Table Of ContentCoherent control of light field with electromagnetically induced transparency in a
dark state Raman coherent tripod system
YongYao Li1,2, HuaRong Zhang1, YongZhu Chen1,3, and JianYing Zhou1∗
1State Key Laboratory of Optoelectronic Materials and Technologies,
Sun Yat-sen University, Guangzhou 510275, China
2Department of Applied Physics,South China Agricultural University, Guangzhou 510642, China
3School of Electro-Mechanic Engineering,Guangdong Polytechnic Normal University, Guangzhou 510665, China
(Dated: February 2, 2008)
The coherent superposition of two-atomic levels induced by coherent population trapping is em-
ployed in a standard Λ type scheme to form a tripod-like system. A weak probe pulse scanning
across the system is shown to experience a crossover from absorption to transparent and then to
amplifcation. Consequently the group velocity of the probe pulse can be controlled to propagate
8
either as a subluminal, a standard, a superluminal or even a negative speed. It is shown that the
0
propagation behavior of the light field is entirely determined and controlled by the initial states of
0
2 thecoherent superposition.
n
PACSnumbers: ValidPACSappear here
a
J
2 I. INTRODUCTION below:
]
h |Di=cosθ|1i−sinθ|2i (1)
p Coherentpopulationtrapping(CPT)[1,2,3]andElec-
-
t tromagnetically induced transparency (EIT)[4, 5] are Here
n
two different quantum coherence processes which have
ua led to the observation of many new physics effects in cosθ = Ω1
q quantum optics and atomic physics. CPT is a prepa- pΩ21+Ω22
[ ration of atoms in a coherent superposition of ground Ω
and metastable state sublevels, which is called dark sinθ = 2 (2)
2 state, for the reason that this state is immune to ex- pΩ21+Ω22
v
citation by a two-component laser radiation under the
9
5 two-photon resonance condition. EIT can also be clas- By scaning a weak probe field through the system, the
2 sified as a dark state. It can modify the optical prop- system exhibits a rich optical properties under differ-
4 erties of a medium and results in making a resonant ent initial conditions because of the Raman coherence
1. opaquemediumtransparent,whichhasbeenobservedin of the dark state. We found that not only double EIT
channels[20]andleft-handedmaterial[15]butalsoalotof
1 awiderangeofatomic,molecular,andcondensedmatter
7 systems[6, 7, 8]. In the past two decades, potential ap- other effects such as Raman amplification[21], refractive
0 plications of EIT to coherentcontrol for groupvelocities index enhancement[22], slow light[9, 10] and superlumi-
v: of the probe field in quantum information and quantum nal light[23] can be simply achieved by adiabatically[24,
i communication[9, 10, 11, 12], to left-handedmaterialfor 25] changing the initial state.
X
negative refraction[13, 14, 15] and to enhanced nonlin- The mixing angle θ in Eq. (2) is determined by the
ar ear optics for frequency conversion[16, 17] have been re- relative intensity of Ω1 and Ω2. This dark state plays
ported. ThedistinctfeatureofEITandCPTisthatEIT the role of the original ground state. Here the system
causes not only a modification of optical property of the resembles a three-level Λ-type system. The probe field
medium but also the optical fields themselves. then experiences a transition from the dark state |Di to
the upper state |4i. A strong resonantcoherentfield Ω
C
is employed to couple states |3i and |4i.
