Plasmon-Induced Quantum Interference near Carbon Nanostructures

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Plasmon-Induced Quantum Interference near Carbon Nanostructures Vasilios D. Karanikolas, and Emmanuel Paspalakis J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02703 • Publication Date (Web): 04 Jun 2018 Downloaded from http://pubs.acs.org on June 4, 2018

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The Journal of Physical Chemistry

Plasmon-Induced Quantum Interference near Carbon Nanostructures Vasilios Karanikolas∗ and Emmanuel Paspalakis∗ Materials Science Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece E-mail: [email protected]; [email protected]

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Abstract We investigate the spontaneous emission properties of a V -type quantum emitter when interacting with either a graphene monolayer or a single-wall carbon nanotube. Quantum interference (QI) effects in spontaneous emission arise due to the anisotropic Purcell effect in both cases. For a graphene monolayer the degree of QI takes values close to 0.333. For the carbon nanotube the confinement of light in one dimension enhances significantly the degree of QI. The high degree of QI is obtained for different orientations of the respective transition dipoles and for a wide range of the transition energy, the chemical potential and the nanotube radius.

Introduction The spontaneous emission (SE) rate of a quantum emitter (QE) can be enhanced several orders of magnitude under the interaction with a conducting nanostructure, by exciting the surface plasmon modes, at specific energies. 1,2 Carbon nanostructures, such as graphene and single-wall carbon nanotubes are materials that can support surface plasmon modes and their interaction with QEs is a field of intensive research over the last years, both theoretically and experimentally. 3–28 Investigating the interaction between QEs and carbon nanostructures is important for a vast amount of applications, such as biosensors, 29 light harvesting devices, 30 and quantum computing. 31 In this work we investigate the quantum interference (QI) in SE of a V -type QE interacting with carbon nanostructures. The phenomenon of QI in SE was initially proposed by Agarwal in 1974 32 and has been extensively studied over the years. 33 The SE of this QE exhibits interference effects due the anisotropic Purcell effect, i.e. the different response of the electromagnetic modes along mutually perpendicular directions. 33,34 This phenomenon has been studied in various photonic structures, including periodic dielectrics, 35 negative refractive index metamaterials, 36–38 negative permeability metamaterial slabs, 39 metasurfaces, 40 hyperbolic metamaterials, 41 metallic and metallo-dielectric nanostructures, 42–45 and semiconductor microcavities. 46 This work extends the QI in SE in carbon nanostructures, namely a planar graphene layer 17 and a single-wall car2


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bon nanotube 18 (see Fig. 1). These nanostructures confine the light in two and one dimensions, respectively, and support propagating surface plasmon modes. We find that the degree of QI, p, which can, in general, take values from −1 to 1, with p = ±1 defining the maximum QI and p = 0 no QI, for a graphene monolayer is flat, having almost a constant value of p ≈ 0.333. This property remains for a wide range of values of the emission energy of the QE and the chemical potential, regardless of the excitation of surface plasmon mode. For a carbon nanotube high values of the degree of QI are obtained for different combinations of the transition dipole orientations for a wide range of parameters of the system in the mid-wavelength and short-wavelength infrared regimes of the spectrum. Moreover, p reaches high values, well above 0.9, for two different combinations of the dipole orientations, when the surface plasmon mode is excited. This behavior occurs since for the tangential orientation of the transition dipole moment with respect to the surface of the nanostructure there is no coupling to the surface plasmon mode. Moreover, due to the geometry of the carbon nanotube, there is one more parameter, its radius, that can be tuned. Additionally, the surface plasmon modes, supported by the graphene nanostructures, can be dynamically tuned by the value of the chemical potential, which characterizes the surface conductivity. Thus opening the possibility to envision switching devices, which can be dynamically tuned, at the telecommunication wavelengths. 47 Furthermore, the level of QI can also be used as a spectroscopic technique to estimate the optical properties of the environment of the V −type QE. 48 The paper is organized as follows. In the next section we present the QE and the calculation of the SE rates for the QE near a graphene monolayer and a single-wall carbon nanotube. Then, in the following section we present results for the degree of QI for various systems parameters and also present results for the population dynamics of the QE for different initial conditions. A summary of our results is also provided. We also include an Appendix for details on the calculation of the Green’s tensor for the graphene layer and the carbon nanotube.

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Quantum System and Calculation of the SE Rates We consider a V -type system with two degenerate Zeeman sublevels for the upper states |2⟩ and |3⟩, and one lower state |1⟩. The quantum system is located in vacuum in the position rQE , at distance d away from the surface of the carbon nanostructure (see Fig. 1). The dipole moment operator ( ) √ is taken as p = ℘(|2⟩ ⟨1| n− + |3⟩ ⟨1| n+ ) + H.c., where n± = n⊥ ± in∥ / 2 describes the rightrotating (n+ ) and left-rotating (n− ) unit vectors and ℘ is assumed to be real. Both excited levels |2⟩ and |3⟩ decay spontaneously to the lower level |1⟩ and the emission energy is h¯ ω . The spontaneous decay dynamics in the above system is described by a density-matrix approach. By considering solely SE effects, the time-dependent density matrix equations describing the interaction of the QE with its environment, in the rotating wave and Born-Markov approximations, are given by 35–40,42,43

