Transient gain-absorption of the probe field in asymmetric semiconductor quantum wells

Optics Communications 283 (2010) 2552–2556

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Transient gain-absorption of the probe field in asymmetric semiconductor quantum wells Zhiping Wang Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China

a r t i c l e

i n f o

Article history: Received 3 November 2009 Received in revised form 4 February 2010 Accepted 4 February 2010

Keywords: Transient gain-absorption property Semiconductor quantum wells

a b s t r a c t Under the weak-probe approximation, we theoretically investigate the transient gain-absorption property of the probe field in a four-level asymmetric semiconductor quantum well system. We find that the strength of Fano interference and the energy splitting affect the transient gain-absorption property of the weak continuous-wave (CW) probe field or Gaussian-pulse probe field dramatically. The dependence of transient gain-absorption property of the probe field on the intensity and the frequency detuning of the strong coupling field is also given. Our study is much more practical than its atomic counterpart due to its flexible design and the controllable interference strength. Thus, it may provide some new possibilities for technological applications. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction In the past decade there has been growing interest in the study of quantum optical phenomena based on coherence and quantum interference in quantum optical media [1–20]. Examples are Coherent Population Trapping [1,2], Electromagnetically Induced Transparency [3–6], Lasing Without Inversion [7–9], Light Storage [10,11], Multi-Wave Mixing [12–14], enhancing Kerr Nonlinearity [15], Optical Soliton [16,17], Optical Bistability/Multistability [18–20], etc. It should be worth pointing out that many kinds of nonlinear quantum optical phenomena based on the quantum interference and coherence have also been extensively studied in the semiconductor quantum wells (SQWs) [21–36], which include Gain Without Inversion [21–23], Electromagnetically Induced Transparency [24–27], Optical Bistability/Multistability [28,29], Kerr Nonlinearity [30], Optical Soliton [31], and Four-Wave Mixing [32]. The reason for this is mainly the phenomena in the SQWs have many potentially important applications in optoelectronics and solid-state quantum information science. Otherwise, devices based on intersubband transitions in the SQWs have many inherent advantages that the atomic systems do not have, especially 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, and the transition energies and the dipoles as well as the symmetries can also be engineered as desired [29]. E-mail addresses: [email protected], [email protected], zhipwang@ 126.com 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.02.009

In this work, we theoretically investigate the transient gain-absorption property of the probe field in a four-level semiconductor quantum well system. It is found that some physical parameters affect the transient gain-absorption property of the weak continuous-wave (CW) probe field or Gaussian-pulse probe field dramatically. Our study and the four-level system are based on the Refs. [29,30,36], however, our results are different from those papers. Firstly, we are mainly interested in studying the controllability of the transient gain-absorption property the probe field via the different physical parameters. Secondly, a very important advantage of our scheme is that our system is much more practical than its atomic counterpart [37–40] due to its flexible design, controllable interference intensity and wide adjustable parameters. Thirdly, we also show the effects of the strength of Fano interference and the energy splitting on the transient property of the Gaussian-pulse probe field, which have never been investigated in this semiconductor quantum well system to our best knowledge. Our paper is organized as follows: in Section 2, we present the theoretical model and establish the corresponding dynamic equations. Our numerical results and physical analysis are shown in Section 3. In Section 4, a simple conclusion is given. 2. The model and the dynamic equations The asymmetric double semiconductor QW structure is shown in Fig. 1 [29]. The j ai; j bi are the first subband of the shallow well and the second subband of the deep well, respectively, which are separated by a narrow barrier (see Fig. 1(a)). Due to the strong coherent coupling via the thin barrier, the levels split into a

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By adopting the standard approach [22,23,27–29,33,34], the density-matrix equations of motion under the electric dipole and rotating-wave approximations for this system can be written as follows:

iq_ 22 ¼ Xp ðq21  q12 Þ þ Xc ðq24  q42 Þ þ ic42 q44  ic2 q22

g

(a)

(b)

iq_ 33

 i ðq23 þ q32 Þ; 2 ¼ Xp ðq31  q13 Þ þ Xc ðq34  q43 Þ þ ic43 q44  ic3 q33

g

Fig. 1. (a) Energy level diagram of a double quantum well structure. It consists of two quantum wells and a collector region separated by thin tunnelling barriers. Subband jai of the shallow well is resonant with the second subband jbi of the deep well. (b) Due to the strong coherent coupling via the thin barrier, the levels split into a doublet j3i and j2i, which are coupled to a continuum by a thin tunnelling barrier adjacent to the deep well. The lower subband in the deep well is denoted by level j1i, and the higher subband in the shallow well is represented by level j4i. xs is the splitting between the two upper levels, xc is the strong coupling field, and xp is the weak-probe filed.

