Electromagnetically induced transparency with degenerate atomic levels

0.3 MB·English

by  V. A. Reshetov

Electromagnetically induced transparency with
degenerate atomic levels
3 V. A. Reshetov, I. V. Meleshko
1
0
Department of General and Theoretical Physics, Tolyatti State University, 14
2
Belorousskaya Street, 446667 Tolyatti, Russia

n
a Abstract
J
ForthecoherentlydrivenΛ-typethree-levelsystemsthegeneralready-
6
to-calculateexpressionforthesusceptibilitytensoratthefrequencyofthe
2
weak probe field is obtained for the arbitrary polarization of the strong

coupling laser field and arbitrary values of the angular momenta of reso-
]
s nantatomiclevels. Thedependenceoftherelativedifferenceinthegroup
c
velocities of the two polarization components of the probe field on the
i
t polarization of the coupling field is studied.
p
o
s. 1 Introduction
c
i
s The phenomenon of the electromagnetically induced transparency (EIT), when
y
the optical properties of the media for the weak probe field are coherently con-
h
p trolled by the strong coupling field, has been extensively studied in the recent
[ yearsandprovidedanumberofattractiveapplications[1]. Themostprominent

feature of EIT is a spectacular reduction of the group velocity of light pulses
1
in the resonant media [2, 3]. The strong coupling field with the definite polar-
v
4 ization produces also the optical anisotropy for the probe field, which reveals
6 itself in the electromagneticallyinduced birefringenceand polarizationrotation
2 of the probe field [4, 5]. Among the most promising applications of EIT is
6
the implementation of quantum memory [6, 7, 8]. Such memory based on the
.
1 controlledadiabaticdecelerationandaccelerationofsingle-photonpulses inthe
0 resonant media was suggested in [9, 10], and soon the possibility of storage of
3
light pulses was realized in rubidium vapor in the experiment [11]. The recent
1
experiments [12, 13, 14] on EIT-based quantum memory demonstrate the con-
:
v
tinuously improving efficiency and fidelity. There are different ways to encode
i
X the single-photonqubit, for example, in twospectralcomponents ofthe photon
r pulse,asitwasrecentlyproposedin[15],butthemostnaturalwayforqubiten-
a coding is providedby the photon two polarizationdegreesof freedom, as it was
implementedintheexperiments[12,13]. Tostorethephotonpolarizationqubit
its both polarization components must be effectively stopped in the medium,
howeverthe groupvelocitiesofthese twocomponentsmaydifferessentiallydue
to the optical anisotropyinduced by the polarizationof the coupling field. The
experiments on EIT are performed on the the three-level Λ-type systems with
degenerate levels, which are in many cases the hyperfine structure components
of alkali atoms degenerate in the projections of the atomic total angular mo-
mentum on the quantization axis. The theoretical treatment of EIT on such
1
(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
(cid:1)
(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:1)
(cid:1)
(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
(cid:1) (cid:1)
(cid:1)
(cid:1)
(cid:1) (cid:5)(cid:13)(cid:3)(cid:14)(cid:11)(cid:1)(cid:10)(cid:7)(cid:11)(cid:6)(cid:12)(cid:1) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:1)(cid:10)(cid:7)(cid:11)(cid:6)(cid:12)(cid:1)
(cid:1) ω(cid:1) ω
(cid:1)
(cid:1) (cid:1)
(cid:1)
(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:1)(cid:1)(cid:1)
(cid:3)
(cid:1)
(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
(cid:4)
Figure 1: The level diagram.
levels involves the solution of the equations for the atomic density matrix and
the number of elements of this matrix increases drastically with the increase of
the level angular momentum values. The objective of the present article is to
obtainthegeneralexpressionforthelinearsusceptibilitytensoratthefrequency
of the probe field, which describes the polarization properties of EIT, and to
studythedependenceoftherelativedifferenceinthegroupvelocitiesofthetwo
polarization components of the probe field on the polarization of the coupling
field. The essential point about this expression is that the susceptibility tensor
may be easily calculated by means of standard matrix operations for the arbi-
trary polarization of the driving field and for the arbitrary values of the level
angular momenta.
