Optics Communications 356 (2015) 510–514

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Optics Communications

http://d0030-40

n CorrE-mURL

journal homepage: www.elsevier.com/locate/optcom

Study of a four-level system in vee þ ladder configuration

Vineet Bharti, Vasant Natarajan n

Department of Physics, Indian Institute of Science, Bangalore 560012, India

a r t i c l e i n f o

Article history:Received 22 July 2015Received in revised form12 August 2015Accepted 16 August 2015

Keywords:Electromagnetically induced absorptionFour-level systemCoherent control

x.doi.org/10.1016/j.optcom.2015.08.04218/& 2015 Elsevier B.V. All rights reserved.

esponding author.ail address: vasant@physics.iisc.ernet.in (V. Na: http://www.physics.iisc.ernet.in/�vasant (V

a b s t r a c t

We present the results of a theoretical study of a four-level atomic system in vee þ ladder configurationusing a density matrix analysis. The absorption and dispersion profiles are derived for a weak probe fieldand for varying strengths of the two strong control fields. For specificity, we choose energy levels of 87Rb,and present results for both stationary atoms and moving atoms in room temperature vapor. An elec-tromagnetically induced absorption (EIA) peak with negative dispersion is observed at zero probe de-tuning when the control fields have equal strengths, which switches to electromagnetically inducedtransparency (EIT) with positive dispersion (due to splitting of the EIA peak) when the control fields areunequal. There is significant linewidth narrowing in thermal vapor.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

The phenomenon of electromagnetically induced transparency(EIT), where the absorption of a weak probe laser on one transitionis reduced using a strong control laser on an auxiliary transition, iswell-studied in three-level systems, in all the three configurationsnamely lambda (Λ), vee (V), and ladder (Ξ) [1–3]. But the relatedphenomenon of electromagnetically induced absorption (EIA)—enhanced absorption of the probe laser—is possible only whenthere are at least four levels, so that additional control lasers canbe used. EIA has been studied—both theoretically and experi-mentally—in N-type ( VΛ + ) four-level systems [4–6]. The otherfour-level systems, namely Y-type ( Ξ Ξ+ ), inverted Y-type( Λ Ξ+ ), and tripod-type ( Λ Λ+ ), only show enhanced EIT.

In this work, we present theoretical analysis of a new kind offour-level system formed by the combination of V and Ξ three-level systems, which shows EIA resonances. This configurationopens up the possibility of the experimental study of EIA in animportant class of atoms called Rydberg atoms, because suchatoms have a highly excited atomic state allowing the formation ofa Ξ-type system. Rydberg atoms have thus been used to study EITphenomena [7,8], and the present analysis allows the extension tothe observation of EIA in such atoms.

For application to a real system, we choose energy levels of87Rb, and present results for both stationary atoms and movingatoms in room temperature vapor. An EIA peak with negativedispersion is observed at zero probe detuning when the controlfields have equal strengths. It switches to an EIT peak with positive

tarajan).. Natarajan).

dispersion (due to splitting of the EIA peak) when the controlfields are unequal. Thus, the atomic medium can be switched fromEIA and super-luminal light propagation to EIT and sub-luminallight propagation.

2. Theoretical considerations

The four-level system considered in the present work is shownin Fig. 1. The ground state 1| ⟩ is coupled with state 2| ⟩ by a weakprobe field. Two strong control fields are present—control 1 be-tween levels 1| ⟩ and 3| ⟩, and control 2 between levels 2| ⟩ and 4| ⟩.The control fields have detunings Δc1 and Δc2, while the frequencyof the probe field is scanned with variable detuning Δp. The in-tensities in the various beams are given in terms of the respectiveRabi frequencies, which are Ωp, Ωc1, and Ωc2.

The total Hamiltonian after carrying out the rotating waveapproximation (RWA) is

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

H2

1 2 1 3 2 4 h. c.

2 2 3 3 4 4 1

p c c

p c p c

1 2

1 2( )Ω Ω Ω

Δ Δ Δ Δ

= ℏ | ⟩⟨ | + | ⟩⟨ | + | ⟩⟨ | +

+ ℏ | ⟩⟨ | + | ⟩⟨ | + + | ⟩⟨ | ( )

where h.c. is the Hermitian conjugate of the preceding off-diag-onal terms. As expected, in the absence of the control 1 (or thecontrol 2) field, the four-level system reduces to a three-levelladder (or vee) system. The time evolution of the system is givenby the standard Liouville equation for the density matrix ρ. Thecoupled density matrix equations for this four-level system aregiven in the appendix.

