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Effects of atomic decay and dephasing on the formation of adiabatons in a Λ-configuration medium
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CQa
15 March 1996
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ELSEVIER
OPTICS COMMUNICATIONS Optics Communications 124 (1996) 602-615
Full length article
Effects of atomic decay and dephasing on the formation of adiabatons in a A-configuration medium Jamal T. Manassah, Barry Gross Photonics Engineering
Center, Department
of Electrical Engineering,
City College ofNew York, New York, NY 10031, USA
Received 10 July 1995; revised version received 27 September 1995; accepted 26 October 1995
Abstract
The formation of the adiabatons solutions in a A-configuration atomic medium are numerically investigated in the presence of nonzero inter-level dephasing and inhomogeneous broadening. We show that stringent limits on the values of these widths are necessary for the formation of adiabatons. Details of the dynamics of the two pulses propagation are illustrated and compared for the different values of the parameters.
1. Introduction Electromagnetic Induced Transparency and Amplification without Inversion have been the topics of some very exciting research [ l-61. Some of the recent work [7-121 focused on such diverse topics as matched pulse generation, normal modes, dressed field states and adiabatons. Grobe, Eberly and their collaborators [9-l I] discovered novel two-pulse coupled nonlinear wave equations that are exactly integrable. The solutions of these equations, called adiabatons, have an unusual property: depending only on the input fields at the entry surface of the medium they can take arbitrary shapes that are invariant upon propagation. Adiabatons [ lo] were shown to develop in a A-configuration (doubly degenerate ground state) atomic medium, under certain conditions, inter alia, the only decay width assumed is that associated with the transition from the upper state to states external to the A-system. In this paper, we study the effects of including other decay and dephasing mechanisms on the formation of the adiabatons. In particular, we show that including inter 0030-4018/96/$12.00 0 1996 Eisevier Science B.V. All rights reserved SsDIOO30-4018(95)00689-3
level dephasing, due to transitions between the lower states, and the inhomogeneous width, resulting from the Doppler effect, can modify significantly the propagation dynamics. For certain values of these widths the formation of adiabatons is inhibited. Details of the dynamics of the two pulses propagation are illustrated and compared for different values of the system’s decay and dephasing rates.
2. Maxwell-Bloch equations The original work [lo] on adiabatons used Schr& dinger’s Equations to represent the dynamics of the Asystem. This formalism is adequate if the only decay present in the problem is that from the upper level to states outside the A-system. In this paper, we use the coupled Maxwell-Bloch equations to describe the dynamics of the A-configuration, shown in Fig. 1. These are given in the slowly-varying-envelopeapproximation, and in the presence of probe and pump fields, by:
J.T. Manassah, B. Gross /Optics
a
!
r
0
A
::’
Y,
i’ i
cp ac \.
i'T
-.
'*_
1
F
(Pdz=
603
124 (1996) 602-615
p,,(Z;
7, 3
exp(-
-$--dc.
--m
Y,
\
!
b
! \ i i i
Communications
(9)
‘!,
i i i i i \
(Henceforth, we will omit writing the subscript E with the ensemble averaging. This averaging will be indicated only by the ( ) brackets.) The normalized decay constants are given by: C
Fig. 1. The A-configuration
Ybb= 7CC= 7’ 9
‘yaa= 7C+ % + 7,X,,
atomic medium level diagram.
Yab=Yac=~(~~+3/b+Yext+3/1),
%x=Y’
7
where rii = yijrR and where yext is the normalized decay rate for the irreversible transition between the upper state and states‘not belonging to the A-system. The normalized detunings are given by 2 = A TR
=
( wprobe
-
%d
7R ,
s=8~R=(opump-wac)7R~
and the normalized 4 = OVrR =
(4)
ap,,lar= - (Y,,-iS+iC)p,, - icp,,(p,, - PA + kbpbc
(5)
T
apJar= - (ybC-iS+ib)pbC (6)
a4o,,laz= (2ip,,),,
(7)
apJaz= (Zp,,),,
(8)
f= i L’
coordinates
t’ 7=-z-
t-zlu
rR
TR
s ?i
rR
9
where 9 is the dipole matrix element between a and b, and a and c, assumed here equal, and rR is the natural time-scale in the propagation problem (This quantity corresponds to the two-level superradiant lifetime [ 131 of the completely inverted system if L was small). It is given by: rR = 2hhlNY2L7r,
+ iqZpnC - icp,,p,*b T
where the normalized
electrical fields are given by:
are given by:
where L denotes the length of the sample, and u is the group velocity at the pulses’ central frequency. The averaging over the inhomogeneous profile is achieved through averaging over a Gaussian distribution with the normalized inhomogeneous linewidth as defining parameter, for instance:
N is the atomic number density and h is the wavelength corresponding to the a-, b or a+ c transitions, assumed here equal. To elucidate further the meaning of the different relaxation rates and broadening widths, we write below the density matrix element (pat), if the system was excited only by an on-resonance cw pump: (P,J
= (i%,~~J+%J
[(r/B)
exp(B2)erfc(4
1, (10)
where the erfc refers to the complementary tion and the quantity B is given by: B2=(%+sI~00,,12)/2%i,. The saturation parameter is given by:
error func-
604
J.T. Manassah, B. Gross/Optics
Communications
124 (1996) 602-615
the input and output planes. quantities:
3. Conservation
I [I rp,,(t= 1,
AF,,=
laws
Defining
the following
7) I‘- I ~a,(Z=o, 4 I*1 dr,
-cc
The following fromEqs. (l)-(8).
conservation
(12)
laws can be derived with similar expression for the Q, field, and 1
3.1. Conservation of probability
Rii = (J0
Adding Eqs. ( 1, 2, and 3)) we obtain:
ap,,la7+apbbla7+apcrla~= -y,,,p,, ifyext=O,Eq.
dF [p;i(Z; T= +m) -pii(Z; 7~ -“)I
(11)
,
(13) where i = a, b or c. The Poynting’s Theorem takes the following forms for the particular cases: (a) For yext # 0, ya = yb = y’ = 0 (Grobe and Eberly case), we have:
(11) leadstop,,+p,,+p,,=l.
3.2. Poynting ‘s theorem This is an energy conservation law. It relates the energy that gets stored in the atomic system with the difference of the electromagnetic fields’ energy flux at
$AF,,=R,,
~AF~~=R~~.
(b) For yb= y,#O,
( 1-M
yext= y’ = 0, we have:
b
0.01
P ac
paJ i
0.005.
i 0
-0.005
-0.02
0-A
2
4
6
8 x 10.'
t'
-0.01 0
2
4
6
t’
Fig. 2. The fields and their sources are plotted as function of the retarded time for different values of the normalized y’=O, y,*=O, (i) 2=0, (ii) 2=0.5, (iii) F=l. 7/e.,=108, x=yC=o,
a
Y10” distance of propagation.
J. T. Manassah, B. Gross/Optics
i(AF,,-
( lab)
AFad =&c - Rbb
while for the general case (i.e. yeX,# 0, 3/b= yC# 0, y’ # 0) , Poynting’s Theorem is given by: $(AF,,+hF,,)
= 2
(Rbb+Rcc) + ya0- “C R,. rext (14c)
We note that the general case’s expression, Eq. ( 14~)) involves both fields because the dephasing rate between the lower states is nonzero.
4. Numerical algorithm
This last equation is obtained by multiplying Eq. (7) by && and Eq. (8) by cp,*,,and from this finding the expression for the partial Z-derivative of [(%b(2+ l~,,(21rthenusingEqs. (4), (5), (6),and ( 11) to obtain:
Next, use the expression for r,, and integrate both sides of the above equation over z from 0 to 1, and over r from - 02 to a.
