Non-Perturbative Approaches to Linear and Nonlinear Responses of Atoms, Molecules, and Molecular Aggregates: A Theoretical Approach to Molecular Quantum Information and Quantum Biology

Abstract

Non-perturbative approaches to linear and nonlinear responses (NLR) of atoms, molecules, and molecular aggregates are reviewed in relation to low and high harmonic generations (HG) by laser fields. These response properties are effective for the generation of entangled light pairs for quantum information processing by spontaneous parametric downconversion (SPDC) and stimulated four-wave mixing (SFWM). Quasi-energy derivative (QED) methods, such as QED Møller–Plesset (MP) perturbation, are reviewed as time-dependent variational methods (TDVP), providing analytical expressions of time-dependent linear and nonlinear responses of open-shell atoms, molecules, and molecular aggregates. Numerical Liouville methods for the low HG (LHG) and high HG (HHG) regimes are reviewed to elucidate the NLR of molecules in both LHG and HHG regimes. Three-step models for the generation of HHG in the latter regime are reviewed in relation to developments of attosecond science and spectroscopy. Orbital tomography is also reviewed in relation to the theoretical and experimental studies of the amplitudes and phases of wave functions of open-shell atoms and molecules, such as molecular oxygen, providing the Dyson orbital explanation. Interactions between quantum lights and molecules are theoretically examined in relation to derivations of several distribution functions for quantum information processing, quantum dynamics of molecular aggregates, and future developments of quantum molecular devices such as measurement-based quantum computation (MBQP). Quantum dynamics for energy transfer in dendrimer and related light-harvesting antenna systems are reviewed to examine the classical and quantum dynamics behaviors of photosynthesis. It is shown that quantum coherence plays an important role in the well-organized arrays of chromophores. Finally, applications of quantum optics to molecular quantum information and quantum biology are examined in relation to emerging interdisciplinary frontiers.

1. Introduction

The main purpose of this paper is to review the fundamental and crucial concepts of quantum coherence in chemical and biological systems, as a contribution to the special issue on open-shell systems. Over the past decades, open-shell systems have attracted great interest in several fields of science. Synthetic, spectroscopic, and theoretical approaches to open-shell systems have been summarized in over twenty papers in the memorial issue for Prof. Nakano. In a previous contributed review to the special issue [1], theoretical approaches to strongly correlated electron systems (SCES) have been examined for reliable computations of relative energies of the ground and low-lying excited states of open-shell atoms, molecules, and molecular aggregates. In particular, the energy gaps between singlet (S) and triplet (T) states of diradical species such as phenarenyl compounds have been investigated by the broken-symmetry (BS) unrestricted Hartree–Fock (UHF), unrestricted density functional theory (UDFT), and hybrid UHF-UDFT methods, followed by an approximate spin projection procedure (AP). The resonating BS (RBS) methods have also been reviewed to obtain the S-T gaps of SCES, supporting the size-consistent AP procedure. The natural orbital (NO) analysis of the BS methods has also been performed to elucidate the natural orbitals (UNO) or localized NO (ULO) and their occupation numbers (n) of SCES, which have been used for the definitions of several chemical indices such as diradical character (y). UNO (ULO) configuration interaction (CI) and coupled-cluster (CC) are also examined as beyond BS methods for open-shell species such as diradicals. The perturbative computational schemes of the nonlinear optical susceptibilities, such as second-order hyperpolarizability (γ) by Prof. Nakano and collaborators [1], were also reviewed in relation to theoretical investigations of γ-values of diradical species, which are well correlated with their y-values. The large γ-values are indispensable for the generation of the so-called quantum lights, such as the squeezed states, which are used to generate entangled states of light for quantum sensing and other applications.

However, fundamental concepts of quantum optics, such as quantum phase and quantum information, and progress in quantum measurements, such as the concept of disturbance, were not reviewed in the previous review [1], indicating the necessity of an extended review including them in this special issue. One hundred years have already passed since the discovery of quantum mechanics [2], which provides fundamental concepts such as wave–particle duality, quantum wavefunction, exclusion principle, quantum entanglement, Bell inequality, observation or measurement, quantum information, etc. [1]. Over the past decades, the so-called measurement problem has been investigated in several fields, such as quantum optics [1] and continuous variable (CV) quantum computation. The dual (particle and wave) nature of quantum matter has been investigated on both theoretical and experimental grounds. The dual property has been explained by Born’s stochastic and statistical explanation of the wavefunction [3]. The famous phrase “God does not play dice” by Einstein is noteworthy, and he is a great contributor to our understanding of quantum mechanics. The uncertainty relation by Heisenberg [4] also greatly impacted our understanding of quantum mechanics.

Quantum (or vacuum state) fluctuation is one of the fundamental concepts in quantum mechanics. The quantum fluctuation σq of an observable, for example, position <x> in quantum mechanics, is defined by the standard deviation σx: (σx)² = <(x − <x>)²> (=valance), and the deviation σp of the conjugated observable momentum <p> is similarly defined by (σp)² = <(p − <p>)²>. Quantum mechanics by Dirac is based on the commutation relation for conjugated variables expressed by a commutator [x,q] = px − xp = ih/2π (h: Planck constant) [5], which sharply contrasts with classical mechanics. This commutation relation entails the well-known Kennard inequality for the quantum fluctuation [6]: σxσp ≥ h/4π in quantum mechanics. Robertson [7] has provided the more general inequality for conjugated observables A and B; σAσB $\ge {\displaystyle \frac{1}{2}}|<\psi \left[A, B\right]\psi >|$ in quantum mechanics. It was also formulated in terms of entropy by Hirschman [8]. These relations are independent of the measurements examined by Heisenberg [4].

On the other hand, developments of experimental techniques and devices have required deep consideration and investigation of the uncertainty of real measurements [9,10,11,12,13,14] in relation to the original uncertainty relation by Heisenberg [4]. In fact, Ozawa [11] has derived an extended formula for it by including error of “measurement”, where ε_A represents the error (inaccuracy) of a measurement of an observable A and η_B the disturbance produced on a subsequent measurement of the conjugate variable B by the former measurement of A. The Ozawa inequality [11] is given as follows

$${\mathsf{\epsilon }}_q{\eta }_p+ {\mathsf{\epsilon }}_q{\mathsf{\sigma }}_p+ {\mathsf{\sigma }}_q{\eta }_p\ge \mathrm{h}/4\mathrm{\pi }$$

(1)

$${\mathsf{\epsilon }}_A{\eta }_B+ {\mathsf{\epsilon }}_A{\mathsf{\sigma }}_B+ {\mathsf{\sigma }}_A{\eta }_B \ge {\displaystyle \frac{1}{2}}|<\psi \left[A, B\right]\psi >|$$

(2)

where the first term in Equation (1) is responsible for Heisenberg’s uncertainty relation [4], and ${\mathsf{\sigma }}_Q$ is defined by (<Q²> − <Q>²)^1/2 (Q = A, B), like in the Robertson inequality [7]. Recently, Ozawa’s inequality [11] has been confirmed by spin system (A, B: x and y-component of neutron spin) experiments by Hasegawa et al. [12]. Thus, examinations of classic ideas and thinkings are in progress in several fields of science [13,14]. The Ozawa inequality provides one of the guiding principles for the design of quantum devices such as a gravitational wave detector [15]. The situation is the same for the developments of quantum devices for quantum information (such as MBQC) and quantum biology.

Classical and quantum optics have been significantly developed [1], providing fundamental information about light and the foundations of optical devices. The uncertainty relation has also been expressed by using the information entropy in the quantum information theory [8]. Historically, the dual nature of light has been elucidated with several experiments, such as the photoelectric effect experiment [16] (particle nature of photon) and the double slit experiment [17] (interference; wave nature of light) (c.f. wave nature of electron [18]). The fluctuation of energy (ΔE) and time (Δt) exhibits a well-known example of the uncertainty relation ΔEΔt ≥ h/4π, which was examined using the “Einstein’s Photon Box” in debates with Bohr. The fluctuations of the number density (Δn) of the photon and quantum phase (Δϕ) exhibit a well-known example of the Kennard–Heisenberg Uncertainty Principle ΔnΔϕ ≥ h/4π [2,6]. However, even under this limitation, many optical devices have already been developed in several fields of science and engineering.

On the other hand, the so-called Einstein–Podolsky–Rosen (EPR) paradox [19] “Can quantum mechanical description of physical reality be considered complete?” was presented to examine the non-local nature of quantum mechanics. Superposition and entanglement of quantum light have been considered as fundamental concepts for understanding the EPR paradox, opening up the quantum information science [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Recently, the quantum nature of light has indeed been accepted, and it is of great interest in the development of the optical quantum computer [20], quantum sensing [21], etc. The merits of the optical quantum devices are that they do not require freezer and vacuum equipment, and have the possibility for scaling up or quantum internet. Very recently, loop-based optical computing using continuous variables by squeezed light [1] for the generation of its entanglement state and time-division multiplexing is now developing [20]. Linear and nonlinear optical responses of atoms and molecules have also been investigated for the generation and construction of quantum light and entangled states [1]. A number of experimental and theoretical works for molecular quantum information science have been reported in chemistry journals.

In a previous review [1] concerning work by the late Prof. Nakano and collaborators, we have summarized mainly the perturbative approaches (see Appendix A) to the optical responses of organic materials such as organic diradicals. The three-states model based on the approaches has been found to be effective for elucidating the third-order hyperpolarizability (γ) of open-shell species such as organic diradicals (see Appendix A) [22]. However, the perturbative model is often insufficient for understanding and explaining the frequency-dependent optical properties of π-conjugated systems, for which the contribution of the higher harmonic generation (HHG) is not negligible. The time-dependent Schrödinger equation (TDSE) provides the theoretical foundations for the elucidation and understanding of such nonlinear responses. The difficulty of the exact solution of TDSE for molecules with moderate sizes has entailed developments of several approximate methods such as the quasi-energy derivative (QED) methods [23,24,25,26]. The QED methods have been extended to elucidate optical responses of open-shell species such as diradicals [1]. On the other hand, spectroscopic experiments provide response properties with relaxation effects [1], indicating the necessity of the density matrix description. To this end, Nakano et al. [27] have developed the numerical Liouville approach (NLA) without the rotational wave approximation (RWA) to the theoretical investigations of such nonlinear responses involving environmental effects. The non-perturbative NLA approach has provided a general expression of the higher harmonic generation (HHG) expected for atoms and molecules by using strong laser fields.

In the 1980s, a generation of the HHG spectra [28,29], such as extremely high orders (13–31 orders) [30], was discovered in the photo-ionization processes of rare gases such as Ne, Xe, etc. by using a strong low-frequency laser field (Ti-sapphire) [31,32], indicating a breakdown of the perturbative approach. Moreover, developments of laser technologies for HHG have provided attosecond laser pulses, which are now applied to elucidate attosecond motions of electrons (for example, 150 attosecond motion of an electron in the Bohr model of a hydrogen atom) in atoms and molecules, opening up the attosecond science and spectroscopy [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. The atomic and molecular orbital tomography [49,68] has also been developed to observe the amplitude and phase of wave functions in the moment space, for which the Fourier transformation provides the shapes and phase of wave functions in real space. The orbital phase of molecular orbitals plays an important role in understanding and explaining chemical reaction mechanisms. Thus, the numerical Liouville approach (NLA) is supported by the new orbital tomography experiment [49,68].

However, the name of a phase in quantum theory means several physical contexts. Conventionally, phase denotes the global phase for the wavefunction <e^iθ|ψ>, which diminishes in the case of observation; <μ> = <e^−iθψ*|μ|ψe^iθ> = <ψ*|μ|ψ>, where μ means the dipole moment operator in quantum optics. Therefore, from the observation point of view, we may ignore global phase factors as discussed in many papers of quantum science, and we can only discuss probabilities of finding an electron and the observed values of quantum operators. On the other hand, there is another kind of phase known as the relative phase in quantum information science [49,68], including Bell states in the previous review, which has quite a different meaning [1]; the magnitude of the amplitude of the wave function is the same, but the signs are different, namely 1 and −1. In fact, the orbital tomography method has revealed both the amplitude and phase of wave functions of molecules such as N₂ and O₂. The in-phase and out-of-phase combinations in the wavefunction are directly related to the formation of chemical bonds in quantum chemistry. Thus, recent attosecond chemistry arising from HHG has opened the door for observation of the wave function instead of the square of the wave function (namely, density), which has been detected by the X-ray diffraction method.

Nowadays, orbital tomography is established to investigate the wave function itself [49,68]. Therefore, this technique is expected to provide indispensable information on surface science and catalysts, as in the case of scanning tunnel microscope (STM) experiments [69,70]. However, serious questions regarding the observation of “orbitals” [71,72,73] have been presented for the STM experiments [69,71] and orbital tomography [49,68]. Dyson orbital concepts are now inevitable for theoretical descriptions of wavefunctions [74,75,76,77,78,79] and understanding the observed images by STM [69,70] and orbital tomography [49,68]. Orbital phase and coherence are now important for macroscopic quantum systems such as Bose–Einstein condensates (BEC) of Rb atom gas, superconducting boxes, and light-harvesting antenna systems, as reviewed in this article.

In this review, we would like to revisit the non-perturbative approaches [23,24,25,26,27] to both low and high harmonic generation (HG) in optical responses of atoms and molecules. The quasi-energy derivative (QED) [23,24,25,26] and numerical Liouville (NLA) [27] methods for open-shell and excited states are reviewed as non-perturbative methods in Section 2 and Section 3, providing theoretical expressions of the n-th order HG. The high harmonic generation (HHG) obtained by above-threshold ionizations (ATI) [30,33,34] is reviewed in Section 4 in relation to the generation of attosecond pulses for attosecond spectroscopy [42,43,44] and molecular orbital tomography [49,68]. Important roles of quantum phases and possible explanations of “orbitals”, such as HOMO, SOMO, LUMO, in several models in quantum chemistry [74,75,76,77,78,79] are examined in relation to recent experiments such as STM and molecular orbital tomography. Dyson orbitals [74,75,76] are also reviewed in relation to the ionized states of diradicals. Interactions between molecules and quantum lights are also reviewed in relation to quantum dynamics and quantum information, providing indispensable information for the active controls of the quantum phase of quantum devices such as quantum computers [20] and quantum sensing devices [21], as shown in Section 5. The time-dependent master equation approach to quantum effects, such as coherent states for excitons (singlet diradical in the excited state) and electron energy transfers (EET) in light-harvesting systems and model dendrimers, is also reviewed in relation to photosynthesis in Section 6. Applications of quantum optics to quantum coherent controls and quantum biology are reviewed on the basis of recent developments, such as the upconversions of low-energy lights in Section 7. Finally, future prospects of quantum information and quantum biology are given briefly.

2. Nonlinear Responses of Molecular Materials

2.1. Linear and Nonlinear Responses of Molecular Materials for Electronic and Magnetic Fields

Here, the historical developments of the chemistry of open-shell species are briefly revisited. In the 1970s, local spins in open-shell species such as diradicals were of early interest in quantum chemical investigations of symmetry-forbidden reactions [1]. In the 1980s, quantum chemistry of molecular aggregates of closed-shell species such as water molecules had been developed extensively. On the other hand, the quantum chemistry of aggregates of open-shell species, such as diradicals, was still in an early stage, and therefore, we proposed several types of organic magnetic materials for chemical synthesis. In the 1990s, molecule materials using local spins were developed significantly because of the discoveries of organic ferromagnets with long-range Neel order [1]. In 2000, single molecular magnets, quantum spins, and molecular qubits were interesting targets in relation to quantum effects, even in the field of chemistry. In 2010, molecular-based materials were investigated in relation to the generation of qubits for quantum computing [20] and quantum sensing [21].

