Einstein-Podolsky-Rosen paradox and quantum steering in pulsed optomechanics

We describe how to generate an Einstein-Podolsky-Rosen (EPR) paradox between a mesoscopic mechanical oscillator and an optical pulse. We find two types of paradox, defined by whether it is the oscillator or the pulse that shows the effect Schrödinger called “steering”. Only the oscillator paradox addresses the question of mesoscopic local reality for a massive system. In that case, EPR's “elements of reality” are defined for the oscillator, and it is these elements of reality that are falsified (if quantum mechanics is complete). For this sort of paradox, we show that a thermal barrier exists, meaning that a threshold level of pulse-oscillator interaction is required for a given thermal occupation $n_0$ of the oscillator. We find there is no equivalent thermal barrier for the entanglement of the pulse with the oscillator or for the EPR paradox that addresses the local reality of the optical system. Finally, we examine the possibility of an EPR paradox between two entangled oscillators. Our work highlights the asymmetrical effect of thermal noise on quantum nonlocality.

Figure 1

Measurement of the EPR paradox between an oscillator and a pulse: (a) Entangling a pulse with a mechanical oscillator. Following HWAH, an “entangling” blue-detuned pulse interacts with an optomechanical system. The output pulse amplitudes $X_c^{\mathrm{out}}$, $P_c^{\mathrm{out}}$ are EPR correlated with the final quadratures $X_m^{\mathrm{out}}$, $P_m^{\mathrm{out}}$ of the mechanical oscillator, according to $X_c^{\mathrm{out}}\sim -P_m^{\mathrm{out}}$ and $P_c^{\mathrm{out}}\sim -X_m^{\mathrm{out}}$, in the limit of a large squeezing parameter $r$.(b) To verify the EPR paradox. The output pulse amplitudes $X_c^{\mathrm{out}}$, $P_c^{\mathrm{out}}$ are measured by homodyne detection. The quadratures of the oscillator can be measured by interacting the cavity with a second “verifying” red-detuned pulse.

Figure 2

Entanglement ${\Delta }_{g,\mathrm{ent}}$ plotted vs the squeezing parameter $r$ for $n_0=0,5,10,50$, where $n_0$ is initial occupation number of the oscillator: Entanglement between the oscillator and pulse is observed when ${\Delta }_{\mathrm{ent}}<1$. Strong entanglement occurs when ${\Delta }_{\mathrm{ent}}\to 0$. The optimal $g$ to minimize ${\Delta }_{g,\mathrm{ent}}$ is shown in the inset. The results indicate presence of entanglement, even for large $n_0$.

Figure 3

EPR paradox and quantum steering between the oscillator and the pulse: An EPR paradox and quantum steering of the oscillator by the pulse is detected when $E=E_{m|c}<1$. [The black upper set of lines intercept with the line $E=1$ at the value for $r$ given by $r_{\mathrm{epr}}$ of Eq. (14)]. An EPR paradox and the steering of the pulse by the oscillator is detected when $E=E_{c|m}<1$ (lower set of lines, blue online). The lowest two superposed curves show $E_{m|c}$ and $E_{c|m}$ with $n_0=0$ (dotted). The inset shows the optimal $g$ to minimize$E_{m|c}$.

Figure 4

Entangling two oscillators: (a) Generation of the entanglement takes place when the output of the first cavity is injected into the second cavity, as red-detuned. The final states of the two oscillators at $A$ and $B$ will be entangled. (b) The entanglement can be verified, at a later stage, using two red-detuned pulses and the homodyne scheme set-up as depicted, to measure the conditional inference variances ${\left\{{\Delta }_{\inf }{X}_{mj}\right\}}^2$ and ${\left\{{\Delta }_{\inf }{P}_{mj}\right\}}^2$.

Figure 5

Detecting an EPR paradox between two mechanical oscillators prepared using the scheme of Eqs. (16) with $r=r^{\prime }$: (a) The threshold squeeze parameter $r_{\mathrm{epr}}$ for observation of the EPR paradox $E_{m2|m1}<1$, vs the thermal occupation number of the oscillator $m2$. The curves are for $n_{m1}=0$ (solid), $n_{m1}=1$ (dashed), $n_{m1}=1.5\times {10}^6$ (dotted). (b) The threshold squeeze parameter $r_{\mathrm{epr}}$ for observation of the EPR paradox $E_{m1|m2}<1$, vs the thermal occupation number of the oscillator $m2$. The curves are for $n_{m1}=0$ (solid), $n_{m1}=1$ (dashed), $n_{m1}=1.5\times {10}^6$ (dotted). (c) The threshold squeeze parameter $r_{\mathrm{epr}}$ for observation of the EPR paradox $E_{m1|m2}<1$, vs the thermal occupation number of the oscillator $m1$. The curves are for $n_{m2}=0$ (solid), $n_{m2}=10$ (dashed), $n_{m2}=100$ (dotted).