††institutetext: Departamento de Física Teórica and IPARCOS, Universidad
Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain

# States of low energy in the Schwinger effect

Álvaro Álvarez-Domínguez  Luis J. Garay  Mercedes Martín-Benito  Rita B. Neves
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

States of low energy in cosmology minimise the energy density when smeared in
a chosen time interval. We extend such construction to generic homogeneous
(possibly anisotropic) particle creation settings. Focusing on the Schwinger
effect, we study the role played by the support of the smearing function and
identify the vacua obtained in the limiting cases of small and large time
intervals. We also analyse the spectral properties of the power spectrum and
the number of created particles, which are complementary in characterising the
vacuum, and investigate the multipolar contributions coming from the
anisotropies.

††preprint: IPARCOS-UCM-23-020

##  1 Introduction

Many particle creation phenomena are an effect of the action of a classical,
time-dependent external agent which excites a quantum matter field. This is
the case, for instance, in cosmological pair production due to an evolving
spacetime geometry Parker1969 ; Ford1987 ; Weinberg2008 , and in the Schwinger
effect resulting from a strong electric field Sauter1931 ; Schwinger1951 .

In this context, it is commonly assumed that the external agents are strong
enough so that backreaction of the quantum test fields can be neglected. Then
a mean-field approximation is considered. See Pla2021 ; Pla2022  for a recent
study on the validity of the semiclassical approach in the Schwinger effect.
In this case, for strengths of the electric field below the critical Schwinger
limit, which is of the order of $10^{18}\ \text{V/m}$, particle production is
exponentially suppressed Schwinger1951 ; Yakimenko2018 . In the literature the
possibility of reaching such critical strength with intense laser facilities
is still under discussion as this would produce fermionic pair cascades which
could deplete the electric field Fedotov2010 ; Bulanov2010 ; Gonoskov2013 .
Nevertheless, recent works prove that these obstacles might be overcome,
making supercritical electric fields attainable in the future Ilderton2023 ;
Vincenti2019 . For a recent review see Fedotov2023 . In addition, the
Schwinger effect has recently been observed in an analogue mesoscopic
experiment in graphene Schmitt2023 .

Pair production effects are often explained in the context of quantum field
theory in curved spacetimes. The Hamiltonians describing these systems depend
on time, and thus when canonically quantizing each classical theory one
encounters the possibility of constructing infinitely many different quantum
theories. Equivalently, there is freedom in the choice of the annihilation and
creation operators, and therefore of the quantum vacuum. One might then
compare two different selections and find that one particular vacuum is
excited with respect to another choice of vacuum: particles have been
produced.

Different quantum theories predict different expectation values for physical
observables. For example, in standard cosmology there is a natural choice of
vacuum for cosmological perturbations, the Bunch-Davies vacuum Birrell1982 ,
as it is usually fixed at the onset of inflation, a quasi-de Sitter regime.
However, in models with pre-inflationary dynamics, such as Loop Quantum
Cosmology (LQC) or other bouncing models, it is well-known that different
choices of vacua provide different power spectra (see e.g. Fahn2018 ;
Elizaga2019 ; Agullo2015 ; DeBlas2016 ; Elizaga2021 ). Since this is a
quantity that is directly related with observations, it is considered the
relevant magnitude in cosmology. On the other hand, works about the Schwinger
effect usually study the number of created particles as the magnitude of
physical relevance. As a consequence of the ambiguities, in the literature
there is still an open discussion about the physical interpretation of the
time evolution of the number of created particles Alvarez2023 ; Ilderton2022 ;
Yamada2021 . Here, we argue that we need both the power spectrum and the
particle number to completely characterise our choice of vacuum. This is why
we introduce the notion of power spectrum in the Schwinger effect. In
addition, the study of these magnitudes will allow us to learn about the
anisotropies introduced by the electric field. Concretely, we find that
anisotropies do not play an important role in the power spectrum neither in
the infrared nor in the ultraviolet, whereas at intermediate scale multipoles
do contribute. The same applies to the particle number.

Many authors have proposed a wide range of choices of vacua depending on the
physical properties that they want to imprint on the quantum theory. Adiabatic
vacua, first put forward by Parker in Parker1969  and later formalised by
Lüders and Roberts in Luders1990 , are among the most common. Based on the WKB
approximation, they naturally extend to a slowly changing external agent the
usual notion of plane waves defining the Minkowski vacuum in flat spacetime.
Other options include the diagonalization of the Hamiltonian Fulling1979 ;
Fahn2018 ; Elizaga2019 , instantaneous minimization of the renormalised
stress-energy tensor Agullo2015 ; Handley2016 , vacua minimizing oscillations
of the primordial power spectrum in cosmological scenarios DeBlas2016 ;
Elizaga2021 , or of the time evolution of the particle number in the Schwinger
effect Dabrowski2014 ; Dabrowski2016 , unitary implementation of the dynamics
Cortez2015 ; Cortez2020 ; Garay2020 ; Alvarez2021 , among many others.

In this work, we explore the so-called States of Low Energy (SLEs) in the
Schwinger effect. Their definition in cosmology was motivated by Fewster2000 ,
which showed that the renormalised energy density smeared along a time-like
curve is bounded from below as a function of the state. Then, in Olbermann2007
, this result was applied to generic cosmological models considering smearing
functions supported on the wordline of an isotropic observer, and a procedure
was devised in order to explicitly obtain such a state. Thus, SLEs are states
that minimise the smeared energy density. Here, we investigate the role of the
support of the smearing function in the context of the Schwinger effect, and
find asymptotic regimes for sufficiently small and large supports, thus
providing physical interpretation to the ambiguities in the choice of smearing
function.

