# The spectral weight of hole doped cuprates across the pseudogap critical
point

B. Michon  Department of Quantum Matter Physics, University of Geneva, 24 Quai
Ernest-Ansermet, 1211 Geneva 4, Switzerland  A.B. Kuzmenko  Department of
Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211
Geneva 4, Switzerland  M.K. Tran  Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland  B.
McElfresh  Department of Quantum Matter Physics, University of Geneva, 24 Quai
Ernest-Ansermet, 1211 Geneva 4, Switzerland  S. Komiya  Central Research
Institute of Electric Power Industry, Materials Science Research Laboratory,
2-6-1 Nagasaka, Yokosuka, Kanagawa, Japan  S. Ono  Central Research Institute
of Electric Power Industry, Materials Science Research Laboratory, 2-6-1
Nagasaka, Yokosuka, Kanagawa, Japan  S. Uchida  Department of
Superconductivity, University of Tokyo, Yayoi 2-11-16, Bunkyo-ku, Tokyo 113,
Japan  D. van der Marel<EMAIL_ADDRESS>Department of Quantum
Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4,
Switzerland

###### Abstract

One of the most widely discussed features of the cuprate high $T_{c}$
superconductors is the presence of a pseudogap in the normal state [1, 2].
Recent transport and specific heat measurements have revealed an abrupt
transition at the pseudogap critical point, denoted $p^{*}$, characterized by
a drop in carrier density [3, 4, 5] and a strong mass enhancement [6, 7]. In
order to give more details about this transition at $p^{*}$, we performed low-
temperature infrared spectroscopy in the normal state of cuprate
superconductors La2-xSrxCuO4 (LSCO) and La1.8-xEu0.2SrxCuO4 (Eu-LSCO) for
doping contents across the pseudogap critical point $p^{*}$ (from $p=0.12$ to
$0.24$). Through the complex optical conductivity $\sigma$, we can extract the
spectral weight, $K^{*}$, of the narrow Drude peak due the coherent motion of
the quasi-particles, and the spectral weight enclosed inside the mid-infrared
(MIR) band, $K_{MIR}$, caused by coupling of the quasi-particles to collective
excitations of the many-body system. $K^{*}$ is smaller than a third of the
value predicted by band calculations, and $K_{MIR}$ forms a dome as a function
of doping. We observe a smooth doping dependence of $K^{*}$ through $p^{*}$,
and demonstrate that this is consistent with the observed doping dependence of
the carrier density and the mass enhancement. We argue that the
superconducting dome is the result of the confluence of two opposite trends,
namely the increase of the density of the quasi-particles and the decrease of
their coupling to the collective excitations as a function of doping.

###### pacs:

74.25.F, 74.45.+c, 74.70.Tx

##  I Introduction

Figure 1:  Real part of the $ab$-plane optical conductivity,
$\sigma_{1}(\omega)$ of La2-x-yEuySrxCuO4 single crystals
($x=0.12,0.15,0.21,0.24$, $y=0.0,0.2$).

The rich phase diagram of hole doped cuprates is a major challenge in
condensed matter physics. In addition to superconductivity with the highest
superconducting $T_{c}$ at ambient pressure ($T_{c}\sim 100$ K) it contains
several other competing phases including the pseudogap that is most prominent
for very low carrier concentrations. While many proposals have been made about
the nature of the pseudogap phase, the microscopic description thereof is not
universally agreed upon [1, 2]. Moreover, the superconducting pairing
mechanism as well as its link to the pseudogap phase has not been established
beyond any doubt. When the carrier density exceeds the value where the highest
$T_{c}$ is observed (the so-called “optimal doping”), a critical point $p^{*}$
is reached where the pseudogap vanishes. At this critical doping $p^{*}$ two
important signatures have been observed: firstly the carrier density $n$
observed by Hall effect [3, 4, 8] and thermal transport [5] jumps from $p$ to
$1+p$ indicating a major Fermi surface reconstruction, secondly the density of
states and associated effective mass $m^{*}$ observed by NMR [9] and specific
heat [6, 7] is strongly enhanced at $p^{*}$ and exhibits a logarithmic
divergence in temperature. The jump in $n$ and the strong renormalization of
$m^{*}$ at the critical doping $p^{*}$ underline that the pseudogap phase
terminates at this point by a quantum phase transition. The close proximity of
$p^{*}$ to optimal doping further suggests a key role in the mechanism of
superconductivity in the cuprates [10].

Two additional ordering phenomena occur at a doping level close to the point
where the pseudogap vanishes: (i) A change of crystal structure from
tetragonal at high doping to orthorhombic (approximately a
$\sqrt{2}\times\sqrt{2}$ superstructure) at low doping [11, 12, 13]. (ii) A
Lifshitz transition of the Fermi surface topology: at low doping it is a hole-
like pocket centered at the Brillouin zone corner, at high doping it is an
electron-like pocket around the Brillouin zone center [14, 15, 16]. For
La2-xSrxCuO4 (LSCO) both the structural transition [12] and the Lifshitz
transition [15] have been situated at $p=0.20$. For La1.8-xEu0.2SrxCuO4 (Eu-
LSCO) the structural phase diagram is more complex due to three different
structural variants [13], but the electronic specific heat is strongly peaked
at $p=0.235$ [6]. At approximately the same doping also the orthorhombic
distortion disappears [13]. Approximately the same critical doping is observed
for the Lifshitz transition, the vanishing of the pseudogap and the structural
transition. In the sequel, we will for each of the two types of compounds of
the present study refer to a single critical doping, namely $p^{*}(LSCO)=0.20$
and $p^{*}(Eu-LSCO)=0.235$.

