Noble classical and quantum approach to model the optical properties of metallic nanoparticles to enhance the sensitivity of optoplasmonic sensors

The bright light obtained from the quantum principle has a key role in the construction of optical sensors. Yet, theoretical and experimental work highlights the challenges of overcoming the high cost and low efficiency of such sensors. Therefore, we report a metallic nanoparticle-based metasurface plasmons polariton using quantum and classical models. We have investigated the material properties, absorption cross-section, scattering cross-section, and efficiency of the classical model. By quantizing light–matter interaction, the quantum features of light – degree of squeezing, correlation, and entanglement are quantified numerically and computationally. In addition, we note the penetration depth and propagation length from a hybrid model in order to enhance the optoplasmonic sensor performance for imaging, diagnosing, and early perception of cancer cells with label-free, direct, and real-time detection. Our study findings conclude that the frequency of incident light, size, shape, and type of nanoparticles has a significant impact on the optical properties of metallic nanoparticles and the nonlinear optical properties of metallic nanoparticles are dynamic, enhancing the sensitivity of the optoplasmonic sensor. Moreover, the resulting bright light shows the systematic potential for further medical image processing.


Introduction
Physicists consider the world we experience in terms of light and matter. 1,2 This can be expressed by classical and quantum models for different applications. The classical model considers light as electromagnetic radiation and matter as a charged particle, in order to drive the propagation length and penetration depth of a surface plasmon polariton (SPP). While the quantum mechanical model describes phenomena by a quantization scheme 3-6 for analytical and computational manipulations of quantum nature, such as quantum decoherence, quantum correlations (QCs), and quantum entanglement. [7][8][9][10] The combinations of both models can help with the study of light-matter interactions for the enhancement of optical-based sensors. 11,12 When light and metallic nanoparticles (MNPs) interact on the metal surface (MS), free electrons transit from the ground state to the excited state and vice versa (jgi 4 jei) by an amount u, depending on the intensity of incident light and meta-stable state of MNPs and MS. The collective oscillation of free electrons inside and outside of MS results from dipole oscillation. An oscillating dipole radiates ngerprint frequency since the particles have different numbers of electrons bound in different congurations. 13,14 The resulting collective excitation of free electrons that propagate along a metal-dielectric interface is called a surface plasmonic polariton (SPP), which is 50% light and 50% particle, has a small mass, travels fast, and interacts strongly. [15][16][17] The strong interaction makes both classical and quantum models best for expressing the optical properties of SPP. [18][19][20] Linear and nonlinear interactions have been introduced for propagation length and interaction time enhancement. 21,22 Using MNPs for difference frequency generation to induce extra noise and enhance propagation length. 23,24 Enhancement of SPP properties has potential application in nanophotonics, security, super-resolution imaging, and optical biosensors. [25][26][27] Plasmonic-based optical sensors (OPS) have high specicity, sensitivity, small size, and cost-effective advantages over conventional sensors. Their application has been used to measure cancer cell mechanics such as extracellular matrix, nucleoskeleton, and cytoskeleton; applying both the classical and quantum models of SPP properties. [28][29][30][31] Recently, researchers have addressed different types of optical sensors for cancer treatment. [32][33][34][35] Zakaria et al. 32 have reported, a gas sensor to detect volatile organic compounds (VOCs) emitted from exhaled breath for early lung cancer diagnosis; their work provides high performance in distinguishing lung cancer from breast cancer with limited detection of other cancer cells. In the same year, Zhang et al. 33 developed metamaterial biosensors in the terahertz frequency for theoretical and experimental investigation of apoptosis cancer cells, but developing metamaterials makes the work more difficult. Two years later Vural et al. 34 proposed electrochemical biosensors using quantum dot (QD) nanocomposite materials for analyzing various cell types, but this requires the preparation of additional uid. Adams et al. 35 have reported electrical impedance spectroscopy to study the dynamics of cancer cells. All the above work encountered difficulties due to the complexity of the experimental work, cost-effectiveness, and specicity. [36][37][38][39] Currently, many pioneers have focused on OPS for imaging, early diagnostics and post diagnostics of cancer cells, for its simplicity, high sensitivity, selectivity, and low cost: Bellassai et al. developed surface plasmon resonance (SPR) and localized SPR (LSPR)-based platforms for the detection of different classes of cancer biomarkers, but their work addresses only liquid-based samples and requires additional work to be costcompetitive, robust, and sensitive. 40 One-year later Sojic et al. worked on enhancing the sensitivity of OPs using gold-coated micropillar-etched tips. 41 However, all reported work on OPs is expensive, difficult to function, and less sensitive and selective for cancer diagnoses.
Therefore, this work aims to study the classical and quantum models of light-matter interactions to optimize OPS for high sensitivity using theoretical and computational expression. With MNPs arranged on a metal surface (Fe 2 O 3 ) and ejecting intense coherent light, the results consider interactions by classically applying Maxwell equations and the modied Drude model; and quantum mechanically by quantization of the eld, MNPs, and interactions. From the classical model the scattering cross-section, absorption cross-section extinction coefficient, and their efficiency are expressed. From the quantum mechanical model, we derive all the quantum properties of SPP by applying the Hamiltonian system. The combination of both models leads to the construction of a simple, cost-effective, highly sensitive, and selective OPS for cancer diagnosis.

