a r X i v :q u a n t -p h /0609206v 1 27 S e p 2006
Light propagation through closed-loop atomic media
beyond the multiphoton resonance condition
Mohammad Mahmoudi 1,2,?and J¨o rg Evers 1,?
1
Max-Planck-Institut f¨u r Kernphysik,Saupfercheckweg 1,D-69117Heidelberg,Germany
2
Physics Department,Zanjan University,P.O.Box 45195-313,Zanjan,Iran
(Dated:February 1,2008)
The light propagation of a probe ?eld pulse in a four-level double-lambda type system driven by laser ?elds that form a closed interaction loop is studied.Due to the ?nite frequency width of the probe pulse,a time-independent analysis relying on the multiphoton resonance assumption is insu?cient.Thus we apply a Floquet decomposition of the equations of motion to solve the time-dependent problem beyond the multiphoton resonance condition.We ?nd that the various Floquet components can be interpreted in terms of di?erent scattering processes,and that the medium response oscillating in phase with the probe ?eld in general is not phase-dependent.The phase dependence arises from a scattering of the coupling ?elds into the probe ?eld mode at a frequency which in general di?ers from the probe ?eld frequency.We thus conclude that in particular for short pulses with a large frequency width,inducing a closed loop interaction contour may not be advantageous,since otherwise the phase-dependent medium response may lead to a distortion of the pulse shape.Finally,using our time-dependent analysis,we demonstrate that both the closed-loop and the non-closed loop con?guration allow for sub-and superluminal light propagation with small absorption or even gain.Further,we identify one of the coupling ?eld Rabi frequencies as a control parameter that allows to conveniently switch between sub-and superluminal light propagation.
PACS numbers:42.50.Gy,42.65.Sf,42.65.An,32.80.Wr
I.INTRODUCTION
The propagation of a light pulse through a dispersive medium has been extensively investigated [1,2,3,4].Re-cently the amplitude and the phase control of the group velocity in a transparent media have attracted much at-tention.It is well known that the propagation velocity of a light pulse can be slowed down [5,6],can become greater than the light propagation velocity in vacuum,or even negative in transparent media [7,8].Superlu-minal light propagation is a phenomenon in which the group velocity of an optical pulse in a dispersive medium is greater than of the light in vacuum.Despite the name superluminal light,it is generally believed that no infor-mation can be sent faster than light speed c in vacuum as explained by Chiao [9].Thus,a group velocity faster than c as reported here does not violate Einstein principle of special relativity.The superluminal light propagation has been investigated for many potential uses,not only as a tool for studying a very peculiar state of matter,but also for developing quantum computers,high speed optical switches and communication systems [10].There have been only few experimental and theoretical studies which realized both superluminal and subluminal light propagation in a single system.Talukder et al.have shown femtosecond laser pulse propagation from super-luminal to subluminal velocities in an absorbing dye by changing the dye concentration [11].Shimizu et al.were
2
3 II.THEORETICAL ANALYSIS
A.The Model
We consider a four-level atomic systems in a double-
Λcon?guration as depicted in Figure1.The scheme
consists of two metastable lower states|1 ,|2 and two
excited states|3 ,|4 .The electric-dipole allowed tran-
sitions|1 ?|3 ,|2 ?|3 and|2 ?|4 are driven by
three coherent laser?elds,while a weak tunable coherent
probe?eld with central frequencyωp=ω41is applied
to the dipole-allowed transition|1 ?|4 .The sponta-
neous decay rates from level|i (i∈{3,4})to the levels
|j (j∈{1,2})are denoted by2γji.The electromagnetic
driving?elds applied to transition|i ?|j can be written
as
E ij=E ij?e ij e?i(ωij t? k ij r+φij)+c.c.,(1)
with amplitude E ij,unit polarization vectors?e ij,fre-
quenciesωij,wave vectors k ij and absolute phaseφij.
Note that we do not assume absolute phase control in
our calculations,but only relative phase control between
the di?erent driving?elds.
The Hamiltonian in dipole and rotating wave approx-
imation is given by[31,32]
H=
4
j=1
E j|j j|
?
