Robust-to-loss entanglement generation using a quantum plasmonic nanoparticle array

If we take a single arm, say the left-hand side of figure 1(a), then depending on the state of the QD, both the transmission and reflection coefficients of the metal nanoparticle array exhibit different characteristics. When the QD is in the state |m〉, i.e. when it is decoupled from the adjacent nanoparticle due to the large detuning (see appendix A.1), the characteristics of the transmission and reflection coefficients of the source nanowire reveal a similar behaviour to those of a single array of nanoparticles (see [13]), where the plasmonic field is transmitted across the array when it is on-resonance with one of the eigenmodes. On the other hand, when the QD is in the state |g〉, i.e. when it is coupled to the adjacent nanoparticle, the propagating plasmons cannot pass through the adjacent nanoparticle and are completely reflected back into the source nanowire due to a phenomenon called dipole induced reflection (DIR), which is an alternative form of DIT [25]. Here, DIR occurs when the operating frequency ω of the plasmons is resonant with the |g〉–|e〉 transition frequency (as shown in appendix A.1). The entanglement generation scheme then works as follows. Consider that light in the form of plasmon excitations are injected into the system via both of the source nanowires at the same time and that the QDs are prepared in a superposition, $\frac {1}{\sqrt {2}}(\vert {g}\rangle _1+\vert {m}\rangle _1)\otimes \frac {1}{\sqrt {2}}(\vert {g}\rangle _2+\vert {m}\rangle _2)$. If a plasmon is detected coming out from drain 1 for given input states at the sources, for example coherent states, then the plasmon came from the cases where either one of QDs is in |m〉, or both are in |m〉. A destructive interference of the case |m〉₁⊗|m〉₂ can be induced by controlling the phases of the initial coherent states, so that only states |m〉₁⊗|g〉₂ and |g〉₁⊗|m〉₂ lead to the detection event and consequently a superposed state of |g〉₁⊗|m〉₂ and |m〉₁⊗|g〉₂ is generated after the detection, i.e. $\vert {\psi ^-}\rangle =\frac {1}{\sqrt {2}}(\vert {m}\rangle _1\otimes \vert {g}\rangle _2-\vert {g}\rangle _1\otimes \vert {m}\rangle _2)$, which is an entangled state of the QDs. Likewise, the detection of a plasmon coming from drain 2 can be also used for generating the entangled state |ψ⁻〉. This requires different phases of the initial states compared to the case of the detection at drain 1, similar to the case of a bulk beam splitter which has the transformation $\vert {\alpha }\rangle \vert {\beta }\rangle \rightarrow \vert {(\alpha +\beta )/\sqrt {2}}\rangle \vert {(\alpha -\beta )/\sqrt {2}}\rangle$ so the output port is controlled by the input phases. Here, we focus on the case of the detection at drain 1. This is the basic concept for generating entanglement between two QDs using the plasmonic modes of a metal nanoparticle array in the ideal case when there is no loss. Here, we have made use of the characteristics of the transmission and reflection coefficients depending on the internal state of the QDs and an appropriate detection event.

Before describing the entanglement generation scheme in more detail and showing its robustness to loss from the nanoparticles, we first briefly introduce the model to describe the system in figure 1(a). We begin with the Hamiltonian for the nanoparticle system which is given by

$$\begin{equation} \hat{H}_{\rm np}=\sum_{i=1}^{2n+2}\hbar \omega_{i} \hat{a}_{i}^{\dagger} \hat{a}_{i} +\sum_{[i,j]}\hbar g_{i,j}\left( \hat{a}^{\dagger}_i \hat{a}_j+\hat{a}^{\dagger}_j \hat{a}_i \right), \end{equation} \tag{ 1 }$$

where ω_i is the natural frequency of the field oscillation at the ith nanoparticle, g_i,j is the coupling strength between the fields of the ith and jth nanoparticles, and [i,j] denotes a summation over adjacent neighbours j for a given nanoparticle i. The natural frequency ω_i satisfies the Fröhlich criterion Re [_m(ω_i)] = −2_d [12, 31], which considers the nanoparticles to be small enough compared to the operating wavelength such that only dipole-active excitations are important [32]. The dielectric function of the metal _m(ω) is given by a Drude–Sommerfeld model, which gives rise to a best fit to experimental data at frequencies corresponding to free space wavelengths λ₀ ≳ 350 (530) nm for silver (gold) [33]. For simplicity we take all nanoparticles to have the same permittivity _m and radius R, thus the local frequencies are set to be equal, ω_i = ω₀, for i = 1,...,2n + 2. The operators $\hat {a}^{\dagger} _i$ ($\hat {a}_i$) represent creation (annihilation) operators associated with a dipole-field excitation at the ith nanoparticle, which obey bosonic commutation relations $[\hat {a}_i,\hat {a}^{\dagger} _j]=\delta _{ij}$. Here, a macroscopic quantization of the fields is used, where the field modes are defined as localized solutions to Maxwell's equations satisfying the boundary conditions of the metal–dielectric interface [34]. In this case, the electron response in the metal is contained within the dielectric function of the metal, _m(ω) [13, 35, 36].

The interaction term in equation (1) involves two approximations for the centre-to-centre distance, d, between nanoparticles: a point-dipole approximation (3R ⩽ d), where multipolar interactions are negligible [37], and a near-field approximation, (d ≪ λ), where λ is the wavelength of the nanoparticle dipole field [10, 14], where the nearest-neighbour interaction is dominant via the Föster fields with a d⁻³ distance dependence [38]. These two approximations for d are covered simultaneously by a weak-coupling approximation, where the couplings between nanoparticles, g_i,j, are much less than the natural frequency, ω₀, [12, 14], i.e. |g_i,j| ≪ ω₀ for which we set max|g_i,j| = 0.1ω₀ throughout the work.

The Hamiltonians for the QDs are then given by

$$\begin{eqnarray*} &&\hat{H}_{\rm QD_{1}}^{g~(m)} = \frac{1}{2} \hbar \Omega_{1}^{g~(m)} \hat{\sigma}_{z,1}^{g~(m)} +\hbar J_{1}^{g~(m)} \left( \hat{a}_{1} \hat{\sigma}^{g~(m) \dagger}_{1} + \hat{a}_{1}^{\dagger} \hat{\sigma}_{1}^{g~(m)} \right),\\ &&\hat{H}_{\rm QD_{2}}^{g~(m)} = \frac{1}{2} \hbar \Omega_{2}^{g~(m)} \hat{\sigma}_{z,2}^{g~(m)} +\hbar J_{2}^{g~(m)} \left( \hat{a}_{2n+2} \hat{\sigma}^{g~(m) \dagger}_{2} + \hat{a}_{2n+2}^{\dagger} \hat{\sigma}_{2}^{g~(m)} \right), \end{eqnarray*}$$

where Ω^g(m)₁ is the transition frequency between |g〉 and |e〉 (|m〉 and |e〉) of QD₁ on the left-hand side, $\hat {\sigma }_{1}^{g~(m)}=\vert {g}\rangle \langle {e}\vert ~(\vert {m}\rangle \langle {e}\vert )$, $\hat {\sigma }_{1}^{g~(m) \dagger }=\vert {e}\rangle \langle {g}\vert ~(\vert {e}\rangle \langle {m}\vert )$ and $\hat {\sigma }_{z,1}^{g~(m)}=\vert {e}\rangle \langle {e}\vert -\vert {g}\rangle \langle {g}\vert ~(\vert {e}\rangle \langle {e}\vert -\vert {m}\rangle \langle {m}\vert )$. The vacuum Rabi frequency of QD₁ when coupled to the first nanoparticle dependent on the individual directions of the pairwise dipole moments of |g〉–|e〉 and |m〉–|e〉 transitions of the QD, J^g(m)₁, is obtained within the rotating-wave approximation and the dipolar approximation, where the distance between the nanoparticle and QD₁ is assumed to be larger than the radius of the nanoparticle [34, 39]. Similar denotations and considerations as given above are used for QD₂ on the right-hand side.

