Control of the probe absorption via incoherent pumping fields in asymmetric semiconductor quantum wells

Annals of Physics 326 (2011) 340–349

Contents lists available at ScienceDirect

Annals of Physics journal homepage: www.elsevier.com/locate/aop

Control of the probe absorption via incoherent pumping fields 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 20 October 2009 Accepted 3 September 2010 Available online 19 September 2010 Keywords: Probe absorption Incoherent pumping fields Semiconductor quantum wells

a b s t r a c t In a three-level asymmetric semiconductor quantum well system, owing to the effects that result from the incoherent pumping fields, the probe absorption of probe field can be effectively controlled. The result is achieved by applying the two incoherent pumping fields, so it is different from the conventional way in ordinary laser-driven schemes that coherent driving fields are necessary to control the probe absorption. Otherwise, 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. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction In the past few decades, a great deal of quantum optical phenomena based on coherence and quantum interference have attracted a lot of attention of many researchers in the quantum coherent 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) [21–35], such as gain without inversion [21–23], electromagnetically induced transparency [24–27,33], optical bistability/multistability [28,29], Kerr nonlinearity [30], optical soliton [31], four-wave mixing [32], and so on. The reason for this is mainly that 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 E-mail address: [email protected] 0003-4916/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.09.002

341

Z. Wang / Annals of Physics 326 (2011) 340–349

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, and the transition energies and the dipoles as well as the symmetries can also be engineered as desired. In this work, we numerically investigate the absorption of the probe field in a three-level asymmetric semiconductor quantum well system. It is found that the probe absorption can be manipulated by properly adjusting the rates of the incoherent pumping fields. Our study and the system are mainly based on the Refs. [29–31,36,38], however, which are drastically different from those works. First and foremost is that we are interested in studying the controllability of probe absorption via different parameters. Secondly, an important advantage of our scheme is that our system is much more practical than its atomic counterpart due to its flexible design, the controllable interference intensity and the wide adjustable parameters. Thirdly, the effect of incoherent pumping field on the probe absorption of the probe field in this three-level asymmetric semiconductor quantum well system has never been investigated to our best knowledge, which motivates us to carry out the current work. Our paper is organized 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 the dynamic equations The semiconductor double QW structure consisting of two quantum wells that are separated by a narrow barrier is shown in Fig. 1 [21,23]. The jai and jbi are the first subband of the shallow well and the second subband of the deep well, respectively, which are resonant (see Fig. 1(a)). Due to the strong coherent coupling via the thin barrier, the levels split into a doublet levels j1i and j2i, and

a

b

2

ωs

1 ωp

0

0

(a)

(b) 2

ωs

1 Λ1 Λ 2

ωp

2

ωs 1

Λ1

Λ2

0

(c)

ωp 0

(d)

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 j1i and j2i, which are coupled to a continuum by a thin tunnelling barrier adjacent to the deep well. xs is the splitting between the two upper levels and xp is the weak probe laser. (c) Schematic diagram of the quantum well system applied by two incoherent pumping fields K1 and K2. (d) Schematic of the energy level arrangement for the quantum well system under study.

342

Z. Wang / Annals of Physics 326 (2011) 340–349

pffiffiffi pffiffiffi j1i ¼ ðjai  jbiÞ= 2; j2i ¼ ðjai þ jbiÞ= 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]. A low intensity pulsed laser field xp (amplitude Ep) is applied to the transitions j0i M j1i and j0i M j2i simultaneously with the respective Rabi frequencies Xp1 = l10Ep/ 2 h and Xp2 = l20Ep/2 h, and l10, l20 are the relevant intersubband dipole moments. The small signal absorption of the weak-probe field propagating through such a system can be computed in the steady state. By adopting the standard approach [27–29,33,34], the density-matrix equations of motion in dipole and rotating-wave approximations for this system can be written as follows:

@ q11 i ¼ Xp2 q10  Xp2 q01  ic1 q11  gðq12 þ q21 Þ; 2 @t @q i i 22 ¼ Xp1 q20  Xp1 q02  ic2 q22  gðq12 þ q21 Þ; 2 @t @q xs c i q  Dp q01  i 10 q01  gq02 ; i 01 ¼ Xp2 q00  Xp2 q11  Xp1 q21 þ 2 @t 2 01 2 @q xs c i q  Dp q02  i 20 q02  gq01 ; i 02 ¼ Xp1 q00  Xp1 q22  Xp2 q12  2 @t 2 02 2 @q c i i 12 ¼ Xp1 q10  Xp2 q02  xs q12  i 21 q12  gðq11 þ q22 Þ 2 @t 2 i

ð1Þ

  >constrained by qnm ¼ qmn ðm; n ¼ 0; 1; 2Þ; i @t@ q00 ¼ i @t@ q11 þ @t@ q22 , and q00 + q11 + q22 = 1. Here we assume Xp1 and Xp2 are real. Dp = x0  xp is the detuning between the frequency of the probe laser and the average transition frequency x0(x0 = (x1 + x2)/2). xs = E2  E1 is the energy splitting between the two upper levels, 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]. The population decay rates for subband jii, denoted by ci, are due primarily to longitudinal optical (LO) phonon emission events at low temperature. The total decay rates cij (i – j) are dph dph dph given by c10 ¼ c1 þ cdph determined by electron– 10 ; c20 ¼ c2 þ c20 , and c21 ¼ c1 þ c2 þ c21 , where cij electron, interface roughness, and phonon scattering processes, is the dephasing decay rate of the pffiffiffiffiffiffiffiffiffiffi quantum coherence of the jii M jji transition. The g ¼ c1 c2 represents cross coupling between the states j1i and j2i via the longitudinal optical (LO) phonon decay, it describes the process in which a phonon is emitted by subband j1i and is recaptured by subband j2i. These cross coupling terms can be obtained if tunnelling is present, e.g., through an additional barrier next to the deeper well. As mentioned above, levels j1i and j2i are both the superpositions of the resonant states jai and jbi. Because the subband jbi is strongly coupled to a continuum via a thin barrier, the decay from state jbi 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 cross coupling terms. The probe absorption can be controlled 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]. The intensity of the Fano interference [21,35], defined by p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g= c10 c20 ¼ c1 c2 =c10 c20 , and the values p = 0 and p = 1 correspond to no interference (there is no negligible coupling between j1i and j2i. This means that Fano interference does not generate) and per¼ 0), respectively. It is worth noting that the abovefect interference (no dephasing decay rates cdph ij described parameter p is mainly controlled via the population decay rates ci (c1, c2) and dephasing dedph dph dph cay rates cdph ij ðc10 ; c20 , and c21 Þ, and it appears when the cross coupling term is present (g – 0) but disappears when the cross coupling term is absent (g = 0). In order to study the effects of the incoherent pumping fields on the quantum well system, two broadband polarized fields K1 and K2 (can be provided by the diode laser that had a broad variable linewidth) that serve as the incoherent pumping fields and apply to the transitions j0i M j1i and j0i M j2i, respectively (see Figs. 1(c) and 1(d)). Then the Eq. (1) becomes as follows:

Z. Wang / Annals of Physics 326 (2011) 340–349

~ 11 @q @t ~ 22 @q i @t ~ 01 @q i @t ~ 02 @q i @t

i

343

i ~ 12 þ q ~ 21 Þ þ iKq ~ 10  Xp2 q ~ 01  ic1 q ~ 11  gðq ~ 00  iKq ~ 11 ; ¼ Xp2 q 2 i ~ 12 þ q ~ 21 Þ þ iKq ~ 20  Xp1 q ~ 02  ic2 q ~ 22  gðq ~ 00  iKq ~ 22 ; ¼ Xp1 q 2 x ~ c ~ i ~ 00  Xp2 q ~ 11  Xp1 q ~ 21 þ s q ~ 01  i 10 q ~ 02  iKq ~ 01 ; ¼ Xp2 q 01  Dp q 01  gq 2 2 2 x c i ~ 00  Xp1 q ~ 22  Xp2 q ~ 12  s q ~ 02  i 20 q ~ 01  iKq ~ 02 ; ~ 02  Dp q ~ 02  gq ¼ Xp1 q 2 2 2

