Numerous mathematical and physical studies have been devoted to the construction of robust framework for solving the time-dependent Schrödinger equation of quantum systems  . In this regard, several approaches were developed, among them there are the usual perturbation theory , Lie algebraic method, generalized invariant operator and path integral methods    , which have been applied to some special cases of time-dependent quantum systems, to find the time evolution operator, which in turn enables to write the explicit expressions of the wave functions and the correspondent energies of these systems. In particular, in the regime of strong laser-matter interaction problems, a non-perturbative approach, based on the quantization version of the Floquet theorem has been established and applied to schemes of physical systems with periodic Hamiltonians   . Notably, this was used to give explanation of multiphoton processes in intense laser fields , selective excitation of molecular vibrational states using short laser pulses . It is worth noticing that mathematical contributions of the above-mentioned researches and physical challenge are dealing with the interpretation of experimental results and the description of the nature of various physical phenomena such as quantum optic, quantum chemistry, cosmology   , etc.
Our motivation consists to the application of this non-perturbative approach, especially to a system driven by a periodic time-varying frequency. Such a system has already been studied by Lemos to give a theoretical approach of the expansion of the universe , and by Paul to describe the motion of a charged particle in an oscillating electric field   . Floquet theory is an efficiency tool for strong time-periodic interaction, and consequently for which the usual stationary states, solutions to the time-dependent Schrödinger equation when no external field acts, disappear. The adiabatic invariance principle formula established by Breuer and Holthaus , is then used to give a physical interpretation to the derived quantum states. These quantum steady states or Floquet states constitute the most probable states, and provide a useful dynamical description induced by the time-periodic interaction.
In previous works, we have applied our non-perturbative method, to the cases of the forced harmonic oscillator, and to some driven anharmonic potentials models  . Indeed, the RAM  provides a useful tool for constructing the evolution operators in a whole resonance zone, which enabled us to obtain readily the wave functions and the associated quasi-energies of these systems. Therefore, some physical properties of these results were explored and exploited. The aim of the present paper is to apply this approach, to first and second ameliorated orders approximation, to the quantum Mathieu oscillator (QMO) model. We have determined the time-evolution operators, the Floquet states, the wave functions and the corresponding energies, and we have shown that the Floquet states of the same parity are coupled in the presence of the interaction Hamiltonian. Moreover, we have verified the Heisenberg uncertainty principle for all time and we have performed numerical evaluations, for the operational parameters that appeared in the analytical expressions.
The paper is structured as follows. In Section 2, we give a review of the theoretical strategy of our method. Section 3, is focused its application to the QMO, and gives the established analytics results, as well as the numerical simulations for the parameters values. In Section 4, we give the conclusions and outlook.
2. Theoretical Strategy  
The Schrödinger equation of a quantum system with a time-dependent Hamiltonian can be written in terms of the time-evolution operator such as,
where is the Hamiltonian of the unperturbed system, and the interaction Hamiltonian, which amplitude is taken as being very small, and is such that
where describes the system state at the time t, and is an arbitrary initial state.
In the case of a time-periodic interaction, the Floquet theorem proves the existence of a couple of operators , where R is a constant Hermitian operator, and is a periodic unitary operator of the same period as so that the formulation of the time-evolution operator , solution of Equation (1), is given in the form   
and the initial condition .
Replacing the unitary transformation on Equation (1) gives rise to the usual time-independent Schrödinger equation of the operator R (so-called the reduced representation of the system), such as
where are the eigenvectors of the operator R, corresponding to the eigenvalues , and where are the eigenstates of the Hamiltonian .
The resolution of Equation (5) leads to obtain the Floquet states, which are canonical solutions of the time-dependent Schrödinger equation of the system ; they are defined as being
These derive quantum states constitute a complete set of bound states of the system, and do not depend on the choice of the couple . One shows otherwise that these are the invariant adiabatic states. Then, the global state describing the quantum system can be expanded in terms of a linear combination with time-independent coefficients of the .
But, no general method is valid to calculate the operators R and . To determine these two operators, we used the technique based on the resonating averages method (RAM) elaborated by Lochak . This method is applied in the interaction picture of which is written with the summation of the
averaging term and the oscillating term . It permits to find an
approximate solution of Equation (1), to a certain order in power of . This method may be compared to the customary rotating wave approximation .
Determination of first and second orders ameliorated solutions and a comparison with the formulation of Equation (4), enable us to obtain the first and second orders of Floquet operators couples, thence we can use Equation (5) and Equation (6) to set up Floquet states, and the correspondents quasi-energies respectively.
3. Application to the Quantum Mathieu Oscillator
3.1. Analytical Results
The differential equation of motion without damping of the spatial coordinates (in one dimension), for a charged particle submitted to a time varying electric field is given in classical mechanics in the general form   
By choosing the potential function (or time varying frequency ) such that,
where is the unperturbed oscillator frequency, is the driven oscillation frequency, and where is the perturbation amplitude taken as been very small.
This model supposes that a charged particle is governed by a Mathieu equation also called Mathieu oscillator, is a particularly important case, which enables to describe the motion of a charged particle in a linear radio frequency Paul trap device .