With the Hamiltonians introduced, the total system including the nanoparticle baths (for modelling loss) and the nanowires (sources and drains) can then be described in terms of Heisenberg equations of motion by using input–output formalism [13, 40, 41]. The Heisenberg equations for the left-hand side of the system (from first to (n + 1)th nanoparticle and QD₁) are given by

$$\begin{equation} \fl \frac{\mathrm{d}\hat{a}_{1}}{\mathrm{d}t} = -\left(\mathrm{i} \omega_{0} + \frac{g_{\rm s_{1}}}{2}+\frac{\Gamma_{1}}{2}\right) \hat{a}_{1} - \mathrm{i} g_{1,2} \hat{a}_{2} + \sqrt{g_{\rm s_{1}}}\hat{s}_{{\rm in},1} + \sqrt{\Gamma_{1}}\hat{b}_{{\rm in},1} - \mathrm{i} J_{1}^{g~(m)} \hat{\sigma}_{1}^{g~(m)}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \fl \frac{\mathrm{d}\hat{a}_{j}}{\mathrm{d}t} = -\left(\mathrm{i} \omega_{0} +\frac{\Gamma_{j}}{2}\right) \hat{a}_{j} - \mathrm{i} g_{j,j-1} \hat{a}_{j-1} - \mathrm{i} g_{j,j+1} \hat{a}_{j+1} + \sqrt{\Gamma_{j}}\hat{b}_{{\rm in},j} ~~~~ (j=2,3,\ldots,n-1), \end{equation} \tag{ 3 }$$

$$\begin{equation} \fl \frac{\mathrm{d}\hat{a}_{n}}{\mathrm{d}t} = -\left(\mathrm{i} \omega_{0} +\frac{\Gamma_{n}}{2}\right) \hat{a}_{n} - \mathrm{i} g_{n,n-1} \hat{a}_{n-1} - \mathrm{i} g_{n,n+1}\hat{a}_{n+1} - \mathrm{i} g_{n,n+3}\hat{a}_{n+3} + \sqrt{\Gamma_{n}}\hat{b}_{{\rm in},n}, \end{equation} \tag{ 4 }$$

$$\begin{equation} \fl \frac{\mathrm{d}\hat{a}_{n+1}}{\mathrm{d}t} = -\left(\mathrm{i} \omega_{0} + \frac{g_{\rm d_{1}}}{2} +\frac{\Gamma_{n+1}}{2}\right) \hat{a}_{n+1} - \mathrm{i} g_{n,n+1}\hat{a}_{n} - \mathrm{i} g_{n+1,n+2}\hat{a}_{n+2} + \sqrt{g_{\rm d_{1}}}\hat{d}_{{\rm in},1} + \sqrt{\Gamma_{n+1}}\hat{b}_{{\rm in},n+1}, \end{equation} \tag{ 5 }$$

$$\begin{equation} \fl \frac{\mathrm{d}\hat{\sigma}_{1}^{g~(m)}}{\mathrm{d}t} =-\left(\mathrm{i}\Omega_{1}^{g~(m)} +\frac{\gamma_{1}^{g~(m)}}{2}\right)\hat{\sigma}_{1}^{g~(m)}+\mathrm{i} J_{1}^{g~(m)}\hat{a}_{1} \hat{\sigma}_{z,1}^{g~(m)} +\hat{f}_{1}^{g~(m)}. \end{equation} \tag{ 6 }$$

Similar Heisenberg equations can be written for the right-hand side of the system (from (n + 2)th to (2n + 2)th nanoparticle and QD₂). The couplings between the nanowires and their adjacent nanoparticles are represented by g_s1, g_s2, g_d1 and g_d2. They are also assumed to obey the weak-coupling approximation, |g_s1|, |g_s2|, |g_d1| and |g_d2| ≪ ω₀, as for the interparticle coupling strengths g_i,j. This is imposed in our model for the nanowires, as the field profiles at the nanowire tips are similar in form to those of the nanoparticles, so that the model is a consistent description [13]. The damping rate Γ for each metal nanoparticle depends on its size, and is given by Matthiessen's rule [12]: Γ = v_F/λ_B + v_F/R, where λ_B is the bulk mean-free path of an electron, v_F is the velocity at the Fermi surface and R is the radius. By considering all the nanoparticles to have the same size and permittivity enables us to set Γ_i = Γ₀ for i = 1,...,2n + 2. The decay rate γ^g(m) of a QD from |e〉 to |g〉 (|e〉 to |m〉) is given by the standard Wigner–Weisskopf spontaneous emission rate of the dipole, where it is assumed that the nanoparticle does not modify it. The operators $\hat {s}_{{\mathrm { in}},1(2)}$, $\hat {d}_{{\mathrm { in}},1(2)}$, and $\hat {b}_{{\mathrm { in}},i}$ represent the input fields of the source nanowires, the drain nanowires and the ith nanoparticle's bath mode, respectively. The output fields of the nanowires, $\hat {s}_{{\mathrm { out}},1(2)}$ and $\hat {d}_{{\mathrm { out}},1(2)}$, and the nanoparticles' baths, $\hat {b}_{{\mathrm { out}},i}$, are related to the input fields by $\hat {a}_{1(2n+2)}=\frac {1}{\sqrt {g_{\mathrm{s_{1}}(\mathrm{s}_{2})}}}(\hat {s}_{{\mathrm { in}},1(2)}+\hat {s}_{{\mathrm { out}},1(2)})$, $\hat {a}_{n+1(n+2)}=\frac {1}{\sqrt {g_{\mathrm { d_{1}}(d_{2})}}}(\hat {d}_{{\mathrm { in}},1(2)}+\hat {d}_{{\mathrm { out}},1(2)})$ and $\hat {a}_{i}=\frac {1}{\sqrt {\Gamma _{0}}}(\hat {b}_{{\mathrm { in}},i}+\hat {b}_{{\mathrm { out}},i})$, for i = 1,...,2n + 2, respectively. Metal losses in the nanowires are not included in the above Heisenberg equations, but will be included and discussed later. The damping rates of the QDs are assumed to be relatively small compared to all other rates, so that the noise operators, $\hat {f}_{i}^{g~(m)}$ (i = 1,2), can be neglected in that they do not significantly affect the quantum coherence of the system.

The entanglement generation scheme works in the weak excitation regime, where the dipole is assumed to be predominantly in the initial state |g〉 or |m〉 (the validity of this regime is discussed in appendix B, where we consider the pulse profiles of the inputs). In this limit, $\hat {\sigma }_{z}^{g~(m)}$ can be replaced by its mean value, $\langle \hat {\sigma }_{z}^{g~(m)} \rangle \approx -1$, so that the coupled Heisenberg equations of motion give rise to four scattering matrices, one for each of the four internal states of the QDs (|mm〉, |gm〉, |mg〉 and |gg〉), with the general form

$$\begin{eqnarray} &&\fl \hat{s}_{{\rm in},1}^{\dagger}(\omega)=t_{\rm s_{1}s_{1}}(\omega) \hat{s}_{{\rm out},1}^{\dagger}(\omega)+t_{\rm s_{1}s_{2}}(\omega)\hat{s}_{{\rm out},2}^{\dagger}(\omega) +t_{\rm s_{1}d_{1}}(\omega) \hat{d}_{{\rm out},1}^{\dagger}(\omega)+t_{\rm s_{1}d_{2}}(\omega)\hat{d}_{{\rm out},2}^{\dagger}(\omega)+\hat{f}_{{\rm out}, s_{1}}^{\dagger}(\omega), \nonumber\\\\ &&\fl \hat{s}_{{\rm in},2}^{\dagger}(\omega)=t_{\rm s_{2}s_{1}}(\omega) \hat{s}_{{\rm out},1}^{\dagger}(\omega)+t_{\rm s_{2}s_{2}}(\omega)\hat{s}_{{\rm out},2}^{\dagger}(\omega) +t_{\rm s_{2}d_{1}}(\omega) \hat{d}_{{\rm out},1}^{\dagger}(\omega)+t_{\rm s_{2}d_{2}}(\omega)\hat{d}_{{\rm out},2}^{\dagger}(\omega)+\hat{f}_{{\rm out}, s_{2}}^{\dagger}(\omega),\nonumber \end{eqnarray} \tag{ 7 }$$