F
605
Communications 124 (1996) 602-615
The numerical algorithm used here was previously described [ 141. It can be summarized as follows: We initialize the system by assigning to the density matrix off-diagonal elements and all diagonal elements except &, the value zero. The states of the ensemble of atoms are assumed to have a distribution in their energy centered at the isolated fixed atom energy and having a width equal to the inhomogeneous (Doppler) width. For the homogeneously broadened case, we assume that the Doppler width is zero. The input probe is taken to be: $&b(.?=O, 7) =&,
f?Xp[ -
(T-_k))2/A?]
and the input pump is taken to be:
a ab
(15)
-
0.262 -
. . __
0.26 0.256 -
_.._
._
'__
0.256-
._._ ii. _ .
0.254 0.252 0
I 0.1
0.2
I 0.3
/ 0.4
I 0.5
1 0.6
I 0.7
. .._
_.
.__
/
,
0.8
0.9
_. 1
z 43.
/
I
-i- - -
-,- - - _ ii
_,W(
b
42-
I: ab
4140-
3938
/// /
/ 370 --
0.1 I
0.2 I
--C-I--.L. 0.3 0.4
I_-_ 0.5
0.6
0.7
0.6
_dLP_ 09
1
z Fig. 3. The
probe
y,>=.yc=O,
y’=O,
integrated energy flux (a) and area, 8, (b) are plotted as function of the normalized yim=O; (ii) yexy=O, yb=yc=lOs, y’=O, .Yiti=O.
dhme
of Propagation.
(i) Ymt= 108,
606
J.T. Manassah, B. Gross/Optics
pao,,(Z=O,
r)=~E~exp[
--
0 ao
q&
-(r-r~~))2/A?]
nc 9 T(1)
ar
exp[ - (r-
)
r~~‘)*/A?]
Q$
, (16)
where Ar= 4lyb or 41 Text as the case may be, CP~,= 5 ( ybrR), and &, = 3.75 ( ybrR), unless otherwise stated, and 10-* ybrR
$‘z------
=
3.75
3
rab
175000 =--- 16
?
100000 (*) _ 1200000 I6 7 r*c _16 ’
The above numbers would correspond to atoms with decay half-life of 10 ns, an atomic number density of
Communications
124 (1996) 602415
2.7 X 10” cmP3, and a sample that is 10 cm long. The pump plateau is 2 ps long and the probe width A r is 40 ns. In the following simulations, the dephasing time and the inhomogeneous width are the changing parameters. The steps in the algorithm proceeds as follows: (i) We fix Z, starting at Z= 0, where pjj(Z= 0, r= - cc),?), &o,b(z= 0, r) and CJJ~,,( f = 0, 7) are given, and calculate pii(z, r, C) over an adaptive r-grid by solving Eqs. ( 1)-( 6) using the Runga-Kutta (RK) technique. During this part of the calculation (~a~(t 7) and CP,,(.Z,T) on the adaptive grid is obtained from an interpolation of their values on the uniform grid; (ii) we average the values of p,(Z, 7, C) over the E distribution; (iii) we increase the value of z by AZ and use Eqs. (7), (8) with RK to calculate the value of 4p,b(Z+ AZ, 7) and so,,(z+ AZ,7);
4.5 ii
P aa
41 3.5 32.5-
Fig. 4. The upper state population is plotted, at the exit plane, as a function of the retarded time. (0 yext = lo’, */b= x= 0, y’ = 0, ‘yinh= 0; (ii) ?/ext= 0, +,I~ = yC= lo*, y’ = 0, yiyinn = 0. (In this graph and subsequent graphs, the faint dotted vertical lines represent the position of the probe peak at the input and output planes for the Grobe-Eberly case.)
.T. Manassah, B. Gross/Optics
Communications
(iv) we return to step (i) except we replace z by z+ AZ. We continue the loop until we reach the final value Z= 1. The accuracy of each of our results is verified by using the conservation of probability and Poynting’s theorem. The finesse of our grids is always adjusted such that Eq. ( 14) is everywhere accurate to better than one part per thousand. Another algorithm that we believe is capable of the same numerical accuracy has been recently described in Ref. [ 151.
5. Numerical results As was pointed out by Grobe and Eberly [ lo], the counterintuitive pulse sequencing, i.e. the pump field is turned on before the probe, leaves the system in the absorptionless state and is an essential condition for the adiabaton formation. Consequently, we assume this sequencing in all simulations that follow. In the following numerical analysis, we will (i) summarize the 8
I24 (1996) 602615
607
Grobe-Eberly case and discuss it within the density-matrix formalism; (ii) analyze the case that only j$, = 7, # 0, while all other y’s are zero, and compare it to the previous results; (iii) analyze the changes from (i) when ‘yinh# 0; (iv) analyze the changes from (ii) when 7’ # 0; and (v) analyze the case that both r,, # 0 and 7’ # 0 and compare these results to (iv).
We have chosen the initial fields strong enough to ensure the formation of the adiabaton pair. If the probe and pump initial Rabi frequencies fall below the atomic decay rate, no adiabaton pair is formed. In Fig. 2, we show the time evolution of the fields and their source terms for different distances of propagation. As pointed out by Grobe and Eberly, the adiabaton-pair (consisting of the dip in the pump and the broadened probe) travels loss-free distances which exceed the weak probe absorption length (here equal to Z= 0.0027) by several
x 10"
7
7 6 5
a
4 3 2 1 0I 0 X
t’
1O-6
0.017 0.016 0.015 0.014 : 0.013/ 0.012
t
0.011 t
Fig. 5. The
probe (a) and the pump (b) fields are plotted, at the exit plane, as function of the retarded time for different inhomogeneous
broadening.
(i) yexr= 1O*,y,=y,=O,y’=O,y,,=O,(ii)
yi,=33X108.(iii)
yinh=100X108.
608
J. T. Manassah, B. Gross/Optics
orders of magnitude without change of shape and at reduced speed. We note the following: - The small absorption, when the probe pulse is present, can be explained by the large coherence between the lower levels created by the simultaneous action of the pump and probe and is related to the absorption inhibition discussed in Ref. [ 11. - The pump, after the probe passes, remains in an absorptionless state since r’ = 0. This ensures the constant plateau required in the pump amplitude in the adiabaton solution. - The existence of lower levels coherence before the arrival of the probe and a finite upper-level population are necessary conditions for AWI [ 61. However both conditions are not simultaneously met here, so amplification is also not possible in this case. - The mechanism responsible for the reduced speed is the absorption of the leading edge of the probe followed by the amplification of its trailing edge. This can be verified by examining the plot of p&, and noting that it changes sign at the position of the probe’s maximum. 0.031
Communicafions
124 (I 996) 602-615
This is similar to the mechanism of velocity reduction present in self-induced-transparency in two-level systems. In Fig. 3 (solid line), we show the total energy flux and the area of the probe field as a function of the normalized distance of propagation. We note that: - The energy flux for the probe field is almost constant over the entire sample. The small variation observed is a measure of the deviation of the physical system results from the exact adiabaton solution due to r,,, + 0. - The area of the probe increases with distance until it reaches an asymptote, the distance at which the area reaches its asymptote defines the adiabaton’s formation length [lO,ll]. cii>
%
=
yc #
0,
‘Yext =
y’
=
‘Yinh =
o
Here, we study the physical case whereby the dominant decay mechanism is that between the upper state
I
-0.02 t
3
/ 4 t’
I
I
5
6
I
7
8 x IO
-7
Fig. 6. Im(p,,,,( E) ) is plotted, at the input plane, as function of the retarded time for different detunings within the inhomogeneously y’=O, n*= 50X 108, (i) 8=0, (ii) 8=0.5, (iii) 8= 1.0, (iv) 8= 1.5. line. -yCxt= 108, y,,=‘yc=o,
broadened
J. T. Manassah, B. Gross / Optics Communications
(a) and the ground state doublet (b and c) . The time profiles of the fields and the density matrix elements for this case follow closely those of (i) , except for paa. In Fig. 4, we compare pcln at the output plane for the two cases (we choose the value of -&,and 7, here to be the same as that of r,,, in (i) ) . We note that the transient values of pna are larger here. This is a result of having now the upper level not irreversibly loosing its excitation to levels outside the A-configuration. In Fig. 3 (dashed line), both the energy flux and area of the fields are plotted for this case. Here, we have larger absorption for both probe and pump, as well as some degradation in the values of the areas. This is explained by the fact that we have in this case two decay paths from state a, and thus larger absorption. This, of course, leads to larger deviations than in (i) from the exact absorptionless adiabaton solution. (iii) 3/eX,#0, yinh#O, yb= yc= y’=O
I24 (I 996) 602-615
609
As the most likely candidate media for experimental observation of adiabaton pair formation are gases, we will investigate next the effects of Doppler broadening on the adiabaton pair formation. We investigate the problem for yid in the range l-10 GHz and compare the results to the case j&, = 0 discussed in (i) above. Since the adiabaton pair solution is expected to hold only when the detuning is smaller than the Rabi frequencies of either probe or pump, we would expect large deviations from this solution when j&, is larger than either Rabi frequency. We plot in Fig. 5, at the output plane, the time development of the fields for different values of yinh. We observe that: - The main maximum of the probe’s amplitude decreases and its width increases as the value of j&, increases. Correspondingly, the dip in the pump’s field follows similarly. - As Yinhincreases, the period of oscillations in the probe’s field increases.
