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If phase-randomizing collisions form a Poisson process of mean interval ${\tau }_c$, field correlation survives only when no such collision occurs during the delay. Thus its magnitude decays as $e^{-|\tau |/{\tau }_c}$. The Wiener-Khinchin theorem gives the normalized Lorentzian line shapeIts full width at half maximum is $2/{\tau }_c$ in angular frequency or $1/(\pi {\tau }_c)$ in ordinary frequency. A population lifetime is a different time constant: lifetime-limited amplitude decay at half the population rate introduces the familiar factor of two. The definition of the supplied lifetime must therefore accompany any numerical linewidth.

Cristiano Fantini coauthored the 2015 graphene isotope study using Raman spectroscopy; the paper gives his UFMG physics affiliation.^[ref]

A useful measurement issue is how a spectral width relates to time-dependent motion. For an original model, suppose a vibrational coherence decays as $e^{-t/T_2}$. Its corresponding homogeneous Lorentzian line shape has frequency full width at half maximum $\Delta \nu =1/(\pi T_2)$, or wavenumber widthWith stipulated $\Gamma =10\mathrm{c}{\mathrm{m}}^{-1}$ and $c=2.998\times 10^{10}\mathrm{cm}{\mathrm{s}}^{-1}$, the coherence time is about $1.06\mathrm{ps}$. The units of $c$ must match the inverse-centimetre linewidth.

A measured linewidth does not automatically give an energy-relaxation lifetime. In a simple relaxation model, $1/T_2=1/(2T_1)+1/T_{\varphi }$, where $T_1$ describes population relaxation and $T_{\varphi }$ pure dephasing. Only when the latter contribution is negligible does $T_1=T_2/2$. Instrumental broadening or spatially varying frequency can additionally widen a recorded Raman spectrum.

This example explains why spectral fitting, instrument characterization and the physical relaxation model matter together. Its linewidth and times are hypothetical; they are not measured values or individually assigned results from Fantini's paper.

A curve of growth describes the equivalent width $E(m)=\int (1-e^{-m{\kappa }_{\nu }})d\nu$ as a function of absorber mass path or column density. Its weak-line limit is $E\simeq mS$. Saturation makes growth sublinear: a pure Gaussian core gives $E\sim 2{\Delta }_D\sqrt{\log {\tau }_0}$, while saturated Lorentzian line shape wings give $E\sim 2\sqrt{mS{\Delta }_L}$. The spectral coordinate and coefficient convention must be consistent.

Two delayed copies of the same laser field probe its first-order optical coherence. With equal detected intensities and matched polarizations, the interferometric visibility is $V=|g^{(1)}(\tau )|$. A Lorentzian line shape of frequency full width at half maximum $\Delta \nu$ gives $V=e^{-\pi \Delta \nu |\tau |}$. For a fibre length difference $\Delta l$, the delay is $\tau =n_g\Delta l/c$, where $n_g$ is the group index. Hence $\Delta \nu =-\log V/(\pi |\tau |)$. Replacing $n_g$ by the effective phase index requires negligible dispersion. Polarization mismatch, unequal intensities and uncontrolled phase drift can reduce visibility independently of linewidth.

Construct an original illustrative spectrum using ordinary frequency detuning $x={\nu }_L-{\nu }_0$. Write the longitudinal Doppler shift in frequency units as $u=v/{\lambda }_0$, where ${\lambda }_0=c/{\nu }_0$. Choose a normal distribution with standard deviation ${\sigma }_D$ and a weak-probe Lorentzian line shape with half width at half maximum ${\gamma }_{\nu }$:Thus $B(0)=1$. Choose ${\sigma }_D=120\mathrm{MHz}$ and natural linewidth $\Gamma /(2\pi )=6\mathrm{MHz}$, so ${\gamma }_{\nu }=3\mathrm{MHz}$. The Gaussian-only full width at half maximum is $2\sqrt{2\ln 2}{\sigma }_D\simeq 282.58\mathrm{MHz}$; the averaged profile also includes the finite Lorentzian line shape width.

Add a deliberately phenomenological Lamb dip:Here $A_s$ is the absorption coefficient normalized to the pump-off line-centre value. The factor $0.24$ is chosen, not inferred from populations or a fit. The dip's assigned full width at half maximum $w_s$ borrows the single-atom power broadening scale purely to illustrate contrast versus width. It is not the preceding overlap integral or a fit to historical spectra. There are no crossover features in this isolated-transition toy spectrum.