ρ˙ 22 = −2γρ22 − κ (ρ23 + ρ32 ) ,

(1)

ρ˙ 33 = −2γρ33 − κ (ρ32 + ρ23 ) ,

(2)

ρ˙ 23 = −2γρ23 − κ (ρ22 + ρ33 ) ,

(3)

∗ . Also, γ = (Γ + Γ )/2, κ = pγ , where p is the degree of with ρ11 + ρ22 + ρ33 = 1 and ρnm = ρmn ⊥ ∥

QI is defined as p=

Γ⊥ − Γ∥ , Γ⊥ + Γ∥

(4)

with Γi , i =⊥, ∥, being the SE of the V -type QE for the perpendicular and parallel transition dipole moments with respect to the carbon nanostructure surface. Note that κ , and therefore p, is an important term as it describes the coupling coefficient between states |2⟩ and |3⟩ due to anisotropic Purcell effect, 34 and it is responsible for the effects of QI. The optical response of the carbon nanostructure is described through its in-plane surface conductivity, σ , in the random phase approximation. This quantity is mainly determined by electronhole pair excitations, which can be divided into intraband and interband transitions. It can be

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Figure 1: Schematic of V-type QE on top of a free-standing graphene sheet (upper figure) and next to single-wall carbon nanotube (lower figure).

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(a)

(b)

Figure 2: (a) The real and imaginary parts of the dielectric permittivity for a graphene monolayer, varying the energy h¯ ω , for different values of the chemical potential µ . (b) The propagation length, LSP , and penetration depth, δSP , for varying energy h¯ ω , for different values of the chemical potential µ .

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expressed via the chemical potential µ and the temperature T as 49,50

σ = σintra + σinter ,

(5)

where the intraband and interband contributions are, [ ( )] 2ie2 kB T µ ln 2 cosh σintra = , h¯ π (¯hω + i¯h/τ ) 2kB T ( [ ) ] h¯ ω − 2µ i (¯hω + 2µ )2 e2 1 1 σinter = + arctan − ln , 4¯h 2 π 2kB T 2π (¯hω − 2µ )2 + (2kB T )2

(6) (7)

with σintra describing a Drude model response corrected for scattering by impurities through a term containing τ , the relaxation time. The intraband contribution accounts for the surface plasmon generation, which lifetime is corrected to become finite by including the relaxation lifetime τ . In this paper we use a value of τ = 1 ps which is consisted with experimental measurements of the DC mobility, 51 where for µ = 1eV the mobility is M = 104 cm/(sV), using the relation

τ = M µ /ev2F . Better quality graphene samples have been prepared, leading to larger plasmon mobilities. 52 In order to better understand the behaviour of the surface conductivity, σ , we plot the dielectric permittivity, ε , using the relation ε = 1 + iσ /ε0 ω . In Fig. 2(a) we observe that for energies h¯ ω < 1.5µ the real part of the dielectric permittivity is largely negative. At these energies the behavior is matching the one described by the Drude model, thus behaves as a metal, where graphene surface plasmons modes can propagate along a graphene layer. At energies around h¯ ω ∼ 2µ the interband contributions dominates; these losses are connected with the generation of electron-hole pairs. At even higher energies the real part of the dielectric permittivity goes to zero and the imaginary part goes to a constant value, which gives the 2.3% of absorption of the graphene in the visible part of the spectrum. At these energies the carbon nanostructures cannot support transverse magnetic plasmonic modes, but support the rather loosely confined transverse electric modes. Throughout this paper we focus at the room temperature T = 300 K and we vary the value of chemical potential, µ .

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In Fig. 2(b) we present the propagation length, LSP = 1/Im(kSP ), and the penetration depth,

δSP = 1/kz,SP , of the surface plasmon mode confined in an infinite graphene layer, for varying the energy, h¯ ω , for different values of the chemical potential µ . Here, kSP is the surface plasmon wave √ 2 . The graphene plasmon propagation length, L , gives the distance vector and kz,SP = k02 − kSP SP at which the plasmon mode has the 1/e amplitude, and it is well defined when the intraband contribution dominates. The LSP can reach values as large as hundreds of microns at low frequencies. Moreover, we observe that the LSP has a shoulder structure, which blueshifts as the value of µ is increased. The existence of the shoulder can be explained by the plot of ε in Fig. 2(a), where we observe that at energies 0.1 eV < h¯ ω < 1.5µ the imaginary part of ε is zero, thus the graphene layer has metallic behavior. The penetration depth is an important quantity when we consider the interaction between a quantum emitter and the graphene layer, as it gives the length up to which the quantum emitter can efficiently excite the surface plasmon mode for a given energy. Both LSP and δSP values decrease rapidly, for a given value of µ , because the emission energy from a QE has enough energy to generate electron-hole pair and the dielectric permittivity is dominated by the interband contributions. In this paper we concentrate on carbon nanotube radius, R, that are larger than 5 nm. For these radii of the carbon nanotube the surface conductivity describing the zig-zag, armchair and chiral nanotubes can be approximated by the expression we use Eq. (5). More details can be found in ref. 53 Furthermore, in ref. 10 there is a theoretical analysis examining the optical response of the carbon nanotube for plane-wave excitation considering various values of the radii, showing that Eq. (5) works quite well. The normalized SE rates in the presence of carbon nanostructures is given by Γi √ 6π c = ε1 + Im(ni · G(rQE , rQE , ω ) · ni ) , i =⊥, ∥ , Γ˜i = Γ0 ω

