We demonstrate a strong coherent backward wave oscillation using forward
propagating fields only. This is achieved by applying laser fields to an
ultra-dispersive medium with proper chosen detunings to excite a molecular
vibrational coherence that corresponds to a backward propagating wave. The
physics then has much in common with propagation of ultra-slow light.
Applications to coherent scattering and remote sensing are discussed

Rostovtsev, Yuri V.

Sariyanni, Zoe-Elizabeth

Scully, Marlan O.

English

arXiv

Texas A&M University

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Electromagnetically induced coherent backscattering
Yuri V. Rostovtsev1, Zoe-Elizabeth Sariyanni1, and Marlan O. Scully1,2,3
1Institute for Quantum Studies and Department of Physics, Texas A&M University, TX 77843
2Princeton Institute for the Science and Technology of Materials and Department
of Mechanical & Aerospace Engineering, Princeton University, NJ 08544
3Max-Planck-Institute fu¨r Quantenoptik, D-85748 Garching, Germany
(Dated: April 5, 2018)
We demonstrate a strong coherent backward wave oscillation using forward propagating fields
only. This is achieved by applying laser fields to an ultra-dispersive medium with proper chosen
detunings to excite a molecular vibrational coherence that corresponds to a backward propagating
wave. The physics then has much in common with propagation of ultra-slow light. Applications to
coherent scattering and remote sensing are discussed.
PACS numbers: 32.80.Qk, 42.65.Dr, 42.50.Hz
Quantum coherence [1, 2] has been shown to result
in many counter-intuitive phenomena. The scattering
via a gradient force in gases [3], the forward Brillouin
scattering in ultra-dispersive resonant media [4, 5], elec-
tromagnetically induced transparency [6, 7, 8, 9], slow
light [10, 11, 12, 13], Doppler broadening elimination [14],
light induced chirality in nonchiral medium [15], a new
class of entanglement amplifier [16] based on correlated
spontaneous emission lasers [17, 18] and the coherent Ra-
man scattering enhancement via maximal coherence in
atoms [19] and biomolecules [20, 21] are a few examples
that demonstrate the importance of quantum coherence.
In this Letter, we predict strong coherent backward
scattering via excitation of quantum coherence between
atomic or molecular levels. The developed approach can
also be used to control the direction of the signal gener-
ated in coherent Raman scattering and other four-wave
mixing (FWM) schemes.
Let us consider the four-wave mixing in a 3-level atomic
medium. The pump and Stokes fields E1 and E2 (whose
Rabi frequencies are defined as Ω1 = ℘1E1/h¯ and Ω2 =
℘2E2/h¯, where ℘1 and ℘2 are the dipole moments of the
corresponding transitions) with wave vectors k1 and k2
and angular frequencies ν1 and ν2 induce a coherence
grating in the medium (see Fig. 1) given by [2]
ρcb ∼ −Ω1Ω
∗
2 (1)
Let us stress that the ρcb coherence grating has an
exp[i(k1 − k2)z] spatial dependence. In an ultra-
dispersive medium (see Fig. 2) where fields propagate
with a slow group velocity, the two co-propagating fields
have wavevectors given by
k1 ≃ k1(ωab) +
∂k1
∂ν1
(ν1 − ωab) = ωab/c+ (ν1 − ωab)/Vg,
(2)
where Vg is the group velocity of the first wave, ωab is the
