Phase modulation of transient evolution of atomic response in open lambda-type system with spontaneously generated coherence

Optics Communications 283 (2010) 1810–1816

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Phase modulation of transient evolution of atomic response in open lambda-type system with spontaneously generated coherence Xijun Fan a,*, Bian Liang a, Zhendong Wang a, Dianmin Tong b a b

College of Physics and Electronics, Shandong Normal University, Jinan 250014, China School of Physics and Microelectronics, Shandong University, Jinan 250100, China

a r t i c l e

i n f o

Article history: Received 13 October 2009 Received in revised form 22 December 2009 Accepted 23 December 2009

Keywords: Open system Atomic response Transient evolution Relative phase Gain without inversion

a b s t r a c t We study the effects of the relative phase, U, between the probe and driving fields on the transient evolution of the atomic response from different respects in an open lambda-type system with spontaneously generated coherence. We find that: variation of value of U will obviously change the transient evolution process, transient and stationary values of gain, absorption and dispersion but has no any effect on the transient evolution of populations. For the same value of U, the stationary gain without inversion (GWI) when the incoherent pumping exists is much larger than that when the incoherent pumping is absent, the time needed to reach at the stationary state when the incoherent pumping exists is much shorter than that when the incoherent pumping is absent. The initial condition varying has remarkable effects on the phase-dependent transient evolution process, transient value of GWI but doesn’t vary size of stationary value of GWI. Varying the ratio C of the atomic injection rates and exit rate c0 has evident effects on the phase-dependent transient evolution process, transient and stationary values of GWI. Our study result shows that the transient evolution of the atomic response in the open system presents some considerable difference from that in the corresponding closed system, specially in the open system we can get much larger GWI than that in the corresponding closed system. In addition, we give a brief discussion about effects of the relative phase on the transient evolution of the atomic response in the open system without SGC, and obtain some results much different from those obtained in the open system with SGC. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Quantum coherence and interference in atomic systems have led to a number of important consequences, such as lasing without inversion (LWI) [1], coherent population trapping (CPT) [2], potentiality for sensitive measurements of magnetic fields [3], and electromagnetically induced transparency (EIT) [4], in which LWI has attracted much more attention (for recent articles see, for example, Refs. [5–7] and references therein). There are many ways to generate quantum coherence and interference. Generally, they can be realized by coherent driving fields or by initial coherence injections, the interference between different spontaneous emissions paths also can lead to the generation of coherent superposition state. The interference effect caused by spontaneous emission is called as spontaneously generated coherence (SGC). There have been considerable interests in the studying SGC [8–27]. It has been shown that atomic systems with SGC are sensitive to the relative phase (RP) of the applied fields [28–44]. Most of the previous

works related to SGC were focused on the steady-state response of the medium. There are several works on the transient properties of the three-level systems [45–48], but all these works did not include the effect of SGC. There only few works on the transient properties of the three-level systems. Wu et al. found that the transient response in closed lambda-type [18] and V-type [34] threelevel systems was greatly affected by SGC. In this paper, we will study the effect of the relative phase on the transient evolution of the atomic response in an open lambda-type three-level system with SGC and compare with the case in the corresponding closed system [18].

2. Motion equations Consider an open lambda-type three-level atomic system with an upper state j1i and two lower close-lying states j2i and j3i, as illustrated in Fig. 1. Transition between j1i and j2i is driven *

*

by a strong coherent driving field Ec ¼ ec eixc t þ c:c:, frequency xc * Corresponding author. E-mail address: [email protected] (X. Fan). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.12.062

and Rabi frequency Xc ¼ a

pumping

rate

2K

*

*

e c  d12 =2h. An incoherent pump with

and

a

weak

coherent

probe

field

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X. Fan et al. / Optics Communications 283 (2010) 1810–1816

