Transition from electromagnetically induced transparency to Autler–Townes splitting in cold cesium atoms

Electromagnetically induced transparency (EIT) and Autler–Townes splitting (ATS) are two similar yet distinct phenomena that modify the transmission of a weak probe field through an absorption medium in the presence of a coupling field, featured in a variety of three-level atomic systems. In many applications it is important to distinguish EIT from ATS splitting. We present EIT and ATS spectra in a three-level cascade system, involving cold cesium atoms in the 35 S 1 / 2 Rydberg state. The EIT linewidth, γEIT, defined as the full width at half maximum of the transparency window, and the ATS splitting, γATS, defined as the peak-to-peak distance between AT absorption peaks, are used to delineate the EIT and ATS regimes and to characterize the transition between the regimes. In the cold-atom medium, in the weak-coupler (EIT) regime γEIT ≈ A + B( Ω c 2 + Ω p 2 ) / Γ eg , where Ωc and Ωp are the coupler and probe Rabi frequencies, Γeg is the spontaneous decay rate of the intermediate 6P3/2 level, and parameters A and B that depend on the laser linewidth. We explore the transition into the strong-coupler (ATS) regime, which is characterized by the relation γATS ≈ Ωc. The experiments are in agreement with numerical solutions of the Master equation. Our analysis accounts for non-ideal conditions that exist in typical realizations of Rydberg-EIT, including laser-frequency jitter, Doppler mismatch of the utilized two-color Rydberg EIT system, and strong probe fields. The obtained criteria to distinguish cold-atom EIT from ATS are readily accessible and applicable in practical implementations.


Introduction
Electromagnetically induced transparency (EIT) [1] is a quantum interference effect in which the absorption of a weak probe laser, interacting resonantly with an atomic transition, is reduced in the presence of a coupling laser, which (near-)resonantly couples the upper probe level to a third level. EIT is, for instance, crucial in optically controlled slowing of light [2] and optical storage [3]. Autler-Townes splitting (ATS) [4], proposed by Autler and Townes, is a linear (resonant) AC Stark effect in a two-level system that is strongly driven at resonance, and can be measured via a weak probe from a third level. ATS was observed originally in the microwave and later the light domain. EIT and ATS have been extensively investigated experimentally and theoretically in Λ−, V− and cascade-type three-level atoms [1,[5][6][7][8][9][10].
EIT and ATS may phenomenologically look similar, but they are different in nature, one being a quantum interference and the other a linear AC Stark effect. This leads to an interest in the establishment of criteria to discern them. In [9], threshold values for distinguishing EIT and ATS, Ω t , are defined via the coherence decay rates in the Master equation of the various systems, according to which ATS is observed in four different threelevel systems in a strong-coupling-field regime (Ω c / Ω t > 1), whereas EIT is observed only in Λand cascade EIT Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
configurations in a weak-coupling-field regime (Ω c / Ω t <1). Anisimov et al [10] also propose an objective method based on Akaike's information criterion (AIC), to discern ATS from EIT in experimental data of a Λtype three-level atom. Recently, a series of experiments [11][12][13] has been performed in which the AIC has been used to discriminate EIT from ATS in Λ-type three-level systems free of Doppler effects. Zhang et al [14] provided a theoretical model using a five-level system and investigated EIT and ATS based on the AIC. Another area of interest, in which EIT and ATS are important, is spectroscopy of highly exited Rydberg levels using alloptical methods. Rydberg-EIT, a cascade EIT case, was first observed in a vapor cell [15] and later in a rubidium MOT [16]. In Rydberg-EIT work, the phenomena under investigation are often due to strong Rydberg-atom interactions that are distinct from the optical couplings that give rise to EIT or ATS. For instance, Rydberg-EIT has been used to realize a single-photon transistor [17] and a single-photon source [18,19] by employing a blockade effect [20][21][22], which results from the strong Rydberg-atom interactions. ATS has also been investigated in systems that have involved Rydberg states of rubidium [23,24] and cesium [25,26]. Exploiting the fact that Rydberg atoms strongly couple to radio-frequency(RF) fields, Rydberg EIT and ATS effects are used to realize precise measurements of such fields [27][28][29][30][31]. Holloway et al [32] have investigated the relationship between the Rabi frequency of resonant RF transitions between Rydberg states and the resultant ATS splitting in Rydberg-EIT spectra measured in room-temperature atomic vapor. In related work, the enhancement and suppression of multi-wave mixing (MWM) in five-level systems [33] has been studied, and the effect of Rydbergatom interactions was explored [34]. Spectra and spatial images of probe and MWM signals were obtained for six-wave [35] and eight-wave implementations [36]. The role of Rydberg interactions as well as the cooccurrence of Rydberg-EIT and ATS were discussed in these works.
