A system and a method for satellite relayed quantum communication

Abstract

A satellite relayed quantum communication system comprises a network of one or more chains of near-synchronously moving satellites, equipped with a telescope module that effectively creates a satellit

Description

A SYSTEM AND A METHOD FOR SATELLITE RELAYED QUANTUM COMMUNICATION FIELD OF INVENTION

[0001] The present invention relates to the field of quantum communication. More particularly, the present invention relates to a system and a method for enabling robust, multi-mode quantum communication by controlling photon loss using a network of satellites with one or more chains of nearly synchronously moving satellites as 'satellite lenses'. BACKGROUND OF THE INVENTION

[0002] The creation of a global quantum network has the potential to revolutionize communication and scientific research by enabling the transfer of quantum information across any two points on Earth. The benefits of a functioning quantum network include secure communication through global quantum key distribution (QKD), distributed quantum computing, and entanglement-based precision quantum sensing. However, the biggest challenge in building a quantum network is the loss of photons, as amplification is not possible due to the no-cloning theorem. Although optical fibers are the default choice for qubit transmission, even the small loss in these fibers are very problematic for single photon transmission over long distances due to exponential absorption loss. To address this issue, two strategies are being explored for building quantum networks over longer distances. One approach involves using quantum repeater protocols with quantum memories to overcome the compounding effect of photon loss through storage when transmitting photons through optical fibers. Another approach is to use orbiting satellitesto send and receive photons, which has gained prominence in recent years.

[0003] The main obstacle to implementing quantum repeaters is the need for high- performance quantum memories, among other issues. Research into quantum memories has been ongoing for nearly two decades, resulting in memories with high efficiency, high storage time, and moderate multimode capacity, but achieving all of these characteristics together is still challenging for practical quantum repeaters. As a result, functional quantum repeaters are currently limited to distances under 100 km. Furthermore, quantum memories do not yet have simple and robust designs that can be easily deployed commercially, often requiring sophisticated setups and cryogenic temperatures. In contrast, photon transmission through quantum satellites has seen significant success in recent years, with the Micius satellite in Low Earth Orbit demonstrating entanglement distribution up to 1200 km on Earth. This achievement is possible because photon transmission through space experiences diffraction loss, which scales quadratically with distance, as opposed to the exponential scaling of absorption loss in fibers.

[0004] It may be noted that the quantum information cannot be transmitted over long distances using either fiber-based repeaters or low earth orbit (LEO) satellites beyond 2000 km. At these distances, quantum repeaters become very complicated and require many links, although some proposals exist. While LEO satellites are low in elevation (200-2000 km) compared to the radius of the Earth (around 6400 km), they still face the curvature of the Earth quickly. To reach further distances (2000-20,000 km), proposals for using higher orbit satellites (e.g., geostationary satellites at 36,000 km elevation) or combining quantum memory with satellites have been made. However, both of these proposals have limited transmission rates due to either photon loss in diffraction or long storage times in the memory.SUMMARY OF THE INVENTION

[0005] In light of the requirements mentioned in the previous section, the following summary is provided to facilitate an understanding of some of the innovative features unique to the present invention and is not intended to be a full description. A full appreciation of the various aspects of the invention can be gained by taking the entire specification and drawings as a whole.

[0006] Embodiments of the present disclosure propose a system and a method for satellite relayed quantum communication.

[0007] The present invention relates to a satellite-relayed quantum communication system designed to enable the transmission of quantum information over variable distances, including across the curvature of the Earth, other celestial bodies or in deep space. The system comprises a network of satellites organized into one or more chains of nearly synchronously moving satellites. Each satellite in the chain comprises a telescope module with one or more telescopes effectively creating a satellite lens, wherein the satellite lens comprises a plurality of mirrors configured to focus light, thereby containing beam divergence and directing the light to the next satellite in the chain or to another fixed or mobile station for transmitting quantum information across variable distances.

[0008] A tracking module is configured for tracking between satellite chains, to fixed or mobile stations, and residual tracking within each satellite chain wherein the satellites are moving nearly synchronously, thereby making tracking needs within a satellite chain minimal.

[0009] Transmission of quantum information across variable distances on Earth is achievedby directing the light along a path that follows the curvature of the Earth. The relationship between satellite lens diameter, consequently telescope diameters, satellite separation, and wavelength of light used for minimizing solely diffraction loss is determined using both an analytical approach and numerical modeling, whereby as an example figure, a 60 cm diameter satellite lens necessitates a satellite separation smaller than 120 km for 810 nm wavelength light in Gaussian mode, in case of diffraction-limited light propagation.

[0010] At least one of the front or back mirrors of the satellite lens is a high reflectivity Bragg mirror to minimize reflection loss. The satellite lens is constructed using a telescope assembly setup comprising at least one of: off-axis telescopes, on-axis telescopes, or telescopes without front mirrors. Beacon lasers are coupled through the same path as the quantum signal for tracking, including pointing and point-ahead, in satellite-to-satellite and satellite-to-fixed or mobile station links.

[0011] A network of spacecraft relays performs quantum communication in deep space instead of satellites around a celestial body. Similarly, a network of spacecraft-based relay architecture can establish quantum communication in deep space too.

[0012] The system further comprises a quantum light source located within the satellite or at a station to facilitate quantum communication protocols, including direct transmission from one station to another through the satellite relay and entanglement distribution either from a satellite-based quantum light source or from a source at a station.

[0013] Quantum memories are integrated into the satellite relay system, either in a satellite or in a fixed or mobile station, to enable either higher efficiency of the system or further capabilities or both.

[0014] Frequency multiplexing modules are located in either the satellites or in fixed or mobile stations or both to increase the rate of quantum information transfer through frequency multiplexing. At least one of the satellite lenses has an adjustable focal length to control the focusing of incoming light beams as necessary for mitigating diffraction losses associated with long-distance photon transmission and ensuring minimal photon loss.

[0015] Satellite separation inside a satellite chain, satellite lens diameter, and telescope mirror design are chosen to minimize total photon loss including diffraction loss, tracking loss, light reflection loss, and other overhead losses, these choices being determined based on both theoretical analysis and numerical modeling of light propagation, including the calculation of optimal focal lengths to align with theoretical and simulated propagation models.

[0016] Each satellite's telescope module employs large diameter mirrors to reduce beam truncation loss significantly, enhancing the system's efficiency in photon transmission. Satellite separation and satellite lens focal length are designed to be non-uniform within a satellite chain, this design being determined based on both theoretical approaches and numerical modeling of light propagation to align with theoretical and simulated models for minimizing photon loss.

[0017] The system is configured to support quantum key distribution (QKD) by enabling quantum information transfer between any two points on Earth, using the satellite relay network.

[0018] The invention also includes a method for transmitting quantum information across variable distances. The method comprises deploying a network of one or more chains of nearly synchronously moving satellites, each satellite comprising a telescope modulefunctioning as a satellite lens with a plurality of mirrors. The method involves focusing light with the mirrors on each satellite lens to contain beam divergence, directing the focused light to either the next satellite in the chain or to another fixed or mobile station, and transmitting quantum information via the directed light.

[0019] The method further comprises quantum information transfer as a quantum light signal encoded in different forms, comprising time-bin, frequency-bin, or continuous variable forms.

[0020] Establishing quantum communication between any two points on Earth involves connecting two different satellite chains with a single inter-chain link, enabling communication between satellites moving at high-speed relative to each other in the inter- chain link, and optionally adjusting the focal length of the satellite lenses in real-time to minimize loss during the inter-chain communication.

[0021] Optimizing for efficient long-distance transmission of quantum information and entanglement distribution enables quantum technology applications like quantum key distribution (QKD), quantum sensing, distributed quantum computing, blind quantum computing, and quantum internet functionalities optionally incorporating quantum memories.

[0022] The reduction of diffraction loss, tracking loss, and reflection loss enables the transfer of quantum information over very long distances.

[0023] According to an embodiment of the present disclosure, a network of low Earth orbit (LEO) satellites moves closely together to transmit photonic qubits directly through space. This arrangement of the LEO satellite chain eliminates the need for quantum memories orrepeater protocols, and this arrangement is termed as the All-Satellite Quantum Network (ASQN).

[0024] According to an embodiment of the present disclosure, photons are reflected from one satellite to another using satellite telescopes in the ASQN. The LEO satellite chain allows photons to be transmitted by bending light along the curvature of the earth, which helps to mitigate the huge diffraction losses faced in transmission from higher orbit satellites.

[0025] The diffraction being only beam divergence can be controlled using optical elements, and the telescope mirrors in satellites can be effectively used as lenses to focus incoming light beams.

[0026] According to an embodiment of the present invention, the satellites are chosen to be co-moving in the same orbit i.e., they are stationary relative to each other, to ensure that the ASQN doesn’t require dynamic tracking of individual satellites and hence works effectively. The chain of satellites behaves as a set of lenses on an optical table that periodically converges the light to completely contain beam diffraction. Each satellite effectively behaves as a lens, and they are referred to as "satellite lenses".

[0027] According to an embodiment of the present invention, an analytical treatment of light propagation is used to determine the focal lengths of the satellite lenses. For a small enough satellite separation distance (approximately 120km) and large enough telescope diameter or "satellite lens" size (approximately 60 cm), diffraction loss can be eliminated.

[0028] According to an embodiment of the present disclosure, a numerical modeling involving simulation of diffraction loss by considering beam truncation due to the finitesize of the telescope mirrors shows that the photons can be transmitted to even large global distances (upto 20,000 km) with almost no diffraction loss. Therefore, ASQN becomes the most important for long-range (5,000 to 20,000 km) quantum communication. In the absence of diffraction loss other losses become dominant like mirror reflection loss, satellite setup errors etc. We showed through detailed investigation that the detrimental effects of these losses can be contained or avoided. If these other losses are limited to 2% in each satellite loss in ASQN can be only 30 dB even in global distances of 30,000 km. Moreover, even in short distances of around 200 km ASQN can already achieve significantly less loss than the optical fiber loss in that distance (around 30 dB with 0.15 dB / km loss) as ASQN loss is also smaller at smaller distances. Hence, ASQN can be the preferred quantum communication protocol with least loss over almost the whole distance range in Earth.

[0029] This summary is provided merely for the purposes of summarizing some example embodiments, to provide a basic understanding of some aspects of the subject matter described herein. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following detailed description and figures.

[0030] The abovementioned embodiments and further variations of the proposed invention are discussed further in the detailed description. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] FIG. 1a is a schematic view of a qubit transmission quantum communication protocol using a chain of reflector satellites according to the embodiments of the present disclosure.

[0032] FIG. 1b is a schematic view of an entanglement distribution quantum communication protocol using the chain of reflector satellites according to the embodiments of the present disclosure.

[0033] FIG.2a, 2b and 2c is a schematic diagrams illustrating a plurality of lens setups to completely contain beam divergence due to diffraction indefinitely according to the embodiments of the present disclosure.

[0034] FIG. 3a is a schematic diagram illustrating the entangled pair source containing ‘satellite lens’ in the middle and the ground links at both ends according to the embodiments of the present disclosure.

[0035] FIG. 3b is a graph illustrating a numerical simulation of diffraction loss showing entanglement distribution probability at 20,000 km for different telescope diameters and satellite separations, without considering ground link according to the embodiments of the present disclosure.

[0036] FIG. 3c is a graph illustrating the same plot as Fig. 3b for diffraction loss with different scales i.e., in units of decibel, according to the embodiments of the present disclosure.

[0037] FIG.3d and 3e is a graph illustrating the diffraction simulated using the ground link to estimate the total diffraction loss according to the embodiments of the present disclosure.

[0038] FIG.4a is a schematic diagram illustrating the schematics of the qubit transmissionprotocol with lenses and apertures according to the embodiments of the present disclosure.

[0039] FIG.4b is a graph illustrating the diffraction loss for qubit transmission at 20,000 km for different telescope radius (d) and satellite separation values (L0) without considering the ground link according to the embodiments of the present disclosure.

