Download Power quantum control of odd-order multiwave

Zheng et al.
Vol. 31, No. 6 / June 2014 / J. Opt. Soc. Am. B
1263
Power quantum control of odd-order multiwave
mixing in an electromagnetically
induced transparency window
Huaibin Zheng,1,2 Ning Li,1 Zhaoyang Zhang,1 Zhenkun Wu,1 Chengjun Lei,1
Yiqi Zhang,1 and Yanpeng Zhang1,*
1
Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Laboratory
of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China
2
School of Science, Xi’an Jiaotong University, Xi’an 710049, China
*Corresponding author: [email protected]
Received January 17, 2014; revised April 11, 2014; accepted April 15, 2014;
posted April 15, 2014 (Doc. ID 204910); published May 12, 2014
We investigate the power control behavior of odd-order multiwave mixing in an electromagnetically induced transparency window. We successfully obtain the evolution from pure enhancement (bright state), half-enhancement,
and half-suppression, to pure suppression (dark state) in a four-wave mixing (FWM) channel. The correlations
of two bright and two dark states of FWM are studied under four different kinds of power configurations. Moreover,
the power-controlled quantum interference among multiple dark states is also studied in FWM and six-wave
mixing channels. Such selective switching among multiple frequency channels could have potential applications
in optical switching, optical communication, and quantum information processing. © 2014 Optical Society of
America
OCIS codes: (190.4223) Nonlinear wave mixing; (270.1670) Coherent optical effects; (030.1670) Coherent
optical effects.
http://dx.doi.org/10.1364/JOSAB.31.001263
1. INTRODUCTION
Electromagnetically induced transparency (EIT) [1,2] has
very important applications, including lasing without inversion [3], slow light [4], photon controlling and information
storage [5], quantum communications [6], and nonlinear optics [7]. In the EIT window, absorption of the medium can
be greatly reduced through atomic coherence and efficiently
lead to the generation of four-wave mixing (FWM) [8–12] and
six-wave mixing (SWM) [13–15] signals. Due to the advantages
of EIT, research on EIT and related topics are continuously
carried forward. Among the progress, the correlation between
double-dark states and splitting of dark states in a four-level
atomic system via atomic coherence was reported, which may
find applications in optical communication and quantum information processing [16–19]. In addition, switches between
the dark state (EIT) and the bright state [electromagnetically
induced absorption (EIA)] were also reported in rubidium
atomic vapors [20,21]. Based on such resonances, FWM processes can be effectively controlled by the polarization of the
involved laser beams [22–25].
In this paper, we theoretically and experimentally demonstrate that the transition of dark and bright states can be obtained by changing the powers of involved laser beams in
multilevel atomic systems. In a ladder-type three-level system,
the probe transmission, FWM, and fluorescence signals will be
investigated to verify such transitions. In a Y-type four-level
system, the enhancement and suppression of FWM will be
also investigated via controlling the dressing effect by the field
power. Moreover, the power-controlled quantum interference
0740-3224/14/061263-10$15.00/0
among multiple dark states will be involved in FWM and SWM
processes.
The paper is organized as follows. Section 2 presents the
basic theory and experimental configuration. Section 3 shows
the doubly dressed FWM signal and fluorescence in an EIT
window. Section 4 discusses the transition of double-bright
and double-dark states. Section 5 gives the power-controlled
quantum interference between multiple dark states. Section 6
is the conclusion.
2. EXPERIMENTAL SETUP AND BASIC
THEORY
The experiment is performed in a five-level atomic system
consisting of 5S 1∕2 F 3j0i, 5P 3∕2 j1i, 5D5∕2 j2i,
5S 1∕2 F 2j3i, and 5D3∕2 j4i of 85 Rb. A weak laser beam
E 1 drives the lower transition j0i↔j1i, two strong coupling
beams E2 & E 02 connect j1i↔j2i, an additional strong coupling
beam E 4 drives j1i↔j4i, and two strong pumping beams E 3 &
E 03 drive j3i↔j1i. The laser beams are aligned spatially as
shown in Fig. 1(a), in which E 1 counterpropagates through
the rubidium vapor cell (with a temperature of 60°C) with
other five beams, and there is a small angle (∼0.3°) between
any two of the beams. In the experiment, the probe beam E 1
(wavelength 780.245 nm, frequency ω1 , wave vector k1 , Rabi
frequency G1 , and frequency detuning Δ1 ) is from an external
cavity diode laser (ECDL) (Toptica DL100L) with horizontal
polarization. E 2 (775.978 nm, ω2 , k2 , G2 , and Δ2 ) and E 02
(775.978 nm, ω2 , k02 , G02 , and Δ2 ) with orthogonal polarizations
are split from an ECDL (Hawkeye Opto-quantum). E 4
© 2014 Optical Society of America
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J. Opt. Soc. Am. B / Vol. 31, No. 6 / June 2014
Generally, the output intensities of the probe transmission,
MWM, and fluorescence signals are related to the expressions
of the different-order density-matrix elements, which can be
obtained by solving the density-matrix master equation
2
y
E4
E3
E′2
E2
E1
EF
∂ρ∕∂t −i∕ℏH; ρ ∂ρ∕∂tinc ;
2
(b)
4
2
4
E2
E′2
E4
E4
ω1
1
ρ0
00 !ρ10 ;
1
1
E1
E1
EF
ES
0
0
(c)
E3
E′3
3
(d)
Fig. 1. (a) Spatial beam geometry used in the experiment. Diagrams
of (b) a ladder-type three-level, (c) a Y-type four-level, and (d) a K-type
five-level atomic system.
(776.157 nm, ω4 , k4 , G4 , and Δ4 ) is from a tapered-amplifier
diode laser (UQEL100). E 3 (780.235 nm, ω3 , k3 , G3 , and Δ3 )
and E03 (780.235 nm, ω3 , k03 , G03 , and Δ3 ) with equal power
and orthogonal polarizations are split from the fourth ECDL.
Here, the frequency detuning is defined as Δi Ωi − ωi , in
which Ωi is the resonant frequency of transition Ei derives.