(8)

where ε1 is the permittivity of the host medium and Γ0 (ω ) = ω 3℘2 /3πε0 h¯ c3 is the free space SE rate. Also, G(r, r′ , ω ) is the induced electromagnetic Green’s tensor due to the carbon nanostruc-

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ture calculated at the position of the QE, rQE . The Green’s tensor is the main quantity needed to calculate the SE rates and the degree of QI. For achieving this, we use the method of scattering superposition, where more details can be found in the Supplementary Information. For a QE above a free-standing graphene layer we use Cartesian coordinates (x, y, z). The graphene layer lies on the xy-plane and the z-axis is perpendicular to it. The SE rates of a QE, positioned at rQE = (0, 0, z), for the z- and x- orientations of the transition dipole moments are given by the expressions

Γ˜ z

Γ˜ x

( ∫∞ ) √ ks3 11 2ikz1 z 3c = ε1 + Im i dks R e , 2ω kz1 k12 N  0  ( ) ∫∞ 2 kz1 11 2ikz1 z √ ks 3c  , = ε1 + Im i dks R11 e M + 2 RN 4ω k1 k1

(9)

(10)

0

√ where ks is the in-plane wavevector and kzi =

ki2 − ks2 the perpendicular wavevector component.

For the single-wall carbon nanotube, due to the symmetry imposed by the geometry we use cylindrical coordinates (ρ , θ , z). The central axis of the nanotube coincides with the z-axis. We consider a QE positioned outside a free-standing carbon nanotube, with radius R, where for rQE = (ρ , 0, 0), ρ > R. The SE rates of the QE, for the ρ , θ and z orientations of the transition dipole moments, have the form: Γ˜ ρ = + Γ˜ θ = + Γ˜ z =

[ +∞ ∫ +∞ ( √ i(2 − δn0 ) n2 11 3c ε1 + Im ∑ dkz R H H∗ 2 2 MM 1n 1n 2ω ρ k 0 s n=0 ρ1 )] 2 kρ 1 11 ′ ′∗ ) kz kρ 1 n ( 11 11 ′∗ RNM + RMN Hn1 Hn1 + R H H , k1 ρs k2 NN 1n 1n [ +∞ ∫ +∞ ( √ i(2 − δn0 ) 2 11 ′ ′∗ 3c ε1 + Im ∑ dkz kρ 1 RMM H1n H1n 2ω kρ2 1 n=0 0 )] ) kz kρ 1 n ( 11 n2 kz2 11 11 ′∗ ∗ RNM + RMN H1n H1n + 2 2 RNN H1n H1n , k1 ρs ρs k1 ] [ +∞ ∫ +∞ 2 √ i(2 − δn0 ) kρ 1 11 3c ∗ ε1 + Im ∑ dkz R H1n H1n , 2ω kρ2 1 k22 NN n=0 0

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(11)

(12) (13)


′ = H ′ (k ρ ) is its derivawhere H1n = H1n (kρ 1 ρ ) is the Hankel function of the first kind and H1n 1n ρ 1

tive, with respect to the argument. Ri and Ri j , for i, j = M, N, are the generalized reflection coefficients for the planar and nanotube carbon nanostructures, respectively, where M are the transverse electric and N is the transverse magnetic modes. For the carbon nanotube we observe that the transverse electric and transverse magnetic modes are hybridized for the ρ and θ orientations of the transition dipole moment.

Results We first consider the interaction between a QE and a free-standing graphene layer. In Fig. 3(a) we present the SE rates for the z- and x- orientations of the transition dipole moment and the degree of QI, p, in the presence of a gated free-standing graphene layer for different emission energies, h¯ ω , of the QE at a fixed position rQE = (0, 0, 10 nm), for a chemical potential value µ = 0.5 eV of the graphene layer. We observe that at emission energies of the QE up to the value of the chemical potential h¯ ω = 0.5 eV the SE rate is enhanced several orders of magnitude compared with the free-space value, for both z and x transition dipole orientations, due to the excitation of the surface plasmon mode. Up to these energies the QI, p, is constant with a value of p = 0.333, as SP ΓSP ¯ ω > µ = 0.5 eV, z /Γx = 2. For emission energies above the value of the chemical potential, h

the lossy surface wave (LSW) is the main path of relaxation for the QE. At these emission energies the ΓLSW /ΓLSW ≈ 2 still holds, so p ≈ 0.333 practically holds the whole spectrum. z x In Fig. 3(b) we present a contour plot of p for the QE placed at fixed position rQE = (0, 0, 10 nm) and varying its emission energy and the chemical potential value. The emission energy (y-axis) is normalized to the chemical potential value, µ , in order to have the same value span for direct comparison. The QI has a value very close to p = 0.333 over the whole range of values of the chemical potential and the emission energy. There is a small variance of p, only for higher values of the chemical potential, µ > 0.9 eV, at emission energies h¯ ω /µ to 2¯hω /µ where the interband contributions are dominating. There, the condition ΓLSW /ΓLSW ≈ 2 does not hold and the value of z x

