# Symplectic circuits, entanglement, and stimulated Hawking radiation in
analog gravity

Anthony J. Brady<EMAIL_ADDRESS>Department of Electrical and
Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA  Ivan
Agullo<EMAIL_ADDRESS>Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA 70803, U.S.A.  Dimitrios Kranas<EMAIL_ADDRESS>Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
LA 70803, U.S.A.

###### Abstract

We introduce a convenient set of analytical tools (the Gaussian formalism) and
diagrams (symplectic circuits) to analyze multi-mode scattering events in
analog gravity, such as pair-creation a lá Hawking by black hole and white
hole analog event horizons. The diagrams prove to be valuable ansatzes for the
scattering dynamics, especially in settings where direct analytic results are
not straightforward and one must instead rely on numerical simulations. We use
these tools to investigate entanglement generation in single- and multi-
horizon scenarios, in particular when the Hawking process is stimulated with
classical (e.g., thermal noise) and non-classical (e.g., single-mode squeezed
vacuum) input states—demonstrating, for instance, that initial squeezing can
enhance the production of entanglement and overcome the deleterious effects
that initial thermal fluctuations have on the output entanglement. To make
further contact with practical matters, we examine how attenuation degrades
quantum correlations between Hawking pairs. The techniques that we employ are
generally applicable to analog gravity setups of (Gaussian) bosonic quantum
systems, such as analog horizons produced in optical analogs and in Bose-
Einstein condensates, and should be of great utility in these domains. We show
the applicability of these techniques by putting them in action for an optical
system containing a pair white-black hole analog, extending our previous
analysis of [Phys. Rev. Lett. 128, 091301 (2022)].

###### Contents

  1. I Introduction
  2. II Gaussian states and symplectic transformation for bosonic systems of finite modes
    1. II.1 Symplectic transformations and Gaussian states
    2. II.2 Quantum entanglement
    3. II.3 Simple yet practical noise models
  3. III Example 1: Black hole
    1. III.1 Physical picture, symplectic-circuit, and example calculations
    2. III.2 Quantum correlations and entanglement degradation
  4. IV Simple recipe for circuit construction
  5. V Example 2: White-black hole pair
    1. V.1 Physical picture, symplectic circuit, and example calculations
    2. V.2 Quantum correlations
    3. V.3 Accurateness of the white-black hole circuit
  6. VI Entanglement enhancement via the stimulated effect
    1. VI.1 General remarks
    2. VI.2 Example: Seeding with single-mode squeezed vacuum
  7. VII Conclusion
  8. A Two examples of Gaussian evolution: two-mode squeezers and beam-spliters
    1. A.1 Two-mode squeezer
    2. A.2 Beam-splitter
  9. B Further comments on symplectic-circuit construction
  10. C Aspects of the entanglement witness $\Delta$ for analog horizons
  11. D Formulae

##  I Introduction

In the 1970’s, Stephen Hawking discovered that black holes are not completely
black but, instead, spontaneously emit thermal radiation as black-bodies [1,
2, 3]—now famously referred to as Hawking radiation. This result originated
from applying quantum field theory on the background space-time of a
collapsing, massive body. Since quantum mechanics of closed quantum-systems
relies on unitary evolution, as is the case here, this presents an apparent
contradiction: how can the final, outgoing quantum-state of the Hawking
radiation be thermal—and therefore a mixed (as opposed to pure) quantum
state—when the initial quantum state was pure (vacuum, in the case of
spontaneous emission)? The contradiction is resolvedI.1I.1I.1Though leads to a
true puzzle (the information-loss paradox [3]) once the evaporation of the
black hole is considered. We do not discuss this true puzzle here. by
observing one intriguing fact that follows from a more detailed analysis:
there exists a counter-part gas of blackbody radiation falling into the black
hole, which is entangled with the thermal outgoing Hawking radiation, thus
purifying the full quantum state and avoiding the apparent contradiction [4].
The quantum entanglement between the outgoing Hawking radiation and the in-
falling partner-radiation is a crucial feature of the process, highlighting
the genuine ‘quantumness’ of the effect.

The spontaneous generation of Hawking radiation (and the entangled partner-
radiation) by an event horizon—a process which we shall generally refer to as
the Hawking process or Hawking effect—was originally thought to be a
mysterious property of black holes alone, but analog-gravity systems have
changed that perspective entirely (e.g., in fluids [5, 6], optical systems [7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], and Bose-Einstein condensates [19,
20, 21, 22, 23, 24, 25, 26, 27, 28]), revealing that the formation of an
analog causal-barrier (event horizon) generically generates entangled Hawking
quanta spontaneously from the vacuum with thermal properties [29, 30, 31, 32,
33]. This has stimulated a tremendous amount of research aimed at discovering
physical systems which support such analog event-horizons, potentially leading
to detectable signatures of Hawking quanta—and their entanglement—in the lab
[34, 35, 36, 37, 38, 39, 40].

Many detailed theoretical analyses have been done, primarily focusing on the
microphysics of the Hawking process in a multitude of physically distinct
platforms. From our perspective, it would be beneficial to have a set of
universal tools and simple intuitive diagrams, which are applicable to various
microphysical settings and which, more importantly, highlights the essential
physics arising therein. Such may be especially relevant for the multi-mode
scattering scenarios that are ever-present in generic dispersive media.
Moreover, as experimental efforts strive to observe the Hawking effect and
entanglement in the lab, some theoretical handle on practical
considerations—such as, e.g., noise and system inefficiencies—would be useful
when modeling such systems. Addressing these matters, in a comprehensive yet
efficient way, is one goal of this paper.

A crucial aspect of the Hawking process—and one that we pay particular
attention to—is the entanglement generated therefrom, whether that be from the
spontaneous or stimulated Hawking effect. From our viewpoint, it is unclear
from previous works what the stimulated Hawking process implies about the
quantumness of the effect, since such is commonly regarded as having purely
classical origin. We are thus motivated to clarify what is quantum and what is
not in the stimulated Hawking process. We do so by evaluating entanglement in
a precise and operationally-meaningful manner (via the logarithmic negativity
[41, 42]), which allows us to describe in detail the harmful role that thermal
noise and ineffiencies play, as well as show how some stimulating
radiation—such as a single-mode squeezed vacuum—can actually enhance
entanglement and even help to vanquish the detrimental effects of thermal
noise.

When appropriate, we compare the logarithmic negativity to another indicator
of entanglement prolifically utilized in the analog-gravity literature (such
as a particular Cauchy-Schwarz inequality originally introduced in Refs. [25,
26]), and point out subtle but important distinctions between the two when
making physically meaningful statements about the entanglement within the
system. In particular, we discuss how the explanatory power of such Cauchy-
Schwarz inequalities are limited and can lead to false conclusions about
entanglement if they are pushed beyond their range of applicability. We
highlight this point with a few simple examples relevant to practical
realizations of the analog Hawking effect. Portions of this article contain a
detailed and extended analysis of the ideas presented succinctly in [43].

This paper is organized as follows. In Section II, we summarize the tools we
use throughout this article to deal with Gaussian states of linear systems.
The tone of this section is pedagogical. We defend the power and usefulness of
these techniques, and the visual tools associated with them, to describe the
physics behind the Hawking effect in analog systems. In Section III, we show
the utility of these tools by applying them to the Hawing effect in
gravitationally-produced, spherically symmetric black holes. Section IV
describes how to build symplectic circuits to describe the Hawking effect.
Section IV applies these tools to a second example: the Hawking effect in
optical systems containing a pair white-black hole, and uses them to analyze
in some detail the production of entanglement in this multi-mode system. This
section also includes a detailed comparison with numerical simulations, in
order to evaluate the accuracy of the proposed symplectic circuit to describe
the system. Section VI discusses ideas to amplify the generation of
entanglement in the Hawking process, in particular the use of quantum inputs
such as single-mode squeezed states. We also evaluate the way ambient noise
and losses affect the entanglement in the final state. Section VII contains an
overview of the main results in this article and discusses them from a broader
perspective. The appendices contain material of a more technical nature
helpful to support some of the claims made along this article. We work with
units in which $c=\hbar=1$.

##  II Gaussian states and symplectic transformation for bosonic systems of
finite modes

###  II.1 Symplectic transformations and Gaussian states

#### Symplectic transformations:

Our focus is on scattering processes between a finite and well-defined set of
“in” modes to a well-defined set of “out” modes, with an interaction region in
between. We presume that the interaction region is described by a Hamiltonian
that is quadratic in the basic variables, with coupling coefficients that may
otherwise be complicated functions of space and time. Since the interaction is
quadratic, the relations between the in-modes and out-modes is linear. We then
wish to provide some sort of a scattering matrix from the in-modes to the out-
modes (and vice versa by unitarity). Fortunately, there exists a general
formalism—the Gaussian formalism—that succinctly describes the evolution of a
finite number of bosonic modes under the influence of a quadratic Hamiltonian.
Gaussian states include vacua, coherent, thermal, and squeezed states and,
therefore, although the formalism is restricted to Gaussian states, it is
sufficiently general to describe most of the states one can easily create and
manipulate in the laboratory. To make this paper self-contained, we provide a
general discussion and some details surrounding the properties of such
transformations. For explicit details and extensive derivations, see Refs.
[44, 45].

To describe linear transformations, we work in the effective phase-space of
the modes. Consider a finite set of in-modes and out-modes, with annihilation
operators $\hat{a}^{\rm(in)}_{J}$ and $\hat{a}^{\rm(out)}_{J}$ with
$J\in\\{1,2,\dots,N\\}$, where $N$ is the number of relevant interacting
modes. (An example of a mode is an electromagnetic wave with frequency
$\omega$ and wavenumber $\vec{k}$.) Define the canonical operators (or
quadrature operators, as they are often called),

$\displaystyle\hat{Q}_{J}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\left(\hat{a}_{J}+\hat{a}_{J}^{\dagger}\right),$
(II.1) $\displaystyle\hat{P}_{J}$
$\displaystyle\equiv\frac{i}{\sqrt{2}}\left(\hat{a}_{J}^{\dagger}-\hat{a}_{J}\right),$
(II.2)

which we define for both the in- and out-modes and which, from the standard
commutation relations between annihilation and creation operators, obey the
canonical commutation relations
$[\hat{Q}_{J},\hat{P}_{K}]=\text{i}\delta_{JK}$. Now, define the (column)
vector of canonical operators as the direct sum of the canonical pairs for
each mode, i.e.

$\hat{\bm{R}}\equiv(\hat{Q}_{1},\hat{P}_{1},\dots,\hat{Q}_{N},\hat{P}_{N})^{\top},$
(II.3)

where the transpose is with respect to the implicitly introduced vector space,
$\mathbb{R}^{2N}$, which is the phase-space of the modes. A point that we wish
to emphasize in this section is that, in the Gaussian setting, quantum
dynamics reduces to the dynamics of vectors and matrices in this
$2N$-dimensional phase-space, without reference to the (infinite dimensional)
Hilbert space or Schrödinger equations. This is a significant simplification.

In terms of the vector $\hat{\bm{R}}$, the canonical commutation relations can
be succinctly written as

$[\hat{\bm{R}}^{i},\hat{\bm{R}}^{j}]=\text{i}\,\bm{\Omega}^{ij},\mbox{\quad\quad}\bm{\Omega}\equiv\bigoplus_{N}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},$ (II.4)

where $i,j,\cdots\in\\{1,\dots,2N\\}$ are indices running from 1 to $2N$
(contrary to capitalized latin labels $I,J,\cdots$, which we choose to run
from 1 to $N$), and we have defined the $2N\times 2N$ anti-symmetric matrix,
$\bm{\Omega}$, which is the (inverse of the) symplectic form.

Given any unitary (i.e. invertible) transformation generated by a quadratic
Hamiltonian, which takes in-modes to out-modes, one can describe such a
transformation by a $2N\times 2N$ symplectic matrix, $\bm{S}$, such that,

$\hat{\bm{R}}^{\rm(out)}=\bm{S}\cdot\hat{\bm{R}}^{\rm(in)}.$ (II.5)

One can think about this transformation as the familiar Heisenberg evolution
of the canonical operators—which is linear because we are assuming the
Hamiltonian is quadratic. The matrix $\bm{S}$ is symplectic in the sense that
it preserves the symplectic form, i.e.
$\bm{S}\bm{\Omega}\bm{S^{\top}}=\bm{\Omega}$, which is the only condition that
$\bm{S}$ must satisfy. The set of such matrices $\bm{S}$ forms the symplectic
group $Sp(\mathbb{R},2N)$; they are made of the subset of canonical
transformations that are also linear. The matrix $\bm{S}$ encodes the relevant
dynamical processes within the interaction region and it thus serves as a
scattering matrix. Appendix A describes two examples of symplectic
transformation we use extensively in this paper: two-mode squeezing and beam-
splitters.

#### Gaussian states:

We restrict our analyses to a class of quantum states known as Gaussian
quantum states (or just Gaussian states). Physically, pure Gaussian states are
states which one can generate with symplectic transformations applied to the
ground state of some quadratic Hamiltonian. More generally, acting with
symplectic transformations on a thermal state of some quadratic Hamiltonian,
one can build all mixed Gaussian states. A Gaussian state has the property
that it is completely and uniquely described in terms of its first and second
moments (defined below) of the basic variables (e.g. the $Q$’s and $P$’s);
higher-order moments can be derived from the second moments, in exactly the
same way that the statistical moments of a Gaussian probability distribution
are all determined from the first and second moments (hence the name
Gaussian).

We define the mean vector (the vector of first moments) and covariance matrix
(the matrix of second moments) for a generic quantum state as,

$\displaystyle\bm{\mu}^{i}$
$\displaystyle\equiv\expectationvalue*{\hat{\bm{R}}^{i}},$ (II.6)
$\displaystyle\bm{\sigma}^{ij}$
$\displaystyle\equiv\expectationvalue{\left\\{\hat{\bm{R}}^{i}-\bm{\mu}^{i},\hat{\bm{R}}^{j}-\bm{\mu}^{j}\right\\}},$
(II.7)

where the expectation value $\expectationvalue{\cdot}$ is taken with respect
to the quantum state under consideration (either pure or mixed), and
$\\{\cdot,\cdot\\}$ denotes the symmetric, anti-commutator. [One subtracts
$\mu_{i}$ to avoid having redundant information in the first and second
moments. The focus on the symmetric part of the second moments is because the
anti-symmetric part is fully determined by the canonical commutation relations
and is thus state independent. Therefore, the pair
$(\bm{\mu},\bm{\sigma}^{ij})$ is the minimum information needed to fully
characterize a Gaussian state.] Note that these definitions apply to any
quantum state whatsoever. It is only for Gaussian quantum states that such
quantities are sufficient in describing the quantum state in its entirety.

To give a few examples, the vacuum of a set of $N$ oscillators is
characterized by $\bm{\mu}=\bm{0},\,\bm{\sigma}=\bm{\bm{I}}_{2N}$
($\bm{I}_{2N}$ is the $2N\times 2N$ identity matrix); a coherent state by
$\bm{\mu}\neq\bm{0},\,\bm{\sigma}={\bm{I}}_{2N}$ (same covariance matrix as
vacuum, but different first moments; for this reason coherent states are
called displaced vacua); a squeezed state in general has
$\bm{\sigma}\neq{\bm{I}}_{2N}$, and a thermal state
$\bm{\mu}=\bm{0},\,\bm{\sigma}=\bigoplus_{I}(1+2\,n_{I})\,{\bm{I}}_{2}$, where
$n_{I}$ is the mean number of thermal quanta in the mode $I=1,\cdots,N$.
Thermal states are mixed states.

Many invariant properties of a Gaussian state can be extracted directly from
$\bm{\sigma}$. For instance, Heisenberg uncertainty relations are completely
and elegantly captured in the relation $\bm{\sigma}+\text{i}\bm{\Omega}\geq
0$;II.1II.1II.1More explicitly,
$(\bm{\sigma}^{ij}+\text{i}\bm{\Omega}^{ij})\bar{{\bm{v}}}_{i}{\bm{v}}_{j}$ is
a non-negative real number for all vectors $\bm{v}\in\mathbb{C}^{2N}$ (the bar
denotes complex conjugation). a Gaussian state is pure, if and only if the
eigenvalues of the matrix $\bm{\sigma}^{ik}\bm{\Omega}_{kj}$ are all equal to
$\pm\text{i}$ [44, 45]; etc.

