Compared with manipulation of microparticles with optical tweezers and control of atomic motion with atom cooling, the manipulation of nanoscale objects is challenging because light exerts a significantly weaker force on nanoparticles than on microparticles. The complex interaction of nanoparticles with the environmental solvent media adds to this challenge. In recent years, optical manipulation using electronic resonance effects has garnered interest because it has enabled researchers to enhance the force as well as sort nanoparticles by their quantum mechanical properties. Especially, a precise observation of the motion of nanoparticles irradiated by resonant light enables the precise measurement of the material parameters of single nanoparticles. Conventional spectroscopic methods of measurement are based on indirect processes involving energy dissipation, such as thermal dissipation and light scattering. This study proposes a theoretical method to measure the nonlinear optical constant based on the optical force. The nonlinear susceptibility of single nanoparticles can be directly measured by evaluating the transportation distance of particles through pure momentum exchange. We extrapolate an experimentally verified method of measuring the linear absorption coefficient of single nanoparticles by the optical force to determine the nonlinear absorption coefficient. To this end, we simulate the third-order nonlinear susceptibility of the target particles with the kinetic analysis of nanoparticles at the solid–liquid interface incorporating the Brownian motion. The results show that optical manipulation can be used as nonlinear optical spectroscopy utilizing direct exchange of momentum. To the best of our knowledge, this is currently the only way to measure the nonlinear coefficient of individual single nanoparticles.
1. Introduction
Light carries momentum, and therefore exerts a force on materials as they scatter or absorb it. This force is called the optical force or radiation pressure. With this principle, Ashkin et al., proposed a method to manipulate the motion of microparticles and atoms [1–4]. The technology using the method to trap microparticles is called “optical tweezer,” and is used in various disciplines, especially in biochemistry.
However, the manipulation of nanoparticles with optical force is considered difficult because the force exerted on the particles of the material is significantly weaker than that exerted on microparticles. Moreover, overcoming the complex interaction with solvent media is challenging. An example of the strategies used to address this problem is the application of a photoelectronic field with a steep gradient. A strongly localized and enhanced field appears around the sharp tip of metallic structures, or nanoscale gaps owing to surface plasmon resonance. Several studies have reported nanoparticle entrapment with this strategy [5–9]. Another unique strategy uses resonance with the transition energy between quantized electronic levels within nanostructures. Generally, the resonance effect as well as the optical force strengthens the optical response. Studies have also demonstrated the entrapment of single molecules aiming at the absorption lines of the molecules, and improved the trapping efficiency [10,11]. Another study [12] theoretically proposed the resonance enhancement of the optical force on semiconductor quantum dots and experimentally demonstrated it at cryogenic temperatures [13]. A fascinating aspect of the resonant optical response is that it not only enhances the force but also allows for selective manipulation according to the quantum mechanical properties of individual nanoparticles. For example, Ajiki et al. theoretically confirmed the selective transportation and trapping of single-walled carbon nanotubes (SWCNs) according to their diameter and chirality [14]. Spesyvtseva et al. experimentally demonstrated the selective transportation of SWCNs sustained in microcapillary, and observed the emanation of Raman signals from the selectively transported SWCNs with targeted chirality [15]. Further study for selective optical manipulation based on the electronic resonance in nanoparticles, combined with recently developed techniques of optical sorting [16–18], will open the scheme to investigate quantum optical properties of single nanoparticles. Recently, we had proposed a method to select nanoparticles in which the resonance peak of the optical spectra is cloaked by the background contributions of non-targeted particles [19]. In this method, counter-propagating laser waves cancelled the background contributions of the force and allowed the selective transportation of only those particles with the target resonance level. This proposal was experimentally demonstrated with a tapered nanofiber [20], where we transported nanodiamonds containing NV-centers and pristine diamonds in the opposite directions, respectively. A notable feature of this method is that it cancels the forces from the background contributions by the two light waves, facilitating motion control solely by the resonance contribution. With this feature, the precise information on the relation between the observed motion and the momentum transferred from the light to the nanoparticle can be derived. This enables the measurement of the absorption coefficient of single nanoparticles. Consequently, we obtained the absorption coefficient of single nanodiamonds containing NV-centers. Unlike the state-of-the-art techniques of single nanoparticle spectroscopy [21,22], this measurement technique does not involve dissipation processes such as fluorescence, scattering, or heating. The absorption coefficient can be evaluated with the motion caused purely by momentum transfer, because the momentum of light absorbed by the particle is independent of the mode of energy dissipation after the exertion of the optical force. We call this technique optical force spectroscopy (OFS) [20].
Nonlinear resonant optical manipulation is a significant direction of research on the resonant optical force. As the incident laser required for the optical manipulation of nanoparticles is generally strong, the nonlinear optical effect is readily observed under the resonant condition. For example, a research [23] revealed a peculiar trapping potential with the double minimum in the optical trapping of Au nanoparticles by a pulsed laser beam. Further, a few exotic effects in molecular entrapment were explained by considering the hypothesis of optical nonlinearity. Based on this hypothesis, studies have proposed unconventional techniques of optical manipulation achieved by the nonlinear effect [24,25]. Thus, the nonlinear effect has important applications in various optical manipulation techniques.
Therefore, the application of OFS for measuring the nonlinear optical coefficient of single nanoparticles must be examined. Several studies have reported the measurement of nonlinear optical coefficient of nanoparticles [26–37]. However, the measurement of the nonlinear optical coefficient of individual single nanoparticles is challenging. Especially, two-photon absorption processes involve states that cannot be reached by one-photon absorption because of the optical selection rule. To the best of the authors’ knowledge, no studies have measured the transition energy and relaxation constant of such dark states of individual single nanoparticles. As these parameters are included in the two-photon absorption term in the third-order susceptibility, such parameters can be determined if the two-photon absorption coefficient can be measured using the optical force. Such information is crucial not only from the spectroscopic viewpoint but also from the optical manipulation perspective, because the optical force is the phenomenon reflecting the optical properties of the individual single particles. Therefore, the method of employing OFS to measure the nonlinear coefficient is an objective of this study for the relevant fields treating single molecules and nanostructures.
Thus, in this study, we propose a technique to measure the two-photon absorption coefficient with OFS, assuming the measurement precision of particle motion by the optical force from our previous study [20]. The method is conducted with the help of OFS for linear response. By using the measured parameters related to the one-photon absorption process, we evaluated the parameters included in nonlinear susceptibility by fitting the energy spectrum of the particle transportation distance using the expression of the third-order nonlinear susceptibility. As model particles, we assumed a dye-embedded polymer. We calculated the particle’s optical force and obtained its transportation distance spectrum by kinetic simulation. As the transportation distance induced by the resonant optical force is proportional to the imaginary part of susceptibility, Im$\chi$, we can consider that the transportation distance spectrum of a particle is equivalent to the Im$\chi$ spectrum.
This study contributes a novel spectroscopy method using the optical force that can quantify the nonlinear optical constants of single nanoparticles.
2. Theory
This section summarizes the theoretical methods.
2.1 Optical force and evaluation of polarizations
We calculated the optical force using the following general expression of the time-averaged optical force [38]:
(1)$$\left \langle \boldsymbol{F} \right \rangle = \frac{1}{2} \mathrm{Re} \left[ \sum_{\omega} \int_{V} d\boldsymbol{r} (\nabla \boldsymbol{E}^{{\ast}}(\boldsymbol{r},\omega)) \cdot \boldsymbol{P}(\boldsymbol{r},\omega) \right],$$
where $\boldsymbol {E}$ and $\boldsymbol {P}$ are the time-harmonic electric field and induced polarization, $\omega$ is the angular frequency of incident light, and $\boldsymbol {r}$ is the particle position. The optical force is generally classified into gradient and dissipative forces. The latter includes the scattering and absorbing forces. The above expression includes all types of forces.In the present case, the effect of response electric field on the optical force is negligible because the dye molecules are sufficiently small [24]. Hence, we used the incident electric field $\boldsymbol {E}$ in Eq. (1). In contrast, we calculated the induced polarization $\boldsymbol {P}$ by solving the density matrix equation for the matter system including the phenomenological relaxation constants as follows:
(2)$$i\hbar \frac{\partial \rho}{ \partial t} = \left[ H,\rho \right] - i\hbar\Gamma \rho ,$$
where $\rho$, $H$ and $\Gamma$ denote the density matrix of the system, the total Hamiltonian to be represented as $H= H_0 + V$, and the phenomenological relaxation constants.Here, $H_0$ denotes the unperturbed component, and $V$ denotes the light–molecule interaction energy given by $V = - \boldsymbol {\mu } \cdot \boldsymbol {E}(\boldsymbol {r},t)$. $\boldsymbol {\mu }$ denotes the matrix element of the transition dipole moment, where we assume that the dye molecule is sufficiently smaller than the light wavelength and can be described by a three-level system, as described below. Therefore, $\rho$ and $\boldsymbol {\mu }$ can be written in a $3 \times 3$ matrix format. Additionally, assuming that the molecule is symmetric, the diagonal element of $\boldsymbol {\mu }$ representing the permanent dipole moment is zero.
