## Giant Kerr Nonlinearity, Controlled Entangled Photons and Polarization Phase Gates in Coupled Quantum-Well Structures |

Optics Express, Vol. 19, Issue 23, pp. 23364-23376 (2011)

http://dx.doi.org/10.1364/OE.19.023364

Acrobat PDF (2132 KB)

### Abstract

We study linear and nonlinear propagations of probe and signal pulses in a multiple quantum-well structure with a four-level, double Λ-type configuration. We show that slow, mutually matched group velocities and giant Kerr nonlinearity of the probe and the signal pulses may be achieved with nearly vanishing optical absorption. Based on these properties we demonstrate that two-qubit quantum polarization phase gates can be constructed and highly entangled photon pairs may be produced. In addition, we show that coupled slow-light soliton pairs with very low generation power can be realized in the system.

© 2011 OSA

## 1. Introduction

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

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

4. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. **93**, 143904 (2004). [CrossRef] [PubMed]

5. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” *Phys. Rev. E*72, 016617 (2005). [CrossRef]

6. M. D. Lukin and A. Imamoǧlu, “Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons,” Phys. Rev. Lett. **84**, 1419 (2000). [CrossRef] [PubMed]

7. 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]

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

*N*-type scheme can be overcame. Our work includes: (i) By a calculation based on density-matrix formulation, we show that the absorption of both the probe and signal fields can be largely suppressed due to the TIT effect in the quantum well, and the group velocities of the both fields in the double Λ-type scheme may be reduced significantly, and can be mutually matched very well in a fairly large region of system parameters; (ii) We calculate the Kerr coefficients of the both fields, and show that giant cross-Kerr nonlinearity can be achieved in the system; (iii) Based on the slow, matched group velocities and the greatly enhanced cross-Kerr nonlinearity, which allow long-time interaction and effective entanglement between the probe and signal photons, we show that two-qubit QPGs can be constructed and highly entangled photon pairs may be produced easily. (iv) We study the nonlinear propagation of the probe and signal pulses, and demonstrate that a stable propagation of coupled slow-light soliton pairs with very low generation power are possible in the system. The results presented in this work are helpful for the application of optical and quantum information processing and transmission based on solid systems.

## 2. The MODEL

_{0.2}Ga

_{0.8}As barriers. (Fig. 1). In this structure, the first electron level in conduction band of the wide well can be energetically aligned with the first electron level of the narrow well through applying a static electric field with a given polarity, whereas the first hole levels in the valence bands of both the wide and narrow wells are never aligned for this polarity of the static electric field (for detail, see Ref. [26

26. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, and D. A.B. Miller, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. **68**, 2216 (1992). [CrossRef] [PubMed]

17. H. Sun, S. Gong, Y. Niu, R. Li, S. Jin, and Z. Xu, “Enhancing Kerr nonlinearity in an asymmetric double quantum well via Fano interference,” Phys. Rev. B **74**, 155314 (2006). [CrossRef]

18. 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 (2005). [CrossRef] [PubMed]

19. Y. Xue, X.-M. Su, G. W. Wang, Y. Chen, and J.-Y. Gao, “Photon switch in quantum well by quantum interference in interband transitions,” Opt. Commun. **249**, 231–237 (2005). [CrossRef]

21. C. Zhu and G. Huang, “Slow-light solitons in coupled asymmetric quantum wells via interband transitions,” Phys. Rev. B **80**, 235408 (2009). [CrossRef]

26. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, and D. A.B. Miller, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. **68**, 2216 (1992). [CrossRef] [PubMed]

27. 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 (2004). [CrossRef]

28. We adopt phenomenological few-level model to study the optical response of SQWs. On the relation between such approach and microscopic theory, see N. H. Kwong, I. Rumyantsev, R. Binder, and A. L. Smirl, “Relation between phenomenological few-level models and microscopic theories of the nonlinear optical response of semiconductor quantum wells,” Phys. Rev. B **72**, 235312 (2005). [CrossRef]

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

*ω*, wavenumber

_{p}*k*=

_{p}*ω*/

_{p}*c*, polarization vector

**e**

*, and amplitude*

_{p}*E*) propagates in

_{p}*z*-direction and interacts with the double quantum-well and induces the transition |1〉 ↔ |3〉 and |1〉 ↔ |4〉 with the respective half Rabi frequencies (