There exists two ladder-type EIT subsystems in such a fivelevel system, i.e., j0i↔j1i↔j2i and j0i↔j1i↔j4i, both of which
satisfy the two-photon Doppler-free conditions. When we
block E 3 & E03 and E 4 [as shown in Fig. 1(b)], one vertically
polarized FWM signal EF satisfying the phase-matching condition kF k1 k2 − k02 is generated in j0i↔j1i↔j2i. The
dressed FWM signal can be observed by applying the dressing
field E 4 , as shown in Fig. 1(c). If we turn on all beams but turn
off E 02 [as shown in Fig. 1(d)], two horizontally polarized SWM
signals E S2 kS2 k1 k3 − k03 k2 − k2 and E S4 kS4 k1 k3 − k03 k4 − k4 are generated in j0i↔j1i↔j2i and
j0i↔j1i↔j4i EIT windows, respectively. All the multiwave
mixing (MWM) signals propagate along the same direction,
which deviate an angle θ from E 1 . The MWM signals and
the transmitted probe are detected by an avalanche photodiode detector and a photodiode, respectively. Two types
of nondirectional fluorescence signals are also generated
and fall into the EIT windows. Due to the spontaneous emission, the single-photon fluorescence signal R0 (780 nm) will be
generated in the decay of photons from j1i to j0i, and the
two-photon fluorescence signal R1 (776 nm) will be generated
in the decay from j2i to j1i. The fluorescence signals are
detected by another photodiode.
In the interaction picture, we can obtain the Hamilton for
such system as
H ℏΔ1 j1ih1j ℏΔ1 Δ2 j2ih2j
ℏΔ1 − Δ3 j3ih3j ℏΔ1 Δ4 j4ih4j
− ℏG1 j1ih0j G2 G02 j2ih1j
G3 G03 j3ih1j G4 j4ih1j H:c::
(2)
where the first term results from the coherence interaction,
and the second term represents the dampings due to the spontaneous emission and other irreversible processes. Here, we
use the perturbation chain method [13,23] to find the solutions
that we are concerned with. The perturbation chains corresponding to each signal are
0
(a)
E2
E′2
x
EM
E1
E2
1
E′3
z
Zheng et al.
(1)
probe transmission;
ω1 1 ω2 2 −ω2 3
ρ0
FWM;
00 !ρ10 !ρ20 ! ρ10 ;
0 ω1 1 ω3 2 −ω3 3 ω2 4 −ω2 5
ρ00 !ρ10 !ρ30 ! ρ10 !ρ20 ! ρ10S2 ;
ω1 1 ω3 2 −ω3 3 ω4 4 −ω4 5
ρ0
00 !ρ10 !ρ30 ! ρ10 !ρ40 ! ρ10S4 ;
ω1 1 −ω1 2
ρ0
R0 fluorescence;
00 !ρ10 ! ρ11 ;
ω1
−ω1
ω4
−ω4
1
2
3
4
ρ0
00 !ρ10 ! ρ11 !ρ21 ! ρ22 ;
SWM2;
SWM4;
R1 fluorescence;
where we list only one probable channel for each signal for
example. Thus, we have
ρ1
10 iG1 ∕d1 ;
probe transmission signal;
2
ρ3
10 GF ∕d1 d2 ;
FWM signal;
(3a)
(3b)
3
ρ5
10S2 GS2 ∕d1 d2 d3 ;
SWM2 signal;
(3c)
3
ρ5
10S4 GS4 ∕d1 d4 d3 ;
SWM4 signal;
(3d)
2
ρ2
11 jG1 j ∕d1 Γ11 ;
R0 fluorescence signal;
2
2
ρ4
22 jG1 j jG2 j ∕Γ22 d1 d2 d21 ;
(3e)
R1 fluorescence signal;
(3f)
where d1 Γ10 iΔ1 , d2 Γ20 iΔ1 Δ2 , d3 Γ30 iΔ1 − Δ3 , d4 Γ40 iΔ1 Δ4 , d21 Γ21 iΔ2 , GF −iG1 G2 G02 expikF · r, GS2 iG1 G2 G2 G0
3 G3 expikS2 · r,
GS4 iG1 G4 G4 G0
G
expik
·
r,
and
Γ
S4
ij is the transverse
3 3
relaxation rate between jii and jji. The notes in parentheses
indicate the related signals in the current system.
3. DOUBLY DRESSED FWM AND
FLUORESCENCE
The doubly dressed probe spectrum, FWM E F , and fluorescence signals in j0i↔j1i↔j2i are investigated in this section.
Under the doubly dressing effect induced by E 1 and E 2
(both strong sufficiently), Eqs. (3a), (3b), (3e), and (3f) are
modified as
2
2
ρ1
10 iG1 ∕d1 jG1 j ∕d5 jG2 j ∕d2 ;
(4a)
Zheng et al.
Vol. 31, No. 6 / June 2014 / J. Opt. Soc. Am. B
2
2
2
ρ3
10 GF ∕d2 d1 jG1 j ∕d5 jG2 j ∕d2 ;
(4b)
2
2
2
ρ2
11 jG1 j ∕Γ11 d1 jG1 j ∕d5 jG2 j ∕d2 ;
(4c)
2
2
2
ρ4
22 jG1 j jG2 j ∕Γ22 d1 d2 jG2 j ∕d1 d21 ;
(4d)
where d5 Γ11 jG2 j2 ∕d21 .
First, Figs. 2(a1)–2(a3) present the intensities of the doubly
dressed probe transmission, FWM signal EF , and fluorescence
signals versus Δ2 (scanning range ∼300 MHz) with different
Δ1 , respectively. Note that, the experimentally obtained spectra of the E F signal behave as a mixture of two independent
effects: a two-photon resonant peak without the dressing effect (“classical emission part”), and the enhanced peak or
suppressed dip due to the dressing effect. The fluorescence
signal also includes two fluorescence processes: R0 and R1 .
Figures 2(b1)–2(b3) are the theoretical calculations corresponding to Figs. 2(a1)–2(a3), respectively.
When we scan Δ2 at different Δ1 , the probe transmission
shows the evolution from pure EIA, to first EIA and next
Signal intensity
(a1)
(a2)
(a3)
-80
-40
-20
0
20
40
80
∆1(MHz)
(a)
Signal intensity
(b1)
(b2)
(b3)
-80
-40
-20
0
20
40
80
∆1(MHz)
(b)
2
G2 +
G 1&2 +
G 1&2 −
G2 −
1
0
(c1)
(c2)
(c3)
(c4)
(c5)
(c)
Fig. 2. (a) Measured intensities (from left to right) of (a1) the probe
transmission, (a2) the FWM and (a3) fluorescence signals versus Δ2 at
Δ1 −80, −40, −20, 0, 20, 40, and 80 MHz, respectively. The powers of
the laser beams are I 1 2 mW, I 2 15 mW, and I 02 2 mW, respectively. (b1)–(b3) Theoretical calculations corresponding to (a1)–(a3).
