^{1,a)}, Eric R. Smith

^{1,b)}, Wei Qian

^{1,c)}and David M. Jonas

^{1,d)}

### Abstract

By analogy to the Raman depolarization ratio, vibrational quantum beats in pump-probe experiments depend on the relative pump and probe laser beam polarizations in a way that reflects vibrational symmetry. The polarization signatures differ from those in spontaneous Raman scattering because the order of field-matter interactions is different. Since pump-probe experiments are sensitive to vibrations on excited electronic states, the polarizationanisotropy of vibrational quantum beats can also reflect electronic relaxation processes. Diagrammatic treatments, which expand use of the symmetry of the two-photon tensor to treat signal pathways with vibrational and vibronic coherences, are applied to find the polarizationanisotropy of vibrational and vibronic quantum beats in pump-probe experiments for different stages of electronic relaxation in square symmetric molecules. Asymmetric vibrational quantum beats can be distinguished from asymmetric vibronic quantum beats by a phase jump near the center of the electronic spectrum and their disappearance in the impulsive limit. Beyond identification of vibrational symmetry, the vibrational quantum beatanisotropy can be used to determine if components of a doubly degenerate electronic state are unrelaxed, dephased, population exchanged, or completely equilibrated.

I. INTRODUCTION

II. CALCULATION OF WITHOUT RELAXATION

A. Ground state vibrations (GSB)

B. Excited state vibrations in emission (ESE)

C. Excited state vibrations in absorption (ESA)

D. Total anisotropy

III. ANISOTROPY WITH RELAXATION

IV. DISCUSSION

A. Vibrational symmetry

B. Tuning dependence

C. Sensitivity to electronic dynamics

V. CONCLUSIONS

### Key Topics

- Quantum beats
- 176.0
- Anisotropy
- 145.0
- Excited states
- 63.0
- Polarization
- 53.0
- Ground states
- 30.0

## Figures

Energy level Feynman diagram contributions for pump-probe compared to corresponding diagrams from various nonlinear Raman active signals. The arrows represent transfer of probability amplitude proportional to the applied excitation fields. Solid arrows represent density matrix ket amplitude and dashed arrows represent density matrix bra amplitude. The wavy lines represent the radiated signal field, which changes the amplitudes for both the upper and lower states connected by the wave. Time increases from left to right. (U) and (R) represent the pump and probe polarizations for pump-probe, femtosecond Raman, and CARS, respectively. (U) and (R) represent the excitation and detection polarizations for spontaneous Raman (resonant or off-resonant) scattering measurements. Because the laboratory frame optical electric field polarization sequences match, pump-probe and femtosecond Raman produce equivalent polarization signatures. Because the laboratory frame polarization sequence is different, CARS and spontaneous Raman have different polarization signatures. If the detected signal field (S) polarization in CARS is set to (R), the CARS and spontaneous Raman polarization signatures become equivalent. (Both equivalences assume that the frequency differences cause no change in the molecular response.) Note that in spontaneous Raman scattering, the first detection field arises from the vacuum electromagnetic field. Its polarization vector must equal the detected polarization vector.

Energy level Feynman diagram contributions for pump-probe compared to corresponding diagrams from various nonlinear Raman active signals. The arrows represent transfer of probability amplitude proportional to the applied excitation fields. Solid arrows represent density matrix ket amplitude and dashed arrows represent density matrix bra amplitude. The wavy lines represent the radiated signal field, which changes the amplitudes for both the upper and lower states connected by the wave. Time increases from left to right. (U) and (R) represent the pump and probe polarizations for pump-probe, femtosecond Raman, and CARS, respectively. (U) and (R) represent the excitation and detection polarizations for spontaneous Raman (resonant or off-resonant) scattering measurements. Because the laboratory frame optical electric field polarization sequences match, pump-probe and femtosecond Raman produce equivalent polarization signatures. Because the laboratory frame polarization sequence is different, CARS and spontaneous Raman have different polarization signatures. If the detected signal field (S) polarization in CARS is set to (R), the CARS and spontaneous Raman polarization signatures become equivalent. (Both equivalences assume that the frequency differences cause no change in the molecular response.) Note that in spontaneous Raman scattering, the first detection field arises from the vacuum electromagnetic field. Its polarization vector must equal the detected polarization vector.

