## Abstract

The coherent transport of surface plasmons with nonlinear dispersion relations on a metal nanowire coupled to two-level emitters is investigated theoretically. Real-space Hamiltonians are used to obtain the transmission and reflection spectra of the surface plasmons. For the single-dot case, we find that the scattering spectra can show completely different features due to the non-linear quadratic dispersion relation. For the double-dot case, we obtain the interference behavior in transmission and reflection spectra, similar to that in resonant tunneling through a double-barrier potential. Moreover, Fano-like line shape of the transmission spectrum is obtained due to the quadratic dispersion relation. All these peculiar behaviors indicate that the dot-nanowire system provides a one-dimensional platform to demonstrate the bandgap feature widely observed in photonic crystals.

©2010 Optical Society of America

## 1. Introduction

In the realization of quantum network [1], coherent single-photon transport is of central issue, in which the photons conveying information (signal field) are controlled by other photons (gate field). Much experimental [2–6] and theoretical [7–13] work has been focused on the photon transport properties. With the development of technologies, a coupled system comprising of a single metal nanowire with a quantum dot (QD) has been fabricated successfully [14, 15]. This leads to the possibilities of investigating cavity quantum electrodynamics [16–19] and coherent single surface plasmon (SP) transport [20–22] within such a device.

Inspired by the work mentioned above, we consider here the scattering properties of nanowire SP coupled to QDs. Based on the findings that the dispersion relations of metal nanowire SPs are parabola-like for higher excitation modes [17, 18], we investigate the transmission and reflection properties of SPs by approximating the dispersion relations with a quadratic form. With reasonable assumption of strong coupling between QDs and SPs, we consider the coherent transport of surface plasmons, i.e. the incoherent scattering can be neglected [9, 10]. As will be seen, the transmission (or reflection) spectrum for the single-dot case is found to have double peaks due to the quadratic form of the dispersion relation. For the double-dot case, the interference curves, similar to resonant tunneling phenomenon, are obtained in the scattering spectra. In addition, the transmission spectrum can reveal a Fano-like line shape because of the non-linear dispersion relation.

## 2. Model and formulas

Let us consider now two QDs placed close to a metal nanowire, such that the SPs are evanescently coupled to the dots as shown in Fig. 1(a). The Hamiltonian of the SPs and two two-level QDs, separated by a distance *d*, is written as

$$\phantom{\rule[-0ex]{2em}{0ex}}+\sum _{k}\overline{h}\sqrt{\genfrac{}{}{0.1ex}{}{2\pi}{\overline{h}{\omega}_{k}V}}\Omega \phantom{\rule[-0ex]{.2em}{0ex}}\mathbf{D}\xb7{\mathbf{e}}_{k}[\left({a}_{k}{\sigma}_{{e}_{1},{g}_{1}}+{a}_{k}{e}^{\mathrm{ikd}}{\sigma}_{{e}_{2,}{g}_{2}}\right)+h.c.],$$

where *ω _{k}* is the frequency (dispersion relation) of the SPs with wave vector

*k*,

*a*

_{k}^{†}(

*a*) is the creation (annihilation) operator of the plasmon mode, Ω is the two-level energy spacing of the QDs, and

_{k}*σ*

_{ej,ej}(

*σ*

_{ej,gj}) = |

*e*〉〈

_{j}*e*|(|

_{j}*e*〉〈

_{j}*g*|) is the diagonal (off-diagonal) element of the QD operator. Here, ${V}_{k}\equiv \sqrt{\genfrac{}{}{0.1ex}{}{2\pi}{\overline{h}{\omega}_{k}V}}\Omega \mathbf{D}\xb7{\mathbf{e}}_{k}$ describes the coupling strength between SPs and the QD, where

_{j}**D**is the dipole moment of the QD,

*V*is the quantization volume, and

**e**

*is the polarization vector of the SPs. For Simplicity, we first assume that these two QDs are equal in distance to the nanowire, which means that their couplings to the nanowire SPs are the same. In addition, we also assume the couplings between QDs and SPs are strong, such that the incoherent scattering can be neglected. The validity of this assumption will be clarified later.*

_{k}The dispersion relations of nanowire SP in Fig. 1(b) are quoted from Refs. [17, 18]. As seen, the dispersion relations for *n* ≠ 0 modes show highly non-linear behaviors. Let us now focus on the *n* = 1 mode (red line). One immediately observes that there is a local minimum around the value of the wavevector **K**(= *kc*/*ω _{p}*)~ 15, where

