Unraveling quantum pathways using optical 3D Fourier-transform spectroscopy

Abstract

Predicting and controlling quantum mechanical phenomena require knowledge of the system Hamiltonian. A detailed understanding of the quantum pathways used to construct the Hamiltonian is essential for deterministic control and improved performance of coherent control schemes. In complex systems, parameters characterizing the pathways, especially those associated with inter-particle interactions and coupling to the environment, can only be identified experimentally. Quantitative insight can be obtained provided the quantum pathways are isolated and independently analysed. Here we demonstrate this possibility in an atomic vapour using optical three-dimensional Fourier-transform spectroscopy. By unfolding the system's nonlinear response onto three frequency dimensions, three-dimensional spectra unambiguously reveal transition energies, relaxation rates and dipole moments of each pathway. The results demonstrate the unique capacity of this technique as a powerful tool for resolving the complex nature of quantum systems. This experiment is a critical step in the pursuit of complete experimental characterization of a system's Hamiltonian.

Figure 1. 3DFT spectra of a potassium vapour.

(a) Time ordering of the three pulses. A rephasing (non-rephasing) spectrum is obtained with the conjugated pulse A* arriving first (second). (b) Relevant energy levels of a potassium atom. (c) Experimental rephasing 3DFT spectrum obtained with pulse A* arriving first. (d) Experimental non-rephasing 3DFT spectrum obtained with pulse A* arriving second. In c and d the solid red isosurface represents a higher amplitude than the semi-transparent red surface. Blue dashed lines are diagonal lines (ω_t=|ω_τ|, ω_T=0).

Figure 2. Projections and quantum pathways in the 3DFT spectrum.

(a) Experimental rephasing 3DFT spectrum from Figure 1c with two-dimensional projections on three planes. (b) Projection on the bottom plane, equivalent to a rephasing 2DFT spectrum at T=0. (c) Projection on the left back plane, equivalent to a zero-quantum 2DFT spectrum. (d) Projection on the right back plane, not accessible by conventional 2DFT spectroscopy. (e) Double-sided Feynman diagrams representing all quantum pathways in the system and the corresponding peaks in the spectra. Overlapping pathways in 2DFT spectra are isolated in the 3DFT spectrum.

Figure 3. Lineshape analysis of a single peak.

(a) A single 3DFT spectral peak 3E with projections on three planes. The peak represents a single quantum pathway. The dynamics of the associated process can be understood by analysing lineshape and projections. (b) Profiles of peak 3E along three frequency directions. Dotted lines are data and red lines are fits to a square root of Lorentzian function. Linewidths of the profiles determine the relaxation rates.

Figure 4. TFWM experiment with four beams in the box geometry.

Three pulses are incident on the sample to generate a TFWM signal, which is later combined with the reference (Ref) beam for spectral interferometry. The dashed line shows the tracer beam, which is blocked during acquisition of a 3DFT spectrum, but is used to calibrate the phase of the reference. The inset shows the titanium vapour cell.