Without loss of generality of the model, we adopt the parameter of the trap .
The Hamiltonian of such oscillator according to Equation (7) and Equation (8) is given as follows,
By using the SU(1, 1) generators group elements and , which are expressed in terms of the conjugate variables q and p and in terms of the creation and the annihilation operators according to
Leads to write the Hamiltonian of this system under the form 
where q and p satisfy the commutation relation .
Using Equation (10) and Equation (11), yields to write in Equation (2) as
Note that the expression of is similar to that published by Profilo et al.  by doing the first order development of given in Eqution (8), and using the invariant operator method in the framework of the Lie Group.
The RAM applied to the interaction picture form   of gives
and where is the complex conjugate of .
Using the expressions of the evolution operator given by the (RAM)  , enabled us to obtain the first and second ameliorated solutions terms of quasi-energies, Floquet states and wave-functions as
The different coefficients are given by
is the wave function of the simple oscillator, are the Hermite polynomials and the parameter .
We note that the correction effects on quasi-energies appear only from the second order term of the approximation and depend on the quantum number n and the parameters ( and ) of the variable frequency . Therefore, the Floquet states expression shows that the transitions between these Floquet states ( and ) are suppressed when the levels do not have the same parity (i.e. ).
The difference between two adjacent second order Floquet levels is given by,
The Equation (23) shows that all levels are shifted equally; therefore the energy levels of the system remain equidistant.
By using some identities and Equation (18), we can easily compute the expectation values of q, p, and for any level. Neglecting the coefficients of ( and ), permits to write the corresponding fluctuations in q and p as follows,
Then one deduce the second order uncertainty relation such as
It is easily seen that the fluctuation and product shows that for all times t and for any level, the Heisenberg uncertainty principle is satisfied, as well as presents an oscillating correction term due to the form of .
3.2. Numerical Results
As an illustration we carried out numerical simulations of the established analytical expressions, for different values of the parameters. From Equation (19), we can compute the expressions of the probability density associated to the wave-packet of the system. As an example, for the values of the parameters taken to be: , and .
On Figures 3-5, we give the simulation plots of , and as functions of t, for diferent values of , ( ), where the constants factor and are given by and .
Figure 1 and Figure 2 show that, when departing from an initial coherent state, the wave-packets and propagate with small displacement of the maximums for different instants, keeping its form and that the curve presents the regular nodal character. Therefore, the dynamic localization of the wave-packet is verified.
an increase of , as functions of time and for different values of and indicate that the maximum of these fluctuations varies as a function of . Thus the squeezing property of Floquet states is apparent here. Figure 5 also indicates that the uncertainty product is periodic in time and increases strongly to reach its maximums when . Thence, the trapped charged particles
Figure 1. The probability density vs. q and , for and .
Figure 2. The evolution of the probability density vs. q and t, for and .
Figure 3. The variation of and vs. t for and for different values of .
Figure 4. The variation of and vs. t for and for different values of .
oscillate harmonically along each spatial axis at the secular frequency . Our results are agreed with those published by other authors    .
Figure 5. The variation of vs. t for different values of .
4. Conclusions and Outlook
Floquet formalism and the RAM have been successfully used to solve the quantum Mathieu oscillator problem. Floquet states and the corresponding eigenstates and eigenvalues have been expressed to first and second orders approximation. We have shown that the transitions are valid only between Floquet states with the same parity; this is due to the form of the interaction Hamiltonian of the system. This approach seems for us easier in constructing the quantum states of the studied system in comparison with others   . Moreover, we have expressed the uncertainty principle and verified its validity for all time. Numerical computations have also been performed to illustrate the dynamic localization of the wave packet of the system.
Let us argue that our analysis indicates that the effect of the periodic variation of the interaction is to generate the coherence and the squeezed proprieties of the wave-packet of this system and that the effect of the periodic variation of involves an oscillating correction of the uncertainty relation. In addition a divergent character of this correction at a primary resonance frequency of the system is observed.
Let us note that in the absence of the perturbation, we recover the uncertainty relation of the simple oscillator. Let us also note that if we consider the parameter , we recover the results established for the forced harmonic oscillator , signifying that the trapped charged particles can be seen as harmonic oscillator with a time modulated frequency . Our studies argued that the Paul trapped ions may constitute a good promising candidate for physical realization of a quantum computer.
Comparisons of our results, with works published by other authors who have used different methods     reveal a good concordance. Our results may play an interesting complementary role in the treatment of various harmonically confined system. We project to apply our approach to a nonlinear driven Duffing oscillator.
: Reduced Planck’s constant.
: Eigenstates of the operator R.
: Eigenvalues of R (Quasi-energies).
: Sates of the unperturbed system.
: Floquet states.
p: Impulsion operator.
q: Position operator.
m: Mass of the particle.
: Parameter of the trap chamber.
: Unperturbed oscillator frequency.
a: Annihilation operator.
a+: Creation operator.
: Imaginary part of .
: Real part of .
: Modulus of .
: Complex conjugate of .
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