4986
OPTICS LETTERS / Vol. 39, No. 17 / September 1, 2014
Dynamics of plasmonic field polarization induced by quantum coherence in quantum dot–metallic nanoshell structures S. M. Sadeghi Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA ([email protected]) Received June 27, 2014; revised July 22, 2014; accepted July 22, 2014; posted July 24, 2014 (Doc. ID 214892); published August 18, 2014 When a hybrid system consisting of a semiconductor quantum dot and a metallic nanoparticle interacts with a laser field, the plasmonic field of the metallic nanoparticle can be normalized by the quantum coherence generated in the quantum dot. In this Letter, we study the states of polarization of such a coherent-plasmonic field and demonstrate how these states can reveal unique aspects of the collective molecular properties of the hybrid system formed via coherent exciton–plasmon coupling. We show that transition between the molecular states of this system can lead to ultrafast polarization dynamics, including sudden reversal of the sense of variations of the plasmonic field and formation of circular and elliptical polarization. © 2014 Optical Society of America OCIS codes: (160.3900) Metals; (160.6000) Semiconductor materials; (260.5430) Polarization; (270.0270) Quantum optics; (320.4240) Nanosecond phenomena; (160.4236) Nanomaterials. http://dx.doi.org/10.1364/OL.39.004986
Polarization analysis of plasmonic fields can offer us unique opportunities to investigate the fundamental physics and applications of metallic nanoparticles (MNPs) [1]. These analyses have been used to study various nanodevices and subplasmonic components [2–6]. However, when a MNP is in the vicinity of a semiconductor quantum dot (QD) and interacts with a laser field the plasmonic field of the MNP can be replaced by a coherent-plasmonic (CP) field [7,8]. Such a field is the result of coherent exciton–plasmon coupling, leading to normalization of the plasmonic field of the MNP by quantum coherence. This coupling process can generate unique features in QD-MNP systems. These include formation of collective molecular states (metastates) and resonances (plasmonic metaresonances, or PMR) that are different from excitons or plasmons [9]. The exciton– plasmon coupling can also induce Rabi oscillations [9–11], enhance second harmonic generation [12,13], generate bistability [14–16], etc. It can also alter for wave mixing and reverse the course of energy transfer between the QDs and MNPs [17,18]. Our objective in this Letter is to investigate the application of quantum coherence in a system consisting of a metallic nanoshell (NS) and a QD to dynamically control the states of polarization of its CP field. For this we study how at each point around the NS the collective molecular properties of the QD-NS system can present a unique dynamic polarization state specified by the coordinates of that point. We investigate the Stokes vectors and the polarization ellipses of the collective states of this system, demonstrating their fast changes as the system undergoes transition from one of these states (dark state) to another (bright state). Our results demonstrate reversal of the sense of variations of the near-field and position-dependent dynamics that can include ultrafast transition between circular and elliptical polarization states. These results suggest that quantum coherence in QD-MNP systems can offer a high degree of control over the states of polarization of near fields. In this Letter we assume the pure dephasing rate of the QD is 1 ps−1 . This suggests that the quantum coherence effects 0146-9592/14/174986-04$15.00/0
discussed here can occur at elevated temperatures where the quantum decoherence rate is high. This is an important feature for potential applications of this research. Consider a QD-NS system interacting with a visible laser field (E1 E 0 t cosωl t) resonant with the fundamental exciton transition of the QD (the 1-2 transition) with the frequency ω12 . We assume this field has frequency ωl and is polarized along the z axis (Fig. 1). We also assume the NS is made of gold with the inner and outer radii R1 and R2 . The center-to-center distance of the QD-NS is considered to be R and the dielectric constant of the NS core is εc . The dielectric constants of the environments around the QD and the NS are shown by ε0q and ε0m , respectively. The interaction between the QD and the NS is characterized by the electric-dipole moments of the QD and plasmons of the NS. The dielectric function of the NS (εm ω) includes both the d-electrons and the Drude contributions. The total electric field at a given point around the NS, Pr; θ, can be written as Er; θ E z eˆ z E y eˆ y , with eˆ z and eˆ y referring to the unit vectors along the z and y axes, respectively. Each component of this field can be written as E i r; θ E iNS E iQD E iapp (i z; y), wherein E iNS ,
