Time-Continuous Bell Measurements

We combine the concept of Bell measurements, in which two systems are projected into a maximally entangled state, with the concept of continuous measurements, which concerns the evolution of a continuously monitored quantum system. For such time-continuous Bell measurements we derive the corresponding stochastic Schrödinger equations, as well as the unconditional feedback master equations. Our results apply to a wide range of physical systems, and are easily adapted to describe an arbitrary number of systems and measurements. Time-continuous Bell measurements therefore provide a versatile tool for the control of complex quantum systems and networks. As examples we show that (i) two two-level systems can be deterministically entangled via homodyne detection, tolerating photon loss up to 50%, and (ii) a quantum state of light can be continuously teleported to a mechanical oscillator, which works under the same conditions as are required for optomechanical ground-state cooling.

Figure 1

Schematic setups for (a) time-continuous teleportation and (b) entanglement swapping. The systems may take the form of (c) a harmonic oscillator (e.g., an optomechanical cavity), or (d) a two-level system (e.g., a single spin).

Figure 2

Performance of continuous entanglement swapping: (a) Entanglement (logarithmic negativity [83]) of the stationary state versus $z$ parametrizing the light-matter interaction $\propto i[(\sqrt{z(1+z)}{\sigma }^{+}-\sqrt{1-z}{\sigma }^{-})a^{†}(t)-\mathrm{H}.\mathrm{c}.]$, with transmissivity $\eta$ as indicated. (b) Entanglement versus $z$ and $\eta$ for optimized gains. (c) Entanglement versus transmissivity for optimized $z$. Nonzero entanglement can be achieved even for losses approaching 50%.

Figure 3

Mechanical squeezing $\zeta$ against cooperativity $C$ for varying mechanical bath occupation $\overline{n}=0$, $1/10$, $1/2$, $\infty$ (represented by different colors/gray levels) and sideband resolution $\kappa /{\omega }_m=1(10)$ [solid (dashed) lines]. The solid black line at $\zeta =-6\mathrm{dB}$ shows the squeezing level of the input light (corresponding to $N\approx 0.56$). The dashed vertical line shows the value of $C_{\mathrm{crit}}=1/[\sqrt{N(N+1)}-N]\approx 2.7$ above which mechanical squeezing is achievable for any $\overline{n}$.