1. Introduction

The effect of electromagnetically induced transparency (EIT) involving Rydberg-state atoms is of great interest currently. The Rydberg-state atoms exhibit the strong electric dipole-dipole interaction (DDI) among each other [1–5]. On the other hand, the EIT effect not only provides high optical nonlinearity for the atom-light interaction, but also gives rise to slow, stored, and stationary light for long interaction time [6–14]. Thus, the combination of the strong DDI of Rydberg atoms and the high optical nonlinearity of EIT can efficiently mediate the interaction between photons via Rydberg polaritons in the dipole blockade regime, where the Rydberg polariton is the collective excitation involving the light and the atomic coherence between the ground and Rydberg states [15,16]. The Rydberg-EIT mechanism can lead to the applications of quantum optics and quantum information processing [17–32].

To our knowledge, most of the present theoretical models dealt with the Rydberg-EIT system in the strongly-interacting regime, i.e., $r_B^{3}$ is comparable to $r_a^{3}$, where $r_B$ is the blockade radius and $r_a$ is the half mean distance between Rydberg polaritons. In Ref. [17], J. D. Pritchard et al. utilized the $N$-atom model to analyze experiment phenomena of the optical nonlinearity and attenuation in the Rydberg-EIT system. In Ref. [18], D. Petrosyan et al. modeled the propagation of light field in strongly-interacting Rydberg-EIT media by considering the superatoms with the volume of the blockade sphere. In Ref. [19], A. V. Gorshkov et al. proposed a theory for the propagation of few-photon pulses in the system of strongly-interacting Rydberg polaritons. In Ref. [24], M. Moos et al. utilized a one-dimensional model to describe the time evolution of Rydberg polaritons and analyze many-body phenomena in the strongly-interacting regime. In Ref. [28], J. Ruseckas et al. proposed a method to create two-photon states by making pairs of Rydberg atoms entangled during the storage.

In this article, we considered the weakly-interacting Rydberg-EIT system, and developed a mean field model to describe the attenuation and phase shift of the output probe field induced by the DDI effect. The Rydberg-EIT system is depicted in Fig. 1(a), and the weakly-interacting condition requires $r_B^{3} \ll r_a^{3}$ [see Fig. 1(b)]. Under such condition, the system of Rydberg polaritons can be considered as nearly the ideal gas. Thus, the nearest-neighbor distribution (NND) shown by Ref. [33] is utilized in our model. The DDI-induced frequency shift between nonuniformly-distributed Rydberg excitations results in the effective phase shift and attenuation of light field. With the probability function of NND and the atom-light coupling equations of EIT system, we calculated the mean field results of transmission and phase shift spectra, and further derived the analytical formulas of the DDI-induced attenuation coefficient and phase shift. The theoretical predictions from the formulas are in good agreement with the experimental data in Ref. [34]. In the experiment of Ref. [34], we utilized the Rydberg state of a low principal number, the laser-cooled ensemble of a moderate atomic density, and the weak probe field of a low photon flux to make the mean number of Rydberg polaritons within the blockade sphere lower than 0.1. The good agreement verifies our model.

 

Fig. 1. (a) Transition diagram of the Rydberg-EIT system. $| 1 \rangle$, $| 2 \rangle$, and $|3 \rangle$ represent the ground, Rydberg, and intermediate states. The weak probe and strong coupling fields form the ladder-type EIT configuration. (b) Top and bottom figures depict the systems of strongly- and weakly-interacting Rydberg polaritons. Red and blue balls represent atoms with and without Rydberg excitations; dashed circles indicate the blockade spheres. The strong- and weak-interaction systems are characterized by $r_{B}^{3}/r_{a}^{3} \rightarrow 1$ and $r_{B}^{3}/r_{a}^{3} \ll 1$, respectively, where $r_B$ is the blockade radius and $r_a$ is the half mean distance between Rydberg excitations or polaritons. As an example, let us consider 8 photons in both systems. There are 8 Rydberg excitations in the weak-interaction system, but only 4 in the strong-interaction system due to the dipole blockade effect.

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Rydberg polaritons are regarded as bosonic quasi-particles, and the DDI-induced phase shift and attenuation coefficient can infer the elastic and inelastic collision rates in the ensemble of these particles [35]. Weakly-interacting Rydberg polaritons assisted by a long interaction time of the EIT effect can be employed in the study of many-body physics such as the Bose-Einstein condensation of polaritons [35–38]. The mean field theory developed in this work provides a useful tool to conceive ideas relevant to weakly-interacting EIT-based Rydberg polaritons and to evaluate feasibilities of experiments.

We organize the article as follows. In Sec. 2, the theoretical model based on the probability function of NND, the atom-light coupling equations of the EIT system, and the ensemble average of the DDI-induced frequency shift are introduced. We obtain the mean field results of the real and imaginary parts of the steady-state absorption cross section of the probe field. In Sec. 3, we numerically evaluate the integrals corresponding to the mean field results and present the spectra of transmission and phase shift of the output probe field. The DDI-induced phenomena observed from the spectra are discussed and explained. In Sec. 4, we derive the analytical formulas of the DDI-induced attenuation coefficient and phase shift. From the formulas, one can see how the DDI effects depend on the system parameters such as the optical depth, coupling and probe Rabi frequencies, coupling detuning, two-photon detuning, and decoherence rate. It is interesting to note that the DDI effects exhibit the asymmetric behavior with respect to the coupling detuning. In Sec. 5, we briefly describe the experimental condition and data in Ref. [34], and calculate the predictions corresponding to the experimental condition from the analytical formulas. The predictions are in good agreement with the data. Finally, we give a summary in Sec. 6.

