David Wilcox Purdue University Department of Chemistry 560 Oval Dr. West Lafayette, IN

2D CP-FTMW spectroscopy surpasses several limitations of previous waveguide methods:2D CP-FTMW spectroscopy surpasses several limitations of previous waveguide methods: Pulse shapingPulse shaping Sequence modeSequence mode Increased bandwidth of detectionIncreased bandwidth of detection Differences in the methods:Differences in the methods: Phase-cyclingPhase-cycling Aliasing with multiple coherencesAliasing with multiple coherences Selection rules for dipole-forbidden coherencesSelection rules for dipole-forbidden coherences Vogelsanger and Bauder used the density matrix formalism to explain three-level systems.Vogelsanger and Bauder used the density matrix formalism to explain three-level systems. The goal of this work is to extend the formalism of the three-level system to an N-level system.The goal of this work is to extend the formalism of the three-level system to an N-level system.

= Time independent rigid rotor Hamiltonian = Transition dipole vector (dipole moment) = Electric field, treated classically and sinusoidally Off-diagonal Matrix Elements

Purely Progressive (Ladder Configuration) Purely Regressive (W Type)

Populations over quantum and statistical mechanical probabilities along the diagonal.

Single coherent superposition of selection rule allowed energy levels. The coherence oscillates at the characteristic resonant frequency of the transition. a b c d

a b c d Multiple photons access dipole-forbidden transitions. This study determined that 3 rd and higher order coherences are not detectable, and thus the approximate phenomenological density matrix selection rule states: Set to zero

a b c d Similar development of the density matrix for a regressively connected energy level scheme. Dipole forbidden quantum beats oscillate with a difference of adjacent transitions sharing a common energy level. This restriction yields the approximate density matrix selection rule: Set to zero

After the Hamiltonian and density matrix are defined, the rotating wave approximation is made and the solutions to the Liouville-von Neumann equation are solved numerically.After the Hamiltonian and density matrix are defined, the rotating wave approximation is made and the solutions to the Liouville-von Neumann equation are solved numerically. The four general periods of a two-pulse 2D experiment are preparation (A), t 1 evolution (B), mixing (C), and t 2 detection (D). The state of the system at the end of each period serves as initial conditions for the subsequent period.The four general periods of a two-pulse 2D experiment are preparation (A), t 1 evolution (B), mixing (C), and t 2 detection (D). The state of the system at the end of each period serves as initial conditions for the subsequent period. pump probe t1t1 A B CD t2t2 preparation t 1 evolution mixing t 2 detection

ΔJ = 21: MHz 488 MHz ΔJ = 2←1: MHz → 488 MHz ΔJ = 32: MHz 267 MHz ΔJ = 3←2: MHz → 267 MHz Sampling at the Nyquist rate requires too many data points to practically record. Intentional under-sampling at 1 ns step size gives 500 MHz of bandwidth in t 1 detection, so shifts in transition frequencies are observed. Energy Level Scheme pump probe (Scan) t1t

AA B C C C C C C *Intensities derived from Boltzmann probabilities at thermal equilibrium 267 MHz A: Direct coherence information (267 & 488 MHz) B: Double quantum coherence (Quantum mixing) C: Classical mixing off of carrier frequency Energy Level Scheme 488 MHz

0 00 F= F=1.5 F=0.5 F=2.5 Energy Level Scheme ΔF = : MHz 120 MHz ΔF = 0.5←1.5: MHz → 120 MHz ΔF = : MHz 128 MHz ΔF = 2.5←1.5: MHz → 128 MHz ΔF = : MHz 138 MHz ΔF = 1.5←1.5: MHz → 138 MHz pump probe (Scan) t1t1

CFE Energy Level Scheme Comparison of 2D plot vs. 1D slice of the same plot. Small spectral features are accentuated and much more spectral detail is seen in the 1D slice which is lost in larger plot F=1.5 F=2.5

CFE energy level scheme of the transition, ΔF = 2.5← B AC A: Fundamental (Coherence) B: Quantum mixing (Beats) C: Classical mixing (Harmonics)

Coherences from adjacent regressive transitions are present in 1D slice. F=1.5 F=0.5 F=2.5

120 MHz: ΔF= 0.5←1.5 Parent 128 MHz: ΔF= 2.5←1.5 Coherence 138 MHz: ΔF= 1.5←1.5 Weak * * MHz: ΔF= 0.5←1.5 Weak 128 MHz: ΔF= 2.5←1.5 Coherence 138 MHz: ΔF= 1.5←1.5 Parent ΔF= 0.5←1.5 ΔF= 1.5←1.5 *Classical Mixing

Three-level system of Vogelsanger and Bauder has been extended to describe peaks in 1D slices of higher order systems accessible with CP-FTMW spectroscopy.Three-level system of Vogelsanger and Bauder has been extended to describe peaks in 1D slices of higher order systems accessible with CP-FTMW spectroscopy. Phenomenological density matrix selection rules are needed to account for quantum mixing peaks.Phenomenological density matrix selection rules are needed to account for quantum mixing peaks. Formalism used to implement quantum computing logic gates using broadband rotational spectroscopy.Formalism used to implement quantum computing logic gates using broadband rotational spectroscopy. Defer further applications to Kelly Hotopp’s talk, TC12.Defer further applications to Kelly Hotopp’s talk, TC12.

Purdue University Camille and Henry Dreyfus Foundation Kelly Hotopp Amanda Shirar Brian Dian