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The use of energy conservation arguments is ubiquitous in understanding the process of high harmonic generation, yet a complete quantum optical description of photon exchange remained elusive. Here, we solve this gap in description by introducing the energy conserving subspace in high harmonic generation in which many photons of the driving field are absorbed to generate a single photon of higher energy. The presented solution to energy conservation in quantum optical high harmonic generation naturally results in highly entangled states of light with non-classical properties in their marginals. This further allows to explain recent experimental results for quantum state engineering, for which we provide analytical bounds on the fidelity with a high photon number optical cat state., Comment: 6 pages, 2 figures, comments are welcome

In the domain of quantum metrology, cat states have demonstrated their utility despite their inherent fragility with respect to losses. Here, we introduce noise robust optical cat states which exhibit a metrological robustness for phase estimation in the regime of high photon numbers. These cat states are obtained from the intense laser driven process of high harmonic generation (HHG), and show a resilience against photon losses. Focusing on a realistic scenario including experimental imperfections we opt for the case in which we can maximize the lower bound of the quantum Fisher information (QFI) instead of analyzing the best case scenario. We show that the decrease of the QFI in the lossy case is suppressed for the HHG-cat state compared to the even and odd counterparts. In the regime of small losses of just a single photon, the HHG-cat state remains almost pure while the even/odd cat state counterparts rapidly decohere to the maximally mixed state. More importantly, this translates to a significantly enhanced robustness for the HHG-cat against photon loss, demonstrating that high photon number optical cat states can indeed be used for metrological applications even in the presence of losses., Comment: 8 pages (6 figures), extended results

Squeezing of optical fields, used as a powerful resource for many applications, and the radiation properties in the process of high harmonic generation have thus far been considered separately. In this Letter, we want to clarify that the joint quantum state of all the optical field modes in the process of high harmonic generation is in general entangled and squeezed. We show that this is already the case in the simplest scenario of driving uncorrelated atoms by a classical laser light field. The previous observation of product coherent states after the high harmonic generation process is a consequence of the assumption that the ground state depletion can be neglected, which is related to vanishing dipole moment correlations. Furthermore, we analyze how the resulting quadrature squeezing in the fundamental laser mode after the interaction can be controlled and explicitly show that all field modes are entangled., Comment: 4 pages (2 figures)

High-harmonic generation has emerged as a pivotal process in strong-field physics, yielding extreme ultraviolet radiation and attosecond pulses with a wide range of applications. Furthermore, its emergent connection with the field of quantum optics has revealed its potential for generating non-classical states of light. Here, we investigate the process of high-harmonic generation in semiconductors under a quantum optical perspective while using a Bloch-based solid-state description. Through the implementation of quantum operations based on the measurement of high-order harmonics, we demonstrate the generation of non-classical light states similar to those found when driving atomic systems. These states are characterized using diverse quantum optical observables and quantum information measures, showing the influence of electron dynamics on their properties. Additionally, we analyze the dependence of their features on solid characteristics such as the dephasing time and crystal orientation, while also assessing their sensitivity to changes in driving field strength. This study provides insights into HHG in semiconductors and its potential for generating non-classical light sources., Comment: 22 pages (16 main text + 6 appendix), 7 figures (5 main text + 2 appendix)

The optical phase of the driving field in the process of high harmonic generation and the coherence properties of the harmonics are fundamental concepts in attosecond physics. Here, we consider to drive the process by incoherent classical and non-classical light fields exhibiting an undetermined optical phase. With this we introduce the notion of quantum optical coherence into high harmonic generation, and show that high harmonics can be generated from incoherent radiation despite having a vanishing electric field. We explicitly derive the quantum state of the harmonics when driven by carrier-envelope phase unstable fields and show that the generated harmonics are incoherent and exhibiting zero electric field amplitudes. We find that the quantum state of each harmonic is diagonal in its photon number basis, but nevertheless has the exact same photon statistics as the widely considered coherent harmonics. From this we conclude that assuming coherent harmonic radiation can originate from a preferred ensemble fallacy. These findings have profound implications for attosecond experiments and how to infer about the harmonic radiation properties., Comment: 5 pages, accepted version

