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We control the atomic coherence and the population transfer among Rb hyperfine atomic levels by the fractional stimulated Raman adiabatic passage (F-STIRAP) in a Λ-type configuration, and verify the theoretical predictions. Applying this technique, we are able to prepare the atoms with maximal coherence to enhance coherent anti-Stokes Raman scattering (CARS) signal. In our experiment by scanning the frequency of one laser from 794.9839 nm to 794.9844 nm with the 794.9698 nm laser frequency fixed to generate a maximum of the CARS signal we are able to obtain the energy level diagram of the sample.
©2007 Optical Society of America
1. Introduction
Quantum coherence and interference have led to the observation of many new effects and techniques in quantum optics and atomic physics. In recent years, electromagnetically induced transparency (EIT) due to atomic coherence effect attracted a great deal of attention. The nonlinear interaction of EIT provides the grounds for optical frequency conversion and Raman generation, which has established its efficiency in identifying constituents of molecular and atomic media [1, 2]. CARS has recently become a favorable method for nonlinear depth-resolved microscopy. It is able to analyze the energy-level diagram of atoms and complex molecules. Coherent control provides an efficient approach to generate or enhance CARS signal. By tailoring the probe pulse, one can obtain a narrow-band coherent anti-Stokes Raman spectroscopy resonant signal from broad-band probe pulse [3].
Stimulated Raman adiabatic passage (STIRAP)[4, 5], one of the applications of quantum coherence and interference, provides a simple and robust technique for transferring population between two nondegenerate metastable levels, making use of two pulses, termed the pump pulse and the Stokes pulse. F-STIRAP [6] is different with the STIRAP, creating the maximal coherence and delaying the time of coherence. The enhancement of the coherent Raman scattering under the condition of maximal coherence between Zeeman sublevels of 87Rb 5S 1/2(F = 2) prepared by F-STIRAP [7] have been implemented by Scully group. They also have demonstrated that the optimization of CARS can be achieved by using the F-STIRAP technique among Zeeman sublevels [8].
In this paper, we report an experimental implementation of F-STIRAP in 87Rb vapor cell where lower level coherence is prepared between two separate atomic levels instead of between two generated Zeeman sublevels (as shown in Fig. 1). In our experiment, light pulses at 794.9842 nm and 794.9698 nm are obtained by two extended cavity diode lasers to prepare the Rb atoms with maximal coherence between hyperfine levels of which frequency shift is 6.8 GHZ to generate CARS signal. The usual situation within conventional CARS is that the ground state coherence is not a maximum. This paper describes the experimental results of optimization of CARS signal by maximal coherence prepared by F-STIRAP. The CARS signal is investigated using adiabatic rapid passage in a nanosecond time scale, which is smaller than the lifetime of first excited Rb. Working with ultra-short laser pulses enable us to tailor the pulse sequence as to overcome some restrictions in the application of conventional CARS. The key point of this paper is that by generating a maximum of the CARS signal we are able to directly determine the energy level diagram of the sample. And that we have proved the optimization of CARS signal band-width is about 94.94 MHz.
2. Theoretical basis
Fig. 1. Energy-level diagram of the Λ-type system in 87Rb. Ωc1, Ωc2, Ωp, ΩCARS are the Rabi frequency of the first, second coupling, probe and CARS signal, respectively.
We consider a three-level system, as shown in Fig. 1. The first coupling laser with Rabi frequency Ωc1, couples the ground state |3〉 and the excited state |2〉 with a frequency detuning of Δs, and the second coupling laser with Rabi frequency Ωc2, couples the lower ground state |1〉 and the excited state |2〉 with a frequency detuning of Δp, adiabatically establish a maximal atomic coherence of the Raman transition defined as the situation that the density matrix elements satisfy ρ 11 ≈ ρ 33 ≈ |ρ 13| ≈ 0.5. The probe laser turned to the |3〉 → |2〉 transition with Rabi frequency Ωp, and mixes with the coherence to generate a fourth field described as the CARS signal with Rabi frequency ΩCARS. The interaction Hamiltonian in the rotating wave and dipole moment approximations for the three-level system is
in the Eq. (1), Ωi, = |℘i|Ei/h¯ (i = c1,c2,p,CARS) is the Rabi frequency corresponding to the respective fields; ℘i is the electrical dipole matrix element; Ei is the amplitude of the respective laser field; Δp = ω21 - ω c2 and Δs = (ω 23 - ω c1 are the laser detuning from the atomic resonance, ω c1, ω c2 are the frequency of fields. The elements of the density matrix are given by the Liouville equation:
where Γ21 and Γ23 are the spontaneous emission from the state |2〉 to the states |1〉 and |3〉. Γ31 is the relaxation caused by atoms collision or other reason from the state |3〉 to |1〉, which is less than Γ21 and Γ23. The off-diagonal density matrix elements satisfy ρij = ρji *, and the conservation relation of the population is ρ11 + ρ22 + ρ33 = 1. The equations for the fields
where η 23 = ω c1 N|℘ 23|2/ε 0 ch¯, η 21 = ω c2 N|℘21|2/ε 0 ch¯ are the coupling constants, N is the density of medium, ε 0 is the permittivity of the vacuum, and c is the speed of light in vacuum.
