- Email: [email protected]
Controlling the optical bistability in a quantum-well nanostructure via indirect incoherent pump field
Optik 127 (2016) 1822–1826
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Controlling the optical bistability in a quantum-well nanostructure via indirect incoherent pump field H. Jafarzadeh ∗ Sama Technical and Vocational Training College, Islamic Azad University, Tabriz Branch, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 27 March 2015 Accepted 4 November 2015 Keywords: Quantum well molecules Optical bistability Incoherent pumping field
a b s t r a c t We investigate the optical bistability (OB) behavior in an asymmetric two-coupled quantum well structure inside a unidirectional ring cavity. By controlling the assisting coherent driven field and frequency detuning of the driven laser field, the threshold of OB can be controlled, and even OB converted to optical multistability (OM) by adjust driven pump field detuning. In addition, when we apply an incoherent indirect pump field, we find that the appearance and disappearance of OB and its threshold can easily be controlled by incoherent indirect pump rate. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Control of light by light is essential in all-optical communication and optical computing. In the past three decades, optical transistors, all-optical switching, and all-optical storage devices based on optical bistability (OB) in two-level atomic systems have extensively been studied both experimentally and theoretically [1,2]. The OB in three-level atomic system confined in an optical ring cavity has theoretically [3] and experimentally [4] been studied. It has been shown that the field-induced transparency and quantum interference effects could significantly decrease the threshold of OB [5]. The effects of phase fluctuation [6] and squeezed state fields [7] on the OB have subsequently been studied. It has been found that the OB could appear for small cooperation parameters due to the present of squeezed vacuum field [8]. On the other hand, many kinds of nonlinear quantum optical phenomena based on the quantum interference and quantum coherence in the semiconductor quantum wells (SQWs) have extensively been studied in recent years [9–11]. These investigations are include gain without inversion [12], electromagnetically induced transparency [13], coherent population trapping [14], enhanced index of refraction without absorption [15], ultrafast all optical switching [16], optical bistability [17], Kerr nonlinearity [18], optical soliton [19], ac stark splitting [20], and so on. The fundamental reason for this studies is the potentially applications in optoelectronics and solid-state quantum information science. Otherwise, the devices based on intersubband transitions in the SQWs
∗ Tel.: +98 4133311590. E-mail address: h [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.11.090 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
have many inherent advantages that the atomic systems do not have; such as the large electric dipole moments due to the small effective electron mass, the great flexibilities in devices design by choosing the materials and structure dimensions, the high nonlinear optical coefficients. In addition, the transition energies and the dipoles as well as the symmetries can also be engineered as desired. The implementation of EIT in semiconductor-based devices is very attractive from an application view-point. It is worth pointing out that the bistable behaviors in a semiconductor QW that interacts with a strong driving electromagnetic field under two-photon resonant condition are recently analyzed, and the results show that the threshold for switching to upper branch of the bistable curve can be reduced due to the presence of quantum interference [21]. The bistable behaviors via tunable Fano-type interference in asymmetric semiconductor quantum well with three-subband V-configuration are also studied [22]. In this work, we investigate the optical bistability, optical multistability, and absorption properties in a three-level asymmetric semiconductor quantum well system. We find that these properties can be efficiently controlled by changing the rate of incoherent indirect pump field. An important advantage of our scheme is that the optical bistability can be converted to optical multistability in our quantum well system. The control of the optical bistability and generate multistability are achieved by applying a coherent driven field and its adjustable parameters. It is well known that the OM also plays a crucial role in nonlinear quantum optics, which will have more advantages than the OB in some applications where more than two states are needed. We are mainly interested in studying the controllability of the optical bistability and absorption properties via the rate of indirect incoherent pumping field, which have never been investigated to our best knowledge. Note
H. Jafarzadeh / Optik 127 (2016) 1822–1826
1823
Fig. 1. (a) Schematic diagram of the three-level ladder-type quantum well system. (b) Schematic diagram of the energy level arrangement for the three-level ladder type quantum well under study.
Fig. 3. (a) Behavior of output-input field intensity for R = 0.0, 0.1, 0.2, 0.3, (b) for R = 1, 1.5, 2, 2.5. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, ˝c = 3, c = 0.0, p = 0.0.
