Cutting Feynman Loops in Ultrastrong Cavity QED: Stimulated Emission and Reabsorption of Virtual Particles Dressing a Physical Excitation

In quantum field theory, bare particles are dressed by a cloud of virtual particles to form physical particles. The virtual particles affect properties such as the mass and charge of the physical particles, and it is only these modified properties that can be measured in experiments, not the properties of the bare particles. The influence of virtual particles is prominent in the ultrastrong-coupling regime of cavity quantum electrodynamics (QED), which has recently been realized in several condensed-matter systems. In some of these systems, the effective interaction between atom-like transitions and the cavity photons can be switched on or off by external control pulses. This offers unprecedented possibilities for exploring quantum vacuum fluctuations and the relation between physical and bare particles. Here we show that, by applying external electromagnetic pulses of suitable amplitude and frequency, each virtual photon dressing a physical excitation in cavity-QED systems can be converted into a physical observable photon, and back again. In this way, the hidden relationship between the bare and the physical excitations becomes experimentally testable. The conversion between virtual and physical photons can be clearly pictured using Feynman diagrams with cut loops.


I. INTRODUCTION
In quantum field theory, the creation and annihilation operators in the Lagrangian describe the creation and destruction of bare particles which, however, cannot be directly observed in experiments (see, e.g., Refs. [1,2]). Bare particles, due to the interaction terms in the Lagrangian, are actually dressed by virtual particles and become real physical particles which can be detected. The interaction modifies the properties of the particles, e.g., giving rise to the Lamb shift of electronic energy levels [3,4] and affecting the charge, mass, and magnetic moment of the electron [1,5,6]. The predictions of the theory must be expressed in terms of the properties of the physical particles, not of the non-interacting (or bare) particles [1,2]. The relations between the bare and the physical particles, like the bare particles themselves, are unobservable.
The need to account for virtual particles in the USC regime of cavity QED is exemplified by the fact that the correct description of the output photon flux from the cavity, as well as of higher-order Glauber normal-order correlation functions, requires a proper generalization of input-output theory [15]. Due to the contribution from counter-rotating terms in the interaction Hamiltonian, the ground state |E 0 of the system contains a finite number of photons [32], i.e., E 0 â †â E 0 = 0, whereâ andâ † are the annihilation and creation operators for the cavity mode. However, the ground state cannot emit energy, so the output photon flux cannot be proportional to â †â , as in standard input-output theory. Instead, it has been shown [15,33] that the cavity output (which can be detected by a photo-absorber) is proportional to x −x+ , wherex + is the positive frequency component of the quadrature operatorx =â+â † andx − = (x + ) † . Since E 0 |x −x+ | E 0 = 0, the photons that contribute to the ground state are not physical observable particles, but virtual ones. Furthermore, also the (physical) system excitations are enriched by unobservable virtual particles. For instance, the first excited state, corresponding to a single physical particle, may contain contributions from an odd number of virtual particles. All these virtual contributions, however, are significant only in the USC regime, not at weaker coupling strengths.
In QED, the modifications of the electron properties due to the interaction with virtual particles are known as radiative corrections [1]. In addition to the diagrams describing the processes in lowest order of perturbation theory, the Feynman diagrams representing the radiative corrections to a process contain additional vertices (loop diagrams), corresponding to the emission and re-absorption of virtual photons. Here we show that, analogously, the energy corrections to the ground state and to the excited states of a cavity-QED system in the USC regime are described by loop diagrams corresponding to the emission and reabsorption of virtual photons.
An interesting feature of these condensed-matter systems is that the effective interaction between atom-like transitions and the cavity field can be switched on and off by applying external drives. This offers the opportunity to convert the virtual excitations into real particles which can be detected. Both spontaneous [7] and stimulated [34] conversion of virtual photons from the ground state of a cavity QED system in the USC regime have recently been analyzed . Also, virtual photon pairs are converted into real ones in the dynamical Casimir effect (DCE) [35], which has been analyzed for [36][37][38] and experimentally demonstrated [39] in circuit QED. Potentially, a proper modulation of the mirror in a DCE setup could also allow for absorption of photon pairs [40]. In contrast to these previous works, we here show how to convert various numbers of virtual photons into real ones and back, both for the dressed vacuum state and for a dressed excited states. We show that the corresponding Feynman diagrams can be obtained by cutting the loop diagrams describing the energy correction of a physical excitation. Specifically, conversion of virtual photons dressing a physical excitation into real ones is described by the first half of cut loop-diagrams (photon emission). Similarly, the conversion of real photons back into virtual ones bound to a physical excitation corresponds to the second half (photon absorption). Moreover, the proposed scheme, does not need ultrafast modulation of boundary conditions and it can give rise to a conversion probability close to one.

