1. Introduction

Due to the strong interaction ($\propto n^{11}$ with $n$ principal quantum number) [1], Rydberg atoms provides an ideal platform to implement quantum information and quantum simulation [2–5] and investigate interaction induced cooperative optical nonlinearities [6]. The optical nonlinear effects are induced by Rydberg atom interactions [7,8], i.e., van der Waals (vdW) interactions [9–11] and dipole-dipole interactions [12,13]. Strong atomic interactions can be effectively mapped onto photon-photon interactions via electromagnetically induced transparency (EIT) [6]. Due to the cooperative effects, the optical nonlinearity can be greatly enhanced [6,14–16]. Based on this, Rydberg EIT experiments are employed to measure a radio-frequency electric field [17] with a room-temperature cell, and to realize few-photon optical nonlinearities [18–21] with an ultracold sample, such as the efficient single photon generation [18], entanglement generation between light and atomic excitations [20], single-photon switches [16,19,22] and transistors [21,23,24]. The resonant dipole-dipole interaction between two individual Rydberg atoms [25] is angular dependent. Recently, the anisotropic Rydberg interaction have been adopted to investigate Rydberg polaritons by using Rydberg $nD$-state [26].

In this work, we present a Rydberg EIT spectrum of a cascade three-level cesium atom involving $80D_{5/2}$ state in a dipole trap. Under the two-photon resonance condition, we observe that EIT transmission displays a slow decrease with time. This could be a signature of interaction induced metastability [27,28]. Due to the large dipole matrix elements, the cesium $nD$-state atom has strong dipole interactions with energetically close $(n+1)P$ state. We find that, strong dipole interactions between the $nD$ and $(n+1)P$ state leads to a fast decay of $nD$ to $(n+1)P$ state, and the dephasing of the transmission spectrum. A theoretical model is built to understand EIT dephasing mechanism. The nonlinear dependence on dephasing rate has also been investigated.

The remainder of the article is arranged as follows. In Sec. 2, we introduce our experimental setup. In Sec. 3, we reveal Rydberg EIT spectrum and its dephasing experimentally. In Sec. 4, we present a simple theoretical model to reveal EIT dephasing mechanism. In Sec. 5, we measure the fast decay process on $80D_{5/2}$ state. Finally, we summarize the main results obtained in this work.

2. Experimental setup

Our experiment is performed in a cesium magneto-optical trap (MOT) with optical dipole trap (ODT). The beam waist of dipole trap is 45 $\mu$m. A schematic of relevant levels and the experimental setting are shown in Fig. 1(a) and (b). A three-level system, shown in Fig. 1(a), consists of a ground state $|6S_{1/2}, F = 4\rangle$ ($|g\rangle$), intermediate state $|6P_{3/2}, F' = 5 \rangle$ ($|e\rangle$) and Rydberg state $|80D_{5/2}\rangle$ ($|r\rangle$). A weak probe beam (Rabi frequency $\Omega _p$, 852-nm laser with a 100-kHz linewidth), provided by an external cavity diode laser (Toptica, DLpro), drives a lower transition and the frequency is stabilized to the $|6S_{1/2}, F = 4\rangle$ $\to$ $|6P_{3/2}, F' = 5 \rangle$ transition using the polarization spectroscopy method. The coupling beam (Rabi frequency $\Omega _c$) provided by a commercial laser (Toptica, TA-SHG110) with linewidth 1 MHz drives Rydberg transition, $|6P_{3/2}, F'=5 \rangle$ $\to$ $|80D_{5/2}\rangle$. The coupling laser frequency is stabilized to the Rydberg transition using a Rydberg EIT reference signal obtained from a cesium room-temperature vapor cell [29]. The weak probe laser and strong coupling laser, with respective Gaussian radius of 9 $\mu$m and 30 $\mu$m, are overlapped and counter-propagated through the MOT center, see Fig. 1(b). The probe laser is scanned using a double-passed acousto-optic modulator (AOM) that covers the lower transition. The transmission of the probe laser, Rydberg EIT spectrum, is detected with a single photon counting module (SPCM) and processed with Labview program. The glass MOT is surrounded by three pairs of field-compensation Helmholtz coils, which allow us to reduce stray magnetic fields via EIT Zeeman splitting, corresponding stray field less than 5 mG. In our experiment, the peak density of atomic cloud about 10$^{11}$ cm$^{-3}$ is measured by shadow imaging and the temperature of atomic cloud is about 100 $\mu$K. The estimated Rydberg density is 2.4 $\times$ 10$^{8}$ cm$^{-3}$.

