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.
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.
A factorization scheme will be introduced that potentially permits any N00N or M&M state to be generated and detected. Explicit results for N00N states are presented for N=3. Internal loss within the entanglement generator and external loss due to atmosphere, detectors and targets are modeled. A method using these approaches for quantum entanglement based imaging is provided that gives N times classical resolution, where N is the number 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. Explicit entanglement generator and detector designs are provided in terms of linear and nonlinear photonics devices. The fundamental role of post-selection measurement for generating entanglement is included. A general factorization scheme for M&M states is provided. 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, 45, and 45 degree 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 generating N00N, M&N, linear combinations of M&N states as well as more complicated quantum entangled and quantum hyper-entangled states will be considered. Quantum hyper-entanglement refers to quantum entanglement in more than one degree of freedom, e.g. energy-time, polarization, orbital angular momentum, etc. Internal noise and loss within the entanglement or hyper-entanglement generators and external noise and loss due to atmospheric effects and detectors are modeled. Analysis related to the devices that generate these entangled or hyperentangled states will be provided. The following will be derived: closed form expressions for wave function normalization, wave function, density operator, reduced density operator, phase error bound, the symmetrized logarithmic derivative, the quantum Fisher information, the quantum Cramer-Rao lower bound, the relevant projection operators and the related probability of detection expressions. Generation and detection of the entangled or hyperentangled states will be considered. The entanglement generators will use linear and nonlinear optical devices. Optimization criteria for the quantum states, generation and detection schemes and designs optimal with respect to the criteria will be discussed. Applications of the generated states for producing super sensitivity and super resolution will be discussed. The fundamental role of coincidence measurement for generating entanglement is included. Hyperentanglement offers N times classical resolution, where N is a quantum number associated with the system.
An atmospheric imaging system based on quantum hyper-entanglement has been developed. Hyper-entanglement refers to entanglement in more than one degree of freedom, e.g. polarization, energy-time and orbital angular momentum (OAM). The system functions at optical or infrared frequencies. Only the signal photon propagates in the atmosphere, the ancilla photon is retained within the detector. This results in loss being essentially classical, giving rise to stronger forms of entanglement. Bell state generation and Bell state measurement, i.e. the ability to distinguish the various Bell states is discussed. A mathematical representation of Bell state generation and the Bell state analyzer, including a projection operator describing the measurement process is provided. Signatures for unique detection of the various Bell states are developed. A method and design for creating states hyper-entangled in polarization, energy-time, OAM and the radial quantum number is examined. Hermite-Gaussian (HG) modes, Laguerre-Gaussian (LG) modes, OAM dependence of the LG modes and a method of mode conversion are discussed. A projection operator that represents the combined measurements between the different degrees of freedom is provided. A model of generation and detection efficiencies for the different degrees of freedom and the implications for signal to noise ratio (SNR), signal to interference ratio (SIR), the quantum Cramer-Rao lower bound, and the measurement time are provided in closed form. A formula describing how hyper-entanglement greatly improves the maximum detection range of the system is derived. The formalism permits random noise and entangled or non-entangled sources of interference to be modeled.
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.
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.
A hyper-entanglement based atmospheric imaging/detection system involving only a signal and an ancilla photon will be considered for optical and infrared frequencies. Only the signal photon will propagate in the atmosphere and its loss will be classical. The ancilla photon will remain within the sensor experiencing low loss. Closed form expressions for the wave function, normalization, density operator, reduced density operator, symmetrized logarithmic derivative, quantum Fisher information, quantum Cramer-Rao lower bound, coincidence probabilities, probability of detection, probability of false alarm, probability of error after M measurements, signal to noise ratio, quantum Chernoff bound, time-on-target expressions related to probability of error and resolution will be provided. The effect of noise in every mode will be included as well as loss. The system will provide the basic design for an imaging/detection systems functioning at optical or infrared frequencies that offer better than classical angular and range resolution. Optimization for enhanced resolution will be included. The signal to noise ratio will be increased by a factor equal to the number of modes employed during the hyper-entanglement process. Likewise, the measurement time can be reduced by the same factor. The hyper-entanglement generator will typically make use of entanglement in polarization, energy-time, orbital angular momentum, etc. Mathematical results will be provided describing the system’s performance as a function of loss mechanisms and noise.