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 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 a outcome of the MI the formation of nonlocal optical solitons is also possible in the system. Different from earlier studies, 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 the strength of the microwave field. Our work 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.

On the other hand, in the past two decades a large amount research works were focused on the investigation of cold Rydberg atomic gases [61,62] working under condition of electromagnetically induced transparency (EIT). EIT is an important quantum destruction interference effect occurring typically in resonant three-level atomic systems, by which the absorption of a probe laser field can be largely eliminated by a control laser field [63]. Due to strong, long-range atom-atom interaction (also called Rydberg-Rydberg interaction), such systems are desirable nonlinear optical media with strong, nonlocal Kerr nonlinearity if the Rydberg-Rydberg interaction is suitably mapped to photon-photon interaction via EIT [64,65]. In an interesting work, Sevinli et al. [66] reported a self-organized hexagonal optical pattern via a MI of plane-wave probe beam in a cold Rydberg atomic gas with a repulsive Rydberg-Rydberg interaction.
In this work, we propose and analyze a scheme for realizing various self-organized optical structures and their structural phase transition in a cold Rydberg atomic gas via a Rydberg-EIT [67,68]. By exploiting a microwave dressing (i.e., a microwave field couples two electrically excited Rydberg states) [69][70][71][72][73][74][75][76][77][78][79][80][81][82][83], we show that the nonlocal Kerr nonlinearity of the Rydberg gas (which has only a repulsive Rydberg-Rydberg interaction in the absence of the microwave field) is significantly modified, and its strength and sign can be tuned actively. Based on such nonlocal Kerr nonlinearity, we demonstrate that a homogeneous (plane wave) state of probe laser field can undergo MI and hence spontaneous symmetry breaking, which may result in the formation of various ordered optical patterns.
Through detailed analytical and numerical analysis, we find that a homogeneous state of the probe field is firstly transited into a hexagonal lattice pattern (which is the only lattice pattern when the microwave dressing is absent). Interestingly, this hexagonal lattice pattern may undergo a structural phase transition and develop into several types of square lattice patterns when the microwave field is applied and its strength is increased. Moreover, as a outcome of the MI the formation of nonlocal spatial optical solitons is also possible by a suitable choice of system parameters. Different from the results reported before, the optical patterns and nonlocal optical solitons found here can be flexibly manipulated via the adjustment of the effective probe-field intensity, nonlocality degree of the Kerr nonlinearity, and the strength of the microwave field. Our study opens a way for actively controlling the self-organization and structural phase transition of optical patterns through microwave-dressing on Rydberg gases, which are not only of fundamental interest, but also useful for potential applications in optical information processing and transmission.
The remainder of the article is arranged as follows. In Sec. II, we describe the physical model, discuss the modification and enhancement of the Kerr nonlinearity contributed by the microwave field, and derive the nonlinear 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 Ωp, Ωc, and Ωm drive the transitions |1 ↔ |2 , |2 ↔ |3 , and |3 ↔ |4 , respectively; States |1 and |2 are respectively ground and excited states, both |3 and |4 are highly excited Rydberg states; ∆ 2 , ∆ 3 , and ∆ 4 are respectively the one-, two-, and three-photon detunings; Γ 12 , Γ 23 , and Γ 24 are the spontaneous emission decay rates from |2 to |1 , |3 to |2 and |4 to |2 , respectively. Two Rydberg atoms locating respectively at position r and r ′ interact through van der Waals potential V l vdw (r ′ − 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. envelope equation of the probe field. In Sec. III, we consider the MI of a plane-wave state, investigate the formation and structural phase transitions of optical patterns controlled by the microwave field, effective probe-field intensity, the nonlocal Kerr nonlinearity and its nonlocality degree. The result on the formation of nonlocal spatial optical solitons is also presented. The last section (Sec. IV) gives a summary of our main research results.

A. Physical model
We consider an ensemble of lifetime-broadened fourlevel atomic gas with an ladder-type level configuration, shown schematically in Fig. 1(a). Here, the weak probe laser field with central angular frequency ω p , wavevector k p , and half Rabi frequency Ω p drives the transition from atomic ground state |1 to intermediate state |2 , and the strong control laser field with central angular frequency ω c , wavevector k c , and half Rabi frequency Ω c drives the transition |2 to the first highly excited Rydberg state |3 . This ladder-type three level EIT is dressed by a microwave field with central angular frequency ω m , wavevector k m , and half Rabi frequency Ω m , which couples the transition between the Rydberg state |3 and another Rydberg state |4 . The total electric fields acting in the atomic system can be written as E(r, t) = j e j E j e i(kj ·r−ωj t) + H.c., with e j and E j respectively the polarization unit vector and the envelope of j-th laser field (j = p, c, m). ∆ 2 , ∆ 3 , and ∆ 4 are respectively the one-, two-, and three-photon detunings; Γ 12 , Γ 23 , and Γ 24 are the spontaneous emission decay rates from |2 to |1 , |3 to |2 , and |4 to |2 , respectively. The microwave field employed here is to realize a microwave-dressed Rydberg-EIT [69][70][71][72][73][74][75][76][77][78][79][80][81][82][83] and thus to modify the Rydberg-Rydberg interaction, which, in turn, can manipulate the interaction strength and sign for the photons in the probe field and hence realize self-organized optical structures not discovered before.
