Abstract

We theoretically investigate the linear susceptibility and propagation of light in a three-dimensional (3-D) Rydberg gas under conditions of electromagnetically induced transparency. Rydberg atoms with two relevant S states are coupled via exchange interactions. When the gas is initially prepared in an entangled spin-wave state, this coupling induces a strong, nonlocal susceptibility whereby the photon field at one point of the medium acts as a source at a distant position. The nonlocal propagation occurs not only in the propagation direction but also in the paraxial direction. We discuss the absorption features and numerically simulate the 3-D propagation of probe laser light. Combined with the long-range exchange interaction, we show that the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Gases of interacting Rydberg atoms [1,2] have recently attracted theoretical [39] and experimental [1015] attention. The atoms in a Rydberg gas undergo long-range dipole-dipole interactions within a certain blockade distance [1618]. Under conditions of electromagnetically induced transparency (EIT) [19], strong atom-light interactions occur without absorption, and we can probe novel quantum phenomena such as many-body phenomena with strongly interacting photons [20,21], and nonlinear and nonlocal light-matter interactions [2225]. Here, a physical mechanism is an essentially classic density-density interaction between the Rydberg states [26]. However, in some cases, these systems may exhibit resonant dipole-dipole interactions [27] and consequent excitation exchanges [28,29]. The exchange interaction establishes coherence between distant atoms, which leads to the linear susceptibility of the medium acquiring a nonlocal term, and the system’s (nonlocal) optical response is dramatically altered.

The authors of Ref. [30] have studied a Rydberg-atom system with exchange interactions initially prepared in a spin-wave state [31,32]. After discussing the underlying mechanism in a two-atom model, they have numerically investigated a one-dimensional atomic gas. Here, we expand the one-dimensional atomic gas model to a three-dimensional (3-D) ensemble of atoms in a spatially homogeneous spin wave and analytically discuss the linear optical response of this strongly interacting Rydberg-EIT medium. The exchange interactions between Rydberg-state atoms induce strong nonlocal interactions between photons, whereby the photon field at one point of the medium can act as a source at a distant position. The "nonlocal" propagation occurs not only in the propagation direction, as discussed in [30], but also in the paraxial direction. We discuss the absorption features, particularly the interplay of the one-photon detuning and phase-matching relations of the involved light fields. Combined with the long range exchange interaction, the Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and the sign of the nonlocal interaction kernel can be widely tuned. To demonstrate the potential of this medium, we numerically simulate the propagation of probe laser light. Our results indicate that the medium enhances the absorptive properties of Rydberg gases.

The remainder of this paper is structured as follows. Section 2 introduces a generic interacting many-body system in the EIT configuration. Section 3 discusses the optical response of the 3-D system and derives linear susceptibility in k space. The effective potential is investigated, and light propagation is numerically simulated in a 3-D gas. The main results and conclusions are summarized in Section 4.

2. System

We consider a beam with wave number $\mathbf {k}_p$ and amplitude $E_p(\mathbf {r},t)$ passing through an atomic medium. Figure 1 depicts the level structure of the medium, in which $|1\rangle$ and $|2\rangle$ are the ground state and low-lying excited state, respectively, and the upper levels $|3\rangle$ and $|4\rangle$ are Rydberg S states. The transition between $|1\rangle$ and $|2\rangle$ is driven by a weak probe field with detuning $\delta$ and a Rabi frequency of $\Omega _p(\mathbf {r},t) = \mu _{21}E(\mathbf {r},t)\exp [i(\mathbf {k}_p\cdot \mathbf {r})]$ where $\mu _{21}$ is the transition dipole moment. The transition between $|2\rangle$ and $|3\rangle$ is coupled by a strong control laser field with Rabi frequency $\Omega _c(\mathbf {r})=\Omega _c \exp [i(\mathbf {k}_c \cdot \mathbf {r})]$. Together, these transitions induce a typical EIT configuration. In the dipole and rotating wave approximation, the corresponding single-particle Hamiltonian is

$$H_j ={-}\delta (\sigma_{22}^j+\sigma_{33}^j)-\frac{1}{2}[\Omega_{p}(\mathbf{r}_j)\sigma_{21}^j + \Omega_{c}(\mathbf{r}_j)\sigma_{32}^j + H.c.],$$
where the atomic transition operators $\sigma _{ab}=|a\rangle \langle b|$. We consider two Rydberg $S$ states $|3\rangle$ and $|4\rangle$, located at positions $\mathbf {r}_j$ and $\mathbf {r}_k$, respectively. The core element of this research is the initial state of the atomic gas: only one atom is in the state $|4\rangle$, and all other atoms are in the ground state $|1\rangle$. When we only consider single photon probe field pulses, there will always be at most one atom in state $|4\rangle$ and one in state $|3\rangle$. Therefore, the self-interactions of state $|4\rangle$ and state $|3\rangle$ are not considered, and only two interactions exist: The van der Waals interaction $V^d_{jk} = C_d \sigma _{33}^j \sigma _{44}^k/|\mathbf {r}_j - \mathbf {r}_i|^6$ and the exchange interaction $V^e_{jk} = C_e (\sigma _{34}^j \sigma _{43}^k + \mathrm {H.c.})/|\mathbf {r}_j - \mathbf {r}_i|^6$. Here, $C_{d,e}$ are the dispersion coefficients [30].

 figure: Fig. 1.

Fig. 1. Level scheme and transitions in the atomic medium.

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The density matrix $\rho$ of the system is evolves under the following quantum master equation:

$$\frac{d}{dt} \rho ={-} i [H, \rho] + \gamma \sum_{j=i}^N \mathcal{D}(\sigma_{12}^j) \rho$$
where $\mathcal {D}(\sigma )\rho = \sigma \rho \sigma ^{\dagger } - \frac {1}{2}\{\sigma ^{\dagger }\sigma , \rho \}$ is the dissipator, $H = \sum _{j=1}^{N} H_j + \sum _{j>k}^N (V_{jk}^d + V_{jk}^e)$, and $\gamma$ is the dissipative decay of the short-lived state $|2\rangle$ into the ground state $|1\rangle$. The decay from Rydberg states $|3\rangle$ and $|4\rangle$ is neglected.