2γ 1

Ec ,ωc 2Λ

r_ 23 ¼ ðK þ iDp  iDc Þr23 þ 2ðc1 c2 Þ1=2 gr11 cosh þ iXc r13  iXp r21 :

|1>

Δc

Δp

γ0

ð1fÞ

2γ 2

Ep ,ωp

J2 |2>

J3

γ0

|3>

γ0 Fig. 1. An open lambda-type three-level atomic system. *

*

Ep ¼ ep eixp t þ c:c: with the frequency xp and Rabi frequency *

The above equations are constrained by rij ¼ r where rij ði–jÞ is the atomic polarization between states jii and jji, rii (i ¼ 1; 2; 3) is the population of the state jii, Dp ¼ x13  xp and Dc ¼ x12  xc are the detunings of the probe and driving fields from their relevant atomic transitions, respectively. 2ðc1 c2 Þ1=2 cos hgr11 represents the quantum interference effect resulting from the crosscoupling between the spontaneous emission j1i $ j3i and j1i $ j2i, that is the SGC effect. If levels j2i and j3i lie so close that the SGC*effect has to be taken into account, then g ¼ 1; otherwise * g ¼ 0. d13 and d12 are not orthogonal, which is necessary for the existence of the SGC effect; h represents the angle between the two dipole moments. When g ¼ 1, J 2 = J 3 = c0 = 0, Eqs. (1) reduce to the equations for a closed K -type three-level atomic system with SGC [18]. The existence of the SGC effect makes the system become quite sensitive to relative phase (U) of the probe and driving fields, and then the Rabi frequencies have to be treated as complex parameters. Let Up and Uc denote phases of the probe and driving fields, respectively, writing Xp ¼ Gp expðiUp Þ, Xc ¼ Gc expðiUc Þ, q13 ¼ r13 expðiUp Þ, q12 ¼ r12 expðiUc Þ, q23 ¼ r23 expðiUÞ, qii ¼ rii , where U ¼ Up  Uc , Gp and Gc are real, then we get the motion equations about qij which are found to be identical with Eqs. (1) except that g, Xp and Xc are replaced by g expðiUÞ, Gp and Gc , respectively. When g ¼ 1 and J 2 = J 3 = c0 = 0, the equations about qij describe a closed lambda-type three-level atomic system with SGC [18]. The probe gain (absorption) and dispersion correspond to the imaginary and real parts of q31 , respectively. If Imðq31 Þ > 0, the system exhibits gain for the probe field; if Imðq31 Þ < 0, the probe field is attenuated. When q11 < q22 < q33 and Imðq31 Þ > 0, the gain without inversion (GWI) and then LWI can be realized. The dispersion is determined by Reðq31 Þ; Reðq31 Þ > 0 corresponds to the red shift of the probe field frequency, and Reðq31 Þ < 0 to the blue shift [49].  ji ,

*

Xp ¼ ep  d13 =2h are applied between levels j1i and j3i. 2c1 and 2c2 denote the spontaneous decay rates from level j1i to levels j3i and j2i respectively. J 2 and J 3 are the atomic injection rates for levels j2i and j3i respectively; the ratio of the atomic injection rates is C = J 3 / J 2 ; c0 is the atomic exit rate from the cavity. We also assume that the number of interacting atoms is constant, which means that J 2 + J 3 = c0 . Using the rotating-wave and the electric dipole approximations, the density matrix equations of motion of this system can be written as [39]:

r_ 11 ¼ 2ðc1 þ c2 Þr11 þ iðXp r31  Xp r13 Þ þ 2Kr33 þ iðXc r21  Xc r12 Þ  c0 r11 ;

ð1aÞ

r_ 22 ¼ 2c2 r11 þ iðXc r12  Xc r21 Þ þ J2  c0 r22 ;

ð1bÞ

r_ 33 ¼ 2c1 r11  2Kr33 þ iðXp r13  Xp r31 Þ þ J3  c0 r33 ;

ð1cÞ

r_ 12 ¼ ðc1 þ c2 þ iDc Þr12 þ iXp r32  iXc ðr11  r22 Þ;

ð1dÞ

r_ 13 ¼ ðc1 þ c2 þ K þ iDp Þr13 þ iXc r23  iXp ðr11  r33 Þ;

