Quantum Beat Spectroscopy

If two or more closely spaces molecular levels are simultaneously excited by a short laser pulse, the time-resolved total fluorescence intensity emitted from these coherently prepared levels shows a modulated exponential decay. The effect is known as quantum beats is due to interference between the fluorescence amplitudes emitted from these coherently excited states.

If the pulse excitation occurs at time $t=0$, the excited state wavefunction can be written as a linear superposition of the sublevel wave functions:

$$\Psi(0) = c_1\Psi_1(0)+c_2\Psi_2(0).$$ (98)

At the time $t$ the excited state evolves as

$$\Psi(t) = c_1\Psi_1(0)e^{-i\left(E_1/\hbar+\gamma/2\right)t}+c_2\Psi_2(0)e^{-i\left(E_2/\hbar+\gamma/2\right)t}.$$ (99)

Having in mind that the fluorescence intensity $I(t)$ is proportional to the square of the dipole moment matrix element

$$I(t) = C \left\vert\langle \Psi(t)\vert{\bf e\cdot d}\vert\Psi_i\rangle\right\vert^2,$$ (100)

we get

$$I(t) = C e^{-\gamma t} (A+B\cos\omega_{12}t),$$ (101)

Thus, the fluorescence signal is modulated with the frequency, corresponding to the energy separation between the states $1$ and $2$, $\omega_{12} = E_{12}/\hbar$. The physical explanation of the interference effect is that the paths through the state $\vert 1>$ and $\vert 2>$ into the final state $\vert f>$ cannot be distinguished from each other.

The Quantum Beat Spectroscopy is a universal tool for determinations of the excited state structure in gases, liquids, and solid states.

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