a
7t
bad
-
4 \
543-
\
\ \ \ \ i \ \
l20 0
0.1
0.2
1,
/
0.1
0.2
0.3
OL ‘1 0
0.3
0.4
0.5
0.6
/
1
0.4
0.5
0.6
0.7
\ 0.6
0.7
0.6
\ '..
I 0.9
1 1
0.9
1
I
I
x IO'& t'
Fig. 7. The magnitude of the probe (a) and pump (b) fields are plotted, at the exit plane, as function of the retarded time for different values of (i) y’=O; (ii) y’=yJlOO; (iii) y’=2y,/lOO; (iv) y’=5yb/100. (Thetracesaf the lower states dephasing. yeXt=0, -yb=yC=108, y,,=O, (iii) and (iv) are in this scale superimposed on the horizontal axis.)
J.T. Manassah,
610
B. Gross /Optics
Communications
has a negative imaginary part. For large detuning ( EB Tat), the time for which that statement holds must satisfy the condition Q >> 11 reXf.The basis vectors that span the system’s state, following the passage of the probe and assuming that the detuning is much larger than the pump Rabi frequency, are given by:
To understand the above observations, we first analyze for a specific Yinhrthe contributions to the source term of the probe’s field from different components of the inhomogeneous distribution. In Fig. 6, we plot, at Z= 0, Im( Prrb) for different values of Z= C//.yih (the Maxwellians’s weighing factor for the velocity averaging is expressed as exp( - ?/2) in this variable). The Re(p,,) is not considered since it is odd in E”and thus averages to zero. We note that as E”increases, the position of the first maximum is further delayed. Furthermore, the period of oscillations increases. This behaviour can be physically explained by examining the quantum state of the atom-pump system in the presence of a small probe field. From the solution of the Schrodinger’s equations (valid here) it can be shown, through slowly-varying time-development analysis, that if the system was originally in the state where all the population is in the b state, and assuming that the probe field is well-behaved, then the system, following the passage of the probe is in a state that is orthogonal to the eigenstate of the Hamiltonian whose eigenvalue
-._
124 (1996) 602-615
where we have used the notation that in the column, the elements refer respectively to the b, a, c states of the free atom. For time T> 70, where 7. is a retarded time that is after the passage of the probe, the state of the system can be written as: l~W~))=a~(~~)
& exp[i~(~-70)l
+u~(T~) f?Zexp[i(E+2&JC)(~-~0)]
(18)
where a, ( 70) and a2( TV) are the wavefunction’s components at 7. and the factors multiplying the difference
i
--___
--_ -._--_
F ab
_.
‘.
..\
-- -_ --_
‘.., \ \ \
.
0.15-
‘.
‘. \
\
z-.-y it ----__
\\
\
0.1 -
\
\
\
iii ‘,
I \ \ I \ \ \ i
0.05-
0’
0
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I
I
I
0.6
0.7
0.8
0.9
\
1
z Fig. 8. The integrated Fig. 7.
energy flux for the probe is plotted as function of the normalized
distance of propagation
for the same parameters
as in
J. T. h4unassnh, B. Gross /Optics
Communications
of time in the exponentials are the approximate eigenvalues corresponding to the respective eigenvectors. From the above, we can deduce the b, a, c amplitudes at T. These are respectively:
u,(7) = -+T, oc
.
(19)
This leads to the conclusion that only the slow oscillation mode amplitude of pnb= a,*ab is nonzero for large detuning. The frequency of this mode is inversely proportional to the detuning, if the detuning is large. Consequently, for large detuning the period of oscillations of the probe’s field increases as the detuning increases. It should be noted that the slow and fast oscillation modes are a result of respectively subtracting or adding the system’s eigenvalues. This situation
611
124 (1996) 602415
would not arise in a two-level system, where the oscillations period would decrease with the detuning. The behaviour of the probe’s field with increasing values for j$,,, can now be directly deduced. If pi,,,, increases, the unnormalized value of the detuning corresponding to a given E”increases, while its weighting factor remains the same. As a consequence, we deduce that for (pab): (i) the spread for the separate velocity components in the position of the first maximum increases with an increase in yinh, thus leading to a broader first maximum, and (ii) the period of oscillations in the amplitude of (Im(pab)) increases with an increase in vinh. We remark here that the details (not shown here) of the fields time-profile are qualitatively unchanged if instead of having the above conditions on the Ts, we have everything the same but now j$,= r,# 0 and r,,, = 0.
Pbb
0.5
I
I
I
I
,
/
/
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.9
1
Pbb
- - -._ _ _
Fig. 9. The b state population
0.7
0.8
is plotted, at the input (a) and exit (b) planes, as a function of the retarded time for the same parameters
as Fig.
612
J.T. Ivfanassah, B. Gross /Optics
In Fig. 7, we plot at the output plane the time development of the fields for different values of r’, including the value zero discussed in (ii) above. In Fig. 8, we plot the energy flux of the probe field as function of the normalized propagation distance for the same parameters as in Fig. 7. The increased absorption of both fields with the value of 7’ can be understood by noting that for 7’ = 0, the pump, before the arrival of the probe, will only excite the system’s absorptionless eigenstate leaving all the initial atomic population in the b state. The condition that only the b state is occupied before the on-set of the probe is a necessary condition for adiabaton pair formation. As the dephasing mechanism is added, pbb is not equal to 1 as the probe enters the sample, and the system is in a superposition state with the population out of the b state increasing with an increase in the value of 7’. In Fig. 9, we show, at the input and output planes the time-development of pbbto illustrate this loss of population at times earlier than the probe’s arrival, we observe further that the deviation of pbbfrom 1 is larger for larger values of 7’. From further analysis of this case, we also note that:
Communications
124 (1996) 602-615
-The velocity of the probe decreases as 7 increases. This is a consequence of the general result in Ref. [ 171, where it was shown that the velocity of the probe decreases as the absorption increases. - The magnitude of the lower states maximum coherence’s value remains close to l/2 as long as the system remains in the nonlinear regime. See Fig. 10. That the coherence remains unaltered while both the pump and probe are absorbed is understood by noting that the degree of coherence depends only on the ratio of the probe and pump fields. -The cw plateau of the pump experiences absorption since the steady state of Im(p,,) is positive when 7’ # 0. The absorption increases as 7’ increases. -As was pointed out by Grobe and Eberly, the amplitude of the probe in the adiabaton pair is directly proportional to the pump’s plateau value. Therefore, as the amplitude of the plateau decreases so will the amplitude of the probe. - For sufficiently large $?, the absorption causes the fields to weaken until the atom-fields systems is linear. The onset of the linear regime occurs when the depend-
cA-0.05
-0.1
7
t
1
-0.15 t -0.2 $ CL
-0.25 -0.3 -0.35 -0.4 i -0.45 -0.5 0’
0.1
0.2
0.3
0.4
0.5 t’
Fig. 10. The lower states coherence
0.6
0.7
0.8
0.9 x 10
-6
is plotted, at the exit plane, as function of the retarded time for the same parameters
as Fig. 7.
