Frequency Spectra of Isolated Laser Pulse Envelopes

TL;DR

The paper investigates how finite-duration, isolated laser pulse envelopes shape the high-frequency content and the statistics of time-averaged quadratic observables. It builds a rate-equation model to generate isolated pulses and analyzes the envelope Fourier transform, revealing a decay of the form $|\hat{f}(\omega)| \sim e^{- b (\tau \omega)^{\alpha}}$ with small $\alpha$ (typically 0.1–0.2), far below the pump's $\alpha_{\rm in}=1/2$. This slow decay implies non-Gaussian, heavy-tailed distributions for large fluctuations of quadratic operators, potentially enhancing zero-point fluctuation effects in condensed-matter systems probed by such pulses. The results motivate experimental scenarios to observe enhanced density fluctuations and related radiation-pressure effects using tailored, isolated pulses.

Abstract

This paper will deal with isolated laser pulses, those which last for a finite time interval and whose envelope function is strictly zero outside of this interval. We numerically calculate the Fourier transform of this function and study its asymptotic behavior at high frequencies. This work is motivated by recent results on the probability distributions of quadratic operators in second quantized systems. An example is the density of a material which is subject to zero point fluctuations in the phonon vacuum state. These distributions can decrease very slowly, leading to a relatively high probability for large fluctuations. If the operator is measured by a laser pulse, the rate of decrease of the distribution mirrors the rate of decrease of the pulse envelope Fourier transform. We describe a model for the creation of isolated pulses in which this Fourier transform falls as an exponential of a fractional power of frequency and find examples where this fraction is in the range 0.1 to 0.2. The probability distribution for large fluctuation has the same functional form and implies a significant probability for fluctuations very large compared to the variance.

Figures (7)

• Figure 1: The function $f_1(t)$ is illustrated. Note that although $f_1(t) \not= 0$ for $0 < t < 1$, its actual value is small outside of $0.2 \alt t \alt 0.8$.
• Figure 2: The photon number for the case $N_0 = 10^7$, $C=10^{-4}$, $r = 2$ is plotted with no re-absorption, $\eta = 0$, and for strong re-absorption, $\eta = 1$
• Figure 3: The number of excited atoms is plotted for the same choices of parameters as in Fig. \ref{['fig:photon-no-1']}.
• Figure 4: The behavior of the photon number, $x(N_0, t)$, as a function of time is illustrated near the switch-on time for different values of $N_0$. Part (a) shows the cases $N_0 = 10^7, 10^8,$ and $10^9$. Because the photon number scales with $N_0$, the cases $N_0 = 10^8$ and $N_0 = 10^9$ have been rescaled by factors of $\frac{1}{10}$ and $\frac{1}{100}$, respectively, so that all three cases could be shown on the same plot. Part (b) gives a similarly rescaled plot for the cases $N_0 = 10^{10}, 10^{11},$ and $10^{12}$. Note that as $N_0$ increases, the initial switch-on time moves closer to $t= 0.5$, the midpoint of the pump pulse. Otherwise, the rescaled graphs all have the same form. In all the cases illustrated here, we have set $\eta = 1$.
• Figure 5: The Fourier transforms of the envelope functions plotted in Fig. \ref{['fig:photon-no-1']}, where $N_0 = 10^7$, $C=10^{-4}$, and $r = 2$, are plotted for the cases $\eta =1$ in parts (a) and (b) for different frequency ranges, and for the case $\eta =0$ in part (c). All of these are log-log plots of $\log(F)$ as a function of $\omega$. We find a straight line in (a) and (b), but a more complicated oscillatory behavior in (c). In the latter case, the value of $\alpha$ is estimated from the line connecting the maxima of the oscillations. Our estimates for the slopes in each case are: $(a)\; \alpha \approx 0.12$, $(b)\; \alpha \approx 0.18$, $(c)\; \alpha \approx 0.097$.