Fig. 1. Schematic illustration of the QD-NS hybrid system. Pr NS ; θ and P 0 r NS ; −θ are two points in the plane with the same distance from the center of the NS (r NS ) but with opposite angles (θNS ). © 2014 Optical Society of America
September 1, 2014 / Vol. 39, No. 17 / OPTICS LETTERS
(2)
E zQD −1–3 cos2 θQD E QD r QD ;
(3)
E yQD 3 sinθQD cosθQD E QD r QD :
(4)
In these equations E NS r NS ε βNSr3
om NS
h
i 21 ρ21 E 0 4μ and 3 ε R eff1
EQD r QD ε 2με21 ρ21r3 , wherein ρ21 refers to the coherence 0m eff1 QD
term or off-diagonal element of the density matrix of the QD. Since we assume the applied field is along the 2ε0q εs , z axis, E zapp E 0 ∕2ε0m and E yapp 0. Here εef f 1 3ε 0q wherein εs is the dielectric constant of the QD material and μ12 is the dipole moment associated with the 1-2 transition. βNS is the polarizability of the NS given by [19] βNS
R32 R31 εc − εm ε0m 2εm − R32 AB : 2R31 A−εc εm R32 B2ε0m εm
(5)
Here A ε0m − εm and B εc 2εm . Since E z and E y can be complex, we can write them as Ez E 0z eiδz and E y E 0y eiδy . Here δz and δy are the phase terms with the difference ϕzy δz − δy . Having Er; θ one can find the plasmonic field enhancement factor as P enh r; θ jEj2 ∕jE 0 j2 . The effective field intensity experienced by the QD is then given by I eff P enh I 0 , wherein I 0 is the applied field intensity. Additionally, as noted in [20], for evanescent waves wherein fields cannot necessarily be described by two orthogonal field components, one needs to use modified Stokes parameters. However, in our case, because of the symmetry, we consider S 0 Ey E y E z E z , S 1 E y E y − E z E z , S 2 E y E z E z E y , and S 3 iE y E z − E z E y . Here S 1 ∕S 0 refers to the dominance of y to z linear polarization, S 2 ∕S 0 to that of 45° to −45° linear polarization, and S 3 ∕S 0 to that of right to left circular polarization. To calculate ρ21 one needs to find the density matrix of the QD (ρ) from ρ_ − ℏi H ↔ ρ; ρt £ρ↔ ρ [7,11]. Here £ρ↔ ρ refers to the decay term of the excitons and H ↔ ρ is the Hamiltonian of the QD, wherein “↔“ refers to the self-normalization process caused by its dependency on ρ. H ↔ ρ, which is given as H ↔ ρ
X j1;2
ℏωj σ ij ℏΩr12 ρσ 21 H:C:;
(6)
(7)
ρ_ 22 2 ImΩeff ρ21 − ΣF − Γ2 ρ22 ;
(8)
ρ_ 21 −iΔeff Λeff ρ21 − iΩeff δ:
(9)
Here Δeff ℏω12 − ReηNS δ − ℏωl is the effective detuning of the QD transition from the laser field and Λeff ΓF δ γ 12 is the effective broadening of this transition. δ ρ11 − ρ22 and ΓF ImηNS refer to the rate of Forster resonance energy transfer (FRET) from the QD to the NS in the absence of quantum coherence. These show that the detuning and damping of the QD in the QD-NS system are normalized by δ. Additionally, Eqs. (7) and (8) show that the FRET process can act as a nonradiative decay channel for the excitons in the QD, represented by ΣF 2ΓF jρ21 j2 . This suggests that, in the presence of quantum coherence, FRET only happens efficiently when jρ21 j2 has noticeable values [21]. To maintain quantum coherence at a high QD decoherence rate (1 ps−1 ), we assume the QD is made of CdSe-based material, with εs 6.25 embedded in ZeSe with ε0q 7.122 [22]. We also consider εc 2.16 (as silica) and R1 4 and R2 6 nm, setting the NS plasmon resonance energy to be 2.195 eV. Additionally, ℏω12 is considered to be 2.27 eV and ωl is adjusted such that Δ ℏω12 − ωl −5 ns−1 . We assume R 9 nm, ε0m 1.77, and the applied laser intensity (I 0 ) changes according to the inset of Fig. 2, i.e., it turns on at 100 ns of the scale and then it remains at 6.5 W∕cm2 . We first study I eff at the typical angle of θNS π∕5 (Fig. 1). Figure 2 shows the way I eff changes when r NS 8, 10, 12, and 15 nm. The results show a time delay in I eff with respect to the rise time of the applied field by about 250 ns. We also see a significant amount of field enhancement for r NS 8 nm. The reason behind the time delay, as discussed in detail in [9,23], is the coherent normalization of the broadening (Λeff ) and transition energy (Δeff ) of the QD. Since these parameters linearly depend on δ for low-carrier excitation, they are significant. Therefore, they hamper engagement of the laser field 80 10 70
f I0
E yNS 3 sinθNS cosθNS E NS r NS ;
ρ_ 11 −2 ImΩeff ρ21 ΣF Γ2 ρ22 − Γ1 ρ11 ;
60 (W/cm2)
(1)
0
50
g
h
i
5 8 0 100
500 e
40
eff
E zNS −1–3 cos2 θNS E NS r NS ;
frequency of the QD in the presence of pure plasmonic effects (without any coherence effects) and in the absence of the NS (very large R). Considering these, the density matrix equations are obtained as
I
EiQD , and E iapp are, respectively, the fields generated by the NS, QD, and the applied field. They can be given as follows:
4987
30 10 20
σ ij jiihjj, and ℏωj refer to the energy of jji. Ωr12 ρ Ωeff ηNS ρ21 is the coherently normalized Rabi 4μ212 βNS ℏε2eff1 R6
, and Ωeff frequency of the QD. Here ηNS μ12 E 0 and Ω012 − 2ℏε are, respectively, the Rabi Ω012 1 2βRNS 3 eff
a
10 0
b
c
d
12 15
0
100
200
300 400 Time (ns)
500
600
Fig. 2. Variation of I eff as a function of time for different r NS (numbers close to each curve in nm).