J.T. Manassah, B. Gross /Optics
Communications
613
124 (1996) 602-615
b
I
0.005-
‘_
1
j
-
-
-
--
-
-.-
-
I, 11 ;:
0
iv 0.5
0
1
1.5 x 10-B
t'
0.02 d
x 1o-3
8, 7
e
Icp acl 0.01
0.005
0 iJ'J 0
0.5
1
I 1.5
Fig. 11.The pump and probe fields are plotted, at the exit plane, as function of the retarded times both y’ and -yilulare scanned. Y_~=O, ly~,=y~=lO’. (a,b) j&,=20$. (c,d) j&,=33Tb, (e,f) yj,=50j&, (g,h) j&,=100$,, (i) y’=O, (ii) y’=yb/lOO,(iii) y’=2yJlOO,(iv) y’=5y,,llOO.
614
J.T. Manassah, B. Gross/Optics
ence of the energy flux on the distance of propagation follows Beer’s law. - In the linear regime, the lower levels coherence is zero and the atomic population is equally divided between the b and c levels for large time.
This is the most common configuration in a gaseous atomic A-medium. It is the most general case and as such, different competing effects discussed earlier are simultaneously at play, namely: (a) As 7 is increased, the pump plateau amplitude is decreased. This in turn leads to a decrease in the probe’s amplitude. (b) 7 # 0 leads to a redistribution of population in the different states of the system prior to the arrival of the probe and thus the probe evolves into a superposition of eigenstates not all absorptionless. (c) As rinh increases while maintaining r’ fixed, the absorption coefficient for the pump is decreased. (d) As j&, increases, the position of the maximum of different velocity components in the probe’s source is smeared thus leading to a broadening in the maximum of the probe. Also the period of the oscillations of the probe’s field is increased. Depending on the relative values of the different parameters, we expect that different physical mechanisms dominate for the different cases. The problem is further complicated since the different effects depend differently on the distance of propagation. We shall consider different regimes to illustrate the different cases. To illustrate, we plot at the output plane in Figs. 1 I the fields for different 7’ as we scan the values of Yinh.
We note the following: - As can be deduced from Eq. ( lo), the magnitude of the pump plateau decreases with an increase of 7’ irrespective of the value of Tint,. Related to this, we find that, for larger r’, there is a steeper decrease of the pump’s energy flux as function of the distance of propagation. - As can also be deduced from Eq. ( lo), the pump absorption for a particular 7’ can be substantially reduced as -&, is increased. - For low values of Tinhr the magnitude of the probe at the exit plane decreases as 7 increases, while for high values of Yinh,the opposite occurs. Related to this,
Communications
124 (1996) 602415
we also find that the rate of decrease of the probe’s energy flux, as function of the distance of propagation, increases with larger 7’ for small yi*, while the opposite occurs for large yi*. - The velocity of the probe depends on the value of 7’. For very low -&, this velocity decreases with an increase in the value of 7. On the other hand, for higher j&, the velocity increases with larger $7. In this later regime, for the same y’, the velocity increases with an increase in the value of yi*. - The observed reversal in the ordering of the magnitude of the probe and in the velocity of its peak, as indexed by the value of 7, do not occur for the same value of Yinh.
6. Conclusion Through extensive numerical simulations, we showed that the effects of the decay and dephasing rates on the formation of an adiabaton pair in an atomic hconfiguration medium can be very substantial. As the values of different parameters are scanned, we observe distinct physical regimes for the pump-probe dynamics. It is interesting to note that for typical gas parameters, the large value of Yinh,as compared to the internal decay constant, permits a larger flexibility in the values of r’, as compared to the fixed atoms case, that allow the formation of a ‘ ‘quasi-adiabaton’ ’ pair structure. However, it should be noted that the main result of this paper is that in most physical regimes of interest, where dephasing and inhomogeneous broadening can not be neglected, the adiabaton pair does not exhibit the same degree of stability usually associated with a soliton-like structure. This should, for example, be contrasted to a Raman soliton [ 181 “which is a robust manifestation of quantum noise, that is resilient to detuning, diffraction and phase matching” [ 191.
References []
0. Kocharovska and Ya.1. Khanin, Pis’ma Zh. Eksp. Teor. Fiz. 48 (1988) 581. [JETP Lett. 48 (1988) 6301; 0. Kocharovskaya and P. Mandel, Phys. Rev. A 42 ( 1990) 523. [2] S. Harris, Phys. Rev. Lett. 62 (1989) 1022; A. Imamoglu and S. Harris, Optics Lett. 14 ( 1989) 1344.
.I. T. Manassah,
B. Gross/Optics
[3] M. Scully, S. Zhu and A. Gavrieliedes, Phys. Rev. Lett. 62 (1989) 2813. [4] A. Lyras, X. Tang, P. Lambropoulos and J. Zhang, Phys. Rev. A40 (1989) 4131; S. Basile and P. Lambropoulos, Optics Comm. 78 (1990) 163. [5] M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. ]6] Atomic Coherence and Interference, special issue of Quantum Optics 6 ( 1994) 203-389. ]7] S.E. Harris, Phys. Rev. Lett. 70 (1993) 552. [S] S.E. Harris, Phys. Rev. Lett. 72 (1994) 52. ]9] J.H. Eberly, M.L. Pons and H.R. Haq, Phys. Rev. Lett. 72 (1994) 56. [ lo] R. Grobe,F.T. Hioe and J.H. Eberly, Phys. Rev. Lett. 73 ( 1994) 3183. [ 111 R. Grobe and J.H. Eberly, Laser Phys. 5 (1995) 542. [ 121 A. Kasapi, M. Jain, G.Y. Yin and S.E. Harris, Phys. Rev. L&t. 74 (1995) 2447. [ 131 R.H. Dicke, Phys. Rev. 93 (1954) 99; R. Friedberg and S. Hartmann, Phys. Rev. A 13 (1976) 495.
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124 (1996) 602615
615
[ 141 J.T. Manassah and B. Gross, Optics Comm. 113 (1994) 213, and references to authors work cited therein. [15] A.S. Manka, M. Scalora, J.P. Dowling and C.M. Bowden, Optics Comm. 115 ( 1995) 283, and references to authors work cited therein. [ 161 S.L. McCall and E.L. Hahn, Phys. Rev. Lett. 18 (1967) 908. [ 171 S.E. Harris, J.E. Field and A. Kasapi, Phys. Rev. A 46 (1992) R29. [ 181 F.Y.F. Chu and C. Scott, Phys. Rev. A 12 (1975) 2060; N. Tan-no, T. Shiiahata and K. Yokoto, Phys. Rev. A 12 (1975) 159; K.J. Druhl, J.L. Carlsten and R.G. Wenzel, Phys. Rev. Lett. 5 1 (1983) 1171; J. Stat. Phys. 39 (1985) 615,621; J.C.Englund and C.M. Bowden, Phys. Rev. A42 ( 1990) 2870; 46 (1992) 578; M. Scalora, S. Singh and CM. Bowden, Phys. Rev. Len. 70 (1993) 1248. [ 191 Anonymous reviewer.