While numerous efforts have already dealt with discriminating EIT from ATS [9][10][11][12][13][14], as reported in the above overview, many of these earlier works are limited to Λ-type systems, deal with Doppler-effect-free coldatom systems or all-optical equivalents of those, assume that the probe is in the weak-field limit, assume that the system is in steady-state, assume a vanishing laser linewidth, and/or assume that there is no interaction between the atoms (or their analogs) that exhibit EIT or ATS. In the present work we explore the EIT/ATS transition in a case where none of these ideal conditions are necessarily satisfied. We use cascade EIT and ATS, with a Rydberg upper-level state, in a cesium magneto-optical trap (MOT) as an experimental test system. We introduce two parameters, the Rydberg-EIT linewidth, γ EIT , and the AT spilitting, γ ATS , to characterize the EIT to ATS transition. In our model, we calculate spectra using methods provided in [32] to study the dependence of γ EIT and γ ATS on the Rabi frequencies of both the probe and coupling transitions. The transition between EIT and ATS effect is linked to a qualitative change in behavior of γ EIT and γ ATS as a function of coupler and probe Rabi frequencies. In our analysis, we expand the above cited previous studies to a cascade Rydberg EIT/ATS system. In the numerical methods used, we do not make any weak-field approximations. Also, the Doppler effect, residual Doppler mismatch, averaging over the Maxwell velocity distribution, and inhomogeneous broadening due to laser-frequency jitter are accounted for. The model does not include Rydberg-atom interactions, which are not relevant under the conditions (Rydberg quantum numbers, atom densities and interaction times) used in the experiment. We obtain readily accessible criteria to distinguish EIT from ATS.

Theoretical model
We consider the cesium cascade three-level system shown in figure 1(a). The coupling laser drives the upper transition, P The weak probe laser couples the lower transition, S The respective wavelengths and Rabi frequencies are λ c and Ω c , and λ p and Ω p . In the rotating-wave approximation and the field picture, the Hamiltonian of the three-level atom represented in the space 1 , 2 , 3 ñ ñ ñ {| | | }is where Δ c and Δ p are the detunings of the coupling and probe beams, respectively. To account for decay and dephasing, the system is described using the Lindblad equation for the density matrix ρ where L is the Lindblad operator that accounts for the decay processes in the atom. In the space 1 , 2 , 3 where γ 2 and γ 3 are the dephasing rates of the intermediate and Rydberg states, respectively. It is γ 2 =γ e +Γ eg , where γ e is a collision-induced dephasing rate of level 2ñ | , and γ e =Γ eg =2π×5.2MHz. Further, γ 3 =γ r + Γ re . For the Rydberg level the population decay rate Γ re is, typically, smaller than the dephasing γ r , because Rydberg-atom lifetimes are long (lifetimes are ∼n 3 and on the order of 100 μs) and interactions between Rydberg atoms in cold-atom clouds are often strong (van der Waals interactions scale as n 11 ).
The spectrum is given by the probe-power transmission, P P L exp 0 a = -( ), with the probe-laser absorption coefficient, α=2πIm(χ)/λ p , the MOT size, L, and the susceptibility of the medium seen by the probe laser, χ. The susceptibility, χ, is where N is the average atomic density, μ 12 is the dipole moment of transition 1 2 ñ  ñ | | , E p is the amplitude of the probe, ò 0 is the vacuum permittivity, and ρ 12 is the density matrix element between 1ñ | and 2ñ | . We numerically solve the equations (1)-(3) to obtain the absorption coefficient α for a range of values of Ω c and Ω p , including the case of a strong probe, Ω p Γ eg . The result is averaged over the thermal velocity distribution in the gas [32]; under our conditions (T=100-200 μK) the thermal motion is marginally important. In the calculation we also assume γ r =0, which is admissible due to our very low experimental probe intensities and the low principal quantum number of the utilized Rydberg state. Further, in our initial discussion we assume zero laser linewidth; this assumption is dropped later-on.
In figure 2(a) we show α as a function of Ω c and Δ p , for the case of a fixed Ω p =2π×1.05MHz and Δ c =0. The shape of the spectrum clearly indicates two qualitatively different regimes, a regime of low Ω c in which the overall linewidth is fixed at ≈Γ eg and there is a very narrow transparency window whose width scales quadratically in Ω c . This regime is the EIT regime. As Ω c increases, the overall linewidth increases and the system enters the ATS regime. There, the line breaks up into two features that are narrower than the coupler-free absorption line and the peak-to-peak separation scales linearly in Ω c . The wide transparency region between the AT lines is almost as wide as the peak-to-peak separation itself.
