Abstract

We propose a method to achieve significant optical signal delays exploiting the effect of Autler–Townes splitting (ATS) in an inhomogeneously broadened quantum dot medium. The absorption and slowdown effects are compared for three schemes i.e., Ξ, V, and Λ, corresponding to different excitation configurations. Qualitative differences of the V scheme compared to the Ξ and Λ schemes are found, which show that features of (ATS) are only revealed in the V scheme. The underlying physical mechanisms causing this discrepancy are analyzed and discussed. Finally we compare field propagation calculations of the schemes showing significantly larger achievable signal delays for the V scheme despite finite absorption of the coupling field. This opens the possibility for using waveguide structures for both coupling and probe fields, thus significantly increasing the achievable signal delays.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [CrossRef]
  4. C. Liu, Z. Dutton, C. Behroozi, and L. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. M. Bajcsy, A. Zibrov, and M. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
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  7. M. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. Zibrov, and M. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837–841 (2005).
    [CrossRef] [PubMed]
  8. N. S. Ginsberg, S. R. Garner, and L. V. Hau, “Coherent control of optical information with matter wave dynamics,” Nature 445, 623–626 (2007).
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  30. R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
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    [CrossRef]
  35. R. Heitz, I. Mukhametzhanov, O. Stier, A. Madhukar, and D. Bimberg, “Enhanced polar exciton-lo-phonon interaction in quantum dots,” Phys. Rev. Lett. 83, 4654–4657 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  40. E. A. Zibik, T. Grange, B. A. Carpenter, N. E. Porter, R. Ferreira, G. Bastard, D. Stehr, S. Winner, M. Helm, H. Y. Liu, M. S. Skolnick, and L. R. Wilson, “Long lifetimes of quantum-dot intersublevel transitions in the terahertz range,” Nature Mater. 8, 803–807 (2009).
    [CrossRef]
  41. M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
    [CrossRef]
  42. S. Rodt, A. Schliwa, K. Pötschke, F. Guffarth, and D. Bimberg, “Correlation of structural and few-particle properties of self-organized InAs/GaAs quantum dots,” Phys. Rev. B 71, 155325 (2005).
    [CrossRef]
  43. M. C. Phillips and H. Wang, “Exciton spin coherence and electromagnetically induced transparency in the transient optical response of GaAs quantum wells,” Phys. Rev. B 69, 115337 (2004).
    [CrossRef]
  44. P. K. Nielsen, H. Thyrrestrup, J. Mørk, and B. Tromborg, “Numerical investigation of electromagnetically induced transparency in a quantum dot structure,” Opt. Express 15, 6396–6408 (2007).
    [CrossRef] [PubMed]
  45. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  46. G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
    [CrossRef]
  47. J. Tidström, P. Jänes, and L. M. Andersson, “Delay-bandwidth product of electromagnetically induced transparency media,” Phys. Rev. A 75, 053803 (2007).
    [CrossRef]
  48. Z. Shi and R. W. Boyd, “Discretely tunable optical packet delays using channelized slow light,” Phys. Rev. A 79, 013805 (2009).
    [CrossRef]

2009 (8)

J.B.Khurgin and R.S.Tucker, eds., Slow Light-Science and Applications (Taylor & Francis, 2009).

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[CrossRef] [PubMed]

J. Mørk, F. Öhman, M. van der Poel, Y. Chen, P. Lunnemann, and K. Yvind, “Slow and fast light: Controlling the speed of light using semiconductor waveguides,” Laser Photonics Rev. 3, 30–44 (2009).
[CrossRef]

P. Lunnemann and J. Mørk, “Reducing the impact of inhomogeneous broadening on quantum dot based electromagnetically induced transparency,” Appl. Phys. Lett. 94, 071108 (2009).
[CrossRef]