frequency of transition between levels a and b, and k2 =
ν2/c. Thus these two fields create a coherence grating
Ω1
a
2
d
c
b
Ω
3Ω
Ω
4
k1
k2
k3
k4
(a)
k1
k2
k4
k3
(b)
FIG. 1: (a) Co-propagating fields 1 and 2 induce coherent
grading inside the medium. The field 3 propagating in the
same direction will be scattered in the opposite direction be-
cause the coherence excited by fields 1 and 2 is propagating
in the opposite direction (see Fig.2). Level scheme, double-Λ
(b), for implementation of coherent back scattering.
in the medium with spatial phase determined by k1 −
k2 = ωcb/c + (ν1 − ωab)/Vg which depends strongly on
the detuning δ = ν1 − ωab. By properly choosing the
detuning, δ, one can make k1 − k2 negative.
After the coherence ρbc is induced in the medium, a
probe field E3, with Rabi frequency Ω3 = ℘3E3/h¯ and
2ν2k   =2 __c
ω
ab
__
c
k1
ω
ab
__
c
_____
Vg
ν − ω
ab+k =
ν
k
ωab
δ
ν ν12
FIG. 2: Dispersion k(ν) of ultra-dispersive medium. Choos-
ing δ = ν1 −ωab = −Vgωcb/c, we can have k1 − k2 < 0 even if
ν1 > ν2, thus the third field can be scattered opposite to the
direction of propagation of the first two fields.
wave vector k3, scatters off that coherence to produce the
signal field, Ω4. The signal field depends on the coherence
and the input fields as
∂
∂z
Ω4 ∼ ρcbΩ3 ∼ Ω1Ω
∗
2Ω3 ∼ e
i(k1−k2+k3−k4)z (3)
That is, the propagation direction of Ω4 depends on the
spatial phase of the ρbc coherence through the phase-
matching condition k4 = k1 − k2 + k3 [22] while its fre-
quency is determined by ν4 = ν1 − ν2 + ν3.
We here show that for dispersive media one can obtain
a strong signal in the backward direction even when all
three input fields propagate forward. This is contrary to
the usual non-dispersive media results, where the phase-
matching in the backward direction cannot be achieved
for E1, E2, and E3 counter-propagating with respect to
E4 [22].
To demonstrate this result, we write the interaction
Hamiltonian of the system as
VI = −h¯[Ω2e
−iωact|a〉〈c|+Ω1e
−iωabt|a〉〈b|+ h.c.] (4)
−h¯[Ω3e
−iωdbt|d〉〈b|+Ω4e
−iωdct|d〉〈c|+ h.c.] (5)
where Ω4 = ℘4E4/h¯ is the Rabi frequency of the signal
field and ωab, ωac, ωdb, ωdc are the frequency differences
between the corresponding atomic or molecular energy
levels. The time-dependent density matrix equations are
given by
∂ρ
∂τ
= −
i
h¯
[VI , ρ]−
1
2
(Γρ+ ρΓ), (6)
where Γ is the relaxation matrix. A self-consistent system
also includes the field propagation equations
∂Ω1
∂z
= −iη1ρab,
∂Ω2
∂z
= −iη2ρac, (7)
∂Ω3
∂z
= −iη3ρdb,
∂Ω4
∂z
= +iη4ρdc, (8)
where
ηj = νjN℘j/(2ǫ0c) (9)
are the coupling constants (j = 1, 2, 3, 4), N is the parti-
cle density of the medium, ǫ0 the permitivity in vacuum.
The equations of motion for the density matrix ele-
ments of the polarization ρab and the coherence ρcb are
given by
ρ˙ab = −Γabρab + iΩ1(ρaa − ρbb)− iρcbΩ
∗
2, (10)
ρ˙cb = −Γcbρcb + iρcaΩ1 − iρabΩ2. (11)
where Γab = γab + i(ωab − ν1); Γca = γca − i(ωac − ν2);
Γcb = γcb + i(ωcb − ν1 + ν2); ωcb is the frequency of
c − b transition, and γαβ are the relaxation rates at the
corresponding atomic transitions. In the steady state
regime, and assuming that |Ω2| >> |Ω1|, almost all of
the population remains in the ground level |b〉, ρbb ≃ 1.
Let us consider the fields as plane waves: Ω1(z, t) =
