Sum-frequency generation through a unique Feynman diagram formalism: the case of bipartite organic/inorganic complexes

TL;DR

This paper introduces a novel Feynman diagram formalism tailored for bipartite organic/inorganic complexes, enabling accurate modeling of their nonlinear optical responses, bridging quantum electrodynamics and condensed matter physics.

Contribution

It presents a new method using classical Feynman diagrams to analyze light-matter interactions in complex hybrid systems, overcoming limitations of traditional double-sided diagrams.

Findings

Derived quantum expressions for second-order optical responses.

Validated the formalism against experimental results.

Provided a pedagogical approach for complex systems.

Abstract

In quantum electrodynamics, optical processes are theoretically described by double-sided Feynman diagrams. This formalism is powerful in the case of molecules but proves inappropriate to account for light-matter interactions within complex hybrid systems constituted of organic and inorganic matter. The double-sided Feynman diagrams do not easily enable to implement the coupling between the electronic properties of the former and the vibrational response of the latter. Here we present a new general method bridging optics and condensed matter physics in order to properly account for the underlying fundamental process thanks to the classical Feynman diagrams dedicated to solid state physics, instead of the double-sided diagrams commonly used in nonlinear optics. In a manner both rigorous and pedagogical, we especially show how Feynman diagrams can be used to analytically derive the quantum expressions of second-order optical response functions which prove to be in complete agreement with previously established experimental results.

Figures

Equations

$$\mathcal{H}=\sum _n{ϵ}_n{c}_n^{†}{c}_n.$$

$${\mathcal{H}}_{\text{QD}}=\sum _{p\in (\mathcal{E})}{ϵ}_p^x{x}_p^{†}{x}_p,$$

$${ϵ}_p^x=E_g+\frac{(\hslash {\alpha }_p^e{)}^2}{2m_e{R}^2}+\frac{(\hslash {\alpha }_p^h{)}^2}{2m_h{R}^2}.$$

$${\mathcal{H}}_{\text{mol.}}=\sum _{\sigma \in (\mathcal{E},\mathcal{V})}{ϵ}_{\sigma }{d}_{\sigma }^{†}{d}_{\sigma }.$$

$${\chi }_{ij}^{(1)}(ı\omega )=-\frac{1}{\beta }\sum _{n,m,\nu }|{\mathbf{p}}_{mn}^i||{\mathbf{p}}_{nm}^j|{\mathfrak{G}}_{mm}(ı{\omega }_{\nu }){\mathfrak{G}}_{nn}(ı{\omega }_{\nu }+ı\omega ).$$

$${\chi }_{ij}^{(1)}(ı\omega )=-\sum _{n,m}|{\mathbf{p}}_{mn}^i||{\mathbf{p}}_{nm}^j|\frac{\rho ({\omega }_m)-\rho ({\omega }_n-ı\omega )}{{\omega }_m-{\omega }_n+ı\omega }.$$

$${\chi }_{ij}^{(1)}(\omega )=-\sum _n\left(\frac{|{\mathbf{p}}_{0n}^i||{\mathbf{p}}_{n0}^j|}{\omega -{\omega }_n+ı0^{+}}-\frac{|{\mathbf{p}}_{0n}^i||{\mathbf{p}}_{n0}^j|}{\omega +{\omega }_n+ı0^{+}}\right).$$

$$\text{upmu}=\sum _{p,p^{\prime }}\left({\text{upmu}}_{p{p}^{\prime }}{x}_p^{†}{x}_{p^{\prime }}+{\text{upmu}}_{p^{\prime }p}{x}_{p^{\prime }}^{†}{x}_p\right),$$

$$\text{and}\mathbf{m}=\sum _{\sigma ,{\sigma }^{\prime }}\left({\mathbf{m}}_{\sigma {\sigma }^{\prime }}{d}_{\sigma }^{†}{d}_{{\sigma }^{\prime }}+{\mathbf{m}}_{{\sigma }^{\prime }\sigma }{d}_{{\sigma }^{\prime }}^{†}{d}_{\sigma }\right).$$

