ 10.1103/PhysRevLett.125.123602http://dx.doi.org/ 10.1103/PhysRevLett.125.123602http://dx.doi.org/10.1098/rsta.1959.0008http://dx.doi.org/10.1098/rspa.1971.0049http://dx.doi.org/10.1103/PhysRevLett.21.838

8

Supplementary Materials

Polarization dependence of the effective length scale

We here mention that the effective length scale ξg , which characterizes the light-matter interaction strength in the asymptoti-cally decoupled (AD) frame, in general depends on a polarization of an electromagnetic mode coupled to a many-body system.To demonstrate this, we consider a two-dimensional many-body system coupled to a circularly polarized electromagnetic modeas an illustrative example:

 = A(eâ+ e∗â†

), e =

1√2

[1i

]. (S1)

In this case, there are no terms that are proportional to ââ or â†â† in the Â2

term; this diamagnetic term simply renormalizesthe photon frequency without performing the Bogoliubov transformation. Thus, the resulting light-matter Hamiltonian in theCoulomb gauge is given by

ĤC =

∫dx ψ̂†x

(−~

2∇2

2m+ V (x)

)ψ̂x − gxωc

∫dx ψ̂†x(−i~∇)ψ̂x ·

(eâ+ e∗â†

)+ ~Ωâ†â, (S2)

where g = qA√ωc/m~ and we introduce

xωc =

√~

mωc, Ω = ωc

(1 +

Ng2

ω2c

). (S3)

Note that the renormalized photon frequency depends on g in a different way from the linearly polarized case discussed in themain text, for which Ω =

√ω2c + 2Ng

2.To asymptotically decouple the light and matter degrees of freedom, we can use a unitary transformation

Û = exp

[−iξg

∫dx ψ̂†x(−i∇)ψ̂x · π̂

], π̂ = i

(e∗↠− eâ

), (S4)

where we define the effective length scale ξg by

ξg =gxωc

Ω= xωc

g/ωc1 +Ng2/ω2c

. (S5)

This length scale for a circularly polarized case asymptotically vanishes in the strong-coupling limit with the scaling ∝ 1/g,which is faster than the linearly polarized case ∝ 1/g1/2 [see Fig. S1]. For the sake of completeness, we also show the fullexpression of the transformed Hamiltonian ĤU = Û†ĤCÛ in the present case:

ĤU =

∫dx ψ̂†x

[−~

2∇2

2m+ V (x+ ξgπ̂)

]ψ̂x+~Ωâ†â−

~2g2

mΩ2

[∫dx ψ̂†x(−i∇)ψ̂x

]2+

∫dxdx′

q2ψ̂†xψ̂†x′ ψ̂x′ ψ̂x

4π�0|x− x′|. (S6)

coupling strength g/ωc

ξ g

Circular polarizationLinear polarization

∝1/g1/2∝1/g

FIG. S1. Effective length scale ξg for a circularly polarized light (black solid curve) and a linearly polarized light (red dashed curve) againstthe bare light-matter coupling g. This length characterizes the effective interaction strength in the asymptotically decoupled frame. We setωc = ~ = m = 1.

9

Enhancement of the effective mass

We here demonstrate that the enhanced effective mass, which was discussed in the main text for a single-particle sector, in thecase of a general N -particle system. For the sake of simplicity, we consider a 1D translationally invariant many-body systemconsisting of N particles coupled to an electromagnetic mode, and write its Hamiltonian in the first-quantization form:

ĤC =

N∑j=1

(p̂j − qÂ)2

2m+∑j

10

x→ −x and π̂ → −π̂ and the lowest states reside in the even parity sector. Since the lowest-band Wannier state w(x) respectsthe even parity symmetry, a photon wavefunction must also be symmetric against π̂ → −π̂. This fact leads to 〈sin(Kξgπ̂)〉 = 0,where 〈· · · 〉 represents an expectation value with respect to a photon wavefunction with the even parity. Thus, after performingthe tight-binding approximation and taking into account the leading contributions, the projection results in the matrix elements

