A new time-frequency method to reveal quantum dynamics of atomic hydrogen in intense laser pulses: Synchrosqueezing Transform

This study introduces a new adaptive time-frequency (TF) analysis technique, synchrosqueezing transform (SST), to explore the dynamics of a laser-driven hydrogen atom at an {\it ab initio} level, upon which we have demonstrated its versatility as a new viable venue for further exploring quantum dynamics. For a signal composed of oscillatory components which can be characterized by instantaneous frequency, the SST enables rendering the decomposed signal based on the phase information inherited in the linear TF representation with mathematical support. Compared with the classical type TF methods, the SST clearly depicts several intrinsic quantum dynamical processes such as selection rules, AC Stark effects, and high harmonic generation.

💡 Research Summary

The paper presents a pioneering application of the synchrosqueezing transform (SST), an adaptive time‑frequency (TF) analysis method, to the quantum dynamics of a hydrogen atom driven by an intense laser pulse. While TF techniques such as the Gabor (short‑time Fourier) transform, Morlet wavelet transform, and Wigner‑Ville distribution have been widely used in classical systems, their application to quantum systems has been limited due to intrinsic blurring, window‑induced artifacts, and insufficient frequency resolution.

The authors first outline the mathematical foundation of SST. Starting from a linear TF representation (either STFT or CWT) denoted R(t,ω), they extract the instantaneous phase information to define a reassignment rule ω_f(t,η)=−i∂tV_f(t,η)/(2πV_f(t,η)). This rule relocates the energy from the original frequency bin η to a more accurate instantaneous frequency ξ, using a Gaussian‑like kernel h(t). The method relies on the concept of intrinsic mode type (IMT) functions—signals with slowly varying amplitude and frequency—and on the space B{ε,d} of superpositions of well‑separated IMT components. Under these conditions, SST can mathematically guarantee the accurate recovery of each component’s instantaneous amplitude (IA) and instantaneous frequency (IF).

For the physical system, the authors solve the time‑dependent Schrödinger equation for a hydrogen atom in a linearly polarized laser field with wavelength 1064 nm, peak intensity 10¹³ W/cm², and a sin² envelope spanning 60 optical cycles. The wavefunction ψ(r,t) is obtained via a time‑dependent generalized pseudospectral method, and the induced dipole in the length gauge d_L(t) is computed. The power spectrum of d_L(t) shows only odd harmonics, as expected from parity symmetry, but the detailed substructure of each harmonic is not evident from a simple Fourier transform.

Applying the conventional TF methods, the authors observe that the Gabor and Morlet transforms produce broadened harmonic lines and artificial interference patterns, while the Wigner‑Ville distribution reveals spurious inter‑harmonic signals that violate parity. These artifacts obscure subtle physical effects such as near‑resonant absorption and AC Stark shifts.

When SST is applied, the TF representation becomes sharply resolved. In the vicinity of the ninth harmonic (H9), SST reveals a distinct line at 8.756 ω₀, corresponding precisely to the 1s→2p transition energy (ΔE = ½(1−1/2²) in atomic units). As the laser intensity rises, this line shifts to 8.700 ω₀, reflecting the AC Stark effect that lifts and splits the 2p sublevels. The authors corroborate these shifts with Floquet calculations of the dressed‑state energies for 1s→2s, 1s→2p_x, 1s→2p_z, and 1s→2p_y transitions, confirming that only the 1s→2p_z transition is allowed under the chosen z‑polarization (selection rules). Moreover, SST detects a weak line at 10.38 ω₀, attributable to the 1s→3p transition, which is barely visible in the other TF representations. At the end of the pulse, SST still resolves residual lines at 8.756 ω₀, 10.377 ω₀, and 12.07 ω₀, the latter possibly arising from higher‑order state superpositions.

The study demonstrates that SST eliminates the intrinsic blurring of linear TF methods, provides superior frequency resolution, and faithfully captures quantum‑mechanical phenomena such as selection rules, AC Stark shifts, and high‑order harmonic generation. By directly revealing instantaneous frequencies of the induced dipole, SST offers a powerful diagnostic tool for strong‑field physics, attosecond spectroscopy, and potentially for other quantum dynamical contexts like nuclear magnetic resonance or ultrafast chemical reactions. The authors anticipate that SST will stimulate further research into the detailed optical processes governing light‑matter interaction at the quantum level.