Three-player polaritons: nonadiabatic fingerprints in an entangled atom–molecule–photon system

Within the framework of QED and the electric dipole representation, the Hamiltonian of a molecule and N_a atoms interacting with a single cavity mode can be written as [44]

$$\begin{equation}{\hat{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\hat{H}}_{\mathrm{m}\mathrm{o}\mathrm{l}}+\sum _{i=1}^{{N}_{\text{a}}}\hslash {\omega }_{\mathrm{a}}{\hat{\sigma }}_{i}^{{\dagger}}{\hat{\sigma }}_{i}+\hslash {\omega }_{\mathrm{c}}{\hat{a}}^{{\dagger}}\hat{a}-\sqrt{\frac{\hslash {\omega }_{\mathrm{c}}}{2{{\epsilon}}_{0}V}}\hat{\mathbf{d}}\hat{\mathbf{e}}\left({\hat{a}}^{{\dagger}}+\hat{a}\right),\end{equation} \tag{ 1 }$$

where Ĥ_mol is the field-free molecular Hamiltonian, ${\hat{{\sigma }_{i}}}^{{\dagger}}$ and $\hat{{\sigma }_{i}}$ are excitation and deexcitation operators, respectively, for the ith atom, ω_a is the atomic transition frequency, â^† and â are photon creation and annihilation operators, respectively, ω_c is the frequency of the cavity mode, ℏ is Planck's constant divided by 2π, ₀ is the electric constant, V is the volume of the electromagnetic mode, $\hat{\mathbf{e}}$ is the polarization vector of the cavity mode, and $\hat{\mathbf{d}}={\hat{\mathbf{d}}}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}+{\hat{\mathbf{d}}}^{\left(\mathrm{a}\right)}$ is the sum of the molecular and atomic dipole moments.

It is useful to inspect the structure of the Hamiltonian of equation (1), when represented using |α⟩|N⟩|I⟩ product basis functions, where α and I are the molecular electronic and atomic quantum numbers, respectively, and N stands for the photon number. In our notation I = 0 indicates that all atoms are in their ground state, and I = n indicates that the nth atom is in an excited state, multiple atomic excitations are omitted in our model. The resulting Hamiltonian for a diatomic molecule reads (for simplicity we show the case when α = 1 or 2, N = 0 or 1, and I = 0, 1 or 2, however, larger N and I values occur in the simulations below)

$$\begin{equation}\hat{H}=\left[\begin{bmatrix}{cccccc}{\hat{H}}_{\mathrm{m}}\hfill & \hat{A}\hfill & 0\hfill & \hat{B}\hfill & 0\hfill & \hat{B}\hfill \\ {\hat{A}}^{{\dagger}}\hfill & {\hat{H}}_{\mathrm{m}}+\hslash {\omega }_{\mathrm{c}}\hfill & \hat{B}\hfill & 0\hfill & \hat{B}\hfill & 0\hfill \\ 0\hfill & {\hat{B}}^{{\dagger}}\hfill & {\hat{H}}_{\mathrm{m}}+\hslash {\omega }_{\mathrm{a}}\hfill & \hat{A}\hfill & 0\hfill & 0\hfill \\ {\hat{B}}^{{\dagger}}\hfill & 0\hfill & {\hat{A}}^{{\dagger}}\hfill & {\hat{H}}_{\mathrm{m}}+\hslash {\omega }_{\mathrm{a}}+\hslash {\omega }_{\mathrm{c}}\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\hat{B}}^{{\dagger}}\hfill & 0\hfill & 0\hfill & {\hat{H}}_{\mathrm{m}}+\hslash {\omega }_{\mathrm{a}}\hfill & \hat{A}\hfill \\ {\hat{B}}^{{\dagger}}\hfill & 0\hfill & 0\hfill & 0\hfill & {\hat{A}}^{{\dagger}}\hfill & {\hat{H}}_{\mathrm{m}}+\hslash {\omega }_{\mathrm{a}}+\hslash {\omega }_{\mathrm{c}}\hfill \end{bmatrix}\right],\end{equation} \tag{ 2 }$$

where

$$\begin{equation}{\hat{H}}_{\mathrm{m}}=\left[\begin{bmatrix}{cc}\hfill \hat{T}\hfill & 0\hfill \\ \hfill 0\hfill & \hfill \hat{T}\hfill \end{bmatrix}\right]+\left[\begin{bmatrix}{cc}\hfill {V}_{1}\left(R\right)\hfill & 0\hfill \\ 0\hfill & \hfill {V}_{2}\left(R\right)\hfill \end{bmatrix}\right],\end{equation} \tag{ 3 }$$

