1. Introdução

The evolution of time-independent Hamiltonian can be easily evaluated by diagonalizing the Hamiltonian and writing the wave function as a sum of exponential terms. This can be done analytically for system with small Hilbert space and numerically for larger system, as described in many textbooks of quantum mechanics [^{1}, ^{2}]. Once we have the time-dependent wave function we can easily obtain the expectation values and its time average by integrating it over long times. The time integration for system with larger Hilbert space usually takes a lot of computation resources as it requires a time integration over long time sequences. The average occupation of a quantum states can give good insight of the best parameters that an experimentalist needs to manipulate the system, or the best parameters needed to simulate the system when a time-dependent field is applied, and a density matrix is needed to solve. We have being using the average occupation in Refs. [[^{5}
^{6}
^{7}
^{8}-^{9}]], usually doing the numerical integration which uses a lot of computation resources.

In this paper we derive an analytical expression for the time average occupation of a quantum state of time-independent Hamiltonians. This analytical expression can be used for system with large Hilbert space, speeding up its numerical computation. As an example, we will use it to find the best parameters of an external field needed to excite an atom in a cavity. Here we will use an Jaynes-Cummings Hamiltonian [^{3}, ^{4}] with the addition of an extra term to describe an external pumping by a laser field, as in Ref. [^{[9]}].

To derive our analytical expression for the average occupation, let's begin with the time-dependent Schrödinger equation

where

where

is the initial state written in the original basis. If the Hamiltonian is diagonal, we have the stationary states in the original basis. For non-diagonal Hamiltonian matrix, we can compute the exponential of the matrix by diagonalizing it. In this case, the time evolution of the wave function can be written as

where

where

being

and in the original basis

with

where

which we find in many quantum mechanics text books.

Now that we have the time-dependent wave-function we can compute the probability to find the system in any state

which can be written as

Here we have used that for a Hermitian matrix the eigenvectors are real, so the changing basis matrix are also real. Finally, we want to obtain the average occupation of the quantum state in the original basis, which we will define as

This integral can be easily done, as the only time-dependence is the cosine function. In the limit of

for any

This analytical solution only depends on the initial state component of the original basis (

To exemplify this result, let assume the Jaynes-Cummings Hamiltonian describing a two-level system with frequency ^{[10]} can be written as (

where

Assuming

where

Once we have a time-independent Hamiltonian we can use our average occupation approach to find a set of parameters that allow us, for example, to excite the system to very specific state final state. The advantage of our approach is that it allows us to vary different parameters in a large range with a low computer cost. In Fig. 2 we plot the average occupation probability for the ground state of our system, as a function of the laser-cavity detuning (

To find the exact parameters, we can make a cut in the 2D color plot. This can be seen in Fig. 3, where we plot the average occupation of all states, considering vertical cut at

With this set of parameters (

In summary, in this paper we have presented a simple way to compute the average occupation of a time-independent Hamiltonian. Our approach allows us to obtain an analytical expression that can be easily implemented using standard matrix diagonalization method. This speed-up the computation and allows to obtain the average occupation for systems with large Hilbert space and find suitable parameter for a more complex simulation using density matrix or for experimental investigations.

We would like to acknowledge CAPES, CNPq and FAPEMIG by the financial support.