Bipartite temporal Bell inequality for squeezed coherent state of inflationary perturbations

TL;DR

This paper explores the use of bipartite temporal Bell inequalities to probe quantum features of primordial cosmological perturbations, finding no violation for coherent initial states but noting subtle differences from vacuum states.

Contribution

It derives an analytical expression for the bipartite temporal Bell operator expectation value in cosmology and highlights unique phase dependence absent in spatial Bell inequalities.

Findings

No temporal Bell violation for coherent initial states.

Differences between squeezed coherent and vacuum states become noticeable at large squeezing.

Temporal Bell inequality depends on an imaginary phase factor of the wave function.

Abstract

We investigate the role of the bipartite temporal Bell inequality, an analogue of the spatial Bell inequality, in probing the quantum imprints of primordial perturbations when the initially chosen Bunch-Davies vacuum is replaced by a coherent state. Although it is based on the same principles of locality and realism, its primary advantage lies in the fact that it does not require two distinct set of observable for its construction. Instead, measurements performed on a single component of the pseudo-spin operator at different times are sufficient. Consequently, it is particularly well suited for cosmological scenarios, where observational constraints typically allow access to only one component of the pseudo-spin operator. Assuming a coherent state as the initial condition, we derive an analytical expression for the expectation value of the bipartite temporal Bell operator and demonstrate the absence of temporal Bell violation in such a scenario. Interestingly, the results for squeezed coherent state is found to differ - albeit slightly - from those of squeezed vacuum state for large values of the squeezing parameter. This suggests that the ability to distinguish among different initial states of primordial perturbations does not rely on the violation of temporal Bell inequality. Furthermore, the dependence of the temporal Bell inequality on a purely imaginary phase factor of the wave function appears to be an unique feature, which is entirely absent in the context of spatial Bell inequalities.