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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 11944–11950
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Influence of quantum coherence on propagation of a pulsed light in a triple quantum well

Aixi Chen  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 11944-11950 (2011)
http://dx.doi.org/10.1364/OE.19.011944


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Abstract

In a triple semiconductor quantum well structure coupled by two external fields, we investigate the influence of atomic coherence induced by external fields and decay interference on the absorption and dispersion of a weak pulsed light, and slow light can be achieved in this system. Quantum well structure behaves as “artificial atom” and its advantage of easy integration makes it has some practical applications.

© 2011 OSA

In a semiconductor quantum well structure, quantum coherence and interference produced due to intersubband transitions between the states of an electron confined in this structure have induced a lot of interesting and unexpected physical consequences. Some of these consequences are electromagnetically induced transparency (EIT) [1

1. M. Phillips and H. Wang, “Electromagnetically induced transparency due to intervalence band coherence in a GaAs quantum well,” Opt. Lett. 28(10), 831–833 (2003). [CrossRef] [PubMed]

], pulsed-induced quantum interference [2

2. T. Müller, W. Parz, G. Strasser, and K. Unterrainer, “Influence of carrier-carrier interaction on time-dependent intersubband absorption in a semiconductor quantum well,” Phys. Rev. B 70(15), 155324 (2004). [CrossRef]

], coherent population trapping [3

3. J. F. Dynes, M. D. Frogley, J. Rodger, and C. C. Phillips, “Optically mediated coherent population trapping in asymmetric semiconductor quantum wells,” Phys. Rev. B 72(8), 085323 (2005). [CrossRef]

], carrier-envelope-phase dependent coherence [4

4. W. X. Yang, X. X. Yang, and R. K. Lee, “Carrier-envelope-phase dependent coherence in double quantum wells,” Opt. Express 17(18), 15402–15408 (2009). [CrossRef] [PubMed]

], gain without inversion [5

5. C. R. Lee, Y. Li, F. K. Men, C. Pao, Y. Tsai, and J. Wang, “Model for an inversionless two-color laser,” Appl. Phys. Lett. 86(20), 201112 (2005). [CrossRef]

]. In addition, the technology of quantum coherence and interference is expected to have potentially important application in various fields such as ultrafast optical switches [6

6. H. Schmidt and R. J. Ram, “All-optical wavelength converter and switch based on electromagnetically induced transparency,” Appl. Phys. Lett. 76(22), 3173–3175 (2000). [CrossRef]

] and quantum switches [7

7. B. S. Ham, “Potential applications of dark resonance to subpicosecond optical switches in hyper-terahertz repetition rates,” Appl. Phys. Lett. 78(22), 3382–3384 (2001). [CrossRef]

], quantum information storage and retrieval [8

8. J. M. Tang, J. Levy, and M. E. Flatté, “All-electrical control of single ion spins in a semiconductor,” Phys. Rev. Lett. 97(10), 106803 (2006). [CrossRef] [PubMed]

], and preparation of entangled state [9

9. X. Lü and J. Wu, “Three-mode entanglement via tunneling-induced interference in a coupled triple-semiconductor quantum-well structure,” Phys. Rev. A 82(1), 012323 (2010). [CrossRef]

].

As a class of quantum coherence and interference, spontaneously generated coherence (SGC) has been studied in atomic system [10

10. A. Fountoulakis, A. Terzis, and E. Paspalakis, “Coherence phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73(3), 033811 (2006). [CrossRef]

] and semiconductor dot system [11

11. S. E. Economou, R. B. Liu, L. J. Sham, and D. G. Steel, “Unified theory of consequences of spontaneous emission in a Λ system,” Phys. Rev. B 71(19), 195327 (2005). [CrossRef]

]. We know semiconductor quantum well structures possess of intersubband transitions and behave as “artificial atoms” [12

12. J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005). [CrossRef] [PubMed]

]. Due to longitudinal optical phonon emission events at low temperature [3

3. J. F. Dynes, M. D. Frogley, J. Rodger, and C. C. Phillips, “Optically mediated coherent population trapping in asymmetric semiconductor quantum wells,” Phys. Rev. B 72(8), 085323 (2005). [CrossRef]

], the population decay occurs. Some research groups have investigated spontaneously generated coherence in quantum well structure [13

13. H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70(25), 3455–3457 (1997). [CrossRef]

]. SGC is the interference of spontaneous emission channels, and the existence of such decay interference requires the nonorthogonality of the two dipole moments. The recent studies show that decay interference can affect the EIT, shot-pulse propagation, dark state, etc [14

14. E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82(10), 2079–2082 (1999). [CrossRef]

,15

15. P. R. Berman, “Spontaneously generated coherence and dark states,” Phys. Rev. A 72(3), 035801 (2005). [CrossRef]

]. To the best of our knowledge, no further theoretical and experimental work has been carried out to investigate the influence of SGC on transmission of a pulsed light in a triple semiconductor quantum well driven by external light fields, which motivates us to investigate effect of decay interference.