For $s=1$, the subtracted dip has depth $0.12$ and width $8.49\mathrm{MHz}$; for $s=9$, depth $0.216$ and width $18.97\mathrm{MHz}$. Its half-depth points are $x=\pm w_s/2$. Measure that width relative to the curved background $B(x)$, not a horizontal line through the total absorption minimum.

For a chosen cell optical depth ${\tau }_s(x)=0.4A_s(x)$, the line-centre pump-off transmission is $e^{-0.4}\simeq 0.6703$. At $s=1$ it is $e^{-0.352}\simeq 0.7033$. The transmitted probe therefore increases by about $4.92$ percent relative to pump off. Absorption contrast, transmission contrast and spectral width are three different quantities.

A hard wall at the origin and an attractive delta potential at distance $a$ form a leaky quantum cavity. For unit incident amplitude from outside, the interior standing wave has coefficient $A=-i\lambda x/[1-e^{2ix}+i\lambda x]$, where $x=ka$ and $\lambda ={\hslash }^2/(mqa)>0$. Its modulus squared is the title expression. A resonance pole $p_n=x_n-i{\gamma }_n$ produces an enhancement with a locally Lorentzian line shape, proportional to $[(x-x_n{)}^2+{\gamma }_n^2{]}^{-1}$. At fixed low $n$ and small $\lambda$, $x_n\simeq n\pi (1+\lambda /2+{\lambda }^2/4)$ and ${\gamma }_n\simeq (\lambda n\pi /2{)}^2$.

The coefficient can exceed one because it measures a standing-wave amplitude, while the net interior probability current is zero. Conservation of probability current gives unit reflected flux: no current passes through the hard wall. Near zero, $|A{|}^2\to {\lambda }^2/(2-\lambda {)}^2$ when $\lambda \ne 2$; the threshold case $\lambda =2$ requires a separate limit.

Broadening shared by otherwise identical emitters, often described by a common optical coherence time $T_2$. Exponential field-coherence decay produces a Lorentzian line shape with full angular-frequency width $2/T_2$. Population decay, dephasing and collisions can all contribute, with their time constants distinguished.

The Lamb-Mössbauer factor $f$ is the fraction in the coherent elastic line. In the harmonic-lattice approximation it is a Debye–Waller factor. Write the displacement of the emitting or absorbing atomic nucleus relative to its equilibrium site as $\mathbf{u}$, put $u_{\parallel }=\hat{\mathbf{k}}\cdot \mathbf{u}$, and define the angular wavenumber $k=E_{\gamma }/(\hslash c)$. Choose the equilibrium site so that $⟨u_{\parallel }⟩=0$. Its directional displacement variance is ${\sigma }_{\parallel }^2=⟨u_{\parallel }^2⟩$. A harmonic solid in thermal equilibrium has a Gaussian distribution for this displacement, whose characteristic function gives the elastic amplitudeTaking the squared magnitude gives the intensity fractionThe factor $1/2$ belongs to the amplitude exponent; it disappears when the amplitude is squared. For an isotropic solid, $⟨u_{\parallel }^2⟩=⟨|\mathbf{u}{|}^2⟩/3$, so an equivalent expression is $f=\exp [-k^2⟨|\mathbf{u}{|}^2⟩/3]$. Directional and three-dimensional mean squared displacements must not be substituted for one another without this factor.

An Einstein solid approximation replaces the vibrational spectrum by one positive angular frequency $\omega$. For the directional quantum harmonic oscillator, with the same effective mass $M$ as in $E_R$,Here $k_B$ is the Boltzmann constant and $T$ is temperature. This follows by taking the expected value of $u_{\parallel }^2$ in each oscillator energy eigenstate, $\hslash (n+1/2)/(M\omega )$, and averaging with the Boltzmann distribution. Substitution givesThe zero-temperature limit retains zero-point energy and a nonzero displacement variance. Cooling therefore increases the recoil-free fraction towards a limit below one for finite $E_R/(\hslash \omega )>0$. At high temperature, the exponent tends to $-2E_R{k}_{B}T/(\hslash \omega {)}^2$. This is a single-frequency approximation to a harmonic lattice's coherent elastic weight, not a complete description of all zero-net-energy processes of an isolated oscillator. Real vibrational spectra, directional binding and anharmonic motion can change its predictions.

