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

Jiahua Li; Rong Yu

doi:10.1364/OE.19.020991

1. Introduction

In recent years, electromagnetically induced grating (EIG) [1, 2] 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]. Afterward, it was demonstrated that such an EIG effect can be used to achieve tunable photonic band gap [4], to store probe pulses in a vapor of rubidium atoms [5], to implement optical routing [6, 7], to devise a dynamic controlled cavity [8], etc. However, as shown in [1], 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 [9–18]. A substantial increase in the coupling strength g_sp between the surface-plasmon modes and any proximal emitter with a dipole-allowed transition can be achieved because $g_{sp} \propto 1 / \sqrt{V_{eff}}$ where the effective mode volume V_eff for the plasmons can be significantly decreased [10]. An effective Purcell factor P = Γ_pl/Γ′ can exceed 10³ in realistic systems according to the theoretical results of Refs. [16, 17], where Γ_pl 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].

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 [19–26]. 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]. 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 [1–3]. 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]. As a result, the ground state is spin triplet and labelled as ³A, with the levels m_s = ±1 nearly degenerated and a zero-field splitting D_gs = 2.88 GHz between the states m_s = 0 and m_s = ±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, 29], 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], but is resonantly coupled to the excited state |e〉 via a classical standing-wave control field with central frequency ω_c and position-dependent coupling strength Ω_c (z). The states |g〉 and |e〉 are coupled with strength g_sp via the nanowire surface plasmon modes of frequency ω which is described by annihilation operations â_R,_ω and â_L,_ω [30, 31]. The states |g〉, |s〉, and |e〉 have the energy ω_g = 0 (the energy origin), ω_s, and ω_e, respectively. The standing-wave control field satisfies the resonance condition: ω_s + ω_c = ω_e. Because the coupling strength g_sp is broadband, it can be assumed to be frequency-independent [10, 16, 17]. Under these circumstances, the Hamiltonian of the whole system in the rotating-wave approximation (RWA) can be written as

\begin{array}{l} \hat{ℋ} & = & \bar{h} ω_{e} {\hat{σ}}_{ee} + \bar{h} ω_{s} {\hat{σ}}_{ss} - \bar{h} [Ω_{c} (z) e^{- i ω_{c} t} {\hat{σ}}_{es} + Ω_{c}^{*} (z) e^{i ω_{c} t} {\hat{σ}}_{se}] \\ + \int_{- \infty}^{+ \infty} \bar{h} ω {\hat{a}}_{R, ω}^{†} {\hat{a}}_{R, ω} d ω + \int_{- \infty}^{+ \infty} \bar{h} ω {\hat{a}}_{L, ω}^{†} {\hat{a}}_{L, ω} d ω \\ - \bar{h} g_{s p} \int_{- \infty}^{+ \infty} ({\hat{a}}_{R, ω} {\hat{σ}}_{e g} + {\hat{a}}_{R, ω}^{†} {\hat{σ}}_{g e}) d ω \\ - \bar{h} g_{s p} \int_{- \infty}^{+ \infty} ({\hat{a}}_{L, ω} {\hat{σ}}_{e g} + {\hat{a}}_{L, ω}^{†} {\hat{σ}}_{g e}) d ω, \end{array}

with 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_es/2h̄, with μ_es 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 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.

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. g_sp is the coupling strength between the NV center and metal nanowire, and Ω_c (z) is the position-dependent coupling strength between the NV center and the standing-wave control field. D_gs = 2.88 GHz is the zero-field splitting between the ground state sublevels m_s = 0 and m_s = ±1 of the NV center (m_s = ±1 are degenerate at zero magnetic field).