 

Fig. 1. (a) Atomic level scheme. A weak probe laser field with Rabi frequency $\Omega _{p}$ drives the lower transition, $|g\rangle = |6 S_{1/2}, F=4\rangle$ $\to$ $|e\rangle =|6P_{3/2}, F'=5\rangle$. The strong coupling laser (Rabi frequency $\Omega _c$) couples the transition $|e\rangle$ $\to$ $|r\rangle = |80D_{5/2}\rangle$. (b) Experiment setup. The coupling and probe beams are counter-propagated through the MOT center and overlap with the dipole trap beam. The transmission of probe beam is detected with a single photon counting module (SPCM). The inset shows EIT dephasing behaviors (red) and probe absorption (black; without coupling beam). The data are taken by the sum of 3000 experimental cycles. (c) Experimental timing. After switching off the MOT and dipole trap beams, Rydberg-EIT coupling and probe lasers are turned on for 20$~\mu$s, during which the probe laser frequency is ramped through the $|g\rangle$ $\to$ $|e\rangle$ transition over $\pm$15 MHz by a double-passed AOM.

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The experimental timing is shown in Fig. 1(c) with the whole time 200 ms, corresponding to a repetition rate of 5 Hz. In each cycle, after turning off the trap beams, we switch on the coupling and probe lasers for 20 $\mu$s, during which the probe-laser frequency is swept across the $|6S_{1/2}, F = 4\rangle$ $\to$ $|6P_{3/2}, F' = 5 \rangle$ transition to observe the EIT spectra in Fig.2 (a) or fixed on resonance to investigate the dephasing in Fig. 1(b) and Fig. 2(b), meanwhile the power is fixed using a proportional-integral-derivative controller (PID) feedback loop that controls the radio-frequency power supplied to the 852-nm AOM. The data shown in Fig. 1(b) is taken by the sum of 3000 cycles.

 

Fig. 2. (a) Rydberg-EIT spectra with a coupling laser, $\Omega _c /2\pi$ =10.6 MHz, resonant with the $|6P_{3/2}, F'= 5\rangle$ $\to$ $|80D_{5/2}\rangle$ transition, and a probe frequency scanning across the lower transition, $|6S_{1/2}, F = 4\rangle$ $\to$ $|6P_{3/2}, F' = 5 \rangle$, at a probe-photon rate $R_{\rm in}$ = 21 photons/$\mu$s and 170 photons/$\mu$s, respectively. The solid lines are the fittings using the density matrix equation of a three-level atom. The EIT transmission displays a strong suppression and blue shift with increasing probe $R_{\rm in}$. (b) EIT transmissions for the indicated probe photon input rates at two-photon resonance condition. The transmission remains same for $R_{\rm in}$ = 21 photons/$\mu$s, whereas decrease with $R_{\rm in}$ at the beginning and then decay with the EIT maintain time for larger $R_{\rm in}$ cases. The solid lines denote the exponential fittings. For higher $R_{\rm in}$ the characteristic time are $\tau _{0({\rm EIT})}$ = 4.04 $\pm$ 0.10 $\mu$s, 4.50 $\pm$ 0.06 $\mu$s and 4.49 $\pm$ 0.05 $\mu$s, respectively.

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3. Experimental observation of Rydberg EIT and transmission dephasing

Due to the quantum interference effect [30], the probe transmission $T$ increases when the coupling and probe laser frequencies satisfy the two-photon resonance [see Fig. 1(a)]. Using the rotating wave approximation and in the interaction picture [31], the eigenstate of three-level Hamiltonian can be obtained with $|D\rangle \propto \Omega _c^{*} |g\rangle - \Omega _p |r\rangle$ at two-photon resonance. For conventional EIT, the system works in dark state and the probe pulse suffers little optical absorption. For the three-level scheme, all atoms are initially prepared in state $|g\rangle$. The EIT system can evolve to the dark state $|D\rangle$ with time $1/\gamma _2$. However, as shown in Fig. 1(b), in $nD_{5/2}$-Rydberg EIT system, the EIT transmission slowly decreases with time. Interesting, the time scale of the dephasing is much longer than the lifetime of intermediate state $|e\rangle$. The dipolar interaction can lead to many-body dephasing [26,32] in case of Rydberg $nD$ state.