where the noise operators for the nanoparticles are $\hat {f}_{{\mathrm { out}}, s_{1}}^{\dagger }(\omega )=\sum _{i}^{2n+2} t_{\mathrm { s_{1}}b_{\it{i}}}(\omega )\hat {b}_{{\mathrm { out}},i}^{\dagger }(\omega )$ and $\hat {f}_{{\mathrm { out}}, s_{2}}^{\dagger }(\omega )=\sum _{i}^{2n+2} t_{\mathrm { s_{2}}b_{\it{i}}}(\omega )\hat {b}_{{\mathrm { out}},i}^{\dagger }(\omega )$. The coefficient $t_{\mathrm { \square \lozenge }}$ denotes the transition amplitude from an input mode '□' to an output mode '$\lozenge$'. According to the state of the QDs (|gg〉,|gm〉,|mg〉 and |mm〉), the transition amplitude from '□' to '$\lozenge$' when the states of the two QDs are in |x〉⊗|y〉, for x,y∈{m,g} is represented by the coefficient $t_{\mathrm { \square \lozenge }}^{xy}$. For the sake of generality we have defined a dipole operator $\hat {\sigma }_{1(2)}^{m}$ for the dipole state |m〉, with the |m〉–|e〉 transition frequency Ω^m₁₍₂₎ and damping rate γ^m₁₍₂₎. However, in what follows we will treat the system when QD₁₍₂₎ is in state |m〉 as if there is no QD₁₍₂₎, i.e. J^m₁₍₂₎ = 0, since it is effectively decoupled from the nanoparticle in this case. For simplicity, we choose all the couplings between nanoparticles in the two arms to be equal, g_j,k = g_np (at a fixed distance d, array orientation and nanoparticle size [12]), and the nanowire to nanoparticle couplings, g_s1 = g_s2 = g_d1 = g_s2 = g_in–out. The weak-coupling approximations for these coupling strengths are equivalent to |g_np| ≪ ω₀ and |g_in–out| ≪ ω₀. We impose these by setting max|g_np| = 0.1ω₀ and max|g_in–out| = 0.1ω₀, which are achieved by varying the distance between the nanoparticles, and between the nanowire tips and their adjacent nanoparticles, respectively, for given polarizations of electron-charge density oscillations [13]. In this work we will mainly focus on the case of Δω = ω − ω₀ = 0, which is approximately the same as the case of using an optical pulse with a small enough bandwidth around ω₀ such that all coefficients in equation (7) are slowly varying. The transitions from the ground (metastable) states to the excited states for the QDs are detuned by δ^g(m)₁ and δ^g(m)₂ from the resonant frequency ω₀ of the nanoparticles, as shown in figure 1(b).

From equation (7), the relation of the operators for the plasmons at resonance, ω = ω₀, can be written as a reduced matrix for each of the four internal states of the QDs to give

$$\begin{equation} \left(\begin{array}{@{}c@{}} \hat{s}_{{\rm in},1}^{\dagger}(\omega_{0}) \\ \hat{s}_{{\rm in},2}^{\dagger}(\omega_{0}) \end{array}\right) =\left(\begin{array}{@{}cc@{}} t_{\rm s_{1}d_{1}}(\omega_{0}) & t_{\rm s_{1}d_{2}}(\omega_{0}) \\ t_{\rm s_{2}d_{1}}(\omega_{0}) & t_{\rm s_{2}d_{2}}(\omega_{0}) \\ \end{array}\right) \left(\begin{array}{@{}c@{}} \hat{d}_{{\rm out},1}^{\dagger}(\omega_{0}) \\ \hat{d}_{{\rm out},2}^{\dagger}(\omega_{0}) \end{array}\right), \end{equation} \tag{ 8 }$$

where |t_s1(2)d1(ω₀)|² + |t_s1(2)d2(ω₀)|² ⩽ 1 as the fields are reduced due to the losses from the metal nanoparticles and the QDs, as well as transmissions (or back reflections) to the source nanowire modes $\hat {s}_{{\mathrm { out}},1(2)}^{\dagger }$. The equality holds only if there is no dissipation and back reflections. A 50/50 beam splitter of lossy nanoparticles is achieved by setting g_n,n+3 = g_n+1,n+2 = g_h = (g_in–out + Γ₀)/2 and $g_{n,n+1}=g_{n+2,n+3}=g_{\mathrm { v}}= \sqrt {2} g_{\mathrm { h}}$, while taking care of the polarization-dependence of the couplings between nanoparticles, such that t_s1d2(ω₀) = it_s1d1(ω₀) and t_s2d1(ω₀) = it_s2d2(ω₀) in equation (8) regardless of the internal states of the QDs. Here we assume no cross couplings between nanoparticles at the sites n and (n + 2) or (n + 1) and (n + 3) or (n − 1) and (n + 1) as they are relatively small compared to g_v and g_h with a d⁻³ distance dependence [38]. The coupling values for the 50/50 beam splitter set the transition amplitudes t_s1s2(ω₀) = t_s2s1(ω₀) = 0 in equation (7), which implies that any plasmonic field coming from the left-hand (right-hand) side source nanowire is not influenced by the state of the QD on the right-hand (left-hand) side, i.e. we have $t_{\mathrm { s_{1}} \lozenge }^{xy}=t_{\mathrm { s_{1}} \lozenge }^{xy'}$ and $t_{\mathrm { s_{2}} \lozenge }^{xy}=t_{\mathrm { s_{2}} \lozenge }^{x'y}$. Note that the configuration of four nanoparticles we use here provides the correct symmetric splitting operation [14].

The role of the 50/50 beam splitter in our scheme is to erase the 'which-way' information of the two pathways associated with the QD internal states |gm〉 and |mg〉. Hence, the amount of energy that leaks into other output modes at the beam splitter is not important as long as we can perfectly erase the which-way information by having the same transition probabilities for the two pathways. Such a lossy beam splitter of nanoparticles is based on the lossless beam splitter proposed by Yurke and Kuang [14]. For the case Δω ≠ 0, the above choice of coupling values for g_h and g_v does not guarantee that the nanoparticle configuration performs as a 50/50 beam splitter, i.e. it leads to a failure in the ability of the beam splitter to erase the which-way information. This introduces decoherence into the generated entangled state. In figure 2, concentrating on the left arm and the beam splitter of nanoparticles, we plot the transmission and reflection spectral profiles for plasmons injected into the left source nanowire, |t_s1s1|², |t_s1s2|², |t_s1d1|² and |t_s1d2|² as Δω/g_np is varied, with g_in–out/g_np = 0.5, Γ₀/g_np = 0.1, J^g₁/g_np = 0.3 and γ^g₁/g_np = 0.001 chosen as an example, when QD₁ is in |m〉 or |g〉. Here, we focus on the case of n = 2 (the minimum number of nanoparticles in our scheme) and 3 in the left arm as the representative of even or odd number of nanoparticles because the dependence of the transmission and reflection amplitude on the in–out coupling strength g_in–out is different for even and odd due to the resonances of the array (as shown in appendix A.2). For the case of QD₁ in state |m〉 (figures 2(a) and (b)), on resonance (Δω = 0), the left arm of nanoparticles follows the characteristics of a single array of nanoparticles and just transfers the plasmon field through the array to the beam splitter where it is split equally. The reason for the large backscattering (|t_s1s1|² probability) in the case of n = 2 (figure 2(a)) is due to the eigenmodes of the array (from the source to each drain) not being on resonance for Δω = 0, whereas for n = 3 (figure 2(b)) there is a resonant eigenmode at Δω = 0 for each drain [13]. For Δω ≠ 0 the various eigenmodes of the system come into resonance at particular frequencies of the input field and cause the oscillatory behaviour seen on either side of Δω = 0. On the other hand, for the case of QD₁ in state |g〉 (figures 2(c) and (d)), the first nanoparticle is coupled to QD₁ and DIR occurs (as shown in appendix A), regardless of the resonant eigenmodes in the nanoparticle system. Thus in both figures 2(c) and (d) at Δω = 0 the backscattered probability |t_s1s1|² goes to unity. Away from Δω = 0 the transition probabilities follow the same behaviour as for the case of QD₁ in state |m〉, as the QD becomes off resonant and so cannot induce any dipole-based interference effect.

