Eur. Phys. J. D 63, 489–494 (2011) DOI: 10.1140/epjd/e2011-10687-1
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Three-coupled-quantum-well nanostructures as a source of far-infrared entangled light X.Y. L¨ u1,a , L.L. Zheng1 , C.L. Ding2 , and X. Yang2 1 2
School of Physics, Ludong University, Yantai 264025, P.R. China Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P.R. China Received 3 December 2010 / Received in final form 11 April 2011 c EDP Sciences, Societ` Published online 28 June 2011 – a Italiana di Fisica, Springer-Verlag 2011 Abstract. A scheme is proposed to achieve the two-mode entanglement in an asymmetric semiconductor three-coupled-quantum-well (TCQW) system based on the intersubband transitions (ISBTs). In the present scheme, the TCQW structure is trapped into a doubly resonant cavity, and the required quantum coherence effects is induced by the corresponding ISBTs, which is the key of realising entanglement. By numerically simulating the dynamics of the system, we show that the entangled cavity modes with the far-infrared wavelengh can be realised in this TCQW system. The present research provides an efficient approach to achieve far-infrared entangled light in the semiconductor nanostructures, which may have significant impact on the progress of solid-state quantum information theory.
1 Introduction Quantum entanglement as the core of quantum information theory has attracted much interests due to its wide applications in quantum teleportation, quantum computation and quantum cryptography [1–14]. In particular, it has been shown that the continuous-variable (CV) entanglement is attracting crescent attention [15–22] due to its limitless conditions for the implementations of many quantum information processes [23]. For example, Xiong et al. [20] proposed a scheme for realising the two-mode CV entanglement in a correlated emission laser (CEL), in which the source of entangled light is a coherence atomic system. However, the required atomic coherence in their scheme is obtained by coupling the dipole forbidden transition with a strong field, which requires a higherorder process and increases the experimental difficulty. Subsequently, a large number of CEL-based schemes have been proposed for realising the CV entanglement by improving the coherence entangled source [24–26]. Summing up the previous studies, we can find that most of them employ the coherence atomic system as the entangled source. However, the atom is gaseous medium, and there are some defects in the flexibility of device fabrication for a gaseous medium. In recent years, it has been shown that the semiconductor quantum wells (QWs) and coupled quantum wells (CQWs) have many atomic-like properties, such as a
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the discrete energy levels and controlled quantum coherence. In addition, it also has many inherent advantages compared to an atomic system, such as large electric dipole moments, high non-linear optical coefficients, and a great flexibility in device design by choosing the materials and structure dimensions. Motivated by the above inherent advantages of the QWs and CQWs, several quantum optical coherence and interference effects have been investigated theoretically and experimentally in the QWs and CQWs system, such as coherently controlled photon-current generation [27], gain without inversion [28–30], coherent population trapping (CPT) [31], electromagnetically induced transparency (EIT) [32,33], and tunnelling-induced transparency (TIT) [34–36], etc. All the mentioned characteristics of SQD medium offer a feasible platform to realise quantum entanglement in a semiconductor solid-state system, which is significant for the progress of quantum information theory. In this paper, we theoretically study the generation of a two-mode CV entanglement in a CEL, where the source of generating entangled light is the asymmetric TCQW structure. Such a TCQW system has been intensively studied previously [37–41] but in different contexts. In the present scheme, the corresponding quantum coherence is obtained by inducing the ISBTs with classical driving fields. The major advantages of our scheme are as follows. (i) The TCQW medium studied here is a solid, which has nice flexibility of device fabrication and the wide adjustable parameters. For example, the transition energies and dipole moments can be manipulated well
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Fig. 1. (Color online) (a) Possible arrangement of experimental apparatus. A TCQW structure considered here is trapped into a doubly resonant cavity. (b) Schematic energy-band diagram of a single period of the asymmetric AlInAs/GaInAs TCQW structure. The thickness of the layers in the TCQW regions are respectively, from left to right, 42 ˚ A (GaInAs well), 16 ˚ A (AlInAs barrier), 20 ˚ A (GaInAs well), 16 ˚ A (AlInAs barrier), 18 ˚ A (GaInAs well). The positions of the calculated energy subbands and the corresponding modulus squared of the electronic wave functions are also displayed. (c) Schematic of the energy level arrangement under study. Two classical laser fields induce the ISBTs |2 ↔ |3 and |1 ↔ |4 with Rabi frequencies Ωd1 , and Ωd2 , respectively. The ISBTs |4 ↔ |3 and |2 ↔ |1 couple with two nondegenerate cavity modes ν1 and ν2 .