We present EIT spectra with 80$D_{5/2}$ Rydberg state in Fig. 2(a). The probe field is adopted as incident photon rates $R_{\rm in}$ = 21 photons/$\mu$s (black dashed line) and 170 photons/$\mu$s (red dashed line) with $\Omega _c$ = $2\pi \times 10.6$ MHz. It is found that the transmission decreases with increasing $R_{\rm in}$ and accompanies with the EIT-peak shift. The EIT transmission rate is around 40% for $R_{\rm in}$ = 21 photons/$\mu$s, and decreases to 20% when $R_{\rm in}$ increases to 170 photons/$\mu$s. We attribute the reduction of optical transmission to atomic interactions between Rydberg states. Besides, when $R_{\rm in}$ = 170 photons/$\mu$s, the EIT peak has a blue shift about $2.5$ MHz compared with the case for $R_{\rm in}$ = 21 photons/$\mu$s. For Rydberg EIT, dark-state polariton is very sensitive to other Rydberg excitation in the strong interaction regime. This is because when two Rydberg polaritons propagate inside the medium with a distance $r$, they experience an interaction induced energy shift $U(r)$. It gives rise to a non-vanishing Im$(\chi )$ of probe beam where $\chi$ optical susceptibility of system and therefore leads to strong absorption and a shift on EIT spectra [6,14,15].

To further investigate the dephasing feature for $nD$ state, we vary the probe-photon rate $R_{\rm in}$ with fixed coupling field $\Omega _c /2\pi$ =10.6 MHz. Both the coupling and probe lasers frequencies are on resonance. As shown in Fig. 2(b), the time dependence of the transmission is plotted with different $R_{\rm in}$. For low probe photon rates (i.e., $R_{\rm in}$ = 21 photons/$\mu$s), the transmission is almost a constant, but when increasing $R_{\rm in}$, transmission exhibits a slow decrease with time $t$. By fitting the data in panel (b) with the exponential function $T = A \exp (-t / \tau _{0({\rm EIT})}) + T_0$, the decay time $\tau _{0({\rm EIT})}$ = 4.04 $\pm$ 0.10 ${\rm \mu s}$, 4.50 $\pm$ 0.06 ${\rm \mu s}$ and 4.49 $\pm$ 0.05 ${\rm \mu s}$ can be extracted for different $R_{\rm in}$. The decay time is much longer than the lifetime in state $|e\rangle$ (i.e., $1/\gamma _2\sim 0.03{\rm \mu s}$).

