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arXiv:1405.2400 · Physical Review A

Estimating Franck-Condon factors using an NMR quantum processor

Sharad Joshi, Abhishek Shukla, Hemant Katiyar, Anirban Hazra, T. S. Mahesh

Published Aug 4, 2014• 20 citations
arXivHTMLPDF

Abstract

The interaction of molecules with light may lead to electronic transitions and simultaneous vibrational excitations. Franck-Condon factors (FCFs) play an important role in quantifying the intensities of such vibronic transitions occurring during molecular photoexcitations. In this article, we describe a general method for estimating FCFs using a quantum information processor. The method involves the application of a translation operator followed by the measurement of certain projections. We also illustrate the method by experimentally estimating FCFs with the help of a three-qubit nuclear magnetic resonance quantum information processor. We describe two methods for the measurement of projections: (i) using the Moussa protocol and (ii) using diagonal tomography. The experimental results agree fairly well with the theory.

DOI
10.1103/physreva.90.022303

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The interaction of molecules with light may lead to electronic transitions and simultaneous vibrational excitations. Franck-Condon factors (FCFs) play an important role in quantifying the intensities of such vibronic transitions occurring during molecular photoexcitations. In this article, we describe a general method for estimating FCFs using a quantum information processor. The method involves the application of a translation operator followed by the measurement of certain projections. We also illustrate the method by experimentally estimating FCFs with the help of a three-qubit nuclear magnetic resonance quantum information processor. We describe two methods for the measurement of projections: (i) using the Moussa protocol and (ii) using diagonal tomography. The experimental results agree fairly well with the theory.
The probabilities of vibronic transitions in molecules are referred to as Franck-Condon factors (FCFs). Although several approaches for calculating FCFs have been developed, such calculations are still challenging. Recently it was shown that there exists a correspondence between the problem of calculating FCFs and boson sampling. However, if the output photon number distribution of boson sampling is sparse, then it can be classically simulated. Exploiting these results, we develop a method to approximately reconstruct the distribution of FCFs of certain molecules. We demonstrate its proof of concept by applying it to formic acid and thymine at 0 K. In our method, we first obtain the marginal photon number distributions for pairs of modes of a Gaussian state associated with the molecular transition. We then apply a compressive sensing method called polynomial-time matching pursuit to recover FCFs.
We show that the Franck-Condon Factor (FCF) associated to a transition between initial and final vibrational states in two different potential energy surfaces, having NNN and MMM vibrational quanta, respectively, is equivalent to calculating the number of perfect matchings of a weighted graph with loops that has P=N+MP = N+MP=N+M vertices. This last quantity is the loop hafnian of the (symmetric) adjacency matrix of the graph which can be calculated in O(P32P/2)O(P^3 2^{P/2})O(P32P/2) steps. In the limit of small numbers of vibrational quanta per normal mode our loop hafnian formula significantly improves the speed at which FCFs can be calculated. Our results more generally apply to the calculation of the matrix elements of a bosonic Gaussian unitary between two multimode Fock states having NNN and MMM photons in total and provide a useful link between certain calculations of quantum chemistry, quantum optics and graph theory.
The quantum Fourier transform is a key factor in achieving exponential speedup relative to classical algorithms. We implemented the phase estimation algorithm, which is the very basic application example of the quantum Fourier transform, on a three-bit nuclear magnetic resonance quantum-information processor. The algorithm was applied to one-bit Grover operators to estimate eigenvalues. We also demonstrated the counting algorithm with the same operators to obtain the number of the marked states. The results of projective measurements required by the algorithms were obtained from the split peak positions in the spectra.
In this work, a direct quantum implementation of the Doktorov formulas for calculating the vibronic spectrum of molecules under the harmonic approximation is presented. It is applied to the three-atom molecules H 2 O, SO 2, ClO 2, HS 2, and ZnOH. The method solves the classically hard problem of estimating the Franck–Condon (FC) factors by using the Duschinsky matrices as the only input via the Doktorov quantum circuit. This has the advantage of avoiding basis changes, artificial squeezing parameters, and symmetry dependencies. In other words, it is a general method for three-atom molecules that can easily be generalized to bigger molecules. The results are compared with other quantum algorithms and classical anharmonic algorithms. Furthermore, the circuit requirements are studied in order to estimate its applicability on real superconducting quantum hardware.
Coherent multidimensional spectroscopy enables detailed investigations of vibronic effects in molecular and solid-state systems. We present explicit analytical expressions for multidimensional nonlinear response functions in the presence of Herzberg-Teller (non-Condon) coupling, within the displaced harmonic oscillator model. The formulation applies to electronic systems with an arbitrary number N of electronic states and to response functions of arbitrary order M in the light-matter interaction. We show that Herzberg-Teller coupling introduces additional oscillatory factors in the time-domain response functions, leading, upon Fourier transformation, to replicas of the Franck-Condon multidimensional spectra shifted by integer multiples of the vibrational frequencies. The present results provide a general analytical framework for the interpretation of non-Condon effects in coherent multidimensional spectroscopies. A Python code that implements the present approach and simulates multidimensional spectra for N-level systems is available on GitHub.
Using second-order perturbation theory in the light-matter interaction, we derive an analytical approximation for the vibronic populations of a diatomic system excited by ultrabroadband frequency-entangled photons and evaluate the population dynamics for different degrees of entanglement between photon pairs. Our analytical approach makes the same predictions as previously derived via numerical solutions of the complete Schrödinger equation [H. Oka, ], with the added advantage of providing clear physical insights into the vibronic selectivity as a function of the degree of photon correlations while requiring significantly reduced computational effort. Specifically, our analytical expression for the probability of vibronic excitation includes a factor which predicts the enhancement of vibrational selectivity as a function of the degree correlation between the entangled photon pairs, the targeted vibrational energy level, and the vibrational molecular structure encoded in the Franck-Condon factors. Our results illustrate the importance of going beyond the usual approximations in second-order perturbation theory to capture the relevance of the vibrational structure of the molecular system of interest in order to gain a deeper understanding of the possible quantum-enhancement provided by the interaction between quantum light and matter.