Event horizons are tunable factories of quantum entanglement

Ivan<EMAIL_ADDRESS>corresponding author

Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
LA 70803, USA
Anthony J<EMAIL_ADDRESS>
Department of Electrical and Computer Engineering, University of Arizona,
Tucson, 85721, USA
Dimitrios<EMAIL_ADDRESS>
Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
LA 70803, USA

That event horizons generate quantum correlations via the Hawking effect is
well known. We argue, however, that the creation of entanglement can be
modulated as desired, by appropriately illuminating the horizon. We adapt
techniques from quantum information theory to quantify the entanglement
produced during the Hawking process and show that, while ambient thermal noise
(e.g., CMB radiation) degrades it, the use of squeezed inputs can boost the
non-separability between the interior and exterior regions in a controlled
manner. We further apply our ideas to analog event horizons concocted in the
laboratory and insist that the ability to tune the generation of entanglement
offers a promising route towards detecting quantum signatures of the elusive
Hawking effect.

Essay written for the Gravity Research Foundation 2022 Awards for Essays on
Gravitation

Submitted on March 26, 2022

The allure of black holes has captivated physicists for nearly a century,
partly due to the fact that their internal mechanisms are completely concealed
by a dark cloak —the black hole’s event horizon. The mystique of black holes
was amplified when, in a set of seminal papers in the early 1970’s [1, 2],
Stephen Hawking showed that, once quantum fluctuations are accounted for, a
black hole is not actually black but, instead, emits radiation as a hot body,
gradually losing its mass in what has been dubbed as the Hawking evaporation
process. Even more, Hawking’s calculations imply that the evaporation products
are quantum mechanically entangled with the bowels of the black hole.

Understanding the generation of entanglement by a black hole, with relation to
its surrounding (perhaps even “noisy”) environment, elicits deeper knowledge
about the black-hole evaporation phenomenon. In the past, entanglement entropy
between the interior and exterior of the event horizon has been extensively
used for these purposes [3], but this quantity only quantifies entanglement
when the global state of the system is pure. This is not the case, for
instance, if a black hole is immersed in a thermal bath, e.g., the cosmic
microwave background (CMB). In this essay, we leverage techniques from quantum
information theory to compute and interpret the entanglement generated during
the Hawking process for an evaporating black hole in a nontrivial environment.

Specifically, we apply the theory of Gaussian states for continuous variable
systems with a finite number of interacting modes [4]. At first glance, our
task may seem barren, since the Hawking effect is formulated in the context of
field theory with its infinitely many degrees of freedom. In Hawking’s
original derivation, this is manifested in the fact that the Hawking mode
reaching future null infinity (as a normalized wave-packet sharply peaked on a
positive frequency mode $e^{-i\,w\,u}$), when propagated backwards in time to
past null infinity, consists of a superposition of modes
$e^{-i\,\tilde{w}\,v}$ over all frequencies $\tilde{w}$ ($u$ and $v$ are the
standard retarded and advanced null coordinates, respectively). In other
words, the evolution mixes infinitely many “in” modes with well defined
frequencies $\tilde{w}$ to produce one “out” mode with frequency $w$. However,
as was noticed in [5], by appropriately combining “in” modes
$e^{-i\,\tilde{w}\,v}$ with positive frequency, one can find the progenitors
of the Hawking modes. These progenitors are two normalized modes $F_{I}(w)$
and $F_{II}(w)$ at past null infinity, which define the same “in” vacuum and,
conveniently, have the property that their evolution produces exactly a single
“out” Hawking mode $e^{-i\,w\,u}$ and a single partner mode falling into the
black hole. The explicit form of $F_{I}(w)$ and $F_{II}(w)$ is not important
for our purposes and can be found in [5, 6].444We remark that both $F_{I}(w)$
and $F_{II}(w)$ are made mostly of modes $e^{-i\,\tilde{w}\,v}$ with
ultrahigh-frequencies $\tilde{w}$ at past null infinity [5, 7]. This is the
origin of the well-known trans-Planckian problem [8, 9]. However an important
observation —not often appreciated— is that this choice of “in” modes
factorizes the evolution into uncoupled $w$-sectors, in the sense that pairs
of modes $F_{I}(w)$ and $F_{II}(w)$ do not mix with other pairs labelled by
different $w$. This decoupling of $w$-modes allows one to straightforwardly
apply techniques from Gaussian quantum information theory, as we now explain.