Zoom In Zoom Out Reset image size

To generate entanglement between the QDs, both are initialized to be in an equal superposition of qubit states |g〉 and |m〉, i.e. $\vert {+}\rangle =\frac {1}{\sqrt {2}}(\vert {g}\rangle +\vert {m}\rangle )$. This can be achieved by first driving the QDs into the lowest-energy state and then rotating them by either a direct π/2 transition, or a Raman transition [42], depending on the specifics of the QD internal energy level structure. After initializing the QDs, a coherent field |α〉 with frequency ω₀ is injected into the left source, $\hat {s}_{{\mathrm { in}},1}^{\dagger }$, and simultaneously another coherent field |β〉 with same polarization and frequency as |α〉 is injected into the right source, $\hat {s}_{{\mathrm { in}},2}^{\dagger }$. The initial state of the whole system is then

$$\begin{eqnarray*} &&\vert{\Psi_{\mathrm{i}}}\rangle=\frac{1}{2}(\vert{g}\rangle_{1}+\vert{m}\rangle_{1}) (\vert{g}\rangle_{2}+\vert{m}\rangle_{2}) \vert{\alpha}\rangle_{\mathrm{s}_{1}} \vert{\beta}\rangle_{\mathrm{s}_{2}} \vert{0}\rangle_{\mathrm{d}_{1}} \vert{0}\rangle_{\mathrm{d}_{2}}. \end{eqnarray*}$$

The input plasmon fields interact with the QDs only if they are in the state |g〉. In the case that one (or both) of the QDs is in the state |m〉, a field is transferred along one (or both) of the arms, then mixed at the beam splitter and goes out through one of the drain nanowires, $\hat {d}_{{\mathrm { out}},1(2)}^{\dagger }$. Entanglement can be generated by the detection of an excitation in drain 1, which could have come from either the left or right arm, when the QDs were in |mg〉 and |gm〉 respectively (the case when both arms supply an excitation, with the QDs in |mm〉, is suppressed as described below). Thus, the nanowire mode $\hat {d}_{{\mathrm { out}},1}^{\dagger }$ is connected to a detector in order to postselect the successful case when the which-way information has been erased and the QDs have been entangled as a consequence. Alternatively, an on-chip detector could be directly coupled to the (n + 1)th nanoparticle so that the drain nanowire is not needed. Note that the total plasmonic system only requires a classical source of light in order to generate entanglement.

To see how the scheme works, it is useful to consider sufficiently weak coherent states (α ≪ 1, β ≪ 1) when there is no loss present. We may expand |Ψ_i〉 to first order in the plasmonic excitation number and approximately drop all other output modes. Then, as the scheme is based on the detection of an excitation we can drop the vacuum terms so that $\vert {\Psi _{\mathrm {i}}}\rangle \approx \frac {1}{2}(\vert {g}\rangle _{1}+\vert {m}\rangle _{1}) (\vert {g}\rangle _{2}+\vert {m}\rangle _{2}) (\alpha \vert {1}\rangle _{\mathrm {s}_{1}} +\beta \vert {1}\rangle _{\mathrm {s}_{2}})$. Perfect DIR leads to t^gg_s1d1 = t^gm_s1d1 = 0 and t^gg_s2d1 = t^mg_s2d1 = 0, and using equation (7) for each of the internal states of the QDs, together with a postselection of the state when an excitation is present in drain 1, we have the unnormalized state of the QDs as

$$\begin{equation} \vert{\Psi}\rangle_{\rm QDs} = \textstyle\frac{1}{2} \left\{ (\alpha t_{\rm s_{1}d_{1}}^{mm} +\beta t_{\rm s_{2}d_{1}}^{mm})\vert{mm}\rangle+\beta t_{\rm s_{2}d_{1}}^{gm}\vert{gm}\rangle+ \alpha t_{\rm s_{1}d_{1}}^{mg} \vert{mg}\rangle \right\}. \end{equation} \tag{ 9 }$$

Here, the beam splitter enables t^mm_s1d1 = t^mg_s1d1 and t^mm_s2d1 = t^gm_s2d1, i.e. the plasmon injected in each source is not influenced by the QD in the opposite arm. If the phase of the amplitude β is chosen such that αt^mm_s1d1 + βt^mm_s2d1 = 0, we have a normalized ideal entangled state as $\vert {\Psi }\rangle _{\mathrm { QDs}}=\frac {1}{\sqrt {2}}( \vert {mg}\rangle -\vert {gm}\rangle )\equiv \vert {\psi ^{-}}\rangle$ with probability |αt^mg_s1d1|². This is the basic idea of the scheme for the limiting case of weak α and β.

We now turn to the more general scenario, starting with the initial state |Ψ_i〉 without any restrictions on α and β. In this case the detection of excitations at drain 1 is modelled by the projection operator ${P}_{\mathrm { d_{1}}}=\sum _{n=1}^{\infty }{\vert {n}\rangle _{\mathrm { d_{1}}}\langle {n}\vert }= \mathbb {I} - \vert {0}\rangle _{\mathrm { d_{1}}}\langle {0}\vert$ that projects the state of the system onto a subspace containing at least one excitation in mode $\hat {d}_{{\mathrm { out}},1}^{\dagger }$. This projection models the measurement performed by an ideal non-photon number resolving detector, which registers a detection event as long as there is at least one excitation at the detector. Later we will also investigate the effects of detection inefficiency in the scheme. As a result of lifting the restriction on the input field amplitudes and the accuracy of the detection, the state of the QDs becomes mixed and can be written as

$$\begin{equation} \rho_{\rm QDs}=\frac{\mathrm{Tr}_{(\mathrm{all~fields})}[{P}_{\rm d_{1}}\vert{\Psi_{\mathrm{f}}}\rangle\langle{\Psi_{\mathrm{f}}}\vert{P}_{\rm d_{1}}]}{\mathrm{Tr}_{(\mathrm{QDs})}[\mathrm{Tr}_{(\mathrm{all~fields})}[{P}_{\rm d_{1}}\vert{\Psi_{\mathrm{f}}}\rangle\langle{\Psi_{\mathrm{f}}}\vert{P}_{\rm d_{1}}]]}. \end{equation} \tag{ 10 }$$

