EP4684484A1 - Vacuum lens guide - Google Patents

Abstract

A quantum network, comprising at least a first node comprising a first quantum system, and a second node comprising a second quantum system; and an optical guide in optical communication with the at least the first and the second nodes, wherein the optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure.

Description

UCT-00625 UC 23-T-074 VACUUM LENS GUIDE CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims the benefit of U.S. Provisional Application No. 63/453,348, filed March 20, 2023, which is hereby incorporate by reference in its entirety. GOVERNMENT SUPPORT [0002] This invention was made with government support under grant no. W911NF-21- 1-0325 awarded by the Army Research Office, grants nos. FA9550-19-1-0399, FA9550-21- 1-0209, FA8649-21-P-0781 awarded by the Air Force Office of Scientific Research, grant nos. OMA-1936118, ERC-1941583, OMA-2137642 awarded by the National Science Foundation, and Q-NEXT awarded by the Department of Energy. The government has certain rights in this invention. BACKGROUND OF THE INVENTION [0003] It is an outstanding challenge to build a low-loss quantum communication channel for global scale quantum networks, which enable transformative applications of secure quantum communication, distributed quantum sensing, and network-based quantum computation . The key challenge is the media loss for optical channels, with characteristic attenuation length limited to tens of kilometers for both fiber and free-space, resulting in an exponential decrease in the direct quantum communication rate over long distances, and which implies that an overhead of ^^⁵⁰ will be suffered or communication over 1000 ^^ ^^. To overcome this challenge, there are two existing approaches: (1) quantum repeaters using a backbone of quantum nodes to store intermediate quantum entangled states or perform active quantum error detection and correction to suppress loss and operation errors, (2) satellite- based quantum communication taking advantage of outer space that are essentially vacuum. Quantum repeaters are still under active investigation they require long-lived quantum memory or active quantum error correction; so far there is no scalable demonstration of quantum repeaters. The quantum satellite was launched in 2016, but it is very expensive, with limited bandwidth and weather dependent. Hence, it is highly desirable to develop an ultra- FH11893796.1 UCT-00625 low loss optical channel – with attenuation length of 1000 ^^ ^^ – so that a reliable ultra-fast large-scale quantum network can be built. SUMMARY OF THE INVENTION [0004] In an example embodiment, the present invention is a quantum network, comprising: at least a first node comprising a first quantum system, and a second node comprising a second quantum system; and an optical guide in optical communication with the at least the first and the second nodes, wherein the optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure. [0005] In another example embodiment, the present invention is a method of transmitting a quantum state between nodes of a quantum network. The method comprises: providing a first node comprising a first quantum system, and a second node, comprising a second quantum system; causing the first quantum system to generate at least one electromagnetic (EM) optical mode, the first quantum system being in a quantum state, the optical mode encoding a quantum state; directing the optical mode into an optical guide, the optical guide being in optical communication with the second node, wherein the optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure, and causing the optical mode to interact with the second quantum system, thereby imparting the quantum state onto the second quantum system. BRIEF DESCRIPTION OF THE DRAWINGS [0006] The foregoing will be apparent from the following more particular description of example embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments of the present invention. [0007] Fig.1A is an example map of a series of VBG backbone networks over long distances, according to embodiments of the present disclosure. [0008] Fig.1B is an example map of a regional network of fiber or free-space quantum channels, according to embodiments of the present disclosure. [0009] Fig.1C is an exemplary design of a VBG, according to embodiments of the present disclosure. [0010] Figs.2A-D each is a graph of an effective rate as a function of wavelength under various conditions, according to embodiments of the present disclosure. FH11893796.1 UCT-00625 [0011] Figs.3A-B each is a plot of an effective rate of a quantum channel as a function of transmission distance, according to embodiments of the present disclosure. [0012] Fig.4 is a schematic diagram of an optical guide that can be used by the devices and methods disclosed herein. [0013] Fig.5 is a bar plot showing the power distribution among the modes after the last iteration. [0014] Fig.6 is a conversion tree. [0015] Fig.7 is a bar plot showing the loss rate at the last iteration. [0016] Fig.8 is a plot showing the loss versus iteration number ^^. Curve fit with ^^_^^ = ^^ ^^^{^}. [0017] Fig.9A and Fig.9B are the plots showing that the distance travelled by an optical mode improves with the focal length of the lenses (A) and with the lens radius (B). [0018] Fig.10 is a table listing the adjustable parameters. [0019] Fig.11A is a plot showing the dependency of the distance travelled by an optical mode as a function of its center wavelength. [0020] Fig.11B is a plot showing the wavelength-dependent loss rate (in units of dB/km) as a function of wavelength and pressure. [0021] Fig.11C is a plot showing the dependency of the loss rate on the wavelength is shown for the misalignment conditions described in the text, assuming the air pressure is assumed to be 10^ିଷ ^^ ^^. [0022] Fig.11D is a plot showing quantum channel capacity at particular wavelength. [0023] Fig.12 is a schematic diagram of an example embodiment of a vacuum enclosure. [0024] Fig.13A is a plot illustrating the reflective loss of a two-layer ARC on silica substrate. [0025] Fig.13B is a plot illustrating Q vs distance under losses. [0026] Fig.14 is a plot illustrating a characteristic length under various configurations. [0027] Fig.15 is a schematic diagram of an example of a computing node. DETAILED DESCRIPTION OF THE INVENTION [0028] A description of example embodiments of the invention follows. [0029] The vacuum beam guide (VBG) presents a completely different solution for quantum channels to overcome the limitations of existing fiber and satellite technologies for long-distance quantum communication. With an array of aligned lenses spaced kilometers apart, the VBG offers ultra-high transparency over a wide range of optical wavelengths. With FH11893796.1 UCT-00625 realistic parameters, the VBG can outperform the best fiber by three orders of magnitude in terms of attenuation rate. Consequently, the VBG can enable long-range quantum communication over thousands of kilometers with quantum channel capacity beyond 10¹³ qubit/sec, orders of magnitude higher than the state-of-the-art quantum satellite communication rate. Remarkably, without relying on quantum repeaters, the VBG can provide a ground-based, low-loss, high-bandwidth quantum channel that enables novel distributed quantum information applications for computing, communication, and sensing. [0030] Significant progress has been made in extending the communication distances for quantum networks, including satellite-based quantum entanglement distribution over 1200 km and memory-enhanced quantum communication beyond the repeater-less bounds. However, satellite-based quantum channels are fragile, expensive, and restricted by local weather conditions. Furthermore, quantum repeaters without full quantum error correction still suffer from a polynomial decrease in communication rate over long distances. Hence, it is highly desirable to establish a reliable quantum channel capable of directly transmitting quantum signals over vast distances, such as the continental scale of 10⁴ km (with an attenuation rate of at the level of 10⁻⁴ dB/km) for a wide range of optical frequencies. [0031] To overcome this challenge, the present disclosure provides a vacuum beam guide (VBG) —which uses an array of lenses in an evacuated tube to guide light, as opposed to relying on total reflection induced by fiber. The large vacuum spacing between lenses significantly reduces the effective travel length of light in optical materials, thus eliminating the problem of material absorption. Inspired by quantum communication satellites, the VBG channel is set up within a vacuum chamber tube, which eliminates air absorption and effectively isolates the channel from the outer environment, ensuring robustness against environmental