In order to reveal the dephasing mechanism, we define the effective optical depth (OD) of the medium as the logarithm of transmission [i.e., OD = -ln($T$)] [26]. The time evolution of OD is shown in Fig. 3(a) [corresponding to the results in Fig. 2(b)]. One sees that OD approximately linearly increases with time before t = 5 $\mu$s. By neglecting saturation effects, we can redefine $OD = \gamma _{\rm OD}t$ where $\gamma _{\rm OD}$ reflects creation rate of optical density by decoupled impurities [26]. At small probe photons (i.e., $R_{\rm in} \lesssim$ 340 $\mu$s$^{-1}$), the extracted rate $\gamma _{\rm OD}$ displays a linear increase with $R_{\rm in}$ and then saturates for large $R_{\rm in}$ [see Fig. 3(b)]. This is because the system is not fully in a blockade regime at small probe photons. Therefore, with increasing photon number, more Rydberg atoms are excited, which increase the dipole-dipole interaction, leading to increased $\gamma _{\rm OD}$. However, after the system enter fully blockade regime, we can’t excite more Rydberg atoms so that the dephasing rate $\gamma _{\rm OD}$ shows saturation. This is a signature of many-body dephasing. We calculate the group velocity of probe photon is around $3920$ m/s under our experiment condition with $\Omega _c=10.6$ MHz. By considering the length of our atomic cloud $1$ mm, the propagation time of photon through the cloud is around $0.26$ ${\rm \mu s}$. The Rydberg blockade radius is about 10 ${\rm \mu m}$ for 80D$_{5/2}$. Under this condition, almost $100$ atoms can be excited to Rydberg state. Thus we can calculate that in one microsecond, the maximum number of Rydberg excitation is around 385, where $100\,{\rm atom}/0.26$ ${\rm \mu s}$ $\simeq 385\,/{\rm \mu s}$. Thus, the critical value for $R_{\rm in}$ is around 385 /${\rm \mu s}$. When the photon incidence rates $R_{\rm in}$ is smaller than 385 /${\rm \mu s}$, the system is not in the blockade regime. When increasing the intensity of the probe laser, the system can works in the full blockade regime. This estimation is agreed with the dephasing rates $\gamma _{\rm OD}$ in Fig. 3(b). We also change $\Omega _c$ and measure EIT dephasing rates $\gamma _{\rm OD}$ versus $R_{\rm in}$. The results show a similar nonlinear dependence on $R_{\rm in}$ [see Fig. 3(b)].

 

Fig. 3. (a) The optical depth of the 80$D_{5/2}$ EIT transmission taken the logarithm of spectra in Fig. 2(b) for indicated photon incidence rates $R_{\rm in}$ at fixed $\Omega _c$ = 2$\pi \times$10.6 MHz. The solid lines are linear fittings before t = 5 $\mu$s to the data to extract the dephasing rates $\gamma _{\rm OD}$. (b) The dephasing rates $\gamma _{\rm OD}$ as a function of $R_{\rm in}$ for coupling Rabi frequency $\Omega _c/2\pi$ = 10.6, 8.2 and 5.2 MHz, respectively.

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4. Analysis of EIT dephasing mechanism

The vdW interaction between $nD_{5/2}$ pair leads to energy level shifts and dipole interaction of $nD_{5/2}$ and nearest Rydberg states yield state transfer of $nD_{5/2}$ to other Rydberg states. We have found [33] that $nD$-$(n+1)P$ transition is the strongest in our experiment conditions (see Sec. 5 for details). Hence this leads to a two stage processes. Rydberg $nD_{5/2}$ state will decay to $(n+1)P_{3/2}$ through spontaneous decay. The dephasing can then be induced in that regime where dipole-dipoles interactions couple nearly degenerate Rydberg pair states [26,32]. A full model to describe the dephasing is rather complicated. In this section, we will focus on the dephasing effects with a simplified model, which nonetheless captures the main effects.

Considering the three-level scheme in Fig. 1(a), the dephasing rate of Rydberg state $\gamma _{3}$ is around $\gamma _{r}$ + $\Gamma _{re}$ $\thickapprox \gamma _{r}$. Rydberg atom has long lifetimes ($1/\Gamma _{re} \sim n^3$) on the order of 100 $\mu$s. $\gamma _{r}$ represents the dephasing of the atomic coherence (originated from atomic collisions, residue Doppler effect, dipole-dipole interaction between the Rydberg atoms, finite laser linewidth). The dephasing rate of the intermediate state $|e\rangle$ denotes $\gamma _{2}$ = $\gamma _{e}$ + $\Gamma _{eg}$ with spontaneous decay rate $\Gamma _{eg}$ and interaction induced decay $\gamma _{e}$. In our physical system, $\gamma _{e}$ is much smaller than $\Gamma _{eg}\simeq 2\pi \times 5.2$ MHz, thus $\gamma _{2}$ $\thickapprox \Gamma _{eg}$. The collective dissipation can emerge in dense atomic gases, typically through two-body dipolar couplings [32]. Here we adopt the effective dephasing $\gamma _3^{\rm eff}$ (i.e., $\gamma _{r}=\gamma _3^{\rm eff}$), and seek the relation between the transmission and dipolar interaction induced dephasing.