(c) Dressed energy level diagrams of the doubly dressing FWM signal
at Δ1 −80, −20, 0, 20, and 80 MHz, respectively.
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EIT, to double-peak EIT, to pure EIT (single peak), to doublepeak EIT, to first EIT and next EIA, finally to pure EIA, as
shown in Fig. 2(a1). The baseline height of each curve represents the probe transmission versus Δ1 without the dressing
effect of E 2 . The peak and dip on each baseline represent the
EIT and EIA, respectively. Correspondingly, the FWM signal
shows evolution from single-peak to asymmetrical double
peaks, to symmetrical double peaks, to asymmetrical double
peaks, finally to one peak successively, as shown in Fig. 2(a2).
The peaks on each baseline represent the undressed FWM
emission signal with enhancement, while the dips represent
the suppression due to the dressing effect. For the detected
fluorescence signal, the baseline represents R0 without dressing effect of E 2 , the suppression dip in each baseline represents R0 is further suppressed by E 2 , and the peak within
the dip is R1 . With Δ1 far away from the resonant, the fluorescence signal with only single-peak structure can be observed.
The phenomenon mentioned above can be explained by the
3
2
doubly dressing effect. In ρ1
10 and ρ10 , the term jG1 j ∕d5 represents the dressing effect induced by E 1 & E 2 , while jG2 j2 ∕d2
represents the dressing effect induced only by E 2 . Fields E 1 &
E 2 will split j1i into two dressed states jG1&2 i, and E2 creates two dressed states jG2 i from j1i, as shown in Fig. 2(c).
The resonance of the probe field with these dressed (bright)
states will bring EIA for the probe transmission signal, and
enhancement for FWM and the fluorescence signal, while resonance with dark states will bring EIT for the probe transmission signal and suppression for FWM and the fluorescence
signal.
When Δ2 is scanned at Δ1 0, the dressing term jG2 j2 ∕d2
represents that the two-photon resonance will occur if the
condition Δ1 Δ2 0 is satisfied, and the probe field does
not resonant with jG2 i. On the other hand, due to the
bright-state condition Δ1 Δ2 jG1 j2 ∕jG2 j2 0 and dark-state
condition Δ2 0 based on the dressing term jG1 j2 ∕d5 , the
bright and dark states will overlap with each other at Δ2 0 [Fig. 2(c3)]. Since E2 is much stronger than E1 , the dressing
effect induced by E2 is stronger than that induced by E 1 & E2 ,
thus the probe transmission as well as FWM signal will show
the dark state dominantly, which leads to pure EIT in the
probe transmission and pure suppression in the FWM signal.
In the region Δ1 < 0, when jΔ1 j is small, two bright states and
two dark states can be obtained. As shown in Fig. 2(c2), when
Δ2 is scanned from negative to positive, first a dark-state
condition Δ2 0, next the bright-state condition Δ1 q
Δ2 Δ22 4jG2 j2 ∕2 0, then Δ1 Δ2 0, and at last
Δ1 Δ2 jG1 j2 ∕jG2 j2 0 will be satisfied. Considering E 1 is
quite weak, the last bright state would be almost invisible.
Thus, the probe transmission as well as the FWM signal successively undergo dark-bright-dark states, which behave as
double-peak EIT with an EIA between them in the probe transmission signals [Fig. 2(a1)] and asymmetrical double-peaks in
the FWM signal [Fig. 2(a2)]. In such asymmetrical doublepeak structure, the left peak represents the resonance of
the bright state created by E 2 while the right one originates
from the classical emission part. When jΔ1 j is large, the effects
of the bright state and dark state created by E1 & E 2 will get
weaker, and the probe transmission shows merely a left-bright
state and a right-dark state, which are determined by the resonance of the dressed states jG2 i and the dark state induced
by E 2 , respectively. Correspondingly, the FWM signal has the
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J. Opt. Soc. Am. B / Vol. 31, No. 6 / June 2014
Zheng et al.
same behavior as the ordinary case with the singly dressing
effect induced by E 2 . When Δ2 is scanned with Δ1 far away
from the resonance point, the probe field can only resonate
with jG2 i [Fig. 2(c1)]. In this case, the probe transmission
and FWM reveal an almost bright state, as shown in
Figs. 2(a1) and 2(a2). Similar phenomena can be obtained
with Δ1 > 0.
Since there is a mutual term d1 jG2 j2 ∕d2 in ρ2
11 , the intensity of R0 is suppressed into its minimum at Δ1 Δ2 0. With
Δ1 set far away from resonance, such suppression also reduces, so the dip is almost invisible at large detuning
[Fig. 2(a3)]. Moreover, it is obvious that the fluorescence
peaks of R1 within the suppression dip of R0 in the curves
in Fig. 2(a3) locate the positions of EIA satisfying Δ1 Δ2 q
Δ22 4jG2 j2 ∕2 0 in the curves in Fig. 2(a1) (marked by
the dashed lines). According to d2 jG2 j2 ∕d1 in ρ4
22 , such
peaks also get stronger with increasing Δ1 . For R1 , it is suppressed more significantly around Δ1 0. The experimental
result agrees well with the theoretical results in Fig. 2(b3).
Next, the dependence of the signal intensities on I 1 with
I 2 10 mW is studied, as shown in Figs. 3(a1)–3(a3). The corresponding theoretical results are shown in Figs. 3(a4)–3(a6).
By setting Δ1 −22.8 MHz, when I 1 is low (0.5 mW),
double-peak EIT is visible on the probe transmission signal.
As I 1 increases to 3 mW, an EIA dip first sinks and then
strengthens. When I 1 gets larger, the EIA dip broadens until
the two EITs are almost invisible. The whole curve will behave
pure EIA when the probe power is sufficiently large (at
I 1 7 mW), as shown in Fig. 3(a1). As for the FWM signal,
a double-peak structure can be observed when I 1 is low. With
increasing I 1 , the right peak grows to be taller than the left
peak. When I 1 gets further larger, the left peak catches up
with the right one. Finally, the left peak will become much
taller than the right peak when I 1 is sufficiently large, as
shown in Fig. 3(a2).
