## Strongly interacting photons in asymmetric quantum well via resonant tunneling |

Optics Express, Vol. 20, Issue 8, pp. 8485-8495 (2012)

http://dx.doi.org/10.1364/OE.20.008485

Acrobat PDF (1362 KB)

### Abstract

We propose an asymmetric quantum well structure to realize strong interaction between two slow optical pulses. The essential idea is the combination of the advantages of inverted-Y type scheme and resonant tunneling. We analytically demonstrate that giant cross-Kerr nonlinearity can be achieved with vanishing absorptions. Owing to resonant tunneling, the contributions of the probe and signal cross-Kerr nonlinearities to total nonlinear phase shift vary from destructive to constrictive, leading to nonlinear phase shift on order of *π* at low light level. In this structure, the scheme is inherent symmetric for the probe and signal pulses. Consequently, the condition of group velocity matching can be fulfilled with appropriate initial electron distribution.

© 2012 OSA

## 1. Introduction

2. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. **79**, 135–174 (2007). [CrossRef]

*Q*cavity. Hence the major obstacle to construct scalable and efficient quantum computation with photonic qubits is the absence of giant cross-Kerr nonlinearity capable of entangling pairs of photons. A promising avenue has been opened by studies of enhanced nonlinear coupling via electromagnetically induced transparency (EIT) [3

3. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. **82**, 4611–4614 (1999). [CrossRef]

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

*N*-type scheme, it was proposed that the ultrahigh sensitivity of EIT dispersion to the two-photon Raman detuning in the vicinity of an absorption minimum can be used to enhance cross-Kerr nonlinearity between two weak optical fields [4

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

6. A. André, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. **94**, 063902 (2005). [CrossRef] [PubMed]

*π*phase shift and high-fidelity implementation of the controlled phase gate has been suggested by trapping the weak signal pulse in a photonic band gap [7

7. I. Friedler, G. Kurizki, and D. Petrosyan, “Giant nonlinearity and entanglement of single photons in photonic bandgap structures,” Europhys. Lett. **68**, 625–631 (2004). [CrossRef]

9. D. Petrosyan and G. Kurizki, “Photon-photon correlations and entanglement in doped photonic crystals,” Phys. Rev. A **64**, 023810 (2001). [CrossRef]

10. D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A **65**, 033833 (2002). [CrossRef]

12. C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Cross phase modulation in a fiveClevel atomic medium: semiclassical theory,” Eur. Phys. J. D **40**, 281–296 (2006). [CrossRef]

13. S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A **70**, 032317 (2004). [CrossRef]

17. Y. Guo, S. S. Li, and L. M. Kuang, “Large cross-phase shifts among three slow weak pulses via triple electromagnetically induced transparency,” J. Phys. B **44**, 065501 (2011). [CrossRef]

18. A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A **72**, 062319 (2005). [CrossRef]

19. Y. F. Bai, W. X. Yang, and X. Q. Yu, “Controllable Kerr nonlinearity with vanishing absorption in a four-level inverted-Y atomic system,” Opt. Commun. **283**, 5062–5066 (2010). [CrossRef]

11. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. **90**, 197902 (2003). [CrossRef] [PubMed]

12. C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Cross phase modulation in a fiveClevel atomic medium: semiclassical theory,” Eur. Phys. J. D **40**, 281–296 (2006). [CrossRef]

20. B.-W. Shiau, M.-C. Wu, C.-C. Lin, and Y.-C. Chen, “Low-light-level cross-phase modulation with double slow light pulses,” Phys. Rev. Lett. **106**, 193006 (2011). [CrossRef] [PubMed]

21. Y. Li, C. Hang, L. Ma, and G. X. Huang, “Controllable entanglement of lights in a five-level system,” Phys. Lett. A **354**, 1–7 (2006). [CrossRef]

22. B. P. Hou, L. F. Wei, G. L. Long, and S. J. Wang, “Large cross-phase-modulation between two slow pulses by coupled double dark resonances,” Phys. Rev. A **79**, 033813 (2009). [CrossRef]

*π*for two single-photon pulses is also attainable [23

23. I. Friedler, D. Petrosyan, M. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A **72**, 043803 (2005). [CrossRef]

24. E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A **83**, 033806 (2011). [CrossRef]

25. L. Deng and M. G. Payne, “Gain-assisted large and rapidly responding Kerr effect using a room-temperature active Raman gain medium,” Phys. Rev. Lett. **98**, 253902 (2007). [CrossRef] [PubMed]

27. C. Hang and G. X. Huang, “Highly entangled photons and rapidly responding polarization qubit phase gates in a room-temperature active Raman gain medium,” Phys. Rev. A **82**, 053818 (2010). [CrossRef]