by accurately tailoring their shapes and sizes whereas they can hardly be found in the models for cold atom medium [28–30]. (ii) In the previous schemes the atoms are injected into the cavity at a constant rate and expected to stay there only for brief time, which enhances the difficulty of experimental implementation. However, the TCQW medium is prepared inside a doubly resonant in the present scheme. (iii) According to corresponding experimental parameters in reference [38], the frequencies of ISBTs in TCQW are usually in the far-infrared scope. So, the proposed system can be served as the source of generating far-infrared entangled lights (that is λ1 10.4 μm, λ2 10 μm in the present scheme), which may have a wide application in quantum communications.
2 Model and the master equations As shown in Figure 1a, an asymmetric TCQW structure is prepared inside a doubly resonant. The TCQW consists of a wide well (WW) and two narrow wells (NW), and all possible transitions in this system are dipole allowed because of the asymmetry that breaks the parity of the wave functions. The quantum well system interacts with two classical laser fields with Rabi frequencies Ωd1 , Ωd2 , and two (non-degenerate) cavity modes ν1 , ν2 with coupling constant g1 and g2 , respectively. Δ = ω43 − ν1 = ω41 − ν2 denotes the corresponding frequency detuning. The present asymmetric semiconductor TCQW structure consists of 40 coupled-well periods with the material Al0.48 In0:52 As/Ga0:47 In0:53 As, which is grown by the molecular-beam epitaxy (MBE) lattice matched to a semiinsulating (100) InP substrate and separated from each
˚ undoped AlInAs barrier. Then, the other by a 150 A sample considered here can be designed to have the energy levels as E1 = 151 meV, E2 = 270 meV, E3 = 386 meV, and E4 = 506 meV, respectively. The energy levels and wave functions (see Fig. 1b) are calculated by self-consistently Schr¨odinger solving and Poisson’s equation in the envelope-function formalism [42]1 using the material parameters of reference [37]. In the present scheme, we use the following conditions. The electron density is on the order of 1011 cm−2 or smaller, and the actual lattice temperature is about 10 K [28–30]. Then, the electronelectron effects have a very small influence on our results. Consequently we assume the many body effects arising from electron-electron interactions are not included in our study, which is consistent with previous experimental [44,45] and theoretical [46] work. Then, under the dipole and rotating wave approximation, the interaction Hamiltonian of our system in the interaction picture can be written as ( = 1) [47–50] HI = Δ|44| − (Ωd2 |41| + Ωd1 |32| + h.c.) + (g1 a1 |43| + g2 a2 |21| + h.c.),
(1)
where the symbol h.c. means the Hermitian conjugate. Ωd1 = |Ωd1 |e−iφd1 = μ32 Ed1 /2, Ωd2 = |Ωd2 |e−iφd2 = μ41 Ed2 /2 (φd1 , φd2 are the phases) and g1 , g2 denote the one-half Rabi frequencies of two classical fields and atom-field coupling constants, respectively. According to the standard method of the laser theory [51], we treat the transitions |4 ↔ |3 and |2 ↔ |1 quantum-mechanically 1
Nonparabolicities were taken into account using the method of [43].
