⇦Chinese Physics Letters, 2016, Vol. 33, No. 12, Article code 123201 Electromagnetically Induced Transparency in a Cold Gas with Strong Atomic Interactions * Yue-Chun Jiao(焦月春)1,2, Xiao-Xuan Han(韩小萱)1,2, Zhi-Wei Yang(杨智伟)1,2, Jian-Ming Zhao(赵建明)1,2**, Suo-Tang Jia(贾锁堂)1,2 Affiliations 1State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006 2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006 Received 13 September 2016 *Supported by the National Basic Research Program of China under Grant No 2012CB921603, Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China under Grant No IRT13076, the State Key Program of the National Natural Science of China under Grant No 11434007, the National Natural Science of China under Grant Nos 11274209, 61475090, 60378039 and 61378013, and Shanxi Scholarship Council of China (2014-009).
**Corresponding author. Email: zhaojm@sxu.edu.cn Citation Text: Jiao Y C, Han X X, Yang Z W, Zhao J M and Jia S T 2016 Chin. Phys. Lett. 33 123201 Abstract Electromagnetically induced transparency (EIT) is investigated in a system of cold, interacting cesium Rydberg atoms. The utilized cesium levels $6S_{1/2}$, $6P_{3/2}$ and $nD_{5/2}$ constitute a cascade three-level system, in which a coupling laser drives the Rydberg transition, and a probe laser detects the EIT signal on the $6S_{1/2}$ to $6P_{3/2}$ transition. Rydberg EIT spectra are found to depend on the strong interaction between the Rydberg atoms. Diminished EIT transparency is obtained when the Rabi frequency of the probe laser is increased, whereas the corresponding linewidth remains unchanged. To model the system with a three-level Lindblad equation, we introduce a Rydberg-level dephasing rate $\gamma_{3}=\kappa \times (\rho_{33}/{\it \Omega}_{\rm p})^2$, with a value $\kappa$ that depends on the ground-state atom density and the Rydberg level. The simulation results are largely consistent with the measurements. The experiments, in which the principal quantum number is varied between 30 and 43, demonstrate that the EIT reduction observed at large ${\it \Omega}_{\rm p}$ is due to the strong interactions between the Rydberg atoms. DOI:10.1088/0256-307X/33/12/123201 PACS:32.80.Ee, 42.50.Gy, 42.50.Nn © 2016 Chinese Physics Society Article Text Atoms in highly excited Rydberg states (principal quantum number $n\gg1$) exhibit rich many-body behavior due to their strong interactions,[1] which lead to excitation blockade effects.[2-5] Strong mutual interactions between the Rydberg atoms make the Rydberg atoms attractive candidates to implement quantum information and logic gates.[6-9] The electromagnetically induced transparency (EIT)[10] involving Rydberg states provides a non-destructive optical detection of Rydberg states.[11] The EIT can also be used to map the strong interaction between the Rydberg atoms onto a strong optical transition,[11,12] to demonstrate a giant dc Kerr effect.[13] The EIT in the strong interacting Rydberg atoms is of great technological importance, which enables the realization of single-photon transistor[14,15] and single-photon sources[16,17] for quantum information processing. The strong interaction between the Rydberg atoms yields the dephasing of the three-level system. The dephasing rate of the Rydberg level, $\gamma_3$, is used to explain the lineshape change of EIT[18] or Autler–Townes (A-T)[19] spectra involving the Rydberg level. Here we report the experimental studies of EIT involving Rydberg states $nD_{5/2}$ ($n=30$–43) in a magneto-optical trap (MOT). The dependence of the Rydberg EIT spectrum on the Rabi frequency of the probe and the coupling laser is investigated. The EIT lineshape and peak transmission are modified by the strong interactions between the Rydberg atoms. The interactions can be controlled by both the Rydberg density and the principal quantum number $n$. The results are simulated with density matrix theory accounting for dephasing due to the Rydberg interactions. Our Rydberg EIT experiment is performed in a standard cesium MOT with temperature $\sim$100 μK and the peak density $\sim$10$^{10}$ cm$^{-3}$. During the experiment the MOT magnetic field is on. Figure 1 shows a schematic diagram of the experiment (Fig. 1(a)) and the relevant energy levels (Fig. 1(b)). A cascade three-level system consists of the ground state $|6S_{1/2},F=4\rangle(|1\rangle)$, the intermediate state $|6P_{3/2}, F'=5 \rangle (|2\rangle)$ and the Rydberg state $|nD_{5/2}\rangle(|3\rangle)$. The probe and coupling beams are counter-propagated through a cold cesium sample. The strong coupling laser, provided by a commercial laser (Toptica TA-SHG110) with wavelength 510 nm and linewidth 1 MHz, drives the Rydberg transition $|2\rangle\to|3\rangle$. The Gaussian radius of the coupling beam in the MOT is $\omega_{0}=110$ μm. The frequency of the coupling laser is stabilized to the Rydberg transition by using a Rydberg EIT reference signal obtained from a cesium room-temperature vapor cell.[20] The probe laser is produced by a diode laser (DLpro, Toptica) with wavelength 852 nm and linewidth 100 kHz. The weak probe beam has a Gaussian waist of $\omega_{0}=85$ μm. The EIT signal is observed by measuring the transmitted probe-beam power using a photodiode. To avoid the fluctuation of probe power, we lock the probe power using a power feedback PID. The power stability of the probe laser is less than 3% after the PID is switched on.
cpl-33-12-123201-fig1.png
Fig. 1. Schematic diagram of the experiment (a) and relevant three levels (b) for Rydberg EIT. The coupling laser with $\lambda_{\rm c}=510$ nm and the Rabi frequency ${\it \Omega}_{\rm c}$ and the probe laser with $\lambda_{\rm p}=852$ nm and the Rabi frequency ${\it \Omega}_{\rm p}$ counter-propagate through a cold cesium sample in an MOT. The coupling laser is resonant with the Rydberg transition $|6P_{3/2}, F'=5\rangle(|2\rangle)\to|nD_{5/2}\rangle(|3\rangle)$, while the frequency of the probe laser is positively chirped (frequency detuned from low frequency to high frequency) over the ground-state transition $|6S_{1/2}, F=4\rangle(|1\rangle)\to|6P_{3/2}, F'=5\rangle(|2\rangle)$ at a range of $\pm$25 MHz in 0.5 ms. The EIT signal is detected as a function of the probe laser detuning, ${\it \Delta}_{\rm p}$.
cpl-33-12-123201-fig2.png
Fig. 2. Measured (black samples) and calculated (red line) EIT spectra for the coupling Rabi frequency ${\it \Omega}_{\rm c}=2\pi\times2.4$ MHz and the probe Rabi frequency ${\it \Omega}_{\rm p}=2\pi\times2.8$ MHz. The calculations are based on the three-level model with ${\it \Gamma}_{\rm re}/2\pi=0.01$ MHz and ${\it \Gamma}_{\rm eg}/2\pi=5.2$ MHz, $\gamma_{2}={\it \Gamma}_{\rm eg}$, and $\gamma_{3}=\kappa \times (\rho_{33}/{\it \Omega}_{\rm p})^2$ with $\kappa/2\pi=90$ μs$^{-3}$. The inset presents the comparison of transmission spectra of the probe laser for positive (blue solid line) and negative (red dashed line) chirp over the $|1\rangle \rightarrow |2\rangle$ transition.
