Photon-Assisted Perfect Conductivity Between Arrays of Two-Level Atoms

We investigate interactions between two (parallel) arrays of two-level atoms (2LA) via photons through quantum electrodynamical interaction with one array (the source array) connected to a particle source, and we study the (photo-)resistivity of the other array (the measured array). The wave function of the interacted photon propagating in an array is a Bloch wave with a gap in its eigenvalue (the photonic dispersion). Due to interactions between arrayed 2LA and the dressed photonic field with non-linear dispersion, the conduction behaviors of the measured array can be very diversified according to the input energy of the particle source connected to the source array, and their relative positions. As a result, the resistivity of the measured array can be zero or negative, and can also be oscillatory with respect to the incoming energy of the particle source of the source array, and the separation between arrays.


Results
In this paper, we study our system at temperature T = 0. The EM wave is assumed to be uniform along y, z-directions, and to move in x. Therefore, in radiation gauge (∇ ⋅ → = A 0), the vector potential → A can be written as . The photonic annihilation operator A, and the photonic creation operator A † are defined as, Then the Hamiltonian for the free photon is   with a the lattice spacing, p ( )  the famous Bloch spectrum, and natural units is used ≡ ≡ c ( 1 )  64 . On the RHS of Eq. (3), the second term describes hoppings of electron from one site to its adjacent sites. Here the field operator of electron at site j on the upper (lower) energy level, the excited state (the ground state), is denoted as c j 2, (c j 1, ) with ε 2 (ε 1 ) the corresponding energy. And both the upper & lower fields can hop to their nearest-neighbor sites with λ 2 as the hopping coefficient. At low temperatures, there are two channels of conduction through transportations of particles of the upper field, and holes corresponding to vacancies of particles of the lower field. In Eq. (3), δ is small and positive. The Hamiltonian LA 2 (4)  describes the on-site hard-core interaction between the upper & lower fields when U is set to ∞ to avoid the possibility of both the upper field particle and the lower field particle at the same site.
The interaction between the arrayed 2LA and photons is int j j j j 2, 1, Model. An array of N two-level atoms with spacing a (x-direction) interacts with another parallel array of N two-level atoms connected with a particle source/sink through emitting and absorbing photons. Inter-array hopping of electrons is prohibited.
www.nature.com/scientificreports www.nature.com/scientificreports/ And the Hamiltonian is the light-matter interaction adopted in QED in which a lower field can absorb a photon to become the upper field, and vice versa; the coupling constant ∼ ∼ g e c / 1/ 137 2  is the coupling between bare electrons and bare photons as is widely adopted in field theory literatures, e.g., ref. 64 . Please note that the collective behavior of this coupling constant between electrons and photons in different geometry or confinements results in different effective 65 coupling strength in quantum optics.