2 Basic equations and relations
We considerthe weakprobe fieldpropagatingalongthe sample axisZ with the
carrierfrequency ω, whichis in resonancewith the frequency ω of an optically
0
allowed transition J J between the ground state J and the excited state
a c a
→
J , and the strong coherent coupling field propagating in the same direction
c
with the carrier frequency ω , which is in resonance with the frequency ω of
c c0
an optically allowed transition J J between the long-livedstate J and the
b c b
→
same excited state J (Fig.1). Here J , J and J are the values of the angular
c a b c
momenta of the levels. The electric field strength of the coupling field (inside
thesampleaswellasoutside)andthatoftheprobefieldincidentonthesample
border z =0 may be put down as follows:
E =e l e−iωc(t−z/c)+c.c., (1)
c c c
E =e e−iω(t−z/c)+c.c., (2)
0 0
2
where e and l are the constant amplitude and the unit polarization vector
c c
of the coupling field, while e is the constant vector amplitude of the incident
0
probe field. At some sample point z, where the vector amplitude of the probe
field is e, the evolution of the atomic slowly-varying density matrix ρˆ in the
rotating-waveapproximation is described by the equation:
dρˆ i dρˆ
= Vˆ,ρˆ + , (3)
dt 2h i (cid:18)dt(cid:19)rel
Vˆ =2(∆Pˆ +δPˆ )+Ω (gˆ +gˆ†)+Gˆ+Gˆ†, (4)
c b c c c
gˆ =gˆ l∗, Gˆ =(2d/¯h)gˆe∗. (5)
c c c | |
Here∆=ω ω and∆ =ω ω arethefrequencydetuningsfromresonance
0 c c c0
of the probe−and of the coupli−ng fields, while δ = ∆ ∆ , Pˆ is the projector
c α
−
on the subspace of the atomic level J (α = a,b,c), Ω = 2d e /¯h is the
α c c c
| |
reduced Rabi frequency for the coupling field, d = d(J J ) and d = d(J J )
a c c b c
being the reduced matrix elements of the electric dipole moment operator for
the transitions J J and J J , while gˆ and gˆ are the dimensionless
a → c b → c c
electric dipole moment operators for the transitions J J and J J .
a c b c
→ →
The matrix elements of the circular components gˆ and gˆ (q =0, 1)of these
q cq
±
vector operators are expressed through Wigner 3J-symbols [16]:
J 1 J
(gˆ )ac =( 1)Ja−ma a c , (6)
q mamc − (cid:18) ma q mc(cid:19)
−
J 1 J
(gˆ )bc =( 1)Jb−mb b c . (7)
cq mbmc − (cid:18) mb q mc(cid:19)
−
Finally, the term (dρˆ/dt) in the equation (3) describes the irreversible relax-
rel
ation. Initially the atoms are at the ground state a the atomic density matrix
being
ρˆ(0)=ρˆ .
a
Generallytheatomsareattheequilibriumgroundstatewithequallypopulated
Zeeman sublevels, however they may be prepared at some special state [13].
In the linear approximation for the probe field we obtain from the equations
(3)-(4) for the elements of the steady-state (dρˆ/dt = 0) atomic density matrix
the following relations:
i
(γ i∆)ρˆca = Ω gˆ†ρˆba+Gˆ†ρˆ , (8)
− 2(cid:16) c c a(cid:17)
i
(Γ iδ)ρˆba = Ω gˆ ρˆca, (9)
c c
− 2
where
ρˆαβ =Pˆ ρˆPˆ , α,β =a,b,c,
α β
3
while the irreversible relaxation is simply characterized by the two real relax-
ation rates – γ for the optically allowed transition J J and Γ for the
a c
→
optically forbidden transition J J :
a b
→
ca ba
dρˆ dρˆ
= γρˆca, = Γρˆba.