Inthispaper,weanalysetheopticalpropertyofafive-
level atomic configuration illustrated in FIG. 1. Two The technique to establish the coherent preparation
coherentfieldswithcomplexRabi-frequenciesΩ andΩ of a dark state was realized in many systems[1]. In our
1 2
create a dark state superposition of states |1i and |2i[1]. case, the atoms are trapped in state |1i and state |2i
By introducing a coherent superprosition in the system, withtheprobabilityofcos2θandsin2θrespectively. The
moredegreesoffreedomtothesystemcanbeadded,then probability is decided by the mixing angle θ, which can
wewillgainacontrolovermorephysicalvariables[18,19]. be tuned adiabatically by changing the relative intensity
The coherentsuperprositionof the dark state is givenas of Ω1 and Ω2. Because none of the atoms is prepared at
thestate|eiandtheprobefieldisweak,thecontribution
from level|ei to the system can be neglected safely. The
energy levels (|1i, |2i, |3i and |4i) can be viewed as a
∗Electronicaddress: [email protected] tripod level configuration[25, 26, 27, 28, 29]. .
2
varying amplitude approximation are:
i i i
ρ˙ =−(i△ +γ )ρ + Ω ρ + Ω ρ + Ω ρ
41 1 41 41 P 11 P 21 C 31
2 2 2
i
ρ˙ =−i(△ −△ )ρ + Ω∗ρ
31 1 C 31 2 C 41
i i i
ρ˙ =−(i△ +γ )ρ + Ω ρ + Ω ρ + Ω ρ
42 2 42 42 P 22 P 12 C 32
2 2 2
i
ρ˙ =−i(△ −△ )ρ + Ω∗ρ (6)
FIG. 1: Energy-Level scheme of the system: two coherent 32 2 C 32 2 C 42
laser field with Rebi-frequency Ω1 and Ω2 coupling of levels
|1i, |2i and |ei create the dark state |Di, which construct The off-diagonal decay rates γij are given by γij =
a Λ type scheme with |3i and |4i. A probe field E transits (γ +γ )/2 denoting the total coherence relaxationrates
i j
fromdarkstate|Ditotheupperlevel|4iandacouplingfield between states |ii and |ji. The detuings are defined as
transits with |3i and |4i △ =ω −ν ,△ =ω −ν , △ =ω −ν . As shown
1 41 P 2 42 P C 43 C
in the system, one finds that:
II. EQUATIONS AND SOLUTIONS △ −△ =ω (7)
1 2 21
The Hamiltonian of this four-level system can be ob- The steady-state solutions of ρ41 and ρ42 are given by:
tainedinthe rotatingframeby introducingthe rotating-
wave approximation: ρ = 2iΩP(cos2θ−cosθsinθ)(△1 −△C)
41 (i△ +γ )(△ −△ )−iΩ2/4
1 41 1 C C
H = X4 ~ωiσii− ~2[Ω1Pe−iνPtσ41+Ω2Pe−iνPtσ42 ρ42 = 2iΩ(iP△(sin+2γθ−)(c△osθ−si△nθ))(−△2iΩ−2△/4C) (8)
i=1 2 42 2 C C
+Ω e−iνCtσ +H.C] (3)
C 43 The polarization P in the slowly varying frame and the
susceptibility χ are related to each other with the equa-
Here ω is the eigenfrequency of the energy level and
i tion [30]:
σ = |iihj|, Rabi-frequency Ω = ℘ E /~, Ω =
ij 1P 41 P 2P
℘41EP/~ and ΩC = ℘43EC/~, where ℘ij is the electric P =ǫ χ(ω)E =2N(℘ ρ +℘ ρ ) (9)
dipole moment from |ii to |ji, E and E is the slowly 0 P 14 41 24 42
P C
varying amplitude of the probe field and the coupling where N is the total number of atoms.