*h̄ω*= 3.76 eV is the plasma energy of bulk silver. The dispersion relation around this local minimum can be analogous to the case when a two-level is put in a photonic crystal. The density of states becomes singular near the bandedge and the dispersion relation can be approximated as a parabolic curve. This parabolic dispersion curve results from the strong interaction between the two-level atom and its own localized radiation [18, 19, 23–25]. In our model, the interactions between QDs and SPs can be also very strong, resulting from a similar feature of local extremum in the dispersion curve. We thus approximate the dispersion relation

_{p}*ω*around this bottom with a quadratic form:

_{k}*ω*+

_{c}*A*(

*k*-

*k*

_{0})

^{2}, where

*k*

_{0}is the wavevector for the local minimum frequency

*ω*. Adopting the units in Refs [16, 17], the value of

_{c}*A*is about 0.0001, which is very small. Therefore, one can expand

*ω*

_{k}^{-1/2}in

*V*to the lowest-order term in

_{k}*k*, such that ${V}_{k}\approx \sqrt{\genfrac{}{}{0.1ex}{}{2\pi}{\overline{h}{\omega}_{c}V}}\Omega \mathbf{D}\xb7{\mathbf{e}}_{k}\left[1-\genfrac{}{}{0.1ex}{}{A{\left(k-{k}_{0}\right)}^{2}}{2{\omega}_{c}}\right]$. Since

*A*again is very small compared to

*ω*, one can further drop the second term in the square bracket. The Hamiltonian,

_{c}*H*=

*H*+

_{SP}*H*+

_{int}*H*, can now be written as

_{QD}$${H}_{\mathrm{int}}=\sum _{k}\overline{h}g\left[\left({a}_{k}{\sigma}_{{e}_{1},{g}_{1}}+{a}_{k}^{\u2020}{\sigma}_{{g}_{1},{e}_{1}}\right)+\left({a}_{k}{e}^{\mathrm{ikd}}{\sigma}_{{e}_{2},{g}_{2}}+{a}_{k}^{\u2020}{e}^{-\mathrm{ikd}}{\sigma}_{{g}_{2},{e}_{2}}\right)\right]$$

$${H}_{\mathrm{QD}}=\sum _{j=\mathrm{1,2}}\overline{h}\Omega {\sigma}_{{e}_{j},{e}_{j}},$$

where $g=\sqrt{\genfrac{}{}{0.1ex}{}{2\pi}{\overline{h}{\omega}_{c}V}}\Omega \mathbf{D}\xb7{\mathbf{e}}_{k}$. Transforming the Hamiltonian into real-space [9, 10], it reads

$$\phantom{\rule[-0ex]{3em}{0ex}}-2\overline{h}{k}_{0}\left[-i{C}_{R}^{\u2020}\left(x\right)\genfrac{}{}{0.1ex}{}{\partial}{\partial x}{C}_{R}\left(x\right)+i{C}_{L}^{\u2020}\left(x\right)\genfrac{}{}{0.1ex}{}{\partial}{\partial x}{C}_{L}\left(x\right)\right]$$

$$\phantom{\rule[-0ex]{3em}{0ex}}+\overline{h}A\left[\genfrac{}{}{0.1ex}{}{\partial}{\partial x}{C}_{R}^{\u2020}\left(x\right)\genfrac{}{}{0.1ex}{}{\partial}{\partial x}{C}_{R}\left(x\right)+\genfrac{}{}{0.1ex}{}{\partial}{\partial x}{C}_{L}^{\u2020}\left(x\right)\genfrac{}{}{0.1ex}{}{\partial}{\partial x}{C}_{L}\left(x\right)\right]\}$$

$${\tilde{H}}_{\mathrm{int}}=\int \mathrm{dx}\{\sqrt{2\pi}g\sum _{j=\mathrm{1,2}}\delta \left(x-\left(j-1\right)d\right)[{C}_{R}^{\u2020}\left(x\right){\sigma}_{{g}_{j},{e}_{j}}$$

$$\phantom{\rule[-0ex]{3em}{0ex}}+{C}_{R}\left(x\right){\sigma}_{{e}_{j},{g}_{j}}+{C}_{L}^{\u2020}\left(x\right){\sigma}_{{g}_{j},{e}_{j}}+{C}_{L}\left(x\right){\sigma}_{{e}_{j},{g}_{j}}\left]\right\}$$

$${\tilde{H}}_{\mathrm{QD}}=\sum _{j=\mathrm{1,2}}\left({E}_{e}{\sigma}_{{e}_{j},{e}_{j}}+{E}_{g}{\sigma}_{{g}_{j},{g}_{j}}\right),$$

where *C _{R}*

^{†}(

*x*) [

*C*

_{L}^{†}(

*x*)] is a bosonic operator which creates a right-going (left-going) photon at real-space position

*x*.