For a quantitative analysis, we extract the full width at half maximum (FWHM) of the dip in the absorption, γ EIT , as well as the peak-to-peak spacing of the line pair, γ ATS , from the absorption coefficient data. Both parameters γ EIT and γ ATS can be calculated in both the EIT and the ATS regime. However, as detailed in the following, γ ATS yields little useful information in the EIT regime and vice versa. In figure 2(b), both parameters using a Doppler-free polarization spectroscopy setup, and is scanned over the resonance. (b) Sketch of the experimental setup. The laser beams are separated via a dichroic mirror (not shown), and the probe light is detected using a single-photon counter module (SPCM).
are shown for a typical spectrum in the EIT regime, where Ω p Ω c =2π×2.03 MHz < Γ eg . In EIT, the main interest lies in how narrow the EIT transparency window is, in comparison with the natural linewidth Γ eg . Therefore, in the EIT regime the parameter γ EIT is most important (hence the name); it measures the bandwidth over which the EIT quantum interference mechanism leading to reduced probe absorption is experimentally observable. In the EIT regime, the separation between the absorption maxima, γ ATS , carries little useful information, as it just returns a value on the order of the overall coupler-free linewidth. Figure 2(c) shows the spectrum calculated for Ω c =2π×13.01MHz>Γ eg , which is the ATS regime. Since ATS is a linear AC Stark effect, the interest mainly is in how far the AT-split line centers are separated from each other. Therefore, in the ATS regime a better parameter to quantify the behavior is the peak-to-peak spacing of the line pair, γ ATS . As seen in figures 2(a) and (c), in the weak probe limit and for Δ c =0, in the ATS regime the peaks tend to have a width of Γ eg /2=2π×2.6 MHz, and a spacing γ ATS ≈Ω c . (The Rydberg states are very long-lived (Γ re =Γ eg ) and do not contribute to the width of the AT peaks here). Figure 3(a) shows the calculated γ EIT as a function of both Ω c and Ω p . In the EIT domain, characterized by Ω c , Ω p Γ eg and outlined by the dashed line, the separation between the contour lines decreases as a function of distance from the origin, reflecting that in the EIT regime . Further out, as the system enters the ATS regime, the separation between the contour lines tends to become fixed. Figure 3(b) shows that in the AT regime, characterized by Ω c Γ eg , the γ ATS contour lines are equidistant circles, reflecting the fact that g » W + W in that regime. In figure 3(b), near the x-axis, and in figure 2(c) it is Ω p = Ω c and therefore γ ATS ≈Ω c . These trends for γ ATS are also borne out by calculating the separation between the pair of non-dark eigenstates of the (decay-free) Hamiltonian in equation (1), when setting all detunings to zero. Calculations similar to those shown in figures 2 and 3 can be performed at any atom temperature(see [32] and references therein), and for both cascade and Λ-type three-level atoms. In the hot-atom cases (i.e., in vapor cells), the difference between coupler and probe wavelengths determines the size of residual Doppler shifts, which in turn determines the visibilities and detailed shapes of EIT and ATS spectra. In hot gases γ EIT generally tends to be larger than in the cold-atom case, in particular in weak coupler and probe fields, and follows a different scaling (see figure 7 in [32]).

Experimental measurement
The EIT and ATS experiments are performed in a standard MOT with temperature ∼100μK and atomic density up to ∼10 10 cm −3 . The coupling and probe lasers have linear and parallel polarizations and counter-propagate through the cold-atom cloud, as seen in figure 1(b). The details of the experiment have been described previously [38]. The probe beam is derived from a diode laser (DLpro, Toptica) that is locked to the ground-state transition, 1ñ | → 2ñ | , using polarization spectroscopy [39]; the beam has a Gaussian waist ω p0 =10μm at the MOT center. The strong coupling laser (Toptica TA-SHG110) has a Gaussian waist ω c0 =30μm and drives the Rydberg transition 2 3 ñ  ñ | | . The frequency of the coupling laser is stabilized to the Rydberg transition using a Rydberg-EIT signal obtained from a cesium room-temperature vapor cell [40]. In each experimental cycle, after turning off the trap beams, we switch on the coupling and probe lasers for 25μs. During the probe pulse the probe-laser frequency is swept across the S F 6 , 4 transition using a double-pass acousto-optic modulator (AOM) over a range of±10MHz relative to the transition center.