Equivalent transformation between quantum spin and quantum light was a guiding principle for our group [1]. Since the 1980s, material-oriented quantum chemistry has been one of the main interests of the theoretical chemistry group at Osaka University [1]. Optical and magnetic responses of molecules and molecular crystals have attracted great interest for the group because of several reasons [1]. Nakano and his collaborators have obtained academic interest in fundamental problems of nonlinear responses of molecular materials [21,22,27]. They are also interested in optical and magnetic properties of open-shell molecules, such as diradicals, for material applications. Therefore, the linear and nonlinear responses of materials to electronic and magnetic fields are briefly revisited to introduce and understand their work and underlying basic concepts and computational procedures [1,22]. The energy changes of molecular materials under the electro (F)-magnetic(B) fields are generally expanded as follows [80]:

$$\begin{array}{ll}\Delta E=& -{\mu }_i{F}_i-{\displaystyle \frac{1}{2}}{\alpha }_{ij}{F}_i{F}_j-{\displaystyle \frac{1}{6}}{\beta }_{ijk}{F}_i{F}_j{F}_k-{\displaystyle \frac{1}{24}}{\gamma }_{ijkl}{F}_i{F}_j{F}_k{F}_l-\cdots \\ & -m_i{B}_i-{\displaystyle \frac{1}{2}}{x}_{ij}{B}_i{B}_j-{\displaystyle \frac{1}{6}}{Y}_{ijk}{B}_i{B}_j{B}_k-{\displaystyle \frac{1}{24}}{\varsigma }_{ijkl}{B}_i{B}_j{B}_k{B}_l-\cdots \\ & -G_i{F}_i{B}_j-{\displaystyle \frac{1}{2}}{\xi }_{ijk}{F_{i}B}_j{B}_k-{\displaystyle \frac{1}{2}}{X}_{ijk}{F}_i{F}_j{B}_k-{\displaystyle \frac{1}{4}}{\eta }_{ijkl}{F}_i{F}_j{B}_k{B}_l-\cdots \end{array}$$

(3)

where i, j, k, l denote the x, y, z coordinates, and μ_i, α_ij, β_ijk, and γ_ijkl are referred to as dipole moment, polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The magnetic responses, such as the magnetic dipole moment (m) and magnetic susceptibility (x_ij), are also given in Equation (3). The optical responses are examined in this memorial issue. The first hyperpolarizability β_ijk (−2ω; ω, ω) is the origin of second harmonic generation (SHG) [1,22,23,24,25,26,27,80]. The second hyperpolarizability γ_ijkl (−3ω; ω, ω, ω) is the origin of third harmonic generation (THG). SHG and THG are applicable to investigate optical responses of molecular materials [27,80]. The final term arising from both electronic and magnetic fields is responsible for the Cotton–Mouton effect [1] of molecular materials.

In this review, the macroscopic optical properties are mainly explained in relation to quantum entanglement and quantum information processing, as illustrated in Figure 1. The macroscopic polarizability is defined by

$$P^I=\sum {{\chi }_{IJ}}^{\left(1\right)}{F}^J\left({\omega }_1\right)+\sum {{\chi }_{IJK}}^{\left(2\right)}{F}^J\left({\omega }_1\right)F^K\left({\omega }_2\right)+\sum {{\chi }_{IJKL}}^{\left(3\right)}{F}^J\left({\omega }_1\right)F^K\left({\omega }_2\right)F^L\left({\omega }_3\right)+\cdots$$

(4)

where P^I denotes the I-th component of polarizability at the experimental coordinate, and ω is the frequency of the polarization. The macroscopic observables χ_IJ⁽¹⁾, χ_IJK⁽²⁾, and χ_IJKL⁽³⁾ are referred to as linear, second-order, and third-order optical susceptibilities, respectively. The origin of the macroscopic polarization in Equation (4) is the microscopic polarization of atoms and molecules in molecular aggregates under examination [1]. Polarization pⁱ at the microscopic level is therefore defined by the difference of the dipole moment with and without the electronic field as

$$\begin{array}{ll}{p}^i& =-{\mu }_{tot}^i-{\mu }_0^i\\ & =\sum {{\alpha }_{ij}}^{\left(1\right)}{F}^j\left({\omega }_1\right)+\sum {{\beta }_{ijk}}^{\left(2\right)}{F}^j\left({\omega }_1\right)F^k\left({\omega }_2\right)+\sum {{\gamma }_{ijkl}}^{\left(3\right)}{F}^j\left({\omega }_1\right)F^k\left({\omega }_2\right)F^l\left({\omega }_3\right)+\cdots \end{array}$$

(5)

where Fⁱ denotes the local electronic field for the molecule. The ${\mu }_{tot}^i$ and ${\mu }_0^i$ denote the i-component of the total and permanent dipole moments, respectively.

Macroscopic second-order χ_IJK⁽²⁾ susceptibility disappears as shown from Equations (3) and (4), in the case of molecular crystals with the inversion center, even if the microscopic β_ijk is not zero. In fact, the β_ij value is large for para-nitroaniline, but its crystal exhibits no χ_IJK⁽²⁾ susceptibility because of the crystal symmetry. This property is also observed for high harmonic generation (HHG) discussed in Section 4. This in turn indicates the utility of the SHG method for investigating the structure and bonding of molecules on surfaces, etc. [81,82,83]. The third-order susceptibility χ_IJKL⁽³⁾ is directly related to γ_ijklf (local field correction coefficient). Details of f are not touched on in this review [1]. These nonlinear response crystals are indispensable for the generation of quantum lights [20,21].

2.2. Non-Perturbative Methods of Linear and Nonlinear Responses of Materials

Response properties of molecules and molecular aggregates are highly sensitive to computational methods and basis sets employed. Theoretical computational schemes of linear and nonlinear optical responses [1] are generally classified into four methods: (i) sum-over-states (SOS); (ii) response theoretical methods; (iii) finite field (FF) method; (iv) numerical quantum Liouville equation method [27]. In a previous review, we have examined the SOS and FF methods [22]. On the other hand, time-dependent (TD) variation principle (VP) methods based on the semi-classical method [22,80,81,82,83,84] using the classical electro-magnetic field and quantum electronic states of molecules are used for several types of response theories in the type (ii) approach.

Several types of non-perturbative methods have been reported for computations of linear and nonlinear optical responses [1,80,81,82,83,84,85,86,87,88,89]. Here, we review the quasi-energy derivative (QED) methods developed by Sasagane, Aiga, and Itoh [23,24,25,26]. They have derived response properties (ii) by the Taylor expansion forms of QED; it is noteworthy that this name is different from the quantum electrodynamic (QED) used in the quantum field theory [1]. The QED formula at the Hatree–Fock (HF) level of theory is reduced to the TD HF (TDCHF, TDCPHF, RPA) method [84,85,86,87,88]. However, the HF method does not involve the electron correlation effect [89,90]. Kobayashi et al. [90,91,92] have used the Møller-Plesett (MP2) perturbation method for the QED computations of both closed and open-shell molecules, indicating the importance of correlation correction. More extensive methods, such as the coupled cluster (CC) methods, have been examined in a recent review [1]. Aiga et al. [26] have developed a QED-CC method.

2.3. Quasi-Energy Derivative Methods of Linear and Nonlinear Responses

In this section, essential features of the QED methods based on the time-dependent (TD) variation principle are given [23,24,25,26]. The frequency-dependent responses are given by the interaction with the time-dependent external field as

$$F_i\left(t\right)={\sum }_{i=-n}^n{\epsilon }_a\left(\omega \right)e^{-i{\omega }_{a}t}, a=x, y, z$$

(6)

where ${\epsilon }_a\left(\omega \right)$ denotes a component of the intensity of the Fourier component of the external field. The interaction operator of a molecule with the external field is approximately given by that of the dipole moment operator and the external field as

$${\mathit{H}}_i\left(t\right)={\sum }_{i=-n}^n{\mathit{\mu }}_i{\epsilon }_a\left(\omega \right)e^{-i{\omega }_{i}t}.$$

(7)

To calculate the responses, the time-dependent (TD) Schrödinger equation is solved as follows.

$$\left\{\mathit{H}\left(r, t\right)-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathrm{t}}}\right\}\mathrm{\Psi }\left(r,t\right)=0 (\mathrm{\hslash }=1)$$

(8)

where the total Hamiltonian is defined as

$$\mathit{H}\left(r,t\right)={\mathit{H}}_0\left(r\right)+{\mathit{H}}_1\left(r,t\right).$$

(9)

The second term denotes the time-dependent interaction Hamiltonian. On the other hand, the first part denotes the time-independent Hamiltonian. Therefore, the Schrödinger equation for the time-independent Hamiltonian is solved to obtain the stational energy and wavefunction as follows.

$$\left\{{\mathit{H}}_0\left(\mathrm{r}\right)-{{\rm E}}_0\right\}{\Phi }_0\left(\mathrm{r}\right)=0$$

(10)

where ${\mathit{H}}_0\left(r\right)$ is a non-relativistic Born–Oppenheimer electronic Hamiltonian. This equation is solved by various quantum mechanical methods discussed in the previous review [1]. The phase factor and the other part are separated to obtain the wave function as

$$\Psi \left(r,t\right)=\Phi \left(r,t\right)\mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{i}{\int }_{t_0}^{t}W\left(t\mathrm{\prime }\right)\mathrm{d}t\mathrm{\prime })$$

(11)

where W(t’) is referred to as the quasi energy. From Equations (8) and (11), we obtain the TD Schrödinger equation for the real part $\Phi \left(r, t\right).$

$$\left\{\mathit{H}\left(r,t\right)-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathrm{t}}}-W\left(r,t\right)\right\}\Phi \left(r, t\right)=0$$

(12)

where the quasi-energy W(r, t) is given by

$$W\left(r, t\right)=<\Phi \left(r, t\right)\left| \mathit{H}\left(r, t\right)- {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t }}\right|\Phi \left(r, t\right)>.$$

(13)

The exact wavefunction of the Schrödinger equation in Equation (8) is hardly obtained for many electron molecules. Therefore, an approximate wavefunction is obtained by the TDVP as follows

$$\delta <\psi (r, t) \left| \mathit{H}\left(r, t\right)-i {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\right|\psi (r, t) >+ i{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}<\psi \left(r, t\right)|\delta \psi (r, t)> =0$$

(14)

where $\delta \psi (r, t)$ denotes the variation of the approximate wavefunction $\psi (r, t)$ and the approximate quasi-energy $w\left(r,t\right)$ (see Equation (15)) is defined as the approximation for W(r, t).

Within the above approximation, Equations (11) and (13) are also applicable for $w\left(r,t\right)$ and $\varphi (r, t)$, where $\varphi (r, t)$ is an approximation of Φ(r, t) in Equation (13). Moreover, the time-dependent (TD) dipole moment in Equation (5) is also available as an expectation value of the dipole moment operator under the assumption of the TD Hellman-Feynman theorem [92] as

$${\mu }_{\mathrm{A}}\left(\epsilon ,t\right)=-\left\{{\displaystyle \frac{\mathsf{\partial }w\left(t\right)}{\mathsf{\partial }{\epsilon }_{\mathrm{A}}\left(\mathrm{w}\right)}}+i{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\mathrm{t}}}<\varphi \left(r,t\right)|{\displaystyle \frac{\mathsf{\partial }\varphi \left(r,t\right)}{\mathsf{\partial }{\epsilon }_{\mathrm{A}}\left(\mathrm{w}\right)}}>\right\}\mathrm{e}\mathrm{x}\mathrm{p} (i\omega t)$$

(15)

Sasagane et al. [23] have found the condition to vanish the second term in Equation (15), providing the computational scheme like Equation (5), which is expanded as a Taylor series of the electronic fields. According to their theory, the response properties are expressed with the derivatives of the quasi-energy derivatives with the external fields as

$$\begin{array}{l}{\chi }_{A_1{A}_2{A}_3{A}_4\dots ..A_{n-1}{A}_n}^n\left(-{\omega }_{\sigma } : {\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4,\dots {\omega }_n\right)\\ ={\displaystyle \frac{{\mathsf{\partial }}^{n+1}w(t)}{\mathsf{\partial }{\epsilon }_A\left(-{\omega }_{\sigma }\right)\mathsf{\partial }{\epsilon }_{A_1}\left({\omega }_1\right)\mathsf{\partial }{\epsilon }_{A_2}\left({\omega }_2\right)\mathsf{\partial }{\epsilon }_{A_3}\left({\omega }_3\right)\dots \mathsf{\partial }{\epsilon }_{A_n}\left({\omega }_n\right)}}\end{array}$$

(16)

where ε_x(ω) denotes the x-component of the electronic field. Some of the nonlinear responses by the quasi-energy derivatives (QED) are summarized in

Table 1. Response properties, optical processes, and formulas to the external electromagnetic field.

Response Property	Optical Process	Formula
${\beta }_{ijk}\left(0;\mathrm{0,0}\right)$	Second-order response to the electrostatic field	${\beta }_{ijk}\left(0;\mathrm{0,0}\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{3}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\mathsf{\partial }{\epsilon }_j\mathsf{\partial }{\epsilon }_k}}\right|}_{\epsilon =0}$
${\beta }_{ijk}\left(-\omega ;\omega ,0\right)$	Pockels effect	${\beta }_{ijk}\left(-\omega ;\omega ,0\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{3}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-\omega \right)\mathsf{\partial }{\epsilon }_j\left(\omega \right)\mathsf{\partial }{\epsilon }_k}}\right|}_{\epsilon =0}$
${\beta }_{ijk}\left(-2\omega ;\omega ,\omega \right)$	Second harmonic generation (SHG)	${\beta }_{ijk}\left(-2\omega ;\omega ,\omega \right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{3}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-2\omega \right)\mathsf{\partial }{\epsilon }_j\left(\omega \right)\mathsf{\partial }{\epsilon }_k\left(\omega \right)}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(0;\mathrm{0,0},0\right)$	Third-order response to the electrostatic field	${\gamma }_{ijkl}\left(0;\mathrm{0,0},0\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\mathsf{\partial }{\epsilon }_j\mathsf{\partial }{\epsilon }_k\mathsf{\partial }{\epsilon }_l}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(-3\omega ;\omega ,\omega ,\omega \right)$	Third harmonic generation (THG)	${\gamma }_{ijkl}\left(-3\omega ;\omega ,\omega ,\omega \right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-3\omega \right)\mathsf{\partial }{\epsilon }_j\left(\omega \right)\mathsf{\partial }{\epsilon }_k\left(\omega \right)\mathsf{\partial }{\epsilon }_l\left(\omega \right)}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(-2\omega ;\omega ,\omega ,0\right)$	Electric-field-induced second harmonic generation (ESHG)	${\gamma }_{ijkl}\left(-2\omega ;\omega ,\omega ,0\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-2\omega \right)\mathsf{\partial }{\epsilon }_j\left(\omega \right)\mathsf{\partial }{\epsilon }_k\left(\omega \right)\mathsf{\partial }{\epsilon }_l}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(-\omega ;\omega ,\omega ,-\omega \right)$	Degenerate four-wave mixing (DFWM)	${\gamma }_{ijkl}\left(-\omega ;\omega ,\omega ,-\omega \right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-\omega \right)\mathsf{\partial }{\epsilon }_j\left(\omega \right)\mathsf{\partial }{\epsilon }_k\left(\omega \right)\mathsf{\partial }{\epsilon }_l\left(-\omega \right)}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(-\omega ;\mathrm{0,0},\omega \right)$	dc-Kerr effect	${\gamma }_{ijkl}\left(-\omega ;\mathrm{0,0},\omega \right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-\omega \right)\mathsf{\partial }{\epsilon }_j\mathsf{\partial }{\epsilon }_k\mathsf{\partial }{\epsilon }_l\left(\omega \right)}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(-2{\omega }_1+{\omega }_2;{\omega }_1,{\omega }_1,{-\omega }_2\right)$	Coherent anti-Stokes Raman scattering (CARS)	${\gamma }_{ijkl}\left(-2{\omega }_1+{\omega }_2;{\omega }_1,{\omega }_1,{-\omega }_2\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-2{\omega }_1+{\omega }_2\right)\mathsf{\partial }{\epsilon }_j\left({\omega }_1\right)\mathsf{\partial }{\epsilon }_k\left({\omega }_1\right)\mathsf{\partial }{\epsilon }_l\left({\omega }_2\right)}}\right|}_{\epsilon =0}$
${\gamma }_{ijkl}\left(-{\omega }_2;{\omega }_1,-{\omega }_1,{\omega }_2\right)$	ac-Kerr effect	${\gamma }_{ijkl}\left(-{\omega }_2;{\omega }_1,-{\omega }_1,{\omega }_2\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-{\omega }_2\right)\mathsf{\partial }{\epsilon }_j\left({\omega }_1\right)\mathsf{\partial }{\epsilon }_k\left({-\omega }_1\right)\mathsf{\partial }{\epsilon }_l\left({\omega }_2\right)}}\right|}_{\epsilon =0}$
${\eta }_{ij,mn}\left(-\omega ;\omega ,\mathrm{0,0}\right)$	Cotton–Mouton effect	${\eta }_{ij,mn}\left(-\omega ;\omega ,\mathrm{0,0}\right)=-{\left.{\displaystyle \frac{{\mathsf{\partial }}^{4}w\left(t\right)}{\mathsf{\partial }{\epsilon }_i\left(-2\omega \right)\mathsf{\partial }{\epsilon }_j\left(\omega \right)\mathsf{\partial }{b}_m\mathsf{\partial }{b}_n}}\right|}_{\epsilon ,b=0}$

.