$${\mathcal{H}}_{\text{dip-dip}}={\text{upmu}}^{\text{T}}\mathbf{V}\mathbf{m}=V_{ij}|{\text{upmu}}^i||{\mathbf{m}}^j|,$$

$$V_{ij}|{\text{upmu}}_{p{p}^{\prime }}^i||{\mathbf{m}}_{\sigma {\sigma }^{\prime }}^j|x_p^{†}{x}_{p^{\prime }}{d}_{\sigma }^{†}{d}_{{\sigma }^{\prime }},$$

$${\chi }_{ijk}^{(2)}(ı{\omega }_1,ı{\omega }_2)$$

$${\chi }_{ijk}^{(2)}(ı{\omega }_1,ı{\omega }_2)=-{\xi }_{hj}(ı{\omega }_1)V^h{}_l\sum _{e,v}\frac{|{\mathbf{m}}_{v0}^k||{\mathbf{m}}_{0e}^i||{\mathbf{m}}_{ev}^l|}{(ı{\omega }_2-{\omega }_v)(ı{\omega }_3-{\omega }_e)},$$

$$\sum _e\frac{|{\mathbf{m}}_{0e}^i||{\mathbf{m}}_{ev}^l|}{ı{\omega }_3-{\omega }_e}$$

$${\chi }_{ijk}^{(2)}({\omega }_1,{\omega }_2)=-\sum _v\frac{\hslash }{2{\omega }_v}\frac{{\partial }_v{\alpha }_{il}({\omega }_3)V^{hl}{\xi }_{hj}({\omega }_1){\partial }_v{m}_k}{{\omega }_2-{\omega }_v+ı0^{+}},$$

$${\mathbf{\chi }}^{(2)}({\omega }_1,{\omega }_2)=-\sum _v\frac{\hslash }{2{\omega }_v}\frac{{\partial }_v\mathbf{\alpha }({\omega }_3)\cdot \mathbf{V}\cdot {\mathbf{\chi }}_{\text{QD}}^{(1)}({\omega }_1)\otimes {\partial }_v\mathbf{m}}{{\omega }_2-{\omega }_v+ı0^{+}}.$$

$$\mathbf{m}=\mathbf{\alpha }{\mathbf{E}}_{\ell }+{\mathbf{m}}_0,$$

$${\mathbf{E}}_{\ell }=\mathbf{\kappa }\text{upmu}=\mathbf{\kappa }\mathbf{\xi }\mathbf{E}.$$

$${\kappa }_{ij}(r)=\frac{3r_i{r}_j-{\delta }_{ij}{r}^2}{4\pi {\epsilon }_0{r}^5}\sim \frac{1}{r^3},$$

$${\mathbf{\chi }}_v^{(2)}({\omega }_1,{\omega }_2)\propto \frac{{\partial }_v\mathbf{\alpha }({\omega }_1)\cdot \mathbf{\kappa }\cdot {\mathbf{\chi }}_{\text{QD}}^{(1)}({\omega }_1)\otimes \mathbf{g}}{{\omega }_2-{\omega }_v+ı0^{+}},$$

$$V_{ij}(r)=\frac{{\delta }_{ij}{r}^2-3r_i{r}_j}{4\pi {\epsilon }_0{r}^5}.$$

$$\mathbf{P}(\omega )={\mathbf{\chi }}^{(1)}(\omega )\mathbf{E}(\omega ).$$

$${\mathbf{P}}_{\text{NL}}(t)={\mathbf{\chi }}^{(2)}\ast \mathbf{E}\otimes \mathbf{E}(t).$$

$$G_{nm}(\omega )={\mathfrak{G}}_{nm}(\omega +ı0^{+}).$$

$${\mathfrak{G}}_{nm}(ı{\omega }_{\nu })=\frac{{\delta }_{nm}}{ı{\omega }_{\nu }-{\omega }_n},$$

$$\mathbf{E}=ı\sum _{\mathbf{\kappa },i}\sqrt{\frac{\hslash \omega }{2V{\epsilon }_0}}\left({\text{e}}^{(i)}{a}_{\mathbf{\kappa }}^{(i)}-{\bar{\text{e}}}^{(i)}{a_{\mathbf{\kappa }}^{(i)}}^{†}\right),$$

