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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 18 — Aug. 31, 2009
  • pp: 15402–15408
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Carrier-envelope-phase dependent coherence in double quantum wells

Wen-Xing Yang, Xiaoxue Yang, and Ray-Kuang Lee  »View Author Affiliations


Optics Express, Vol. 17, Issue 18, pp. 15402-15408 (2009)
http://dx.doi.org/10.1364/OE.17.015402


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Abstract

By analyzing the interaction of a few-cycle laser pulse within an asymmetric semiconductor double quantum well structure, we show that the transient coherence thus produced is strongly dependent on the carrier-envelope-phase (CEP) and significantly enhanced due to the Fano-type interference. A method to determine the CEP is proposed by directly mapping the CEP dependent coherence to the quantum beat signals.

© 2009 Optical Society of America

In this letter, we theoretically investigate the effects of CEP on the transient coherence produced by an ultrashort laser pulse of a few cycles in an asymmetric double quantum well structures. We demonstrate that the coherent effect is strongly dependent on the CEP, and the magnitude of transient coherence can be enhanced significantly due to the Fano-type interference. We also show that the coherence thus produced can also be mapped into the signal of quantum beats and hence might be used to determine the CEP of few-cycle pulses.

The schematic energy-level diagram of a GaAs/AlxGa1-xAs coupled quantum well structure are shown in Fig. 1(a): a AlxGa1-xAs shallow well and a GaAs deep well separated by a thick AlyGa1-yAs tunnel barrier. This barrier will couple the excited state of deep well with the ground state of shallow well to create a doublet states |2〉 and |3〉. One external light field is used to illuminate the system, and acts both on the transitions |1〉↔ |3〉 and |1〉↔|2〉 simultaneously. Tunneling to a continuum of energies takes place from states |2〉 and |3〉 through the thin barrier on the right of the deep well. The probability amplitude for the absorption of a photon can be thought as the superposition of two absorption paths, one via level |2〉 and one via level |3〉, both decaying by tunneling to the same continuum. Fano-type destructive interference between the two absorption paths may then occur so as to cancel the absorption altogether. Nearly vanishing absorptions due to the Fano effect have already been predicted [23

23. H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductor intersubband transitions,” Opt. Commun. 131, 333–338 (1996).

] and observed [3

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

, 4

4. J. Faist, F. Capasso, C. Sirtori, K. W. West, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature (London) 390, 589–591 (1997).

, 24

24. J. Faist, F. Capasso, C. Sirtori, A. L. Hutchinson, K. W. West, and L. N. Pfeiffer, “Itersubband emission in double-well structure with quantum interference in absorption,” Appl. Phys. Lett. 71, 3477–3479 (1997).

]. As shown in Fig. 1(b), we consider an ultrashort optical pulse of the electric field E(t)=-∂A(t)/∂t with the vector potential A(t)=A0e(t2τ)2τ2sin(ωt+ϕ) [16

16. T. Nakajima and S. Watanabe, “Effects of the carrier-encelope phase in the multiphoton ionization regime,” Phys. Rev. Lett. 96, 213001(1–4) (2006).

, 17

17. T. Nakajima and S. Watanabe, “Phase-dependent excitation and ionization in the multiphoton ionization regime,” Opt. Lett. 31, 1920–1922 (2006). [PubMed]

, 18

18. Y. Wu and X. Yang, “Carrier-envelope phase-dependent atomic coherence and quantum beats,” Phys. Rev. A 76, 013832(1–4) (2007).

, 19

19. Y. Wu and X. Yang, “Strong-Coupling Theory of Periodically Driven Two-Level Systems,” Phys. Rev. Lett. 98, 013601(1–4) (2007).