*μ*

_{31}·

**e**

*)*

_{p}*E*/

_{p}*h̄*≡ Ω

*and (*

_{p}*μ*

_{41}·

**e**

*)*

_{p}*E*/

_{p}*h̄*=

*f*

_{1}Ω

*, where*

_{p}*f*

_{1}= (

*μ*

_{41}·

**e**

*)/(*

_{p}*μ*

_{31}·

**e**

*),*

_{p}*μ*

_{31}and

*μ*

_{41}are corresponding interband electric dipole moments. At the same time a weak signal optical field (with angular frequency

*ω*, wavenumber

_{s}*k*=

_{s}*ω*/

_{s}*c*, polarization vector

**e**

*, and amplitude*

_{s}*E*) propagates also in

_{s}*z*-direction and interacts with the double quantum-well and induces the transition |2〉 ↔ |3〉 and |2〉 ↔ |4〉 with the respective half Rabi frequencies (

*μ*

_{32}·

**e**

*)*

_{s}*E*/

_{s}*h̄*≡ Ω

*and (*

_{s}*μ*

_{42}·

**e**

*)*

_{s}*E*/

_{s}*h̄*=

*f*

_{2}Ω

*, where*

_{s}*f*

_{2}= (

*μ*

_{42}·

**e**

*)/(*

_{s}*μ*

_{32}·

**e**

*),*

_{s}*μ*

_{32}and

*μ*

_{42}are corresponding interband electric dipole moments. Detunings are defined by

*ω*+ Δ

_{p}_{1}= Δ

_{3}+ (

*E*

_{3}–

*E*

_{1})/

*h̄*,

*ω*+ Δ

_{p}_{4}+ Δ

_{1}= (

*E*

_{4}–

*E*

_{1})/

*h̄*,

*ω*+ Δ

_{s}_{2}= Δ

_{3}+ (

*E*

_{3}–

*E*

_{2})/

*h̄*, and

*ω*+ Δ

_{s}_{4}+ Δ

_{2}= (

*E*

_{4}–

*E*

_{2})/

*h̄*.

*E*

_{4}–

*E*

_{3})/

*h̄*, we can express the detunings as Δ

_{3}= Δ/2 +

*δ*, Δ

_{4}=

*δ*– Δ/2 with

*δ*=

*ω*– (

_{p}*E*

_{3}+

*E*

_{4}– 2

*E*

_{1})/(2

*h̄*), here

*E*is the eigen energy of the state |

_{j}*j*〉 (

*j*= 1 – 4).

*σ*in interaction picture read where dot above

*σ*denotes time derivative,

_{jl}*d*≡ Δ

_{jl}*– Δ*

_{j}*+*

_{l}*iγ*(

_{jl}*j*,

*l*= 1 – 4),

*being the rates at which population decays from the state |*

_{jl}*l*〉 to the state |

*j*〉, and

*σ*(

_{jl}*j*≠

*l*), which may originate not only from electron-electron scattering and electron-phonon scattering, but also from inhomogeneous broadening due to scattering on interface roughness. The Fano-type quantum interference is reflected by the terms proportional to

*U*≡

*iκ*with

^{2}

**E**– (1/

*c*

^{2})

*∂*

^{2}

**E**/

*∂t*

^{2}=(1/

*ɛ*

_{0}

*c*

^{2})

*∂*

^{2}

**P**/

*∂t*

^{2}, where

**P**=

**P**

*+*

_{p}**P**

*with*

_{s}**P**

*≡*

_{p}*N*{

*μ*

_{31}

*σ*

_{13}exp[

*i*(

*k*–

_{p}z*ω*)] +

_{p}t*μ*

_{41}

*σ*

_{14}exp[

*i*(

*k*–

_{p}z*ω*)]} and

_{p}t**P**

*≡*

_{s}*N*{

*μ*

_{32}

*σ*

_{23}exp[

*i*(

*k*–

_{s}z*ω*)] +

_{s}t*μ*

_{42}

*σ*

_{24}exp[

*i*(

*k*–

_{s}z*ω*)] + c.c.}. Here

_{s}t*N*is carrier density in the conduction band. For simplicity, we assume both the probe and signal fields are homogeneous in the transverse (i.e.