perturbations. Although beam waveguides have been proposed for classical optical communication, communications were previously taken over by low-cost optical fiber sufficient for classical communication, despite its intrinsic loss. For quantum communication, the design of the vacuum beam guide has the potential to achieve ultra-low-loss long-distance communication. [0032] In embodiments of the present disclosure, the VBG is deployed as the back-bone quantum channel that can be a part of a global quantum network with a hierarchy structure as shown in Fig.1A-B. The system can be built by estimating the upper bound of the attenuation dominated by residue air absorption, optical losses introduced by lenses, and misalignment of the beam guide. The estimation demonstrates the VBG as state-of-the-art FH11893796.1 UCT-00625 under a practical and exemplifying con-figuration, establishing it as one of the most practical and potentially useful quantum communication techniques at a global scale. [0033] A hierarchical structure of large-scale quantum networks connected by vacuum beam guides (VBGs) is shown in Figs.1A-C. In Fig.1A, the VBG backbone network shon on map 100 can transfer quantum information with high capacity (with TeraQubits/sec) over long distances, connecting regional quantum networks shown in Fig.1B. Fig.1B depicts an example regional network 102 that comprises fiber or free-space quantum channels designed to distribute quantum information to branch nodes and quantum terminals (with GigaQubits/sec) across urban scales. [0034] Ultra-High Transmission of VBG. The VBG is a long vacuum chamber tube that comprises an array of ^^_௧^௧ lenses spaced ^^_^ apart, which enables the connection of quantum terminals 110 separated by ^^_௧^௧ = ^^_௧^௧ ^^_^, as illustrated in Fig.1C. The vacuum 112 has a typical pressure below ∼ 1 Pascal, which ensures low absorption from the remaining gas at room temperature. The lenses 108 are shielded from seismic vibrations and are optically aligned with adaptive feedback. In this analysis, a feasible and robust confocal design is considered with a spacing of ^^_^ = 4km and a focal length of ^^ = ^^_^/2. The beam waist is ^^_^ = _గ ≈ 3cm for the telecom-band wavelength ^^. The lens radius of R = 10 cm is sufficient to achieve negligible diffraction loss. [0035] Fig.1C illustrates the design of the VBG 106, placing lenses 108 with identical focus length f and radius R at regular intervals within a vacuum chamber tube 112, constructing a VBG with ^^_௧^௧ sections. The positions of the lenses 108 are stabilized by an advanced alignment system. In the VBG, the fundamental Gaussian mode with waist ^^_^ can travel at an extremely low loss over a distance of ^^_௧^௧. Quantum states can be transferred from quantum node A 110 at one end of the VBG to quantum node B 110 at the other end of the VBG. [0036] Optical signals encoding quantum information can transmit through the VBG with little loss, even after passing through an array of aligned lenses over thousands of kilometers. The effective attenuation rate (in units of dB/km) characterizes the VBG transmission loss, with three major contributions associated with the lens, residual gas, and imperfect alignment: ^^_௧^௧ = ^^_^^^^ + ^^_^^^ + ^^_^^^^^. (1)The conditions to achieve an attenuation rate at the level of 10⁻⁴ dB/km for all three contributions are discussed below. FH11893796.1 UCT-00625 [0037] The lens loss, ^^_^^^^, is associated with absorption, scattering, reflection, and diffraction. As illustrated in Fig.2A, a lens radius of R ≥ 10 cm is adequate to suppress diffraction loss for L0 = 4 km. Fig.2A shows a graph illustrating attenuation rate from the lenses, ^^_^^^^, with lens radius R = 8, 10, 12 cm. For a large radius (R ≥ 10 cm), the lens loss is limited by residual reflection and absorption from the AR coating. By adding a multi-layer anti-reflective coating, the total loss per lens can be reduced to less than 10⁻⁴, resulting in an effective loss rate of ^^_^^^^< 10⁻⁴ dB/km at the wave-lengths λ ≈ 1550 nm, which can match the telecom band. [0038] Gas loss in the VBG is primarily due to the absorption from residual air in the vacuum chamber tube. As shown in Fig.2B, the attenuation rate ^^_^^^ is computed at various levels of gas pressure with the components of air based on the HITRAN database. Fig.2B shows a graph illustrating attenuation rate due to residual gas, α_gas, with moderate vacuum pressures: P = 0.01 Pa or 1 Pa. At a pressure of 1 Pascal, the attenuation rate is ^^_^^^ < 10⁻⁴ dB/km for optical wavelengths within the selected telecom bands. Further reducing the air pressure below 10⁻² Pascal is sufficient to achieve negligible air absorption by reducing attenuation rate below 10⁻⁴ dB/km almost over the entire spectrum. [0039] An upper bound can be derived for the effective attenuation rate induced by imperfect in the confocal ௪_బ ^మ బ where ^^_^ and ^^_^బ are the magnitudes of the fluctuations of transverse and longitudinal displacements for each lens, and ^^_^ is the deviation of the focal length. Since ^^_^ ≪ ^^_^, f, the fluctuating transverse displacement is the dominant cause of attenuation, while the other two contributions can be sufficiently small, , ^ఙ _^ ^ିଷ ^ < 10 , but achievable. [0040] As illustrated in Fig.2C, it is essential to have good transverse alignment (e.g., ^^_^ < 0.2 mm) to achieve a low effective attenuation rate ( ^^_^^^^^< 10⁻⁴ dB/km). Fig.2C is a graph illustrating the misalignment attenuation rate, ^^_^^^^^, with random transverse displacements ^^ = 0.06, 0.1, 0.2, 0.6 m ^ఙ ^^{బ ఙ} _{^ ^} m while fixing _^బ = _^ = 0.1%. Even with relatively poor alignment (e.g., ^^_^= 0.6 mm), ^^_^^^^^< 10−2 dB/km can still be achieved, which is much better than the attenuation rate of fiber by at least an order of magnitude. Practically, each lens will be actively positioned using slow actuators, based on standard alignment sensing systems, bringing the residual miscentering to < 0.1 mm. FH11893796.1 UCT-00625 [0041] By summing up all three contributions, the total attenuation rate can be plotted along with the individual contributions in Fig.2D, assuming practical parameters such as R = 10 cm, P = 1 Pascal, and ^^_^ = 0.1 mm. The VBG can achieve an attenuation level as low as 3 × 10⁻⁵ dB/km for an optimized choice of wavelength within the atmospheric window, which corresponds to an effective attenuation length of 80,000 km, more than three orders of magnitude better than state-of-the-art fiber technology. Fig.2D illustrates the total attenuation rate of the VBG compared to the advanced fiber under the configuration of L₀ = 4 km, R = 10 cm, P = 1 Pascal, and ^^_^ = 0.1 mm. [0042] Quantum Channel Capacities of VBG. The VBG can be used as a highly transparent bosonic quantum channel to transmit quantum information over long distances. The VBG has a transmission efficiency at wave-length λ of: ^^[ ^^] = 10^{ି^.^^^^^ఈ[ఒ]} (3) which can be used for various quantum communication protocols. For one-way quantum communication (e.g., from Alice to Bob only), the one-way pure-loss capacity for a single wave packet at a λ is _ଶ which vanishes for ^^[ ^^] ≤ 1/2. If the pure-loss VBG chan-nel is further assisted by two- way classical communication (between Alice and Bob), the corresponding two-way pure-loss capacity at a λ is: which is finite for all η[λ] > 0. [0043] As shown in the insets of Fig.3A-B, the VBG can have a broad band with a large one-way or two-way channel capacity over long distances. In the low-loss regime with ^^_௧^௧ ≪ 1/ ^^, efficient multi-mode encoding techniques can be used to approach the asymptotic scaling of one-way channel capacity. [0044] The frequency-integrated channel capacity can be calculated, which provides the maximum transmission rate of information, over the ఒ_మ for one-way and two-way quantum communication protocols, respectively. The VBG has a significant advantage over other techniques, particularly due to its ultra-low attenuation rate over a wide range of wavelengths, leading to an extremely large quantum capacity exceeding 10¹³ qubits/second over a distance of 10⁴ km under one-way quantum communication FH11893796.1 UCT-00625 protocol as shown in Fig.3A. While the estimation in Fig.3B suggests that even with a baseline coupling loss as large as 50%, where the one-way protocol fails, it is still possible to achieve a Tera-level qubit rate via two-way protocols for continental scale communication. This points out a practically feasible pathway toward ultra-fast global quantum networks. [0045] The quantum channel capacity ^^_^(^^ଶ) at 1 Pascal as a function of transmission distance with ^^_^ = 0.1, 1 mm when fixing σL0 = σf = 0.1% compared to the ideal alignment is shown in Figs.3A-B. Fig.3A illustrates a one-way frequency-integrated quantum capacity with perfect couplings. Fig.3B