[0040] FIG.4c is a graph illustrating photon transmission probabilities with a distance with d = 60 cm and L0 = 120 km for both qubit transmission and entanglement distribution according to the embodiments of the present disclosure.

[0041] FIG. 5a and 5b is a graph illustrating total loss for entanglement distribution with 2% absorption loss for each satellite according to the embodiments of the present disclosure.

[0042] FIG. 5c is a graph illustrating the same plot as Fig. 5b with 5% satellite loss and showing points up to 100 dB loss according to the embodiments of the present disclosure.

[0043] Fig.5d is a graph illustrating entangled distribution loss with distance for different (d, L0) values and diffraction loss values (satellite loss 2% of 5%) for a total propagation of 20,000 km according to the embodiments of the present disclosure.

[0044] Fig. 5e is a graph illustrating Qubit transmission protocol total loss values (including both uplink and downlink loss, satellite loss of 2% and other losses) are shown for different (d, L0) values according to the embodiments of the present disclosure.

[0045] FIG. 5f is a graph illustrating qubit transmission loss shown with distance for d = 60 cm, L0 = 120 km, and 2% satellite loss according to the embodiments of the present disclosure.

[0046] FIG. 6a is a schematic illustration of atmospheric turbulence modeled using successive phase screens, completely fragments the initial Gaussian beam by the time it reaches the satellite according to the embodiments of the present disclosure.

[0047] FIG. 6b, 6c and 6d is a schematic representation illustrating a large fragmented beam (6b), focused beam (6c) and the final beam (d) according to the embodiments of the present disclosure.

[0048] FIG. 6e is a graph illustrating a plot of average light propagation loss with propagation distance according to the embodiments of the present disclosure.

[0049] FIG.6f is a schematic illustration of a network of satellites arranged in a 2D mesh (or more generally a 3D structure), consisting of individual satellite chains as previously detailed, facilitates quantum communication across the globe according to the embodiments of the present disclosure.

[0050] FIG.6g is a schematic illustration of quantum information being transmitted from point A to point B using a single link between two satellite chains (specifically between satellites C and D), with the remainder of the transmission occurring along the respective satellite chains according to the embodiments of the present disclosure.

[0051] FIG. 6h is a schematic illustration of distance of the connection between the two different chains will dynamically change over time, as depicted here at a slightly later time according to the embodiments of the present disclosure.

[0052] FIG.7a, 7b, 7c, 7d, and 7e illustrate a plurality of possible telescope setups suitable for a chain of satellite reflectors showing focusing and bending according to the embodiments of the present disclosure.

[0053] FIG.7f illustrates a simulation of entangled photon pair propagation through 20,000 km in vortex beam profile through the on-axis system showing nearly identical initial and final beams (after 10,000 km propagation by each photon) according to the embodiments of the present disclosure.

[0054] FIG.7g is a graph illustrating a plot of entangled pair transmission probability with distance according to the embodiments of the present disclosure.

[0055] FIG. 8a is a schematic diagram illustrating quantum internet by entanglement distribution with entangled source on satellite (S1) and on ground station (S2) according to the embodiments of the present disclosure.

[0056] FIG. 8b is a schematic diagram illustrating the quantum internet using a repeater scheme according to the embodiments of the present disclosure.

[0057] FIG. 8c is a diagram illustrating multiple teleportations without using quantum memories according to an embodiment of the present disclosure.

[0058] FIG.9a is a schematic diagram illustrating the effect of ‘satellite lens’ focal length (f) and position (z or xy) error on beam propagation, (i.e., on diffraction loss) is shown according to an embodiment of the present disclosure.

[0059] FIG. 9b is a graph illustrating the effect of f error according to an embodiment of the present disclosure.

[0060] FIG.9c is a graph illustrating the effect of ^ error according to an embodiment of the present disclosure.

[0061] FIG.9d is a graph illustrating the effect of xy error according to an embodiment ofthe present disclosure.

[0062] FIG.9e is a graph illustrating the effect of total error according to an embodiment of the present disclosure.

[0063] Fig. 10 is a system for satellite relayed quantum communication according to the embodiments of the present disclosure.

[0064] Fig.11 is a method for transmitting quantum information across variable distances according to the embodiments of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0065] In the following description of the embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and which are shown by way of illustration of specific embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that other embodiments may be utilized and that changes may be made without departing from the scope of the present invention.

[0066] The specification may refer to “an”, “one” or “some” embodiment(s) in several locations. This does not necessarily imply that each such reference is to the same embodiment(s), or that the feature only applies to a single embodiment. Single feature of different embodiments may also be combined to provide other embodiments.

[0067] As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well unless expressly stated otherwise. It will be further understood that the terms “includes”, “comprises”, “including” and / or “comprising” when used in this specification, specify the presence of stated features, integers, steps, operations, elementsand / or components, but do not preclude the presence or addition of one or more other features integers, steps, operations, elements, components, and / or groups thereof. As used herein, the term “and / or” includes any and all combinations and arrangements of one or more of the associated listed items.

[0068] Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure pertains. It will be further understood that terms, such as those defined in commonly used dictionaries should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.

[0069] The utility of the devices described herein will be explained further in detail in the following sections of this document referring to the figures. Specific terms used herein do not restrict the scope of the present disclosure.

[0070] Embodiments of the present disclosure propose a system 1000 and a method 1100 for satellite relayed quantum communication.

[0071] According to an embodiment of the present invention, the satellite relayed quantum communication system 1000 as illustrated in Fig.10 comprises a network of satellites 1002 with one or more chains of nearly synchronously moving satellites. Further, each satellite in a chain comprises a telescope module 1004 with one or more telescopes 1006 effectively creating a satellite lens, wherein the satellite lens comprises a plurality of mirrors configured to focus light, thereby containing beam divergence and directing the light to the next satellite in the chain or to another fixed or mobile station 1016 for transmitting quantum information across variable distances.

[0072] The system 1000 further comprises a tracking module 1008 configured for tracking between satellite chains, to fixed or mobile stations and residual tracking within each satellite chain wherein the satellites are moving nearly synchronously, thereby making tracking needs within a satellite chain minimal.

[0073] According to an embodiment of the present invention, the transmission of quantum information across variable distances on Earth is achieved by directing the light along a path that follows the curvature of the Earth.

[0074] According to an embodiment of the present invention, the relationship between satellite lens diameter, consequently telescope diameters, satellite separation and wavelength of light used for minimizing solely diffraction loss is determined using both an analytical approach and numerical modelling. In a non-limited example figure, a 60 cm diameter satellite lens necessitates a satellite separation smaller than 120 km for 810 nm wavelength light in gaussian mode, in case of diffraction-limited light propagation.

[0075] According to an embodiment of the present invention, at least one of the front or back mirrors of the satellite lens is a high reflectivity Bragg mirror to minimize reflection loss.

[0076] According to an embodiment of the present invention, the satellite lens is constructed using a telescope assembly setup comprising at least one of: off-axis telescopes, on-axis telescopes, or telescopes without front mirrors.

[0077] According to an embodiment of the present invention, the beacon lasers are coupled through the same path as the quantum signal for tracking, including pointing and point- ahead, in satellite-to-satellite and satellite to a fixed or mobile station links.

[0078] According to an embodiment of the present invention, a network of spacecraft relay performs quantum communication in deep space instead of satellites around a celestial body.

[0079] According to an embodiment of the present invention, further comprises a quantum light source 1010 located within the satellite or at a station to facilitate quantum communication protocols, including direct transmission from one station to another through the satellite relay and entanglement distribution either from a satellite-based quantum light source 1010 or from a source at a station 1012.

[0080] According to an embodiment of the present invention, the system 1000 further comprises quantum memories 1012 integrated into the satellite relay system 1000, either in a satellite or in a fixed or mobile station, to enable either higher efficiency of the system 1000 or further capabilities or both.

[0081] According to an embodiment of the present invention, the system 1000 comprises frequency multiplexing modules 1014 located in either the satellites or in fixed or mobile stations or both to increase the rate of quantum information transfer through frequency multiplexing.

[0082] According to an embodiment of the present invention, at least one of the satellite lenses possess adjustable focal length to control the focusing of incoming light beams as necessary for mitigating diffraction losses associated with long-distance photon transmission and ensuring minimal photon loss.

[0083] According to an embodiment of the present invention, wherein satellite separation inside a satellite chain, satellite lens diameter and telescope mirror design are chosen tominimize total photon loss including diffraction loss, tracking loss, light reflection loss and other overhead losses, these choices being determined based on both theoretical analysis and numerical modelling of light propagation, including the calculation of optimal focal lengths to align with theoretical and simulated propagation models.

[0084] According to an embodiment of the present invention, wherein each satellite's telescope module 1004 employs large diameter mirrors to reduce beam truncation loss significantly, enhancing the system's 1000 efficiency in photon transmission.

[0085] According to an embodiment of the present invention, wherein satellite separation and satellite lens focal length are designed to be non-uniform within a satellite chain, this design being determined based on both theoretical approaches and numerical modelling of light propagation to align with theoretical and simulated models for minimizing photon loss.

[0086] According to an embodiment of the present invention, the system 1000 is further configured to support quantum key distribution (QKD) by enabling quantum information transfer between any two points on Earth, using the satellite relay network.

[0087] According to an embodiment of the present invention, the system 1000 further comprises a light collection unit 1018 with a detector 1020, wherein the light collection unit 1018 is primarily present in the fixed or mobile stations 1016 and also alternatively in the satellites as well.

[0088] According to an embodiment of the present invention, the method 1100 as illustrated in Fig. 11 for transmitting quantum information across variable distances comprises deploying a network of one or more chains of nearly synchronously movingsatellites, each satellite comprising a telescope module 1004 functioning as a satellite lens with a plurality of mirrors at step 1102. Next at step 1104, focusing light with the mirrors on each satellite lens to contain beam divergence and then at step 1106 directing the focused light to either the next satellite in the chain or to another fixed or mobile station. Next at step 1108 transmitting quantum information via the directed light.

[0089] According to an embodiment of the present invention, the method 1100 further comprises quantum information transfer as quantum light signal encoded in different forms, comprising time-bin, frequency-bin or continuous variable forms.

[0090] According to an embodiment of the present invention, the method 1100 further comprises establishing quantum communication between any two points on Earth by connecting two different satellite chains with a single inter-chain link. Next enabling communication between satellites moving at high-speed relative to each other in the inter- chain link and then optionally adjusting the focal length of the satellite lenses in real-time to minimize loss during the inter-chain communication.

[0091] According to an embodiment of the present invention, wherein optimizing for efficient long-distance transmission of quantum information and entanglement distribution enables quantum technology applications like quantum key distribution (QKD), quantum sensing, distributed quantum computing, blind quantum computing and quantum internet functionalities optionally incorporating quantum memories 1012.

[0092] According to an embodiment of the present invention, wherein the reduction of diffraction loss, tracking loss, and reflection loss enables the transfer of quantum information over very long distances.

[0093] According to an embodiment of the present invention, when sending a photonic qubit from one location on Earth to another through space, two issues arise that are intertwined. The first issue is diffraction loss, which is in itself not a limiting problem even at global distances (around 20,000 km) when only two telescopes can be used in a straight line. However, the second issue, the curvature of the Earth, quickly increases diffraction loss. To transmit light from one point on Earth to another, multiple reflectors in space are needed to guide the light along the curvature of the Earth. However, these reflector satellites cause more diffraction due to beam truncation at each satellite, leading to even greater beam divergence. To mitigate this issue, beam truncation must be reduced to decrease both beam truncation loss and beam divergence stemming from the truncated beam. This is achieved by the present disclosure providing a large chain of satellites with a smaller telescope (diameters in 40-60 cm) separated by a much smaller distance.

[0094] Though very large telescopes separated by longer distances can be employed as an alternative to using the large chain of satellites, the same is not preferred since large telescopes are heavy, difficult to manufacture and expensive.