2. Theoretical model

In the system of Rydberg polaritons, the DDI induces a frequency shift of the Rydberg state. Since Rydberg excitations are nonuniformly distributed, the Rydberg-state frequency shift is not a constant in the medium. The system of low-density Rydberg excitations can be considered as nearly the ideal gas, in which the nearest-neighbor distribution (NND) is given by [33]

 

Fig. 2. (a) Probability density $P(r)$ as a function of distance $r$ in the nearest-neighbor distribution. Units of $r_a$ defined by Eq. (2) is the half mean distance between particles. (b) Probability density $P(\omega )$ as a function of frequency shift $\omega$. Units of $\omega _a$ defined by Eq. (5) is the frequency shift corresponding to $r_a$.

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The NND is the consequence of each particle being randomly distributed. In Appendix VII of Ref. [33], Eqs. (669)-(671) and the relating descriptions explain how the NND is derived. The derivation is summarized in the following: The probability $P(r)dr$ satisfies the equation of $P(r)dr = [(4\pi r^{2} dr)n_R]\times \left [ 1-\int _{0}^{r}P(r')dr' \right ],$ where $(4 \pi r^{2} dr)n_R$ is the probability of finding one particle within the volume of $4 \pi r^{2} dr$ under the particle’s density $n_R$, and $1-\int _{0}^{r}P(r')dr'$ is that of all the remaining particles locating outside a sphere of the radius $r$. By eliminating $dr$ and taking the derivative on both sides of the equation, we obtain $(d/dr) \left [ P(r)/(4 \pi r^{2} n_R) \right ] = -P(r)$. The solution of this differential equation gives Eq. (1). Thus, as long as the interaction between the Rydberg excitations is weak enough to maintain the nature of random distribution, Eq. (1) is valid for the Rydberg-EIT system.

The frequency shift of a Rydberg state induced by the DDI is $C_6/r^{6}$, where $C_6$ is the van der Waals coefficient [39] and $r$ represents the distance between two particles. In the ensemble of Rydberg excitations, the Rydberg-state frequency shift consists of two parts. The first part is $C_6/r^{6}$ contributed from the nearest-neighbor Rydberg excitation at the distance $r$, and the second part is $n_R \int _{r}^{\infty } (C_6/r'^{6}) 4\pi r'^{2} dr'$ contributed from all the other Rydberg excitations outside the sphere of the radius $r$. Here, we consider the particles in the second part are uniformly distributed. Thus, the Rydberg-state frequency shift is the following:

In the EIT system shown in Fig. 1(a), the weak probe field drives the transition between the ground state $|1\rangle$ and the intermediate state $|3\rangle$, and the strong coupling field drives that between $|3\rangle$ and the Rydberg state $|2\rangle$. We consider the steady-state continuous-wave case in this work. As the system reaches its steady state, a given amount of Rydberg excitations with the density $n_R$ are produced and distributed according to Eq. (1). The weak probe field propagates through the system consisting of the atoms with their Rydberg-state levels shifted by the existing Rydberg excitations via the DDI [40]. We will first use the optical Bloch equation (OBE) to calculate the optical coherence of the probe transition, which determines the susceptibility of the probe field. Then, the susceptibility will be averaged over the frequency shift $\omega$ according to the distribution function in Eq. (4), producing a factor of $n_R$ in the averaged susceptibility. Because $n_R$ is proportional to the square of the probe-field amplitude or Rabi frequency, this gives rise to the nonlinearity in the system. Finally, we will employ the averaged susceptibility in the Maxwell-Schrödinger equation (MSE) to obtain the attenuation and phase shift of the probe field caused by the DDI-shifted Rydberg-state levels.

We utilize the OBE of the atomic density matrix and the MSE of the probe field of the EIT system in the theory. The complete OBE and MSE are shown below, but their time-derivative terms are dropped in the calculation because we consider the steady-state case.

We will determine the optical coherence, $\rho _{31}$, which is responsible for the attenuation coefficient and phase shift of the probe field. Equations (6) and (7) without the time-derivative terms are used to obtain the steady-state solution of $\rho _{31}$ given by

 

Fig. 3. Imaginary and real parts of $(\rho _{31}/\Omega _p)\Gamma$ as functions of the frequency shift $\omega$ at the two-photon resonance. We calculate the spectra by making the substitutions of first $\Delta _c \rightarrow \Delta _c + \omega$ and then $\Delta _p \rightarrow -\Delta _c$ in Eq. (12) with $\Omega _c = 1.0\Gamma$, $\gamma _0 = 0$, and $\Delta _c = 1.0\Gamma$ in (a,d), 0 in (b,e), and $-1.0\Gamma$ in (c,f).