We wrote the density matrix equation for the off-diagonal terms with the relaxation constant as follows:
(3)$$i\hbar \frac{\partial \rho_{\mathrm{mn}}}{ \partial t} = \left[ H,\rho \right]_{\mathrm{mn}} - i\gamma_{\mathrm{mn}} \rho_{\mathrm{mn}} ,$$
where $m$ and $n$ are integers corresponding to the respective energy level numbers, which in this study were either 1, 2, or 3.To calculate linear susceptibility $\chi ^{(1)}$, we substituted the result of the perturbation expansion of Eq. (3) into the following equation,
(4)$$\begin{aligned}\left \langle {P^{(1)}} \right \rangle &= \mathrm{Tr} \{ \rho^{(1)} \mu \}\\ &= \rho^{(1)}_{12} \mu_{21} + \rho^{(1)}_{13} \mu_{31} + \rho^{(1)}_{23} \mu_{32} + \mathrm{C.C.}, \end{aligned}$$
and we get (5)$$\begin{aligned} \chi^{(1)} = & \frac{\mu_{21}\mu_{12}}{\hbar} \left( \frac{1}{\Omega_{21} - \omega - i \gamma_{12}} + \frac{1}{\Omega_{21} + \omega + i \gamma_{12}} \right)\\ + & \frac{\mu_{31}\mu_{13}}{\hbar} \left( \frac{1}{\Omega_{31} - \omega - i \gamma_{13}} + \frac{1}{\Omega_{31} + \omega + i \gamma_{13}} \right). \end{aligned}$$
where $\Omega _{mn}$ denotes the energy difference of $m$–$n$. For the explicit relation between $\left \langle {P^{(1)}} \right \rangle$ and $\chi ^{(1)}$, see Eq. (56) and Eq. (57) in Appendix A. By using this expression for fitting in OFS, we determined $\mu _{12}$, $\Omega _{12}$, and $\gamma _{12}$. However, note that if the transition between $n$=1 to $n$=3 is optically forbidden (namely, $n$=3 state is the one-photon dark state), $\mu _{13}$ vanishes, and $\Omega _{13}$ and $\gamma _{13}$ cannot be determined by OFS.To calculate the third-order nonlinear susceptibility $\chi ^{(3)}$, we substituted the result of the perturbation expansion of Eq. (3) to the third order into the following equation,
(6)$$\begin{aligned}\left \langle {P^{(3)}} \right \rangle &= \mathrm{Tr} \{ \rho^{(3)} \mu \}\\ &= \rho^{(3)}_{\mathrm{12}} \mu_{\mathrm{21}} + \rho^{(3)}_{\mathrm{13}} \mu_{\mathrm{31}} + \rho^{(3)}_{\mathrm{23}} \mu_{\mathrm{32}} + \mathrm{C.C.}, \end{aligned}$$
and the result is represented as: (7)$$\begin{aligned} {\chi}^{(3)} = & \frac{\mu_{21}\mu_{12}\mu_{23}\mu_{32}}{\hbar^{3}} \left\{ \frac{1}{\Omega_{21} - \omega - i \gamma_{12}} \right\} \left\{ \frac{1}{\Omega_{31} - 2 \omega - i \gamma_{31}} \right\} \left\{ \frac{1}{\Omega_{21} - \omega - i \gamma_{12}} -\frac{1}{\Omega_{32} - \omega - i \gamma_{32}} \right\}\\ + & \frac{\mu_{21}\mu_{12}\mu_{23}\mu_{32}}{\hbar^{3}} \left\{ \frac{1}{\Omega_{21} + \omega + i \gamma_{12}} \right\} \left\{ \frac{1}{\Omega_{13} - 2 \omega - i \gamma_{13}} \right\} \left\{ \frac{1}{\Omega_{12} - \omega - i \gamma_{12}} -\frac{1}{\Omega_{23} - \omega - i \gamma_{23}} \right\}. \end{aligned}$$
For the explicit relation between $\left \langle {P^{(3)}} \right \rangle$ and $\chi ^{(3)}$, read Appendix A.4. Here, note that, by using this expression in the fitting in OFS for nonlinear optical process, $\mu _{23}$, $\Omega _{13}$, and $\gamma _{13}$ can be determined with the help of OFS for a linear response that provides information on $\mu _{12}$, $\Omega _{12}$, and $\gamma _{12}$. Namely, OFS for the nonlinear optical process can determine the transition energy and relaxation constant of the one-photon dark state of the single nanoparticle by using the optical force to move them.
As mentioned above, these susceptibility values were used for fitting the transported-distance spectrum. Derivations of these expressions are detailed in Appendix A. As the dissipative force that transports a particle in the direction of light is proportional to the imaginary part of susceptibility, the incident light energy spectrum of the particle’s transportation distance is equal to the energy spectrum of the optical force, and so is the energy spectrum of the imaginary part of susceptibility. Therefore, by fitting the transportation distance spectrum of the particle with the imaginary part of susceptibility, the optical constants related to the absorption of the particle can be evaluated as a fitting parameter [19].
Regarding the background polarization $\boldsymbol {P}_{\mathrm {b}}$ responsible for the scattering force, we used the Clausius–Mosotti equation including the radiation reaction effect as [39,40],
(8)$$\begin{array}{lll} \boldsymbol{P}_{\mathrm{b}}& =&a_{\mathrm{CMRR}}\boldsymbol{E}{(\boldsymbol{r},\omega)},\\ a_{\mathrm{CMRR}} &= &\frac{a_{\mathrm{CM}}}{1-iq^{3}\frac{a_{\mathrm{CM}}}{6\pi \epsilon_0 \epsilon_2}},\\ a_{\mathrm{CM}} &= &4\pi a^{3} \epsilon_0 \epsilon_2 \frac{m-1}{m+2}, \end{array}$$
where $\epsilon _0$ is the electric constant, $m=\epsilon _1/\epsilon _2$, $\epsilon _1$ and $\epsilon _2$ are the dielectric constants of the target substance and surroundings, respectively, $q$ is the wavenumber of incident light wave, and $a$ is the radius of target particle. We employ this formula for the scattering force by the polymer, and this provides a good approximation if the particle is not largely deformed from the perfect spherical shape.2.2 Incident field
We used the following relationship to evaluate the intensity of the incident field ${\boldsymbol E}_{0}$,:
(9)$$\begin{array}{lll}\boldsymbol{E} &=& \boldsymbol{E}_0 \exp \bigl[ i (\boldsymbol{q}\cdot \boldsymbol{r} - \omega t) \bigr]\\ |\boldsymbol{E}_0|^{2} &=& \frac{2I}{c \epsilon_2 \epsilon_0}, \end{array}$$
where $I$ is the intensity of the incident light wave, $c$ is the speed of light. In addition, the evanescent wave at the solid–liquid interface is given by (10)$$\boldsymbol{E}_t = \boldsymbol{A}_t \exp \left[ -\frac{\omega}{c} \sqrt {n_s^{2} \sin^{2} \theta_i - n_1^{2}} x + i \left(\frac{\omega}{c}n_s \sin \theta_i z - \omega t\right) \right].$$
where $\boldsymbol {A}_t$ is the transmitted electric field coefficient, $n_s$ and $n_1$ are the solid and liquid refractive indices, and $\theta _i$ is the incident angle. This equation is general and for any polarization. We used this equation to treat the evanescent wave for the incident light with TE and TM modes.2.3 Kinetic simulation
To investigate the feasibility of selective transportation using counter-propagating light waves, we performed a kinetic analysis with the Brownian motion in a solvent using the Langevin equation, [41]:
(11)$$m \frac{d^{2} \boldsymbol{r}(t)}{dt^{2}}= \boldsymbol{F}-g \frac{d\boldsymbol{r}(t)}{dt}+\boldsymbol{F}_{\mathrm{random}},$$
where ${\boldsymbol {r}}$ is the position of the target particle of mass $m$, $\boldsymbol {F}$ is the optical force, $g=6\pi {\eta }a$ ($\eta$: viscosity of media, $a$: radius of particle), and $\boldsymbol {F}_{\mathrm {random}}$ is a random force. We used $\eta =0.89 \times 10^{-3}$ as the experimental value of the viscosity of water as the surrounding medium ($\mathrm {T}=300 \mathrm {K}$) [42].3. Model and procedure
To measure the absorption coefficient, we used the fact that the absorption momentum transferred by the resonant object is proportional to the transportation distance. As detailed in Ref. [19], the transported distance is proportional to the light exposure time $t$, and the diffusion length by the Brownian motion is the square root of $t$. Thus, for this measurement, a long transportation distance is conducive to reducing the effect of fluctuations. However, even if the distance is small, where fluctuation by the Brownian motion has an effect on the measurement, averaging over several iterations can reduce it. This point will be discussed later by examining the results. The laser light should not exceed a certain intensity to avoid quenching of the dye. Therefore, we considered using low-intensity light with a longer exposure time. For the kinetic simulation, given the limitation of computational resources, we assumed a high-intensity laser light (not used in the actual experiment) to reduce the simulation time. However, we can easily deduce the result for the case of the weaker laser used in the experiment by converting the theoretical results. This is because a straightforward equation can evaluate the relationship between external force $F$ and particle transport distance $X_h$.
(12)$$X_{\mathrm{h}} = \frac{F}{6 \pi \eta a}t,$$
where $t$ is a transport time.3.1 Procedure of measurement
Here, we summarized the OFS procedure for measuring the nonlinear absorption coefficient.
• Step 1: First, we measured the linear absorption coefficient of a nanoparticle. One laser wavelength was set as the non-resonant one that induces the scattering force of the mother matrix material. Another laser was set as the resonant wavelength to aim at embedded resonant particles. Its intensity is controlled so that its scattering force is balanced with that by non-resonance light. (See, Ref. [
20], for example.) We should note that the intensity should be controlled to avoid quenching and heating of the dye in the case of one-photon absorption. When the energy of the resonant light was scanned, and the intensity of the non-resonant light was tuned to fully cancel the scattering force by the mother matrix material. The intensity required to cancel the scattering force for each energy level can be obtained by the Clausius–Mosotti Eq. (
8), or from the experimental result obtained using the empty mother matrix of the same size. Thus, the particle moves only by the absorption force induced by the resonant laser. Hence, the absorption line profile can be obtained from the transportation distance spectra. By fitting this line profile with the expression of linear susceptibility (Eq. (
5)), we determined the transition energy and the relaxation constant of the embedded resonant particles. By applying a similar method as in Ref. [
20] to stop the motion of the nanoparticles through force balance, we obtained the absolute value of the absorption coefficient that determines the dipole moment.