Extend the stipulated teaching system by choosing $\hslash \omega =0.020\mathrm{eV}$, so $\eta =E_R/(\hslash \omega )=0.1$. Then $f(0)=0.90484$, while $f=0.80542$ at $k_{B}T=\hslash \omega$. If source and absorber are independent, the fraction of events eligible for recoil-free emission followed by recoil-free absorption contains the product $f_s{f}_a$; this gives $0.81873$ for two zero-temperature copies. It is not the total probability of absorption, which also depends on the number of resonant atomic nuclei, geometry and spectral line overlap.

Consider an original illustrative resonance model, with all numerical settings chosen here. On one isolated branch approximate the transition frequency by $\nu (B)={\nu }_0+aB$, with known slope $a=20MHz/mT$ and drive frequency ${\nu }_d=1300\mathrm{MHz}$. Choose ${\nu }_0=1100\mathrm{MHz}$, hence resonance field $B_r=10\mathrm{mT}$. This local linear model omits hyperfine structure and nonlinear level mixing; it is not the historical hydrogen Zeeman splitting calculation.

Use a Lorentzian line shape for a weak-drive current dip,Choose baseline $C=1$, depth $A=0.25$ in arbitrary detector signal units, and full width at half maximum $w_{\nu }=100\mathrm{MHz}$, so $w_B=5\mathrm{mT}$. The half-depth points are $B_r\pm w_B/2$; this checks the factor of four. If an exponential decay with lifetime $\tau$ alone set this width, natural linewidth would give $w_{\nu }=1/(2\pi \tau )$ and $\tau \simeq 1.59\mathrm{ns}$. The chosen width is an explanatory input, not a measurement of a historical lifetime.

Generate 65 equally spaced fields from $2$ to $18\mathrm{mT}$ and add independent Gaussian noise with known standard deviation ${\sigma }_I=0.004$ signal units. A simultaneous nonlinear regression fit of $C,A,B_r,w_B$ gives $B_r\simeq 10.0007\mathrm{mT}$, $w_B\simeq 4.9305\mathrm{mT}$ and ${\nu }_0\simeq 1099.986\mathrm{MHz}$. With residual derivatives $J_{ik}=\partial I(B_i)/\partial {\theta }_k$ and $\theta =(C,A,B_r,w_B)$, the local covariance matrix estimate isIt gives centre standard deviation $0.01427\mathrm{mT}$, or $0.285\mathrm{MHz}$ in the inferred intercept. This uses the stipulated noise variance and a local linear approximation, rather than treating the width as the centre error. Repeated noisy samples can locate a broad line's centre much more closely than its width.

Include a common magnetic calibration offset with standard deviation $u_b=0.020\mathrm{mT}$, a fractional gain error $u_g=0.001$, and independent drive-frequency error $u_d=0.100\mathrm{MHz}$. To first order the inferred variance isHere $u$ denotes a standard uncertainty, not a frequency. The four contributions in $\mathrm{MHz}$ are $0.285$, $0.400$, $0.200$, and $0.100$; adding their squares gives $u({\nu }_0)\simeq 0.540\mathrm{MHz}$. The calibration offset shifts all sampled fields together, so increasing the sample count cannot average it away. Uncertainty in $a$, unresolved components, asymmetric line distortion, and errors in the assumed field dependence would require additional terms or a richer model. The historical inference requires the appropriate Zeeman splitting branches, rather than simply drawing a straight line through one resonance.

In the absence of additional dephasing, a transition whose excited-state population has lifetime $T_1$ has a field-coherence decay time $2T_1$. Its Lorentzian line shape has angular-frequency half-width $1/(2T_1)$ and full width $1/T_1$. Population decay time, amplitude decay time and linewidth conventions must therefore be distinguished. Extra dephasing shortens the optical coherence time without necessarily changing the population lifetime.

