Nonperturbative Quantum Electrodynamics in the Cherenkov Effect

Charles Roques-Carmes,1,* Nicholas Rivera,2 John D. Joannopoulos,2 Marin Soljačić,1,2 and Ido Kaminer3,†1Research Laboratory of Electronics, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139, USA2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA3Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

(Received 1 April 2018; revised manuscript received 18 July 2018; published 17 October 2018)

Quantum electrodynamics (QED) is one of themost precisely tested theories in the history of science, givingaccurate predictions to a wide range of experimental observations. Recent experimental advances allow for theability to probe physics on extremely short attosecond timescales, enabling ultrafast imaging of quantumdynamics. It is of great interest to extend our understanding of short-time quantumdynamics toQED,where thefocus is typically on long-timeobservables such asSmatrices, decay rates, and cross sections.That said, solvingthe short-time dynamics of the QED Hamiltonian can lead to divergences, making it unclear how to arrive atphysical predictions. We present an approach to regularize QED at short times and apply it to the problem offree-electron radiation into amedium, known asCherenkov radiation. Our regularizationmethod,which can beextended to otherQEDprocesses, is performed by subtracting the self-energy in free space from the self-energycalculated in the medium. Surprisingly, we find a number of previously unknown phenomena yieldingcorrections to the conventional Cherenkov effect that could be observed in current experiments. Specifically,the Cherenkov velocity threshold increases relative to the famous conventional theory. Thismodification to theconventional theory, which can be non-negligible in realistic scenarios, should result in the suppression ofspontaneous emission in readily available experiments. Finally, we reveal a bifurcation process creatingradiation into new Cherenkov angles, occurring in the strong-coupling regime, which would be realizable byconsidering the radiation dynamics of highly charged ions. Our results shed light on QED phenomena at shorttimes and reveal surprising new physics in the Cherenkov effect.

DOI: 10.1103/PhysRevX.8.041013 Subject Areas: Optics, Photonics, Quantum Physics

I. INTRODUCTION

One of the early achievements of quantum electrody-namics (QED) was the accurate match of the predictedhydrogen-energy-level shifts with the experiments byLamb and Retherford [1]. This significant accomplishmentwas allowed by the work of Tomonaga [2], Bethe [3],Schwinger [4], and Feynman [5] and the later unification byDyson [6]. They eliminated the problematic divergenceof the Lamb-Retherford shift due to the photon loopcorrection to the atom’s propagator. Renormalization ofthe electron mass also led to a very precise agreementwith the experimentally measured values of the Lamb-Retherford line shift [1], the anomalous magnetic momentof the electron [7], and the anomalous hyperfine splitting ofthe ground state of the hydrogen atom [8]. Since these early

successes, QED has been one of the most accurate theoriesof modern physics and bolstered fundamental develop-ments, from quantum field theory to grand unified theories.Central to calculations in QED and other field theories isthe S matrix, representing the long-time transition ampli-tude between an initial state and some final state [9–11].From the S matrices, one can derive many importantobservables such as decay rates, cross sections, and self-energies, which have been successfully used to calculate agreat variety of effects in particle physics, condensed matterphysics, atomic physics, and many other fields. Despitethese successes, S-matrix methods are essentially used tomake predictions at infinite times. Altogether in QED, veryfew studies consider the physics at short times (e.g.,Refs. [12–14]) and almost none when considering thecoupling to a continuum of electromagnetic modes. Westill lack the basic methods to describe the short-timedynamics in such systems, where it remained unknownhow to apply renormalization at finite times.One of the emblematic problems in light-matter inter-

action that can be used as a testing ground for new ideas inQED is the Vavilov-Cherenkov effect (abbreviated as theCherenkov effect)—the radiation by a charged particlepropagating in a dispersive medium at a speed larger than

*chrc@mit.edu†kaminer@technion.ac.il

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

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the phase velocity of light in the medium. It is the leading-order process in the radiation by a free electron in a mediumand is the fundamental building block for any diagramdescribing the interaction of charges with photons in amedium. This effect was originally observed by Vavilovand Cherenkov in 1934 [15,16] and later explained byFrank and Tamm [17]. Most of the quantum treatment ofthis problem, which originated from Ginzburg’s Ph.D.thesis [18], did not result in a significant deviation fromthe original prediction by Frank and Tamm [17]. Even latertheories taking into account higher-order processes did notfind regimes where the quantum nature of the interactingparticles would result in a significant mismatch from theoriginal prediction of Frank and Tamm [17–19]. Onlyrecently was it theoretically predicted that quantum effectscould emerge in the optical regime when taking intoaccount the quantum recoil of the electron or shapingthe electron wave function [20].Despite the many years that have passed since the original

discovery of the Cherenkov effect, it continues to lead to newdiscoveries in different systems including heavy-ion jets[21–26], nonlinear phase-matched systems [27–33], movingvortices in Josephson junctions [34,35], condensed-matteranalogue models of gravitational physics, and movingdipoles showing the hybridization of the Cherenkov andDoppler effects [36–38]. The concept of the Cherenkovvelocity threshold is also found in the theory of superfluidityand Landau damping of highly confined plasmons [39,40],two topics of major interest these days. New physics in theCherenkov effect, such as backward Cherenkov radiation[41], was unveiled with free electrons propagating incomplex nanophotonic systems: photonic crystals [42],metamaterials [43–45], plasmonic systems [46], and gra-phene [47]. This continued interest in explaining the theoryof Cherenkov radiation in novel settings originates in itswide range of applications, from high-energy particlecharacterization [48] to biomedical imaging [49].The Cherenkov effect follows the Frank-Tamm formula

that gives the rate of photon emission per unit frequency∂εΓrad ¼ ½ðαZβÞ=ℏ�f1 − ½1=ðnεβÞ2�g. In this expression,αZ ¼ Z2α is the effective fine-structure constant, α ≈1=137 being the fine-structure constant and Z the particlecharge; β ¼ v=c, v being the electron speed and nε therefractive index of the medium (function of the photonenergy ε ¼ ℏω). From the Frank-Tamm formula and itsgeneralizations to nanostructured optical systems oranomalous cases [42,50], one sees that, in the opticalregime, the timescale at which a photon is emitted in themedium is between picoseconds and femtoseconds.Physics at these timescales was unveiled at the turn ofthe millennium with the development of ultrafast optics,light-wave electronics, and high harmonic generation [51].Today, the integration of electron optics with opticaltechnologies allows the probing of atomic transitions[51], molecular bonding [52], and even more recently

dynamics of plasmonic systems [53–59]. Given thesenew experimental techniques, it is timely to ask how thelong-standing physics of the Cherenkov effect changes atvery short timescales.In this article, we develop a general framework for time-

dependent QED, which is free of short-time divergences,and use our framework to reveal new underlying phenom-ena in Cherenkov radiation. This process yields concreteclosed-form solutions of the power spectrum of photonemission and a surprising increase in the Cherenkovvelocity threshold. Applying our theory to probe the timedependence of the emitted radiation from a charged particlepropagating in a medium, we observe new quantumphenomena in the Cherenkov effect. Even when theelectron and photon are weakly coupled (α ≪ 1), thewell-known Cherenkov dispersion relation is modifiedfrom the known one on account of a nonzero self-energyassociated with the virtual emission and reabsorption ofphotons in the medium. When pushing the limits of ourformalism to strongly coupled regimes ðαZ2 ≳ 1Þ, we findevidence of regimes where the electron experiences anonexponential decay and the Cherenkov dispersion rela-tion changes drastically. In these strongly coupled settings,we predict the emergence of multiple Cherenkov lines thatcan be tested in existing experiments such as heavy-ioncolliders. More generally, the Cherenkov effect provides uswith a powerful playground to develop new general toolsfor time-dependent QED. Our work suggests applicationsin the study of fundamental effects in attosecond physics,such as high harmonic generation, and new experimentalendeavors of Cherenkov radiation and its analogues instrongly coupled settings.

II. TIME-DEPENDENT NONPERTURBATIVETHEORY OF CHERENKOV RADIATION

We consider the system shown in Fig. 1. A chargedparticle, such as a single electron or a highly charged ion,is propagating at speed v ¼ βc inside a lossless materialdefined by its refractive index nε. According to the conven-tional Cherenkov theory, if the particle velocity surpassesthe speed of light in the medium β > c=nε, it emits aphoton of energy ε at angle θC relative to the electrontrajectory, satisfying the condition [Fig. 1(a)] cos θC ¼½1=ðnεβÞ� [17]. This energy-angle relation is a directconsequence of energy-momentum conservation [60],while the photon emission rate per unit energy can bederived from Fermi’s golden rule [18].We solve for the time dependence of the time-evolution

operator corresponding to the QED Hamiltonian using theresolvent method [10]. This method allows us to performresummations of partial sets of the perturbation seriesin a compact algebraic formulation while maintainingdynamical information of the correlated electron-photonwave function. The method works as follows: We refor-mulate the Schrödinger equation in terms of the resolvent

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operator GðzÞ ¼ ðz −HÞ−1, which is the complex-variableFourier transform of the unitary time-evolution operator,used to determine the time-dependent electron-photonwave function. The matrix elements of G represent theFourier transform of transition amplitudes between differ-ent states. We first derive a matrix equation for G0 ≡hp; 0jGjp; 0i (for more details on the formalism and theQED Hamiltonian, we refer to Appendix A). Inverting thematrix equation and moving back into the time domain, wefind the following expressions for the evolution operatorprojected on the zero-photon stateU0ðtÞ and the probabilityof observing a photon p1ðtÞ:

U0ðtÞ ¼1

πe−iEit=ℏ

ZRdz e−izt=ℏ

ΓðzÞ2

½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2; ð1Þ

p1ðtÞ ¼1

ℏ2

Zdz

ΓðzÞ2

����Z

t

0

dt0e−izt0=ℏeiEit0=ℏU0ðt0Þ����2; ð2Þ

where Ei (respectively, Ef) is the charged particle’sinitial (final) energy and p0 ¼ jU0j2 the probability ofthe electron being in a state with zero photons. Equation (1)can be seen as a modification of the exponential decay law.In particular, when ΔðzÞ and ΓðzÞ are constant, thiscorresponds to the usual exponential decay law for sponta-neous emission which is valid at intermediate times.Similar modifications have been studied in the case oftwo-level systems [10] and for quasistationary states [61].From Eqs. (1) and (2), one can derive time-dependentphysical observables such as the spectrum of emittedphotons per unit angle and per unit frequency∂2p1=∂ε∂u, which can also be used to determine the rateof photon emission. Figure 2 shows several examples ofsuch spectral emission rates. In Eq. (1), ΔðzÞ and ΓðzÞ,which fully specify the dynamics of the system, are the realand imaginary parts, respectively, of the self-energy of the

electron fðzÞ [also known as the radiative shift in atomic,molecular, and optical (AMO) physics]:

fðzþ i0∓Þ ¼ αZβ2

2π

Z∞

0

dεZ

1

−1du

nεεð1 − u2Þzþ i0∓ − εð1 − nεβuÞ

≔ ΔðzÞ � iΓðzÞ2

: ð3Þ

The real and imaginary parts of the self-energy function canbe written analytically as

ΔðzÞ ¼ αZβ

2π

Z∞

0

dε�2ðz − εÞnεεβ

�

þ�1 −

�z − ε

nεεβ

�2�log

���� zþ ε½nεβ − 1�z − ε½nεβ þ 1�

����; ð4Þ

ΓðzÞ2

¼ αZβ

2

Z∞

0

dε

�1−�ε− zεnεβ

�2���

ε− zεnεβ

�2

≤ 1

�; ð5Þ

respectively, where Θ½.� ¼ 1 if the condition betweenbrackets is satisfied, Θ½.� ¼ 0 otherwise.Δð0Þ and Γð0Þ exactly match the real and imaginary part,

respectively, of the radiative shift calculated via the second-order perturbation theory in the weak-coupling regime.More generally, ΔðzÞ and ΓðzÞ correspond to the second-order perturbation of the energy, shifted by the complexfrequency variable z. Our approach is nonperturbative inthe sense that the self-energy function models absorptionand reemission cycles of the photon before a final photonemission, thus taking into account an infinite sum of single-loop Feynman diagrams (see Fig. S4 in SupplementalMaterial [62]). We add additional diagrams altering theelectron after the photon emission, showing quantitativecorrections but no qualitative changes to our predictions(see Ref. [62], Sec. I). Our theory could be extended to take

(a) (b) (c)

FIG. 1. The quantum Cherenkov effect. (a) Conventional Cherenkov effect. (b) Regularized quantum Cherenkov effect for an electron(weak coupling), where the particle’s regularized self-energy is a small correction. (c) Regularized quantum Cherenkov effect for ahighly charged ion, where the self-energy results in large corrections.

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into account higher-order processes with a more sophisti-cated numerical treatment [63].

III. THE RADIATIVE SHIFT OF THEFREE ELECTRON

Upon a first evaluation of the self-energy, we find thatthe real part features a UV divergence as a result of theunbounded photonic density of states at high energies [64],even when the index of refraction is too low for Cherenkovradiation. Nevertheless, we regularize the self-energyfunction in a way that agrees with physical predictions,by subtracting its free-space part (nε ≡ 1):

fðzÞ → fðzÞ − fðz; nε ≡ 1Þ: ð6Þ

To get the conservation of probability, we substitute theregularized form of ΓðzÞ in Eq. (2), which yieldsp0ðtÞ þ p1ðtÞ≡ 1. Regularizing the spectral angular den-sity of the emitted photon also suppresses unphysicalemission in free space (see Appendix B and Ref. [62],Sec. I, for further discussion of this point). This approachcould be extended to any optical medium, as it takes intoaccount material dispersion, and can be directly generalizedto arbitrary geometries of photonic structures, by takinginto account the eigenmodes of Maxwell’s equations forsuch structures. Thus, our approach can be used in awide range of phenomena stemming from the Cherenkoveffect, including Cherenkov variants in 2D materials[47,65], in circuit QED [34,35], and in heavy-ion physics[22]. Moreover, this regularization approach can

be extended to other types of electron-photon interactions.For example, we propose a regularization scheme to givewell-defined time dynamics for electron-photon (Compton)scattering in vacuum in Ref. [62], Sec. III.When the coupling to light is weak—as in the case of an

electron moving through a typical optical medium—wefind that the contribution of the self-energy function inEqs. (1) and (2) can be approximated by its value taken atthe zero of ΔðzÞ − z, which we denote as z0 ¼ Δðz0Þ. Thisapproximation, known as the flat-continuum approxima-tion, retains validity when the decay rate or linewidth of theelectron-emission process associated with Cherenkov radi-ation is much narrower than the frequency variation of thephotonic density of states.We find that the real part of the self-energy—known as

the radiative shift z0 [10]—is typically of the order of−αβ2=ð2πÞΔε ∼ −0.001 eV for a single electron in theoptical regime [see Appendix C, Eq. (C12)]. This free-electron radiative shift is significantly greater than itsatomic counterpart—the Lamb-Retherford shift [1]—and,as a result, leads to potentially large corrections to quantumobservables for small photon energies or particle velocitiesclose to the conventional Cherenkov threshold (nεβ → 1).In particular, it modifies the well-established Cherenkovrate of emission to become

∂εΓrad ¼αZβ

ℏ

�1 −

�ε − z0εnεβ

�2�: ð7Þ

(We denote with a bar sign every quantum observable andstate probability from our regularized theory—taking into

(a) (b)

(c) (d)

FIG. 2. Nonperturbative self-energy-induced modifications to Cherenkov radiation in weakly and strongly coupled regimes.(a) [respectively, (c)] An electron propagating at velocity β ¼ 0.9999 (respectively, β ¼ 0.7) in a chamber filled with CF4 gas(respectively, in a material of constant index equal to 4) emits a photon with spectral angular density ½ð∂2p1Þ=ð∂ε∂uÞ�ðtÞ, whereu ¼ cos θ shown in (b) [respectively, (d)] at different times. The gradual narrowing of the Cherenkov energy-angle relation for longtimes is a direct consequence of the time-energy uncertainty principle.

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account the particle-regularized self-energy—in contrast tothe conventional Cherenkov theory.) Additional correctionscan come up from our quantum formalism. They typicallyinclude the electron recoil due to the Cherenkov emissionof a single photon [20,47], which depends on the photonmomentum nεε=c relative to the electron momentumβEi=c. The radiation-angle correction is proportional tothe radiative shift z0 and to the electron recoil. In scenarioswhere the electron recoil is large [47], we expect to observeadditional large self-energy corrections. Even withoutelectron recoil, the radiative shift z0 still significantlymodifies the Cherenkov decay rate. This modificationmakes the new quantum corrections that we find far biggerthan any previously found quantum effects in Cherenkovradiation [18,20].Figure 3 shows that the account of the nonperturbative

self-energy can shift the well-known Cherenkov threshold:1=ðnεβÞ ≤ ε=ðε − z0Þ ≈ 1þ z0=ε for small negative valuesof z0. As a consequence, the regularized rate of emission∂εΓrad converges to zero when approaching the regularized

Cherenkov threshold: In an experiment, this convergencetranslates into arbitrarily fewer photons emitted, comparedto the conventional prediction. The most extreme deviationis found when the particle velocity is above the conven-tional threshold and below the regularized threshold, wherethe conventional theory predicts a finite rate of emission,while the regularized theory states that the spontaneousemission of a photon is forbidden.Far above the conventional Cherenkov velocity thresh-

old, the values of the regularized decay rate Γðz0Þ ¼ Γ0 andof the conventional decay rate Γðz ¼ 0Þ agree, as can beseen in Fig. 3(b). However, the divergence of 1=Γ0 aroundβ ≳ βth results in arbitrarily small decay rates that directlytranslate on the dynamics of p0ðtÞ, as illustrated inFig. 4(a). While the conventional theory predicts a totalabsence of decay below the conventional Cherenkovthreshold, the regularized probability of the zero-photonstate decays to a value of 1=f1 − ½dΔðzÞ=dz�jz0g2 slightlybelow unity [Fig. 3(b)]. This result could be interpreted asthe electron transitioning to a modified ground state of itsenergy dressed by the Cherenkov coupling, or it may

(a)

(b)

FIG. 3. The modified Cherenkov velocity threshold. (a) Regu-larized decay rate (blue curve) as a function of the distance to theconventional Cherenkov threshold βth, above this threshold.(b) Probability of no Cherenkov emission p0ðþ∞Þ as a functionof the distance to the conventional Cherenkov threshold βth,below this threshold. The system considered for this example is asingle electron traversing an index window defined by nε ¼2 � 1ðε ∈ ½0; 5 eV�Þ [where 1ðε ∈ AÞ is the indicator function ofa subset of energies A]; however, similar effects occur regardlessof the choice of particle or transparent index medium.