• Step 2: We measured the two-photon absorption coefficient. The resonant laser was tuned to the two-photon absorption line and scanned around the spectral peak. The intensity and energy of the non-resonant light were controlled so that the scattering force could be cancelled in the same way as with the measurement of one-photo absorption. Then, we obtained the transportation distance spectra under only the effect of the two-photon resonant contribution. As both the laser frequencies were one-photon non-resonant, we used sufficiently strong laser intensities to move the particle by two-photon absorption without the problem of quenching.
• Step 3: We fitted the parameters by using the expression of the third-order nonlinear susceptibility responsible for the two-photo absorption (Eq. (
7)). The two-photon absorption spectra obtained from the transportation distance spectra include the hem of the one-photon absorption. Thus, in the fitting process, this contribution was subtracted by linear interpolation. The third-order nonlinear susceptibility includes the parameters of transition energy and relaxation constant determined in Step 1. Thus, we used these parameters to determine the two-photon absorption coefficient, including the transition energy and relaxation constant of the one-photon dark state. If we wish to determine the absolute value of absorption coefficient as in Ref. [
20], the dipole moment determined in OFS for one-photon absorption can be used.
3.2 Model of nanoparticles and laser setup for simulation
To simulate the OFS calculation of the nonlinear absorption coefficient through the above procedure, we assumed the following model and laser setup: To calculate the optical force on the target particles, we set the parameters for the target particles and the solid–liquid interface where they were manipulated as follows: the target particles were assumed to be polystyrene nanoparticles as the matrix embedded with 5,000 dye molecules. Here, the radius of the polystyrene matrix particles was assumed to be 75 nm, and the dielectric constant was assumed to be $\epsilon _b = 2.5$. The solid–liquid interface introduced to limit the degrees of freedom of particle motion was assumed as illustrated in Fig. 1. The angle of incidence was set to $\theta _i$ = 1.2 rad > $\theta _c$ = 1.1 rad, and the refractive indices of the glass substrate and water to $1.5$ and $1.33$, respectively. Generally, dye molecules in a polymer have a large, inhomogeneous line width. Thus, in general spectroscopy, the observed spectra are fitted using the Voigt function, which is the convolution of the Gaussian and Lorentzian functions. This method is effective if the inhomogeneous line width is less than approximately 100 times the homogeneous line width. Generally, the inhomogeneous and homogeneous line widths of dye molecules in polymers obey this condition. The core objective of this simulation is to ensure the accuracy of parameters determining the homogeneous line. If this accuracy is ensured, the targeted parameters can be determined despite the presence of inhomogeneous broadening by fitting them with the Voigt function. As the fitting procedure is inessential for this demonstration, we performed the fitting assuming only the homogeneous line width. First, we prove the energy dependence of the optical force on the dye embedded in the target polymer nanoparticle for simple planewave irradiation as in Fig. 2. To highlight the two-photon absorption peak, we assumed high laser intensity, i.e., 50 MW/cm$^{2}$. The inset of Fig. 2 illustrates the assumed level scheme of the dye molecule we set for the calculation. The relaxation constants between each level were set to $\gamma _{31}$ = 2 meV, $\gamma _{21}$ = 20 meV and $\gamma _{32}$ = 20 meV. For this intensity, although the two-photon absorption peak is distinct, the one-photon absorption peak is saturated due to the nonlinear effect. Therefore, the laser must have a significantly lower intensity for the measurement of the linear absorption coefficient. Hence, for the kinetic simulation of one-photon absorption, we set wave1: 3.38 - 3.42 eV, 100 kW/cm$^{2}$ and wave2: 1.16 eV as the resonant incident light conditions. The intensity of wave2 was determined using the Clausius–Mosotti Eq. (8) to have the non-resonant contribution to fully cancel the scattering force, and it was adjusted to match the energy change of wave1. In the experiment, we limited the degrees of freedom of motion of the nanoparticles using micro or nanocapillaries, or tapered nanofiber as in Refs. [15,20,43]. In this simulation, we assumed a simple glass–water interface to reduce the computational load for the simulation in the case of multi-level molecules. Thus, to sufficiently capture the particles, we additionally assumed the TE-polarized, non-resonant counterparts wave1’: 1.21 eV, 116 MW/cm$^{2}$ and wave2’: 1.16 eV, 100 MW/cm$^{2}$. Because maintaining this intensity over a long distance would be difficult in the experiment, we used micro or nanocapillaries, or nanofibers, as mentioned above.
In contrast, the optical force due to the two-photon absorption is feeble. Therefore, light of a certain intensity is required to transport the particle. Therefore, we set wave1: 2.08 - 2.12 eV, 50 MW/cm$^{2}$ and wave2: 1.16 eV to simulate for two-photon absorption. (As mentioned above, the experiment will employ micro or nanocapillaries, or tapered nanofiber, to maintain this light intensity over a long distance.) Because the energy is one-photon off-resonance, we dismiss the probability of dye quenching in this case.
These settings of the incident counter-propagating waves enabled the extraction of the resonant component of the optical force and the kinetic analysis of particle transportation. Note that the optical force of two-photon absorption is affected by the hem of the broad spectrum of the one-photon-absorption optical force. Regarding the fitting of two-photon absorption, the contribution is removed by linear interpolation to eliminate the hem of the resonant optical force caused by one-photon absorption.
4. Results
We simulated the transport of nanoparticles by varying the incident light energy conditions. The particles were transported multiple times for each light energy, and the average amount of transported distance was calculated. Figure 3 illustrates the results of 50 iterations. The transportation time of each trial was 20 s for (a) one-photon absorption and (b) two-photon absorption. First, by fitting with the first-order linear susceptibility $\chi ^{(1)}$ (in Eq. (5)), as observed in Fig. 3(a)), we estimated $\omega _{21}$ = 3.400 eV and $\gamma _{21}$ = 20.91 meV as the fitting parameters. Next, using these values of $\omega _{21}$ and $\gamma _{21}$, we performed fitting, as observed in Fig. 3(b) with the third-order nonlinear susceptibility $\chi ^{(3)}$ (in Eq. (7)). Consequently, we obtained $\omega _{31}$ = 4.2 eV and $\gamma _{31}$ = 2.025 meV.
These values concurred with the dye parameters assumed for the calculations: $\omega _{21}$ =3.4 eV, $\omega _{31}$ = 4.2 eV, $\gamma _{21}$ = 20 meV and $\gamma _{31}$ =2 meV. This concurrence is because the spectra of the transportation distance corresponded with the shapes of the susceptibility curves, even though the simulations considered the noise of the Brownian motion (Fig. 3). This result indicates that the experiment can be used to determine the value of the optical coefficients with high accuracy.
Next, we examine the relation between accuracy and the number of times the particles are transported (number of iterations) and the simulation time (transportation time). These two types of relationships are illustrated in two different figures: Fig. 4 depicts the case where the transportation time is maintained at 20 s and the trial number is varied. In contrast, Fig. 5 illustrates the case where the trial number is maintained at 50 times and the transportation time is varied. As depicted in Fig. 4, the transportation distance spectrum becomes closer to the susceptibility spectrum by executing a sufficiently large number of trials. In contrast, the transportation time does not have a significant influence on accuracy, as observed in Fig. 5. The relative error in the estimation accuracy of each parameter for the change in accuracy corresponding to Fig. 4 and Fig. 7 for the result corresponding to Fig. 5 is summarized in Fig. 6.
These results indicate that averaging over a sufficient number of trials will ensure the accuracy of fit even if a relatively weak optical force was used for measuring the two-photon absorption coefficient, overcoming the Brownian motion effect. The parameters $\omega _{12}, \gamma _{12}$ determined by OFS for one-photon absorption are less accurate than $\omega _{13}, \gamma _{13}$ by OFS for two-photon absorption, because of the absorption saturation by optical nonlinearity for the assumed light intensity. A subtle change of the peak shape of one-photon absorption causes the estimated values of $\omega _{12}, \gamma _{12}$ to deviate by approximately 5%. Interestingly, this deviation did not affect the accuracy of estimation of the latter parameters because $\chi ^{(3)}$ is insensitive to $\omega _{12}, \gamma _{12}$ around the two-photon resonance region, as revealed by Eq. (7). Thus, the parameters of one-photon absorption used in OFS for two-photon absorption need not be highly accurate.
5. Summary and conclusion
We theoretically demonstrated the measurement of nonlinear susceptibility of single nanoparticles using OFS. We numerically simulated the transportation of dye-embedded nanoparticles on the solid–liquid interface, where the optical force from the absorption of the embedded dyes is extracted by canceling the scattering force from the mother matrix polymer using counter-propagating light waves. The resonance energy and relaxation constant of a single nanoparticle were accurately determined by fitting the transported distance spectra with the first- and third-order nonlinear susceptibilities. Further, the accuracy of measurement depends on the transportation time and the number of trials.
This method can measure the transition energy and relaxation constant of one-photon dark states of individual single nanoparticles, because the motion of the nanoparticles due to the optical force reflects the momentum exchange between individual single nanoparticles. The transportation distance spectrum is equivalent to the spectrum of the imaginary part of susceptibility. Hence, in OFS for two-photon absorption, the information of optical constants can be determined even for the optical forbidden state. Although various spectroscopic methods exist for nanosized materials in the region of nonlinear optical response, to the best of the authors’ knowledge, no study has provided a method to measure the optical constant of the dark state of individual single nanoparticles.
In the simulations, we assumed the solid–liquid interface reduced the computational load. However, in practical experiments, the degrees of freedom of motion of the nanoparticles must be constrained using micro or nanocapillaries or tapered nanofibers, as in Refs. [15,20,43]. These devices can also maintain a strong laser intensity over a long distance in OFS for two-photon absorption.
We performed a numerical simulation to demonstrate the concept of OFS of nonlinear optical response by assuming a simplified model and limiting conditions. However, future experimental studies must be based on the simulations employing more sophisticated models and methods.
