Electromagnetically Induced Transparency and Absorption of A Monochromatic Light Controlled by a Radio Frequency Field

Transcription

1 Commun. Theor. Phys. 6 (2015) Vol. 6, No. 2, February 1, 2015 Electromagnetically Induced Transparency and Absorption of A Monochromatic Light Controlled by a Radio Frequency Field CAI Xun-Ming ( ) College of Information Engineering, Guizhou Minzu University, Guiyang , China (Received August 4, 2014; revised manuscript received October 11, 2014) Abstract Electromagnetically induced transparency and absorption of a monochromatic light controlled by a radio frequency field in the cold multi-zeeman-sublevel atoms are theoretically investigated. These Zeeman sublevels are coupled by a radio frequency (RF) field. Both electromagnetically induced transparency and electromagnetically induced absorption can be obtained by tuning the frequency of RF field for both the linear polarization and elliptical polarization monochromatic lights. When the transfer of coherence via spontaneous emission from the excited state to the ground state is considered, electromagnetically induced absorption can be changed into electromagnetically induced transparency with the change of intensity of radio field. The transparency windows controlled by the RF field can have potential applications in the magnetic-field measurement and quantum information processing. PACS numbers: Gy, i, 2.80.Qk Key words: electromagnetically induced transparency, electromagnetically induced absorption, absorption spectrum 1 Introduction Since coherent population trapping (CPT), electromagnetically induced transparency (EIT), and electromagnetically induced absorption (EIA) phenomena have been demonstrated in various three-level atomic systems, many new CPT-related and EIT-related phenomena have been discovered and studied. [1 5 One interesting effect is the dual-dark-state resonances studied in two coupled three-level EIT systems. [6 7 The dual-dark-states resonances phenomenon can be used in studying multi-channel optical communication, researching the interaction of dark states and enhancing nonlinear effect. CPT in various Zeeman sublevels has been used in atomic clock and atom magnetometer. Usually two different frequencies of light fields are used. Two light fields are coupled to different Zeeman sublevels. CPT is built when the twophoton resonant condition is met. Microwaves coupling the ground states can manipulate the characteristics of the EIT medium. [8 12 Microwave interaction has been applied to excite the Raman trapped state and to influence the CPT in a Λ system. Fast and slow light phenomena in the EIT medium controlled by microwave have been theoretically predicted. [1 15 Constructive and destructive interference in the presence of a microwave field have been experimentally realized based on a V-type system in solid (Pr+YALO) [16 and in Λ-type atom vapor. [17 The microwave field can be used to provide additional control to the light storage system. [18 19 But for CPT, two different frequency lights are needed, and the phases of two lights need to be locked. It is a complex question for locking the phases of lights in the CPT magnetometer. EIA is shown to be due to the transfer of coherence (TOC) via spontaneous emission from the excited state to the ground state. [20 22 In early optical pump magnetometer, a radio frequency field is used to balance population among Zeeman sublevels. When the frequency of RF field is equal to the frequency interval of two adjoining Zeeman sublevels, the pump light will be absorbed most. The CPT phenomenon induced by the interaction of a linearly polarized monochromatic light and an RF field with neon atoms has been studied. [2 The monochromatic light is positioned on the 2p 5 s P 1 (J = 1) 2p 5 p P 0 (J = 