Signal Intensity
0.5mW
1mW
2mW
3mW
4mW
5mW
Finally, the dependence of these signal intensities on the
coupling power I 2 with I 1 6 mW is investigated, as shown
in Fig. 3(b). As I 2 increases, the shape of the curve still maintains the double-peak style, but the magnitudes of the right
EIT and EIA are strengthened a little. Correspondingly, the
left peak of the FWM signal gets higher than the right one.
One can conclude that the left and right EIT peaks in the
two cases have different power dependences, which can
verify that they are derived from different processes. Actually,
they are caused by jG1 j2 ∕d5 and jG2 j2 ∕d2 , respectively. We
find that the two pairs of bright and dark states created by
E 1 & E 2 and E 2 can be controlled by I 1 and I 2 , respectively.
As I 1 is set low and I 2 is high, double-peak EIT will be obtained, which is composed of two dark states and one bright
state. When we increase I 2 , the bright as well as dark state
related to E2 show a strengthening trend without any influence on the state related to E 1 & E2 . While I 1 is increased,
the bright state created by E1 & E 2 is strengthened. Due to
the interaction among these states, the two dark states are
weakened, which makes the probe transmission transit
from double EIT to EIA. Moreover, according to
Δ1 Δ2 jG1 j2 ∕jG2 j2 0, the bright state jG1&2 i shifts from
right to left with the increasing of I 1 along the Δ2 axis. This
shifting can be observed both in probe transmission and FWM.
Now, we turn to the fluorescence signal. When we change
I 1 with I 2 fixed, the suppression dip of R0 in each curve in
Fig. 3(a3) deepens, but the peak within the dip remains the
same. When I 2 increases, the peak and the dip gradually become higher and deeper, respectively [Fig. 3(b3)]. This interesting characteristic of the fluorescence signal can be well
4
explained according to ρ2
11 and ρ22 . In fact, since R0 related
2
to ρ11 is modulated by both of the dressing effects from E1 and
E 2 , increasing both I 1 and I 2 would make the suppression dip
deeper. The fluorescence peak R1 is mainly modulated by I 2
according to the numerator in ρ4
22 which makes it get larger
with increased I 2 . Note that the above discussion does not include all broadening mechanisms, such as the residual
Doppler effect and power broadening, which make the experimental data in general broader than the theoretical
predictions.
6mW
7mW
4. TRIPLY DRESSED FWM
0
-200
200
-200
(a1)
0
200 -200
(a2)
0
200 -200
(a3)
∆ 2 (MHz)
0
200 -200
(a4)
0
200 -200
(a5)
0
200
(a6)
(a)
Signal Intensity
2mW
4mW
6mW
8mW
10mW
12mW
14mW
16mW
-200
0
(b1)
200 -200
0
(b2)
200 -200
0
200 -200
(b3)
∆ 2 (MHz)
0
(b4)
200 -200
0
(b5)
200
-200
0
200
(b6)
(b)
Fig. 3. (a) Measured intensities (from top to bottom) of (a1) the
probe transmission, (a2) the FWM signal, and (a3) fluorescence signals versus Δ2 with I 2 10 mW and I 02 2 mW at Δ1 −22.8 MHz
for I 1 0.5, 1, 2, 3, 4, 5, 6, and 7 mW, respectively. (a4)–(a6) Theoretical calculations corresponding to (a1)–(a3). (b) Figure setup is as
(a) but with I 1 2 mW at Δ1 −22.8 MHz, for I 2 I 02 2, 4, 6, 8,
10, 12, 14, and 16 mW, respectively.
In this section, the enhancement and suppression of FWM will
be investigated directly by scanning Δ4 with Δ2 0 and different Δ1 in a Y-type four-level system.
When we change I 1 and I 4 from low to high, respectively,
the FWM signal will show the evolution from double-bright
states (enhancement) to the dark state (suppression). Figure 4
shows that the experimental results correspond to the
power combinations low I 1 low I 4 [Fig. 4(a1)], high I 1 low
I 4 [Fig. 4(a2)], low I 1 high I 4 [Fig. 4(a3)] and high I 1 high
I 4 [Fig. 4(a4)], respectively. And Figs. 4(b1)–4(b4) are the
corresponding theoretical results. The peaks and dips in
the baseline of each curve represent the enhancement and
suppression of the FWM, while the profiles [dashed curves
in Figs. 4(a1)–4(b4)] composed of the baselines reveal
Autler–Townes splitting structure due to the dressing effect
of E 2 .
Under the triply dressing effect induced by E1 , E2 , and E4 ,
ρ3
10 will be modified into
Vol. 31, No. 6 / June 2014 / J. Opt. Soc. Am. B
(a1)
(b1)
(a2)
(b2)
Signal intensity
Signal intensity
Zheng et al.
(a3)
(b3)
(a4)
-80
1267
(b4)
-40
-20
0
20
40
80
-80
-40
-20
0
20
∆1(MHz)
∆1(MHz)
(a)
(b)
40
80
4
2
G4 +
G1&4 +
1
G1&4 −
G4 −
0
(c1)
(c2)
(c3)
(c4)
(c5)
(c)
Fig. 4. (a) Intensity of the suppression and enhancement of the FWM signal EF versus Δ4 at different Δ1 when (a1) I 1 3 mW and I 4 4 mW,
(a2) I 1 8 mW and I 4 4 mW, (a3) I 1 3 mW and I 4 40 mW, and (a4) I 1 8 mW and I 4 40 mW with I 2 I 02 8 mW. (b) Theoretical
calculations corresponding to (a). (c) Dressed energy level diagrams of the triply dressing FWM signal at Δ1 −80, −20, 0, 20, and 80 MHz,
respectively.
2
2
2
2
ρ3
10 GF ∕d2 d1 jG1 j ∕d6 jG2 j ∕d2 jG4 j ∕d4 ;
(5)
where d6 Γ11 jG4 j2 ∕Γ41 iΔ4 . The term jG1 j2 ∕d6 represents the dressing effect from E 1 & E4 , while jG4 j2 ∕d4 from E 4 .
With these dressing effects, state j1i can be split into four
dressed states jG1&4 i and jG4 i, as shown in Fig. 4(c).
When I 1 and I 4 are low (I 1 3 mW and I 4 4 mW),
double-peak enhancement can be observed. With Δ1 changing
from negative to positive, the dressed FWM signal shows the
evolution from weak enhancement, to strong enhancement, to
weak enhancement, and then increasing enhancement, to finally weak enhancement successively shown as Fig. 4(a1).
