Structural phase transitions of optical patterns in atomic gases with microwave-controlled Rydberg interactions

Spontaneous symmetry breaking and formation of self-organized structures in nonlinear systems are intriguing and important phenomena in nature. Advancing such research to new nonlinear optical regimes is of much interest for both fundamental physics and practical applications. Here we propose a scheme to realize optical pattern formation in a cold Rydberg atomic gas via electromagnetically induced transparency. We show that, by coupling two Rydberg states with a microwave field (microwave dressing), the nonlocal Kerr nonlinearity of the Rydberg gas can be enhanced significantly and may be tuned actively. Based on such nonlocal Kerr nonlinearity, we demonstrate that a plane-wave state of a probe laser field can undergo a modulation instability (MI) and hence spontaneous symmetry breaking, which may result in the emergence of various self-organized optical patterns. Especially, we find that a hexagonal lattice pattern (which is the only optical pattern when the microwave dressing is absent) may develop into several types of square lattice ones when the microwave dressing is applied; moreover, as an outcome of the MI the formation of nonlocal optical solitons is also possible in the system. The optical patterns and nonlocal optical solitons found here can be flexibly manipulated by adjusting the effective probe-field intensity, nonlocality degree of the Kerr nonlinearity, and strength of the microwave field. Our paper opens a route for versatile controls of self-organizations and structural phase transitions of laser light, which may have potential applications in optical information processing and transmission.

Figure 1

Schematics of the model. (a) Ladder-type four-level atomic configuration for realizing the microwave-dressed Rydberg-EIT. Here, the weak probe laser field (blue), strong control laser field (red), and strong microwave field (green) with half Rabi frequencies ${\mathrm{\Omega }}_p, {\mathrm{\Omega }}_c$, and ${\mathrm{\Omega }}_m$ drive the transitions $|1\rangle \leftrightarrow |2\rangle , |2\rangle \leftrightarrow |3\rangle$, and $|3\rangle \leftrightarrow |4\rangle$, respectively. States $|1\rangle$ and $|2\rangle$ are, respectively, ground and excited states. Both $|3\rangle$ and $|4\rangle$ are highly excited Rydberg states. ${\mathrm{\Delta }}_2, {\mathrm{\Delta }}_3$, and ${\mathrm{\Delta }}_4$ are, respectively, the one-, two-, and three-photon detunings. ${\mathrm{\Gamma }}_{12}, {\mathrm{\Gamma }}_{23}$, and ${\mathrm{\Gamma }}_{24}$ are the spontaneous emission decay rates from $|2\rangle$ to $|1\rangle , |3\rangle$ to $|2\rangle$, and $|4\rangle$ to $|2\rangle$, respectively. Two Rydberg atoms locating, respectively, at position $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ interact through van der Waals potential $\hslash {\mathcal{V}}_{\mathrm{vdw}}^l({\mathbf{r}}^{\prime }-\mathbf{r})$ ($l=s,d,e$; see text). (b) Possible experimental geometry, where small solid circles denote atoms and large dashed circles denote Rydberg blockade spheres. (c) Emergence of an optical pattern via modulation instability.