$${\mathcal{H}}_{\text{light-mat.}}=-\mathbf{p}\cdot \mathbf{E},$$

$$\mathbf{p}=\sum _{n,m}\left({\mathbf{p}}_{nm}{c}_n^{†}{c}_m+{\mathbf{p}}_{mn}{c}_m^{†}{c}_n\right).$$

$$|{\mathbf{p}}_{nm}^i|c_n^{†}{c}_m{a}_{\mathbf{\kappa }}^{(i)}\text{and}|{\mathbf{p}}_{mn}^i|c_m^{†}{c}_n{a_{\mathbf{\kappa }}^{(i)}}^{†}.$$

Full text

Results

Second quantization of simple and hybrid systems. Within the formalism introduced in Methods, the linear and nonlinear optical response functions $\chi_{ij}^{(1)}$ and $\chi_{ijk}^{(2)}$ of any simple or complex systems are expanded on the base of imaginary time Green’s functions. To show the interest and the effectiveness of the approach, we consider in this article the cases of organic (e.g. molecules, Figure 1b & e), inorganic (e.g. quantum dots, Figure 1c) and hybrid systems (e.g. ligand-conjugated quantum dots, Figure 1f). In Figure 1 and the following, the notations $(\mathcal{E})$ and $(\mathcal{V})$ depict any set of electronic and vibrational states, respectively. In all generality, we thus describe organic/inorganic hybrid systems as made of an electronic subsystem $(\mathcal{E})$ and a vibrational subsystem $(\mathcal{V})$. Hence, to be relevant, the method first requires to derive the proper quantum description of the two interacting subsystems $(\mathcal{E})$ and $(\mathcal{V})$. In order to present a universal description, we consider here canonical hamiltonians in the framework of the second quantization formalism.

Each quantum system is characterized by its eigenenergies $\epsilon_{n}$, whether its states are electronic or vibrational. By introducing the associated creation and annihilation operators, $c_{n}^{\dagger}$ and $c_{n}$, the hamiltonian takes the standard form:

[TABLE]

For instance, the excitonic hamiltonian of QDs can be written as:

[TABLE]

where $x_{p}^{\dagger}$ and $x_{p}$ respectively creates and annihilates an exciton of energy Haug :

[TABLE]

For a spherical QD of radius $R$, $\alpha_{p}^{e/h}$ depicts the zeros of the spherical Bessel functions associated to the wave functions of the electron, $e$, and the hole, $h$, which compose the exciton Haug . $m_{e}$ and $m_{h}$ correspond to their respective effective masses. In the following, we assume that $p$ is ordered by increasing energy, i.e. $\epsilon_{p}^{x}<\epsilon_{p+1}^{x}$, and that $p=0$ depicts the ground state, i.e. vacuum of exciton, with zero energy.

For molecules, we make the choice to itemize the quantum states, both electronic and vibrational, by the integer $\sigma$, and to describe them by the fermion operators $d_{\sigma}^{\dagger}$ and $d_{\sigma}$:

[TABLE]

Here again, we assume that the eigenenergies $\epsilon_{\sigma}$ are known, precondition required for identifying the imaginary Green’s functions.

Given their generality, equations (2) and (4) do not only describe the case of QD/ligand hybrid systems but can be used to account for all subsystems $(\mathcal{E})$ and $(\mathcal{V})$.

Derivation of linear response functions. For any simple system described by equation (1), we consider the Feynman diagram drawn in Figure 2c. In virtue of the second quantization of light-matter interactions reminded in Methods, this diagram is built with the vertices associated to the absorption of a $j$-polarized photon of Matsubara frequency $\imath\omega$ and the emission of an $i$-polarized photon of Matsubara frequency $\imath\omega$. As evidenced by the following derivation, this picture gives access to the linear susceptibility $\bm{\chi}^{(1)}$ of a given system of dipole moment $\mathbf{p}$. Indeed, from Feynman rules Zagoskin ; Mahan , the diagram is converted into:

[TABLE]

From equation (26), in Methods, and thanks to Matsubara’s theorem (see Supplementary Note 1):

[TABLE]

As $\imath\omega$ is the Matsubara frequency of a photon, that is a boson: $\rho(\omega_{n}-\imath\omega)=\rho(\omega_{n})$. Assuming that the system is in its ground state $|0\rangle$ at equilibrium, i.e. $\rho(\omega_{n})=\delta_{n0}$, and setting the ground state energy to zero:

[TABLE]

According to the nature of the system, $n$ describes electronic and/or vibrational states. Thus, we retrieve the common formula of $\chi_{ij}^{(1)}$ obtained within the density matrix formalism Shen and the time-dependent perturbation theory Haug .

