Inhibition of Two-Photon Absorption in a Four-Level Atomic System with Closed-Loop Configuration

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1 Commun. Theor. Phys. Beijing, China) ) pp c International Academic Publishers Vol. 47, No. 5, May 15, 007 Inhibition of Two-Photon Absorption in a Four-Level Atomic System with Closed-Loop Configuration JIN Li-Xia, LI Jia-Hua, and QI Chun-Chao Department of Physics and National Key Laboratory for Laser Technique, Huazhong University of Science and Technology, Wuhan , China Received June 14, 006) Abstract We theoretically investigate the features of two-photon absorption in a coherently driven four-level atomic system with closed-loop configuration. It is found that two-photon absorption can be completely suppressed just by properly adjusting the relative phase of four coherent low-intensity driving fields and the atomic system becomes transparent against two-photon absorption. From a physical point of view, we explicitly explain these results in terms of quantum interference induced by two different two-photon excitation channels. PACS numbers: 4.50.Gy, 4.50.Hz, 3.80.Qk Key words: two-photon absorption, quantum interference, closed-loop configuration Effects of atomic coherence and quantum interference are currently becoming an important and intensive tool in resonant nonlinear optical physics. The inhibition of two-photon and multiphoton absorption based on these effects in nonlinear optical regime has potential applications in high-efficiency generation of short-wave length coherent radiation at pump intensities approaching the single-photon level, 1 7 trapping and manipulating photon states in atomic ensembles, 8,9 nonlinear spectra at very low light intensity, ultraslow optical solitons in highly resonant media, 19,0 and pulse-preserving propagation in dissipative media, 1, and so on. In particular, nonlinear optics assisted by electromagnetically induced transparency EIT) under few-photon optical field due to extremely large Kerr nonlinearities and refractiveindex enhancement without absorption has been already observed, 3 and this opens up the possibilities for a new regime of quantum nonlinear optics. For example, Harris et al. proposed the use of EIT to suppress absorption of the short-wavelength light generated in a fourwave mixing FWM) scheme and showed that the FWM efficiency could be greatly enhanced. 1 Agarwal and Harshawardhan showed that in a four-level Y-type system, two-photon absorption could be selectively suppressed or enhanced, 10 and later Gao et al. reported the experimental observation of the electromagnetically induced inhibition of two-photon absorption with atomic sodium vapor as a test medium. 11 In a recent experiment, Yan et al. demonstrated that EIT in a standard Λ-type configuration could be used to suppress both single-photon and twophoton absorptions simultaneously. 4,5 Later on, Wu et al. investigated and discussed a four-wave-mixing FWM) scheme in a five-level atomic system and hyper-raman scattering HRS) in resonant coherent media by the use of EIT, which led to suppressing both two-photon and threephoton absorptions in both FWM and HRS schemes and enabling the four-wave mixing to proceed through real, resonant intermediate states without absorption loss. Quite recently, he and his coworkers again analyzed a lifetime-broadened four-state FWM scheme in the ultraslow propagation regime and put forward a new type of induced transparency resulted from multiphoton destructive interference. 