[0095] According to an embodiment of the present invention, each of the satellites acts as a reflector, with only mirror reflection contributing to photon absorption loss. The curvature of the telescopes is used to focus the light, essentially turning the satellites into effective 'satellite lenses', wherein this chain of 'satellite lenses' eliminates diffraction loss. The protocols include qubit transmission, entanglement distribution, and quantum internet. Quantum internet protocols are more complex and may require quantum memories to execute complex operations. This is discussed in detail later.

[0096] FIG. 1a is a schematic view of a qubit transmission quantum communicationprotocol using a chain of reflector satellites 106 according to the embodiments of the present disclosure. According to an embodiment of the present disclosure, the qubit transmission protocol involves sending photons from the quantum source 102 in the ground station towards the nearest satellite 106 using a telescope 108. The receiving telescope on the satellite 106 collects the light and reflects it towards the other telescope on the same satellite 106, which then transmits the light to the next satellite 106 towards the destination. The two telescopes on the satellite 106 work together to focus the light to the appropriate amount and minimize beam divergence due to diffraction. The succeeding satellite 106 retrieves the photons and transmits them to the subsequent one, continuing this operation until the destination point is reached. At that point, the final satellite 106 directs the photons towards the ground station, using its telescope 108 to aim downwards. The photons are gathered by the telescope located at the ground station and can be detected by the detector 104 (as shown in Figure 1a) to conduct end-to-end secure Quantum Key Distribution (QKD) or used for other objectives.

[0097] According to an embodiment of the present invention, the qubit transmission protocol offers several advantages by having both the source 102 and detector 104 on the ground. These advantages comprise of the ability to design and perform new experiments such as sending or teleporting different quantum states (e.g., continuous variable states, squeezed states, multiphoton states, etc.) simply by visiting the source 102 and detector 104 ground stations, without the need for new satellite launches. Future developments and maintenance of the source 102 and detector 104 can also be easily incorporated. Additionally, the larger physical space and electric power available at ground stations may allow for greater frequency multiplexing capabilities. The use of large cryogenic coolers and other sophisticated techniques is also feasible on the ground. These advantages maynot be achievable when the source 102 or detector 104 is onboard a satellite due to limited access, resources, or space. Even if possible on a satellite 106, many of these capabilities may be prohibitively expensive.

[0098] However, qubit transmission protocol faces a challenge with one uplink and one downlink transmission, unlike entanglement distribution protocol from a satellite-based source which faces two downlink transmissions. The uplink transmission experiences a significantly larger atmospheric turbulence loss compared to the downlink, resulting in a greater impact on qubit transmission. Turbulence causes the beam size to increase, leading to higher total loss and reduced transmission rates. The amount of turbulence loss depends on two critical factors - increased propagation length after the atmosphere ends (approximately 20 km from the ground) and a small receiving telescope size on-board the satellite. However, in ASQN of the present disclosure, the low-elevation satellites (about 200 km – 2000 km) equipped with moderately-sized telescopes (40-60 cm), reduces the turbulence loss.

[0099] According to an embodiment of the present disclosure, the decoy state QKD can be performed using weak coherent pulse (WCP) sources in qubit transmission protocol. WCP sources have high rates and are simple to develop. Another potential benefit of qubit transmission is the possibility of frequency multiplexing. While frequency multiplexing can also be achieved in entanglement distribution, physical space and other resource limitations in a relatively small orbiting LEO satellite like ASQN may restrict its implementation. However, since the source 102 remains in the ground station in qubit transmission protocol, these limitations can be avoided. This advantage may lead to the development of larger frequency multiplexing capabilities in the free space-based ASQN protocol. Although qubit transmission protocol faces extra loss due to turbulence, theadvantages of using WCP sources and the possibility of large multiplexing capabilities may compensate for this loss.

[0100] In addition to qubit transmission, the protocol shown in Fig. 1(a) can also be used for entanglement distribution by incorporating a quantum memory. To achieve this, one of the photons from an entangled pair needs to be stored in a quantum memory while the other photon is sent using the setup depicted in Fig.1(a). However, a quantum memory with long storage time and large multimode capacity is required for this process. Although high memory efficiency is not necessary as the entanglement distribution rate will decrease merely by the efficiency factor, developing such a quantum memory remains a challenging task.

[0101] FIG. 1b is a schematic view of an entanglement distribution quantum communication protocol using a chain of reflector satellites 106 according to the embodiments of the present disclosure. According to an embodiment of the present disclosure, the entanglement distribution protocol of the present disclosure does not require quantum memory.

[0102] According to an embodiment of the present disclosure, an entangled pair of photons is sent along two directions from roughly around the mid-point between the two places where entanglement need to be transferred. The entangled photon source can be located either on a satellite or on the ground, as depicted by S1 and S2 in Fig. 1(b) respectively. The source satellite (S1 in Fig.1(b)) functions as a reflector satellite with an entangled source, which can be remotely removed from the beam path, thereby transforming the source satellite into a reflector satellite.

[0103] According to an embodiment of the present disclosure, the entanglementdistribution protocol and the qubit transmission protocol (Fig. 1(a)) experience similar diffraction loss in space, as diffraction is controlled by effective "satellite lenses" in both cases. However, the major difference in loss occurs in the ground links, especially due to uplink turbulence. Qubit transmission faces uplink turbulence once, while the entangled source in satellite (S1) faces none and the entangled source in ground (S2) faces it twice. Hence, placing the entangled source in the satellite results in much smaller loss than placing it on the ground. Nevertheless, despite the larger loss, a ground source can still be a viable alternative option due to the potential to design new experiments involving different types of entangled states and larger multiplexing capabilities.

[0104] According to an embodiment of the present invention, the satellite-relayed quantum communication leverages a chain of reflecting satellites acting as an optical relay, eliminating the need for quantum memory and thus simplifying the system significantly, as illustrated in Fig.1c (figure not to scale). The system utilizes a series of satellites equipped with reflectors (telescopes) that act as an optical relay, positioned in a chain to enable the transmission of quantum information by reflecting photons from one satellite to the next. Each relay satellite only needs reflectors, designed to focus and direct the quantum signals accurately to the next satellite in the chain, minimizing potential points of failure and reducing overall photon loss. These relay satellites effectively act like lenses, focusing the incoming light to control beam divergence and ensuring that the quantum signals remain aligned over long distances, by aligning the mirrors that constitute these effective lenses. To keep photon loss to an absolute minimum, only essential optical elements are included in the light path, maintaining the highest possible efficiency in photon transmission. By employing a chain of reflecting satellites moving in the same orbit, the system needs minimal real-time satellite tracking within the satellite chain. Hence, the system cancompletely contain diffraction loss through the satellite lenses, allowing the light to be bent around the curvature of the Earth and maintaining optical beam over vast distances. Theoretical and Numerical Analysis of ASQN for the qubit transmission protocol and the entanglement distribution protocol

[0105] It is to be noted that the rate of quantum communication is dependent on both the source rate and the loss incurred during transmission. Loss in the ASQN network is a combination of several factors including beam divergence loss due to diffraction, air transmission loss caused by atmospheric absorption and turbulence, and individual satellite losses which involve factors such as reflection loss, beam pointing error, and focal length error. Sometimes, diffraction loss and satellite loss are interdependent, where beam pointing errors can result in diffraction loss.

[0106] According to an embodiment of the present disclosure, photon propagation is used interchangeably with beam transmission, and the fraction of total intensity is used to represent single photon transmission probability.

[0107] According to an embodiment of the present disclosure, each satellite in the ASQN network has a system of telescopes that collects and transmits photons to the next satellite through reflection. This system of telescopes can be represented as a group of lenses, and the entire system is modeled as one effective lens, known as the "satellite lens." The focal length of the satellite lens depends on the separation and the individual focal lengths of the telescope mirrors. In some cases, the effective satellite lens can have an infinite focal length, meaning that the telescope setup on the satellite only acts as an aperture. Different types of telescope systems can be used, including on-axis and off-axis telescopes with various designs. A detailed analysis of these telescope systems is illustratedin Figure 7 with a corresponding detailed description.

[0108] FIG. 2a, 2b and 2c is a schematic diagrams illustrating a plurality of lens setups to completely contain beam divergence due to diffraction indefinitely according to the embodiments of the present disclosure.

[0109] Before discussing the numerical calculations, a simple theoretical analysis is presented using Figures 2a, 2b and 2c. This analysis ignores beam truncation caused by the finite lens size and assumes infinite-sized lenses. The change in beam size over distance is investigated, and the fraction of light intensity contained within a circle of radius r of a Gaussian beam is given by) . Truncation effects become negligible around w0 ~ d / 4. Almost all of the intensity (99.96%) is contained within a circle of radius r=2w0.

[0110] For lenses that do not truncate the beam, it can be shown that by choosing a proper set of focal lengths, beam diffraction can be completely controlled for small enough lens separation (L0). According to an embodiment of the present invention, the case of an incident Gaussian laser beam with beam waist (w0) at the position of the first lens is considered for simplicity. This assumption is made for convenience in the calculations and is not essential.

[0111] The Rayleigh range ( ^R) is defined as ^R = πw2 / λ for a Gaussian beam with wavelength λ and beam waist w0. The present disclosure demonstrates that diffraction can be completely controlled for a lens system with lens separation L0 = ^R and focal lengths (f) of successive lenses as ^R, ^R / 2, ^R / 2.., and so on. Such a system would preserve the beam size and cause no beam divergence indefinitely, even at infinite distance. This can be directly verified using the ABCD matrix formulation for Gaussian laser beams.

[0112] Fig.2 demonstrates three stages to explain the phenomenon. If a Gaussian laser beam is focused using a lens with focal length f1 at its beam waist, and place another lens with focal length f2 = f1 at double the focusing distance (Lf with Lf = L0 / 2), the beam will return to its beam waist after the second lens due to the symmetry of the setup. Note that the focusing distance Lf is not the same as the lens focal length f1 because the input beam is a Gaussian laser beam and not a constant wave-front. At this point, the original input state is restored, and another set of lenses with focal length f1 and double focusing distance (2Lf) separation, exactly as in Fig.2(a), would result in the beam returning to its beam waist after another 2Lf distance apart (as shown in Fig. 2(b)). This process can be repeated indefinitely to make the beam return to its beam waist at infinitely far away distances. The effective focal length of the middle lenses would be f1 / 2 when combining two lenses of focal length f1. Hence, a lens configuration of f1 = 2 f2 = 2 f3 = ... = 2 fn−1 = fn would indefinitely preserve the beam size and not cause any beam divergence when placed at uniform separations of L0 = 2Lf, as shown in Fig.2(c).

[0113] According to an embodiment of the present invention, the maximum possible lens separation for present disclosure’s setup - L0 = ^R - can be found as following. When a Gaussian beam with Rayleigh range ^R is focused by a lens of focal length f placed at beam waist, the beam is focused at a focusing distance Lf = f / (1 + f2 / ^2), which is different from the lens focal length f . The focusing distance Lf is maximum at f = ^R with Lf (max) = ^R / 2. If L0 = ^R, the beam is focused at the midpoint between the first and second lens (Lf (max) = ^R / 2) and hence the beam profile is symmetric around the midpoint.

[0114] According to an embodiment of the present invention, it is not necessary for the incident light to be at the beam waist on the first lens with focal length f = ^R. In fact, it can begin with a diverging input beam and simply start from the second lens, consideringit as the first lens. More generally, the scheme can accommodate light with any curvature as long as the beam spot matches the beam waist and the focal length of the lens is adjusted accordingly. For instance, in the case of entanglement distribution from a satellite source (as shown in Fig. 3(a)), an extremely divergent beam emanating from an optical source located in the same satellite can be collimated using the telescope mirrors of the satellite.

