Abstract

We investigate the optical Kerr nonlinearities of an ensemble of cold Rydberg atoms under the condition of electromagnetically induced transparency (EIT). By using an approach beyond mean-field theory, we show that the system possesses not only enhanced third-order nonlinear optical susceptibility, but also giant fifth-order nonlinear optical susceptibility, which has a cubic dependence on atomic density. Our results demonstrate that both the third-order and the fifth-order nonlinear optical susceptibilities consist of two parts, contributed respectively by photon-atom interaction and 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. The results obtained may have promising applications in light and quantum information processing and transmission at weak-light level.

© 2016 Optical Society of America

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 ωp (half Rabi frequency Ωp) couples to the transition between |1〉 and |2〉, and a strong control field of angular frequency ωc (half Rabi frequency Ωc) couples to the transition between |2〉 and |3〉. The electric field of the system can be written as E(r, t) = Ep(r, t) + Ec(r, t) with Ep(r, t) = ep p exp[i(kp · rωpt)] + c.c. and Ec(r, t) = ec c exp[i(kc · rωct)] + c.c., where c.c. represents complex conjugate and kp, ep and p (kc, ec and 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〉 = |60S1/2〉 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 ĤH = 𝒩a ∫ d3rℋ̂H (r, t), with 𝒩a the atomic density and ℋ̂H (r, t) the Hamiltonian of the atom at position r of the form

^H(r,t)=α=13h¯ωαS^αα(r,t)h¯[ΩpS^12(r,t)+Ωp*S^21(r,t)+ΩcS^23(r,t)+Ωc*S^32(r,t)]+𝒩ad3rS^33(r,t)h¯V(rr)S^33(r,t),
where h̄ωα the eignenergy of the state |α〉, Ωp = (ep · p21)p/h̄ and Ωc = (ec · p32)c/h̄ are respectively the half Rabi frequencies of the probe and control fields with pαβ the electric dipole matrix element associated with the transition from |β〉 to |α〉, Ŝαβ = |β〉 〈α|ei[(kβkα)·r−(ωβωαβ−Δα)t] are transition operators (α, β = 1, 2, 3) satisfying the commutation relation
[S^αβ(r,t),S^μν(r,t)]=(δανS^μβ(r,t)δμβS^αν(r,t))δrr,
where δαβ is Kronecker symbol. The last term on the right side of Eq. (1) is the contribution of the Rydberg-Rydberg interaction, i.e. the Rydberg atom at position r interacts with the Rydberg atom at position r′ described by the long-range interaction potential V(r′r). Because the Rydberg-Rydberg interaction results in a Rydberg blockade [17–19], the integration region of the radial coordinate r′ in the integral in the last line of Eq. (1) is from 2Rb to infinity, where Rb is the radius of Rydberg blockade sphere [see Fig. 1(c)].

 

Fig. 1 (a) Excitation scheme of the three-level ladder system, in which the probe field with angular frequency ωp and half Rabi frequency Ωp couples the levels |1〉 and |2〉, and the control field with angular frequency ωc and half Rabi frequency Ωc couples the levels |2〉 and |3〉. Δ2 and Δ3 are one- and two-photon detunings, respectively; Γ1223) is the spontaneous emission decay rate from |2〉 to |1〉 (|3〉 to |2〉). (b) The long-range interaction potential of 87Rb atoms V(rij)=C6/rij6 (red solid line) as a function of rij = |rirj|, describing the interaction between the atom at ri and the atom at rj (represented by yellow spheres), both of which are at the Rydberg state |3〉 = |60S1/2〉. (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 Rb. 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]

itρ11iΓ12ρ22Ωpρ12+Ωp*ρ21=0,
itρ22iΓ23ρ33+iΓ12ρ22+Ωpρ12Ωp*ρ21Ωcρ23+Ωc*ρ32=0,
itρ33+iΓ23ρ33+Ωcρ23Ωc*ρ32=0,
(it+d21)ρ21Ωp(ρ22ρ11)+Ωc*ρ31=0,
(it+d31)ρ31Ωpρ32+Ωcρ21𝒩ad3rV(rr)ρρ33,31(r,r,t)=0,
(it+d32)ρ32Ωp*ρ31Ωc(ρ33ρ22)𝒩ad3rV(rr)ρρ33,32(r,r,t)=0,
where ραβ = 〈Ŝαβ〉 [48] is the one-body density matrix element, dαβ = Δα − Δβ + αβ1 = 0; α, β = 1, 2, 3; αβ), Δ2 = ωp − (ω2ω1) and Δ3 = ωp + ωc − (ω3ω1) are respectively the one-photon and two-photon detunings, γαβ=(Γα+Γβ)/2+γαβcol with Γβ = ∑α<β Γαβ. Here Γαβ denotes the spontaneous emission decay rate from the state |β〉 to the state |α〉 and γαβcol represents the dephasing rate reflecting the loss of phase coherence between |α〉 and |β〉.

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) ρρ33,3α (r′, r, t) ≡ 〈Ŝ33(r′, t)Ŝ3α (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]

|1=|5s2S1/2,F=2,|2=|5p2P3/2,F=3,|3=|ns2S1/2,Γ12=2π×6MHz,Γ23=2π×3kHz,
with n principle quantum number and other parameters taken as Ωc = 2π × 32 MHz, Δ2 = 2π × 160 MHz. All calculations given below will be based on these realistic physical parameters. The long-range interaction potential between two Rydberg atoms has the form as V(rij)=C6/rij6 [23], where rij = |rirj| is the distance between the ith and jth Rydberg atoms, represented by the yellow spheres in Fig. 1(b). The red solid line in Fig. 1(b) is the curve of the long-range interaction potential C6/rij6 as a function of rij for n = 60, with the dispersion parameter C6 ≃ −2π × 140GHzμm6, adopted from [23].

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) ≥ δEIT, where δEIT is the linewidth of EIT transmission spectrum (i.e. the width of EIT transparency window), defined by Ωc2/γ12 for Δ2 = 0 and Ωc2/|Δ2| for |Δ2| ≫ γ12 (we assumed −Δ2/C6 > 0). Thus the blockade sphere has the radius Rb = (|C6/δEIT|)1/6 ≃ 5.29μm [23, 31, 50]. Comparing this to the average interatomic separation obtained by R¯=(5/9)𝒩a1/32.5μm for 𝒩a = 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 2Rb [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 P=𝒩aα,β=13pαβρβαexp{i[kβkβr(ωβωα+ΔβΔα)t]}, the electric polarization intensity of the probe field reads Pp = 𝒩a{p12ρ21exp[i(kp · rωpt)] + c.c.}, by which one can obtain the optical susceptibility χp of the probe field by using the formula Pp = ε0χpeppexp[i(kp · rωpt)] + c.c., which yields

χp=𝒩a(epp12)ρ21ε0p.

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

(it+d31+iΓ23V(rr))S^33S^31+Ωc(S^23S^31+S^33S^21)Ωc*S^32S^31ΩpS^33S^32𝒩ad3rS^33(r,t)S^33(r,t)S^32(r,t)V(rr)=0,
(it+iΓ23+d32V(rr))S^33S^32+Ωc(S^23S^32S^33S^33+S^33S^22)Ωc*S^32S^32Ωp*S^33S^31𝒩ad3rS^33(r,t)S^33(r,t)S^32(r,t)V(rr)=0,
where r′r″ and ŜαβŜμν in the terms without integration means Ŝαβ (r′, t)Ŝμν (r, t).

Eqs. (5a) and (5b) have the following features. (i) The equations for two-body correlators 〈Ŝ33Ŝ3α〉 (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 〈ŜαβŜμν〉 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]

Sαβ(r)Sμν(r)Sαβ(r)=Sαβ(r)Sμν(r)Sαβ(r)+Sαβ(r)Sμν(r)+Sαβ(r)Sμν(r)Sαβ(r),+Sαβ(r)Sαβ(r)Sμν(r)2Sαβ(r)Sμν(r)Sαβ(r),
As a special case, when the atomic density is low and the interaction between atoms is weak so that the correlation between atoms are negligible, one has 〈Sαβ (r″)Sμν (r′)Sα′β′ (r)〉 → 〈Sαβ (r″)〉 〈Sμν (r′)〉 〈Sα′β′ (r)〉, corresponding to a mean-field approximation. We stress that the mean-field approximation is not valid for the Rydberg gases even at lower atomic density because of the strong Rydberg-Rydberg interaction. On the other hand, to acquire a giant nonlinear optical effect, a higher atomic density is usually needed and hence one must adopt a method beyond the mean-field approximation [52]. The factorization method stated above is an effective approach for dealing with interacting multi-body problems and has been widely adopted in nonlinear laser spectroscopy [51] and Bose-condensed gases [53], by which the equations of motion for one-body and two-body correlators are closed and hence can be solved by using some suitable techniques.

