---
title: "Spectral Diffusion and Frequency Memory Interactive Lab | Sandbox Physics"
description: "How long does a molecule remember its vibration frequency? A molecule remembers its frequency until its surroundings change enough to shift the vibration."
canonical_url: "https://sandboxphysics.com/labs/spectral-diffusion-inference/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/spectral-diffusion-inference/"
---

FR16-C · Molecules forget their frequency

# Spectral Diffusion and Frequency Memory

**How long does a molecule remember its vibration frequency?** A molecule remembers its frequency until its surroundings change enough to shift the vibration. A two-dimensional peak loses its diagonal tilt as that memory fades. Measurements at several waiting times reveal the decay and let a fitted model predict later spectra.

Watch a molecular environment change, then reconstruct a waiting-time series from saved complex signals. Fit the early measurements and predict the later ones with two competing correlation models.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Synthetic signals describe an isolated positive band with Gaussian frequency fluctuations. Frequencies are frozen during each coherence period. A calibrated Gaussian spectral acceptance and known homogeneous decay modify the signals; additive quadrature noise mimics detection. The selected band excludes excited-state absorption. Molecular motion is illustrative; no structural trajectory is reconstructed. Fits assume known linewidth, bandwidth and phase calibration. Exchange, non-Gaussian fluctuations, overlapping bands and pulse overlap require richer models.

Synthetic complex signals → spectra → prediction

## Does the fitted memory survive a later measurement?

Absorptive band from the saved complex record

Frequency correlation and two competing predictions

The selected band is reconstructed by Fourier transformation. Filled points train both fits; open points test them.

Fit results and reconstruction details

Reconstructed center-line slope $0.7$

$\rho(T)=0.18+0.82e^{-T/(1.4\,\mathrm{ps})}$ $\rho(T)=0.8e^{-T/(1.4\,\mathrm{ps})}+0.2e^{-T/(10\,\mathrm{ps})}$

Early / held-out RMS, decay plus offset $0.003\,/\,0.005$

Early / held-out RMS, two decay times $0.003\,/\,0.005$

Signals contain forty-eight samples per coherence axis, spaced by 0.08 picoseconds. The rotating-frame carrier is 1650 inverse centimeters. A Gaussian apodization of four inverse centimeters smooths the finite-window Fourier reconstruction. Every waiting-time signal and both quadratures are included in the download.

[Fenn and Fayer (2011), Eq. 12 and the limits of overlapping bands](https://web.stanford.edu/group/fayer/articles/407.pdf)

[Study the full course on multidimensional spectroscopy](https://sandboxphysics.com/stories/multidimensional-spectroscopy/)

Physics tutorial

## Watching a frequency lose its memory

**Background** A molecular vibration feels its surroundings. Two molecules can absorb at different frequencies, and the same molecule can drift to a new frequency after its solvent rearranges.

**Why it matters** A broad line alone hides this motion. A sequence of two-dimensional spectra asks whether a molecule that started on one side of the line is still there later.

Start with the essentials

- **Focus question**: How long does a molecule remember its vibration frequency?
- **One-sentence intuition**: Frequency memory links the initial and final frequencies, tilting a two-dimensional peak along its diagonal.

### Core mathematical model

#### A simple correlation model

$$
\rho(T)=\frac{C(T)}{C(0)}=b+(1-b)e^{-T/\tau_c}
$$

Waiting time and correlation time are in picoseconds. The static fraction stays correlated throughout this measurement. A second model replaces that static part with another exponential decay.

#### The Gaussian peak and its center line

$$
p(y\mid x)\propto\exp\!\left[-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right],\qquad y_{\max}=\rho x
$$

Here the two frequency shifts are divided by their common standard deviation. The ideal Gaussian peak has a center-line slope equal to its correlation. Finite observation windows, homogeneous decay and spectral acceptance change the reconstructed slope; the fit uses calibrated time signals.

### Common difficulties

#### A round peak does not identify a molecular mechanism

**Typical misconception** The fitted decay time uniquely measures one solvent rearrangement.

**Better mental model** The model describes frequency correlation. Multiple microscopic processes, overlapping bands and exchange can change similar spectra. Additional polarization, temperature or independent structural measurements are needed to identify the mechanism.

### Run the experiment

1. 01

   #### Follow one peak through time

   Choose A lasting memory. Drag the sample horizontally, or move Waiting-time frame from the first to the last frame.

   **What to observe:** The peak loses its tilt as the frequency correlation decreases. The colored molecules illustrate correlated frequency pairs, and the heatmap comes from the saved complex signals.

2. 02

   #### Ask the later measurements

   Choose Two hidden clocks and compare the solid fitted curves at the three open held-out points.

   **What to observe:** A curve that matches the early points can miss later points. Prediction makes the consequences of the assumed correlation model visible.

3. 03

   #### Shorten the experiment

   Choose Stop measuring early. Compare both fits and their fitted decay times, then extend the maximum waiting time under Calibration and noise.

   **What to observe:** A short window makes a slow decay resemble a constant offset. Longer measurements help distinguish the models when the signal precision is sufficient.

Physical conclusion

Use a waiting-time series to test a frequency-memory model, and keep a later measurement outside the fit.

## Related experiments and articles

Multidimensional spectroscopy · Reading path

### Vibrations, echoes and frequency memory

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Reconstruct a 2D spectrum from a complex record**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/signal-to-spectrum/)
2. [Lab **Coupled Vibrations in Two-Dimensional IR** Read signed peaks from an exported record](https://sandboxphysics.com/labs/coupled-vibrations-2d-ir/)
3. [Article **How a molecule forgets its earlier frequency**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/spectral-diffusion/)
4. [Article **Read a centre-line paper and test frequency memory**](https://sandboxphysics.com/stories/multidimensional-spectroscopy/reading-and-inference/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)