The expectation values of the coherent π−electron angular momentum and the ring current operators in an aromatic molecule are generally expressed as 〈 O ^ ( t ) 〉 = n ∫ d 3 r 1 ⋯ d 3 r n Ψ ∗ ( t ) O ^ ( r ) Ψ ( t ) . (1)

Here, O ^ ( r ) is a single-electron operator for the angular momentum or current. Ψ ( t ) is the wave function of the π−electrons in laser field F(t) at time t, n denotes the number of electrons, and r i express the ith electron coordinates. Since the optically-allowed electronic excited states of the conjugated aromatic rings are of our interest, the electronic wave function can be expanded by the two electronic configurations, i.e., the ground configuration Φ 0 , and the singly excited ones Φ α as Ψ ( t ) = c 0 ( t ) Φ 0 + ∑ α c α ( t ) Φ α , (2) where Φ 0 is defined by a single Slater determinant as Φ 0 ( r 1 , ⋯ , r n ) = ‖ ϕ 1 ⋯ ϕ a ⋯ ϕ b ⋯ ϕ n ‖ with ϕ n ≡ ϕ n ( r n ) . Here, ϕ a and ϕ b are the occupied orbitals, Φ α is the electronic wave function for a single electron excited configuration α : a → a ′ , i.e., the single electron transition from the occupied molecular orbital (MO) a to the unoccupied MO a ′ , when Φ α ( r 1 , ⋯ , r n ) = ‖ ϕ 1 ⋯ ϕ a ' ⋯ ϕ b ⋯ ϕ n ‖ .

The coherent and incoherent temporal behaviors of the electrons induced by a laser field F ( t ) can be obtained directly by solving the coupled electronic equations of motion expressed by the density matrix elements ρ α β ( t ) under the initial population conditions ρ 00 ( 0 ) = 1 and ρ α α ( 0 ) = 0 for α ≠ 0 and ρ α β ( 0 ) = 0 for α ≠ β , that is, there is no electronic coherence at the initial time such that d ρ α β ( t ) d t = − i ℏ ∑ γ ( V α γ ( t ) ρ γ β ( t ) − ρ α γ ( t ) V γ β ( t ) ) − ( i ω α β + γ α β ) ρ α β ( t ) . (3)

Here, ρ α β ( t ) ≡ c α ( t ) c β ∗ ( t ) is the density matrix element, and V α γ ( t ) is the coupling matrix element between states α and γ via the molecule-laser interaction V ^ ( t ) = − μ ⋅ F ( t ) , where μ denotes the transition dipole moment operator, γ α β ( = 1 2 ( γ α + γ β ) + γ α γ ( d ) ) are the dephasing constants in the Markov approximation [38, 39] and ω β α is the frequency difference between the two electronic states α and β. Here, γ α ( γ β ) is the non-radiative transition rate constant of the electronic state α(β), and γ α β ( d ) are the pure dephasing constants induced by the elastic interaction between the heat bath and molecule of interest.

Because we are interested in the coherent behaviors of dipole-allowed quasi-degenerate π-electronic excited states in the visible or UV region of an aromatic ring molecule, Eq. 1 can be rewritten in terms of singly excited configurations { Φ κ } if κ ≠ 0 as 〈 O ( t ) 〉 = n ∫ d 3 r 1 ⋯ d 3 r n Tr ( ρ ( t ) O ( r ) ) , (4) where O α β ( r ) = 〈 Φ α | O ^ ( r ) | Φ β 〉 .

In Eq. 4, the coherence between singly excited state configurations is considered, and the coherence between the ground state and the excited state is neglected because this coherence is much shorter compared to that between the singly excited state configurations. In this case, the coherence time is proportional to the inverse of the energy gap between the two electronic states.

The coherent π-electron angular momentum is formulated in the quantum chemical MO theory, and the π-orbital ϕ k associated with the optical transition is expanded in terms of a linear combination of atomic orbitals χ i as ϕ k = ∑ i c k , i χ i ,   ( k = a ,   a ′ ,   b ,   b ′ ) , (5) where χ i denotes the atomic orbital, and c k , i indicates the molecular orbital coefficient.

