Optical parametric amplification protocol for quantum nondemolition measurements

Abstract

[Solution] Methods and systems are presented that use optical parametric amplifiers in various ways to enhance the native second-order coupling strength, thereby measuring or otherwise observing photonic components of interest without destroying the components of interest at the system output. For example, such components may include a number of signal Bogoliubov excitations, pump modular quadratures, or signal quadratures squared.

Description

[Technical Field]

[0001] Priority is claimed to U.S. Patent No. 63 / 403,217 ("Quantum Nondemolition Measurements With Optical Parametric Amplifiers For Ultrafast Fault-Tolerant Universal Quantum Information Processing"). [Brief explanation of the drawings]

[0002] [Figure 1] FIG. 1 shows a schematic diagram of a system in which a first encoding unit includes one or more optical parametric amplifiers (OPAs) configured for nonlinearity enhancement of at least one photonic component incorporating one or more improvement techniques. [Figure 2] FIG. 2 similarly illustrates a system suitable for nonlinearity enhancement, in which one or more enhancement techniques may be incorporated. [Figure 3] FIG. 3 illustrates a system featuring at least one phase-mismatched OPA for nonlinear enhancement incorporating one or more improvement techniques. [Figure 4] Figure 4 plots the positive-operator-valued measure (POVM) purity as a function of the pump homodyne result for a quantum nondemolition (QND) measurement protocol that incorporates one or more improvement techniques. [Figure 5]Figure 5 plots the relative weight of the squeezed-Fock-state projector in the POVM as another function of the homodyne result for various values ​​of Na, where one or more refinement techniques can be incorporated. [Figure 6] FIG. 6 illustrates a system featuring at least one out-of-phase OPA suitable for nonlinear enhancement, in which one or more improvement techniques may be incorporated. [Figure 7] FIG. 7 shows the trajectory of signal excitation and pump displacement as a function of interaction time, in which one or more improvement techniques may be incorporated. [Figure 8] FIG. 8 shows the signal x-quadrature squeezing level versus the corresponding quadrature noise level, in which one or more improvement techniques may be incorporated. [Figure 9] FIG. 9 illustrates a system featuring at least one phase-matched OPA suitable for nonlinear enhancement, in which one or more improvement techniques may be incorporated. [Figure 10] FIG. 10 summarizes the results of numerical simulations showing a deterministic cubic-phase state generating system featuring at least one OPA and in which one or more improvement techniques may be incorporated. [Figure 11] FIG. 11 shows a plot of nonlinear squeezing as a function of initial EPR squeezing for various values ​​of τ, in which one or more improvement techniques may be incorporated. DETAILED DESCRIPTION OF THE INVENTION

[0003] This invention was made with government support under grant numbers CCF1918549, ECCS1846273, PHY2011363, and ARO grant W911NF-23-1-0048 awarded by the National Science Foundation, and with support from the NASA Jet Propulsion Laboratory. The government has certain rights in this invention. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] R. Yanagimoto, R. Nehra, R. Hamerly, E. Ng, A. Marandi, and H. Mabuchi, Quantum Nondemolition Measurements with Optical Parametric Amplifiers for Ultrafast Universal Quantum Information Processing, PRX Quantum 4, 010333 (2023) (hereinafter "PRX paper") [Non-patent document 2] .Yanagimoto, R. Nehra, R. Hamerly, E. Ng, A. Marandi, and H. Mabuchi, Engineering Cubic Quantum Nondemolition Hamiltonian with Mesoscopic Optical Parametric Interactions, Quantum Physics (quant-ph), arXiv:2305.03260 [quant-ph], See https: / / doi.org / 10.48550 / arXiv.2305.03260 (2023) (hereinafter “Oph paper”).

[0005] The following detailed description is presented primarily in terms of processes and symbolic representations of operations by conventional computer components, including a processor, memory storage devices for the processor, connected display devices, and input devices. Furthermore, some of these processes and operations may utilize conventional computer components in a heterogeneous distributed computing environment, including remote file servers, computer servers, and memory storage devices.

[0006] The terms used in the following description are intended to be interpreted in the broadest reasonable manner, even when used in conjunction with a detailed description of certain exemplary embodiments. Certain terms may be emphasized below, but terms that are intended to be interpreted in a limiting manner are expressly and specifically defined as such.

[0007] The phrases "in one embodiment," "in various embodiments," "in some embodiments," etc. are used repeatedly. Such phrases do not necessarily refer to the same embodiment. The terms "comprising," "having," and "including" are synonymous unless the context dictates otherwise.

[0008] <Translation notes (regarding notation in the translated text)> If you add the symbols "^" or " ~ " cannot be displayed in the fonts supported by the Japan Patent Office, so symbols such as "^" and " ~ " is written by shifting it forward. For example, "^χ b " indicates that the symbol is a "χ" with a "^" directly above it. Also, in the case of "√" which indicates a square root, the bar above the root symbol is sometimes omitted. Also, there are symbols such as "φ" whose appearance changes depending on the font, but they represent the same symbol.

[0009] "above," "add," "allow," "between," "cat state," "calculate," "combine," "break," "effective," "encoding," "enhanced," "established," "first," "Gaussian," "general The terms "dyne," "generation," "GKP," "Hamiltonian," "homodyne," "implemented," "comprises," "indirect," "input," "intact," "intra-cavity," "larger," "measured," "mismatched," "further," "native," "nonlinear," "nonnegative," "of," "optical," "other," "parametric," "photonic," "ponderomotive," "quadratic," "quantum," "said," "as," "squeezed," "ultrafast," "universal," "here," "wide," "none," or other such descriptors are used in the ordinary sense of yes or no, and not simply as terms of degree, unless the context requires otherwise. In light of this disclosure, those skilled in the art will understand from context the meaning of "remote" and other such location descriptors as used herein. Similarly, those skilled in the art will understand the meaning of "based in part on" or other such descriptions of dependent computational variables / signals. As used herein, "multiple" refers to two dozen or more. As used herein, "immediate" means having a duration of less than two seconds, unless the context requires otherwise. As used herein, unless the context requires otherwise, a circuit is "invoked" when a voltage state transition is made and a digital signal is requested to be transmitted from or through it. Software is "invoked" herein when it is executed / triggered, unless the context requires otherwise. A value is "on the order of" another value if the values ​​differ by less than one order of magnitude (i.e., less than a factor of 10), unless the context requires otherwise. "Cause," as used herein, is not limited to proximate cause, but also includes the realization, combination, or other actual cause of an event or phenomenon. "Instances" of an item, as used herein, may or may not be identical to one another.

[0010] Terms such as "processor," "center," "unit," and "computer" are used herein in their ordinary sense to refer to an inanimate structure. These terms do not include any person, regardless of their location, employment, or other relationship to the described object, unless the context dictates otherwise. Additionally, "for" is typically used descriptively to identify special-purpose software or structures, rather than to indicate a mere intended purpose, as in "circuitry for" or "instruction for." "Specific," "given," and "particular" are not intended to provide nuanced substantive descriptions related to specification, gift, or particle. Rather, these adjectives individualize items or materials to clearly distinguish them from similar or other items or materials in a given context, without the use of hierarchical terms such as "first."

[0011] Reference will now be made in detail to the description of the embodiments illustrated in the drawings. While the embodiments are described in connection with the drawings and associated description, there is no intention to limit the scope to the embodiments disclosed herein. On the contrary, the intention is to cover all alternatives, modifications, and equivalents. In alternative embodiments, additional devices or combinations of devices illustrated may be added or combined without limiting the scope to the embodiments disclosed herein.

[0012] Referring now to FIG. 1, a system 100 is shown in which a first encoding unit 140 includes one or more optical parametric amplifiers (OPAs) 160 . Each OPA 160 includes one or more photonic components 176 (e.g., signal quadrature squared 161, pump modular quadrature 162, or a number of signal Bogoliubov excitations 163) and exhibits a first decoherence rate (κ) 177 and a nonlinear coupling strength (g) 183. The “native” nonlinear coupling strength (g) 183 of the nonlinearity enhancement coupling can be significantly enhanced (e.g., 100% or more) by various configurations described herein, depending on which configuration of the OPA(s) 160 and photonic component(s) 176 is used. Such significant enhancement allows many such configurations to utilize the primary photonic component(s). It is possible to leave the signal processing component 176 intact at one output port while obtaining measurements 178 or other useful results via another output port.

[0013] In some contexts, for example, the photonic component 176 may be a shift 138, a state 137, a mode 136, or a pump field quadrature (χ) that characterizes the field 131 of the external pump 130. b ) 134 to encoding unit 140, whereby one or more features 195 of a given input component 176 (e.g., indicative of pump modular quadrature 162) can be monitored as signal output 191 (e.g., via one or more operations 197 instead of detector 170A) to ensure that a given input component 176 continues at pump output 192 over long distances (e.g., distances greater than 100 kilometers) rather than being destroyed during measurement.

[0014] In some contexts, a photonic component 176 may be provided to the encoding unit 140 as an element 165 or mode 166 of the input signal state 111A, such that the number of signal Bogoliubov excitations (SBEs) 163 or other input components 176 can be monitored via a pump output 192 so that the corresponding input component 176 continues to the fiberoptic-born signal output 191 rather than being destroyed.

[0015] Referring now to FIG. 2, a schematic representation of system 200 (e.g., as an example of system 100) is shown in which external optical input 232 arrives via port 208A and weakly nonlinear OPA 160A and can then be combined with input signal state 111B (e.g., number of SBEs 264) received via port 208B via dichroic mirror 209A. The entangled input then reaches medium 282, which contains optical parametric amplifier 160B, characterized by additional strength 253, displacement 261, field 271, and coupling 272, as further described below. As shown, second dichroic mirror 209B then separates pump output 192, which exits through port 208C, from signal output 291, which exits through port 208D.

[0016] In some variations of system 200, another photonic component 176 comprises a pump modular quadrature 162 passing through a corresponding (instance of) phase-mismatched OPA 160B, which has a first quadratic coupling strength 183, which is then enhanced with a larger additional quadratic coupling strength 253. This allows such a primary photonic component 176 to pass through intact via output port 208C, while obtaining measurements 178B or other accessible encodings via other output port 208D instead of pump power detector 170A. For a variation in which the primary photonic component 176 of the pump input or similar external input 232 passes directly through the system 100, 200 as (a component of) the pump output 192 or similar result 292 via a feedforward operator, see Figure 6 in conjunction with the accompanying description below.

[0017] Similarly, in some variations of one or more systems 100, 200, a first particular photonic component 176 has a number 264 of signal Bogoliubov excitations 163 that pass through (an instance of) a corresponding phase-mismatched OPA 160B, which has a first quadratic coupling strength 183 that is enhanced with a larger additional quadratic coupling strength 253. This allows such primary photonic component 176 to pass through intact via output port 208D while obtaining state-indicative inference / measurement 178A or other accessible encoding 278 via another output port 208C instead of detector 170B. Preferably, large additional quadratic coupling strengths 253 as described herein can retain the fundamental harmonic primary photonic component 176 passing through the signal outputs 191, 291, 391. See FIG.

[0018] 3, a schematic depiction of system 300 is shown, which in some variations may include or resemble system 100 or system 200 (or both). A phase-mismatched optical parametric amplifier 160C as shown receives a fundamental harmonic mode 385A (e.g., number 264 of SBEs 163) corresponding to state 311A ​​and a second harmonic mode 385B corresponding to state 311B, causing harmonic waves 385A-B to undergo entanglement 322 and selective nonlinear coupling strength enhancement. See paper PRX

[0019] Figure 1 in the paper PRX corresponds to Figure 3 in this specification, which has been adapted to comply with the PCT manuscript preparation standards. Figure 3 shows our PNR QND measurement scheme using the nonlinear quantum behavior of the OPA, and the phase space representation (i.e., Wigner function) of the system state at each step of the protocol is shown using numerical data. In the numerical simulation, the cosine component axis 315A (x a Quadrature) with respect to the sine component axis 314A(p a The initial coherent signal state |φ a Consider (0)>=|α=0.7>. The apparently circular zone 302 (shown as a solid black line) indicates a quasi-probability value 377A of approximately 0.2 or greater. The dashed contour 301 indicates a smaller positive quasi-probability value 377A for this state 311A. The p-squeezed vacuum state 311B of width w=1 / 4 is plotted along axis 315B (xb ) with respect to axis 314B(p b ) is assumed as the initial pump state 311B to be plotted.