Energy ladder subdiagrams for vibrational quantum beats in the GSB contribution to the pump-probe signal. The diagrams shown include all GSB pathways through states of and . The solid and dashed arrows correspond to changes in the density matrix ket and bra indices, respectively (Ref. 47). Time increases from left to right. The number 3 on the left of each subdiagram label represents the general type of diagram (GSB, bra amplitude transfer first) as labeled in Refs. 56 and 58. The right subscript (1–4) on each subdiagram label specifies the path through states and (assigned according to the scheme of Refs. 41 and 42).

Energy ladder subdiagrams for vibrational quantum beats in the GSB contribution to the pump-probe signal. The diagrams shown include all GSB pathways through states of and . The solid and dashed arrows correspond to changes in the density matrix ket and bra indices, respectively (Ref. 47). Time increases from left to right. The number 3 on the left of each subdiagram label represents the general type of diagram (GSB, bra amplitude transfer first) as labeled in Refs. 56 and 58. The right subscript (1–4) on each subdiagram label specifies the path through states and (assigned according to the scheme of Refs. 41 and 42).

Electronic displacements along asymmetric and symmetric vibrational coordinates. Asymmetric, JT active vibrations (B) break the degeneracy of and . Totally symmetric vibrations (A) maintain degeneracy.

Electronic displacements along asymmetric and symmetric vibrational coordinates. Asymmetric, JT active vibrations (B) break the degeneracy of and . Totally symmetric vibrations (A) maintain degeneracy.

Energy ladder sub-subdiagrams for vibrational quantum beats in the GSB. The diagrams shown represent GSB pathways with two steps and two steps, where the two steps may involve different electronic levels. These quantum beats contribute at order . The time ordered set of molecular frame transition moments [e.g., for ] and product of vibrational overlap integrals [e.g., for ] are shown below each subdiagram for both modes and modes, as indicated at right.

Energy ladder sub-subdiagrams for vibrational quantum beats in the GSB. The diagrams shown represent GSB pathways with two steps and two steps, where the two steps may involve different electronic levels. These quantum beats contribute at order . The time ordered set of molecular frame transition moments [e.g., for ] and product of vibrational overlap integrals [e.g., for ] are shown below each subdiagram for both modes and modes, as indicated at right.

Energy ladder subdiagram for spontaneous Raman scattering. Subdiagrams are presented in this format for comparison with four-wave mixing signals from coherent fields. (U) and (R) represent the excitation and detection polarizations for spontaneous Raman (resonant or off-resonant). In resonance Raman, subdiagrams and only contribute when transitions to and both lie within the excitation pulse bandwidth. The second arrow (counting from the left) represents an interaction with the electromagnetic vacuum field similar to that in spontaneous emission. Depolarization ratios can be calculated from orientational integrals for parallel and perpendicular (, ) polarization configurations. The orientational integrals are shown for subdiagrams and for each of the four Raman active vibrational symmetries. (The subdiagrams for and produce the same orientational averages for both parallel and perpendicular signals.) is , , , and ∞ for , , , and vibrations, respectively.

Energy ladder subdiagram for spontaneous Raman scattering. Subdiagrams are presented in this format for comparison with four-wave mixing signals from coherent fields. (U) and (R) represent the excitation and detection polarizations for spontaneous Raman (resonant or off-resonant). In resonance Raman, subdiagrams and only contribute when transitions to and both lie within the excitation pulse bandwidth. The second arrow (counting from the left) represents an interaction with the electromagnetic vacuum field similar to that in spontaneous emission. Depolarization ratios can be calculated from orientational integrals for parallel and perpendicular (, ) polarization configurations. The orientational integrals are shown for subdiagrams and for each of the four Raman active vibrational symmetries. (The subdiagrams for and produce the same orientational averages for both parallel and perpendicular signals.) is , , , and ∞ for , , , and vibrations, respectively.

Energy ladder subdiagrams for vibrational quantum beats in the ESE contribution to the pump-probe signal. The diagrams shown include only ESE pathways through (on the ground electronic state). Diagrams in which the first field interaction involves a or transition are labeled with a right subscript or , respectively. If only these diagrams were considered, the anisotropy of and modes would be equivalent and both vibrational quantum beat amplitudes would be of order .

Energy ladder subdiagrams for vibrational quantum beats in the ESE contribution to the pump-probe signal. The diagrams shown include only ESE pathways through (on the ground electronic state). Diagrams in which the first field interaction involves a or transition are labeled with a right subscript or , respectively. If only these diagrams were considered, the anisotropy of and modes would be equivalent and both vibrational quantum beat amplitudes would be of order .