In Fig. 2, we show the measured EIT spectra with the coupling laser driving the transition to the Rydberg state, $30D_{5/2}$, and the corresponding Rabi frequency ${\it \Omega}_{\rm c}=2\pi\times2.4$ MHz and ${\it \Omega}_{\rm p}=2\pi\times2.8$ MHz. The EIT peak height $H$ and full width at half maximum (FWHM) of the EIT signal, obtained by fitting the EIT spectra with Gaussian functions, are used to characterize the features of the EIT. The deduced FWHM of the EIT spectrum in Fig. 2 is $\gamma_{\rm EIT}=2\pi\times5.2$ MHz. The EIT FWHM is larger than the expected EIT linewidth, $\gamma_{\rm expect}=({\it \Omega}_{\rm c}^2+{\it \Omega}_{\rm p}^2)/{\it \Gamma}_{\rm eg}=2\pi\times2.6$ MHz, including the intrinsic EIT linewidth, ${\it \Omega}_{\rm c}^2/{\it \Gamma}_{\rm eg}$, and saturation broadening due to the probe laser. To understand the obtained result, we consider a ladder three-level system involving the Rydberg state in Fig. 1(b). The Hamiltonian of the system can be written, in the field picture, as $$ H=\frac{\hbar}{2}\left(\begin{matrix} 0&{\it \Omega}_{\rm p}&0\\ {\it \Omega}_{\rm p}&-2{\it \Delta}_{\rm p}&{\it \Omega}_{\rm c}\\ 0&{\it \Omega}_{\rm c}&-2({\it \Delta}_{\rm p}+{\it \Delta}_{\rm c}) \end{matrix}\right),~~ \tag {1} $$ where ${\it \Delta}_{\rm c}$ and ${\it \Delta}_{\rm p}$ are the detunings of the coupling and probing lasers, respectively. Since in experiments the coupling laser is frequency locked to the Rydberg transition, $|6P_{3/2}, F'=5\rangle\rightarrow |nD_{5/2}\rangle$, we set ${\it \Delta}_{\rm c}=0$. To account for decay and dephasing processes, we use the Lindblad equation to describe the evolution of the density matrix $\rho$ as follows: $$ \dot{\rho}=\frac{i}{\hbar}{[H,\rho]+\mathfrak{L}},~~ \tag {2} $$ where $\mathfrak{L}$ describes the decay and dephasing of the three-level system[21] $$ \mathfrak{L}=\left(\begin{matrix} {\it \Gamma}_{\rm eg}\rho_{22}&-\frac{1}{2}\gamma_{2}\rho_{12}&-\frac{1}{2}\gamma_{3}\rho_{13}\\ -\frac{1}{2}\gamma_{2}\rho_{21}&-{\it \Gamma}_{\rm eg}\rho_{22}+{\it \Gamma}_{\rm re}\rho_{33}&-\frac{1}{2}(\gamma_{2}+\gamma_{3})\rho_{23}\\ -\frac{1}{2}\gamma_{3}\rho_{31}&-\frac{1}{2}(\gamma_{2}+\gamma_{3})\rho_{32}&-{\it \Gamma}_{\rm re}\rho_{33} \end{matrix}\right).~~ \tag {3} $$ Here $\mathfrak{L}$ includes the decays, ${\it \Gamma}_{\rm eg}$ and ${\it \Gamma}_{\rm re}$, and dephasing rates, $\gamma_2$ and $\gamma_3$, of two upper levels $|2\rangle$ and $|3\rangle$, and ${\it \Gamma}_{\rm eg}$ is the natural decay rate of the 6$P_{3/2}$ state, which is 2$\pi\times5.2$ MHz.[22] The Rydberg-state decay rate ${\it \Gamma}_{\rm re}$ is much lower than ${\it \Gamma}_{\rm eg}$ and the dephasing rates of the system. The dephasing rate due to the state $|2\rangle$, $\gamma_2$ is set to ${\it \Gamma}_{\rm eg}$, and the collision-induced dephasing of that state is negligible. The dephasing rate $\gamma_{3}$ of the Rydberg level is several orders of magnitude larger than ${\it \Gamma}_{\rm re}$ and mainly arises from the interactions between the Rydberg atoms. The van der Waals interaction energy between