Figure 1: Various depictions of the Hawking process. The main ingredients —a
two-mode squeezer associated with pair-production at the event horizon and a
beam-splitter induced by the potential barrier surrounding the horizon— are
emphasized in the middle and right panels. We illustrate the difference
between low and high frequency modes by their wavelengths. The right panel
represents the process in the form of a quantum circuit.

The evolution of the $F_{I}(w)$ and $F_{II}(w)$ “in” modes is made of two
contributions of distinct physical origin, which we wish to differentiate. Let
$\hat{a}^{\rm in}_{I}(w)$ and $\hat{a}^{\rm in}_{II}(w)$ be annihilation
operators defined from normalized wave-packets sharply peaked on the modes
$F_{I}(w)$ and $F_{II}(w)$ at past null infinity, and $\hat{a}^{\rm out}_{w}$
and $\hat{a}^{\rm int}_{I}(w)$ be annihilation operators similarly defined
from Hawking modes of frequency $w$ at future null infinity and their partner
crossing the horizon, respectively. The first contribution —which constitutes
the core of the Hawking effect— is the transformation

$\displaystyle\hat{a}^{\rm in}_{I}$ $\displaystyle\to$ $\displaystyle a^{\rm
int}_{I}\,\cosh{r_{H}}-\hat{a}^{\rm out\,\dagger}_{w}\,\sinh{r_{H}}\,,$
$\displaystyle\hat{a}^{\rm in}_{II}$ $\displaystyle\to$
$\displaystyle\hat{a}^{\rm out}_{w}\,\cosh{r_{H}}-a^{\rm
int\,\dagger}_{I}\,\sinh{r_{H}}\,,$ (1)

where $r_{H}=\tanh^{-1}{e^{-\frac{w}{2\,T_{H}}}}$ and $T_{H}$ is Hawking’s
temperature. Here, we have omitted the label $w$ and the angular quantum
numbers $\ell$ and $m$ for brevity. In the terminology of quantum optics, this
transformation is precisely a process of two-mode squeezing [10], which is
responsible for the creation of entangled Hawking pairs.

A second contribution of the Hawking process is “back-scattering”, which
occurs as Hawking radiation tries to escape to infinity. In doing so, the
outgoing radiation meets and scatters off the gravitational potential barrier
surrounding the black hole —leading to a portion of the wave-packet being
reflected (or scattered back) into the black hole; the remaining portion gets
transmitted to future null infinity. This classical scattering phenomenon
involves a third “in” mode, made of wave-packets centered on
$e^{-i\tilde{w}\,v}$ at past null infinity, with frequency $\tilde{w}\approx
w$ (i.e., back-scattering does not involve an ultrahigh blue-shift), and a
second “int” mode falling into the horizon, centered on
$e^{-i\,\tilde{w}\,v}$, also with $\tilde{w}\approx w$ (see Fig. 1). If we
denote by $\hat{b}^{\rm in}_{w}$ and $\hat{b}^{\rm int}_{w}$ the annihilation
operators defined for these wave-packets, respectively, back-scattering
induces the transformation,

$\displaystyle\hat{b}^{\rm in}_{w}$ $\displaystyle\to$
$\displaystyle\hat{b}^{\rm int}_{w}\,\cos{\theta}+\hat{a}^{\rm
out}_{w}\,\sin{\theta}\,,$ $\displaystyle\hat{a}^{\rm out}_{w}$
$\displaystyle\to$ $\displaystyle\hat{a}^{\rm
out}_{w}\,\cos{\theta}-\hat{b}^{\rm int}_{w}\,\sin{\theta}\,,$ (2)

where $\Gamma=\cos^{2}\theta$ is the probability to transmit across the
potential barrier, which depends on $w$, $\ell$, $m$ and the spin $s$ of the
field under consideration. In the terminology of quantum optics, this is the
action of a beam splitter.

