Open Access
Add to BibSonomy
Abstract
In this study, we investigate 420-nm yoked superfluorescence (YSF) emitted from the atomic vapor of rubidium (Rb) by driving the Rb 5S – 5D two-photon transition with an ultrashort pulsed laser. When the pump pulse is close to its transform limit (~ 100 fs) or down-chirped up to around 200 fs, the 420-nm YSF appears as a low-divergence beam with a ring-shaped radial profile. Although such a beam profile is less sensitive to the vapor pressure of Rb in a cell, its diameter rigorously varies as a function of the pump-pulse power. By numerically solving a time-dependent Schrödinger equation for a single-Rb atom, we well reproduce our experimental observation, indicating that a single-atom Rabi oscillation is responsible for the spatial beam profile of the 420-nm emission.
© 2017 Optical Society of America
1. Introduction
Four-wave mixing (FWM) has for a long time been identified as a powerful tool for frequency conversion; moreover, further experimental pursuits and measurements have revealed a number of its novel features, most of which are demonstrated to be alternative techniques useful in nonlinear optics. Some of these techniques include generating ultrashort pulses [1], correlated photon pairs [2], and squeezed light [3].
In many FWM experiments with gaseous media, alkali-metal atoms are frequently used owing to their high number density at relatively low temperatures. More importantly, alkali-metal atoms have simple quasi-hydrogenic energy-level structures, and they can be optically excited from the ground state into one of the low-lying excited states using visible or infrared coherent light [4,5]. Among others, the rubidium (Rb) atom is of particular interest because the optical two-photon excitation of a Rb atom from the 5S ground state to the 5D excited states can result in a FWM process with simultaneous coherent radiations at 5.2 µm and 420 nm (blue light) during its radiative decay process (see Fig. 1(a)). In more detail, once the population inversion is achieved between the 5D – 6P states by the 5S – 5D optical excitation, the 5D state is quickly depopulated by the superfluorescence (SF) decay with emission at 5.2-µm corresponding to the 5D–6P line. If this decay process occurs within a characteristic coherence time between the 5S and 5D states, the 6P–5S transition is driven instantaneously, which results in simultaneous radiation at 420 nm. The emission angle of the 5.2-µm SF is determined by the shape of the sample. On the other hand, the blue light is emitted along the phase-matched direction. Such a specific FWM process, which is found in the four-level systems of Rb and cesium (Cs), has been extensively studied in the last two decades using various experimental approaches [6–13].
Fig. 1 (a) Energy diagram of Rb and related transitions. (b) Schematic of the experimental setup. (c) The blue-light power plotted as a function of Rb number density n, which is fitted by a function proportional to n2.
Brownell et al. first observed the abovementioned coupled two-photon emission from Cs vapor and named it as yoked superfluorescence (YSF) [6]. Applying a noncollinear excitation method to a quasi-spherical sample, Lvovsky et al. indirectly demonstrated an omnidirectional character of the 5.2-µm SF from Rb vapor, and at the same time, clarified the crucial role of a phase-matching condition in this time-delayed FWM process [7]. To date, further experimental results have been reported with regard to the YSF, such as ringing properties [8], temporal coherent control [9–11], and saturation effects during excitations [12, 13]. Furthermore, dependences of the blue-light generation in the FWM process on pump laser power, detuning, and polarization have been investigated using low-power two-color continuous-wave (cw) diode lasers [14–19] or a single high-power cw diode laser [20]. More recently, it has been demonstrated that using a ring cavity, the power of the blue light can be enhanced and its spectral linewidth can be reduced [21]. It should be noted that the cw blue-light emission is not directly related to the YSF, and it is usually called a parametric FWM process.
In contrast to the extensive study on the temporal profiles of the YSF mentioned earlier, few reports exist on their spatial properties. In general, spatially resolved signals contain detailed information about signal sources, which are usually lost owing to spatial averaging. Extracting this information should be important for applications such as standoff spectroscopy of remote vapors [11, 18, 19, 22]. Here we report results on the novel observation of spatial beam profiles of the YSF. The blue light shows a ring-shaped beam pattern when Rb atoms are excited by an irradiation of ultrashort laser pulses that drives the 5S – 5D transition of Rb atoms in a gas cell. Using numerical simulations based on a single-atom model, we concluded that the ring-shaped pattern can be attributed to two-photon resonant Rabi oscillations in a single atom.