~ 12 @q c ~ i ~ 11 þ q ~ 22 Þ  iKq ~ 10  Xp2 q ~ 02  xs q ~ 12  i 21 q ~ 12 ; ¼ Xp1 q 12  gðq @t 2 2 where we have set K = K1 = K2. i

ð2Þ

By the straightforward semiclassical analysis, the above nonlinear density-matrix equations can be used to calculate the total complex susceptibility v of the probe transition, i.e.,

~ 01 þ q ~ 02 Þ Njlj2 ðq ~ 01 þ q ~ 02 : /q ð3Þ 2he0 Xp Here we assume l01 = l02 = l and Xp1 = Xp2 = Xp, N is the electron density and e0 is the permittivity of

v¼

free space. It is well known that the absorption and dispersion are related to the susceptibility of the system [21,38]. Therefore, the probe absorption–dispersion coefficient coupled to the transitions ~ 01 þ q ~ 02 . The imaginary parts Imðq ~ 01 þ q ~ 02 Þ correj0i M j1i and j0i M j2i is proportional to the terms q ~ 01 þ q ~ 02 Þ correspond to dispersion. If spond to absorption or amplification, yet the real parts Reðq ~ 01 þ q ~ 02 Þ > 0, the probe laser will be amplified; on the contrary, the probe laser will be absorbed. Imðq 3. Numerical results Now we give some numerical studies under the steady-state conditions, as shown in Figs. 2–5. In the following numerical calculations, 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 based on experimental results from reference [21]. 3.1. The case K = 0 When we do not add the incoherent pumping fields, it is found that the strength of the Fano interference and the energy splitting can affect the probe absorption efficiently, as illustrated in Figs. 2 and 3. In the Fig. 2, we analyze how the strength of the Fano interference p modifies the probe absorption while keeping all other parameters fixed. It can be easily seen from Fig. 2 that the probe absorption at the line center Dp = 0 reduces dramatically due to the increasing strength of the Fano interference. The reason can be qualitatively explained as follows. It is obvious the quantum interference parameter g dph remains fixed when c1 = 5.6 meV and c2 = 7 meV, but the varying dephasing decay rates cdph ðcdph 10 ; c20 , ij dph and c21 Þ influence the strength of the Fano interference p, which affects the probe absorption. In order to gain deeper insight into the above problem, we plot the probe absorption versus the detuning Dp for the different values of the energy splitting xs in Fig. 3. For the small strength of the Fano interference, say p = 0.46 as shown in Fig. 3(a), we find that the probe absorption is very large at the line center and the increasing energy splitting xs can reduce the probe absorption at the line center Dp = 0, while increasing p = 0.77 further to large value, say as shown in Fig. 3(b), the probe absorption is very small at the line center Dp = 0 and the energy splitting xs can also reduce the probe absorption at the line center. Therefore, the behavior of the probe field can be tuned by appropriately adjusting the tunnelling barrier. 3.2. The case K – 0 When we add the two incoherent pumping fields, the results are extremely different, which can be easily seen from Figs. 4 and 5.

