Transport of Photonic Bloch Wave in Arrayed Two-Level Atoms

In a quantum system of arrayed two-level atoms interacting with light, the interacted (dressed) photon is propagating in a periodic medium and its eigenstate ought to be of Bloch type with lattice symmetry. As the energy of photon is around the spacing between the two atomic energy levels, the photon will be absorbed and is not in the propagating mode but the attenuated mode. Therefore an energy gap exists in the dispersion relation of the photonic Bloch wave of dressed photon in addition to the nonlinear behaviors due to atom-light interactions. There follows several interesting results which are distinct from those obtained through a linear dispersion relation of free photon. For example, slow light can exist, the density of state of dressed photon is non-Lorentzian and is very large around the energy gap; the Rabi oscillations become monotonically decreasing in some cases; and besides the superradiance occurs at long wavelengths, the spontaneous emission is also very strong near the energy gap because of the high density of state.


Results
Model. Following our previous work 32 , we consider an array (x-direction) of N two-level atoms (σ i 's) with the distance between adjacent atoms a (lattice constant) interacting with a quantized EM field A → through a quantum interaction that two-level atoms can be excited (de-excited) by absorbing (emitting) photons as is shown in Fig. 1. Assuming that the EM wave is propagating in the x-direction and uniform along ŷ, ẑ, we can then write the vector potential A → as , in the radiation gauge ( A 0 ∇ ⋅ → = ). The Hamiltonian H em for the EM which describe annihilation and creation of one photon, respectively. To describe transitions between the ground state |g〉 j and the excited state |e〉 j of the two-level atom on the jth site, the raising and lowering operators are defined, is the Hamiltonian of N two-level atoms with excitation energy ν of each atom (ground state energy is 0); and is the Hamiltonian for the quantum interaction between atom and photon with the coupling constant g e 1/137 ∼ ∼ . In this paper, we use natural units by putting ħ = c = 1 as is widely adopted in field theory literatures, e.g., ref. 39 , p. 88. (Nevertheless, we shall put ħ & c back, if necessary.) Hence both the resonant angular frequency ν and the resonant energy ħν of the two-level atom are expressed as ν in the natural units.
Following the calculations in ref. 32 , the free propagator of the two-level atom is The Green's function of the EM field G(x, t; x′, t′) satisfies the Dyson's equation as, or can be expressed in the following way in the momentum space,   is the free propagator of the EM field ( 0 → + ), h's the reciprocal lattice vectors (h = 2nπ/a, a the lattice constant), and Π(ω) is 2 (0) which represents the modification to the propagator (self-energy) of the EM wave due to atom-photon interaction. And it contains both real part and imaginary part which is originated from δ. Then the dispersion relation can be obtained; and k ω vs. ω is depicted in Fig. 2. In fact, the eigenstate |Ψ k 〉 corresponding to the eigenenergy (excitation spectrum) ω k is a photonic Bloch state 33 . In Fig. 2, for 0.914ν < ω < 1.089ν, there is no real k which can satisfy the dispersion relation ω(k) = ω. Therefore, an energy gap appears in the range between 0.914ν & 1.089ν. Within this range, photons will be absorbed. During the revision of this paper, we were informed that there is an interesting similarity between our dispersion relation Eq. (6) and that of the system of surface plasmon polaritons 41 . For example, in ref. 41 , there is also a gap in their dispersion relation between the plasmon frequency and the surface plasmon frequency. It needs further explorations to study the connections between these two systems of light-matter interactions. The index of refraction n(ω) can then be obtained 42 (putting c back), and is depicted in the inset of Fig. 2. The function Π(ω) (Eqs (3) and (5)) plays an important role in the behavior of the index of refraction n(ω). By its definition, Π(ω) is proportional to the Green's function of the two-level which is the response of the two-level atoms to the impulse from other field, the photonic field in this case. Therefore, by the principle of action and reaction, Π(ω) represents the impact on the photonic field from the two-level atoms as is shown in the photonic dispersion relation ω k (Eq. (6)). Were there no Π(ω), ω k would become linear as the dispersion relation of the free photon. And it can be seen that undergoes rapid and significant changes near the resonant angular frequency ν. We define λ ω π ω = c ( ) 2 / to be the wavelength of the free photon. As a result of significant changes of Π(ω) near the resonant angular frequency, the wavelength of the photonic Bloch wave λ(ω) varies from a value less than ( ) λ ω to a value greater than it. Consequently, the index of refraction ω λ ω λ ω = − n( ) ( )/ ( ) becomes less than unity as ω > ν, or the corresponding wavelength λ < λ 0 , with λ 0 the wavelength at the resonant angular frequency ν (Fig. 2). This phenomenon also appears in other single-resonance media 43 . The figure of the index of refraction n vs. λ/λ 0 (=ν/ω) shown in the inset of Fig. 2 is qualitatively similar to the first part of Figure 5.6-4 of ref. 43 , p. 188.
