^{1,2,a)}and K. Birgitta Whaley

^{2,3,b)}

### Abstract

We propose a two-step protocol for inverting ultrafast spectroscopy experiments on a molecular aggregate to extract the time-evolution of the excited state density matrix. The first step is a deconvolution of the experimental signal to determine a pump-dependent response function. The second step inverts this response function to obtain the quantum state of the system, given a model for how the system evolves following the probe interaction. We demonstrate this inversion analytically and numerically for a dimer model system, and evaluate the feasibility of scaling it to larger molecular aggregates such as photosynthetic protein-pigment complexes. Our scheme provides a direct alternative to the approach of determining all Hamiltonian parameters and then simulating excited state dynamics.

We thank Jahan Dawlaty and Yuan-Chung Cheng for helpful discussions. This work was supported in part by DARPA under Award No. N66001-09-1-2026. S.H. is a DOE Office of Science Graduate Fellow.

I. INTRODUCTION

II. RECIPE FOR PUMP-PROBE SPECTROSCOPY

A. Detection scheme and probe convolution

B. Pump-probe response function

III. INVERSION PROTOCOLS

A. Deconvolution of the pump-probe signal

B. Obtaining the quantum state

IV. EXAMPLE: DIMER MODEL

A. Analytical calculation of pump-probe response

B. Numerical example

C. Response function inversion

D. State tomography

V. SCALING TO LARGER SYSTEMS

VI. CONCLUSIONS

### Key Topics

- Tomography
- 30.0
- Excited states
- 28.0
- Pump probe experiments
- 23.0
- Polarization
- 20.0
- Quantum tomography
- 16.0

## Figures

Absorptive (left) and dispersive (right) parts of the pump-probe response function (top) and the corresponding heterodyne detected signal *S*(ω, τ) (bottom) for our dimer model system. The dashed line indicates the two exciton transition energies in this system. Only the absorptive part (left) is revealed directly by a pump-probe experiment. Obtaining the dispersive part (right) requires a transient grating setup with heterodyne detection, as described in Sec. II A .

Absorptive (left) and dispersive (right) parts of the pump-probe response function (top) and the corresponding heterodyne detected signal *S*(ω, τ) (bottom) for our dimer model system. The dashed line indicates the two exciton transition energies in this system. Only the absorptive part (left) is revealed directly by a pump-probe experiment. Obtaining the dispersive part (right) requires a transient grating setup with heterodyne detection, as described in Sec. II A .

(a) Example reconstruction of the pump-probe response at fixed probe-frequency ω_{α} for an instance of the high-noise test problem. (b) Errors in the estimated pump-probe response obtained by the direct and Tikhonov inversion methods for a single example of the low and high noise test problems. The error is given by the absolute value squared of the difference between the estimated and actual response function, .

(a) Example reconstruction of the pump-probe response at fixed probe-frequency ω_{α} for an instance of the high-noise test problem. (b) Errors in the estimated pump-probe response obtained by the direct and Tikhonov inversion methods for a single example of the low and high noise test problems. The error is given by the absolute value squared of the difference between the estimated and actual response function, .

Results of quantum state tomography for our dimer test problem. (a) Original (solid) and reconstructed (dotted) values for each element of the Bloch state vector for the reconstruction with static disorder of standard deviation 40 cm^{−1}. Normalization is omitted since the state vector elements are rescaled such that *r* _{0} = 1 fixed for all times following initial excitation. (b) Worst- and average-case fidelities for the reconstructions ρ_{ e }(τ) for delay times τ in the range 50 fs to 1 ps as a function of the width (standard deviation) of the distribution of static disorder. Results are obtained from an ensemble average over 10^{6} samples for each point.

Results of quantum state tomography for our dimer test problem. (a) Original (solid) and reconstructed (dotted) values for each element of the Bloch state vector for the reconstruction with static disorder of standard deviation 40 cm^{−1}. Normalization is omitted since the state vector elements are rescaled such that *r* _{0} = 1 fixed for all times following initial excitation. (b) Worst- and average-case fidelities for the reconstructions ρ_{ e }(τ) for delay times τ in the range 50 fs to 1 ps as a function of the width (standard deviation) of the distribution of static disorder. Results are obtained from an ensemble average over 10^{6} samples for each point.