344

Z. Wang / Annals of Physics 326 (2011) 340–349

0 -0.02

Probe Absorption

-0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -40

Λ=0 meV p=0.77 p=0.63 p=0.46 -30

-20

-10

0

10

20

30

40

Δp Fig. 2. The probe absorption as a function of the probe field frequency detuning Dp for different strength of the Fano dph dph dph dph interference p: red line ðp ¼ 0:77Þ cdph 10 ¼ 1:5 meV; c20 ¼ 2:3 meV; c21 ¼ 1:9 meV; green line ðp ¼ 0:63Þ c10 ¼ 3 meV; c20 ¼ dph dph dph 4:6 meV; cdph 21 ¼ 3:8 meV; blue line ðp ¼ 0:46Þ c10 ¼ 6 meV; c20 ¼ 9:2 meV; c21 ¼ 7:6 meV. The other parameters are Xp = 0.5 meV, xs = 17.6 meV, c1 = 5.6 meV, c2 = 7 meV, K = 0 meV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

We show the dependence of the probe absorption on the rate K in Fig. 4. For the small value of K = 2 meV, the probe absorption at the line center is large (see red line in Fig. 4(a)). However, when K = 8 meV, the probe field can not be amplified but the probe absorption is obviously suppressed by the increasing rate of the pumping fields (see blue line in Fig. 4(a)). For further show explicitly the influence of the rate K on the absorption of the probe field, we give the plot of probe absorption as a function of the rate K for different strength of the Fano interference p in Fig. 4(b). Obviously, the increasing the rate K leads to a significant decrease of the probe absorption in Fig. 4(b). Also, from Fig. 4(b), we can see that the dependence of the probe absorption on the strength of the Fano interference p becomes very small when value of K is large (K P 10 meV). According to the above discussion, one can realize the incoherent pumping fields play important roles in manipulating the probe absorption and can be used to reduce the effect of the strength of the Fano interference p on the probe absorption. In order to test the validity of the analysis described above, we carry out extensive numerical calculations. In the following numerical calculations, under the condition that the incoherent pumping fields are considered, we study the effects of the strength of the Fano interference p and the energy splitting xs on the probe absorption in Fig. 5. As shown in Fig. 5(a), one can find that the probe absorption spectra changes monotonically at Dp = 0 and the order of magnitude of the probe absorption becomes very smaller. At the same time, in Fig. 5(b), it is found that the probe absorption depends so sensitively on the energy splitting xs, but the effect of the strength of the Fano interference on the probe absorption becomes lower with increasing energy splitting. Specifically, for the case when the energy splitting reaches a certain range 14 meV [ xs [ 18 meV (see the inset in Fig. 5(b)), the probe absorption is insensitive to the strength of the Fano interference. It should be worth pointing out that these results can not be seen when the incoherent pumping fields are not considered, which are also verified the comments in Fig. 4. Before ending this section, we would like to mention some key points for the present study: (i) when the rate K reaches a reasonable value, the effect of the strength of the Fano interference p on the probe absorption can be ignored. This means that we do not need to consider influences of dephasing decay

345

Z. Wang / Annals of Physics 326 (2011) 340–349

0 -0.01

Probe Absorption

-0.02 -0.03 -0.04 -0.05 -0.06

Λ=0 meV, p=0.46

-0.07

ωs =17.6 meV

-0.08

ωs =19 meV

-0.09 -40

ωs =21 meV -30

-20

-10

0

10

20

30

40

10

20

30

40

Δp

(a) 0 -0.02

Probe Absorption

-0.04 -0.06 -0.08 -0.1

Λ=0 meV, p=0.77 ωs =17.6 meV

-0.12

ωs =19 meV ωs =21 meV

-0.14 -40

-30

-20

-10

0

Δp

(b) Fig. 3. The probe absorption as a function of the probe field frequency detuning Dp for different energy splitting xs: (a) p = 0.77 and dph dph (b) p = 0.46. The other parameters are cdph 10 ¼ 1:5 meV; c20 ¼ 2:3 meV; c21 ¼ 1:9 meV; Xp ¼ 0:5 meV; c1 ¼ 5:6meV; c2 ¼ 7 meV; K ¼ 0 meV.

dph dph rates cdph ðcdph 10 ; c20 , and c21 Þ on the probe absorption. It is well known that the dephasing decay rates ij dph cij are determined by electron–electron, interface roughness, and phonon scattering processes. So we can also ignore these processes, which make our scheme much more convenient in experimental realization; (ii) by applying two pumping fields on the intersubband transitions j0i M j1i and j0i M j2i, the absorption of the probe field can be efficiently manipulated without need to resort the strong