The dynamics of the system is controlled by the Hamil-tonianĤ = N a d 3 rĤ(r, t), whereĤ(r, t) is Hamiltonian density and N a is atomic density. Under the electricdipole and rotating-wave approximations, the Hamiltonian density in interaction picture readŝ where HamiltonianĤ 1 describes unperturbed atoms as well as the interaction between the atoms and the laser fields,Ĥ vdW describes the Rydberg-Rydberg interaction, respectively given bŷ (2b) Here ; the one-, two-, and three-photon detuings are respectively given by , with E α = ω α the eigenenergy of the atomic state |α . The half Rabi frequencies of the probe, control, and microwave fields are, respectively, Ω p = (e p ·p 21 )E p / , Ω c = (e c ·p 32 )E c / , and Ω m = (e m · p 43 )E m / , with p αβ the electric dipole matrix element associated with the transition between the states |α and |β .
The time evolution of the atoms in the system is governed by the optical Bloch equation where ρ(r, t) = Ŝ (r, t) [84] is a 4 × 4 density matrix (with density matrix elements ρ αβ (r, t) = Ŝ αβ (r, t) ; α, β = 1, 2, 3, 4) describing the atomic population and coherence, Γ is a 4 × 4 relaxation matrix describing the spontaneous emission and dephasing. Explicit expressions of ρ αβ (r, t) are presented in Appendix A.
The propagation of the probe field is controlled by Maxwell equation, which under paraxial and slowlyvarying envelope approximations is reduced into [65] i ∂ ∂z where ∇ 2 ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 describes diffraction, κ 12 = N a ω p |(e p · p 12 )| 2 /(2ǫ 0 c ) is a parameter describing the coupling between the atoms and the probe field, and c is the light speed in vacuum. Without loss of generality, we assume the probe field propagates along z direction, i.e., k p = (0, 0, ω p /c); to suppress Doppler effect, the microwave field is along the z direction but the control field is along the negative z direction [i.e., k m = (0, 0, ω m /c) and k c = (0, 0, −ω c /c)]. A possible experimental arrangement is given in Fig. 1(b).

B. Enhanced Kerr nonlinearity by the microwave dressing
We first consider the modification of Kerr nonlinearity of the system induced by the microwave field based on the physical model described above. For simplicity, we assume that the control and microwave fields are strong enough, so that they are not depleted during the propagation of the probe field. Since the probe field is weak, a perturbation expansion can be applied to solve the Maxwell-Bloch (MB) equations (3) and (4) by taking Ω p as a small expansion parameter. Generalizing the approach developed in Refs. [86][87][88][89], where MB equations for Rydberg atomic gases without microwave dressing are solved beyond mean-field approximation in a selfconsistent and effective way, we can obtain the solutions of the Bloch Eq. (3) using the perturbation expansion up to third-order approximation. In particular, the result of the one-body density matrix element ρ 21 can be obtained analytically (see the Appendix A for detail).
With the expression of ρ 21 and the definition of probefield susceptibility, i.e. χ = N a (e · p 12 )ρ 21 /(ǫ 0 E p ), it is easy to obtain the optical susceptibility of the probe field, which reads where χ (1) is the linear susceptibility; χ loc and χ nloc are local and nonlocal third-order nonlinear (Kerr) susceptibilities, originated respectively from non-zero two-photon detuning (i.e. ∆ 3 = 0) [86][87][88][89][90][91]) and from the Rydberg-Rydberg interaction in the system. Expressions of χ (1) , χ loc , and χ nloc are much smaller than their real parts, which is due to the EIT effect contributed by the control field; moreover, the nonlocal Kerr nonlinearity is three orders of magnitude larger than the local one, which is due to the strong Rydberg-Rydberg interaction together with the microwave dressing.
It is helpful to reveal how the microwave-dressing modify the Kerr effect of the system. Fig. 2(a) shows the real part Re(χ loc as a function of the half Rabi frequency Ω m of the microwave field. Shown in Fig. 2(b) is same as that in Fig. 2(a) but for the nonlocal nonlinear susceptibility χ Thus, microwave-dressing can be used to modify the Kerr effect of the system greatly.
