= Two-state vibrational frequency memory
\b[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 are
$$
p_A=\frac{b}{a+b},\qquad p_B=\frac{a}{a+b}.
$$
For spectroscopic wavenumbers $\nu_A,\nu_B$, the stationary mean is $\overline\nu=p_A\nu_A+p_B\nu_B$ and the <variance> is
$$
C(0)=p_Ap_B(\nu_B-\nu_A)^2.
$$
The centered state values form a decaying mode of the two-state transition generator, with rate $a+b$. Consequently,
$$
C(T)=p_Ap_B(\nu_B-\nu_A)^2e^{-(a+b)T},\qquad
\tau_c=\frac1{a+b}.
$$
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 $\nu_A=1000$, $\nu_B=1012\,\mathrm{cm^{-1}}$, $a=0.1$ and $b=0.3\,\mathrm{ps^{-1}}$. Then $p_A=0.75$, $p_B=0.25$, $\overline\nu=1003\,\mathrm{cm^{-1}}$, $C(0)=27\,\mathrm{cm^{-2}}$ 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 is
$$
P_{AA}(T)=p_A+p_Be^{-(a+b)T}.
$$
At $2.5\,\mathrm{ps}$ it is about $0.842$, while the probability of never leaving A is only $e^{-aT}\simeq0.779$. 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.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction and Section II.A, frequency labeling and correlation-function description]{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.
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