II. RESULTS
A. The Rabi model The simplest cavity-QED model beyond the RWA is the quantum Rabi model [41,42].
The Hamiltonian is ( = 1)Ĥ R =Ĥ 0 +V , whereĤ 0 = ω câ †â + ω e |e e| + ω g |g g| is the bare Hamiltonian in the absence of interaction. Here,â andâ † are the photon destruction and creation operators for the cavity mode with resonance frequency ω c , |g and |e are the ground and excited atomic states, respectively, and ω e(g) are the corresponding energy eigenvalues. The interaction Hamiltonian iŝ where Ω R is the coupling strength andσ x =σ + +σ − = |e g| + |g e|. When ω c ≈ ω eg ≡ ω e − ω g , the interaction Hamiltonian can be separated into a resonant and a nonresonant part:V =V r +V nr , whereV r = Ω R (â †σ − +âσ + ), andV nr = Ω R (â †σ + +âσ − ). The nonresonant terms do not conserve energy nor the number of excitations. They can be neglected when The interaction Hamiltonian has a structure which is very similar to that of the QED interaction potential, although it is less complicated. The Rabi model can be viewed as a very simple QED system, where there is only a single photon mode and a two-state electron.
As a consequence, we would expect that Feynman diagrams for the Rabi Hamiltonian will be a simplified version of QED diagrams. One such diagram, for the nonresonant transition |g, 0 → |e, 1 (the second entry in the ket denotes the photon number), is shown in Fig. 1a.
However, some care must be taken when drawing diagrams for processes involving more than one photon in the same mode [43], which occur in cavity QED. Stimulated emission [44], the mechanism behind laser action, is one such process. It is a one-photon process |e, n → |g, n+1 , where the n photons in the initial state stimulate the downward transition of the atom, affecting the transition rate which becomes proportional to n + 1. This factor must be included in the rules for the diagrams in order for calculations to be correct. An example of a diagram showing the stimulated-emission process |e, 1 → |g, 2 is presented in

B. Bare vs physical excitations
Owing to the presence ofV nr in the Rabi Hamiltonian, the operator describing the total number of excitations,N =â †â + |e e|, does not commute withĤ R and as a consequence the eigenstates ofĤ R do not have a definite number of excitations [32]. WhenV nr can be neglected, the Hamiltonian becomes block-diagonal and easy to diagonalize (this is the Jaynes-Cummings (JC) model [45]). The resulting eigenstates can be labelled according to their definite number of excitations n. The ground state (zero excitations) is simply |E 0 = |g, 0 , and the n ≥ 1 excitation states |E ± n , obtained by diagonalization of 2 × 2 subspaces, can be written as where C n and S n are amplitudes determined by Ω R and the detuning ω c −ω eg . The eigenstates |E i of the full Rabi Hamiltonian, however, are expressed as a superposition of bare states with varying numbers of bare excitations (see, e.g., Ref. [33]): where the coefficients c i g,k and d i e,k are determined by Ω R , ω c and ω eg . When Ω R ω c , ω eg , the Rabi eigenstates reduce to the JC ones. Note that whileN is not conserved with the Rabi Hamiltonian, the parity (even or odd number of excitations) still is.
The mean photon number for the system in the ground state is However, these ground-state photons are virtual and cannot be detected. Otherwise the system, emitting a continuous stream of photons from its ground state, would be a perpetualmotion machine. A proper treatment shows that the output emission rate from a single-mode resonator is not proportional to â †â , but to x −x+ , wherex + is the positive frequency component of the quadrature operatorx =â+â † andx − = (x + ) † [15,33]. For weak coupling, â †â and x −x+ coincide, but in the USC regime they can differ markedly.
The componentsx + andx − are obtained in the eigenvector basis ofĤ R asx if the eigenstates ofĤ R are labelled according to their eigenvalues such that E k > E j for k > j. As expected, we find that E 0 |x − (t)x + (t)| E 0 = 0, which demonstrates that the photonic Fock states enriching the Rabi ground state are actually virtual. This reasoning can be generalized to the excited states of the system. For the first excited state, the one-photon correlation is different from zero ( E 1 |x − (t)x + (t)| E 1 = 0). However, the output coincidence rate from this state, proportional to the physical two-photon correlation function E 1 |(x − ) 2 (x + ) 2 | E 1 , is equal to zero. On the contrary, the correlation functions for n ≥ 2 bare photons in the first excited state are different from zero (e.g., E 1 (â † ) 2 (â) 2 E 1 = 0). We can conclude that |E 1 is a single physical excitation which, however, is enriched by a larger number of virtual photons.

