Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum

Subscriber access provided by University of South Dakota

C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum Emitters and Metallic Nanoparticle Hybrids Mahi R. Singh, and Kevin James Black J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b06352 • Publication Date (Web): 24 Oct 2018 Downloaded from http://pubs.acs.org on October 26, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 19 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

The Journal of Physical Chemistry

Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum Emitters and Metallic Nanoparticle Hybrids Mahi R Singh*, Kevin Black Department of Physics and Astronomy, The University of Western Ontario, London, Canada

Abstract: We have developed a theory for the photoluminescence (PL) and scattering cross section of a core-shell hybrid, where the core is the metallic nanoparticle and the shell is made of an ensemble of quantum emitters. A probe field is applied to calculate the scattering cross section of the core-shell hybrid. The surface plasmon polariton field in the metallic nanoparticle is calculated by solving the Maxwell equations in the quasi-static approximation. Dipoles are induced in the ensemble of quantum emitters due to the probe field and surface plasmon polariton field. Therefore, the dipole of one quantum emitter interacts with dipoles of other quantum emitters in the ensemble and hence there is the dipole-dipole interaction (DDI) between quantum emitters. We discovered an anomalous DDI, which is induced by the surface plasmon polaritons. It is shown that the strength of the DDI can be controlled by the surface plasmon polariton frequency and it plays a dominant role in the phenomenon of the PL and scattering cross section. The surface plasmon polariton field can also interact with excitons of the quantum emitters via the exciton- surface plasmon polariton interaction. Using the density matrix method, the PL and scattering cross section are evaluated. It is found that the spectrum of the PL and the scattering cross section splits from one peak into two peaks mainly due to the strong coupling between the excitons and anomalous dipole-dipole interactions. It means that the PL and scattering spectrums can be switched ON (one peak) and OFF (two-peaks). This finding is consistent with the experimental data of the PL and scattering cross section of the J-aggregate and silver core-shell hybrid. We have found that the splitting and height of the two peaks can be increased or decreased by controlling mainly the strength of the anomalous dipole-dipole interaction. The anomalous dipole-dipole interactions can be controlled by applying an external pulse pressure and pulse control laser. Hence the present findings can be used for fabricating nanosensors and nanoswitches for applications in nanotechnology and nanomedicine.

1 ACS Paragon Plus Environment

The Journal of Physical Chemistry 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

I. Introduction Recent hybrid nanostructures have been fabricated by combining quantum emitters (QEs) and metallic nanoparticles (MNPs).1-16 When light falls on QEs, electron-hole pairs (excitons) are created in QEs. On the other hand, when light falls on a MNP, surface plasmon polaritons are created at the interface of the MNP. SPPs are spatially confined and have significantly enhanced fields at the nanoparticle surface relative to the incident light. The SPP resonances can be manipulated in these structures by changing the size and shape of MNPs. Similarly, exciton energies can be modified by size and shape of QEs. Due to their desirable SPP and exciton resonances, these nanostructures can be important building blocks for the study of light-matter interactions and their applications in nanomedicine and nanotechnology. These structures also have important technological applications in areas of chemical and biological sensors. The exciton-SPP interaction between QEs and MNPs plays a key role in the application of these materials. The enhanced SPP field can strongly modify the optical properties of QEs. When the exciton-SPP interaction is strong, the energies of both the SPP and the exciton are modified by their mutual interaction.1-6 This strong interaction can generate a new hybrid state called plexcitonic nanoparticle under the strong coupling regime. For example, Wersall et al.1 synthesized QE-MNP hybrid nanoparticles by self-assembly of an ensemble of J-aggregate dye molecules over a Ag nanoprism surface. They measured the photoluminescence (PL) and scattering spectrum of these hybrids. They observed splitting not only in scattering spectrum, but also in the PL spectrum on the core-shell nanoparticle. Fofang et al.2 have fabricated hybrids from J-aggregate molecules and Au-nanoshells to study the coherent coupling between excitons from J-aggregate molecules and SPPs from metallic nanoshells. The metallic nanoshells are made of a spherical silica core coated with a thin, uniform Au layer. They measured the extinction coefficient by varying the core size/shell thickness ratio. They observed splitting of the extinction spectrum peak due to the strong exciton-SPP coupling. The strong exciton-SPP interaction has also been observed in a Ag nanosphere and J-aggregate molecule hybrid.3 In this paper, we have developed a theory for the PL and scattering cross section for a core-shell hybrid where the core is the MNP and the shell is made of an ensemble of QEs. A probe field is applied to calculate the scattering cross section of the core-shell hybrid. The surface plasmon polariton’s electric field in the MNP is calculated by solving the Maxwell equation in the quasi-static approximation. Dipoles are induced in the ensemble of quantum emitters due to the probe field and SPP field. Therefore, there is a dipole-dipole interaction (DDI) between quantum emitters. The DDI induced by the SPP field is called anomalous DDI since its strength can be controlled by the SPP frequency. The SPP field also interacts with the excitons in the QEs via the exciton-SPP interaction. Using the density matrix method in the presence of the anomalous DDI and the exciton-SPP interaction, the PL and scattering cross section are evaluated. It is found that the PL and scattering spectra with one peak splits into two peaks due mainly to the strong anomalous DDI. These findings are consistent with the PL and scattering cross section experiments of the J-aggregate molecules and metallic nanoparticle core-shell hybrids.1 We have also found that the splitting and height of the two peaks can be controlled by the strength of the anomalous DDI. These findings can be used for fabricating nanosensors and nanoswitches.


Page 2 of 19

Page 3 of 19 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


II. Anomalous dipole-dipole interactions. We consider a hybrid that is fabricated by embedding a MNP in an ensemble of QEs. This hybrid can be called core-shell, where the core is the MNP and the shell is the ensemble of QEs. These types of hybrids have been fabricated by Wersall et al.,1 where the core is made from a Ag nanoprism and the shell is made from an ensemble of J-aggregate molecules. A schematic diagram of the core-shell hybrid is depicted in figure 1. We consider that this hybrid is surrounded by the biological cell with a dielectric constant 𝜖𝑏. The dielectric constant of the core and shell are denoted as 𝜖𝑐 and 𝜖𝑠, respectively. The radius of the core is taken as 𝑅𝑐 and the radius of the shell is denoted as 𝑅𝑠, respectively. The thickness of the shell is represented as 𝑡𝑠 and this gives 𝑅𝑠 = 𝑅𝑐 + 𝑡𝑠. Let the radius of the QE be 𝑅𝑞 and its dielectric constant 𝜖𝑞.