Equation 4 can be expressed by using the Eq. 5 as 〈 O ^ ( t ) 〉 = 2 n ∑ α < β Im ρ β α ( t ) ∑ i j ( δ a b c a ′ i ∗ c b ′ j + δ a ′ b ′ c a i ∗ c b j ) ∫ d 3 r χ i ∗ i O ^ ( r ) χ j . (6)

Here, the temporal behavior of the expectation value 〈 O ^ ( t ) 〉 can be expressed using the off-diagonal density matrix element ρ β α ( t ) . The suffixes (a, a’) ((b, b’)) corresponds to the electronic configurations α(β), respectively.

Consider the spatially fixed aromatic ring molecule with N aromatic rings. The electron angular momentum operator is defined as O ^ ( r ) = l Z = ∑ K O ^ K ( r ) with O ^ K ( r ) = l Z , K = − i ℏ ( x K ∂ / ∂ y K − y K ∂ / ∂ x K ) n ⊥ , K . Here l Z , K is the electronic angular momentum operator of the component perpendicular to ring K. Coordinates x K and y K are defined on the aromatic ring K, and n ⊥ , K is the unit vector perpendicular to the aromatic ring. The expectation value of Kth angular momentum operator is given in terms of a 2p _z carbon atomic orbitals (AOs) as l Z , K ( t ) ≡ ∫ d 3 r K 〈 Ψ ( t ) | l Z , K | Ψ ( t ) 〉 = − 2 n e ℏ   n ⊥ , K ∑ α < β Im ρ β α ( t ) ∑ i j ∈ K ( δ a b c a ′ i ∗ c b ′ j + δ a ′ b ′ c a i ∗ c b j ) × ∫ d 3 r χ i ∗ ( x ∂ ∂ y − y ∂ ∂ x ) χ j , (7) where n _e is the total number of electrons in the system.

Note that in Eq. 7 the dependence of the electron angular momentum on the laser intensity is reflected in the imaginary part of the off-diagonal density matrices Im ρ β α ( t ) .

From Eq. 6, the perpendicular component of the time-dependent electric current flowing through a surface S (See Figure 1) is generally defined as 〈 J ( t ) 〉 = 2 n e ∑ α < β Im ρ β α ( t ) ∑ i j ( δ a b c a ′ i ∗ c b ′ j + δ a ′ b ′ c a i ∗ c b j ) ∫ d 3 r χ i ∗ i J ^ ( r ) χ j . (8)

Here J ^ ( r ) = e ℏ 2 m e i ( ∇ → − ∇ ← ) is the current density operator. ∇ → ( ∇ ← ) is the nabla operating on the atomic orbital on the right-hand side (left-hand side).

Equation 8 can be expressed as 〈 J ( t ) 〉 = ∑ i < j J i j ( t ) , (9) where J i j ( t ) = 2 n e ∑ α < β Im ρ β α ( t ) ( δ a b ( c a ′ i ∗ c b ′ j − c b ′ i ∗ c a ′ j ) + δ a ' b ' ( c a i ∗ c b j − c b i ∗ c a j ) ) J i j , (10a) with J i j = ∫ S d 2 r ⊥ χ i ∗ n ⊥ , S ⋅ ∇ → χ j , (10b) where n ⊥ , S is a unit vector perpendicular to a surface S, which is given as n ⊥ , S = r j − r i | r j − r i | , and the surface S is set at the center of the carbon bond C _i − C _j . The surface integrations in Eqs. 8, 10b are carried out over the half-plane S (see Figure 1). Using Slater type AOs for { χ i } , J i j in Eq. 10b can be expressed in analytical form [40].

We introduce the bridge bond current J B , p q ( t ) which is a specific case of an interatomic bond current, and is defined as the current bridging two aromatic rings P and Q, passing from the nearest neighbor carbon atom at site p to q. Here, each carbon atom C _p , or C _q belongs to a different aromatic ring P or Q which are in the neighbors. (see Figure 1).