[0115] According to an embodiment of the present invention, in ASQN, the curvature of the Earth is not accounted for directly, as lenses are considered in a straight line for both theory and simulation. However, this does not significantly affect the overall calculation. Firstly, the angle of the light is bent at each satellite, and hence at each "satellite lens," but this angle is very small - less than half a degree for satellite separations of approximately 100 km. In Fig.7, it is demonstrated how telescope mirrors can be used to impart this angle to the light beam. This change in angle does not provide any extra focusing, similar to how plane mirrors are used in optics experiments to change the angle of a laser beam. Therefore, this does not have any effect on diffraction.

[0116] FIG. 3a is a schematic diagram illustrating the entangled pair source containing ‘satellite lens’ in the middle and the ground links at both ends according to the embodiments of the present disclosure.

[0117] FIG. 3b is a graph illustrating a numerical simulation of diffraction loss showing entanglement distribution probability at 20,000 km for different telescope diameters and satellite separations, without considering ground link according to the embodiments of the present disclosure.

[0118] FIG. 3c is a graph illustrating the same plot as Fig. 3b for diffraction loss with different scales i.e., in units of decibel, according to the embodiments of the presentdisclosure.

[0119] FIG. 3d and 3e is a graph illustrating the diffraction simulated using the ground link to estimate the total diffraction loss according to the embodiments of the present disclosure.

[0120] Next, the diffraction loss is numerically simulated by considering beam truncation. The numerical simulation begins by examining the scenario where entanglement distribution occurs with the source on a satellite (referred to as S1 in Figure 1(b)). The entanglement distribution arrangement used in the simulation is depicted in Figure 3(a), with the source satellite located in the center and the ground links at either end.

[0121] Figure 3(b) displays the diffraction loss for photons with a wavelength of 800 nm over a distance of 20,000 km, for various values of the lens diameter d and satellite separation distance L0. To conduct the simulation, a beam waist of w0 = √L0λ / π is used. A lens system of L0, L0 / 2, L0 / 2, and so on, was then used to contain the beam diffraction. If the lens diameter d is large enough to encompass the beam, no diffraction loss will occur, as per theoretical expectations. However, for smaller lenses (i.e., telescopes), the beam will be truncated, resulting in loss. If the truncation is significant, beam divergence due to diffraction will occur, and the loss will increase quickly. For example, for L0 = 120 km, with a telescope diameter of 60 cm, has a loss of only 0.67 dB when only 0.28% of the beam is truncated. However, if the telescope diameter is reduced to 35 cm, loss increases dramatically to 324 dB, as 13.47% of the beam is truncated. It is important to note that any loss other than diffraction loss (e.g. turbulence loss) is not included in this analysis.

[0122] Figure 3(b) illustrates the transmission probability of an entangled photon pair over a distance of 20,000 km. To generate this graph, photon transmission is simulatedover a distance of 10,000 km (excluding the final ground link) and then squared the result to account for the photon pair. The figure demonstrates that, for a particular combination of satellite lens diameter and distance between two satellites, transmission can be achieved with minimal loss. As expected, diffraction loss is highest for small values of d and large L0 values, while it is lowest for large values of d and small L0 values. Diffraction is constant in regions where L0 is proportional to d2, which is also expected since L0 = zR =and diffraction loss in ASQN is determined by the ratio of w0 and d. The region with low loss is evident in Figure 3(b). Figure 3(c) provides the same plot for diffraction loss in decibels, showing clearly the region with high loss. At this distance of 20,000 km, the loss can be as high as 324 dB for the smallest lens diameter of 35 cm and the largest satellite separation of 120 km.

[0123] Fig.3(d) depicts the total diffraction loss, including the ground link, for the entanglement distribution system. For this simulation, a ground telescope with a diameter of 60 cm and a satellite elevation of 200 km were used. The same lens configuration as before (L0, L0 / 2, L0 / 2,…), was employed, but with a shrinking region of maximum intensity. The probability of entanglement distribution in the region with small L0 values was reduced as the lens configuration diverges beams more at small L0 and then they have to travel 200km distance in the ground link. However, this issue can be addressed through optimization, as shown in Fig.3(e), where the focal length of the last two lenses before the ground link is adjusted to increase the area of maximum intensity. It is noteworthy that even without optimization, in Fig.3(d), the intensity did not decrease significantly for large L0 values, which is the desired regime. This is shown in Fig. 5 while considering total loss. Therefore, this regime can function without adjustable focal lengths.

[0124] If the source and detector are both on the ground (as in Fig. 1(a)), thediffraction loss for qubit transmission would be similar to that for entanglement distribution if the effects of atmospheric turbulence can be completely neglected. However, the source being on the ground means that the photons have to pass through atmospheric turbulence first, resulting in a significant beam divergence when the photons reach the satellite. The large beam spot at the satellite means that the portion of the beam captured by the satellite telescope can be considered as having a constant wavefront, although this is not entirely accurate since the beam gets fragmented.

[0125] FIG. 4a is a schematic diagram illustrating the schematics of the qubit transmission protocol with lenses and apertures according to the embodiments of the present disclosure.

[0126] FIG. 4b is a graph illustrating the diffraction loss for qubit transmission at 20,000 km for different telescope radius (d) and satellite separation values (L0) without considering the ground link according to the embodiments of the present disclosure.

[0127] FIG. 4c is a graph illustrating photon transmission probabilities with a distance with d = 60 cm and L0 = 120 km for both qubit transmission and entanglement distribution according to the embodiments of the present disclosure.

[0128] To control diffraction caused by a constant wavefront, a two-step strategy can be employed. Firstly, the wavefront can be focused using the first lens onto a later lens (not necessarily the next one), as illustrated in Fig. 4(a). When a constant wavefront is focused by a lens, an Airy disk pattern is formed at the focus due to Fraunhofer diffraction, and the majority of the light energy is contained within the central disk, which resembles a Gaussian beam with similar-sized beam waist. Based on this assumption, the size of the central Airy disk at the focus can be determined using the equation wAiry = (nL01.22λ) / d,where n is the number of lenses after which the beam is focused (n = 3 in Fig.4(a)). The number n is selected to ensure that the size of the Airy disk is larger than a corresponding Gaussian beam waist with Rayleigh length L0, i.e., wAiry > w0 =√L0λ / π, which enables diffraction to be controlled using the "satellite lenses". The lenses between the first lens and the target lens (i.e. the nth lens) would be removed, and they would act as apertures of diameter d. As the Airy disk pattern at the focus is similar to a Gaussian beam waist, the original lens configuration of L0, L0 / 2, L0 / 2,… can be employed for the following lenses, starting from the nth lens. The simulation results confirm the validity of this scheme.

[0129] Figure 4(b) displays a numerical simulation of diffraction loss for qubit transmission at global distances of 20,000 kilometers using different values of d and L0. The simulation follows the configuration of Figure 4(a) without considering the ground link. Unlike the entanglement distribution case, in qubit transmission, the probability of light transmission does not reach unity asymptotically, even without considering the ground link. There could be several reasons for this, such as the non-Gaussian Airy disk beam profile, which can cause errors during propagation, or the diffraction effects of the initial apertures (where the lenses are removed) on the light beam. Due to the combined effect of these factors, light intensity drops during the initial propagation of the beam before stabilizing. This drop in intensity is evident in Figure 4(c), where the light intensity is plotted at different lengths for the cases of d=60 cm, L0=120 km, and λ=800 nm for both entanglement distribution and qubit transmission. The graph shows the total intensity captured at each lens. However, any loss other than diffraction loss, such as turbulence loss, is not included in this analysis. In both cases, the ground link is included after the last two lenses, and the focal lengths fN-1 and fN are optimized for the ground link transmission. Except for the initial drop discussed above, the intensity decreases similarly for qubittransmission as it does for entanglement distribution.

[0130] The lens configurations for entanglement distribution (L0, L0 / 2, L0 / 2,…) and qubit transmission (nL0, ∞ , ∞ ,…, L0, L0 / 2, L0 / 2,…) only work optimally in certain cases, such as the case with large d where there is no beam truncation. Optimization over the whole lens configuration or a part of it, considering only a few lenses, can potentially improve diffraction loss, particularly in cases where loss is starting to increase. With an optimized lens configuration, it possible to have a much smaller maximum loss than the ~300 dB loss seen in Fig. 3(c). An example of an optimized lens configuration is a lens waveguide system, where lenses are configured to confine light within them and at the end, the initial beam profile is returned. Such lens waveguide systems are commonly used to confine light along a curved path.

[0131] It may be noted that there are other types of losses that are not related to diffraction. One of these losses is air transmission loss, which consists of absorption in air and atmospheric turbulence. These losses only affect the satellite-ground links and are not present in satellite-to-satellite transmission. In satellite-to-satellite transmission, the absorption loss becomes negligible at high elevations due to the low air density. However, in ground links, atmospheric absorption loss depends heavily on the optical frequency and angle of transmission. Absorption losses increase exponentially with distance, and grazing incidence is harmful to quantum information transfer.

[0132] In addition to absorption loss, air turbulence also contributes to loss in satellite-ground links. Turbulent eddies in the atmosphere cause beam wander and beam spreading, leading to loss in both uplink and downlink transmission along with diffraction. Turbulence is much more significant in the uplink than in the downlink transmissionbecause the dephased beam emanating from the turbulent atmosphere has to travel a long way to reach the satellite, where it spreads further due to diffraction. However, in the downlink, there is no propagation after the atmosphere, so turbulence loss is much lower. To reduce loss due to turbulence in the uplink, less propagation distance can be allowed by using low-elevation satellites. This is feasible in ASQN because of the small separation (around 100 km) between satellites in the chain, which does not require a large field of view for each satellite. Low-elevation satellites also decrease diffraction loss in the ground link. Further, turbulence loss can be decreased by using larger diameter receiving telescopes and shrinking the initial beam waist to limit spreading.

[0133] Another significant contributor to photon loss is called satellite loss, which encompasses all losses caused by one satellite while reflecting the photon towards the next one. This loss has two parts: intrinsic loss and satellite errors. Intrinsic loss, such as mirror reflection loss, grows exponentially with the number of satellites and must be kept small, or the number of satellites needs to be reduced dramatically. The reflection loss at each satellite mirror is the most significant of the exponential scaling losses, especially as there are multiple mirrors needed on one satellite itself. Standard metal reflectors (gold or silver- coated mirrors) have reflectivities of at most 99.5% and these high reflectivities are available only in wavelengths above 1 µm. More sophisticated dielectric mirrors or distributed Bragg reflectors, or simply Bragg mirrors, can have very high reflectivity by using multiple thin layers of different refractive index glasses. They can have a refractive index as high as 99.9999% and can be manufactured for almost any frequency. These mirror systems are discussed in detail later.

[0134] There can be several other factors contributing to satellite loss, such as aberration in mirror reflection, mirror positioning error, mirror angular position error, andsatellite positioning error, all of which may cause beam deviation or focal length error. Moreover, assuming monochromatic light for the calculation, the finite frequency width of a photon may also influence diffraction.

[0135] FIG.5a and 5b is a graph illustrating total loss for entanglement distribution with 2% absorption loss for each satellite according to the embodiments of the present disclosure.

[0136] FIG.5c is a graph illustrating the same plot as Fig.5b with 5% satellite loss and showing points up to 100 dB loss according to the embodiments of the present disclosure.

[0137] Fig. 5d is a graph illustrating entangled distribution loss with distance for different (d, L0) values and diffraction loss values (satellite loss 2% of 5%) for a total propagation of 20,000 km according to the embodiments of the present disclosure.

[0138] Fig. 5e is a graph illustrating Qubit transmission protocol total loss values (including both uplink and downlink loss, satellite loss of 2% and other losses) are shown for different (d, L0) values according to the embodiments of the present disclosure.

[0139] FIG. 5f is a graph illustrating qubit transmission loss shown with distance for d = 60 cm, L0 = 120 km, and 2% satellite loss according to the embodiments of the present disclosure.