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(a)

(b)

(c)

Figure 3: (a) Normalized SE rates in the presence of a graphene monolayer, considering the z and x orientations of the transition dipole moments and the degree of QI, p, varying the emission energy of the QE, h¯ ω . The value of the chemical potential is µ = 0.5 eV. The position of the QE is at r = (0, 0, 10 nm). (b-c) Contour plot of p: (b) varying h¯ ω and the chemical potential, µ , of the graphene monolayer keeping fixed the QE position at r = (0, 0, 10 nm), (c) varying the emission energy of the QE, h¯ ω , and the position of the QE, r = (0, 0, zQE ), for fixed value of the chemical potential, µ = 0.5 eV.

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p drops. In Fig. 3(c) we present a contour plot of p considering a fixed value for the chemical potential of the graphene layer µ = 0.5 eV, and vary the emission energy of the QE and its position, rQE = (0, 0, zQE ). We observe that for the whole range of the emission energies at small QE-graphene monolayer separations the QI has the value p = 0.333. As the separation increases above 35 nm, for emission energies above h¯ ω > µ = 0.5 eV, where the LSWs contribution dominates, the QI value goes to zero, p → 0. Due to the fact that the LSW are only excited from the near field of the QE and at large distances decouples and fails to excite them. For emission energies h¯ ω < µ = 0.5 eV where the surface plasmon waves are excited, the penetration depth dictates the interaction between QEgraphene monolayer. This is the reason that the QI has a constant value of p = 0.333 for distances above 100 nm for emission energies that excite the surface plasmon modes. We now investigate the single-wall carbon nanotube as being the environment of the QE. In Fig. 4 we present the normalized SE rates, for the ρ , θ and z transition dipole moments, and the degree of QI for the three combinations of transition dipole orientations ρ , θ , named pρθ , z, θ , named pzθ , and ρ , z, named pρ z in the presence of a free standing carbon nanotube. We vary the emission energy of the QE, which is placed at rQE = (15 nm, 0, 0), in cylindrical coordinates, for a fixed radius R = 5 nm of the carbon nanotube (10 nm above the surface of the nanotube) and a fixed value of the chemical potential, µ = 0.5 eV. We observe in Fig. 4(a) that all normalized SE rates are enhanced several orders of magnitude, compared with the free space value of the SE rate. The ρ component of the Green’s tensor is discontinuous at the interface of the nanotube, resulting in an enhancement of the field created by a dipole source. Furthermore, for the θ orientation transition dipole moment the QE can only poorly couple, through its near field, to the surface plasmon mode. The z orientation of the transition dipole moment is continuous, like the θ orientation, at the interface of the nanotube but it can excite efficiently the surface plasmon modes, supported by the nanotube. Moreover, the z orientation has a purely transverse magnetic mode nature. In the range where the surface plasmon modes are excited, h¯ ω < µ = 0.5 eV, the SE rate for the ρ and z orientations is enhanced more than one

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(a)

(b)

Figure 4: (a) Normalized SE rates in the presence of a carbon nanotube with R = 5 nm, considering the ρ , θ and z orientations of the transition dipole moment, and (b) the degrees of QI, pρθ , pzθ , and pρ z , varying the emission energy of the QE. The value of the chemical potential is µ = 0.5 eV and the QE position rQE = (15 nm, 0, 0).

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order of magnitude compared to the θ orientation. For smaller emission energies, h¯ ω < 0.15 eV, the QE-carbon nanotube separations are well within the near field of the QE, and the QE relaxes through the LSW path. Moreover, for emission energies h¯ ω > µ the electron-hole pair dominates, there are no surface plasmon modes, and the poor coupling for the θ orientation of the transition dipole is evident. Regarding the degrees of QI we observe in Fig. 4(b) that there is considerable dispersion. At emission energies h¯ ω < 0.15 eV the QI values for pρθ are small but when the surface plasmon mode is excited it reaches values above 0.95. However, pzθ takes large negative values for very low emission energies h¯ ω < 0.05 eV and very large positive values when the surface plasmon mode is excited. Also, pρ z takes large values for small emission energies but for larger emission energies it takes very small values. For the transition dipole moments of the QE oriented along ρ and z there are excitations of the carbon nanotube surface plasmon waves, while for θ -oriented dipole there is not. For the case where the intraband transition dominate the surface conductivity, σ , the LSW relaxation path dominates for the ρ and z orientations while for the θ orientation there is only a poor coupling. Finally, the ρ and z transition dipole moments of the QE can both excite the surface plasmon mode, thus at these emission energies the pρ z is close to zero. For small emission energies, the coupling of the LSW for a QE with z orientation of the transition dipole moment is poorer than for the ρ one, Fig. 4(a), this leads to large negative values for the pρ z . In Figs. 5(a) and 5(c) we present a contour plot of pρθ [Fig. 5(a)] and pzθ [Fig. 5(c)], where we vary the emission energy of the QE and the chemical potential of the carbon nanotube. The radius of the carbon nanotube is R = 5 nm and the position of the QE is kept fixed at rQE = (15 nm, 0, 0). We observe that for a wide range of parameters both pρθ and pzθ have values above 0.9 (the green curves defines these areas), while pzθ for certain regions takes large negative values, pzθ < −0.8. The enclosed green areas are also the areas where the surface plasmon modes are excited. For the higher values of the chemical potential we observe that the QI is more dispersed and can be varied form values from 0.3 to close to 1 (the maximum is about 0.975) for pρθ and from −0.4 to close to 1 (the maximum is about 0.97) for the pzθ showing the importance of the modulation