*E*(

_{e}*E*) is the energy of the QD’s excited (ground) state, and

_{g}*E*-

_{e}*E*=

_{g}*h̄*Ω.

The stationary state |*E _{k}*〉 of the system with energy

*E*can be described as

_{k}$$\phantom{\rule[-0ex]{3em}{0ex}}+\sum _{j=\mathrm{1,2}}{e}_{{k}_{j}}{a}_{{e}_{j}}^{\u2020}{a}_{{g}_{j}}\mid 0,{g}_{1},{g}_{2}\u3009$$

where |0,*g*
_{1},*g*
_{2}〉 means that the QDs are both in the ground state with no SP present, and *e _{kj}* is the probability amplitude of the j-th QD in the excited state. For a SP incident from the left,

*ϕ*

^{†}

_{k,R}(

*x*) =

*e*[

^{ikx}*θ*(-

*x*)+

*aθ*(

*x*)

*θ*(

*d*-

*x*)+

*tθ*(

*x*-

*d*)] and

*ϕ*

^{†}

_{k,L}(

*x*) ≡

*e*

^{-ikx}[

*r θ*(-

*x*)+

*b θ*(

*x*)

*θ*(

*d*-

*x*)], where

*t*and

*r*are the transmission and reflection amplitudes respectively. Here,

*a*and

*b*are the probability amplitudes of the SP between

*x*= 0 and

*d*, and

*θ*(

*x*) is the step function. By solving the eigenvalue equation

*H*|

*E*〉 =

_{k}*E*|

_{k}*E*〉, one can obtain the exact forms of the transmission and reflection coefficients:

_{k}and

Here, we have defined the function *F*(*k*) ≡ *h̄A*(*k* + 2*k*
_{0})[*δ* +*A*(*k* - *k*
_{0})^{2}] and *D*(*k*) ≡ {8*g*
^{4}[*F*(*k*)]^{2} ≡ [*F*(*k*)]^{4} - 32*g*
^{6}
*F*(*k*)sin(2*kd*) - 8*g*
^{4}(4*g*
^{4} - [*F*(*k*)]^{2})cos(2*kd*)+32*g*
^{8}, where *δ* = *ω _{c}* - Ω. is the detuning between

*ω*and the two-level energy spacing.

_{c}## 3. Results and discussions

Let us first study how the quadratic dispersion relation affects the transmission/reflection. For simplicity, the inter-dot distance *d* in Fig. 2 is assumed to be zero, i.e. the single-dot case. As seen, the main feature in Fig. 2 is the double peak when the detuning, *δ* = *ω _{c}* - Ω, is negative. To explain this, we note in the limit of

*d*→ 0,

*T*and

*R*can be reduced to

and

From Eqs. (7) and (8) one realizes that, for negative *δ*, it yields two zeros (unities) in *T* (*R*), corresponding to the two dips (peaks) in the curves. For the case of *δ* > 0, the incident SP is affected slightly for positive detunings. To understand this clearly, one recalls that the approximation of quadratic dispersion relation with a bandgap is commonly used for photons in photonic crystals [19, 23–25]. The propagation of photons is determined by whether the energy of photons is above the bandgap. The effects in Eqs. (7) and (8) are analogous to the transport of a photon in a photonic crystal with a bandgap. This means the QD-SP system provides a one-dimensional platform to demonstrate the bandgap feature.

Figure 3 displays the transmission and reflection spectra, *T* and *R*, for different inter-dot distance *d* with *g* = 1 and *δ* = 0. We can see that, for *d* = 0, the results return to the ordinary single dot scattering spectrum. When increasing *d*, the jiggling behavior becomes more obvious. Mathematically, the zero-reflection (R=0) occurs when the two functions

and

coincide with each other. Figure 4(a) displays the transmission and reflection spectra for the conditions of *d* = 6, *g* = 1, *δ* = 0, and *k*
_{0}. Figure 4(c) demonstrates the two functions *X* and *Y* numerically. One can identify that the intersections of *X* and *Y* are the zeros of *R*. However, for the case of *K* = *k*
_{0}, *R* is not zero, but unity. This is because for this particular choice, the denominator of *R* is also zero. Therefore, one can not determine *R* only from the numerator. The limit of *R* at *k* = *k*
_{0} is identified to be unity by applying the L’Hopital’s rule. One could interpret this as the phenomenon of resonant tunneling of an electron through double barriers in quantum mechanics.

Figure 5 shows how the coupling *g* affects *R* and *T* under the conditions of *d* = 6 and *δ* = 0. When increasing the coupling, the region (light green area) for forbidden propagation of SPs becomes wider. The coupling strength g can be controlled by the dot-wire separation and the polarization of the QD exciton. In reality, it is difficult to alter the dot-wire separation. However, one can still vary the coupling g by applying an external field to orient the direction of the dipole moment of the QD exciton.

