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We report experimental observation of large group index across the Lamb dips of ground hyperfine states in Doppler-broadened 87Rb vapor. By sweeping the laser frequency through each hyperfine transition we measure the saturated absorption and optical phase shift using a phase-locked Mach-Zehnder interferometer. Our measurements provide a direct demonstration of the theoretical prediction by Agarwal et al. [G. S. Agarwal and T. N. Dey, Phys. Rev. A 68, 063816, (2003)] for the first time. An enhancement factor as large as 1005 in group index was observed for Rb vapor at temperature of 85 °C. The experimental data are in good agreement with the theory.
© 2015 Optical Society of America
1. Introduction
In the past decades intensive studies to control the speed of a light pulse in systems, from atomic vapors to condensed matter and photonic media, were carried out by tailoring the refractive index [1–5]. The subluminal or superluminal light is interesting in it’s own right and has many potential and practical applications in imaging spectroscopy [6], high-precision interferometers [7–9], telecommunication and quantum information science [3,10].
The fundamental physics lies on this effect is the wave nature of light. It is known when an EM wave of frequency ν passes a dispersive medium the repetitive interference between the scattered primary and rescattered secondary waves manifests it an overall phase shift of the resultant transmitted wave and induces a change in phase velocity from vacuum c to c/n(ν), where n is the refractive index, a function of the spanned light frequency [11]. Nevertheless, for a light pulse, a superposition of many waves whose frequencies are distributed in a small range, its speed depends on the derivative of n(ν) with respect to ν and is determined by the group velocity vg = c/ng(ν), where ng(ν) is known as the group index [12]. In general, by measuring the dispersive curve of refractive index it allows to estimate ng(ν).
So far, light speed can be manipulated by means of the available techniques using electromagnetically induced transparency (EIT) [13] and electromagnetically induced absorption (EIA) [14], population oscillations [15], nonlinear magneto-optical effect [16], gain resonances [17], and photonic crystals [4]. In most experiments superluminal and subluminal light were observed by directly measuring the time delay of a light pulse in medium with respect to in vacuum. Some groups measured the beat notes using another reference beam to infer the phase shift and thus obtain the refractive index as well as the group index [18, 19]. To do this, the probe beam frequency must be locked during each beat note measurement. Since the magnitude of group index varies with the slope of refractive index versus frequency, to obtain the group index at a certain frequency it thus requires a full knowledge of the refractive index over a sufficient large bandwidth centered at the desired frequency. Therefore, in this regard, it is somewhat time consuming using the beat note measurement.
Agarwal et al. [20] and Perdian et al. [21] proposed the possibility to produce slow light in the Doppler-broadened atomic vapors by applying saturation absorption spectroscopy to burn holes in the inhomogeneous spectrum. Camacho et al. [22] observed large fractional pulse delays in hot Rb atoms using an experimental scheme similar to [20], while also accompanying with frequency modulation on the pumping beam.
In this paper we present the experimental observation of large group index across the Lamb dips of the ground state hyperfine transitions in Doppler-broadened 87Rb vapor and show a direct demonstration of the proposals in [20] and [21]. In our work we measure the group index, instead of pulse delay as in the previous experiment [22], by combining a probe beam which going through the atomic vapor with a separate reference beam in a Mach-Zehnder interferometer (MZI). We will describe our experiment, including the working principle and method, to measure the optical phase shift and hence the group index. In addition, we will show our observation, by the detailed comparison with a theoretical model, and the potential applications using this experimental scheme below.
