Tomographic State Reconstruction and Time Resolved Surface Enhanced Coherent Anti-Stokes Raman Scattering in the Single Molecule Limit

Time-resolved, surface-enhanced, coherent anti-Stokes Raman spectroscopy (tr-SECARS) isideally suited for preparing and probing vibrational coherences in molecules. By enhancingthe local response of a single molecule with a dipolar nano-antenna, vibrational dynamicshave been measured at the single molecule limit. In contrast with tr-CARS measurementsin ensembles, the vibrational coherence of a single molecule is not subject to pure dephasing.It exhibits characteristic phase and amplitude noise, which allows the statistical distinctionbetween single, few, and many molecule sources to be determined. To build on the cur-rent work, by using three unique pulses to spectrally lter the response of the molecule,the characteristic noise can be isolated and measured background-free. If the probing of asuperposition state is carried out over a real resonance, then it is possible to tomographicallyreconstruct the complete description of quantum dynamics in phase space representation viathe Wigner Distribution Function(WDF). The WDF can be reconstructed from either thewavepacket via Wigner Transform, or an experimentally measured density, via an InverseRadon Transform. The calculations presented here highlight the necessary conditions inorder to reconstruct the WDF with delity from a proposed experiment and compare thedensity derived WDF with that of the wavepacket. The principle is rstly demonstrated us-ing a Kerr gated detection of emission from an evolving state on a bound harmonic potentialenergy surface. The model is then explained in the case of a proposed spectrally resolvedtransient grating experiment (SRTG). The WDFs generated from the limiting conditionsshow that the reproduction delity of the experimentally derived WDF are dependent onthe probe, utilized to measure the evolving superposition, and the curvature, or the vibra-tional frequency of the potential energy surfaces, and the dephasing time of the vibrationalsuperposition states. Given two potentials, I show that it is possible to optimize probepulse parameters to improve the delity of the state reconstruction. Due to the variationalprinciple, the negative volume of the WDF, or the Wigner hole, can only be reduced viameasurements - the pulse parameters can be optimized iteratively even when the molecularpotentials are not known