State Reconstruction of the Kicked Rotor

We propose two experimentally feasible methods based on atom interferometry to measure the quantum state of the kicked rotor.

Figure 1

Methods of self-interference (left) and holography (right) to reconstruct the wave function of the kicked rotor. The method of self-interference considers a three-level atom with a laser field detuned half between the two hyperfine levels $|1⟩$ and $|2⟩$. The periodic potentials due to the light shifts experienced by the individual levels are out of phase. The method of holography uses a phase modulator to put sidebands on the transition frequency whereas the frequency ${\omega }_2$ itself is filtered out. This symmetric detuning around ${\omega }_2$ leads to a constant potential for the lower state $|2⟩$ but a periodic potential for the upper state $|1⟩$. In both methods atoms in state $|1⟩$ play the role of the kicked rotor whereas the atoms in state $|2⟩$ serve as our reference.

Figure 2

Monte Carlo simulation (left column) of the holographic method for the state reconstruction of the kicked rotor after $N=1,2$, and $5$ kicks. For each value of $N$ we represent the state by the amplitude (top) and the phase (bottom) of the momentum wave function. For comparison we also show the exact functions (right column). For the reconstruction of the momentum wave function we use a grid of 4096 points and $M={10}^6$ simulated measurement events.

Figure 3

Comparison between the methods of self-interference and holography based on the fidelity for the state of the kicked rotor after nine kicks. Shown is the mean fidelity (averaged over 25 realizations) of the reconstructed state versus the accuracy $a$ of the measured distributions. The accuracy is defined by $a=1/\Delta W$ with $\Delta W$ being the relative uncertainty of the measured distributions. The reciprocal of $a$ is the relative uncertainty of the distributions.