Previous optical studies on bismuth and mercury cuprate families [17, 18, 19]
have demonstrated that the free carrier spectral weight is far below the band
calculations and approaches zero at zero doping. These studies have also
demonstrated the coupling to high energy degrees of freedom far exceeding
those of phonons supporting the possibility of an electronic pairing
mechanism. However, the strongly overdoped side of the phase diagram has
remained unexplored and the high $T_{c}$ of the Hg1201 and Bi2212 families has
limited the accessible temperature range where the normal state properties
could be analyzed.

To elucidate this puzzle, we measured infrared optics and extracted the
complex optical conductivity $\sigma$ in the normal state of two cuprates,
LSCO and Eu-LSCO for four dopings across the pseudogap critical point p∗:
0.12, 0.15, 0.21 and 0.24 (a total of six samples). In LSCO and Eu-LSCO, the
pseudogap line delimited by $T^{*}$ is well defined both by transport
measurements [20] and ARPES [21]. Compared to other cuprates, the $T_{c}$ of
LSCO and Eu-LSCO is relatively low, and thus interesting for extracting the
normal state properties down to low temperatures without using external
magnetic fields.

The parent compound of LSCO and Eu-LSCO is La2CuO4. Substituting La by Sr
introduces holes amounting to a hole concentration $p=x$. Substituting La with
Eu does not introduce holes or electrons, but it modifies the maximum of the
superconducting critical temperature ($T_{c}^{max}$): in Eu-LSCO $T_{c}^{max}$
= 15-20 K versus $T_{c}^{max}$ = 30-40 K in LSCO. The high temperature crystal
structure of LSCO and Eu-LSCO for all Sr concentrations is body centered
tetragonal (space group $I4/mmm$) with lattice parameters $a=b=3.78$ Å and
$c=13.2$ Å. For LSCO with doping $p<p^{*}(LSCO)$, the low temperature
structure is single-face-centered orthorhombic (space group $Abma$) with, for
the undoped parent compound, $a^{\prime}=5.42$ Å, $b^{\prime}=5.34$ Å, and
$c=13.1$ Å [11, 12, 13]. The orthorhombic distortion corresponds to a rotation
of the CuO6 octahedra such as to buckle the Cu-O squares along the
$a^{\prime}$ direction, causing the lattice parameter $a^{\prime}$ to be
smaller than $b^{\prime}$ [22]. Accordingly, in-plane anisotropy has been
observed for the magnetic susceptibility [23], the DC resistivity [24] and the
optical conductivity [25] of untwinned crystals with doping concentrations
$p<0.04$. The LSCO and Eu-LSCO crystals used for the present study were
prepared as described in Refs. 26, 27, 6, 28. Since in the orthorhombic phase,
these crystals were $ab$-plane micro twinned, the properties reported in the
present study represent the effective medium average of $a$-axis and $b$-axis
properties. X-ray diffraction was used to align and select the surface, which
for part of the samples was chosen parallel to the $ab$ plane and for the
other ones perpendicular to $ab$ such as to obtain surface dimensions of at
least 0.5 mm2. A polarizer was used for selecting the optical response
parallel to the CuO2 planes.

##  II Optical conductivity

We measured the infrared reflectivity from 2 meV to 0.5 eV using a Fourier
transform spectrometer with a home design UHV optical flow cryostat and in-
situ gold evaporation for calibrating the signal. In the energy range from 0.5
eV to 5 eV we measured real and imaginary parts of the dielectric function
using a home-design UHV cryostat installed in a Visible-UV ellipsometer.
Combining the ellipsometry and reflectivity data and using the Kramers-Kronig
relations between the reflectivity amplitude and phase, provided for each
measured temperature the complex optical conductivity spectrum
$\sigma(\omega,T)=\sigma_{1}(\omega,T)+i\sigma_{2}(\omega,T)$ in the range
from 2 meV to 5 eV.

The optical conductivity spectra of LSCO and Eu-LSCO hole-doped from $p=0.12$
to $p=0.24$ are presented in Fig. 1. The doping content $p=0.15$ corresponds
to the “optimal” doping where $T_{c}$ is maximum [29, 30]. In these figures,
we clearly observe a broad MIR band located around 0.1-0.3 eV, which is more
pronounced in underdoped and optimally doped samples compared to the Drude
response. Below $T_{c}$, a depletion at low energy is present, corresponding
to the superconducting gap.