Thus, for each individual frequency $w$, the Hawking process can be understood
as an evolution from three modes to three modes, $\hat{a}^{\rm
in}_{I},\hat{a}^{\rm in}_{II},\hat{b}^{\rm in}_{w}\to\hat{a}^{\rm
out}_{w},\hat{a}^{\rm int}_{I},\hat{b}^{\rm int}_{w}$ —only one of which
escapes to infinity— made by concatenating a two-mode squeezer and a beam
splitter, as depicted in Fig. 1. The beam-splitter divides both the intensity
and the entanglement generated by the squeezer, in such a way that the
$\hat{a}^{\rm out}_{w}$ mode reaching infinity is generically entangled with
both, the high frequency mode $\hat{a}^{\rm int}_{I},$ and the low frequency
one $\hat{b}^{\rm int}_{w}$ (frequencies measured by freely falling observers
crossing the horizon). Describing the Hawking process in this way is
advantageous, as such allows us to compute the resulting evolution of various
input Gaussian states in a few simple lines.

Recall that the information in a quantum Gaussian state is exhaustively
encoded in its first moments and its covariance matrix (defined below). For
each “in” mode, define a pair of canonically conjugate operators
$\hat{x}_{\alpha}=\frac{1}{\sqrt{2}}(\hat{a}_{\alpha}+\hat{a}_{\alpha}^{\dagger})$,
$\hat{p}_{\alpha}=\frac{-i}{\sqrt{2}}(\hat{a}_{\alpha}-\hat{a}_{\alpha}^{\dagger})$,
where the index ${\alpha}$ labels the three “in” modes. Let
$\hat{r}^{i}=(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2},\hat{x}_{3},\hat{p}_{3})$
be the vector of canonical operators. The first moments of a Gaussian state
$\hat{\rho}$ are $\mu^{i}={\rm Tr}[\hat{\rho}\,\hat{r}^{i}]$, and its
covariance matrix $\sigma^{ij}={\rm
Tr}[\hat{\rho}\,\\{\hat{r}^{i}-\mu^{i},\hat{r}^{j}-\mu^{j}\\}]$, where the
curly brackets indicate an anti-commutator.

The Hawking process is a linear evolution, as is evident from (S0.Ex1) and
(S0.Ex2); hence it preserves Gaussianity: i.e., the evolution maps an initial
Gaussian state with $(\vec{\mu}^{\rm in},\sigma^{\rm in})$ to another Gaussian
state with $(\vec{\mu}^{\rm out},\sigma^{\rm out})$. Specifying the input
moments and the evolution, we can obtain any desired information about the
Hawking effect. For instance, the mean number of quanta on any “out” mode is
$\langle\hat{n}_{\alpha}\rangle=\frac{1}{4}{\rm Tr}[\sigma^{\rm out}_{\rm
red}]+\vec{\mu}^{\rm out\,\top}_{\rm red}\cdot\vec{\mu}^{\rm out}_{\rm
red}-\frac{1}{2}$, where the subscript “red” stands for “reduced”, and
indicates the components corresponding to the concrete mode under
consideration.

Another quantity that we are particularly interested in is the entanglement
between the interior and exterior regions of the black hole. Entanglement can
be conveniently quantified by means of the logarithmic negativity (LogNeg)
[11, 12]. LogNeg has important advantages for our purposes, as compared to
other entanglement quantifiers or witnesses. On the one hand, it can be easily
computed from the covariance matrix $\sigma^{AB}_{\rm red}$ of any bi-partite
system. On the other hand, if either subsystem A or B is made of a single
mode, LogNeg is a faithful quantifier of entanglement, even for mixed states
(see, e.g., [4] for further details). We note that entanglement is encoded
entirely in the covariance matrix, with no reference to the first moments.

We apply these tools to a few examples. The simplest case is when the “in”
state consists of only vacuum fluctuations, in which case the first moments
are $\vec{\mu}^{\rm in}_{\rm vac}=\vec{0}$ and the covariance matrix is
$\sigma^{\rm in}_{\rm vac}=\mathbb{I}_{6}$, where $\mathbb{I}_{6}$ is the
$6\times 6$ identity matrix. Straightforward application of (S0.Ex1) and
(S0.Ex2) produces $\langle\hat{n}_{\rm
out}(w)\rangle=\Gamma(w)\,\sinh^{2}r_{H}(w)$ for the out mode reaching
infinity, where we can distinguish the contribution from both the two-mode
squeezer and the beam splitter. This is a well known result for the
spontaneous Hawking effect [2].

