Control of Self-Kerr Nonlinearity in a Driven Coupled Semiconductor

Subscriber access provided by ECU Libraries

C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Control of Self-Kerr Nonlinearity in a Driven Coupled Semiconductor Quantum Dot – Metal Nanoparticle Structure Spyridon G. Kosionis, and Emmanuel Paspalakis J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10817 • Publication Date (Web): 04 Mar 2019 Downloaded from http://pubs.acs.org on March 7, 2019

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 28 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

Control of Self-Kerr Nonlinearity in a Driven Coupled Semiconductor Quantum Dot – Metal Nanoparticle Structure Spyridon G. Kosionis and Emmanuel Paspalakis* Materials Science Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece Corresponding author: [email protected]

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

Page 2 of 28

Abstract The self-Kerr nonlinear optical response of a probe field is investigated in the presence of a near resonant pump field in coupled metal nanoparticle (MNP) - semiconductor quantum dot (SQD) nanostructures. The study of the spectra is based on the solution of the resulted density matrix equations for the case of a colloidal CdSe-based SQD(Au)MNP molecule, as well as for an epitaxial SQD structure coupled to an Au nanoparticle. It is demonstrated that, below a critical value of the interparticle distance, the susceptibility of each distinct component of the structure exhibits a characteristic single-resonance spectrum, while, above this value, the spectrum has three resonances. This phenomenon can be understood in terms of the theory of exciton-plasmon quantum metastates. In addition, the conditions under which optical transparency occurs with concurrent positive self-phase-modulation coefficient are also identified.

Introduction In recent years, hybrid structures combining semiconductor quantum dots (SQDs) and metal nanoparticles (MNPs) have become appealing hosts for the investigation of novel optical effects in nanoscience. When a MNP is excited by an electromagnetic field, it exhibits collective oscillations of its conduction electrons (localized surface plasmons) 1. It has been shown that when the SQDs are in the vicinity of the MNPs, the localized surface plasmon resonances interact with the excitonic resonances, forming coupled excitations2–4. The exciton–plasmon interaction has created a new momentum for the investigation of novel effects with interdisciplinary applications, ranging from ultrasensitive sensing and ultrafast switching to quantum technology and even medical applications. The optical effects that are induced fall within quantum plasmonics, a relatively new area of nanophotonics. These effects include enhanced emission5,6, induction of Rabi oscillations with tunable period and controlled population transfer7-11, formation of Fano-type absorption features12-17, tunable gain without inversion18-22, plasmonically-induced

optical

transparency

accompanied

with

slow

light23-25,

significantly modified four-wave mixing26-31, four-wave parametric amplification32, as well as, self-Kerr nonlinearity33-38. The most extensively studied nanostructure in this research area is composed of a SQD, which is modelled by a two-level quantum system, and a spherical MNP7,10-14,16-20,23,2632,35.

The interaction of this hybrid system with a weak probe field of varying frequency 2 ACS Paragon Plus Environment

Page 3 of 28 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


and a strong pump field of fixed frequency has been the subject of several studies18,20,23,2632,

and the phenomena that have been investigated include the pump-controlled probe

absorption and dispersion of the structure18,20,23, as well as the study of cross-Kerr nonlinearity26-32 that leads to four-wave mixing28,31 and four-wave parametric amplification32. In addition, the strong plasmonically-induced modification of self-Kerr nonlinearity of a weak probe field in a coupled SQD-MNP structure has been only considered either in a two-level SQD without the application of the pump field35, or in three-level and four-level models of the SQDs that exhibit vacuum-induced transparency due to the presence of the plasmonic nanostructure33, combined electromagnetically induced transparency effects and plasmonically induced effects36-38, or even combined vacuum-induced transparency and electromagnetically induced transparency near a plasmonic nanostructure34. We note that in the studies which include electromagnetically induced transparency effects34,36-38, the weak probe and the strong pump fields couple to different optical transitions. Previous studies have considered the problem of control of the self-Kerr nonlinearity of a probe field by a strong pump field in a two-level atomic system39 and in a semiconductor quantum well two-subband system40. In these works, the probe and pump fields drive simultaneously the same optical transitions. Interestingly, the potential of control of selfKerr nonlinearity of a weak probe field in a SQD which is modelled by a two-level system by combining a strong pump field, which drives the same transition as the probe field, and a MNP remains unexplored. This problem is studied in the present article. Specifically, the purpose of the present study is to examine the self-Kerr effect of a lowintensity probe beam interacting with a strongly driven SQD-MNP hybrid system. We show that the self-Kerr nonlinearity of the probe field can be strongly affected and controlled by the intensity and frequency of the pump field and depends strongly on the distance between the SQD and the MNP. In particular, we find that depending on the intensity and frequency of the pump field and the distance between SQD and MNP, the total system can be found in either of the two distinct (‘dark’ and ‘bright’) states of a coherent exciton–plasmon coupling, usually called plasmonic metaresonances7,18, which determine the behavior of the self-Kerr nonlinearity. We also investigate the conditions under which this compound nanostructure acts as a non-absorptive medium, yet with an appreciable self-phase modulation coefficient. The paper is organized as follows: in the next section, we consider that the SQD-MNP structure interacts simultaneously with a probe and a pump field and present the details for the derivation of the relevant density matrix equations that describe the system 3 ACS Paragon Plus Environment


