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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20991–21002
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Single-plasmon scattering grating using nanowire surface plasmon coupled to nanodiamond nitrogen-vacancy center

Jiahua Li and Rong Yu  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20991-21002 (2011)
http://dx.doi.org/10.1364/OE.19.020991


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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

In recent years, electromagnetically induced grating (EIG) [1

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

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

] have attracted intensive interests owing to its potential applications. It is known that EIG, which is attributed to destructive quantum interference, is a phenomenon that the absorption of a weak probe beam coupled to an atomic transition can be spatially modulated (reduced absorption at the peaks of the standing-wave field and high absorption at the nodes) by applying a strong standing wave that is coupled to another atomic transition in a three-level atomic system. EIG was originally observed in cold three-level atomic vapors [3

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

]. Afterward, it was demonstrated that such an EIG effect can be used to achieve tunable photonic band gap [4

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

], to store probe pulses in a vapor of rubidium atoms [5

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]

], to implement optical routing [6

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

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]

], to devise a dynamic controlled cavity [8

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]

], etc. However, as shown in [1

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]

], low efficiency of the EIG due to weak interactions between single atoms and photons in a three-level atomic medium restricts its practical uses.

A new approach based on surface plasmons in metal nanowire to reach the strong-coupling regime on a chip has been intensively studied theoretically and experimentally [9

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

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]

]. A substantial increase in the coupling strength gsp between the surface-plasmon modes and any proximal emitter with a dipole-allowed transition can be achieved because gsp1/Veff where the effective mode volume Veff for the plasmons can be significantly decreased [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]

]. An effective Purcell factor P = Γpl/Γ′ can exceed 103 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

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]

], where Γpl 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

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]

].

On the other hand, the nitrogen-vacancy (NV) center in diamond nanocrystal have recently emerged as an excellent test bed for solid-state quantum physics experiments and quantum information processing because they possess long-lived spin triplets at room temperature [19

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

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]

]. Owing to the potential key roles of NV center in quantum information and nanophotonics, interactions of surface plasmons and NV center represent a topic of great interest. Recent experimental investigations have addressed the controlled coupling of a single NV center in diamond nanocrystal to a surface plasmon mode propagating along a silver nanowire [18

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]

]. Motivated by these considerations, we demonstrate that such a system consisting of a single plasmon and a NV center coupled to a metal nanowire can be employed to act as a kind of scattering grating, where alternating regions of high reflection and absorption in a left-scattering direction as well as high transmission and absorption in a right-scattering direction can be caused by a classical standing-wave control field. The model shows an obvious effect which has a direct analogy with the phenomenon of EIG in atomic systems [1

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

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

]. The proposed scheme may find applications in chip-integrated grating, switcher and multi-channel drop filter.

The paper is organized as follows. In Section 2, we establish the physical model and its theoretical description. By solving the coupled amplitude equations of motion for a metal nanowire and a three-level Λ-type NV center in the frequency domain, we derive explicit analytical expressions of the left-going plasmon reflection and right-going plasmon transmission functions for the output fields. In Section 3, we devote to analyzing and demonstrating in details optical scattering grating in this device. At the same time, we also present the principal mechanism behind the scattering grating. Finally, our main conclusions are summarized in Section 4.

2. Description of the model system and observables

The configuration of the system considered in this paper is exhibited in Fig. 1. An NV center is placed close to a metal nanowire. The NV center is a point defect in the diamond lattice, which consists of a substitutional nitrogen atom (N) plus a vacancy (V) in an adjacent lattice site. The NV center addressed in this study is negatively charged with two unpaired electrons located at the vacancy, usually treated as electron spin-1 [27

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]

]. As a result, the ground state is spin triplet and labelled as 3A, with the levels ms = ±1 nearly degenerated and a zero-field splitting Dgs = 2.88 GHz between the states ms = 0 and ms = ±1 due to spin-spin interactions (see the bubble of Fig. 1). The excited state 3E 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

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]

], i.e., an excited state |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]

], but is resonantly coupled to the excited state |e〉 via a classical standing-wave control field with central frequency ωc and position-dependent coupling strength Ωc (z). The states |g〉 and |e〉 are coupled with strength gsp via the nanowire surface plasmon modes of frequency ω which is described by annihilation operations âR,ω and âL,ω [30

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

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]

]. The states |g〉, |s〉, and |e〉 have the energy ωg = 0 (the energy origin), ωs, and ωe, respectively. The standing-wave control field satisfies the resonance condition: ωs + ωc = ωe. Because the coupling strength gsp is broadband, it can be assumed to be frequency-independent [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]