X.Y. L¨ u et al.: Three-coupled-quantum-well nanostructures as a source of far-infrared entangled light
up to the second order in the coupling constants g1 and g2 , and all orders for the strong driving fields Ωd1 and Ωd2 . The equation of motion for the density operator ρf of the cavity modes can be written as ρ˙ f = − α11 (a†1 a1 ρf − a1 ρf a†1 ) − α12 (a†1 a†2 ρf − a†2 ρf a†1 ) − β11 (a†1 ρf a1 − α22 (a†2 a2 ρf −
β22 (a†2 ρf a2 −
−
− −
ρf a1 a†1 ) a2 ρf a†2 )
ρf a2 a†2 )
κ1 (a†1 a1 ρf −
− −
β12 (a†1 ρf a†2 α21 (a†2 a†1 ρf
−
β21 (a†2 ρf a†1
−
ρf a†1 a1
κ2 (a†2 a2 ρf
−
−
− ρf a†2 a†1 ) − a†1 ρf a†2 )
ρf a†1 a†2 )
(2)
where the expressions αij and βij (i, j = 1, 2) are given in the Appendix. κ1 and κ2 denote the damping constants of two cavity modes, respectively. It should be pointed out that, in the above calculations, the population decay rates and dephasing decay rates of TCQW have been added phenomenologically. Specifically, γij (i = j) denote the population decay rates from subband |i to subband |j, which are due primarily to longitudinal optical (LO) phonon emission events at low temperature. The total dedph cay rates Γij are given by Γ43 = γ43 + γ32 + γ43 , Γ42 = dph dph dph γ43 + γ21 + γ42 , Γ41 = γ43 + γ41 , Γ32 = γ32 + γ21 + γ32 , dph dph dph Γ31 = γ32 + γ31 , Γ21 = γ21 + γ21 , where γij , determined by electron-electron, interface roughness, and phonon scattering processes, is the dephasing decay rate of the quantum coherence of the |i ↔ |j transition. Here it should be pointed out that the TCQW is placed in a doubly resonant throughout the operation, which exposes the TCQW to noise fluctuations and broadening due to dph heating. This will enhance the decay rates γij and γij in the present scheme.
3 Generation of entanglement In this section, we will study the generation of two-mode CV entanglement based on the sufficient inseparability criterion proposed by Duan et al. [18]. A quantum system is said to be entangled if and only if it is inseparable. That is, the density operator for the state ρ can not be written as, (1) (2) ρ= p j ρ j ⊗ ρj (3) j
(1)
(2)
with pj ≥ 0 and j pj = 1. ρj and ρj as the normalised states of two field modes, respectively. According to the criterion derived in paper [18], the state of the system is entangled if the total variance of two Einstein-PodolskyRosen (EPR) type operators u and υ of the two modes satisfy the inequality (Δˆ u)2 + (Δˆ (4) υ )2 < 2, where u ˆ=x ˆ1 + x ˆ2 , υˆ = pˆ1 − pˆ2 ,
√ √ with xj = (aj + a†j )/ 2 and pj = (aj − a†j )/ 2i (j = 1, 2) are the quadrature operators for the modes 1 and 2. By substituting equation (5) into equation (4), we can express the total variance of the operators uˆ and νˆ in terms of the operators aj and a†j and get (Δˆ u)2 + (Δˆ υ )2 = 2[1 + a†1 a1 + a†2 a2 + a1 a2 +a†1 a†2 − a1 a†1 − a2 a†2
+ h.c.
− 2a1 ρf a†1 ) ρf a†2 a2 − 2a2 ρf a†2 ),
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(5)
−a1 a2 − a†1 a†2 ].