(cid:18)dt(cid:19) − (cid:18)dt(cid:19) −
rel rel
From (8)-(9) it follows immediately:
i
ρˆca = Uˆ−1Gˆ†ρˆ , (10)
a
2(γ i∆)
−
where
Ω2
Uˆ =Pˆ + c gˆ†gˆ . (11)
c 4(γ i∆)(Γ iδ) c c
− −
The medium polarization component at point z
P=pe−iωt+c.c.
with the frequency ω of the probe field is expressedthroughthe density matrix
ρˆca defined by (10)-(11):
p=n dTr gˆρˆca , (12)
0 A
| | { }
wheren istheconcentrationofresonantatomsandthetraceiscarriedoutover
0
atomic states. With the two orthonormal vectors l in the polarization plane
i
XY (l l∗ =δ , j,k =1,2) the equation (12) with an account of (10)-(11) and
j k jk
(5) may be expressed through the susceptibility tensor χ :
jk
2
p =ε χ e , (13)
j 0 jk k
kX=1
where
iχ
χ = 0 Tr Uˆ−1gˆ†ρˆ gˆ , (14)
jk 1 i(∆/γ) A k a j
n o
−
n d2
χ = 0| | , gˆ =gˆl∗. (15)
0 ε ¯hγ j j
0
With the introduction of the orthonormal set of eigenvectors c > with non-
n
|
negative eigenvalues c2 of the hermitian operator gˆ†gˆ :
n c c
gˆ†gˆ c >=c2 c >,n=1,...,2J +1,
c c| n n| n c
the susceptibility tensor (14) may be transformed to the expression:
iχ 2Jc+1<c gˆ†ρˆ gˆ c >
χ = 0 n| k a j| n , (16)
jk
1 i(∆/γ) λ
− nX=1 n
4
c2Ω2
λ =1+ n c . (17)
n
4(γ i∆)(Γ iδ)
− −
Now let χ be the two eigenvalues and
k
2
v = v l , k =1,2, (18)
k jk j
Xj=1
be the corresponding two eigenvectors of the 2 2 susceptibility matrix χˆ =
×
χ :
jk
{ }
χˆv =χ v , k =1,2.
k k k
Then, the vector amplitude e of the incident probe field after the passage of
0
the distance z in the medium is transformed to
e=Sˆe , Sˆ=vˆTˆvˆ−1, (19)
0
where vˆ = v is the 2 2-matrix, defined by the equation (18), while the
jk
{ } ×
diagonal matrix Tˆ is determined by the eigenvalues χ of the susceptibility
k
matrix:
T =δ ei(ω/c)znj, n = 1+χ , (20)
jk jk j j
p
as it may be obtained in a usual way from the Maxwell equation
∂2E 1 ∂2E 1 ∂2P
=
∂z2 − c2 ∂t2 ε c2 ∂t2
0
forthe electric fieldstrengthE of the probe field. The intensity I andthe 2 2
×
polarizationmatrixσˆ = σ oftheprobefieldafterthepassageofthedistance
jk
{ }
z in the medium are related to the intensity I and the polarization matrix σˆ
0 0
of the incident probe field by the equations:
I Sˆσˆ Sˆ†
=tr Sˆσˆ Sˆ† , σˆ = 0 , (21)
0
I0 (cid:16) (cid:17) tr Sˆσˆ Sˆ†
0
(cid:16) (cid:17)
where transformation matrix Sˆ is defined by (19)-(20).
The electromagneticallyinduced transparencyis associatedwith the signifi-
cantreductionofthegroupvelocityoftheprobefieldduetothesteepdispersion
at the transparencywindow. Since the dispersion is different for the two polar-
ization components of the probe field, these two components will be differently
slowed down in the medium. Let us calculate the relative difference of the
group velocities for these two polarization components. For the observation of
the electromagnetically induced transparencythe coupling field must be strong
enough:
Ω2 Γγ.