In order to avoid Rydberg excitation blockade and interaction effects, and to be able to reach high coupling Rabi frequencies, Ω c , we have chosen a state with a low principal quantum number n (35S 1/2 ). Also, the MOT density is reduced to about ∼10 8 cm −3 by reducing the repumping laser power. Further, to ensure that the effect of radiation pressure [38,41] on the EIT and ATS spectral profiles is negligible, we use a single-photon counter module (SPCM) for probe-light detection, allowing us to use a very low probe power, P 852 =200pW. Under these conditions, the radiation-pressure-induced velocity change during the probe pulse has an upper limit of ≈7 cm s −1 , corresponding to Doppler shifts <100 kHz (which is negligible in the present work). The EIT and ATS spectra are recorded using a data acquisition card (NI-PCI-6542) and processed with a Labview program.
The absorption coefficient, α, is obtained from the measured SPCM count number, P(Δ p ), and the offresonant (absorption-free) count number, P 0 , using the relation L P P ln p 0 a = -D ( ( ) ), where L is the effective MOT diameter along the probe beam path. In figure 4, we present experimental EIT and ATS sample spectra with indicated linewidth parameters γ EIT and γ ATS . We have determined γ EIT and γ ATS from experimental spectra covering a range of the coupling Rabi frequency Ω c , for fixed Ω p . In figure 5 we show the measured (hollow symbols) and calculated (lines) results for γ EIT and γ ATS versus Ω c .
To compare experimental spectra with calculated ones, it is important to account for the laser linewidth in the model. The utilized diode laser systems exhibit a frequency jitter on the order of 1MHz FWHM, whereas high-frequency laser phase noise is negligible. The frequency jitter causes inhomogeneous broadening of the spectra, because in every realization of the experiment the lasers have a slightly different frequency, that does not change significantly over the duration of a single realization. To account for the inhomogeneous broadening, subsequent to solving the Master equation and obtaining calculated spectra such as the ones in figure 2, the spectra are convoluted with a Gaussian, whose width is given by the laser-frequency jitter. It is noted that highfrequency laser noise would cause homogeneous broadening, which would have to be accounted for differently, namely via additional dephasing terms in the off-diagonal terms in equation (3). The γ EIT and γ ATS -values are obtained from the convoluted calculated spectra. In the present case, the calculations for 1.5MHz FWHM of the laser-frequency jitter (bold solid lines in figure 5) reproduce the measurements best (also see discussion of figure 6 below). We first discuss the EIT linewidth γ EIT . In the EIT domain, characterized by Ω c , Ω p Γ eg , the value of γ EIT in figure 5 exhibits a quadratic behavior, as expected from figure 3(a). For the experimental data in figure 5, we find . Measurements of EIT and ATS absorption spectra for Ω p =2π×1.05MHz and the indicated Rabi frequencies of the coupling laser, Ω c =2π×0MHz (top), 3.48MHz (middle) and 13.8MHz (bottom). The solid lines show the results of Lorentzian multi-peak fits. The coupler-free linewidth (top curve) is 2π×(4.5 ± 0.3)MHz, which is close to the expected value of 2π×5.2MHz. The FWHM EIT linewidth, γ EIT , and the ATS, γ ATS , are obtained from the fit functions, as indicated. The depth of the EIT dip (or, in the ATS regime, the depth of the valley between the AT peaks), H, is defined as the difference between the peak absorption, averaged over the two peaks, and the absorption minimum between the peaks. g » + W + W G ( ) , as shown by the black thin dashed line. The fitting parameters, which are introduced to account for the inhomogeneous line broadening caused by the laser-frequency jitter, are found to be A=2π×(1.42±0.10) MHz and B=0.44±0.03. Without the inhomogeneous broadening, it would be A≈0 and B≈1. It is seen that the laser jitter reduces the pre-factor B to a value significantly less than its ideal value of 1, and it adds an additive constant A on the order of the laser linewidth. In the AT regime, characterized by Ω c Γ eg , the γ EIT -value follows a trend 2 3.2MHz c EIT g p » W -´. (In the absence of laser-line broadening, in the AT regime it would be 2 c e g EIT g » W -G =Ω c −2π×2.6 MHz). Over the entire Ω c -range studied, the measured dependence γ EIT (Ω c ) agrees quite well with a calculation in which a Gaussian laser-frequency jitter of 1.5MHz FWHM is assumed.
