Ultrafast quantum optical system and method for communication

Abstract

Methodology of generation, with the use of a degenerate four-wave mixing nonlinear process, of ultrafast synthesized quantum light pulses with attosecond resolution (as demonstrated - at petaHertz and sub-petaHertz frequencies) and that exhibit amplitude squeezing. Demonstration of controllability and tunability of quantum uncertainty of light in real time via switching between amplitude and phase squeezing. Example of an attosecond quantum encryption protocol leveraging squeezed synthesized light for secure digital communication at petahertz-speeds.

Description

International Patent Application Attorney Docket No.: 122170.00263(UA25-152)ULTRAFAST QUANTUM OPTICAL SYSTEM AND METHOD FOR COMMUNICATIONCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of US Provisional Patent Application No.63 / 730,562 filed on December 11, 2024 and titled “Ultrafast Quantum Optics and Communication”, the entire disclosure of which is incorporated by reference herein.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under Grant No. FA95502210494 awarded by the Air Force Material Command. The government has certain rights in the invention.RELATED ART

[0003] Over the past few decades, significant advances in quantum optics (particularly the generation of quantum light, see for example Loudon, R. et al., in Journal of Modern Optics 34, 709-759, 1987 or Qin, W. et al., in Physics Reports 1078, 1-59, 2024) have played a pivotal role in enhancing the sensitivity of gravitational wave measurements by the Laser Interferometer Gravitational-Wave Observatory (LIGO) 10. (See The LIGO Scientific Collaboration et al.Advanced LIGO. Classical and Quantum Gravity 32, 074001, 2015). Concurrently, the evolution of ultrafast science, attosecond physics, and the development of cutting-edge tools — such as femtosecond high-power lasers and XUV pulses — have spurred new frontiers in both fundamental research and technological applications. These breakthroughs, acknowledged by the Nobel Prizes in Physics in 2018 and 2023, have enabled detailed investigations into light-matter interactions at the classical level, thereby providing profound insights into molecular, atomic, and electron dynamics. For instance, attosecond science has leveraged the high-harmonic generation (HHG) to probe electron motion on timescales as short as attoseconds (Corkum, P. B. et al., in Nature Physics 3, 381-387, 2007; Luu, T. T. etal., in Nature 521, 498-502, 2015, to name just a few). Recent experimental and theoretical studies have explored the impact and generation of squeezed light on the HHG process, revealing that the underlying physics of HHG with squeezed light deviates significantly from classical descriptions (see, for example, Rasputnyi, A. et al., in Nature Physics,1QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)20, 1960-1965, 2024). Theoretical models predict shifts in both the electronic trajectories driven by the field and the emitted photon energy compared to those observed under coherent laser fields. Experimental demonstrations of HHG in solid-state materials using squeezed light have also shown promising results, yielding higher photon fluxes. Additionally, the control of electron and XUV photon statistics through different quantum light modes has been reported, suggesting the potential for tailored quantum control in ultrafast processes. These findings raise critical questions: How do light-matter interactions and ultrafast phenomena evolve when triggered by squeezed light? How do the physics of ultrafast nonlinear optics differ when using quantum light? Addressing these questions calls for novel not yet existing tools — specifically, the ones generating and / or utilizing ultrafast quantum light pulses — that can offer a new window into ultrafast science through the lens of quantum physics.SUMMARY OF THE INVENTION

[0004] Embodiments of the invention discussed herein present a novel approach for generating ultrafast quantum-synthesized few-light-cycle pulses produced via a degenerate nonlinear four-wave mixing process. The resulting quantum light pulse exhibits intensity squeezing at the cost of phase uncertainty, as confirmed by both experimental and theoretical analyses. With a pulse duration of several femtosecond (in one illustrated example - 5.3 fs), these pulses represent a promising tool for a range of advanced applications, including ultrafast quantum optics and spectroscopy aimed at exploring light-matter interactions at the quantum level. Moreover, the amplitude squeezing inherent in these pulses enhances the signal-to-noise ratio, surpassing the performance of classical light sources (Lemieux, S. et al., in Nature Photonics 19, 767-771, 2025). with promising applications in ultrafast quantum spectroscopic studies of biological samples.

[0005] As the skilled person will readily appreciate, the discussed approach allows the user to control the temporal and spectral properties of the squeezed light, thereby enabling the probing of the real-time dynamics of the squeezed light with attosecond resolution. The discussed experimental results reveal that the uncertainty in amplitude-squeezed light is dynamic, that is varying with the system's state and its interactions. This ability to manipulate and observe the change of uncertainty in real time opens new venues for exploring the fundamental nature of quantum mechanics and its implications for quantum control-based technologies.2QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0006] As an initial demonstration of the potential significance of the demonstrated quantum synthesized light pulses, an embodiment of the secure ultrafast quantum communication scheme it presented, which leverages the digital encoding on ultrafast waveforms. Beyond the inherent security of squeezed light in quantum communication, the demonstrated synthesized quantum pulses introduce an additional layer of digital encryption where the data are carried on the squeezed ultrafast light waveform in the binary format of “ones and zeros” (1 & 0).

[0007] Embodiments of the invention provide an apparatus that includes an optical arrangement or device and specific programmable electronic circuitry or processor operably connected with such optical arrangement or device. The optical arrangement is configured or structured (a) to receive, at an input of the arrangement, an input beam of coherent light that carries an input pulse of the coherent light (such light having an input spatial distribution and including the coherent light in multiple spectrally distinct from one another spectral bands) and (b) to transform this input beam into multiple intermediate spatial distributions of the coherent light that are spatially distinct from one another and (c) to direct each of the so-formed multiple intermediate spatial distributions of the coherent light towards a predetermined location to have it spatially overlap with every other of the multiple intermediate spatial distributions and to interact with a medium present at the predetermined spatial location to generate a pulse of squeezed quantum light containing quantum light squeezed in a chosen quadrature. The programmable electronic circuitry or processor is operably cooperated with a tangible non-transitory storage medium that contains program code, which the program code (when loaded to the programmable electronic circuitry) enables or governs the electronic circuitry to perform specific actions or operations. Such specific actions or operations include at least the following: based on information characterizing the pulse of squeezed light received by the electronic circuitry, to generate first indicia representing uncertainties of at least one of a phase and an intensity of the squeezed quantum light and second indicia representing a temporal profile of an electric field and / or a temporal profile of an intensity of the pulse of the squeezed quantum light. In at least one specific implementation, the optical arrangement may include a light mask spatially structured or configured to transform the input spatial distribution into the multiple intermediate spatial distributions of the coherent light that have substantially equal light powers and / or the optical arrangement may be configured or structured to generate the pulse of the squeezed quantum light containing the quantum light squeezed in a phase quadrature.3QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0008] Alternatively or in addition - and substantially in every implementation of the apparatus - the optical arrangement may include a reflector that is disposed between a first location at which the input spatial distribution of the coherent light is transformed into the multiple intermediate spatial distributions of the coherent light and the predetermined spatial location; and that includes first and second spatially distinct from one another mirrors structured to move with respect to one another. Optionally, an embodiment of the apparatus may have the optical arrangement to include a light-collecting optical system that contains at least an optical detection device (such optical detection device being positioned and / or oriented or disposed to receive only the pulse of the squeezed quantum light and to produce an image representing an optical parameter of the pulse of the squeezed quantum light) while the programmable electronic circuitry is configured to acquire an output containing said image (or imaging data representing such image) from the light-collecting optical system. Substantially in every implementation of the apparatus, the optical arrangement may be structured to carry out or implement a four- wave mixing process and / or to have the medium (at the predetermined spatial location) that includes a piece of substantially non-electrically-conducting material. Optionally - and substantially in every implementation of the apparatus - the program code may be configured to govern the electronic circuitry to generate indicia of acquisition of an optical field at an optical detection device if and / or when intensity of the optical field of the squeezed quantum light exceeds a pre-determined optical signal threshold. (At least in the specific case of the latter, the program code may be configured to govern the electronic circuitry to define the optical signal threshold to be registered with a light-collecting optical system that includes at least an optical detection device disposed to receive only said pulse of the squeezed quantum light and to produce an image representing an optical parameter of said pulse of the squeezed quantum light, while the optical signal threshold may be defined such that indicia of acquisition is generated only if and / or when an intensity of an optical field of the squeezed quantum light acquired at the light-collecting system exceeds such optical signal threshold.) Alternatively or in addition - and substantially in every implementation - the apparatus may be configured to generate the squeezed quantum light in a form of a train of light pulses in a petaHertz frequency range. Optionally, an embodiment of the apparatus may include a source of light configured to generate the input beam of coherent light carrying the input light pulse having a femtosecond duration or a sub-femtosecond duration, and / or to be configured or structured such that a number of the multiple intermediate input beams is equal to a number of distinct spectral bands. (Here, the 4QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)source of light may include a multi -spectral channel light field synthesizer configured to generate respective beams of the coherent light carrying corresponding pulses of light in each of the multiple distinct spectral bands and to form the input beam by substantially collinearly overlapping the respective beams spatially and the corresponding pulses - temporally. Optionally, the source of light may be further configured to introduce a delay between the corresponding pulses of light in each of the multiple distinct spectral bands.)

