Violation of Smooth Observable Macroscopic Realism in a Harmonic Oscillator

We study the emergence of macrorealism in a harmonic oscillator subject to consecutive measurements of a squeezed action. We demonstrate a breakdown of dynamical realism in a wide parameter range that is maximized in a scaling limit of extreme squeezing, where it is based on measurements of smooth observables, implying that macroscopic realism is not valid in the harmonic oscillator. We propose an indirect experimental test of these predictions with entangled photons by demonstrating that local realism in a composite system implies dynamical realism in a subsystem.

Figure 1

(a) A plot of the LG expectation value as a function of the time interval $\varphi$ between measurements (in units of oscillator period), for the step parameter $s=0.99$ and the squeezing parameter $\gamma \approx 7.6$ that gives maximal violation for this $s$. The dashed red line corresponds to the bound $⟨\mathrm{LG}⟩=2$ imposed on realistic theories. (b) A plot of $\underset{\varphi }{max}⟨\mathrm{LG}⟩$ as a function of $s$ and $\gamma$ (only values greater than 2 are shown). The LGI violation is stronger for $s$ close to 1 and $\gamma$ large.

Figure 2

(a) A plot similar to Fig. 1a with data shown for several values of $s$ between $s_{\mathrm{cr}}\approx 0.983$ (the minimal $s$ allowing LGI violation) and $s=1–{10}^{-7}$ and $\gamma$ that yields the maximal LGI violation for each $s$. $s$ is increasing in order from bottom to top. The global maximum of $⟨\mathrm{LG}⟩$, $\approx 2.106$, is shown in the upper dashed purple line. (b) The limit reached by $⟨\mathrm{LG}⟩$, as $s\to 1$ and $\gamma \to \infty$, $\varphi \to 0$, as a function of the scaling variable $x={\varphi }^2/(2{\gamma }^3(1-s{)}^2)$. The maximum is obtained for $x\approx 0.24$.

Figure 3

A schematic proposal for testing the LGI indirectly with a pair of entangled photons. The photon pair emitted from the source is in an entangled state $\sum _n{c}_n|nn⟩$ with a fully correlated photon number, where $c_n$ is the $n$-photon amplitude in the squeezed vacuum state. The number-resolving photon counter in the top arm effectively performs the first measurement in the LG scheme on the photon state in the bottom arm, which then undergoes the second measurement in the bottom photon counter. Different correlation functions can be measured by tuning the squeezing strength in the second mode.