As an example of the entanglement in the final state for vacuum input, the
LogNeg between the Hawking mode $a^{\rm out}_{w}$ and the two modes falling
into the black hole is given by the following expression, which, although not
particularly illuminating, shows that this quantity can be analytically
expressed in closed form

$\displaystyle{\rm LogNeg}[\hat{a}^{\rm out}_{w}|(\hat{a}^{\rm
int}_{I},\hat{b}^{\rm int}_{w})]={\rm Max}\Bigg{[}0,$ $\displaystyle-{\rm
Log}_{2}\Big{[}\frac{1}{\sqrt{2}}\Big{(}\Big{|}-2\cos
4\theta\,\sinh^{4}r_{H}+\cosh^{2}r_{H}-2\cos
2\theta\,\sinh^{2}2r_{H}-3\cosh^{2}r_{H}\,\cosh 2r_{H}$
$\displaystyle+\sqrt{-4+\big{(}2\cos
4\theta\,\sinh^{4}r_{H}+\cosh^{2}r_{H}\,(-1+3\cosh 2r_{H}+8\cos
2\theta\,\sinh^{2}r_{H})\big{)}^{2}}\Big{|}\Big{)}^{1/2}\Big{]}\Bigg{]}.$

By plotting this expression, one can check that the presence of the potential
barrier degrades the entanglement carried out to infinity. For example,
entanglement vanishes in the limit $\theta\to\pi/2$ ($\Gamma\to 0$), since, in
that limit, the barrier completely blocks the outgoing Hawking radiation. This
in turn implies that, for a fixed frequency $w$, modes with the lowest angular
multipoles $\ell$ carry most of the entanglement (as well as most of the
energy).

If we replace the initial vacuum by an excited state, we are in the realm of
the stimulated Hawking process. A simple scenario here is when the initial
state is a coherent state. Coherent states are “displaced vacua”, in the sense
that they are Gaussian states with the same covariance matrix as the vacuum,
but have different first moments: $(\vec{\mu}_{\rm
coh}\neq\vec{0},\,\sigma_{\rm coh}=\mathbb{I}_{6})$. Using the expression for
$\langle\hat{n}_{\alpha}\rangle$ given above, we quickly see that the mean
number of quanta in the Hawking mode $a^{\rm out}_{w}$ gets amplified, as
expected. However, the entanglement structure in the final state remains
exactly the same as for vacuum input, as the covariance matrices for each are
identical. Hence, seeding the process with coherent states does not change
quantum aspects of the final state. This justifies the common lore about the
intrinsically classical character of the stimulated Hawking effect. We now
argue, contrary to the common lore, that such is not true for other input
states.

By illuminating the black hole with, e.g., a single-mode squeezed state in
either of the modes $\hat{a}^{\rm in}_{I}$ or $\hat{a}^{\rm in}_{II}$, we find
that both the number of final quanta in the Hawking modes $\hat{a}^{\rm
out}_{w}$ and its entanglement with the interior modes are amplified. Consider
a squeezed vacuum in the $\hat{a}^{\rm in}_{I}$ mode, $(\vec{\mu}_{\rm
sqz}=\vec{0},\,\sigma_{\rm
sqz}=e^{\sigma_{z}\,2\,r_{I}}\,\oplus\mathbb{I}_{4})$, where $\sigma_{z}$ is
the familiar $z$-Pauli matrix and $r_{I}$ is the initial squeezing intensity.
Fig. 2 shows that the entanglement between the black hole interior and the
exterior (formally quantified by ${\rm LogNeg}[a^{\rm out}_{w}|(a^{\rm
int}_{I},b^{\rm int}_{w})]$), for a given frequency $w$, grows with $r_{I}$.
Note that a single-mode squeezed state does not contain any entanglement among
each of the three “in” modes, so the final entanglement is entirely generated
during pair-production in the Hawking process. Hence, by tuning the initial
squeezing $r_{I}$, one can enhance the entanglement generated by black holes.
There are multiple reasons that make the implementation of this idea
impossible for astrophysical black holes; the most prominent being that the
two “in” modes $\hat{a}^{\rm in}_{I}$ and $\hat{a}^{\rm in}_{II}$ are made of
ultrahigh-frequency modes at past null infinity. However, as we argue below,
the implementation in analog event horizons produced in the lab is feasible
with present technology, since the blue-shift between frequencies of “in” and
“out” modes is not exponentially large, as in the astrophysical case [13].