H. Olbermann proved in Olbermann2007  one appealing property of SLEs in
Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological backgrounds: they are
of Hadamard type. This relates to the ultraviolet behaviour of the two-point
function, and guarantees that computations such as that of the stress-energy
tensor are well defined Fewster2013 . There are some studies on the Hadamard
property in static electric backgrounds Wrochna2012  and under time-dependent
external potentials Finster2016 . Along these lines, we will see in this work
that the ultraviolet behaviour of SLEs in the Schwinger effect is compatible
with the Hadamard condition, although this property remains to be rigorously
proven. The properties of SLEs for cosmological models were further
investigated in Niedermaier2020 , where they were found to have the same
infrared behaviour up to a constant factor for any smearing function and the
same ultraviolet behaviour independently of the smearing function. In the
context of cosmological perturbations, these authors found that these states
are suitable candidates for vacua in models with a period of kinetic dominance
prior to inflation, as they provide the correct infrared and ultraviolet
behaviours for perturbations at the end of inflation. This prompted the
proposal of SLEs as vacua of cosmological perturbations in the context of LQC
Martin-Benito2021-1 ; Martin-Benito2021-2 . These two works have shown an
interesting dependence of SLEs on the smearing function: they are independent
of it as long as it is wide enough around the bounce of LQC Martin-
Benito2021-1 , but very sensitive to whether the moment of the bounce is
included in the support of the smearing function Martin-Benito2021-2 . The
notion of SLEs has been recently extended to fermionic fields in Nadal2023 ,
where they are applied in a radiation-dominated CPT-invariant Universe.

The structure of the paper is as follows. In section 2 we start by reviewing
the issue of quantum vacuum ambiguities, and we extend the construction of
SLEs to general homogeneous settings, considering especially the Schwinger
effect. In section 3 we investigate the dependence on the smearing function.
Section 4 is dedicated to anisotropies. Here, we define the power spectrum in
the Schwinger effect and investigate the multipolar contributions. Finally, in
section 5 we study the number of created particles for different choices of
the smearing function. Section 6 is dedicated to conclusions and closing
remarks.

##  2 SLEs in general homogeneous settings

In Olbermann2007  H. Olbermann introduced a procedure to define and compute
the SLEs in FLRW cosmological backgrounds. In this section our aim is to
extend this method to generic spatially homogeneous backgrounds, paying
special attention to the Schwinger effect. For completeness, let us first
summarise how we can parameterise the ambiguities in the choice of quantum
vacua.

###  2.1 Quantum vacuum ambiguities

For concreteness, we consider the motion of a classical charged scalar field
$\phi(t,\textbf{x})$ in flat spacetime coupled to an external spatially-
homogeneous time-dependent electric field. The dynamics of this matter field
is governed by the Klein-Gordon equation

$\left[(\partial_{\mu}+iqA_{\mu})(\partial^{\mu}+iqA^{\mu})+m^{2}\right]\phi(t,\textbf{x})=0,$
(1)

where $q$ and $m$ are the charge and the mass of $\phi(t,\textbf{x})$, and
$A_{\mu}(t,\textbf{x})$ is the electromagnetic potential. We use the temporal
gauge $A_{\mu}(t)=(0,\textbf{A}(t))$, as this is the only choice which
explicitly translates homogeneity to the equations of motion, and thus, to the
quantum theory. Each complex Fourier mode

$\phi_{\textbf{k}}(t)=\frac{1}{(2\pi)^{3/2}}\int d^{3}\textbf{x}\
e^{-i\textbf{k}\cdot\textbf{x}}\phi(t,\textbf{x})$ (2)

satisfies a harmonic oscillator equation

$\ddot{\phi}_{\textbf{k}}(t)+\omega_{\textbf{k}}(t)^{2}\phi_{\textbf{k}}(t)=0$
(3)

with time-dependent frequency

$\omega_{\textbf{k}}(t)=\sqrt{|\textbf{k}+q\textbf{A}(t)|^{2}+m^{2}}.$ (4)

These equations are decoupled for different wavevectors k. In addition, note
that the real and imaginary parts of each mode
($\phi_{\textbf{k}}^{\text{R}}(t)$ and $\phi_{\textbf{k}}^{\text{I}}(t)$,
respectively) also satisfy the same equation, so we can deal with them
independently.

Matter fields coupled to other external time-dependent spatially homogeneous
agents different from an electric field are also governed by harmonic
oscillator equations of the type (3). All the information on the external
field is encoded in the time-dependent frequency. An example is the case of
scalar and tensor gauge invariant perturbations in FLRW backgrounds and in
LQC, where now the gravitational field plays the role of the electric field.
The following formalism can be applied to these models as well as to particle
creation settings which can be described as above. Thus, we will consider a
generic frequency, which we denote as $\Omega_{\textbf{k}}(t)$ from now on,
reserving $\omega_{\textbf{k}}(t)$ for the particular case of the Schwinger
effect (4).

We now wish to canonically quantise the matter field $\phi(t,\textbf{x})$.
First, we choose a basis of solutions
$\\{\varphi_{\textbf{k}}(t)\\}_{\textbf{k}}$ to the dynamic equations (3).
Note that in general we could consider different basis for
$\phi_{\textbf{k}}^{\text{R}}(t)$ and $\phi_{\textbf{k}}^{\text{I}}(t)$, but
for simplicity we are fixing the same basis for both of them. In order to
preserve the Poisson bracket structure, this basis has to be normalised with
respect to the Klein-Gordon product:

$\varphi_{\textbf{k}}(t)\dot{\varphi}^{*}_{\textbf{k}}(t)-\varphi_{\textbf{k}}^{*}(t)\dot{\varphi}_{\textbf{k}}(t)=i.$
(5)

Then, we can define the quantum field operators
$\hat{\phi}^{\alpha}_{\textbf{k}}(t)$ ($\alpha=\text{R},\text{I}$) as the
linear combination

$\hat{\phi}^{\alpha}_{\textbf{k}}(t)=\hat{a}^{\alpha}_{\textbf{k}}\varphi_{\textbf{k}}(t)+\hat{a}^{\alpha\dagger}_{\textbf{k}}\varphi_{\textbf{k}}^{*}(t).$
(6)