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of the chemical potential for tuning the QE-nanotube interaction. For small emission energies and small values of the chemical potential a QE, oriented along θ and positioned close to the nanotube, can excite more efficiently the LSW modes than when the z orientation is considered [see also Fig. 4(a)]. Hence, the pzθ can get large negative values. (a)

(b)

(c)

(d)

Figure 5: Contour plot of pρθ for (a) and (b) and pzθ for (c) and (d): (a), (c) varying the emission energy, h¯ ω , and the chemical potential, µ , keeping fixed the QE position, r = (15 nm, 0, 0) with R = 5 nm, (b), (d) varying h¯ ω , and the radius R of the carbon nanotube, the position of the QE is r = (R + 10 nm, 0, 0), for fixed value of the chemical potential, µ = 0.5 eV. The green lines enclose the areas where p > 0.9.

In Figs. 5(b) and 5(d) we present a contour plot of pρθ [Fig. 5(b)] and pzθ [Fig. 5(d)], varying its emission energy and the carbon nanotube radius, R. We keep fixed the value of the chemical potential, µ = 0.5 eV, and the position of the QE, 10 nm away from the carbon nanotube surface, rQE = (R + 10 nm, 0, 0). We observe that there are optimum values of the emission energy of the QE where both pρθ and pzθ have values above 0.9. These are given by the enclosed green curves 15



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in Figs. 5(b) and 5(d). At these energies the QE can efficiently excite the surface plasmon mode, supported by the carbon nanotube. The position of the QE also influences the optimum values of the emission energy, for a given nanotube radius, by the surface plasmon penetration depth. Comparing Figs. 5(b) and 5(d) we observe that the QI, pρθ , can sustain values above 0.9 for radius up to 22 nm, while for the pzθ is up to 16 nm. This effect is due to the stronger coupling with the surface plasmon mode for a dipole with ρ orientation of the transition dipole moment, compared with one with z orientation. In Fig. 5(d), for small emission energies of the QE, where the QEnanotube separation is 10 nm, we are well within its near field where the main path of relaxation is the LSW mode. At these energies Γθ > Γz , so pzθ < 0, where values pzθ < −0.8 can be exhibited. We observe in Figs. 5(b) and 5(d) that as the radius of the nanotube, R, increases the QI decreases. In the limit of R → ∞ we will be again in the graphene monolayer case, where the QI has a value of up to p = 0.333. Thus, there are different ways to tune the interaction between a V -type QE and a carbon nanotube, resulting to dispersed values of p′ s for different dipole orientations, using the chemical potential and the radius of the carbon nanotube. Furthermore, the confinement of light from 2D, for the graphene monolayer, to 1D, for the carbon nanotube, breaks the barrier of having a limited value of p about 0.333 to values well above 0.9. In the case where only one of the upper states is initially excited, for example ρ22 (0) = 1,

ρ33 (0) = 0, ρ23 (0) = 0, the analytical formulae for the populations from Eqs. (1)-(3) are given by 43 )2 1 ( −Γ∥t e + e−Γ⊥t 4 )2 1 ( −Γ∥t ρ33 (t) = e − e−Γ⊥t . 4

ρ22 (t) =

(14) (15)

If no QI is considered, p = κ = 0, and

ρ22 (t) = e−(Γ∥ +Γ⊥ )t ,

ρ33 (t) = 0 .

(16)

An example of population dynamics for the V -type QE next to a graphene layer is shown in Fig. 16


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(a) 1.0 Ρ22 HtL, Ρ33 HtL

0.8 0.6 0.4 0.2 0.0 0

0.00002 0.00004 0.00006 0.00008 G 0t

(b) 1.0 0.8 Ρ22 HtL, Ρ33 HtL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2 0.0 0.0000

0.0002

0.0004

0.0006

0.0008

G 0t

Figure 6: Population dynamics of states |2⟩ (solid curve with QI and dot-dashed curve without QI, i.e. with p = κ = 0) and |3⟩ (dashed curve with QI and dotted curve without QI), when the QE is initially in state |2⟩. (a) The QE is at position rQE = (0, 0, 10 nm) from a graphene monolayer, (b) the QE is at r = (15 nm, 0, 0) from a carbon nanotube with R = 5 nm for the combination ρ , θ of the dipole orientations. In all plots h¯ ω = 0.4 eV and µ = 0.5 eV.