The recent progress in the quantum optical formulation of the process of high harmonic generation has reached a point where the successful semi-classical model shows its limitations. So far the light source which drives the process was considered to be provided by a laser, in agreement with the classical picture. However, quantum optics allows to consider light fields beyond the classical realm, such as bright squeezed vacuum of photon number states. Both field states have vanishing mean electric field amplitudes, but can still lead to the generation of high harmonic radiation for sufficiently high intensities. This poses new questions about the range of validity of the semi-classical picture, and allows to extend the domain of questions which could possibly be asked., Comment: 3 pages, slightly extended version

The demanding experimental access to the ultrafast dynamics of materials challenges our understanding of their electronic response to applied strong laser fields. For this purpose, trapped ultracold atoms with highly controllable potentials have become an enabling tool to describe phenomena in a scenario where some effects are more easily accessible and twelve orders of magnitude slower. In this work, we introduce a mapping between the parameters of attoscience platform and atomic cloud simulators, and propose an experimental protocol to access the emission spectrum of high harmonic generation, a regime that has so far been elusive to cold atom simulation. As we illustrate, the benchmark offered by these simulators can provide new insights on the conversion efficiency of extended and short nuclear potentials, as well as the response to applied elliptical polarized fields or ultrashort few-cycle pulses., Comment: Accepted version of the manuscript

Optical Schr\"odinger "cat" states created by superpositions of coherent light states, correspond to an optical analog of the Schr\"odinger's cat in his $\textit{Gedankenexperiment}$. These non-classical light states are generated by means of quantum state engineering methods, and they are considered as one of the main resources for fundamental tests of quantum theory and the development of new quantum technologies. However, the power of existing optical "cat" state sources is limited by their low average photon number, which prevents their use in nonlinear optics. Here, we demonstrate the generation of a femtosecond duration optical "cat" state in the infrared spectral range, with mean photon number orders of magnitude higher than those delivered by current available sources. These states exhibit intensities sufficient to induce nonlinear processes in matter. This is shown using the process of second harmonic generation in an optical crystal, in which the infrared-frequency photons of an optical "cat" state are up-converted into blue-frequency photons. We create the light states driving the second harmonic generation process, by means of conditioning operations applied on the quantum state of an intense infrared femtosecond laser field after its nonlinear interaction with atoms. Due to the presence of quantum interference between the coherent states composing the optical "cat" state, the quantum properties of the state are imprinted in the measured second-order interferometric autocorrelation traces. The findings introduce the optical "cat" states into the realm of nonlinear quantum optics, opening up exciting new paths in quantum information science., Comment: 8 pages (5 main text + 3 Methods), 5 figures (3 in main text + 2 in Methods)

We start this short note by remembering the beginnings of the Warsaw School of Quantum Optics, evidently stimulated by Iwo Bialynicki-Birula at the Warsaw University, and then Centre for Theoretical Physics of Polish Academy of Sciences, and Adam Kujawski and Zofia Bialynicka-Birula at the Institute of Physics of Polish Academy of Sciences. In the theoretical approaches of the Warsaw School Quantum Field Theory was always present, and Quantum Optics was considered to be Applied Quantum Electrodynamics (QED). All of us who grew up in this fantastic community have carried and are still carrying the gospel to others. In particular, now QED began her run on the red carpet of Super Instense Laser Matter Interactions, Attosecond-physics, and Ultrafast Laser Physics, in general. We will elaborate on the recent progress in this direction, and on the open questions towards future investigations. This paper celebrates the 90th birthday of Prof. Iwo Bialynicki-Birula, our QED guru!, Comment: This article belongs to the special issue of Acta Physica Polonica A printed in honor of Professor Iwo Bialynicki-Birula on the occasion of his 90th birthday (Ed. Tomasz Sowinski, DOI:10.12693/APhysPolA.143.S0)