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 Ωp [35] to solve these nonlinear equations in a systematic way. In fact, when EIT systems are weakly driven, the half Rabi frequency Ωp 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] ρα1=Ωpl=0ρα1(2l+1)|Ωp|2l, ρ32=l=1ρ32(2l)|Ωp|2l, ρββ=l=0ρββ(2l)|Ωp|2l with ρββ(0)=δβ1δβ1 (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 Ωp, we obtain a set of approximated equations for ραβ(l), which are listed in Appendix A. The approximated equations for the two-body density matrix (correlator) elements ρραβ,μν(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 ρ21(1)=d31/D and ρ31(1)=Ωc/D, with D = |Ωc|2d21d31. For the second order (l = 2), one obtains the solution

ρ11(2)=[iΓ232|Ωc|2M]NiΓ12(|Ωc|2D*d32*|Ωc|2Dd32)Γ12Γ23iΓ12|Ωc|2M,
ρ33(2)=1iΓ12(NiΓ12ρ11(2)),
ρ32(2)=1d32(ΩcD+2Ωcρ33(2)+Ωcρ11(2)),
where M=1/d321/d32*, N=d31*/D*d31/D. At the third order (l = 3), the solution reads ρ21(3)=a21(3)+𝒩ab21(3) and ρ31(3)=a31(3)+𝒩ab31(3), with
a21(3)=Ωc*ρ32(2)+d31(2ρ11(2)+ρ33(2))|Ωc|2d21d31,
b21(3)=Ωc*d3rρρ33,31(3)(rr)V(rr)|Ωc|2d21d31,
a31(3)=(2ρ11(2)+ρ33(2))Ωc+d21ρ32(3)|Ωc|2d21d31,
b31(3)=d3rρρ33,31(3)(rr)V(rr)d21|Ωc|2d21d31,
where a general expression of ρρ33,31(3) is given in Appendix B [see Eq. (18) ]. Because we have assumed the probe field to be weak, the Rydberg-Rydberg interaction gives the contribution to the solution starting only from the third order approximation.

At the fourth order (l = 4), the solution is given by ρ11(4)=a11(4)+𝒩ab11(4), ρ33(4)=a33(4)+𝒩ab33(4), and ρ32(4)=a32(4)+𝒩ab32(4), with

a11(4)=[iΓ232|Ωc|2M](a21*(3)a21(3))+iΩcd32*1Γ12a31*(3)iΩc*d321Γ12a31(3)Γ12Γ23iΓ12|Ωc|2M,
b11(4)=[iΓ232|Ωc|2M](b21*(3)b21(3))+iΩcd32*1Γ12b31*(3)iΩc*d311Γ12b31(3)Γ12Γ23iΓ12|Ωc|2M+d3riΓ12Ωcd32*1ρρ33,32*(4)(rr)V(rr)iΓ12Ωc*d321ρρ33,32(4)(rr)V(rr)Γ12Γ23iΓ12|Ωc|2M,
a33(4)=ia21*(3)/Γ12a11(4)+ia21(3)/Γ12,
b33(4)=ib21*(3)/Γ12b11(4)+ib21(3)/Γ12,
a32(4)=1d32(a31(3)+2Ωca33(4)+Ωca11(4)),
b32(4)=1d32(b31(3)+2Ωcb33(4)+Ωcb11(4))+d3rρρ33,32(4)(rr)V(rr)d32.
With the above solutions, we go to the fifth order (l = 5). The solution at this order reads ρ21(5)=a21(5)+𝒩ab21(5)+𝒩a2c21(5) and ρρ33,31(5)=aa33,31(5)+𝒩abb33,31(5) with
a21(5)=Ωc*a32(4)+d31(2a11(4)+a33(4))|Ωc|2d21d31,
b21(5)=Ωc*b32(4)+d31(2b11(4)+b33(4))|Ωc|2d21d31+Ωc*d3raa33,31(5)(rr)V(rr)|Ωc|2d21d31,
c31(5)=Ωc*d3rbb33,31(5)(rr)V(rr)|Ωc|2d21d31,
where the expressions of ρρ33,32(4), aa33,31(5) and bb33,31(5) in Eqs. (9) and (10) have been given in Appendix B [see Eqs. (20), (24), and (25)].

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 ρ21ρ21(1)Ωp+ρ21(3)|Ωp|2Ωp+ρ215|Ωp|5Ωp+, where ρ21(j) (j = 1, 3, 5,...) are independent of Ωp and their explicit expressions have been given in the previous section. Using the formula (4) and the definition Ωp = (ep · p21)p/h̄, we have

χpχp(1)+χp(3)|p|2+χp(5)|p|4,
where χp(1), χp(3), and χp(5) are respectively the first-order (linear), the third-order and the fifth-order (nonlinear) optical susceptibilities of the probe field, defined by
χp(1)=𝒩a|p12|2ε0h¯d31D,
χp(3)=χp1(3)+χp2(3),
χp(5)=χp1(5)+χp2(5),
with
χp1(3)=𝒩a|p12|4ε0h¯31D[Ωc*ρ32(2)+d31(2ρ11(2)+ρ33(2))],
χp2(3)=𝒩a2|p12|4ε0h¯3Ωc*Dd3rρρ33,31(3)(rr)V(rr),
χp1(5)=𝒩a|p12|6ε0h¯51D[Ωc*a32(4)+d31(2a11(4)+a33(4))],
χp2(5)=𝒩a2|p12|6ε0h¯5Ωc*b32(4)+d31(2b11(4)+b33(4))D+𝒩a2|p12|6ε0h¯5Ωc*Dd3raa33,31(5)(rr)V(rr)+𝒩a3|p12|6ε0h¯5Ωc*Dd3rbb33,31(5)(rr)V(rr),
where χp1(3) and χp1(5) are the third-order and the fifth-order nonlinear optical susceptibilities arising from the interaction between the probe field and the atoms (i.e. by the photon-atom interaction), χp2(3) and χp2(5) are third-order and fifth-order nonlinear optical susceptibilities arising and with from the Rydberg-Rydberg interaction (i.e. by the atom-atom interaction). From the expressions (13a) and (13c), we see that χp1(3) and χp1(5) have a linear dependence on the atomic density 𝒩a. Differently, from the expressions (13b) and (13d) we observe that χp2(3) has a quadratic dependence on the atomic density (i.e. on 𝒩a2) and χp2(5) has not only quadratic but also cubic dependence on the atomic density (i.e. on 𝒩a2 and 𝒩a3), which implies that the nonlinear optical susceptibilities in the Rydberg-EIT system are very sensitive to the change of the atomic density 𝒩a.

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, 𝒩a = 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 χpα(3)=Re(χpα(3))+iIm(χpα(3)) (α = 1, 2) and the fifth-order nonlinear optical susceptibilities χpα(5)=Re(χpα(5))+iIm(χpα(5)) (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 χp1(3) (i.e. Re(χp1(3))=4.4×1011m2V2), and the value at the second row and the third column is the imaginary part of χp1(3) (i.e. Im(χp1(3))=1.3×1013m2V2), etc. In the last column of Table I, the physical origins of various nonlinear optical susceptibilities of the system are given.

Tables Icon

Table 1. Real part Re(χpα(j)) and imaginary part Im(χpα(j)) (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.

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. 𝒩a = 3 × 1010 cm−3), the nonlinear optical susceptibilities contributed by the Rydberg-Rydberg interaction (i.e. χp2(3) and χp2(5)) are three orders of magnitude greater than those contributed by the photon-atom interaction (i.e. χp1(3) and χp1(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 Re(χp1(3)) (black dashed-dotted line) and Re(χp2(3)) (red solid line) as functions of 𝒩a. When plotting the figure, the parameters used are the same as those used for getting the results in the Table I except for 𝒩a, 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 𝒩a around 108 cm−3, and both of them are increasing functions of 𝒩a. (ii) For 𝒩a 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 Re(χp1(3))=Re(χp2(3))=1.3×1013m2V2 at 𝒩a = 0.9 × 108 cm−3. (iii) When 𝒩a 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 𝒩a increases.

 

Fig. 2 (a) Re(χp1(3)) (dashed-dotted line) and Re(χp2(3)) (red solid line) as functions of atomic density 𝒩a. (b) Re(χp1(5)) (dashed-dotted line) and Re(χp2(5)) (red solid line) as functions of atomic density 𝒩a. (c) Total nonlinear optical susceptibility Re(χN)=Re(χp(3)|p|2+χp(5)|p|4) as a function of |p|. Lines from 1 to 3 correspond to 𝒩a= 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 Re(χp1(5)) (black dashed-dotted line) and Re(χp2(5)) (red solid line) as functions of 𝒩a, which originate respectively from the photon-atom and the Rydberg-Rydberg interactions. We observe that both Re(χp1(5)) and Re(χp2(5)) are comparable, and they are decreasing functions of 𝒩a. Furthermore, for 𝒩a 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 𝒩a = 1.86 × 108 cm−3, the both optical susceptibilities become equal to have the value Re(χp1(5))=Re(χp2(5))=1.67×1018m4V4. When 𝒩a is larger than 1.86 × 108 cm−3 Re(χp2(5)) becomes to be smaller than Re(χp1(5)).