From Eq. (II.5), given a generic set of in-moments
$(\bm{\mu}^{\rm(in)},\bm{\sigma}^{\rm(in)})$ and a symplectic transformation
$\bm{S}$, the out-moments are obtained by simple multiplication with $\bm{S}$,
as follows

$\displaystyle\bm{\mu}^{\rm(out)}$ $\displaystyle=\bm{S}\bm{\mu}^{\rm(in)},$
(II.8) $\displaystyle\bm{\sigma}^{\rm(out)}$
$\displaystyle=\bm{S}\bm{\sigma}^{\rm(in)}\bm{S}^{\top}.$ (II.9)

This is a direct consequence of the linearity of evolution. Let us assume the
initial state is a Gaussian quantum state with in-moments
$(\bm{\mu}^{\rm(in)},\bm{\sigma}^{\rm(in)})$. These quantities completely
determine the initial quantum state. Then, since symplectic transformations
map Gaussian states to Gaussian states and since the first and second moments
completely determine all properties of a Gaussian quantum state, Eqs. (II.8)
and (II.9) completely determine all properties of the out quantum state in
terms of the in quantum state. Hence, the full quantum dynamics of the system
of $N$ bosonic modes can be reduced to the dynamics of vectors and matrices in
the $2N$-dimensional phase-space, $\mathbb{R}^{2N}$. Since we are evolving
expectation values, there is no need to differentiate between Schrödinger or
Heisenberg pictures here.

Finally, let us state a simple and useful formula for the mean number of
quanta for any Gaussian state (pure or mixed) of $N$ modes with mean
$\bm{\mu}$ and covariance matrix $\bm{\sigma}$,

$\expectationvalue{\hat{n}}=\frac{1}{4}\text{Tr}\\{\bm{\sigma}\\}+\frac{1}{2}\bm{\mu}^{\top}\bm{\mu}-\frac{1}{2}N.$
(II.10)

In particular, the mean $\bm{\mu}$ and covariance matrix $\bm{\sigma}$ could
describe the reduced moments of an $N$-dimensional subsystem of an $N+M$
Gaussian quantum state. (See Appendix A for additional details.)

###  II.2 Quantum entanglement

An essential feature of the Hawking process is the quantum correlations
(entanglement) generated therefrom. For pure quantum states, the most popular
entanglement-quantifier is the von Neumann entropy. However, for mixed quantum
states (as generic sub-systems of a pure quantum state), the von Neumann
entropy for a bi-partition no longer captures quantum correlations alone.
There exists several measures, witnesses, and quantifiers of entanglement,
with increasing levels of technicality and computational difficulty. We
consider a few such quantifiers and witnesses of entanglement here. See
Appendix A for some example applications.

#### Logarithmic Negativity

One easily-computable measure of entanglement for pure states and mixed states
alike is the logarithmic negativity (LN) [41, 42]. A non-zero value of the LN
is in one-to-one correspondence with the violation of the Positivity of
Partial Transpose (PPT) criterion for quantum states [42]—a criterion that
separable quantum states faithfully obey. In simpler words, a non-zero value
of LN implies the existence of entanglement (or non-separability), however,
the reverse is not generically true—some entangled states have zero LN. On the
other hand, when restricting to Gaussian states and when one of the subsystems
is made of a single mode ($N_{A}=1$), regardless of the size of the other
subsystem, the LN is different from zero if and only if the state is
entangled. Importantly, the LN is an entanglement monotone; it is a faithful
quantifier of entanglement, in the sense that higher LN means more
entanglement. Later in this paper, we will use the LN to quantify entanglement
precisely.

For Gaussian quantum states (and other types of quantum states), the value of
the LN has an operational meaning as the exact cost (where the currency is
Bell pairs or entangled bits, ebits) that is required to prepare or simulate
the quantum state under consideration [46, 47]. Hence, if a quantum state
$\rho$ has a larger LN value than a quantum state $\tau$, then one can
quantitatively and confidently claim that $\rho$ possesses more entanglement
than $\tau$, as it is more expensive to simulate $\rho$ than it is to simulate
$\tau$. This is a useful fact. For consider that $\rho$ and $\tau$ differ by a
continuous change in a parameter $\vartheta$, such that $\rho(\vartheta)$ and
$\tau=\rho(\vartheta+\delta\vartheta)$. Then, one can make definitive
statements about the role that $\vartheta$ plays in entanglement by monitoring
the changes it induces in the LN. For instance, if the LN monotonically
increases with the parameter $\vartheta$, then this parameter quantifies a
valuable resource (such as squeezing). On the other hand, if the LN
monotonically decreases with $\vartheta$, then this parameter is obviously a
nuisance (such as loss and thermal noise) and should be minimized as much as
possible.

The LN for a Gaussian quantum state can be directly computed from its
covariance matrix (i.e., the LN does not dependent on the first moments
$\bm{\mu}$). Consider an $N+M$-mode Gaussian quantum state $\rho_{AB}$ with
covariance matrix $\bm{\sigma}_{AB}$, where the sub-system $A$ consists of $N$
modes and $B$ consists of $M$ modes. The LN for the bi-partition can be
computed from the symplectic eigenvaluesII.2II.2II.2The symplectic eigenvalue
of a covariance matrix $\bm{\sigma}$, are defined as the absolute value of the
eigenvalues of the matrix $\bm{\sigma}^{ik}\bm{\Omega}_{kj}$. The properties
of $\bm{\sigma}$ guarantee that the $2N$ eigenvalues of this matrix are purely
imaginary and they come in pairs, $\pm\text{i}\nu_{I}$, with $I=1,\cdots,N$
and $\nu_{I}$ real for all $I$. Hence, by taking the absolute values, we are
left with $N$ distinct symplectic eigenvalues equal to $\nu_{I}$. of the
covariance matrix $\widetilde{\bm{\sigma}}_{AB}$ of the partially transposed
quantum state $\rho_{AB}^{\top_{B}}$, which can be written in terms of
$\bm{\sigma}_{AB}$ as

$\widetilde{\bm{\sigma}}_{AB}=\bm{T}\bm{\sigma}_{AB}\bm{T},$ (II.11)

where $\bm{T}=\bm{I}_{2N}\oplus\bm{\Sigma}_{M}$ and
$\bm{\Sigma}_{M}=\oplus_{M}\sigma_{z}$ is a direct sum of $M$ $2\times 2$
Pauli-z matrices; see, for instance, Ch. 7.1 of Ref. [45] for details, and
Appendix A for concrete calculations in a simple situation. Let
$\\{\tilde{\nu}_{j}\\}_{J=1}^{M+N}$ denote the symplectic eigenvalues of
$\widetilde{\bm{\sigma}}_{AB}$. The LN for the Gaussian quantum state
$\rho_{AB}$ is then given as [44, 45],

$LN(\rho_{AB})=\sum_{J=1}^{M+N}\max\left[0,-\log_{2}(\tilde{\nu}_{J})\right].$
(II.12)

Observe that a sufficient condition for quantum entanglement is
$\min\\{\tilde{\nu}_{J}\\}<1$. For instance, we use this condition later on to
evaluate precise, yet physically-motivated entanglement-criteria based on
relationships between the Hawking temperature of an analog-gravity system and
the temperature of initial thermal fluctuations.

#### A Cauchy-Schwarz inequality:

In the analog gravity literature, there has been much focus on a particular
Cauchy-Schwarz inequality (to be defined precisely below) to evaluate
entanglement, which was first introduced in Refs. [25, 26] in the context of
BECs; see, e.g., Refs. [48, 49, 37] for further usage. We describe such below.

Consider two modes, labelled $\hat{a}$ and $\hat{b}$, occupying the two-mode
quantum state $\rho_{AB}$. Define the mean occupation numbers
$n_{a}=\expectationvalue*{\hat{a}^{\dagger}\hat{a}}$ and
$n_{b}=\expectationvalue*{\hat{b}^{\dagger}\hat{b}}$ and the parameter
$c_{ab}\equiv\expectationvalue*{\hat{a}\hat{b}}$, where the expectation is
taken with respect to the state $\rho_{AB}$. It was shown in Refs. [25, 26]
that the following Cauchy-Schwarz (CS) inequality,

$\Delta\equiv n_{a}n_{b}-\absolutevalue{c_{ab}}^{2}<0,$ (II.13)

is a sufficient condition for the two modes, $\hat{a}$ and $\hat{b}$, to be
quantum mechanically entangled in the state under consideration.

The inequality for $\Delta$ above is simple and can be evaluated by measuring
only a handful of observables, leading to an approachable method for
witnessing entanglement in the lab. However, it is important to be aware of
its limitations. The witness $\Delta$ does not quantify entanglement. In other
words, a larger violation of the inequality (a more negative $\Delta$) does
not imply more entanglement is present within the system. Moreover, the
inequality is only sufficient to witness entanglement—not sufficient and
necessary; this implies that for some entangled states we may find $\Delta>0$.
In Ref. [26], the inequality was shown to be necessary and sufficient for a
certain class of two-mode quantum states (so-called ‘stationary states’; cf.
Refs. [25, 26]), though this does not generally hold. Hence, even when
restricting to Gaussian states and to situations when one subsystem is made of
a single mode (conditions under which the LN is a necessary and sufficient
condition for entanglement), a non-negative value of $\Delta$ does not
guarantee the absence of entanglement (see Appendix A for an explicit
example); $\Delta$ is faithful only when the additional condition of
‘stationarity’, defined in [25, 26], is met.

As pointed out in Appendix A of Ref. [26], the inequality $\Delta<0$ is
actually subsumed by a more generic inequality based on the PPT criterion
[50]. Consider the following covariance matrix for a two-mode Gaussian state,
$\rho_{AB}$,

$\bm{\sigma}_{AB}=\begin{pmatrix}\bm{A}&\bm{C}\\\
\bm{C}^{\top}&\bm{B}\end{pmatrix},$ (II.14)

where $\bm{A}$ ($\bm{B}$) is the covariance matrix for the sub-system $A$
($B$), which we assume to be made of a single mode, and $\bm{C}$ describes the
correlations among them. Now define the PPT parameter,II.3II.3II.3Our
definition here differs from Refs. [50, 26] by a numerical factor of 1/16 due
to our definition of the covariance matrix, Eq. (II.7).

$\mathcal{P}_{-}\equiv\det\bm{A}\det\bm{B}+\left(1-\absolutevalue{\det\bm{C}}\right)^{2}\\\
-\Tr{\bm{A}\bm{\Omega}_{1}\bm{C}\bm{\Omega}_{1}\bm{B}\bm{\Omega}_{1}\bm{C}^{\top}\bm{\Omega}_{1}}-\det\bm{A}-\det\bm{B},$
(II.15)

where $\bm{\Omega}_{1}=\big{(}\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\big{)}$ is the single-mode symplectic form. Actually,
$\Delta$ happens to be equal to the first line of the last equation, namely

$\Delta=\det\bm{A}\det\bm{B}+\left(1-\absolutevalue{\det\bm{C}}\right)^{2}\,.$
(II.16)

A necessary and sufficient condition for a Gaussian two-mode quantum state to
be entangled is $\mathcal{P}_{-}<0$, which is derived from the PPT criterion
as applied to Gaussian states; see Ref. [50] for explicit details. Note also
that $\mathcal{P}_{-}$ is a binary indicator of entanglement, not a
quantifier. The drawback to the more general quantity $\mathcal{P}_{-}$, as
opposed to $\Delta$, is that computing the former requires access to the
entire covariance matrix of the quantum state. For Gaussian states, this is
equivalent to performing quantum-state tomography on the two-mode system (see,
e.g., Ref. [51] for details about quantum-state tomography of optical
systems). However, as shown in Ref. [26], for a certain class of Gaussian
quantum states (‘stationary states’), ${\mathcal{P}_{-}<0\iff\Delta<0}$.
Hence, for this sub-family of states, the simpler quantity $\Delta$ is as
useful as $\mathcal{P}_{-}$. [Again, we insist that neither is a quantifier of
entanglement; see Appendix A.]

#### Remarks:

The LN precisely quantifies the amount of entanglement within a system under
the circumstances described above (Gaussian states and when one of the
subsystems is made of a single mode). However, computing it in practice
requires knowledge of the entire covariance matrix, which amounts to knowing
the quantum state of said system (i.e. via quantum-state tomography [51]).
Entanglement witnesses, such as $\Delta$, on the other hand, can relieve the
burden of full quantum-state tomography, as such are typically computed by
measuring only a few observables—often appearing in the form of Cauchy-Schwarz
inequalities. The trade-off, however, is that entanglement witnesses have a
binary outcome, telling us only if a state is entangled or not (they do not
quantify entanglement), and they are not necessarily always faithful.

In this paper, we utilize the LN to genuinely quantify entanglement and
analyze how certain parameters (e.g., thermal noise, inefficiencies) affect
quantum entanglement within a system of modes. We briefly consider
entanglement witnesses discussed in previous paragraphs, due to their
practical appeal, and point out discrepancies between such and the LN.

###  II.3 Simple yet practical noise models

Since there is currently much experimental effort to observe Hawking radiation
in the lab with analog systems, it seems appropriate to include potential
imperfections in a hypothetical experimental setup which may distort or
degrade certain features of the Hawking effect. In this paper, we focus on two
potential sources of noise:

  * •

Background temperature (i.e. initial thermal fluctuations), $T_{\rm env}$

  * •

Attenuation or losses, $\eta$

where $T_{\rm env}$ is the temperature of the ‘environment’ and $0\leq
1-\eta\leq 1$ can represent the probability that a mode scatters into some set
of inaccessible modes (attenuation); or $\eta$ can be the efficiency to which
one can couple outgoing modes to a detector setup.

The former is relevant since, generally, the Hawking temperature of analog-
gravity systems is meager, and thus, there will be a competition between the
quantum fluctuations produced in the Hawking effect versus the initial thermal
fluctuations (more on this in Section III.2).

We include the effects of a background temperature within our model by simply
taking the initial state be in thermal equilibrium at temperature $T_{\rm
env}$. In the Gaussian context, this means that the initial first and second
moments of each mode, prior to any interactions or operations, are
$\bm{\mu}_{\Theta}=\bm{0}$ and $\bm{\sigma}_{\Theta}=(1+2n_{{\rm
env}})\bm{I}_{2}$ (and different modes are uncorrelated) where $n_{{\rm env}}$
is the mean occupation-number of a harmonic oscillator (e.g. a mode of a
photon gas) at temperature $T_{\rm env}$. Generally, each mode can be of
different frequency, as is the case for analog models with Bose-Einstein
condensates [52], and for them $n_{{\rm env}}\equiv 1/(\text{e}^{\omega/T_{\rm
env}}-1)$ with $\omega$ the frequency of the mode. It is straightforward to
generalize to noise which may have different spectral profiles than simple
black body-like occupation numbers.

A second source of noise, which plagues all experimental platforms, is
unwanted scattering of the system modes into a set of inaccessible modes
(i.e., attenuation), which could arise from, e.g., inefficiencies within the
setup. This leads to a loss of quanta, which in turn reduces quantum
correlations between the outgoing-modes of the system and, as we show below,
it may cause serious issues if the goal is to measure entanglement generated
during the Hawking process. We provide a simple model for such in the form of
a homogeneous, thermal-loss channel (also known as an attenuation map),
$\mathcal{L}_{\eta}^{N}$, which preserves the Gaussianity of the state, and
maps the mean $\bm{\mu}$ and covariance matrix $\bm{\sigma}$ of an $N$-mode
system to

$\displaystyle\bm{\mu}\overset{\mathcal{L}_{\eta}^{N}}{\longrightarrow}\,$
$\displaystyle\sqrt{\eta}\,\bm{\mu}$ (II.17)
$\displaystyle\bm{\sigma}\overset{\mathcal{L}_{\eta}^{N}}{\longrightarrow}\,$
$\displaystyle\eta\,\bm{\sigma}+N_{\rm noise}\,(1-\eta)\bm{I}_{2N},$ (II.18)

where $1-\eta$ is the attenuation parameter quantifying the inefficiencies
(the probability to scatter into an inaccessible channel; $\eta=1$ is no
attenuation). The presence of the parameter $N_{\rm noise}\geq 1$ allows for
the possibility that the the inaccessible scattering channel is not merely a
pure losses channel, but it also injects noise to the system. This model
assumes that the inaccessible modes, to which the system modes scatter into,
act homogeneously on the system and are uncorrelated. These assumptions allow
us to discuss generic features of unwanted scattering processes in simple
terms and to evaluate the impact of losses on quantum aspects of the Hawking
process. As an aside, our analyses can be extended by introducing a more
elaborate model for the open-system dynamics, perhaps leveraging the input-
output formalism for open systems of bosonic modes (see for instance Ch. 6 of
Ref. [45]).