〈Ψj |ĤU |Ψi〉 = 〈Ψj |p̂2

2meff+ V (x) + v cos (Kx) [cos (Kξgπ̂)− 1] + ~Ωb̂†b̂|Ψi〉

' tg (δi,j+1 + δi,j−1) + µgδi,j +[t′g (δi,j+1 + δi,j−1) + µ

′gδi,j

]〈δ̂g〉+ ~Ωδi,j〈b̂†b̂〉, (S16)

where we introduce the renormalized tight-binding parameters depending on g as

tg =

∫dk

K�k,ge

ikd ∈ R, µg =∫dk

K�k,g, (S17)

t′g = v

∫dxw∗i−1 cos(Kx)wi ∈ R, µ′g = v

∫dxw∗i cos(Kx)wi, (S18)

and the operator describing the electromagnetically induced fluctuation by

δ̂g = cos (Kξgπ̂)− 1. (S19)

After transforming to the second quantization notation, we obtain the tight-binding Hamiltonian in the AD frame, which providesEq. (9) in the main text

ĤTBU =(tg + t

′g δ̂g

)∑i

(ĉ†i ĉi+1 + h.c.

)+(µg + µ

′g δ̂g

)∑i

ĉ†i ĉi + ~Ωb̂†b̂, (S20)

where the annihilation operator should be understood in terms of the expansion ψ̂x =∑j wj(x)ĉj . We remark that, while

this tight-binding description is valid when low-energy equilibrium properties are of interest, one may have to include furthercorrection terms for analyzing nonequilibrium dynamics. For instance, the contribution from the sin(Kx) term in Eq. (S12) canbe relevant when excitations to higher bands are nonnegligible.

We here note that the calculation of matrix elements of the shifted potential term V (x + ξgπ̂) has been separated into lightand matter parts. This simplification has its origin in the translationally invariance of the potential V (x), and can be transferredto a general periodic potential. While one needs to work with the operator-valued term such as cos(Kξgπ̂), in practice, it canstill efficiently be calculated since it usually suffices to set a photon-number cutoff to at most ∼ 100 in the AD frame, for whichπ̂ is just a matrix with a small dimension [cf. Fig. S2]. This is the reason why we did not need to rely on the approximative form(Eq. (10) in the main text) obtained by the expansion in ξgπ̂, but only on the tight-binding approximation, resulting in Eq. (S20).

Meanwhile, when we are interested in a nonperiodic potential, the simplification at a large coupling can be made possible byperforming the Taylor expansion and truncating at a finite order as discussed in the main text. Said differently, the calculationof the shifted potential can be challenging if the potential is nonperiodic and singular such that the truncation cannot be well-justified and the coupling strength lies in the intermediate regime g/ωc ∼ 1 for which the effective length ξg is rather large [cf.Fig. 1 in the main text].

For the sake of comparison, we next explain the construction of the tight-binding Hamiltonian in the Coulomb gauge. Westart from the Coulomb-gauge Hamiltonian

ĤC =p̂2

2m+ V (x)− gxΩp̂(b̂+ b̂†) + ~Ωb̂†b̂. (S21)

Similar to the above discussion, we consider the lowest-band Wannier states for the single-particle Hamiltonian with the baremass m: [

p̂2

2m+ V (x)

]ψ̃k = �̃kψ̃k, w̃j(x) =

∫dk

Ke−ikjdψ̃k(x), (S22)

and consider a manifold spanned by the following light-matter states

|Ψ̃j〉 =∫dx w̃j(x)|x〉 ⊗ |ψphoton〉. (S23)

11

We note that the dispersion �̃k is independent of g as we here consider the bare massm. The projection of ĤC onto this manifoldresults in the matrix elements

〈Ψ̃j |ĤC|Ψ̃i〉 = 〈Ψ̃j |p̂2

2m+ V (x)− gxΩp̂(b̂+ b̂†) + ~Ωb̂†b̂|Ψ̃i〉

' t̃ (δi,j+1 + δi,j−1) + µ̃δi,j −(λ̃gδi,j+1 + λ̃

∗gδi,j−1

)〈b̂+ b̂†〉+ ~Ωδi,j〈b̂†b̂〉, (S24)

where 〈· · · 〉 represents an expectation value with respect to an arbitrary photon state and the tight-binding parameters are definedby

t̃ =

∫dk

K�̃ke

ikd ∈ R, µ̃ =∫dk

K�̃k, λ̃g = ~gxΩ

∫dx w̃∗i−1(−i∂x)w̃i ∈ iR. (S25)

We again emphasize that, in contrast to the AD-frame case above, the tight-binding parameters are defined in terms of thesingle-particle states with the bare mass m; thus, in particular, t̃, µ̃ are independent of the light-matter coupling g.

In the second quantization notation, the tight-binding Hamiltonian can be written as

ĤTBC =∑i

([t̃− λ̃g

(b̂+ b̂†

)]ˆ̃c†i

ˆ̃ci+1 + h.c.)

