Optical bistability via quantum interference from incoherent pumping and spontaneous emission

Journal of Luminescence 131 (2011) 2395–2399

Contents lists available at ScienceDirect

Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin

Optical bistability via quantum interference from incoherent pumping and spontaneous emission M. Sahrai a,n, S.H. Asadpour a, R. Sadighi-Bonabi b a b

Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Department of Physics, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

abstract

Article history: Received 26 March 2011 Received in revised form 18 May 2011 Accepted 30 May 2011 Available online 13 June 2011

We theoretically investigate the optical bistability (OB) in a V-type three-level atomic system confined in a unidirectional ring cavity via incoherent pumping field. It is shown that the threshold of optical bistability can be controlled by the rate of an incoherent pumping field and by interference mechanism arising from the spontaneous emission and incoherent pumping field. We demonstrate that the optical bistability converts to optical multi-stability (OM) by the quantum interference mechanism. & 2011 Elsevier B.V. All rights reserved.

Keywords: Optical bistability Optical multi-stability Incoherent pumping field Spontaneously generated coherence

1. Introduction It is well known that atomic coherence can be created by the coherent interaction of a multi-level atomic system with strong coherent laser fields. Atomic coherence has attracted considerable attention in the last two decades, since it gives such interesting phenomena as electromagnetically induced transparency [1], lasing without population inversion [2], ultra- slow [3], and ultra-fast [4] group velocities in a transparent medium. In fact, the most important key to successful experiments on creation of atomic coherence lies in its ability to control optical properties of a medium by a laser field. It is well known that the radiation decay of two closely lying upper levels to a common third level creates coherence between these levels [5,6]. In order to have the parallel dipole moments and to produce interference between decay channels, however, the near-degenerate levels must have the same j and mj quantum numbers. There is no real atomic system to meet the necessary conditions. To solve this difficulty, Fleischhauer et al. [7] proposed an incoherent pump processes to create the quantum interference. Two near-degenerate levels do not need to have the same j and mj quantum numbers always. The use of magnetic split upper levels with different mj quantum numbers does conserve the quantum interference if one applies linear polarized light. In contrast to the case of interference radiation decays, the level spacing only needs to be smaller than

n

Corresponding author. Tel.: þ98 411 3393010; fax: þ98 411 3347050. E-mail address: [email protected] (M. Sahrai).

0022-2313/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2011.05.059

the spectral width of the incoherent pump field. There are a lot of interests to the quantum interference of incoherent pumping processes in recent years. Fleischhauer et al. [7,8] demonstrated that the interference of incoherent pump processes leads to gain without inversion and enhancement of refraction index without absorption. The effect of an incoherent pump process on quenching of spontaneous emission is also proposed [9]. Quantum interference arising from incoherent pumping processes has been used for modifying spontaneous emission [10,11], phase dependence of collection fluorescence [12], and coherent population trapping [13]. The effect of incoherent pumping field on phase control of group velocity [14] and dynamical behavior of the absorption and dispersion [15] have also been studied. Moreover, the effect of quantum interference arising from incoherent pump fields on controlling of a light pulse from subluminal to superluminal has been investigated [16]. Another quantum interference mechanism has been raised from the decay processes. Spontaneous emission is usually believed to result in the decay but not building the coherence between atomic levels. However, it is shown that the spontaneous emission can also be used to produce atomic coherence as long as there exist two closely lying levels with non-orthogonal dipoles in an atomic system. Atomic coherence based on spontaneous emission is usually referred to as vacuum-induced coherence or spontaneously generated coherence (SGC) [17]. It is demonstrated that the combine effect of SGC and quantum interference of incoherent pumping fields make the system phase dependent; thus the optical properties of the medium such as amplification without population inversion, dispersion, absorption, and group velocity of a weak probe field