Strong laser physics is a research direction that relies on the use of high-power lasers and has led to fascinating achievements ranging from relativistic particle acceleration to attosecond science. On the other hand, quantum optics has been built on the use of low photon number sources and has opened the way for groundbreaking discoveries in quantum technology, advancing investigations ranging from fundamental tests of quantum theory to quantum information processing. Despite the tremendous progress, until recently these directions have remained disconnected. This is because, the majority of the interactions in the strong-field limit have been successfully described by semi-classical approximations treating the electromagnetic field classically, as there was no need to include the quantum properties of the field to explain the observations. The link between strong laser physics, quantum optics, and quantum information science has been developed in the recent past. Studies based on fully quantized and conditioning approaches have shown that intense laser--matter interactions can be used for the generation of controllable entangled and non-classical light states. This achievement opens the way for a vast number of investigations stemming from the symbiosis of strong laser physics, quantum optics, and quantum information science. Here, after an introduction to the fundamentals of these research directions, we report on the recent progress in the fully quantized description of intense laser--matter interaction and the methods that have been developed for the generation of non-classical light states and entangled states. Also, we discuss the future directions of non-classical light engineering using strong laser fields, and the potential applications in ultrafast and quantum information science., Comment: 60 pages, 20 figures. Comments are welcome

The development of sources delivering non-classical states of light is one of the main needs for applications of optical quantum information science. Here, we demonstrate the generation of non-classical states of light using strong-laser fields driving a solid-state system, by using the process of high-order harmonic generation, where an electron tunnels out of the parent site and, later on, recombines on it emitting high-order harmonic radiation, at the expense of affecting the driving laser field. Since in solid-state systems the recombination of the electron can be delocalized along the material, the final state of the electron determines how the electromagnetic field gets affected because of the laser-matter interaction, leading to the generation of entanglement between the electron and the field. These features can be enhanced by applying conditioning operations, i.e., quantum operations based on the measurement of high-harmonic radiation. We study non-classical features present in the final quantum optical state, and characterize the amount of entanglement between the light and the electrons in the solid. The work sets the foundation for the development of compact solid-state-based non-classical light sources using strong-field physics., Comment: We present a different formulation to that of the previous version, more in line with the approach followed in our previous works. 12 pages (8 main text + 4 Methods), 4 figures. Comments are welcome

Light-matter entanglement plays a fundamental role in many applications of quantum information science. Thus, finding processes where it can be observed is an important task. Here, we address this matter by theoretically investigating the entanglement between light and electrons generated in above-threshold ionization (ATI) process. The study is based on the back-action of the ATI process on the quantum optical state of the system, and its dependence on the kinetic energy and direction of the emitted photoelectrons. Taking into account the dynamics of the process, we demonstrate the creation of hybrid entangled states. The amount of entanglement has been studied in terms of the entropy of entanglement. Additionally, we use the Wigner function of the driving field mode to motivate the entanglement characterization when considering electrons propagating in opposite directions., Comment: 20 pages (13 main text + 7 supplementary material), 8 figures (6 main text + 2 supplementary material). Comments are welcome

Intense laser-matter interactions are at the center of interest in research and technology since the development of high power lasers. They have been widely used for fundamental studies in atomic, molecular, and optical physics, and they are at the core of attosecond physics and ultrafast opto-electronics. Although the majority of these studies have been successfully described using classical electromagnetic fields, recent investigations based on fully quantized approaches have shown that intense laser-atom interactions can be used for the generation of controllable high-photon-number entangled coherent states and coherent state superpositions. In this tutorial, we provide a comprehensive fully quantized description of intense laser-atom interactions. We elaborate on the processes of high harmonic generation, above-threshold-ionization, and we discuss new phenomena that cannot be revealed within the context of semi-classical theories. We provide the description for conditioning the light field on different electronic processes, and their consequences for quantum state engineering of light. Finally, we discuss the extension of the approach to more complex materials, and the impact to quantum technologies for a new photonic platform composed by the symbiosis of attosecond physics and quantum information science., Comment: 43 pages, 19 figures, Tutorial including new results