28. B. He, A. MacRae, Y. Han, A. I. Lvovsky, and C. Simon, “Transverse multimode effects on the performance of photon-photon gates,” Phys. Rev. A **83**, 022312 (2011). [CrossRef]

29. B. He, Y. Han, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A **83**, 053826 (2011). [CrossRef]

48. H. Sun, Y. P. Niu, R. X. Li, S. Q. Jin, and S. Q. Gong, “Tunneling-induced large cross-phase modulation in an asymmetric quantum well,” Opt. Lett. **32**, 2475–2477 (2007). [CrossRef] [PubMed]

48. H. Sun, Y. P. Niu, R. X. Li, S. Q. Jin, and S. Q. Gong, “Tunneling-induced large cross-phase modulation in an asymmetric quantum well,” Opt. Lett. **32**, 2475–2477 (2007). [CrossRef] [PubMed]

*π*can be realized at low light level. For the probe and signal pulses, the structure is an inherent symmetric configuration. Hence the condition of group velocity matching can be easily satisfied by adjusting the initial electron distribution.

## 2. Structure and linear optical properties

_{0.50}Ga

_{0.50}As barrier is followed by an Al

_{0.10}Ga

_{0.90}As layer with thickness of 8.8 nm (shallow well). This shallow well is separated from a 6.9 nm GaAs layer (deep well) on the right by a 3.8 nm Al

_{0.50}Ga

_{0.50}As potential barrier. Finally, a thin (2.4 nm) Al

_{0.50}Ga

_{0.50}As barrier separates the deep well from the last Al

_{0.40}Ga

_{0.60}As thick layer on the right. In this structure, one would observe the ground subbands of the right deep well |1〉 and the left shallow well |2〉 with energies 57.2 meV and 123.1 meV, respectively. The eigenenergy of the second excited subband of the left shallow well |5〉 is 385.9 meV. Two new subbands |3〉 and |4〉 with eigenenergies 224.1 meV and 231.4 meV are, respectively, created by mixing the first excited subbands of the shallow (|

*se*〉) and deep (|

*de*〉) wells by tunneling. Their corresponding wave functions are symmetric and antisymmetric combinations of |

*se*〉 and |

*de*〉, i.e.,

*ω*and a weak pulsed signal field with frequency

_{p}*ω*to drive the transitions |1〉 ↔ |3〉, |1〉 ↔ |4〉 and |2〉 ↔ |3〉, |2〉 ↔ 4〉, respectively. The subbands |3〉 and |4〉 are coupled with |5〉 by a continuous-wave control field with angular frequency

_{s}*ω*. Thus, an inverted-Y-type configuration with two-fold degenerate middle subbands is realized. Under the dipole and rotating-wave approximations (RWA), this structure is governed by a set of density matrix equations given below, where

_{c}*d*

_{21}= Δ

*– Δ*

_{p}*+*

_{s}*iγ*

_{21},

*d*

_{31}= Δ

*+*

_{p}*iγ*

_{31},

*d*

_{41}= Δ

*–*

_{p}*δ*+

*iγ*

_{41},

*d*

_{51}= Δ

*+ Δ*

_{p}*– Δ*

_{c}*+*

_{s}*iγ*

_{51},

*d*

_{32}= Δ

*+*

_{s}*iγ*

_{32},

*d*

_{42}= Δ

*–*

_{s}*δ*+

*iγ*

_{42},

*d*

_{52}= Δ

*+*

_{c}*iγ*

_{52}with Δ

*, Δ*

_{p}*, and Δ*

_{s}*being the detunings of the probe, signal and control fields with the corresponding transitions, and they are defined as Δ*

_{c}

_{p,}_{(}

_{s,c}_{)}=

*ω*

_{p,}_{(}

_{s,c}_{)}– (

*ω*

_{3,(3,5)}–

*ω*

_{1,(2,3)}).

*δ*=

*ω*

_{4}–

*ω*

_{3}≃ 7.3 meV denotes the energy difference between the subbands |3〉 and |4〉. Halves of the Rabi frequencies of the probe, signal and control fields are Ω

*=*

_{p}*μ*⃗

_{13}·

*E*⃗

*/2*

_{p}*h̄*, Ω

*=*

_{s}*μ*⃗

_{23}·

*E⃗*/2

_{s}*h̄*, and Ω

*=*

_{c}*μ*⃗

_{53}·

*E⃗*/2

_{c}*h̄*with

*μ*⃗

*being electric dipole momentum between subbands |*

_{ij}*i*〉 and |

*j*〉 (

*i*,

*j*= 1 – 5 and

*i*≠

*j*), while

*m*=

*μ*

_{41}/

*μ*

_{31}= −0.73,

*q*=

*μ*

_{42}/

*μ*

_{32}= 1.2, and

*k*=

*μ*

_{54}/

*μ*

_{53}= 2.3 give the ratios between the relevant subband transition dipole momentum.