##  III Example 1: Black hole

In this section, we apply the formalism presented in Section II to the (usual)
Hawking process triggered by the formation of a black hole by the
gravitational collapse of a spherically symmetric body, and introduce a simple
symplectic circuit associated with this process. The motivation is to
illustrate the use of these tools in a simple context.

Figure 1: Conventional illustrations of the Hawking process. (Left) Space-time
diagram of a black hole formed by gravitational collapse depicting ingoing
modes, interior modes, and outgoing Hawking radiation. (Right) A close-up of
the different scattering events. The origin-story of the long-wavelength
Hawking radiation in the asymptotic future typically follows by tracing the
evolution of the outgoing Hawking radiation ‘back in time’ along geometric
rays to the asymptotic past. Doing so, one finds that the Hawking radiation
has contributions from particle-pair creation at the event horizon (Hawking
effect) as well as contributions from a classical scattering process (back-
scattering) at a gravitational potential barrier. In absence of back-
scattering, the outgoing Hawking radiation has a blackbody spectrum due to the
Hawking effect. However back-scattering modulates the blackbody spectrum by a
grey-body factor, which is just the probability to transmit through the
barrier.

###  III.1 Physical picture, symplectic-circuit, and example calculations

#### Physical picture:

Imagine a scattering experiment consisting of a well-defined and finite set of
in-modes and out-modes, with an interaction region in between that is well-
described by a quadratic interaction Hamiltonian, with coupling coefficients
that are dependent on space and time. The interaction then leads to a linear
transformation between the in- and out-modes, as in Section II.

One usually derives the astrophysical Hawking effect in precisely this
context, in which case there is a free (bosonic) quantum field, described by a
quadratic Hamiltonian, evolving on the spacetime-dependent (classical) metric
of a collapsing body. The relevant in-modes are the modes which define the
standard vacuum in the asymptotic in-region, at past infinity (past null
infinity for massless fields, which are the type of fields that dominate the
Hawking effect [53]), while the out-modes consist of the Hawking radiation
emitted by the black hole, which make it to the asymptotic out-region at
future infinity, asymptotically far from the black hole, together with the
‘Hawking-partner’ modes, as well as back-scattered modes, which fall through
the event horizon and into the black hole. The conventional story is
illustrated in Fig. 1.

Though there is generically an infinite number of modes to deal with in a
quantum field theory in curved space-time, the Hawking process can be reduced
to interactions between only a handful of modes, as heuristically alluded to
above and illustrated in Figs. 1 and 2, such that one can directly apply the
Gaussian formalism of Section II.1 to the process. This can be done by using
the “Wald’s basis” [54] (see also [55]). We sketch the essential ingredients
here and provide a reconstructive argument starting with Hawking’s final
result [2] and working backwards (see [56, 55, 57] for detailed derivations).
We start by ignoring back-scattering, which we introduce later. The arguments
below apply to any field that is emitted in the Hawking process, but for
concreteness, we focus on massless fields (e.g., electromagnetic radiation).

Let $\mathscr{H}_{\rm out}$ be the Hilbert space of the field at future
infinity. The Hawking process leads to a quantum state in this Hilbert space
corresponding to an outgoing flux of uncorrelated blackbody-radiation,
beginning with only vacuum fluctuations in the asymptotic past [1, 2]. Such
state has a density matrix of the form

$\hat{\Theta}_{\rm out}=\bigotimes_{\omega}\hat{\Theta}_{\rm out,{\omega}},$
(III.1)

where

$\hat{\Theta}_{\rm
out,{\omega}}=\frac{1}{1+n_{\omega}}\sum_{N=1}^{\infty}\left(\frac{n_{\omega}}{1+n_{\omega}}\right)^{N}\outerproduct{N}{N}_{\rm
out,{\omega}},$ (III.2)

and $n_{\omega}\equiv 1/(e^{\omega/T_{H}}-1)$ is the Bose-Einstein
distribution characterized by the Hawking temperature $T_{H}$, which is
frequency independent. Here, $\omega$ labels positive frequency modes—of which
there are infinitely many—and, hence, $\ket{N}_{\rm out,{\omega}}$ is a Fock
state of $N$ quanta in the out frequency-mode $\omega$ reaching future null
infinity. The quantity $n_{\omega}$ can be precisely interpreted as the
(average) number of quanta emitted by the black hole per frequency per time.

In words, Eq. (III.1) asserts that the frequency modes in the outgoing
Hawking-flux are completely uncorrelated among each other, while Eq. (III.2)
asserts that, for an individual frequency, the quantum state
$\hat{\Theta}_{\rm out,\omega}$ is completely mixed, in the sense that its
photon-statisticsIII.1III.1III.1By photon-statistics, we are referring to the
statistical fact that $N$ is a discrete random-variable, governed by the
probability distribution
$\frac{1}{1+n_{\omega}}\left(\frac{n_{\omega}}{1+n_{\omega}}\right)^{N}$,
which is often referred to as a completely mixed (or ‘thermal’) distribution
in the quantum optics literature for any form of $n_{\omega}$. is dictated by
the probability distribution given in Eq. (III.2). Moreover, since
$n_{\omega}$ follows a Bose-Einstein distribution, the entire “out” state,
$\hat{\Theta}_{\rm out}$, is precisely that of an uncorrelated quantum-gas of
black-body radiation, characterized by a single number—the Hawking
temperature, $T_{H}$.

The state $\hat{\Theta}_{\rm out}$ is obviously a mixed state, but this is
only because we are restricting ourselves to the “out” Hilbert space
$\mathscr{H}_{\rm out}$, whereas the entire Hilbert space is $\mathscr{H}_{\rm
int}\otimes\mathscr{H}_{\rm out}$, with $\mathscr{H}_{\rm int}$ the Hilbert
space of modes falling into the balck hole horizon. By introducing the “int”
degrees of freedom, we can purify the out-state, such that $\hat{\Theta}_{\rm
out}\rightarrow\hat{\Psi}_{\rm int,out}$ and $\hat{\Psi}_{\rm
int,out}\in\mathscr{H}_{\rm int}\otimes\mathscr{H}_{\rm out}$ is the pure
quantum state which arises from evolving the in-vacuum. The correct purifying
modes where discovered in Ref. [54]. Their explicit form is not important for
the arguments written below, and they can be found in Refs. [54, 56, 55, 57].
We will denote the creation operators defined from them (or, as usual, by
wave-packets built from them [2, 54]) by $\hat{a}_{\omega}^{({\rm int})}$,
labeled by the out-frequency $\omega$ (note that the int-modes in Wald’s basis
do not have well defined frequency with respect to any natural notion of time
translation at the horizon). The total state in $\mathscr{H}_{\rm
int}\otimes\mathscr{H}_{\rm out}$ is the simplest purification
$\hat{\Psi}_{\rm int,out}$ of the out-state $\hat{\Theta}_{\rm out}$, namely

$\hat{\Psi}_{\rm int,out}=\bigotimes_{\omega}\hat{\Psi}_{\rm int,out;\omega},$
(III.3)

where

$\hat{\Psi}_{\rm
int,out;\omega}=\frac{1}{\sqrt{1+n_{\omega}}}\sum_{N=1}^{\infty}\left(\frac{n_{\omega}}{1+n_{\omega}}\right)^{N/2}\ket{N,N}_{\rm
int,out;\omega},$ (III.4)

and $\\{\ket{N}_{\rm int}\\}$ is the number state associated with the number
operator defined from $\hat{a}_{\omega}^{({\rm int})}$. This is a ‘thermo-
field double’ state [4] (although we need to keep in mind that at the horizon
there is no natural notion of time which allows us to physically interpret the
interior state as black body radiation). Using Eqs. (III.3) and (III.4), it is
easy to check that $\hat{\Theta}_{\rm out}=\Tr_{\rm int}(\hat{\Psi}_{\rm
int,out})$, justifying that $\hat{\Psi}_{\rm int,out}$ is indeed a
purification of $\hat{\Theta}_{\rm out}$. In words, Eqs. (III.1)-(III.4) tell
us that there exists two quantum-gases of blackbody radiation: the Hawking
radiation escaping to spatial infinity and another gas falling into the black
hole. Moreover, these quantum gases are entangled, per Eqs. (III.3) and
(III.4).

The state $\hat{\Psi}_{\rm int,out}$ is dynamically related to the “in”
vacuum, $\ket{\rm vac}_{\rm in}\in\mathscr{H}_{\rm in}$. There exists a basis
of in-modes [54] such that this dynamical relation is just a direct-product of
two-mode transformations (one for each $\omega$). We denote Wald’s in-modes by
$\hat{a}_{\omega;1}^{\rm(in)}$ and $\hat{a}_{\omega;2}^{\rm(in)}$. They are
related with the int- and out-modes via [54]

$\displaystyle\hat{a}_{\omega}^{\rm(out)}$
$\displaystyle=\frac{1}{\sqrt{1-\text{e}^{-\omega/T_{H}}}}\left(\hat{a}_{\omega;1}^{\rm(in)}+\text{e}^{-\omega/2T_{H}}\hat{a}_{\omega;2}^{\rm(in)\dagger}\right)$
(III.5) $\displaystyle\hat{a}_{\omega}^{\rm(int)}$
$\displaystyle=\frac{1}{\sqrt{1-\text{e}^{-\omega/T_{H}}}}\left(\hat{a}_{\omega;2}^{\rm(in)}+\text{e}^{-\omega/2T_{H}}\hat{a}_{\omega;1}^{\rm(in)\dagger}\right).$
(III.6)

Wald’s in modes are labeled by the out-frequency $\omega$ that they are the
progenitors of. They do not have well defined in-frequency but, rather, are
made by combining positive frequency in-modes, without contribution from
negative-frequency in-modes. This guarantees that Wald’s basis is an explicit
basis to define the unique in-vacuum—i.e., ${\hat{a}_{\omega;k}^{(\rm
in)}\ket{\rm vac}_{\rm in}=0}$, $k=1,2$. The advantage of Wald’s basis is that
it makes the dynamical relation between in- and int-out-modes a direct-product
of two-mode transformations (one for each $\omega$). We can therefore focus on
one $\omega$ at a time and think of the Hawking-process as just a two-mode
interaction mapping two “in” modes to the “out” and “int” modes. The full
interaction is then a direct-product over all the modes $\omega$.

Back-scattering can then be introduced by noticing that the out-modes do not
actually propagate to infinity, since they are partially scattered back by the
gravitational potential barrier surrounding the black hole. Only a portion
$\Gamma_{\ell,\omega}$ of the initial out-mode makes it to infinity, while the
rest falls back into the black hole.

#### Symplectic circuit:

We now show how the previous results can be described in an exact and
efficient manner using the Gaussian formalism summarized in Section II. The
reformulation is simple, and allows us to extend the calculation to any
initial Gaussian state, and to compute all aspects of the final state in an
efficient manner, including entanglement between any subsystem of modes. The
reformulation comes together with useful diagrams, which we refer to as
symplectic circuits, which help to derive non-trivial qualitative information
about the Hawking process. The name ‘symplectic circuit’ is motivated by the
fact that the diagrams depict the dynamical evolution by a concatenation of
elementary symplectic transformations (squeezers, beam-splitters and phase
shifters). They describe both the classical and the quantum evolution.

For the Hawking effect, we notice that Eqs. (III.5) above, which constitute
the core of the Hawking effect, correspond precisely to a process of two-mode
squeezing (see Appendix A), while the process of back scattering is described
by a beam-splitter (see below). The physics of the Hawking process can be
described diagrammatically by the symplectic circuit depicted in Fig. 2, which
consists of a discrete concatenation of physical processes in a concrete
manner. The circuit is simple: First, the in-modes $a_{\omega;1}^{\rm(in)}$
and $a_{\omega;2}^{\rm(in)}$ pass through a two-mode squeezer, which we label
as $\bm{S}_{\rm H}$ (H stands for Hawking), leading to particle-pair creation.
Formally, this process corresponds to the two-mode symplectic squeezing
transformation, written in $2\times 2$ blocks as

$\bm{S}_{H}=\begin{pmatrix}\cosh r_{H}\,\bm{I}_{2}&\sinh
r_{H}\,\bm{\sigma}_{z}\\\ \sinh r_{H}\,\bm{\sigma}_{z}&\cosh
r_{H}\,\bm{I}_{2}\end{pmatrix},$ (III.7)

which acts on the canonical operators as described in Section II.1 (note that
this matrix is the particularization of Eq. (A.2) to $r=r_{H}$ and $\phi=0$).
Here, $\bm{\sigma}_{z}$ is the standard Pauli-z matrix, and $r_{H}$ are
frequency dependent real numbers given by
$\tanh^{2}r_{H}\equiv\text{e}^{-\omega/T_{H}}$, where $T_{H}$ is the Hawking
temperature of the system. The expression for $r_{H}$ accounts for the black-
body spectrum at the Hawking temperature $T_{H}$ of the outgoing radiation.

Figure 2: Symplectic-circuit of the Hawking process. Three in-modes
—$\hat{a}_{\omega;1}^{\rm(in)}$, $\hat{a}_{\omega;2}^{\rm(in)}$, and a back-
scattering mode $\hat{b}_{\omega}^{\rm(in)}$—scatter to three out-modes. The
outgoing Hawking radiation occupies the mode $\hat{a}_{\omega;2}^{\rm(in)}$.
The other two modes, $\hat{a}_{\omega}^{\rm(int)}$ and
$\hat{b}_{\omega}^{\rm(int)}$, propagate into the interior of the black hole.
The former is the Hawking-partner mode while the latter, the back-scattered
mode, carrying Hawking radiation that was reflected back into the black hole
by a potential barrier.

Following pair-creation in the near horizon region, one mode of the pair, the
one described by the int-mode $a_{\omega;1}^{\rm(int)}$ (the Hawking-partner
mode), falls into the black hole, while the other outgoing mode
$a_{\omega;2}^{\rm(out)}$ (the long wavelength, outgoing Hawking radiation)
must climb the potential barrier on its way to infinity. This process of back-
scattering is described completely by a simple process in which two-modes
scatter to two-modes. The incoming modes are the mode
$a_{\omega;2}^{\rm(out)}$ produced in the near horizon region (this mode, is
in many references, called the “up” mode; see e.g. [55, 57]) and one long-
wavelength incoming mode from past infinity, which we will denote by the
annihilation operator $\hat{b}^{\rm(in)}_{\omega}$. The result of the
scattering process are the out-modes which make it to the asymptotic out-
region as Hawking radiation, together with long-wavelength modes back-
scattered into the black hole, which we will denote as
$\hat{b}_{\omega}^{\rm(int)}$ (see Fig. 1 for an illustration). Formally, the
back-scattering process corresponds to a beam-splitter, described by an
orthogonal-symplectic transformation $\bm{O}_{\theta}$, written in $2\times 2$
blocks as,

$\bm{O}_{\theta}=\begin{pmatrix}\cos\theta_{\ell,\omega}\,\bm{I}_{2}&\sin\theta_{\ell,\omega}\,\bm{I}_{2}\\\
-\sin\theta_{\ell,\omega}\,\bm{I}_{2}&\cos\theta_{\ell,\omega}\,\bm{I}_{2}\end{pmatrix},$
(III.8)

where the angle $\theta_{\ell,\omega}$ encodes the grey-body factor
$\Gamma_{\ell,\omega}$ through the transmission probability
$\cos^{2}\theta_{\ell,\omega}=\Gamma_{\ell,\omega}$. We represent this
classical scattering in the circuit by a beam-splitter element [to simplify
our notation we will omit the labels $\ell$ (the angular quantum number) and
$\omega$ in $\theta$ and $\Gamma$]. At the level of annihilation operators,
the beam-splitter imposes the following transformation (see Appendix A for
further details)

$\displaystyle\hat{a}_{\omega;2}^{\rm(out)}$
$\displaystyle\longrightarrow\cos\theta_{\omega}\,\hat{a}_{\omega;2}^{\rm(out)}+\sin\theta_{\omega}\,\hat{b}_{\omega}^{\rm(in)}\,,$
(III.9) $\displaystyle\hat{b}_{\omega}^{\rm(in)}$
$\displaystyle\longrightarrow\cos\theta_{\omega}\,\hat{b}_{\omega}^{\rm(in)}-\sin\theta_{\omega}\,\hat{a}_{\omega;2}^{\rm(out)}\,.$
(III.10)

Observe that this process does not create particles, easily seen by the fact
that there are no creation operators present in the output modes.