], where A 0, τ,ω, and ϕ are the amplitude, pulse width, carrier-envelope frequency, and the phase of the vector potential, respectively. Let us assume that the electronic wave function in the form of |ψ〉=a 1|1〉+a 2|2〉+a 3|3〉, then the time evolution equation for |ψ〉 is governed by the Schrödinger equation, with which we can have the corresponding differential equations for the probability amplitudes aj as follows:

a.1=iΩξ(t)[a2(t)eiΔt+qa3ei(Δ+δ)t],
(1)
a.2=γ2a2+iΩξ(t)a1eiΔt+pγ2γ3a3eiδt,
(2)
a.3=γ3γ3+iΩξ(t)qa1ei(Δ+δ)t+pγ2γ3a2eiδt,
(3)

Fig. 1. (a) Schematic diagram of our proposed GaAs/AlxGa1-xAs QW structure illuminated by an ultrashort few-cycle laser pulse in (b), where the electric field E(t) of the ultrashort pulse versus time t is shown for ϕ=π/2.
Fig. 2. The transient coherence ρ 23×104 versus the CEP, ϕ, (solid curves for the real part; dashed curves for the imaginary part; and dotted curves for the absolute value, respectively) at the time t=4τ for different widths, τ, and Rabi frequencies, Ω, of the pulse with other parameters h̄ω=125 meV, q=1.2, Δ=0, 2δ=7.6 meV, γ2l=5.6 meV, γ3l=7.0 meV, γ2d=4.13 meV, and γ3d=5.35 meV.

As an example for the numerical calculations, we consider the structure design of the asymmetric double quantum-well: a 68 Å thick Al0.15Ga0.85As shallow well and a 70 Å thick GaAs deep well separated by a 20 Å thick Al0.3Ga0.7As tunnel barrier. The doublet states (|2〉 and |3〉) are both coupled to the continuum by a 15 Å thin Al0.3Ga0.7As barrier, which produces the decay-induced coherence. Note that, for temperature up to 10 K with electron sheet densities smaller than 1012 cm-2, the dephasing rates γid can be estimated [3

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

] to be γ2d=4.13 meV and γ3d=5.35 meV. The population-decay rates can be calculated [31

31. D. Ahn and S. L. Chuang, “Exact calculations of quasibound states of an isolated quantum well with uniform electric field: Quantum-well stark resonance,” Phys. Rev. B 34, 9034–9037 (2008).

]: upon solving the effective mass Schrödinger equation with outgoing waves at infinity, we obtain a set of complex eigenvalues whose real and imaginary parts yield, respectively the quasibound state energy levels and resonance widths. For our asymmetric double quantum well structure, the population-decay rates turn out to be γ2l=5.6 meV and γ3l=7.0 meV. In such a scenario, a coupling ultrashort laser can produce the oscillation between the doublet states. Sequentially the induced oscillation is strongly dependent on the CEP of a few-cycle pulse, which produce a CEP dependent transient coherence for |ρ23|=|a 2(t)a*3(t)|. Direct numerical calculations for the solutions of Eqs. (1–3) demonstrate that the CEP of ultrashort laser pulses with only a few cycles has indeed significant effects on the coherence ρ 23 in the weak field regime (α=Ω/ω≪1). Figure 2 illustrates this point via some typical examples. The real, imaginary, and absolute values of the transient coherence ρ 23 is shown with the dependence of the CEP (ϕ) at the time t=4τ for two different pulse widths (τ=9/ω, τ=18/ω) and for two different Rabi frequencies (Ω=ω/20, Ω=ω/5), under the initial conditions a 1(0)=1 and a 2(0)=a 3(0)=0.

This result can be explained physically by the time-dependent perturbation theory with a small parameter α≪1. Under the initial conditions a 2,3(0)=0 and a 1(0)=1, taking aj=∑ka(k)j with a(k)j=𝒪(αk), we can see from Eqs. (1–3) that a 2,3(t)=𝒪(α) and a 1(t)=𝒪(α0), thus ρ 23=𝒪(α2). Clearly CEP dependence has been produced even for the low Rabi frequency i.e., Ω=ω/20. Just as shown in Fig. 2, the dependent amplitude become pronounced as the Rabi frequency increases and the pulse width becomes narrow. The low Rabi frequencies induce less transient coherence and hence are obviously non-favorable from the viewpoint of the experimental measurement. The lower limit for Rabi frequency depends on the precision of the technique in measurement. With state-of-the-art technologies to handle the weak light-QW interaction, relative effects of the QW system considered here can be measured in low temperature (10 K) [2

2. H.C. Liu and F. Capasso, Intersubband Transitions in Quantum Wells: Physics and Device Applications (Academic Press, San Diego, 2000).

