# Energy conservation in Kapitza-Dirac diffraction?

In Kapitza-Dirac diffraction, a standing wave of light (wavevector of single wave $k$) is pulsed on for a very short period of time ($\sim \mu s$) onto a bunch of cold atoms. This results in the atoms receiving momentum kicks of $2n\hbar k$.

How energy conservation work? Where does the energy come from?

## 1 Answer

Kapitza-Dirac diffraction can be described in two different pictures:

Photon picture

Describe the pulsed lattice as two strong, counter-propagating lasers with photon momentum $$\hbar k_{\text{lat}}$$ and $$-\hbar k_{\text{lat}}$$, and energy $$\hbar\omega$$. The lattice is off-resonant so there are no one-photon transitions, but two-photon (= Bragg) transitions are possible. As pointed out by you, the atoms acquire $$2n\hbar k_{\text{lat}}$$ and corresponding kinetic energies $$(2n\hbar k_{\text{lat}})^2/(2m)$$. But since all photons have the same energy $$\hbar\omega$$, the final energy states are energetically off-resonant. Nevertheless, the photons will drive an off-resonant Rabi oscillation to higher momentum states. You need a quite high intensity to see the oscillations.

Lattice picture

Starting from a stationary atom at momentum $$p=0$$ we quench on an optical lattice for a certain duration. The initial state of the atom (for example a gaussian) is therefore projected onto the lattice eigenstates (Bloch states), leading to a population of a number of Bloch states, including states of the second, third, and higher Bloch bands. The subsequent unitary evolution again gives a Rabi oscillation at quasimomentum $$q=0$$ between the different Bloch states until the ToF image is taken and Bloch states are mapped again onto real momentum. These now have contributions from $$2n\hbar k_{\text{lat}}$$ momenta.

In real space, the initial state (e.g. a gaussian or even a plane wave with momentum zero) is rather smooth, corresponding to a low curvature of the wave function and low kinetic energy $$-\frac{(\hbar\nabla)^2}{2m}$$. Contrary, the Bloch states have many 'wiggles' in real-space which reflect the periodicity of the lattice. The lattice basically imprints its wiggles on the initial smooth wave function, thus increasing its energy. One could say, the energy 'comes from the wiggles of the lattice'.