We now combine these elements to write a scattering matrix corresponding to
the circuit of Fig. 2. Let us choose the following order for the vector of
canonical operators

$\displaystyle\hat{\bm{R}}^{\rm(in)}$ $\displaystyle\equiv$
$\displaystyle\left(\hat{Q}_{\omega;1}^{\rm(in)},\hat{P}_{\omega;1}^{\rm(in)},\hat{Q}_{\omega;b}^{\rm(in)},\hat{P}_{\omega;b}^{\rm(in)},\hat{Q}_{\omega;2}^{\rm(in)},\hat{P}_{\omega;2}^{\rm(in)}\right)^{\top},$
$\displaystyle\hat{\bm{R}}^{\rm(out)}$ $\displaystyle\equiv$
$\displaystyle\left(\hat{Q}_{\omega;1}^{\rm(int)},\hat{P}_{\omega;1}^{\rm(int)},\hat{Q}_{\omega;b}^{\rm(int)},\hat{P}_{\omega;b}^{\rm(int)},\hat{Q}_{\omega;2}^{\rm(out)},\hat{P}_{\omega;2}^{\rm(out)}\right)^{\top},$

where, e.g.,
$\hat{Q}_{\omega;1}^{\rm(in)}=(\hat{a}_{\omega;1}^{\rm(in)}+\hat{a}_{\omega;1}^{{\rm(in)}\dagger})/\sqrt{2}$
and
$\hat{P}_{\omega;1}^{\rm(in)}=-i\,(\hat{a}_{\omega;1}^{\rm(in)}-\hat{a}_{\omega;1}^{{\rm(in)}\dagger})/\sqrt{2}$,
and similarly for the other pairs of canonical variables. Under this ordering,
the full symplectic-transformation taking the in-modes to the out/int-modes is
easily found as

$\bm{S}_{\rm BH}=\begin{pmatrix}\bm{I}_{2}&0&0\\\
0&\cos\theta_{\omega}\bm{I}_{2}&-\sin\theta_{\omega}\bm{I}_{2}\\\
0&\sin\theta_{\omega}\bm{I}_{2}&\cos\theta_{\omega}\bm{I}_{2}\end{pmatrix}\cdot\begin{pmatrix}\cosh
r_{H}\bm{I}_{2}&0&\sinh r_{H}\bm{\sigma}_{z}\\\ 0&\bm{I}_{2}&0\\\ \sinh
r_{H}\bm{\sigma}_{z}&0&\cosh
r_{H}\bm{I}_{2}\end{pmatrix}=\begin{pmatrix}\cosh{r_{H}}\bm{I}_{2}&0&\sinh{r_{H}}\,\bm{\sigma}_{z}\\\
\sin{\theta_{\omega}}\sinh{r_{H}}\,\bm{\sigma}_{z}&\cos\theta_{\omega}\bm{I}_{2}&\sin\theta_{\omega}\cosh{r_{H}}\,\bm{I}_{2}\\\
\cos\theta_{\omega}\sinh{r_{H}}\,\bm{\sigma}_{z}&-\sin\theta_{\omega}\,\bm{I}_{2}&\cos\theta_{\omega}\cosh{r_{H}}\,\bm{I}_{2}\end{pmatrix}.$
(III.11)

We note that a similar parameterization was given in Ref. [49, 28] in the
context of an analog black hole formed in BECs, but no diagrammatic
decomposition (nor generalizations thereof to multi-mode scattering scenarios)
was discussed.

From the $S$-matrix, and given the first and second moments of a Gaussian
initial state, one can derive all aspects of the final state in a simple
manner. We emphasize that the circuit simply provides a re-formulation of
Hawking’s original derivation [2] and, in particular, the circuit does not
introduce any extra assumption or approximation.

Let us compute, for instance, the particle density for the outgoing Hawking
radiation, assuming a vacuum input. The vacuum state is a three-mode Gaussian
state with moments $\bm{\mu}_{\rm vac}^{\rm(in)}=\bm{0}$ and $\bm{\sigma}_{\rm
vac}^{\rm(in)}=\bm{I}_{6}$. The out/int moments can then be found by acting
with the scattering matrix of Eq. (III.11) on the in-moments, using the
general transformations of Eqs. (II.8) and (II.9), namely $\bm{\mu}_{\rm
vac}^{\rm(out)}=\bm{S}_{\rm BH}\,\bm{\mu}_{\rm vac}^{\rm(in)}$ and
$\bm{\sigma}_{\rm vac}^{\rm(out)}=\bm{S}_{\rm BH}\,\bm{\sigma}_{\rm
vac}^{\rm(in)}\,\bm{S}_{\rm BH}^{\top}$. One then finds the quantum state of
the outgoing Hawking radiation from these results by simply taking the reduced
moments of the single-mode phase-space corresponding to the mode
$a_{w;2}^{\rm(out)}$ (just extract the relevant matrix coefficients of the
final out/int moments). After a little algebra, the mean vector and covariance
matrix for the outgoing Hawking mode are

$\displaystyle\bm{\mu}_{w;2}^{\rm(out)}$ $\displaystyle=\bm{0},$ (III.12)
$\displaystyle\bm{\sigma}_{w;2}^{\rm(out)}$
$\displaystyle=\left(1+2\cos^{2}\theta_{\omega}\sinh^{2}r_{H}\right)\bm{I}_{2}.$
(III.13)

This is the covariance matrix of a thermal state with mean number of quanta,

$\expectationvalue{\hat{n}_{H}}=\frac{1}{4}\text{Tr}\bm{\sigma}_{w;2}^{\rm(out)}-\frac{1}{2}=\cos^{2}\theta_{\ell,\omega}\sinh^{2}r_{H}=\frac{\Gamma_{\ell,\omega}}{\text{e}^{\omega/T_{H}}-1},$
(III.14)

where we have used Eq. (II.10), the correspondence
$\cos^{2}\theta_{\ell,\omega}=\Gamma_{\ell,\omega}$ (and restored the labels
$\ell$ and $\omega$ in this expression) and the Hawking relation
$\tanh^{2}{r_{H}}=\text{e}^{-\omega/T_{H}}$. Eq. (III.14) is the standard
result for the outgoing Hawking-flux in the asymptotic out-region, physically
describing a quantum gas of blackbody radiation modulated by a grey-body
coefficient, $\Gamma_{\ell,\omega}$.

Figure 3: Entanglement (quantified by LN) between the ‘Hawking-pair’ of modes
$\hat{a}_{\omega;2}^{(\rm int)}$ and $\hat{a}_{\omega;1}^{(\rm out)}$ in the
Hawking process. Left panel: Variation of entanglement across system-
parameters, $(\Gamma,\omega/T_{H})$, assuming ideal conditions of zero
background temperature, $T_{\rm env}=0$, and perfect efficiency, $\eta=1$
(zero attenuation). Right panel: Variation of entanglement in noisy parameter
space $(1-\eta,T_{\rm env}/T_{H})$ at $\omega/T_{H}=1$ and $\Gamma=1$. This
plot assumes isotropic thermal noise (an unphysical assumption)—i.e., the same
number of thermal quanta in the three in modes. Null-space indicates zero
entanglement. Observe the boundary curve where entanglement vanishes
identically. Behavior does not change with choice of $\omega/T_{H}$.

###  III.2 Quantum correlations and entanglement degradation

Next, we use the black hole circuit to analyze the entanglement between the
modes in the final state. From the symplectic circuit, we can obtain
analytical expressions for the entanglement in any bipartition or any pair of
modes in the 3-mode network, using techniques described in Sections II.1 and
II.2. In an analog-gravity experiment, it is likely that entanglement will
only be measured pair-wise for the outgoing radiation, as measuring
entanglement between any-or-all bipartitions requires multi-mode measurements
(or multi-mode quantum-state tomography)—an experimentally challenging feat.
In this section, we therefore restrict our attention to the modes,
$a_{\omega;2}^{(\rm int)}$ and $a_{\omega;1}^{(\rm out)}$ (the “original
Hawking pairs”), and quantify the entanglement by the LN. (We discuss multi-
mode entanglement in Section V.2.)

We also incorporate the effects of initial thermal fluctuations and
attenuation. Defining efficiency $\eta$ ($1-\eta$ measures attenuation), the
covariance matrix of the process is,

$\bm{\sigma}^{(\rm out)}=\eta\,\bm{S}_{\rm BH}\bm{\sigma}^{(\rm
in)}\bm{S}_{\rm BH}^{\top}+(1-\eta)\,\bm{I}_{6},$ (III.15)

where $\bm{S}_{\rm BH}$ is the symplectic matrix of Eq. (III.11) and
$\bm{\sigma}^{(\rm in)}=(1+2n_{{\rm env}})\,\bm{I}_{6}$ is an initial thermal
state at e.g. temperature $T_{\rm env}$. Notice that we have assumed isotropic
initial thermal fluctuations in the inputs for simplicity (i.e., $n_{{\rm
env}}$ is the same for the three input modes). In fact, isotropic thermal
noise is highly unphysical, since the in-modes $\hat{a}_{\omega;1}^{(\rm in)}$
and $\hat{a}_{\omega;2}^{(\rm in)}$ have support only on ultra-high
frequencies as measured by an inertial observer at past infinity, while the
back-scattering mode $\hat{b}_{\omega}^{(\rm in)}$ has frequency of the same
order as the Hawking quanta (of the order of the Hawking temperature $T_{H}$).
Therefore, for any realistic thermal environment for astrophysical black holes
(e.g. the cosmic microwave background radiation), the in-modes
$\hat{a}_{\omega;1}^{(\rm in)}$ and $\hat{a}_{\omega;2}^{(\rm in)}$ would be
meekly populated due to their ultra-high frequencies. However, since our main
goal in this section is to illustrate the effects that thermal noise have in
the generation of entanglement and to extract conclusion for analog black
holes for which all in-modes can be realistically thermally populated, we
consider the situation of isotropic thermal noise. Though, it is
straightforward to relax this assumption.III.2III.2III.2We can generalize this
to non-isotropic noise and also include noise added from the loss channel,
$\bm{\sigma}^{(\rm out)}=\eta\,\bm{S}_{\rm
BH}(\oplus_{i=1}^{3}N_{k_{i}}\bm{I}_{2})\bm{S}_{\rm
BH}^{\top}+(1-\eta)\,(\oplus_{i=1}^{3}N_{k_{i}}\bm{I}_{2})$, where
$N_{k_{i}}=1+2n_{k_{i}}$ and $n_{k_{i}}$ is the number of noisy quanta in the
$i$th mode. Here, we have assumed the added quanta from the loss channel is
equal to the initial noise population. Similarly, the physical meaning of the
attenuation parameter $\eta$ is more clear for analog black holes—where $\eta$
is related to the peculiarities of the experimental set up or to the
efficiency of detectors—than for astrophysical black holes. Anyway, we study
the effects that varying $\eta$ has on the entanglement of the final state as
a pedagogical preparation for the section V.

The covariance matrix $\bm{\sigma}^{(\rm out)}$ is a $6\times 6$ matrix for
the three modes of the black hole circuit (corresponding to a mixed 3-mode
Gaussian quantum state), from which we take the reduced covariance matrix for
the modes $a_{\omega;2}^{(\rm int)}$ and $a_{\omega;1}^{(\rm out)}$.

Given the reduced covariance matrix for Hawking pairs in the final state, we
compute the LN as a function of all the open parameters and plot the result;
see Fig. 3. The left panel is the entanglement in the system-parameter space,
$(\omega/T_{H},\Gamma)$, for ideal operating conditions, $n_{\rm env}=0$ and
$\eta=1$. We show this to simply illustrate that there is entanglement in this
pair of modes, over all system parameters (vanishing only asymptotically as
$\Gamma\rightarrow 0$ or $\omega/T_{H}\rightarrow\infty$). Note that the
entanglement decreases with the greybody factor, $\Gamma$, as some quanta are
lost to the back-scattering channel, $\hat{b}_{\omega}^{(\rm int)}$. However,
in the absence of environmental thermal fluctuations (characterized by $n_{\rm
env}$), there is always residual entanglement between the modes
$\hat{a}_{\omega}^{(\rm int)}$ and $\hat{a}_{\omega}^{(\rm out)}$, no matter
the amount of back-scattering $\Gamma>0$. The entanglement within the entire
system of three modes, however, does not change with the grey-body factor, as
back-scattering simply shifts quanta around. On the other hand, we note that,
if initial thermal fluctuations are present, non-negligible back-scattering
will generally degrade (or even destroy) entanglement between Hawking-pair
modes if the thermal fluctuations are strong enough (not shown here).

The right panel of Fig. 3 illustrates the effects of noise due to thermal
fluctuations and attenuation, assuming $\Gamma=1$ and $\omega/T_{H}=1$. Both
of these effects are harmful to the entanglement generated in the Hawking
process. For instance, ignoring attenuation for the moment (i.e., restricting
to the bottom line at $\eta=1$), once we “turn on” the Hawking effect, the
Hawking process will cause initially independent thermal fluctuations to
spread amongst the modes, leading to (classically) correlated noise within the
system, but these correlations have nothing to do with entanglement. Even
worse, such classical-correlations can dominate over the genuine quantum-
correlations produced during the Hawking process—rendering any entanglement
null, as depicted by the white-space in the right panel of Fig. 3.

For near-unit efficiency, a critical condition is found $T_{\rm env}/T_{H}<2$,
above which entanglement is zero (under the assumption of isotropic thermal
noise). This is an equivalent condition as that discussed in the BEC context
of Ref. [52]. We extend these prior results to include attenuation—thus
obtaining a family of critical-conditions, $T_{\rm env}/T_{H}<f(\eta)$, where
$0\leq f(\eta)\leq 2$, as seen by the boundary curve in Fig. 3. For instance,
there is entanglement for all $T_{\rm env}/T_{H}<1$ for $1-\eta\approx.3$. We
remark that this family of conditions are independent of the ratio
$\omega/T_{H}$ in this simple setup.

The LN explicitly shows the behavior of entanglement in the noise parameter
space $(1-\eta,T_{\rm env}/T_{H})$ (see Fig. 3): the entanglement is
monotonically decreasing in both the attenuation, $1-\eta$, and in the
environment temperature, $T_{\rm env}$, as intuitively expected. Hence, the
take home message is that the entanglement generated in the Hawking process is
fragile to both loss and ambient noise; this is an important lesson to keep in
mind in analog scenarios discussed below.

