Scattering of nanowire surface plasmons coupled to quantum dots with azimuthal angle difference

Coherent scatterings of surface plasmons coupled to quantun dots have attracted great attention in plasmonics. Recently, an experiment has shown that the quantum dots located nearby a nanowire can be separated not only in distance, but also an angle ϕ along the cylindrical direction. Here, by using the real-space Hamiltonian and the transfer matrix method, we analytically obtain the transmission/reflection spectra of nanowire surface plasmons coupled to quantum dots with an azimuthal angle difference. We find that the scattering spectra can show completely different features due to different positions and azimuthal angles of the quantum dots. When additionally coupling a cavity to the dots, we obtain the Fano-like line shape in the transmission and reflection spectra due to the interference between the localized and delocalized modes.

represents the diagonal element of the j th QD operator with a atomic resonance frequency ω 0 and σ = | 〉〈 | + e g j j j ( ) represents the rasing operator. Here, † a k n , (a k,n ) is the creation (annihilation) operator of the SP. We assume a SP is incident from the left with energy E k = v g k for the n th mode. Here, v g and k are the group velocity and wave number of the incident SP, respectively. Since the SPs are confined on the surface of the cylindrical nanowire, the summation of n in Eq. (1) stands for the contributions from all the possible n modes, and g is the coupling constant between the SP and QD exciton. Note that Γ ′ ≡ γ 0 + Γ 0 is the total dissipation including the decay rate into free space γ 0 and other dissipative channels Γ 0 . By using the Fourier transform, each term in Eq. (1) can be easily represented in real space is a bosonic operator creating a right-going (left-going) SP at x and ϕ. The stationary state of the above QDs-NW coupled system with the energy matching condition E k = v g k can be written as j where |g 1 , g 2 〉 |0〉 sp denotes that both the QDs are in their ground states with zero SP state, and ξ k j is the probability amplitude that the j th QD jumps to its excited state. Suppose that a SP is incident from the left, the scattering amplitudes ψ + k n R , , and ψ + k n L , , take the forms where t and r are the transmission and reflection amplitude, respectively. Here, a and b represent the probability amplitudes of the SP between x = 0 and d, ϕ = 0 and φ, respectively. Besides, θ(x) is the unit step function. From the eigenvalue equation, H|E k 〉 = E k |E k 〉 , one can obtain the following relations for the coefficients: is the detuning between the incident SP energy with E k and the QD exciton energy ω 0 . By solving Eq. (5), the exact forms of the transmission and reflection amplitudes, t and r, are given by where d (j) is the distance between the first dot and j th dot, and φ j is the angle of j th QD with respect to the first QD along the φ direction when setting d 1 and φ 1 being zero. On the other hand, the scattering property of a nanowire coupled to N identical QDs can also be studied by applying the transfer-matrix method. Let us briefly review the transmission amplitude t and the reflection amplitude r for the case of a single-dot coupled to the nanowire:  where F has been defined in Sec. II. By making use of the transmission and reflection coefficients in Eq. (8), the transfer matrix T q of the NW coupled to a single-QD can be written as Thus, the transfer matrix τ for the entire system is determined by (1) . Consequently, the total reflect and transmit amplitudes with N QDs can be obtained: 2 when kd 1 + nφ 2 = mπ with m being an odd (red) or even (black) integer. (b) When m is an odd integer and I 1 is close to I 2 with appropriate value of ϵ/Γ pl , the typical transmission spectra present more distinct Fano resonance for the first case. For Γ ′ = 0 and κ = 0, we plot the transmission probabilities T 1 (red-dashed), T 2 (green-solid), T 3 (blue-dot-dashed), T 4 (black-dotted) with the detuning ϵ/Γ pl = 0.05, 0.6, 1.2, 3 when = .
. In order to make a comparison to the double-dot case, we specifically consider the three-dot case 58 as shown in Fig. 3. By solving the eigenvalue equation with N = 3 in Eq. (7) or Eq. (12), the transmission and reflection amplitudes can be obtained. For simplicity, we only show the transmission amplitudes where we have defined the phase terms ζ ≡ 2nφ 2 , α ≡ 2[kd 3 + n(φ 2 + φ 3 )], β ≡ 2(kd 2 + 2nφ 2 ) and γ ≡ 2[k(d 3 − d 2 ) + nφ 3 ], respectively. Here, we are interested in the scattering spectra resulting from the varying angles of QD-2 and QD-3. Figure 4 shows the scattering spectra as functions of the angles φ 2 and φ 3 . We find that the transmission (reflection) coefficient reveals the periodic maximum (minimum) value 1 (0), when keeping one QD fixed at the certain angle along the φ direction.

