[@kiefer92]. In the Fourier decomposition, the functional Schrödinger equation becomes the Schrödinger equation for decoupled, time-dependent oscillators, and the quantum state of the field is the product of the time-dependent wave function for each oscillator. It has long been known that an oscillator with time-dependent frequency and/or mass has a quantum invariant, known as the Lewis-Riesenfeld invariant, whose eigenstate provides an exact solution of the Schrödinger equation up to a time-dependent phase factor [@lewis-riesenfeld69]. Hence, in the former case of electric fields, we may employ the time-dependent annihilation and the creation operators, also quantum invariants, and construct not only excited states for each time-dependent oscillator [@MMT70; @kim-kim99; @kim-page01] but also a thermal state [@kim-lee00; @kim-page13]. In the second case of magnetic fields, however, the Hamiltonian for the functional Schrödinger equation is equivalent to coupled, time-dependent oscillators, and hence, we should employ the invariants for coupled, time-dependent oscillators [@leach77; @ji-hong98; @dodonov00; @lee01].

The quantum invariant method for time-dependent oscillators is very useful in constructing various quantum states from the vacuum state, ranging from excited states to coherent states and even to thermal states (for review and references, see Refs. [@dodonov-manko89; @lohe09]). By analogy with time-independent oscillators, the time-dependent annihilation and the 