To support the above conclusion, a calculation is carried out for the interaction potential of two Rydberg atoms located respectively at positions r and r ′ , which may occupy in the Rydberg states |3 and |4 . In the absence of the microwave field (i.e., Ω m = 0), the basis set of the such two-atom system consists of states |33 = |3 1 3 2 , |44 = |4 1 4 2 , and |34 ± = 1/ √ 2(|3 1 4 2 ± |3 2 4 1 ), with the subscript 1 and 2 representing atom 1 and 2, respectively. The (bare state) eigen ener-gies of the system are 98 MHz. Since antisymmetric state |34 − is nearly not coupled to laser field, one can disregard it if the microwave field is present (i.e., Ω m = 0). Then, the Hamiltonian in the two-atom basis set {|33 , |34 + , |44 } takes the form After diagonalization, we can obtain the energies E 1 , E 2 , and E 3 of the Hamiltonian (6). Potential-energy curves of E 1 , E 2 , and E 3 as functions of the interatomic separation r = |r ′ − r| for Ω m = 10 MHz are shown in Fig. 2(c). For comparison, the bare potential-energy curves E 33 , E 44 , and E + 34 (for Ω m = 0) are also shown. We see that, compared with the case without the microwave field, the potential-energy curves are modified largely by the introduction of the microwave field, especially for small interatomic separation r. The reason is that the microwave dressing brings a coupling between the Rydberg states |3 and |4 , and thereby a modification of the Rydberg-Rydberg interaction. It is the use of the microwave dressing that brings the significant change and enhancement of the nonlocal Kerr nonlinearity.

C. Nonlinear envelope equation and the property of nonlinear response function
We now derive the envelope equation which controls the dynamics of the probe field. By substituting the solution of ρ 21 into the Maxwell Eq. (4) and making a local approximation along the z direction on the nonlocal nonlinear response function (see the Appendix B), we obtain the following three-dimensional (3D) nonlocal nonlinear Schrödinger (NNLS) equation The third and forth terms on the left hand side of this equation describe two types of self-phase modulations of the probe field, contributed respectively by the local Kerr nonlinearity (originated from non-zero two-photon detuning (i.e., ∆ 3 = 0) [86][87][88][89][90][91]) and the nonlocal Kerr nonlinearity (originated from the Rydberg-Rydberg interaction). In the integral of the forth term of the NNLS equation, G is a reduced nonlinear response function, taking the form is contributed by the interaction of the atoms lying in the same Rydberg state |3 (|4 ); the third (forth) part G d is contributed by the interaction of the atoms lying in the different Rydberg states |3 and |4 .
For the convenience of later discussions and numerical calculations, we rewrite the 3D NNLS Eq. (7) into the non-dimensional form 0 /c is the typical diffraction length, which is 1.61 mm in our system; U 0 is the typical Rabi frequency of the probe field; R 0 is the typical beam radius of the probe field; the non-dimensional nonlinear response function is defined by Note that in writing Eq. (8) we have neglected the term related to W 1 because the local Kerr nonlinearity is much smaller than the nonlocal one [88].
The property of the nonlocal Kerr nonlinearity of the system is characterized by the nonlinear response function ℜ( ζ). Comparing with the case without the microwave field (Ω m = 0), ℜ( ζ) is largely modified and can be manipulated by the use of the microwave field (Ω m = 0). To demonstrate this, the normalized response function ℜ/ℜ max as a function for ξ = x/R 0 is shown in Fig. 2(d), where the dotted black line, solid red line, and solid blue line are for Ω m = 0, Ω m = 5 MHz, and Ω m =15 MHz, respectively. We see that, due to the role played by the microwave field, the shape of ℜ/ℜ max is changed significantly. Especially, for a larger microwave field, the ℜ/ℜ max curve becomes negative for the small value of ξ (not shown here), consistent with the result obtained in Ref. [73]. Plotted in the inset of the figure is the normalized response function in momentum spacẽ ℜ/R max (i.e., the Fourier transformation of ℜ/ℜ max ) as a function of the non-dimensional wavenumber β 1 = R 0 k x (k x is the wavenumber in x direction) with Ω m = 0, 5, and 15 MHz, respectively. One sees thatR/R max has only one change in sign for Ω m = 0; however, more changes in sign arise when Ω m takes nonzero values. Such behavior ofR/R max is due to the joint action by the nonlocal Kerr nonlinearity and the microwave field, through which the MI of the plane-wave probe field may occur (see the next section).
