Fractional squeezing: spectra and dynamics from generalized squeezing Hamiltonian with fractional orders

TL;DR

This work extends generalized squeezing to fractional orders $n$ by constructing a truncated, real-$n$ Hamiltonian in a basis of multiples of $n$ and diagonalizing to study near-zero eigenvalues. It uncovers a spectral transition around $n=2$, with the asymptotic near-zero spectrum obeying $E_{min, inf} \approx \sqrt{\Gamma(n+1)}$ at large $n$, and reveals distinct scaling regimes for the renormalized photon-number $\langle m\rangle$ that saturate for $n\gtrsim 5$. The results show the spectrum is continuous for $n<2$ and discrete for $n>2$, and provide a quantitative framework $E_{min} = E_{min, inf} + \delta E_{min} N^{-\alpha}$ and $\langle m\rangle \approx A N^{-\alpha} + B$ to describe finite-size effects, with a hierarchical toy-model interpretation that explains the large-$n$ behavior and parity sensitivity. The findings contribute to understanding high-order nonlinearities in quantum optics and lay groundwork for studying related multi-photon models.

Abstract

We generalize the generalized-squeezing problem to include fractional values of the squeezing order $n$. This approach allows us to determine the locations of critical points at which qualitative changes in behaviour occur and accurately predict the behaviour at these critical points, which are challenging for conventional computational methods. Based on our numerical calculations, we identify with a high degree of confidence the point at which the spectrum turns from continuous to discrete and the point at which oscillations turn from having asymptotically infinite amplitudes to finite amplitudes. Furthermore, we numerically investigate the behaviour in the large $n$ regime and provide an intuitive explanation that coincides with the numerical results.

Figures (5)

• Figure 1: (a) Smallest positive eigenvalue of the Hamiltonian $\hat{H}(n)$, which we denote as $E_{\rm min}$, as a function of the squeezing order $n$. The different lines (from top to bottom, i.e. red, green, blue, ...) correspond to to truncation sizes $N=250, 500, 1000, ..., 250 \times 2^7$. The inset zooms in on the range $2 \leq n \leq 3$. The progression of these lines shows that for small values of $n$ (visually, $n\lesssim 2$), the eigenvalue has not converged yet and appears to have the asymptotic values zero, while for large values of $n$ ($n\gtrsim 3$) the eigenvalue has clearly converged to its asymptotic value. As a further visual depiction of the convergence behaviour, Panel (b) shows $E_{\rm min}$ as a function of the Hamiltonian truncation size $N$ for $n=2, 2.1, 2.2,..., 3$ (bottom to top, i.e. red, green, blue, ...).
• Figure 2: (a) The asymptotic value of $E_{\rm min}$, obtained by extrapolating the data in Fig. \ref{['Fig:EigenvalueVsSqueezingOrderAndTruncationSize']} to $N\to\infty$ and denoted by $E_{\rm min,\infty}$, as a function of the squeezing order $n$. Starting from $n=0$, $E_{\rm min,\infty}$ remains essentially at zero up to around $n=2$, after which it rises rapidly. The dependence of $E_{\rm min,\infty}$ on $n$ is examined further in the inset, which uses a logarithmic scale for the $y$ axis. The green line is given by $\sqrt{\Gamma(n+1)}$, i.e. the smallest nonzero matrix element in the Hamiltonian. We will explain in Sec. \ref{['Sec:Discussion']} why this function gives the asymptotic value of $E_{\rm min,\infty}$ in the large-$n$ limit. (b) The exponent $\alpha$ obtained from fitting the $(N,E_{\rm min})$ data to the function in Eq. (\ref{['Eq:EminFittingFunction']}). Away from $n=2$, $\alpha$ clearly follows linear functions of $n$, one for $n<2$ and one for $n>2$. The linear dependence is represented by the dotted lines. As we approach $n=2$, we see deviations from the linear functions. This deviation is likely caused by the inevitable numerical errors in our finite numerical simulations. Both linear functions go through the point $(n=2,\alpha=0)$ when extrapolated.
• Figure 3: (a) Expectation value of the renormalized photon number operator $\left\langle \hat{m} \right\rangle$ for the eigenstate that corresponds to the eigenvalue $E_{\rm min}$ as a function of squeezing order $n$. Panel (b) is a similar plot for the operator $n \hat{m}$, which we express as the usual photon number operator $\hat{a}^{\dagger} \hat{a}$. Both panels show that the plotted quantities seem to increase indefinitely as functions of $N$ when $n<4$. When $n>5$, there is no discernible dependence on $N$ for our simulation data, in which $N$ covers the range $[250,250\times 2^7]$. The functional dependence is analyzed in more detail in Figs. \ref{['Fig:EigenstateSizeVsTruncationSize']} and \ref{['Fig:EigenstateSizeFittingParametersVsSqueezingOrder']}.
• Figure 4: $\left\langle \hat{m} \right\rangle$ as a function of the truncation size $N$ for a few values of $n$. Panel (a) shows the data in a log-log plot, while Panel (b) shows the data in a log-linear plot. Both panels show that $\left\langle \hat{m} \right\rangle$ increases indefinitely as a function of $N$ up to $n=4$, while it converges to a finite value when $n=5,6$. The fact that each data set up to $n=3$ follows a straight line in the log-log plot indicates that the data follows a power law. The case $n=4$ was analyzed in Ref. Ashhab2025, where it was found that the scaling is logarithmic. The logarithmic scaling cannot be seen clearly in this figure. The inset in Panel (b) shows that $\left\langle \hat{m} \right\rangle$ converges to finite values for $n=5,6$. This point will be shown more clearly in Fig. \ref{['Fig:EigenstateSizeFittingParametersVsSqueezingOrder']}.
• Figure 5: Fitting parameters obtained by fitting the $(N,\left\langle \hat{m} \right\rangle)$ data to the function $A\times N^{-\alpha}+B$. As can be seen in Panel (b), the exponent $\alpha$ remains essentially constant from $n=0$ to $n=2$. Then it increases linearly. The line interruption and change in color at $n=4$ signal that the point $n=4$ represents a demarcation point. When $n<4$, $\alpha$ is negative, which means that $\left\langle \hat{m} \right\rangle$ increases indefinitely and has the asymptotic value infinity. When $n>4$, $\alpha$ is positive, which means that $\left\langle \hat{m} \right\rangle$ approaches a finite asymptotic value. For this reason, it makes sense to focus on the parameter $A$ (i.e. the scaling prefactor) for $n<4$ and to focus on the parameter $B$ (i.e. the asymptotic value) for $n>4$, as shown in Panel (a). The parameter $B$ approaches 0.5 in the large-$n$ limit, as will be discussed in Sec. \ref{['Sec:Discussion']}.