Next we discuss γ ATS . In the AT regime, Ω c Γ eg , and for Ω p = Ω c , it is γ ATS ≈Ω c . In the EIT regime, Ω c , Ω p Γ eg , the value of γ ATS tends to be larger than Ω c ; for Ω c 2π×2 MHz γ ATS cannot be determined because the spectrum ceases to split into a separated line pair (also see figure 6). Inhomogeneous broadening due to the laser-frequency jitter generally increases the measured γ ATS , because the outer wings of the split lines are wider than the inside wings, hence the spectral averaging will pull the line centers in figure 2(b) outward. The absorption minimum at Δ p =0 disappears entirely when Ω c drops below a critical value, which is related to the laser linewidth and other broadening mechanisms. In our case this critical value is ≈2π×2 MHz. Similar nonlinear behavior of the AT splitting is observed in [32], where an RF-induced ATS is used to measure a microwave electric field.
We note that Ω c has a different significance in the EIT and AT domains. In the AT regime, Ω c splits the lines but does not broaden them, whereas in the EIT regime Ω c broadens the EIT line, as γ EIT increases with (Ω c 2 +Ω p 2 ). Conversely, laser jitter and other inhomogeneous broadening mechanisms also have a different significance in the two domains. In the EIT domain the spectra are very sensitive to inhomogeneous broadening, because even moderate laser jitter already strongly affects the parameters A and B in the fit function γ EIT ;A +B( c the broadening has a comparatively small effect on γ EIT , and an even smaller one on γ ATS . Generally, the agreement between theory and experiment in figure 5 is slightly better for γ ATS than it is for γ EIT . We believe this is because the details of the line broadening are harder to model than the AT splitting of the line centers. In figure 5 we have included a similar measurement with 100 μs probe-and coupler-pulse sweeps (filled diamonds in figure 5). It is seen that γ ATS for the longer pulses is less than in the 25 μs case. We attribute the reduction in γ ATS in part to dephasing caused by Rydberg-atom interactions, which could be considered via the dephasing terms in equation (3). Longer pulses will generally lead to a higher Rydberg-atom number in the atom-field interaction volume, causing dephasing by Rydberg-Rydberg interactions. The dependence of the ATS splitting on Rydberg interactions is detailed in [26]; in our present work interactions are avoided by using a low principal quantum number, a low atom density and a relatively short interaction time. Longer times will also increase the likelihood of Penning and thermal ionization. Any ions in the sample would contribute to Rydberglevel dephasing via the ion electric fields. With increasing pulse duration, the spectra are also increasingly affected by radiation-pressure-induced Doppler shifts, causing changes in the γ ATS -values derived from the spectra.
We finally consider the depth of the transparency window, which is an important measure for any practical application of EIT and ATS. The depth of the transparency window, H, is defined as the difference between the absorption coefficients on the peaks and at the center of the spectrum, Δ p =0 (see middle curve in figure 4). The depth H increases with Ω c in the EIT regime and plateaus at a laser-jitter-dependent maximal value in the ATS regime, as shown by the calculated curves in figure 6. Assuming a laser jitter with 1.5MHz FWHM, we achieve good agreement between experiment and calculation. This best-fit laser linewidth agrees well with manufacturer estimates for the utilized laser systems.

Conclusion
We have presented measured and calculated cold-atom EIT and AT absorption spectra of a cascade three-level atom involving the S 35 1 2 Rydberg state. The measurements show good agreement with calculations. The spectra exhibit two regimes, an EIT regime for weak Ω c and Ω p , and an ATS regime for large Ω c . While similar in appearance, the EIT and AT spectra are different in physical interpretation [9,10]. We have defined widths γ EIT and γ ATS that facilitates making the distinction between EIT and ATS. In the cold-atom EIT regime, Ω c , Ω p Γ eg , we find A B c p eg EIT 2 2 g = + W + W G (( ) ), where the parameters A and B depend on inhomogeneous broadening due to laser jitter. The MOT magnetic field, which is always left on, may add to the inhomogeneous broadening. In the ATS regime, Ω c Γ eg , it is γ ATS ≈Ω c and γ EIT =γ ATS −C, with a constant CΓ eg /2. The lower limit of C is realized in the absence of inhomogeneous broadening. An emphasis of our analysis has been to include the effects of laser jitter, saturation of the probe transition, and Doppler mismatch caused by non-zero temperature and unequal probe and coupling laser wavelengths.
Practical criteria to distinguish between EIT and ATS, as developed in this work, are valuable in a wide variety of atomic-physics, quantum-optics and quantum information applications of these schemes. For instance, the details of ATS splitting and line broadening effects are important in applications that deal with quantitative, atom-based microwave field measurements using Rydberg-EIT and microwave-coupled ATS [27,32]. Cold-atom, narrow-linewidth EIT and ATS are helpful in improving the accuracy and resolution of the atom-based field measurements.