Except for the significant dependence on the microwave field Ω m , the property of the response function depends also on another parameter, i.e., the nonlocality degree of the Kerr nonlinearity, defined by where R b is the radius of Rydberg blockade sphere, given by R b = |C s 33 /δ EIT | 1/6 [62,64,65], with δ EIT the width of EIT transparency window. One has δ EIT = |Ω c | 2 /γ 21 for ∆ 2 = 0, and δ EIT = |Ω c | 2 /∆ 2 for ∆ 2 ≫ γ 21 . With the system parameters used here, we have R b ≈ 8.34 µm. In the next section, we shall show that the structural phase transitions of the optical patterns of the system depend strongly not only on the microwave field Ω m but also on the nonlocality degree σ of the Kerr nonlinearity.

III. MODULATIONAL INSTABILITY, EMERGENCE OF OPTICAL PATTERNS AND SOLITONS
A. Modulation instability MI is a nonlinear instability of constant-amplitude continuous waves under long-wavelength perturbations, occurring in a variety of contexts where Kerr nonlinearity is attractive and local [19,22]; it can also arise in systems with repulsive but nonlocal Kerr nonlinearity when the perturbations have both long [36,38] and short [43,44,66] wavelengths. To explore the MI in the our system, we consider the MI of the plane-wave solution of the NNLS Eq. (8), i.e., where A 0 is a real number. Since any perturbation can be expanded as a superposition of many Fourier modes, we make the MI analysis of the plane wave by taking only a periodic mode as the perturbation, i.e., where a 1 and a 2 are small complex amplitudes of the perturbation and β = (β 1 , β 2 ) (β 1 ≡ R 0 k x , β 2 ≡ R 0 k y ; k x k y are wavenumbers in x and y directions, respectively) is non-dimensional 2D wavevector and λ is the growth rate of the perturbation, to be determined yet.
Substituting the perturbation solution (11) into Eq. (8) and keeping only linear terms of a 1 and a 2 , it is easy to obtain the expression of the growth rate where β = β 2 1 + β 2 2 andR( β) is the response function in momentum space [i.e., the Fourier transformation of The property of the growth rate λ depends on the plane-wave intensity A 2 0 , the shape of the response func-tionR where the microwave field Ω m plays an important role. Shown in Fig. 3(a) is the curve of −λ 2 as a function of the non-dimensional wavenumber β for the microwave That is to say, MI occurs in these shadow regions and hence the plane-wave state of the probe field is unstable. The MI will lead to a symmetry breaking of the system and hence a phase transition to new states. As a result, new optical self-organized structures (or pattern formation) appear in the system (see next section). We note that, different from the cases reported in Refs. [36,38] but similar to those considered in Refs. [43,44,66], the MI in the present system arises for the perturbation of short wavelengths.
To obtain a further understanding of the MI, Fig. 3(b) shows the real part of the growth rate, Re(λ), as a function of β and the effective probe-field intensity for Ω m = 10 MHz, where α = − ℜ( ζ)d 2 ζ is a parameter characterizing the role by the nonlocal Kerr nonlinearity. The colorful region in the figure is the one where Re(λ) > 0 and hence MI occurs. Fig. 3(c) shows Re(λ) as a function of β and Ω m for I eff = 20, with the colorful region denoting the one where the MI happens. From these results we see that the MI depends not only on the effective probe-field intensity I eff but also on the microwave field Ω m , which provides ways to manipulate the MI and thereby the emergence of the optical patterns in the system.

B. Pattern formation controlled by the Kerr nonlinearity and the microwave field
We now turn to consider the outcome of the MI in the system. Note that in the absence of the mi-crowave field the system is reduced to a three-level one (i.e. conventional Rydberg-EIT) and the atomatom interaction HamiltonianĤ vdw owns only the term N a d 3 r ′Ŝ 33 (r ′ , t)V s 33 (r ′ − r)Ŝ 33 (r, t); however, in the presence of the microwave field, the state |4 may have a significant population and hence it plays an important role for the dynamics of the probe field. In this caseĤ vdw owns four terms, which may be comparable through the tuning of the system parameters. As a result, the nonlinear response function G in the envelope Eq. (7)  are repulsive, but the one contributed by G s 44 is attractive. Therefore, depending on system parameters and based on the competition among these four terms in G, the total Kerr nonlinearity of the system may be type of self-defocusing or self-focusing, which means that the system may support very rich nonlinear structures after the occurrence of the MI, including the emergence of various optical patterns and solitons. Generally, when the repulsive part (contributed by G s 33 , G d 34 , and G e 34 ) plays a dominant role over the attractive part (contributed by G s 44 ), the MI results in the formation of optical patterns; on the contrary, when the attractive part is dominant over the repulsive part, the MI gives rise to the formation of bright solitons.
