Quantumness and its hierarchies in PT-symmetric down-conversion models

TL;DR

This work analyzes how PT-symmetric dynamics and reservoir noise shape the hierarchy of quantum correlations in a two-mode bosonic down-conversion system. Using a Gaussian-state Langevin–Heisenberg approach, it quantifies local/global nonclassicality, entanglement, steering, and Bell nonlocality across standard, passive, and active PTSSs. The key finding is that the passive PTSS consistently yields the strongest nonclassical states and quantum correlations, while the standard PTSS offers limited advantages and the active PTSS is generally less favorable due to amplified noise; Bell nonlocality, in particular, is most readily achieved without amplification. The results have practical implications for designing nonclassical light sources, suggesting that implementing passive PTSS configurations in χ^(2) down-conversion within resonators provides robust routes to highly nonclassical light suitable for quantum technologies.

Abstract

We investigate the hierarchy of quantum correlations in a quadratic bosonic parity-time-symmetric system (PTSS) featuring distinct dissipation and amplification channels. The hierarchy includes global nonclassicality, entanglement, asymmetric quantum steering, and Bell nonlocality. We elucidate the interplay between the system physical nonlinearity -- which serves as a source of quantumness -- and the specific dynamics of bosonic PTSSs, which are qualitatively influenced by their damping and amplification characteristics. Using a set of quantifiers -- including local and global nonclassicality depths, negativity, steering parameters, and the Bell parameter -- we demonstrate that the standard PTSS typically exhibits weaker quantumness than its counterparts affected solely by damping or solely by amplification. Both the maximum values attained by these quantifiers and the speed and duration of their generation are generally lower in the standard PTSS. A comparative analysis of three two-mode PTSSs -- standard, passive, and active -- with identical eigenvectors and real parts of eigenfrequencies, but differing in their damping and amplification strengths, reveals the crucial role of quantum fluctuations associated with gain and loss. Among them, the passive PTSS yields the most strongly nonclassical states. Nevertheless, under suitable conditions, the standard PTSS can also generate highly nonclassical states. The supremacy of the passive PTSS is further supported by its fundamental advantages in practical realizations.

Figures (8)

• Figure 1: Schematics of the 2-mode bosonic system analyzed under different conditions: (a) standard $\mathcal{PT}$-symmetric system, where mode 1 is damped ($\gamma_1 > 0$) and mode 2 is amplified ($\gamma_2 < 0$); (b) system with only mode 1 damped ($\gamma_1 > 0$, $\gamma_2 = 0$); (c) system with only mode 2 amplified ($\gamma_2 < 0$, $\gamma_1 = 0$); (d) passive $\mathcal{PT}$-symmetric system with mode 1 doubly damped ($2\gamma_1 > 0$, $\gamma_2 = 0$); (e) active $\mathcal{PT}$-symmetric system with mode 2 doubly amplified ($2\gamma_2 < 0$, $\gamma_1 = 0$). Various parameters calculated for these systems are consistently distinguished in the following figures using the superscripts: (a) ad, (b) d, (c) a, (d) dd, and (e) aa.
• Figure 2: (a) Nonclassicality depth $\tau^{\rm ad}$ and (b) the corresponding mean photon number $n_{\tau}^{\rm ad}$, (c) [(d)] local nonclassicality depth $\tau_{1}^{\rm ad}$ [$\tau_{2}^{\rm ad}$] of mode 1 [2] of standard PTSS as they depend on model parameters $\gamma/\epsilon$ and $\kappa/\epsilon$. The values of the drawn parameters are compared with those originating in the model with considered only damping (superscript d) and only amplification (superscript a). In white areas, $\tau_{2}^{\rm ad} = 0$. Solid [dashed] black curves identify positions of EPs in PTSS [systems with only damping and only amplification]. The superscript notation is explained in Fig. \ref{['fig1']}.
• Figure 3: (a) Negativity $E_N^{\rm ad}$ and (b) the corresponding mean photon number $n_{E}^{\rm ad}$, (c) [(d)] steering parameter $S_{1\rightarrow 2}^{\rm ad}$ [$S_{2\rightarrow 1}^{\rm ad}$] of standard PTSS as they depend on model parameters $\gamma/\epsilon$ and $\kappa/\epsilon$. The values of the drawn parameters are compared with those originating in the model with considered only damping (superscript d) and only amplification (superscript a). In white areas, $S_{1\rightarrow 2}^{\rm ad} = 0$. Solid [dashed] black curves identify positions of EPs in PTSS [systems with damping and amplification]. The superscript notation is explained in Fig. \ref{['fig1']}.
• Figure 4: (a) Bell parameter $B_{\rm Bell}^{\rm d}$, (b) [(c)] ratio $(B_{\rm Bell}^{\rm ad}-2)/(B_{\rm Bell}^{\rm d}-2)$ [$(B_{\rm Bell}^{\rm a}-2)/(B_{\rm Bell}^{\rm d}-2)$] of Bell parameters, (d) Bell parameter $B_{\rm Bell}^{\rm dd}$ and the corresponding mean photon number $n_{\rm Bell}^{\rm dd}$ as they depend on model parameters $\gamma/\epsilon$ and $\kappa/\epsilon$. In white areas, the Bell inequalities are not violated ($B_{\rm Bell} = 2$). Solid [dashed] black curves identify positions of EPs in PTSS as well as systems with doubled damping and amplification [systems with damping and amplification]. The superscript notation is explained in Fig. \ref{['fig1']}.
• Figure 5: (a) Nonclassicality depth $\tau^{\rm ad}$ and (b) the corresponding mean photon number $n_{\tau}^{\rm ad}$, (c) [(d)] local nonclassicality depth $\tau_{1}^{\rm ad}$ [$\tau_{2}^{\rm ad}$] of mode 1 [2] of standard PTSS relative to the values of passive PTSS with only doubled damping (superscript dd) and active PTSS with only doubled amplification (superscript aa) as they depend on model parameters $\gamma/\epsilon$ and $\kappa/\epsilon$. In white areas, $\tau_{2}^{\rm ad} = 0$. Solid black curves identify positions of EPs in PTSS as well as systems with only doubled damping and only doubled amplification. The superscript notation is explained in Fig. \ref{['fig1']}.