Figure 2

Kerr nonlinearity enhancement, interaction potential energy, and normalized nonlinear-response function $\Re$ of the probe field in the presence of the microwave dressing. (a) Real part $\mathrm{Re}\left({\chi }_{\mathrm{loc}}^{\left(3\right)}\right)$ (solid red line) and imaginary part $\mathrm{Im}\left({\chi }_{\mathrm{loc}}^{\left(3\right)}\right)$ (dotted blue line) of the local nonlinear susceptibility ${\chi }_{\mathrm{loc}}^{\left(3\right)}$ as a function of the half Rabi frequency ${\mathrm{\Omega }}_m$ of the microwave field. (b) The same as (a) but for the nonlocal nonlinear susceptibility ${\chi }_{\mathrm{nloc}}^{\left(3\right)}$. (c) Potential-energy curves $E_1$ (solid blue line), $E_2$ (solid black line), and $E_3$ (solid red line) of two Rydberg atoms as functions of the interatomic distance $r=|{\mathbf{r}}^{\prime }-\mathbf{r}|$ for ${\mathrm{\Omega }}_m=10\mathrm{MHz}$ and $\mathrm{\Delta }=13.98\mathrm{MHz}$; $E_{33}$ (dashed blue line), $E_{44}$ (dashed black line), and $E_{34}^{+}$ (dashed red line) are for the case without the microwave field. (d) Normalized response function $\Re /{\Re }_{\max }$ as a function of the dimensionless coordinate $\xi =x/R_0$ ($R_0$ is the typical transverse beam radius of the probe field) with ${\mathrm{\Omega }}_m=0$ (dotted black line), 5 MHz (solid red line), and 15 MHz (solid blue line), respectively. Inset: The normalized response function $\tilde{\Re }/{\tilde{\Re }}_{\max }$ in momentum space (i.e., the Fourier transformation of $\text{Re}/{\text{Re}}_{\max }$) as a function of the dimensionless wave number ${\beta }_1=R_0{k}_x$ ($k_x$ is the dimensional wave number in the $x$ direction) with ${\mathrm{\Omega }}_m=0$, 5, and 15 MHz, respectively.

Figure 3

Modulation instability and its manipulation for the plane-wave probe field in the Rydberg atomic gas with the microwave dressing. (a) $-{\lambda }^2$ ($\lambda$ is the growth rate) as a function of $\beta =\sqrt{{\beta }_1^2+{\beta }_2^2}$ [${\beta }_j\equiv R_0{k}_j$; $k_j$ is the wave number along the $j\mathrm{th}$ direction ($j=x,y$); $R_0$ is the typical transverse beam radius of the probe field], for the microwave field ${\mathrm{\Omega }}_m=0$ (dotted black line), 5 MHz (dashed red line), and 15 MHz(solid blue line), respectively; shadow regions are ones where MI occurs. (b) Real part of the growth rate $\mathrm{Re}(\lambda$) as a function of $\beta$ and the effective probe-field intensity $I_{\mathrm{eff}}=\alpha A_0^2$ [with $A_0$ the amplitude of the plane wave and $\alpha =-\int \Re \left(\vec{\zeta }\right)d^2\zeta$ ], for the microwave field ${\mathrm{\Omega }}_m=10\mathrm{MHz}$. The colorful region is the one for $\mathrm{Re}\left(\lambda \right)>0$, where MI occurs. (c) $\mathrm{Re}(\lambda$) as a function of $\beta$ and ${\mathrm{\Omega }}_m$ for $I_{\mathrm{eff}}=20$; the colorful region is the one where MI occurs.

Figure 4

Pattern formation and phase diagram controlled by the effective intensity of the probe field $I_{\mathrm{eff}}=\alpha A_0^2$ and the microwave field ${\mathrm{\Omega }}_m$, for the nonlocality degree $\sigma \equiv R_b/R_0=1$. Here, $A_0$ is the amplitude of the plane-wave state; $\alpha =-\int \Re \left(\vec{\zeta }\right)d^2\zeta$; $R_b$ and $R_0$ are the Rydberg blockade radius and the transverse radius of the probe beam, respectively. (a) Phase diagram of the structural transition of optical patterns, where different regions (phases) are obtained by changing the values of $I_{\mathrm{eff}}$ and ${\mathrm{\Omega }}_m$. Region 1, the homogeneous (i.e., the plane-wave) state; region 2, the hexagonal lattice; region 3, the type-I square lattice; region 4, the type-II square lattice. (b) The hexagonal lattice of the normalized probe-field amplitude $\left|u\right|$ as a function of $\xi =x/R_0$ and $\eta =y/R_0$ for ${\mathrm{\Omega }}_m=10\mathrm{MHz}$ and $I_{\mathrm{eff}}=15$, corresponding to the region 2 in panel (a). (c) The type-I square lattice of $\left|u\right|$ as a function of $\xi$ and $\eta$ for ${\mathrm{\Omega }}_m=13\mathrm{MHz}$ and $I_{\mathrm{eff}}=25$, corresponding to the region 3 in panel (a). (d) The type-II square lattice of $\left|u\right|$ as a function of $\xi$ and $\eta$ for ${\mathrm{\Omega }}_m=15\mathrm{MHz}$ and $I_{\mathrm{eff}}=35$, corresponding to the region 4 in panel (a).