Figure 2: Left Panel: ${\rm LogNeg}[\hat{a}^{\rm out}_{w}|(\hat{a}^{\rm
int}_{I},\hat{b}^{\rm int}_{w})]$ measuring the entanglement between the
Hawking mode reaching infinity and the interior of a Schwarzschild black hole,
for photons with $w=6.25\,T_{H}$, $\ell=1$, $m=0$ (the mode which emits the
maximum amount of energy in electromagnetic radiation, and for which
$\Gamma=0.4689$ [14]) as a function of the squeezing intensity $r_{I}$ in the
“in” mode $\hat{a}^{\rm in}_{I}$. Right Panel: Same quantity versus the
environment temperature (no initial squeezing; $r_{I}=0$).

Let us next consider thermal input radiation. This is of obvious interest
since all astrophysical black holes are immersed in the CMB. Thermal states
are mixed Gaussian states, with zero mean, no correlations among modes, and
covariance matrix for each individual mode equal to
$(2\,n^{\alpha}_{env}+1)\,\mathbb{I}_{2}$, where $n^{\alpha}_{\rm env}$ is the
number of thermal quanta in the mode $\alpha$. Since the modes $\hat{a}^{\rm
in}_{I}$ or $\hat{a}^{\rm in}_{II}$ are ultrahigh-frequency modes (see
footnote 4), and assuming the temperature is small compared to the Planck
scale, it is reasonable to substitute $n_{env}\approx 0$ for them. The
relevant thermal quanta in the initial state occupy only the low frequency
mode $\hat{b}^{\rm in}_{w}$. The initial state is thus characterized by
$\vec{\mu}^{\rm in}_{\rm th}=\vec{0}$ and $\sigma_{\rm
th}\approx\mathbb{I}_{4}\oplus(2\,n_{\rm env}+1)\,\mathbb{I}_{2}$, from which
we find $\langle\hat{n}_{\rm out}\rangle=n_{\rm
env}\,(1-\Gamma)+\,\Gamma\,\sinh^{2}{r_{H}}$ quanta reaching infinity. This is
the same result that we would obtain for vacuum by simply adding the thermal
quanta scattered back to infinity by the potential barrier. From here, we see
that if $n_{\rm env}$ is equal to $\sinh^{2}r_{H}$ (i.e., $T_{H}=T_{env}$) we
have $\langle\hat{n}_{\rm out}\rangle-n_{\rm env}=0$ for all $w$, and the
black hole is in equilibrium, as expected from thermodynamical arguments. Note
that the black hole loses mass only if $T_{H}>T_{\rm env}.$

It is intriguing to wonder how the entanglement between the interior and
exterior regions of the black hole is modified when the black hole is immersed
in a thermal bath; so we compute the entanglement in this scenario and plot
the results in Fig. 2. We see that thermal fluctuations of the bath are
detrimental to the quantum entanglement between the emitted Hawking quanta and
the black hole. Hence, although the mode $\hat{b}^{\rm in}_{w}$ is only
involved in the process of back-scattering, thermal fluctuations present in
this mode degrade the entanglement produced in the Hawking process. A physical
consequence of this is that, since astrophysical black holes have Hawking
temperatures several orders of magnitude lower than the CMB temperature, the
faint Hawking quanta they emit are barely entangled with the black hole’s
interior. This can be seen quantitatively in the right panel of Fig. 2, where
we observe a sharp transition in the entanglement as $T_{\rm env}\gtrsim
T_{H}$. Remarkably though, for arbitrarily high environmental temperatures,
the entanglement saturates to a small —but still non-zero— value. (We note
that astrophysical black holes are also immersed in noisy backgrounds of other
fields that contribute significantly to the Hawking process, like the cosmic
background of neutrinos and the stochastic background of gravitational waves.
Both will generically degrade the generation of entanglement for these fields,
especially for neutrinos due to their fermionic character.)