The bridge bond current is given in terms of the interatomic current, J p q ( t ) , as J B , p q ( t ) = − J p q ( t ) cos ⁡ θ d . (11)

Here, θ d is the dihedral angle between the two rings P and Q [39].

We now define an averaged ring current along ring K, J K ( t ) , which is defined by taking the average of all bond currents as J K ( t ) ≡ 1 n K ∑ ( i < j ) ⊂ K n K J i j ( t ) , (12) where n K is the number of bonds in ring K.

(P)-2,2’-biphenol is one of the typical nonplanar chiral aromatic ring molecules, which has two aromatic rings connected through the C-C bridge bond. For convenience, hereafter, we denote L (R) for the left- (right-) hand side phenol ring of (P)-2,2’-biphenol. Figure 2A exhibits the geometrical structure of (P)-2,2’-biphenol with the transition dipole moments vectors between the ground and excited states. (P)-2,2’-biphenol was assumed to be fixed on a surface by a non-conjugated chemical bond or was oriented in the space by the molecular orientation techniques by laser [44–46]. The laboratory-fixed Y-axis was set parallel to the single chemical bond bridging two phenol molecules, and the rotational axis of the molecule belonging to point group C₂ was set along the laboratory-fixed Z-axis which was parallel to the surface normal. The ground state geometry of (P)-2,2’-biphenol was optimized using the DFT-B3LYP level theory with the 6-31G+(d,p) basis set in the GAUSSIAN09 code [47]. The dihedral angle θ d between the two phenol rings, was found to be 108.9 ∘ based on the density functional theory (DFT) calculations. The calculated geometrical structures are provided in Refs. [39, 40].

To create the coherent angular momenta and ring currents in (P)-2,2’-biphenol, we focused on the three dipole-allowed electronic excited states (a, b ₁, and b ₂) as shown in Figure 2B. The transition energies from the ground (g) to the a, b ₁, and b ₂ states, which were calculated under the optimized ground state geometry using the TD-DFT B3LYP level of theory [39, 40] were 6.67, 6.78, and 6.84 eV, respectively.

The objective functional to be maximized, is defined as [59–61]. J [ F ] = 〈 Ψ ( T ) | O ^ T | Ψ ( T ) 〉 − α 0 ∫ 0 T d t ( F ( t ) ) 2 − 2 Im [ ∫ 0 T d t 〈 ξ ( t ) | i ℏ ∂ ∂ t − ( H 0 − μ ⋅ F ( t ) ) | Ψ ( t ) 〉 ] , (16) where H 0 is a Hamiltonian in absence of field, and Ψ ( t ) is the time-dependent wave function in the electric field F ( t ) . Here, the target operator O ^ T is expressed as O ^ T = | Ψ T > < Ψ T | , where Ψ T is the target state wave function at the final time T, defined as Ψ T = ∑ a c a ( T ) Φ a , and is equal to Ψ ( T ) under the optimal condition, O ^ T = | ∑ a c a ( T ) Φ a 〉 〈 ∑ b c b ( T ) Φ b | . The penalty factor α 0 is introduced to suppress the magnitude of the electric field F ( t ) . ξ ( t ) is the time-dependent Lagrange multiplier. The third term in Eq. 16 is added to decouple the boundary conditions of the equations for Ψ ( t ) and ξ ( t ) as indicated in Eq. 17. Taking the variational condition δ J [ F ] = 0 , the following coupled equations are obtained, i ℏ ∂ ∂ t Ψ ( t ) = ( H 0 − μ ⋅ F ( t ) ) Ψ ( t ) , (17a) i ℏ ∂ ∂ t ξ ( t ) = ( H 0 − μ ⋅ F ( t ) ) ξ ( t ) , (17b) where F ( t ) = − 1 α 0 Im 〈 ξ ( t ) | μ | Ψ ( t ) 〉 . (17c)

Here, Ψ ( t ) satisfies the initial condition Ψ ( 0 ) = Φ 0 , and ξ ( T ) satisfies the condition, ξ ( T ) = O ^ T Ψ ( T ) at final time T. Note that by solving Eq. 17, we obtain the optimal solution Ψ ( T ) , which is equal to Ψ T .