CQa
15 March 1996
‘!B
ELSEVIER
OPTICS COMMUNICATIONS Optics Communications 124 (1996) 602-615
Full length article
Effects of atomic decay and dephasing on the formation of adiabatons in a A-configuration medium Jamal T. Manassah, Barry Gross Photonics Engineering
Center, Department
of Electrical Engineering,
City College ofNew York, New York, NY 10031, USA
Received 10 July 1995; revised version received 27 September 1995; accepted 26 October 1995
Abstract
The formation of the adiabatons solutions in a A-configuration atomic medium are numerically investigated in the presence of nonzero inter-level dephasing and inhomogeneous broadening. We show that stringent limits on the values of these widths are necessary for the formation of adiabatons. Details of the dynamics of the two pulses propagation are illustrated and compared for the different values of the parameters.
1. Introduction Electromagnetic Induced Transparency and Amplification without Inversion have been the topics of some very exciting research [ l-61. Some of the recent work [7-121 focused on such diverse topics as matched pulse generation, normal modes, dressed field states and adiabatons. Grobe, Eberly and their collaborators [9-l I] discovered novel two-pulse coupled nonlinear wave equations that are exactly integrable. The solutions of these equations, called adiabatons, have an unusual property: depending only on the input fields at the entry surface of the medium they can take arbitrary shapes that are invariant upon propagation. Adiabatons [ lo] were shown to develop in a A-configuration (doubly degenerate ground state) atomic medium, under certain conditions, inter alia, the only decay width assumed is that associated with the transition from the upper state to states external to the A-system. In this paper, we study the effects of including other decay and dephasing mechanisms on the formation of the adiabatons. In particular, we show that including inter 0030-4018/96/$12.00 0 1996 Eisevier Science B.V. All rights reserved SsDIOO30-4018(95)00689-3
level dephasing, due to transitions between the lower states, and the inhomogeneous width, resulting from the Doppler effect, can modify significantly the propagation dynamics. For certain values of these widths the formation of adiabatons is inhibited. Details of the dynamics of the two pulses propagation are illustrated and compared for different values of the system’s decay and dephasing rates.
2. Maxwell-Bloch equations The original work [lo] on adiabatons used Schr& dinger’s Equations to represent the dynamics of the Asystem. This formalism is adequate if the only decay present in the problem is that from the upper level to states outside the A-system. In this paper, we use the coupled Maxwell-Bloch equations to describe the dynamics of the A-configuration, shown in Fig. 1. These are given in the slowly-varying-envelopeapproximation, and in the presence of probe and pump fields, by:
J.T. Manassah, B. Gross /Optics
a
!
r
0
A
::’
Y,
i’ i
cp ac \.
i'T
-.
'*_
1
F
(Pdz=
603
124 (1996) 602-615
p,,(Z;
7, 3
exp(-
-$--dc.
--m
Y,
\
!
b
! \ i i i
Communications
(9)
‘!,
i i i i i \
(Henceforth, we will omit writing the subscript E with the ensemble averaging. This averaging will be indicated only by the ( ) brackets.) The normalized decay constants are given by: C
Fig. 1. The A-configuration
Ybb= 7CC= 7’ 9
‘yaa= 7C+ % + 7,X,,
atomic medium level diagram.
Yab=Yac=~(~~+3/b+Yext+3/1),
%x=Y’
7
where rii = yijrR and where yext is the normalized decay rate for the irreversible transition between the upper state and states‘not belonging to the A-system. The normalized detunings are given by 2 = A TR
=
( wprobe
-
%d
7R ,
s=8~R=(opump-wac)7R~
and the normalized 4 = OVrR =
(4)
ap,,lar= - (Y,,-iS+iC)p,, - icp,,(p,, - PA + kbpbc
(5)
T
apJar= - (ybC-iS+ib)pbC (6)
a4o,,laz= (2ip,,),,
(7)
apJaz= (Zp,,),,
(8)
f= i L’
coordinates
t’ 7=-z-
t-zlu
rR
TR
s ?i
rR
9
where 9 is the dipole matrix element between a and b, and a and c, assumed here equal, and rR is the natural time-scale in the propagation problem (This quantity corresponds to the two-level superradiant lifetime [ 131 of the completely inverted system if L was small). It is given by: rR = 2hhlNY2L7r,
+ iqZpnC - icp,,p,*b T
where the normalized
electrical fields are given by:
are given by:
where L denotes the length of the sample, and u is the group velocity at the pulses’ central frequency. The averaging over the inhomogeneous profile is achieved through averaging over a Gaussian distribution with the normalized inhomogeneous linewidth as defining parameter, for instance:
N is the atomic number density and h is the wavelength corresponding to the a-, b or a+ c transitions, assumed here equal. To elucidate further the meaning of the different relaxation rates and broadening widths, we write below the density matrix element (pat), if the system was excited only by an on-resonance cw pump: (P,J
= (i%,~~J+%J
[(r/B)
exp(B2)erfc(4
1, (10)
where the erfc refers to the complementary tion and the quantity B is given by: B2=(%+sI~00,,12)/2%i,. The saturation parameter is given by:
error func-
604
J.T. Manassah, B. Gross/Optics
Communications
124 (1996) 602-615
the input and output planes. quantities:
3. Conservation
I [I rp,,(t= 1,
AF,,=
laws
Defining
the following
7) I‘- I ~a,(Z=o, 4 I*1 dr,
-cc
The following fromEqs. (l)-(8).
conservation
(12)
laws can be derived with similar expression for the Q, field, and 1
3.1. Conservation of probability
Rii = (J0
Adding Eqs. ( 1, 2, and 3)) we obtain:
ap,,la7+apbbla7+apcrla~= -y,,,p,, ifyext=O,Eq.
dF [p;i(Z; T= +m) -pii(Z; 7~ -“)I
(11)
,
(13) where i = a, b or c. The Poynting’s Theorem takes the following forms for the particular cases: (a) For yext # 0, ya = yb = y’ = 0 (Grobe and Eberly case), we have:
(11) leadstop,,+p,,+p,,=l.
3.2. Poynting ‘s theorem This is an energy conservation law. It relates the energy that gets stored in the atomic system with the difference of the electromagnetic fields’ energy flux at
$AF,,=R,,
~AF~~=R~~.
(b) For yb= y,#O,
( 1-M
yext= y’ = 0, we have:
b
0.01
P ac
paJ i
0.005.
i 0
-0.005
-0.02
0-A
2
4
6
8 x 10.'
t'
-0.01 0
2
4
6
t’
Fig. 2. The fields and their sources are plotted as function of the retarded time for different values of the normalized y’=O, y,*=O, (i) 2=0, (ii) 2=0.5, (iii) F=l. 7/e.,=108, x=yC=o,
a
Y10” distance of propagation.
J. T. Manassah, B. Gross/Optics
i(AF,,-
( lab)
AFad =&c - Rbb
while for the general case (i.e. yeX,# 0, 3/b= yC# 0, y’ # 0) , Poynting’s Theorem is given by: $(AF,,+hF,,)
= 2
(Rbb+Rcc) + ya0- “C R,. rext (14c)
We note that the general case’s expression, Eq. ( 14~)) involves both fields because the dephasing rate between the lower states is nonzero.
4. Numerical algorithm
This last equation is obtained by multiplying Eq. (7) by && and Eq. (8) by cp,*,,and from this finding the expression for the partial Z-derivative of [(%b(2+ l~,,(21rthenusingEqs. (4), (5), (6),and ( 11) to obtain:
Next, use the expression for r,, and integrate both sides of the above equation over z from 0 to 1, and over r from - 02 to a.