, 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

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]

]. Under these circumstances, the Hamiltonian of the whole system in the rotating-wave approximation (RWA) can be written as
^=h¯ωeσ^ee+h¯ωsσ^ssh¯[Ωc(z)eiωctσ^es+Ωc*(z)eiωctσ^se]++h¯ωa^R,ωa^R,ωdω++h¯ωa^L,ωa^L,ωdωh¯gsp+(a^R,ωσ^eg+a^R,ωσ^ge)dωh¯gsp+(a^L,ωσ^eg+a^L,ωσ^ge)dω,
(1)
with σ̂mn = |m〉〈n| (m,n = g,s,e) for mn, are the electronic transition or projection operators between the states |m〉 and |n〉 and σ̂mm = |m〉〈m| (m = s,e) represent the electronic population operators involving the levels of the NV center [see also the bubble of Fig. 1]. Ωc(z) stands for Rabi frequency of the standing-wave control field for the |s〉 ⇔ |e〉 transition, i.e., Ωc(z) = Ec(z)μes/2, with μes 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 |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 Ωc (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 gsp.

Fig. 1 Schematic diagram of a metal nanowire coupled to a NV center in diamond nanocrystal. A single surface plasmon injected from the left is coherently scattered by the NV center. The bubble shows energy configuration of the NV center. gsp is the coupling strength between the NV center and metal nanowire, and Ωc (z) is the position-dependent coupling strength between the NV center and the standing-wave control field. Dgs = 2.88 GHz is the zero-field splitting between the ground state sublevels ms = 0 and ms = ±1 of the NV center (ms = ±1 are degenerate at zero magnetic field).

To remove the time dependence in ℋ̂, we choose the proper free Hamiltonian ℋ̂0 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]

], that is,
^int=ei^0t/h¯^resei^0t/h¯,
(2)
^0/h¯=ωsσ^ss+ωeσ^ee++ωe(a^R,ωa^R,ω+a^L,ωa^L,ω)dω,
(3)
^res/h¯=+Δω(a^R,ωa^R,ω+a^L,ωa^L,ω)dω[Ωc(z)eiωctσ^es+Ωc*(z)eiωctσ^se]gsp+[(a^R,ω+a^L,ω)σ^eg+(a^R,ω+a^L,ω)σ^ge]dω,
(4)
where Δω = ωωe is the frequency detuning. The resulting interaction Hamiltonian in the interaction picture can be then reexpressed as follows
^int/h¯=+Δω(a^R,ωa^R,ω+a^L,ωa^L,ω)dω[Ωc(z)σ^es+Ωc*(z)σ^se]gsp+[(a^R,ω+a^L,ω)σ^eg+(a^R,ω+a^L,ω)σ^ge]dω.
(5)

It should be pointed out that the present coupled system described by this Hamiltonian, under optical excitation and the interaction of the surface plasmon with NV center, has two invariant Hilbert subspaces (IHS), with the bases {|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)〉 = Ag|g,vac〉 + AIHSIHS (t)〉 in the interaction picture, where
|ΨIHS(t)=+[αR,ω(t)a^R,ω|g,vac+αL,ω(t)a^L,ω|g,vac]dω+αe(t)|e,vac+αs(t)|s,vac.
(6)

Firstly, by making use of the interaction Hamiltonian operator [see Eq. (5)] and the well-known Schrödinger equation ih¯t|ΨIHS(t)=H^int|ΨIHS(t) in the interaction picture, the time evolution of the amplitudes for the surface plasmon modes in metal nanowire and the NV center in diamond nanocrystal are
α˙R,ω(t)=iΔωαR,ω(t)+igspαe(t),
(7)
α˙L,ω(t)=iΔωαL,ω(t)+igspαe(t),
(8)
α˙e(t)=Γe2αe(t)+iΩc(z)αs(t)+igsp+[αR,ω(t)+αL,ω(t)]dω,
(9)
α˙s(t)=Γs2αs(t)+iΩc*(z)αe(t),
(10)
where Γe denotes the decay rate of the excited state of the NV center in diamond and Γs 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.

Integrating Eq. (7) and Eq. (8) from an initial time t0 < t formally yields
αR,ω(t)=αR,ω(t0)eiΔω(tt0)+igspt0tαe(t)eiΔω(tt)dt,
(11)
αL,ω(t)=αL,ω(t0)eiΔω(tt0)+igspt0tαe(t)eiΔω(tt)dt,
(12)
where αR(L),ω (t0) denotes the value of αR(L),ω (t) at t = t0.

