Broad-Band Negative Refraction via Simultaneous Multi-Electron Transitions

We analyze different factors which influence the negative refraction in solids and multi-atom molecules. We find that this negative refraction is significantly influenced by simultaneous multi-electron transitions with the same transition frequency and dipole redistribution over different eigenstates. We show that these simultaneous multi-electron transitions and enhanced transition dipole broaden the bandwidth of the negative refraction by at least one order of magnitude. This work provides additional connection between metamaterials and Mobius strips.

Generally, it is expensive to fabricate optical metamaterials, as most of these require producing a huge number of split-ring resonators, which are of sub-optical-wavelength. Also, the size of a resonator in classical metamaterials is of the order of the sub-wavelength of the electromagnetic field [31,32]. However, since in a molecular medium the electromagnetic response results from quantum transitions, the size of a molecule can be intrinsically smaller by 2 orders of magnitude than the size of a resonator for classical metamaterials. Furthermore, molecular media are crystals of such molecules and thus can be easily fabricated by crystallization. Therefore, these observations motivate us to consider fabricating metamaterials using chemical molecules.
In multi-atom molecules and solids, there are generally more than one electron which can respond to applied electromagnetic fields. In previous investigations [9, 17, 23-25, 30, 33-38], only transitions for a single electron were considered. Because each atom contributes a π-electron to the molecule, there are N π-electrons in a molecule with N atoms. Thus, there would be N possible transitions for the π-electrons in the molecule. Since two or more transitions can effectively overlap with each other, it would be reasonable to expect a broadened bandwidth of the negative refraction by simultaneous multi-electron transitions. On the other hand, Möbius molecules with nontrivial topology have been synthesized [39][40][41][42][43] and proposed to realize, e.g., metamaterials [28,30,44], quantum devices [45][46][47], dual-mode resonators and bandpass filters [48], topological insulators [49], molecular knots and engines [50], and artificial light harvesting [51,52].
As a concrete example for demonstrating the above principle, here we consider multi-electron transitions in a double-ring Möbius molecule [30,47] with N 2 π-electrons. Every energy level could be filled with two electrons of different spins, due to Pauli's exclusion principle. Because the lower energy level has a higher probability to be filled with electrons, half of the N energy levels with lower energy can be filled. Furthermore, electrons can transit from a highest occupied molecular orbital (HOMO) to a lowest unoccupied molecular orbital (LUMO). Because multi-electrons can be involved in the transitions of nearby frequencies, the bandwidth of the negative refraction would be significantly broadened compared to previous findings [23,24,30,34,36], as schematically illustrated in figure 1.
This paper is organized as follows: In the next section, we briefly introduce the relative permittivity and permeability for simultaneous multi-electron transitions by linear-response theory. In section 3, we summarize the selection rules for optical transitions in Möbius molecules from the transition matrices of electric and magnetic dipoles, as given in appendix A. Then, we show the significantly-broadened bandwidths of the negative refraction by simultaneous multi-electron transitions in section 4. Finally, the main results are summarized in section 5.

Simultaneous multi-electron transitions
The electric displacement field  D and magnetic induction  B can be written as [31,32] e e e = + = where  E is the applied electric field,  P is the polarization, ε 0 and e e « r 0 are, respectively, the permittivity of the vacuum and medium,  H is the applied magnetic field,  M is the magnetization, μ 0 and m m « r 0 are, respectively, the permeability of the vacuum and medium. In order to simulate the electromagnetic response of the Möbius medium in the presence of applied fields, we employ linear-response theory [53] to calculate the electric permittivity and magnetic permeability. When there is an electric field applied on the Möbius molecule, the molecule is polarized as ò w p w w á ñ = Here, the dipole-dipole correlation function reads where ρ(0) is the initial state of the molecule. Because of the dipole approximation [31,32], the molecular electric dipole can be calculated as where v 0 is the volume of the molecule. In obtaining (10), (11), we assume that all molecules are identical, and every molecule responds equally to the applied electromagnetic field. Due to the response of the molecules, the actual electromagnetic field inside the medium is different from that in the vacuum. However, it can be proven by Lorentzian local-field theory that both the center and bandwidth of the negative refraction will not be modified significantly [22][23][24]30]. Only the magnitudes of permittivity and permeability are significantly modified. Furthermore, the dipole-dipole interaction between molecules can be omitted as long as it is smaller than either the decoherence rate of the molecular excited states [56,57] or the static disorder [58].

