Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas

We propose a scheme to realize parity-time ($\mathcal{PT}$) symmetry and nonlocal optical solitons in a cold Rydberg atomic system with electromagnetically induced transparency. We show that a two-dimensional (2D) periodic optical potential with $\mathcal{PT}$ symmetry can be obtained for the propagation of probe laser field by using an incoherent population pumping between two low-lying levels and spatial modulations of control and assistant laser fields. We also show that, based on the giant nonlocal Kerr nonlinearity originated from the strong, long-range atom-atom interaction, the system supports 2D nonlocal gap solitons with very low light intensity. In particular, we find that the degree of the nonlocality of the Kerr nonlinearity, which can be actively tuned in our system, can be used to manipulate the phase transition of the $\mathcal{PT}$ symmetry and the behavior of the nonlocal optical solitons. Our study opens a route for developing non-Hermitian nonlinear optics, especially for realizing and controlling high-dimensional weak-light optical solitons through adjustable $\mathcal{PT}$ symmetry and giant nonlocal optical nonlinearity.

Figure 1

(a) Level diagram and excitation scheme of the Rydberg-dressed EIT. States $|1\rangle , |2\rangle , |3\rangle$ constitute a standard $\mathrm{\Lambda }$-type EIT configuration, where the probe field (half Rabi frequency ${\mathrm{\Omega }}_p$) couples the transition $|1\rangle \leftrightarrow |3\rangle$, and the control field (half Rabi frequency ${\mathrm{\Omega }}_c$) couples the transition $|2\rangle \leftrightarrow |3\rangle$. ${\mathrm{\Delta }}_j$ are detunings and ${\mathrm{\Gamma }}_{jl}$ are the spontaneous-emission decay rate from $|l\rangle$ to $|j\rangle$. The $\mathrm{\Lambda }$-type EIT is dressed by a high-lying Rydberg state $|4\rangle$, which is far off-resonantly (${\mathrm{\Delta }}_3+{\mathrm{\Delta }}_4\gg {\mathrm{\Omega }}_a$) coupled to the state $|3\rangle$ through an assisted laser field (half Rabi frequency ${\mathrm{\Omega }}_a$). An incoherent pumping (with pumping rate ${\mathrm{\Gamma }}_{21}$) is used to pump atoms from $|1\rangle$ to $|2\rangle$, providing a gain for the probe field. The control and the assisted fields are assumed to be spatially modulated. The interaction between Rydberg atoms is described by the van der Waals potential $V_{\mathrm{vdw}}\equiv \hslash V({\mathbf{r}}^{\prime }-\mathbf{r})$, with $V({\mathbf{r}}^{\prime }-\mathbf{r})=C_6/{|{\mathbf{r}}^{\prime }-\mathbf{r}|}^6$. (b) Populations ${\rho }_{jj}^{\left(0\right)}$ ($j=1,2,3,4$) of atoms at different levels as functions of the incoherent population pumping rate ${\mathrm{\Gamma }}_{21}$. For ${\mathrm{\Gamma }}_{21}=0$, all populations are in the ground state $|1\rangle$; however, when ${\mathrm{\Gamma }}_{21}\ne 0$, we have ${\rho }_{33}^{\left(0\right)}\ne 0$. (c) Spatial distributions of the real and imaginary parts of the nonlocal nonlinear coefficient $W_2$, i.e., $\mathrm{Re}\left(W_2\right)$ and $\mathrm{Im}\left(W_2\right)$, in the plane of dimensionless spatial coordinates $\xi =x/R_{\perp }$ and $\eta =\eta /R_{\perp }$.

Figure 2

Realization of the 2D $\mathcal{PT}$-symmetric optical potential. (a) $\mathcal{PT}$-symmetric optical potential (7) with $(V_1,V_2)=(0.1,0.1)$ of the Rydberg-dressed EIT system as a function of $\xi =x/R_{\perp }$ at $\eta =y/R_{\perp }=0$. The real part of optical potential $\mathrm{Re}\left(V\right)$ (black solid line) and the imaginary part of optical potential $\mathrm{Im}\left(V\right)$ (red dashed line) versus $\xi$ are plotted. (b),(c) 2D spatial distributions of ${\mathrm{\Omega }}_c$ [panel (b)] and ${\mathrm{\Omega }}_a$ [panel (c)], required to obtain the $\mathcal{PT}$-symmetric period potential.

