#  Studies on particle creation during the universe expansion with a laser
system

De-Chang Dai  Corresponding author<EMAIL_ADDRESS>Department of
Physics, national Dong Hwa University, Hualien, Taiwan, Republic of China
CERCA, Department of Physics, Case Western Reserve University, Cleveland OH
44106-7079  Changbo Fu  Key Laboratory of Nuclear Physics and Ion-beam
Application (MoE), Institute of Modern Physics, Fudan University, Shanghai
200433, China

###### Abstract

While two highly intensive laser beams collide, they create a region where the
refractive index varies so quickly that photons are created. The variance of
the refractive index is analog to the universe scale factor variance.
Therefore, this laser system can be an analog to the expansion of the
universe. We find that several hundreds of photons can be created under
feasible conditions. This system can demonstrate the particle creation during
inflation or other similar periods.

##  I introduction

A black hole is undoubtedly among the most interesting objects in the
universe. Classically, it absorbs everything since nothing can escape from the
horizon. However, when quantum mechanics is included, Hawking found that a
black hole is a thermal object that radiates energy at a certain
temperature[1, 2]. A black hole’s temperature is inversely proportional to its
mass. A large astrophysical black hole is too massive to have an observable
radiation. Because of this, Unruh proposed to use non-gravitational systems to
mimic the behavior of the gravitational ones [3]. These systems are known as
analog black holes.

At first, such analog systems were constructed using water waves [4, 5, 7, 8,
9, 10], and then the methods were extended to Bose-Einstein condensates [11,
12, 13, 14, 15, 16] and optical systems [17, 18, 19, 20, 21, 22, 23, 6].
Today, analog black holes are found in other systems as well, see for example
reviews in [25, 24, 26].

Though there are huge successes in constructing analog black holes,
registering Hawking radiation is a subtler issue. For example, it was claimed
that Hawking radiation has been found in an optical system[18], but it has
also been pointed out that this radiation was just Cerenkov radiation[27, 28].
More effort is required to clarify this issue.

It is unlikely to reproduce inflation in a laboratory. However, it is possible
to create a system analogous to an expanding or shrinking universe. We propose
that a collision of two laser beams can be analogous to the universe
expansion. Several hundreds of photons are created when two pulse lasers cross
each other. This particle creation is also an analog to a Schwinger
effect[29]. In the following, we review the optical analog system and then
demonstrate that this setup is feasible with the present experimental
technology.

##  II Particle creation in nonlinear dielectric material

Optical analog black holes are created in nonlinear dielectric systems, where
the material polarization responds to the external electric field as

$\vec{P}=\epsilon_{0}\left[\chi^{(1)}\vec{E}+\chi^{(3)}(\vec{E}\cdot\vec{E})\vec{E}\right]+\ldots,$
(1)

here, $\vec{E}\cdot\vec{E}$ is proportional to the intensity, $I$, of the
laser. The propagation of an electromagnetic wave is given by

$\nabla\times(\nabla\times\vec{E})+\mu_{0}\partial_{t}^{2}\vec{P}+\partial_{t}^{2}\vec{E}=0,$
(2)

which depends on its intensity. The speed of light is set to $c=1$ for
convenience. Using
$\nabla\times(\nabla\times\vec{E})\approx-\nabla^{2}\vec{E}$, this equation
can be simplified to [30]

$\partial_{x}^{2}\vec{E}+\partial_{\perp}^{2}\vec{E}-\mu_{0}\partial_{t}^{2}\vec{P}-\partial_{t}^{2}\vec{E}=0.$
(3)

In general, the refractive index,
$n^{2}=(1+\chi^{(1)}+\chi^{(3)}|\vec{E_{p}}|^{2})$, is controlled by an
intensive pump field, $\vec{E}_{p}$, which then induces a small probe field,
$\vec{E}_{s}$. If the probe field’s frequency is much larger than the variance
of the intensity, then the probe field’s equation of motion is approximated to

$(\partial_{x}^{2}+\partial_{\perp}^{2}-n^{2}\partial_{t}^{2})\vec{E}_{s}=2\chi^{(3)}\partial_{t}^{2}(\vec{E}_{p}\cdot\vec{E}_{s})\vec{E}_{p}.$
(4)

The probe field has two polarizations. The equation of motion for the probe
field perpendicular to the pump field, $\vec{E}_{p}\cdot\vec{E}_{s}=0$, can be
further simplified to

$(\partial_{x}^{2}+\partial_{\perp}^{2}-n^{2}\partial_{t}^{2})\vec{E}_{s}=0.$
(5)

Based on this equation, an analog metric is introduced [40]

$ds^{2}=\frac{1}{n^{2}}dt^{2}-dx^{2}-dx_{\perp}^{2}.$ (6)

The photon creation process can be studied from the Lagrangian.