[0020] The signal and pump states interact through a frequency-detuned OPA 160C, whose dynamics induce conditional p displacements of the pump field that depend on the number 264 of signal Bogoliubov excitations 163. Simultaneously, the OPA dynamics b also causes a conditional rotation of the signal Bogolyubov excitation 163 depending on the axis 335B(x a ) with respect to axis 334B(p a ) yields a phase spread of the final unconditional signal state 331A characterizing the signal output 391. Note, however, that for each state 375 where Na equals 1 to 3, some zones 303 have ellipses or annuli that exhibit pseudo-probability values ​​377A of about -0.2 or less.

[0021] The complete p-homodyne measurement 370 on the final pump state 331B serves as a QND measurement of ̂Na and projects the signal mode onto a squeezed photon-number state, which is an eigenstate of ̂Na. The final pump state 331B is denoted by the p-quadrature distribution P(pb). The ensemble-averaged signal state 375 is ~ gt(Na-1)≦^pb≦ ~ Conditioned on the results of homodyne measurements in gt(Na+1). System parameters are Δ / g=150,~ Use g / g=1 and total interaction time gt=1.

[0022] The realization of room-temperature ultra-fast photon-number-resolving (PNR) quantum non-demolition (QND) measurements has important implications for photonic quantum information processing (QIP), enabling, for example, deterministic quantum computation in discrete-variable architectures. However, the difficulty of implementing sufficiently strong coupling has hindered the development of scalable implementations. In the paper PRX, we present a quadratic (i.e., χ (2) We propose and analyze a nonlinear optical route to PNR QND using a phase-mismatched (i.e., frequency-detuned) OPA. We show that a coherent pump field driving a phase-mismatched (i.e., frequency-detuned) OPA experiences a favorably conditioned displacement of the signal Bogoliubov excitations. Thus, measurement of the pump displacement provides a QND measurement of the signal Bogoliubov excitations, projecting the signal mode onto the squeezed photon-number state. We then show how our nonlinear OPA dynamics can be utilized to deterministically generate Gottesman-Kitaev-Preskill states via one or more additional Gaussian resources, providing an all-optical route for fault-tolerant QIP in continuous-variable systems. Finally, we place these QND schemes in a more classical context by highlighting the analogy between phase-mismatched optical parametric oscillators and multilevel atom-cavity QED systems. Our analysis shows that our proposal will soon be able to (2) We demonstrate its feasibility in nonlinear nanophotonics, highlighting the high potential of the OPA160 as a universal tool for ultrafast non-Gaussian quantum state engineering and quantum computing.

[0023] Quantum information science and engineering holds great potential to revolutionize many fields, including computing, communications, and metrology. Among the various physical systems that have been experimented with to encode and process quantum information, photonics offers significant advantages in terms of scalability and ultrafast operation at room temperature. Optical photons cover the terahertz bandwidth and can be transmitted over long distances with little decoherence, making them ideal carriers of quantum information. Photonic quantum computing can encode and process information in both discrete-variable (DV) and continuous-variable (CV) architectures. However, the lack of strong optical nonlinearities has hindered the realization of deterministic two-qubit entanglement gates in DV architectures and non-Gaussian resources such as Gottesman-Kitaev-Preskill (GKP) states in CV architectures, both of which are essential for building universal, fault-tolerant quantum information processors. Although the limitations of weak optical nonlinearities can be circumvented by measurement-based nonlinear operations using photon-number discrimination (PNR) measurements, the inherent stochastic nature of these operations and the slow speed of conventional single-photon detectors with complex cryogenic systems (e.g., superconducting nanowires and superconducting transition edge sensors) severely limit the scalability and computational clock rates of these architectures.

[0024] In this context, the realization of ultrafast room-temperature PNR QND measurements is of great importance for both DV and CV systems, where information about the photon number is encoded in auxiliary probes and the backaction is limited to a (partial) projection onto the corresponding photon number eigenstates. Such ultrafast QND measurements can not only replace conventional superconducting PNR detectors, but also directly realize deterministic two-qubit entanglement gates that enable deterministic DV optical quantum computation. Furthermore, the QND properties of this measurement offer unique opportunities in quantum engineering, communications, and metrology. To realize PNR QND measurements with separable single-photon energy shifts, strong coupling, where g / κ>1 (where g is the coherent coupling rate and κ is the decoherence rate), is essential. Since the pioneering work in atomic cavity quantum electrodynamics (QED), strong coupling has been demonstrated in a variety of physical systems. However, simultaneous realization of QND measurements in a scalable, high-bandwidth, room-temperature platform has yet to be developed.

[0025] In the PRX paper, we propose and analyze a nonlinear optical route to PNR QND measurements and all-optical quantum state engineering of GKP states using a quadratic optical parametric amplifier (OPA). Compared to previous proposals for PNR QND measurements using third-order nonlinearities and GKP state generation schemes, our proposal using the OPA160 utilizes a much stronger second-order nonlinearity, offering a more experimentally viable route. Recently, g / κ~0.01 has been demonstrated in second-order nonlinear nanophotonic resonators, but g / κ~10 is also conceivable for ultrafast pulses.

[0026] In the following, we first show that the pump field of a phase-mismatched OPA experiences conditional displacements that depend on the number of signal Bogoliubov excitations (^Na), while ^Na is substantially conserved under the OPA dynamics. As a result, we can perform PNR QND measurements of ^Na by measuring the pump displacement. We then demonstrate that nonlinear OPA dynamics can be exploited to perform modulo-quadrature QND measurements of the pump mode. This demonstrates the nearly deterministic generation of GKP states of the pump mode with only additional Gaussian resources, demonstrating a nonlinear optical route to universal fault-tolerant CVQIP. Finally, we bridge the physics of these QND schemes to a more traditional context by establishing an analogy between phase-mismatched optical parametric oscillators (OPOs) and multilevel atomic-cavity QED systems. We observe conditional localization of the intracavity states into a squeezed Fock state ladder, which can be experimentally inferred from the pump homodyne record without observing any loss of signal photons using quantum filters.

[0027] We consider a phase-matched single-mode second-order nonlinear Hamiltonian. TIFF2025534214000001.tif857(1) where ^a and ^b represent the annihilation operators of the signal (i.e., fundamental harmonic) and pump (i.e., second harmonic) modes, respectively, and g>0 is the nonlinear coupling strength. See "Temporal trapping: a route to strong coupling and deterministic optical quantum computation" by Ryotatsu Yanagimoto et al. in Optica Vol. 9, No. 11 (November 2022) (hereinafter the "Optica paper"). All papers mentioned herein further provide useful context that reinforces this disclosure and should ideally be considered for their useful context.

[0028] Without loss of generality, we assume a phase mismatch between the signal and pump 130 with a non-negative δ≧0. It is worth noting that a variety of photonic systems can be described by equation (1), including high-Q microring resonators, photonic-crystal cavities, temporally trapped ultrashort pulses, and superconducting microwave circuits, and our results here are consistent with any of these variants.

[0029] To treat the pump coherent amplitude (which can be large in many practical scenarios) parametrically, we transform to a displaced frame given by the unitary, where the mean field of the pump modes is hereafter "factored out". TIFF2025534214000002.tif953(2) where |ψ(t)> and |φ(t)> are the system states in the lab frame and the displaced frame, respectively. Without loss of generality, we assume that β is real and positive. Physically, |φ(t)> describes quantum fluctuations around the mean field, and its dynamics is as follows: TIFF2025534214000003.tif1455(3) Here, the Hamiltonian TIFF2025534214000004.tif873(4) consists of a third-order nonlinear term and a second-order nonlinear term. TIFF2025534214000005.tif11112(5) Note that r = 2gβ. From here on, we will assume that we are in the displaced frame unless otherwise specified. OPA160 in initial state TIFF2025534214000006.tif899(6) The pump state is realized for ^H, where the pump state is a coherent state with displacement β in the laboratory frame. The conventional approach to the analysis of OPA is to use the undepleted pump approximation, where the pump state is invariant during the dynamics. This approximation is D ^H NL This is equivalent to ignoring the σ, leading to single-mode squeezing of the signal state. This is the behavior expected for OPAs in the Gaussian quantum optics regime. See R. Yanagimoto, E. Ng, A. Yamamura, T. Onodera, LG Wright, M. Jankowski, MM Fejer, PL McMahon, and H. Mabuchi, Onset of Non-Gaussian Quantum Physics in Pulsed Squeezing with Mesoscopic Fields, Optica 9, 379 (2022).

[0030] Under strong nonlinearities where the pump-depletion approximation breaks down, the nonlinear contributions give rise to non-Gaussian quantum features195, such as signal-pump entanglement, for which a qualitative physical description is critically lacking. In what follows, we present a concise description of the nonlinear quantum behavior of a phase-mismatched OPA160 as a key facilitator of QND measurements178 of signal photons in the squeezed photon number basis. Our analysis employs the Hamiltonian transformation recently introduced in W. Qin, A. Miranowicz, and F. Nori, "Beating the 3 dB Limit for Intracavity Squeezing and Its Application to Nondemolition Qubit Readout," Phys. Rev. Lett. 129, 123602 (2022).

[0031] Assuming a relatively large phase mismatch δ>r, ^H Q can be rewritten as follows: TIFF2025534214000007.tif1294(7) where ^A=^acoshu+^a † sinhu corresponds to the annihilation operator 163 of the Bogoliubov excitation, and Δ=√(δ 2 -r 2 ) and u=tanh -1 (r / δ) / 2. Intuitively, ^A can be interpreted as the annihilation operator of photon excitations in the squeezed photon number basis. The nonlinear Hamiltonian can be rewritten in terms of the Bogoliubov operator as TIFF2025534214000008.tif2792(8) For the rest of the work, ^H Q " is ^H NL Dominant in, i.e., ge 2uAssume that ≪Δ. This can always be achieved by appropriate choices of δ and r (i.e., β). Under these conditions, ^A 2 and ^A †2 Contributions from rapidly rotating terms, including , are averaged out and a rotating-wave approximation can be made. Thus, TIFF2025534214000009.tif1496(9)

[0032] In the Heisenberg picture, the dynamics of the operators can be analytically solved under the above equations as follows: TIFF2025534214000010.tif14112(10) where ^p b =(^b-^b † ) / 2i is the p-quadrature operator of the pump mode. From equation (10), the pump mode ^p b Note that experiences a displacement conditioned on the value of ^Na, leading to a specific signal-pump entanglement structure. Furthermore, [^HD, ^Na] ≈ 0 ensures that the value of ^Na is not disturbed during the evolution of the system. As a result, ^p b By homodyne measurement of , ^Na can be estimated without destructive measurement of the signal mode, and QND measurement of ^Na is realized. b Depending on the measurement result of , the signal state is an eigenstate of ^Na with eigenvalue Na, i.e., a squeezed photon number state TIFF2025534214000011.tif757(11) is projected onto This situation is summarized in Figure 3.

[0033] The performance of our PNR QND measurements is bdepends on the measurement accuracy of , which is limited by the quadrature fluctuations of the probe pump state. Intuitively, the conditional displacement d= ~ gt is the width of the p-quadrature fluctuations w=√(<φ b |^p b 2 |φ b >-<φ b |^p b |φ b > 2 ), which is sufficiently large compared to the initial squeezed-vacuum pump state w=1 / 4 that the value of ^Na can be reliably estimated. Figure 3 shows the results of a full quantum simulation of the nonlinear OPA dynamics with an initial squeezed-vacuum pump state w=1 / 4. The final pump state exhibits multiple Gaussian peaks in phase space separated by a distance d, each corresponding to a different number of Bogoliubov excitations ^Na in the signal. The parameters used in this figure are d / w=4 (d=1, w=1 / 4), so that ^p b Conditioning the measurement results allows us to project the signal state onto a squeezed photon-number state with greater than 90% fidelity for the assumed system parameters.

[0034] Referring now to FIG. 4, the pump homodyne result 401(p bA plot of the POVM purity of the QND measurement protocol as a function of |φ ( / d) is shown. See R. Nehra, M. Eaton, C. Gonzalez-Arciniegas, M. Kim, and O. Pfister, Loss tolerant quantum state tomography by number-resolving measurements without approximate displacements, arXiv:1911.00173 [quant-ph] (2019). We consider Gaussian probe pump states |φ with various widths w. b (0)>. Here, w below the vacuum level w0=1 / 2 is |φ b (0)> is a squeezed vacuum.