4
l=3
2
m=1
g lm e?iαlm|l m|+h.c. ,(2)
where the corresponding Rabi frequencies are denoted by g ij=E ij(?e ij d ij)/ with d ij as the atomic dipole moment of the corresponding transition.The energies of the in-volved states are denoted E i(i∈{1,...,4}),and we de?ne the transition frequenciesˉωij=(E i?E j)/ .The arguments of the exponential functions are given by
αij=ωij t? k ij r+φij.(3) In a suitable reference frame,the Hamiltonian can be written as
V= (?32??31)?ρ22? ?31?ρ33
+ (?32??31??42)?ρ44
? (g31?ρ31+g32?ρ32+g42?ρ42
+g41?ρ41e?iΦ+h.c. .(4) Here,we have de?nedρij=|i j|and the corresponding operator in the new reference frame is denoted by?ρij (i,j∈{1,...,4}).It is interesting to note that now the residual time dependence along with the laser?eld phases appears only together with the probe?eld Rabi frequency g41in the parameterΦgiven by
Φ=?t? K r+φ0,(5a)
?=(?32+?41)?(?31+?42),(5b)
K=( k
32
+ k41)?( k31+ k42),(5c)φ0=(φ32+φ41)?(φ31+φ42).(5d) The parameters?, K andφ0are known as the multi-photon resonance detuning,wave vector mismatch and initial phase di?erence,respectively.Note that in gen-eral it is not possible to?nd a reference frame where the explicit time dependence due to?vanishes from the Hamiltonian,such that for?=0no stationary long-time limit can be https://www.doczj.com/doc/295339766.html,ing the notations Eqs.(5)and Eq.(3),the transformation of the operatorρ41,which will be of particular interest later on,to the chosen in-teraction picture can be written as
?ρ41=e?iΦe iα41ρ41.(6) We further de?ne?ρ41as the coherenceρ41in a reference frame oscillating in phase with the probe?eld E41as
?ρ41=e iα41ρ41=e iΦ?ρ41.(7) From the Hamiltonian Eq.(4),and including the spon-taneous decay in Born-Markov approximation,the den-sity matrix equations of motion can be derived as ?
?t
?ρ22=ig?32?ρ32?ig32?ρ23+ig?42?ρ42
?ig42?ρ24+2γ23?ρ33+2γ24?ρ44,(8b)?
?t
?ρ12=i(?32??31)?ρ12+ig?31?ρ32?ig32?ρ13
+ig?41?ρ42e iΦ?ig42?ρ14?Γ12?ρ12,(8d)?
?t
?ρ14=i(?32??31??42)?ρ14+ig?41e iΦ(?ρ44??ρ11)?ig?42?ρ12+ig?31?ρ34?Γ14?ρ14,(8f)?
?t
?ρ24=?i?42?ρ24+ig?42(?ρ44??ρ22)?ig?41?ρ21e iΦ+ig?32?ρ34?Γ24?ρ24,(8h)?
4?ρ44=1??ρ11??ρ22??ρ33.(8j)
We have further de?nedγj=γ1j+γ2j,and our chosen
level scheme impliesγ1=γ2=0.Γij=(2γi+2γj)/2
(i∈{1,2}and j∈{3,4})are the damping rates of the
coherences on transitions|i ?|j .
Due to the explicit time dependence of Eqs.(8)via
the parameter?inΦ,it is clear that in general the
system does not have a constant steady state solution.If,
however,the so-called multiphoton resonance condition
?=0, K=0is ful?lled,then the equations of motion(8)
have constant coe?cients and thus a stationary solution
in the long-time limit can be found,which for a suitable
choice of parameters depends on the constant relative
phaseφ0.
B.Linear susceptibility and group velocity
The linear susceptibility of the weak probe?eld can be written as[31,32]
χ(ωp)=
2Nd41
1+2πχ′(ωp)+2π?
