## Single-plasmon scattering grating using nanowire surface plasmon coupled to nanodiamond nitrogen-vacancy center |

Optics Express, Vol. 19, Issue 21, pp. 20991-21002 (2011)

http://dx.doi.org/10.1364/OE.19.020991

Acrobat PDF (813 KB)

### Abstract

We investigate the scattering properties of a single surface plasmon in metal nanowire coupled to a nitrogen-vacancy (NV) center in diamond nanocrystal under optical excitation. We demonstrate that, by spatially modulating a classical control beam, alternating regions of high reflection and absorption as well as high transmission and absorption of a single plasmon can be created in the left- and right-going directions that act as a kind of scattering grating. Such approach to induce grating gets out the well investigating region in which the weak interactions between single atoms and light is often used. The proposal may be used for chip-integrated grating, switcher and multi-channel drop filter.

© 2011 OSA

## 1. Introduction

1. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A **57**, 1338–1344 (1998) and references therein. [CrossRef]

2. J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett. **98**, 081108 (2011). [CrossRef]

3. M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A **59**, 4773–4776 (1999). [CrossRef]

4. A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. **89**, 143602 (2002). [CrossRef] [PubMed]

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

6. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. **30**, 699–701 (2005). [CrossRef] [PubMed]

7. J. W. Gao, J. H. Wu, N. Ba, C. L. Cui, and X. X. Tian, “Efficient all-optical routing using dynamically induced transparency windows and photonic band gaps,” Phys. Rev. A **81**, 013804 (2010). [CrossRef]

8. J. H. Wu, M. Artoni, and G. C. La Rocca, “All-optical light confinement in dynamic cavities in cold atoms,” Phys. Rev. Lett. **103**, 133601 (2009). [CrossRef] [PubMed]

1. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A **57**, 1338–1344 (1998) and references therein. [CrossRef]

9. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature (London) **450**, 402–406 (2007). [CrossRef]

18. A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. **106**, 096801 (2011). [CrossRef] [PubMed]

*g*between the surface-plasmon modes and any proximal emitter with a dipole-allowed transition can be achieved because

_{sp}*V*for the plasmons can be significantly decreased [10

_{eff}10. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. **3**, 807–812 (2007). [CrossRef]

*P*= Γ

*/Γ′ can exceed 10*

_{pl}^{3}in realistic systems according to the theoretical results of Refs. [16

16. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006). [CrossRef] [PubMed]

17. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B **76**, 035420 (2007). [CrossRef]

*is the spontaneous emission rate into the surface plasmons and Γ′ describes contributions both from emission into free space and non-radiative emission via ohmic losses in the conductor. Furthermore, unlike the strong coupling based on cavity quantum electrodynamics (CQEDs), this strong coupling is broadband [10*

_{pl}10. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. **3**, 807–812 (2007). [CrossRef]

19. F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate,” Phys. Rev. Lett. **93**, 130501 (2004). [CrossRef] [PubMed]

26. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) **466**, 730–734 (2010). [CrossRef]

18. A. Huck, S. Kumar, A. Shakoor, and U. L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. **106**, 096801 (2011). [CrossRef] [PubMed]

1. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A **57**, 1338–1344 (1998) and references therein. [CrossRef]

3. M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A **59**, 4773–4776 (1999). [CrossRef]

## 2. Description of the model system and observables

27. N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B **74**, 104303 (2006). [CrossRef]

^{3}

*A*, with the levels

*m*= ±1 nearly degenerated and a zero-field splitting

_{s}*D*= 2.88 GHz between the states

_{gs}*m*= 0 and

_{s}*m*= ±1 due to spin-spin interactions (see the bubble of Fig. 1). The excited state

_{s}^{3}

*E*is also a spin triplet, associated with a broadband photoluminescence emission with the resonant zero phonon line (ZPL) of 637 nm (1.945 eV), which allows optical detection of individual NV defects using confocal microscopy. By combining a coherent laser irradiation (

*σ*

^{+}circularly polarized) with the nanowire surface-plasmon modes (

*π*polarized), one can model the NV center as a Λ-type three-level structure [28

28. C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A.D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. **97**, 247401 (2006). [CrossRef]

29. P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. **10**, 045004 (2008). [CrossRef]

*e*〉 and two lower states |

*g*〉 and |

*s*〉. Specifically, the state |

*s*〉 is decoupled from the surface plasmons owing to a different orientation of its associated dipole moment [10

10. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. **3**, 807–812 (2007). [CrossRef]

*e*〉 via a classical standing-wave control field with central frequency

*ω*and position-dependent coupling strength Ω

_{c}*(*

_{c}*z*). The states |

*g*〉 and |

*e*〉 are coupled with strength

*g*via the nanowire surface plasmon modes of frequency

_{sp}*ω*which is described by annihilation operations

*â*and

_{R,}_{ω}*â*[30

_{L,}_{ω}30. J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A **79**, 023837 (2009). [CrossRef]