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To remove the time dependence in ℋ̂, we choose the proper free Hamiltonian ℋ̂₀ to transform to the interaction picture [32], that is,

{\hat{ℋ}}_{int} = e^{i {\hat{ℋ}}_{0} t / \bar{h}} {\hat{ℋ}}_{res} e^{- i {\hat{ℋ}}_{0} t / \bar{h}},

{\hat{ℋ}}_{0} / \bar{h} = ω_{s} {\hat{σ}}_{s s} + ω_{e} {\hat{σ}}_{e e} + \int_{- \infty}^{+ \infty} ω_{e} ({\hat{a}}_{R, ω}^{†} {\hat{a}}_{R, ω} + {\hat{a}}_{L, ω}^{†} {\hat{a}}_{L, ω}) d ω,

\begin{array}{l} {\hat{ℋ}}_{res} / \bar{h} & = & \int_{- \infty}^{+ \infty} Δ ω ({\hat{a}}_{R, ω}^{†} {\hat{a}}_{R, ω} + {\hat{a}}_{L, ω}^{†} {\hat{a}}_{L, ω}) d ω \\ - [Ω_{c} (z) e^{- i ω_{c} t} {\hat{σ}}_{e s} + Ω_{c}^{*} (z) e^{i ω_{c} t} {\hat{σ}}_{s e}] \\ - g_{s p} \int_{- \infty}^{+ \infty} [({\hat{a}}_{R, ω} + {\hat{a}}_{L, ω}) {\hat{σ}}_{e g} + ({\hat{a}}_{R, ω}^{†} + {\hat{a}}_{L, ω}^{†}) {\hat{σ}}_{g e}] d ω, \end{array}

where Δω = ω – ω_e is the frequency detuning. The resulting interaction Hamiltonian in the interaction picture can be then reexpressed as follows

\begin{array}{l} {\hat{ℋ}}_{int} / \bar{h} & = & \int_{- \infty}^{+ \infty} Δ ω ({\hat{a}}_{R, ω}^{†} {\hat{a}}_{R, ω} + {\hat{a}}_{L, ω}^{†} {\hat{a}}_{L, ω}) d ω - [Ω_{c} (z) {\hat{σ}}_{e s} + Ω_{c}^{*} (z) {\hat{σ}}_{s e}] \\ - g_{s p} \int_{- \infty}^{+ \infty} [({\hat{a}}_{R, ω} + {\hat{a}}_{L, ω}) {\hat{σ}}_{e g} + ({\hat{a}}_{R, ω}^{†} + {\hat{a}}_{L, ω}^{†}) {\hat{σ}}_{g e}] d ω . \end{array}

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)〉 = A_g|g,vac〉 + A_IHS|Ψ^IHS (t)〉 in the interaction picture, where

\begin{array}{l} | Ψ^{IHS} (t) 〉 & = & \int_{- \infty}^{+ \infty} [α_{R, ω} (t) {\hat{a}}_{R, ω}^{†} | g, vac 〉 + α_{L, ω} (t) {\hat{a}}_{L, ω}^{†} | g, vac 〉] d ω \\ + α_{e} (t) | e, vac 〉 + α_{s} (t) | s, vac 〉 . \end{array}

Firstly, by making use of the interaction Hamiltonian operator [see Eq. (5)] and the well-known Schrödinger equation $i \bar{h} \frac{\partial}{\partial t} | Ψ^{IHS} (t) 〉 = {\hat{H}}_{int} | Ψ^{IHS} (t) 〉$ 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

{\dot{α}}_{R, ω} (t) = - i Δ ω α_{R, ω} (t) + i g_{s p} α_{e} (t),

{\dot{α}}_{L, ω} (t) = - i Δ ω α_{L, ω} (t) + i g_{s p} α_{e} (t),

{\dot{α}}_{e} (t) = - \frac{Γ_{e}}{2} α_{e} (t) + i Ω_{c} (z) α_{s} (t) + i g_{s p} \int_{- \infty}^{+ \infty} [α_{R, ω} (t) + α_{L, ω} (t)] d ω,

{\dot{α}}_{s} (t) = - \frac{Γ_{s}}{2} α_{s} (t) + i Ω_{c}^{*} (z) α_{e} (t),

where Γ_e denotes the decay rate of the excited state of the NV center in diamond and Γ_s 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

α_{R, ω} (t) = α_{R, ω} (t_{0}) e^{- i Δ ω (t - t_{0})} + i g_{s p} \int_{t_{0}}^{t} α_{e} (t^{'}) e^{- i Δ ω (t - t^{'})} d t^{'},

α_{L, ω} (t) = α_{L, ω} (t_{0}) e^{- i Δ ω (t - t_{0})} + i g_{s p} \int_{t_{0}}^{t} α_{e} (t^{'}) e^{- i Δ ω (t - t^{'})} d t^{'},

where α_R(L),ω (t₀) denotes the value of α_R(L),ω (t) at t = t₀.