A. Appendix
A.1. Solution of matter density matrix equations
We begin wtih the equation of motion of the density matrix:
(13)$$\frac{\partial \rho}{\partial t} ={-}\frac{i}{\hbar} \left[H, \rho \right] - \Gamma \rho.$$
We demonstrate methods to solve these equations using perturbation expansion and simultaneous equations assuming steady state. Assuming a three-level system, the density matrix $\rho$ can be written as follows,
(14)$$\begin{aligned}\rho = \begin{pmatrix} \rho_{11} & \rho_{12} & \rho_{13} \\ \rho_{21} & \rho_{22} & \rho_{23} \\ \rho_{31} & \rho_{32} & \rho_{33} \\ \end{pmatrix}. \end{aligned}$$
The dipole moment $\boldsymbol {\mu }$ is also written in a similar form. Consequently, the equation of motion of the density matrix for each matrix element can be written for diagonal and non-diagonal terms as,
(15)$$\frac{\partial \rho_{nm}}{\partial t} = ({-}i \Omega_{nm} - \gamma_{nm}) \rho_{nm} - \frac{i}{\hbar} \left[V, \rho \right]_{nm}\ (n \ne m)$$
(16)$$\frac{\partial \rho_{nn}}{\partial t} ={-} \frac{i}{\hbar} \left[V, \rho \right]_{nn} + \sum_{p>n} \Gamma_{pn}\rho_{pp} - \sum_{p<n} \Gamma_{np}\rho_{nn} \ (p \neq n \ p = 1,2,3),$$
where the integers
$l,m,n,p$ correspond to the number of levels,
$\Omega _{nm}$ is the energy difference of
$n-m$, and
$\Gamma _{nm}, \gamma _{nm}$ is the phase relaxation constants and population relaxation constants, respectively, between each level.
$V= -\boldsymbol {\mu } \cdot \boldsymbol {E}$ is handled specifically in the middle of the calculation.
Note that the diagonal term of the density matrix represents the occupancy of each level and satisfies
(17)$$\rho_{11} + \rho_{22} + \rho_{33} = 1 .$$
In the process of deriving the first- and third-order nonlinear susceptibility, the higher order diagonal terms of $\rho$ were discarded, and the diagonal term of $\boldsymbol {\mu }$ was assumed to be zero. Note that, if the matter system is geometrically asymmetric, the permanent dipole moment is involved, and hence the diagonal term of $\mu$ should be included in the calculation.
A.2. First-order perturbation expansion
Perturbation expansion for $\rho _{21}$ is written as follows,
(18)$$ \frac{\partial \rho^{(0)}_{21}}{\partial t} = ({-}i \Omega_{21} - \gamma_{21}) \rho^{(0)}_{21} $$
(19)$$ \frac{\partial \rho^{(1)}_{21}}{\partial t} = ({-}i \Omega_{21} - \gamma_{21}) \rho^{(1)}_{21} - \frac{i}{\hbar} \left[V, \rho^{(0)} \right]_{21} $$
(20)$$ \frac{\partial \rho^{(j)}_{21}}{\partial t} = ({-}i \Omega_{21} - \gamma_{21}) \rho^{(j)}_{21} - \frac{i}{\hbar} \left[V, \rho^{(j-1)} \right]_{21} $$
These equations sequentially represent zero-, first-, and j-order terms. To solve Eq. (20), we assumed that the solution of Eq. (19) is,
(21)$$\rho_{21}^{(1)}(t) = g_{21}^{(1)}(t) e^{({-}i \Omega_{21} - \gamma_{21})t},$$
and the first-order term can be expressed as follows,
(22)$$g_{21}^{(1)}(t) = \int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{21}e^{-({-}i\Omega_{21}-\gamma_{21})t^{^{\prime}}}dt^{^{\prime}}, $$
(23)$$\rho_{21}^{(1)}(t) = \int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{21}e^{(i\Omega_{21}+\gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}}. $$
The diagonal terms can be treated in the same way.
(24)$$\frac{\partial \rho_{11}^{(0)}}{\partial t} = \Gamma_{21}\rho_{22}^{(0)}+\Gamma_{31}\rho_{33}^{(0)} $$
(25)$$\frac{\partial \rho_{11}^{(j)}}{\partial t} = \Gamma_{21}\rho_{22}^{(j)}+\Gamma_{31}\rho_{33}^{(j)}-\frac{i}{\hbar}[{V,\rho^{(j-1)}}]_{11} $$
(26)$$\frac{\partial \rho_{22}^{(0)}}{\partial t} = \Gamma_{32}\rho_{33}^{(0)}-\Gamma_{21}\rho_{22}^{(0)} $$
(27)$$\frac{\partial \rho_{22}^{(j)}}{\partial t} = \Gamma_{32}\rho_{33}^{(j)}-\Gamma_{21}\rho_{22}^{(j)}-\frac{i}{\hbar}[{V,\rho^{(j-1)}}]_{22} $$
(28)$$\frac{\partial \rho_{33}^{(0)}}{\partial t} ={-}\Gamma_{31}\rho_{33}^{(0)}-\Gamma_{32}\rho_{33}^{(0)} $$
(29)$$\frac{\partial \rho_{33}^{(j)}}{\partial t} ={-}\Gamma_{31}\rho_{33}^{(j)}-\Gamma_{32}\rho_{33}^{(j)}-\frac{i}{\hbar}[{V,\rho^{(j-1)}}]_{33}. $$
For $\rho _{33}$, we assumed the solution of
(30)$$\frac{\partial \rho_{33}^{(1)}}{\partial t} ={-}\Gamma_{31}\rho_{33}^{(1)}-\Gamma_{32}\rho_{33}^{(1)}-\frac{i}{\hbar}[{V,\rho^{(0)}}]_{33}$$
where,
(31)$$\rho_{33}^{(1)} = g_{33}^{(1)}(t)e^{-(\Gamma_{31}+\Gamma_{32})t},$$
and
$\rho _{33}$ is calculated as follows,
(32)$$g_{33}^{(1)}(t) = \int_{-\infty}^{t}-\frac{i}{\hbar}[{V,\rho^{(0)}}]_{33}e^{(\Gamma_{31}+\Gamma_{32})t^{^{\prime}}}dt^{^{\prime}}, $$
(33)$$\rho_{33}^{(1)}(t) = \int_{-\infty}^{t}-\frac{i}{\hbar}[{V,\rho^{(0)}}]_{33}e^{(\Gamma_{31}+\Gamma_{32})(t^{^{\prime}}-t)}dt^{^{\prime}}. $$
The above equations can be summarized as follows,
(34)$$ \rho_{21}^{(1)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{21}e^{(i\Omega_{21}+\gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(35)$$\rho_{31}^{(1)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{31}e^{(i\Omega_{31}+\gamma_{31})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(36)$$\rho_{32}^{(1)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{32}e^{(i\Omega_{32}+\gamma_{32})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(37)$$\rho_{12}^{(1)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{12}e^{(i\Omega_{12}+\gamma_{12})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(38)$$\rho_{13}^{(1)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{13}e^{(i\Omega_{13}+\gamma_{13})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(39)$$\rho_{23}^{(1)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(0)}}]_{23}e^{(i\Omega_{23}+\gamma_{23})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(40)$$\rho_{33}^{(1)}(t) = \int_{-\infty}^{t}-\frac{i}{\hbar}[{V,\rho^{(0)}}]_{33}e^{(\Gamma_{31}+\Gamma_{32})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(41)$$\rho_{22}^{(1)}(t) = \int_{-\infty}^{t}\left[-\frac{i}{\hbar}[{V,\rho^{(0)}}]_{22}+\Gamma_{32}\rho_{33}^{(1)}\right]e^{(\Gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(42)$$\rho_{11}^{(1)}(t) = \int_{-\infty}^{t}\left[-\frac{i}{\hbar}[{V,\rho^{(0)}}]_{11}+\Gamma_{31}\rho_{33}^{(1)}+\Gamma_{21}\rho_{22}^{(1)}\right]dt^{^{\prime}}, $$
For $\rho _{21}^{(1)}$, we integrated Eq. (34) as,
(43)$$\begin{array}{ll}[{V,\rho^{(0)}}]_{21} &=\sum_{n}^{3} \left[ V_{2n}\rho_{n1}^{(0)} - \rho_{2n}^{(0)}V_{n1} \right]\\&= V_{21}\rho_{11}^{(0)} + V_{22}\rho_{21}^{(0)} + V_{23}\rho_{31}^{(0)} - \rho_{21}^{(0)}V_{11} - \rho_{22}^{(0)}V_{21} - \rho_{23}^{(0)}V_{31} \\&= V_{21}.\end{array}$$
Here, the initial condition was set to $\rho _{11}^{(0)}=1, \rho _{12}^{(0)}=\rho _{23}^{(0)}=\rho _{13}^{(0)}=\rho _{22}^{(0)}=\rho _{33}^{(0)}=0$. By substituting
(44)$$\rho_{21}^{(1)}(t) = \int_{-\infty}^{t} -\frac{i}{\hbar}V_{21}e^{(i\Omega_{21}+\gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(45)$$V_{21} ={-}\mu_{21}\left[E_{0}(\omega)e^{{-}i\omega t} + E_{0}(-\omega)e^{i\omega t} \right], $$
into Eq. (
34), we obtained
(46)$$\begin{aligned}\rho_{21}^{(1)}(t)&=\int_{-\infty}^{t} -\frac{i}{\hbar}\left\{ -\mu_{21}\left[E_{0}(\omega)e^{{-}i\omega t} + E_{0}(-\omega)e^{i\omega t} \right]\right\}e^{(i\Omega_{21}+\gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}}\\ &=\mu_{21}\frac{i}{\hbar}\int_{-\infty}^{t} \left[E_{0}(\omega)e^{{-}i\omega t} + E_{0}(-\omega)e^{i\omega t} \right]e^{i(\Omega_{21}-i\gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}}\\ &=\mu_{21}\frac{i}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{i(\Omega_{21}-\omega-i\gamma_{21})} + \frac{E_{0}(-\omega)e^{i\omega t}}{i(\Omega_{21}+\omega-i\gamma_{21})} \right]\\ &=\frac{\mu_{21}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{(\Omega_{21}-\omega-i\gamma_{21})} + \frac{E_{0}(-\omega)e^{i\omega t}}{(\Omega_{21}+\omega-i\gamma_{21})} \right]. \end{aligned}$$
Similarly, solving for the other components gives
(47)$$\rho_{21}^{(1)}(t)=\frac{\mu_{21}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{21}-\omega-i\gamma_{21}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{21}+\omega-i\gamma_{21}} \right], $$
(48)$$\rho_{12}^{(1)}(t)=\frac{\mu_{12}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{21}+\omega+i\gamma_{12}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{21}-\omega+i\gamma_{12}} \right], $$
(49)$$\rho_{31}^{(1)}(t)=\frac{\mu_{31}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{31}-\omega-i\gamma_{31}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{31}+\omega-i\gamma_{31}} \right], $$
(50)$$\rho_{13}^{(1)}(t)=\frac{\mu_{13}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{31}+\omega+i\gamma_{13}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{31}-\omega+i\gamma_{13}} \right], $$
(51)$$\rho_{32}^{(1)}(t)=0, $$
(52)$$\rho_{23}^{(1)}(t)=0, $$
(53)$$\rho_{11}^{(1)}(t)=0, $$
(54)$$\rho_{22}^{(1)}(t)=0, $$
(55)$$\rho_{33}^{(1)}(t)=0. $$
Here, the diagonal terms of the first or higher orders are assumed to be zero. Finally, the expected value of polarization obtained by the first-order perturbation expansion can be calculated as follows,