3. Light propagation in a 3-D gas

3.1 Optical response of a gas

We first consider a two-atom system initially existing in a spin-wave or super-atom state $|\Psi (\mathbf {r}_1,\mathbf {r}_2)\rangle = (|14\rangle + e^{i\mathbf {k}_s\cdot \mathbf {r}_{12}}|41\rangle )/\sqrt 2$, where $\mathbf {r}_{12} = \mathbf {r}_1 - \mathbf {r}_2$. The system will evolve in the two-atom basis $\{|14\rangle , |24\rangle , |41\rangle , |42\rangle , |34\rangle , |43\rangle \}\equiv \{|1\rangle , |2\rangle , |3\rangle , |4\rangle , |5\rangle , |6\rangle \}$. The optical response of atom $i$ positioned at $\mathbf {r}_i$, $\langle \sigma _{21}(\mathbf {r}_i)\rangle = \mathrm {Tr} [\rho \sigma _{21}^i]$, is given by the coherence between states $|1\rangle$ and $|2\rangle$. To capture the feature of the optical response, we calculate the susceptibility using the first-order perturbation method, considering the first-order coupling in terms of $\Omega _p$. We assume $\rho \simeq \rho _0 + \rho _1$, where $\rho _0$ and $\rho _1$ are the zeroth-order and first-order corrections of the density matrix, respectively. When $\Omega _c \gg \Omega _p$, our initial state $|\Psi (\mathbf {r}_1,\mathbf {r}_2)\rangle$ is a metastable state in the metastable decoherence-free subspace [29,30]. We assume $\rho _0 = |\Psi (\mathbf {r}_1,\mathbf {r}_2)\rangle \langle \Psi (\mathbf {r}_1,\mathbf {r}_2)|$, i.e., the nonzero matrix elements in $\rho _0$ are constant and the other matrix elements (coupling the initial state) evolve over time. The two-atom Hamiltonian can be written as $H = H_0 + H_1$, where $H_0 = \sum _{j=1,2} [\delta (|2\rangle _j \langle 2| + |3\rangle _j \langle 3|) + \frac {\Omega _c(\mathbf {r}_j)}{2}\sigma _{32}^{j} + h.c. ] + V_{12}^d + V_{12}^e$ excludes the probe field, and $H_1 = \sum _{j=1,2}[\frac {\Omega _p(\mathbf {r}_j)}{2}\sigma _{21}^j + h.c.]$. The master equation for $\rho _1$ is then given by

$$\dot \rho_1 \simeq{-} i [H_0, \rho_1] - i[H_1, \rho_0] + \gamma \sum_{j=1,2} (\sigma_{12}^j \rho^1 \sigma_{21}^j - \frac{1}{2} \{ \sigma_{21}^j \sigma_{12}^j, \rho^j \}).$$
In the limit of weak probe light $[\Omega _p(\mathbf {r}_i) \ll \Omega _c]$, we perturbatively compute the stationary state of the master equation to first order in $\Omega _p/\Omega _c$. For this purpose, we set $\dot {\rho } = \dot {\rho _1} = 0$, and thus obtain:
$$\dot{\rho_{21}} ={-}(i\delta + \frac{1}{2} \gamma)\rho_{21} -i \frac{\Omega_c}{2}\rho_{61} - i\frac{\Omega_{p}(\mathbf{r}_1)}{4} = 0,$$
$$\dot{\rho_{61}} ={-}i(\delta + V_{12}^d)\rho_{61} -i V_{12}^e\rho_{51} - i\frac{\Omega_{c}}{2} \rho_{21} = 0,$$
$$\dot{\rho_{51}} ={-}i(\delta + V_{12}^d )\rho_{51} -i V_{12}^e\rho_{61} - i\frac{\Omega_{c}}{2} \rho_{41} = 0,$$
$$\dot{\rho_{41}} ={-}(i\delta + \frac{1}{2} \gamma )\rho_{41} -i \frac{\Omega_c}{2}\rho_{51} - i\frac{\Omega_{p}(\mathbf{r}_2)}{4}e^{i\mathbf{k}_s\cdot \mathbf{r}_{12}} = 0.$$
By solving these equations, we get
$$\langle \sigma_{21}(\mathbf{r}_1)\rangle ={-} \frac{\mu_{21}}{2} \left[\alpha_l(\mathbf{r}_{12}) E(\mathbf{r}_1) + \alpha_n(\mathbf{r}_{12}) e^{i\mathbf{K}\cdot \mathbf{r}_{12}} E(\mathbf{r}_2)\right]$$
where
$$\begin{aligned}\alpha_l(\mathbf{r}_{12}) = & \frac{\delta-C_d-Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d-C_e)}+\frac{\delta-C_d+Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d+C_e)}\\ =&\frac{2\delta}{\Omega_c^2-2i\Gamma\delta} + \frac{\Omega_c^2}{2i\Gamma(2i\Gamma\delta-\Omega_c^2)} (A_+(\mathbf{r}_{12})+ A_-(\mathbf{r}_{12})), \end{aligned}$$
$$\begin{aligned}\alpha_n(\mathbf{r}_{12}) =& \frac{\delta-C_d-Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d-C_e)}-\frac{\delta-C_d+Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d+C_e)}\\ =& \frac{\Omega_c^2}{2i\Gamma(2i\Gamma\delta-\Omega_c^2)} (A_+(\mathbf{r}_{12})- A_-(\mathbf{r}_{12})), \end{aligned}$$
with $\Gamma = \gamma - 2 i \delta$, $R_b^{\pm } = [2i\Gamma (C_d \pm C_e)/(\Omega _c^2 - 2i\Gamma \delta )]^{1/6}$, $A_{\pm }(r) = 1/\left (1+\left (\frac {r}{R_b^{\pm }}\right )^6\right )$, and the relative wave vector $\mathbf {K} = \mathbf {k}_s - \mathbf {k}_c - \mathbf {k}_p$ [30].

Here we are interested in a three-dimensional ensemble of atoms. The atoms are in a spatially homogeneous spin wave, i.e., $|\Psi (\mathbf {r}_1,\ldots ,\mathbf {r}_N)\rangle = \sum _{j}c_j e^{i\mathbf {k}_s\cdot \mathbf {r}_j}|1\ldots 4_j \ldots 1\rangle$, with the probability amplitudes $c_j$ and the spin-wave vector $\mathbf {k}_s$. In the continuum limit, we set continuous densities $n(\mathbf {r})$, and normalized them by $\int n(\mathbf {r})d \mathbf {r} = 1$, thus obtain

$$\sigma_{21}(\mathbf{r}) ={-}\frac{\mu_{21}N}{2}\left[\int \alpha_l (\mathbf{r} - \mathbf{r}')n(\mathbf{r}') d\mathbf{r}' E(\mathbf{r}) + \int \alpha_n (\mathbf{r} - \mathbf{r}')n(\mathbf{r}')e^{i \mathbf{K}\cdot (\mathbf{r - r'})} E(\mathbf{r}')d\mathbf{r}'\right].$$

Unlike the optical response in conventional EIT, the coherence in Eq. (8) involves a nonlocal term describing the dependence of an atoms response not only on the local field $E(\mathbf {r}_i)$, but also on the electric fields at more distant locations. As shown in Ref. [30], the nonlocal term emerges through the exchange interaction and initial entangled state. Note that if the exchange interaction between the atoms is absent, then $A_{+} = A_-$ so $\alpha _n=0$.