6 In Ref. 7, Zou et al. investigated two-photon absorption in a three-level ladder atomic system driven by a pair of bichromatic fields of equal frequency differences and demonstrated the reduction of twoabsorption absorption. In this paper, we put forward a new approach to efficiently suppress two-photon absorption only via modulating the relative phase of four coherent driving fields at low intensity in a closed-loop four-level atomic system, unlike the previous schemes based on EIT effect The proposed scheme requires four driving fields but is more convenient in its experimental realization compared with the scheme proposed in Ref. 7. Let us consider a four-level atomic system driven by four coherent laser fields, as shown in Fig. 1. The atomic states are labelled as 1,, 3, and 4, respectively. Two probe laser fields E p1 and E p with carrier frequencies ω p1, ω p and Rabi frequencies Ω p1, Ω p ) are applied to the transitions 1 and 3 1 to serve as the first step of the two-photon excitation of upper atomic state 4. Two signal laser fields E s1 and E s with carrier frequencies ω s1, ω s and Rabi frequencies Ω s1, Ω s ) are coupled to the transitions 4 and 4 3 to complete the two-photon excitation of atomic state 4. The transitions own a closedloop configuration. In the interaction picture and under the electro-dipole interaction and rotating-wave approximation, with the assumption of = 1, the semiclassical The project supported in part by National Natural Science Foundation of China under Grant Nos , , and Author to whom correspondence should be addressed, huajia li@163.com

2 No. 5 Inhibition of Two-Photon Absorption in a Four-Level Atomic System with Closed-Loop Configuration 917 Hamiltonian describing the atom-field interaction for the system under study can be written as V = ) 4 4 Ω p1 1 + Ω s1 4 + Ω p Ω s h.c.), 1) where h.c. means Hermitian conjugation and for the sake of simplicity we have taken the ground state 1 as the energy origin. The quantities Ω n n = p1, p, s1, s) stand for one-half Rabi frequencies for the respective transitions, i.e., Ω p1 = µ 1 E p1 / ), Ω p = µ E p / ), Ω s1 = µ 4 E s1 / ), and Ω s = µ 43 E s / ), where µ ij = µ ij ê L where ê L is the polarization unit vector of the laser field) denotes the dipole moment for the transition between levels i and j. 1 = ω 1 ω p1, = ω ω p, 4 = ω 4 ω s1, and 43 = ω 43 ω s are the frequency detunings of the four coherent fields from the corresponding two-level transitions see Fig. 1). The decay rates from the states 4 to, 4 to 3, to 1, and 3 to 1 are γ 4, γ 43, γ 1, and γ, respectively. In deriving Eq. 1), we have assumed that the carrier frequencies of the four fields satisfy ω p1 +ω s1 = ω p +ω s for the sake of simplification, the relation = + 43 can be obtained. Fig. 1 Schematic diagram of four-level atoms in a coherent medium interacting with two probe lasers with Rabi frequencies Ω p1, Ω p and two signal lasers with Rabi frequencies Ω s1, Ω s. The atomic states are labelled as 1,, 3, and 4, respectively. The transitions own a closed-loop configuration. 1,, 4, and 43 are the frequency detunings of the four coherent laser fields, see text for details. In what follows, using the density-matrix formalism we begin to describe the dynamic response of the resonant coherent medium under study. By the standard approach, 8 we can easily obtain the time-dependent density matrix equations of motion as follows: ρ 11 = γ 1 ρ + γ ρ 33 + iω p1ρ 1 + iω pρ iω p1 ρ 1 iω p ρ 13, ρ = γ 4 ρ 44 γ 1 ρ + iω p1 ρ 1 + iω s1ρ 4 iω p1ρ 1 iω s1 ρ 4, ρ 33 = γ 43 ρ 44 γ ρ 33 + iω sρ 43 + iω p ρ 13 iω s ρ 34 iω pρ, ρ 44 = γ 43 + γ 4 ) ρ 44 + iω s1 ρ 4 + iω s ρ 34 iω s1ρ 4 iω sρ 43, ρ 1 = i 1 γ ) 1 ρ 1 + iω p1ρ ρ 11 ) + iω pρ 3 iω s1 ρ 14, ρ 13 = i γ ) ρ 13 + iω p ρ 33 ρ 11 ) + iω p1ρ 3 iω s ρ 14, ρ 14 = i ) γ 43 + γ 4 ρ 14 + iω p1ρ 4 + iω pρ 34 iω s1ρ 1 iω sρ 13, ρ 3 = i 1 ) γ 1 + γ ρ 3 + iω p1 ρ 13 + iω s1ρ 43 iω s ρ 4 iω pρ 1, ρ 4 = i 4 γ 43 + γ 4 + γ 1 ρ 4 + iω s1ρ 44 ρ ) + iω p1 ρ 14 iω sρ 3, ρ 34 = i ) γ 43 + γ 4 + γ ρ 34 + iω sρ 44 ρ 33 ) + iω p ρ 14 iω s1ρ 3, ) where the dots denote the derivative with respect to time 4 t. Closure of this atomic system requires that ρ jj = 1 j=1 and ρ ij = ρ ji. The appearance of the closed-loop configuration makes the system becomes quite sensitive to phases of the four coherent fields, thus the Rabi frequencies have to be treated as complex-valued parameters. We use φ p1, φ p, φ s1, and φ s to denote relevant phases of two pair of probe and signal fields, then we reexpress Ω p1 = G p1 e iφp1, Ω p = G p e iφp, Ω s1 = G s1 e iφs1, and Ω s = G s e iφs, where G p1, G p, G s1, and G s are real parameters, and other density matrix elements σ ii = ρ ii i = 1 4), σ 1 = ρ 1 e iφp1, σ = ρ e iφp, σ 4 = ρ 4 e iφs1, σ 43 = ρ 43 e iφs, σ 3 = ρ 3 e iφp φp1), and σ 14 = ρ 14 e iφp1+φs1). Then we can obtain the corresponding density matrix equations about σ ij from Eq. ) as follows: σ 11 = γ 1 σ + γ σ 33 + ig p1 σ 1 σ 1 ) + ig p σ σ 13 ), σ = γ 4 σ 44 γ 1 σ + ig p1 σ 1 σ 1 ) + ig s1 σ 4 σ 4 ), σ 33 = γ 43 σ 44 γ σ 33 + ig s σ 43 σ 34 ) + ig p σ 13 σ ), σ 44 = γ 43 + γ 4 ) σ 44 + ig s1 σ 4 σ 4 ) + ig s σ 34 σ 43 ), σ 1 = i 1 γ ) 1 σ 1 + ig p1 σ σ 11 ) + ig p σ 3 ig s1 σ 14,

