Entanglement, quantum coherence and quantum Fisher information of two qubit-field systems in the framework of photon-excited coherent states

Abstract

Based on coherent state theory, the photon-excited coherent states associated with pseudo-harmonic oscillators (PE-CSPHOs) are considered to investigate the interaction between a system of two qubits (two-level atoms) and a radiation field. By considering the dipole–dipole Hamiltonian and Ising Hamiltonian, we solve the Schrödinger equation to examine the influence of the qubit-qubit interaction (Q–QI) on the dynamics of the Q–Q entanglement and the entanglement between the two qubits and field. Furthermore, we study the dynamics of norm coherence and Fisher information, based on symmetric logarithmic derivative, with respect to the physical parameters of the system and explore the relation among the quantifiers during quantum dynamics.

Appendix A

Acting the Hamiltonian operator ${\hat{\mathbf{H}}}$ Eq. (5) on the wave function $\left| {{{\varvec{\uppsi}}}\left( {\mathbf{t}} \right)} \right\rangle$ and applying the Schrödinger equation,

$${\mathbf{i}}\hbar \frac{{\partial \left| {{{\varvec{\uppsi}}}\left( {\mathbf{t}} \right)} \right\rangle }}{{\partial {\mathbf{t}}}} = {\hat{\mathbf{H}}}\left| {{{\varvec{\uppsi}}}\left( {\mathbf{t}} \right)} \right\rangle ,$$

(17)

hence this allows us to determine ${\mathbf{Y}}_{{\mathbf{j}}} \left( {{\mathbf{n}}, {\mathbf{t}}} \right),$ ${\mathbf{for}}\; \hbar = 1,$ that are verifying the system of coupled ode

$$\frac{{{\mathbf{dY}}_{1} }}{{{\mathbf{dt}}}} = - {\mathbf{i}}{\varvec{\lambda}}_{{\varvec{N}}} {\mathbf{Y}}_{{\mathbf{1}}} - {\mathbf{ib}}\sqrt {{\mathbf{n}} + {\mathbf{m}} + 1} \left( {{\mathbf{Y}}_{{\mathbf{2}}} + {\mathbf{Y}}_{{\mathbf{3}}} } \right)$$

(18)

$$\frac{{{\mathbf{dY}}_{{\mathbf{2}}} }}{{{\mathbf{dt}}}} = - {\mathbf{i}}\left( {{\varvec{\lambda}}_{{\varvec{N}}} {\mathbf{Y}}_{{\mathbf{2}}} + {\varvec{\lambda}}_{{\varvec{D}}} {\mathbf{Y}}_{{\mathbf{3}}} } \right) - {\mathbf{ib}}\left[ {\sqrt {{\mathbf{n}} + {\mathbf{m}} + {\mathbf{1}}} {\mathbf{Y}}_{{\mathbf{1}}} + \sqrt {{\mathbf{n}} + {\mathbf{m}} + {\mathbf{2}}} {\mathbf{Y}}_{{\mathbf{4}}} } \right]$$

(19)

$$\frac{{{\mathbf{dY}}_{{\mathbf{3}}} }}{{{\mathbf{dt}}}} = - {\mathbf{i}}\left( {{\varvec{\lambda}}_{{\varvec{N}}} {\mathbf{Y}}_{{\mathbf{3}}} + {\varvec{\lambda}}_{{\varvec{D}}} {\mathbf{Y}}_{{\mathbf{2}}} } \right) - {\mathbf{ib}}\left[ {\sqrt {{\mathbf{n}} + {\mathbf{m}} + {\mathbf{1}}} {\mathbf{Y}}_{{\mathbf{1}}} + \sqrt {{\mathbf{n}} + {\mathbf{m}} + {\mathbf{2}}} {\mathbf{Y}}_{{\mathbf{4}}} } \right]$$

(20)

$$\frac{{{\mathbf{dY}}_{{\mathbf{4}}} }}{{{\mathbf{dt}}}} = - {\mathbf{i}}{\varvec{\lambda}}_{{\varvec{N}}} {\mathbf{Y}}_{{\mathbf{4}}} - {\mathbf{i}}{\varvec{b}}\sqrt {{\mathbf{n}} + {\mathbf{m}} + {\mathbf{2}}} \left( {{\mathbf{Y}}_{{\mathbf{2}}} + {\mathbf{Y}}_{{\mathbf{3}}} } \right).$$

(21)

The above coupled system (18)–(21) is solved numerically under the initial conditions

$${\text{Y}}_{1} \left( 0 \right) = {\text{Y}}_{4} \left( 0 \right) = 0$$

(22)

$${\text{Y}}_{2} \left( 0 \right) = {\text{Y}}_{3} \left( 0 \right) = \frac{{{\text{U}}_{{\text{n}}} }}{{\sqrt {2\mathop \sum \nolimits_{{{\text{n}} = 0}}^{\infty } \left| {{\text{U}}_{{\text{n}}} } \right|^{2} } }}$$

(23)

where ${\text{U}}_{{\text{n}}}$ is given in Eq. (8).