Selection rules for transitions
As a specific case, we consider a Möbius molecule with double rings consisting of N 2 carbon atoms [41], as shown in figure 1. The jth atomic position of ring a (b) is where the radius and width of the Möbius molecule are, respectively, R and W 2 , with W the radius of the carbon atom, j j =jδ, and δ=2π/N. According to the Hückel molecular orbital theory [59], the Hamiltonian of the Möbius molecule for the single electron is [30,47] òis the on-site energy difference between the two rings, and V (ξ) is the inter-ring (intra-ring) resonant integral. Because of the Möbius boundary condition, the last atom of each ring is linked to the first atom of the other ring, i.e. a 0 =b N , and b 0 =a N . The Möbius boundary condition can be canceled by a local unitary transformation, i.e. º = where c jσ is the annihilation operator of an electron at jth atomic site with σ pseudo-spin label, Therefore, the Hamiltonian with periodic boundary condition can be rewritten as By using the Fourier transformation, = å -B e C j k ikj k , the Hamiltonian of the Möbius molecule is diagonalized as the eigenenergies of the upper and lower bands, respectively.
In (10), (11), we apply linear-response theory to rewrite the relative permittivity and permeability in terms of the transition matrix elements of the electric and magnetic dipoles, i.e.

 d if and
 m if . According to (10), (11), we would expect negative permittivity and permeability close to the transition frequencies Δ fi =(E f −E i )/ÿ. The selection rules of transitions are explicitly provided by the matrix elements of the electric and magnetic dipoles as given in appendix A.
Using the dipole approximation, the interaction Hamiltonian with the electric field is written as [31,32] w where the molecule is subject to an electric field with amplitude = x y z 0 0 0 0 and frequency ω. The electricdipole-induced transition is allowed if the corresponding matrix element is nonzero. By using the relation js jj ss [59], the selection rules for the electric-dipole-induced transitions are summarized as x y z , , x y , where the superscripts over the arrows indicate the polarizations of the electric fields. In the following calculations, we only consider the transitions from the lower band to the upper band because of the rotating-wave approximation [60,61] and initial condition. However, it has been shown that the counter-rotating terms will essentially influence the quantum dynamics in combination with some specific initial states [60,62]. Here, we remark that when the temperature is sufficiently high or the initial state is prepared on purpose, the transition from the upper band to the lower band will also participate in the response of metamaterials to the applied electromagnetic fields. Under the dipole approximation, the interaction Hamiltonian with the magnetic field is written as [31,32] w x y z 0 0 0 0 . A straightforward calculation in appendix A shows that the selection rules for the magnetic-dipole-induced transitions are the same as (3).
The transitions allowed by both electric and magnetic dipole couplings are depicted in figure 2, in combination with initial and final conditions. Because a transition can take place from the HOMOs, denoted by solid symbols, to the LUMOs, denoted by empty symbols. The initial state ñ |i of the molecule is occupied byn i the total number of π-electrons. Particularly, we obtain that n i =1 when , denoted by the blue solid squares, or ñ = ñ 2 3 , denoted by the green squares.

Broad-band negative refraction
According to the selection rules, there are four possible transitions from a given initial state. In order to investigate the effects of simultaneous multi-electron transitions on the negative refraction, we explore the relative permittivity and permeability for different detunings Δω=ω−Δ fi . In (21), the lower-band  E k, is symmetric with respect to k=0, while the upper-band  E k, is symmetric with respect to k=δ/2. In Figure 2, the following four pairs of transitions possess the same transition frequencies, respectively: , , denoted by the green curves; , , denoted by the magenta curves; 2, , denoted by the black curves; , , denoted by the blue curves, which also fulfill the selection rules (3) for the transitions in Möbius molecules.
Generally speaking, the permittivity is a second-rank tensor with nine matrix elements, i.e.        In figure 3, we show the widest bandwidth for the negative refraction which corresponds to the transitions labeled by the thick green curves in figure 2. The bandwidth of the negative refraction is 80μeV, which is about 20 times that of [30]. To explore the underlying broadening mechanism, we investigate the relevant contributions from all simultaneous transitions. Generally, the magnetic response is much smaller than the electric response. In appendix B, it was proven that for E-polarized incident fields, the bandwidth is determined by the zeros of smallest principal value of the negative permeability. Therefore, the bandwidth of the negative refraction is mainly given by bandwidth of the negative permeability. In figure 4, we investigate the effect of simultaneous transitions on the negative refraction by the relative permeability. At the transition frequency, there are two degenerate transitions, denoted by the thick green curves in figure 2. One is from the initial state - p , 2 to the final state - p , 3 , denoted by the green circles in figure 4. The other transition is from  p , 2 to  p , 2 , denoted by the black curve. The summation of the two transitions, denoted by the red dotted curve, is nearly the same as that of all contributions.