(56)$$\begin{aligned} \langle P^{(1)} \rangle &=\mathrm{Tr}\{\rho \mu\}\\ &=\rho_{21}^{(1)}\mu_{12} +\rho_{31}^{(1)}\mu_{13} +\mathrm{C.C.}\\ &=\frac{\mu_{21}\mu_{12}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{21}-\omega-i\gamma_{21}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{21}+\omega-i\gamma_{21}} \right]\\ &+\frac{\mu_{31}\mu_{13}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{31}-\omega-i\gamma_{31}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{31}+\omega-i\gamma_{31}} \right]\\ &+ \mathrm{C.C.}\ . \end{aligned}$$
With this expression, we wrote the susceptibility $\chi ^{(1)}(\omega )$ in the following form,
(57)$$\chi^{(1)}(\omega) = \frac{\mu_{21}\mu_{12}}{\hbar} \frac{1}{\Omega_{21}-\omega-i\gamma_{21}} + \frac{\mu_{31}\mu_{13}}{\hbar} \frac{1}{\Omega_{31}-\omega-i\gamma_{31}} + \mathrm{C.C.}\ .$$
A.3. Second-order perturbation expansion
With the results of the first-order perturbation expansion, we followed the same steps to perform the second-order perturbation expansion.
(58)$$ \rho_{21}^{(2)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(1)}}]_{21}e^{(i\Omega_{21}+\gamma_{21})(t^{^{\prime}}-t)}dt^{^{\prime}},$$
(59)$$\rho_{31}^{(2)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(1)}}]_{31}e^{(i\Omega_{31}+\gamma_{31})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(60)$$\rho_{32}^{(2)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(1)}}]_{32}e^{(i\Omega_{32}+\gamma_{32})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(61)$$\rho_{12}^{(2)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(1)}}]_{12}e^{(i\Omega_{12}+\gamma_{12})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(62)$$\rho_{13}^{(2)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(1)}}]_{13}e^{(i\Omega_{13}+\gamma_{13})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(63)$$\rho_{23}^{(2)}(t)=\int_{-\infty}^{t} -\frac{i}{\hbar}[ {V,\rho^{(1)}}]_{23}e^{(i\Omega_{23}+\gamma_{23})(t^{^{\prime}}-t)}dt^{^{\prime}}, $$
(64)$$\rho_{33}^{(2)}(t) = 0 $$
(65)$$\rho_{22}^{(2)}(t) = 0 $$
(66)$$\rho_{11}^{(2)}(t) = 0, $$
For $\rho _{21}^{(2)}$, $[ {V,\rho ^{(1)}}]_{21}$ is calculated as,
(67)$$\begin{array}{ll}[{V,\rho^{(1)}}]_{21} &=\sum_{n}^{3} \left[ V_{2n}\rho_{n1}^{(1)} - \rho_{2n}^{(1)}V_{n1} \right]\\ &= V_{21}\rho_{11}^{(1)} + V_{22}\rho_{21}^{(1)} + V_{23}\rho_{31}^{(1)} - \rho_{21}^{(1)}V_{11} - \rho_{22}^{(1)}V_{21} - \rho_{23}^{(1)}V_{31}\\ &= V_{23}\rho_{31}^{(1)}. \end{array}$$
By substituting
\begin{align*} \rho_{31}^{(1)}(t)&=\frac{\mu_{31}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{31}-\omega-i\gamma_{31}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{31}+\omega-i\gamma_{31}} \right],\\ V_{23} &={-}\mu_{23}\left[E_{0}(\omega^{\prime})e^{{-}i\omega^{\prime} t} + E_{0}(-\omega^{\prime})e^{i\omega^{\prime} t} \right], \end{align*}
into Eq. (
58), we obtained
(68)$$[{V,\rho^{(1)}}]_{21}={-}\mu_{23}\left[E_{0}(\omega^{\prime})e^{{-}i\omega^{\prime} t} + E_{0}(-\omega^{\prime})e^{i\omega^{\prime} t} \right] \frac{\mu_{31}}{\hbar}\left[ \frac{E_{0}(\omega)e^{{-}i\omega t}}{\Omega_{31}-\omega-i\gamma_{31}} + \frac{E_{0}(-\omega)e^{i\omega t}}{\Omega_{31}+\omega-i\gamma_{31}} \right].$$
Thus, we have
(69)$$\begin{aligned}\rho_{21}^{(2)}(t)&=\frac{\mu_{23}\mu_{31}}{\hbar^{2}} \left[ \frac{E_{0}(\omega^{\prime}) E_{0}( \omega)e^{{-}i(\omega^{\prime} + \omega)t}}{\Omega_{21}-(\omega^{\prime}+\omega)-i\gamma_{21}}\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\right.\\& +\frac{E_{0}(-\omega^{\prime}) E_{0}( \omega)e^{ i(\omega^{\prime} - \omega) t}}{\Omega_{21}+(\omega^{\prime}-\omega)-i\gamma_{21}}\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\& +\frac{E_{0}( \omega^{\prime}) E_{0}(-\omega)e^{{-}i(\omega^{\prime} - \omega) t}}{\Omega_{21}-(\omega^{\prime}-\omega)-i\gamma_{21}}\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\left.+\frac{E_{0}(-\omega^{\prime}) E_{0}(-\omega)e^{ i(\omega^{\prime} + \omega) t}}{\Omega_{21}+(\omega^{\prime}+\omega)-i\gamma_{21}}\frac{1}{\Omega_{31}+\omega-i\gamma_{31}} \right].\end{aligned}$$
Similarly, for $\rho _{31}^{(2)}$,
(70)$$\begin{array}{ll}[{V,\rho^{(1)}}]_{31} &= V_{31}\rho_{11}^{(1)} + V_{32}\rho_{21}^{(1)} + V_{33}\rho_{31}^{(1)} -\rho_{31}^{(1)}V_{11} - \rho_{32}^{(1)}V_{21} - \rho_{33}^{(1)}V_{31}\\ &= V_{32}\rho_{21}^{(1)}, \end{array}$$
Thus,
(71)$$\begin{aligned}\rho_{31}^{(2)}(t) &=\frac{\mu_{32}\mu_{21}}{\hbar^{2}} \left[ \frac{E_{0}( \omega^{\prime}) E_{0}( \omega)e^{{-}i(\omega^{\prime} + \omega) t}}{\Omega_{31}-(\omega^{\prime}+\omega)-i\gamma_{31}}\frac{1}{\Omega_{21}-\omega-i\gamma_{21}}\right.\\& +\frac{E_{0}(-\omega^{\prime}) E_{0}( \omega)e^{ i(\omega^{\prime} - \omega) t}}{\Omega_{31}+(\omega^{\prime}-\omega)-i\gamma_{31}}\frac{1}{\Omega_{21}-\omega-i\gamma_{21}}\\ &+\frac{E_{0}( \omega^{\prime}) E_{0}(-\omega)e^{{-}i(\omega^{\prime} - \omega) t}}{\Omega_{31}-(\omega^{\prime}-\omega)-i\gamma_{31}}\frac{1}{\Omega_{21}+\omega-i\gamma_{21}}\\&\quad\left. +\frac{E_{0}(-\omega^{\prime}) E_{0}(-\omega)e^{ i(\omega^{\prime} + \omega) t}}{\Omega_{31}+(\omega^{\prime}+\omega)-i\gamma_{31}}\frac{1}{\Omega_{21}+\omega-i\gamma_{21}} \right].\end{aligned}$$
For $\rho _{32}^{(2)}$,
(72)$$\begin{array}{ll}[{V,\rho^{(1)}}]_{32} &= V_{31}\rho_{12}^{(1)} + V_{32}\rho_{22}^{(1)} + V_{33}\rho_{32}^{(1)} -\rho_{31}^{(1)}V_{12} - \rho_{32}^{(1)}V_{22} - \rho_{33}^{(1)}V_{32}\\ &= V_{31}\rho_{12}^{(1)} - \rho_{31}^{(1)}V_{12}, \end{array}$$
(73)$$\begin{aligned}\rho_{32}^{(2)}(t) &=\frac{\mu_{31}\mu_{12}}{\hbar^{2}} \left[ \frac{E_{0}( \omega^{\prime}) E_{0}( \omega)e^{{-}i(\omega^{\prime} + \omega) t}}{\Omega_{32}-(\omega^{\prime}+\omega)-i\gamma_{32}}\{\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime}) E_{0}( \omega)e^{ i(\omega^{\prime} - \omega) t}}{\Omega_{32}+(\omega^{\prime}-\omega)-i\gamma_{32}}\{\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\}\\&\quad +\frac{E_{0}( \omega^{\prime}) E_{0}(-\omega)e^{{-}i(\omega^{\prime} - \omega) t}}{\Omega_{32}-(\omega^{\prime}-\omega)-i\gamma_{32}}\{\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\}\\&\quad\left. +\frac{E_{0}(-\omega^{\prime}) E_{0}(-\omega)e^{ i(\omega^{\prime} + \omega) t}}{\Omega_{32}+(\omega^{\prime}+\omega)-i\gamma_{32}}\{\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\} \right].\end{aligned}$$
For $\rho _{12}^{(2)}$,
(74)$$\begin{array}{ll}[{V,\rho^{(1)}}]_{12} &= V_{11}\rho_{12}^{(1)} + V_{12}\rho_{22}^{(1)} + V_{13}\rho_{32}^{(1)} -\rho_{11}^{(1)}V_{12} - \rho_{12}^{(1)}V_{22} - \rho_{13}^{(1)}V_{32}\\ &={-} \rho_{13}^{(1)}V_{32}, \end{array}$$
(75)$$\begin{aligned}\rho_{12}^{(2)}(t) &={-}\frac{\mu_{32}\mu_{13}}{\hbar^{2}} \left[ \frac{E_{0}( \omega^{\prime}) E_{0}( \omega)e^{{-}i(\omega^{\prime} + \omega) t}}{\Omega_{12}-(\omega^{\prime}+\omega)-i\gamma_{12}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime}) E_{0}( \omega)e^{ i(\omega^{\prime} - \omega) t}}{\Omega_{12}+(\omega^{\prime}-\omega)-i\gamma_{12}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime}) E_{0}(-\omega)e^{{-}i(\omega^{\prime} - \omega) t}}{\Omega_{12}-(\omega^{\prime}-\omega)-i\gamma_{12}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad\left. +\frac{E_{0}(-\omega^{\prime}) E_{0}(-\omega)e^{ i(\omega^{\prime} + \omega) t}}{\Omega_{12}+(\omega^{\prime}+\omega)-i\gamma_{12}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}} \right].\end{aligned}$$
For $\rho _{13}^{(2)}$,
(76)$$\begin{array}{ll}[{V,\rho^{(1)}}]_{13} &= V_{11}\rho_{13}^{(1)} + V_{13}\rho_{23}^{(1)} + V_{13}\rho_{33}^{(1)} -\rho_{11}^{(1)}V_{13} - \rho_{12}^{(1)}V_{23} - \rho_{13}^{(1)}V_{33}\\ &={-} \rho_{12}^{(1)}V_{23}, \end{array}$$
(77)$$\begin{aligned}\rho_{13}^{(2)}(t) &={-}\frac{\mu_{23}\mu_{12}}{\hbar^{2}} \left[ \frac{E_{0}( \omega^{\prime}) E_{0}( \omega)e^{{-}i(\omega^{\prime} + \omega) t}}{\Omega_{13}-(\omega^{\prime}+\omega)-i\gamma_{13}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime}) E_{0}( \omega)e^{ i(\omega^{\prime} - \omega) t}}{\Omega_{13}+(\omega^{\prime}-\omega)-i\gamma_{13}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime}) E_{0}(-\omega)e^{{-}i(\omega^{\prime} - \omega) t}}{\Omega_{13}-(\omega^{\prime}-\omega)-i\gamma_{13}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad\left. +\frac{E_{0}(-\omega^{\prime}) E_{0}(-\omega)e^{ i(\omega^{\prime} + \omega) t}}{\Omega_{13}+(\omega^{\prime}+\omega)-i\gamma_{13}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}} \right].\end{aligned}$$
For $\rho _{23}^{(2)}$,
(78)$$\begin{array}{ll}[{V,\rho^{(1)}}]_{23} &= V_{21}\rho_{13}^{(1)} + V_{22}\rho_{23}^{(1)} + V_{23}\rho_{33}^{(1)} -\rho_{21}^{(1)}V_{13} - \rho_{22}^{(1)}V_{23} - \rho_{23}^{(1)}V_{33}\\ &= V_{21}\rho_{13}^{(1)} - \rho_{21}^{(1)}V_{13}, \end{array}$$
