# Matter-wave interference from free falling cold atoms

and another exam question, this is about current research:

Interference of matter waves has been studied using ultra-cold atoms. The phase of a matter wave for free-falling cold-atoms at time $t$ and height $y$ is given as $$\phi(y,t) = \phi(y_0,t) + Et - \int_{y_0}^y dy' k(y')$$ where $E$ is the total energy of individual coherent atoms and $\hbar k(y)$ is the semi-classical momentum of falling atoms at height $y$. A recent experiment observed coherent matter waves continuously emitted from two vertical micro-traps (you can neglect the size of the traps) with height $-\lambda/2$ and $\lambda/2$. The atoms are emitted at rest from the traps.

A) For a given height $y$ below the two micro-traps, derive the condition of constructive and destructive interference.

Well, from basic wave mechanics I know that the phase difference of the two waves has to be an even (constructive) or odd (destructive) multiple of $\pi$, but apart from that I find the question poorly worded: What does $t$ refer to? Total time in laboratory frame, or time since leaving the trap? And what about the function $\phi(y_0,t)$? If I don't know its form, can I say anything meaningful about the interference pattern at all?

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The formula you indicated above is for ONE wave. For two waves you will have superposition, i.e. ϕ_1(y,t) + ϕ_2(y,t), where in ϕ_1(y,t) = ϕ_1(y_{0,1},t_1) + E_1(t-t_1) + ∫dy'k_1(y') with the integral from y_{0,1}, to y, and ϕ_2(y,t) = ϕ_2(y_{0,2},t_2) + E_1(t-t_2) + ∫dy'k_2(y') . – Sofia Nov 20 '14 at 20:30