In the context of previous research in strong-field physics, the laser field has frequently been regarded as a classical field, largely overlooking the quantum effects of the field. The advent of specific quantum optical technologies has resulted in the emergence of quantum light sources that meet the requisite standards for strong-field physics. Consequently, there is a necessity to consider the quantum effects of the laser field. Recent studies have demonstrated that strong-field processes driven by quantum optical sources exhibit new phenomena. For example, high harmonic driven by squeezed state light is squeezed light. However, there are numerous phenomena related to strong-field physics under quantum optical drive remain unexplored, and significant gaps in our understanding of the quantum effects involved persist. This study examines the momentum distribution and energy spectrum of hydrogen atoms under coherent state light (quasi-classical light) and bright squeezed vacuum state light (quantum light lacking a classical counterpart), providing a comprehensive investigation of the phenomenon of above-threshold ionization.

In the interaction between atoms and quantum light, the density matrix of the atom-quantum light system satisfies the complete quantum time-dependent Schrödinger equation. The linear nature of the density matrix, as dictated by the Schrödinger equation, allows any quantum light field to be decomposed into a linear combination of coherent states. This decomposition is achieved by transforming the Schrödinger equation into a summation of solutions corresponding to coherent state light. Consequently, the solution of the density matrix is a non-coherent superposition of the solutions of the Schrödinger equation for coherent state light. By separately solving the Schrödinger equation for each coherent state, the Schrödinger equation for the density matrix of the atom-quantum light system can be indirectly obtained. This approach effectively employs numerical methods to explore the three-dimensional time-dependent Schrödinger equation and investigate phenomena such as strong-field ionization of hydrogen atoms under the influence of a quantum light field.

We demonstrate the broadening effect observed in the photoelectron spectrum under the influence of bright squeezed vacuum state light. It combines the quantization of the light field with solutions derived from the time-dependent three-dimensional Schrödinger equation for hydrogen atoms (Figs. 2 and 5). Additionally, this work discusses and elucidates this phenomenon (Figs. 4 and 6). Our investigation delves into how the quantum properties of the light field impact the interference structure observed in the photoelectron momentum distribution. Our findings highlight that when exposed to bright squeezed vacuum state light, the photoelectron energy spectrum exhibits higher cutoff energy compared to coherent state light. Moreover, the photon statistics of the quantum light field have a notable impact on the interference patterns of the photoelectrons (Figs. 3 and 7). Specifically, the intra-cycle interference fringes are diminished under the influence of bright squeezed vacuum state light, while the inter-cycle interference, known as the above-threshold ionization ring, persists. Remarkably, the holographic interference of photoelectrons, characterized by interference fringes between directly ionized electrons and forward re-scattered electrons, remains observable. We enhance our comprehension of the quantum effects induced by optical fields on strong-field ionization processes. Moreover our research holds the promise for providing additional insights into imaging atomic and molecular structures, as well as probing ultrafast dynamics through strong-field ionization.

Our research aims to analyze disparities in the photoelectron energy spectra and momentum distributions of hydrogen atoms driven by light at 400 nm and 800 nm. Regardless of the wavelength, the photoelectron energy spectra driven by BSV light exhibit a broader distribution compared to those driven by coherent state light. This is primarily due to the broader quasi-probability distribution of BSV light compared to coherent state light. Furthermore, under 400 nm coherent state light, the momentum distribution of the photoelectrons reveals interferences caused by inter-cycle channel (ATI), which is preserved by BSV light. In the 400 nm case, the ATI in BSV light is attributed to the narrower final ionization amplitude distribution, which reduces the level of smearing of the periodic interferences. Conversely, under 800 nm coherent state light, the momentum distribution of the photoelectrons exhibits multiple structures. In contrast, BSV light preserves only the fork-like structure while other interference patterns are attenuated. This observation underscores the quantum nature of the light field and enhances comprehension of the strong-field ionization process within the domain of quantum optics.