The final state of the output fields, |Ψ_f〉, is found by transforming |Ψ_i〉 according to the four sets of scattering matrices from equation (7) depending on the state of the QDs. It is written as

$$\begin{eqnarray} &&\fl \vert{\Psi_{\mathrm{f}}}\rangle =\textstyle\frac{1}{2} \left( \vert{gg}\rangle\vert{\psi^{gg}}\rangle_{\rm fields}+\vert{mm}\rangle\vert{\psi^{mm}}\rangle_{\rm fields} + \vert{gm}\rangle\vert{\psi^{gm}}\rangle_{\rm fields}+ \vert{mg}\rangle\vert{\psi^{mg}}\rangle_{\rm fields} \right), \end{eqnarray} \tag{ 11 }$$

where the state of the fields at the output modes is represented by a product of coherent states

$$\begin{eqnarray} &&\vert{\psi^{xy}}\rangle_{\rm fields}=\vert{\xi_{1}^{xy}}\rangle_{\rm s_{1}} \vert{\xi_{2}^{xy}}\rangle_{\rm s_{2}} \vert{\mu_{1}^{xy}}\rangle_{\rm d_{1}} \vert{\mu_{2}^{xy}}\rangle_{\rm d_{2}} \otimes_{j=1}^{2n+2} \vert{\chi_{j}^{xy}}\rangle_{\rm b_{\it{j}}} \end{eqnarray} \tag{ 12 }$$

with amplitudes ξ^xy_j = αt^xy_s1sj + βt^xy_s2sj (j = 1,2) at the source nanowires, μ^xy_j = αt^xy_s1dj + βt^xy_s2dj (j = 1,2) at the drain nanowires, and χ^xy_j = αt^xy_s1bj + βt^xy_s2bj (j = 1,...,2n + 2) for the nanoparticle bath modes. Each coherent field consists of an interference of two pathways, one from source 1 and the other from source 2 with respective transition amplitudes. Throughout this work, we impose the matching condition αt^mm_s1d1 + βt^mm_s2d1 = 0 in order to exclude the possibility of the case of |mm〉 leading to a detection, i.e. the amplitude for the coherent state in drain 1 is μ^mm₁ = 0. As mentioned previously, this can be achieved by properly adjusting the amplitude β of the coherent field injected into source 2. In order to measure how well the QDs are entangled, we employ the fidelity as a figure of merit, defined as the overlap between the desired final state, |ψ⁻〉, and the actual final state of the system, ρ_QDs, and given by F = 〈ψ⁻| ρ_QDs |ψ⁻〉. The fidelity allows us to quantify how close the final state is to the desired state, and at the same time provides a lower bound on the concurrence, C, a typical entanglement measure for two qubits, i.e. C ⩾ max(0,2F − 1). Such a lower bound implies that entanglement is always found when the fidelity is greater than 1/2 [43, 44]. The fidelity is given as $F=\langle {\psi ^{-}}\vert \rho _{\mathrm { QDs}} \vert {\psi ^{-}}\rangle =\frac {1}{2}(\rho _{22}+\rho _{33}-\rho _{23}-\rho _{32})$, where ρ_pq is the entry in the pth row and the qth column of the matrix for ρ_QDs that is spanned by the basis {|gg〉,|gm〉,|mg〉, |mm〉}. The entries are given by

$$\begin{equation} \fl \rho_{22} = \frac{1}{4\eta} \Big(1-\langle \mu_{1}^{gm} |0 \rangle_{\rm d_{1}}\langle 0 |\mu_{1}^{gm} \rangle_{\rm d_{1}} \Big), \quad\rho_{33} = \frac{1}{4\eta} \Big(1-\langle \mu_{1}^{mg} |0 \rangle_{\rm d_{1}}\langle 0 |\mu_{1}^{mg} \rangle_{\rm d_{1}} \Big), \end{equation} \tag{ 13 }$$

$$\begin{eqnarray*} \fl \rho_{23}& = &\frac{1}{4\eta} \Big(\langle \mu_{1}^{gm} |\mu_{1}^{mg} \rangle_{\rm d_{1}}-\langle \mu_{1}^{gm} |0 \rangle_{\rm d_{1}}\langle 0 |\mu_{1}^{mg} \rangle_{\rm d_{1}} \Big) \\ &&\times \langle \mu_{2}^{gm} |\mu_{2}^{mg} \rangle_{\rm d_{2}} \langle \xi_{1}^{gm} |\xi_{1}^{mg} \rangle_{\rm s_{1}} \langle \xi_{2}^{gm} |\xi_{2}^{mg} \rangle_{\rm s_{2}} \otimes_{j=1}^{2n+2} \langle \chi_{j}^{gm} |\chi_{j}^{mg} \rangle_{\rm b_{\it{j}}} = \rho_{32}^{*}. \end{eqnarray*}$$

The efficiency (or success probability for generating an entangled state) is the probability to detect photons at drain 1 and is given by the denominator of equation (10) as

$$\begin{eqnarray} \eta&=&\mathrm{Tr}_{(\mathrm{QDs})}[\mathrm{Tr}_{(\mathrm{all~fields})}[{P}_{\rm d_{1}} \vert{\Psi_{\mathrm{f}}}\rangle\langle{\Psi_{\mathrm{f}}}\vert{P}_{\rm d_{1}}]] \nonumber\\ &=&1-\textstyle\frac{1}{4}\left[\vert\langle 0 |\mu_{1}^{gg} \rangle\vert^{2}+\vert\langle 0 |\mu_{1}^{mm} \rangle\vert^{2}+\vert\langle 0 |\mu_{1}^{gm} \rangle\vert^{2}+\vert\langle 0 |\mu_{1}^{mg} \rangle\vert^{2}\right]. \end{eqnarray} \tag{ 14 }$$

Thus, by inspecting equation (13), a high fidelity is obtained in the limiting scenario where we have μ^gm₁ = −μ^mg₁ with all the other field modes for the case when the QDs are in |gm〉 approximately equal to the field modes for the case when the QDs are in |mg〉, so that they are effectively factored out in equation (11). In this case, the state of the total system becomes |Ψ_f〉 = (μ^gm₁|gm〉 + μ^mg₁|mg〉)⊗|all fields〉. However, this limiting scenario is hard to achieve for arbitrary fields injected into the system as higher-order photon number contributions act as a decoherence mechanism on the final state. The impact of higher-order photon number contributions on the value of the fidelity and efficiency are shown figure 3. Here, a trade-off between fidelity and efficiency can be seen as α is increased (note: β = iα by the matching condition αt^mm_s1d1 + βt^mm_s2d1 = 0) [45]. The fidelity decays quickly with increasing α due to the presence of higher-order excitations in the plasmonic system. It asymptotically approaches 0.5 in the limit α ≫ 1, indicating that the higher-order excitation contributions have completely decohered the state of the QDs. On the other hand, the fidelity reaches close to one when α ≪ 1, which indicates that an ideal entangled state is generated in the limit of the weak field injection case, as described in the previous subsection. Note that higher-order excitations are inevitable but not a practical problem as the amplitude α is easily controllable, so that we can decrease the magnitude of α as low as required (or increased) for a reasonable fidelity and efficiency.

Zoom In Zoom Out Reset image size

We first examine the fidelity and efficiency of our scheme with respect to loss in the metal nanoparticles. For this purpose, we choose the cases of n = 2 and 3 as representatives of even and odd numbers of nanoparticles in each arm. Figure 4 shows the fidelity and efficiency as the nanoparticle loss Γ₀/g_np increases and the in–out coupling strength g_in–out/g_np is varied for α = 0.5 (with β = iα). Here, we have assumed that the |g〉–|e〉 transition of each QD is resonant with its adjacent nanoparticle such that δ^g₁₍₂₎ = 0, while δ^m₁₍₂₎ is chosen for the QD |m〉–|e〉 transition to be completely decoupled from the dynamics.

