Mathematical analysis is presented related to novel quantum entangled states useful for interferometry. For interferometry applications, the approach based on entanglement presented here, even in the presence of loss offers a quadratic reduction in the number of photons required while maximizing phase sensitivity and visibility. Reducing the number of photons decreases the effects of certain mechanisms that degrade results. The new interferometer is an improvement on a previous interferometer based on linear combination of M&N states (LCMNS). LCMNS have been shown to be much more robust in the presence of environmental effects than plain M&N states or N00N states. Adaptive optics techniques are used to adjust the coefficients of the LCMNS. Closed form expressions for the minimum phase error, visibility, and coefficients are derived. Environmental effects are included using open systems theory. Numerical results are provided.
Mathematical analysis is presented related to novel quantum entangled states useful for interferometry. For interferometry applications, the approach based on entanglement presented here, even in the presence of loss offers a quadratic reduction in the number of photons required while maximizing phase sensitivity and visibility. Reducing the number of photons decreases the effects of certain mechanisms that degrade results. The new interferometer is an improvement on a previous interferometer based on linear combination of M&N states (LCMNS). LCMNS have been shown to be much more robust in the presence of environmental effects than plain M&N states or N00N states. Adaptive optics techniques are used to adjust the coefficients of the LCMNS. Closed form expressions for the minimum phase error, visibility, and coefficients are derived. Environmental effects are included using open systems theory. Numerical results are provided.
Novel multiphoton entangled states have been developed, which offer utility for sensors. These states are linear combinations of M&N states (LCMNS). Closed form expressions for the coefficients have been derived that result in states offering sensitivities far greater than those found in the literature. Adaptive optics procedures have been conceived that permit these coefficients to be determined. LCMNS can also be used to generate high quality hyperentangled states. Various states generated by this process are developed. The utility of these procedures for quantum interferometry and RF signal detection is examined. The susceptibility to loss of the various interferometry and RF detection approaches is examined using an open systems analysis. The novel quantum interferometer uses adaptive optics, which offers a huge improvement in phase sensitivity over previous interferometers, while greatly reducing the number of photons used. Large numbers of photons can contribute to vibration reducing sensitivity. The approach based on entanglement presented here, even in the presence of loss can reduce the number of photons used significantly, while enhancing phase sensitivity. The new interferometer simultaneously minimizes phase error, maximizes visibility and greatly reduces the number of photons used. When the interferometer employing LCMNS is combined with an atomic RF antenna, E-field sensitivity is increased to a value nearly 400,000 times that found in the literature. Extensive numerical results are provided comparing phase sensitivity, visibility and E-field sensitivity for LCMNS, N00N states, and non-entangled states. LCMNS offer huge improvements in sensitivity compared to other approaches.
With the goal of designing interferometers and interferometer sensors, e.g., LADARs with enhanced sensitivity, resolution, and phase estimation, states using quantum entanglement are discussed. These states include N00N states, plain M and M states (PMMSs), and linear combinations of M and M states (LCMMS). Closed form expressions for the optimal detection operators; visibility, a measure of the state’s robustness to loss and noise; a resolution measure; and phase estimate error, are provided in closed form. The optimal resolution for the maximum visibility and minimum phase error are found. For the visibility, comparisons between PMMSs, LCMMS, and N00N states are provided. For the minimum phase error, comparisons between LCMMS, PMMSs, N00N states, separate photon states (SPSs), the shot noise limit (SNL), and the Heisenberg limit (HL) are provided. A representative collection of computational results illustrating the superiority of LCMMS when compared to PMMSs and N00N states is given. It is found that for a resolution 12 times the classical result LCMMS has visibility 11 times that of N00N states and 4 times that of PMMSs. For the same case, the minimum phase error for LCMMS is 10.7 times smaller than that of PMMS and 29.7 times smaller than that of N00N states.
