Engineering asymmetric steady-state Einstein-Podolsky-Rosen steering in macroscopic hybrid systems

Generation of quantum correlations between separate objects is of significance both in fundamental physics and in quantum networks. One important challenge is to create the directional “spooky action-at-a-distance” effects that Schrödinger called “steering” between two macroscopic and massive objects. Here, we analyze a generic scheme for generating steering correlations in cascaded hybrid systems in which two distant oscillators with effective masses of opposite signs are coupled to a unidirectional light field, a setup which is known to build up quantum correlations by means of quantum back-action evasion. The unidirectional coupling of the first to the second oscillator via the light field can be engineered to enhance steering in both directions and provides an active method for controlling the asymmetry of steering. We show that the resulting scheme can efficiently generate unconditional steady-state Einstein-Podolsky-Rosen steering between the two subsystems, even in the presence of thermal noise and optical losses. As a scenario of particular technological interest in quantum networks, we use our scheme to engineer enhanced steering from an untrusted node with limited tunability (in terms of interaction strength and type with the light field) to a trusted, highly tunable node, hence offering a path to implementing one-sided device-independent quantum tasks.

Figure 1

A cascaded hybrid system composed of two oscillators with effective masses of opposite signs and individual decoherence rates ${\tilde{\gamma }}_{j,0}$, $j\in \{+,-\}$, coupled with a unidirectional propagating light field. The light-oscillator interaction is induced by a coherent laser drive (assumed to be quantum limited) and is parametrized by the interaction strength ${\mathrm{\Gamma }}_j$ and angle ${\theta }_j$ controlling the interaction type. Additionally, the second oscillator is driven by an uncorrelated vacuum field ${\widehat{b}}_{in}^{\prime }$ due to the presence of the transmission (power) loss $\epsilon$ between the subsystems. In order to verify EPR steering, independent local measurements are made on the individual subsystems (top part of figure). These are combined using gain factors $g_{x,p}$, which can be optimized to minimize the product of the uncertainties (standard deviations) $\mathrm{\Delta }({\widehat{X}}_{+}-g_x{\widehat{X}}_{-})$ and $\mathrm{\Delta }({\widehat{P}}_{+}+g_p{\widehat{P}}_{-})$ entering the steering criteria.

Figure 2

(Left column) $E_{+\mid -}$ (<0.5, in the region delineated by the dashed contour line) and $E_{-\mid +}$ (<0.5, solid contour line) as a function of the quantum cooperativities $C_{\pm }$. One-way steering from the first $(-)$ to the second $(+)$ subsystem exists in the light pink region, and one-way steering of the opposite direction exists in the dark blue region. (Right column) $E_{+\mid -}$ (thick blue, dashed), $E_{-\mid +}$ (thick blue, solid) and the bare variances ${\mathrm{\Delta }}^2{\widehat{X}}_{+0}$ (thin orange, dashed), ${\mathrm{\Delta }}^2{\widehat{X}}_{-0}$ (thin orange, solid) of the two subsystems as a function of the ratio $C_{+}/C_{-}$ when assuming $C_{-}=50$. The remaining parameters are fixed to the following values: ${\theta }_{-}=0.35\pi$, $\epsilon =0$, the intrinsic linewidth ${\gamma }_0=2\pi \times 0.1\text{Hz}$, and the thermal occupation number $\overline{n}={10}^5$. The thermal decoherence rate resulting from these thermal parameters is ${\tilde{\gamma }}_0\approx 2\pi \times 10\text{kHz}$.

Figure 3

(Left column) $E_{+\mid -}$ (<0.5, in the region delineated by the dashed contour) and $E_{-\mid +}$ (<0.5, solid contour) as a function of interaction angles ${\theta }_{\pm }$ for (a) $C_{+}=C_{-}$ and (c) $C_{+}=2C_{-}$. One-way steering from the first $(-)$ to the second $(+)$ subsystem exists in the light pink region, and one-way steering of the opposite direction exists in the dark blue region. (Right column) $E_{+\mid -}$ (thick blue, dashed), $E_{-\mid +}$ (thick blue, solid) and the bare variances in the two subsystems ${\mathrm{\Delta }}^2{\widehat{X}}_{+0}$ (thin orange, dashed), ${\mathrm{\Delta }}^2{\widehat{X}}_{-0}$ (thin orange, solid) as a function of ${\theta }_{+}$ when assuming ${\theta }_{-}=0.35\pi$; again, we consider the two cases (b) $C_{+}=C_{-}$ and (d) $C_{+}=2C_{-}$. The other parameter values used are $C_{-}=50$, transmission loss $\epsilon =0$, intrinsic linewidth ${\gamma }_0=2\pi \times 0.1\text{Hz}$, and thermal occupation number $\overline{n}={10}^5$, resulting in the decoherence rate ${\tilde{\gamma }}_0\approx 2\pi \times 10\text{kHz}$.