Page 4 of 28

dynamics. Specifically, we consider that the pump field is relatively weak and thus it can be treated up to third order, while the strong pump field can be treated to all orders. Then, in the coming section, we solve the density matrix equations numerically and present results for the real and the imaginary parts of the third-order self-Kerr optical susceptibility of the SQD, the MNP and the total hybrid structure, against the pumpprobe field detuning, for different values of the interparticle distance, as well as, for different pump field detunings. We also find conditions under which the total system acts as an almost non-absorptive medium with large self-phase modulation. Furthermore, in the final section, we summarize our findings. We also provide Supporting Information with the equations of the density matrix elements of the different orders and a method for calculating the critical distance between the SQD and MNP.

Theory We consider a hybrid structure composed of a spherical MNP of radius  and a spherical SQD with radius  (  ) . The distance between the centers of the two particles, denoted by R, is higher than the radius of the MNP ( R   ), as shown if Fig. 1. Here,  S is the dielectric constant of the SQD,  m ( ) is the dielectric function describing the MNP, which is treated as a classical dielectric particle, and  env is the dielectric constant of the environment of the hybrid nanostructure. This system interacts with a bichromatic electromagnetic field, polarized along the zˆ direction, which constitutes of a combination of a pump field, with amplitude Ea and angular frequency a and a probe field with amplitude Eb and angular frequency b . The total linearly polarized oscillating electric field is given by r E (t )  [ Ea cos(a t )  Eb cos(bt )]zˆ ,

(1)

and excites the interband transition between the two energy levels of the SQD 1  2 , with exciton energy equal to h0 . We assume that only these two energy levels contribute to the dynamics of the system and an oscillating dipole moment is induced. The electromagnetic field also provides a strong continuous spectral response, since it is responsible for plasmonic excitations on the surface of the MNP. The plasmons induced are coupled to the excitons, an effect which is responsible for the Förster energy transfer from the SQD to the MNP41.


Page 5 of 28 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


Taking into account the symmetry of the SQD, the Hamiltonian of the system takes the form

H  h0 2 2   ESQD  1 2  2 1  ,

(2)

where h0 is the energy difference between the SQD states,  represents the dipole moment of the SQD, and ESQD denotes the total electric field induced inside the SQD, which in the quasi-static approximation, is explicitly written as12,14,28

ESQD (t ) 

h

  e

 n  a ,b 

n

 in t

  Gn %21 (t )  *n eint  Gn* % 12 (t )  .

(3)

In Eq. (3), we have introduced the density matrix elements % ij (t ) , as well as the parameters

 En  sa n 3  n  1  , 2h effS  R3  Gn 

1 4 env

(4)

sa2 n 3  2 , 2 h effS R6

(5)

with  n   m (n )   env   m (n )  2 env  and  effS   2 env   S  3 env . Here, the Rabi frequency  n includes the direct coupling to the applied field (first term), as well as the coupling to the field from the MNP that is induced by the applied field (second term). For

n  a , it stands for the Rabi frequency of the pump field, while, for n  b , it represents the Rabi frequency of the probe field. The parameter Gn represents the self-interaction of the SQD12,14,41, since the MNP responds to the polarization field of the SQD and in turn produces a field felt back at the SQD. In Eq. (5), sa is set equal to 2, which is the case for an electric field applied with polarization along the axis on which the centers of the MNP and the SQD are placed (zaxis in our case). In the following equations we introduce the parameter G , which is defined as the sum of the two distinct contributions of the applied fields, Ga  Gb . At this point, after defining the slowly varying matrix elements  nm as 00 (t )  % 00 (t ) , i t  i t * % 11 (t )  % (and thus 10 (t )  % , since  mn   nm ) and 11 (t ) ,  01 (t )   01 (t )e 10 (t )e a

omitting the terms that contain the elements e 2iat , e 2ibt , e

a

 i a b t

in the density matrix

equations. In other words, we apply the rotating wave approximation (RWA) and derive the following density matrix equations: 5 ACS Paragon Plus Environment




&12 (t )    i (t )  

1 * * * i t  12 (t )  iG w(t ) 12 (t )  i    b e  w(t ) , T2 

w&(t )  2i    b e  i t  12 (t )  2i  *  *b ei t   21 (t )  4GI 12 (t )  21 (t ) 

Page 6 of 28

(6)

w(t )  1 . (7) T2

In Eqs. (6) and (7), w(t )  11 (t )   22 (t ) is the population difference in the SQD,

 a  0  a is the detuning of the pump field from the excitonic resonance,   b  a is the detuning between the two applied fields,  a , b are, respectively, the Rabi frequencies of the pump and probe fields, and GI is the imaginary part of the parameter

G . Moreover, T1 is the longitudinal population relaxation time and T2 is the transverse population relaxation time. Now, we assume that the pump field is strong and all the orders of its interaction with the hybrid nanostructure will be considered, while the probe field is comparatively weak and we just maintain the terms of up to the third order while studying its interaction with the system. After substituting the following third order expansion of the terms 12 (t ) and w(t ) :