We report on the appearance of a structure at low energies in the photo-electron momentum distribution of the hydrogen atom exposed to two-color counter-rotating bi-circular laser fields. These structures, which arise due to AC-Stark shifted resonances, break the three-fold symmetry, typical for the $\omega-2\omega$ bi-circular fields. We discuss the physical origin of this structure in terms of partial-wave interference between direct ionization channels and a resonant pathway, that passes through the AC-Stark shifted state and show how the underlying Rydberg state population depend on the field strength and pulse duration., Comment: 7 pages

Quantum information science and intense laser matter interaction are two apparently unrelated fields. Here, we introduce the notion of quantum information theory to intense laser driven processes by providing the quantum mechanical description of measurement protocols for high harmonic generation in atoms. This allows to conceive new protocols for quantum state engineering of light. We explicitly evaluate conditioning experiments on individual optical field modes, and provide the corresponding quantum operation for coherent states. The associated positive operator-valued measures are obtained, and give rise to the quantum theory of measurement for the generation of high dimensional entangled states, and coherent state superposition with controllable non-classical features on the attosecond timescale. This establish the use of intense laser driven processes as a novel quantum technology platform for quantum information processing., Comment: 6 pages

Recently, intensely driven laser-matter interactions have been used to connect the fields of strong laser field physics with quantum optics by generating non-classical states of light. Here, we make a further key step and show the potential of strong laser fields for generating controllable high-photon-number coherent-state superpositions. This has been achieved by using two of the most prominent strong-laser induced processes: high-harmonic generation and above-threshold ionization. We show how the obtained coherent-state superpositions change from an optical Schr\"odinger "cat" state to a "kitten" state by changing the atomic density in the laser-atom interaction region, and we demonstrate the generation of a 9-photon shifted optical "cat" state which, to our knowledge, is the highest photon number optical "cat" state experimentally reported. Our findings anticipate the development of new methods that naturally lead to the creation of high-photon-number controllable coherent-state superpositions, advancing investigations in quantum technology., Comment: Revised version submitted to Physical Review A

We present a theoretical demonstration on the generation of entangled coherent states and of coherent state superpositions, with photon numbers and frequencies orders of magnitude higher than those provided by the current technology. This is achieved by utilizing a quantum mechanical multimode description of the single- and two-color intense laser field driven process of high harmonic generation in atoms. It is found that all field modes involved in the high harmonic generation process are entangled, and upon performing a quantum operation, leads to the generation of high photon number optical cat states spanning from the far infrared to the extreme-ultraviolet spectral region. This provides direct insights into the quantum mechanical properties of the optical field in intense laser matter interaction. Finally, these states can be considered as a new resource for fundamental tests of quantum theory, quantum information processing or sensing with non-classical states of light., Comment: 5 pages (3 figures) + 4 pages supplement (2 figures)

Recently, using conditioning approaches on the high-harmonic generation process induced by intense laser-atom interactions, we have developed a new method for the generation of optical Schr\"odinger cat states (M. Lewenstein et al., arXiv:2008.10221 (2020)). These quantum optical states have been proven to be very manageable as, by modifying the conditions under which harmonics are generated, one can interplay between $\textit{kitten}$ and $\textit{genuine cat}$ states. Here, we demonstrate that this method can also be used for the development of new schemes towards the creation of optical Schr\"odinger cat states, consisting of the superposition of three distinct coherent states. Apart from the interest these kind of states have on their own, we additionally propose a scheme for using them towards the generation of large cat states involving the sum of two different coherent states. The quantum properties of the obtained superpositions aim to significantly increase the applicability of optical Schr\"odinger cat states for quantum technology and quantum information processing., Comment: 13 pages (12 main text + 1 appendix), 8 figures. This paper has been submitted to the JCEL Special Issue on Wigner Functions in Computational Electronics and Photonics. Comments are welcome