(79)$$\begin{aligned}\rho_{23}^{(2)}(t) &={-}\frac{\mu_{21}\mu_{13}}{\hbar^{2}} \left[ \frac{E_{0}( \omega^{\prime}) E_{0}( \omega)e^{{-}i(\omega^{\prime} + \omega) t}}{\Omega_{23}-(\omega^{\prime}+\omega)-i\gamma_{23}}\{\frac{1}{\Omega_{12}-\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime}) E_{0}( \omega)e^{ i(\omega^{\prime} - \omega) t}}{\Omega_{23}+(\omega^{\prime}-\omega)-i\gamma_{23}}\{\frac{1}{\Omega_{12}-\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\}\\&\quad +\frac{E_{0}( \omega^{\prime}) E_{0}(-\omega)e^{{-}i(\omega^{\prime} - \omega) t}}{\Omega_{23}-(\omega^{\prime}-\omega)-i\gamma_{23}}\{\frac{1}{\Omega_{12}+\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\}\\&\quad\left. +\frac{E_{0}(-\omega^{\prime}) E_{0}(-\omega)e^{ i(\omega^{\prime} + \omega) t}}{\Omega_{23}+(\omega^{\prime}+\omega)-i\gamma_{23}}\{\frac{1}{\Omega_{12}+\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\} \right].\end{aligned}$$
We used these results to perform the third-order perturbation expansion.
A.4. Third-order perturbation expansion
We derived the third-order nonlinear susceptibility from the second-order perturbation expansion. The intermediate equations are omitted because they are the same as those for the first and second orders.
For $\rho _{21}^{(3)}$, we calculated $[ {V,\rho ^{(2)}}]_{21}$ by
(80)$$\begin{aligned}[{V,\rho^{(2)}}]_{21} &= V_{21}\rho_{11}^{(2)} + V_{22}\rho_{21}^{(2)} + V_{23}\rho_{31}^{(2)} - \rho_{21}^{(2)}V_{11} - \rho_{22}^{(2)}V_{21} - \rho_{23}^{(2)}V_{31}\\ &= V_{23}\rho_{31}^{(2)} - \rho_{23}^{(2)}V_{31}, \end{aligned}$$
where,
$$V_{31} ={-}\mu_{31}\left[E_{0}(\omega^{\prime\prime})e^{{-}i\omega^{\prime\prime} t} + E_{0}(-\omega^{\prime\prime})e^{i\omega^{\prime\prime} t} \right],$$
By substituting the second-order perturbation result, we obtained,
(81)$$\begin{aligned}\rho_{21}^{(3)}(t) &=\frac{\mu_{23}\mu_{32}\mu_{21}}{\hbar^{3}} \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{31}-(\omega^{\prime}+\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\right.\\& \quad+\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{31}-(\omega^{\prime}+\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\\& \quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{31}+(\omega^{\prime}-\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\\& \quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{31}+(\omega^{\prime}-\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\\& \quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{31}-(\omega^{\prime}-\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}+\omega-i\gamma_{12}}\\& \quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{31}-(\omega^{\prime}-\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}+\omega-i\gamma_{12}}\\&\quad+\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{31}+(\omega^{\prime}+\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}+\omega-i\gamma_{12}}\\&\quad\left. +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{31}+(\omega^{\prime}+\omega)-i\gamma_{31}\}}\frac{1}{\Omega_{21}+\omega-i\gamma_{12}} \right]\\& -\frac{\mu_{31}\mu_{21}\mu_{13}}{\hbar^{3}}\\& \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{23}-(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}-\omega-i\gamma_{21}} )\right.\\&\quad+\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{23}-(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}-\omega-i\gamma_{21}} )\\&\quad+\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{23}+(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}-\omega-i\gamma_{21}} )\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{23}+(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}-\omega-i\gamma_{21}} )\\& \quad+\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{23}-(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}+\omega-i\gamma_{21}} )\\& \quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{23}-(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}+\omega-i\gamma_{21}} )\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{21}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{21}\}\{\Omega_{23}+(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}+\omega-i\gamma_{21}} )\\& \quad\left. +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{21}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{21}\}\{\Omega_{23}+(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}-\frac{1}{\Omega_{21}+\omega-i\gamma_{21}} ) \right],\end{aligned}$$
where we used
(82)$$E_{0}( \omega^{\prime\prime}) E_{0}( \omega^{\prime}) E_{0}( \omega) = E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega).$$
For $\rho _{31}^{(3)}$,
(83)$$\begin{array}{ll}[{V,\rho^{(2)}}]_{31} &= V_{31}\rho_{11}^{(2)} + V_{32}\rho_{21}^{(2)} + V_{33}\rho_{31}^{(2)} -\rho_{31}^{(2)}V_{11} - \rho_{32}^{(2)}V_{21} - \rho_{33}^{(2)}V_{31}\\ &= V_{32}\rho_{21}^{(2)} - \rho_{32}^{(2)}V_{21} \end{array}$$
(84)$$\begin{aligned}\rho_{31}^{(3)}(t) &=\frac{\mu_{32}\mu_{23}\mu_{31}}{\hbar^{3}} \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{21}-(\omega^{\prime}+\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{21}-(\omega^{\prime}+\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{21}+(\omega^{\prime}-\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\&\quad+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{21}+(\omega^{\prime}-\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\&\quad+\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{21}-(\omega^{\prime}-\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{21}-(\omega^{\prime}-\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{21}+(\omega^{\prime}+\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\quad\left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{21}+(\omega^{\prime}+\omega)-i\gamma_{21}\}}\frac{1}{\Omega_{31}+\omega-i\gamma_{31}} \right]\\&-\frac{\mu_{31}\mu_{12}\mu_{21}}{\hbar^{3}}\\& \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{32}-(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}} )\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{32}-(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}} )\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{32}+(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}} )\\&\quad+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{32}+(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}} )\\&\quad+\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{32}-(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}} )\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{32}-(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}} )\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{31}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{31}\}\{\Omega_{32}+(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}} )\\&\quad\left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{31}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{31}\}\{\Omega_{32}+(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}} ) \right]\end{aligned}$$
For $\rho _{32}^{(3)}$,
(85)$$\begin{array}{ll}[{V,\rho^{(2)}}]_{32} &= V_{31}\rho_{12}^{(2)} + V_{32}\rho_{22}^{(2)} + V_{33}\rho_{32}^{(2)} -\rho_{31}^{(2)}V_{12} - \rho_{32}^{(2)}V_{22} - \rho_{33}^{(2)}V_{32}\\ &= V_{31}\rho_{12}^{(2)} - \rho_{31}^{(2)}V_{12} \end{array}$$
(86)$$\begin{aligned}\rho_{32}^{(3)}(t) &={-}\frac{\mu_{31}\mu_{13}\mu_{32}}{\hbar^{3}} \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{12}-(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\right.\\&\quad+\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{12}-(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{12}+(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{12}+(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{12}-(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{12}-(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{12}+(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{12}+(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}} \right]\\& -\frac{\mu_{31}\mu_{12}\mu_{21}}{\hbar^{3}}\\& \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{31}-(\omega^{\prime}+\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{31}-(\omega^{\prime}+\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{31}+(\omega^{\prime}-\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{31}+(\omega^{\prime}-\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}-\omega-i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{31}-(\omega^{\prime}-\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}+\omega-i\gamma_{12}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{31}-(\omega^{\prime}-\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}+\omega-i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{32}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{32}\}\{\Omega_{31}+(\omega^{\prime}+\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}+\omega-i\gamma_{12}}\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{32}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{32}\}\{\Omega_{31}+(\omega^{\prime}+\omega)-i\gamma_{31}\}} \frac{1}{\Omega_{21}+\omega-i\gamma_{12}} \right]\end{aligned}$$