ð1eÞ

0.08

0.08 Λ =0 η=1

Φ =π/2 Φ =π/4

0.04

Φ =9π/8

Re(ρ31)

Im(ρ31)

Φ =π/8

0.00

Φ =0,Φ =π

Φ =9π/8

-0.04

Φ =5π/4

open

0

1

2

τγ1

3

4

Φ =π

0.04

(a) 5

Φ =π/4

Φ =π/2,Φ =3π/2

closed

Φ =0

0

1

2

τγ1

3

open

2

4

τγ1

3

(b)

4

Φ = π/2

5

Φ =π/8

0.00 Φ =9π/8

-0.02

5

-0.06

Φ =0,Φ =π

Φ =5π/4

-0.04

(c)

Λ =0 η =1

Φ =π/4

0.02

Φ =π/8

-0.04

1

0.04 Re(ρ31)

-0.02

0

0.06

0.00

-0.06

-0.08

Φ =5π/4

0.02

Φ =π/4 Φ =π/2

Λ =0 η=1

Φ =9π/8

Φ =0,Φ =π

Φ =π/8

-0.04

Φ =3π/2

0.06

Im(ρ31)

Λ =0 η=1

Φ =5π/4

0.04

0.00

-0.08

Φ =3π/2

Φ =3π/2

0

1

2 τγ

closed 3

4

(d) 5

1

Fig. 2. Transient evolution of Imðq31 Þ and Reðq31 Þ for different values of U in the open system and corresponding closed system. The parameters values are Dp = Dc =0. g ¼ 1, c1 ¼ 1, h = p=4; Gc = 8sin h; Gp = 0.01 sin h; c2 = 2.0, c0 = 0.8, C = J3 / J 2 = 7, K ¼ 0. The initial condition is q22 ð0Þ = q33 ð0Þ = 0.5 and others qij ð0Þ ¼ 0 (i; j ¼ 1; 2; 3).

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0.06

0.08

Φ =0,Φ =π Φ =9π/8 Φ =5π/4

-0.04 Φ =3π/2

-0.08

Im(ρ31)

Φ =π/4 Φ =π/8

0.00

0

1

2

Λ =0.4γ1 η =1

Φ =9π/8

0.02 Φ =5π/4

0.00

Φ =π/ 2 ,Φ =3π/2

Φ =π/4

-0.02

open

τγ1

Φ =π

0.04

η =1

Φ =π/2

0.04 Im(ρ31)

Λ =0.4γ1

3

-0.04

(a)

4

-0.06

5

Φ =π/8 Φ =0

0

1

2

τγ1

closed (b)

3

4

5

Fig. 3. Transient evolution of Imðq31 Þ for different values of U in the open system and corresponding closed system when the incoherent pump presents. The initial condition and parameters values are same as those in Fig. 2 but K ¼ 0:4c1 .

3. The analysis of the transient evolution of the atomic response In the following, we investigate effects of relative phase on the transient evolution of the atomic response from different respects by using the numerical result from the density matrix motion equations about qij .

tain the largest values. If U1  U2 ¼ p, the behaviors of Imq31 and Req31 corresponding to U1 and U2 are just opposite to those in the open system. t Comparing with the corresponding closed system, in the open system the gain is larger and the time needed to reach the stationary values of the gain (absorption) and dispersion is shorter. 3.2. Transient evolution of Imq31 when the incoherent pump presents

3.1. Transient evolution of Imq31 and Req31 when the incoherent pump is absent First we consider the transient evolution of the gain (absorption) Imq31 and dispersion Req31 in the open system and corresponding closed system respectively when the incoherent pump is absent. Fig. 2 shows that the transient gain (absorption) and dispersion properties can be greatly affected by the relative phase U: r In the open system, when U ¼ 0 and U ¼ p, the transient and steady values of Imq31 and Req31 are near to zero, the electromagnetically induced transparency (EIT) occurs. When U ¼ p=2 (U = 3p/2), the transient and steady gain (absorption) can get the largest values. If U1  U2 ¼ p, Imq31 corresponding to U2 is always positive, the system exhibits gain for the probe field, Req31 corresponding to U2 is always negative and this corresponds to the blue shift of the probe field frequency; while Imq31 corresponding to U1 is always negative, the system exhibits absorption for the probe field, Req31 corresponding to U2 is always positive and this corresponds to the red shift. s In the corresponding closed system, when U ¼ p=2 and U ¼ 3p=2, Imq31 equals always to zero, there is no gain (absorption) for the probe field; while Req31 > 0 for U ¼ p=2 and this corresponds to the red shift, Req31 < 0 for U ¼ 3p=2 and this corresponds to the blue shift, so EIT cannot occur. When U ¼ p (U ¼ 0), the transient and steady gain (absorption) can ob-