As a first step, we focus on the case of pattern formation, for which the whole Kerr nonlinearity must be type of self-defocusing. This can be realized by choosing suitable system parameters to make the repulsive part in G (i.e., G s 33 , G d 34 , and G e 34 ) is larger than the attractive part (i.e., G s 44 ). In fact, the system parameters given at the final part of Sec. II A fulfill such requirement. Except for these parameters, other three parameters, i.e., I eff (the effective probe-field intensity), σ (the nonlocality degree of the Kerr nonlinearity), and Ω m (the microwave field), play significant roles for determining the types of optical patterns in the system. Based on such consideration and for obtaining the optical patterns, we seek the groundstate solution of the system by a numerical simulation solving Eq. (8) via an imaginary evolution and split-step Fourier methods [92], for which the total energy of the system is minimum. The initial condition used in the simulation is the plane wave (10), perturbed by a random noise. Shown in Fig. 4(a) is the phase diagram describing the phase transition of self-organized optical structures, which are controlled by the effective intensity of the probe field I eff = αA 2 0 and the microwave field Ω m . The dashed lines in the figure are boundaries of different phases. When obtaining the phase diagram, the nonlocality degree of the Kerr nonlinearity, i.e., σ = R b /R 0 , is fixed to be 1. are changed in the following ways: (i) from the homogeneous state 1 to the hexagonal lattice 2 ; (ii) from the hexagonal lattice 2 to the type I square lattice 3 ; (iii) from the type I square lattice 3 to the type II square lattice 4 ). Here 1 , 2 , 3 , and 4 represent regions of the homogeneous state, hexagonal lattice, type I square lattice, and type II square lattice, respectively.
To be more concrete, we give several examples for illustrating the optical lattice patterns that correspond to the self-organized structures indicated in the different regions of Fig. 4(a). Fig. 4(b) shows a hexagonal lattice pattern, where the amplitude |u| of the probe field is normalized as a function of ξ = x/R 0 and η = y/R 0 ; it is obtained by taking Ω m = 10 MHz and I eff = 15, located in the region 2 of Fig. 4(a). Such hexagonal lattice pattern was found by Sevinli et al. [66] where no microwave dressing is used (i.e., Ω m = 0); in this case the hexagonal lattice pattern is the only one that can be obtained via the MI of the homogeneous (plane wave) state.
Plotted in Fig. 4(c) is the optical pattern by taking |u| as a function of ξ and η, for Ω m = 13 MHz and I eff = 25 [which locates in the region 3 of Fig. 4(a)]. We see that in this case a new optical structure, called the type I square lattice, emerges. Obviously, such new optical structure, which does not exist if the microwave dressing is absent, arises due to the symmetry breaking induced by the introduction of the microwave field. Fig. 4(d) gives the result for the optical pattern with increasing microwave field and the effective intensity of the probe field, by taking Ω m = 15 MHz and I eff = 35 [which is in the region 4 of Fig. 4(a)]. One sees that in this situation another type of optical structure, called the type II square lattice, appears. By comparing the type I square lattice pattern of Fig. 4(c), we see that there is an angle difference (around 45 • ) between the type I and type II square lattices; furthermore, there are also differences for the normalized probe-field amplitudes and the lattice constants between these two types of square lattice patterns (for detail, see Table I below).

C. Pattern formation controlled by the nonlocality degree of the Kerr nonlinearity and the microwave field
To explore the structural phase transition of the optical patterns further, we now fix the effective probe-field intensity (I eff = 35) but take the nonlocality degree of the Kerr nonlinearity σ and the microwave field Ω m as control parameters. Similar to the last subsection, we seek the spatial distribution of the probe field for which the total energy (14) of the system is minimum, through a numerical simulation of Eq. (8).
Shown in Fig. 5(a) is the phase diagram of the structural transition of optical patterns, where different regions (phases) are obtained by changing the values of σ and Ω m , separated by dashed lines (i.e., boundaries of different phases). We see that several structural transitions [i.e., from the homogeneous state 1 to the hexagonal lattice 4 , from the hexagonal lattice 4 to the type I square lattice 3 , and from the type I square lattice 3 to the type II square lattice 2 ] of the optical patterns arise when σ and Ω m are varied.
We also give several examples for illustrating the optical patterns corresponding to the self-organized structures indicated in the different regions of Fig. 5(a). Fig. 5(b) shows a hexagonal lattice pattern, obtained by taking the normalized amplitude of the probe field |u| as a function of ξ = x/R 0 and η = y/R 0 , for Ω m = 10 MHz and σ = 2 [located in the region 4 of Fig. 5(a)]. Shown in Fig. 5(c) is the optical pattern for Ω m = 12 MHz and σ = 1 [located in the region 3 of Fig. 5(a)]; one sees that in this case the lattice pattern is a type I square lattice, which is absent without microwave field.  Fig. 5(d) is the optical pattern with Ω m = 18 MHz and σ = 1, which is in the region 2 of Fig. 5(a); in this case the type II square lattice structure appears.