3 918 JIN Li-Xia, LI Jia-Hua, and QI Chun-Chao Vol. 47 σ 13 = i γ ) σ 13 + ig p σ 33 σ 11 ) + ig p1 σ 3 ig s e iφ σ 14, σ 14 = i ) γ 43 + γ 4 σ 14 + ig p1 σ 4 + ig p e iφ σ 34 ig s1 σ 1 ig s e iφ σ 13, σ 3 = i 1 ) γ 1 + γ σ 3 + ig p1 σ 13 + ig s1 e iφ σ 43 ig s e iφ σ 4 ig p σ 1, σ 4 = i 4 γ 43 + γ 4 + γ 1 σ 4 + ig s1 σ 44 σ ) + ig p1 σ 14 ig s e iφ σ 3, σ 34 = i ) γ 43 + γ 4 + γ σ 34 + ig s σ 44 σ 33 ) + ig p e iφ σ 14 ig s1 e iφ σ 3, 3) where Φ = φ p1 + φ s1 φ p φ s is the overall phase difference between two -photon pathways. It is quite obvious from the general structure of Hamiltonian 1) that two possible pathways from state 1 to state 4 exists: 1 Ωp1 Ωs1 4 and 1 Ωp 3 Ωs 4. The dynamics and the role of the phase Φ on the two-photon absorption in the closed-loop four-level system can be explained from quantum interference induced by these two channels. As a result, the relative phase Φ can be used as a control parameter to investigate the features of the two-photon absorption spectra. As it is well known, the two-photon absorption for the pathway 1 3 ) 4 is proportional to the population distribution in the excited state 4, i.e., σ 44. In the following, we begin with investigating the response of two-photon absorption by solving the time-dependent density matrix equations 3) numerically via a nice mathematica code. Note that, in this paper, all involving parameters are scaled γ, which should be in the order of MHz for rubidium or sodium atoms. First of all, we will analyze how the relative phase of the four driving fields modifies the two-photon transient absorption spectra via the numerical calculations based on Eq. 3). In Fig., we plot the time evolution of the two-photon absorption by modulating the relative phase Φ under the initial conditions of σ 11 0) = 1 and other σ ij 0) = 0 i, j = 1 4) when the four driving fields are respectively tuned to the resonant interaction with the corresponding transitions. It is clearly shown from Fig. that, two-photon transient absorption is quite sensitive to the relative phase Φ. Specifically, for the case that Φ = 0, two-photon transient absorption grows quickly with the time firstly, and then it reaches a large steady-state value accompanied by a slight reduction. For the case that Φ = π/, the behavior of the two-photon absorption is similar to the case that Φ = 0 except for the considerable reduction of its amplitude. However, for the case that Φ = π, two-photon absorption is completely suppressed all the time and the atomic system becomes transparent against two-photon absorption. Fig. Two-photon absorption versus normalized time for the different relative phase Φ under the initial conditions σ 110) = 1 and other σ ij0) = 0. The parameters are chosen as G p1 = G p = G s1 = G s = 0.γ, 1 = = 4 = 43 = 0, γ 1 = γ = γ, and γ 4 = γ 43 = 0.15γ, respectively. Fig. 3 One- and two-photon absorption versus frequency detuning for a) Φ = 0 and b) Φ = π. The parameters are chosen as G p1 = G p = G s1 = G s = 0.γ, 4 = 1, = 43 = 0, γ 1 = γ = γ, and γ 4 = γ 43 = 0.15γ, respectively. In Fig. 3, we plot one- and two-photon absorption as a function of frequency detuning 1 in the steady-state limit for the cases that both Φ = 0 and Φ = π. For the case that Φ = 0, two-photon absorption is enhanced

