TwoDimensional ChirpedPulse Fourier Transform Microwave Spectroscopy Modeling Coherence

Two-Dimensional Chirped-Pulse Fourier Transform Microwave Spectroscopy: Modeling Coherence Transfer David Wilcox Purdue University Department of Chemistry 560 Oval Dr. West Lafayette, IN 47907 -2084 65 th Ohio State University International Symposium on Molecular Spectroscopy 6/22/2010

Motivation • 2 D CP-FTMW spectroscopy surpasses several limitations of previous waveguide methods: • Pulse shaping • Sequence mode • Increased bandwidth of detection • Differences in the methods: • Phase-cycling • Aliasing with multiple coherences • Selection rules for dipole-forbidden coherences • 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. B. Vogelsanger and A. Bauder, J. Chem. Phys. 92, 4101 (1990) OSU Molecular Symposium 6/22/2010

Modeling Coherence Transfer with the Liouville-von Neumann Equation: Evolution of the Density Matrix = Time independent rigid rotor Hamiltonian = Transition dipole vector (dipole moment) = Electric field, treated classically and sinusoidally Off-diagonal Matrix Elements OSU Molecular Symposium 6/22/2010

The Hamiltonian Matrix: 4 Level System Purely Progressive (Ladder Configuration) Purely Regressive (W Type) OSU Molecular Symposium 6/22/2010

The Density Matrix: The Progressive 4 Level System First Term: Energy level populations Populations over quantum and statistical mechanical probabilities along the diagonal. OSU Molecular Symposium 6/22/2010

The Density Matrix: The Progressive 4 Level System Second Term: Dipole allowed single quantum coherences d c b a Single coherent superposition of selection rule allowed energy levels. The coherence oscillates at the characteristic resonant frequency of the transition. OSU Molecular Symposium 6/22/2010

The Density Matrix: The Progressive 4 Level System Third Term: Double quantum coherences described by the density matrix Set to zero d c b a 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: OSU Molecular Symposium 6/22/2010

The Density Matrix: The Regressive 4 Level System Set to zero d c b 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: a OSU Molecular Symposium 6/22/2010

Modeling Coherence Transfer • 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 2 D 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. B t 1 evolution C mixing probe pump A preparation D t 2 detection OSU Molecular Symposium 6/22/2010

Demonstrating the Validity with a 3 Level System: Trifluoropropyne Autocorrelation Pulse Sequence 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. Transition Frequency → Observed Frequency 3 ΔJ = 2← 2 1: 11511 MHz → 488 MHz ΔJ = 3← 3 2: 17267 MHz → 267 MHz Energy Level Scheme t 1 (Scan) probe 1 Autocorrelation Pulse Sequence pump 2 OSU Molecular Symposium 6/22/2010

TFP Autocorrelation 1 D Spectrum: Identifying Peak Types 3 C A 488 MHz A 2 C B C 267 MHz C C C 1 Energy Level Scheme A: Direct coherence information (267 & 488 MHz) B: Double quantum coherence (Quantum mixing) C: Classical mixing off of carrier frequency *Intensities derived from Boltzmann probabilities at thermal equilibrium OSU Molecular Symposium 6/22/2010

Regressive 4 Level System: 1 -Chloro-1 Fluoroethylene Autocorrelation Experiment Transition Frequency → Observed Frequency ΔF = 0. 5← 0. 5 1. 5: 14120 MHz → 120 MHz ΔF = 2. 5← 2. 5 1. 5: 14128 MHz → 128 MHz F=1. 5 F=2. 5 F=0. 5 111 ΔF = 1. 5← 1. 5: 14138 MHz → 138 MHz Energy Level Scheme 000 t 1 (Scan) probe F=1. 5 pump Autocorrelation Pulse Sequence OSU Molecular Symposium 6/22/2010

Raw 1 D Slice of CFE: Enhancement of Spectral Features 138 128 F=2. 5 120 F=1. 5 CFE Energy Level Scheme Comparison of 2 D plot vs. 1 D slice of the same plot. Small spectral features are accentuated and much more spectral detail is seen in the 1 D slice which is lost in larger plot. OSU Molecular Symposium 6/22/2010

Additional Spectral Features of 1 D Slice: A Closer Look A C 138 128 B 120 CFE energy level scheme of the 111 -000 transition, ΔF = 2. 5← 1. 5 A: Fundamental (Coherence) B: Quantum mixing (Beats) C: Classical mixing (Harmonics) OSU Molecular Symposium 6/22/2010

Closer-er View of the Coherence Peaks Coherences from adjacent regressive transitions are present in 1 D slice. 128 120 138 F=1. 5 F=2. 5 138 128 F=0. 5 120 F=1. 5 OSU Molecular Symposium 6/22/2010

1 D Slices of CFE Hyperfine Transitions ΔF= 0. 5← 1. 5 138 120 MHz: ΔF= 0. 5← 1. 5 Parent 120 128 128 MHz: ΔF= 2. 5← 1. 5 Coherence * 120 138 ΔF= 1. 5← 1. 5 138 MHz: ΔF= 1. 5← 1. 5 Weak *Classical Mixing 128 138 120 MHz: ΔF= 0. 5← 1. 5 Weak 128 120 * 120 128 MHz: ΔF= 2. 5← 1. 5 Coherence 138 MHz: ΔF= 1. 5← 1. 5 Parent OSU Molecular Symposium 6/22/2010

Conclusion and Future Direction • Three-level system of Vogelsanger and Bauder has been extended to describe peaks in 1 D slices of higher order systems accessible with CP-FTMW spectroscopy. • 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. • Defer further applications to Kelly Hotopp’s talk, TC 12. OSU Molecular Symposium 6/22/2010

Acknowledgments Funding: Dian Research Group: Purdue University Kelly Hotopp Amanda Shirar Brian Dian Camille and Henry Dreyfus Foundation OSU Molecular Symposium 6/22/2010