OPTICS LETTERS / Vol. 39, No. 17 / September 1, 2014
(a)
0.05
8 0
200
0
400
0
600
8
−0.1
0.15 0.1
10
0.05
12 15
(c)
(d)
0
500 Time (ns)
0
z
a f e d
−0.1 (a)
rNS=12
f e d
(b)
−0.2 0.2 −0.2
0
0.05
0
f e a
a 0
−0.05
0.2
rNS=15
d
−0.05 (c) −0.05
0 E
0.05
(d) −0.05
0 E
y
0.05
y
Fig. 4. Dynamics of polarization ellipse of the CP field at r NS 8 (a), 10 (b), 12 (c), and 15 nm (d). The letters close to each ellipse refer to the sampled times introduced in Fig. 2.
considering the result in Fig. 3(a). For r NS 8 nm, one can see that at the end of the time delay ReE z becomes negative. This leads to a dramatic change in the phase of the electric field. To see this in Fig. 5 we show ϕzy for r NS 8, 10, 12, and 15 nm. The circles refer to the a to i sampling points highlighted in Fig. 2. Note that for r NS 8, at around 380 ns ϕzy undergoes a 180 deg shift, changing the sense and orientation of the polarization ellipse. For other r NS ’s the change of ϕzy is rather smooth. The mapping of the polarization dynamics of the CP field becomes more evident if we consider the evolution of their Stokes vectors (S). Figure 6 shows the normalized component of this vector for r NS 8 (a) 10 (b) 12 (c) and 15 (d) nm. The circles show the sampled points introduced in Fig. 2. Note that Fig. 6(a) shows a dramatic rotation of S over a short period of time, corresponding to the distinct transition between the metastates of the QD-MNP system. As r NS increases the amount of rotation reduces. Figure 7 shows that the time evolution of the polarization of the CP field for different θNS when r NS 8 nm. For θNS 0 the QC field becomes purely linearly polarized along the z axis. Under this condition, only the amplitude of E z changes with time. For θNS π∕4 we do not see any polarization change as seen in the case of Fig. 4(a). This is because at this angle no phase change 0
−50
8
0
0
500
y
Re[Ey]
−0.05
d a
−0.2 −0.2
0.05
rNS=10
0.1
−0.1
12 15
0
f
0
0.2
15 12 10
10
0.1
e
0
10 12
−100
zy
0
8 8 10 12 15
0.2
=8
NS
(degree)
10
Im[Ez]
12
0.02
Im[E ]
Re[Ez]
0.04
(b)
0.1
15
0.06
r
15
φ
0.08
0.2
E
with the QD-MNP at the rise time of the applied field, causing the time delay. Additionally, the values of I eff during the time delay and after suggest two different collective states (or metastates) of the QD-NS system. Since during the time delay the QD is nearly nonemissive but after that it emits efficiently [9], we refer to these states as OFF and ON, respectively. To study the phase of the CP field, in Fig. 3 we show variations of E z and E y corresponding to the results shown in Fig. 2. For r NS 8 nm, within the delay period ReE z is quite significant, but after that it becomes negative [Fig. 3(a)]. On the other hand, ImE z increases significantly after the delay, demonstrating dramatically different dynamics [Fig. 3(b)]. As shown in Figs. 3(c) and 3(d), ReE y and ReE y also show very different dynamics before and after the time delay, while the former remains negative at all times. As r NS increases, except for ReE z , all other components are suppressed, suggesting the CP field at large distances is dominated by this component. This is expected, since for large r NS the field of the NS diminishes and ReE z is mostly determined by the applied field, which is polarized along z. The polarization states of the near field of the NS become clearer when we plot E z versus E y . Figure 4 shows the resulting polarization ellipses at different times and r NS . For this we chose certain times as labeled in Fig. 2 (a to i). In Fig. 4 we show the time corresponding to a, d, e, and f, as they depict the overall evolution of the system. When r NS 8 nm the polarization ellipse of the near field undergoes dramatic dynamics. As shown in Fig. 4(a), from a to e we see the variation of the shape of the ellipse in a counterclockwise sense. However, around f the ellipse suddenly switches while reversing its sense. This process suggests that at this r NS the polarization of the CP field changes when the system goes from one metastate to another. For the other r NS [Figs. 4(b)–4(d)], although such transition occurs, no such switching seems visible in the CP field. At r NS 12 nm the polarization ellipse evolves from a very elliptical one (represented by a) to a nearly left-handed circular one (indicated by f). The reason behind the switching and reversing of the sense of electric-field variation can be explored
Ez
4988
−150
500 Time (ns)
Fig. 3. Variation of the real and imaginary parts of Ez ((a) and (b)) and Ey ((c) and (d)) as a function of time for r NS 8, 10, 12, and 15 nm (numbers close to the curves). All specifications are the same as those in Fig. 2.