F
605
Communications 124 (1996) 602-615
The numerical algorithm used here was previously described [ 141. It can be summarized as follows: We initialize the system by assigning to the density matrix off-diagonal elements and all diagonal elements except &, the value zero. The states of the ensemble of atoms are assumed to have a distribution in their energy centered at the isolated fixed atom energy and having a width equal to the inhomogeneous (Doppler) width. For the homogeneously broadened case, we assume that the Doppler width is zero. The input probe is taken to be: $&b(.?=O, 7) =&,
f?Xp[ -
(T-_k))2/A?]
and the input pump is taken to be:
a ab
(15)
-
0.262 -
. . __
0.26 0.256 -
_.._
._
'__
0.256-
._._ ii. _ .
0.254 0.252 0
I 0.1
0.2
I 0.3
/ 0.4
I 0.5
1 0.6
I 0.7
. .._
_.
.__
/
,
0.8
0.9
_. 1
z 43.
/
I
-i- - -
-,- - - _ ii
_,W(
b
42-
I: ab
4140-
3938
/// /
/ 370 --
0.1 I
0.2 I
--C-I--.L. 0.3 0.4
I_-_ 0.5
0.6
0.7
0.6
_dLP_ 09
1
z Fig. 3. The
probe
y,>=.yc=O,
y’=O,
integrated energy flux (a) and area, 8, (b) are plotted as function of the normalized yim=O; (ii) yexy=O, yb=yc=lOs, y’=O, .Yiti=O.
dhme
of Propagation.
(i) Ymt= 108,
606
J.T. Manassah, B. Gross/Optics
pao,,(Z=O,
r)=~E~exp[
--
0 ao
q&
-(r-r~~))2/A?]
nc 9 T(1)
ar
exp[ - (r-
)
r~~‘)*/A?]
Q$
, (16)
where Ar= 4lyb or 41 Text as the case may be, CP~,= 5 ( ybrR), and &, = 3.75 ( ybrR), unless otherwise stated, and 10-* ybrR
$‘z------
=
3.75
3
rab
175000 =--- 16
?
100000 (*) _ 1200000 I6 7 r*c _16 ’
The above numbers would correspond to atoms with decay half-life of 10 ns, an atomic number density of
Communications
124 (1996) 602415
2.7 X 10” cmP3, and a sample that is 10 cm long. The pump plateau is 2 ps long and the probe width A r is 40 ns. In the following simulations, the dephasing time and the inhomogeneous width are the changing parameters. The steps in the algorithm proceeds as follows: (i) We fix Z, starting at Z= 0, where pjj(Z= 0, r= - cc),?), &o,b(z= 0, r) and CJJ~,,( f = 0, 7) are given, and calculate pii(z, r, C) over an adaptive r-grid by solving Eqs. ( 1)-( 6) using the Runga-Kutta (RK) technique. During this part of the calculation (~a~(t 7) and CP,,(.Z,T) on the adaptive grid is obtained from an interpolation of their values on the uniform grid; (ii) we average the values of p,(Z, 7, C) over the E distribution; (iii) we increase the value of z by AZ and use Eqs. (7), (8) with RK to calculate the value of 4p,b(Z+ AZ, 7) and so,,(z+ AZ,7);
4.5 ii
P aa
41 3.5 32.5-
Fig. 4. The upper state population is plotted, at the exit plane, as a function of the retarded time. (0 yext = lo’, */b= x= 0, y’ = 0, ‘yinh= 0; (ii) ?/ext= 0, +,I~ = yC= lo*, y’ = 0, yiyinn = 0. (In this graph and subsequent graphs, the faint dotted vertical lines represent the position of the probe peak at the input and output planes for the Grobe-Eberly case.)
.T. Manassah, B. Gross/Optics
Communications
(iv) we return to step (i) except we replace z by z+ AZ. We continue the loop until we reach the final value Z= 1. The accuracy of each of our results is verified by using the conservation of probability and Poynting’s theorem. The finesse of our grids is always adjusted such that Eq. ( 14) is everywhere accurate to better than one part per thousand. Another algorithm that we believe is capable of the same numerical accuracy has been recently described in Ref. [ 151.
5. Numerical results As was pointed out by Grobe and Eberly [ lo], the counterintuitive pulse sequencing, i.e. the pump field is turned on before the probe, leaves the system in the absorptionless state and is an essential condition for the adiabaton formation. Consequently, we assume this sequencing in all simulations that follow. In the following numerical analysis, we will (i) summarize the 8
I24 (1996) 602615
607
Grobe-Eberly case and discuss it within the density-matrix formalism; (ii) analyze the case that only j$, = 7, # 0, while all other y’s are zero, and compare it to the previous results; (iii) analyze the changes from (i) when ‘yinh# 0; (iv) analyze the changes from (ii) when 7’ # 0; and (v) analyze the case that both r,, # 0 and 7’ # 0 and compare these results to (iv).
We have chosen the initial fields strong enough to ensure the formation of the adiabaton pair. If the probe and pump initial Rabi frequencies fall below the atomic decay rate, no adiabaton pair is formed. In Fig. 2, we show the time evolution of the fields and their source terms for different distances of propagation. As pointed out by Grobe and Eberly, the adiabaton-pair (consisting of the dip in the pump and the broadened probe) travels loss-free distances which exceed the weak probe absorption length (here equal to Z= 0.0027) by several
x 10"
7
7 6 5
a
4 3 2 1 0I 0 X
t’
1O-6
0.017 0.016 0.015 0.014 : 0.013/ 0.012
t
0.011 t
Fig. 5. The
probe (a) and the pump (b) fields are plotted, at the exit plane, as function of the retarded time for different inhomogeneous
broadening.
(i) yexr= 1O*,y,=y,=O,y’=O,y,,=O,(ii)
yi,=33X108.(iii)
yinh=100X108.
608
J. T. Manassah, B. Gross/Optics
orders of magnitude without change of shape and at reduced speed. We note the following: - The small absorption, when the probe pulse is present, can be explained by the large coherence between the lower levels created by the simultaneous action of the pump and probe and is related to the absorption inhibition discussed in Ref. [ 11. - The pump, after the probe passes, remains in an absorptionless state since r’ = 0. This ensures the constant plateau required in the pump amplitude in the adiabaton solution. - The existence of lower levels coherence before the arrival of the probe and a finite upper-level population are necessary conditions for AWI [ 61. However both conditions are not simultaneously met here, so amplification is also not possible in this case. - The mechanism responsible for the reduced speed is the absorption of the leading edge of the probe followed by the amplification of its trailing edge. This can be verified by examining the plot of p&, and noting that it changes sign at the position of the probe’s maximum. 0.031
Communicafions
124 (I 996) 602-615
This is similar to the mechanism of velocity reduction present in self-induced-transparency in two-level systems. In Fig. 3 (solid line), we show the total energy flux and the area of the probe field as a function of the normalized distance of propagation. We note that: - The energy flux for the probe field is almost constant over the entire sample. The small variation observed is a measure of the deviation of the physical system results from the exact adiabaton solution due to r,,, + 0. - The area of the probe increases with distance until it reaches an asymptote, the distance at which the area reaches its asymptote defines the adiabaton’s formation length [lO,ll]. cii>
%
=
yc #
0,
‘Yext =
y’
=
‘Yinh =
o
Here, we study the physical case whereby the dominant decay mechanism is that between the upper state
I
-0.02 t
3
/ 4 t’
I
I
5
6
I
7
8 x IO
-7
Fig. 6. Im(p,,,,( E) ) is plotted, at the input plane, as function of the retarded time for different detunings within the inhomogeneously y’=O, n*= 50X 108, (i) 8=0, (ii) 8=0.5, (iii) 8= 1.0, (iv) 8= 1.5. line. -yCxt= 108, y,,=‘yc=o,
broadened
J. T. Manassah, B. Gross / Optics Communications
(a) and the ground state doublet (b and c) . The time profiles of the fields and the density matrix elements for this case follow closely those of (i) , except for paa. In Fig. 4, we compare pcln at the output plane for the two cases (we choose the value of -&,and 7, here to be the same as that of r,,, in (i) ) . We note that the transient values of pna are larger here. This is a result of having now the upper level not irreversibly loosing its excitation to levels outside the A-configuration. In Fig. 3 (dashed line), both the energy flux and area of the fields are plotted for this case. Here, we have larger absorption for both probe and pump, as well as some degradation in the values of the areas. This is explained by the fact that we have in this case two decay paths from state a, and thus larger absorption. This, of course, leads to larger deviations than in (i) from the exact absorptionless adiabaton solution. (iii) 3/eX,#0, yinh#O, yb= yc= y’=O
I24 (I 996) 602-615
609
As the most likely candidate media for experimental observation of adiabaton pair formation are gases, we will investigate next the effects of Doppler broadening on the adiabaton pair formation. We investigate the problem for yid in the range l-10 GHz and compare the results to the case j&, = 0 discussed in (i) above. Since the adiabaton pair solution is expected to hold only when the detuning is smaller than the Rabi frequencies of either probe or pump, we would expect large deviations from this solution when j&, is larger than either Rabi frequency. We plot in Fig. 5, at the output plane, the time development of the fields for different values of yinh. We observe that: - The main maximum of the probe’s amplitude decreases and its width increases as the value of j&, increases. Correspondingly, the dip in the pump’s field follows similarly. - As Yinhincreases, the period of oscillations in the probe’s field increases.
