1. Intruduction

The study of optical Kerr effect, i.e. nonlinear response of optical materials to applied light field, is one of main topics in nonlinear optics because it is essential for the realization of most nonlinear optical processes [1]. Optical Kerr effect has also found many new applications, including nonlinear and quantum controls of light fields, quantum nondemolition measurement, all-optical deterministic quantum logic, single-photonic switches and transistors, and so on [2, 3]. However, Kerr effect is usually produced in passive optical media such as glass-based optical fibers, in which far-off resonance excitation schemes are employed to avoid serious optical absorption. As a result, the Kerr nonlinearity in passive optical media is weak and hence to obtain a significant Kerr effect a long propagation distance or a high light intensity is required.

In recent years, many efforts have focused on the study of electromagnetically induced transparency (EIT) [4, 5]. Light propagation in EIT media possesses many striking features, including the suppression of optical absorption, the reduction of group velocity, and an enhancement of Kerr nonlinearity [4–6], by which many important applications (e.g. quantum memory, highly efficient four-wave mixing, optical clocks, and slow-light solitons, etc.) are possible [4–13]. However, the largest Kerr nonlinearity, obtained in conventional EIT media [14], is still too small for nonlinear optics at single-photon level [15].

Recently, much attention has been paid to the investigation of cold Rydberg gases [15–17], i.e. highly excited atoms with very large principal quantum number. Due to their long lifetime and large electric dipole moment, Rydberg atoms have many practical applications [18]. Especially, the strong and controllable atom-atom interaction in Rydberg gases (called Rydberg-Rydberg interaction for short [17–19]) brings many intriguing aspects that can be used to design quantum gates and simulate strongly correlated quantum many-body systems, etc [15–20].

Since the first experiment reported in 2007 [21], considerable achievements have been made on the EIT in cold Rydberg gases. Experimental [22–27] and theoretical [28–35] works showed that EIT can be used not only for coherent optical detection of Rydberg atoms, but also for obtaining giant Kerr nonlinearity [36]. Different from conventional EIT media, the giant Kerr nonlinearity in Rydberg-EIT systems comes from the strong Rydberg-Rydberg interaction between atoms, which can be many orders of magnitude larger than those obtained before. These studies [21–35,35] opened a new and important avenue for the nonlinear optics at single-photon level [3, 37, 38].

However, all studies up to now on the Kerr nonlinearity in Rydberg-EIT systems are limited to the third-order one. Because of the requirement of many applications in quantum and nonlinear optics, such as highly efficient six-wave mixing [39, 40], three-photon phase gates [41, 42], multi-photon entangled states and Schrödinger cat states of light [43, 44], and stabilization of spatial optical solitons [6, 45], it is necessary to find a giant high-order Kerr nonlinearity that can be realized at very weak light level [46].

In this article, we make a systematic theoretical investigation on the optical Kerr effect in a cold Rydberg atomic system via EIT. By using an approach beyond mean-field theory [30, 35, 47] on the correlators of one-body, two-body, and three-body based on a second-order ladder approximation, we show that the system possesses not only an enhanced third-order nonlinear optical susceptibility, but also a giant fifth-order nonlinear optical susceptibility, which has a cubic dependence on atomic density and can be arrived at the order of magnitude 10⁻¹¹ m⁴V⁻⁴. Our results demonstrate that both the third-order and the fifth-order nonlinear optical susceptibilities consist of two parts. One part is contributed by photon-atom interaction and another part comes from the Rydberg-Rydberg interaction. The Kerr nonlinearity induced by the Rydberg-Rydberg interaction plays a leading role at high atomic density. We find that the fifth-order nonlinear optical susceptibility in the Rydberg-EIT system may be five orders of magnitude larger than that obtained in traditional EIT systems, which may have promising applications in light and quantum information processing and transmission at weak-light level.

