Study On The Dynamics Properties In A Mesoscopic Circuit Under The Effect Of Environment

The effect of dissipation and the quantum state of environment is very important to investigate the dynamical properties of mesoscopic circuit, because that resistance always exists in practical mesoscopic circuit, and the circuit is always in certain condition. So, in this paper, we investigate the quantum properties of mesoscopic dissipation circuit by quantum statistics.We investigate the evolution of the fluctuations of charge and current with time under the effect of dissipation, while the squeezed state is the initial state. In the case of over-damping circuit, the fluctuations of charge and current evolve with time by hyperbolic functions. Because of the strong dissipation, the fluctuations of charge and current evolve with time is non-periodic, and they will tend to certain values respectively. In the case of under damped circuit, the fluctuations of charge and current damply oscillate with time by sinusoidal functions. Because of the dissipation exists in the circuit, the fluctuations of charge and current will not tend to infinite but to certain values respectively in the limit of infinitely long times.Based on the investigation of the evolution of the quantum system under the effect of the initial state of the circuit, we investigate the effect of the environment-phonon bath to the mesoscopic circuit further. In the limit of infinitely long times ( t→∞): when the environment is in thermal equilibrium state, the average values of current and charge depend only on their initial average values, and have noting to do with the environment. When the environment is in number state, the evolution of the average values of current and charge is the same as in thermal equilibrium state. It indicates that the average values of current and charge have nothing to do with the occupation numbers of the quantum state of the environment. When the environment is in the coherent state, the evolution of the average values of current and charge are different from the values in the above states, it shows that the average values of current and charge depend only on the certain ensemble of the coherent state. In general, the fluctuations of charge and current are depend on the initial state of the system and the environment that the system exists in. Finally, we investigate the evolution of the quantum state of the time-dependent mesoscopic coupled circuit by the theory of Lewis—Risesenfeld quantum invariant operator. We have obtained not only the exact solution of relevant Schr?dinger equation, but also the geometric phase in the system of the circuit.