+ µ̃∑i

ˆ̃c†iˆ̃ci + ~Ωb̂†b̂, (S26)

where the annihilation operator is defined in terms of the Wannier function with the bare mass, ψ̂x =∑j w̃j(x)

ˆ̃cj . Its eigen-spectrum can analytically be given by

�̃TBk,n,C = 2t̃ cos (kd) + µ̃−4λ̃2gΩ

sin2 (kd) + ~Ωn, (S27)

where n = 0, 1, 2 . . . The results plotted in Fig. 2 in the main text correspond to the n = 0 sector of this dispersion. It is evidentfrom Eq. (S27) that the tight-binding spectrum in the Coulomb gauge is completely independent of g at k = 0,±π/d, whichclearly indicates difficulties of level truncations in the Coulomb gauge.

Finally, we remark that the one-dimensional tight-binding model acquires additional contributions in the case of the circularlypolarized light. To see this, it is sufficient to consider the following single-particle continuum model [see Eq. (S3) for thedefinitions of the microscopic parameters]:

ĤC =p̂2

2m+ V (x) +

mΩ2yy2

2− gxωc p̂ ·

(eâ+ e∗â†

)+ ~Ωâ†â, (S28)

where e = 1√2[1, i]T is the polarization vector, Ω = ωc(1 + g2/ω2c ), and the electron is tightly localized in the transverse y

direction via the potential mΩ2yy2/2, while it is subject to the periodic potential V (x) in the x direction. Using the unitary

transformation (S4), we obtain

ĤU =p̂2

2meff+ V

(x+ iξg(â

† − â)/√

2)

+mΩ2y

[y + ξg(â+ â

†)/√

2]2

2+ ~Ωâ†â, (S29)

where meff = m[1 + (g/ωc)2] and ξg = gxωc/Ω = xωcg/(ωc + g2/ωc). It is now clear that, even when we are interested in

1D electron dynamics in the x direction and aim to derive the tight-binding model along this direction, we must in general takeinto account the contribution from the third term in the RHS of Eq. (S29) that arises from the coupling between the transversemotion and the circularly polarized light. Nevertheless, the simple 1D description along the x direction analogous to Eq. (5)in the main text (i.e., neglecting the orbital motion along the y direction) can still be recovered when (i) Ωy is large so thatmotional excitation in the y direction is suppressed and (ii) the coupling g is large in the sense that Ωy � g2/ωc (which meansmΩ2yξ

2g � ~Ω) in such a way that the light-matter coupled term can be neglected compared to the dressed photon term. We note

that the condition (ii) can in principle be satisfied for any finite Ωy provided that g is sufficiently large.

Photon-number cutoff dependence of the low-energy spectra

We here briefly mention the photon-number cutoff dependence of the low-energy spectra in different frames. We compare thespectra for the double-well potential V = −λx2/2 + µx4/4 in the AD frame ĤU = Û†ĤCÛ with Û = exp(−iξgp̂π̂/~) andthe Power-Zienau-Woolley (PZW) frame ĤPZW = Û

†PZWĤCÛPZW with ÛPZW = exp(iqxÂ/~):

ĤU =p̂2

2meff+ V (x+ ξgπ̂) + ~Ωb̂†b̂, (S30)

ĤPZW =p̂2

2m+ V (x) +mg2x2 + ig

√m~ωcx(↠− â) + ~ωcâ†â. (S31)

12

AD framePZW frame

nc

E - E

0

photon-number cutoff

ener

gy

nc

E - E

0

photon-number cutoff

ener

gy

g/ω =2c

nc

E - E

0

photon-number cutoff

ener

gy

g/ω =10c

g/ω =5c

nc

E - E

0

photon-number cutoff

ener

gy

g/ω =50c

FIG. S2. Comparisons of convergence of low-energy spectra in the AD frame (blue solid curves) and the PZW frame (red dotted curves)with respect to the photon-number cutoff nc at different coupling strengths g. The AD-frame energies efficiently converge at low nc, whilethe results in the PZW frame require an increasingly large cutoff at stronger g, and do not converge for g/ωc & 10 at least in the plotted scale.We set ωc =

√~/mωc = 1 and choose the parameters λ=3, µ=3.85.