2. Experiment
A schematic of the experimental setup is shown in Fig. 1(b). We use a 100-fs Ti:sapphire laser at a center wavelength of 782 nm with a repetition rate of 1 kHz and a maximum pulse energy of 0.5 mJ. Although the center wavelength is detuned from the 778-nm two-photon resonant wavelength (see Fig. 1(a)), the 5S–5D transition occurs owing to the broad spectral width of the laser pulse, which is ~ 10 nm at full width at half-maximum (FWHM). In the experiments, the pulse width is controlled by slightly (< 1 mm) changing the distance between the grating pair in the compressor. The pulse width is monitored using a commercial autocorrelator. The output beam from the laser system is vertically polarized, and the power is varied with a combination of a half-wave plate and a Glan–Thompson polarizer. The half-wave plate is attached on a motorized rotation stage for precisely controlling the output power of the laser. A telescope is used to spatially collimate the pump beam. With a complementary metal-oxide semiconductor (CMOS) camera (Thorlabs DCC1645C), the spatial profile of the collimated beam is monitored to be slightly elliptical, i.e., FWHM is 1.07 mm in the vertical direction and 1.33 mm in the horizontal direction. The beam enters a Pyrex-glass cell containing Rb vapor, the temperature of which is changeable in the range of 130 – 200 °C in the present experiments. The angle between the propagation axes of the pump beam and the blue light, which is generated inside the cell, is 1.1 mrad. A bandpass filter (Thorlabs FB420-10) is placed behind the cell that blocks the pump beam but transmits ~ 45% power of the blue light. Spatial profiles, temporal profiles, and average power of the transmitted blue light are monitored by the CMOS camera, a 1.2-GHz bandwidth photodiode (Electro-Optics Technology ET-2030), and a power meter (Ophir PD300-UV), respectively.
3. Results and discussion
In the present experiment, we have unexpectedly observed that the radial profile of the blue light has a ring-shaped pattern when the pump pulse is set close to its transform limit (TL) or down-chirped up to around 200 fs with its average power ranging from 50 to 200 mW. Typical images of the blue light showing ring-shaped profiles are given in Figs. 2(b) and 2(c). To clarify that the ring-shaped blue light originates in the YSF during the FWM process in Rb vapor, we first measured the divergence of the ring-profiled beam, which is estimated to be ~ 0.5 mrad in the vertical direction and ~ 1 mrad in the horizontal direction. Such a small divergence of the beam is consistent with those reported in the previous observation of the 420-nm YSF [8, 13]. Next, we confirmed that the blue light is emitted almost simultaneously with the pump pulse, indicating that the temporal delay of the blue light to the pump pulse is less than the resolution time of our photodiode (with a rise/fall time of 300 ps). In fact, from a recent study using a high-speed streak camera [8], the time delays are expected to be several tens of ps under our experimental conditions. Overall, we consider the observed ring-shaped blue light to most likely originate in the YSF. We also monitored the blue-light power while changing the temperature of the cell. Figure 1(c) shows a typical plot of the measured power against the Rb atomic density, which is deduced from a pressure curve of Rb [23] and the cell temperature. The plotted points are successfully fitted by a function proportional to the square of Rb atomic density. This result might provide valuable information regarding the generation process of the YSF, the rigorous simulation of which still remains a challenging task [8, 24].
Fig. 2 (a)–(c) The blue-light images observed at the different pump-pulse power indicated in (d). (d) Dependence of the ring diameter on the pump power fitted with the function given in Eq. (2). (e) Illustration to explain the origin of ring-shaped blue light. Radial intensity distributions of the pump pulses are depicted for three different cases, where the peak intensity of the pump pulse is smaller than IM for the left one, whereas it is larger for the middle and right ones.
