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One-photon wavepacket interacting with two separated atoms in a one-dimensional waveguide: Influence of virtual photons

We present a theoretical study of one photon wave-packet scattered by two atoms in one dimensional waveguide. We investigate the role of non rotating wave approximation terms to take into account correctly for the effects of the virtual photons that are exchanged between the atoms. These terms are shown to influence drastically the reflected and the transmitted fields, imposing strict constraints on their temporal envelopes.


I-Introduction:
The control of the interaction between light and matter is a research area undergoing continuous evolution because of the appearance of ever new challenges. A recent issue is the realization of all-optical quantum devices in one-dimensional waveguide for quantum information purposes [1][2][3]. Recent experimental progress in the designing of these systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the possibility to reach the strong interaction regime between photons and atoms (or artificial atoms) open new perspectives, allowing the controllable transport of the flying qubits (photon) and the realization of fundamental quantum information operations [1][2][3][18][19][20][21][22]. Beside these challenging, the interaction of light and a collection of atoms in such systems represents by its own an interesting new theoretical problem. The photon scattering by a single atom in a 1-d waveguide has been studied by Domokos et al. in a two-level system [23] using a Heisenberg approach whereas spectral studies involving different experimental configurations have also been realized in [2,18], and the case for three-level atoms has been studied by Witthaut and Sorensen [24]. The extension of these studies to systems with two artificial atoms and an array of N artificial atoms have been also investigated [25][26][27][28][29][30]. However, all these studies were restricted to the regime where the rotating wave approximation (RWA) is done. Introduction of frequencies cut-off, extension to negative frequencies are some procedures that are generally invoked to justify the neglect of far resonance frequencies or to recover finite coupling. In the case of two atoms in a dispersionless waveguide, non-RWA contributions cannot be neglected and are essential for a correct treatment of the problem and a deep understanding of virtual photon effects on the system dynamics. These features are well known in the field of superadiance since the exhaustive works of Friedberg, Hartmann and Manassah [31,32], Milloni and Knight [33] and others [34,35] following the pioneering work of Dicke [36]. The influence of virtual photons on the collective spontaneous emission of a photon wave packet by a (3-d) cloud of dense atoms has recently received a great deal of attention [37][38][39]. This problem is particularly rich in new striking quantum effects like collective lamb-shift, collective encoding, entanglement and directive photon reemission [40][41][42][43]. This interaction is associated with atomic shifts that modify the dynamics even for large samples.
Here, we present a detailed study of the scattering of a one photon wave packet by a system of two atoms in a lossless and dispersionless 1-d waveguide, taking into account the effects of the virtual photons. RWA is not done and we are interested in both the temporal and the spectral behavior of the scattered field. We show that both the atomic and the field dynamics dependent strongly on the nature (real or virtual) of photons exchanged by the atoms and we 3 clarify the role of each. Moreover, we establish the expression of the effective coupling between atoms and we show that it results from a subtle interference effect between parts of virtual photons. We discuss the consequences of using RWA and introducing artificial frequencies cut-off. We demonstrate the important result that the central wave packet frequency is always reflected only if non-RWA terms are taken into account. An additional feature in our approach is the development of a "time-dependent" point of view for the interaction. We show that the total reflection of the resonant frequency is related to the specific behavior of the temporal envelopes of the reflected and transmitted fields. Moreover, the transmitted wavepacket obeys a strong constraint that forces the electric field to distort so that its pulse-area (e.g. integral of the electric field envelope) vanishes, whereas for the reflected field the pulse area is opposite to the incident one. This feature was already pointed in our previous study of photon scattering by a single atom in 1-d waveguide [44] and turn to be fruitful to understand straightforwardly some temporal shaping effects. The time dependent approach is only little addressed in quantum optics, in contrast with semiclassical optics, where intensive studies have been carried out leading to fascinating experiments for optical control and manipulation of quantum systems [45].

II-The theoretical model :
We consider two identical atoms that interacts resonantly with a one-photon wave-packet propagating in the +z direction of an infinite lossless waveguide (figure 1). The transverse dimension of the waveguide d is assumed to be much smaller than 0 λ (the resonant wavelength) and the interatomic distance l (e.g. 0 , d l λ

<<
). An important consequence is that the electrostatic dipole-dipole interaction between the atoms is strongly inhibited in the waveguide and will be neglected through the paper [46,47]. Moreover, the atoms no longer radiate outside the z direction and the field remains uniform in the longitudinal direction of propagation [23,24]. The confinement of light in this waveguide ensures also that the strong interaction regime between the atoms and the photons can be realized.  [31-35, 37-42, 48].
With initially the atoms in the ground state, and for the second order in the interaction Hamiltonian, the wavefunction ( ) t ψ of the whole system (atoms+field) can be formally expanded as : The two first terms corresponds to states with an excitation number equal to one. In the first term, we have states with one photon in the field and both atoms in the ground level whereas in the second term, we have states with only one atom (j) in the excited state and no photons in the field. The last two terms correspond to an excitation number of three. The third term describes the situation where both atoms are excited and there is one photon in the field, whereas the last term corresponds to the situation with one excited atom (j) and two photons in the field. These states are necessary for the correct treatment of virtual photon and the collective Lamb-shift effects [37][38]42].
The evolution of the system is determined by the Schrödinger equation  (1), we obtain the following set of equations for the amplitudes: These equations show that states with excitation number equal to one (e.g. 1 2 , ,1