Ω˜1(z, t) exp(ik1z), Ω2(z, t) = Ω˜2(z, t) exp(ik2z), where
Ω˜1(z, t) and Ω˜2(z, t) are the slowly varying in envelopes of
the fields Ω1 and Ω2 in space, while k1 = ν1[1+χab(ν1)]/c
and k2 = ν2[1+χac(ν2)]/c. The succeptebilities are χab =
η1ρab/Ω1 = (ν1 − ωab)/2πVg and χac = η2ρac/Ω2 ≃ 0.
By solving the self-consistent system of Maxwell’s equa-
tions (7,8) and the density matrix equations (10,11), we
obtain Eq. (2) for the wavevectors, where Vg ≃ c/η1|Ω2|
2
is the group velocity of the optical field Ω1. Thus, the
spatial dependence of ρcb is determined by
∆k = k1 − k2 =
ν1 − ν2
c
+
ν1 − ωab
Vg
. (12)
The signal field Ω4 is generated by the polarization ρdb of
the transition it couples (Eq. 8). The equation of motion
for this polarization element reads
ρ˙db = −Γdbρdb + iΩ4(ρdd − ρbb)− iρcbΩ
∗
3, (13)
where Γdb = γdb + i(ωdb − ν4), and ν4 is the frequency
of generated field. In the steady-state regime and for
|Ω4| << |Ω3|, the field Ω4 at the output of the cell is
given by
Ω4 =
∫ L
0
dzei(k4−∆k−k3)z
η4ρcbΩ˜
∗
3
Γdb
(14)
where L is the length of the cell. Note here that Eq.(14)
is valid if the field |Ω3| does not change coherence ρcb via
power broadening which is true if |Ω3|
2 ≪ |Ω1|
2 + |Ω2|
2.
Hence, after integrating Eq.(14), we obtain for the
scattered field Ω4
Ω4 =
[
sin(k4 − k3 −∆k)L
(k4 − k3 −∆k)L
]
η4LΩ3
Γdb
(15)
3where the expression in the brackets describes the phase-
matching and determines the direction in which the sig-
nal field is generated.
The most interesting effect following from Eq.(15)
is the coherent backscattering. Indeed, even when all
three input fields propagate forward, one may observe a
backscattered signal field by satisfying the condition
k3 +∆k = k3 +
ν1 − ωab
Vg
< 0, (16)
and for appropriate detuning, ν1−ωab < 0, this inequality
can be meet. That is, in order to obtain phase-matching
in the backward direction, we have to satisfy
k3 +∆k < −k4, (17)
which can be rewritten as
−
ν1 − ωab
Vg
> k3 + k4. (18)
Hence, in order to demonstrate the effect, the detuning
δ should meet the condition
δ = −(k3 + k4)Vg ≃ −2k4Vg. (19)
It is useful to rewrite the condition in terms of succepti-
bility for the probe field, indeed,
k1 =
ν1
c
n1 ≃
ν1
c
(1 + c
ν1 − ωab
ν1Vg
) =
ν1
c
(1 − c
2k4
ν1
), (20)
then,
χab = 2(n1 − 1) = −4
λab
λdb
, (21)
for gases χab ≪ 1, so λab ≪ λdb, i.e. the effect can be
implemented for scattering of IR fields. Then, for the
Doppler broadened EIT media as shown in [23, 24], we
can write
χab(δ) ≃
3λ3abN
8π2
(
γrδ
|Ω2|2
+ i
γr∆Dδ
2
|Ω2|4
)
, (22)
where ∆D is the Doppler width; γr is the radiative decay
rate. Thus, for detuning smaller than the EIT width
|δ| ≤ |Ω2|
2/
√
γr∆D, absorption can be neglected, and
3λ2abNγrδ
16π|Ω|2
= −2k4, (23)
then, the atomic or molecular density is given by
N =
32πk4
3λ2ab
|Ω2|
2
γr|δ|
≃
32πk4
3λ2ab
√
∆D
γr
. (24)
There are several schemes to demonstrate the effect.
For example, the double-Lambda scheme can be imple-
mented in molecular rotational levels (see Fig. 1a). More-
over, the effect can be implemented in the ladder-Λ using
molecular vibrational levels (see Fig. 3a). The phase-
matching condition should be slightly modified for the
scheme as k4 = k1 − k2 − k3. Also, the phenomenon can
be demonstrated in a V-Λ scheme that can be realized