Figure 2:  (a) Sketch of the two components of the free carrier optical
conductivity: Zero-energy mode with spectral weight $K^{*}$ and mid-infrared
band with spectral weight $K_{MIR}$. (b) Total free carrier spectral weight
$K=K^{*}+K_{MIR}$ in a broad range of hole doping for LSCO (red squares) and
Eu-LSCO (blue circles). Black squares and black circles are $K$ from
previously published optical data [31, 32, 33, 34]. The solid black curve
represents the theoretical value $K_{th}$ extracted from band calculations.
(c) Spectral weight of the zero-energy mode $K^{*}$. Solid purple line:
superconducting spectral weight $K_{sc}^{*}$ calculated from the London
penetration depth [35, 36]. (d) Effective mass in units of the free electron
mass $m_{e}$ obtained from the linear term of the specific heat (black stars
[7]) compared to the effective mass from $K^{*}$ assuming a Fermi liquid state
($1+p$ free carriers, blue squares) and a doped Mott insulator ($p$ free
carriers, red circles). The red and blue vertical lines indicate respectively
the critical dopings $p^{*}$ for LSCO and for Eu-LSCO separating the
orthorhombic ($p<p^{*}$) and tetragonal ($p>p^{*}$) crystallographic phases.

The optical conductivity probes the collective response due to the free charge
carriers, phonons, and inter-band transitions. For a non-interacting Fermi-
gas, the free charge response is described by a zero-energy mode (the Drude
peak) with a spectral weight $K$ and the width corresponding to the (frequency
independent) scattering rate. If we take into account coupling to phonons or
collective excitations of the electron many-body system, part of the Drude
spectral weight is transferred to a mid-infrared band (MIR band) at higher
energy, carrying a spectral weight $K_{MIR}$. These two components,
schematically sketched in Fig. 2(a), are clearly visible in the experimental
spectra shown in Figs. 1(a-f). The residual Drude response carries a spectral
weight $K^{*}$. The total spectral weight becomes

$K=K^{*}+K_{MIR}.$ (1)

A convenient tool for the analysis of the optical conductivity of an
interacting electron liquid is provided by the extended Drude model, where
both the scattering rate $1/\tau$ and the effective mass ratio $m^{*}/m$ are
frequency dependent functions [37, 38, 39]. For two-dimensional materials,
like cuprates, the extended Drude model can be written as

$\sigma(\omega)=\frac{e^{2}}{d_{c}\hbar^{2}}\frac{K}{1/{\tau}(\omega)-i\omega\,m^{*}(\omega)/m},$
(2)

where $d_{c}$ represents the distance between two CuO2 planes, and $m$ is the
band mass. In this expression the relation between spectral weight $K$,
density $n$ and mass $m$ of the free charge carriers is provided by the
expression $K=d_{c}\hbar^{2}{n}/{m}$. A slight rearrangement of terms in Eq. 2
provides

$\sigma(\omega)=\frac{e^{2}}{d_{c}\hbar^{2}}\frac{Km/m^{*}(\omega)}{1/{\tau^{*}}(\omega)-i\omega},$
(3)

which is a particularly useful expression in the context of a Fermi liquid. In
this case, the scattering rate and effective mass of the quasi-particles are
given by $\lim_{\omega\rightarrow 0}1/\tau^{*}(\omega)$ and
$\lim_{\omega\rightarrow 0}m^{*}(\omega)$ respectively. The coherent Drude
spectral weight $K^{*}$ can be defined as

$K^{*}=K\frac{m}{m^{*}}.$ (4)

Experimentally, we obtain $K$ from an integration of the real part of the
optical conductivity $\sigma_{1}$

$\int_{0}^{\omega_{c}}\sigma_{1}(\omega)d\omega=\frac{e^{2}\pi}{2d_{c}\hbar^{2}}K,$
(5)

with $\omega_{c}$ the cut-off frequency chosen below the interband transitions
and above the MIR band (1 eV in the present case). An equivalent way to obtain
$K$, $K^{*}$ and $K_{MIR}$, providing the same values, is fitting the spectra
with a linear superposition of Lorentzians, and extracting the different
spectral weights associated to each intra-band transition: $K^{*}$ for the
narrow Drude response and $K_{MIR}$ for the broad MIR band.

##  III Spectral weight analysis

Figure 3:  (a) $K_{MIR}$ as a function of p. (b) $K_{MIR}$ as a function of
$K^{*}$. Black squares and black circles are from Refs. 31, 32, 33, 34, and
the $T_{c}$ dome for LSCO in green line is extracted from Ref. 40.

In the experimental optical conductivities shown in Fig. 1, the two main
contributions in the total spectral weight $K$ are from the coherent Drude
response $K^{*}$ and the broad MIR band $K_{MIR}$. Phonons contribute less
than 10% to the total spectral weight $K$. In Figs. 2(b,c) and Fig. 3(a), we
report the value of $K$, $K^{*}$ and $K_{MIR}$ as a function of doping. Our
data in blue circles (Eu-LSCO) and red squares (LSCO) correspond to the lowest
temperature we can reach in the normal state (slightly above $T_{c}$). Black
squares and black circles are values extracted from published data in low
doped and highly doped samples [31, 32, 33, 34]. The red dashed curve is a
guide to the eye.

