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DOI: 10.1039/C9SC04992D
(Edge Article)
Chem. Sci., 2020, Advance Article

Bing Gu*^{a} and
Shaul Mukamel*^{b}
^{a}Department of Chemistry, University of California, Irvine, CA 92697, USA. E-mail: bingg@uci.edu
^{b}Department of Chemistry, Physics and Astronomy, University of California, Irvine, CA 92697, USA. E-mail: smukamel@uci.edu

Received
4th October 2019
, Accepted 12th December 2019

First published on 12th December 2019

Optical cavities hold great promise to manipulate and control the photochemistry of molecules. We demonstrate how molecular photochemical processes can be manipulated by strong light–matter coupling. For a molecule with an inherent conical intersection, optical cavities can induce significant changes in the nonadiabatic dynamics by either splitting the pristine conical intersections into two novel polaritonic conical intersections or by creating light-induced avoided crossings in the polaritonic surfaces. This is demonstrated by exact real-time quantum dynamics simulations of a three-state two-mode model of pyrazine strongly coupled to a single cavity photon mode. We further explore the effects of external environments through dissipative polaritonic dynamics computed using the hierarchical equation of motion method. We find that cavity-controlled photochemistry can be immune to external environments. We also demonstrate that the polariton-induced changes in the dynamics can be monitored by transient absorption spectroscopy.

Cavity photochemistry happens in a very different way under strong coupling. Without the cavity, the nonadiabatic molecular dynamics after photoexcitation is dictated by the Born–Oppenheimer adiabatic potential energy surfaces (APES). For many photochemical reactions, the APESs contain conical intersections (CI), which offer a funnel for the nuclear wavepacket to relax nonradiatively to lower electronic states. In the strong coupling regime, the photoreactions are determined by the polaritonic potential energy surfaces (PPES). These are defined similarly to the APES but with the polaritonic states playing the role of adiabatic states. As demonstrated in recent theoretical studies,^{1,2,12,28} the PPESs can look very different from the APESs and may contain more complex features, in particular when many molecules interact with the same cavity mode.^{29}

While previous efforts had focused on light-induced conical intersections in optical cavities,^{1,29,30} studies on how strong light–matter interaction influence the nonadiabatic dynamics of molecules with inherent CIs are scarce.^{31} Here we investigate how optical cavities can alter the nonadiabatic dynamics through a conical intersection, how external dissipative environments affect the photodynamics, and how polaritonic dynamics can be probed by nonlinear spectroscopy. We first simulate the exact real-time quantum dynamics of a three-state two-mode model of pyrazine interacting with a single cavity photon mode with a complete basis set in the polaritonic space. We then simulate the dissipative dynamics of this polaritonic system coupled to an external environment using the hierarchical equation of motion approach.^{32,33} Finally we compute the transient absorption spectrum of this polaritonic system.

The pyrazine model contains three electronic states {|ψ_{k}〉, k = 0, 1, 2} and a conical interaction (CI) exist between |ψ_{1}〉 and |ψ_{2}〉 that leads to ultrafast non-radiative relaxation of electronic excitation. When tuned to the |ψ_{0}〉 ↔ |ψ_{1}〉 transition (case I), the optical cavities can split the pristine CI in the APESs into two novel polaritonic CIs in the PPESs. This leads to significant changes of the nonadiabatic dynamics, including creating a channel for direct electronic relaxation to the ground state that can only occur by radiative emission in pristine molecules. In turn, when the cavity photon couples to the |ψ_{1}〉 ↔ |ψ_{2}〉 transition (case II), it can create avoided crossings in the PPESs while leaving the pristine CI intact. This cavity-induced crossing provides an additional relaxation channel leading to faster relaxation dynamics.

To monitor the nonadiabatic dynamics of the nuclear wavepacket moving through the CI, we use transient absorption spectroscopy (TAS). This pump–probe technique can reveal the level-populations in the course of non-adiabatic molecular dynamics. Our simulations demonstrate that the cavity-induced changes in the non-adiabatic dynamics can be observed by TAS.