Figure 1. (Left Side) Schematic diagram of a hybrid, which consists of a MNP (Blue), embedded in an ensemble of QEs (Orange). (Right Side) A schematic diagram of a three-level QE. Energy levels are denoted as |𝑎⟩, |𝑏⟩, |𝑐⟩. We consider that the probe field 𝐸𝑃 acts only on the transition |𝑎⟩↔|𝑏⟩. The SPP field 𝐸𝑆𝑃𝑃, and the DDI field, are acting on the |𝑎⟩↔|𝑐⟩ transition only. The radius of the core is taken as 𝑅𝑐 and the radius of the shell is denoted as 𝑅𝑠. The thickness of the shell is represented as 𝑡𝑠 and this gives 𝑅𝑠 = 𝑅𝑐 + 𝑡𝑠.

A probe field with frequency/wavelength 𝜔/𝜆 and amplitude 𝐸𝑃 is applied in the hybrid. The frequency and wavelength are related as 𝜔 = 2𝜋𝑐/𝜆 where 𝑐 is the speed of light. When the probe falls on the QE, induced dipoles are created in the QEs due to the probe field. The dipole of one QE interacts with dipoles of other QEs. This is called the dipole-dipole interaction (DDI). Following reference,17 the DDI Hamiltonian is written as H DDI =

1 N 1 J ij pi .p j , J ij   2 i j b 0 rij3


(1)


Page 4 of 19

Where pi and pj are the induced dipole moments in the ith-QE and jth-QE, respectively. Here Jij is the DDI coupling constant and rij is the distance between the ith and the jth QEs. In the mean field approximation,17-22 the DDI Hamiltonian can be rewritten as H DDI = pi E DDI , E DDI  i

1  Jp 2 ji ij j

(2)

Where EDDI is the average dipole electric field created by all QEs on the ith-QE. The average in the right side of equation of EDDI has been evaluated in references17-22 by using the method of Lorentz23. The expression of EDDI has been evaluated as EDDI =

QE pi

(3)

3

Where pi 

PQE

(4)

4 b Rq3

Where 𝜆𝑄𝐸 is the DDI constant and the typical value 𝜆𝑄𝐸 is taken as unity.17 Here ⟨𝑝𝑖⟩ is the average polarization of the ith-QE and 𝑃𝑄𝐸 is the polarization of a QE. The polarization of the QE and an electric field produced by 𝑃𝑄𝐸 are calculated as follow. Polarization and Electric field in the Quasi-Static Approximation: When the probe field falls on a QE, an induce polarization occurs in the QE. It is denoted as produce 𝑃𝑄𝐸 which in turn produces a dipole electric field denoted as 𝐸𝑄𝐸. The electric field produced by the QE can be calculated using the quasi-static approximation.24,25 The size of QE is much smaller than the wavelength of the electromagnetic field in the optical region. For example, the wavelength of light in the visible region is of the order of 600 nm whereas the size of the QE is of the order of 10 nm. In this case one can consider a situation that the amplitude of electric field is constant over the nanoparticle. This condition is known as quasi-static approximation.24,25 When the probe field 𝐸𝑃 falls on a QE, it induces a polarization 𝑃𝑄𝐸, which in turn produces a dipole electric field 𝐸𝑄𝐸. Solving Maxwell's equations in the quasi-static approximation, one can derive the following expression for the electric field produced by the QE as follows 24,25

EQE =

PQE

(5)

4 b r 3

Where the expression for 𝑃𝑄𝐸 is found as PQE  4 b Rq3 gl QE EP ,  QE 


q  b q 2 b

(6)

Page 5 of 19 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


The constant gl is called the polarization parameter and it has values gl =2 and gl = -1 for 𝑃𝑄𝐸 ∥ 𝐸𝑃 and 𝑃𝑄𝐸 ⊥ 𝐸𝑃.24 For the calculations in this paper, we consider gl =-1. Similarly, when electric field falls on the MNP, an induce polarization occurs in the MNP. It is denoted as 𝑃𝑆𝑃𝑃. The induced polarization produces a surface plasmon polariton (SPP) field denoted as denoted as 𝐸𝑆𝑃𝑃. There are two electric fields, namely 𝐸𝑃 and 𝐸𝑄𝐸 which are falling on the MNP. Solving Maxwell's equations in the quasi-static approximation, one can derive the following expression for the SPP electric field produced by the MNP. ESPP =

PSPP 4 b r 3

(7)

Where the expression for 𝑃𝑆𝑃𝑃 is found as PSPP  4 b  Rc  ts  gl SPP  EP  EQE  3

(8)

We want to make comment on distance r being used in both eqn. (5) for EQE and eqn. (7) for ESPP. Here the distance r is the same in the both equations, because it is the distance from the center of the QE to the center of the MNP. The parameter 𝜁𝑆𝑃𝑃 appearing in eqn. (8) is called the SPP polarization. It is found as  SPP 

sc  b sc 2 b

3    s  2  Rc   c  c 2 s sc s 3 3     Rc  ts   2  Rc   c s  c 2 s

 Rc  ts 

3

     

(9)

In the above expression, 𝜖𝑐 is the dielectric constant of metal and we have considered 𝜖𝑐 = 𝜖∞ ― 𝜔2𝑝/𝜔(𝜔 + 𝑖/𝜏).26 Where 𝜔𝑝 (𝜆𝑝) is the plasmon frequency/wavelength and 𝜖∞ is the dielectric constant of metal when light frequency is very large. Note that the real part of 𝜖𝑐 has negative value when 𝜔 < 𝜔𝑝. It is interesting to find that when 𝜖𝑐 has negative value the denominator of the polarization factor 𝜁𝑆𝑃𝑃 becomes zero at certain value of the frequency. Let us call this value 𝜔 = 𝜔𝑠𝑝. This means that the polarization and SPP field has a singularity when 𝜔 = 𝜔𝑠𝑝. The frequency and wavelength are related as 𝜔𝑠𝑝 = 2𝜋𝑐/𝜆𝑠𝑝 where 𝑐 is the speed of light. Let us express eqns. (5) and (7) in simple forms. To get a simple expression for 𝐸𝑄𝐸, we substitute eqn. (6) into eqn. (5) and we get



4 b Rq3 gl QE

EQE =

EQE =

4 b r 3

QE Rq3 r3

EP 

 EP  ,

Page 6 of 19

Rq3 gl QE r3

EP

QE  gl QE

(10)

Similarly, to get a simple expression for the SPP field, we substitute eqn. (8) into eqn. (7) and we get