Fig. 2. Numerical simulations of the population transfer, coherence and the propagation of laser pulses by the technique of F-STIRAP. (a) The first coupling, the second coupling and probe pulses at the entrance of the sample cell. (b) The output pulses and the CARS signal at the exit of the cell. (c) Population transfer between |1〉 and |3〉, and the coherence between states |1〉 and |3〉.
We performed numerical simulations of the above theory for the case of maximal atomic coherence between hyperfine levels in 87Rb vapor cell by the technique of F-STIRAP. Fig. 2(a) shows the input pulse envelopes with the first coupling pulse preceding the second coupling pulse and the probe pulse coming behind of them at the entrance of the medium. The variation of the first coupling and probe pulses at the exit of the cell is shown in Fig. 2(b). It also shows that the output of the second coupling pulse and CARS signal. Fig. 2(c) shows the population transfer as a function of time. The initial population is equally distributed between state |1〉 and |3〉. After the first coupling pulse is applied, all population is driven into state |1〉, leaving |3〉 empty. Then application of the second coupling pulse transfer a half of population from state |1〉 to |3〉 via F-STIRAP processes. The coherence term ρ 13, as illustrated in Fig. 2(c), reaches its maximum value for the interval time when states |1〉 and |3〉 are half/half populated.
3. Experiment
Now, we turn to a description of the experiment. An energy levels diagram of the experiment is shown in Fig. 1. The first coupling pulse with Rabi frequency Ωc1(λ = 794.9842 nm) is right circularly polarized, the second coupling pulse with Rabi frequency Ωc2(λ = 794.9698 nm) is left circularly polarized. Hyperfine levels |52 S 1/2,F = 1,MF = -1,0〉, showed as state |1〉, and levels |52 S 1/2,F = 2,MF = +1,+2), showed as state |3〉, serve as the lower states while the upper state is the hyperfine levels |52 P 1/2,F′ = 1,MF′ = 0, +1), showed as level |2〉. The power of the first coupling pulse is 7.5 mw, with duration 150 ns, which can drive all population of |3〉 into state |1〉 before the second coupling pulse is applied. The power of the second coupling pulse is 8 mw, and its duration of 20 ns is shorter than the lifetime of the excited state which is required by the STIRAP. The two coupling pulses with the same back edge drive all the atoms of the ensemble into a maximally coherent superposition between the |1〉 and |3〉 states. The time delay between the end of the first coupling pulse and beginning of the probe pulse is 80 ns. The duration of the probe pulse is 20 ns.
The experimental arrangement is shown in Fig. 3, which is a modified version of that described in Ref. [5, 7, 8]. Both ECDL1 and ECDL2 are the external-cavity diode lasers with linearly polarized output beams. The acousto-optic modulator AOM1 with the frequency shift 200 MHZ, driven by a pulse generator, is used to switch on and off of the ECDL1 laser to generate two pulses as the first coupling and the probe pulses. Considering the rise time of the AOM, the focus length of L1 and L2 is 5 cm. The polarization of the first coupling and the probe pulses is rotated by 90° after passing through λ/2 wave plate. Connecting with a digital delay/pulse generator, the AOM2 is used to turn on and off ECDL2 laser to generate the second coupling pulse. The focus length of L3, L4 is 5 cm. The two beams are combined with a polarizing beam splitter. They propagate collinearly through a λ/4 wave plate, which results in opposite circular polarization of the two beams. Then the two beams are focused by a lens (focus length 30 cm) into the atomic Rb vapor cell which is 3.5 cm long, 2.5 cm in diameter. The temperature of the Rb vapor cell was set to 85 ~ 98°C, corresponding to atomic density of 1012 cm-3. After the cell, the beams with opposite circular polarization can be separated by another polarizing beam splitter after they pass through another λ/4 wave plate and become linearly polarized with orthogonal polarization, respectively. The first coupling and the probe laser are detected by PD1, and the second coupling pulse and the CARS signal are detected by PD2.