2. Experiment
To measure the group index across the Lamb dips of ground state hyperfine transitions in 87Rb atoms, we set up the saturation spectroscopy in a Mach-Zehnder interferometer. Two homemade external cavity diode lasers (ECDLs) [23] are used to generate linearly polarized Gaussian beams for saturation spectroscopy and phase tuning purposes, respectively. One beam picked from the first ECDL (ECDL 1) with wavelength λ and size (1/e2 radius) of 1.0 mm is shined through a 50% – 50% beam splitter (BS1) and divided into probe and reference beams along the two arms, respectively. The probe beam propagates into the Rb cell of length L along one arm of the interferometer and overlaps with a counter-propagating pumping beam, generating from the same ECDL but expanding to 1.7 mm to fully cover the probe beam, as shown in Fig. 1. Both windows on the cell are AR coated to minimize the reflection.
Fig. 1 Schematic of the experimental setup. A collimated phase-tuning beam (blue) with wavelength λL and size 1.9 mm is injected into the interferometer and with few mm separation from the probe beam (black). The interference signal from the phase-tuning beam is sensed by another photodiode (not shown) and further processed to give real-time feedback to the PZT glued behind mirror 1 for relative phase stabilization of the interferometer. The pumping beam is shown in orange.
The reference beam passes along another arm of the interferometer and recombines with the probe beam after the second 50% – 50% beam splitter (BS2). The ECDL 1 laser is frequency calibrated and permits linear scan for almost 1 GHz by ramping a voltage to the external cavity PZT behind which a grating is glued [23]. A separate phase-tuning beam made from another ECDL (ECDL 2) is tuned far from the resonances of Rb atoms and injected into the interferometer to set the relative optical phase between the two arms to the desired value. In this experiment, the optical phase is stabilized to ±0.5° similar to the work in [24].
To carry out the experiment using this scheme, we first measure the Lamb dip absorption signal directly from the probe beam by blocking the reference beam under a certain cell temperature Tc. The ECDL 1 laser frequency ν is swept across the hyperfine transitions from the ground 5S1/2 F = 2 to 5P3/2 F = 1,2,3 states. For an incident probe beam with intensity Ip before the Rb cell, the output intensity at the photodiode, following directly from the Beer’s law, is simply
by integrating through the cell length, where α is the absorption coefficient which will be discussed below and αe is the effective absorption coefficient.The corresponding refractive index n(ν) is obtained by measuring the optical phase shift Δϕ(ν) induced from the atoms using the interference signal while unblocking the reference beam. The reference beam intensity Ir is set equal to Ip in all measurements. If the relative phase difference of the bare interferometer is locked at 90° the photodiode signal is [24, 25]
By measuring the interference signal it allows to know the induced optical phase shift Δϕ(ν) and retrieve the refractive index as n(ν) = Δϕ(ν)c/2πνL + 1. Hence, the group index is given by ng(ν) = n(ν)/[1 +ν · dn(ν)/dν] [12]. Clearly, the group index depends on the cell length and absorption coefficient α, and varies with the cell temperature as well as pumping beam intensity, which we will describe in the next section.Though the interferometer is actively phase stabilized, the arm length difference ΔL might still produce extra phase difference with a magnitude (2π/c) · Δν · ΔL when the laser frequency is tuned by Δν. To suppress the possible phase picked up during frequency scan, ΔL must be set to less than 1 mm. As a result, the induced phase shift due to the arm-length difference is small and can be neglected in this experiment.
3. Results and discussions
In our experiment, the absorption and phase shift measurements are made by sweeping the frequency of the ECDL 1 laser, starting with 825 MHz red-detuned from the F = 2 → F = 2,3 cross-over transition, for total tuning range about 1.06 GHz. The initial large detuning for the frequency scan also provides convenient calibration of the relative phase due to the very small absorption by the atoms. The probe beam intensity is set to 0.25Is through all the measurements presented in this paper, with Is the saturation intensity. Figure 2 shows the absorption signal versus sweeping frequency under the condition L = 3 mm, Tc = 85 °C, and Ipp = 1.5Is. Figure 3 is the corresponding optical phase shift signal under the same condition in Fig. 2. To give quantitative fitting with the experimental data we develop a theoretical model for the absorption signal [25], based on the theory in [26]. Each fitted absorption curve is subsequently transformed to its counterpart of refractive index using the Kramers-Kronig relations [27] and compared with the measured data shown in Fig. 3.
