Tomasz Wasak

Optimal measurements in phase estimation: simple examples

We identify optimal measurement strategies for phase estimation in different scenarios in which the interferometer acts on two-mode symmetric states. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed, this set reduces to merely two possible measurements. For two-qubit symmetric Werner state, we find the optimal measurement and show that estimation from the population imbalance is optimal only if the state is pure. We also determine the optimal measurements for a wide class of symmetric N-qubit Werner-like states. Finally, for a pure symmetric state of N qubits, we find under which conditions the estimation from the full N-body correlation and from the population imbalance is optimal.

Cauchy-Schwarz inequality and particle entanglement

The Glauber-Sudarshan

Bogoliubov theory for atom scattering into separate regions

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Four-wave mixing in a parity-time (PT)-symmetric coupler

-representation is used in quantum optics to distinguish between semi-classical and genuinely quantum electromagnetic fields. We employ the analog of the

Stabilization of solitons under competing nonlinearities by external potentials

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Cauchy---Schwarz inequality for general measurements as an entanglement criterion

-representation to show that the violation of the Cauchy-Schwarz inequality for the second-order correlation function is a proof of entanglement between identical massive bosons. The presented derivation is valid both in systems with fixed and fluctuating number of particles. Thanks to the recent advances in techniques of detecting positions of separate particles, the violation of the Cauchy-Schwarz inequality can be used as a simple entanglement criterion in various many-body quantum systems.

Interferometry with independently prepared Bose-Einstein condensates

We review the Bogoliubov theory in the context of recent experiments, where atoms are scattered from a Bose‐Einstein condensate into two well-separated regions. We find the full dynamics of the pair-production process, calculate the first and second order correlation functions and show that the system is ideally number-squeezed. We calculate the Fisher information to show how the entanglement between atoms from the two regions changes in time. We also provide a simple expression for the lower bound of the useful entanglement in the system in terms of the average number of scattered atoms and the number of modes they occupy. We then apply our theory to a recent ‘twin-beam’ experiment (Bucker et al 2011 Nature Phys. 7 608). The only numerical step of our semianalytical description can be easily solved and does not require implementation of any stochastic methods.

Four-wave mixing with Bose-Einstein condensates in nonlinear lattices

Parity-time (PT) symmetry allows for implementing controllable matching conditions for the four-wave mixing in 1D coupled waveguides. Different types of the process involving energy transition between slow and fast modes are established. In the case of defocusing Kerr media, the degenerated four-wave mixing is studied in detail. It is shown that unbroken PT symmetry supports the process existing in the conservative limit and, at the same time, originates new types of matching conditions, which cannot exist in the conservative system. In the former case, a slow beam splits into two fast beams, with nearly conserved total power, while in the latter case, one slow beam and one fast beam are generated. In the last process, the energy of the input primary slow beam is not changed and growth of the energy of the generated slow beam varies due to gain and loss of the medium. The appreciable generation of the fifth mode, i.e., the effect of the secondary resonant interactions, is observed.

Quantum-enhanced interferometry and the structure of twisted states

We report results of the analysis for families of one-dimensional (1D) trapped solitons, created by competing self-focusing (SF) quintic and self-defocusing (SDF) cubic nonlinear terms. Two trapping potentials are considered, the harmonic-oscillator (HO) and delta-functional ones. The models apply to optical solitons in colloidal waveguides and other photonic media, and to matter-wave solitons in Bose-Einstein condensates loaded into a quasi-1D trap. For the HO potential, the results are obtained in an approximate form, using the variational and Thomas-Fermi approximations, and in a full numerical form, including the ground state and the first antisymmetric excited one. For the delta-functional attractive potential, the results are produced in a fully analytical form, and verified by means of numerical methods. Both exponentially localized solitons and weakly localized trapped modes are found for the delta-functional potential. The most essential conclusions concern the applicability of competing Vakhitov-Kolokolov (VK) and anti-VK criteria to the identification of the stability of solitons created under the action of the competing SF and SDF terms.

Tradeoffs for number squeezing in collisions of Bose-Einstein condensates

Considering the broadest set of measurements allowed by quantum mechanics, we demonstrate that the violation Cauchy–Schwarz inequality for any-order correlation function signals the entanglement among bosons. Our result is general—it applies to any system of bosons, even when the number of particles is not fixed, provided that there is no coherence between different number states.