346

Z. Wang / Annals of Physics 326 (2011) 340–349

0 -0.005

Probe Absorption

-0.01 -0.015 -0.02 -0.025 -0.03 p=0.77 -0.035

Λ=2 meV Λ=4 meV Λ=8 meV

-0.04 -0.045 -40

-30

-20

-10

0

10

20

30

40

Δp

(a) 0 -0.005

Probe Absorption

-0.01 -0.015 -0.02 -0.025 p=0.77 p=0.63 p=0.46

-0.03 -0.035

0

2

4

6

8

10

12

14

16

18

20

Λ

(b) Fig. 4. (a) The probe absorption as a function of the probe field frequency detuning Dp for different K. The other parameters are dph dph cdph (b) The probe 10 ¼ 1:5 meV; c20 ¼ 2:3 meV; c21 ¼ 1:9 meV; Xp ¼ 0:5 meV; c1 ¼ 5:6 meV; c2 ¼ 7 meV; xs ¼ 17:6 meV. absorption as a function of the K for different strength of the Fano interference p: red line ðp ¼ 0:77Þ cdph 10 ¼ dph dph dph 1:5 meV; cdph green line ðp ¼ 0:63Þ cdph blue line 20 ¼ 2:3 meV; c21 ¼ 1:9 meV; 10 ¼ 3 meV; c20 ¼ 4:6 meV; c21 ¼ 3:8 meV; dph dph ðp ¼ 0:46Þ cdph 10 ¼ 6 meV; c20 ¼ 9:2 meV; c21 ¼ 7:6 meV. The other parameters are Xp = 0.5 meV, xs = 17.6 meV, c1 = 5.6 meV, c2 = 7 meV, Dp = 0 meV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fano-interference effect, which is also simple in experimental arrangements; and (iii) the SQW sample used in this investigation is very much similar to the one reported in reference [21] so that we can keep

347

Z. Wang / Annals of Physics 326 (2011) 340–349 -3

-2

x 10

-3

Probe Absorption

-4

-5

-6

Λ=10 meV p=0.77 p=0.63 p=0.46

-7

-8 -40

-30

-20

-10

0

10

20

30

40

Δp

(a) 0 p=0.77 p=0.63 p=0.46

-0.002

Probe Absorption

-0.004

-0.006 -3

-6

-0.008

x 10

-6.5

-0.01

-7 -7.5

-0.012

-8 14

-0.014

0

10

20

30

15

40

16

50

17

60

18

70

80

ωs

(b) Fig. 5. (a) The probe absorption as a function of the probe field frequency detuning Dp for different strength of the Fano dph dph dph dph interference p: red line ðp ¼ 0:77Þ cdph 10 ¼ 1:5 meV; c20 ¼ 2:3 meV; c21 ¼ 1:9 meV; green line ðp ¼ 0:63Þ c10 ¼ 3 meV; c20 ¼ dph dph dph 4:6 meV; cdph 21 ¼ 3:8 meV; blue line ðp ¼ 0:46Þ c10 ¼ 6 meV; c20 ¼ 9:2 meV; c21 ¼ 7:6 meV. The other parameters are Xp = 0.5 meV, xs = 17.6 meV, c1 = 5.6 meV, c2 = 7 meV, K = 10 meV. (b) The probe absorption as a function of the energy dph dph splitting xs for different strength of the Fano interference p: red line ðp ¼ 0:77Þ cdph 10 ¼ 1:5 meV; c20 ¼ 2:3 meV; c21 ¼ 1:9 meV; dph dph dph dph dph green line ðp ¼ 0:63Þ cdph 10 ¼ 3 meV; c20 ¼ 4:6 meV; c21 ¼ 3:8 meV; blue line ðp ¼ 0:46Þ c10 ¼ 6 meV; c20 ¼ 9:2 meV; c21 ¼ 7:6 meV. The other parameters are Xp = 0.5 meV, c1 = 5.6 meV, c2 = 7 meV, Dp = 0 meV, K = 10 meV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