(b)

(a)

FIG. 4. Modified time dynamics in weakly and stronglycoupled regimes. (a) Comparison of the regularized and conven-tional evolution of p0ðtÞ for an electron very close to theregularized Cherenkov threshold β − βth ≈ 3 × 10−5. Here, anelectron propagating at a speed of β ¼ 0.5004 emits a photon(e.g., its zero-photon state probability decays) at a rate that is 2orders of magnitude less than the conventional prediction. (b) Instrongly coupled regimes, the presence of several poles of thefunction fðzÞ can result in a pseudoperiodic decay of p0ðtÞ.

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complete a full decay when introducing higher-orderdiagrams including additional, multiple, photons. Farbelow the conventional Cherenkov threshold, the behaviorsof the conventional and regularized theories match again.We observe a discontinuous transition in the behaviorabove and below the Cherenkov velocity threshold, whichis essentially due to a discontinuous derivative of ΓðzÞ.However, we expect higher-order processes, that areneglected in the current formalism, to smoothen thistransition: For multiphoton emission processes, the phasespace for emission is larger, allowing for the emission ofphotons that do not individually satisfy the conventionalCherenkov relation.This modification of the Cherenkov threshold has direct

implications on the time-dependent spectrum of Cherenkovradiation, compared to the conventional theory. Our theoryreveals ultrafast dynamics of Cherenkov radiation, for whichwe already observe significant modifications to the well-known observables: its energy-angle relation and the corre-sponding photon rate of emission. Specifically, Fig. 2(b)shows the implications of the time-energy uncertaintyprinciple broadening the Cherenkov dispersion at very shorttimes. These new effects can be seen through the flat-continuum approximation [10], where the radiative shift z0results in an effective decay rate that we denote asΓ0 ¼ Γðz0Þ. This approximation can be applied toEq. (1), as it resembles the Fourier transform of aLorentzian peaked at z ¼ z0. The Fourier transform yieldsan exponential decay with the rate Γ0=ℏ: U0ðtÞ ≈e−iðEiþz0Þt=ℏe−Γ0t=2ℏ [p0ðtÞ ≈ e−Γ0t=ℏ] with ∂2p1ðtÞ=∂ε∂upeaked around Ef − Ei ¼ z0, corresponding to the regular-ized Cherenkov dispersion relation shown in Fig. 1(b), witha finite width in the angular domain proportional to Γ0

[Fig. 2(b)]. However, our predictions remain qualitativelyvalid only for times comparable to or less than the effectivedecay time of the zero-photon state ℏ=Γ0, after which it is nolonger legitimate to assume a single photon is emitted. Toaccount for longer times, it is necessary to extend thedimension of the initial Hilbert space, allowing the emissionof more than a single photon (such an approach is proposedand discussed in Ref. [62], Sec. I). For dynamics at shorttimescales of the Cherenkov effect, which are of primaryinterest, this restriction is unimportant, and our theoryremains predictive.

IV. BIFURCATION BETWEEN WEAKLY ANDSTRONGLY COUPLED REGIMES

Intriguing consequences of our nonperturbative formal-ism arise when pushing the limits of our formalism to thestrong-coupling regime in which the effective fine-structureconstant significantly increases, αZ ¼ Z2α≳ 1, leading tothe collapse of the flat-continuum approximation, whichwas accurately describing the physics above but does nothold here. Note that while in typical quantum field theories

the coupling constants scale like αZ, the coupling relevantto the Cherenkov effect actually scales like αZ2, because itis an effect of a single vertex diagram. Physically, the strongcoupling could be realized in the Cherenkov dynamics ofhighly charged ions. In the strongly coupled case, we findthat multiple zeros, zi, of ΔðziÞ − zi ¼ 0 exist, allowing fora multibranched Cherenkov dispersion relation, withmultiple Cherenkov angles θi satisfying cos θi ¼ ½ðϵ − ziÞ=ðnϵϵβÞ� for multiple zi, as illustrated in Fig. 1(c). InFig. 2(c), we consider the specific case of an Fe26þ ionmoving at v ¼ 0.7c, corresponding to a kinetic energy of20 GeV, as might be realized for various particles in largeaccelerator facilities [67] and high-intensity plasma wake-field accelerators [68]. Coincidentally, laser acceleration ofiron atoms has recently reached nearby energy scales [69].In this particular case, where the effective fine-structureconstant is about 4.9, we see that, at long times, the spectraldistribution of emitted photons is highly concentratedaround two angle-frequency relations, both substantiallydifferent from the conventional Cherenkov relation. Thesephotons are emitted over a short timescale, ti ∼ 1=ΓðziÞ, ofthe order of femtoseconds. Interestingly, we find that, forsome emission frequencies, the emission angles can bebackwards relative to the direction of electron motion,which is impossible in the conventional Cherenkov effectwithout a negative index of refraction.Considering the detailed time dynamics of the charge,

we find that these multiple poles can translate into apseudoperiodic decay of the zero-photon state, as can beseen in Fig. 4(b) for a Fe26þ. Pushing the effective charge toeven larger values, we predict Rabi oscillations of the zero-photon probability, when the imaginary part of one of thepoles cancels out (see Ref. [62], Sec. I). These dampedRabi oscillations are similar to those experienced by anatom in strong coupling with a lossy cavity mode [70].Quantum Rabi oscillations have recently been investigatedin ultrastrong-coupling regimes of cavity [71] and circuitQED [72], where a method for treating multiple discretephotonic modes has been proposed. Importantly, unlikebound charge systems that have discrete energy levels, herewe work with a charged particle that can take a continuumof energy values. Moreover, the energy scale of free-chargetransitions is not limited by the energy levels of AMOsystems (can reach the x- or gamma-ray scale and not onlyvisible or IR or below). We should expect new effects thathave not been observed in cavity QED, such as the excitingperspective of observing ultrastrong-coupling physics froma single particle in the optical regime. These considerationsmake the appearance of AMO-physics-inspired phenomenalike Rabi oscillation appealing, as no work has everdemonstrated free-charge vacuum Rabi oscillations andrelated phenomena. Our method could bridge the gapbetween AMO physics and free-charge physics, allowingus to systematically translate ground-breaking results inAMO to light-matter interactions with free charges.

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To reveal the underlying physics, we analyze the relationbetween the new nonperturbative Cherenkov dynamics andthe pole structure of the regularized self-energy. We showin Fig. 5 how an abrupt change in the pole structure signalsa bifurcation process between the conventional Cherenkovdynamics and the strong-coupling physics reported inFig. 2(c). This transition is characteristic of a bifurcationprocess, as a small change of a parameter (here, the electronvelocity β or the particle charge Z) can result in theemergence of new branches of poles or the cancellationof their imaginary part. Physically, these abrupt changesin the pole structure result, respectively, in new multipledecay modes and Rabi-like oscillations of the decay rate. Inorder to show the parameter space in which the reportedeffects are expected to occur, we study the detaileddependence of the poles of fðzÞ as a function of the ionenergy [Fig. 5(a)] and as a function of the ion charge Zaltering the effective fine-structure constant αZ [Fig. 5(b)].Stronger-coupling regimes can be effectively achieved

either by increasing the velocity of a particle of a givencharge Z [Fig. 5(a)] or by increasing the particle’s effectivecharge, at a given speed β [Fig. 5(b)]. In both cases, wedenote the emergence of new poles at a given value of β orZ, corresponding to critical coupling. The emerging polesalways appear in pairs and split along two branches forlarger β or Z. An even larger effective charge results in theemergence of two supplementary poles before cancelingthe real part of the original pole Γ0 ¼ 0 for Z ≥ 56, whichresults in the emergence of a Dirac delta function in Eq. (1)and, thus, in the previously mentioned Rabi oscillations.Cherenkov radiation from highly charged ions in these

ranges of energies has been experimentally reported inheavy-ion colliders [25,67]. It is worth mentioning thatanomalous Cherenkov radiation in these settings has beenobserved and attributed, at the time, to superluminalparticles—tachyons [26]. The account of the electronrelativistic recoil [19,73] was proposed as an explanationfor the observation of seemingly superluminal particles.Interestingly, we also predict that the Cherenkov angle ofphotons radiated by highly charged ions would verify1=ðnϵ cos θiÞ > 1 at some frequencies, which is the behav-ior expected from superluminal particles [26].QED interactions are conventionally limited by the size

of the fine-structure constant. By increasing the effectivefine-structure constant, we explore strong-coupling physicsin QED and reveal the presence of bifurcation processes inthe Cherenkov effect. The bifurcation is a strong qualitativeindicator of new phenomena occurring in these ranges ofparameters that may occur at quantitatively differentparameters due to other strong-coupling corrections.These findings point to the possibility of studying thephysics of strong-coupling field theories like quantumchromodynamics [9], using some of the most commonlystudied and relatively accessible effects of QED. Thepresence of a critical parameter (effective fine-structureconstant or electron velocity) with a splitting of the polebranch is reminiscent of a phase transition. However,we should include higher-order processes in the calculationof the nonperturbative self-energy to trust quantitativepredictions in these (ultra)strongly coupled settings.Additional corrections occur due to electron-positroncreation and annihilation diagrams that dress such high-energy particles in their steady states (corresponding toresummations at all orders in αZ [23], which we can safelyneglect when αZ < 1). Further corrections also occur dueto interactions between the ion and the medium that cannotbe described through the macroscopic permittivity of themedium and require a more extensive microscopic descrip-tion (e.g., bremsstrahlung). For longer times, higher-orderdiagrams should be taken into account in our calculations;however, it is hard to predict their relative contribution instrongly coupled regimes: For instance, it has recently beenpredicted that the rate of spontaneous photon emission byan atom in the ultrastrong-coupling regime can actually

(a)

(b)

FIG. 5. Cherenkov radiation in the strongly coupled regime:bifurcation process causing abrupt changes in Cherenkov angles.New regimes of behaviors allowed by strong coupling can beexplained by studying the poles of the function fðzÞ. (a) Real(solid lines) and imaginary (dashed lines) parts of self-energyfðzÞ as a function of the reduced speed β of a highly charged ionFe26þ. (b) Real (circles) and imaginary (rectangles) parts of thepoles of fðzÞ as a function of the effective charge number of anion Z, propagating in a dielectric medium defined by an indexwindow equal to 4. Definitions of weak and strong coupling areconsistent with Ref. [10].

