We discuss a quantum-metrology protocol designed to estimate a physical parameter in a Bose-Einstein condensate of N atoms, and we show that the measurement uncertainty can decrease faster than 1/N. The 1/N scaling is usually thought to be the best possible in any measurement scheme. From the perspective of quantum information theory, we outline the main idea that leads to a measurement uncertainty that scales better than 1/N. We examine in detail some potential problems and challenges that arise in implementing such a measurement protocol using a Bose-Einstein condensate. We discuss how some of these issues can be dealt with by using lower-dimensional condensates trapped in nonharmonic potentials.Comment: 32 pages, 1 figure, updated reference

Chains of quantum dots coupled to superconductors are promising for the realization of the Kitaev model of a topological superconductor. While individual superconducting quantum dots have been explored, control of longer chains requires understanding of interdot coupling. Here, double quantum dots are defined by gate voltages in indium antimonide nanowires. High transparency superconducting niobium titanium nitride contacts are made to each of the dots in order to induce superconductivity, as well as probe electron transport. Andreev bound states induced on each of dots hybridize to define Andreev molecular states. The evolution of these states is studied as a function of charge parity on the dots, and in magnetic field. The experiments are found in agreement with a numerical model.

The interface between the two complex oxides LaAlO 3 and SrTiO 3 has remarkable properties that can be locally reconfigured between conducting and insulating states using a conductive atomic force microscope. Prior investigations of "sketched" quantum dot devices revealed a phase in which electrons form pairs, implying a strongly attractive electron-electron interaction. Here, we show that these devices with strong electron-electron interactions can exhibit a gate-tunable transition from a pair-tunneling regime to a singleelectron (Andreev bound state) tunneling regime where the interactions become repulsive. The electronelectron interaction sign change is associated with a Lifshitz transition where the d xz and d yz bands start to become occupied. This electronically tunable electron-electron interaction, combined with the nanoscale reconfigurability of this system, provides an interesting starting point towards solid-state quantum simulation.

We analyze a proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving sensitivity scaling better than 1/N in a nonlinear Ramsey interferometer that uses a two-mode Bose-Einstein condensate (BEC) of N atoms. We present numerical simulations that confirm the analytical predictions for the effect of the spreading of the BEC ground-state wave function on the ideal 1/N 3/2 scaling. Numerical integration of the coupled, time-dependent, two-mode Gross-Pitaevskii equations allows us to study the several simplifying assumptions made in the initial analytic study of the proposal and to explore when they can be justified. In particular, we find that the two modes share the same spatial wave function for a length of time that is sufficient to run the metrology scheme.

We investigate the emergence of three-dimensional behavior in a reduced-dimension Bose-Einstein condensate trapped by a highly anisotropic potential. We handle the problem analytically by performing a perturbative Schmidt decomposition of the condensate wave function between the tightly confined (transverse) direction(s) and the loosely confined (longitudinal) direction(s). The perturbation theory is valid when the nonlinear scattering energy is small compared to the transverse energy scales. Our approach provides a straightforward way, first, to derive corrections to the transverse and longitudinal wave functions of the reduced-dimension approximation and, second, to calculate the amount of entanglement that arises between the transverse and longitudinal spatial directions. Numerical integration of the three-dimensional Gross-Pitaevskii equation for different cigar-shaped potentials and experimentally accessible parameters reveals good agreement with our analytical model even for relatively high nonlinearities. In particular, we show that even for such stronger nonlinearities the entanglement remains remarkably small, which allows the condensate to be well described by a product wave function that corresponds to a single Schmidt term.

Abstract. We show how a generalized quantum metrology protocol can be implemented in a twomode Bose-Einstein condensate of n atoms, achieving a sensitivity that scales better than 1/n and approaches 1 /n^'^ for appropriate design of the condensate.Keywords: quantum metrology, nonUnear interferometry, Bose-Einstein condensate PACS: 03.65.Ta, 03.75.Nt, 03.75.Mn In a separate paper elsewhere in this volume [1], we showed that two-body couplings between the n qubits that make up the quantum probe in a parameter-estimation scheme can lead to measurement sensitivities that scale as I/M^/^ even when the initial state of the probe is unentangled. A Bose-Einstein condensate (BEC) is a physical system in which effective two-body couplings exist between all the atoms in the condensate. We examine in this contribution how a BEC of n atoms can be turned into a quantum probe to measure a parameter with a sensitivity scaling that approaches l/rP^^.The many-body Hamiltonian, in second-quantized notation, for a dilute Bose gas in which the inter-particle spacing is much larger than the scattering length, a, is [2,3,4,5] H = I dr(-^V^*(r) -2^ \i/"{r)a^, where y^"{r) is the single-particle ground-state wave function and a* is operator that creates atoms with this wave function. So far we have assumed that all the atoms in the BEC are in the same atomic state, but to use the atoms as components of a quantum probe, we want them to be two-level systems or qubits. We thus consider two-mode BECs in which the atoms can occupy one of two internal states, labeled 11) and |2). These two states are typically hyperfine levels of the atoms. In practice, the atoms are cooled to form the BEC while they are all in the same internal state, and then an external field is used to drive transitions between the two levels to achieve the desired coherent superposition of the two levels. We assume that

We study the mean-field dynamics and the reduced-dimension character of two-mode Bose-Einstein condensates (BECs) in highly anisotropic traps. By means of perturbative techniques, we show that the tightly confined (transverse) degrees of freedom can be decoupled from the dynamical equations at the expense of introducing additional effective three-body, attractive, intra-and inter-mode interactions into the dynamics of the loosely confined (longitudinal) degrees of freedom. These effective interactions are mediated by changes in the transverse wave function. The perturbation theory is valid as long as the nonlinear scattering energy is small compared to the transverse energy scales. This approach leads to reduced-dimension mean-field equations that optimally describe the evolution of a two-mode condensate in general quasione-dimensional (1D) and quasi-two-dimensional geometries. We use this model to investigate the relative phase and density dynamics of a two-mode, cigarshaped 87 Rb BEC. We study the relative-phase dynamics in the context of a nonlinear Ramsey interferometry scheme, which has recently been proposed as a novel platform for high-precision interferometry. Numerical integration of the coupled, time-dependent, three-dimensional, two-mode Gross-Pitaevskii equations for various atom numbers shows that this model gives a considerably 3

Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the (mean-field) product-state Ansatz , in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of N = l × n bosons is derived by symmetrizing the n-fold product of an l-particle quantum state. Quantum correlations of the l-particle state "spread out" to any subset of the N bosons by symmetrization. The particle-correlated states can be simulated efficiently for large N , because their parameter spaces, which depend on l, do not grow with n. Here we formulate and develop in great detail the pure-state case for l = 2, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time O(N ). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-N limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-N approximate form, and comparing with the exact ground state. For N = 1 000, the relative errors of the ground-state energy for both cases are within 10 −5 over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.