The convergence between our method based on imaginary time Green’s functions and the common approaches that we have just mentioned indicates that our initiative is at least as appropriate as the techniques based on double-sided diagrams. The relevance of the method in the first case of linear optics strongly suggests it is likely to give rise to a brand new way of thinking and describing nonlinear optics. It especially proves that the condensed matter picture of Feynman diagrams is well suited for performing analytical computations of optical response functions, and that imaginary time Green’s functions are not only dedicated to solid state physics: they are of great interest for optics too. Consequently, this observation naturally leads us to examine the case of higher-order response functions. As a matter of fact, our method releases all its effectiveness for nonlinear optics.

Nonlinear sum-frequency generation. Probing matter with intense light allows studying its behaviour beyond the linear dielectric regime. One of the highest stakes then lies in the theoretical understanding of the nonlinear response of organic/inorganic hybrid systems. To date, there is not any analytical description, whereas the experiment evidences the existence of vibroelectronic couplings within these systems, thanks to SFG spectroscopy Humbert2015 ; Sengupta2017 ; Noblet2018 . Here, we show that our Feynman diagram-based method takes up the challenge and enables to compute the associated second-order susceptibility $\bm{\chi}^{(2)}=(\chi_{ijk}^{(2)})$. In particular, we choose to illustrate our approach with the case of ligand-capped CdTe QDs Noblet2018 . As a second-order response function, each $\chi_{ijk}^{(2)}$ is a Green’s function which combines two frequencies, $\omega_{1}$ and $\omega_{2}$ Shen ; Boyd . We thus need to draw a Feynman diagram which represents how the hybrid system mixes $\omega_{1}$ and $\omega_{2}$ to generate $\omega_{3}=\omega_{1}+\omega_{2}$. Figure 1d illustrates the SFG process. The aim of the paper is then to build a Feynman diagram, based on the imaginary time Green’s functions associated to the two subsystems $(\mathcal{E})$, from the QDs, and $(\mathcal{V})$, from the ligands, in order to relate $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$. The complete Green’s function computed from this new diagram will be therefore interpreted as $\chi_{ijk}^{(2)}(\omega_{1},\omega_{2})$.

In this context, Figures 1e & f explicitly show the difference of point of view between (e) purely molecular and (f) hybrid systems. In the first case, the electronic and the vibrational states are coupled within the same molecule, and the double-sided Feynman diagrams efficiently allow the calculation of $\bm{\chi}^{(2)}(\omega_{1},\omega_{2})$ Huang1994 . In the second case, $(\mathcal{E})$ is associated to the inorganic subsystem, while $(\mathcal{V})$ corresponds to the organic one. In the following, we dwell on this unconventional hybrid configuration, but a complete derivation of $\bm{\chi}^{(2)}(\omega_{1},\omega_{2})$ in the conventional case of molecular systems is achieved in Supplementary Note 2, thanks to our diagrammatic method.

Dipole-dipole interaction between $\bm{(\mathcal{E})}$ and $\bm{(\mathcal{V})}$. The diagrammatic connection between subsystems $(\mathcal{E})$ and $(\mathcal{V})$ requires the knowledge of their interaction hamiltonian. In our case, QDs and ligands are expected to interact through dipolar coupling Noblet2018 . This is why we detail hereafter the quantum description of such an interplay between electric dipoles.