12 (t )  12(0) (t )  b 12(1) (t )ei t  *b 12( 1) (t )ei t  b2 12(2) (t )e2i t   b

2

f12(0) (t )  *b 2 12( 2) (t )e 2i t  3b 12(3) (t )e 3i t 

2

(8)

2

( 3) 3i t  b b f12(1) (t )e  i t  b *b f12( 1) (t )ei t  *3 , b 12 (t )e

w(t )  w(0) (t )  b w(1) (t )e  i t  *b w( 1) (t )ei t  b2 w(2) (t )e 2i t  2

 b y (0) (t )  *b 2 w( 2) (t )e 2i t  3b w(3) (t )e 3i t  2

(9)

2

( 3)  b b y (1) (t )e  i t  b *b y ( 1) (t )ei t  *3 (t )e3i t b w

in Eqs. (2) and (3), we obtain the differential equations for the density matrix elements of different orders. The actual form of the resulted density matrix equations are given in the Supporting Infromation, Eqs. (A1)-(A13). These equations will be solved numerically, due to the nonlinear terms, for long times, in order to ensure that the system has reached steady state and the resulted terms can be used for the calculation of the different susceptibilities. The calculation of the third order self-Kerr nonlinear susceptibility constitutes the main *

objective of the present work. Thus, we are specifically interested in the term  f 12( 1)  , since it is proportional to the coefficient  . By applying a series expansion on the SQD (3)

susceptibility 6 ACS Paragon Plus Environment

Page 7 of 28 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


2

 SQD  

(1) SQD

 3

(3) SQD

 Eb     ... ,  2 

(10)

we extract the formula for the calculation of the third-order, self-Kerr, susceptibility of the SQD: 2



(3) SQD

4 * 4 * 2  2  i t ( 1)  f 12( 1) (t  T )  ,  b b e  f 12 (t  T )   i t 2 3 V  0 Eb e 3Eb V 3h  0

(11)

where, T is a sufficient time length for the interaction of the hybrid system with the electromagnetic fields, so as the system to reach steady state behavior. Also, Γ is the optical confinement factor, V is the volume of the SQD and   1  sa b 3 R 3   effS . To derive an expression for the optical susceptibility of the MNP, we should first find the polarization of the MNP, which is given by14,19,35: (  )  ib t MNP

PMNP  P

e

* E 1 sa   12 (t )    ibt b  c.c.  3 env b    c.c. , e  effS R 3   2 4 env

(12)

where c.c. stands for the complex conjugate of the first term. We define the susceptibility of the MNP as

 MNP 

2 () , PMNP  0 Eb

(13)

and obtain

 MNP 

3 b env

0



3 b sa  *  (t  T )  . 3  12 2 Eb 0 effS R

(14)

The total susceptibility of the MNP is related to the total susceptibility of the SQD by19,35

 MNP 

3 b env

0



V 3 b sa  SQD .  4 effS R 3

(15)

Thus, the third-order susceptibility of the MNP with respect to the weak probe field can be calculated given the third-order susceptibility of the SQD, by the following expression: (3)  MNP 

V 3 b sa (3)  SQD . 3  4 effS R

(16)

So, the third-order self-Kerr nonlinear susceptibility of the MNP is obtained by multiplying the respective susceptibility of the SQD by the factor 3V  b sa (4 effS R 3 ) . The total third-order self-Kerr nonlinear susceptibility of the coupled SQD–MNP system is calculated by the summation of the third-order nonlinear susceptibilities of the SQD and the MNP: 7 ACS Paragon Plus Environment


(3) (3)  tot(3)   SQD   MNP .

Page 8 of 28

(17)

Finally, we introduce two important quantities that will be useful for the determination of the position and the magnitude of the spectral resonances, in the next section. The first one is the effective Rabi frequency of the pump field28 *

eff   a  G  12(0) SS  ,

(18)

that determines the spectral intensity, as well as, the value of the critical distance Rcr at which an abrupt switch from a region of triple-resonance to a region of single-resonance spectra is observed. The second one is and the effective detuning of the pump field42

 eff   a  GR w(0)SS ,

(19)

where 12(0) SS and w(0)SS are respectively the values of the density matrix elements 12(0) (t ) (0) and w (t ) , in the steady state. We note that the quantity eff does not constitute a

regular pump-field Rabi frequency, since it contains the element 12(0) SS .