Different choices of solutions $\varphi_{\textbf{k}}(t)$ translate into
different annihilation and creation operators $\hat{a}^{\alpha}_{\textbf{k}}$
and $\hat{a}^{\alpha\dagger}_{\textbf{k}}$, and vice versa. The quantum vacuum
$\ket{0}$ associated with a given choice is defined as the state annihilated
by all the annihilation operators; i.e.,
$\hat{a}^{\alpha}_{\textbf{k}}\ket{0}=0$, for all k. Thus, constructing a
particular quantum theory is equivalent to choosing one solution
$\varphi_{\textbf{k}}(t)$ to equation (3) for each k. In other words, all we
need to do is to impose initial conditions
$(\varphi_{\textbf{k}}(t_{0}),\dot{\varphi}_{\textbf{k}}(t_{0}))$ to that
equation at a certain time $t_{0}$. Using the normalisation condition (5), we
can parameterise the different possible choices (up to an irrelevant phase) as
follows Mottola2000 :

$\varphi_{\textbf{k}}(t_{0})=\frac{1}{\sqrt{2W_{\textbf{k}}(t_{0})}},\qquad\dot{\varphi}_{\textbf{k}}(t_{0})=\sqrt{\frac{W_{\textbf{k}}(t_{0})}{2}}\left[Y_{\textbf{k}}(t_{0})-i\right],$
(7)

where $W_{\textbf{k}}(t_{0})>0$ and $Y_{\textbf{k}}(t_{0})$ are independent
real quantities which determine the particular vacuum that we select.

This raises a non-trivial question. If each choice defines a different quantum
theory, each leading to different theoretical predictions, which one should we
select? Depending on the particular system that we are studying, one can find
in the literature many vacuum proposals. For example, in the simplest case in
which we have a matter field in flat spacetime where no external agent is
present, the frequency $\Omega_{\textbf{k}}$ is constant and the Minkowski
vacuum is the only one preserving Poincaré symmetry. This well-known vacuum is
defined by a basis $\\{\varphi_{\textbf{k}}(t)\\}_{\textbf{k}}$ which are
positive-frequency plane waves, i.e.,

$W_{\textbf{k}}(t_{0})=\Omega_{\textbf{k}},\qquad Y_{\textbf{k}}(t_{0})=0.$
(8)

However, when we introduce a time-dependent external agent in the system, it
breaks Poincaré invariance and the classical group of symmetries is severely
reduced. Then, the criterion of preservation of the classical symmetries in
the quantum theory is not strong enough to select one unique vacuum. One
possibility is to find the vacuum which minimises the energy density. However,
as the Hamiltonian is time-dependent, this criterion has ambiguities. One
option is to choose a particular time $t_{0}$ and find the vacuum that
instantaneously minimises the expectation value of the energy density at
$t_{0}$. This prescription defines the so-called instantaneous lowest-energy
state (ILES) at $t_{0}$, given by Mukhanov2007

$W_{\textbf{k}}(t_{0})=\Omega_{\textbf{k}}(t_{0}),\qquad
Y_{\textbf{k}}(t_{0})=0.$ (9)

It is important to remark that the minimum of the energy density only exists
if $\Omega_{\textbf{k}}(t_{0})^{2}$ is positive, so we restrict our study to
this case.111Note that in the Schwinger case (4) this condition is always
verified. Note that at different times $t_{0}$ we have different notions of
instantaneous lowest-energy states as long as the frequency
$\Omega_{\textbf{k}}(t)$ depends on time. In addition, the ILES has another
interesting property: it instantaneously diagonalises the Hamiltonian in the
basis of the vacuum and its excited states.

###  2.2 SLEs in spatially homogeneous backgrounds

We introduce in the following the SLEs. With the construction of these states,
one aims to minimise the energy density in a finite time interval, instead of
at an exact instant of time. More precisely, Fewster Fewster2000  proved in
gravitational scenarios that if we smear the energy density along a timelike
curve, it is bounded from below. Then, one can find the particular choice of
vacua that provide this minimum: the states of low energy. In Olbermann2007 ,
H. Olbermann computed these vacua in FLRW backgrounds and proved that they
have the Hadamard property.

Here we propose a direct generalisation of Olbermann’s procedure to systems
characterised by modes $\phi_{\textbf{k}}(t)$ satisfying harmonic oscillator
equations with time-dependent frequencies $\Omega_{\textbf{k}}(t)$. Let $f(t)$
be a smearing function of compact support $[t_{1},t_{2}]$. Each mode
$\phi_{\textbf{k}}(t)$ contributes to the total smeared energy as

$E[\phi_{\textbf{k}}]=\frac{1}{2}\int dt\
f(t)^{2}\left[|\dot{\phi}_{\textbf{k}}(t)|^{2}+\Omega_{\textbf{k}}(t)^{2}|\phi_{\textbf{k}}(t)|^{2}\right].$
(10)

The aim is to find, for each k, the solution $S_{\textbf{k}}(t)$ which
minimises this energy density. The strategy is as follows. First, we provide a
fiducial solution $F_{\textbf{k}}(t)$ to the equation of motion. Then, the
problem translates into finding complex constants $\lambda_{\textbf{k}}$ and
$\mu_{\textbf{k}}$ such that the solution $S_{\textbf{k}}(t)$ is written as
the linear combination

$S_{\textbf{k}}(t)=\lambda_{\textbf{k}}F_{\textbf{k}}(t)+\mu_{\textbf{k}}F_{\textbf{k}}(t)^{*}.$
(11)

Note that this is actually a Bogoliubov transformation, so in order to
preserve the Poisson algebra of the corresponding annihilation and creation
operators, the Bogoliubov coefficients should satisfy
$|\lambda_{\textbf{k}}|^{2}-|\mu_{\textbf{k}}|^{2}=1$. On the other hand, the
phase of the solution $S_{\textbf{k}}(t)$ is irrelevant, so without loss of
generality we can assume that $\mu_{\textbf{k}}$ is a positive real constant.
Substituting (11) in the smeared energy density (10), we can write