c ≫
Then, close to the two-photon resonance δ Γ the general formula (16)-(17)
≤
for the susceptibility tensor is simplified to the following expression:
4χ γ
0
χ = (δ+iΓ)a , (22)
jk Ω2 jk
c
5
(cid:3)(cid:4)(cid:6)
(cid:3)(cid:4)(cid:5)
(cid:9)(cid:10)(cid:11)(cid:12) (cid:3)(cid:4)(cid:1)
(cid:8)
(cid:3)(cid:4)(cid:2)
(cid:3)
−(cid:1) −(cid:2) (cid:3) (cid:2) (cid:1)
(cid:2)(cid:7)γ
Figure 2: The imaginary part of susceptibility Im(χ /χ ) (dashed line) and
xx 0
Im(χ /χ ) (solid line) versus dimensionless frequency detuning ∆/γ on the
yy 0
transitions with J = 0, J = J = 1 at Ω /γ = 1, Γ/γ = 0.01 and l = l ,
a b c c c x
∆ =0.
c
where
<c gˆ†ρˆ gˆ c >
a = n| k a j| n , (23)
jk c2
Xn n
is a hermitian 2 2 matrix. The summation in (23) is carried out over all
×
eigenvectors c > of the operator gˆ†gˆ with non-zero eigenvalues c2 > 0. The
| n c c n
twoeigenvectorsv (18)ofthesusceptibilitymatrix χ coincidewiththetwo
k jk
{ }
orthonormaleigenvectors of the hermitian matrix a , while its eigenvalues
jk
{ }
4χ γ
0
χ = (δ+iΓ)a ,
k Ω2 k
c
are expressed through the real eigenvalues a of matrix a . The group ve-
k jk
locities Vgr of the two probe field components polarized{alon}g eigenvectors v
k k
are determined by the refraction indices n′ =Re √1+χ :
k k
(cid:0) (cid:1)
c
Vgr = .
k n′ +ω(dn′/dω)
k k
Since χ 1, in the case of steep dispersion
k
| |≪
χ γω Ω2,
0 ≫ c
we obtain for the group velocities:
c 2χ γωa
Vgr = , ngr = 0 k. (24)
k ngr k Ω2
k c
The relative difference of these two group velocities
Vgr Vgr a
ε= 2 − 1 =1 2 (25)
Vgr − a
2 1
6
(cid:3)(cid:4)(cid:1)
(cid:3)(cid:4)(cid:2)
(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:3)
−(cid:3)(cid:4)(cid:2)
−(cid:3)(cid:4)(cid:1)
−(cid:1) −(cid:2) (cid:3) (cid:2) (cid:1)
(cid:2)(cid:5)γ
Figure 3: The real part of susceptibility Re(χ /χ ) (dashed line) and
xx 0
Re(χ /χ ) (solid line) versus dimensionless frequency detuning ∆/γ on the
yy 0
transitions with J = 0, J = J = 1 at Ω /γ = 1, Γ/γ = 0.01 and l = l ,
a b c c c x
∆ =0.
c
isdeterminedbythedifferenceofthetwoeigenvaluesa ofmatrix a ,a be-
k jk 1
ingthe largestofthemandthe groupvelocityVgr ofthe probefield{ com}ponent
1
polarized along the corresponding eigenvector v being the smallest of two.
1
3 Discussion
The (2J +1) (2J +1) matrices gˆ and (2J +1) (2J +1) matrices gˆ
b c c a c j
× ×
(j = 1,2) of the electric dipole moment operators may be easily calculated for
any reasonable values of the angular momenta J , J and J of the resonant
a b c
atomiclevelsusingtheknownformulaeforWigner3J-symbols,andfurthercal-
culations ofthe susceptibility tensor χˆ (14) and ofthe transformationmatrix Sˆ
(19), involving standard matrix operations, are rather simple. In the following
numerical calculations we shall assume the coupling field to be rather strong:
Ω /γ =1andexactlyresonant∆ =0,andthe relaxationrateofthe forbidden
c c
transition to be rather small: Γ/γ = 0.01, as it is the case for the electromag-
netically induced transparency, while the atoms are initially at the equilibrium
state:
Pˆ
a
ρˆ = .