[0009] Embodiments of the invention additionally provide a method that includes at least the following steps: receiving the input beam of the coherent light at the optical arrangement of (substantially any embodiment) of the apparatus alluded to above; irradiating a target of substantially non-electrically-conducting material, disposed at the pre-determined spatial location, with multiple pulses of the coherent light carrying substantially equal light pulses and independently and substantially contemporaneously delivered thereto along spatially distinct and substantially parallel to one another optical paths to generate said pulse of the squeezed quantum light; and, with the programmable electronic circuitry of the embodiment of the apparatus, at least generating the first indicia representing uncertainties of each of the phase and the intensity of the squeezed quantum light. An embodiment of the method may additionally include a step of transforming the input beam into multiple intermediate spatial distributions of the coherent light that are spatially distinct from one another, while the step of irradiating includes directing each of the multiple intermediate spatial distributions of the coherent light to the pre-determined spatial location.Alternatively or in addition - and substantially in every implementation - the method may include a process of performing (with the programmable electronic circuitry) one or more of the following steps: (a) governing an optical detector of the apparatus to acquire the squeezed quantum light to identify the temporal profile of the electrical field thereof and / or the temporal profile of the intensity thereof, (b) generating the second indicia representing the temporal profile of the electrical field and / or the temporal profile of the intensity, and (c) generating third indicia representing an amplitude of the electrical field at the optical detector when said amplitude exceeds a predetermined signal threshold. Further, substantially every embodiment of the method may include one or more of the following steps: (i) delivering only the squeezed quantum light to an optical detector of the apparatus at least in part by spatially filtering out the squeezed quantum light from other light transmitted through the target, and (ii) determining a goal degree of squeezing of the squeezed quantum light based at least on spectral interference between temporally delayed with 5QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)respect to one another first and second pulses of the coherent light that contain light in respective different spectral bands of the multiple distinct spectral bands. Optionally, substantially every embodiment of the method may additionally include interpreting the generating of the third indicia by the programmable circuitry of the apparatus as one of "0" and "1" of a binary number system. Optionally (and, in particular, when then irradiating the target produces multiple pulses of the squeezed quantum light that include quantum light squeezed in the chosen quadrature, an embodiment of the method may include a step of transmitting a communication beam of light containing the multiple pulses of the squeezed quantum light from an output of the apparatus configured to generate the squeezed quantum light towards a receiver. (If this is the case, the embodiment of the method may also include one or more of (i) altering a degree of squeezing of the squeezed quantum light received at the receiver as compared with the goal degree of squeezing by interacting with the communication beam of light at an intermediate location between said output and the receiver; and (ii) when such interacting includes forming secondary beam of light by redirecting a portion of the communication beam of light away from the receiver at the intermediate location, producing a redirected communication beam of light that carries such portion of the communication beam and that possesses a degree of squeezing different from the target degree of squeezing. Further, optionally, the method may include (a) with the use of an electronic circuitry of the receiver, issuing an alert signal representative of interaction with the communication beam at an intermediate location between said output and the receiver; and / or (b) ceasing the transmitting from the output of the apparatus in response to the alert signal.)

[0010] Embodiments of the invention additionally provide a method that includes delivering information by passing, between a transmitter system and a receiver system, a communication beam of light carrying pulses of squeezed quantum light with pulse frequency at least in a petaHertz range (here, (a) a degree of squeezing of light, of the pulses of squeezed quantum light, that is received at the receiver system is varied, from a planned degree of squeezing of such light that has been achieved in a chosen quadrature at the transmitter system, by interacting with the communication beam of light performed at an intermediate location between the transmitter system and the receiver system, and / or (b) when such interacting includes forming a secondary beam of light carrying pulses of squeezed quantum light by redirecting a portion of the communication beam of away from the receiver system at the intermediate location, producing a redirected communication beam of the light that carries the portion of the communication beam and 6QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)that possesses a degree of squeezing different from the planned degree of squeezing). An embodiment of the method may further include 1) a step of sharing the threshold value and the planned degree of squeezing between the transmitter system and the receiver system and / or 2) determining the degree of squeezing that is received at the receiver system to obtain a received degree of squeezing at least in part by measuring stability of intensity of light of the pulses of squeezed quantum light between different of said pulses, and maintaining the transmitter system being aware of the received degree of squeezing and / or generating an alert signal if the received degree of squeezing substantially differs from the planned degree of squeezing of the light received, at the receiver system, from the transmitter system. (At least in the latter case, the implementation of the method may additionally include a step of ceasing or stopping the process of passing the communication beam at the transmitter system if and / or when the received degree of squeezing substantially differs from the planned degree of squeezing.) Alternatively or in addition - and substantially in every implementation - the method may contain a step of generating the pulses of squeezed light from a first portion of a coherent source light carrying light pulses in each of multiple pre-determined distinct spectral bands (where the light pulses in all of the multiple pre-determined distinct spectral bands are substantially temporally overlapped), and determining an intensity uncertainty of the squeezed light by measuring a variance of an overall intensity of a pulse of squeezed quantum light with the use of a first optical detection system, and determining an intensity uncertainty of the coherent source light by measuring a variance of an overall intensity of a light pulse of the coherent source light with the use of a second optical detection system, and generating indicia of amplitude squeezing of the squeezed light when and / or if the intensity uncertainty of the squeezed light is smaller than the intensity uncertainty of the coherent source light.

[0011] Embodiments of the invention further provide a computer program product (for digital data encoding and quantum communication) that includes a computer usable tangible non-transitory storage medium having computer readable program code therein such that the computer readable program code contains program code(s) configured to implement substantially every step of substantially every implementation of embodiments of the method(s) alluded to above. As a nonlimiting example, the computer readable program code may include one or more of the following: (i) program code for determining an actual degree of squeezing of pulsed squeezed quantum light, generated at least at a petaHertz pulse rate at a first location from a coherent source light carrying light pulses in each of multiple pre-determined distinct spectral bands, wherein the light pulses in all 7QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)of the multiple pre-determined distinct spectral bands are substantially temporally overlapped; (ii) program code for transmitting information by passing a communication beam of light carrying said pulsed squeezed quantum light and the actual degree of squeezing from the first location to a second location; and (iii) program code for ceasing said transmitting in response to having received, at the first location from the second location, indicia of the information having been received at the second location with a non-compliant degree of squeezing (here, the non-compliant degree of squeezing differs from the actual degree of squeezing). Alternatively or in addition - and again, in a non-limiting example - the program code for determining the actual degree of squeezing may include program code for determining the actual degree of squeezing by measuring stability of intensity of light of pulses of the pulsed squeezed quantum light with the optical detection system.BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The invention will be more fully understood by referring to the following Detailed Description of Specific Embodiments in conjunction with the Drawings, of which:

[0013] FIGs. 1A, 1B, 1C, 1D, and 1E address an embodiment of methodology of generation of ultrafast squeezed light pulse synthesized with attosecond resolution. FIG. 1A:Schematic of an embodiment of the overall apparatus of the invention that includes, at the input, a light field synthesizer (LFS) setup that possesses three spectral channels (and supporting the respectively corresponding to such channels light pulses) and that is configured to generate synthesized waveform pulses. The output beam carrying the output pulse is delivered to the two-arm light-transforming device or portion of the apparatus (that includes the arms 150A and 150B) and split into two beams: one is a classical (or classic) coherent light pulse propagating in arm 150A of the of the light-transforming device portion (and serving as a reference), and the second beam directed to undergo a four- wave mixing process in a SiO₂ sample to generate a squeezed light pulse in the arm 150B of the light-transforming device or portion. The phase and intensity quadrature uncertainties for each of these beams were measured using spectrometers 1 and 2. FIGS.1A, 1C: Spectra of the broadband classical light pulse (FIG. IB) and the spectra of light propagating in its constituent LFS channels (FIG. 1C). FIG. ID: Spectrum of the generated squeezed light pulse. FIG. IE: Spectra of the squeezed light generated by the three LFS pulses. Insets in FIGs. 1C, IE show the spectral interference fringes formed by light portions propagating in the LFS channels for both the classical and squeezed light pulses, respectively.8QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0014] FIGs. 2A, 2B, 2C, 2D present results of measurements of squeezed light phase and intensity uncertainties. Measured phase (FIG. 2A) and intensity (FIG. 2B) uncertainties between the near-IR and visible pulses of the LFS, retrieved by averaging 2800 spectra for classical light (FIGs.2AI, 2BI) and squeezed light (FIGs. 2aII, 2BII). FIGs. 2C, 2D: Results of corresponding phase and intensity uncertainty measurements for the visible and ultraviolet pulses of the LFS, presented in the same order as in FIGs. 2 A and 2B.

