From a quantum statistical viewpoint, four typical quantum states are Fock, Sub-Poissonian, Poissonian and SuperPoissonian states. Quantum interactions are focus among Fock and Poissonian states. Using quantum statistics, model and simulation, this paper proposes two models: matrix and variant transformations: 1. MT Matrix Transformation – eigenvalue states; 2. VT Variant Transformation – invariant states to analyze three random sequences: 1) random; 2) conditional random in a constant; 3) periodic pattern. Four procedures are proposed. Fast Fourier Transformation FFT is applied as one of MT schemes and two invariant scheme of VT schemes are applied, three random sequences are in M segments and each segment has a length m to generate a measuring sequence. Shifting operations are applied on each random sequence to create m+1 spectrum distributions. For FFT, a pair of eigenvalues are selected as the output. Two types of 1D and 2D variant maps are generated to illustrate multiple parameter selections to generate a series of results. Since sequences 1) and 3) are related simple, more cases are focus on sequences 2). Better than FFT, VT distinguishes various Fock, Sub-Poissonian, Poissonian states in random analysis to distinguish three random sequences as three levels of statistical ensembles: Micro-canonical, Canonical, and Grand-Canonical ensembles. Applying two transformations, quantum statistics, model and simulation of modern quantum theory and applications can be explored.
In modern photon statistics, classical and quantum behavior can be distinguished by various quantum states of photon statistical distributions: Poisson (coherent/semi-classical wave behavior), and sub-Poisson (compressed state/particle behavior). Since this type of measurement mechanism is often associated with advanced laser/optical or photonic techniques, can this type of distribution model be modeled using discrete 0-1 sequences? In this paper, several sets of simulation modes are designed, and FFT transformation is used to extract relevant eigenvalues. Following the processing methods in the variant construction, special filters are constructed using the quantum random sequence provided by ANU (Australian national university), and conditional random sub-sequences are collected as input sequences. Multiple segments are separated from a random sequence, and relevant eigenvalues of FFT are selected to form a special set of eigenvalues. The shift operations are used to transform each sequence, showing obvious non-stationary random effects on various maps.
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