We consider the task of performing quantum state tomography on a $d$-state spin qudit, using only measurements of spin projection onto different quantization axes. By an exact mapping onto the classical problem of signal recovery on the sphere, we pr
ove that full reconstruction of arbitrary qudit states requires a minimal number of measurement axes, $r_d^{mathrm{min}}$, that is bounded by $2d-1le r_d^{mathrm{min}}le d^2$. We conjecture that $r_d^{mathrm{min}}=2d-1$, which we verify numerically for all $dle200$. We then provide algorithms with $O(rd^3)$ serial runtime, parallelizable down to $O(rd^2)$, for (i) computing a priori upper bounds on the expected error with which spin projection measurements along $r$ given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol, for which we find that using more measurement axes can yield substantial benefits that plateau after $rapprox3d$.
We propose to simulate dynamical phases of a BCS superconductor using an ensemble of cold atoms trapped in an optical cavity. Effective Cooper pairs are encoded via internal states of the atoms and attractive interactions are realized via the exchang
e of virtual photons between atoms coupled to a common cavity mode. Control of the interaction strength combined with a tunable dispersion relation of the effective Cooper pairs allows exploration of the full dynamical phase diagram of the BCS model, as a function of system parameters and the prepared initial state. Our proposal paves the way for the study of non-equilibrium features of quantum magnetism and superconductivity by harnessing atom-light interactions in cold atomic gases.
Macroscopic arrays of cold atoms trapped in optical cavities can reach the strong atom-light collective coupling regime thanks to the simultaneous interactions of the cavity mode with the atomic ensemble. In a recent work we reported a protocol that
takes advantage of the strong and collective atom-light interactions in cavity QED systems for precise electric field sensing in the optical domain. We showed that it can provide between $10$-$20$~dB of metrological gain over the standard quantum limit in current cavity QED experiments operating with long-lived alkaline-earth atoms. Here, we give a more in depth discussion of the protocol using both exact analytical calculations and numerical simulations, and describe the precise conditions under which the predicted enhancement holds after thoroughly accounting for both photon loss and spontaneous emission, natural decoherence mechanisms in current experiments. The analysis presented here not only serves to benchmark the protocol and its utility in cavity QED arrays but also sets the conditions required for its applicability in other experimental platforms such as arrays of trapped ions.
