3.1 Interferometric geometry with 4 atomic ( $^{N}$= 4 ) sites 3.1.1 Conductance-energy characteristics In Fig. 2, we plot conductance g as a function of the injecting electron energy E for the interferom- eter considering φ = 1, where (a), (b), (c) and (d) correspond to Δ φ = 0 . 2, 0 . 4, 0 . 6 and 0 . 8, respec- Figure 2: (Color online). g - E curves in the weak- (black) and strong-coupling (red) limits for the in- terferometer with four atomic sites ( N = 4) con- sidering φ = 1. (a) Δ φ = 0 . 2, (b) Δ φ = 0 . 4, (c) Δ φ = 0 . 6 and (d) Δ φ = 0 . 8. GLYPH<Minus> 8 GLYPH<Minus> 4 0 4 8 E 0 1 2 g GLYPH<LParen1> E GLYPH<RParen1> GLYPH<LParen1> d GLYPH<RParen1> GLYPH<Minus> 8 GLYPH<Minus> 4 0 4 8 E 0 1 2 g GLYPH<LParen1> E GLYPH<RParen1> GLYPH<LParen1> c GLYPH<RParen1> GLYPH<Minus> 8 GLYPH<Minus> 4 0 4 8 E 0 1 2 g GLYPH<LParen1> E GLYPH<RParen1> GLYPH<LParen1> b GLYPH<RParen1> GLYPH<Minus> 8 GLYPH<Minus> 4 0 4 8 E 0 1 2 g GLYPH<LParen1> E GLYPH<RParen1> GLYPH<LParen1> a GLYPH<RParen1> 5 tively. The black curves represent the results for the weak-coupling limit, while the results for the strong- coupling limit are shown by the red curves. In the limit of weak-coupling, conductance shows fine reso- nant peaks for some particular energies, while it ( g ) drops to zero almost for all other energies. At these resonances, conductance reaches the value 2, and therefore, the transmission probability T becomes unity, since the relation g = 2 T is satisfied from the Landauer conductance formula (see Eq. (1) with e = h = 1). The transmission probability of getting an electron across the interferometer significantly depends on the quantum interference of electronic waves passing through the different arms of the in- terferometer, and accordingly, the probability am- plitude becomes strengthened or weakened. Now all the resonant peaks in the conductance spec- tra are associated with the energy eigenvalues of the interferometer, and thus it is emphasized that the conductance spectrum reveals itself the elec- tronic structure of the interferometer. The situation becomes quite interesting as long as the coupling strength of the interferometer to the electrodes is increased from the weak regime to the strong one. In the strong-coupling limit, all the resonances get substantial widths compared to the weak-coupling limit. The contribution for the broadening of the resonant peaks in this strong-coupling limit appears from the imaginary parts of the self-energies Σ$_{S}$ and Σ$_{D}$, respectively [46]. Hence, by tuning the cou- pling strength from the weak to strong regime, elec- tronic transmission across the interferometer can be obtained for the wider range of energies, while a fine scan in the energy scale is needed to get the electron conduction across the bridge in the limit of weak-coupling. These results provide an impor- tant signature in the study of current-voltage ( I - V ) characteristics. Another interesting feature ob- served from the conductance spectra is the existence of the anti-resonant states. The positions of the anti-resonance states can be clearly noticed from the red curves, compared to the black curves since the widths of these curves are too small, where they sharply drop to zero for the respective energy val- ues associated with the different values of Δ φ (see Figs. 2(a)-(d)). Such anti-resonant states are spe- cific to the interferometric nature of the scattering and do not occur in conventional one-dimensional scattering problems of potential barriers [7, 8, 9].