2396

M. Sahrai et al. / Journal of Luminescence 131 (2011) 2395–2399

can be controlled by the relative phase of applied fields [14,18]. Optical bistability, on other hand, has extensively been studied from the both experimental and theoretical points of view [19,20] in various atomic media. Optical bistability has been developed due to its potential application in all optical switching and optical transistors, which are necessary for quantum computing and quantum communications. Most of the experimental and theoretical studies in OB have been devoted to two-level atoms in an optical resonator [19,21]. One recent study is a pumped-probe two-level atomic system in which the behavior of the optical bistability under the scattering of the pump field into the probe field frequency is presented [22]. However, multi-level atomic systems have also been proposed to investigate the bistable and multi-stable behavior of the atomic systems [23–27]. One major advantage of using three-level, instead of two-level, atoms as a nonlinear medium inside an optical cavity is to make the atomic coherence induced in three-level atomic system, which can greatly modify the absorption, dispersion, and nonlinearity of the medium [28]. In some studies, it has been observed that the optical properties of a multi-level atomic system can be controlled by coherent driving fields. In such cases, quantum coherence and quantum interference are the basic mechanisms for modifying the optical response of the medium. On the basis of these investigations, Harshawadhan and Agarwal [29] demonstrated that the quantum coherence and interference can effectively reduce the threshold intensity of optical bistability. Now, intriguing question arises whether one can create optical bistability and optical multistability via quantum interference of incoherent pumping field and quantum interference of spontaneous emission. In this paper we show that the optical bistability can be controlled and optical multi-stability can be created by the quantum interference mechanisms of incoherent pumping and spontaneous emission processes. No coherent laser field is used at the pumping processes of the system. It is shown that the quantum interference of incoherent pumping field can dramatically modify the nonlinear response of the medium, thus optical bistability and multi-stability can be controlled by the quantum interference of incoherent pumping field and spontaneous emission. Moreover, we find that the rate of an incoherent pumping field reduces the threshold of optical bistability. The paper is organized as follows: in Section 2, we present the model and equations. The results are discussed in Section 3, and the conclusion can be formed in Section 4.

2. Model and equations Consider a three- level atomic system in a V configuration as shown in Fig. 1. The closely spaced doublets 93S and 92S separated in frequency by o23 are coupled by a weak probe field to the ground state 91S. The upper levels decay into the ground level with the rates g1 and g2, respectively. An incoherent polaraized pumping field, ep(t), with pumping rates r1 and r2 is applied between lower level 91S to doublet levels 92S and 93S. It is assumed that the spectral width of incoherent pumping field is greater than the separation of the two closely lying doublet states. Therefore, the

In the above equations we have set r1 ¼r2 ¼r, and detuning parameter is defined as d ¼ D þðo32 =2Þ, where parameter D ¼ op ððo21 þ o31 Þ=2Þ measures the common detuning of probe field from the middle of doublet states. Rabi-frequency of the probe field is defined as Op1,2 ¼ ðep Y1j =2_Þ ðj ¼ 2,3Þ, where ep is the amplitude of probe field. In addition, we assume that the dipole matrix elements for both transitions j1i-j2i and j1i-j3i are both     !  !  equal, i.e.  Y 12  ¼  Y 13  ¼ Y. So, the Rabi-frequency of probe field is defined as Op1 ¼ Op2 ¼ O ¼ ðep Y=2_Þ. The parameter P ¼ ! ! ! ! ðY 13 Y 12 Þ=ð9Y13 99Y12 9Þ and K ¼ ðY 13 Y 12 Þ=ð9Y13 99Y12 9Þ denote the alignment of the matrix elements of two dipole moments, which correspond to the interference of the incoherent pumping process and spontaneous emission, respectively. In principle the ! ! dipole moments Y 13 and Y 12 have different directions. When the two electric dipole moments are parallel we have maximum interference, while for the perpendicular electric dipole moments the interference term vanishes. The P (or K)¼1 shows the maximum interference, while P (or K)¼0 corresponds to no interference case. Now, we consider a medium of length L composed of the above described atomic system immersed in unidirectional ring cavity as shown in Fig. 2. Intensity reflection and transmission coefficients of mirrors 1 and 2 are R and T (with R þT¼1), respectively. We assume that both the mirrors 3 and 4 are prefect reflectors. The total electromagnetic field can be written as E ¼ Ep eiop t þcc,