Before preceding, we note that third-order Kerr nonlinearity was considered in [30, 35, 47] where nonlinearity is estimated by using the approach beyond mean-field theory, and in [47] where a second-order ladder approximation is adopted to investigate coherent population trapping in Rydberg atoms. Furthermore, the interaction between Rydberg atoms via EIT was also suggested in [31, 32]. However, our work is different from [30–32, 35, 47]. First, no fifth-order Kerr nonlinearity was considered in [30–32, 35, 47] (see also recent review [15]). Second, our study (see below) shows that both the photon-atom interaction and the Rydberg-Rydberg interaction have significant contributions to the Kerr nonlinearities (including third-order and fifth-order ones), but the contribution of the photon-atom interaction was overlooked in [30–32, 35, 47].

The remainder of the article is arranged as follows. In Sec. 2, the physical model of the Rydberg-EIT system under study is described. In Sec. 3, a perturbation expansion is used to solve the equations of motion of many-body correlators. In Sec. 4, explicit expressions of the nonlinear optical susceptibilities are presented. Finally, the last section contains a summary of the main results of our work.

2. Model

We consider an ensemble of lifetime-broadened three-level atomic gas with a ladder-type level configuration, shown schematically in Fig. 1(a). We assume the atomic gas are loaded into a magneto-optical trap and works at a ultracold temperature so that their center-of-mass motion is negligible. A weak probe field of angular frequency ω_p (half Rabi frequency Ω_p) couples to the transition between |1〉 and |2〉, and a strong control field of angular frequency ω_c (half Rabi frequency Ω_c) couples to the transition between |2〉 and |3〉. The electric field of the system can be written as E(r, t) = E_p(r, t) + E_c(r, t) with E_p(r, t) = e_p ℰ_p exp[i(k_p · r − ω_pt)] + c.c. and E_c(r, t) = e_c ℰ_c exp[i(k_c · r − ω_ct)] + c.c., where c.c. represents complex conjugate and k_p, e_p and ℰ_p (k_c, e_c and ℰ_c) are respectively the wavevector, polarization unit vector and amplitude of the probe field (control) field. The upper state |3〉 is chosen as a Rydberg state, which is assumed to be |3〉 = |60S_1/2〉 for simplicity. The atoms in Rydberg states (i.e. Rydberg atoms) have many exaggerated properties, including long radiative lifetime, large electric dipole moment, strong Rydberg-Rydberg interaction, and so on [16].

Under electric-dipole and rotating-wave approximations, in the Heisenberg picture the Hamiltonian of the atomic gas including the Rydberg-Rydberg interaction is given by Ĥ_H = 𝒩_a ∫ d³rℋ̂_H (r, t), with 𝒩_a the atomic density and ℋ̂_H (r, t) the Hamiltonian of the atom at position r of the form

 

Fig. 1 (a) Excitation scheme of the three-level ladder system, in which the probe field with angular frequency ω_p and half Rabi frequency Ω_p couples the levels |1〉 and |2〉, and the control field with angular frequency ω_c and half Rabi frequency Ω_c couples the levels |2〉 and |3〉. Δ₂ and Δ₃ are one- and two-photon detunings, respectively; Γ₁₂ (Γ₂₃) is the spontaneous emission decay rate from |2〉 to |1〉 (|3〉 to |2〉). (b) The long-range interaction potential of ⁸⁷Rb atoms $V(r_{ij})=-C_6/r_{ij}^6$ (red solid line) as a function of r_ij = |r_i − r_j|, describing the interaction between the atom at r_i and the atom at r_j (represented by yellow spheres), both of which are at the Rydberg state |3〉 = |60S_1/2〉. (c) Schematic of Rydberg blockade. The long-range interaction between Rydberg atoms blocks the excitation of the atoms within blockade spheres (i.e. the ones with the boundary indicated by the yellow dashed lines) of radius R_b. In each blocked sphere only one Rydberg atom (small yellow sphere) is excited and other atoms (small blue spheres) are prevented to be excited. The orange (blue) arrow indicates the propagating direction of the probe (control) field.

