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Taking into account the interaction between electrons and phonons, in the case without-rotating-wave approximation, we study the entangling property between the mesoscopic circuit and environment at coherent state or equilibrium state. The result indicates that, in long time limit t → ∞, the averages of charge and current in the circuit only depend on the average of the system at the initial state when the environment is initially at thermal equilibrium. However, when the environment is initially at coherent state, the average of charge and current in the circuit is determined by the specific coherent state ensemble. Generally speaking, the entanglement between the circuit and environment will lead to the quantum state purity declining of the circuit, then the circuit emerges decoherent phenomenon, and so a mixed state appears. Purity changes are related to the initial quantum state of environment and circuit. With the further evolution of time, coherence will be gradually restored, but cannot return to 1.

Based on the information theory, we present a density matrix to discuss coherent and squeezed states in a mesoscopic LC circuit with time-dependent frequency. With the relevant operators included in the density matrix, a connection between the appearance of coherent and squeezed states is established, i.e., the quantum state evolution of the system is closely related to the initial state. Generally speaking, due to the effect of environment temperature, the mesoscopic LC circuit will evolve to a squeezed state when it initially lies in an excited state. In particular, at a low temperature, step changes of circuit parameters will result in a squeezed minimum uncertainty state if the resonance frequency remains the same after the change.

Under the Born–von-Karmann periodic boundary condition, we propose a quantization scheme for non-dissipative distributed parameter circuits (i.e. a uniform periodic transmission line). We find the unitary operator for diagonalizing the Hamiltonian of the uniform periodic transmission line. The unitary operator is expressed in a coordinate representation that brings convenience to deriving the density matrix ρ(q,q',β). The quantum fluctuations of charge and current at a definite temperature have been studied. It is shown that quantum fluctuations of distributed parameter circuits, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals and temperature T. The higher the temperature is, the stronger quantum noise the circuit exhibits.

We discuss the relationship between external magnetic field and persistent currents of three mesoscopic-coupled rings with charge discreteness. Through numerical calculation and analysis, it is found that the properties of the persistent currents depend on both external magnetic flux and coupled factors. The positive coupling between the rings will decrease the persistent currents when 0

Under the Born–von-Karmann periodic boundary condition, we propose a quantization scheme for a non-dissipative distributed parameter circuit (a uniform transmission line). Quantum fluctuations of charge and current in the vacuum state are investigated by adopting the canonical transformation and unitary transformation method. Our results indicate that in distributed parameter circuits, quantum fluctuations, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals.

The quantum theory of the mesoscopic RLC circuit and the condition for Coulomb blockade are given by using canonical quantization and a unitary transformation from the classical equation of motion. Our results show that there is a threshold voltage eT in the circuit. The threshold voltage is related not only to the junction capacitance and inductance, but also to the resistance of the circuit. Generally speaking, the larger the resistance, the larger the threshold voltage. This clarifies the phenomenon of the Coulomb blockade of the dissipative mesoscopic circuit.

Using the path integral method we derive quantum wave function and quantum fluctuations of charge and current in the mesoscopic RLC circuit. We find that the quantum fluctuation of charge decreases with time, oppositely, the quantum fluctuation of current increases with time monotonously. Therefore there is a squeezing effect in the circuit. If some more charge devices are used in the mesoscopic-damped circuit, the quantum noise can be reduced. We also find that uncertainty relation of charge and current periodically varies with the period π/2 in the under-damped case.

In this paper we consider the movement of an electron in the single electron tunnel process through a mesoscopic capacitor. The results show that, due to the Coulomb force, there is a threshold voltage Vt in the mesoscopic LC circuit. When the external voltage is lower than the threshold voltage, the tunnel current value is zero and the Coulomb blockade phenomenon arises. Furthermore, considering that the mesoscopic dimension is comparable to the coherence length in which charge carriers retain the phase remembrance, a weak coupling can be produced through the proximity effect of the normal metal electrons of both electrodes of a mesoscopic capacitor. By varying the external voltage, we can observe the Shapiro current step on the current-voltage characteristic curve of a mesoscopic LC circuit.