A normalized retarded amplitude $G^R(\mathbf{k},t)=-i\Theta (t)e^{-i{\Omega }_{\mathbf{k}}t-{\Gamma }_{\mathbf{k}}t}$, with constant ${\Gamma }_{\mathbf{k}}>0$, has Fourier transform $G^R(\mathbf{k},\omega )=(\omega -{\Omega }_{\mathbf{k}}+i{\Gamma }_{\mathbf{k}}{)}^{-1}$. Consequently $A=-\mathrm{Im}{G}^R/\pi$ is the displayed positive Lorentzian line shape, of unit frequency integral. Its maximum is at ${\Omega }_{\mathbf{k}}$ and its half width at half maximum is ${\Gamma }_{\mathbf{k}}$. The energy half-width is $\hslash {\Gamma }_{\mathbf{k}}$. Amplitude decays on time $1/{\Gamma }_{\mathbf{k}}$ whereas probability decays on time $1/(2{\Gamma }_{\mathbf{k}})$. A general interacting electron has quasiparticle residue $Z$ and incoherent spectral weight in addition to this normalized local pole model; frequency-dependent width or self-energy can invalidate a global Lorentzian.

Let $v$ be the source velocity towards the absorber, positive for approach along the photon direction. A first-order Lorentz transformation of the photon energy gives $E^{\prime }=E+vp$; hence, for $|v|\ll c$,Moving the source shifts its emission line through the stationary absorption line. A minimum in transmitted photon counts marks their strongest overlap. The absorber is the narrow energy discriminator; the downstream particle detector only needs to count the transmitted gamma rays.

For a thin absorber with unsplit Lorentzian line shapes, the overlap is a convolution of the source and absorber profiles. The combined full width at half maximum is $W={\Gamma }_s+{\Gamma }_a$. A simple expected transmission model isHere $I_0$ is the off-resonance intensity, $v_0$ encodes a relative shift of the two transition energies, and $A$ includes the resonant absorber strength and $f_s{f}_a$. The convolution result can be checked by multiplying the Fourier transforms of two Cauchy distributions: their exponential decay constants add. The width of the absorption dip in velocity is $cW/E_{\gamma }$, rather than the width of one individual spectral line converted to velocity.

For the teaching system with ${\Gamma }_s={\Gamma }_a={\Gamma }_E$ and $v_0=0$, the dip's full width at half maximum is $2.9979\mathrm{mm}{\mathrm{s}}^{-1}$. Stipulate $A=0.25$ at zero temperature and $10,000$ expected off-resonance counts per equal-duration bin: the central bin then has $7,500$ expected counts, and the half-depth bins have $8,750$. These are model expectations, without sampled counting noise. Holding all other parameters fixed while heating both lattices to $k_{B}T=2\hslash \omega$ multiplies $A$ by $[f(T)/f(0){]}^2$. This isolates how a falling recoil-free fraction weakens the dip.

The thin-absorber model is deliberately limited. A thick absorber can saturate and broaden the dip; unresolved hyperfine structure can create multiple components. Differences in chemical surroundings can shift the centers, and uncontrolled motion can smear them. These effects must be modeled before interpreting an observed width as a natural linewidth. Their sensitivity also makes the Mössbauer effect useful: a small displacement of a narrow dip can reveal a small change in the local nuclear environment.

A simple isolated particle resonance model starts with a survival amplitude proportional toIts survival probability is $|a(t){|}^2=e^{-\Gamma t/\hslash }$, giving mean rest-frame lifetime $\tau =\hslash /\Gamma$. The Fourier transform of this one-sided amplitude is proportional to $(\Gamma /2-i(E-E_0){)}^{-1}$; its squared modulus gives a Lorentzian line shape. Write its normalized density asAt $E-E_0=\pm \Gamma /2$ the density is half its peak, so its intrinsic full width at half maximum is $\Gamma$. This is a local isolated-line approximation, extending the offset coordinate over the real line; physical production thresholds, interference and energy-dependent decay widths can require a richer quantum scattering model.

Now stipulate a centered normal distribution for the measurement response, with standard deviation $\sigma$. A measured coordinate $x$ has densityThis convolution describes the smearing of an underlying line, with constant acceptance and no background. For a pair spectrometer the response summarizes invariant-mass resolution; for an idealized collider scan it summarizes incoming center-of-mass energy spread. These are different experimental sources of uncertainty even when the same mathematical response is useful.

Choose an arbitrary scale $E_{\ast }$, intrinsic decay width $\Gamma =0.20E_{\ast }$ and response standard deviations $0.40E_{\ast }$ and $1.20E_{\ast }$. Numerical integration gives apparent full width at half maximum values of approximately $1.05E_{\ast }$ and $2.93E_{\ast }$, respectively. The state and its rest-frame lifetime are identical in both cases. The normal distribution's width alone is $2\sqrt{2\ln 2}\sigma$; adding that width to $\Gamma$ in quadrature is not an exact rule for this convolution.