We finish this essay by arguing that the ideas thus presented have direct
applicability for analog event horizons manufactured in the laboratory. We
highlight this with a popular scenario to recreate the physics of the Hawking
process: optical systems [13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27]. In an optical analog, an electromagnetic pulse in a dielectric material
locally modifies the refractive index via the so-called Kerr effect. Hence, by
introducing a strong pulse in a dielectric material, the speed of weak probes
propagating thereon can be tuned. Probes that are initially faster than the
pulse will slow down when trying to overtake it, and if the pulse is strong
enough, its rear end acts as an (moving) impenetrable barrier. This is the
analog of the horizon of a white hole. Similarly, an analog black hole horizon
appears in the front end of the pulse. The creation of these white-black holes
has been experimentally carried out, and the stimulated radiation with
coherent inputs has been recently demonstrated in [23]. Optical systems offer
a great advantage to generate, manipulate, and observe quantum states as well
as their entanglement structure [28] —tasks that are routinely done with
present technology.

Figure 3: Left panel: Illustration of the structure of modes for an optical
analog white-black hole in the frame comoving with the pulse; the modes
$k_{i}$ shown are the solutions to the dispersion relation $k(w)$ for a fixed
frequency $w$. Right Panel: LogNeg between the outgoing white-hole Hawking-
pair $k^{\rm out}_{1}$ and $k^{\rm out}_{4}$ versus the initial squeezing
intensity $r_{k_{3}}^{\rm in}$, for $w=2T_{H}$. We assume isotropic thermal
noise, $\bar{n}_{\rm env}=(e^{w/T_{\rm env}}-1)^{-1}$ with ${T_{\rm
env}=T_{H}}$, and $10\%$ losses.

The decomposition of the Hawking process in terms of two-mode squeezers and
beam-splitters, as described in this essay, can be repeated for these optical
systems [29], and although the scenario is richer than that of astrophysical
black holes, due to non-trivial dispersion relations of the media, the
conclusions are similar. Namely, the generation of entanglement in the Hawking
process can be tuned, either degrading it by illuminating the system with
noisy thermal photons or amplifying it using squeezed inputs. Some
quantitative results of our recent detailed analysis [29] are shown in the
right panel of Fig. 3. These calculations incorporate the effects of ambient
noise and losses, both ubiquitous in real experiments. The ability of tuning
the input quantum state provides a sharp tool to test the two defining aspects
of the Hawking process, namely the generation of quanta with a black body
distribution and a concrete entanglement structure, and a protocol to observe
these features has been put forth in [29].

Acknowledgments. We have benefited from discussions with A. Ashtekar, D.
Bermudez, M. Jacquet, A. Fabbri, J. Pullin, J. Olmedo, O. Magana-Loaiza and J.
Wilson. We thank A. del Rio for assistance with the evaluation of the grey
body factors. This work is supported by the NSF grant PHY-2110273, and by the
Hearne Institute for Theoretical Physics. A.J.B. also acknowledges support
from the Defense Advanced Research Projects Agency (DARPA) under Young Faculty
Award (YFA) Grant No. N660012014029.