$\displaystyle L_{o}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
dtdx^{3}(\vec{E}\cdot\vec{D}-\vec{B}\cdot\vec{H})$ (7) $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dtdx^{3}\Big{(}\epsilon(\partial_{t}\vec{A})^{2}-(\nabla\times\vec{A})^{2}\Big{)},$

here $\vec{E}=\partial_{t}\vec{A}$ and $B=\nabla\times\vec{A}$. Since this
equation is valid only for one polarization, the probe field, $\vec{E}_{s}$,
may be replaced by a scalar field $\phi_{s}$. The effective Lagrangian is

$L_{o}\approx\frac{1}{2}\int
dtdx^{3}\Big{(}n^{2}(\partial_{t}\phi_{s})^{2}-(\nabla\phi_{s})^{2}\Big{)}.$
(8)

When the high intensity pump field is inputted in a material whose background
refractive index is $n_{0}=\sqrt{1+\chi^{(1)}}$, the free particle Lagrangian
is

$L_{f}\approx\frac{1}{2}\int
dtdx^{3}\Big{(}n_{0}^{2}(\partial_{t}\phi_{s})^{2}-(\nabla\phi_{s})^{2}\Big{)},$
(9)

and the interaction Hamiltonian is

$H_{I}\approx\xi\Pi_{s}^{2}.$ (10)

Here, $\xi=\frac{1}{2}(\frac{1}{n^{2}}-\frac{1}{n_{0}^{2}}$).
$\Pi_{s}=n_{0}^{2}\partial_{t}\phi$ is the conjugate momentum. This
interaction term represents two photons created by a background field which is
converting two virtual photons to real photons from the background (fig. 1).
These photon pairs are highly correlated.

Figure 1:  Two photons are created by the background fluctuation. The cross
circle represents the background field, while vertical lines represent two
background electric fields which then become two real photons.

The free particle quantization is [31, 32, 33, 35, 34, 36, 37, 38],

$\displaystyle\phi_{s}$ $\displaystyle=$
$\displaystyle\int\frac{dk^{3}}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{\vec{k}}}}(a_{\vec{k}}e^{i\vec{k}\cdot\vec{r}}+a_{\vec{k}}^{\dagger}e^{-i\vec{k}\cdot\vec{r}}),$
(11) $\displaystyle\Pi_{s}$ $\displaystyle=$
$\displaystyle-i\int\frac{dk^{3}}{(2\pi)^{3}}\sqrt{\frac{\omega_{\vec{k}}}{2}}(a_{\vec{k}}e^{i\vec{k}\cdot\vec{r}}-a_{\vec{k}}^{\dagger}e^{-i\vec{k}\cdot\vec{r}}),$
(12)

here, $\omega_{\vec{k}}=\frac{k}{n_{0}}$, and

$[a_{\vec{k}},a_{\vec{k}^{\prime}}^{\dagger}]=(2\pi)^{3}\delta(\vec{k}-\vec{k}^{\prime}).$
(13)

It is well known that a strong field can generate particles. The particle
creation amplitude [31, 32, 33, 35, 34, 36, 37, 38] is given by

$\displaystyle A_{\vec{k};\vec{k}^{\prime}}$ $\displaystyle=$
$\displaystyle\left\langle\vec{k};\vec{k}^{\prime}\left|\exp(-i\int
H_{I}dx^{4})\right|0\right\rangle$ (14) $\displaystyle\approx$
$\displaystyle-i\omega_{k}\omega_{k^{\prime}}\tilde{\xi}(k+k^{\prime}),$

where,

$\tilde{\xi}(k)=\int\xi(x,t)\exp(-ikx)dx^{4}.$ (15)

The created photon number is [31, 32, 33, 35, 34, 36, 37, 38]

$N=\int\frac{dk^{3}}{(2\pi)^{3}}\frac{dk^{\prime
3}}{(2\pi)^{3}}\omega_{k}\omega_{k^{\prime}}|\tilde{\xi}(k+k^{\prime})|^{2}.$
(16)

Only one polarization mode is included in the calculation, which is different
from the literature [31, 32, 33, 35, 34, 36, 37, 38].