[0015] FIGs. 3A, 3B, 3C, and 3D: Wigner functions of the ultrafast squeezed light pulses. FIGs. 3 A, 3B: Results of optimization for the IR-VIS and UV-VIS datasets as a function of the number of modes. The dashed curves indicate the absolute error between the theoretical and experimental variances for intensity (curves A) and phase (curves B). For reference, the experimental variances are represented by corresponding solid lines. FIGs. 3C, 3D: The Wigner functions of the resulting squeezed states for the IR-VIS and UV-VIS measurements (for N = 1)), respectively. In these plots, x (abscissa) and p (ordinate) represent the optical quadrature, and for representational purposes, the distributions have been rotated by 45° relative to the p-axis.

[0016] FIGs. 4A, 4B: Ultrafast uncertainty dynamics in real time. FIG. 4A: The measured amplitude uncertainty at different delay τ times between the three input photons in the FWM nonlinear process (all are on the same scale). FIG. 4B: The amplitude uncertainty as evolves in time. The measured amplitude standard deviation “uncertainty” is shown in black dotes connected with a dashed line 410 (the smoothing is presented in dashed red line as a guide to the eye).

[0017] FIGs. 5A, 5B: Ultrafast synthesized squeezed light pulse. FIG. 5 A: Retrieved electric field; FIG. 5B: intensity temporal profile of the ultrafast squeezed light pulse generated by the four-wave mixing nonlinear process of the superimposed three spectral channels of the LFS.

[0018] FIG. 6 & FIG. 6 (continued): Petahertz digital-encoded secure quantum communication. A: Alice encodes digital data onto squeezed synthesized light waveforms (panels al-alll) using the LFS. These waveforms have predefined amplitude thresholds, indicated by the line THR: signals above the threshold represent “one” (1), and signals below represent “zero” (0). The encrypted data are carried by ultrafast squeezed light pulses, which Alice sends to Bob. Alice also shares with Bob the squeezing degree between channels, the predefined threshold, and the encryption key. B: Upon receiving the data, Bob first checks the squeezing level to confirm the security of the communication. Bob then decodes the data by sampling the waveform of the synthesized squeezed light pulses. C: Eva attempts to intercept the data using an optical9QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)beamsplitter, for example, inserted across the beam of light carrying the data from Alice to Bob. Eva’s intervention alters the squeezing level, alerting Alice and Bob to potential tampering. Since Eva does not know either the predefined threshold or the data decoding key, Eva’s attempt to decode the information produces erroneous results (results in faulty decoding), as shown in panels cl-clll.

[0019] FIGs. 7A, 7B, 7C: Amplitude squeezing of ultraviolet (UV), visible (VIS), and near-infrared (NIR) pulses in LFS. Comparison of intensity stability between coherent and squeezed light for (FIG. 7A) ultraviolet, (FIG. 7B) visible, and (FIG. 7C) near-infrared pulses.

[0020] FIG. 8 refers to the Shot noise measurement. The measured ΔI standard deviation as a function of the coherent light (diamond points connected with line “C”) and amplitude squeezed light (red circle pointed connected with line “S”).

[0021] FIGs. 9A, 9B, and 9C: Illustrates parameters of phase squeezed light. FIG. 9A (panels I and II): Measured phase uncertainty for coherent and phase squeezed light, between the near-IR and visible pulses of the LFS, retrieved by averaging 2000 spectra, respectively. FIG. 9B (panels I and II): Results of the intensity uncertainty measured for coherent, and phase squeezed light, respectively, corresponding to panels I and II of FIG. 9A. FIG. 9C: The plot of amplitude uncertainty of the phase squeezed light as it evolves in time. The measured standard deviation (uncertainty) of the amplitude is shown in black dotes, with the smoothed dependency illustrated with a dashed line 910 as a guide to the eye).

[0022] FIGs. 10A, 10B, 10C, and 10D: The temporal profiles of the LFS channels and output pulses. The intensity profile and phase measured by the FROG approach for Ultraviolet (FIG. 10A), Visible (FIG. 10B), Near-infrared (FIG. 10C), and Output pulses from the LFS (FIG. 10D). The corresponding spectral phase is presented in the same graphs.

[0023] FIGs. 11 A, 11B The dark intensity noise of the spectrometer devices of the set-up of FIG. 1 A. The measured intensity dark noise level of Spectrometer #1 (FIG. 11 A, and Spectrometer #2 (FIG. 11B) used to measure the intensity stability of the classical light and the squeezed lights presented in FIGs. 2A-2D. The range of abscissa values is the same as in FIGs. 2-2D. The insets are “zoom in” to show the measured dark-noise level on increasing scale.10QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)DETAILED DESCRIPTION

[0024] In accordance with non-limiting embodiments of the present invention, methods and apparatus are disclosed for generation of pulsed squeezed quantum light with a pulse rate of at least petaHertz or higher, and the use of same in a secured quantum communication scheme.

[0025] As known in related art, the term "classical light" refers to light described by classical physics, where light is treated as a continuous electromagnetic wave (that is, oscillating electric and magnetic fields propagating through space) that has properties such as wavelength and amplitude and without considering its quantized nature, thereby allowing for explanations of phenomena such as interference and diffraction using wave-based calculations. Classical light is associated with highly coherent light sources such as lasers, where all photons have a consistent phase relationship thereby allowing for predictable wave-like behavior. In comparison, the term “quantum light” most generally refers to light considered from a quantum mechanical perspective, meaning it is described as being composed of individual particles (photons), each carrying a discrete amount of energy and exhibiting both wave-like and particle-like properties and capable of displaying quantum phenomena such as superposition and entanglement.

[0026] Harnessing the quantum properties of light for ultrafast applications requires the ability to generate squeezed light pulses with precise temporal and spectral control. This disclosure for the first time, to the best knowledge of the inventors, addresses practical demonstration of fewcycle squeezed light pulses in the ultraviolet, visible, and near-infrared (NIR) spectral ranges.Non-limiting Example of an Experimental Setup

[0027] Embodiments of the proposed methodology are based on a degenerate four-wave mixing (FWM) process in a nonlinear crystal in the system 100 that utilizes an embodiment of the light field synthesizer apparatus 104 (referred to herein as LFS, some practical examples of which were discussed, for example, in US Patent 12,092,519 or by Hui, D. D. et al., in Nature Photonics 16, 33-37, 2022, each of the disclosures of which is incorporated herein by reference) and the lighttransforming or light-squeezing portion or apparatus or optical arrangement 150 (which, in turn, includes a reference arm 150A and the light-squeezing arm or device 150B). In comparison with the embodiment of a four-channel LFS discussed in US 12,092,519, for example, the LFS 104 of11QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)FIG. 1 A is a three-spectral channel LFS that processes the input coherent broadband beam of light 108 characterized by supercontinuum spectrum (as further discussed below in the section on Materials and Methods (Additional)') into the output beam 112.

[0028] The output beam 112 was further split into two portions 112A and 112B, the former of which was opted out as a coherent reference beam that was focused (with lens 114) onto and measured with the Spectrometer #1 in the first output arm 150A of the light-transforming device 150 of the system 100. The portion 112B of the light of the output beam 112, however, was further directed into a second output arm 150B of the device 150, where it was spatially reconfigured into multiple intermediate spatial distributions of light that are distinct from one another and each of which is directed towards a predetermined location to spatially overlap with every other of such multiple intermediate spatial distributions and to interact with a medium present at such predetermined location to generate a pulse of squeezed quantum light containing quantum light squeezed in a chosen quadrature. In reference to the specific non-limiting example of the systemlOO of FIG. 1A, a light mask 116 (for example, a substantially optically opaque screen with multiple holes in it - in this example, three holes forming a three-hole light mask) to divide the beam 112B into three substantially identical spatially distinct beams of light 120 A, 120B, 120C that propagate substantially colinearly towards a reflector 124 that is formed by two D-shaped curved focusing mirrors (at least one of which is controlled by a highly precise piezoelectric delay stage, to be repositionable independently from one another; see, for example, the relevant portion of US Patent 12, 174, 116, in particular that pertaining to FIG. 2 and delay unit 240, the disclosure of which patent is incorporated by reference herein). The reflector 124 focuses the beams 120A - 120C onto a target 130 structured in this example as an approximately 100 μm-thick SiO₂ substrate, where the FWM optical signal is generated. The embodiment of the system 100 is configured to maximize the phase matching between the three beams 120A, 120B, 120C which can be controlled by changing the angle of incidence of these beams on the target 130 (by rotating the SiO₂ target). Such arrangement enables generation of an FWM signal 136 from few-light-cycle long laser pulses carried by respectively corresponding beams 120A, 120B, and 120C that span three different spectral regions: near-infrared (NIR, 1000-690 nm), visible (VIS, 715-500), and ultraviolet (UV, 515-350 nm). The generated FWM signal 136 is further spatially isolated with a spatial filter 140 such as, for example, a one-hole light mask and the conversion efficiency was estimated to be on the order of 0.1%, depending on the nonlinear material of the target 130, its thickness, and intensity of 12QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)the beam(s) incident onto the target 130. As the skilled artisan will readily appreciate from the further disclosure, the pulses of the so-generated FWM signal 136 (shown being redirected with a reflector 144 towards the focusing lens 148 and then to the Spectrometer #2) exhibit squeezing in the amplitude or phase quadratures, which can be controlled easily in our setup light squeezer setup by changing the phase matching condition (by changing the tiling angle of the nonlinear medium). The combination of the components 116, 124, 140 of the arm 150B of the system 100 can be interchangeably referred to herein as a “light squeezer”.

