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arXiv:cond-mat/0307453 · Journal of Physics B Atomic Molecular and Optical Physics

Phase engineering of controlled entangled number states in a single component Bose–Einstein condensate in a double well

Khan W. Mahmud, Heidi Perry, William P. Reinhardt

Published Aug 13, 2003• 43 citations
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Abstract

We propose a model for the creation of entangled number states (Schrödinger cat states) of a Bose–Einstein condensate in a double well through simple phase engineering. We show that a π-phase imprinted condensate in a double well evolves, with a simultaneous change of barrier height, to number states with well defined and controlled entanglement. The cat state generation is understood in terms of the underlying classical phase space dynamics of a π-phase displaced coherent state put at the hyperbolic fixed point of the separatrix of a physical pendulum. The extremity and sharpness of the final cat state is determined by the initial barrier height and the rate at which it is ramped during the evolution.

DOI
10.1088/0953-4075/36/17/102

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We propose a model for the creation of entangled number states (Schrödinger cat states) of a Bose–Einstein condensate in a double well through simple phase engineering. We show that a π-phase imprinted condensate in a double well evolves, with a simultaneous change of barrier height, to number states with well defined and controlled entanglement. The cat state generation is understood in terms of the underlying classical phase space dynamics of a π-phase displaced coherent state put at the hyperbolic fixed point of the separatrix of a physical pendulum. The extremity and sharpness of the final cat state is determined by the initial barrier height and the rate at which it is ramped during the evolution.
We provide a scheme for the generation of controlled entangled number states of Bose-Einstein condensates in multiple wells, and also provide a novel method for the creation of squeezed states without severe adiabatic constraints on barrier heights. The condensate in a multiple well trap can be evolved, starting with a specific initial phase difference between the neighboring wells, to a tunable entangled state or a squeezed state. We propose a general formula for the initial phase difference between the neighboring wells that is valid for any number of wells, even and odd.
We present a quantum phase-space model of the Bose-Einstein condensate (BEC) in a double-well potential. In a quantum two-mode approximation we examine the eigenvectors and eigenvalues and find that the energy correlation diagram indicates a transition from a delocalized to a fragmented regime. Phase-space information is extracted from the stationary quantum states using the Husimi distribution function. We show that the mean-field phase-space characteristics of a nonrigid physical pendulum arises from the exact quantum states, and that only 4--8 particles per well are needed to reach the semiclassical limit. For a driven double-well BEC, we show that the classical chaotic dynamics is manifest in the dynamics of the quantum states. Phase-space analogy also suggests that a $\ensuremath{\pi}$ phase-displaced wave packet put on the unstable fixed point on a separatrix bifurcates to create a superposition of two pendulum rotor states---a macroscopic superposition state of BEC. We show that the choice of initial barrier height and ramping, following a $\ensuremath{\pi}$ phase imprinting on the condensate, can be used to generate controlled entangled number states with tunable extremity and sharpness.
We provide an experimentally viable scheme for the generation of entangled number states of Bose–Einstein condensates in multiple wells with cyclic pairwise connectivity. The condensate ground state in a multiple well trap can self-evolve, when appropriately phase engineered with specific phase differences between the neighbouring wells, to a macroscopic superposition state with controllable entanglement – as for example the multiple well generalization of double well NOON states. We demonstrate, through numerical simulations, the creation of entangled states of small numbers of particles in three and four wells and then explore the creation of entangled states where there are larger numbers of particles or a larger number of wells. In the four well case we study a novel, initially unexpected, type of entanglement produced as the particle numbers, or interaction strengths, increase.
We propose and numerically validate a scheme for the realization of Schrödinger cat-like states in a two-component Bose-Einstein condensate, emphasizing their twinning across components under tunable intra- and inter-species interactions within the miscibility regime. Wigner phase-space analysis reveals sub-Planck-scale interference fringes, confirming the nonclassical character of the states and their potential utility in quantum-enhanced metrology. The dynamical response of the system to a weak linear gravitational-like perturbation further demonstrates cooperative enhancement: while the directly perturbed component retains its cat-like features, the coupled partner exhibits a pronounced population imbalance and a distinct phase-space rotation, providing a sensitive detection channel absent in single-component condensates. These results establish binary condensates as a versatile platform for engineering macroscopic quantum superpositions and exploiting their twinning dynamics for precision measurements.
We present a scheme to prepare generalized coherent states in a system with two species of Bose-Einstein condensates. First, within the two-mode approximation, we demonstrate that a Schrodinger cat-like can be dynamically generated and, by controlling the Josephson-like coupling strength, the number of coherent states in the superposition can be varied. Later, we analyze numerically the dynamics of the whole system when interspecies collisions are inhibited. Variables such as fractional population, Mandel parameter and variances of annihilation and number operators are used to show that the evolved state is entangled and exhibits sub-Poisson statistics.
Three different schemes for producing Schrödinger cat states in Bose-Einstein condensates are outlined and the effects of loss in each of them compared. The first scheme involves coupled interacting condensates and proves to be very fragile to loss. This is improved upon with a second scheme which first evolves a cat state in phase space and then rapidly transforms it to a number cat state. Finally a third scheme is discussed which makes use of number correlated condensates and is remarkably robust to loss. It may prove to be valuable for experimentally creating such states.