B. D. Gerardot, D. Brunner, P. A. Dalgarno, K. Karrai, A. Badolato, P. M. Petroff, and R. J. Warburton, “Dressed excitonic states and quantum interference in a three-level quantum dot ladder system,” New J. Phys. 11, 013028 (2009).
[CrossRef]

D. Barettin, J. Houmark, B. Lassen, M. Willatzen, T. R. Nielsen, J. Mørk, and A.-P. Jauho, “Optical properties and optimization of electromagnetically induced transparency in strained InAs/GaAs quantum dot structures,” Phys. Rev. B 80, 235304 (2009).
[CrossRef]

E. A. Zibik, T. Grange, B. A. Carpenter, N. E. Porter, R. Ferreira, G. Bastard, D. Stehr, S. Winner, M. Helm, H. Y. Liu, M. S. Skolnick, and L. R. Wilson, “Long lifetimes of quantum-dot intersublevel transitions in the terahertz range,” Nature Mater. 8, 803–807 (2009).
[CrossRef]

Z. Shi and R. W. Boyd, “Discretely tunable optical packet delays using channelized slow light,” Phys. Rev. A 79, 013805 (2009).
[CrossRef]

2008 (4)

M. Kroner, C. Lux, S. Seidl, A. W. Holleitner, K. Karrai, A. Badolato, P. M. Petroff, and R. J. Warburton, “Rabi splitting and ac-stark shift of a charged exciton,” Appl. Phys. Lett. 92, 031108 (2008).
[CrossRef]

S. Marcinkevičius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett. 92, 041113 (2008).
[CrossRef]

G. Jundt, L. Robledo, A. Hogele, S. Falt, and A. Imamoglu, “Observation of dressed excitonic states in a single quantum dot,” Phys. Rev. Lett. 100, 177401 (2008).
[CrossRef] [PubMed]

A. Muller, W. Fang, J. Lawall, and G. S. Solomon, “Emission spectrum of a dressed exciton-biexciton complex in a semiconductor quantum dot,” Phys. Rev. Lett. 101, 027401 (2008).
[CrossRef] [PubMed]

2007 (4)

X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Coherent optical spectroscopy of a strongly driven quantum dot,” Science 317, 929–932 (2007).
[CrossRef] [PubMed]

N. S. Ginsberg, S. R. Garner, and L. V. Hau, “Coherent control of optical information with matter wave dynamics,” Nature 445, 623–626 (2007).
[CrossRef] [PubMed]

P. K. Nielsen, H. Thyrrestrup, J. Mørk, and B. Tromborg, “Numerical investigation of electromagnetically induced transparency in a quantum dot structure,” Opt. Express 15, 6396–6408 (2007).
[CrossRef] [PubMed]

J. Tidström, P. Jänes, and L. M. Andersson, “Delay-bandwidth product of electromagnetically induced transparency media,” Phys. Rev. A 75, 053803 (2007).
[CrossRef]

2006 (2)

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, “Influence of wetting-layer wave functions on phonon-mediated carrier capture into self-assembled quantum dots,” Phys. Rev. B 74, 195342 (2006).
[CrossRef]

P. Borri, S. Schneider, W. Langbein, and D. Bimberg, “Ultrafast carrier dynamics in InGaAs quantum dot materials and devices,” J. Opt. A, Pure Appl. Opt. 8, S33–S46 (2006).
[CrossRef]

2005 (4)

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[CrossRef] [PubMed]

M. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. Zibrov, and M. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837–841 (2005).
[CrossRef] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

S. Rodt, A. Schliwa, K. Pötschke, F. Guffarth, and D. Bimberg, “Correlation of structural and few-particle properties of self-organized InAs/GaAs quantum dots,” Phys. Rev. B 71, 155325 (2005).
[CrossRef]

2004 (4)

M. C. Phillips and H. Wang, “Exciton spin coherence and electromagnetically induced transparency in the transient optical response of GaAs quantum wells,” Phys. Rev. B 69, 115337 (2004).
[CrossRef]

W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Radiatively limited dephasing in InAs quantum dots,” Phys. Rev. B 70, 033301 (2004).
[CrossRef]

J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnain, “Slow light using semiconductor quantum dots,” J. Phys. Condens. Matter 16, S3727–S3735 (2004).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, 2004).