Following similar calculations done in ref. 33 , the dressed photon propagator can be obtained as,  ω ∼ is the Fourier transform of the time evolution operator t ( )  , δ p q , specifies crystal momentum conservation (i.e., δ = 1 , otherwise); and |Ψ k 〉 is the photonic Bloch state, k k a k 0 . Since |Ψ k′ 〉 and |Ψ k′+2nπ/a 〉 are the same, without loss of generality, we can require the Bloch state indices k′ and l′ in the above equation (Eq. (8)) to be in the same Brillouin zone as k and l.
Density of state. The dressed propagator of the two-level atom at the ith site Δ i (t;t′) = Δ i (t′ − t) satisfies the following Dyson's equation 32 , Please notice that by incorporating the renormalized photon propagator in Eq. (13), the renormalized propagator of the two-level atom Δ i (t' − t) includes all those amplitudes that photon is emitted at site i and repeatedly absorbed/emitted at other sites and finally absorbed at site i. The above Dyson's equation can also be expressed in the following way in the momentum space, where the last term is self-energy, in the language of field theory, and it can be viewed as the renormalization correction δν(ω) from atom-photon interactions to the energy ν, Eq. (14) can also be written as, By comparing the above equation with the conventionally adopted density of state 44 ρ E (ω), and by Eq.
is the number of plane wave state |k ω 〉 between ω & ω + dω, and k k 2 |〈 |Ψ 〉| ω ω the probability of overlapping between the plane wave state |k ω 〉 and the corresponding photonic Bloch state |Ψ 〉 ω k . Thus ρ E (ω)dω is the number of (photonic) Bloch state |Ψ 〉 ω k between ω & ω + dω, and ρ E (ω) is the density of state (DOS) which is depicted in Fig. 3. It is non-Lorentzian and is very large around the energy gap. A lot of physical behaviors of photon propagations, e.g. Rabi oscillations and Spontaneous emissions, depend on the DOS ρ E (ω). And we will show this later in the Discussion section.
It should be noted that as ω is within the energy gap (k ω is complex), the state |k ω 〉 is corresponding to an attenuated wave, and 〈 |Ψ 〉 ω x k is not a periodic Bloch wave. Therefore, it is beyond our previous discussions based on the (periodic) Bloch wave, and ρ E (ω) does not carry the meaning of density of periodic photonic Bloch state in the energy gap. where, |i〉 = |g 1 , g 2 , …, g i−1 , e i , g i+1 , …, g N 〉 is the state with the ith two-level atom being in its excited state (|e〉) and other atoms being in their ground states (|g〉). Since k 0 appears in the form as ± e ik x i 0 in this model, without loss of generality, we can restrict k 0 to be within the 1st Brillouin zone.
Rabi oscillations before renormalization. From Fig. 4(a,b), and by Eq. (8), with the time evolution operator  t ( ), the amplitude of an initial Dicke state remains unchanged after a time period t can then be expressed as For ω lies outside the energy gap, the corresponding k ω is real, and we have Therefore, by Eqs (19-24), we have, Spontaneous emissions before renormalization. From Fig. 4(c), and by Eqs (8) and (21), the amplitude of finding a photon at site j at time t from a Dicke state initially is Because of the many-body interactions in our system, it is necessary to do renormalization in our field-theoretical treatment. And it is shown in the Method section.