illustrates a two-way frequency-integrated quantum capacity assuming 50% coupling efficiency. The insets show the relation between channel capacity ^^_^(^^ଶ) as a function of wavelength at L = 100, 1000, 10000 km. The inset of 10000 km in Fig.3B is in log scale to show finite two-way quantum capacity over a wide range of parameters. [0046] Compared to state-of-the-art satellite-ground links for quantum communication, the ground-based VBG offers high reliability as it is operational at all times, and provides an extremely high throughput, at least eight orders of magnitude higher in terms of achievable quantum communication rate. The concept of a beam guide was investigated to address diffraction loss in satellite-relayed quantum communication. Nevertheless, that protocol remains susceptible to atmospheric and turbulence-induced losses, thereby affecting reliability, transmission rate, channel capacity, and communication latency (due to the limitation of two-way quantum communication). Additionally, unlike quantum repeaters, the VBG only requires passive optical lenses to focus the optical beams and does not need any quantum memory or active quantum devices to perform quantum error detection or correction. Therefore, the VBG is feasible to implement with current technology. [0047] The VBG can be designed to operate with visible light, which would benefit from a larger air transmission window and low-loss lens materials if sufficiently small lens roughness is achievable, while better transversal alignment is required to match the reduced భ beam waist ^^_^ ∝ ^^ ^మ with shorter wavelengths. In practice, there is a trade-off between the VBG’s performance and cost, and the design parameters can be optimized to achieve the desired balance. [0048] The present disclosure relates to the use of an optical guide implemented as a vacuum lens guide (VLG) in quantum communications. An example embodiment 400 of a suitable VLG is shown in Fig.4. The VLG shown in Fig.4 comprises a vacuum enclosure FH11893796.1 UCT-00625 402 and a plurality of lenses 404a, 404b, 404c, etc. In the example embodiment shown, the lenses have the same focal length f and are disposed at a distance D = 2f from each other. [0049] The VLG shown in Fig.4 can be used for ultra-low loss optical channel to achieve attenuation length beyond 10³ km (or even 10⁴ km), which will enable achieving continental (or global) scale ground-based ultra-low-loss quantum communication channel. [0050] The VLG such as the one shown in Fig.4 has the following merits: [0051] It can be employed in a ground-based long-range quantum channel, which is strongly preferred because of the low cost and high reliability. [0052] An ultra-low loss: it can achieve effective attenuation length of above 10³ kilometers, orders of magnitude larger than the attenuation length of air or state-of-the-art fiber. [0053] A high bandwidth: it can have much higher bandwidth than the optical fiber because optical signals are propagating mostly in vacuum with little absorption. The dispersion in optical lens can be compensated using low-dispersion material or compensating elements. [0054] Weather resistance: unlike satellite mediated optical channel, VLG can operate in all weather conditions. [0055] The proposed VLG is compatible with existing quantum technology of quantum cryptography (QKD), entanglement generation, quantum routers, as well as various classical communication protocols. [0056] As used herein, the term “network node” or “node” refers to a redistribution point or a communication endpoint within a network, whether that network is classical or quantum. For example, a physical network node may either be data communication equipment (DCE) such as a modem, hub, bridge or switch; or data terminal equipment (DTE) such as a digital telephone handset, a printer or a host computer. If the network in question is a local area network (LAN) or wide area network (WAN), every LAN or WAN node that participates on the data link layer must have a network address, typically one for each network interface controller it possesses. Examples are computers, a DSL modem with Ethernet interface and wireless access point. If the network in question is the Internet or an intranet, many physical network nodes are host computers, also known as Internet nodes, identified by an IP address, and all hosts are physical network nodes. [0057] Quantum networks facilitate the transmission of information in the form of qubits between physically separated quantum processors. The basic structure of a quantum network is analogous to a classical network. A quantum network may include one or more end nodes, FH11893796.1 UCT-00625 which may include one or more quantum processors. A quantum network also includes one or more communication lines, for example utilizing optical guides such as telecom fibers or free space transmission. Some quantum networks further include optical switches capable of delivering qubits to an intended endpoint. Both switches and communication lines must preserve quantum coherence in order to be suitable for use in a quantum network. In particular, since qubits cannot be copied, classical signal amplification is not possible. [0058] I. Principle of Lens Guide [0059] All the calculations below were done under the confocal condition, satisfying D = 2f, with distance between lens D and focus length f. In practice, it is also possible operate with other lens spacing choices, but for simplicity the confocal condition is assumed below. [0060] I.1 The optical mode [0061] The optical mode is assumed to be the Laguerre-Gauss function instead of hyper spheroidal function under relatively large Fresnel number ^^. [0062] The Laguerre-Gaussian modes are used as the basis. The mode function is given where integer ^^ ≥ 0 is the radial index, ^^ is the azimuthal index, ^^ = ^^ cos ^^, ^^ = ^^ sin ^^, ^^ is along the propagating direction with ^^ = 0 at the location of the waist, ^^ is the focus length, and [0063] It is known that the diffraction loss is ^^ For mode ( ^^, ^^), ^^. ^^. ^^ ^{^} ^_^ ^{^}, where ^^ is the Fresnel variable defined by ^^ = ^{^^ ^^} ^_^ = ^^ ^^ ^^. Here ^^ is the radius of the lenses, ^^ is the distance between the lenses and ^^ is the wavelength of the light. [0064] I.2 Perturbation caused by random offset [0065] Only displacement perturbation is considered here, as it is known that tilt of the lens will lead to higher order perturbation. It is assumed that all the perturbations are Gaussian. The definitions are (for each lens): FH11893796.1 UCT-00625 1_{) Transversal displacement:} ^{^} _^^ ^{ଶ^} _{= ^^^} ^ଶ 2) Parallel displacement: ^^ ^^ ^^( ^^) = ^^_^ ^ଶ ^^ 3) Fluctuation of focus length: ^^ ^^ ^^( ^^) = ^^ ^^ ^^ ^^{^ ଶ} ^^ [0066] It is known that after ^^ the above perturbations will lead to 1) Beam center fluctuation: ^ ^^_^ ^ଶ^ = ^ 2) Waist parameter enlargement: ௪_{మ ଶ ఙ^} ^మ _ାఙವ ^మ . [0067] It has been estimated that the diffraction loss caused by perturbation has a trivial effect and need not be accounted for. [0068] II. Numerical Results [0069] II.1 Basic Attenuation Length Estimation [0070] To illustrate the performance of the VLG, the following configuration was considered: 1) Wavelength: ^^ = 1064 ^^ ^^ 2) Len radius: ^^ = 20 ^^ ^^ 3) Focus length: ^^ = 2 ^^ ^^ = ^{^} ଶ ^^ 4) Absorption loss and reflection loss of each lens: ≤ 101ppm 5) Other Loss Irrelevant to the Lens (Air turbulence…): 0.01% [0071] Under these assumptions, the Gaussian beam propagating in the ideal VLG will have the following configuration: 1) Light waist: ^^_^ = 2.60 ^^ ^^ 2) Spot size at lens: ^^_^ = 3.68 ^^ ^^ [0072] These assumptions lead to the following “ideal” attenuation length: ^_మ ^^_^^௫ = 39796 ^^ ^^. [0073] For a more realistic estimation, consider the perturbation of the displacement of each lens to be Gaussian and i.i.d. as follows: 1_{) Transversal displacement: ^^^} ^ଶ ₌ ⁽ _{100 ^^ ^^} ^)ଶ 2) Parallel displacement: ^^_^ = 1% 3) Fluctuation of the focus length: ^^_^ = 1% [0074] Under these assumptions, the estimation of the propagation are: FH11893796.1 UCT-00625 1) The expected enlarged waists in the last section: ^^_^ ^ᇱ = 2.63 ^^ ^^ 2) The expected enlarged spot size at the last lens: = 3.72 ^^ ^^ 3) The attenuation length under perturbation: a) Consider the conversion back from ^^_^ ^{^}, ^^_^ ^{^}, ^^_^ ^ଶ, ^^_^ ^ଷ, ^^_^ ^{^}, ^^_^ ^ସ, ^^_^ ^ଶ, ^^_ଶ ^{^} modes: ^_௫ The loss for the last iteration: ^^_^ ^{^^} = 0.0355% b) No conversion back: ^^^ே ^^{^} ^_௫ = 22212 ^^ ^^ The loss for the last iteration: ^^_^ ^{ே^} = 0.0360% The above results are valid for confocal case, i.e., | ^^_^ ^^_^ − 2| < ^^_^ ^ଶ + ^^_^ ^ଶ [0075] II.2 Power Distribution After the Last Iteration [0076] Upon considering the conversions, the total power ratio of the tracking modes is 82.8%. It can be concluded that the power is not lost