2D
+
iγ3(|g31|2+|g32|2+|g42|2)|g32|2g41e?iφ0
2D2
,(13) where
D=|g31|2·|g42|2+γ2 |g31|2+|g32|2+|g42|2 .(14) Transferring to the frame rotating in phase with the probe?eld using Eqs.(6)and(7),we obtain
?ρ41=
?iγg?32g31g42e iφ0
2D2
?
iγ|g31|2|g32|2|g42|2g?41e2iφ0
FIG.2:(Color online)Interpretation of the di?erent contributions to the probe?eld susceptibility in terms of transition pathways.(a)represents the interaction loop leading to a scattering of the driving?elds into the probe?eld mode.(b)is the direct scattering of the probe?eld o?of the probe transition.(c)shows a counter-rotating term.The solid red arrows indicate coupling?eld transitions,the dashed blue line is a probe?eld interaction.
via di?erent laser?elds is involved,this contribution does
not depend on the relative
phase
φ
.The last contribu-
tion in Eq.(15)proportional to g?14depends on twice the
relative?eld phaseφ0can be interpreted as a counter-
rotating contribution.This preliminary interpretation of
the individual contributions will become more transpar-
ent in the time-dependent analysis without the assump-
tion of multiphoton resonance in the following Sec.II D.
D.Beyond the multi-photon resonance
approximation
We now drop the multi-photon resonance condition
and evaluate the equations of motion Eqs.(8)for the
general case of time-dependent coe?cients.The Eqs.(8)
can be written in a compact form
?
?t
?R+Σ
+ˉg41e?i?tΣ+1+ˉg?41e i?tΣ?1
=(M0+ˉg41e?i?t M1+ˉg?41e i?t M?1)?R.(21)
It is important to note that the time dependence of
Eqs.(8)only arises from the parameter?,such that the
coe?cients are periodic in time.According to Floquet’s
theorem[33],the solution?R therefore has only contri-
butions oscillating at harmonics of the detuning?.The
higher-order harmonics in this frequency expansion are
suppressed by powers of the probe?eld Rabi frequency
g41relative to the other frequencies involved in the sys-
tem.Since we are interested in the case where the probe
?eld is weak,we truncate the Floquet expansion after the
leading?rst order and obtain as ansatz for the solution
?R:
?R=?R
+ˉg41e?i?t?R1+ˉg?41e i?t?R?1.(22)
By using Eq.(22)in Eq.(21)and equating the coe?cients
oscillating at di?erent harmonics of?,we obtain the
solutions for?R0and?R±1as
?R
=M?1
Σ0,(23a)
?R
1
=(M0+i?)?1(Σ1?M1?R0),(23b)
?R
?1
=(M0?i?)?1(Σ?1?M?1?R0).(23c)
The medium response is determined by the13th com-
ponent?ρ41of?R.Transferring back?ρ41to the operator
?ρ41in the frame oscillating in phase with the probe?eld
using Eqs.(6)and(7),we obtain
?ρ41=[?R0]13e iΦ+g41[?R1]13+g?41[?R?1]13e2iΦ.(24)
Here,[x]13denotes the13th component of x.It can be
seen that the contribution proportional to[?R1]13oscil-
lates in phase with the probe beam,and thus contributes
to the probe beam susceptibility independent of the fre-
quency of the incident driving?eld.For?=0,the two
other contributions proportional to[?R0]13and[?R?1]13
oscillate at di?erent frequencies due to the time depen-
dence ofΦand thus do not contribute to the probe beam
susceptibility.However,even in the case?→0,the term [?R1]13is distinct in that the two other contributions in general propagate di?erently due to the wave vector mis-match K.
Using the result Eq.(24),we are now also in the po-sition to in detail understand the expression Eq.(15)we previously derived assuming?=0and K=0.For this, we explicitly evaluate[?R0]13,[?R1]13and[?R?1]13using the parameters Eq.(12)as well as?=0and obtain [?R0]13e iΦ=
?iγg?32g31g42
2D2
, (25b)
g?41[?R?1]13e2iΦ=
?iγ(|g31|2|g32|2|g42|2)g?41
-4
-2
2
4
-4
-2024-4
-2
2
4
-0.4
-0.200.20.40.60.8
8
-4
-2
2
4
-0.6
-0.4-0.200.20.40.6FIG.6:(Color online)Real (blue solid line)and imaginary (red dashed line)parts of the contribution to the Floquet de-composition representing the direct scattering of the probe
?eld o?of the probe transition,g 41[?R
1]13[see Eq.(24)].The susceptibility is plotted against the probe ?eld detun-ing ?41.This corresponds to a calculation relevant for the evaluation of the group velocity,but violates the multipho-ton resonance condition and thus requires a Floquet anal-ysis.The parameters in (a)are ?31=?32=?42=0,2γ13=2γ14=2γ23=2γ24=γ0,g 31=1.8γ0,g 32=0.2γ0,g 42=0.5γ0,and g 41=0.01γ0.Sub?gure (b)shows the case ?31=10γ0,?32=?42=0,2γ13=2γ14=2γ23=2γ24=γ0,g 31=g 32=0.1γ0,g 42=0.5γ0,and g 41=0.01γ0.