## References

  * [1] S. W. Hawking, “Black hole explosions?” Nature 248 no. 5443, (1974) 30–31.
  * [2] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975) 199–220. [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * [3] D. N. Page, “Information in black hole radiation,” Phys. Rev. Lett. 71 (Dec, 1993) 3743–3746.
  * [4] A. Serafini, Quantum continuous variables: a primer of theoretical methods.  CRC press, 2017.
  * [5] R. M. Wald, “On Particle Creation by Black Holes,” Commun. Math. Phys. 45 (1975) 9–34.
  * [6] R. M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics.  Chicago Lectures in Physics. University of Chicago Press, Chicago, IL, 1995.
  * [7] A. Fabbri and J. Navarro-Salas, Modeling black hole evaporation.  World Scientific, 2005.
  * [8] T. Jacobson, “Black hole evaporation and ultrashort distances,” Phys. Rev. D 44 (1991) 1731–1739.
  * [9] S. Corley and T. Jacobson, “Hawking spectrum and high frequency dispersion,” Phys. Rev. D 54 (1996) 1568–1586, arXiv:hep-th/9601073.
  * [10] C. Gerry, P. Knight, and P. Knight, Introductory Quantum Optics.  Cambridge University Press, 2005.
  * [11] A. Peres, “Separability criterion for density matrices,” Physical Review Letters 77 no. 8, (1996) 1413.
  * [12] M. B. Plenio, “Logarithmic negativity: A full entanglement monotone that is not convex,” Physical Review Letters 95 no. 9, (2005) 090503.
  * [13] Philbin, Thomas G and Kuklewicz, Chris and Robertson, Scott and Hill, Stephen and König, Friedrich and Leonhardt, Ulf, “Fiber-optical analog of the event horizon,” Science 319 no. 5868, (2008) 1367–1370.
  * [14] D. N. Page, “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole,” Phys. Rev. D 13 (1976) 198–206.
  * [15] Demircan, A and Amiranashvili, Sh and Steinmeyer, G, “Controlling light by light with an optical event horizon,” Physical Review Letters 106 no. 16, (2011) 163901.
  * [16] Rubino, Elenora and Lotti, A and Belgiorno, F and Cacciatori, SL and Couairon, Arnaud and Leonhardt, Ulf and Faccio, D, “Soliton-induced relativistic-scattering and amplification,” Scientific Reports 2 no. 1, (2012) 1–4.
  * [17] Petev, Mike and Westerberg, Niclas and Moss, Daniel and Rubino, Elenora and Rimoldi, C and Cacciatori, SL and Belgiorno, F and Faccio, D, “Blackbody emission from light interacting with an effective moving dispersive medium,” Physical Review Letters 111 no. 4, (2013) 043902.
  * [18] S. Finazzi and I. Carusotto, “Quantum vacuum emission in a nonlinear optical medium illuminated by a strong laser pulse,” Physical Review A 87 no. 2, (2013) 023803.
  * [19] F. Belgiorno, S. L. Cacciatori, and F. Dalla Piazza, “Hawking effect in dielectric media and the Hopfield model,” Phys. Rev. D 91 no. 12, (2015) 124063, arXiv:1411.7870 [gr-qc].
  * [20] Linder, Malte F and Schützhold, Ralf and Unruh, William G, “Derivation of Hawking radiation in dispersive dielectric media,” Physical Review D 93 no. 10, (2016) 104010.
  * [21] D. Bermudez and U. Leonhardt, “Hawking spectrum for a fiber-optical analog of the event horizon,” Phys. Rev. A 93 no. 5, (2016) 053820, arXiv:1601.06816 [gr-qc].
  * [22] F. Belgiorno, S. L. Cacciatori, F. Dalla Piazza, and M. Doronzo, “Hopfield-Kerr model and analogue black hole radiation in dielectrics,” Phys. Rev. D 96 no. 9, (2017) 096024, arXiv:1707.01663 [hep-th].
  * [23] Drori, Jonathan and Rosenberg, Yuval and Bermudez, David and Silberberg, Yaron and Leonhardt, Ulf, “Observation of stimulated Hawking radiation in an optical analogue,” Physical Review Letters 122 no. 1, (2019) 010404.
  * [24] Jacquet, Maxime J and König, Friedrich, “Analytical description of quantum emission in optical analogs to gravity,” Physical Review A 102 no. 1, (2020) 013725.
  * [25] M. Jacquet and F. Koenig, “The influence of spacetime curvature on quantum emission in optical analogues to gravity,” SciPost Physics Core 3 no. 1, (Sep, 2020) .
  * [26] Rosenberg, Yuval, “Optical analogues of black-hole horizons,” Philosophical Transactions of the Royal Society A 378 no. 2177, (2020) 20190232\.
  * [27] R. Aguero-Santacruz and D. Bermudez, “Hawking radiation in optics and beyond,” Phil. Trans. Roy. Soc. Lond. A 378 no. 2177, (2020) 20190223, arXiv:2002.07907 [gr-qc].
  * [28] A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Reviews of Modern Physics 81 no. 1, (2009) 299.
  * [29] I. Agullo, A. J. Brady, and D. Kranas, “Quantum Aspects of Stimulated Hawking Radiation in an Optical Analog White-Black Hole Pair,” Phys. Rev. Lett. 128 no. 9, (2022) 091301, arXiv:2107.10217 [gr-qc].