The observable in our work is the response of atoms to theweak probe field. The susceptibility of probe field is determined by

Fig. 1. Four-level system formed by the combination of three-level vee and laddersystems.

V. Bharti, V. Natarajan / Optics Communications 356 (2015) 510–514 511

the coherence between levels 1| ⟩ and 2| ⟩—ρ12—and is defined as [9]

k kN

where2p

1212

2

0χ ρ

με Ω

= =| |

( )

Here N is the atom number density in the medium and μ12 is thedipole matrix element for the probe transition. The dispersion ofthe probe field is proportional to Re 12ρ{ }, and its absorption isproportional to Im 12ρ{ }.

The group velocity of the probe field is given by

vc

112

Re2

Re

3

gp

pχ

ω χω

=+ + ∂

∂ ( )

where c is the velocity of light and ωp is the probe frequency. Theabove equation shows that vg is inversely proportional to the slopeof probe dispersion—positive dispersion (EIT in the imaginarypart) gives rise to sub-luminal propagation, while negative dis-persion (EIA in the imaginary part) gives rise to super-luminalpropagation.

The various density-matrix elements are solved following theprocedure given in Ref. [10]. The steady-state solution for ρ12 isgiven by

⎛⎝⎜⎜

⎞⎠⎟⎟

i i

2 81

4 4 4

p p c c c12

11

12

12

11 33

12 31 32

22

32 34

22

14 34

ρΩ ρ

γ βΩ Ω ρ ρ

γ γ γ βΩγ γ α

Ωγ γ α

= − +| | ( − )

−| |

−| |

( )

where

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎜ ⎟

⎡⎣⎢⎢

⎤⎦⎥⎥

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

i

i i

i

i

14 4 16

1 1,

14 4

,2

,

2,

2,

2,

2.

c c c c

c cp

c p c

c p

p c c

12

12 32

22

12 14

12

22

12 34 32 14

2

12

14 34

22

32 3412

2

313

1 144

2 32

2 31

343 4

2 1

βΩγ γ

Ωγ γ

Ω Ωγ γ α γ γ

αΩγ γ

Ωγ γ

γΓ

Δ

γΓ

Δ γΓ

Δ Δ γ

Γ ΓΔ Δ

γΓ Γ

Δ Δ Δ

= +| |

+| |

−| | | |

+

= +| |

+| |

= − +

= − − = − + ( + )

= −+

− ( − )

= −+

+ ( + − )

Also, the population difference between states 1| ⟩ and 3| ⟩ is givenby [11]

⎡

⎣

⎢⎢⎢⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

1

24 5

c

c

11 331

2

32

12

1

ρ ρΩ

ΓΔ

− = +| |

+( )

−

As expected, the above equation yields the solutions for three-le-vel Ξ and V-systems by considering that only the correspondingcontrol fields are non-zero [12,13].

For specificity, we have chosen energy levels of the 87Rb atom—

level 1| ⟩ is the F¼1 hyperfine level of the 5S1/2 ground state; level2| ⟩ is the F 2= hyperfine level of the 5P3/2 state; level 3| ⟩ is theF 2= hyperfine level of the 5P 1/2 state; and level 4| ⟩ is the F 1=hyperfine level of the 5D5/2 state. The corresponding decay ratesare 01Γ = (because it is a ground state), 2 6.1 MHz2Γ π= × ,

2 5.9 MHz3Γ π= × , and 2 0.68 MHz4Γ π= × . Since the probe field isweak, it ensures that 022ρ = , and hence ρ44 is also 0. The popu-lation cycles between 1| ⟩ and 3| ⟩ (closed transition) according toEq. (5). The energy levels considered here do not allow the for-mation of the well-known diamond configuration dealt with inreferences [14,15], because the 5D5/2 state does not couple to the5P1/2 intermediate state.

The above analysis is also applicable to Rydberg atoms withminor changes in parameters for level 4| ⟩. For example, level 4| ⟩ canbe the D44 5/2 state considered in Ref. [16] with a decay rate of

2 0.3 MHz4Γ π≈ × .