From the above results, we see that there exist various, synergetic optical nonlinearities in the Rydberg-EIT system. Due to the 𝒩a2- and 𝒩a3-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 𝒩a. From Fig. 2(a) and Fig. 2(b) we also see that the sign of χpα(3) is opposite to that of χpα(5) (α = 1, 2) and χpα(5) grows faster than χpα(3), which means that there exists a competition between χpα(3) and χpα(5) when 𝒩a 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 𝒩a are illustrated, in Fig. 2(c) we show Re(χN)=Re(χp(3))|p|2+Re(χp(5))|p|4, i.e. the real part of the total nonlinear optical susceptibility of the probe field, as a function of |p| (p is the envelope of the probe field) for several different atomic density. Lines from 1 to 3 in the figure are for 𝒩a taken to be 6 × 1010 cm−3, 4 × 1010 cm−3, and 3 × 1010 cm−3, respectively. We observe that, for all 𝒩a, Re(χN) grows fast initially, then arrives a peak value, and finally decreases as |p| increases. We also observe that the higher the atomic density, the faster Re(χN) 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 |pmax| = 120 V/m, which is within the validity domain of the perturbation theory used above because |Ωpmaxc| ≃ 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 χp(3) ( χp(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 𝒩a = 6 × 1010 cm−3, which agrees fairly with reported experimental and theoretical results [57]. If the atomic density increases to 𝒩a = 5.0 × 1012 cm−3 (used in [14]), we obtain χp(3)=4.6×103m2V2. 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 χp(3)=7×108m2V2 [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 𝒩a = 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 χp(3) ( χp(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 𝒩a = 3 × 1010 cm−3, Ωc = 2π × 16 MHz, Δ2 = 1 kHz and Δ3 = 0 (the blockade sphere radius Rb = 3.26μm) and the other parameters the same as those given above we obtain χp(3)=(7.07+i3.4)×1010m2V2 and χp(5)=(2.03+0.8)×1014m4V4. In this situation, imaginary parts of χp(3) and χp(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 χp(3) and χp(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 𝒩a2-dependence, but also a giant fifth-order nonlinear optical susceptibility, which has 𝒩a2- and 𝒩a3-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.

Appendix

A. Expansion of the one-body density matrix equation

Under the perturbation expansion ρα1=Ωpl=0ρα1(2l+1)|Ωp|2l, ρ32=l=1ρ32(2l)|Ωp|2l, ρββ=l=0ρββ(2l)|Ωp|2l with ρββ(0)=δβ1δβ1 (α = 2, 3; β = 1, 2, 3), the one-body density matrix equation (3) becomes

(it+d21)ρ21(l)+1+Ωc*ρ31(l)=A(l),
(it+d31)ρ31(l)+Ωcρ21(l)=B(l),
itρ11(l)+iΓ12(ρ11(l)+ρ33(l))=ρ12(l1)ρ21(l1),
itρ33(l)+iΓ23ρ33(l)+Ωcρ23(l)Ωc*ρ32(l)=0,
(it+d32)ρ32(l)Ωc(2ρ33(l)+ρ11(l))=C(l).
Here A(1) = A(2) = B(1) = B(2) = C(1) = 0, A(l)=2ρ11(l1)+ρ33(l1) (l = 3, 4, 5), B(l)=ρ32(l1)+𝒩ad3rV(rr)ρρ33,31(l)(r,r,t) (l = 3, 4, 5), C(2)=ρ31(1), C(3)=ρ31(2), and C(l)=ρ31(l1)+𝒩ad3rV(rr)ρρ33,32(l)(r,r,t) (l = 4, 5).

B. Expansion of the equations of the two-body correlators

By a simple inspection on the order of magnitude for the two-body density matrix (correlator) elements 〈ŜαβŜμν〉 ≡ ρραβ,μν based on the weak driven EIT condition (i.e. the probe-field is weak), we have the expansion ρραβ,μν=ρραβ,μν(2)Ωp2+ρραβ,μν(4)Ωp2|Ωp|2+. By a detailed and careful calculation, we obtain the following equations of motion for the two-body correlators from second-order to fifth-order approximations:

  • (i) Second-order approximation (l = 2). For the two-body correlators, the lowest-order approximation starts from ε2-order. We obtain the equations
    [2d2102Ωc*02d31V2ΩcΩcΩc*d21+d31][ρρ21,21(2)ρρ31,31(2)ρρ31,21(2)]=[2d31D0ΩcD],
    and
    [d21+d120ΩcΩc*Ωc*Ωc*d21+d1300d31+d13ΩcΩc*ΩcΩc0d21*+d13*][ρρ21,12(2)ρρ31,13(2)ρρ21,13(2)ρρ21,13*(2)]=[d31Dd31*D*Ωc*D*0ΩcD],
  • (ii) Third-order approximation (l = 3). At this order, we have the equations
    [M31Ωc*iΓ230Ωc*Ωc00ΩcM320iΓ2300Ωc*Ωc00M33Ωc*Ωc*Ωc0000ΩcM3400Ωc*ΩcΩc0Ωc0M350Ωc*0Ωc*0Ωc*00M360Ωc*0Ωc0ΩcΩc0M3700Ωc*0Ωc*0Ωc0M38][ρρ22,21(3)ρρ22,31(3)ρρ33,21(3)ρρ33,31(3)ρρ32,21(3)ρρ21,23(3)ρρ32,31(3)ρρ31,23(3)]=[ρρ21,12(2)+ρρ21,21(2)ρ22(2)ρρ31,12(2)+ρρ21,31(2)ρ33(2)0ρρ21,31(2)ρ32(2)ρ32*(2)ρρ21,13(2)ρρ31,31(2)ρρ31,13(2)],
    where M31 = iΓ12 + d21, M32 = iΓ12 + d31, M33 = iΓ23 + d21, M34 = d31 + iΓ23V, M35 = d32 + d21, M36 = d23 + d21, M37 = d32 + d31V and M38 = d23 + d31. The general expression of ρρ33,31(3) reads
    ρρ33,31(3)=P0+P1V(rr)+P2V(rr)2Q0+Q1V(rr)+Q2V(rr)2+Q3V(rr)3,
    Here Pn and Qn (n = 0, 1, 2, 3) are functions of the spontaneous emission decay rate γμν, detunings Δμ and half Rabi frequency Ωc. The third-order nonlinear susceptibility χp2(3) can be obtained by integrating ρρ33,31(3) (see Eq. (8)).
  • (iii) Fourth-order approximation (l = 4). At this order, one has the equations
    [iΓ120iΓ23Ωc*000Ωc000iΓ230000Ωc*0Ωc00iΓ23M43Ωc*00Ωc*ΩcΩc0Ωc0ΩcM44Ωc*ΩciΓ230000002ΩcM4502Ωc000000Ωc*0M46Ωc*ΩcΩc00ΩcΩc0Ωc*ΩcM47000Ωc*0Ωc*00Ωc*0M48iΓ23Ωc0Ωc*Ωc*00Ωc*00M49Ωc00000002Ωc*2Ωc*M40][ρρ22,22(4)ρρ33,33(4)ρρ33,22(4)ρρ22,32(4)ρρ32,32(4)ρρ23,32(4)ρρ33,32(4)ρρ22,23(4)ρρ33,23(4)ρρ23,23(4)]=[ρρ22,21(3)ρρ22,21*(3)0ρρ33,21(3)ρρ33,21*(3)ρρ32,21(3)+ρρ22,31(3)ρρ21,23*(3)2ρρ32,31(3)ρρ31,23(3)ρρ32,13(3)ρρ33,31(3)ρρ21,23(3)ρρ32,21*(3)ρρ22,31*(3)ρρ33,31*(3)2ρρ32,31*(3)],
    where M43 = iΓ12 + iΓ23, M44 = d32 + iΓ12, M45 = 2d32V, M46 = d32 + d23, M47 = iΓ23 + d32V, M48 = d23 + iΓ12, M49 = iΓ23 + d23 +V, M40 = 2d23 +V. A general expression for the radial dependence of ρρ33,32(4) is given by
    ρρ33,32(4)=n=06KnV(rr)nn=07JnV(rr)n,
    where Kn and Jn are functions of the spontaneous emission decay rate γμν, detunings Δμ and half Rabi frequency Ωc. The fourth order of atomic population ραα(4) (α = 1, 2, 3) can be calculated by integrating ρρ33,32(4).

    Another part of equations at the fourth-order reads

    [2d2102Ωc*02d31V2ΩcΩcΩc*d21+d31][ρρ21,21(4)ρρ31,31(4)ρρ31,21(4)]=[4ρρ22,21(3)+2ρρ33,21(3)2a21(3)2ρρ32,31(3)2ρρ22,31(3)+ρρ33,31(3)+ρρ32,21(3)a31(3)]+Na[2b21(3)2d3r(2Ωcρρ33,31(3)D+ρ33(2)ρρ31,31(2)Ωc2ρ33(2)D2)V(rr)b31(3)+d3r(d31ρρ33,31(3)DΩcρρ33,21(3)D+ρ33(2)ρρ31,21(2)+Ωcd31ρ33(2)D2)V(rr)],
    and
    [d21+d120ΩcΩc*0d31+d13ΩcΩc*Ωc*Ωc*d21+d130ΩcΩc0d21*+d13*][ρρ21,12(4)ρρ31,13(4)ρρ21,13(4)ρρ21,13*(4)]=[2(ρρ22,21*(3)ρρ22,21(3))+ρρ33,21*(3)ρρ33,21(3)+b21(3)b21*(3)ρρ31,23*(3)ρρ31,23(3)2ρρ22,31*(3)+ρρ33,31*(3)a31*(3)ρρ21,23(3)2ρρ22,31(3)+ρρ33,31(3)a31(3)ρρ21,23*(3)]+𝒩a[b21(3)b21*(3)0b31*(3)+d3r(d31ρρ33,31*(3)DΩc*ρρ33,21(3)D*+ρ33(2)ρρ21,13(2)+ρ33(2)Ωc*d31|D|2)V(rr)b31(3)+d3r(d31*ρρ33,31(3)D*Ωcρρ33,21*(3)D+ρ33(2)ρρ21,13*(2)+ρ33(2)Ωcd31*|D|2)V(rr)],
    The solution of the Eqs. (21) and (22) is given by ρρα1,β1(4)=aaα1,β1(4)+𝒩abbα1,β1(4), and ρρα1,1β(4)=aaα1,1β(4)+𝒩abbα1,1β(4) (α, β = 2, 3). The explicit expressions of aaα1,β1(4), aaα1,1β(4), bbα1,β1(4) and bbα1,1β(4) can be easily obtained by using Cramer’s rule, which are lengthy and omitted here for saving space.