Illustrated in
To see clearly the differences between the two types of square lattice patterns, a quantitative comparison is made for the normalized probe-field amplitude |u|, microwave field Ω m , effective probe-field intensity I eff , and lattice constant l (i.e., the distance between the maximums of two adjacent optical spots) between the type I and type II square lattice patterns obtained in Fig. 4 and Fig. 5 by taking the nonlocality degree of the Kerr nonlinearity σ = 1, with the result presented in Table I. We see that: (i) the lattice constant l of the type I square lattice pattern is larger than that of the type II one; (ii) comparing Fig. 5(c) with Fig. 5(d) [Fig. 4(c) with Fig. 4(d)], the lattice constant l is larger for smaller microwave field Ω m . The physical reason for such differences is that the nonlocal nonlinear response function ℜ/ℜ max has a sig-  Fig. 4 and Fig. 5, which are the key results of this work, we see that, in the parameter domains considered here, the system supports three types of selforganized optical structures (i.e., the hexagonal lattice, the type I and type II square lattices), and their phase transitions can be controlled by actively manipulating the microwave field (Ω m ), the effective probe-field intensity (I eff ), and the nonlocality degree of the Kerr nonlinearity (σ). The basic physical mechanism of the MI and the formation of the optical patterns found here may be understand as follows. When the plane-wave probe field with a finite amplitude is applied to and propagates in the Rydberg atomic gas along the z direction, the nonlocal Kerr nonlinearity coming from the Rydberg-Rydberg interaction brings a phase modulation to the probe field; due to the role played by the diffraction in the transverse (i.e., x and y) directions, the phase modulation is converted into amplitude modulation. Because of the joint effect of the phase and amplitude modulations, in some parameter domains the probe field undergoes MI and reorganizes its spatial distribution and hence the formation of optical patterns occurs. The emergence of the different self-organized structures (i.e. the hexagonal and square lattice optical patterns) and related phase changes are originated from the spatial symmetry breaking of the system. To understand this and illustrate furthermore the differences between various optical lattice patterns, a detailed theoretical analysis on the ground-state energy of the system for different spatial distributions of the probe-field intensity is given in Appendix C.

D. Formation of nonlocal spatial optical solitons
Spatial optical solitons, i.e., localized nonlinear optical structures resulting from the balance between nonlinearity and diffraction, can form through the MI of plane waves [33,93]. However, a necessary condition for the formation of an optical soliton is that the Kerr nonlin-earity in the system should be of the type of self-focusing. As indicated in Sec. III B, due to the microwave dressing in our system there exist four kinds of nonlocal Kerr nonlinearities, which are described by the four response functions (i.e., G s 33 , G s 44 , G d 34 , G e 34 ), and one of them (i.e., G s 44 ) is attractive. Therefore, it is possible to make the total Kerr nonlinearity of the system to be a self-focused one if suitable system parameters are chosen.
To confirm the MI, a numerical simulation based on an imaginary-time propagation method is carried out by solving the NNLS equation (8) with the above parameters. Shown in Fig. 6(b) is the spatial distribution of the probe-field envelope when it propagates 10 diffraction length (i.e., s ≡ z/2L diff = 5), by taking |u| (the normalized probe-field amplitude) as a function of nondimensional coordinates ξ = x/R 0 and η = y/R 0 , for the microwave field Ω m = 10 MHz and the nonlocality degree of the Kerr nonlinearity σ = 1. The initial condition used in the simulation is |u| = 1.3 sech[(ξ 2 + η 2 ) 1/2 ] [ Fig. 6(a)]. We see that a nonlocal spatial optical soliton can indeed form in the system and it is quite stable during propagation. Note that solitons can be also generated by using random initial conditions. Fig. 6(d) shows the spatial distribution of a soliton when it is created and propagates 10 diffraction length, for which the initial condition used is of the form |u| = 1 + 0.05f , where f is Gaussian noise [Fig. 6(c)].