Contrariwise, one cannot faithfully deduce such behavior from the entanglement
witnesses $P_{-}$ or $\Delta$ of Eq. (II.13) for the same set of parameters,
as they are witnesses of entanglement and not measures of it. Indeed, there
are regions in the parameter space $(1-\eta,T_{\rm env}/T_{H})$ which exhibit
a larger violation of the CS inequality, $\Delta<0$, when the noise is
increased (see Fig. 21 in Appendix C). Although $\Delta$ cannot be used as a
quantifier of entanglement, we point out that, in this simple set up,
$\Delta=0$ genuinely highlights the critical line in the $(1-\eta,T_{\rm
env}/T_{H})$ plane where entanglement vanishes, consistent with the vanishing
of the LN (the boundary line in Fig. 3), as should be the case for a good
entanglement witness. [On the other hand, Appendix C shows other situations
where $\Delta=0$ does not faithfully capture the boundary in the
$(1-\eta,T_{\rm env}/T_{H})$ plane where entanglement vanishes.]

##  IV Simple recipe for circuit construction

From the simple circuit decomposition detailed in the previous section, we
ascertain a few rules-of-thumb that we can follow to assess more generic
settings (e.g., multi-mode settings with multiple event horizons) in order to
find effective circuit descriptions therein. The rules are: (1) Insert a two-
mode squeezer, with squeezing intensity $r_{H}$ characterized by the Hawking
temperature $T_{H}$ via $\tanh^{2}r_{H}=\text{e}^{-\omega/T_{H}}$, for each
Hawking-pair creation mechanism; (2) Insert a two-mode beam-splitter for each
classical scattering-event between two modes; and finally, (3) Arrange circuit
elements appropriately to highlight the physics at play. If white hole and
black hole horizons are both present, then one must take into consideration
the inverse nature of the white hole scattering process and the order of
scattering events (see next section).

In a more mathematical language, a pair creation occurs when a mode with
positive symplectic normIV.1IV.1IV.1Given a complex solution
$\varphi(\vec{x},t)$ to the equations of motion describing one mode of the
system, its symplectic norm is defined from the symplectic product
$(\varphi_{1},\varphi_{2})\equiv
i\,\int_{\Sigma}d^{3}\Sigma\,n^{\mu}\,(\bar{\varphi}_{1}\nabla_{\mu}\varphi_{2}-\nabla_{\mu}\bar{\varphi}_{1}\,\varphi_{2})$,
where $d^{3}\Sigma$ is the volume element of the Cauchy hypersurface $\Sigma$
and $n^{\mu}$ its fututre-oriented unit normal (see, e.g. [57, 56]). The name
‘symplectic’ is motivated from the fact that such product is defined from the
symplectic structure of the classical phase space. mixes with a mode with
negative symplectic norm (the mixing is, of course, dictated by the equations
of motion). Such interaction is described, therefore, by a two-mode squeezer.
On the contrary, when two modes with the same sign of their symplectic norm
interact, the process is described by a beam-splitter. Free propagation
corresponds to phase shifters. Therefore, if one knows the dynamics, it is
simple to translate it to a simpler and more intuitive circuit.

When the details of the dynamics are unknown, the rules-of-thumb (1)-(3)
enumerated above are heuristic and not necessarily
algorithmic,IV.2IV.2IV.2Though, using physical arguments, one can be slightly
more rigorous and, e.g., place bounds on the number of free parameters
required to specify/construct the desired circuit; see Appendix B for further
comments on this matter. nor are such rules entirely general. For instance,
the role of phases are not mentioned in the rules above, which can lead to
important consequences in certain scenarios where resonance effects matter
(such as lasing in a white-black hole [58, 59]). Moreover, our circuit
diagrams are not a substitute for the micro-physics, since a micro-physical
description is required in order to specify, e.g., the functional form of the
greybody factor, $\Gamma$, as well as the Hawking temperature, $T_{H}$.
Nevertheless, with some ingenuity, such a circuit diagram proves to be
immensely useful as an explanatory tool and supplementary guide, especially
when an analytical description is not straightforward, as we demonstrate in
the next section for an optical-analog, white-black hole generated by a strong
electromagnetic pulse.

##  V Example 2: White-black hole pair

In this section we put the techniques described above in action for the more
complicated scenario where a black hole event-horizon and a white hole event-
horizon are both present and interacting. We construct a symplectic circuit
using the recipe given above and derive analytical expressions for different
aspects of the out-state therefrom (regarding both particle creation and
multi-mode entanglement). Finally, in order to quantitatively assess the
accuracy of the circuit in describing the underlying physics, we compare the
results derived from such with numerical solutions to the equations of motion
for the particular case of optical systems.

###  V.1 Physical picture, symplectic circuit, and example calculations

#### Physical picture:

Consider the circumstance where a black hole event-horizon and a white hole
event-horizon share a mutual interior region. This is common in optical setups
where the analog space-time is generated from a strong pulse with a rising
tail (corresponding to the white hole) and a descending front (corresponding
to the black hole) [34, 39]. For concreteness, the analysis of this section
will have this system in mind, but we expect a similar scenario for any
background flow with a rise-then-fall profile and a (approximate)
trapped/anti-trapped region in between.

The “white-black hole”, as we shall call it, partitions two asymptotically
flat space-time regions. The exterior region of the white hole (ideally) has
no access whatsoever to the exterior region of the black hole, due to the
causal-impossibility of traversing the white hole event horizon. However, the
converse is not true. Anything that falls into the black hole emerges from the
white hole, though slightly perturbed by the Hawking pair-creation mechanisms
occurring near each event horizon. Figure 4 illustrates this configuration,
together with the structure of modes involved in the process, separated in
three groups: incoming, outgoing, and interior. In optical systems, these are
the modes associated with a single frequency $\omega$ but different wavenumber
$k$ (as defined in the frame comoving with the white-black hole pair), and
modes with different frequencies do not interact (more details below, and in
Ref. [12]). The discussion in this section applies to any system with a
similar mode structure, regardless of its physical origin.

Figure 4: Illustration of the structure of $in$, $int$, and $out$ modes for an
optical analog white-black hole in the comoving frame [43]. The analog white-
black hole is generated by a strong electromagnetic pulse via the Kerr effect.
There are four in-modes (three arriving at the black hole horizon and one at
the white hole horizon), and four outgoing-modes. There are two real,
propagating interior modes (int-modes) between horizons; the other two modes
are evanescent (and thus exponentially suppressed) within this region.

The Hawking process of the white hole is the time-reverse process of the black
hole. Instead of emitting bland, thermal Hawking-radiation, the white hole
emits quantum-correlated Hawking pairs of modes $k^{(\rm out)}_{1}$ and
$k^{(\rm out)}_{4}$. However, for the tangled white-black hole system
colloquially described above, the white hole pair-creation process is not
spontaneous, as the infalling Hawking-partner generated at the black hole
event horizon stimulates the white hole. Thus, the Hawking-pairs generated by
the white hole are slightly tainted and in a mixed state—purified only by the
outgoing Hawking radiation in the exterior region of the black hole.

#### Symplectic circuit:

For simplicity, we assume that the white hole and black hole event-horizons
are identical. In other words, the pulse generating the white-black hole is
symmetric about its center. One can straightforwardly generalize this to
include asymmetries in the flow profile induced by the pulse.

To describe the modes involved in the scattering process, it is simpler to
work in the frame comoving with the pulse, in which the white-black hole pair
is at rest and stationary. The frequency $\omega$ of waves defined in this
frame is conserved because of stationarity, and modes with different
frequencies do not mix with each other. Therefore, the evolution factorizes
into uncoupled $\omega$-sectors, reducing the problem to a finite set of
interacting modes for each $\omega$. The analysis of the dispersion relation
for this system (see e.g. [12] for an analytical description), shows that, far
away from the strong pulse producing the white-black hole system (in the
asymptotic regions), there exist four different modes for each frequency
$\omega$, which we label as $k_{1},k_{2},k_{3},k_{4}$. Three of them move to
the left, and the fourth moves to the right (see Fig. 4 to see our
conventions). Hence, considering both the white and black hole sides of the
pulse, we have four modes coming in, and four modes going out, which is a
slightly more complicated setup than that of astrophysical black holes, where
we had only three in-modes and three out-modes and a single event horizon. In
spite of such complications, the analysis is conceptually similar.

We choose the following order for the in-vector of canonical operators,

$\hat{\bm{R}}^{\rm(in)}\equiv\\\
\left(\hat{Q}_{k_{1}}^{\rm(in)},\hat{P}_{k_{1}}^{\rm(in)},\hat{Q}_{k_{2}}^{\rm(in)},\hat{P}_{k_{2}}^{\rm(in)},\hat{Q}_{k_{3}}^{\rm(in)},\hat{P}_{k_{3}}^{\rm(in)},\hat{Q}_{k_{4}}^{\rm(in)},\hat{P}_{k_{4}}^{\rm(in)}\right)^{\top},$
(V.1)

where, e.g., $\hat{Q}_{k_{3}}^{\rm(in)}$ is the only in-mode to the white hole
and is the time-reverse of the outgoing Hawking mode emitted by the black hole
(thus, $\hat{Q}_{k_{3}}^{\rm(in)}$ is the “ingoing Hawking mode”). We choose a
similar order for the out-modes.

Figure 5: Minimal ansatz for the symplectic-circuit of the Hawking process for
the white-black hole. The squeezer and beam-splitter on the left corresponds
to the black hole, while the elements to the right correspond to the white
hole horizon. The white hole scattering process is the time reverse of the
black hole scattering process; hence the in-modes of the black hole are the
out-modes of the white hole and vice versa. $\bm{S}_{H}$ is a two-mode
squeezer with squeezing intensity $r_{H}$, while beam-splitters are labeled by
the parameter $\theta_{\omega}$.

In Ref. [43], we introduced a symplectic-circuit to describe the Hawking
process for a white-black hole (shown here in Fig. 5), which serves as a
powerful explanatory tool to describe the physics behind the distinct
scattering processes of a white-black hole. Here, we provide a more detailed
analysis of the circuit description. This circuit is built following the
recipe provided in the previous section; namely, each event horizon introduces
a two-mode squeezer accounting for the pair-creation and a beam-splitter to
implement a possible process of back-scattering. As discussed in Refs. [60,
12], back-scattering primarily affects long-wavelength modes ($k_{2}$ and
$k_{3}$ in this scenario), while it can be neglected for the short-wavelength
modes $k_{1}$ and $k_{4}$. Observe that the symplectic circuit of Fig. 5
arises by simply gluing together the circuit for the black hole alone (Fig. 2)
with its mirror image (because a white hole is the time reversal of a black
hole). In the next section, we show that this simple circuit captures the
underling physics with great accuracy in the regime where the analogy with the
Hawking effect occurs—i.e., in the low frequency regime where dispersive
effects do not dominate.

The next step is to convert the symplectic circuit into a scattering matrix,
from which we can derive analytical expressions depending on the parameters of
the circuit. Let us label each symplectic operation in the circuit by
$\bm{S}_{i}$, where $i\in\\{1,2,3,4\\}$ labels the order of operations in time
(from left to right in Fig. 5). For instance, $\bm{S}_{1}$ is the black hole
two-mode squeezer which, with this mode ordering, has a matrix description

$\bm{S}_{1}=\begin{pmatrix}\cosh{r_{H}}\,\bm{I}_{2}&0&0&\sinh{r_{H}}\,\bm{\sigma}_{z}\\\
0&\bm{I}_{2}&0&0\\\ 0&0&\bm{I}_{2}&0\\\
\sinh{r_{H}}\,\bm{\sigma}_{z}&0&0&\cosh{r_{H}}\,\bm{I}_{2}\end{pmatrix}.$
(V.2)

Next, we have a scattering process in the exterior of the black hole horizon,
described by

$\displaystyle\bm{S}_{2}$ $\displaystyle=\begin{pmatrix}\bm{I}_{2}&0&0&0\\\
0&\cos{\theta_{\omega}}\bm{I}_{2}&0&-\sin\theta_{\omega}\bm{I}_{2}\\\
0&0&\bm{I}_{2}&0\\\
0&\sin\theta_{\omega}\bm{I}_{2}&0&\cos{\theta_{\omega}}\bm{I}_{2}\end{pmatrix},$
(V.3)

followed by the counterpart of these two operations for the white-hole,

$\displaystyle\bm{S}_{3}$ $\displaystyle=\begin{pmatrix}\bm{I}_{2}&0&0&0\\\
0&\cos{\theta_{\omega}}\bm{I}_{2}&\sin\theta_{\omega}\bm{I}_{2}&0\\\
0&-\sin\theta_{\omega}\bm{I}_{2}&\cos{\theta_{\omega}}\bm{I}_{2}&0\\\
0&0&0&\bm{I}_{2}\end{pmatrix},$ (V.4) $\displaystyle\bm{S}_{4}$
$\displaystyle=\begin{pmatrix}\cosh{r_{H}}\bm{I}_{2}&0&-\sinh{r_{H}}\bm{\sigma}_{z}&0\\\
0&\bm{I}_{2}&0&0\\\ 0&0&0&\bm{I}_{2}\\\
-\sinh{r_{H}}\bm{\sigma}_{z}&0&\cosh{r_{H}}\bm{I}_{2}&0\end{pmatrix}.$ (V.5)

The relative minus signs between $S_{1}$ and $S_{4}$, and $S_{2}$ and $S_{3}$
are due to the inverse character of the white hole relative to the black hole.
The full scattering matrix describing the transformation from the four in-
modes to the four out-modes is then found by matrix multiplication, which we
write formally as $\bm{S}_{\rm WB}=S_{4}S_{3}S_{2}S_{1}$ where the subscript
WB refers to “white-black” hole. Written out fully,

$\bm{S}_{\rm
WB}=\begin{pmatrix}(1+\cos^{2}\theta\sinh^{2}r_{H})\bm{I}_{2}&\cos\theta\sin\theta\sinh
r_{H}\bm{\sigma}_{z}&-\cos\theta\sinh
r_{H}\bm{\sigma}_{z}&\cos^{2}\theta\cosh{r_{H}}\sinh{r_{H}}\bm{\sigma}_{z}\\\
-\cos\theta\sin\theta\sinh{r_{H}}\bm{\sigma}_{z}&\cos^{2}\theta\bm{I}_{2}&\sin\theta\bm{I}_{2}&-\cos\theta\sin\theta\cosh{r_{H}}\bm{I}_{2}\\\
\cos\theta\sinh{r_{H}}\bm{\sigma}_{z}&\sin\theta\bm{I}_{2}&0&\cos\theta\cosh{r_{H}}\bm{I}_{2}\\\
-\cos^{2}\theta\cosh{r_{H}}\sinh{r_{H}}\bm{\sigma}_{z}&-\cos\theta\sin\theta\cosh{r_{H}}\bm{I}_{2}&\cos\theta\cosh{r_{H}}\bm{I}_{2}&(\sin^{2}\theta-\cos^{2}\theta\sinh^{2}r_{H})\bm{I}_{2}\end{pmatrix}$
(V.6)

Due to the multi-mode structure of the setup and the many physically distinct
operations in play, it is instructive to perform some simple calculations to
gain some insight into the dynamics of the network. We will thus begin by
restricting ourselves to the ideal scenario where all modes initially contain
only vacuum fluctuations. Then, the output covariance matrix for the 4-mode
pure state is given by $\bm{\sigma}^{(\rm out)}_{\omega}=\bm{S}_{\rm
WB}\cdot\bm{\sigma}_{\omega}^{(\rm in)}\cdot\bm{S}_{\rm WB}^{\top}$, with
$\bm{\sigma}^{(\rm in)}_{\omega}=\bm{I}_{8}$; the mean of the state is zero.
All properties of the quantum state of the modes can then be written in terms
of the elements of $\bm{\sigma}_{\omega}^{(\rm out)}$.

We first calculate the outgoing spectrum of Hawking particles emitted by the
black hole. This can be computed by taking the reduced covariance matrix of
$\bm{\sigma}_{\omega}^{(\rm out)}$ of the single-mode sub-space corresponding
to the mode $a_{k_{3}}^{(\rm out)}$, $\bm{\sigma}_{\omega;k_{3}}^{(\rm out)}$.
It is straightforward to see that this covariance matrix describes a thermal
state, $\bm{\sigma}_{H}^{(\rm out)}=(1+2\expectationvalue*{\hat{n}_{H}^{(\rm
out)}})\,\bm{I}_{2}$, as expected. That is, the outgoing Hawking radiation
from the black hole is completely oblivious to the presence of the white hole,
as required by causality (or an approximate form thereof for analog horizons).

