## Macroscopic entanglement and violation of Bell’s inequalities between two spatially separated quantum dots in a planar photonic crystal system

Optics Express, Vol. 17, Issue 14, pp. 11505-11514 (2009)

http://dx.doi.org/10.1364/OE.17.011505

Acrobat PDF (456 KB)

### Abstract

We present and apply a medium-dependent quantum optics formalism for describing the exciton dynamics of two spatially-separated quantum dots on-chip, in the regime of coupled-cavity quantum electrodynamics. With each dot placed in a spatially-separated cavity and coupled through a periodic waveguide channel, the quantum dot excitons behave as a composite entangled pair, exhibiting pronounced entanglement over distances of 300*µ*m and more. The computed light spectra above the two cavities show clear signatures of pronounced photon coupling including increased vacuum Rabi splitting and cavity-induced transmission and absorption. The macroscopic entanglement is confirmed by investigating the Bell inequality, which is shown to be violated for hundreds of picoseconds.

© 2009 Optical Society of America

## 1. Introduction

7. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. **95**, 013904 (2005).
[CrossRef] [PubMed]

6. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot cavity system,” Nature **445**, 896 (2007).
[CrossRef] [PubMed]

8. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature **432**, 200 (2004).
[CrossRef] [PubMed]

10. E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett. **95**, 067401 (2005).
[CrossRef] [PubMed]

11. S. Hughes and H. Kamada, “Single-quantum-dot strong coupling in a semiconductor photonic crystal nanocavity side coupled to a waveguide,” Phys. Rev. B **70**, 195313 (2004).
[CrossRef]

12. D. Englund, A. Faraon, I. Fushman, N. Stoltz, P. Petroff, and J. Vučković, “Controlling cavity reflectivity with a single quantum dot,” Nature **450**, 857 (2007).
[CrossRef] [PubMed]

13. S. Hughes, “Modified spontaneous emission and qubit entanglement from dipole-coupled quantum dots in a photonic crystal nanocavity,” Phys. Rev. Lett. **94**, 227402 (2005)
[CrossRef] [PubMed]

*et al*. through investigating the QD emission spectra [14

14. M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science **291**, 451 (2001).
[CrossRef] [PubMed]

15. G. Bester, A. Zunger, and J. Shumway “Broken symmetry and quantum entanglement of an exciton in InxGa_{1-x}AsGaAs quantum dot molecules,” Phys. Rev. B **71**, 075325 (2005).
[CrossRef]

*macroscopically*separated QDs on-chip. We demonstrate that, with suitably excited QDs in a planar PC medium, long lived entangled states can be maintained, showing the violation of the Bell inequality over hundreds of picoseconds. More fundamentally, this coupled QD system gives rise to rich coupling effects including modified strong coupling regimes of cavity-QED. The structure of interest is depicted above in Fig. 1, which includes two single-mode nanocavities and one coupling waveguide. The nanocavities can be formed by localized defects created within the PC, for example by removing and rearranging two of the air holes [16

16. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944 (2003).
[CrossRef] [PubMed]

*L*and no direct coupling is assumed. The PC waveguide and cavity resonances are considered to be deep inside the PC photonic bandgap and the propagating waveguide mode is below the light line (lossless).

## 2. Theory

*et al*. [17

17. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A **70**, 053823 (2004).
[CrossRef]

*a*and

*b*, which are embedded in cavities 1 and 2, respectively. Subsequently, we obtain the multipolar-coupling Hamiltonian:

*â*represents the field mode operators and

_{λ}σ ^

_{m}^{+/-}are the Pauli operators of the QD excitons; we assume one exciton in the spectral region of interest per QD. In addition, Ω

*is the resonant frequency of each QD, and*

_{m}*ω*is the eigenfrequency corresponding to the transverse modes of the system (

_{λ}**f**

*(*

_{λ}**r**)), excluding the dots;