[0029] To verify the quantum squeezing characteristics of the generated pulses 136, a new metrology approach was developed that differs from that conventionally used in related art for continuous wave (CW) lasers. According to the idea of the invention, an embodiment of a method enabling the verification of quantum squeezing characteristics involved measuring the spectral interference between temporally delayed pulses of light (NIR vs VIS and VIS vs UV, for example) to extract the phase uncertainty (ΔΦ). Additionally, the intensity uncertainty (Al) was determined by evaluating the variance in the overall spectral intensity of the squeezed light pulses 136, which was then compared to that of the coherent light 112 A.

[0030] To this end, the measured spectra of light delivered to the output beam 112 by the three spectrally different channels of the LFS 104 (UV, VIS, and NIR spectral channels) and carried by the coherent light pulses 112A are shown in FIG. 1C. The characteristics of the squeezed light pulses 136 (produced upon the transformation of the light 112B in the light squeezer) are determined with the use of spectrometer #2 in each of the spectral channels present in the light 136. The corresponding spectra are shown in FIG. ID. Thereafter, by analyzing the interference fringes within the spectral overlap region (insets in FIGs. 1C and ID) between the light pulses of different spectral channels, the phase uncertainty (ΔΦ) for the coherent and squeezed portions 112A, 136 of light could be determined. The intensity uncertainty (Al) was obtained by measuring the variance in the overall intensity of a given light pulse. Amplitude squeezing was confirmed when the intensity fluctuations of the squeezed light (ΔIs) were smaller than those of the coherent light (ΔIc), while the phase jittering of the squeezed light (ΔΦs) exceeded that of the coherent light (ΔΦc).

[0031] The results produced by metrology measurements, presented in FIGs. 2A, 2B, 2C, and 2D, confirms the presence of amplitude squeezing. The phase uncertainty is higher in the light 136 (FIG. 2 A, panel (all) and FIG. 2C, panel (ell)) as compared to that in light 112A (FIG. 2 A, panel (al) and FIG. 2C panel (cl)), while the amplitude of the light 136 (FIG. 2B panel (bll) and 13QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)FIG. 2D panel (dll)) is more stable than that of the light 112A. These results evidence that the degenerate FWM process through which light passing through the light squeezer portion of the arm 150B of the device 150 of the apparatus 100 transforms the portion 112B of the otherwise substantially coherent light into the amplitude-squeezed light 136 at the expense of increased phase uncertainty. To assess amplitude stability for individual pulse components, the constituent pulses were spectrally filtered to retrieve a parameter of the intensity or amplitude stability. The reader is referred to FIGs. 7A, 7B, and 7C, which indicate that the amplitude stability of the pulses of squeezed light 136 and that of the pulses of substantially coherent light in different spectral ranges (here, chosen as UC, VIS, and NIR spectral ranges defined by the construction of the LFS 104 of FIG. 1A) are comparable. Additionally, the Shot noise was assessed for both the classic light 112A and the squeezed light 136 by measuring the standard deviation of light intensity I (GI) as a function of the beam power; shown in FIG. 8. This result provides additional proof that the ultrafast pulses of light 136 are pulses of the amplitude squeezed light.

[0032] Theoretical modeling was carried out to investigate the nature of the measured phase and amplitude uncertainties and compare the results with those of a squeezed state. Then, the Wigner function was calculated as a visual representation of the generated state of light. As the skilled artisan will readily appreciate, the Wigner function is a quasi-probability distribution that provides a visual representation of the quantum state of in a phase space. Our analysis reveals that the squeezed states of light corresponded to a Gaussian distribution of light that has been squeezed along one optical quadrature and expanded along the conjugate quadrature (see FIGs. 3 A, 3B, 3C, 3D and Section on Supplementary Information, below).

[0033] From a theoretical point of view, the FWM process under co-linear phase matching conditions and within the parametric approximation produces squeezed coherent states of the form ⊗Ni=1Ŝ(ri)|αi⟩. In this expression, N denotes the number of modes, S(r) = exp[ra2− r*a†2] is the squeezing operator, and |ct) represents a coherent state of light. To determine the compatibility of the experimental results with these theoretical squeezed states, the variances in phase and intensity obtained experimentally, ΔΦ2expand ΔI2exp, were compared with the theoretical predictions, ΔΦ2th(x) and ΔI2th(x), (see Freyberger, M. & Schleich, W. Phase uncertainties of a squeezed state, in Physical Review A 49, 5056-5066, 1994R; and the Supplementary Information section below for14QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)detailed calculations), the latter being parametrized by x = {(ri, αi)}Ni=1. Specifically, the following function is defined:C(x) = A[ΔI2th(x) − ΔI2exp]2+ B[ΔΦ2th(x) − ΔΦ2exp]2, (1A)

[0034] which serves as a measure of the distance between the experimental observations and the theoretical expectations, where a value of C(x) = 0 would represent a perfect match between the two. Consequently, the comparison involves finding an optimal set of parameters x* that minimizes Eq. (1 A) as much as possible. To achieve this, the parameters A and B have to be chosen carefully to ensure simultaneous minimization of both terms within the function. In practice, the optimization was performed across a range of values for A and £?, and the best result was retained (see Supplemental Information for more details). Furthermore, given that the experimental data already suggested the presence of amplitude-squeezing, as was alluded to above, the search-space was limited to real values of riand αiwhen solving this optimization problem, so as to reduce the number of variables over which to optimize. The results from this optimization are shown in FIGs. 3A, 3B as a function of the number of modes N, where FIG. 3A represents the IR-VIS dataset and FIG. 3B represents the UV-VIS dataset. The dashed curves show the absolute difference between the experimental and theoretical variances after optimization, while the solid horizontal lines represent the experimental variance values. In all cases, the skilled person will observe that the dashed curves lie below their respective solid horizontal lines, thereby demonstrating a good agreement between theory and experiment, as the absolute error remains around one or two orders of magnitude below the experimental values. Notably, this trend persists across all values of N considered in the study, suggesting that the experimental data is compatible with the presence of squeezed states along multiple modes. (It is appreciated that generally, for both the IR-Vis and UV-VIS datasets, an increase in the number of modes generally corresponds to reduced accuracy in the absolute error, particularly pronounced in the intensity variance.) Overall, the skilled person will observe that the best results were achieved in the N=1 case, yielding a total estimated squeezing of 13.03 dB for the IR-Vis dataset and 8.81 dB for the UV-VIS dataset. The difference in the squeezing levels may be attributed to the alignment optimization in two measurements to achieve the optimized FWM output signal. The Wigner functions (see Schleich, W. P. Quantum Optics in Phase Space. John Wiley & Sons, Hoboken, NJ, United States, 2015) of the resulting squeezed 15QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)states are shown in FIGs. 3C and 3D, respectively, where χ and p represent the optical quadratures. The functions have been rotated by 45° relative to the p-axis for representational purposes. These functions correspond to a Gaussian distribution that has been squeezed along one optical quadrature and expanded along the conjugate quadrature.Attosecond Control of the Quantum Light Uncertainty Dynamics

[0035] Having established the generation of ultrafast squeezed light pulses 136, the question becomes: How does amplitude uncertainty is changing in real-time? To address this, the amplitude uncertainty al of the squeezed light was directly measured as a function of time. Here, the temporal control between the three input beams in the FWM process was exploited to introduce a variable delay (T) between one photon and the other two photons using the D-shape mirrors-containing reflector 124 (see FIG. 1 A). By measuring al for different values of r, one can track the evolution of amplitude uncertainty as the system interacts and evolves in time. The results of the measurements at different time delays T are shown in FIG. 4A. The results, presented in FIG. 4B (black circles connected by the black dashed line, and the red dashed line 410 as a guide to the eye), show that the al had higher values at the positive and negative delays in time. The minimum al was achieved when all three input photons were overlapping time. These results demonstrate that al was not static but, to the contrary, varied in real-time and was influenced by the system's state and its interactions.