3. Results and discussion

3.1. Stationary atoms

We first consider the results for stationary atoms. The ima-ginary and the real parts of the solution—corresponding to ab-sorption and dispersion, respectively—as a function of probe de-tuning are shown in Fig. 2. The two control fields are taken to beon resonance, i.e. 0c c1 2Δ Δ= = . Parts (a) and (b) show the resultswhen only one control field is present, i.e. corresponding to ladderand V-type three-level systems respectively. As expected, probeabsorption shows an EIT dip near line center, because absorptionsplits into an Autler-Townes doublet at the locations of the dressedstates created by the control field, i.e. when /2p cΔ Ω= ± . Thecorresponding dispersion profile has a positive slope near zerodetuning, and hence can be used for sub-luminal light propagation.

Now look at what happens when both control fields are ap-plied. The results for fixed value of 6c1 2Ω Γ= and varying c2Ω areshown in part (c)–(f) of the figure. When the two control fields areequal, the medium switches from transmitting to absorbing, andhas an EIA peak at line center. The corresponding dispersionprofile has negative slope near zero detuning, which can thereforebe used for super-luminal light propagation. On either side of thisequality condition, the central EIA peak splits into two, and showsa transparency dip at line center. The corresponding dispersionprofile has positive slope. Therefore, the medium can be switchedfrom transparency (slow light) to absorption (fast light) by chan-ging the relative strengths of the two control fields.

3.2. Thermal averaging for moving atoms

We now consider the results of Doppler averaging for an atomicvapor at room temperature, which is the typical condition for Rbatoms used for an experimental realization. Since the probe andcontrol beams are taken to be co-propagating or counter-propa-gating, the relevant distribution is the one-dimensional Maxwell–Boltzmann velocity distribution. For an atom of mass M at atemperature T and moving with a velocity v, this is given by

Fig. 2. Calculated probe absorption ( Im 12ρ{ }, solid), and dispersion (Re 12ρ{ }, da-shed) versus probe detuning for (a) 0c1Ω = , 6c2 2Ω Γ= ; (b) 6c1 2Ω Γ= , 0c2Ω = ; (c)

6c1 2Ω Γ= , 4c2 2Ω Γ= ; (d) 6c1 2Ω Γ= , 6c2 2Ω Γ= ; (e) 6c1 2Ω Γ= , 8c2 2Ω Γ= ; (f) 6c1 2Ω Γ= ,12c2 2Ω Γ= .

Fig. 3. Effect of Doppler averaging on probe absorption at room temperature. (a)0c1Ω = , 6c2 2Ω Γ= ; (b) 6c1 2Ω Γ= , 0c2Ω = ; (c) 6c1 2Ω Γ= , 4c2 2Ω Γ= ; (d) 6c1 2Ω Γ= ,6c2 2Ω Γ= ; (e) 6c1 2Ω Γ= , 8c2 2Ω Γ= ; (f) 6c1 2Ω Γ= , 12c2 2Ω Γ= . Solid curves are for all

atoms, dashed curves are for atoms moving with positive velocities, and dottedcurves are for atoms moving with negative velocities.

V. Bharti, V. Natarajan / Optics Communications 356 (2015) 510–514512

⎛⎝⎜

⎞⎠⎟f v dv

Mk T

Mvk T

dv2

exp2B B

2

π( ) = −

where kB is the Boltzmann constant. If the velocity is along thebeam direction, then the detuning of the field will change by theDoppler shift of kv± , where k is the photon wavevector and thesign depends on the relative direction of the beam with respect tothe velocity. The thermal averaging over the Maxwell–Boltzmanndistribution is done by assuming that the probe field and controlfield 1 are co-propagating, and control field 2 is counter-propagating.

The absorption profiles after thermal averaging, shown in Fig. 3,confirm our earlier observation of linewidth reduction in Ref. [17].The main results are the same as that seen for stationary atoms—

an EIT dip for a three-level system when one of the control fieldshas zero strength; an EIA peak when the two control fields areequal; and the EIA peak splitting into two when the control fieldshave unequal strengths. The linewidth reduction is because mov-ing atoms fill in the transparency window. This can be seen in thefigure, where the profile for atoms moving only with positive ve-locities is shown with a dashed line while the profile for atomsmoving only with negative velocities is shown with a dotted line—these profiles are calculated after Doppler averaging for one sign ofvelocity only.