iq_ 44 iq_ 12

 i ðq23 þ q32 Þ; 2 ¼ Xc ðq42  q24 þ q43  q34 Þ  ic4 q44 ;   xs C12 ¼ Xp ðq11  q22  q32 Þ þ Xc q14 þ q12  Dp  i 2 2

g

 i q13 ; 2 iq_ 13 ¼ Xp ðq11  q33  q23 Þ þ Xc q14 



xs 2

C13

þ Dp þ i



q13

2

g

 i q12 ; 2

pffiffiffi doublet states pffiffiffi j 2i and j 3i, and j 2i ¼ ðj ai j biÞ= 2; j 3i ¼ ðj aiþ j biÞ= 2 (see Fig. 1(b)). The splitting xs on resonance is given by the coupling intensity and can be controlled by adjusting the height and width of the tunnelling barrier with applied bias voltage [21]. The lower subband in the deep well is denoted by level j 1i, and the higher subband in the shallow well is represented by level j 4i. A low intensity pulsed laser field xp (amplitude Ep ) is applied to the transitions j 2i $j 1i and j 3i $j 1i simultah and neously with the respective Rabi frequencies Xp1 ¼ l21 Ep =2 Xp2 ¼ l31 Ep =2h, and l21 ; l31 are the relevant intersubband dipole moments, while a strong coupling field xc (amplitude Ec ) is applied to the transitions j 4i $j 2i and j 4i $j 3i simultaneously with the h and Xc2 ¼ l43 Ec =2 h, respective Rabi frequencies Xc1 ¼ l42 Ec =2 and l42 ; l43 are the relevant intersubband dipole moments.

  C14 iq_ 14 ¼ Xc ðq12 þ q13 Þ  Xp ðq34 þ q24 Þ  Dc þ Dp þ i q14 ; 2   C23 q23 iq_ 23 ¼ Xp ðq21  q13 Þ þ Xc ðq24  q43 Þ  xs þ i 2

g

 i ðq22 þ q33 Þ; 2 iq_ 24 ¼ Xc ðq22  q44 þ q23 Þ  Xp q14 



xs 2

þ Dc þ i

C24 2



q24

g

 i q34 ; 2 iq_ 34 ¼ Xc ðq33  q44 þ q32 Þ  Xp q14 þ



xs 2

 Dc  i

C34 2



q34

g

 i q24 ; 2

ð1Þ

-3

0.01

6

x 10

Ω c =2γ

Δc =0 4

Ω c =6γ

0.005

Ω c =10γ

Δc =3γ Δc =6γ

2

0

Im(ρ12+ ρ13)

Im(ρ12+ ρ13)

0 -0.005

-0.01

-2 -4 -6

-0.015

-8 -0.02 -10

(a ) -0.025

0

1

2

3 tγ

4

5

(b) -12

6

0

1

2

3 tγ

4

5

6

-3

-3

0

x 10

0

(c 2) Im(ρ12+ρ13)

Im(ρ12+ ρ13)

(c1) -1 -2 -3 -4

-1 -2 -3 -4

0

x 10

1

2

3

4

5 Ω c /γ

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

Δc/γ

Fig. 2. (a) The time evolution of Imðq12 þ q13 Þ for different Xc . The other parameters are Xp ¼ 0:1c; xs ¼ 8:72c; Dp ¼ Dc ¼ 0; c2 ¼ 3:47c; c3 ¼ 4:13c; c42 ¼ c43 ¼ 0:4c; dph dph dph dph dph cdph 12 ¼ 0:68c; c13 ¼ 0:8c; c14 ¼ 0:5c; c23 ¼ 0:74c; c24 ¼ 0:59c, and c34 ¼ 0:65c. (b) The time evolution of Imðq12 þ q13 Þ for different Dc , the parametric conditions are the same as in the panel (a) except for Xc ¼ 10c. (c) The steady-state studies for CW probe field, the parametric conditions of the panel (c1) are the same as in the panel (a), and the parametric conditions of the panel (c2) are the same as in the panel (b).

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constrained by qnm ¼ qmn ðm; n ¼ 1; 2; 3; 4Þ and q11 þ q22 þ q33 þ q44 ¼ 1. Here we assume Xp1 ¼ Xp2 ¼ Xp and Xc1 ¼ Xc2 ¼ Xc are real. DpðcÞ ¼ x0pð0cÞ  xpðcÞ is the detuning between the frequency of the probe (coupling) field and the average transition frequency   x0pð0cÞ x0pð0cÞ ¼ ðx2ð42Þ þ x3ð43Þ Þ=2 . xs ¼ E3  E2 is the energy splitting between the states j 3i and j 2i, given by the coherent coupling intensity of the tunnelling. The population decay rates and dephasing decay rates are added phenomenologically in the above equations [34,35]. The population decay rates for subband j ii, denoted by ci , are due primarily to longitudinal optical (LO) phonon emission events at low temperature. The total decay rates Cij ði – jÞ are given