For $\rho _{12}^{(3)}$,
(87)$$\begin{array}{ll}[{V,\rho^{(2)}}]_{12} &= V_{11}\rho_{12}^{(2)} + V_{12}\rho_{22}^{(2)} + V_{13}\rho_{32}^{(2)} -\rho_{11}^{(2)}V_{12} - \rho_{12}^{(2)}V_{22} - \rho_{13}^{(2)}V_{32}\\ &= V_{13}\rho_{32}^{(2)} - \rho_{13}^{(2)}V_{32}\end{array}$$
(88)$$\begin{aligned}\rho_{12}^{(3)}(t) &=\frac{\mu_{12}\mu_{23}\mu_{32}}{\hbar^{3}} \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{13}-(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{13}-(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{13}+(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{13}+(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{13}-(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{13}-(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{13}+(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{13}+(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}} \right]\\&+\frac{\mu_{13}\mu_{31}\mu_{12}}{\hbar^{3}}\\& \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{32}-(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}})\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{32}-(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}})\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{32}+(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}})\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{32}+(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}-\omega-i\gamma_{31}})\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{32}-(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}})\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{32}-(\omega^{\prime}-\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}})\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{12}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{12}\}\{\Omega_{32}+(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}})\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{12}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{12}\}\{\Omega_{32}+(\omega^{\prime}+\omega)-i\gamma_{32}\}} (\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}-\frac{1}{\Omega_{31}+\omega-i\gamma_{31}}) \right] \end{aligned}$$
For $\rho _{13}^{(3)}$,
(89)$$\begin{array}{ll}[{V,\rho^{(2)}}]_{13} &= V_{11}\rho_{13}^{(2)} + V_{13}\rho_{23}^{(2)} + V_{13}\rho_{33}^{(2)} -\rho_{11}^{(2)}V_{13} - \rho_{12}^{(2)}V_{23} - \rho_{13}^{(2)}V_{33}\\ &= V_{12}\rho_{23}^{(2)} - \rho_{12}^{(2)}V_{23} \end{array}$$
(90)$$\begin{aligned}\rho_{13}^{(3)}(t) &=\frac{\mu_{13}\mu_{32}\mu_{23}}{\hbar^{3}} \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{12}-(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{12}-(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{12}+(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{12}+(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}+\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{12}-(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{12}-(\omega^{\prime}-\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{12}+(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{12}+(\omega^{\prime}+\omega)-i\gamma_{12}\}}\frac{1}{\Omega_{31}-\omega+i\gamma_{13}} \right]\\& -\frac{\mu_{12}\mu_{21}\mu_{13}}{\hbar^{3}}\\& \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{23}-(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}-\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}+\omega+i\gamma_{13}})\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{23}-(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}-\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}+\omega+i\gamma_{13}})\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{23}+(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}-\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}+\omega+i\gamma_{13}})\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{23}+(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}-\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}+\omega+i\gamma_{13}})\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{23}-(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}+\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}-\omega+i\gamma_{13}})\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{23}-(\omega^{\prime}-\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}+\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}-\omega+i\gamma_{13}})\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{13}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{13}\}\{\Omega_{23}+(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}+\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}-\omega+i\gamma_{13}})\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{13}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{13}\}\{\Omega_{23}+(\omega^{\prime}+\omega)-i\gamma_{23}\}} (\frac{1}{\Omega_{12}+\omega-i\gamma_{21}}-\frac{1}{\Omega_{31}-\omega+i\gamma_{13}}) \right]\end{aligned}$$
For $\rho _{23}^{(3)}$,
(91)$$\begin{array}{ll}[{V,\rho^{(2)}}]_{23} &= V_{21}\rho_{13}^{(2)} + V_{22}\rho_{23}^{(2)} + V_{23}\rho_{33}^{(2)} -\rho_{21}^{(2)}V_{13} - \rho_{22}^{(2)}V_{23} - \rho_{23}^{(2)}V_{33}\\ &= V_{21}\rho_{13}^{(2)} - \rho_{21}^{(2)}V_{13} \end{array}$$
(92)$$\begin{aligned}\rho_{23}^{(3)}(t) &={-}\frac{\mu_{21}\mu_{12}\mu_{23}}{\hbar^{3}} \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{13}-(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{13}-(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{13}+(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{13}+(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}+\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{13}-(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{13}-(\omega^{\prime}-\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{13}+(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}}\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{13}+(\omega^{\prime}+\omega)-i\gamma_{13}\}}\frac{1}{\Omega_{21}-\omega+i\gamma_{12}} \right]\\&\quad +\frac{\mu_{23}\mu_{31}\mu_{13}}{\hbar^{3}}\\& \left[ \frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{21}-(\omega^{\prime}+\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\right.\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{21}-(\omega^{\prime}+\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime}, \omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{21}+(\omega^{\prime}-\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime}, \omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{21}+(\omega^{\prime}-\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}-\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime}, \omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}+\omega^{\prime}-\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}+\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{21}-(\omega^{\prime}-\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}(-\omega^{\prime\prime}, \omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}-\omega^{\prime}+\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}-\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{21}-(\omega^{\prime}-\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\quad +\frac{E_{0}( \omega^{\prime\prime},-\omega^{\prime},-\omega)e^{{-}i(\omega^{\prime\prime}-\omega^{\prime}-\omega) t}}{\{\Omega_{23}-(\omega^{\prime\prime}-\omega^{\prime}-\omega)-i\gamma_{23}\}\{\Omega_{21}+(\omega^{\prime}+\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}+\omega-i\gamma_{31}}\\&\quad \left.+\frac{E_{0}(-\omega^{\prime\prime},-\omega^{\prime},-\omega)e^{ i(\omega^{\prime\prime}+\omega^{\prime}+\omega) t}}{\{\Omega_{23}+(\omega^{\prime\prime}+\omega^{\prime}+\omega)-i\gamma_{23}\}\{\Omega_{21}+(\omega^{\prime}+\omega)-i\gamma_{21}\}} \frac{1}{\Omega_{31}+\omega-i\gamma_{31}} \right]\end{aligned}$$
While considering these derivations, the conditions
(93)$$E_{0}(-\omega^{\prime\prime}={-}\omega,\omega^{\prime}=\omega,\omega=\omega),$$
were imposed to extract the terms related to the resonance of two-photon absorption between levels 3 and 1 as follows,
(94)$$\begin{aligned}\chi^{(3)} &= \frac{\mu_{12}\mu_{23}\mu_{32}\mu_{21}}{\hbar^{3}}\left[ \frac{1}{(\Omega_{21}-\omega-i\gamma_{12})(\Omega_{31}-2\omega-i\gamma_{31})}\left\{ \frac{1}{\Omega_{21}-\omega-i\gamma_{21}} - \frac{1}{\Omega_{32}-\omega-i\gamma_{32}} \right\}\right.\\&\quad\left. +\frac{1}{(\Omega_{21}+\omega+i\gamma_{12})(\Omega_{31}-2\omega-i\gamma_{31})}\left\{ \frac{1}{\Omega_{12}-\omega-i\gamma_{12}} - \frac{1}{\Omega_{23}-\omega-i\gamma_{23}} \right\} \right] .\end{aligned}$$
Funding
Japan Society for the Promotion of Science (JP16H06504, JP21H05019, JP21J13476).