$$\begin{equation}\hat{A}=\left[\begin{bmatrix}{cc}\hfill {g}_{11}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R,\theta \right)\hfill & \hfill {g}_{12}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R,\theta \right)\hfill \\ \hfill {g}_{21}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R,\theta \right)\hfill & \hfill {g}_{22}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R,\theta \right)\hfill \end{bmatrix}\right]\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad \hat{B}=\left[\begin{bmatrix}{cc}\hfill {g}^{\left(\mathrm{a}\right)}\hfill & 0\hfill \\ 0\hfill & \hfill {g}^{\left(\mathrm{a}\right)}\hfill \end{bmatrix}\right],\end{equation} \tag{ 4 }$$

with

$$\begin{equation}{g}_{ij}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R,\theta \right)=-\sqrt{\frac{\hslash {\omega }_{\mathrm{c}}}{2{{\epsilon}}_{0}V}}{d}_{ij}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R\right)\mathrm{cos}\left(\theta \right)\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {g}^{\left(\mathrm{a}\right)}=-\sqrt{\frac{\hslash {\omega }_{\mathrm{c}}}{2{{\epsilon}}_{0}V}}{d}^{\left(\mathrm{a}\right)},\end{equation} \tag{ 5 }$$

where R is the internuclear distance, V_i(R) is the ith potential energy curve (PEC), $\hat{T}$ is the nuclear kinetic energy operator, ${d}_{ij}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R\right)$ is the transition dipole moment matrix element between the ith and jth molecular electronic states, d^(a) is the transition dipole between the atomic ground and excited state, and θ is the angle between the electric field polarization vector and the molecular transition dipole vector, assumed to be parallel to the molecular axis. In equation (2) no rotating-wave approximation is applied, the accuracy of the model is only limited by the dipole approximation, the neglection of dissipative processes, and the limited number of electronic excitations considered for the molecule and atoms. Naturally, the maximum value of the photon number N should be increased in practical applications until convergence is reached, resulting in a larger Hamiltonian and appropriate $\sqrt{N}$ factors for the N > 1 $\hat{\mathrm{A}}$ and $\hat{B}$ coupling terms [44].

Based on the Hamiltonian of equation (1), the left panels of figure 1 show the potential energy landscape of a Na₂ molecule and two-state atoms interacting with a single cavity radiation mode. For Na₂ the V₁(R) and V₂(R) PECs correspond to the ${\mathrm{X}}^{1}{{\Sigma}}_{\text{g}}^{+}$ and the ${\mathrm{A}}^{1}{{\Sigma}}_{\text{u}}^{+}$ electronic states, respectively, taken from reference [45]. For ${\mathrm{N}\mathrm{a}}_{2}\;{d}_{11}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R\right)={d}_{22}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R\right)=0$ and the ${d}_{12}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R\right)$ transition dipole was taken from reference [46], while d^(a) = 1 a.u. was used for the atoms. Figure 1 demonstrates that the coupling with the radiation field leads to the formation of three 'bright' polaritonic surfaces, and as shown in the bottom left panel, N_a − 1 'dark states'. Although these 'dark states' can affect excited state dynamics [35], they usually have no significant contribution to the spectrum [3, 47]. With increasing atom number the effective coupling with the field increases and the polaritonic surfaces become more pronounced.

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The left panels of figure 2 show the polaritonic states formed for different atomic and photonic excitation energies. Note that the 590 nm wavelength for atomic excitation shown in the lower left panel of figure 2 stands for the ²P ←²S transition of Na atoms. It is clear from figures 1 and 2 that the presence of atoms, and their indirect coupling with molecules through the radiation mode, can considerably change the polaritonic landscape of molecules confined in cavities, providing an additional degree of freedom to modify and manipulate the excited state properties and dynamics of molecules.

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Now we turn to the weak-field absorption spectrum of the composite system. Similar to previous works [40, 48], the spectrum is determined by first computing the field-dressed states, i.e., the eigenstates of the full 'molecule + atoms + radiation field' system, and then computing the dipole transition amplitudes between the field-dressed states with respect to a probe pulse. The probe pulse is assumed to be weak, inducing primarily one-photon processes, implying that the standard approach [49] of using first-order time-dependent perturbation theory to compute the transition amplitudes should be adequate.

