Augusto Smerzi

Entanglement, Nonlinear Dynamics, and the Heisenberg Limit

We show that quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in an 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 useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the nonlinear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.

Dynamical superfluid-insulator transition in a chain of weakly coupled bose-Einstein condensates.

We predict a dynamical classical superfluid-insulator transition in a Bose-Einstein condensate trapped in an optical and a magnetic potential. In the tight-binding limit, this system realizes an array of weakly coupled condensates driven by an external harmonic field. For small displacements of the parabolic trap about the equilibrium position, the condensates coherently oscillate in the array. For large displacements, the condensates remain localized on the side of the harmonic trap with a randomization of the relative phases. The superfluid-insulator transition is due to a discrete modulational instability, occurring when the condensate center of mass velocity is larger than a critical value.

Twin Matter Waves for Interferometry Beyond the Classical Limit

An entangled state of up to 10,000 atoms is used to enhance the resolution of an atomic interferometer. Interferometers with atomic ensembles are an integral part of modern precision metrology. However, these interferometers are fundamentally restricted by the shot noise limit, which can only be overcome by creating quantum entanglement among the atoms. We used spin dynamics in Bose-Einstein condensates to create large ensembles of up to 104 pair-correlated atoms with an interferometric sensitivity −1.61−1.1+0.98decibels beyond the shot noise limit. Our proof-of-principle results point the way toward a new generation of atom interferometers.

Fisher information and multiparticle entanglement

The Fisher information F gives a limit to the ultimate precision achievable in a phase estimation protocol. It has been shown recently that the Fisher information for a linear two-mode interferometer cannot exceed the number of particles if the input state is separable. As a direct consequence, with such input states the shot-noise limit is the ultimate limit of precision. In this work, we go a step further by deducing bounds on F for several multiparticle entanglement classes. These bounds imply that genuine multiparticle entanglement is needed for reaching the highest sensitivities in quantum interferometry. We further compute similar bounds on the average Fisher information F for collective spin operators, where the average is performed over all possible spin directions. We show that these criteria detect different sets of states and illustrate their strengths by considering several examples, also using experimental data. In particular, the criterion based on F is able to detect certain bound entangled states.

Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light.

We show that the phase sensitivity Deltatheta of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit Deltatheta approximately 1/N(T), where N(T) is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, Deltatheta approximately 1/square root[|alpha|(2)e(2r)+sinh(2)r], can be saturated for arbitrary values of the squeezing parameter r and the amplitude of the coherent mode alpha by using a Bayesian phase inference protocol.

Fisher information and entanglement of non-Gaussian spin states

Subtle entanglement in an atomic cloud In the quantum world, atoms can be correlated with each other—“entangled”—which reduces the uncertainty in the knowledge of some of their properties. Physicists then use this reduced uncertainty to perform precision measurements. Strobel et al. made an unusual type of entangled state consisting of hundreds of ultracold Rb atoms. These methods may in the future be able to generate states that will be more useful in precision measurement. Science, this issue p. 424 An unconventional entangled state is created out of a mesoscopic number of ultracold rubidium atoms. Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here, we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement, which we confirm by Bayesian phase estimation with sub–shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems.

Nonlinear tight-binding approximation for Bose-Einstein condensates in a lattice

The dynamics of Bose-Einstein condensates trapped in a deep optical lattice is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity and the intersite hopping rates are retrieved from a nonlinear tight-binding approximation taking into account the effective dimensionality of each condensate. We derive analytically the Bloch and the Bogoliubov excitation spectra and the velocity of sound waves emitted by a traveling condensate. Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of localized wave packets. We calculate the ground-state atomic distribution in the presence of a harmonic confining potential, the frequencies of small amplitude dipole, and quadrupole oscillations. We finally quantize the DNL, recovering an extended Bose-Hubbard model.

Insulating Behavior of a Trapped Ideal Fermi Gas

We investigate theoretically and experimentally the center-of-mass motion of an ideal Fermi gas in a combined periodic and harmonic potential. We find a crossover from a conducting to an insulating regime as the Fermi energy moves from the first Bloch band into the band gap of the lattice. The conducting regime is characterized by an oscillation of the cloud about the potential minimum, while in the insulating case the center of mass remains on one side of the potential.

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.

Entanglement and Sensitivity in Precision Measurements with States of a Fluctuating Number of Particles

The concepts of separability, entanglement, spin squeezing, and the Heisenberg limit are central in the theory of quantum-enhanced metrology. In the current literature, these are well established only in the case of linear interferometers operating with input quantum states of a known fixed number of particles. This manuscript generalizes these concepts and extends the quantum phase estimation theory by taking into account classical and quantum fluctuations of the particle number. Our analysis concerns most of the current experiments on precision measurements where the number of particles is known only on average.