2003 (4)

C. Chang-Hasnain, P.-C. Ku, J. Kim, and S.-l. Chuang, “Variable optical buffer using slow light in semiconductor nanostructures,” in Proc. IEEE 91, 1884–1897 (2003).
[CrossRef]

P.Michler, ed., Single Quantum Dots: Fundamentals, Applications, and New Concepts, Vol. 90 of Topics in Applied Physics (Springer-Verlag, 2003).

M. Bajcsy, A. Zibrov, and M. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

B. Bransden and C. Joachain, Physics of Atoms and Molecules, 2nd ed. (Prentice Hall, 2003).

2002 (1)

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[CrossRef]

2001 (5)

R. Heitz, H. Born, F. Guffarth, O. Stier, A. Schliwa, A. Hoffmann, and D. Bimberg, “Existence of a phonon bottleneck for excitons in quantum dots,” Phys. Rev. B 64, 241305 (2001).
[CrossRef]

C. Liu, Z. Dutton, C. Behroozi, and L. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
[CrossRef] [PubMed]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001).
[CrossRef] [PubMed]

P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. 87, 157401 (2001).
[CrossRef] [PubMed]

D. Birkedal, K. Leosson, and J. M. Hvam, “Long lived coherence in self-assembled quantum dots,” Phys. Rev. Lett. 87, 227401 (2001).
[CrossRef] [PubMed]

1999 (6)

R. Heitz, I. Mukhametzanov, H. Born, M. Grundmann, A. Hoffmann, A. Madhukar, and D. Bimberg, “Hot carrier relaxation in InAs/GaAs quantum dots,” Physica B 272, 8–11 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

O. Stier, M. Grundmann, and D. Bimberg, “Electronic and optical properties of strained quantum dots modeled by 8-band k⋅p theory,” Phys. Rev. B 59, 5688–5701 (1999).
[CrossRef]

R. Heitz, I. Mukhametzhanov, O. Stier, A. Madhukar, and D. Bimberg, “Enhanced polar exciton-lo-phonon interaction in quantum dots,” Phys. Rev. Lett. 83, 4654–4657 (1999).
[CrossRef]

Y. Nakata, Y. Sugiyama, and M. Sugawara, Self-Assembled InGaAs/GaAs Quantum dots (Academic, 1999), Chap. 2, pp. 117–154.
[CrossRef]

P. Borri, W. Langbein, J. Mørk, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Dephasing in InAs/GaAs quantum dots,” Phys. Rev. B 60, 7784–7787 (1999).
[CrossRef]

1998 (1)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

1997 (2)

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1997), Vol. 36.

G. S. Agarwal, “Nature of the quantum interference in electromagnetic-field-induced control of absorption,” Phys. Rev. A 55, 2467–2470 (1997).
[CrossRef]

1996 (1)

D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, “Fine structure splitting in the optical spectra of single GaAs quantum dots,” Phys. Rev. Lett. 76, 3005–3008 (1996).
[CrossRef] [PubMed]

1989 (1)

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

1955 (1)

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Nature of the quantum interference in electromagnetic-field-induced control of absorption,” Phys. Rev. A 55, 2467–2470 (1997).
[CrossRef]

Agrawal, G.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

Andersson, L. M.

J. Tidström, P. Jänes, and L. M. Andersson, “Delay-bandwidth product of electromagnetically induced transparency media,” Phys. Rev. A 75, 053803 (2007).
[CrossRef]

Andre, A.

M. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. Zibrov, and M. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837–841 (2005).
[CrossRef] [PubMed]

Autler, S. H.

