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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 12467–12472
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Slow light and voltage control of group velocity in resonantly coupled quantum wells

Pavel Ginzburg and Meir Orenstein  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 12467-12472 (2006)
http://dx.doi.org/10.1364/OE.14.012467


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Abstract

We analyze slow light propagation in a coupled semiconductor quantum wells system exhibiting tunneling induced transparency. A group index as high as 85, for simple applicable GaAs/AlGaAs quantum wells structures, is predicted. Using DC voltage, the resonant tunneling rate can be altered and the related group index can be controlled over a broad range.

© 2006 Optical Society of America

1. Introduction

The generic EIT system is doubly driven and controlled by external electromagnetic fields. The slow light control (as well as the transparency window) is accomplished by changing the amplitude of the pump electromagnetic field [2

2. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997). [CrossRef]

]. In order to realize an autonomous system that reduces the group velocity of a primary light field without a pump one, we employ here an alternative coupling mechanism to replace the pump light field. This scheme, namely tunneling induced transparency was previously proposed by Imamoglu group in relations to implementation of inversionless lasers and Fano interference [11

11. P. Palinginis, F. Sedgwick, S. Crankshaw, M. Moewe, and C. Chang-Hasnain, “Room temperature slow light in a quantum-well waveguide via coherent population oscillation,” Opt. Express 13, 9909–9915 (2005). [CrossRef] [PubMed]

]. It is based on two resonant quantum wells exhibiting a periodic Rabi oscillation of the electron due to the resonant tunneling effect – thus can serve as the ’second‘ arm in the V or Λ configurations of the EIT scheme. It should be emphasized that the tunneling scheme is not a time varying perturbation (such as the pump field) but a stationary ingredient of the quantum structure.

We propose and analyze the exploitation of the tunneling configuration for slow light generation and the application of a DC voltage to control the group velocity magnitude - a scheme that is favorable for implementation in semiconductor materials. Tunneling control by voltage is a phenomenon exploited in devices such as resonant tunneling diodes [12

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

] - but here, it is proposed for a very efficient and simple control of the group propagation velocities of slow light. Reports on slow light in solid state media (by coherent population trapping or by unique nonlinearities [13

13. H. Mizuta and T. Tanoue, The Physics and Applications of Resonant Tunnelling Diodes (Cambridge, MA: Cambridge, 1995). [CrossRef]

]) all required a second electromagnetic field and thus are substantially different from our proposed scheme.

2. Anomalous dispersion of double quantum well configuration

The tunneling induced transparency system proposed for inversionless lasers in Ref. [14

14. H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004). [CrossRef]

] is comprised of two different quantum wells – one supporting two energy states and the second a single state, resonant with the excited state of the first well. For many material systems, the well supporting the one state must be very narrow, thus is usually inapplicable in a realizable practical scheme. In our case we used two identical coupled quantum wells [Fig. 1(a)], each one is supporting two quantum states, ground |1>, |3> and exited |2>, |4>. The potential wells are designed such that under zero applied voltage – state |2> (|1>) is resonant (equal-energy) with state |4> (|3>). We exploited the much larger localization of the ground state, such that the inter-well coupling efficiency of the excited states is dominant and the ground states coupling can be ignored (the ratio of the coupling constants is about an order of magnitude). Thus the four (effectively three) levels scheme of the EIT is achieved in a simple, established structure of identical coupled wells. We denote the coupling coefficients by α′s and depict the in-plane wave functions in Fig. 1(b). We assume that the incident light polarization is in the direction of the epitaxial growth, enabling only the intraband transitions. Note that we used the intraband electronic transitions only as the most basic demonstration of the concept, while similar results in the visible – near IR regime, using interband transitions will be detailed in a forthcoming publication [15

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

].

We employ here the density matrix formalism [16

16. P. Ginzburg and M. Orenstein, “Visible -Near IR controllable slow light by interband transitions in coupled quantum well structures,” in preparation.

], for a system described by a Hamiltonian expressed by the eigen functions of the separated wells. This different approach for solving this quantum system is more transparent to the similarity between the tunneling and external field.