Note that the dark states created by E 1 & E 2 will influence
the FWM process quite weakly in contrast to the one of
E 1 & E4 . The bright states jG4 i will be shifted close to each
other, which makes the dark state created by E4 too weak to
be revealed. In this case, only the bright state can be observed.
With Δ4 scanned in Δ1 < 0 (Δ1 > 0), the states jG1&4 i
(jG1&4 −i) and jG4 i (jG4 −i) can be obtained when the
bright-state conditions Δ1 Δ4 jG1 j2 ∕jG4 j2 0 and Δ1 Δ4 q
q
Δ24 4jG4 j2 ∕2 0 [Δ1 Δ4 − Δ24 4jG4 j2 ∕2 0] are
satisfied, respectively. Thus, the whole signal will show double-bright states. In addition to E 4 , due to the self-dressing effect of E 2 , the peak will be suppressed around Δ1 0 when
the dark-state condition Δ1 Δ2 0 is satisfied. While the
q
bright-state condition Δ1 Δ2 Δ22 4jG2 j2 ∕2 0 is satisfied, the peak will be enhanced around Δ1 G2 . The
theoretical results in Fig. 4(b1) fit the experimental results
in Fig. 4(a1) perfectly.
When I 4 4 mW and I 1 8 mW, peaks as well as dips can
be observed on each baseline. As we scan Δ4 with different
Δ1 , the dressed FWM signal shows the evolution from pure
enhancement, to first enhancement and next suppression,
to asymmetric double-peak enhancement, to symmetric double-peak enhancement, to asymmetric double-peak enhancement, to first suppression and next enhancement, to pure
enhancement successively shown as Fig. 4(a2).
2
As for ρ3
10 , increasing I 1 will magnify the value of jG1 j ∕d6
and therefore makes ρ3
decrease,
thus
the
bright
states
10
jG1&4 i will be weakened. Moreover, the states jG4 i will
move, which makes the dark state created by E4 able to be
observed.
When Δ4 is scanned with Δ1 0, both the dark-state condition Δ1 Δ4 0 and bright-state condition Δ1 Δ4 jG1 j2 ∕jG4 j2 0 can be satisfied. The two states will be obtained simultaneously, with their centers overlapping with
each other [Fig. 4(c3)]. Since the bright state is stronger than
the dark state, the curve reveals symmetric double-peak enhancement, as shown in Fig. 4(a2). In Δ1 < 0, when jΔ1 j is
small, two bright states and one dark state can be obtained.
When Δ4 is scanned from negative to positive, first it will resonate with jG4 i, and the bright-state condition Δ1 Δ4 q
Δ24 4jG4 j2 ∕2 0 will be satisfied. Then, a dark state appears when Δ1 Δ4 0 is satisfied. At last, when it resonates
with jG1&4 i [Fig. 4(c2)], Δ1 Δ4 jG1 j2 ∕jG4 j2 0 will be
1268
J. Opt. Soc. Am. B / Vol. 31, No. 6 / June 2014
Zheng et al.
satisfied. Thus, the FWM signal exhibits a bright-dark-bright
structure, in which the bright state jG1&4 i have been
weakened, so the signal shows asymmetrical double-peak
enhancement, as shown in Fig. 4(a2). With increasing jΔ1 j,
the weakened state jG1&4 i decays more quickly than
jG4 i. When jΔ1 j is large, jG1&4 i is nearly unable to be obtained. Since only one bright state and one dark state can be
observed, the FWM signal shows left-bright state and rightdark state. When Δ4 is scanned with Δ1 far away from resonance point, the probe field can only resonate with the state
jG4 i [Fig. 4(c1)]. The FWM signal presents an almost allbright state, as shown in Fig. 4(a2). A similar phenomenon
can be obtained in Δ1 > 0. The theoretical results in Fig. 4(b2)
fit the experimental results in Fig. 4(a2) perfectly.
When I 1 3 mW and I 4 40 mW, the dressed FWM signal
shows the evolution from pure enhancement, to first enhancement and next suppression, to double-peak enhancement, to
first suppression and next enhancement, to pure suppression,
to first enhancement and next suppression, to double-peak
enhancement, to first suppression and next enhancement,
to pure enhancement, as shown in Fig. 4(a3).
For high I 4 , the state jG4 i will be shift by E4 , which will
weaken the bright states and strengthen the dark states. Thus,
the dark state is stronger than the bright state, and the curve
shows suppression, as shown in Fig. 4(a3). When Δ1 < 0, only
the dark state and the bright state can be observed. Because of
the weakening of jG4 i, when jΔ1 j is small, jG1&4 i is
stronger when compared with jG4 i, so the left peak of
the signal is much taller than the right one. When jΔ1 j is large,
jG1&4 i decays more quickly than jG4 i. The FWM signal
shows left-bright state and right-dark state, similar to the case
shown in Fig. 4(a2). Similar behavior can be found with
Δ1 > 0. The experimental results in Fig. 4(a3) and the theoretical results in Fig. 4(b3) match well with each other.
When I 1 8 mW and I 4 40 mW [Fig. 4(a4)], the intensity
of the dressed FWM shows suppression with a similar changing rule to that of enhancement in Fig. 4(a1). It is obvious that
the evolution of the dressed FWM signal shows a symmetrical
pattern, and the magnitude of the suppression can be very
large under such a special power condition.
In theory, since jG1&4 i and jG4 i are weakened, the signal will show a dominating dark state. The intensity of the suppression dip is modulated by the self-dressing effect of E2 , the
same as the characteristic of the enhancement peak shown in
Fig. 4(a1). The experimental results in Fig. 4(a4) agree well
with the theoretical results in Fig. 4(b4).
In order to investigate the correlations of two bright and
two dark states mentioned above in more detail, we study
the dependence of the FWM signals on I 4 with fixed I 1 , as
shown in Fig. 5(a). By setting Δ1 −30 MHz and I 1 3 mW,
the signal versus Δ4 shows the evolution from double-peak
enhancement, to first suppression then enhancement, and
finally to pure suppression [Fig. 5(a1)]. When we set
I 1 8 mW, such evolution will change from first enhancement then suppression to pure suppression [Fig. 5(a3)].
