**5. Conclusion**

Nonclassical features of superpositions of coherent states and squeezed states for electromagnetic field in linear media whose electromagnetic parameters vary with time are examined. The expansion of Maxwell equations in charge-source free medium gives second order differential equations for both position function **u**(**r**) and time function *q*(*t*). The Hamiltonian associated with the classical equation of motion for *q*(*t*) varies with time. Among several methods that are useful in managing time-dependent Hamiltonian systems, the quantum invariant operator method is used in order to solve quantum solutions of the system. The annihilation and the creation operators related to quantum invariant operator satisfy boson commutation relation. The wave functions in number state are derived by taking advantage of the annihilation and the creation operators.

Coherent state is obtained from the expansion of the wave functions of number state. By solving the eigenvalue equation of squeeze operator, squeezed state is also obtained. We can confirm from Figs. 4, 6 and 7, that the detailed structure of probability densities for superposition states are somewhat complicated due to the interference between the two component states. We cannot observe nonclassical properties of the coherent state from WDF, because the value of WDF for a single coherent state is always positive. However, a minor nonclassicality of the coherent state has been reported (Johansen, 2004): A particular quantum distribution function for the coherent state of simple harmonic oscillator, the so-called Margenau-Hill distribution, can take negative values in some regions, but the negative values are relatively very small. This appearance reflects the nonclassicality demonstrated in weak measurements which are, in general, performed under the situation where the coupling of a measuring device to the measured system is very weak. The average values obtained from the weak measurement reveal a time-symmetric dependence on initial and final conditions (Shikano & Hosoya, 2010), providing a natural definition of conditional probabilities in quantum mechanics and, consequently, enabling a more complete description for quantum statistics. However, the interpretation of the results of weak measurements is somewhat controversial on account of its peculiar feature that the measured (weak) value can take strange ones which are outside the range of the eigenvalues of a target observable and may even be complex. The detailed analysis of the strange features of weak measurements may provide a better understanding for the essential differences between quantum and classical statistics.

From Figs. 5 and 8, we can see the interference structure produced between the two main bells. If we consider Eqs. (38) and (39), this determines the pattern of interference fringe in real space of *q* and *p*. Though two main bells are always positive, interference structure takes positive and negative values in turn, where the appearance of the negative values is an important signal for the nonclassicality of the system. In fact, nonclassical quantum states are general and ubiquitous. Not only any pure state of the harmonic oscillator can be represented in terms of nonclassical quantum states but also even the number state is a class of nonclassical quantum states (Kis et al., 2001). Nonclassical states in physical fields such as various optical systems, ion motion in a Paul trap, and quantum dot can be applied to fundamental problems in modern technology ranging from high-resolution spectroscopy to low-noise communication and quantum information processing (Kis et al., 2001). In particular, nonclassical properties of correlated quantum systems are expected to play the key role to overcome some limitations relevant to information processing in classical computer system and classical communication.