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100, 703–722 (1955).
[CrossRef]

Badolato, A.

B. D. Gerardot, D. Brunner, P. A. Dalgarno, K. Karrai, A. Badolato, P. M. Petroff, and R. J. Warburton, “Dressed excitonic states and quantum interference in a three-level quantum dot ladder system,” New J. Phys. 11, 013028 (2009).
[CrossRef]

M. Kroner, C. Lux, S. Seidl, A. W. Holleitner, K. Karrai, A. Badolato, P. M. Petroff, and R. J. Warburton, “Rabi splitting and ac-stark shift of a charged exciton,” Appl. Phys. Lett. 92, 031108 (2008).
[CrossRef]

Bajcsy, M.

M. Bajcsy, A. Zibrov, and M. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

Barettin, D.

D. Barettin, J. Houmark, B. Lassen, M. Willatzen, T. R. Nielsen, J. Mørk, and A.-P. Jauho, “Optical properties and optimization of electromagnetically induced transparency in strained InAs/GaAs quantum dot structures,” Phys. Rev. B 80, 235304 (2009).
[CrossRef]

Bastard, G.

E. A. Zibik, T. Grange, B. A. Carpenter, N. E. Porter, R. Ferreira, G. Bastard, D. Stehr, S. Winner, M. Helm, H. Y. Liu, M. S. Skolnick, and L. R. Wilson, “Long lifetimes of quantum-dot intersublevel transitions in the terahertz range,” Nature Mater. 8, 803–807 (2009).
[CrossRef]

Bayer, M.

M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, “Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots,” Phys. Rev. B 65, 195315 (2002).
[CrossRef]

Behroozi, C.

C. Liu, Z. Dutton, C. Behroozi, and L. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
[CrossRef] [PubMed]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Berg, T. W.

T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, “Influence of wetting-layer wave functions on phonon-mediated carrier capture into self-assembled quantum dots,” Phys. Rev. B 74, 195342 (2006).
[CrossRef]

Berman, P. R.

X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Coherent optical spectroscopy of a strongly driven quantum dot,” Science 317, 929–932 (2007).
[CrossRef] [PubMed]

Bimberg, D.

P. Borri, S. Schneider, W. Langbein, and D. Bimberg, “Ultrafast carrier dynamics in InGaAs quantum dot materials and devices,” J. Opt. A, Pure Appl. Opt. 8, S33–S46 (2006).
[CrossRef]

R. Seguin, A. Schliwa, S. Rodt, K. Pötschke, U. W. Pohl, and D. Bimberg, “Size-dependent fine-structure splitting in self-organized InAs/GaAs quantum dots,” Phys. Rev. Lett. 95, 257402 (2005).
[CrossRef] [PubMed]

S. Rodt, A. Schliwa, K. Pötschke, F. Guffarth, and D. Bimberg, “Correlation of structural and few-particle properties of self-organized InAs/GaAs quantum dots,” Phys. Rev. B 71, 155325 (2005).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Three EIT schemes Ξ (left), V (middle), and Λ (right) as implemented for a QD confinement potential. Dashed blue and solid red arrows indicate the probe and coupling transition, respectively.

Fig. 2
Fig. 2

Density of resonant transitions as a function of frequency for the three considered transitions indicated by the level diagram insets. The IHB FWHM is in this work referred to the ground state transition (middle peak).

Fig. 3
Fig. 3

Top: Normalized absorption as a function of coupling intensity for the three schemes Ξ (red), V (black), and Λ (blue), with (solid lines) and without (dashed lines) IHB. The absorption is normalized by the absorption without an applied coupling field. Note that the curves for Ξ and Λ overlap. Bottom: Corresponding slowdown factor S. Material parameters are presented in Table 1.