As we set I 1 3 mW, when I 4 is low, jG1&4 i and jG4 i are
stronger than the dark states, which make the signal behave as
double-peak enhancement. When I 4 gets higher, jG4 i will
move far away from the resonant point, which weakens
jG4 i and strengthens the dark state. So, the left peak of enhancement will be weakened and a dip will appear between the
Fig. 5. (a) Measured intensity of the FWM signal versus Δ4 at Δ1 −30 MHz when (a1) I 1 3 mW and (a3) I 1 8 mW for I 4 4, 10, 15,
20, 25, 30, 35, and 40 mW. (a2) and (a4) Theoretical calculations corresponding to (a1) and (a3). (b) Figure setup is as (a) but at Δ1 −30 MHz when (b1) I 2 4 mW and (b3) I 2 40 mW, for I 1 0.5,
1, 2, 3, 4, 5, 6, and 7 mW. (c) Figure setup is as (a) but at (c1) Δ1 −60 MHz and (c3) Δ1 −120 MHz when I 1 3 mW, for I 4 4, 10,
15, 20, 25, 30, 35, and 40 mW. The other parameters are
I 2 I 02 11.3 mW.
two peaks, as shown in the Fig. 5(a1). When I 4 keeps increasing, jG4 i will become too weak to be observed, so the signal
shows first suppression and then enhancement. As I 4 gets sufficiently high, the dark state will be so strong that the enhancement peak on the right induced by jG1&4 i will be lowered in
contrast and almost unable to be obtained. When we fix
I 1 8 mW, since jG1&4 i have be weakened, the dressing effect induced by E1 & E 4 is hardly observed. When I 4 is low, the
original double-peak enhancement will become asymmetric,
and one suppression dip appears between the two peaks. When
I 4 is higher, the dark state will be strengthened and jG4 i will
Zheng et al.
be too weak to be observed, so the signal shows first enhancement then suppression. If I 4 is sufficiently high, jG4 i will be
too weak to be observed, which makes the curve behaves as
pure suppression, as shown in Fig. 5(a3). The experimental results match well with the theoretical results in Figs. 5(a2) and
5(a4), respectively.
When we fix I 4 and change I 1 , the dressing effect induced
by E1 & E4 can be investigated, and the results are shown in
Fig. 5(b). By setting Δ1 −30 MHz and I 4 4 mW, the signal
shows the evolution from double-peak enhancement, to first
enhancement then suppression, finally to pure suppression
[Fig. 5(b1)]. And when we set I 4 40 mW, the signal will
change from first suppression then enhancement, to pure suppression [Fig. 5(b3)].
When I 4 4 mW is fixed, two bright states and one dark
state can be observed. As I 1 is low, both jG4 i and jG1&4 i
are strong in comparison with the dark states, which makes
the signal exhibit a double peak. When I 1 gets higher, jG1&4 i
is weakened, which makes the enhancement effect induced by
E 1 & E4 unattainable. Moreover, because jG4 i shifts, the
dressing effect induced by E4 will tend to lead to suppression.
For the weakening of a bright state and strengthening of a
dark state, the signal will show enhancement first and suppression later. When I 1 is sufficiently high, the dark state created by E4 would be so strong that the whole curve behaves as
dominating pure suppression. When I 4 40 mW, jG4 i will
be far away from resonant point, that only jG1&4 i and the
dark state can be observed, which makes the signal behave
as first suppression then enhancement, as shown in Fig. 5(b3).
As I 1 increases, jG1&4 i become weaker, and the dark state
becomes stronger. The whole curve behaves as dominating
pure suppression. The experimental results shown in
Figs. 5(b1) and 5(b3) match well with the theoretical results
shown in Figs. 5(b2) and 5(b4).
Moreover, when we change I 4 from low to high with I 1
fixed at low value, the shift of the bright state jG1&4 i can
be investigated with Δ1 −60 MHz and Δ1 −120 MHz. At
Δ1 −60 MHz, jG1&4 i, jG4 i, and the dark state can be obtained, in which jG4 i will be weakened with increasing I 4 .
According to the term Δ1 Δ4 jG1 j2 ∕jG4 j2 0; jG1&4 i will
shift to the left with increasing I 4 . Since Δ1 is much larger,
this shift phenomenon will be more obvious than that in
the case with little detuning. In fact, as we can see from
Fig. 4(c1), jG4 i even shifts to the dark state and even overlaps with the other bright state. So, the signal shows first
enhancement then suppression when I 4 is large. At
Δ1 −120 MHz, jG4 i can shift much further. The separation
between the two overlapped bright states can be observed, as
shown in Fig. 5(c3). The experimental results in Figs. 5(c1)
and 5(c3) and the theoretical results in Figs. 5(c2) and
5(c4) agree well with each other.
In a comparison between the curves in the Figs. 5(a3) and
5(b3), one would found that the signal can be switched to the
dark state by the increment of I 4 as well as I 1 , due to the correlations between the bright and dark states created by E 4 ,
and between the bright states created by E 1 and the dark
states created by E 4 . Obviously, the latter interaction is much
stronger than the former one, since the necessary transition I 1
is lower than I 4 , and the obtained suppression dip is much
deeper in high I 1 . To get a dark state, increasing I 1 is more
efficient. In a comparison between the signals in Figs. 5(a1)
Vol. 31, No. 6 / June 2014 / J. Opt. Soc. Am. B
1269
and 5(b1), one can find that when I 1 is low, no matter how
strong I 4 is there still exists enhancement. The signal cannot
be switched to the dark state by changing I 4 singly. While I 1
increases, the dark state can be obtained even when I 4 is quite
low. Moreover, if we compare the signals in Figs. 5(a1), 5(c1),
and 5(c3), the bright states created by E 1 & E 4 almost maintain their positions with the increment of I 4 when jΔ1 j is small.
But as jΔ1 j is large, this state would shift from right to left and
finally overlap with the other bright state. If jΔ1 j is larger, the
separation of the two overlapped states will be obtained with
the same power. The phenomena mentioned above demonstrate that not only field power but also jΔ1 j can shift the
bright state created by E 1 & E4 .