##  III An expanding-space-like metric

Suppose the refractive index is a function of time only, $n(t)$. The
corresponding analog metric is that of an expanding universe (characterized by
some time-dependent scale factor $a(t)$). In this case, particles are also
generated. The corresponding metric is chosen to be

$d\tau^{2}=\frac{1}{n^{3}}dt^{2}-\frac{1}{n}d\vec{x}^{2},$ (17)

which gives the same equation of motion as eq. (5) and cannot be re-scaled to
a flat space. This is a model giving by Parker and Toms[41, 42, 43] , in which
the cosmological scale factor $a=n^{-1/2}$. The refraction index, $n(t)$, is
considered to be $n_{0}$ as $t\rightarrow\pm\infty$ for a relevant optical
system (fig.2).

The scalar field can be redefined as

$\phi_{k}=\frac{e^{i\vec{k}\cdot\vec{x}}}{(2\pi)^{\frac{3}{2}}}\chi_{k}(t)$
(18)

Then, the equation of motion reduces to 1-dimensional Schroding’s equation
form

$-\partial_{t}^{2}\chi_{k}+U(t)\chi_{k}=\omega^{2}\chi_{k}$ (19)

here, $U(t)=k^{2}(\frac{1}{n_{0}^{2}}-\frac{1}{n^{2}})$ and $\omega=k/n_{0}$.
If $\chi_{k}$ is chosen to be

$\chi_{k}^{(in)}\sim\frac{e^{-i\omega_{in}t}}{\sqrt{2\omega_{in}}}\text{ ,
}t\rightarrow-\infty$ (20)

The field is quantized in the form,

$\hat{\phi}=\int\frac{dk^{3}}{(2\pi)^{3}\sqrt{2\omega_{in}}}a_{k}\chi_{k}^{(in)}e^{i\vec{k}\cdot\vec{x}}+a^{\dagger}_{k}\chi_{k}^{(in)*}e^{i\vec{k}\cdot\vec{x}}$
(21)

here, $\omega_{in}=k/n_{0}$ and
$[a_{k},a_{k}^{\prime\dagger}]=\delta(\vec{k}-\vec{k}^{\prime})$. The $in$
vacuum is defined as.

$a_{k}\ket{in;0}=0$ (22)

One the other hand, if $\chi$ is chosen to be

$\chi_{k}^{(out)}\sim\frac{e^{-i\omega_{out}t}}{\sqrt{2\omega_{out}}}\text{ ,
}t\rightarrow\infty,$ (23)

where,

$\hat{\phi}=\int\frac{dk^{3}}{(2\pi)^{3}\sqrt{2\omega}}b_{k}e^{-i\omega
t}e^{i\vec{k}\cdot\vec{x}}+b^{\dagger}_{k}e^{i\omega
t}e^{i\vec{k}\cdot\vec{x}},$ (24)

with $\omega_{out}=k/n_{0}$ and
$[b_{k},b_{k}^{\prime\dagger}]=\delta(\vec{k}-\vec{k}^{\prime})$. The $out$
vacuum is defined as

$b_{k}\ket{out;0}=0$ (25)

The required quantization condition is

$\chi_{k}\frac{d\chi^{*}_{k}}{dt}-\chi_{k}^{*}\frac{d\chi_{k}}{dt}=i$ (26)

The Bogoliubov transformation is applied to study the particle creation.