  • (ii) Fifth-order approximation (l = 5). An this order, we have the equations
    [M31Ωc*iΓ230Ωc*Ωc00ΩcM320iΓ2300Ωc*Ωc00M33Ωc*Ωc*Ωc0000ΩcM3400Ωc*ΩcΩc0Ωc0M350Ωc*0Ωc*0Ωc*00M360Ωc*0Ωc0ΩcΩc0M3700Ωc*0Ωc*0Ωc0M38][ρρ22,21(5)ρρ22,31(5)ρρ33,21(5)ρρ33,31(5)ρρ32,21(5)ρρ21,23(5)ρρ32,31(5)ρρ31,23(5)]=[2ρρ22,22(4)+ρρ33,22(4)aa21,12(4)+aa21,21(4)a22(4)aa31,21(4)aa31,12(4)+ρρ22,32(4)2ρρ22,33(4)+ρρ33,33(4)a33(4)ρρ33,32(4)2ρρ22,32(4)+ρρ33,32(4)+aa31,21(4)a32(4)2ρρ22,32*(4)+ρρ33,32*(4)aa21,13(4)a32*(4)ρρ32,32(4)+aa31,31(4)ρρ32,23(4)aa31,13(4)]+𝒩a[b22(4)bb21,12(4)+bb21,21(4)bb31,21(4)bb31,12(4)+N12b33(4)N14bb31,21(4)b32(4)+N15b32*(4)bb21,13(4)N16bb31,31(4)+N17bb31,13(4)],
    where N12=d3r(ρρ33,31(3)ρ22(2)+ρρ22,21(3)ρ33(2)ρρ22,33(4)Ωc/D+ρ33(2)ρ22(2)Ωc/D)V(rr), N14=d3r(2ρρ33,31(3)ρ33(2)ρρ33,33(4)Ωc/D+ρ33(2)ρ33(2)Ωc/D)V(rr), N15=d3r(ρρ32,21(3)ρ33(2)+ρ33,21(3)ρ32(2)+ρρ33,32(4)d31/Dρ33(2)ρ32(2)d31/D)V(rr), N16=d3r(ρρ23,21(3)ρ33(2)+ρρ33,21(3)ρ32*(2)+ρρ33,32*(4)d31/Dρ33(2)ρ32*(2)d31/D)V(rr), N17=d3r[ρρ33,31(3)(ρ32(2)+ρ22(2))+(ρρ32,31(3)+ρρ22,31(3))ρ33(2)(ρρ33,32(4)+ρρ33,22(4))Ωc/D+ρ33(2)(ρ32(2)+ρ22(2))Ωc/D]V(rr).

The solution of Eq. (23) is given by ρραα,β1(5)=aaαα,β1(5)+𝒩abbαα,β1(5), ρρ32,α1(5)=aa32,α1(5)+𝒩abb32,α1(5) and ρρα1,23(5)=aaα1,23(5)+𝒩abbα1,23(5) (α, β = 2, 3). General expressions for the radial dependence of aa33,31(5) and bb33,31(5) are given as:

aa33,31(5)=n=08WnV(rr)nn=09YnV(rr)n,
bb33,31(5)=X0+X1V(rr)Z0+Z1V(rr)+Z2V(rr)2,
were Wn, Yn, Xn and Zn are functions of the spontaneous emission decay rate γμν, detunings Δμ and half Rabi frequency Ωc. The fifth-order nonlinear susceptibility χp2(5) can be obtained by integrating these general forms analytically. The explicit expressions of aaαα,β1(5), aa32,α1(5), aaα1,23(5), bbαα,β1(5), bb32,α1(5) and bbα1,23(5) are omitted here.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants No. 11174080 and No. 11474099, and by the Chinese Education Ministry Reward for Excellent Doctors in Academics under Grant No. PY2014009.

References and links

1. R. W. Boyd, Nonlinear Optics (3rd edition) (Academic, Elsevier, 2008).

2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

3. D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014). [CrossRef]  

4. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]  

5. K. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (CRC, 2009).

6. H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006). [CrossRef]   [PubMed]  

7. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009). [CrossRef]  

8. R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005). [CrossRef]  

9. T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006). [CrossRef]  

10. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef]   [PubMed]  

11. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005). [CrossRef]  

12. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008). [CrossRef]  

13. Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014). [CrossRef]  

14. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]  

15. J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013). [CrossRef]  

16. T. F. Gallagher, Rydberg Atoms (Cambridge University, 2008).

17. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010). [CrossRef]  

18. J. D. Pritchard, Cooperative Optical Non-Linearity in a Blockaded Rydberg Ensemble (Springer, 2012). [CrossRef]  

19. S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011). [CrossRef]  

20. N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010). [CrossRef]   [PubMed]  

21. A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007). [CrossRef]   [PubMed]  

22. A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008). [CrossRef]  

23. J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010). [CrossRef]  

24. J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011). [CrossRef]  

25. V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012). [CrossRef]  

26. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013). [CrossRef]   [PubMed]  

27. C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013). [CrossRef]   [PubMed]  

28. K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005). [CrossRef]  

29. C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011). [CrossRef]  

30. S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011). [CrossRef]   [PubMed]  

31. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011). [CrossRef]   [PubMed]  

32. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011). [CrossRef]  

33. D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013). [CrossRef]  

34. A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013). [CrossRef]   [PubMed]  

35. J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013). [CrossRef]  

36. The optical nonlinearities in three-level Rydberg gases of ladder-type level configuration can be classified as dissipative [30] or dispersive [25,35] ones, corresponding to a large (zero) single-photon detuning Δ2 (see Fig. 1(a)). In the case of the dissipative (dispersive) nonlinearity, the imaginary (real) part of nonlinear optical susceptibilities plays a leading role.

37. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012). [CrossRef]   [PubMed]  

38. O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013). [CrossRef]   [PubMed]  

39. S. Saltiel, S. Tanev, and A. D. Boardman, “High-order nonlinear phase shift caused by cascaded third-order processes,” Opt. Lett. 22, 148–150 (1997). [CrossRef]   [PubMed]  

40. V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009). [CrossRef]  

41. M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002). [CrossRef]  

42. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006). [CrossRef]  

43. A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003). [CrossRef]   [PubMed]  

44. S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2006). [CrossRef]  

45. Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008). [CrossRef]  

46. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997). [CrossRef]  

47. H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010). [CrossRef]   [PubMed]  

48. Here 〈Ô〉 represents the average of the operator Ô in the Heisenberg picture.

49. D. A. Steck, Rubidium 87 D Line Data, http://steck.us/alkalidata/.

50. Because Rb is independent of atomic density 𝒩a, the increase of 𝒩a results in only an increase of atoms (which are not in the Rydberg state) in the blockade sphere.

51. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).

52. Researches [23, 35, 47] showed that if the atomic density 𝒩a is large than 4 × 108cm−3 the mean-field approximation is broken down.

53. A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University, 2009). [CrossRef]  

54. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

55. G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008). [CrossRef]  

56. C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010). [CrossRef]   [PubMed]  

57. The third-order nonlinear optical susceptibilities in Rydberg atomic gases can arrive at the order of magnitude of 10−7 m2V−2 and has a cubic dependence on atomic density, which have been reported in experment (see Refs. [15, 23–25]) and theory (see Refs. [30, 32, 35]).

58. Since these high-order nonlinear optical susceptibilities are much smaller than the third- and fifth-order ones, the discussion on them is omitted here.