]. Besides, we note that α~E characterizes the electric field E(t) with the period 2π for the CEP ϕ. In such a case, the relation ρ 23=𝒪(α 2) implies that ρ 23 should approximately have the period π, as illustrated in Fig. 2.

Fig. 3. The transient coherence |ρ 23|×104 versus the CEP ϕ for the case of (a, c) p=0 and (b, d) p=1 at the time t=4τ for different widths, τ, and Rabi frequencies, Ω, of the pulse with other parameters h̄ω=125 meV, q=1.2, Δ=0, 2δ=17.6 meV, γ2l=0.31 meV, γ3l=0.26 meV, γ2d=0.031 meV, and γ3d=0.026 meV.

It should be noted that the interference induced by the resonant tunneling have been included in plotting Fig. 2. According to the decay-rate values (γ2l=5.6 meV, γ3l=7.0 meV, γ2d=4.13 meV, and γ3d=5.35 meV), we can obtain the cross coupling strength between |2〉 and |3〉 p=0.54. In order to examine the effect of the interference induced by the resonant tunneling on the CEP dependent coherence, we consider a similar GaAs/AlGaAs asymmetric double quantum well structure consists of two quantum wells (55 Å Al0.3Ga0.7As shallow well and 57 Å GaAs deep well) separated by a 35 Å Al0.5Ga0.5As tunneling barrier. Aluminum is added to the shallow well in order to reduce the contribution of interface roughness scattering. The energy splitting between the upper levels is calculated to be 2δ=7.6 meV. For a sheet carrier density of 1012 cm-2 in the quantum wells, we can obtain the LO-phonon decay rates γ2l=0.31 meV and γ3l=0.26 meV, and the dephasing rates can be estimated to be γ2d=0.031 and γ3d=0.026 meV. Thus, the cross coupling strength is estimated as p=0.90. This is close to the ideal value p=1 and corresponds to a large tunneling efficiency leading to a strong Fano-type interference effect. With new parameter values of this QW structure, we show in Fig. 3 the transient coherence |ρ 23| versus the CEP ϕ at the time t=4τ under the same initial conditions as in Fig. 2, and it demonstrates that the amplitude of the transient coherence is enhanced. This interesting result is produced from the perfectly interference induced by the resonant tunneling. The large amplitude is obviously favorable from the viewpoint of the experimental measurement in the weak-field regime. More interestingly, the parameters of the electron subbands in QWstructures can be engineered to give a desired amplitude of coherence by utilizing so-called structure coherent control in design [2

2. H.C. Liu and F. Capasso, Intersubband Transitions in Quantum Wells: Physics and Device Applications (Academic Press, San Diego, 2000).

].

We now study the quantum beats due to the coherence ρ23 produced by a few-cycle ultrafast pulse for the time interval T>t with the initial time t=t 0=4τ. The quantum beat note signal I can be given as [32

32. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997).

]

I=ψ(T)Ê1()(T)Ê2(+)(T)ψ(T)+c.c.,
(4)

with the state of our system |ψ(T)〉 satisfying |ψ(T)〉=∑jaj|j,0〉+b 2|1,1ω21〉+b 3|1,1ω31〉. Here |n,0〉, |1,1ω j1〉 describe the levels |n〉(n=1,2, 3) with no photon, and ground state |1〉 with one photon in the field mode j characterizing the transition |0〉→| j〉(j=2, 3), respectively. Ê1()(T)=1â1eiω21(Tt) and Ê2(+)(T)=2â2e31(Tt) denote the electric field per photon for the mode j. Inserting Hamiltonian H=h̄∑jgj(âj|j〉〈1|+â j|1〉〈j|) into the Schrödinger equation, i.e. |ψ(T)〉/∂T=-i(H/h̄)|ψ(T)〉, we obtain

i(ddT+γj)ajpγ2γ3(a3δj,2+a2δj,3)=gjbj,
(5)
idbjdTgjaj=0,j=2,3,
(6)

with gj0 jj/(2h̄). By solving Eqs. (5,6) under the initial conditions of b 1,2(t)=0, the quantum beat signals can be calculated as

I=I0(ϕ)cos[2δ(Tt)+η(t)],
(7)

