1. Introduction

The understanding of classical and quantum systems in different areas of physics that can be described by the same mathematical formalism is a relevant topic of study and has always attracted the attention of physicists. This study is recognized to be of great utility because solutions of problems in different branches of physics that present similar behavior and properties help us understand more about others.

It is well known that superconductors are materials that exhibit no electrical resistance below a temperature that is known as its critical temperature. Their unusual properties have made them key components in many areas of physics, such as quantum computation, quantum optical systems, nanoelectronic devices, particle accelerators and NMR magnets ^[1-9]. The main advantages of devices made from superconductors are low power dissipation, high-speed operation, and high sensitivity. The superconductivity was discovered by Onnes in 1911 ^[10], and for many years afterwards, it was thought to consist simply of a complete disappearance of its electrical resistance below the its critical temperature ^[9]. A major advance for the study of suprconductors was given, twenty years after its discovery, by Meissener and Ochsenfeld ^[11]. These researchers showed that when a magnetic field is applied to a superconductor, the applied field is excluded from its interior, except in a thin penetration region near the surface ^[9]. This is the so-called Meissener-Ochsenfeld. Shortly after the discovery that magnetic fields are expelled from superconductors the brothers Fritz and Heinz London in 1935 ^[12] proposed a phenomenological theory to describe the electromagnetic dynamics of superconductivity. Their proposed equations are consistent with the Meissner-Ochsenfeld effect and can be used with Maxwell’s equations to predict how the magnetic field and surface current vary with distance from the surface of a superconductor. Further, a microscopic theory of superconductivity, the famous BCS theory, was presented in 1957 by Bardeen, Cooper and Schrieffer ^[9]. Since then, many physicists have contributed to the study of superconductor materials ^[13-18]. In this work we investigate the classical and quantum electromagnetic dynamics of a conventional superconductor based on the phenomenological London equations^[12].

Over the last decades, a great deal of attention has been paid to the study of quantum effects of mesoscopic circuits^[17-23]. This interest is mainly motivated by the fact that their applications in nanoscience and specially in nanoelectronics seem endless. In this context an LC (inductance L and capacitance C) represents a typical and fundamental circuit. In history, Louisell^[24] was the first physicits who proposed a quantization scheme for this circuit. Another, more complicated, mesoscopic electromagnetic oscillation system is the RLC circuit. For this case, one has to consider the effect of the resistance R on the circuit, that is, the dissipation. The quantization scheme and quantum properties of the RLC circuit are certainly of great theoretical and experimental physical interest. In fact, in recent years, many works on the quantum behavior of this circuit have been published in the literature^{[19,22,23,25,26},^26-28].

The main purpose of this paper is to discuss the connection between the London superconductor and a mesoscopic RLC circuit. We demonstrate that the classical and quantum dynamics of these two different systems are similar and both can be described by similar Hamiltonians. Based on the quantum invariant theory introduced by Lewis and Riesenfeld ^[29], together with Fock states, we easily solve the time-dependent Schrödinger equation for our problem and use its solutions to construct coherent states for the quantized RLC circuit. Finally, we evaluate various quantum properties of these systems, such as the expectation values of the charge and magnetic flux, their quantum fluctuations and the uncertainty principle.

We organize this paper in the following order. In Sec. 2 we discuss the classical dynamics of the London superconductor and of the mesoscopic RLC circuit. In Sec. 3, we present the quantum dynamics of the RLC circuit. In Sec. 4, we construct coherent states for the quantized RLC circuit and calculate some physical properties of this system. We conclude the paper with a short summary in Sec. 5.

2 Classical dynamics analysis

3 Quantum dynamics analysis

In order to obtain the quantum description of the mesoscopic RLC circuit (or, equivalently, of the London superconductor) we must solve the Schrödinger equation associated with the Hamiltonian (21) which is given by

where q1 and Φ1 are now operators with [q1,]=iℏ and Φ1=-iℏ∂/∂q1. In this paper we use the invariant theory introduced by Lewis and Riesenfeld^[29] to find the solutions of the time-dependent Schrödinger (26). Thus, if the system described by the Hamiltonian (21) admits an exact dynamical invariant operator I(t) that satisfies the relation

the solutions of the equation (26) can be written in terms of the eigenstates of I(t) and a time-dependent phase. Then, we can write the solutions of (26) in the form

with

where the |ϕn,t〉 form a complete orthonormal set 〈ϕn',t|ϕn,t〉=δn'n. The phase functions βn(t) are obtained from

Now, it is known that the Hamiltonian (21) admits an invariant of the form ^[22,37-39]

where ρ(t) is a real function satisfying the equation ^[38-41]

with

In what follows, let us now return to the eigenvalue equation (29). To obtain the solutions of this equation we will use the Fock states. For this purpose, we define annihilation and creation-type operators a(t) and a†(t) as^[22,29,37]

with

If we perform the inverse transformations of Eqs. (34) and (35), it can be shown that the invariant (31) can be written in the form

Hence, using (36) and (37) we can solve the eigenvalue equation for I(t)just as for the time-independent mechanical harmonic oscillator by employing the Fock states n,t. Then, writing the Hermitian number operator N=a†a and N|n,t〉=n|n,t〉, we get

Consider next, the phase functions (30). By making |ϕn,t〉=|n,t〉 we find, after a little of algebra, that

Notice that a particular solution of (32) can be written as

for which (41) reduces to

Then, the solutions of the Schrödinger (26) are (see (28))

with βn(t) given by (43) Therefore, the general Schrödinger state can be written as |Ψ,t〉=∑n‍cn|ψn,t〉, where the c_n are time-independent.