H0=ћ(ω10000ω20000ω30000ω4)Vpert=(0μ12E00μ12*E*0α00α*0μ43E00μ43*E*0)
(1)

where ħωi is the self energy of the i-th QW level, µij is the dipole moment of the i-j transition, E is light field and α1, α2 are the tunneling constants, depicted on Fig. 1(a) and discussed further.

Fig. 1. The coupled quantum wells configuration (a) Energy-level diagram. (b) In plane electronic wave functions

Applying the rotating-wave and electric-dipole approximations and phenomenology introducing the dephasing rates (γ′s) we obtain for the slowly varying density matrix [1

1. P. W. Milonni, Fast Light, Slow light and Left-Handed Light, (Bristol, England: Institute of Physics, c2005).

]:

(i(ωω21)γ21iα2ћ00iα1ћi(ωω41)γ410000i(ωω23)γ23iα2ћ00iα1ћi(ωω43)γ43)×
×(σ12σ14σ32σ34)=iћ(μ12E(ρ11ρ22)00μ34E(ρ33ρ44))
(2)

Solving Eq. (2) easily may be obtained, that the four level system decouples to the pair of inverse λ tunneling-EIT schemes. For the harmonic solution, we get the complex material susceptibility:

χ=χbaciN(μ122(ρ11ρ22)ћ1α1α2ћ2(i(ωω41)γ41)+i(ωω21)γ21++μ342(ρ44ρ33)ћ1α1α2ћ2(i(ωω23)γ23)+i(ωω43)γ43)
(3)

ngroup(ωs)=nr(ωs)+ωs(dnrdω)ω=ωsvgroup=cngroup
(4)

These results replicate the outcome of conventional EIT configuration. The excited states coupling to the resonant neighboring well – splits the upper level, thus opening a transmission window which is characterized by a substantially large group index.

Fig. 2. The permeability (0 bias voltage) (a) Dispersion curve (b) Group index

3. Results for GaAs–Al0.3Ga0.7As coupled wells

We explored a well known heterostructure GaAs–Al0.3Ga0.7As as a simple example to demonstrate a medium exhibiting significant light slowing. Many other variants of material families and coupled structures exhibit the same phenomena and can be optimally engineered for applicable devices. The parameters were taken from experimental data in Ref. [19

19. D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective Intersubband Excitations in Quantum Wells: Coulomb Interaction versus Subband Dispersion,” Phys. Rev. Lett. 79, 4633–4636 (1997). [CrossRef]

]: well depth - ΔEc=230meV, width 10nm, barrier width 6nm, effective masses - ma=0.067m0 and mb=0.097m0, the volume density of charge 3·1023m-3. Dephasing rates of the dipolar transitions were chosen to be γ1234=1012 rad/sec and 2·γ14=2·γ23=γ· [20

20. P. Basu, Theory of optical processes in semiconductors: bulk and microstructures, (Oxford: Clarendon Press, c1997).

]·χbac is the average between the constituents and its value is ~13 and it should be noted that it is significantly different compared to atomic EIT, where the background susceptibility is 1 - due to the differences in the media density. This results in the asymmetry of the group index curve [Fig. 2(b)], which is absent in atomic vapor EIT.

When applying a field on the system the very well known tilting of the heterostructure band diagram [19

19. D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective Intersubband Excitations in Quantum Wells: Coulomb Interaction versus Subband Dispersion,” Phys. Rev. Lett. 79, 4633–4636 (1997). [CrossRef]

] occurs. While explicit analytical solutions for infinite triangular well are the Airy functions [19

19. D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective Intersubband Excitations in Quantum Wells: Coulomb Interaction versus Subband Dispersion,” Phys. Rev. Lett. 79, 4633–4636 (1997). [CrossRef]

]; the tilted finite quantum well wave function solutions are much more complex. We treated the applied DC field in terms of a first order perturbation theory to find corrections for energy levels and electron wave functions. The validity of the perturbative approach is manifested in Fig. 3 - exhibiting the results of numerical calculations of the specific system under study. Both the small energy shift and especially the very small modification of the wavefunctions validate the first order perturbation calculation.