Keeping SP-QD1 coupling (*g*
_{1}) fixed, Fig. 6 shows the transmission and reflection spectra for different SP-QD2 couplings (*g*
_{2}). When decreasing *g*
_{2}, the feature of Fano-like resonance becomes more evident as seen in Fig. 6(d). Fano resonance [26] is a general property whenever there is interference between localized and delocalized states (channels). The interference between these two channels leads to an asymmetric line shape universally found in many physical systems. To illustrate the feature here is related to Fano resonance, we further deduce the explicit forms of the transmission and reflection coefficients for *δ*
_{1} = 0, *d* = 0, and *g*
_{1} = 1:

$$\phantom{\rule[-0ex]{2em}{0ex}}+\left[4+{A}^{4}{\left(k-{k}_{0}\right)}^{4}{\left(k+2{k}_{0}\right)}^{2}\right]{\Delta}_{2}^{2}$$

and

$$\phantom{\rule[-0ex]{2em}{0ex}}+\left[4+{A}^{4}{\left(k-{k}_{0}\right)}^{4}{\left(k+2{k}_{0}\right)}^{2}\right]{\Delta}_{2}^{2}$$

where Δ_{2} ≡ *δ*
_{2}+*A*(*k*-*k*
_{0})^{2}. The zeros of the reflection coefficient R (totally transmitted) are

From Eq. (12) and (13), one knows the condition for the presence of the double dips in R is *δ*
_{2} < 0. This means the appearance of Fano-like resonance in our case is due to the quadratic dispersion relation, which produces a photonic band-gap similar to those in photonic crystals [23, 24]. The effect of QD1 (QD2) is just like a delocalized (localized) channel for the SP passing through it. One notes that the Fano-like line shape due to the non-linear dispersion was also found in Refs. [11–13]. However, the origins of the non-linear dispersion relations are different. In Refs. [11–13], the non-linear dispersion relation comes form the tight-binding consideration of the of the cavity array. In our case, the non-linear dispersion is solely due to the geometric nature of the cylindrical wire, i.e. a single wire is enough to produce the non-linearity. Furthermore, a single dot coupled to the wire can not reveal the Fano-like line shape. Two dots with proper energy differences to the bandgap (*ω _{c}*) are needed to show this phenomenon. Our work clearly demonstrating the interference between the localized and de-localized channels due to the bandgap.

A few remarks on the validity of our assumption for real parameters and experimental readout should be given here. From the numerical calculations for CdSe QD coupled to Ag nanowire in Refs. [16] and [17], one realizes the corresponding energy scale for the case of *g* = 1 is about 0.376meV. If the QD is 10 nm away from the metal nanowire, the corresponding coupling strength in this special case is: *g* ≈ 1.6. Compared with the dissipated energy scale of the QD exciton (*h̄*/lifetime≈ 1*μ*eV), the QD-SP coupling is two to three orders of magnitude larger. This validates our assumption of strong coupling. In real experiments, surface plasmons inevitably experience dissipations like Ohmic losses during propagation. To preserve coherence, we propose a setup to carry out the experiments: Instead of using an infinite long nanowire, we consider two separate wires with finite length evanescently coupled to a phase-matched dielectric waveguide [5, 20]. The two QDs are coupled to these two wires as shown in Fig. 7. In this case, one can have both the advantages of strong coupling from SP and long-distance transport by the dielectric waveguide. With this setup, one can measure the transmission and reflection spectra of SP by scanning the wave vector spectrum of the incident SP. In addition, one can also apply the detecting technique developed in Ref. [21], such that the surface plasmons in metal wire can be converted into electron-hole pairs in semiconductor wire. This would allow one to detect the surface plasmons with an electrical way.

## 4. Summary

In summary, real-space Hamiltonians with nonlinear quadratic dispersion relation are used to obtain the transport properties of SPs propagating on the surface of a silver nanowire coupled to two QDs. For the single-QD case, the incident SP is affected slightly for positive detunings, while it shows two peaks (dips) in reflection (transmission) spectra for negative detunings. For the double-QD case, the region for forbidden propagation of SPs becomes wider when increasing the SP-QD coupling *g*. In addition, we also found the Fano-like resonance can appear if the SP-QD coupling for each dot is different (*g*
_{1} ≠ *g*
_{2}). The reasons of these phenomena are attributed to the quadratic dispersion relation and related to the band-edge effect in photonic crystals.

## Acknowledgements

This work is supported partially by the National Science Council, Taiwan under the grant number NSC 98-2112-M-006-002-MY3.

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