in atomic levels (see Fig. 3b, for Rb atoms, b = 5S1/2,
c = 7D3/2,5/2, a = 5P1/2,3/2, d = 8P1/2,3/2), and phase-
matching condition has a form k4 = k1 + k2 − k3. Let
us note that the requirement for detuning in all cases is
δ/Vg = −2k4 and the use of Eq. (24) to estimate molec-
ular or atomic density is still valid.
Examples of systems to this effect could be seen we
mention molecules NO (a resonant transition at 236 nm,
A2Σ+−X2Π), NO2 (a resonant transition at wavelength
337 nm), and atomic Rb vapor (EIT and CPT have
been recently demonstrated for molecules, see [26]). The
required molecular density of NO and NO2 molecules
is N ≃ 1.2 · 1013 cm−3 if one can use transition be-
tween rotational levels ≃ 10 cm−1. Using vibrational IR
transitions for NO (vibration frequency of 1900 cm−1)
at 5.26 µm and for NO2 (vibrational frequency of 750
cm−1) 13.3 µm, the densities are N = 8 · 1015 cm−3
and N = 1.4 · 1015 cm−3, correspondingly. For atomic
Rb vapor, wavelengths are λ1 = 780 nm, λ2 = 565 nm,
λ3 = 335 nm, λ4 = 23.4 µm, and the atomic density is
N = 1.4 · 1013 cm−3.
The intensity needed for EIT is determined by condi-
tion |Ω|2 ≫ γbc∆D which corresponds to laser intensity
of the order of 1 mW/cm2 for atoms and of the order
of 10 W/cm2 for molecules because the dipole moment
is two orders of magnitude smaller for molecules. These
conditions are realistic and well-suited for an experimen-
tal implementation.
Several applications of the effect can be envisioned, like
in nonlinear CARS microscopy [27], while the controlling
of coherent backscattering could provide a new tool for
creating an image. A variation in the molecular density
would modify the intensity of the signal in both the for-
ward and the backward direction. Additionally, Eq.(15)
also allows one to control the direction of the generated
signal field and thus provide an all-optical control when
scanning an optical field over an object.
In conclusion, we theoretically predict strong coher-
ent scattering in the backward direction while using only
forward propagating fields. This is achieved by exciting
atomic or molecular coherence by properly detuned fields,
in such a way that the resulting coherence has a spatial
phase corresponding to a backward, counter-propagating
wave. Applications of the technique to coherent scatter-
ing and remote sensing are discussed. The method holds
promise for observation induced scattering in a backward
direction with application to CARS microscopy.
We thank J. Giordmaine, K. Hakuta, N. Kroo´, G. Kur-
izki, K.K. Lehmann, R. Miles, H. Walther, and X.S. Xie
for useful discussions and gratefully acknowledge the sup-
port from the Defense Advanced Research Projects, the
Office of Naval Research under Award No. N00014-03-1-
4Ω 4
Ω1
3Ω
Ω2
4Ω
3Ω
Ω2
Ω1
a
b
c
d
b
c
a
d
(a) (b)
k1
k2
k3
k4
k4
k3 k1
k2
FIG. 3: Implementation: molecular systems, (a) vibrational levels; co-propagating fields 1 and 2 induce coherent between
vibrational levels. The field 3 propagating in the same direction will be scattered in the opposite direction. (b) Atomic Rb
scheme for implementation of coherent back scattering.
0385, the Robert A. Welch Foundation (Grant #A1261).
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Physical Review Letters