*g*

*m*is the field-dot coupling coefficient, defined through

_{λ}*=*

**µ**_{m}

*n**is the optical dipole moment of the QD electron-hole pair.*

_{m}µ_{m}*=-*Ô ˙

_{i}*ih̄*

^{-1}[

*Ô*,

_{i}*Ĥ*], yielding dynamical equations for

*â*,

_{λ}*â*

^{†}

*,*

_{λ}σ ^

_{m}^{+/-}, and

σ ^

*=*

_{mz}σ ^

^{+}

_{m}σ ^

^{-}

*-*

_{m}*-*σ ^

_{m}^{-}

σ ^

_{m}^{+}. In an electron picture,

σ ^

^{+}=

*cĉ*

^{†}

_{e}*ĉ*creates an electron in the excited state (conduction band) and

_{g}σ ^

^{-}=

*ĉ*

_{e}*ĉ*

^{†}

*destroys an electron in the excited state [or creates an electron in the ground state (valence band)]; similarly,*

_{g}σ ^

^{+}

σ ^

^{-}=

*ĉ*

^{†}

_{e}*ĉ*, where

_{e}*ρ*

_{aa/bb}=〈

σ ^

^{+}

_{aa/bb}

σ ^

^{-}

_{aa/bb}〉 is the excited state population in each QD. After carrying out the Laplace transform of operators [17

17. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A **70**, 053823 (2004).
[CrossRef]

*Ô*(

_{i}*ω*)=∫

^{∞}

_{0}

*Ô*(

*t*)

*e*, then

^{iωt}dt*Ê*(

_{µ}**r**

*,*

_{m}*ω*)=

**Ê**(

**r**

*,*

_{m}*ω*) ·

**n**

*, and the symbol ‘∗’ represents a convolution operator. Using the formal definition of electric-field operator,*

_{m}*a*/

*b*is located at

**r**

_{a/b}, and we have introduced the quantum dipole source

*QD-modified*Green function for the medium:

**K**̃(

**r**,

**r**

*;*

_{m}*ω*)=

**K**(

**r**,

**r**

*;*

_{m}*ω*)/[1-

**n**

*·*

_{m}**K**(

**r**

*,*

_{m}**r**

*;*

_{m}*ω*) ·

**n**

_{m}*α*(

_{m}*ω*)].We stress that

**K**

^{(2)}

*exactly*includes the full dynamical coupling effects between the two QDs, and fully covers both weak and strong coupling regimes in a self-consistent way. Similar expressions have been derived classically for the medium-dependent QD polarizabilities [19

19. S. Hughes, H. Gotoh, and H. Kamada, “Classical and quantum optical correlation effects between single quantum dots: the role of the hopping photon,” Phys. Rev. B **74**, 115334 (2006).
[CrossRef]

17. M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A **70**, 053823 (2004).
[CrossRef]

18. S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. **98**, 083603 (2007).
[CrossRef] [PubMed]

**K**

*) can be derived by assuming*

_{wcc}*weakly coupled*cavity-to-waveguide channels and employing a matrix inversion technique [11

11. S. Hughes and H. Kamada, “Single-quantum-dot strong coupling in a semiconductor photonic crystal nanocavity side coupled to a waveguide,” Phys. Rev. B **70**, 195313 (2004).
[CrossRef]

18. S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. **98**, 083603 (2007).
[CrossRef] [PubMed]

20. A. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E **68**, 46606 (2003).
[CrossRef]

**K**(

**r**,

**r**′;

*ω*)=∑

*〈*

_{αβ}**r**|

**K**

*|*

^{αβ}**r**′〉, where

**K**

*is the projection of*

^{αβ}**K**on |

**E**

*〉 from the left and 〈*

_{α}**E**

*| from the right, and we use the notation*

_{β}**f**

*(*

_{α}**r**)=〈

**r**|

**E**

*〉 and*

_{α}**f***

*(*

_{α}**r**)=〈

**E**

*|*

_{α}**r**〉. Assuming a positive group velocity for the waveguide mode (

*v*>0), example projections include [18

_{g}18. S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. **98**, 083603 (2007).
[CrossRef] [PubMed]