4 0b  Rc  ts  gl SPP 3

ESPP = ESPP =

4 0b r 3

 SPP  Rc  ts 

3

r3

E

P

E

P

 EQE  ,

 EQE 

3 Rc  ts  gl SPP  

r3

E

P

 EQE  (11)

 SPP  gl SPP

Now we calculate the 𝐸𝑆𝑃𝑃 produced by the MNS. We substitute the expression for 𝐸𝑄𝐸 from eqn. (10) into eqn. (11) and we get ESPP =

 SPP  Rc  ts  r3

QE  SPP  Rc  ts  Rq3 3

3

EP 

r6

EP

(12)

Now we substitute the values of 𝛽𝑄𝐸 and 𝛽𝑆𝑃𝑃 from eqns. (10) and (11), respectively into eqn. (12) which reduces to ESPP =

gl SPP  Rc  ts 

EP 

r3

gl gl QE SPP  Rc  ts  Rq3 3

3

r6

(13)

EP

The above expression can further be rewritten in terms of physical parameters 𝛱𝑃𝑆𝑃𝑃 of 𝛱𝑄𝐸 𝑆𝑃𝑃 as follows





(14)

P ESPP =  SPP   QE SPP EP

Where 

P SPP

=

gl  Rc  ts  r3

gl gl  Rc  ts  Rq3 3

3

 SPP ,



QE SPP



r6

 QE SPP

(15)

𝑃 It is worth mentioning that the SPP field depends on two terms 𝛱𝑃𝑆𝑃𝑃 and 𝛱𝑄𝐸 𝑆𝑃𝑃. The first 𝛱𝑆𝑃𝑃 -term is the SPP field, and it is induced by the probe field. Similarly, the second 𝛱𝑄𝐸 𝑆𝑃𝑃-term is the SPP field, and it is induced by the QE dipole field. Note that the first term depends on 𝑟 ―3 and the second term depends on 𝑟 ―6. Therefore, the SPP field induced by the probe field is stronger than the SPP field induced by the QE field.


Page 7 of 19 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


DDI Electric field: Now we calculate the DDI electric field found in eqn. (3). We substitute eqn. (4) into eqn. (3) and we get EDDI =

QE

PQE

(16)

3 4 b Rq3

Note that two electric fields namely 𝐸𝑃 and 𝐸𝑆𝑃𝑃 are falling on the QEs. Putting the expression of 𝑃𝑄𝐸 from eqn. (6) into the above eqn. (16) and we get

QE 4 b Rq3 gl QE  EP  ESPP  EDDI = 3 4 b Rq3 EDDI =

QE gl QE 3

(17)

 EP  ESPP 

Putting the expression of 𝐸𝑆𝑃𝑃 from eqn. (14) into the above eqn. (17) and we get the expression of the DDI field as

EDDI =

QE gl QE 3

EP 

QE gl QE 3



P SPP



  QE SPP EP

(18)

The final expression for the DDI electric field is written in terms of DDI parameters 𝛷𝑃 and 𝛷𝑆𝑃𝑃 as follows EDDI =   P   SPP  EP ,

P  SPP   SPP   QE SPP

(19)

Where

P 

QE gl QE 3

,

P  SPP 

QE gl QE 3

  , P SPP

 QE SPP 

QE gl QE 3

  QE SPP

(20)

Note that the DDI found in eqn. (19) is made of two terms 𝛷𝑃 and 𝛷𝑆𝑃𝑃. The first term 𝛷𝑃 is the DDI term induced by the probe field. We call it the classical DDI field induced by the external probe field. The second term 𝛷𝑆𝑃𝑃 is the DDI term induced by the SPP field. We call 𝑃 this term the anomalous DDI. This term is further made of two terms 𝛷𝑃𝑆𝑃𝑃 and 𝛷𝑄𝐸 𝑆𝑃𝑃. The 𝛷𝑆𝑃𝑃 depends on 𝛱𝑃𝑆𝑃𝑃 which is given by eqn. (15). This term depends on 𝑟 ―3. On the other hand, the ―6 𝑄𝐸 . In other words, 𝛷𝑄𝐸 𝑆𝑃𝑃 depends on 𝛱𝑆𝑃𝑃 which is given by eqn. (15). This term depends on 𝑟 ―6 𝑄𝐸 the 𝛷𝑆𝑃𝑃-term depends on 𝑟 . This means that the anomalous DDI term due to 𝛷𝑃𝑆𝑃𝑃-term is 𝑃 dominant compared to the 𝛷𝑄𝐸 𝑆𝑃𝑃-term. This is because the 𝛷𝑆𝑃𝑃-term is due to the SPP field induced by the probe field and it depends on the SPP polarization factor, 𝜁𝑆𝑃𝑃, which has a large value at 𝜔 = 𝜔𝑠𝑝. Whereas, the 𝛷𝑄𝐸 𝑆𝑃𝑃-term is induced by the QE electric field which is a very weak field.



Page 8 of 19

Further, we want to make a comment on the anomalous DDI. In the condensed matter physics literature, the classical dipole-dipole interaction (DDI) is calculated as follows. When light falls on the ensemble of particles, induced dipoles are created in the particles. These induced dipoles interact with each other and this is called the DDI. In this paper, we call it the classical DDI. It is important to note that this classical DDI is induced by the external (probe) light. On the other hand, in the present paper, we have invented a new type of DDI and we call it anomalous DDI to distinguish it from the classical DDI. In this paper, the external (probe) field induces the surface plasmon polariton (SPP) field in the MNP. This SPP field falls on the ensemble of QEs and induces dipoles in the QEs. Therefore, the induced dipoles in the ensemble of QEs interact with each other which we call the anomalous DDI. One can see from eqn. (19) that the DDI field is made of two terms, (i) the classical DDI induced by the external (probe) field denoted by Φ𝑃, and (ii) the anomalous DDI induced by the SPP field denoted by Φ𝑆𝑃𝑃 and it is a very strong field. One of the interesting points of our theory is that it provides a method to calculate the emission rate of a strongly coupled system between J-aggregates and plasmonic nanoparticles.