field respectively. Assume that the electric dipole mo-
The expression of the susceptibility is given as below:
ment ℘ ≈ ℘ = ℘ and the value of ℘ is real, which
41 42
results that Ω1P ≈Ω2P =ΩP. f(θ)(△ −△ )
1 C
The evolution equation of density matrix reads: χ(ω,θ)= iK[
(i△ +γ )(△ −△ )−iΩ2/4
1 41 1 C C
i g(θ)(△2 −△C)
ρ˙ =− [H,ρ]−[Γ,ρ] (4) + (10)
~ + (i△ +γ )(△ −△ )−iΩ2/4
2 42 2 C C
Thematrixelementhi|Γ|ji=γiδij,whereγi isthedecay Where K = N℘2/ǫ0~, and f(θ) = cos2θ − cosθsinθ,
ratedesignatingthepopulationdampingfromtheenergy g(θ)=sin2θ−cosθsinθ
level|ii. Tosimplify,weassumethatΩ1 andΩ2 arereal, Assume that γ41 ≈ γ42 = γ and △C= 0, the real part
andγ4 ≫γ3 ≈γ2 ≈γ1 ≈0. Withtheseapproximations, and imaginary part of susceptibility are given as below:
the population of the atoms are prepared at the energy
levels |1i, |2i and |3i and the atoms are initially set to: △ (△2 −Ω2/4)
Re[χ]= K[f(θ) 1 1 C
γ2 △2 +(△2 −Ω2/4)2
ρ(101) =cos2θ,ρ(202) =sin2θ △ 1(△2 −1Ω2/4C)
ρ(0) =ρ(0) =−cosθsinθ (5) +g(θ)γ2 △22 +(2△2 −CΩ2/4)2] (11)
21 12 2 2 C
The initial conditions indicate that cos2θ and sin2θ de-
notes population of the atoms prepared at the energy γ △2
level |1i and |2i respectively. While −cosθsinθ denotes Im[χ]= K[f(θ)γ2 △2 +(△21−Ω2/4)2
the dark state Raman coherence between levels |1i and 1 1 C
γ △2
|2i. +g(θ) 2 ] (12)
The equations of density matrix elements in slowly γ2 △22 +(△22 −Ω2C/4)2
3
1.0
2
0.8 f ()=cos2-cossin
g ()=sin -cossin
0.6
0.4
0.2
0.0
-0.2
0.00 0.25 0.50 0.75 1.00 1.25 1.50
FIG. 2: Function of f(θ) (solid line) and g(θ) (dash line) in
[0,π/2]theminimumofthetwoexpressionappearatθ=π/8
and θ=3π/8
(a)
III. ANALYSIS OF THE SOLUTION
Eq. (10) and Eq. (11) show that the susceptibility
is mainly determined by f(θ) and g(θ). f(θ) affects the
dispersion relationship in the vicinity of the resonance
with △ = 0 and g(θ) affects the relationship near the
1
resonance at △ = 0. The term of cos2θ and sin2θ in
2
f(θ) and g(θ), denotes the population of the atoms at
the energy level |1i or |2i respectively and relates with
the absorption to the probe field. While the term of
−cosθsinθ denotesthe darkstate Ramancoherencebe-
tween |1i and |2i and gives rise Raman amplification to
the probe field. The behavior of these two functions at
the range of [0,π/2] are plotted in FIG. 2.
It is shown from FIG. 2 that f(θ) > 0 and g(θ) < 0
(b)
for θ < π/4. This indicates that the probe experiences
absorptioninthevicinityof△ =0andgaininthe vicin-
1
ity of △ =0. For θ >π/4, on the other hand, the probe 0.5 0.5
experien2ces gain in the vicinity of △1=0 and absorption 00..34 HG RIme[[]] 00..34 RIme[[]]
in the vicinity of △2=0. Therefor, the dark state super- 0.2 0.2
position creates mixtures of active and passive optical 0.1 0.1
0.0
0.0
materials at frequencies near their resonances[31]. -0.1
-0.1 -0.2
Particularly, when the mixing angle θ = π/4, we have
-0.2 -0.3 LG
χ(ω) = 0. In this case, both the real part and the -4 -2 0 2 4 6 8 10 -4 -2 0 2 4 6 8 10
imaginary part of susceptibility are vanished. Raman
(c) (d)
coherence of the dark states induces transparency at all
rangeoffrequency,and the probe fieldpropagatesinthe
FIG. 3: Real and imaginary parts of the susceptibility as a
medium just as it propagates through the vacuum.