4. The analysis of experimental results
Fig. 4. (a). The pulses at the entrance of the Rb cell, the duration of Ωc1, Ωc2, Ωp are 150 ns, 20 ns, 20 ns, respectively. (b)The pulses after Rb cell, ΩCARS is the generated signal. All the signals are normalized to the intensity of the probe pulse before entering the cell.
The time evolution of the laser pulses is shown in Fig. 4. The direct comparison between the Fig. 2 and Fig. 4 shows that the experiments are in good agreement with numerical simulations. Figure 4(a) gives the experiment demonstration of all laser pulses before the Rb cell. The duration of the first coupling pulse with Rabi frequency Ωc1, the second coupling pulse with Rabi frequency Ωc2 are 150 ns and 20 ns, respectively. The probe delays from the first coupling pulse by τp = 80 ns, and its duration is 20 ns. Figure 4(b) displays the laser pulses after passing through the cell. One can see that the intensity of the generated CARS is 0.12 of the intensity of the initial probe pulse, which is less than the generated field of Ref. [7]. The experiment of Ref. [7] prepare coherence between Zeeman levels using single laser, which create coherence being better than that between difficult hyperfine states using two independent ECDLs. From Fig. 4, one can see that the coupling and probe fields experiences a strong absorption. The maximum coherence is not maintained on propagation, as the second coupling field is absorbed more than the first one. Only a portion of the whole atoms reach maximum coherence. The peak at the back of the first coupling pulse arises from stimulated Raman resonance scattering of the two-photon process.
Fig. 5. The intensity of the signal versus the wavelength of the first coupling pulse with the second coupling laser frequency fixed at 794.9698 nm. Squares are the experimental results, the solid curve is the theoretical one.
We tune and lock the second coupling beam at the resonant frequency of the coupling transition (|52 S 1/2,F = 1,MF = -1,0〉 → |52 P 1/2,F′ = 1,MF′ = 0,+1〉), e.g., Δp ≈ 0. Then, we turn the first coupling pulse to be at resonance with the transition |52 S 1/2,F = 2,MF = + 1,+2〉 to |52 P 1/2,F′ = 1,MF′ = 0, +1〉, and scan it a few MHZ above or below resonance. The main result is shown in Fig. 5. The experimentally obtained signal intensity of CARS as a function of the frequency of the first coupling and probe pulses is shown in it. Experimental data are described with squares, and the error bars represent the standard deviation of data points. By scanning the frequency of the first coupling pulse from 794.9839 nm to 794.9844 nm, one can see that there is a maximum of the CARS signal corresponding to the two-photon resonance. Then we are able to obtain the energy level diagram of the sample. It is worth to note that the generation of the signal occurs only in a narrow range of the frequency of the first coupling and probe pulses. The solid curve is the result of numerical simulations. Note that half-width of the curve is about 0.0002nm which is about 94.94 MHz. It offers a new method to obtain energy levels which is different with other techniques.
5. Conclusion
In summary, the paper demonstrated the generation of anti-Stokes Raman signal via maximal atomic coherence prepared by F-STIRAP in the Rb vapor cell between separate hyperfine levels instead of generated levels. And we have proved the feasibility to enhance CARS signal. Our technique is different from conventional CARS technique. By generating a maximum of the CARS signal we are able to directly determine the energy level diagram of the sample. The potential application of our experiment technique is the application in various environments ( solids, liquids, and gases) either atoms or molecules to determine the configuration of energy levels.
Acknowledgments
The authors acknowledge the financial support from the NSFC (Grant No.10334010), from the doctoral program foundation of institution of High Education of China, and from the National Basic Research Program (Grant No.2006CB921103).
References and links
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