[0035] To more quantitatively link the measurement performance and the squeezing of the probe-pump quadrature fluctuations, we express the Kraus operators of the QND measurement protocol. From equation (9), the Kraus operators can be expressed as follows: TIFF2025534214000012.tif1766(12) where C Na (p b )=e -iΔNat <p b -d(N a +1 / 2)|φ b > is the complex probability amplitude of the measurement result, and |p b > is the eigenvalue p b ^p with b (See Appendix B of the PRX paper for a complete derivation.) The Claus operator is associated with a positive operator-valued measure (POVM) which has the following elements: TIFF2025534214000013.tif1594(13) Physically, the result of the complete pump homodyne measurement is pb obeys the probability distribution P(p b ) = <φ a |^F(p b )|φ a >. Given the result p b , the post - measurement signal state is as follows. TIFF2025534214000014.tif1164(14) Up to normalization.

[0036] Note that the POVM is not completely selective with respect to Na. This is because ^F(p b ) is not composed only of the single squeezed - Fock - state projector |Na><Na|. To characterize the mixture of the POVM, it is useful to consider the weights relative to the squeezed - Fock - state projector. TIFF2025534214000015.tif2074(15) Intuitively, this can be interpreted as the weight applied to |Na> based on the homodyne result (see Appendix B below for a complete discussion). In particular, W Na (p b ) = 1 means that the post - measurement state conditioned on the homodyne outcome p b is the pure squeezed Fock state |Na>.

[0037] In Figure 4, the purity of the POVM is plotted against the homodyne measurement result p bThe figure shows the ρ as a function of . Here, we assume squeezed vacuum states with width w as the initial pump state. As can be seen, using pump-probe states with smaller w improves the purity of the POVM for a given d, and allows the signal to be projected into a squeezed photon-number state with higher fidelity. From an experimental point of view, squeezing the pump quadrature allows PNR QND measurements to be performed with shorter nonlinear interaction times, which may result in lower transmission losses.

[0038] Referring now to FIG. 5, the corresponding homodyne result 501(p) for Na of 0, 1, 2, and 3 at w / w = 0.5 is b / d) as a function of the squeezed Fock state projector W in the POVM. Na (p b ) is plotted 500. In the plots 400 and 500, d= ~ Assume a conditional displacement of gt=1.0.

[0039] In contrast to the phase-insensitive photon-number tomography achievable in conventional PNR QND measurements, our system 300 can perform PNR QND measurements in any squeezed photon-number basis, enabling phase-sensitive squeeze tomography, from which phase information about the state under tomographic reconstruction can be obtained. Here, introducing a complex phase into the pump displacement β changes the basis rotation angle, and the ratio r / δ determines the squeezing factor. The measurement basis can be further squeezed as r / δ → 1, resulting in a nonlinear coupling ~ In the other limit as r / δ → 0, the measurement basis converges to a (non-squeezed) photon number state basis, which is where the effective nonlinear coupling vanishes. ~ At the expense of g / g → 0. We note that adding Gaussian operations197 allows for flexible control of the measurement basis without compromising the nonlinear coupling. For this purpose, we use ^H D The signal states before and after evolving under a and ^S † a can be applied. This means that ^N eff =^A † eff ^A eff ^A eff =^S † a ^S a Transform the measurement base to be measured by ^A. eff =^a and ^S a By choosing a =^a †This achieves a QND measurement of ^a. Such a pair of squeezing and antisqueezing operations has been experimentally demonstrated in pulsed nonlinear nanophotonics, reported in R. Nehra, R. Sekine, L. Ledezma, Q. Guo, RM Gray, A. Roy, and A. Marandi, Few-cycle vacuum squeezing in nanophotonics, Science 377, 1333 (2022). A complete analysis of the impact of losses on the external squeezing operation is given in Appendix E. Quantum State Engineering of Gottesman-Kitaev-Preskill States

[0040] So far we have focused on QND measurements of signal excitations163, but here we show that the same physics of nonlinear OPA dynamics can also be used to perform QND measurements of pump field quadratures. To do so, we make use of the operator dynamics of equation (9) as follows: TIFF2025534214000016.tif1079(16) where 2 ~ gt^x b The information about -Δt is encoded in the phase of ^A modulo 2π. Therefore, measuring the phase of ^A, for example in a general-dyne measurement, yields ^x modulo μ=π / (~gt'). b The value is indirectly estimated by assigning the pump mode to the phase measurement of φ. b =x φ (mod μ), where x φ =(φ+Δt) / (2 ~ gt)(modμ). Pump Quadrature^x b Itself is [^x b , ^H D]≈0, ensuring the QND property of measurements. Such modular quadrature measurements play a central role in modern CVQIP, e.g., deterministic generation, stabilization, and quantum error correction using GKP states. Below, we demonstrate the generation of approximate GKP states using the nonlinear dynamics of an OPA160, where additional Gaussian resources (e.g., Gaussian initial states, measurements, and feedforward operations) are used. Our proposal for generating GKP states builds on the work of D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001) and D. J. Weigand and B. M. Terhal, Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit QED, Phys. Rev. A 101, 053840 (2020). However, we provide technical differences that are important for some implementations due to the nonlinear dynamics of the phase-mismatched / frequency-detuned OPA 160.

[0041] In the following discussion, we denote the coherent excitation of the Bogoliubov signal mode as |A>. Physically, |A> is a displaced squeezed state, an eigenstate of the operator ^A with eigenvalue A. As shown in Figure 6, we prepare an initial signal state |A0> with A0>0 as the "meter" state for the phase shift. For the initial pump state, we assume a p-squeezed vacuum with width w along the p-quadrature.

[0042] After propagation through nonlinear OPA 160D for time t, a complete general-dyne measurement 178A measures the phase of ̂A, which projects the signal mode onto a measurement basis of displaced squeezed states. TIFF2025534214000017.tif938(17) This may occur, for example, when system 600 implements an instance of system 100, 200 without detector 170A so as not to corrupt the pump output 192, 692 of system 100, 200, 600, where the measurement basis is parameterized by radius (A0 + ε) ≥ 0 and phase φ. See Appendix F of the PRX paper for details on implementing general-dyne measurements 178B. The performance of the phase measurement can be further improved by applying measurement schemes such as HM Wiseman, Adaptive Phase Measurements of Optical Modes: Going Beyond the Marginal Q Distribution, Phys. Rev. Lett. 75, 4587 (1995) or MA Armen, JK Au, JK Stockton, AC Doherty, and H. Mabuchi, Adaptive Homodyne Measurement of Optical Phase, Phys. Rev. Lett. 89, 133602 (2002). To prepare the GKP state, moduloquadrature measurements with modulus μ = √(2π) are preferred, since they are based on the interaction time. ~ Set gt=√(2π / 2).

[0043] If the magnitude of the meter state A0 is much larger than the vacuum noise level, the measurement results are expected to be exponentially localized around |ε|≪A0. Assuming this condition is met, the post-measurement pump state is approximately TIFF2025534214000018.tif1485(18) This is done by generating an approximate GKP logical state via one or more displacement operations 197, TIFF2025534214000019.tif1787(19) (See Appendix C below and the PRX paper for details), where TIFF2025534214000020.tif54 is the floor function, and |κ> is the width along the x-quadrature κ=√(<^x b 2 >-<^x b > 2 ) = 1 / (2√π)A0. It is worth noting that this GKP generation is nearly deterministic because the extra displacement^D induced by the probabilistic phase readout φ b (^x φ ) is approximately compensated by the feedforward displacement operation 197. The resulting GKP state is symmetric when w = κ is true, corresponding to A0 = 1 / (2√π)w.

[0044] FIG. 6 shows a schematic representation of system 600, along with numerical simulation results demonstrating the generation of a symmetric GKP state with a squeezing level of 15 dB (above an error correction threshold of ∼10 dB). System 600 may, in some variations, include or resemble system 100 or system 200 (or both). A phase-mismatched optical parametric amplifier 160D as shown receives a signal input 685A (e.g., number 264 of SBEs 163) having state 611A and a pump input 685B having state 611B, such that inputs 685A-B experience entanglement 622 and selective nonlinear coupling strength enhancement.

[0045] Figure 6 shows the PNRQND measurement scheme using the nonlinear quantum behavior of the OPA160D (also shown in Figure 3 of the PRX paper), with the phase-space representation (i.e., Wigner function) of the system state at each step of the protocol shown with numerical data. In the numerical simulation, the initial coherent signal state |φ is shown in state 611A, plotting the sine component axis 614A (pa quadrature) against the cosine component axis 615A (xa quadrature). a Consider (0) >= |α=0.7>. Elliptical zones 602A-B (shown by solid black lines) indicate pseudo-probability values ​​(QPVs) 677B of approximately 0.2 or greater.

[0046] Dashed contour 601A indicates the smaller positive QPV 677B of state 631A in an annular zone of weakly positive QPV surrounding the elliptical zone of negligible QPV. Dashed contour 601B similarly indicates the smaller positive QPV 677B of state 631B in several eccentric elliptical zones of negligible QPV. Initial pump state 611B plots axis 614B (pb) against axis 615B (xb). The signal state and pump state interact through frequency-detuned OPA 160D, whose dynamics respond to information-bearing pump modular quadrature 162 as described above.

[0047] A general dyne detector 670 in the final signal state 631A acts as a QND measurement and provides a feedforward 691 to one or more displacement operators 673, which modulate the entangled pump state 631B and generate a displacement along axis 655 (x b ) with respect to axis 654(p b ) to generate a pump output 692 having a state 631C that plots As shown, the resulting pattern of state 631C provides alternating columns and rows of black zones (each signaling a QPV of approximately 0.2 or greater) and matrices of non-black zones 603 (each signaling a QPV of approximately -0.2 or less).

[0048] Our results demonstrate that the nonlinear OPA160D is a sufficient building block for realizing universal nonlinear optical QC, as the at-will supply of GKP states enables fault-tolerant universal quantum computation simply by adding Gaussian resources. Compared with existing nonlinear optical GKP state generation schemes using cross-phase modulation (XPM), our approach employs much stronger quadratic nonlinearities, which we believe will offer great potential for non-Gaussian state engineering at room temperature. Nonlinear quantum fluctuations in OPO dynamics

[0049] An important application of parametric interactions is the OPO (optical parametric oscillator). An OPO is realized by pumping a quadratic nonlinear resonator with an external drive field. In the absence of signal loss, a phase-matched OPO (i.e., δ = 0) has two transient states: odd and even signal cat states, which consist of a quantum superposition of π-phase-shifted coherent states. In the presence of finite signal loss, the parity of the cat states spontaneously switches, transforming the cat state into an incoherent mixture of the original coherent states. This is reminiscent of the spontaneous quantum jumps observed in two-level atom-cavity QED systems. Here we show that phase-mismatched OPOs exhibit behavior reminiscent of multilevel atom-cavity QED, where loss of a signal photon induces quantum jumps between signal states in a squeezed Fock state ladder.

[0050] Hamiltonian term ^H drive =iλ(^b † We introduce an external pump drive for the OPO given by (-^b), and the outcoupling pump loss is expressed by the Lindblad operator ^L b =√κ b (^b+β) (^Lb=√κ in lab frame) b In the absence of signal loss, the pump operator dynamics is TIFF2025534214000021.tif1263(20) where ^N a remains constant. λ=(κ b β) / 2, we obtain the steady state TIFF2025534214000022.tif960(21) where |β Na > is displacement β Na =2iκ b -1~ g(N a +1 / 2) coherent pump state. Na is N a , the pump photons emitted from the OPO depend on N a propagates information about ^N aTherefore, by monitoring the outcoupled pump field 131, the system (pump-signal) state is conditionally at steady state |N a >|β Na >It is expected to collapse into one of the following.

[0051] Now consider the effect of finite signal loss. If the signal photon is |β Na >, the intracavity signal state becomes |N a >→|Na>, resulting in a quantum jump, where the signal mode is TIFF2025534214000023.tif11106(22) This means that the loss of a signal photon corresponding to photon subtraction from the squeezed photon number state is a →N a This means that it causes discrete jumps in both the positive and negative directions of ±1. Note that the flow is biased towards the negative direction because coshu>sinhu.

[0052] 7-8, the stochastic master equation quantum trajectory of the OPO dynamics as revealed by continuous pump-homodyne measurements is shown. Plot 700 in FIG. 7 shows the signal excitation 702<^N a > and pump displacement 703<^p b The trajectory of > as a function of interaction time is shown as the plateau <^p b >=Im(β Na) compared to the expected level. Plot 800 of FIG. 8 illustrates the trajectory of signal x-quadrature squeezing compared to the quadrature noise level of vacuum (0 dB dotted line) and the squeezing limit of the OPO steady-state (-3 dB dotted line). System parameters, Δ / g=100, ~ g / g=1.5, κ a / g=0.03, and κ b Equation (9) was used with / g=3.0.