One of the subtle features observed in our experiments is that the beam profiles of the blue light are significantly changed at different pump power. In Fig. 2, we show the typical profiles taken at pump power of (a) 21, (b) 51, and (c) 108 mW. The data were obtained when the pump pulse was down-chirped with a duration of 160 fs and the cell temperature was set at 140 °C. In general, low power generates a single small bright spot (Fig. 2(a)), whereas increasing the power results in a ring-shaped profile with a larger diameter (Figs. 2(b) and 2(c)). To systematically observe this behavior, the vertical and horizontal ring diameters are measured as a function of the pump power. The results are shown in Fig. 2(d).
To figure out the intensity dependence of the spatial profile of the blue light described earlier, we first assume that the pump-pulse beam has a Gaussian spatial profile with a radial intensity distribution of
where IC is the intensity at the beam center and w is the beam diameter at FWHM (see Fig. 2(e)). If we further anticipate that the blue-light intensity becomes maximum at some pump-pulse intensity of IM, ring-shaped profiles are expected to appear when the IC exceeds the IM (see Fig. 2(e)). Substituting I(r) = IM in Eq. (1), the ring diameter is calculated to be where PIn and PM are defined as the input pump power and the pump power when IC = IM, respectively. Fits of Eq. (2) to the data using a least-squares algorithm are represented as solid (vertical direction) and dashed (horizontal direction) curves in Fig. 2(d), in which w and PM are used as the fitting parameters. The retrieved values of both the vertical and horizontal beam diameters, w, are 1.3 mm. The vertical value is slightly larger than that measured experimentally. This could be attributed to the discrepancy between the actual beam profile and the ideal Gaussian beam. The retrieved values of PM are 41 mW (vertical) and 31 mW (horizontal). They are well within the experimentally measured lower and upper limits indicated by (a) and (b) in Fig. 2(d), respectively.Although we could reasonably reproduce the intensity dependence of the ring diameter of the blue light, whether the blue-light intensity indeed peaks at some pump-pulse intensity of IM remains unclear. Taking into account the strong relation between the blue-light intensity and the population of the 5D states [8, 25], the analysis mentioned earlier indicates an oscillatory dependence of the excitation probability to the 5D states on the pump-pulse intensity. To confirm this, we further calculate the time evolution of a Rb electronic wave function for a single atom under irradiation of the pump pulses by numerically solving a time-dependent Schrödinger equation. The calculation details are given in [26]. Briefly, we include five states in total; 5S, 5P3/2, 5P1/2, 5D5/2, and 5D3/2. We treat the radiation interaction in the dipole approximation. Dipole matrix elements are represented as a product of Wigner 3-j and a reduced matrix element [23, 27], and we use the values for the reduced matrix elements given in [28]. We choose a quantization axis along the polarization direction of the pump pulse so that the magnetic quantum number of mJ remains unchanged according to the selection rule of ΔmJ = 0.
Figure 3(a) shows the results simulated under irradiation by the TL or frequency-chirped pump pulses, the analytical expressions for which are given in [29]. In the figure, populations of the 5D5/2 state after the excitations are plotted as a function of the temporal peak intensity of the TL pulse. Note that at the same peak intensity, spectral amplitudes are the same for all pulses, thus enabling us to extract the chirp effects. Because the populations of the 5D3/2 state are one order of magnitude smaller than those of the 5D5/2 state, the effects of the 5D3/2 state are assumed to be relatively small and are not discussed here. As observed in Fig. 3(a), our calculation suggests that the 5D5/2 state is populated more effectively by the down-chirped pulses than the TL or up-chirped pulses. This is consistent with the previous reports on adiabatic excitations using counterintuitively ordered double pulses [25, 30]. For the present system, the frequency component which is resonant to the 5P3/2 – 5D5/2 transition should precede the frequency component which is resonant to the 5S – 5P3/2 transition (see Fig. 1(a)). Such a pulse corresponds to the down-chirped pulse, where the instantaneous frequency decreases with time. Note that strong two-photon resonant Rabi oscillations appear for the down-chirped and TL pulses. These Rabi oscillations are more profound for the shorter down-chirped pulses and become less prominent for the longer pulses, where the excitations are more likely explained by the adiabatic picture. For the TL or down-chirped pulses, peaks can be found in the Rabi oscillation curves at around 15 and 70 GW/cm2, whereas they are less clear when the pulses are up chirped.