Now, by substituting α_R,ω (t) and α_L,ω (t) from Eq. (11) and Eq. (12) into Eq. (9), we can obtain

{\dot{α}}_{e} (t) = - (\frac{Γ_{e}}{2} + Γ_{pl}) α_{e} (t) + i Ω_{c} (z) α_{s} (t) + i \sqrt{2 π} g_{s p} [α_{R, in} (t) + α_{L, in} (t)],

where

Γ_{p l} = 2 π g_{s p}^{2}

is the decay rate into the surface plasmons. The input right-going (left-going) surface plasmon is defined as

α_{R (L), in} (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} α_{R (L), ω} (t_{0}) e^{- i Δ ω (t - t_{0})} d ω

. Above, we have applied the relationships

\frac{1}{2 π} \int_{- \infty}^{+ \infty} e^{- i Δ ω (t - t^{'})} d ω = δ (t - t^{'})

and

\int_{t_{0}}^{t} α_{e} (t^{'}) δ (t - t^{'}) = \frac{1}{2} α_{e} (t)

(Here, notice that we integrate over half the delta-function, which results in a factor of

\frac{1}{2}

).

Similarly, we integrate Eq. (7) and Eq. (8) up to a final time t₁ > t to get

α_{R, ω} (t) = α_{R, ω} (t_{1}) e^{- i Δ ω (t - t_{1})} - i g_{s p} \int_{t}^{t_{1}} α_{e} (t^{'}) e^{- i Δ ω (t - t^{'})} d t^{'},

α_{L, ω} (t) = α_{L, ω} (t_{1}) e^{- i Δ ω (t - t_{1})} - i g_{s p} \int_{t}^{t_{1}} α_{e} (t^{'}) e^{- i Δ ω (t - t^{'})} d t^{'},

where α_R(L),ω (t₁) denotes the value of α_R(L),ω (t) at t = t₁. Carrying out the same procedure as above, we can achieve

{\dot{α}}_{e} (t) = - (\frac{Γ_{e}}{2} - Γ_{p l}) α_{e} (t) + i Ω_{c} (z) α_{s} (t) + i \sqrt{2 π} g_{s p} [α_{R, out} (t) + α_{L, out} (t)],

where

α_{R (L), out} (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} α_{R (L), ω} (t_{1}) e^{- i Δ ω (t - t_{1})} d ω

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

α_{R, out} - α_{R, in} = i \sqrt{2 π} g_{s p} α_{e},

α_{L, out} - α_{L, in} = i \sqrt{2 π} g_{s p} α_{e} .

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 $ℱ (Δ ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} ℱ (t) e^{i Δ ω t} dt$ on the amplitudes α_s and α_e in Eq. (10) and Eq. (13), respectively. After carrying out some algebraic calculations, the analytical solution of the amplitude α_e can be found as

α_{e} (z) = \frac{- i \sqrt{2 π} g_{s p} (α_{R, in} + α_{L, in})}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}} .

According to the input-output formalism $α_{R, out} - α_{R, in} = i \sqrt{2 π} g_{s p} α_{e}$ and $α_{L, out} - α_{L, in} = i \sqrt{2 π} g_{s p} α_{e}$ , finally we can readily derive the relationship between the input optical field and the output optical field

α_{R, out} (z) = \frac{i Δ ω - \frac{Γ_{e}}{2} + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}} α_{R, in} + \frac{Γ_{p l}}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}} α_{L, in},

α_{L, out} (z) = \frac{Γ_{p l}}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}} α_{R, in} + \frac{i Δ ω - \frac{Γ_{e}}{2} + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}} α_{L, in} .