$\chi^{(in)}_{k}\rightarrow\alpha_{k}\chi^{out}_{k}+\beta_{k}\chi_{k}^{out*}$
(27)

Based on the perturbation method[44, 45, 46], the coefficients are

$\displaystyle\alpha_{k}$ $\displaystyle\approx$
$\displaystyle(1-\frac{i}{2\omega_{in}}\int^{\infty}_{-\infty}Udt)$ (28)
$\displaystyle\beta_{k}$ $\displaystyle\approx$
$\displaystyle\frac{i}{2\omega_{in}}\int^{\infty}_{-\infty}Ue^{-2i\omega_{in}t}dt=\frac{i\hat{U}(2\omega_{in})}{2\omega_{in}}$
(29)

here, $\hat{U}(\omega)=\int^{\infty}_{-\infty}\xi(t)e^{-i\omega t}dt$ is the
Fourier tranform of $U(t)$.

From the Bogolyubov transformation, the number density of the created
particles are

$\displaystyle N_{e}$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\int
d^{3}k|\beta_{k}|^{2}$ (30) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int
d^{3}k\frac{|\hat{U}(2\omega_{in})^{2}|}{4\omega_{in}^{2}}$

Figure 2:  The refractive index changing with time: The black curve represents
that the reflective index is a constant at $t\rightarrow-\infty$. It varies
for a while, settles down, and returns to the original value. The left arrow
represents a positive energy wave at the beginning. It then becomes a negative
energy and positive energy in the future. This process induces particle
creation.

##  IV Experimental setup to demonstrate particle creation in an expanding
universe

It has been pointed out earlier that if an electric pulse varies in time much
faster than in space, the resulting equation might resemble an expanding
universe in a small region[39]. Here, we demonstrate that two colliding laser
beams have a component that satisfies this requirement.

Figure 3:  Two cos-wave laser beams which collide head to head.

Consider two laser beams with electric fields $E_{1}$ and $E_{2}$ as

$E_{i}=A_{i}\exp\left[-\frac{\rho^{2}}{2\sigma_{i}^{2}}-\frac{(x\pm
v_{g}t)^{2}}{2\sigma_{g}^{2}}\right]\cos(k_{0}x\pm\omega_{0}t).$ (31)

$E_{1}$ and $E_{2}$ are left and right moving waves, respectively.
$\sigma_{i}$ is the width of the Gaussian wavepacket in the y and z direction,
and $\sigma_{g}$ is the width of the Gaussian wavepacket in the propagating
direction (x direction). $\rho^{2}=y^{2}+z^{2}$. The main angular frequency
and wave number are $\omega_{0}$ and $k_{0}$. We take $k_{0}\sigma_{g}\gg 1$
and $k_{0}\sigma_{i}\gg 1$ to satisfy the expanding-universe-like requirement.
The particles are created by the fluctuation of the reflective index,
$\xi\approx-\frac{\chi^{(3)}}{2(1+\chi^{(1)})^{2}}|E|^{2}$. The fluctuation of
the refractive index is proportional to the square of the electric field. When
the laser beams collide, the electric field is in the superposition

$(E_{1}+E_{2})^{2}=E_{1}^{2}+2E_{1}E_{2}+E_{2}^{2},$ (32)

where the polarization directions of the two beams are the same. The first and
third terms on the right side do not contribute to radiation if the motion is
subluminal [27, 28]. If the system is superluminal, it may be related to
Cerenkov radiation or Hawking-like radiation. It is highly concentrated in a
particular direction, which is not what we focus on right now. We focus on the
second term or the cross-term. The cross term is

$2E_{1}E_{2}=A_{1}A_{2}e^{-\frac{\rho^{2}}{\sigma_{i}^{2}}-\frac{x^{2}}{\sigma_{g}^{2}}-\frac{(v_{g}t)^{2}}{\sigma_{g}^{2}}}\left[\cos(2k_{0}x)+\cos(2\omega_{o}t)\right].$
(33)

Based on eq. (15), the Fourier transform for term with $\cos(2k_{o}x)$ is

$\tilde{E}_{12c}=A_{1}A_{2}\sigma_{i}^{2}\sigma_{g}^{2}v_{g}^{-1}\pi^{2}e^{-\frac{k_{\rho}^{2}\sigma_{i}^{2}}{4}}e^{-\frac{\left[(2k_{0}\pm
k)^{2}+\frac{\omega^{2}}{v_{g}^{2}}\right]\sigma_{g}^{2}}{4}}.$ (34)

This term generates particles colinear to laser beams. These particles have
the same energy as the laser beams. It is challenging to distinguish them from
the background. Therefore, it is also not what we are interested in.