−200 100
8 200
300 400 Time (ns)
500
600
Fig. 5. Dynamics of the phase difference between Ey and Ez for r NS 8, 10, 12, and 15 nm (numbers close to each curve).
September 1, 2014 / Vol. 39, No. 17 / OPTICS LETTERS i
i
(a)
10
3
S /S
S3/S0
0
8
(b)
0
a
0
1
S /S 2
a
S /S
0
0
0 0 i
0 0
i
(c)
(d)
3
S /S
S3/S0
15
0
12
S1/S0
S2/S0
a
0 S /S 2
0
S1/S0
S2/S0
0 0
0 0
Fig. 6. Time evolution of the Stokes vector for different r NS (legends in nm) for the system studied in Fig. 2.
0.2 θNS=0
0.2
f
0
0
−0.1
−0.1
E
z
0.1
−0.2 (a) −0.2
0.2
z
−0.2 0.2
a
0
0.2 f
θ
=π/2
NS
0.1
0.1 E
θNS=π/15
−0.2 (b) 0
0.2 θ =π/4 NS
a
0
e f
ae
0 −0.1
−0.1 −0.2 (c) −0.2
f e
0.1
−0.2 (d) 0 Ey
0.2
−0.2
0 Ey
In conclusion, we showed that polarization of the plasmonic field of a NS can undergo unique time evolution once it is in the presence of a QD and driven by a laser field. We demonstrated that such dynamics are the result of the coherent exciton–plasmon coupling, causing characteristic changes in the phase and amplitude of the near field and sudden reversal of the sense and form of the polarization ellipse. These results may have applications in the investigation of chiral properties of single molecules and nanodevices based on plasmonic fields. The author is grateful for U.S. National Science Foundation support under grant no. CMMI 1234823.
a
0 S1/S0
4989
0.2
Fig. 7. Dynamics of polarization ellipse for different θNS and r NS 8 nm. All other specifications are the same as in Fig. 2.
(labeled with 8) happens as seen in Fig. 5. When θNS π∕2 the polarization of the QD field becomes elliptical with high ellipticity. Note that when the pure dephasing rate is less than 1 ps−1 , there are less stringent conditions for the effects discussed above. With a higher rate, these effects starts to smear out. We found that, for the QD-NS structure considered here, when this rate is about 5 ps−1 the quantum effects mostly disappear. Moreover, these effects depend on Δ in an asymmetric way. For Δ > 0, the time delay is infinite and the QD-NS system is in the OFF state. For Δ less than −10 nm, these effects start to smear out as the impact of the laser weakens.