Quantum-optical spectrometry is a recently developed shot-to-shot photon correlation-based method, namely using a quantum spectrometer (QS), that has been used to reveal the quantum optical nature of intense laser-matter interactions and connect the research domains of quantum optics (QO) and strong laser-field physics (SLFP). The method provides the probability of absorbing photons from a driving laser field towards the generation of a strong laser-field interaction product, such as high-order harmonics. In this case, the harmonic spectrum is reflected in the photon number distribution of the infrared (IR) driving field after its interaction with the high harmonic generation medium. The method was implemented in non-relativistic interactions using high harmonics produced by the interaction of strong laser pulses with atoms and semiconductors. Very recently, it was used for the generation of non-classical light states in intense laser-atom interaction, building the basis for studies of quantum electrodynamics in strong laser-field physics and the development of a new class of non-classical light sources for applications in quantum technology. Here, after a brief introduction of the QS method, we will discuss how the QS can be applied in relativistic laser-plasma interactions and become the driving factor for initiating investigations on relativistic quantum electrodynamics.

We quantify the disturbance of a quantum state undergoing a sequence of observations, and particularly focus on a weak measurement followed by post-selection and compare these results to the projective counterpart. Taking into account the distinguishability of both, the system and the device, we obtain the exact trade-off between the system state disturbance and the change of the device pointer state. We show that for particular post-selection procedures the coupling strength between the system and the device can be significantly reduced without loosing measurement sensitivity, which is directly transferred to a reduced state disturbance of the system. We observe that a weak measurement alone does not provide this advantage but only in combination with post-selection a significant improvement in terms of increased measurement sensitivity and reduced state disturbance is found. We further show that under realistic experimental conditions this state disturbance is small, whereas the exact post-selection probability is considerably larger than the approximate value given by the overlap of the initial and final state when neglecting the system state disturbance.

The physics of intense laser-matter interactions is described by treating the light pulses classically, anticipating no need to access optical measurements beyond the classical limit. However, the quantum nature of the electromagnetic fields is always present. Here, we demonstrate that intense laser-atom interactions may lead to the generation of highly non-classical light states. This was achieved by using the process of high-harmonic generation in atoms, in which the photons of a driving laser pulse of infrared frequency are up-converted into photons of higher frequencies in the extreme ultraviolet spectral range. The quantum state of the fundamental mode after the interaction, when conditioned on the high-harmonic generation, is a so-called Schr\"odinger cat state, which corresponds to a superposition of two distinct coherent states: the initial state of the laser and the coherent state reduced in amplitude that results from the interaction with atoms. The results open the path for investigations towards the control of the non-classical states, exploiting conditioning approaches on physical processes relevant to high-harmonic generation., Comment: We dedicate this work to the memory of Roy J. Glauber, the inventor of coherent states

Measurements on the strong-field ionization of carbonyl sulfide molecules by short, intense, 2~\um wavelength laser pulses are presented from experiments where angle-resolved photoelectron distributions were recorded with a high-energy velocity map imaging spectrometer, designed to reach a maximum kinetic energy of 500~eV. The laser-field-free elastic-scattering cross section of carbonyl sulfide was extracted from the measurements and is found in good agreement with previous experiments, performed using conventional electron diffraction. By comparing our measurements to the results of calculations, based on the quantitative rescattering theory (QRS), the bond lengths and molecular geometry were extracted from the experimental differential cross sections to a precision better than $\pm5$~pm and in agreement with the known values.

Recording molecular movies on ultrafast timescales has been a longstanding goal for unravelling detailed information about molecular dynamics. We present the direct experimental recording of very-high-resolution and -fidelity molecular movies over more than one-and-a-half periods of the laser-induced rotational dynamics of carbonylsulfide (OCS) molecules. Utilising the combination of single-quantum-state selection and an optimised two-pulse sequence to create a tailored rotational wavepacket, an unprecedented degree of field-free alignment, $\langle \cos^{2}{\theta_{2D}}\rangle=0.96$ ($\langle \cos^{2}{\theta}\rangle=0.94$) was achieved, exceeding the theoretical limit for single-pulse alignment. The very rich experimentally observed quantum dynamics is fully recovered by the angular probability distribution obtained from solutions of the time-dependent Schr\"odinger equation with parameters refined against the experiment. The populations and phases of rotational states in the retrieved time-dependent three-dimensional wavepacket rationalised the observed very high degree of alignment., Comment: 9 Figures