a
7t
bad
-
4 \
543-
\
\ \ \ \ i \ \
l20 0
0.1
0.2
1,
/
0.1
0.2
0.3
OL ‘1 0
0.3
0.4
0.5
0.6
/
1
0.4
0.5
0.6
0.7
\ 0.6
0.7
0.6
\ '..
I 0.9
1 1
0.9
1
I
I
x IO'& t'
Fig. 7. The magnitude of the probe (a) and pump (b) fields are plotted, at the exit plane, as function of the retarded time for different values of (i) y’=O; (ii) y’=yJlOO; (iii) y’=2y,/lOO; (iv) y’=5yb/100. (Thetracesaf the lower states dephasing. yeXt=0, -yb=yC=108, y,,=O, (iii) and (iv) are in this scale superimposed on the horizontal axis.)
J.T. Manassah,
610
B. Gross /Optics
Communications
has a negative imaginary part. For large detuning ( EB Tat), the time for which that statement holds must satisfy the condition Q >> 11 reXf.The basis vectors that span the system’s state, following the passage of the probe and assuming that the detuning is much larger than the pump Rabi frequency, are given by:
To understand the above observations, we first analyze for a specific Yinhrthe contributions to the source term of the probe’s field from different components of the inhomogeneous distribution. In Fig. 6, we plot, at Z= 0, Im( Prrb) for different values of Z= C//.yih (the Maxwellians’s weighing factor for the velocity averaging is expressed as exp( - ?/2) in this variable). The Re(p,,) is not considered since it is odd in E”and thus averages to zero. We note that as E”increases, the position of the first maximum is further delayed. Furthermore, the period of oscillations increases. This behaviour can be physically explained by examining the quantum state of the atom-pump system in the presence of a small probe field. From the solution of the Schrodinger’s equations (valid here) it can be shown, through slowly-varying time-development analysis, that if the system was originally in the state where all the population is in the b state, and assuming that the probe field is well-behaved, then the system, following the passage of the probe is in a state that is orthogonal to the eigenstate of the Hamiltonian whose eigenvalue
-._
124 (1996) 602-615
where we have used the notation that in the column, the elements refer respectively to the b, a, c states of the free atom. For time T> 70, where 7. is a retarded time that is after the passage of the probe, the state of the system can be written as: l~W~))=a~(~~)
& exp[i~(~-70)l
+u~(T~) f?Zexp[i(E+2&JC)(~-~0)]
(18)
where a, ( 70) and a2( TV) are the wavefunction’s components at 7. and the factors multiplying the difference
i
--___
--_ -._--_
F ab
_.
‘.
..\
-- -_ --_
‘.., \ \ \
.
0.15-
‘.
‘. \
\
z-.-y it ----__
\\
\
0.1 -
\
\
\
iii ‘,
I \ \ I \ \ \ i
0.05-
0’
0
I
I
I
I
0.1
0.2
0.3
0.4
0.5
I
I
I
0.6
0.7
0.8
0.9
\
1
z Fig. 8. The integrated Fig. 7.
energy flux for the probe is plotted as function of the normalized
distance of propagation
for the same parameters
as in
J. T. h4unassnh, B. Gross /Optics
Communications
of time in the exponentials are the approximate eigenvalues corresponding to the respective eigenvectors. From the above, we can deduce the b, a, c amplitudes at T. These are respectively:
u,(7) = -+T, oc
.
(19)
This leads to the conclusion that only the slow oscillation mode amplitude of pnb= a,*ab is nonzero for large detuning. The frequency of this mode is inversely proportional to the detuning, if the detuning is large. Consequently, for large detuning the period of oscillations of the probe’s field increases as the detuning increases. It should be noted that the slow and fast oscillation modes are a result of respectively subtracting or adding the system’s eigenvalues. This situation
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124 (1996) 602415
would not arise in a two-level system, where the oscillations period would decrease with the detuning. The behaviour of the probe’s field with increasing values for j$,,, can now be directly deduced. If pi,,,, increases, the unnormalized value of the detuning corresponding to a given E”increases, while its weighting factor remains the same. As a consequence, we deduce that for (pab): (i) the spread for the separate velocity components in the position of the first maximum increases with an increase in yinh, thus leading to a broader first maximum, and (ii) the period of oscillations in the amplitude of (Im(pab)) increases with an increase in vinh. We remark here that the details (not shown here) of the fields time-profile are qualitatively unchanged if instead of having the above conditions on the Ts, we have everything the same but now j$,= r,# 0 and r,,, = 0.
Pbb
0.5
I
I
I
I
,
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/
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0.2
0.3
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0.5
0.6
0.7
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0.9
1
0
0.1
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1
Pbb
- - -._ _ _
Fig. 9. The b state population
0.7
0.8
is plotted, at the input (a) and exit (b) planes, as a function of the retarded time for the same parameters
as Fig.
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In Fig. 7, we plot at the output plane the time development of the fields for different values of r’, including the value zero discussed in (ii) above. In Fig. 8, we plot the energy flux of the probe field as function of the normalized propagation distance for the same parameters as in Fig. 7. The increased absorption of both fields with the value of 7’ can be understood by noting that for 7’ = 0, the pump, before the arrival of the probe, will only excite the system’s absorptionless eigenstate leaving all the initial atomic population in the b state. The condition that only the b state is occupied before the on-set of the probe is a necessary condition for adiabaton pair formation. As the dephasing mechanism is added, pbb is not equal to 1 as the probe enters the sample, and the system is in a superposition state with the population out of the b state increasing with an increase in the value of 7’. In Fig. 9, we show, at the input and output planes the time-development of pbbto illustrate this loss of population at times earlier than the probe’s arrival, we observe further that the deviation of pbbfrom 1 is larger for larger values of 7’. From further analysis of this case, we also note that:
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-The velocity of the probe decreases as 7 increases. This is a consequence of the general result in Ref. [ 171, where it was shown that the velocity of the probe decreases as the absorption increases. - The magnitude of the lower states maximum coherence’s value remains close to l/2 as long as the system remains in the nonlinear regime. See Fig. 10. That the coherence remains unaltered while both the pump and probe are absorbed is understood by noting that the degree of coherence depends only on the ratio of the probe and pump fields. -The cw plateau of the pump experiences absorption since the steady state of Im(p,,) is positive when 7’ # 0. The absorption increases as 7’ increases. -As was pointed out by Grobe and Eberly, the amplitude of the probe in the adiabaton pair is directly proportional to the pump’s plateau value. Therefore, as the amplitude of the plateau decreases so will the amplitude of the probe. - For sufficiently large $?, the absorption causes the fields to weaken until the atom-fields systems is linear. The onset of the linear regime occurs when the depend-
cA-0.05
-0.1
7
t
1
-0.15 t -0.2 $ CL
-0.25 -0.3 -0.35 -0.4 i -0.45 -0.5 0’
0.1
0.2
0.3
0.4
0.5 t’
Fig. 10. The lower states coherence
0.6
0.7
0.8
0.9 x 10
-6
is plotted, at the exit plane, as function of the retarded time for the same parameters
as Fig. 7.