Fig. 2 Absorption signal versus sweeping frequency at Tc = 85 °C, corresponding a number density of 3.8×1011 atoms/cm3. The measured data are presented in red. The fitting curve calculated from the model is shown in blue.
Fig. 3 Optical phase shift induced by the Rb atoms versus sweeping frequency. The red curve is taken from the measurements and the blue one is directly transformed from the fitted absorption spectrum in Fig. 2 using Kramers-Kronig relations.
In general, the Lamb dips are characterized by two main categories, the on-resonance and cross-over transitions. We adopt the theory proposed by Pappas et al. [26] to model the Lamb dip spectrum of 87Rb ground hyperfine transitions from F = 2 to F = 1,2,3 states in our experiment. For a probe beam with detuning Δij = 2π(ν −νij) to the transition between the two hyperfine states |i⟩ (lower state) and |j⟩ (upper state), corresponding a resonance frequency νij, under a given pumping beam intensity Ipp and lower state number density Ni, the Lamb dip absorption coefficient is shown to be
where , describing the dip depth, and , showing the dip width, with , τij = 1/Γij the radiative lifetime of the upper state, , T the transit time, the on-resonance cross section, μij the electric dipole matrix element of the two-level transition [28], kij = 2πνij/c; the Doppler-broadened cross section, with k = 2πν/c, and u the average velocity of the atoms [26]. The number density Ni can be estimated by knowing the temperature, abundance of the atomic sample, and level degeneracy [28]. The transit time depends on the probe beam diameter d and average velocity, and is estimated to be T ∼ d/u.The model is also applied for the cross-over transitions simply extending the above condition to a double two-level system with a common ground state. Consider the 87Rb ground hyperfine cross-over transitions from the lower state |i⟩ = |F = 2⟩ to the upper states |j⟩ and |l⟩, with j, l = 1,2,3 and l > j. For the two resonance transitions, from |i⟩ to |j⟩ and |i⟩ to |l⟩, corresponding the resonance frequencies νij and νil, respectively, the cross-over transition locates exactly at the midway in between and has the resonance frequency νi,jl = (νij +νil)/2. Since there are two velocity groups centered at v=±|π(νij −νil)/k contributing to the Lamb dip, therefore the absorption coefficient for |i⟩ to |j⟩ and |l⟩ cross-over transition is given by
where Δij = 2π(ν −νij) and Δil = 2π(ν −νil); Δi,jl = 2π(ν −νi, jl); , , Sij, Sil, wij, wil are defined in the same way as in Eq. (3), for each respective pair of states labeled by the superscripts/subscripts.The overall absorption coefficient α(ν) is thus obtained by considering all the related on-resonance and cross-over transitions using Eq. (3) and Eq. (4), i.e. α(ν) = ∑jαij(ν) + ∑j,l αi,jl(ν). Knowing the incident probe intensity Ip, cell length and temperature, and taking α(ν) into Eq. (1) it allows to obtain the absorption spectrum from the probe beam. In our experiment, in order to fit the measured absorption data the nearby absorption from the 85Rb D2-line hyperfine transitions, F = 3 to F = 2,3,4 states, is crucial and must be taken into account as well. The typical fitting values are τ ~ 30 ns, T ~ 1 μs, σ0 ~ 1.94 × 10−9 cm2, and Is ~ 2.5 mW/cm2.