For the initial given input fields α_R,in ≠ 0 and α_L,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 = \frac{α_{L, out}}{α_{R, in}} = \frac{Γ_{p l}}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + S_{c} (z)},

t = \frac{α_{R, out}}{α_{R, in}} = 1 + r = \frac{i Δ ω - \frac{Γ_{e}}{2} + S_{c} (z)}{i Δ ω - (\frac{Γ_{e}}{2} + Γ_{p l}) + S_{c} (z)},

where

S_{c} (z) = \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}

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 α_R,in = 0 and α_L,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 Ω_c (z) = 0, we have S_c = 0. In this case Eq. (22) and Eq. (23) can reduce to the same expressions appearing in Ref. [10]. Thirdly, it is shown that the NV center would be detuned from the surface plasmon by an optical Stark shift S_c 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 |}^{2} and T = {| t |}^{2} .

3. Plasmon scattering properties and grating effect

In the calculations, the choices of the system parameters are based on the result of Ref. [10]. The Purcell factor (P ≡ Γ_pl/Γ_e) 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, 17]. 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., Γ_pl.

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 $S_{c} (z) = \frac{{| Ω_{c} (z) |}^{2}}{i Δ ω - \frac{Γ_{s}}{2}}$ 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 S_c, 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 – 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 Δω = 0, with R(Δω = 0) ≈ 0.91 whereas T(Δω = 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 Ω_c(z) is located at the antinode (Ω_c(z) = 0.5Γ_pl, see Fig. 2(b)), the reflection R decreases rapidly and transmission T increases quickly near the resonance point, where R(Δω = 0) ≈ 0 whereas T(Δω = 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 = Ω_c and Δω = −Ω_c, 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 Ω_c gives rise to the splitting of the upper level |e〉 into two dressed sublevels $| \pm 〉 (| + 〉 = \frac{1}{\sqrt{2}} (| e 〉 + | s 〉)$ and $| - 〉 = \frac{1}{\sqrt{2}} (| e 〉 - | s 〉))$ , which correspond to the energy separation λ_± = ±Ω_c. These two sideband peaks/dips are located at λ_± = ±Ω_c, 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 [Ω_c (z)/Γ_pl = 0]. Panel (b) is produced with the control field [Ω_c (z)/Γ_pl = 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 Γ_e/Γ_pl = 0.1 and Γ_s/Γ_pl = 0.001, respectively. The inset of (b) shows an enlarged view of the spectral line-shapes near resonant point Δω = 0.

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In a word, for the situations considered in Figs. 2(a) and 2(b), conditioned on the position of the standing-wave control field Ω_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_pl at the position of antinode. That is to say, a standing-wave control field triggers the change {R(Δω = 0) ≈ 0.91,T(Δω = 0) ≈ 0} for Ω_c(z) = 0 ⇔ {R(Δω = 0) ≈ 0,T(Δω = 0) ≈ 1} for Ω_c(z) = 0.5Γ_pl. 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 Ω_c. From this figure, we can find interesting and useful phenomena: (i) When the control field Ω_c is switched off, the input plasmon field α_R,in transmits into α_L,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 Ω_c 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 Ω_c 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 α_R,in ⇒ α_L,out is switched on whereas right-scattering channel α_R,in ⇒ α_R,out is switched off. Instead, the control-field intensity Ω_c ≠ 0 (e.g., Ω_c = 0.5Γ_pl) indicates left- scattering channel α_R,in ⇒ α_L,out is switched off whereas right- scattering channel α_R,in ⇒ α_R,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 Ω_c. 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 Ω_c/Γ_pl for Δω = 0. The other system parameters are chosen as Γ_e/Γ_pl = 0.1 and Γ_s/Γ_pl = 0.001, respectively.

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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 S_c. 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: $Ω_{c} (z) = Ω_{c 0} sin (\frac{π z}{Λ})$ , where Λ is the spatial period of the standing wave along the z direction. The system parameters are chosen as Ω_c₀/Γ_pl = 0.5, Γ_e/Γ_pl = 0.1, Γ_s/Γ_pl = 0.001, and Λ = 1 μm, respectively.

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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.

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

Abstract

1. Introduction

2. Description of the model system and observables

3. Plasmon scattering properties and grating effect

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Equations (24)

Optics Express