--- title: "Why a spectral line broadens" description: "Use phase arrows to distinguish coherence, lifetime and two broadening mechanisms, then derive a Lorentzian line and convert its width into time." canonical_url: "https://sandboxphysics.com/stories/multidimensional-spectroscopy/coherence-and-linewidth/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/multidimensional-spectroscopy/coherence-and-linewidth/" last_modified: "2026-10-28" --- [Multidimensional spectroscopy](https://sandboxphysics.com/stories/multidimensional-spectroscopy/) L1 · Chapter 2 # Why a spectral line broadens Sandbox Physics · 2026-10-28 A broad line can mean that each molecule loses coherence quickly. It can also mean that many molecules have slightly different frequencies. The first mechanism acts within an individual time evolution; the second spreads the phases across an ensemble. An echo separates these contributions by testing which phase spread can refocus. ## Before you begin Read chapter one. Think of a complex number as an arrow in a plane: its length is amplitude and its angle is phase. An integral adds contributions from different times. [Review the preceding chapter](https://sandboxphysics.com/stories/multidimensional-spectroscopy/vibrational-fingerprints/) - Distinguish population lifetime from coherence lifetime. - Explain homogeneous and inhomogeneous broadening. - Convert linewidths between angular frequency and wavenumber. ## Motion can persist after phases spread A short pulse prepares related phases across many molecules. Represent each vibration by a rotating arrow. Their vector sum gives the macroscopic polarization, the collective electric-dipole response that radiates from the sample. Aligned arrows add strongly; dispersed arrows can nearly cancel. A coherence describes an amplitude with a definite phase relation between quantum states. A population describes how much occupation an energy level has. Let $T_1$ denote population relaxation time and $T_2$ coherence decay time. Population loss contributes half its decay rate to the coherence rate. $$ \frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_2^*} $$ This is a simple two-level Markov model. The starred time denotes pure dephasing; no separate orientational relaxation is included. Reading: Andrei Tokmakoff · Fenn and Fayer ## The same broad peak can have different origins If every molecule loses coherence at the same exponential rate, the broadening is homogeneous. A distribution of long-lived resonance frequencies instead causes inhomogeneous broadening: individual arrows keep rotating, but their sum decays. In liquids, those frequencies can change as environments evolve. We will call that spectral diffusion. For a static Gaussian frequency distribution, write the angular-frequency offset as $\delta\omega$ and its standard deviation as $\sigma_\omega$. Averaging the phase arrows gives a Gaussian envelope. An independent homogeneous decay multiplies that envelope. $$ \left\langle e^{-i\delta\omega t}\right\rangle=e^{-\sigma_\omega^2t^2/2},\qquad P(t)\propto e^{-t/T_2}e^{-\sigma_\omega^2t^2/2} $$ Offsets in radians per picosecond require times in picoseconds. The two factors describe different mechanisms. ## Transform a decay into a line Take the causal response $e^{-t/T_2}e^{-i\omega_0t}$ and integrate over positive time. At resonance, successive contributions add in phase. Away from resonance, their phases spread. The real part of the following integral gives an absorptive line. $$ \operatorname{Re}\int_0^\infty e^{-t/T_2}e^{i(\omega-\omega_0)t}\,dt=\frac{1/T_2}{(1/T_2)^2+(\omega-\omega_0)^2} $$ This Lorentzian has angular-frequency half width equal to the inverse coherence time. Converting its full width to wavenumber introduces the speed of light and a full-turn factor. Reading: Andrei Tokmakoff · Fenn and Fayer ## Worked example for one picosecond of coherence Set $T_2=1\,\mathrm{ps}$. The angular-frequency full width is $2\,\mathrm{rad\,ps}^{-1}$, or about $10.6\,\mathrm{cm}^{-1}$. If $T_1=2\,\mathrm{ps}$, population loss contributes $0.25\,\mathrm{ps}^{-1}$ to the coherence decay rate. The remaining $0.75\,\mathrm{ps}^{-1}$ is pure dephasing. Adding a static frequency distribution broadens the line further. The inverse total linewidth can no longer be read directly as $T_2$. An echo tests which part of the phase spread can return. $$ \Delta\widetilde\nu_{\mathrm{FWHM}}=\frac{1}{\pi cT_2}=10.6\,\mathrm{cm}^{-1} $$ This conversion applies to the exponential coherence decay used here. A Gaussian distribution follows a different linewidth formula. ## Try the prediction ### [Spread the phases and try to refocus them](https://sandboxphysics.com/labs/photon-echo-pulse-sequence/) 1. Increase static frequency spread and observe the faster loss of collective polarization. 2. Keep that spread fixed and compare rephasing with nonrephasing. 3. Shorten coherence lifetime and explain why the echo also becomes weaker. ## Exercises and explanations 1. For a purely Lorentzian line, how does doubling coherence time change the wavenumber full width? Read the explanation It halves the width because the coherence time enters the denominator. The conclusion assumes exponential decay and does not directly describe static Gaussian broadening. 2. Can the macroscopic polarization nearly vanish while individual molecules still oscillate? Read the explanation Yes. Polarization is a phase-sensitive vector sum. Opposing contributions can cancel without each molecular vibration losing its energy. [Previous: What one infrared peak can tell us](https://sandboxphysics.com/stories/multidimensional-spectroscopy/vibrational-fingerprints/) [Next: How three pulses build a third-order response](https://sandboxphysics.com/stories/multidimensional-spectroscopy/third-order-response/) ## Sources and calculation scope 1. [Andrei Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (June 2011, v1)](https://knowledge.uchicago.edu/records/3tv7m-5ha50) Author lecture notes, nonlinear-spectroscopy section: density-matrix perturbation theory; printed pp. 23–25, Eqs. (51)–(55), for rephasing and nonrephasing; pp. 29–30 for the vibrational harmonic limit. The course examples are new teaching calculations. 2. [Fenn and Fayer, Extracting 2D IR frequency-frequency correlation functions from two component systems (2011)](https://web.stanford.edu/group/fayer/articles/407.pdf) Journal of Chemical Physics 135, 074502, publisher version, DOI 10.1063/1.3625278. Read Figs. 1–3 and Eqs. (1)–(4), (12), (14)–(15). The bounded numerical benchmark is the single-component Gaussian slope relation in Eq. (12), with original synthetic parameters; it is not a fit to the paper’s water measurements.