Now, by substituting αR,ω (t) and αL,ω (t) from Eq. (11) and Eq. (12) into Eq. (9), we can obtain
α˙e(t)=(Γe2+Γpl)αe(t)+iΩc(z)αs(t)+i2πgsp[αR,in(t)+αL,in(t)],
(13)
where Γpl=2πgsp2 is the decay rate into the surface plasmons. The input right-going (left-going) surface plasmon is defined as αR(L),in(t)=12π+αR(L),ω(t0)eiΔω(tt0)dω. Above, we have applied the relationships 12π+eiΔω(tt)dω=δ(tt) and t0tαe(t)δ(tt)=12αe(t) (Here, notice that we integrate over half the delta-function, which results in a factor of 12).

Similarly, we integrate Eq. (7) and Eq. (8) up to a final time t1 > t to get
αR,ω(t)=αR,ω(t1)eiΔω(tt1)igsptt1αe(t)eiΔω(tt)dt,
(14)
αL,ω(t)=αL,ω(t1)eiΔω(tt1)igsptt1αe(t)eiΔω(tt)dt,
(15)
where αR(L),ω (t1) denotes the value of αR(L),ω (t) at t = t1. Carrying out the same procedure as above, we can achieve
α˙e(t)=(Γe2Γpl)αe(t)+iΩc(z)αs(t)+i2πgsp[αR,out(t)+αL,out(t)],
(16)
where αR(L),out(t)=12π+αR(L),ω(t1)eiΔω(tt1)dω is the output right-going (left-going) surface plasmon. Combining Eq. (13), Eq. (16) and the symmetry of the surface plasmon modes, we can arrive at the input-output formalism
αR,outαR,in=i2πgspαe,
(17)
αL,outαL,in=i2πgspαe.
(18)

In the following, we are interested in the reflection and transmission properties in the frequency domain of the coupled system, therefore we perform the Fourier transformations (Δω)=12π+(t)eiΔωtdt on the amplitudes αs and αe in Eq. (10) and Eq. (13), respectively. After carrying out some algebraic calculations, the analytical solution of the amplitude αe can be found as
αe(z)=i2πgsp(αR,in+αL,in)iΔω(Γe2+Γpl)+|Ωc(z)|2iΔωΓs2.
(19)

According to the input-output formalism αR,outαR,in=i2πgspαe and αL,outαL,in=i2πgspαe, finally we can readily derive the relationship between the input optical field and the output optical field
αR,out(z)=iΔωΓe2+|Ωc(z)|2iΔωΓs2iΔω(Γe2+Γpl)+|Ωc(z)|2iΔωΓs2αR,in+ΓpliΔω(Γe2+Γpl)+|Ωc(z)|2iΔωΓs2αL,in,
(20)
αL,out(z)=ΓpliΔω(Γe2+Γpl)+|Ωc(z)|2iΔωΓs2αR,in+iΔωΓe2+|Ωc(z)|2iΔωΓs2iΔω(Γe2+Γpl)+|Ωc(z)|2iΔωΓs2αL,in.
(21)

For the initial given input fields αR,in ≠ 0 and αL,in = 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
r=αL,outαR,in=ΓpliΔω(Γe2+Γpl)+Sc(z),
(22)
t=αR,outαR,in=1+r=iΔωΓe2+Sc(z)iΔω(Γe2+Γpl)+Sc(z),
(23)
where Sc(z)=|Ωc(z)|2iΔωΓs2 represents the optical Stark shift created by the coupling between the standing-wave control field and the |s〉 ↔ |e〉 transition of the NV center. Here, three points need to be emphasized. Firstly, for the case that αR,in = 0 and αL,in ≠ 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 Ωc (z) = 0, we have Sc = 0. In this case Eq. (22) and Eq. (23) can reduce to the same expressions appearing in Ref. [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]

]. Thirdly, it is shown that the NV center would be detuned from the surface plasmon by an optical Stark shift Sc when the control field is applied, and it leads to a significant change of the reflection and transmission coefficients (r and t) in the left- and right-scattering directions, as will be shown in Section 3.

Finally, the intensity reflection and transmission can be given by
R=|r|2andT=|t|2.
(24)

3. Plasmon scattering properties and grating effect

In the calculations, the choices of the system parameters are based on the result of Ref. [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]

]. The Purcell factor (P ≡ Γple) is taken to be 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

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]

]. In this case, the NV center is positioned near the silver nanowire about 78 nm. The wire diameter and the wire length are about D ≃ 42 nm and L ≃ 1.3 μm, respectively. All the parameters used in this paper are scaled by the decay rate into the surface plasmons, i.e., Γpl.

