Nonperturbative Quantum Electrodynamics in the Cherenkov Effect

Charles Roques-Carmes, Nicholas Rivera, John D. Joannopoulos, Marin Soljačić, and Ido Kaminer Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel


I. INTRODUCTION
One of the early achievements of quantum electrodynamics (QED) was the accurate match of the predicted hydrogen-energy-level shifts with the experiments by Lamb and Retherford [1]. This significant accomplishment was allowed by the work of Tomonaga [2], Bethe [3], Schwinger [4], and Feynman [5] and the later unification by Dyson [6]. They eliminated the problematic divergence of the Lamb-Retherford shift due to the photon loop correction to the atom's propagator. Renormalization of the electron mass also led to a very precise agreement with the experimentally measured values of the Lamb-Retherford line shift [1], the anomalous magnetic moment of the electron [7], and the anomalous hyperfine splitting of the ground state of the hydrogen atom [8]. Since these early successes, QED has been one of the most accurate theories of modern physics and bolstered fundamental developments, from quantum field theory to grand unified theories. Central to calculations in QED and other field theories is the S matrix, representing the long-time transition amplitude between an initial state and some final state [9][10][11]. From the S matrices, one can derive many important observables such as decay rates, cross sections, and selfenergies, which have been successfully used to calculate a great variety of effects in particle physics, condensed matter physics, atomic physics, and many other fields. Despite these successes, S-matrix methods are essentially used to make predictions at infinite times. Altogether in QED, very few studies consider the physics at short times (e.g., Refs. [12][13][14]) and almost none when considering the coupling to a continuum of electromagnetic modes. We still lack the basic methods to describe the short-time dynamics in such systems, where it remained unknown how to apply renormalization at finite times.
One of the emblematic problems in light-matter interaction that can be used as a testing ground for new ideas in QED is the Vavilov-Cherenkov effect (abbreviated as the Cherenkov effect)-the radiation by a charged particle propagating in a dispersive medium at a speed larger than the phase velocity of light in the medium. It is the leadingorder process in the radiation by a free electron in a medium and is the fundamental building block for any diagram describing the interaction of charges with photons in a medium. This effect was originally observed by Vavilov and Cherenkov in 1934 [15,16] and later explained by Frank and Tamm [17]. Most of the quantum treatment of this problem, which originated from Ginzburg's Ph.D. thesis [18], did not result in a significant deviation from the original prediction by Frank and Tamm [17]. Even later theories taking into account higher-order processes did not find regimes where the quantum nature of the interacting particles would result in a significant mismatch from the original prediction of Frank and Tamm [17][18][19]. Only recently was it theoretically predicted that quantum effects could emerge in the optical regime when taking into account the quantum recoil of the electron or shaping the electron wave function [20].
The Cherenkov effect follows the Frank-Tamm formula that gives the rate of photon emission per unit frequency ∂ ε Γ rad ¼ ½ðα Z βÞ=ℏf1 − ½1=ðn ε βÞ 2 g. In this expression, α Z ¼ Z 2 α is the effective fine-structure constant, α ≈ 1=137 being the fine-structure constant and Z the particle charge; β ¼ v=c, v being the electron speed and n ε the refractive index of the medium (function of the photon energy ε ¼ ℏω). From the Frank-Tamm formula and its generalizations to nanostructured optical systems or anomalous cases [42,50], one sees that, in the optical regime, the timescale at which a photon is emitted in the medium is between picoseconds and femtoseconds. Physics at these timescales was unveiled at the turn of the millennium with the development of ultrafast optics, light-wave electronics, and high harmonic generation [51]. Today, the integration of electron optics with optical technologies allows the probing of atomic transitions [51], molecular bonding [52], and even more recently dynamics of plasmonic systems [53][54][55][56][57][58][59]. Given these new experimental techniques, it is timely to ask how the long-standing physics of the Cherenkov effect changes at very short timescales.
In this article, we develop a general framework for timedependent QED, which is free of short-time divergences, and use our framework to reveal new underlying phenomena in Cherenkov radiation. This process yields concrete closed-form solutions of the power spectrum of photon emission and a surprising increase in the Cherenkov velocity threshold. Applying our theory to probe the time dependence of the emitted radiation from a charged particle propagating in a medium, we observe new quantum phenomena in the Cherenkov effect. Even when the electron and photon are weakly coupled (α ≪ 1), the well-known Cherenkov dispersion relation is modified from the known one on account of a nonzero self-energy associated with the virtual emission and reabsorption of photons in the medium. When pushing the limits of our formalism to strongly coupled regimes ðαZ 2 ≳ 1Þ, we find evidence of regimes where the electron experiences a nonexponential decay and the Cherenkov dispersion relation changes drastically. In these strongly coupled settings, we predict the emergence of multiple Cherenkov lines that can be tested in existing experiments such as heavy-ion colliders. More generally, the Cherenkov effect provides us with a powerful playground to develop new general tools for time-dependent QED. Our work suggests applications in the study of fundamental effects in attosecond physics, such as high harmonic generation, and new experimental endeavors of Cherenkov radiation and its analogues in strongly coupled settings.