The propagation of the probe field is governed by the Maxwell equation. Under the slowly varying amplitude approximation, this equation reads

$$\left(-\frac{i}{2k_p}\nabla_\bot^2 + \frac{\partial}{\partial z} + \frac{1}{c} \frac{\partial}{\partial t}\right)E(\mathbf{r}) = i\frac{k_p}{2\epsilon_0}\mathcal{P}(\mathbf{r})$$
where $\epsilon _0$ is the vacuum permittivity, $\nabla _\bot$ accounts for the transverse dynamics with respect to the axial coordinate $\mathbf {r}_\bot = (x,y)$, and the propagation direction is $z$. The relevant medium properties are determined by the polarization $\mathcal {P}(\mathbf {r})$,
$$\mathcal{P}(\mathbf{r}) = 2\mu_{12}\sigma(\mathbf{r}) = \epsilon_0 \left[ \chi_l (\mathbf{r}) E(\mathbf{r}) + \int \chi_n (\mathbf{r} - \mathbf{r'}) E(\mathbf{r}) d\mathbf{r}' \right]$$
where
$$\chi_l (\mathbf{r}) = g \int \alpha_l (\mathbf{r} - \mathbf{r}')n(\mathbf{r}')d\mathbf{r}',$$
$$\chi_n (\mathbf{r} - \mathbf{r'}) = g \alpha_n (\mathbf{r} - \mathbf{r}')n(\mathbf{r}') e^{i \mathbf{K}\cdot (\mathbf{r - r'})}$$
with $g = |\mu _{21}|^2 N/\epsilon _0$. As shown in Eq. (10), the polarization $\mathcal {P}(\mathbf {r})$ of the electric field at position $\mathbf {r}$ depends not only on the electric field at position $\mathbf {r}$ but also on the fields at the other positions $\mathbf {r}'$. We thus call $\chi _l(\mathbf {r})$ in Eq. (10) the local susceptibility and $\chi _n(\mathbf {r}-\mathbf {r'})$ the nonlocal susceptibility.

The second term on the left-hand side of Eq. (9) describes the diffraction effect of the probe field. For convenience and without loss of generality, we assume a long time duration of the probe field. The system then works in a steady state, and the time derivative in the Maxwell Eq. (9) can be neglected (i.e., $\partial /\partial t = 0$) [33,34]. The wave equation in Eq. (9) is then written as

$$(-\frac{i}{2k_p}\nabla_\bot^2 + \frac{\partial}{\partial z})E(\mathbf{r})= \frac{i k_p}{2}[\chi_l(\mathbf{r})E(\mathbf{r}) +\int \chi_n(\mathbf{r}-\mathbf{r'}) E(\mathbf{r}')d \mathbf{r}'].$$

3.2 Susceptibility in $k$ space

To understand the basic propagation physics, it is convenient to compute the susceptibility (local part $\chi _l (\mathbf {r})$ and nonlocal part $\chi _n (\mathbf {r} - \mathbf {r'})$) in $k$ space through the following Fourier transform (see Supplement 1):

$$\chi_l(\mathbf{r})E(\mathbf{r}) = \mathcal{F}^{{-}1}\{\mathcal{F}[\chi_l(\mathbf{r})E(\mathbf{r})]\}= \mathcal{F}^{{-}1}\{\mathcal{F}[\chi_l(\mathbf{r})]\ast \mathcal{F}[E(\mathbf{r})]\}= \mathcal{F}^{{-}1} [\tilde \chi_l \tilde E(\mathbf{k})]$$
$$\int \chi_n(\mathbf{r}-\mathbf{r'}) E(\mathbf{r}')d \mathbf{r}' = \mathcal{F}^{{-}1}\{\mathcal{F}[\int \chi_n(\mathbf{r}-\mathbf{r'}) E(\mathbf{r}')d \mathbf{r}']\}=\mathcal{F}^{{-}1} [\tilde \chi_n(\mathbf{k}) \tilde E(\mathbf{k})],$$
with
$$\tilde \chi_l=A_0 + A_1 \left((R_b^{+})^3+(R_b^{-})^3\right),$$
$$\tilde \chi_n(\mathbf{k}) = A_1\left[(R_b^{+})^3f(|\mathbf{k}+\mathbf{K}|R_b^+) - (R_b^{-})^3f(|\mathbf{k}+\mathbf{K}|R_b^-)\right],$$
where $A_0 = 2g\delta /(\Omega _c^2-2i\Gamma \delta )$, $A_1 = \pi g\Omega _c^2 n /(6i\Gamma (2i\Gamma \delta -\Omega _c^2))$, and $f(x)=(e^{-x}-Be^{-B^*x} -B^*e^{-B^*x})/x$. In the last expression, $B=(1+i\sqrt 3)/2$. $\tilde {\chi }_l$ corresponds to the (k independent) local term, and $\tilde \chi _n (\mathbf {k})$ is (k dependent) nonlocal term.

In Figs. 2(a) and 2(b), we plot the nonlocal susceptibility $\tilde \chi _n (\mathbf {k})$ as function of $\mathbf {k}$ in the absence and presence of detuning $\delta$. It shows that the nonlocal susceptibility depends on both the wave number $\mathbf {k}$ and the detuning $\delta$. Its imaginary part goes negative at certain values of k, where the probe field gains strength. To show this feature, we consider a Gaussian probe field pulse in $\mathbf {k}$ space and plot the $k_z$-integrated of the probe field $\mathcal {E}(k_x,k_y)=\int \mathrm {Im}[\tilde \chi _n(\mathbf {k})]\tilde E(\mathbf {k})dk_z/ \int \tilde E(\mathbf {k})dk_z$. For comparison, we consider the phase-matched case ($\mathbf {K} = 0$) in the absence and presence of detuning $\delta$ (Figs. 2(c) and 2(d), respectively). We find that $\mathcal {E}(k_x,k_y)$ non-monotonically decays for a Gaussian probe field pulse in the $\mathbf {k}$ space. The increase of $\mathcal {E}(k_x,k_y)$ at some points directly results from the negative imaginary part of the nonlocal susceptibility and the resulting gain.

 figure: Fig. 2.

Fig. 2. (a-b) Real (dashed line) and imaginary (solid line) parts of the nonlocal susceptibility $\tilde \chi _n(\mathbf {k})$ for $\delta = 0$ (a) and $\delta = \gamma$ (b). (c-d) $\mathcal {E}(k_x,k_y)$ in the k-space when $\delta = 0$ (c) and $\delta = \gamma$ (d) in the phase-matched situation ($\mathbf {K} =0$). The other parameters are $\Omega _c=2\gamma , g= 0.1\gamma , L= 28.1\mu$m, $C_d=9.7\times 10^6\mathrm {GHz}\mu \mathrm {m}^6, C_e=7.5\times 10^6\mathrm {GHz}\mu \mathrm {m}^6$.