Fig. 2(b) shows the doping dependence of the total spectral weight
$K=K^{*}+K_{MIR}$. The black curve represents the spectral weight of the sheet
conductance obtained from the calculated bandstructure [41] using the
expression [17, 42]

$K_{th}=\frac{d_{c}}{V}\sum\limits_{k,\sigma}n_{k,\sigma}\frac{\partial^{2}\varepsilon_{k,\sigma}}{\partial
k^{2}}$ (6)

where $n_{k,\sigma}$ is the occupancy of the state with momentum $k$ and spin
$\sigma$, $\varepsilon_{k,\sigma}$ is the energy-momentum dispersion, $d_{c}$
is the interplanar lattice spacing and $V$ is the sample volume. The
bandstructure calculation of Ref. 41 was done for the high temperature
tetragonal crystal structure of La2CuO4. Here we assume that the band
dispersion is doping independent. These model assumptions do not (and are not
intended to) account for the low temperature orthorhombic structure and/or
emerging electronic order in part of the phase diagram. For a two-dimensional
free electron gas, Eq. 6 provides $K_{th}=E_{F}/\pi$. The effect of
interactions is to transfer part of this spectral weight to interband
transitions, which typically spread over a range of 10 eV. An additional
effect of electron-electron interactions and electron-phonon coupling is to
distribute the free carrier optical conductivity over a “coherent” zero-energy
peak and an “incoherent” mid-infrared band. For $p<p^{*}$, the unit cell is
doubled and consequently we expect folding of the band structure to occur.
This will normally cause a change of the free carrier spectral weight roughly
in proportion to the $1+p$ to $p$ reduction of the number of free charge
carriers. As stated above, this effect is not taken into account in the
bandstructure calculation shown in Fig. 2(b). A recent ab initio band
structure calculation [43] has shown that, while antiferromagnetic
correlations have a strong effect on the band dispersion, the orthorhombic
distortion causes band splittings of order 0.1 eV. The spectral weight removed
from the $K^{*}$ then reappears in the interband transitions in the mid-
infrared range. For this reason, the calculated $K_{th}$ shown in Fig. 2(b)
should be compared to the experimental $K=K^{*}+K_{MIR}$ (i.e. instead of
$K^{*}$).

For $p<p^{*}(LSCO)$, the experimental value of $K$ initially increases roughly
proportional to the hole doping $p$. Then, between $p=0.10$ and $0.30$, $K$
remains at an almost constant value of about 125 meV, which falls a factor 3
below $K_{th}$. This significant reduction of the free carrier spectral weight
can be explained by the fact $K\propto n/m$, where $n$ is small inside the
pseudogap state. Above $p=0.30$, $K$ finally increases again, and the
extrapolation of the experimental data reaches the band value $K_{th}$ around
$p=0.40$.

In Fig. 2(c) the narrow Drude spectral weight $K^{*}$ is presented as a
function of doping. Similar as for $K$, $K^{*}=0$ in the insulating phase and
it increases gradually when the system gains more and more holes. It increases
continuously, approximately as a quadratic function of doping, up to $p=0.34$,
without any noticeable anomaly (i.e. a kink or a jump) at $p^{*}$. In a free
electron approximation where $K^{*}\propto n/m^{*}$, our observations for
$K^{*}$ would be in conflict with the sharp transition in carrier density
observed in the DC transport data [3, 4, 5]. However, for an interacting Fermi
liquid, the Drude spectral weight $K^{*}$ is obtained by replacing
$\varepsilon_{k,\sigma}$ in Eq. 6 with the quasiparticle energy
$\varepsilon^{*}_{k,\sigma}$. This expression mixes two different quantities,
$n_{k,\sigma}$ and
$1/m_{k,\sigma}^{*}=\hbar^{-2}\partial^{2}\varepsilon^{*}_{k,\sigma}/\partial
k^{2}$, both of which undergo important changes around $p^{*}$. A detailed
description of the doping dependence of $K^{*}$ requires ad hoc assumptions
about the evolution of the electronic structure in the pseudogap phase.
Instead, staying close to the experimentally observed quantities, we present
in Fig. 2(d) a comparison between the effective mass obtained from the linear
term of the specific heat $\gamma$ [7] using the relation
$m^{*}_{sh}=3d_{c}\hbar^{2}\gamma/(\pi k_{B}^{2})$, and the effective mass
from the spectral weight of the zero-energy mode assuming a Fermi liquid using
$m^{*}_{F}=d_{c}\hbar^{2}(1+p)/K^{*}$ (blue squares) and a doped Mott
insulator using $m^{*}_{M}=d_{c}\hbar^{2}p/K^{*}$ (red circles). We see that
$m^{*}_{sh}$ matches $m^{*}_{M}$ for $p<p^{*}(LSCO)$, and $m^{*}_{F}$ for
$p>p^{*}(LSCO)$. This comparison puts in evidence a transition in the carrier
density around $p=p^{*}(LSCO)$, in agreement with the transition observed by
transport measurements.