By comparing Fig. 3 with Fig. 2 we can see that transient evolution of the gain (absorption) when the incoherent pump presents is similar to that when the incoherent pump is absent; however, when the incoherent pump presents, the gain (absorption) can more rapidly reach the stationary value, the oscillatory amplitudes in the transient evolution process are smaller, the steady gain values increase obviously except U ¼ 0 and U ¼ p for the open system and U ¼ p=2 and U ¼ 3p=2 for the closed system. 3.3. Transient evolution of the populations when the incoherent pump presents and doesn’t Fig. 4 shows that the transient evolution of the population distribution when the incoherent pump presents is very similar to that when the incoherent pump is absent; the steady populations of the upper levels 1 and 2 are larger than those without incoherent pump, respectively, the steady population of the lowest level 1 is much smaller than that without incoherent pump, and this is due to that the incoherent pump makes the transition probability from lower level to upper level increasing, but q11 < q22 < q33 is still satisfied. Therefore, there is no population inversion on the probe transition j1i $ j3i. For different values of U, the population distribution curves are completely superposable, that is to say, the

0.8

1.0

ρ 33

0.6

Λ =0 η =1

ρ22

0.4

Population

Population

0.8

ρ11

0.2

0.6

ρ33 0.4

Λ =0.4γ1

ρ22

η=1

ρ11

0.2

(a)

0.0

(b)

0.0 0

1

2

τγ1

3

4

5

0

1

2

τγ1

3

4

5

Fig. 4. Transient evolution of the populations in the open system when the incoherent pump present and doesn’t. (a) K ¼ 0 and g ¼ 1, (b) K ¼ 0:4c1 and g ¼ 1. The initial condition and other parameters values are the same as those in Fig. 2. The population curves corresponding to different values of U overlap completely.

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X. Fan et al. / Optics Communications 283 (2010) 1810–1816

0.0025

0.14

0.0015

(2)

0.0010

Λ =0 η =1 Φ =0

(3)

0.0005

0

1

2

τγ1

3

0.06 (2)

0.04

4

-0.02

5

(1)

0

(b) 1

2

τγ 1

3

4

5

0.12

0.0008

(2)

0.0004

Im(ρ31)

(1) Λ =0.4γ1

(c) 1

2

τγ1

0.06

(2)

0.02

0.0000 0

η =1 Φ =π/2

0.08

0.04

η =1 Φ =0

(3)

3

Λ =0.4γ1

(3)

0.10

0.0012 Im(ρ31)

0.08

0.00

(a)

0.0016

-0.0004

Λ =0 η =1 Φ = π /2

0.02

0.0000 -0.0005

(3)

0.10 Im(ρ31)

Im(ρ31)

0.12

(1)

0.0020

4

(1)

(d)

0.00 5

0

1

2

τγ1

3

4

5

Fig. 5. Transient evolution of Imðq31 Þ in the open system under different initial conditions. (a) g ¼ 1, K ¼ 0, U ¼ 0, (b) g ¼ 1, K ¼ 0, U ¼ p=2, (c) g ¼ 1, K ¼ 0:4c1 , U ¼ 0 and (d) g ¼ 1, K ¼ 0:4c1 , U ¼ p=2. Other parameters values are same as those in Fig. 2. Curves: (1) q33 ð0Þ ¼ 1; others qij ð0Þ ¼ 0 ði; j ¼ 1; 2; 3Þ; q22 ð0Þ ¼ q33 ð0Þ ¼ 0:5; others qij ð0Þ ¼ 0 ði; j ¼ 1; 2; 3Þ; (3) q22 ð0Þ ¼ 1, others qij ð0Þ ¼ 0 ði; j ¼ 1; 2; 3Þ.

transient evolution of the population distribution can’t be affected by varying values of the relative phase U.