ð2Þ

where Ep is the amplitude of probe field, which circulate in the ring cavity. Note that incoherent pumping field is not circulating field [30]. Under a slowly varying envelope approximation, the

E

1 0

3

δ

ω

incoherent pumping field has a broad spectrum with effective d-like correlation function [11], i.e. o enp ðtÞep ðt0 Þ 4 ¼ Gp dðtt0 Þ. Threfore, the interference from incoherent pumping field may arise. The density matrix equations of motion in the rotating wave approximation and in rotating frame are   r32 ¼  1=2ðg1 þ g2 Þ þ io23 r32 þ iOp1 r12 iOp2 r31 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi g1 g2 r1 r2 ðr22 þ r33 ÞK P ðr22 þ r33 Þ þ K r1 r2 r11 , 2 2   r31 ¼  1=2g1 þ id þr r31 iOp1 ðr33 r11 Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi g1 g2 r r r21 K 1 2 r21 , iOp2 r32 P 2 2   r21 ¼  1=2g2 þ iðdo23 Þ þ r r21 iOp1 r23 þ iOp2 ðr11 r22 Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi g1 g2 r r r31 K 1 2 r31 , P 2 2 pffiffiffiffiffiffiffiffiffiffi g g  r22 ¼ iOp2 ðr12 r21 Þg2 r22 þr r11 P 1 2 ðr32 þ r23 Þ 2 pffiffiffiffiffiffiffiffiffi r1 r2 K ðr32 þ r23 Þ, 2 pffiffiffiffiffiffiffiffiffiffi g g r33 ¼ iOp1 ðr13 r31 Þg1 r33 þr r11 P 1 2 ðr32 þ r23 Þ 2 pffiffiffiffiffiffiffiffiffi r1 r2 K ð1Þ ðr32 þ r23 Þ, r11 þ r22 þ r33 ¼ 1, 2

L

2

E

2 r r

γ

γ

Ω

1 Fig. 1. Energy levels diagram of a three-level V-type atomic system interacting with incoherent pump fields and a weak probe field.

4

3

Fig. 2. Unidirectional ring cavity with sample of length L. EIp and ETp are the incident and transmitted fields, respectively.

M. Sahrai et al. / Journal of Luminescence 131 (2011) 2395–2399

2397

field ETP are

dynamical equation of the probe field is described by @Ep @Ep iop þc ¼ Pðop Þ, @t @z 2e0

ð3Þ

where c and e0 are the speed of light and permittivity of free space, respectively. P(op) is the induced polarization in transition 91S-92S and 91S-93S that is given by Pðop Þ ¼ NYðr21 þ r31 Þ,

ð4Þ

here N denotes the density number of atom in the cavity. Substituting Eq. (4) into Eq. (3), we obtain the field amplitude relation in the steady state as

ETp Ep ðLÞ ¼ pffiffiffi T, Ep ð0Þ ¼

ð6Þ

pffiffiffi I T EP þ REp ðLÞ,

ð7Þ

where L is the length of the atomic sample. Note that R is the feedback mechanism due to the reflection from mirror M2. It is responsible for the bistable behavior, so we do not expect any bistability when R¼0 in Eq. (7). According to the mean-field limit [31] and using the boundary condition, the steady state solution of transmitted field is given by y ¼ 2xiCðr31 þ r21 Þ:

@Ep N op Y ¼i ðr31 þ r21 Þ: 2ce0 @z

ð5Þ

The coherent field EIP enters through mirror M1, interacts with the atomic sample of length L, circulates in the cavity, and partially comes out of the mirror M2 as ETP . The probe field at the start of atomic sample is Ep(0) and propagates to the end of the atomic sample to be Ep(L) in the single pass transition. For a perfectly tuned cavity, the boundary conditions in the steady state limit between the incident field EIP and transmitted