t-matrix.
To handle the hard-core interaction  LA 2 (4) (Eq. (5)), we can apply the method of binary collision which was developed in 1959 [66][67][68] . We add up all the repeated scatterings between the upper and the lower field (ladder diagrams (Fig. 2)) to get a finite effective coupling λ t( ) between them at low energy. We define 〈 |Γ | 〉 q P P p ( , ) 0 to be the sum of the ladder diagrams (in Fig. 2) which is the amplitude of repeated scatterings between upper field (1) and lower field (2)  2 , and P 0 is the total energy, P the total momentum 69 .
Following ref. 69 , we can obtain that q 0 2, 1, 0 , And by setting → ∞ U , we obtain the following equation, 0 00 0 that is, the hard-core interaction between the upper field particle and the lower field particle is equivalent to a soft-core one, and is independent of both the incoming & outgoing relative momenta (p & q). Thus, by the binary collision method 66,68,69 (Eq. (8)), and Eqs (4) and (5), the following Hamiltonian can be obtained, and it is the low-energy soft-core effective Hamiltonian for the hard-core interaction,   (5)). Please be noticed that the hard-core scatterings between upper & lower fields at low energy is summarized in v P (Fig. 2).
(α = 1, 2) (with hard-core interaction taken into account) are diagrammatically represented in Fig. 3, and they satisfy the following Dyson's equation (by Eqs. (3) and (9)), (1) 1 1 (1) where the + sign is for particle propagator going forward in time, and the − sign is for propagator going backward in time (hole), and the effective mass is (14) That is, the renormalized energy levels of both the upper & lower fields are shifted by Σ (0, 0) 1 (1) due to hard-core interaction. And these two effective masses can be well incorporated into our theory by redefining the energy levels to be ε ε . It follows that the energy difference (resonance energy of the 2LA) It should be remarked that the renormalized propagators of the α field (α = 2 for the upper field, & 1 for the is the Fourier transform of the time-evolution amplitude 〈Ψ | |Ψ 〉 ∼ ∼ α α t ( )  of the renormalized eigenstate corresponding to the α field |Ψ 〉 ∼ α which carries the information of the hard-core interaction (or equivalently, the effective soft-core interaction) between the bare upper & lower fields, and  t ( ) is the time-evolution operator. Here, the renormalized eigenkets |Ψ 〉 ∼ 1 & |Ψ 〉 ∼ 2 are orthogonal to each other, and they form a basis of eigenkets that diagonalizes the Hamiltonian  .
(thick single-line) represented by the sum of free propagators of the upper field (thin single-line). The black dot "•" is the effective coupling v P (Eq. (12)).
www.nature.com/scientificreports www.nature.com/scientificreports/ photon propagator. In this model, the lattice spacing will be denoted as a. Following ref. 70 , the photonic  is the propagator of free photon with  being an infinitesimal positive number, π = h n a 2 / is the reciprocal lattice vector (for all integer values of n), and which is the renormalization correction of self-mass to the photonic propagator from the light-matter interaction. We can then obtain the photonic dispersion relation through a calculation similar to that done in refs 70 & 71 as, 2 as is shown in Fig. 4. (Please notice that our self-mass ω Π k ( , ) (Eq. (18)) is not exactly the same as that in refs 70 & 71 ). It is nonlinear with an energy gap near the energy spacing ν, around there the momenta are complex corresponding to attenuated light waves. By ref. 71 , the dressed photon propagator is , otherwise; and the  . The amplitude that a photon with momentum k x emitted from the source array (s) and propagating to the measured array (m) (a distance y afar) with momentum ′ k x after a time t evolution has its Fourier transform as x xs m y i k y x y x y y  And its integration over k x & ′ k x , denoted as ω Λ ∼ y ( , ), is very important in the multi-varied features of the (photo-)resistance of the measured array, is the photonic density of state (DOS) 71 dominated by energies around the energy gap (Fig. 4). Without causing confusion, we shall depict in Fig. 5 the renormalized propagators of upper & lower fields of electron, and renormalized propagator of photon by thin double-line, single-line, and wavy-line, respectively, for brevity; and the term renormalized propagator will be simplified as propagator in the following paragraphs.
Dc conductivity. At zero temperature, the (renormalized) ground states |Ψ 〉 ∼ 1 's of the arrayed 2LA are occupied. Thus, the electron in the ground state can not transport unless it is raised to the (renormalized) excited state |Ψ 〉 ∼ 2 , or its neighboring electrons are excited leaving holes there. Please be noted that the renormalized ground state is orthogonal to the renormalized excited state 〈Ψ |Ψ 〉 = ∼ ∼ 0 1 2 (within the approximation of binary collision). Before reaching the end, it is hard for electrons in the (renormalized) excited states in the measured array to drop to the (renormalized) ground states for being almost occupied. Therefore, it would be a good approximation to assume that electrons in the (renormalized) ground state |Ψ 〉 ∼ 1 of the measured array will not be excited to |Ψ 〉 ∼ 2 twice by absorbing and emitting and then absorbing again emitted photons from the source array during the process of transportation from one end to the other. It follows that we can calculate the Feynman diagram of www.nature.com/scientificreports www.nature.com/scientificreports/ interactions between the source array and the measured array (Fig. 5) to get the retarded current-current correlation 〈 ′ − ′ 〉 j q j q ( ) ( ) 0 0 to the leading order; then we obtain the DC conductivity through the Kubo formula, where the superscript R represents the retarded propagators for the fields, and which is a source of particle with energy ω e located at the left end of the array = − x Na/2. Here we shall define a function ζ ω′ ′ q ( , ) 0 as, Then the modification to the conductivity from another parallel array of 2LA is, ( , ) ( ) Im ( , ) If there is no external source, the measured array is independent of the source array, and its retarded current-current correlation and the DC conductivity for the measured array can be obtained through calculation similar to that done in ref. 58   It is demonstrated in Fig. 6 that DC resistivity can reach zero; and in some regions, it can even be negative. The property of zero resistance has been found in many experiments in 2DEG systems irradiated by microwave on relatively pure samples 1,4,[8][9][10][11][12][13][14][15][16][17][18][19][20][21] . And negative resistance arises in the experimental work of ref. 11 . We shall investigate the diverse behaviors of resistivity in our system shown in Figs 6 and 7 in more details in the Discussion section.