Consider the ring current localization to a designated ring χ in a PAH, as shown in Figure 6. From Eqs. 10a, 12, the ring current on ring κ at the target time T is expressed as J κ ( T ) ≡ 1 n κ ∑ α = 1 n ∑ β = 1 n ∑ ( i < j ) ⊂ κ n κ J i j , α β ( T ) = ∑ α = 1 n ∑ β = 1 n J κ , α β Im ( c α ( T ) c β ∗ ( T ) ) = J κ ( c 1 , c 2 , ⋅ ⋅ ⋅ c n ) , ( 1 ≤ κ ≤ m ) (18a) with c i ≡ c i ( T ) , and J κ , α β = 2 n e e ℏ m e ∑ ( i < j ) ⊂ κ n κ { ( δ a b ( c a ′ i ∗ c b ′ j − c b ′ i ∗ c a ′ j ) + δ a ′ b ′ ( c a i ∗ c b j − c b i ∗ c a j ) ) J i j } . (18b)

Here, n is the number of electronic excited states. Hereafter, we write c α ( T ) as c α for simplicity.

The target state Ψ T = ∑ α c α Φ α can be determined by applying the Lagrange multiplier method to the ring currents at the target time T in Eq. 18a, which provides the coupled equations in terms of the configuration interaction coefficients with { Re c α , Im c α }. The target operator is given as O ^ T = | Ψ T > < Ψ T | .

The coupled equations for the ring current localization to aromatic ring χ can be expressed as ∑ β = 1 n J χ , β α Im c β + ∑ κ ≠ χ m ∑ β = 1 n λ κ J κ , β α Im c β + λ χ Re c α = 0 ,   ( 1 ≤ α ≤ n ) (19a) − ∑ β = 1 n J χ , β α Re c β − ∑ κ ≠ χ m ∑ β = 1 n λ κ J κ , β α Re c β + λ χ Im c α = 0 ,   ( 1 ≤ α ≤ n ) (19b) ∑ α = 1 n ( ( Re c α ) 2 + ( Im c α ) 2 ) − 1 = 0 , (19c) and ∑ β = 1 n ∑ α = 1 n J κ , α β ( Im c α Re c β − Re c α Im c β ) = 0   . (19d)

A brief derivation of Eq. 19 is summarized in Appendix A.

The coefficients { Re c α , Im c α } with λ l included are numerically determined by applying the Newton-Raphson method to the coupled equations in Eq. 19. The target operator for the localization is obtained as O ^ T = | ∑ α c α Φ α 〉 〈 ∑ β c β Φ β | .

Now consider the bridge bond currents J B l ′ shown in Figure 6, and the perimeter ring current of a PAH with m aromatic rings, J P ≡ 1 m ∑ l = 1 m J l , which is defined as the average of the π-electron ring currents at each aromatic ring site. For the perimeter ring current, the bridge bond currents J B l ' ( 1 ≤ l ′ ≤ m − 1 ) flowing among the nearest neighbor aromatic rings, should be zero at the target time. The target operators for the bridge bond currents, J B l ' (   1 ≤ l ′ ≤ m − 1 ), and perimeter ring current, J P , can be derived from Eq. 19 by replacing J κ , β α with J B l ' , β α and J P , β α , respectively, because J B l ' and J P can be written in terms of { Re c α , Im c α } as J P = ∑ β = 1 n ∑ α = 1 n J P , α β ( Im c α Re c β − Re c α Im c β ) , (20a) and J B l ′ = ∑ β = 1 n ∑ α = 1 n J B l ′ , α β ( Im c α Re c β − Re c α Im c β ) , (20b) respectively.

Similar to Eq. 19, the coupled equations for the perimeter ring current are derived in Appendix B.