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decrease as the coupling is increased, suggesting some kindof destructive interference of the bare spontaneous emis-sion diagram with higher-order diagrams that terminatewith the emission of a single photon [74]. The smallerdecay rate observed in our prediction [Fig. 4(b)] alreadysuggests partially destructive interference between thefundamental Cherenkov diagram and our nonperturbativesummation. Higher-order processes and all-order develop-ments in αZ2 could be integrated in our theory byadequately modifying the coupling between different termsof the resolvent operator.

V. CONCLUSION

In summary, we present a method to regularize time-dependent quantum observables in systems where light andmatter interact. Our theory accounts for the nonperturbativeself-energy in a time-dependent way. Applied to theCherenkov effect, we predict readily observable modifica-tions to the conventional theory in weakly coupled regimesand propose a pathway to reveal new physics in stronglycoupled regimes that can be effectively achieved withhighly charged ions. The observed bifurcation process inthe strong-coupling regime originates from the behavior ofthe self-energy function fðzÞ. Thus, more generally, weshould expect to see similar bifurcation processes in anysystem where strong coupling can be effectively achieved[10]. This observation is especially interesting, sincethere are Cherenkov analogues in platforms, such assuperconducting qubits, where strong-coupling physics isusually observed [34,35], and therefore our predictionscould be tested there.An experiment involving Cherenkov radiation in the

strong-coupling regime may also furnish new tests ofradiation reaction in dielectric media [75], which is basedon the observation that radiation-reaction effects are man-ifested in the self-energy of the system of the chargedparticle and the quantized electromagnetic field [76,77].Further physical understanding of the self-energy canalso be provided by adequately transforming the QEDHamiltonian via the Pauli-Fierz transformation, thus clearlyexhibiting the interplay between the particle transverse fieldand mass correction [10].Other directions to induce stronger-coupling regimes

can be investigated by our formalism. Couplings can beenhanced by using systems that display strong light-matterinteractions, like photonic cavities, or polaritonic media,which support modes of an effectively high index ofrefraction. Examples of such polaritonic media are thinmetallic films, graphene, and thin polar dielectrics, each ofwhich can potentially support optical modes with aneffective mode index of refraction of about 100 [78,79],thus being a promising paradigm for enhanced light-matterinteraction [47,65,66]. Another interesting option forachieving free-electron–photon strong coupling is to con-fine a free electron into a photonic cavity which supports a

broadband resonance phase matched to the electron phasevelocity, allowing the electron to coherently emit andreabsorb the cavity photons, thus experiencing Rabi oscil-lations with free electrons.Additionally, our predictions create new opportunities

for the design of ultrafast Cherenkov probes. The theo-retical ability to probe short-time dynamics in systemswhere multiple particles interact pushes towards experi-mental advances in pump-probe-like experiments or time-dependent electron energy-loss spectroscopy [59,80–82].More specifically, our findings coincide with the recentdevelopment of time-resolved imaging techniques combin-ing electron optics with state-of-the-art spectroscopy[83–86]. We believe these theoretical findings are showingnew prospects by which free electrons and other chargedparticles can provide a platform for the exploration ofquantum light-matter interactions and strong-couplingphysics.

ACKNOWLEDGMENTS

The authors acknowledge Robert Jaffe (MIT), JeffreyGoldstone (MIT), Jean Dalibard (College de France),Eric Akkermans (Technion), Morgan Lynch (Technion),and Liang Jie Wong (SIMTech) for helpful discussions.N. R. was supported by a Department of Energy fellowshipNo. DE-FG02-97ER25308. I. K. acknowledges support ofthe Azrieli Faculty Fellowship, supported by the AzrieliFoundation and by the Marie Curie Grant No. 328853-MC-BSiCS. This material is based upon work supported inpart by the U.S. Army Research Laboratory and the U.S.Army Research Office through the Institute for SoldierNanotechnologies, under Contract No. W911NF-18-2-0048.

APPENDIX A: GENERAL TIME-DEPENDENTTHEORY OF THE QUANTUM

CHERENKOV EFFECT

In this work, we consider the following system: asingle electron propagating at speed v ¼ βc (momentump ¼ mv), interacting with a continuum of photons withmomenta q ¼ ℏk. We consider the following Hilbert space:

H ¼ Hel ⊗ Hph; ðA1Þ

whereHel (respectively,Hph) is the Hilbert space describingthe quantum states of a single electron (respectively, thefull Fock space of photons), defined by its momentum jpi(respectively, by the momentum of every photonjq; q0; q00;…; qðNÞ;…i).We consider the usual QED Hamiltonian to describe the

interaction of a spin-1=2 particle with the electromagneticfield [9]:

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HQED ¼ eZ

d3xψ=Aψ ðA2Þ

¼ eZ

d3xψ†γμAμψ ; ðA3Þ

where Aμ is the electromagnetic vector potential andfγμgμ∈f0;1;2;3g are the Dirac matrices. In this work, weassume that the photon energies are much smaller than theenergy of the moving charge, which enables us to neglectspin effects and the charge recoil due to photon emission.Assuming the Coulomb gauge ∇ · A ¼ 0 in Eq. (A3), wecan show that the coupling Hamiltonian is equal toV ¼ A · v, where v is the electron velocity [47].In the following, we define α as the fine-structure

constant

α ¼ e2

4πϵ0ℏc; ðA4Þ

and the coupling Hamiltonian V.Additionally, only the polarization in the ðp; qÞ plane

contributes to the coupling. These assumptions allow us tocompute the following coupling matrix element:

hp0; qjVjp; 0i ¼ eβ sin θ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiℏc2

2Ωϵ0ωq

s Zd3x

e−ip0·xffiffiffiffiΩ

p e−iq·xeip·xffiffiffiffiΩ

p

ðA5Þ

¼ ev sin θ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiℏ

2Ωϵ0ωq

sð2πÞ3Ω

δð−p0 − qþ pÞ;

ðA6Þ

where jp; 0i is the quantum state of a single electronpropagating with momentum p in the medium, with nophoton present. Here, ωq ¼ 1

2½∂=ð∂ωÞ�ðω2

qεωÞ ¼c2q½ð∂qÞ=ð∂ωÞ� is the modified normalization factor ina dispersive medium [87] and Ω the normalizationvolume factor.

1. First-order perturbation theory

We first quickly review the result from the application ofFermi golden rule to this quantum system, to predict theintensity spectrum of Cherenkov radiation. A similarderivation can be found in Ref. [87]. We chose similarnotations as in the main text of this article: Ω is thenormalization volume factor, ℏ is the reduced Planckconstant, ω is the photon energy, and (Ei, p) [respectively,(Ef, p0)] are the (energy, momentum) of the initial (respec-tively, final) states of the electron.We compute the decay rate of the initial state, given by

Fermi’s golden rule (FGR):

ΓFGR ¼ 2π

ℏ

Zd3q

ð2πÞ3=ΩZ

d3pð2πÞ3=Ω

× jhp0; qjVjp; 0ij2δ½ℏω − ðEi − EfÞ�: ðA7Þ

Using Eq. (A6), we get the following:

ΓFGR ¼ 2π

ℏ

Zd3q

ð2πÞ3=ΩZ

d3pð2πÞ3=Ω

×

����ev sin θffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ℏ2Ωε0ωq

sð2πÞ3Ω

δð−p0 − qþ pÞ����2

× δ½ℏω − ðEi − EfÞ� ðA8Þ

¼ αβ

Z þ∞

0

dω

�1 −

�1

nωβ

�2���

1

nωβ

�2

≤ 1

�:

ðA9Þ

We then find the original Frank-Tamm formula:

dΓFGR

dω¼ αβ

�1 −

�1

nωβ

�2���

1

nωβ

�2

≤ 1

�: ðA10Þ

where Θ½.� ¼ 1 if the condition between bracket issatisfied, Θ½.� ¼ 0 otherwise, and the subscript ωin nω denotes the frequency dependence of the refractiveindex. We also use the notation nε in the future, whereε ¼ ℏω is the photon energy.

2. Resolvent theory applied to the quantumCherenkov effect

A comprehensive introduction to the resolvent theoryapplied to atomic systems in QED can be found inRef. [10]. The only information we need about the totalHamiltonian H ¼ H0 þ V (whereH0 is the Hamiltonian ofthe unperturbed system) is

(i) the eigenstates and energies of the unperturbedHamiltonian H0 and

(ii) the coupling matrix elements of the perturbationterm V, previously used in our derivation of theFermi golden rule.