$E[S_{\textbf{k}}]=\left(1+2\mu_{\textbf{k}}^{2}\right)E[F_{\textbf{k}}]+2\mu_{\textbf{k}}\Re{\lambda_{\textbf{k}}C[F_{\textbf{k}}]},$
(12)

where the complex constant $C[F_{\textbf{k}}]$ depends on the fiducial
solution $F_{\textbf{k}}(t)$ as

$C[F_{\textbf{k}}]=\frac{1}{2}\int dt\
f(t)^{2}\left[\dot{F}_{\textbf{k}}(t)^{2}+\Omega_{\textbf{k}}(t)^{2}F_{\textbf{k}}(t)^{2}\right].$
(13)

Direct inspection of (12) reveals that the minimum of $E[S_{\textbf{k}}]$ is
reached for the most negative value that the quantity
$\Re{\lambda_{\textbf{k}}C[F_{\textbf{k}}]}$ can attain. This is achieved when
the principal arguments satisfy
$\text{Arg\,}\lambda_{\textbf{k}}+\text{Arg\,}C[F_{\textbf{k}}]=\pi$. Then,
using the relation $|\lambda_{\textbf{k}}|^{2}-|\mu_{\textbf{k}}|^{2}=1$ we
can write $E[F_{\textbf{k}}]$ in (12) only in terms of the Bogoliubov
coefficient $\mu_{\textbf{k}}$. Finally, we minimise $E[F_{\textbf{k}}]$ with
respect to $\mu_{\textbf{k}}$ and obtain

$\mu_{\textbf{k}}=\sqrt{\frac{E[F_{\textbf{k}}]}{2\sqrt{E[F_{\textbf{k}}]^{2}-|C[F_{\textbf{k}}]|^{2}}}-\frac{1}{2}},\qquad\lambda_{\textbf{k}}=-e^{-i\arg{C[F_{\textbf{k}}]}}\sqrt{\mu_{\textbf{k}}^{2}+1}.$
(14)

These two coefficients define the SLE $S_{\textbf{k}}(t)$ through the
Bogoliubov transformation (11).

The construction of the SLE that we presented here seems to be explicitly
dependent on the fiducial solution $F_{\textbf{k}}(t)$. However, the SLE is
independent of this choice Niedermaier2020 . On the other hand, the previous
deduction does not determine a unique vacuum but a family of SLEs, each one
associated with a particular smearing function $f(t)$. In section 3 we study
this dependence in detail. A natural question is whether there exists a
particular SLE which minimises the smeared energy density for all smearing
functions $f(t)$. Reference Olbermann2007  analysed this question for FLRW
spacetimes and proved that such a state exists only when the scale factor is
constant, but not otherwise. For a state of minimal energy to exist in a
generic background, the frequencies $\Omega_{\textbf{k}}(t)$ of the harmonic
oscillator equations must be time-independent. Indeed, according to the
Bogoliubov transformation (11), a solution $F_{\textbf{k}}(t)$ is a SLE if and
only if the coefficient $C[F_{\textbf{k}}]$ given in (13) vanish. Moreover, if
we impose that $F_{\textbf{k}}(t)$ is a SLE for all smearing functions, then
$\dot{F}_{\textbf{k}}(t)^{2}+\Omega_{\textbf{k}}(t)^{2}F_{\textbf{k}}(t)^{2}=0$,
which is compatible with the equation of motion if and only if the frequency
is constant. Note that, in the Schwinger effect, the frequency (4) is constant
only when the electric field vanishes. In other words, a notion of state of
minimal energy does not exist when we apply an electric field.

###  2.3 The Schwinger effect

In the following sections we focus our analysis on the Schwinger effect. In
particular, we consider the so-called Sauter-type potential Sauter1931 , i.e.,
an electric field potential of the form

$\textbf{A}(t)=E_{0}\tau\left[\tanh\left(t/{\tau}\right)+1\right]\
\textbf{e}_{z}.$ (15)

As it is shown in figure 1(a), it models a Pöschl-Teller electric pulse of
maximum amplitude $E_{0}$ at time $t=0$ Poschl1933 . It vanishes
asymptotically, and the characteristic width of the pulse is given by $\tau$.

(a)

(b)

Figure 1: Illustrative plots of (a) Sauter-type electric field of time width
$\tau$ and maximum amplitude $E_{0}$ and (b) smearing function (18) of compact
support $[-T,T]$ and slope of length $\delta$.

This particular potential allows us to find an expression for the in-solution
$\varphi_{\textbf{k}}^{\text{in}}(t)$ to equation (3), which behaves as a
plane wave of frequency
$\omega_{\textbf{k}}^{\text{in}}~{}=~{}\sqrt{k^{2}+m^{2}}$ for
$t\rightarrow-\infty$, where $k=|\textbf{k}|$. According to BeltranPalau2019 ,
this in-solution can be written in terms of hypergeometric functions
Abramowitz1965 :

$\varphi_{\textbf{k}}^{\text{in}}(t)=\frac{1}{\sqrt{2\omega_{\textbf{k}}^{\text{in}}}}e^{-i\omega_{\textbf{k}}^{\text{in}}t}\left(1+e^{2t/\tau}\right)^{(1-i\delta)/2}{}_{2}F_{1}\left(\rho_{\textbf{k}}^{+},\rho_{\textbf{k}}^{-};1-i\tau\omega_{\textbf{k}}^{\text{in}};-e^{-2t/\tau}\right),$
(16)

where

$\delta=\sqrt{(2qE_{0}\tau^{2})^{2}-1},\qquad\rho_{\textbf{k}}^{\pm}=\frac{1}{2}\left[1-i\tau\left(\omega_{\textbf{k}}^{\text{in}}\pm\omega_{\textbf{k}}^{\text{out}}\right)-i\delta\right],$
(17)

and the out-frequency
$\omega_{\textbf{k}}^{\text{out}}=\sqrt{|\textbf{k}+2qE_{0}\tau\textbf{e}_{z}|^{2}+m^{2}}$
corresponds to the asymptotic limit $t\rightarrow+\infty$ of
$\omega_{\textbf{k}}(t)$ in (4).