For the transitions
For the transitions 2 ), the three eigenvalues of m In both cases, the lowest eigenvalue is m r . The bandwidth of the negative refraction is determined by the zeros of the lowest eigenvalue of m    , and the black curve are for   p p  , , 2 2 , and the red dotted curve sums up the two resonant transitions.
where h¢ ¢ k k , is of the same form for different transitions but with a different central frequency. Among four possible a ¢ k k , ʼs, the maximum of α k,k and α k,k+δ are generally larger than those of α k,k−δ and α k,k+2δ . Assuming that ε=V, we have In [30], negative refraction was considered for the single transition Since a a a = -- with two different electronic spins,the bandwidth of the negative refraction is about 20 times of our previous observation in [30]. Compared to other discoveries as listed in table 1,our proposed bandwidth is about: 10 5 times that of [23,24],4×10 5 times that of [36],and 121 times that of [34].
As shown in figure 2, there are 20 possible transitions on account of the initial condition, where there is an electron occupying the HOMOs, and the final condition, where there is no electron occupying the LUMOs. By scanning over all transition frequencies, we obtain the bandwidths of the negative refraction for all transitions, as listed in table 2. Due to simultaneous multi-electron transitions, the bandwidths of the negative refraction are generally larger than the corresponding single-electron transition and that in [30]. Interestingly, the bandwidths of five transitions are larger than the previous observation by one order of magnitude.

Discussion and conclusions
In the previous sections, we use the Möbius molecule as an example to demonstrate broad-band metamaterials by simultaneous multi-electron transitions. Although the currently broadened bandwidth, 22.2GHz, is much narrower than that of the traditional split-ring resonator, 10THz [64], we show that this scheme might be possibly applied to other solids. In [22], due to anisotropic electric response to the applied electromagnetic fields, the diamond with NV centers can realize hyperbolic dispersion relation and thus negative refraction with bandwidth about 10GHz at the room temperature. The experimental observation of the NV center suggested a six-electron model [65,66], where the electrons consist of the five unpaired electrons of the nearest-neighbour nitrogen and carbon atoms to the vacancy and an additional trapped electron. The manifold of the electronic ground state includes spin-triplet states, i.e. , ;1, s . When the static magnetic fields and strain fields are absent, there are two sets of two transitions with the same transition frequencies, respectively. Notice that the bandwidth is inversely proportional to the square decoherence rates of excited states, cf equations (10), (11). The bandwidth can be increased by nearly 2 orders of magnitude if the temperature is decreased to 10K due to a sharp decline of the , -ñ -ñ 15 ñ ñ , -ñ -ñ , -ñ -ñ decoherence rate from 0.1GHz to 16.2MHz [67]. If the density of NV centers is further increased to 10ppm, a bandwidth of negative refraction about 45.4THz may be observed.
To summarize, we explore the possibility of significantly broadening the negative refraction of metamaterials by simultaneous multi-electron transitions. As a specific example, we calculate the optical properties of Möbius molecules based on the Hückel molecular orbital method and linear-response theory. The simultaneous multielectron transitions provide parallel contributions to the overlapping negative refraction. Furthermore, in each transition, the negative refraction is significantly broadened by the enhanced transition dipole. As a result of these two broadening effects, the bandwidth for the negative refraction is larger than the previous discoveries in [23,34] by at least two orders of magnitude. Therefore, we clearly show that the contributions of multi-electron transitions and enhanced transition dipoles can lead to a large broadening of the negative-index interval. Moreover, as metamaterials consisting of Möbius molecules are based on quantum transitions, their size is smaller by 2 orders of magnitude than in classical metamaterials. Instead of producing a huge number of split-ring resonators, metamaterials of Möbius molecules are chemically synthesized and self-assembled by crystallization. In conclusion, this work provides a possible route for broad-band negative refractive-index of metamaterials.    Generally, b ¢ kk is smaller than α kk′ by one order of magnitude. To the zeroth order of β kk′ , the relative permeability is approximated as When an E-polarized electromagnetic field, = , which is consistent with our analysis presented in the main text.