4 No. 5 Inhibition of Two-Photon Absorption in a Four-Level Atomic System with Closed-Loop Configuration 919 and one-photon absorption is suppressed at 1 = 0. Adjusting Φ = π, two-photon absorption is completely suppression and one-photon absorption is greatly enhanced at 1 = 0. In order to further explicitly show the dependence of the one- and two-photon absorption on the relative phase, in Fig. 4 we also plot the one- and two-photon absorption as a function of the relative phase Φ in the steady-state limit. From the figure, it can be seen clearly that the complete inhibition of two-photon absorption can be observed by choosing the relative phase appropriately, e.g., Φ = k + 1)π where k is an integer). While when adjusting Φ = kπ, the two-photon absorption is greatly enhanced. The feature of the one-photon absorption is just opposite to that of the two-photon absorption. Alternatively, it is clear that one- and two-photon absorption is a periodical function of the relative phase Φ with the period of π. The reason for the above results can be interpreted in terms of quantum interference as follows. The sign of the interference constructive or destructive) between double coherent two-photon excitation paths depends on the phase differences between the paths. To be specific, for the case that Φ = kπ, constructive interference between the two different two-photon channels occurs, resulting in great enhancement of the two-photon absorption. In contrast, for the case that Φ = k + 1)π, the two excitation pathways interfere destructively. Such a robust two-photon destructive interference thus leads to complete suppression of the two-photon absorption. For the case that kπ < Φ < k +1)π, the degree of quantum interference lies between the above two circumstances. Before concluding, we give a brief discussion on the possible experimental realization of our proposed scheme by means of alkali-metal atoms and appropriate diode lasers. Specifically, we consider for instance the cold atoms 87 Rb nuclear spin I = 3/) on the 5S 1/ 5P 3/ 5D 5/ transitions as a possible candidate. The designated states can be chosen as follows: 1 = 5S 1/, F = 1, = 5P 3/, F = 1, 3 = 5P 3/, F =, and 4 = 5D 5/, F =, respectively. In this case, two pairs of coherent probe and signal laser radiations, whose wavelengths are, respectively, 780. nm 1 with a σ + field and 1 3 with a σ field) and nm 4 with a σ field and 4 3 with a σ + field) can be obtained from external cavity diode lasers. Moreover, in order to eliminate the Doppler broadening effect, atoms should be trapped and cooled by the magnetooptical trap MOT) technique. Fig. 4 One- and two-photon absorptions versus relative phase. The parameters are chosen as G p1 = G p = G s1 = G s = 0.γ, 1 = = 4 = 43 = 0, γ 1 = γ = γ, and γ 4 = γ 43 = 0.15γ, respectively. In conclusion, we have analyzed and discussed the phase control of the two-photon absorption in a four-level closed-loop atomic system driven by four low-intensity laser fields. Via numerical simulations we clearly show that two-photon absorption can be completely suppressed just by properly choosing the relative phase of four coherent driving fields and the atomic system becomes transparent against two-photon absorption. From a physical point of view, we well explain these results in terms of quantum interference induced by two different two-photon excitation channels. According to our analyses, these interesting phenomena should be observable in realistic experiments by using alkali-metal atoms e.g., cold Rb or Na atoms) and appropriate diode lasers. Clearly this control of the two-photon absorption should be of importance in the context of related issues like two-photon lasing, pulse-preserving propagation in dissipative media and two-photon entanglement in quantum computing and information processing. Acknowledgments The authors would like to thank professor Ying WU for helpful discussion and his encouragement. References 1 S.E. Harris, J.E. Field, and A. Imamoğlu, Phys. Rev. Lett ) Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A ) ; Y. Wu, L. Wen, and Y. Zhu, Opt. Lett ) 6. 3 X.X. Yang, Z.W. Li, and Y. Wu, Phys. Lett. A ) 30; Y. Wu and X. Yang, Phys. Rev. A ) A.S. Zibrov, M.D. Lukin, L. Hollberg, and M.O. Scully, Phys. Rev. A 65 00) R); A.S. Zibrov, A.B. Matsko, and M.O. Scully, Phys. Rev. Lett ) S.E. Harris and L.V. Hau, Phys. Rev. Lett ) Z. Li, D.Z. Cao, and K. Wang, in Proceedings of SPIE: Quantum Optics and Applications in Computing and

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