Species associated spectra, defined by the contribution of the marked density matrix elements to the pump-probe response, for the FMO complex at 77 K (blue) and 300 K (red), obtained as the average of 1000 samplings over static disorder. Labels indicate the contributing density matrix element in the excitonic basis. Shaded regions indicate central 95% confidence intervals obtained from 1000 additional samplings over Hamiltonian uncertainty, as described in the text.

Species associated spectra, defined by the contribution of the marked density matrix elements to the pump-probe response, for the FMO complex at 77 K (blue) and 300 K (red), obtained as the average of 1000 samplings over static disorder. Labels indicate the contributing density matrix element in the excitonic basis. Shaded regions indicate central 95% confidence intervals obtained from 1000 additional samplings over Hamiltonian uncertainty, as described in the text.

Normalized singular values from a singular value decomposition of the real valued pump-probe map given by Eq. (20) for the FMO complex under various conditions. The top line is from the spectra of a single monomer at 77 K, from the combination of measurements in all independent polarization configuration. Subsequent lines add additional constraints, which apply cumulatively: ensemble measurement (over static disorder), the isotropic average of the signal, only the absorptive (real) part of the signal and finally performing the measurement at room temperature.

Normalized singular values from a singular value decomposition of the real valued pump-probe map given by Eq. (20) for the FMO complex under various conditions. The top line is from the spectra of a single monomer at 77 K, from the combination of measurements in all independent polarization configuration. Subsequent lines add additional constraints, which apply cumulatively: ensemble measurement (over static disorder), the isotropic average of the signal, only the absorptive (real) part of the signal and finally performing the measurement at room temperature.

Maximum amplitude over probe frequencies of the species associated spectra for a FMO monomer at 77 K for (a) the isotropic average and (b) each independent polarization configuration of the probe and local oscillator, including the ensemble average over static disorder. The labeling of each species matches that used in Figure 4 : all entries including and above the diagonal correspond to the real part of the matching density matrix element (in the excitonic basis), and all entries below the diagonal correspond to the imaginary part. The cartesian coordinates were chosen arbitrarily, matching those used in an assignment of the crystal structure.

Maximum amplitude over probe frequencies of the species associated spectra for a FMO monomer at 77 K for (a) the isotropic average and (b) each independent polarization configuration of the probe and local oscillator, including the ensemble average over static disorder. The labeling of each species matches that used in Figure 4 : all entries including and above the diagonal correspond to the real part of the matching density matrix element (in the excitonic basis), and all entries below the diagonal correspond to the imaginary part. The cartesian coordinates were chosen arbitrarily, matching those used in an assignment of the crystal structure.

## Tables

Summary of deconvolution performance over 1000 instances of simulated experimental noise. RMSE (root-mean-squared-error) is given by the sum of the absolute difference between the estimated and actual response functions, . Improvement is the multiple of the reduction in RMSE compared to the naive approach. Uncertainties indicate one standard deviation in the empirical distribution.

Summary of deconvolution performance over 1000 instances of simulated experimental noise. RMSE (root-mean-squared-error) is given by the sum of the absolute difference between the estimated and actual response functions, . Improvement is the multiple of the reduction in RMSE compared to the naive approach. Uncertainties indicate one standard deviation in the empirical distribution.

Regularization performance for different penalty operators and parameter selection techniques for 1000 instances of random noise with relative magnitude 10^{−2} or 10^{−3}. Numbers are the mean plus or minus one standard deviation. Improvement is the multiple of the reduction in mean-squared-error for the reconstructed response function using Tikhonov regularization over the error associated with the naive impulse-probe estimate.

Regularization performance for different penalty operators and parameter selection techniques for 1000 instances of random noise with relative magnitude 10^{−2} or 10^{−3}. Numbers are the mean plus or minus one standard deviation. Improvement is the multiple of the reduction in mean-squared-error for the reconstructed response function using Tikhonov regularization over the error associated with the naive impulse-probe estimate.

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