With the goal in mind of designing radars, interferometers and other sensors based on quantum entanglement the virtues of N00N states, plain M and M states (PMMSs) and linear combinations of M and M states (LCMMS) are considered. A derivation of the closed form expression for the detection operator that is optimal subject to constraints is provided. The raising and lowering properties of the detection operator and its square are developed. The expectations of the optimal detection operator and its square are derived. The expression for the visibility, the maximum expectation of the optimal detection operator, is developed. From the expectation of the square of the detection operator and the visibility, the phase error and the minimum phase error for the detection operator are derived. The optimal resolution for the maximum visibility and minimum phase error are found. For the visibility, comparisons between PMMSs, LCMMS and N00N states are provided. For the minimum phase error comparisons between LCMMS, PMMSs, N00N states, separate photon states (SPSs), the shot noise limit (SNL), and the Heisenberg limit (HL) are provided. A representative collection of computational results illustrating the superiority of LCMMS when compared to PMMSs and N00N states is given. It is found for a resolution 12 times the classical result LCMMS has visibility 11 times that of N00N states and four times that of PMMSs. For the same case, the minimum phase error for LCMMS is 10.7 times smaller than that of PMMS and 29.7 times smaller than that of N00N states.
A quantum LIDAR for improving resolution using quantum entanglement in the polarization degree of freedom is described. A thorough mathematical analysis of this device is provided. A mathematical discussion of how to generate other more robust entangled states is developed. Internal loss within the entanglement generator and external loss due to atmosphere, detectors, and targets are modeled. A method using these approaches for imaging is provided giving N times classical resolution, where N is the number of photons entangled with explicit results exhibited for N = 3. Closed form expressions for the wave function, normalization, density matrix, reduced density matrix, visibility, and probabilities of detection of one through three photons using detectors with general polarization characteristics are provided. An explicit entanglement generator and detector designs are provided in terms of linear and nonlinear photonics devices. The fundamental role of postselection measurement for generating entanglement is included. Discussions of entanglement devices that will produce general M&M states at near-visible frequencies are given. A discussion of a bearing measurement device that exhibits both super sensitivity and resolution is provided. Computational results are provided that compare probabilities of detection for three single photon detectors with −45-deg, 45-deg, and 45-deg linear polarization. Results for detecting one to three photons or the vacuum state are compared. Computational results for detecting three photons with these detectors for various values of internal and atmospheric loss are provided. Resolution improvements born of quantum entanglement are shown not to degrade with loss. Loss degrades probability of detection not resolution.
Methods of improving quantum LADARs and related sensors are developed based on quantum entanglement and hyperentanglement. Multiple single photon states are used to obtain a multiphoton entangled state. These states can be N00N states, M&N states (M&N), or a linear combination of M&N states (LCMNS) or generalized states kindred to LCMNS. The procedure for doing this derives from the fundamental theorem of algebra. Various states generated by this process are developed. A diagram of a device for producing such states is examined. They are related to a simpler version of this concept introduced in the seminal experiment by Mitchell-Lundeen and Steinberg. A certain class of states obtained through the Schrodinger kitten process are shown to be effective for generating states hyper-entangled in polarization and energy-time. A diagram of a device for producing such states is considered. Alternate methods of generating hyperentanglement in polarization and energy-time are discussed. Improvements offered by networks are discussed. The utility of these procedures for sensing and communications is examined. An open systems analysis based on density operator theory is conducted including both noise and loss mechanisms. The susceptibility to noise and loss of the various hyper-entanglement procedures is examined. Various measures of effectiveness (MOEs) are derived to quantize system performance. MOEs include but are not limited to, SNR, signal-to-interference ratio, quantum Cramer Rao’ lower bound, quantum Chernoff bound, measurement time, the Holevo bound, sensing range, and resolution. A summary table with expanded MOEs results drawing from multiple papers is provided.
Abstract Quantum phenomena such as entanglement can improve fundamental limits on the sensitivity of a measurement probe. In optical interferometry, a probe consisting of N entangled photons provides up to a N N enhancement in phase sensitivity compared to a classical probe of the same energy. Here, we employ high-gain parametric down-conversion sources and photon-number-resolving detectors to perform interferometry with heralded quantum probes of sizes up to N = 8 (i.e. measuring up to 16-photon coincidences). Our probes are created by injecting heralded photon-number states into an interferometer, and in principle provide quantum-enhanced phase sensitivity even in the presence of significant optical loss. Our work paves the way toward quantum-enhanced interferometry using large entangled photonic states.