To compute the SLEs we could take this in-solution as the fiducial solution
$F_{\textbf{k}}(t)$, and thus we could write the SLEs in terms of integrals of
hypergeometric functions. However, recall that the construction of the SLEs is
independent of the fiducial solution chosen, and thus one may choose any
convenient one. For example, for our numerical computations we took the
numerical solution with zeroth-order adiabatic initial conditions Birrell1982
at $t=0$.

In the following sections we use the natural units $c=\hbar=1$, time will be
always expressed in units of the time width $\tau$ of the Sauter-type pulse,
and frequencies in units of $\tau^{-1}$. On the other hand, in our plots, we
fix the value of the mass to $m=\tau^{-1}$ and the maximum amplitude of the
electric field to $qE_{0}=\tau^{-2}$. This corresponds to the critical
Schwinger limit $qE_{0}=m^{2}$ Yakimenko2018 . For lower strengths, the
probability of pair production is exponentially suppressed. Only for electric
fields of this order does the Schwinger effect become physically relevant. In
practice, the qualitative behaviour of the system for stronger electric fields
is the same, and provides no additional information that is relevant to this
work. Furthermore, we are interested in studying the physical differences
between choices of vacua. Considering larger intensities than the Schwinger
limit makes these differences less clear. Finally, it is worth noting that the
Schwinger limit is not attainable experimentally yet, although recent works in
laser facilities are optimistic about achieving this in the near future
Vincenti2019 ; Ilderton2023 ; Ilderton2013 .

##  3 Role of the smearing function

In the previous section we saw that each SLE minimises the energy density
smeared with a certain compact support function $f(t)$. We are interested in
studying the physical interpretation of choosing different supports for the
smearing function, each defining a particular notion of SLE. As the Sauter
potential (15) is symmetric around its maximum at $t=0$, it will be useful to
consider smearing functions with compact support $[-T,T]$, where $T>0$. In
particular, we are going to use smooth window functions as shown in figure
1(b). We will describe them in terms of regularised step functions
$\Theta_{\delta}(t)$ of width $\delta$, such that in the limit $\delta\to 0$
we recover the discontinuous Heaviside step function. The function
$\Theta_{\delta}(t)$ interpolates between $0$ and $1$ for
$t\in(-\delta/2,\delta/2)$ and it is constant outside. We choose for the
interpolating function $\bm{(}1+\tanh\\{\cot[\pi(1/2-t/\delta)]\\}\bm{)}/2$
although the results will not qualitatively depend on this particular
selection. Then, we can write the smearing functions as

$f(t)^{2}=\left[\Theta_{\delta}(t+T-\delta/2)+\Theta_{\delta}(-t+T-\delta/2)\right]/2.$
(18)

We fix a small step width of $\delta=10^{-4}\tau$ for all the figures in this
work. For supports smaller than this width (i.e., $T<\delta$), we readapt the
parameter by setting $\delta=T/2$ so that it is still smooth.

For simplicity, we are choosing to maintain the shape of the test function,
considering only the effects of changing its support. In principle its shape
may also be relevant to the resulting SLE. However, for large enough supports
the SLEs should be fairly insensitive to the form of the test function, as
long as it is reasonably behaved, as is indeed corroborated in Martin-
Benito2021-1 . Furthermore, even when the form of the test function may be
relevant, different shapes would simply translate to more or less weight being
given to specific time periods when computing the smeared energy density.
Therefore, we may understand the physics behind the consequences of different
shapes by understanding the physical interpretation of the support first.
Besides, one may also argue that more intricate shapes are less natural
choices that would require additional motivation.

We saw in the previous section that the freedom in the choice of vacuum is
parameterised by $W_{\textbf{k}}(t_{0})$ and $Y_{\textbf{k}}(t_{0})$, which
define via (7) the initial conditions of the selected basis of solutions at
time $t_{0}$. We are going to fix $t_{0}=0$, the instant at which the Sauter-
type electric field reaches its maximum, and consider the smearing functions
(18), varying $T$. In addition, in this section we focus on modes whose
wavevectors k are parallel to the direction of the electric field.
Anisotropies will be analysed in detail in the following sections.

In figure 2 we show $W_{\textbf{k}}(t_{0})$ and $Y_{\textbf{k}}(t_{0})$ for an
infrared mode with $k=10^{-5}\tau^{-1}$ as functions of the support of the
smearing functions. We identify a transition regime around the time scale
$\tau$, which is the characteristic length of the Sauter-type electric pulse,
where the dependence on the support is not monotonic. It separates the
behaviours of the SLEs for small and large supports. We have verified that
this happens independently of the strength of the electric field.

(a)

(b)

Figure 2: Dependence of (a) $W_{\textbf{k}}(t_{0}=0)$ and (b)
$Y_{\textbf{k}}(t_{0}=0)$ defining the SLEs on the support $[-T,T]$ of the
smearing functions (18). We show the infrared mode whose wavevector k is
parallel to the electric field and $k=10^{-5}\tau^{-1}$. We use units
$\tau=1$.

When the support is small ($0<T\ll\tau$), the SLEs asymptotically approach the
values (9) that characterise the ILES at $t_{0}$. The physical justification
of this fact resides in the definition of the ILES at $t_{0}$, which minimises
the instantaneous energy density obtained identifying the smearing function
$f(t)^{2}$ with the Dirac delta $\delta(t-t_{0})$ in (10). We might then say
that the ILES at $t_{0}$ is the limit for small supports around $t_{0}$ of the
SLEs. However, note that this limit is singular in the sense that the Dirac
delta is not a smooth compact-support function, so ILESs are not a particular
example of a SLE. These conclusions are also valid for other times different
from $t_{0}$, as we have verified numerically.