II. TIME-DEPENDENT NONPERTURBATIVE THEORY OF CHERENKOV RADIATION
We consider the system shown in Fig. 1. A charged particle, such as a single electron or a highly charged ion, is propagating at speed v ¼ βc inside a lossless material defined by its refractive index n ε . According to the conventional Cherenkov theory, if the particle velocity surpasses the speed of light in the medium β > c=n ε , it emits a photon of energy ε at angle θ C relative to the electron trajectory, satisfying the condition [ Fig. 1(a)] cos θ C ¼ ½1=ðn ε βÞ [17]. This energy-angle relation is a direct consequence of energy-momentum conservation [60], while the photon emission rate per unit energy can be derived from Fermi's golden rule [18].
We solve for the time dependence of the time-evolution operator corresponding to the QED Hamiltonian using the resolvent method [10]. This method allows us to perform resummations of partial sets of the perturbation series in a compact algebraic formulation while maintaining dynamical information of the correlated electron-photon wave function. The method works as follows: We reformulate the Schrödinger equation in terms of the resolvent operator GðzÞ ¼ ðz − HÞ −1 , which is the complex-variable Fourier transform of the unitary time-evolution operator, used to determine the time-dependent electron-photon wave function. The matrix elements of G represent the Fourier transform of transition amplitudes between different states. We first derive a matrix equation for G 0 ≡ hp; 0jGjp; 0i (for more details on the formalism and the QED Hamiltonian, we refer to Appendix A). Inverting the matrix equation and moving back into the time domain, we find the following expressions for the evolution operator projected on the zero-photon state U 0 ðtÞ and the probability of observing a photon p 1 ðtÞ: where E i (respectively, E f ) is the charged particle's initial (final) energy and p 0 ¼ jU 0 j 2 the probability of the 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, this corresponds to the usual exponential decay law for spontaneous emission which is valid at intermediate times.
Similar modifications have been studied in the case of two-level systems [10] and for quasistationary states [61]. From Eqs. (1) and (2), one can derive time-dependent physical observables such as the spectrum of emitted photons per unit angle and per unit frequency ∂ 2 p 1 =∂ε∂u, which can also be used to determine the rate of photon emission. Figure 2 shows several examples of such spectral emission rates. In Eq. (1), ΔðzÞ and ΓðzÞ, which fully specify the dynamics of the system, are the real and imaginary parts, respectively, of the self-energy of the electron fðzÞ [also known as the radiative shift in atomic, molecular, and optical (AMO) physics]: The real and imaginary parts of the self-energy function can be written analytically as respectively, where Θ½. ¼ 1 if the condition between brackets is satisfied, Θ½. ¼ 0 otherwise. Δð0Þ and Γð0Þ exactly match the real and imaginary part, respectively, of the radiative shift calculated via the secondorder perturbation theory in the weak-coupling regime. More generally, ΔðzÞ and ΓðzÞ correspond to the secondorder perturbation of the energy, shifted by the complex frequency variable z. Our approach is nonperturbative in the sense that the self-energy function models absorption and reemission cycles of the photon before a final photon emission, thus taking into account an infinite sum of singleloop Feynman diagrams (see Fig. S4 in Supplemental Material [62]). We add additional diagrams altering the electron after the photon emission, showing quantitative corrections but no qualitative changes to our predictions (see Ref. [62], Sec. I). Our theory could be extended to take into account higher-order processes with a more sophisticated numerical treatment [63].