Numerical Results In Figs. 2 to 6, the spectral forms of the real (first column) and the imaginary (second (3) column) parts of the third-order, self-Kerr, nonlinear susceptibility of the SQD,  SQD , the

(3) (3) MNP,  MNP , and the total system,  tot , with respect to a weak probe field, are plotted, in

the presence of a strong pump field, as a function of the pump-probe field detuning  . Also, the dependence of these quantities on the center-to-center distance between the MNP and SQD is examined. First, in Figs 2-5, we examine the hybrid system for colloidal SQDs (typically CdSebased SQDs). The value of the matrix element for the interband optical transition  is equal to 0.65  e nm , the energy of the transition resonance h0 is 2.5 eV and the value of the parameter  / V is taken 5 1023 m 3 . The population relaxation time T1 and dephasing time T2 are, respectively, set equal to 0.8 ns and 0.3 ns , as in previous studies10,12,13,19,35. The dielectric constant for the SQD is  S  6 0 , where  0 is the vacuum permittivity, while for the MNP we use the dielectric function  m ( ) corresponding to the case of gold from Ref. 43. The whole system is placed in vacuum and thus  env   0 . In all the figures, we present calculations, solving numerically Eqs. (A1)-(A13), using a fourth-order Runge-Kutta method, assuming that the SQD is initially 8 ACS Paragon Plus Environment

Page 9 of 28 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) found in the ground state, which means that w (0)  1 , and the rest of the density matrix

elements are zero. Here, the intensity of the probe field I b is assumed to be four orders of 2 magnitude lower than the intensity of pump field I a  100 W / cm .

In Fig. 2, we study the case of exact pump-field resonance ( a  0 ). As it is described in detail in Ref. 20, when the value of the interparticle distance lies in the area above a critical point, the system is found in a ‘bright’ state, a specific conjugation of SQD excitonic and MNP plasmonic excitations for which the exchange of energy between the MNP and the SQD is strongly suppressed. In this case, the SQD exhibits a tripleresonance self-Kerr spectrum. A method for calculating the critical distance, following Ref. 28 is presented in the Supporting Information. For the parameters of the system studied here, this critical distance Rcr (  15.305 nm) separating the MNP from the SQD. (3) In Fig. 2, we observe that the real and the imaginary part of  SQD present Rabi sidebands

roughly found at   eff and an exact zero at   0 . Although for a weak SQD-MNP (3)  is antisymmetric, the interaction [blue solid curve in Fig. 2 (a): R=100 nm], Re   SQD

(3) imaginary part of  SQD appears to be symmetric about the vertical axis at   0 .

Furthermore, in Figs. 2(a) and (b), we note that when the components of the hybrid system start approaching each other, the spectral symmetry is broken, the intensity of the resonances gradually increases, and their position is progressively displaced away from the spectral center   0 . In Figs. 2(c) and (d), we examine the optical response of the MNP, for interparticle (3) distances above Rcr . While, for high values of R (blue solid curve),  MNP is zero, since

the analogy coefficient 3V  n sa (4 effS R 3 ) of Eq. (16) tends to zero, a stronger interaction between two components of the molecule (low R, but yet higher than Rcr ), (3) leads to a non-negligible dependence of  MNP on the pump-probe field detuning, because (3) (3) now  MNP is directly proportional to  SQD , with a significant analogy coefficient. Also,

(3) as we note in Figs. 2(e) and (f), the total self-Kerr coefficient is dominated by  SQD .

(3)  MNP 

V 3 b sa (3)  SQD 3  4 effS R

When R  Rcr , a case presented in Fig. 3, the hybrid nanostructure is found in a ‘dark’ state20. In this metastate, the vast percentage of the energy of the pump field is absorbed 9 ACS Paragon Plus Environment


Page 10 of 28

by the MNP and emitted back to the environment. Thus, the SQD fundamentally interacts with a weak probe field and exhibits a characteristic single-resonance spectrum, similar to the one presented in Ref. 35, where the SQD-MNP was not driven by a strong pump field. In Figs. 3(e) and (f), it becomes clear that the spectrum of the total third-order (3) susceptibility is effectively defined by the shape of  SQD , since the intensity of the

contribution of the MNP is comparatively small. Furthermore, we observe that as the (3) ] (blue solid interparticle distance decreases, the asymmetric Fano resonance of Re[  SQD

curve: R= 15 nm), becomes Lorentzian-shaped (crimson dashed curve: 13.5 nm and yellow dotted curve: 12 nm), while, at the same time, the position of the resonances for (3) (3) ] and Im[  SQD ] is shifted towards lower values of the pump-probe field both Re[  SQD

detuning. We also note that the tuning of the pump field to a frequency far from resonance, i.e.  a  5 ns 1 , does not induce any prominent modification to the Kerreffect spectra (not shown here), which maintain the profile presented in Fig. 3. On the other hand, when the interparticle distance takes values above the critical distance, the adjustment of the value of the pump field detuning can strongly reshape the Kerreffect spectral forms, as we observe in Fig. 4 (for  a  5 ns 1 ) and Fig. 5 (for (3) ]  a  5 ns 1 ). Here, as in the case of exact resonance, the spectral profiles of Re[  SQD (3) ] for the SQD [Figs. 4, 5 (a) and (b)] almost coincide with the corresponding and Im[  SQD

spectra of the total structure [Figs. 4, 5 (e) and (f)], where the positions of the resonance 2  eff sidebands are given by the formula     eff

2

. It is noteworthy that the

(3) (3) symmetry of Re[  SQD ] and Im[  SQD ] , for an exactly resonant pump field, at R= 100 nm