We take the experimental parameters used for measurements shown in Fig. 2 into the model to obtain the theoretical absorption curve. As we can see, the fitted curve is in good agreement with the measured absorption spectrum. Subsequently, the fitted absorption coefficient αT(ν) is further transformed to the refractive index by the Kramers-Kronig relation and hence gives the induced phase shift from the atoms as
where P.V. denotes the Cauchy principal value of the integral. Ideally, the lower and upper bounds for the integral should be a = 0 and b = ∞, respectively. However, since we only have an approximately 1 GHz wide absorption spectrum available, due to the limitation of the experiment, integration directly from the narrow bandwidth spectrum results in significant distortion on Δϕ, especially around the two wings. To solve this problem, the nearby absorption from the 85Rb D2-line hyperfine transitions, F = 3 to F = 2,3,4 states, must be included to fit the absorption spectrum as well. By doing so, it allows to expand the theoretical absorption spectrum to ν2,23 ∓ 4 GHz, with ν2,23 the F = 2 → F = 2,3 cross-over resonance frequency, and is much wider than the minimal required bandwidth of 4 GHz for a good fit if using our model.The theoretical phase shift spectrum is then compared with the data measured by interferometer, and is presented in Fig. 3. We also see the observed data reasonably agree with the theory. Both measured data and fitting curve on the phase shift are further transformed into group index. The result is presented in Fig. 4, where a group index as large as 1005 is observed near the F = 2 → F = 2,3 cross-over transition, and directly demonstrates the theoretical prediction in [20] and [21].
Fig. 4 The group index ng versus frequency, obtained from the measurements in Fig. 3, under the cell temperature Tc = 85 °C. The inset is a zooming for the F = 2 → F = 2,3 cross-over transition. The red curve is calculated from the measured data and the blue one is obtained from the fitting curve in Fig. 3.
For the atomic sample with L = 3 mm, a group index ng = 1005 implies a group delay τd = L(ng − 1)/c = 10 ns. From the absorption spectrum in Fig. 2, the spectral hole bandwidth of the F = 2 → F = 2,3 cross-over transition is estimated to be 12 MHz. If replacing the cw probe beam by a light pulse with frequency bandwidth well within the spectral hole, i.e. pulse duration longer than ~ 100 ns, the given scheme will achieve 0.1 fractional delay, corresponding to a delay bandwidth product (DBP) about 0.12 which is consistent with the theory prediction for the systems [29]. For applications in optical delay lines, much larger fractional delays are more desirable. By frequency modulating the pumping beam in the similar scheme to further broaden the transmission window while still burning deep enough spectral holes, Camacho et al. were able to achieve 10 fractional delays in the hot Rb vapor [22].
The group index enhancement showing in the Doppler-broadened system using a saturating counter-propagating pumping beam is established by the spectral burning hole. Larger group index requires steeper Lamb dip in the absorption spectrum. As seen in Eq. (3), at higher atomic density, corresponding higher Tc, the Lamb dip is deeper, under the price of larger pumping beam intensity attenuation, leading to nonuniform optical pumping along the propagation direction. Besides, to burn a spectral hole, sufficient optical pumping is needed. However, higher Ipp might cause significant broadening on the dip and reduce the group index. Therefore, to control ng in an atomic cell, it requires to manage and make somewhat trade-off among the atomic density, pumping beam intensity, and cell length [20, 22].
We thus do an experiment to study the temperature dependence of ng. This is done in a Rb cell with L = 10 mm, under the similar condition as used in Fig. 2, but only varying the cell temperature. When carrying out the measurements, Tc is stabilized to within ±1.5°C. This causes density uncertainty by 3%, and leading to an uncertainty in group index measurement by about 1%. The measured data and the theoretical fittings are presented in Fig. 5, showing the group index grows up with the temperature and indicating higher atomic density is more favorable under the given experimental condition. However, there must have a upper bound on the temperature under which the pumping beam is seriously absorbed and it can no longer sustain a deeper hole.
Fig. 5 Group index ng versus temperature for L = 10 mm, under the pumping beam intensity Ipp = 1.5Is. The blue fitting curve is obtained from our theoretical model.
In addition, when the pumping beam passes the atomic sample it also accompanies intensity attenuation, with the magnitude exponentially proportional to its propagation distance. For a thin cell, the pumping beam intensity can be treated as constant all the way through the cell. This is, however, not true when intensity attenuation is not negligible as the cell is thick, under which the pumping beam intensity is attenuated with the propagation distance and the induced group index thus varies accordingly.
