Bridging chemistry and Gaussian boson sampling: A photonic hierarchy of approximations for molecular vibronic spectra

Abstract

Simulating vibronic spectra is a central task in physical chemistry, offering insight into important properties of molecules. Recently, it has been experimentally demonstrated that photonic platforms based on Gaussian boson sampling (GBS) are capable of performing these simulations. However, whether an actual GBS approach is required depends on the molecule under investigation. To develop a better understanding on the requirements for simulating vibronic spectra, we explore connections between theoretical approximations in physical chemistry and their photonic counterparts. Mapping these approximations into photonics, we show that for certain molecules the GBS approach is unnecessary. We place special emphasis on the linear coupling approximation, which in photonics corresponds to sampling from multiple coherent states. By implementing this approach in experiments, we demonstrate improved similarities over previously reported GBS results for formic acid and identify the particular attributes that a molecule must exhibit for this, and other approximations, to be valid. These results highlight the importance in forming deeper connections between traditional methods and photonic approaches.

Figures (4)

• Figure 1: Pictorial representation of effects on energy levels occurring upon molecular excitation. First, a change of the geometrical structure of the molecules translates to a shift of the potential energy surface by $\Delta$$q_i$ along the normal coordinate $q_i$ (left). Second, squeezing accounts for changes in the potential energy surface leading to alteration of energy levels from $\hbar\textit{$\omega_i$}/2$ to $\hbar\textit{$\omega_i^{\prime}$}/2$ (middle). Finally, the shape of vibrations can change resulting in mixing of normal modes described by the Duschinsky rotation $q_i^\prime$=U$q_i$ (right).
• Figure 2: General schematic for implementing Duschinsky (left), parallel (middle) and linear coupling (right) approximations using photonics. Note that the displacement is applied after the interferometer, while Huh et al. shifted it to the beginning huh2015boson and Zhu et al. encoded the displacement by using additional squeezed vacuum modes zhu2024large.
• Figure 3: Schematic of the experimental setup for sampling vibronic spectra under linear coupling approximation. A source of coherent states (laser) is attenuated by a variable attenuator (VA) to reduce the overall mean photon number. The programmable beam-splitter tree splits the beam into multiple outputs with specific splitting ratios to set the required mean photon number for all the vibronic modes (conceptually shown). Photon-number resolving detectors are used to measure the exact number of photons for every sample (pulse).
• Figure 4: Measurement results for vibronic spectra in linear coupling approximation (blue) and simulation results for parallel approximation (gray) compared to the Duschinsky/GBS simulation (orange) as a benchmark. The insets show the geometric structures of the molecules in their initial electronic states: gray - carbon; white - hydrogen; red - oxygen; blue - nitrogen. Formic acid (a) and p-benzyne (b) show a similarity of $98.4$ % and $99.5$ % between linear coupling and GBS, respectively. In the case of Formaldehyde (c), although reaching a similarity of $96.5$ %, linear coupling fails to show the same structure for the spectrum. However, the parallel approximation is sufficient for this molecule and reaches a similarity of $99.6$ %. For pyridazine (d) neither linear coupling nor parallel approximation correctly identify the values of the FCFs, yielding similarities of $75.9$ % and $85.4$ % respectively, therefore requiring the GBS approach. Note that the plots only show $FCFs \geq 0.005\cdot FCF_{max}$ for the sake of visibility. Assuming Poissonian errors, the uncertainty on the similarities is $0.04$ % for all molecules. A short summary for the choice of molecular parameters can be found in the Methods section. More details on molecular parameters as well as a detailed discussion of the impact of imperfect displacements on the measured similarities are given in the Supplementary.