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The dipole blockade effect is that an atom inside the blockade sphere cannot be excited to the Rydberg state, where the blockade sphere centering with a Rydberg excitation has the radius $r_B \equiv (2 C_6 \Gamma /\Omega _c^{2})^{1/6}$ [20]. This effect has already been included in the integrals of Eqs. (16) and (17). In the weakly-interacting system considered here, i.e., $r_{B}^{3} \ll r_a^{3}$, the average number of Rydberg excitations per volume of the blockade sphere is far less than one, and thus the dipole blockade appears rarely.

To evaluate Eqs. (16) and (17), one needs to know the value of $\omega _a$ in $P(\omega )$. According to the definition of $\omega _a$ in Eq. (5) and that of $r_a$ in Eq. (2), we can relate $\omega _a$ to the Rydberg-polariton density, $n_R$, as $\omega _a = |C_6| [(4\pi /3) n_R]^{2}$. The product of the atomic density, $n_\textrm {atom}$, and the average Rydberg-state population, $\bar {\rho }_{22}$, gives $n_R$, and therefore $\omega _a = |C_6| [(4\pi /3) n_\textrm {atom} \bar {\rho }_{22}]^{2}$. The DDI-induced nonlinear and many-body effects make $\bar {\rho }_{22}$ no longer be the steady-state solution of the OBE shown in Eqs. (6)-(10). Nevertheless, one can phenomenologically associate $\bar {\rho }_{22}$ to the steady-state solution of Rydberg-state population at the input, $\rho _{22,\textrm {in}}$, by introducing a parameter $\varepsilon$. Substituting $\varepsilon \rho _{22,\textrm {in}}$ for $\bar {\rho }_{22}$, we obtain

3. Predictions of transmission and phase-shift spectra

The spectra of probe transmission and phase shift under the DDI effect are obtained by numerically evaluating the integrals of Eqs. (16) and (17) with the value of $\omega _a$ given by Eq. (18). Figures 4(a)–4(c) show the probe transmission versus the probe detuning at the coupling detunings of $+1\Gamma$, 0, and $-1\Gamma$; similarly, Figs. 4(d)–4(e) show the probe phase shift. The spectra without and with the DDI are calculated with Eq. (14) [or Eq. (15)] and Eq. (16) [or Eq. (17)], respectively.

 

Fig. 4. (a-c) Transmissions of the probe field as functions of the probe detuning, $\Delta _p$. (d-f) Phase shifts of the probe field as functions of $\Delta _p$. The left two, middle two, and right two figures correspond to the coupling detunings, $\Delta _c$, of $+1.0\Gamma$, 0, and $-1.0\Gamma$, respectively. The vertical axes of the top three figures have the same scale, and those of the bottom three figures also have the same scale. Black lines represent predictions without DDI. Red, green, and blue lines represent predictions with DDI at $\Omega _{p,\rm {in}} =$ 0.05$\Gamma$, 0.1$\Gamma$, and 0.2$\Gamma$. All the predictions are calculated with $\alpha$ = 81, $\Omega _c$ = 1.0$\Gamma$, and $\gamma _0 = 0$ in Eqs. (16) and (17), and $|C_6|[(4\pi /3)n_\textrm {atom}\varepsilon ]^{2} = 0.35\Gamma$ in Eq. (18).

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The DDI-induced phenomena exhibited in the transmission and phase shift spectra are summarized as follows: (1) A larger probe intensity results in a smaller transmission or larger attenuation. (2) A larger probe intensity results in a larger phase shift. (3) With the same probe intensity, the EIT peak transmission at a positive coupling detuning (e.g., $\Delta _c = +1\Gamma$) is larger than that at a negative coupling detuning (e.g., $\Delta _c = -1\Gamma$), where the positive and negative detunings have the same magnitude. (4) With the same probe intensity, the phase shift of a positive coupling detuning (e.g., $\Delta _c = +1\Gamma$) is larger than that of a negative coupling detuning (e.g., $\Delta _c = -1\Gamma$), where the positive and negative detunings have the same magnitude. (5) The position of the EIT peak transmission at $\Delta _c = +1\Gamma$ changes very little and locates around $\delta = 0$; that at $\Delta _c = -1\Gamma$ shifts away from $\delta = 0$ significantly and a larger probe intensity induces a greater shift. We will explain the five phenomena below.

First of all, the peak transmission decreases against the probe Rabi frequency [see Figs. 4(a), 4(b), and 4(c)]. This is expected, because the Rydberg-state population is proportional to the probe intensity or Rabi frequency square. A larger Rydberg-state population or Rydberg-polariton density makes $\omega _a$ larger as shown by Eq. (18). The probability density $P(\omega )$ with the larger $\omega _a$ has a broader width and a longer tail as demonstrated by Fig. 2(b). Under the broader $P(\omega )$, more atoms have the Rydberg-state frequency shifted away from the EIT resonance condition, reducing the peak transmission more. Secondly, the phase shift increases against the probe Rabi frequency [see Figs. 4(d), 4(e), and 4(f)]. The explanation is similar to that in the first phenomenon.