but stored in the higher order mode. In fact, the system is far from steady, the power distribution after the last iteration is as depicted in FIG.2. [0077] It was found that most of the power is stored in the mode. It can be expected from the conversion matrix that ^^_^ ^{^} is more robust than any other modes, é ^{^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^} ^_{^ ^^} ù ê _{^^ ^^ ^^ ^^ ^^ ^^ ^^} ^^ ^^ ^^ ^^ ^^ ú ê ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^ ú ê ^ú ^^ ^^ ^^ − ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^: ê ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ú ê ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ú ê ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ú ê ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ú ë _{^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^} ^û ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^: ^^ ^^ ^^ ^^^ ^^, ^^, ^^, ^^ ^^, ^^, ^^, ^^ ^^, ^^ ^^, ^^ ^^^ [0078] The second column corresponds to the conversion out from E₀ ¹ while the second column corresponds to the conversion into E0¹. It was found that at each iteration, E0¹ mode takes four portions of energy from E0⁰ and six portions of energy from other higher modes while only giving three portions out. This is illustrated in the conversion tree shown in FIG.3 (note that ^^, ^^ is flipped in the tree). [0079] In addition, it is also possible to investigate the loss for each tracking mode at the last iteration, as shown in Fig.7. The negative losses indicate some of the modes are actually gaining energy from lower order mode [0080] III. Accumulated Loss Grown With Iteration FH11893796.1 UCT-00625 [0081] The accumulated loss ^^_^^ = 10 log verse iteration ^^ is dominant by the conversion and is expected to grow around ^^⁽ ^^⁾. The numerical result is plotted as in Fig.8. The curve fit with function ^^_^^( ^^) = ^^ ^^^{^} is estimated as ^^_^^( ^^) = 0.0016 ^^^{^.ଽଽ^}, which agrees well as our expectation. [0082] III.1 Better Performance With Longer Section [0083] The diffraction loss is determined by the Fresnel variable c. However, as long as c is not that small, the loss is dominant by the mode conversion, which is determined by λ and D. Therefore, the strategy is to use a relatively large wavelength and large focus length (and thus longer section), while also increasing R to maintain a relatively large c. This also helps with reducing the number of iterations. [0084] If the value of c is maintained at 59 (the value obtained in the above calculation), the focal length is taken from 1 ^^ ^^ to 15 ^^ ^^ (the section length will be from 2 ^^ ^^ to 20 ^^ ^^), and the maximum radius of the lens is taken for 15 ^^ will be 55 ^^ ^^, then the performance can be illustrated by the plots shown in Figs.9A and 9B. It can be concluded that for ^^ = 15 ^^ ^^, the distance will be about 16000 ^^ ^^, which is approaching half of the circumference of the equator. [0085] The Table shown in Fig.10 lists the adjustable parameters that can be used to fine tune the performance of the VLG described above. [0086] IV. Exemplary Aspects of the Design [0087] In one exemplary embodiment, the VLG described herein can have the following characteristics: - Center wavelength: 1064 ^^ ^^ - _{Lens focus length: f^ = f} ⁽ _1064nm ⁾ _{= 2km} - Lens interval: ^^ = 2 ^^_^ = 4 ^^ ^^ Lens Radius: ^^ = 20 ^^ ^^ - Lens loss: ≤ 110 ^^ ^^ ^^ (SOTA) Other loss: ≤ 100 ^^ ^^ ^^ - Environment: T = 300K, P ≈ 1Pa [0088] In another example embodiment, the VLG described herein can have the following design tolerances: - Focus length: Δ ^^ = ^^_^ ^^ = ±0.1% ^^ ≈ ±2 ^^ - Lens interval: Δ ^^ = ^^_^ ^^ = ±0.1% ^^ = ±4 ^^ - Lens transversal offset: ^^ = ±0.1 ^^ ^^ - Lens tilting: Higher order effects FH11893796.1 UCT-00625 [0089] In another example embodiment, the VLG described herein can have the loss per iteration of approximately 0.0262% for center wavelength, which gives an attenuation length of 30,524 ^^ ^^. [0090] In another example embodiment, the VLG described herein can support a wide linewidth of wavelength, increasing wavelength gives rise to even larger attenuation length envelop. This is illustrated in Fig.11A(peaks are due to residue gas absorption). [0091] In certain example embodiments, the vacuum beam guide (the VLG) can reduce the loss rate by four orders of magnitude (~10⁴) compared to that of an optical fiber. This permits achieving global scale direct quantum networks. This is illustrated in Fig.11B which shows the wavelength-dependent loss rate (in units of dB/km) for different wavelength under various pressure for an exemplary beam guide described herein having the parameters listed below:. [0092] ^^ = 2 ^^ ^^, ^^ = 20 ^^ ^^, ^{^^} ^ = ^^ ^ = 0.1%, Δ ^^ = 0.1 ^^ ^^ [0093] It is also possible to compute the loss rate under different misalignment configuration. Referring to Fig.11C, the dependency of the loss rate on the wavelength is shown for the misalignment conditions described below, assuming the air pressure is assumed to be 10^ିଷ ^^ ^^. Three levels of configuration were considered: • Low: ^{Δ ^^} ^_^ = Δ ^^ ^_^ = 0.03%, Δ ^^ = 0.03 ^^ ^^ • Mid: ^{Δ ^^} ^_^ = Δ ^^ ^_^ = 0.1%, Δ ^^ = 0.1 ^^ ^^ • High: ^{Δ ^^} ^_^ = Δ ^^ ^_^ = 0.3%, Δ ^^ = 0.3 ^^ ^^ [0094] It is also possible to compute quantum channel capacity (QCC) under various misalignment configurations. The QCC (q2) shows the maximal quantum information (in qubits) that can be transmitted per two-way pure loss channel usage. It is given by the following formula [0095] ^^₂[ ^^] = − log₂൫ ^^( ^^)൯, where ^^( ^^) is the loss rate. [0096] The plot reproduced in Fig.11D shows the QCC as a function of the transmission length for different configurations of misalignment at the wavelength ^^ = 1064 ^^ ^^ under the pressure of 1 ^^ ^^. The parameters of the misalignment conditions are the same as above. [0097] The materials of the vacuum enclosure can be selected from a wide variety of materials, for example, from a stainless steel, carbon steel, aluminum, or titanium. [0098] The lenses employed by the VLG described herein can be obtained from a commercial provider, for example from Laboratoire des Materiaux Avances (LMA). For FH11893796.1 UCT-00625 example, the lenses can be made of ultrapure fused silica (such as Suprasil 3002 available from Heraeus Quartz Glass), and optionally include a coating, such as Ti-Ta2O5∕ SiO2 IBS multilayer. [0099] An example design of the vacuum enclosure and the means for obtaining the vacuum are illustrated in Fig.12. In this embodiment, the vacuum enclosure 1200 includes an outer envelope 1202 that can be made of a mild steel, and an inner envelope 1204, made out of a low-outgassing material. The vacuum enclosure 1200 further includes vacuum pumps 1206 (only one pump is labelled). V. Optical Losses Due to the Lens Array [00100] In various embodiments, a quantum network is paired with a classical network, which may include one or more classical computing nodes. The major sources of optical loss (and thereby decoherence) in the VBG are due to the lens array. In the lens array, the major losses are ^^_^^, the absorption loss in the lens substrate as well as ARC coating, ^^_^^^, the scattering loss from the lens surface (this includes scattering into low order Laguerre-Gauss modes that scatter back into the mean beam), and ^^_^^^, the reflection loss due to the non-zero reflectivity of practical anti-reflection coat-ings (it is assumed here that reflected light has a random phase and does not interfere with the signal beam). Notice that a different quantity generally denoted by l is used to analyze the loss for convenience instead of using the effective attenuation rate α used in the main text. They are related to each other by ^^ = − ^{^^} ^_బ ^^ ^^ ^^ (1 − ^^). [00101] For a perfectly aligned confocal VBG with lenses of finite radius, the exact optical mode is described by the hyper-spheroidal function with a diffraction loss per section for rotation mode number m and radial mode num-ber n in power ratio approximately given by: (Eq.1) 102] where ^^ = ^{^ మ} [00 ^ோ ^_బ is the Fresnel variable. Such a variable is assumed to be large enough (c ∼ 10) so that the hyper-spheroidal function can be replaced with the Laguerre- Gauss function to describe the optical mode in the misalignment analysis. The total loss for the fundamental mode introduced by the lenses can be written as: ^^_^^^^ = ^^_ௗ,^^ + ^^_^^ + ^^_^^^ + ^^_^^^ FH11893796.1 UCT-00625 (Eq.2) where approximate values for the losses are shown in Table I. [00103] For a large enough radius (R ≥ 10 cm), lens loss is limited by residual ARC reflection and absorption. Commercially available telecom band ARC can achieve loss at the 100 ppm level, which can be further improved. Table 1: The approximate values of different sources contributing to lens loss. † 10^ିସ ppm