al., discussing various nonclassical properties in connection with quantum number distribution, purity, quadrature squeezing, W-function, etc. . Methods and results Simplification via unitary transformation Let us consider two loops of RLC circuit, whose elements are nanosized, that are coupled with each other via inductance and resistance as shown in Figure selleck inhibitor 1. Using Kirchhoff’s law, we obtain the classical equations of motion for charges of the system : (1) Figure 1 Electronic
circuit. This is the diagram of a two-dimensional electronic circuit composed of nanoscale elements. (2) where q j (j=1,2; hereafter, this convention will be used for all j) are charges stored in the capacitances C j , respectively, and is an arbitrary time-varying voltage source connected in loop 1. If we consider not only the existence of but also the mixed appearance of q 1 and q 2 in these two equations, it may be not an easy task to treat the system directly. If the scale of resistances are sufficiently large, the system is described by an overdamped harmonic oscillator, whereas
the system becomes an underdamped harmonic oscillator in the case of small resistances. In this paper, we consider only the underdamped see more case. For convenience, we suppose that R 0/L 0=R 1/L 1=R 2/L 2≡β. Then, the classical Hamiltonian of the system can be written as (3) where p j are canonical currents of the system, and k j =(1/L j )(1/L 0+1/L 1+1/L 2)−1/2. From Hamilton’s equations, we can easily see that p j are given by (4) (5) If we replace classical variables q j and p j in Equation 3 with their corresponding operators, and , the classical Hamiltonian becomes quantum 17-DMAG (Alvespimycin) HCl Hamiltonian: (6) where . Now, we are going to transform into a simple form using the unitary transformation method, developed in  for a two-loop LC circuit, in
order to simplify the problem. Let us first introduce a unitary operator (7) where (8) (9) with (10) Using Equation 7, we can transform the Hamiltonian such that (11) A straightforward algebra after inserting Equation 6 into the above equation gives (12) where (13) with (14) (15) One can see from Equation 13 that the coupled term involving in the original Hamiltonian is decoupled through this transformation. However, the Hamiltonian still contains linear terms that are expressed in terms of , which are hard to handle when developing a quantum theory of the system. To remove these terms, we introduce another unitary operator of the form (16) (17) (18) where q j p (t) and p j p (t) are classical particular solutions of the firstly transformed system described by in the charge and the current spaces, respectively.