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3.3 Light propagation through a 3-D gas

In the following we investigate the wave equation (13) on a space-time grid. Assuming that the spatial length of the probe pulse in the $z$ direction is much larger than the range of atom-atom interactions, we make a local approximation along the z direction [23,24,33]. The last term on the right-hand side of Eq. (13) then reduces to $\int \chi _n(\mathbf {r}-\mathbf {r'}) E(\mathbf {r}')d \mathbf {r}'\simeq \int \chi _n'(\mathbf {r}_\bot -\mathbf {r'}_\bot ) E(\mathbf {r}'_\bot ,z)d \mathbf {r}'_\bot$, with $\chi _n'(\mathbf {r}_\bot -\mathbf {r'}_\bot )=\int \chi _n(\mathbf {r}-\mathbf {r'})dz'$. We also define the scaled time $\tau = z/(k_p |R_b^+|^2)$, $\bar {\mathbf {r}}_\bot = \mathbf {r}_\bot /|R_b^+|$ and the dimensionless probe amplitude $\bar E$ normalized to $\int d\bar {\mathbf {r}_\bot } |\bar E(\bar {\mathbf {r}_\bot },\tau )|^2 = 1$. Assuming $\Phi _{\mathbf {rr'}}\approx 2\pi \sqrt {|\bar {\mathbf {r}}_\bot -\bar {\mathbf {r}}'_\bot )|^2+\bar z^2}$ and, for simplicity, a homogeneous atom density, we get

$$i\frac{\partial}{\partial \tau}\bar E(\bar{\mathbf{r}}_\bot,\tau) ={-}\frac{1}{2}\bar{\nabla}_{\bot}^2 \bar E(\bar{\mathbf{r}}_\bot,\tau) + \int d\bar{\mathbf{r}'}_\bot U_l(\bar{\mathbf{r}}_\bot-\bar{\mathbf{r}'}_\bot)\bar E(\bar{\mathbf{r}}_\bot,\tau) +\int d\bar{\mathbf{r}'}_\bot U_n(\bar{\mathbf{r}}_\bot-\bar{\mathbf{r}'}_\bot)\bar E(\bar{\mathbf{r}'}_\bot,\tau)$$
where
$$U_l(\bar{r}_\bot) = \beta\int_{-\bar L_z}^{\bar L_z} g \alpha_l (\bar{r}_\bot,\bar z)d\bar z,$$
$$U_n(\bar{r}_\bot)= \beta\int_{-\bar L_z}^{\bar L_z} g \alpha_n (\bar{r}_\bot,\bar z)e^{i 2\pi\sqrt{\bar{r}_\bot^2+\bar z^2}}d\bar z,$$
with $\beta = -k_p^2|R_b^+|^5 n/2$. We call $U_l$ and $U_n$ the dimensionless axial local and nonlocal susceptibilities, respectively. The real part of the potential represents dispersion, and the imaginary part reflects the absorption properties. Obviously, according to Eqs. (20) and (21), when $r_\bot \gg R_b$, $A_{\pm }(r)\rightarrow 0$ so $U_l(r_{\bot } \gg R_b)\simeq 2\beta A_0 L_z/|R_b^+|$ and $U_n(r_{\bot } \gg R_b)\rightarrow 0$. That is, when the field point is far from the spin-wave source, the nonlocal affect can be ignored. Figures 3(a)–3(d) plot the axial potential as a function of detuning $\delta$. As shown in the Figs. 3(c)–3(d), we can change the sign of the nonlocal potential by setting different $\delta$, thus modulating the output intensity profiles.

 figure: Fig. 3.

Fig. 3. (a-b) Local potential changes with $\delta$ and $r_\bot$ showing (a) the imaginary part and (b) the real part of the local potential. (c-d) Nonlocal potential changes with $\delta$ and $r_\bot$: (c) the imaginary part and (b) the real part of the nonlocal potential. The other parameters are listed in the caption of Fig. 2.

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We then simulated the 3-D propagation of a Gaussian probe field pulse. We also assume a Gaussian distribution of atom densities. Figure 4 presents the calculated output intensity profiles for different detunings in the phase-matched case. The shape of the probe field changed throughout the propagation. In particular, the intensity increased at some points and decreased at other points. For example, when $\delta$ was 0 (no detuning), the intensity spread along the transverse direction forming a ring. When $\delta$ was $\delta =0.3\gamma$, the intensity was focused at the center of the transverse direction. When $\delta =-0.4\gamma$, the intensity was localized at certain points. In the phase-unmatched case (Fig. 5), the phase parameter comes from the nonlocal potential, and the differences between the plots reflect the effect of the nonlocal terms on the probe light propagation. The features of the light propagation strongly depended on the one-photon detuning and phase-matching relation among the involved light fields. Combined with the long-range exchange interaction, the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.

 figure: Fig. 4.

Fig. 4. Intensity profiles in the phase-matched situation ( $\mathbf {K} =0$) for different detuning values: (a) $\delta =0$, (b) $\delta =0.3\gamma$, (c) $\delta =-0.4\gamma$ and $\bar z=0\ (1), \bar z=1/3 \bar L_z\ (2), \bar z=2/3 \bar L_z\ (3)\ \texttt {and}\ \bar z=\bar L_z\ (4)$, respectively. The other parameters are $\Omega _c=\gamma , g= 0.1\gamma , V_d=0.3\gamma , \bar L_z=3 |R_b^+|$. For color coding each distribution has been normalized by the actual maximum intensity $|E_{max}|^2$.

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 figure: Fig. 5.

Fig. 5. (a1-f1): The Effective nonlocal potential ($\times 10^3$) in the phase-nonmatched situation with (a) $K_x=1, K_y=0$ (b) $K_x=K_y=1$, (c) $K_x=2, K_y=0$, (d) $K_x=K_y=2$, (e) $K_x=3, K_y=0$ and (f) $K_x=K_y=3$ with $\delta =-0.4\gamma$. (a2-f2) The corresponding compressed output intensity profile. The other parameters and the color coding is identical to Fig. 4.

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4. Summary and conclusions

In this work, we studied nonlocal light 3-D propagation in Rydberg EIT with an initial spin-wave state. We derived analytical expressions for the optical susceptibility of the medium and showed its nonlocal and k-dependent properties, through which light absorption or transmission enhances at some positions. The discussed features are expected in other initial states such as multiple Rydberg excitations. As an interesting future problem, we could investigate the coupled collective dynamics of an ensemble [35] and a propagating probe field in Rydberg CEIT experiments utilizing two interacting Rydberg states [12,28,36].

Funding

National Natural Science Foundation of China (12074433, 11871472); National Key Basic Research Program For Youth (2016YFA0301903); Natural Science Foundation of Hunan Province (2018JJ2467).

Acknowledgments

The author would like to thank Weibin Li for the fruitful discussions.