We further explore the effects of external environments on the cavity control. It is found that the environment can have a deleterious effects on the polaritonic dynamics in case I, while in case II, the cavity control is immune to environmental effects. This different behavior of the environments can be attributed to the decoherence timescale.

H = H_{M} + H_{C} + H_{CM} + H_{LM}(t).
| (1) |

The linear-vibronic coupling model of pyrazine developed in ref. 34 includes three electronic states and two vibrational modes that strongly couple to the electronic motion. It has been widely used to study the nonadiabatic dynamics of pyrazine through the CI.^{35,36} In the diabatic representation, the molecular Hamiltonian reads

(2) |

The cavity Hamiltonian is given by

(3) |

(4) |

(5) |

The classical light–matter interaction H_{LM}(t) contains the interaction between the molecule and light pulses used in the TAS optical measurement – an actinic and a probe pulse (see the ESI†).

|kn_{cav}n_{c}n_{t}〉 = |ψ_{k}〉 ⊗ |n_{cav}〉 ⊗ |n_{c}〉 ⊗ |n_{t}〉
| (6) |

(7) |

H_{p}(r,q;R)|Φ_{n}(R)〉 = E_{n}(R)|Φ_{n}(R)〉
| (8) |

Alternatively, one can define a potential energy surface in the joint photon-nuclear space

H_{p}(r;q,R)|Φ_{n}(q,R)〉 = E_{n}(q,R)|Φ_{n}(q,R)〉.
| (9) |

We introduce an external environment consisting of a collection of harmonic oscillators with frequencies {ω_{α}} described by the creation a^{†}_{α} and annihilation operators a_{α}. The molecule-bath coupling describing the decoherence effects between two electronic states |ψ_{i}〉 and |ψ_{j}〉 is given by

(10) |

J(ω) = 2λωγ/(ω^{2} + γ^{2})
| (11) |

To account for the non-Markovian effects of the environment, we use the non-perturbative hierarchical equation of motion method.^{42} In this method, the reduced dynamics of the primary polaritonic system is described by a set of coupled differential equations (ℏ = 1; see the ESI† for a summary of this technique)

(12) |

(13) |

(14) |

We first assume that the cavity photon mode couples to the |ψ_{0}〉 ↔ |ψ_{1}〉 transition. The cavity resonant frequency ω_{cav} = 4.3 eV is close to the transition energy |ψ_{0}〉 → |ψ_{1/2}〉 at the CI point. The polaritonic surfaces are shown in Fig. 1c for the cavity–molecule coupling strength g_{01} = 124 meV. The electronic state population dynamics at different cavity–molecule coupling strength is shown in Fig. 2. The molecule is prepared at electronic state |ψ_{2}〉. The population dynamics of states |ψ_{1}〉 and |ψ_{0}〉 are shown in Fig. 2a and b, respectively, and the average value of the tuning mode is shown in Fig. 2c. Without the cavity, the nuclear wavepacket traverses through the CI and, due to the strong electron-nuclear coupling, the electronic excitation relaxes to |ψ_{1}〉 within 30 fs, see also Fig. 1b for the APESs of the model pyrazine without cavity–molecule coupling. The ground electronic state |ψ_{0}〉 is not populated during the dynamics as it is decoupled from the two excited electronic states.