QDs-NW System Coupled To Cavity.
Recently, hybrid quantum system (HQS) has attracted renewed attention for its prospect of applications in future quantum devices. Here, we consider the HQS of the QDs (with nanowire) coupled to a metal-nanoparticle (MNP) as shown in Fig. 5. it was reported 82-87 that, for the very small separation between a quantum emitter and a metal nanoparticle, the spectral density of the surface electromagnetic fields of the nanoparticle becomes Lorentzian. This indicates that the emitter-nanoparticle system can form an effective cavity quantum electrodynamics (QED) system. We therefore study the scattering spectra of two kinds of HQS comprising the cavity coupled to two QDs. For the first case, we assume both QD-1 and QD-2 are coupled to the same cavity as shown in Fig. 6. In real space, the Hamiltonian of the cavity photon with a loss rate κ can be written as  I  I  I I  I I  I  I   I  I  I I  I  Here, we have defined the function , and the phase term The detuning between the incident SP energy (with E k ) and the cavity resonant frequency (ω c ) is labeled by the symbol ε. For the second case, we study the configuration that each QD is individually coupled to its own cavity as shown in Fig. 7. Here, we have assumed the two cavities are identical for simplicity. The Hamiltonian of the composite system can be rewritten as  I  I  I  I  I I   I  I  I  I  I  We plot in Figs 8, 9 and 10 the transmission probabilities T = |t| 2 (dashed lines) and reflection probabilities R = |r| 2 (solid lines) as a function of the detuning for the both cases. In plotting Fig. 8, we find that when kd 1 + nφ 2 = πm with m being an integer and I 1 being close to I 2 with appropriate value of ϵ/Γ pl , the transmission and reflection spectra have a more distinct Fano-type line shapes. In Fig. 9, when increasing the detuning ϵ/Γ pl , the position of the Fano-type line shapes would be shifted from the right to the left along the δ/Γ pl axis. For the second case, however, we can only observe two peaks with the absence of asymmetric Fano-type line shape as shown in Fig. 10. When increasing the detuning ϵ, the inter-peak separation is reduced rapidly.

Discussion
Since the Fano resonance only occurs in the first case, it is interesting to ask: What makes the two cases different? To answer this, let us note that, in Fig. 8(b), the stronger coupling strength of the two QDs to the cavity, the larger detuning ϵ is required to form the Fano-type line shapes for the first case. Contrarily, when J 1 coincides with J 2 , the Fano resonance vanishes rapidly. In this regard, the Fano resonance arises from the constructive and destructive interference between the localized and delocalized channels by the virtue of the coupling of the two QDs to the same cavity. Here, the localized channel represents the single QD mode, and the delocalized channel denotes the hybridization mode of the cavity photon and the two dots 88 . The surface plasmons passing through the two channels carry different phases and result in the interference. On the other hand, we can easily control the position of each peak along the δ/Γ pl axis by adjusting the coupling strength between each QD to the cavity in the second case as shown in Fig. 10. When J 1 = J 2 , the overlapping of two peaks makes the two QDs collectively act like a single QD. The notable feature of these results indicates that the Fano-type line shape can't be created due to the individual coupling to each own cavity. In other words, the difference between J 1 and J 2 is the primary cause of the Fano resonance.
In conclusion, the real-space Hamiltonians and transfer-matrix method are used to obtain the transport properties of SPs propagating on the surface of a silver NW coupled to QDs. The transmission and reflection spectra of the SPs depend not only on the position, but also on the azimuthal angle of the QDs. For the double-dot case, even the two QDs are placed at the same position in the x-axis, changing the angle of a QD along φ direction also affects the reflection (transmission) spectra. For the triple-dot case, the transmission (reflection) coefficient reveals the periodic maximum (minimum) value when keeping one QD fixed at the certain angle along the φ direction. Moreover, when there is an additional cavity coupled to QDs, the Fano-type line shape can be created if both the QDs are coupled to the same cavity. The appearance of Fano resonances is attributed to the interference between the localized and delocalized modes.