In a recent study of LSCO [15], the linear term in the specific heat,
$\gamma$, which is proportional to the density of states at $E_{F}$, was
compared to the value expected from the electronic band structure
$\varepsilon_{k,\sigma}$ measured with angle resolved photoelectron
spectroscopy (ARPES), with the result that $\varepsilon_{k,\sigma}$ measured
with ARPES gives a $\gamma$ in overdoped LSCO well below the specific heat
value. It was speculated that the mass enhancement of $\gamma$ in the
electronic specific heat results from quantum criticality emerging from the
pseudogap collapse. The good agreement Fig. 2(d) between the optical data and
specific heat on the overdoped side would suggest that the renormalization of
the spectral function near the critical doping $p=p^{*}(Eu-LSCO)$, is more
easily resolved with optics than with ARPES.

For comparison, the penetration depth $1/\lambda_{L}^{2}$ is plotted in Fig.
2(c) as a solid purple line [35, 36]. As $1/\lambda_{L}^{2}\propto n/m^{*}$
inside the superconducting state, we can calculate the superconducting
spectral weight $K_{sc}^{*}$ of the condensed Cooper pairs. From $p=0$ to the
optimally doped sample $p=0.15$, $K_{sc}^{*}=K^{*}$, which is what we should
expect if the pair breaking gap $\Delta$ is much bigger than the residual
scattering rate $\hbar/\tau_{0}$. This observation does not in itself allow to
distinguish between the BCS limit where $\Delta$ is much smaller than the
Fermi energy $E_{F}$, or the opposite case corresponding to Bose-Einstein
condensation of preformed pairs [44, 45, 46, 47, 48]. On the overdoped side,
the observation that $K_{sc}^{*}$ goes to zero while $K^{*}$ keeps increasing,
indicates that $\Delta<\hbar/\tau_{0}$. This is a natural consequence of
$\Delta$ becoming smaller for reasons that we will discuss below.

By knowing $K$ and $K^{*}$, we can directly extract $K_{MIR}=K-K^{*}$, the
spectral weight enclosed inside the MIR band corresponding to the electronic
correlations in the extended Drude model. We show $K_{MIR}$ as a function of
doping in Fig. 3(a) and compare it with $T_{c}$ (solid green curve [40]). We
give another representation of this behaviour in Fig. 3(b), where $K_{MIR}$ is
plotted as a function of $K^{*}$, which is a monotonously increasing function
of doping. The three colors represent the different doping regions: underdoped
in blue, optimally doped in green and overdoped in red. It is interesting to
note that this representation results in a smoother doping dependence,
indicating that any scatter in the doping value obtained from sample
stoichiometry (for example oxygen off-stoichiometry) is absent when we use
$K^{*}$ as a measure of the charge carrier density. We observe that the MIR
spectral weight forms a dome with a maximum around $p=0.15$, i.e. $K^{*}=26$
meV. This doping value $p=0.15$ matches the optimal doping for $T_{c}$
obtained in Refs. 35, 36. This correlation of the MIR spectral weight and
$T_{c}$ as a function of doping suggests that the strength of the MIR band
reveals an important part of the superconducting pairing interaction.

In Ref. 17, a linear relation between $p$ and $K^{*}$, $K^{*}=K_{0}p$ was
reported for HgBa2CuO4+δ (Hg1201), Bi2Sr2CaCu2O8-δ (Bi-2212) and Bi2Sr2CuO6+δ
(Bi-2201) with $K_{0}=0.5$ eV. In contrast, the results for LSCO and Eu-LSCO
shown in Fig. 2(c) follow a superlinear dependence. The aforementioned linear
relation would give $K^{*}\approx 0.2$ eV at $p=p^{*}(LSCO)$, which is two
times higher than the value for LSCO shown in Fig 2(c). Given that the $ab$
planes of LSCO are buckled whereas Hg1201, Bi-2212, and Bi-2201 have flat $ab$
planes, we postulate that the buckling of the $ab$ planes of LSCO causes a
further suppression of $K^{*}$ in addition to the effect of the pseudogap that
is common to all cuprate high $T_{c}$ superconductors.

##  IV $T_{c}$ and the spectral weight

Figure 4:  (a) Spectral weight ratio $K_{MIR}/K^{*}$ as a function of hole
doping - (b) $T_{c}$ as a function of $K_{MIR}$. The spectral weight ratio
$K_{MIR}/K^{*}$ vanishes around $p=0.27$ exactly where the superconducting
dome ends.

For the overdoped samples where no band folding is present ($n=1+p$), the
ratio $K_{MIR}/K^{*}$ describes the mass enhancement
$\lambda=m^{*}(0)/m-1=K_{MIR}/K^{*}$ due to the coupling to electronic and
vibrational collective excitations. At the underdoped side of the phase
diagram where $n=p$, $K^{*}$ is reduced due to two effects, namely the
reconstruction of the Fermi surface due to band folding, as well as mass
enhancement due to many-body effects. In Fig. 4(a), $K_{MIR}/K^{*}$ is
represented as a function of doping content in red squares. The black dashed
line is a guide to the eye. In the underdoped region, $K_{MIR}/K^{*}$ is very
high and goes to infinity down to $p=0$. This is consistent with the notion
that the charge in a Mott-insulator is fully localized, hence bound at a non-
zero energy. In the overdoped region, $K_{MIR}/K^{*}$ becomes
indistinguishable from zero within the measurement accuracy for $p\geq 0.27$,
coinciding with the upper doping limit of the superconducting dome. In Fig.
3(a), the $T_{c}$-like dome of $K_{MIR}$ puts in evidence a potential link
between $K_{MIR}$ and $T_{c}$. For this reason, in Fig. 4(b), we plot $T_{c}$
as a function of $K_{MIR}$. Underdoped samples are in blue squares, optimally
doped in green square and overdoped samples in red squares. For $p\geq 0.10$,
we find a proportionality between these two quantities (with the familiar
exception at $1/8$ doping), which implies that the broad MIR band not only
results from coupling of the electrons to collective (spin, charge and
vibrational) excitations, but also that this coupling mediates superconducting
pairing. For the strongly underdoped samples ($p\leq 0.08$) $T_{c}(p)$ is much
lower, indicating that in the underdoped region the coupling to collective
excitations contributes only in part to the superconducting pair formation.