For a purely illustrative $E_{\ast }=1\mathrm{MeV}$, taking $\hslash \simeq 6.5821\times 10^{-22}\mathrm{MeV}\mathrm{s}$ gives $\tau \simeq 3.2911\times 10^{-21}\mathrm{s}$. These are invented line parameters, not J/psi meson measurements. Substituting the larger apparent width into $\hslash /\Gamma$ would incorrectly shorten the inferred lifetime by nearly a factor of fifteen. An unresolved width can support an upper bound on intrinsic decay width, and consequently a lower bound on lifetime, only after the instrumental response and model assumptions have been assessed.

The discovery peak and its later interpretation answer different questions. The J/psi meson is understood as charmonium, a bound charm quark–charm antiquark system. Richter's 1976 lecture distinguishes the initial discovery from subsequent tests of competing explanations, including the related level structure and later evidence for particles carrying uncancelled charm. A narrow peak alone did not establish all these properties in 1974.^[ref]

In the classical thermal model, a coordinate in a harmonic potential obeys $\gamma \dot{x}+kx=\xi$, where $\gamma$ is the drag coefficient and $⟨\xi (t)\xi (t^{\prime })⟩=2\gamma k_{B}T\delta (t-t^{\prime })$. Its stationary one-sided spectral density per hertz is the displayed Lorentzian line shape. The integral over positive frequencies equals $k_{B}T/k$, consistent with the equipartition theorem. On a log-log plot the spectrum is flat for $f\ll f_c$ and has slope $-2$ for $f\gg f_c$. The crossover is determined by the restoring stiffness and drag; an added detector response multiplies the mechanical spectrum by the squared magnitude of its transfer function.

With $\Omega =\gamma B_x/2$, the Born rule applied to the supplied down-spin probability amplitude givesFor nonzero transverse field, the maxima occur atThe first maximum is reached at $\pi /{\omega }_R$; on resonance this is a Rabi pi pulse. The resonance envelope of a Rabi transition is a Lorentzian line shape, centred on ${\omega }_{\mathrm{res}}={\omega }_z$ under the orientation convention above. It has half its peak value at $|\omega -{\omega }_z|=|\Omega |$, so its full width at half maximum isIf resonance width is instead defined as the half-width, the ratio differs by a factor of two but has the same requested proportionality. The quality factor is large precisely in the regime where the rotating-wave approximation is justified. The envelope is obtained by choosing the best time separately at each drive frequency; a measurement at one fixed pulse duration has a different line profile.

Let the upper-state population lifetime be $T_1=1/\Gamma$, and take the lower energy level to be stable. Spontaneous emission gives survival probability $e^{-\Gamma t}$, hence a field or excited-state probability amplitude proportional to $e^{-\Gamma t/2}{e}^{-i{\omega }_{0}t}$ for $t\ge 0$. Its Fourier transform at angular frequency $\omega$ is proportional toThe squared modulus therefore has the Lorentzian line shapeHere $L$ is normalized over the full mathematical frequency axis. The full width at half maximum in ordinary frequency is $1/(2\pi T_1)$. If both energy levels decay, their population decay rates add in the natural linewidth; pure dephasing adds to the amplitude-decay rate instead.

Impact-regime pressure broadening provides another example. Independent phase-randomizing collisions with rate ${\gamma }_c$ give optical correlation proportional to $e^{-{\gamma }_c|t|}$, whose Fourier transform is a Lorentzian line shape of full width at half maximum $2{\gamma }_c$. This assumes short, uncorrelated impacts: a general static distribution of perturbing fields need not give a Lorentzian line shape.

Use first-order time-dependent perturbation theory, setting $c_1=1$ in the equation for $c_2$ and retaining the generally complex matrix element $d=⟨{\psi }_2|{\hat{D}}_0|{\psi }_1⟩$. With $\delta ={\omega }_0-\omega$ and $\Sigma ={\omega }_0+\omega$, the probability amplitude isFor $t>0$, split the integral at the pulse maximum. For either $x=\delta$ or $x=\Sigma$,Thus the full first-order answer, without discarding the counter-rotating contribution, isAt long times it givesThis full expression is not exactly one squared Lorentzian line shape. To obtain the intended near-resonance spectral line profile, additionally assume $b,|\delta |\ll {\omega }_0$, so the pulse has a narrow positive-frequency lobe and $F(\Sigma ,t)$ is negligible. ThenThe weak-excitation condition near resonance is $|d|/(\hslash b)\ll 1$. The squared Lorentzian line shape occurs because the symmetric exponential pulse has a Lorentzian line shape amplitude Fourier transform, which is then squared to obtain probability.