*E*,

_{p}*E*, and

_{s}*E*are, respectively, the slowly varying electric field amplitudes of the probe, signal and control fields. The half linewidths are, respectively, given by

_{c}*γ*

_{3}(

*γ*

_{4},

*γ*

_{5}) is the electron decay rate of subband |3〉, (|4〉, |5〉) and

48. H. Sun, Y. P. Niu, R. X. Li, S. Q. Jin, and S. Q. Gong, “Tunneling-induced large cross-phase modulation in an asymmetric quantum well,” Opt. Lett. **32**, 2475–2477 (2007). [CrossRef] [PubMed]

49. 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**, R9034–R9037 (1986). [CrossRef]

^{12}cm

^{−2},

31. 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). [CrossRef]

44. H. Sun, X. L. Feng, S. Q. Gong, and C. H. Oh, “Giant cross-Kerr nonlinearity in carbon nanotube quantum dots with spin-orbit coupling,” Phys. Rev. B **79**, 193404 (2009). [CrossRef]

26. C. Hang and G. X. Huang, “Giant Kerr nonlinearity and weak-light superluminal optical solitons in a four-state atomic system with gain doublet,” Opt. Express **18**, 2952–2966 (2010). [CrossRef] [PubMed]

*|, |Ω*

_{p}*| ≪ |Ω*

_{s}*|, Δ*

_{c}_{1}, Δ

_{2}, Δ

_{3},

*δ*and solve the density matrix equations [Eqs. (1)–(7)] in the non-depletion approximation (

*σ*

_{11}+

*σ*

_{22}≈ 1) together with Maxwell’s equations and expand the linear dispersion relations as Taylor series around their center frequency. The group velocities of the probe and signal pulses are, respectively, given by with

*N*being the electron volume density,

**32**, 2475–2477 (2007). [CrossRef] [PubMed]

*σ*

_{31}and

*σ*

_{41}(

*σ*

_{32}and

*σ*

_{42}). The initial electron distribution governs the evolutions of these coherent terms, and hence the steady state solutions [details can be seen in Eqs. (11)–(15)]. The electron decay rates are

*γ*

_{3}≈

*γ*

_{4}= 0.5 meV,

*γ*

_{5}= 0.2 meV (corresponding intrasubband relaxation time

*T*

_{1}∼ 10 ps) [50

50. J. B. Williams, M. S. Sherwin, K. D. Maranowski, and A. C. Gossard, “Dissipation of intersubband plasmons in wide quantum wells,” Phys. Rev. Lett. **87**, 037401 (2001). [CrossRef] [PubMed]

31. 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). [CrossRef]

*= 1.5 meV and Δ*

_{c}*= −5.3 meV. With Δ*

_{c}*= Δ*

_{p}*= 3.0 meV (around the center of their transparency windows) and*

_{s}*N*= 5 × 10

^{−17}cm

^{−3}, Fig. 2 illustrates the dependence of the group velocities of the probe and signal pulses on the initial electron distribution

51. S. Q. Jin, S. Q. Gong, R. X. Li, and Z. Z. Xu, “Coherent population transfer and superposition of atomic states via stimulated Raman adiabatic passage using an excited-doublet four-level atom,” Phys. Rev. A **69**, 023408 (2004). [CrossRef]

12. C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Cross phase modulation in a fiveClevel atomic medium: semiclassical theory,” Eur. Phys. J. D **40**, 281–296 (2006). [CrossRef]

*m*↔

*q*, the expression of

*= 3.0 meV in Fig. 3(a) and Δ*

_{s}*= 2.995 meV in Fig. 3(b). The other parameters are the same as those in Fig. 2. With this set of parameters, the dispersion of the probe and signal pulses around the center of the transparent window are linearly proportional to their detunings. This means that the probe and signal pulses will propagate with comparable and small group velocities, and the influence of group velocity dispersion can be neglected within the region considered. At the center of the transparency window, the linear absorptions of the probe and signal fields are very small because of the destructive interference between transition pathes, and can be safely ignored.*

_{p}## 3. Strongly interacting photons with resonant tunneling

*T*(

_{αβ}*α*=

*p,s*,

*β*= 1,3,4,5). In the QW structure under consideration, the symmetric and asymmetric wave functions of subbands |3〉 and |4〉 lead to

*m*≠

*q*≠

*k*, which indicates that the resonant tunneling can modify the optical nonlinearity such as cross-Kerr effect. In Figs. 4(a) and 4(b), we illustrate the evolutions of

*γ*

_{2}is dominantly determined by the electron dephasing rate of a long-lived ground-subband coherence, smaller

*γ*

_{2}can be attained by decreasing the temperature. Hence, the present QW structure yields the strength of cross-Kerr nonlinearities that are two orders of magnitude higher than that in Ref. [48