out the pulse propagation.Therefore,the propagation of a pulse cannot be described assuming the multiphoton resonance condition,and our numerical results Fig.3-5presented so far are not su?cient to evaluate the group velocity.
Therefore,in Fig.6(a),we show the probe ?eld sus-ceptibility against the probe ?eld detuning ?41,while the coupling ?eld detunings are ?31=?32=?42=0.This corresponds to a calculation relevant for the eval-uation of the group velocity,but violates the multipho-ton resonance condition ?=0for most values of ?41shown in the ?gure and thus requires the use of the time-dependent Floquet analysis.Consequently,in Fig.6,we
only show the component [?R
1]13since this part of the Floquet decomposition is the only one which oscillates in phase with the probe ?eld under these conditions.It can be seen in Fig.6(a)that around ?41=±2γ0,the real part of the susceptibility has positive slope,while the imaginary part is strongly negative.This indicates sub-luminal light propagation with gain.At about ?41=0,the real part of the susceptibility has negative slope over a wide frequency range,together with a small positive
-4
-2
2
4
-0.2
-0.100.10.2FIG.7:(Color online)Real (a)and imaginary (b)parts of the scaled contribution to the Floquet decomposition repre-senting the direct scattering of the probe ?eld o?of the probe
transition,[?R
1]13[see Eq.(24)].The susceptibility is plotted against the probe ?eld detuning ?41with other detunings chosen as ?31=?42=0.One of the Rabi frequencies is chosen as zero,g 32=0,such that the setup corresponds to the one shown in Fig 1(b).The solid blue line is for g 31=0.7γ0,the red dashed line for g 31=0.85γ0,and the green dash-dotted line for g 31=1.5γ0.The other parame-ters are 2γ13=2γ14=2γ23=2γ24=γ0,g 42=0.2γ0,and g 41=0.01γ0.
or even negative imaginary part.This corresponds to superluminal propagation with small absorption or gain.If,however,the parameters and the laser ?eld align-ments are such that for a particular value of ?41within the resonance structure in the susceptibility the multi-photon resonance condition is ful?lled,then the coupling ?elds may be scattered into the probe ?eld mode.This would give rise to a huge,very narrow contribution to the probe ?eld susceptibility [see Fig.3],which is likely to completely distort the probe pulse shape.In Fig.6(a),this could occur at ?41=0if the laser ?eld wavevectors are chosen suitably.Therefore for probe pulse propa-gation,parameters may be favorable which suppress the scattering of the coupling ?elds into the probe ?eld mode.But under such conditions,the system is not phase-dependent.
In Fig.6(b),we show a case where the multiphoton res-onance condition ?=0is violated for all shown values of ?41.This avoids the scattering of the coupling ?elds into the probe ?eld mode,and is accomplished by choos-ing the detunings as ?31=10γ0,and ?32=?42=0.Around ?41=0,subluminal light propagation with lit-tle absorption can be achieved.The fact that this and
9 similar result can be obtained for a rather large detuning
?31=100g31as compared to the corresponding Rabi
frequency then suggest that a closed-loop con?guration
of the laser?elds may not be necessary at all.