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The equations of motion for one-body density matrix ρ is given by [19, 35]

From the above equations we see that there are two evident nonlinear characters in the system: (i) There is a photon-atom interaction due to the resonant coupling between the probe field and the atoms even when the Rydberg-Rydberg interaction is absent. (ii) There is an atom-atom interaction reflected by the last terms on the left hand side of Eq. 3(e) and Eq. 3(f), i.e. the two-body density matrix elements (or the two-body correlators) ρρ_33,3α (r′, r, t) ≡ 〈Ŝ₃₃(r′, t)Ŝ_3α (r, t)〉 (α = 1, 2) contributed from the Rydberg-Rydberg interaction. It is just these two different nonlinear characters that make the Rydberg-EIT system possess very interesting nonlinear optical properties. Especially, two different types of Kerr nonlinearities (one is resulted from the photon-atom interaction and another one is resulted from the Rydberg-Rydberg interaction) occur in the system, as will be illustrated below.

Our model can be easily realized by experiment. One of candidates is the laser-cooled ⁸⁷Rb atomic gas with the atomic states shown in Fig. 1(a) assigned as [23, 49]

Due to the Rydberg-Rydberg interaction, an atom in the state |3〉 would induce an energy-shift V(R) of the state |3〉 of another atom separated by distance R, which translates into an effective two-photon detuning. Then the long-range interaction energy-shift will block the excitation of all the atoms for which V(R) ≥ δ_EIT, where δ_EIT is the linewidth of EIT transmission spectrum (i.e. the width of EIT transparency window), defined by ${\mathrm{\Omega }}_c^2/{\gamma }_{12}$ for Δ₂ = 0 and ${\mathrm{\Omega }}_c^2/\left|{\mathrm{\Delta }}_2\right|$ for |Δ₂| ≫ γ₁₂ (we assumed −Δ₂/C₆ > 0). Thus the blockade sphere has the radius R_b = (|C₆/δ_EIT|)^1/6 ≃ 5.29μm [23, 31, 50]. Comparing this to the average interatomic separation obtained by $\overline{R}=(5/9){𝒩}_a^{-1/3}\simeq 2.5\mu \text{m}$ for 𝒩_a = 10¹⁰cm⁻³, the blockade effect can be obviously observed, as shown in Fig. 1(c). The system can be divided into many blockade spheres (represented by the spheres with the boundary indicated by yellow dashed line in Fig. 1(c)) and each blockade sphere contains only one Rydberg atom (represented by the small yellow sphere in Fig. 1(c)). Hence, the spatial coarse-graining distance between two nearest Rydberg atoms has size 2R_b [32].

We are interested in the optical Kerr effects, especially the third-order and fifth-order nonlinear optical susceptibilities of the system. To this aim, we need the relation between the optical susceptibility of the probe field and the density matrix elements. Since the total electric polarization intensity of the system is given by $\mathbf{P}={𝒩}_a{\sum }_{\alpha ,\beta =1}^3{\mathbf{p}}_{\alpha \beta }{\rho }_{\beta \alpha }\text{exp}\left\{i\left[{\mathbf{k}}_{\beta }-{\mathbf{k}}_{\beta }\cdot \mathbf{r}-\left({\omega }_{\beta }-{\omega }_{\alpha }+{\mathrm{\Delta }}_{\beta }-{\mathrm{\Delta }}_{\alpha }\right)t\right]\right\}$, the electric polarization intensity of the probe field reads P_p = 𝒩_a{p₁₂ρ₂₁exp[i(k_p · r − ω_pt)] + c.c.}, by which one can obtain the optical susceptibility χ_p of the probe field by using the formula P_p = ε₀χ_pe_pℰ_pexp[i(k_p · r − ω_pt)] + c.c., which yields

To obtain the explicit expression of ρ₂₁, we must solve Eqs. (3a)–(3f). However, due to the Rydberg-Rydberg interaction, we must also solve the motion of equations for the two-body correlators 〈Ŝ₃₃Ŝ₃₁〉 and 〈Ŝ₃₃Ŝ₃₂〉 simultaneously