Fig. 2. Unidirectional ring cavity containing a QW sample of length L; EpI and EpT are the incident and the transmitted field, respectively. Ec represents the coupling field which is noncirculating in the cavity.
that this scheme is much more practical than its atomic counterpart due to its flexible design, controllable interference intensity, and the wide tunable parameters. These may provide the possibility for obtaining potential applications on solid-state quantum computation, quantum communication. The remainder of our paper is organized into three parts as follows: in Section 2, we present the theoretical model and establish the corresponding equations. Our numerical results and physical analysis are shown in Section 3. In Section 4, some simple conclusions are given. 2. The model and equations We consider a three-level ladder-type quantum well system as shown in Fig. 1(a) [17]. Two laser fields and an indirect incoherent field are applied to drive the medium. A weak probe field with amplitude Ep , angular frequency ωp , and Rabi-frequency ˝p =
is applied to 1 → 2 transition. Another control laser field with amplitude Ec , angular ωc , and Rabi-frequency ˝c = frequency 23 Ec is applied to the 2 → 3 transition. Here ij (i, j = 1,2,3) are the corresponding atomic dipole moments. Therefore the total electromagnetic fields can be written as E = Ep e−iωp t + Ec e−iωc t + c . c. In order to reduce the absorption at the probe field transition frequency, an incoherent pumping field with suitable polarization is applied to the transition 1 ↔ 3 of a double QW system. By adopting the standard density matrix equations of motion in dipole 12 Ep
and rotating-wave approximations for this system can be written as follows: ˙ 21 = −(ip +
12 + R)21 − i˝p (22 − 11 ) + i˝c 31 , 2
˙ 32 = −(ic +
23 )32 − i˝c (33 − 22 ) − i˝p 31 , 2
˙ 31 = −[i(p + c ) +
13 + R]31 − i˝p 32 + i˝c 21 , 2
(1)
˙ 22 = −21 22 + 32 33 − i˝c (23 − 32 ) − i˝p (21 − 12 ), ˙ 33 = −(31 + 32 )33 + i˝c (23 − 32 ) + 2R11 . where ij = ji∗ (i, j = 1, 2, 3), and constrained by 11 + 22 + 33 = 1.
The p = ω12 − ωp and c = ω23 − ωc are detuning frequencies of probe and control fields, respectively. Also, ω12 and ω23 are resonant frequencies, which associates with the corresponding optical transitions 1 ↔ 2 and 2 ↔ 3 . Here R is an indi-
rect incoherent pump rate that applied between 1 and 3 transition according to Fig. 1(b). The population decay rates and dephasing decay rates are added phenomenologically in the above density matrix equations [20], which are comprised of a population decay contribution as well as a dephasing contribution. The population decay rates from subband i to subband
j denoted by ij (i =/ j) is mainly due to longitudinal optical (LO) photon emission at low temperature. The total decay rates dph / j) are given by 12 = 12 + 13 + 23 + 12 ; 23 = 23 + ij (i = dph
dph
dph
23 and 13 = 12 + 13 + 13 , where ij (i = / j) determined by electron–electron, interface roughness, and phonon scattering
1824
H. Jafarzadeh / Optik 127 (2016) 1822–1826
Fig. 4. (a) Absorption as a function of probe field detuning for R = 0.0, 0.1, 0.2, 0.3, (b) for Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, ˝c = 3, c = 0.0, ˝p = 0.01.
processes isthe dephasing decay rate of the quantum coherence of the i ↔ j transition. Now, we enclose this QW sample inside a unidirectional optical cavity comprising of four mirrors, as shown in Fig. 2. For simplicity, we assume that mirrors 3 and 4 have 100% reflectivity, and the intensity reflection and transmission coefficient of mirrors 1 and 2 are R and T (with R + T = 1), respectively. The behavior of the above-described three-level ladder-type quantum well system will be investigated. The probe field circulates in the ring cavity, while the control laser field does not circulate in the ring cavity. So, the dynamics of the probe field in the optical cavity is governed by Maxwell’s equation, where, under slowly varying envelope approximation is given by:
∂Ep ∂Ep ωp P(ωp ). +c =i 2ε0 ∂t ∂z
(2)
Here, P(ωp ) = N12 21 is the slowly oscillating of the induced polarization. N is the number density of the electrons in the sample, and c and ε0 are the light speed and permittivity of free space, respectively. In the steady state, the term ∂Ep /∂t in Eq. (2) equal to zero. Then Eq. (2) can easily be given as follows:
∂Ep Nωp 12 21 . =i 2cε0 ∂z
(3)
For a perfectly tuned ring cavity, in the steady-state limit, the boundary conditions impose the following conditions between the incident field EpI and the transmitted field EpT [1] √ Ep (L) = EpT / T , (4)
Fig. 5. (a) Behavior of output-input field intensity, (b) absorption as a function of probe field detuning for c = 0, 2, 4, 6. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, ˝c = 3, R = 0.1, ˝p = 0.01, p = 0.0.
Ep (0) =
√
T EpI + REp (L).