*r*

_{1}(

*ω*)=

*iω*Γ

^{w}_{1}/[

*ω*

^{2}

_{1}-

*ω*

^{2}-

*iω*(Γ

^{0}

_{1}+Γ

*w*

_{1})] is the waveguide mode reflection coefficient from cavity 1, Γ

*is the cavity-to-waveguide coupling rate whose explicit form depends on the overlap between the cavity mode and the waveguide mode [11*

^{w}_{c}11. S. Hughes and H. Kamada, “Single-quantum-dot strong coupling in a semiconductor photonic crystal nanocavity side coupled to a waveguide,” Phys. Rev. B **70**, 195313 (2004).
[CrossRef]

^{0}≪Γ

*is the bare cavity coupling rate caused by coupling to radiation modes above the light line. We note that these functions are not simple Lorenzian lineshapes, and the cavity decay process is a complicated function of frequency and waveguide length*

^{w}_{c}*L*.

*ψ*(

*t*)〉=∑

_{m=a,b}

*C*(

^{u}_{m}*t*)|

*a*,0〉+∑

^{e}_{m}_{m=a,b}

*C*

^{g}_{m,λ}(

*t*) |

*a*

^{g}*,*

_{m}*λ*〉, we define the amplitude coefficient of the QD upper excited state as

*C*

^{u}_{a/b}(

*t*)=∫

^{∞}

_{-∞}〈0|

σ ^

^{-}

_{a/b}(

*ω*)|

*ψ*(

*t*=0)〉

*e*

^{-iωt}

*dω*, with |0〉 the lower level eigenstate of the QDs. The upper state population of each QD is then obtained from

*ρ*=|

_{mm}*C*|

^{u}_{m}^{2}.

*exact analytical expressions for the light spectrum and the entanglement of formation*[21

21. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. **80**, 2245 (1998).
[CrossRef]

*S*(

**r**,

*ω*)=〈(

**Ê**(

*ω*))

^{†}

**Ê**(

*ω*)〉, yielding

21. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. **80**, 2245 (1998).
[CrossRef]

*Con*(

*t*), through

*Con*=|〈

*ψ*|

*| (tilde represents a spin flip) [21*ψ ˜

21. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. **80**, 2245 (1998).
[CrossRef]

*ρ*, which is obtained by tracing out the electric field modes, where the elements of the wave function |

*ψ*〉 are obtained from the QD operators, i.e., Eqs. (12) and (8). For our system of interest, where the initial excitation resides in the QDs, the concurrence becomes simply

*Con*=2|

*C*‖

^{u}_{a}*C*|.

^{u}_{b}## 3. Quantum dynamics, entanglement of formation, and cavity-emitted spectra

*Q*

^{0}

_{1/2}(=

*ω*

_{1/2}/Γ

^{0}

_{1/2})=20000, and a cavity-waveguide quality factor of

*Q*

^{w}_{1/2}(=

*ω*

_{1/2}/Γ

*w*

_{1/2})=500 for both cavities. Increasing

*Q*

^{0}

_{1/2}actually has little influence on our results because the decay of PC system is dominated by the coupling between the cavities and the waveguide; and for the regime of coupled-cavity QED [18

**98**, 083603 (2007).
[CrossRef] [PubMed]

*/Γ*

^{w}^{0}or

*Q*

^{0}/

*Q*. In addition, we choose

^{w}*ω*

_{1/2}=Ω

_{a/b},

*h̄ω*

_{1}=

*h̄*Ω

*=0.95eV, effective cavity mode volume*

_{a}*V*

_{c1/c2}=0.07

*µ*m

^{3}, and a background dielectric constant of the PC slab,

*ε*=12; for the QD dipole moments we use parameters similar to those in experiments [22