In this paper, we consider the effect of quantum interference in a triple semiconductor quantum well with one ground subband and three excited subbands when driven by a pump field and a control field. The influence of atomic coherence produced by coherent driving of external fields on the absorption and dispersion of the pulsed field is investigated. In addition, due to longitudinal optical phonon emission, coupling to the lattice phonon spectrum, and interface roughness scattering, these excited subbands take place decay. According to the theory of decay interference, we theoretically study the influence of decay interference on the absorption and dispersion of a pulsed light in a quantum well. The effect of slow light is investigated in this system. Being an attractive quantum coherent medium, semiconductor quantum well structure has its own advantages: strong electron-electron interactions can produce a collective oscillation, which can behave as a single quantum object, and its large transition dipoles give rise to sizeable Rabi frequencies which are large enough to overcome dephasing. In addition, compared with atomic system, semiconductor quantum well structure has an advantage of easy integration, so the manipulation of absorption of light in this system has a more practical value. Our work is based on existing physical model [5

5. C. R. Lee, Y. Li, F. K. Men, C. Pao, Y. Tsai, and J. Wang, “Model for an inversionless two-color laser,” Appl. Phys. Lett. 86(20), 201112 (2005). [CrossRef]

,9

9. X. Lü and J. Wu, “Three-mode entanglement via tunneling-induced interference in a coupled triple-semiconductor quantum-well structure,” Phys. Rev. A 82(1), 012323 (2010). [CrossRef]

] and maybe our theoretical studies have some reference value for the future experiments.

We consider a semiconductor quantum well structure that consists of a deep well and two shallow wells. This triple quantum well is shown in Fig. 1(a)
Fig. 1 Band diagram of a triple semiconductor quantum well.
, where the first excited state in the deep well and the respective ground states in the two shallow wells mix to form three new excited states of this system, and the ground state in the deep well and three excited states created newly are expressed as |0, |1, |2 and |3, respectively, which refer to the electronic wave functions (subbands). For the excited states, there are two different types of dephasing rates; one is population decay rate which is represented by γjp (j=1,2,3), the other is pure dipole dephasing rate which is stood for γjd. For the population decay rate, it is due primarily to longitudinal optical (LO) phonon emission events at low temperature, which is different from atomic system where the excited state spontaneously decays to a lower lying level, that is, population decay rates for atomic system are pure radiative decay. For the pure dipole dephasing rate, it results from a combination of quasielastic interface roughness scattering or acoustic phonon scattering, and this kind of dephasing rate is akin to atomic collisional broadening in gases. The total dephasing rates are expressed as γj and are given by γj=γjp+γjd.

For the dephasing rate in quantum well structure, the dominant phonon contributions come from piezoelectric and Frohlich coupling to acoustic and optical phonon branches, respectively. In the two branches of coupling, the intersubband transition couples via the phonon’s oscillating electric field [16

16. B. K. Ridley, Quantum Processes in Semiconductors (Clarendon Press, 1988).

], so the rates scale with Dij2 (Dij is transition dipole), just as natural atoms. The dipole dephasing rates of this structure are strongly dependent upon the electronelectron scattering process. For simplicity, we assume our calculation is performed at zero temperature, so the pure dipole dephasing rate due to scattering is negligible compared with the population decay rates due to longitudinal optical phonon emission events at low temperature.

In this paper, we consider the triple quantum well interacts with a weak pulsed laser field (with central frequency ωp and amplitude Ep) that couples transition between states |0 and |1, simultaneously, |1|2 and |1|3 transitions are coupled by a strong control field (with frequency ωc and amplitude Ec) and a pump field (with frequency ωb and amplitude Eb), respectively. The schematic of the system interacting with three light fields is showed in Fig. 1(b).

