The optical properties of composite nanostructures including metal are strongly affected by surface plasmon polaritons’ (SPPs) collective oscillations of conduction electrons with respect to the positive ion background [1-5]. SPPs are the propagating excitations of charge-density waves, and their associated intense electromagnetic fields are generated in the vicinity of the surface of a conductor. They are bound to and propagate along the metal/dielectric interface. Recently, great attention has been paid to the study of the strong coupling between individual emitters and SPPs in metallic nanowires [6-9]. The emission properties of nanoscale optical emitters can be significantly modified by the proximity of a nanowire that supports surface plasmons because an SPP is generally very intense and highly localized [5,7,10]. This subwavelength localization is accompanied by a dramatic condensation of optical fields. A substantial increase in the coupling strength g∝1/√
Near-field coupling of excited colloidal nanocrystals (NCs) to SPPs in a metallic nanowire (NW) is a challenging problem since nanocrystals are randomly dispersed relative to the metallic nanowire . In this paper, we investigate the spontaneous emission (SE) of the excited energy of one NC in NC aggregation to the SPPs in a nearby metallic nanowire for some specified configurations and for specific transition dipolar orientation of NCs. The radius of the NW is to be so small that it is sufficient to consider only the fundamental TE SPP mode in our calculations. In this work, we analyze the cooperative effect in the spontaneous decay of an excited NC in an NC aggregate into a SPP mode with and without the dipole-dipole interaction between NCs . The spontaneous emission rate into surface plasmons can be suppressed or enhanced for specially arranged NCs, depending on their relative positions and transition dipolar orientations due to their quantum interference between the interactions. This variable emission rate can be characterized by cooperativeness, which for thin wires and closely spaced NCs is predicted to be large. In the next section, a theoretical modeling is employed to calculate the transition amplitude of the spontaneous emission under the perturbative expansion scheme. Results of the numerical evaluation for the transition probability (TP) of the spontaneous emission are presented in Sec. III followed by a discussion on its characteristics. A brief conclusion is given in Sec. IV.
The system of interest in this paper is schematically shown in Fig. 1. NCs are supposed to be aggregated around the silver nanowire along the axis, where only one of the NCs is excited. First, we briefly review the properties of SPPs propagating along the surface of metallic NW [19,20], which have been observed experimentally [21-24]. We are especially interested in azimuthally independent SPPs, which are supported under the condition that the radius of the wire is small enough such that only the fundamental TM surface plasmon polariton mode exists while all other higher order modes are cut off . In this case the electric and magnetic fields of the fundamental TM surface plasmon polariton mode are given by 
[FIG. 2.] Dispersion relation of the fundamental SPP mode on lossless silver, where ？ ∞ = 9.6 for silver, ？ ∞ = 5.3 for dielectric (GaN), ω p = 3.76 eV/h, γ = 0, and the radius of the NW is set to be a = 5 nm
[FIG. 3.] Electric field outside the wire. (a) as a function of wave number for ρ = 1.2a and (b) as a function of radial distance from the wire center for k = 107/cm. Arbitrary unit is used for vertical axis. The solid and dashed lines are for the radial and axial components of the field, respectively. The radial component is real and the axial component is imaginary.
electric field outside the nanowire as a function of wave number is shown in Fig. 3(a); solid and dashed lines are for the radial and axial components of the field, respectively. Fig. 3(b) is the electric field as a function of radial distance from the wire center. Dramatic concentration of optical fields is well manifested.