III. THE RADIATIVE SHIFT OF THE FREE ELECTRON
Upon a first evaluation of the self-energy, we find that the real part features a UV divergence as a result of the unbounded photonic density of states at high energies [64], even when the index of refraction is too low for Cherenkov radiation. Nevertheless, we regularize the self-energy function in a way that agrees with physical predictions, by subtracting its free-space part (n ε ≡ 1): To get the conservation of probability, we substitute the regularized form of ΓðzÞ in Eq. (2), which yields p 0 ðtÞ þ p 1 ðtÞ ≡ 1. Regularizing the spectral angular density of the emitted photon also suppresses unphysical emission in free space (see Appendix B and Ref. [62], Sec. I, for further discussion of this point). This approach could be extended to any optical medium, as it takes into account material dispersion, and can be directly generalized to arbitrary geometries of photonic structures, by taking into account the eigenmodes of Maxwell's equations for such structures. Thus, our approach can be used in a wide range of phenomena stemming from the Cherenkov effect, 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 give well-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-we find that the contribution of the self-energy function in Eqs. (1) and (2) can be approximated by its value taken at the zero of ΔðzÞ − z, which we denote as z 0 ¼ Δðz 0 Þ. This approximation, known as the flat-continuum approximation, retains validity when the decay rate or linewidth of the electron-emission process associated with Cherenkov radiation is much narrower than the frequency variation of the photonic density of states.
We find that the real part of the self-energy-known as the radiative shift z 0 [10]-is typically of the order of −αβ 2 =ð2πÞΔε ∼ −0.001 eV for a single electron in the optical regime [see Appendix C, Eq. (C12)]. This freeelectron radiative shift is significantly greater than its atomic counterpart-the Lamb-Retherford shift [1]-and, as a result, leads to potentially large corrections to quantum observables for small photon energies or particle velocities close to the conventional Cherenkov threshold (n ε β → 1). In particular, it modifies the well-established Cherenkov rate of emission to become (We denote with a bar sign every quantum observable and state probability from our regularized theory-taking into account the particle-regularized self-energy-in contrast to the conventional Cherenkov theory.) Additional corrections can come up from our quantum formalism. They typically include the electron recoil due to the Cherenkov emission of a single photon [20,47], which depends on the photon momentum n ε ε=c relative to the electron momentum βE i =c. The radiation-angle correction is proportional to the radiative shift z 0 and to the electron recoil. In scenarios where the electron recoil is large [47], we expect to observe additional large self-energy corrections. Even without electron recoil, the radiative shift z 0 still significantly modifies the Cherenkov decay rate. This modification makes the new quantum corrections that we find far bigger than any previously found quantum effects in Cherenkov radiation [18,20]. Figure 3 shows that the account of the nonperturbative self-energy can shift the well-known Cherenkov threshold: 1=ðn ε βÞ ≤ ε=ðε − z 0 Þ ≈ 1 þ z 0 =ε for small negative values of z 0 . As a consequence, the regularized rate of emission ∂ εΓrad converges to zero when approaching the regularized Cherenkov threshold: In an experiment, this convergence translates into arbitrarily fewer photons emitted, compared to the conventional prediction. The most extreme deviation is found when the particle velocity is above the conventional threshold and below the regularized threshold, where the conventional theory predicts a finite rate of emission, while the regularized theory states that the spontaneous emission of a photon is forbidden.
Far above the conventional Cherenkov velocity threshold, the values of the regularized decay rate Γðz 0 Þ ¼ Γ 0 and of the conventional decay rate Γðz ¼ 0Þ agree, as can be seen in Fig. 3(b). However, the divergence of 1=Γ 0 around β ≳β th results in arbitrarily small decay rates that directly translate on the dynamics of p 0 ðtÞ, as illustrated in Fig. 4(a). While the conventional theory predicts a total absence of decay below the conventional Cherenkov threshold, the regularized probability of the zero-photon state decays to a value of 1=f1 − ½dΔðzÞ=dzj z 0 g 2 slightly below unity [ Fig. 3(b)]. This result could be interpreted as the electron transitioning to a modified ground state of its energy dressed by the Cherenkov coupling, or it may (b) Probability of no Cherenkov emission p 0 ðþ∞Þ as a function of the distance to the conventional Cherenkov thresholdβ th , below this threshold. The system considered for this example is a single electron traversing an index window defined by n ε ¼ 2 Ã 1ðε ∈ ½0; 5 eVÞ [where 1ðε ∈ AÞ is the indicator function of a subset of energies A]; however, similar effects occur regardless of the choice of particle or transparent index medium. Here, an electron 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 2 orders of magnitude less than the conventional prediction. (b) In strongly coupled regimes, the presence of several poles of the function fðzÞ can result in a pseudoperiodic decay of p 0 ðtÞ. complete a full decay when introducing higher-order diagrams including additional, multiple, photons. Far below the conventional Cherenkov threshold, the behaviors of the conventional and regularized theories match again. We observe a discontinuous transition in the behavior above and below the Cherenkov velocity threshold, which is essentially due to a discontinuous derivative of ΓðzÞ. However, we expect higher-order processes, that are neglected in the current formalism, to smoothen this transition: For multiphoton emission processes, the phase space for emission is larger, allowing for the emission of photons that do not individually satisfy the conventional Cherenkov relation.