348

Z. Wang / Annals of Physics 326 (2011) 340–349

the same parametric conditions here, and the three-level system of electronic subbands can be grown on the GaAs/AlGaAs substrate. Thus, the experimental feasibility of the scheme is evident. 4. Conclusions In the present study, we only focus on the condition of low temperatures up to 10 K, and have neglected other many-body effects such as the depolarization effect, which renormalizes the free-carrier and carrier-field contributions. These contributions and their interplay have been investigated quite thoroughly in the literature [41,42]. Note that, due to the small carrier density considered here, these effects only give a small correction. In conclusion, we have theoretically investigated the absorption of the probe field in a three-level asymmetric semiconductor quantum well system. We find that the incoherent pumping fields can affect the absorption of the probe field dramatically, which can be used to manipulate efficiently the probe absorption. The result is achieved in the absence of any external coherent driving field, so it is different from the different from the conventional way in ordinary laser-driven schemes that coherent driving fields are necessary to control the absorption of the probe field. By comparing the studies in the atomic system [36–40], our study is much more practical than its atomic counterpart due to its flexible design and the controllable interference strength. As a result, 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 help and encouragement, and this work is partially supported by National Natural Science Foundation of China under Grant No. 10874174. The author also acknowledges helpful advice and comments from the referees. References [1] E. Arimondo, G. Orriols, Nuovo Cimento Lett. 17 (1976) 333. [2] G. Alzetta, A. Gozzini, L. Moi, G. Oriols, Nuovo Cimento B 36 (1976) 5. [3] M.O. Scully, M. Fleischhauer, Science 263 (1994) 337; G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O. Scully, Phys. Rev. Lett. 82 (1999) 5229. [4] S.E. Harris, Phys. Today 50 (1997) 36. [5] M. Fleischhauer, A. Imamog˘lu, J.P. Marangos, Rev. Mod. Phys. 77 (2005) 633. [6] M. Xiao, Y.Q. Li, S.Z. Jin, J. Gea-Banacloche, Phys. Rev. Lett. 74 (1995) 666. [7] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. [8] A.S. Zibrov, M.D. Lukin, D.E. Nikonov, L. Hollberg, M.O. Scully, V.L. Velichansky, H.G. Robinson, Phys. Rev. Lett. 75 (1995) 1499. [9] Y.F. Zhu, A.I. Rubiera, M. Xiao, Phys. Rev. A 53 (1996) 1065. [10] D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, M.D. Lukin, Phys. Rev. Lett. 86 (2001) 783. [11] A. Lezama, A.M. Akulshin, A.I. Sidorov, P. Hannaford, Phys. Rev. A 73 (2006) 033806. [12] Y. Wu, J. Saldana, Y. Zhu, Phys. Rev. A 67 (2003) 013811; Y. Wu, X. Yang, Phys. Rev. A 71 (2005) 053806; Y. Wu, X. Yang, Phys. Rev. B 76 (2007) 054425. [13] Y. Zhang, B. Anderson, M. Xiao, Phys. Rev. A 77 (2008) 061801; Y. Zhang, U. Khadka, B. Anderson, M. Xiao, Phys. Rev. Lett. 102 (2009) 013601. [14] L. Deng, M.G. Payne, W.R. Garrett, Phys. Reports 429 (2006) 123. [15] Y.P. Niu, S.Q. Gong, Phys. Rev. A 73 (2006) 053811. [16] Y. Wu, Phys. Rev. A 71 (2005) 053820; Y. Wu, L. Deng, Phys. Rev. Lett. 93 (2004) 143904. [17] G. Huang, L. Deng, M.G. Payne, Phys. Rev. E 72 (2005) 016617. [18] W. Harshawardhan, G.S. Agarwal, Phys. Rev. A 53 (1996) 1812. [19] Z. Wang, M. Xu, Opt. Commun. 282 (2009) 1574. [20] A. Joshi, M. Xiao, Phys. Rev. Lett. 91 (2003) 143904; A. Joshi, A. Brown, H. Wang, M. Xiao, Phys. Rev. A 67 (2003) 041801(R). [21] A. Imamog˘lu, R.J. Ram, Opt. Lett. 19 (1994) 1744; H. Schmidt, K.L. Campman, A.C. Gossard, A. Imamog˘lu, Appl. Phys. Lett. 70 (1997) 3455. [22] C.R. Lee, Y.C. Li, F.K. Men, C.H. Pao, Y.C. Tsai, J.F. Wang, Appl. Phys. Lett. 86 (2005) 201112. [23] J. Faist, F. Capasso, C. Sirtori, K. West, L.N. Pfeiffer, Nature 390 (1997) 589; M.D. Frogley, J.F. Dynes, M. Beck, J. Faist, C.C. Phillips, Nature Mater. 5 (2006) 175.