[Fig. 2 (a) and (b)], is broken for an off-resonant pump field [Figs. 4 and 5 (a) and (b)]. Furthermore, for a positive detuning (Fig. 4), as the distance between the MNP and the SQD increases, the intensity of the comparatively weaker left sidebands is enhanced, while, at the same time, the more prominent right-sided resonances are suppressed. However, for a negative detuning (Fig. 5), as the interparticle distance approaches the critical value 15.305 nm , a very narrow resonance arises on the right side of the self-Kerr spectra. This is prominent even with R=16.5 nm (green dash-dotted curve). This resonance becomes extremely sharp, as we approximate the transition point from the ‘bright’ to the ‘dark’ metastate. As the R parameter decreases, the intensity of the


Page 11 of 28 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


resonance observed on the left side of the spectra (   0 ), initially increases, it reaches a maximum (for a center-to-center distance close to 17 nm) and then starts decreasing. Figs. 4 and 5, show significant spectral similarities with Mollow spectra44, and it is similar to what was found in the self-Kerr nonlinearity of a probe field in strongly driven two-level atomic systems39 and semiconductor quantum well two-subband systems40. In Fig. 6, we explain the Mollow-shaped nonlinear response that the different counterparts of the hybrid system exhibit in the ‘Dark’ state regime ( R  Rcr ), following the dressed state approach. More explicitely, we present the possible excitation pathways that may be followed for the probe field, for a negative [Fig. 6(a)] and a positive [Fig. 6(b)] pump field detuning. In the dressed-state approach, the strong pump field is responsible for the splitting of the energy levels of the SQD into doublets. The first dressed state is a linear combination of the first (blue) and the third (green) levels, while the second dressed state is a linear combination of the second (orange) and the fourth (red) levels. In the case of a negative/positive detuning, the majority of the electrons occupy the higher/lower-energy level of each doublet. Thus, the electron population transition rate from the second/first to the third/fourth energy level, in the dressed-state diagram, is much higher than the population transition rate from the fourth/third to the first/second energy level. For  a  0 , in the steady-state, the Im[χ(3)] spectra present an absorption valley at a   2eff   eff

2

and a gain peak of higher magnitude, at a   2eff   eff

However, for  a  0 , the Im[χ(3)] spectra present a gain valley at a   2eff   eff an absorption peak of lower magnitude, at a   2eff   eff

2

2

2

.

and

.

In Fig. 7, we investigate the dependence of the third-order nonlinear susceptibility on the interparticle distance, for a set of parameters corresponding to an epitaxial SQD(Au)MNP, at a low temperature regime13,19,35. Here, the radius a of the MNP is equal to 10 nm, the population relaxation time T1 and the dephasing time T2 satisfy the relations T1  T2 / 2 , with h / T2  2 meV , leading to T2  0.329 ps, the electric dipole matrix

element is   0.6  e nm and the exciton energy of the SQD is h0  1.546 eV , a value approximately equal to the plasmon resonance frequency, as the MNP is embedded in a material with a high refractive index, also equal to the refractive index of the SQD 6 2 ( env   S  12 0 ) . Finally, the pump field intensity I a is taken equal to 5 10 W / cm ,



Page 12 of 28

while for the intensity of the probe field we assume that I b  I a /104 . Here, we consider that the pump field is at exact resonance with the SQD energy levels ( a  0) . First, we note that for all the cases studied the broadening of the self-Kerr spectra are three orders of magnitude higher than the broadening obtained for a colloidal CdSe-based SQD-(Au)MNP structure (Figs. 2, 5, 6). This occurs due to the modification of the value of the dephasing time T2 . While, in an epitaxial-SQD structure, for high values of the center-to-center distance (blue solid curve: R= 100 nm), the real and the imaginary parts (3) (3) of  SQD and  MNP [Figs. 2 (e) and (f)] exhibit a symmetric spectral response, when the

value of R decreases, this symmetry is only slightly distorted unlike that of a colloidalSQD molecule (Fig. 2). We note that for the epitaxial-SQD structure, the shape of the total self-Kerr optical susceptibility is practically determined by the MNP component and not by the SQD component as in a colloidal SQD-MNP complex [compare with Figs. 2 (e) and (f)]. Moreover, for the epitaxial SQD-MNP structures, the decrease of the R parameter displaces the sideband resonances towards higher values of δ. A characteristic that both colloidal and epitaxial SQD-MNP structures share in common is the increase of spectral intensity along with the decrease of the interparticle distance. In Fig. 8, we investigate the potential of simultaneous nihilation of the first and third order absorption coefficients, Im   (1)  and Im   (3)  , while at the same time Re   (3)   0 . Under these conditions, the propagation of a weak probe field through the

medium can be achieved without absorption (optical transparency), yet with a positive and relatively large self-phase modulation coefficient, which can be important for applications, such as squeezed light generation by self-phase modulation, propagation of spatial solitons and optical switching39. A case for which the conditions presented above are almost met is the one of a hybrid SQD-MNP with colloidal SQDs and interparticle distance equal to 18 nm (magenta curves), interacting with a resonant pump field and a pump-probe field detuning equal to  ; 9.57 ns -1 . However, for even lower or higher interparticle distances (yellow curves: R=16.5 nm, green curves: R=20 nm and blue curves: R=100 nm), we move further away from the accomplishment of the transparency conditions.