References 1. O. Schubert, J. Becker, L. Carbone, Y. Khalavka, T. Provalska, I. Zins, and C. Sönnichsen, Nano Lett. 8, 2345 (2008). 2. P. Biagioni, M. Savoini, J.-S. Huang, L. Duò, M. Finazzi, and B. Hecht, Phys. Rev. B 80, 153409 (2009). 3. E. Öğüt and K. Şendur, Appl. Phys. Lett. 96, 141104 (2010). 4. J. Li, S. Chen, P. Yu, H. Cheng, X. Duan, and J. Tian, Opt. Express 21, 10342 (2013). 5. A. Roberts and L. Lin, Opt. Lett. 37, 1820 (2012). 6. A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, Opt. Lett. 38, 513 (2013). 7. R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, Phys. Rev. B 83, 235406 (2011). 8. S. M. Sadeghi, Nanotechnology 20, 225401 (2009). 9. S. M. Sadeghi, Phys. Rev. B 79, 233309 (2009). 10. M.-T. Cheng, S.-D. Liu, H.-J. Hao, and Q.-Q. Wang, Opt. Lett. 32, 2125 (2007). 11. M.-T. Cheng, S.-D. Liu, and Q.-Q. Wang, Appl. Phys. Lett. 92, 162107 (2008). 12. J. D. Cox, M. R. Singh, C. von Bilderling, and A. V. Bragas, Adv. Opt. Mater. 1, 460 (2013). 13. M. R. Singh, Nanotechnology 24, 125701 (2013). 14. R. D. Artuso and G. W. Bryant, Nano Lett. 8, 2106 (2008). 15. A. V. Malyshev and V. A. Malyshev, Phys. Rev. B 84, 035314 (2011). 16. J.-B. Li, N.-C. Kim, M.-T. Cheng, L. Zhou, Z.-H. Hao, and Q.-Q. Wang, Opt. Express 20, 1856 (2012). 17. E. Paspalakis, S. Evangelou, S. G. Kosionis, and A. F. Terzis, J. Appl. Phys. 115, 083106 (2014). 18. S. G. Kosionis, A. F. Terzis, V. Yannopapas, and E. Paspalakis, J. Phys. Chem. C 116, 23663 (2012). 19. A. O. Govorov and I. Carmeli, Nano Lett. 7, 620 (2007). 20. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002). 21. S. M. Sadeghi, Appl. Phys. Lett. 101, 213102 (2012). 22. O. Fedorych, C. Kruse, A. Ruban, D. Hommel, G. Bacher, and T. Kümmell, Appl. Phys. Lett. 100, 061114 (2012). 23. S. M. Sadeghi, Phys. Rev. B 82, 035413 (2010).
OPTICS LETTERS / Vol. 39, No. 17 / September 1, 2014
Dynamics of plasmonic field polarization induced by quantum coherence in quantum dot–metallic nanoshell structures S. M. Sadeghi Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA ([email protected]) Received June 27, 2014; revised July 22, 2014; accepted July 22, 2014; posted July 24, 2014 (Doc. ID 214892); published August 18, 2014 When a hybrid system consisting of a semiconductor quantum dot and a metallic nanoparticle interacts with a laser field, the plasmonic field of the metallic nanoparticle can be normalized by the quantum coherence generated in the quantum dot. In this Letter, we study the states of polarization of such a coherent-plasmonic field and demonstrate how these states can reveal unique aspects of the collective molecular properties of the hybrid system formed via coherent exciton–plasmon coupling. We show that transition between the molecular states of this system can lead to ultrafast polarization dynamics, including sudden reversal of the sense of variations of the plasmonic field and formation of circular and elliptical polarization. © 2014 Optical Society of America OCIS codes: (160.3900) Metals; (160.6000) Semiconductor materials; (260.5430) Polarization; (270.0270) Quantum optics; (320.4240) Nanosecond phenomena; (160.4236) Nanomaterials. http://dx.doi.org/10.1364/OL.39.004986
Polarization analysis of plasmonic fields can offer us unique opportunities to investigate the fundamental physics and applications of metallic nanoparticles (MNPs) [1]. These analyses have been used to study various nanodevices and subplasmonic components [2–6]. However, when a MNP is in the vicinity of a semiconductor quantum dot (QD) and interacts with a laser field the plasmonic field of the MNP can be replaced by a coherent-plasmonic (CP) field [7,8]. Such a field is the result of coherent exciton–plasmon coupling, leading to normalization of the plasmonic field of the MNP by quantum coherence. This coupling process can generate unique features in QD-MNP systems. These include formation of collective molecular states (metastates) and resonances (plasmonic metaresonances, or PMR) that are different from excitons or plasmons [9]. The exciton– plasmon coupling can also induce Rabi oscillations [9–11], enhance second harmonic generation [12,13], generate bistability [14–16], etc. It can also alter for wave mixing and reverse the course of energy transfer between the QDs and MNPs [17,18]. Our objective in this Letter is to investigate the application of quantum coherence in a system consisting of a metallic nanoshell (NS) and a QD to dynamically control the states of polarization of its CP field. For this we study how at each point around the NS the collective molecular properties of the QD-NS system can present a unique dynamic polarization