59. S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015). [CrossRef]  

60. C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015). [CrossRef]   [PubMed]  

References

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  1. R. W. Boyd, Nonlinear Optics (3rd edition) (Academic, Elsevier, 2008).
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  3. D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014).
    [Crossref]
  4. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  5. K. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (CRC, 2009).
  6. H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006).
    [Crossref] [PubMed]
  7. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009).
    [Crossref]
  8. R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
    [Crossref]
  9. T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
    [Crossref]
  10. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
    [Crossref] [PubMed]
  11. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
    [Crossref]
  12. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
    [Crossref]
  13. Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014).
    [Crossref]
  14. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [Crossref]
  15. J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013).
    [Crossref]
  16. T. F. Gallagher, Rydberg Atoms (Cambridge University, 2008).
  17. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
    [Crossref]
  18. J. D. Pritchard, Cooperative Optical Non-Linearity in a Blockaded Rydberg Ensemble (Springer, 2012).
    [Crossref]
  19. S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
    [Crossref]
  20. N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010).
    [Crossref] [PubMed]
  21. A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007).
    [Crossref] [PubMed]
  22. A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
    [Crossref]
  23. J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
    [Crossref]
  24. J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
    [Crossref]
  25. V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
    [Crossref]
  26. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
    [Crossref] [PubMed]
  27. C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
    [Crossref] [PubMed]
  28. K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
    [Crossref]
  29. C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011).
    [Crossref]
  30. S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
    [Crossref] [PubMed]
  31. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
    [Crossref] [PubMed]
  32. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
    [Crossref]
  33. D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
    [Crossref]
  34. A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013).
    [Crossref] [PubMed]
  35. J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
    [Crossref]
  36. The optical nonlinearities in three-level Rydberg gases of ladder-type level configuration can be classified as dissipative [30] or dispersive [25,35] ones, corresponding to a large (zero) single-photon detuning Δ2 (see Fig. 1(a)). In the case of the dissipative (dispersive) nonlinearity, the imaginary (real) part of nonlinear optical susceptibilities plays a leading role.
  37. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
    [Crossref] [PubMed]
  38. O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
    [Crossref] [PubMed]
  39. S. Saltiel, S. Tanev, and A. D. Boardman, “High-order nonlinear phase shift caused by cascaded third-order processes,” Opt. Lett. 22, 148–150 (1997).
    [Crossref] [PubMed]
  40. V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009).
    [Crossref]
  41. M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
    [Crossref]
  42. C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
    [Crossref]
  43. A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
    [Crossref] [PubMed]
  44. S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2006).
    [Crossref]
  45. Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
    [Crossref]
  46. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
    [Crossref]
  47. H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
    [Crossref] [PubMed]
  48. Here 〈Ô〉 represents the average of the operator Ô in the Heisenberg picture.
  49. D. A. Steck, Rubidium 87 D Line Data, http://steck.us/alkalidata/ .
  50. Because Rb is independent of atomic density 𝒩a, the increase of 𝒩a results in only an increase of atoms (which are not in the Rydberg state) in the blockade sphere.
  51. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).
  52. Researches [23, 35, 47] showed that if the atomic density 𝒩a is large than 4 × 108cm−3 the mean-field approximation is broken down.
  53. A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University, 2009).
    [Crossref]
  54. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  55. G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
    [Crossref]
  56. C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010).
    [Crossref] [PubMed]
  57. The third-order nonlinear optical susceptibilities in Rydberg atomic gases can arrive at the order of magnitude of 10−7 m2V−2 and has a cubic dependence on atomic density, which have been reported in experment (see Refs. [15, 23–25]) and theory (see Refs. [30, 32, 35]).
  58. Since these high-order nonlinear optical susceptibilities are much smaller than the third- and fifth-order ones, the discussion on them is omitted here.
  59. S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
    [Crossref]
  60. C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
    [Crossref] [PubMed]

2015 (2)

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

2014 (2)

D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014).
[Crossref]

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014).
[Crossref]

2013 (7)

J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013).
[Crossref] [PubMed]

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

2012 (2)

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

2011 (6)

J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
[Crossref]

C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011).
[Crossref]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

2010 (5)

N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010).
[Crossref] [PubMed]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010).
[Crossref] [PubMed]

2009 (2)

V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009).
[Crossref]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

2008 (4)

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

2007 (1)

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007).
[Crossref] [PubMed]

2006 (3)

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006).
[Crossref] [PubMed]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

2005 (4)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
[Crossref]

2004 (1)

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

2003 (1)

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

2002 (1)

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

1997 (1)

Adams, C. S.

J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
[Crossref]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007).
[Crossref] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Amthor, T.

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Arimondo, E.

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

Ates, C.

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011).
[Crossref]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref] [PubMed]

Auffeves, A.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Bai, Z.

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014).
[Crossref]

Barredo, D.

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

Bason, M. G.

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

Behrrozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Bienias, P.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

Bimbard, E.

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Blatt, S.

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

Boardman, A. D.

Boyd, M. M.

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (3rd edition) (Academic, Elsevier, 2008).

Browaeys, A.

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

Brune, M.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Büchler, H. P.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

Busche, H.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

Butscher, B.

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

Chang, D. E.

D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014).
[Crossref]

Chen, Y.

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Côté, R.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
[Crossref]

Cui, C.

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

Deng, L.

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

DePaola, B. D.

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Evers, J.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

Firstenberg, O.

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Fleischhauer, M.

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Gallagher, T. F.

T. F. Gallagher, Rydberg Atoms (Cambridge University, 2008).

Gärttner, M.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

Gauguet, A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
[Crossref]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

Giese, C.

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Gleyzes, S.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Gorniaczyk, H.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

Gorshkov, A. V.

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

Grangier, P.

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Greene, C. H.

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

Griffin, A.

A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University, 2009).
[Crossref]

Günter, G.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Güunter, G.

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

Hang, C.

C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010).
[Crossref] [PubMed]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

Haroche, S.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2006).
[Crossref]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Henkel, N.

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref] [PubMed]

N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010).
[Crossref] [PubMed]

Hilliard, A. J.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Hofferberth, S.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Hofmann, C. S.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Huang, G.

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014).
[Crossref]

C. Hang and G. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express 18, 2952–2966 (2010).
[Crossref] [PubMed]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Ido, T.

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Jackson, T. R.

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007).
[Crossref] [PubMed]

Jarutis, V.

V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009).
[Crossref]

Jones, M. P. A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

Labuhn, H.

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

Lahaye, T.

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

Li, Y.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

Liang, Q.-Y.

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Liu, Y.

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

Ludlow, A. D.

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

Lukin, M. D.

D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

Lvovsky, A. I.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

Ma, L.

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

Maioli, P.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Matsko, A. B.

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

Maxwell, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

Meunier, T.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Michinel, H.

H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006).
[Crossref] [PubMed]

Mirgorodskiy, I.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

Mohapatra, A. K.

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007).
[Crossref] [PubMed]

Mølmer, K.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

Mukamel, S.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).

Nath, R.

A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013).
[Crossref] [PubMed]

N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Nikuni, T.

A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University, 2009).
[Crossref]

Nogrette, F.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Nogues, G.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Otterbach, J.

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

Ourjoumtsev, A.

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Paredes-Barato, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

Parigi, V.

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Payne, M. G.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Paz-Alonso, M. J.

H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006).
[Crossref] [PubMed]

Pentaris, D.

V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009).
[Crossref]

Perez-Garcia, V. M.

H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006).
[Crossref] [PubMed]

Petrosyan, D.

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

Peyronel, T.

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Pohl, T.

A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref] [PubMed]

C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011).
[Crossref]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010).
[Crossref] [PubMed]

Pritchard, J. D.

J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
[Crossref]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

J. D. Pritchard, Cooperative Optical Non-Linearity in a Blockaded Rydberg Ensemble (Springer, 2012).
[Crossref]

Raimond, J. M.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

Raimond, J.-M.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2006).
[Crossref]

Ravets, S.

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

Robert-de-Saint-Vincent, M.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

Saffman, M.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

Saliba, S. D.

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Saltiel, S.

Sanders, B. C.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

Santra, R.

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

Schempp, H.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Scully, M. O.

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
[Crossref]

Sevincli, S.

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref] [PubMed]

C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011).
[Crossref]

Singer, K.

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
[Crossref]

Song, L.

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

Stanojevic, J.

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
[Crossref]

Szwer, D. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

Tanev, S.

Tittel, W.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

Tresp, C.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

Tualle-Brouri, R.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Vaicaitis, V.

V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009).
[Crossref]

Vuletic, V.

D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Walker, T. G.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

Weatherill, K. J.

J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
[Crossref]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

Weber, S.

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

Weidemüller, M.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
[Crossref]

Wen, F.

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Whitlock, S.

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

Wu, J. H.

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

Wu, Y.

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Wu, Z.

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Xiao, M.

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Yan, D.

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

Ye, J.

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

Yuan, C.

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Zanon-Willette, T.

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

Zaremba, E.

A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University, 2009).
[Crossref]

Zhang, Y.

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Zheng, H.

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

Zubairy, M. S.

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
[Crossref]

Annu. Rev. Cold At. Mol. (1)

J. D. Pritchard, K. J. Weatherill, and C. S. Adams, “Nonlinear optics using cold Rydberg atoms,” Annu. Rev. Cold At. Mol. 1, 301–350 (2013).
[Crossref]

J. Phys. B: At. Mol. Opt. Phys (1)

J. D. Pritchard, A. Gauguet, K. J. Weatherill, and C. S. Adams, “Optical non-linearity in a Rydberg gas,” J. Phys. B: At. Mol. Opt. Phys 44, 184019 (2011).
[Crossref]

J. Phys. B: At. Mol. Opt. Phys. (2)

S. Sevincli, C. Ates, T. Pohl, H. Schempp, C. S. Hofmann, G. Güunter, T. Amthor, M. Weidemüller, J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Quantum interference in interacting three-level Rydberg gases: coherent population trapping and electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 44, 184018 (2011).
[Crossref]

K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interaction between alkali Rydberg atom pairs correlated to the ns-ns, np-np, nd-nd asymptotes,” J. Phys. B: At. Mol. Opt. Phys. 38, S295–S307 (2005).
[Crossref]

Nat. Photon. (2)

D. E. Chang, V. Vuletić, and M. D. Lukin, “Quantum nonlinear optics – photon by photon,” Nat. Photon. 8, 685–694 (2014).
[Crossref]

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical Quantum Memory,” Nat. Photon. 3, 706–714 (2009).
[Crossref]

Nature (3)

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behrrozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Nature Phys. (1)

A. K. Mohapatra, M. G. Bason, B. Butscher, K. J. Weatherill, and C. S. Adams, “Giant electro-optic effect using polarizable dark states,” Nature Phys. 4, 890–894 (2008).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (10)

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

S. Ravets, H. Labuhn, D. Barredo, T. Lahaye, and A. Browaeys, “Measurement of the angular dependence of the dipole-dipole interaction between two individual Rydberg atoms at a Förster resonance,” Phys. Rev. A 92, 020701(R) (2015).
[Crossref]

G. Huang, C. Hang, and L. Deng, “Gain-assisted superluminal optical solitons at very low light intensity,” Phys. Rev. A 77, 011803(R) (2008).
[Crossref]

Z. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2008).
[Crossref]