In conclusion, we have studied the generation of transient coherence induced by few-cycle laser pulses in an asymmetric semiconductor double QW structure, and shown that the coherence thus produced strongly depends on the carrier-envelope phase of the ultrashort laser pulses. Importantly, the amplitude of the CEP dependent transient coherence can be greatly enhanced due to the Fano-type interference. Besides, we also shown that the CEP-dependent coherence can be mapped into the signal of quantum beats, thus one can determine the CEP by measuring the quantum beat signals. We believe that the CEP dependent coherence in our proposed QW structure will also manifest itself in other quantum interference phenomena as well, and hence our study might open up an avenue to explore and utilize the CEP dependent coherent effects and could be exploited in real solid-state devices as high speed optical modulators and switches.

This work is supported by National Natural Science Foundation (NSF) of China under Grants No. 10704017 and No. 10874050, and also partially supported by the National Basic Research Program of China (973 program), Grant No. 2007CB936300 and No. 2005CB724508. We thank Prof. Y. Wu and Ite. Yu for their helpful discussions.

References and links

1.

Z. Ficek and S. Swain, Quantum Interference and Coherence (Springer, New York, 2004).

2.

H.C. Liu and F. Capasso, Intersubband Transitions in Quantum Wells: Physics and Device Applications (Academic Press, San Diego, 2000).

3.

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

4.

J. Faist, F. Capasso, C. Sirtori, K. W. West, and L. N. Pfeiffer, “Controlling the sign of quantum interference by tunnelling from quantum wells,” Nature (London) 390, 589–591 (1997).

5.

G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. 84, 1019–1021 (2000). [PubMed]

6.

L. Silvestri, F. Bassani, G. Czajkowski, and B. Davoudi, “Electromagnetically induced transparency in asymmetric double quantum wells,” Eur. Phys. J. B 27, 89–102 (2002).

7.

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, 155324(1–5) (2004).

8.

S. M. Sadeghi, S. R. Leffler, and J. Meyer, “Quantum interference and nonlinear optical processes in the conduction bands of infrared-coupled quantum wells,” Phys. Rev. B 59, 15388–15394 (1999).

9.

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nature Materials 5, 175–178 (2006).

10.

E. Paspalakis, M. Tsaousidou, and A. F. Terzis, “Coherent manipulation of a strongly driven semicondutor quantum well,” Phys. Rev. B 73, 125344(1–5) (2006).

11.

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, 157403(1–4) (2005).

12.

B. S. Williams, B. Xu, Q. Hu, and M. R. Melloch, “Narrow-linewidth terahertz intersubband emission from three-level systems,” Appl. Phys. Lett. 75, 2927–2929 (1999).

13.

T. M. Frontier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, “Carrier-Envelope Phase-Controlled Quantum Interference of Injected Photocurrents in Semiconductors,” Phys. Rev. Lett. 92, 147403(1–4) (2004).

14.

K. A. Pronin and A. D. Bandrauk, “Coherent Control of Harmonic Generation in Superlattices: Single-Mode Response,” Phys. Rev. Lett. 97, 020602(1–4) (2006).

15.

C. Van Vlack and S. Hughes, “Carrier-Envelope-Offset Phase Control of Ultrafast Optical Rectification in Resonantly Excited Semiconductors,” Phys. Rev. Lett. 98, 167404(1–4) (2007).

16.

T. Nakajima and S. Watanabe, “Effects of the carrier-encelope phase in the multiphoton ionization regime,” Phys. Rev. Lett. 96, 213001(1–4) (2006).

17.

T. Nakajima and S. Watanabe, “Phase-dependent excitation and ionization in the multiphoton ionization regime,” Opt. Lett. 31, 1920–1922 (2006). [PubMed]

18.

Y. Wu and X. Yang, “Carrier-envelope phase-dependent atomic coherence and quantum beats,” Phys. Rev. A 76, 013832(1–4) (2007).

19.

Y. Wu and X. Yang, “Strong-Coupling Theory of Periodically Driven Two-Level Systems,” Phys. Rev. Lett. 98, 013601(1–4) (2007).

20.