Eqs. (5a) and (5b) have the following features. (i) The equations for two-body correlators 〈Ŝ₃₃Ŝ_3α〉 (α = 1, 2) involve many other two-body correlators (e.g. 〈Ŝ₂₃Ŝ₃₁〉, etc.). Thus one also have to solve additional equations of other two-body correlators. An explicit list of the equations of motion for two-body correlators 〈Ŝ_αβŜ_μν〉 of the system is too long and omitted here. (ii) The equations for the two-body correlators involve three-body correlators (e.g. 〈Ŝ₃₃Ŝ₃₃Ŝ₃₁〉, etc.), which obey the equations of motion for three-body correlators (which are lengthy and not listed here) and also have to be solved. Similarly, the equations of motion of the three-body correlators involve four-body correlators. Finally, one obtains an infinite hierarchy of equations of motion for the correlators of one-body, two-bodies, three-bodies, and so on. Obviously, to make the problem tractable one must truncate the hierarchy of the equations for many-body correlators by using an appropriate method. Here we adopt a second-order ladder approximation, such that for moderate atomic density the three-body correlation terms in the two-body correlator equations are factorized in the following way [30, 35, 47, 51]

3. Solutions based on perturbation expansion

Although by using the factorization method stated above the equations of motion for one-body and two-body correlations can be made to be closed and their number becomes finite, they are still nonlinear due to the coupling with the applied laser field. Fortunately, since the probe field in the EIT-based experiments [21–27] is weak we hence can make a perturbation expansion of the correlators in the powers of the Rabi frequency of probe field Ω_p [35] to solve these nonlinear equations in a systematic way. In fact, when EIT systems are weakly driven, the half Rabi frequency Ω_p is a natural expansion parameter for investigating many weak nonlinear phenomena, including ultraslow and weak-light solitons in EIT-based systems [11–13].

To investigate the optical Kerr effects in the present Rydberg-EIT system, we make the perturbation expansion [35] ${\rho }_{\alpha 1}={\mathrm{\Omega }}_p{\sum }_{l=0}{\rho }_{\alpha 1}^{(2l+1)}{\left|{\mathrm{\Omega }}_p\right|}^{2l}$, ${\rho }_{32}={\sum }_{l=1}{\rho }_{32}^{(2l)}{\left|{\mathrm{\Omega }}_p\right|}^{2l}$, ${\rho }_{\beta \beta }={\sum }_{l=0}{\rho }_{\beta \beta }^{(2l)}{\left|{\mathrm{\Omega }}_p\right|}^{2l}$ with ${\rho }_{\beta \beta }^{(0)}={\delta }_{\beta 1}{\delta }_{\beta 1}$ (α = 2, 3; β = 1, 2, 3). Substituting this expansion into the Eq. (3) for the one-body density matrix elements and comparing the expansion parameter of each power Ω_p, we obtain a set of approximated equations for ${\rho }_{\alpha \beta }^{(l)}$, which are listed in Appendix A. The approximated equations for the two-body density matrix (correlator) elements $\rho {\rho }_{\alpha \beta ,\mu \nu }^{(l)}$ after using the factorization formula (6) can also be obtained, which are listed in Appendix B. In order to acquire third-order and fifth-order nonlinear optical susceptibilities, we must solve the expansion equations from the first order to the fifth order. Although these expansion equations are lengthy and complicated, they become linear after the above expansion and thus can be solved analytically order by order in a systematical and clear way. Notice that in this work we are interested in static (or instantaneous) nonlinear optical susceptibilities, both the probe and control fields are assumed to be continuous waves. Thus the operator ∂/∂t in all equations of the correlators can be put into zero.

At the first order (l = 1), we obtain the solution ${\rho }_{21}^{(1)}=d_{31}/D$ and ${\rho }_{31}^{(1)}=-{\mathrm{\Omega }}_c/D$, with D = |Ω_c|² − d₂₁d₃₁. For the second order (l = 2), one obtains the solution

At the fourth order (l = 4), the solution is given by ${\rho }_{11}^{(4)}=a_{11}^{(4)}+{𝒩}_a{b}_{11}^{(4)}$, ${\rho }_{33}^{(4)}=a_{33}^{(4)}+{𝒩}_a{b}_{33}^{(4)}$, and ${\rho }_{32}^{(4)}=a_{32}^{(4)}+{𝒩}_a{b}_{32}^{(4)}$, with