ð8Þ

Transmitted field depends on the incident probe field and the coherence terms r31 , r21 via Eq. (8). So, the bistable behavior of the medium can be determined by the atomicpvariables through pffiffiffi ffiffiffi r31 , r21 : Here y ¼ YEIp =_ T and x ¼ YETp =_ T represent the normalized input and output field, respectively. The parameter C ¼ N op LY2 =2_e0 cT is the cooperatively parameter for atoms in a ring cavity. We set the time derivatives @rij =@t ¼ 0 ði,j ¼ 1,2,3Þ in the above density matrix Eqs. (1) for the steady state and solve the corresponding density matrix equations together with the coupled field Eq. (8), and then we can arrive at the steady state solution.

10 9

r=0.6γ r=0.7γ r=0.8γ

8

0.03 0.025

6

0.02

5

Absorption

Output (ΩTp)

7

4 3 2 1 0

r=0.6γ r=0.7γ r=0.8γ

0.015 0.01 0.005

0

2

4

6 Input

8

10

12

0 -5

I (Ωp)

-4

-3

-2

-1

0

1

2

3

4

5

2

3

4

5

δ 10

0

9 8 7

-0.005

6

Absorption

Input (ΩpI )

r=1.2γ r=1.4γ r=1.8γ

r=1.2γ r=1.4γ r=1.8γ

5 4 3

-0.01

2 1 0

0

2

4

6

8

10

12

Output (ΩTp) Fig. 3. Behavior of output–input field intensity: (a) r¼ 0.6g (solid line), r¼ 0.7g (dashed line), and r ¼0.8g (dotted line); (b) r¼ 1.2g (solid line), r ¼1.4g (dashed line), and r¼ 1.8g (dotted line). Selected parameters are g1 ¼ g2 ¼ g, d ¼0.5g, K¼ 0, P ¼0, and o23 ¼ g.

-0.015 -5

-4

-3

-2

-1

0

1

δ Fig. 4. Plot of probe absorption versus probe field detuning for (a) weak pumping rate r¼ 0.6g (solid line), r¼ 0.7g (dashed line), and r¼ 0.8g (dotted line); (b) strong pumping rate r ¼1.2g (solid line), r ¼1.4g (dashed line), and r ¼1.8g (dotted line). Other parameters are same with Fig. 3.

2398

M. Sahrai et al. / Journal of Luminescence 131 (2011) 2395–2399

3. Result and discussion

0.7

r=0.6γ r=0.7γ r=0.8γ

0.6 0.5 Absorption

We summarize our results for the steady state behavior of the output field intensity versus the input field intensity for various parameters illustrated in Figs. 3–7. We assume g1 ¼ g2 ¼ g, and all the figures are in the unit of g. First, we show the behavior of output–input fields for various rate of incoherent pumping field in the absence of quantum interference mechanisms, i.e., P¼0 and K ¼0. Fig. 3(a) shows that the threshold of optical bistability decreases by increasing the rate of incoherent pumping field. For a strong incoherent pumping field the threshold of the optical bistability increases by enhancing the rate of incoherent pumping field (Fig. 3(b)). The physical mechanism of such unexpected behavior can be understood by the absorption spectrum. Fig. 4(a) implies that for a weak incoherent pumping rate, the linear absorption decreases by increasing the rate of incoherent pumping field. In fact, for r ¼0.6g the absorption peak located at d ¼0.0 reaches to 0.02686. However, for a strong incoherent pumping field the absorption spectrum is converted to a gain and by enhancing the rate of incoherent pumping field the gain will be increased (Fig. 4(b)). Physically, increasing the rate of incoherent pumping field may reduce the probe field absorption

