Dark Bogolon-Excitons in a Linear Atomic Super-Lattice

Dark and bright excitons are shown to appear naturally in a linear atomic super-lattice with two atoms per unit cell. In bringing the super-lattice into a strong coupling regime with a one-dimensional nanophotonic waveguide, bright excitons and photons are coherently mixed to form polaritons. Treating excitons as bosons implies a mechanism that forbids two excitations from being at the same atomic state, which is included here through a bosonization procedure with kinematic interactions. Interestingly these interactions couple dark and bright excitons, and which we exploit as a new tool for exciting dark states in a controllable way. We suggest a pump-probe experiment where two polaritons scatter into two dark excitons that found to be correlated and are represented as dark bogolon-excitons. The results can be adapted for any super-lattice of active materials, e.g., of organic molecules.


Introduction
Collective electronic excitations (excitons) have been a subject of research since their introduction by Frenkel in 1931 [1]. Frenkel excitons are of importance for fundamental physics, as they explain various electrical and optical properties in molecular crystals, e.g. in organic and noble atom crystals [2,3]. This type of excitation has been extensively investigated and has applications in optoelectronic devices [4,5]. The mechanism behind its appearance is through the delocalization of electronic excitations among the crystal atoms or molecules via electrostatic interactions, mainly resonant dipole-dipole interactions, and in exploiting the lattice symmetry. Such coherent states can appear spontaneously among electronic states of a huge number of atoms without the aid of external fields. It should be mentioned that coherence among electronic states inside the same atom implies external laser fields, as for electromagnetic-induced transparency and population trapping [6].
The progress in the fabrication of optical lattices provides artificial crystals with controllable parameters [7,8], which are adopted to simulate a wide range of complex condensed matter models and present a deep understanding of puzzling solid state effects [9]. Of importance here are ultracold atoms in optical lattices at the Mott insulator phase with one atom per site, which have been proposed for simulating Frenkel excitons and unveiled physical phenomena that are masked by thermal excitations and impurities in solid crystals [10,11].
On the other hand, optical lattices have been introduced as the active material for cavity QED [10]. It was shown that electronic excitations and cavity photons in the strong coupling regime are naturally mixed to form polaritons as the system collective excitations [11]. Several configurations of optical lattices within a cavity have been suggested with diverse dimensionality and geometry [12,13]. Of special interest is that of a onedimensional optical lattice formed by a nanophotonic waveguide [14][15][16][17], which was realized for cesium atoms using tapered nanofibers [18,19]. The optical lattice is localized outside and parallel to the nanofiber by evanescent fields of counter-propagating colored fiber beams. The atomic lattice can be directly interrogated by a resonant beam that is sent through the fiber and observed via linear optical spectra [20].
In the present paper we address an optical super-lattice with two sites per unit cell and that includes one atom per site, where the distance between the two in cell atoms is taken to be smaller than the distance between the centers of two neighboring unit cells. This configuration gives rise to symmetric and antisymmetric entangled states, in which an electronic excitation can be delocalized among the two atoms inside the same unit cell. The symmetric states are bright and super-radiant with a damping rate twice that of single excited atoms, while the antisymmetric states are dark with zero damping rate and decouple to the light [21]. It is a significant challenge to excite dark states in an efficient and controllable way. In localizing the super-lattice parallel to a nanophotonic waveguide at a fixed distance, only the symmetric excitons coherently couple to the fiber photons in order to form one-dimensional polaritons.
Excitons in an extended system behave as bosons, but then they acquire a mechanism that forbids double excitations in the same atom [22,23]. In previous work we presented a bosonization procedure that transforms two-level atoms into bosons with additional kinematic interactions that forbid two excitations from being localized in the same atom, and which was found to be a good approximation at a low intensity of excitations [24]. Here we apply such a procedure to optical super-lattices, and we interestingly show that kinematic interactions can couple symmetric and antisymmetric states. We exploit this mechanism as a controllable mean for exciting dark excitons. In order to achieve our goal we proposed a pump-probe experiment, where two pumped polaritons with momentum k are scattered into two dark excitons of momenta + k p and − k p. This parametric amplification of dark states is induced by a probe field at either + k p or − k p momenta, and subjected to conservation of energy. The scattering of two polaritons into a state of two excited atoms in the same unit cell can be excluded due to conservation of energy. In the frame of Bogoliubov mean field theory the dark exciton pairs are strongly correlated and can be represented by bogolon excitations that are termed bogolon excitons here [25].
The paper is organized as follows. In section 2 we introduce dark and bright excitons into an optical superlattice. An atomic super-lattice strongly coupled to a nanophotonic waveguide is presented in section 3 by using the language of polaritons. Kinematic interactions are derived in section 4 with emphasis on the symmetricantisymmetric mixing. Pump-probe experiments appear in section 5 where a process for exciting dark states is proposed. Section 6 includes Bogoliubov mean field theory with dark bogolon excitons. A summary is given in section 7.