We determine the field-dressed states of the Hamiltonian in equation (1) by diagonlizing its matrix representation obtained with |αvJ⟩|N⟩|I⟩ product basis functions, where v and J are the molecular vibrational and rotational quantum numbers, respectively. The field-free molecular rovibronic states |αvJ⟩ are computed using 200 discrete variable representation (DVR) spherical-DVR basis functions [50] for each PEC with the related grid points placed in the internuclear coordinate range (0, 10) bohr. All states with J < 16 and an energy not exceeding the zero point energy of the respective PEC by more than 2000 cm⁻¹ were included into the product basis. The maximum photon number in the cavity mode was set to two.

The ith field-dressed state can then be expressed as

$$\begin{equation}\vert {{\Psi}}_{i}^{\mathrm{F}\mathrm{D}}\rangle =\sum _{\alpha ,v,J,N,I}{C}_{i,\alpha vJNI}\vert \alpha vJ\rangle \vert N\rangle \vert I\rangle ,\end{equation} \tag{ 6 }$$

where the C_i,αvJNI coefficients are the eigenvector components obtained from the diagonalization. Using first-order time-dependent perturbation theory, the transition amplitude between two field-dressed states, induced by the weak probe pulse, is proportional to [44, 51]

$$\begin{equation}\langle {{\Psi}}_{i}^{\mathrm{F}\mathrm{D}}\vert \langle M\vert \hat{\mathbf{d}}\hat{\mathbf{E}}\vert {M}^{\prime }\rangle \vert {{\Psi}}_{j}^{\mathrm{F}\mathrm{D}}\rangle =\left(\langle {{\Psi}}_{i}^{\mathrm{F}\mathrm{D}}\vert {\hat{d}}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\mathrm{cos}\left(\theta \right)\vert {{\Psi}}_{j}^{\mathrm{F}\mathrm{D}}\rangle +\langle {{\Psi}}_{i}^{\mathrm{F}\mathrm{D}}\vert {\hat{d}}^{\left(\mathrm{a}\right)}\vert {{\Psi}}_{j}^{\mathrm{F}\mathrm{D}}\rangle \right)\langle M\vert \hat{E}\vert {M}^{\prime }\rangle .\end{equation} \tag{ 7 }$$

In equation (7), $\hat{E}$ is the electric field operator of the weak probe pulse, assumed to have a polarization axis identical to that of the cavity mode, leading to M = M' ± 1 for the probe pulse. Thus, equation (7) accounts for single-photon absorption or stimulated emission. When the field-dressed states are expanded according to equation (6), many terms in equation (7) become zero due to the selection rules of the atomic and molecular transitions involved. For a homonuclear diatomic molecule, such as Na₂, the T_j←i transition probability is proportional to

$$\begin{equation}{\left\vert \sum _{\alpha vJNI}\left(\sum _{{\alpha }^{\prime }{v}^{\prime }{J}^{\prime }}{C}_{i,\alpha vJNI}^{{\ast}}{C}_{j,{\alpha }^{\prime }{v}^{\prime }{J}^{\prime }NI}\langle v\vert {d}_{\alpha {\alpha }^{\prime }}^{\left(\mathrm{m}\mathrm{o}\mathrm{l}\right)}\left(R\right)\vert {v}^{\prime }\rangle \langle J\vert \mathrm{cos}\left(\theta \right)\vert {J}^{\prime }\rangle +\sum _{{I}^{\prime }\ne I}{C}_{i,\alpha vJNI}^{{\ast}}{C}_{j,\alpha vJN{I}^{\prime }}{d}^{\left(\mathrm{a}\right)}\right)\right\vert }^{2}.\end{equation} \tag{ 8 }$$

Equation (8) demonstrates that in the general field-dressed case the spectral peaks cannot be evaluated as the sum of field-dressed atomic and field-dressed molecular peaks, because of the interference between atomic and molecular transitions. Thus, fingerprints of the coherent mixing of atomic and molecular states can appear in the spectrum of the composite system. Naturally, in the limit of the cavity field strength going to zero (and simultaneously the molecule–atom interaction going to zero) the transitions should reflect a spectrum composed as the sum of the molecular spectrum and the atomic spectrum. However, when molecular and atomic couplings with the cavity are strong enough for polaritonic states to be formed, the spectrum should reflect transitions corresponding to field-dressed states, which are a coherent mixture of states containing atomic, molecular, and photonic excitations.