Fig. 3 (a) The calculated results of Rb excitation dynamics under irradiation by the linearly chirped pump pulses. Populations of the 5D5/2 are plotted as a function of the temporal peak intensity of the TL pulse. The peak intensities estimated from the measured pump-pulse power as well as the observed blue-light image are indicated by the dashed line and the solid line, respectively. (b) The observed (upper row) and simulated (lower row) blue-light images at several pump-pulse conditions indicated at the bottom of each column.
The elucidated intensity dependence on the 5D-state population under several pump-pulse conditions can be compared with the experimental results. The observed images of the blue light are given in the upper row in Fig. 3(b). The corresponding pump-pulse conditions are indicated at the bottom of each column. The pump power is fixed at 77 mW, and the cell temperature is set at 130 °C. We estimate the peak intensity at the beam center as 41 GW/cm2, indicated by a dashed line in Fig. 3(a), under an assumption that the pump-beam profile is a Gaussian beam with a diameter of 1.20 mm, which is an average of the measured values. For the case of the down-chirped 130-fs pulse, the ring profile is clearly visible, which we attribute to the peak at around 15 GW/cm2 in Fig. 3(a). Assuming that the blue-light intensity is proportional to the square of 5D5/2-state population, we simulate spatial profiles of the blue light. The results are shown in the lower row in Fig. 3(b). The simulated results well reproduce the observed ring profiles for the down-chirped 130-fs pulse as well as the TL pulse. As for the down-chirped 250-fs pulse, the ring is obscured owing to the poor contrast in the Rabi oscillation in Fig. 3(a), which agrees with the experimental result. As for the up-chirped 110-fs pulse, two rings are observed at different diameters in the simulated result. While the outer dilute ring is not obvious in the experiment, the inner bright ring is clearly observed. Next, we retrieve the peak intensity of the pump pulse from the observed blue-light image and consider if the above analysis is quantitatively reasonable. The observed ring diameters for the down-chirped 130-fs pulse are 1.05 mm and 1.49 mm in the vertical and horizontal directions, respectively. They are extrapolated to be 0.93 mm and 1.24 mm at the cell center, giving an average value of 1.08 mm. As mentioned earlier, we assume the pump-beam profile as the Gaussian beam with a diameter of 1.20 mm. Then the attribution of the observed ring for the down-chirped 130-fs pulse to the peak at around 15 GW/cm2 in Fig. 3(a) results in the pump-pulse peak intensity of 33 GW/cm2, which is indicated by a solid line in Fig. 3(a). The retrieved value is in good agreement with the one estimated from the measured pump-pulse power.
As mentioned above, the observed blue-beam profiles are successfully explained from the viewpoint of the 5D population. We also attempted to explain our observation from the 5S – 5D coherence, as is discussed in [25]; however, we could not obtain a quantitative agreement in our experimental conditions. The YSF is a cooperative process, and the result that a single-atom model well reproduces the experimental observation needs some explanations. Because the time delay of the YSF is far larger than the width of the pump pulses used in the experiments, we believe that it is reasonable to apply a single-atom model to calculate the 5D atom density right after the excitations, which could be a crucial parameter of the subsequent YSF. In future work, for further arguments, rigorous simulations are necessary by numerically solving Maxwell–Schrödinger equations [8, 24].
In summary, nonlinear population dynamics of Rb atoms are retrieved from spatial profiles of the blue light. This unprecedented result owes to the characteristic feature of the blue light as mentioned below. In general, a detection of Rabi oscillations, in other words, ultrafast nonlinear dynamics of atomic populations suffers from spatial averaging due to intensity distributions of pump pulses [31]. On the other hand, the blue light generated at different transverse spatial positions reach different positions on the detector. In other words, a spatial intensity distribution of the pump pulse is directly imprinted on a beam profile of the blue light. It should be emphasized that our method does not need any scan, and it can be extended to detect nonlinear population dynamics in a single shot. Furthermore, our method might be applied to pulse-shape characterization. In fact, we observed that the blue-beam profiles dramatically change depending on the pump-pulse width, especially when the pulse is slightly up-chirped from the TL pulse. We also confirmed by simulations that the blue-beam profiles are quite sensitive to a center wavelength of the pump pulse, and change enough when the center wavelength is shifted by ±1 nm.
