squeezing operator can therefore be expressed as161616If a further global
phase is inserted at the right-hand side of Eq. (B.7) the modulus of $z$
remains the same but its final phase gets modified by the phase possibly
introduced in Eq. (B.7).

$\Sigma(z)=e^{z^{\ast}\widehat{a}\,\widehat{b}\,\,-z\,\,\widehat{a}^{\dagger}\,\widehat{b}^{\dagger}}=S^{(1)}(z)\,S^{(2)}(z),$
(B.8)

where the second equality follows by inserting into $\Sigma(z)$ the explicit
form of $\widehat{a}$ and $\widehat{b}$ in terms of $\widehat{a}_{1}$ and
$\widehat{a}_{2}$ as dictated by Eq. (B.7). It is also easy to verify that the
combinations of the group generators still obey the $SU(1,1)$ algebra; more
precisely we have that

$\displaystyle
L_{+}=L_{+}^{(1)}+L_{+}^{(2)}=\widehat{a}^{\dagger}\,\widehat{b}^{\dagger},\qquad
L_{-}=L_{-}^{(1)}+L_{-}^{(2)}=\widehat{a}\,\widehat{b},$ $\displaystyle
L_{0}=L_{0}^{(1)}+L_{0}^{(2)}=\frac{1}{2}[\widehat{a}^{\dagger}\widehat{a}+\widehat{b}\,\widehat{b}^{\dagger}].$
(B.9)

The combinations of the single-mode generators lead to the results of Eq.
(B.9) via the transformation (B.7); moreover, as expected, $L_{\pm}$ and
$L_{0}$ satisfy the commutation relations of the $SU(1,1)$ Lie algebra

$[L_{+},\,L_{-}]=-2L_{0},\qquad[L_{0},L_{\pm}]=\pm L_{\pm}.$ (B.10)

Thanks to Eq. (B.7) we can also verify that the two-mode Hamiltonian is the
sum of the two single-mode Hamiltonians, namely:

$\widehat{H}=\widehat{H}^{(1)}+\widehat{H}^{(2)}=2k\,L_{0}+\lambda\,L_{+}+\lambda^{\ast}L_{-}.$
(B.11)

The factorization of the two-mode squeezing operator in terms of the Baker-
Campbell-Hausdorff decomposition follows from the techniques of Refs. [120,
121] and [122, 123]:

$\Sigma(z)=e^{z^{\ast}\,\,L_{-}-z\,\,L_{+}}=e^{-\Gamma
L_{+}}e^{-2\ln{\cosh{|z|}}L_{0}}e^{\Gamma^{\ast}L_{-}},\qquad\Gamma=\frac{z}{|z|}\tanh{|z|}.$
(B.12)

As in the case of Eq. (B.5) the two-mode squeezing operator can be viewed as
the generator of a unitary transformation involving the two-mode creation and
annihilation operators:

$\widehat{c}=\Sigma^{\dagger}(z)\,\widehat{a}\,\Sigma(z)=\cosh{r}\widehat{a}\,-\,e^{i\theta}\,\sinh{r}\,\widehat{b}^{\dagger}.$
(B.13)

As stressed in Ref. [78, 79] the relation between two-mode operators and
(degenerate) single-mode operators is technically useful but it can be
physically misleading. In our problem the two modes of $\Sigma(z)$ correspond
to gravitons with opposite three-momenta.

##  Appendix C Explicit form of squeezing parameters in notable limits

The unitary transformation of Eq. (4.35) coincides with the one already
introduced in Eqs. (4.6)–(4.7) and the complex functions
$u_{k,\,\alpha}(\tau)$ and $v_{k,\,\alpha}(\tau)$ can then be directly
expressed in terms of the two phases $\delta_{k,\,\alpha}$,
$\theta_{k,\,\alpha}$ supplemented by the amplitude $r_{k,\,\alpha}$:

$u_{k,\,\alpha}(\eta)=e^{-i\,\delta_{k,\,\alpha}}\cosh{r_{k,\,\alpha}},\qquad\qquad
v_{k,\,\alpha}(\eta)=e^{i(\theta_{k,\,\alpha}+\delta_{k,\,\alpha})}\sinh{r_{k,\,\alpha}}.$
(C.1)