A deep 7.1-nm-thick GaAs well is coupled, on one side, to two shallow 6.8-nm-thick Al0.2Ga0.8As wells by a 2.5-nm-thick Al0.4Ga0.6As barrier. The two shallow wells are separated by a 2.0-nm-thick Al0.4Ga0.8As barrier. Both sides of quantum well contact with 36 nm Al0.4Ga0.6As. The electronic wave functions of the ground state of the deep well and the three excited states are shown with respective energies of 52.8 (|0), 197.1 (|1), 206.2 (|2), and 219.4 (|3) meV. (b) the schematic of the system interacting with three light fields.

In the interaction picture, the wave function of the triple semiconductor quantum well can be written in the form (at time t)|Ψ(t)=C0(t)|0+C1(t)|1+C2(t)|2+C3(t)|3, and the corresponding time-dependent Schrödinger equation is (we let =1)
(C˙0(t)C˙1(t)C˙2(t)C˙3(t))T=iVI(C0(t)C1(t)C2(t)C3(t))T,
(1)
where superscript T denotes transpose, and the Hamiltonian VI describes the four-level system interacting with three light fields. By using the completeness relation k=03|kk|=I and considering the effect of decay interference, the matrix form of the Hamiltonian can be written as
VI=(0Ωp*00ΩpΔpiγ1/2Ωc*Ωb*0ΩcΔcΔpiγ2/2ip23γ2γ3/20Ωbip23γ2γ3/2ΔbΔpiγ3/2),
(2)
where Ωp=D10Ep/2, Ωc=D21Ec/2, and Ωb=D31Eb/2 are the half Rabi frequency for the intersubband transitions |0|1, |1|2 and |1|3, respectively. Δp, Δb, and Δc are the detunings of the pulsed light, the pump light and the control field from their respective optical transitions. The parameter p23stands for the alignment of two decay emission dipole matrix elements and is defined as p23=D31D21/(|D31||D21|)=cosθ with θ being the angle between the two dipole elements. In experiment, the decay interference requires the levels |2 and |3 are close in our system, that is, they are two close-lying levels, and the corresponding dipole matrix elements are not orthogonal. Theses rigorous conditions are rarely met in real atoms. In the quantum well, energy spacing between the two higher excited states are easily adjusted by changing the structure of well. Furthermore, in order to obtain significant decay interference, the system satisfies the restriction that each field acts only on one transition. In our paper, we assume choice of parameter p is theoretically unrestricted (p[0,1]), and our aim is investigating the influence of decay interference on the propagation of a pulsed light. In the slowly varying envelope approximation, Maxwell’s equation that denotes the dynamic response of the pulsed light is described by Ep/t+cE/z=iNωpD01C1C0*. According to Ωp=D10Ep/(2), the Maxwell’s equation is rewritten as
Ωpz+1cΩpt=iκC1C0*,
(3)
where κ=Nωp|D10|2/(2c), N is the electron density in the coupled quantum well sample.

We now assume that the pulsed light is weak enough so that we can perform a perturbation expansion of the amplitude, so the first order of the pulsed light Ωp and the amplitudes C1,2,3 are retained [17

17. Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71(5), 053820 (2005). [CrossRef]

]. In addition, we assume that the system of quantum well is initially in the state |0 (C01). Under the condition of the weak field approximation, we carry out Fourier transformations C1,2,3(t)=12πα1,2,3(ω)eiωtdω, with the Fourier transformation variable ω for Eq. (1) and Ωp(t)=12πΛp(ω)eiωtdω for Eq. (3), we obtain
α1=(p232γ2γ3/4(Δp+Δb+iω+iγ3/2)(Δp+Δc+iω+iγ2/2))Λp((Δpiωiγ1/2)(p232γ2γ3/4(Δp+Δb+iω+iγ3/2)(Δp+Δc+iω+iγ2/2))|Ωc|2(Δp+Δb+iω+iγ3/2)+(Ωb*ΩcΩc*Ωb)p23γ2γ3/2i|Ωb|2(Δp+Δc+iω+iγ2/2)),
(4)
ΛpziωcΛp=iκα1.
(5)
Substituting Eq. (4) into Eq. (5), we then get Λp/z=iK(ω)Λp, where the frequency-dependent propagation factor K(ω) is given as K(ω)=ω/cκα1/Λp. In order to obtain the properties of the propagation for the pulsed light, the propagation factor is expected into a Taylor series as a function of the angular frequency ω, so we obtain
K(ω)=K(0)+K(0)ω+K(0)ω2/2+O(ω3),
where the real part of K(0) denotes the phase shift per unit length, the imaginary part of K(0) stands for energy absorption, Re(1/K(0)) is the group velocity vg, K(0) is called as the group velocity dispersion which contributes to the input pulsed shape change and additional loss of the pulsed intensity, and O(ω3) are the terms of high order.