4. Giant third-order and fifth-order nonlinear optical susceptibilities

Collecting the first-order to the fifth-order solutions of ρ₂₁ obtained in the last section, we obtain ${\rho }_{21}\simeq {\rho }_{21}^{(1)}{\mathrm{\Omega }}_p+{\rho }_{21}^{(3)}{\left|{\mathrm{\Omega }}_p\right|}^2{\mathrm{\Omega }}_p+{\rho }_{21}^5{\left|{\mathrm{\Omega }}_p\right|}^5{\mathrm{\Omega }}_p+\dots$, where ${\rho }_{21}^{(j)}$ (j = 1, 3, 5,...) are independent of Ω_p and their explicit expressions have been given in the previous section. Using the formula (4) and the definition Ω_p = (e_p · p₂₁)ℰ_p/h̄, we have

We now calculate the numerical values of the third-order and the fifth-order nonlinear optical susceptibilities in the system based on the experimental parameters as given above. The other system parameters are selected as n = 60, 𝒩_a = 3 × 10¹⁰ cm⁻³ and Δ₃ = 2π × 0.8MHz. By using the solutions presented in Sec. 3 and the susceptibility formulas given in Eq. (13), we obtain the third-order nonlinear optical susceptibilities ${\chi }_{p\alpha }^{(3)}=\text{Re}\left({\chi }_{p\alpha }^{(3)}\right)+i\text{Im}\left({\chi }_{p\alpha }^{(3)}\right)$ (α = 1, 2) and the fifth-order nonlinear optical susceptibilities ${\chi }_{p\alpha }^{(5)}=\text{Re}\left({\chi }_{p\alpha }^{(5)}\right)+i\text{Im}\left({\chi }_{p\alpha }^{(5)}\right)$ (α = 1, 2) of the system, which are listed in Table I. Note that the value at the second row and the second column in the table is the real part of ${\chi }_{p1}^{(3)}$ (i.e. $\text{Re}\left({\chi }_{p1}^{(3)}\right)=-4.4\times {10}^{-11}{\text{m}}^2{\text{V}}^{-2}$), and the value at the second row and the third column is the imaginary part of ${\chi }_{p1}^{(3)}$ (i.e. $\text{Im}\left({\chi }_{p1}^{(3)}\right)=-1.3\times {10}^{-13}{\text{m}}^2{\text{V}}^{-2}$), etc. In the last column of Table I, the physical origins of various nonlinear optical susceptibilities of the system are given.

Table 1. Real part $\text{Re}\left({\chi }_{p\alpha }^{(j)}\right)$ and imaginary part $\text{Im}\left({\chi }_{p\alpha }^{(j)}\right)$ (j = 3, 5; α = 1, 2) of the third-order and the fifth-order optical susceptibilities of the Rydberg-EIT system obtained for the realistic system parameters given in the text.

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From Table I, we can obtain the following conclusions: (i) The nonlinear optical susceptibilities in the present Rydberg-EIT system are much larger than those obtained with conventional optical media such as optical fibers. They are also larger than that obtained by using conventional EIT [14]. (ii) For the given atomic density (i.e. 𝒩_a = 3 × 10¹⁰ cm⁻³), the nonlinear optical susceptibilities contributed by the Rydberg-Rydberg interaction (i.e. ${\chi }_{p2}^{(3)}$ and ${\chi }_{p2}^{(5)}$) are three orders of magnitude greater than those contributed by the photon-atom interaction (i.e. ${\chi }_{p1}^{(3)}$ and ${\chi }_{p1}^{(5)}$). Thus at this atomic density the Rydberg-Rydberg interaction plays a leading role for the contribution of the nonlinear optical susceptibilities in the system. In particular, the fifth-order nonlinear optical susceptibility originating from the Rydberg-Rydberg interaction can reach the order of magnitude of 10⁻¹² m⁴V⁻⁴. (iii) The imaginary parts of the all nonlinear optical susceptibilities are much smaller than their corresponding real parts, which means that the nonlinear absorption can be suppressed in the nonlinear optical processes of the system. The physical reason for such suppression of the nonlinear absorption is the quantum interference effect induced by the control field (i.e. EIT effect) and also the introduction of the larger one-photon detuning Δ₂, which makes the system have the giant optical nonlinearity of dispersive type [36].