Fig. 4
Fig. 4

Electric susceptibility [in units of Γ μ 13 2 / ( V ε 0 ) ] for QDs with η = 1 . Left middle and right columns are for Ξ, V, and Λ, respectively. Top: Average real (dashed line) and imaginary (solid line) parts of the susceptibility as functions of the normalized probe detuning. Middle: Real part of the electric susceptibility as a function of the normalized probe detuning Δ p and spectral shift Δ i h . Bottom: Imaginary part of the electric susceptibility as a function of the normalized probe detuning Δ p and spectral shift Δ i h . The asymptote of the primary and secondary resonances is indicated by dashed-dotted and dashed lines, respectively.

Fig. 5
Fig. 5

Top: Absorption as a function of coupling field intensity for the three cases: [(a), solid line] lifetime limited dephasing rates with no intraband population decay rate, [(b), dashed line] lifetime limited dephasing rates with a large intraband population decay rate, and [(c), dashed-dotted line] being equivalent to (a) except the intraband dephasing is set large. Bottom: Corresponding slowdown factors.

Fig. 6
Fig. 6

Plot of real (top) and imaginary (bottom) parts of the electric susceptibility χ p [in units of Γ μ 13 2 / ( V ε 0 ) ] as functions of the scaled probe detuning Δ p / Γ 13 and intraband decay rate Γ 12 / Γ 13 .

Fig. 7
Fig. 7

Calculated electric susceptibility using η = 1 . Top: Average real (dashed line) and imaginary (solid line) parts of the electric susceptibility [normalized by Γ μ 13 2 / ( V ε 0 ) ] for the alternative V scheme (gray) and V scheme from Fig. 1 (gray). Bottom: Associated imaginary part of the electric susceptibility for the alternative V scheme as a function of scaled probe detuning Δ p / Γ 13 and spectral shift Δ i h / Δ 13 . The inset illustrates the excitation configuration of the alternative scheme.

Fig. 8
Fig. 8

(a) Asymmetric QD potential with symmetry axis (dashed lines) along the crystal planes [110] and [ 1 1 ¯ 0 ] . (b) Fine structure splitting based schemes. The two exciton states | X and | X are coupled with linearly polarized light parallel to the symmetry axis.

Fig. 9
Fig. 9

Electric susceptibility [in units of Γ μ 13 2 / ( V ε 0 ) ] of the V (left) and Λ (right) scheme based on the FSS configuration. Bottom: Imaginary part of the electric susceptibility as a function of the normalized probe detuning Δ p and spectral shift Δ i h . The asymptote of the primary and secondary resonances is indicated by dashed-dotted and dashed lines, respectively. Top: Corresponding calculated average real (dashed) and imaginary part (solid) parts of the electric susceptibility. Key parameters are Γ p = Γ c and Γ u c = 0 , where Γ p , Γ c , and Γ u c denote the population decay rate of the probe, coupling, and uncoupled transition, respectively. Dephasing rates were assumed lifetime limited.

Fig. 10
Fig. 10

Calculated delay (surface plot) and transmission (isocurves) as functions of injected coupling intensity and propagation length z for the Λ (top), V (middle), and Ξ (bottom) schemes. The transmission curves are labeled with the corresponding transmission in decibels. Note from the color bar the different scales of the delay. Decay and material parameters are chosen as in Table 1.

Fig. 11
Fig. 11

Delay as a function of the injected coupling intensity at a fixed transmission of −10 dB for the V (solid) and Ξ (dashed) schemes. Corresponding propagation lengths are indicated by arrows for the V scheme. Inset shows a zoom-in of the calculated delay for the Ξ scheme. Decay and material parameters are chosen as in Table 1.