Summary


Page 13 of 28 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


We studied theoretically self-Kerr nonlinear optical phenomena in a hybrid system composed of a SQD and a spherical MNP interacting simultaneously with a weak probe field and a strong pump field. We verified that the entire hybrid structure, as well as its separate components, exhibit a self-Kerr optical response that is governed by two collective molecular states, the so-called ‘dark’ and ‘bright’ states, which arise, respectively, above and below a certain critical interparticle distance. Above this critical distance, the MNP is screened and the SQD shows a common third-order optical response that resembles the one of an isolated SQD, while, below this critical distance, the pump field experienced by the SQD is screened dramatically, due to the strong exciton– plasmon coupling. It was also demonstrated that, in the case of a colloidal CdSe-based SQD-(Au)MNP structure, the presence of the SQD bears important impact on the Kerr response of the MNP, while the total Kerr response of the nanostructure is dependent on both the SQD and the MNP contributions, with the SQD having the most important contribution. However, for an epitaxial structure, the MNP component basically determines the self-Kerr susceptibility of the hybrid nanostructure. We also identify a set of parameters that leads to a non-absorptive medium with large self-phase modulation coefficient, for a colloidal SQD – (Au)MNP structure.

Supporting Information The equations of the density matrix elements of the different orders and a method for calculating the critical distance between the SQD and MNP are presented in the Supporting Information.

References (1) Törmä, P.; Barnes, W.L. Strong Coupling between Surface Plasmon Polaritons and Emitters: a review. Rep. Prog. Phys. 2015, 78, 013901. (2) Marquier, F.; Sauvan, C.; Greffet, J.-J. Revisiting Quantum Optics with Surface Plasmons and Plasmonic Resonators. ACS Photon. 2017, 4, 2091-2101. (3) Vasa, P.; Lienau, C.; Strong Light–Matter Interaction in Quantum Emitter/Metal Hybrid Nanostructures. ACS Photon. 2018, 5, 2-23. (4) Moilanen, A. J.; Hakala, T. K.; Törmä, P. Active Control of Surface Plasmon–Emitter Strong Coupling. ACS Photon. 2018, 5, 54-64. (5) Chang, D. E.; Sorensen, A. S.; Hemmer, P. R.; Lukin, M. D. Quantum Optics with Surface Plasmons. Phys. Rev. Lett. 2006, 97, 053002. 13 ACS Paragon Plus Environment


Page 14 of 28

(6) Trügler, A.; Hohenester, U. Strong Coupling between a Metallic Nanoparticle and a Single Molecule. Phys. Rev. B 2008, 77, 115403. (7) Sadeghi, S. M. The Inhibition of Optical Excitations and Enhancement of Rabi Flopping in Hybrid Quantum Dot–Metallic Nanoparticle Systems. Nanotechnology 2009, 20, 225401; Plasmonic Metaresonances: Molecular Resonances in Quantum Dot– Metallic Nanoparticle Conjugates. Phys. Rev. B 2009, 79, 233309. (8) Antón, M. A.; Carreño, F.; Melle, S.; Calderón, O. G.; Cabrera Granado, E.; Cox, J.; Singh, M. R. Plasmonic Effects in Excitonic Population Transfer in a Driven Semiconductor–Metal Nanoparticle Hybrid System. Phys. Rev. B 2012, 86, 155305. (9) Antón, M. A.; Carreño, F.; Melle, S.; Calderón, O. G.; Cabrera Granado, E.; Singh, M. R.; Optical Pumping of a Single Hole Spin in a p-doped Quantum Dot Coupled to a Metallic Nanoparticle. Phys. Rev. B 2013, 87, 195303. (10) Paspalakis, E.; Evangelou, S.; Terzis, A. F. Control of Excitonic Population Inversion in a Coupled Semiconductor Quantum Dot–Metal Nanoparticle System. Phys. Rev. B 2013, 87, 235302. (11) Yang, W.-X.; Chen, A.-X.; Huang, Z.; Lee, R.-K. Ultrafast Optical Switching in Quantum Dot-Metallic Nanoparticle Hybrid Systems. Opt. Express 2015, 23, 13032. (12) Zhang, W.; Govorov, A. O.; Bryant, G. W. Semiconductor-Metal Nanoparticle Molecules: Hybrid Excitons and the Nonlinear Fano Effect, Phys. Rev. Lett. 2006, 97, 146804. (13) Yan, J.-Y.; Zhang, W.; Duan, S.; Zhao X.-G.; Govorov, A. O. Optical Properties of Coupled Metal-Semiconductor and Metal-Molecule Nanocrystal Complexes: Role of Multipole Effects. Phys. Rev. B 2008, 77, 165301. (14) 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, 195419. (15) Singh, M. R.; Schindel, D. G.; Hatef, A. Dipole-Dipole Interaction in a Quantum Dot and Metallic Nanorod Hybrid System. Appl. Phys. Lett. 2011, 99, 181106. (16) Kosionis, S. G.; Terzis, A. F.; Yannopapas V.; Paspalakis, E. Nonlocal Effects in Energy Absorption of Coupled Quantum Dot−Metal Nanoparticle Systems. J. Phys. Chem. C 2012, 116, 23663. (17) Liu, X.-N.; Yue, Q.; Yan, T.-F.; Li, J.-B.; Yan, W.; Ma, J.-J.; Zhao, C.-B.; Zhang, X.-H. Competition between Local Field Enhancement and Nonradiative Resonant Energy Transfer in the Linear Absorption of a Semiconductor Quantum Dot Coupled to a Metal Nanoparticle. J. Phys. Chem. C 2016, 120, 18220. 14 ACS Paragon Plus Environment