IV. SUMMARY
In this work, we have proposed a scheme for the realization of optical pattern formation and spatial solitons via a Rydberg-EIT. Through the use of a microwave dressing, we have shown that the nonlocal Kerr nonlinearity of the system can be manipulated actively and its magnitude can be enhanced significantly. Based on such nonlocal and tunable Kerr nonlinearity, we have demonstrated that a plane-wave probe field can undergo MI and spontaneous symmetry breaking, and thereby var- ious self-organized optical patterns may emerge in the system. In particular, we have found that a hexagonal lattice pattern, which appears after the MI when the repulsive part of the nonlocal nonlinear response function is larger than its attractive part, may develop into several types of square lattice patterns when the microwave field is applied and tuned actively. Furthermore, through the MI the formation of nonlocal spatial optical solitons has also been found when the attractive part of the nonlocal nonlinear response function is dominant over its repulsive part. Different from the results reported before, the optical patterns and nonlocal optical solitons discovered here can be flexibly adjusted and controlled through the change of the effective probe-field intensity, nonlocality degree of the Kerr nonlinearity, and the strength of the microwave field. Our work opens a way for a versatile control of the self-organizations and structural phase transitions of laser light based on microwave-dressed Rydberg gases, which may have potential applications in optical information processing and transmission. The optical Bloch equation (3) for the one-body density matrix (DM) elements ρ αβ (r ′ , t) = Ŝ αβ (r ′ , t) [84] reads for diagonal matrix elements, and , and ∆ 4 = ω c + ω p + ω m − (ω 4 − ω 1 ) the one-, two-, and three-photon detunings, respectively.
From left hand side of the above equations, we see that, different from conventional EIT, there are many terms coming from the Rydberg-Rydberg interaction. One class of them involves the van der Waals interaction between the two atoms located respectively at positions r ′ and r and excited to the same Rydberg state [i.e., V s 33 (r ′ − r) and V s 44 (r ′ − r)]; the other class involves the direct non-resonant van der Waals interaction and the resonant exchange dipole-dipole interaction between the two atoms excited to different Rydberg states [i.e., V d 34 (r ′ − r) and V e 34 (r ′ − r)]. Notice that, although the above equations describe the time evolution of the one-body DM elements ρ αβ (r, t), they involve two-body DM elements ρ αβ,µν (r ′ , r, t) = Ŝ αβ (r ′ , t)Ŝ µν (r, t) due to the Rydberg-Rydberg interaction. Similarly, equations of motion for the two-body DM elements (not shown here for saving space) involves three-body DM elements, etc. To solve such many-body problem, a suitable truncation for the infinite equation chain concerning many-body correlations is necessary, and a self-consistent calculation beyond mean-field approximation is needed, which have been developed recently [86][87][88][89]. Based on such approach, we can solve the MB Eqs. (3) and (4) by using an asymptotic expansion in a standard way.
Notice also that in these equations there exists two types of nonlinearities. One of them (characterized by the terms like Ω * p ρ 31 ) arises from the photon-atom interaction due to the coupling between the probe field and atoms, which results in local Kerr nonlinearity if the two-photon detuning ∆ 3 is not zero [86][87][88][89][90][91]; another one arises from the Rydberg-Rydberg interaction (characterized by the terms involving the two-body interactions potentials V s 33 , V s 44 , V d 34 , and V e 34 ), which results in nonlocal Kerr nonlinearity. It is the nonlocal Rydberg-Rydberg interaction that makes the Rydberg-EIT interesting and typical on the study of nonlocal nonlinear optics.

Solutions for density matrix elements
We are interested in stationary states of the system, and hence the time derivatives in the MB equations (3) and (4) can be neglected (i.e., ∂/∂t = 0), which is valid if the probe, control, and microwave fields have large time durations. We adopt the method developed in Refs. [86][87][88][89] to firstly solve the Bloch equation (3) under the condition of Rydberg-EIT. We assume that all the atoms are initially prepared in the ground state |1 .
With these results, the third-order equations of the twobody DM elements (which are too lengthy thus omitted here) can be solved, which have the solution of the form ρ αβ,µν are functions of r ′ − r. Notice that, when solving the equations of the twobody DM elements, some three-body DM elements are involved. To make the equations closed, the method developed in Ref. [86][87][88][89] has been exploited to factorize the three-body DM elements into one-and two-body ones.
Solutions of the equations of density matrix elements above the third-order approximation can also be obtained in a similar way. However, we can stop here since in this work we are interested only in the Kerr effect up to third order. Thereby, based on the first-, second-, and third-order solutions given above, we may obtain the explicit expression of ρ21 = ρ  The optical susceptibility of the probe field is defined by χ = Na(e · p 12 )ρ21/(ǫ0Ep). Based on the result in the Appendix A, we have χ = χ (1) + χ respectively, where Dm and A are given by (A4) and (A7). Due to the Rydberg-Rydberg interaction, the system also supports the nonlocal third-order nonlinear susceptibility where V l αβ ≡ V l αβ (r ′ ) (l = s, d, e) and a αβ,µν ≡ a αβ,µν (r ′ , t). Substituting the result of ρ21 obtained in the Appendix A into the Maxwell equation (4), we obtain the envelope equation for Ωp, which has the form of the nonlocal nonlinear Schrödinger (NNLS) equation where W1 = Aκ12/D and R(r) = α=3,4 R s αα (r) + γ=d,e R γ 34 (r) with R s 33 (r) = κ12Ωcd41Naa which are contributed by V s 33 , V s 44 , V d 34 , and V e 34 , respectively. The coefficient W1 characterizes the local self-phase modulation and the nonlinear response function R(r) characterizes nonlocal self-phase modulation of the probe field.
