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Journal of the Physical Society of Japan

Thermofield Dynamics for Two-Dimensional Dissipative Mesoscopic Circuit Coupled to a Power Source

Jeong Ryeol Choi

Published Mar 25, 2010• 2 citations

Abstract

Exact quantum solutions of two-dimensional dissipative mesoscopic circuit coupled to a power source are derived using two-step unitary transformation approach. With the choice of appropriate unitary operators, the original Hamiltonian of the system is transformed to a simple form whose quantum behavior is well known. From inverse transformation of the quantum solutions associated to transformed system, we obtained the full wave functions in the original system. Thermofield dynamics is employed in order to investigate thermal states of the system. The vacuum correlations between the so called tilde and non-tilde variables produce the temperature effects. The fluctuations and uncertainty relations for charges and currents are derived in thermal states. Due to the effect of energy dissipation generated in the resistances of circuit, the fluctuation of charges decreases exponentially with time whereas that of currents exhibits exponential increase. Accordingly, their products which gives the uncertainty relation does not depend on such exponential functions and always larger than or equal to \(\hbar/2\) so that the uncertainty principle holds.

DOI: 10.1143/jpsj.79.044402

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Exact quantum solutions of two-dimensional dissipative mesoscopic circuit coupled to a power source are derived using two-step unitary transformation approach. With the choice of appropriate unitary operators, the original Hamiltonian of the system is transformed to a simple form whose quantum behavior is well known. From inverse transformation of the quantum solutions associated to transformed system, we obtained the full wave functions in the original system. Thermofield dynamics is employed in order to investigate thermal states of the system. The vacuum correlations between the so called tilde and non-tilde variables produce the temperature effects. The fluctuations and uncertainty relations for charges and currents are derived in thermal states. Due to the effect of energy dissipation generated in the resistances of circuit, the fluctuation of charges decreases exponentially with time whereas that of currents exhibits exponential increase. Accordingly, their products which gives the uncertainty relation does not depend on such exponential functions and always larger than or equal to (\hbar/2) so that the uncertainty principle holds.

By means of a two-step unitary transformation, the exact quantum mechanical solution of two-dimensional mesoscopic circuit coupled via inductance is derived. The wavefunction, propagator and expectation value of Hamiltonian are evaluated. The fluctuations of charges and currents are calculated and uncertainty products between charges and their conjugate currents are obtained. We apply our results to the system with special cases of the power source such as the sinusoidal and the sawtooth power source and show that the probability densities oscillate with time due to the power source.

Starting from Kirchhoff's equation for each mesoscopic coupled circuit, we investigated the quantum fluctuations of a dissipative mesoscopic capacitance–resistance–inductance coupled circuit using canonical transformation and unitary transformation. It is shown that the quantum fluctuations of charges and currents are not only related to the parameters of components in the self-circuit and coupled part, but also influenced by the parameters of other circuit. The uncertainty relations of the charges and currents are analyzed briefly.

When a time-dependent harmonic system is further driven by an external source, it presents significant challenges to obtaining analytical solutions for the system. This paper applies the Lewis-Riesenfeld invariant method to a mesoscopic time-dependent dissipative inductor-capacitor (RLC) circuit with an external source. By constructing quantum invariant Hermitian operators and introducing auxiliary equations that can describe the effects of external sources and resistance, we analytically solve the Schrödinger equation of the system. Taking an alternating current (AC) voltage source as an example, we derive the generalized coherent states of a mesoscopic time-dependent RLC circuit with an external source, calculate the quantum fluctuations of charge and current, and obtain their corresponding uncertainty relations. The results indicate that the time-dependent inductance and resistance disrupt the minimum uncertainty relation of the Glauble coherent states in driven time-dependent mesoscopic RLC circuits. This research contributes to the establishment of a quantum theory for mesoscopic circuits.

In this paper, we use Hermitian linear invariants and the Lewis and Riesenfeld invariant method to obtain the general solution of the Schrödinger equation for a mesoscopic RLC circuit with time-dependent resistance, inductance, capacitance and a power source and represent it in terms of an arbitrary weight function. In addition, we construct Gaussian wave packet solutions for this electromagnetic oscillation circuit and employ them to calculate the quantum fluctuations of the charge and the magnetic flux as well as the associated uncertainty product. We also show that the width of the Gaussian packet and the fluctuations do not depend on the external power.