The Hamiltonian of this composite system is written by
is the transition dipole moment of the
is the electric field at the location of the
are the distance and the directional unit vector between
The transition amplitude of the SE is calculated from perturbation theory. Within the dipole approximation, the first order transition amplitude of the spontaneous emission from an excited
being the electric field of the surface plasmon mode of wave number
The even-order perturbation terms are null, since the coupling is linear to the dipole moment. Since the NC’s transition energy is detuned from the SPPs in the metallic nanowire and sin(Δ
When all NCs are not identical, the third order perturbative transition amplitude of the SE is similarly calculated, but in this case the detuning parameter Δ
On the other hand, if the direct DDI between NCs cannot be ignored, the second order transition amplitude of the SE term survives. From the calculation carried out for the case when all NCs are identical with only the nearest neighbor interaction, the transition amplitude of the SE is
The first order TP,
for the SE of an excited NC to a SPP of mode
the TP would have the cross product contributions between the first and third order transition amplitudes of the SEs. However, neither do they show any substantial collective effect on the same footing as the first order contribution alone. Then, the most important contribution with respect to the cooperative effect comes from the square of
From Eq. (10), the third order TP for the SE of an NC to an SPP of mode
is obtained by
The cooperativeness in the third order TP is manifested by the phase Φ
Now we consider the collective effect of the SE in the third order TP. As was pointed out in the previous section, the terms with (sin Δ
[FIG. 4.] NC-SPP coupling constant gj,k. (a) The coupling strength ?gj,k? as a function of the dipolar orientation and the wave number for ρ = 1.2a. (b) The phase of the coupling constant θj,k as a function of the dipolar orientation αand the wave number k.
which is plotted as a function of
and is shown in Fig. 5(b), for comparison. No noticeable
[FIG. 5.] Transition probability of the spontaneous emission to the k mode of SPP without temporal part as a function of time and wave number for ρ =1.2a. (a) The third order transition probability of the SE under resonant approximation. (b) The first order transition probability of the SE.
change for TP is observed with regard to including multiple interactions between NC and SPP.
Now we analyze the cooperative effect due to direct DDI. When all NCs are identical, the second order TP of the SE to the k mode of SPP,
The cooperativeness of NCs is involved in the phase Φas the case without DDI. As described above, the phase difference caused by the axial distance between NCs, is order of
When the radial distances of NCs from the wire center are almost the same among the NC aggregates (？
In this case, the phase difference Δ
[FIG. 6.] The second order transition probability of the SE to the k mode of SPP when all DDI coupling is the same and dipolar directions are all the same, where Δz = zj 1- zj = α , and f(t) is set to be unity. (a) TP as a function of k and α . (b) TP as a function of for α = 0. Solid and dashed lines are for μm = ρ and μ m = z , respectively. (c) TP as a function of k and Δ z for α=π /4. The range of the inter-NC distance Δz is from a to 6α .
suppressed, as expected. And when all NCs are arranged along the NW, the TP of the SE will decrease as the distance between the NCs and the NW increases as seen from Fig. 3(b).
In practice, the phase difference Δ
the second order TP as a function of
the transition probability of the SE as a function of
[FIG. 7.] The second order transition probability of the SE to the k mode of SPP when ρj ？ρ with r mj= z, μ 1 = ρ, μ2 ？z = cosθand Δz = a. (a) TP as a function of k and α when μ 3=z . (b) TP as a function of k and αwhen μ3 = ρ.
excited. Meanwhile, if two or more NCs are excited, then the TP of the SE will strongly depend on the initially prepared state. For the case without the interaction between excitons, if the NCs are excited in the form of a superradiative (subradiative) state, the emission rate will be enhanced (suppressed) .
In summary, we have analyzed the characteristics of the cooperative feature of the spontaneous emission of one excited NC among NC aggregate into a mode of SPP in a metallic nanowire. In order to see the collective effect of the SE, higher order perturbation expansions of the transition probability of the SE have been derived. In particular, the influence of the dipolar orientation and the axial and radial arrangement of NCs has been considered. The first-order transition probability of the SE does not show collective behavior. Also the collective effect disappears in the thirdorder process of the spontaneous emission when direct DDI is not involved under the resonant coupling approximation between SPPs. However, the second-order transition probability of the SE yields a cooperative effect of the SE. In particular, the spontaneous emission of the excited NC energy is suppressed or enhanced depending on the relative orientation of the dipoles and the arrangement of NCs.