Two Mode Photon Added Schr\"odinger Cat States: Nonclassicality and Entanglement

The concept of photon added two-mode Schr\"odinger cat states in which both modes are independent is introduced, their non-classical properties and entanglement are studied. The introduced states emerge as the eigenstates of $f_1f_2a_1a_2$, where $f_1, f_2$ are nonlinear functions of the number operator and $a_1, a_2$ are annihilation operators. We study the evolution of these states under the canonical transformation using the parity operator for the case of standard coherent states of the harmonic oscillator. The non-classical properties of these states are evaluated especially by considering sub-Poissonian photon statistics and photon number distribution. Interestingly, the addition of photons leads to shifting the region in which photon number distribution shows oscillatory behavior. In addition, the entanglement of introduced states has been quantitatively analyzed using concurrence. We observe that the state approaches the maximum entanglement more rapidly after the addition of photons.