2012 International Conference on Optical Engineering (ICOE)

978-1-4673-2463-2/12/$31.00 ©2012 IEEE

σ +

σ + π

π

σ -

σ -

6.835GHz

72 MHz

157 MHz

267 MHz

282 MHz

Δc

ΔP

1

3 4

2

5

Polarization Dependence of Electromagnetic Induced Transparency in Multilevel Cascade System

Vineet Bharti and Ajay Wasan Department of Physics

Indian Institute of Technology, Roorkee Roorkee, India

vineet26iitr@gmail.com

Abstract—A theoretical model using density matrix approach has been presented to show the polarization dependence of Electromagnetically Induced Transparency (EIT) in cascade multilevel atomic system. EIT has already been investigated both theoretically and experimentally by using three-level system in lambda (Λ), vee (V), and cascade (Ξ) configurations. However there is asymmetry in absorption measurement. To understand this asymmetry we introduce hyperfine levels in EIT cascade system. We present a five-level atomic system, which is a simple case of 87Rb. This study has been extended to investigate the polarization dependence of probe and control lasers on EIT. It has been observed that there is significant change in transparency window for different laser polarizations.

Keywords- Electromagnetic Induced Transparency, EIT cascade system, five-level model.

I. INTRODUCTION In last two decades, there have been several theoretical

and experimental studies to show that strong resonant field in three-level system can modify the absorption properties of weak probe field [1,2], such as the phenomenon of Electromagnetic induced transparency (EIT), in which a strong control field is resonant with an atomic transition is used to make the atomic medium transparent for a weak probe field with an atomic transition sharing same excited state [3,4]. EIT technique have several practical applications in lasing without inversion [5], probe amplification [6], high resolution hyperfine spectroscopy [7]. EIT also led to demonstration of reduction of group velocity of light [8]. Over recent years great deal of progress has been made in EIT based quantum memories [9].

So far all the EIT’s theoretical and experimental models usually rely on three-level system. This three-level system can be in lambda (Λ), vee (V) and cascade (Ξ) configurations. But all the EIT experiments are performed in ensembles of alkali metal atoms. In this case many hyperfine levels are present and level structure is more complex than simple three-level approximation.

In this present paper, we have introduced hyperfine levels, to investigate the effect of multiple excited states on EIT cascade system. We present a general analysis of EIT in five-level system. We consider five possible polarization

configurations of probe and control lasers and compare our results with three-level approximation.

This paper is organized as: In section II, theoretical model is presented. In section III, theoretical results are discussed for absorption profile for three and multiple excited states with different lasers polarization. Finally in section IV, we give some conclusions of this theoretical work.

II. THEORATICAL MODEL In this section, we present our theoretical model. Consider

5S1/2 → 5P3/2 → 7S1/2 multilevel transition of 87Rb as shown in Fig. 1.

Fig. 1. Multilevel system of 87Rb under consideration.

In rubidium, the closely spaced hyperfine levels are also present. To study the effect of these hyperfine levels, we consider a simplest five-level model. Ground state 1 F 1= = is coupled with excited states 2 F 0= = , 3 F 1= = , 4 F 2= = by a weak probe laser and excited states

7 S1/2

5 S1/2

5 P3/2

3 2 1 0

2 1

2 1

mF values -3 -2 -1 0 1 2 3

Probe

Control

2012 International Conference on Optical Engineering (ICOE)

2 , 3 , 4 are coupled with state 5 F 1= = . We also consider different polarizations (i.e π, σ +, σ – ) of probe and control lasers. The selection rules for π, σ +, σ – polarizations are Δ mF = 0, +1, –1 respectively [10]. Depending upon these laser configurations there can be five different EIT cascade systems (with mF = 0 for 5P3/2 excited state), as shown in Fig.1. The detuning of the probe with frequency ωp and control field with frequency ωc from the atomic transition 1 2→ and 2 5→ will be denoted, Δp = ωp – ω21 and Δc =

ωc – ω52, respectively. Here ij i i(E E ) /ω = − is the atomic transition frequency between levels i and j (i > j), and Ei is energy of unperturbed atomic state i . The decay rate from state 5P3/2 and 7S1/2 are Γ2 = 6.1 MHz, Γ5 = 1.8 MHz respectively [11]. This system can be considered as three-level system when only levels 1 , 2 and 5 are present.

In our model, we have also include the branching ratios of excited states. The branching ratio for transition from intermediate state 5P3/2 to ground state 5S1/2 is 1 and branching ratio for transition from excited state 7S1/2 to intermediate state 5P3/2 is 0.41. So atoms can decay from excited state to ground state via different decay channels. But we can consider our system closed or cyclic, as it has been shown that EIT is independent of branching ratios [12].