Utilizing the general formula (II.10) for the mean number of quanta, the mean
number of quanta in the mode $k^{\rm(out)}_{3}$ is

$\expectationvalue{\hat{n}_{H}^{(\rm
out)}}=\cos^{2}\theta_{\omega}\,\sinh^{2}r_{H}=\frac{\Gamma_{\omega}}{\text{e}^{\omega/T_{H}}-1},$
(V.7)

where we have used the definition
$\Gamma_{\omega}\equiv\cos^{2}\theta_{\omega}$, and we have assumed the
Hawking relation $\tanh^{2}r_{H}=\text{e}^{-\omega/T_{H}}$ holds (see [12] for
a justification using an analytical model, valid in the low frequency regime
where dispersive effects do not dominate; see below for a numerical
confirmation). We see that the outgoing Hawking spectrum is a modulated
blackbody spectrum, analogous to the original Hawking effect, with modulation
coming from back-scattering via the greybody factor $\Gamma_{\omega}$.

Beyond the Hawking spectrum alone, another interesting quantity is the total
number of quanta generated in the full process, i.e., the quanta emitted by
both the black and white hole (per unit frequency and unit time). Using the
circuit and the relations above, we find

$\sum_{\rm all\,modes}\expectationvalue{\hat{n}_{i}^{(\rm out)}}=\\\
2\cos^{2}\theta_{\omega}\sinh^{2}r_{H}\left(2+\cos^{2}\theta_{\omega}\sinh^{2}r_{H}\right).$
(V.8)

Interestingly, this is a decreasing function of the greybody factor,
$\Gamma_{\omega}=\cos^{2}\theta_{\omega}$. Hence, back-scattering, if present,
acts as a self-stabilising mechanism for the white-black hole. We further note
that this result is agnostic to individual phases which might accumulate
between the black hole and white hole pair creation processes in Fig. 5. [We
do not include such phases in our circuit decomposition, but it is
straightforward to check this claim.] In other words, adding a phase to each
mode in the middle of the circuit, which preserves the space-time symmetry of
the diagram (a symmetry which is induced by the symmetry of the pulse), does
not change the total particle number at the output.V.1V.1V.1This is in
contrast to, e.g., white-black hole lasers which support a resonant
enhancement of particle creation [58, 59]. Though, we note that the laser
setup is quite different than the white-black hole here, since, in the laser
system, the output of the white hole seeds the black hole and vice versa,
leading to a cascaded succession of pair-creation events.

The reduction in the total number of particles by the grey-body factor is
unique to the white-black hole case (it does not arise for a single horizon)
and has an interesting story behind which can be easily understood by simple
inspection of the circuit. In short, if $\Gamma_{\omega}\to 0$ (completely
reflecting potential barrier for the long-wavelength modes $k_{2}$ and
$k_{3}$) the white hole undoes exactly the pair-production at the black hole
horizon. In more detail, consider the $a_{k_{4}}^{(\rm in)}$ mode approaching
the black hole in Fig. 5). This mode converts into outgoing Hawking radiation
via the Hawking process, while its entangled partner falls into the black
hole. On the way out, the Hawking radiation meets a barrier and scatters back
into the black hole as a $k^{\rm(int)}_{2}$ mode. This back-scattered Hawking
radiation then traverses through the interior region of the white-black hole,
emerging from the white hole unperturbed (the back-scattered mode mixes
negligibly with the mode $k_{1}$ in the interior). Following its escape from
the white hole horizon, the back-scattered radiation meets another barrier in
the exterior region of the white hole; at which point, it reflects back
towards the white hole—now as an ingoing Hawking-mode, $k^{\rm(in)}_{3}$,
seeding the white hole pair-creation mechanism. This “reflective-seeding”
process undoes the Hawking process initiated by the black hole, since the
white hole and black hole unitary dynamics are time-reversals of one
another—thus resulting in an overall reduction in the total number of
particles. One can see this directly by treating each of the beam-splitter
elements in Fig. 5 as perfect mirrors (such that ingoing quanta bounce off the
beam-splitters and cannot traverse through them) and tracing the paths of the
modes through the circuit. Of course, the exact cancellation we just described
only occurs under the assumption of symmetric pulse, which amounts to saying
that the white hole is the exact time-reversal of the black hole.

This example illustrates the usefulness of symplectic circuits to understand
complex aspects of system by mere visual inspection.

Figure 6: Entanglement predicted by the circuit of Fig. 5 in all possible bi-
partitions and mode-pairs for the out-modes of the analog white-black hole, as
measured by the logarithmic negativity (LN); assuming only initial vacuum
fluctuations are present and taking
$\Gamma_{\omega}=1\,\forall\,\omega/T_{H}$.

###  V.2 Quantum correlations

We now investigate the quantum correlations generated by the circuit depicted
in Fig. 5. We will show results for $\Gamma_{\omega}=1$ (negligible back-
scattering), not only for pedagogical reasons, but also because in the
numerical simulations of this systems shown below (see also [43]), we find
that $\Gamma_{\omega}$ is indeed very close to one for all frequencies of
interest. For the squeezing intensity, as mentioned above, we use
$\tanh^{2}{r_{H}}=\text{e}^{-\omega/T_{H}}$, with $T_{H}$ a single number
(i.e., independent of frequency). The output covariance matrix can be found
from the symplectic elements introduced in the previous section, namely
$\bm{S}_{\rm WB}\cdot\bm{\sigma}^{(\rm in)}_{\omega}\cdot\bm{S}_{\rm
WB}^{\top}$, with $\bm{S}_{\rm WB}$ given in Eq. (V.6). We start by discussing
vacuum input, for which $\bm{\sigma}^{(\rm in)}_{\omega}=\bm{I}_{8}$.

Due to the entangling effects of the two-mode squeezers, there is genuine
multi-mode entanglement within the network. To assess the entanglement, we
consider pairwise entanglement for a reduced set of modes as well as bi-
partite entanglement for the entire system of modes. Since the scattering
intensity is $\Gamma_{\omega}\approx 1$, the mode $k_{2}$ weakly mixes with
the rest and does not get entangled with the other modes. We are left with
$k_{1}$, $k_{3}$ and $k_{4}$, out of which there are 2 entangled mode-pairs
corresponding toV.2V.2V.2Entanglement is contained only in mode pairs of
opposite symplectic norm.
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$ and
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{3}})$. There are also 3
multi-mode bi-partitions corresponding to
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{4}},\hat{a}^{\rm(out)}_{k_{3}})$,
$(\hat{a}^{\rm(out)}_{k_{4}}|\hat{a}^{\rm(out)}_{k_{3}},\hat{a}^{\rm(out)}_{k_{1}})$,
and
$(\hat{a}^{\rm(out)}_{k_{3}}|\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$.
Figure 6 depicts the entanglement between all such mode-pairs and bi-
partitions; the entanglement in the pair
$(\hat{a}^{\rm(out)}_{k_{4}},\hat{a}^{\rm(out)}_{k_{3}})$ is identically zero
(see below for an explanation) and is thus not shown. [The circuit produces
analytical expressions for all the curves shown in Fig. 6, but they are
lengthy and provide no real insight.]

Figure 7: Entanglement between “Hawking pairs” emitted by the white hole
horizon, $(\hat{a}_{k_{1}}^{(\rm out)},\hat{a}_{k_{4}}^{(\rm out)})$, at
$\omega/T_{H}=1$ for varying environmental temperatures $T_{\rm env}/T_{H}$;
assumes initial, isotropic thermal fluctuations of temperature $T_{\rm env}$
in the comoving frame for all modes.

Note that the bi-partition
$(\hat{a}^{\rm(out)}_{k_{3}}|\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$
physically represents the partition between the black hole and white hole
exterior regions. By looking at the circuit, it is easy to understand that the
entanglement across the black hole/white hole bi-partition is formally
equivalent to that of a single two-mode squeezed vacuum state, with squeezing
governed by the Hawking relation $\tanh^{2}{r_{H}}=\text{e}^{-\omega/T_{H}}$.
This is because the white hole squeezer entangles the modes
$\hat{a}^{\rm(out)}_{k_{1}}$ and $\hat{a}^{\rm(out)}_{k_{4}}$, but this is a
local transformation within one subsystem in the bi-partition
$(\hat{a}^{\rm(out)}_{k_{3}}|\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$
and as such it does not alter the entanglement in the bi-partition (recall
that entanglement is invariant under local transformations—where ‘local’ here
means within a subsystem of modes). Furthermore, Fig. 6 shows that the black
hole/white hole bi-partition hosts the least amount of entanglement out of all
3-mode bi-partitions, which is due to the fact that the white hole squeezer
does not contribute.

It is interesting to point out that, for vacuum input, the black hole horizon
generically produces more entanglement than the white hole one. This is
because, as it is manifest by simple inspection of the circuit, the white hole
horizon receives thermal radiation emitted by the black hole in a flux of
Hawking partners. Hence, the white hole emission is always stimulated by
thermal radiation, and we know from the analysis of the previous section and
also from Appendix A that such thermal input degrades the production of
entanglement. It is important to emphasize that this, as well as all the
features regarding entanglement discussed above, can be easily understood by
simple inspection of the circuit. More than that, the circuit provides
analytical expressions for all these quantities.

Generally, the entanglement between bi-partitions of the 3 modes is greater
than or equal to the entanglement between the mode-pairs because the LN is an
entanglement monotone [42]. This is quantitatively manifest in Fig. 6. Also,
observe that the bi-partition with the greatest amount of entanglement is
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{4}},\hat{a}^{\rm(out)}_{k_{3}})$,
since the mode $\hat{a}^{\rm(out)}_{k_{1}}$ participates in each pair-creation
mechanism this system experiences. The mode-pair with the largest amount of
entanglement are the Hawking-pairs emitted from the white hole,
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$. Oddly, the maximal
amount of entanglement between the original Hawking pair
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{3}})$ (the pair generated
by the black hole) carries, at most, precisely 1 bit of
entanglement,V.3V.3V.3The LN is in units of entangled bits or ebits. which is
maximal at about one Hawking wavelength.

Figure 8: Entanglement between Hawking pairs emitted by the white hole
horizon, $(\hat{a}_{k_{1}}^{(\rm out)},\hat{a}_{k_{4}}^{(\rm out)})$, at
$\omega/T_{H}=1$ and $\Gamma=1$ for varying environmental temperatures $T_{\rm
env}/T_{H}$ and attenuation, $1-\eta$; assumes initial, isotropic thermal
fluctuations of temperature $T_{\rm env}$ in the comoving frame.

We now include initial thermal fluctuations assuming attenuation is
negligible. We consider (the unphysical but simplistic assumption of)
isotropic thermal noise in the comoving frame—i.e., the same amount of noisy
quanta in all input modes—and parameterize the thermal noise by a single
temperature $T_{\rm env}$, which determines the number of noisy quanta by
means of $n_{\rm env}=[{\rm exp}(\omega/T_{\rm env})-1]^{-1}$. We show in Fig.
7 the LN for the sub-system
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{4}})$ (again, this is the
Hawking-pair emitted by the white hole) versus $T_{\rm env}/T_{H}$ for
$\omega/T_{H}=1$. In order to discuss one problem at a time, we consider no
losses for the moment ($\eta=1$). The plot shows the fragility of entanglement
in noisy environments. This fragility is actually frequency dependent, and we
report in Table 1 the values of $T_{\rm env}$ for which the entanglement in
different bi-partitions is lost, for low ($\omega/T_{H}\rightarrow\infty$) and
high ($\omega/T_{H}\rightarrow 0$) particle emission regimes. In the low
particle regime (large frequency), almost all the entanglement conditions
collapse to $T_{\rm env}<2T_{H}$ [with the exception of the
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{4}},\hat{a}^{\rm(out)}_{k_{3}})$
bi-partition], which is the entanglement-condition for a single horizon in
BECs found by Bruschi et al [52], and can also be deduced from the convergence
of curves in Fig. 6 as $\omega/T_{H}\rightarrow\infty$. This simple condition
is thus a good heuristic proxy for entanglement in bosonic analogue-gravity
systems, at least in the low-particle limit. The situation changes, however,
as frequencies approach the typical Hawking frequency (high-particle limit),
which one can observe by scanning the far-right column of Table 1 or the
curves in Fig. 6 as $\omega/T_{H}\rightarrow 0$. We stress, however, that
these results correspond to an isotropic temperature in the comoving frame of
the white-black hole. For optical analogs, the conclusions drawn henceforth
can drastically change due to large boosts between the lab frame and the
comoving frame, as we discuss briefly below.

Note that these results represent the best-case scenario, when attenuation is
negligible ($\eta=1$), but such can be extended to include attenuation,
leading to a family of critical conditions parameterized by
$(1-\eta,T_{H},\omega)$, similar to the black hole case discussed in Section
III.2. This information is shown in Fig. 8 for the entanglement in the Hawking
pair emitted by the white hole,
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$, for
$\omega/T_{H}=1$ and $\Gamma=1$.

Sub-systems | $(\omega/T_{H}\rightarrow\infty)$ | $(\omega/T_{H}\rightarrow 0)$
---|---|---
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{3}})$ | $T_{\rm env}<2T_{H}$ | 0
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{4}})$ | $T_{\rm env}<2T_{H}$ | $T_{\rm env}<T_{H}$
$(\hat{a}^{\rm(out)}_{k_{3}}|\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$ | $T_{\rm env}<2T_{H}$ | $T_{\rm env}<2T_{H}$
$(\hat{a}^{\rm(out)}_{k_{4}}|\hat{a}^{\rm(out)}_{k_{3}},\hat{a}^{\rm(out)}_{k_{1}})\ \ $ | $T_{\rm env}<2T_{H}$ |  $T_{\rm env}<4T_{H}(\frac{T_{H}}{\omega})$
$(\hat{a}^{\rm(out)}_{k_{1}}|\hat{a}^{\rm(out)}_{k_{4}},\hat{a}^{\rm(out)}_{k_{3}})\ \ $ | $T_{\rm env}<\frac{2T_{H}}{1-\ln(2)T_{H}/\omega}$ |  $T_{\rm env}<4T_{H}(\frac{T_{H}}{\omega})$
Table 1: Necessary criteria for entanglement between different sub-systems of
out-modes for the white-black hole, in different particle-number regimes,
($\omega/T_{H}\rightarrow\infty$) and ($\omega/T_{H}\rightarrow 0$). Assuming
initial, homogenous thermal fluctuations at temperature $T_{\rm env}$ in the
comoving frame and no loss ($\eta=1$).

A situation of practical interest is when the thermal bath is at rest in the
laboratory frame. This case is more subtle since, on the one hand, the modes
$\\{k_{i}\\}$ have different populations (due to having different lab
frequencies; their comoving frequencies are the same) and, on the other hand,
the large Lorentz boost between the lab and comoving frames implies that
higher temperatures are needed for the thermal noise to appreciably affect
entanglement. We find that the threshold lab temperature
$T^{\rm(lab)\,\star}_{\rm env}$ above which entanglement is extinguished is
$T^{\rm(lab)\,\star}_{\rm env}/T_{\rm H}\sim\order{10^{3}}$—significantly
higher than the Hawking temperature, as qualitatively alluded to in previous
studies of optical analogs [34]. This is a subtle and distinctly interesting
point that warrants focused attention, which we leave for future study.

###  V.3 Accurateness of the white-black hole circuit

The circuit of Fig. 5 has been proposed using physical arguments resting on
the analogy with the Hawking effect in gravitational black holes. A legitimate
question is whether this circuit actually captures correctly the physics of
optical white-black hole pairs. The goal of this section is to quantitatively
address this query. Optical white-black hole pairs rest on complex physics;
even in situation where the effect of higher order non-linearities can be
neglected, the dispersive character of these systems may introduce additional
complications, which can break the analogy with the (gravitational) Hawking
effect. Our circuit has been built assuming the analogy works well. Therefore,
any deviation from the predictions of the circuit are actually signaling a
break down of the analogy between the physics of these optical systems and the
Hawking effect. We show in this subsection that there is a region in the
parameter space where the analogy with the Hawking effect is on solid ground
and our circuit describes the system with great accuracy. We also find regions
in the parameter space where dispersive effects lead to a breakdown of the
analogy with the Hawking effect and identify physical origins of such.