Collective excitations in linear super-lattices
We consider a system of a one-dimensional optical lattice with a basis of two sites (super-lattice). Namely, we have a lattice with a unit cell of two trapping minima, as seen in figure (1) . The lattice constant is a, which is the distance between the center of two neighboring unit cells, and the distance between the two sites within the same unit cell is R. The optical super-lattice can be realized using counter-propagating laser beams to obtain a standing wave super-lattice [14][15][16][17]. Ultracold or cold atoms are loaded into such a system, where the atoms experience optical super-lattice potential. We treat the case of one atom per site with two atoms per unit cell. Here we concentrate on a system of a nanophotonic waveguide, as besides the off-resonance light used to produce the super-lattice it provides a strong resonant coupling of light to the atomic super-lattice. The recent set-up of tapered nanofibers has the potential for realizing such a system [18,19].
The atoms are taken to be of two-level systems with atomic transition energy E A . In using the second quantization language, the excitation Hamiltonian reads where α S n † and α S n are the creation and annihilation operators of an electronic excitation at atom α in site n, respectively. Here α β = ( , 1, 2)stand for the two atoms at the same unit cell, and = ± ⋯ ± n m M ( , 0, 1, , ) for the summation over the super-lattice sites, with = + N M 2 1the number of unit cells. The Hamiltonian includes resonant dipole-dipole interactions between the atoms that allow for energy transfer, in which an excited atom decays to the ground state and another atom is excited. The interaction parameter among α and β atoms at sites n and m is αβ J nm , which is  The primes indicate that in the summations we have ≠ n m ( )and α β ≠ ( ). The H n Hamiltonian can be diagonalized using the transformation Note that due to the dependence of J 0 on θ, the polarization direction can provide a control parameter for the symmetric-antisymmetric splitting energy. In using the above transformation into symmetric and antisymmetric operators in H nm , we get explicitly Let us assume interactions are only among nearest-neighbor unit cells, which is a good assumption in optical lattices. Between the n and + n 1 sites we have The antisymmetric states are on-site localized, and the mixing terms between the symmetric and antisymmetric states vanish. Only the symmetric states can be delocalized among the supper-lattice sites.
In the limit of ≪ R a, the Hamiltonian reads n k ikz k . In place of the discrete energy levels we get an energy band. Excitons are collective electronic excitations that behave as quasi-particles [2,3]. The combination of electrostatic interactions and lattice symmetry gives rise to energy waves that propagate with wave numbers that are good quantum numbers.

Super-lattice excitations coupled to photons
The optical supper-lattice is localized parallel to the nanophotonic waveguide at a distance b, as in figure 2. We consider one-dimensional propagating photons that are represented by the Hamiltonian ph q ph q q † where a q † and a q are the creation and annihilation operators of a photon of mode q, respectively. The wave number q takes the values π = q l L 2 , where = ± ± ⋯ ±∞ l 0, 1, 2, , , and L is the waveguide length. The photon dispersion can be given by  where ϵ is an effective dielectric constant, and q 0 is the confinement wave number. The excitation-photon coupling in the electric dipole approximation is written as The photon electric field operator is given by where V is the photon effective volume, e is the photon unit vector polarization, and u(r) is the photon mode function. The interaction Hamiltonian in the rotating wave approximation, and in the Schr ö dinger picture, reads I qn I qn qn q n qn n q , † * † with the coupling parameters The interatomic distance R can provide a significant control parameter for the interaction of the photons to the symmetric and antisymmetric excitons. This mechanism can be used to exchange the states from dark into bright and vice versa. The total Hamiltonian now reads Due to translational symmetry the Hamiltonian is separated in momentum space, and the interaction is between excitons and photons with the same wave number.
In the limit of small k, that is where μ μ = e ( · ). We used = V SNā and L = Na, with S the photon effective cross-section. The coupling of symmetric excitons to the photons is much larger than the antisymmetric ones.
In neglecting the antisymmetric weak coupling part, the Hamiltonian is written as  [21]. At this point we replace the two-level operators by bosonic ones, that is → In the next section we justify this step and present a bosonization procedure that goes beyond this simple replacement. In the strong coupling regime where the symmetric exciton and photon line widths are smaller than the coupling parameter, the excitons and photons are coherently mixed to form two polariton branches [26]. The Hamiltonian is diagonalized by using the upper and lower polariton operators The exciton-photon detuning can be changed in order to control the mixing amplitude around small wave number polaritons. Now we present the results using some typical numbers. For the transition energy we have = E eV 1.5 A , for the lattice constant we use = Å a 1000 , and for the interatomic distance we have = Å R 100 . The transition dipole is μ = Å e 2.5 , the mode function is taken to be = u b ( ) 0.25, and the photon effective area is π = S a 2 , with the effective dielectric constant ϵ = 2.
In figure 3 we plot the polariton energies − θ =°90 can intersect the lower polariton branch at a finite wave number. The intersection point approaches small k as θ approaches 90 o . This regime will be of interest later. The polariton dispersions are plotted in figure 4 as a function of k. Here we choose θ =°80 , which is a reasonable angle in real experiments. We plot also the photon dispersion and the symmetric-antisymmetric energies. The dark state now intersects the lower polariton branch at a given wave number. In figure 5 we plot the excitation and photon fractions in the upper and lower polariton branches as a function of k, for θ =°80 . In the upper branch, the full line is for the photon fraction| | + Y 2 and the dashed line for the excitation fraction| | + X 2 , and vice versa in the lower branch.