**32**, 2475–2477 (2007). [CrossRef] [PubMed]

*and Δ*

_{p}*, which can be seen from Eqs. (19) and (20). For certain detunings, for example Δ*

_{s}*= 2.9995 meV and Δ*

_{p}*= 3.0 meV, we have*

_{s}*m*=

*q*=

*k*= 0. In this case, the subband |4〉 is decoupled, and the system can hence be described as an inverted-Y-type configuration. In order to see the role of resonant tunneling more clearly, we choose Δ

*= 0 and Ω*

_{c}*≈ 2.65 meV (similar transparency windows as those with tunneling). The other parameters are the same as those in Figs. 4(a) and 4(b). Within the transparency windows, the enhancement of cross-Kerr nonlinearities can still be achieved, while*

_{c}*= 0.5*

_{p}*μ*eV and Δ

*= 0 meV, we have*

_{s}*π*nonlinear phase shift at low light level. For Gaussian probe and signal pulses of time durations

*τ*, and with peak Rabi frequencies

_{p,s}**40**, 281–296 (2006). [CrossRef]

*ζ*can be obtained from

_{s}*ζ*upon interchanging

_{p}*p*↔

*s. l*is the length of the QW structure, and erf(

*ζ*) represents the error function.

11. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. **90**, 197902 (2003). [CrossRef] [PubMed]

*l*= 1.0 mm. With resonant tunneling,

*= Ω*

_{p}*= Ω ≈ 6.22 × 10*

_{s}^{−4}meV. The probe and signal pulses can have a mean amplitude of about one photon when these beams are focused or propagate in a tightly confined waveguide. With these parameters, the corresponding intensities of the probe and signal pulses are, respectively, given by

*I*≈ 4.35 mW cm

_{p}^{−2}and

*I*≈ 6.94 mW cm

_{s}^{−2}. We remark that the intensities of a single probe and signal photons per 1.0 ns on the area of 1

*μ*m

^{2}are

*I*≈ 2.73 mW cm

_{p}^{−2}and

*I*≈ 2.25 mW cm

_{s}^{−2}, respectively. The numerical results indicate that our semiconductor QW structure can indeed realize a

*π*-nonlinear phase shift at low light level. In Fig. 5(a), we also illustrate the positive effect of the control field on

*= Ω*

_{p}*= Ω ≈ 6.5 × 10*

_{s}^{−3}meV (

*I*≈ 0.48 W cm

_{p}^{−2}and

*I*≈ 0.76 W cm

_{s}^{−2}), i.e., more than one hundred of photons.

## 4. Conclusion

46. 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). [CrossRef] [PubMed]

**32**, 2475–2477 (2007). [CrossRef] [PubMed]

46. 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). [CrossRef] [PubMed]

47. X. Y. Hao, L. G. Si, C. L. Ding, P. Huang, J. H. Li, and X. X. Yang, “Polarization qubit phase gate between two far-infrared pulses in three-coupled quantum wells,” J. Opt. Soc. Am. B **27**, 1792–1798 (2010). [CrossRef]

*π*at low light level. Due to the symmetry of the scheme with respect to the probe and signal pulses, their group velocities could be equal by modulating the initial distribution of electron. We believe that the present study is useful for guiding experimental realization of electroptically modulated devices and facilitating more practical applications in solid quantum information processing.

## Acknowledgments

## References and links

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45. | J. J. Li, W. He, and K. D. Zhu, “All-optical Kerr modulator based on a carbon nanotube resonator,” Phys. Rev. B |

46. | W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express |

47. | X. Y. Hao, L. G. Si, C. L. Ding, P. Huang, J. H. Li, and X. X. Yang, “Polarization qubit phase gate between two far-infrared pulses in three-coupled quantum wells,” J. Opt. Soc. Am. B |

48. | H. Sun, Y. P. Niu, R. X. Li, S. Q. Jin, and S. Q. Gong, “Tunneling-induced large cross-phase modulation in an asymmetric quantum well,” Opt. Lett. |

49. | 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 |

50. | J. B. Williams, M. S. Sherwin, K. D. Maranowski, and A. C. Gossard, “Dissipation of intersubband plasmons in wide quantum wells,” Phys. Rev. Lett. |

51. | S. Q. Jin, S. Q. Gong, R. X. Li, and Z. Z. Xu, “Coherent population transfer and superposition of atomic states via stimulated Raman adiabatic passage using an excited-doublet four-level atom,” Phys. Rev. A |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 7, 2012

Revised Manuscript: February 27, 2012

Manuscript Accepted: February 27, 2012

Published: March 27, 2012

**Citation**

H. Sun, S. L. Fan, X. L. Feng, C. F. Wu, S. Q. Gong, G. X. Huang, and C. H. Oh, "Strongly interacting photons in asymmetric quantum well via resonant tunneling," Opt. Express **20**, 8485-8495 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8485

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