Appendix C: Ground-state energy analysis on the hexagonal and square lattice patterns Here we make a detailed analysis to illustrate the reason why the system supports only the hexagonal, type I, and type II square lattices, found in the parameter range we used. To this end, we assume that the solution of the NNLS Eq. (8) is a superposition of many Fourier modes, i.e., where βj = (β1j , β2j ) are non-dimensional wavenumbers, and Aj (j = 1, 2, ..., N ) are complex amplitudes which are functions of s, ξ, and η when the Kerr nonlinearity of the system plays a significant role.
Because the system has a rotation symmetry, generally the value of N can take a very large value. However, due to the joint action of the Kerr nonlinearity and diffraction, the system undergoes MI for some wavenumbers and hence symmetry breaking, by which only several modes are kept at end. To see this, we assume all the modes in (C1) are unstable ones, and satisfy | βj | ≈ βcr where βcr is the first wavenumber of the unstable band of the MI [e.g., in the shadow regions of Fig. 3(a)]. (b) Square lattice, for which β 1 · β 2 = 0; (c) Hexagonal lattice, for which β 1 + β 2 + β 3 = 0. All the wavevectors have the same module (i.e. | β j | ≈ βcr; βcr is the first wavenumber of the unstable band of the MI).
Any complicated periodic pattern are made of the most periodic basic pattern, i.e., an array of parallel stripes (or called rolls). For example, a square lattice pattern is made of two kinds of parallel stripes with equal amplitudes but different orientations (the angle difference between the directions of the two kinds of parallel stripes is 90 • ), a hexagonal pattern is made of three kinds of parallel stripes with equal amplitudes but different orientations (the angle difference between any two kinds of parallel stripes is 120 • ).
Shown in panels (a), (b) and (c) of Fig. 7 are nondimensional wavevectors for the parallel stripes, square lattice (for which β1 · β2 = 0), and hexagonal lattice (for which β1 + β2 + β3 = 0). Note that here all the wavevectors have the same module (i.e., | βj | ≈ βcr; βcr is the first wavenumber of the unstable band of the MI).
Note that the response function ℜ has an imaginary part, which however is very small due to the EIT effect and hence is negligible in the calculation below. For hexagonal lattice pattern, we assume the solution of the NNLS equation (8)  (Dj e i β j · ζ + c.c.) , where ρ0 = A 2 0 ; µ = − ℜ( ζ)d 2 ζ; Dj (j = 1, 2, 3) are complex constants called modulation amplitudes; the wavevectors βj fulfill the condition β1 + β2 + β3 = 0 with |β1| = |β2| = |β3| = βcr. Note that for obtaining the solution with minimal energy, the phase φ must be homogeneous (i.e. a real constant) [95,96]. Inserting (C3) and (C4) into Eq. (C2) and expanding Dj in Taylor series, we obtain the energy of the system for the |Dj | 2 + high order term, whereR(βcr) is the value of the response function ℜ in momentum space for β = βcr, V = d 2 ζ is the volume of the system. The ground-state energy can be acquired by solving the equations ∂EHex/∂Dj = 0 (j = 1, 2, 3). Then, we obtain its expression for D1 = D2 = D3 = D. The ground-state energy for square lattice pattern can be obtained in a similarly way, which reads ESqu = V ρ0β 2 cr (D 2 + 3D 4 ) + V ρ 2 0R (βcr)D 2 .
(C7) Shown in Fig. 8(a) is the ground-state energy E as a function of modulation amplitude D, with parameters βcr = 4.3, I eff = 15, Ωm = 10 MHz, and σ = 1. The solid red line in the figure is for the ground-state energy EHex of the hexagonal lattice; the dotted blue line is the groundstate energy ESqu for the square lattice. We see that the minimal energy of the system occurs at D = 0.32 (where Emin = EHex,min = −0.78×10 3 ), which means that the hexagonal lattice pattern is preferred to emerge in the system. The case shown in Fig. 8(b) is similar to Fig. 8(a), but the parameters are taken as βcr = 4.3, I eff = 25, and Ωm = 15 MHz. We see that in this case the minimal energy of the system occurs for D = ±0.28 (where Emin = ESqu,min = −0.81 × 10 3 ), which means that the system favors the emergence of the square lattice pattern. These results are consistent with the ones found in Fig. 4 and Fig. 5.