Energy ladder subdiagrams for vibrational quantum beats in ESE. The diagrams are representative of ESE pathways involving only on the electronic ground state (superscript ) or involving both and on the electronic ground state (superscript ). All diagrams originate from on the electronic ground state. Due to cancellation between subdiagrams of types and , vibronic and pure vibrational quantum beats contribute at lowest order . Pure vibrational quantum beats also contribute at order , but vibronic beats contribute at . Instead of canceling, the vibronic subdiagrams of types and add constructively due to sign flips in the vibrational overlap integrals [Eqs. (16)–(27)] for modes. Therefore, vibronic quantum beats dominate the anisotropy of quantum beats .

Energy ladder subdiagrams for vibrational quantum beats in ESE. The diagrams are representative of ESE pathways involving only on the electronic ground state (superscript ) or involving both and on the electronic ground state (superscript ). All diagrams originate from on the electronic ground state. Due to cancellation between subdiagrams of types and , vibronic and pure vibrational quantum beats contribute at lowest order . Pure vibrational quantum beats also contribute at order , but vibronic beats contribute at . Instead of canceling, the vibronic subdiagrams of types and add constructively due to sign flips in the vibrational overlap integrals [Eqs. (16)–(27)] for modes. Therefore, vibronic quantum beats dominate the anisotropy of quantum beats .

Energy ladder subdiagrams for vibrational quantum beats in the ESA contribution to the pump-probe signal for an symmetry DES. The diagrams shown include ESA pathways through the level of the DES (labeled 2 at right). Subdiagrams in which the first field interaction involves a or transition have an additional subscript or , respectively. Along the coordinate, the phase shift of ESA quantum beats is manifested as a sign flip between the diagrams for the ESA offset (which have a positive product of two squared vibrational overlap integrals) and the vibrational quantum beats (which have a negative product of four different vibrational overlap integrals). If ESA transitions to on the DES are considered, the product of four different vibrational overlap integrals that determines the sign of the quantum beat is positive.

Energy ladder subdiagrams for vibrational quantum beats in the ESA contribution to the pump-probe signal for an symmetry DES. The diagrams shown include ESA pathways through the level of the DES (labeled 2 at right). Subdiagrams in which the first field interaction involves a or transition have an additional subscript or , respectively. Along the coordinate, the phase shift of ESA quantum beats is manifested as a sign flip between the diagrams for the ESA offset (which have a positive product of two squared vibrational overlap integrals) and the vibrational quantum beats (which have a negative product of four different vibrational overlap integrals). If ESA transitions to on the DES are considered, the product of four different vibrational overlap integrals that determines the sign of the quantum beat is positive.

Energy ladder subdiagrams for vibrational quantum beats in ESA. The diagrams are representative of ESA pathways ending in and , superscripted and , respectively. Due to a partial cancellation between diagrams with ESA to and , vibronic (subscript 4) and pure vibrational (subscript 2) quantum beats both contribute at order . Pure vibrational quantum beats also contribute at order , but vibronic beats contribute at . Therefore, vibronic coherence dominates the anisotropy of quantum beats and .

Energy ladder subdiagrams for vibrational quantum beats in ESA. The diagrams are representative of ESA pathways ending in and , superscripted and , respectively. Due to a partial cancellation between diagrams with ESA to and , vibronic (subscript 4) and pure vibrational (subscript 2) quantum beats both contribute at order . Pure vibrational quantum beats also contribute at order , but vibronic beats contribute at . Therefore, vibronic coherence dominates the anisotropy of quantum beats and .

Schematic for calculation of quantum beat anisotropy with PT during . The new subdiagram transfers bra and ket amplitudes to the other potential surface ( or ) during . Population relaxation between states of equal energies leads to signal amplitudes from each diagram on the right multiplied by .

Schematic for calculation of quantum beat anisotropy with PT during . The new subdiagram transfers bra and ket amplitudes to the other potential surface ( or ) during . Population relaxation between states of equal energies leads to signal amplitudes from each diagram on the right multiplied by .