The fundamental equation on the resolvent operator G isthe following:

ðz −H0ÞGðzÞ ¼ 1þ VGðzÞ; ðA11Þ

where z is a complex energy variable and GðzÞ is theresolvent operator defined in Ref. [10]. It can be understoodas the Fourier transform of the evolution operator UðtÞ.They can be mapped as follows:

UðtÞ ¼ 1

2πi

ICþþC−

dzGðzÞe−izt=ℏ; ðA12Þ

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GðzÞ ¼ 1

z −H; ðA13Þ

with Cþ þ C− a complex contour defined as in Fig. 6.Assuming the system can emit a maximum of one

photon, we restrict the photon Hilbert space. It followsthat we can get a set of algebraic equations on differentmatrix elements of the operator GðzÞ, by multiplyingEq. (A11) by adequate bra and ket:

ðz − EiÞG0ðzÞ ¼ 1þZ

d3qð2πÞ3=Ω

d3rð2πÞ3=Ω

× Vðp;0Þ;ðr;qÞGðr;qÞ;ðp;0ÞðzÞ; ðA14Þ

ðz − Er − ℏωqÞGðr;qÞ;ðp;0ÞðzÞ

¼Z

d3pð2πÞ3=ΩVðr;qÞ;ðp;0ÞGðp;0Þ;ðp;0ÞðzÞ; ðA15Þ

where, for every operator A, we denote Al;r ¼ hljAjri. Thekey to solving the dynamics of the system is to identify theself-energy function, which appears in various places alongthe derivation.In Ref. [62], Sec. I, we derive another set of fundamental

equations on the resolvent operator, by further extendingthe Hilbert space to more-than-one photon emission. Let usemphasize that the derivation of the self-energy function(and its regularization) is independent of the approach.The simplest way to find the self-energy is by inserting

Eq. (A15) in Eq. (A14), and, solving for G0ðzÞ, we get thefollowing expression (momentum conservation is enforcedby the coupling Hamiltonian Vðr;qÞ;ðp;0Þ):

G0ðzþ EiÞ ¼1

z − fðzÞ ; ðA16Þ

where

fðzÞ ¼ αβ2

2π

Z∞

0

dεZ

1

−1du

nεεð1 − u2Þz − εð1 − nεβuÞ

: ðA17Þ

To find UðtÞ, we are interested in values of fðzÞ arbitrarilyclose to the real axis, of which we can compute the real andimaginary parts. To get the imaginary part, we use the factthat Im½1=ðxþ i0þÞ� ¼ −πδðxÞ:

fðzþ i0�Þ ¼ ΔðzÞ ∓ iΓðzÞ2

; ðA18Þ

ΔðzÞ ¼ αβ

2π

Z∞

0

dε

�2ðz − εÞnεεβ

�

þ�1 −

�z − ε

nεεβ

�2�log

���� zþ εðnεβ − 1Þz − εðnεβ þ 1Þ

����; ðA19Þ

ΓðzÞ2

¼ αβ

2

Z∞

0

dε

�1 −

�ε − zεnεβ

�2���

ε − zεnεβ

�2

≤ 1

�:

ðA20Þ

We note that Γð0Þ ¼ ΓFGR and that Δð0Þ matches with thesecond-order perturbation-theory calculation of the energyshift of jp; 0i due to the electromagnetic coupling. Thus,Eqs. (A18)–(A20) represent a nonperturbative generaliza-tion of Cherenkov radiation and the Lamb (radiative) shiftof an electron in a medium. These equations are key todeveloping the results of this paper, and we explore theirconsequences.

APPENDIX B: REGULARIZATION OF f ðzÞIN FREQUENCY DOMAIN

This Appendix presents the regularization technique wedevelop for the purpose of this paper, in order to derivetime-dependent regularized observables of the Cherenkoveffect. We can readily notice that the real part of fðzÞdiverges and that it is related to the imaginary part of fðzÞbeing unbounded. Splitting the imaginary part of fðzÞ intotwo terms, we observe the following:

ΓðzÞ2

¼ Γ1ðzÞ2

þ Γ2ðzÞ2

; ðB1Þ

Γ1ðzÞ2

¼ αβ

2

Znεβ>1

dε

�1 −

�ε − zεnεβ

�2���

ε − zεnεβ

�2

≤ 1

�;

ðB2Þ

Γ2ðzÞ2

¼ αβ

2

Znεβ≤1

dε

�1 −

�ε − zεnεβ

�2���

ε − zεnεβ

�2

≤ 1

�:

ðB3Þ

We can simplify the integration domain of these twofunctions:

Γ1ðzÞ2

¼8<:

αβ2

Rdεh1 −

�ε−zεnεβ

2iΘ�ε ≥ z

1−nεβ

if z ≥ 0;

αβ2

Rdεh1 −

�ε−zεnεβ

2iΘ�ε ≤ z

1þnεβ

if z ≤ 0;

ðB4Þ

FIG. 6. Complex integration to Fourier transform the resolventinto the time domain. Complex contour to Fourier invert GðzÞ toUðtÞ, in the limit η → 0. The dashed line in the middle representsthe real axis ImðzÞ ¼ 0.

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Γ2ðzÞ2

¼αβ

2

Rdεh1−�

ε−zεnεβ

2i�

z1þnεβ

≤ ε≤ z1−nεβ

if z≥0;

0 if z≤0.

ðB5Þ(i) When z → þ∞, the Heaviside condition in Γ1ðzÞ

forces ε to be very large. In this limit, all materialsmust have nε → 1 (thus, nεβ < 1), and so theintegration domain of Γ1ðzÞ goes to zero. Weobserve a similar behavior when z → −∞:

limz→�∞

Γ1ðz; nεÞ ¼ 0: ðB6Þ

(ii) Because the Heaviside function and the integrationdomain of Γ2 have zero intersection when z < 0,Γ2ðz; nεÞ is zero for z < 0. However, when z → þ∞,the integration domain of Γ2ðzÞ is unbounded, andthis integral will diverge:

Γ2ðz ≤ 0; nεÞ ¼ 0; limz→þ∞

Γ2ðz; nεÞ ¼ þ∞: ðB7Þ

The fact that ΓðzÞ is unbounded results in a misdefinitionof ΔðzÞ, as real and imaginary parts of the analyticalfunction fðzÞ are related by Kramers-Kronig relations:

ΔðzÞ ¼ 1

2πPZ

Γðz0Þz0 − z

dz0: ðB8Þ

A natural way to regularize fðzÞ in the case of theCherenkov effect is to subtract the contribution of freespace to the integral nε ≡ 1, as a free electron propagatingin free space at a constant velocity cannot decay and emita photon. In the following, we define fðzÞ as the free-space-regularized function:

fðzÞ ≔ fðzÞ − fðz; nε ≡ 1Þ: ðB9Þ

We find that the regularized self-energy function fðzÞ iswell behaved for realistic refractive indices nε. Let us showthat with an example, given a linear lossless material,together with the assumption of causality. Under suchconditions, it is possible to prove [88] that the index scaleslike approximately 1 − ð1=ε2Þ for large ε (here, ε is still thephoton energy). We thus get nε ¼ 1 − ð1=ε2Þ þOð1=ε2Þwhen ε → þ∞ (it is a direct consequence of the Kramers-Kronig relations [88]). We can thus expand the integrand ofthe function fðzÞ − fðz; nε ≡ 1Þ when ε → þ∞:

ð1 − 1ε2Þε

z − ε½1 − ð1 − 1ε2Þβu� −

ε

z − εð1 − βuÞ ¼ −1

ε½z − εð1 − βuÞ�

∼ε→þ∞

1

ε2; ðB10Þ

which is integrable when ε → þ∞ (and the integrand isalso integrable when ε → 0þ). Thus, our regularizationtechnique is naturally backed up by fundamental propertiesof the refractive index nε in realistic materials.We also notice that the renormalized imaginary part

ΓðzÞ=2 is always positive:

ΓðzÞ2

≔Z

dε

χðε; nεÞΘ

��unε

�2�− χðε; 1ÞΘ½u2�

�;

ðB11Þ

χðε; nεÞ ¼αβ

2

�1 −

�ε − zεnεβ

�2�; ðB12Þ

u ¼ ε − zε

; ðB13Þ

ΘðvÞ ¼ 1½0;1�ðvÞ: ðB14Þ

If u ≤ 1, the integrand is χðε; nεÞ − χðε; 1Þ ≥ 0. If u=nε ≤ 1but u ≥ 1, the integrand is χðε; nεÞ ≥ 0. Overall, theintegrand is positive; thus, ΓðzÞ ≥ 0 for all z.

APPENDIX C: DERIVATION OF REGULARIZEDTIME-DEPENDENT QUANTUM OBSERVABLES

In Ref. [62], Sec. I, we differentiate between twomethods, relying on two different assumptions on theHilbert space in which the quantum state representingour system evolves. The first method is presented in thisAppendix. However, in both formulations, the resolvent ofthe zero-photon state remains the same [given byEq. (A16)], which results in the same time-dependentevolution of U0ðtÞ ¼ hp; 0jUðtÞjp; 0i. We show that, byFourier transforming the resolvent matrix element, one cancompute the time evolution of quantum states and relevantobservables. In the following, integral limits will often beleft implicit.