When the cavity photon mode is coupled to the molecular transition |ψ_{0}〉 ↔ |ψ_{1}〉, the electronic excitation directly relaxes back to the ground state as shown in Fig. 2b, in contrast to the nonadiabatic dynamics of the bare molecule. This implies that the strong light–matter coupling creates a channel for such relaxation process. The relaxation to the ground state is accompanied by the decrease of populations in the first electronic state with the overall relaxation dynamics accelerated by the strong coupling. The significant cavity-induced changes in the nonadiabatic dynamics can be understood from the shape of the PPESs. As shown in Fig. 3, the cavity–molecule coupling creates two new CIs in the PPESs. Distinct from the bare CI in the APES, these CIs are between a purely electronic state |ψ_{2}〉 and another polaritonic state with significant hybridization of matter and photon, as reflected in the color scheme which encodes the photon component in polaritonic states. More precisely, it corresponds to the expected photon number N_{ph} = 〈Φ(R)|a^{†}_{cav}a_{cav}|Φ(R)〉with the polaritonic state |Φ(R)〉, which is a superposition of states |ψ_{0}〉|1_{cav}〉 and |ψ_{1}〉|0_{cav}〉. The polaritonic CIs play similar roles in the nonadiabatic polaritonic dynamics as CIs play in the nonadiabatic dynamics of pristine molecules. While CIs in APESs induce transitions between electronic states, polaritonic CIs induce transitions in the polaritonic surfaces. The nuclear wavepacket first traverses through one of the polaritonic CIs that is closer to the Franck–Condon point R ≡ (Q_{c}, Q_{t}) = (0, 0), part of the nuclear wavepacket relaxes to the third PPES (the ground surface is not shown in Fig. 3). Because the polaritonic states of this PPES consist of the contribution from |ψ_{0}〉|1_{cav}〉, the electronic ground state gains population during the course of dynamics. This picture is further confirmed by the expectation value of the position operator of the tuning mode. As the cavity–molecule coupling strength increases, the polaritonic CI is pushed further away from the pristine CI, thus explaining the increase of the average position of the nuclear wavepacket at t ∼ 25 fs.

As the polaritonic CI is closer to the Franck–Condon point, the geometric phase effect is clearly observed in Fig. 4, which compares the nuclear probability density at t = 32 fs in electronic state |S_{1}〉 between the bare dynamics and the polaritonic dynamics (g = 240 meV). The nuclear probability density is obtained by tracing over the electronic and photonic degrees of freedom in the total wavefunction projected on the electronic subspace i.e., The geometric phase effect is clearly seen in the vanishing of the probability along the Q_{c} = 0 line in the polaritonic dynamics (Fig. 4b). The main difference is that ρ_{n}(R) is less spread over the configuration space than the bare dynamics, meaning that less excitation energy flows into the vibrational modes. This is expected because part of the energy is contained in the cavity photon mode.

It is interesting to compare the novel polaritonic CI to the light-induced CI identified previously.^{1,31} In the latter case, the conical intersection is between two product states between the molecule and the cavity at configurations where the cavity–molecule interaction (transition dipole moments) vanishes. In other words, |g〉 ⊗ |1_{cav}〉 and |e〉 ⊗ |0_{cav}〉 are degenerate when μ_{eg}·e = 0, where μ_{eg} is the transition dipole between states |e〉 and |g〉 and e is the polarization vector of the cavity photon mode, and the transition energy is resonant with the cavity photon energy. When one of the nuclear coordinates is a rotation, this CI can be realized at the angle where the transition dipole is orthogonal to the polarization of the cavity photon. It does not require a pristine CI in the molecule and the CI geometry is not affected by the light–matter coupling strength. In contrast, the novel pair of polaritonic CIs are between two hybrid polaritonic states. This emerges when the molecule itself contains a CI in the APESs and when the cavity mode couples to the transition between one of the APESs forming the bare CI and a third electronic state. Moreover, the polaritonic CIs geometry can be tuned by the strength of light–matter coupling strength. As the cavity–molecule coupling is increased to g_{cav} = 240 meV, the positions of the two polaritonic CIs are further apart. This change of the location of polaritonic CI can also be seen in the average position of the tuning mode, see Fig. 2c. Population relaxation to the ground electronic state is further increased.

We now turn to a different scenario where the cavity mode couples to the transition |ψ_{1}〉 ↔ |ψ_{2}〉 transition. The PPESs are shown in Fig. 1d for ω_{cav} = 0.62 eV (slightly lower than the transition frequency at the Frank–Condon point) and g_{12} = 120 meV. In this case, the ground electronic state is not playing any role in the dynamics as it is decoupled from the other electronic states. Upon turning on the light–matter interaction, we first observe a much faster electronic relaxation as shown in Fig. 5a. The relaxation time reduces from 30 fs without the cavity to ∼15 fs at g_{12} = 120 meV. Interestingly, as can be seen from the average position of the tuning mode in Fig. 5b, the nuclear wavepacket does not even reach the pristine conical intersection. Similar to the previous analysis, we interpret the nonadiabatic polaritonic dynamics from the shape of PPESs shown in Fig. 6. The main feature of the polaritonic surfaces is the emergence of a light-induced (avoided) crossing. This offers an additional electronic relaxation channel other than the pristine CI. Because the light-induced crossing is much closer to the Franck–Condon point in this case, the relaxation happens mainly through this channel and consequently, the relaxation happens much faster. As the coupling strength is enhanced, the relaxation rate is further increased.

