The Exercise

A system has the orthonormal basis $|0\rangle, |1\rangle, |2\rangle, \dots$, and its state $\rho$ has the matrix elements $\rho_{nm} = \langle n|\rho|m\rangle$. A phase kick through the angle $\theta$ is the unitary $D(\theta)$ with $D(\theta)|n\rangle = e^{-in\theta}|n\rangle$. The system receives one such kick, with $\theta$ drawn at random from a normal distribution of mean zero and variance $2\lambda$. The exercise is to show that the average over the kick multiplies each matrix element $\rho_{nm}$ by $e^{-\lambda(n-m)^2}$.^✦

The Kick and Its Average

The phase kicked element of a complex matrix $\rho = (\rho_{nm})_{n,m \ge 0}$ at the angle $\theta$ and the position $(n, m)$ is the complex number $$\langle n|D(\theta)\,\rho\,D(\theta)^\dagger|m\rangle = e^{-in\theta}\,\rho_{nm}\,e^{im\theta}.$$(1)^✦

The phase damped element $\langle n|\mathcal{E}(\rho)|m\rangle$ of $\rho$ is the average of the kicked element over an angle with the normal distribution of mean zero and variance $2\lambda$: $$\langle n|\mathcal{E}(\rho)|m\rangle = \frac{1}{\sqrt{4\pi\lambda}} \int_{-\infty}^{\infty} \langle n|D(\theta)\,\rho\,D(\theta)^\dagger|m\rangle\, e^{-\theta^2/4\lambda}\,d\theta .$$(2)^✦

The decay factor $\gamma_\lambda(k)$ at the separation $k$, for real numbers $\lambda$ and $k$, is the real number $$\gamma_\lambda(k) = e^{-\lambda k^2}.$$(3)^✦

The Claim

The average (2) is $\rho_{nm}$ times the Fourier transform of the normal density at $n - m$, and the Fourier transform of a Gaussian is again a Gaussian.

What Is Checked

The claim is stated over the phase damped element (2), the Gaussian average of the phase kicked element (1), and over the decay factor (3).

About these constructions the library proves (4), which evaluates the Gaussian integral (2) in closed form for every $\lambda > 0$.