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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11. G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014). [CrossRef]  

12. J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017). [CrossRef]  

13. I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020). [CrossRef]  

14. M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020). [CrossRef]  

15. D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020). [CrossRef]  

16. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000). [CrossRef]  

17. M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001). [CrossRef]  

18. M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020). [CrossRef]  

19. M. Fleischhauer and J. P. Marangos, “Electromagnetically induced transparency : Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]  

20. D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008). [CrossRef]  

21. S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020). [CrossRef]  

22. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011). [CrossRef]  

23. S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011). [CrossRef]  

24. Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020). [CrossRef]  

25. D. Petrosyan, “Dipolar exchange induced transparency with Rydberg atoms,” New J. Phys. 19(3), 033001 (2017). [CrossRef]  

26. J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018). [CrossRef]  

27. J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012). [CrossRef]  

28. H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016). [CrossRef]  

29. K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017). [CrossRef]  

30. W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014). [CrossRef]  

31. Y. O. Dudin and A. Kuzmich, “Strongly Interacting Rydberg Excitations of a Cold Atomic Gas,” Science 336(6083), 887–889 (2012). [CrossRef]  

32. Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012). [CrossRef]  

33. C. Hang and G. Huang, “Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas,” Phys. Rev. A 98(4), 043840 (2018). [CrossRef]  

34. S.-L. Xu, H. Li, Q. Zhou, G.-P. Zhou, D. Zhao, M. R. Belić, J.-R. He, and Y. Zhao, “Parity-time symmetry light bullets in a cold Rydberg atomic gas,” Opt. Express 28(11), 16322–16332 (2020). [CrossRef]  

35. V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020). [CrossRef]  

36. C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020). [CrossRef]  

References

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  1. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
    [Crossref]
  2. T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
    [Crossref]
  3. G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
    [Crossref]
  4. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107(13), 133602 (2011).
    [Crossref]
  5. J. Otterbach, “Wigner Crystallization of Single Photons in Cold Rydberg Ensembles,” Phys. Rev. Lett. 111(11), 113001 (2013).
    [Crossref]
  6. D. Paredes-Barato and C. S. Adams, “All-Optical Quantum Information Processing Using Rydberg Gates,” Phys. Rev. Lett. 112(4), 040501 (2014).
    [Crossref]
  7. J. Wang, J. N. Byrd, I. Simbotin, and R. Côté, “Tuning Ultracold Chemical Reactions via Rydberg-Dressed Interactions,” Phys. Rev. Lett. 113(2), 025302 (2014).
    [Crossref]
  8. F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
    [Crossref]
  9. F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett. 124(4), 043402 (2020).
    [Crossref]
  10. A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
    [Crossref]
  11. G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
    [Crossref]
  12. J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
    [Crossref]
  13. I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
    [Crossref]
  14. M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
    [Crossref]
  15. D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
    [Crossref]
  16. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
    [Crossref]
  17. M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
    [Crossref]
  18. M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
    [Crossref]
  19. M. Fleischhauer and J. P. Marangos, “Electromagnetically induced transparency : Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
    [Crossref]
  20. D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
    [Crossref]
  21. S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
    [Crossref]
  22. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011).
    [Crossref]
  23. S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
    [Crossref]
  24. Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020).
    [Crossref]
  25. D. Petrosyan, “Dipolar exchange induced transparency with Rydberg atoms,” New J. Phys. 19(3), 033001 (2017).
    [Crossref]
  26. J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
    [Crossref]
  27. J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
    [Crossref]
  28. H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
    [Crossref]
  29. K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
    [Crossref]
  30. W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
    [Crossref]
  31. Y. O. Dudin and A. Kuzmich, “Strongly Interacting Rydberg Excitations of a Cold Atomic Gas,” Science 336(6083), 887–889 (2012).
    [Crossref]
  32. Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012).
    [Crossref]
  33. C. Hang and G. Huang, “Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas,” Phys. Rev. A 98(4), 043840 (2018).
    [Crossref]
  34. S.-L. Xu, H. Li, Q. Zhou, G.-P. Zhou, D. Zhao, M. R. Belić, J.-R. He, and Y. Zhao, “Parity-time symmetry light bullets in a cold Rydberg atomic gas,” Opt. Express 28(11), 16322–16332 (2020).
    [Crossref]
  35. V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020).
    [Crossref]
  36. C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020).
    [Crossref]

2020 (11)

F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
[Crossref]

F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett. 124(4), 043402 (2020).
[Crossref]

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020).
[Crossref]

S.-L. Xu, H. Li, Q. Zhou, G.-P. Zhou, D. Zhao, M. R. Belić, J.-R. He, and Y. Zhao, “Parity-time symmetry light bullets in a cold Rydberg atomic gas,” Opt. Express 28(11), 16322–16332 (2020).
[Crossref]

V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020).
[Crossref]

C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020).
[Crossref]

2018 (3)

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

C. Hang and G. Huang, “Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas,” Phys. Rev. A 98(4), 043840 (2018).
[Crossref]

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

2017 (3)

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

D. Petrosyan, “Dipolar exchange induced transparency with Rydberg atoms,” New J. Phys. 19(3), 033001 (2017).
[Crossref]

2016 (1)

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

2014 (5)

W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
[Crossref]

D. Paredes-Barato and C. S. Adams, “All-Optical Quantum Information Processing Using Rydberg Gates,” Phys. Rev. Lett. 112(4), 040501 (2014).
[Crossref]

J. Wang, J. N. Byrd, I. Simbotin, and R. Côté, “Tuning Ultracold Chemical Reactions via Rydberg-Dressed Interactions,” Phys. Rev. Lett. 113(2), 025302 (2014).
[Crossref]

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

2013 (1)

J. Otterbach, “Wigner Crystallization of Single Photons in Cold Rydberg Ensembles,” Phys. Rev. Lett. 111(11), 113001 (2013).
[Crossref]

2012 (3)

Y. O. Dudin and A. Kuzmich, “Strongly Interacting Rydberg Excitations of a Cold Atomic Gas,” Science 336(6083), 887–889 (2012).
[Crossref]

Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012).
[Crossref]

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

2011 (3)

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011).
[Crossref]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (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(13), 133602 (2011).
[Crossref]

2010 (2)

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
[Crossref]

2008 (1)

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

2005 (1)

M. Fleischhauer and J. P. Marangos, “Electromagnetically induced transparency : Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

2001 (1)

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

2000 (1)

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

Adams, C. S.

C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020).
[Crossref]

D. Paredes-Barato and C. S. Adams, “All-Optical Quantum Information Processing Using Rydberg Gates,” Phys. Rev. Lett. 112(4), 040501 (2014).
[Crossref]

Andrijauskas, T.

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Assemat, F.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Ates, C.

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
[Crossref]

Bai, Z.

Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020).
[Crossref]

Balewski, J. B.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

Bariani, F.

Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012).
[Crossref]

Barredo, D.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Belic, M. R.

Bienias, P.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Boninsegni, M.

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
[Crossref]

Boulier, T.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Brandner, K.

F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
[Crossref]

Browaeys, A.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Brune, M.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Büchler, H. P.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Butscher, B.

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

Byrd, J. N.

J. Wang, J. N. Byrd, I. Simbotin, and R. Côté, “Tuning Ultracold Chemical Reactions via Rydberg-Dressed Interactions,” Phys. Rev. Lett. 113(2), 025302 (2014).
[Crossref]

Cantat-Moltrecht, T.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Cantu, S. H.

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

Carollo, F.

F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
[Crossref]

Chang, D.