The third phenomenon observed in the spectra is the asymmetry in the peak transmissions of Fig. 4(a) and Fig. 4(c). With the same value of $\Omega _{p,\rm {in}}^{2}$, the DDI-induced reduction of the peak transmission at $\Delta _c =$ $+1\Gamma$ is less than that at $\Delta _c =$ $-1\Gamma$. This can be explained with the help of Figs. 3(a) and 3(c), which show $\textrm {Im}[\rho _{31}/\Omega _p]$ as functions of $\omega$ at the two-photon resonance for $\Delta _c = +1\Gamma$ and $-1\Gamma$, respectively. To obtain the probe transmission, the integration of Eq. (16) is performed only for the region of $\omega >0$. In Fig. 3(a), the large sharp absorption peak, corresponding to the two-photon transition, in the spectrum of $\textrm {Im}[\rho _{31}/\Omega _p]$ locates at the left to $\omega < 0$ and plays no role in the DDI-induced effect. The value of $\textrm {Im}[\rho _{31}/\Omega _p]$ is always small for $\omega > 0$, producing a smaller value of $\int _{0}^{\infty } d\omega P(\omega ) \textrm {Im}[\rho _{31}/\Omega _p]$, i.e., a higher probe transmission. On the other hand, in Fig. 3(c), the large sharp absorption peak locates at the right to $\omega$. This peak produces a larger value of $\int _{0}^{\infty } d\omega P(\omega ) \textrm {Im}[\rho _{31}/\Omega _p]$, and reduces the transmission significantly. Thus, the location of the two-photon-transition peak with respect to $\omega = 0$ in the spectrum of $\textrm {Im}[\rho _{31}(\omega )/\Omega _p]$ is responsible for the asymmetry that at the same probe Rabi frequency the peak transmission in Fig. 4(a) is larger than that in Fig. 4(c).

The fourth phenomenon observed in the spectra is that the probe intensity or $\Omega _{p,\rm {in}}^{2}$ has a much larger effect on the phase shift of $\delta = 0$ at $\Delta _c =$ 1$\Gamma$ as shown by Fig. 4(d) than that at $\Delta _c =$ $-1\Gamma$ as shown by Fig. 4(f). This can be explained with the help of Figs. 3(d) and 3(f), which show $\textrm {Re}[\rho _{31}/\Omega _p]$ at $\Delta _c =$ 1$\Gamma$ and $-1\Gamma$, respectively. To obtain the phase shift, the integration of Eq. (17) is performed only for the region of $\omega >0$. In Fig. 3(d), the value of $\textrm {Re}[\rho _{31}/\Omega _p]$ is always positive for $\omega > 0$, resulting in a larger value of $\int _{0}^{\infty } d\omega P(\omega ) \textrm {Re}[\rho _{31}/\Omega _p]$, i.e., a larger phase shift. On the other hand, in Fig. 3(f), $\textrm {Re}[\rho _{31}/\Omega _p]$ has both positive and negative values for $\omega > 0$, because the resonance of the two-photon transition locates at $\omega > 0$. The cancellation between positive and negative values of the integrand makes $\int _{0}^{\infty } d\omega P(\omega ) \textrm {Re}[\rho _{31}/\Omega _p]$ nearly zero, i.e., almost no phase shift. Therefore, with the same value of $\Omega _{p,\rm {in}}$, the phase shift at $\Delta _c =$ 1$\Gamma$ shown by Fig. 4(d) is significant, and that at $\Delta _c =$ $-1\Gamma$ shown by Fig. 4(f) is little.

The fifth phenomenon is that at $\Delta _c = +1\Gamma$ the EIT peak positions of different $\Omega _p$ are all very close to $\delta = 0$ as shown in Fig. 4(a), and at $\Delta _c = -1\Gamma$ those of $\Omega _p = 0.1\Gamma$ and $0.2\Gamma$ shift away from $\delta = 0$ significantly as shown in Fig. 4(c). Furthermore, in Fig. 4(c) a larger value of $\Omega _p$ results in a larger shift. Because of $C_6 < 0$, all the shifts should be negative as expected. Please refer to Figs. 3(a) and 3(c) plotted at $\Delta _c = +1\Gamma$ and $-1\Gamma$, respectively. In each of the two plots, a negative shift (or a smaller value of $\Delta _p$) makes the entire solid line move to the right. In Fig. 3(a), the magnitude of the shift can only be small, otherwise the big sharp absorption peak moves toward $\omega = 0$, and the result of the integration from $\omega = 0$ to $\infty$ becomes larger, i.e., the probe transmission decreases. On the other hand, in Fig. 3(c) the magnitude of the shift can be large such that the big sharp absorption peak moves further away from $\omega = 0$, and the result of the integration becomes smaller, i.e., the probe transmission increases. Thus, the magnitude of the shift is asymmetric with respect to the one-photon detuning. In the Appendix, we will derive an analytical formula to quantitatively predict the DDI-induced frequency shift of the EIT peak.