a
2J +1
a
We shall also considerthe linearly polarizedcoupling field, then without loss of
generalityitspolarizationvectormaybedirectedalongtheCartesianaxisX: l
c
= l , and we shall choose the Cartesian basis l = l , l l in the polarization
x 1 x 2= y
plane XY for the calculation of the elements of the susceptibility tensor. With
this choice of the Cartesian basis the polarization matrix σˆ (21) of the probe
field may be expressed through the Stokes parameters ξ (n=1,2,3):
n
1 1+ξ ξ iξ
σ = 3 1− 2 ,
2(cid:18)ξ1+iξ2 1 ξ3 (cid:19)
−
7
(cid:3)
(cid:31)
(cid:21)(cid:22)(cid:16)(cid:10)(cid:22)(cid:23)(cid:12)(cid:16)(cid:24)(cid:14)(cid:21)(cid:14)(cid:25)(cid:22)(cid:11)(cid:14)(cid:11)(cid:10)(cid:26)(cid:27)(cid:10)(cid:10)(cid:14)(cid:28)(cid:29)(cid:14)(cid:15)(cid:28)(cid:30)(cid:25)(cid:27)(cid:12)(cid:18)(cid:25)(cid:16)(cid:12)(cid:28)(cid:22)(cid:14) (cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:4)(cid:4)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)
(cid:1)
(cid:1) (cid:2) (cid:3)(cid:1) (cid:3)(cid:2)
(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:9)(cid:14)(cid:11)(cid:10)(cid:15)(cid:16)(cid:17)(cid:14)(cid:18)(cid:19)(cid:20)
Figure 4: The relative intensity I/I (dashed line) and the degree of polariza-
0
tion P (solid line) of the initially unpolarized probe field versus dimensionless
medium depth z/L (L=1/kχ ) on the transitions with J =0, J =J =1 at
0 a b c
Ω /γ =1, Γ/γ =0.01 and l =l , ∆=∆ =0.
c c x c
P = ξ2+ξ2+ξ2
1 2 3
q
being the total degree of polarization,
P = ξ2+ξ2
l 1 3
q
being the degree of linear polarization, and
P = ξ
c 2
| |
being the degree of circular polarization.
Let us consider the transitions with the low values of angular momenta
J = 0, J = J = 1, which may be realized in thallium vapor for example.
a b c
With the linear polarization l = l of the coupling field the susceptibility ten-
c x
sorbecomesdiagonalintheCartesianbasis. Thedependenciesoftheimaginary
Im(χ /χ ),Im(χ /χ )andrealRe(χ /χ ),Re(χ /χ )partsofthesuscep-
xx 0 yy 0 xx 0 yy 0
tibility eigenvalues versus the dimensionless frequency detuning ∆/γ, obtained
from the calculations according to the formula (16), are presented in the Fig-
ures2and3. Asitfollowsfromthesedependenciesthestronglinearlypolarized
(along the axis X) coupling field opens the ”transparency window” (solid line)
for the polarization component of the probe field linearly polarized in the per-
pendicular direction (along the axis Y), while for the component of the probe
field collinearly polarized with the coupling field (along the axis X) this ”win-
dow” remains shut (dashed line), which gives rise to rather strong dichroism
for the resonant (∆ = 0) probe field. This happens because the X-polarized
component of the probe field interacts with a single substate
1
c >= (m = 1> m =1>)
x c c
| √2 | − −|
8
(cid:1)(cid:2)(cid:7)(cid:3)
(cid:1)(cid:2)(cid:7)
(cid:10)(cid:11)(cid:12)(cid:13) (cid:1)(cid:2)(cid:1)(cid:6)
(cid:9)
(cid:1)(cid:2)(cid:1)(cid:5)
(cid:1)(cid:2)(cid:1)(cid:4)
−(cid:1)(cid:2)(cid:3) (cid:1) (cid:1)(cid:2)(cid:3)
(cid:2)(cid:8)γ
Figure 5: The imaginary part of susceptibility Im(χ /χ ) (dashed line) and
xx 0
Im(χ /χ ) (solid line) versus dimensionless frequency detuning ∆/γ on the
yy 0
transitions with J = 1, J = J = 2 at Ω /γ = 1, Γ/γ = 0.01 and l = l ,
a b c c c x
∆ =0.
c
of the excited level c, which is not affected by the coupling field:
gˆ c >=0.
c x
|
For the initially unpolarized probe field:
1 1 0
σˆ = ,
0 2(cid:18)0 1(cid:19)
the dependencies of its relative intensity I/I (dashed line) and the degree of
0
polarization P (solid line) versus the dimensionless medium depth z/L, where
L = 1/kχ , are presented in the Figure 4. As it may be seen in this figure,
0
the probe field becomes almost fully polarized with the degree of polarization
P =0.96(linearlyalongthe axisY), while its intensity is reducedbythe factor
of 0.19.