[0036] This dynamical uncertainty of the amplitude can be explained as follows: the amplitude squeezing was achieved by utilizing the quantum correlations arising from the nonlinear four-wave mixing interaction among three input photons. In our case, the angles of incidence of these beams onto the non-medium 130 was kept very small (< 5°). As a result, the good phase matching - which can be optimized by tilting the nonlinear medium - occurred and ensured constructive interference when the three pulses arrive simultaneously in time. Consequently, the nonlinear signal generation window was temporally confined to the time of overlap between / among the pulses (T = —1.5 to 1.5 fs, from FIG. 4B), thereby reducing the uncertainty in the number and amplitude of generated photons. Conversely, if one of the input pulses arrived with a16QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)certain delay (r) relative to the other two, the nonlinear signal generation window widened, thereby increasing randomness and, in turn, intensity uncertainty.

[0037] To further confirm this interpretation, the measurements were repeated with a squeezed phase light, obtained by altering the tilting angle of the nonlinear medium 130 (see FIGs.9A, 9B, 9C). In that case, the behavior of al as a function of time, shown in FIG. 9C (black circles connected by black dashed line, blue dashed line 910 is a guide to the eye) is opposite to that in the case of amplitude squeezing (FIG. 4B), thereby confirming the dynamic behavior of uncertainty.Example of Application of Synthesized Quantum Light in petaHertz Digital Data Encoding and Quantum CommunicationRemarkably, by the spatiotemporally overlapping the UV, VIS, and NIR pulses, as demonstrated the squeezed light pulse was obtained on the order of 5.3 fs duration (FIGs. 5A, 5B). Moreover, by controlling the delay and the amplitude of at least some of the UV, VIS, and NIR pulses in the LFS portion 104 of the system 100, the attosecond temporal resolution provided by the LFS 104 (see, for example, Hassan, M. T. et al. Optical attosecond pulses and tracking the nonlinear response of bound electrons. Nature 530, 66-70, 2016)) is now transferred to the synthesis of the broadband squeezed light waveforms with the same attosecond resolution. The ability to generate and control ultrafast squeezed light pulses unlocks exciting new possibilities for quantum technologies. One particularly promising application is ultra-secured quantum communication.

[0038] Traditional continuous-variable quantum key distribution (CV-QKD) relies on encoding information into the quadratures of quantum states, such as amplitude (ΔI) and phase (ΔΦ), using a squeezed or coherent light source. The receiver measures one of these quadratures using a local oscillator (LO) to project the quantum state onto the desired quadrature, chosen at random. The security of CV-QKD arises from the inability of an eavesdropper (Eve) to intercept the quantum state without introducing detectable noise, owing to the Heisenberg uncertainty principle.

[0039] In one embodiment of the petahertz communication scheme that employs the synthesis of the broadband squeezed light waveforms with the same attosecond resolution (as discussed above), the ultrafast nature of the squeezed light pulses 136 can be used to encode17QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)quantum-encrypted digital data at unprecedented speeds. This is schematically illustrated in FIGs. 6 and 6 (Continued). In particular:

[0040] As schematically depicted in FIGs. 6 and 6(Continued), assume that Alice synthesizes and encodes digital data onto amplitude-squeezed light waveforms 136 using the LFS 104 ( Hassan, M. T. Lightwave Electronics: Attosecond Optical Switching. ACS Photonics 11, 334-338, 2024).

[0041] Examples of the measured waveforms shown in FIG. 6(Continued) panels (al-alll) are the average of three scans, and despite the phase uncertainty of these waveforms, the waveforms substantially maintain their shapes. Alice then sets a predefined intensity threshold THR (for example, 30%) for the modulation, encrypting digital data within the squeezed waveforms. Alice then sends the encoded squeezed light beam 610 to Bob (FIG. 6), sharing the degree of squeezing and the threshold information.

[0042] As in conventional CV-QKD, any eavesdropping attempt by Eva (FIG. 6) to intercept - for example, by tapping the beam of light sent from Alice to Bob - and decode the squeezed light waveform would alter the degree of squeezing of the squeezed light (its squeezing degree), thereby revealing the intrusion to both Alice and Bob. Bob can check the signal's squeezing degree by measuring its intensity stability between different pulses (for example, using the methodology discussed above, with exemplary results presented in FIGs. 2A through 2D) to confirm the security of communication with Alice.

[0043] According to yet another idea of the invention, the security of communication between Alice and Bob, one of the three spectral pulses — for example, the NIR pulse from the LFS 104 - can be employed as a quantum-correlated local oscillator (see, for example, Qi, B. et al. Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection. Physical Review X 5, 041009, 2015). Such arrangement eliminates the need for an external classical LO and simplifies synchronization.

[0044] One example of such protocol, according the idea of the invention, may include the following actions by Bob. At the receiver station, after sampling the waveform and decoding the digital data, Bob introduces a controllable relative time delay between the NIR reference pulse and the other pulses. This delay adjusts the interference conditions at specific wavelengths, thereby allowing Bob to effectively select quadrature projections by inducing constructive or destructive interference. By varying the introduced delay, Bob dynamically selects which quadrature18QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)component (amplitude or phase) to measure at a given wavelength. Bob then measures the noise at specific wavelengths within the spectral overlap region, where spectral interference exists between / among the different channels. Depending on the interference condition at that wavelength, Bob measures a specific quadrature and obtains the squeezing level of the selected quadrature projection. This protocol enhances security by making it difficult for an eavesdropper to predict Bob's quadrature choices or replicate the quantum-correlated states without introducing detectable noise. This approach avoids challenges associated with classical LOs, enables dynamic quadrature selection, and leverages quantum correlations to strengthen the security and efficiency of CV-QKD.

[0045] Remarkably, the example of the proposed protocol introduces additional layers of security, in analogy to the typical CV-QKD protocol, protecting the communication from potential tampering. Specifically, the proposed protocol provides multiple safeguards against Eve's interference: (i) Eve would need the decoding key to correctly interpret the transferred information, (ii) Eve cannot accurately decode the data without knowing the predefined intensity threshold (here, THR=30%), and (iii) even if Eve knows the threshold, her measurement of the squeezed light waveform using a tapping beamsplitter inserted into the beam 610 will inevitably disturb the squeezing, thereby introducing errors in the decoded data (“faulty decoding” in FIG. 6) due to the altered squeezing degree. This disturbance of squeezing of light increases the tolerance error of the predefined threshold THR and amplifies the likelihood of faulty decoding, as shown in FIG. 6 (continued) panels (cl-clll), dTHR. Consequently, embodiment of the proposed methodology not only secures the communication channel but also protects the transferred data from unauthorized retrieval.Materials and Methods (Additional)

[0046] In the experimental setup (the example 100 of which is illustrated in FIG. 1 A) fewcycle laser pulses centered at 750 nm (passively stabilized carrier-envelope phase) are generated by an Optical Parametric Chirped-Pulse Amplification (OPCPA)-based laser system with a repetition rate of 20 kHz (not shown). These light pulses propagate nonlinearly through a hollow-core fiber filled with neon gas (at 3.5 bar), not shown, generating a supercontinuum that spans from the ultraviolet to the near-infrared spectral range. The light 108 characterized by this supercontinuum is directed into a light field synthesizer (LFS) device (the principles of structure and operation of 19QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)which are discussed elsewhere), where it is split into three spectral channels using dichroic beam splitters: Ultraviolet (UV, from about 515 nm to about 350 nm), Visible (VIS, from about 715 nm to about 500 nm), and Near-Infrared (NIR, from about 1000 nm to about 690 nm). Each of these spectral channels within the LFS 104 is spatially independent and distinct from the other, and light in each of these spectral channels is compressed (using six pairs of chirped mirrors) to approach the corresponding Fourier limit. Examples of the practical temporal profiles of the light pulses in each channel are shown in FIGs. 10A, 10B, 10C, and 10D. The full width at half maximum (FWHM) pulse durations are 10, 9, and 8.5 fs for the UV, VIS, and NIR light pulses in the corresponding channels, respectively. The light pulses from all three of the spectral channels are recombined using similar dichroic beam splitters and coherently and colinearly superimposed at the output of the LFS 104 to form a substantially two-light-cycle laser pulse output 112. This pulse waveform 112 is finely controlled by adjusting the relative delay between the three LFS light pulses with a high-resolution piezoelectric linear stage in the UV and NIR beam paths. See FIG. 1 A. Neutral density filters were used where appropriate to control the relative intensities of the constituent light pulses in the three spectral channels of the LFS 104. The total power of the output beam of light 112 from the LFS is about one Watt.