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6(a) when the QE is initially in state |2⟩. In this case, the QI has a weak influence to the population dynamics of state |2⟩, and state |3⟩ is very weakly excited at short times. The dynamics is different when the QE is next to a carbon nanotube, as it is shown in Fig. 6(b) for the combination ρ , θ of the transition dipole orientations. There, the population transfer to state |3⟩ is significant and after an initial fast decay of state |2⟩, which is determined mainly by Γρ , both populations decay simultaneously with a decay that is determined by 2Γθ . This is in contrast to the case that the QI is zero, where the population of the initially excited state evolves exponentially with decay Γρ + Γθ , and the other state is completely unexcited.

Summary In summary, we studied the interaction of a V -type QE with a graphene monolayer and a carbon nanotube and showed that these systems may give QI in SE due to anisotropic Purcell effect. For the graphene monolayer the degree of QI takes values close 0.333 for a wide range of parameters. Decreasing the dimensionality of the nanostructure to the carbon nanotube, the degree of QI for the different transition dipole orientations is significantly enhanced. Also, there is much larger control for tuning the interaction of the QE with the carbon nanotube using various system parameters. The emergence of QI in SE affects strongly in a positive manner quantum and nonlinear optical phenomena 33,43,44,46,54 and has even the potential to enhance the power generated by a photovoltaic device. 55–58 Therefore, the studied coupled QE-carbon nanostructures may find useful applications in quantum information and other emerging technologies.

Supplementary Information The supplementary information presents: 1. Details on the calculation of the Green’s tensor for a graphene layer and a carbon nanotube. 2. Field distributions created from a QE for the graphene layer and the carbon nanotube ge18


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ometries 3. Results for the QI term pρ z for a carbon nanotube. 4. Results for the population dynamics for initial coherent superposition states.

References (1) Barnes, W. L.; Dereux, A.; Ebbesen, T. W. Surface Plasmon Subwavelength Optics. Nature 2003, 424, 824-830. (2) Marquier, F.; Sauvan, C.; Greffet, J.-J. Revisiting Quantum Optics with Surface Plasmons and Plasmonic Resonators. ACS Photonics 2017, 4, 2091-2101. (3) Shafran, E.; Mangum, B. D.; Gerton, J. M. Energy Transfer from an Individual Quantum Dot to a Carbon Nanotube. Nano Lett. 2010, 10, 4049-4054. (4) Koppens, F. H. L.; Chang, D. E.; García de Abajo, F. J. Graphene Plasmonics: A Platform for Strong Light−matter Interactions. Nano Lett. 2011, 11, 3370-3377. (5) Cox, J. D.; Singh, M. R.; Gumbs, G.; Antón, M. A.; Carreño, F. Dipole−Dipole Interaction Between a Quantum Dot and a Graphene Nanodisk. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 125452. (6) Manjavacas, A.; Nordlander, P.; García de Abajo, F. J. Plasmon Blockade in Nanostructured Graphene. ACS Nano 2012, 6, 1724-1731. (7) Grigorenko, A. N.; Polini, M.; Novoselov, K. S. Graphene Plasmonics. Nature Photon. 2012, 6, 749-758. (8) Biehs, S.-A.; and Agarwal, G. S. Large Enhancement of Förster Resonance Energy Transfer on Graphene Platforms. Appl. Phys. Lett. 2013, 103, 243112.

19



(9) Agarwal, G. S.; and Biehs, S.-A. Highly Nonparaxial Spin Hall Effect and its Enhancement by Plasmonic Structures. Opt. Lett. 2013, 38, 4421-4424. (10) García de Abajo, F. J. Graphene Plasmonics: Challenges and Opportunities. ACS Photonics 2014, 1, 135-152. (11) Low, T.; Avouris, P. Graphene Plasmonics for Terahertz to Mid-infrared Applications. ACS Nano 2014, 8, 1086-1101. (12) Forati, E.; Hanson, G. W.; Hughes, S. Graphene as a Tunable THz Reservoir for Shaping the Mollow Triplet of an Artificial Atom via Plasmonic Effects. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 085414. (13) Mazzamuto, G.; Tabani, A.; Pazzagli, S.; Rizvi, S.; Reserbat-Plantey, A.; Schädler, K.; Navickaite, G.; Gaudreau, L.; Cataliotti, F. S.; Koppens, F. H. L.; et al. Single−Molecule Study for a Graphene-Based Nanoposition Sensor. New J. Phys. 2014, 16, 113007. (14) Kort-Kamp, W. J. M.; Amorim, B.; Bastos, G.; Pinheiro, F. A.; Rosa, F. S. S.; Peres, N. M. R.; Farina, C. Active Magneto−Optical Control of Spontaneous Emission in Graphene. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 205415. (15) Shi, Z.; Hong, X.; Bechtel, H. A.; Zeng, B.; Martin, M. C.; Watanabe, K.; Taniguchi, T.; Shen, Y.-R.; Wang, F. Observation of a Luttinger−Liquid Plasmon in Metallic Single−Walled Carbon Nanotubes. Nature Photon. 2015, 9, 515-519. (16) Tielrooij, K. J.; Orona, L.; Ferrier, A.; Badioli, M.; Navickaite, G.; Coop, S.; Nanot, S.; Kalinic, B.; Cesca, T.; Gaudreau, L.; et al. Electrical Control of Optical Emitter Relaxation Pathways Enabled by Graphene. Nat. Phys. 2015, 11, 281-287. (17) Karanikolas, V. D.; Marocico, C. A.; Bradley, A. L. Dynamical Tuning of Energy Transfer Efficiency on a Graphene Monolayer. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91, 125422. 20