III. Photoluminescence, Scattering Cross Section and Density Matrix Method. Next, we calculate the interaction between excitons in a QE with the SPPs in the MNP and the DDI field in the ensemble of QE’s. We consider a three-level QE whose energy levels are denoted as |𝑎⟩, |𝑏⟩ and |𝑐⟩. Two excitons are induced in the QE with frequency/wavelength ( 𝜔𝑎𝑏/ λab) and (𝜔𝑎𝑐/ λac) due to transitions |𝑎⟩↔|𝑏⟩ and |𝑎⟩↔|𝑐⟩, respectively. A schematic diagram of the QE is shown in fig. 1. It is important to note that we have used two levels |𝑎⟩ and |𝑏⟩ to calculate the PL and SCS spectra. We applied the probe field 𝐸𝑝 between the transition |𝑎⟩↔|𝑏⟩ to monitor the PL and scaterring cross section. Following the method of references25,26 we have evaluated the PL and scattering cross section for the QE in the hybrid system as,     I PL  QQE Im  ab ab ab  EQE  2 EP 

 scatt

ab  ab 4  4 2 6 c 0 E p

2

2

(21)

Parameter 𝑄𝑄𝐸 is called the PL efficiency factor and is taken as unity for simplicity. Here 𝜌𝑎𝑏 and 𝜇𝑎𝑏 represent the density matrix element and dipole moment of the QE for transition |𝑎⟩↔|𝑏⟩. The density matrix element 𝜌𝑎𝑏 for two levels |𝑎⟩ and |𝑏⟩ is calculated as follows. We consider that the SPP frequency 𝜔𝑠𝑝 lies closely to the exciton energy 𝜔𝑎𝑐. Hence the SPP electric field (𝐸𝑆𝑃𝑃) and the DDI electric field (𝐸𝐷𝐷𝐼) are acting only between the transition |𝑎⟩↔|𝑐⟩. Note that three fields are falling on the QE and they are written as 𝐸𝑄𝑇 = 𝐸𝑃 + 𝐸𝐷𝐷𝐼 + 𝐸𝑆𝑃𝑃, where 𝐸𝑆𝑃𝑃 and 𝐸𝐷𝐷𝐼 are given by eqns. (14) and (19), respectively. These three fields induce dipoles in the QE and the induced dipoles interact with these three fields. In the dipole and the rotating wave approximation,29,30 the interaction Hamiltonian of the QE in the hybrid is expressed as 8 ACS Paragon Plus Environment

Page 9 of 19 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


(22) Where h.c. stands for the Hermitian conjugate. Parameter 𝛺𝑛 = 𝜇𝑛𝑎𝐸𝑃/ℏ (n = b, c) is called the Rabi frequency associated with the transitions |𝑛⟩↔|𝑎⟩. Here 𝜎𝑛𝑎 = |𝑛⟩⟨𝑎| is the exciton creation operator. The physical meaning of each term appearing in eqn. (22) is as follows. The first term is the interaction between excitons and the probe field. The second term is the exciton interaction with the classical DDI field induced by the probe field. The third term is the exciton interaction with the SPP field produced by the MNP. The last term is the exciton interaction with the anomalous DDI field. This term is called the anomalous DDI term since the dipoles in the QE are induced by the SPP field. This term is large near the SPP frequency 𝜔𝑠𝑝. Equations of motion for different density matrix elements are found with the help of the density matrix equation method29,30 and Hamiltonian eqn. (22) as follows.  d cb P QE P QE P P QE  cb cb  i  abc  SPP  i        ca  QE     c SPP SPP SPP SPP SPP SPP SPP dt d ba P P  ba ba  i( bb   aa ) b   b   P   i cbc  SPP  QE   SPP  QE SPP SPP dt d ca P P  ca  ca  i( cc   aa ) c  SPP  i cb  b  b   P   QE   SPP  QE SPP SPP dt























(23)



Where 𝛯𝑐𝑎 = 𝛾𝑐 +𝑖𝛿𝑐, 𝛯𝑏𝑎 = 𝛾𝑏 +𝑖𝛿𝑏, 𝛯𝑐𝑏 = (𝛾𝑐 + 𝛾𝑏)/2 + 𝑖(𝛿𝑐 ― 𝛿𝑏).

(24)

Here 𝛿𝑏 = 𝜔𝑎𝑏 ―𝜔 and 𝛿𝑐 = 𝜔𝑎𝑐 ―𝜔 are called the probe field detunings. The physical quantities 𝛾𝑏 and 𝛾𝑐 appearing in eqn. (24) represent the exciton decay rates from |𝑎⟩ to levels |𝑏⟩ and |𝑐⟩, respectively. An analytical expression for PL and the SCS can be obtained in the steady state if we consider that 𝜌𝑎𝑎 > 𝜌𝑏𝑏 and 𝜌𝑎𝑎 > 𝜌𝑐𝑐. For simplicity, we also consider that 𝜇𝑏𝑎 = 𝜇𝑐𝑎, which gives us 𝛺𝑏 = 𝛺𝑐. An analytical expression for 𝜌𝑏𝑎 is obtained from eqn. (23) as

ba 

i b   b   P    ac  i ab 





P P  QE   SSP  QE / 2  ab  i ab   ac  i ab   c SSP SSP SSP 

2

(25)

Substituting eqn. (25) into eqn. (21), with the simplification Ω𝑏 = Ω𝑐, we get the equations of the PL and scattering cross section as



I PL

Page 10 of 19





2   i b  b   P     ac  i ab  1   P    QQE hab b Im  2  P P QE      ab  i ab   ac  i ab   c  SPP   QE SPP   SPP   SPP / 2    



(26)



2

 scatt 

b  b   P     ac  i ab  (27) 6 c h  02    i    i     P   QE   P   QE / 2  2 ab ab ac ab SPP SPP SPP  c SPP 

 

4 2 ab sp 4 2 2 b





Note that the PL and scattering cross section depend on the anomalous DDI terms 𝛷𝑃𝑆𝑃𝑃 + 𝛷𝑄𝐸 𝑆𝑃𝑃. 𝑃 𝑄𝐸 It also depends on the exciton-SPP interaction (𝛱𝑆𝑃𝑃-term and 𝛱𝑆𝑃𝑃-term).

IV. Results and Discussions We compare our theory with the experimental data of the scattering cross section and the PL for the core-sell hybrid which is fabricated by embedding Ag MNP in an ensemble of J-aggregate dye molecules1. Here dye molecules form a nanoshell and the Ag metallic nanoprism acts as the core. The numerical simulations are performed using wavelength 𝜆 rather than frequency ω. Almost all the parameters used in our theory can be taken from the plasmonic and quantum optics literature1,6,26,27,31-33. The references are provided beside each physical parameter. The plasmon frequency and the relaxation time for Ag are taken as 𝜔𝑝 = 8.7 𝑒𝑉 and 𝜏 = 1.45 ∗ 10 ―14 𝑠, which are taken from reference Nash et al. 27 and Jiang et al.31. The value of the dielectric constant 𝜖∞ is taken as 𝜖∞ = 6 from Yang et al.26 We have taken 𝑅𝑐 = 25 𝑛𝑚,1,6 𝑡𝑠 = 3 𝑛𝑚,1,6 and 𝑅𝑞 = 0.5 𝑛𝑚,1,6,32. For a core of radius 𝑅𝑐 = 25 𝑛𝑚, the volume is given by 𝑉𝑐 4𝜋𝑅3𝑐

