Letter

pubs.acs.org/JPCL

Cavity Femtochemistry: Manipulating Nonadiabatic Dynamics at Avoided Crossings Markus Kowalewski,* Kochise Bennett, and Shaul Mukamel*

Chemistry Department, University of California, Irvine, California 92697-2025, United States

ABSTRACT: Molecular potential energy surfaces can be actively manipulated by light. This is usually done by strong classical laser light but was recently demonstrated for the quantum field in an optical cavity. The photonic vacuum state of a localized cavity mode can be strongly mixed with the molecular degrees of freedom to create hybrid field-matter states known as polaritons. We simulate the avoided crossing of sodium iodide in a cavity by incorporating the quantized cavity field into the nuclear wave packet dynamics calculation. The quantized field is represented on a numerical grid in quadrature space, thus avoiding the limitations set by the rotating wave approximation (RWA) when the field is expanded in Fock space. This approach allows the investigation of cavity couplings in the vicinity of naturally occurring avoided crossings and conical intersections, which is too expensive in the fock space expansion when the RWA does not apply. Numerical results show how the branching ratio between the covalent and ionic dissociation channels can be strongly manipulated by the optical cavity.

he photochemistry of molecules can be significantly level system linearly coupled to a quantized radiation field is − T influenced by specially tailored electromagnetic fields.1 7 given by the quantum Rabi model27 While control is usually achieved with classical laser fields, a ̂ strong coupling with the vacuum state of a cavity utilizes the Hec=++HHH e c I quantum nature of the electromagnetic mode. The underlying ℏω = 0 σσ̂††††̂−+ℏ ω̂̂+ℏ ̂+ ̂ σ̂+ σ̂ theoretical framework, cavity quantum electrodynamics, has (2 1)caa ga ( a )( ) 8−11 2 been well developed in theory and experiment for atoms (1) 12 and has been applied to atomic trapping and cooling. Its ℏω ℏ ω − ω ff 13,14 where 0 = ( e g) is the energy di erence between the potential applications to molecular cooling, as a tool to | ⟩ | ⟩ σ̂ | ⟩⟨ | 15,16 17−20 exited state e and the ground state g , = g e is the probe larger molecules, to enhance vibrational spectra, (̂†) 21 annihilation operator for the bare electronic excitation, and a for use with electromagnetically induced transparency, and to annihilates (creates) a cavity photon of mode frequency ω . fi 22,23 c expedite cavity-modi ed photochemistry have been objects The vacuum Rabi-frequency g = n με/ℏ, the coupling of intensive studies. Strong cavity coupling in molecular eg c systems has been demonstrated recently for electronic between the photon mode and the electronic degrees of 24 18,25 freedom, depends on the electronic transition dipole moment transitions and for vibrational transitions. In this regime, μ and the vacuum field amplitude the dynamics is described using joint photon-matter states eg called polaritons. The inclusion of the internal nuclear degrees ℏωc of freedom, not present in atoms, gives rise to nonadiabatic εc = V ϵ dynamics.23,26 The light field can strongly mix the nuclear and 0 (2) electronic degrees of freedom, creating avoided curve crossings This amplitude is determined by the active volume V of the and conical intersections (CIs) between the polariton potential cavity mode. Nanoscale cavities can thus induce very strong energy surfaces. Virtually all photochemical and photophysical coupling. The cavity coupling is further enhanced by a factor of processes are controlled by the presence of naturally occurring √n if an ensemble of n molecules contributes in a coherent π ω CIs in the bare electronic energy landscape, opening up fast, fashion in a cavity smaller than the optical wavelength 2 c/ c. nonradiative relaxation pathways by which the molecule is Different approximations can be used to find the eigenvalues funneled back to the ground state. of eq 1 depending on the magnitude of g. Perturbative solutions In this contribution, we theoretically investigate how a are possible when g is much smaller than all other system − molecule that already possesses a curve crossing is affected by frequencies. If the time scale g 1 is faster than possible decay strong coupling to a nanoscale cavity. To set the stage, we first processes, we enter the strong coupling regime though the neglect the nuclear degrees of freedom and only introduce them at a later point. This will allow us to introduce the basic Received: April 21, 2016 structure of the Hamiltonian and the cavity coupling. The Accepted: May 17, 2016 Hamiltonian describing the light-matter interaction of a two-

© XXXX American Chemical Society 2050 DOI: 10.1021/acs.jpclett.6b00864 J. Phys. Chem. Lett. 2016, 7, 2050−2054 The Journal of Physical Chemistry Letters Letter

σ †σ† Ψ =|⟩+|⟩ψϕψϕ counter rotating terms (a and a ) in the Hamiltonian can be ggee(,xq ) (, xq ) (6) neglected. This is known as the Jaynes-Cummings (JC) model9 and approximates solutions to the Hamiltonian (eq 1) in the Note that the wave function of the cavity mode and the nuclear ψ form of dressed states, which are then expressed in the basis of degrees of freedom are generally nonseparable ( k(x, q)). The the photon number states: matrix elements of the cavity-molecule Hamiltonian in the basis M of the bare adiabatic states are then