is obtained by setting R = 12cm while 2 ppm is obtained by the default 10cm configuration. ∗ 100 ppm is rather pessimistic while commercially available. And it can be further reduced, as illustrated in ^^ ^^ ^^. ^^ ^^ ^^ and measured in LIGO. [00104] Fig.13A illustrates the reflective loss of two-layer MgF2/ZrO2 ARC on silica substrate optimized around 1300nm and 1550nm compared with a commercially available telecom band ARC. The thickness of layers for 1300nm and 1550nm designs are 169/270nm and 371/45nm respectively. The refractive indexes used in the simulation are obtained from the empirical formula known in the art. Fig.13B illustrates Q1 vs distance under losses levels of ^^_^^^^ = 1 to 103 ppm with ^^_^ = 4 km, R = 10 cm, ^^_^ = 0.1mm, and ^{ఙ^బ} ^_బ = ^ఙ _{^ ^} = 0.1% at 1 Pascal. The red curve shows that even with a commercial-level ARC loss, it is still possible to achieve significant low loss up to 10⁴ km. Fig.13B shows the one-way quantum capacity Q1 under different levels of lens loss. Even with a 100-ppm level lens loss (red curve), it is still possible to achieve a non-zero communication rate over 10 thousand kilometers. Additionally, the maximum distance saturates when the lens loss reaches 1 ppm, with the performance mainly limited by misalignment. VI. Non-Confocal Design of VBG [00105] Practically, it is challenging for the VBG to fulfill the confocal condition over a wide range of wavelengths due to dispersion even using dispersion compensation. Fortunately, applying the well-known equivalent optical resonator method, it can be shown that the non-confocal VBG fulfills the stability criteria, FH11893796.1 UCT-00625 is uniquely equivalent to a confocal VBG with an equivalent focal length and an equivalent Fresnel variable where ^^ = _^బ . The diffraction loss should now be calculated using the equivalent Fresnel number. Let ( ^^, ^^, ^^) denoted cylindrical coordinate along the VBG with the original point at the middle of the the field inside each section of VBG is thus [00106] which provides the fundamental mathematical description for the analysis of VBG. An upper bound can now be obtained for the effective attenuation rate induced by imperfect alignment for parameters deviated from the confocal design as: [00107] where ^^ = ≈ 1, and ^^_^ and ^^_^బ are the magnitudes of the fluctuations of transverse and longitudinal displacements for each lens, and ^^_^ is the deviation of the focal length. VII. Misalignment in VBG [00108] The misalignment will result in the mode conversion and thus cause the loss in the fundamental mode. Here, only the misalignment loss due to the displacement of the lens as well as the fluctuation of the focal length of the lens caused by imperfect manufacture is considered. The tilting of lenses has been shown to be the higher-order correction and thus can be neglected. Therefore, the loss in power ratio per section caused by misalignment can be written as a sum as follows when each term is small enough, FH11893796.1 UCT-00625 where ^^_^, ^^_^, and ^^_^బ denote the losses caused by transversal displacement, parallel displacement and the fluctuation of the focal length of the lenses, and only depend on the misalignment parameter ^^_^, ^^_^బ and ^^_^ respectively. In general, the loss may not necessarily be a constant for each section due to the power conversion back from higher order mode. However, neglecting such contributions will always give an upper bound of the loss. A. Loss Caused by Parallel Displacement [00109] If the field is in perfect fundamental mode while the next lens is shifted for a distance of ^^_^బ , then the field of the fundamental mode right before the next lens can be simply written as ^^_^′ ≡ ^^_^ ^{^}( ^^, ^^, ^^ ^/2 + . When calculating the loss such field can be expanded in the basis of the Beam Guide modes Enm(r, ϕ, L0/2), so that is always kept of the perfect fundamental mode components, ^^_^ ≡ ^^_^ ^{^}( ^^, ^^, ^^ ^/2 + ., emitting at the next lens. Thus, the loss is simply given by: After some routine calculations, the upper bound is calculated as: B. Loss Caused by Imperfect Focal Length [00110] Supposing that the focal length of the current lens is shifted by ^^_^ , and such shifting results in a new phase factor of the lens as: Then the new phase factor is plugged in and use the inner product method to derive the loss. Equivalently, the perturbation method can be adapted here for a more convenient derivation. The focal length deviation gives rise to an additional phase factor as: which slightly converts the fundamental mode into the first-order mode: FH11893796.1 UCT-00625 while the conversion to other modes is higher order in σ_f /f. Therefore, the fraction of excitation loss from the fundamental mode to the first order mode (1, 0) is: C. Loss Caused by Transversal Displacement [00111] For transversal displacement, a naive error model is that assuming there is a “correct” position for each lens determined by geodesic connecting two end-point of VBG, along which each lens’s transversal displacement is fluctuating with variance ^^_^. To proceed with the calculation, techniques are used to identify the additional phase shift as: Repeating the procedure again, it is seen that: following which, the loss is: [00112] The above error model is widely applied while discussing the effect of misalignment, while it requires the ability to identify the “correct” position for each lens over such a large-scale system as VBG. In principle, this can be done by scanning the offset of each lens by advanced sensing technique many times to minimize the error, after which the variance shall be dominated by adjustment precision of the optical bench. [00113] However, a more common practice is that the previous lens’s location is used as reference to align the next lens. In this model, it is assumed that each lens has a random offset of ^^_^ referred to the location of the previous lens and the fundamental mode is tracked in the coordinate “attached” to each lens instead of the ”correct” position. In this model, it can be shown that the additional phase is replaced by: FH11893796.1 UCT-00625 Thus, loss is calculated as: It can be observed that the loss is even smaller by a factor of approximately 2. This can be understood from the as-pect of geometrical optics: when the waist of the light is shifted down from the center of the lens, the image of the waist should shift up from the center. And thus the new power center is closer to the reference line of the lens’ s center instead of the shifted incident beam center. [00114] Finally, a comment can be made on the misalignment parameter selected as default.0.1% of relative error of D and f will result in an absolute standard deviation of meter level, which shall be easily achieved. Alignment techniques have been demonstrated that make a standard deviation, ^^_^ = 0.1mm, practically feasible. VIII. Effective Characteristic Length [00115] As can be see from Eq. (1) and Fig.2, the performance of VBG is limited by the lens loss ( ^^ ) and the alignment loss ( ^^ ), because the ^_{^^^ ^^^^^} gas loss at telecom-band wavelength can be negligible when the pressure is reduced below 1 Pa. [00116] The transverse displacement is the dominant contribution to the alignment loss, since the achievable longitudinal displacement and focal-length deviations can be sufficiently small ( ^^_^బ , ^^_^ < 10−3). Hence, according to Eq. (2), ^^_^^^^^ ≃ _௪ బ^మ , with transversal displacement ^^_^, spacing ^^_^=4km and beam waist ^^_^ ≈ 3cm. [00117] To see how these two factors affect the performance of VBG, the effective characteristic length ^^_ொ^ is defined as the distance at which the one-way quantum capacities of VBG vanishes, i.e. η = 50% for the corresponding wavelength. A 2-D contour plot is constructed of characteristic length ^^_ொ^ with respect to the lens loss ^^_^^^^ (or ^^_^^^^ defined in Eq. (2)) and transversal displacement ^^_^. Using Eq. (1) and Eq. (2), one can approximate the characteristic length as: FH11893796.1 UCT-00625 where ^^_^^^^ includes all sources of lens-related loss as listed in Table 1 and ^^_^௧^^^^ ≈ 2 ppm denotes some other losses including parallel perturbation and residue gas absorp-tion. [00118] The characteristic length of VBG is plotted under various lens and misalignment configurations in Fig.14. VBG is quite sensitive to transversal misalignment, in order to have a characteristic length over 104 km, the transversal misalignment must be controlled to submillimeter level while a total lens loss of a few hundred ppm is sufficient. It is also noted that based on the configuration achievable by current techniques with 1 ppm lens loss, VBG has a saturated performance with characteristic length over around 10⁵ km, which is larger than the perimeter of the earth. [00119] Fig.14 illustrates a characteristic length under