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

Choi, J.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Choi, S.

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

Cirac, J. I.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

Cooper, A.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Cortiñas, R.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Cote, R.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

Côté, R.

J. Wang, J. N. Byrd, I. Simbotin, and R. Côté, “Tuning Ultracold Chemical Reactions via Rydberg-Dressed Interactions,” Phys. Rev. Lett. 113(2), 025302 (2014).
[Crossref]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

Covey, J. P.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

de Léséleuc, S.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Demler, E. A.

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

Dotsenko, I.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Duan, L. M.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

Dudin, Y. O.

Y. O. Dudin and A. Kuzmich, “Strongly Interacting Rydberg Excitations of a Cold Atomic Gas,” Science 336(6083), 887–889 (2012).
[Crossref]

Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012).
[Crossref]

Endres, M.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Epple, G.

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Euser, T. G.

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Fedorov, I. A.

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

Fleischhauer, M.

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107(13), 133602 (2011).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011).
[Crossref]

M. Fleischhauer and J. P. Marangos, “Electromagnetically induced transparency : Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

Fortágh, J.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

Gaj, A.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

Gambetta, F. M.

F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett. 124(4), 043402 (2020).
[Crossref]

F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
[Crossref]

Garrahan, J. P.

F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
[Crossref]

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

Gleyzes, S.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Gorniaczyk, H.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Gorshkov, A. V.

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107(13), 133602 (2011).
[Crossref]

Grimmel, J.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

Gritsev, V.

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

Grünwald, P.

V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020).
[Crossref]

Günther, A.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

Hang, C.

C. Hang and G. Huang, “Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas,” Phys. Rev. A 98(4), 043840 (2018).
[Crossref]

Haroche, S.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

He, J.-R.

Henkel, N.

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
[Crossref]

Hofferberth, S.

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
[Crossref]

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

Hu, Y.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Huang, G.

Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020).
[Crossref]

C. Hang and G. Huang, “Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas,” Phys. Rev. A 98(4), 043840 (2018).
[Crossref]

Ilzhöfer, P.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

Jaksch, D.

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

Jelenkovic, B.

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

Jia, S.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Jing, M.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Jolicoeur, T.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Joly, N. Y.

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Juzeliunas, G.

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Kale, A.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Kleinbach, K. S.

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Krupp, A. T.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

Kudriašov, V.

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Kurz, M.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

Kuzmich, A.

Y. O. Dudin and A. Kuzmich, “Strongly Interacting Rydberg Excitations of a Cold Atomic Gas,” Science 336(6083), 887–889 (2012).
[Crossref]

Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012).
[Crossref]

Lahaye, T.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Lesanovsk, I.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Lesanovsky, I.

F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett. 124(4), 043402 (2020).
[Crossref]

F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, “Nonequilibrium quantum many-body rydberg atom engine,” Phys. Rev. Lett. 124(17), 170602 (2020).
[Crossref]

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
[Crossref]

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
[Crossref]

Li, H.

Li, W.

F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett. 124(4), 043402 (2020).
[Crossref]

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
[Crossref]

Liang, Q.-y.

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

Lienhard, V.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Low, R.

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Löw, R.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

Lukin, M. D.

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107(13), 133602 (2011).
[Crossref]

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

Ma, J.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Macieszczak, K.

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

Madjarov, I. S.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Marangos, J. P.

M. Fleischhauer and J. P. Marangos, “Electromagnetically induced transparency : Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Mekys, A.

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Micheli, A.

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
[Crossref]

Mirgorodskiy, I.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Mølmer, K.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

Morigi, G.

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

Nguyen, T. L.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Nicholson, T. L.

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

Nipper, J.

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

Nold, R.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

Otterbach, J.

J. Otterbach, “Wigner Crystallization of Single Photons in Cold Rydberg Ensembles,” Phys. Rev. Lett. 111(11), 113001 (2013).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107(13), 133602 (2011).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011).
[Crossref]

Paredes-Barato, D.

D. Paredes-Barato and C. S. Adams, “All-Optical Quantum Information Processing Using Rydberg Gates,” Phys. Rev. Lett. 112(4), 040501 (2014).
[Crossref]

Paris-Mandoki, A.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Petrosyan, D.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

D. Petrosyan, “Dipolar exchange induced transparency with Rydberg atoms,” New J. Phys. 19(3), 033001 (2017).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011).
[Crossref]

Pfau, T.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

Pichler, H.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Pohl, T.

V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020).
[Crossref]

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (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(13), 133602 (2011).
[Crossref]

Pritchard, J. D.

C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020).
[Crossref]

Pupillo, G.

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
[Crossref]

Raimond, J. M.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Rolston, S. L.

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

Roux, G.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Ruseckas, J.

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Russell, P. S.

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Saffman, M.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

Sayrin, C.

T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Cortiñas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, “Towards Quantum Simulation with Circular Rydberg Atoms,” Phys. Rev. X 8(1), 011032 (2018).
[Crossref]

Schkolnik, V.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Schmelcher, P.

A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, “Alignment of D -state Rydberg molecules,” Phys. Rev. Lett. 112(14), 143008 (2014).
[Crossref]

Schmidt-Kaler, F.

F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett. 124(4), 043402 (2020).
[Crossref]

Scholl, P.

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

Sevincli, S.

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
[Crossref]

Shaffer, J. P.

C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020).
[Crossref]

Shaw, A. L.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Simbotin, I.

J. Wang, J. N. Byrd, I. Simbotin, and R. Côté, “Tuning Ultracold Chemical Reactions via Rydberg-Dressed Interactions,” Phys. Rev. Lett. 113(2), 025302 (2014).
[Crossref]

Stecker, M.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

Steinert, L.-M.

M. Stecker, R. Nold, L.-M. Steinert, J. Grimmel, D. Petrosyan, J. Fortágh, and A. Günther, “Controlling the dipole blockade and ionization rate of rydberg atoms in strong electric fields,” Phys. Rev. Lett. 125(10), 103602 (2020).
[Crossref]

Thompson, J. D.

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

Tresp, C.

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

Venkatramani, A. V.

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

Viscor, D.

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
[Crossref]

Vuletic, V.

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

Walker, T. G.

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010).
[Crossref]

Walther, V.

V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020).
[Crossref]

Wang, J.

J. Wang, J. N. Byrd, I. Simbotin, and R. Côté, “Tuning Ultracold Chemical Reactions via Rydberg-Dressed Interactions,” Phys. Rev. Lett. 113(2), 025302 (2014).
[Crossref]

Williams, J. R.

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

Xiao, L.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Xu, S.-L.

Xu, W.

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

Yu, I. A.

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Zhang, H.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Zhang, L.

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

Zhang, Q.

Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020).
[Crossref]

Zhao, D.

Zhao, Y.

Zhou, G.-P.

Zhou, L.

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

Zhou, Q.

Zhou, Y.-L.

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

Zoller, P.