Page 20 of 26

Page 21 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(18) Martin-Moreno, L.; García de Abajo, F. J.; Garcia-Vidal, F. J. Ultraefficient Coupling of a Quantum Emitter to the Tunable Guided Plasmons of a Carbon Nanotube. Phys. Rev. Lett. 2015, 115, 173601. (19) Cuevas, M.; Riso, M.A.; Depine, R.A. Complex Frequencies and Field Distributions of Localized Surface Plasmon Modes in Graphene−Coated Subwavelength Wires. J. Quant. Spectr. Rad. Transfer 2016, 173, 26-33. (20) de Vega, S.; Cox, J. D.; García de Abajo, F. J. Plasmons in Doped Finite Carbon Nanotubes and their Interactions with Fast Electrons and Quantum Emitters. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 94, 075447. (21) Nikitin, A. Y.; Vélez, S.; Mastel, S.; Centeno, A.; Pesquera, A.; Zurutuza, A.; Casanova, F.; Hueso, L. E.; Koppens, F. H. L.; Hillenbrand, R. Real-Space Mapping of Tailored Sheet and Edge Plasmons in Graphene Nanoresonators. Nat. Photon. 2016, 10, 239-243 . (22) Rivera, N.; Kaminer, I.; Zhen, B.; Joannopoulos, J. D.; Soljaˇcić, M. Shrinking Light to Allow Forbidden Transitions on the Atomic Scale. Science 2016, 353, 263-269. (23) Falk, A. L.; Chiu, K.-C.; Farmer, D. B.; Cao, Q.; Tersoff, J.; Lee, Y.-H.; Avouris, P.; Han, S.-J. Coherent Plasmon and Phonon−Plasmon Resonances in Carbon Nanotubes. Phys. Rev. Lett. 2017, 118, 257401. (24) Chang, C.-H.; Rivera, N.; Joannopoulos, J. D.; Soljaˇcić, M.; Kaminer, I. Constructing “Designer Atoms” via Resonant Graphene-Induced Lamb Shifts. ACS Photonics 2017, 4, 30983105. (25) Ma, Z.-H.; Cai, W.; Xiang, Y.-X.; Ren, M.-X.; Zhang, X.-Z.; Xu, J.-J. Dynamic Spontaneous Emission Control of an Optical Emitter Coupled to Plasmons in Strained Graphene. Opt. Express 2017, 25, 23070-23081.

21



(26) Cuevas, M. Spontaneous Emission in Plasmonic Graphene Subwavelength Wires of Arbitrary Sections. J. Quant. Spectr. Rad. Transfer 2018, 206, 157-162. (27) Sloan, J.; Rivera, N.; Soljaˇcić, M.; Kaminer, I.; Tunable UV-Emitters Through Graphene Plasmonics. Nano Lett. 2018, 18, 308-313. (28) Ho, P.-H.; Farmer, D. B.; Tulevski, G. S.; Han, S.-J.; Bishop, D. M.; Gignac, L. M.; Bucchignano, J.; Avouris, P.; Falk, A. L.; Intrinsically Ultrastrong Light-matter Interactions in Crystalline Films of Carbon Nanotubes, 2018, arXiv:cond-mat.mes-hall/1803.01087. arXiv.org e-Print archive. https://arxiv.org/abs/1803.01087 (accessed May 14, 2018). (29) Marini, A.; Silveiro, I.; García de Abajo, F. J. Molecular Sensing with Tunable Graphene Plasmons. ACS Photon. 2015, 2, 876-882. (30) Lin, C.-C.; Wang, D.-Y.; Tu, K.-H.; Jiang, Y.-T.; Hsieh, M.-H.; Chen, C.-C.; Chen, C.-W. Enhanced Infrared Light Harvesting of Inorganic Nanocrystal Photovoltaic and Photodetector on Graphene Electrode. Appl. Phys. Lett. 2011, 98, 263509. (31) Zeng, X.-D.; Liao, Z.; Al-Amri, M.; Suhail Zubairy, M. Controllable Waveguide via Dielectric Cylinder Covered with Graphene: Tunable Entanglement. Europhys. Lett. 2016, 115, 14002. (32) Agarwal, G.S. Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches. Springer-Verlag, Heidelberg, Germany 1974. (33) Kiffner, M.; Macovei, M.; Evers, J.; Keitel, C. H. Vacuum−induced Processes in Multi−level Atoms. Progress in Optics 2010, 55, 85-197. (34) Agarwal, G. S. Anisotropic Vacuum−induced Interference in Decay Channels. Phys. Rev. Lett. 2000, 84, 5500. (35) Li, G. X.; Li, F.-L.; Zhu, S.-Y. Quantum Interference Between Decay Channels of a

