Ultrafast pump-probe phase-randomized tomography

Abstract

Measuring fluctuations in matter's low-energy excitations is the key to unveiling the nature of the non-equilibrium response of materials. A promising outlook in this respect is offered by spectroscopic methods that address matter fluctuations by exploiting the statistical nature of light-matter interactions with weak few-photon probes. Here we report the first implementation of ultrafast phase randomized tomography, combining pump-probe experiments with quantum optical state tomography, to measure the ultrafast non-equilibrium dynamics in complex materials. Our approach utilizes a time-resolved multimode heterodyne detection scheme with phase-randomized coherent ultrashort laser pulses, overcoming the limitations of phase-stable configurations and enabling a robust reconstruction of the statistical distribution of phase-averaged optical observables. This methodology is validated by measuring the coherent phonon response in α-quartz. By tracking the dynamics of the shot-noise limited photon number distribution of few-photon probes with ultrafast resolution, our results set an upper limit to the non-classical features of phononic state in α-quartz and provide a pathway to access non-equilibrium quantum fluctuations in more complex quantum materials.

Fig. 1. Ultrafast pump-probe spectroscopy of fluctuations measuring the quantum optical statistics of ultrashort laser pulses.

We investigate the possibility of accessing the system fluctuations distinguishing the effects that the light-matter interaction produces in the photon number distribution of weak probe pulses, altering the classical coherent state statistics

Fig. 2. Reconstruction of the photon number distribution with a phase-randomized pump-probe experiment.

a Experimental scheme. The signal and idler outputs of an optical parametric amplifier (OPA), such that ω_SIG = 2ω_IDL, are used in combination with a second harmonic generation process to set up a phase-averaged pump-probe heterodyne detection sensitive to the random laser CEP (details in text). b Wigner distribution of the phase-averaged coherent state resulting from the randomization of the CEP-dependent LO-probe phase. c The detection output is the distribution of the phase-averaged quadrature. d Applying the tomography procedure to the quadrature data we obtain the probe photon number distribution

Fig. 3. Pump-probe modulation of the photon number distribution induced by the coherent phonon excitation.

a Phase-averaged quadrature distribution for a probe pulse with a mean photon number of 13.8 photons per pulse. Left: histogram of the equilibrium phase-averaged quadrature distribution. Right: histogram map describing the time-resolved dynamics of the quadrature distribution. b Applying the tomography algorithm to the quadrature data we study the evolution of the photon number distribution. Equilibrium (left) and time-resolved evolution (right) are shown. We also consider a weaker probe beam with an average of 2.9 photons per pulse and report the relative evolution for the quadrature (c) and photon number (d) distributions. We observe in both cases the presence of coherent oscillations at the phonon frequency

Fig. 4. Phonon-dependent evolution of the photon distribution parameters.

a–c We simulate the optical response for coherent (a), thermal (b), and squeezed (c) phonon states. The top panel compares the average phonon displacement and its variance. The middle panel reports the optical response (average and variance of the photon number). The Mandel parameter Q (bottom) quantifies the deviations from the coherent state Poissonian statistics. d The experimental mean photon number and variance oscillate at the 4 THz phonon frequency, as shown by the Fourier Transform analysis of the positive times (insert). The data are consistent with the detector response Q_det (green line), which describes a Poissonian behavior (Q = 0) corrected considering the intensity-dependent classical excess noise (see Supplementary). The pink area accounts for the error calculated as the standard deviation of repeated measurements of Q (2σ)