_{b}22. K. L. Silverman, R. P. Mirin, S. T. Cundiff, and A. G. Norman, “Direct measurement of polarization resolved transition dipole moment in InGaAs/GaAs quantum dots,” Appl. Phys. Lett. **82**, 4552 (2003).
[CrossRef]

23. T. H. Stievater, Li Xiaoqin, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single quantum dots,” Phys. Rev. Lett. **87**, 133603 (2001).
[CrossRef] [PubMed]

*effective*dipole moment

*µ*=30D(Debye) and

_{a}*µ*=60D (this asymmetry helps to ensure maximum entanglement for QD

_{b}*a*initially excited). In practise, the effective dipole moments can be achieved from different orientations, or, equivalently, by different spatial positioning with respect to the cavity field antinode positions; though we could also easily change other parameters within the formalism. We note that, according to the recent fabrication improvements [6

6. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoǧlu, “Quantum nature of a strongly coupled single quantum dot cavity system,” Nature **445**, 896 (2007).
[CrossRef] [PubMed]

*g*

^{exp}

*>0.9*

_{c}*g*

^{max}

*), and to within this accuracy, we have tested that our proposed scheme is robust. Necessarily taking causality into consideration, the chromatic dispersion of the waveguide is introduced linearly and the group velocity*

_{c}*v*=

_{g}*dω*/

*dk*=

*c*/10; also, unless stated otherwise, the cavity-cavity separation

*L*=3-300

*µ*m, which yield waveguide retardation times of 0.1 ps to 10 ps.

*b*removed, and set

*ω*

_{1}=

*ω*

_{2}=Ω

*. In Figs. 2(a,c,e), we show the excited QD dynamics for various cavity separation distances of*

_{a}*L*=3,30, and 300

*µ*m, respectively. In Fig. 2(a), we clearly see strong coupling phenomena and vacuum Rabi oscillations, which is enhanced over the one cavity result because cavity 2 acts to increase the local density of photon states at the position of QD

*a*(in cavity 1) [18

**98**, 083603 (2007).
[CrossRef] [PubMed]

*L*, the Rabi oscillation period increases (smaller effective cavity coupling), and we eventually recognize weak retardation oscillations in the excited state population dynamics; these oscillations are caused by the single photon being recycled between the two cavities with a period that corresponds to the round trip time of the coupled cavity system. The corresponding emitted spectra from cavity 1 and cavity 2 are shown in Figs. 2(b,d,f), which show that the vacuum Rabi doublet decreases in width as a function of waveguide length

*L*. In this latter case, we highlight that the length

*L*=300

*µ*m would correspond to around 700–800 PC unit cells. A clear advantage of our formalism is the ease with which we can study various structural parameters such as waveguide length, where direct numerical approaches would be impractical and cumbersome.

*b*. We study the resonant condition of ω1(=Ω

*)=*

_{a}*ω*

_{2}(=Ω

*), and also the case with detuning,*

_{b}*ω*

_{1}(=Ω

_{1})≠

*ω*

_{2}(=Ω

*). For both cases, only QD a is initially excited (*

_{b}*C*(0)=1,

^{u}_{a}*C*(0)=0). The resonant case is shown in Fig. 3, which demonstrates that pronounced entanglement values can be realized (see Fig. 3(a), green dashed curve), reaching a maximum value of around 0.67 ebits for

^{u}_{b}*L*=3

*µ*m. The oscillations in the QD dynamics are caused by both the dot-dot photon exchange interactions and the cavity-dot interactions, and we find that the vacuum Rabi oscillations are much faster than the one QD scenario (cf. red curves in 2(a,c) and 3(a,c)). For the emitted spectra, in addition to the expected vacuum Rabi doublet (now with a significantly larger splitting than before), we also obtain a pronounced constructive or destructive center peak that originates from the photon emission from the excited dot in the

*opposite*cavity propagating back and resonantly scattering from the cavity (e.g., the dot in cavity 2 can act as a source for pumping the dot in cavity 1); this is analogous to dipole-induced transparency or dipole-induced absorption [11

