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Sandbox Physics
Multidimensional spectroscopy

L2 · Chapter 3

How three pulses build a third-order response

Sandbox Physics ·

Each pulse changes the molecular state once. The first creates a phase relation, the second can turn it into a population, and the third makes the sample radiate again. Keeping the order, phase and direction of those interactions tells us which response reaches the detector.

One table holds population and phase

For a ground and an excited state, a two-by-two density matrix ρ\rho stores both. Its diagonal entries are populations and sum to one. Its off-diagonal entries are coherences, with magnitude and phase. This description also handles a mixed ensemble whose members do not all share one state.

The electric field E(t)E(t) acts through the dipole operator μ\mu with interaction energy Hint=−μE(t)H_{\mathrm{int}}=-\mu E(t). A commutator is [A,B]=AB−BA[A,B]=AB-BA. Changing the order of operator multiplication can change the result.

ρ=(ρggρgeρegρee),ρ˙=−iℏ[H,ρ]\rho=\begin{pmatrix}\rho_{gg}&\rho_{ge}\\\rho_{eg}&\rho_{ee}\end{pmatrix},\qquad \dot\rho=-\frac{i}{\hbar}[H,\rho]
This is unitary evolution. A specified relaxation model must be added when populations or coherences decay.

Reading: Andrei Tokmakoff · Park

Calculate the first interaction

Start entirely in the ground state and choose a real transition dipole dd. Multiply these matrices row by column, then subtract the reversed product. A diagonal population becomes an off-diagonal contribution: the first pulse creates coherence.

ρ0=(1000),μ=(0dd0)\rho_0=\begin{pmatrix}1&0\\0&0\end{pmatrix},\quad\mu=\begin{pmatrix}0&d\\d&0\end{pmatrix}
Basis order is ground state followed by excited state.
[μ,ρ0]=(0−dd0)[\mu,\rho_0]=\begin{pmatrix}0&-d\\d&0\end{pmatrix}
Field factors, time integration and evolution convert this contribution into the first-order density matrix.

What makes the response third order

For a weak field, we expand the state in powers of the field. One interaction supplies first order; three supply third order. The number counts field interactions, not energy levels. A third-order signal can be measured with a small perturbation of the sample.

Starting with equilibrium state ρ0\rho_0, each interaction adds a commutator with the dipole. Free evolution propagates the state between interactions. Absorb that evolution into time-labelled operators to write the response kernel below. The final dipole reads out the polarization; the trace adds the diagonal entries of the resulting matrix.

R(3)=(iℏ)3Tr⁡ ⁣{μ4[μ3,[μ2,[μ1,ρ0]]]}R^{(3)}=\left(\frac{i}{\hbar}\right)^3\operatorname{Tr}\!\left\{\mu_4[\mu_3,[\mu_2,[\mu_1,\rho_0]]]\right\}
Subscripts follow interaction time. The full polarization requires convolution with three pulse envelopes; the impulsive approximation treats those interactions as instantaneous.

Three intervals carry different information

The first interval t1t_1 encodes excitation frequency. The waiting interval TT separates the second and third interactions and probes population or environmental evolution. The final interval t3t_3 encodes detection frequency. All intervals are positive and measured in picoseconds.

The rotating-wave approximation drops rapidly oscillating, off-resonant terms. The remaining pathways can evolve through opposite phase directions in the first and final intervals, or through the same direction. These are the rephasing and nonrephasing contributions. The next chapter calculates their difference.

Pulse 1Pulse 2Pulse 3EmissionCoherenceWaitingDetection
t1t_1TTt3t_3
Three interactions leave two coherence intervals around one waiting interval. The local oscillator reads the emitted field.

Select a pathway by direction and phase

Write the spatial and controlled phase of input jj as kj⋅r+ϕj\mathbf k_j\cdot\mathbf r+\phi_j. A rephasing contribution conjugates the first field and multiplies it by the other two. Its wavevector and phase inherit the corresponding signs. A suitable geometry separates that emission direction. Collinear experiments can also select phase components by phase cycling.

kR=−k1+k2+k3,ϕR=−ϕ1+ϕ2+ϕ3\mathbf k_{\mathrm R}=-\mathbf k_1+\mathbf k_2+\mathbf k_3,\qquad \phi_{\mathrm R}=-\phi_1+\phi_2+\phi_3
The rephasing combination. Actual phase matching also depends on dispersion and finite sample geometry.
kNR=k1−k2+k3,ϕNR=ϕ1−ϕ2+ϕ3\mathbf k_{\mathrm{NR}}=\mathbf k_1-\mathbf k_2+\mathbf k_3,\qquad \phi_{\mathrm{NR}}=\phi_1-\phi_2+\phi_3
The nonrephasing combination exchanges the first two signs. A schematic collector must lie on the corresponding outgoing beam.

Reading: Andrei Tokmakoff · Park

Worked example with one phase rotation

Hold the last two phases fixed and increase the first phase by π/2\pi/2. The rephasing signal rotates by −π/2-\pi/2; the nonrephasing signal rotates by +π/2+\pi/2. Relative to a fixed local oscillator, their quadratures rotate in opposite directions.

Signal intensity alone misses this change: phase can rotate without changing magnitude. Mixing the weak field with a phase-controlled local oscillator exposes it. Chapter six uses that complex field to reconstruct a frequency spectrum.

Try the prediction

Select the echo by phase cycling

  1. Compare rephasing and nonrephasing time-domain signals.
  2. Inspect the saved phase-cycle samples and compare the unweighted average with the selected phase component.
  3. Use the phase combination above to explain the difference between an unweighted average and a selected phase component.

Exercises and explanations

  1. Does third order require three energy levels?

    Read the explanation

    It counts three appearances of the field in a perturbation expansion. A two-level system can have third-order response. Testing the harmonic vibrational limit requires the second excited level so that cancellation is represented correctly.

  2. What happens to a rephasing signal when the second phase increases by π\pi?

    Read the explanation

    Its complex amplitude changes sign because the second phase enters with a positive sign. The heterodyne cross term reverses relative to a fixed reference; the isolated signal intensity stays unchanged.

Sources and calculation scope

  1. Andrei Tokmakoff, Nonlinear and Two-Dimensional Spectroscopy (June 2011, v1)

    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. Park, Kwak and Fayer, Ultrafast 2D-IR vibrational echo spectroscopy (2007)

    Laser Physics Letters 4, 704–718, DOI 10.1002/lapl.200710046. Experimental review: pulse sequence, local oscillator and absorptive phasing in Sections 2–3 and Fig. 3. Its center-line convention uses the inverse slope for the opposite slice direction; the course states its own axes before defining a slope.

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