First of all, we briefly analyze the principal mechanism behind single-plasmon scattering grating. It follows from Eq. (22) and Eq. (23) that the standing-wave-induced Stark shift Sc(z)=|Ωc(z)|2iΔωΓs2 plays a crucial role in creating the alternating regions of both high reflection and absorption as well as high transmission and absorption of a single plasmon, which can be used for a type of scattering grating. That is to say, when the coupling field is applied, the NV center would be detuned from the nanowire surface plasmon by an optical Stark shift Sc, 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 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.

In what follows, we illustrate explicitly these points above by comparing the variation of the spectral line-shapes for the reflection R, transmission T and loss (loss = 1 – RT) 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 Ωc(z) is located at the node (Ωc(z) = 0, see Fig. 2(a)) for frequencies lying near the resonance point Δω = 0, with Rω = 0) ≈ 0.91 whereas Tω = 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 Δω = 0. In contrast, when the standing-wave control field Ωc(z) is located at the antinode (Ωc(z) = 0.5Γpl, see Fig. 2(b)), the reflection R decreases rapidly and transmission T increases quickly near the resonance point, where Rω = 0) ≈ 0 whereas Tω = 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 Δω = Ωc and Δω = −Ωc, respectively. One central dip in the reflection line-shape and one central peak in the transmission line-shape lie at the resonance point Δω = 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 Ωc gives rise to the splitting of the upper level |e〉 into two dressed sublevels |±(|+=12(|e+|s) and |=12(|e|s)), which correspond to the energy separation λ± = ±Ωc. These two sideband peaks/dips are located at λ± = ±Ωc, which correspond to the dress-state transitions |g〉 ↔ |+〉, |g〉 ↔ |−〉 for the surface plasmon. One central dip/peak at resonant point λ0 = 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〉 ↔ |−〉.

Fig. 2 Reflection R, transmissions T and loss versus Δω. Panel (a) is produced without the control field [Ωc (z)/Γpl = 0]. Panel (b) is produced with the control field [Ωc (z)/Γpl = 0.5]. Note that, the intensity of the control field which is used for producing panels (a) and (b), corresponds to the peak intensity of a standing-wave field. The other system parameters are chosen as Γepl = 0.1 and Γspl = 0.001, respectively. The inset of (b) shows an enlarged view of the spectral line-shapes near resonant point Δω = 0.

In a word, for the situations considered in Figs. 2(a) and 2(b), conditioned on the position of the standing-wave control field Ωc(z), the reflection and transmission characteristics of the coupled system can transmit from those of Fig. 2(a) for the case Ωc(z) = 0 at the position of node to those of Fig. 2(b) for Ωc(z) = 0.5Γpl at the position of antinode. That is to say, a standing-wave control field triggers the change {Rω = 0) ≈ 0.91,Tω = 0) ≈ 0} for Ωc(z) = 0 ⇔ {Rω = 0) ≈ 0,Tω = 0) ≈ 1} for Ωc(z) = 0.5Γpl. 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.

In order to gain a deeper insight into the dependence of the reflection and transmission spectra of the surface-plasmon-NV-center coupled system on the control field at resonance Δω = 0, Figure 3 displays the reflection R and transmission T versus the control-field intensity Ωc. From this figure, we can find interesting and useful phenomena: (i) When the control field Ωc is switched off, the input plasmon field αR,in transmits into αL,out. In this case, the intensity reflection 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 Ωc is switched on, the reflection 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 Ωc at the output of the nanowire (see dashed curve). Obviously, we can see that the control-field intensity Ωc = 0 suggests left-scattering channel αR,inαL,out is switched on whereas right-scattering channel αR,inαR,out is switched off. Instead, the control-field intensity Ωc ≠ 0 (e.g., Ωc = 0.5Γpl) indicates left- scattering channel αR,inαL,out is switched off whereas right- scattering channel αR,inαR,out 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 Ωc. By this way, we can design an efficient single-plasmon all-optical switching.

Fig. 3 Reflection R (solid line) and transmissions T (dashed line) versus the standing-wave control field Ωcpl for Δω = 0. The other system parameters are chosen as Γepl = 0.1 and Γspl = 0.001, respectively.

Finally, we show how single-plasmon scattering grating can be implemented with a surface plasmon in a metal nanowire coupled to an NV center in a diamond nanocrystal. In Fig. 4(a), the reflection function 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 Sc. This leads to a periodic amplitude modulation across the output profile of the nanowire, a phenomenon reminiscent of the amplitude grating.

