An important mathematical tool for studying open quantum system theory, which studies the dynamics of a reduced system, is the completely positive and trace-preserving dynamical linear map parameterized by a special parameter-time. Counter-intuitively, akin to the Fourier transform of a signal in time-sequence to its frequency distribution, the time evolution of a reduced system can also be studied in the frequency domain. A recent proposed idea which studies the representation of dynamical processes in the frequency domain, referred to as canonical Hamiltonian ensemble representation (CHER), proved its capability of characterizing the noncalssical traits of the dynamics. Here we elaborate in detail the theoretical foundation within a unified framework and demonstrate several examples for further studies of its properties. In particular, we find that the thermal fluctuations are clearly manifested in the manner of broadening CHER, and consequently rendering the CHER less nonclassical. We also point out the discrepancy between the notions of nonclassicality and non-Markovianity, show multiple CHERs beyond pure dephasing, and, finally, to support the practical viability, propose an experimental realization based upon the free induction decay measurement of nitrogen-vacancy center in diamond.
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