Outcoupling from a Bose-Einstein condensate with squeezed light to produce entangled-atom laser beams

We examine the properties of an atom laser produced by outcoupling from a Bose-Einstein condensate with squeezed light. We model the multimode dynamics of the output field and show that a significant amount of squeezing can be transferred from an optical mode to a propagating atom laser beam. We use this to demonstrate that two-mode squeezing can be used to produce twin atom laser beams with continuous variable entanglement in amplitude and phase.

Figure 1

Internal energy levels of a three-level atom. A condensate of state ∣1⟩ atoms confined in a trapping potential is coupled to free space via a Raman transition affected by a probe beam (annihilation operator ${\widehat{a}}_{13}$) which is detuned from the excited state ∣3⟩ by an amount ${\Delta }_1$, and a pump field (annihilation operator ${\widehat{a}}_{23}$) which is detuned from the excited state by an amount ${\Delta }_2$.

Figure 2

${\mid f(k,k^{\prime },t=0.11s)\mid }^2$ for the values of parameters indicated in the text. $k_{\mathit{\text{kick}}}$ is the kick acquired due to the Raman transition, i.e., $k_{\mathit{\text{kick}}}=k_{23}-k_{13}$. The function was discretized for numerical calculation by replacing $\delta (k-k^{\prime })$ with ${\delta }_{k,k^{\prime }}∕\Delta k$, where $\Delta k$ is the grid spacing. The dip in the function near the $k=k_{\mathit{\text{kick}}}$ resonance shows how the quantum state of the atoms has been affected by the interaction with the optical fields. The region of $f(k,k^{\prime })$ sufficiently far away from $k_{\mathit{\text{kick}}}$ remains diagonal, indicating that the quantum state of the atomic field far from $k_{\mathit{\text{kick}}}$ has not been affected by the outcoupling process.

Figure 3

(Color online) ${\mid g(k,t)\mid }^2$ as found numerically for the values of parameters indicated in the text. This shows that atoms are created around $k=k_{\mathit{\text{kick}}}$ with a quantum state related to the initial state of the probe field.

Figure 4

${\mid p\left(t\right)\mid }^2$ as found numerically for the values of parameters indicated in the text. The decay of $p\left(t\right)$ represents the decay of the initial optical state due to the outcoupling process.

Figure 5

(Color online) ${\mid q(k^{\prime },t)\mid }^2$ as found numerically for the values of parameters indicated in the text. $\int q(k^{\prime },t)d{k}^{\prime }$ represents the amount of vacuum that is added to the initial state of the optical field due to the outcoupling process.

Figure 6

(Color online) $⟨{\widehat{\Psi }}^{†}\left(x\right)\widehat{\Psi }\left(x\right)⟩$ for $⟨{\widehat{a}}_s^{†}{\widehat{a}}_s⟩=1000$.

Figure 7

The relative variance $v\left(\widehat{N}\right)=V\left(\widehat{N}\right)∕⟨\widehat{N}⟩$ for the outcoupled atoms versus time for different states of the optical field. The solid line represents the initial optical mode in a coherent state $\mid \alpha ⟩$ with ${\mid \alpha \mid }^2=1000$, the dotted line represents a squeezed state $\mid \alpha ,r⟩$ with ${\mid \alpha \mid }^2=1000$, $r=1.38$, and the dashed line represents a Fock state $\mid n⟩$ with $n=1000$.

Figure 8

Flux of outcoupled atoms at a point in the atomic beam $(x=1.5\mathrm{mm})$ for $⟨{\widehat{a}}_s^{†}{\widehat{a}}_s⟩=1000$.

Figure 9

$v\left(\widehat{J}\right)$ at a point in the path of the atomic beam $(x=1.5\mathrm{mm})$ for $\Omega =18$ (dotted line), 144 (solid line), and $270\mathrm{rad}{\mathrm{s}}^{-1}$ (dashed line). Coupling weakly produces a long pulse, but the variance in the flux is almost unaffected by the statistics of the optical state. Coupling too strongly causes significant back coupling from the output field to the photonic state.

Figure 10

Minimum value of $v\left(\widehat{J}\right)$ versus $\Omega$. The shot noise is below the vacuum noise for all values of $\Omega$ when using a Fock state to outcouple. For low coupling strengths, very little of the optical state is transferred to the out coupled atoms, so the squeezing is minimal. When the coupling strength is too strong, there is significant back coupling of atoms which limits the flux, and hence degrades the squeezing. The finite momentum spread of the condensate means that there is a spread in the time taken for an atom to leave the outcoupling region, so even at the optimum Rabi frequency, there will not be complete transfer of the optical state to the atomic field. This limits the amount of squeezing that can be transferred to the atomic field.

Figure 11

(Color online) Twin atom laser beams produced by outcoupling with two-mode squeezed light. An OPO is driven by a classical, nondepletable driving field $\left(\beta \right)$. The ${\chi }^{\left(2\right)}$ process produces two optical modes ${\widehat{a}}_1$ and ${\widehat{a}}_2$, which are used to outcouple the atom laser beams.

Figure 12

(Color online) Density of outcoupled atoms versus time. Outcoupling using light from a nondegenerate OPO produces two atomic beams in opposite directions, with steady flux.

Figure 13

Variance in the flux difference. The fluctuations are eight times smaller than for uncorrelated atoms.

Figure 14

Product of the inferred variances $V^{\mathit{inf}}\left({\widehat{X}}_{-}\right)V^{\inf }\left({\widehat{Y}}_{-}\right)$ versus time. As the system goes to steady state, the requirement for entanglement is satisfied.