It is well known that the interaction of atoms with a (near) resonant electromagnetic field leads to a Rabi splitting of the energy levels in the excited state manifold. This mechanism leads to splittings in the spectral peaks of the light-dressed system, often referred to in spectroscopy as Autler–Townes splitting [52]. When N_a atoms simultaneously interact with the same mode of an electromagnetic field, a pair of splitted states appear, reflecting an effective coupling strength increased by $\sqrt{{N}_{\text{a}}}$, accompanied by N_a − 1 dark states with no shift appearing in their energy with respect to the field-free atomic states [47]. Depending on the homogeneity of the atom–field interactions, with inhomogeneity caused either by inhomogeneities in the field or the atomic transition frequencies, the energy level structure and the absorption spectrum could become more complex, as detailed in reference [47]. As for molecules interacting with a cavity radiation mode, Autler–Townes-type splittings, as well as 'intensity borrowing' effects [49, 53] and the formation of polaritonic potential energy surfaces, are reflected in the spectrum [40]. The right panels of figure 1 show the light-dressed spectra of an Na₂ molecule and two-state atoms interacting with a single cavity mode. All plotted peaks represent transitions from the ground state of the entangled 'atom + molecule + cavity' system, primarily composed of |100⟩|0⟩|0⟩, to higher-lying field-dressed states, formed as a quantum superposition of molecular, atomic and photonic excited states. The composition of each mixed state is indicated by the color of its respective transition line, see details in the footnote of figure 1. In order to help identify overall intensity changes in the spectra, in addition to the peak splittings, an envelope for each spectrum was generated by convolving the peaks with a Gaussian function having a standard deviation of σ = 50 cm⁻¹.

The upper right panel of figure 1 demonstrates that at lower coupling strengths a light-dressed spectrum of the molecule, and a well defined, separate atomic transition peak at 14 800 cm⁻¹ can be observed. The peaks between 15 000 − 17 000 cm⁻¹ mostly reflect transitions similar to the field-free molecular transitions |2v1⟩ ← |100⟩, however, the presence of the coupling between the molecule and the cavity field is also visible in peak splittings and in the colors of some transitions deviating from red. With increasing coupling strength the molecular, atomic, and photonic states become entangled and three bright polaritonic states are formed, which is reflected in the three groups of peaks in the spectrum (see middle panels of figure 1). The splitting of the atomic peak at 14 800 cm⁻¹ in the middle right panel of figure 1 as well as the appearance of purple colored peaks near 15 100 cm⁻¹ indicates that a considerable mixing of the atomic and molecular states occurred. Increasing the atom number to N_a while keeping the cavity field strength fixed leads to (1) an $\sqrt{{N}_{\text{a}}}$ times increased effective atom–cavity coupling, resulting in a shift in the atomic transition frequency, (2) an N_a times increased atomic line strengths, and (3) the appearance of N_a − 1 dark states, see lower left panel of figure 1. Our numerical results show, as expected, that these dark states have no significant contribution to the spectrum (see lower right panel of figure 1).

A striking feature in figure 1 is that the roughly 1.3 overall peak intensity of the split atomic transition in the middle right panel is significantly different from the 1.0 intensity of the unsplit atomic peak of the upper right panel. This indicates an intensity borrowing effect in the atomic spectrum, i.e., the intensity of the atomic transition peak is modified, due to the contamination of the atomic excited state with molecular excited rovibronic states. This striking new phenomenon in the atomic spectrum is caused by the indirect coupling with the molecule, attributed to the radiation mode of the cavity.

The right panels of figure 2 show the absorption spectra obtained for the polaritonic states presented in the left panels of figure 2. The position and the composition of the dressed states, represented by the color of their transition peaks, reflect well the corresponding polaritonic landscapes. For example, peaks with various colors between 15 600 − 17 000 cm⁻¹ in the upper right panel of figure 2 show the fact that in that energy region the polaritonic states (see upper left panel of figure 2) are composed of a strong mixture of molecular, atomic, as well as photonic excited states, represented by basis functions of the form |2vJ⟩|0⟩|0⟩, |1vJ⟩|0⟩|I ≠ 0⟩, and |1vJ⟩|1⟩|0⟩, respectively. On the other hand, the red peaks between 15 000 − 16 000 cm⁻¹ in the lower right panel of figure 2 indicate that the corresponding polaritonic state includes mostly molecular excitation. Figure 2 demonstrates that the response of molecules (atoms) to light can be significantly modified and manipulated by the presence of atoms (molecules), and their simultaneous interaction with a cavity radiation mode.