Zoom In Zoom Out Reset image size

In figures 4(a) and (b), we show that the fidelity varies only slightly with increasing Γ₀/g_np, implying a remarkable robustness against metal loss. Such a robustness in fidelity can be understood from the observation that the reflection coefficient of each arm predominantly determines the fidelity for the case of resonant QDs (δ^g₁ = δ^g₂ = 0), as we explain in detail in appendix C. This indicates that the fidelity mostly depends on the behaviour of the reflection coefficient of each arm with metal loss Γ₀/g_np and g_in–out/g_np; the reflection coefficient varies only slightly with Γ₀/g_np since the energy never enters the array, and has a monotonic trend with g_in–out/g_np (see appendix A), resulting in robustness of the fidelity against metal loss and a general decrease in the fidelity with respect to g_in–out/g_np, respectively. Contrary to the case of the fidelity, we show that the efficiency of generating entanglement is decreased with Γ₀/g_np in figures 4(c) and (d). This is because the efficiency is determined by how much energy is transferred from the sources to the drains, indicating that the efficiency is mostly related to the transmission coefficient of each arm. In contrast to the reflection, the transmission coefficient is more sensitive to metal loss, and has different dependence on g_in–out/g_np (as shown in more detail in appendix A).

In this subsection, we chose α = 0.5 as an example. However, if we choose a larger value of α then higher-order excitations will reduce the fidelity and increase the efficiency on average in figure 4, as already shown in the previous subsection. Note that this is not a practical problem as the amplitude α is easily controllable, so that it can be decreased as low as required (or increased) for a reasonable fidelity and efficiency. Thus, using coherent states as initial states is a key merit of the scheme, which allows us to use a metal nanoparticle array supporting lossy plasmonic modes.

In terms of the total amount of loss present in the entire system, the effect of increasing the loss of each nanoparticle, for a fixed number of nanoparticles, can be regarded as equivalent to increasing the number of nanoparticles in each arm for a fixed amount of loss. In figure 5, for a fixed amount of metal loss, Γ₀/g_np = 0.1, and input amplitude α = 0.5, the fidelity and efficiency are shown as g_in–out/g_np is varied with increasing number of nanoparticles. As expected, the behaviour of the fidelity and the efficiency as the length of nanoparticle array is increased is very similar to the behaviour seen in figure 4. It is quite remarkable that one can achieve a robust high fidelity (and thus entanglement) over a reasonably long array of metal nanoparticles, even though it has so far been believed that loss limits the length of an array of nanoparticles for quantum information processing purposes.

Zoom In Zoom Out Reset image size

Furthermore, we have checked that the number of nanoparticles can be increased arbitrarily with the fidelity staying consistently above 0.8 at the maximum efficiency optimized over g_in–out/g_np for α = 0.5 but that at the same time this maximum efficiency asymptotically approaches zero as the loss becomes much more dominant in the overall dynamics. Therefore the overall performance is limited by the repetition rate of the pulses used for entangling. Meanwhile, g_np can be arbitrarily changed as long as the value of g_in–out/g_np is kept within the necessary limits, but if the distance between nanoparticles increases too much (g_np decreases), then their coupling becomes so weak that the radiative damping rate becomes important [12]. According to a typical example of a nanoparticle array [12], where nanoparticles with radius of 25 nm are separated by a centre-to-centre distance 75 nm and our approximations are satisfied, the size of the total array is about n × 150 nm (for example, if n = 40, the total size is about 6 μm).

A major challenge when using solid-state emitters is inhomogeneous broadening, typically caused by emitter size variation and strain fields in the host material. This means that two emitters usually have non-identical emission wavelengths and thus the two QDs in our scheme cannot be easily assumed to have the same resonant transition frequencies in a realistic scenario. Here, as one of the merits of the scheme, we show the robustness of fidelity against detunings of the QDs. Such robustness has been pointed out already in [45], but here we provide a more detailed analysis for the case of a plasmonic nanoparticle array system. In this section, we set δ^g₁ = δ₀ + Δδ and δ^g₂ = δ₀ − Δδ, and assume that γ^g₁ = γ^g₂ and J^g₁ = J^g₂. In figure 6 the fidelities are shown as the average of the detunings, δ₀ = (δ^g₁ + δ^g₂)/2, and the difference in the detunings, Δδ = (δ^g₁ − δ^g₂)/2, are varied for α = 0.5. Here, we have taken the matching condition used in the previous section, αt^mm_s1d1 + βt^mm_s2d1 = 0. Note that this matching condition does not guarantee the optimal fidelity⁸ but it is still sufficiently useful for observing the robustness of the fidelity against the detunings of the QDs. An optimization of the amplitude β to reach a higher fidelity than simply using the matching condition might lead to more robustness of the fidelity against the detunings of the QDs. This, however, involves numerical calculations that make it complicated to understand the origin of the robustness and is not considered in this work. In order to understand the behaviour of the fidelities as the detunings vary, as shown in figures 6(a) and (b), it is helpful to analyse two extremal cases. The first is when the respective detunings of the two QDs have different signs while their average is zero, i.e. δ₀/g_np = 0 and Δδ/g_np ≠ 0, labelled as line A (see figure 6(c) for level diagram), corresponding to a cut along the y-axis. The second is when the respective detunings of the two QDs are the same as each other, i.e. δ₀/g_np ≠ 0 and Δδ/g_np = 0, labelled as line B (see figure 6(c) for level diagram), corresponding to a cut along the x-axis. The trends of the fidelity along two lines shown in figures 6(d) and (e) are explained in detail in appendix D. Here, we only discuss an interesting behaviour displayed by the fidelity around the line B.

Zoom In Zoom Out Reset image size

The fidelity change in line B is such that the greater |δ₀/g_np| becomes, the more robust the fidelity is against changes in |Δδ/g_np|. That is, the higher the overall detuning δ₀, the better the fidelity behaves around Δδ = 0. This robustness can be understood by looking at how sensitive the various amplitudes are as they change from their values that lead to the best fidelity in line B when we vary Δδ. Consider two examples, δ₀/g_np = 0.02 and 0.12, for which the change in the amplitudes as Δω/g_np varies is shown in figures 6(f) and (g), for both n = 2 and 3. Here, it can be seen that by varying δ₀/g_np (solid and dashed lines represent two different values) the resonant peaks and dips in the transmission and reflection spectrums are shifted. In addition, for a fixed value of Δδ/g_np for the two chosen detunings of δ₀/g_np, a variation in the spectrums is induced. As we are interested in the difference in the detunings, Δω, one can see the values of the spectrums at Δω/g_np = 0 are changed less sensitively by a slight variation of Δδ/g_np for δ₀/g_np = 0.12 than for δ₀/g_np = 0.02. Such characteristics of transmission and reflection spectrums result in the robustness of the fidelity against the QD detunings Δδ/g_np.