C. Energy corrections and loop diagrams
The analytical spectrum of H R is defined in terms of the power series of a transcendental function [42]. Approximate forms, which may provide more insight, can be derived by a perturbative approach (see, e.g., Ref. [46]). Let us consider the correction to the ground state energy ∆ 0 ≡ E 0 − E 0 . The lowest-order (in the nonresonant potential) contribution can be expressed as A direct inspection of the terms in the series shows that only the terms with an even number of V r are different from zero. It is possible to associate a diagram with each of the terms in the series appearing in Eq. (26). Figure 2a shows the first three diagrams providing a nonzero contribution. The first corresponds to the first term in the r.h.s. of Eq. (26). The second diagram describes the third term in the series: Each bubble diagram, corresponding to a matrix element ofĜ 0 , describes intermediate virtual excitations. All the resulting bubble diagrams contain at most two photon waves, since we considered only the lowest-order corrections in the nonresonant potential. Four and more photon waves arise when going beyond second-order perturbation theory.
This approach can also be applied to the excited states. Considering the first excited state, we obtain The mean value over the state |g, 1 in Eq.

D. Three-level atom
We now consider a system consisting of a single-mode cavity interacting with the upper two levels |e and |g of a three-level atom. The energy difference E gs between the bottom level |s and the middle level |g is assumed to be much larger than the cavity-mode resonance frequency such that the cavity does not interact with the atom in the lowest energy state |s (see Fig. 3a). As we will show, the additional state |s enables an effective on/off-switch of the atom-cavity interaction. The system Hamiltonian is simplyĤ C =Ĥ R + ω s |s s|. This Hamiltonian is block-diagonal and its eigenstates can be separated into a non-interacting sector |s, n , with energy ω s + nω c , where n labels the cavity photon number, and dressed atom-cavity states |E i , resulting from the diagonalization of the Rabi Hamiltonian (see The direct excitation of the atom by applied electromagnetic pulses is described by the |s, 3i whereV sg = µ sg (|g s| + |s g|),V se = µ se (|e s| + |s e|), and µ sg and µ se are the dipole moments (here assumed to be real) for the transitions |s ↔ |g and |s ↔ |e , respectively.
We consider quasi-monochromatic pulses ThusĤ d applied to a dressed state is able to convert the virtual photons enriching the physical excitations into real ones which can be detected. This is possible becauseĤ d induces transitions from the atomic states |g and |e (coupled to the cavity) to the noninteracting state |s . Of course, the transitions only occur if the driving-field frequency ω is resonant with the corresponding transition frequency. In the absence of counter-rotating terms, a JC eigenstate with n excitations can only undergo transitions towards states with n photons:

E. Stimulated emission and reabsorption of virtual particles
We first consider the system prepared in the ground state |E 0 of the Rabi Hamiltonian.
corresponding to a stimulated emission process (see Fig. 3b). The corresponding matrix element s, 2 V sg E 0 = µ sg c 0 g,2 , determining the transition probability, is proportional to the probability amplitude c 0 g,2 that in the Rabi ground state there are two virtual photons. By exploiting second-order perturbation theory, this matrix element can be expressed as It is even more interesting to undress the excited states of the Rabi model. This can provide access to the relationship between bare and physical excitations. Let us consider the lowest-energy excited state |E 1 which, as we have shown above, is a single-particle state.
Following the same steps as used in obtaining the series in Eq. (10), the diagrams in Fig. 4b can be drawn. According to the Fermi golden rule, an input pulse of central frequency 3 is proportional to the probability amplitude that in the state |E 1 there are three virtual photons. By applying second-order perturbation theory, it can be expressed as The We consider the system initially prepared in the state |E 0 (preparation starting from the ground state |s, 0 can be easily achieved by sending a suitable π pulse). Then, a Gaussian pulse with central frequency ω = E 0 −ω s −2ω c induces the transition |E 0 → |s, 2 .
Specifically, the pulse area required to obtain a complete transition is π/ s, 2 V sg E 0 . The pulse arrival-time corresponds to the time when the loops in the Feynman diagrams are cut. Figure 4c displays the dynamics of the intracavity mean excitation number x −x+ , which is directly related to the output photon flux Φ out (t) = γ c x − (t)x + (t) (where γ c is the photon escape rate through the cavity boundary), as well as the equal-time second-order correlation Before the arrival of the Gaussian pulse (shaded red curve), the output photon flux is zero, since E 0 |x −x+ | E 0 = 0. After the arrival of the pulse, the photon flux becomes nonzero and G (2) (t) x −x+ , confirming that a two-photon state is actually generated as expected from the diagrams in Fig. 4a. When a second pulse is sent, the two photons are reabsorbed into the Rabi ground state: |s, 2 → |E 0 (diagrams in Fig. 4a with the leftward time-arrow). Figure 4c shows that a residual small excitation remains in the system after the arrival of the second pulse. This can be attributed to the influence of damping and to a non-negligible transition probability to higher-energy levels induced by the tails of the pulse spectrum. Figure 4d displays the dynamics starting from the system prepared in the state |E 1 . We observe that, after the arrival of the Gaussian pulse (with central frequency ω = E 1 −ω s −3ω c and area π/ s, 3 V sg E 1 ), the initial zero third-order correlation function G (3) approaches 6, the value corresponding to a three-photon state. This result confirms the occurrence of the transition |E 1 → |s, 3 . Also in this case, the emitted photons are reabsorbed by sending an additional identical Gaussian pulse. We observe that, within the standard RWA, Figure 4d at t = 0 displays a higher value. This is a peculiar effect of the USC regime, where the intracavity mean excitation number is quadraturedependent. In particular, it increases forx measurements and decreases for measurements of the conjugate quadratureŷ = i(â † −â).
Having studied above processes withV se , we now turn to those involvingV se instead. Figure 5 shows these processes, with the action ofV se represented in the diagrams by blue crosses. These processes are able to break one-photon loops, as illustrated in Fig. 5a, which shows the diagrams associated with the transition |E 0 → |s, 1 , where a cavity photon is emitted (rightward time-arrow) and reabsorbed (leftward time-arrow). Figure 5b  In complete analogy with what was shown in Fig. 4c and Fig. 4d, we present in Fig. 5c  and Fig. 5d nonperturbative numerical calculations describing the dynamics of the undressing and re-dressing of the Rabi vacuum and of the Rabi lowest-energy excitation, taking into account the presence of dissipation channels, the presence of higher-energy levels, and the non-monochromaticity of the driving pulses. The dynamics of the intracavity mean excitation number x −x+ , which becomes close to 1 shown in Fig. 5c, and the equal-time second-order correlation function G (2) (t) x −x+ , shown in Fig. 5d, confirm that onephoton and two-photon states are actually generated as expected from the diagrams in Fig. 5a and Fig. 5b, respectively. We observe that both in Fig. 4 and have been realized [49].

III. DISCUSSION
The results presented here show that the USC regime of cavity QED can be exploited to observe, in a direct way, how interactions dress observed particles by a cloud of virtual particles. Such particle dressing is a general feature of quantum field theory and many-body quantum systems. We have shown that, by applying external electromagnetic pulses of suitable amplitude and frequency, each virtual photon enriching a physical excitation can be converted into a physical observable photon. In this way, the hidden relationship between the bare and physical excitations can be unravelled and becomes experimentally testable.
We have shown that the Feynman diagrams describing the photon emission from a phys- We limited our analysis to the dressed vacuum and to a one-particle state. It can be easily extended to study higher-energy excitations. Moreover, we considered only processes up to second-order perturbation theory. The present analysis can be generalized to describe higher-order processes, involving more than three photons, which can take place if the lightmatter interaction is sufficiently strong [49].
The most promising candidates for an experimental realization of the proposed stimulated conversion effects are superconducting quantum circuits and intersubband quantum-well polaritons. In particular, phase-biased flux qubits can reach the USC regime in circuit QED [50], as has been shown in experiments [23,49,51]. By adjusting the externally applied magnetic flux, these artificial atoms can acquire both the quantized level structure and the transition matrix elements required for the observation of the stimulated emission and reabsorption of virtual particles [7]. The USC regime can also be reached for intersubband transitions in undoped quantum wells [52]. In this system, an optical resonator in the terahertz spectral range is resonantly coupled to transitions between the two-lowest energy conduction subbands of a large number of identical undoped quantum wells. In this case, the upper valence subband plays the role of the lowest energy state |s (see Fig. 3). Ultrafast optical pulses can induce transitions between the valence and conduction subbands prompting the conversion from virtual to real photons and vice versa. Such experiments, being able to look inside the loops of Feynman diagrams, would provide deep insight into fundamental aspects of interaction processes in QFT.