Fig. 4 Reflection R (panel (a)) and transmissions T (panel (b)) a function of position z. The standing-wave control field: Ωc(z)=Ωc0sin(πzΛ), where Λ is the spatial period of the standing wave along the z direction. The system parameters are chosen as Ωc0pl = 0.5, Γepl = 0.1, Γspl = 0.001, and Λ = 1 μm, respectively.

Likewise, the transmission function 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

In summary, we have demonstrated how single-plasmon scattering grating, induced by the classical standing-wave control field, can be implemented with the surface plasmon in metal nanowire coupled to the NV center in diamond nanocrystal. Such a approach to induce the grating gets out the well investigating region in which the weak coupling interactions between single atoms and light is often used. The whole process proposed here is implemented in metal nanowire and diamond nanocrystal, so it is compatible with the modern semiconductor fabrication. The proposal may be used for the chip-integrated grating, switcher and can be a promising prototype for multi-channel drop filter. Finally, it is pointed out that we consider only a single NV color center in diamond nanocrystal coupled to a surface plasmon in metal nanowire in our treatment. For a practical application, a lot of NV color centers with the same spatial distance to the metal nanowire are required to make up the grating efficiently.

Acknowledgment

We would like to thank Professor Ying Wu for his encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under Grants No. 11004069, No. 11104210, and No. 91021011, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081, by the National Basic Research Program of China under Contract No. 2012CB922103, and by the Fundamental Research Funds from Huazhong University of Science and Technology (HUST) under Grant No. 2010MS074.

References and links

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]

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]

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]

11.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007). [CrossRef] [PubMed]

12.

G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008). [CrossRef] [PubMed]

13.

A. Gonzalez-Tudela, F. J. Rodríguez, L. Quiroga, and C. Tejedor, “Dissipative dynamics of a solid-state qubit coupled to surface plasmons: From non-Markov to Markov regimes,” Phys. Rev. B 82, 115334 (2010). [CrossRef]

14.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B 82, 075427 (2010). [CrossRef]

15.

W. Chen, G. Y. Chen, and Y. N. Chen, “Coherent transport of nanowire surface plasmons coupled to quantum dots,” Opt. Express 18, 10360–10368 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-10-10360. [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]

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]

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]

20.

R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond,” Nat. Phys. 1, 94–98 (2005). [CrossRef]

21.

T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006). [CrossRef]

22.

M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science 316, 1312–1316 (2007). [CrossRef]

23.

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320,, 352–355 (2008). [CrossRef] [PubMed]

24.

Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006). [CrossRef] [PubMed]

25.

M. Larsson, K. N. Dinyari, and H. Wang, “Composite optical microcavity of diamond nanopillar and silica microsphere,” Nano Lett. 9, 1447–1450 (2009). [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]

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]

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]

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]

32.

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

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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References

  1. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A57, 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. A59, 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. A81, 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]
  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]
  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]
  11. C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett.7, 2784–2788 (2007). [CrossRef] [PubMed]
  12. G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett.33, 2212–2214 (2008). [CrossRef] [PubMed]
  13. A. Gonzalez-Tudela, F. J. Rodríguez, L. Quiroga, and C. Tejedor, “Dissipative dynamics of a solid-state qubit coupled to surface plasmons: From non-Markov to Markov regimes,” Phys. Rev. B82, 115334 (2010). [CrossRef]
  14. D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Greens function approach,” Phys. Rev. B82, 075427 (2010). [CrossRef]
  15. W. Chen, G. Y. Chen, and Y. N. Chen, “Coherent transport of nanowire surface plasmons coupled to quantum dots,” Opt. Express18, 10360–10368 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-10-10360 . [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. B76, 035420 (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]
  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]
  20. R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D. Awschalom, “Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond,” Nat. Phys.1, 94–98 (2005). [CrossRef]
  21. T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys.2, 408–413 (2006). [CrossRef]
  22. M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science316, 1312–1316 (2007). [CrossRef]
  23. R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science320,, 352–355 (2008). [CrossRef] [PubMed]
  24. Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett.6, 2075–2079 (2006). [CrossRef] [PubMed]
  25. M. Larsson, K. N. Dinyari, and H. Wang, “Composite optical microcavity of diamond nanopillar and silica microsphere,” Nano Lett.9, 1447–1450 (2009). [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]
  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. B74, 104303 (2006). [CrossRef]
  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]
  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. A79, 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]
  32. Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A54, 1586–1592 (1996). [CrossRef] [PubMed]

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