[0140] Figures 3 and 4 show the effect of diffraction loss only. In addition to diffraction loss, loss due to every satellite (exponential satellite loss), atmospheric absorption, turbulence, satellite chain setup error, detector loss, etc., is considered as they produce a constant overhead loss. In Fig.5a, the total loss is plotted with 2% exponentialloss for each satellite, which scales exponentially with the number of satellites. This is due to diffraction loss due to truncation, which also scales almost exponentially when multiple satellites are used. In Fig.5b, the same loss is plotted in decibel units up to a certain loss value (45 dB). The loss increases quickly with decreasing L0 due to the increasing number of satellites required, which contributes to satellite loss. In Fig.5c, it shows the effect of a much higher loss at 5% exponential satellite loss. Here, much more loss occurs for the same (d, L0) values, and loss values up to 100 dB are shown. For a certain lens diameter value d, one can find an optimum loss value at a particular L0. The present disclosure shows total intensity with propagation distance (L) at these optimum d and L0 values in Fig.5d for both 2% and 5% satellite loss. The best intensity scaling with distance is seen for (d; L0 = 60cm, 120km) values for 2% satellite loss. However, even for much smaller d values, reasonable intensity scaling is seen for (45cm, 80 km) for 2% satellite loss and (60cm, 120 km) for 5% satellite loss. Although there is more than 40 dB loss at the full global distance of 20,000 km, at the more intermediate distances (around 10,000 km), there is much less loss. Similar to entanglement distribution, the effect of total loss was shown for qubit transmission in Fig.5e, and in Fig.5f, the same exponential trend of Fig.5d is shown along with the initial diffraction loss seen at the first few apertures for qubit. Simulation Method

[0141] According to the protocols of the present disclosure, the propagation of light is simulated through various optical elements using a Python module called Lightpipe. To evaluate the propagated field using the Fresnel approximation, Lightpipe functions are utilized that employ Fast Fourier transform (FFT). The angular spectrum A(α, β, z = 0) is related to the electric field distribution U(x, y, z) through Fourier transform.

[0142] In the Fresnel approximation, the propagated angular spec- trum A(α, β, z) is related to the initial angular spectrum A(α, β, z = 0) by the following relation.

[0143] This algorithm implements light propagation by calculating the angular spectrum A(α, β, z = 0), given the field distribution U(x, y, 0) at z = 0, using FFT as

[0144] First, A(α, β, z = 0) is computed using Eq. (3), and then A(α, β, z) is obtained using Eq. (2). The inverse Fourier transform is then applied to A(α, β, z) to calculate U(x, y, z), using FFT. The algorithm simulates beam propagation on a finite grid with periodic boundary conditions, which mimics light propagation in a waveguide with the grid size. Reflective behavior is observed if the field approaches the grid edge during propagation, which is prevented by using a sufficiently large grid size. In simulation of the present disclosure, after each beam propagation iteration, either an aperture or a thin lens is used. A thin lens is implemented using an aperture, due to the finite size of the lens, and by multiplying the field with the phase shift given by the thin lens formula.

[0145] To calculate total loss in light propagation in Fig. 5, after considering the mentioned satellite loss for each satellite the present disclosure also considers the following additional losswhere d is diameter of the receiving telescope and wLT is the long-term beam width. η0 here consists of three important factors – ηe which is transmission probability in presence of error calculated in Section VII B,which is atmospheric absorption and ηd contains other losses like detector efficiency. They are related by the following expression:

[0146] Hence, the above additional loss factor contains the uplink loss due to turbulence, atmospheric absorption loss, effect of setup errors and other losses at source and detector. The η0 factor produced an additional 10 dB of loss which is the constant overhead loss for ASQN as it can be clearly seen in Fig.5(d). Sources

[0147] The rate of quantum communication is influenced by several factors such as source, detector, and electronics rates. The rate of Quantum Key Distribution (QKD) varies for qubit transmission using weak coherent pulses (WCP) and entanglement distribution, depending on the sources used. Loss is the principal limiting factor at large distances. Entanglement distribution protocols generally use probabilistic entangled photon-pair generation such as spontaneous parametric down conversion (SPDC) sources. On the other hand, qubit transmission can use single photons or weak coherent pulses to perform QKD.

[0148] Quantum satellite experiments mostly used polarization encoded qubits, but they can decohere due to highly oblique reflection from spherical mirrors. Time-bin and frequency-bin qubits can be used as alternatives for quantum communication in this case. Time-bin qubits should not be affected by atmospheric turbulence or reflections from the moving satellites if the two bins are separated in time closely enough.

[0149] Overall, the choice of sources is a critical factor when comparing between the two protocols. Loss is the principal limiting factor at large distances, and time-bin qubits can be used as alternatives to polarization-encoded qubits to prevent decoherence from highly oblique reflection. Turbulences

[0150] Quantum communication is affected by atmospheric turbulence, particularly in uplink transmission, where the contribution to turbulence mainly comes from the 20 km of atmosphere closest to the Earth's surface. However, in downlink transmission, the beam does not have to travel through the turbulent atmosphere, resulting in negligible turbulence loss compared to diffraction loss. Therefore, turbulence loss is usually not considered in downlink transmission, but this may not be accurate for downlink optimized beam profiles that are not necessarily flat beams. While there may be small effects of downlink turbulence in special cases, they would not be significant. Diffraction loss is still calculated and included as ground link loss. In uplink transmission, however, the loss due to turbulence is significant as the distorted beam has to travel a long distance before reaching the receiver telescope on the satellite, resulting in a broad and fragmented beam at the satellite. The size of the beam spot at the satellite can be quantified by the long-term beam waist WLT, withwhere w0 is the beam waist emanating from the ground telescope, L = satellite elevation in meters, k = 2π / λ and z0 = π(w02 / λ) with light wavelength λ and r0 as Fried parameter or coherence length. The Fried parameter r0 can be calculated using,whereis the atmospheric structure constant with A = 1.7 x 10-14 and v = 21 m / s. Upon calculating WLT, the uplink loss can be calculated from Eq. (5). WLT is almost independent of w0 - as long as it is above a certain threshold - as the turbulence term (2[4L / kr0]2) dominates over the diffraction term in the uplink. Hence, the choice of w0 is not very significant. The present disclosure has taken w0 = 25cm.

[0151] In Figure 5(e), the uplink turbulence loss is included for ASQN, with parameters such as satellite telescope diameter (d), satellite to ground distance (Lsg = 200 km), and wavelength (λ = 800 nm). The loss due to uplink turbulence is lower when using low elevation satellites (i.e., small Lsg) and large diameter telescopes (i.e. large d), which are naturally present in ASQN. However, these loss calculations do not consider adaptive optics corrections, which can partially compensate for turbulence losses at the cost of moresophisticated systems like segmented mirror telescopes and laser guide stars.

[0152] Despite the detrimental effects of atmospheric turbulence on the beam, the qubit transmission scheme in Fig.1(a) and Fig.4(a) works well, as shown through detailed numerical modeling. The turbulence is simulated using phase screens constructed following Kolmogorov’s theory, implemented by the python module AOtools. Although the highly fragmented and spread out final beam profile is created due to the turbulence, the qubit transmission proposal is not significantly affected. This is because part of the uplink transmitted beam captured through the aperture is considered a constant wavefront, and the fragmented beam is focused, generating a Gaussian-like shaped beam. The ASQN is flexible with focal lengths, even with large apertures, allowing even a distorted beam to be faithfully transmitted through the lens system over global distances.

[0153] FIG.6a is a schematic illustration of atmospheric turbulence modeled using successive phase screens, completely fragments the initial Gaussian beam by the time it reaches the satellite according to the embodiments of the present disclosure.

[0154] FIG. 6b, 6c and 6d is a schematic representation illustrating a large fragmented beam (6b), focused beam (6c) and the final beam (d) according to the embodiments of the present disclosure.

[0155] FIG.6e is a graph illustrating a plot of average light propagation loss with propagation distance according to the embodiments of the present disclosure.

[0156] The simulations in Fig. 6(b)-(d) are carried out for one particular case of turbulence, and to find out the average loss due to turbulence, the simulation is repeated 300 times and averaged. The average propagation loss for transmission up to 20,000 km isshown in Fig. 6(e), which clearly shows that the bulk of the loss is due to the initial turbulence effect, while afterwards there is only a small loss over the lateral 20,000 km propagation. This supports the reasoning behind the qubit transmission scheme described in section III B.

[0157] However, the detrimental effect of beam fragmentation still exists and influences the lens diameter and separation relationship. In the presence of turbulence, complete light confinement by the lens system can only be achieved by using larger diameter or smaller separation lenses than needed otherwise. For example, 60 cm diameter 'satellite lenses' need to be separated by 80 km instead of the 120 km separation used in Fig. 4(b) when the turbulence effect was not considered. This effect occurs due to the distorted beam profile, which contains higher order modes and has a shorter Rayleigh range compared to an ideal Gaussian beam.

[0158] FIG.6f is a schematic illustration of a network of satellites arranged in a 2D mesh (or more generally a 3D structure), consisting of individual satellite chains as previously detailed, facilitates quantum communication across the globe according to the embodiments of the present disclosure.

[0159] FIG.6g is a schematic illustration of quantum information being transmitted from point A to point B using a single link between two satellite chains (specifically between satellites C and D), with the remainder of the transmission occurring along the respective satellite chains according to the embodiments of the present disclosure.

[0160] FIG.6h is a schematic illustration of distance of the connection between the two different chains will dynamically change over time, as depicted here at a slightly later time according to the embodiments of the present disclosure.

[0161] Although tracking for all the individual satellites is not necessary for in the present protocol, it requires need tracking (pointing, and point-ahead) in at least two cases. One is tracking towards the ground stations. For ground tracking in the case of Micius, fine tracking accuracy of 0.4 μrad has already been achieved.

[0162] Another scenario where real-time tracking (pointing, and point-ahead) capabilities are necessary is when establishing a network of satellites arranged in 2D (or more generally 3D) constellations, as depicted in Fig.6f. Such a constellation will enable comprehensive global coverage for quantum communication between any two locations on Earth. Within this network, tracking is required to direct photons towards a satellite in a different chain, as illustrated in Fig.6g. However, only one link is needed for this purpose, as quantum information can be transmitted from point A to point B using a single link between two satellite chains (specifically between satellites C and D), with the remainder of the transmission occurring along the respective satellite chains, as shown.

[0163] Another complexity in a 2D satellite network is that the distance between satellites at the link connecting the two chains can vary over time (Fig. 6(h)), potentially causing some transmission loss. Therefore, real-time adjustment of the satellite lens focal length (by dynamically adjusting the focal lengths of the mirrors used in the satellites) is beneficial. However, even without this adjustment, the additional diffraction loss should not be significant since it will only affect a single satellite link. Influences of Different Factors Mirror Reflectivity

[0164] To ensure the success of the protocols, it is crucial to minimize absorptionloss during reflection from the telescope system. Absorption due to reflection increases exponentially with the number of satellites, so it is important to maintain minimal absorption at each reflection. Various simulations were conducted to determine the total loss for 2% and 5% satellite absorption loss and found that each telescope mirror must have absorption loss of less than 0.5% or 1.25% (i.e., 99.5% or 98.75% reflectivity) to achieve total satellite loss below 2% or 5%, respectively, if all four mirrors are made from the same material.

[0165] While the reflectivity of gold and silver mirrors is high at higher wavelengths, care must be taken in choosing the reflective coating and wavelength used for the beam. Atmosphere absorption and Rayleigh length are dependent on wavelength, so the appropriate wavelength is chosen for specific reflective materials. Metal coatings are delicate and require protective coating, which can affect their reflectivity. Bragg mirrors are an alternative option and can achieve very high reflectivity (as high as 99.9999%) through the principle of constructive and destructive interference, but their fabrication can be challenging due to their large size. Bragg mirrors with a diameter of 10 cm are commercially available, and they have excellent reflectivity for different spectral ranges. The angle of incidence must be kept low to achieve high enough reflectivity and low enough polarization aberration, especially for wavelength specific Bragg mirrors. There are however broadband Bragg mirrors available, which can allow wider incidence angle. By using mirrors with high reflectivity, mirror reflectivity loss can be eliminated, even at global distances of 20,000 km. Moreover, smaller Bragg mirrors can be used as front mirrors along with larger metal back mirrors (as shown in Fig. 7) and the reflectivity requirement of metal mirrors would be less stringent. Alignment, Tracking and Beam Deviation

[0166] In the ASQN protocol, reflector satellites move in a chain within the same orbit, making them co-moving and stationary with respect to each other. This eliminates the need for dynamic tracking for light propagation, as the satellites only need to be aligned. This is a significant advantage, as tracking over a large number of rapidly moving satellites would be infeasible due to significant beam deviation losses. However, there may still be some relative motion between the satellites due to possible slight differences in their orbit, which may require some tracking. This would be slower and less stringent requirement than dynamic tracking, and beam deviation losses may primarily be due to alignment losses.