The same linewidth reduction is also seen for the dispersionprofiles, shown in Fig. 4. As in the case of stationary atoms, theslope near line center changes from positive for EIT to negative forEIA, demonstrating the potential of switching the speed of light

Fig. 4. Effect of Doppler averaging on probe dispersion at room temperature. (a)0c1Ω = , 6c2 2Ω Γ= ; (b) 6c1 2Ω Γ= , 0c2Ω = ; (c) 6c1 2Ω Γ= , 4c2 2Ω Γ= ; (d) 6c1 2Ω Γ= ,6c2 2Ω Γ= ; (e) 6c1 2Ω Γ= , 8c2 2Ω Γ= ; (f) 6c1 2Ω Γ= , 12c2 2Ω Γ= . Solid curves are for all

atoms, dashed curves are for atoms moving with positive velocities, and dottedcurves are for atoms moving with negative velocities.

V. Bharti, V. Natarajan / Optics Communications 356 (2015) 510–514 513

propagation.

4. Conclusion

In conclusion, we have investigated a four-level atomic systemin vee þ ladder configuration, with a weak probe field and twostrong control fields. For application to a real system, we considerrelevant low-lying energy levels in 87Rb. The coupled density-matrix equations are solved in steady state to calculate the ab-sorption and dispersion profiles of the probe beam. The calculationis done for both stationary atoms and for atoms moving in roomtemperature vapor. In both cases, probe response shows an EIAwindow with negative dispersion at line center when the controlfields have equal strengths. The induced absorption peak splitswhen the control fields are unequal, resulting in an EIT window

with positive dispersion. For moving atoms, Doppler averagingleads to significant linewidth narrowing for both absorption anddispersion profiles. These results can therefore be used for appli-cations such as switching between sub- and super-luminal lightpropagation inside an atomic medium.

Acknowledgments

This work was supported by the Department of Science andTechnology, India. V.B. acknowledges financial support from a DSKothari post-doctoral fellowship of the University Grants Com-mission, India.

Appendix

In our analysis, the coupled density matrix equations in rotat-ing frame are calculated by using the definition of density matrixand the expression for Hamiltonian. The time evolution of system—

by incorporating the decay of atoms from each level and re-population from excited levels—is given by

i i2 2

;p p c c11 2 22 3 33 12 21 1 13 1 31( ) ( )ρ Γ ρ Γ ρ Ω ρ Ω ρ Ω ρ Ω ρ = + + − + −⁎ ⁎

i i2 2

;p p c c22 2 22 4 44 21 12 2 24 2 42( ) ( )ρ Γ ρ Γ ρ Ω ρ Ω ρ Ω ρ Ω ρ = − + + − + −⁎ ⁎

i

i2

;

2;

c c

c c

33 3 33 1 31 1 13

44 4 44 2 42 2 24

( )( )

ρ Γ ρ Ω ρ Ω ρ

ρ Γ ρ Ω ρ Ω ρ

= − + −

= − + −

⁎

⁎

⎜ ⎟⎛⎝

⎞⎠i

i i2 2 2

;p p c c122

12 11 22 1 32 2 14( )ρΓ

Δ ρ Ω ρ ρ Ω ρ Ω ρ = − + + ( − ) − − ⁎

⎜ ⎟⎛⎝

⎞⎠i

i i2 2 2

;c c p133

1 13 1 11 33 23ρΓ

Δ ρ Ω ρ ρ Ω ρ = − + + ( − ) −

⎜ ⎟⎛⎝

⎞⎠i

i2 2

;p c c p c144

2 14 2 12 24 1 34( )ρΓ

Δ Δ ρ Ω ρ Ω ρ Ω ρ = − + ( + ) + − −

⎜ ⎟⎛⎝

⎞⎠i

i2 2

;c p c p c232 3

1 23 1 21 13 2 43( )ρΓ Γ

Δ Δ ρ Ω ρ Ω ρ Ω ρ = −+

+ ( − ) + − −⁎

⎜ ⎟⎛⎝

⎞⎠i

i i2 2 2

;c c p242 4

2 24 2 22 44 14ρΓ Γ

Δ ρ Ω ρ ρ Ω ρ = −+

+ + ( − ) − ⁎

⎜ ⎟⎛⎝

⎞⎠i

i2 2

;p c c c c343 4

2 1 34 2 32 1 14( )ρΓ Γ

Δ Δ Δ ρ Ω ρ Ω ρ = −+

+ ( + − ) + − ⁎

The positive decay terms in iiρ (i 1, 2, 3, 4= ) equations are dueto the fact that there is an increase in population of lower levelsdue to the decay of atoms from higher levels.

Appendix A. Supplementary data

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.optcom.2015.08.042.

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