##  V Conclusions

In this study, the optical conductivity $\sigma(\omega,T)$ was measured in the
normal state of La2-xSrxCuO4 and La1.8-xEu0.2SrxCuO4 at four doping contents
$p=0.12,0.15,0.21$ and $0.24$ across the pseudogap critical point $p^{*}$. The
free carrier spectral weight exhibits the following trends as a function of
doping: (i) the total spectral weight $K$ is much smaller than the band
calculation, illustrating the consequences of the pseudogap state and the
strong electronic correlations in the underdoped region. (ii) The coherent
spectral weight $K^{*}$ increases continuously and smoothly from $p=0$ to
$0.34$ without any noticeable transition. Due to a compensating effect in
$K^{*}$, which is proportional to the ratio of carrier density over effective
mass, this is consistent with anomalies near p∗ observed in the specific heat
and the Hall constant. (iii) The spectral weight of the broad MIR band forms a
dome looking alike the superconducting dome and T${}_{c}\propto K_{MIR}$,
putting in evidence a candidate for the pairing mechanism revealed by the MIR
band. In particular, $T_{c}$ goes to zero at the same doping value where the
intensity of the MIR band becomes negligible. (iv) $K^{*}_{sc}=K^{*}$ in the
underdoped region and $K^{*}_{sc}\ll K^{*}$ in the overdoped region.

Taken together, the superconducting dome in the phase diagram of LSCO and Eu-
LSCO cuprates is the result of two competing phenomema, namely (i) the density
of charge carriers available for superconductivity characterized by $K^{*}$,
which vanishes for doping $p\rightarrow 0$ and (ii) the pairing interaction
revealed by $K_{MIR}$, which drops to zero at high doping.

The datasets generated and analyzed during the cur-rent study are available in
Ref. 49 and will be preserved for 10 years.

## Acknowledgments

We thank Antoine Georges and Christophe Berthod for fruitful discussions, and
Bernd Büchner, Sunseng Pyon, Tomohiro Takayama, and Hidenori Takagi for
providing samples. This project was supported by the Swiss National Science
Foundation through project 200020-179157.

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# Maximizing the Validity of the Gaussian Approximation for the biphoton State
from Parametric Downconversion

Baghdasar Baghdasaryan<EMAIL_ADDRESS>Theoretisch-
Physikalisches Institut, Friedrich-Schiller-Universität Jena, 07743 Jena,
Germany  Helmholtz-Institut Jena, 07743 Jena, Germany  Fabian Steinlechner
<EMAIL_ADDRESS>Fraunhofer Institute for Applied Optics and
Precision Engineering IOF, 07745 Jena, Germany  Abbe Center of Photonics,
Friedrich-Schiller-University Jena, 07745 Jena, Germany  Stephan Fritzsche
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,
07743 Jena, Germany  Helmholtz-Institut Jena, 07743 Jena, Germany  Abbe Center
of Photonics, Friedrich-Schiller-University Jena, 07745 Jena, Germany

###### Abstract

Spontaneous parametric down-conversion (SPDC) is widely used in quantum
applications based on photonic entanglement. The efficiency of photon pair
generation is often characterized by means of a $\operatorname{sinc}(L\Delta
k/2)$-function, where $L$ is the length of the nonlinear medium and $\Delta k$
the phase mismatch between the pump and down-converted fields. In theoretical
investigations, the sinc behavior of the phase mismatch has often been
approximated by a Gaussian function $\exp{(-\alpha x^{2})}$ in order to derive
analytical expressions for the SPDC process. Different values have been chosen
in the literature for the optimization factor $\alpha$, for instance by
comparing the widths of sinc and Gaussian functions or the momentum of down-
converted photons. As a consequence, different values for $\alpha$ provide
different theoretical predictions for the same setup. Therefore, an informed
and unique choice of this parameter is necessary. In this work, we present a
choice of $\alpha$ which maximizes the validity of the Gaussian approximation.
Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian
approximations as practical alternatives with improved predictive power for
experiments.

###### pacs:

Valid PACS appear here

##  I Introduction

In spontaneous parametric down-conversion (SPDC), a nonlinear-responding
quadratic crystal is pumped by a laser field, in order to convert (high-
energy) photons into entangled photon pairs. Entangled states generated by
SPDC have provided an experimental platform for fundamental quantum
technologies, such as quantum cryptography Ursin  _et al._ (2007), quantum
teleportation Bouwmeester  _et al._ (1997) or optical quantum information
processing Kok  _et al._ (2007) and by including recent milestone experiments
in photonic quantum computing Zhong  _et al._ (2020).