Based on eq. (15), the Fourier transform for term with $\cos(2\omega_{o}t)$ is

$\tilde{E}_{12s}=A_{1}A_{2}\sigma_{i}^{2}\sigma_{g}^{2}v_{g}^{-1}\pi^{2}e^{-\frac{k_{\rho}^{2}\sigma_{i}^{2}}{4}}e^{-\frac{\left[\frac{(2\omega_{0}\pm\omega)^{2}}{v_{g}^{2}}+k^{2}\right]\sigma_{g}^{2}}{4}}.$
(35)

The term emits photon pairs in all directions. The photon pairs are created
with energy of about $w_{0}$, and propagate in the opposite directions. They
behave like scattered photons and can be separated from the background since
the radiation is in all directions.

The total number of emitted photons is [31, 32, 33, 35, 34, 36, 37, 38].

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{I_{1}I_{2}n_{1}^{2}\sigma_{i}^{4}\sigma_{g}^{4}\pi^{4}}{n_{0}^{6}v_{g}^{2}}\int\frac{\omega_{k_{1}}\omega_{k_{2}}dk_{1}^{3}dk_{2}^{3}}{(2\pi)^{6}}e^{-\frac{k_{\rho}^{2}\sigma_{i}^{2}+(\frac{(2\omega_{0}\pm\omega)^{2}}{v_{g}^{2}}+k^{2})\sigma_{g}^{2}}{2}}$
(36) $\displaystyle\approx$
$\displaystyle\frac{I_{1}I_{2}n_{1}^{2}\sigma_{i}^{2}\pi}{4n_{0}^{6}\sigma_{g}^{2}}(k_{0}\sigma_{g})^{4},$

here, $\chi^{(3)}E_{i}^{2}=2n_{0}n_{1}I_{i}$.
$\vec{k}=\vec{k}_{1}+\vec{k}_{2}$ and $\omega=\omega_{k_{1}}+\omega_{k_{2}}$.
$I_{i}$ is intensity. Quartz’s refractive index related to the intensity is
$n=n_{0}+n_{1}I=1.5+3\times 10^{-16}I[cm^{2}/W]$. Consider a laser with
wavelength $\lambda=800\ nm$ in vacuum and waist $\sigma_{i}=10\ \mu m$. If
the pulse length is about the same as the width, $\sigma_{g}=\sigma_{i}$ and
intensity is $10^{13}\ W/cm^{2}$, then the photon number is about 120 per
pulse, compared with the total photon number of about $2\times 10^{12}$ per
pulse in the pump laser beams. Even though the signal photon pairs have the
same frequency as the pump laser’s, they have an almost isotropic angular
distribution different from the laser beams. The signal-to-noise ratio with
order $10^{10}$ is achievable when detecting them at angles away from the
original lasers. In a real experiment, the two colliding lasers may be aligned
to a angle not equal to 180∘ to avoid damage of the optical parts.

##  V Conclusion

It is widely accepted that a fluid dynamical system can exhibit analogous
physical quantities to a system in curved space. Additionally, it is possible
to construct a black-hole-like object and observe Hawking-like radiation in
such systems. We choose a different matrix representing an expanding universe
and keep the equation of motion the same as that of particle creation as the
refractive index varies. Two head-to-head collision laser beams can construct
the refractive index variance. This collision creates a small region where the
refractive index varies very quickly with time. The refractive index variance
then represents the universe expansion in the collision region. We demonstrate
that the requirement is reachable under equipment precision and can be
preceded in the laboratory.