Figures 4–5 depict the real and imaginary parts of permeability for two levels of a DC applied field, along with the derived group velocity index. The applied electric field tilts the energy levels of the structure such that |2> and |4> move out of resonance reducing the coupling efficiency. For high voltage – the two wells are tilted enough such that virtually no coupling occurs – regressing to isolated two-level absorption in the quantum wells.

Fig. 3. Applying 106 eV DC field (a) Energy-level diagram tilting (dashed line: original energy levels) (b) In plane electronic wave functions tilting (dashed line: original wavefunctions)
Fig. 4. Applying 104 eV DC field (a) Dispersion curve (b) Group index
Fig. 5. Applying 105 eV DC field (a) Dispersion curve (b) Group index

4. Electrostatically controllable slowing down factor

Fig. 6. Group index as a function of applied DC field (a) Zoomed at positive group indices (b) Extended voltage region

5. Conclusion

We proposed a realizable semiconductor based scheme for slowing and controlling the light group velocity – the latter is achieved by an applied electrostatic field. Instead of the cumbersome employment of external light control for slow light generation, the resonant tunneling induced transparency was used. The structure is easily controllable (tunable), because employment of external voltage tilts the wells potential - reducing their coupling strength and modifies the group velocity. About two orders of magnitude in the slow down factor are achievable and controlled. In addition, the large dephasing in solids flattens the group velocity frequency dependence [Fig. 2(b)] permitting many GHz bandwidth of information modulation.

References and links

1.

P. W. Milonni, Fast Light, Slow light and Left-Handed Light, (Bristol, England: Institute of Physics, c2005).

2.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997). [CrossRef]

3.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005). [CrossRef] [PubMed]

4.

S. Sarkar, Y. Guo, and H. Wang, “Tunable optical delay via carrier induced exciton dephasing in semiconductor quantum wells,” Opt. Express 14, 2845–2850 (2006). [CrossRef] [PubMed]

5.

J. E. Heebner, R. W. Boyd, and Q-Han Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002). [CrossRef]

6.

H. Altuga and J. Vuèkoviæb “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111102 (2005). [CrossRef]

7.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxatoin,” Phys. Rev. Lett. 83, 1767 (1999). [CrossRef]

8.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

9.

P. -C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. -W. Chang, and S. -L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. 29, 2291–2293 (2004). [CrossRef] [PubMed]

10.

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]

11.

P. Palinginis, F. Sedgwick, S. Crankshaw, M. Moewe, and C. Chang-Hasnain, “Room temperature slow light in a quantum-well waveguide via coherent population oscillation,” Opt. Express 13, 9909–9915 (2005). [CrossRef] [PubMed]

12.

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

13.

H. Mizuta and T. Tanoue, The Physics and Applications of Resonant Tunnelling Diodes (Cambridge, MA: Cambridge, 1995). [CrossRef]

14.

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004). [CrossRef]

15.

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

16.

P. Ginzburg and M. Orenstein, “Visible -Near IR controllable slow light by interband transitions in coupled quantum well structures,” in preparation.

17.

L. Allen and J. H. Eberly, Optical resonance and two-level atoms, (New York, Wiley-Interscience, c1975).

18.

A. M. Fox, D. A. B. Miller, G. Livescu, J. E. Cunningham, and W. Y. Jan, “Excitonic effects in coupled quantum wells,” Phys. Rev. B 44, 6231–6242 (1991). [CrossRef]

19.

D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective Intersubband Excitations in Quantum Wells: Coulomb Interaction versus Subband Dispersion,” Phys. Rev. Lett. 79, 4633–4636 (1997). [CrossRef]

20.

P. Basu, Theory of optical processes in semiconductors: bulk and microstructures, (Oxford: Clarendon Press, c1997).

21.