SUPPLEMENTARY MATERIAL
In this Supplementary Material, we first restate some properties of the Rabi model and its diagrammatic representation, expanding on the discussion in the main text. We then proceed to explicitly calculate analytically the second-order correction to the lowest energy eigenvalues and comparing them to full numerical calculations. We also calculate matrix elements associated with the external drive used to stimulate the emission and reabsorption of the virtual particles dressing the excitations in the system.

A. Hamiltonian and basic diagrams
The interaction Hamiltonian of the Rabi model iŝ where Ω R is the coupling strength andσ x =σ + +σ − = |e g| + |g e|. Referring to the case ω c ≈ ω eg ≡ ω e − ω g , the interaction Hamiltonian can be separated into a resonant and a nonresonant contribution:V =V r +V nr , whereV r = Ω R â †σ − +âσ + , and V nr = Ω R â †σ + +âσ − . This interaction term has a structure which is very similar to that of the QED interaction potential, although it is simpler. The Rabi model can be viewed as a prototypical QED system where there is only one photon mode and a two-state electron. Therefore, we expect that the Feynman diagrams for the Rabi Hamiltonian will be a simplified version of the QED diagrams.
As in QED, there is only one vertex type with three lines: One wavy (photonic) line, one solid line with an incoming arrow, and one solid line with an outgoing arrow. The vertices (of the same type) corresponding to the four terms in the interaction Hamiltonian are displayed in Fig. 6. The upper diagram in Fig. 6a describes the spontaneous emission process and the lower one the absorption process. Starting from these four building blocks, it is possible to describe higher-order processes as in QED. However, in cavity QED there are processes that are not described in a complete way by Feynman diagrams directly derived from this form of the interaction Hamiltonian. Specifically, the presence of a resonator supporting discrete modes opens up the possibility of observing processes involving more than one photon in the same mode. Stimulated emission, the process underlying laser action, is one of these.
It is a one-photon process |e, n → |g, n + 1 , where, however, the n photons in the initial state stimulate the downward transition of the atom, affecting the transition rate which becomes proportional to n + 1. The Feynman diagram describing the process is the same one describing spontaneous emission (n = 0), shown in Fig. 6a. However, the transition rate for stimulated emission is n + 1 times larger than that of spontaneous emission. Hence the Feynman diagram in the absence of additional rules is not able to determine uniquely the transition amplitude for this process.
A possible solution is to expand the photon creation and destruction operators in Eq. (12) in the Fock basis. The resulting interaction operator iŝ whereα (n) + =â † |n n| = √ n + 1 |n + 1 n|, andα  Fig. 7b. In this case, the vertices will have n in = n incoming and n out = n ± 1 outgoing wavy lines. Each vertex (full/empty circle) contributes with a factor √ n V r/nr , where n = max(n in , n out ).
The Green's function for the system in the absence of interaction, where ω qg = ω q − ω g , with q = e, g, corresponds to a loop diagram with n wavy lines and one straight arrow. In Fig. 8, we show the two loop diagrams corresponding to G (2) e and G (1) g .