If we now calculate the expectation values and quantum fluctuations of q₁ and Φ₁ in the Fock states, we obtain

whence

From the above relations we see that the expectation values of the charge and the magnetic flux are zero, but their quantum fluctuations are not zero. Further, it follows from (48) and (49) that

which, by making use of (33) and (42), becomes

Here, it is worth noticing that the uncertainty principle (51) depends on the electric resistance R. Yet, if the usual dissipation effect due to the electric resistance is null, that is, R = 0 the Eq. (51) is converted into

which represents the uncertainty principle of a time-independent mechanical harmonic oscillator with frequency ω₁ (see Eqs. (18) and (21)) with L playing the role of the mass m. We also notice that for this case, i.e., R = 0, the particular solution (42) becomes ρ=1/(Lω1)1/2 and the Hamiltonian (21) and the annihilation and creation operators (34) and (35) are reduced to that of the standard harmonic oscillator.

We end this section remarking that we can proceed in the same way to discuss the quantum behavior of the London superconductor, since, as we have proved in the previous section, from a mathematical point of view the analysis of the electromagnetic dynamics of these two different systems is identical.

4 Coherent states for the RLC circuit

In this section, we are going to construct coherent states for the mesoscopic RLC circuit. It is well-known that coherent states for time-dependent quantum systems described by Hamiltonians-like (21) are given by^[42]

where α is an arbitrary complex constant. Of course, the states |α,t〉 are eigenstates of the annihilation operator a(t) with eigenvalue αe2iβ0(t) being β0(t)=-(Ω1t/2).

The calculation of the expectation value of q₁ in the coherent states |𝛼,𝑡〉 yields

where ξ is the phase of α ^[42]. Here, we observe that this result does indeed agree with that of Eq. (19), so that it is according to the Schrödinger quantum-classical correspondence for the coherent states ^[43].

The quantum fluctuations in q₁ and Φ₁ in the state |𝛼,𝑡〉 are given by

whence

where we have used the (33) and the solution (42). By comparing (51) and (57) we see that the uncertainty principle in the coherent states is exactly the same as the minimum value of that in the Fock states. It may be helpful, at this point, to note that these uncertainty principles do not depend on time and that their values become larger when the resistance increases. We also observe that the uncertainty principle (57), in general, does not attain its minimum value. This occurs because the states |α,t〉 are not minimum-uncertainty states and correspond to the well-known squeezed states^[44-47]. Further, it is worth noticing that when the resistance is null, that is, R = 0 the uncertainty principle attains its minimum value because in this case the states |α,t〉 reduce to the coherent states of the ordinary mechanical harmonic oscillator model. In short, we observe that one can follow the same steps of this section to construct coherent states for the London superconductor.

To end this section, let us make some comments. In Ref^.[48] we have demonstrated that the electromagnetic dynamics of a London superconductor and a time-dependent LC circuit with inductance and capacitance modulated exponentially at a constant rate are equivalent and both can be described in the same mathematical framework. In this case, the time-derivative of the inductance L˙, causes a damping similar to that produced by the conductivity σWhat is more, in Refs^.[22,23] have been shown that a time-dependent mesoscopic LC circuit with inductance and capacitance increasing exponentially with time is equivalent to the time-independent RLC circuit. For this case, the time differentiation of the inductance L˙, produces a dissipation similar to that produced by the resistance R. Now, in the present paper, we have demonstrated that the London superconductor and the time-independent RLC circuit are equivalent and can be described through the same mathematical formalism, in both the classical and quantum contexts. Therefore, we can confirm that these three completely different systems, that is, the London superconductor, the time-dependent LC circuit and the time-independent RLC circuit share amazing similarities so that their electromagnetic dynamics can be analyzed in the same mathematical framework. This result is interesting and important since, as we have already mentioned previously, the study of classical and quantum dynamics of physical systems that present similar behavior and properties is a topic of special interest in the context of theoretical physics. Finally, it is worth mentioning that, to the best of our knowledge, the mathematical equivalence between the London superconductor and the time-independent mesoscopic RLC circuit has not reported in the literature yet.

5 Summary

In this work, we have established a simple and elegant connection between the London superconductor and the mesoscopic RLC circuit. We have demonstrated that they share similar behavior, both classically and quantum mechanically and can be described in terms of genuine damped harmonic oscillators which are generated by similar Hamiltonians. Further, by using the invariant method, appropriated annihilation and creation-type operators and Fock states, we have easily solved the time-dependent Schrödinger equation for the RLC circuit (the same procedure can be made for the London superconductor) and employ its solutions to construct coherent states for the quantized RLC circuit. Yet, we have calculated expectation values of the charge and magnetic flux, their quantum fluctuations as well as the uncertainty principle in both states, namely, Fock and coherent states. We also have seen that the uncertainty principle in the coherent states is equal to the minimum value of that in the Fock states. In addition, we have found that the uncertainty principle in the coherent states does not attain its minimum value. This latter result occurs because the coherent state constructed in this work correspond to the well-known squeezed states. Finally, we would like to observe that Schrödinger states for a London superconductor with time-dependent conductivity has been obtained in Ref^.[49].