state specified by the coordinates of that point. We investigate the Stokes vectors and the polarization ellipses of the collective states of this system, demonstrating their fast changes as the system undergoes transition from one of these states (dark state) to another (bright state). Our results demonstrate reversal of the sense of variations of the near-field and position-dependent dynamics that can include ultrafast transition between circular and elliptical polarization states. These results suggest that quantum coherence in QD-MNP systems can offer a high degree of control over the states of polarization of near fields. In this Letter we assume the pure dephasing rate of the QD is 1 ps−1 . This suggests that the quantum coherence effects 0146-9592/14/174986-04$15.00/0
discussed here can occur at elevated temperatures where the quantum decoherence rate is high. This is an important feature for potential applications of this research. Consider a QD-NS system interacting with a visible laser field (E1 E 0 t cosωl t) resonant with the fundamental exciton transition of the QD (the 1-2 transition) with the frequency ω12 . We assume this field has frequency ωl and is polarized along the z axis (Fig. 1). We also assume the NS is made of gold with the inner and outer radii R1 and R2 . The center-to-center distance of the QD-NS is considered to be R and the dielectric constant of the NS core is εc . The dielectric constants of the environments around the QD and the NS are shown by ε0q and ε0m , respectively. The interaction between the QD and the NS is characterized by the electric-dipole moments of the QD and plasmons of the NS. The dielectric function of the NS (εm ω) includes both the d-electrons and the Drude contributions. The total electric field at a given point around the NS, Pr; θ, can be written as Er; θ E z eˆ z E y eˆ y , with eˆ z and eˆ y referring to the unit vectors along the z and y axes, respectively. Each component of this field can be written as E i r; θ E iNS E iQD E iapp (i z; y), wherein E iNS ,
Fig. 1. Schematic illustration of the QD-NS hybrid system. Pr NS ; θ and P 0 r NS ; −θ are two points in the plane with the same distance from the center of the NS (r NS ) but with opposite angles (θNS ). © 2014 Optical Society of America
September 1, 2014 / Vol. 39, No. 17 / OPTICS LETTERS
(2)
E zQD −1–3 cos2 θQD E QD r QD ;
(3)
E yQD 3 sinθQD cosθQD E QD r QD :
(4)
In these equations E NS r NS ε βNSr3
om NS
h
i 21 ρ21 E 0 4μ and 3 ε R eff1
EQD r QD ε 2με21 ρ21r3 , wherein ρ21 refers to the coherence 0m eff1 QD
term or off-diagonal element of the density matrix of the QD. Since we assume the applied field is along the 2ε0q εs , z axis, E zapp E 0 ∕2ε0m and E yapp 0. Here εef f 1 3ε 0q wherein εs is the dielectric constant of the QD material and μ12 is the dipole moment associated with the 1-2 transition. βNS is the polarizability of the NS given by [19] βNS
R32 R31 εc − εm ε0m 2εm − R32 AB : 2R31 A−εc εm R32 B2ε0m εm
(5)
Here A ε0m − εm and B εc 2εm . Since E z and E y can be complex, we can write them as Ez E 0z eiδz and E y E 0y eiδy . Here δz and δy are the phase terms with the difference ϕzy δz − δy . Having Er; θ one can find the plasmonic field enhancement factor as P enh r; θ jEj2 ∕jE 0 j2 . The effective field intensity experienced by the QD is then given by I eff P enh I 0 , wherein I 0 is the applied field intensity. Additionally, as noted in [20], for evanescent waves wherein fields cannot necessarily be described by two orthogonal field components, one needs to use modified Stokes parameters. However, in our case, because of the symmetry, we consider S 0 Ey E y E z E z , S 1 E y E y − E z E z , S 2 E y E z E z E y , and S 3 iE y E z − E z E y . Here S 1 ∕S 0 refers to the dominance of y to z linear polarization, S 2 ∕S 0 to that of 45° to −45° linear polarization, and S 3 ∕S 0 to that of right to left circular polarization. To calculate ρ21 one needs to find the density matrix of the QD (ρ) from ρ_ − ℏi H ↔ ρ; ρt £ρ↔ ρ [7,11]. Here £ρ↔ ρ refers to the decay term of the excitons and H ↔ ρ is the Hamiltonian of the QD, wherein “↔“ refers to the self-normalization process caused by its dependency on ρ. H ↔ ρ, which is given as H ↔ ρ
X j1;2
ℏωj σ ij ℏΩr12 ρσ 21 H:C:;
(6)
(7)
ρ_ 22 2 ImΩeff ρ21 − ΣF − Γ2 ρ22 ;
(8)
ρ_ 21 −iΔeff Λeff ρ21 − iΩeff δ:
(9)