[0053] ^N a and ^p b Since there is a quantum correlation between , the occurrence of such a quantum jump can be estimated from the recording of the pump homodyne measurement without monitoring any signal loss photons. To emulate this situation, we performed a numerical simulation of the stochastic master equation (SME) implied by the pump p-homodyne measurement without monitoring the signal loss photons. As shown in Figure 7, a > and <^p b > correlated spontaneous jumps are observed, with multilevel plateaus corresponding to the creation of squeezed photon number states, which can only be deduced from pump homodyne recordings. Such discrete behavior emerges from a continuous-variable system where only continuous observables are monitored, and demonstrates the inherent quantum nature of photons. a=0 >|β Na=0> , the signal state becomes a squeezed vacuum state, and its squeeze level 802 can conditionally exceed the -3 dB limit for steady-state squeezing in an OPO cavity (see Figure 8). This strong signal squeezing phenomenon has been demonstrated in W. Qin, A. Miranowicz, and F. Nori, Beating the 3 dB Limit for Intracavity Squeezing and Its Application to Nondemolition Qubit Readout, Phys. Rev. Lett.129, 123602 (2022), where squeezing of over 3 dB was achieved in the pump mode of an OPO. Experimental Outlook

[0054] We discuss experimental features for realizing PNR QND measurements in the single-photon regime. For this purpose, we assume large squeezing factors for all fields involved in the dynamics (in this case, including the signal Bogoliubov excitation163 and the probe pump state137) in order to study the possibility of squeezing enhancing the effective nonlinear coupling. We assume that the signal and pump losses and squeezing factors are on the same level, i.e., κ a ~κ b and w~e -u Assuming that ≪1, the experimental characteristics for several variants of our scheme are as follows: TIFF2025534214000024.tif1015(23) (See Appendix D of the PRX paper for a full discussion.) Here, the wavy lines represent approximate equations (inequalities) that are correct to a factor of the order of unity. We can see that squeezing the probe pump mode reduces g / κ. For example, applying 15 dB of squeezing to the initial pump reduces g / κ. aThe constraints are approximately w -1 A promising realization of the nonlinear optical system of Eq. (1) is via high-Q microring resonators, where g / κ a A g / κ of ~0.01 has recently been achieved in indium gallium phosphide nanophotonics and thin-film lithium niobate nanophotonics. Furthermore, ultrafast pulse operations, enabled by advanced dispersion engineering, can further enhance nonlinear coupling by simultaneously exploiting temporal and spatial field confinements, leading to g / κ of ~0.01. a If realized in a single-path, such implementations with ultrashort pulses may enable PNR QND measurements at terahertz slew rates. These figures suggest that this proposed scheme may soon achieve χ (2) This provides a bright picture of what can be achieved in nonlinear nanophotonics.

[0055] In accordance with the above disclosure, we propose and analyze a scheme for PNR QND measurement and quantum state engineering utilizing the nonlinear quantum behavior of the OPA 160. The pump modes driving the phase-mismatched OPA 160 are the number of signal Bogoliubov excitations 163^N a Depending on the condition, we experience a conditional displacement 261 and non-destructively measure it by pump homodyne detection. a Such PNR QND measurements enable efficient ultrafast PNR measurements (replacing conventional slow superconducting detectors) and deterministic implementation of photon-photon entangling gates, providing all the ingredients necessary for deterministic room-temperature DV optical quantum computing at ultrafast clock rates.

[0056] Next, we demonstrate that nonlinear OPA dynamics can be exploited to realize modular quadrature QND measurements of pump modes via signal phase measurements, which naturally provides a method for deterministically generating optical GKP states with additional fully Gaussian resources. Our results unlock many promising possibilities for room-temperature ultrafast universal quantum computation using GKP states in a CV architecture. It is also worth noting that our GKP state generation protocol employs Gaussian quadrature measurements, which can be purified using the recently demonstrated high-gain linear OPA160 amplification technique prior to inefficient generalyne measurements, thereby providing a method for generating highly pure GKP states. Finally, we extend the discussion to OPO physics and show that continuous homodyne monitoring of the outcoupled pump field131 leads to conditional localization of the signal mode on the squeezed photon number state, thereby highlighting the unique opportunity to synthesize and characterize intracavity nonclassical states in real time.

[0057] The above embodiments provide a clear path to overcoming the long-standing challenges of nonlinear optical PNR QND schemes based on cross-phase modulation (XPM), whose inherent self-phase modulation introduces detrimental phase noise into the probe field, because they do not rely on materials with cubic nonlinearity. Our work establishes a concise description of nonlinear-optical parametric interactions that goes beyond the conventional semiclassical picture, thereby pointing a practical route to large-scale, ultrafast, fault-tolerant universal photonic quantum information processors at room temperature.

[0058] Referring again to FIG. 3, a system 300 is shown that implements a squeezed cat-state generation scheme using cubic QND measurement with optical parametric interactions. The Wigner functions of the quantum states at each stage of the protocol are shown using data from a full quantum simulation. As the initial state, ω a The FH mode 385A and the SH mode 385B are prepared in the p-squeezed vacuum state 311A ​​and the vacuum state 311B, respectively, with the sine wave frequency (P(p)) = √5 / 2. After propagating through one or more external squeezers 381A-B and one or more nonlinear media 382, ​​the final unconditional FH mode 331A and the SH mode (P(p b ) to obtain the illustrated state 331B with a marginal p-quadrature distribution 335B corresponding to state 375. Depending on the results of SH homodyne measurements 178, the FH modes are projected into squeezed Schrodinger's cat states 375. Each color band in (d) corresponds to the SH homodyne measurement results p that yield an ensemble average state with a color corresponding to state 375 with probability P.b Represents an interval. To generate cat states of size ∈{√4,√8,√12,√16}, the intervals are set as follows: TIFF2025534214000025.tif9134(24) The 10dB power gain allows you to adjust the squeezer a 2 =r b 2 =10. Appendix A: Derivation of the Rotating Frame Hamiltonian

[0059] To derive Hamiltonian (1), we use the single-mode χ (2) We start with the Hamiltonian. TIFF2025534214000026.tif989(25) We move to the rotating system by the following unitary: TIFF2025534214000027.tif1367(26) This transforms the Hamiltonian into: TIFF2025534214000028.tif1962(27) where frequency detuning δ=ω a -ω b / 2. APPENDIX B: PNR DETECTION KRAUS OPERATORS

[0060] In this section, we consider the Hamiltonian ^H D We derive the Kraus operators for PNR QND measurement using p bFor a pump p-homodyne outcome of , the post-measurement signal state is: TIFF2025534214000029.tif1081(28) Up to normalization, where |p b > is the eigenvalue p b ^p b is an eigenstate of target signal state TIFF2025534214000030.tif1754(29) For , we obtain the following. TIFF2025534214000031.tif3597(30) where γ Na =id(N a +1 / 2), TIFF2025534214000032.tif1098(31) Equation (30) can be summarized as follows: TIFF2025534214000033.tif951(32) Using Kraus operators, TIFF2025534214000034.tif1562(33) Pump-probe state |φ b If we assume a squeezed vacuum with width w along a p-quadrature as (0)>, we can write down the complex probability amplitude analytically. TIFF2025534214000035.tif2089(34) This is because the center is p b =d(N a +1 / 2), which is a Gaussian function of width w.

[0061] QND measurement protocol^F(pb The positive-operator-valued measure (POVM) of ) is easily obtained from the Claus operator. TIFF2025534214000036.tif2571(35) POVM is normalized under the normalization condition ∫dp b ^F(p b )=1 a Note that satisfies.

[0062] Since the POVM(35) consists of a mixture of multiple squeezed-Fock-state projectors, N a It should be noted that the POVM is not completely selective for . To quantitatively characterize this mixing property of the POVM, we introduce the relative weights of the squeezed Fock state projectors. TIFF2025534214000037.tif1661(36) W Na (p b To understand the physical interpretation of (29), it is useful to consider what the squeezed photon number distribution in the premeasurement state looks like: TIFF2025534214000038.tif1053(37) is the homodyne result p b Using equation (32), the squeezed photon number distribution in the postmeasurement state can be expressed as follows: TIFF2025534214000039.tif980(38) N is a normalization constant. Comparing equations (37) and (38), {W Na (p b )} is the squeezed photon number distribution |αN of the input statea | 2 can be interpreted as a conditional weight multiplied by In particular, there is N a W Na (p b ) = 1, the postmeasurement state is a pure squeezed photon number state |N a >This becomes: Appendix C: Fully Gaussian generation of GKP states

[0063] In this section, we introduce a scheme for generating GKP states by modular quadrature measurements that exploits the nonlinear quantum behavior of the OPA 160. In some variations, we incorporate protocols using ponderomotive interactions as presented in D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001) and DJ Weigand and BM Terhal, Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit QED, Phys. Rev. A 101, 053840 (2020).

[0064] In the following discussion, we denote the coherent excitation of the signal Bogoliubov excitation by |A>. Physically, |A> is a displaced squeezed state, an eigenstate of ^A with eigenvalue A. We consider the following as the initial state of the system: TIFF2025534214000040.tif1373(39) A0>0, and φ b (xb b ) denotes the x-quadrature amplitude of the initial pump state. After passing through the phase-mismatched OPA 160 at time t, the phase of the signal mode is measured by a general-dyne measurement. This is done by measuring the phase of the signal mode at state |e iφ It projects onto a measurement basis spanned by (A0 + ε)> and is parameterized by the radius A0 + ε ≥ 0 and the phase φ. Details of the construction of the general-dyne measurement are given in Appendix F of the PRX paper.

[0065] For given measurements ε and φ, the post-measurement pump state is: TIFF2025534214000041.tif25107(40) As a result, the Kraus operators that represent the measurement protocol can be written as follows: TIFF2025534214000042.tif16101(41) where: TIFF2025534214000043.tif32105(42) is the complex amplitude.

[0066] If we adopt a "meter" signal state with an amplitude much larger than the vacuum noise level, we expect the signal measurement results to be exponentially localized near |ε|≪A0. Assuming this condition is met, equation (42) can be approximated as follows: TIFF2025534214000044.tif2197(43) where xn=nμ and x φ =(Δt+φ) / (2 ~ gt) (mod μ), μ=π / ( ~ gt). Note that equation (43) represents multiple Gaussian peaks with width κ=1 / (2√π)A0 and spaced apart by the same distance μ.

[0067] To generate the GKP state, we consider a p-squeezed pump state. TIFF2025534214000045.tif1967(44) The width along the p-quadrature is w, and the interaction time is ~ Set gt = √(π / 2) and μ = √(2π). With these parameters, the post-measurement pump state is as follows: TIFF2025534214000046.tif73115(45) where we have neglected the overall normalization constant, where |κ> is the x-squeezed vacuum with width κ along the x-quadrature. If we assume that |κ> is strongly squeezed, we can make the following approximation: TIFF2025534214000047.tif6996(46) where: TIFF2025534214000048.tif54 is the floor function, which allows the post-measurement state to be rewritten as follows: TIFF2025534214000049.tif1280(47) where: TIFF2025534214000050.tif1782(48) is the approximate GKP logical state. Note that Equation (47) can be transformed into an approximate GKP state by feedforward displacement operations based on the general-dyne measurement result. Appendix D: Experimental design characteristics of PNR QND measurements

[0068] In this section, we study the experimental features and conditions for realizing the PNR QND measurement scheme in the single-photon regime. In the presence of dissipation, the density matrix of the system state obeys the following master equation: TIFF2025534214000051.tif20107(49) Here, {^O1, ^O2}=^O1^O2+^O2^O1 is the anti-commutator. In this paper, we have assumed that the dynamical time scale Δ of the phase rotation of the Bogoliubov excitation dominates the nonlinear coupling rate, i.e., Δ≫g. Here, we further assume that Δ dominates the time scale of dissipation, i.e., Δ≫κa, κb. If this assumption holds, then by the rotating wave approximation, ^A in Eq. (49) 2 and ^A †2 This justifies ignoring the contribution from fast rotating terms, including . Specifically, for the term describing signal loss, we obtain TIFF2025534214000052.tif14102(50) where ^L + =√κ a sinh(u)^A † and ^L- =√κ a cosh(u)^A † This result shows that under the rotating wave approximation, the original signal Lindblad operator ^L a =√κ a The effect of ^a is expressed as two Lindblad operators ^L + and ^L - Show that it can be decomposed into

[0069] In the following discussion, for the sake of concreteness, we consider the squeezed single-photon state |N_a=1}$ as the initial signal state. For the initial pump state, we assume a p-squeezed vacuum with width w along the p-quadrature. For successful QND measurements of PNR, the probability of a quantum jump occurring in the signal mode needs to be sufficiently low. In the low-loss limit, the quantum jump probability is approximately given by TIFF2025534214000053.tif2989(51) This means that the characteristic timescale of the loss-induced quantum jump is t jump ~1 / (cosh 2 (u)κ a )

[0070] "High" reliability ^N a To be able to measure t jumpThe conditional displacement occurring on the time scale of must be larger than the characteristic width of the pump state. Here, in the presence of a finite but small pump loss, the width of the final pump state along the p-quadrature is TIFF2025534214000054.tif2854(52) Here we consider κ b We assumed t ≪ 1. As a result, the experimental condition for the success of our scheme is ~ gt jump TIFF2025534214000055.tif33w'(t jump ), where the wavy symbol is used to represent approximate equalities up to factors of the order of unity. Here, we assume strong squeezing for our purposes (i.e., signal Bogoliubov excitation and pump states). We also assume similar loss and squeezing for both the signal and the pump, i.e., κ a ~κ b and w~e -u ≪1. Under these conditions, w'(t jump ) is a factor of unity larger than w, and w'(t jump )~w. As a result, the experimental design features of our scheme can be expressed in a simple formula as follows: TIFF2025534214000056.tif1014(53)

[0071] Appendix E of the PRX describes a loss analysis of external squeezers, where the loss of each squeezer is modeled by a pair of equal beam splitters placed before and after the squeezer.