We applied the quantum optimal control (QOC) method described in the preceding section to anthracene, one of the smallest PAHs [42]. Anthracene (D_2h) was assumed to be fixed on a surface (the XY plane), or oriented spatially by lasers The molecular geometry was optimized in the MP2/6-311+g(d,p) level theory using the GAUSSIAN09 code [47]. As demonstrated in Figure 7A, anthracene consists of three aromatic rings, which are called L-, M-, and R- rings respectively.

Consider the two types of π-electron localized ring currents of anthracene: the perimeter current flowing along the outside chemical bonds of anthracene, and the middle ring current localized to the M-ring. Both types of localized ring currents belong to the irreducible representation B_1g of the D_2h point group, which are symmetry-adapted. The excited states contributing to the two ring currents need to belong to the B_3u and B_2u representations, because each corresponding electronic coherence created by the two excited states belongs to the B_1g representation. The excited state for the localized ring current is S₃ with B_3u representation, and the other two excited states S₅ and S₆ with B_2u representation as shown in Figure 7B. This defines a “symmetry-adapted ring current” [41, 42].

The excited state energies in Figure 7B were calculated to be E 3 = 5.10  eV, E 5 = 5.71  eV, and E 6 = 5.78  eV. Here, S₄ (B_3u) with E 4 = 5.13  eV was excluded because the oscillator strength was negligibly small (f = 0.0013) in our numerical simulation. The non-zero transition dipole moments relevant to the coherent π-electron ring current control were calculated using the time-dependent density functional theory (TDDFT) method as, μ S0,S 3 = ( − 3.98 , 0,0 )   , μ S0,S 5 = ( 0 , − 0.64 , 0 ) , and μ S0,S 6 = ( 0 , − 0.44 , 0 )   (in a.u.) for the ground and excited states. The vapor absorption spectrum of anthracene shows the strongest absorption peak at 5.30 eV [62], which corresponds to E 3 = 5.10  eV in the TDDFT calculation.

Figure 8 shows the QOC results for the generation of an anti-clockwise perimeter ring current in anthracene. The target state is expressed from the results obtained using the Lagrange multiplier method as Ψ AAA = 0.717 i Φ S 3 + 0.581 Φ S 5 + 0.385 Φ S 6 . (21)

Here, the subscripts specifying the target state, for example, ACA for Ψ ACA , indicates the anti-clockwise ring currents in two aromatic rings, L and R, and a clockwise ring current in aromatic ring M. It should be noted that the target state for a generation of any coherent ring current is expressed in terms of its complex form. The control target time was set to T = 60 fs. The matrix elements of π-electron ring currents J Χ , α β (X = L, M, R) and the bond current J B i , α β (i = 1, 2), for two excited states α, β are presented in Table 2. Here the π-electron ring current flowing in an anticlockwise direction is defined as positive, whereas the bond current flowing toward the Y direction is defined as positive (See Figure 6). Figure 8A shows the temporal evolutions of the coherent ring currents J χ for the three aromatic rings of anthracene, χ = L ,   M ,   R , and those for B1 and B2. It can be observed from Figure 8A that the two bond currents J B 1 , α β and J B 2 , α β vanish at the target control time of T = 60 fs, and all ring currents in three aromatic rings exhibit positive. This indicates that the coherent π-electrons rotate clockwise along the outside (perimeter) of the aromatic ring. The laser-controlled π-electrons includes a ring current averaged over the three aromatic rings, J p ≡ ( J L + J M + J R ) / 3 = 89.0  μA , which is the perimeter ring current.

Figure 8B shows the temporal behavior of the ground state S₀, and three excited states S₃ (B_3u), S₅ (B_2u), and S₆ (B_2u) populations during the QOC process, which are induced by the two control lasers. Figure 8C shows the X and Y-components of the electric field F ( t ) generated by the control lasers, and Figure 8D shows the Fourier transformed spectra F X ( ω ) and F Y ( ω ) in the ultraviolet (UV) frequency domain. By analyzing the temporal behavior of the electric field F ( t ) of the control lasers in Figure 8C and the Fourier transformed spectra of the control laser fields in Figure 8D, the mechanisms of the laser-controlled ring currents in anthracene can be clarified. It is evident that from the modulation in Figure 8C the two linearly (Y-) polarized lasers with a relative phase zero induce the electronic coherence between the two excited states S₅ (B_2u) and S₆ (B_2u), and that the linearly polarized laser pulse parallel to the X-axis creates the perimeter ring current on the molecular plane.