Classical approach to model SPP properties
To model the properties of SPP classically, we applied Maxwell equations and Drude-Lorentz models. From the properties we determine, intensity, absorption cross-section, scattering crosssection, extinction cross-section and efficiency, with these parameters being tools to measure the capacity of a sensor.
The electric eld resulting from light-matter interactions is given by the summation of the incident and output eld. The output eld is the difference between the incident eld, absorbed, and transmitted eld, dened by 42,43 where the propagation of the incident laser eld to the right and le (E in (u,t) ¼ E + in (u,t) + E : in (u,t)) takes the form, Here, r ¼x + yĵ + zk is the position in 3D, u and E 0 are the frequency and amplitude of the incident eld, respectively, k ¼ u c is wave number and c is speed of light in a vacuum.
The output eld (for details see Appendix A) takes the form, where, m a ¼ À q 2 3 0 m and m b ¼ À q 2 u 4 p 3 0 m are the transmission and absorbance dipole moment, respectively; plasma frequency, , is calculated using 3 0 permittivity of free space, m e mass of electron, q charge and G ¼ rN a M n e ; in which, r is density, N a is Avogadro's number, M is the atomic mass of the particle and n e is the number of free electrons. Incorporating eqn (3) into eqn (1), the SPP eld gives, Here, the decay rate of a polariton is a result of the incident coherent eld 2.1.1 Linear and nonlinear response of light-matter interactions. Linear light-matter interactions occur when a single incident eld produces a single photon as a result of a weak interactions. In other cases, if the matter is pumped by an intense incident eld, more than a single photon is created i.e. a nonlinear response of light. Excitation of matter by either a weak or intense eld results in the polarization of light, given by a summation of the linear and the nonlinear term 44 for and where, rst-order susceptibility (c (1) ) describes the linear optical properties including absorption and reection. While c (2) , c (3) , and c (n) are second, third, and n th order susceptibility, respectively, used for frequency conversion processes. Thus, the collective movement of free electrons inside and outside of MS resulting from light-matter interactions is expressed by the modied Drude model in terms of polarization as reported by, 45 rst order derivation of polarization (damping factor). While, g ab and u 0 are the damping coefficient and characteristic frequency, respectively. Substituting eqn (4) into eqn (7) and applying differentiation with respect to time we nd Applying simple rearrangement, the rst order susceptibility takes the form In addition, c (2) ¼ 0 for crystals with a center of symmetry and optically isotropic optical materials, and c (3) ¼ b(c (1) ) 4 as dened in ref. 46 so third order susceptibility can be dened as where b ¼ 1 Â 10 À7 is the mean value constant.

Dielectric function of materials.
The dielectric function of materials is derived using Maxwell's relation which connects the electric displacement eld D with material properties as From this relation we can establish Using linear (eqn (9)) and the nonlinear susceptibility (eqn (10)) relation, the linear dielectric function is given by while, the nonlinear dielectric function gives, The light-matter interaction encounters either loss of light by transforming to heat, called the absorption cross-section (d abs ), or is redirected in a different direction, called the scattering cross-section (d sca ). They can be dened in terms of the dielectric function of materials using the Clausius-Mossotti relation as, 46 and Here, k is the wave number, is the polarizability of the material with V ¼ 4/3pr 3 , the volume of spherically shaped MNPs (r is the radius of MNPs), 3 NPs is the linear (eqn (13)) or nonlinear (eqn (14)) dielectric function of MNPs, 3 MS is the dielectric function of the medium or metallic surface and J stands for an imaginary term. With this, the total radiant ux area (extinction cross-section) is dened by a superposition of absorption and scattering cross-section as 47 The sensitivity of the optical sensor depends on the efficiency of the SPP. That is dened as the ratio of scattering to extinction cross-section, and absorption cross-section to extinction cross-section, mathematically dened by, 48,49 and where, h sca , h abs and h ext are scattering efficiency, absorption efficiency and extinction efficiency, respectively.