a pair of the Rydberg atoms at separation $R$ is given by $V_{\rm vdw}=C_{6}/R^{6}$, where the dispersion coefficient $C_{6}$ is calculated using the method in Ref. [23]. The distance $R$ follows $R^{3}=3/(4\pi N_{\rm R})$, where $N_{\rm R}$ is the Rydberg-atom density. Considering the strong interaction between the Rydberg atoms proportional to $N_{\rm R}^2$, we define a dephasing rate as $\gamma_{3}=\kappa \times (\rho_{33}/{\it \Omega}_{\rm p})^2$ in Eq. (3), with a constant $\kappa$ in units of μs$^{-3}$. It is worth noting that the dephasing rate is dependent on $(\rho_{33}/{\it \Omega}_{\rm p})^2$. We numerically solve Eq. (2) and obtain the EIT spectrum as shown in Fig. 2 with the solid red line for ${\it \Omega}_{\rm c}=2\pi\times2.4$ MHz and ${\it \Omega}_{\rm p}=2\pi\times2.8$ MHz. The simulation reproduces the experimental EIT spectrum well for ${\it \Delta}_{\rm p} \lesssim 2\pi \times 10$ MHz. The difference between the measured and simulated EIT spectra on the blue-detuned side in Figs. 2 and 4 is attributed to the radiation pressure of the probe laser, due to the fact that atoms become accelerated at the resonance. In the experiments, the probe laser frequency is chirped from lower frequency to higher frequency (positive chirp) through the $|1\rangle \to |2\rangle$ transition over a time interval of 0.5 ms. In the vicinity of the zero-detuning point the atoms resonantly interact with the probe laser and pick up momentum in the direction of the probe beam. This entails a Doppler shift of the $|1\rangle\to|2\rangle$ transition, due to the fact that the atoms stay close to resonance somewhat longer during the time the frequency sweep proceeds. In effect, the positive frequency chirp of the probe laser partially compensates for the increasing Doppler shift of the atoms, which enhances the absorption of the atoms on the blue-detuned side in Figs. 2 and 4. In the case of a negative frequency chirp, the radiation-pressure-induced Doppler shift and the frequency sweep are in opposite directions, leading to a faster decrease in absorption after passage through the zero-detuning point. The inset in Fig. 2 shows the probe absorption as a function of time for the positive (blue solid) and negative (red dashed line) frequency chirps, which clearly shows the effect of the radiation pressure on the absorption spectra.
cpl-33-12-123201-fig3.png
Fig. 3. Comparison of measurements (red circles) and calculations (black squares) of the peak height $H$ (a) and FWHM (b) of EIT spectra as a function of the probe laser Rabi frequency, ${\it \Omega}_{\rm p}$ for ${\it \Omega}_{\rm c}=2\pi \times 4.1$ MHz. In the calculation, $\kappa/2\pi=168$ μs$^{-3}$ is used to achieve good agreement between experiments and calculations. The EIT peak height $H$ decreases with ${\it \Omega}_{\rm p}$. This trend is well matched by the simulation. The measured FWHM of the EIT peak remains approximately constant, while the simulated result slightly increases. (c) The calculations of $\gamma_3$ and $\rho_{33}$ versus ${\it \Omega}_{\rm p}$ under our experimental condition in (a).