*x*and

*y*) directions. Then under the slowly-varying envelope approximation, the Maxwell equation reduces into with

*B*

_{1}=

*Nω*|

_{p}*μ*

_{31}|

^{2}/(2

*h̄ɛ*

_{0}

*c*) and

*B*

_{2}=

*Nω*|

_{s}*μ*

_{32}|

^{2}/(2

*h̄ɛ*

_{0}

*c*). Here

*c*is the light speed in vacuum.

## 3. Linear and nonlinear optical susceptibilities

### 3.1. Suppression of absorption and slowdown and matching of group velocities

*σ*

_{11}and

*σ*

_{22}are constants satisfying

*σ*

_{11}+

*σ*

_{22}≈ 1. Taking

*σ*

_{31},

*σ*

_{41}and Ω

*to be proportional to exp[*

_{p}*i*(

*K*–

_{p}z*ωt*)] and

*σ*

_{32},

*σ*

_{42}and Ω

*to be proportional to exp[*

_{s}*i*(

*K*–

_{s}z*ωt*)], by using the Maxwell-Bloch Eqs. (2) and (1) we obtain

*K*(

_{p}*ω*) (“

*p*” denotes the probe field) and

*K*(

_{s}*ω*) (“

*s*” denotes the signal field) can be Taylor expanded around

*ω*= 0, which corresponds to the center frequency of both the probe and the signal field [29], i.e.

*ϕ*and

^{p,s}*β*describing respectively the phase shift per unit length and the absorption,

^{p,s}_{1}= Δ

_{2}, the curves for both the probe and signal fields will almost coincide with each other. Such similarity of the two linear dispersion relations

*K*(

_{p}*ω*) and

*K*(

_{s}*ω*) is due to the symmetric configuration between the probe and signal fields we have chosen in the system.

*ω*for Δ = 1.2 × 10

^{12}s

^{−1}. Other parameters are the same those given above. If taking Δ

_{1}= Δ

_{2}the group velocity curves for the probe and signal fields nearly coincide. In this case, we can obtain the greatly reduced, mutually matched group velocities for both the probe and signal fields at

*ω*= 0 (where absorption is minimum) [32] From the figure, we see that, different from Ref. [22

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

### 3.2. Giant Kerr nonlinearity

**P**

*(for the probe field) and*

_{p}**P**

*(for the signal field) can be obtained analytically. In order to obtain the self- and cross-Kerr nonlinear coefficients, we assume both the probe and the signal fields are weak, so that a Taylor expansion with respect to*

_{s}*E*and

_{p}*E*can be made. Keeping to the second-order approximation, one obtains expressions of the optical susceptibility

_{s}*χ*(for the probe field) and

_{p}*χ*(for the signal field) where

_{s}*α*=

*p*) and the signal field (for

*α*=

*s*) respectively which read with

*κ*=

*N*|

*μ*

_{31}|

^{2}|

*μ*

_{32}|

^{2}/(

*ɛ*

_{0}

*h̄*

^{3}),

*D*

_{1}=

*U*

^{2}–

*d*

_{31}

*d*

_{41}and

*D*

_{2}=

*U*

^{2}−

*d*

_{32}

*d*

_{42}.

*E*and the signal field

_{p}*E*as functions of the detuning

_{s}*δ*(defined above Eq. (1)). We have chosen Δ

_{1}= Δ

_{2}= 1.5 × 10

^{10}s

^{−1}so that the curves for both the probe and signal fields coincide with each other. The other parameters in all panels are the same as those in Fig. 2. we see that the self-phase modulation (SPM) determined by

*δ*= 0). Simultaneously, the nonlinear absorption from the SPM determined by

*E*and

_{p}*E*. One sees that in the TIT transparency window the cross-phase modulation (CPM) determined by

_{s}*δ*. In this way we can obtain an enhanced cross-Kerr coefficient with a very small imaginary part.

_{1}= Δ

_{2}= 1.5 × 10

^{10}s

^{−1}and

*δ*= 5 × 10

^{10}s

^{−1}. We see that the cross-Kerr susceptibilities have very small imaginary parts, and their real parts are two orders of magnitude larger than those of the self-Kerr susceptibilities.