C. Ates, S. Sevincli, and T. Pohl, “Electromagnetically induced transparency in strongly interacting Rydberg gases,” Phys. Rev. A 83, 041802(R) (2011).
[Crossref]

D. Yan, C. Cui, Y. Liu, L. Song, and J. H. Wu, “Normal and abnormal nonlinear electromagnetically induced transparency due to dipole blockade of Rydberg excitation,” Phys. Rev. A 87, 023827 (2013).
[Crossref]

J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, “Dispersive optical nonlinearities in a Rydberg electromagnetically-induced-transparency medium,” Phys. Rev. A 88, 053845 (2013).
[Crossref]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89, 023835 (2014).
[Crossref]

Phys. Rev. E (1)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Phys. Rev. Lett (1)

H. Schempp, G. Günter, C. S. Hofmann, C. Giese, S. D. Saliba, B. D. DePaola, T. Amthor, and M. Weidemüller, “Coherent Population Trapping with Controlled Interparticle Interactions,” Phys. Rev. Lett 104, 173602 (2010).
[Crossref] [PubMed]

Phys. Rev. Lett. (17)

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a Mesoscopic Field with an Atom Induced by Photon Graininess in a Cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[Crossref] [PubMed]

V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical Third-Harmonic Generation in Normally Dispersive Media,” Phys. Rev. Lett. 103, 103901 (2009).
[Crossref]

C. Tresp, P. Bienias, S. Weber, H. Gorniaczyk, I. Mirgorodskiy, H. P. Büchler, and S. Hofferberth, “Dipolar Dephasing of Rydberg D-State Polaritons,” Phys. Rev. Lett. 115, 083602 (2015).
[Crossref] [PubMed]

H. Michinel, M. J. Paz-Alonso, and V. M. Perez-Garcia, “Turning Light into a Liquid via Atomic Coherence,” Phys. Rev. Lett. 96, 023903 (2006).
[Crossref] [PubMed]

R. Santra, E. Arimondo, T. Ido, C. H. Greene, and J. Ye, “High-Accuracy Optical Clock via Three-Level Coherence in Neutral Bosonic 88Sr,” Phys. Rev. Lett. 94, 173002 (2005).
[Crossref]

T. Zanon-Willette, A. D. Ludlow, S. Blatt, M. M. Boyd, E. Arimondo, and J. Ye, “Cancellation of Stark Shifts in Optical Lattice Clocks by Use of Pulsed Raman and Electromagnetically Induced Transparency Techniques,” Phys. Rev. Lett. 97, 233001 (2006).
[Crossref]

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

A. V. Gorshkov, R. Nath, and T. Pohl, “Dissipative Many-Body Quantum Optics in Rydberg Media,” Phys. Rev. Lett. 110, 153601 (2013).
[Crossref] [PubMed]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-Photon Interactions via Rydberg Blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetic ally Induced Transparency with Rydberg Atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Cooperative Atom-Light Interaction in a Blockaded Rydberg Ensemble,” Phys. Rev. Lett. 105, 193603 (2010).
[Crossref]

N. Henkel, R. Nath, and T. Pohl, “Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates,” Phys. Rev. Lett. 104, 195302 (2010).
[Crossref] [PubMed]

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency,” Phys. Rev. Lett. 98, 113003 (2007).
[Crossref] [PubMed]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and Measurement of Interaction-Induced Dispersive Optical Nonlinearities in an Ensemble of Cold Rydberg Atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref] [PubMed]

C. S. Hofmann, G. Günter, H. Schempp, M. Robert-de-Saint-Vincent, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, “Sub-Poissonian Statistics of Rydberg-Interacting Dark-State Polaritons,” Phys. Rev. Lett. 110, 203601 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

Other (17)

J. D. Pritchard, Cooperative Optical Non-Linearity in a Blockaded Rydberg Ensemble (Springer, 2012).
[Crossref]

T. F. Gallagher, Rydberg Atoms (Cambridge University, 2008).

K. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (CRC, 2009).

R. W. Boyd, Nonlinear Optics (3rd edition) (Academic, Elsevier, 2008).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

The optical nonlinearities in three-level Rydberg gases of ladder-type level configuration can be classified as dissipative [30] or dispersive [25,35] ones, corresponding to a large (zero) single-photon detuning Δ2 (see Fig. 1(a)). In the case of the dissipative (dispersive) nonlinearity, the imaginary (real) part of nonlinear optical susceptibilities plays a leading role.

The third-order nonlinear optical susceptibilities in Rydberg atomic gases can arrive at the order of magnitude of 10−7 m2V−2 and has a cubic dependence on atomic density, which have been reported in experment (see Refs. [15, 23–25]) and theory (see Refs. [30, 32, 35]).

Since these high-order nonlinear optical susceptibilities are much smaller than the third- and fifth-order ones, the discussion on them is omitted here.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
[Crossref]

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2006).
[Crossref]

Here 〈Ô〉 represents the average of the operator Ô in the Heisenberg picture.

D. A. Steck, Rubidium 87 D Line Data, http://steck.us/alkalidata/ .

Because Rb is independent of atomic density 𝒩a, the increase of 𝒩a results in only an increase of atoms (which are not in the Rydberg state) in the blockade sphere.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).

Researches [23, 35, 47] showed that if the atomic density 𝒩a is large than 4 × 108cm−3 the mean-field approximation is broken down.

A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperature (Cambridge University, 2009).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (3)

Fig. 1
Fig. 1 (a) Excitation scheme of the three-level ladder system, in which the probe field with angular frequency ωp and half Rabi frequency Ωp couples the levels |1〉 and |2〉, and the control field with angular frequency ωc and half Rabi frequency Ωc couples the levels |2〉 and |3〉. Δ2 and Δ3 are one- and two-photon detunings, respectively; Γ1223) is the spontaneous emission decay rate from |2〉 to |1〉 (|3〉 to |2〉). (b) The long-range interaction potential of 87Rb atoms V ( r i j ) = C 6 / r i j 6 (red solid line) as a function of rij = |rirj|, describing the interaction between the atom at ri and the atom at rj (represented by yellow spheres), both of which are at the Rydberg state |3〉 = |60S1/2〉. (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 Rb. 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.
Fig. 2
Fig. 2 (a) Re ( χ p 1 ( 3 ) ) (dashed-dotted line) and Re ( χ p 2 ( 3 ) ) (red solid line) as functions of atomic density ��a. (b) Re ( χ p 1 ( 5 ) ) (dashed-dotted line) and Re ( χ p 2 ( 5 ) ) (red solid line) as functions of atomic density ��a. (c) Total nonlinear optical susceptibility Re ( χ N ) = Re ( χ p ( 3 ) | p | 2 + χ p ( 5 ) | p | 4 ) as a function of |p|. Lines from 1 to 3 correspond to ��a= 4 × 1010 cm−3, 2 × 1010 cm−3 and 1 × 1010 cm−3, respectively.
Fig. 3
Fig. 3 (a) ((b)) Third-order (Fifth-order) nonlinear optical susceptibility χ p ( 3 ) ( χ 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.

Tables (1)

Tables Icon

Table 1 Real part Re ( χ p α ( j ) ) and imaginary part Im ( χ p α ( j ) ) (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.

Equations (53)