[0047] The beam 112 (carrying the approximately two-light-cycle broadband laser pulse) is first split using a beamsplitter BS, with the reflected beam 112A (about 8% of the total power) representing the classic coherent synthesized pulse of light. This reflected beam 112A is delivered to the arm 150 of the light-transforming section 150 of the apparatus 100 and focused into spectrometer #1 (Ocean Optics HR 4000, which has an estimated typical quantum efficiency of 50-70%) with a 5-cm focal length lens 114 (the intensity loss through the optics and optical fiber is estimated to be < 5%). Overall, 2800 spectra were measured to assess the average intensity stability of the light pulse 112A. First, each of these spectra was integrated to get the total intensity (In), where n is the number of iterations. Then, Al was calculated by normalizing each spectrum to the average intensity of all the spectra and then calculating the percentage stability. The intensity standard deviation (ol) percentage was calculated from the average and standard deviation of the n intensities for n iterations. On the other hand, the interference fringes between the spectral channels (inset of FIGs. 1C, 1E) were analyzed, and their corresponding Fourier transforms were computed to extract phase jittering (A ) between the individual pulses.20QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0048] The transmitted portion of the beam 112 (after passing through the beamsplitter), portion 112B, is directed into the light-squeezing arm 150B of the light-transforming section 150 through a 3-hole light mask 116. The three beams 120A, 120B, 120C emerging through the mask 116, which are substantially identical in power (~ 162.5 mW), were focused using two D-shape focusing mirrors (forming, as a combination, the reflector 124) onto a fused silica sample 130 (with the beam diameter / spot of about 50 pm). One of these D-shape mirrors of the reflector 124 was mounted on the piezo stage to control the relative delay between one of the beams 120A, 120B, 120C with respect to the other two beams. Then, a four-wave mixing process occurred, and a nonlinear light signal was generated, which was amplitude-squeezed with about 153 pW power). A 1-hole mask 144 filters out this squeezed light as beam 136, which is then focused into spectrometer #2 (Ocean Optics HR 4000CG) for characterization of its intensity, phase stability, and jitter. The LFS constituent pulses combined in the beam 112 can be appropriately delayed to generate better interference fringes at the spectral boundary of each pulse (see insets of FIGs. 1C, 1E). Dark noise from the spectrometers 1 and 2 of the light-transforming section 150 was negligible (FIGs. 11A, 1 IB), specifically - two orders of magnitude lower than the observed amplitude fluctuations.

[0049] For the time-resolved intensity uncertainty measurements, 1000 spectra were acquired at each time delay (T) between one of the beams 120A, 120B, 120C with respect to the other two beams, which interact with the SiO₂ substrate or plate to generate the nonlinear signal. The deviation of intensity σI for each time delay τ is plotted in FIGs 4A, 4B. The relative delay between the LFS channels was also adjusted to synthesize the squeezed light waveform, which was sampled using an all-optical light field sampling technique, reported previously in, for example, Alqattan, H. et al. Attosecond light field synthesis, in APL Photonics 7, 041301 (2022) or Hassan, M. T. Lightwave Electronics: Attosecond Optical Switching, in ACS Photonics 11, 334-338 (2024), the disclosures of which are incorporated by reference herein. (In the case at hand, this technique involved focusing the squeezed light onto the sample (200 pm thick silica plate) alongside a probe beam. By measuring the modulation of the probe beam's transmission induced by the squeezed light, the waveform was extracted by analyzing the recorded spectrum intensity as a function of the delay between the squeezed light and the probe pulse.)21QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)Supplemental Information

[0050] Extracting the variance from the experimental data. Let 0 represent the physical observable of interest — in this case, phase or intensity — and let o( / ) denote the value of the physical observable obtained in the t-th experimental run. Defining Mter as the total number of experimental iterations (corresponding to the x- axis in Figs. 9A and 9B), the experimental variance can be calculated as follows

[0051] (1)

[0052] where ⟨O⟩ denotes the average value of the physical observable O. Effectively, if / Viter is infinitely large, one can write

[0053] - lim < — — V (o( / )2- 2o(i){O} + (O)2)? “ (O2} - {O}2, / V itey ■ j(2)

[0054] thus recovering the definition for the variance.

[0055] Quantum optical state after four-wave mixing. The Hamiltonian describing the four-wave mixing (FWM) interaction within the nonlinear crystal can generally be written as

[0056] H ~ * "U)2L S zC W'+ h*c‘] ’

[0057] where ^(3) is the material’s third-order susceptibility, and on and an are the frequencies of light entering the nonlinear medium, while on and o>4 are those of light generated by the FWM interaction. However, the experimental design ensures that, before the nonlinear medium,22QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)the wavevectors ki and k2 align in a quasi-collinear configuration, and after the interaction, only specific spatial directions are analyzed — specifically those where wavevectors ka and k4 lie within a narrow cone. The setup of FIG. 1A implies that m3 = m4, thereby allowing the

[0058] Hamiltonian to be simplified as

[0059] “ (w{+w2) / 2

[0060] Furthermore, in the parametric approximation — assuming the FWM drivers are instrong coherent states that get barely depleted due to the interaction such that<xf<’t!j‘ with i E { 1, 2}, the Hamiltonian is further simplified to

[0061]

[0062] Considering that the initial state of the output modes is generally in a coherent stateproduct formw given that %(1) processes induce a displacement in the field degrees of freedom, the final state after the FWM interaction can be expressed as

[0063] w(6),S' ( / ■ I

[0064] where ' -?denotes the squeezing operator.

[0065] The primary objective of the theoretical analysis is to determine whether, for a specificset of parametersmpus^a^es of form above can replicate the experimental variances observed for both intensity and phase.23QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0066] Single mode theoretical variances for intensity and phase. Initially, the variance in the intensity operator is considered and, for simplicity, the analysis is restricted to a single optical mode. The intensity of a single optical mode is defined asM<: J.. where ~~ isthe photon number operator acting on the mode m, while 6 is a parameter that sets the intensity units and also accounts for potential imperfections in the measurement devices.

[0067] Taking into account that for a single-mode squeezed coherent state with r / -

[0068] («<■>) - |<r< J2cosher, j + (1 + |<r(t.|2) sinlr(r(j - coshOyj sinh(rw)(7)

[0069] h?2) - [k*< J2+ kM4] cosh4(rw) - [(2|cyv_, |2+ 2)(a’r*Ml' + at2?“'*“)] cosh3(rui) suih(rw) + [8kr< R + 4iaw|4+ <r4e~l20‘” + + 2] cosh2(r<u) sinlTh;,.) - [(2|a,yp + 4)(a*^enA" + cosh(rw) sinh-’(rw) + [ 1 + 3|awk + inyj4] siDh“(r<u),

[0070] such that the theoretical variance in intensity reads

[0071] A / t (o / l = t ' i’S' ~x 7J= e2khj2cosh4(r0z) ~ cos(^) cosir (rw) sinhfr^)+ (6!^ J2+ 2) cosh2(r<u) sinh2(rw) - 4|jat.;|2cos(^v) cosh(rfe)) sinh*5(r<£>) + |aaJ2sinh4(rw)(9)rAA^A1, with a(<)=

[0072] where24QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0073] To account for phase fluctuations, one must take a different approach, however. In theory, there is no standard way to define a phase operator, which prevents one from computing the variance as it was done for the intensity (as shown above). In this case, the variance is calculated by following the geometrical approach from Rivera-Dean, J. et al. (in Non-classicality induces recombination in high-harmonic generation with circularly polarized fields; available at https: / / arxiv.org / abs / 2411.11042vl (2024). For a single optical mode, this approach defines the variance as

[0074] AX2,£t.> &<£>&(&») =(10)and X2M)= ie(d^ - d(!j)

[0075] where are the operators representing the optical quadratures. For squeezed coherent states, one now finds that

[0076] ~ [ cosh(rw) - e'(,“ sinh(rw) + [ cosh(r^) - sinh(rw)]a*r(11)

[0077] and

[0078] = cosh2^) + tf (rj - 2cos(^) cosh(r) sinh(r)....

[0079] Multimode theoretical variances for intensity and phase. Although (as was alluded to above) the single-mode squeezing was sufficient to reproduce the experimental results, an additional investigation was conducted to figure out whether the results were compatible with the presence of squeezing in multiple modes. For the intensity, the total intensity operator was / - V / introduced aswith a mean value given by / >.... y F

[0080] 1~~ (13)25QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0081] and the mean value of its square, * by

[0082] whereWas taken into account (since these operators act on separate Hilbert spaces). Furthermore, given that the states on which these operators are evaluated have a product-state structure, we can rewrite the previous expression as

[0083]

[0084] such that the variance becomes

[0085] Following similar arguments, one can write - for the vari.ance of1- ” -k' Y

[0086] W = X |{X-J - «■_„}-]w (17)s ™ AX- / ( V

[0087] and define the phase variance for the multimode scenario as1

[0088] Numerical optimization. To perform the numerical optimization, we first benchmark the value of e, which jointly determines the intensity units and the detector’s efficiency. Here, we assume the “classical” data (obtained before the FWM occurs) originates from coherent states, allowing us to theoretically expressmV un, where / tot is the total intensity, andA®,h= VA / VTT> thereby resulting in

[0089] By comparing with the experimental data, one can estimate E™ (z^ / eXpAeKp) / x^26QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0090] Secondly, a function for optimization must (1) compare the experimental fluctuations with theoretical predictions, and (2) simultaneously optimize over both phase and intensity fluctuations. One suitable function that meets these requirements is

[0091] C(x) = >4[d / 2h(x) - d / 2xp]2+ - 4< D2xp]2(18)X " TT < / ' ' I

[0092] where "1This function can be interpreted as a measure of distance between experimental data and theoretical results: if C(x) = 0, one achieves a perfect match. The function is weighted by two parameters A and B, chosen ad-hoc to ensure that both terms in Eq. (18) are optimized jointly. The optimization problem to solve is formulated asC(x:) = minxC(x)!). To red,uce t.he number or variables, we restri.c+t both / v;andare restricted to real values. This choice is motivated by experimental results suggesting the presence ofamplitude squeezing, which can be achieved by67; however, negative values were also considered for completeness.