3.4. Transient evolution of Imq31 under different initial conditions Generally speaking, the transient evolution of the atomic response is relative to the initial condition [42]. We have just discussed the transient evolution of Imq31 for different values of U under the initial condition q22 ð0Þ ¼ q33 ð0Þ ¼ 0:5 and others qij ð0Þ ¼ 0 (i; j ¼ 1; 2; 3). Now we investigate the transient evolution of Imq31 only for U ¼ 0 and U ¼ p=2 under different initial conditions: (1) q33 ð0Þ ¼ 1, others qij ð0Þ ¼ 0 ði; j ¼ 1; 2; 3Þ; (2) q22 ð0Þ ¼ q33 ð0Þ ¼ 0:5, others qij ð0Þ ¼ 0 ði; j ¼ 1; 2; 3Þ; (3) q22 ð0Þ ¼ 1, others qij ð0Þ ¼ 0 ði; j ¼ 1; 2; 3Þ. From Fig. 5 we can see that the initial condition varying can affect the transient evolution process of Imq31 but doesn’t affect steady value of Imq31 . If U ¼ 0, the transient evolution of gain with the incoherent pump is similar to that without the incoherent pump. The value of the transient gain is the biggest under the initial condition (1), the transient gain is the smallest under the initial conditions (3), but the steady values of the gain are same for different initial conditions, and the steady value (about 0.00215) for the case without the coherent pump is obviously larger than that (about 0.001) for the case with the coherent pump. If U ¼ p=2, like to U ¼ 0, the transient evolution of gain with the incoherent pump is also similar to that without the incoherent pump, however the effects of the initial conditions on the transient gain are different from that when U ¼ 0. The value of transient gain is the biggest under the initial condition (3), the smallest under the initial conditions (1). The steady values of the gain also are same for different initial conditions, but the steady value (about 0.0086) for the case without the coherent pump is much smaller than that (about 0.04039) for the case with the coherent pump. For both cases U ¼ 0 and U ¼ p=2, the system can reaches the steady state more rapidly when the incoherent pump presents.

3.5. Transient evolution of Imq31 under different atomic exit rate and ratio of injection rates Fig. 6 shows that the transient evolution of Imq31 depends on not only the relative phase, U, but also the ratio of the atomic injection rates, C, and the atomic exit rate, c0 . Fig. 6(a) and (b) represents the transient evolution of Imq31 for different values of C and the same value of c0 . With value of C increasing, when U ¼ 0, the gain of the probe field increases gradually; however, when U ¼ p=2, the gain decreases gradually. Fig. 6(c) and (d) give the transient evolution of Imq31 for different values of c0 and the same value of C. With value of c0 increasing, when U ¼ 0, the gain of the probe field decreases gradually; while when U ¼ p=2, the gain increases gradually. It is obviously that for a same value of U, the effects of variation of the atomic exit rate and ratio of the injection rates on the transient evolution of Imq31 is just opposite. 4. A brief discussion for effect of the relative phase in the open system without SGC We have already analyzed in detail the effects of the relative phase between the probe field and driving field on the transient evolution of the atomic response from different respects in the open lambda-type system when SGC presents. In the following we will briefly discus effects of the relative phase on the transient evolution of the atomic response when SGC is absent. We assume that phases of the probe field and driving field are Up and Uc , respectively, and Uc ¼ 0. Then the relative phase between the probe field and driving field is U ¼ Up  Uc ¼ Up . The Rabi frequencies of the probe field and driving field can written as Xp ¼ Gp expðiUp Þ ¼ Gp expðiUÞ and Xc ¼ Gc expðiUc Þ ¼ Gc , respectively, Gp and Gc are real. When SGC is absent and the relative phase is not considered, the density matrix equations of motion of the open system is same as Eqs. (1) except that Eq. (1f) becomes following form