[0072] Appendix F of the PRX paper describes the construction of a general-dyne measurement using two balanced homodyne detectors and one ancillary vacuum state. The overall measurement protocol projects the input state |φ>1 into a measurement basis spanned by displaced squeezed states. The results of the general-dyne measurement are related to those of the homodyne detectors via x+ip=secθx1+icscθp2. The choice of beam splitter (BS) transmittance allows the squeezing level of the measurement basis ξ=tanθ in the configuration to be set. See OPH paper

[0073] We propose a scheme to realize a third-order cubic quantum nondemolition (QND) Hamiltonian with optical parametric interactions. (2)We show that strongly squeezed fundamental and second harmonic fields propagating through nonlinear media can be efficiently evolved under a third-order QND Hamiltonian. We highlight the versatility offered by such Hamiltonians for designing non-Gaussian quantum states, such as Schrödinger cat states and cubic phase states. We show that these generation schemes are highly tolerant to various loss sources, including detector inefficiencies and outcoupling losses in off-chip measurements. Our proposal involves operating the parametric interaction in the mesoscopic photon-number regime, which significantly enhances the effective nonlinear coupling from the native single-photon coupling rate and provides a powerful tool against photon loss. Experimental figures show that this scheme is feasible in the near future, especially in pulsed nonlinear nanophotonics.

[0074] Engineering nonclassical states of light is a central challenge in photonic quantum information processing and engineering, enabling new architectures that transcend classical limits in various fields, including metrology, sensing, communications, and computing. As the discovery of one-way optical quantum computation (QC) attests, in some variants, the generation of an initial nonclassical resource state can be the only nontrivial step toward universal quantum operations. In continuous-variable (CV) systems, any unitary operation is only realizable by adding additional Gaussian (i.e., linear-optical) resources, giving access to non-Gaussian resource states, such as Schrödinger's cat state, Gottesman-Kitaev-Preskill (GKP) state, or cubic phase states.

[0075] A conventional approach to non-Gaussian quantum state engineering is to exploit the nonlinearities induced by photon-number-resolving (PNR) measurements, which allow for the engineering of highly non-classical states using complex optical circuits. However, the inherent stochastic nature of these operations and the cryogenic temperature requirements of conventional PNR detectors (e.g., superconducting nanowires and superconducting transition-edge sensors) significantly limit the overall scalability of the architecture.

[0076] In this study, we investigate the third-order cubic quantum nondemolition (QND) Hamiltonian ∝^x using optical parametric interactions. 2 a ^x bWe present an engineering scheme for CV quantum information engineering and propose a means to circumvent the limitations of conventional approaches, where the operator ^x a and ^x b are the amplitude quadrature operators for the fundamental and second-harmonic fields, respectively. The cubic QND Hamiltonian can play a versatile role in non-Gaussian quantum engineering. First, it directly enables deterministic implementation of cubic QND gates, completing the universal gate set for CVQC. Second, it can efficiently generate non-Gaussian quantum states with only additional Gaussian operations197 and measurements178. To emphasize the latter point, we introduce schemes for generating Schrödinger cat states and cubic phase states and analyze their performance. Our protocol employs only homodyne conditioning without photon counting, making it compatible with pre-amplification schemes that are robust to losses at the detection stage, such as detector inefficiencies and outcoupling losses in off-chip measurements. Furthermore, our method naturally encompasses mesoscopic photon populations, improving the effective nonlinear coupling by orders of magnitude and providing a remedy for photon loss. Experimental results demonstrate that our approach is feasible in the near future, especially with pulsed nonlinear nanophotonics.

[0077] Resonant single-mode χ with the Hamiltonian (2) Consider a nonlinear system. TIFF2025534214000057.tif1048(54) where g>0 is the nonlinear coupling constant, and ^a and ^b are the annihilation operators for the FH and SH modes, respectively. The Hamiltonian in equation (54) can be realized in a variety of systems, including microcavities, ultrashort pulses trapped in time, and superconducting microwave circuits. Our results do not depend on a specific physical implementation of the Hamiltonian. The initial state of the system is TIFF2025534214000058.tif959(55) Before and after the state evolves under the Hamiltonian of Eq. (54), a pair of orthogonal squeezing operations, ^S a ^S b and ^S † a ^S † b As a result, the whole system evolves and TIFF2025534214000059.tif13103(56) Here, the effective Hamiltonian ^H eff ^H's ^a→^S † a ^a^S a and ^b → ^S † b ^b^S b In the following discussion, ^S † c ^c^S c =r c ^x c +ir c -1 ^p c , ^x c =(^c+^c † ) / 2, ^p c =(^c-^c † ) / 2, and for c∈{a, b}, the field gain r c ≧ 1. As a result, we obtain the following. TIFF2025534214000060.tif18109(57) geff=ra 2 r b g, and the cubic QND Hamiltonian ^H eff ∝^x a ^x b effectively achieve this. Remarkably, such a third-order QND Hamiltonian allows for a universal gate set for CVQC, and our scheme provides a deterministic realization. c Assuming ≫1, ^H eff The time evolution under can be approximately solved in the Heisenberg picture as follows: TIFF2025534214000061.tif21105(58) where the normalized interaction time τ=g eff t and the SH quadrature operator ^p b ^x a 2 This means that the eigenvalues ​​undergo a conditional displacement that depends on the value of [^H eff , ^x a 2 ]≒0 is ^x a 2 is constant during the evolution of the system, and the homodyne measurement b By measuring the quadratic quadrature ^x a 2 Note that this allows for QND measurements to be made.

[0078] A schematic of a system 900 implementing the squared quadrature QND measurement protocol is shown in Figure 9, which also summarizes the results of our numerical simulations shown in Figure 1 of the Oph paper. System 900 may, in some variations, include or be similar to system 100 or system 200 (or both). Each medium 981A-B and at least one phase-matched optical parametric amplifier 160 (e.g., including medium 982) receives a fundamental harmonic 985A having a state 911A and a second harmonic 985B having a state 911B. State 911A plots the sine component axis 914A (pa quadrature) against the cosine component axis 915A (xa quadrature). An elliptical zone 902 (shown as a solid black line) within the plot exhibits quasi-probability values ​​(QPV) 377C of approximately 0.2 or greater. State 911B similarly plots the axis 915B (xa quadrature) within the low eccentricity elliptical zone 902 of high QPV 377C. b ) with respect to axis 914B(p b ) to plot.

[0079] Downstream of the additional media 983A-B as shown are states 931A-B with a zone 901 of high QPV377C (shown in black) and weakly positive QPV377C (bounded on the outside by the dashed contour 901 and on the inside by the solid black zone 902). Homodyne conditioning 965 applies p a Axis 954 corresponds to x a Each condition 975 plotted against axes 955A-B is applied as shown.

[0080] The Claus operator that characterizes the QND measurement scheme is the SHp-homodyne measurement result p b is given as a function of TIFF2025534214000062.tif1471(59) Here, the complex amplitude C pb (x a )=φ b (p b -τx a 2 ) is given as a function of the initial probe SH state. TIFF2025534214000063.tif968(60) where |p b > is the eigenvalue p b ^p with b is an eigenstate of (|x a Physically, the homodyne result p b The probability distribution of is given by the Born rule. TIFF2025534214000064.tif9133(61) is the non-normalized post-measurement FH state. For a general discussion of optical implementations of nonlinear quantum measurements readers, see also J.M. Epstein, K. Birgitta Whaley, and J. Combes, Quantum limits on noise for a class of nonlinear amplifiers, Phys. Rev. A 103, 052415 (2021).

[0081] The resolution of QND measurements depends crucially on the p-quadratur fluctuations of the probe SH state, which are the fluctuations that cause the p-squeezed vacuum to move towards the probe state |φ b (0)> can be used to improve the situation. b The squeezing of (0)> is the initial SH squeezing operation^S b can be absorbed into |φ without loss of generality. b We can assume that |φ(0)> = |0>. Also, the imbalance between the first and second SH squeezing operations can be taken into account by a trivial scaling of the final SH p-homodyne readout. Therefore, in the following, we use |φ b Assume (0)> =|0>.

[0082] Vacuum probe state |φ b (0)> =|0>, and C pb (x a )=(2 / π) (1 / 4) e(-(p b -τx a 2 ) 2 ) This means that p b When is much larger than vacuum fluctuations, it can be approximated as the sum of two Gaussian distributions. TIFF2025534214000065.tif1051(62) In addition, TIFF2025534214000066.tif1134. The separation and width of the Gaussian peaks are ξ = 2√(p b / τ) and w=(2τξ) -1Intuitively, equation (62) is b The measurement result of |^x up to the uncertainty of w a This means estimating |=ξ / 2, which projects the FH mode into a coherent superposition of displaced squeezed states.

[0083] In the following, we analyze the squared quadrature QND measurement to generate the squeezed Schedinger cat state. As the initial FH state, we use a squared quadrature QND measurement with width w along the x-quadrature. a =√(<^x a 2 >-<^x a > 2 Consider a p-squeezed vacuum state with p b Subject to a measurement result of >0, the post-measurement FH state is approximately: TIFF2025534214000067.tif1386(63) Here we a 2 We assume that ≫ξw (see Appendix B of the Oph paper for a complete discussion). Note that (63) is a coherent superposition of two x-squeezed states, each with width w and separated by a distance ξ, which is a squeezed cat state. Figure 9 shows the results of a full quantum simulation, where the initial FH squeezed vacuum is transformed into the SH homodyne measurement result p b is projected to a non-Gaussian state that depends on p b In the large region, the post-measurement FH state becomes a highly non-classical squeezed cat state.

[0084] The realization of a cubic QND Hamiltonian suggests more general non-Gaussian quantum state engineering. To emphasize this point, we introduce the deterministic generation of cubic phase states. A schematic of this system 600 is shown in Figure 6. As an initial state, we consider the correlation ^x a (0)-^x b (0)≒0 and ^p a (0)+^p b Consider the EPR state (referring to Einstein, Podolsky, and Rosen) with (0) ≈ 0. Equation (59) allows us to solve the dynamics of the FH quadrature operator as follows: TIFF2025534214000068.tif1794(64)

[0085] Here, the first and second terms are approximately 3τ^x a 2 (0)≒3τ^x a 2 (τ) and 0. χ (2) After propagation through a nonlinear medium 682, a p-quadrature measurement is performed on the SH mode, and the third term is expressed as a real p b As a result, applying the FH p-displacement operation to correct for this change results in ^p a (τ)=3τ^x a 2 (τ), which indicates that the final FH state is a cubic phase state675.

[0086] Referring now to FIG. 10, a deterministic cubic-phase state generation system 1000 using optical parametric interaction (via medium 1082 configured as OPA 160) is shown. The phase-space portrait (Wigner function) of the generated state is shown using 10 dB squeezing and an initial EPR pair with τ = 0.2. This results in nonlinear quadrature squeezing Δ NL 2 =0.255 was obtained.

[0087] Referring now to FIG. 11, the initial EPR squeezing Δ for various values ​​of τ in the context of the system 1000 of FIG. EPR 2 Nonlinear (NL) squeezing Δ as a function of axis 1101 NL 2 A log-log plot 1100 is shown with the axis 1102. The dashed black line indicates Δ EPR 2 =Δ NL 2 In Figure 9-11, r a 2 =r b 2 =10 is used.