The most interesting property of the nonlinear optical susceptibilities in the system is their dependence on the atomic density and the probe-field intensity when the other physical parameters are fixed [23]. Fig. 2(a) shows the real part of the third-order nonlinear optical susceptibilities $\text{Re}\left({\chi }_{p1}^{(3)}\right)$ (black dashed-dotted line) and $\text{Re}\left({\chi }_{p2}^{(3)}\right)$ (red solid line) as functions of 𝒩_a. When plotting the figure, the parameters used are the same as those used for getting the results in the Table I except for 𝒩_a, which is now taken as a variable. From the figure we see that: (i) The Kerr nonlinearities contributed by the photon-atom interaction and the Rydberg-Rydberg interaction are comparable with atomic density 𝒩_a around 10⁸ cm⁻³, and both of them are increasing functions of 𝒩_a. (ii) For 𝒩_a less than 0.9 × 10⁸ cm⁻³, the Kerr nonlinearity by the Rydberg-Rydberg interaction is smaller than that by the photon-atom interaction. However, the both Kerr nonlinearities arrive at the same value $\text{Re}\left({\chi }_{p1}^{(3)}\right)=\text{Re}\left({\chi }_{p2}^{(3)}\right)=-1.3\times {10}^{-13}{\text{m}}^2{\text{V}}^{-2}$ at 𝒩_a = 0.9 × 10⁸ cm⁻³. (iii) When 𝒩_a is larger than 0.9 × 10⁸ cm⁻³ the Kerr nonlinearity by the Rydberg-Rydberg interaction surpasses that by the photon-atom interaction and grows rapidly as 𝒩_a increases.

 

Fig. 2 (a) $\text{Re}\left({\chi }_{p1}^{(3)}\right)$ (dashed-dotted line) and $\text{Re}\left({\chi }_{p2}^{(3)}\right)$ (red solid line) as functions of atomic density 𝒩_a. (b) $\text{Re}\left({\chi }_{p1}^{(5)}\right)$ (dashed-dotted line) and $\text{Re}\left({\chi }_{p2}^{(5)}\right)$ (red solid line) as functions of atomic density 𝒩_a. (c) Total nonlinear optical susceptibility $\text{Re}({\chi }_N)=\text{Re}\left({\chi }_p^{(3)}{\left|{ℰ}_p\right|}^2+{\chi }_p^{(5)}{\left|{ℰ}_p\right|}^4\right)$ as a function of |ℰ_p|. Lines from 1 to 3 correspond to 𝒩_a= 4 × 10¹⁰ cm⁻³, 2 × 10¹⁰ cm⁻³ and 1 × 10¹⁰ cm⁻³, respectively.

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Shown in Fig. 2(b) are the real parts of the fifth-order nonlinear optical susceptibilities $\text{Re}\left({\chi }_{p1}^{(5)}\right)$ (black dashed-dotted line) and $\text{Re}\left({\chi }_{p2}^{(5)}\right)$ (red solid line) as functions of 𝒩_a, which originate respectively from the photon-atom and the Rydberg-Rydberg interactions. We observe that both $\text{Re}\left({\chi }_{p1}^{(5)}\right)$ and $\text{Re}\left({\chi }_{p2}^{(5)}\right)$ are comparable, and they are decreasing functions of 𝒩_a. Furthermore, for 𝒩_a less than 1.86 × 10⁸ cm⁻³, the fifth-order nonlinear optical susceptibility by the Rydberg-Rydberg interaction is larger than that by the photon-atom interaction. At 𝒩_a = 1.86 × 10⁸ cm⁻³, the both optical susceptibilities become equal to have the value $\text{Re}\left({\chi }_{p1}^{(5)}\right)=\text{Re}\left({\chi }_{p2}^{(5)}\right)=1.67\times {10}^{-18}{\text{m}}^4{\text{V}}^{-4}$. When 𝒩_a is larger than 1.86 × 10⁸ cm⁻³ $\text{Re}\left({\chi }_{p2}^{(5)}\right)$ becomes to be smaller than $\text{Re}\left({\chi }_{p1}^{(5)}\right)$.

