Dissipative versus conditional generation of Gaussian entanglement and spin squeezing

Spin squeezing of collective atomic spins can be achieved conditionally via probing with light and subsequent homodyne detection, as is done in a quantum nondemolition measurement. Recently it has been shown that squeezing can also be created unconditionally by a properly designed dissipative dynamics. We compare the two approaches in a Gaussian description and optimize over all Gaussian light-matter interactions. We find that in the optimal unconditional scheme based on dissipation the level of squeezing scales with optical depth as $d^{-1/2}$. In contrast, the optimal conditional scheme based on measurement of light—which in fact is not a quantum nondemolition measurement—can provide squeezing which scales as $d^{-1}$. Our results apply directly also to the creation of entanglement in the form of nonlocal spin squeezing of two atomic ensembles.

Figure 1

One (a) or two (b) bosonic modes (collective atomic spins) couple sequentially to a 1D field, which might be subject to homodyne detection after the interactions.

Figure 2

Unconditional squeezed variance $V_{\mathrm{u}}$ [Eq. (19)] versus interaction parameter $\theta$. The dynamics is stable only for $\theta >{\theta }_c$ where ${\theta }_c$ is given in (21). $V_{\mathrm{u}}$ exhibits a minimum for an optimal choice of ${\theta }_{\mathrm{u}}^{\mathrm{opt}}$ given in (22) in a region where the beam splitter (BS) interaction dominates over the two-mode-squeezing (TMS) interaction. Curves for the following parameter values are shown: $d=5,n=0$ is the blue (dark gray) line, $d=50,n=0$ is the green (light gray) line, and $d=50,n=n_{\mathrm{c}}$ is the red (top) line.

Figure 3

Optimal interaction parameter $\theta$ versus optical depth. Solid curves show conditional (the lower blue line) ${\theta }_{\mathrm{c}}^{\mathrm{opt}}$ and unconditional (the upper green line) ${\theta }_{\mathrm{u}}^{\mathrm{opt}}$ optimal interaction parameters. Dashed lines are asymptotics. For large optical depths the optimal interaction for dissipative generation of squeezing approaches the QND interaction ($\theta =0$), while the best choice for conditional generation of squeezing becomes the two-mode-squeezing interaction ($\theta =-\pi /4$).

Figure 4

Conditional squeezed variance $V_{\mathrm{c}}$ (solid) and unsqueezed variance $U_c$ (dashed) versus interaction parameter $\theta$. The beam-splitter reflectivity is $\epsilon =0$. Curves for the following parameter values are shown: $d=5,n=0$ is the blue (dark gray) solid line for $V_{\mathrm{c}}$ and dashed for $U_{\mathrm{c}}$, $d=50,n=0$ is the green (light gray) solid line for $V_{\mathrm{c}}$ and dashed for $U_{\mathrm{c}}$, and $d=50,n=n_{\mathrm{c}}$ is the red (top) solid line for $V_{\mathrm{c}}$.

Figure 5

Squeezed unconditional variance $V_{\mathrm{u}}$ [green (light gray)], conditional variance $V_{\mathrm{c}}$ [blue (dark gray)], and unsqueezed conditional variance $U_{\mathrm{c}}$ [blue (dark gray) dashed] versus interaction parameter $\theta$ for an optical depth $d=5$, minimal atomic noise $n=0$, and beam-splitter reflectivity $\epsilon =0.05$. Interaction parameter $\theta =0$ corresponds to a QND interaction, for $0<\theta <\pi /2$ the beam-splitter interaction dominates, and for $-\pi /2<\theta <0$ the two-mode-squeezing interaction is dominant. The grayed region corresponds to the unconditionally stable dynamics.

Figure 6

Same as Fig. 5 but for critical (effective) temperature corresponding to decay to a thermal state with mean occupation number $n=n_c=d(\sqrt{2}-1)/4$. It is impossible to unconditionally achieve spin squeezing or entanglement for this or higher occupation numbers.

Figure 7

Squeezed and unsqueezed variances vs optical depth for minimal noise $n=0$. Blue (dark gray), conditional variance for the optimal ${\theta }_{\mathrm{c}}^{\mathrm{opt}}$; green (lower light gray), unconditional squeezed variance for optimized interaction parameter ${\theta }_{\mathrm{u}}^{\mathrm{opt}}$; yellow (upper light gray), unconditional unsqueezed variance for optimized interaction parameter ${\theta }_{\mathrm{u}}^{\mathrm{opt}}$; red (top), conditional unsqueezed variance for the optimal ${\theta }_{\mathrm{c}}^{\mathrm{opt}}$. Dashed lines: Blue (dark gray), asymptotic scaling $2/d$ for conditional dynamics; green (light gray), $2/\sqrt{d}$ for the unconditional dynamics; red (top), asymptotic of the conditional unsqueezed variance $1/\epsilon$.