Entanglement, Non-linear Dynamics, and the Heisenberg Limit

We show that the quantum Fisher information provides a sufficient condition to recognize multi-particle entanglement in a $N$ qubit state. The same criterion gives a necessary and sufficient condition for sub shot-noise phase sensitivity in the estimation of a collective rotation angle $\theta$. The analysis therefore singles out the class of entangled states which are {\it useful} to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the non-linear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.Comment: Phys. Rev. Lett. 102, 100401 (2009

Localized and extended states in a disordered trap

We study Anderson localization in a disordered potential combined with an inhomogeneous trap. We show that the spectrum displays both localized and extended states, which coexist at intermediate energies. In the region of coexistence, we find that the extended states result from confinement by the trap and are weakly affected by the disorder. Conversely, the localized states correspond to eigenstates of the disordered potential, which are only affected by the trap via an inhomogeneous energy shift. These results are relevant to disordered quantum gases and we propose a realistic scheme to observe the coexistence of localized and extended states in these systems.Comment: Published versio

Mach-Zehnder Interferometry at the Heisenberg Limit with coherent and squeezed-vacuum light

We show that the phase sensitivity $\Delta \theta$ of a Mach-Zehnder interferometer fed by a coherent state in one input port and squeezed-vacuum in the other one is i) independent from the true value of the phase shift and ii) can reach the Heisenberg limit $\Delta \theta \sim 1/N_T$, where $N_T$ is the average number of particles of the input states. We also show that the Cramer-Rao lower bound, $\Delta \theta \propto 1/ \sqrt{|\alpha|^2 e^{2r} + \sinh^2r}$, can be saturated for arbitrary values of the squeezing parameter $r$ and the amplitude of the coherent mode $|\alpha|$ by a Bayesian phase inference protocol.Comment: 4 pages, 4 figure

Rabi Interferometry and Sensitive Measurement of the Casimir-Polder Force with Ultra-Cold Gases

We show that Rabi oscillations of a degenerate fermionic or bosonic gas trapped in a double-well potential can be exploited for the interferometric measurement of external forces at micrometer length scales. The Rabi interferometer is less sensitive, but easier to implement, than the Mach-Zehnder since it does not require dynamical beam-splitting/recombination processes. As an application we propose a measurement of the Casimir-Polder force acting between the atoms and a dielectric surface. We find that even if the interferometer is fed with a coherent state of relatively small number of atoms, and in the presence of realistic experimental noise, the force can be measured with a sensitivity sufficient to discriminate between thermal and zero-temperature regimes of the Casimir-Polder potential. Higher sensitivities can be reached with spin squeezed states

Phase Sensitivity of a Mach-Zehnder Interferometer

The best performance of a Mach-Zehnder interferometer is achieved with the input state |N_T/2 + 1>|N_T/2-1 > + |N_T/2 - 1>|N_T/2+1>, being N_T the total number of atoms/photons. This gives: i) a phase-shift error confidence C_{68%}=2.67/N_T with ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling ~ 1/N_T but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.Comment: 4 figure

Entanglement and Extreme Spin Squeezing for a Fluctuating Number of Indistinguishable Particles

We extend the criteria for $k$-particle entanglement from the spin squeezing parameter presented in [A.S. S{\o}rensen and K. M{\o}lmer, Phys. Rev. Lett. {\bf 86}, 4431 (2001)] to systems with a fluctating number of particles. We also discuss how other spin squeezing inequalities can be generalized to this situation. Further, we give an operational meaning to the bounds for cases where the individual particles cannot be addressed. As a by-product, this allows us to show that in spin squeezing experiments with cold gases the particles are typically distinguishable in practise. Our results justify the application of the S{\o}rensen-M{\o}lmer bounds in recent experiments on spin squeezing in Bose-Einstein condensates

Matter Wave Transport and Anderson Localization in Anisotropic 3D Disorder

We study quantum transport in anisotropic 3D disorder and show that non rotation invariant correlations can induce rich diffusion and localization properties. For instance, structured finite-range correlations can lead to the inversion of the transport anisotropy. Moreover, working beyond the self-consistent theory of localization, we include the disorder-induced shift of the energy states and show that it strongly affects the mobility edge. Implications to recent experiments are discussed.Comment: Text unchanged, Reference added: EPL 99 (2012) 5000

Useful Multiparticle Entanglement and Sub-Shot-Noise Sensitivity in Experimental Phase Estimation

We experimentally demonstrate a general criterion to identify entangled states useful for the estimation of an unknown phase shift with a sensitivity higher than the shot-noise limit. We show how to exploit this entanglement on the examples of a maximum likelihood as well as of a Bayesian phase estimation protocol. Using an entangled four-photon state we achieve a phase sensitivity clearly beyond the shot-noise limit. Our detailed comparison of methods and quantum states for entanglement enhanced metrology reveals the connection between multiparticle entanglement and sub-shot-noise uncertainty, both in a frequentist and in a Bayesian phase estimation setting.Comment: 4 pages, 4 figure