Quantum state of interacting primordial inhomogeneities: de-squeezing and decoherence

TL;DR

The paper addresses how interactions during inflation modify the quantum state of primordial scalar perturbations. By modeling decoherence with a Lindblad equation in a Caldeira–Leggett-like setup, it shows that environmental coupling causes both purity loss and de-squeezing, which jointly suppress quantum correlations beyond mere purity reductions. Using the covariance-matrix formalism, the authors quantify changes in squeezing parameters $r_k$ and $\varphi_k$ and compute quantum discord, logarithmic negativity, and Bell-inequality violations in the late-time, de Sitter regime, uncovering universal behavior governed by the time dependence $a^{p-3}$ and identifying critical values of the parameter $p$ for entanglement and Bell violations. The results demonstrate that purity is not a reliable proxy for quantum character and provide general insights into decoherence processes affecting primordial fluctuations, with implications for interpreting quantum signatures in cosmology.

Abstract

We investigate how interactions affect the quantum state of scalar perturbations during inflation and the quantum correlations they may exhibit. Focusing on the case of scalar perturbations in single-field inflation, we model interactions using a Lindblad equation with a non-unitary contribution quadratic in the scalar perturbations, and of parametrisable amplitude and time dependence. We compute the quantum state of these interacting perturbations, which is fully described by its purity and squeezing parameters. First, we show that, in most of the parameter space, not only the purity but also the squeezing parameter is significantly reduced by interactions. Second, we show that this de-squeezing induced by the interactions, on top of the purity loss, causes a further suppression of quantum correlations. We thus emphasise that the quantum or classical character of the correlations exhibited by the perturbations cannot be correctly determined by computing the effect of interactions on the purity alone. Since the phenomenological framework adopted in this paper encompasses a wide class of possible interactions, our results provide general insights into the nature of decoherence processes in primordial fluctuations.

Figures (8)

• Figure 1: Squeezing ellipses representing the points where the Wigner function of the mode $\bm{k}, \mathrm{s}$ given in Eq. \ref{['def:Wigner_1mode']} reaches $e^{-1/2}$ times its maximum value. The yellow circle corresponds to the vacuum state for whihc $r_k= \varphi_k=0$ and $p_k=1$. The blue ellipse is a 2-mode squeezed vacuum state with $r_k=1$, $\varphi_k= \pi/4$ and $p_k=1$. The red ellipse is a 2-mode squeezed thermal state with $r_k=1$, $\varphi_k= \pi/4$ and $p_k=1/4$.
• Figure 2: Evolution of the squeezing parameters as a function of the number of $e$-folds in the absence of interaction. The red line shows the values of $\ln (r_k^{\rm w.o.})$ computed using Eq. \ref{['eq:no_deco_rk']}. The blue line shows the values of $\varphi_k^{\rm w.o.}$ computed using Eq. \ref{['eq:no_deco_phik']}.
• Figure 3: Values of the purity $p_k$ as a function of $p$ and $\log_{10}{(k_{\Gamma}/k)}$ after $N=30.0$ (left) and $N=60$ (right) $e$-folds of inflation.
• Figure 4: Evolution of the squeezing parameter $r_k$ as a function of the number of $e$-folds in the presence of interaction for $k_{\Gamma}/k = 0.1$, $p=2.1$ (left panel) and $p=9.1$ (right panel). The full red line is obtained using Eq. \ref{['eq:rk_exact']} with the covariance-matrix elements $\gamma_{ij}$ computed with Eqs. \ref{['eq:gij:exact']} where the integrals of Eq. \ref{['def:integrals_gammaij']} are numerically evaluated. We stop plotting this line around $N=7$$e$-folds when it starts to show numerical error for $p=9.1$. The full light blue line is computed using the late-time approximation \ref{['eq:latetime_rk']} of $r_k$ with the covariance-matrix elements evaluated from Eqs. \ref{['eq:latetime_gammaij']}. The yellow dotted line is computed using Eq. \ref{['eq:cases_rk']}.
• Figure 5: Values of $\delta r_k/ r_k^{\rm w.o.}=r_k/r_k^{\rm w.o.}-1$ in percentage as a function of $p$ and $\log_{10}{(k_{\Gamma}/k)}$ for $N=30.0$ (left) and $N=60$ (right) $e$-folds.