(6)
d With the help of equation (2) and the fact that dt b(t) = Tr(ρ˙ f b) [21] (b denote the field operators in Eq. (6)). So, the equations of motion for the expected values of the field operators in equation (6) can be written as
d a1 = −(A11 + κ1 )a1 − A12 a†2 , dt d † a = −A∗21 a1 − (A∗22 + κ2 )a†2 , dt 2
(7a) (7b)
d † a a1 = −B11 a†1 a1 − A∗12 a1 a2 − A12 a†1 a†2 dt 1 ∗ ), (7c) −(β11 + β11 d † a a2 = −B22 a†2 a2 − A∗21 a1 a2 − A21 a†1 a†2 dt 2 ∗ ), (7d) −(β22 + β22 d a1 a2 = −A21 a†1 a1 − A12 a†2 a2 − B12 a1 a2 dt (7e) −(α12 + α∗21 ), d † † ∗ a a = −A∗21 a†1 a1 − A∗12 a†2 a2 − B12 a†1 a†2 dt 1 2 (7f) −(α∗12 + α21 ), ∗ + 2κi where Aij = αij + βij , Bii = αii + α∗ii + βii + βii (i, j = 1, 2) and B12 = α11 + β11 + κ1 + κ2 . Associating with equations (6) and (7), we give a few numerical results for the time evolution of the total variance of the operators u ˆ, υˆ and the total mean photon ˆ (N ˆ = a† a1 + a† a2 ) for different values numbers N 1 2 of parameters, as illustrated in Figures 2−5. Note that, in the corresponding numerical calculations, the choice of the parameters are based on the investigation of references [37,38]. In the following paragraphs, we will mainly discuss the generation and evolution of two-mode CV entanglement from two aspects. On the one hand, we will demonstrate that the two-mode CV entanglement with the frequency of each entangled mode in the far-infrared range can be realised in the present TCQW based on ISBTs. At the same time, we will also discuss the gain property of our system and show that the bipartite entanglement amplifier can be realised in our scheme. On the other hand, we will discuss the influence of system parameters on the generated entangled state and provide corresponding clues to optimise the generated two-mode CV entangled state. In Figure 2, we plot the time evolution of (Δˆ u)2 + 2 (Δˆ υ ) for different values of |Ωd1 | (Fig. 2a) and |Ωd2 | (Fig. 2b), when the cavity field is initially in a vacuum
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Fig. 3. (Color online) The entanglement period T versus |Ωd1 | (panel (a)) and |Ωd2 | (panel (b)). The parameters are same as in Figure 2.
Fig. 2. (Color online) The time evolution of (Δˆ u)2 + (Δˆ υ )2 for different values |Ωd1 | (panel (a): |Ωd2 | = 2γ) and |Ωd2 | (panel (b): |Ωd1 | = 15γ) when the cavity field is initially in a vacuum state. The other parameters are γ43 = γ32 = γ21 = 0.2 meV, Γ43 = Γ42 = Γ41 = 1 meV, Γ31 = Γ32 = 1.2 meV, dph dph dph dph Γ21 = 3.2 meV, γ43 = γ42 = 0.6 meV, γ41 = γ32 = dph dph 0.8 meV, γ31 = 1 meV, γ21 = 3 meV, g1 = g2 = 0.02 meV, κ1 = κ2 = 10−6 meV, Δ = 5 meV, and φd2 − φd1 = π/2.
state. Firstly, it is clearly seen from the Figure 2 that the two-mode CV entanglement can be realised during a proper time interval (entanglement period), which corresponds to the range when (Δˆ u)2 + (Δˆ υ )2 is smaller than two. In addition, our number results also show the entanglement period can be prolonged effectively when the intensities of two classical fields |Ωd1 | and |Ωd2 | are enhanced. The above property is consistent with the previous study [20] and it can expain the increases of quantum interference effects between the ISBTs |4 ↔ |3 and |2 ↔ |1. In order to show explicitly show the effects of the intensities of driving fields on the entanglement period T more distinctly, we also plot T as a function of |Ωd1 | and |Ωd2 | in Figure 3. From this figure, we can find that the entanglement period approximately manifests linear increase property as the intensity of two classical driving fields, which is consistent with our above discussion. Here, it also should be pointed out that, according to corresponding experimental parameters in reference [38], the wavelengh of two generated entangled modes are in the far-infrared range (that is λ1 = 10 μm, λ2 = 10.4 μm) in Figure 2. As a result, the present system can be served as the source of far-infrared entangled light.
Fig. 4. (Color online) The time evolution of mean photon numbers for two cavity mode Nˆ1 and Nˆ2 for different values |Ωd1 | (panel (a)) and |Ωd2 | (panel (b)). When the cavity field is initially in a vacuum state. The parameters are same as in Figure 2.