Starting from the fermion operators defined above, the second quantization formalism enables us to expand the respective dipole moments of $(\mathcal{E})$ and $(\mathcal{V})$:

[TABLE]

[TABLE]

The dipole-dipole interaction between the two subsystems is then described by:

[TABLE]

where $\bm{V}=(V_{ij})$ is the dipolar interaction matrix between the two dipole moments, given in Ref. Craig . In view of equations (8) and (9), the terms of the interaction hamiltonian have the following form:

[TABLE]

and thus involve four-particle interactions. In terms of Feynman diagrams, such an interaction is depicted by the four-particle vertex which is presented in Figure 3. This dipole-dipole vertex is the crux issue. It is the building block which enables us to formally couple $(\mathcal{E})$ and $(\mathcal{V})$, that is QDs and molecules, throughout the analytical derivation of the second-order susceptibility $\bm{\chi}^{(2)}$.

Diagrammatic calculation of $\bm{\chi^{(2)}}$. Given the dipole-dipole and light-matter vertices featured in Figures 2 & 3, we propose to describe the nonlinear SFG coupling between QDs and molecules with the Feynman diagram drawn in Figure 4a. Within this picture, $\omega_{1}$ and $\omega_{2}$ respectively play the roles of the visible and infrared frequencies $\omega_{\text{vis}}$ and $\omega_{\text{ir}}$, as displayed in Figure 1f. Since we are interested in the influence of QDs on the SFG signal of molecules, i.e. the molecular SFG response, the Feynman diagram is built in such a way that the sum-frequency $\omega_{3}=\omega_{1}+\omega_{2}$ is generated by the molecule. First, as presented in Figure 4a & b, the QD is excited by a $j$-polarized photon of frequency $\omega_{1}$ (Fig. 4b, vertex 1) and the molecule is excited by a $k$-polarized photon of frequency $\omega_{2}$ (Fig. 4b, vertex 3). Second, these excited states interact through dipole-dipole coupling (Fig. 4b, vertex 2), which translates into the exchange of a virtual photon of frequency $\omega_{1}$. Third and last, the molecule relaxes and emits an $i$-polarized photon of frequency $\omega_{3}$ (Fig. 4b, vertex 4). As a result, this Feynman diagram shows how a visible excitation (polarized along $j$) of QDs, combined to an infrared excitation (polarized along $k$) of molecules, leads to the molecular SFG response (polarized along $i$) of the hybrid system. Intuitively, this is thus expected to give access to the Green’s function $\chi_{ijk}^{(2)}(\omega_{1},\omega_{2})$.

Naturally, a second diagram can be proposed, in which the SFG signal is generated by the QD. This possibility is presented in Figure 4c. As shown in Supplementary Note 3, this Feynman diagram results in a non-zero susceptibility only if the QD is non-centrosymmetric itself, which is a restrictive condition that is not necessary for the first diagram. Besides, as discussed further in this section, the expansion of $\chi_{ijk}^{(2)}(\omega_{1},\omega_{2})$ from the first diagram leads to a factorized form which involves the Raman and the infrared efficiencies of the molecule. This classical result, obtained so far in the conventional case of purely molecular systems, cannot arise from the diagram of Figure 4c (see Supplementary Note 3). Hence, the molecular SFG response is well and truly modeled by the Feynman diagram of Figure 4a.

As a second-order Green’s function, $\chi_{ijk}^{(2)}$ is supposed to be expanded on the base of first-order Green’s functions Doniach ; Zagoskin ; Kadanoff ; Mahan . This is the true meaning of Feynman diagrams, which actually represent perturbative expansions. As a consequence of the Feynman rules which drive such an expansion Zagoskin ; Mahan , the diagram of Figure 4a results in:

[TABLE]