Kinematic interactions
Kinematic interactions were intensively studied by us in a previous work, and found to be a significant nonlinear mechanism for collective excitations in optical lattices [22,23,27]. In applying a bosonization procedure we transformed the excitation spin-half operators into boson ones with additional terms that gave rise to  interactions between bosons [24]. In solving the scattering problem we derived an effective potential that we aim to use in the following.
The operators α S n † and α S n are of spin-half type. They obey the Fermi anti-commutation relation in the same atom, that is n n n n n n n n † † † † and the Bose commutation relation between different atoms, that is Then spin-half operators on a lattice have mixed statistics. They are fermions on-site and bosons among different sites, where they are usually termed paulions. We now apply the bosonization transformation [24]   We get kinematic interactions between symmetric states and between antisymmetric states, as appear in the first two terms Interestingly, the last three terms represent interactions among symmetric and antisymmetric states.
Here two symmetric states can be scattered into two antisymmetric ones, and vice versa. Moreover the last term represents scattering among symmetric and antisymmetric states, and they will be our main concern in the rest of the paper. Writing the interaction Hamiltonian in terms of polariton operators yields ∑ ∑ We concentrate only on the lower polariton branch, hence we can drop the polariton branch indices. Furthermore, we are interested in processes with small k, so we can neglect the dependence of X k on k and replace it by X 0 at k = 0. We obtain where α β ≠ for n = m. The interaction is between two excited atoms and is of higher order than the dipoledipole interaction; e.g., of van der Waals type. As the lattice constant is relatively large, we neglect dynamical interactions among atoms in different unit cells, and we include only interactions between two excited atoms in the same unit cell. Now we have

