Inflationary spectra and partially decohered distributions

It is generally expected that decoherence processes will erase the quantum properties of the inflationary primordial spectra. However, given the weakness of gravitational interactions, one might end up with a distribution which is only partially decohered. Below a certain critical change, we show that the inflationary distribution retains quantum properties. We identify four of these: a squeezed spread in some direction of phase space, nonvanishing off-diagonal matrix elements, and two properties used in quantum optics called non-$P$ representability and nonseparability. The last two are necessary conditions to violate Bell’s inequalities. The critical value above which all these properties are lost is associated with the “grain“ of coherent states. The corresponding value of the entropy is equal to half the maximal (thermal) value. Moreover it coincides with the entropy of the effective distribution obtained by neglecting the decaying modes. By considering backreaction effects, we also provide an upper bound for this entropy at the onset of the radiation dominated era.

Figure 1

Decoherence of a one-mode squeezed state in the coherent state basis. Given the 2-dimensional character of this sheet of paper, it is simpler to represent the phase space of a one-mode squeezed state than that of a two-mode squeezed state. To this end, one should first decompose the two-mode squeezed states we are dealing with into two copies of one-mode squeezed states, as exposed in Appendices . The large red ellipse represents the $1-\sigma$ contour of a one-mode squeezed state in $(q_1,p_1)$ space, with $\omega =1$. The squeezed state is characterized by the squeezing parameters $r_k$ and ${\varphi }_k$, where $n_k={\mathrm{sh}}^2{r}_k$ and ${\varphi }_k=\arg (c_k)/2$. The small green circles represent the $1-\sigma$ contour of coherent states, which have no privileged direction in phase space. When replacing ${\widehat{\rho }}_{\mathrm{in}}$ by ${\widehat{\rho }}_{\min }$, two modifications obtain. First, the squeezed spread in the direction of the small axis of the ellipse is replaced by that of the vacuum. Second, and concomitantly, the macroscopic quantum coherence along the big axis, see Eq. (42), is removed and replaced by that of coherent states.

Figure 2

The inverse width ${\sigma }_v^{-2}$ of the off-diagonal matrix elements as a function of the degree of coherence of the distribution for $n={10}^1$ (lower curves, blue), $n={10}^2$ (middle curves, red), and $n={10}^3$ (upper curves, black). Left panel, as a function of $\zeta =\delta /n$, one clearly sees the extreme sensitivity of ${\sigma }_v$ with respect to $\delta$ for small values. On the right panel, the same spread as a function of $x=\arctan (\ln (\delta ))$. This further shows that, for all values of $n\gg 1$, the critical regime is centered around $\delta =1$, i.e. the “grain“ of coherent states.

Figure 3

The product of the width ${\sigma }_v^2$ with the power of the subfluctuant mode, for $n={10}^1$ (upper curve, blue), $n={10}^2$ (middle curve, red), and $n={10}^3$ (lower curve, black). For the horizontal axis, we used the nonlinear scale $x=\arctan (\ln (\delta ))$ of Fig. 2. For the vertical axis, we used the nonlinear scale $y=\arctan ({\sigma }_v^2⟨\{{\Pi }_{\mathbf{k}},{\Pi }_{-\mathbf{k}}\}⟩)$. One clearly sees that for all $n\gg 1$, the product of Eq. (D16) stays between $1/4$ and $1/2$ as long as $\delta \le 1$. One also verifies that it is asymptotically $n$ independent for all values of $\delta \ll n$. Finally we recall that the power of the decaying mode coincides, to leading order, with $⟨\{{\Pi }_{\mathbf{k}},{\Pi }_{-\mathbf{k}}\}⟩$, see Eq. (D17).

Figure 4

The loss of violation as decoherence increases. The function is normalized to its maximal classical value $=2$. The variable $x$ gives the square field amplitude. The occupation number is $n=100$, and the three values of $\delta$ are 0 (upper curve, red), the pure state, 0.05 (middle curve, blue), and 0.1 (lower curve, brown), the border regime. It is also worth to point out that the function $(n+1)\mathcal{C}(x)$ is independent of $n$ in the large $n$ limit as what is found in Figs. 3, 5, 6.

Figure 5

Entropy as a function degree of coherence for three values of $n$. The function is normalized to it maximal value. The occupation number is $n={10}^2$ (lower curves, blue), $n={10}^3$ (middle curves, red), and $n={10}^5$ (upper curves, black). On the left panel, as a function of the ratio $\zeta =|c|/(n+1/2)$, one clearly sees the sensitivity for $\zeta \to 1^{-}$. On the right panel, as a function of $x=\arctan (\ln (\delta ))$, one verifies that the critical value $S_{\max }/2$ corresponds to $\delta =1$ for all $n$.

Figure 6

Parametric curves. We plot the product ${\sigma }_v^2⟨\{{\Pi }_{\mathbf{k}},{\Pi }_{-\mathbf{k}}\}⟩$ versus the entropy. The latter (horizontal axis) has been normalized to its maximum value $S_{\max }$. The occupation number is $n={10}^2$ (upper curves, blue), $n={10}^3$ (middle curves, red), and $n={10}^5$ (lower curves, black). On the left panel, $\delta$ goes from 0 to $n$ and the product ${\sigma }_v^2⟨\{{\Pi }_{\mathbf{k}},{\Pi }_{-\mathbf{k}}\}⟩$ has been normalized to its maximum value $n$. On the right panel, we show a zoom on the region $\delta \le 1$, and the product is not normalized. One sees the two regimes. First as long as the entropy has not reached half its maximum value, the product stays nearly constant. On the contrary, for $S>S_{\max }/2$, the product explodes and reaches $n$. One can also admire the fact that the curves meet near $(1/2,1/2)$ irrespectively of the value of $n$. Moreover in the limit $n\to \infty$ the curve becomes a Heaviside function, thereby clearly separating the two regimes: the first one where quantum properties are progressively lost, and the second one where the power of the decaying mode linearly grows with $\delta$.