## Tables

Anisotropy of vibrational quantum beats and the population offset contributions to the GSB. Vibrational wavepackets on the ground electronic state modulate the GSB contribution to the pump-probe signal. The transition dipole pattern and anisotropy for the unmodulated background caused by depopulation of the ground state (pop) match that of the totally symmetric vibrations. For finite pulses, symmetric vibrations and asymmetric vibrations of either or symmetry have nonzero amplitude under the Condon approximation. Within the Condon approximation, all vibrational quantum beats in GSB vanish in the limit of impulsive excitation. Both parallel and perpendicular signal strengths are identically zero for vibrational quantum beats so long as and are exactly degenerate. The breakdown of the Condon approximation allows simultaneous excitation of and vibrations.

Anisotropy of vibrational quantum beats and the population offset contributions to the GSB. Vibrational wavepackets on the ground electronic state modulate the GSB contribution to the pump-probe signal. The transition dipole pattern and anisotropy for the unmodulated background caused by depopulation of the ground state (pop) match that of the totally symmetric vibrations. For finite pulses, symmetric vibrations and asymmetric vibrations of either or symmetry have nonzero amplitude under the Condon approximation. Within the Condon approximation, all vibrational quantum beats in GSB vanish in the limit of impulsive excitation. Both parallel and perpendicular signal strengths are identically zero for vibrational quantum beats so long as and are exactly degenerate. The breakdown of the Condon approximation allows simultaneous excitation of and vibrations.

Vibrational anisotropy for quantum beats and the population offset contributions to ESA and ESE after initial excitation, ED of states along the coordinate, PT between surfaces through the coordinate, and both ED and PT. Various DES symmetries have been considered. PT between states with respect to the coordinate is equivalent to ED along the coordinate. Therefore, PT in the absence of ED and ED alone should produce the same anisotropy when the relative symmetry of DES and the active vibrational coordinate are taken into account. Quantum beats that survive in the impulsive limit, and therefore have large amplitudes in the near impulsive limit, are labeled with an asterisk .

Vibrational anisotropy for quantum beats and the population offset contributions to ESA and ESE after initial excitation, ED of states along the coordinate, PT between surfaces through the coordinate, and both ED and PT. Various DES symmetries have been considered. PT between states with respect to the coordinate is equivalent to ED along the coordinate. Therefore, PT in the absence of ED and ED alone should produce the same anisotropy when the relative symmetry of DES and the active vibrational coordinate are taken into account. Quantum beats that survive in the impulsive limit, and therefore have large amplitudes in the near impulsive limit, are labeled with an asterisk .

Parallel pump-probe amplitude for quantum beats in ESA and ESE after initial excitation, ED of states along the coordinate, PT between surfaces through the coordinate, and both ED and PT. Various DES symmetries have been considered. PT between states with respect to the coordinate is equivalent to ED along the coordinate. Therefore, PT in the absence of ED and ED alone should produce the same anisotropy when the relative symmetry of DES and the active vibrational coordinate are taken into account. Vibronic quantum beats that survive in the impulsive limit, and therefore dominate other contributions in the near impulsive limit, are marked with an asterisk . Diagrams with basis for the excited state could not be used to directly calculate amplitudes after both PT and ED and are marked a plus . In this case, amplitudes were calculated using symmetry relationships between and modes. The signs of the ESA amplitudes for pure vibrational quantum beats assume that transitions involving on the DES dominate over transitions involving and should be reversed if transitions involving on the DES dominate. The signs of the ESA amplitudes for vibronic quantum beats are not dependent on whether DES transitions involving or dominate.

Parallel pump-probe amplitude for quantum beats in ESA and ESE after initial excitation, ED of states along the coordinate, PT between surfaces through the coordinate, and both ED and PT. Various DES symmetries have been considered. PT between states with respect to the coordinate is equivalent to ED along the coordinate. Therefore, PT in the absence of ED and ED alone should produce the same anisotropy when the relative symmetry of DES and the active vibrational coordinate are taken into account. Vibronic quantum beats that survive in the impulsive limit, and therefore dominate other contributions in the near impulsive limit, are marked with an asterisk . Diagrams with basis for the excited state could not be used to directly calculate amplitudes after both PT and ED and are marked a plus . In this case, amplitudes were calculated using symmetry relationships between and modes. The signs of the ESA amplitudes for pure vibrational quantum beats assume that transitions involving on the DES dominate over transitions involving and should be reversed if transitions involving on the DES dominate. The signs of the ESA amplitudes for vibronic quantum beats are not dependent on whether DES transitions involving or dominate.

