Quantum-field dynamics of expanding and contracting Bose-Einstein condensates

We analyze the dynamics of quantum statistics in a harmonically trapped Bose-Einstein condensate, whose two-body interaction strength is controlled via a Feshbach resonance. From an initially noninteracting coherent state, the quantum field undergoes Kerr squeezing, which can be qualitatively described with a single mode model. To render the effect experimentally accessible, we propose a homodyne scheme, based on two hyperfine components, which converts the quadrature squeezing into number squeezing. The scheme is numerically demonstrated using a two-component Hartree-Fock-Bogoliubov formalism.

Figure 1

(Color online) Evolution of the condensate density and ground state population for scenarios I (a),(c) and II (b),(d), as described in the text. The ground-state population in panels (c) and (d) is defined by $N_0=\int d^3\mathbf{x}{\phi }_0{\left(\mathbf{x}\right)}^{\ast }\varphi \left(\mathbf{x}\right)$. In either case the mean field undergoes visible change of shape but nonetheless most of the population remains in the ground state (c),(d). Note the different time-scales for the two scenarios, and that the spatial axis is a radial coordinate.

Figure 2

Minimum quadrature variances $V={\text{min}}_{\theta }{\left[\Delta {\widehat{X}}^{\theta }\right]}^2$ corresponding to maximal squeezing versus evolution time. We compare HFB simulations [$\times$, Eq. (A4)] with truncated Wigner results [solid line, Eq. (12), the dotted lines indicate the sampling error]. (a) Scenario I. (b) Scenario II. Both panels also include the corresponding analytical result for the single mode Kerr effect (dashed). We use Eq. (1) of Ref. [1], with $\chi =0.0134$.

Figure 3

(a) Initial shape of “squeezing” condensate ${\left|{\varphi }_1\left(r\right)\right|}^2$ (solid) and local oscillator ${\left|{\varphi }_2\left(r\right)\right|}^2$ (dashed) and the corresponding (equal) densities after the mixing at $t=0.4\text{ms}$ (dash-dotted). (b) Number variance in condensate component 1 after the mixing step. For $\text{arg}\left(\Omega \right)=0.66\pi$ the Kerr squeezing in component one with ${\text{min}}_{\theta }{\left[X_0^{\theta }\right]}^2=0.86$ is largely converted into detectable number squeezing. The thin line indicates the value of 0.86 for full conversion of the quadrature squeezing.

Figure 4

Quantum state evolution of a two component BEC. Panels (a),(c) show component $|1⟩$, (b),(d) component $|2⟩$. The variance of the trap ground state for the quadrature with maximal squeezing, Eq. (9), is shown in panels (a),(b). The variance of the total number in each component, Eq. (10), is shown in panels (c),(d) as a fraction of the total atom number in each component. For the first 0.4 ms, $\Omega =0$, followed by a pulse of length $f\times t_{\text{pulse}}=f\times 5\mu \text{s}$ with $\Omega =\hslash \pi /\left(4t_{\text{pulse}}\right)$, where $f\lesssim 1$ is adjusted to achieve full mixing. Subsequently, we again have $\Omega =0$. The two curves use a different reference phase: (solid) $\text{arg}\left(\Omega \right)=0.66\pi$, (dashed) $\text{arg}\left(\Omega \right)=0.16\pi$. The grayed region shows the value achieved after the component mixing. We did not evolve the system past $t_{\text{evolve}}+t_{\text{pulse}}=0.405\text{ms}$, but continued the graph assuming constant variances until 0.6 ms as a visualization aid.