On the other hand, we also find an asymptotic constant behaviour for large
supports ($T\gg\tau$). This is consistent with the fact that the leading
contributions to the smeared energy density are for times in the interval
$[-\tau,\tau]$, and that the electric pulse decreases asymptotically. This
limit defines a precise vacuum with a well-defined interpretation: the state
which minimises the energy density when it is smeared over the entire pulse.

For other values of $k$ we also distinguish analogue behaviours of
$W_{\textbf{k}}(t_{0})$ and $Y_{\textbf{k}}(t_{0})$ for small and large
supports. However, as we increase $k$, the dependence on the support
decreases. Indeed, the limit $k\rightarrow\infty$ corresponds to local flat
spacetime with no electric field, thus all vacua tend to the Minkowski vacuum
defined in (8). Nevertheless, how fast or slow we reach the Minkowski vacuum
strongly depends on each particular vacuum. We will analyse this in more
detail in section 5 when studying the number of created particles.

Finally, one might wonder why the noticeable dependence of the SLEs on the
support of the test function for supports of the order of the characteristic
length of the electric pulse seems absent in the case of LQC Martin-
Benito2021-1 . Indeed, in that work, SLEs are described as independent of the
support as long as it is large enough, which agrees with the large support
convergent behaviour we observe. Let us then clarify that, although in LQC the
equivalent to our potential is different, it also has a characteristic time
scale (around the bounce), in which the variations of the potential are most
important. This scale plays the same role as our $\tau$, and there it should
be less than a hundredth of a Planck second.222If we approximate the time-
dependent mass in the equation of motion of cosmological perturbations in LQC
by a Pöschl-Teller potential, the equivalent to $\tau$ is easily found as the
time after the bounce at which the potential reduces to half its maximum.
Therefore, the considered supports in Martin-Benito2021-1  were already quite
larger than this scale and the dependence of the SLE on them was minimal, and
achieved convergence quickly. In general, the behaviour of SLEs in LQC most
likely displays an intermediate regime as we observe in the Schwinger effect,
though it corresponds to very small supports around the bounce, which are not
physically interesting within the context of cosmology.

##  4 Anisotropic power spectrum

In this section we consider the extension of the common notion of power
spectrum in cosmological scenarios to the Schwinger effect. In addition, we
are interested in studying in detail the anisotropies present in this electric
background. Motivated by the works in anisotropic cosmologies as Bianchi I
Agullo2022 , we introduce an expansion of the power spectrum in Legendre
polynomials and analyse its multipolar contributions.

The Wightman function is defined as the vacuum expectation value

$W(t,\textbf{x};t^{\prime},\textbf{x}^{\prime})=\bra{0}\hat{\phi}^{\dagger}(t,\textbf{x})\hat{\phi}(t^{\prime},\textbf{x}^{\prime})\ket{0}=2\int\frac{d^{3}\textbf{k}}{(2\pi)^{3}}\
e^{i\textbf{k}\cdot(\textbf{x}-\textbf{x}^{\prime})}\varphi_{\textbf{k}}(t)\varphi^{*}_{\textbf{k}}(t^{\prime}),$
(19)

where we followed the notation of section 2. In the last equality we used the
definition of the quantum field operator $\hat{\phi}(t,\textbf{x})$ in terms
of the chosen basis $\varphi_{\textbf{k}}(t)$ given by (6). Note that the
Wightman function only depends on the position vectors through the difference
$\textbf{x}-\textbf{x}^{\prime}$ because the electric field is spatially
homogeneous. Writing the integral in (19) in spherical coordinates, we can
integrate out the azimuthal angle. Indeed, we are assuming that the electric
field is applied in the $z$ direction and thus it introduces anisotropy only
in the polar angle $\theta$. In addition, taking the limit of coincidence
$t\rightarrow t^{\prime}$ yields

$\lim_{t\rightarrow
t^{\prime}}W(t,\textbf{x};t^{\prime},\textbf{x}^{\prime})=\int\frac{dk}{k}\int
d(\cos{\theta})\
e^{i\textbf{k}\cdot(\textbf{x}-\textbf{x}^{\prime})}\mathcal{P}(t,\textbf{k}),$
(20)

where we defined the power spectrum as

$\mathcal{P}(t,\textbf{k})=\frac{k^{3}}{2\pi^{2}}|\varphi_{\textbf{k}}(t)|^{2}.$
(21)

The power spectrum (21) depends on the solutions $\varphi_{\textbf{k}}(t)$
that we choose to construct the quantum theory. More precisely, at a precise
time $t_{0}$ it knows about the ambiguities in the selection of
$W_{\textbf{k}}(t_{0})$ in (7), although it is oblivious to
$Y_{\textbf{k}}(t_{0})$. As we are going to see in section 5, the number of
created particles does depend on both $W_{\textbf{k}}(t_{0})$ and
$Y_{\textbf{k}}(t_{0})$. Thus, the power spectrum at a fixed time does not
encode all the information about the vacuum. Furthermore, compared with the
spectrum of $W_{\textbf{k}}(t_{0})$, the infrared power spectrum blurs the
differences between different vacua as a consequence of the factor of $k^{3}$
in its definition (21).

We show in figure 3(a) the power spectrum $\mathcal{P}(t_{0},\textbf{k})$
divided by the factor $k^{3}/2\pi^{2}$. This magnitude is computed for SLEs
with smearing functions of the type (18) of sufficiently small
($T=10^{-2}\tau$) and sufficiently large ($T=10^{2}\tau$) supports.333These
are chosen according to figure 2. This figure refers to a particular infrared
mode, but we have verified that the two supports considered here are also
sufficiently small and sufficiently large for intermediate and ultraviolet
modes as well. We see that all SLEs have the same infrared behaviour except
for a constant. This is in agreement with reference Niedermaier2020 . In the
ultraviolet, all vacua see a vanishing electric field at sufficiently short
scales. Accordingly, they all converge to the same Minkowski vacuum at all
times.