Note that similar ring-shaped beam profiles have been previously observed in the YSF [7, 14] and in the parametric FWM [14]. Reference [7] reports FWM experiments with ultrashort double pulses applied in a noncollinear configuration. In this case, the ring-shaped beam is divergent to fulfill the phase-matching condition, which is clearly different from our observation. In the experiments reported in [14], two-color cw lasers are applied as pump sources. The ring-shaped blue light appeared by adjusting the excitation wavelengths of the two lasers. The authors attribute these profiles to high-order transverse modes in a light-induced waveguide in a Rb cell, which does not have any obvious relation with our explanation based on a single-atom model.
4. Conclusion
The YSF at 420 nm resulting from the 5S – 5D excitation of Rb atoms has been investigated using ultrashort laser pulses. The spatial profile of the blue light was highly sensitive to the chirp rate of the pump pulse, which is reproduced by the simulated results of the excitation dynamics for a single atom. In particular, the observed ring-shaped pattern is attributed to the two-photon resonant Rabi oscillations. These results indicate that nonlinear population dynamics of Rb atoms can be retrieved from spatial profiles of the blue light, where the spatial intensity distribution of the pump pulse is directly imprinted. Since our method does not need any scan, it could be extended to investigate nonlinear population dynamics in a single shot as well as to measure the chirp rate of ultrashort laser pulses.
Funding
This work was partially supported by the Grant-in-Aid for Young Scientists (B) (Grant No. 15K17806) from JSPS, Japan.
References and links
1. S. Baker, I. A. Walmsley, J. W. G. Tisch, and J. P. Marangos, “Femtosecond to attosecond light pulses from a molecular modulator,” Nat. Photonics 5, 664 (2011). [CrossRef]
2. V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation of Paired Photons with Controllable Waveforms,” Phys. Rev. Lett. 94, 183601 (2005). [CrossRef]
3. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of Squeezed States Generated by Four-Wave Mixing in an Optical Cavity,” Phys. Rev. Lett. 55, 2409 (1985). [CrossRef] [PubMed]
4. M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Suppression of Amplified Spontaneous Emission by the Four-Wave Mixing Process,” Phys. Rev. Lett. 55, 1086 (1985). [CrossRef] [PubMed]
5. F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration,” Phys. Rev. A 78, 013834 (2008). [CrossRef]
6. J. H. Brownell, X. Lu, and S. R. Hartmann, “Yoked Superfluorescence,” Phys. Rev. Lett. 75, 3265 (1995). [CrossRef] [PubMed]
7. A. I. Lvovsky, S. R. Hartmann, and F. Moshary, “Omnidirectional Superfluorescence,” Phys. Rev. Lett. 82, 4420 (1999). [CrossRef]
8. G. O. Ariunbold, M. M. Kash, V. A. Sautenkov, H. Li, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, “Observation of picosecond superfluorescent pulses in rubidium atomic vapor pumped by 100-fs laser pulses,” Phys. Rev. A 82, 043421 (2010). [CrossRef]
9. D. Felinto, L. H. Acioli, and S. S. Vianna, “Temporal coherent control of a sequential transition in rubidium atoms,” Opt. Lett. 25, 917 (2000). [CrossRef]
10. G. O. Ariunbold, V. A. Sautenkov, and M. O. Scully, “Temporal coherent control of superfluorescent pulses,” Opt. Lett. 37, 2400 (2012). [CrossRef] [PubMed]