Fig. 6 Same as Fig. 3 but with the cavity photon mode coupling to the transition |ψ_{1}〉 ↔ |ψ_{2}〉 and g ≡ g_{12}. The characteristic feature of the polaritonic surfaces is the light-induced crossing providing an additional channel for electronic relaxation. |

Interestingly, this light-induced avoided crossing is always associated with a light-induced CI. This is because one can always rotate the molecule such that the transition dipole at the configuration R*, where the electronic transition is in resonance with the cavity photon energy, is orthogonal to the electric field polarization. Since we can always find such molecular geometry R* in an avoided crossing, the light-induced CI is guaranteed to exist.

For case II, the counterrotating terms that are neglected in the rotating-wave approximation [eqn (5)] may become important as the coupling strength becomes comparable to the cavity resonance (so-called ultrastrong coupling). To investigate the influences of counterrotating terms, we simulate the polaritonic dynamics employing the full interaction as in eqn (4). As shown in Fig. S2† where the dynamics with and without the RWA are contrasted, when g = 120 meV [Fig. S2(a and b)†], there is only minor changes in the polaritonic dynamics without invoking the RWA. For g = 240 meV [Fig. S2(c and d)†], the counterrotating terms start to play a role in the long-time dynamics t > 40 fs, but the intial relaxation dynamics (t ∈ [0, 40] fs) remains almost the same.

Because the formation of cavity polaritons relies on the coherence between electronic states, the environment is expected to diminish the cavity control. This is indeed seen in the population dynamics of electronic state |ψ_{2}〉 in case I when g = 120 meV (Fig. 7a). When the cavity–molecule coupling is stronger, this deleterious effect becomes less significant (Fig. 7b). Another effect of the environment is reflected in case II where the coherent population transfer between electronics states |ψ_{2}〉 and |ψ_{1}〉 at g = 240 meV during the time periods 0–20 fs and 40–60 fs is reduced (Fig. 8b). The polaritonic dynamics at g = 120 meV in case II is almost immune to the external environment (Fig. 8a), the dissipative polaritonic dynamics for all system–bath coupling strengths λ/g = 0, 1, 2 shows the same timescale ∼20 fs for the electronic relaxation in comparison to 40 fs in the bare molecule case. This implies that the relaxation channel provided by the light-induced crossing is robust to the environment.

Fig. 8 Same as Fig. 7 but for both the cavity mode and the external environment coupled to the transition |ψ_{1}〉 ↔ |ψ_{2}〉. The cavity parameters are ω_{cav} = 0.62 eV, the cavity–molecule couplings are (a) g = 120 meV and (b) g = 240 meV. |

We further explored the environmental effects on the polaritonic dynamics and demonstrated that even when the reorganization energy is larger than the cavity–molecule coupling, the optical cavities can still be employed to manipulate the photodynamics of molecules. Moreover, we showed that the TAS can be a useful spectroscopic technique to monitor the polaritonic dynamics. These results provide novel schemes to control the non-adiabatic dynamics with strong light–matter coupling, enriching the toolbox to use optical cavities to alter molecular properties and dynamics. Considering the rapid developments in nanophotonics,^{49,50} we expect that polaritonic systems can be experimentally investigated in the forseeable future. The strong coupling can be also achieved through the collective interaction between many molecules and a single cavity mode, extending the proposed control schemes to photochemistry with many molecules will be investigated in the future.

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## Footnote |

† Electronic supplementary information (ESI) available: (i) Theory and computational protocol of transient absorption spectroscopy for polaritonic chemistry, and (ii) the hierarchical equations of motion method, and relation between spectral density and environment correlation function [eqn (13)]. See DOI: 10.1039/c9sc04992d |

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