If the explicit evolution of $u_{k,\,\alpha}(\tau)$ and $v_{k,\,\alpha}(\tau)$
given in Eqs. (4.8) is combined with Eq. (C.1), the evolution of the squeezing
parameters in the different physical regimes can be determined. Let us first
consider, for the sake of illustration, an expanding de Sitter phase where
$\eta\to\tau$, $b(\eta)\to a(\tau)$ and ${\mathcal{F}}\to{\mathcal{H}}$.
Solving then Eq. (4.2) with the appropriate initial conditions we obtain for
$u_{k}(\tau)$ and $v_{k}(\tau)$:

$u_{k}(\tau)=\biggl{(}1-\frac{i}{2k\tau}\biggr{)}\,e^{-ik\tau},\qquad
v_{k}(\tau)=-\frac{i}{2k\tau}\,e^{-ik\tau}.$ (C.2)

According to the dictionary implicitly established by Eq. (C.1) we can easily
obtain the accurate expressions for $r_{k}$, $\theta_{k}$ and $\delta_{k}$:

$\displaystyle r_{k}$ $\displaystyle=$
$\displaystyle-\ln{(-k\tau)}+k^{2}\tau^{2}+{\mathcal{O}}(k^{4}\tau^{4}),$
(C.3) $\displaystyle\delta_{k}$ $\displaystyle=$
$\displaystyle-k\tau-\frac{\pi}{2}+\frac{8}{3}k^{3}\tau^{3}+{\mathcal{O}}(k^{5}\tau^{5}),$
(C.4) $\displaystyle\theta_{k}$ $\displaystyle=$ $\displaystyle
2k\tau+\pi-\frac{8}{3}k^{3}\tau^{3}+{\mathcal{O}}(k^{5}\tau^{5}),$ (C.5)

where the polarization index has been suppressed since the result holds for
both polarization states. The limit $|k\tau|\ll 1$ appearing in Eqs.
(C.3)–(C.4) and (C.5) gives the evolution of the squeezing parameters when the
relevant wavelengths are larger than the Hubble radius. The expression for the
squeezing parameters after the various wavelengths have reentered the Hubble
radius follows from Eqs. (4.24)–(4.25); in those formulas
$A_{k}(\tau_{ex},\tau_{re})$ and $B_{k}(\tau_{ex},\tau_{re})$ are both complex
and if we now introduce their real and imaginary parts as

$A_{k}(\tau_{ex},\tau_{re})=Q_{A}(\tau_{ex},\tau_{re})-iP_{A}(\tau_{ex},\tau_{re}),\qquad
B_{k}=Q_{B}(\tau_{ex},\tau_{re})-iP_{B}(\tau_{ex},\tau_{re}),$ (C.6)

we can obtain a simplified form of $u_{k}(\tau)$ and $v_{k}(\tau)$. We remind,
in this respect, that $\tau_{ex}$ and $\tau_{re}$ denote the times at which
each of the different wavelengths exited and reentered the Hubble radius; for
this reason both quantities ultimately depend on the modulus of the comoving
three-momentum, i.e. $\tau_{ex}=\tau_{ex}(k)$ and $\tau_{re}=\tau_{re}(k)$.
Recalling then Eq. (C.6) the explicit expressions of $u_{k}(\tau)$ and
$v_{k}(\tau)$ given in Eqs. (4.24)–(4.25) become:

$\displaystyle
u_{k}(\tau)=\frac{1}{2}\biggl{\\{}\biggl{[}(Q_{A}+P_{B})\cos{\beta}+(Q_{B}-P_{A})\sin{\beta}\biggr{]}+i\biggl{[}(Q_{B}-P_{A})\cos{\beta}-(Q_{A}+P_{B})\sin{\beta}\biggr{]}\biggr{\\}},$
(C.7) $\displaystyle
v_{k}(\tau)=\frac{1}{2}\biggl{\\{}\biggl{[}(P_{B}-Q_{A})\cos{\beta}-(Q_{B}+P_{A})\sin{\beta}\biggr{]}-i\biggl{[}(Q_{B}+P_{A})\cos{\beta}+(P_{B}-Q_{A})\sin{\beta}\biggr{]}\biggr{\\}},$
(C.8)

where, for the sake of conciseness, the arguments of the functions
$(Q_{A},\,P_{A})$ and $(Q_{B},\,P_{B})$ have been momentarily dropped. In Eqs.
(C.7)–(C.8) $\beta=k\tau$ while the explicit expressions of $(Q_{A},\,P_{A})$
and $(Q_{B},\,P_{B})$ are:

$\displaystyle Q_{A}(\tau_{ex},\,\tau_{re})$ $\displaystyle=$
$\displaystyle\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}\biggl{[}1-q_{ex}\,{\mathcal{I}}(\tau_{ex},\tau_{re})\biggr{]},\qquad
P_{A}(\tau_{ex},\,\tau_{re})=\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}\,{\mathcal{I}}(\tau_{ex},\tau_{re}),$
$\displaystyle Q_{B}(\tau_{ex},\,\tau_{re})$ $\displaystyle=$ $\displaystyle
q_{re}\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}\biggl{[}1-q_{ex}\,{\mathcal{I}}(\tau_{ex},\tau_{re})\biggr{]}-q_{ex}\,\biggl{(}\frac{a_{ex}}{a_{re}}\biggr{)},$
$\displaystyle P_{B}(\tau_{ex},\,\tau_{re})$ $\displaystyle=$ $\displaystyle
q_{re}\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}+\biggl{(}\frac{a_{ex}}{a_{re}}\biggr{)}.$
(C.9)

Three auxiliary quantities have been introduced in Eq. (C.9), namely

$q_{ex}=a_{ex}\,H_{ex}/k,\qquad
q_{re}=a_{re}\,H_{re}/k,\qquad{\mathcal{I}}(\tau_{ex},\tau_{re})=k\,\int_{\tau_{ex}}^{\tau_{re}}\biggl{[}\frac{a_{ex}}{a(\tau)}\biggr{]}^{2}\,d\tau.$
(C.10)

It is a bit lengthy (but conceptually straightforward) to verify from Eqs.
(C.7)–(C.8) and (C.9)–(C.10) that, indeed,
$|u_{k}(\tau)|^{2}-|v_{k}(\tau)|^{2}=1$. As stressed in section 4, the problem
of approximating the expressions of $u_{k}(\tau)$ and $v_{k}(\tau)$ ultimately
rests on the enforcement of unitarity. For instance in Eq. (C.9)
$(a_{ex}/a_{re})\ll 1$ in the case of expanding backgrounds. Should we
systematically neglect all the terms proportional to $(a_{ex}/a_{re})\ll 1$?
If we would do that the results of Eqs. (C.7)–(C.8) would simply imply that
$u_{k}^{2}(\tau)\simeq v_{k}^{2}(\tau)$ with an explicit violation of
unitarity. The correct approach is instead to use an approximation scheme that
is fully consistent with unitarity when we evaluate the expansion to a given
order in the small parameter of the problem; this is the rationale for the
result of Eq. (4.29) which is incidentally consistent with the strategies that
are customarily employed in quantum optics and quantum communication.
Recalling that the turning points $\tau_{re}(k)$ and $\tau_{ex}(k)$ are
determined as a solution of the equation $k^{2}\simeq
a^{2}\,H^{2}[2-\epsilon(a)]$, when a given wavelength exits the Hubble radius
during inflation we have that $q_{ex}={\mathcal{O}}(1)$ since $\epsilon\ll 1$
and the turning points of the WKB approximation are fixed, in practice, by the
condition $k^{2}=a^{2}\,H^{2}$. Conversely when the reentry takes place far
from the radiation stage the same observation can be made since
$q_{re}=a_{re}\,H_{re}/k={\mathcal{O}}(1)$. However if the reentry occurs when
$\epsilon_{re}={\mathcal{O}}(2)$ the turning point is not regular and
$q_{re}\gg 1$ in all the expressions that appear in Eqs. (C.7)–(C.8) and
(C.9).