Figure 3

Active manipulation of the $\mathcal{PT}$ phase transition by the nonlocal optical nonlinearity. Shown in (a)–(e) are the real part $\mathrm{Re}\left(b\right)$ and the imaginary part $\mathrm{Im}\left(b\right)$ ($b$ is propagation constant) as functions of the lattice momentum $q=\sqrt{q_x^2+q_y^2}$ for the $\mathcal{PT}$-symmetric potential (7), for different light power $P$ and degree of the nonlocality $\sigma$, with $(V_1,V_2)=(1,0.6)$. (a) $\mathrm{Re}\left(b\right)$ and $\mathrm{Im}\left(b\right)$ versus $q$ along points $M, X, \mathrm{\Gamma }, M$ for $(P,\sigma )=(7,0)$. (b) $\mathrm{Re}\left(b\right)$ and $\mathrm{Im}\left(b\right)$ for $(P,\sigma )=(8,0)$. (c) $\mathrm{Re}\left(b\right)$ and $\mathrm{Im}\left(b\right)$ for $(P,\sigma )=(9,0)$. (d) $\mathrm{Re}\left(b\right)$ and $\mathrm{Im}\left(b\right)$ for $(P,\sigma )=(9,0.5)$. (e) $\mathrm{Re}\left(b\right)$ and $\mathrm{Im}\left(b\right)$ for $(P,\sigma )=(9,1)$. Shaded domains in (a)–(e) indicate the region where the $\mathcal{PT}$ symmetry is broken. (f) Phase diagram of the $\mathcal{PT}$ symmetry in the plane of $(\sigma ,P)$, determined by the maximum of $\mathrm{Im}\left(b\right)$. The white dashed line represents the $\mathcal{PT}$ phase transition. The system works in the full (broken) $\mathcal{PT}$-symmetric phase when $\sigma$ is small (large) and $P$ is large (small).

Figure 4

Nonlocal optical gap solitons and their stability. (a) Curve of $P-\mu$ ($\mu \equiv -b$; $b$ is propagation constant) for degree of nonlocality $\sigma =0.5$ and the optical potential parameters $(V_1,V_2)=(1,0.3)$. Nonlocal gap solitons exist in some regions of the semi-infinite gap [i.e., in the region $\mu >-1$; the shaded region ($\mu <-1$) is the first energy band]. Solid part (where point “A” locates) and dashed part (where point “B” locates) in the red curve represent stable and unstable regions of nonlocal gap solitons, respectively. Inset: dimensionless wave shape $\left|U\right|=|{\mathrm{\Omega }}_p/U_0|$ for a typical 2D nonlocal optical soliton with $P=10$. (b) Stability spectrum for small perturbations around the optical gap soliton with $P=10$ (upper panel) and with $P=14$ (lower panel). $\mathrm{Re}(\lambda$) and $\mathrm{Im}(\lambda$) are real part and imaginary part of the eigenvalue $\lambda$ of the small perturbations, respectively. (c),(d) Propagation of the nonlocal gap soliton under small perturbations as functions of $\xi =x/R_{\perp }$ and $\zeta =z/z_0$, for $P=10$ [panel (c)] and $P=14$ [panel (d)]. The soliton propagates robustly against the perturbations in the $\mathcal{PT}$-symmetry phase [panel (c)], but it breaks up in the broken $\mathcal{PT}$-symmetry phase [panel (d)].

Figure 5

Active control of the nonlocal gap solitons. (a) $\mathcal{PT}$ phase diagram in the plane of $(P,\sigma )$ for the optical potential (7) with $(V_1,V_2)=(1,0.6)$. The red solid line represents the $\mathcal{PT}$ phase transition, above (below) which is the region of full (broken) $\mathcal{PT}$ symmetry. A nonlocal gap soliton with light power $P=9$ and the degree of the nonlocality $\sigma =0.25$ (marked by the point “A”) is stable; however, a nonlocal gap soliton with the same light power but $\sigma =0.5$ (marked by the point “B”) is unstable. (b) Propagation of the nonlocal gap soliton (as functions of $\xi =x/R_{\perp }$ and $\zeta =z/z_0$) by changing the value of $\sigma$ during the propagation. The soliton loses its stability at $\zeta \approx 5$, where $\sigma$ is increased rapidly from 0.25 to 0.5. (c) Dimensionless amplitude of the output soliton $\left|u\right|=|{\mathrm{\Omega }}_p/U_0|$ at $\zeta =10$, where a significant deformation and attenuation occurs. Dimensionless amplitude of the input soliton at $\zeta =0$ cm is also given, as shown in the inset.