At half the maximum, $[1+(\delta /b{)}^2{]}^2=2$, soThe full width at half maximum refers to varying the drive frequency through the positive-frequency resonance. For a concrete failure of the unqualified squared-Lorentzian line shape claim, set $b=\omega ={\omega }_0$ while keeping $|d|/(\hslash b)\ll 1$. The full first-order resonant probability is $36/25$ times the one-lobe prediction. Weak radiation alone therefore does not justify dropping the second Fourier transform lobe.

Natural broadening is about $8\times 10^2$ times narrower than the thermal Doppler width and is negligible for this comparison. The supplied Doppler broadening formula is a half width at half maximum. Using the integer molecular mass $m=28m_{\mathrm{u}}$,For an excited-state population lifetime $\tau$, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Lorentzian line shape has natural linewidthEquivalently, compare full widths $2.11\mathrm{GHz}$ and $2.65\mathrm{MHz}$. Using the same linewidth convention in both expressions matters more than the small isotope-mass correction. This estimate assumes that the lower state's lifetime is long and leaves any pressure broadening as a separate contribution.

Let $I_0$ be the intensity from either fibre alone at the observation point, assuming equal output intensities, matching polarizations and negligible drift. The screen samples the relative phase $\Phi (X)$ between the two outputs. For an ideal monochromatic laser, first-order optical coherence is complete andThe interference fringes are approximately equally spaced in the small-angle far field, with a slowly varying diffraction envelope. If the output separation is $a$ and the screen distance is $D$, the interference fringe spacing is approximately ${\lambda }_{0}D/a$. Equal phases at the two input ends do not ensure a central bright fringe: the extra fibre contributes carrier phase $2\pi n_{\mathrm{eff}}(l_2-l_1)/{\lambda }_0$ in addition to the free-space optical path difference.

For a finite linewidth, the time-averaged interference signal becomesHere $g^{(1)}$ may be defined for the slowly varying field envelope, with the carrier phase already included in $\Phi$. A realistic laser therefore gives reduced interferometric visibility; for $V=1/2$, its extrema are $I_0$ and $3I_0$.

For a Lorentzian line shape of frequency full width at half maximum $\Delta \nu$, temporal optical coherence givesThe relevant index is the group index. Using the intended nondispersive approximation $n_g\simeq n_{\mathrm{eff}}=2$,This delayed-fibre linewidth measurement assumes the loss of contrast comes from linewidth. Unequal output intensities, polarization mismatch or mechanical phase fluctuations also lower interferometric visibility. If dispersion matters, the stated phase index alone does not specify the necessary group index.

Interpret the stated ${\tau }_1=1\mathrm{ns}$ as the coherence-decay time, or the mean interval between fully phase-randomizing collisions in the simple impact model of pressure broadening. Then each line has correlation $e^{-|\tau |/{\tau }_1}{e}^{-i{\omega }_j\tau }$. The collisional phase relaxation and Lorentzian linewidth transform givesThe intrinsic spectrum is the sum of two such Lorentzian line shapes, still centred $5.0\mathrm{GHz}$ apart. With equal weights and balanced arms, its normalized visibility isFor the intensity normalization of the first source plot, the corresponding model interferogram isThe constant background remains $4$; only the oscillatory contrast decays. The beating zeros and recurrence spacing remain fixed, but successive visibility revivals shrink, reaching a factor $e^{-0.2}\simeq 0.819$ at $|\tau |=200\mathrm{ps}$ and $e^{-1}$ at $1\mathrm{ns}$. Long-delay fringes disappear toward the background.

The sketch chooses a carrier frequency for legibility; the envelope and linewidth follow from the inferred spacing and specified coherence time. A finite experimental delay window additionally convolves these intrinsic Lorentzians with the instrumental line profile. If ${\tau }_1$ were instead a population lifetime, the lifetime-limited field decay rate would be $1/(2{\tau }_1)$ and the corresponding linewidth would be half the boxed value. The source does not define that time convention explicitly; pressure-collision coherence decay is the interpretation used here.