Parametric excitation of the dark states
It is a great experimental challenge to excite dark antisymmetric states as they are weakly coupled to the light. Usually they are populated through the decay of higher states, or by multi-photon nonlinear processes. Here we exploit the result of the previous section to suggest kinematic interactions as a mechanism for significantly exciting such dark states. The above kinematic Hamiltonian results in mixing interactions that allow scattering among bright polaritons and dark excitons.
We exploit the kinematic scattering of polaritons into dark excitons as a mechanism for populating the dark states. In order to achieve this task in a controllable way we present the following scenario that implies two external sources in the pump-probe experiment [28]. An external pump is used to strongly excite polaritons with a fixed wave number, say k. A probe field is applied in order to weakly excite dark states at a different wave number, say + k q or − k q, or both. This probe field is needed in order to stimulate scattering of bright polaritons into these dark states. Therefore, for the lower branch we consider only polaritons with a fixed wave number k. The upper polariton branch is excluded due to conservation of energy. As two polaritons of wave number k can scatter into two dark excitons, the conservation of momentum implies two dark excitons with momenta + k q and − k q. Due to the fact that dark excitons are dispersionless and on-site localized, any pair of dark states is allowed in the scattering with wave numbers + k q and − k q. Here we consider the pump and probe fields to have the same energy E, where we choose = = E E k E ( ) pol a . The probe field at wave numbers + k q or − k q are off resonance with polaritons of energy E a . On the other hand, the pump field can directly and weakly excite dark excitons at k, and in the following we neglect the scattering of these dark excitons into the + k q and − k q states. This configuration is used in order to excite dark excitons with energies separated from the polariton in an efficient and controllable manner, as presented in figure 6.
In light of the above discussion, we consider the interaction Hamiltonian with only A k polaritons, that is he first term represents interactions among polaritons. The second term gives interactions among dark excitons, and as dark states are weakly excited we will neglect this term in the following. The second line includes scattering among polaritons and dark excitons, where the first term describes the scattering of two excitons into two polaritons, and vice versa for the second. The third line represents the scattering of an exciton and a polariton. We concentrate on the interaction Hamiltonian  We added three external fields, one field pumps the polaritons through the waveguide at wave number k, and the probe fields weakly excite the dark excitons at wave numbers + k q and − k q. The pump and probe fields are . For a strong pump field we can neglect the contribution of the dark excitons scattering into polaritons, hence we neglect terms that include dark excitons in the first equation. Moreover, we define the renormalized exciton and polariton energies by . The mean field equations of motion read We concentrate here on the case of one probe field at + k q, that is In order to examine whether two such resonances can be experimentally resolved we include the finite lifetime for both the exciton and the polariton phenomenologically in changing the energies into complex parameters where Γ a is the dark exciton damping rate, which is a relatively small number. Γ k pol is the polariton damping rate, and is defined by , where Γ s is the symmetric exciton damping rate and Γ ph is the photon damping rate. The polariton and dark exciton average numbers are written as  . In figure 7 we plot the scaled intensity −   k q probe as a function of − E E a . We pump at k and probe at + k q of energy E a , and hence the dark excitons get excited at − k q with blue shift energy. As dark excitons have small Γ a , that is much smaller than the interaction parameter Δ the spectra split into two peaks at + Ẽ and − Ẽ . We get parametric amplification of the dark excitons both at + k q and − k q with blue shifts, which can be experimentally resolved.

Dark bogolon excitons
We now aim to get a much deep understanding of the results of the previous section for the dark states. In order to achieve this target we follow a different direction for treating pair excitations of dark excitons in adopting Bogoliubov mean field theory [25].
We start again from the Hamiltonian (41). Now we assume the strong pump of the polaritons at wave number k to be a classical field, which fits with the previous mean field theory treatment. Hence the first term becomes a number. As before, the dark states are weakly excited so we can neglect the second term of their mutual interactions. These assumptions are in the spirit of Bogoliubov mean field theory [25]. Using 〈 〉 =  We conclude that in the case of a single probe field at + k q, the two dark excitons, which appear through the scattering of two k polaritons, are spontaneously correlated and form two coherent states that are separated by the interaction energy V in the optical spectra.

Conclusions
In the present paper we showed how coherent states of electronic excitations are spontaneously formed only through resonant dipole-dipole interactions and in exploiting the lattice symmetry, without the aid of additional external fields. The study is presented for an optical super-lattice with two atoms per unit cell in which dark and bright excitons appear. In localizing the one-dimensional atomic super-lattice parallel to a nanophotonic waveguide the bright excitons are strongly coupled to the waveguide photons to form polaritons as the system natural collective excitations.
Excitons behave as bosons at low intensity of excitations, but as the atomic transitions are of two-level systems a mechanism is required to forbid two excitations from being localized in the same atom. A bosonization procedure from two-level systems into bosons was suggested by us in other work that excludes double excitation in the same atom by including kinematic interactions among excitons. In adopting this procedure for the present atomic super-lattice the kinematic interactions interestingly couple dark and bright excitons.
We used the kinematic interactions in order to excite dark excitons in an efficient and controllable way. We suggested a pump-probe experiment in which two pumped polaritons of a given momentum k can be scattered into two dark excitons of momenta + k p and − k p, where the process is induced by a probe field. The two dark excitons are found to be strongly correlated and can be represented as dark bogolon excitons.
Bogolon excitons can be implemented as qubits for quantum information processing, and their long lifetime allows them to serve as memory states. The above parametric amplification scenario can be used to write into these states, and the addition of asymmetry into the system can be used to read out. Furthermore, achieving a fixed population of dark bogolon excitons in a controllable way is an important step toward achieving the long awaited Bose-Einstein condensation of excitons.
The consideration in the present paper even though presented for an atomic super-lattice can be also realized for a one-dimensional organic crystal with two molecules per unit cell. Here one obtains an anisotropic crystal and the orientation of the molecules relative to the axis gives rise to two Davydov exciton branches, in which both can be bright or one bright and the other dark. This fact can enrich the present physical properties as the polarization direction provides an extra degree of freedom.