Thus far, we have considered the QOC procedure for generating an anti-clockwise perimeter current. The QOC procedure for generating a clockwise perimeter current can be carried out in the same manner as described above (See Eq. 21) by considering the target state, Ψ CCC = Ψ AAA ∗ = − 0.717 i Φ S 3 + 0.581 Φ S 5 + 0.385 Φ S 6 . (22)

We carry out the QOC procedure to generate the anticlockwise ring current localized to the middle ring of anthracene. The target state is expressed as Ψ 0A0 = 0.707 Φ S 3 − 0.440 i Φ S 5 + 0.553 i Φ S 6 . (23)

The target state for the middle ring current with the clockwise flow is given by the relationship Ψ 0C0 = Ψ 0 A 0 ∗ . Figure 9 exhibits the QOC results, where Figure 9A displays the temporal evolutions of ring current localization control to the three aromatic rings, indicating that J _M is a ring current with 64.4 μA at the target control time of 60 fs, but on the other hand the ring currents J _L and J _R of the other two aromatic rings vanish. This indicates that the ring current localized to the middle aromatic ring is created by the control laser pulses presented in Figures 9C,D. The temporal behaviors of the populations shown in Figure 9B are nearly the same as those of the perimeter ring currents as shown in Figure 8B. As presented in Figure 9C, the two linearly (Y-) polarized lasers induce the electronic coherence between the two excited states S₅ and S₆ with a definite relative phase π, in contrast to the results presented in Figure 8C. In the mechanism of generation between the perimeter and the middle ring currents there is no difference except the phase difference between the two excited states S₅ and S_6, because the temporal behavior in both Figure 8B and Figure 9B and the temporal behavior in the X-polarized laser pulse in both Figure 8C and Figure 9C are similar to each other.

Main difference between the temporal behaviors of the Y-polarized lasers in Figure 8C and Figure 9C can be explained briefly in the following. The two coherent states created by the Y-polarized lasers can be expressed as F + ( t ) = F S 5 s i n ( ω S 5 t ) + F S 6 s i n ( ω S 6 t ) , (24a) and F − ( t ) = F S 5 s i n ( ω S 5 t ) − F S 6 s i n ( ω S 6 t ) . (24b)

For simplicity, the amplitudes of the two coherent states are assumed to have the same value, i.e., F S 5 = F S 6 ≡ F Y . The above expressions can then be simplified to F + ( t ) = F Y s i n [ ( ω S 5 + ω S 6 ) t 2 ] cos [ ( ω S 5 − ω S 6 ) t 2 ] , (25a) and F − ( t ) = F Y c o s [ ( ω S 5 + ω S 6 ) t 2 ] sin [ ( ω S 5 − ω S 6 ) t 2 ] . (25b)

Here, the frequency difference | ω S 5 − ω S 6 | / 2 yields an oscillation beating period of 120 fs, which is nearly the same as that of the quantum beat frequency observed in Figure 8C and Figure 9C.

Having clarified the control mechanism of the coherent ring current generation, we can semi-quantitatively reproduce the above results in Figures 8, 9 by using an analytical method [42]. This indicates, in principle, that two types of ring current localizations, the perimeter ring current and the middle ring current in linear PAHs, can be generated in experiment using three coherent UV lasers without a sophisticated QOC device.

For QOC numerical simulations of the ring current localization control in anthracene, we have only considered the perimeter and the middle ring currents generations, which are symmetry-adapted, while we did not consider a ring current localization to the L- or R-ring of anthracene, which are created by a symmetry-broken procedure, as demonstrated for naphthalene in Ref. [41]. That is, other excited state(s) with gerade symmetry must be considered in addition to the excited states with ungerade symmetry as described in the previous subsection.