As mentioned above, QED methods are applicable to the restricted Hatree–Fock (RHF), MP2, limited CI, CC, and others [1,23,24]. Karna proposed the TDVP approach to the unrestricted Hatree–Fock method (UHF), namely TDUHF which was applied to calculate nonlinear responses of open-shell species [86,87]. The QED method is also applicable to the more general Hatree–Fock (GHF) method [93] for complex radicals and the general Hartree–Fock–Bogoriuvov (GHFB) method of open-shell systems [94]. TD Hatree–Fock (HF) methods, such as TDRHF, TDUHF, and TDGHF, are applicable for relatively large systems. TD density functional theory (DFT) methods are also useful for such systems, as shown below [95,96,97,98].

2.4. TD HF and TD DFT Computations of Nonlinear Responses

The QED method is reduced to the TD HF (TDCHF, TDCPHF) when the HF wavefunction is employed as an approximation to the many-electron wave function. The TDHF is also equivalent to the random phase approximation (RPA) in solid-state physics [89,99,100,101,102]. The exponential type formulation of the wavefunction is often used for the derivation of the response properties as

$$|\varphi \left(\mathrm{t}\right)>=\mathrm{e}\mathrm{x}\mathrm{p}[\mathit{K}\left(\mathrm{t}\right)\left]\right|\mathrm{H}\mathrm{F}>$$

(17)

where the excitation operator is expressed with excitation and de-excitation operators as

$$\mathit{K}(t)={\sum }_p\left\{ k_p(t)q_v^{\dotplus }-k_v^{*} q_v\right\}.$$

(18)

The $q_v^{\dotplus }$ and $q_v$ denote the creation and annihilation operators defined by using the occupied (s) and virtual (r) orbitals [1] as

$$q_v^{\dotplus }=q_{rs}^{\dotplus }=a_r^{\dotplus }{a}_s, q_v=q_{rs}=a_s^{\dotplus }{a}_r.$$

(19)

The excitation operator $\mathit{K}\left(t\right)$ is determined by the TDVP HF calculation. The expectation value of the dipole moment operator is given by the $\mathit{K}\left(t\right)$ operator as

$${\mu }_A\left(\mathrm{\epsilon },\mathrm{t}\right)=<\mathsf{\varphi }\left(\mathrm{t}\right)\left|{\mathit{\mu }}_A\right|\mathsf{\varphi }\left(\mathrm{t}\right)> = <\mathrm{H}\mathrm{F}\left|\mathrm{e}\mathrm{x}\mathrm{p}\right[-\mathit{K}\left(t\right)\left]{\mathit{\mu }}_A\mathrm{e}\mathrm{x}\mathrm{p}\right[\mathbf{K}\left(\mathrm{t}\right)\left]\right|\mathrm{H}\mathrm{F}>$$

(20)

$$=<\mathrm{H}\mathrm{F}\left|\left\{{\mathit{\mu }}_A+1/2\left[\left[{\mathit{\mu }}_A,\mathit{K}\left(t\right)\right], \mathit{K}\left(t\right)\right]+\dots \right\}\right|\mathrm{H}\mathrm{F}>.$$

(21)

Therefore, the optical responses are analytically expressed with orbital energies and other factors at the HF level of the approximation because of Equations (17) and (21). The long explicit expressions of them are here slipped for brevity. They are given in the original papers [103,104]. To obtain the spectra expressions of the responses, the diagonalization of [$q_v^{\dotplus }$, $q_v]$ is performed to obtain the diagonalized excitation operators ${[O}_p^{\dotplus }$, $O_p] \mathrm{a}\mathrm{s}$

$$O_p^{\dotplus }={\sum }_v\left\{q_v^{\dotplus }{X}_{vp} + q_v{Y}_{vp}\right\}$$

(22)

$$O_p={\sum }_v\left\{q_v^{\dotplus }{Y}_{vp}^{*} + q_v{X}_{vp}^{*}\right\}$$

(23)

where the transformation vectors are obtained using the random-phase-approximation (RPA) equations [101] as

$$\left[\left(\begin{array}{cc}A& B\\ B^{*}& A^{*}\end{array}\right)-{\omega }_p\left(\begin{array}{cc}S& ∆\\ {-∆}^{*}& {-S}^{*}\end{array}\right)\right]\left(\begin{array}{c}{X}_p\\ Y_p\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$

(24)

where the explicit expressions of the matrix elements A, B, etc. are given by the HF (RHF, UHF [27], GHF [93], GHFB [94]) orbital energies and wavefunctions. The explicit expressions of the integrals are given in the original papers [22,27]. Karna [87] has indeed implemented the frequency-dependent (F-D) polarizabilities and first hyperpolarizabilities using the time-dependent (TD) spin-unrestricted Hatree–Fock (TDUHF) theory. Similar equations are also derived for the DFT (RDFT, UDFT, GDFT) solutions [95,96,97,98], where the matrix elements are evaluated by the DFT methods. Aiga et al. [98] have derived the QED-DFT method.

2.5. Computations of Nonlinear Responses for Open-Shell Atoms and Molecules

Both spin-restricted (ROHF) [90,92] and unrestricted (UHF) [86,87] methods are employed to calculate linear and nonlinear optical responses of atoms and molecules. Kobayashi et al. [90] have derived the QED formula of frequency-dependent (F-D) polarizability (α) based on the spin-restricted open-shell Hatree–Fock (ROHF) method. They have implemented the computational programs, for which the smooth-tempered basis sets (spd) by Sasagane et al. [23] are used for quantitative computations. They have calculated α-values of small radicals: BeH, MgH, CaH, CN, and NH₂ [91]. The computational results for NH₂ [92] are summarized in

Table 2. Polarizabilities of NH₂ radical by QED-ROHF and QED-ROHF + MP2 methods.

Method	0.000	0.005	0.015	0.020	0.025	0.035	0.045
αzz (QED-ROHF)	11.74	11.74	11.75	11.76	11.77	11.80	11.84
αaver (QED-ROHF) (a)	11.17	11.17	11.19	11.20	11.22	11.29	11.39
αzz (QED-ROHF + MP2)	13.18	13.18	13.20	13.21	13.22	13.26	13.32
αaver (QED-ROHF + MP2)	12.21	12.22	12.23	12.25	12.28	12.34	12.45
αzz (MP2 Contribution) (b)	10.95	10.95	10.97	10.99	11.01	11.06	11.13
αaver (MP2 Contribution)	8.57	8.57	8.57	8.57	8.67	8.56	8.55

^(a) αaver = (αxx + αyy + αzz)/3, ^(b) contribution by MP2 (={α(QED)[ROHF + MP2] [91]; −ROHF}/α(QED)[ROHF + MP2] is expressed with %.

. From Table 2, the contribution of the dynamical correlation corrections to the calculated F-D ORHF polarizability is in the range 8–11% for NH₂. However, it becomes 10–21% for CN radical, indicating the increase in the correlation correction contribution with the increase in the strength of the frequency (ω). The QED-MP2 method is reliable for open-shell systems without strong non-dynamical correlations [91].

Kobayashi et al. [92] have also derived the QED formula of frequency-dependent (F-D) nonlinear responses based on the spin-restricted open-shell Hatree–Fock (ROHF) method, and they have implemented the in-house QED-ROHF program for computations of open-shell species [1]. They have performed the QED-ROHF calculations of F-D second hyperpolarizabilities of the ground doublet Li, Na, Ka and quartet nitrogen atoms using very large basis sets: third harmonic generation [γ (−3ω; ω, ω, ω)], (static) electric field-induced second harmonic generation (ESHG) [γ (−2ω; ω, ω, 0)], degenerate four-wave mixing (DFWM) [γ (−ω; ω, ω, −ω)], the dv-Kerr effect [γ (−ω; 0, 0, ω), etc. Many computational results [90,91,92,93,94] are abbreviated in this review.

The multi-determinant methods have also been used for open-shell systems [103] such as diradical systems. Albertsen et al. [104] have performed the MCSCF time-dependent (MCTDHF) calculations of F-D polarizability of molecular oxygen, indicating the utility of MCTDHF for open-shell species with non-dynamical correlations. Dynamical correlations are further necessary for quantitative purposes, as shown below.

2.6. Coupled-Cluster Approaches for Nonlinear Responses

Electron correlation effects play important roles in nonlinear responses of molecules. Historically, Rice and Handy [105] calculated the frequency-dependent (F-D) first hyperpolarizabilities for the electro-optic Pockets effect (EOPE) [β (−ω; ω, 0)] and second harmonic generation (SHG) [β (−2ω; ω, ω)] using the MP2 perturbation theory based on the pseudo-energy derivative (PED) method. Applications of the PED-MP2 to nonlinear response properties have also been performed by Bishop et al. [106]. On the other hand, Hättig et al. [107,108,109] have reported the first application of coupled-cluster (CC) single (S) and double (D) quadratic response (CCSDQR) approach to the F-D first hyperpolarizabilities of the FH molecule and presented the first implementation of the F-D second hyperpolarizabilities using CCSD cubic response (CCSDCR) theory. Gauss et al. [110] reported the implementation of the first F-D hyperpolarizabilities at the CC3 level in the framework of CC response theory. However, in the CC response calculations by Hättig et al. [107,108,109] and Gauss et al. [110], the orbital relaxation in the presence of the electronic field is not explicitly treated but only implicitly included by the single excitation part of the cluster operator. Rozyczko and Bartlett [111] derived the F-D hyperpolarizabilities of any order and any process recursively based on the equation-of-motion (EOM) CC method, where normalization terms were neglected as pointed out by Hättig et al. [112]. After the criticisms by Hättig et al. [112], their EOM-CC formulation of the F-D hyperpolarizabilities has been modified to include the renormalization terms [113]. The size-consistent EOM-CC approach is nowadays a standard dynamical correlation-included scheme for F-D nonlinear responses for quantitative purposes [114]. The computational results for a parent molecule by EOM-CC are also used as references of useful functionals of the TD DFT computations of related large molecules.

Kobayashi et al. [91] have performed the QED-MP2 calculations of the dipole moment, static polarizability, and first hyperpolarizability of FH, H₂O, CO, and NH₃ in comparison with the CC approaches. The computational results for NH₃ are summarized in

Table 3. Dipole moment, static polarizabilities, and first hyperpolarizability energies of NH₃ by QED-MP2 method.

Method	μz	αzz	αave	βzzz	βzxx	βyxx	β||
RHF SCF (a)	0.636	13.27	12.93	−6.980	−6.728	9.410	−12.26
RHF SCF (b)	0.636	13.32	12.96	−11.67	−7.293	8.665	−15.76
RMP2 (a)	0.600	15.71	14.41	−31.15	−7.553	9.086	−27.75
RMP2 (b)	0.599	15.82	14.47	−37.93	−8.235	6.911	−32.64
EOM-CCSD (c)	---	---	---	−41.0	−9.1	7.2	−35.9
RCCSD(T) (d)	0.590	15.71	14.38	−39.6	−8.8	---	−34.3
Exp (e)	0.578	---	14.56	---	---	---	---

^(a) [5s3p2d/3s2p] (ref. [1]), ^(b) [5s3p2df/3s2p2d] (ref. [1]), ^(c) [5s3p2d/3s2p] (ref. [91]), ^(d) [5s3p2d/3s2p]. (ref. [91]), ^(e) ref. [92].

. From Table 3, the dipole moments calculated by RHF are a little larger than those of the RHF MP2 (RMP2) values, which are also close to the calculated value by RHF CCSD(T) (RCCSD(T)) [115] and the experimental value [116]. Static polarizabilities by RHF are smaller than the corresponding RMP2 and RCCSD(T) values, which are consistent with the experimental value. The magnitude of the first hyperpolarizability β_zzz by RMP2 is about three times larger than that of RHF, indicating a significant contribution of the dynamical correlation effect. The RMP2 value with the large basis set is compatible with those of the EOM-CCSD and RCCSD(T). Thus, the MP2-type method with reliable basis sets is useful for a first step to the correlation approach. As a next step, the CC-type methods are necessary for quantitative computations of the β-values (β_zzz, β_||). Dynamical correlation corrections are also expected to be important for the above γ-values. Indeed, Neogrády et al. [117] performed the ROHF-CCSD(T) computations of triplet molecular oxygen, one of the open-shell molecules in this special issue. Nowadays, EOM-CCSD(T) methods are implemented in several program packages [118].

3. Numerical Liouville Approaches to Frequency-Dependent Nonlinear Responses

3.1. Nonperturbative Approach for Higher Nonlinear Responses

The perturbative theory in a previous review [1] often breaks down in the case of irradiation of strong laser fields. On the other hand, the nonperturbative approach is applicable to the examination of higher harmonic generation (HHG) [119,120,121,122,123,124]. In the 1980s, partial high-order harmonic generations were reported because of developments in the ionization processes by the intense laser techniques [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. Wang and Chu [120] have developed a nonperturbative method named the Floquet-Liouville supermatrix (FLSM) approach, which can give an analytical expression for the intensity-dependent nonlinear optical susceptibility of a two-state system in polychromatic fields valid for arbitrary field intensities, detuning, and relaxation. The method is based on the exact transformation of the time-dependent Liouville equation [119] for the density matrix of quantum systems interacting with the classical external fields into an equivalent time-independent non-Hermitian Floquet-Liouvillian eigenvalue problem [120]. For these analytical approaches, however, the contributions from the antirotating parts of the external fields are practically incorporated by using the perturbation theory, in contrast to those from rotating parts that are incorporated nonperturbatively.

In 1994, Nakano et al. [27,122,123,124] proposed a new numerical nonperturbative approach to the n-th susceptibilities ${\chi }_g^n$(w) in the n-th harmonic generation (NHG), which is quite different from the FLSM approach [120]. This method, referred to as the numerical Liouville approach (NLA), is based on the Fourier transformation of numerically exact solutions of the Liouville equation [122,123,124], so that it can provide both real and imaginary nonlinear optical spectra valid for arbitrary laser intensities, frequencies, and relaxation as illustrated in Figure 2A. This method is closely related to the three-step procedures of molecular orbital tomography [49,68] reviewed in Section 4. The numerical stability of the ${\chi }_g^n$(w) with respect to the parameters used are also examined with a minimum interval of time (Dt = T/L; T = period of the external electric field and L is a division number of the period) in the time evolution of polarization P(t) and the region of time series of P(t) used in the discrete Fourier transformation. The NLA method is developed to calculate nonlinear susceptibilities summarized in Table 1.

The NLA method [27,122,123] is briefly revisited in this review. The macroscopic polarization density P(t) in Figure 1 can be written with the dipole moment operator μ and the density matrix operator ρ(t)

$$P\left(t\right)=N_{0}Tr\left[\mu \rho \left(t\right)\right]$$

(25)

where $N_0$ denotes the number density of the ensemble. The macroscopic polarization density P(t) can be expanded as a Fourier series in the external frequencies in the steady state as follows.

$$P(t)={\sum }_{m_1,m_2,m_3\dots .m_M}{P}_{m_1,m_2,m_3\dots .m_M}\left(\omega \right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{-i\left(m_1{\omega }_1+m_2{\omega }_2+\dots m_M{\omega }_M\right)t\right\}$$

(26)

where $P_{m_1,m_2,m_3\dots .m_M}\left(\omega \right)$ is the Fourier component at frequency $\omega$ = $m_1{\omega }_1+m_2{\omega }_2+\dots m_M{\omega }_M$.

