Atom chips provide a versatile quantum laboratory for experiments with ultracold atomic gases. They have been used in diverse experiments involving low-dimensional quantum gases, cavity quantum electrodynamics, atom-surface interactions, and chip-based atomic clocks and interferometers. However, a severe limitation of atom chips is that techniques to control atomic interactions and to generate entanglement have not been experimentally available so far. Such techniques enable chip-based studies of entangled many-body systems and are a key prerequisite for atom chip applications in quantum simulations, quantum information processing and quantum metrology. Here we report the experimental generation of multi-particle entanglement on an atom chip by controlling elastic collisional interactions with a state-dependent potential. We use this technique to generate spin-squeezed states of a two-component Bose-Einstein condensate; such states are a useful resource for quantum metrology. The observed reduction in spin noise of -3.7 +/- 0.4 dB, combined with the spin coherence, implies four-partite entanglement between the condensate atoms; this could be used to improve an interferometric measurement by -2.5 +/- 0.6 dB over the standard quantum limit. Our data show good agreement with a dynamical multi-mode simulation and allow us to reconstruct the Wigner function of the spin-squeezed condensate. The techniques reported here could be directly applied to chip-based atomic clocks, currently under development.

We study the truncated Wigner method applied to a weakly interacting spinless Bose condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ(r) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve each classical field with the Gross-Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work [Jour. of Mod. Opt. 47, 2629-2644] and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three dimensional spatially homogeneous Bose condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev-Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross-Pitaevskii equation, thermalises to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy ǫmax of the Bogoliubov modes in the simulation does not exceed a few kBT .

We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas. DOI: 10.1103/PhysRevLett.87.210404 PACS numbers: 03.75.Fi, 05.10.Gg, 42.50. -p Since the first experimental demonstrations of BoseEinstein condensation in atomic gases [1], the role played by finite temperature effects in the physics of BoseEinstein condensed alkali gases has drawn increasing attention. For example, in thermal equilibrium, both the spatial density of the condensate atoms [2] and the distribution of number of particles in the condensate [3] are modified due to the presence of the thermal atoms. Similarly, we must take into account the noncondensed atoms in order to explain time dependent phenomena such as the damping and frequency shifts of collective modes [4,5] or the evolution of the recently created vortices [6], where the dissipation of the condensate motion is provided by the noncondensed atoms [7].The widely used Gross-Pitaevskii equation, the nonlinear Schrödinger equation for the condensate wave function, is not able to describe these effects since it neglects the interaction between the condensate and the noncondensed atoms [8]. One possibility to go beyond the GrossPitaevskii equation is to use Bogoliubov theory, which is a perturbative method valid for a small noncondensed fraction. The Bogoliubov method can be applied to thermal equilibrium but also to time dependent situations: in a U(1) symmetry breaking point of view, to zeroth order one solves the time dependent Gross-Pitaevskii equation for the condensate field c 0 ͑ r, t͒, to first order one linearizes the quantum field equations around the classical field c 0 ͑ r, t͒ to get the dynamics of noncondensed particles, and to second order one includes the backaction of noncondensed particles on the condensate. However, in this way one can predict only small corrections to the Gross-Pitaevskii equation. Additionally, if the number of noncondensed particles increases during the evolution of the system, the Bogoliubov approach is valid only for short times [9]. Another existing approach is the mean field HartreeFock-Bogoliubov approximation. This approach is known however to present consistency problems and is still the object of research [10].In this Letter we propose an alternative method to study the time evolution of Bose condensed gases perturbed from an initial thermal equilibrium, based on the classical field approximation in the Wigner representation, the so-called truncated Wigner approximation. The classical field approximation amounts to evolving a set of initially randomly distributed atomic fields w...

The problem of spin squeezing with a bimodal condensate in the presence of particle losses is solved analytically by the Monte Carlo wave function method. We find the largest obtainable spin squeezing as a function of the one-body loss rate, the two-body and three-body rate constants, and the s-wave scattering length.