Taking into consideration the interactions between electrons and phonons, we have studied the temporal evolution of the average charge and current in a dissipative mesoscopic RLC circuit. Our results show that a mesoscopic RLC circuit can be treated as an interactive system between an electromagnetic harmonic oscillator and many lattice harmonic oscillators; this is called the bathing of the harmonic oscillators. The results also show that the quantum equation of motion of the linear mesoscopic RLC circuit is identical in form to its classical equation of motion, the only difference between them being their respective meanings. In order to thoroughly study the quantum properties of a dissipative mesoscopic circuit, we have to consider not only the electromagnetic energy of the circuit, but also the crystal lattice vibration energy and the interactive energy between electrons and phonons.

We study the quantum fluctuations of the charge and current of two L-C dissipative mesoscopic circuit with the mutual inductance in the vacuum state. Our results show that the system state will evolve to a squeezed coherent state under the effect of external source. We find that the squeezing amplitude parameter is relative to the parameters of circuit and the mutual-inductance coefficient in the existence of dissipation. When the circuit has no dissipation or there is complete coupling between two meshes, the squeezing amplitude parameter only depends on the capacitance's ratio.

Taking into account the interaction between electrons and phonons, we study the dynamical behavior of a mesoscopic dissipative circuit for two classes of pure initial states (grand canonical ensemble and number state) of phonon bath modes. In the long time limit t→∞: when the environment is initially at thermal equilibrium, the average values of current and charge in the circuit system will only depend on the average values of the circuit in initial quantum state, and they are not related with the environment; when the environment is initially in number state, the time evolution of the average values of current and charge are just the same as that in thermal equilibrium, which indicates that the average values of current and charge are related to certain set of occupied numbers. The quantum fluctuations of current and charge in the circuit not only depend on the initial state of the system but also depend on the quantum state and temperature of the environment. Generally, the entanglement between the circuit system and environment will lead to the reduction of the purity of the quantum state. The mesoscopic circuit will remain in a pure quantum state only when the environment is at very low temperature and at thermal equilibrium.

According to the LR invariant theory, the mesoscopic RLC circuit driven by a alternating current source was discussed. The mutual coupling of electron wave functions between the two plates of capacitor and the resistance of the circuit were taken into account. In our conclusion, the quantum states which dominate the dynamic property of the system and the time evolution operator of the mesoscopic circuit were obtained. The wave function of the mesoscopic RLC system will evolve from an arbitrary initial state to a generalized squeezed state.

Based on the fact that a mesoscopic capacitor can be regarded as a mesoscopic tunnel junction, we quantized the mesoscopic LC circuit by using the rotating -wave approximation.The study shows: Considering the coupled energy of the meso scopic capacitor, the system of the mesoscopic LC circuit will be equivalent to a Kerr system. We also used the thermal field dynamic theory to investigate the thermal Kerr state at a definite temperature and calculated the quantum fluc tuation of both charge and magnetic flux, and discussed the results.

Based on the discreteness of electric charge, the influences of an external magnetic filed on the persistent current of the mesoscopic coupling metallic rings are investigated. The results obtained indicate that there is an additional persistent current in each mesoscopic metallic ring. The additional persistent currents are related to the circuit parameter and coupling coefficient. If the external magnetic flux varies according to sine law, the effect of double-frequency and division-frequency appears in these rings.

A scheme is presented for preparing the superposition of coherent states,i.e.,Schr?dinger cat states. These states can be generated by the action of an external impulse functional signal on initial vaccum state of a mesoscopic LC circuit. The nonclassical squeezing properties appear in the circuit under the Schr?dinger cat states.

In this paper, the influences of an external pulse on the variation of the quantum state of a mesoscopic RLC circuit are investigated emphatically. Our research result indicates that the transition probabilities of the variation of the quantum state are adjusted by controlling the variation of the pulse parameters when the parameters of the mesoscopic circuit are constant. We point out that the quantum state of a system is stable when the width of the pulse is equal to the integral times of the minimum fixed value. The minimum fixed value is not related to the altitude of pulse but the circuit parameters. The larger the resistance, the wider the minimum fixed width of the external pulse.

By keeping the frequency of non-dissipative mesoscopic LC circuit a constant and its parameter varying with the step function, the quantum state of the circuit evolves to a squeezed even coherent state from the initial even coherent state. The nonclassical properties, not only squeezing but also antibunching effects, appear in the circuit by adjusting its inductor parameter under the squeezed even coherent state.