various configurations of ^^_^^^^ and ^^ with ^^ = 4 km, R = 10 cm, and ^{ఙ^బ} = ^ఙ _{^ ^ ^ ^బ ^} = 0.1% at 1 Pascal. The green dots show the configurations calculated in Fig.1B. V. Exemplary Classical Communication and Computation [00120] In various embodiments, a quantum network is paired with a classical network, which may include one or more classical computing nodes. [00121] Referring now to Fig.15, a schematic of an example of a computing node is shown. Computing node 10 is only one example of a suitable computing node and is not intended to suggest any limitation as to the scope of use or functionality of embodiments described herein. Regardless, computing node 10 is capable of being implemented and/or performing any of the functionality set forth hereinabove. [00122] In computing node 10 there is a computer system/server 12, which is operational with numerous other general purpose or special purpose computing system environments or configurations. Examples of well-known computing systems, environments, and/or configurations that may be suitable for use with computer system/server 12 include, but are not limited to, personal computer systems, server computer systems, thin clients, thick clients, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputer systems, mainframe computer systems, and distributed cloud computing environments that include any of the above systems or devices, and the like. [00123] Computer system/server 12 may be described in the general context of computer system-executable instructions, such as program modules, being executed by a computer system. Generally, program modules may include routines, programs, objects, components, FH11893796.1 UCT-00625 logic, data structures, and so on that perform particular tasks or implement particular abstract data types. Computer system/server 12 may be practiced in distributed cloud computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed cloud computing environment, program modules may be located in both local and remote computer system storage media including memory storage devices. [00124] As shown in Fig.15, computer system/server 12 in computing node 10 is shown in the form of a general-purpose computing device. The components of computer system/server 12 may include, but are not limited to, one or more processors or processing units 16, a system memory 28, and a bus 18 that couples various system components including system memory 28 to processor 16. [00125] Bus 18 represents one or more of any of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, and a processor or local bus using any of a variety of bus architectures. By way of example, and not limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, Peripheral Component Interconnect (PCI) bus, Peripheral Component Interconnect Express (PCIe), and Advanced Microcontroller Bus Architecture (AMBA). [00126] Computer system/server 12 typically includes a variety of computer system readable media. Such media may be any available media that is accessible by computer system/server 12, and it includes both volatile and non-volatile media, removable and non- removable media. [00127] System memory 28 can include computer system readable media in the form of volatile memory, such as random access memory (RAM) 30 and/or cache memory 32. Computer system/server 12 may further include other removable/non-removable, volatile/non-volatile computer system storage media. By way of example only, storage system 34 can be provided for reading from and writing to a non-removable, non-volatile magnetic media (not shown and typically called a "hard drive"). Although not shown, a magnetic disk drive for reading from and writing to a removable, non-volatile magnetic disk (e.g., a "floppy disk"), and an optical disk drive for reading from or writing to a removable, non-volatile optical disk such as a CD-ROM, DVD-ROM or other optical media can be provided. In such instances, each can be connected to bus 18 by one or more data media interfaces. As will be further depicted and described below, memory 28 may include at least FH11893796.1 UCT-00625 one program product having a set (e.g., at least one) of program modules that are configured to carry out the functions of embodiments of the disclosure. [00128] Program/utility 40, having a set (at least one) of program modules 42, may be stored in memory 28 by way of example, and not limitation, as well as an operating system, one or more application programs, other program modules, and program data. Each of the operating system, one or more application programs, other program modules, and program data or some combination thereof, may include an implementation of a networking environment. Program modules 42 generally carry out the functions and/or methodologies of embodiments as described herein. [00129] Computer system/server 12 may also communicate with one or more external devices 14 such as a keyboard, a pointing device, a display 24, etc.; one or more devices that enable a user to interact with computer system/server 12; and/or any devices (e.g., network card, modem, etc.) that enable computer system/server 12 to communicate with one or more other computing devices. Such communication can occur via Input/Output (I/O) interfaces 22. Still yet, computer system/server 12 can communicate with one or more networks such as a local area network (LAN), a general wide area network (WAN), and/or a public network (e.g., the Internet) via network adapter 20. As depicted, network adapter 20 communicates with the other components of computer system/server 12 via bus 18. It should be understood that although not shown, other hardware and/or software components could be used in conjunction with computer system/server 12. Examples, include, but are not limited to: microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data archival storage systems, etc. VI. Exemplary Embodiments [00130] As used herein, a “quantum system” refers to an information device that exploits quantum mechanical phenomena to enhance the computational power, communication security, sensor sensitivity, etc. [00131] As used herein, a “lens cascade” refers to an optical train comprising at least two lenses sharing a common optical axis and disposed along the common optical axis at the distance approximately equal to the sum of the focal lengths of the two adjacent lenses. In example embodiments, a lens cascade comprises lenses having the same focal lengths, where any two adjacent lenses are disposed at a distance approximately equal to twice the focal length. FH11893796.1 UCT-00625 [00132] It will be appreciated that lenses will have dispersion in general and thus the focal length for light of different wavelength may vary. Thus, multiple wavelength systems may be confocal for a certain wavelength, but non-confocal for other wavelengths of the system. Accommodating configurations that are not precisely confocal thus allows more flexibility. The below examples describe the performance of configurations that are not precisely confocal. In particular, this is shown in the misalignment loss formula provided above. Under the same relative level of misalignment, having an ^^ factor larger than 1 (where ^^ = 1 in the confocal design) may reduce the misalignment loss and thus result in stronger stability. However, there will be extra diffusion loss caused by the deviation from confocal design which leads to a trade-off for such implementations. [00133] As used herein a “qubit” refers to a unit of quantum information, analogous to a bit in classical computing. A qubit can exist in a superposition of two states, usually referred to as 0 and 1, and can be manipulated using quantum algorithms. For example, a single optical mode can be in a superposition of |vacuum> and |single excitation> state, which can store one qubit of quantum information. In another example, two optical models (an earlier pulse and a later pulse) can be used, sharing a single excitation, which can be in a superposition state over |earlier excitation> and |later excitation>. [00134] In a first example embodiment, the present invention is a quantum network. [00135] In a 1^st aspect of the 1^st embodiment, the quantum network comprises at least a first node comprising a first quantum system, and a second node comprising a second quantum system; and an optical guide in optical communication with the at least the first and the second nodes, wherein the optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure. [00136] In a 2^nd aspect of the 1^st embodiment, the first quantum system is in a quantum state and is configured to generate at least one electromagnetic (EM) optical mode encoding the quantum state; the first node is configured to direct the optical mode into the optical guide; and the second node is configured to cause an interaction of