31. J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. **30**, 2001–2003 (2005). [CrossRef] [PubMed]

*g*〉, |

*s*〉, and |

*e*〉 have the energy

*ω*= 0 (the energy origin),

_{g}*ω*, and

_{s}*ω*, respectively. The standing-wave control field satisfies the resonance condition:

_{e}*ω*+

_{s}*ω*=

_{c}*ω*. Because the coupling strength

_{e}*g*is broadband, it can be assumed to be frequency-independent [10

_{sp}**3**, 807–812 (2007). [CrossRef]

16. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006). [CrossRef] [PubMed]

17. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B **76**, 035420 (2007). [CrossRef]

*σ̂*= |

_{mn}*m*〉〈

*n*| (

*m,n*=

*g,s,e*) for

*m*≠

*n*, are the electronic transition or projection operators between the states |

*m*〉 and |

*n*〉 and

*σ̂*= |

_{mm}*m*〉〈

*m*| (

*m*=

*s,e*) represent the electronic population operators involving the levels of the NV center [see also the bubble of Fig. 1]. Ω

*(*

_{c}*z*) stands for Rabi frequency of the standing-wave control field for the |

*s*〉 ⇔ |

*e*〉 transition, i.e., Ω

*(*

_{c}*z*) =

*E*(

_{c}*z*)

*μ*/2

_{es}*h̄*, with

*μ*denoting the dipole moment for the relevant driven transition. In the Hamiltonian [see Eq. (1)], the first and second terms represent the energies of the states |

_{es}*e*〉 and |

*s*〉. For simplicity, the energy of the ground state |

*g*〉 is set as zero. In the third term, a classical standing-wave control field resonantly drives the transition |

*s*〉 ↔ |

*e*〉 of the NV center with position-dependent coupling strength Ω

*(*

_{c}*z*). The fourth and fifth terms are the energies of the right- and left-going surface plasmon modes with frequency

*ω*. In the sixth and seventh terms, the transition |

*g*〉 ↔ |

*e*〉 of the NV center is coupled to the right- and left-going surface plasmon modes with coupling strength

*g*.

_{sp}_{0}to transform to the interaction picture [32

32. Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A **54**, 1586–1592 (1996). [CrossRef] [PubMed]

*ω*=

*ω*–

*ω*is the frequency detuning. The resulting interaction Hamiltonian in the interaction picture can be then reexpressed as follows

_{e}*g,vac*〉} and {|

*s,vac*〉,|

*e,vac*〉,|

*g*,

*ω*〉}, respectively, where in |

*m,n*〉,

*m*=

*g,s,e*denotes the state of three-level Λ-type NV center in diamond and

*n*denotes the number of plasmons in the surface plasmon modes, i.e., |

*ω*〉 denotes the one-plasmon Fock state of the surface plasmon mode with frequency

*ω*,

*|vac*〉 describes the vacuum state of the surface plasmon mode. So the evolution of the whole system can be generally described by the wave function |Ψ(

*t*)〉 =

*A*|

_{g}*g,vac*〉 +

*A*|Ψ

_{IHS}*(*

^{IHS}*t*)〉 in the interaction picture, where

*denotes the decay rate of the excited state of the NV center in diamond and Γ*

_{e}*is the dephasing decay rate between the ground-state coherence. In general, the dephasing rate of the ground-state coherence is smaller than the decay rate from excited to ground state.*

_{s}*t*

_{0}<

*t*formally yields where

*α*

_{R(L),ω}(

*t*

_{0}) denotes the value of

*α*

_{R(L),ω}(

*t*) at

*t*=

*t*

_{0}.

*α*(

_{R,ω}*t*) and

*α*(

_{L,ω}*t*) from Eq. (11) and Eq. (12) into Eq. (9), we can obtain where

*t*

_{1}>

*t*to get where

*α*

_{R(L),ω}(

*t*

_{1}) denotes the value of

*α*

_{R(L),ω}(

*t*) at

*t*=

*t*

_{1}. Carrying out the same procedure as above, we can achieve where

*α*and

_{s}*α*in Eq. (10) and Eq. (13), respectively. After carrying out some algebraic calculations, the analytical solution of the amplitude

_{e}*α*can be found as

_{e}*α*≠ 0 and

_{R,in}*α*= 0 in the left- and right-scattering channels, from Eq. (20) and Eq. (21) the reflection and transmission coefficients of a single surface plasmon can be respectively defined by where

_{L,in}*s*〉 ↔ |

*e*〉 transition of the NV center. Here, three points need to be emphasized. Firstly, for the case that

*α*= 0 and

_{R,in}*α*≠ 0, it is easy to check from Eq. (20) and Eq. (21) that the reflection and transmission coefficients of the coupled system has the same form as Eq. (22) and Eq. (23) due to the symmetry. Secondly, for the case that Ω

_{L,in}*(*

_{c}*z*) = 0, we have

*S*= 0. In this case Eq. (22) and Eq. (23) can reduce to the same expressions appearing in Ref. [10

_{c}**3**, 807–812 (2007). [CrossRef]

*S*when the control field is applied, and it leads to a significant change of the reflection and transmission coefficients (

_{c}*r*and

*t*) in the left- and right-scattering directions, as will be shown in Section 3.