0) transition of neon. In the literature, Stokes and anti-stokes optical fields were generated due to stimulated Raman scattering of both the radio-frequency field on the probe-induced optical coherence and the probe field on the Zeeman coherence. The polarization direction of light is parallel with magnetic field. Only the m = 0 m = 0 transition between the upper and lower level is probed. So the transparency degree is only about 1%. In this article, we investigate both the enhanced transmission and the enhanced absorption of linear polarization and elliptical polarization monochromatic lights in the cold multi-zeemansublevel atoms controlled by a radio frequency field. The monochrome elliptically polarized or linearly polarized light is positioned on the 87 Rb5S 1/2 F = 2 5P 1/2 F = 1 transition. The RF field is used to couple the Zeeman sublevels. When TOC via spontaneous emission from the Supported by the Science Foundation of Guizhou Province under Grant Nos. LKM(201)19 and(2014)2090 Corresponding author, caixm1997@aliyun.com c 2015 Chinese Physical Society and IOP Publishing Ltd

2 20 Communications in Theoretical Physics Vol. 6 excited state to the ground state is considered, EIT can be changed to EIA with the change of intensity of radio field. We focus on the transition peak of the monochromatic polarized light in the transmission spectra by tuning the frequency of RF field. level F = 2, m i and e j represents the excited state level F = 1, m j. 2 Model and Equations We consider an elliptically or linearly polarized laser turned to the 87 Rb5S 1/2 F = 2 5P 1/2 F = 1 transition. As is shown in Fig. 1, the elliptic polarization or linear polarization direction is perpendicular to that of the magnetic field. It can be decomposed into the left-rotated circularly polarized light (σ ) and the right-rotated circularly polarized light (σ + ). An RF field is used to couple the Zeeman sublevels. One can see that the transition studied in Fig. 1 is not a cycling transition. Here 1 = ω ω 2, 1, 2 = ω ω 1,0, = ω ω 0,1, 4 = ω ω 0, 1, 5 = ω ω 1,0, 6 = ω ω 2,1 are light frequency detuning. ω is the frequency of light and ω f is the frequency of Radio field. ω i,j is the transition frequency from the atomic levels 87 Rb5S 1/2 F = 2m i to the 5P 1/2 F = 1m j. f is the detuning of Radio field with the ground state Zeeman levels. The ground state Zeeman splitting is δ and the excited state Zeeman splitting is δ/. δ = ω f + f. g i represents the ground state Fig. 1 (Color online) Atomic levels of 87 Rb. A linearly polarized light is positioned on the transition 87 Rb5S 1/2 F = 2 5P 1/2 F = 1 (red dotted line). An RF field is tuned to couple different Zeeman levels (blue line). The Hamiltonian for this system in the rotating wave approximation can be described by ( H = δ g 2 g 2 + 2δ g g + δ g 4 g 4 + 4δ g 5 g 5 + ω 2,1 e 1 e 1 + ω 2,1 + δ ) e 2 e 2 + (ω 2,1 + 2 ) δ e e [Ω f ( g 1 g 2 exp(iω f t) + g 2 g 1 exp( iω f t)) + Ω f ( g 2 g exp(iω f t) + g g 2 exp( iω f t)) + Ω f ( g g 4 exp(iω f t) + g 4 g exp( iω f t)) + Ω f ( g 4 g 5 exp(iω f t) + g 5 g 4 exp( iω f t)) + Ω f ( e 1 e 2 exp(iω f t) + e 2 e 1 exp( iω f t)) + Ω f ( e 2 e exp(iω f t) + e e 2 exp( iω f t)) + Ω 1 ( g 1 e exp(iωt) + e g 1 exp( iωt)) + Ω 2 ( g 2 e 2 exp(iωt) + e 2 g 2 exp( iωt)) + Ω ( g e 1 exp(iωt) + e 1 g exp( iωt)) + Ω 4 ( g e exp(iωt) + e g exp( iωt)) + Ω 5 ( g 4 e 2 exp(iωt) + e 2 g 4 exp( iωt)) + Ω 6 ( g 5 e 1 exp(iωt) + e 1 g 5 exp( iωt)), (1) where the energy of ground state g 1 = F = 2, m i = 2 is set to 0 and the Planck constant is set to 1. Ω 1 = 1/2Ω, Ω2 = 1/4Ω, Ω = 1/12Ω, Ω 4 = 1/12ηΩ, Ω 5 = 1/4ηΩ, Ω 6 = 1/2ηΩ are the Rabi frequencies of the left rotation circularly polarized light and right rotation circularly polarized light for various transitions among different Zeeman sublevels (j = 2, 1, 0, 1, 2 and i = 1, 0, 1), where the Rabi frequency is Ω = µ g,e E/. µ g,e is the dipole moment and E is the electric field amplitude. η is the amplitude ratio of the left rotation circularly polarized light and the right rotation circularly polarized light. Ω f is the Rabi frequency of the Radio field. The differences of the Clebsch Gordan (C G) coefficients for various transitions among different Zeeman sublevels are considered. In the interaction picture, the evolution of the atomic variables in the system is governed by the master equation: [20 ˆρ t = i [Ĥint, ˆρ Γ{ ˆP e, ˆρ} + Γ q=1,2, Q + q ˆρQ q, (2) where Γ is the total spontaneous decay rate of one of the excited Zeeman levels. P e = e 1 e 1 + e 2 e 2 + e e is the projector onto the excited level. The second term on the right-hand side of Eq. (2) having the structure of the anticommutator describes the radiative damping of the excited-level populations and optical coherences. The last term on the right-hand side corresponds to the transfer of populations and low-frequency coherences from excited

3 No. 2 Communications in Theoretical Physics 21 level to the ground level. The operators Q q are given by 1 Q 1 = 5 e g e 2 g e 1 g, 1 Q 2 = 10 e g + 10 e 2 g e 1 g 5, ρ g1,e Q = 2 10 e g e 2 g + 10 e 1 g 4. This model involves 8 sublevels, and therefore requires 64 equations. By simplifying these equations, the 6 equations related to the absorption are given: [( 2 ) δ ρ g1,e Ω f ρ g1,e 2 e 2iω f t Ω 1 ρ g1,g 1 Ω 4 ρ g1,g e 2iω f t + Ω f ρ g2,e ρ g2,e 2 + Ω 1 ρ e,e γ g1,e ρ g1,e, [( 2 ) δ ρ g2,e 2 Ω f ρ g2,e 1 e 2iω f t Ω f ρ g2,e e 2iω f t Ω 2 ρ g2,g 2 Ω 5 ρ g2,g 4 e 2iω f t + Ω f ρ g1,e 2 (a) ρ g,e 1 ρ g,e ρ g4,e 2 + Ω f ρ g,e 2 + Ω 2 ρ e2,e 2 γ g2,e 2 ρ g2,e 2, (b) [( ) 2δ ρ g,e 1 Ω f ρ g,e 2 e 2iω f t Ω ρ g,g Ω 6 ρ g,g 5 e 2iω f t + Ω f ρ g2,e 1 + Ω f ρ g4,e 1 e 2iω f t + Ω ρ e1,e 1 + Ω 4 ρ e,e 1 e 2iω f t γ g,e 1 ρ g,e 1, (c) [( 4 ) δ ρ g,e Ω f ρ g,e 2 Ω 1 ρ g,g 1 e 2iω f t Ω 4 ρ g,g + Ω f ρ g2,e e 2iω f t + Ω f ρ g4,e + Ω ρ e1,e e 2iω f t + Ω 4 ρ e,e γ g,e ρ g,e, (d) [( 8 ) δ ρ g4,e 2 Ω f ρ g4,e 1 e 2iω f t Ω f ρ g4,e e 2iω f t Ω 2 ρ g4,g 2 e 2iω f t Ω 5 ρ g4,g 4 + Ω f ρ g,e 2 e 2iω f t + Ω f ρ g5,e 2 + Ω 5 ρ e2,e 2 γ g4,e 2 ρ g4,e 2, (e) ρ g5,e 1 [( 4δ )ρ g5,e 1 Ω f ρ g5,e 2 e 2iω f t Ω ρ g5,g e 2iω f t Ω 6 ρ g5,g 5 + Ω f ρ g4,e 1 + Ω 6 ρ e1,e 1 γ g5,e 1 ρ g5,e 1. (f) The 64 equations can be solved numerically using a fourthorder Runge Kutta algorithm. One can see that Eq. () is time dependent with time-dependent exponentials. So the solutions of Eq. () periodically oscillate with time. The mean values of the solutions of Eq. () can be obtained over a period of time. The results shown in the Figures in the article are the mean values of solutions. The expressions for the susceptibilities of the atomic medium are given by [24 χ gi,e j = 2Nµ g i,e j ε 0 E ρ g i,e j. (4) N is the density of atoms and µ gi,e j are the dipole moments. The real and imaginary parts of the susceptibility χ = χ + iχ lead to the dispersion and absorption characteristics of the atomic medium in the usual way, i.e., the intensity absorption coefficient is proportional to χ. So the absorption of left rotation circularly polarized component (α ) and right rotation circularly polarized component (α + ) can be written as: α imag(µ g,e ρ g,e + µ g4,e 2 ρ g4,e 2 + µ g5,e 1 ρ g5,e 1 ), α + imag(µ g1,e ρ g1,e + µ g2,e 2 ρ g2,e 2 + µ g,e 1 ρ g,e 1 ). Numerical Results and Discussion Equations () can be solved numerically. γ gi,e j are the decay rates of coherences. The total spontaneous decay rate of one of the excited Zeeman levels Γ is set to 1. The other physical quantities are in unit of Γ. γ gi,e j = 1/2. γ gi,g j and γ ei,e j are the ground-state collisional decay rates and the excited-state collisional