Quantum model of light-matter interactions
Enhancing light-matter interactions is required to enhance optoplasmonic biosensors. This section deals with the quantum mechanical model of emitted SPP (see Fig. 1) to study its quantum properties by employing the Hamiltonian of the system and quantization of both light and matter, from which all dynamical equations of a system are dened. Fig. 1 illustrates a model for three level light-matter interactions denoted by jei, jgi and jg*i that represent the excited state, rst ground state and second ground state, respectively, with possible transitions between jei 4 jgi and jei 4 jg*i, but forbidden for jgi 4 jg*i.
The Hamiltonian of a system through the quantization of light and matter was stated by Jaynes and Cummings as 50 Here,Ĥ F ,Ĥ M ,Ĥ Int are the eld, matter and light-matter interactions Hamiltonian, and using Appendix B we can set the Hamiltonian of the system aŝ where gives the coupling between the eld and MNPs, with V the quantized volume of MNPs that results ingf 1 R where R is the radius of MNPs, implying that the smaller the radius the stronger the coupling with strong quantum properties. The plasma frequency u p 2 ¼ Gq 2 3 0 m in which G ¼r 12 +r 32 .

Dynamical equations of a system
The dynamical equations are given by the Heisenberg-Langevin equations of evolution of atomic operators applying a quantized system as reported in ref. 51, where k is the cavity damping constant and applying commutation relations, Langevin equations for evolution of the atomic operator 52 take the form (for details see Appendix C), where,X a ¼ and

Quantum properties of MNP based SPPs
SPPs that have a quantum nature have enhanced performance for microscopy, target detection, and phase estimation. 53 The basic quantum features are dened by degrees of squeezing, quantum correlation, and preservation of entanglement 54 as stated in the following subsections.
2.4.1 Squeezing properties. The squeezing properties of light are described by quadrature squeezing and quadrature variance given by 55 whereĉ Here,ĉ ¼â +b. Then using eqn (29) and (30), eqn (28) can be rewritten as, On account of eqn (23) and (24) we also have, Then, incorporating eqn (25)-(27) into eqn (31), plus and minus quadrature take the form From quadrature variance, we can derive quadrature squeezing as

Entanglement properties.
The rst inseparability condition proposed by Simon and Duan is the sufficient conditions for quantication of entanglement in a two-mode CV system. The criterion suggest that if the two modes are separable, they should satisfy the inequality: 56 . Substituting eqn (23) and (24) into eqn (35) the Duan-Giedke-Cirac-Zoller (DGCZ) criterion takes the form, 2.4.3 Second order correlation function. The second order correlation function is the major feature to distinguish nonclassical, anti-bunching light sources from classical light. 57 It plays an instrumental role in the construction of advanced biosensors since it is used for quantum information, cold atomic cloud, spectroscopy of quantum dots and uorescence correlation. 58 Mathematically, the second order correlation function is dened by 59 g ða;bÞ 2 ð0Þ ¼ Since the expectation of both beams of light are Gaussian variable with zero means, eqn (37) takes the form On account of eqn (23) and (24) the second order correlation function is given by, Then, incorporating eqn (25)- (27) into eqn (39) the second order correlation function takes the form g ða;bÞ 2 ð0Þ

Results and discussion
The classical and quantum model of the SPP was studied using Maxwell equations, the Drude model, and by quantizing the resulting eld from laser light and Ag/Fe 2 O 3 , Au/Fe 2 O 3 , Al/Fe 2 O 3 interactions. The classical model result revealed that the nonlinear optical properties of materials have more intensity than the linear (as illustrated in Fig. 2), since more variation of the induced electronic polarization occurs. Additionally, out of all nobel MNPs, Au has more intensity due to more free electrons and so the result is in agreement with the experimental work of ref. 64. Further, employing the nonlinear susceptibility of materials (eqn (14)) we have investigated the absorption cross section (see The additional measurement of rate at which a particular light-matter interaction occurs has a scattered cross section dened by eqn (15). Fig. 4 displays the scattering cross section of Au (Fig. 4A), Ag (Fig. 4B) and Al (Fig. 4C) under different spherical MNP sizes. Fig. 4 clearly shows that the scattering cross section increases with the size of MNPs, and Ag has a larger scattering cross section than Au or Al (Table 1). Fig. 5 shows the extinction cross section and it exceeds both the absorption and scattering cross sections since it is the sum of the two. This work is in agreement with other theoretical reports. 67 Therefore, on account of eqn (18)- (20) and Fig. 3-5 we can summarize that Ag-based SPPs have more overall efficiency than Au and Al.