Figure 4

$E_{+\mid -}$ (dashed) and $E_{-\mid +}$ (solid) as a function of $C_{-}$ with optimal interaction types ${\theta }_{\pm }$ and quantum cooperativity $C_{+}$ when $f=0$ and $f\ne 0$, in the cases (a) $\epsilon =0$ and (b) $\epsilon =0.2$, i.e., without and with transmission losses. The two sets of thermal parameters are (thick blue curves) intrinsic linewidth ${\gamma }_{\pm ,0}=2\pi \times 0.1\text{Hz}$, thermal occupation number ${\overline{n}}_{\pm }={10}^5$; (thin orange curves) ${\gamma }_{-,0}=2\pi \times 0.1\text{Hz}$, ${\gamma }_{+,0}=2\pi \times 20\text{kHz}$, ${\overline{n}}_{-}={10}^5$, and ${\overline{n}}_{+}=0$.

Figure 5

$E_{+\mid -}$ as a function of the transmission loss $\epsilon$ evaluated for optimized ${\theta }_{\pm }$ and $C_{+}$ when $C_{-}=2$ (solid), 10 (dashed), and 100 (dot-dashed). The two sets of fixed thermal parameters are (thick blue curves) intrinsic linewidth ${\gamma }_{\pm ,0}=2\pi \times 0.1\text{Hz}$ and thermal occupation number ${\overline{n}}_{\pm }={10}^5$; (thin orange curves) ${\gamma }_{-,0}=2\pi \times 0.1\text{Hz}$, ${\gamma }_{+,0}=2\pi \times 20\text{kHz}$, ${\overline{n}}_{-}={10}^5$, and ${\overline{n}}_{+}=0$.

Figure 6

The optimal interaction angles $\left({\theta }_{\pm }\right)$ for the minimized $E_{+\mid -}$ (dashed) and $E_{-\mid +}$ (solid) as a function of $C_{-}$ for the fixed thermal parameters as intrinsic linewidth ${\gamma }_{\pm ,0}=2\pi \times 0.1\text{Hz}$ and thermal occupation number ${\overline{n}}_{\pm }={10}^5$ when $f=0$ (thin red curves) and $f\ne 0$ (thick blue curves), as shown in panels (a) and (b). Panels (c) and (d) represent the scenario of asymmetric thermal parameters with intrinsic linewidths ${\gamma }_{-,0}=2\pi \times 0.1\text{Hz}$, ${\gamma }_{+,0}=2\pi \times 20\text{kHz}$ and thermal occupation numbers ${\overline{n}}_{-}={10}^5$, ${\overline{n}}_{+}=0$, when $f=0$ (thin black curves) and $f\ne 0$ (thick orange curves). Note that in panel (c) when minimizing $E_{-\mid +}$ (solid), the optimal values are ${\theta }_{-}=\pi /2$ and ${\theta }_{+}=\pi /4$ when $C_{-}$ is small, and then the optimal values of ${\theta }_{\pm }$ switch, i.e., ${\theta }_{-}=\pi /4$ and $\pi /4<{\theta }_{+}\le \pi /2$ with increasing $C_{-}$.

Figure 7

The optimal ratio of quantum cooperativities $(C_{+}/C_{-})$ for the minimized $E_{+\mid -}$ (dashed) and $E_{-\mid +}$ (solid) as a function of $C_{-}$ for the fixed thermal parameters as intrinsic linewidth ${\gamma }_{\pm ,0}=2\pi \times 0.1\text{Hz}$, thermal occupation number ${\overline{n}}_{\pm }={10}^5$ when $f=0$ (thin red curves) and $f\ne 0$ (thick blue curves), as shown in panels (a) and (b). Panels (c) and (d) represent the scenario of asymmetric thermal parameters with intrinsic linewidths ${\gamma }_{-,0}=2\pi \times 0.1\text{Hz}$, ${\gamma }_{+,0}=2\pi \times 20\text{kHz}$ and thermal occupation numbers ${\overline{n}}_{-}={10}^5$, ${\overline{n}}_{+}=0$, when $f=0$ (thin black curves) and $f\ne 0$ (thick orange curves).

Figure 8

The relative improvement $d$ of the optimal $E_{+\mid -}$ (dashed) and $E_{-\mid +}$ (solid) as a function of $C_{-}$ for the fixed thermal parameters as intrinsic linewidth ${\gamma }_{\pm ,0}=2\pi \times 0.1\text{Hz}$ and thermal occupation number ${\overline{n}}_{\pm }={10}^5$ (thick blue curves). The thin orange curves are plotted based on the asymmetric thermal parameters: intrinsic linewidths ${\gamma }_{-,0}=2\pi \times 0.1\text{Hz}$, ${\gamma }_{+,0}=2\pi \times 20\text{kHz}$ and thermal occupation numbers ${\overline{n}}_{-}={10}^5$, ${\overline{n}}_{+}=0$.