the optical mode with the second quantum system, thereby imparting the quantum state onto the second quantum system. The remainder of the features and example features of the 2^nd aspect are as described above with respect to the 1^st aspect of the 1^st embodiment. [00137] In a 3^rd aspect of the 1^st embodiment, the first quantum system and the second quantum system each comprises a qubit. The remainder of the features and example features of the 3^rd aspect are as described above with respect to the 1^st and 2^nd aspects of the 1^st embodiment. FH11893796.1 UCT-00625 [00138] In a 4^th aspect of the 1^st embodiment, the first and the second quantum systems, each independently comprises a quantum processor, a quantum memory, a quantum sensor, or a quantum simulator. The remainder of the features and example features of the 4^th aspect are as described above with respect to the 1^st through 3^rd aspects of the 1^st embodiment. [00139] In a 5^th aspect of the 1^st embodiment, the first and the second quantum systems, each independently comprises a neutral atom array, a trapped ion array, or a solid state emitter. The remainder of the features and example features of the 5^th aspect are as described above with respect to the 1^st through 4^th aspects of the 1^st embodiment. [00140] In a 6^th aspect of the 1^st embodiment, the first and the second nodes, each independently comprises an optical mode quantum emitter. The quantum emitter can emit quantum states (e.g., single photons or entangled photons). The remainder of the features and example features of the 6^th aspect are as described above with respect to the 1^st through 5^th aspects of the 1^st embodiment. [00141] In a 7^th aspect of the 1^st embodiment, the optical mode quantum emitter comprises one or more of a single photon source, a squeezed photon source, or a spin-photon entangled source. The remainder of the features and example features of the 7^th aspect are as described above with respect to the 1^st through 6^th aspects of the 1^st embodiment. [00142] In an 8^th aspect of the 1^st embodiment, the first and the second nodes, each independently comprises an optical mode quantum detector. A quantum detector can projectively measure in photon number basis. The remainder of the features and example features of the 8^th aspect are as described above with respect to the 1^st through 7^th aspects of the 1^st embodiment. [00143] In a 9^th aspect of the 1^st embodiment, the optical mode quantum detector is a single photon detector, a number resolving photon detector, a homodyne photon detector, or a heterodyne photon detector. The remainder of the features and example features of the 9^th aspect are as described above with respect to the 1^st through 7^th aspects of the 1^st embodiment. [00144] In a 10^th aspect of the 1^st embodiment, the first and the second nodes are interconnected by a classical network. The remainder of the features and example features of the 10^th aspect are as described above with respect to the 1^st through 9^th aspects of the 1^st embodiment. [00145] In an 11^th aspect of the 1^st embodiment, the classical network is an optical signal or a radio frequency (RF) signal network. The remainder of the features and example features of the 11^th aspect are as described above with respect to the 1^st through 10^th aspects of the 1^st embodiment. FH11893796.1 UCT-00625 [00146] It is contemplated that the optical guide (the lens cascade and/or the reduced pressure enclosure) can be aligned and stabilized. For example, the optical guide can be divided into segments, and the segments can be individually aligned. For example, a 10⁴ km optical guide can be divided into 100 segments, each having 25 lenses separated by 4 km. The time of travel of an optical signal across such a segment is estimated to be 0.3 ms, so that the noise can suppressed within each segment with bandwidth up to a few kHz. The misalignments between sequential segments can then be performed using, for example, active optical elements disposed between neighboring segments. A suitable example of the alignment using the global positioning system (GPS) is described, for example, in Althouse et al. Review of scientific instruments 72, 3086-3094 (2001). A high-precision dual-frequency differential GPS can achieve millimeter level position precision (e.g., less than 5 mm deviation over 4 km. The relevant teachings of this publication is incorporated herein by reference. [00147] It is also contemplated that once the system is aligned, various stabilization protocols can be employed to locally adjust misaligned elements. For example, the relative alignment between neighboring lenses can be measured, and the lens positions adjusted. One example of lens stabilization is lens self-alignment. [00148] A suitable example of self-aligning lens cascade is provided in R. Christian et al., “Self-Aligning Optical Beam Waveguides” IEEE J. of Quantum Electronics, vol. QE-3, No. 11, November 1967. The relevant teachings of this publication are incorporated herein by reference. Briefly, a plurality of lenses is mechanically adjustable under the control of a sending device. The sensing device indicates the displacement of the beam from the lens center. The displacement information is used to move the preceding lens in such a direction that the beam displacement is reduced. The self-aligned lenses are probed sequentially and the corresponding preceding lens is correctively moved. [00149] In a 12^th aspect of the 1^st embodiment, the lens cascade comprises a plurality of lenses, and wherein at least two lenses are stabilized. The remainder of the features and example features of the 12^th aspect are as described above with respect to the 1^st through 11^th aspects of the 1^st embodiment. [00150] In a 13^th aspect of the 1^st embodiment, at least two lenses are self-aligned. The remainder of the features and example features of the 13^th aspect are as described above with respect to the 1^st through 12^th aspects of the 1^st embodiment. [00151] In a 14^th aspect of the 1^st embodiment, the lens cascade comprises at least one lens having a focal length of greater than 1 km. The remainder of the features and example FH11893796.1 UCT-00625 features of the 14^th aspect are as described above with respect to the 1^st through 13^th aspects of the 1^st embodiment. [00152] In a 15^th aspect of the 1^st embodiment, the lens cascade comprises a plurality of lenses, each lens having the same focal distance. The remainder of the features and example features of the 15^th aspect are as described above with respect to the 1^st through 14^th aspects of the 1^st embodiment. [00153] In a 16^th aspect of the 1^st embodiment, the lens cascade comprises a plurality of lenses, each lens of the lens cascade having a focal length tolerance of about 2 m or less; lenses of the lens cascade being spaced apart at an interval, the interval having a tolerance of about 4 m or less; any two adjacent lenses of the lens cascade having a transversal offset of about 0.1 mm or less; each lens of the lens cascade having a radius of about 15 cm or more; and each lens of the lens cascade having a lens loss of about 210 ppm or less. The remainder of the features and example features of the 16^th aspect are as described above with respect to the 1^st through 15^th aspects of the 1^st embodiment. [00154] In a 17^th aspect of the 1^st embodiment, the lens cascade is configured to operate at a temperature of about 300 K and wherein the reduced pressure enclosure is configured to operate at a pressure of about 10 Pa or less. [00155] The remainder of the features and example features of the 17^th aspect are as described above with respect to the 1^st through 16^th aspects of the 1^st embodiment. [00156] In a 2^nd example embodiment, the present invention is a method of transmitting a quantum state between nodes of a quantum network. [00157] In a 1^st aspect of the 2^nd embodiment, the method comprises: providing a first node comprising a first quantum system, and a second node, comprising a second quantum system; causing the first quantum system to generate at least one electromagnetic (EM) optical mode, the first quantum system being in a quantum state, the optical mode encoding a quantum state; directing the optical mode into an optical guide, the optical guide being in optical communication with the second node, and causing the optical mode to interact with the second quantum system, thereby imparting the quantum state onto the second quantum system. The optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure. [00158] In a 2^nd aspect of the 2^nd embodiment, the first quantum system and the second quantum system each comprises a qubit. [00159] The remainder of the features and example features of the 2^nd aspect are as described above with respect