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Human and model-generated texts can be distinguished by examining the magnitude of likelihood in language. However, it is becoming increasingly difficult as language model's capabilities of generating human-like texts keep evolving. This study provides a new perspective by using the relative likelihood values instead of absolute ones, and extracting useful features from the spectrum-view of likelihood for the human-model text detection task. We propose a detection procedure with two classification methods, supervised and heuristic-based, respectively, which results in competitive performances with previous zero-shot detection methods and a new state-of-the-art on short-text detection. Our method can also reveal subtle differences between human and model languages, which find theoretical roots in psycholinguistics studies. Our code is available at https://github.com/CLCS-SUSTech/FourierGPT, Comment: 13 pages, 12 figures

The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators ${\bf X}$ and ${\bf Y}$, according to the Campbell-Baker-Hausdorff-Dynkin theorem, ${\rm e}^{{\bf X}+{\bf Y}}$ is not equivalent to ${\rm e}^{\bf X}{\rm e}^{\bf Y}$, but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators $\{{\bf X,Y,Z}\}$, such that $[{\bf X}, {\bf Y}] = \kappa{\bf Z}$ for scalar $\kappa$ and cyclic permutations, here it is proven that ${\rm e}^{a{\bf X}+b{\bf Y}}$ is equivalent to ${\rm e}^{p{\bf Z}}{\rm e}^{q{\bf X}}{\rm e}^{-p{\bf Z}}$ for scalar $p$ and $q$. Extensions for ${\rm e}^{a{\bf X}+b{\bf Y}+c{\bf Z}}$ are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields., Comment: 6 pages,0 figures

In order to probe the transitions of a nuclear spin $s$ from one of its substate quantum numbers $m$ to another substate $m'$, the experimenter applies a magnetic field ${\bm B}_0$ in some particular direction, such along $\hat{\bm z}$, and then applies an weaker field ${\bm B}_1(t)$ that is oscillatory in time with the angular frequency $\omega$, and is normally perpendicular to ${\bm B}_0$, such as ${\bm B}_1(t)=B_1\hat{\bm x}\cos(\omega t)$. In the rotating wave approximation, ${\bm B}_1(t)=B_1[\hat{\bm x}\cos(\omega t)+\hat{\bm y}\sin(\omega t)]$. Although this problem is solved for spin $\frac{1}{2}$ in every quantum mechanics textbook, for the general spin $s$ case, its general solution has been published only for the overall probability of a transition between the states, but the time dependence of the probability of finding the nucleus in each of the substates has not previously been published. Here we present an elementary method to solve this problem exactly, and present figures for the time dependencies of the various substates states for a variety of initial substate probabilities for a variety of $s$ values. We found a new result: unlike the $s=\frac{1}{2}$ case, for which if the initial probability of finding the particle in one of the substates was 1, and the time dependence of the probabilities of each of the substates oscillates between 0 and 1, for higher spin values, the time dependencies of the probabilities finding the particle in each of its substates, which periodic, is considerably more complicated., Comment: 9 pages 8 figures