Up to now, we have shown that the two-mode CV entanglement can be realised in the TCQW system during the entanglement period. Then, in Figure 4, we plot the time evolution of mean photon numbers N1 and N2 for different values of |Ωd1 | and |Ωd2 |. Firstly, the corresponding curves show that the mean photon number of each entangled mode increases rapidly with the evolvement
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easily seen from Figures 5a and 5b that the entanglement period between two cavity modes becomes smaller with the increasing of frequency detuning Δ, which implies that the frequency detuning Δ should be chosen properly for obtaining the entanglement with a long period. Summing up the above discussion, we can make the conclusion that the intensities of two driving fields |Ωd1 |, |Ωd2 | and frequency detuning Δ should be chosen properly (large |Ωd1 |, and small |Ωd2 |, Δ) in order to obtain the optimum CV entangled state (entanglement with long period and high intensity for each entangled mode at the same time) in the present scheme.
4 Conclusion
Fig. 5. (Color online) The time evolution of (Δˆ u)2 + (Δˆ υ )2 for different values Δ (panel (a)) and the entanglement period T versus Δ (panel (b)), when the cavity field is initially in a vacuum state. The other parameters are same as in Figure 2 except for |Ωd1 | = 15 meV and |Ωd2 | = 2 meV.
of time, which implies that the intensities of entangled lights can be amplified effectively during the entanglement period. This phenomenon can be owed to the processes of a†
a†
1 2 ISBTs |4 −→ |3, and |2 −→ |1, which are stronger than the processes of corresponding cavity modes decay due to the presence of gain property in the proposed TCQW system. As a result, the CEL considered here can be served as a bipartite entanglement amplifier. In addition, Figure 4 also demonstrate the influence of two classical field intensities on N1 and N2 . It is shown that the mean photon numbers N1 and N2 increase (decrease) along with the enhancing of |Ωd1 | (|Ωd2 |). This property can be qualitatively explained as follows. On the one hand, the ISBTs
a†
Ω
a†
1 2 d1 |3 −→ |2 −→ |1 will become more inprocesses |4 −→ tensive with the increasing of Rabi frequency Ωd1 , which leads to the mean photon number of each entangled mode becoming larger. On the contrary, the above processes will be weakened with the increasing of Rabi frequency Ωd2 Ωd2 due to the increasing of ISBTs process |4 −→ |1. Lastly, in order to obtain optimum CV entangled state, we also study the effects of frequency detuning Δ on the time evolution of entanglement in Figure 5. Here it should be pointed out that the Rabi frequency |Ωd1 | and |Ωd2 | have been set as determinate values in Figure 5. It can be
In conclusion, we have studied the generation and evolution of two-mode CV entanglement in the TCQW system based on the corresponding ISBTs. By numerically simulating the dynamics of system, we demonstrate that the proposed TCQW system can be served as the source of farinfrared entangled light. In addition, we also have studied the influences of classical fields on the generation of entanglement and have shown that the entanglement period can be prolonged via enhancing the intensities of classical fields in our scheme. As a result, the present investigation provides a clue for achieving two-mode CV entanglement in a semiconductor solid-state medium, which may be used to design solid-state quantum information devices. The research was supported in part by the Natural Science Foundation of China (Grants No. 11005057, 10904033, and No. 10975054), by the National Basic Research Program of China (Contract No. 2005CB724508), and by the Foundation from the Ministry of the National Education of China (Grant No. 200804870051), and by the School Scientific Research Foundation of Ludong University (Grant No. LY2011001). We would like to thank Professor Ying Wu for helpful discussion and encouragement.