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, “Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics,” Phys. Rev. Lett. 104(22), 223002 (2010).
[Crossref]

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87(3), 037901 (2001).
[Crossref]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000).
[Crossref]

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

C. S. Adams, J. D. Pritchard, and J. P. Shaffer, “Rydberg atom quantum technologies,” J. Phys. B: At., Mol. Opt. Phys. 53(1), 012002 (2020).
[Crossref]

Nat. Commun. (2)

H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsk, and S. Hofferberth, “Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances,” Nat. Commun. 7(1), 12480 (2016).
[Crossref]

G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. S. Russell, and R. Low, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5(1), 4132 (2014).
[Crossref]

Nat. Phys. (4)

I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, “High-fidelity entanglement and detection of alkaline-earth Rydberg atoms,” Nat. Phys. 16(8), 857–861 (2020).
[Crossref]

M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020).
[Crossref]

D. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4(11), 884–889 (2008).
[Crossref]

S. H. Cantu, A. V. Venkatramani, W. Xu, L. Zhou, B. Jelenkovic, and M. D. Lukin, “Repulsive photons in a quantum nonlinear medium,” Nat. Phys. 16(9), 921–925 (2020).
[Crossref]

Nature (1)

J. D. Thompson, T. L. Nicholson, Q.-y. Liang, S. H. Cantu, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, and M. D. Lukin, “Symmetry-protected collisions between strongly interacting photons,” Nature 542(7640), 206–209 (2017).
[Crossref]

New J. Phys. (1)

D. Petrosyan, “Dipolar exchange induced transparency with Rydberg atoms,” New J. Phys. 19(3), 033001 (2017).
[Crossref]

Opt. Express (1)

Phys. Rev. A (4)

C. Hang and G. Huang, “Parity-time symmetry along with nonlocal optical solitons and their active controls in a Rydberg atomic gas,” Phys. Rev. A 98(4), 043840 (2018).
[Crossref]

J. Ruseckas, V. Kudriašov, A. Mekys, T. Andrijauskas, I. A. Yu, and G. Juzeliunas, “Nonlinear quantum optics for spinor slow light,” Phys. Rev. A 98(1), 013846 (2018).
[Crossref]

Z. Bai, Q. Zhang, and G. Huang, “Quantum reflections of nonlocal optical solitons in a cold Rydberg atomic gas,” Phys. Rev. A 101(5), 053845 (2020).
[Crossref]

K. Macieszczak, Y.-L. Zhou, S. Hofferberth, J. P. Garrahan, W. Li, and I. Lesanovsky, “Metastable decoherence-free subspaces and electromagnetically induced transparency in interacting many-body systems,” Phys. Rev. A 96(4), 043860 (2017).
[Crossref]

Phys. Rev. Lett. (18)

W. Li, D. Viscor, S. Hofferberth, and I. Lesanovsky, “Electromagnetically Induced Transparency in an Entangled Medium,” Phys. Rev. Lett. 112(24), 243601 (2014).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with rydberg atoms,” Phys. Rev. Lett. 107(21), 213601 (2011).
[Crossref]

S. Sevincli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal Nonlinear Optics in Cold Rydberg Gases,” Phys. Rev. Lett. 107(15), 153001 (2011).
[Crossref]

Y. O. Dudin, F. Bariani, and A. Kuzmich, “Emergence of Spatial Spin-Wave Correlations in a Cold Atomic Gas,” Phys. Rev. Lett. 109(13), 133602 (2012).
[Crossref]

J. Nipper, J. B. Balewski, A. T. Krupp, B. Butscher, R. Löw, and T. Pfau, “Highly resolved measurements of stark-tuned förster resonances between rydberg atoms,” Phys. Rev. Lett. 108(11), 113001 (2012).
[Crossref]

V. Walther, P. Grünwald, and T. Pohl, “Controlling exciton-phonon interactions via electromagnetically induced transparency,” Phys. Rev. Lett. 125(17), 173601 (2020).
[Crossref]

D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       Susceptibility in k space

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)

Fig. 1.
Fig. 1. Level scheme and transitions in the atomic medium.
Fig. 2.
Fig. 2. (a-b) Real (dashed line) and imaginary (solid line) parts of the nonlocal susceptibility $\tilde \chi _n(\mathbf {k})$ for $\delta = 0$ (a) and $\delta = \gamma$ (b). (c-d) $\mathcal {E}(k_x,k_y)$ in the k-space when $\delta = 0$ (c) and $\delta = \gamma$ (d) in the phase-matched situation ($\mathbf {K} =0$). The other parameters are $\Omega _c=2\gamma , g= 0.1\gamma , L= 28.1\mu$m, $C_d=9.7\times 10^6\mathrm {GHz}\mu \mathrm {m}^6, C_e=7.5\times 10^6\mathrm {GHz}\mu \mathrm {m}^6$.
Fig. 3.
Fig. 3. (a-b) Local potential changes with $\delta$ and $r_\bot$ showing (a) the imaginary part and (b) the real part of the local potential. (c-d) Nonlocal potential changes with $\delta$ and $r_\bot$: (c) the imaginary part and (b) the real part of the nonlocal potential. The other parameters are listed in the caption of Fig. 2.
Fig. 4.
Fig. 4. Intensity profiles in the phase-matched situation ( $\mathbf {K} =0$) for different detuning values: (a) $\delta =0$, (b) $\delta =0.3\gamma$, (c) $\delta =-0.4\gamma$ and $\bar z=0\ (1), \bar z=1/3 \bar L_z\ (2), \bar z=2/3 \bar L_z\ (3)\ \texttt {and}\ \bar z=\bar L_z\ (4)$, respectively. The other parameters are $\Omega _c=\gamma , g= 0.1\gamma , V_d=0.3\gamma , \bar L_z=3 |R_b^+|$. For color coding each distribution has been normalized by the actual maximum intensity $|E_{max}|^2$.
Fig. 5.
Fig. 5. (a1-f1): The Effective nonlocal potential ($\times 10^3$) in the phase-nonmatched situation with (a) $K_x=1, K_y=0$ (b) $K_x=K_y=1$, (c) $K_x=2, K_y=0$, (d) $K_x=K_y=2$, (e) $K_x=3, K_y=0$ and (f) $K_x=K_y=3$ with $\delta =-0.4\gamma$. (a2-f2) The corresponding compressed output intensity profile. The other parameters and the color coding is identical to Fig. 4.

