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arXiv:0806.3877 · Physical Review A

Optimizing number squeezing when splitting a mesoscopic condensate

Julian Grond, Jörg Schmiedmayer, Ulrich Hohenester

Published Feb 9, 2009• 98 citations
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Abstract

We optimize number squeezing when splitting a mesoscopic Bose-Einstein condensate. Applying optimal control theory to a realistic description of the condensate allowed us to identify a form of the splitting ramp, which drastically outperforms the adiabatic splitting. The results can be interpreted in terms of a generic two-mode model mapped onto a parametric harmonic oscillator. This optimal route to squeezing paves the way to a much longer phase coherence and atom interferometry close to the Heisenberg limit.

DOI
10.1103/physreva.79.021603

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We optimize number squeezing when splitting a mesoscopic Bose-Einstein condensate. Applying optimal control theory to a realistic description of the condensate allowed us to identify a form of the splitting ramp, which drastically outperforms the adiabatic splitting. The results can be interpreted in terms of a generic two-mode model mapped onto a parametric harmonic oscillator. This optimal route to squeezing paves the way to a much longer phase coherence and atom interferometry close to the Heisenberg limit.
We theoretically analyze atom interferometry based on trapped ultracold atoms and employ optimal control theory in order to optimize number squeezing and condensate trapping. In our simulations, we consider a setup where the confinement potential is transformed from a single to a double well, which allows us to split the condensate. To avoid in the ensuing phase-accumulation stage of the interferometer dephasing due to the nonlinear atom-atom interactions, the atom-number fluctuations between the two wells should be sufficiently low. We show that low number fluctuations (high number squeezing) can be obtained by optimized splitting protocols. Two types of solutions are found: in the Josephson regime we find an oscillatory tunnel control and a parametric amplification of number squeezing, while in the Fock regime squeezing is obtained solely due to the nonlinear coupling, which is transformed to number squeezing by peaked tunnel pulses. We study splitting and squeezing within the frameworks of a generic two-mode model, which allows us to study the basic physical mechanisms and the multiconfigurational time-dependent Hartree for bosons method, which allows for a microscopic modeling of the splitting dynamics in realistic experiments. Both models give similar results, thus highlighting the general nature of these two solution schemes. We finally analyze our results in the context of atom interferometry.
We theoretically investigate the creation of squeezed states of a Bose-Einstein condensate (BEC) trapped in a magnetic double-well potential. The number or phase squeezed states are created by modulating the tunnel coupling between the two wells periodically with twice the Josephson frequency, i.e., through parametric amplification. Simulations are performed with the multiconfigurational time-dependent Hartree method for bosons. We employ optimal control theory to bring the condensate to a complete halt at a final time, thus creating a highly squeezed state (squeezing factor of 0.12, ${\ensuremath{\xi}}_{S}^{2}=\ensuremath{-}18$ dB) suitable for atom interferometry.
We theoretically study atom number fluctuations between the halves of a double-well trap containing a Bose-Einstein condensate. The basic tool is the two-mode approximation, which assumes that only two one-particle states are involved. An analytical harmonic-oscillator-like model is developed and verified numerically for both stationary fluctuations in the ground state of the system, and for the fluctuations resulting from splitting of a single trap by dynamically erecting a barrier in the middle. With increasing strength of the atom-atom interactions and/or increasing height of the potential barrier, the fluctuations tend to evolve from Poissonian to sub-Poissonian. Limits of validity of the two-mode model and its relations to the phase-atom-number approach of Leggett and Sols [A. J. Leggett and F. Sols, Found. Phys. 21, 353 (1991)] are discussed in detail.
We measure the relative phase of two Bose-Einstein condensates confined in a radio frequency induced double-well potential on an atom chip. We observe phase coherence between the separated condensates for times up to approximately 200 ms after splitting, a factor of 10 longer than the phase diffusion time expected for a coherent state for our experimental conditions. The enhanced coherence time is attributed to number squeezing of the initial state by a factor of 10. In addition, we demonstrate a rotationally sensitive (Sagnac) geometry for a guided atom interferometer by propagating the split condensates.
We study the dynamic process of splitting a condensate by raising a potential barrier in the center of a harmonic trap. We use a two-mode model to describe the phase coherence between the two halves of the condensate. Furthermore, we explicitly consider the spatial dependence of the mode funtions, which varies depending on the potential barrier. This allows us to get the tunneling coupling between the two wells and the on-site energy as a function of the barrier height. Moreover, we can get some insight into the collective modes that are excited by raising the barrier. We describe the internal and external degrees of freedom by variational ansatz. We distinguish the possible regimes as a function of the characteristic parameters of the problem and identify the adiabaticity conditions.
We report manipulation of the atom number statistics associated with Bose-Einstein condensed atoms confined in an array of weakly linked mesoscopic traps. We used the interference of atoms released from the traps as a sensitive probe of these statistics. By controlling relative strengths of the tunneling rate between traps and atom-atom interactions within each trap, we observed trap states characterized by sub-Poissonian number fluctuations and adiabatic transitions between these number-squeezed states and coherent states of the atom field. The quantum states produced in this work may enable substantial gains in sensitivity for atom interference-based instruments as well as fundamental studies of quantum phase transitions.