The quantum theory of down-converted pairs is well known and plays a crucial
role in the modeling of SPDC experiments. The state of such a SPDC pair is
also known as biphoton state. Within the paraxial approximation, where the
longitudinal and transversal components of the wave vector are treated
separately $\bm{k}=\bm{q}+k_{z}(\omega)\bm{z}$, the biphoton state can be
written as Walborn  _et al._ (2010); Karan  _et al._ (2020); Baghdasaryan  _et
al._ (2022)

$\displaystyle\ket{\Psi}=N\iint$ $\displaystyle
d\bm{q}_{s}\>d\bm{q}_{i}\>d\omega_{s}\>d\omega_{i}\>\overbrace{\mathrm{V}(\bm{q}_{s}+\bm{q}_{i})\,\mathrm{S}_{p}(\omega_{s}+\omega_{i})}^{Pump}$
$\displaystyle\times\underbrace{\operatorname{sinc}\biggl{(}\frac{L\Delta
k_{z}}{2}\biggr{)}}_{Phase\medspace
Matching}\,\hat{a}^{\dagger}_{s}(\bm{q}_{s},\omega_{s})\>\hat{a}^{\dagger}_{i}(\bm{q}_{i},\omega_{i})\ket{vac}.$
(1)

In the expression (I), $N$ is the normalization factor, $V_{p}(\bm{q}_{p})$ is
the spatial and $S_{p}(\omega_{p})$ is the spectral distribution of the pump
beam, $L$ is the length of the nonlinear crystal, $\ket{vac}$ is the vacuum
state, $\omega_{s,i}$ and $\bm{q}_{s,i}$ are the energies and transverse
components of wave vectors of down-converted fields (signal and idler), and
$\hat{a}^{\dagger}_{s,i}(\bm{q}_{s,i})$ are the corresponding creation
operators.

The expression (I) can be simplified, if we approximate the sinc function in
terms of Gaussian function $\operatorname{sinc}(x^{2})\approx\exp(-x^{2})$.
Many useful analytical expressions can be then defined for SPDC within this
approximation. A good example is the expression for the Schmidt number of the
biphoton state presented in Law and Eberly (2004).

Already in Ref. Law and Eberly (2004), the calculations within the Gaussian
approximation delivered a small deviation from experimentally measured values
for the Schmidt number. It has often been suggested in the literature to
optimize this approximation by presenting an optimization factor $\alpha$ in
the exponential $\exp{(-\alpha x^{2})}$. The value of $\alpha$ has been chosen
for example, by matching the widths of Gaussian and sinc functions Joobeur
_et al._ (1996); Valencia  _et al._ (2007); Miatto  _et al._ (2012), by
matching second-order momenta Schneeloch and Howell (2016), by comparing
momentum correlations in SPDC Gómez  _et al._ (2012) or the coincidence and
single-particle spectral widths of the biphoton state Fedorov  _et al._
(2009).

Obviously, different values of $\alpha$ deliver different theoretical
predictions for the same setup. A unique choice of $\alpha$, which should
minimize the error of the approximation, is lucking. In order to solve this
problem, we look in this work at states themselves, instead of comparing just
the sinc and Gaussian functions or other observable. Eventually, the goal is
to make the distance of sinc\- and Gaussian-like states as small as possible.
We use an appropriate distance measure for this purpose such as fidelity
Nielsen and Chuang (2000) and find the particular $\alpha$ that maximizes it.
Except for the Gaussian approximation, we will also discuss the cosine\- and
super-Gaussian approximations and will compare them with the usual Gaussian
approximation.

##  II Theory and results

We first should determine the pump and phase matching characteristics from the
expression (I), in order to calculate the fidelity of the sinc\- and Gaussian-
like states. The pump beam is usually fixed by the experimental setup, and its
function in the expression (I) is well known. The phase-mismatch in the
$z$-direction $\Delta k_{z}=k_{p,z}-k_{s,z}-k_{i,z}$ requires more careful
determination, which depends on many characteristics of experiment, such the
crystal type, the polarization of interacting beams, the geometry of the
setup. The derivation of $\Delta k_{z}$ has been already reported for a very
general experimental scenario in Ref. Baghdasaryan  _et al._ (2022). In
comparison to Ref. Baghdasaryan  _et al._ (2022), here, we additionally assume
degenerate momentum vectors for signal and idler photons $k_{p}\approx
2k_{s}$, which allows us to apply the Gaussian approximation. With this in
mind, the phase-mismatch $\Delta k_{z}$ can be written as Baghdasaryan  _et
al._ (2022)

$\Delta
k_{z}=\frac{\Omega_{s}+\Omega_{i}}{u_{p}}-\frac{\Omega_{s}}{u_{s}}-\frac{\Omega_{i}}{u_{i}}+\frac{\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{2k_{p}},$
(2)

where $\Omega_{j}$ is the deviation from the central frequency $\omega_{j,0}$,
$\omega_{j}=\omega_{j,0}+\Omega_{j}$ with the assumption
$\Omega_{j}\ll\omega_{j,0}$, and $u_{j}=1/(\partial k_{j}/\partial\Omega)$ is
the group velocity evaluated at the central frequency.