Tables (1)

Tables Icon

Table 1 QD Parameters Used in the Calculations Unless Otherwise Stated a

Equations (28)

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ε n l m ( r , ς ) = 2 2 m i ( ( Z n l r ) 2 + ( m π ς ) 2 ) ,
δ ε n l m = 2 m e 1 r 3 [ Z n l 2 + ( m π η ) 2 ] δ r .
δ ε 101 2 r 3 m e 115.4 δ r ,
δ ε 111 2 r 3 m e 62.17 δ r .
Ξ : Δ i h , c = δ ( ε 111 ε 101 ) δ ε 101 Δ i h , p κ Δ i h , p 0.077 Δ i h , p ,
V : Δ i h , c = δ ε 111 δ ε 101 Δ i h , p κ Δ i h , p 1.08 Δ i h , p ,
Λ : Δ i h , c = δ ( ε 111 ε 101 ) δ ε 111 Δ i h , p κ Δ i h , p 0.072 Δ i h , p ,
V : χ p ( Δ ̃ p , Δ ̃ c ) = Γ V μ 13 2 ε 0 2 { γ 23 [ i Γ 13 Γ 2 2 Γ 1 ( Δ p + i γ 12 ) ] + Δ c ( Γ 13 Γ 2 + 2 γ 23 Γ 1 ) } Ω c 2 2 ( δ + i γ 12 ) ζ [ 4 ( δ + i γ 12 ) ( i γ 13 + Δ ̃ p ) Ω c 2 ] [ ζ + γ 23 ( 2 Γ 13 + Γ 12 ) Ω c 2 ] ,
Ξ : χ p ( Δ ̃ p , Δ ̃ p ) = Γ V μ 13 2 ε 0 2 ( δ + + i γ 12 ) Ω c 2 4 ( δ + + i γ 12 ) ( Δ ̃ p + i γ 13 ) ,
Λ : χ p ( Δ ̃ p , Δ ̃ c ) = Γ V μ 13 2 ε 0 2 δ + i γ 12 Ω c 2 4 ( δ + i γ 12 ) ( Δ ̃ p + i γ 13 ) ,
χ ( Δ p ) = f ( Δ i h ) χ ( Δ p Δ i h , Δ c κ Δ i h ) d Δ i h ,
n g = Re ( 1 + χ b g + χ p ) + ω ( Re [ 1 + χ b g + χ p ] ) ω ,
H ̂ = ( Δ 1 0 Ω p 0 Δ 2 Ω c Ω p Ω c 0 ) ,
λ 1 = Δ 1 ,     λ ± = 1 2 ( Δ 2 ± 4 Ω c 2 + Δ 2 2 ) ,
Ξ , Λ : Δ p = Δ i h + λ ± ,     V : Δ p = Δ i h λ ± .
( [ 2 ± κ ] 2 κ 2 ) Δ i h 2 = 4 Ω c 2 ,
Ξ : κ > 1 ,     V , Λ : κ < 1.
I : Δ i h = Δ p ,     II : Δ i h = ( 1 ± κ ) 1 Δ p ,
V : Δ i h , c = δ ( ε X ± 1 2 ε FSS ) δ ( ε X 1 2 ε FSS ) Δ i h , p
= ( 1 ± δ ε FSS δ ( ε X 1 2 ε FSS ) ) Δ i h , p
( 1 ± δ ε FSS δ ε X ) Δ i h , p ,
Ξ : Δ i h , c δ ε X X δ ε X Δ i h , p
= ( 1 + δ ( ε X X ε X ) δ ε X ) Δ i h , p .
E ( z , Ω ) z = i ( Ω u g + ω 0 2 n b g c χ p ( z ) ) E ( z , Ω ) ,
E ( z , Ω ) = E ( 0 , Ω ) exp [ i z ( Ω u g + 1 z Ω + ω p 2 n b g c 0 z χ p ( z ̃ , ω ) d z ̃ ) ] .
n g ( ω ) = c u g + 1 z 1 2 n 0 0 z ( χ q d ( z ̃ , ω ) + ω χ q d ( z ̃ , ω ) ω ) d z ̃ ,
α = 1 z ω p 2 n b g c 0 z χ p ( z ̃ , ω ) d z ̃ .
Δ t = z c ( n g ( z ) n b g ) .

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