To simplify the notations in equation (13-13), the imaginary time Green’s functions are itemized by a unique integer: $\mathfrak{G}_{n}(\imath\omega_{\nu})=\mathfrak{G}_{nn}(\imath\omega_{\nu})=(\imath\omega_{\nu}-\omega_{n})^{-1}$, from equation (26), in Methods. In virtue of equation (5), the sum over $p$, $p^{\prime}$ and $\nu$ is identified to the linear susceptibility $\xi_{hj}=[\bm{\chi}^{(1)}_{\text{QD}}]_{hj}$ of QDs at frequency $\omega_{1}$. In equation (13), the sum over $\lambda$ can be reduced according to Matsubara’s theorem Zagoskin ; Mahan , reminded in Supplementary Note 1, and the sum over $\sigma$, $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ can be simplified by following these four assumptions: (i) the energy of the molecular ground state is set to zero ; (ii) the molecules remain in their ground state $|\sigma=0\rangle$ at equilibrium, i.e. $\rho(\omega_{\sigma})=\delta_{\sigma,0}$ ; (iii) the molecular states $|\sigma\rangle$ can be split into electronic and vibrational states, itemized by $|\sigma=e\rangle$ and $|\sigma=v\rangle$ ; (iv) the molecular SFG response is dominated by vibrational resonances with respect to the infrared frequency so that we only keep the terms with $(\imath\omega_{2}-\omega_{v})^{-1}$. Under these hypotheses, we obtain:

[TABLE]

thanks to a calculation detailed in Supplementary Note 4. The sum over the molecular electronic states is known to give the molecular polarizability $\bm{\alpha}=(\alpha_{il})$ according to Lin1994 :

[TABLE]

where $Q_{v}$ is the normal coordinate of vibration mode $|v\rangle$. For simplicity, we rewrite the derivative with respect to $Q_{v}$ as $\partial_{v}\alpha_{il}$. With the same notations, $|\mathbf{m}_{v0}^{k}|=\sqrt{\hbar/2\omega_{v}}\ \partial_{v}m_{k}$. Eventually, the second-order susceptibility reads:

[TABLE]

that is, in terms of tensors:

[TABLE]

For real systems, $\imath 0^{+}$ is replaced by the damping constant $\imath\gamma_{v}$ associated to vibration mode $|v\rangle$.

Here, the coupling between QDs and molecules explicitly appears through matrix $\bm{V}$, which drives their mutual dipolar interaction. In particular, this matrix couples the Raman response of molecules, via $\partial_{v}\bm{\alpha}(\omega_{3})$, to the excitonic response of QDs, via $\bm{\xi}(\omega_{1})=\bm{\chi}^{(1)}_{\text{QD}}(\omega_{1})$. The infrared cross section also intervenes with $\partial_{v}\mathbf{m}$. In respect of molecules, we notice that the SFG process combines their Raman and infrared cross sections. As announced above, we thus retrieve the classical result obtained in the case of purely molecular systems Lin1994 . For the hybrid system we are interested in, an additional contribution associated to QDs modulates the SFG process, whether these QDs are centrosymmetric or not, that is in spite of their ability to generate SFG on their own. The amplitude of each vibration mode $|v\rangle$ is driven by $\partial_{v}\bm{\alpha}(\omega_{3})\cdot\bm{V}\cdot\bm{\chi}_{\text{QD}}^{(1)}(\omega_{1})\otimes\partial_{v}\mathbf{m}$ and directly depends on the linear response function of QDs in the visible range.

Discussion

We experimentally evidenced this coupling between QDs and molecules in the work reported in Ref. Noblet2018 . Within this previous article, a classical model based on dielectrics was proposed. It consists in assuming that the molecular dipole moment $\mathbf{m}$ experiences the local electric field $\mathbf{E}_{\ell}$ generated by the QD:

[TABLE]

where $\mathbf{m}_{0}$ is the dipole moment in absence of QD exciton. The local electric field $\mathbf{E}_{\ell}$ is related to the dipole moment $\bm{\upmu}$ of the QD, itself related to the excitation field $\mathbf{E}$ of the incident light through its linear susceptibility $\bm{\xi}=\bm{\chi}^{(1)}_{\text{QD}}$:

[TABLE]

The matrix $\bm{\kappa}$ contains the spatial dependence of $\mathbf{E}_{\ell}$. For instance, in the static regime:

[TABLE]

wherein the origin is set at the center of the QD. Thanks to Fourier analysis, this model therefore results in Noblet2018 :