decay rates. We set γ gi,g j = 0, γ ei,e j = 0, 2 = 0. The population is initially evenly distributed in the sublevels of 5S 1/2. The ground state level Zeeman splitting is δ = 1. In Fig. 2, the Rabi frequency of light field is chosen as Ω = 1. From Figs. 2(a), 2(b), we can see that the absorption of light is restrained to some extent when the frequency of RF field is resonant with the Zeeman sublevels. The depth of the transparency window in the absorption spectrum is related to both the Rabi frequency of RF field and the amplitude ratio of the left circularly component and the right circularly component of light. The reason is that there is interference between the transition channels of left-handed and right-handed lights. So like as the phenomenon of CPT, there is a transparent window in the absorption spectrum of light. Although the coherence in the scheme is not as good as CPT, only the frequency of the RF field needs to be tuned. There is a big difference of absorption between the left-handed and righthanded lights, which can be understood in two ways. One is that the detunings of the left-handed and right-handed lights are very different. As shown in Fig. 1, they are 1 = 4δ/, 2 = 0, = 4δ/ for the right-handed

4 22 Communications in Theoretical Physics Vol. 6 light. That is, the right-handed light is resonant with the transition F = 2, m i = 1 F = 1, m i = 0, and is red-detuned from the transition F = 2, m i = 2 F = 1, m i = 1, and is blue-detuned from the transition F = 2, m i = 0 F = 1, m i = 1. For the left-handed light, the detunings are 4 = 2δ/, 5 = 2δ, 6 = 10δ/, which means the left-handed light is bluedetuned from all the three transitions. A larger C-G coefficient is corresponding to a larger detuning for the left-handed light. So the atom-light coupling strengths are very different for the right circularly component and the left circularly component. On the other hand, the effect of radio field and TOC add the uniformity of population among the ground state Zeeman sublevels. To understand this, the absorption of light without TOC is shown in Fig.. The populations of ground state sublevels are compared with and without TOC for f = 0. Fig. 2 The absorption spectra of light against the detuning of the RF field. (a) The Rabi frequency of RF field is Ω f = 0.0, η = 1. (b) The Rabi frequency of RF field is Ω f = 0.0, η = 1. Fig. The absorption spectra of light against the detuning of the RF field, Ω = 1. (a) The Rabi frequency of RF field is Ω f = 0.0, η = 1. (b) The Rabi frequency of RF field is Ω f = 0.0, η = 1.. One can see from Fig. that the difference of absorption between the left-handed light and the right-handed light is greatly reduced without TOC. For f = 0 and η = 1, the populations of ground state sublevels are ρ g1g 1 = , ρ g2g 2 = , ρ gg = , ρ g4g 4 = 0.065, ρ g5g 5 = without TOC, and are ρ g1g 1 = 0.157, ρ g2g 2 = , ρ gg = , ρ g4g 4 = 0.264, ρ g5g 5 = with TOC. The populations of ground state sublevels are more uniform when the TOC is considered. So light is more absorbed from the right-handed light transition channels than the left-handed light transition. The case that the light is resonant with the transition F = 2, m i = 0 F = 1, m i = 0 is considered in Fig. 4. The detunings are 1 = 7δ/, 2 = δ, = δ/, 4 = δ/, 5 = δ, 6 = 7δ/. As almost the same coupling strengths for the right circularly component and the left circularly component, the absorption of the right circularly component and the left circularly component are much the same in Fig. 4. But the coherence of system are reduced in most cases compared to the case 2 = 0. From Figs. 5 to 7, the detuning is set to 2 = 0. From Fig. 5, one can see that EIA can be changed into EIT if the Rabi frequency of RF field is increased from 0.01 to But the Rabi frequency of RF field can not be always increased, or else the coherence for the left-handed and right-handed lights in the multi Zeeman sublevels will be destroyed. In Fig. 5, the Rabi frequency of light field is chosen as Ω = 0.1