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X. Fan et al. / Optics Communications 283 (2010) 1810–1816

0.0025

0.08

C=7 C=3 C=1

0.0015

C=1/3

0.0010

Λ=0 η=1 Φ =0

0.0005 0.0000

0

1

2

(a) 3

τγ1

4

0.00 5

C=1/3 C=1 C=3 C=7

(b)

0

1

2

0.0010

0

1

2

3

τγ1

5

0.04 γ 0=0.6

Λ =0 η=1 Φ =0

0.0005

Im(ρ31)

γ0=0.4

γ0=0.6

4

Λ =0 η=1 Φ =π/2

0.06

0.0015

τγ1

3

0.08

γ0=0.2

0.0020

Im(ρ31)

0.04 0.02

0.0025

0.0000

Λ =0 η=1 Φ =π/2

0.06 Im(ρ31)

Im(ρ31)

0.0020

0.02

(c) 4

0.00 5

γ 0=0.4

(d)

γ 0=0.2

0

1

2

τγ 1

3

4

5

Fig. 6. Transient evolution of Imq31 under different the atomic exit rate and ratio of the injection rates in the open system without the incoherent pump. (a) and (b) c0 = 0.8, (c) and (d) C = J3 /J2 = 1. Other parameters values are same as those used in Fig. 2.

r_ 23 ¼ ðK þ iDp  iDc Þr23 þ iXc r13  iXp r21 ;

Physical meaning of r0i;j ði; j ¼ 1; 2; 3Þ in Eqs. (2) is same as that of qi;j ði; j ¼ 1; 2; 3Þ used above when we study the effects of the relative phase in the open system with SGC. In the following, we use numerical result from Eqs. (2) to give a brief discussion for effect of relative phase when SGC is absent. Fig. 7 illustrates transient evolution of the probe gain (absorption) Im ðr031 Þ and dispersion Re ðr031 Þ for different values of the relative phase U in the open system without SGC when the incoherent pump is absent. The other parameters values are same as those in Fig. 2. By comparing Fig. 7 with Fig. 2(a) and (b) we can see that: r For the same value of U, transient evolution rule of the probe dispersion and specially the probe gain (absorption) when SGC is absent is much different from that in the case with SGC. Regardless of that U takes what value, transient and steady values of the probe gain, absorption and dispersion when SGC is absent always are much smaller than those in the case with SGC, but the relative modulation role of some values of U on these values when SGC is absent is more evident than that in the case with SGC. s

i.e. the term representing SGC effect is moved from original Eq. (1f). When SGC is absent but the relative phase is considered, the density matrix equations of motion of the open system can be written as

r_ 011 ¼ 2ðc1 þ c2 Þr011 þ iGp ½expðiUÞr031  expðiUÞr013  þ 2Kr033 þ iGc ðr021  r012 Þ  c0 r011 ;

ð2aÞ

r_ 022 ¼ 2c2 r011 þ iGc ðr012  r021 Þ þ J2  c0 r022 ;

ð2bÞ

r_ 033 ¼ 2c1 r011  2Kr033 þ iGp ½expðiUÞr013  expðiUÞr031  þ J3  c0 r033 ; ð2cÞ

r_ 012 ¼ ðc1 þ c2 þ iDc Þr012 þ iGp expðiUÞr032  iGc ðr011  r022 Þ; ð2dÞ

r_ 013 ¼ ðc1 þ c2 þ K þ iDp Þr013 þ iGc r023  iGp expðiUÞðr011  r033 Þ;

In the case with SGC, when U ¼ 0 and U ¼ p, EIT occurs; but when SGC is absent, there isn’t any value of U to make EIT arising. t Different from the case with SGC, when SGC is absent, transient evo-

ð2eÞ

r_ 023 ¼ ðK þ iDp  iDc Þr023 þ iGc r013  iGp expðiUÞ0 r21 :