For example, the Fourier component $P_{\mathrm{3,0},0}\left(3\omega \right)$ in Equation (26) indicates the generation of a polarization wave with frequency $\omega$ = $3{\omega }_1$ with respect to three incident fields with frequency ${\omega }_1$. In general, $P_{m_1,m_2,m_3\dots .m_M}$ can be related to the intensity independent perturbative l-th order susceptibility ${\chi }^{\mathrm{l}}$ as

$$\begin{array}{l}{P}_{\mathrm{3,0},0\dots .0}\left(3{\omega }_1\right)={\chi }^{\left(3\right)}(-3{\omega }_1;{\omega }_1,{\omega }_1,{\omega }_1){\epsilon }_1^3\left({\omega }_1\right)\\ +{\chi }^{\left(5\right)}(-3{\omega }_1;{\omega }_1,{\omega }_1,{\omega }_1,{\omega }_1,-{\omega }_1){\epsilon }_1^4\left({\omega }_1\right){\epsilon }_1^{*}\left({\omega }_1\right)\end{array}$$

(27)

where

$${\epsilon }_i\left({n\omega }_i\right)=[{\epsilon }_i\left({\omega }_i\right){]}^{n_i}, n_i\ge 0 ; [{\epsilon }_i^{*}\left({\omega }_i\right){]}^{n_i}, n_i<0$$

(28)

P(${\omega }_i$) is the Fourier transformation of the external electric field with a frequency ${\omega }_i$. In the weak external electric fields, the main contribution to the Fourier component $P_{m_1,m_2,m_3\dots .m_M}\left(\omega \right)$ is given by the term with the lowest order susceptibility. The intensity-dependent generalized nonlinear optical susceptibility by Wang and Chu [119,120] is given by

$${\chi }_{m_1,m_2,m_3\dots .m_M}\left(\omega \right)={\displaystyle \frac{P_{m_1,m_2,m_3\dots .m_M }(\omega )}{{\epsilon }_1^{m_1}\left({\omega }_1\right){\epsilon }_2^{m_2}\left({\omega }_2\right){\epsilon }_3^{m_3}\left({\omega }_3\right)\dots {\epsilon }_M^{m_M}({\omega }_M)}}$$

(29)

where $\omega$ = $m_1{\omega }_1+m_2{\omega }_2+\dots m_M{\omega }_M$. For the strong fields, the higher-order contributions (${\chi }^{\left(5\right)}$, ${\chi }^{\left(7\right)}$, …) to ${\chi }^{\left(3\right)}$ emerge as shown in Equation (27).

On the other hand, the third Harmonic response $P\left(3{\omega }_1\right)$ by Nakano et al. [122,123,124] is given by

$$\begin{array}{c}P\left(3\mathsf{\omega }\right)=P_{\mathrm{1,1},1}\left(3\mathsf{\omega }\right)+P_{\mathrm{2,1},0}\left(3\mathsf{\omega }\right)+P_{\mathrm{1,2},0}\left(3\mathsf{\omega }\right)+P_{\mathrm{0,1},2}\left(3\mathsf{\omega }\right)+P_{\mathrm{0,2},1}\left(3\mathsf{\omega }\right)+P_{\mathrm{2,0},1}\left(3\mathsf{\omega }\right)+P_{\mathrm{1,0},2}\left(3\mathsf{\omega }\right)\\ ={27\chi }^{\left(3\right)}(-3\mathsf{\omega };\mathsf{\omega },\mathsf{\omega },\mathsf{\omega }){\epsilon }^3\left(\mathsf{\omega }\right)+1215{\chi }^{\left(5\right)}(-3\mathsf{\omega };\mathsf{\omega },\mathsf{\omega },\mathsf{\omega },\mathsf{\omega },-\mathsf{\omega }){\epsilon }^2\left(\mathsf{\omega }\right)|\mathsf{\epsilon }(\mathsf{\omega }{\left)\right|}^2+\cdots \end{array}$$

(30)

where the Fourier component $P_{m_1,m_2,m_3\dots .m_M }(\omega )$ satisfy the condition: m₁ + m₂ + m₃ = 3. Therefore, the intensity-dependent third-order nonlinear susceptibility is defined as

$${\chi }_g^{\left(3\right)}\left(3\omega \right)(\mathrm{T}\mathrm{H}\mathrm{G})={\displaystyle \frac{P\left(3\omega \right)}{27{\epsilon }^3(\omega )}}={\displaystyle \frac{P\prime \left(3\omega \right)}{{\epsilon \prime }^3(\omega )}}$$

(31)

where ${\epsilon }^{\prime \left(\omega \right)}=3\epsilon \left(\omega \right)$ and $P^{\prime }\left(3\omega \right)$ are obtained by the Fourier transformation of the time-dependent series of the solution of the Liouville equation, as illustrated in Figure 1. Other response properties in Table 2 can also be expressed by the non-perturbative forms like Equations (30) and (31). For the case of the n-th-order harmonic generation (NHG), where the polarization with frequency nω is generated by n incident beams with an amplitude $\epsilon \left(\omega \right)$ like the HHG experiments. The n-th-order nonlinear susceptibility ${\chi }_g^{\left(n\right)}$ is defined as

$${\chi }_g^{\left(n\right)}\left(n\omega \right)={\displaystyle \frac{P\left(n\omega \right)}{n^n{\epsilon }^n\left(\omega \right)}}={\displaystyle \frac{P\left(n\omega \right)}{{\epsilon \prime }^n(\omega )}}$$

(32)

where ${{\epsilon }^{\prime }}^n\left(\omega \right)=n\epsilon \left(\omega \right)$ is the amplitude of the incident field.

3.2. Numerical Liouville Approaches to Frequency-Dependent Nonlinear Responses

In this section, the NLA approach to obtain ${\chi }_g^{\left(n\right)}\left(n\omega \right)$ [27] in Figure 2 is explained briefly. The first step is to obtain the time evolution of an N-state quantum system interacting with the M linearly polarized external fields as

$${\rm E}\left(t\right)={\sum }_i^M{\epsilon }_i\left({\omega }_i\right) \left(e^{i{\omega }_t}+e^{-i{\omega }_t}\right)={\sum }_i^M{E}_i\left({\omega }_i\right)cos{\omega }_{i}t$$

(33)

where ${\epsilon }_i$ and ${\omega }_i$ are the amplitude and frequency of the i-th classical field, respectively.

The time-development of the system is governed by the Liouville equation expressed as

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\mathrm{\rho }\left(t\right)=-{\displaystyle \frac{i}{\hslash }}\left[H\left(t\right), \mathrm{\rho }\left(t\right)\right]-{\displaystyle \frac{i}{\hslash }}\left[R, \mathrm{\rho }\left(t\right)\right]$$

(34)

where $\hslash =h/2\pi$ is the Dirac constant ($h$ is Plank’s constant), and $\mathrm{\rho }\left(t\right)$ is the reduced density matrix operator averaged over the reservoir. $H\left(t\right)$ is the total Hamiltonian defined by Equation (9), which consists of the time-independent Hamiltonian $H_0$ in Equation (10) and the time-dependent Hamiltonian $H_1$ (=−μ$\mathrm{E}\left(t\right))$ for the interaction between dipole moment (μ) and the time-dependent external field $\mathrm{E}\left(t\right)$ in Equation (9).

The second term in the right-hand side of Equation (30) represents the relaxation processes in the Markov approximation. The relaxation processes can be considered as two types of mechanisms as

$${\left[R, \mathrm{\rho }\left(t\right)\right]}_{\alpha \alpha }=-{\mathrm{\Gamma }}_{\alpha \alpha }{\mathrm{\rho }}_{\alpha \alpha }+ {\sum }_{\beta (\ne \alpha )}^N{\gamma }_{\beta \alpha }{\mathrm{\rho }}_{\beta \beta } (T_1 \mathrm{process})$$

(35)

$${\left[R, \mathrm{\rho }\left(t\right)\right]}_{\alpha \beta }=-{\mathrm{\Gamma }}_{\alpha \beta }{\mathrm{\rho }}_{\alpha \beta } \beta (\ne \alpha ) (T_2 \mathrm{process})$$

(36)

where T₁ and T₂ processes represent the population and coherent damping mechanisms, respectively, in Figure 2B. The feeding parameters ${\gamma }_{\beta \alpha }$ ($\ne {\gamma }_{\alpha \beta }$) provides the inelastic transition rate from $|\beta ⟩$ to $|\alpha ⟩$. The off-diagonal decay rates ${\mathrm{\Gamma }}_{\alpha \beta }$ are related by

$${\mathrm{\Gamma }}_{\alpha \beta }={\displaystyle \frac{1}{2}}\left({\mathrm{\Gamma }}_{\alpha \alpha }+{\mathrm{\Gamma }}_{\beta \beta }\right)+{\mathrm{\Gamma }}_{\alpha \beta }^{\prime } \left({\mathrm{\Gamma }}_{\alpha \beta }={\mathrm{\Gamma }}_{\beta a}\right)$$

(37)

where ${\mathrm{\Gamma }}_{\alpha \beta }^{\prime }$ is the pure dephasing factor that arises from the phase-changing collisions. Here, we set the factor ${\mathrm{\Gamma }}_{\alpha \beta }^{\prime }=0$ and assume a closed system.

We solve numerically the Liouville equation in Equations (33)–(37) by the use of the fourth-order Runge-Kutta method in order to obtain the time series of the polarization $P\left(t\right)$= <μ> in Equation (25) as illustrated in Figure 2A. The second step is to calculate the polarization $P\left(\omega \right)$ in the frequency domain using the time series of $P\left(t\right)$. To this end, the Fourier transformation is used as follows

$$P(\omega )={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }P(t)e^{i\omega t}dt.$$

(38)

The Fourier transformation is carried out numerically by the discrete Fourier transformation of the time series of $\mathrm{P}\left(\mathrm{t}\right)$ using the following formula

$$P\left({\omega }_j\right)={\displaystyle \frac{1}{N}} {\sum }_{k=O}^{N-1}P\left(t_k\right)\exp \left\{i\left|{\displaystyle \frac{2\pi }{N}}jk\right|\right\}, \left(\begin{array}{c}j=0, 1, \dots , N-1\\ k=0, 1, \dots , N-1\end{array}\right).$$

(39)

The number used for the time-series data is N, the k-th discrete time is $t_k$ = (L/N)k, and the j-th discrete frequency is ω_j = (2π/L)j where the minimum t value (t₀) is 0 and the maximum t value (t_N−1) is L. Similarly, the external field ε(ω_j) in the frequency domain is calculated from the time series ε(t_k). Using the ratio between these frequency domain quantities, we calculate the intensity-dependent n-th-order nonlinear susceptibility ${\chi }_g^{\left(n\right)}\left(n\omega \right)$ in Equation (32).

3.3. Numerical Liouville Approaches to Frequency-Dependent Nonlinear Responses

Optical properties of π-conjugated systems have attracted great interest in relation to optical visions in biological systems and light-harvesting systems in photosynthesis (see Section 6). Third-order optical properties of π-conjugated molecules [124] have also attracted great interest in the fields of material sciences. In a previous paper [1], we performed the perturbative approach on them. Here, as an example, the non-perturbative NLA computations of third-order optical properties of trans (t) octa-tetraene are revisited as shown in Figure 3A, where the excitation energy (E₂₁) between the ground (1 with ¹A_g) and first excited (2 with ¹B_u) state and the excitation energy (E₃₂) between the first excited (2) and second excited (3 with ¹A_g) state, and the transition dipole moments (D) μ₂₁ and μ₃₂ are illustrated. The power density of the external field is assumed to be 100 MW/cm² under the assumption of the low-energy region (10~1000 MW/cm²). The excitation damping factor ${\mathrm{\Gamma }}_{ii}$ = ${\mathrm{\alpha }\mathrm{E}}_{i1}$(($\mathrm{\alpha }$ = 0.02), i = 2 or 3) was assumed for the computations to obtain ${\chi }_g^{\left(3\right)}\left(3\omega \right)$ as shown in Figure 3B.

Figure 3B illustrates the real and imaginary parts of ${\chi }_g^{\left(3\right)}\left(\mathrm{T}\mathrm{H}\mathrm{G}\right)$ dispersion spectra obtained by both perturbative and nonperturbative (NLA) methods. The nonlinear susceptibilities obtained by the NLA method involve two characteristic effects, which cannot be involved in the conventional perturbative values. One is the intensity-dependent effects originating from higher-order terms involved in the definitions of the intensity-dependent susceptibilities. The other is the non-perturbative effect caused by the one- and/or multiphoton resonant processes among the allowed excited and ground states. The calculated spectra in the one-photon resonance region (E₂₁) in Figure 3B clearly indicate the non-perturbative effect. Nakano et al. [123,124] have also examined other processes such as EFISH and DFWM in detail, demonstrating the utility of the NLA methods for theoretical investigations of nonlinear optics.

4. Discovery of High-Order Harmonic Generation and Developments of Attosecond Chemistry

4.1. Discovery of High-Order Harmonic Generation

The weak laser field examined in Section 3 cannot generate higher Harmonic generations (HHG), formally discussed in Equation (7). The high-harmonic generation (HHG) defined in the previous section was partly realized in the 1980s because of the developments of generation methods of strong and stable laser fields 10¹¹ MW/cm² [31,32], as illustrated in Figure 4. L’Huillher et al. [30] have discovered the generation of HHG in the above-threshold ionization (ATI) processes of atoms by strong IR (1064 nm) laser fields, which indicates absorption of more photons than the necessary number of photons in the ionization, as illustrated in Figure 5. L’Huillher et al. [30] have detected the generation of odd-order higher-order (n = 13~31 in Equation (28)) HHG in the ATI process. The odd-order is responsible for inversion symmetry as mentioned in Section 2. The intensity distributions of the observed HHG have indicated a plateau where HHG have similar intensity and a cutoff at the 31st order (see Figure 5). During 2001–2006 [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56], great developments in HHG methods were made to open up attosecond spectroscopy. In this section, historical developments of attosecond sciences are briefly revisited to understand the observation of the phase of wave functions obtained by the Schrödinger equation for atoms and molecules [49,68].

In 1986, the titanium–sapphire (Ti:Al₂O₃) laser was discovered [31,32], providing a very strong Laser field for the generation of the HHG. Brabec and Krausz et al. have summarized the frontiers of nonlinear optics in relation to HHG [28,29,30,31,32,33,34,35,36]. The efficiency of the ATI process is examined by the Keldysh parameter [28], which is given by ($I_p/2U_p{)}^{1/2}$, where $I_p$ and $U_p$ denote ionization energy and pondermotive energy, respectively. The cut-off energy (maximum n-value) is given by $I_p$ + 3.2$U_p$, indicating the breakdown of the perturbative approach to the HHG. The HHG experiments clearly indicated the necessity of the exact solution of the time-dependent Schrödinger equation (TDSE).

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\Psi \left(r, t\right)=-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2\mathrm{\Psi }\left(r,t\right)+\left[V\left(r\right)+\mu E\left(t\right)\right]\Psi \left(r,t\right)$$

(40)

where $V\left(r\right)$ denotes the atomic potential, and the last term in Equation (38) is the dipole interaction with the Laser field in Equation (5). Kulander and Shore [38] have solved the TDSE numerically to explain the HHG generation. The time-dependent dipole moment $\mu \left(t\right)=⟨\Psi \left(r,t\right)\left|\mu \right|\Psi \left(r, t\right)⟩$, given by Equation (40), is Fourier-transformed to obtain the frequency-dependent series of an HHG like Equation (32) under the strong field approximation. The TDSE results are found to be compatible with available experiments.

In 1993, a theoretical breakthrough for lucid understanding and explanation of HHG was made under the assumption of the classical electronic field (semi-classical model) by two groups, the Kulander and Shafer group [38] and Corkum [34,35,36,37], providing the semi-classical three-step model, as illustrated in Figure 6. The three-step model consists of three stages: (i) ionization of atoms and molecules via a tunnel process; (ii) recombination process of liberated electron; (iii) emission of HHG light. In step (i), the atomic potential V(R) is strongly distorted under the strong Laser field, and a bound electron may tunnel through the distorted potential barrier for ionization. The total wavefunction ψ is expressed with the superposed state of the ground state wave function ψ_g and the ejected electron wave function ψ_c, where the potential barrier plays the role of the beam splitter in quantum optics. In the second step, the liberated electron is pushed back toward the ionized core because of the laser field, which provides kinetic energy to the electron, and the excess energy obtained is emitted in the form of a high-energy photon (HHG) in the last step. The three-step model is also supported by the quantum model in Equation (40) by Lewenstein et al. [34,35,36,37].