Page 15 of 28 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) Sadeghi, S. M. Gain without Inversion in Hybrid Quantum Dot-Metallic Nanoparticle Systems. Nanotechnology 2010, 21, 455401. (19) Kosionis, S. G.; Terzis, A. F.; Sadeghi, S. M.; Paspalakis, E. Optical Response of a Quantum Dot–Metal Nanoparticle Hybrid Interacting with a Weak Probe Field. J. Phys.: Condens. Matter 2013, 25, 045304. (20) Sadeghi, S. M. Ultrafast Plasmonic Field Oscillations and Optics of Molecular Resonances Caused by Coherent Exciton-Plasmon Coupling. Phys. Rev. A 2013, 88, 013831. (21) Zhao, D.-X.; Gu, Y.; Zhang, J.-X.; Zhang, T.-C.; Gerardot, B. D.; Gong, Q.-H. Quantum-Dot Gain without Inversion: Effects of Dark Plasmon-Exciton Hybridization. Phys. Rev. B 2014, 89, 245433. (22) Carreño, F.; Antón, M. A.; Yannopapas, V.; Paspalakis, E. Control of the Absorption of a Four-Level Quantum System near a Plasmonic Nanostructure. Phys. Rev. B 2017, 95, 195410. (23) Lu Z.; Zhu, K.-D. Slow Light in an Artificial Hybrid Nanocrystal Complex. J. Phys. B 2009, 42, 015502. (24) Evangelou, S.; Yannopapas, V.; Paspalakis, E. Transparency and slow light in a four-level quantum system near a plasmonic nanostructure. Phys. Rev. A 2012, 86, 053811. (25) Wang, L.; Gu, Y.; Chen, H.; Zhang, J.-Y.; Cui, Y.; Gerardot, B. D.; Gong, Q.-H. Polarized

Linewidth-Controllable

Double-Trapping

Electromagnetically

Induced

Transparency Spectra in a Resonant Plasmon Nanocavity. Sci. Rep. 2013, 3, 2879. (26) Lu Z.; Zhu, K.-D. Enhancing Kerr Nonlinearity of a Strongly Coupled Exciton– Plasmon in Hybrid Nanocrystal Molecules. J. Phys. B 2008, 41, 185503. (27) Li, J.-B.; Kim, N.-C.; Cheng, M.-T.; Zhou, L.; Hao, Z.-H.; Wang, Q.-Q. Optical Bistability and Nonlinearity of Coherently Coupled Exciton-Plasmon Systems. Opt. Express 2012, 20, 1856-1861. (28) Paspalakis, E.; Evangelou, S.; Kosionis, S. G.; Terzis, A. F. Strongly Modified FourWave Mixing in a Coupled Semiconductor Quantum Dot-Metal Nanoparticle System. J. Appl. Phys. 2014, 115, 083106. (29) Li J.-J.; Zhu, K.-D. Recent Advances of Light Propagation in Surface Plasmon Enhanced Quantum Dot Devices. Crit. Rev. Solid State Mater. Sci. 2014, 39, 25-45. (30) Li, J.-B.; Liang, S.; He, M.-D.; Chen, L.-Q.; Wang, X.-J.; Peng, X.-F. A Tunable Bistable Device Based on a Coupled Quantum Dot–Metallic Nanoparticle Nanosystem. Appl. Phys. B 2015, 120, 161-166. 15 ACS Paragon Plus Environment


Page 16 of 28

(31) Singh, S. K.; Abak, M. K.; Tasgin, M. E. Enhancement of Four-Wave Mixing via Interference of Multiple Plasmonic Conversion Paths. Phys. Rev. B 2016, 93, 035410. (32) Li, J.-B.; He, M.-D.; Chen, L.-Q. Four-Wave Parametric Amplification in Semiconductor Quantum Dot-Metallic Nanoparticle Hybrid Molecules. Opt. Express 2014, 22, 24734. (33) Evangelou, S.; Yannopapas, V.; Paspalakis, E. Modification of Kerr Nonlinearity in a Four-Level Quantum System near a Plasmonic Nanostructure. J. Mod. Opt. 2014, 61, 1458-1464. (34) Chen, H.; Ren, J.; Gu, Y.; Zhao, D.-X.; Zhang, J.; Gong, Q. Nanoscale Kerr Nonlinearity Enhancement Using Spontaneously Generated Coherence in Plasmonic Nanocavity. Sci. Rep. 2015, 5, 18315. (35) Terzis, A. F.; Kosionis, S. G.; Boviatsis, J.; Paspalakis, E. Nonlinear Optical Susceptibilities of Semiconductor Quantum Dot – Metal Nanoparticle Hybrids. J. Mod. Opt. 2016, 63, 451-461. (36) Ren, J.; Chen, H.; Gu, Y.; Zhao, D.-X.; Zhou, H.; Zhang, J.; Gong, Q. PlasmonEnhanced Kerr Nonlinearity via Subwavelength-Confined Anisotropic Purcell Factors. Nanotechnology 2016, 27, 425205. (37) Tohari, M. M.; Lyras, A.; AlSalhi, M. S. Giant Self-Kerr Nonlinearity in the Metal Nanoparticles-Graphene Nanodisks-Quantum Dots Hybrid Systems Under Low-Intensity Light Irradiance. Nanomaterials 2018, 8, 521. (38) Gupta, P. Surface Plasmon-Assisted Modification of χ(3) Non-Linearity in a ThreeLevel

