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Journal of Xiamen University

Relation Between the Wehrl Entropy and the Quantum Effects of Mesoscopic Circuit

Xing-Lei Xu

Published Jan 1, 2008• 0 citations
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Abstract

By using the quantum canonic ensemble theory,the density matrix and the Wehrl entropy of a harmonic oscillator are derived in a thermal field.The mesoscopic RLC series circuit is equivalent to a harmonic oscillator.The relation between the Wehrl entropy and the quantum effects of mesoscopic circuit is proposed at a finite temperature.It is shown that the Wehrl entropy of a harmonic oscillator not only depends on the eigenfrequency,but also relies on the temperature.The quantum fluctuations of charge,magnetic flux and the corresponding uncertainty relation increase monotonously with the Wehrl entropy,thereby increase monotonously with the temperature,and consistent with results of the thermal field dynamics(TFD).So the Wehrl entropy can be regarded as a measurement of the quantum effects of the mesoscopic circuit.

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By using the quantum canonic ensemble theory,the density matrix and the Wehrl entropy of a harmonic oscillator are derived in a thermal field.The mesoscopic RLC series circuit is equivalent to a harmonic oscillator.The relation between the Wehrl entropy and the quantum effects of mesoscopic circuit is proposed at a finite temperature.It is shown that the Wehrl entropy of a harmonic oscillator not only depends on the eigenfrequency,but also relies on the temperature.The quantum fluctuations of charge,magnetic flux and the corresponding uncertainty relation increase monotonously with the Wehrl entropy,thereby increase monotonously with the temperature,and consistent with results of the thermal field dynamics(TFD).So the Wehrl entropy can be regarded as a measurement of the quantum effects of the mesoscopic circuit.
By using the information-theoretic Wehrl entropy,the Wehrl entropy of a harmonic oscillator is obtained in thermal field.The mesoscopic RLC electric circuit is quantized.The relation between the Wehrl entropy and the quantum effects of mesoscopic RLC electric circuit is researched in thermal field.These results show that the Wehrl entropy is related to not only the engenfrequency,but also the temperature.And the quantum fluctuation and the corresponding uncertainty relation of charge and magnetic flux grow with the Wehrl entropy increasing and become small with the Wehrl entropy decreasing.So the Wehrl entropy can be regarded as a best measurement of the quantum fluctuation and the corresponding uncertainty of mesoscopic RLC electric circuit.
By using of the quantum canonic ensemble theory,the quantum effects of mesoscopic RLC series circuit have been studied in the energy mixed eigenstates.The relation between the quantum effects of mesoscopic circuit and the temperature has been obtained.The results show that the quantum fluctuations of mesoscopic RLC series circuit are not only related to engenfrequency of the circuit,but also the temperature.The quantum noise grows up with the temperature.This result agrees with that obtained by thermal field dynamics(TFD) .However the quantum canonic ensemble theory is easily understood and applied.
When the inelastic coherence length of the charge carrier reaches the Fermi wavelength,quantum effects in electronic devices and circuit should be taken into account.Due to the existence of finite environmental temperature and damped resistance,practical circuits are always working in damped thermal states at finite temperature.With the rapid development of nanotechnology and microeletronics,the quantum effects of mesoscopic circuit have become an active problem now.In this paper,mesoscopic series-parallel connection RLC circuit was quantized by the method of damped harmonic oscillator quantization.Quantum fluctuatioins of the current and voltage of each branch were studied by thermo-field dynamics(TFD) theory in thermal vacuum state,thermal coherent state and thermal squeezed state.The results show that the quantum fluctuations of the current and voltage of each branch are related to not only parameters of the circuit,but also squeezed coeficients,squeezed angle and envionmental temperature.Because of the existence of current Joule heat,environmental temperature and damped resistances,the fluctuations increase with the increasing of temperature and decay with time.The study of these quantum fluctuation is significant for the design of miniature circuits and the reduction of quantum noise.
By using canonical transformation,the quantization scheme of mesoscopic RLC circuit is given.With the aid of numerical calculation,the regulation is investigated about how the Wigner function of the thermal vacuum state varies with the temperature.Meanwhile, the quantum effects of the charge and self-conductance magnetic flux in the thermal vacuum state are discussed by using Weyl-Wigner correspondence.It is found that,Wigner function of the thermal vacuum state at low temperature is a stable wave packet,but it diffuses with the temperature increasing and the quantum effects of charge and self-conductance magnetic flux in the thermal vacuum state vary with the temperature and time,besides the circuit parameters.
Mesoscopic damped mutual inductance coupled double resonance RLC circuit is quantized by the method of damped harmonic oscillator quantization and linear transformation. The energy levels of this circuit are given. By thermo-field dynamics (TFD), the quantum fluctuations of the current and voltage of each loop are researched in the thermal vacuum state, thermal coherent state and thermal squeezed state. It is shown that the quantum fluctuations of the current and voltage are related not only to the circuit inherent parameter and coupled magnitude of mutual inductance, but also squeezed coefficients, squeezed angle, environmental temperature and damped resistance. Furthermore, because of environmental temperature and damped resistance, the quantum fluctuations increase with the increase of temperature and decay along with time.