Electromagnetically Induced Coherent Backscattering

Rostovtsev, Y.

Sariyanni, Zoe-Elizabeth, Z.

Scully, M.

MPG.PuRe

Texas A&amp;M Repository

arXiv:quant-ph/0606044v1  6 Jun 2006Electromagnetically induced coherent backscatteringYuri V. Rostovtsev1, Zoe-Elizabeth Sariyanni1, and Marlan O. Scully1,2,31Institute for Quantum Studies and Department of Physics, Texas A&M University, TX 778432Princeton Institute for the Science and Technology of Materials and Departmentof Mechanical & Aerospace Engineering, Princeton University, NJ 085443Max-Planck-Institute fu¨r Quantenoptik, D-85748 Garching, Germany(Dated: April 5, 2018)We demonstrate a strong coherent backward wave oscillation using forward propagating fieldsonly. This is achieved by applying laser fields to an ultra-dispersive medium with proper chosendetunings to excite a molecular vibrational coherence that corresponds to a backward propagatingwave. The physics then has much in common with propagation of ultra-slow light. Applications tocoherent scattering and remote sensing are discussed.PACS numbers: 32.80.Qk, 42.65.Dr, 42.50.HzQuantum coherence [1, 2] has been shown to resultin many counter-intuitive phenomena. The scatteringvia a gradient force in gases [3], the forward Brillouinscattering in ultra-dispersive resonant media [4, 5], elec-tromagnetically induced transparency [6, 7, 8, 9], slowlight [10, 11, 12, 13], Doppler broadening elimination [14],light induced chirality in nonchiral medium [15], a newclass of entanglement amplifier [16] based on correlatedspontaneous emission lasers [17, 18] and the coherent Ra-man scattering enhancement via maximal coherence inatoms [19] and biomolecules [20, 21] are a few examplesthat demonstrate the importance of quantum coherence.In this Letter, we predict strong coherent backwardscattering via excitation of quantum coherence betweenatomic or molecular levels. The developed approach canalso be used to control the direction of the signal gener-ated in coherent Raman scattering and other four-wavemixing (FWM) schemes.Let us consider the four-wave mixing in a 3-level atomicmedium. The pump and Stokes fields E1 and E2 (whoseRabi frequencies are defined as Ω1 = ℘1E1/h¯ and Ω2 =℘2E2/h¯, where ℘1 and ℘2 are the dipole moments of thecorresponding transitions) with wave vectors k1 and k2and angular frequencies ν1 and ν2 induce a coherencegrating in the medium (see Fig. 1) given by [2]ρcb ∼ −Ω1Ω∗2 (1)Let us stress that the ρcb coherence grating has anexp[i(k1 − k2)z] spatial dependence. In an ultra-dispersive medium (see Fig. 2) where fields propagatewith a slow group velocity, the two co-propagating fieldshave wavevectors given byk1 ≃ k1(ωab) +∂k1∂ν1(ν1 − ωab) = ωab/c+ (ν1 − ωab)/Vg,(2)where Vg is the group velocity of the first wave, ωab is thefrequency of transition between levels a and b, and k2 =ν2/c. Thus these two fields create a coherence gratingΩ1a2dcbΩ3ΩΩ4k1k2k3k4(a)k1k2k4k3(b)FIG. 1: (a) Co-propagating fields 1 and 2 induce coherentgrading inside the medium. The field 3 propagating in thesame direction will be scattered in the opposite direction be-cause the coherence excited by fields 1 and 2 is propagatingin the opposite direction (see Fig.2). Level scheme, double-Λ(b), for implementation of coherent back scattering.in the medium with spatial phase determined by k1 −k2 = ωcb/c + (ν1 − ωab)/Vg which depends strongly onthe detuning δ = ν1 − ωab. By properly choosing thedetuning, δ, one can make k1 − k2 negative.After the coherence ρbc is induced in the medium, aprobe field E3, with Rabi frequency Ω3 = ℘3E3/h¯ and2ν2k   =2 __cωab__ck1ωab__c_____Vgν − ωab+k =νkωabδν ν12FIG. 2: Dispersion k(ν) of ultra-dispersive medium. Choos-ing δ = ν1 −ωab = −Vgωcb/c, we can have k1 − k2 < 0 even ifν1 > ν2, thus the third field can be scattered opposite to thedirection of propagation of the first two fields.wave vector k3, scatters off that coherence to produce thesignal field, Ω4. The signal field depends on the coherenceand the input fields as∂∂zΩ4 ∼ ρcbΩ3 ∼ Ω1Ω∗2Ω3 ∼ ei(k1−k2+k3−k4)z (3)That is, the propagation direction of Ω4 depends on thespatial phase of the ρbc coherence through the phase-matching condition k4 = k1 − k2 + k3 [22] while its fre-quency is determined by ν4 = ν1 − ν2 + ν3.We here show that for dispersive media one can obtaina