## 3. Plasmon scattering properties and grating effect

**3**, 807–812 (2007). [CrossRef]

*P*≡ Γ

*/Γ*

_{pl}*) is taken to be*

_{e}*P*= 10. In the presence of an NV-center, the coupling rate to the surface plasmon mode can be achieved within a vacuum wavelength around 637 nm via using the method in Refs. [16

16. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**, 053002 (2006). [CrossRef] [PubMed]

17. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B **76**, 035420 (2007). [CrossRef]

*D*≃ 42 nm and

*L*≃ 1.3

*μ*m, respectively. All the parameters used in this paper are scaled by the decay rate into the surface plasmons, i.e., Γ

*.*

_{pl}*S*, and this will lead to a significant change of the reflection and transmission intensities in the left-and right-scattering directions. Because, on the one hand, the standing-wave control field has an amplitude and space period, on the other hand, the reflection and transmission functions

_{c}*R*(

*z*) and

*T*(

*z*) depend sensitively on the intensity of the control field, they are expected to change periodically as the standing wave changes from nodes to antinodes across

*z*dimension.

*R*, transmission

*T*and loss (loss = 1 –

*R*–

*T*) in the absence of the control field (see Fig. 2(a)) with the ones in the presence of the control field (see Fig. 2(b)). Notice that, here, the intensity of the control field which is used for producing the curves in Figs. 2(a) and 2(b), corresponds to the peak intensity of a standing-wave field. The reflection

*R*increases rapidly and transmission

*T*decreases quickly when the standing-wave control field Ω

*(*

_{c}*z*) is located at the node (Ω

*(*

_{c}*z*) = 0, see Fig. 2(a)) for frequencies lying near the resonance point Δ

*ω*= 0, with

*R*(Δ

*ω*= 0) ≈ 0.91 whereas

*T*(Δ

*ω*= 0) ≈ 0. Thus the reflection spectral line-shape exhibits a single ‘peak’ structure whereas the transmission spectral line-shape exhibit a single ‘dip’ structure at the resonance point Δ

*ω*= 0. In contrast, when the standing-wave control field Ω

*(*

_{c}*z*) is located at the antinode (Ω

*(*

_{c}*z*) = 0.5Γ

*, see Fig. 2(b)), the reflection*

_{pl}*R*decreases rapidly and transmission

*T*increases quickly near the resonance point, where

*R*(Δ

*ω*= 0) ≈ 0 whereas

*T*(Δ

*ω*= 0) ≈ 1. It is easy to find from Fig. 2(b) that, the reflection line-shape around the resonance point behaves a transition from a single ‘peak’ structure to a multiple ‘peak-dip-peak’ structure as the control field changes from the absence to the presence whereas the transmission line-shape behaves a transition from a single ‘dip’ structure to a multiple ‘dip-peak-dip’ structure. Two sideband peaks in the reflection line-shape and two sideband dips in the transmission line-shape are always located at Δ

*ω*= Ω

*and Δ*

_{c}*ω*= −Ω

*, respectively. One central dip in the reflection line-shape and one central peak in the transmission line-shape lie at the resonance point Δ*

_{c}*ω*= 0. These results can be qualitatively explained in terms of the dressed states created by the control field. The coupling of the NV center to the control field Ω

*gives rise to the splitting of the upper level |*

_{c}*e*〉 into two dressed sublevels

*λ*

_{±}= ±Ω

*. These two sideband peaks/dips are located at*

_{c}*λ*

_{±}= ±Ω

*, which correspond to the dress-state transitions |*

_{c}*g*〉 ↔ |+〉, |

*g*〉 ↔ |−〉 for the surface plasmon. One central dip/peak at resonant point

*λ*

_{0}= 0 appears due to the destructive interference for left-going surface plasmon and the constructive interference for right-going surface plasmon between two excitation pathways to these dressed states from the ground state |

*g*〉: |

*g*〉 ↔ |+〉 and |

*g*〉 ↔ |−〉.