###### Acknowledgements.

D.C. Dai is supported by the National Science and Technology Council (under
grant no. 111-2112-M-259-016-MY3) C.B. Fu is supported by the National Nature
Science Foundation of China (under grant no. 12235003)

## References

  * [1] S. W. Hawking, Nature 248, 30-31 (1974) doi:10.1038/248030a0
  * [2] S. W. Hawking, Commun. Math. Phys. 43, 199-220 (1975) [erratum: Commun. Math. Phys. 46, 206 (1976)] doi:10.1007/BF02345020
  * [3] W. G. Unruh, Phys. Rev. Lett. 46, 1351-1353 (1981) doi:10.1103/PhysRevLett.46.1351
  * [4] G. Rousseaux, C. Mathis, P. Maissa, T. G. Philbin and U. Leonhardt, New J. Phys. 10, 053015 (2008) doi:10.1088/1367-2630/10/5/053015 [arXiv:0711.4767 [gr-qc]].
  * [5] G. Jannes, R. Piquet, P. Maissa, C. Mathis and G. Rousseaux, Phys. Rev. E 83, 056312 (2011) doi:10.1103/PhysRevE.83.056312 [arXiv:1010.1701 [physics.flu-dyn]].
  * [6] M. Elazar, V. Fleurov, S. Bar-Ad, Phys. Rev. A, 86, 063821 (2012). doi:10.1103/PhysRevA.86.063821
  * [7] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh and G. A. Lawrence, Phys. Rev. Lett. 106, 021302 (2011) doi:10.1103/PhysRevLett.106.021302 [arXiv:1008.1911 [gr-qc]].
  * [8] L. P. Euvé, F. Michel, R. Parentani and G. Rousseaux, Phys. Rev. D 91, no.2, 024020 (2015) doi:10.1103/PhysRevD.91.024020 [arXiv:1409.3830 [gr-qc]].
  * [9] L. P. Euvé, F. Michel, R. Parentani, T. G. Philbin and G. Rousseaux, Phys. Rev. Lett. 117, no.12, 121301 (2016) doi:10.1103/PhysRevLett.117.121301 [arXiv:1511.08145 [physics.flu-dyn]].
  * [10] T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford and S. Weinfurtner, Nature Phys. 13, 833-836 (2017) doi:10.1038/nphys4151 [arXiv:1612.06180 [gr-qc]].
  * [11] O. Lahav, A. Itah, A. Blumkin, C. Gordon and J. Steinhauer, Phys. Rev. Lett. 105, 240401 (2010) doi:10.1103/PhysRevLett.105.240401 [arXiv:0906.1337 [cond-mat.quant-gas]].
  * [12] J. Steinhauer, Nature Phys. 10, 864 (2014) doi:10.1038/NPHYS3104 [arXiv:1409.6550 [cond-mat.quant-gas]].
  * [13] J. Steinhauer, Nature Phys. 12, 959 (2016) doi:10.1038/nphys3863 [arXiv:1510.00621 [gr-qc]].
  * [14] J. R. Muñoz de Nova, K. Golubkov, V. I. Kolobov and J. Steinhauer, Nature 569, no.7758, 688-691 (2019) doi:10.1038/s41586-019-1241-0 [arXiv:1809.00913 [gr-qc]].
  * [15] W. C. Syu, D. S. Lee and C. Y. Lin, Phys. Rev. D 106, no.4, 044016 (2022) doi:10.1103/PhysRevD.106.044016 [arXiv:2204.10790 [hep-th]].
  * [16] W. C. Syu, D. S. Lee and C. Y. Lin, Phys. Rev. D 99, no.10, 104011 (2019) doi:10.1103/PhysRevD.99.104011 [arXiv:1901.11187 [gr-qc]].
  * [17] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig and U. Leonhardt, Science 319, 1367-1370 (2008) doi:10.1126/science.1153625 [arXiv:0711.4796 [gr-qc]].
  * [18] F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V. G. Sala and D. Faccio, Phys. Rev. Lett. 105, 203901 (2010) doi:10.1103/PhysRevLett.105.203901 [arXiv:1009.4634 [gr-qc]].
  * [19] E. Rubino, J. McLenaghan, S. C. Kehr, et al. Phys. rev. Lett. 108, 253901 (2012). doi:10.1103/PhysRevLett.108.253901
  * [20] H. S. Nguyen, D. Gerace, I. Carusotto, et al., Phys. Rev. lett., 114, 036402 (2015). doi:10.1103/PhysRevLett.114.036402