G. L. Kamta and A. D. Bandrauk, “Phase Dependence of Enhanced Ionization in Asymmetric Molecules,” Phys. Rev. Lett. 94, 203003(1–4) (2005).

21.

W. Yang, X. Song, S. Gong, Y. Cheng, and Z. Xu, “Carrier-Envelope Phase Dependence of Few-Cycle Ultrashort Laser Pulse Propagation in a Polar Molecule Medium,” Phys. Rev. Lett. 99, 133602(1–4) (2007).

22.

C. Zhang, X. Song, W. Yang, and Z. Xu, “Carrier-envelope phase control of carrier-wave Rabi flopping in asymmetric semiparabolic quantum well,” Opt. Express 16, 11487–1496 (2008).

23.

H. Schmidt and A. Imamoglu, “Nonlinear optical devices based on a transparency in semiconductor intersubband transitions,” Opt. Commun. 131, 333–338 (1996).

24.

J. Faist, F. Capasso, C. Sirtori, A. L. Hutchinson, K. W. West, and L. N. Pfeiffer, “Itersubband emission in double-well structure with quantum interference in absorption,” Appl. Phys. Lett. 71, 3477–3479 (1997).

25.

I. Waldmüller, J. Förstner, S.-C. Lee, A. Knorr, M. Woerner, K. Reimann, R. A. Kaindl, T. Elsaesser, R. Hey, and K. H. Ploog, “Optical dephasing of coherent intersubband transitions in a quasi-two-dimensional electron gas,” Phys. Rev. B 69, 205307(1–9) (2004).

26.

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

27.

E. Paspalakis, N. J. Kylstrra, and P. L. Knight, “Transparency of a short laser pulse via decay interference in a closed V-type system,” Phys. Rev. A 61, 045802(1–4) (1999).

28.

J. H. Wu, J. Y. Gao, J. H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast All Optical Switching via Tunable Fano Interference,” Phys. Rev. Lett. 95, 057401(1–4) (2005).

29.

W. X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16, 17161–17170 (2008). [PubMed]

30.

W. X. Yang, J. M. Hou, and R.-K. Lee, “Ultraslow bright and dark solitons in semiconductor quantum wells,” Phys. Rev. A 77, 033838(1–7) (2008).

31.

D. Ahn and S. L. Chuang, “Exact calculations of quasibound states of an isolated quantum well with uniform electric field: Quantum-well stark resonance,” Phys. Rev. B 34, 9034–9037 (2008).

32.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997).

OCIS Codes
(270.1670) Quantum optics : Coherent optical effects
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

ToC Category:
Quantum Optics

History
Original Manuscript: April 23, 2009
Revised Manuscript: July 24, 2009
Manuscript Accepted: August 11, 2009
Published: August 17, 2009

Citation
Wen-Xing Yang, Xiaoxue Yang, and Ray-Kuang Lee, "Carrier-envelope-phase dependent coherence in double quantum wells," Opt. Express 17, 15402-15408 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15402