Now, based on the various physical parameters of the system, we investigate the probe absorption and dispersion and propagation of the pulsed light. We first consider influences of atomic coherence induced by external fields. We choose physical parameters Δb=Δc=0, that is, the system is in the two-photon resonance. The curves of the probe absorption and dispersion are shown in Fig. 2(a)
Fig. 2 Curves of absorption (dashed line) and dispersion (solid line) of pulsed light. Physical parameters: κ=0.5γ1, γ2=γ1, γ3=1.5γ1, Ωc=6γ1, Δc=Δb=0. For (a), decay interference p=0, Ωb=6γ1 is for black line, and Ωb=2γ1 is for red line; For (b), p=0.96 and Ωb=2γ1.
. From this figure, we find the system shows one transparent window for the probe field, that is, EIT occurs, which is similar to phenomena produced in cascade-type three-level atomic system [18

18. Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71(5), 053806 (2005). [CrossRef]

]. In addition, if the intensity of the control field is kept unchangeable and intensity of the pump field is changed, the width of transparent window changes: decreasing the intensity of the pump field, the window becomes narrow (see black dashed line and red dashed line in Fig. 2(a)). Under the condition of two-photon resonance, if we consider the decay interference simultaneously, where we choose strong effect of decay interference and let p=0.96, the plot of absorption is shown in Fig. 2(b). This figure shows transparent window disappears. In the place of zero detuning, a weak absorption peak appears, which can be explained as follows: decay interference destroys the atomic coherence induced by external fields and destructive interference of the system becomes weak, so the medium does not become transparent.

Next, we consider the case where Δb=Δc0 and the control field and pump field have same Rabi frequency. Firstly, the effect of atomic interference based on induction of external fields is considered. The curves of absorption and dispersion versus dimensionless detuning Δp/γ1 may be plotted as shown in Figs. 3(a)
Fig. 3 Curves of absorption (dashed line) and dispersion (solid line) of pulsed light. κ=0.5γ1, γ2=γ1, γ3=1.5γ1, Ωc=Ωb=6γ1, p=0. For (a), Δb=2γ1=Δc; For (b), Δb=4γ1=Δc.
and 3(b). The two figures show that two EIT windows occur. With the increase of detuning Δb and Δc, the central peak of absorption become higher, and both sides of absorption peaks become lower. In this case, the occurrences of two EIT windows result from the quantum interference effect induced by the control field and pump field. Similar phenomena can also occur in the four-level atomic system [19

19. S. Yelin, V. A. Sautenkov, M. Kash, G. Welch, and M. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003). [CrossRef]

,20

20. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

]. If we consider the effect of decay interference, the curve of probe absorption obviously changes (see Figs. 4(a)
Fig. 4 Curves of absorption (dashed line) and dispersion (solid line) of pulsed light. The parameters are same as Fig. 3(b) but with p=0.6 for (a) and p=0.96 for (b).
and 4(b)), where the parameter p which denotes decay interference has different value without changing other parameters noted in Fig. 3(b). Comparing Fig. 4 with Fig. 3(b), we can find locations of transparent windows move significantly and the width of windows narrows when the decay interference intensifies from p=0 to p=0.96, and it is also to be seen the central absorption peak increases obviously and both sides of absorption are inhibited significantly. In fact, in the case of p=0.96 which means strong decay interference, the minimum value of curve of absorption approaches zero and do not reach zero (see Fig. 4(b)), that is, strong decay interference affects destructive interference produced by external fields, which makes weak absorption appear.