(5)
where L is the length of the quantum well sample, and the second term on the right-hand side of Eq. (5) describes a feedback mechanism due to the mirror, which is essential to give rise to OB. There limit [23],√normalizing the is no OB when R = 0. In the mean-field √ fields by letting y = 12 EpI / T and x = 12 EpT / T and using the boundary conditions, we can obtain the input–output relationship: y = x − iC21 (x)
(6)
where C = ωp L|12 |2 N/2ε0 cT is the usual electronic cooperation parameter. The second term on the right-hand side of Eq. (6) is very important to achieve the OB. 3. Numerical results 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. 3–6. We assume 12 = 13 = 23 = , and all the figures are plotted in the unit of . In addition, we assume that all subbands have the same effective mass, and the electron–electron effects have very small influence on our results. First, we show the behavior of output–input fields for various rate of indirect incoherent pumping field. Fig. 3(a) shows that the threshold of optical bistability decreases by increasing the rate of weak indirect incoherent pumping field. For a strong indirect incoherent pumping field the optical bistability disappears (Fig. 3(b)). The physical mechanism of such behavior can be understood by the absorption spectrum. Fig. 4(a) implies that for a weak indirect incoherent pumping rate, the linear absorption decreases by increasing
H. Jafarzadeh / Optik 127 (2016) 1822–1826
1825
In this regime, i.e., ˝p2 → 0, we seek the corresponding steady state solution for 21 in the presence of indirect incoherent pumping field. In this case probe transition coherence becomes ˝p
21 =
A(B2 + C 2 )
where
{[BD − C(E + F)] − i[DC + B(E + F)]}.
⎛
⎞
⎜
2c
⎛ ⎜
+
2c +
23
2
+
23 p 2
2
D = (G − A) c
12
2
2
+R
13
13
23 2
+R
2
12
+R
2
13
+R
2
⎛
G = 2 ⎝21 + 232 +
+ R + (p + c )
F = 2˝c2 (21 − 32 )R
⎜
+ R + (p + c )
2 2
12
13
E = (G − A) −c (p + c ) + Fig. 6. (a) Behavior of output-input field intensity, (b) absorption as a function of probe field detuning for ˝c = 1, 3, 5. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, R = 0.1, c = 0.0, ˝p = 0.01, p = 0.0.
+R
+ R + (p + c )
C = c −p (p + c ) +
13
⎞
⎟ 2 ⎠ + 2R32 + 4R
23 −p (p + c ) + 2 − c p
23
2
˝c2 23
+ 32 ⎝21 +
⎟ 2 ⎠
˝c2 23
A = 21 ⎝32 + 2R +
B=
(7)
13
2
12
2 23 2 +R
˝c2 23 2c +
2
+ ˝c2
+R
23
2
23 2
+ ˝c2 c
⎞ ⎟ 2 ⎠ R
3˝c2 23 2c +
23
2 (8)
The response of the quantum system to the applied fields is determined by the susceptibility = + i , which is corresponding to the probe transition coherence 21 . The imaginary part of correspond to the absorption of the weak probe field. In our notation if Im[ ] 0, the system exhibits gain for the probe field, while for Im[ ] 0, the probe field will be attenuated. Note that for [DC + B(E + F)] = 0, the imaginary parts of 21 are equal to zero, i.e., Im(21 ) = 0. According to the this equation, the necessary value for switching the absorption to gain is approximately given by Fig. 7. Behavior of output-input field intensity for C = 250, 300, 350 mev. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, R = 0.1, ˝c = 3, c = 0.0, p = 0.0.
the rate of indirect incoherent pumping field. In fact, for R = 0.1 the absorption peak located at ı =0.0 reaches to 0.0125 value. However, for a strong indirect incoherent pumping field the absorption spectrum is converted to again, and by enhancing the rate of incoherent pumping field the gain will be increased (Fig. 4(b)). Physically, increasing the rate of indirect incoherent pumping field may reduce the probe field absorption and thus enhance the Kerr nonlinearity of QW sample, which makes the cavity field easier to reach saturation. For further discussion, in order to understand the physical mechanism of the results, we introduce an analytical expression for the steady state solutions of Eqs. (1) in the weak field approximation.