To test our model for the dependence of $\gamma_3$ on the probe laser Rabi frequency ${\it \Omega}_{\rm p}$, we have performed a series of measurements in which ${\it \Omega}_{\rm p}$ has been varied. Figure 3 shows the measurements (red circles) and calculations (black squares) of the height $H$ and the FWHM of the EIT peaks as a function of ${\it \Omega}_{\rm p}$. It is found that the peak height diminishes with increasing ${\it \Omega}_{\rm p}$. We attribute this decreasing behavior to the fact that our measurement was performed with the Rydberg atoms, which are prone to strong mutual interactions. Our calculations (black squares in Fig. 3(a)) reproduce the experimentally observed suppression of the EIT resonance at larger values of ${\it \Omega}_{\rm p}$ very well. The measured EIT FWHM remains fixed at $\gamma_{\rm EIT} \simeq 5.1$ MHz when the probe Rabi frequency ${\it \Omega}_{\rm p}$ is varied, as shown in Fig. 3(b). Due to the absence of any broadening of the EIT feature with increasing ${\it \Omega}_{\rm p}$, we can rule out inhomogeneous line broadening caused by the Stark shifts due to the electric fields of ions or by the van-der-Waals level shifts of the Rydberg state.[24] In Fig. 3(c), we present the calculations of the dependences of the Rydberg populations $\rho_{33}$ and the dephasing rate $\gamma_{3}$, on the probe Rabi frequency ${\it \Omega}_{\rm p}$. It is shown that both the Rydberg atom density and the dephasing rate increase at the center of the EIT resonance with the increasing probe Rabi frequency. The large dephasing rate induced due to the strong interaction suppresses the EIT, see Fig. 3(a). On the other hand, the dephasing rate begins to trend to saturation after ${\it \Omega}_{\rm p}>3.0$ MHz, meaning that as the Rydberg atom density increases, the system evolves into the blockaded domain, where only one atom can be excited into the Rydberg state within each blockade volume and contribute to the EIT. All the other ground-state atoms located within blockade volumes that already carry a Rydberg excitation are subjected to a coupler field that is off-resonant, due to the van-der-Waals level shifts, leaving those atoms to act as plain two-level systems that scatter light from the probe field. It is therefore the Rydberg blockade that leads to the suppression of the EIT transparency at large ${\it \Omega}_{\rm p}$. In our experiments, the blockade radius is calculated to be 3.4 μm for the $43D_{5/2}$ state, and the corresponding atomic density is $\sim$7.0$\times10^{10}$. It is seen from Fig. 3(b) that in the measurements the suppression of EIT is not accompanied by any line broadening, while the FWHM of the calculated EIT spectra does exhibit a small increase of the FWHM by $ < 1$ MHz. This difference between calculations and measurements is attributed to the fact that our simulations only account for the Rydberg-atom interactions via a density-enhanced dephasing rate, $\gamma_{3}=\kappa \times (\rho_{33}/{\it \Omega}_{\rm p})^2$. In this model, we do not account for the blockade effect. Our simulations suggest that the pure density-enhanced dephasing in $\gamma_{3}$, induced by the van-der-Waals interaction between the Rydberg atoms, would lead to a broadening of the EIT resonance. Considering the blockade effect in our simulations of the EIT spectrum will require future investigation. As mentioned above, in our model we include a dephasing term $\gamma_{3}$ which is attributed to the strong Rydberg long-range van der Waals interactions, $V_{\rm vdw}= C_{6}/R^{6}$. The dispersion coefficient $C_{6}$ scales as $n_{\rm eff}^{11}$ ($n_{\rm eff}$ is the effective quantum number). We employ a scaling $\gamma_{3} \propto \rho_{33}^2$ in our model. The Rydberg-atom interactions can be controlled by varying the Rydberg-atom density (change of the excitation Rabi frequency) or by using different principal quantum numbers. The dephasing rate $\gamma_{3}$ is expected to increase with $n$, due to the $n_{\rm eff}^{11}$ scaling of $C_{6}$. To explore the resultant behavior, we keep the Rabi frequency of the probe and coupling lasers fixed at ${\it \Omega}_{\rm c}=2\pi\times 1.5$ MHz and ${\it \Omega}_{\rm p}=2\pi\times 2.8$ MHz, respectively, for a series of probe absorption measurements for $n$ ranging of 30–43. Figure 4 shows the EIT spectra for the indicated quantum states $nD_{5/2}$. It is found that the EIT peak becomes greatly suppressed with increasing $n$. This behavior is due to the increasing van-der-Waals interactions between the Rydberg atoms.
cpl-33-12-123201-fig4.png
Fig. 4. Measurements (solid lines) and calculations (dashed lines) of EIT spectra for the indicated Rydberg state $|nD_{5/2}\rangle$ with the positive probe-frequency chirp. The Rabi frequencies ${\it \Omega}_{\rm p}=2\pi\times 2.8$ MHz and ${\it \Omega}_{\rm c}=2\pi\times 1.5$ MHz are kept to be constant when tuning the coupling laser field into resonance with different Rydberg states (i.e., the coupler-beam power is scaled proportional to $n^3$). By comparing the measured EIT spectra with calculated spectra, the dephasing rates that yield the best agreement between experiment and model are found to be $\gamma_{3}=2\pi\times 2.07$ MHz ($30D_{5/2}$), 2$\pi\times 2.78$ MHz ($35D_{5/2}$), 2$\pi\times 3.90$ MHz ($38D_{5/2}$) and 2$\pi\times 4.34$ MHz ($43D_{5/2}$).