4.2. Generation of Attosecond Pulses and Attosecond Spectroscopy

In 1994, Corkum et al. [36] predicted the possibility of the generation of sub-femtosecond pulses by controlling step (iii), namely, re-collision processes, in the three-step model. Constant et al. [37] have also proposed the measurement methods of the duration of high-harmonic pulses. In 2001, Paul et al. indeed observed a train of attosecond pulses from High Harmonic Generation [41]. The interference of the pulses produces the phase shift, which provides a short pulse width (250 attoseconds). Their method is often referred to as RABITT (Reconstruction of Attosecond Beating by Interference of Two-photon Transition) [41]. On the other hand, Krausz et al. [39] have developed the single attosecond pulse (650 attoseconds), opening up attosecond spectroscopy [42]. The extensive reviews have already been published, providing important information on the principles, methods, and applications of attosecond lights to physics, chemistry, and biology. Recently, the generation of squeezed (quantum) HHG has been developing, indicating possible quantum applications. Therefore, historical developments of laser technologies [39] are skipped in this review, only discussing the molecular orbital topography, which is closely related to the quantum chemistry of open-shell species such as diradicals.

4.3. Molecular Orbital Tomography

Attosecond light pulses have been applied to several fields of science. Molecular orbital tomography for observation of the wave function instead of its square value (electron density) is one of the most important applied fields of attosecond spectroscopy in chemistry. This problem is closely related to the fundamental problem of quantum mechanics and includes Born’s stochastic explanations of wave functions [3] examined in Section 1. Molecular orbitals are obtained for molecules with many electrons under the mean field approximation and are found in models such as the Hatree–Fock (HF), Kohn–Sham, and Dyson models. Under these models, the electrons in the highest-lying orbitals (HOMO) in energy are usually ejected in the ionization process by strong laser fields [49]. Two methods have been used to image HOMO. The electron momentum spectroscopy is a scattering technique that can determine the radially averaged density of the outermost valence electrons. The scanning tunneling microscopy (STM) [69,70,71] also gives the electron density, distorted by surface states.

Itatani et al. [49] initiated a new tomography technique, where the full three-dimensional structure of a molecular orbital can be imaged by using high harmonics (HHG) generated from intense femtosecond laser pulses focused on aligned molecules, and a tomographic reconstruction of the orbital is accomplished by the Fourier transformation procedure discussed in Section 3 (see Figure 2). The tomographic imaging of a molecular orbital is achieved in three crucial steps: (i) non-adiabatic alignment of the molecular axis in the laboratory frame; (ii) selective ionization of the orbital; (iii) projection of a coherent set of plane waves. In the first step (i), gas-phase molecules can be aligned using a laser pulse [39]. High harmonics are produced by ionizing atoms or molecules [34,35,36,37]. The orbital with the lowest ionization potential is preferentially selected in the second step (ii). The generated wave packet, ψ_c, is approximately given by the superpositions of plane waves; ΣA(p)exp[iωt − ipr]. The vibrational dipole d induced by the re-collision step (iii) is approximately given by d = [<ψ_c|r|ψ_g> + c.c]. The HHG spectra are obtained by changing the alignment angles of molecules. The Fourier transformation of these spectra in the momentum space p to the coordinate vector r by the computed tomography technique provides 3D wave functions in the real space.

Figure 7 illustrates wave functions of molecular nitrogen (N₂) and molecular oxygen (O₂). The generations of the HHG become the maximum and minimum, respectively, along the parallel and perpendicular alignments of N₂ for the direction of the high-field laser (E), indicating the angle dependency of cos²θ. On the other hand, the generations of the HHG become the minimum along the direction of E in the case of O₂, indicating the different behavior (sin²θ) with the maximum direction of θ = π/4. These differences arise from the difference in the orbital symmetries between the σ-type bonding orbital (Figure 7A) of N₂ and π-type anti-bonding orbitals (Figure 7B) of O₂. Thus, the orbital phases of molecules are imaged by attosecond spectroscopy. The orbital tomography has been performed for several molecules [49].

The ground state of molecular oxygen is a typical triplet diradical with the degenerate π_x- and π_y-type singly occupied MOs (SOMOs) as shown in Figure 7. This means that the degenerated doublet cation radical O₂⁺ is described at least by two configurations, providing the so-called Dyson-type MO (see later) ψ = C_xψ_x + C_yψ_y, where C_x and C_y denote the superposition coefficients of the π_x- and π_y-SOMOs. The weights of the coefficients may be dependent on the alignments of molecular oxygen against the strong laser field (E), as shown in Figure 7. The linear CO₂ molecule is a closed-shell molecule with two degenerated π_x- and π_y-HOMOs. Therefore, the Dyson orbital of the CO₂⁺ cation radical may be expressed by the above mixing form under the HHG ionization process. Thus, the HHG orbital tomography of singlet or triplet diradicals, molecules with degenerated HOMO, etc., will be an interesting problem for elucidation of the scope and applicability of the one-electron orbital model. To this end, computations of Dyson equations are necessary, as shown in several papers [74,75,76].

4.4. Observations of the Real and Imaginary Parts of Wave Functions

The wave nature of electrons is described with the amplitude and phase as discussed in the introduction of this review. Recently, attosecond spectroscopy has been developed to obtain the images of both degrees of freedom: the amplitude (real part of the wave function) and phase (imaginary part of the wave function) [68]. To this end, several methods have been developed to accomplish the interference of wave packets responsible for electronic states. Niikura et al. [68] have developed a two-path interference scheme using an attosecond pulse train to obtain a phase and amplitude distribution of photoelectrons in a momentum space, like in the double-slit experiment. For example, they have performed the ionization of Ne gas by an extreme ultraviolet (XUV) attosecond pulse train, which consists of two selected paths: (A) 14th HHG one-photon ionization and (B) 13th HHG + IR ionization process. The photoelectron momentum distributions via the interference between A and B processes were recorded as the velocity map imaging by changing the attosecond time difference (τ) between the XUV and IR pulses. The time-dependent momentum distributions obtained are mapped into two-dimensional momentum space (k_x, k_y). The mapped images were responsible for the phase and amplitude of the f-atomic orbitals in accord with the photon energy of HHG. Thus, attosecond spectroscopy using an attosecond pulse train opened the door for both the phase and amplitude of the wave function in the momentum space. Therefore, this method is also applicable to investigate the wave functions of solids [68].

4.5. Dyson Orbitals of Ionized Diradicals and Related Species

The one-electron orbital has been derived under the mean-field approximation for many electron atoms and molecules. The orbital energy levels of molecular orbitals are approximately responsible for the ionization potentials (Ip) obtained by the ionizations by using a strong laser field. Dyson orbitals [74,75,76] are defined as overlaps between initial N-electron states and final N-(or +) −1 electrons and therefore are useful for understanding and explaining observed orbitals by the HHG ionization process discussed in Section 4.4. According to theoretical investigations by Ortiz [76], Dyson orbitals are often similar to the Kohn–Sham (KS) DFT orbitals, indicating the utility of the DFT orbitals for understanding and explaining the ionizations of molecules by the HHG experiments.

Over the past decades, the scope and applicability of the one-electron orbital concepts [77,78] based on the mean field approximations such as Hückel models, Hatree–Fock (HF), KS DFT, and hybrid HF + (KS) DFT models have been examined from beyond mean-field models such as MR CI(CC), etc. [1]. In this review, we do not discuss the details of such discussions [71,72,73], which are already summarized in the excellent review article by Ortiz et al. [76]. However, it is noteworthy that the concepts of the orbital phase continuity and discontinuity [1] have been used for symmetry-allowed concerted and symmetry-forbidden diradical reactions, respectively. The observation of the orbital phase by attosecond spectroscopy [49] has indeed revealed the importance of the orbital phase of the wave function of molecules. The quantum phase is also important for material science [39]. As discussed in the previous review [1], the mesoscopic superconductors have been regarded as a macroscopic quantum coherence (MQC) state [125], for which the Josephson junction (JC) has been used for active control of the quantum phase for quantum computations. The Bose–Einstein condensation (BEC) of Rydberg atoms is also very important for the same reason [1]. Recently, quantum coherence for light harvesting in photosynthesis is a current topic (see Section 6). Thus, the active control of the quantum phase in the native and artificial quantum systems is now of current interest because of several reasons [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

5. Quantum Phase Dynamics and Entangled Entropy for Quantum Material Sciences

5.1. Quantum Dynamics Simulations and Level Models for Quantum Optics of Molecules

Here, the fundamental concepts touched on in Section 1 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] are revisited briefly. The wavefunction is used for the pure quantum state, but it is not applicable to the mixed states different from the superposed state, as shown by Landau [126]. Von Neumann introduced the density matrix (ρ) for the pure and mixed states [127], providing the entropy concept: S = −trace (ρ ln ρ) as a measure of the statistical uncertainty where ln denotes the matrix version of the natural logarithm. Dirac first introduced the first-order density matrix for solving the Hatree–Fock solution [128]. In 1940, Husimi (presenter of the Kagome lattice [1] at Osaka University) investigated the reduced density matrix of any order at a finite temperature [129]. Löwdin has developed the density matrix approach to quantum chemical problems, such as the rapid convergence of the configuration interaction (CI) expansion [130]. Fano has developed the density matrix methods for several fields of science, including spectroscopy [131]. Since then, many density matrix formulations have been developed in quantum chemistry.

On the other hand, phase space representation plays an important role in understanding the relationship between classical and quantum mechanics, providing fundamental concepts for quantum optics [132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156]. Wigner [132] has elucidated the correspondence between the density matrix and the distribution function in the phase space of classical statistical mechanics, presenting the quasiprobability distribution via Wigner-Weyl transformation [132,133,134]. However, the Wigner distribution often indicates a negative value in the quantum interference effect. The time evolution of the density matrix is also given by the correspondence between Liouville (classical) [135] and von Neumann-Liouville (quantum) [127] equations. The scope and applicability of the Wigner distribution functions have been examined in several papers [129,132,135,136]. Husimi first introduced the Husimi function H(P, Q) (Q-representation) in statistical physics [129], which was reduced to the Maxwell–Boltzmann distribution function at the classical limit (h = 0) [135]. The Husimi Q function exhibits a positive value in contrast to the Wigner distribution, indicating the distribution function responsible for the coherent state (minimum uncertainty state) [137,138,139,140,141,142]. The Q-function is also used for the derivation of the concept of entropy [127,129,139]. The relationship between the Wigner and Husimi distribution functions is elucidated in several papers [139,140,141,142]. The Husimi Q function is now extended to several fields of science and engineering [144,145,146] and quantum chemistry [147,152,153,156].

Recently, theories and experiments of quantum measurements have been developed significantly, as shown in Section 1 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,147,148,149,150,151,152,153,154,155,156]. Interactions of atoms, molecules, and molecular aggregates with quantized photon fields [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] have indeed attracted great interest from both fundamental science and application to quantum engineering. As discussed in a previous review [1], the simplest model of such interactions [151] is the Jaynes–Cummings (JC) model [125], which describes the interaction between a two-state atom and a one-mode photon field. Nakano et al. [147,148,149,150,151,152,153,154,155,156] have performed quantum dynamic simulations of molecule–photon interacting systems, providing important information to the design of molecular quantum devices such as quantum sensors [21] and quantum optical computers [20]. To this end, they have developed the exact numerical Liouville dynamics [122,123,124] simulation methods to examine concepts and several distribution functions for quantum optics [126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146]. It was found that various extensions of the JC model [125] provide guiding principles for examining quantum coherence in quantum materials such as laser-cooled atom aggregates (BEC) [157,158,159] and the Cooper pair box [160], and quantum biological systems such as light-harvesting antennae. Thus, a unified quantum picture [154] for both materials and biological systems has been essential for understanding and explaining the new finding. Aggregates of neutral atoms such as Rb are nowadays used as a new-type neutral atom quantum computer [161,162,163,164,165,166,167,168,169,170,171].

In this section, theoretical models for interactions between molecules and quantum light are briefly revisited. The Hamiltonian for such interactions [150] is schematically given by

$$H=H_{mol}+H_{cfield}+H_{mol-cfield}+H_R+H_{cfield-R}$$

(41)

where five partial interaction Hamiltonians are obtained as illustrated in Figure 8. The Hamiltonian for a molecular system is expressed with the second quantization form as

$$H_{mol}={\sum }_i{E}_i a_i^{+}{a}_i$$

(42)

where $E_i$ denotes the energy of the i-th state, $a_i^{+}$ and $a_i$ are the creation and annihilation operators of state i. The Hamiltonian for the one-mode photon field within the cavity in Figure 8 is defined by

$$H_{cfield}=\left( b_i^{+}{b}_i+{\displaystyle \frac{1}{2}}\right)\hslash \omega$$

(43)

where $b_i^{+}$ and $b_i$ are the creation and annihilation operators of a photon, and $\hslash \omega$ is the energy of a photon. The Hamiltonian for the interaction between the molecule and the photon field is defined by

$$H_{mol-cfield}=\sqrt{{\displaystyle \frac{\hslash \omega }{2{V\epsilon }_0}}}{\sum }_{i,j}{d}_{i,j} a_i^{+}{a}_j\left( b_i^{+}+b_i \right)$$

(44)

where $d_{i,j}$ is the transition moment between i- and j-states of molecule, and V is the volume of the cavity. The Hamiltonian for the reservoir is approximated with the harmonic oscillator models as

$$H_R={\sum }_j\left( r_j^{+}{r}_j+{\displaystyle \frac{1}{2}}\right)\hslash {\omega }_R.$$

(45)

The Hamiltonian for the interaction between the cavity field and reservoir is therefore given by

$$H_{cfield-R}={\sum }_j \left( k_j^{*}b{r}_j^{+} + k_j{b}^{+}{r}_j \right)\hslash$$

(46)

where $k_i$ denotes the coupling constant between the cavity photon field and j-th reservoir mode.

These model Hamiltonians are used for quantum dynamics simulations of molecules in the cavity in Figure 8A. Nakano et al. [147,148,149,150,151,152,153,154,155,156] have examined several types of photon fields for quantum simulations, providing the Wigner and Husimi distribution functions for several model systems. The detailed results are not reproduced in this review. It is noteworthy that the present formulation is directly extended to the macroscopic quantum systems responsible for decoherence [156,157,158,159,160]. Nakano et al. [147,148,149,150,151,152,153,154,155,156] have examined several key concepts in the quantum phase space approach, as shown in Figure 8B.

5.2. Quantum Master Equation and Reduced Density Matrix

Here, we revisit the quantum master equation for the quantum dynamics of a molecular system under the Markov approximation in Figure 8. To this end, the reduced density matrix ${\mathrm{\rho }}_S\left(t\right)$ for a molecular part under investigation is obtained by tracing out the contribution of the reservoir from the total density matrix $\mathrm{\rho }\left(t\right)$ [147,148,149,150,151,152,153,154,155,156]

$${\mathrm{\rho }}_S\left(t\right)={Tr}_S\left[\mathrm{\rho }\left(t\right)\right].$$

(47)

The Schrödinger equation for the total density matrix is given by

$$i\hslash {\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\mathrm{\rho }\left(t\right)=\left[H\left(t\right), \mathrm{\rho }\left(t\right)\right]= \left[H_S+H_R+H_{SR} , \mathrm{\rho }\left(t\right)\right]=\left[H_0+H_{SR} , \mathrm{\rho }\left(t\right)\right]$$

(48)

where $H_0$ = $H_S + H_R$, and $H_{SR}$ denotes the interaction between the system and reservoir.

The quantum dynamics of the target system is considered as illustrated in Figure 8. The interaction is assumed to be zero at the initial time, providing the product state as

$$\mathrm{\rho }\left(t_0\right)={\mathrm{\rho }}_S\left(t_0\right){\mathrm{\rho }}_R\left(t_0\right).$$

(49)

Several approximations, such as the Markoff approximation [155], are introduced to obtain the time-dependent equation of ${\mathrm{\rho }}_S\left(t\right)$ instead of Equation (48), providing the quantum master equation as follows

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}{\mathrm{\rho }}_S\left(t\right)=-{\displaystyle \frac{i}{\hslash }}\left[H_{mol}+H_{cfield}+H_{mol-cfield} , {\mathrm{\rho }}_S\left(t\right)\right] + H_{relax}$$

(50)

where the relaxation term $H_{relax}$ is given under the assumption of the dense reservoir mode ($n_R$) as

$$H_{relax}={\displaystyle \frac{\gamma }{2}}\left( 2b{\rho }_S{b}^{+}-b^{+}b{\rho }_S-{\rho }_S{b}^{+}b\right)+\gamma n_R\left( b^{+}{\rho }_{S}b+b{\rho }_S{b}^{+}-b^{+}b{\rho }_S-{\rho }_{S}b{b}^{+}\right)$$

(51)

where $\gamma$ denotes the relaxation parameter given by the coupling constant between the system S and the reservoir R.