1. Time-domain derivation of thezero-photon state decay

To solve for the time evolution dynamics of the system,this section first presents the probability of having nophoton emission as a function of time.The value of fðzÞ above and below the axis are complex

conjugated; thus,

U0ðtÞ ¼1

2πi

Idz e−izt=ℏG0ðzÞ ðC1Þ

¼ 1

πe−iEit=ℏ

ZRdz e−izt=ℏ

ΓðzÞ2

½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2: ðC2Þ

And the decay probability of the zero-photon state isgiven by

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p0ðtÞ ¼ jU0ðtÞj2: ðC3Þ

We readily notice that the real frequency z0 verifyingΔðz0Þ ¼ z0 is of particular importance to describe the decayof the zero-photon state. For negligible variations of fðzÞaround z0 (flat-continuum approximation), the integrandcan be approximated by a Lorentzian peaked at z0, Δðz0Þbeing an energy shift, while Γðz0Þ=2 is the decay rate.

U0ðtÞ ≈ exp½−iðEi þ z0Þt=ℏ� · exp�−1

2Γðz0Þt=ℏ

�: ðC4Þ

Because our expression of the regularized function fðzÞmatches with first- and second-order perturbation theory,the regular quantum prediction is given by

U0ðtÞ≈expf−i½EiþΔð0Þ�t=ℏg · exp�−1

2Γð0Þt=ℏ

�: ðC5Þ

When Γðz0Þ ¼ 0, part of the integrand becomes aDirac distribution and the zero-photon state onlypartially decays to a nonzero value jU0ðþ∞Þj2 ¼f1=½1 − (dΔðzÞ=dz)jz¼z0 �g2:

U0ðtÞ ¼ e−iEpt=ℏ

e−iz0t=ℏ

1 − dΔðzÞdz jz¼z0

þ 1

π

Z þ∞

z0þηdz e−izt=ℏ

ΓðzÞ=2½z − ΔðzÞ�2 þ ðΓðzÞ

2Þ2!: ðC6Þ

In every case, it is obvious that U0ð0Þ ¼ 1.

2. Estimation of z0 in weakly coupled regime

In practice, we observe that z0 takes a small negativevalue for single electrons in optical regimes, which can beseen through the following expansion for small z, in thecase of an index constant over an energy range Δε:

fðzÞ ≈ αβ2

2π

�nΔε

Zduð1 − u2Þnβu − 1

− zZ

dεduð1 − u2Þεðnβu − 1Þ2

�:

ðC7Þ

Using this expansion around z0, we get

2π

αβ2z0 ≈

2π

αβ2½fðz0Þ − fðz0; nε ≡ 1Þ� ðC8Þ

≈ΔεZ

duð1 − u2Þ�

nnβu − 1

−1

βu − 1

�ðC9Þ

−z0Z

dεdu ð1 − u2Þ�

nεðnβu − 1Þ2 −

1

εðβu − 1Þ2�

ðC10Þ

The second term, proportional to z0, is of the order of 1,while ½ð2πÞ=ðαβ2Þ� > 850 for a single electron; thus, wecan safely neglect it. We thus get the following scaling lawfor z0:

z0 ∼αβ2

2πΔε�Z

duð1 − u2Þ�

nnβu − 1

−1

βu − 1

��ðC11Þ

∼ −αβ2

2πΔε; ðC12Þ

as we can compute the integral term and check that it willtake a negative value on the order of −1 when nβ ≳ 1 in theoptical regime.

3. Spectral angular density of the first emittedphoton (closed Hilbert space formulation)

We insert the expression ofG0ðzÞ into Eq. (A15). We usethe momentum conservation enforced by the couplingHamiltonian Vðr;qÞ;ðp;0Þ to reduce the number of degreesof freedom on Gðr;qÞ;ðp;0ÞðzÞ ¼ Gðp−q;qÞ;ðp;0ÞðzÞ. Doing so,the volume factors cancel out, and we get the followingexpression for Gðr;qÞ;ðp;0ÞðzÞ:

Gqðzþ EpÞ ≔ Gðr;qÞ;ðp;0Þðzþ EpÞ

¼ ð−ev sin θÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ℏ2Ωϵ0ωq

s1

z − fðzÞ

×1

z − ℏωqð1 − βnω cos θÞ: ðC13Þ

From there, we can readily convert this expression to thetime domain and then get the probability of the first emittedphoton, by integrating over possible photon momenta q:

UqðtÞ ¼1

2πi

Idz e−izt=ℏGqðzÞ ðC14Þ

¼ ð−ev sin θÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ℏ2Ωϵ0ωq

se−iEpt=ℏ

1

π

Zdz e−izt=ℏ

ΓðzÞ2

f½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2g½z − ℏωqð1 − βnω cos θÞ�

; ðC15Þ

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p1ðtÞ ¼Z

d3qð2πÞ3=Ω jUqðtÞj2 ðC16Þ

¼ αβ2

2π

Zdεdu εnεð1 − u2Þ

���� 1πZ

dz e−izt=ℏΓðzÞ2

f½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2g½z − εð1 − βnεuÞ�

����2

ðC17Þ

¼Z

dz0Γðz0Þ2

���� 1πZ

dz e−izt=ℏΓðzÞ2

f½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2gðz − z0Þ

����2

: ðC18Þ

Equation (C18) gives a frequency-domain expression for p1ðtÞ. We can get another expression for p1ðtÞ, by converting thefundamental Eq. (A15) to the time domain. This conversion is affected through the operation ½1=ð2πiÞ� H dz (still assumingmomentum conservation r ¼ p − q):

−iℏe−iðErþℏωqÞt=ℏ ddt

ðUqeiðErþℏωqÞt=ℏÞ ¼ ðev sin θÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ℏ2Ωε0ωq

sU0ðtÞ; ðC19Þ

which gives us, using the expression of ΓðzÞ,

p1ðtÞ ¼1

ℏ2

Zdz

ΓðzÞ2

����Z

t

0

dt0e−izt0=ℏeiEpt0=ℏU0ðt0Þ����2: ðC20Þ

In both expressions, we notice that the expression of the imaginary part of fðzÞ intervenes. We here notice that replacingΓðzÞ by its regularized form ΓðzÞ ≔ ΓðzÞ − Γðz; nε ≡ 1Þ is of paramount importance to ensure that the total probability inthe Hilbertian space is conserved and equal to 1:

p0ðtÞ þ p1ðtÞ ¼ 1 ∀ t: ðC21Þ

p1ðtÞ regularization also results in the regularization of the spectral angular density f½∂2p1ðtÞ�=∂ε∂ug using theexpression derived from the frequency domain:

∂2p1ðε; u; tÞ∂ε∂u ≔

αβ2

2πεnεð1 − u2Þ

���� 1πZ

dz e−izt=ℏΓðzÞ2

f½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2g½z − εð1 − βnεuÞ�

����2

−αβ2

2πεð1 − u2Þ

���� 1πZ

dz e−izt=ℏΓðzÞ2

f½z − ΔðzÞ�2 þ ðΓðzÞ2Þ2g½z − εð1 − βuÞ�

����2

: ðC22Þ

And similarly, from the time-domain expression [Eq. (C20)], we get

∂2p1ðtÞ∂ε∂u ≔

1

ℏ2

αβ2

2πnεεð1 − u2Þ

����Z

t

0

dt0eiεð1−nεβuÞt0=ℏe−iEpt0=ℏU0ðt0Þ����2 − 1

ℏ2

αβ2

2πεð1 − u2Þ

����Z

t

0

dt0eiεð1−βuÞt0=ℏe−iEpt0=ℏU0ðt0Þ����2:

ðC23Þ

For convenience, we use the latter expression derived from the time domain in our numerical simulations. In the flat-continuum approximation, we can readily notice that the spectral angular density is enhanced for ðε; uÞ satisfying

εð1 − nεβuÞ ¼ z0 ⇔ε − z0nεβu

¼ 1; ðC24Þ

which coincides with the conventional Cherenkov relation in the case z0 ¼ 0.This new dispersion relation also allows the existence of backward Cherenkov radiation in this simple setting when

z0 > 0 for ε < z0 [see, for instance, Fig. 2(d)].

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APPENDIX D: ADDITIONAL INFORMATION

Additional information can be found in SupplementalMaterial [62], including a nested Hilbert space formulation,which can take into account several photons, influence ofthe permittivity function on our predictions, and thegeneralization of our regularization method to vac-uum QED.

[1] W. E. Lamb and R. C. Retherford, Fine Structure of theHydrogen Atom by a Microwave Method, Phys. Rev. 72,241 (1947).

[2] S. Tomonaga and J. Robert Oppenheimer, On Infinite FieldReactions in Quantum Field Theory, Phys. Rev. 74, 224(1948).

[3] H. A. Bethe, The Electromagnetic Shift of Energy Levels,Phys. Rev. 72, 339 (1947).

[4] J. Schwinger, Quantum Electrodynamics. I. A CovariantFormulation, Phys. Rev. 74, 1439 (1948).

[5] R. P. Feynman, A Relativistic Cut-Off for ClassicalElectrodynamics, Phys. Rev. 74, 939 (1948).

[6] F. J. Dyson, The Radiation Theories of Tomonaga,Schwinger, and Feynman, Phys. Rev. 75, 486 (1949).

[7] P. Kusch and H. M. Foley, The Magnetic Moment of theElectron, Phys. Rev. 74, 250 (1948).

[8] A. Bohr, On the Hyperfine Structure of Deuterium, Phys.Rev. 73, 1109 (1948).

[9] M. Peskin and D. Schroeder, An Introduction to QuantumField Theory (Perseus, Cambridge, MA, 1995).

