Pump-driven transitions between non- and near-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric steady states in polariton condensates

We propose an approach for realizing transitions between non- and near-$\mathcal{PT}$-symmetric systems through controlled modulation of a nonresonant optical pump in exciton-polariton condensates. Specifically, we first construct a $(2+1)$-dimensional $\mathcal{PT}$-symmetric Gaussian local potential and discover several non-$\mathcal{PT}$-symmetric nonlinear steady states with unique profiles that are excited by the $\mathcal{PT}$-symmetric potential and a homogeneous pump. Since the gain-loss coefficients corresponding to these steady states follow specific spatial distribution functions, the polariton model operates in a non-$\mathcal{PT}$ symmetry regime, leading to inherently non-$\mathcal{PT}$-symmetric steady states. However, we find that for the non-$\mathcal{PT}$-symmetric Gaussian steady states, the gain-loss coefficients in their corresponding models exhibit a Gaussian-like distribution, which can be balanced by an incoherent composite Gaussian pump. Therefore, the symmetry of the system can be effectively modulated through appropriate adjustments of the pump parameters. By selecting four types of non-$\mathcal{PT}$-symmetric Gaussian steady states as initial conditions, we numerically demonstrate that the gain-loss coefficients of the system are effectively balanced in space after introducing the composite Gaussian pump. Once the numerical distribution of the gain-loss coefficients approaches zero across the entire space, the non-$\mathcal{PT}$-symmetric Gaussian steady states transform into near-$\mathcal{PT}$-symmetric Gaussian steady states, facilitating the successful activation of the near-$\mathcal{PT}$-symmetric polariton model. Additionally, we define an equilibrium rate to effectively characterize the physical process that leads to the balance of gain-loss coefficient. Finally, we analyze the Poynting vectors within these near-$\mathcal{PT}$-symmetric Gaussian steady states. Similar to previous reports, transverse power flow within the state similar to a bright soliton is also observed. The proposed pumping scheme proves a promising pathway for generation of $\mathcal{PT}$-symmetric states of polariton condensates.