The minimum of the two functions appear at θ =π/8 function of the probe field detuning △1 /γ and the mixing
angleθ. Weassumethatthematerialiscoldatomicgaswith
and θ = 3π/8, so that the max gain in vicinity of two N ∼1.0×1012cm−3, γ ∼10MHz, ΩC ≈ 2γ, ω21 ≈ 5γ and
resonances is appearing respectively correspond to θ at
K ≈ γ. (a) and (b) 3D graph about the real part and the
these points. We plot the real part and the imaginary imaginary of the susceptibility. The point of 0 at the axes of
part of the susceptibility in FIG. 3. The special points △/γ is related to the resonance of △1=0 and the point of 5
of θ =π/8 and θ =3π/8 are also plotted respectively in is relate to the resonance of △2= 0. (c) and (d) Dispersion
FIG. 3. relationship when θ = π/8 and θ = 3π/8. ’HG’ is a high
FIG.3(a)and(b)showthatthepopulationistrapped refractive index without absorption and ’LG’ is a negative
in level |1i for θ = 0. In the vicinity of △ = 0, the real refractive index without absorption.
1
and the imaginary part of the susceptibility exhibit a
typical EIT phenomenon in a standard Λ system. Simi-
larly, for the case of θ = π/2, the population is trapped
4
Here g = ℘p2ω41/ǫ0~, and we define that tan2ϕ =
7 g2N/Ω2C. If the mixing angle θ is not variable as a func-
tion of time. Eq. (15) can be simplified to:
6 2
tan =1
5 2 ∂ ∂
tan =3 ( +V )Ω (z,t)=0 (16)
4 tan2 =6 ∂t g∂z P
c/Vg 3 And
2 c
V = (17)
1 g 1+f(θ)tan2ϕ
0 It is interesting to compare Eq. (17) to the standard
0.00 0.25 0.50 0.75 1.00 1.25 1.50 expressionofgroupvelocityinthedispersionmediagiven
by:
c
V = (18)
FIG.4: Groupvelocityasafunctionofθ(θ∈[0,π/2])insome g n(ω)+ωdn
different mixing angle of ϕ dω
Because the carrier frequency which the light field in
near resonance with level |1i and |4i, according to the
in level |2i and a typical EIT phenomenon is appeared
near △ = 0. When the mixing angle θ increases from 0 propertyoftheEIT,therefractiveindexofthelightfield
to π/2,2the real part of the susceptibility in vicinity of n(ω)|△1=0 = p1+Re{χ(ω)}|△1=0 ≈ 1. The compari-
resonance at △1= 0 changes from normal dispersion to sionshows ω(ddωn)|△1=0 ∼f(θ)tan2ϕ. Then we candraw
a conclusion that the sign of f(θ) determines the light
abnormaldispersionand contrarysimilar processcan be
observed in vicinity of △ = 0. As the imaginary parts pulse propagate in the medium for a normal dispersion
2
or an abnormal dispersion.
of the susceptibility near this two resonances are var-
For example, at θ > π/4, and with near resonant ex-
ied proportionalto the function of f(θ) and g(θ) respec-
citation at △ =0, we have the normal dispersion. How-
tively. Especially, one can find that both the real part 1
ever, when θ > π/4, it is an abnormal dispersion with
andtheimaginarypartofthesusceptibilityareviewedas
f(θ) < 0 , leading to a superluminal light pulse [23].
astraightlineinthesetwofiguresforthecaseofθ =π/4.