22


Page 22 of 26

Page 23 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Three−level Atom in a Multilayer Dielectric Medium. Phys. Rev. A: At., Mol., Opt. Phys. 2001, 64, 013819. (36) Yang, Y.-P.; Xu, J.-P.; Chen, H.; Zhu, S.-Y. Quantum Interference Enhancement with Left−handed Materials. Phys. Rev. Lett. 2008, 100, 043601. (37) Li,

G.-X.;

Evers,

J.;

Keitel,

C. H. Spontaneous Emission Interference in

Negative−Refractive−Index Waveguides. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 045102. (38) Xu, J.-P.; Yang, Y.-P. Quantum Interference of V−type Three−Level Atom in Structures Made of Left−handed Materials and Mirrors. Phys. Rev. A: At., Mol., Opt. Phys. 2010, 81, 013816. (39) Zeng, X.-D.; Xu, J.-P.; Yang, Y.-P. Spontaneous Emission Interference Enhancement with a

µ −negative Metamaterial Slab. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 84, 033834. (40) Jha, P. K.; Ni, X.; Wu, C.; Wang, Y.; Zhang, X. Metasurface−enabled Remote Quantum Interference. Phys. Rev. Lett. 2015, 115, 025501. (41) Sun, L.; Jiang, C. Quantum Interference in a Single Anisotropic Quantum Dot near Hyperbolic Metamaterials. Opt. Express 2016, 24, 7719-7727. (42) Yannopapas, V.; Paspalakis, E.; Vitanov, N. V. Plasmon−induced Enhancement of Quantum Interference Near Metallic Nanostructures. Phys. Rev. Lett. 2009, 103, 063602. (43) Evangelou, S.; Yannopapas, V.; Paspalakis, E. Simulating Quantum Interference in Spontaneous Decay near Plasmonic Nanostructures: Population Dynamics. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 83, 055805. (44) Gu, Y.; Wang, L.; Ren, P.; Zhang, J.-X.; Zhang, T.-C.; Martin, O. J. F.; Gong, Q.H. Surface−plasmon-induced Modification on the Spontaneous Emission Spectrum via Subwavelength-confined Anisotropic Purcell Factor. Nano Lett. 2012, 12, 2488-2493. 23



(45) Thanopulos, I.; Yannopapas, V.; Paspalakis, E. Non−Markovian Dynamics in Plasmoninduced Spontaneous Emission Interference. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 95, 075412. (46) Hughes S.; Agarwal, G. S. Anisotropy−induced Quantum Interference and Population Trapping Between Orthogonal Quantum Dot Exciton States in Semiconductor Cavity Systems. Phys. Rev. Lett. 2017, 118, 063601. (47) Chang, D. E.; Sorensen, A. S.; Demler, E. A.; Lukin, M. D. A Single−Photon Transistor Using Nanoscale Surface Plasmons. Nat. Phys. 2007, 3, 807-812. (48) Basov, D. N.; Fogler, M. M.; Garcia de Abajo, F. J. Polaritons in van der Waals materials. Science 2016, 354, 6309, aag1992 1-8. (49) Wunsch, B.; Stauber, T.; Sols, F.; Guinea, F. Dynamical Polarization of Graphene at Finite Doping. New J. Phys. 2006, 8, 318. (50) Jablan, M.; Buljan, H.; Soljaˇcić, M.; Plasmonics in Graphene at Infrared Frequencies. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 245435. (51) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666-669. (52) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Two−dimensional Gas of Massless Dirac Fermions in Graphene. Nature 2005, 438, 197-200. (53) Slepyan, G. Ya.; Maksimenko, S. A.; Lakhtakia, A.; Yevtushenko, O.; Gusakov, A. V. Electrodynamics of Carbon Nanotubes: Dynamic Conductivity, Impedance Boundary Conditions, and Surface Wave Propagation. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 60, 17136. 24


Page 24 of 26

Page 25 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(54) Ficek, Z.; Swain, S. Quantum Interference and Coherence. Springer, New York, NY, USA 2004. (55) Scully, M. O. Quantum Photocell: Using Quantum Coherence to Reduce Radiative Recombination and Increase Efficiency. Phys. Rev. Lett. 2010, 104, 207701. (56) Svidzinsky, A. A.; Dorfman, K. E.; Scully, M. O. Enhancing Photovoltaic Power by Fano−induced Coherence. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 84, 053818. (57) Dorfman, K. E.; Voronine, D. V.; Mukamel, S.; Scully, M. O. Photosynthetic Reaction Center as a Quantum Heat Engine. PNAS 2013, 110, 2746-2751. (58) Creatore, C.; Parker, M. A.; Emmott, S.; Chin, A. W. Efficient Biologically Inspired Photocell Enhanced by Delocalized Quantum States. Phys. Rev. Lett. 2013, 111, 253601.

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