dph dph C12 ¼ c2 þ cdph 12 ; C13 ¼ c3 þ c13 ; C14 ¼ c4 þ c14 ; C23 ¼ dph C24 ¼ c2 þ c4 þ c24 , and C34 ¼ c3 þ c4 þ cdph 34 ðc4 ¼

by

dph 23 ;

c2 þ c3 þ c c43 þ c42 Þ, where cdph determined by electron–electron, interface ij roughness, and phonon scattering processes, is the dephasing decay rate of the quantum coherence of the j ii $j ji transition. The pffiffiffiffiffiffiffiffiffiffi g ¼ c2 c3 represents a coupling term between the states j 3i and j 2i via the LO phonon decay, it describes the process in which a phonon is emitted by subband j 1iand is recaptured by subband j 2i. These coupling terms can be obtained if tunnelling is present, e.g., through an additional barrier next to the deeper well [21]. As

mentioned above, states j 3i and j 2i are both the superpositions of the resonant states j ai and j bi. Because the subband j bi is strongly coupled to a continuum via a thin barrier, the decay from state j bi to the continuum inevitably results in these two dependent decay pathways: from the excited doublet to the common continuum. That is to say, the two decay pathways are related: the decay from one of the excited doublets can strongly affect the neighbouring transition, resulting in the interference characterized by those coupling terms. The probe absorption can be canceled due to the Fano interference between the two decay paths. Such interference is similar to the decay induced coherence in atomic systems with two closely lying energy states and occurs due to quantum interference in the electronic continuum [21,22]. The intensity of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the Fano interference [21,29,30,36], defined by p ¼ g= C12 C13 , and the values p ¼ 0 and p ¼ 1 correspond to no interference (no dephasing) and perfect interference (negligible coupling between state j 3i and state j 2i), respectively. It is well known that the absorption and dispersion are related to the susceptibility of the system. Therefore, the probe absorption-dispersion coefficient coupled to the transitions j 2i $j 1i and j 3i $j 1i is proportional to the terms q12 þ q13 . The imaginary parts Imðq12 þ q13 Þ correspond to gain-absorption coefficient, yet -3

-3

6

x 10

8

ωs =8.72γ 4

ωs =8.72γ

6

ωs =14γ

ωs =14γ

4

ωs =20γ

2

x 10

ωs =20γ

2 Im(ρ12+ ρ13)

Im(ρ12+ ρ13)

0 -2 -4

0 -2 -4

-6

-6

-8

-8

-10 -12

-10

(a) 0

0.5

1

1.5

2 tγ

2.5

3

3.5

-12 4

(b) 0

0.5

1

1.5

2 tγ

2.5

3

3.5

4

-4

-2.5

x 10

-3 -3.5

Im(ρ12+ρ13)

-4 -4.5 -5 -5.5 -6 -6.5 0

(c) 5

10

15

20

25

ωs /γ Fig. 3. (a) The time evolution of Imðq12 þ q13 Þ for different xs . The other parameters are Xp ¼ 0:1c; Xc ¼ 10c; Dp ¼ Dc ¼ 0; c2 ¼ 3:47c; c3 ¼ 4:13c; c42 ¼ c43 ¼ 0:4c; dph dph dph dph dph 2 2 cdph 12 ¼ 0:68c; c13 ¼ 0:8c; c14 ¼ 0:5c; c23 ¼ 0:74c; c24 ¼ 0:59c, and c34 ¼ 0:65c. (b) Study for the Gaussian-pulse probe field Xp ðtÞ ¼ X0 expðt =s Þ (the s is the Gaussianpulse width, and s ¼ c1 ), the parametric conditions are the same as in the panel (a) except for X0 ¼ 0:1c. (c) The steady-state studies for CW probe field, the parametric conditions are the same as in the panel (a).

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-3

6

x 10

8 p=0.84 p=0.72 p=0.56

4

x 10

p=0.84 p=0.72 p=0.56

6 4

2 2 Im(ρ12+ ρ13 )

Im(ρ12+ ρ13 )

