[Show abstract][Hide abstract] ABSTRACT: We investigate the reduced dimensionality of highly anisotropic Bose-Einstein
condensates (BECs) in connection to the entanglement between its spatial
degrees of freedom. We argue that the reduced-dimensionality of the BEC is
physically meaningful in a regime where spatial correlations are negligible. We
handle the problem analytically within the mean-field approximation for general
quasi-one-dimensional (-1D) and -two-dimensional (-2D) geometries, and obtain
the optimal reduced-dimension, pure-state description of the condensate mean
field. We give explicit solutions to the case of harmonic potentials, which we
compare against exact numerical integration of the three-dimensional (3D)
Gross-Pitaevskii equation.
Physical Review A 10/2013; 90(1). DOI:10.1103/PhysRevA.90.013605 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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
quasi-1D and quasi-2D geometries. We use this model to investigate the relative
phase and density dynamics of a two-mode, cigar-shaped $^{87}$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 more refined analytical account of
the mean-field evolution than an idealized quasi-1D description.
New Journal of Physics 07/2012; 15(2). DOI:10.1088/1367-2630/15/2/023008 · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review A 07/2011; 84(5). DOI:10.1103/PhysRevA.84.053606 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate the effects of the emergence of three-dimensional
behavior on a quasi-one-dimensional Bose-Einstein condensate (BEC)
trapped by a highly elongated potential. By analytically performing the
Schmidt decomposition of the condensate wave function in the
perturbative regime, we derive corrections to the 1D approximation due
to the reshaping of the BEC in the tightly confined direction with
increasing nonlinearity strength. This approach provides a straight-
forward way to redefine the transverse and longitudinal wave functions
as well as to calculate the amount of entanglement that arises between
the two spatial directions. Numerical integration of the
three-dimensional Gross-Pitaevskii equation for different trapping
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 separable wave function that
corresponds to a single Schmidt term.
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review A 08/2010; 82(5). DOI:10.1103/PHYSREVA.82.053636 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review A 09/2009; 80(3). DOI:10.1103/PhysRevA.80.032103 · 2.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We analyze in detail the recently proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving better than 1/N scaling in a quantum metrology protocol using a two mode Bose-Einstein condensate of N atoms. There were several simplifying assumptions in the original proposal that made it easy to see how a scaling approaching 1/N^3/2 may be obtained. We look at these assumptions in detail to see when they may be justified. We present numerical results that confirm our theoretical predictions for the effect of the spreading of the BEC wave function with increasing N on the scaling. Numerical integration of the coupled Gross-Pitaevskii equations for the two mode BEC also shows that the assumption that the two modes share the same spatial wave function is justified for a length of time that is sufficient to run the metrology scheme.
[Show abstract][Hide abstract] ABSTRACT: Questions about quantum limits on measurement precision were once viewed from the perspective of how to reduce or avoid the effects of quantum noise. With the advent of quantum information science came a paradigm shift to proving rigorous bounds on measurement precision. These bounds have been interpreted as saying, first, that the best achievable sensitivity scales as 1/n, where n is the number of particles one has available for a measurement and, second, that the only way to achieve this Heisenberg-limited sensitivity is to use quantum entanglement. We review these results and show that using quadratic couplings of n particles to a parameter to be estimated, one can achieve sensitivities that scale as 1/n^2 if one uses entanglement, but even in the absence of any entanglement at any time during the measurement protocol, one can achieve a super-Heisenberg scaling of 1/n^(3/2)
[Show abstract][Hide abstract] ABSTRACT: We show how a generalized quantum metrology protocol can be implemented in a two-mode Bose-Einstein condensate of n atoms, achieving a sensitivity that scales better than 1/n and approaches 1/n^(3/2) for appropriate design of the condensate.