This modification of the Cherenkov threshold has direct implications on the time-dependent spectrum of Cherenkov radiation, compared to the conventional theory. Our theory reveals ultrafast dynamics of Cherenkov radiation, for which we already observe significant modifications to the wellknown observables: its energy-angle relation and the corresponding photon rate of emission. Specifically, Fig. 2 shows the implications of the time-energy uncertainty principle broadening the Cherenkov dispersion at very short times. These new effects can be seen through the flatcontinuum approximation [10], where the radiative shift z 0 results in an effective decay rate that we denote as Γ 0 ¼ Γðz 0 Þ. This approximation can be applied to Eq. (1), as it resembles the Fourier transform of a Lorentzian peaked at z ¼ z 0 . The Fourier transform yields an exponential decay with the rate Γ 0 =ℏ: corresponding to the regularized Cherenkov dispersion relation shown in Fig. 1(b), with a finite width in the angular domain proportional to Γ 0 [ Fig. 2(b)]. However, our predictions remain qualitatively valid only for times comparable to or less than the effective decay time of the zero-photon state ℏ=Γ 0 , after which it is no longer legitimate to assume a single photon is emitted. To account for longer times, it is necessary to extend the dimension of the initial Hilbert space, allowing the emission of more than a single photon (such an approach is proposed and discussed in Ref. [62], Sec. I). For dynamics at short timescales of the Cherenkov effect, which are of primary interest, this restriction is unimportant, and our theory remains predictive.

IV. BIFURCATION BETWEEN WEAKLY AND STRONGLY COUPLED REGIMES
Intriguing consequences of our nonperturbative formalism arise when pushing the limits of our formalism to the strong-coupling regime in which the effective fine-structure constant significantly increases, α Z ¼ Z 2 α ≳ 1, leading to the collapse of the flat-continuum approximation, which was accurately describing the physics above but does not hold here. Note that while in typical quantum field theories the coupling constants scale like αZ, the coupling relevant to the Cherenkov effect actually scales like αZ 2 , because it is an effect of a single vertex diagram. Physically, the strong coupling could be realized in the Cherenkov dynamics of highly charged ions. In the strongly coupled case, we find that multiple zeros, z i , of Δðz i Þ − z i ¼ 0 exist, allowing for a multibranched Cherenkov dispersion relation, with multiple Cherenkov angles θ i satisfying cos θ i ¼ ½ðϵ − z i Þ= ðn ϵ ϵβÞ for multiple z i , as illustrated in Fig. 1(c). In Fig. 2(c), we consider the specific case of an Fe 26þ ion moving at v ¼ 0.7c, corresponding to a kinetic energy of 20 GeV, as might be realized for various particles in large accelerator facilities [67] and high-intensity plasma wakefield accelerators [68]. Coincidentally, laser acceleration of iron atoms has recently reached nearby energy scales [69]. In this particular case, where the effective fine-structure constant is about 4.9, we see that, at long times, the spectral distribution of emitted photons is highly concentrated around two angle-frequency relations, both substantially different from the conventional Cherenkov relation. These photons are emitted over a short timescale, t i ∼ 1=Γðz i Þ, of the order of femtoseconds. Interestingly, we find that, for some emission frequencies, the emission angles can be backwards relative to the direction of electron motion, which is impossible in the conventional Cherenkov effect without a negative index of refraction.
Considering the detailed time dynamics of the charge, we find that these multiple poles can translate into a pseudoperiodic decay of the zero-photon state, as can be seen in Fig. 4(b) for a Fe 26þ . Pushing the effective charge to even larger values, we predict Rabi oscillations of the zerophoton probability, when the imaginary part of one of the poles cancels out (see Ref. [62], Sec. I). These damped Rabi oscillations are similar to those experienced by an atom in strong coupling with a lossy cavity mode [70]. Quantum Rabi oscillations have recently been investigated in ultrastrong-coupling regimes of cavity [71] and circuit QED [72], where a method for treating multiple discrete photonic modes has been proposed. Importantly, unlike bound charge systems that have discrete energy levels, here we work with a charged particle that can take a continuum of energy values. Moreover, the energy scale of free-charge transitions is not limited by the energy levels of AMO systems (can reach the x-or gamma-ray scale and not only visible or IR or below). We should expect new effects that have not been observed in cavity QED, such as the exciting perspective of observing ultrastrong-coupling physics from a single particle in the optical regime. These considerations make the appearance of AMO-physics-inspired phenomena like Rabi oscillation appealing, as no work has ever demonstrated free-charge vacuum Rabi oscillations and related phenomena. Our method could bridge the gap between AMO physics and free-charge physics, allowing us to systematically translate ground-breaking results in AMO to light-matter interactions with free charges.