Discussion
For convenience, we shall define a parameter ω′ e which is the energy difference between the external frequency and the renormalized ground state energy, ω ω ε ′ ≡ − ′ e e (16)) that most of the contributions in the integration over the photonic energy ω′ are from the region

. Then in Eq. (29), it can be seen from
. And seeing from ω when ω e is around ε′ 2 , or ω′ e is around ν ε ε = ′ − ′  (31)). In the following subsections, we shall study the behaviors of the resistivity of the measured array vs. the external frequency ω e (or ω′ e ), and separations between arrays, in addition to phenomena of zero & negative resistivity shown in Fig. 6. They can be understood by investigating ω |Λ ′ | ∼ y ( , ) 2 (Eq. (24)) which is the probability of photon emitted from the source array to the measured array, and

As the input frequency of the external source is small (
, and there are not many photons emitted from the source array. Therefore, the resistivity of the measured array is almost not changed as if it is alone.  Fig. 4, many photons with energy close to the difference between the excited state energy and the ground state energy (ν) are emitted from the source array. In the measured array, originally most atoms are in the (renormalized) ground state at zero temperature. Being almost fully occupied, ground state electrons can hardly hop to their neighbors, and the conductivity is poor. Once photons with frequency near ν emitted from the source array are absorbed in the measured array, electrons can be raised to the (renormalized) excited state. They whence can easily hop to their neighbors and transport. Furthermore, excited electrons are boosted by the absorbed photons with additional momentum = ω′ k k x e . As a result, the electron conductivity is significantly modified, and zero resistance appears 58 For very small δ, the DC conductivity of the measured array would become very large, and meanwhile, the resistance goes to zero (Fig. 6). 4. For negative resistivity around ω ν ′ ∼ , the momentum ω′ K of the corresponding photon is complex (Fig. 4) and the light wave is attenuated.
K Im , the real part of the square bracket in Eq. (28) is When ω ν ′ ∼ , for some λ, the RHS of Eq. (32) becomes negative. For example, when parameters take values shown in Fig. 4, as ω ν ′ = .
1 114 e & ω ω ′ ∼ ′ e , the above numerator is negative and the corresponding resistivity of the measured array is ρ ρ (Fig. 6). From a physical point of view, briefly speaking, the appearance  Fig. 7(b)) depicts the DC resistivity when the incoming frequency of the external source ω ν ε = .
+ ′ 1 30 www.nature.com/scientificreports www.nature.com/scientificreports/ of negative conductivity is related to the Bloch wave functions of electrons and photons. For an electron with momentum > k 0 transmitting in a lattice and interacting with photons on lattice sites, its eigenfunction is a Bloch wave function which is a linear combination of plane waves with momenta π + k n a 2 / 's, for all integer values of n. Among these plane waves, the = n 0 & the = − n 1 components dominate and represent the principal forward & backward scatterings, respectively 72,73 . As the energy of the dressed photon is around the energy gap (ω ν ′ ∼ ), under certain circumstances (as we illustrated above), the amplitude of the = n 0 component of the Bloch wave of the electron being excited by absorbing a dressed photon with frequency ω′ is small 73 , and the forward scattering diminishes. As a result, the backward scattering dominates and negative conductivity appears.
Behaviors of resistivity vs. the separation between two arrays from 0 to Y. In Fig. 7(a), as the input frequency ω ν ′ < e , the average of the resistivity increases monotonically with the separation between arrays (y). The periodic oscillations of the resistivity are from occurrences of standing waves, and therefore, the separation between peaks Δy satisfies π Δ ∼ ω ′ y c e . 0 0075 as are presented in the inset of Fig. 7(a).
For ω ν ′ ≥ e , the resistivity is shown in Fig. 7(b), and there are periodic oscillations due to standing wave as before. Nevertheless, the average of the resistivity goes up and down before it increases monotonically with the separation between arrays. This is originated from the energy gap (in the photonic dispersion relation) around where the DOS dominates (Fig. 4). The photonic energy satisfies ω ω ω x & k y 2 are the (kinetic) energies associated with the x-momentum k x & the y-momentum k y of the emitted photon, respectively. As the photonic frequency ω ω ν ′ ∼ ′ ≥ e , ω k x can be larger than the lower edge of the energy gap for small k y . Most of the contributions to the amplitude of the emitted photon ω Λ ′ ∼ y ( , ) e are from photons with ω k x close to the energy gap, i.e., ∼ k k with a spreading δω k x0 of the energy gap around which the DOS dominates 71 . In terms of the parameters listed in Fig. 4, the spreading δω ν ∼ .
0 03 k x0 (see Fig. 4). Accordingly, we can then understand behaviors of the resistivity with respect to the separation between arrays as are shown in Fig. 7(b) through the following discussions.

