Time-delayed coincidence technique for subnatural-width spectroscopy as an interference phenomenon

A single photon, emitted in a transition between two states, has a frequency distribution of intensity that is given by a Lorentzian if the transition is only naturally broadened and the period of observation $T$ is long compared to the lifetime $T_1$ of the excited state. However, when the observation time $T$ is short or comparable to $T_1$, the frequency spectrum is appreciably broadened. If only the delayed part of the emitted radiation field is detected, then the radiation spectrum does not change. However, if the radiation field is transmitted through a resonant absorber and then detected, the transmission spectrum of the delayed radiation field is narrowed. We show that this narrowing is due to the interference of the spectral components of the incident and coherently scattered fields. Experimental spectra of absorption of Mössbauer radiation, obtained by the coincidence technique, confirm this conclusion.

Figure 1

Power spectrum of the source radiation field, seen by the frequency selective detector in a thought experiment. The opening time $T$ of the shutter is $10T_1$ (solid red line), $5T_1$ (dotted blue line), $2T_1$ (dashed black line), and $T_1$ (dash-dotted blue line).

Figure 2

Dependence of the number of counts on the detuning $\mathrm{\Delta }$, which are collected in different time windows $n_{\text{tw}}\left(\mathrm{\Delta }\right)$. Here the index “tw” denotes time window. The number of counts is normalized to unity for $n_{\infty }(\pm \infty )$. The dashed black line shows the case when the time window is $(0,\infty )$. The solid red line and the dotted blue line show the case when the time window $(0,T)$ is short, i.e., $T=T_1/2=1/2\mathrm{\Gamma }$. Spectra are plotted for ${\mathrm{\Gamma }}_A={\mathrm{\Gamma }}_S=\mathrm{\Gamma }$ and optical thickness $d=4$. The solid red line is plotted in accord with Eq. (21), while the dotted blue line corresponds to Eq. (27).

Figure 3

Time spectrum collected in a short time interval $(0,T)$, where $T=T_1/2$. The effective thickness of the absorber is $d=4$. The analytical approximation, Eq. (34), is shown by a red solid line, and the exact numerical calculation by a blue dotted line.

Figure 4

Time spectrum collected in a time interval $(T,\infty )$, where (a) $T=T_1$, (b) $T=2T_1$, and (c) $T=3T_1$. Effective thickness of the absorber is $d=4$ in (a)–(c) and $d=0.04$ in (c). The analytical approximation, Eq. (37), is shown by a red solid line, and the exact numerical calculation by a blue dotted line.

Figure 5

Simplified scheme of the experimental setup. TAC is a time-to-amplitude converter. PHA is a pulse-height analyzer. TA is a timing amplifier. SA is a spectroscopy amplifier. SCA is a single-channel analyzer. DFG-MD is the Mössbauer driving unit and function generator. HV is a high-voltage supply.

Figure 6

Decay curve of the radiation field from the source along with time windows ${\delta }_n(t_1-t_2)$ where photon counts were collected to plot delayed spectra. The solid red line is a theoretical fitting by an exponent with $T_1=141$ ns. Blue dots are experimental data.

Figure 7

Transemission spectra obtained with different delay times $t_1$ and $t_2$ and different durations of data collection time windows. They are (a) $t_1=2$ ns, $t_2=40$ ns, and ${\delta }_1=38$ ns; (b) $t_1=40$ ns, $t_2=120$ ns, and ${\delta }_2=80$ ns; (c) $t_1=120$ ns, $t_2=200$ ns, and ${\delta }_3=80$ ns; (d) $t_1=200$ ns, $t_2=280$ ns, and ${\delta }_4=80$ ns; and (e) $t_1=280$ ns, $t_2=480$ ns, and ${\delta }_5=200$ ns. Conventional spectrum is shown in (f).

Figure 8

Transemission spectra of GO containing Fe nanoparticles, which are obtained by conventional Mössbauer spectroscopy (a) and by collecting $\gamma$-photon counts in a time window $(t_1,t_2)$, where $t_1=280$ ns and $t_2=550$ ns (b).

Figure 9

(a) Dependence of the number of counts on the detuning $\mathrm{\Delta }$, which are collected in infinite time windows. The number of counts is normalized to unity for $\mathrm{\Delta }=\pm \infty$. The solid red line shows the exact result, Eq. (20), and the dotted blue line demonstrates the approximation Eq. (A1). Spectra $n_{0T}\left(\mathrm{\Delta }\right)$ for $T=2T_1$ (b) and $T=T_1/2$ (c), which are calculated with the help of exact equations (21), (28), and (29) (solid red line) and approximation Eq. (A4) (dotted blue line). The optical thickness of the absorber is $d=1$.

Figure 10

Time-delayed spectra. Time $T$ when count collection starts is $0.3T_1$ (a), $T_1$ (b), and $2T_1$ (c). The solid red line corresponds to the exact result. The dotted blue line is plotted with the help of approximation (A14). Optical thickness is $d=1$.