4. Analytical formulas of the DDI-induced attenuation coefficient and phase shift

We now derive the analytical formulas for the DDI-induced attenuation coefficient, $\Delta \beta$, and phase shift, $\Delta \phi$, at the condition of $\gamma _0 = 0$ and $\delta = 0$ (or $\Delta _p = -\Delta _c$). Here, $\Delta \beta$ (or $\Delta \phi$) is defined as the difference between the values of $\beta$ (or $\phi$) with and without the DDI effect, i.e., $\Delta \beta \equiv \beta -\beta _0$ and $\Delta \phi \equiv \phi -\phi _0$.

At $\gamma _0 = 0$ and $\delta = 0$, Eq. (12) gives $\beta _0 = 0$ and $\phi _0 = 0$, and thus $\Delta \beta$ = $\beta$ and $\Delta \phi$ = $\phi$. Replacing $\Delta _p$ by $-\Delta _c$ in $\beta$ of Eq. (16) and in $\phi$ of Eq. (17), we obtain $\Delta \beta$ and $\Delta \phi$ as follows:

In Fig. 5, we compare the results of the above two analytical formulas with those of the numerical integrations of Eqs. (19) and (20) without the approximation of $P(\omega )$. The agreement between the results of the analytical formulas and numerical integrations is satisfactory except the line of $\Delta \phi$ at $\Omega _c = 1.0\Gamma$ in the region of $\Omega _{p,\rm {in}}^{2} >$ 0.02$\Gamma ^{2}$. In this region, $(\omega _a/\Delta \omega _\textrm {EIT})^{1/2} \ll 1$ is no longer well satisfied, and the deviation between the analytical formula and the numerical integration becomes observable. Figures 5(a) and 5(c) demonstrate that both of $\Delta \beta$ and $\Delta \phi$ are proportional to $\Omega _{p,\rm {in}}^{2}/\Omega _c^{3}$. Figure 5(b) [or 5(d)] shows the asymmetric phenomenon that the value of $\Delta \beta$ (or $\Delta \phi$) at the coupling detuning of $|\Delta _c|$ is smaller (or larger) than that at the coupling detuning of $-|\Delta _c|$.

 

Fig. 5. (a,c) The DDI-induced attenuation coefficient $\Delta \beta$ and phase shift $\Delta \phi$ as functions of $\Omega _{p,\rm {in}}^{2}$ under $\Delta _p = \Delta _c =0$. (b,d) $\Delta \beta$ and $\Delta \phi$ as functions of $\Delta _c$ under $\delta = 0$ and $\Omega _{p,\rm {in}} = 0.1\Gamma$. The horizontal axes of the left two figures have the same scale, so do those of the right two figures. Red, cyan, and green solid lines represent the numerical evaluations of the integrals in Eqs. (19) and (20) at $\Omega _c =$ 1.0$\Gamma$, 1.4$\Gamma$, and 2.0$\Gamma$. Dashed lines are the results of the analytical formulas given by Eqs. (23) and (24). All the predictions are calculated with $\alpha$ = 81, $\gamma _0 = 0$, and $|C_6|[(4\pi /3)n_\textrm {atom}\varepsilon ]^{2} = 0.35\Gamma$.

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In reality, there exist a nonzero decoherence rate $\gamma _0$ and the two-photon detuning $\delta$ in the system. We need to consider the corrections of $\gamma _0$ and $\delta$ to the analytical formulas. Under the condition of $\Omega _c^{2} \gg \gamma _0 \Gamma ,\delta \Gamma$, the attenuation coefficient and phase shift without the DDI effect, $\beta _0$ and $\phi _0$, are approximately given by

5. Simulation of the experimental data

To verify the mean field theory developed in this work, we systematically measured the attenuation coefficient, $\beta$, and phase shift, $\phi$, of the output probe field as shown in Fig. 2 of Ref. [34]. The experiment was carried out in cold $^{87}$Rb atoms with the temperature of 350 $\mu$K. The ground state $|1\rangle$, Rydberg state $|2\rangle$, and excited state $|3\rangle$ in the EIT system here correspond to $|5S_{1/2}, F=2, m_F=2\rangle$, $|32D_{5/2}, m_J=5/2\rangle$, and $|5P_{3/2}, F=3, m_F=3\rangle$ in the experiment. We set $\Omega _c = 1.0\Gamma$ and the Rydberg state has $C_6 = -2\pi \times$260 MHz$\cdot \mu$m$^{6}$. The values of $\Omega _c$ and $C_6$ result in $r_B^{3} \approx 9.3 \mu m^{3}$. Furthermore, the atomic density $n_\textrm {atom}$ was about 0.05 $\mu$m$^{-3}$ and $\Omega _{p,\textrm {in}} \leq 0.2\Gamma$. The values of $n_\textrm {atom}$, $\Omega _c$, and $\Omega _{p,\textrm {in}}$ give $r_a^{3} \geq 120$ $\mu$m$^{3}$. Thus, $r_B^{3}/r_a^{3} \leq 0.08$, showing that the Rydberg polaritons are weakly-interacting in the experiment. With a given Rydberg state, a low value of $n_\textrm {atom} \Omega _{p,\textrm {in}}^{2}/\Omega _c^{3}$ make Rydberg polaritons weakly interacting. Nevertheless, to observe the DDI effect in the weak-interaction regime, a high OD is the necessary condition. The OD of the cold atom cloud was about 81 in the experiment. Other experimental details can be found in Ref. [34].