FIG. 4 shows that the group velocity as a function of θ
FIG. 3 (c) and (d) show that two points which is la-
in some of the mixing angle ϕ.
beled as ’HG’ and ’LG’ . The point of ’HG’ has an high
Especially, when
refractive index with zero absorption at θ = π/8, while
the ’LG’ has an negative refractive index with zero ab-
1
sorption at θ =3π/8. tan2ϕ>| |≈4.83 (19)
f (θ)
min
Assuming a plus with Rabi-frequency ΩP(z,t)e−iνt the group velocity can become negative. So that one
propagation in this medium. The Maxwell-Bloch equa- can achieve a desirable group velocity simply by setting
tion in one-dimensional slowly-varying envelope approx- different mixing angle θ and ϕ in this system.
imation is[30]:
An interesting extension of the topic is to vary Ω or
1
∂ ∂ ω41℘ Ω2 in the space[32, 33]. This treating leads to a spa-
(∂t +c∂z)ΩP(z,t)=−2ǫ ~Im(2Nρ41+2Nρ42) tiallydependentpopulationdistributionatlevels|1iand
0
|2i and hence a spatially dependent of the dispersion re-
(13)
lation. If the variation is periodic in the space, it can
Let the carrier frequency of the plus ν = ω , the effect produce a photonic band gap (PBG) structure in the
41
of the Nρ on the right hand side of the Eq. (13) can medium[34, 35].
42
be neglected. With the approximation as that adopted
in Ref [12], one can obtain that:
IV. CONCLUSIONS
2 ∂
ρ =−i ρ
41 31
Ω ∂t
C Controlled group velocity of light in the tripod sys-
ΩP(z,t) tem with EIT is an interesting topic which leads to rich
ρ ≈− f(θ) (14)
31
ΩC phenomena[36]. In this contribution, coherent superpo-
sition of a dark state is proposed in a four-level tripod
We assume that Ω is real and constant, then the prop-
C
system to control the probe propagationthrough EIT in
agation equation is changed to:
the medium. It is found that the dark state superposi-
∂ ∂ g2N ∂ tion creates mixtures of active and passive optical mate-
( +c )Ω (z,t)=− Ω (z,t)f(θ) (15)
∂t ∂z P Ω2 ∂t P rialsatfrequenciesneartheirresonancesandleadingtoa
C
5
because these medium have abundant level structures.
20
4 Ref. [15] gives an example[41] of this system in practice.
15 RIme{{’’}} 3 RIme{{’’}} If increasing the density of the material, the local field
10 2 effect[42] must take into account. The electric field re-
5 1
places to the microscopic field E =E+P/3ǫ and the
0 0 m 0
dielectric constant is changed as below:
-1
-5
-2
-10 -4 -2 0 2 4 6 8 10 -4 -2 0 2 4 6 8 10
Nα
(a) (b) ǫ−1= (20)
1−Nα/3
FIG. 5: Real and imaginary parts of thesusceptibility in the
local field corrections. We chose N ∼ 1.0×1013cm−3, γ ∼ where α is the total polarizability. FIG. 5 plots the
10MHz, ΩC ≈ 2γ, ω21 ≈ 5γ and K ≈ 10γ. (a) and (b): susceptibility with θ = π/8 and θ = 3π/8 in the local
Dispersion relationship inthelocal field corrections with θ= field correction.
π/8 and θ=3π/8.
Acknowledgments
controllable the group velocity either as a sublimunal, a
standard, a superlimunal or even a negative speed. And
the propagation behavior is entirely determined by the YongYao Li thanks Dr JingFeng Liu & GuiHua
mixing angle θ and ϕ of the initial states. Chen and Prof WeiMin Kuang & Jing Cheng for use-
The energy levels in the present paper can be es- ful discussion. HuaRong Zhang thanks for the advice
tablished in a wide range of atomic, molecular, and from Prof XiangYang Yu. This work is supported by
condensed matter. Rb vapor, metastable neon, rare- the National Key Basic Research Special Foundation
earthatoms,donorboundelectronsandboundexcitonsin (NKBRSF) (G2004CB719805), Chinese National Natu-
semiconductors[9, 37, 38, 39, 40] are appropriate sample ral Science Foundation (60677051,10774193).
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