Rnecessary =
21 + 31 , (with consideration : 21 = 32 ) 2
(9)
The analytical calculation shows that the indirect incoherent pumping rate is an important parameter to changing the probe absorption to the probe gain. For R Rnecessary , Im(21 ) around zero probe field detuning is negative corresponding to a probe field amplification, while for R Rnecessary , Im(21 ) becomes positive corresponding to probe absorption. Note that for R = Rnecessary the imaginary part of 21 , i.e. Im(21 ), definitely becomes zero, thus no OB appears. According to the above discussion, when R = Rnecessary = 1.0, the OB disappears. In Fig. 4(a) and (b) we show the probe absorption in the presence of indirect incoherent pumping field. For R Rnecessary „ the incoherent pumping field is not able to establish the population inversion in probe transition and absorption around zero detuning. For R = Rnecessary the saturation effect appears and absorption
1826
H. Jafarzadeh / Optik 127 (2016) 1822–1826
vanishes. By increasing the incoherent pumping field, the population inversion is established. The absorption is negative and the absorption converted to gain. We will analyze how the intensity and the frequency detuning of the control field affect the optical bistability behavior, while keeping all other parameters fixed. Fig. 5(a) demonstrates the dependence of the optical bistability and multistability behaviors on the frequency detuning of the control field. It can be easily seen from Fig. 5(a) that the threshold and the hysteresis cycle shape change obviously due to the variational frequency detuning of the control field. The reason for the above result can be qualitatively explained as follows. The variational frequency detuning of control field can dramatically modify the absorption of the electronic medium according Fig. 5(b). So, in Fig. 5(a) we can see that the variational frequency detuning of control field lead to the OB converted to the OM. In Fig. 6(a), we plot the input–output field curves for different intensity of the control field. It is shown that the system is sensitive to intensity of the control field. This is the result of the influence of the controlling-field intensity on the absorption and dispersion properties of the electronic sample that is smaller than the non-resonance condition. It is easily seen, when intensity of the control field increases, the bistable threshold value reduces. In Fig. 6(b) we show that by increasing the intensity of the control field, the absorption of the probe field in the sample can be decreased. Finally, the effect of the electronic cooperation parameter C on optical bistability is depicted for a fixed value of p,c = 0 in Fig. 7. We observe that by increasing electronic cooperation parameter, the threshold of OB goes up. From the term C = ωp L|12 |2 N/2ε0 cT, we can see that the cooperation parameter is directly proportional to the number density of electrons. So, the enhancement in the absorption of the sample as the number density of electrons increases could account for the raise of the threshold intensity with respect to the cooperation parameter. We have assumed that the three level excitation scheme is valid under the defined conditions, which is justified in view of the experimental measurements and the theoretical simulation of that work.
4. Conclusion To sum up, we theoretically investigated optical bistability behavior in a three-level ladder type quantum well system inside a unidirectional ring cavity. It is shown that the threshold of optical bistability can be controlled by the rate of indirect incoherent pumping field. We find that the frequency detuning of the control field can affect for converting the optical bistability to multistability. In addition, the control field intensity can also be used to manipulate efficiently the threshold intensity and the hysteresis loop. Our scheme may be used for building more efficient all-optical switches and logic-gate devices for optical computing and quantum information processing. References [1] L.A. Lugiato, in: E. Wolf (Ed.), Progress in Optics, vol. 21, 1984, pp. 71, NorthHolland, Amsterdam. [2] H.M. Gibbs, S.L. McCall, T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. [3] P.D. Drummond, D.F. Walls, J. Phys. A 13 (1980) 725. [4] A. Joshi, M. Xiao, Phys. Rev. Lett. 91 (2003) 143904. [5] W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. [6] X. Hu, Z. Xu, J.B. Opt, Quantum Semiclass. Opt. 3 (2001) 35. ˜ Phys. Lett. A 311 (2003) 297. [7] M.A. Antón, O.G. Calderón, F. Carreno, [8] Z. Chen, C. Du, S. Gong, Z. Xu, Phys. Lett. A 259 (1999) 15. [9] A. Joshi, Phys. Rev. B 79 (2009) 115315. [10] J.F. Dynes, E. Paspalakis, Phys. Rev. B 73 (2006) 233305. [11] J.H. Li, Phys. Rev. B 75 (2007) 155329. [12] C.R. Lee, Y.C. Li, F.K. Men, C.H. Pao, Y.C. Tsai, J.F. Wang, Appl. Phys. Lett. 86 (2005) 201112. [13] M. Phillips, H. Wang, Opt. Lett. 28 (2003) 831. [14] J.F. Dynes, M.D. Frogley, J. Rodger, C.C. Phillips, Phys. Rev. B 72 (2005) 085323. [15] S.M. Sadeghi, H.M. vanDriel, J.M. Fraser, Phys. Rev. B 62 (2000) 15386. [16] J.H. Wu, J.Y. Gao, J.H. Xu, L. Silvestri, M. Artoni, G.C. LaRocca, F. Bassani, Phys. Rev. Lett. 95 (2005) 057401. [17] A. Joshi, M. Xiao, Appl. Phys. B 79 (2004) 65. [18] H. Sun, Y. Niu, R. Li, S. Jin, S. Gong, Opt. Lett. 32 (2007) 2475. [19] W.X. Yang, J.M. Hou, R.K. Lee, Phys. Rev. A 77 (2008) 033838. [20] J.F. Dynes, M.D. Frogley, M. Beck, J. Faist, C.C. Phillips, Phys. Rev. Lett. 94 (2005) 157403. [21] A. Joshi, M. Xiao, Appl. Phys. B: Lasers Opt. 79 (2004) 65. [22] J. Li, X. Yang, Eur. Phys. J. B 53 (2006) 449. [23] M.A. Anton, O.G. Calderon, F. Carreno, Phys. Lett. A 311 (2003) 297.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Controlling the optical bistability in a quantum-well nanostructure via indirect incoherent pump field H. Jafarzadeh ∗ Sama Technical and Vocational Training College, Islamic Azad University, Tabriz Branch, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 27 March 2015 Accepted 4 November 2015 Keywords: Quantum well molecules Optical bistability Incoherent pumping field
