3 results considering a ring with N = 20. The upper panel presents the time variation of magnetic flux with ampli- tude φ$_{0}$/ 2 whose mathematical form is given in Eq. (1). The variation of conductance g as a function of time t is illustrated in the middle panel. Here we determine the typical conductance for a particular energy E = 2 . 5. It shows that the conductance oscillates periodically as a function of ωt exhibiting π periodicity and it gets the amplitude g$_{max}$ = 2. This reveals that the transmis- sion amplitude T becomes unity since we get the relation g = 2 T from the Landauer conductance formula in our chosen unit c = e = h = 1. Now we try to justify the os- partitioned into sub-matrices corresponding to the indi- vidual sub-systems and the effective Green’s function for the ring can be written in the form [30, 31], where, H$_{R}$ describes the Hamiltonian of the ring. Within the non-interacting picture, the tight-binding Hamilto- nian of the ring can be expressed like, G$_{R}$ = ( E − H$_{R}$ − Σ$_{S}$ − Σ$_{D}$) − 1 (7) H$_{R}$ = ∑ i ϵ$_{i}$c † $_{i}$c$_{i}$ + ∑ <ij> v ( c † $_{i}$c$_{j}$ e iθ + c † $_{j}$c$_{i}$ e − iθ ) (8) where, ϵ$_{i}$ and v correspond to the site energy and nearest- neighbor hopping strength, respectively. c † i ( c$_{i}$ ) is the creation (annihilation) operator of an electron at the site i and θ = 2 πφ/Nφ$_{0}$ is the phase factor due to the flux φ enclosed by the ring consists of N atomic sites. A simi- lar kind of tight-binding Hamiltonian is also used, except the phase factor θ , to describe the electrodes where the Hamiltonian is parametrized by constant on-site poten- tial ϵ ′ and nearest-neighbor hopping integral t $^{′}$. The hop- ping integral between the ring and source is τ$_{S}$ , while it is τ$_{D}$ between the ring and drain. In Eq. (7), Σ$_{S}$ and Σ$_{D}$ are the self-energies due to the coupling of the ring to the source and drain, respectively, where all the information of the coupling are included into these self-energies. To determine current, passing through the mesoscopic ring, we use the expression [30, 31], where, f$_{S}$$_{(}$$_{D}$$_{)}$ = f ( E − µ$_{S}$$_{(}$$_{D}$$_{)}$ ) gives the Fermi distribu- tion function with the electrochemical potential µ$_{S}$$_{(}$$_{D}$$_{)}$ = E$_{F}$ ± eV/ 2 and E$_{F}$ is the equilibrium Fermi energy. For the sake of simplicity, we take the unit c = e = h = 1 in our present calculations. cillating behavior of conductance with time t . The prob- ability amplitude of getting an electron from the source to drain across the ring depends on the quantum inter- ference effect of the electronic waves passing through the upper and lower arms of the ring. For a symmetrically connected ring (upper and lower arms are identical to each other), penetrated by a magnetic flux φ , the proba- bility amplitude of getting an electron across the ring be- comes exactly zero ( T = 0) for the typical flux, φ = φ$_{0}$/ 2. This vanishing behavior of transmission probability can be shown very easily by simple mathematical calculation as follows. For a symmetrically connected ring, the wave functions passing through the upper and lower arms of the ring are given by, To illustrate the numerical results, we begin our dis- cussion by mentioning the values of different parameters used for our calculations. In the mesoscopic ring, the on- site energy ϵ$_{i}$ is fixed to 0 for all the atomic sites i and nearest-neighbor hopping strength v is set to 3. While, for the side-attached electrodes the on-site energy ( ϵ $^{′}$) and nearest-neighbor hopping strength ( t $^{′}$) are chosen as 0 and 4, respectively. The hopping strengths τ$_{S}$ and τ$_{D}$ are set as τ$_{S}$ = τ$_{D}$ = 2 . 5. The equilibrium Fermi energy E$_{F}$ is fixed at 0. where, γ$_{1}$ and γ$_{2}$ are used to indicate the two different paths of electron propagation along the two arms of the ring. ψ$_{0}$ denotes the wave function in absence of magnetic flux φ and it is same for both upper and lower arms as the The modulation action for the circuit configuration I is clearly illustrated in Fig. 3, where we compute all the A. Responses in circuit configuration I III. NUMERICAL RESULTS AND DISCUSSION I ( V ) = 2 e h ∞ ∫ −∞ ( f$_{S}$ − f$_{D}$ ) T ( E ) dE (9) ψ$_{1}$ = ψ$_{0}$e$^{ie}$ ¯ hc ∫ γ 1 ⃗ A. ⃗ dr ψ$_{2}$ = ψ$_{0}$e$^{ie}$ ¯ hc ∫ γ 2 ⃗ A. ⃗ dr (10) FIG. 3: (Color online). Responses in circuit configuration I. Upper, middle and lower panels describe the time depen- dences of flux φ , conductance g and current I as a function of time t . Conductance is calculated at the energy E = 2 . 5 and current is determined at the typical bias voltage V = 2 . 5. The ring size is fixed at N = 20. The amplitudes are: φ max = 0 . 5, g max = 2 and I max = 2 . 54. 0 Π 2 Π 4 Π 5 Π 6 Π 3 Π GLYPH<RArrow> Ω t 0 I g 0 0 Φ