The analysis of this subsection rests on a numerical simulation of the optical
systems described above (see [8, 9, 61, 62, 21, 25, 22, 23, 24, 6, 27] for
previous numerical efforts in similar systems). Our simulations are based on
the microscopic analytical model proposed in Ref. [12], building on previous
work [63], and based on the Hopfield model in which the dielectric material is
modeled by a collection of oscillators [64]. The interaction with a strong
electromagnetic pulse modifies the natural frequency of the oscillators, and
this modification, in turn, is experienced by weak electromagnetic probes
propagating thereon as a change in the effective index of refraction. As
described in Ref. [12], the dynamics of weak probes in this model is governed
by a 4th order ordinary differential equation in the comoving coordinates (Eq.
(11) in Ref. [12]). For each (comoving) frequency $\omega$, we numerically
solve this equation and compute the evolution of in-wavepackets which,
asymptotically far from the horizons, are peaked on each of the four
wavenumbers $k_{1},k_{2},k_{3},k_{4}$ (these are the four solutions of the
dispersion relations for a fixed $\omega$). We propagate these wavepackets and
track the evolution to outgoing wavepackets propagating asymptotically far
away from the strong pulse. From this, we obtain the $S$-matrix relating in-
and out-modes. [A companion paper [65], which goes over the details of this
microphysical model and of the numerical simulations, is in preparation.]

In our numerical simulations, we model the perturbation of the refractive
index as ${\delta n(x,t)=\delta
n_{0}\,\sech^{2}\left(\frac{t-x/u}{D}\right)}$, a profile which is a common
and convenient choice [34, 39], where $u$ is the group velocity of this
perturbation, and $x$ and $t$ are space-time coordinates in the lab frame.
This profile is parameterized by two real positive numbers, $\delta n_{0}$ and
$D$; they determine the amplitude and width of the perturbation, respectively.
We have performed simulations for $\delta n_{0}$ in the interval $0.01$ to
$0.1$, and $D$ ranging from $2$ fs to $10$ fs, respectively. These ranges are
chosen by demanding that analog white and black hole event horizons are
present even for the fastest mode (since the dispersion relation of this
system is sub-luminal, low frequency modes are faster). This is also the range
of parameters for which the analogy with the Hawking effect is closer, as we
discuss below.

Our numerical calculations are independent of the circuit of Fig. 5. The
inputs of our numerics are the parameters $\delta n_{0}$ and $D$, and the
output is an $8\times 8$ matrix of numbers representing the $S$-matrix (there
is an $S$-matrix for each value of the frequency $\omega$). We say that the
circuit describes well the physics of the system when there exist values of
the free parameters of the circuit, $r_{H}(\omega)$ and $\theta_{\omega}$ for
which the analytical $S$-matrix associated with the circuit predicts values of
physical quantities of interest that are equal to those obtained from the
numerical $S$-matrix. This is a non-trivial demand. A symplectic evolution of
four modes generically requires 36 free parameters (this is the dimension of
the symplectic group in four dimensions), though this can be reduced to 16
using information from the dispersion relation (see Appendix B for more
discussion on this)—whereas our circuit ansatz contains only three free
parameters.

We do not expect, however, our circuit to describe every single aspect of the
dynamics since, as discussed above, we have not included in the circuit, e.g.,
“single-mode phase shifters” to account for possible phases that different
modes acquire during the propagation. In other words, we expect our circuit to
describe the system “up to phases”. It is not difficult to include these
phases, but we consider unnecessary to add this complication for the following
reason. We have found that the quantities we are interested in—namely, the
mean number of quanta in each out-mode and the entanglement between modes—are
insensitive to phases. In particular, none of the quantities we plot below
depends on, for instance, the squeezing phase $\phi$ in our circuit. Hence, we
will leave this phase unspecified.

Keeping this in mind, to compare both calculations we will focus attention on
their physical predictions, rather than merely comparing the $S$-matrices. We
proceed as follows. For each frequency $\omega$, we determine the two circuit-
parameters—$r_{H}(\omega)$ and $\theta_{\omega}$—by demanding that two
components of the circuit $S$-matrix matches the same components of the
numerical $S$-matrix. More concretely, we obtain $\theta_{\omega}$ from the
component (3,3) of the numerical $S$-matrix, which, according to the circuit,
is equal to $\cos^{2}\theta_{\omega}$ [see Eq. (V.6)]; $r_{H}(\omega)$ is then
obtained from the component (7,5), which corresponds to
$\cos\theta_{\omega}\,\cosh r_{H}(\omega)$, according to the circuit. In this
way, we determine $r_{H}(\omega)$ and $\theta_{\omega}$ (we repeat the the
calculation for each value of $\omega$ in the regime of interest). Next, we
substitute these values into the $S$-matrix written in Eq. (V.6), and use it
to evaluate quantities of interest. The values for physical quantities
obtained in this way are then compared with the results derived with the
numerically computed $S$-matrix. If both calculations agree, we say the
circuit captures the correct physics. We insist that this is a non-trivial
check, since the circuit-calculation uses inputs from just two-components of
the numerical $S$-matrix, while we demand the two calculations to agree in
quantities which depend on complicated combinations of all components. This
summarizes the strategy we follow. We now show the results.

Figure 9: The dots correspond to the numerically determined values of
$\ln(\coth^{2}r_{H})$ versus $\omega$, for a strong pulse with $D=6\,\rm{fs}$
and $\delta n_{0}=0.05$. We also show a straight line, $\omega/T_{H}$, which
is a good fit to the data for $T_{H}=3.51\,{\rm K}$. This plot shows that the
squeezers representing each horizon produce radiation with a frequency
spectrum in good agreement with a black-body distribution at temperature
$T_{H}=3.51\,{\rm K}$. Stronger deviations from thermality emerge at high
frequencies.

First, we plot the two-circuit parameters $r_{H}(\omega)$ and
$\theta_{\omega}$ versus frequency $\omega$ (in this simulation we use
$D=6\,{\rm fs}$ and $\delta n_{0}=0.05$). More concretely, Figs. 9 and 10 show
$\ln(\coth^{2}r_{H})$ and $1-\cos^{2}\theta$, respectively. On the one hand,
$\ln(\coth^{2}r_{H})$ helps us to evaluate whether the squeezers associated
with both white and black hole horizons emit radiation with a black-body
frequency spectrum (as expected from the analogy with the Hawking effect)
since, in such a situation, $r_{H}$ must depend on frequency as $r_{H}={\rm
arcoth}\,(e^{\omega/2T_{H}})$, with $T_{H}$ a frequency independent real
number. This in turn implies that $\ln(\coth^{2}r_{H})=\omega/T_{H}$ is a
linear function of $\omega$. Figure 9 confirms that this is in fact the case
to a great approximation. From this plot, we can read the value of the Hawking
temperature, from which we obtain $T_{H}=3.51\,{\rm K}$. This value agrees
with analytical approximations derived in Ref. [12] within a few percent.

The quantity $\cos^{2}\theta$ informs us about the transmission probability of
the potential barrier (recall, $\Gamma_{\omega}=\cos^{2}\theta_{\omega}$).
Fig. 10 shows that $\cos^{2}\theta$ is very close to one for all frequencies
in the range of interest, and it is closer to one for large $\omega$. Since
the beam splitters in the circuit are the means by which the mode $k_{2}$
interacts with the rest, these results tell us that $k_{2}$ evolves almost
entirely in solitude, with little interaction with the rest of modes.
[Physically, this is because the wavelength of this mode in the lab frame is
much larger than the rest, and therefore this mode probes the system at
different scales.]

Figure 10: Numerically determined value of $1-\cos^{2}\theta$ versus
$\omega/T_{H}$, for a strong pulse with $D=6\,\rm{fs}$ and $\delta
n_{0}=0.05$. The plot shows that the transmission probability of the beam-
splitters ($\cos^{2}\theta$) of the circuit is very close to one. This implies
the mode $k_{2}$ weakly interacts with the other modes in the system.

Figure 11: Expectation values of the number operator of the out-modes
$k^{\rm(out)}_{1}$, $k^{\rm(out)}_{2}$, $k^{\rm(out)}_{3}$, and
$k^{\rm(out)}_{4}$ resulting from evolving the vacuum state (spontaneous
creation of particles), for a strong pulse with $D=6\,\rm{fs}$ and $\delta
n_{0}=0.05$. The red squares represent the predictions of our circuit, while
the black dots correspond to the results of our numerical code. The number of
quanta produced in the mode $k^{\rm(out)}_{2}$ is several order of magnitude
smaller than in the rest of modes. We observe great agreement between the
predictions of the circuit and our numerical code for the modes
$k^{\rm(out)}_{1}$, $k^{\rm(out)}_{3}$ and $k^{\rm(out)}_{4}$, which are the
ones involved in the Hawking process. Discrepancies appear in the mode
$k^{\rm(out)}_{2}$; the circuit underestimates the quanta produced in this
mode. This indicates a (weak) coupling between the modes $k^{\rm(out)}_{1}$
and $k^{\rm(out)}_{2}$ not captured by the circuit.

Next, we compare the predictions for the mean number of quanta in the out-
modes $\hat{a}^{\rm(out)}_{k_{1}}$, $\hat{a}^{\rm(out)}_{k_{2}}$,
$\hat{a}^{\rm(out)}_{k_{3}}$, and $\hat{a}^{\rm(out)}_{k_{4}}$, starting from
vacuum input. This is shown in different panels of Fig. 11. A few interesting
messages emerge from these plots. The number of quanta produced in the mode
$\hat{a}^{\rm(out)}_{k_{2}}$ is much smaller than the rest, by about four
orders of magnitude. This confirms that the mode $k_{2}$ is a spectator in
this process. The symplectic circuit also produces results in great agreement
with the numerical code for the number of quanta in the modes
$\hat{a}^{\rm(out)}_{k_{1}}$, $\hat{a}^{\rm(out)}_{k_{3}}$ and
$\hat{a}^{\rm(out)}_{k_{4}}$ (differences are of order of one part in
thousand, or smaller). However discrepancies do appear for the mode
$\hat{a}^{\rm(out)}_{k_{2}}$.

Figure 12: Logarithmic negativity between pairs of modes in the out-state
corresponding to vacuum input, for a strong pulse with $D=6\,\rm{fs}$ and
$\delta n_{0}=0.05$. The red squares represent the predictions of our circuit,
while the black dots correspond to the results of our numerical code. The plot
shows great agreement for the pairs ($k^{\rm(out)}_{1}$, $k^{\rm(out)}_{3}$)
and ($k^{\rm(out)}_{1}$, $k^{\rm(out)}_{4}$), which are the pairs that
dominate the entanglement in the final state. The value of LN for the pair
($k^{\rm(out)}_{1}$, $k^{\rm(out)}_{2}$) is several orders of magnitude
smaller and is not well captured by our circuit.

We show in Figs. 12 the entanglement between pair-wise out-modes. First of
all, such entanglement is exactly zero unless the mode
$\hat{a}^{\rm(out)}_{k_{1}}$ is a member of the pair. This is true both in our
circuit as well as in our numerical calculations, and it has a simple
explanation: entanglement can only be generated if modes with different
symplectic norms interact. The mode $\hat{a}_{k_{1}}$ is the only one having
negative symplectic norm [12]. The remaining modes have positive norm (which
is a consequence of the the dispersion relation of the system [12]). Our
circuit respects this fact, since both squeezers involve a positive- and a
negative-norm mode. Hence, in Figs. 12, we only show the non-zero pair-wise
entanglements. We see that the entanglement in the pairs
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{3}})$ and
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{4}})$ shows again a great
agreement between the outcomes of the circuit and the numerical calculations.
The entanglement in the pair
$(\hat{a}^{\rm(out)}_{k_{1}},\hat{a}^{\rm(out)}_{k_{2}})$ predicted by our
circuit is several orders of magnitude too small.

The discrepancy in the predictions involving the mode
$\hat{a}^{\rm(out)}_{k_{2}}$ for both the number of quanta and entanglement,
have a common origin—namely, the absence of a direct coupling between the
modes $\hat{a}^{\rm(out)}_{k_{1}}$ and $\hat{a}^{\rm(out)}_{k_{2}}$ in our
circuit. Since these modes have symplectic norms of different signs, such
coupling would imply a source of particle-pair production, which would account
for the extra quanta in the mode $\hat{a}^{\rm(out)}_{k_{2}}$ shown in the
numerical simulation, as well as the extra entanglement between the modes
$\hat{a}^{\rm(out)}_{k_{1}}$ and $\hat{a}^{\rm(out)}_{k_{2}}$. However, the
fact that our circuit captures very well the behavior of the mode
$\hat{a}^{\rm(out)}_{k_{1}}$ implies that such coupling is very weak and does
not significantly affect the rest of the modes in the system. [For very large
frequencies, $\omega\gg T_{H}$, this coupling has a larger relative effect,
since the number of quanta and entanglement in the rest of modes fall of
exponentially with $\omega$, due to its black-body character, while the
coupling of the mode $k_{2}$ with the rest, falls at a slower rate.] It is not
difficult to modify our circuit to account for the missing coupling between
modes $\hat{a}^{\rm(out)}_{k_{1}}$ and $\hat{a}^{\rm(out)}_{k_{2}}$, by simply
adding an extra squeezer. Though, we find this additional complication
unnecessary, since such coupling is unrelated to the Hawking effect and, more
importantly, it produces negligible effects for the modes actually involved in
the Hawking process for frequencies in the range $\omega\in(0,10\,T_{H})$. A
similar dispersive effect was previously identified in [60], in a different
system.

Our numerical simulations reveal another dispersive effect, which indicates a
second departure from the analogy with the Hawking effect and the predictions
of our circuit. We identify the physical origin of this effect as being due to
the tunneling of the mode $k_{3}$ (the ingoing Hawking mode) from outside the
white hole to the exterior of the black hole horizon—hence, indicating that
these are not perfect event horizons. This limitation is intrinsic to this
(and other) analog models, and originates from the fact that, in between
horizons, there actually exist modes propagating in both directions, but the
modes propagating from the white hole to the black hole have complex
wavenumbers [60, 12]. [These modes are not drawn in Fig. 4]. They are
evanescent modes (not propagating modes), in the sense that their amplitude
falls off exponentially in the interior region. If the pulse generating the
horizons is narrow enough, a significant portion of the mode can emerge on the
black hole side. This is not very different from the familiar tunneling effect
in wave mechanics. From general arguments, this effect is expected to be more
important for narrow (small $D$) and weaker pulses (small $\delta n_{0}$), as
well as for long wavelength modes. In agreement with this expectations, we
observe this effect only for the mode $k_{3}$, since this is the lone, longest
wavelength mode propagating to the right (with the conventions of Fig. 4) and
for low frequencies. This effect is sizable already for $D=4$ fs, and $\delta
n_{0}=0.01$ (although only for very low frequencies) and quickly disappears
when these two parameters are increased.

Figure 13: Tunneling probability, $\absolutevalue{t_{33}}^{2}$, across the
white-black hole. In the ideal case, when the analogy to gravity is firm, the
probability to tunnel through the white-black hole is zero and the mode
$k_{3}^{(\rm out)}$ (the outgoing Hawking radiation from the black hole) has
no contribution from the mode $k_{3}^{(\rm in)}$ (the ingoing mode to the
white hole). However, as the height of the strong pulse, which generates the
effective horizons, decreases and the width narrows, transmission through the
white hole horizon increases. The parameters of the strong pulse used in this
figure are $\delta n_{0}=.01$ and $D=4$ fs. An (approximate) Hawking
temperature can still be found for these parameter values, $T_{H}\approx 1.14$
K.

