Mth Coherent State Induces Patterns in the Interaction of a Two-Level Atom in the Presence of Nonlinearities

Abstract

Recently, the Mth coherent state (Mth CS) has been defined to be the resultant state of applying the number operator M times on the coherent state (CS) (Othman, Int. J. Theor. Phys. 58: 2451, 2019). In this article, we study the interaction between a two-level atom and a quantized single-mode field with intensity-dependent coupling in a Kerr medium. The Mth CS is assumed to be the interacting field with the atom. We study the dynamical behavior of the inversion, Q Mandel, linear squeezing, amplitude squared squeezing, and field entropy for different parameters and M values. We find that a certain pattern in the dynamics is induced with the Mth CS that is not present with the CS. We term this pattern as the “resolution phenomenon”. It is related to a semi-regular pattern being apparent in a specific region while after or before this region the pattern is overlapping. We also discover, in general, that the Mth CS of large M values for a given dynamic tends to have more periodicity, wider peaks, less growing/decaying rates, and more covering its range. Many other classical/nonclassical features are discussed for the concerning measurements. Finally, in the end, we propose some potential applications.

Appendix

The Hamiltonian of the system as explained earlier is given as

$$ \hat{H}/\hbar = \omega_{1} |1 \rangle \langle 1|+ \omega_{2} |2 \rangle \langle 2|+ {\Omega} \hat{a}^{\dagger} \hat{a}+ \lambda (\hat{A}^{\dagger} |1 \rangle \langle 2|+ \hat{A} |2 \rangle \langle 1|) +\chi \hat{a}^{\dagger 2} \hat{a}^2. $$

(A.1)

Operating in the interaction picture, this Hamiltonian can be separated into two parts; the natural part $\hat {H}_{0}$ and the interaction part $\hat {H}_{I}$ as $\hat {H}= \hat {H}_{0}+\hat {H}_{I}$ with

$$ \hat{H}_0/\hbar=\omega_{1} |1 \rangle \langle 1|+ \omega_{2} |2 \rangle \langle 2|+ {\Omega} \hat{a}^{\dagger} \hat{a}, \qquad \hat{H}_I/\hbar= \lambda (\hat{A}^{\dagger} |1 \rangle \langle 2|+ \hat{A} |2 \rangle \langle 1|) +\chi \hat{a}^{\dagger 2} \hat{a}^{2}. $$

(A.2)

The interaction Hamiltonian can be calculated from $\hat {{\mathscr{H}}}= e^{i\hat {H}_{0} t/\hbar } \hat {H}_{I} e^{-i \hat {H}_{0} t/\hbar }$. Employing this identity

$$ \hat{\mathcal{H}}= e^{i\hat{H}_0 t} \hat{H}_{I} e^{-i \hat{H}_0 t}= \hat{H}_I+ \left( \frac{i t}{\hbar} \right) \left[ \hat{H_0}, \hat{H}_{I} \right]+ \frac{1}{2!} \left( \frac{it}{\hbar} \right)^{2} \left[\hat{H}_0, \left[ \hat{H_0}, \hat{H}_{I} \right] \right]+ \cdots, $$

(A.3)

the interaction Hamiltonian $\hat {{\mathscr{H}}}$ becomes

$$ \hat{ \mathcal{H}}/\hbar = \lambda \left( \hat{A} e^{-i {\Delta} t} |2\rangle \langle 1|+\hat{A}^{\dagger} e^{i{\Delta} t} |1\rangle \langle 2| \right) +\chi \hat{a}^{\dagger 2} \hat{a}^{2}, $$

(A.4)

where Δ is the detuning parameter and equals ω₂ − ω₁ −Ω. Then, we apply the Schrödinger equation (SE) in the interaction picture

$$ \frac{\partial|\psi\rangle}{\partial t} = -\frac{i \hat{\mathcal{H}}}{\hbar} |\psi\rangle. $$

(A.5)

The structure of the wave function is expressed as

$$ |\psi \rangle = \sum\limits_{n=0}^{\infty} q_{n} \left[ A_{n+1} |1; n+1 \rangle + B_n |2 ; n \rangle \right]. $$

(A.6)

Then, implementing the SE into this wave function, and after that in the first time multiply the SE to 〈1|〈1 + n| to receive the first differential equation (DE) and in the second time multiply the SE to 〈2|〈n| to receive the second DE. The two DEs are

$$ \begin{array}{@{}rcl@{}} \dot{A}_{n+1}&=& -i \left[ \lambda e^{i {\Delta} t} B_n \sqrt{n+1} f(n+1)+ \chi n(n+1) A_{n+1} \right], \\ \dot{B}_n &=& -i \left[ \lambda e^{-i{\Delta} t} A_{n+1} \sqrt{n+1} f(n+1) + \chi n(n-1) B_n \right] \end{array} $$

(A.7)

Utilizing the rotation frame as A_n+ 1 → A_n+ 1e^−iΔt, the DEs become

$$ \begin{array}{@{}rcl@{}} \dot{A}_{n+1}&=& -i \left[ \lambda B_n \sqrt{n+1} f(n+1)+(\chi n(n+1)+{\Delta}) A_{n+1} \right], \\ \dot{B}_n &=& -i \left[ \lambda A_{n+1} \sqrt{n+1} f(n+1) + \chi n(n-1) B_n \right]. \end{array} $$

(A.8)

The solution to these two equations with considering the atom in the excited state [A_n+ 1(0) = 0,B_n(0) = 1] are written in (4, 5).