From the above results, we see that there exist various, synergetic optical nonlinearities in the Rydberg-EIT system. Due to the ${𝒩}_a^2$- and ${𝒩}_a^3$-dependence, the nonlinear optical susceptibilities contributed by the Rydberg-Rydberg interaction are sensitive to the atomic density and hence they can exceed the optical nonlinearities contributed by the photon-atom interaction for large 𝒩_a. From Fig. 2(a) and Fig. 2(b) we also see that the sign of ${\chi }_{p\alpha }^{(3)}$ is opposite to that of ${\chi }_{p\alpha }^{(5)}$ (α = 1, 2) and ${\chi }_{p\alpha }^{(5)}$ grows faster than ${\chi }_{p\alpha }^{(3)}$, which means that there exists a competition between ${\chi }_{p\alpha }^{(3)}$ and ${\chi }_{p\alpha }^{(5)}$ when 𝒩_a becomes larger.

Different from Fig. 2(a) and Fig. 2(b), where the dependence of the third-order and the fifth-order nonlinear optical susceptibilities on the atomic density 𝒩_a are illustrated, in Fig. 2(c) we show $\text{Re}({\chi }_N)=\text{Re}\left({\chi }_p^{(3)}\right){\left|{ℰ}_p\right|}^2+\text{Re}\left({\chi }_p^{(5)}\right){\left|{ℰ}_p\right|}^4$, i.e. the real part of the total nonlinear optical susceptibility of the probe field, as a function of |ℰ_p| (ℰ_p is the envelope of the probe field) for several different atomic density. Lines from 1 to 3 in the figure are for 𝒩_a taken to be 6 × 10¹⁰ cm⁻³, 4 × 10¹⁰ cm⁻³, and 3 × 10¹⁰ cm⁻³, respectively. We observe that, for all 𝒩_a, Re(χ_N) grows fast initially, then arrives a peak value, and finally decreases as |ℰ_p| increases. We also observe that the higher the atomic density, the faster Re(χ_N) arrives to its peak value, which is due to the effect coming from the fifth-order nonlinear susceptibilities. We stress that the maximum value of the probe field used in Fig. 2(c) is |ℰ_pmax| = 120 V/m, which is within the validity domain of the perturbation theory used above because |Ω_pmax/Ω_c| ≃ 0.1.

For comparison, in Fig. 3(a) (Fig. 3(b)) we show the third-order (fifth-order) nonlinear optical susceptibility obtained for several typical physical systems, including optical fibers [54], conventional EIT system [14], active Raman gain (ARG) system [55,56], and the present Rydberg EIT system, with the value of ${\chi }_p^{(3)}$ ( ${\chi }_p^{(5)}$) indicated by the yellow solid circle, green solid circle, blue solid circle, and red solid circle, respectively. The black vertical line at each solid circle indicates the range of the nonlinear optical susceptibility for the atomic density varying from 10⁹ cm⁻³ to 6 × 10¹⁰ cm⁻³. The Grey shaded area in the lower part of the figure symbolizes the range of the nonlinear optical susceptibilities for optical fibers. We see that the third-order and the fifth-order nonlinear optical susceptibilities obtained by using the present Rydberg EIT system have the highest values in comparison with the other systems [14,54–56]. Especially, the third-order nonlinear optical susceptibilities in the present Rydberg EIT system can reach the order of magnitude of 10⁻⁷ m²V⁻² for atomic density 𝒩_a = 6 × 10¹⁰ cm⁻³, which agrees fairly with reported experimental and theoretical results [57]. If the atomic density increases to 𝒩_a = 5.0 × 10¹² cm⁻³ (used in [14]), we obtain ${\chi }_p^{(3)}=4.6\times {10}^{-3}{\text{m}}^2{\text{V}}^{-2}$. Thus the third-order nonlinear optical susceptibility in the present Rydberg-EIT system is five orders of magnitude larger than that obtained in the conventional EIT system, where ${\chi }_p^{(3)}=7\times {10}^{-8}\hspace{0.17em}{\text{m}}^2{\text{V}}^{-2}$ [14]. Furthermore, The fifth-order nonlinear optical susceptibilities in the present Rydberg EIT system can reach the order of magnitude of 10⁻¹¹ m⁴V⁻⁴, which is five orders of magnitude larger than that of the conventional EIT system and the ARG systems for the peak atomic density 𝒩_a = 6 × 10¹⁰ cm⁻³. The physical reasons for such giant third-order and fifth-order optical Kerr effects obtained in the present Rydberg-EIT system are the cooperative response of a large number of atoms, the quantum interference contribution from the EIT, and the Rydberg-Rydberg interaction in the system. Such giant third-order and the fifth-order optical nonlinearities are very promising for the investigation of many nonlinear optical processes not possible by using conventional optical media up to now.