[10] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,Atom-Photon Interactions: Basic Processes andApplications (Wiley, Weinheim, 1992).

[11] V. L. Ginzburg, Applications of Electrodynamics inTheoretical Physics and Astrophysics (Gordon and Breach,New York, 1989), p. 475.

[12] S. John and T. Quang, Spontaneous Emission Near the Edgeof a Photonic Band Gap, Phys. Rev. A 50, 1764 (1994).

[13] R. E. Wagner, Q. Su, and R. Grobe, Time-Resolved Comp-ton Scattering for a Model Fermion-Boson System, Phys.Rev. A 82, 022719 (2010).

[14] L. Garziano, R. Stassi, V. Macrì, A. F. Kockum, S. Savasta,and F. Nori, Multiphoton Quantum Rabi Oscillations inUltrastrong Cavity QED, Phys. Rev. A 92, 063830 (2015).

[15] P. A. Cherenkov, Visible Emission of Clean Liquids byAction of γ Radiation, Dokl. Akad. Nauk SSSR 2, 451(1934).

[16] S. I. Vavilov, On the Possible Causes of Blue γ-Glow ofLiquids, CR Dokl. Akad. Nauk. SSSR. 2, 8 (1934).

[17] I. Frank and I. Tamm, Coherent Visible Radiation from FastElectrons Passing Through Matter, C.R. Acad. Sci. USSR(1937).

[18] V. L. Ginzburg, Quantum Theory of Radiation of ElectronUniformly Moving in Medium, Zh. Eksp. Teor. Fiz. 10, 589(1940); J. Phys. USSR 2, 441 (1940).

[19] A. Sokolow, Quantum Theory of Cherenkov Effect, Dokl.Akad. Nauk SSSR (1940).

[20] I. Kaminer, M. Mutzafi, A. Levy, G. Harari, H. H. Sheinfux,S. Skirlo, J. Nemirovsky, J. D. Joannopoulos, M. Segev, and

M. Soljacic,Quantum Cerenkov Radiation: Spectral Cutoffsand the Role of Spin and Orbital Angular Momentum, Phys.Rev. X 6, 011006 (2016).

[21] O. Bogdanov, E. Fiks, and Y. Pivovarov, Angular Distri-bution of Cherenkov Radiation from Relativistic Heavy IonsTaking into Account Deceleration in the Radiator, J. Exp.Theor. Phys. 115, 392 (2012).

[22] V. Koch, A. Majumder, and X. N. Wang, CherenkovRadiation from Jets in Heavy-Ion Collisions, Phys. Rev.Lett. 96, 172302 (2006).

[23] A. V. Volotka, D. A. Glazov, G. Plunien, and V. M. Shabaev,Progress in Quantum Electrodynamics Theory of HighlyCharged Ions, Ann. Phys. (Berlin) 525, 636 (2013).

[24] E. Fiks, O. Bogdanov, and Y. Pivovarov, New Peculiaritiesin Angular Distribution of Cherenkov Radiation fromRelativistic Heavy Ions Caused by their Stopping in Radi-ator: Numerical and Theoretical Research J. Phys. Conf.Ser. 357, 012002 (2012).

[25] A. S. Vodopianov, Y. I. Ivanshin, V. I. Lobanov, I. I.Tatarinov, A. A. Tyapkin, I. A. Tyapkin, A. I. Zinchenko,V. P. Zrelov, A. A. Bogdanov, V. A. Kaplin, A. I. Karakash,M. F. Runtzo, M. N. Strikhanov, V. M. Biryukov, Y. A.Chesnokov, S. B. Nurushev, J. Ruzicka, P. Chochula, M.Ciljak, and A. Hrmo, Crystal-Assisted Extraction of AuIons from RHIC and Application of the Au Beam for theSearch of Anomalous Cherenkov Radiation, Nucl. Instrum.Methods Phys. Res., Sect. B 201, 266 (2003).

[26] A. S. Vodopianov, V. P. Zrelov, and A. A. Tyapkin, Analysisof the Anomalous Cherenkov Radiation Obtained in theRelativistic Lead Ion Beam at CERN SPS, Joint Institute forNuclear Research Report No. JINR–2-99-2000, 2000.

[27] T. Wulle and S. Herminghaus, Nonlinear Optics of BesselBeams, Phys. Rev. Lett. 70, 1401 (1993).

[28] C.D’Amico,A.Houard,M.Franco,B.Prade,A.Mysyrowicz,A. Couairon, and V. T. Tikhonchuk, Conical Forward THzEmission from Femtosecond-Laser-Beam Filamentation inAir, Phys. Rev. Lett. 98, 235002 (2007).

[29] S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski,A. Arie, and Y. S. Kivshar, Generation of Second-HarmonicConical Waves via Nonlinear Bragg Diffraction, Phys. Rev.Lett. 100, 103902 (2008).

[30] Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming,Nonlinear Čerenkov Radiation in Nonlinear PhotonicCrystal Waveguides, Phys. Rev. Lett. 100, 163904(2008).

[31] Y. Sheng, W. Wang, R. Shiloh, V. Roppo, Y. Kong, A. Arie,and W. Krolikowski, Čerenkov Third-Harmonic Generationin χð2Þ Nonlinear Photonic Crystal, Appl. Phys. Lett. 98,241114 (2011).

[32] K. Vijayraghavan, Y. Jiang, M. Jang, A. Jiang, K.Choutagunta, A. Vizbaras, F. Demmerle, G. Boehm, M.C. Amann, and M. A. Belkin, Broadly Tunable TerahertzGeneration in Mid-Infrared Quantum Cascade Lasers, Nat.Commun. 4, 2021 (2013).

[33] M. A. Belkin and F. Capasso, New Frontiers in QuantumCascade Lasers: High Performance Room TemperatureTerahertz Sources, Phys. Scr. 90, 118002 (2015).

[34] A. Wallraff, A. V. Ustinov, V. V. Kurin, I. A. Shereshevsky,and N. K. Vdovicheva, Whispering Vortices, Phys. Rev.Lett. 84, 151 (2000).

CHARLES ROQUES-CARMES et al. PHYS. REV. X 8, 041013 (2018)

041013-14

[35] J. Pfeiffer, M. Schuster, A. A. Abdumalikov, and A. V.Ustinov, Observation of Soliton Fusion in a JosephsonArray, Phys. Rev. Lett. 96, 034103 (2006).

[36] V. Frolov and V. Ginzburg, Excitation and Radiation of anAccelerated Detector and Anomalous Doppler Effect, Phys.Lett. A 116, 423 (1986).

[37] G. Calajó and P. Rabl, Strong Coupling Between MovingAtoms and Slow-Light Cherenkov Photons, Phys. Rev. A 95,043824 (2017).

[38] X. Shi, X. Lin, I. Kaminer, F. Gao, Z. Yang, J. D.Joannopoulos, M. Soljačić, B. Zhang, and B. Zhang, Super-light Inverse Doppler Effect, Nat. Phys. 14, 1001 (2018).

[39] P. Leboeuf and S. Moulieras, Superfluid Motion of Light,Phys. Rev. Lett. 105, 163904 (2010).

[40] I. Carusotto and G. Rousseaux, The Cerenkov EffectRevisited: From Swimming Ducks to Zero Modes inGravitational Analogues, Analogue Gravity Phenomenol-ogy (Springer, Cham, 2013), pp. 109–144.

[41] V. G. Veselago, The Electrodynamics of Substances withSimultaneously Negative Values of ε and μ, Sov. Phys. Usp.10, 509 (1968).

[42] C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos,Cerenkov Radiation in Photonic Crystals, Science 299, 368(2003).

[43] A. V. Kats, S. Savel’ev, V. A. Yampol’skii, and F. Nori, Left-Handed Interfaces for Electromagnetic Surface Waves,Phys. Rev. Lett. 98, 073901 (2007).

[44] S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B. I. Wu,J. A. Kong, and M. Chen, Experimental Verification ofReversed Cherenkov Radiation in Left-Handed Metamate-rial, Phys. Rev. Lett. 103, 194801 (2009).

[45] F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, and W.Zhang, Integrated Cherenkov Radiation Emitter Eliminat-ing the Electron Velocity Threshold, Nat. Photonics 11, 289(2017).

[46] S. Liu, P. Zhang, W. Liu, S. Gong, R. Zhong, Y. Zhang, andM. Hu, Surface Polariton Cherenkov Light RadiationSource, Phys. Rev. Lett. 109, 153902 (2012).

[47] I. Kaminer, Y. T. Katan, H. Buljan, Y. Shen, O. Ilic, J. J.López, L. J. Wong, J. D. Joannopoulos, and M. Soljačić,Efficient Plasmonic Emission by the Quantum CerenkovEffect from Hot Carriers in Graphene, Nat. Commun. 7,ncomms11880 (2016).

[48] IceCube Collaboration, Evidence for High-EnergyExtraterrestrial Neutrinos at the IceCube Detector, Science

342,1242856)2013 ).[49] T. M. Shaffer, E. C. Pratt, and J. Grimm, Utilizing the Power

of Cerenkov Light with Nanotechnology, Nat. Nanotechnol.12, 106 (2017).

[50] C. Kremers, D. N. Chigrin, and J. Kroha, Theory ofCherenkov Radiation in Periodic Dielectric Media:Emission Spectrum, Phys. Rev. A 79, 013829 (2009).