For those photons with y-momentum
, and are on the right (left) edge of the gap. (24)). Therefore, those photons around the left edge of the energy gap are out of phase with those around the right. This would reduce to the most extent the amplitude of the emitted photon to the measured array. Then we have the location of the first peak of the average of the resistivity at π π ω ν = ′ ∼ ∼ ′ − y y k /2/ / 2/ ( )  Fig. 7(b). 2. As the separation y between the two arrays increases from ′ y 1 , not all photons around the left edge of the energy gap are out of phase with those around the right; and the amplitude of the emitted photon to the measured array grows. Therefore, the average of the resistivity decreases. 3. As the separation y between the two arrays increases to ′ y 2 , the average of the resistivity decreases to its minimum. This can be understood by looking at the phases of those photons around the energy gap at and are around the energy gap. At the distance = ′ y y 2 , those emitted photons with ω k x located within the spreading of the energy gap are all in phase with each other, i.e., . To find the spreading δk y0 around k y0 so as to get ′ y 2 , we have the spreading δω k x0 around the energy gap ω ν = k x0 as is shown in Fig. 7(b). 4. As the separation y between the two arrays increases from ′ y 2 and further, more photons around the spreading of the energy gap (or around k y0 ) get out of phase with each other, and the resistivity increases. (2019) 9:13033 | https://doi.org/10.1038/s41598-019-49606-y www.nature.com/scientificreports www.nature.com/scientificreports/ In summary, we explored interactions between two (parallel) arrays of 2LA through emitting and absorbing photons via QED interaction. We calculate the t-matrix of the two fields, upper & lower fields, for electrons. The t-matrix summarizes the ladder diagrams of binary collisions between upper & lower fields interacting with each other through hard-core interaction. We take the t-matrix at low energy as the (finite) effective coupling between upper & lower fields. And we find renormalized propagators of the upper & lower fields. Their corresponding renormalized eigenkets are orthogonal to each other. Then we include in our calculations the interactions between photons and electrons through diagrammatic techniques in terms of renormalized propagators.
Due to transportation with repeated scatterings in the source array which is a linear lattice of 2LA, the emitted photons are Bloch waves 58 with a nonlinear dispersion relation which has a gap around the spacing between 2LA energy levels. This significantly modifies the group velocity and the DOS of the photonic field. In addition, standing waves can occur for photonic Bloch wave as it propagates from the source to the measured array. It follows that the conduction behaviors of the measured array can be very diversified according to the input frequency of the source and the separation between arrays. As a result, the resistivity of the measured array can be zero or negative, and can also show oscillations when we change the incoming frequency of the source array and the separation between arrays.
The theoretical scheme that we investigated in this work can be experimentally realized in many two-level-system arrays such as superconducting-qubit array [74][75][76][77] , trapped atom array [59][60][61][62][63] , and gate-control dot array 78 . For example, the source array can be achieved experimentally in the semiconductor quantum-dot array 78 . By connecting to the source and drain reservoirs, the electron in the source-reservoir with energy around (renormalized) excited energy of the array can transport through the quantum-dot array to form a source array. The energy of the electron tunneling out can be further tuned by changing the applied bias voltage 79 between source and drain reservoirs. The scheme has also the potential for measuring the photoresisitance version of the quantum interference, such as super-radiance 80 , the quantum phase transition 81 , and the optical non-linearity [82][83][84] . Furthermore, the scheme can be applied for reading out the quantum memory 85 .