In this quantum coherence medium, if we choose γ1=1.0×105s1, κ=0.5γ1, γ2=γ1, γ3=1.5γ1, Ωc=Ωb=6γ1, Δb=4γ1=Δc, for different values of decay interference p, the ratio vg/c of group velocity of pulsed light to the velocity of light is plotted in Fig. 5
Fig. 5 The plot of the ratio vg/c as a function of variable Δp/γ1. p=0 for dashed line, p=0.96 for solid line, and other parameters are shown in text.
, where the dashed curve means that decay interference is not considered (p=0), and the influence of decay interference on the group velocity is shown by solid curve (p=0.96). We find, in the places of absorption peaks (Δp/γ1=0,±5 for solid line, Δp/γ1=0,±10 for dashed line), the pulsed light propagates in the coherent medium with minimum velocity that is far less than the velocity of light (vg/c~104). In the locations of transparent windows (Δp/γ1~±2.5 for solid line, Δp/γ1~±5 for dashed line), we find values of vg/c of solid line are less than ones of dashed line, which means we can get slower light in transparent medium if we consider effect of decay interference.

In summary, we have investigated and analyzed the influence of atomic coherence induced by external fields and decay interference on the probe absorption and dispersion in a triple semiconductor quantum well structure. One EIT window or two EIT windows can be obtained by changing parameters of external fields. Strong decay interference can affect atomic coherence and weakens destructive interference due to atomic coherence, and ultimately transparent windows disappear or weaken. In addition, a new scheme to achieve slow propagation of the pulsed field has been proposed. Compared with the case in atomic system, semiconductor quantum well may easily be designed and integrated, so it has more practical value. Slow light proposed in present work may lead to important applications such high fidelity optical delay lines and optical buffers.

Acknowledgments

We thank theoretical physics group in Centre for Atom Optics and Ultrafast Spectroscopy of Swinburne University for useful discussions. Financial support by the National Natural Science Foundation of China under Grant No. 11065007 and the Foundation of Talent of Jinggang of Jiangxi Province, China, Grant No. 2008DQ00400, is acknowledged.

References and links

1.

M. Phillips and H. Wang, “Electromagnetically induced transparency due to intervalence band coherence in a GaAs quantum well,” Opt. Lett. 28(10), 831–833 (2003). [CrossRef] [PubMed]

2.

T. Müller, W. Parz, G. Strasser, and K. Unterrainer, “Influence of carrier-carrier interaction on time-dependent intersubband absorption in a semiconductor quantum well,” Phys. Rev. B 70(15), 155324 (2004). [CrossRef]

3.

J. F. Dynes, M. D. Frogley, J. Rodger, and C. C. Phillips, “Optically mediated coherent population trapping in asymmetric semiconductor quantum wells,” Phys. Rev. B 72(8), 085323 (2005). [CrossRef]

4.

W. X. Yang, X. X. Yang, and R. K. Lee, “Carrier-envelope-phase dependent coherence in double quantum wells,” Opt. Express 17(18), 15402–15408 (2009). [CrossRef] [PubMed]

5.

C. R. Lee, Y. Li, F. K. Men, C. Pao, Y. Tsai, and J. Wang, “Model for an inversionless two-color laser,” Appl. Phys. Lett. 86(20), 201112 (2005). [CrossRef]

6.

H. Schmidt and R. J. Ram, “All-optical wavelength converter and switch based on electromagnetically induced transparency,” Appl. Phys. Lett. 76(22), 3173–3175 (2000). [CrossRef]

7.

B. S. Ham, “Potential applications of dark resonance to subpicosecond optical switches in hyper-terahertz repetition rates,” Appl. Phys. Lett. 78(22), 3382–3384 (2001). [CrossRef]

8.

J. M. Tang, J. Levy, and M. E. Flatté, “All-electrical control of single ion spins in a semiconductor,” Phys. Rev. Lett. 97(10), 106803 (2006). [CrossRef] [PubMed]

9.

X. Lü and J. Wu, “Three-mode entanglement via tunneling-induced interference in a coupled triple-semiconductor quantum-well structure,” Phys. Rev. A 82(1), 012323 (2010). [CrossRef]

10.

A. Fountoulakis, A. Terzis, and E. Paspalakis, “Coherence phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73(3), 033811 (2006). [CrossRef]

11.

S. E. Economou, R. B. Liu, L. J. Sham, and D. G. Steel, “Unified theory of consequences of spontaneous emission in a Λ system,” Phys. Rev. B 71(19), 195327 (2005). [CrossRef]

12.

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005). [CrossRef] [PubMed]

13.

H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70(25), 3455–3457 (1997). [CrossRef]

14.

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82(10), 2079–2082 (1999). [CrossRef]

15.