[0167] Although tracking for all individual satellites is not necessary, it is required for two specific cases: tracking towards the ground stations and sending photons towards a satellite in a different chain, which is needed for a 2D network of satellites. Moreover, to achieve a complete global quantum communication protocol using a network of 2D satellites, the focal lengths of the mirror used in the satellite can be dynamically adjusted to compensate for changes in satellite-to-satellite distance over time. However, even achieving 1D transmission to global distances using one chain of satellites is a substantial achievement.

[0168] To track and align the satellites, a high precision acquiring, pointing, and tracking (APT) system is needed. In ASQN, if only one tracking beam is used the tracking beam must pass through the same path as the transmitted photons, going through multiple satellites from the source to collection points. Dichroic mirrors or beam splitters with very small reflectivity can be used for this purpose, with these optical elements being dynamically brought in or out of the light path as needed to minimize loss during operation. However, for alignment using asymmetric beam splitters, large power may be necessary for the tracking beam. Artificial intelligence may also be useful in handling complicatedtracking and alignment issues quickly. Telescope setups, vortex beam and focal length

[0169] FIG.7a, 7b, 7c, 7d and 7e illustrate a plurality of possible telescope setups suitable for a chain of satellite reflectors showing focusing and bending according to the embodiments of the present disclosure.

[0170] FIG.7f illustrates a simulation of entangled photon pair propagation through 20,000 km in vortex beam profile through the on-axis system showing nearly identical initial and final beams (after 10,000 km propagation by each photon) according to the embodiments of the present disclosure.

[0171] FIG. 7g is a graph illustrating a plot of entangled pair transmission probability with distance according to the embodiments of the present disclosure.

[0172] According to an embodiment of the present disclosure, Figure 7 illustrates various possible telescope setups for a reflector satellite. In particular, Figs. 7(a), (b) and (c) depict off-axis telescopes, while Fig.7(d) shows an on-axis telescope. Figs.7(a) and (d) employ four mirrors and have both front and back mirrors for each of the two telescopes. The back mirrors in these setups are significantly curved, enabling them to focus light at a short distance of around 1 meter. On the other hand, Fig. 7(b) is a two-mirror setup with slightly curved mirrors, having a focal length of about 70 meters. Each of these three setups has its own advantages and disadvantages, which will be explored in detail in this subsection.

[0173] Off-axis telescopes typically face more severe beam aberration compared to on-axis telescopes. However, off-axis telescope setups also have their own benefits. InASQN, a slight bending of the light is required at each satellite to facilitate light propagation along the curvature of the Earth. This slight bending can be naturally achieved in off-axis setups, as shown in Figs. 7(a) and (b), without the need for additional mirrors. On-axis telescopes, however, require additional plane mirrors (or fold mirrors) in the middle to provide the necessary bending. The use of additional mirrors would result in more loss and polarization aberration, although these effects can be mitigated by using Bragg mirrors and small incidence angles.

[0174] Another significant advantage of off-axis telescopes is that they do not obstruct the incoming light beam, unlike the front mirror in an on-axis telescope. This makes off-axis telescopes a more suitable choice for the ASQN protocol, as successive obstruction can cause photon loss. Nonetheless, an on-axis telescope can still be used by employing a vortex beam, as depicted in Figs. 7(e)-(g) and explained below. All the simulations presented previously assume off-axis telescope setups, as no light obstruction due to the front mirror is assumed.

[0175] An on-axis telescope would obstruct the central portion of a Gaussian beam. To transmit the entire beam, a vortex beam can be used instead. Vortex beams have a doughnut-shaped intensity profile, as shown in Fig. 7(e), and can be represented by the following equation:with vortex charge m. w0is the beam waist radius, is the spot size parameter, E0 is the electric field amplitude at origin (ρ =0, z = 0), R( ^) = ^[1 + (( ^ / ^R)2)] isradius of curvature at position z and n( ^) = arctan ( ^ / ^r) is the Gouy phase while ^R =isthe Rayleigh range for the beam.

[0176] The arrangement depicted in Fig.7(d), known as the On-axis setup, can be represented using a lens placed between two screens, as illustrated in Fig.7(d). The screens symbolize the hindrance caused by the front mirrors to the light, while the lens imitates the effective focusing due to the four mirror surfaces, similar to the off-axis setups. In the proposed model, the bending of light is not taken into account, as in the off-axis models depicted in Fig. 3 and Fig. 4. As a result, the folding plane mirrors are not included. A sequence of such satellites with on-axis telescopes is modeled by a sequence of screen- lens-screen setups (as shown in Fig.7(e)), which simulate the propagation of the beam. By adjusting the beam and telescope parameters, the vortex beam can be transmitted flawlessly. For a telescope back mirror diameter (d) of 60 cm, satellite separation (L0) of 80 km, vertex charge (m) of 2, and front mirror diameter of (d / 10) 6 cm, the transmission probability of the vortex beam is almost perfect, with a value of 0.6 at 20,000 km (excluding the ground link). The initial and final profiles of the vortex beam are presented in Fig.7(e). Only diffraction loss is taken into account here. The final profile is slightly dimmer and truncated on the inner edge, with very small differences compared to the initial one, thus confirming the faithful propagation of the beam. In Fig.7(f), the decrease in transmission probability with distance follows a similar trend as that of Gaussian beams in Fig. 4(c). However, the beam parameters differ because the Rayleigh range is no longer zRwhich is only valid for an ideal Gaussian beam. For higher-order modes, the Rayleigh range decreases, and therefore, the satellite separation (L0) needs to decrease as well to contain diffraction loss.

[0177] When modeling the transmission of vortex beams through an on-axis setup, the front mirror holders are not taken into account. These holders would obstruct the light path if they were present, unlike in the off-axis telescope case, which would affect diffraction at each satellite along with intensity loss. However, since the satellites would be weightless in LEO, the front mirror holders are not entirely necessary. During final transmission, the front mirror holders can be detached, and the alignment of the front mirror can be physically adjusted using the holder after detachment but before complete withdrawal. Alternatively, remote alignment through electromagnetism could be attempted. Any minor misalignment could also be corrected by adjusting the plane fold mirrors' alignment.

[0178] Another important factor to consider is how the telescopes in Figure 7(a)- (c) need to be adjusted to achieve the required focusing. The effective focal length produced by the mirror setups can be changed by adjusting the positions of the mirrors. In the setups of Figure 7(a) and Figure 7(c), it is possible to adjust mirror positions to maintain the parallelism of rays, owing to the inherent reflection symmetry of the setups. Thus, these setups behave effectively as apertures or lenses with an infinite focal length. However, changing the mirror positions would change the focal lengths, which can be calculated as follows. The four mirror setups in Figure 7(a) and Figure 7(c) can be modeled by two lenses if the two front mirrors are considered as plane mirrors. If this assumption is not made, the analysis would be more complex, but the result would be similar. Curved mirrors are equivalent to lenses. The two back mirrors are treated as two lenses with focal lengths f1 and f2 separated by a distance d. The compound or effective focal length of this lens system measured from the back mirror (with focal length f2) is given by fe = f2(d - f1) / (d - (f1+f2)). If parallel rays are kept parallel, fe = infinity, implying d = (f1 + f2). The two curved backmirrors (and hence the lenses) are assumed to have the same focal length f1 = f2 = f and d = 2f. If the separation is changed by Δ, i.e., from d to d + Δ, the effective focal length becomes fe = f(f + Δ) / Δ = f2 / if f >>Δ . In Figure 7(a) and Figure 7(c), the separation between back mirrors can be increased by shifting front mirrors only. To achieve fe = 50 km with f = 1 m, one must have Δ = 20µm. Although it may seem challenging to shift the front mirror (with a diameter of approximately 10 cm and a weight of 5 kg) by such a small amount, commercial piezo motors can accomplish this regularly on Earth. However, even with this technique, there would still be some errors in the focal length. Fortunately, the effects of a focal length error grow slowly, and a 10% focal length error would only decrease transmission probability by 4.4 dB.

[0179] Now, let us consider the case of Figure 7(b), where f1 and f2 have opposing signs as the mirrors have opposite curvatures. However, one can still achieve d = f1 + f2 with f1 and f2 having slightly different values if |f1| and |f2| >> d. For small shifts inkm, one may set f = 70 m and Δ = 1 cm. Thus, it is possible to have near-zero curvature mirrors while making the mirror position adjustment easier.

[0180] This can be pushed this further by using mirrors with even smaller curvature for the telescopes in the Fig.7(b). In such a setup, the effective focal length will be nearly independent of the telescope separation \(d\). To achieve this, the difference between the magnitudes of the two focal lengths (\(f_1\) and \(f_2\)) needs to be much larger than the telescope separation \(d\), i.e., \(||f_1| - |f_2|| = \alpha f_2 \gg d\), where \(\alpha\) denotes a fraction like 0.1 or 0.01. In that case, \(f_e \approx \frac{f_2f_1}{\alpha f_2} \approx \frac{f_1}{\alpha}\).

[0181] For example, with \(\alpha = \frac{1}{50}\) and \(f_e = 50 \text{ km}\), we would have \(f_1 = \frac{f_e}{50} = 1 \text{ km}\). In this case, \(||f_1| - |f_2|| = \alpha f_2 = 20 \text{ m} \gg d\) if \(d \approx 1 \text{ m}\). This is just an example, and other values of \(\alpha\) and hence \(f_1\) can be chosen. Here, the satellite lens focal length will be robust against telescope position fluctuations, although we would lose the ability to adjust the telescope focal lengths, and mirrors with large focal lengths of 1 km would need to be manufactured.

[0182] One can have telescope setups with even larger focal lengths, like the two concave mirror setup shown in Fig 7 (a) to (c). The focusing capability of this telescope setup is practically independent of the position fluctuation of the mirrors. If the individual focal lengths of the two concave telescope mirrors are equal (\(f_1 = f_2 = f\)), then the effective focal length is \(f_e = f / 2\), independent of \(d\) as long as \(f \gg d\). The challenge of this design is, of course, creating the extremely large focal length mirrors with \(f = 2 f_e = 100 \text{ km}\) for \(f_e = 50 \text{ km}\).

[0183] According to an embodiment of the present invention, we can push this further by using mirrors with even smaller curvature for the telescopes in the Fig.7(b). In such a setup, the effective focal length will be nearly independent of the telescope separation \(d\). To achieve this, the difference between the magnitudes of the two focal lengths (\(f_1\) and \(f_2\)) needs to be much larger than the telescope separation \(d\), i.e., \(||f_1| - |f_2|| = \alpha f_2 \gg d\), where \(\alpha\) denotes a fraction like 0.1 or 0.01. In that case, \(f_e \approx \frac{f_2f_1}{\alpha f_2} \approx \frac{f_1}{\alpha}\).

[0184] For example, with \(\alpha = \frac{1}{50}\) and \(f_e = 50 \text{ km}\), we would have \(f_1 = \frac{f_e}{50} = 1 \text{ km}\). In this case, \(||f_1| - |f_2|| = \alpha f_2= 20 \text{ m} \gg d\) if \(d \approx 1 \text{ m}\). This is just an example, and other values of \(\alpha\) and hence \(f_1\) can be chosen. Here, the satellite lens focal length will be robust against telescope position fluctuations, although we would lose the ability to adjust the telescope focal lengths, and mirrors with large focal lengths of 1 km would need to be manufactured.

