-
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”.
∗ [email protected][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:[email protected]://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