In Fig. 3 in the main text, we show the results at the large photon-number cutoff nc = 100, and demonstrate that the PZW framefails to capture the key features in the extremely strong coupling (ESC) regime, such as the level degeneracy and narrowing.This is caused by the slower convergence of the PZW results at larger coupling g with respect to the photon-number cutoff nc.To see this explicitly, we plot the low-lying energies (subtracted by the lowest eigenvalue) in different frames in Fig. S2. Whilethe results in the AD frame efficiently converge already for low cutoff nc ∼ 5−10 at any coupling strength g, the convergencein the PZW frame becomes worse as g is increased. In particular, in the ESC regime (roughly corresponding to g/ωc & 10), thePZW results typically fail to converge within a tractable value of the photon-number cutoff. This difficulty stems from the rapidincrease of the mean-photon number in an energy eigenstate due to large entanglement among high-lying levels present in thePZW frame.

Derivation of the multimode generalization of the unitary transformation

We provide details about the derivation of the multimode generalization of our formalism presented in the main text. We startfrom the light-matter Hamiltonian including multiple spatially varying electromagnetic modes in the Coulomb gauge:

ĤC =p̂2

2m+ V (x)− q

2m

(p̂ · Â(x) + h.c.

)+q2Â

2(x)

2m+∑kλ

~ωkâ†kλâkλ, (S32)

13

where we consider the vector potential expanded by plane waves

Â(x) =∑kλ

�kλAk(âkλe

ik·x + h.c.), k · �kλ = 0, �kλ · �kν = δλν (S33)

with λ denoting polarization. To generalize the asymptotically decoupling unitary transformation Û to this multimode case, wefirst introduce the field operators

X̂kλ(x) ≡√

~2ωk

(âkλe

ik·x + â†kλe−ik·x

), P̂kλ(x) ≡

√~ωk

2i(â†kλe

−ik·x − âkλeik·x), (S34)

and define the coupling as

gk = qAk√ωkm~

. (S35)

We then rewrite the quadratic photon part of the Hamiltonian (aside the constant) as

q2Â2(x)

2m+∑kλ

~ωkâ†kλâkλ =∑kλ

P̂ 2kλ(x)

2+

1

2

∑kλk′λ′

(δkλ,k′λ′ω

2k + 2gkgk′�kλ · �k′λ′

)X̂kλ(x)X̂k′λ′(x)

=1

2

∑α

(P̂ 2α(x) + Ω

2αX̂

2α(x)

), (S36)

where we define the diagonalized basis labeled by α via

X̂kλ(x) =∑α

Okλ,αX̂α(x) (S37)

with Okλ,α being an orthogonal matrix.We next introduce the x-dependent annihilation operators via

b̂α(x) ≡√

Ωα2~X̂α(x) +

i√2~Ωα

P̂α(x), (S38)

and also define the vector-valued variables labeled by α as

ζα = xΩα∑kλ

�kλgkOkλ,α, (S39)

where xΩα =√

~mΩα

. We now introduce the unitary transformation in the multimode case by

Û = exp

[−i p̂

~·∑α

ξαπ̂α(x)

], π̂α(x) = i

(b̂†α(x)− b̂α(x)

), ξα =

ζαΩα

. (S40)

We note that, since the electromagnetic modes now explicitly depend on the position x, they do not commute with the momentumoperator p̂ in the transformation Û , and thus, the transformed Hamiltonian in general acquires additional contributions comparedto the simple expression obtained in the single-mode case [cf. Eq. (S30)]. Nevertheless, significant simplification can occur whenthe field variation is small compared with the effective length scale:

k|ξ| � 1 for |∇b̂| ∼ kb̂. (S41)

We emphasize that this condition is independent of system size and thus much less restrictive than the standard dipole approx-imation. In particular, Eq. (S41) can, in principle, be attained for any k if the coupling g is taken to be sufficiently strong suchthat the effective length scale |ξ| is short enough to satisfy this condition. Under this condition, the derivative terms of the fieldoperators b̂(x), π̂(x) can be neglected, resulting in the simple transformed Hamiltonian:

ĤU = Û†ĤCÛ '

p̂2

2m−∑α

(p̂ · ζα)2

~Ωα+ V

(x+

∑α

ξαπ̂α (x)

)+∑α

~Ωαb̂†α(x)b̂α(x), (S42)

which provides Eq. (12) in the main text.

Cavity Quantum Electrodynamics at Arbitrary Light-Matter Coupling StrengthsAbstract Acknowledgments References Polarization dependence of the effective length scale Enhancement of the effective mass Derivation of the tight-binding models Photon-number cutoff dependence of the low-energy spectra Derivation of the multimode generalization of the unitary transformation