**70**, 195313 (2004).
[CrossRef]

24. E. Waks and J. Vučković, “Dipole induced transparency in drop-filter cavity-waveguide systems,” Phys. Rev. Lett. **96**, 153601 (2006).
[CrossRef] [PubMed]

25. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys.Rev. **188**, 1969 (1969).
[CrossRef]

*L*=300

*µ*m) which is caused by the dynamic photon coupling that takes place from cavity to cavity, as discussed above. The latter spectral doublet give clear evidence to the formation of a macroscopic molecule. Again for the longer cavity separations, the vacuum Rabi oscillations (coherent oscillations) are suppressed, and for

*L*=300

*µ*m, a clear retardation feature of around 10 ps is observed. When

*L*is increased to a larger value of 300

*µ*m, the entanglement peak is still reasonable (~0.3). While the usual coupled-QD entanglement distance between excitons is only about 10 nm, here we obtain a

*macroscopic*entanglement by

*indirect*cavity-cavity coupling, via the integrated waveguide. Furthermore, this macroscopic entanglement has the added advantage that the coupling phenomena can be probed by measuring the individual cavity-emitted light spectra.

*T*=4K, even for regular (non-cavity) structures, the exciton relaxation is typically dominated by radiative decay, and the non-radiative decay times (rates) are around 2 ns (~2

*µ*eV) [26

26. 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]

*-Ω*

_{a}*=25*

_{b}*µ*eV and repeat the same investigation as shown in Fig. 3. While the entanglement and photon coupling effects are reduced, we still obtain significant coupling between the QDs via the coupling waveguide.

## 4. Bell inequalities

28. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. **47**, 777 (1935).
[CrossRef]

29. A. Beige, W. J. Munro, and P. L. Knight, “Bell’s inequality test with entangled atoms,” Phys. Rev. A **62**, 052102 (2000).
[CrossRef]

*B*=|

_{s}*E*(

*θ*,

_{a}*θ*)-

_{b}*E*(

*θ*,

_{a}*θ*′

*)+*

_{b}*E*(

*θ*′

*,*

_{a}*θ*)-

_{b}*E*(

*θ*′

*,*

_{a}*θ*′

*)|≤2, where the correlation function*

_{b}*=*σ ^

_{mx}σ ^

^{+}

*+*

_{m}σ ^

_{m}^{-},

*=-*σ ^

_{my}*i*σ ^

^{+}

*+*

_{m}*i*σ ^

_{m}^{-}and

*θ*(

_{m}*θ*′

*) is a real-angle parameter. Thus, if one chooses*

_{m}*θ*=

*θ*-

_{a}*θ*=

_{b}*θ*-

_{b}*θ*′

*=*

_{a}*θ*′

*-*

_{a}*θ*′

*, then above inequality will be simplified to*

_{b}*B*=|3

_{s}*E*(

*θ*,0)-

*E*(3

*θ*,0)|≤2, where

*B*>2.

_{s}*π*), which then excite the pair of QDs. More elaborate schemes could, in principle, send in an incident field through a leaky waveguide mode, that could prepare the QDs pair in the correct superposition state. Inspecting the symmetry properties of Eq. (12), is can be recognized that

*C*(

^{u}_{a}*t*)=±

*C*(

^{u}_{b}*t*) if cavity 1 is symmetric with cavity 2 and QD

*a*is resonant with QD

*b*, and their evolution will be governed by the term

**K**(

**r**

_{a/b},

**r**

_{a/b};

*ω*)±

**K**(

**r**

_{b/a},

**r**

_{a/b};