Appendix A: Coefficients Here we provide the explicit express of the coefficients αij and βij (i, j = 1, 2) in equation (A.1) of motion for the density operator ρf of the cavity modes (Eq. (2)): α11 = 40g12|Ωd2 |2 [6γ|Ωd2 |2 +(5γ − 2iΔ)(24γ 2 + |Ωd1 |2 )]/M, (A.1a) 2 2 2 β11 = 8g1 |Ωd2 | [−(35γ + 2iΔ)(20γ − 8iγΔ + |Ωd2 |2 ) −(20γ − 12iΔ)|Ωd1 |2 ]/M, (A.1b) ∗ 2 α12 = 40g1g2 Ωd2 Ωd1 |Ωd2 | (11γ − 2iΔ)/M, (A.1c) ∗ 2 β12 = −2g1 g2 Ωd2 Ωd1 [(200γ − 32iΔ)|Ωd2 | +(5γ − 2iΔ)(151γ 2 + 12iγΔ +4Δ2 + 4|Ωd1 |2 )]/M, (A.1d) α22 = g22 [3γ(25γ 2 + 4Δ2 )2 + (10γ + 4iΔ)|Ωd2 |2 ×(145γ 2 − 48iγΔ + 4Δ2 + 16|Ωd2 |2 ) +4(75γ 2 + 12γΔ2 + 70γ|Ωd2|2 +4iΔ|Ωd2|2 ))|Ωd1 |2 ]/M ∗ , (A.1e)
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β22 = −20g22 |Ωd2 |2 [(10γ + 4iΔ)|Ωd2 |2 + 3γ(25γ 2 + 4Δ2 + 4|Ωd1 |2 )]/M ∗ , α21 =
∗ 2g1 g2 Ωd2 Ωd1 [16(15γ 2
(A.1f)
2
+ 2iΔ)|Ωd2 |
− (5γ − 2iΔ)(25γ + 4Δ2 + 4|Ωd1 |2 )]/M ∗ , (A.1g) ∗ |Ωd2 |2 (11γ + 2iΔ)/M ∗ , β21 = −40g1 g2 Ωd2 Ωd1
(A.1h)
where M = (25γ 2 + 4Δ2 + 80|Ωd2 |2 ) × [4|Ωd2 |4 + 2|Ωd2 |2 (55γ 2 − 10iγΔ − 4|Ωd1 |2 ) + (24γ 2 + |Ωd1 |2 )(25γ 2 + 4Δ2 + 4|Ωd1 |2 )].
(A.2)
It should be pointed in this section, for simplicity, we have set the parameters γ43 = γ32 = γ21 = γ, Γ43 = Γ42 = Γ41 = 5γ, Γ32 = Γ31 = 6γ, and Γ21 = 16γ in this section for simplicity.
References 1. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000) 2. C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993) 3. D.P. Divincerzo, Science 270, 225 (1995) 4. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Nature 390, 575 (1997) 5. J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, A. Zeilinger, Nature 403, 515 (2000) 6. G.-X. Li, H. Tan, M. Macovei, Phys. Rev. A 76, 053827 (2004) 7. G.-X. Li, S. Ke, and Z. Ficek, Phys. Rev. A 79, 033827 (2009) 8. X.M. Hu, J.H. Zou, Phys. Rev. A 78, 045801 (2008) 9. X. Li, X.M. Hu, Phys. Rev. A 80, 023815 (2009) 10. M. Feng, Phys. Rev. A 66, 054303 (2002) 11. Z.J. Deng, M. Feng, K.L. Gao, Phys. Rev. A 75, 024302 (2007) 12. Z.-X. Man, Y.-J. Xia, N.B. An, Phys. Rev. A 75, 052306 (2007) 13. S.B. Zheng, G.C. Guo, Phys. Rev. A 73, 032329 (2006) 14. Y.-F. Xiao, Z.-F. Han, G.-C. Guo, Phys. Rev. A 55, 1239 (1997) 15. A. Fang, Y. Chen, F.-L. Li, H. Li, P. Zhang, Phys. Rev. A 81, 012323 (2001) 16. Z.Y. Ou, S.F. Pereira, H.J. Kimble, K.C. Peng, Phys. Rev. Lett. 68, 3663 (1992) 17. R. Simon, Phys. Rev. Lett. 84, 2726 (2000) 18. L.M. Duan, G. Giedke, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 84, 2722 (2000) 19. G.X. Li, Y.P. Yang, K. Allaart, D. Lenstra, Phys. Rev. A 69, 014301 (2004)