In Fig. 6, we made the predictions with Eqs. (27), (28), (33), and (34) for the comparison with the experimental data in Fig. 2 of Ref. [34]. The calculation parameters of OD, coupling Rabi frequency, two-photon detuning, and decoherence rate were determined experimentally. As for the value of $\sqrt {|C_6|} n_\textrm {atom} \varepsilon$ used in the calculation, $n_\textrm {atom}$ mentioned above is estimated from the experimental condition, and $\varepsilon$ is determined by fitting the experimental data of the slopes of $\beta$ and $\phi$ versus $\Delta _c$. In the fitting, $\varepsilon$ is the only fitting parameter and the best fits give $\varepsilon = 0.43$. Figs. 6(a) and 6(c) show the attenuation coefficient, $\beta$, and phase shift, $\phi$, of the output probe field as functions of $\Omega _{p,\rm {in}}^{2}$, where $\Omega _{p,\rm {in}}$ (and also $\Omega _c$) is the Rabi frequency at the center of the input Gaussian beam in the experiment. In the derivation of Eqs. (33) and (34), we do not consider the Gaussian intensity profiles of the probe and coupling fields. Nevertheless, as shown in Eqs. (18) and (22) the phenomenological parameter $\varepsilon$ relates the average Rydberg-state population $\bar {\rho }_{22}$ in the medium to the value of $\Omega _{p,\textrm {in}}^{2}/\Omega _c^{2}$. The parameter $\varepsilon$ can account for the correction factor for the effect of nonuniform intensity profiles of the light fields and that of decay of the probe field in the medium. Figure 6(b) [or 6(d)] shows the slope of the straight line of $\beta$ (or $\phi$) versus $\Omega _{p,\textrm {in}}^{2}$ as a function of $\Delta _c$. Note that the decoherence rate, $\gamma _0$, of 0.012$\Gamma$ makes the $y$-axis interception, i.e., $\beta _0$ or $\phi _0$, becomes nonzero according to Eqs. (27) and (28), and changes the slopes very little according to Eqs. (33) and (34).

 

Fig. 6. Simulation of the experimental data shown in Fig. 2 of Ref. [34]. In the simulation, $\alpha$ = 81, $\Omega _c =$ 1.0$\Gamma$, $\delta = 0$, $\gamma _0 =$ 0.012$\Gamma$, and $|C_6|[(4\pi /3)n_\textrm {atom}\varepsilon ]^{2} = 0.35\Gamma$. (a,c) Attenuation coefficient $\beta$ and phase shift $\phi$ as functions of $\Omega _{p,\rm {in}}^{2}$ at $\Delta _c = -2\Gamma$ (black), $-1\Gamma$ (red), 0 (blue), 1$\Gamma$ (magenta), and 2$\Gamma$ (olive). (b,d) Slope of $\beta$ versus $\Omega _{p,\rm {in}}^{2}$ and that of $\phi$ versus $\Omega _{p,\rm {in}}^{2}$ as functions of $\Delta _c$.

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In Fig. 2 of Ref. [34], the circles are the experimental data and the lines are their best fits. One can clearly observe the important characteristics of asymmetry in the data of slope versus $\Delta _c$. The consistency between the theoretical predictions in Fig. 6 here and the experimental data in Fig. 2 of Ref. [34] is satisfactory. The discrepancies in the $y$-axis interceptions of straight lines between the predictions and best fits are minor, and can be explained by the uncertainties or fluctuations of $\delta$ and $\gamma _0$ in the experiment. Therefore, the mean field theory developed in this work is confirmed by the experimental data.