Here Δeff ℏω12 − ReηNS δ − ℏωl is the effective detuning of the QD transition from the laser field and Λeff ΓF δ γ 12 is the effective broadening of this transition. δ ρ11 − ρ22 and ΓF ImηNS refer to the rate of Forster resonance energy transfer (FRET) from the QD to the NS in the absence of quantum coherence. These show that the detuning and damping of the QD in the QD-NS system are normalized by δ. Additionally, Eqs. (7) and (8) show that the FRET process can act as a nonradiative decay channel for the excitons in the QD, represented by ΣF 2ΓF jρ21 j2 . This suggests that, in the presence of quantum coherence, FRET only happens efficiently when jρ21 j2 has noticeable values [21]. To maintain quantum coherence at a high QD decoherence rate (1 ps−1 ), we assume the QD is made of CdSe-based material, with εs 6.25 embedded in ZeSe with ε0q 7.122 [22]. We also consider εc 2.16 (as silica) and R1 4 and R2 6 nm, setting the NS plasmon resonance energy to be 2.195 eV. Additionally, ℏω12 is considered to be 2.27 eV and ωl is adjusted such that Δ ℏω12 − ωl −5 ns−1 . We assume R 9 nm, ε0m 1.77, and the applied laser intensity (I 0 ) changes according to the inset of Fig. 2, i.e., it turns on at 100 ns of the scale and then it remains at 6.5 W∕cm2 . We first study I eff at the typical angle of θNS π∕5 (Fig. 1). Figure 2 shows the way I eff changes when r NS 8, 10, 12, and 15 nm. The results show a time delay in I eff with respect to the rise time of the applied field by about 250 ns. We also see a significant amount of field enhancement for r NS 8 nm. The reason behind the time delay, as discussed in detail in [9,23], is the coherent normalization of the broadening (Λeff ) and transition energy (Δeff ) of the QD. Since these parameters linearly depend on δ for low-carrier excitation, they are significant. Therefore, they hamper engagement of the laser field 80 10 70
f I0
E yNS 3 sinθNS cosθNS E NS r NS ;
ρ_ 11 −2 ImΩeff ρ21 ΣF Γ2 ρ22 − Γ1 ρ11 ;
60 (W/cm2)
(1)
0
50
g
h
i
5 8 0 100
500 e
40
eff
E zNS −1–3 cos2 θNS E NS r NS ;
frequency of the QD in the presence of pure plasmonic effects (without any coherence effects) and in the absence of the NS (very large R). Considering these, the density matrix equations are obtained as
I
EiQD , and E iapp are, respectively, the fields generated by the NS, QD, and the applied field. They can be given as follows:
4987
30 10 20
σ ij jiihjj, and ℏωj refer to the energy of jji. Ωr12 ρ Ωeff ηNS ρ21 is the coherently normalized Rabi 4μ212 βNS ℏε2eff1 R6
, and Ωeff frequency of the QD. Here ηNS μ12 E 0 and Ω012 − 2ℏε are, respectively, the Rabi Ω012 1 2βRNS 3 eff
a
10 0
b
c
d
12 15
0
100
200
300 400 Time (ns)
500
600
Fig. 2. Variation of I eff as a function of time for different r NS (numbers close to each curve in nm).
OPTICS LETTERS / Vol. 39, No. 17 / September 1, 2014
(a)
0.05
8 0
200
0
400
0
600
8
−0.1
0.15 0.1
10
0.05
12 15
(c)
(d)
0
500 Time (ns)
0
z
a f e d
−0.1 (a)
rNS=12
f e d
(b)
−0.2 0.2 −0.2
0
0.05
0
f e a
a 0
−0.05
0.2
rNS=15
d
−0.05 (c) −0.05
0 E
0.05
(d) −0.05
0 E
y
0.05
y
Fig. 4. Dynamics of polarization ellipse of the CP field at r NS 8 (a), 10 (b), 12 (c), and 15 nm (d). The letters close to each ellipse refer to the sampled times introduced in Fig. 2.
considering the result in Fig. 3(a). For r NS 8 nm, one can see that at the end of the time delay ReE z becomes negative. This leads to a dramatic change in the phase of the electric field. To see this in Fig. 5 we show ϕzy for r NS 8, 10, 12, and 15 nm. The circles refer to the a to i sampling points highlighted in Fig. 2. Note that for r NS 8, at around 380 ns ϕzy undergoes a 180 deg shift, changing the sense and orientation of the polarization ellipse. For other r NS ’s the change of ϕzy is rather smooth. The mapping of the polarization dynamics of the CP field becomes more evident if we consider the evolution of their Stokes vectors (S). Figure 6 shows the normalized component of this vector for r NS 8 (a) 10 (b) 12 (c) and 15 (d) nm. The circles show the sampled points introduced in Fig. 2. Note that Fig. 6(a) shows a dramatic rotation of S over a short period of time, corresponding to the distinct transition between the metastates of the QD-MNP system. As r NS increases the amount of rotation reduces. Figure 7 shows that the time evolution of the polarization of the CP field for different θNS when r NS 8 nm. For θNS 0 the QC field becomes purely linearly polarized along the z axis. Under this condition, only the amplitude of E z changes with time. For θNS π∕4 we do not see any polarization change as seen in the case of Fig. 4(a). This is because at this angle no phase change 0
−50
8
0
0
500
y
Re[Ey]
−0.05
d a
−0.2 −0.2
0.05
rNS=10
0.1
−0.1
12 15
0
f
0
0.2
15 12 10
10
0.1
e
0
10 12
−100
zy
0
8 8 10 12 15
0.2
=8
NS
(degree)
10
Im[Ez]
12
0.02
Im[E ]
Re[Ez]
0.04
(b)
0.1
15
0.06
r
15
φ
0.08
0.2
E