Tunneling by the $k_{3}$ mode (from the white hole exterior to the black hole
exterior) obviously breaks the analogy with the Hawking effect, since nothing
should emerge from the interior of an actual black hole. Not surprisingly,
this effect also produces significant differences between the predictions of
our circuit and the numerical simulations. These differences manifest
primarily in a non-zero tunneling probability, $\absolutevalue{t_{33}}^{2}$,
from $k_{3}^{(\rm in)}$ to $k_{3}^{(\rm out)}$ (which is identically zero in
the perfect analog case; see Fig. 5). In Fig. 13, we plot the
$\absolutevalue{t_{33}}^{2}$ for a pulse characterized by $D=4\,\rm{fs}$ and
$\delta n_{0}=0.01$. The tunneling probability is higher for low-frequency
modes ($\omega/T_{H}\lesssim 1$), as expected on physical groundsV.4V.4V.4Low
frequency modes have longer wavelengths, and thus the pulse is relatively
narrower for such modes., and is non-negligible
($\absolutevalue{t_{33}}^{2}\sim.001-.1$) for this choice of pulse parameters.
As a comparison, for $D=6\,\rm{fs}$ and $\delta n_{0}=0.05$, where the analogy
to the Hawking effect holds well, we find $\absolutevalue{t_{33}}^{2}\lesssim
10^{-6}$ in the frequency range of interest. We are currently developing a
more detailed analysis of tunneling, which will appear in a later paper [65],
as such is prominent in realistic optical platforms [34, 8, 9, 39].

In summary, for a strong pulse with $D=6\,\rm{fs}$ and $\delta n_{0}=0.05$ (or
larger), we find that our circuit accounts very well (with differences of
order of one part in thousand, or smaller) for the evolution of the modes
actually involved in the Hawking effect, namely modes $k^{\rm(out)}_{1}$,
$k^{\rm(out)}_{3}$ and $k^{\rm(out)}_{4}$. The analogy with the Hawking effect
is on firm ground. The mode $k^{\rm(out)}_{2}$ interacts very weakly with the
rest. We have identified two dispersive effects which break the analogy with
the Hawking effect and induce deviations from the predictions of our circuit.
On the one hand, dispersive effects produce a coupling between the mode
$k^{\rm(out)}_{1}$ and $k^{\rm(out)}_{2}$ which induce production of pairs of
quanta in the modes $k^{\rm(out)}_{1}$ and $k^{\rm(out)}_{2}$. Such production
is not thermal and is unrelated to the Hawking effect, and it can be neglected
for low frequencies $\omega<10\,T_{H}$. On the other hand, if the strong pulse
producing the horizon is too weak or too narrow (approximately under
$D=6\,\rm{fs}$ and $\delta n_{0}=0.05$), the white- and black hole horizons
are imperfect and modes can propagate in both directions, breaking the
analogy. These are intrinsic limitations caused by the dispersion-relation
underlying this—and other—analog systems. It is worth mentioning that these
threshold values for $D$ and $\delta n_{0}$, at which tunneling becomes
relevant, depend on the properties of the material.

##  VI Entanglement enhancement via the stimulated effect

###  VI.1 General remarks

As discussed above, practical constraints—such as initial thermal fluctuations
and attenuation—are a detriment to the quantum correlations generated during
the Hawking process. These are serious hurdles for experimental platforms to
overcome, in order to genuinely observe the quantumness of the Hawking
process. Of course, one can always seek to improve the operating conditions
(low $T_{\rm env}$ and $\eta\approx 1$). However, this requires operating at
temperatures near the Hawking temperature of the system, which is generally
quite low, as well as having control of unwanted scattering channels (i.e.,
increasing efficiency of the setup). An alternative way to overcome noise is
to utilize quantum resources at the inputs. We discuss this possibility for
two specific strategies:

  1. (i)

Entanglement resonance: One can, in some sense, revitalize entanglement in the
face of noise by injecting a controllable two-mode squeezed-vacuum state into
the analog system, which is phase-matched (in resonance) with the Hawking
process. The resonance condition allows for the entanglement in each source
(the input and the Hawking process, respectively) to add constructively, thus
boosting the total amount of entanglement in the output. This can be used to
bypass the temperature conditions discussed previously. The proviso is that
one must be able to phase-match the controllable two-mode squeezer to the
Hawking process of the analog system in order to observe constructive effects;
otherwise, one could induce destructive interference, potentially
disentangling the particles. Another aspect to bear in mind here is the
ability to distinguish between the entanglement generated by the input
resource and the Hawking process; although this may be possible with
sufficient control over the input resource and sufficient knowledge of the
analog system. We do not explore this idea further, as such has already been
discussed in the context of BECs in Ref. [52].

  2. (ii)

Leverage a single-mode resource: Likewise, injecting a single-mode squeezed
vacuum into one of the modes can enhance the entanglement generated during the
Hawking process and help to subdue background thermal noise [43]. The benefit
here (contrasted with the entanglement-resonance scheme above) is that the
single-mode squeezing approach does not rely on a phase-matching condition
(the entanglement varies only with the amount of squeezing), and thus no fine-
tuning is necessary. The drawback, however, is that this approach is more
sensitive to losses. Nevertheless, we provide a quantitative assessment of
this strategy in the context of an analog white-black hole in the next
section, and discuss where it may be beneficial.

The amplification of the entanglement in the stimulated Hawking process may
come as a surprise at first, since the stimulated process is usually regarded
as a purely classical effect. We insist that this intuition is true if the
stimulation is done with a ‘classical’ state, namely a coherent state or
mixtures thereof. The covariance matrix of a coherent state is the
identity—the same as for vacuum—and since entanglement is derived exclusively
from the covariance matrix (with no reference to the first moments),
entanglement in the output state cannot distinguish between initial vacuum or
coherent input. In other words, by illuminating the system with a coherent
state, we can amplify the output intensities, but there is no amplification of
entanglement in the final state. This conclusion, however, ceases to be true
if non-classical inputs are used to stimulate the system, even if the input is
separable (not entangled). Appendix A shows an example of this mechanism for a
simple two-mode squeezing interaction.

Entanglement-enhancement strategies are promising, however we must also be
careful of the conclusions we draw about the Hawking process itself when non-
classical—even non-entangled—resources are in play. The reason being that
there are passive operations that can be done on non-classical, separable
inputs which generate quantum entanglement. For instance, it is known that a
network of passive elements (consisting only of orthogonal symplectic
transformations, such as the beam-splitter discussed in Appendix A) generally
generates multi-mode entanglement starting from single-mode squeezed vacuum at
the input [66, 67]. A very simple example is the scattering of a single-mode
squeezed vacuum by a potential barrier: the transmitted and the reflected
beams are quantum mechanically entangled, even when the input is separable;
the barrier is able to transform initial single-mode squeezing into two-mode
entanglement. If we now consider the network as a blackbox, with elements
unknown to us, would we deduce that the box is intrinsically quantum by
nature, or is the quantumness of an output a feature solely of the input
resource?

Hence, is there a way to extract the quantum features of the output that are
products of the Hawking process and not necessarily due to our input
resources? One indication would be that the analog system works as an
amplifier (creates particles) for the entangled modes under question and that
the amount of entanglement increases with the number of particles generated by
the system (holding everything else fixed). Thus, concurrently observing
amplification of the modes as well as quantum correlations would provide
support for the quantumness of the stimulated Hawking process in the presence
of extrinsic non-classical resources. Though, more sophisticated methods for
distinguishing such features may be desired in practice. Sufficient control of
the inputs is thus a necessity.

###  VI.2 Example: Seeding with single-mode squeezed vacuum

We illustrate the strategy of stimulating the system with a single-mode
squeezed vacuum with the example of the optical white-black hole discussed
above. These results have been reported in [43]; here we add further details
omitted there and generalize the strategy to non-isotropic noise. Our strategy
is to illuminate the white hole (i.e., populate the mode
$a_{k_{3}}^{\rm(in)}$; see Fig. 4) with a single-mode squeezed-vacuum state
and monitor the entanglement in the Hawking-pairs emitted by the white hole,
i.e. the modes $a_{k_{1}}^{\rm(out)}$ and $a_{k_{4}}^{\rm(out)}$, as well as
the output intensities. As we shall show, illuminating the white hole with a
single-mode squeezed vacuum allows one to tune the output entanglement as a
function of the input squeezing, and extract the symplectic-circuit parameters
(squeezing amplitudes and beam splitter angles) of the white-black hole from
the intensities and entanglement, even in the presence of thermal fluctuations
and (a mild amount of) attenuation. Before moving forward, we emphasize again
that such entanglement enhancement is not possible by stimulating the process
with classical states, such as coherent states (or more generally, a convex
combination of coherent states, e.g. a thermal state) for the reasons
discussed in the previous subsection and in Appendix A. [All results described
below can be understood in the simpler context of a two-mode squeezer
described in detail in Appendix A.]

The single-mode covariance matrix for the initially squeezed mode
$a_{k_{3}}^{\rm(in)}$, written in terms of the squeezing strength $s$, is
given by $N_{\rm env}\,\text{e}^{2\,s\,\bm{\sigma}_{z}}$, with
$\bm{\sigma}_{z}$ the familiar $z$-Pauli matrix, where we have assumed
squeezing along the quadrature $P$ (and, consequently, anti-squeezing along
$Q$ quadrature)—although the direction of squeezing does not alter the
quantities that we are interested in. Assuming all input modes are populated
by thermal fluctuations, each with individual noise factor $N_{k_{i}}$,
$i=1,2,3,4$, the input covariance matrix is given by the direct sum of single-
mode covariance matrices

$\bm{\sigma}^{\rm(in)}=N_{k_{1}}\bm{I}_{2}\oplus N_{k_{2}}\bm{I}_{2}\oplus
N_{k_{3}}\,e^{2s\bm{\sigma}_{z}}\oplus N_{k_{4}}\,\bm{I}_{2}\,.$ (VI.1)

Given this input to the white-black hole and including attenuation effects,
the output covariance matrix is formally given by,

$\bm{\sigma}_{\rm WB}^{\rm(out)}=\eta\,\bm{S}_{\rm
WB}\,\bm{\sigma}^{\rm(in)}\,\bm{S}_{\rm WB}^{\top}+(1-\eta)\,\bm{I}_{6},$
(VI.2)

(the first moments ${\bm{\mu}}$ of the initial state—and hence of the final
state too—are chosen to be zero). From this covariance matrix all quantities
of interest can be calculated.

Figure 14: Entanglement between the photon-pairs emitted by the white hole,
$(a_{k_{1}}^{\rm(out)},a_{k_{4}}^{\rm(out)})$, versus initial squeezing (solid
curves), for various efficiency values $\eta=1,.75,.5$ (from top to bottom)
and the value ${T_{\rm env}^{(\rm lab)}=100T_{H}}$ has been taken for the
initial thermal fluctuations in the lab frame. Dashed lines represent zero
initial squeezing, similarly with $\eta=1,.75,.5$ (from top to bottom).

#### Entanglement enhancement with squeezing:

In the near-ideal scenario of perfect efficiency, $\eta=1$, we have found that
single-mode squeezing can always be used to overcome the degrading effects of
initial thermal fluctuations. In other words, for a given frequency
$\omega/T_{H}$ and arbitrary thermal noise, there exists a value of the
squeezing strength $s^{\star}(\omega)$ such that the LN is non-zero. Moreover,
the entanglement is shown to increase with more squeezing, $s$ (see the curve
corresponding to $\eta=1$ in Fig. 14). Importantly, the entanglement is not
generated by the squeezing resource itself per se; in the sense that, for a
set squeezing strength $s$, the entanglement between the
$(a_{k_{1}}^{\rm(out)},a_{k_{4}}^{\rm(out)})$ modes increases with the
Hawking-squeezing parameter, $r_{H}$, and vanishes entirely as $r_{H}$ goes to
zero. In this sense, the origin of the entanglement can be attributed to
Hawking pair production.

We now discuss the effects of attenuation ($\eta<1$). As shown in Fig. 14,
attenuation limits the entanglement-enhancement gained by initially squeezing
the white hole input, which also depends on the initial thermal fluctuations
in the lab frame. However, enhancements are to be found even for very
inefficient setups; for instance, initial squeezing can help even for
$\eta=.5$ and $T_{\rm env}^{(\rm lab)}=100T_{H}$. The thermal fluctuations (in
the lab frame) do not cause much harm to the entanglement generated in the
system, even when efficiency effects are accounted for, thanks to the large
Lorentz boost between the lab frame and the comoving frame. Our squeezing
protocol is thus more robust to efficiency effects than we anticipated in Ref.
[43].

It is worth mentioning that, upon stimulating the white hole with a single-
mode squeezed vacuum, the entanglement criterion imposed by the parameter
$\Delta$ [$\Delta<0$; Eq. (II.13)] is not sufficient to witness entanglement
in our setup across all regions of parameter space; see Appendix C and Fig. 22
for more details.

Figure 15: Logarithmic negativity between modes $\hat{a}_{k_{1}}^{(\rm out)}$
and $\hat{a}_{k_{4}}^{(\rm out)}$ forming the Hawking pairs emitted by the
white hole horizon, versus the gain of the initial single-mode squeezing,
$e^{2s}$. Curves correspond to different possible values of the Hawking
temperature $T_{H;i}$, at a given comoving frequency $\omega$. This plot is
obtained for a transmission coefficient of the potential barrier
$\Gamma=0.999929$.

#### Circuit parameters from intensities and LN:

An interesting by-product of squeezing is that it also allows us to extract
the functional form of the symplectic-circuit parameters from the output
intensities of the white hole only, assuming our squeezing source is
characterized and tunable. Using Eq. (VI.2), we obtain an analytical
expression for the mean-occupation number for each mode, and find that they
all have the simple form,
$\expectationvalue*{\hat{n}^{\rm(out)}_{k_{i}}}=m_{i}\sinh^{2}s+b_{i}$, where
$\sinh^{2}s$ is the number of initially squeezed photons and the intercepts
$b_{i}$ are independent of the squeezing parameter $s$. The slopes, $m_{i}$,
encode the circuit parameters via the relations,

$\displaystyle m_{1}$
$\displaystyle=N_{k_{3}}\,\eta\,\cos^{2}\theta_{\omega}\sinh^{2}r_{H},$
$\displaystyle m_{2}$
$\displaystyle=N_{k_{3}}\,\eta\,\sin^{2}\theta_{\omega},$ $\displaystyle
m_{3}$ $\displaystyle=0,$ $\displaystyle m_{4}$
$\displaystyle=N_{k_{3}}\,\eta\,\cos^{2}\theta_{\omega}\cosh^{2}r_{H},$ (VI.3)

where, recall, $N_{k_{3}}$ is the noise factor in the initial state of the
mode $k^{\rm(in)}_{3}$. The expressions for $b_{i}$ are lengthy, and will not
be used in this section (for completeness, we report them in expressions (D)
in Appendix D). Note that all slopes $m_{{i}}$ are proportional to the thermal
noise in the input mode $k_{3}$ and to the attenuation parameter $\eta$. For a
given frequency, we can monitor the output intensities of each mode as we vary
the squeezing parameter $s$. If we then plot the output intensities versus the
intensity of the initial squeezing $\sinh^{2}s$, we will find straight lines
with slopes given as above. Taking ratios of these slopes we cancel out the
factors $N_{k_{3}}$ and $\eta$, allowing us to map out the frequency dependent
forms of the Hawking intesity $r_{H}(\omega)$ from the ratio,

$\frac{m_{1}}{m_{4}}=\tanh^{2}r_{H}=\text{e}^{-\omega/T_{H}},$ (VI.4)

from which the Hawking temperature may also be derived.

The value of the Hawking temperature, $T_{H}$ (or equivalently, Hawking’s
squeezing intensity, $r_{H}$) can be independently checked against the size of
the entanglement between, say, the Hawking pair $k_{1}$ and $k_{4}$ emitted by
the white hole, as described in [43]. As mentioned previously, using the
circuit, we can obtain analytical expressions for the LN between these two
modes (and any other bi-partition), in terms of the parameters of the circuit
and the initial state. These expressions are lengthy, but they can be easily
obtained using any software for symbolic calculus. Rather than writing the
lengthy expressions for LN, we plot its value in terms of the squeezing