5 No. 2 Communications in Theoretical Physics Fig. 4 The absorption spectra of light against the detuning of the RF field. The parameters are Ω = 1, Ωf = 0.0, η = 1 in (a), and Ω = 0.1, Ωf = 0.01, η = 1 in (b). Fig. 5 The absorption spectra of light against the detuning of the RF field. The parameters are Ωf = 0.01, η = 1 in (a), and Ωf = 0.01, η = 1. in (b), and Ωf = 1, η = 1 in (c), and Ωf = 1, η = 1. in (d), and Ωf = , η = 1 in (e), and Ωf = , η = 1. in (f). 2

6 24 Communications in Theoretical Physics Vol. 6 Fig. 6 The energy levels in the dressed state picture for a two-level system. Figures 5(a), 5(b) show that the TOC can lead to EIA in the multi-zeeman levels system for both the linear polarization and elliptical polarization monochromatic lights. [25 26 From Figs. 5(e), 5(f), one can see that the EIT windows are centered at the detuning of the RF field f = δ Ω f, where δ is the ground state Zeeman splitting. To understand the relation between the EIT windows and the detuning of the RF field, the dynamical Stark effect of the strong RF field should be investigated. Due to the symmetry of the ground state multi-zeeman levels, we can reduce to a simple two-level model to investigate the Alter Townes splitting of the Zeeman levels caused by an RF field. The formula of Alter Townes splitting of a two-level system can be obtained from the existing literatures. [27 In Fig. 6, the detuning of RF field is f = δ ω f and the energy spacing parameter is Ω = 2 f /4 + Ω2. E mα, E mβ, E (m 1)α, E (m 1)β are the dressed state energy levels. If the RF field resonates with the dressed state energy levels, the frequency of RF field satisfies ω f = Ω. In the case of a small detuning, the detuning f can be ignored in the formula of Ω. So the detuning of the RF field meets f = δ Ω f. The coherence of the ground state multi-zeeman levels achieves a maximum if the RF field resonates with the dressed state energy levels. So the EIT windows centered at the detuning of the RF field f = δ Ω f can be understood. Fig. 7 The absorption spectra of light against the detuning of the RF field. (a) The Rabi frequency of RF field is Ω f = η = 1. (b) The Rabi frequency of RF field is Ω f = η = 1.. In Fig. 7, the Rabi frequency of light field is Ω = From Fig. 7, one can see that there are two small transparent windows around f = 0 for the right circularly component of light. The reason is the dynamical Stark effect. In the case of a weak light field and a weak RF field, the frequency shift of Zeeman sublevels can be regarded as equally spaced. The coherence in the atomic system led to the decrease of absorption of light when the RF field is resonant with the sublevels. 4 Conclusions EIT and EIA of a monochromatic light controlled by a radio frequency field in the cold multi-zeeman-sublevel atoms are theoretically investigated. The absorption of light is restrained to some extent when the frequency of RF field is resonant with the Zeeman sublevels. EIA can be changed to EIT with the change of intensity of radio field. The reason is attributed to the interference between the transition channels of left-handed and right-handed lights. The shift of the EIT window in the absorption spectrum is related to the Rabi frequency of RF field. This can be understood from the dynamical Stark effect of RF field. When the light field is a weak field, there are two small transparent windows for the right circularly component of light around the zero detuning of the RF field. The reason is the dynamical Stark splitting.

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