The time evolution of the system in rotating wave approximation is given by using density matrix equation

[ ]i H,ρ= ρ + relaxations where the relaxation terms are added phenomenologically [13]. The strength of coupling between states i and j is given by using the definition of Rabi frequency i.e. ij E /Ω = μ , where μ is dipole moment of transition. The dipole moments of all the transitions in our five-level system have been calculated using Clebsch-Gordan coefficients for hyperfine levels [10]:

F'1 L ' S J J ' I meg

F F '

e ( 1) 'L ' r L

(2J 1) (2J ' 1) (2F 1) (2F' 1)

F 1 F'L' J ' S J ' F' Im q mJ L 1 F J 1

+ + + + + −μ = − α α

× + + + +

⎛ ⎞⎧ ⎫ ⎧ ⎫× ⎨ ⎬ ⎨ ⎬ ⎜ ⎟−⎩ ⎭ ⎩ ⎭ ⎝ ⎠

Where L, S, J, I and F represents the orbital, electron spin,

total angular momenta, nuclear spin angular momenta and angular momenta for hyperfine levels, respectively. (...) and {...} denotes the 3J– and 6J– symbols. 'L ' r Lα α is reduced matrix element. Also q= mF’ – mF and for 87Rb atom I=3/2. The Rabi frequencies of various transitions with different polarizations considered in our model are given in table I.

We have used the usual method to solve density matix equations for steady-state solution ( ij 0ρ = ) using weak probe conditions. As we know that 11 22 33 44 55 1ρ + ρ + ρ + ρ + ρ = and under weak probe conditions all the population is in ground state i.e. 11 22 33 44 551 and , , , 0.ρ ≈ ρ ρ ρ ρ ≈

TABLE I.

Rabi frequencies for all the transitions used in our model

Probe field

π - polarization Ωp13 0 Ωp14 – Ωp12

σ+ - polarization Ωp13 1.1182 Ωp12 Ωp14 0.5002 Ωp12

σ – - polarization Ωp13 – 1.1182 Ωp12 Ωp14 0.5002 Ωp12

Control

field

π - polarization Ωc35 0 Ωc45 – Ωc25

σ+ - polarization Ωc35 –1.1182 Ωc25 Ωc45 0.5002 Ωc25

σ – - polarization Ωc35 1.1182 Ωc25 Ωc45 0.5002 Ωc25

The solution to the five-level density matrix equations yields:

*p12 c 25 c 35 p13 c 45 p14

12 12 12 12 15 12 15 13 13 14 14

2 2 2 2p12 p13 p14 c 45

15 1415 25 15 35 15 45 14 15 15

2 2 2c35 c35 c 45

13 2 213 15 15 13 14 15 14 15

i i;2 8

where

1 ; 1 ;4 4 4 4

1 ;4 16

⎛ ⎞Ω Ω Ω Ω Ω Ω⎜ ⎟ρ = − + +⎜ ⎟γ β γ γ β β γ β γ β⎝ ⎠

Ω Ω Ω Ωβ = + + + β = +

γ γ γ γ γ γ γ γ β

Ω Ω Ωβ = + −

γ γ β γ γ γ β β

β2 2 2 2

c 25 c 25 c 35 c 4512 2 2

12 15 15 13 13 14 1412 15 15

3212 p 13 p 32

5414 p 42 15 p c

2 5 325 c 35

1 ;4 16

and

i ; i ( ) ;2 2

i( ) ; i ( ) ;2 2

( ) (i ;

2

⎛ ⎞Ω Ω Ω Ω⎜ ⎟= + − +⎜ ⎟γ γ β γ β γ βγ γ β ⎝ ⎠

ΓΓ ⎛ ⎞⎛ ⎞γ = − − Δ γ = − − Δ −ω⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

ΓΓ ⎛ ⎞⎛ ⎞γ = − − Δ −ω γ = − − Δ + Δ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

Γ +Γ Γ +⎛ ⎞γ = − − Δ γ = −⎜ ⎟⎝ ⎠

5c 32

4 545 c 42

)i ( ) ;

2

( )i ( ) .

2

Γ⎛ ⎞− Δ + ω⎜ ⎟⎝ ⎠

Γ + Γ⎛ ⎞γ = − − Δ + ω⎜ ⎟⎝ ⎠

If we apply above solution to three-level system ( 1 , 2 and 5 ), then it takes following form:

p1212 2

c2512

15

i / 2| |4

Ωρ = −

Ωγ +γ

2012 International Conference on Optical Engineering (ICOE)

This is well known solution for three-level cascade system. So we can say that solution form of five-level system is in right direction.

III. RESULTS AND DISCUSSIONS We show the numerical solution of imaginary part of ρ12

with detuning ΔP of probe field for three-level system and five-level system with different polarizations with various values of control Rabi frequencies (Ωc25). One clearly sees that for three-level case, if one applies a control field that is resonant with corresponding transition 2 5→ , then the absorption peak split into a doublet. For Ωc25 < Γ2, we obtain a sharp EIT window with a line width much narrower than Γ2. This modification in absorption can be explained using quantum interference phenomena [7]. For Ωc25 > Γ2, the absorption split into Autler-Townes doublet [14], and shows a classic transparency window in the center. The doublet is attributed to the fact that the resonant control field dresses the level 2 and 5 into the superposition of these states. Both of these cases are shown in Fig 2(a) and 2(b), respectively.