0.4 0.3 0.2 0.1 0

-5

-4

-3

-2

-1

0

1

2

3

0.045

Absorption

0.03

10

r=0.6γ r=0.7γ r=0.8γ

9 8

0.025 0.02 0.015

7 Output (ΩTp)

5

r=1.2γ r=1.4γ r=1.8γ

0.04 0.035

0.01

6 5

0.005

4

0

3 2 1 0

4

δ

0

10

20

30

40

50

60

70

-5

-4

-3

-2

-1

0 δ

1

2

3

4

5

Fig. 6. Plot of probe absorption versus probe field detuning for (a) weak pumping rate r¼ 0.6g (solid line), r ¼ 0.7g (dashed line), and r¼ 0.8g (dotted line); (b) strong pumping rate r ¼1.2g (solid line), r¼ 1.4g (dashed line), and r ¼1.8g (dotted line). Other parameters are same with Fig. 5.

Input (ΩIp) 10

10

9

9 r=1.2γ r=1.4γ r=1.8γ

8

8 7

6

Output (ΩTp)

Output (ΩTp)

7 5 4 3

6 5 4 3

2

2

1 0

P=K=0 P=K=0.5 P=K=0.7

1

0

2

4

6 Input

8

10

12

(ΩIp)

Fig. 5. Behavior of output–input field intensity: (a) r¼ 0.6g (solid line), r¼ 0.7g (dashed line), and r ¼0.8g (dotted line); (b) r¼ 1.2g (solid line), r ¼1.4g (dashed line), and r¼ 1.8g (dotted line). Selected parameters are g1 ¼ g2 ¼ g, P ¼ K¼1, o23 ¼ g, and d ¼0.5g.

0

0

2

4

6 Input

8

10

12

(ΩIp)

Fig. 7. Behavior of output–input field intensity for different interference parameters. P¼ K¼0, (solid line), P ¼K¼ 0.5, (dotted line), and P¼ K¼ 0.7, (dashed line). Other parameters are g1 ¼ g2 ¼ g, d ¼ 0.25g, o23 ¼ 0.7g, and r¼ 0.5g.

M. Sahrai et al. / Journal of Luminescence 131 (2011) 2395–2399

and thus enhance the Kerr nonlinearity of atomic medium, which makes the cavity field easier to reach saturation. However, atomic coherence between levels 92S and 93S will be destroyed by the strong incoherent pumping from lower level 91S to upper levels 92S and 93S. Consequently, population in levels 92S and 93S cannot be trapped leading to enhancement of probe gain. So, enhancement of gain in the medium causes the field hard to reaches saturation. This is to say that the linear absorption (or gain) has a critical role in reduction or enhancement of bistability threshold. This also implies that the nonlinear behavior of the atomic medium may be controlled by the rate of an incoherent pumping field. It is worth noting that in the presence scheme if the interference from incoherent pumping and spontaneous emission is included, the situation will be quite different. The effect of quantum interference arising from decay processes and incoherent pumping field is displayed in Fig. 5. It is realized that by increasing the rate of incoherent pumping field the threshold of optical bistability decreases (Fig. 5(a) and (b)). In this case by increasing the rate of incoherent pumping field the probe absorption substantially decreases leading to reduction of optical bistability threshold (Fig. 6(a) and (b)). In the presence of interference mechanism arising from spontaneous emission and incoherent pumping field, i.e., K ¼P¼1, the population will be trapped in upper levels 92S and 93S. So, the probe field absorption will be reduced by increasing the rate of incoherent pumping field. Therefore, the gain cannot be appeared due to the existence of interference mechanisms. We emphasize that above results are derived under the condition that interference from incoherent pumping field is included. The interference from incoherent pumping will destroy the coherence created by the vacuuminduced coherence, i.e. SGC, so gain becomes impossible [32]. Now we show that for weak values of incoherent pumping rate, i.e., r ¼0.5g, quantum interference of spontaneous emission and incoherent pumping field have a major role on creation of optical bistability and optical multi-stability, (Fig. 7). We realize that for P¼K ¼0 (solid line) the optical bistability appears. By increasing the quantum interference parameters, the optical bistability converts to optical multi-stability. Therefore, the optical bistability can be converted to the optical multi-stability with the quantum interference mechanism for a weak incoherent pumping rate.

