 Two-state vibrational frequency memory

Two-state vibrational frequency memory models a probe that switches between two frequency environments while retaining a time-dependent relation to its earlier state. It distinguishes state occupancy, a vibrational frequency-state residence time and decay of the vibrational frequency-frequency correlation function.

Let a continuous-time Markov chain switch from state A to B at rate $a>0$ and from B to A at rate $b>0$. Its stationary probabilities areFor spectroscopic wavenumbers ${\nu }_A,{\nu }_B$, the stationary mean is $\overline{\nu }=p_A{\nu }_A+p_B{\nu }_B$ and the variance isThe centered state values form a decaying mode of the two-state transition generator, with rate $a+b$. Consequently,The vibrational frequency-state residence times are instead $1/a$ in A and $1/b$ in B. Correlation decay includes paths that leave and return, so it is not identical to survival without a switch.

An original example uses ${\nu }_A=1000$, ${\nu }_B=1012\mathrm{c}{\mathrm{m}}^{-1}$, $a=0.1$ and $b=0.3\mathrm{p}{\mathrm{s}}^{-1}$. Then $p_A=0.75$, $p_B=0.25$, $\overline{\nu }=1003\mathrm{c}{\mathrm{m}}^{-1}$, $C(0)=27\mathrm{c}{\mathrm{m}}^{-2}$ and ${\tau }_c=2.5\mathrm{ps}$. The residence times are $10$ and approximately $3.33\mathrm{ps}$.

Starting in A, the probability of being in A later isAt $2.5\mathrm{ps}$ it is about $0.842$, while the probability of never leaving A is only $e^{-aT}\simeq 0.779$. Their difference includes return trajectories. At long times, the stationary probability of matching the initial state approaches $p_A^2+p_B^2=0.625$, while normalized frequency covariance tends to zero. A nonzero same-state baseline therefore need not mean persistent covariance.

Primary two-dimensional infrared spectroscopy theory connects waiting-time-dependent spectral geometry with frequency memory under response approximations.^[ref] The rates and wavenumbers above are illustrative. A real assignment also needs spectral resolution, pulse characterization and a population-relaxation model; agreement with one exponential decay does not uniquely establish two microscopic states.