## 4. Two-Qubit Polarization Phase Gates and highly entangled photons

7. 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]

*σ*

^{−}〉 (corresponding to the signal field) and |

*σ*

^{+}〉 (corresponding to the probe field) to encode binary information for each qubit. The scheme shown in Fig. 1 is completely implemented only if both the probe and signal fields have the “right” polarization states. When the both fields have “wrong” polarizations, there is no sufficient close excited states to which levels |1〉 and |2〉 can couple, and hence the probe and signal fields will only acquire the trivial vacuum phase shift

*k*≡

_{j}*ω*/

_{j}*c*(

*j*=

*p*,

*s*), and

*L*denotes the length of the medium. When one of the two fields have “wrong” polarization state, say for a

*σ*

^{−}-polarized probe field, there is no sufficiently close excited state to which levels |1〉 can couple. Thus the signal field experiences a self-Kerr effect and acquires a nontrivial phase shift

*j*=

*p, s*). Here

*ξ*(

_{j}*ω*) being a Gaussian frequency distribution of incident wave packet centered at frequency

*ω*. The photon field operators undergo a transformation while propagating through the medium of length

_{j}*L*, i.e.

*n*

_{±}(

*ω,z*) (the real part of the refractive index) varies slowly over the bandwidth of the wave packet centered at

*ω*, one gets

_{j}*τ*

_{j}_{′}being the width of the pulse. If group velocity matching is satisfied, i.e.

*ξ*

_{jj′}→ 0, erf(

*ξ*

_{jj′})/

*ξ*

_{jj′}reaches its maximum value

33. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A **61**, 052306 (2000). [CrossRef]

*h*(

*x*) = −

*x*log

_{2}(

*x*) – (1 –

*x*)log

_{2}(1 –

*x*) is Shannon’s entropy function,

*C*is the concurrence given by

*C*(

*ρ̂*) = max{0,

*λ*

_{1}–

*λ*

_{2}–

*λ*

_{3}–

*λ*

_{4}}. Here

*λ*’s are square roots of eigenvalues of the matrix

_{i}*ρ̂*can be directly obtained by using Eq. (7), the quantity

*ρ*̂, and

*σ̂*denotes the

_{y}*y*-component of the Pauli matrix.

*L*. The system parameters are taken the same as those given in section 3.2 except for Δ = 1.9×10

^{12}s

^{−1}. We see that a nearly 100% degree of entanglement can be obtained at

*L*= 6.0

*μ*m. Our result is different from that obtained in Ref. [22

**16**, 17161 (2008). [CrossRef] [PubMed]

**16**, 17161 (2008). [CrossRef] [PubMed]

34. K.-P. Marzlin, Z.-B. Wang, S. A. Moiseev, and B. C. Sanders, “Uniform cross-phase modulation for nonclassical radiation pulses,” J. Opt. Soc. Am. B **27**, A36–A45 (2010). [CrossRef]

## 5. Coupled slow-light soliton pairs

*i, j*= 1 to 3) and

*ɛ*is a small parameter characterizing the amplitude of the both fields. To obtain a divergence-free expansion, all quantities on the right hand side of the expansion are considered as functions of the multi-scale variables

*z*=

_{l}*ɛ*(

^{l}z*l*= 0 to 2) and

*t*=

_{l}*ɛ*(

^{l}t*l*= 0 to 1). Substituting above expansion into Eqs. (1), we obtain a series of equations for

*i, j*=1 to 4;

*n*=1 to 3), which can be solved order by order.

*s*=

*z/L*,

_{D}*u*

_{1}= (Ω

*/*

_{p}*U*) exp (−

_{p}*iK̃*),

_{p}z*u*

_{2}= (Ω

*/*

_{s}*U*) exp (−

_{s}*iK̃*) (

_{s}z*K̃*≡ Re(

_{p,s}*K*)),

_{p,s}*α*= Im(

_{p,s}*K*),

_{p,s}*g*sgn(

*δ*)

*L*/

_{D}*L*,

_{δ}*g*

_{11}=

*W*/|

_{pp}*W*|,

_{ss}*g*

_{22}= −1

*g*

_{12}=

*W*/|

_{ps}*W*|,

_{ss}*g*

_{21}=

*W*/|

_{sp}*W*|, with

_{ss}*τ*

_{0}being the characteristic pulse length of the probe and signal fields.