 

Fig. 3 (a) ((b)) Third-order (Fifth-order) nonlinear optical susceptibility ${\chi }_p^{(3)}$ ( ${\chi }_p^{(5)}$) for optical fibers [54] (yellow solid circle), conventional EIT [14] (green solid circle), ARG system [55,56] (blue solid circle), and the present Rydberg-EIT system (red solid circle), respectively. The black vertical line at each solid circle indicates the range of the nonlinear optical susceptibility for the atomic density varying from 10⁹ cm⁻³ to 6×10¹⁰ cm⁻³. The Grey shaded area symbolizes the range of the nonlinear optical susceptibilities for optical fibers.

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Note that when taking Δ₂ ≪ Γ₁₂, we can also gain another type of optical nonlinearity of the system. For instance, if we choose 𝒩_a = 3 × 10¹⁰ cm⁻³, Ω_c = 2π × 16 MHz, Δ₂ = 1 kHz and Δ₃ = 0 (the blockade sphere radius R_b = 3.26μm) and the other parameters the same as those given above we obtain ${\chi }_p^{(3)}=\left(-7.07+i3.4\right)\times {10}^{-10}\hspace{0.17em}\hspace{0.17em}{\text{m}}^2{\text{V}}^{-2}$ and ${\chi }_p^{(5)}=(2.03+0.8)\times {10}^{-14}{\text{m}}^4{\text{V}}^{-4}$. In this situation, imaginary parts of ${\chi }_p^{(3)}$ and ${\chi }_p^{(5)}$ have the same orders of magnitude as their corresponding real parts, i.e. the system displays an optical nonlinearity of dissipative type. The large imaginary part in ${\chi }_p^{(3)}$ and ${\chi }_p^{(5)}$ will result inevitably in high photon loss for the nonlinear behavior in the system. Notice that the dissipative-type third-order nonlinear optical susceptibility was considered in [30], in which the photon-atom interaction has negligible contribution to optical susceptibilities. The reasons are the following. (i) The Rydberg state has a long lifetime (i.e. Γ₂₃ is very small); (ii) The two-photon detuning Δ₃ was taken to be zero. As a result, in [30] only the Rydberg-Rydberg interaction contributes to the nonlinear optical susceptibilities. In our work, both the third-order and the fifth-order nonlinear optical susceptibilities have been considered by taking a non-zero Δ₃, and hence both the photon-atom interaction and the Rydberg-Rydberg interaction play significant roles for the optical nonlinearity of the system.

5. Summary

In this article, we have investigated the optical Kerr effects in an ensemble of cold Rydberg atoms via EIT. By using an approach beyond mean-field approximation, we have proved that the system can possess not only an enhanced third-order nonlinear optical susceptibility, which has a ${𝒩}_a^2$-dependence, but also a giant fifth-order nonlinear optical susceptibility, which has ${𝒩}_a^2$- and ${𝒩}_a^3$-dependence. We have demonstrated that both the third-order and the fifth-order nonlinear optical susceptibilities consist of two parts, which are contributed respectively by the photon-atom interaction and the strong Rydberg-Rydberg interaction. The Kerr nonlinearity induced by the Rydberg-Rydberg interaction plays a leading role for high atomic density. We have found that the fifth-order nonlinear susceptibility in the present Rydberg-EIT system may be five orders of magnitude larger than that obtained in traditional EIT systems, which may have promising applications in light and quantum information processing and transmission at few photon level. The theoretical method proposed here is systematic and can be used to calculate other high-order (e.g. the seventh-order, ninth-order, etc.) nonlinear optical susceptibilities [58], and for other Rydberg states (e.g. nD states for which the dipole-dipole interaction is angular dependent [59, 60]). It can be also generalized to investigate non-instantaneous optical Kerr effects in cold, interacting Rydberg systems.