with the QD-MNP at the rise time of the applied field, causing the time delay. Additionally, the values of I eff during the time delay and after suggest two different collective states (or metastates) of the QD-NS system. Since during the time delay the QD is nearly nonemissive but after that it emits efficiently [9], we refer to these states as OFF and ON, respectively. To study the phase of the CP field, in Fig. 3 we show variations of E z and E y corresponding to the results shown in Fig. 2. For r NS 8 nm, within the delay period ReE z is quite significant, but after that it becomes negative [Fig. 3(a)]. On the other hand, ImE z increases significantly after the delay, demonstrating dramatically different dynamics [Fig. 3(b)]. As shown in Figs. 3(c) and 3(d), ReE y and ReE y also show very different dynamics before and after the time delay, while the former remains negative at all times. As r NS increases, except for ReE z , all other components are suppressed, suggesting the CP field at large distances is dominated by this component. This is expected, since for large r NS the field of the NS diminishes and ReE z is mostly determined by the applied field, which is polarized along z. The polarization states of the near field of the NS become clearer when we plot E z versus E y . Figure 4 shows the resulting polarization ellipses at different times and r NS . For this we chose certain times as labeled in Fig. 2 (a to i). In Fig. 4 we show the time corresponding to a, d, e, and f, as they depict the overall evolution of the system. When r NS 8 nm the polarization ellipse of the near field undergoes dramatic dynamics. As shown in Fig. 4(a), from a to e we see the variation of the shape of the ellipse in a counterclockwise sense. However, around f the ellipse suddenly switches while reversing its sense. This process suggests that at this r NS the polarization of the CP field changes when the system goes from one metastate to another. For the other r NS [Figs. 4(b)–4(d)], although such transition occurs, no such switching seems visible in the CP field. At r NS 12 nm the polarization ellipse evolves from a very elliptical one (represented by a) to a nearly left-handed circular one (indicated by f). The reason behind the switching and reversing of the sense of electric-field variation can be explored
Ez
4988
−150
500 Time (ns)
Fig. 3. Variation of the real and imaginary parts of Ez ((a) and (b)) and Ey ((c) and (d)) as a function of time for r NS 8, 10, 12, and 15 nm (numbers close to the curves). All specifications are the same as those in Fig. 2.
−200 100
8 200
300 400 Time (ns)
500
600
Fig. 5. Dynamics of the phase difference between Ey and Ez for r NS 8, 10, 12, and 15 nm (numbers close to each curve).
September 1, 2014 / Vol. 39, No. 17 / OPTICS LETTERS i
i
(a)
10
3
S /S
S3/S0
0
8
(b)
0
a
0
1
S /S 2
a
S /S
0
0
0 0 i
0 0
i
(c)
(d)
3
S /S
S3/S0
15
0
12
S1/S0
S2/S0
a
0 S /S 2
0
S1/S0
S2/S0
0 0
0 0
Fig. 6. Time evolution of the Stokes vector for different r NS (legends in nm) for the system studied in Fig. 2.
0.2 θNS=0
0.2
f
0
0
−0.1
−0.1
E
z
0.1
−0.2 (a) −0.2
0.2
z
−0.2 0.2
a
0
0.2 f
θ
=π/2
NS
0.1
0.1 E
θNS=π/15
−0.2 (b) 0
0.2 θ =π/4 NS
a
0
e f
ae
0 −0.1
−0.1 −0.2 (c) −0.2
f e
0.1
−0.2 (d) 0 Ey
0.2
−0.2
0 Ey
In conclusion, we showed that polarization of the plasmonic field of a NS can undergo unique time evolution once it is in the presence of a QD and driven by a laser field. We demonstrated that such dynamics are the result of the coherent exciton–plasmon coupling, causing characteristic changes in the phase and amplitude of the near field and sudden reversal of the sense and form of the polarization ellipse. These results may have applications in the investigation of chiral properties of single molecules and nanodevices based on plasmonic fields. The author is grateful for U.S. National Science Foundation support under grant no. CMMI 1234823.
a
0 S1/S0
4989
0.2
Fig. 7. Dynamics of polarization ellipse for different θNS and r NS 8 nm. All other specifications are the same as in Fig. 2.
(labeled with 8) happens as seen in Fig. 5. When θNS π∕2 the polarization of the QD field becomes elliptical with high ellipticity. Note that when the pure dephasing rate is less than 1 ps−1 , there are less stringent conditions for the effects discussed above. With a higher rate, these effects starts to smear out. We found that, for the QD-NS structure considered here, when this rate is about 5 ps−1 the quantum effects mostly disappear. Moreover, these effects depend on Δ in an asymmetric way. For Δ > 0, the time delay is infinite and the QD-NS system is in the OFF state. For Δ less than −10 nm, these effects start to smear out as the impact of the laser weakens.
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