To reveal the underlying physics, we analyze the relation between the new nonperturbative Cherenkov dynamics and the pole structure of the regularized self-energy. We show in Fig. 5 how an abrupt change in the pole structure signals a bifurcation process between the conventional Cherenkov dynamics and the strong-coupling physics reported in Fig. 2(c). This transition is characteristic of a bifurcation process, as a small change of a parameter (here, the electron velocity β or the particle charge Z) can result in the emergence of new branches of poles or the cancellation of their imaginary part. Physically, these abrupt changes in the pole structure result, respectively, in new multiple decay modes and Rabi-like oscillations of the decay rate. In order to show the parameter space in which the reported effects are expected to occur, we study the detailed dependence of the poles of fðzÞ as a function of the ion energy [ Fig. 5(a)] and as a function of the ion charge Z altering the effective fine-structure constant α Z [ Fig. 5(b)]. Stronger-coupling regimes can be effectively achieved either by increasing the velocity of a particle of a given charge Z [ Fig. 5(a)] or by increasing the particle's effective charge, at a given speed β [ Fig. 5(b)]. In both cases, we denote the emergence of new poles at a given value of β or Z, corresponding to critical coupling. The emerging poles always appear in pairs and split along two branches for larger β or Z. An even larger effective charge results in the emergence of two supplementary poles before canceling the real part of the original pole Γ 0 ¼ 0 for Z ≥ 56, which results 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 in heavy-ion colliders [25,67]. It is worth mentioning that anomalous Cherenkov radiation in these settings has been observed and attributed, at the time, to superluminal particles-tachyons [26]. The account of the electron relativistic recoil [19,73] was proposed as an explanation for the observation of seemingly superluminal particles. Interestingly, we also predict that the Cherenkov angle of photons radiated by highly charged ions would verify 1=ðn ϵ cos θ i Þ > 1 at some frequencies, which is the behavior expected from superluminal particles [26].
QED interactions are conventionally limited by the size of the fine-structure constant. By increasing the effective fine-structure constant, we explore strong-coupling physics in QED and reveal the presence of bifurcation processes in the Cherenkov effect. The bifurcation is a strong qualitative indicator of new phenomena occurring in these ranges of parameters that may occur at quantitatively different parameters due to other strong-coupling corrections. These findings point to the possibility of studying the physics of strong-coupling field theories like quantum chromodynamics [9], using some of the most commonly studied and relatively accessible effects of QED. The presence of a critical parameter (effective fine-structure constant or electron velocity) with a splitting of the pole branch is reminiscent of a phase transition. However, we should include higher-order processes in the calculation of the nonperturbative self-energy to trust quantitative predictions in these (ultra)strongly coupled settings. Additional corrections occur due to electron-positron creation and annihilation diagrams that dress such highenergy particles in their steady states (corresponding to resummations at all orders in αZ [23], which we can safely neglect when αZ < 1). Further corrections also occur due to interactions between the ion and the medium that cannot be described through the macroscopic permittivity of the medium and require a more extensive microscopic description (e.g., bremsstrahlung). For longer times, higher-order diagrams should be taken into account in our calculations; however, it is hard to predict their relative contribution in strongly coupled regimes: For instance, it has recently been predicted that the rate of spontaneous photon emission by an atom in the ultrastrong-coupling regime can actually decrease as the coupling is increased, suggesting some kind of destructive interference of the bare spontaneous emission diagram with higher-order diagrams that terminate with the emission of a single photon [74]. The smaller decay rate observed in our prediction [ Fig. 4(b)] already suggests partially destructive interference between the fundamental Cherenkov diagram and our nonperturbative summation. Higher-order processes and all-order developments in αZ 2 could be integrated in our theory by adequately modifying the coupling between different terms of the resolvent operator.

V. CONCLUSION
In summary, we present a method to regularize timedependent quantum observables in systems where light and matter interact. Our theory accounts for the nonperturbative self-energy in a time-dependent way. Applied to the Cherenkov effect, we predict readily observable modifications to the conventional theory in weakly coupled regimes and propose a pathway to reveal new physics in strongly coupled regimes that can be effectively achieved with highly charged ions. The observed bifurcation process in the strong-coupling regime originates from the behavior of the self-energy function fðzÞ. Thus, more generally, we should expect to see similar bifurcation processes in any system where strong coupling can be effectively achieved [10]. This observation is especially interesting, since there are Cherenkov analogues in platforms, such as superconducting qubits, where strong-coupling physics is usually observed [34,35], and therefore our predictions could be tested there.