II-1 Atomic coupling:
The system of equations (2) can be considerably simplified because of the presence of a continuum of modes. In annex 1, we show that when 0 , / , c l ω >> Γ ∆ , a Markovian approximation can be used leading to the fundamental equation for the amplitudes Using the the mathematical relations: where ℘ designs the Cauchy principal part of the integral, we obtain : The integrals appearing in (6)  for the coupling elements: iCi l c π ε ± = ± + → and its imaginary part diverges ( ε is an artificial infrared frequency cutoff). This is not surprising since the atom-photon coupling is and diverges also in the infrared. However, only the sum of these integrals is involved in the integral in (3)

II-2 Quantum paths
The field and the atomic dynamic can be also understood from equation (3) in terms of photon exchange between atoms. Moreover, we represent in figure 2 the paths corresponding to all terms of equation (3)  The forward photon is then absorbed by atom 2 that relaxes to the ground state. The 2 M contribution is the same process than 4 M but with the absorption of the backward photon by atom 2. It's worthy here to notice that although the photon emitted in the forward direction fly away from atom 1 ( 3 M contribution) or 2 ( 2 M contribution) it can interact with these atoms because the coupling diverges for long wavelengths explaining for non vanishing contributions of these terms.

II-3 Real and virtual photons interplay:
The coupling term and paths associated with j M contributions can also be related to the virtual and real characters of the photons involved in the process. According to the common signification of these expressions, the real photons are those created in resonant physical processes that conserve the bare energy (e.g. energy without atom-radiation coupling) whereas virtual photons are the ones that are created in non-resonant processes that do not conserve this energy [51]. Thus, the real part expresses population modifications for the atoms (e.g. transitions) whereas the imaginary part is associated with a frequency shift of the atomic resonances.

II-3 Importance of non-RWA contribution:
The expression of Only the real part is correct with respect to the true value of M . The imaginary part diverges.
The non-RWA contributions are thus necessary for convergence of the coupling parameter (without the need of any frequency cutoff). However, it's worthy to notice that this is not the only role these terms play. Indeed, the addition of non-RWA parts 2 4 , M M also introduces an However, the divergence of the interaction parameter in the infrared domain (responsible for the 1/ ω term in the integrals (6-7) ) leads to an efficient interaction between atoms even if 0 l λ > , thus violating causality. These contributions have necessarily to be compensated in the expression of any measurable physical quantity to fulfill causality principle.
A frequently situation considered also is the case where RWA is used and the frequency variation of the coupling g ω is neglected (e.g. Finally, another procedure used in some works [52], is to perform RWA with the extension of integration in the coupling parameter to negative frequencies (e.g. in (6-a, c) the integration is performed from −∞ to +∞ ). This gives a coupling parameter matches with the true coupling M . This procedure was introduced in previous works to preserve causality in photodetection processes [53] and found here its justification in our case.

III-a Photoelectric signal:
The field behaviour is modified by the interaction with the atoms. An important feature already mentioned in the one atom case [2,18,[23][24] is the reflection of the resonant frequency of the field. This property leads to a transmission of a electric field that distorts temporally such as its algebraic area vanishes (pulse-area theorem [44]). This important feature is the key point to understand the field dynamics. Here, we show that the presence of non-RWA terms is necessary to preserve this feature and is the consequence of the compensation between parts of virtual photons contributions.
We consider the mean field intensity at a photodetector located at a distance z from atom 1. We assume that the photodetector is fast enough to resolve the temporal variation of the entering field. In Glauber theory of photodetection (RWA done in the detector), the photodetector signal -in Coulomb jauge-is then given by is the detector spectral bandwidth satisfying 0 , ∆ >> ∆ Γ to ensure the spectral collection of all emitted photons. We also assume that 1 1 , j z z c c − − − >> Γ ∆ to ensure that the field emission is complete before its entry in the photodetector. Using the above definitions and expression (1) of the wavefunction, we find the following expression for the mean field intensity 1 2 3 ( , ) I t z I I I = + + with , eff A given by : I represents the intensity due to the incident field and the field radiated through RWA processes. The intensity expression is similar to that obtained in the classical regime with an effective potential vector field eff A . 2 I and 3 I are associated with fields radiated trough non-RWA processes and are thus exclusively due to virtual photons. In annex 2, we show that 2 3 , 0 I I  as long as λ ω − ≤ = ), a non-RWA treatment of the whole interaction (atoms + photodetector) is needed [53,54]. The effective field eff A in relation (11) can be related to the population amplitude j β of excited states. In annex 3, we show that in 13 the limit of Markovian approximation, the effective field can be decomposed in three propagating parts as follows:

A t z h z A t z c e h z A t z c e h z l A t z c e
With :

III-2 Transmitted and reflected wavepackets : Pulse-area theorem.
We establish in this section that the algebraic pulse-area of the transmitted and reflected pulse obey to strict conditions. Indeed, the transmitted potential field is given by (14-b) and Note that this result is valid whatever is the coupling Γ , the pulse width ∆ and the distance l between the atoms. The pulse area can be identified in the spectral domain with the spectrum at resonance. Thus, equations (16) means that the central frequency is always totally reflected.
Moreover, the atoms radiates in both backward and forward directions but for the resonance frequency the interference between radiated fields is always destructive (constructive) in the forward (backward) direction.
An important remark has to be done at this level. The pulse-area theorem holds only when equations (15) are valid. As noticed above, this is the consequence of the interference between consequence. In the same manner, we obtain from (15) for large coupling parameter / Γ ∆ . In (c), we represent the spectrum of the transmitted field for corresponding values of the coupling parameter. We see that the central frequency is never transmitted and is hence totally reflected. The spectrum exhibits also a profound dip whose width increases with the coupling parameter.

IV-Conclusion:
A detailed study of the scattering of an incident photon wave-packet by two atoms in one dimensional waveguide has been presented. We clarify the role and importance of non-RWA terms to account correctly for virtual photons contributions. Moreover, we have shown (2-d) using the approximation . We obtain :