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^ H ( r , t ) = α = 1 3 h ¯ ω α S ^ α α ( r , t ) h ¯ [ Ω p S ^ 12 ( r , t ) + Ω p * S ^ 21 ( r , t ) + Ω c S ^ 23 ( r , t ) + Ω c * S ^ 32 ( r , t ) ] + 𝒩 a d 3 r S ^ 33 ( r , t ) h ¯ V ( r r ) S ^ 33 ( r , t ) ,
[ S ^ α β ( r , t ) , S ^ μ ν ( r , t ) ] = ( δ α ν S ^ μ β ( r , t ) δ μ β S ^ α ν ( r , t ) ) δ r r ,
i t ρ 11 i Γ 12 ρ 22 Ω p ρ 12 + Ω p * ρ 21 = 0 ,
i t ρ 22 i Γ 23 ρ 33 + i Γ 12 ρ 22 + Ω p ρ 12 Ω p * ρ 21 Ω c ρ 23 + Ω c * ρ 32 = 0 ,
i t ρ 33 + i Γ 23 ρ 33 + Ω c ρ 23 Ω c * ρ 32 = 0 ,
( i t + d 21 ) ρ 21 Ω p ( ρ 22 ρ 11 ) + Ω c * ρ 31 = 0 ,
( i t + d 31 ) ρ 31 Ω p ρ 32 + Ω c ρ 21 𝒩 a d 3 r V ( r r ) ρ ρ 33 , 31 ( r , r , t ) = 0 ,
( i t + d 32 ) ρ 32 Ω p * ρ 31 Ω c ( ρ 33 ρ 22 ) 𝒩 a d 3 r V ( r r ) ρ ρ 33 , 32 ( r , r , t ) = 0 ,
| 1 = | 5 s 2 S 1 / 2 , F = 2 , | 2 = | 5 p 2 P 3 / 2 , F = 3 , | 3 = | n s 2 S 1 / 2 , Γ 12 = 2 π × 6 MHz , Γ 23 = 2 π × 3 kHz ,
χ p = 𝒩 a ( e p p 12 ) ρ 21 ε 0 p .
( i t + d 31 + i Γ 23 V ( r r ) ) S ^ 33 S ^ 31 + Ω c ( S ^ 23 S ^ 31 + S ^ 33 S ^ 21 ) Ω c * S ^ 32 S ^ 31 Ω p S ^ 33 S ^ 32 𝒩 a d 3 r S ^ 33 ( r , t ) S ^ 33 ( r , t ) S ^ 32 ( r , t ) V ( r r ) = 0 ,
( i t + i Γ 23 + d 32 V ( r r ) ) S ^ 33 S ^ 32 + Ω c ( S ^ 23 S ^ 32 S ^ 33 S ^ 33 + S ^ 33 S ^ 22 ) Ω c * S ^ 32 S ^ 32 Ω p * S ^ 33 S ^ 31 𝒩 a d 3 r S ^ 33 ( r , t ) S ^ 33 ( r , t ) S ^ 32 ( r , t ) V ( r r ) = 0 ,
S α β ( r ) S μ ν ( r ) S α β ( r ) = S α β ( r ) S μ ν ( r ) S α β ( r ) + S α β ( r ) S μ ν ( r ) + S α β ( r ) S μ ν ( r ) S α β ( r ) , + S α β ( r ) S α β ( r ) S μ ν ( r ) 2 S α β ( r ) S μ ν ( r ) S α β ( r ) ,
ρ 11 ( 2 ) = [ i Γ 23 2 | Ω c | 2 M ] N i Γ 12 ( | Ω c | 2 D * d 32 * | Ω c | 2 D d 32 ) Γ 12 Γ 23 i Γ 12 | Ω c | 2 M ,
ρ 33 ( 2 ) = 1 i Γ 12 ( N i Γ 12 ρ 11 ( 2 ) ) ,
ρ 32 ( 2 ) = 1 d 32 ( Ω c D + 2 Ω c ρ 33 ( 2 ) + Ω c ρ 11 ( 2 ) ) ,
a 21 ( 3 ) = Ω c * ρ 32 ( 2 ) + d 31 ( 2 ρ 11 ( 2 ) + ρ 33 ( 2 ) ) | Ω c | 2 d 21 d 31 ,
b 21 ( 3 ) = Ω c * d 3 r ρ ρ 33 , 31 ( 3 ) ( r r ) V ( r r ) | Ω c | 2 d 21 d 31 ,
a 31 ( 3 ) = ( 2 ρ 11 ( 2 ) + ρ 33 ( 2 ) ) Ω c + d 21 ρ 32 ( 3 ) | Ω c | 2 d 21 d 31 ,
b 31 ( 3 ) = d 3 r ρ ρ 33 , 31 ( 3 ) ( r r ) V ( r r ) d 21 | Ω c | 2 d 21 d 31 ,
a 11 ( 4 ) = [ i Γ 23 2 | Ω c | 2 M ] ( a 21 * ( 3 ) a 21 ( 3 ) ) + i Ω c d 32 * 1 Γ 12 a 31 * ( 3 ) i Ω c * d 32 1 Γ 12 a 31 ( 3 ) Γ 12 Γ 23 i Γ 12 | Ω c | 2 M ,
b 11 ( 4 ) = [ i Γ 23 2 | Ω c | 2 M ] ( b 21 * ( 3 ) b 21 ( 3 ) ) + i Ω c d 32 * 1 Γ 12 b 31 * ( 3 ) i Ω c * d 31 1 Γ 12 b 31 ( 3 ) Γ 12 Γ 23 i Γ 12 | Ω c | 2 M + d 3 r i Γ 12 Ω c d 32 * 1 ρ ρ 33 , 32 * ( 4 ) ( r r ) V ( r r ) i Γ 12 Ω c * d 32 1 ρ ρ 33 , 32 ( 4 ) ( r r ) V ( r r ) Γ 12 Γ 23 i Γ 12 | Ω c | 2 M ,
a 33 ( 4 ) = i a 21 * ( 3 ) / Γ 12 a 11 ( 4 ) + i a 21 ( 3 ) / Γ 12 ,
b 33 ( 4 ) = i b 21 * ( 3 ) / Γ 12 b 11 ( 4 ) + i b 21 ( 3 ) / Γ 12 ,
a 32 ( 4 ) = 1 d 32 ( a 31 ( 3 ) + 2 Ω c a 33 ( 4 ) + Ω c a 11 ( 4 ) ) ,
b 32 ( 4 ) = 1 d 32 ( b 31 ( 3 ) + 2 Ω c b 33 ( 4 ) + Ω c b 11 ( 4 ) ) + d 3 r ρ ρ 33 , 32 ( 4 ) ( r r ) V ( r r ) d 32 .
a 21 ( 5 ) = Ω c * a 32 ( 4 ) + d 31 ( 2 a 11 ( 4 ) + a 33 ( 4 ) ) | Ω c | 2 d 21 d 31 ,
b 21 ( 5 ) = Ω c * b 32 ( 4 ) + d 31 ( 2 b 11 ( 4 ) + b 33 ( 4 ) ) | Ω c | 2 d 21 d 31 + Ω c * d 3 r a a 33 , 31 ( 5 ) ( r r ) V ( r r ) | Ω c | 2 d 21 d 31 ,
c 31 ( 5 ) = Ω c * d 3 r b b 33 , 31 ( 5 ) ( r r ) V ( r r ) | Ω c | 2 d 21 d 31 ,
χ p χ p ( 1 ) + χ p ( 3 ) | p | 2 + χ p ( 5 ) | p | 4 ,
χ p ( 1 ) = 𝒩 a | p 12 | 2 ε 0 h ¯ d 31 D ,
χ p ( 3 ) = χ p 1 ( 3 ) + χ p 2 ( 3 ) ,
χ p ( 5 ) = χ p 1 ( 5 ) + χ p 2 ( 5 ) ,
χ p 1 ( 3 ) = 𝒩 a | p 12 | 4 ε 0 h ¯ 3 1 D [ Ω c * ρ 32 ( 2 ) + d 31 ( 2 ρ 11 ( 2 ) + ρ 33 ( 2 ) ) ] ,
χ p 2 ( 3 ) = 𝒩 a 2 | p 12 | 4 ε 0 h ¯ 3 Ω c * D d 3 r ρ ρ 33 , 31 ( 3 ) ( r r ) V ( r r ) ,
χ p 1 ( 5 ) = 𝒩 a | p 12 | 6 ε 0 h ¯ 5 1 D [ Ω c * a 32 ( 4 ) + d 31 ( 2 a 11 ( 4 ) + a 33 ( 4 ) ) ] ,
χ p 2 ( 5 ) = 𝒩 a 2 | p 12 | 6 ε 0 h ¯ 5 Ω c * b 32 ( 4 ) + d 31 ( 2 b 11 ( 4 ) + b 33 ( 4 ) ) D + 𝒩 a 2 | p 12 | 6 ε 0 h ¯ 5 Ω c * D d 3 r a a 33 , 31 ( 5 ) ( r r ) V ( r r ) + 𝒩 a 3 | p 12 | 6 ε 0 h ¯ 5 Ω c * D d 3 r b b 33 , 31 ( 5 ) ( r r ) V ( r r ) ,
( i t + d 21 ) ρ 21 ( l ) + 1 + Ω c * ρ 31 ( l ) = A ( l ) ,
( i t + d 31 ) ρ 31 ( l ) + Ω c ρ 21 ( l ) = B ( l ) ,
i t ρ 11 ( l ) + i Γ 12 ( ρ 11 ( l ) + ρ 33 ( l ) ) = ρ 12 ( l 1 ) ρ 21 ( l 1 ) ,
i t ρ 33 ( l ) + i Γ 23 ρ 33 ( l ) + Ω c ρ 23 ( l ) Ω c * ρ 32 ( l ) = 0 ,
( i t + d 32 ) ρ 32 ( l ) Ω c ( 2 ρ 33 ( l ) + ρ 11 ( l ) ) = C ( l ) .
[ 2 d 21 0 2 Ω c * 0 2 d 31 V 2 Ω c Ω c Ω c * d 21 + d 31 ] [ ρ ρ 21 , 21 ( 2 ) ρ ρ 31 , 31 ( 2 ) ρ ρ 31 , 21 ( 2 ) ] = [ 2 d 31 D 0 Ω c D ] ,