By fitting the measured EIT spectra with the simulation results, the dephasing values for the Rydberg states $|3\rangle=nD_{5/2}$ are found to be $\gamma_{3}=2\pi\times 2.07$ MHz ($30D_{5/2}$), 2$\pi\times 2.78$ MHz ($35D_{5/2}$), 2$\pi\times3.90$ MHz ($38D_{5/2}$) and 2$\pi\times 4.34$ MHz ($43D_{5/2}$). While the dephasing rate $\gamma_{3}$ increases with the principal quantum number $n$, the increase with $n$ is less steep than the $n_{\rm eff}^{11}$-scaling of $C_6$. This fact may be attributed to the Rydberg excitation blockade, which suppresses the Rydberg-atom excitation at high $n$. In conclusion, we have investigated the EIT involving Rydberg states in a cesium MOT. An increase of the probe Rabi frequency yields a suppression of EIT, without broadening of the EIT linewidth. This suppression of EIT without linewidth broadening is attributed to a combination of strong interactions between the Rydberg atoms with the Rydberg excitation blockade. When the probe Rabi frequency is increased, the system evolves into a blockaded domain at the EIT resonance, where the Rydberg atom density peaks and atomic interactions shift a fraction of the atoms out of resonance. This destroys the EIT condition for part of the atomic ensemble, leading to the suppression of the EIT. This interpretation is corroborated by measurements at several principal quantum numbers in the range of 30–43, which have shown the enhanced EIT suppression at high $n$. The radiation pressure caused by the probe laser leads to significant atom acceleration and the Doppler shifts in the range of several MHz. To avoid this effect in future work, we will use shorter laser pulses and weaker probe intensities. Using the Rydberg EIT, one can map the strong interaction of the atoms onto the probe laser field. Therefore, EIT can be used to probe the properties of the interacting Rydberg ensembles and to generate single-photon sources. References Dipole blockade in a cold Rydberg atomic sample [Invited]Local Blockade of Rydberg Excitation in an Ultracold GasDipole Blockade at Förster Resonances in High Resolution Laser Excitation of Rydberg States of Cesium AtomsElectric-Field Induced Dipole Blockade with Rydberg AtomsFast Quantum Gates for Neutral AtomsDipole Blockade and Quantum Information Processing in Mesoscopic Atomic EnsemblesInformation and computation: Classical and quantum aspectsDemonstration of a Neutral Atom Controlled-NOT Quantum GateObservation of electromagnetically induced transparencyCoherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced TransparencyCooperative Atom-Light Interaction in a Blockaded Rydberg EnsembleA giant electro-optic effect using polarizable dark statesSingle-Photon Transistor Mediated by Interstate Rydberg InteractionsSingle-Photon Transistor Using a Förster ResonanceElectromagnetically induced transparency of a single-photon in dipole-coupled one-dimensional atomic cloudsStrongly Interacting Rydberg Excitations of a Cold Atomic GasDipolar Dephasing of Rydberg D -State PolaritonsAutler-Townes spectroscopy with interaction-induced dephasingLaser frequency locking based on Rydberg electromagnetically induced transparencyInvestigation of dephasing rates in an interacting Rydberg gasPrecision lifetime measurements of Cs 6 p 2 P 1 / 2 and 6 p 2 P 3 / 2 levels by single-photon countingLong-range interactions between alkali Rydberg atom pairs correlated to the n s– n s, n p– n p and n d– n d asymptotesSuperradiance: An essay on the theory of collective spontaneous emission
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