strong signal in the backward direction even when allthree input fields propagate forward. This is contrary tothe usual non-dispersive media results, where the phase-matching in the backward direction cannot be achievedfor E1, E2, and E3 counter-propagating with respect toE4 [22].To demonstrate this result, we write the interactionHamiltonian of the system asVI = −h¯[Ω2e−iωact|a〉〈c|+Ω1e−iωabt|a〉〈b|+ h.c.] (4)−h¯[Ω3e−iωdbt|d〉〈b|+Ω4e−iωdct|d〉〈c|+ h.c.] (5)where Ω4 = ℘4E4/h¯ is the Rabi frequency of the signalfield and ωab, ωac, ωdb, ωdc are the frequency differencesbetween the corresponding atomic or molecular energylevels. The time-dependent density matrix equations aregiven by∂ρ∂τ= −ih¯[VI , ρ]−12(Γρ+ ρΓ), (6)where Γ is the relaxation matrix. A self-consistent systemalso includes the field propagation equations∂Ω1∂z= −iη1ρab,∂Ω2∂z= −iη2ρac, (7)∂Ω3∂z= −iη3ρdb,∂Ω4∂z= +iη4ρdc, (8)whereηj = νjN℘j/(2ǫ0c) (9)are the coupling constants (j = 1, 2, 3, 4), N is the parti-cle density of the medium, ǫ0 the permitivity in vacuum.The equations of motion for the density matrix ele-ments of the polarization ρab and the coherence ρcb aregiven byρ˙ab = −Γabρab + iΩ1(ρaa − ρbb)− iρcbΩ∗2, (10)ρ˙cb = −Γcbρcb + iρcaΩ1 − iρabΩ2. (11)where Γab = γab + i(ωab − ν1); Γca = γca − i(ωac − ν2);Γcb = γcb + i(ωcb − ν1 + ν2); ωcb is the frequency ofc − b transition, and γαβ are the relaxation rates at thecorresponding atomic transitions. In the steady stateregime, and assuming that |Ω2| >> |Ω1|, almost all ofthe population remains in the ground level |b〉, ρbb ≃ 1.Let us consider the fields as plane waves: Ω1(z, t) =Ω˜1(z, t) exp(ik1z), Ω2(z, t) = Ω˜2(z, t) exp(ik2z), whereΩ˜1(z, t) and Ω˜2(z, t) are the slowly varying in envelopes ofthe fields Ω1 and Ω2 in space, while k1 = ν1[1+χab(ν1)]/cand k2 = ν2[1+χac(ν2)]/c. The succeptebilities are χab =η1ρab/Ω1 = (ν1 − ωab)/2πVg and χac = η2ρac/Ω2 ≃ 0.By solving the self-consistent system of Maxwell’s equa-tions (7,8) and the density matrix equations (10,11), weobtain Eq. (2) for the wavevectors, where Vg ≃ c/η1|Ω2|2is the group velocity of the optical field Ω1. Thus, thespatial dependence of ρcb is determined by∆k = k1 − k2 =ν1 − ν2c+ν1 − ωabVg. (12)The signal field Ω4 is generated by the polarization ρdb ofthe transition it couples (Eq. 8). The equation of motionfor this polarization element readsρ˙db = −Γdbρdb + iΩ4(ρdd − ρbb)− iρcbΩ∗3, (13)where Γdb = γdb + i(ωdb − ν4), and ν4 is the frequencyof generated field. In the steady-state regime and for|Ω4| << |Ω3|, the field Ω4 at the output of the cell isgiven byΩ4 =∫ L0dzei(k4−∆k−k3)zη4ρcbΩ˜∗3Γdb(14)where L is the length of the cell. Note here that Eq.(14)is valid if the field |Ω3| does not change coherence ρcb viapower broadening which is true if |Ω3|2 ≪ |Ω1|2 + |Ω2|2.Hence, after integrating Eq.(14), we obtain for thescattered field Ω4Ω4 =[sin(k4 − k3 −∆k)L(k4 − k3 −∆k)L]η4LΩ3Γdb(15)3where the expression in the brackets describes the phase-matching and determines the direction in which the sig-nal field is generated.The most interesting effect following from Eq.(15)is the coherent backscattering. Indeed, even when allthree input fields propagate forward, one may observe abackscattered signal field by satisfying the conditionk3 +∆k = k3 +ν1 − ωabVg< 0, (16)and for appropriate detuning, ν1−ωab < 0, this inequalitycan be meet. That is, in order to obtain phase-matchingin the backward direction, we have to satisfyk3 +∆k < −k4, (17)which can be rewritten as−ν1 − ωabVg> k3 + k4. (18)Hence, in order to demonstrate the effect, the detuningδ should meet the conditionδ = −(k3 + k4)Vg ≃ −2k4Vg. (19)It is useful to rewrite the condition in terms of succepti-bility for the probe field, indeed,k1 =ν1cn1 ≃ν1c(1 + cν1 − ωabν1Vg) =ν1c(1 − c2k4ν1), (20)then,χab = 2(n1 − 1) = −4λabλdb, (21)for gases χab ≪ 1, so λab ≪ λdb, i.e. the effect can beimplemented for scattering of IR fields. Then, for theDoppler broadened EIT media as shown in [23, 24], wecan writeχab(δ) ≃3λ3abN8π2(γrδ|Ω2|2+ iγr∆Dδ2|Ω2|4), (22)where ∆D is the Doppler width; γr is the radiative decayrate. Thus, for detuning smaller than the EIT width|δ| ≤ |Ω2|2/√γr∆D, absorption can be neglected, and3λ2abNγrδ16π|Ω|2= −2k4, (23)then, the atomic or molecular density is given byN =32πk43λ2ab|Ω2|2γr|δ|≃32πk43λ2ab√∆Dγr. (24)There