[0093] For the numerical implementation, Python was used along with the minimization tools from the SciPy package. Specifically, the following steps were followed:

[0094] Step 1. Define a set of coefficients A = {1, 10⁻¹, 10⁻²,..., 10⁻⁸} while keeping B = 1. (Here, values of A > B were not considered, as preliminary tests indicated that they prioritized intensity fluctuations over phase fluctuations, leading to suboptimal results.)

[0095] Step 2. For each data set (IR- Vis and Vis-UV) and for a fixed number of modes N, optimize the cost function across each value of A value in the set. (Here, optimization was carried out using SciPy’ s minimize function with the Nelder-Mead algorithm, a local optimization method that iterates based on the local environment of an initial point xo. To avoid local minima, for each combination of N and A, 20000 optimizations were initiated using different initial points generated using the Halton sequence, which uniformly distributed points across the parameter space. An upper bound on 10000 total iterations of the Nelder-Mead algorithm for each point was used.)

[0096] Step 3. After completing all optimizations, identify the best results for each IV,specifically those that minimized both'!■ and27QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)I ( th ( “ ( ^ex o ) i (The final selection was done over the different considered values ofDiscussion

[0097] Overall, implementations of the ideas of the invention generated the petaHertz-scale quantum light waveforms and leveraged such waveforms to implement methodology of secure quantum communication, thereby marking a significant step toward the realization of high-speed, encrypted communication networks.

[0098] The proposed ultrafast (petaHertz) quantum communication protocol operates offering the potential for ultra-secure, high-speed communication networks due to the intrinsic sensitivity of squeezed light to any eavesdropping attempt. Potential practical challenges, such as maintaining low-loss transmission or pulse dispersion distortion of squeezed pulses over long distances in dispersive media / optical fibers - possibly can be pre-compensated by controlling the relative delay between the LFS pulses. Moreover, the proposed methodology would be promising for space-space quantum communication, where the light pulse is not affected by dispersion as it propagates in vacuum. Furthermore, scaling up the LFS and the input coherent laser pulse power to generate a high -power squeezing synthesized pulse could overcome the transmission loss problem and support robust ultrafast quantum communication. Furthermore, achieving quantum communication at petaHertz frequency speeds is currently limited only by the laser repetition rate and the delay stage speed in the used LFS.

[0099] The demonstrated approach can be potentially extended to the light-matter coupled in QED cavity and to quantum spin squeezing.

[0100] Additional use of the proposed implementations can be found in ultrafast spectroscopy, for example to probe the dynamics of chemical reactions or biological processes on the attosecond timescale. In quantum computing, the demonstrated implementations of generating ultrafast squeezed light could be used to manipulate qubits or perform quantum logic operations at unprecedented speeds, thus paving the way for the development of faster and more powerful quantum computers.28QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)

[0101] For the purposes of this disclosure and the appended claims, the use of the terms "substantially", "approximately", "about" and similar terms in reference to a descriptor of a value, element, property or characteristic at hand is intended to emphasize that the value, element, property, or characteristic referred to, while not necessarily being exactly as stated, would nevertheless be considered, for practical purposes, as stated by a person of skill in the art. These terms, as applied to a specified characteristic or quality descriptor means "mostly", "mainly", "considerably", "by and large", "essentially", "to great or significant extent", "largely but not necessarily wholly the same" such as to reasonably denote language of approximation and describe the specified characteristic or descriptor so that its scope would be understood by a person of ordinary skill in the art. In one specific case, the terms "approximately", "substantially", and "about", when used in reference to a numerical value, represent a range of plus or minus 20% with respect to the specified value, more preferably plus or minus 10%, even more preferably plus or minus 5%, most preferably plus or minus 2% with respect to the specified value. As a non-limiting example, two values being "substantially equal" to one another implies that the difference between the two values may be within the range of + / - 20% of the value itself, preferably within the + / - 10% range of the value itself, more preferably within the range of + / - 5% of the value itself, and even more preferably within the range of + / - 2% or less of the value itself.

[0102] The use of these terms in describing a chosen characteristic or concept neither implies nor provides any basis for indefiniteness and for adding a numerical limitation to the specified characteristic or descriptor. As understood by a skilled artisan, the practical deviation of the exact value or characteristic of such value, element, or property from that stated falls and may vary within a numerical range defined by an experimental measurement error that is typical when using a measurement method accepted in the art for such purposes. Other specific examples of the meaning of the terms "substantially", "about", and / or "approximately" as applied to different practical situations may have been provided elsewhere in this disclosure.

[0103] References throughout this specification to "one embodiment," "an embodiment," "a related embodiment," or similar language mean that a particular feature, structure, or characteristic described in connection with the referred to "embodiment" is included in at least one embodiment of the present invention. Thus, appearances of the phrases "in one embodiment," "in an embodiment," and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment. It is to be understood that no portion of disclosure, taken on its own and in possible 29QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)connection with a figure, is intended to provide a complete description of all features of the invention.

[0104] The term “image” refers to an ordered representation of detector signals corresponding to spatial positions. For example, an image may be an array of values within an electronic memory, or, alternatively, a visual image may be formed on a display device X such as a video screen or printer. The term “A and / or B” or a similar term is defined to cover “A, B, or a combination of A and B”.

[0105] Embodiments of the invention have been described as including a programmable processor or electronic circuitry or computer controlled by instructions stored in a memory. The memory may be random access memory (RAM), read-only memory (ROM), flash memory or any other memory, or combination thereof, suitable for storing control software or other instructions and data. Those skilled in the art should also readily appreciate that instructions or programs defining the functions of the present invention may be delivered to such processor / electronic circuitry / computer in many forms, including, but not limited to, information permanently stored on non-writable storage media (e.g. read-only memory devices within a computer, such as ROM, or devices readable by a computer I / O attachment, such as CD-ROM or DVD disks), information alterably stored on writable storage media (e.g. floppy disks, removable flash memory and hard drives) or information conveyed to a computer through communication media, including wired or wireless computer networks. In addition, while the invention may be embodied in software, the functions necessary to implement the invention may optionally or alternatively be embodied in part or in whole using firmware and / or hardware components, such as combinatorial logic, Application Specific Integrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs) or other hardware or some combination of hardware, software and / or firmware components.

[0106] While the invention is described through the above-described specific non-limiting embodiments, it will be understood by those of ordinary skill in the art that modifications to, and variations of, the illustrated embodiments may be made without departing from the inventive concepts disclosed herein. For example, computer program product configured to govern the operation of the embodiment of the apparatus of the invention and / or to perform steps of an embodiment of the method and / or to implement digital data encoding and quantum communication process, to name just a few, remain within the scope of the invention. The disclosed aspects may be30QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)combined in ways not listed above. Accordingly, the invention should not be viewed as being limited to the disclosed embodiment(s).31QB\122170.00263\99812540.1

Claims

International Patent Application Attorney Docket No.: 122170.00263(UA25-152)CLAIMS1. An apparatus comprising:an optical arrangement that is configured:(a) to receive, at an input thereof, an input beam of coherent light that carries an input pulse of the coherent light having an input spatial distribution and including the coherent light in multiple spectrally distinct from one another spectral bands,(b) to transform said input beam into multiple intermediate spatial distributions of the coherent light that are spatially distinct from one anotherand(c) to direct each of the multiple intermediate spatial distributions of the coherent light towards a predetermined location to spatially overlap with every other of the multiple intermediate spatial distributions and to interact with a medium present at the predetermined spatial location to generate a pulse of squeezed quantum light containing quantum light squeezed in a chosen quadrature;anda programmable electronic circuitry that is operably cooperated with a tangible non-transitory storage medium containing program code thereon, wherein the program code, when loaded to the programmable electronic circuitry, enables the electronic circuitry:based on information characterizing the pulse of squeezed light received by the electronic circuitry, to generate first indicia representing uncertainties of at least one of a phase and an intensity of the squeezed quantum light and second indicia representing a temporal profile of an electric field and / or a temporal profile of an intensity of said pulse of the squeezed quantum light.32QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)2. An apparatus according to claim 1, wherein the optical arrangement includes a light mask configured to transform the input spatial distribution into the multiple intermediate spatial distributions of the coherent light that have substantially equal light powers.