[51] F. Krausz and M. Ivanov, Attosecond Physics, Rev. Mod.Phys. 81, 163 (2009).

[52] A. H. Zewail, Femtochemistry: Atomic-Scale Dynamics ofthe Chemical Bond Using Ultrafast Lasers (Nobel Lecture),Angew. Chem. Int. Ed. 39, 2586 (2000).

[53] A. Kubo, N. Pontius, and H. Petek, Femtosecond Micros-copy of Surface Plasmon Polariton Wave Packet Evolutionat the Silver/Vacuum Interface, Nano Lett. 7, 470 (2007).

[54] M. Bauer, C. Wiemann, J. Lange, D. Bayer, M. Rohmer, andM. Aeschlimann, Phase Propagation of Localized SurfacePlasmons Probed by Time-Resolved PhotoemissionElectron Microscopy, Appl. Phys. A 88, 473 (2007).

[55] M. I. Stockman, M. F. Kling, U. Kleineberg, and F. Krausz,Attosecond Nanoplasmonic-Field Microscope, Nat.Photonics 1, 539 (2007).

[56] P. Kahl, S. Wall, C. Witt, C. Schneider, D. Bayer, A. Fischer,P. Melchior, M. Horn-von Hoegen, M. Aeschlimann, andF.-J. Meyer zu Heringdorf, Normal-Incidence Photoemis-sion Electron Microscopy (NI-PEEM) for Imaging SurfacePlasmon Polaritons, Plasmonics 9, 1401 (2014).

[57] G. Spektor, D. Kilbane, A. K. Mahro, B. Frank, S. Ristok, L.Gal, P. Kahl, D. Podbiel, S. Mathias, H. Giessen, F. J. MeyerZu Heringdorf, M. Orenstein, and M. Aeschlimann,Revealing the Subfemtosecond Dynamics of OrbitalAngular Momentum in Nanoplasmonic Vortices, Science355, 1187 (2017).

[58] P. Kahl, D. Podbiel, C. Schneider, A. Makris, S. Sindermann,C. Witt, D. Kilbane, M. H.-v. Hoegen, M. Aeschlimann, andF. M. zu Heringdorf, Direct Observation of Surface PlasmonPolariton Propagation and Interference by Time-ResolvedImaging in Normal-Incidence Two Photon PhotoemissionMicroscopy, Plasmonics 13, 239 (2018).

[59] G. M. Vanacore, I. Madan, G. Berruto, K. Wang, E.Pomarico, R. J. Lamb, D. McGrouther, I. Kaminer, B.Barwick, F. J. G. de Abajo, and F. Carbone, From Atto-second to Zeptosecond Coherent Control of Free-ElectronWave Functions Using Semi-Infinite Light Fields, Nat.Commun. 9, 2694 (2018).

[60] R. T. Cox, Momentum and Energy of Photon andElectron in the Čerenkov Radiation, Phys. Rev. 66, 106(1944).

[61] L. A. Khalfin, Contribution to the Decay Theory of a Quasi-Stationary State, Sov. Phys. JETP 6, 1053 (1958).

[62] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.8.041013 for the nestedHilbert space formulation, which can take into accountseveral photons, the influence of the permittivity function onour predictions, and the generalization of our regularizationmethod to vacuum QED.

[63] A method is proposed in Ref. [62] to take into accountdiagrams of the successive absorption and reemission of avirtual photon after the emission of the first photon. Takinginto account higher-order diagrams (e.g., two-photon emis-sion) requires the computation of the resolvent matrixelements projected on these higher-order states.

[64] K. G. Wilson, The Renormalization Group: Critical Phe-nomena and the Kondo Problem, Rev. Mod. Phys. 47, 773(1975).

[65] L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, andM. Soljačić, Towards Graphene Plasmon-Based Free-Electron Infrared to X-ray Sources, Nat. Photonics 10,46 (2016).

[66] N. Rivera, I. Kaminer, B. Zhen, J. D. Joannopoulos, and M.Soljačić, Shrinking Light to Allow Forbidden Transitions onthe Atomic Scale, Science 353, 263 (2016).

[67] M. Harrison, T. Ludlam, and S. Ozaki, RHIC ProjectOverview, Nucl. Instrum. Methods Phys. Res., Sect. A499, 235 (2003).

NON-PERTURBATIVE QUANTUM ELECTRODYNAMICS IN … PHYS. REV. X 8, 041013 (2018)

041013-15

[68] I. Blumenfeld, C. E. Clayton, F. J. Decker, M. J. Hogan, C.Huang, R. Ischebeck, R. Iverson, C. Joshi, T. Katsouleas, N.Kirby, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, E. Oz,R. H. Siemann, D. Walz, and M. Zhou, Energy Doubling of42 GeV Electrons in a Metre-Scale Plasma WakefieldAccelerator, Nature (London) 445, 741 (2007).

[69] M. Nishiuchi, H. Sakaki, T. Z. Esirkepov, K. Nishio, T. A.Pikuz, A. Y. Faenov, I. Y. Skobelev, R. Orlandi, H. Sako,A. S. Pirozhkov, K. Matsukawa, A. Sagisaka, K. Ogura, M.Kanasaki, H. Kiriyama, Y. Fukuda, H. Koura, M. Kando, T.Yamauchi, Y. Watanabe et al., Acceleration of HighlyCharged GeV Fe Ions from a low-Z Substrate by IntenseFemtosecond Laser, Phys. Plasmas 22, 033107 (2015).

[70] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley,J. M. Raimond, and S. Haroche, Quantum Rabi Oscillation:A Direct Test of Field Quantization in a Cavity, Phys. Rev.Lett. 76, 1800 (1996).

[71] O. D. Stefano, R. Stassi, L. Garziano, A. F. Kockum, S.Savasta, and F. Nori, Feynman-Diagrams Approach to theQuantum Rabi Model for Ultrastrong Cavity QED: Stimu-lated Emission and Reabsorption of Virtual Particles Dress-ing a Physical Excitation, New J. Phys. 19, 053010 (2017).

[72] M. F. Gely, A. Parra-Rodriguez, D. Bothner, Y. M. Blanter,S. J. Bosman, E. Solano, and G. A. Steele, Convergence ofthe Multimode Quantum Rabi Model of Circuit QuantumElectrodynamics, Phys. Rev. B 95, 245115 (2017).

[73] G. Afanasiev, M. Lyubchenko, and Y. P. Stepanovsky, Finestructure of the Vavilov-Cherenkov Radiation, Proc. R. Soc.A 462, 689 (2006).

[74] S. De Liberato, Light-Matter Decoupling in the Deep StrongCoupling Regime: The Breakdown of the Purcell Effect,Phys. Rev. Lett. 112, 016401 (2014).

[75] T. N. Wistisen, A. Di Piazza, H. V. Knudsen, and U. I.Uggerhøj, Experimental Evidence of Quantum RadiationReaction in Aligned Crystals, Nat. Commun. 9, 795 (2018).

[76] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji,Vacuum Fluctuations and Radiation Reaction : Identifica-tion of Their Respective Contributions, J. Phys. (Paris) 43,1617 (1982).

[77] P. W. Milonni, The Quantum Vacuum: An Introductionto Quantum Electrodynamics (Academic, Boston, 1994),p. 522.

[78] D. N. Basov, M. M. Fogler, and F. J. García De Abajo,Polaritons in van der Waals Materials, Science 354,aag1992 (2016).

[79] T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P.Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F.Koppens, Polaritons in Layered 2D Materials, Nat. Mater.16, 182 (2017).

[80] Y. Morimoto and P. Baum, Diffraction and Microscopy withAttosecond Electron Pulse Trains, Nat. Phys. 14, 252 (2018).

[81] B. Barwick, D. J. Flannigan, and A. H. Zewail, Photon-Induced Near-Field Electron Microscopy, Nature (London)462, 902 (2009).

[82] A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin,S. Schäfer, and C. Ropers, Quantum Coherent OpticalPhase Modulation in an Ultrafast Transmission ElectronMicroscope, Nature (London) 521, 200 (2015).

[83] F. J. García De Abajo, Optical Excitations in ElectronMicroscopy, Rev. Mod. Phys. 82, 209 (2010).

[84] L. Piazza, T. T. Lummen, E. Quiñonez, Y. Murooka, B. W.Reed, B. Barwick, and F. Carbone, Simultaneous Obser-vation of the Quantization and the Interference Pattern of aPlasmonic Near-Field, Nat. Commun. 6, 6407 (2015).

[85] T. T. Lummen, R. J. Lamb, G. Berruto, T. Lagrange, L. DalNegro, F. J. García De Abajo, D. McGrouther, B. Barwick,and F. Carbone, Imaging and Controlling Plasmonic In-terference Fields at Buried Interfaces, Nat. Commun. 7,13156 (2016).

[86] E. Pomarico, I. Madan, G. Berruto, G. M. Vanacore, K.Wang, I. Kaminer, J. García de Abajo, and F. Carbone, meVResolution in Laser-Assisted Energy-Filtered TransmissionElectron Microscopy, ACS Photonics 5, 759 (2017).

[87] E. Harris, A Pedestrian Approach to Quantum Field Theory(Dover, New York, 2014).

[88] L. Davidovich, L. D. Landau, E.M. Lifshitz, and L. P.Pitaevskii,Electrodynamics of ContinuousMedia (Pergamon,New York, 1984), p. 460.

CHARLES ROQUES-CARMES et al. PHYS. REV. X 8, 041013 (2018)

041013-16