As we are interested in the mapping of the input fields at the source tips to the output fields at the drain tips in the scheme, we have assumed so far that the source and drain excitations experience no loss when propagating into and out of the tip regions. However, in a realistic system, the losses in the nanowires would affect the overall performance. Here, we consider two kinds of losses in the nanowires: (i) insertion loss, which occurs when the light is transferred from a far-field source into the source nanowires and the resulting surface plasmons propagate along the nanowires up to the first or (2n + 2)th nanoparticle and (ii) outcoupling loss, which occurs when surface plasmons propagate along the drain 1 nanowire and are extracted into the far-field to be detected (or equivalently the detection efficiency of an on-chip detector directly connected to (n + 1)th nanoparticle). For insertion loss, this is not a problem for the scheme as we can simply compensate it by increasing the intensity of the input coherent states as much as we need for the plasmon excitations at the first and (2n + 2)th nanoparticles. In this sense, one of the advantages of the scheme is the use of coherent states as initial 'seed' states for the generation of entanglement. These enable the scheme to be stable no matter how much loss the initial coherent fields experience on input to the array system. On the other hand the second type of loss, the outcoupling loss, more significantly affects the fidelity and the efficiency. The outcoupling loss can be included as a 'lumped detection efficiency', κ, by replacing P_d1 with $\tilde {P}_{\mathrm { d_{1}}}$, where $\tilde {P}_{\mathrm { d_{1}}}=\mathbb {I} - \sum _{n=0}^{\infty }(1-\kappa )^n {\vert {n}\rangle _{\mathrm { d_{1}}}\langle {n}\vert }$, which corresponds to the case of n-photons being detected by the non-photon number resolving detectors with a probability 1 − (1 − κ)ⁿ [44]. Such a treatment of outcoupling loss is valid due to the fact that for an output coherent field at drain 1, say |μ〉, the expectation value of the measurement of an ideal detection with outcoupling loss amplitude $\sqrt {\kappa }$ is equivalent to that of a lumped detection without outcoupling loss, i.e. $\langle {\mu }\vert \tilde {P}_{\mathrm { d_{1}}}\vert {\mu }\rangle = 1 - \mathrm {e}^{-\kappa \vert \mu \vert ^{2}} =\langle {\sqrt {\kappa }\mu }\vert {\mathrm { P_{d_{1}}}}\vert {\sqrt {\kappa }\mu }\rangle$. This compatibility between nanowire loss and detection efficiency allows us to treat them in the same manner. To show how much the output coupling loss (or detection efficiency) affects the entanglement generation scheme, in figure 7 we plot the fidelity and the efficiency as the detection efficiency, κ, is varied and the number of nanoparticles, n, increases for α = 0.5 and g_in–out/g_np = 0.5. As the detection efficiency κ increases, the fidelity and the efficiency become generally higher. The gradients of the fidelity and the efficiency as κ varies depend on the magnitude of the input amplitude, α. For very small α (or large α) the gradient as κ varies is small, whereas for α having an intermediate value, the gradient is big.

Zoom In Zoom Out Reset image size

In the scheme, the QDs are initially prepared in an equal superposition of |g〉 and |m〉. However, here we show that an equal superposition is not a strict requirement for the QD initialization. When the QDs are prepared in arbitrary states, the initial state of the system is given by

$$\begin{equation} \vert{\Psi_{\mathrm{i}}^{'}}\rangle = (c_{g1} \vert{g}\rangle_{1}+c_{m1}\vert{m}\rangle_{1}) (c_{g2}\vert{g}\rangle_{2}+c_{m2}\vert{m}\rangle_{2}) \vert{\alpha}\rangle_{\mathrm{s}_{1}} \vert{\beta}\rangle_{\mathrm{s}_{2}} \vert{0}\rangle_{\mathrm{d}_{1}} \vert{0}\rangle_{\mathrm{d}_{2}}, \end{equation} \tag{ 15 }$$

where |c_g1|² + |c_m1|² = 1 and |c_g2|² + |c_m2|² = 1. In the ideal case of the scheme, the state |Ψ'_i〉 is transformed into |Ψ'〉_QDs = { c_m1c_m2(αt^mm_s1d1 + βt^mm_s2d1)|mm〉 + c_g1c_m2βt^gm_s2d1|gm〉 + c_m1c_g2αt^mg_s1d1|mg〉} , and by choosing β to satisfy the matching condition αt^mm_s1d1 + βt^mm_s2d1 = 0, we have the final state

$$\begin{equation} \vert{\Psi^{'}}\rangle_{\rm ent.QDs}=\frac{1}{\sqrt{N}}( c_{m1} c_{g2} \vert{mg}\rangle - c_{g1} c_{m2} \vert{gm}\rangle), \end{equation} \tag{ 16 }$$

where the normalization is given by N = |c_m1c_g2|² + |c_g1c_m2|². Equation (16) shows that |Ψ'〉_ent.QDs is the desired ideal entangled state |ψ⁻〉 as long as c_m1c_g2 = c_g1c_m2 is satisfied. In other words, we can achieve a high fidelity even when the two QDs are not prepared in an equal superposition. An equal superposition at the initialization stage only guarantees a higher efficiency η, but is not required for achieving a high fidelity. This property of the scheme provides greater flexibility in the initialization of the QDs.

We first consider the case of n = 1 in order to provide a basic understanding of the physical mechanism for DIR. Arbitrary n is considered in the next section. For a given input field from the source nanowire we have

$$\begin{equation} t_{\rm s}(\Delta\omega)=\frac{g_{\rm in\mbox{--}out}(\frac{\gamma}{2} + \mathrm{i}(\delta-\Delta\omega))}{J^{2} + (\frac{\gamma}{2}+\mathrm{i}(\delta-\Delta\omega))(g_{\rm in\mbox{--}out}+\frac{\Gamma_0}{2}-\mathrm{i}\Delta\omega)}, \end{equation} \tag{ A.6 }$$

$$\begin{equation} r_{\rm s}(\Delta\omega)=-\frac{J^{2}+(\frac{\gamma}{2}+\mathrm{i}(\delta-\Delta\omega))(\frac{\Gamma_0}{2} - \mathrm{i}\Delta\omega)}{J^{2} + (\frac{\gamma}{2}+\mathrm{i}(\delta-\Delta\omega))(g_{\rm in\mbox{--}out} +\frac{\Gamma_0}{2}-\mathrm{i}\Delta\omega) }, \end{equation} \tag{ A.7 }$$

$$\begin{equation} b_{\rm s,b_{1}}(\Delta\omega)=\frac{\sqrt{g_{\rm in\mbox{--}out}}\sqrt{\Gamma_{0}} (\frac{\gamma}{2} + \mathrm{i}(\delta-\Delta\omega))}{J^{2} + (\frac{\gamma}{2} + \mathrm{i}(\delta-\Delta\omega))(g_{\rm in\mbox{--}out} +\frac{\Gamma_0}{2}-\mathrm{i}\Delta\omega)}, \end{equation} \tag{ A.8 }$$

where Δω = ω − ω₀ and δ = Ω − ω₀ denote the detunings of the input field from the nanoparticle resonance and the dipole resonance from nanoparticle resonance, respectively. If there is no dipole in the proximity of the nanoparticle, i.e. J = 0, in the ideal case (Γ₀ = 0), the field is entirely transmitted on resonance (Δω = ω − ω₀ = 0), i.e. r_s(Δω = 0) = 0 and t_s(Δω = 0) = 1. Once the loss in the metal nanoparticle is included, i.e. Γ₀ ≠ 0, the field is no longer entirely transmitted on resonance, i.e., t_s(0) ≠ 1, as shown in figure A.1(b) as dashed lines. However, if there is a dipole in the proximity of the nanoparticle, i.e. J ≠ 0, then the transmission and reflection behave differently from the case of no dipole as we discuss next.

A.1.1. Dipole and single nanoparticle in tune (δ = 0)

Consider the case where the dipole is resonant with the nanoparticle (δ = 0), equations (A.6)–(A.8) can be rewritten as

$$\begin{equation} t_{\rm s}(\Delta\omega) =\frac{g_{\rm in \raise -1.5pt\hbox{--}out} (\frac{\gamma}{2} - \mathrm{i} \Delta\omega)}{J^{2} + (\frac{\gamma}{2} - \mathrm{i} \Delta\omega)(g_{\rm in\hbox{--}out} +\frac{\Gamma_{0}}{2} - \mathrm{i}\Delta\omega) }, \end{equation} \tag{ A.9 }$$

$$\begin{equation} r_{\rm s}(\Delta\omega) =- \frac{J^{2} + (\frac{\gamma}{2} - {\rm i} \Delta\omega)(\frac{\Gamma_{0}}{2} - {\rm i}\Delta\omega) }{J^{2} + (\frac{\gamma}{2} - {\rm i} \Delta\omega)(g_{\rm in\hbox{--}out} +\frac{\Gamma_{0}}{2} - {\rm i}\Delta\omega) }, \end{equation} \tag{ A.10 }$$

$$\begin{equation} b_{\rm s,b_{1}}(\Delta\omega) = \frac{ \sqrt{g_{\rm in\hbox{--}out}} \sqrt{\Gamma_{0}} (\frac{\gamma}{2} - \mathrm{i} \Delta\omega)}{J^{2} + (\frac{\gamma}{2} - \mathrm{i} \Delta\omega)(g_{\rm in\hbox{--}out} +\frac{\Gamma_{0}}{2} - \mathrm{i}\Delta\omega) }. \end{equation} \tag{ A.11 }$$