Active

Medium

Near

Gold

Grating,

Plasmonics,

https://link.springer.com/article/10.1007/s11468-018-0820-5, in press, (2018). (39) Bennink, R. S.; Boyd, R. W.; Stroud, C. R.; Wong, Jr., and V. Enhanced Self-Action Effects by Electromagnetically Induced Transparency in the Two-Level Atom, Phys. Rev. A 2001, 63, 033804. (40) Kosionis, S. G.; Terzis, A. F.; Paspalakis, E. Kerr Nonlinearity in a Driven TwoSubband System in a Semiconductor Quantum Well. J. Appl. Phys. 2011, 109, 084312. (41) Sadeghi S. M.; West, R. G. Coherent Control of Forster Energy Transfer in Nanoparticle Molecules: Energy Nanogates and Plasmonic Heat Pulses. J. Phys.: Condens. Matter 2011, 23, 425302. (42) Malyshev A. V.; Malyshev, V. A. Optical Bistability and Hysteresis of a Hybrid Metal-Semiconductor Nanodimer. Phys. Rev. B 2011, 84, 035314. (43) Johnson P. B.; Christy, R. W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6, 4370-4379. 16 ACS Paragon Plus Environment

Page 17 of 28 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


(44) Mollow B. R. Stimulated Emission and Absorption near Resonance for Driven Systems. Phys. Rev. A 1972, 5, 2217-2222.



Page 18 of 28

Figure Captions Figure 1. Schematic diagram of the SQD-MNP hybrid system. Here, α and β are the radii of SQD and MNP, respectively and R is the center-to-center distance between SQD and the MNP. Figure 2. The real (first column) and the imaginary part (second column) of the thirdorder self-Kerr susceptibility for the SQD [(a), (b)], the MNP [(c), (d)] and the total hybrid SQD-MNP system [(e), (f)], as a function of the pump-probe detuning  , for a 2 2 pump filed detuning  a  0 , a pump filed intensity I a  10 W / cm and interparticle

distances R= 100 nm (blue solid curve), 20 nm (crimson dashed curve), 18 nm (yellow dotted curve) and 16.5 nm (green dash-dotted curve). These curves have been plotted for the case of a colloidal SQD. Figure 3. The same as in Fig. 2, but with R= 15 nm (blue solid curve), 13.5 nm (crimson dashed curve) and 12 nm (yellow dotted curve). Figure 4. The same as in Fig. 2, but with  a  5 ns 1 . Figure 5. The same as in Fig. 2, but with  a  5 ns 1 . Figure 6. The dressed states for (a)  a  0 , (b)  a  0 . Left arrow: three-photon resonance, Central arrows: Rayleigh resonances, Right arrow: ac-Stark-shifted atomic resonance. Figure 7. The same as in Fig. 2, but for a set of parameters corresponding to an epitaxial SQD–MNP (Au), at a low temperature regime, with  a  0 and R= 100 nm (blue solid curve), 25 nm (crimson dashed curve), 18 nm (yellow dotted curve) and 14 nm (green dash-dotted curve). Figure 8. The real (solid curves) and imaginary parts (dashed curves) of linear (a) and third-order

self-Kerr

nonlinear

susceptibilities

(b)

for

pump

filed

intensity

I a  102 W / cm 2 and pump filed detuning  a  0 . Caption (c) depicts Re   (3)  (solid 18 ACS Paragon Plus Environment

Page 19 of 28 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


curves), Im   (3)  (dashed curves) and Im   (3)  (dash-dotted curves), in the area of our interest. The color of the curves corresponds to different interparticle distances (blue curves: R= 100 nm, green curves: 20 nm, magenta curves: 18 nm and yellow curves: 16.5 nm).



Page 20 of 28

Figures

Fig. 1


Page 21 of 28 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


Fig. 2



Page 22 of 28

Fig. 3


Page 23 of 28 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


Fig. 4



Page 24 of 28

Fig. 5


Page 25 of 28 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


Fig. 6



Page 26 of 28

Fig. 7


Page 27 of 28 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


Fig. 8



Page 28 of 28

Graphical TOC


Control of Self-Kerr Nonlinearity in a Driven Coupled Semiconductor

Recommend Documents