a b s t r a c t We investigate the optical bistability (OB) behavior in an asymmetric two-coupled quantum well structure inside a unidirectional ring cavity. By controlling the assisting coherent driven field and frequency detuning of the driven laser field, the threshold of OB can be controlled, and even OB converted to optical multistability (OM) by adjust driven pump field detuning. In addition, when we apply an incoherent indirect pump field, we find that the appearance and disappearance of OB and its threshold can easily be controlled by incoherent indirect pump rate. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Control of light by light is essential in all-optical communication and optical computing. In the past three decades, optical transistors, all-optical switching, and all-optical storage devices based on optical bistability (OB) in two-level atomic systems have extensively been studied both experimentally and theoretically [1,2]. The OB in three-level atomic system confined in an optical ring cavity has theoretically [3] and experimentally [4] been studied. It has been shown that the field-induced transparency and quantum interference effects could significantly decrease the threshold of OB [5]. The effects of phase fluctuation [6] and squeezed state fields [7] on the OB have subsequently been studied. It has been found that the OB could appear for small cooperation parameters due to the present of squeezed vacuum field [8]. On the other hand, many kinds of nonlinear quantum optical phenomena based on the quantum interference and quantum coherence in the semiconductor quantum wells (SQWs) have extensively been studied in recent years [9–11]. These investigations are include gain without inversion [12], electromagnetically induced transparency [13], coherent population trapping [14], enhanced index of refraction without absorption [15], ultrafast all optical switching [16], optical bistability [17], Kerr nonlinearity [18], optical soliton [19], ac stark splitting [20], and so on. The fundamental reason for this studies is the potentially applications in optoelectronics and solid-state quantum information science. Otherwise, the devices based on intersubband transitions in the SQWs
∗ Tel.: +98 4133311590. E-mail address: h [email protected] http://dx.doi.org/10.1016/j.ijleo.2015.11.090 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
have many inherent advantages that the atomic systems do not have; such as the large electric dipole moments due to the small effective electron mass, the great flexibilities in devices design by choosing the materials and structure dimensions, the high nonlinear optical coefficients. In addition, the transition energies and the dipoles as well as the symmetries can also be engineered as desired. The implementation of EIT in semiconductor-based devices is very attractive from an application view-point. It is worth pointing out that the bistable behaviors in a semiconductor QW that interacts with a strong driving electromagnetic field under two-photon resonant condition are recently analyzed, and the results show that the threshold for switching to upper branch of the bistable curve can be reduced due to the presence of quantum interference [21]. The bistable behaviors via tunable Fano-type interference in asymmetric semiconductor quantum well with three-subband V-configuration are also studied [22]. In this work, we investigate the optical bistability, optical multistability, and absorption properties in a three-level asymmetric semiconductor quantum well system. We find that these properties can be efficiently controlled by changing the rate of incoherent indirect pump field. An important advantage of our scheme is that the optical bistability can be converted to optical multistability in our quantum well system. The control of the optical bistability and generate multistability are achieved by applying a coherent driven field and its adjustable parameters. It is well known that the OM also plays a crucial role in nonlinear quantum optics, which will have more advantages than the OB in some applications where more than two states are needed. We are mainly interested in studying the controllability of the optical bistability and absorption properties via the rate of indirect incoherent pumping field, which have never been investigated to our best knowledge. Note
H. Jafarzadeh / Optik 127 (2016) 1822–1826
1823
Fig. 1. (a) Schematic diagram of the three-level ladder-type quantum well system. (b) Schematic diagram of the energy level arrangement for the three-level ladder type quantum well under study.
Fig. 3. (a) Behavior of output-input field intensity for R = 0.0, 0.1, 0.2, 0.3, (b) for R = 1, 1.5, 2, 2.5. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, ˝c = 3, c = 0.0, p = 0.0.
Fig. 2. Unidirectional ring cavity containing a QW sample of length L; EpI and EpT are the incident and the transmitted field, respectively. Ec represents the coupling field which is noncirculating in the cavity.