Additionally, to investigate the anisotropies we represent modes parallel and
antiparallel to the direction of the electric field (i.e., $\theta=0$ and
$\theta=\pi$, respectively). Both the infrared and ultraviolet behaviours are
oblivious to the direction of k. This is rooted in the angular dependence of
the frequency in (4),
$\omega_{\textbf{k}}(t)^{2}=k^{2}+2qA(t)k\cos\theta+q^{2}A(t)^{2}+m^{2}$.
Indeed, for $k~{}\ll~{}(q^{2}A(t)^{2}~{}+~{}m^{2})~{}/~{}|2qA(t)|$ and $k\gg
2|qA(t)|$ the angular contribution is negligible. Conversely, this defines an
intermediate regime where the dependence on $\theta$ is important.
Accordingly, in figure 3(a) the difference between parallel and antiparallel
modes is significant at these intermediate scales. Note that in this regime
the effects of the anisotropy are much more relevant than that of different
choices of SLE. Furthermore, the curves for $\theta=\pi$ are non-monotonic in
contrast with those for $\theta=0$. Indeed, for positive $\cos\theta$,
$\omega_{\textbf{k}}(t_{0})^{2}$ grows monotonously as $k$ increases, leading
to a power spectrum that monotonously decreases. On the other hand, for
negative $\cos\theta$, ${\omega_{\textbf{k}}}(t_{0})^{2}$ presents a minimum
at $k=qA(t_{0})|\cos\theta|$, which translates into a maximum in the power
spectrum around that point (in our case, $k=\tau^{-1}$). Note that in this
work we have chosen $q$ and $A(t)$ to have same sign. Had we chosen them with
opposite signs, the roles of $\theta=0$ and $\theta=\pi$ would have been
interchanged.

(a)

(b)

Figure 3: (a) Power spectrum divided by $k^{3}/(2\pi^{2})$ at $t=0$ for a mode
parallel (solid) and antiparallel (dashed) to the electric field and for SLEs
with different supports. The power spectra for ILES coincide with those for
the SLE of smallest support. (b) Absolute value of the contributions
$g_{\ell}$ of the multipoles $\ell$ with respect to the monopole at $t=0$ for
a SLE with support $T=10^{2}\tau$. Note that $g_{\ell}$ are negative for odd
values of $\ell$ and positive for even $\ell$. We use units $\tau=1$.

We now expand the power spectrum (21) in the orthonormal basis of square-
integrable functions in $[-1,1]$ formed by the Legendre polynomials,
$P_{\ell}(\cos{\theta})$:

$\mathcal{P}(t,\textbf{k})=\sum_{\ell=0}^{\infty}\mathcal{P}_{\ell}(t,k)P_{\ell}(\cos{\theta}),$
(22)

where the multipoles are given by

$\mathcal{P}_{\ell}(t,k)=\frac{2\ell+1}{2}\int_{0}^{\pi}d{(\cos{\theta})}\
\mathcal{P}(t,\textbf{k})P_{\ell}(\cos{\theta}).$ (23)

Let us now consider the multipolar contributions $\ell\geq 1$ with respect to
the isotropic monopole $\ell=0$, i.e., the coefficients

$g_{\ell}(t,k)=\mathcal{P}_{\ell}(t,k)/\mathcal{P}_{0}(t,k).$ (24)

We show in figure 3(b) how these coefficients depend on the module $k$ of the
wavevector for the SLE with large support $T=10^{2}\tau$. We verified that
similar behaviours are obtained for smearing functions with different
supports. We observe that the maximum contribution of all the multipoles with
respect to the monopole happens precisely for the same scale, which is in the
aforementioned intermediate regime identified also in figure 3(a). In
addition, we confirm that the contribution of multipoles decreases
asymptotically in both the infrared and the ultraviolet.

##  5 Number of created particles

As mentioned in the previous section, the power spectrum at a fixed time does
not fully encapsulate all the information on the vacuum. In cosmology, this is
usually the only relevant quantity as it is the only one that can be related
with observations of the CMB. However, in general and especially in the
context of the Schwinger effect, this can be complemented with the number of
created particles in one vacuum with respect to a reference one.

We will consider as the reference vacuum $\ket{0}^{\textrm{in}}$ the one
determined by the in-solution $\varphi_{\textbf{k}}^{\textrm{in}}(t)$ in (16),
which is a positive-frequency plane wave in the asymptotic past. Any other
choice of basis of solutions $\varphi_{\textbf{k}}(t)$ defines another quantum
theory, with its corresponding annihilation and creation operators
$\hat{a}^{\alpha}_{\textbf{k}}$ and $\hat{a}^{\alpha\dagger}_{\textbf{k}}$
according to (6). The number of particles in this vacuum with respect to the
in-vacuum is

$N^{\alpha}_{\textbf{k}}={}^{\textrm{in}}\\!\bra{0}\hat{a}^{\alpha\dagger}_{\textbf{k}}\hat{a}^{\alpha}_{\textbf{k}}\ket{0}^{\textrm{in}}=|\beta_{\textbf{k}}|^{2},$
(25)

where the $\beta$-coefficients quantify the differences between the two
solutions:

$\beta_{\textbf{k}}=i\left[\varphi_{\textbf{k}}(t)\dot{\varphi}_{\textbf{k}}^{\textrm{in}}(t)-\dot{\varphi}_{\textbf{k}}(t)\varphi_{\textbf{k}}^{\textrm{in}}(t)\right].$
(26)

Noticeably, at each time $t$ this depends on $\varphi_{\textbf{k}}(t)$ as well
as its derivative, and therefore encodes information on both
$W_{\textbf{k}}(t)$ and $Y_{\textbf{k}}(t)$ of the parameterisation (7).
However, it is still not fully descriptive of the vacuum, as it only depends
on a combination of these two functions. As such, it may be used in addition
to the power spectrum in order to characterise a given vacuum at a given time.