11. G. O. Ariunbold, V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Ultrafast laser control of backward superfluorescence towards standoff sensing,” Appl. Phys. Lett. 104, 021114 (2014). [CrossRef]
12. B. Gai, S. Hu, J. Liu, R. Cao, J. Guo, Y. Jin, and F. Sang, “The response speed and fatigue characteristics of a pulsed 778 nm→420 nm conversion in rubidium vapor,” Opt. Commun. 374, 142 (2016). [CrossRef]
13. C. V. Sulham, G. A. Pitz, and G. P. Perram, “Blue and infrared stimulated emission from alkali vapors pumped through two-photon absorption,” Appl. Phys. B 101, 57 (2010). [CrossRef]
14. A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002). [CrossRef]
15. T. Meijer, J. D. White, B. Smeets, M. Jeppesen, and R. E. Scholten, “Blue five-level frequency-upconversion system in rubidium,” Opt. Lett. 31, 1002 (2006). [CrossRef] [PubMed]
16. A. M. Akulshin, R. J. McLean, A. I. Sidorov, and P. Hannaford, “Coherent and collimated blue light generated by four-wave mixing in Rb vapour,” Opt. Express 17, 22861 (2009). [CrossRef]
17. A. Vernier, S. Franke-Arnold, E. Riis, and A. S. Arnold, “Enhanced frequency up-conversion in Rb vapor,” Opt. Express 18, 17020 (2010). [CrossRef] [PubMed]
18. A. Akulshin, D. Budker, and R. McLean, “Directional infrared emission resulting from cascade population inversion and four-wave mixing in Rb vapours,” Opt. Lett. 39, 845 (2014). [CrossRef] [PubMed]
19. J. F. Sell, M. A. Gearba, B. D. DePaola, and R. J. Knize, “Collimated blue and infrared beams generated by two-photon excitation in Rb vapor,” Opt. Lett. 39, 528 (2014). [CrossRef] [PubMed]
20. E. Brekke and L. Alderson, “Parametric four-wave mixing using a single cw laser,” Opt. Lett. 38, 2147 (2013). [CrossRef] [PubMed]
21. R. F. Offer, J. W. C. Conway, E. Riis, S. Franke-Arnold, and A. S. Arnold, “Cavity-enhanced frequency up-conversion in rubidium vapor,” Opt. Lett. 41, 2177 (2016). [CrossRef] [PubMed]
22. A. Dogariu, J. B. Michael, M. O. Scully, and R. B. Miles, “High-gain backward lasing in air,” Science 331, 442 (2011). [CrossRef] [PubMed]
23. P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines:comparison between theory and experiment,” J. Phys. B 41, 155004 (2008). [CrossRef]
24. L. Yuan, B. H. Hokr, A. J. Traverso, D. V. Voronine, Y. Rostovtsev, A. V. Sokolov, and M. O. Scully, “Theoretical analysis of the coherence-brightened laser in air,” Phys. Rev. A 87, 023826 (2013). [CrossRef]
25. E. Paradis, B. Barrett, A. Kumarakrishnan, R. Zhang, and G. Raithel, “Observation of superfluorescent emissions from laser-cooled atoms,” Phys. Rev. A 77, 043419 (2008). [CrossRef]
26. K. Kitano, N. Ishii, and J. Itatani, “High degree of molecular orientation by a combination of THz and femtosecond laser pulses,” Phys. Rev. A 84, 053408 (2011). [CrossRef]
27. R. N. Zare, Angular Momentum (Wiley-Interscience, 1986).
28. M. S. Safronova, C. J. Williams, and C. W. Clark, “Relativistic many-body calculations of electric-dipole matrix elements, liftimes, and polarizabilities in rubidium,” Phys. Rev. A 69, 022509 (2004). [CrossRef]
29. A. M. Weiner, Ultrafast Optics (John Wiley & Sons, Inc., 2009). [CrossRef]
30. W. Süptitz, B. C. Duncan, and P. L. Gould, “Efficient 5D excitation of trapped Rb atoms using pulses of diode-laser light in the counterintuitive order,” J. Opt. Soc. Am. B 14, 1001 (1997). [CrossRef]
31. J. Lim, K. Lee, and J. Ahn, “Ultrafast Rabi flopping in a three-level energy ladder,” Opt. Lett. 37, 3378 (2012). [CrossRef]