Quantum feature measurement
In quantum mechanical principles, the full picture of the physical observable cannot be captured in a single measurement, rather the detection has to be performed many times under the same preparation conditions and taking the expected value. Squeezed, correlated and entangled sources of light are used to reduce noise by phase matching techniques to measure elasticity, ionization, and bond type (covalent, ionic, and hydrogen bonds) of a cell. From the classical model we illustrate that Ag-based SPPs have higher efficiency, therefore, in the next subsections we focus on the quantum features of Ag-based SPPs. Fig. 6 shows that squeezing occurs in the minus quadrature, and quadrature variance depends on size and detuning. Quadrature variance increases with detuning and frequency but decreases as the size of MNPs increases, this indicates smaller particles (10-40 nm) have quantum features rather than bulk materials. Fig. 7 shows the direct proportionality of the degree of squeezing with detuning, size of the Ag NPs, and frequency.  (16)) plotted using the parameters presented in Table 1 for different values of r. (A) for Ag, (B) for Au and (C) for Al. Fig. 4 Plots of scattering cross sections (eqn (15)) plotted using the parameters listed in Table 1 for different values of r and l ¼ 632.8 nm: (A) for Au, (B) Ag and (C) for Al. Fig. 8 is an illustration of entanglement in 2D and 3D that shows that the degree of entanglement is proportional to detuning but decreases as the size of MNPs increases. This shows that detuning is proportional to relaxation time, and as the size of MNPs decreases relaxation time increases since the polariton (collective free electron) can propagate through tiny particles freely. Fig. 9 shows that the degree of correlation increases with detuning and decreases as the size of Ag NPs increases, since the size of material increases, the absorption of light leads to the production of thermal energy that results in noise.
Therefore the Rabi splitting with active control expressed by Wen et al. 68 has a signicant impact for controlling temperature, but it is not enough to cool and control the biosensor for study both experimentally and theoretically. We introduce  (17)) plotted using the parameters listed in Table 1 Table 1.
nonlinear-based OPS with ve main components as shown in Fig. 10, which generates bright light. The components are arranged as follows: light source, different types of lenses for controlling light, object (sample holder), detector (interaction of organisms/light with different molecular species) and visual screen (gives the output of measurements). The efficiency of the SPP eld is described by the propagation length (L x ) and penetration depth (L y ) as light propagates along the z-axis, 69-71 expressed mathematically where, E x and E y are the incident eld along the x and y-axis with and Here, R and J are the real and imaginary parts of the dielectric function for and The quantum features of light are strong and sensitive enough to measure cell mechanics. Cell mechanics is the generalized name for the differentiation of a cell, morphological alterations of the cell, and cell cycle. Measurement of cell mechanics under sensitive instruments like SPP-based OPBs helps to distinguish cancer cells from normal ones. The output measurements of light with and without SPPs are presented in Fig. 11. Fig. 11 shows the enhancement of the incident eld with MNPs (Ag) enhancing the sensitivity and selectivity of OPS. We note from this gure that with the brightness of incident light, if any biological sample, especially cancer cells, are added the obtained brightness is enough to show the cell mechanics of cancer cells.

Conclusions
In this work, we have established the Nobel classical and quantum approach to model optical properties of MNPs for the sensitivity enhancement of optical biosensors. With the help of the Maxwell equations and modied Drude model, we have obtained equations of the dielectric function of materials from which relaxation time, absorption efficiency, and extinction efficiency, comparing linear with nonlinear optical properties, of SPPs are derived. Following this, we have obtained the Hamiltonian of a system by quantizing the equation of propagation that leads to obtaining the quantum Langevin equation to test the quantum nature of the resulting SPP from light-matter interactions. From this work, it can be noted that the sensitivity of the optoplasmonic biosensor increases as the size of stimulants (noble MNPs) decreases, Ag NPs are more efficient for sensitivity and selectivity enhancement in OPS compared with other MNPs. In addition, we have also found that nonlinear SPPs have a highefficiency quantum nature that contributes to controlling noise in OPS. Finally we conclude that the nonlinear optical properties of a Ag-based surface plasmon polariton results in bright light with sensitive, selective, and low cost OPS that could be used to image and treat cancer cells.