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arXiv:1002.2504 · Physical Review A

Two-mode Bose gas: Beyond classical squeezing

Cédric Bodet, Jérôme Estève, Markus K. Oberthaler, Thomas Gasenzer

Published Jun 3, 2010• 19 citations
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Abstract

The dynamical evolution of squeezing correlations in an ultracold Bose-Einstein condensate distributed across two modes is investigated theoretically in the framework of the Bose-Hubbard model. It is shown that the eigenstates of the Hamiltonian do not exploit the full region allowed by Heisenberg's uncertainty relation for number and phase fluctuations. The development of nonclassical correlations and relative number squeezing is studied in the transition from the Josephson to the Fock regime. Comparing the full quantum evolution with classical statistical simulations allows us to identify quantum aspects of the squeezing formation. In the quantum regime, the measurement of squeezing allows us to distinguish even and odd total particle numbers.

DOI
10.1103/physreva.81.063605

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Physical Review A•Jan 23, 2008•2 citations
The dynamical evolution of squeezing correlations in an ultracold Bose-Einstein condensate distributed across two modes is investigated theoretically in the framework of the Bose-Hubbard model. It is shown that the eigenstates of the Hamiltonian do not exploit the full region allowed by Heisenberg's uncertainty relation for number and phase fluctuations. The development of nonclassical correlations and relative number squeezing is studied in the transition from the Josephson to the Fock regime. Comparing the full quantum evolution with classical statistical simulations allows us to identify quantum aspects of the squeezing formation. In the quantum regime, the measurement of squeezing allows us to distinguish even and odd total particle numbers.
The dynamical evolution of an ultracold Bose gas distributed across the sites of an optical lattice is investigated theoretically in the framework of the Bose-Hubbard model. First, the focus is set on the evolution of squeezing correlations in the two mode system. It is shown that the eigenstates of the Hamiltonian do not exploit the full region of possible squeezing allowed by Heisenbergs uncertainty relation for number and phase fl uctuations. The development of nonclassical correlations and relative number squeezing is studied at the transition from the Josephson to the Fock regime. Comparing the full quantum evolution with classical statistical simulations allows us to identify quantum aspects of the squeezing formation. In the quantum regime, the measurement of squeezing allows us to distinguish even and odd total particle number states. Then, a far from equilibrium quantum field theory method, the so-called two-particle-irreducible effective action approach, is presented for the description of the dynamics in larger lattices. The resulting dynamics is compared to the classical statistical time evolution. The validity of the quantum field evolution is probed for various initial conditions in the classical regime.
Dynamical evolution of quantum mechanical squeezing and entanglement in a two-mode Bose-Einstein condensate (TBEC) with an adiabatic, time-varying Raman coupling is studied by finding analytical expressions for these quantities. In particular, we study the entanglement between the atoms in the condensate as well as that between the two modes. The nature of the enhanced quantum correlations in TBEC is clarified by considering squeezing and entanglement both with and without the nonlinear interaction turned on; it is found that entanglement approaching maximal value can be achieved with the nonlinear interactions present. Somewhat counterintuitively, greater squeezing is found in the absence of nonlinear interactions. This is due to the collapses and revivals of the TBEC quantum state induced by the nonlinear interactions. In addition, results involving the self-trapping phase state of TBEC indicate potential for creating a dynamically stable, macroscopic entangled quantum state which is relatively robust with respect to atom number fluctuations.
We use a two-mode Bose-Hubbard Hamiltonian to determine the ground state and dynamical evolution for a Bose Josephson junction realized by an ultracold Bose gas in a double-well trap. We identify Mott-like lobes where number fluctuations are suppressed and the interference fringes in the momentum distribution are strongly reduced. Different from superconducting Josephson junctions, the lobes size increases at increasing wells imbalance. Upon a sudden rise of the barrier between the two wells, an initially phase-coherent state evolves into a coherent superposition of phase states, leading to destructive interference in the time-dependent momentum distribution.
Abstract In a previous paper, Das et al (2013 J. Phys. B: At. Mol. Opt. Phys . 46 035501), it was shown that the unitary quantum phase operators play a particularly important role in quantum dynamics of bosons and fermions in a one-dimensional (1D) double-well (DW) when the number of particles is small. In this paper, we define the standard quantum limit (SQL) for phase and number fluctuations, and describe two-mode squeezing for number and phase variables. The usual two-mode number squeezing parameter, also used to describe two-mode entanglement of a quantum field, is defined considering phase as a classical variable. However, when phase is treated as a unitary quantum-mechanical operator, number and phase operators satisfy an uncertainty relation. As a result, the usual definition of number squeezing parameter becomes modified. Two-mode number squeezing occurs when the number fluctuation goes below the SQL at the cost of enhanced phase fluctuation. As an application of number-phase uncertainty, we consider bosons or fermions trapped in a quasi-1D DW potential interacting via a 3D finite-range two-body interaction potential with large scattering length a s . Under tight-binding or two-mode approximation, we describe in detail the effects of the range of interaction on the quantum dynamics and number-phase uncertainty in the strongly interacting or unitarity regime a s → ± ∞ . Our results show intriguing coherent dynamics of number-phase uncertainty with number squeezing for bosons and ph squeezing for fermions. Our results may be important for exploring new quantum interferometry, Josephson oscillations, Bose–Hubbard and Fermi–Hubbard physics with ultracold atoms in DW potentials or DW optical lattices. Particularly interesting will be the question of the importance of quantum phase operators in two-atom interferometry and entanglement.
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.
The squeezing and entanglement properties of a one-component Bose-Einstein condensate trapped in optical lattices are theoretically studied. A two-mode squeezing parameter for the system is found, which is identical with Einstein-Podolsky-Rosen correlation, indicating the quantum correlation between excitation modes. Its dynamics and relation to the Hamiltonian parameters are quantitatively shown, and some constructive relation between squeezing and condensation is pointed out. For entanglement, the entanglement of formation of the two-mode system is also presented. For the probing of such a correlation, relevant experimental issues about atom shot noise are also discussed.