= 3 = 6.5 ∗ 104 𝑛𝑚3, which approximately agrees with the volume of the nanoprism ~1.3 ∗ 104 𝑛𝑚3.1 We have increased the volume of the sphere slightly to get SPP wavelength (frequency) approximately equal to the experimental value of reference1. The size of the quantum emitters 𝑅𝑞 ≈ 0.5 𝑛𝑚 is approximated as a Wannier-Mott exciton as suggested in Baranov et al.32 The shell thickness is taken as 𝑡𝑠 = 3 𝑛𝑚1. The dielectric constant of the surrounding medium is taken as 𝜖𝑏 = 1.14.1 The values of γab/Ω and γac/Ω are shown in the captions of figures 2-5 and are known from reference33. Finally, the surface plasmon wavelength 𝜆𝑠𝑝, two exciton wavelengths 𝜆𝑎𝑏 and 𝜆𝑎𝑐 are taken from reference1 and are written as 𝜆𝑎𝑏 = 𝜆𝑎𝑐 = 𝜆𝑠𝑝 = 588 𝑛𝑚. The relative dielectric constants of J-Aggregates at high energies and the shell made of J-Aggregate are taken as 𝜖𝑠 = 𝜖𝑞 = 2.11,6. Note that there should be some difference between 𝜖𝑠 and 𝜖𝑞. This is because the shell is made of the ensemble of QEs and there is extra space between the close packed QEs. It will be impossible to calculate the dielectric constant of the shell. The difference will be very small, and we found it will not change the physics and calculations of the paper. That is why, in our numerical calculation we have taken the dielectric constant of the shell (𝜖𝑠) and the dielectric constant of the QE (𝜖𝑞) to be the same. It is a good approximation. The results are plotted in figs. 2a and 2b for scattering cross section. In fig. 2a, the open 10 ACS Paragon Plus Environment

Page 11 of 19 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


diamonds and dashed curve represent the scattering data and theoretical calculations for the free MNP. Similarly, in fig. 2b, the open circles and solid curve represent the scattering data and theoretical simulations for the J-aggregate dye molecule in the core-shell hybrid. Note that a good agreement between our theory and experiment is found on the qualitative basis but not on the quantitative basis. The reason for the disagreement on the quantitative basis is because we used a nanosphere MNP whereas a nanoprism was used in the experiment. The nanosphere MNP is used so that we can get an analytical expression for the scattering cross section. The analytical expression for scattering cross section will be useful to fabricate new core-shell nanohybrids and develop new experiments. These expressions can be also used in the fabrication of new nanoswitches and nanosensors.

Figure 2. (a) The scattering cross section is plotted as a function of wavelength for the Ag-MNP alone. The dashed line and open circles represent the theoretical simulations and experimental data respectively. Parameters used are 𝛾𝑠𝑝/Ω = 0.3, Ω = 0.3 𝑒𝑉 and 𝜆𝑠𝑝 = 588 𝑛𝑚 (b) The scattering cross section is plotted as a function of wavelength for the core-shell hybrid made from J-aggregate dye molecules and the Ag- MNP. The solid line is the theoretical curve and the open circles represent experimental data. Parameters used are 𝛾𝑎𝑏/Ω = 0.3, 𝛾𝑎𝑐/Ω = 0.1, Ω = 0.3 𝑒𝑉 and 𝜆𝑎𝑏 = 𝜆𝑠𝑝 = 588 𝑛𝑚. Notice that the peak not only splits from one to two peaks.

We have also compared our PL theory with the experimental data of the J-aggregate dye molecules and core-shell hybrid.1 The results are plotted in figs. 3a and 3b for the PL for free J-aggregate and the core-shell hybrid made from J-aggregate dye molecules and the Ag- MNP, respectively. In fig. 3a, the open diamonds and dashed curve represent the PL data and theoretical simulations for the free J-aggregate. Similarly, in fig. 3b, the open circles and solid curve represent the scattering data and theoretical calculations for the core-shell hybrid. Note that a good agreement between our theory and experiment is found on the qualitative basis but not on the quantitative basis since we used spherical MNP instead of prism MNP.



Figure 3. (a) The PL is plotted as function of wavelength for the J-aggregate dye molecule alone. The dashed line and open circles represent the theoretical simulations and experimental data respectively. Parameters used are 𝛾𝑎𝑏/Ω = 0.126, 𝛾𝑎𝑐/Ω = 0.15, Ω = 0.14 𝑒𝑉 and 𝜆𝑎𝑏 = 588 𝑛𝑚. (b)The solid line is the theoretical curve and the open circles represent experimental data for the core-shell hybrid. The parameters used are 𝛾𝑎𝑏/Ω = 0.22, 𝛾𝑎𝑐/Ω = 0.04, Ω = 0.6 𝑒𝑉 and 𝜆𝑠𝑝 = 𝜆𝑎𝑏 = 588 𝑛𝑚. Notice that the peak splits from one to two peaks.

It is an important note that our theory for the PL and scattering cross section shows that the peak of the free J-aggregate dye molecule splits into two peaks when the J-aggregate dye molecules are part of the core-shell hybrid. The splitting is mainly due to the strong coupling between excitons and the anomalous DDI. Physics of the splitting can be explained in the strong coupling limit and can be explained as follows. In the case of the free J-aggregate dye molecule, there is no coupling between the exciton and the anomalous DDI. Hence, the scattering cross section and PL spectrum have one peak due to the transition |𝑎⟩↔|𝑏⟩. In the presence of the strong anomalous DDI couplings, the dressed states are created in the system. The state |𝑎⟩ splits into two dressed states called |𝑎 ― ⟩ and |𝑎 + ⟩ due to the anomalous DDI coupling. Therefore, two transitions occur, |𝑎 ― ⟩↔|𝑏⟩ and |𝑎 + ⟩↔|𝑏⟩, due to the presence of the dressed states. That is why one peak due to the transition |𝑎⟩↔|𝑏⟩ in the PL and scattering cross section spectrum splits into two peaks due to two transitions |𝑎 ― ⟩↔|𝑏⟩ and |𝑎 + ⟩↔|𝑏⟩. To observe the effect of the anomalous DDI, we have plotted the three-dimensional fig. 4a and fig. 4b for scattering cross section and PL spectrum for the core-shell hybrid, respectively. The results are plotted in fig. 4a and 4b for the scattering cross section and the PL as a function of wavelength and the anomalous DDI coupling parameter, respectively. Note that as the DDI increases, the one peak splits into two peaks. The strength of the splitting increases as the DDI parameter increases. On the other hand, the height of the peaks decreases as the anomalous DDI parameter increases. Note that the splitting in PL is significantly smaller than splitting in scattering for the same nanostructure’ 12 ACS Paragon Plus Environment