that this scheme is much more practical than its atomic counterpart due to its flexible design, controllable interference intensity, and the wide tunable parameters. These may provide the possibility for obtaining potential applications on solid-state quantum computation, quantum communication. The remainder of our paper is organized into three parts as follows: in Section 2, we present the theoretical model and establish the corresponding equations. Our numerical results and physical analysis are shown in Section 3. In Section 4, some simple conclusions are given. 2. The model and equations We consider a three-level ladder-type quantum well system as shown in Fig. 1(a) [17]. Two laser fields and an indirect incoherent field are applied to drive the medium. A weak probe field with amplitude Ep , angular frequency ωp , and Rabi-frequency ˝p =
is applied to 1 → 2 transition. Another control laser field with amplitude Ec , angular ωc , and Rabi-frequency ˝c = frequency 23 Ec is applied to the 2 → 3 transition. Here ij (i, j = 1,2,3) are the corresponding atomic dipole moments. Therefore the total electromagnetic fields can be written as E = Ep e−iωp t + Ec e−iωc t + c . c. In order to reduce the absorption at the probe field transition frequency, an incoherent pumping field with suitable polarization is applied to the transition 1 ↔ 3 of a double QW system. By adopting the standard density matrix equations of motion in dipole 12 Ep
and rotating-wave approximations for this system can be written as follows: ˙ 21 = −(ip +
12 + R)21 − i˝p (22 − 11 ) + i˝c 31 , 2
˙ 32 = −(ic +
23 )32 − i˝c (33 − 22 ) − i˝p 31 , 2
˙ 31 = −[i(p + c ) +
13 + R]31 − i˝p 32 + i˝c 21 , 2
(1)
˙ 22 = −21 22 + 32 33 − i˝c (23 − 32 ) − i˝p (21 − 12 ), ˙ 33 = −(31 + 32 )33 + i˝c (23 − 32 ) + 2R11 . where ij = ji∗ (i, j = 1, 2, 3), and constrained by 11 + 22 + 33 = 1.
The p = ω12 − ωp and c = ω23 − ωc are detuning frequencies of probe and control fields, respectively. Also, ω12 and ω23 are resonant frequencies, which associates with the corresponding optical transitions 1 ↔ 2 and 2 ↔ 3 . Here R is an indi-
rect incoherent pump rate that applied between 1 and 3 transition according to Fig. 1(b). The population decay rates and dephasing decay rates are added phenomenologically in the above density matrix equations [20], which are comprised of a population decay contribution as well as a dephasing contribution. The population decay rates from subband i to subband
j denoted by ij (i =/ j) is mainly due to longitudinal optical (LO) photon emission at low temperature. The total decay rates dph / j) are given by 12 = 12 + 13 + 23 + 12 ; 23 = 23 + ij (i = dph
dph
dph
23 and 13 = 12 + 13 + 13 , where ij (i = / j) determined by electron–electron, interface roughness, and phonon scattering
1824
H. Jafarzadeh / Optik 127 (2016) 1822–1826
Fig. 4. (a) Absorption as a function of probe field detuning for R = 0.0, 0.1, 0.2, 0.3, (b) for Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, ˝c = 3, c = 0.0, ˝p = 0.01.
processes isthe dephasing decay rate of the quantum coherence of the i ↔ j transition. Now, we enclose this QW sample inside a unidirectional optical cavity comprising of four mirrors, as shown in Fig. 2. For simplicity, we assume that mirrors 3 and 4 have 100% reflectivity, and the intensity reflection and transmission coefficient of mirrors 1 and 2 are R and T (with R + T = 1), respectively. The behavior of the above-described three-level ladder-type quantum well system will be investigated. The probe field circulates in the ring cavity, while the control laser field does not circulate in the ring cavity. So, the dynamics of the probe field in the optical cavity is governed by Maxwell’s equation, where, under slowly varying envelope approximation is given by:
∂Ep ∂Ep ωp P(ωp ). +c =i 2ε0 ∂t ∂z
(2)
Here, P(ωp ) = N12 21 is the slowly oscillating of the induced polarization. N is the number density of the electrons in the sample, and c and ε0 are the light speed and permittivity of free space, respectively. In the steady state, the term ∂Ep /∂t in Eq. (2) equal to zero. Then Eq. (2) can easily be given as follows:
∂Ep Nωp 12 21 . =i 2cε0 ∂z
(3)
For a perfectly tuned ring cavity, in the steady-state limit, the boundary conditions impose the following conditions between the incident field EpI and the transmitted field EpT [1] √ Ep (L) = EpT / T , (4)
Fig. 5. (a) Behavior of output-input field intensity, (b) absorption as a function of probe field detuning for c = 0, 2, 4, 6. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, ˝c = 3, R = 0.1, ˝p = 0.01, p = 0.0.
Ep (0) =
√
T EpI + REp (L).
