Why do correlation amplitudes tend to zero as time increases? I'm reading a section about correlation amplitudes in Sakurai. We are considering the correlation amplitude $C(t)$ of state $|\alpha \rangle$ which is in a superposition of energy eigenstates $\{|a'\rangle\}$
$$C(t) = \sum_{a^{\prime}}\left|c_{a^{\prime}}\right|^{2} \exp \left(\frac{-i E_{a^{\prime}} t}{\hbar}\right).$$
I don't understand the following statement by the author:

As we sum over many terms with oscillating time dependence of
different frequencies, a strong cancellation is possible for
moderately large values of $t$. We expect the correlation amplitude
that starts with unity at $t=0$ to decrease in magnitude with time.

My intuition is the following. We see from the correlation amplitude that each contributing term has a complex phase that changes with time but at a different frequency. When enough time has passed, because the phases change with a different frequency, each term will essentially have a random orientation. As a consequence, if there are many energy eigenstates, and therefore many terms in $C(t)$, each contribution coming from them will tend to cancel with another contribution with the opposite orientation. Crucially, this argument seems to only work in the limit $t,\text{number of energy eigenvalues} \to \infty$.
I would really apreciate if someone told me whether my intution is correct and maybe show me a more rigorous version of my arguments.
 A: Yes, your intuition is basically right.
It's useful to look at a few examples.
First, for a finite number of eigenstates in the sum, $C(t)$ will inevitably be periodic. Nevertheless, one still generically expects $C(t)$ to decrease from $1$ to some small value, before being "dragged back" to $1$ by the requirement that $C(t)$ be periodic. As an example, we can consider the situation $E_a = a \hbar \omega$, with the sum going from $a=0$ to $a=N-1$, and $c_a=1/N$ for all $a$. Then we have
\begin{equation}
C(t) = \sum_{a=0}^{N-1} \frac{e^{i a \omega t}}{N}
=  \frac{1}{N} \frac{1 - e^{i N \omega t}}{1-e^{i \omega t}} 
= \frac{e^{i (N-1) \omega t /2}}{N} \frac{\sin\left(\frac{N\omega t}{2}\right)}{\sin\left(\frac{\omega t}{2}\right)}
\end{equation}
First, note that in the limit $t\rightarrow 0$, $C(t)$ goes to $1$, as expected. Additionally, $C(t)$ is periodic with period $2\pi/\omega$, which is the period of the slowest (non-constant) mode. (You might think the right hand side is periodic with period $4\pi/\omega$ based on the sine wave in the denominator, but note that when $t=2\pi/\omega$ the numerator and denominator are both $-1$ for any $N$, and so the final result is $+1$; this agrees with the translational symmetry that is obvious in the mode expansion for $C(t)$).
Nevertheless, there are places where $C(t)$ vanishes. If we just consider the amplitude $|C(t)| \propto \left|\frac{\sin(N \omega t/2)}{\sin(\omega t/2)}\right|$, then the amplitude will be largest when the numerator and denominator are both small and have approximately the same size (near $t \approx 0, 2\pi/\omega, 4\pi/\omega, ...$), and smallest when the denominator is "large" (the most extreme cases being $t\approx \pi/\omega, 3\pi/\omega, ... $. Here is a plot showing the behavior of $|C(t)|$ for $N=\{3,10,100\}$

There are also a few interesting limiting cases to consider.

*

*In the limit $N\rightarrow \infty$, the sum will become a Dirac comb with spacing $2\pi/\omega$. This is a extreme case where $|C(t)|$ is only non-zero when $t=0$, then immediately decoheres to zero, before being "forced back" by periodicity and repeating the cycle.


*If you take the continuum limit so $a$ is a continuous variable and not a discrete index over a finite range of frequencies from $\omega=0$ to $\omega=\Omega$ (this amounts to sending $\omega\rightarrow 0$ but sending $N\rightarrow \infty$ in such a way that we keep the maximum frequency $\Omega=N\omega$ finite), then you will find that $|C(t)|$ is given by a sinc function. In other words, $|C(t)|=1$ for $t=0$, and then falls off to zero over some time scale given by one divided by the bandwidth, $\sim 1/\Omega$.


*If you then take the limit $\Omega\rightarrow \infty$, then $|C(t)|$ will simply be a delta function at $t=0$. It will be non-zero exactly when $t=0$, then decohere immediately and never be zero again.