3. An apparatus according to claim 1, wherein the optical arrangement is configured to generate said pulse of the squeezed quantum light containing the quantum light squeezed in a phase quadrature.

4. An apparatus according to claim 1, wherein the optical arrangement includes a reflector (a) that is disposed between a first location at which the input spatial distribution of the coherent light is transformed into the multiple intermediate spatial distributions of the coherent light and the predetermined spatial location; and(b) that includes first and second spatially distinct from one another mirrors structured to move with respect to one another.

5. An apparatus according to claim 1,wherein the optical arrangement includes a light-collecting optical system comprising at least an optical detection device that is disposed to receive only said pulse of the squeezed quantum light and to produce an image representing an optical parameter of said pulse of the squeezed quantum light, andwherein the programmable electronic circuitry is configured to acquire an output containing imaging data representing said image from the light-collecting optical system.

6. An apparatus according to claim 1, wherein the optical arrangement is structured to carry out a four-wave mixing process.

7. An apparatus according to claim 1, wherein the medium at the predetermined spatial location includes a piece of substantially non-electrically-conducting material.33QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)8. An apparatus according to claim 1, wherein the program code is configured to govern the electronic circuitry to generate indicia of acquisition of an optical field at an optical detection device when intensity of the optical field of the squeezed quantum light exceeds a pre-determined optical signal threshold.

9. An apparatus according to claim 8,wherein the program code is configured to govern the electronic circuitry to define an optical signal threshold to be registered with a light-collecting optical system that includes at least an optical detection device disposed to receive only said pulse of the squeezed quantum light and to produce an image representing an optical parameter of said pulse of the squeezed quantum light,wherein the optical signal threshold is defined such that indicia of acquisition is generated only when an intensity of the optical field of the squeezed quantum light acquired at the lightcollecting system exceeds said optical signal threshold.

10. An apparatus according to claim 1, configured to generate the squeezed quantum light in a form of a train of light pulses in a petaHertz frequency range.

11. An apparatus according to claim 1, wherein:(i) the system further comprises a source of light configured to generate said input beam of coherent light carrying said input light pulse having a femtosecond duration or a sub-femtosecond duration, and / or(ii) wherein a number of said multiple intermediate input beams is equal to a number of distinct spectral bands.

12. An apparatus according to claim 11, wherein the source of light includes a multi-spectral channel light field synthesizer configured to generate respective beams of the coherent light carrying corresponding pulses of light in each of the multiple distinct spectral bands and to form said input beam by substantially collinearly overlapping said respective beams spatially and said corresponding pulses temporally.34QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)13. An apparatus according to claim 12, wherein the source of light is configured to introduce a delay between said corresponding pulses of light in each of the multiple distinct spectral bands.

14. A method comprising:receiving the input beam of the coherent light at the optical arrangement of the apparatus according to claim 1;irradiating a target of substantially non-electrically-conducting material, disposed at the predetermined spatial location, with multiple pulses of the coherent light carrying substantially equal light pulses and independently and substantially contemporaneously delivered thereto along spatially distinct and substantially parallel to one another optical paths to generate said pulse of the squeezed quantum light;with said programmable electronic circuitry, at least generating the first indicia representing uncertainties of each of the phase and the intensity of the squeezed quantum light.

15. A method according to claim 14,further comprising: transforming said input beam into multiple intermediate spatial distributions of the coherent light that are spatially distinct from one another, andwherein said irradiating includes directing each of the multiple intermediate spatial distributions of the coherent light to the pre-determined spatial location.

16. A method according to claim 14, further comprising:performing, with said programmable electronic circuitry, one or more of the following steps:(a) governing an optical detector of the apparatus to acquire said squeezed quantum light to identify the temporal profile of the electrical field thereof and / or the temporal profile of the intensity thereof,35QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)(b) generating the second indicia representing said temporal profile of the electrical field and / or the temporal profile of the intensity, and(c) generating third indicia representing an amplitude of the electrical field at the optical detector when said amplitude exceeds a pre-determined signal threshold.

17. A method according to claim 14, comprising one or more of the following:(a) delivering only the squeezed quantum light to an optical detector of the apparatus at least in part by spatially filtering out the squeezed quantum light from other light transmitted through the target, and(b) determining a goal degree of squeezing of the squeezed quantum light based at least on spectral interference between temporally delayed with respect to one another first and second pulses of the coherent light that contain light in respective different spectral bands of the multiple distinct spectral bands.

18. A method according to claim 14, further comprising interpreting said generating the third indicia by the programmable circuitry of the apparatus as one of "0" and " 1 " of a binary number system.

19. A method according to claim 14,wherein, when said irradiating the target produces multiple pulses of the squeezed quantum light that include quantum light squeezed in the chosen quadrature, the method further comprises:transmitting a communication beam of light containing said multiple pulses of the squeezed quantum light from an output of the apparatus configured to generate the squeezed quantum light towards a receiver.36QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)20. A method according to claim 19, further comprising:(i) altering a degree of squeezing of the squeezed quantum light received at the receiver as compared with the goal degree of squeezing by interacting with the communication beam of light at an intermediate location between said output and the receiver;and / or(ii) when said interacting includes forming secondary beam of light by redirecting a portion of the communication beam of light away from the receiver at the intermediate location, producing a redirected communication beam of light that carries said portion and that possesses a degree of squeezing different from the target degree of squeezing.

21. A method according to claim 19, further comprising:(a) with the use of an electronic circuitry of the receiver, issuing an alert signal representative of interaction with the communication beam at an intermediate location between said output and the receiver; and / or(b) ceasing the transmitting from the output of the apparatus in response to said alert signal.

22. A method comprising:delivering information by passing, between a transmitter system and a receiver system, a communication beam of light carrying pulses of squeezed quantum light with pulse frequency at least in a petaHertz range,wherein:a. a degree of squeezing of light, of the pulses of squeezed quantum light, that is received at the receiver system is varied, from a planned degree of squeezing of said light that has been achieved in a chosen quadrature at the transmitter system, by interacting with said communication beam of light performed at an37QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)intermediate location between the transmitter system and the receiver system, and / orb. when said interacting includes forming a secondary beam of light carrying pulses of squeezed quantum light by redirecting a portion of the communication beam of away from the receiver system at the intermediate location, producing a redirected communication beam of the light that carries said portion and that possesses a degree of squeezing different from the planned degree of squeezing.

23. A method according to claim 22, further comprising:sharing the threshold value and the planned degree of squeezing between the transmitter system and the receiver system.

24. A method according to claim 23, further comprising:determining the degree of squeezing that is received at the receiver system to obtain a received degree of squeezing at least in part by measuring stability of intensity of light of the pulses of squeezed quantum light between different of said pulses;maintaining the transmitter system being aware of the received degree of squeezing and / or generating an alert signal if the received degree of squeezing substantially differs from the planned degree of squeezing of the light received, at the receiver system, from the transmitter system.

25. A method according to claim 24, further comprising:ceasing said passing the communication beam at the transmitter system if the received degree of squeezing substantially differs from the planned degree of squeezing.38QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)26. A method according to claim 22, further comprising:generating said pulses of squeezed light from a first portion of a coherent source light carrying light pulses in each of multiple pre-determined distinct spectral bands, wherein the light pulses in all of the multiple pre-determined distinct spectral bands are substantially temporally overlapped. determining an intensity uncertainty of the squeezed light by measuring a variance of an overall intensity of a pulse of squeezed quantum light with the use of a first optical detection system, determining an intensity uncertainty of the coherent source light by measuring a variance of an overall intensity of a light pulse of the coherent source light with the use of a second optical detection system, andgenerating indicia of amplitude squeezing of the squeezed light when the intensity uncertainty of the squeezed light are smaller than the intensity uncertainty of the coherent source light.

27. A computer program product for digital data encoding and quantum communication, the computer program product comprising a computer usable tangible non-transitory storage medium having computer readable program code therein, the computer readable program code including: program code for determining an actual degree of squeezing of pulsed squeezed quantum light, generated at least at a petaHertz pulse rate at a first location from a coherent source light carrying light pulses in each of multiple pre-determined distinct spectral bands, wherein the light pulses in all of the multiple pre-determined distinct spectral bands are substantially temporally overlapped;program code for transmitting information by passing a communication beam of light carrying said pulsed squeezed quantum light and the actual degree of squeezing from the first location to a second location;program code for ceasing said transmitting in response to having received, at the first location from the second location, indicia of said information having been received at the second location with a non-compliant degree of squeezing, wherein the non-compliant degree of squeezing differs from the actual degree of squeezing.39QB\122170.00263\99812540.1International Patent Application Attorney Docket No.: 122170.00263(UA25-152)28. A computer program product according to claim 27, wherein the program code for determining the actual degree of squeezing includes program code for determining the actual degree of squeezing by measuring stability of intensity of light of pulses of the pulsed squeezed quantum light with the optical detection system.40QB\122170.00263\99812540.1

Images