*ω*). The decay of the state

**K**(

**r**

*,*

_{a}**r**

*;*

_{a}*ω*) is in phase (interferences constructively) with

**K**(

**r**

*,*

_{a}**r**

*;*

_{b}*ω*). On the contrary, if

**K**(

**r**

*,*

_{a}**r**

*;*

_{a}*ω*) is out of phase (interferences destructively) with

**K**(

**r**

*,*

_{a}**r**

*;*

_{b}*ω*), then the decay of the state

*ψ*〉

^{-}

*is shorter than that of the initial condition |*

_{QD}*ψ*〉

^{+}

*, which is caused by the*

_{QD}*destructive*interference between

**K**(

**r**

*,*

_{a}**r**

*;*

_{a}*ω*) and

**K**(

**r**

*,*

_{a}**r**

*;*

_{b}*ω*). Thus, we will only show the initial condition of |

*ψ*〉

^{+}

*.*

_{QD}*L*and QD dipole moments. To better understand these results, we note that the violation of the Bell inequality for the initial condition |

*ψ*〉

^{+}

*is largely insensitive to the distance*

_{QD}*L*, but more sensitive to the dipole moments (or effective dipole moments, as ultimately the coupling is controlled through

*g*). In Fig 5(a–b), we show the QD population dynamics and

_{λ}*B*for a QD dipole of

_{s}*d*

_{a/b}=30D, with

*L*=3

*µ*m (red curve) and

*L*=300

*µ*m (blue curve); the Bell inequality is found to be violated for both waveguide lengths for a timescale of up to 50 ps, with the longer system simply exhibiting more pronounced retardation dynamics (oscillations). When the effective dipole moment reduces from 30D to 8D, the violation time increases from around 50ps to more than 800ps (see Fig. 5(d)).

*µ*eV, then this violation is reduced to around 200–300 ps (see Fig.5(f)), though can be increased by having a larger

*L*. There is also an interesting entanglement revival for the longer waveguide, where the

*B*increases about 400 ps. Note that for sufficiently large

_{s}*L*, one must be careful of disorder-induced losses in the PC waveguide [30

30. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. **94**, 33903 (2005).
[CrossRef]

## 5. Conclusions

## Acknowledgements

## References and links

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13. | S. Hughes, “Modified spontaneous emission and qubit entanglement from dipole-coupled quantum dots in a photonic crystal nanocavity,” Phys. Rev. Lett. |

14. | M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science |

15. | G. Bester, A. Zunger, and J. Shumway “Broken symmetry and quantum entanglement of an exciton in InxGa |

16. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

17. | M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to interatomic interactions and super-radiance in inhomogeneous dielectrics,” Phys. Rev. A |

18. | S. Hughes, “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. |

19. | S. Hughes, H. Gotoh, and H. Kamada, “Classical and quantum optical correlation effects between single quantum dots: the role of the hopping photon,” Phys. Rev. B |

20. | A. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E |

21. | W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. |

22. | K. L. Silverman, R. P. Mirin, S. T. Cundiff, and A. G. Norman, “Direct measurement of polarization resolved transition dipole moment in InGaAs/GaAs quantum dots,” Appl. Phys. Lett. |

23. | T. H. Stievater, Li Xiaoqin, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single quantum dots,” Phys. Rev. Lett. |

24. | E. Waks and J. Vučković, “Dipole induced transparency in drop-filter cavity-waveguide systems,” Phys. Rev. Lett. |

25. | B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys.Rev. |

26. | W. Langbein, P. Borri, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck, “Radiatively limited dephasing in InAs quantum dots,” Phys. Rev. B |

27. | J. S. Bell, “On the problem of hidden variables in quantum mechanics,” Physics |

28. | A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. |

29. | A. Beige, W. J. Munro, and P. L. Knight, “Bell’s inequality test with entangled atoms,” Phys. Rev. A |

30. | S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. |

**OCIS Codes**

(270.5580) Quantum optics : Quantum electrodynamics

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: April 9, 2009

Revised Manuscript: June 12, 2009

Manuscript Accepted: June 18, 2009

Published: June 24, 2009

**Citation**

P. Yao and S. Hughes, "Macroscopic entanglement and violation of Bell’s inequalities between two spatially separated quantum dots in a planar photonic crystal system," Opt. Express **17**, 11505-11514 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11505

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