It is important to consider how to observe the ultrafast coherent ring currents in PAHs. We proposed a method to detect the direction of the atto-second coherent ring current by tracking the femtosecond molecular vibrational motions that can induce the nonadiabatic couplings [30]. Rodriguez and Mukamel [63] have proposed measuring the circular dichroism (CD) of the ring current using the pump-probe method. Recently, two methods have been proposed for the detection of the ultrafast coherent ring currents. One method, recently proposed by Yuan et al., Bandrauk’s group [64], is utilizing the atto-second detection method of the coherent electronic dynamics in molecules with the temporal and spatial resolutions using the circularly polarized ultrashort UV pump and X-ray probe laser pulses. The other method is to utilize the time-resolved scanning microscopy (STM) and the magnetic force microscopy (MFM) to detect the ring current-induced magnetic fields during the ultrashort time [65–67].

It is also expected that to apply our methods to planar or non-planar extended molecular systems, such as graphene sheets. In principle, our method can be extended to these large systems, by considering the symmetry of the molecular system and the constraints on the π-electron ring currents.

Using laser control, it is also essential to maintain the created ring current at the target region (path) at least during one vibrational period of the PAH [68]. Otherwise, the created ring current would dissipate quickly because of the vibronic interactions and/or the nonadiabatic couplings between the two electronic excited states [30, 69–71].

Consider the coherent π-electron angular momentum in an aromatic ring molecule with low symmetry excited by non-resonant, stationary UV/visible lasers. The molecule of our interest is one oriented in a space or attached to a surface, as mentioned in the preceding sections. We adopt a three-electronic state model in the frozen nuclei approximation. The three electronic states including the ground state specified by the energy ε 0 ( ≡ ℏ ω 0 ) , and two excited states specified by ε 1 ( ≡ ℏ ω 1 ) and ε 2 ( ≡ ℏ ω 2 ) , as shown in Figure 10. Here, DSIDES can be formed by two stationary linearly polarized lasers with detuning frequencies Δ 1 = ω a − ω 10 < 0 and Δ 2 = ω b − ω 20 > 0 . The frequency difference between the two excited states is expressed as ω 21 = ω 2 − ω 1 . The dynamic Stark shifts are denoted by the Rabi frequencies Ω₁ and Ω₂, as presented in Figure 10. The wave function of the total system in the stationary lasers is defined through laser-molecular interactions as Φ ( t ) = c 0 ( t ) exp ( − i ω 0 t ) ϕ 0 + c 1 ( t ) exp ( − i ω 1 t ) ϕ 1 + c 2 ( t ) exp ( − i ω 2 t ) ϕ 2 . (27)

Here, the normalization condition for the coefficients c 0 ( t ) , c 1 ( t ) , and c 2 ( t ) are as follows, | c 0 ( t ) | 2 + | c 1 ( t ) | 2 + | c 2 ( t ) | 2 = 1 . (28)

The system Hamiltonian interacting with the electric fields is given as H ( t ) = H 0 + V ( t ) , (29) where H 0 satisfies H 0 ϕ i = ε i ϕ i for i =0, 1, and 2. In Eq. 29, the interaction Hamiltonian V ( t ) between the system and the two electric fields is written as V ( t ) = − μ ⋅ F a ⁡ cos ( ω a t − ζ a ) − μ ⋅ F b ⁡ cos ( ω b t − ζ b ) . (30)

Here μ = − e r is the electronic dipole moment operator, where r means the electron coordinate. F α = F α e α ( α = a , b ) is the electric field with amplitude F α and photo-polarization vector e α , ζ α is the initial phase, and ω α is the central frequency. In Eq. 30, the two electric fields denoted by a and b induce non-resonant transitions from the ground state to the two excited states. The selective transition conditions are set by the choice of the laser polarization vectors ( e a and e b ) satisfying μ 02 ⊥ e a and μ 01 ⊥ e b , respectively.

