2 SOURCE DRAIN 1 2 − + RING−1 RING−2 oscillating output current. A single mesoscopic ring can provide an oscillating current with a particular frequency, while in the case of two rings, oscillating currents with other frequencies can be obtained. These ideas may be generalized further to produce oscillating currents with other frequencies by considering more number of rings. Let us start by referring to Fig. 1. A mesoscopic ring, penetrated by a time varying magnetic flux φ , is attached symmetrically to two semi-infinite one-dimensional (1D) metallic electrodes, namely, source and drain. These two electrodes are directly coupled to the positive and nega- tive terminals of a battery, a source of constant voltage. The time varying magnetic flux passing through the ring where, δ refers to constant phase difference between the two fluxes φ$_{1}$ and φ$_{2}$ . Using this circuit configuration also oscillating current in the output can be achieved, but in this case the frequency of the current gets modified depending on the phase shift δ . A. Circuit configuration I FIG. 2: (Color online). Actual scheme of connection with the battery where two mesoscopic rings, subject to time vary- ing magnetic fluxes φ$_{1}$ and φ$_{2}$ are attached symmetrically to source and drain. The blue arrow indicates current direction in the circuit. FIG. 1: (Color online). Actual scheme of connection with the battery where a mesoscopic ring, subject to a time vary- ing magnetic flux φ , is attached symmetrically to source and drain. The blue arrow indicates current direction in the cir- cuit. SOURCE DRAIN RING + − In this sub-section we will describe the basic theoret- ical formulation for calculation of conductance and cur- rent through a single mesoscopic ring, penetrated by a magnetic flux φ , attached to two source and drain. This similar theory is also used to study electron transport in an array of mesoscopic rings. Using Landauer conductance formula [30, 31] we de- termine two-terminal conductance ( g ) of the mesoscopic ring. At much low temperatures and bias voltage it ( g ) can be written in the form, can be expressed mathematically in the form, C. Theoretical formulation B. Circuit configuration II g = 2 e 2 h T (4) φ ( t ) = φ$_{0}$ 2 sin( ωt ) (1) T = Tr [Γ$_{1}$ G r $_{R}$Γ$_{2}$ G a $_{R}$] (5) G = ( E − H ) − 1 (6) φ$_{1}$ ( t ) = φ$_{0}$ 2 sin( ωt ) (2) φ$_{2}$ ( t ) = φ$_{0}$ 2 sin( ωt + δ ) (3) where, E is the energy of the source electron. Evalua- tion of this Green’s function needs the inversion of an infinite matrix, which is really a difficult task, since the full system consists of the finite size ring and two semi- infinite 1D electrodes. However, the full system can be where, Γ$_{S}$ and Γ$_{D}$ describe the coupling of the ring to the source and drain, respectively. Here, G r R and G a R are the retarded and advanced Green’s functions, respectively, of the ring considering the effects of the electrodes. Now, for the full system i.e., the mesoscopic ring, source and drain, the Green’s function is expressed as, where, T corresponds to the transmission probability of an electron across the ring. In terms of the Green’s func- tion of the ring and its coupling to two electrodes, the transmission probability can be expressed as [30, 31], In Fig. 2 two such mesoscopic rings those are directly coupled to each other, penetrated by time varying mag- netic fluxes φ$_{1}$ and φ$_{2}$ are attached symmetrically to the electrodes, viz, source and drain. A DC voltage source is connected to these two electrodes. The time varying magnetic fluxes are expressed mathematically as, where, φ$_{0}$ = ch/e is the elementary flux-quantum, ω cor- responds to the angular frequency and t represents the time. This electronic circuit provides an oscillating cur- rent in the output though a constant DC input signal is applied which we will describe in the forthcoming sec- tion. The frequency of the current is identical to that of the applied flux φ ( t ).