The dynamics of the effective three-level system can be modeled by the quantum master equation for the many-atom density operator $\rho$:

The effective Hamiltonian in the equation is given by

Due to the dephasing and spectral shift of the transparency resonance [see Fig.2(a)], we employ the theoretical description of individual atoms coupled to a mean field (MF) to analyze the EIT spectrum with strong Rydberg interactions [6,34,35]. In the MF approximation, the many-body density matrix $\rho$ is decoupled into individual ones through $\hat {\rho } \approx \Pi _i$ $\hat {\rho }_i$. In the thermodynamic limit, the optical Bloch equations for the three-level system can be obtained, where elements of the density matrix are represented with $\rho _{ab}=N^{-1}\sum _j\langle \hat {\sigma }_{ab}^j\rangle$. As indicated in our experiment, over a long time evolution, the interactions between Rydberg state leads to a MF shift where $\Delta _c\rightarrow \Delta _c+V\rho _{rr}$ with MF interaction energy $V=N^{-1}\sum _{k\neq j}V_{jk}$.

By numerically solving the MF Bloch equation, one can obtain EIT transmission varies with $\Delta _p$. As shown in Fig. 4(a), there is an increasing blue shift of transparency away from the non-interaction EIT resonance. This is because that the shifted Rydberg state detunes the EIT windows. The similar EIT signal is also observed experimentally [see Fig.2(a)]. To obtain the dependence of the EIT transmission on the effective dephasing induced by Rydberg interactions, we calculate EIT spectra for a series of effective $\gamma _3^{\rm eff}$, accounting for the many-body dephasing by Rydberg atoms. In Fig. 4(b), the EIT transmission as a function of $\gamma _3^{\rm eff}$ is plotted. The system works at two-photon detuning $\delta$ = 0 with the probe $\Omega _p /2\pi$ =1.04 MHz, $\Omega _c /2\pi$ =10.6 MHz and atomic density $N_a=1\times 10^{11}$ cm$^{-3}$. It shows that the EIT transmission decrease with $\gamma _3^{\rm eff}$. The EIT transmission decreases to 50% when $\gamma _3^{\rm eff}=2\pi \times 1.5$ MHz. It is consistent with the trend of our experimental data [see Fig. 2(a)]. For comparing, we also plot the corresponding OD in Fig. 4(b), as expected, calculated OD of the probe beam increase as $\gamma _3^{\rm eff}$. We should note that the presented theoretical model is based on MF equation by simply varying the effective decay rate $\gamma _3^{\rm eff}$. Beyond the present model, many-body quantum model need to be developed to gain better understanding of the metastable dynamics [27,28]. We will discuss it somewhere else.

 

Fig. 4. (a) Theoretical results of EIT transmission varies with $\Delta _p$ for $V=0$ and $V\neq 0$. (b) EIT transmission and corresponding OD versus effective dephasing rate $\gamma _3^{\rm eff}$ of Rydberg state that accounts for the interaction between Rydberg atoms with $\Omega _p /2\pi$ =1.04 MHz, $\Omega _c /2\pi$ =10.6 MHz and atomic density 1$\times$10$^{11}$cm$^{-3}$.

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5. Test of the decay in $80D_{5/2}$ state

The dephasing effects arise from many physical reasons, e.g., atomic collisions, residue Doppler effect, depopulation between the Rydberg atoms, or finite laser linewidth. To test the collective dephasing process, we also conduct the experiment to measure the fast population transfer from $80D_{5/2}$ to $81P_{3/2}$. For 80$D_{5/2}$ state used in this work, the space to nearest 81$P_{3/2}$ state is 1.3124 GHz, corresponding dipole matrix element 5649.1 $ea_0$ with $e$ electron charge and $a_0$ Bohr radius. Therefore, 80$D_{5/2}$ Rydberg state displays strong dipole interactions with 81$P_{3/2}$, resulting to the state transfer of 80$D_{5/2}$ $\to$ 81$P_{3/2}$ [33]. This transformation leads to the decay of 80$D_{5/2}$ Rydberg atoms and further decreases of EIT transmission. In order to verify this conjecture, we carry out more test that is performed in an additional MOT [not shown in Fig. 1(b)], in which Rydberg atoms is detected with a state select field ionization detection. The temperature of the atomic cloud is almost same with the main setup, and the peak density of atomic cloud is $8.0 \times 10^{10} cm^{-3}$, which is comparable with the main setup of $1.0 \times 10^{11} cm^{-3}$. In addition, we make the probe and coupling Rabi frequency on the almost same to that of the main setup so that the Rydberg population is comparable with that of the EIT in the main apparatus. Therefore, we obtain the similar EIT spectra and field ionization signal simultaneously in the test setup. The details of the setup can be seen in our previous work [33,36]. In the test experiments, after switching off the MOT beams, we apply a two-photon excitation pulse with duration 4 $\mu$s for preparing 80$D_{5/2}$ Rydberg atoms, an optional interaction time $t_{\rm INT}$ before the ionization detection allows us to study the decay of 80$D_{5/2}$ Rydberg and state transformation.