*g*= 0. In addition, we also have

_{δ}_{1}= Δ

_{2}= 0.1 × 10

^{11}s

^{−1}, Δ = 1.9 × 10

^{12}s

^{−1},

*δ*= 0.5 × 10

^{11}s

^{−1}and other parameters are the same as those in Fig. 2. With these parameters, we obtain

*W*=

_{pp}*W*= (−1.72+

_{ss}*i*0.03)× 10

^{−21}cm

^{−1}s

^{2},

*W*=

_{ps}*W*= (−1.56 +

_{sp}*i*0.05) × 10

^{−21}cm

^{−1}s

^{2}. Notice that the imaginary parts of these quantities are indeed much smaller than their relevant real parts due to the TIT effect. Then we have

*L*= 3.0

_{D}*μ*m with

*τ*

_{0}= 2 × 10

^{−12}s,

*L*

_{0}= 5.0

*μ*m. The dimensionless coefficients read

*g*≈ −1 (

_{jl}*j,l*= 1,2). The propagating velocity of the coupled soliton pair is after returning to the dimensional variables. We see that the coupled soliton pair propagates with matched, ultraslow group velocity in comparison with

*c*.

*u*

_{1}=

*u*

_{2}= 1.0sech(

*σ*+ 4)exp(

*i*0.5

*σ*), we numerically integrated Eqs. (10a) and (10b). The evolution of the probe-field half Rabi frequency |Ω

*/*

_{p}*U*

_{0}| versus the dimensionless time

*t*/

*τ*

_{0}and the distance

*z*/(2

*L*) is shown in Fig. 5(a) (The evolution of the signal field is the same, and thus not shown). We see that the soliton propagates stably up to

_{D}*z*= 4

*L*. There is however a small attenuation contributed from the small absorption. For comparison, Fig. 5(b) shows the evolution of the probe field half Rabi frequency as a function of the

_{D}*t*/

*τ*

_{0}and

*z*/(2

*L*) by a direct numerical integration based on the Eqs. (1) and (2). One sees that initially the soliton is fairly stable, but it radiates small-amplitude continuous waves when propagating to a large distance. This can be understood as the contribution by the effects of high-order dispersion and high-order nonlinearity that are not included in the approximations when obtaining the coupled NLS Eqs. (10a) and (10b).

_{D}*μ*

_{31}| ≈ |

*μ*

_{32}| = 2.688 × 10

^{−28}cm C and

*S*

_{0}=

*πR*

^{2}=

*π*× 10

^{−7}cm

^{2}(

*R*is the transverse radius of the probe and signal beams), we obtain the peak power We see that to generate the ultraslow coupled optical soliton pair in this semiconductor quantum-well system very low input power is needed. This is drastically different from the optical soliton generation schemes in passive (e.g. fiber-based) media, where much higher input power is required in order to bring out the nonlinear effect required for soliton formation.

## 6. Conclusions

## Acknowledgments

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25. | Z.-B. Wang, K.-P. Marzlin, and B. Sanders, in Quantum Communications and Quantum Imaging, edited by R. E. Meyers, Y. Shih, and K. S. Deacon, |

26. | H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, and D. A.B. Miller, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. |

27. | 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 |

28. | We adopt phenomenological few-level model to study the optical response of SQWs. On the relation between such approach and microscopic theory, see N. H. Kwong, I. Rumyantsev, R. Binder, and A. L. Smirl, “Relation between phenomenological few-level models and microscopic theories of the nonlinear optical response of semiconductor quantum wells,” Phys. Rev. B |

29. | The frequency and wavevector of the probe (signal) field in the quantum well are given by ω and k + _{p}K(_{p}ω) (ω + _{s}ω and k + _{s}K(_{s}ω)), respectively. Thus ω = 0 corresponds to the center frequency of both the probe and signal fields. |

30. | Y. Xue, X.-M. Su, G. W. Wang, Y. Chen, and J.-Y. Gao, “Photon switch in a quantum well by quantum interference in interband transitions,” Opt. Commun. |

31. | A. Neogi, “Transient interband light modulation via intersubband coupling light in undoped semiconductor quantum wells,” Opt. Commun. |

32. | Because |

33. | V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A |

34. | K.-P. Marzlin, Z.-B. Wang, S. A. Moiseev, and B. C. Sanders, “Uniform cross-phase modulation for nonclassical radiation pulses,” J. Opt. Soc. Am. B |

35. | A. Hasegawa and Y. Kodama, |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Nonlinear Optics

**Citation**

Chengjie Zhu and Guoxiang Huang, "Giant Kerr Nonlinearity, Controlled Entangled Photons and Polarization Phase Gates in Coupled Quantum-Well Structures," Opt. Express **19**, 23364-23376 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23364

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