Quantum process tomography on vibrational states of atoms in an optical lattice

Quantum process tomography is used to fully characterize the evolution of the quantum vibrational state of atoms. Rubidium atoms are trapped in a shallow optical lattice supporting only two vibrational states, which we characterize by reconstructing the $2\times 2$ density matrix. Repeating this reconstruction for a complete set of inputs allows us to completely characterize both the system’s intrinsic decoherence and resonant coupling.

Figure 1

(a) A fluorescence image of the state populations in a lattice obtained by continuous adiabatic decrease of the lattice potential over a period of $45\mathrm{ms}$. An adiabatic stepwise decrease of the potential leads to a clearer separation of the states as shown in (b).

Figure 2

Matrix of measured projections. Input density matrices (reading left to right: ground state, mixed state, superposition with real coherence, and superposition with imaginary coherence) are shown along the bottom while the postselected states are listed on the side. Note the decreased contrast in the ${\theta }_x$ and ${\theta }_y$ projections for the superposition states. All populations are unchanged to within experimental error.

Figure 3

Bloch sphere representation of process tomography. (a) The initial Bloch sphere representing the space of pure input states. (b) After one period of free evolution the sphere contracts horizontally due to a loss of coherence. (c) After sine drive showing rotation about the $y$ axis. (d) After cosine drive, showing rotation about the $x$ axis.