ð2fÞ

0.0006

(a)

0.003

Φ=0 Φ=π/8 Φ=π/4 Φ=π/2

0.001 0.000 -0.001

Φ=π Φ=9π/8 Φ=5π/4 Φ=3π/2

Re (σ 31)

31

Im (σ )

(b)

0.0004

0.002

1

2

τγ 1

3

4

5

0 π/8 π/4 π/2

0.0000 Φ= Φ= Φ= Φ=

Φ =0,Φ =π

-0.0004

Λ =0

0

0.0002

-0.0002

-0.002 -0.003

Φ= Φ= Φ= Φ=

Λ =0

-0.0006

0

1

2

τγ 1

3

4

π 9π/8 5π/4 3π/2

5

Fig. 7. Transient evolution of Im ðr031 Þ and Re ðr031 Þ for different values of U in the open system without SGC when the incoherent pump is absent. The other parameters values are same as those in Fig. 2.

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X. Fan et al. / Optics Communications 283 (2010) 1810–1816

0.0015

Φ=0 Φ=π/8 Φ=π/4 Φ=π/2

0.0000 -0.0005

Λ =0.4γ 1

0.0002

-0.0002 -0.0004

Λ =0.4γ 1

0

1

2

τγ 1

3

4

Φ= Φ= Φ= Φ=

0 π/8 π/4 π/2

Φ= Φ= Φ= Φ=

π 9π/8 5π/4 3π/2

0.0000

Φ=0,Φ=π

-0.0010 -0.0015

Φ=π Φ=9π/8 Φ=5π/4 Φ=3π/2

(b)

31

0.0005

Re (σ )

31

Im (σ )

0.0004

(a)

0.0010

0

5

1

2

τγ 1 3

4

5

Fig. 8. Transient evolution of Im ðr031 Þ and Re ðr031 Þ for different values of U in the open system without SGC when the incoherent pump presents. The other parameters values are same as those in Fig. 2.

0.8

1.0

0.6 σ 33

0.6

Λ =0

σ 22

0.4

(a)

Population

Population

0.8

σ 11

0.2

σ 33

Λ =0.4γ1

σ 22

0.4

(b)

σ 11

0.2 0.0

0.0 0

1

2

τγ1

3

4

5

0

1

2

τγ1

3

4

5

Fig. 9. Population transient evolution of populations for different values of U, which corresponds to Figs. 7 and 8. The population curves corresponding to different values of U overlap completely.

lution rule of the probe gain (absorption) is much different from that of the probe dispersion. Above characteristics still keep when the incoherent pump presents, as shown in Fig. 8. Just opposite to the case with SGC (see Fig. 2(a) and Fig. 3(a)) that incoherent pump can obviously increase steady value of the probe gain (absorption), when SGC is absent (see Fig. 7(a) and Fig. 8(a)) incoherent pump will remarkably decrease steady value of the probe gain (absorption). Same as the case with SGC, when SGC is absent, changing value of U also has no any effect on the transient evolution of populations regardless of that the incoherent pump exists or does not as shown in Fig. 9. Fig. 9 is complete same as Fig. 4, and this shows that existence of SGC has no effect on transient evolution of populations. 5. Conclusions In summary, using the rotating-wave and the electric dipole approximations, we have investigated the effects of the relative phase, U, between the probe and driving fields on the transient evolution of the atomic response from different respects in an open lambda-type system with SGC by numerical results and have compared the case with that of the corresponding closed system. We find that: varying value of U will obviously change the transient evolution process, transient and stationary values of gain, absorption and dispersion but has no any effect on the transient evolution of populations. For the same value of U, the stationary gain without inversion (GWI) when the incoherent pumping exists is much larger than that when the incoherent pumping is absent, the time to reach at the stationary state when the incoherent pumping exists is much shorter than that when the incoherent pumping is absent. The initial condition varying has remarkable effects on the phase-dependent transient evolution process, transient value of GWI but doesn’t vary size of stationary value of GWI. Varying the ratio C of the atomic injection rates and exit rate c0 has obvious ef-

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