[0088] In reality, an EPR state can only have finite squeezing, and therefore finite variance Var(^x a (0)-^x b (0))=Var(^p a (0)+^p b (0))=(Δ EPR 2 ) / 4, which reduces the quality of the resultant cubic phase state. To quantify the approximate cubic phase state, Var(^p NL )=Δ NL 2Consider the nonlinear squeezing characterized by the nonlinear quadrature ^p NL =^p a -3τ^x a 2 The plot 1100 shows the variance of Δ NL , Δ EPR and τ, where for a given τ, Δ NL The optimal Δ that minimizes EPR can find

[0089] FIG. 10 shows a phase-space portrait of the cubic phase states generated in corresponding system 1000, summarizing the results of numerical simulations. System 1000 can include or resemble system 100 or system 200 (or both) in some variations. Each of media 1081A-B and at least one phase-matched optical parametric amplifier 160 (e.g., consisting of medium 1082) receives a fundamental harmonic 1085A and a second harmonic 1085B. Additional media 1083A-B, 1084 downstream as shown, provide corresponding x values ​​for various results supporting inferences, as further explained below. a p relative to axis 1055 a There is a detector 1070 conditioning system output 1091 that corresponds to an output state 1075 (characterized by contours 1001 and zones 1002-1003 as described above) represented as an axis 1054.

[0090] In general, in quantum state engineering using measurement-based post-selection, the purity of the resulting states is critically limited by the overall quantum efficiency (QE) of the measurement. In addition to detector inefficiency, photon losses in the setup (e.g., outcoupling losses in nanophotonic implementations) can degrade the overall QE. This problem is particularly severe in photon-number-resolving (PNR) measurements, where a low QE directly impacts the purity of the generated states. On the other hand, imperfect QE can be mitigated in quadrature measurements, e.g., homodyne measurements, by preamplifying the signal using an optical parametric amplifier. Our QND measurement scheme described above relies on a second-stage SH squeezing operation, ^S b † (second-stage SH squeezing operation) already includes such pre-amplification.

[0091] See Figure 3 of the Oph paper for a phase-space representation of heralded squeezed cat states by homodyne detection with a finite QE η. As can be seen from the figure, the cat state generation scheme described here can tolerate fairly large detector imperfections, e.g., η = 80%. By applying additional preamplification with gain G, we can generate highly pure cat states with G = 10 even under larger detector inefficiencies, e.g., η = 20%. Such high robustness against low quantum efficiency is particularly attractive for countering large fixed losses in detector setups (which are common, e.g., outcoupling losses in off-chip detection from nanophotonic waveguides). Note that in general, the higher the gain of a preamplifier, the greater the loss, and in reality, the maximum gain G that can be used is limited.

[0092] Wigner functions of heralded squeezed cat states using cubic QND measurements and homodyne detectors with various QEη. The generation of cat states of size ξ=4 is the result of SH homodyne p b =√(Gη)(τξ 2 ) / 4, where τ = 1.0 is the normalized interaction time and G is the power gain of the preamplifier placed before the detector. The purity of the resulting state (abbreviated as Pur.) is shown at the bottom of each plot. The effect of loss is simulated using the Monte-Carlo wavefunction (MCWF) method with 104 trajectories.

[0093] Another major cause of decoherence is propagation loss within the nonlinear medium. Nominally, the characteristic nonlinear coupling rate g is desired to be larger than the characteristic photon loss rate κ to observe non-Gaussian quantum features,195 leading to the design feature of strong coupling g / κ>1. In our scheme, strong squeezing of the field leads to a mesoscopic number of photons participating in the dynamics, enhancing the effective nonlinear dynamical rate. This allows us to generate highly nonclassical states with a native nonlinear coupling rate at least an order of magnitude smaller than the strong coupling. To see this more concretely, for FH and SH, we consider the same squeezing gain and decoherence rate, i.e., r = r. a =r b and κ = κ a =κ b As Equation (57) suggests, the external squeezing operations reduce the effective nonlinear coupling rate by the field gain g eff =r 3 g, a factor of a cube scaling coefficient. At the same time, the photon loss rate increases linearly with the number of photons, and the effective decoherence rate is κ eff =r 2 As a result, the overall figure of merit g is a factor proportional to the field gain of the squeezer. eff / κ eff=rg / κ is improved, providing resistance to photon loss. Good examples of enhancing nonlinear coupling using amplified quantum fluctuations have been recently published by R. Yanagimoto, T. Onodera, E. Ng, L. G. Wright, P. L. McMahon, and H. Mabuchi, Engineering a Kerr-Based Deterministic Cubic Phase Gate via Gaussian Operations, Phys. Rev. Lett. 124, 240503 (2020); C. Leroux, L. C. G. Govia, and A. A. Clerk, Enhancing Cavity Quantum Electrodynamics via Antisqueezing: Synthetic Ultrastrong Coupling, Phys. 120, 093602 (2018); and W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lu, J. Q. You, and F. Nori, Exponentially Enhanced Light-Matter Interaction, Cooperativities, and Steady-State Entanglement Using Parametric Amplication, Phys.120, 093601 (2018); and in Y. Michael, L. Bello, M. Rosenbluh, and A. Pe'er, Squeezing-enhanced Raman spectroscopy, npj Quantum Inf.5, 1 (2019).

[0094] To verify the enhancement of nonlinearity, we plot (in Figure 4 of the Oph paper) the amount of negativity of the Wigner function of heralded cat states for various squeezing parameters and g / κ. As can be seen from the figure, the strong squeezing operation can improve the quality of the generated cat states for a given value of g / κ. The inset shows the Wigner function of achievable states for g / κ ≈ 0.15 and 20 dB squeezing (i.e., r = 10). We see that the design feature of g / κ that produces a noticeable amount of Wigner function negativity is relaxed by an order of magnitude.

[0095] Figure 4 in the Oph paper shows the amount of Wigner function negativity for cat states generated by third-order QND measurements with various squeezing and losses. Homodyne conditioning is performed there to predict the generation of a cat state of size ξ = 3.5 at τ = 0.55, which nearly maximizes the nonclassicality of the state over the parameter space studied here. The inset shows the Wigner function for the generated state with 20 dB squeezing and g / κ ≈ 0.15. See Appendix C of the Oph paper for a full discussion.

[0096] Experimentally, χ (2)Recent advances in nonlinear nanophotonics have taken us a long way towards the strong coupling regime. Using high-Q micro-ring resonators, g / κ~0.01 has been achieved in thin-film lithium niobate (TFLN) nanophotonics and indium gallium phosphide nanophotonics. With further advances in fabrication techniques that allow material-absorption-limited loss, g / κ~1 is expected. Going beyond conventional continuous-wave devices, g / κ~10 may be possible by exploiting three-dimensional confinement of optical fields using ultrashort pulses. These figures are in line with the next generation of χ (2) We demonstrate that experimental realization of our scheme may be within reach in nanophotonics.

[0097] We propose and analyze the engineering of a cubic QND Hamiltonian using squeezing operations and optical parametric interactions. Such a cubic QND not only directly enables deterministic CVQC but also serves as a versatile tool for efficient non-Gaussian quantum state engineering, e.g., cat states and cubic phase states. The generated resource states are essential building blocks for modern quantum engineering, such as the generation of GKP states and four-component cat states. Compared to existing quantum engineering protocols using cubic nonlinear optics, our approach employs quadratic nonlinear interactions with stronger native coupling rates, potentially providing a more experimentally feasible route. Our work sheds light on the unique capabilities that nonlinear optics can realize in the mesoscopic regime. We hope that our work will contribute to the rapidly developing quantum engineering toolbox of nonlinear photonics, thereby making the most of rapid experimental progress.

[0098] While various operational flows are described in a particular order, it should be understood that various operations may be performed in orders other than those depicted, or may be performed simultaneously. Examples of such alternative orders include overlapping, interleaving, interrupting, reordering, incremental, preparatory, supplemental, simultaneous, reverse, or other variant orders, unless the context dictates otherwise. Furthermore, terms such as past tense adjectives, such as "responsive to" and "related to," are generally not intended to exclude such variants, unless the context dictates otherwise.

[0099] While various system, method, article of manufacture, or other embodiments or aspects have been disclosed above, other combinations of the embodiments or aspects will also be apparent to those skilled in the art in view of the above disclosure. The various embodiments and aspects disclosed above are for purposes of illustration and are not intended to be limiting, with the true scope and spirit being indicated in the final set of claims below.

[0100] In the numbered clauses below, the first combination of aspects and embodiments provides that (1) according to each embodiment, for each instance where a "component" or other such identifier is introduced multiple times within a particular clause chain (e.g., "a" or "an"), such designations may identify the same or different entities; and (2) what may hereinafter be referred to as "dependent" clauses may, in each embodiment, incorporate features of the "independent" clauses to which they refer, or other features described above. term

[0101] Item 1 Quantum detection methods (e.g., using one or more of systems 100, 200) include: configuring (at least) a first quadratic coupling strength 183 within one or more optical media 282 implementing one or more optical parametric amplifiers (OPAs) 160; obtaining a first pump state 137 or other input states 111A-B including one or more photonic components 176; establishing a first nonlinearity enhancement coupling 272 such that a first second-order coupling strength 183 in one or more OPAs 160 is enhanced with (at least) an additional second-order coupling strength 253; transmitting a first output 191-192, 291 (first output) including (at least) the first photonic component 176 of the first input state (e.g., via the first output port 208C or 208D); and A first extraction result 292 (e.g., digital measurement value 178) that encodes the first photonic component 176 of the first input state is transmitted via a first nonlinearity enhancement coupling 272 (e.g., via a second output port 208D or 208C) without destroying the first photonic component 176 of the first output 191-192, 291.

[0102] Section 2 A quantum detection method according to any of the methods above, comprising: In a computing system (100, 200) having a continuous-variable portion thereof, a first nonlinearity enhancement coupling (272) that realizes one or more Gottesman-Kitaev-Preskill (GKP) states is used to trigger ultrafast universal quantum computation.

[0103] Section 3 A quantum detection method according to any of the methods above, comprising: In a continuous-variable portion of the computing system 100, 200, room-temperature universal quantum computation is triggered by one or more GKP states of a first nonlinearity enhancement coupling 272.

[0104] Section 4 A quantum detection method according to any of the methods above, comprising: In the continuous variable portion of the computing system 100, 200, room temperature quantum computation is performed with one or more GKP states via a first nonlinear enhancement coupling 272.

[0105] Section 5 A quantum detection method according to any of the methods above, comprising: obtaining a first Gaussian quadrature measurement 178; A general-dyne measurement 178 is performed after refining the first Gaussian quadrature measurement 178 to generate one or more refined GKP states (e.g., as output feature 195) via a first nonlinear enhancement coupling 272.

[0106] Section 6 A quantum detection method according to any of the methods above, comprising: To achieve the appropriate non-classicality of the generated cat state, we create a first nonlinear enhancement coupling 272 with a cat state size of 3.5±0.1 and a squeeze time of 0.55±0.5.

[0107] Section 7 A quantum detection method according to any of the methods above, comprising: To achieve (at least temporarily) appropriate non-classicality of the generated cat state (e.g., an order of magnitude of the gain or loss of the squeezer as shown in Figure 5 of the Oph paper), we implement a first nonlinear enhancement coupling 272 to the generated cat state with a cat state size of 3.5±1.0 and a squeezing time of 0.55±1.0.

[0108] Section 8 The quantum detection method described in any of the above method sections includes achieving sufficient non-classicality of the generated cat state (e.g., suitable for a wide range of squeezer gains and loss parameters) by allowing the generated cat state to emerge in a first nonlinear enhancement coupling 272 having a cat state size of 3.5±0.2 and a squeeze time of 0.55±2.0.

[0109] Section 9 The first nonlinear enhancement coupling 272 is (At least temporarily) the first ponderomotive (^N a ×^x b ) configured as coupling 272 A quantum detection method according to any of the above method sections.

[0110] Section 10 Configuring a particular photonic component 176 of the one or more photonic components as a signal quadrature squared 161, and configuring the one or more OPAs 160 to include a particular phase-matched OPA 160 receiving the signal quadrature squared 161. The quantum detection method according to any one of the above method sections, comprising:

[0111] Section 11 A quantum detection method according to any of the methods above, comprising: A particular photonic component 176 of the one or more photonic components is configured as a signal quadrature squared 161, and the one or more OPAs 160 are configured to include a particular phase-matched OPA 160 that receives the signal quadrature squared 161, wherein the particular phase-matched OPA 160 has a first quadratic coupling strength enhanced with an additional quadratic coupling strength 253 that is 2-20 times greater than the first quadratic coupling strength.