Equation (48) can be solved by the conventional Runge-Kutta method. Recently, to this end, quantum trajectory and Monte Carlo (damping) wavefunction methods in Figure 2B [155] have also been developed on the basis of the developments of single-molecule spectroscopy, as shown previously in Section 4. The quantum simulation scheme, including nuclear vibration, is illustrated in Figure 2B.

5.3. Jaynes–Cummings (JC) Model and Collapse-Revival Phenomena for Atoms and Molecules

A simple example of the quantum interacting system in Figure 8 is the Jaynes–Cummings (JC) model [125], where a two-state model of an atom is interacting with a one-mode coherent photon field. This system was first found to exhibit the collapse and revival phenomena of population and photon number [125]. Nakano and Yamaguchi [147] performed the quantum dynamic simulations of the two-level model of molecules (the energy gap = 37,800 cm⁻¹ and dipole moment 5D) coupled with one-mode coherent field (ω = 37,800 cm⁻¹), showing the generation of Rabi oscillation in the first place, followed by its reduction via dephasing as illustrated in Figure 9. However, the revival of the Rabi oscillation via the rephasing process is also reproduced on the computational ground in accord with the JC model.

The coherent field |α > [149,150] is expressed with the creation and annihilation operators as

$$|\mathrm{\alpha }>=\mathrm{D}\left(\mathrm{\alpha }\right)|0>=\exp (\alpha a^{+}-{\alpha }^{*}\mathrm{a})|0>$$

(52)

where $\mathrm{D}\left(\mathrm{\alpha }\right)$ denotes the mutation operator, and $\alpha$ is the eigenstate of a. The populations of the photon number are given by the Poisson distribution in the coherent state [149,150], for which the minimum fluctuation relation (uncertainty principle) Δx₁Δx₂ = 1/4 for quadrature amplitude, namely real (x₁) and imaginary (x₂) parts of the phase $\alpha$, is kept as discussed in a previous review [1]. On the other hand, the quantum fluctuations can be squeezed to the broken symmetry state: Δx₁ < Δx₂ (amplitude squeezed state) or Δx₁ > Δx₂ (phase squeezed state) under the minimum uncertainty constraint [152,155]. Thus, the concepts of coherence and squeezing are very important for the characterization of quantum lights. The broken-symmetry (BS) squeezed parameter is given by the ratio between large and small parts (Δx_L/Δx₂).

The collapse and revival phenomena for interaction of the three-states model of molecules with the squeezed photon field (two-photon coherent state) in Figure 3A were also investigated to elucidate different behaviors between the coherent and squeezed fields [151]. The Wigner distributions [132] and Husimi Q-functions [129] were successfully applied to elucidate characteristic behaviors of several quantum model systems [147,148,149,150,151,152,153,154,155,156]. Details of various numerical results are not represented in this review. From these simulations, it was found that the off-diagonal density between the ground and first excited states exhibits quantum coherence, but those of other transitions exhibit no such behavior. This implies the coexistence of the thermal and quantum states in the model systems. This dynamical quantum-classical picture was expected to be very important for understanding and explaining the partial coherent energy transfer in biological systems. Thus, the quantum dynamic simulations for control of quantum phases were expected to provide useful information for designing future quantum devices. Nature may be aware of active quantum control of quantum effects in biological systems, providing important information for the development of artificial bio-inspired systems [172].

5.4. Bose–Einstein Condensation, Quantum Information Sciences and Quantum Computing

In 1925, Einstein [157] predicted the Bose–Einstein condensation (BEC) for dilute gases at low temperature. Laser cooling techniques have been developed to obtain BEC states. In 1995, Anderson et al. [158] observed the BEC for a dilute Rb vaper. In the same year, Davis et al. [159] also observed the BEC for dilute Na gas. These experiments have demonstrated a macroscopic quantum coherence (MQC) [1,160] of dilute gases at low temperature, opening up information science and technology [161,162,163,164,165,166] for the generation of atom waves at low temperature. Nakano et al. performed the above quantum dynamic simulation for a two-component BEC system coupled with a Josephson junction (JJ), showing the collapse and revival quantum phenomena [156]. As mentioned above, Nakamura et al. [160] first consisted of a small superconductivity island (Box) with excess Cooper pair charges connected by a tunneling junction with capacitance and Josephson coupling energy, and they have shown the coherent control of the mesoscopic superconductive systems, opening the door for quantum computations using MQC [160].

As summarized in Figure 23 in the previous review [1], several systems have been proposed for candidates of qubits for quantum computing. Neutral atom qubits [21] have attracted current interest among them because of several reasons [161,162,163,164,165,166,167,168,169,170,171]. Rydberg atoms with large principal quantum numbers (n >> 1) have indeed advantages such as tunable dipole–dipole interactions to implement quantum gates between neutral atom qubits [161,162,163,164,165,166,167,168,169,170,171]. Saffman et al. [161] have summarized advances in experimental and theoretical investigations of Rydberg-mediated quantum information processing investigated until 2010. In the 2010s, Nogrette et al. [162] have optically trapped ⁸⁷Rb atoms in triangular, honeycomb, and Kagome lattices for quantum information processing and quantum simulations. Kauman et al. [163] have developed a new technology to provide a framework for dynamically entangling remote qubits via local operations within a large-scale quantum register. Levine et al. [164] have constructed the universal two- and three-qubit entangling gates on neutral-atom qubits, which are mediated by excitation to strongly interacting Rydberg states. They have realized the controlled-phase gate with high fidelity. Browaeys and Lahaye [165] have demonstrated that different types of interactions between Rydberg atoms allow a natural mapping onto various quantum spin models [154,172] for many-body physics.

In the 2020s, Bluvstein et al. [166] constructed a quantum computer using dynamically reconfigurable neutral atom arrays trapped and transported by optical tweezers in two spatial dimensions: hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation. Graham et al. [167] have demonstrated several quantum algorithms on a programmable gate-model neutral-atom quantum computer in an architecture based on a two-dimensional array of qubits. Evered et al. [169] have achieved the realization of two-qubit entangling gates with 99.5% fidelity on up to 60 atoms in parallel, surpassing the surface-code threshold for error correction. This procedure is also applied to generate large qubit systems [170]. Quantum computing using neutral atom qubits is promising for investigations of strongly correlated systems such as arrays of diradical species [171]. Thus, the concept of entanglement plays a crucial role in quantum computing, as illustrated in Figure 8B.

5.5. Low-Dimensional Arrays for Organic and Molecular Qubits

Over the past decades, diradicals with anti-ferromagnetic interaction, triradicals with spin frustration, and tetraradicals with cubane-like structure have been target molecules for our theoretical investigations, as shown in a previous review [1]. However, chemical syntheses of such molecules were not easy because of their instability. The low-dimensional arrays, such as linear, triangular, and Kagome structures for systems of neutral atom qubits, can now be generated using optical tweezers as examined in the preceding section [166,167,168,169,170,171]. The chemical synthesis of such quantum lattices using open-shell molecules was our dream [172], but its realization is still difficult. The Hamiltonians for these arrays can be equivalently transformed into several kinds of spin Hamiltonians [161,162,163,164,165,166,167,168,169,170,171], supporting the concept of the equivalent transformations for quantum information sciences [1,154]. In the NATO ASI symposium, spin Hamiltonians for atomic and molecular systems are examined based on three factors [173]: (i) lattice dimensionality (1D, 2D, 3D); (ii) spin dimensionality (1D Ising, 2D XY, and 3D Heisenberg models); (iii) scale factors (Single Molecule, Nano clusters, Mesoscopic aggregates, bulk materials, etc.). Based on these guiding principles, the specially promoted project for chemical synthesis of such low-dimensional materials was undertaken during 1994–1996 [172].

In 2003, we summarized the basic concepts and experimental results for the molecular report [172]. One of the theoretical results was the quantum tunneling of 1D clusters such as Cr clusters [174]. Experimental efforts to construct so-called 1D Haldane chains [175], consisting of triplet species such as molecular oxygen, were also performed by using the organometallic conjugated nano-porous systems, which consisted of the Cu-ligated by CO₂ anion group of organic linkers [176]; adsorptions of gases by the nano-porous lattices are now being developed extensively. The molecular dynamic (MD) simulation for the nano-porous lattice [177] revealed the 1D molecular oxygen chain with the disordered structure, indicating the difficulty of the formation of a molecular Haldane system [175].

On the other hand, Mn-oxide clusters were theoretically examined as candidates for molecular qubits [178]. Spin and pseudo-spin models [154] were also examined for understanding of molecular quantum optics, quantum spins, and mesoscopic superconductors in a unified manner based on the spin Hamiltonian models [179]. Several candidates of qubits for quantum devices were also summarized in a book [179]. We have expected developments of bio-inspired molecular devices [154]. However, chemical synthetic efforts for molecular quantum systems were limited at that time [172]. Interestingly, nature has already constructed such quantum systems, such as light-harvesting systems, providing the indispensable information on the possible roles of proteins [172]. Recently, synthetic efforts for molecular materials have been performed again in relation to developing quantum devices. Indeed, many theoretical and experimental results have been reported for molecular qubits [180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208]. Several ideas have been proposed for the elongation of coherent times in molecular qubit systems and for the construction of room-temperature qubits. To this end, organic, inorganic, and hybrid organic-inorganic spins have been examined as candidates for molecular qubits. Several geometric structures with spin frustrations have also been investigated to realize long relaxation times. Several organometallic conjugated systems [172], such as metal-organic framework (MOF) [203], have also been used to construct rigid molecular qubits. Photoexcitation is another degree of freedom for the generation of molecular qubits. The rigid molecular structure of MOF plays an important role in suppression of decoherence. Here, details of these manuscripts are abbreviated for brevity. The interplay between theory and experiments is expected to realize molecular quantum materials for quantum sensing [21], quantum computation [20], etc.

6. Exciton Migration and Coherent Energy Transfers in Biological Systems

6.1. Exciton Dynamics in Dendric Structures and Model for Antenna Systems for Photosystem

Quantum coherence may be important for molecular biology. There are three steps for light reaction in photosynthesis [209]; namely (i) light harvesting step, (ii) water oxidation step to generate proton and electron, and (iii) reduction of NADP+ to NADPH. In our special project for organometallic conjugation [172], biological and bio-mimetic materials such as manganese-oxide clusters and iron-sulfur clusters [210] relating to photosynthesis steps (ii) and (iii) were investigated to elucidate possible roles of proteins and hydrogen-bonding networks. Optical properties of conjugated polyenes were also investigated in the first step (i). In this section, the first step in the light reaction is reviewed in both natural and artificial systems.

Absorption of solar light and transfer of absorption energy to reaction centers in antenna systems are indeed the first steps for native oxygenic photosynthesis. The singlet excited open-shell state (target state of this special issue) of a pigment in the antenna systems is referred to as an exciton in material science. In the 1940s, Oppenheimer and Förster investigated the energy transfer in photosynthesis [205,206,207,208,209,210,211,212]. The excited state consisted of one electron-occupied HOMO, and LUMO (electron-hole pair), which is almost localized within a pigment molecule, is denoted as a Frenkel-type exciton in a molecular solid [213]. On the other hand, the electron-hole pair is largely delocalized over pigment molecules in a Wannier exciton [214]. Several theoretical models were published for exciton movements, energy transfers, and energy relaxations [215,216,217,218,219,220,221,222,223,224,225,226].

In 1985, Tomalia et al. [227] first synthesized topological macromolecules denoted “starburst dendrimers” as illustrated in Figure 10A. Interestingly, efficient energy transfer from the periphery to the core was found in the dendrimer structures [227,228,229,230,231,232,233,234], indicating the importance of aggregate structures and types of dipole–dipole interactions. Thus, dendrimer was considered as an appropriate chemical model of the light-harvesting systems [218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246]. Figure 10A illustrates a geometric structure of the starburst phenyl acetylene dendrimer [227], which is often called the D25 structure. D25 structure indicates two characteristic features: (D25-a) increases in the lengths of generations from the periphery to the core, and (D25-b) destroys the π-electron conjugation between generations at meta-branching points of phenyl rings. Meta-branching is also found to be effective for the realization of the ferromagnetic interactions in magnetic polymers [247]. These structural characteristics are compatible with the Frenkel-type exciton model [213]. The quantum master equation in Section 5.2 [232,233,234,235,236,237,238,239,240,241,242,243,244,245,246] is therefore applicable to examine the dynamical energy transfer processes in D25. To this end, the dipole–dipole (DD) interactions between chromophores are assumed as illustrated in Figure 10B, and the electron–phonon interactions are also included, as shown in the simulation model and in Figure 2B. Quantum phases [154,155,178,179,248,249,250,251,252] in other mesoscopic systems in relation to developments of molecular qubits [1] are also examined on theoretical grounds.

6.2. Master Equation for Exciton Dynamics in Dendric Structures

Historically, fundamental theories for quantum systems with environmental effects have been presented as shown in many papers [211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,253,254,255,256,257,258]. In this section, theoretical models for exciton dynamics are briefly revisited [235,236,237,238,239,240,241,242,243,244,245,246]. The Hamiltonian for molecular aggregates such as D25 and S25 is given by

$$H_S={\sum }_i^N{E}_i\left|i><i\right|+{\sum }_{i,j}^N{J}_{ij}\left|i><j\right|$$

(53)

where $|i>$ indicates the aggregate basis, in which the monomer i is excited, and N is the total number of sites. $J_{ij}$ is given by the dipole–dipole interaction between excitons as a first step approximation for molecular excitons [212]

$$J_{ij}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4\pi {\epsilon }_0 R_{ij}^3$}\right.} {\mu }_i{\mu }_j\{\cos \left({\theta }_{ij}-{\theta }_{ji}\right)-3cos{\theta }_{ij}cos{\theta }_{ji}\}$$

(54)

where ${\theta }_{ij}$ is the angle between the transition moment of monomer i and the vector drawn from monomer i to j, as illustrated in Figure 10B. This dipole approximation is acceptable if the sizes of monomers are smaller than the intermolecular distance R_ij, indicating its applicability even for the antenna systems in photosynthesis. The angle between dipole moments (${\theta }_{ij}$) plays an important role in the control of the intermolecular interaction ($J_{ij}$).

One-exciton states {$|{\psi }_k>$} with excitation energies {${\omega }_k$} is expressed with the following eigenvalue equation

$$H_S|{\psi }_k>={\omega }_k|{\psi }_k>$$

(55)

where

$$|{\psi }_k>={\sum }_i^N\left|i><i\right|{\psi }_k>={\sum }_i^N{C}_{ki}\left|i>\right|(k=2, \dots , M).$$

(56)

where $C_{ki}$ denotes the linear combination coefficient for the exciton distribution in aggregates. The number of excitation states, M, is equal to N + 1 for the one-exciton model. An exciton on monomer i is considered to interact with a nuclear vibration (phonon) as illustrated in Figure 8B, namely a phonon state |$q_i>$ with a frequency {${\mathsf{\Omega }}_{q_i}$}. The Hamiltonian $H_S$ for the phonon is given by

$$H_S={\sum }_i^N{\sum }_{q_i}{\mathsf{\Omega }}_{q_i} c_{i,q_i}^{+}{c}_{i,q_i},$$

(57)

where $c_{i, { q}_i}^{+}$ and $c_{i, { q}_i}$ represent the creation and annihilation operators concerning a phonon state, respectively. The interaction Hamiltonian $H_{SR}$ for the exciton–phonon coupling is given by

$$H_{SR}={\sum }_i^N{\sum }_{q_i}\left|i><i\right|({\kappa }_{i,q_i}^{*} c_{i,q_i}^{+} + {\kappa }_{i,q_i}{c}_{i,q_i}) ,$$

(58)

where ${ \kappa }_{i, { q}_i}$ represents a coupling constant between an exciton on monomer i and a phonon state $|q_i>$.