[TABLE]

for each vibration mode $|v\rangle$. We notice three differences with equation (17): the $\omega_{1}$-dependence of $\partial_{v}\bm{\alpha}$, the matrix $\bm{\kappa}$ and the vector $\mathbf{g}$. It turns out that matrix $\bm{\kappa}$ plays the role of the dipolar interaction matrix $\bm{V}$, which is physically logical. Indeed, $\bm{\kappa}$ characterizes the dipolar interplay between the QD of moment $\bm{\upmu}$ and its ligands of moment $\mathbf{m}$ (see equations (18-19)). In the static regime, $\bm{V}$ and $-\bm{\kappa}$ actually coincide Craig :

[TABLE]

Here, the quantum and the classical approaches are equivalent. Nevertheless, the latter suffers from inaccuracies with respect to the Raman and the infrared cross sections. First, the classical derivation involves a Raman factor $\partial_{v}\bm{\alpha}$ at frequency $\omega_{1}$, whereas it is expected at frequency $\omega_{3}=\omega_{1}+\omega_{2}$ Lin1994 . As shown in equation (17), the quantum and diagrammatic method allows the correction of this first inaccuracy. Second, the infrared efficiency $\partial_{v}\mathbf{m}$ does not appear in equation (21), contrary to equation (17). Instead, the classical derivation requires the introduction of a vector $\mathbf{g}$ related to the geometry of the molecules which is not linked to $\partial_{v}\mathbf{m}$. More specifically, $\mathbf{g}$ depends on the mass-weighted cartesian coordinates of the atoms. Its precise definition is given in Ref. Noblet2018 . Consequently, the diagrammatic method we developed in the present article constitutes the proper way to compute second-order response functions for hybrid systems, and to resolve the deficiencies of classical approaches.

We chose to illustrate the diagrammatic method with SFG spectroscopy in the case of hybrid organic/inorganic systems because the understanding of the vibroelectronic coupling occurring within such systems is a controversial issue Humbert2015 ; Noblet2018 ; Swenson2016 ; Frederick2013 ; Lifshitz2015 . Given that our approach is very fundamental and general, this paves the way for the theoretical study of SHG (second harmonic generation), DFG (difference-frequency generation) and Raman spectroscopies. SHG and DFG are second-order processes, while Raman scattering is a third-order one. For a given system, the computation of SHG and DFG response functions would consist in drawing Feynman diagrams to connect three photons, in a way similar to what is done here for SFG. Besides, SHG is a particular case of SFG, for which $\omega_{1}=\omega_{2}$. Formally, the reasoning we used for the calculation of the SFG response tensor $\bm{\chi}^{(2)}(\omega_{1},\omega_{2})$ could thus be applied to the other second-order optical processes.

As a third-order process, Raman scattering is quite more complex to account for on a theoretical point of view Polavarapu1990 . Especially, when it is performed on molecules deposited on a rough metallic surface, this scattering is enhanced. One talks about surface-enhanced Raman spectroscopy (SERS), whose explanation is still a matter of debate Gersten1980 ; Moskovits1985 ; Otto1992 ; Stiles2008 . In this context, the present work may pave a new way of thinking for Raman and SERS with the difference that the associated Feynman diagrams would consist in combining four photons to compute $\bm{\chi}^{(3)}$ tensors. This may seem a daunting task, but we must keep in mind that the efficiency of the diagrammatic approach lies in mixing solid state and molecular physics with optics, through a unique formalism. For SERS, the question is precisely on the coupling between the metallic surface and the grafted molecules. This is why it is almost certain that our method will also yield positive results for organic/inorganic systems probed by third-order optical spectroscopies.

Fundamentally, Green’s functions are the proper and dedicated language of linear and nonlinear responses theories. Although they may seem tricky to use for complex systems, this difficulty can be overcome thanks to Feynman diagrams. In the case of hybrid systems, our method consists in building such diagrams taking into account the interactions between the different subsystems, and to apply the Feynman rules which enables to translate any diagram into the analytical expression of the associated Green’s function. Here, the case of ligand-conjugated QDs probed by SFG spectroscopy is considered to illustrate the method. Assuming a dipole-dipole interaction between QDs and ligands, the molecular SFG response proves to be modulated by the linear susceptibility of QDs in the visible range. The SFG process is thus expected to be the most efficient when the visible excitation of QDs is maximum. As shown in Ref. Noblet2018 , such a behaviour has been observed. Our theoretical result therefore coincides with the experiment, which encourages the use of this approach in nonlinear optics to describe and understand the behaviour of such organic/inorganic hybrid systems.