Equations (23)

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$$H_j ={-}\delta (\sigma_{22}^j+\sigma_{33}^j)-\frac{1}{2}[\Omega_{p}(\mathbf{r}_j)\sigma_{21}^j + \Omega_{c}(\mathbf{r}_j)\sigma_{32}^j + H.c.],$$
$$\frac{d}{dt} \rho ={-} i [H, \rho] + \gamma \sum_{j=i}^N \mathcal{D}(\sigma_{12}^j) \rho$$
$$\dot \rho_1 \simeq{-} i [H_0, \rho_1] - i[H_1, \rho_0] + \gamma \sum_{j=1,2} (\sigma_{12}^j \rho^1 \sigma_{21}^j - \frac{1}{2} \{ \sigma_{21}^j \sigma_{12}^j, \rho^j \}).$$
$$\dot{\rho_{21}} ={-}(i\delta + \frac{1}{2} \gamma)\rho_{21} -i \frac{\Omega_c}{2}\rho_{61} - i\frac{\Omega_{p}(\mathbf{r}_1)}{4} = 0,$$
$$\dot{\rho_{61}} ={-}i(\delta + V_{12}^d)\rho_{61} -i V_{12}^e\rho_{51} - i\frac{\Omega_{c}}{2} \rho_{21} = 0,$$
$$\dot{\rho_{51}} ={-}i(\delta + V_{12}^d )\rho_{51} -i V_{12}^e\rho_{61} - i\frac{\Omega_{c}}{2} \rho_{41} = 0,$$
$$\dot{\rho_{41}} ={-}(i\delta + \frac{1}{2} \gamma )\rho_{41} -i \frac{\Omega_c}{2}\rho_{51} - i\frac{\Omega_{p}(\mathbf{r}_2)}{4}e^{i\mathbf{k}_s\cdot \mathbf{r}_{12}} = 0.$$
$$\langle \sigma_{21}(\mathbf{r}_1)\rangle ={-} \frac{\mu_{21}}{2} \left[\alpha_l(\mathbf{r}_{12}) E(\mathbf{r}_1) + \alpha_n(\mathbf{r}_{12}) e^{i\mathbf{K}\cdot \mathbf{r}_{12}} E(\mathbf{r}_2)\right]$$
$$\begin{aligned}\alpha_l(\mathbf{r}_{12}) = & \frac{\delta-C_d-Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d-C_e)}+\frac{\delta-C_d+Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d+C_e)}\\ =&\frac{2\delta}{\Omega_c^2-2i\Gamma\delta} + \frac{\Omega_c^2}{2i\Gamma(2i\Gamma\delta-\Omega_c^2)} (A_+(\mathbf{r}_{12})+ A_-(\mathbf{r}_{12})), \end{aligned}$$
$$\begin{aligned}\alpha_n(\mathbf{r}_{12}) =& \frac{\delta-C_d-Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d-C_e)}-\frac{\delta-C_d+Ce}{\Omega_c^2-2i(\gamma-2i\delta)(\delta-C_d+C_e)}\\ =& \frac{\Omega_c^2}{2i\Gamma(2i\Gamma\delta-\Omega_c^2)} (A_+(\mathbf{r}_{12})- A_-(\mathbf{r}_{12})), \end{aligned}$$
$$\sigma_{21}(\mathbf{r}) ={-}\frac{\mu_{21}N}{2}\left[\int \alpha_l (\mathbf{r} - \mathbf{r}')n(\mathbf{r}') d\mathbf{r}' E(\mathbf{r}) + \int \alpha_n (\mathbf{r} - \mathbf{r}')n(\mathbf{r}')e^{i \mathbf{K}\cdot (\mathbf{r - r'})} E(\mathbf{r}')d\mathbf{r}'\right].$$
$$\left(-\frac{i}{2k_p}\nabla_\bot^2 + \frac{\partial}{\partial z} + \frac{1}{c} \frac{\partial}{\partial t}\right)E(\mathbf{r}) = i\frac{k_p}{2\epsilon_0}\mathcal{P}(\mathbf{r})$$
$$\mathcal{P}(\mathbf{r}) = 2\mu_{12}\sigma(\mathbf{r}) = \epsilon_0 \left[ \chi_l (\mathbf{r}) E(\mathbf{r}) + \int \chi_n (\mathbf{r} - \mathbf{r'}) E(\mathbf{r}) d\mathbf{r}' \right]$$
$$\chi_l (\mathbf{r}) = g \int \alpha_l (\mathbf{r} - \mathbf{r}')n(\mathbf{r}')d\mathbf{r}',$$
$$\chi_n (\mathbf{r} - \mathbf{r'}) = g \alpha_n (\mathbf{r} - \mathbf{r}')n(\mathbf{r}') e^{i \mathbf{K}\cdot (\mathbf{r - r'})}$$
$$(-\frac{i}{2k_p}\nabla_\bot^2 + \frac{\partial}{\partial z})E(\mathbf{r})= \frac{i k_p}{2}[\chi_l(\mathbf{r})E(\mathbf{r}) +\int \chi_n(\mathbf{r}-\mathbf{r'}) E(\mathbf{r}')d \mathbf{r}'].$$
$$\chi_l(\mathbf{r})E(\mathbf{r}) = \mathcal{F}^{{-}1}\{\mathcal{F}[\chi_l(\mathbf{r})E(\mathbf{r})]\}= \mathcal{F}^{{-}1}\{\mathcal{F}[\chi_l(\mathbf{r})]\ast \mathcal{F}[E(\mathbf{r})]\}= \mathcal{F}^{{-}1} [\tilde \chi_l \tilde E(\mathbf{k})]$$
$$\int \chi_n(\mathbf{r}-\mathbf{r'}) E(\mathbf{r}')d \mathbf{r}' = \mathcal{F}^{{-}1}\{\mathcal{F}[\int \chi_n(\mathbf{r}-\mathbf{r'}) E(\mathbf{r}')d \mathbf{r}']\}=\mathcal{F}^{{-}1} [\tilde \chi_n(\mathbf{k}) \tilde E(\mathbf{k})],$$
$$\tilde \chi_l=A_0 + A_1 \left((R_b^{+})^3+(R_b^{-})^3\right),$$
$$\tilde \chi_n(\mathbf{k}) = A_1\left[(R_b^{+})^3f(|\mathbf{k}+\mathbf{K}|R_b^+) - (R_b^{-})^3f(|\mathbf{k}+\mathbf{K}|R_b^-)\right],$$
$$i\frac{\partial}{\partial \tau}\bar E(\bar{\mathbf{r}}_\bot,\tau) ={-}\frac{1}{2}\bar{\nabla}_{\bot}^2 \bar E(\bar{\mathbf{r}}_\bot,\tau) + \int d\bar{\mathbf{r}'}_\bot U_l(\bar{\mathbf{r}}_\bot-\bar{\mathbf{r}'}_\bot)\bar E(\bar{\mathbf{r}}_\bot,\tau) +\int d\bar{\mathbf{r}'}_\bot U_n(\bar{\mathbf{r}}_\bot-\bar{\mathbf{r}'}_\bot)\bar E(\bar{\mathbf{r}'}_\bot,\tau)$$
$$U_l(\bar{r}_\bot) = \beta\int_{-\bar L_z}^{\bar L_z} g \alpha_l (\bar{r}_\bot,\bar z)d\bar z,$$
$$U_n(\bar{r}_\bot)= \beta\int_{-\bar L_z}^{\bar L_z} g \alpha_n (\bar{r}_\bot,\bar z)e^{i 2\pi\sqrt{\bar{r}_\bot^2+\bar z^2}}d\bar z,$$