--- title: "How two vibrations meet in a 2D spectrum" description: "Build a weakly anharmonic two-mode Hamiltonian, calculate coupled frequencies, and combine bleaching, emission and excited-state absorption into signed peaks." canonical_url: "https://sandboxphysics.com/stories/multidimensional-spectroscopy/coupled-vibrations/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/multidimensional-spectroscopy/coupled-vibrations/" last_modified: "2026-10-28" --- [Multidimensional spectroscopy](https://sandboxphysics.com/stories/multidimensional-spectroscopy/) L3 · Chapter 5 # How two vibrations meet in a 2D spectrum Sandbox Physics · 2026-10-28 When two vibrations couple, an excitation no longer belongs entirely to one bond. A two-dimensional spectrum connects excitation on one transition with detection on another. Signed peak pairs also reveal that a molecule starting in an excited state encounters different energy gaps from one starting in its ground state. ## Before you begin Read chapters three and four. An eigenstate is a combination that evolves at a definite frequency after coupling is included. Keep only the few levels reached in weak excitation. [Review the preceding chapter](https://sandboxphysics.com/stories/multidimensional-spectroscopy/photon-echoes/) - Build a coupled Hamiltonian from local vibrations. - Explain the three contributions to signed peaks. - Test zero coupling and the harmonic limit. ## Coupling mixes local excitations Specify local fundamentals $\widetilde\nu_1,\widetilde\nu_2$, downward anharmonic shifts $\Delta_1,\Delta_2$ and coupling $J$, all in inverse centimetres. The number operator $n_j$ counts excitation quanta; creation and annihilation operators add or remove one. Divide energy by $hc$ so that eigenvalues are wavenumbers. The coupling transfers one quantum between modes. We neglect off-resonant terms that create or remove two quanta together. $$ \frac{H}{hc}=\sum_{j=1}^{2}\left[\widetilde\nu_jn_j-\frac{\Delta_j}{2}n_j(n_j-1)\right]+J(a_1^\dagger a_2+a_2^\dagger a_1) $$ A low-excitation, weak-anharmonicity model with rotating-wave coupling. Ground, one-quantum and two-quantum manifolds are all needed for excited-state absorption. Reading: Tokmakoff ## Worked example for the one-quantum levels The one-quantum block has two basis states: excitation on the first or second mode. Choose local frequencies $1680,\ 1720\,\mathrm{cm}^{-1}$ and $J=15\,\mathrm{cm}^{-1}$. Setting the two-by-two determinant to zero gives $1675,\ 1725\,\mathrm{cm}^{-1}$. The separation grows from 40 to 50 inverse centimetres. Each new state mixes the local excitations. The anharmonic terms vanish in this block because no mode holds two quanta. They matter in the two-quantum block, where they shift overtone and combination states and hence the negative features. $$ \frac{H_1}{hc}=\begin{pmatrix}1680&15\\15&1720\end{pmatrix} $$ Every entry is in inverse centimetres. $$ (1680-\lambda)(1720-\lambda)-15^2=0\quad\Longrightarrow\quad(\lambda-1700)^2=625 $$ The two roots give the coupled frequencies. $$ \widetilde\nu_\pm=\frac{\widetilde\nu_1+\widetilde\nu_2}{2}\pm\sqrt{\left(\frac{\widetilde\nu_1-\widetilde\nu_2}{2}\right)^2+J^2} $$ These are eigenfrequencies of the two-state block. The numbers are a teaching example, not a fit to a named molecule. ## Why the map has opposite signs Pulses change the ground- and excited-state populations. Removing ground-state population reduces absorption: ground-state bleach. Stimulated emission also reduces net absorption. Excited-state absorption adds a different process, promoting an excited molecule into the two-quantum manifold. Our sign convention shows bleach and stimulated emission as positive, and excited-state absorption as negative. Some papers reverse the overall sign, so inspect the colour scale first. Anharmonicity separates their detection frequencies into signed pairs. Taking an absolute value discards that information. Reading: Golonzka et al. · Tokmakoff ## Two limits test the entire spectrum Set coupling to zero and assume completely independent modes with an additive dipole. Exciting one mode cannot change the response of the other. Summed pathways must cancel the true coherent cross response. Overlapping line tails can still colour pixels away from the diagonal; those pixels alone do not establish a cross peak. Next set every anharmonic shift to zero. A harmonic system with a dipole linear in displacement and consistent damping is linear. Bleach and stimulated-emission weights are cancelled by excited-state absorption because $\mu_{12}=\sqrt2\mu_{01}$ and the transition frequencies coincide. A two-level truncation would fail this test. $$ 2\mu_{01}^{4}-\mu_{01}^{2}\mu_{12}^{2}=0\quad\text{when}\quad\mu_{12}=\sqrt2\mu_{01} $$ Harmonic cancellation requires coincident frequencies, a linear dipole and consistent pathway damping. Electrical anharmonicity or level-dependent damping can break it. Reading: Andrei Tokmakoff ## Dipole directions set polarization weights Each interaction projects a transition dipole onto a field polarization. Four such projections multiply in a third-order signal. Changing the angle between local dipoles changes effective coupled-state dipoles and the relative strengths of pathways. An oriented sample and an isotropic solution require different averages. The experiment states its chosen orientation and polarization model. Inferring a molecular angle from intensity ratios also needs transition moments, spectral coverage and all relevant pathways. Golonzka and colleagues demonstrated this structural sensitivity in 2001; our few-level model exposes its mechanism. Reading: Golonzka et al. · Andrei Tokmakoff ## Try the prediction ### [Test cross features in two limits](https://sandboxphysics.com/labs/coupled-vibrations-2d-ir/) 1. Save the default spectrum, then reduce coupling to zero. Distinguish a cross response from overlapping tails. 2. Restore coupling and reduce anharmonicity to zero. Explain why positive and negative contributions approach and cancel. 3. Restore the default and change dipole angle. Read the orientation and polarization assumption before interpreting peak heights. ## Exercises and explanations 1. Why do resolved one-quantum frequencies not determine all negative-peak locations? Read the explanation Negative peaks involve transitions from one-quantum to two-quantum states. Anharmonicity and combination-state mixing shift those final energies, which are not fixed by the one-quantum levels alone. 2. What is wrong with normalizing a near-zero harmonic signal into a bright map? Read the explanation That would magnify roundoff into an apparently finite physical response. Display the actual near-zero amplitude and explain that the selected linear-dipole model has no third-order harmonic response. [Previous: How dispersed phases form a photon echo](https://sandboxphysics.com/stories/multidimensional-spectroscopy/photon-echoes/) [Next: Reconstruct a 2D spectrum from a complex record](https://sandboxphysics.com/stories/multidimensional-spectroscopy/signal-to-spectrum/) ## 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. [Tokmakoff, Two-dimensional spectroscopy chapter (June 2011)](https://doi.org/10.6082/uchicago.3314) In the same deposited notes, the section whose printed page numbers restart at 1: pp. 4–5, Eqs. (1)–(3), connect response, Fourier transformation and heterodyne detection; pp. 10–11 discuss coupled-vibration peak pairs. No source figure is redistributed. 3. [Golonzka et al., Vibrational Anharmonicities Revealed by Coherent Two-Dimensional Infrared Spectroscopy (2001)](https://doi.org/10.1103/PhysRevLett.86.2154) Physical Review Letters 86, 2154–2157, published 5 March 2001. Publisher abstract checked for paired features and dipole-angle sensitivity. The full paper was not used for a figure or numerical-data reproduction; the explicit model follows the author lecture notes.