Z. Wang / Annals of Physics 326 (2011) 340–349 [24] [25] [26] [27] [28]

[29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

349

D.E. Nikonov, A. Imamog˘lu, M.O. Scully, Phys. Rev. B 59 (1999) 12212. M. Phillips, H. Wang, Opt. Lett. 28 (2003) 831. L. Silvestri, F. Bassani, G. Czajkowski, B. Davoudi, Eur. Phys. J. B 27 (2002) 89. A. Joshi, Phys. Rev. B 79 (2009) 115315. A. Joshi, M. Xiao, Appl. Phys. B 79 (2004) 65; J.H. Li, X. Yang, Eur. Phys. J. B 53 (2006) 449; J.H. Li, Phys. Rev. B 75 (2007) 155329. Z. Wang, Opt. Commun. 282 (2009) 4745; Z. Wang, H. Fan, J. Lumin. 130 (2010) 2084. H. Sun, S. Gong, Y. Niu, S. Jin, R. Li, Z. Xu, Phys. Rev. B 74 (2006) 155314. W.X. Yang, J.M. Hou, R.K. Lee, Phys. Rev. A 77 (2008) 033838. X. Hao, J. Li, X. Yang, Opt. Commun. 282 (2009) 3339. K.I. Osman, S.S. Hassan, A. Joshi, Eur. Phys. J. D 54 (2009) 119. G.B. Serapiglia, E. Paspalakis, C. Sirtori, K.L. Vodopyanov, C.C. Phillips, Phys. Rev. Lett. 84 (2000) 1019; J.F. Dynes, M.D. Frogley, M. Beck, J. Faist, C.C. Phillips, Phys. Rev. Lett. 94 (2005) 157403. J.H. Wu, J.Y. Gao, J.H. Xu, L. Silvestri, M. Artoni, G.C. La Rocca, F. Bassani, Phys. Rev. Lett. 95 (2005) 057401. W.H. Xu, J.H. Wu, J.Y. Gao, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 1461; J.H. Wu, H.F. Zhang, J.Y. Gao, Opt. Lett. 28 (2003) 654. G.X. Li, J.S. Peng, G. Huang, J. Phys. B 33 (2000) 3743. Y. Zhu, Phys. Rev. A 53 (1996) 2742; Y. Zhu, Phys. Rev. A 55 (1997) 4568. J. Wang, Y.F. Zhu, K.J. Jiang, M.S. Zhan, Phys. Rev. A 68 (2003) 063810; H. Kang, L. Wen, Y. Zhu, Phys. Rev. A 68 (2003) 063806. X.M. Hu, Z.Z. Xu, J. Phys. B 35 (2002) 4527. T. Shih, K. Reimann, M. Woerner, T. Elsaesser, I. Waldmüller, A. Knorr, R. Hey, K.H. Ploog, Phys. Rev. B 72 (2005) 195338. D.E. Nikonov, A. Imamog˘lu, L.V. Butov, H. Schmidt, Phys. Rev. Lett. 79 (1997) 4633.