An experiment involving Cherenkov radiation in the strong-coupling regime may also furnish new tests of radiation reaction in dielectric media [75], which is based on the observation that radiation-reaction effects are manifested in the self-energy of the system of the charged particle and the quantized electromagnetic field [76,77]. Further physical understanding of the self-energy can also be provided by adequately transforming the QED Hamiltonian via the Pauli-Fierz transformation, thus clearly exhibiting the interplay between the particle transverse field and mass correction [10].
Other directions to induce stronger-coupling regimes can be investigated by our formalism. Couplings can be enhanced by using systems that display strong light-matter interactions, like photonic cavities, or polaritonic media, which support modes of an effectively high index of refraction. Examples of such polaritonic media are thin metallic films, graphene, and thin polar dielectrics, each of which can potentially support optical modes with an effective mode index of refraction of about 100 [78,79], thus being a promising paradigm for enhanced light-matter interaction [47,65,66]. Another interesting option for achieving free-electron-photon strong coupling is to confine a free electron into a photonic cavity which supports a broadband resonance phase matched to the electron phase velocity, allowing the electron to coherently emit and reabsorb the cavity photons, thus experiencing Rabi oscillations with free electrons.
Additionally, our predictions create new opportunities for the design of ultrafast Cherenkov probes. The theoretical ability to probe short-time dynamics in systems where multiple particles interact pushes towards experimental advances in pump-probe-like experiments or timedependent electron energy-loss spectroscopy [59,[80][81][82]. More specifically, our findings coincide with the recent development of time-resolved imaging techniques combining electron optics with state-of-the-art spectroscopy [83][84][85][86]. We believe these theoretical findings are showing new prospects by which free electrons and other charged particles can provide a platform for the exploration of quantum light-matter interactions and strong-coupling physics. In this work, we consider the following system: a single electron propagating at speed v ¼ βc (momentum p ¼ mv), interacting with a continuum of photons with momenta q ¼ ℏk. We consider the following Hilbert space:

ACKNOWLEDGMENTS
where H el (respectively, H ph ) is the Hilbert space describing the quantum states of a single electron (respectively, the full Fock space of photons), defined by its momentum jpi (respectively, by the momentum of every photon jq; q 0 ; q 00 ; …; q ðNÞ ; …i). We consider the usual QED Hamiltonian to describe the interaction of a spin-1=2 particle with the electromagnetic field [9]: where A μ is the electromagnetic vector potential and fγ μ g μ∈f0;1;2;3g are the Dirac matrices. In this work, we assume that the photon energies are much smaller than the energy of the moving charge, which enables us to neglect spin effects and the charge recoil due to photon emission. Assuming the Coulomb gauge ∇ · A ¼ 0 in Eq. (A3), we can show that the coupling Hamiltonian is equal to where v is the electron velocity [47].
In the following, we define α as the fine-structure constant and the coupling Hamiltonian V.
Additionally, only the polarization in the ðp; qÞ plane contributes to the coupling. These assumptions allow us to compute the following coupling matrix element: where jp; 0i is the quantum state of a single electron propagating with momentum p in the medium, with no photon present. Here,ω q ¼ 1 2 ½∂=ð∂ωÞðω 2 q ε ω Þ ¼ c 2 q½ð∂qÞ=ð∂ωÞ is the modified normalization factor in a dispersive medium [87] and Ω the normalization volume factor.

First-order perturbation theory
We first quickly review the result from the application of Fermi golden rule to this quantum system, to predict the intensity spectrum of Cherenkov radiation. A similar derivation can be found in Ref. [87]. We chose similar notations as in the main text of this article: Ω is the normalization volume factor, ℏ is the reduced Planck constant, ω is the photon energy, and (E i , p) [respectively, (E f , p 0 )] are the (energy, momentum) of the initial (respectively, final) states of the electron.
We compute the decay rate of the initial state, given by Fermi's golden rule (FGR): Using Eq. (A6), we get the following: We then find the original Frank-Tamm formula: where Θ½. ¼ 1 if the condition between bracket is satisfied, Θ½. ¼ 0 otherwise, and the subscript ω in n ω denotes the frequency dependence of the refractive index. We also use the notation n ε in the future, where ε ¼ ℏω is the photon energy.