The master equations for the quantum dynamics [234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252] for exciton movements [201,202,203,204,205,206,207,208,209,210] are given by the extensions of Equations (47)–(51), where $H_0$ = $H_S + H_R$, and $H_{SR}$ denotes the interaction between the system and reservoir as shown in Figure 8

$${\displaystyle \frac{d}{dt}}{\mathrm{\rho }}_{\alpha \beta }=-i\left({\omega }_{\alpha }-{\omega }_{\beta }\right){\mathrm{\rho }}_{\alpha \beta }-{\sum }_{m,n}^M{\mathrm{\Gamma }}_{\alpha \beta ;mn}{\mathrm{\rho }}_{mn}-E{\sum }_n^M{(\mathsf{\mu }}_{\alpha n }{\mathrm{\rho }}_{n\beta }-{\mathrm{\rho }}_{\alpha n }{\mathsf{\mu }}_{n\beta }) (\alpha \ne \beta )$$

(59)

$${\displaystyle \frac{d}{dt}}{\mathrm{\rho }}_{\alpha \alpha }=-{\sum }_m^M{\mathrm{\Gamma }}_{\alpha \alpha ;mm}{\mathrm{\rho }}_{mm}-E{\sum }_n^M{(\mathsf{\mu }}_{\alpha n}{\mathrm{\rho }}_{n\alpha }-{\mathrm{\rho }}_{\alpha n }{\mathsf{\mu }}_{n\alpha }) (\alpha \ne \beta ).$$

(60)

The third term in the right-hand side of Equation (59) and the second term in (60) are responsible for the interactions between the electronic system and the electronic field. E represents the electronic amplitude, and ${\mathsf{\mu }}_{n\alpha }$ is the transition moment between states α and β.

The complex interaction terms are given by the energy gaps between electronic excitations p and q $\left({\omega }_{\alpha } -{\omega }_{\beta }\right)$ and the exciton population coefficient ($C_{pq}$) and others as follows

$$\begin{array}{c}{{\mathrm{\Gamma }}_{\alpha \beta ;mn}=\sum }_k^M{\sum }_q^N\left[{\delta }_{\beta n}{C}_{\alpha i}^{*}{\left|C_{ki}\right|}^2{C_{mi}\gamma }_{(i,i)}\left({\omega }_{\alpha }-{\omega }_{\beta }\right)+{{\delta }_{\alpha n}{C}_{ni}^{*}{{\left|C_{ki}\right|}^{2}C}_{\beta i}\gamma }_{(i,i)}\left({\omega }_n-{\omega }_k\right)\right]\\ {\sum }_i^N\left[{C_{\alpha i}^{*}C}_{mi}{C}_{ni}^{*}{C}_{\beta i} \left\{{\gamma }_{\left(i,i\right)}\left({\omega }_m-{\omega }_{\alpha }\right)+{\gamma }_{\left(i,i\right)}\left({\omega }_n-{\omega }_{\beta }\right)\right\}\right]\end{array}$$

(61)

$${\mathrm{\Gamma }}_{\alpha \alpha ;mm}=2{\delta }_{\alpha m}{\sum }_k^M{\sum }_i^N\left[\right|C_{\alpha i}{|}^2|C_{ki}{|}^2{\gamma }_{(i,i)}\left({\omega }_m-{\omega }_k\right)-2{\delta }_{\alpha n}{\sum }_1^N|C_{\alpha i}{|}^2|C_{mi}{|}^2{\gamma }_{(i,i)}\left({\omega }_n-{\omega }_k\right)]$$

(62)

where

$${\mathrm{\gamma }}_{\left(i, i\right)}(\omega )={\displaystyle \frac{2{\gamma }_{\left(i,i\right)}^0}{1+\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{\omega }{k_{B}T})}}.$$

(63)

The factor ${\mathrm{\gamma }}_{\left(i, i\right)}(\omega )$ is taken to satisfy the thermal equilibrium condition [225,226,227,228,229,230,231,232,233]: ${\gamma }_{\left(i,i\right)}^0$ indicates the zero-temperature limit of ${\mathsf{\gamma }}_{\left(i, i\right)}(\omega )$, and $k_B$ is the Boltzmann constant. The fourth-order Runge-Kutta method was applied to solve Equations (59)–(63). Quantum dynamic theories beyond Markov approximations have been developed in relation to coherent energy transfers in biological systems [253,254,255,256,257,258]. Beyond Markov models are not touched on in this review.

6.3. Characteristic Features of Energy Transfers in Dendrimer Systems

The numerical computations of the master equations in Equations (59)–(63) have been performed to elucidate the energy levels of excitation and dynamics of energy transfers for D25 and S25 in Figure 10, which are crucial for theoretical understanding and explanation of the two characteristics, D25-a and D25-b, of D25, which consisted of two-state monomers. For the D25-a feature, the monomers involved in the linear-leg regions are increased from the periphery to the core, while for the D25-b feature, the intermolecular distances between the adjacent legs at the branching points are increased in comparison with those in the linear-leg regions. By diagonalizing Equations (59) and (60), the excitation energies and transition moments between ground and excited states have been obtained, elucidating several degenerate excitation states with dominant transition moments in a sharp contrast to those of S25 and special distributions of excitons in these excited states [235,236,237,238,239,240,241,242,243,244,245,246].

The exciton dynamics of D25 and S25 were also investigated under the assumptions of the amplitude (E) of the applied field of 10 MW/cm² and the zero-temperature value of ${\gamma }_{\left(i,i\right)}^0$ = 2000 cm⁻¹ and temperature T = 300 K in Equation (63). These parameter settings do not influence the characteristic features of relaxation pathways but only affect the relaxation speed. Figure 11 shows the spatial variations in the exciton distributions for D25 at three different times, I-III. The exciton generated in D25 is first distributed in the peripheral region, as shown in I of Figure 11, and it is displaced in the intermediate region II. Finally, the exciton reaches the central region III. Thus, the exciton for D25 is found to migrate from the periphery to the core via intermediate regions [238]. On the other hand, such variations of the spatial exciton distributions at these three times were not obtained for the S25 structure. This difference between D25 and S25 is mainly ascribed to the overlap between the distributions of the two exciton states, namely $\left|C_{\alpha i}{|}^2\right|C_{\beta i}{|}^2$ arising from the characteristic structure. This tendency was the same in the two-exciton model [239]. Thus, the unique exciton migration observed in the dendric systems of the present concern is realized by satisfying the following two criteria: (i) there exist multi-step exciton states whose spatial distributions are mutually well segmented; (ii) exciton–phonon couplings exist at the blanching points; (iii) there remain partially spatial overlaps of exciton distributions between neighboring exciton states. These conclusions are equivalently transformed into an understanding of exciton and energy transfers in photosynthesis.

6.4. Coherent Energy Transfer and Coherent Control for Quantum Devices

Over the past decades, general theories for quantum systems interacting with external baths have developed in theoretical physics [219,220,221,222,223,224,225,226,253,254,255,256,257,258]. Quantum effects for photochemical excited states, such as coherent energy transfer, are important issues examined on the Bloch and Redfield equations [217] by Hochstrasser and his collaborators [259,260]. They have derived a general formula of the relaxation time of the exciton using the T₁ (population) and T₂ (decoherence) processes in Equations (35) and (36), and the angle between dipole moments in Equation (54). They performed the observations of fluorescence anisotropy decays of bi-fluorenyl (or naphthyl (BN)) systems, demonstrating that the electronic excitation can jump coherently from one chromophore moiety to the other as long as the pure dephasing (T₂′) between the excited states is slower than the inverse resonance splitting (1/β). In early 2000, the Yamazaki group [261,262,263,264,265,266] performed pioneering experimental work on the quantum effects for exciton dynamics and quantum control for quantum phase. They have synthesized meta-1, 2-bis(antheracene-2-yl)benzene (m-DAB) as illustrated in Figure 12, where anthracenes a and b are coupled via the meta-phenylene bridge. The one-electron excitation a or b provides the two different excited local exciton (excited diradical) configurations ${\varphi }_1= a^{\prime }b$ and ${\varphi }_2= a{b}^{\prime }$, which are degenerated in energy. Therefore, the quantum resonance occurs to provide the in- and out-of-phase coherent states ${\psi }_1$ and ${\psi }_2$, providing the energy gap ($E_2$ − $E_1$ = 2β) as illustrated in Figure 12A. The same situation was illustrated in the case of resonating broken-symmetry (RBS) state of local spins, as shown in a previous paper [1]: an example of the equivalent transformation. The pulse laser excitations of this system provide the time-dependent excited state given by the superposed state $\mathrm{\Psi }\left(t\right)$

$$|\Psi \left(t\right)|={\displaystyle \frac{1}{\sqrt{2}}}{\Psi }_1\exp -i{E}_{1}t/\hslash +{\displaystyle \frac{1}{\sqrt{2}}}{\Psi }_1\exp -i{E}_{2}t/\hslash$$

(64)

The probability density is given by

$$|{\Psi \left(\mathrm{t}\right)|}^2={cos}^2(\beta t/\hslash ){\varphi }_1^2+{sin}^2(\beta t/\hslash ){\varphi }_2^2$$

(65)

where an exciton localized on either half of a bichromophore moves back and forth between two chromophores with a time period of h/2$\beta$.

The relaxation terms are essential for realistic dendrimer models. As shown in Equations (35) and (36), m-DAB interacts with the reservoir, as illustrated in Figure 8, suffering the T₁ (population) and T₂ (coherence) dissipation processes. Therefore, the relaxation process of the resonating exciton is expected on theoretical grounds. Yamazaki et al. [261] have developed the fluorescence upconversion observation method, where sum frequency light between sample fluorescence and probe laser light generated using nonlinear optics is selectively observed, and time delays between the two lights are changed to obtain the time-resolved fluorescence. They have observed the decay curve of fluorescence of m-DAB by this method, observing quantum beats responsible for the quantum recurrence of the coherence. The observed T₂ value was 0.7~0.9 picoseconds for m-DAB. The interaction energy (β) was calculated to be 10~30 cm⁻¹ for m-DAB on the basis of the dipole–dipole interaction for the X-ray diffraction structure. Yamazaki et al. have discussed the possibility of quantum devices based on the active control of the quantum phase of excitons [266].

6.5. Quantum Dynamics of Excitons in Dendrimers with and Without Relaxation Term

As a continuation of the previous work [238], Nitta et al. [244] have further examined the quantum dynamics of the dendrimer structure D25 in Figure 10. First of all, they have examined dendrimers, a dimer model with the orthogonal dipole units (two two-state monomers), assuming the exciton excitation energy E = 38,000 cm⁻¹ and transition moment μ = 8D. The diagonalization of the 2 × 2 matrix provided two one-exciton states with E₁ = 37,957.3 cm⁻¹, E₂ = 38,042.7 cm⁻¹ and the energy gap 2β = (E₂ − E₁) = 85.4 cm⁻¹ (see Figure 12). As the next step, they performed the quantum dynamic simulation assuming the power of the applied electronic field = 100 MW/cm², the temperature T = 300 K, and ${\gamma }_{\left(i,i\right)}^0$ value in Equation (63) = 10 cm⁻¹. Figure 13A shows the calculated exciton population dynamics after the irradiation of 300 optical cycles (ca, 263 fs) under the assumption of no relaxation to the ground state, indicating the quantum beats in the population curves: oscillation period is about 400 fs, and it responds to the energy gap (2β).

In early 2000, we theoretically investigated coherent processes in mesoscopic systems in relation to magnetism, superconductivity, photonics, etc. [154,172,179,209,252]. Coherent energy transfers have been important and interesting phenomena in relation to light-harvesting in photosynthesis. The experimental results by Hochstrasser [259,260] and Yamazaki [261,262,263,264,265,266] groups have provided indispensable information to our theoretical study of the optical quantum coherence. We examined two different geometric structures, (A) a uniform interaction model (constant distance = 15 a.u.) and (B) a non-uniform model with a short distance model for the neighboring monomers, as illustrated in Figure 10B, elucidating different exciton dynamics of excitons under the assumption E = 38,000 cm⁻¹ and transition moment μ = 5D. After the diagonalization of Hs in Equation (53), the transition energies and the magnitudes of transition moments between the ground and the one-exciton states were obtained as illustrated in Figure 13B. The energy levels obtained for model B become lower or higher than those of the constituent monomer because of the strong dipole–dipole interactions, indicating the significant energy splitting. The low and higher states exhibit exciton populations in the inner and outer regions, as shown in Figure 13B.

The exciton dynamics were also performed for both (A) and (B) models, assuming the same condition for the dimer in Figure 13A. In order to reveal the effect of the differences in structure and energetics of the (A) and (B) models, they first performed the exciton dynamics without relaxation terms (${\gamma }_{\left(i,i\right)}^0$ = 0). It was found that the oscillations of the exciton population are well localized in each segmented region, namely the external region, because of the applied field (E₁ = 37,301.7 cm⁻¹) responsible for the exciton in this region, as shown in Figure 13C. On the other hand, in addition to the relaxation term, both energy transfer from the external to the internal region and oscillations occur, as shown in Figure 13D. They have also examined the possibility of controlling the coherent exciton motions by tuning the frequency of an external field. This model is considered to be responsible for the active roles of protein environments in the light-harvesting systems.

6.6. Energy Transfers for Phycobilisomes in Cyanobacteria

Over the past decades, light-harvesting antennas have been investigated in relation to photosynthesis. The computational results for energy transfers in the D25 dendrimers with the weak dipole–dipole interaction provide indispensable information on the light-harvesting phycobilisomes (PBSs) in cyanobacteria [267,268,269,270,271,272,273,274,275,276,277,278,279] as illustrated in Figure 14A. Cyanobacteria, glaucophytes, and rhodophytes utilize giant, light-harvesting phycobilisomes (PBSs) for capturing solar energy and conveying it to photosynthesis reaction centers. Recently, cryo-electron microscope (EM) experimental methods have been applied to elucidate three-dimensional (3D) structures of the PBS complex for light absorption and energy transfer in photosynthesis. PBSs have generally hemidiscoidal geometric structures, like half of the dendrimers examined in Figure 10A, and they consist of a core and rod, as illustrated in Figure 14A. The core protein of PBS is allophycocyanine (APC), which absorbs red light (650 nm). On the other hand, rod proteins mainly consist of phycocyanine (PC), which also absorbs 620–630 nm red light. Other proteins, such as phycoerythrocyanine (PEC; 550–620 nm orange lights), phycoerythrin (PE; 490–550 nm blue-to yellow lights), are used in some PSBs. Therefore, judging from the absorption energies and quantum simulation, the energy flow systems in PSBs are generally regarded as PE (450–550 nm) → PEC (550–620 nm) → PC (620–630 nm) → APC (650 nm) → Chlorophyll a (670–680 nm) → Reaction centers of PSII and PSI. Indeed, the 3D structures of PSBs usually consist of the rod (PE → PC) and core (APC). Thus, PSB with a hemidiscoidal shape of dendrimers suggests possibilities of similar mechanisms for light-harvesting as shown in the energy gap between chromophores in Equations (61) and (62).

Recently, cryo-EM experiments have been performed for PBSs of cyanobacteria [263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279]. Kawakami et al. [277] have performed cryo-EM experiments to elucidate 3D structures of phycobilisome (PBS) in the thermophilic cyanobacterium Thermosynechococcus (T) vulcanus, elucidating the refined structures of the core (3.7 Å resolution) and rods (4.2 Å resolution) by the single particle reconstruction as illustrated in Figure 14C. The core of PSB was found to consist of pentacylindrical parts (A₁, A₂, B, C₁, C₂) as shown in Figure 14C. All these parts of the core are constructed with allophycocyanine (APC; 650 nm). On the other hand, the cylinder rod by the refined cryo-EM structure [278] consisted of six (eight in the previous paper [277]) parts: Rb, Rb’, Rt, Rt’, R_S1, R_S1’, R_S2, R_S2’. These rods consisted of PC (620–630 nm). Kawakami et al. have elucidated relative orientations and distances leading to excitation energy transfer pathways (cryo-EM experimental procedures and detailed structural information are available in the supporting materials of ref. [278]). They have also elucidated the organization of linker proteins for efficient energy transfer. Thus, the possible roles of proteins for energy transfers are key factors in the environmental effects in the theoretical model. Theoretical investigations on these complex PBSs remain to be a future task.

6.7. Energy Transfers in Light-Harvesting Systems in Photosynthesis

Several types of antenna systems have been discovered in photosynthetic systems, which are classified into three groups based on two criteria: (a) eubacteria or eucarya because of no archaebacteria; (b) anaerobic (anoxygenic) and aerobic (oxygenic). They are as follows: (i) eubacteria-anoxygenic; (ii) eubacteria-oxygenic; (iii) eucarya-oxygenic types [280,281]. Cyanobacteria examined in Section 6.4 are classified into type (ii), whereas plants are regarded as type (iii). In 1985, the X-ray structure of purple bacteria (type (i); Bchl a, b; 860 nm) using ubiquinone acceptor was first discovered [282], elucidating the geometric structure of the reaction center (RC). In 1995, the X-ray structure of the light-harvesting (LH2) center was also solved for the purple non-sulphur photosynthetic bacterium to elucidate the aggregate structure of the LH2 [283]. Theoretical studies on energy transfers in LH1(2) have been performed [284,285,286,287,288,289,290,291,292,293,294], elucidating the scope and applicability of Förster theory for energy transfers.