Perpendicular pump-probe amplitude for quantum beats ESA and ESA after initial excitation and ED of states along the coordinate. PT between surfaces through the coordinate and both ED and PT. Various DES symmetries have been considered. PT between states with respect to the coordinate is equivalent to ED along the coordinate. Therefore, PT in the absence of ED and ED alone should produce the same anisotropy when the relative symmetry of DES and the active vibrational coordinate are taken into account. Vibronic quantum beats that survive in the impulsive limit, and therefore dominate other contributions in the near impulsive limit, are marked with an asterisk . The signs of the ESA amplitudes for pure vibrational quantum beats assume that transitions involving on the DES dominate over transitions involving and should be reversed if transitions involving on the DES dominate. The signs of the ESA amplitudes for vibronic quantum beats are not dependent on whether DES transitions involving or dominate.

Perpendicular pump-probe amplitude for quantum beats ESA and ESA after initial excitation and ED of states along the coordinate. PT between surfaces through the coordinate and both ED and PT. Various DES symmetries have been considered. PT between states with respect to the coordinate is equivalent to ED along the coordinate. Therefore, PT in the absence of ED and ED alone should produce the same anisotropy when the relative symmetry of DES and the active vibrational coordinate are taken into account. Vibronic quantum beats that survive in the impulsive limit, and therefore dominate other contributions in the near impulsive limit, are marked with an asterisk . The signs of the ESA amplitudes for pure vibrational quantum beats assume that transitions involving on the DES dominate over transitions involving and should be reversed if transitions involving on the DES dominate. The signs of the ESA amplitudes for vibronic quantum beats are not dependent on whether DES transitions involving or dominate.

Total vibrational anisotropy of vibrations present in both degenerate excited and ground electronic states, after initial excitation, ED of states along coordinate, PT between surfaces through the coordinate, and both ED and PT. Calculations consider four possible DES symmetries allowed by dipole selection rules and assume near impulsive excitation. Vibronic quantum beats that survive in the impulsive limit have large amplitudes in the near impulsive limit and are marked with an asterisk . Only ground state beats contribute in some cases .

Total vibrational anisotropy of vibrations present in both degenerate excited and ground electronic states, after initial excitation, ED of states along coordinate, PT between surfaces through the coordinate, and both ED and PT. Calculations consider four possible DES symmetries allowed by dipole selection rules and assume near impulsive excitation. Vibronic quantum beats that survive in the impulsive limit have large amplitudes in the near impulsive limit and are marked with an asterisk . Only ground state beats contribute in some cases .

Experimental vibrational quantum beat anisotropies for -band excitation of SiNc compared to total vibrational quantum beat anisotropies calculated for four stages of electronic relaxation using independently measured isotropic transition strengths. The first row contains vibrational quantum beat frequencies (pop indicates the electronic population offset). The second row contains the most probable experimental vibrational quantum beat anisotropies and the third row specifies error bars (which are smaller than the 95% confidence intervals of Ref. 33 because the error bars do not include correlation). The symmetry assigned is given in the fourth row, with entries split into two columns to indicate the two possibilities for the asymmetric vibrations. The bottom four rows give the calculated anisotropy after initial excitation, ED of states along coordinate, PT between surfaces through the coordinate, and both ED and PT. These calculations assume that all vibrations are present in both the degenerate excited state and the ground electronic states and that the relative transition strengths are in the ratio determined in the Appendix of Ref. 33.

Experimental vibrational quantum beat anisotropies for -band excitation of SiNc compared to total vibrational quantum beat anisotropies calculated for four stages of electronic relaxation using independently measured isotropic transition strengths. The first row contains vibrational quantum beat frequencies (pop indicates the electronic population offset). The second row contains the most probable experimental vibrational quantum beat anisotropies and the third row specifies error bars (which are smaller than the 95% confidence intervals of Ref. 33 because the error bars do not include correlation). The symmetry assigned is given in the fourth row, with entries split into two columns to indicate the two possibilities for the asymmetric vibrations. The bottom four rows give the calculated anisotropy after initial excitation, ED of states along coordinate, PT between surfaces through the coordinate, and both ED and PT. These calculations assume that all vibrations are present in both the degenerate excited state and the ground electronic states and that the relative transition strengths are in the ratio determined in the Appendix of Ref. 33.

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