In this case, when 2J²/γ ≫ g_in–out + Γ₀/2 we have t_s(0) ≈ 0 and r_s(0) ≈ −1, with a π phase shift on resonance (Δω = 0), so that the field is totally reflected, similar to the frequency selective perfect mirror described by Shen and Fan [49]. This is a quantum interference effect called DIR. It is based on the same mechanism as DIT originally introduced by Waks and Vučković [25], in which the cavity–dipole system is driven by an external field and the cavity field destructively interferes with the excited state population of the dipole. Here, a metal nanoparticle serves as a bad cavity, where cavity decay rate Γ₀ is larger than the dipole decay rate γ. Nevertheless, in this bad cavity regime, DIR (or transmission) still provides strong dispersive properties (as shown in figure A.1(b) as solid lines), which is one of the interesting merits of the original DIT scheme, as pointed out by Waks and Vučković [25]. Our plasmonic system for n = 1 is mathematically equivalent to driving a double sided cavity with an incident field.

In order to understand the origin of the dispersive properties, we use the Purcell factor, defined as $F_{\mathrm {p}}=\frac {2J^{2}}{\gamma }\frac {1}{g_{\mathrm{ in\mbox {--}out}}+\Gamma _{0}/2}$ (proportional to the ratio of the dipole decay rate into the nanoparticle, J, to the bare dipole decay rate, γ). In terms of the Purcell factor, the transition amplitudes can be rewritten on resonance as t_s(Δω = 0) = t₀/(F_p + 1), r_s(Δω = 0) = −(F_p − r₀)/(F_p + 1) and b_s,b1(Δω = 0) = a₀/(F_p + 1), where $t_{0}=\frac {g_{\mathrm{{in\raise -1.5pt\hbox{--}out}}}}{g_{\mathrm{{in\raise -1.5pt\hbox{--}out}}}+\Gamma _{0}/2}$, $r_{0}=-\frac {\Gamma _{0}/2}{g_{\mathrm{{in\raise -1.5pt\hbox{--}out}}}+\Gamma _{0}/2}$, and $a_{0}=\frac {\sqrt {g_{\mathrm{{in\raise -1.5pt\hbox{--}out}}}}\sqrt { \Gamma _{0}}}{g_{\mathrm{{in\raise -1.5pt\hbox{--}out}}}+\Gamma _{0}/2}$ are the transmission, reflection and absorption amplitudes for a bare nanoparticle in the absence of the dipole. One can see that in order to have DIR a large Purcell factor is required, i.e. F_p ≫ 1. However, as in DIT, we do not need the full normal mode splitting condition J > g_in–out + Γ₀/2, known as the high-Q cavity regime. We can achieve a large Purcell factor when γ ≪ g_in–out + Γ₀/2 even for much smaller values of J. This is known as the low-Q cavity regime, where the coupling strength J between the nanoparticle and the dipole is less than the nanoparticle decay rate Γ₀.

When the nanoparticle plasmonic excitations and the |g〉–|e〉 transition of the dipole are in tune (δ = 0), the overall spectrum of the reflection amplitude is always symmetric with respect to Δω = 0, at which the local maximum is located, as shown in figure A.1(b). Here, the dipole dissipation rate has a stronger effect on the reflection (transmission) of an on-resonance excitation than the nanoparticle dissipation rate does. When γ = 0, a plasmon is still completely reflected, even with the presence of the nanoparticle loss Γ₀, whereas when γ ≠ 0, the dissipation of the dipole interrupts the quantum interference, so that the nanoparticle is marginally excited for a plasmon at ω = Ω and the loss via the nanoparticle increases. Two minima (maxima) in the reflection (transmission) spectrum are located at Δω ≈ ± J, in the limit of a large Purcell factor F_p ≫ 1, which corresponds to the Rabi-split frequencies ω = Ω ± J. We find that the linewidth of the broadest transmission peak is the nanoparticle linewidth whereas the linewidth of the narrow dip corresponds to the linewidth of the dipole dressed by the nanoparticle excitation. This quantum interference that produces the narrow reflection window with a simultaneously strong dispersion results in a significant enhancement of the group delay and the possibility of a slowdown or 'storage' of light, as in electromagnetic induced transparency [50].

A.1.2. Dipole and single nanoparticle detuned (δ ≠ 0)

We now increase the number of nanoparticles in the array, while keeping the dipole coupled to the first nanoparticle, as shown in figure A.1. As in the case of n = 1, we have transmission, reflection and absorption amplitudes from the scattering matrix given in equations (A.6)–(A.8). If the dipole is decoupled from nanoparticle, the transmission and the reflection amplitudes follow the characteristic of a metal nanoparticle array studied in [13], which have different dependencies on g_in–out. We classify their behaviour into two groups: even and odd numbers of nanoparticles in the array. As a representative of even and odd numbers of nanoparticles, we consider the case of n = 2 and 3. As shown in [13], for an odd number of nanoparticles, there is always a resonance at the natural frequency ω₀, whereas for an even number of nanoparticles, a resonance property at the natural frequency ω₀ depends on the magnitude of g_in–out/g_np. Thus, the transmission properties as g_in–out/g_np is varied at Δω = 0 are different for n = 2 and 3 when the dipole is decoupled from nanoparticle, as shown in figures A.2(a) and (b). Here, the transmission (reflection) has a maximum (minimum) value at g_in–out/g_np = 2 for n = 2, whereas the transmission (reflection) rises (drops) quickly as g_in–out/g_np is increased for n = 3. In addition, in figure A.2 we show the dependence of the transmission and reflection with varying amount of metal loss for n = 2 and 3. The dependence on the metal loss is more sensitive for the transmission than for the reflection, regardless of the parity of the number of nanoparticles, which is the main reason for the robustness against the metal loss mentioned in the main text. As the metal loss is increased, the transmissions become lower; for n = 2 (or any even number) they still have a maximum near g_in–out/g_np ≈ 2, which is reflected in the efficiency as shown in figure 4(c). On the other hand, for n = 3 (or any odd number) the transmissions are still flat as g_in–out/g_np is increased, which is also reflected in the efficiency, as shown in figure 4(d). The reflection amplitudes shown in figures  A.2(c) and (d) are highly related to the transition amplitude t^gm_s1s1 shown in figures C.1(a) and (b), enabling the scheme to be robust to the metal loss.

Zoom In Zoom Out Reset image size

On the other hand, if the nanoparticle is coupled to the dipole whose detuning is zero, δ/g_np = 0, DIR can be observed at Δω = 0 for n = 2 and 3 when F_p ≫ 1. In figure A.2, while the transmission is nearly zero (dashed lines in (a) and (b)), the reflection (dashed lines in (c) and (d)) is nearly one and decreased only slightly with g_in–out/g_np, which is related to the coefficient t^mg_s1s1 in figures C.1(a) and (b). In addition, when DIR occurs, both the transmission and reflection amplitudes are largely insensitive to nanoparticle losses, as expected from the case of n = 1. In figure A.3, the spectral profile of the transmission is presented as Δω/g_np and g_in–out/g_np are varied for n = 2, 3, 4, 5, 6 and 7, in the presence of metal loss Γ₀/g_np = 0.1, as in [13]. Note that on resonance (Δω = 0) DIR can be observed regardless of g_in–out/g_np and the number of nanoparticles, where the transmission becomes nearly zero for Δω/g_np = 0, with the dipole detuning given as δ/g_np = 0.

Zoom In Zoom Out Reset image size