SHAPIRO EFFECT IN INDUCTIVE QUANTUM CIRCUITS WITH CHARGE DISCRETENESS EFECTO SHAPIRO EN CIRCUITOS CUÁNTICOS INDUCTIVOS CON CARGA DISCRETA

As it is known, quantum inductive circuits with charge discreteness show Bloch-like oscillations in electrical current under a dc external voltage. In this paper, the effect of a superimposed ac voltage in the circuit is considered. The Shapiro effect is found to be related to the existence of resonance. Surprisingly, in the limit of low frequency (no resonance), the electrical averaged current exists and has always the same sign. Eventually this allows for an experimental method to measure discrete charge effect in quantum mesoscopic circuits.


INTRODUCTION
Quantum circuits larger than atomic systems are known as mesoscopic circuits.At mesoscopic scale and low temperatures, quantum mechanics plays an important role.Particularly, some effects to considere at this scale are: heat flux quantization, charge discreteness manifestation, Casimir (electrodynamic) effect, Coulomb blockage, persistent current and others, with potential application to electronic devices, including quantum computers.Some of these effects are used in modelling sensing-devices, digital processing, and quantum computers [1].Practical examples are found in the paper by De los Santos [2] who works in the construction of nanoelectromecanics systems (NEMS), with application to circuits in new communication trends and nano-devices.For instance, the Casimir effect could be (after him) used in tunning circuits [2] (RF Varactors).In the explicit subject of charge discreteness, persisten current, Coulomb blockade and Bloch-like oscillations could be considered in the fabrication processes of the nano-devices.
In this paper, the Shapiro effect will be studied.Shapiro effect [3] was originally outlined for a Josephson junction under a dc ( 0 ) and ac (Acos( t)) superimposed voltages.Actually, this was the first experimental demonstration of the Josephson dynamics [4,5,6].As pointed out by Feynman [6], the ac perturbation produces resonances for some characteristic frequency of the system (Josephson dc).Note that Shapiro effect also has been observed in superfluid where pressure plays the role of voltage [7,8].
Quantum inductive circuits with charge discreteness q e [9][10][11][12][13][14] have a formal mathematical equivalence with Josephson junction.In fact, a quantum circuit with inductance L, applied voltage , and charge discreteness q e , has the quantum Hamiltonian: and then quite fast.So, any possible measure of oscillating electrical current is expected to be practically zero due to high fluctuations.In this paper we are concerned with measurable manifestations of charge discreteness.
In following sections, we impose an ac voltage on the electric quantum system.The averaged electrical current is explicitly evaluated.Next, the connection with Stark ladders is considered in a very condensed way.Conclusions are touched in the end section.

AC VOLTAGE ON THE CIRCUIT AND AVERAGED ELECTRICAL CURRENT
The formal similitude between Josephson current and equation ( 2) suggests to consider Shapiro effect [3,5,6] in quantum circuit with charge discreteness.Namely, we will consider the time depending emf given by Acos( t), in the Hamiltonian (1) (see Figure 1).The evolution equation (Heisenberg) for the pseudo-flux becomes For simplicity, choosing the integration constant (t = 0) = 0, the electrical current could be put explicitly as a function of time, namely, Note that with the initial condition (0) = 0 the current becomes proportional to the identity operator and then ( 4) is a scalar equation.If the condition of small ac voltage is assumed, the instantaneous current becomes and then, we conclude (see figure 2): Figure 2 shows the averaged current as a function of the external frequency .Note that for / q e 0 1 the current is always positive ( .This is a physical effect detectable by direct current measurements. Averaged Current (Arbitrary Units) Figure 2. The average current as function of the frecuency .

STARK LADDERS
As pointed out in reference [14], oscillations similar to the one described in the introduction, with B e q 0 are consequence of the discrete energy spectrum of the systems (Stark ladders).These results are shown briefly as follows: the Schrödinger equation for a dc voltage becomes directly from the Hamiltonian (1).In the pseudo-flux representation we have: The condition of charge discreteness ensures that the wavefunction in pseudo-flux space must be periodic.
as conjectured in Ref. [14] for oscillations in electrical current and Stark ladder.
operator Q , with discrete eigenvalues proportional to q e , and the pseudo-flux operator ˆ commute like [ ˆ, ˆ] Q i I , where Î is the identity operator.According to Heisenberg equation of motion d dt Â 1 i [ Â, Ĥ ] , the electrical current in the circuit becomes the pseudo-flux behaves linearly in time when the emf is constant 0 , namely, d dt I ˆˆ 0 .In this way, current oscillations occur in the system with frequency B e q 0[13,14] with formal similarity to Bloch oscillations in Solid State Physics[15,16].Note that the role of the lattice constant in a crystal corresponds to the elementary charge q e in quantum circuits.For typical mesoscopic voltage

Figure 1 .
Figure 1.An electrical schematic model of the quantum circuit with inductance.An electrical schematic model of the quantum circuit with inductance L, dc voltage 0 and ac voltage Acos( t) is shown in Figure 1.The corresponding quantum Hamiltonian is given by (1).
The more surprising fact is that for 'low frequency'