In order to fit the data well for both Fig. 2 and Fig. 5 by our model, we need divide the cell into N sections, with N sufficient large to mimic each section as a thin cell. The overall absorption requires to sum the individual absorption coefficient, calculated according to the attenuated pumping beam intensity in the given section. We find to satisfy the thin sample condition in our theoretical simulations, N must be larger than 10 for measurements in Fig. 2, and 30 in Fig. 5.
Furthermore, to explain and understand why larger group index is not seen in L = 10 mm cell shown in Fig. 5, it requires to show the group index is a function of pumping beam intensity, predicted in [20]. We then carry out a series of measurements for L = 10 mm cell by varying the pumping beam intensity, under the temperature of 80 °C. The results are presented in Fig. 6, showing a clear distribution of ng with respect to Ipp and a maximum at pumping beam intensity around 1.5Is. The observation implies it is unable to efficiently burn a hole in the given scheme when the incident intensity of the pumping beam is below Is/4. Under a modest pumping beam condition as used in Fig. 2, Ipp quickly decays to Is/4 after traveling about 3 mm in the cell. Therefore, as we can see in Fig. 6, using a cell much longer than 3 mm does not help producing even larger ng. The result is also consistent with the observation in [22], that larger ng and hence longer light pulse delay could be achieved more efficiently by using multiple short cells than a single long cell.
Fig. 6 Group index ng versus pumping beam intensity for L = 10 mm and Tc = 80 °C. The maximum appears at the pumping beam intensity close to 1.5Is. The fitting curve is shown in blue.
Our experimental scheme to measure the saturated spectroscopy in a Doppler-broadened Rb vapor shown above demonstrates group index enhancement and its dependence on the pumping beam intensity, cell temperature and length. Since the relative phase of the Mach-Zehnder interferometer is actively stabilized to a desired value, one obvious and immediate advantage in our system is the phase noise associated with the environment is much reduced. Another prominent feature, which could be hard or even impossible to do by the homodyne detection means [30], is the phase measurements using a MZI in our configuration also provide the spatial resolution. This can be done by replacing the photodiode with a CCD camera. In other words, the phase data presented in this letter are one-pixel phase images. With the recent developments on the storage of two-dimensional light information in atomic vapors [31, 32], our scheme should turn useful to measure and diagnose the phase images, group index distributions, and group delays in the transverse directions. Besides, accompanied with the phase-shifting interferometery (PSI), a weak probe beam going through an optically dense atomic gas can be significantly amplified using the given scheme [33]. This feature also benefits the phase measurements for those in which superluminal/subluminal effect plays an important role, and is especially favorable for studies on EIA effects with which much weaker probe signals are expected.
4. Conclusions
In conclusion, we experimentally demonstrate large group index across the Lamb dips of ground state hyperfine transitions in Doppler-broadened 87Rb gas. To our knowledge, our measurements provide a direct experimental demonstration of the theoretical prediction by Agarwal et al. [20, 21] for the first time. We observe an enhancement factor as large as 1005 in group index for Rb vapor in 85°C. We also develop a theoretical model for the absorption and refraction spectra of saturated spectroscopy and show in good agreement with the experimental measurements. Our experimental scheme using a phase-locked Mach-Zehnder interferometer shows robust for both absorption and optical phase shift measurements while with spatial resolution, and can be employed in the EIT and EIA systems in the future.
Acknowledgments
We are grateful to Li-Chung Ha for assistance during the early stage of the experiment, to Prof. Agarwal for valuable information, to Dr. Zhimin Shi for useful discussion, to K.F. Lin and Po-Jui Tseng for data fitting. This project was supported from the National Science Council of Taiwan (R.O.C.) under NSC grant NO. 98-2112-M-194-002-MY3.
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