Rabi oscillations after renormalization.
Having done the renormalization, without causing confusion, we shall omit the superscript (r)'s for renormalized quantities hereafter, and ( ) are the renormalized propagators of the two-level atom and photon defined in Eqs (42) and (43), respectively. Then the probability P D→D (t) of an initial Dicke state k 0 | + 〉 remaining unchanged after some time evolution t can be rewritten from Eqs (19)(20)(21)(22)(23)(24)(25) in terms of the renormalized quantities and renormalization parameters as,  Fig. 4(a,b), the amplitude 〈+| |+〉  t ( ) k k 0 0 is composed of two parts: the direct propagations of renormalized propagators of two-level atom, and indirect propagations from one site to another. The amplitude corresponding to direct propagations does not depend on k 0 . The indirect term can be further divided into two parts, the k 0 -wave part, and the gap part (Eq. (29)). The k 0 -wave part is the amplitude of the interference between two photonic Bloch waves emitted at one site and absorbed at another site; and they carry the same lattice momentum k 0 but different frequencies ω < (k 0 ) & ω > (k 0 ). (Here we have ω < (k 0 ) < ν, and ω > (k 0 ) > ν). This part is much larger than the gap part (two orders of magnitude in amplitude). In addition, the closer ω < (k 0 ) (ω > (k 0 )) is to the edges of the energy gap, the larger its corresponding amplitude would be. It is because that DOS of the photonic Bloch wave ρ E (ω) is very large around the energy gap (Fig. 3). For very small (large) k 0 , there is no corresponding ω < (k 0 ) (ω > (k 0 )), and this can be seen from the photonic dispersion relation (Fig. 2). The behaviors of the evolution probabilities are results of the sum of amplitudes represented by the direct propagation and the two k 0 -waves.
It can be seen from Fig. 5 that the probability P D→D (t) for the Dicke state |+〉 k 0 remaining unchanged shows quite different behaviors for different k 0 's (the momentum at resonance k(ν) = 0.25π/a): a) For very small k 0 (the curve k 0 a = 0.05π in Fig. 5), there is only one k 0 -wave with energy ω ν < k ( ) 0  . In this case, DOS ρ E (ω < ) is small, but the factor 1/ω < in Eq. (29) becomes very large. Therefore, the amplitude corresponding to the k 0 -wave with energy ω < (k 0 ) is not too small compared with that at . But the amplitude corresponding to the direct propagation is very small compared to that of the indirect propagation. Their interference shows small wiggling. b) For k k( ) 0 ν ∼ (the curve k 0 a = 0.25π in Fig. 5), two k 0 -waves exist, and their energies are close to the edges of the energy gap. Therefore, the amplitudes of both k 0 -waves are large because of their high densities of state around the energy gap. As a result, the interference of the two k 0 -waves is notable, and the evolution probability P D→D (t) is significantly oscillatory. c) For k 0 which is not much away from k(ν) (the curve k 0 a = 0.3π in Fig. 5), two k 0 -waves exist and ω < (k 0 ) is close to the energy gap, but ω > (k 0 ) is not. Therefore, the amplitude of the k 0 -wave corresponding to ω > (k 0 ) is small because of its small DOS. Thus, the interference between the two k 0 -waves is small, and P D→D (t) is far less oscillatory. d) For k 0 which is much larger than k(ν) (the curve k 0 a = 0.5π in Fig. 5), there is only one k 0 -wave with energy k ( ) 2 0 ω ν ∼ > . And its amplitude is very small because both the factor 1/ω < & DOS are small. As a result, only the amplitude corresponding to direct propagation is significant; and there is almost no interference at all. We can see that, for different incident momenta k 0 's, the Rabi oscillations show very different oscillatory behaviors in decay. If it were not for the nonlinear photonic dispersion relation with energy gap and the non-Lorentzian DOS of photonic Bloch wave, the interference performances shown in the behaviors of P D→D (t) of Rabi oscillations would not be so rich.