Resolvent theory applied to the quantum Cherenkov effect
A comprehensive introduction to the resolvent theory applied to atomic systems in QED can be found in Ref. [10]. The only information we need about the total Hamiltonian H ¼ H 0 þ V (where H 0 is the Hamiltonian of the unperturbed system) is (i) the eigenstates and energies of the unperturbed Hamiltonian H 0 and (ii) the coupling matrix elements of the perturbation term V, previously used in our derivation of the Fermi golden rule. The fundamental equation on the resolvent operator G is the following: where z is a complex energy variable and GðzÞ is the resolvent operator defined in Ref. [10]. It can be understood as the Fourier transform of the evolution operator UðtÞ. They can be mapped as follows: with C þ 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 follows that we can get a set of algebraic equations on different matrix elements of the operator GðzÞ, by multiplying Eq. (A11) by adequate bra and ket: where, for every operator A, we denote A l;r ¼ hljAjri.
The key to solving the dynamics of the system is to identify the self-energy function, which appears in various places along the derivation. In Ref. [62], Sec. I, we derive another set of fundamental equations on the resolvent operator, by further extending the Hilbert space to more-than-one photon emission. Let us emphasize 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 G 0 ðzÞ, we get the following expression (momentum conservation is enforced by the coupling Hamiltonian V ðr;qÞ;ðp;0Þ ): To find UðtÞ, we are interested in values of fðzÞ arbitrarily close to the real axis, of which we can compute the real and imaginary parts. To get the imaginary part, we use the fact that Im½1=ðx þ i0 þ Þ ¼ −πδðxÞ: ΔðzÞ ¼ αβ 2π We note that Γð0Þ ¼ Γ FGR and that Δð0Þ matches with the second-order perturbation-theory calculation of the energy shift of jp; 0i due to the electromagnetic coupling. Thus, Eqs. (A18)-(A20) represent a nonperturbative generalization of Cherenkov radiation and the Lamb (radiative) shift of an electron in a medium. These equations are key to developing the results of this paper, and we explore their consequences.

APPENDIX B: REGULARIZATION OF f ðzÞ IN FREQUENCY DOMAIN
This Appendix presents the regularization technique we develop for the purpose of this paper, in order to derive time-dependent regularized observables of the Cherenkov effect. We can readily notice that the real part of fðzÞ diverges and that it is related to the imaginary part of fðzÞ being unbounded. Splitting the imaginary part of fðzÞ into two terms, we observe the following: We can simplify the integration domain of these two functions: (i) When z → þ∞, the Heaviside condition in Γ 1 ðzÞ forces ε to be very large. In this limit, all materials must have n ε → 1 (thus, n ε β < 1), and so the integration domain of Γ 1 ðzÞ goes to zero. We observe a similar behavior when z → −∞: (ii) Because the Heaviside function and the integration domain of Γ 2 have zero intersection when z < 0, Γ 2 ðz; n ε Þ is zero for z < 0. However, when z → þ∞, the integration domain of Γ 2 ðzÞ is unbounded, and this integral will diverge: The fact that ΓðzÞ is unbounded results in a misdefinition of ΔðzÞ, as real and imaginary parts of the analytical function fðzÞ are related by Kramers-Kronig relations: A natural way to regularize fðzÞ in the case of the Cherenkov effect is to subtract the contribution of free space to the integral n ε ≡ 1, as a free electron propagating in free space at a constant velocity cannot decay and emit a photon. In the following, we define fðzÞ as the free-spaceregularized function: fðzÞ ≔ fðzÞ − fðz; n ε ≡ 1Þ: We find that the regularized self-energy function fðzÞ is well behaved for realistic refractive indices n ε . Let us show that with an example, given a linear lossless material, together with the assumption of causality. Under such conditions, it is possible to prove [88] that the index scales like approximately 1 − ð1=ε 2 Þ for large ε (here, ε is still the photon energy). We thus get n ε ¼ 1 − ð1=ε 2 Þ þ Oð1=ε 2 Þ when ε → þ∞ (it is a direct consequence of the Kramers-Kronig relations [88]). We can thus expand the integrand of the function fðzÞ − fðz; n ε ≡ 1Þ when ε → þ∞: which is integrable when ε → þ∞ (and the integrand is also integrable when ε → 0 þ ). Thus, our regularization technique is naturally backed up by fundamental properties of the refractive index n ε in realistic materials. We also notice that the renormalized imaginary part ΓðzÞ=2 is always positive: ΘðvÞ ¼ 1 ½0;1 ðvÞ: ðB14Þ If u ≤ 1, the integrand is χðε; n ε Þ − χðε; 1Þ ≥ 0. If u=n ε ≤ 1 but u ≥ 1, the integrand is χðε; n ε Þ ≥ 0. Overall, the integrand is positive; thus, ΓðzÞ ≥ 0 for all z.