  * [21] T. Roger, C. Maitland, K. Wilson, N. Westerberg, D. Vocke, E. M. Wright and D. Faccio, Nature Commun. 7, 13492 (2016) doi:10.1038/ncomms13492 [arXiv:1611.00924 [physics.optics]].
  * [22] R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, Control of light by curved space in nanophotonic structures, Nature Photonics 11, 664 (2017).
  * [23] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg and U. Leonhardt, Phys. Rev. Lett. 122, no.1, 010404 (2019) doi:10.1103/PhysRevLett.122.010404 [arXiv:1808.09244 [gr-qc]].
  * [24] C. Barcelo, Nature Phys. 15, no.3, 210-213 (2019) doi:10.1038/s41567-018-0367-6
  * [25] C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel. 8, 12 (2005) doi:10.12942/lrr-2005-12 [arXiv:gr-qc/0505065 [gr-qc]].
  * [26] Y. Rosenberg, Phil. Trans. Roy. Soc. Lond. A 378, no.2177, 20190232 (2020) doi:10.1098/rsta.2019.0232 [arXiv:2002.04216 [physics.optics]].
  * [27] F. Belgiorno, S. L. Cacciatori, G. Ortenzi, V. G. Sala and D. Faccio, Phys. Rev. Lett. 104, 140403 (2010) doi:10.1103/PhysRevLett.104.140403 [arXiv:0910.3508 [quant-ph]].
  * [28] F. Dalla Piazza, F. Belgiorno, S. L. Cacciatori and D. Faccio, Phys. Rev. A 85, 033833 (2012) doi:10.1103/PhysRevA.85.033833 [arXiv:1201.2354 [quant-ph]].
  * [29] J. S. Schwinger, Phys. Rev. 82, 664-679 (1951) doi:10.1103/PhysRev.82.664
  * [30] E. Rubino, F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, V. G. Sala, M. Kolesik and D. Faccio, New J. Phys. 13, 085005 (2011) doi:10.1088/1367-2630/13/8/085005
  * [31] M. Hillery and L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984)
  * [32] P. D. Drummond Phys. Rev. A 42, 6845 (1990)
  * [33] R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467-491 (1991) doi:10.1103/PhysRevA.43.467
  * [34] B.J. Dalton, E.S. Guerra and P.L. Knight Phys. Rev. A 54, 2292 (1996)
  * [35] E. Schmidt, L. Knoell and D. G. Welsch, [arXiv:quant-ph/9607026 [quant-ph]].
  * [36] R. Schutzhold, G. Plunien and G. Soff, Phys. Rev. A 58, 1783 (1998) doi:10.1103/PhysRevA.58.1783 [arXiv:quant-ph/9801035 [quant-ph]].
  * [37] F. Belgiorno, S. L. Cacciatori and F. Dalla Piazza, Eur. Phys. J. D 68, 134 (2014) doi:10.1140/epjd/e2014-40803-6 [arXiv:1402.2838 [quant-ph]].
  * [38] Raymer, M. G. 2020, Journal of Modern Optics, 67, 196. doi:10.1080/09500340.2019.1706773
  * [39] R. Schutzhold, [arXiv:1110.6064 [quant-ph]].
  * [40] F. Belgiorno, S. L. Cacciatori, G. Ortenzi, L. Rizzi, V. Gorini and D. Faccio, Phys. Rev. D 83, 024015 (2011) doi:10.1103/PhysRevD.83.024015 [arXiv:1003.4150 [quant-ph]]. [41, 42, 43]
  * [41] L. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, (Cambridge University Press, Cambridge, 2009).
  * [42] L. Parker, Nature 261, 20-23 (1976).
  * [43] L. H. Ford, Rept. Prog. Phys. 84, no.11, 116901 (2021) doi:10.1088/1361-6633/ac1b23 [arXiv:2112.02444 [gr-qc]].
  * [44] Y. B. Zeldovich and A. A. Starobinsky, Zh. Eksp. Teor. Fiz. 61, 2161-2175 (1971)
  * [45] Y. B. Zel’dovich and A. A. Starobinsky, JETP Lett. 26, no.5, 252 (1977)
  * [46] N. D. Birrell and P. C. W. Davies, J. Phys. A 13, 2109 (1980) doi:10.1088/0305-4470/13/6/032