Page 12 of 19

Page 13 of 19 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


Figure 4. (a) The scattering coefficient is plotted as a function of the probe wavelength and the DDI constant (𝛷𝑆𝑃𝑃). Parameters used are 𝛾𝑎𝑏/Ω = 0.16, 𝛾𝑎𝑐/Ω = 0.02 Ω = 0.6 𝑒𝑉 and 𝜆𝑠𝑝 = 𝜆𝑎𝑏 = 588𝑛𝑚 (b) The photoluminescence is plotted as a function of the probe wavelength and the DDI constant (𝛷𝑆𝑃𝑃). Parameters used are 𝛾𝑎𝑏/Ω = 0.22, 𝛾𝑎𝑐/Ω = 0.04 Ω = 0.6 𝑒𝑉 and 𝜆𝑠𝑝 = 𝜆𝑎𝑏 = 588𝑛𝑚. In both Figures we have neglected the effect of the EPI. Notice that as the DDI increases, the one peak splits into two peaks. However, the height of the peaks decreases, and the splitting of peaks increases.

Finally, in fig. 5a and fig. 5b we have plotted the effect of the anomalous DDI on PL and SCS spectra respectively. The dotted curves, plotted in both figures, represent when the anomalous DDI is absent (i.e. 𝛷𝑆𝑃𝑃 = 0). The dashed curves are plotted when 𝜆𝑎𝑐 ≠ 𝜆𝑠𝑝 where the experimental value of 𝜆𝑠𝑝 ≈ 588 𝑛𝑚. This means that the SPP wavelength 𝜆𝑠𝑝 is NOT in resonance with the wavelength 𝜆𝑎𝑐 of the exciton transition |𝑎⟩↔|𝑐⟩. The dashed curve corresponds to 𝜆𝑎𝑐 ≠ 𝜆𝑠𝑝 (off resonance). Next, we have plotted the solid curve when the SPP wavelength 𝜆𝑠𝑝 is in resonance with the wavelength of the transition |𝑎⟩↔|𝑐⟩, that is 𝜆𝑎𝑐 ≈ 𝜆𝑠𝑝. Note that when both wavelengths are NOT in resonance with each other, the PL and SCS spectra splitting from one peak to two peaks is very small. This is because when both wavelengths are NOT resonant (i.e. 𝜆𝑎𝑐 ≠ 𝜆𝑠𝑝), the anomalous DDI has a very small value. However, when both wavelengths are in resonance with each other, the splitting in PL and SCS spectra is large from one peak to two peaks. This is because the anomalous DDI has the largest value when both wavelengths are in resonance (i.e. 𝜆𝑎𝑐~𝜆𝑠𝑝). Note peaks are shifted in the presence of the DDI. The shift is due to the DDI coupling. In summary one can say that the anomalous DDI is responsible for splitting the PL and SCS spectra. We have not included the effect of temperature in our theory since its effect is negligible as experimentally found1. It is found as temperature is raised from T=4K to T= 293 K, the Rabi splitting decreases very little in the SCS spectra. See figure 2 (d) in reference.1 The inclusion of the temperature effect makes the theory very complicated and one cannot get an analytical expression for the PL and SCS. However, there is a simple way to include the effect of the temperature dependence in the density matrix formulation by following the method of reference. 13 ACS Paragon Plus Environment


34

According to this reference the effect of temperature can be included by multiplying the decay rate term 𝛾𝑏 by (1 + 𝑛𝐵) where 𝑛𝐵 is the boson distribution function written as 𝑛𝐵 = 1/[exp (ℏ𝜔/𝑘𝐵𝑇) ― 1].34 The temperature dependent part changes the height, the width and locations of the peaks.1 We have shown that the splitting and height of the scattering spectrum can be controlled by the anomalous DDI. Hence core-shell hybrids can be used to develop nanosensors which can measure the splitting and height of the PL and scattering spectra. These hybrids can also be used to develop nanoswitches by switching the spectrum from one peak to two peaks. Here one peak can be thought of as OFF position and two peaks can be thought of as ON position. We showed that the anomalous DDI depends on the thickness and dielectric constant of the shell. The thickness of the shell can be tuned by applying ultra-fast pulse pressure or stress, whereas applying an external intense control laser can modify the dielectric constant of the shell. These very interesting findings and these nanosensors and nanoswitches can be used for applications in nanotechnology and nanomedicine.

Figure 5. (a) The photoluminescence is plotted for different values of 𝜆𝑠𝑝. (b) The scattering cross section is plotted for different values of 𝜆𝑠𝑝. There are three curves. In both figures, the dotted curve is for the anomalous DDI = 0 (the free J-Aggregate (a) and free MNP (b)), the solid line is for 𝜆𝑠𝑝 = 𝜆𝑎𝑏 = 588 𝑛𝑚 (on resonance), and the dashed line is for 𝜆𝑠𝑝 ≠ 𝜆𝑎𝑐(off resonance).

We want to comment on the three-level system used in the present theory. In the absence of the MNP, the PL and SCS have one peak. This means there is one transition |𝑎⟩↔|𝑏⟩ where |𝑏⟩ is the ground state. In the presence of the MNP, PL and SCS spectra have two peaks. This is due to strong coupling between the MNP and QEs due to strong DDI coupling |𝑎⟩↔|𝑐⟩. Due to this strong coupling, dressed states are created in the QEs. The excited state |𝑎⟩ splits into two dressed states called |𝑎 ― ⟩ and |𝑎 + ⟩ due to the strong DDI coupling. According to the quantum mechanics, the ground state does not split. Therefore, two transitions occur, |𝑎 ― ⟩↔|𝑏⟩ and |𝑎 + ⟩↔|𝑏⟩. One can see clearly that in the calculation of the PL and SCS, we calculate 𝜌𝑎𝑏 (eqn. 20) due to two levels |𝑎⟩ and |𝑏⟩. This means basically we are using two-level in the calculation of 14 ACS Paragon Plus Environment

