Unconditional Steady-State Entanglement in Macroscopic Hybrid Systems by Coherent Noise Cancellation

The generation of entanglement between disparate physical objects is a key ingredient in the field of quantum technologies, since they can have different functionalities in a quantum network. Here we propose and analyze a generic approach to steady-state entanglement generation between two oscillators with different temperatures and decoherence properties coupled in cascade to a common unidirectional light field. The scheme is based on a combination of coherent noise cancellation and dynamical cooling techniques for two oscillators with effective masses of opposite signs, such as quasispin and motional degrees of freedom, respectively. The interference effect provided by the cascaded setup can be tuned to implement additional noise cancellation leading to improved entanglement even in the presence of a hot thermal environment. The unconditional entanglement generation is advantageous since it provides a ready-to-use quantum resource. Remarkably, by comparing to the conditional entanglement achievable in the dynamically stable regime, we find our unconditional scheme to deliver a virtually identical performance when operated optimally.

Figure 1

Hybrid system consisting of two oscillators with negative and positive mass, respectively, typically implemented in collective spin ($S$) and motional ($M$) degrees of freedom. These are coupled in cascade to a common unidirectional light field via quadratic interactions induced typically by a strong, classical carrier. For each oscillator, this results in Stokes and anti-Stokes sidebands proportional to rates ${\mathrm{\Gamma }}_{jP/B}$ and of width ${\gamma }_j$, $j\in \{S,M\}$ (see inset); the effective resonance frequency ${\mathrm{\Omega }}_j\equiv \text{sgn}(m_j){\omega }_j$ accounts for, e.g., the fact that energy must be extracted from a negative-mass oscillator to excite it. The joint interaction with the light field is best described by homodyne quadratures ${\widehat{X}}_L$, ${\widehat{P}}_L$ (symmetrized combinations of sidebands), whose initial state ${\widehat{X}}_{L,\mathrm{in}}$, ${\widehat{P}}_{L,\mathrm{in}}$ is vacuum (see lower part of figure). The negative mass system is driven by ${\widehat{X}}_{L,\mathrm{in}}$ only (${\mathrm{\Gamma }}_{SP}={\mathrm{\Gamma }}_{SB}$) and its response is mapped onto ${\widehat{P}}_L$. The positive mass system is likewise coupled to ${\widehat{X}}_{L,\mathrm{in}}$, but also to ${\widehat{P}}_L$ (${\mathrm{\Gamma }}_{MP}<{\mathrm{\Gamma }}_{MB}$) at an adjustable rate $R$, so that the response of the negative mass system drives the positive mass system. Consequently, the response of the positive mass system will interfere destructively with that of the negative mass system in the outgoing field quadrature ${\widehat{P}}_{L,\text{out}}$ as can (optionally) be verified by homodyne detection. Additionally, the oscillators are driven by distinct thermal reservoirs with decoherence rates ${\tilde{\gamma }}_{j,0}$ (wavy arrows).

Figure 2

Entanglement ${\xi }_g$ ($<1$ in the colored region) as a function of quantum cooperativities of the spin ($C_S$) and motional ($C_M$) subsystems for optimized coupling angles ${\theta }_S$ and ${\theta }_M$ while fixing the parameters ${\gamma }_{S,0}=2\pi \times 5\mathrm{kHz}$, ${\overline{n}}_S=1$, ${\gamma }_{M,0}{\overline{n}}_M=2\pi \times 10\mathrm{kHz}$, and assuming no transmission losses, $\epsilon =0$. Optimal $C_M$ for given $C_S$ is indicated by the dashed-dotted curve. Imposing the additional constraint ${\theta }_S={\theta }_M\Rightarrow R=0$, entanglement ${\xi }_g<1$ is only possible in the subregion delineated by the solid contour.

Figure 3

Entanglement ${\xi }_g$ as a function of spin cooperativity $C_S$ for optimized coupling angles ${\theta }_S$, ${\theta }_M$ and motional cooperativity $C_M$ when $R=0$ (thin black curves) and $R\ne 0$ (thick red curves), when transmission loss is absent, $\epsilon =0$ (solid), and present, $\epsilon =0.1$ (dashed). (Inset) Plot of $-2\sqrt{1-\epsilon }R/{\gamma }_M$ (right scale, brighter green curves) as a function of $C_S$ used in evaluating the optimized curves of the main plot, and the relative entanglement improvement (left scale, darker red curves) of the conditional scheme over the optimal unconditional scheme (referenced to the latter) for $\epsilon =0.1$; the conditional performance is evaluated using parameters optimized for the unconditional scheme (dashed) and optimal conditional parameters for QND readout ${\theta }_S={\theta }_M=\pi /4$ (solid). The fixed parameters are ${\gamma }_{S,0}=2\pi \times 5\mathrm{kHz}$, ${\overline{n}}_S=1$, and ${\gamma }_{M,0}{\overline{n}}_M=2\pi \times 10\mathrm{kHz}$.