Correlation Function and the Classical Limit

The correlation function $C(t)$ is the overlap of a state with its own time-evolved self, so it tells you how quickly a quantum state moves away from where it started. In this post I compute it for a state built as a superposition of energy eigenstates with a Gaussian distribution of energies. What comes out at the end is a very concrete version of the energy-time uncertainty relation.

The setup#

Suppose you have prepared a state $|\alpha\rangle$ as a superposition of $N$ energy eigenstates $|w_k\rangle$ with a mean energy $\omega_0$ and a width $\sigma_\omega$ :

c_k \propto e^{-\frac{(\omega_k - \omega_0)^2}{2\sigma_\omega^2}}

|\alpha\rangle = \sum_k c_k |w_k\rangle

We want the normalized state and the correlation amplitude $C(t) = \langle \alpha | U(t) | \alpha \rangle$ .

Normalization#

To normalize the state, we require $\langle \alpha | \alpha \rangle = 1$ :

\langle \alpha | \alpha \rangle = \sum_{k,l} c_k^* c_l \langle w_k | w_l \rangle = \sum_k |c_k|^2 = 1

Given the proportionality $c_k = A e^{-\frac{(\omega_k - \omega_0)^2}{2\sigma_\omega^2}}$ , we find the normalization constant $A$ :

A = \frac{1}{\sqrt{\sum_k e^{-\frac{(\omega_k - \omega_0)^2}{\sigma_\omega^2}}}}

The Correlation Function#

The time evolution operator is $U(t) = e^{-iHt/\hbar}$ . Acting on the state $|\alpha\rangle$ :

U(t) |\alpha\rangle = \sum_k c_k e^{-i\omega_k t} |w_k\rangle

The correlation function $C(t)$ is defined as the overlap between the initial state and the state at time $t$ :

C(t) = \langle \alpha | U(t) | \alpha \rangle = \sum_k |c_k|^2 e^{-i\omega_k t}

Using the normalization and the Gaussian coefficients:

C(t) = \frac{\sum_k e^{-\frac{(\omega_k - \omega_0)^2}{\sigma_\omega^2}} e^{-i\omega_k t}}{\sum_k e^{-\frac{(\omega_k - \omega_0)^2}{\sigma_\omega^2}}}

Classical Limit#

In the limit of a large number of states ( $N \to \infty$ ) and small spacing, we can approximate the sum as an integral:

C(t) \approx \frac{\int e^{-\frac{(\omega - \omega_0)^2}{\sigma_\omega^2}} e^{-i\omega t} d\omega}{\int e^{-\frac{(\omega - \omega_0)^2}{\sigma_\omega^2}} d\omega}

The integral in the numerator is the Fourier transform of a Gaussian:

\int_{-\infty}^{\infty} e^{-a\omega^2 + b\omega} d\omega = \sqrt{\frac{\pi}{a}} e^{b^2/4a}

After completing the square and performing the integration, we find:

C(t) \approx e^{-i\omega_0 t} e^{-\sigma_\omega^2 t^2 / 4}

Interpretation#

The absolute value of the correlation function, $|C(t)| = e^{-\sigma_\omega^2 t^2 / 4}$ , shows a Gaussian decay. The “correlation time” $\tau$ can be defined as the time it takes for $|C(t)|$ to drop to $1/e$ of its initial value:

\tau = \frac{2}{\sigma_\omega}

So $\tau \sigma_\omega = 2$ , which is the energy-time uncertainty relation in about the most concrete form you can get. The broader the spread of energies $\sigma_\omega$ (i.e. the less well-defined the energy, which is the more “classical” direction), the faster the state decorrelates.