[ d 21 + d 12 0 Ω c Ω c * Ω c * Ω c * d 21 + d 13 0 0 d 31 + d 13 Ω c Ω c * Ω c Ω c 0 d 21 * + d 13 * ] [ ρ ρ 21 , 12 ( 2 ) ρ ρ 31 , 13 ( 2 ) ρ ρ 21 , 13 ( 2 ) ρ ρ 21 , 13 * ( 2 ) ] = [ d 31 D d 31 * D * Ω c * D * 0 Ω c D ] ,
[ M 31 Ω c * i Γ 23 0 Ω c * Ω c 0 0 Ω c M 32 0 i Γ 23 0 0 Ω c * Ω c 0 0 M 33 Ω c * Ω c * Ω c 0 0 0 0 Ω c M 34 0 0 Ω c * Ω c Ω c 0 Ω c 0 M 35 0 Ω c * 0 Ω c * 0 Ω c * 0 0 M 36 0 Ω c * 0 Ω c 0 Ω c Ω c 0 M 37 0 0 Ω c * 0 Ω c * 0 Ω c 0 M 38 ] [ ρ ρ 22 , 21 ( 3 ) ρ ρ 22 , 31 ( 3 ) ρ ρ 33 , 21 ( 3 ) ρ ρ 33 , 31 ( 3 ) ρ ρ 32 , 21 ( 3 ) ρ ρ 21 , 23 ( 3 ) ρ ρ 32 , 31 ( 3 ) ρ ρ 31 , 23 ( 3 ) ] = [ ρ ρ 21 , 12 ( 2 ) + ρ ρ 21 , 21 ( 2 ) ρ 22 ( 2 ) ρ ρ 31 , 12 ( 2 ) + ρ ρ 21 , 31 ( 2 ) ρ 33 ( 2 ) 0 ρ ρ 21 , 31 ( 2 ) ρ 32 ( 2 ) ρ 32 * ( 2 ) ρ ρ 21 , 13 ( 2 ) ρ ρ 31 , 31 ( 2 ) ρ ρ 31 , 13 ( 2 ) ] ,
ρ ρ 33 , 31 ( 3 ) = P 0 + P 1 V ( r r ) + P 2 V ( r r ) 2 Q 0 + Q 1 V ( r r ) + Q 2 V ( r r ) 2 + Q 3 V ( r r ) 3 ,
[ i Γ 12 0 i Γ 23 Ω c * 0 0 0 Ω c 0 0 0 i Γ 23 0 0 0 0 Ω c * 0 Ω c 0 0 i Γ 23 M 43 Ω c * 0 0 Ω c * Ω c Ω c 0 Ω c 0 Ω c M 44 Ω c * Ω c i Γ 23 0 0 0 0 0 0 2 Ω c M 45 0 2 Ω c 0 0 0 0 0 0 Ω c * 0 M 46 Ω c * Ω c Ω c 0 0 Ω c Ω c 0 Ω c * Ω c M 47 0 0 0 Ω c * 0 Ω c * 0 0 Ω c * 0 M 48 i Γ 23 Ω c 0 Ω c * Ω c * 0 0 Ω c * 0 0 M 49 Ω c 0 0 0 0 0 0 0 2 Ω c * 2 Ω c * M 40 ] [ ρ ρ 22 , 22 ( 4 ) ρ ρ 33 , 33 ( 4 ) ρ ρ 33 , 22 ( 4 ) ρ ρ 22 , 32 ( 4 ) ρ ρ 32 , 32 ( 4 ) ρ ρ 23 , 32 ( 4 ) ρ ρ 33 , 32 ( 4 ) ρ ρ 22 , 23 ( 4 ) ρ ρ 33 , 23 ( 4 ) ρ ρ 23 , 23 ( 4 ) ] = [ ρ ρ 22 , 21 ( 3 ) ρ ρ 22 , 21 * ( 3 ) 0 ρ ρ 33 , 21 ( 3 ) ρ ρ 33 , 21 * ( 3 ) ρ ρ 32 , 21 ( 3 ) + ρ ρ 22 , 31 ( 3 ) ρ ρ 21 , 23 * ( 3 ) 2 ρ ρ 32 , 31 ( 3 ) ρ ρ 31 , 23 ( 3 ) ρ ρ 32 , 13 ( 3 ) ρ ρ 33 , 31 ( 3 ) ρ ρ 21 , 23 ( 3 ) ρ ρ 32 , 21 * ( 3 ) ρ ρ 22 , 31 * ( 3 ) ρ ρ 33 , 31 * ( 3 ) 2 ρ ρ 32 , 31 * ( 3 ) ] ,
ρ ρ 33 , 32 ( 4 ) = n = 0 6 K n V ( r r ) n n = 0 7 J n V ( r r ) n ,
[ 2 d 21 0 2 Ω c * 0 2 d 31 V 2 Ω c Ω c Ω c * d 21 + d 31 ] [ ρ ρ 21 , 21 ( 4 ) ρ ρ 31 , 31 ( 4 ) ρ ρ 31 , 21 ( 4 ) ] = [ 4 ρ ρ 22 , 21 ( 3 ) + 2 ρ ρ 33 , 21 ( 3 ) 2 a 21 ( 3 ) 2 ρ ρ 32 , 31 ( 3 ) 2 ρ ρ 22 , 31 ( 3 ) + ρ ρ 33 , 31 ( 3 ) + ρ ρ 32 , 21 ( 3 ) a 31 ( 3 ) ] + N a [ 2 b 21 ( 3 ) 2 d 3 r ( 2 Ω c ρ ρ 33 , 31 ( 3 ) D + ρ 33 ( 2 ) ρ ρ 31 , 31 ( 2 ) Ω c 2 ρ 33 ( 2 ) D 2 ) V ( r r ) b 31 ( 3 ) + d 3 r ( d 31 ρ ρ 33 , 31 ( 3 ) D Ω c ρ ρ 33 , 21 ( 3 ) D + ρ 33 ( 2 ) ρ ρ 31 , 21 ( 2 ) + Ω c d 31 ρ 33 ( 2 ) D 2 ) V ( r r ) ] ,
[ d 21 + d 12 0 Ω c Ω c * 0 d 31 + d 13 Ω c Ω c * Ω c * Ω c * d 21 + d 13 0 Ω c Ω c 0 d 21 * + d 13 * ] [ ρ ρ 21 , 12 ( 4 ) ρ ρ 31 , 13 ( 4 ) ρ ρ 21 , 13 ( 4 ) ρ ρ 21 , 13 * ( 4 ) ] = [ 2 ( ρ ρ 22 , 21 * ( 3 ) ρ ρ 22 , 21 ( 3 ) ) + ρ ρ 33 , 21 * ( 3 ) ρ ρ 33 , 21 ( 3 ) + b 21 ( 3 ) b 21 * ( 3 ) ρ ρ 31 , 23 * ( 3 ) ρ ρ 31 , 23 ( 3 ) 2 ρ ρ 22 , 31 * ( 3 ) + ρ ρ 33 , 31 * ( 3 ) a 31 * ( 3 ) ρ ρ 21 , 23 ( 3 ) 2 ρ ρ 22 , 31 ( 3 ) + ρ ρ 33 , 31 ( 3 ) a 31 ( 3 ) ρ ρ 21 , 23 * ( 3 ) ] + 𝒩 a [ b 21 ( 3 ) b 21 * ( 3 ) 0 b 31 * ( 3 ) + d 3 r ( d 31 ρ ρ 33 , 31 * ( 3 ) D Ω c * ρ ρ 33 , 21 ( 3 ) D * + ρ 33 ( 2 ) ρ ρ 21 , 13 ( 2 ) + ρ 33 ( 2 ) Ω c * d 31 | D | 2 ) V ( r r ) b 31 ( 3 ) + d 3 r ( d 31 * ρ ρ 33 , 31 ( 3 ) D * Ω c ρ ρ 33 , 21 * ( 3 ) D + ρ 33 ( 2 ) ρ ρ 21 , 13 * ( 2 ) + ρ 33 ( 2 ) Ω c d 31 * | D | 2 ) V ( r r ) ] ,
[ M 31 Ω c * i Γ 23 0 Ω c * Ω c 0 0 Ω c M 32 0 i Γ 23 0 0 Ω c * Ω c 0 0 M 33 Ω c * Ω c * Ω c 0 0 0 0 Ω c M 34 0 0 Ω c * Ω c Ω c 0 Ω c 0 M 35 0 Ω c * 0 Ω c * 0 Ω c * 0 0 M 36 0 Ω c * 0 Ω c 0 Ω c Ω c 0 M 37 0 0 Ω c * 0 Ω c * 0 Ω c 0 M 38 ] [ ρ ρ 22 , 21 ( 5 ) ρ ρ 22 , 31 ( 5 ) ρ ρ 33 , 21 ( 5 ) ρ ρ 33 , 31 ( 5 ) ρ ρ 32 , 21 ( 5 ) ρ ρ 21 , 23 ( 5 ) ρ ρ 32 , 31 ( 5 ) ρ ρ 31 , 23 ( 5 ) ] = [ 2 ρ ρ 22 , 22 ( 4 ) + ρ ρ 33 , 22 ( 4 ) a a 21 , 12 ( 4 ) + a a 21 , 21 ( 4 ) a 22 ( 4 ) a a 31 , 21 ( 4 ) a a 31 , 12 ( 4 ) + ρ ρ 22 , 32 ( 4 ) 2 ρ ρ 22 , 33 ( 4 ) + ρ ρ 33 , 33 ( 4 ) a 33 ( 4 ) ρ ρ 33 , 32 ( 4 ) 2 ρ ρ 22 , 32 ( 4 ) + ρ ρ 33 , 32 ( 4 ) + a a 31 , 21 ( 4 ) a 32 ( 4 ) 2 ρ ρ 22 , 32 * ( 4 ) + ρ ρ 33 , 32 * ( 4 ) a a 21 , 13 ( 4 ) a 32 * ( 4 ) ρ ρ 32 , 32 ( 4 ) + a a 31 , 31 ( 4 ) ρ ρ 32 , 23 ( 4 ) a a 31 , 13 ( 4 ) ] + 𝒩 a [ b 22 ( 4 ) b b 21 , 12 ( 4 ) + b b 21 , 21 ( 4 ) b b 31 , 21 ( 4 ) b b 31 , 12 ( 4 ) + N 12 b 33 ( 4 ) N 14 b b 31 , 21 ( 4 ) b 32 ( 4 ) + N 15 b 32 * ( 4 ) b b 21 , 13 ( 4 ) N 16 b b 31 , 31 ( 4 ) + N 17 b b 31 , 13 ( 4 ) ] ,
a a 33 , 31 ( 5 ) = n = 0 8 W n V ( r r ) n n = 0 9 Y n V ( r r ) n ,
b b 33 , 31 ( 5 ) = X 0 + X 1 V ( r r ) Z 0 + Z 1 V ( r r ) + Z 2 V ( r r ) 2 ,

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