0 -2 -4 -6

-2 -4 -6

-8

-8

-10 -12

0

-10

(a) 0

1

2

3 tγ

4

5

-12 6

(b) 0

0.5

1

1.5

2 tγ

2.5

3

3.5

4

dph dph dph dph dph Fig. 4. (a) The time evolution of Imðq12 þ q13 Þ for different p. p ¼ 0:84 : cdph 12 ¼ 0:68c; c13 ¼ 0:8c; c14 ¼ 0:5c; c23 ¼ 0:74c; c24 ¼ 0:59c, and c34 ¼ 0:65c; p ¼ 0:72 : dph dph dph dph dph dph dph dph dph dph dph cdph 12 ¼ 1:36c; c13 ¼ 1:6c; c14 ¼ c; c23 ¼ 1:48c; c24 ¼ 1:18c, and c34 ¼ 1:3c. p ¼ 0:56 : c12 ¼ 2:72c; c13 ¼ 3:2c; c14 ¼ 2c; c23 ¼ 2:96c; c24 ¼ 2:36c, and c34 ¼ 2:6c. The other parameters are Xp ¼ 0:1c; Xc ¼ 10c; xs ¼ 8:72c Dp ¼ Dc ¼ 0; c2 ¼ 3:47c; c3 ¼ 4:13c, and c42 ¼ c43 ¼ 0:4c. (b) Study for the Gaussian-pulse probe field Xp ðtÞ ¼ X0 expðt2 =s2 Þ (the s is the Gaussian-pulse width, and s ¼ c1 ), the parametric conditions are the same as in the panel (a) except for X0 ¼ 0:1c.

the real parts Reðq12 þ q13 Þ correspond to dispersion coefficient. If Imðq12 þ q13 Þ > 0, the probe field will be amplified; on the contrary, the probe field will be absorbed.

3. Numerical results and physical analysis Now we show some results of numerical studies, as shown in Figs. 2–4. In the following numerical calculations, we choose the parameters to be units by scaling c and we assume that all subbands have the same effective mass and the electron–electron effects have very small influence on our results. In addition, the choices of the parameters are mainly based on results from Refs. [29,30,36]. In Fig. 2, we show the dependence of the transient gain-absorption property of the probe field on the intensity and the frequency detuning of the strong coupling field. It can be easily seen from Fig. 2(a) and (b) that the intensity and the frequency detuning of the coupling field can affect amplitude of the transient gainabsorption coefficient dramatically, however, when the system reaches the steady state, the increasing intensity of the coupling field leads to a significant decreasing of the gain-absorption coefficient in Fig. 2(a), while the increasing frequency detuning of the coupling field leads to a significant increasing of the gain-absorption coefficient in Fig. 2(b). The reason for the above results can be qualitatively explained as follows. By applying an increasingly strong coupling field between both the subbands j 4i $j 2i and j 4i $j 3i, the absorption for the probe field on the intersubband transitions j 2i $j 1i and j 3i $j 1i of the electronic medium can be reduced dramatically [29,36]. Clearly, the increasing frequency detuning of the coupling field will reduce the coupling effect. So, we can receive the above results. In order to gain deeper insight into the phenomenon, the steady-state study for the CW probe field is given in Fig. 2(c), which is verified the above analysis. In the following, we analyze the influence of the energy splitting on transient gain-absorption coefficient of the probe field in Fig. 3. Obviously, whatever continuous-wave (CW) probe field or Gaussian-pulse probe field, the amplitude of the transient gainabsorption coefficient is insensitive to the energy splitting, but we remain see from Fig. 3(a) and (b) that the energy splitting can suppress the steady-state absorption of the electronic medium weakly. In order to test the validity of the analysis described above,

we carry out the steady-state numerical calculation for the CW probe field in Fig. 3(c), as already verified the above result. At last, the effect of the strength of the Fano interference on transient gain-absorption property of the probe field is given in Fig. 4. We find that the strength of the Fano interference can also reduce the steady-state gain-absorption coefficient of the electronic medium (Fig. 4(a)) and influence the amplitude of the transient gain-absorption coefficient obviously. A reasonable explanation for this is that the strength of the Fano interference can dramatically modify the optical property of the electronic medium, which makes the decreasing of the gain-absorption coefficient. Before ending this section, we would like to mention two key points of the present study. The first one is that we presume that the coupling and probe fields are switched on at t ¼ 0. The second one is that the results for transient gain-absorption property of the weak continuous-wave (CW) probe field (or Gaussian-pulse probe field) are carried out under the initial conditions q11 ð0Þ ¼ 1; q22;33;44 ð0Þ ¼ 0, and qij ð0Þ ¼ 0 for i – j ði; j ¼ 1; 2; 3; 4Þ. 4. Conclusions To sum up, we theoretically investigate the transient gainabsorption property of the probe field in a four-level asymmetric semiconductor quantum well system under weak-probe approximation. We find that the strength of Fano interference, the energy splitting as well as the intensity and the frequency detuning of the strong coupling field affect the gain-absorption property of the probe field dramatically, which can be used to manipulate efficiently the gain-absorption coefficient. By comparing the studies in the atomic system [37–40], our study is much more practical than its atomic counterpart due to its flexible design and the controllable interference strength. Thus, it may provide some new possibilities for technological applications in optoelectronics and solid-state quantum information science. Acknowledgements The author thanks Prof. Hong-Yi Fan for his support and encouragement, and this work was partly supported by National Natural Science Foundation of China under grant 10775097.

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