2399

4. Conclusion The effect of quantum interference of spontaneous emission and incoherent pumping processes on optical bistability and optical multi-stability is investigated in a three-level V-type atomic system. It is shown that the threshold of optical bistability can be controlled by the rate of incoherent pumping field. In addition, the optical bistability can be converted to optical multistability only with quantum interference mechanisms. It is important to note that no laser field is used at the pumping processes of the system.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

S.E. Harris, Phys. Today 50 (1997) 336. Y. Bai, H. Guo, H. Sun, D. Hun, Ch. Li, Xu Cheng, Phys. Rev. A 69 (2004) 043814. L.V. Hau, S.E. Harris, Z. Datton, C.H. Behroozi, Nature (London) 397 (1999) 594. H. Tajalli, M. Sahrai, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) 168. A. Imamouglu, Phys. Rev. A 40 (1989) 2835. M. Sahrai, M. Maleki, R. Hemmati, M. Mahmoudi, Eur. Phys. J. D 56 (2010) 105. M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, Opt. Commun. 87 (1992) 109. M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, B.T. Ulrich, S.Y. Zhu, Phys. Rev. A 46 (1992) 1468. K.T. Kappalle, M.O. Scully, M.S. Zubairy, S.Y. Zhu, Phys. Rev. A 67 (2003) 023804. D. Bullock, J. Everse, C.H. Keitel, Phys. Lett. A 307 (2003) 8. M. Sahrai, R. Hemmati, H. Tajalli, M. Mahmoudi, J. Mod. Opt. 55 (2008) 67. M.A. Macovei, J. Everse, Opt. Commun. 240 (2004) 379. X.M. Hu, J.P. Zhang, J. Phys. B 37 (2004) 345. M. Mahmoudi, M. Sahrai, H. Tajali, Phys. Lett. A 357 (2006) 66. M. Sahrai, Eur Phys. J. Spec. Top. 160 (2008) 383. M. Mahmoudi, M. Sahrai, H. Tajali, J. Phys. B 39 (2006) 1825. J. Javanainen, Eur. Phys. Lett. 1715 (1992) 407. Y. Bai, H. Guo, P. Hun, H. Sun, J. Opt. B Quantum Semiclass. Opt. 7 (2005) 35. A.T. Rosenberger, L.A. Orozco, H.J. Kimble, Phys. Rev. A 28 (1983) 2529. D.E. Grant, H.J. Kimble, Opt. Lett. 7 (1982) 353. H.J. Gibbs, S.L. Mccall, T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. M. Mahmoudi, M. Sahrai, M. Mousavi, J. Lumin. 130 (2010) 877. D. Cheng, C. Liu, S. Gong, Opt. Commun. 263 (2006) 111. Z. Wang, M. Xu, Opt. Commun. 282 (2009) 1574. A. Joshi, W. Yang, M. Xiao, Phys. Rev. A 68 (2003) 015806. S.Q. Gong, S.D. Du, Z.Z. Xu, S.H. Pan, Phys. Lett. A 222 (1996) 237. X.M. Hu, Z.Z. Xu, J. Opt. B: Quantum Semiclass. Opt. 3 (2001) 35. A. Brown, A. Joshi, M. Xiao, Appl. Phys. Lett. 83 (2003) 1301. W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. W. Zhiping, Opt. Commun. 283 (2010) 3291. L.A. Lugiato, E. Wolf, Prog. Opt. (Amsterdam) 21 (1984) 71. W. Hua, J.H. Wu, J.Y. Gao, J. Phys. B 39 (2006) 1461.