Taking into account the observations of the previous paragraph, $u_{k}(\tau)$
and $v_{k}(\tau)$ can be expanded to a given order in $(a_{re}/a_{ex})$ which
is always small if the Universe expands; in doing so we shall keep all the
other quantities generic. As already anticipated, the result of this strategy
gives directly Eq. (4.29). We can immediately verify that, to the given order
in the perturbative expansion, Eq. (4.29) implies that
$|u_{k}(\tau)|^{2}-|v_{k}(\tau)|^{2}=1$ so that unitarity is not lost while
correctly accounting for all the leading terms of the expansion. From Eq.
(4.29) we can also obtain the expression of $\theta_{k}$:

$e^{i\theta_{k}}=\frac{u_{k}(\tau)\,v_{k}(\tau)}{|u_{k}(\tau)\,v_{k}(\tau)|}=e^{-2i\beta_{k}(\tau)}\bigg{\\{}\frac{-i+q_{re}}{i+q_{re}}+{\mathcal{O}}\biggl{[}\biggl{(}\frac{a_{re}}{a_{ex}}\biggr{)}^{2}\biggr{]}\biggr{\\}}.$
(C.11)

We now have two separate possibilities. If the given wavelength reenters in a
decelerated stage different from radiation then $q_{re}={\mathcal{O}}(1)$ and
Eq. (C.11) implies that $\theta_{k}=-2k\tau-\pi/2$. Conversely if the
wavelength reenters precisely during radiation, then $q_{re}\gg 1$ and
$\theta_{k}=-2k\tau$. With similar strategy we can also deduce that,
approximately, $e^{2\,r}\simeq(a_{re}/a_{ex})$. From the evolution equations
of $u_{k,\alpha}$ and $v_{k\,\alpha}$ we can also deduce the equations
governing the squeezing parameters.

$\displaystyle r_{p,\alpha}^{\prime}$ $\displaystyle=$
$\displaystyle-{\mathcal{H}}\cos{\theta_{p,\alpha}},\qquad\delta_{p,\alpha}^{\prime}=p-{\mathcal{H}}\sin{\theta_{p,\alpha}}\tanh{r_{p,\alpha}},$
(C.12) $\displaystyle\theta_{p,\alpha}^{\prime}$ $\displaystyle=$
$\displaystyle-2p+2\frac{{\mathcal{H}}\sin{\theta_{p,\alpha}}}{\tanh{2r_{p,\alpha}}}.$
(C.13)

Equations (C.12)–(C.13) provide useful relations among the squeezing
parameters especially in the limit where $r_{k}\gg 1$. Indeed, by combining
Eqs. (C.12)–(C.13) we get the following relation

$(\theta_{k}+2\delta_{k})^{\prime}=\frac{{\mathcal{H}}\sin{\theta_{k}}}{\sinh{r_{k}}\,\cosh{r_{k}}}.$
(C.14)

When $r_{k}\gg 1$ we have that, in practice,
$(\theta_{k}+2\delta_{k})^{\prime}=0$. But this means that

$\displaystyle\delta_{k}(\tau)=k\tau+\pi/4+\alpha,\qquad\theta=-2k\tau-\pi/2,\qquad
q_{re}={\mathcal{O}}(1).$
$\displaystyle\delta_{k}(\tau)=k\tau+\alpha,\qquad\theta=-2k\tau,\qquad
q_{re}\gg 1,$ (C.15)

where $\alpha$ is an integration constant. It is therefore an
oversimplification to regard the large squeezing limit as a classical limit.
This potentially misleading conclusion follows from the adoption of an
approximation scheme where the inaccuracies of the expansion seem to destroy
the unitarity of the evolution which is however always preserved. The same
argument holds for the quantum coherence of the system. The graviton pairs
created as a result of the quantum theory of parametric amplification are then
entangled and this essential quantum mechanical property is preserved by the
unitary evolution.

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