[0185] One can have telescope setups with even larger focal lengths, like the two concave mirror setup shown in Fig 7(c)). The focusing capability of this telescope setup is practically independent of the position fluctuation of the mirrors. If the individual focal lengths of the two concave telescope mirrors are equal (\(f_1 = f_2 = f\)), then the effective focal length is \(f_e = f / 2\), independent of \(d\) as long as \(f \gg d\). The challenge of this design is, of course, creating the extremely large focal length mirrors with \(f = 2 f_e = 100 \text{ km}\) for \(f_e = 50 \text{ km}\). Aberration

[0186] The deviation of an optical system from its ideal working scenario is called aberration. There are different forms of aberration like geometric, wavefront, chromatic, and polarization aberration, which can be caused by telescope mirrors. Wavefront aberration is caused by the variation of optical path lengths due to the geometry of reflecting or refracting surfaces. Chromatic aberration originates due to frequency-dependent refractive index or reflection coefficient of an optical element. Polarization aberration occurs because of the variation of two polarization components (s and p) in refracted or reflected light depending on the angle of incidence.

[0187] Wavefront aberration consists of different forms like spherical aberration, coma, or astigmatism, while geometric aberrations are beam spatial mode transformationsdue to reflection. These can be compensated by arranging reflecting surfaces properly, although it is very difficult to compensate for all forms of aberration together. One such aberration-compensating system is a three-mirror anastigmat, which corrects coma, astigmatism, and spherical aberration together to a large degree and can be employed in ASQN. Plane mirrors, on the other hand, produce no wavefront aberration, but they do not provide beam focusing, which is necessary for ASQN.

[0188] Numerical simulations did not consider any effects of aberration as the telescope systems are modeled as a single lens. Chromatic aberration can be neglected if the variation of the reflection coefficient is small over the pulse bandwidth. Polarization aberration is an important concern for ASQN setup as it may cause geometric aberration effects that can affect diffraction. Polarization aberration of a reflecting surface depends on two quantities, retardance and diattenuation, which in turn depends on the reflection coefficients. As both retardance and diattenuation vary quadratically with the angle of incidence, polarization aberration can be significantly diminished by reducing the largest angle of incidence.

[0189] In conclusion, different telescope setups would have different advantages and challenges. The off-axis setup of Fig.7(a) would provide no obstruction, easy bending of light in two dimensions, small polarization aberration while there may be detrimental geometric aberration effects and fine mirror position adjustment would be needed for focusing. Mirrors in Fig.7(b) would possibly have smaller geometric aberration effects and crude mirror adjustments would be sufficient for focusing although there may be larger polarization aberration and other difficulties in using a two-mirror setup, e.g., difficulty in aligning for ground link transmission. The on-axis setup in Fig.7(d) would probably suffer the least geometric aberration but to bend the light multiple fold mirrors would be neededwhich would increase both reflection loss and polarization aberration. The obstruction of light by the front mirror would necessitate the use of vortex beam instead of Gaussian beams needed for earlier setups. Considering this myriad of benefits and challenges different telescope setups can be used in different satellites in the same chain. For example, Fig.7(a) telescope setups can be used in special satellites for ground links while Fig.7(b) setups in most of the satellites to control aberrations in propagation.

[0190] Another interesting point to note is that, it is quite possible that even significant aberration effects due to telescope mirrors doesn’t affect the diffraction loss a lot. One promising example in this direction is the propagation of the fragmented uplink turbulent beam shown in Fig.6. Although the fragmented beam was focused, its shape (in Fig. 6(c)) is far from an ideal Gaussian and still the beam propagated to 20,000 km with very less loss under the very same lens configuration with only a little less satellite separation distance (L0). Tabletop Experiments

[0191] In addition to theoretical analysis, laboratory experiments can be used to verify the influence of many factors discussed above. To achieve this, mirror setups similar to those used in actual satellite chains can be used, but with smaller sizes as shown in Figure 3(a), where both the distance between mirrors (L0) and the diameter of the mirrors (d) are decreased. For instance, to restrict the Raleigh range to a manageable distance of 1 meter, an 800 nm laser beam needs to have a beam waist of approximately 0.5 mm. Mirrors of size 2 mm are needed, considering the diameter of the mirrors (lens in Fig. 7(a)) to be 4 times the beam waist (w0). The back mirrors would have this size, while the front mirrors would be even smaller. Micro mirrors of diameter in millimeters or even smaller are usedin various optical device applications. Moreover, ultra-small parabolic mirrors have also been developed, and thus, it should be possible to adapt these mirrors to the setup. If necessary, even larger scale mirrors (centimeters in size) can be used by fitting an appropriately sized aperture next to them. However, this may cause additional diffraction effects depending on the fitting of the aperture to the mirror. The experiment can also be performed using beams with a larger beam waist (around 1cm), but the Rayleigh range would increase to hundreds of meters, possibly necessitating an outdoor experiment. Quantum Internet

[0192] One of the long-term applications of the protocols of the present disclosure is the construction of quantum internet capabilities. Quantum memories may be necessary to implement quantum internet, especially for complicated tasks. However, some of the strict requirements on quantum memories can be relaxed due to the elimination of diffraction loss. The basic building blocks of a quantum internet would be heralded entangled pairs distributed between two distant points. These entangled pairs can then be utilized to interface quantum computers for distributed quantum computing or quantum sensing applications.

[0193] There are two main reasons why memories are needed. Firstly, they are needed to store the qubit while waiting for confirmation on the heralded entangled pairs. For heralding, there are two main strategies, using sophisticated quantum non-demolition (QND) detectors at each end or detecting photons in the middle. In both cases, detection results need to be received to determine which photons are part of an entangled pair. Secondly, memories maybe required for the more fundamental reason of maintaining causality. In general quantum internet protocols would require quantum operationsconditioned on classical communication of measurement results from the other end to obey causality (like in quantum teleportation). Hence, the optical qubits need to be stored during the classical communication. Additionally, memories may help boost the communication rate like repeater protocols. Although that is not necessary in ASQN, it can be useful.

[0194] FIG. 8a is a schematic diagram illustrating quantum internet by entanglement distribution with entangled source on satellite (S1) and on ground station (S2) according to the embodiments of the present disclosure.

[0195] FIG. 8b is a schematic diagram illustrating the quantum internet using a repeater scheme according to the embodiments of the present disclosure.

[0196] FIG. 8c is a diagram illustrating multiple teleportation without using quantum memories according to an embodiment of the present disclosure.

[0197] Using memories, one strategy to create a quantum internet with entanglement distributed from a source in the middle is shown in Figure 8(a). QND detectors are used to non-destructively measure which photons have successfully arrived, and these photons are then stored in quantum memory. For long-distance links, high storage time (≈ 100 ms) quantum memories would be required. High memory efficiency is not essential as there are only two memories involved, in contrast to the many memories needed in a repeater scheme.

[0198] The other strategy to create a functioning quantum internet would be to create one link of a quantum repeater using ASQN, as shown in Figure 8(b). This would not require a QND detector but would require memories with high storage time and multimode capacity. Again, high efficiency would not be a necessity. In this architecture,photons will be sent from the two end stations, and entanglement would be created and heralded by Bell state measurement (BSM) at the midpoint. Optional QND detectors and quantum memories can be used to increase rate.

[0199] Another strategy towards quantum internet would be to use the entanglement distribution by qubit transmission protocol described in Section II. This would not require a repeater protocol. In entanglement distribution by qubit transmission, one photon of an entangled pair is stored in a memory while the other is sent by qubit transmission to the destination ground station. By using another quantum memory in the destination ground station, along with a QND detector, quantum internet capabilities can be enabled.

[0200] Memories may not be required at all in certain cases, as described in Figure 8(c), where a way of creating heralded entangled pairs would be to send qubits redundantly while implementing quantum internet protocols to mitigate the effect of loss. Such redundancy would be important for small losses, such as for entanglement distribution through downlink where turbulence loss can be very small. For very small losses the redundancy approach would essentially be equivalent to heralding approach as even while heralding, quantum memory and the subsequent quantum operations all would have some inherent losses themselves. The other reason to use memories - i.e., causality constraining us to wait for classical communication - can also be circumvented using the following technique. To perform a certain operation with an entangled photon pair, after a photonic qubit is measured at one end, the generated classical communication signal needs to reach the other end before the other qubit (the other photon of the entangled pair) reaches there. This can be achieved in entanglement distribution using a satellite source (S1 in Fig.1(b)) if the two photons emitted from the satellite travel unequal paths with difference betweenthe two optical paths being greater than the classical communication optical path length. This is possible in 2D constellation of quantum satellites where one photon takes intentionally longer path through the 2D mesh, as seen in Fig.8(C). Using this scheme for example, an unknown qubit can be teleported it to a faraway place, certain quantum operations can be done on it and the output qubit can be teleported back. The whole process would not require memory or QND detectors, although some redundancy would be required. However, this approach would be challenging due to the larger redundancy required for multiple qubits. Analyzing different errors influencing diffraction

[0201] FIG.9a is a schematic diagram illustrating the effect of ‘satellite lens’ focal length (f) and position (z or xy) error on beam propagation, (i.e., on diffraction loss) is shown according to an embodiment of the present disclosure.

[0202] FIG. 9b is a graph illustrating the effect of f error according to an embodiment of the present disclosure.

[0203] FIG. 9c is a graph illustrating the effect of ^ error according to an embodiment of the present disclosure.

[0204] FIG. 9d is a graph illustrating the effect of xy error according to an embodiment of the present disclosure.

[0205] FIG. 9e is a graph illustrating the effect of total error according to an embodiment of the present disclosure.

[0206] According to an embodiment of the present invention, the impact of different errors in satellite chain setup on diffraction is analyzed. Specifically, the effectsof lens focal length error (f error) is investigated, as well as lens position errors in the ^ and xy directions (z error and xy error). The errors are illustrated in Fig.9(a). Focal length error is simulated by introducing random errors in focal lengths (Δf), while ^ and xy errors are simulated by introducing random longitudinal (Δ ^) or lateral displacement (Δr, Δθ) errors. The error values are generated from a uniform random distribution within a given interval (e.g.5% f error means choosing random f error values from the interval [-0.05f, 0.05f]). It is to be noted that 100 different error values is used in each case and calculate photon transmission probability at each satellite link for a total distance of 10,000 km, resulting in 100 different photon transmission probabilities for each distance. Further calculate their average and variance, and plot the average photon transmission probability with distance while also showing the standard deviation at each point in each plot. The standard deviation values increase as the effect of the errors increases (i.e., mean probability decreases), either due to large error values or with increasing distance or both. If there is no effect of errors (e.g., for negligibly small error values), there would be identical photon transmission values for all error values and thus variance would become zero.

[0207] In this analysis, the average photon transmission probability is used to assess the effect of the error. First, the effect of each of these errors on diffraction loss in entanglement distribution is considered individually in Fig. 9(b)-(d) and then their combined effect is seen in Fig.9(e). In all cases, d=60 cm, L0=120 km is considered, and a total distance of L=10,000 km, which is half of the global distance (20,000 km) and hence the distance one photon of an entangled pair needs to travel. The diffraction loss without any errors for the above parameters can be deduced to be 0.925, using Fig.3(b).

[0208] First, the effect of f error is considered. In Fig.9(b), two cases with uniform random focal length errors are considered. These errors reside within intervals of ±2.5%and ±10% of focal length (f), respectively. In each case, diffraction is simulated for 100 different runs of focal length (f) errors. Further the mean and standard deviation of photon transmission probability is calculated at each aperture from these 100 runs, which are plotted in Fig. 9(b). It observed that for 2.5% f error probability, the mean drops to 0.87, while for 10% error, it drops to 0.36. These probabilities are respectively 94% and 39% of the original photon transmission probability (0.925) without any errors. It shows that 2.5% f error is quite acceptable, and even a 10% error doesn't have an extreme effect. The focal length error originates due to uncontrollable changes in focal length, which can occur either due to errors in adjusting mirror position for adjustable focal lengths setups or while manufacturing for fixed focal length setups. To achieve a focal length error below 10%, one needs a mirror position movement accuracy of 2 µm for front mirrors in Fig.7(a) and Fig.7(c), while only 1 mm for Fig.7(b), although in the absence of any front mirror, the larger telescope back mirrors need to be moved.