In Fig. 5(a), we present the time of flight (TOF) signals for laser excitation to 80$D_{5/2}$ state. Initially, atoms are populated in 80$D_{5/2}$ state (see the black curve) in the TOF signal. Due to the resonant dipole interaction, atoms in 80$D_{5/2}$ state in light pink shaded region partly transfer to nearby 81$P_{3/2}$ state in the gray shaded region [see the red curve for $t_{\rm INT}$ = 3 $\mu$s]. With further increasing $t_{\rm INT}$, most of 80$D_{5/2}$ state Rydberg atoms transfer to 81$P_{3/2}$ state (see the blue curve for $t_{\rm INT}$ = 6 $\mu$s). The population in state 81$P_{3/2}$ reaches maximal at $t_{\rm INT}$ = 6 $\mu$s, and then decays to other states. For better understanding of the transfer process, we make a series of measurements for different $t_{\rm INT}$. Figure 5(b) displays the measured 80$D_{5/2}$ as a function of $t_{\rm INT}$. It is clear to observe the fast decay process on 80$D_{5/2}$ state within $8 {\rm \mu s}$. To obtain the decay characteristic of 80$D_{5/2}$ state, we fit the experimental data using exponential function [see the solid line of Fig. 5(b)]. The fitted results show decay time $\tau _{0(80D)}$ = 3.15 $\pm$ 0.38 $\mu$s is close to the decay of the EIT transmission rate $\tau _{0(EIT)}$. Therefore, we conclude that the dipole interaction induced the state transfer may be the reason that leads to the decay of 80$D_{5/2}$ Rydberg three-level EIT transmission.

 

Fig. 5. (a) Normalized time of flight (TOF) signals for laser excitation to 80$D_{5/2}$ state with indicated interaction times, $t_{\rm INT}$. The three traces are set vertically offset for clarity, with the respective zero levels shown as horizontal dashed lines. The gates for the 80$D_{5/2}$ and 81$P_{3/2}$ states signals are shown as a light pink and gray shaded regions, respectively. (b) Measurements of the 80$D_{5/2}$ state as a function of the interaction time $t_{\rm INT}$. Initial prepared 80$D_{5/2}$ Rydberg atoms transfer to nearby Rydberg states $|r^\prime \rangle$ during $t_{\rm INT}$ due to the strong resonant dipole interaction. The solid line shows the exponential fitting with the characteristic time $\tau _{0(80D)}$ = 3.15 $\pm$ 0.38 $\mu$s.

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6. Conclusion

We have presented Rydberg EIT spectra in a cascade three-level scheme involving 80$D_{5/2}$ state of Cs atoms. Rydberg EIT spectrum shows strong dependence on the probe incident photon number. The optical transmission displays decay behavior with time. An optical depth of the medium is defined to characterize the transmission. We have shown that the optical depth displays linear increase with the time at onset for fixed probe $R_{\rm in}$. We have further obtained the dephasing rate $\gamma _{\rm OD}$ by redefining OD=$\gamma _{\rm OD}t$. The dephasing rate linearly increases with weak probe field and then saturates for large $R_{\rm in}$. We have shown that the dephasing mechanism is mainly attributed to the strong dipole-dipole interactions, which lead to strong population decay on $|nD_{5/2}\rangle$ state. The experimental setting provides a platform to explore quantum nonlinear optics and quantum information processing, and creates metastable state in quantum many-body systems.