APPENDIX C: DERIVATION OF REGULARIZED TIME-DEPENDENT QUANTUM OBSERVABLES
In Ref. [62], Sec. I, we differentiate between two methods, relying on two different assumptions on the Hilbert space in which the quantum state representing our system evolves. The first method is presented in this Appendix. However, in both formulations, the resolvent of the zero-photon state remains the same [given by Eq. (A16)], which results in the same time-dependent evolution of U 0 ðtÞ ¼ hp; 0jUðtÞjp; 0i. We show that, by Fourier transforming the resolvent matrix element, one can compute the time evolution of quantum states and relevant observables. In the following, integral limits will often be left implicit.

Time-domain derivation of the zero-photon state decay
To solve for the time evolution dynamics of the system, this section first presents the probability of having no photon emission as a function of time.
The value of fðzÞ above and below the axis are complex conjugated; thus, And the decay probability of the zero-photon state is given by We readily notice that the real frequency z 0 verifying Δðz 0 Þ ¼ z 0 is of particular importance to describe the decay of the zero-photon state. For negligible variations of fðzÞ around z 0 (flat-continuum approximation), the integrand can be approximated by a Lorentzian peaked at z 0 , Δðz 0 Þ being an energy shift, while Γðz 0 Þ=2 is the decay rate.
Because our expression of the regularized function fðzÞ matches with first-and second-order perturbation theory, the regular quantum prediction is given by When Γðz 0 Þ ¼ 0, part of the integrand becomes a Dirac distribution and the zero-photon state only partially decays to a nonzero value jU 0 ðþ∞Þj 2 ¼ f1=½1 − (dΔðzÞ=dz)j z¼z 0 g 2 : In every case, it is obvious that U 0 ð0Þ ¼ 1.

Estimation of z 0 in weakly coupled regime
In practice, we observe that z 0 takes a small negative value for single electrons in optical regimes, which can be seen through the following expansion for small z, in the case of an index constant over an energy range Δε: Using this expansion around z 0 , we get 2π αβ 2 z 0 ≈ 2π αβ 2 ½fðz 0 Þ − fðz 0 ; n ε ≡ 1Þ ðC8Þ The second term, proportional to z 0 , is of the order of 1, while ½ð2πÞ=ðαβ 2 Þ > 850 for a single electron; thus, we can safely neglect it. We thus get the following scaling law for z 0 : as we can compute the integral term and check that it will take a negative value on the order of −1 when nβ ≳ 1 in the optical regime.

Spectral angular density of the first emitted photon (closed Hilbert space formulation)
We insert the expression of G 0 ðzÞ into Eq. (A15). We use the momentum conservation enforced by the coupling Hamiltonian V ðr;qÞ;ðp;0Þ to reduce the number of degrees of freedom on G ðr;qÞ;ðp;0Þ ðzÞ ¼ G ðp−q;qÞ;ðp;0Þ ðzÞ. Doing so, the volume factors cancel out, and we get the following expression for G ðr;qÞ;ðp;0Þ ðzÞ: G q ðz þ E p Þ ≔ G ðr;qÞ;ðp;0Þ ðz þ E p Þ ¼ ð−ev sin θÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ℏ 2Ωϵ 0ωq From there, we can readily convert this expression to the time domain and then get the probability of the first emitted photon, by integrating over possible photon momenta q: which gives us, using the expression of ΓðzÞ, In both expressions, we notice that the expression of the imaginary part of fðzÞ intervenes. We here notice that replacing ΓðzÞ by its regularized form ΓðzÞ ≔ ΓðzÞ − Γðz; n ε ≡ 1Þ is of paramount importance to ensure that the total probability in the Hilbertian space is conserved and equal to 1: p 1 ðtÞ regularization also results in the regularization of the spectral angular density f½∂ 2 p 1 ðtÞ=∂ε∂ug using the expression derived from the frequency domain: For convenience, we use the latter expression derived from the time domain in our numerical simulations. In the flatcontinuum approximation, we can readily notice that the spectral angular density is enhanced for ðε; uÞ satisfying εð1 − n ε βuÞ ¼ z 0 ⇔ ε − z 0 n ε βu ¼ 1; which coincides with the conventional Cherenkov relation in the case z 0 ¼ 0.
This new dispersion relation also allows the existence of backward Cherenkov radiation in this simple setting when z 0 > 0 for ε < z 0 [see, for instance, Fig. 2(d)].