Page 14 of 19

Page 15 of 19 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


the PL and SCS. In other words, two levels play the main role in the present theory. In summary, the density matrix method creates the dressed states along with transition matrix elements to calculate the PL and SCS. This theory is very elegant, simple and analytical expressions for the PL and SCS can be obtained. On the other hand, one can also use two levels |𝑎⟩ and |𝑏⟩ in the calculation of the PL and SCS. In this case we have to solve the Schrodinger equation with the strong DDI coupling Hamiltonian and find the dressed states |𝑎 ― ⟩ and |𝑎 + ⟩. In this case we have three states |𝑏⟩, |𝑎 ― ⟩ and |𝑎 + ⟩ (three-level system). After that we have to use the density matrix method to calculate the transition matrix elements 𝜌𝑎 ― 𝑏 (|𝑎 ― ⟩↔|𝑏⟩) and 𝜌𝑎 + 𝑏 (|𝑎 + ⟩↔|𝑏⟩.) to calculate the PL and SCS. This approach will be very messy and complicated mathematically. We tried this method for other systems and we found it very tedious and complicated to get an analytical expression. Therefore, we abandoned this method. We want to make comments on the parameters used in figures Fig 2 and Fig 3. In fig. 2a, we studied the scattering cross section due to the Ag-MNP only. In fig. 2b, we studied the SCS due to the core-shell hybrid made of Ag-MNP and the ensemble of QEs (J-aggregates). In the core-shell hybrid, the excitons in QEs and SPPs in the Ag-MNP interact with each other via the exciton-SPP interaction and also the DDI. The DDI plays a very important role in decay rates of both excitons and SPPs. The DDI also plays a critical role in the splitting of the SCS spectrum. The DDI is absent in fig. 2a since the experiment was performed on Ag-MNP alone. On the other hand, the DDI is present in fig. 2b since the experiment was performed on the core-shell hybrid. That is why parameters used in both figures are different. In fig. 3a, we studied the PL due to the QE (J-aggregates) only. In fig. 3b, we studied the PL due to the core-shell hybrid made from Ag-MNP and the ensemble of QEs (J-aggregates). In the core-shell hybrid, the excitons in QEs and SPPs in the Ag-MNP interact with each other via the DDI. The DDI interaction is absent in fig. 3a since the experiment was performed on QE (J-aggregates) alone. On the other hand, the DDI is present in fig. 3b since the experiment was performed on the core-shell hybrid. That is why decay rates are different in both fig. 3a and fig 3b. We can conclude from the above explanations about fig 2 and 3 that the experimental environment and conditions in fig.2 and 3 are different that is why the parameters used in these figures are also different. We found that the splitting in PL is significantly smaller than splitting in scattering for the core-shell hybrid. In principle these two splitting must be the same if the sample environment and experimental conditions were the same but they are not. The splitting is due to strong DDI. This means that DDI coupling in the SCS experiment in fig. 2b is stronger than that of the PL experiment in fig. 3b. In other words, in reference1 the SCS experiment was performed in the stronger coupling limit than that of the PL experiments. The main aim of the paper is to develop a theory for core-shell structures and to obtain analytical expressions of the PL and SCS so that experimentalists in the plasmonic field can use to explain their experiments and plan new experiments. It will be impossible to get an analytical expression for the PL and SCS for a prism shape MNP. Since we are considering three level quantum emitters and make use of the Maxwell and density matrix equations for the calculation of PL and SCS, it is recommended to use a numerical simulation method such as a Maxwell-Liouville-von Neumann FDTD method35,36 to calculate the PL and SCS for the prism MNP. The most 15 ACS Paragon Plus Environment


important achievement of our theory is that with minor modifications, our theory can be applied for MNPs with different shapes such as single metallic nanoshells, double metallic nanoshells and metallic rods, metallic spheroidal, metallic ellipsoidals and etc. Our theory predicts that there will be splitting in the PL and SCS spectra due to the strong DDI from one peak to two peaks for all shapes of MNP. The only thing which changes for different MNP shapes is position of the peaks, the height of peaks and energy splitting between two peaks. We have developed theories for differently shaped MNPs such as rods,37 ellipsoidal,10 single nanoshell,12 double nanoshells38 and graphene flake,39 except prism. The most fascinating and interesting part of our theory is that the physics of our finding will NOT change and it can be applied to any shape of MNP. For different shapes, the location of SPP frequency will vary. Therefore, in our theory one can substitute the SPP frequency for different shapes. For example, in the present paper we mimic the SPP frequency of prism by changing the shape of spheroidal.

V. Conclusion In conclusion, we have developed a theory for the PL and scattering cross section for core-shell nanohybrids using the density matrix method. We have compared our theory with the scattering cross section experiment of the J-aggregate-Ag MNP core-shell nanohybrid.1 A good agreement between our theory and experiments is found. It is found that the PL and scattering spectrum with one peak splits into two peaks due mainly to the anomalous strong DDI. These findings can be used for fabricating nanosensors and nanoswitches.

Author Information Corresponding Author * e-mail [email protected] (M.R.S.)

Acknowledgment One of the authors (M.R.S) is thankful to the Natural Sciences and Engineering Research Council of Canada for the research grant. We also thank Timur Shegai, Department of Physics, Chalmers University of Technology, Goteborg, Sweden. Without his help, the paper would have not been completed.

References (1) Wersäll, M.; Cuadra, J.; Antosiewicz, T. J.; Balci, S.; Shegai, T. Observation of Mode Splitting in Photoluminescence of Individual Plasmonic Nanoparticles Strongly Coupled to Molecular Excitons. Nano Lett. 2016, 17 (1), 551–558. (2) Fofang, N. T.; Park, T.; Neumann, O.; Mirin, N. A.; Nordlander, P.; Halas, N. J. Plexcitonic Nanoparticles: Plasmon−Exciton Coupling in Nanoshell−J-Aggregate Complexes. Nano Lett. 2008, 8 (10), 3481–3487. (3) Wiederrecht, G. P.; Wurtz, G. A.; Hranisavljevic, J. Coherent Coupling of Molecular Excitons to Electronic Polarizations of Noble Metal Nanoparticles. Nano Lett. 2004, 4 (11), 2121–2125. (4) Schlather, A. E.; Large, N.; Urban, A. S.; Nordlander, P.; Halas, N. J. Near-Field Mediated Plexcitonic Coupling and Giant Rabi Splitting in Individual Metallic Dimers. Nano lett. 2013, 13 (7), 3281–3286. (5) Zengin, G.; Johansson, G.; Johansson, P.; Antosiewicz, T. J.; Käll, M.; Shegai, T. Approaching the Strong Coupling Limit in Single Plasmonic Nanorods Interacting with J-Aggregates. Sci. Rep. 2013, 3, No. 3074. (6) Zengin, G.; Wersäll, M.; Nilsson, S.; Antosiewicz, T. J.; Käll, M.; Shegai, T. Realizing Strong Light-Matter Interactions between Single-Nanoparticle Plasmons and Molecular Excitons at Ambient Conditions. Phys. Rev. Lett.