Acknowledgments
This work was supported in part by JSPS KAKENHI Grant Number JP16H06504 for Scientific Research on Innovative Areas “Nano-Material Optical-Manipulation”, JSPS KAKENHI (Grant Number: JP21H05019), and by JSPS KAKENHI (Grant Number: JP21J13476).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
References
1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]
2. A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978). [CrossRef]
3. A. Ashkin, J. M. Dziedzic, J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef]
4. S. Chu, J. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57(3), 314–317 (1986). [CrossRef]
5. D. V. Verschueren, S. Pud, X. Shi, L. De Angelis, L. Kuipers, and C. Dekker, “Label-free optical detection of dna translocations through plasmonic nanopores,” ACS Nano 13(1), 61–70 (2018). [CrossRef]
6. D. Verschueren, X. Shi, and C. Dekker, “Nano-optical tweezing of single proteins in plasmonic nanopores,” Small Methods 3(5), 1800465 (2019). [CrossRef]
7. T. Shoji, K. Itoh, J. Saitoh, N. Kitamura, T. Yoshii, K. Murakoshi, Y. Yamada, T. Yokoyama, H. Ishihara, and Y. Tsuboi, “Plasmonic manipulation of dna using a combination of optical and thermophoretic forces: separation of different-sized dna from mixture solution,” Sci. Rep. 10(1), 3349 (2020). [CrossRef]
8. H. Yamane, J. Yamanishi, N. Yokoshi, Y. Sugawara, and H. Ishihara, “Theoretical analysis of optically selective imaging in photoinduced force microscopy,” Opt. Express 28(23), 34787–34803 (2020). [CrossRef]
9. J. Yamanishi, H. Yamane, Y. Naitoh, Y. J. Li, N. Yokoshi, T. Kameyama, S. Koyama, T. Torimoto, H. Ishihara, and Y. Sugawara, “Optical force mapping at the single-nanometre scale,” Nat. Commun. 12(1), 3865 (2021). [CrossRef]
10. M. Osborne, S. Balasubramanian, W. Furey, and D. Klenerman, “Optically biased diffusion of single molecules studied by confocal fluorescence microscopy,” J. Phys. Chem. B 102(17), 3160–3167 (1998). [CrossRef]
11. G. Chirico, C. Fumagalli, and G. Baldini, “Trapped brownian motion in single-and two-photon excitation fluorescence correlation experiments,” J. Phys. Chem. B 106(10), 2508–2519 (2002). [CrossRef]
12. T. Iida and H. Ishihara, “Theoretical study of the optical manipulation of semiconductor nanoparticles under an excitonic resonance condition,” Phys. Rev. Lett. 90(5), 057403 (2003). [CrossRef]
13. K. Inaba, K. Imaizumi, K. Katayama, M. Ichimiya, M. Ashida, T. Iida, H. Ishihara, and T. Itoh, “Optical manipulation of cucl nanoparticles under an excitonic resonance condition in superfluid helium,” Phys. Status Solidi B 243(14), 3829–3833 (2006). [CrossRef]
14. H. Ajiki, T. Iida, T. Ishikawa, S. Uryu, and H. Ishihara, “Size-and orientation-selective optical manipulation of single-walled carbon nanotubes: A theoretical study,” Phys. Rev. B 80(11), 115437 (2009). [CrossRef]
15. S. E. S. Spesyvtseva, S. Shoji, and S. Kawata, “Chirality-selective optical scattering force on single-walled carbon nanotubes,” Phys. Rev. Appl. 3(4), 044003 (2015). [CrossRef]
16. Y. Shi, S. Xiong, L. K. Chin, J. Zhang, W. Ser, J. Wu, T. Chen, Z. Yang, Y. Hao, B. Liedberg, P. H. Yap, D. P. Tsai, C.-W. Qiu, and A. Q. Liu, “Nanometer-precision linear sorting with synchronized optofluidic dual barriers,” Sci. Adv. 4(1), eaao0773 (2018). [CrossRef]
17. Y. Shi, S. Xiong, Y. Zhang, L. Chin, Y.-Y. Chen, J. Zhang, T. Zhang, W. Ser, A. Larrson, S. Lim, J. Wu, T. Chen, Z. Yang, Y. Hao, B. Liedberg, P. Yap, K. Wang, D. Tsai, C.-W. Qiu, and A. Liu, “Sculpting nanoparticle dynamics for single-bacteria-level screening and direct binding-efficiency measurement,” Nat. Commun. 9(1), 815 (2018). [CrossRef]
18. Y. Shi, T. Zhu, T. Zhang, A. Mazzulla, D. P. Tsai, W. Ding, A. Q. Liu, G. Cipparrone, J. J. Sáenz, and C.-W. Qiu, “Chirality-assisted lateral momentum transfer for bidirectional enantioselective separation,” Light: Sci. Appl. 9(1), 62 (2020). [CrossRef]
19. T. Wada, H. Fujiwara, K. Sasaki, and H. Ishihara, “Proposed method for highly selective resonant optical manipulation using counter-propagating light waves,” Nanophotonics 9(10), 3335–3345 (2020). [CrossRef]
20. H. Fujiwara, K. Yamauchi, T. Wada, H. Ishihara, and K. Sasaki, “Optical selection and sorting of nanoparticles according to quantum mechanical properties,” Sci. Adv. 7(3), eabd9551 (2021). [CrossRef]
21. P. Kukura, M. Celebrano, A. Renn, and V. Sandoghdar, “Single-molecule sensitivity in optical absorption at room temperature,” J. Phys. Chem. Lett. 1(23), 3323–3327 (2010). [CrossRef]
22. M. Celebrano, P. Kukura, A. Renn, and V. Sandoghdar, “Single-molecule imaging by optical absorption,” Nat. Photonics 5(2), 95–98 (2011). [CrossRef]
23. Y. Jiang, T. Narushima, and H. Okamoto, “Nonlinear optical effects in trapping nanoparticles with femtosecond pulses,” Nat. Phys. 6(12), 1005–1009 (2010). [CrossRef]
24. T. Kudo and H. Ishihara, “Proposed nonlinear resonance laser technique for manipulating nanoparticles,” Phys. Rev. Lett. 109(8), 087402 (2012). [CrossRef]
25. T. Kudo, H. Ishihara, and H. Masuhara, “Resonance optical trapping of individual dye-doped polystyrene particles with blue-and red-detuned lasers,” Opt. Express 25(5), 4655–4664 (2017). [CrossRef]
26. E. Heilweil and R. Hochstrasser, “Nonlinear spectroscopy and picosecond transient grating study of colloidal gold,” J. Chem. Phys. 82(11), 4762–4770 (1985). [CrossRef]
27. D. Ricard, P. Roussignol, and C. Flytzanis, “Surface-mediated enhancement of optical phase conjugation in metal colloids,” Opt. Lett. 10(10), 511–513 (1985). [CrossRef]
28. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n 2 measurements,” Opt. Lett. 14(17), 955–957 (1989). [CrossRef]
29. M. Leuchs and W. Kiefer, “Excitation profile of coherent anti-stokes raman scattering from the mno- 4 ion doped in a kclo4 crystal at low temperature,” J. Chem. Phys. 99(9), 6302–6307 (1993). [CrossRef]
30. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65(1), 96–99 (1990). [CrossRef]
31. R. Ganeev, M. Baba, A. Ryasnyansky, M. Suzuki, and H. Kuroda, “Characterization of optical and nonlinear optical properties of silver nanoparticles prepared by laser ablation in various liquids,” Opt. Commun. 240(4-6), 437–448 (2004). [CrossRef]
32. R. Ganeev and A. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1-2), 295–302 (2006). [CrossRef]
33. A. Ryasnyanskiy, B. Palpant, S. Debrus, U. Pal, and A. Stepanov, “Third-order nonlinear-optical parameters of gold nanoparticles in different matrices,” J. Lumin. 127(1), 181–185 (2007). [CrossRef]
34. Z. Dehghani, S. Nazerdeylami, E. Saievar-Iranizad, and M. M. Ara, “Synthesis and investigation of nonlinear optical properties of semiconductor zns nanoparticles,” J. Phys. Chem. Solids 72(9), 1008–1010 (2011). [CrossRef]
35. H. Baida, D. Christofilos, D. Christofilos, P. Maioli, A. Crut, N. D. Fatti, and F. Vallee, “Ultrafast nonlinear spectroscopy of a single silver nanoparticle,” J. Raman Spectrosc. 42(10), 1891–1896 (2011). [CrossRef]
36. Y.-x. Zhang and Y.-h. Wang, “Nonlinear optical properties of metal nanoparticles: a review,” RSC Adv. 7(71), 45129–45144 (2017). [CrossRef]
37. A. Prakash, B. P. Pathrose, P. Radhakrishnan, and A. Mujeeb, “Nonlinear optical properties of neutral red dye: Enhancement using laser ablated gold nanoparticles,” Opt. Laser Technol. 130, 106338 (2020). [CrossRef]
38. T. Iida and H. Ishihara, “Theory of resonant radiation force exerted on nanostructures by optical excitation of their quantum states: From microscopic to macroscopic descriptions,” Phys. Rev. B 77(24), 245319 (2008). [CrossRef]
39. J. D. Jackson, Classical electrodynamics3rd ed (John wiley & sons, Inc., NewYork, NY, 1999).
40. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25(15), 1065–1067 (2000). [CrossRef]
41. D. L. Ermak and J. A. McCammon, “Brownian dynamics with hydrodynamic interactions,” J. Chem. Phys. 69(4), 1352–1360 (1978). [CrossRef]
42. R. J. Donnelly and C. F. Barenghi, “The observed properties of liquid helium at the saturated vapor pressure,” J. Phys. Chem. Ref. Data 27(6), 1217–1274 (1998). [CrossRef]
43. C. Pin, R. Otsuka, and K. Sasaki, “Optical transport and sorting of fluorescent nanodiamonds inside a tapered glass capillary: optical sorting of nanomaterials at the femtonewton scale,” ACS Appl. Nano Mater. 3(5), 4127–4134 (2020). [CrossRef]
Open Access
Add to BibSonomy