Therefore,in the?nal part,we discuss the dispersion
and absorption spectrum for the system without a closed
interaction loop shown in Fig.1(b).Fig.7shows again
the probe?eld susceptibility due to the direct scatter-
ing of the probe beam o?of the probe transitions,with Rabi frequency g32=0.Sub?gure(a)shows the real part,(b)the imaginary part.The solid blue line is for g31=0.7γ0,the red dashed line for g31=0.85γ0,and the green dash-dotted for g31=1.5γ0.All coupling ?eld detunings are zero,and the other parameters are 2γ13=2γ14=2γ23=2γ24=γ0,g42=0.2γ0,and g41=0.01γ0.If the Rabi frequency g31for transition |1 ?|3 is small,then the results are similar to the sim-ple three-level electromagnetically induced transparency (EIT)case.The slope of the dispersion is positive,and subluminal light propagation appears with an EIT dip in the absorption spectrum,see the solid blue line in Fig.7. With increasing Rabi frequency,the slope of the disper-sion around zero detuning becomes negative and super-luminal light propagation sets in.Together with super-luminal light propagation,zero absorption or even gain is achieved.Thus the intensity of the pump?elds can be used as a simple control parameter to switch the light propagation from subluminal to superluminal.
Fig.8shows another set of results for the system in Fig.1(b),with parameters?42=?5γ0,?31=0, g32=0,2γ13=2γ14=2γ23=2γ24=γ0,g42=0.6γ0, and g41=0.01γ0,and g31=0.7γ0for the solid blue line, g31=0.85γ0for the red dashed line,and g31=0.85γ0for the green dash-dotted line.Due to the di?erent coupling ?eld detuning?42,the interesting probe susceptibility structure is shifted to?41≈?5γ0.In this region,for g13=0.8γ,one?nds subluminal light propagation ac-companied by strong gain.Increasing the Rabi frequency g13,again the light propagation switches to superluminal, still together with gain.
IV.DISCUSSION AND SUMMARY
We have discussed the propagation of a probe pulse through a medium that is driven by coupling and probe laser?elds that form a closed interaction loop.This gives rise to a dependence of the system on the relative phase of the various laser?elds,but in general prohibits the existence of a time-independent steady state of the sys-tem.A stationary steady state only exists if the so-called multiphoton resonance condition?=0[see Eq.(5)]is ful?lled.A probe pulse,however,has a?nite frequency width.Therefore,a time-independent analysis is insu?-cient to correctly describe the pulse propagation through the medium.To solve this problem,we have solved the time-dependent system without assuming the multipho-ton resonance condition by means of a Floquet decom-
-10-50510 -0.2
-0.1
0.1
0.2
FIG.8:(Color online)As in Fig.7,but with parameters g31=0.7γ0for the solid blue line,g31=0.85γ0for the red dashed line,and g31=0.85γ0for the green dash-dotted line. The other parameters are?42=?5γ0,?31=0,g32=0, 2γ13=2γ14=2γ23=2γ24=γ0,g42=0.6γ0,and g41= 0.01γ0.The solid vertical line indicates?41=4γ0.
position of the equations of motion.We found that the di?erent Floquet components can be interpreted in terms of di?erent scattering processes.The phase dependence arises from a scattering of the coupling?elds into the probe?eld mode,and thus only occurs at a speci?c probe ?eld frequency.If this frequency is within the frequency width of the probe pulse,a strong distortion of the pulse shape can be expected.In some cases,it may be possible to?nd probe pulses that are su?ciently narrow in the frequency domain such that they?t into the frequency range where the scattering of the coupling?elds into the probe?eld mode dominates.Especially for shorter and thus broader pulses,however,we conclude that it may be advantageous to avoid the scattering of the coupling?elds into the probe?eld mode.Then,however,the system is no longer phase dependent,and a closed interaction loop may not be necessary at all.We expect that similar re-sults hold for other level schemes which involve a closed laser?eld interaction loop.
Apart from these general considerations,using our time-dependent Floquet analysis,we have shown that for realistic parameter sets both the closed-loop double-Λsystem and the corresponding system without a closed loop allow for sub-and superluminal light propagation with small absorption or even with gain.Further,we have identi?ed the Rabi frequency of one of the coupling ?elds as a convenient control parameter to switch the light propagation between sub-and superluminal light
10
propagation.
Acknowledgments
MM gratefully acknowledges support for this work from the German Science Foundation and from Zanjan University.He also thanks M.Sahrai for useful discus-sions.
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