Figure 5

Pattern formation and phase diagram controlled by the nonlocality degree of the Kerr nonlinearity $\sigma =R_b/R_0$ and the microwave field ${\mathrm{\Omega }}_m$, for the effective probe-field intensity $I_{\mathrm{eff}}=35$. (a) Phase diagram of the structural transition of optical patterns, where different regions (phases) are obtained by changing the values of $\sigma$ and ${\mathrm{\Omega }}_m$. Region 4, the pattern with hexagonal lattice structure; region 3, the pattern with type-I square lattice structure; region 2, the pattern with type-II square lattice structure; region 1, the homogeneous (i.e., the plane-wave) state. (b) The hexagonal structure of the normalized probe-field amplitude $\left|u\right|$ as a function of $\xi =x/R_0$ and $\eta =y/R_0$ for ${\mathrm{\Omega }}_m=10\mathrm{MHz}$ and $\sigma =2$, corresponding to the region 4 in panel (a). (c) The type-I square structure of $\left|u\right|$ as a function of $\xi$ and $\eta$ for ${\mathrm{\Omega }}_m=12\mathrm{MHz}$ and $\sigma =1$, corresponding to the region 3 in panel (a). (d) The type-II square structure of $\left|u\right|$ as a function of $\xi$ and $\eta$ for ${\mathrm{\Omega }}_m=18\mathrm{MHz}$ and $\sigma =1$, corresponding to the region 4 in panel (a).

Figure 6

Formation and propagation of nonlocal spatial optical solitons in the presence of the microwave dressing. (a) Initial condition $\left|u\right|=1.3\text{sech}\left[{({\xi }^2+{\eta }^2)}^{1/2}\right]$ at $s=0$ ($s=z/2L_{\mathrm{diff}}$; $L_{\mathrm{diff}}$ is diffraction length). (b) Spatial distribution of the soliton when propagating to the position $s=5$, by taking the normalized probe-field amplitude $\left|u\right|$ as a function of nondimensional coordinates $\xi =x/R_0$ and $\eta =y/R_0$ ($R_0$ is the typical transverse beam radius of the probe field) for the microwave field ${\mathrm{\Omega }}_m=10\mathrm{MHz}$ and the nonlocality degree $\sigma =1$. (c) Random initial condition $\left|u\right|=1+0.05f$ ($f$ is a Gaussian noise) and (d) the soliton (generated by the random initial condition) when propagating to $s=5$. (e), (f), and (g) are two-, three-, and four-soliton solutions when propagating to $s=5$ for $({\mathrm{\Omega }}_m,\sigma )=(10,1.2)$, (15,1.4), and (18,1.8), respectively.

Figure 7

Schematic diagram of the nondimensional wave vectors ${\vec{\beta }}_j$ for different optical self-organized structures. (a) Parallel stripes. (b) Square lattice, for which ${\vec{\beta }}_1·{\vec{\beta }}_2=0$. (c) Hexagonal lattice, for which ${\vec{\beta }}_1+{\vec{\beta }}_2+{\vec{\beta }}_3=0$. All the wave vectors have the same module (i.e., $|{\vec{\beta }}_j|\approx {\beta }_{\mathrm{cr}}$; ${\beta }_{\mathrm{cr}}$ is the first wave number of the unstable band of the MI).

Figure 8

Ground-state energy for cases of hexagonal and square lattices. (a) Ground-state energy $E$ as a function of modulation amplitude $D$, with parameters ${\beta }_{\mathrm{cr}}=4.3,I_{\mathrm{eff}}=15, {\mathrm{\Omega }}_m=10\mathrm{MHz}$, and $\sigma =1$. The solid red line ($E_{\mathrm{Hex}}$) is for the hexagonal pattern; the dotted blue line ($E_{\mathrm{Squ}}$) is for the square pattern. (b) The same as (a) but for ${\beta }_{\mathrm{cr}}=4.3,I_{\mathrm{eff}}=25, {\mathrm{\Omega }}_m=15\mathrm{MHz}$, and $\sigma =1$.