J.T. Manassah, B. Gross /Optics
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b
I
0.005-
‘_
1
j
-
-
-
--
-
-.-
-
I, 11 ;:
0
iv 0.5
0
1
1.5 x 10-B
t'
0.02 d
x 1o-3
8, 7
e
Icp acl 0.01
0.005
0 iJ'J 0
0.5
1
I 1.5
Fig. 11.The pump and probe fields are plotted, at the exit plane, as function of the retarded times both y’ and -yilulare scanned. Y_~=O, ly~,=y~=lO’. (a,b) j&,=20$. (c,d) j&,=33Tb, (e,f) yj,=50j&, (g,h) j&,=100$,, (i) y’=O, (ii) y’=yb/lOO,(iii) y’=2yJlOO,(iv) y’=5y,,llOO.
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ence of the energy flux on the distance of propagation follows Beer’s law. - In the linear regime, the lower levels coherence is zero and the atomic population is equally divided between the b and c levels for large time.
This is the most common configuration in a gaseous atomic A-medium. It is the most general case and as such, different competing effects discussed earlier are simultaneously at play, namely: (a) As 7 is increased, the pump plateau amplitude is decreased. This in turn leads to a decrease in the probe’s amplitude. (b) 7 # 0 leads to a redistribution of population in the different states of the system prior to the arrival of the probe and thus the probe evolves into a superposition of eigenstates not all absorptionless. (c) As rinh increases while maintaining r’ fixed, the absorption coefficient for the pump is decreased. (d) As j&, increases, the position of the maximum of different velocity components in the probe’s source is smeared thus leading to a broadening in the maximum of the probe. Also the period of the oscillations of the probe’s field is increased. Depending on the relative values of the different parameters, we expect that different physical mechanisms dominate for the different cases. The problem is further complicated since the different effects depend differently on the distance of propagation. We shall consider different regimes to illustrate the different cases. To illustrate, we plot at the output plane in Figs. 1 I the fields for different 7’ as we scan the values of Yinh.
We note the following: - As can be deduced from Eq. ( lo), the magnitude of the pump plateau decreases with an increase of 7’ irrespective of the value of Tint,. Related to this, we find that, for larger r’, there is a steeper decrease of the pump’s energy flux as function of the distance of propagation. - As can also be deduced from Eq. ( lo), the pump absorption for a particular 7’ can be substantially reduced as -&, is increased. - For low values of Tinhr the magnitude of the probe at the exit plane decreases as 7 increases, while for high values of Yinh,the opposite occurs. Related to this,
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we also find that the rate of decrease of the probe’s energy flux, as function of the distance of propagation, increases with larger 7’ for small yi*, while the opposite occurs for large yi*. - The velocity of the probe depends on the value of 7’. For very low -&, this velocity decreases with an increase in the value of 7. On the other hand, for higher j&, the velocity increases with larger $7. In this later regime, for the same y’, the velocity increases with an increase in the value of yi*. - The observed reversal in the ordering of the magnitude of the probe and in the velocity of its peak, as indexed by the value of 7, do not occur for the same value of Yinh.
6. Conclusion Through extensive numerical simulations, we showed that the effects of the decay and dephasing rates on the formation of an adiabaton pair in an atomic hconfiguration medium can be very substantial. As the values of different parameters are scanned, we observe distinct physical regimes for the pump-probe dynamics. It is interesting to note that for typical gas parameters, the large value of Yinh,as compared to the internal decay constant, permits a larger flexibility in the values of r’, as compared to the fixed atoms case, that allow the formation of a ‘ ‘quasi-adiabaton’ ’ pair structure. However, it should be noted that the main result of this paper is that in most physical regimes of interest, where dephasing and inhomogeneous broadening can not be neglected, the adiabaton pair does not exhibit the same degree of stability usually associated with a soliton-like structure. This should, for example, be contrasted to a Raman soliton [ 181 “which is a robust manifestation of quantum noise, that is resilient to detuning, diffraction and phase matching” [ 191.
References []
0. Kocharovska and Ya.1. Khanin, Pis’ma Zh. Eksp. Teor. Fiz. 48 (1988) 581. [JETP Lett. 48 (1988) 6301; 0. Kocharovskaya and P. Mandel, Phys. Rev. A 42 ( 1990) 523. [2] S. Harris, Phys. Rev. Lett. 62 (1989) 1022; A. Imamoglu and S. Harris, Optics Lett. 14 ( 1989) 1344.
.I. T. Manassah,
B. Gross/Optics
[3] M. Scully, S. Zhu and A. Gavrieliedes, Phys. Rev. Lett. 62 (1989) 2813. [4] A. Lyras, X. Tang, P. Lambropoulos and J. Zhang, Phys. Rev. A40 (1989) 4131; S. Basile and P. Lambropoulos, Optics Comm. 78 (1990) 163. [5] M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. ]6] Atomic Coherence and Interference, special issue of Quantum Optics 6 ( 1994) 203-389. ]7] S.E. Harris, Phys. Rev. Lett. 70 (1993) 552. [S] S.E. Harris, Phys. Rev. Lett. 72 (1994) 52. ]9] J.H. Eberly, M.L. Pons and H.R. Haq, Phys. Rev. Lett. 72 (1994) 56. [ lo] R. Grobe,F.T. Hioe and J.H. Eberly, Phys. Rev. Lett. 73 ( 1994) 3183. [ 111 R. Grobe and J.H. Eberly, Laser Phys. 5 (1995) 542. [ 121 A. Kasapi, M. Jain, G.Y. Yin and S.E. Harris, Phys. Rev. L&t. 74 (1995) 2447. [ 131 R.H. Dicke, Phys. Rev. 93 (1954) 99; R. Friedberg and S. Hartmann, Phys. Rev. A 13 (1976) 495.
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[ 141 J.T. Manassah and B. Gross, Optics Comm. 113 (1994) 213, and references to authors work cited therein. [15] A.S. Manka, M. Scalora, J.P. Dowling and C.M. Bowden, Optics Comm. 115 ( 1995) 283, and references to authors work cited therein. [ 161 S.L. McCall and E.L. Hahn, Phys. Rev. Lett. 18 (1967) 908. [ 171 S.E. Harris, J.E. Field and A. Kasapi, Phys. Rev. A 46 (1992) R29. [ 181 F.Y.F. Chu and C. Scott, Phys. Rev. A 12 (1975) 2060; N. Tan-no, T. Shiiahata and K. Yokoto, Phys. Rev. A 12 (1975) 159; K.J. Druhl, J.L. Carlsten and R.G. Wenzel, Phys. Rev. Lett. 5 1 (1983) 1171; J. Stat. Phys. 39 (1985) 615,621; J.C.Englund and C.M. Bowden, Phys. Rev. A42 ( 1990) 2870; 46 (1992) 578; M. Scalora, S. Singh and CM. Bowden, Phys. Rev. Len. 70 (1993) 1248. [ 191 Anonymous reviewer.