(5)
where L is the length of the quantum well sample, and the second term on the right-hand side of Eq. (5) describes a feedback mechanism due to the mirror, which is essential to give rise to OB. There limit [23],√normalizing the is no OB when R = 0. In the mean-field √ fields by letting y = 12 EpI / T and x = 12 EpT / T and using the boundary conditions, we can obtain the input–output relationship: y = x − iC21 (x)
(6)
where C = ωp L|12 |2 N/2ε0 cT is the usual electronic cooperation parameter. The second term on the right-hand side of Eq. (6) is very important to achieve the OB. 3. Numerical results 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. 3–6. We assume 12 = 13 = 23 = , and all the figures are plotted in the unit of . In addition, we assume that all subbands have the same effective mass, and the electron–electron effects have very small influence on our results. First, we show the behavior of output–input fields for various rate of indirect incoherent pumping field. Fig. 3(a) shows that the threshold of optical bistability decreases by increasing the rate of weak indirect incoherent pumping field. For a strong indirect incoherent pumping field the optical bistability disappears (Fig. 3(b)). The physical mechanism of such behavior can be understood by the absorption spectrum. Fig. 4(a) implies that for a weak indirect incoherent pumping rate, the linear absorption decreases by increasing
H. Jafarzadeh / Optik 127 (2016) 1822–1826
1825
In this regime, i.e., ˝p2 → 0, we seek the corresponding steady state solution for 21 in the presence of indirect incoherent pumping field. In this case probe transition coherence becomes ˝p
21 =
A(B2 + C 2 )
where
{[BD − C(E + F)] − i[DC + B(E + F)]}.
⎛
⎞
⎜
2c
⎛ ⎜
+
2c +
23
2
+
23 p 2
2
D = (G − A) c
12
2
2
+R
13
13
23 2
+R
2
12
+R
2
13
+R
2
⎛
G = 2 ⎝21 + 232 +
+ R + (p + c )
F = 2˝c2 (21 − 32 )R
⎜
+ R + (p + c )
2 2
12
13
E = (G − A) −c (p + c ) + Fig. 6. (a) Behavior of output-input field intensity, (b) absorption as a function of probe field detuning for ˝c = 1, 3, 5. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, R = 0.1, c = 0.0, ˝p = 0.01, p = 0.0.
+R
+ R + (p + c )
C = c −p (p + c ) +
13
⎞
⎟ 2 ⎠ + 2R32 + 4R
23 −p (p + c ) + 2 − c p
23
2
˝c2 23
+ 32 ⎝21 +
⎟ 2 ⎠
˝c2 23
A = 21 ⎝32 + 2R +
B=
(7)
13
2
12
2 23 2 +R
˝c2 23 2c +
2
+ ˝c2
+R
23
2
23 2
+ ˝c2 c
⎞ ⎟ 2 ⎠ R
3˝c2 23 2c +
23
2 (8)
The response of the quantum system to the applied fields is determined by the susceptibility = + i , which is corresponding to the probe transition coherence 21 . The imaginary part of correspond to the absorption of the weak probe field. In our notation if Im[ ] 0, the system exhibits gain for the probe field, while for Im[ ] 0, the probe field will be attenuated. Note that for [DC + B(E + F)] = 0, the imaginary parts of 21 are equal to zero, i.e., Im(21 ) = 0. According to the this equation, the necessary value for switching the absorption to gain is approximately given by Fig. 7. Behavior of output-input field intensity for C = 250, 300, 350 mev. Other parameters are 21 = 31 = 32 = , 12 = 13 = 23 = 5, R = 0.1, ˝c = 3, c = 0.0, p = 0.0.
the rate of indirect incoherent pumping field. In fact, for R = 0.1 the absorption peak located at ı =0.0 reaches to 0.0125 value. However, for a strong indirect incoherent pumping field the absorption spectrum is converted to again, and by enhancing the rate of incoherent pumping field the gain will be increased (Fig. 4(b)). Physically, increasing the rate of indirect incoherent pumping field may reduce the probe field absorption and thus enhance the Kerr nonlinearity of QW sample, which makes the cavity field easier to reach saturation. For further discussion, in order to understand the physical mechanism of the results, we introduce an analytical expression for the steady state solutions of Eqs. (1) in the weak field approximation.