5. MULTIDARK STATES OF MWM
When the dressing fields E2 , E 02 , and E4 are all sufficiently
strong, the dressing effects on the FWM process will be affected by quantum interference. In the interaction picture,
we can obtain the Hamilton for the Y-type four-level system
H ℏΔ1 j1ih1j ℏΔ1 Δ2 j2ih2j ℏΔ1 Δ4 j4ih4j
− ℏG1 j1ih0j G2 G02 j2ih1j G4 j4ih1j H:c::
(6)
Here, for convenience, we replace G2 G02 with G2 . Under the
resonant conditions Δ1 Δ2 0 and Δ2 0, we can easily
get three identified dark states [24]
G2 j0i − G1 j2i
G
jD1i p
≈ j0i − 1 j2i;
G2
jG1 j2 jG2 j2
(7a)
G4 j0i − G1 j4i
G
jD2i p
≈ j0i − 1 j4i;
2
2
G4
jG4 j jG1 j
(7b)
G4 j2i − G2 j4i
jD4i p
cos θj2i − sin θj4i;
(7c)
jG2 j2 jG4 j2
p
p
where cos θ G4 ∕ jG2 j2 jG4 j2 , sin θ G2 ∕ jG2 j2 jG4 j2 .
The total dark state amplitude is then given by
jDi jD1i jD2i jD4i
2j0i cos θ − G1 ∕G2 j2i − G1 ∕G4 sin θj4i:
(8)
In order to understand how the three dark states interfere
with each other, we would like to find the population distributions in the “dark” state, jhDjψij2 , where
jψi c0 j0i c1 j1i c2 j2i c3 j4i
(9)
is the wave function of the atom in its bare-state basis. From
Eqs. (8) and (9), we get
2
G 2
G1
jhDjψij2 4ρ00 cos θ − 1 ρ22 sin θ ρ44
G2
G4
G
G1
4 Re cos θ − 1 ρ20 −
sin θ ρ40
G2
G4
G1
G1 (10)
−
sin θ cos θ −
ρ42 ;
G4
G2
1270
J. Opt. Soc. Am. B / Vol. 31, No. 6 / June 2014
Zheng et al.
where
ρ20 iG2
iG1
ρ ;
d2 d1 jG2 j2 ∕d2 jG4 j2 ∕d4 00
(11a)
ρ40 iG4
iG1
ρ ;
d4 d1 jG2 j2 ∕d2 jG4 j2 ∕d4 00
(11b)
ρ21 −
iG1
ρ20 ;
d21 jG4 j2 ∕d7
(11c)
ρ41 −
iG1
ρ40 ;
d41 jG2 j2 ∕d7
(11d)
ρ42 −iG2 ρ41 iG4 ρ12 ∕d7 ;
(11e)
where d41 iΔ4 Γ41 and d7 iΔ4 − Δ2 Γ42 . We also assume the population distributions ρ00 ≈ 1, ρ22 ≈ 0, and ρ44 ≈ 0
in Eq. (10). By considering quantum interference among the
three states j0i, j2i, and j4i, the intensity of the FWM signal is
2
0
2
obtained as I ∝ jN 0 μρ3
10 j , where N N1 − jhDjψij . The
2
term, jhDjψij , describes the population distribution in dark
state jDi, and N is the particle number density It is obvious
that the quantum interference effect can exert an influence
on the FWM signals. In fact, according to Eq. (8), there are
three dark states jD1i, jD2i, and jD4i in the signal. By
changing I 4 , the three dark states can be distinguished for
investigation.
The experimental results are shown in Fig. 6(a), with Δ4
being scanned at different I 4 . The FWM signal under the quantum interference effect can be observed as a multidark state
phenomenon. With I 4 increasing from low to high, the original
Signal intensity
(a1)
(b1)
G2 j0i − G1 j2i
G
≈ j0i − 1 j2i;
jD0 1i p
2
2
G2
jG1 j jG2 j
(12a)
G3 j0i − G1 j3i
G
jD0 2i p
≈ j0i − 1 j3i;
2
2
G3
jG3 j jG1 j
(12b)
G4 j0i − G1 j4i
G
jD0 3i p
≈ j0i − 1 j4i;
G4
jG4 j2 jG1 j2
(12c)
G3 j4i − G4 j3i
jD0 4i p
cos θj4i − sin θj3i;
jG3 j2 jG4 j2
(12d)
(a2)
(b2)
4
8
15
25
35
45
55
0
∆ 4 (MHz)
(a)
(b)
150
(d1)
(c2)
p
p
where cos θ G3 ∕ jG3 j2 jG4 j2 , sin θ G4 ∕ jG3 j2 jG4 j2 .
The total dark state amplitude is then given by
(d2)
(c3)
(d3)
(c4)
5
150
I 4 (mW)
(c1)
Signal intensity
enhancement induced by E 4 is split by the multidark state,
shows the variation from two peaks to three peaks. According
to Eq. (7), the dark states jD2i and jD4i are related to E4 , while
jD1i is independent of E 4 . In this way, the amplitude of jD1i
will not change with I 4 . Even if I 4 is quite low, the suppression
dip induced by jD1i still can be observed, which splits the
enhancement into two peaks [Fig. 6(a) at I 4 4 mW]. With
increasing I 4 , the amplitudes of jD2i and jD4i will grow,
and since E 1 is weaker than E2 , there will be jhD2jψij2 >
jhD4jψij2 which means jD2i is stronger than jD4i. When I 4 increases to 8 mW, the suppression dips induced by jD1i and
jD2i are obtained, which split the enhancement into triple
peaks. As we set I 4 55 mW higher, all three dark states will
appear in the signal, as shown in Fig. 6(b). The three suppression dips represent jD1i, jD2i, and jD4i from right to left, respectively. The theoretical results in Figs. 6(a2) and 6(b2)
respectively agree well with the experimental results in
Figs. 6(a1) and 6(b1).
If we turn off E 02 and turn on E3 & E 03 , the quantum interference of SWM process can be investigated in the K-type
atom system, as shown in Fig. 1(d). According to the Hamilton
[Eq. (1)] of the K-type five-level system, we can easily get four
identified dark states under the resonant conditions
Δ1 Δ2 0, Δ2 0, and Δ3 0 [24]
(d4)
10
30
50
55
60
70
150
0
I 4 (mW)
∆ 4 (MHz)
(c)
(d)
150
Fig. 6. (a) Intensity of (a1) experimental results and (a2) theoretical
calculations of the quantum interference effect in the FWM signal EF
versus Δ4 for I 4 4, 8, 15, 25, 35, 45, and 55 mW. Powers of other all
laser beams are I 1 1 mW and I 2 I 02 20 mW. (b) The FWM signal
taken at I 4 55 mW in (a) versus Δ4 . (c) Measured SWM signals versus Δ4 for I 4 5, 10, 30, 50, 55, 60, and 70 mW. (c1) Signal obtained
with all laser beams on. (c2) Signal obtained with the laser beams E2
blocked. (c3) Sum of the pure dressing effect. (c4) Theoretical calculations corresponding to (c3). Powers of other laser beams are
I 1 3.6 mW, I 2 1 mW, and I 3 I 03 51 mW. (d) The FWM signal
taken at I 4 70 mW in (c) versus Δ4 .
jD0 i jD0 1i jD0 2i jD0 3i jD0 4i
G
G1
G1
3j0i − 1 j2i −
− cos θ j4i −
sin θ j3i: (13)
G2
G4
G3
The wave function of the atom in its bare-state basis is written as
jψi c0 j0i c1 j1i c2 j2i c3 j3i c4 j4i:
From Eqs. (13) and (14), we get
(14)
Zheng et al.