6. Conclusion

In summary, a mean field theory based on the nearest-neighbor distribution is developed to describe the DDI effect in the system of weakly-interacting EIT-Rydberg polaritons. We deal with the steady-state continuous-wave case in this work. As the system driven by the probe and coupling fields reaches its steady state, Rydberg excitations or polaritons of a given density are produced and locate randomly as described by the nearest-neighbor distribution in Eq. (1). The probe field propagates through the system consisting of the atoms with their Rydberg-state levels shifted by the already existing Rydberg excitations via the DDI. We calculate the optical coherence $\rho _{31}$ of the probe transition, and average $\rho _{31}$ over the frequency shift $\omega$ according to the probability density function of $\omega$ in Eq. (4). The averaged $\rho _{31}$ determines the attenuation and phase shift of the probe field caused by the shifted Rydberg-state levels. The numerically-calculated spectra of probe transmission and phase shift are shown in Fig. 4. We explain the DDI-induced phenomena observed from the spectra. To make the theory convenient for predicting experimental outcomes and evaluating experimental feasibility, analytical formulas of the DDI-induced attenuation coefficient, $\Delta \beta$, and phase shift, $\Delta \phi$, are derived. As long as $\omega _a$ is much smaller than the EIT linewidth, the results of analytical formulas are in good agreement with those of numerical calculations. According to the formulas, $\Delta \beta$ and $\Delta \phi$ are linearly proportional to $\Omega _{p,\rm {in}}^{2}$ as demonstrated in Figs. 5(a) and 5(c), and $\Delta \beta$ and $\Delta \phi$ as functions of $\Delta _c$ are asymmetric with respect to $\Delta _c = 0$ as demonstrated in Figs. 5(b) and 5(d). We further consider the existences of nonzero but small decoherence rate and two-photon detuning in the system, and make corrections to the formulas of $\Delta \beta$ and $\Delta \phi$ as shown in Eqs. (33) and (34). Finally, we make the predictions with the parameters determined experimentally and compare them with the experimental data in Ref. [34]. The good agreement between the predictions and data demonstrates the validity of our theory. Here the steady-state density of Rydberg polaritons is given in the present method, and we have not investigated the transient evolution of Rydberg-polariton density. The theoretical method for the study of nonlinear dynamics of Rydberg polaritons, such as transient behavior and pulse propagation, can be referred to Ref. [41]. Rydberg polaritons are regarded as bosonic quasi-particles, and the DDI is the origin of the interaction between the particles. Thus, the DDI-induced phase shift and attenuation coefficient can infer the elastic and inelastic collision rates in the ensemble of these bosonic particles. Our mean field theory provides a useful tool for conceiving ideas relevant to the EIT system of weakly-interacting Rydberg polaritons, and for evaluating experimental feasibility.

Appendix

The DDI effect shifts the position of the EIT peak in the transmission spectrum a little at $\Delta _c = +1\Gamma$ as shown by Fig. 4(a) and significantly at $\Delta _c = -1\Gamma$ as shown by Fig. 4(c). We will derive an analytical formula to quantitatively predict the DDI-induced frequency shift of the EIT peak in this Appendix.

We start with $\rho _{31}/\Omega _p$ in Eq. (12). Since we are interested in the EIT peak position but not transmission, $\gamma _0 = 0$ is used in the derivation for simplicity and without sacrificing the generality. The DDI-induced frequency shift of a Rydberg state results in the replacement of $\Delta _c \rightarrow \Delta _c + \omega$ in Eq. (12). The spectra in Fig. 4 are obtained by sweeping the probe frequency at a given coupling detuning. Thus, $\Delta _p$ is expressed by $-\Delta _c + \delta$ in Eq. (12), and $\Delta _c$ is treated as a fixed parameter. Under the condition of $\delta \ll \Delta \omega _\textrm {EIT}$, we expand $\textrm {Im}[\rho _{31}/\Omega _p]$ with respect to $\delta$ as the followings:

(35)$$\textrm{Im}\left[\frac{\rho_{31}}{\Omega_p}\right] \approx C_0 + C_1 \delta + C_2 \delta^{2},$$

where

(36a)$$\begin{aligned} C_0 = \frac{4\omega^{2}\Gamma} {4\omega^{2}\Gamma^{2}+(\Omega_c^{2}+4\Delta_c\omega)^{2}}, \end{aligned}$$

(36b)$$\begin{aligned} C_1 = \frac{ 8\Gamma(16\omega^{4}\Delta_c+4\omega^{3}\Omega_c^{2}+4\omega^{2}\Delta_c\Omega_c^{2} +\omega\Omega_c^{4})} {[4\omega^{2}\Gamma^{2}+(\Omega_c^{2}+4\Delta_c\omega)^{2}]^{2}}, \end{aligned}$$

(36c)$$\begin{aligned} C_2 = \frac{4 \Gamma}{\Omega_c^{4}}[1 + \xi(\omega, \Delta_c, \Omega_c)], \end{aligned}$$

where $\xi$ is a complicate function of $\omega$, $\Delta _c$, and $\Omega _c$. Because $(\omega _a/\Delta \omega _\textrm {EIT})^{1/2} \ll 1$ is satisfied in the weak-interaction regime, the contribution of $\xi$ to the integration $\int d\omega P(\omega ) C_2$ is negligible, and we can drop $\xi$ from the derivation of the analytical formula of $\delta _\textrm {shift}$ for simplicity. Next, we average $\textrm {Im}[\rho _{31}/\Omega _p]$ over $\omega$ with the NND and obtain $\int d\omega P(\omega ) \textrm {Im}\left [ \rho _{31}/\Omega _p \right ] \approx \left [ \int d\omega P(\omega ) C_0 \right ] +\left [ \int d\omega P(\omega ) C_1 \right ] \delta +\left [ \int d\omega P(\omega ) C_2 \right ] \delta ^{2}$. With the DDI effect, the EIT peak position shifts to $\delta _\textrm {shift}$, which minimizes $\int d\omega P(\omega ) \textrm {Im}\left [ \rho _{31}/\Omega _p \right ]$. Since $\int d\omega P(\omega ) \textrm {Im}\left [ \rho _{31}/\Omega _p \right ]$ is a quadratic function of $\delta$, its minimum locates at

(37)$$\delta_\textrm{shift} =-\frac{\int d\omega P(\omega) C_1}{2 \int d\omega P(\omega)C_2}.$$