Rnecessary =
21 + 31 , (with consideration : 21 = 32 ) 2
(9)
The analytical calculation shows that the indirect incoherent pumping rate is an important parameter to changing the probe absorption to the probe gain. For R Rnecessary , Im(21 ) around zero probe field detuning is negative corresponding to a probe field amplification, while for R Rnecessary , Im(21 ) becomes positive corresponding to probe absorption. Note that for R = Rnecessary the imaginary part of 21 , i.e. Im(21 ), definitely becomes zero, thus no OB appears. According to the above discussion, when R = Rnecessary = 1.0, the OB disappears. In Fig. 4(a) and (b) we show the probe absorption in the presence of indirect incoherent pumping field. For R Rnecessary „ the incoherent pumping field is not able to establish the population inversion in probe transition and absorption around zero detuning. For R = Rnecessary the saturation effect appears and absorption
1826
H. Jafarzadeh / Optik 127 (2016) 1822–1826
vanishes. By increasing the incoherent pumping field, the population inversion is established. The absorption is negative and the absorption converted to gain. We will analyze how the intensity and the frequency detuning of the control field affect the optical bistability behavior, while keeping all other parameters fixed. Fig. 5(a) demonstrates the dependence of the optical bistability and multistability behaviors on the frequency detuning of the control field. It can be easily seen from Fig. 5(a) that the threshold and the hysteresis cycle shape change obviously due to the variational frequency detuning of the control field. The reason for the above result can be qualitatively explained as follows. The variational frequency detuning of control field can dramatically modify the absorption of the electronic medium according Fig. 5(b). So, in Fig. 5(a) we can see that the variational frequency detuning of control field lead to the OB converted to the OM. In Fig. 6(a), we plot the input–output field curves for different intensity of the control field. It is shown that the system is sensitive to intensity of the control field. This is the result of the influence of the controlling-field intensity on the absorption and dispersion properties of the electronic sample that is smaller than the non-resonance condition. It is easily seen, when intensity of the control field increases, the bistable threshold value reduces. In Fig. 6(b) we show that by increasing the intensity of the control field, the absorption of the probe field in the sample can be decreased. Finally, the effect of the electronic cooperation parameter C on optical bistability is depicted for a fixed value of p,c = 0 in Fig. 7. We observe that by increasing electronic cooperation parameter, the threshold of OB goes up. From the term C = ωp L|12 |2 N/2ε0 cT, we can see that the cooperation parameter is directly proportional to the number density of electrons. So, the enhancement in the absorption of the sample as the number density of electrons increases could account for the raise of the threshold intensity with respect to the cooperation parameter. We have assumed that the three level excitation scheme is valid under the defined conditions, which is justified in view of the experimental measurements and the theoretical simulation of that work.
4. Conclusion To sum up, we theoretically investigated optical bistability behavior in a three-level ladder type quantum well system inside a unidirectional ring cavity. It is shown that the threshold of optical bistability can be controlled by the rate of indirect incoherent pumping field. We find that the frequency detuning of the control field can affect for converting the optical bistability to multistability. In addition, the control field intensity can also be used to manipulate efficiently the threshold intensity and the hysteresis loop. Our scheme may be used for building more efficient all-optical switches and logic-gate devices for optical computing and quantum information processing. References [1] L.A. Lugiato, in: E. Wolf (Ed.), Progress in Optics, vol. 21, 1984, pp. 71, NorthHolland, Amsterdam. [2] H.M. Gibbs, S.L. McCall, T.N.C. Venkatesan, Phys. Rev. Lett. 36 (1976) 1135. [3] P.D. Drummond, D.F. Walls, J. Phys. A 13 (1980) 725. [4] A. Joshi, M. Xiao, Phys. Rev. Lett. 91 (2003) 143904. [5] W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. [6] X. Hu, Z. Xu, J.B. Opt, Quantum Semiclass. Opt. 3 (2001) 35. ˜ Phys. Lett. A 311 (2003) 297. [7] M.A. Antón, O.G. Calderón, F. Carreno, [8] Z. Chen, C. Du, S. Gong, Z. Xu, Phys. Lett. A 259 (1999) 15. [9] A. Joshi, Phys. Rev. B 79 (2009) 115315. [10] J.F. Dynes, E. Paspalakis, Phys. Rev. B 73 (2006) 233305. [11] J.H. Li, Phys. Rev. B 75 (2007) 155329. [12] C.R. Lee, Y.C. Li, F.K. Men, C.H. Pao, Y.C. Tsai, J.F. Wang, Appl. Phys. Lett. 86 (2005) 201112. [13] M. Phillips, H. Wang, Opt. Lett. 28 (2003) 831. [14] J.F. Dynes, M.D. Frogley, J. Rodger, C.C. Phillips, Phys. Rev. B 72 (2005) 085323. [15] S.M. Sadeghi, H.M. vanDriel, J.M. Fraser, Phys. Rev. B 62 (2000) 15386. [16] J.H. Wu, J.Y. Gao, J.H. Xu, L. Silvestri, M. Artoni, G.C. LaRocca, F. Bassani, Phys. Rev. Lett. 95 (2005) 057401. [17] A. Joshi, M. Xiao, Appl. Phys. B 79 (2004) 65. [18] H. Sun, Y. Niu, R. Li, S. Jin, S. Gong, Opt. Lett. 32 (2007) 2475. [19] W.X. Yang, J.M. Hou, R.K. Lee, Phys. Rev. A 77 (2008) 033838. [20] J.F. Dynes, M.D. Frogley, M. Beck, J. Faist, C.C. Phillips, Phys. Rev. Lett. 94 (2005) 157403. [21] A. Joshi, M. Xiao, Appl. Phys. B: Lasers Opt. 79 (2004) 65. [22] J. Li, X. Yang, Eur. Phys. J. B 53 (2006) 449. [23] M.A. Anton, O.G. Calderon, F. Carreno, Phys. Lett. A 311 (2003) 297.