Now we will discuss the case of five-level system with different polarization configurations. In the case of five-level system, the position of Autler-Townes absorption doublet for

Fig 2. Probe absorption Vs. Probe detuning for three level system, for control laser Rabi frequency (a) Ωc25 = 0.5 Γ2 and (b) Ωc25 = 5 Γ2.

probe field are strongly modified. Numerical solutions shows that, in this case one of the absorption peak shift towards the center and other absorption peak shift away from the center with different amplitudes. Also overall transparency window

shifts towards negative detuning. This is understood because our five-level system has three different cascade channels with different dipole moments for various transitions. So the

Fig 3. Probe absorption Vs. Probe detuning for five-level system with different polarizations of probe and control lasers (a) σ – – σ+ or σ+– σ –, (b) π – π, (c) π – σ – or π – σ+, (d) σ –– π or σ+– π, (e) σ+–σ+ or σ –– σ –. In present calculations control laser Rabi frequency is Ωc25 = 5 Γ2.

2012 International Conference on Optical Engineering (ICOE)

combined effect of these channels generates asymmetry in absorption amplitudes as well as transparency window shift. The results for five-level system are shown in Fig. 3 (a – e) for five different polarization combinations. These different combinations for probe and control lasers are: 1. σ – – σ+ or σ+– σ –, 2. π – π, 3. π – σ – or π – σ+, 4. σ –– π or σ+– π, 5. σ+–σ+ or σ –– σ –, respectively. It can be seen in Fig. 3 that amplitude of absorption peaks varies with change in polarizations of probe and control lasers.

IV. CONCLUSION We have studied the phenomenon of electromagnetic

induced transparency using multiple excited states in cascade system. We have considered the five-level system which is a simple case of 87Rb. We have investigated the polarization dependence of EIT in five-level cascade system. It has been observed that the presence of multiple excited states shifts the EIT window towards negative detuning with different amplitudes of absorption peaks. The amplitude of absorption peaks in five-level system also depends upon polarization of probe and control lasers. The formulation presented here may be readily extended to other systems with more excited closely spaced levels.

ACKHNOWLEDGMENTS V. B. acknowledges financial support from Ministry of Human Resource Development (MHRD), India.

REFERENCES [1] L. M. Narducci, M. O. Scully, G. L. Oppo, P. Ru, and J. R. Tredicce,

“Spontaneous absorption and emission properties of driven three-level system” Phys. Rev. A, vol. 42 pp 1630, 1990.

[2] G. Vemuri, G. S. Agarwal, and B. D. Nageswara Rao, “Sub Doppler resolution in inhomogenously broadened media usind intenst control fields” Phys. Rev. A, vol. 53 pp2842, 1996.

[3] S. E. Harris, “Electromagnetically induced transparency,” Physics Today, vol. 50 pp 36-42, 1997.

[4] K. J. Boller, A Imamoglu, and S. E. Harris, “Observation of electomagnetic induced trandsprancy” Phys. Rev. A, vol. 83 pp 053809, 1991.

[5] A. S. Zibrib, M. D. Lukin, D. E. Nikonov, L. Hollberg, M. O. Scully, V. L. Velichansky, and H. Robinson,. “Experimental Demonstration of Laser Oscillation without Population Inversion via Quantum Interference in Rb” Phys. Rev. Lett., vol. 75 pp 1630,1995.

[6] S. Menon, and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels” Phys. Rev. A, vol. 61 pp 013807, 1999.

[7] A. Krishna, K. Pandey, A. Wasan, and V. Natarajan, “High resolution hyperfine spectroscopy of excited states using eletromagnetic induced transprency” Europhys. Lett. Vol. 72 pp 221-227, 2002.

[8] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas” nature vol. 397 pp 594-598, 1999.

[9] M. Fleischhauer, A. Imamoglu, and J. P. Marangos , “Electromagnetic Induced Transparency: optics in coherent media” Rev. Mod. Phys, vol. 77 pp 663-673,2005.

[10] H. J. Metcalf, and P. van der Stratten, “ Laser cooling and trapping” springer 1999.

[11] M. S. Safronova and U. I. Safarnova, “Critically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities in 87Rb” Phys. Rev. A, vol. 83 pp 052508, 2011.

[12] H. R. Noh, and H. S. Moon “Discrimination of one photon and two photon coherence parts in electromagnetically induced transparency for a ladder type three level atomic system” Opt. Express, vol.19pp 11128-11137, 2011.

[13] M. O. Scully and M. S. Zubairy, “Quantum optic” Cambridge University Press 1997.

[14] S. Autler, and C. H. Townes “ Stark effect in rapidly varying fields” Phys. Rev., vol. 75 pp 703-722, 1955.