[0112] Section 12 A quantum detection method according to any of the methods above, comprising: A predetermined photonic component 176 of the one or more photonic components is configured as a pump modular quadrature 162, and the one or more OPAs 160 are configured to include a predetermined out-of-phase OPA 160 that receives the pump modular quadrature 162.

[0113] Section 13 A quantum detection method according to any of the methods above, comprising: A predetermined photonic component 176 of the one or more photonic components is configured as a pump modular quadrature 162, and one or more OPAs 160 are configured to include a predetermined phase-mismatched OPA 160 that receives the pump modular quadrature 162, such that the predetermined phase-mismatched OPA 160 has a native quadratic coupling strength 183 and is enhanced with an additional quadratic coupling strength 253 that is at least 50% greater than the native quadratic coupling strength 183 but less than 50 times greater.

[0114] Section 14 A quantum detection method according to any of the methods above, comprising: A particular photonic component 176 of the one or more photonic components is configured as a number of signal Bogolyubov pumps 163, and the one or more OPAs 160 are configured to include a particular phase-mismatched OPA 160 that receives a pump modular quadrature 162.

[0115] Item 15 A quantum detection method according to any of the methods above, comprising: A particular photonic component 176 among the one or more photonic components is configured as a number of signal Bogolyubov excitations 163, and the one or more OPAs 160 are configured to include a particular phase-mismatched OPA 160 receiving the pump modular quadrature 162, such that the particular phase-mismatched OPA 160 has the native quadratic coupling strength 183 enhanced with an additional quadratic coupling strength 253 that is 2-20 times greater than the native quadratic coupling strength 183.

[0116] Section 16 A quantum detection method according to any of the methods above, comprising: A first OPA 160 of the one or more OPAs 160 is designated as a first ponderomotive (^N a ×^x b ) coupling 272.

[0117] Section 17 A quantum detection method according to any of the methods above, comprising: A particular OPA 160 of the one or more OPAs 160 is configured (at least temporarily) as a phase-matched OPA 160 configured to establish a squeezed-cat state.

[0118] Section 18 A quantum detection method according to any of the methods above, comprising: The first quadratic coupling strength 183 in the OPA 160 is greater than or equal to 1, and the first ponderomotive (^N a ×^x b ) coupling 272, as a first ponderomotive (^N a ×^x b ) coupling 272 is established.

[0119] Section 19 A quantum detection method according to any of the methods above, comprising: configuring a quadratic nonlinear resonator as a first nonlinearity enhancement coupling 272; An external drive field 131 with a finite decoherence rate 177 (κ) that develops a quantum superposition of transient signal cat states in a squeezed Fock state ladder excites a first nonlinearity enhancement coupling 272, which acts as an optical parametric oscillator (OPO) whereby signal photon loss induces quantum jumps between signal states in the transient signal cat state.

[0120] Section 20 A quantum detection method according to any of the methods above, comprising: transmitting a first output via the first output port 208D, the first output including (at least) the primary feature 195 of the first input state; At least one element of the first nonlinear enhancement coupling 272 is a first ponderomotive (^N a ×^x b) coupling 272, a first extraction result 292 (e.g., digital measurement 178 or other encoding 278) encoding one or more elements of the first photonic component 176 (e.g., state-indicative modes 166) including the number 264 of the signal Bogolyubov excitations 163 of the first input state is transmitted via the second output port 208C without destroying the first output.

[0121] Section 21 The phase noise induced by the self-phase modulation is sufficiently mitigated to obtain a first extraction result 292 without destroying the first photonic component 176. A quantum detection method according to any of the above method sections.

[0122] Section 22 One or more non-Gaussian quantum states are generated in a Hamiltonian medium 282 and used to obtain a first extracted result 292 without destroying the first photonic component 176. A quantum detection method according to any of the above method sections.

[0123] Section 23 One or more non-Gaussian quantum states are generated in (the optical cavity of) a Hamiltonian medium 282 and used to obtain a first extraction result 292 without destroying the first photonic component 176. A quantum detection method according to any of the above method sections.

[0124] Section 24 A quantum detection method according to any of the methods above, comprising: A Hamiltonian medium 282 is implemented as a component of an optical parametric oscillator (OPO), in which the extraction result 292 has an outcoupled pump field 131 monitored by a homodyne detector 170, allowing the intra-cavity squeezed photon-number state to be deduced without destroying the first photonic component 176.

[0125] Section 25 The number of signal Bogoliubov excitations163 of the first input state (^N a ) induces one or more displacements 261 in the pump mode 136 conditioned to the Bogoliubov excitation 163, which allows a quantum non-demolition measurement 178 of the signal Bogoliubov excitation 163 to be obtained indirectly via a homodyne detector 170. The quantum detection method according to any one of the above method sections, comprising:

[0126] Section 26 The mesoscopic number of photons inside the Hamiltonian medium 282 (of the optical resonator) enables an effective native nonlinear coupling rate of 0.1 < g / κ < 1, thereby enabling the first extraction result 292 to be obtained without destroying the first photonic component 176. The quantum detection method according to any one of the items of the above method.

[0127] Item 27 The phase noise induced by self-phase modulation is sufficiently relaxed, and the first extraction result 292 is obtained without destroying the first photonic component 176. The quantum detection method according to any one of the items of the above method.

[0128] Item 28 The quantum detection method according to any one of the items of the above method, wherein photon-number-resolving (PNR) quantum nondemolition (QND) measurement 178 is obtained using a homodyne detector 170 between 15 °C and 30 °C (e.g., room temperature).

[0129] Item 29 The quantum detection method according to any one of the items of the above method, wherein photon-number-resolving (PNR) quantum nondemolition (QND) measurement 178 is obtained in less than 10 microseconds through a Hamiltonian medium 282 between 15 ° and 30 °C (e.g., room temperature).

[0130] Item 30 The quantum detection method according to any one of the items of the above method, wherein photon-number-resolving (PNR) quantum nondemolition (QND) measurement 178 is obtained in less than 100 nanoseconds (e.g., as an "ultrafast" measurement 178) through a homodyne detector 170 between 0 °C and 55 °C.

[0131] Item 31 The quantum detection method of any of the above method sections, wherein the photon number resolved (PNR) quantum non-demolition (QND) measurement 178 is obtained in less than 100 nanoseconds (e.g., as an "ultrafast" measurement 178) via a homodyne detector 170 at between 15° and 30° C. (e.g., room temperature).

[0132] Section 32 The first ponderomotive (^N a ×^x b ) coupling 272 couples the first input state to one or more pump field quadratures (^x b ) 134 is at least temporarily established between the quantum detection method described in any of the above method sections.

[0133] Item 33 First ponderomotive (^N a ×^x b ) coupling 272 is the number of signal Bogoliubov excitations 163 of the first input state (^N a )264 and 1 or more pump field quadratures (^x b ) 134, A quantum detection method according to any of the above method sections.

[0134] Section 34 Number of signal Bogolyubov excitations 163 (^N a a first preparatory operation of configuring the encoding unit 140 to include at least one phase-mismatched OPA 160B among the one or more OPAs 160 receiving 264; and A second preparatory operation that constitutes an encoding unit 140 in universal photonic quantum information processing (QIP) systems 100, 200 The quantum detection method according to any one of the items of the above method.

[0135] Item 35 Establish the first ponderomotive (^N a ×^x b ) coupling 272 as the first nonlinearity enhancement coupling 272 such that 0.1 < g / κ < 10000 The quantum detection method according to any one of the items of the above method.

[0136] Item 36 Establish the first ponderomotive (^N a ×^x b ) coupling 272 as (at least temporarily) the first nonlinearity enhancement coupling 272 such that 0.3 < g / κ < 3000 The quantum detection method according to any one of the items of the above method. Here, g is the nonlinear coupling constant 176, and κ is the decoherence rate 177(κ) of the first ponderomotive coupling 272. <*

[0137] Item 37 So that 0.1 < g / κ < 1, The first ponderomotive (^N a ×^x b) coupling 272 as the first nonlinearity enhancement coupling 272 (at least temporarily) The quantum detection method according to any one of the above method sections, comprising: where g is its nonlinear coupling constant 176 and κ is its decoherence rate 177 (κ).

[0138] Section 38 0.2 <g / κとなるように、 First ponderomotive (^N a ×^x b ) coupling 272 as the first nonlinearity enhancement coupling 272 The quantum detection method according to any one of the above method sections, comprising:

[0139] Section 39 0.5 <g / κとなるように、 First ponderomotive (^N a ×^x b ) coupling 272 as the first nonlinearity enhancement coupling 272 The quantum detection method according to any one of the above method sections, comprising: where g is its nonlinear coupling constant 176 and κ is its decoherence rate 177 (κ).

[0140] Section 40 Transmitting a measurement 178 or other encoding 278 of the first photonic component of a first input state without losing the first photonic component 176. The quantum detection method according to any one of the above method sections, comprising:

[0141] Section 41 The additional quadratic coupling strength 253 is at least 50% greater than the first quadratic coupling strength 183 and is less than 50 times the first quadratic coupling strength 183. A quantum detection method according to any of the above method sections.

[0142] Section 42 The additional quadratic coupling strength 253 is 2-20 times the first quadratic coupling strength 183 A quantum detection method according to any of the above method sections.

[0143] Section 43 The additional quadratic coupling strength 253 is 4-40 times the first quadratic coupling strength 183 A quantum detection method according to any of the above method sections.

[0144] Section 44 The number of signal Bogoliubov excitations 163 of the first input state (^N a ) 264 The quantum detection method according to any one of the above method sections, comprising:

[0145] Section 45 one or more photonic components 176 including a number of signal quadrature squared 161 or pump modular quadrature 162 or signal Bogoliubov excitations 163; A quantum detection method according to any of the above method sections.

[0146] Section 46 One of the one or more photonic components 176 is configured to include a signal quadrature squared 161 or a pump modular quadrature 162 (or both). The quantum detection method according to any one of the above method sections, comprising:

[0147] Section 47 Configuring one of the one or more photonic components 176 to include a number of signal quadrature squared 161 or signal Bogoliubov excitations 163. The quantum detection method according to any one of the above method sections, comprising:

[0148] Section 48 Configuring one of the one or more photonic components 176 to include a number of pump modular quadratures 162 or signal Bogoliubov excitations 163 (or both). The quantum detection method according to any one of the above method sections, comprising:

[0149] Section 49 A specific photonic component 176 of the one or more photonic components 176 is configured to include a signal quadrature squared 161. The quantum detection method according to any one of the above method sections, comprising:

[0150] Section 50 A given photonic component 176 of the one or more photonic components 176 is configured to include a pump modular quadrature 162. The quantum detection method according to any one of the above method sections, comprising:

[0151] Section 51 A particular photonic component 176 among one or more photonic components 176 is identified by the number of signal Bogoliubov excitations 163 (^N a )264 The quantum detection method according to any one of the above method sections, comprising:

[0152] Section 52 The number of (at least) signal Bogoliubov excitations 163 (^N a ) 264 and transmits the pump output 192 as a component of the first extraction result 292 The quantum detection method according to any one of the above method sections, comprising:

[0153] Section 53 The number of (at least) signal Bogoliubov excitations 163 (^N a ) 264 or other first result 292. The quantum detection method according to any one of the above method sections, comprising:

[0154] Section 54 Obtaining and transmitting a digital measurement 178 of the first photonic component 176 at the first input state in the pump output 192 or other extracted result 292 without degrading the first output by more than 1%. The quantum detection method according to any one of the above method sections, comprising:

[0155] Section 55 (At least) the first ponderomotive (^N a ×^x b ) Coupling 272 is established in a system 200 such as that of FIG. A quantum detection method according to any of the above method sections.

[0156] Section 56 The first ponderomotive (^N a ×^x b ) Coupling 272 is established in system 100 as in FIG. A quantum detection method according to any of the above method sections.

[0157] Section 57 A system 100, 200 configured to perform the method described in any of the method sections above.

[0158] Section 58 A system 100, 200 manufactured by the method according to any one of the above methods.