[0209] The mean photon transmission probability in the presence of xy error is analyzed in Fig. 9(d), where xy error is the error due to a shift in lens centroid in the xy directions. This type of error can also be referred to as r-theta error, as the lens centroid error is calculated by taking random values in r (within a bound) and theta. Two error values are presented in Fig.9(d), given by r = 0:3 cm and 3 cm, which correspond to 1% and 10%, respectively, of the telescope radius R = 30 cm. The photon transmission probability decreases to 0.90 and 0.12 at 10,000 kilometers, which are 97% and 13%, respectively, of the original case with no errors. This large error for 3 cm xy error can be attributed to a shifted lens, which truncates light more on one side and deflects the beam at an angle. The angular deflection causes the beam to go out of path, and as a result, it gets even more truncated by subsequent lenses. One way such xy position error can occur for the effective'satellite lens' is by satellite position error. However, satellite position errors in xy can be easily eliminated by remembering that the 'satellite lenses' are actually made of mirrors that can reflect light at a slightly different angle compensating for the xy deviation. This would result in slightly longer propagation distance and cause z error. Nevertheless, compensating this small xy deviation (say, 1 m) in a large link (say, ^ = 100 km) would cause completely negligible z error (p(105)2 + 1-105) 5×10-6 m = 5 µm). More importantly, satellite xy position error would actually produce an error in satellite velocity as it would change the orbital radius of the satellite. This would again produce z errors and cause some difficulty in satellite alignment and tracking. Beam deviation error is another important source of error related to xy error, which would originate due to errors in telescope mirror alignment in the satellite chain. However, beam deviation error cannot be simulated using the lens system as it originates from the mirrors used in the telescopes, which can deflect light. Nevertheless, xy error is similar to beam deviation error in many ways as shifted lenses do deflect light, and beam deviations also cause beam truncation as beams fall on telescope mirrors of center. It is important to note that beam deviation error is one of the most important concerns in existing quantum satellite experiments too where it manifests as tracking error. For instance, the tracking error for the Micius satellite was 0.41 radians, which would result in a xy error of 4.1 cm for 100 km satellite separation. Considering 0.41 µ rad was dynamic tracking error in a satellite-ground link, alignment error in a relatively stationary satellite chain should be considerably less. Hence, the xy error values used in the simulation above (0.3 cm and 3 cm) were within reasonable limits. Hence, xy error simulation do gives us some ideas about the effect of beam deviation error.

[0210] In Fig.9(e) the combined effect of f, z and xy errors in photon transmission is investigated. In one case, f, z and xy errors of 2.5%, 5%, 1% respectively generated anaverage photon transmission probability of 0.78, while in another case 5%, 10% and 2% respective errors generated a mean probability of 0.48. These mean probabilities of photon transmission (at 10,000 km), in presence of errors, are respectively 84 % (0.76 dB) and 51 % (2.92 dB) of the 92.5 % photon transmission probability, while not considering any error. Thus, the different satellite change setup errors do affect photon transmission probability, although even their combined effect can probably be constrained to only a few dB of extra loss if the above-described system parameter regimes can be achieved. This shows the robustness of ASQN against different forms of errors.

[0211] Overall, the embodiment of the present disclosure discloses a protocol for satellite-relayed space-based quantum communication, which utilizes "satellite lenses" to eliminate diffraction loss and enable low-loss photon transmission even to global distances of 20,000 km. This protocol, known as ASQN, is especially useful for establishing quantum communication, including secure QKD, at large distances between 5,000 and 20,000 km. However, ASQN can also be used for communication over shorter distances, and it has the potential to be the preferred quantum communication protocol with least loss over almost the whole range of distances (200- 20,000 km) on Earth. The protocol does not require the development of quantum memories, which has been a major challenge for quantum networks.

[0212] As ASQN eliminates diffraction loss, other factors such as mirror reflectivity, beam deviation, aberration, turbulence, and more become important in determining photon loss. The present disclosure investigated these effects in detail and discloses possible ways to mitigate their detrimental effects. For example, vortex beam transmission was successfully simulated as an alternative to Gaussian beams obstructed by front mirrors in on-axis telescope setups. Further theoretical analysis and proposedlaboratory experiments would be helpful to confirm the influence of all factors on photon transmission.

[0213] The present disclosure also discloses the "qubit transmission" protocol, which has multiple advantages by having both sources and detectors on the ground. Despite the adverse effect of uplink turbulence, the present disclosure established that "qubit transmission" can deliver comparable or even larger rates than regular entanglement distribution.

[0214] The present invention enables the efficient transmission of quantum information over vast distances, overcoming the limitations of traditional fiber optic networks by utilizing a satellite relay system. This facilitates global-scale quantum communication and expands the reach of quantum technologies.

[0215] The system is designed with a focus on minimizing photon loss during transmission. This is achieved through a multi-pronged approach creating effective satellite lenses for focusing incoming light, incorporating high reflectivity Bragg mirrors, adjustable focal length lenses, optimized satellite positioning, and large diameter mirrors. These features collectively contribute to maintaining signal integrity and ensuring reliable quantum communication over long distances.

[0216] The present invention's versatility allows for the implementation of various quantum communication protocols, including direct transmission, entanglement distribution, and using various forms of encoding like time-bin or frequency-bin qubits or higher dimensional continuous variable encodings. This versatility makes the system adaptable to a wide range of quantum applications, from secure communication through quantum key distribution (QKD), quantum sensing to distributed quantum computing inlong-term future to quantum computing.

[0217] The inclusion of optional quantum memories within the system further enhances its capabilities. These quantum memories have the potential to increase system efficiency and enable enable novel functionalities such as distributed quantum computing and other complex quantum protocols.

[0218] In addition, the optional incorporation of frequency multiplexing modules increases the rate of quantum information transfer by allowing for the simultaneous transmission of multiple signals on different frequencies. This can significantly boost the system's overall throughput and efficiency.

[0219] Examples described herein can also be used in various other scenarios and for various purposes. It may be noted that the above-described examples of the present solution are for the purpose of illustration only. Although the solution has been described in conjunction with a specific embodiment thereof, numerous modifications may be possible without materially departing from the instructions and advantages of the subject matter described herein. Other substitutions, modifications, and changes may be made without departing from the spirit of the present solution. All of the features disclosed in this specification (including any accompanying claims, abstract, and drawings), and / or all of the steps of any method or process so disclosed, may be combined in any arrangement, except combinations where at least some of such features and / or steps are mutually exclusive.

[0220] The present description has been shown and described with reference to the foregoing examples. It is understood, however, that other forms, details, and examples can be made without departing from the spirit and scope of the present subject matter.

Claims

We Claim:

1. A satellite relayed quantum communication system, comprising: a network of satellites with one or more chains of nearly synchronously moving satellites; each satellite in a chain comprises a telescope module with one or more telescopes effectively creating a satellite lens, wherein the satellite lens comprises a plurality of mirrors configured to focus light, thereby containing beam divergence and directing the light to the next satellite in the chain or to another fixed or mobile station for transmitting quantum information across variable distances; and a tracking module configured for tracking between satellite chains, to fixed or mobile stations and residual tracking within each satellite chain wherein the satellites are moving nearly synchronously, thereby making tracking needs within a satellite chain minimal.

2. The system as claimed in claim 1, wherein transmission of quantum information across variable distances on Earth is achieved by directing the light along a path that follows the curvature of the Earth.

3. The system as claimed in claim 1, wherein the relationship between satellite lens diameter, consequently telescope diameters, satellite separation and wavelength of light used for minimising solely diffraction loss is determined using both an analytical approach and numerical modelling, whereby as an example figure a 60 cm diameter satellite lens necessitates a satellite separation smaller than 120 km for 810 nm wavelength light in gaussian mode, in case of diffraction-limited light propagation.

4. The system as claimed in claim 1, wherein at least one of the front or back mirrors of the satellite lens is a high reflectivity Bragg mirror to minimize reflection loss.

5. The system as claimed in claim 1, wherein the satellite lens is constructed using a telescope assembly setup comprising at least one of: off-axis telescopes, on-axis telescopes, or telescopes without front mirrors.

6. The system as claimed in claim 1, wherein beacon lasers are coupled through the same path as the quantum signal for tracking, including pointing and point-ahead, in satellite-to-satellite and satellite to a fixed or mobile station links.

7. The system as claimed in claim 1, wherein a network of spacecraft relay performs quantum communication in deep space instead of satellites around a celestial body.

8. The system as claimed in claim 1, further comprising a quantum light source located within the satellite or at a station to facilitate quantum communication protocols, including direct transmission from one station to another through the satellite relay and entanglement distribution either from a satellite-based quantum light source or from a source at a station.

9. The system as claimed in claim 1, further comprises quantum memories integrated into the satellite relay system, either in a satellite or in a fixed or mobile station, to enable either higher efficiency of the system or further capabilities or both.

10. The system as claimed in claim 1, further comprises frequency multiplexing modules located in either the satellites or in fixed or mobile stations or both to increase the rate of quantum information transfer through frequency multiplexing.

11. The system as claimed in claim 1, wherein at least one of the satellite lenses possess adjustable focal length to control the focusing of incoming light beams as necessary for mitigating diffraction losses associated with long-distance photon transmission and ensuring minimal photon loss.

12. The system as claimed in claim 1, wherein satellite separation inside a satellite chain, satellite lens diameter and telescope mirror design are chosen to minimize total photon loss including diffraction loss, tracking loss, light reflection loss and other overhead losses, these choices being determined based on both theoretical analysis and numerical modelling of light propagation, including the calculation of optimal focal lengths to align with theoretical and simulated propagation models.

13. The system as claimed in claim 1, wherein each satellite's telescope module employs large diameter mirrors to reduce beam truncation loss significantly, enhancing the system's efficiency in photon transmission.

14. The system as claimed in claim 1, wherein satellite separation and satellite lens focal length are designed to be non-uniform within a satellite chain, this design being determined based on both theoretical approaches and numerical modelling of light propagation to align with theoretical and simulated models for minimizing photon loss.

15. The system as claimed in claim 1, configured to support quantum key distribution (QKD) by enabling quantum information transfer between any two points on Earth, using the satellite relay network.

16. A method for transmitting quantum information across variable distances, comprising: deploying a network of one or more chains of nearly synchronously moving satellites, each satellite comprising a telescope module functioning as a satellite lens with a plurality of mirrors; focusing light with the mirrors on each satellite lens to contain beam divergence; directing the focused light to either the next satellite in the chain or to another fixed or mobile station; and transmitting quantum information via the directed light.

17. The method as claimed in claim 16, further comprising quantum information transfer as quantum light signal encoded in different forms, comprising time-bin, frequency- bin or continuous variable forms.

18. The method as claimed in claim 16, further comprises establishing quantum communication between any two points on Earth by: connecting two different satellite chains with a single inter-chain link; enabling communication between satellites moving at high-speed relative to each other in the inter-chain link; andoptionally adjusting the focal length of the satellite lenses in real-time to minimize loss during the inter-chain communication.

19. The method as claimed in claim 16, wherein optimizing for efficient long-distance transmission of quantum information and entanglement distribution enables quantum technology applications like quantum key distribution (QKD), quantum sensing, distributed quantum computing, blind quantum computing and quantum internet functionalities optionally incorporating quantum memories.

20. The method as claimed in claim 16, wherein the reduction of diffraction loss, tracking loss, and reflection loss enables the transfer of quantum information over very long distances.

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