Spontaneous emissions after renormalization. Similarly, the probability
of finding a photon at site j after some time evolution t from a Dicke state k 0 |+〉 initially (Fig. 4(c)) can be rewritten from Eqs (26) and (27) in terms of the renormalized quantities and renormalization parameters as, (31)) is composed of two parts: the k 0 -wave part and the gap part; but they are different from their counterparts in Eq. (29) with the power of the renormalized propagator of the two-level atom ( ) ω Δ ∼ in the integrand to be 1 rather than 2. The k 0 -wave part here is also the interference of two light waves of the same lattice momentum k 0 but with different frequencies ω < (k 0 ) & ω > (k 0 ). This part is even larger than the gap part (three orders of magnitude in amplitude).
From Fig. 6, the spontaneous emission of a Dicke state P t ( ) shows significantly different behaviors for different k 0 's. Most of them can be understood in a similar way from explanations we made with k 0 -waves and the density of photonic Bloch state in the previous subsection except that there is no direct term here (Eq. (31). In summary, we have two things to point out: a) It shows strong radiance for k , but getting weaker notably for k 0 is away from 0 & k(ν). The difference could go up to 5 orders of magnitude. That is, besides the superradiance occurring at long wavelengths (∝1/ω < (k 0 ) see Eq. (31)), the spontaneous emission is also very strong near the energy gap because of the high DOS in this energy range. b) It shows significant oscillatory behavior as ν ∼ k k( ) 0 ; but it is far less oscillatory as k 0 is away from k(ν). Therefore, for different incident momenta k 0 's, the spontaneous emissions show very different behaviors in magnitudes and oscillations. We studied the system of an arrayed two-level atoms (with energy spacing ν) interacting with a photonic field via a quantum interaction. We take into account the multi-scatterings between photon and N two-level atoms in our calculations. For ν ∼ g /100 2 , the obtained photonic dispersion relation is almost linear but with a peak followed by a dip as the photonic energy is around the energy spacing ν. Within such a range of photonic energy (ω ν ∼ ), the corresponding lattice momentum k ω is complex and the photon is in the attenuated mode. In fact, this agrees with our physical intuitions that photon will be absorbed as its energy is around the spacing between two atomic energy levels. Thus, there is an energy gap in the photonic dispersion relation. And for photon propagating in a lattice of two-level atoms, its eigenfunction ought to be Bloch wave rather than plane wave. Consequently, the propagator of the dressed photon is modified significantly. In addition, due to repeated atom-light interactions, there is also a dynamical correction to the atomic energy spacing ν → ν(ω) = ν + δν(ω). Accordingly, as the photonic propagator, the propagator of the two-level atom is modified notably, too. In a field-theoretical treatment like ours, we need to go through the renormalization process to calculate physical results. The renormalization scheme is that we first adopt appropriate renormalization conditions, and from them we solve the renormalization parameters. Afterwards, we present our results with the renormalized quantities. There follows several interesting results which are distinct from those obtained through a linear dispersion relation of free photon. For example, DOS of dressed photon is non-Lorentzian and is very large near the energy gap around which slow light can exist; the Rabi oscillations become monotonically decreasing in some cases; and besides the super-radiance occurs at long wavelengths, the spontaneous emission is also very strong near the energy gap because of the high DOS.
There are systems that mediate interactions between one-dimensional fields and two-level-system array such as an atom array coupled to photonic fields [23][24][25][26]29,30,45,46 , a superconducting qubit array coupled to transmission-line resonator [47][48][49][50] , a gate-control dot array coupled to microwave photons 51 . And they can be realized in experiments. In these systems, our results can be applied to the studies of slow (storage) light and the quantum memory in the atomic medium 31 , the optical nonlinearity 52-54 and the multipartite quantum entanglement 55,56 .

Methods
Renormalization. The scheme of the renormalization process is that we first treat the original field operators