4 as in ring-1 with a phase shift π/ 2. Therefore, ring-1 and ring-2 enclose φ$_{0}$/ 2 flux alternatively in the interval ωt = π/ 2, and accordingly, zero transmission probability is achieved at this interval. In the same footing, here we also describe the variation current I with time t (see lower panel of Fig. 4) to support the oscillatory action observed in this circuit configuration II. The current is computed at the typical bias voltage V = 4. The variation of cur- rent shows π/ 2 periodicity with an amplitude I$_{max}$ = 2 and this periodic nature is well understood from the con- ductance spectrum. From these conductance and current ring is symmetrically coupled to the electrodes. ⃗ A is the vector potential associated with the magnetic field ⃗ B by the relation ⃗ B = ⃗ ∇ × ⃗ A . Hence the probability amplitude of finding the electron passing through the ring can be calculated as, From Eq. (11) it is clearly observed that at φ = φ$_{0}$/ 2 the transmission probability of an electron drops exactly to zero. On the other hand, for all other values of φ i.e., φ = φ$_{0}$/ 2, electron transmission through the ring takes place which provides non-zero value of conduc- tance. Thus, for the particular cases when φ ( t ) becomes maximum (+ φ$_{0}$/ 2) or minimum ( − φ$_{0}$/ 2), conductance drops to zero which is clearly shown from the conduc- tance spectrum (middle panel of Fig. 3). Hence, chang- ing the frequency of time dependent flux φ ( t ), periodicity in conductance can be regulated. To visualize the oscil- latory action more prominently we present the variation of current as a function of ωt in the lower panel of Fig. 3. The current I through the ring is obtained by integrating over the transmission function T (see Eq. ( 9)). Here we compute the current for the typical bias voltage V = 2 . 5. Following the conductance pattern, the oscillatory be- havior of the current is clearly understood, and like the conductance spectrum current exhibits π periodicity pro- viding the amplitude I$_{max}$ = 2 . 54. All these character- istic features suggest that an oscillatory response in the output is obtained though the ring is subject to a DC bias voltage. where, φ = ∮ ⃗ A. ⃗ dr = ∫ ∫ ⃗ B. ⃗ ds is the flux enclosed by the ring. | ψ$_{1}$ + ψ$_{2}$ | 2 = 2 | ψ$_{0}$ | 2 + 2 | ψ$_{0}$ | $^{2}$cos ( 2 πφ φ$_{0}$ ) (11) 0 Π 2 Π 4 Π 3 Π GLYPH<RArrow> Ω t 0 I g 0 0 Φ$_{i}$ FIG. 4: (Color online). Responses in circuit configuration II. Upper, middle and lower panels describe the time depen- dences of two fluxes φ$_{1}$ (orange line) and φ$_{2}$ (magenta line), conductance g and current I as a function of time t . Con- ductance is calculated at the energy E = 2 . 5 and current is determined at the typical bias voltage V = 4. In each ring, total number of atomic sites N is fixed at 8 and we choose δ = π/ 2. The amplitudes are: φ max = 0 . 5, g max = 1 . 9 and I max = 2. spectra it is manifested that in the two ring system which is subject to a DC bias voltage, the oscillatory response can be modulated very easily by tuning the phase differ- ence δ between two time varying magnetic fluxes. Finally, we can say that extending this idea to an ar- ray of multi-ring system in which different rings subject to time varying magnetic fluxes in different phases, os- cillatory responses can be achieved with π/n frequencies, where n corresponds to an integer. Our exact analysis may provide some significant insights in designing nano- electronic circuits. Next, we concentrate on the responses obtained in the circuit configuration II. The results are illustrated in Fig. 4, where total number of atomic sites N in each ring is fixed at 8. In the upper panel, we plot the time depen- dent fluxes φ$_{1}$ ( t ) (orange line) and φ$_{2}$ ( t ) (magenta line) those pass through two different rings. A constant phase shift δ exists between these two fluxes as mathematically expressed in Eqs. (2) and (3). Here we set δ = π/ 2. In the middle panel, we describe the time dependence of conduc- tance g with amplitude g$_{max}$ = 1 . 9, where conductance is evaluated at the typical energy E = 2 . 5. Conductance shows the oscillatory behavior as a function of ωt provid- ing π/ 2 periodicity. Thus, for this circuit configuration II, periodicity becomes exactly half compared to the cir- cuit configuration I. The explanation of π/ 2 periodicity is as follows. For this two ring system, the transmission probability depends on the combined effect of quantum interferences in the two rings. In ring-1, φ$_{1}$ ( t ) is sinu- soidal in form as described mathematically in Eq. (2), while in ring-2, the variation of flux φ$_{2}$ ( t ) is the same In a nutshell, we have addressed the possibilities of current modulation at nano-scale level using mesoscopic rings enclosing a time varying magnetic flux. We have shown that a single mesoscopic ring or two such rings, subject to a DC bias voltage, can support an oscillat- ing output current. A single mesoscopic ring can ex- hibit an oscillating current with a particular frequency associated with the flux φ ( t ), while the frequency of the IV. CONCLUDING REMARKS B. Responses in circuit configuration II