Vol. 31, No. 6 / June 2014 / J. Opt. Soc. Am. B
2
2
G1
G1
ρ22 sin θ ρ33
G2
G3
2
G1
G
− cos θ ρ44 2 Re −3 1 ρ20
G4
G2
G1
G1
−3
sin θ ρ30 − 3
− cos θ ρ40
G3
G4
G G1
G G1
1
sin θ ρ32 1
− cos θ ρ42
G2 G3
G2 G4
G1
G1
sin θ
− cos θ ρ43 ;
G3
G4
jhD0 jψij2 9ρ00 (15)
where
ρ20 iG2
iG1
ρ ;
d2 d1 jG2 j2 ∕d2 jG3 j2 ∕d3 jG4 j2 ∕d4 00
(16a)
ρ30 iG3
iG1
ρ ;
d3 d1 jG2 j2 ∕d2 jG3 j2 ∕d3 jG4 j2 ∕d4 00
(16b)
ρ40 iG4
iG1
ρ ;
d4 d1 jG2 j2 ∕d2 jG3 j2 ∕d3 jG4 j2 ∕d4 00
(16c)
ρ21 −
iG1
ρ20 ;
d21 jG3 j2 ∕d8 jG4 j2 ∕d07
(16d)
ρ31 −
iG1
ρ30 ;
d31 jG2 j ∕d08 jG4 j2 ∕d09
(16e)
ρ41 −
2
d41 jG2
iG1
j2 ∕d
7
jG3 j2 ∕d9
ρ40 ;
(16f)
ρ32 −iG2 ρ31 iG3 ρ12 ∕d08 ;
(16g)
ρ42 −iG2 ρ41 iG4 ρ12 ∕d7 ;
(16h)
ρ43 −iG3 ρ41 iG4 ρ13 ∕d9 ;
(16i)
1271
above, the SWM signal under the quantum interference effect
will show multidark state phenomenon, i.e., the splitting of the
original enhancement signal from one peak to two peaks, to
three peaks finally.
According to Eq. (12), the dark states jD0 3i and jD0 4i are
related to E 4 , while jD0 1i and jD0 2i are independent of E4 .
In this way, the suppression dip induced by jD0 1i and jD0 2i
can be observed at low I 4 , appearing at both sides of the enhancement signal [as shown in Fig. 6(c) at I 4 5 mW]. With
increasing I 4 , the amplitudes of jD0 3i and jD0 4i will grow, and
since E1 is weaker than E3 , there will be jhD0 3jψij2 >
jhD0 4jψij2 which means jD0 3i is stronger than jD0 4i. When
I 4 increases to 10 mW, the suppression dips induced by
jD0 1i, jD0 2i, and jD0 3i are obtained, which split the enhancement into double peaks. As we further increase I 4 , all the four
dark states will appear in the signal, and the signal will be split
into triple peaks. The structure of the signals at I 4 70 mW is
shown in Fig. 6(d). One can clearly distinguish jD0 3i and jD0 4i
from the four suppression dips. Figures 6(c4) and 6(d4) are
the theoretical results, which agree well with the experimental
results in Figs. 6(c3) and 6(d3), respectively.
6. CONCLUSION
In summary, we have investigated the probe transmission,
FWM, and fluorescence signals with dressing field detuning
scanned and experimentally demonstrated the correlation
among the two dark states and two bright states by changing
the powers of the probe and dressing fields, respectively. In
addition, when we investigate the enhancement and suppression signal of FWM under four different kinds of power combinations, the state of FWM switches from double-bright
states to dark states. Moreover, the power-controlled quantum
interference between multiple dark states has been studied
in FWM and SWM processes. We also have demonstrated
that more dark states will appear with the power of the
external-dressing field increasing.
ACKNOWLEDGMENTS
where d31 −iΔ3 Γ31 , d8 iΔ3 Δ2 Γ23 , d07 −i
Δ4 − Δ2 Γ24 , d08 −iΔ3 Δ2 Γ32 , d9 iΔ3 Δ4 Γ43 ,
and d09 −iΔ3 Δ4 Γ34 . In Eq. (15), the assumptions
ρ00 ≈ 1, ρ22 ≈ 0, ρ33 ≈ 0, and ρ44 ≈ 0 are adopted. By considering
the quantum interference among j0i, j2i, j3i, and j4i, the
2
intensity of the SWM signal is obtained as I jN 0 μρ5
10 j , where
N 0 N1 − jhD0 jψij2 .
By changing I 4 , four dark states jD0 1i, jD0 2i, jD0 3i, and jD0 4i
can be distinguished. The experimental results are shown in
Fig. 6(c), with Δ4 being scanned with different I 4 . When all
five laser beams are turned on, two external-dressed SWM
D
signals ED
S2 and E S4 will form simultaneously, as shown in
Fig. 6(c1). When E 2 is blocked, only the measured SWM
signal E S4 remains, as shown in Fig. 6(c2). By subtracting
E S4 [Fig. 6(c2)] and the height of each baseline (representing
E S2 ) from the total signal [Fig. 6(c1)], the sum of the pure
D
dressing effect, i.e., ED
S2 − E S2 and E S4 − E S4 can be obtained,
as shown in Fig. 6(c3). Similar to the FWM signal mentioned
This work was supported by the 973 Program
(2012CB921804), NSFC (11104216, 61308015, 61078002,
61078020, 11104214, 61108017, and 61205112), RFDP
(20110201110006, 20110201120005, and 20100201120031),
FRFCU (2012jdhz05, xjj2011083, xjj2011084, xjj2012080, and
xjj2013089), and CPSF (2012M521773).
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