MethodsGreen’s functions in optics and solid-state physics. Probing matter with light consists in examining how the polarization of matter, $\mathbf{P}$, reacts to an excitation electric field $\mathbf{E}$. In frequency space, the linear response theory teaches that there exists a tensor $\bm{\chi}^{(1)}=(\chi_{ij}^{(1)})$ so that Boyd :

[TABLE]

In time domain, this gives: $\mathbf{P}(t)=\bm{\chi}^{(1)}\ast\mathbf{E}(t)$. To go further, the use of an intense light allows studying the behaviour of matter beyond the linear dielectric regime. In this case, a nonlinear polarization $\mathbf{P}_{\text{NL}}$ is induced in the material which reveals the existence of a second-order response tensor $\bm{\chi}^{(2)}=(\chi_{ijk}^{(2)})$, so that Boyd :

[TABLE]

As response functions, each $\chi_{ij}^{(1)}$ and $\chi_{ijk}^{(2)}$ are Green’s functions and thus belongs to a large function family which exceeds the scope of optics Greiner . In condensed matter physics, Green’s functions are used to describe the propagation of excited states in solids Doniach ; Zagoskin ; Kadanoff ; Mahan . More precisely, for a quantum system characterized by the eigenstates $|n\rangle$, $n\in\mathbb{N}$, and prepared in state $|m\rangle$ at $t=0$, the retarded Green’s function $G_{mn}(t)$ gives the complex amplitude of probability to measure the system in state $|n\rangle$ at time $t$ Zagoskin ; Mahan . In particular, $G_{mm}(t)$ measures the coherence of each state $|m\rangle$ over time. Given those two subfamilies of Green’s functions, our theoretical approach aims to connect both.

Feynman diagrams in Matsubara frequency space. For convenience, we choose to handle imaginary time Green’s functions $\mathfrak{G}_{nm}$ instead of $G_{nm}$ Zagoskin ; Mahan . In frequency domain, both are related through analytical continuity and then constitute two equivalent mathematical pictures Mahan :

[TABLE]

Given the eigenfrequencies $\omega_{n}$ associated to the eigenstates $|n\rangle$, the imaginary time Green’s functions prove to simply read Zagoskin ; Mahan :

[TABLE]

where $\imath\omega_{\nu}$, $\nu\in\mathbb{Z}$, are the Matsubara frequencies, that is the poles of the Fermi-Dirac function $\rho(z)=(1+e^{\hbar\beta z})^{-1}$, $z\in\mathbb{C}$ (see Supplementary Note 1). Within the formalism of Feynman diagrams, each $\mathfrak{G}_{nm}(\imath\omega_{\nu})$ can be represented as an oriented straight line embodying the propagation of $|n\rangle$ with energy $\omega_{\nu}$ Zagoskin ; Mahan . When different lines are crossing at a vertex, it means that several states interact with each other. Thereby, drawing a Feynman diagram consists in representing the propagation of an excited state according to the interactions that it experiences. In virtue of Feynman rules Zagoskin ; Mahan , the Green’s function which drives the whole process is computed as a combination of all the Green’s functions associated to the straight lines which compose the whole diagram.

Light-matter interaction. To draw Feynman diagrams accounting for optical response functions, we need the unit diagrams associated to the interaction between the quantum states of matter and light. As we know, light is described by a quantized electric field Craig :

[TABLE]

where $a_{\bm{\kappa}}^{(i)}$ and ${a_{\bm{\kappa}}^{(i)}}^{\dagger}$ are boson operators associated to $i$-polarized photons of momentum $\bm{\kappa}$. Therefore, the interaction between light and any quantum system described by equation (1) translates into:

[TABLE]

wherein $\mathbf{p}$ is the dipole moment of the system, whose second quantization expansion reads:

[TABLE]

Equation (28) then involves terms of the forms:

[TABLE]

The associated three-particle vertices are presented in Figure 2a & b. These embody light absorption and light emission processes.