The X-ray structure was obtained for the Thermochromatium tepidum at 3.0 Å resolution, elucidating the three-dimensional (3D) structure of the LH1-RC and an important role of the Ca²⁺ ion for absorption regulation [295]. Theoretical investigation was performed to confirm the possible role of the Ca²⁺ ion [272]. Recently, cryo-EM methods have been applied to elucidate structures and functions of the LH1(2)-RC center of several purple bacteria [296,297,298,299,300,301,302]. The 3D ring structures of LH1-RC and LH2 are different from the PBS for cyanobacteria, indicating different characteristics for energy transfers [301]. Exciton dynamics in the LH1 and LH2 are really complex, for which the quantum beats have been observed within the rings, suggesting vibration-assisted energy transfers.

The X-ray [280] and cryo-EM studies for photosystem II (PSII) have been performed extensively [303,304,305,306,307,308,309,310,311,312,313,314,315,316,317]. The light-harvesting systems in PSII (oxygen evolution) exhibit partial similarities to those of LHI(II) (no oxygen evolution). Both spectroscopic and theoretical investigations have been performed for these systems. However, details of the energy transfers in LH1, LH2, and PSII are not touched on here: they have been discussed briefly in relation to macroscopic coherence in Appendix A.3.

6.8. Quantum Effects for Energy Transfers in Fenna–Mattews–Olson (FMO) Proteins

Recently, quantum effects for electronic energy transfers in photosynthesis have attracted renewed interest [318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350] in relation to the quantum coherence observed in model dendrimers [266]. In green sulfur bacteria such as Chlorobium tepidum, the energy transfer between the main chlorosome antenna and the reaction center is mediated by a protein containing bacteriochlorophyll (BChl, 840 nm) molecules called the Fenna–Matthews–Olson (FMO) protein [318]. The green sulfur bacteria undergo no oxygenic oxidation, exhibiting similarity to photosystem I (PSI). Recently, single particle cryo-EM structures of green sulfur bacterium have been elucidated [320,321], providing 3D structures consisting of two FMOs, two accessory protein subunits, and others (P840l F_X, F_A, F_B, etc.) and elucidating the molecular asymmetry-biased energy transfer. The FMO protein is a trimer consisting of identical subunits, each of which contains seven bacteriochlorophyll a (BChls) as shown in Figure 14B.

In 2007, Engel et al. [322] investigated the FMO complex isolated from Chlorobaculum tepidum by means of 2D electronic spectroscopy and succeeded in observing long-lasting quantum beating, providing direct evidence for long-lived electronic coherence. Their finding has raised a fundamental question about the role of the protein environment in protecting the quantum coherence in light-harvesting efficiency [323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350]. Aspuru-Guzik and his collaborators [324,326] have developed an environment-assisted quantum walks model for studying the role of quantum interference effects in energy transfer dynamics of molecular arrays interacting with a thermal bath within the Lindblad formalism [222] under the Born–Markov approximation [246,256,260]. Application of their model [326] to FMO [318] has elucidated that a contribution of coherent dynamics is about 30% in the presence of realistically correlated phonons [323]. They have also shown that quantum transport efficiency can be enhanced by a dynamical interplay of the system Hamiltonian with pure dephasing induced by a fluctuating environment in FMO [318]. Ishizaki and Fleming [328,329] have examined the scope and reliability of the Redfield-type equations [217,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268] with and without the secular approximation, indicating a new theoretical framework of the quantum dynamic simulation [328], demonstrating that the quantum wave-like motion persists for several hundred femtoseconds even at physiological temperature [328,329]. Thus, quantum coherence is found to be one of the important factors for analysis, understanding, and explanation of efficient energy transfers in light-harvesting systems of photosynthesis [241,244,275,328,329]. However, further experimental investigations, such as two-dimensional electronic spectroscopy, remain to obtain the final conclusions on the weights of the quantum coherence and implications of quantum coherence in many other antenna systems [347,348,349,350].

7. Discussions, Future Prospects, and Concluding Remarks

7.1. Down-Conversion and Four-Wave Mixing for Generation of Entangled Light Pair

One of the basic concepts in quantum optics [1] is entanglement, for which two or more particles become correlated in such a way that the state of one particle cannot be described independently of the others [19]. This concept has been experimentally demonstrated through investigations of photon entanglements, indicating potential application to optical quantum computing [20] and quantum sensing [21]. A well-known technique is the spontaneous parametric down-conversion (SPDC) to generate entangled photon pairs. In SPDC, a high-intensity pump beam is passed through a nonlinear optical (second-order χ⁽²⁾) crystal, causing it to split into two lower-energy photons that are entangled in their polarization and momentum, as illustrated in Figure 15A. An alternative process to generate entangled photon pairs is the spontaneous four-wave mixing (SFWM) using a nonlinear optical (third-order χ⁽³⁾) crystal as illustrated in Figure 15B.

The nonlinear optical response materials in Table 1 are thus effective for the generation of entangled light states for quantum information processing [351,352]. Therefore, the search for nonlinear optical materials in Section 2 is directly related to the development of molecular quantum devices. Previously [1], third-order (γ) hyperpolarizabilities were examined for several types of geometric structures of open-shell diradical species, providing guiding principles to obtain open-shell materials with large (γ) values as illustrated in Appendix A. Both theoretical and experimental efforts remain to discover new robust organic crystals with large γ values.

7.2. Decoherence, Error Corrections, and Fault-Tolerant Quantum Computer

Quantum computing is now extensively investigated because of several reasons. Several types of qubits for quantum devices have been proposed, as summarized in Figure 30 of the previous review [1]. The qubits have been used in gate-type quantum computing. On the other hand, the wave nature of a photon provides the continuous variables (CV) such as amplitude and phase (see Section 4) for another type of quantum computing [353,354,355,356,357]. Indeed, CV optical quantum computer (OQC) has been of great interest because of several reasons, such as operation at room temperature. Measurement-based (MB) OQC using large-scale entangled systems is also under current investigation. However, quantum systems more or less suffer quantum decoherence (noise) as illustrated in Figure 8B, exhibiting errors for quantum information processing. Recently, several types of error correcting procedures [168,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368] have been proposed for quantum computing using qubits consisting of superconducting and neural atom (BEC) materials, indicating a new stage for the fault-tolerant quantum computer (FTQC) [358]. CV OQC and MB OQC are also investigated because of the same reason.

Thus, FTQC has been of great interest for practical applications. To this end, the Gotteman–Kitaev–Preskill (GKP) code [369,370,371] has been used to suppress the quantum error in quantum computation. Therefore, several groups have been performing experimental efforts for the generation of GKP qubits in propagating light to realize optical quantum computers with several merits, such as operation at room temperature. However, the generation of this type of GKP qubit has not been realized yet because non-linear materials for the generation of squeezed light with the high broken symmetry (14.8 dB) [369] have not been obtained yet. Judging from recent developments of FTQC, large-scale configuration interactions (LSCI) using UNO(ULO) for large molecular systems [1] may be feasible, providing static energy levels, transition moments, etc., which are essential for successive quantum dynamic simulations in Figure 8A. On the other hand, hybrid quantum and classical systems may be effective for functional materials operating at room temperature. For example, biological light-harvesting systems may not exhibit complete quantum phenomena, but they are still effective for biological functions such as energy transfers examined in Section 6. Recent efforts and developments for optical QC with and without complete error corrections are summarized in several articles.

7.3. Triplet–Triplet Annihilation for Singlet Oxygen Formation and Photochemical Damages

Recently, energy conversions have been attracted great interests because of several reasons. Triplet–triplet annihilation (TTA) is one of the important energy conversion processes for triplet species. Historically, the TTA process was known as the toxic process for plants. The ground state of molecular oxygen is triplet (³O₂), exhibiting no high reactivity for closed-shell molecules [372,373,374,375]. In the 1970s, the TTA process between triplet molecular oxygen (³O₂) and the excited triplet state of chromophore (³CP*) was a useful photochemical method for the generation of singlet molecular oxygen (¹O₂)(¹Δ) [373,374], which exhibits high reactivity [375] for organic molecules. Theoretically, this TTA process is spin exchange-allowed because of the total spin state of the complex (2S + 1 = 0, 1, 2) as shown in Equation (66). The high reactivity of singlet oxygen (¹O₂) is often harmful for biological molecules such as chlorophyll (CHl). The carotenoid compounds (CA) are effective for the quenching of ¹O₂, indicating the reverse process as illustrated in Equation (67). Therefore, CA plays a role as a scavenger of toxic ¹O₂ in the oxygen-evolving center (OEC) of photosystem II (PSII) [210].

³O₂ + ³CP* →     ^1,3,5[³O₂ …³CP*] → ¹O₂ + ¹CP

(66)

¹O₂ + ¹CA →    ^1,3,5[³O₂ …³CA*] → ³O₂ + ³CA*

(67)

7.4. Triplet–Triplet Annihilation, Upconversion and Frequency Doubling

Recently, the TTA process has been extensively investigated for energy upconversion processes. The TTA mechanism is indeed useful for the generation of a singlet excited state by the recombination of two triplet excited states of chromophores (CP) as

³CP₁* + ³CP₂* → ^1,3,5[³CP₁*…³CP₂*] → ¹CP₁* + ¹CP₂

(68)

This TTA process can be applicable to obtain an excited singlet state (¹CHP₁*) from the low-energy triplet state (³CP₁*), indicating upconversion of a low energy (for example, infrared light) to a high-energy (visible) light, which can be used for several purposes such as water oxidation of photosynthesis and medical applications such as photodynamic therapy.

In Section 6, the fluorescence upconversion method was touched on in relation to femtosecond spectroscopy. Recently, up-conversion of photon energy (formally the reverse process of the down-conversion discussed above) has attracted great interest in relation to energy conversion methods for photochemistry and photosynthesis [376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392]. Two-photon absorption and frequency doubling with nonlinear response materials in Section 2 are also useful methods for upconversion of low-energy light to high-energy light [376,377,378,379,380,381,382,383,384,385,386,387]. Other methods have been developed for up-conversions of low-energy lights. Many organic conjugated molecules have been synthesized for developing efficient upconversion systems. Mesoscopic cluster systems are also investigated in relation to upconversions because of several reasons. Indeed, lanthanide systems have been developed for up-conversions in both material and biomedical applications. Gold clusters [393,394,395,396,397,398] have also been investigated for non-toxic upconversion materials. Many excellent reviews have already been presented for organic, inorganic, and hybrid materials for upconversions [376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398]. Recent theoretical and computational results for TTA processes are summarized in several papers [399,400,401].

7.5. Future Prospects of Quantum Effects in Photosynthesis

Many interesting findings have been presented in relation to quantum effects in molecular science. Here, we concentrate on our current interest among them. As mentioned above, the oxygenic photosynthesis consists of three steps: (1) solar energy absorption and energy transfer to the reaction center of PSII and PSI, (2) photo-induced redox reactions for generation of electron-hole pair at oxygen evolving complex (OEC) of PSII and four-holes generation of the catalytic CaMn₄O_x cluster and (3) four electron oxidation of two water molecules, generating molecular oxygen, and four electrons and four protons which are used for reduction of NADP+ into NADPH at PSI [210]. Umena, Kawakami, Shen, and Kamiya [277,305,402] and their collaborators have performed systematic X-ray and cryo-EM experimental studies of these steps of photosynthesis by using the T. vulcanus [306,403]. Interplay between theory and experiments [210] has also contributed to understanding and explanation of the mechanisms of water oxidation at the quantum and molecular levels.

Recently, uphill energy transfers have been found in some green algae, indicating the use of far infra-red light for photosynthesis [404,405,406,407,408,409,410]. Cryo-EM experiments [409] have elucidated geometric structures of 11 rings of chromophores for energy transfer, providing foundations for constructions of realistic model clusters for quantum simulations. Judging from the simulation results in Section 6.5, low-energy absorption for the generation of an exciton and high-energy exciton transfer assisted with protein dynamics may be feasible in biological systems. The master equation approach revisited in this review will be a next theoretical approach for uphill energy transfer in the first step of photosynthesis, namely light-harvesting (see Appendix A.4). To this end, the intermolecular interactions between chromophores can be calculated by the quantum-mechanical methods [1,210], elucidating possible roles of protein matrix for confinements of the ring for the dipole–dipole interactions. As mentioned above, reactive oxygen species (ROS) such as ¹O₂ have been regarded as unwanted species in biological systems. However, recent experimental investigations [411] have elucidated that ROS are crucial regulators in biological systems, exhibiting important signaling functions, namely quantum sensing. Thus, new fields are open for the theoretical modeling of dynamical biological functions, which are interesting with regard to quantum information and quantum biology.

A generation of the attosecond laser pulses using the HHG method [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68] has been reviewed in Section 4 of this review. Very recently, a new attosecond approach has been developed based on the X-ray free electron laser (XFEL), providing an attosecond inner-shell lasing method at Ångström wavelength [412,413,414,415,416,417,418]. The serial femtosecond crystallography (SFX) by XFEL has been successfully applied to elucidate molecular structures of short-lived intermediates generated in metalloenzymes such as photosystems II (PSII). The newly-developed attosecond spectroscopy and imaging by XFEL will open the door for elucidating transient molecular structure and quantum-mechanical motion of electron. These simultaneous measurements provide crucial information for understanding and explanation of dynamical mechanism of enzyme reactions [413], such as, the mechanism of water oxidation in PSII [209].

7.6. Concluding Remarks

As a continuation of a previous review [1], we have summarized the non-perturbative approach to nonlinear optics of atoms and molecules, providing the explicit expressions of their n-th-order optical responses [27,124]. Nonlinear optical materials are crucial for spontaneous parametric down-conversion (SPDC) to generate entangled photon pairs, which are used for continuous variable (CV) quantum computing (QC). Historical developments of the ionization processes by the intense laser fields are reviewed in relation to the n-th-order higher harmonic generations (HHG), which are used to develop the attosecond spectroscopy for energy transfer and electron transfer [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. The orbital tomography methods [49,68] using the attosecond pulses are also reviewed to obtain the optical images of the amplitude and phase of wavefunctions of atoms and molecules, opening up a new field of quantum chemistry. Dyson-type orbitals [74,75,76] are also reviewed in relation to the orbital tomography [49,68]. Recent quantum (squeezed) HHG is not touched on in this review.

The quantum dynamic simulations for quantum systems with interactions between molecules (atoms) and quantum lights are also reviewed in relation to early Wigner [132] and Husimi [129] distribution functions and recent active controls of quantum phase for macroscopic quantum coherence (MQC) and tunneling (MQT) in mesoscopic systems such as Bose–Einstein condensation (BEC) [157] of Rydberg atoms and others [161,162,163,164,165,166,167,168,169]. The master equation approach [217,222,235,236,237,238,239,240,241,242,243,244,245] is also extended to the exciton (singlet electron-hole open-shell pair) dynamics and electron energy transfers to elucidate the mechanisms of energy transfer from the periphery to the core in the dendrimer systems [227,228,229,230,231,232,233]. These mechanisms revealed for the model systems are equivalently transferred for understanding the coherent and non-coherent energy transfers in light-harvesting systems in native photosynthesis, providing guiding principles for the design of artificial energy conversion systems.

Recently, cryo-EM experiments have been extensively performed to elucidate complex structures between light-harvesting systems and core CaMn₄O_x cluster systems for water oxidation, indicating master equation approaches to them. To this end, the scope and reliability of the conventional master equation approaches [248,249,250,251,252] and more extended approaches [253,254,255,256,257,258] will be elucidated to obtain reliable information based on the observed 3D structures of the native antenna systems. Discovery of the native systems using far infra-red light for photosynthesis [393,394,395,396,397,398,399] indicates the necessity of many more experimental searches of photosynthesis systems under extreme conditions, including other than Earth. It may provide a new guiding principle for constructions of robust artificial systems. Large-scale computer simulations are also expected to provide indispensable information for the mechanisms of native photosynthesis, developing the quantum information science for the constructions of quantum devices.