*(*

_{c}*z*), the reflection and transmission characteristics of the coupled system can transmit from those of Fig. 2(a) for the case Ω

*(*

_{c}*z*) = 0 at the position of node to those of Fig. 2(b) for Ω

*(*

_{c}*z*) = 0.5Γ

*at the position of antinode. That is to say, a standing-wave control field triggers the change {*

_{pl}*R*(Δ

*ω*= 0) ≈ 0.91,

*T*(Δ

*ω*= 0) ≈ 0} for Ω

*(*

_{c}*z*) = 0 ⇔ {

*R*(Δ

*ω*= 0) ≈ 0,

*T*(Δ

*ω*= 0) ≈ 1} for Ω

*(*

_{c}*z*) = 0.5Γ

*. As a result, the position-dependent control field is sufficient to well control the scattering of a single propagating surface plasmon in a metal nanowire, and the system therefore can be used as a substantial amplitude modulation. We name this phenomenon single-plasmon scattering grating.*

_{pl}*ω*= 0, Figure 3 displays the reflection

*R*and transmission

*T*versus the control-field intensity Ω

*. From this figure, we can find interesting and useful phenomena: (i) When the control field Ω*

_{c}*is switched off, the input plasmon field*

_{c}*α*transmits into

_{R,in}*α*. In this case, the intensity reflection

_{L,out}*R*in the left-going direction reaches to 0.91 while the intensity transmission

*T*in the right-going direction reaches to 0; (ii) When the control field Ω

*is switched on, the reflection*

_{c}*R*in the left-going direction quickly decreases to a zero steady-state value (

*R*≈ 0) (see solid curve). On the other hand, the intensity transmission

*T*in the right-going direction rapidly increases to a saturation value (

*T*≈ 1) and is independent of the control-field intensity Ω

*at the output of the nanowire (see dashed curve). Obviously, we can see that the control-field intensity Ω*

_{c}*= 0 suggests left-scattering channel*

_{c}*α*⇒

_{R,in}*α*is switched on whereas right-scattering channel

_{L,out}*α*⇒

_{R,in}*α*is switched off. Instead, the control-field intensity Ω

_{R,out}*≠ 0 (e.g., Ω*

_{c}*= 0.5Γ*

_{c}*) indicates left- scattering channel*

_{pl}*α*⇒

_{R,in}*α*is switched off whereas right- scattering channel

_{L,out}*α*⇒

_{R,in}*α*is switched on. Therefore, it is possible for us to efficiently control the propagating path of an input optical field with the external control light Ω

_{R,out}*. By this way, we can design an efficient single-plasmon all-optical switching.*

_{c}*R*(

*z*) as a function of the position

*z*is plotted. At the transverse locations around the nodes of the standing wave, the control-field intensity is very weak, in this case the intensity reflection

*R*in the left-scattering channel reaches to a value of 0.91. In contrast, at the transverse locations around the antinodes of the standing wave, the control-field intensity is quite strong, and the intensity reflection

*R*in the left-scattering channel quickly decreases to a zero value due to the Stark effect

*S*. This leads to a periodic amplitude modulation across the output profile of the nanowire, a phenomenon reminiscent of the amplitude grating.

_{c}*T*(

*z*) in the right-scattering channel versus the position

*z*is plotted in Fig. 4(b). At the transverse locations around the nodes of the standing wave, the intensity transmission

*T*in the right-scattering channel arrives at a zero value because the control field intensity is very weak. Instead, at the transverse locations around the antinodes of the standing wave where the control field intensity is quite strong, the intensity transmission

*T*in the right-scattering channel rapidly increases to a peak value of

*T*≈ 1. So, by this means, we can design another type of periodic amplitude modulation across the output profile of the nanowire. Yet, from the results of Figs. 4(a) and 4(b), it should be pointed out that the characteristics of the amplitude grating in the right-scattering channel is very different from one in the left-scattering channel.

## 4. Conclusion

## Acknowledgment

## References and links

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25. | M. Larsson, K. N. Dinyari, and H. Wang, “Composite optical microcavity of diamond nanopillar and silica microsphere,” Nano Lett. |

26. | E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) |

27. | N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B |

28. | C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A.D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. |

29. | P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, “Spin-flip and spin-conserving optical transitions of the nitrogen-vacancy centre in diamond,” N. J. Phys. |

30. | J. T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom,” Phys. Rev. A |

31. | J. T. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. |

32. | Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A |

**OCIS Codes**

(230.4320) Optical devices : Nonlinear optical devices

(240.6680) Optics at surfaces : Surface plasmons

(270.1670) Quantum optics : Coherent optical effects

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: August 15, 2011

Revised Manuscript: September 16, 2011

Manuscript Accepted: September 20, 2011

Published: October 6, 2011

**Citation**

Jiahua Li and Rong Yu, "Single-plasmon scattering grating using nanowire surface plasmon coupled to nanodiamond nitrogen-vacancy center," Opt. Express **19**, 20991-21002 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20991

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