are several schemes to demonstrate the effect.For example, the double-Lambda scheme can be imple-mented in molecular rotational levels (see Fig. 1a). More-over, the effect can be implemented in the ladder-Λ usingmolecular vibrational levels (see Fig. 3a). The phase-matching condition should be slightly modified for thescheme as k4 = k1 − k2 − k3. Also, the phenomenon canbe demonstrated in a V-Λ scheme that can be realizedin atomic levels (see Fig. 3b, for Rb atoms, b = 5S1/2,c = 7D3/2,5/2, a = 5P1/2,3/2, d = 8P1/2,3/2), and phase-matching condition has a form k4 = k1 + k2 − k3. Letus note that the requirement for detuning in all cases isδ/Vg = −2k4 and the use of Eq. (24) to estimate molec-ular or atomic density is still valid.Examples of systems to this effect could be seen wemention molecules NO (a resonant transition at 236 nm,A2Σ+−X2Π), NO2 (a resonant transition at wavelength337 nm), and atomic Rb vapor (EIT and CPT havebeen recently demonstrated for molecules, see [26]). Therequired molecular density of NO and NO2 moleculesis N ≃ 1.2 · 1013 cm−3 if one can use transition be-tween rotational levels ≃ 10 cm−1. Using vibrational IRtransitions for NO (vibration frequency of 1900 cm−1)at 5.26 µm and for NO2 (vibrational frequency of 750cm−1) 13.3 µm, the densities are N = 8 · 1015 cm−3and N = 1.4 · 1015 cm−3, correspondingly. For atomicRb vapor, wavelengths are λ1 = 780 nm, λ2 = 565 nm,λ3 = 335 nm, λ4 = 23.4 µm, and the atomic density isN = 1.4 · 1013 cm−3.The intensity needed for EIT is determined by condi-tion |Ω|2 ≫ γbc∆D which corresponds to laser intensityof the order of 1 mW/cm2 for atoms and of the orderof 10 W/cm2 for molecules because the dipole momentis two orders of magnitude smaller for molecules. Theseconditions are realistic and well-suited for an experimen-tal implementation.Several applications of the effect can be envisioned, likein nonlinear CARS microscopy [27], while the controllingof coherent backscattering could provide a new tool forcreating an image. A variation in the molecular densitywould modify the intensity of the signal in both the for-ward and the backward direction. Additionally, Eq.(15)also allows one to control the direction of the generatedsignal field and thus provide an all-optical control whenscanning an optical field over an object.In conclusion, we theoretically predict strong coher-ent scattering in the backward direction while using onlyforward propagating fields. This is achieved by excitingatomic or molecular coherence by properly detuned fields,in such a way that the resulting coherence has a spatialphase corresponding to a backward, counter-propagatingwave. Applications of the technique to coherent scatter-ing and remote sensing are discussed. The method holdspromise for observation induced scattering in a backwarddirection with application to CARS microscopy.We thank J. Giordmaine, K. Hakuta, N. Kroo´, G. Kur-izki, K.K. Lehmann, R. Miles, H. Walther, and X.S. Xiefor useful discussions and gratefully acknowledge the sup-port from the Defense Advanced Research Projects, theOffice of Naval Research under Award No. N00014-03-1-4Ω 4Ω13ΩΩ24Ω3ΩΩ2Ω1abcdbcad(a) (b)k1k2k3k4k4k3 k1k2FIG. 3: Implementation: molecular systems, (a) vibrational levels; co-propagating fields 1 and 2 induce coherent betweenvibrational levels. The field 3 propagating in the same direction will be scattered in the opposite direction. (b) Atomic Rbscheme for implementation of coherent back scattering.0385, the Robert A. Welch Foundation (Grant #A1261).[1] E. Arimondo, in Progress in Optics, edited by E. Wolf(Elsevier, Amsterdam, 1996), Vol. XXXV, p. 257; S. E.Harris, Phys. Today 50, 36 (1997).[2] M. O. Scully and M. S. Zubairy, Quantum Optics, Cam-bridge University Press, Cambridge, England, (1997).[3] S. E. Harris, Phys. Rev. Lett. 85, 4032 (2000).[4] A. B. Matsko, Yu. V. Rostovtsev, H. Z. Cummins, andM. O. Scully, Phys. Rev. Lett. 84, 5752 (2000).[5] A. B. Matsko, Y. V. 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Electromagnetically induced coherent backscattering

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Electromagnetically induced coherent backscattering

Abstract

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Texas A&M University

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