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Violation of Bell inequality from a squeezed coherent state of inflationary perturbations

Aurindam Mondal, Rathul Nath Raveendran

TL;DR

This work examines whether primordial inflationary perturbations can exhibit quantum nonlocality detectable via Bell inequalities when the initial state is a coherent state. By formulating a GKMR pseudo-spin Bell test for the continuous-variable perturbations, the authors derive an analytical expression for the Bell operator expectation value in a squeezed coherent state and analyze its evolution in de Sitter inflation. They show that the Bell expectation value asymptotically reaches the Tsirelson bound $2\sqrt{2}$ in the super-Hubble limit, with the approach accelerated by nonzero one-point correlations from the coherent displacement. The study highlights persistent quantum features of inflationary perturbations and provides a pathway to quantify nonlocality in the early universe, though practical observational tests remain challenging due to limited access to momentum correlators.

Abstract

We investigate the quantum nature of primordial perturbations by studying the violation of Bell inequality when the initial state is taken to be a coherent state rather than the usual Bunch-Davies vacuum. As inflation progresses, the coherent state evolves into a squeezed coherent state, and we derive an analytical expression for the expectation value of the Bell operator constructed from pseudo-spin operators. Our analysis shows that although the expectation value of the Bell operator initially deviates from the vacuum case, it asymptotically saturates to the same value. Notably, this saturation occurs more rapidly for non-zero coherent state parameters, indicating that a larger one-point correlation function accelerates the approach to maximal Bell inequality violation.

Violation of Bell inequality from a squeezed coherent state of inflationary perturbations

TL;DR

This work examines whether primordial inflationary perturbations can exhibit quantum nonlocality detectable via Bell inequalities when the initial state is a coherent state. By formulating a GKMR pseudo-spin Bell test for the continuous-variable perturbations, the authors derive an analytical expression for the Bell operator expectation value in a squeezed coherent state and analyze its evolution in de Sitter inflation. They show that the Bell expectation value asymptotically reaches the Tsirelson bound in the super-Hubble limit, with the approach accelerated by nonzero one-point correlations from the coherent displacement. The study highlights persistent quantum features of inflationary perturbations and provides a pathway to quantify nonlocality in the early universe, though practical observational tests remain challenging due to limited access to momentum correlators.

Abstract

We investigate the quantum nature of primordial perturbations by studying the violation of Bell inequality when the initial state is taken to be a coherent state rather than the usual Bunch-Davies vacuum. As inflation progresses, the coherent state evolves into a squeezed coherent state, and we derive an analytical expression for the expectation value of the Bell operator constructed from pseudo-spin operators. Our analysis shows that although the expectation value of the Bell operator initially deviates from the vacuum case, it asymptotically saturates to the same value. Notably, this saturation occurs more rapidly for non-zero coherent state parameters, indicating that a larger one-point correlation function accelerates the approach to maximal Bell inequality violation.
Paper Structure (14 sections, 69 equations, 1 figure)

This paper contains 14 sections, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Inflationary cosmological perturbations
  3. Quantization of perturbations
  4. Choice of initial state
  5. Squeezed coherent state
  6. Correlation functions
  7. Bell Inequality violation
  8. Bell operator in terms of GKMR pseudo-spin operators
  9. Spin correlations
  10. Bell Inequality violation in de Sitter inflation
  11. Conclusion
  12. Wave function of squeezed coherent state
  13. Derivation of the time evolution operator
  14. Detailed computation of spin-xx correlation

Figures (1)

  • Figure 1: One-point correlation of $\hat{q}_{\mathbf{k}}, \hat{\pi}_{\mathbf{k}}$ (on the left column) and the expectation value of Bell operator (on the right column) as a function of $r_{k}$ while coherent state parameters are fixed as, $\alpha_{\mathbf{k}} = 0.001 + i\,0.001$ and $\alpha_{-\mathbf{k}} = 0.002 + i\,0.002$ (on the first row), $\alpha_{\mathbf{k}} = 0.01 + i\,0.01$ and $\alpha_{-\mathbf{k}} = 0.02 + i\,0.02$ (on the second row), $\alpha_{\mathbf{k}} = 0.1 + i\,0.1$ and $\alpha_{-\mathbf{k}} = 0.2 + i\,0.2$ (on the third row). The green curves represent the results of squeezed vacuum state i.e. $\alpha_{\mathbf{k}} = \alpha_{-\mathbf{k}} = 0$. The coherent-state parameters are chosen such that increasing values of $\alpha_{\mathbf{k}}$ correspond to increasing magnitudes of the one-point correlation function.