Page 16 of 19

Page 17 of 19 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


2015, 114 (15), No. 157401. (7) Tame, M.; McEnery, K.; Özdemir, Ş. K.; Lee, J.; Maier, S.; Kim, M. Quantum Plasmonics. Nat. Phys. 2013, 9 (6), 329–340. (8) Törmä, P.; Barnes, W. Strong Coupling between Surface Plasmon Polaritons and Emitters: A Review. Rep. Prog. Phys. 2015, 78 (1), No. 013901. (9) Artuso, R. D.; Bryant, G. W. Strongly Coupled Quantum Dot-Metal Nanoparticle Systems: Exciton-Induced Transparency, Discontinuous Response, and Suppression as Driven Quantum Oscillator Effects. Phys. Rev. B 2010, 82 (19), No. 195419. (10) Cox, J. D.; Singh, M. R.; Gumbs, G.; Anton, M. A.; Carreno, F. Dipole-Dipole Interaction between a Quantum Dot and a Graphene Nanodisk. Phys. Rev. B 2012, 86 (12), No. 125452. (11) Singh, M. R.; Guo, J.; Cid, J. M.; Martinez, J. E. Control of Fluorescence in Quantum Emitter and Metallic Nanoshell Hybrids for Medical Applications. J. Appl. Phys. 2017, 121 (9), No. 094303. (12) Singh, M. R.; Sekhar, C. M.; Balakrishnan, S.; Masood, S. Medical Applications of Hybrids Made from Quantum Emitter and Metallic Nanoshell. J. Appl. Phys. 2017, 122 (3), No. 034306. (13) Politano, A.; Radovic, I.; Borka, D.; Miskovic, Z. L.; Yu, H. K.; Farias, D.; Chiarello, G. Dispersion and Damping of the Interband 𝜋 Plasmon in Graphene Grown on Cu(111) Foils. Carbon 2017, 114, 70-76. (14) Fainberg, B. D.; Rosanov, N. N.; Veretenov, N. A. Light-Induced “Plasmonic” Properties of Organic Materials: Surface Polaritons and Switching Waves in Bistable Organic Thin Films. J. Appl. Phys. 2017, 110, No. 203301. (15) Yannopapas, V.; Paspalakis, J. Giant Enhancement of Dipole-Forbidden Transitions via Lattices of Plasmonic Nanoparticles. J. Mod. Opt. 2015, 62 (17), 1435-1441. (16) Halas, N. J.; Lal, S.; Chang, W.-S.; Link, S.; Nordlander, P. Plasmons in Strongly Coupled Metallic Nanostructures. Chem.Rev. 2011,111(6), 3913−3961. (17) Singh, M. R. Dipole-Dipole Interaction in Photonic-Band-Gap Materials Doped with Nanoparticles. Phys. Rev. A 2007, 75 (4), No. 043809. (18) Mazenko, G. F. Quantum Statisitcal Mechanics; John Wiley & Sons Inc.: New York, 2000; Sec. 5.8. (19) Kittel, C. Introduction to Solid State Physics; Sixth Ed.; John Wiley & Sons Inc.: New York, 1996; Chap. 13. (20) Ali Omar, M. Elementary Solid State Physis; Addison-Wesley: New York, 1993; Sec. 8.11. (21) Gerstein, J. L.; Smith, F. W. The Physics and Chemistry of Materials; John Wiley & Sons Inc.: New York, 2001; Chap. 15. (22) Eyring, H. et al. Statistical Mechanics and Dynamics; Wiley: New York, 1982. (23) Lorentz, H. The Theory of Electrons; Dover: New York, 1952. (24) Ohtsu, M.; Kobayashi, K. Optical Near Fields: Introduction to Classical and Quantum Theories of Electromagnetic Phenomena at the Nanoscale; Springer: Berlin, 2013. (25) Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University Press: Cambridge, 2006; pp 266. (26) Yang, U.; D’Archangel, J.; Sundheimer, M. L.; Tucker, E.; Boreman, G. D.; Raschke, M. B. Optical dielectric function of silver. Phys. Rev. B 2015, 91, 235137. (27) Nash, D. J.; Sambles, J. R. Surface plasmon-polariton study of the optical dieletric function of silver. J. Mod. Optics. 1996, 42, No. 1, 81-91. (28) Singh, M. R.; Cox, J. D.; Brzozowski, M. Photoluminescence and Spontaneous Emission Enhancement in Metamaterial Nanostructures. J. Phys. D: Applied Physics 2014, 47 (8), No. 085101. (29) Singh, M. R. Electronic, Photonic, Polaritonic and Plasmonic Materials; Wiley Custom: Toronto, 2014. (30) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: London, 1997. (31) Jiang, Y.; Pillai, S.; Green, M. A. Realistic Silver Optical Constants for Plasmonics. Sci. Rep. 2016, 6, No. 30605. (32)Baranov, D. G.; Wersäll, M.; Cuadra, J.; Antosiewicz, T. J.; Shegai, T. Novel Nanostructures and Materials for Strong Light-Matter Interactions. ACS Photonics. 2018, 5, 24-42. (33) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: London, 1997, section 7.3 and 7.5. (34) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: London, 1997, section 8.2. (35) Riesch, M.; Tchipev, N.; Senninger, S.; Bungartz, H-J.; Jirauschek, C. Performance evaluation of numerical methods for the Maxwell-Liouville-von Neumann equations. Opt. Quant. Electron. 2018, 50:112, 1-9. (36) Nitzan, A.; Sukharev, M. Optics of exciton-plasmon nanomaterials. J. Phys.: Condens. Matter 2017, 29, 1-34.



(37) Singh, M.; Schindel, D.; Hatef, A. Dipole-dipole interaction in a quantum dot and metallic nanorod hybrid system. Appl. Phys. Lett. 2011, 99, 181106. (38) Guo, J.; Black, K.; Hu, J.; Singh, M. R. Study of Plasmonics in hybrids made from a quantum emitter and double metallic nanoshell dimer. .J. Phys.: Condens. Matter 2018, 30, No. 185301. (39) Singh, M. R.; Brzozowski, M. J.; Apter, B. Effect of phonon-plasmon and surface plasmon polaritons on the energy absorption in quantum dot and graphene nanocomposite deposited on polar crystals. J. Appl. Phys. 2016, 120, 124308.


Page 18 of 19

Page 19 of 19 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


TOC Graphic


Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum

Recommend Documents