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References

  1. Z. Ficek and S. Swain, Quantum Interference and Coherence (Springer, New York, 2004).
  2. H.C. Liu and F. Capasso, Intersubband Transitions in Quantum Wells: Physics and Device Applications (Academic Press, San Diego, 2000).
  3. H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoğlu, "Tunneling induced transparency: Fano interference in intersubband transitions," Appl. Phys. Lett. 70, 3455-3457 (1997).
  4. J. Faist, F. Capasso, C. Sirtori, K. W. West, and L. N. Pfeiffer, "Controlling the sign of quantum interference by tunnelling from quantum wells," Nature (London) 390, 589-591 (1997).
  5. G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, "Laser-induced quantum coherence in a semiconductor quantum well," Phys. Rev. Lett. 84, 1019-1021 (2000). [PubMed]
  6. L. Silvestri, F. Bassani, G. Czajkowski, and B. Davoudi, "Electromagnetically induced transparency in asymmetric double quantum wells," Eur. Phys. J. B 27, 89-102 (2002).
  7. 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, 155324 (2004).
  8. S. M. Sadeghi, S. R. Leffler, and J. Meyer, "Quantum interference and nonlinear optical processes in the conduction bands of infrared-coupled quantum wells, " Phys. Rev. B 59, 15388-15394 (1999).
  9. M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, "Gain without inversion in semiconductor nanostructures," Nature Materials 5, 175-178 (2006).
  10. E. Paspalakis, M. Tsaousidou, and A. F. Terzis, "Coherent manipulation of a strongly driven semicondutor quantum well," Phys. Rev. B 73, 125344 (2006).
  11. 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, 157403 (2005).
  12. B. S. Williams, B. Xu, Q. Hu, and M. R. Melloch, "Narrow-linewidth terahertz intersubband emission from three-level systems," Appl. Phys. Lett. 75, 2927-2929 (1999).
  13. T. M. Frontier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-Envelope Phase-Controlled Quantum Interference of Injected Photocurrents in Semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
  14. K. A. Pronin and A. D. Bandrauk, "Coherent Control of Harmonic Generation in Superlattices: Single-Mode Response," Phys. Rev. Lett. 97, 020602 (2006).
  15. C. Van Vlack and S. Hughes, "Carrier-Envelope-Offset Phase Control of Ultrafast Optical Rectification in Resonantly Excited Semiconductors, " Phys. Rev. Lett. 98, 167404 (2007).
  16. T. Nakajima and S. Watanabe, "Effects of the carrier-encelope phase in the multiphoton ionization regime," Phys. Rev. Lett. 96, 213001 (2006).
  17. T. Nakajima and S. Watanabe, "Phase-dependent excitation and ionization in the multiphoton ionization regime," Opt. Lett. 31, 1920-1922 (2006). [PubMed]
  18. Y. Wu and X. Yang, "Carrier-envelope phase-dependent atomic coherence and quantum beats," Phys. Rev. A 76, 013832 (2007).
  19. Y. Wu and X. Yang, "Strong-Coupling Theory of Periodically Driven Two-Level Systems," Phys. Rev. Lett. 98, 013601 (2007).
  20. G. L. Kamta and A. D. Bandrauk, "Phase Dependence of Enhanced Ionization in Asymmetric Molecules," Phys. Rev. Lett. 94, 203003 (2005).
  21. W. Yang, X. Song, S. Gong, Y. Cheng, and Z. Xu, "Carrier-Envelope Phase Dependence of Few-Cycle Ultrashort Laser Pulse Propagation in a Polar Molecule Medium," Phys. Rev. Lett. 99, 133602 (2007).
  22. C. Zhang, X. Song, W. Yang, and Z. Xu, "Carrier-envelope phase control of carrier-wave Rabi flopping in asymmetric semiparabolic quantum well," Opt. Express 16, 11487-1496 (2008).
  23. H. Schmidt and A. Imamoglu, "Nonlinear optical devices based on a transparency in semiconductor intersubband transitions," Opt. Commun. 131, 333-338 (1996).
  24. J. Faist, F. Capasso, C. Sirtori, A. L. Hutchinson, K. W. West, and L. N. Pfeiffer, "Itersubband emission in double-well structure with quantum interference in absorption," Appl. Phys. Lett. 71, 3477-3479 (1997).
  25. I. Waldmüller, J. F¨orstner, S.-C. Lee, A. Knorr, M. Woerner, K. Reimann, R. A. Kaindl, T. Elsaesser, R. Hey, and K. H. Ploog, "Optical dephasing of coherent intersubband transitions in a quasi-two-dimensional electron gas," Phys. Rev. B 69, 205307 (2004).
  26. E. Paspalakis, N. J. Kylstrra, and P. L. Knight, "Transparency induced via decay interference," Phys. Rev. Lett. 82, 2079-2082 (1999).
  27. E. Paspalakis, N. J. Kylstrra, and P. L. Knight, "Transparency of a short laser pulse via decay interference in a closed V-type system," Phys. Rev. A 61, 045802 (1999).
  28. J. H. Wu, J. Y. Gao, J. H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, F. Bassani, "Ultrafast All Optical Switching via Tunable Fano Interference," Phys. Rev. Lett. 95, 057401 (2005).
  29. W. X. Yang and R.-K. Lee, "Controllable entanglement and polarization phase gate in coupled double quantumwell structures," Opt. Express 16, 17161-17170 (2008). [PubMed]
  30. W. X. Yang, J. M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
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