Let us now consider the transitions with larger values of angular momenta
J = 1, J = J = 2, which were employed in the experiment [4] performed
a b c
in 87Rb vapor. For the linear polarization l = l of the coupling field the
c x
dependencies of the imaginary Im(χ /χ ), Im(χ /χ ) and real Re(χ /χ ),
xx 0 yy 0 xx 0
Re(χ /χ ) parts of the susceptibility eigenvalues versus the dimensionless fre-
yy 0
quency detuning ∆/γ are presented in the Figures 5 and 6. In this case the
”transparencywindows”areopenforbothpolarizationcomponentsoftheprobe
field, however,that for the Y-component (solid line), perpendicular to the cou-
pling field, is larger than for the X-component(dashed line), collinear with the
couplingfield. Inthe caseoflargevaluesoftheangularmomentathe dichroism
is less than in the case of small ones, as it may be seen in the Figure 7. When
the degree of polarization of the initially unpolarized probe field attains the
value of 0.5 its intensity is reduced by the factor of 0.17.
9
(cid:1)(cid:2)(cid:1)(cid:5)
(cid:1)(cid:2)(cid:1)(cid:3)
(cid:7)(cid:8)(cid:9)(cid:10)(cid:11) (cid:1)
−(cid:1)(cid:2)(cid:1)(cid:3)
−(cid:1)(cid:2)(cid:1)(cid:5)
−(cid:1)(cid:2)(cid:3) −(cid:1)(cid:2)(cid:4) (cid:1) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:3)
(cid:2)(cid:6)γ
Figure 6: The real part of susceptibility Re(χ /χ ) (dashed line) and
xx 0
Re(χ /χ ) (solid line) versus dimensionless frequency detuning ∆/γ on the
yy 0
transitions with J = 1, J = J = 2 at Ω /γ = 1, Γ/γ = 0.01 and l = l ,
a b c c c x
∆ =0.
c
The relativedifference ε inthe groupvelocities ofthe two polarizationcom-
ponents of the probe field (25) depends only on the values of total angular
momenta of resonant levels, on the initial atomic state and on the polarization
of the coupling field. Let us consider the coupling field with arbitrary elliptic
polarization in the plane XY:
l =cosαδ sinαδ ,
cq q,−1 q,1
−
where parameter α defines the ratio of the lengthes a and a of the axes of
y x
polarization ellipse along the axes Y and X according to the relation
π a
y
tan α = ,
(cid:12) (cid:16) − 4(cid:17)(cid:12) ax
(cid:12) (cid:12)
(cid:12) (cid:12)
while the sign of tan(α π/4) determines the direction of rotation of the pulse
−
electricfieldvectorintheplaneXY. Fortheatomsatinitiallyequilibriumstate
andwiththe fixedvaluesofthelevelangularmomentathe relativedifferencein
thegroupvelocitiesεdependsonlyontheellipticityparameterαofthecoupling
field. An example of such dependence for the levels with the angular momenta
J =J =1,J =2,employedintheexperiment[13],ispresentedintheFigure
a c b
8. Themaximumdifferenceinthegroupvelocitiesε=0.625isobtainedwiththe
circularly polarized coupling field (α = 0,π/2), while the minimum difference
ε = 0.125 is obtained with the linearly polarized coupling field (α = π/4).
In the case of circularly polarized coupling field the polarization component
with the larger group velocity is circularly polarized in the same direction as
the coupling field, while the polarization component with the smaller group
velocity is circularly polarized in the direction opposite to that of the coupling
field. Inthe caseoflinearlypolarizedcouplingfieldthe polarizationcomponent
with the smaller group velocity is linearly polarized in same direction as the
coupling field, while the polarization component with the larger group velocity
is linearly polarizedin the direction perpendicular to that of the coupling field.
10