Because $C_2$ is independent of $\omega$ after $\xi$ is dropped, the evaluation of $\int d\omega P(\omega ) C_2$ gives $C_2$. In the evaluation of $\int d\omega P(\omega ) C_1$, we approximate $P(\omega )$ as $P'(\omega )$ of Eq. (21) to obtain an analytical expression. Finally, the frequency shift of the EIT peak is given by

(38)$$\delta_\textrm{shift} = -\frac{\pi^{2}\sqrt{|C_6|} n_\textrm{atom} \varepsilon}{12\Gamma\Omega_c} \left[ 3\sqrt{W_c-2\Delta_c} + \left( 2\Delta_c\sqrt{W_c-2\Delta_c} +\Gamma\sqrt{W_c+2\Delta_c} \right) \frac{\Omega_c^{2}}{W_c^{3}} \right] \Omega_{p,\textrm{in}}^{2}.$$

As expected, the frequency shift of the EIT peak is always negative due to $C_6 < 0$. We can make the substitutions of $\delta _\textrm {shift} \rightarrow -\delta _\textrm {shift}$ and $\Delta _c \rightarrow -\Delta _c$ to obtain the formula for $C_6 > 0$.

It can be seen from Eq. (38) that the magnitude of $\delta _\textrm {shift}$ is linearly proportional to $\Omega _{p,\textrm {in}}^{2}$ and independent of the optical depth $\alpha$. Since $W_c \equiv \sqrt {\Gamma ^{2} + 4\Delta _c^{2}}$, the result of $\delta _\textrm {shift}$ as a function of $\Delta _c$ shows the magnitude of $\delta _\textrm {shift}$ at $+|\Delta _c|$ is less than that at $-|\Delta _c|$ as long as $|\Delta _c| \geq \Omega _c/2$. This is consistent with the phenomena shown in Figs. 4(a) and 4(c) that the frequency shift of the EIT peak at $\Delta _c = +1.0\Gamma$ is a little and that at $\Delta _c = -1.0\Gamma$ is significant. Figure 7(a) compares $|\delta _\textrm {shift}|$ predicted by Eq. (38) with that determined from the numerically-calculated spectrum. As long as the condition of $|\delta _\textrm {shift}| \ll \Delta \omega _\textrm {EIT}$ is satisfied, the analytical formula is in the good agreement with the numerical result.

 

Fig. 7. (a) Magnitude of the DDI-induced EIT peak shift, $\delta _\textrm {shift}$, as a function of $\Delta _c$. Solid lines are the predictions of Eq. (38), and circles are those obtained from the spectra numerically calculated by sweeping the probe frequency. (b) Magnitude of $\delta _\textrm {shift}$ as a function of $\Delta _p$. Solid lines are the predictions of Eq. (39), and circles are those obtained from the spectra numerically calculated by sweeping the coupling frequency. In (a) and (b), $\Omega _p = 0.2\Gamma$, and $\Omega _c = 1.0\Gamma$ (red), 1.4$\Gamma$ (cyan), and 2.0$\Gamma$ (green). All the predictions are calculated with $\gamma _0 = 0$ and $|C_6|[(4\pi /3)n_\textrm {atom}\varepsilon ]^{2} = 0.35\Gamma$.

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The Rydberg-EIT spectra can also be obtained by sweeping the coupling frequency at a given probe detuning. An analytical formula for the DDI-induced EIT peak shift in such spectra is useful. We derive the formula by using $\rho _{31}/\Omega _p$ of Eq. (12) again. In Eq. (12), $\Delta _c$ is expressed by $-\Delta _p + \delta$ and $\Delta _p$ is treated as a fixed parameter. We expand $\textrm {Im}[\rho _{31}/\Omega _p]$ with respect to $\delta$, and follow the similar procedure in the paragraph consisting of Eq. (38). Finally, the frequency shift of the EIT peak is given by

(39)$$\begin{aligned} \delta_\textrm{shift} = -\frac{\pi^{2}\sqrt{|C_6|} n_\textrm{atom} \varepsilon}{4\Gamma\Omega_c} \sqrt{W_p+2\Delta_p} \; \Omega_{p,\textrm{in}}^{2}, \end{aligned}$$

(40)$$\begin{aligned}W_p \equiv \sqrt{\Gamma^{2}+4\Delta_p^{2}}. \end{aligned}$$

The behavior of Eq. (39) is similar to that of Eq. (38), except that the dependence of $\Delta _p$ in Eq. (39) quantitatively differs from that of $\Delta _c$ in Eq. (38). Figure 7(b) compares $|\delta _\textrm {shift}|$ predicted by Eq. (39) with that determined from the numerically-calculated spectra. Degrees of consistency between the analytical predictions and the numerical results in Fig. 7(b) are similar to those in Fig. 7(a).

Funding

Ministry of Science and Technology, Taiwan (107-2745-M-007-001, 108-2639-M-007-001-ASP).

Acknowledgments

This work was supported by Grant Nos. 107-2745-M-007-001 and 108-2639-M-007-001-ASP of the Ministry of Science and Technology, Taiwan.

Disclosures

The authors declare no conflicts of interest.

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