Ψc =|⟩+|∑ cgngn,,,, c cen en c⟩ ⎛ ⎞ cc ℏ∂2 1 n=0 (3) H = δωδω⎜TV̂ + ̂− + 22xgx̂⎟ +−(1 ) 2 ℏ̂ kl kl⎝ nuc k2 ∂x2 2 c ⎠ kl c Here n is the number of photons in the cavity mode, and M is c (7) total number of photon number states. The JC model keeps † † only the aσ and a σ terms in the cavity-molecule coupling. The where fi ̂≡ ̂ σ̂†σ̂ total matter+ eld excitation number, N nc + ,isa ̂ ̂ ℏ∂1 2 conserved quantity ([N, H] = 0), reducing the Hamiltonian to T̂ =− ∑ a 2-by-2 block-diagonal form (see Figure 1). Assuming that the nuc 2 m ∂ 2 i i qi (8)

is the nuclear kinetic energy operator, mi the reduced mass of δ the vibrational mode qi, and kl =1ifk = l and zero otherwise. The indices k and l run over the bare electronic states g and e. The coordinate x is expanded in a numerical grid, putting the nuclear coordinates and the cavity mode on an equal footing in a numerical simulation. The second derivative with respect to x can be conveniently calculated by a discrete Fourier trans- form.28 For a , this results in two-dimensional potential energy surfaces for the electronic ground and excited Figure 1. Atom-field Hamiltonian for quantum Rabi problem. The state, each depending on q and x. This yields a more favorable σ† σ † resonant coupling terms a and a act within each block N1, ...,NM. † † scaling of the computational effort of 6(log)nnrather than The non-RWA terms aσ and a σ couple different blocks. 6()n2 in the basis of photon number states. For the case of large cavity coupling, the number of grid points in x necessary 29 cavity is initially in the vacuum state (no photons), the wave is in practice even smaller than the number of Fock states. function in the JC model can thus be expressed by the two The effect of ultrastrong cavity coupling on nonadiabatic dressed states |g,1⟩, |e,0⟩ and the ground state |g,0⟩, i.e., the dynamics is demonstrated on the excited state dynamics of bare ground state with 1 photon and the excited state with 0 sodium iodide, which has been an important landmark in 31 photons. An extension of the JC model to include the nuclear femtochemistry. The adiabatic electronic states of the bare degrees of freedom within the RWA has been presented sodium iodide experience an avoided crossing near q ∼ 8Å recently.23 However, when the RWA is violated, the expansion (Figure 2a), which leads to dissociation into the neutral (eq 3) requires a large number of Fock states to converge, products through the 1X state. Upon photoexcitation, a nuclear becoming intractable for quantum dynamics time propagation wave packet is launched in the 1A state and then oscillates back 32 1 (scaling with 6()M2 ). and forth. Part of the wave packet couples to the X state via To overcome this difficulty we employ an efficient the avoided crossing into the covalent dissociation channel. The corresponding nonadiabatic coupling matrix element, respon- computational scheme obtained by the direct treatment of 1 1 the cavity mode as a quantum harmonic oscillator in quadrature sible for the population transfer between the A and the X fi state, is shown in Figure 2b (red curve). The branching ratio, space. We rst express the annihilation and creation operators 1 of the cavity mode in terms of their quadrature coordinates x i.e., the population transferred to the X state, is determined by and p:27 the momentum of the wave packet, which in turn depends on the wavelength of the pump laser. Zewail had measured a ⎛ ⎞ ≈ ωc i stepwise increase in the covalent state population of 11% at a = ⎜x̂+ p̂⎟ 32 fl 2ℏ ⎝ ω ⎠ each passage through the avoided crossing. To in uence the c (4) dynamics at the avoided crossing, the cavity needs to be set in − ℏ∂ 1 1 with p = i x. The extension of the Hamiltonian to the with the X and the A state in a region where the molecular case is now straightforward: The nuclear degrees of transition dipole moment is significantly large (≈ 2−9 Å; see ≡ ω freedom q (q1,.., qN) are accounted for by treating 0 and g as Figure 2b, blue curve). We will investigate the scenario where ≡ ≈ functions of q (i.e., the potential energy curves Vg Vg (q), Vg the cavity is set in resonance at 6 Å, i.e., before the wave ≡ μ ≡ μ 1 ℏω Ve(q), and the transition dipole curves eg eg(q) packet in the A state reaches the avoided crossing ( c = 815 respectively). The coupled light-molecule Hamiltonian eq 1 meV). then reads: Wave packet calculations were carried out to reveal the possible modifications of the dynamics under the influence of ω 2 0()q † ℏ∂ 1 22 the cavity coupling. This was done by propagating the Hec = (2σσ̂̂−− 1) + ωc x̂ 2 2 ∂x2 2 photonic-nuclear wave packet with a Chebychev propagation † scheme33 on the potential energy curves of NaI (1X and 1A) +ℏgq() 2ωσc x (̂̂+ σ̂ ) (5) using the Hamiltonian from eq 7 and an additional coupling The total wave function expanded in the adiabatic electronic term34,35 to account for the nonadiabatic coupling at the bare |ϕ ⟩ |ϕ ⟩ states g and e becomes state avoided crossing:

2051 DOI: 10.1021/acs.jpclett.6b00864 J. Phys. Chem. Lett. 2016, 7, 2050−2054 The Journal of Physical Chemistry Letters Letter

Figure 3. Selected time traces of the 1X state population following ω ω impulsive excitation. (a) No cavity. (b) g = 0.050 c (red), g = 0.13 c ω ω ω (blue). (c) g = 0.19 c (black), g = 0.25 c (green), g = 0.38 c (magenta). Figure 2. (a) Bare ground (1X) and excited (1A) electronic potential energy surfaces of NaI. Dissociation on the 1X potential leads to the μ a much smaller fraction reaches the original crossing. This neutral reaction products. (b) Transition dipole moment eg (blue) partially coincides with the effect described by Galego et al.:26 and the derivative coupling matrix element f (red). The potential eg for higher coupling strengths, the states become well separated energy curves, the nonadiabatic coupling matrix element, and the − transition dipole moment were calculated with the program package and the Born Oppenheimer approximation regains validity. MOLPRO30 at the MRCI/CAS(6/7)/aug-cc-VQZ level of theory with For larger splittings, the upper polariton state is well separated an effective core potential for Iodine (ECP46MWB). from the lower lying states, effectively suppressing the nonadiabatic population transfer. Moreover, the nonadiabatic nac ℏ coupling at the original intersection is affected by the off- Ĥ = Ĥ+−(1δ ) (2ff∂+∂ ) kl kl kl 2m kl qqkl (9) resonant interaction with the cavity allowing for further suppression of the population transfer into the covalent where f kl is the nonadiabatic coupling matrix element shown in channel. For higher coupling strengths, the dissociation is Figure 2b. The wave function, as well as the potential energy strongly suppressed. surfaces, are represented by a numerical grid with 95 points for The scaling of the dynamics with g can be displayed in terms x and 1200 points for q. The initial condition is a product state of the population in the 1X state after the wave packet has formed by the vacuum state of the cavity |0⟩ and the vibrational reached the crossing point for the first time (≈ 480 fs). Figure ground state of the 1X state |1X, v =0⟩ shifted by 0.63 Å. 4a depicts the population in the covalent channel for varying Figure 3a shows the time-dependent population of the coupling strengths. While an intermediate coupling strength (g ω covalent state of the free molecule (no cavity), mimicking the < 0.1 c) can slightly increase the dissociation probability, the original experimental setup.32 The dynamics of the system is transfer to the 1X state is suppressed for larger values of g. presented by inspecting the time evolution of the population of We have employed an efficient computational protocol for 1 ⟨ψ1 |ψ1 ⟩ − the X state, X X , rather than the pump probe signal, cavity nonadiabatic dynamics, which allows for a seamless which probes the excited state wave packet within a certain integration of quantized field modes into existing schemes at a window of internuclear distance. The stepwise decay observed cost comparable to adding another vibrational mode to the in the pump−probe experiment (e.g., ref 32, Figure 8) is less system. The Hamiltonian used to calculate the dynamics (eq 7) clear for the total population but it better illustrates the overall is in a very general form and can be used for a wide range of effect of the cavity. In Figure 3b, the time evolution in a cavity cavity couplings. It only relies on the dipole approximation, ω is shown for moderate coupling strength (g = (0.05, 0.13) c). making it robust and reliable. We have demonstrated the The time traces are still similar to the uncoupled case but method for an avoided crossing and one vibrational mode, but already show the influence of the cavity coupling. At coupling it can be applied to more than one vibrational mode, which will ≈ ω strengths of g 0.19 c (Figure 3c), the behavior of the time allow addressing conical intersections. Including additional traces begins to change. The dissociation is suppressed, and the cavity modes is also straightforward and only limited by its oscillation period increases. The wave packet becomes trapped computational cost. While a partially analytic solution of the in a more tight effective potential, and its recurrence time non-RWA problem based on tunable coherent states is becomes shorter. The dynamics is mainly influenced by two possible36 and might give insight into the resulting structure modifications of the potential energy surfaces: the new avoided of the diagonalized potential energy surfaces, it might become crossing created by the cavity and the modified avoided intractable for a quantum dynamics time propagation. crossing already present in the bare molecule. With increased Our wave packet simulations show that the photochemistry cavity coupling, the cavity-created crossing generates well- of NaI can be significantly manipulated by the cavity. This is separated states such that a nuclear wave packet is trapped, and caused by two mechanisms: in the strong coupling regime, the

2052 DOI: 10.1021/acs.jpclett.6b00864 J. Phys. Chem. Lett. 2016, 7, 2050−2054 The Journal of Physical Chemistry Letters Letter

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2054 DOI: 10.1021/acs.jpclett.6b00864 J. Phys. Chem. Lett. 2016, 7, 2050−2054