[0159] With respect to the numbered claims set forth below, those skilled in the art will understand that the operations described therein may generally be performed in any order. Also, while various operational flows are shown in a sequential order, it should be understood that various operations may be performed in orders other than those depicted, or may be performed simultaneously. Examples of such alternative orders include overlapping, interleaved, interrupted, reordered, incremental, preparatory, supplemental, concurrent, reversed, or other variant orders, unless the context dictates otherwise. Terms such as "responsive to," "related to," or other such transitive, relational, or other conjunctions generally do not exclude such variants, unless the context dictates otherwise. Moreover, each of the following claims is intended to be given the least restrictive interpretation reasonable to one of ordinary skill in the art.

[0160] The following claims are fully supported by the above description, independently of any documents referenced herein. However, the use of media including color, shading, searchable text, and hyperlink access to related content may result in greater brevity and clarity. Therefore, it is recommended that users refer to color-enhanced online versions of the publications cited herein, whenever possible, to more quickly master the techniques supporting the content of this specification.

Claims

1. configuring a first second-order coupling intensity within one or more optical media implementing one or more optical parametric amplifiers (OPAs); obtaining a first input state including one or more photonic components; establishing a first nonlinear enhancement coupling such that the first second-order coupling strength in the one or more OPAs is enhanced with an additional second-order coupling strength greater than the first second-order coupling strength; transmitting a first output including a first photonic component of the one or more photonic components through a first output port; transmitting a first extraction result encoding the first photonic component of the first input state via the first nonlinear enhancement coupling through a second output port without destroying the first photonic component of the first output. A quantum detection method comprising:

2. The quantum detection method of claim 1 , comprising generating a deterministic cubic phase state using at least one of the first photonic component and the one or more OPAs.

3. 2. The quantum detection method of claim 1, further comprising obtaining and transmitting, via the first nonlinear enhancement coupling, the first extraction result encoding the first photonic component of the first input state via the second output port.

4. 10. The quantum detection method of claim 1, further comprising using the first nonlinear enhancement coupling to trigger ultrafast universal quantum computation to realize one or more Gottesman-Kitaev-Preskill (GKP) states in a computational system.

5. Triggering room temperature universal quantum computation with one or more GKP states of the first nonlinear enhancement coupling.

2. The quantum detection method of claim 1, comprising:

6. triggering room temperature quantum computation in one or more GKP states via the first nonlinear enhancement coupling in the continuous variable portion of the computational system; 2. The quantum detection method of claim 1, comprising:

7. obtaining a first Gaussian quadrature measurement; triggering a generaldyne measurement after refining the first Gaussian quadrature measurement to generate one or more refined GKP states via the first nonlinear enhancement coupling; 2. The quantum detection method of claim 1, comprising:

8. Create a generated cat state with the first nonlinear enhancement coupling having a cat state size of 3.5±0.1 and a squeezing time of 0.55±0.5, achieving nonclassicality of the generated cat state.

2. The quantum detection method of claim 1, comprising:

9. creating a generated cat state with the first nonlinear enhancement coupling having a cat state size of 3.5±1.0 and a squeezing time of 0.55±1.0; Achieving the non-classicality of the generated cat state 2. The quantum detection method of claim 1, comprising:

10. Create a generated cat state with the first nonlinear enhancement coupling having a cat state size of 3.5±0.2 and a squeezing time of 0.55±2.0, and achieve nonclassicality of the generated cat state over a wide range of squeezer gains and losses.

2. The quantum detection method of claim 1, comprising:

11. Configuring the first photonic component as a signal quadrature squared, and configuring the one or more OPAs to include a specific phase-matched OPA that receives the signal quadrature squared.

2. The quantum detection method of claim 1, comprising:

12. The first photonic component is configured as a signal quadrature squared, and the one or more OPAs are configured to include a specific phase-matching OPA that receives the signal quadrature squared, the specific phase-matching OPA having a first second-order coupling strength, the first second-order coupling strength being enhanced with an additional second-order coupling strength that is 2-20 times greater than the first second-order coupling strength.

2. The quantum detection method of claim 1, comprising:

13. Configuring the first photonic component as a pumped modular quadrature and configuring the one or more OPAs to include a predetermined phase-mismatched OPA receiving the pumped modular quadrature.

2. The quantum detection method of claim 1, comprising:

14. Configuring the first photonic component as a pumped modular quadrature, and configuring the one or more OPAs to include a predetermined out-of-phase OPA receiving the pumped modular quadrature, the predetermined out-of-phase OPA having the native second-order coupling strength enhanced with an additional second-order coupling strength that is at least 50% greater than but less than 50 times greater than the native second-order coupling strength.

2. The quantum detection method of claim 1, comprising:

15. The first photonic component is configured as a number of signal Bogoliubov pumps, and the one or more OPAs are configured to include a specific phase-mismatched OPA receiving the pump modular quadrature.

2. The quantum detection method of claim 1, comprising:

16. The first photonic component is configured as a number of single Bogoliubov pumps, and the one or more OPAs are configured to include a specific phase-mismatched OPA receiving the pump modular quadrature, such that the specific phase-mismatched OPA has an additional quadratic coupling that is 2-20 times greater than the native second-order coupling strength, thereby enhancing the native second-order coupling strength.

2. The quantum detection method of claim 1, comprising:

17. The first photonic component is divided into a number of signal Bogolyubov excitations (^N ａ ) and the first nonlinear enhancement coupling is configured as a first ponderomotive (^N ａ ×^x ｂ ) Coupling 2. The quantum detection method of claim 1, comprising:

18. The additional secondary coupling intensity is at least 50% greater than the first secondary coupling intensity and less than 50 times the first secondary coupling intensity.

2. The quantum detection method according to claim 1.

19. A first OPA of the one or more OPAs is connected to the first OPA via a first ponderomotive ( ａ ×^x ｂ ) as a phase-mismatched OPA configured to establish coupling.

2. The quantum detection method of claim 1, comprising:

20. configuring (at least temporarily) certain OPAs of the one or more OPAs as phase-matching OPAs configured to establish a squeezed-cat state; 2. The quantum detection method of claim 1, comprising:

21. The first ponderomotive (^N ａ ×^x ｂ ) coupling as the first nonlinear enhancement coupling; The first second-order coupling strength in the one or more OPAs is ａ ×^x ｂ ) coupling is enhanced by additional second-order coupling strength resulting from 2. The quantum detection method of claim 1, comprising:

22. a second-order nonlinear resonator is configured as the first nonlinear enhancement coupling; By exciting the first nonlinear enhancement coupling with an external driving field having a finite decoherence rate (κ) that develops a quantum superposition of transient signal cat states in a squeezed Fock state ladder, the first nonlinear enhancement coupling becomes an optical parametric oscillator (OPO), and loss of a signal photon induces a quantum jump between the signal states in the transient signal cat state.

2. The quantum detection method of claim 1, comprising:

23. Phase noise induced by self-phase modulation is mitigated to obtain the first extraction result at the first output without destroying the first photonic component.

2. The quantum detection method according to claim 1.

24. One or more non-Gaussian quantum states are generated within a Hamiltonian medium and used to obtain the first extraction result without destroying the first photonic component.

2. The quantum detection method according to claim 1.

25. one or more non-Gaussian quantum states are generated within the Hamiltonian medium; used to obtain the first extraction result without destroying the first photonic component at the first output.

2. The quantum detection method according to claim 1.

26. The Hamiltonian medium is configured as an optical parametric oscillator (OPO) with an outcoupled pump field, the extraction result of which is monitored by a homodyne detector, allowing estimation of the intracavity squeezed photon number state without destroying the first photonic component.

2. The quantum detection method of claim 1, comprising:

27. a non-negative number of signal Bogoliubov excitations of the first input state (^N ａ ) to induce one or more excursions in the pump mode, and a quantum non-demolition measurement of the signal Bogoliubov excitation is obtained indirectly via a homodyne detector.

2. The quantum detection method of claim 1, comprising:

28. The mesoscopic number of photons in a Hamiltonian medium effectively allows for a native nonlinear coupling rate of 0.1<g / κ<1, allowing the first extraction result to be obtained without destroying the first photonic component at the first output.

2. The quantum detection method according to claim 1.

29. Phase noise induced by self-phase modulation is sufficiently mitigated to obtain the first extraction result without destroying the first photonic component.

2. The quantum detection method according to claim 1.

30. Photon-number-discriminating (PNR) quantum non-demolition (QND) measurements are obtained with a homodyne detector between 15°C and 30°C.

2. The quantum detection method according to claim 1.

31. Photon-number-discriminating (PNR) quantum non-demolition (QND) measurements Obtained in less than 10 microseconds through a Hamiltonian medium between 15°C and 30°C 2. The quantum detection method according to claim 1.

32. Photon-number-discriminating (PNR) quantum non-demolition (QND) measurements Obtained in less than a nanosecond via a homodyne detector between 0°C and 55°C 2. The quantum detection method according to claim 1.

33. Photon-number-discriminating (PNR) quantum non-demolition (QND) measurements Obtained in less than a nanosecond via a homodyne detector between 15°C and 30°C 2. The quantum detection method according to claim 1.

34. a first ponderomotive (^N ａ ×^x ｂ ) coupling between the first input state and one or more pump field quadratures (^x ｂ ) is established between 2. The quantum detection method according to claim 1.

35. a first ponderomotive (^N ａ ×^x ｂ ) coupling is a non-negative number (^N ａ ) and one or more pump field quadratures (^x ｂ ) is established between 2. The quantum detection method according to claim 1.

36. The non-negative number of signal Bogoliubov excitations (^N ａ and configuring the encoding unit to include at least one out-of-phase OPA among the one or more OPAs receiving the configuring the encoding unit into a universal optical quantum information processing (QIP) system; 2. The quantum detection method of claim 1, comprising:

37. The first nonlinear enhancement coupling is a first ponderomotive (̂N ａ ×^x ｂ ) Establish coupling 2. The quantum detection method of claim 1, comprising:

38. The first nonlinear enhancement coupling is a first ponderomotive (̂N ａ ×^x ｂ 10. The quantum detection method of claim 1, further comprising: establishing a coupling. where g is the nonlinear coupling constant and κ is the decoherence rate (κ) of the first coupling.

39. The first nonlinear enhancement coupling is a first ponderomotive (̂N ａ ×^x ｂ 10. The quantum detection method of claim 1, further comprising: establishing a coupling. where g is the nonlinear coupling constant and κ is the decoherence rate (κ) of the first coupling.

40. Without destroying elements of the first output that are not the first photonic component, a non-negative number (^N ａ transmitting the first pump power as at least one component of the first extraction result encoding 2. The quantum detection method of claim 1, comprising:

41. A non-negative number (^N) of Bogoliubov excitations of the first input state without destroying modes or other elements of the first output that are not the first photonic component. ａ ) transmitting the pump power encoding 2. The quantum detection method of claim 1, comprising:

42. The non-negative number of signal Bogoliubov excitations (^N ａ ), and configuring the encoding unit such that the one or more OPAs receiving the transmitting the first output from the encoding unit, the first output including the first photonic component of the first input state; 2. The quantum detection method of claim 1, comprising:

43. 10. The quantum detection method of claim 1, comprising transmitting a primary photonic component of said first input state as said first output, said primary photonic component being greater than any other optical characteristic of said first input state.

44. Without destroying the first output, a non-negative number of signal Bogoliubov excitations (^N ａ ) and transmitting the pump power as a component of the first extraction result that encodes 2. The quantum detection method of claim 1, comprising:

45. Without destroying the first output, a non-negative number of signal Bogoliubov excitations (^N ａ ) and transmitting a pump output or other first result that encodes 2. The quantum detection method of claim 1, comprising:

46. The primary of the one or more photonic components is a signal quadrature squared or a pump modular quadrature or a signal Bogoliubov excitation number.

2. The quantum detection method according to claim 1.

47. Configuring the primaries of the one or more photonic components as quadrature-squared or pump-modular quadrature.

2. The quantum detection method of claim 1, comprising:

48. Configuring the primaries of the one or more photonic components as a number of signal quadrature squared or signal Bogoliubov excitations.

2. The quantum detection method of claim 1, comprising:

49. structuring the primaries of the one or more photonic components as a number of pump modular quadratures or signal Bogoliubov excitations; 2. The quantum detection method of claim 1, comprising:

50. 50. A quantum detection system configured to carry out the quantum detection method of any one of claims 1 to 49.

51. means for obtaining a first input state including one or more photonic components; means for establishing a first nonlinear enhancement coupling such that the first second-order coupling strength in the one or more OPAs is enhanced with an additional second-order coupling strength greater than the first second-order coupling strength; means for transmitting a first output including a first photonic component of the one or more photonic components through a first output port; means for transmitting a first extraction result encoding the first photonic component of the first input state via the first nonlinear enhancement coupling through the second output port without destroying the first photonic component of the first output; A quantum detection system comprising:

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