Estimating Franck-Condon factors using an NMR quantum processor

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.

Figure 1

A harmonic potential centered at the origin $\left(V_1\right)$, a displaced harmonic potential centered at $x=2$ and with $\Delta E=0$ $\left(V_2\right)$, and their corresponding wave functions. Both position and energy axes are shown in arbitrary units. The undisplaced eigenfunctions are labelled in the computational basis and the displaced eigenfunctions are indicated with primes.

Figure 2

Molecular structure of iodotrifluoroethylene and its Hamiltonian parameters. The diagonal elements represent relative resonance frequencies (in Hz) and the off-diagonal elements represent the strengths of scalar $\left(J\right)$ couplings (in Hz) of ${}^{19}\mathrm{F}$ spins. The relaxation time constants $(T_1$ and $T_2^{*})$ are also mentioned in the last two columns.

Figure 3

Circuits for (a) Moussa protocol, (b) its extension, and the overall circuit for estimating FCFs. In all the cases the top qubit corresponds to the ancilla and the remaining are system qubits. In (b) $P$ is a projection observable.

Figure 4

Experimental FCFs (circles) corresponding to four-level harmonic oscillators (encoded by two qubits) versus the displacement $b$ (in atomic units) obtained using the Moussa protocol. The simulated FCFs (dotted lines) for the four-level system and analytical FCFs (smooth lines) for infinite-level system (Table 1) are also shown for comparison. The dashed curve at the top of (b) corresponds to the normalization used. The thin vertical dashed lines at $b=2,1+\sqrt{3}$ mark the beginning of classically forbidden regions for $f_{0,0^{\prime }},f_{0,1^{\prime }}$, respectively.

Figure 5

Experimental FCFs (circles) corresponding to lowest four-levels of harmonic oscillators (shown in inset) measured using diagonal density matrix tomography of three qubits. Analytical forms of FCFs (solid line) for infinite-level potentials (Table 1) are shown for comparison. All the FCFs are plotted versus displacement $b$ (in atomic units). Filled circles indicate the FCFs which are entirely due to the overlap of tunneling amplitudes.

Figure 6

Numerical simulation of FCFs to show the effect of truncation of basis (i.e., finite levels of the potentials). The displacement $b$ is plotted in atomic units. Four-basis states (top-panel), eight-basis states (middle-panel), and 16-basis states (bottom-panel) are used to simulate FCFs.

Figure 7

Average standard deviation of simulated FCFs as a function of noise amplitude $\eta$. The blue solid line corresponds to the diagonal-tomography method and the red dashed line corresponds to the Moussa protocol.