J. Li and C. Ning, “Collective excitations in InAs quantum well intersubband transitions,” Physica E 22, 628–631 (2004). [CrossRef]

OCIS Codes
(230.1150) Optical devices : All-optical devices
(270.1670) Quantum optics : Coherent optical effects

ToC Category:
Quantum Optics

History
Original Manuscript: October 16, 2006
Revised Manuscript: November 30, 2006
Manuscript Accepted: December 1, 2006
Published: December 11, 2006

Citation
Pavel Ginzburg and Meir Orenstein, "Slow light and voltage control of group velocity in resonantly coupled quantum wells," Opt. Express 14, 12467-12472 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12467


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References

  1. P. W. Milonni, Fast Light, Slow light and Left-Handed Light, (Bristol, England: Institute of Physics, c2005).
  2. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997). [CrossRef]
  3. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005). [CrossRef] [PubMed]
  4. S. Sarkar, Y. Guo, and H. Wang, "Tunable optical delay via carrier induced exciton dephasing in semiconductor quantum wells," Opt. Express 14, 2845-2850 (2006). [CrossRef] [PubMed]
  5. J. E. Heebner, R. W. Boyd, and Q-Han Park, "Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide," Phys. Rev. E 65, 036619 (2002). [CrossRef]
  6. H. Altuga and J. Vuèkoviæb "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005). [CrossRef]
  7. D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, "Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxatoin," Phys. Rev. Lett. 83, 1767 (1999). [CrossRef]
  8. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, P. R. Hemmer, "Observation of ultraslow and stored light pulses in a solid," Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]
  9. P. -C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. -W. Chang, and S. -L. Chuang, "Slow light in semiconductor quantum wells," Opt. Lett. 29, 2291-2293 (2004). [CrossRef] [PubMed]
  10. J Kim, S L Chuang, P. C. Ku, C J Chang-Hasnain, "Slow light using semiconductor quantum dots," J. Phys. Condens. Matter 16, S3727 - S3735 (2004). [CrossRef]
  11. P. Palinginis, F. Sedgwick, S. Crankshaw, M. Moewe, and C. Chang-Hasnain, "Room temperature slow light in a quantum-well waveguide via coherent population oscillation," Opt. Express 13, 9909-9915 (2005). [CrossRef] [PubMed]
  12. H. Schmidt and A. Imamoglu, "Nonlinear optical devices based on a transparency in semiconductor intersubband transitions," Opt. Commun. 131, 333 (1996). [CrossRef]
  13. H. Mizuta and T. Tanoue, The Physics and Applications of Resonant Tunnelling Diodes (Cambridge, MA: Cambridge, 1995). [CrossRef]
  14. H. Kang, G. Hernandez, and Y. Zhu, "Resonant four-wave mixing with slow light," Phys. Rev. A 70, 061804(R) (2004). [CrossRef]
  15. H. Schmidt, K. L. Campman, A. C. Gossard, and A. Imamoglu, "Tunneling induced transparency: Fano interference in intersubband transitions," Appl. Phys. Lett. 70, 3455-3457 (1997). [CrossRef]
  16. P. Ginzburg and M. Orenstein, "Visible -Near IR controllable slow light by interband transitions in coupled quantum well structures," in preparation.
  17. L. Allen and J. H. Eberly, Optical resonance and two-level atoms, (New York, Wiley-Interscience, c1975).
  18. A. M. Fox, D. A. B. Miller, G. Livescu, J. E. Cunningham, and W. Y. Jan, "Excitonic effects in coupled quantum wells," Phys. Rev. B 44, 6231-6242 (1991). [CrossRef]
  19. D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, "Collective Intersubband Excitations in Quantum Wells: Coulomb Interaction versus Subband Dispersion," Phys. Rev. Lett. 79, 4633-4636 (1997). [CrossRef]
  20. P. Basu, Theory of optical processes in semiconductors: bulk and microstructures, (Oxford: Clarendon Press, c1997).
  21. J. Li, and C. Ning, "Collective excitations in InAs quantum well intersubband transitions," Physica E 22, 628-631 (2004). [CrossRef]

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