20. H. Xiong, M.O. Scully, M.S. Zubairy, Phys. Rev. Lett. 94, 023601 (2005) 21. H.T. Tan, S.Y. Zhu, M.S. Zubairy, Phys. Rev. A 72, 022305 (2005) 22. X.-Y. L¨ u, P. Huang, W.-X. Yang, X. Yang, Phys. Rev. A 80, 032305 (2009). 23. S.L. Braunstein, P. van Loock, Rev. Mod. Phys. 77, 513 (2005) 24. L. Zhou, H. Xiong, M.S. Zubairy, Phys. Rev. A 74, 022321 (2006) 25. M. Kiffner, M.S. Zubairy, J. Evers, C.H. Keitel, Phys. Rev. A 75, 033816 (2007) 26. S. Qamar, M. Al-Amri, S. Qamar, M.S. Zubairy, Phys. Rev. A 80, 033818 (2009) 27. R. Atanasov et al., Phys. Rev. Lett. 76, 1703 (1996) 28. A. Imamoˇ glu, R.J. Ram, Opt. Lett. 19, 1744 (1994) 29. M.D. Frogley, J.F. Dynes, M. Beck, J. Faist, C.C. Phillips, Nature Mater. 5, 175 (2006) 30. C.R. Lee, Y.C. Li, F.K. Men, C.H. Pao, Y.C. Tsai, J.F. Wang, Appl. Phys. Lett. 86, 201112 (2005) 31. J.F. Dynes, M.D. Frogley, J. Rodger, C.C. Phillips, Phys. Rev. B 72, 085323 (2005) 32. D.E. Nikonov, A. Imamoˇ glu, M.O. Scully, Phys. Rev. B 59, 12212 (1999) 33. M. Phillips, H. Wang, Opt. Lett. 28, 831 (2003) 34. H. Schmidt, K.L. Campman, A.C. Gossard, A. Imamoˇ glu, Appl. Phys. Lett. 70, 3455 (1997) 35. J. Faist, F. Capasso, C. Sirtori, K.W. West, L.N. Pfeiffer, Nature 390, 589 (1997) 36. H. Sun, Y. Niu, R. Li, S. Jin, S. Gong, Opt. Lett. 32, 2475 (2007) 37. C. Sirtori, F. Capasso, D.L. Sivco, A.Y. Cho, Phys. Rev. Lett. 68, 1010 (1992) 38. C. Sirtori, F. Capasso, J. Faist, S. Scandolo, Phys. Rev. B 50, 8663 (1994) 39. W.-X. Yang, J.-M. Hou, Y.Y. Lin, R.-K. Lee, Phys. Rev. A 79, 033825 (2009) 40. X. Hao, J. Li, X. Yang, Opt. Commun. 282, 3339 (2009) 41. J. Li, R. Yu, X. Hao, A. Zheng, X. Yang, Opt. Commun. 282, 4284 (2009) 42. G. Bastard, Phys. Rev. B 24, 5693 (1981) 43. D.F. Nelson, R.C. Miller, D.A. Kleinmann, Phys. Rev. B 35, 7770 (1987) 44. G.B. Serapiglia, E. Paspalakis, C. Sirtori, K.L. Vodopyanov, C.C. Phillips, Phys. Rev. Lett. 84, 1019 (2000) 45. E. Paspalakis, M. Tsaousidou, A.F. Terzis, Phys. Rev. B 73, 125344 (2006) 46. H. Schmidt, A. Imamoˇ glu, Opt. Commun. 131, 333 (1996) 47. Y. Wu, L. Wen, Y. Zhu, Opt. Lett. 28, 631 (2003) 48. Y. Wu, X. Yang, Opt. Lett. 28, 1793 (2003) 49. Y. Wu, P.T. Leung, Phys. Rev. A 60, 630 (1999) 50. Y. Wu, X. Yang, Appl. Phys. Lett. 91, 094104 (2007) 51. M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997), Chap. 14, p. 409