B. Second-order correction to the energy eigenvalues
The well-known second-order correction to the nth energy eigenvalue is where the prime in the summation means that the values k = n have to be excluded. For the first-order correction to the eigenfunction we have Following [46], defining the projection operator onto the space orthogonal to |n ,Q n = 1 − |n n|, Eq. (16) becomes whereĜ is the unperturbed Green's function calculated for E 0 n , the unperturbed eigenenergy of the system.
Using the definition of the Green's function from Eq. (18) and the projection operators, Eq. (15) becomes We apply these results to the JC Hamiltonian perturbed by the nonresonant potentialV nr .
In this case, the unperturbed HamiltonianĤ 0 becomesĤ JC =Ĥ 0 +V r , whose eigenvalues and eigenstates are E ± n and |E ± n , respectively. We have |E + n = C n |g, n + S n |e, n − 1 |E − n = −S n |g, n + C n |e, n − 1 .
The action of the nonresonant potential on these eigenstates iŝ andV nr |E − n = Ω R (−S n |e, n + 1 + C n |g, n − 2 ).
From the last two equations, we deduce that the nonresonant potentialV nr determines transitions from the subspace n (spanned by E ± n ) to (n+2) or (n−2) subspaces. As a consequence, we haveQ Owing to this property, Eq. (19) becomes whereĜ is the JC Green's function. Equation (24)  G =Ĝ 0 +Ĝ 0VrĜ0 + . . . . Equation (24) can thus be expanded as The lowest-order (second-order) correction to the ground state |g, 0 energy due to the nonresonant potentialV nr , using Eq. (26), can be expressed as In order to calculate ∆ 0 , we observe that g, 0 V nr e, 1 = e, 1 V nr g, 0 = Ω R . The remaining term, e, 1 Ĝ e, 1 , is a convergent geometric series that is calculated in Sec. III E. We obtain For Ω R /ω c < 1, ∆ 1 can be approximated to second order in Ω R : In Fig. 9, we show the comparison between the exact (numerical) and approximated (diagrammatic) calculation of the correction term to the ground state energy.
This approach can also be applied to the excited states. We consider the first excited state. Using Eq. (24), we are able to calculate the correction up to second order in the Observing thatV nr |e, 0 = 0,V nr |g, 1 = √ 2Ω R |e, 2 , and |E − 1 = −S 1 |g, 1 + C 1 |e, 0 , we havê Equation (32) becomes The energy correction ∆ 1 can be evaluated easily by directly usingĜ(z) or by summing up the infinite contributions arising from the Dyson series, described by the diagrams (see Sec. III E, Eq. (63)).
In the absence of detuning, we have for the first excited state with energy We obtain For Ω R /ω c < 1, ∆ 1 can be approximated to second order in Ω R : A comparison between the approximate analytical energy corrections and the corresponding nonperturbative numerical calculations can be found in Fig. 9.
C. Additional perturbationV sg allowing transitions from |g to |s Using Eq. (15), the correction to the JC eigenstate |E n up to the first order in the nonresonant potential becomes, We now consider the direct excitation of the artificial atom by applied electromagnetic pulses, described by the Hamiltonian whereV sg = µ sg (|g s| + |s g|),V se = µ se (|e s| + |s e|), and µ sg and µ se are the dipole moments (here assumed to be real) for the transitions |s ↔ |g and |s ↔ |e , respectively.
First, we consider the case with the system prepared in the state |E 0 = |g, 0 . The time-dependent perturbation can induce additional transitions whose rate can be evaluated with the Fermi golden rule.
In the absence of the counter-rotating interaction termsV nr ,Ĥ d can induce only zero-  (17)): We obtain The corresponding Dyson series is In Fig. 10 we compare the exact (numerical) and approximated (diagrammatic) calculation of this matrix element.
We now consider the case with the system prepared in the state |E − 1 = 1 √ 2 (−|g, 1 + |e, 0 ), whose energy is E − 1 = ω c − Ω R . The time-dependent perturbation can induce additional transitions whose rate can be evaluated with the Fermi golden rule. In the presence ofV nr , additional transitions, such as |E 1 ↔ |s, 3 , become activated. The matrix element for this transition is s, 3 V sg E 1 . It can be calculated perturbatively inV nr , approximating |E 1 to first order inV nr (see Eq. (17)): Observing thatV where we have used the relationsV nr |g, 1 = √ 2Ω R |e, 2 andV nr |e, 0 = 0, we obtain The corresponding Dyson series is (see Sec. III E and Eq. (64)) We now consider a situation analogous to that analyzed before. In this case, a transition |s ↔ |e is detuned at a much higher energy than the cavity resonance. The part of the time-dependent potential inducing the |s ↔ |e transitions isV (t) = E(t)V se . In the absence of the counter-rotating interaction termsV nr ,V (t) can induce zero-cavity-photon transitions |e, 0 ↔ |s, 0 .
In the presence ofV nr , additional transitions, such as |E 0 ↔ |s, 1 , can be activated. The matrix element for this transition is s, 1 V se E 0 . It can be calculated perturbatively in V nr , approximating |E 0 to the first order inV nr (see Eq. (17)): |E 0 |g, 0 +Ĝ(E 0 )V nr |g, 0 .
Using the results of Sec. III E, the Dyson series calculation of Eq. (49) gives on resonance E. Calculation of theĜ matrix elements using the Dyson equation Here we perform all the calculations for the determination of the correction to the selfenergy to second order inV nr . We have to sum all the elements of the infinite series. The generic matrix element is e, 1 Ĝ 0 (z)[V rĜ0 (z)] n e, 1 .
Hence we may perform the calculation for the self-energy considering only the even-power terms. In addition, we observe that g, n + 1 V r e, n = e, n V r g, n + 1 = √ n + 1 Ω R .