to the 1^st aspect of the 2^nd embodiment. FH11893796.1 UCT-00625 [00160] In a 3^rd aspect of the 2^nd embodiment, the first and the second quantum systems, each independently comprises a quantum processor, a quantum memory, a quantum sensor, or a quantum simulator. The remainder of the features and example features of the 3^rd aspect are as described above with respect to the 1^st and the 2^nd aspects of the 2^nd embodiment. [00161] In a 4^th aspect of the 2^nd embodiment, the first and the second quantum systems, each independently comprises a neutral atom array, a trapped ion array, or a solid state emitter. The remainder of the features and example features of the 4^th aspect are as described above with respect to the 1^st through the 3^rd aspects of the 2^nd embodiment. [00162] In a 5^th aspect of the 2^nd embodiment, the first and the second nodes, each independently comprises an optical mode quantum emitter. The remainder of the features and example features of the 5^th aspect are as described above with respect to the 1^st through the 4^th aspects of the 2^nd embodiment. [00163] In a 6^th aspect of the 2^nd embodiment, the optical mode quantum emitter comprises one or more of a single photon source, a squeezed photon source, or a spin-photon entangled source. The remainder of the features and example features of the 6^th aspect are as described above with respect to the 1^st through the 5^th aspects of the 2^nd embodiment. [00164] In a 7^th aspect of the 2^nd embodiment, the first and the second nodes, each independently comprises an optical mode quantum detector. The remainder of the features and example features of the 7^th aspect are as described above with respect to the 1^st through the 6^th aspects of the 2^nd embodiment. [00165] In an 8^th aspect of the 2^nd embodiment, the optical mode quantum detector is a single photon detector, a number resolving photon detector, a homodyne photon detector, or a heterodyne photon detector. The remainder of the features and example features of the 8^th aspect are as described above with respect to the 1^st through the 7^th aspects of the 2^nd embodiment. [00166] In a 9^th aspect of the 2^nd embodiment, the method further comprises transmitting a classical signal from the first node to the second node. The remainder of the features and example features of the 9^th aspect are as described above with respect to the 1^st through the 8^th aspects of the 2^nd embodiment. [00167] In a 10^th aspect of the 2^nd embodiment, the classical signal is an optical signal or an radio frequency (RF) signal. The remainder of the features and example features of the 10^th aspect are as described above with respect to the 1^st through the 9^rd aspects of the 2^nd embodiment. FH11893796.1 UCT-00625 [00168] While this invention has been particularly shown and described with references to example embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims. FH11893796.1

Claims

UCT-00625 CLAIMS What is claimed is: 1. A quantum network, comprising: at least a first node comprising a first quantum system, and a second node comprising a second quantum system; and an optical guide in optical communication with the at least the first and the second nodes, wherein the optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure. 2. The quantum network of Claim 1, wherein: the first quantum system is in a quantum state and is configured to generate at least one electromagnetic (EM) optical mode encoding the quantum state; the first node is configured to direct the optical mode into the optical guide; and the second node is configured to cause an interaction of the optical mode with the second quantum system, thereby imparting the quantum state onto the second quantum system. 3. The quantum network of any one of Claims 1-2, wherein the first quantum system and the second quantum system each comprises a qubit. 4. The quantum network of any one of Claims 1-3, wherein the first and the second quantum systems, each independently comprises a quantum processor, a quantum memory, a quantum sensor, or a quantum simulator. 5. The quantum network of Claim 4, wherein the first and the second quantum systems, each independently comprises a neutral atom array, a trapped ion array, or a solid state emitter. FH11893796.1 UCT-00625 6. The quantum network of any one of Claims 1-5, wherein the first and the second nodes, each independently comprises an optical mode quantum emitter. 7. The quantum network of Claim 6, wherein the optical mode quantum emitter comprises one or more of a single photon source, a squeezed photon source, or a spin- photon entangled source. 8. The quantum network of any one of Claims 1-7, wherein the first and the second nodes, each independently comprises an optical mode quantum detector. 9. The quantum network of Claim 8, wherein the optical mode quantum detector is a single photon detector, a number resolving photon detector, a homodyne photon detector, or a heterodyne photon detector. 10. The quantum network of any one of Claims 1-9, wherein the first and the second nodes are interconnected by a classical network. 11. The quantum network of Claim 10, wherein the classical network is an optical signal or a radio frequency (RF) signal network. 12. The quantum network of any one of Claims 1 to 11, wherein the lens cascade comprises a plurality of lenses, and wherein at least two lenses are stabilized. 13. The quantum network of Claim 12, wherein at least two lenses are self-aligned. 14. The quantum network of any one of Claims 1 to 13, wherein the lens cascade comprises at least one lens having a focal length of greater than 1 km. 15. The quantum network of any one of Claims 1 to 14, wherein the lens cascade comprises a plurality of lenses, each lens having the same focal distance. 16. The quantum network of any one of Claims 1 to 15, wherein the lens cascade comprises a plurality of lenses, FH11893796.1 UCT-00625 each lens of the lens cascade having a focal length tolerance of about 2 m or less; lenses of the lens cascade being spaced apart at an interval, the interval having a tolerance of about 4 m or less; any two adjacent lenses of the lens cascade having a transversal offset of about 0.1 mm or less; each lens of the lens cascade having a radius of about 15 cm or more; and each lens of the lens cascade having a lens loss of about 210 ppm or less. 17. The quantum network of any one of Claims 1 to 16, wherein the lens cascade is configured to operate at a temperature of about 300 K and wherein the reduced pressure enclosure is configured to operate at a pressure of about 10 Pa or less. 18. A method of transmitting a quantum state between nodes of a quantum network, the method comprising: providing a first node comprising a first quantum system, and a second node, comprising a second quantum system; causing the first quantum system to generate at least one electromagnetic (EM) optical mode, the first quantum system being in a quantum state, the optical mode encoding a quantum state; directing the optical mode into an optical guide, the optical guide being in optical communication with the second node, wherein the optical guide comprises: a reduced pressure enclosure; and a lens cascade disposed in the reduced pressure enclosure, and causing the optical mode to interact with the second quantum system, thereby imparting the quantum state onto the second quantum system. 19. The method of Claim 18, wherein the first quantum system and the second quantum system each comprises a qubit. 20. The method of any one of Claims 18 to 19, wherein the first and the second quantum systems, each independently comprises a quantum processor, a quantum memory, a quantum sensor, or a quantum simulator. FH11893796.1 UCT-00625 21. The method of Claim 20, wherein the first and the second quantum systems, each independently comprises a neutral atom array, a trapped ion array, or a solid state emitter. 22. The method of any one of Claims 18 to 21, wherein the first and the second nodes, each independently comprises an optical mode quantum emitter. 23. The method of Claim 22, wherein the optical mode quantum emitter comprises one or more of a single photon source, a squeezed photon source, or a spin-photon entangled source. 24. The method of any one of Claims 18-23, wherein the first and the second nodes, each independently comprises an optical mode quantum detector. 25. The method of Claim 24, wherein the optical mode quantum detector is a single photon detector, a number resolving photon detector, a homodyne photon detector, or a heterodyne photon detector. 26. The method of any one of Claims 1-25, further comprising transmitting a classical signal from the first node to the second node. 27. The method of Claim 26, wherein the classical signal is an optical signal or an radio frequency (RF) signal. FH11893796.1