Light polarization measurements in tests of macrorealism

According to the world view of macrorealism, the properties of a given system exist prior to and independent of measurement, which is incompatible with quantum mechanics. Leggett and Garg put forward a practical criterion capable of identifying violations of macrorealism, and so far experiments performed on microscopic and mesoscopic systems have always agreed with quantum mechanics. However, a macrorealist can always assign the cause of such violations to the perturbation that measurements effect on such small systems, and hence a definitive test would require using noninvasive measurements, preferably on macroscopic objects, where such measurements seem more plausible. However, the generation of truly macroscopic quantum superposition states capable of violating macrorealism remains a big challenge. In this work we propose a setup that makes use of measurements on the polarization of light, a property that has been extensively manipulated both in classical and quantum contexts, hence establishing the perfect link between the microscopic and macroscopic worlds. In particular, we use Leggett-Garg inequalities and the criterion of no signaling in time to study the macrorealistic character of light polarization for different kinds of measurements, in particular with different degrees of coarse graining. Our proposal is noninvasive for coherent input states by construction. We show for states with well-defined photon number in two orthogonal polarization modes, that there always exists a way of making the measurement sufficiently coarse grained so that a violation of macrorealism becomes arbitrarily small, while sufficiently sharp measurements can always lead to a significant violation.

Figure 1

Sketch of the measurement protocol. Light pulses (following the red path) are prepared in a state with well-defined photon numbers in the $x$ and $y$ linear polarizations. Each detection port acts following the same three steps: (i) the pulses go through a beam splitter with reflectivity $r$; (ii) their polarization components with respect to some linear basis defined by an angle ${\theta }_p$ are separated with a calcite crystal or any other method of choice; and (iii) finally the photon number is measured on each polarization. Violations of macrorealism can be then checked by studying the correlations between measurements on the different ports.

Figure 2

(a) $K$ as a function of ${\theta }_2$ and ${\theta }_3$ for ${\theta }_1=\pi /2$ and for the state $|1,1\rangle$ and ${\mathcal{S}}_1$ measurements. For this type of measurements this is the only one that violates the LGI, and it does so maximally, as $K_{max}=3$. (b) $K_{\text{max}}$ as a function of $N$ and $M$ for ${\mathcal{S}}_2$ measurements. The gray area marks the region where the LGI is preserved, and it can be appreciated that the violation remains for large values of $N$ and $M$.

Figure 3

$K_{max}$ for states $|N,\sim N/6\rangle$ and $|N,N\rangle$ in (a) and (b), respectively, as a function of $N$ for different types of ${\mathcal{S}}_{\omega }$ measurements (as indicated for each line).

Figure 4

(a) $K_{\text{max}}$ for states $|5000,M\rangle$ as function of $M$ for two types of measurements, ${\mathcal{S}}_2$ and ${\mathcal{F}}_2$ (with $r=0.1$ for the last one). (b) Bhattacharyya coefficients $V_{\left(1\right)2}$ and $V_{\left(1\right)23}$ for the same states as a function of $M$ subject to ${\mathcal{S}}_2$ measurements.

Figure 5

Critical value $N_{\text{c}}$ of the photon number above which violations of LGI disappear as a function of the reflectivity $r$ for states of the type $|N,\sim N/6\rangle$ and the types of $\mathcal{B}$ measurements indicated in the figure. Notice the logarithmic scale in both axes.

Figure 6

Region of violation of the LGI in the $(N,r)$ parameter space for states of the $|N,\sim N/6\rangle$ type and ${\mathcal{F}}_4$ measurements.

Figure 7

Bhattacharyya coefficients $V_{\left(1\right)2}$ and $V_{\left(1\right)23}$ as a function of $M$ for states $|5000,M\rangle$ and ${\mathcal{F}}_2$ measurement. We set $r=0.01$.