An interesting property of SLEs is that its predicted number of created
particles per mode is proportional to the relative difference between the
quantum and classical energy. More precisely,

$N^{\alpha}_{\textbf{k}}=\frac{1}{2}\frac{{}^{\textrm{in}}\\!\bra{0}E[\hat{\phi}^{\alpha}_{\textbf{k}}]\ket{0}^{\textrm{in}}-E[\varphi_{\textbf{k}}]}{E[\varphi_{\textbf{k}}]}.$
(27)

This is only true when $\varphi_{\textbf{k}}$ is taken to be a SLE and
$\hat{\phi}^{\alpha}_{\textbf{k}}$ is quantized according to (6). This
property follows from the fact that, as it was commented in section 2.2, SLEs
are the only states such that the constant $C[\varphi_{\textbf{k}}]$ defined
in (13) vanishes.

Figure 4 shows the behaviour of the number of created particles
$N^{\alpha}_{\textbf{k}}$ for modes parallel and antiparalel to the electric
field, as a function of the wavenumber $k$ and for SLEs of sufficiently small
($T=10^{-2}\tau$) and sufficiently large ($T=10^{2}\tau$) supports around
$t_{0}$. Again, we identify the same infrared behaviour for all vacua, which
are distinguished by a constant contribution. In the ultraviolet, however,
each vacuum tends to the Minkowski state at a different rate. For small
supports, the spectral particle number $N^{\alpha}_{\textbf{k}}$ of the SLE
seems to agree with that of the ILES (see section 3). However, for small
enough scales, these states behave differently. To illustrate this separation,
we have also represented a SLE with $T=10^{-1}\tau$, whose
$N^{\alpha}_{\textbf{k}}$ departs from that of the ILES at a lower
(numerically achievable) $k$. This behaviour is compatible with SLEs being of
Hadamard type, while the ILES is not. In fact, Hadamard states are infinite-
order adiabatic vacua Pirk1993 , whose $N^{\alpha}_{\textbf{k}}$ decays with a
power of $k$ proportional to its adiabatic order. Thus, for the ILES the
$N^{\alpha}_{\textbf{k}}$ is not exponentially suppressed, decaying more
slowly than SLEs for sufficiently ultraviolet modes, not depicted in figure 4.
Along these lines, the $N^{\alpha}_{\textbf{k}}$ for the SLE with large
support $T=10^{2}\tau$ must also decay faster than that for the ILES, for
sufficiently ultraviolet modes.444Solutions to the equation of motion are
oscillatory, with increasing frequency after the maximum of the electric
pulse, as well as for increasing $k$. Thus, the computation of the SLE becomes
computationally demanding for large supports and large $k$, as it requires the
integration of oscillations with very short periods.

Finally, figure 4 also shows the intermediate regime where anisotropies are
important. As motivated in the previous section, we verify that in the
infrared and ultraviolet, the particle number is isotropic. For intermediate
scales, parallel modes k to the electric field show a monotonic
$N^{\alpha}_{\textbf{k}}$, in contrast to antiparallel modes.

Figure 4: Number of created particles $N^{\alpha}_{\textbf{k}}$ as a function
of the module $k$ of the wavevector for SLEs with small and large supports,
for modes parallel (solid lines) and antiparallel (dashed lines) to the
electric field. For ILES at $t_{0}$, $N^{\alpha}_{\textbf{k}}$ coincides with
that of the SLE with the smallest support considered. We use units $\tau=1$.

##  6 Conclusions

In Olbermann2007 , SLEs were introduced in general cosmological spacetimes as
the states that minimise the energy density, smeared along the trajectory of
an isotropic observer. They were shown to be Hadamard states, and later proven
to be good candidates for the vacuum of cosmological perturbations in models
with a period of kinetic dominance prior to inflation Niedermaier2020 . Since
then, they have been applied in the context of LQC Martin-Benito2021-1 ;
Martin-Benito2021-2 , where it was found that they heavily depend on the
choice of smearing function only in regards to whether its support includes or
excludes the bounce of LQC. Recently, they have also been applied to fermionic
fields in a radiation-dominated CPT-invariant universe Nadal2023 .

In this work, we have extended the construction of SLEs to general spatially
homogeneous settings, with the emphasis on the Schwinger effect. To
investigate the dependence of these SLEs on the choice of smearing function,
we have considered regularised step-like smearing functions with a wide range
of supports centred at the maximum of a Sauter-type electric pulse. We discern
two asymptotic behaviours of SLEs. In the limit of small supports they behave
as ILESs, which instantaneously minimise the energy density (although ILESs
are not a particular case of SLEs, just a limiting behaviour). For very large
supports the dependence on the support of the smearing function gradually
disappears, thus determining in the limit a vacuum which minimises the smeared
energy density over the entire electric pulse. For supports of the order of
the characteristic time scale of the electric pulse there is a non-trivial
dependence. We have been able to draw parallels with what is observed in
Martin-Benito2021-1 , and conclude that the sizes of the support considered in
that work already corresponded to the large support regime, which is why
convergence is obtained quickly there and no non-trivial dependence on the
smearing function is observed.

We have also calculated the power spectrum in the Schwinger effect,
analogously to the usual definition in cosmology. We have shown that all SLEs
have the same infrared behaviour except for a constant contribution, in
agreement with Niedermaier2020 . In the ultraviolet, all vacua tend to the
Minkowski vacuum although at different rates. As the power spectra only depend
on the configuration of the state, they all converge for large wavenumbers.
However, as the particle number encodes information not only on the
configuration of the state but also on its velocity, each vacuum leads to
different decay rates when approaching short scales. In particular, we observe
that the particle number for all SLEs decays faster than that for the ILES.
This might be an indication of SLEs being Hadamard in the Schwinger effect.

Finally, we have analysed the anisotropy of the system. We find that in both
the ultraviolet and the infrared regions, the anisotropies do not contribute
to either the power spectrum or the number of created particles. An
intermediate regime where they are most important has been identified.

###### Acknowledgements.

This work has been supported by Project. No. MICINN PID2020-118159GB-C44 from
Spain. R. N. acknowledges financial support from Fundação para a Ciência e a
Tecnologia (FCT) through the research grant SFRH/BD/143525/2019.

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