P. R. Berman, “Spontaneously generated coherence and dark states,” Phys. Rev. A 72(3), 035801 (2005). [CrossRef]

16.

B. K. Ridley, Quantum Processes in Semiconductors (Clarendon Press, 1988).

17.

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71(5), 053820 (2005). [CrossRef]

18.

Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71(5), 053806 (2005). [CrossRef]

19.

S. Yelin, V. A. Sautenkov, M. Kash, G. Welch, and M. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003). [CrossRef]

20.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]

OCIS Codes
(270.1670) Quantum optics : Coherent optical effects
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Quantum Optics

History
Original Manuscript: March 7, 2011
Revised Manuscript: April 19, 2011
Manuscript Accepted: May 27, 2011
Published: June 6, 2011

Citation
Aixi Chen, "Influence of quantum coherence on propagation of a pulsed light in a triple quantum well," Opt. Express 19, 11944-11950 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-11944


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References

  1. M. Phillips and H. Wang, “Electromagnetically induced transparency due to intervalence band coherence in a GaAs quantum well,” Opt. Lett. 28(10), 831–833 (2003). [CrossRef] [PubMed]
  2. T. Müller, W. Parz, G. Strasser, and K. Unterrainer, “Influence of carrier-carrier interaction on time-dependent intersubband absorption in a semiconductor quantum well,” Phys. Rev. B 70(15), 155324 (2004). [CrossRef]
  3. J. F. Dynes, M. D. Frogley, J. Rodger, and C. C. Phillips, “Optically mediated coherent population trapping in asymmetric semiconductor quantum wells,” Phys. Rev. B 72(8), 085323 (2005). [CrossRef]
  4. W. X. Yang, X. X. Yang, and R. K. Lee, “Carrier-envelope-phase dependent coherence in double quantum wells,” Opt. Express 17(18), 15402–15408 (2009). [CrossRef] [PubMed]
  5. C. R. Lee, Y. Li, F. K. Men, C. Pao, Y. Tsai, and J. Wang, “Model for an inversionless two-color laser,” Appl. Phys. Lett. 86(20), 201112 (2005). [CrossRef]
  6. H. Schmidt and R. J. Ram, “All-optical wavelength converter and switch based on electromagnetically induced transparency,” Appl. Phys. Lett. 76(22), 3173–3175 (2000). [CrossRef]
  7. B. S. Ham, “Potential applications of dark resonance to subpicosecond optical switches in hyper-terahertz repetition rates,” Appl. Phys. Lett. 78(22), 3382–3384 (2001). [CrossRef]
  8. J. M. Tang, J. Levy, and M. E. Flatté, “All-electrical control of single ion spins in a semiconductor,” Phys. Rev. Lett. 97(10), 106803 (2006). [CrossRef] [PubMed]
  9. X. Lü and J. Wu, “Three-mode entanglement via tunneling-induced interference in a coupled triple-semiconductor quantum-well structure,” Phys. Rev. A 82(1), 012323 (2010). [CrossRef]
  10. A. Fountoulakis, A. Terzis, and E. Paspalakis, “Coherence phenomena due to double-dark states in a system with decay interference,” Phys. Rev. A 73(3), 033811 (2006). [CrossRef]
  11. S. E. Economou, R. B. Liu, L. J. Sham, and D. G. Steel, “Unified theory of consequences of spontaneous emission in a Λ system,” Phys. Rev. B 71(19), 195327 (2005). [CrossRef]
  12. J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005). [CrossRef] [PubMed]
  13. H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoglu, “Tunneling induced transparency: Fano interference in intersubband transitions,” Appl. Phys. Lett. 70(25), 3455–3457 (1997). [CrossRef]
  14. E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency induced via decay interference,” Phys. Rev. Lett. 82(10), 2079–2082 (1999). [CrossRef]
  15. P. R. Berman, “Spontaneously generated coherence and dark states,” Phys. Rev. A 72(3), 035801 (2005). [CrossRef]
  16. B. K. Ridley, Quantum Processes in Semiconductors (Clarendon Press, 1988).
  17. Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71(5), 053820 (2005). [CrossRef]
  18. Y. Wu and X. X. Yang, “Electromagnetically induced transparency in V-, Λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71(5), 053806 (2005). [CrossRef]
  19. S. Yelin, V. A. Sautenkov, M. Kash, G. Welch, and M. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68(6), 063801 (2003). [CrossRef]
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