The time-dependent Schrödinger equation can be written as i ℏ ∂ Φ ( t ) ∂ t = H ( t ) Φ ( t ) . (31)

The coefficients must satisfy the coupled differential equation i ℏ ∂ ∂ t ( c 0 ( t ) c 1 ( t ) c 2 ( t ) ) = H ( t ) ( c 0 ( t ) c 1 ( t ) c 2 ( t ) ) , (32) where the interaction Hamiltonian, H ( t ) = ( 0 V 01 a ( t ) V 02 b ( t ) V 10 a ( t ) 0 0 V 20 b ( t ) 0 0 ) , (33) is applied with V 01 a ( t ) = < ϕ 0 | μ ⋅ F a | ϕ 1 > cos ( ω a t − ζ a ) e x p ( − i ω 10 t ) and V 02 b ( t ) = < ϕ 0 | μ ⋅ F b | ϕ 2 > cos ( ω b t − ζ b ) e x p ( − i ω 20 t ) .

Equation 32 can be rewritten in the rotating approximation (RWA) and solved under the following restriction conditions to obtain the analytical solutions for the time-dependent coefficients { c i ( t ) } . For this purpose, we introduce three conditions that can be set experimentally: i.   V ≡ V 01 a = V 02 b                  for the transition magnitudes , (34a) ii.   Δ ≡ Δ 2 = − Δ 1 > 0          for detuning frequencies , (34b)

And iii.   Δ = Ω − ω 21 2   to induce the degeneracy condition , (34c) where the transition magnitudes are V 01 a ≡ − μ 01 ⋅ F a / ( 2 ℏ ) , and V 02 b ≡ − μ 02 ⋅ F b / ( 2 ℏ ) .

In Eq. 34c, the two dressed states are assumed to have equal energies, i.e., Ω ≡ Ω 1 = Ω 2 ( = Δ 2 + 2 V 2 ) , which is called the Rabi frequency [43]. The three conditions lead to a reduction of the input parameters of the two lasers, such that the two amplitudes (F _a and F _b), and two central frequencies ( ω a and ω b ) are reduced to one input parameter. We take F _a ≡ F hereafter. Analytical expressions for time-dependent coefficients { c i ( t ) } are given in Appendix C.

The time-dependent angular momentum defined as an expectation value of an angular momentum operator L ^ Z = − i ℏ ∂ ∂ φ = ( ℏ l ^ Z ,   l ^ Z = − i ∂ ∂ φ ) , can be expressed, L Z ( t ) = < Φ ( t ) | L ^ Z | Φ ( t ) > = − 2 ℏ Im ( l Z , 12 ) Im [ c 1 ∗ ( t ) c 2 ( t ) e x p ( − i ω 21 t ) ] . (35)

Here, l Z , 12 = < ϕ 1 | l ^ Z | ϕ 2 > = − < ϕ 2 | l ^ Z | ϕ 1 > = − l Z , 21 and l Z , 12 = i Im l Z , 12 .

We calculate L _Z (t) (derived in the preceding subsection) in a real molecule, toluene. The simplest three-electronic state model is applied for toluene because the quasi-degenerate states S₃ (A") and S₄ (A′) in toluene (C_s) correspond to the doubly degenerate state S₃ (E_1u) in benzene (D_6h): Note that the S₃ (E_1u) state is a dipole-allowed excited state in benzene (D_6h), whereas the other two lower excited states, S₂(B_1u) and S₁(B_2u), are dipole-forbidden. The geometry optimization of toluene was carried out with the MP2 level of theory [43]. The C_S symmetry of toluene indicates that one of the hydrogen atoms belonging to the methyl group is perpendicular to an aromatic ring plane (Figure 11A).

Figure 12A shows the calculated time-dependent angular momentum expectation values L _Z (t) with respect to the relative phase ζ = −π/2 (ζ = +π/2) between the two lasers for the left (right) panel. These were calculated using Eq. 35 combined with Eq. 34. Here, the amplitude of the electric field F _a is adopted as the input parameter F (≡ F _a ). Other parameters are shown in Table 3A, while Table 3B shows the time-dependent populations in the three electronic states.