The right-hand side of Eq. (2) can be divided into two parts. The first three
terms represent the spectral and the last term the spatial properties of the
biphoton state. The spectral and spatial degrees of freedom are in general
coupled due to the phase matching characteristics in SPDC Osorio  _et al._
(2008); Gatti  _et al._ (2009); Baghdasaryan  _et al._ (2022); Sevilla-
Gutiérrez  _et al._ (2022). However, under certain approximations, such as the
narrowband Miatto  _et al._ (2011), thin-crystal Yao (2011); Baghdasaryan  _et
al._ (2021) or plane wave approximations Valencia  _et al._ (2007), either
only the first three terms survive (frequency-resolved biphoton state) or the
last one (spatially-resolved biphoton state). In this work, we first develop
the Gaussian approximation for spectral and spatial DOF separately. The
description of the coupling in spectral and spatial domains is more
challenging in the scope of the Gaussian approximation, which is also
discussed in this paper.

###  II.1 Spatially-resolved biphoton state

The signal and idler fields can be assumed monochromatic if narrowband filters
are used in front of detectors. This is called narrowband approximation, which
transforms the biphoton state into

$\displaystyle\ket{\Psi}=N\iint$ $\displaystyle
d\bm{q}_{s}\>d\bm{q}_{i}\mathrm{V}(\bm{q}_{s}+\bm{q}_{i})\>\operatorname{sinc}\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}$
$\displaystyle\times\hat{a}^{\dagger}_{s}(\bm{q}_{s})\>\hat{a}^{\dagger}_{i}(\bm{q}_{i})\ket{vac}.$
(3)

The expression (II.1) has now a simple form so that we can apply the Gaussian,
super-Gaussian or cosine-Gaussian approximations, and calculate the
corresponding fidelity. We will present the detailed calculation only for the
Gaussian approximation since the derivations for the super-Gaussian and
cosine-Gaussian approximations are similar.

####  II.1.1 Gaussian approximation

Similar to the expression (I), the approximated Gaussian-state is written as

$\displaystyle\ket{\Psi_{G}}=N_{G}\iint$ $\displaystyle
d\bm{q}_{s}\>d\bm{q}_{i}\mathrm{V}(\bm{q}_{s}+\bm{q}_{i})\>\exp\biggl{(}-\alpha\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}$
$\displaystyle\times\hat{a}^{\dagger}_{s}(\bm{q}_{s})\>\hat{a}^{\dagger}_{i}(\bm{q}_{i})\ket{vac},$
(4)

where $N_{G}$ is the new normalization constant. The fidelity of the states
from Eqs. (II.1) and (II.1.1), which is simply the scalar product of these
states, is then given by

$\displaystyle\braket{\Psi}{\Psi_{G}}=$ $\displaystyle N\cdot N_{G}\iint
d\bm{q}_{s}\>d\bm{q}_{i}\,\mathinner{\\!\left\lvert\mathrm{V}(\bm{q}_{s}+\bm{q}_{i})\right\rvert}^{2}$
$\displaystyle\times\operatorname{sinc}\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}\>\exp\biggl{(}-\alpha\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}.$
(5)

The integral is difficult to calculate in the momentum space but
mathematically more straightforward in the position space. We use the
notations $\bm{q}_{-}=\bm{q}_{s}-\bm{q}_{i}$ and
$\bm{q}_{+}=\bm{q}_{s}+\bm{q}_{i}$ and rewrite the function from Eq. (II.1.1)
with their Fourier transforms

$\displaystyle\braket{\Psi}{\Psi_{G}}=$ $\displaystyle N\cdot N_{G}\iint
d\bm{q}_{s}\>d\bm{q}_{i}\,\frac{1}{2\pi}\int
d\bm{r}\>\Phi_{F}(\bm{r})\,e^{i\bm{r}\bm{q}_{-}}$
$\displaystyle\times\frac{1}{2\pi}\int
d\bm{r}^{{}^{\prime}}\,\mathrm{V}_{F}(\bm{r}^{{}^{\prime}})\,e^{i\bm{r}\bm{q}_{+}},$

where the transformed functions are given by

$\displaystyle\Phi_{F}(\bm{r})=$ $\displaystyle\frac{1}{2\pi}\int
d\bm{q}_{-}\,\operatorname{sinc}\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{-}}\right\rvert}^{2}}{4k_{p}}\biggr{)}\>\exp\biggl{(}-\alpha\frac{L\mathinner{\\!\left\lvert{\bm{q}_{-}}\right\rvert}^{2}}{4k_{p}}\biggr{)}$
$\displaystyle\times e^{-i\bm{r}\bm{q}_{-}}$ (6)

and

$\mathrm{V}_{F}(\bm{r}^{{}^{\prime}})=\frac{1}{2\pi}\int
d\bm{q}_{+}\mathinner{\\!\left\lvert\mathrm{V}(\bm{q}_{+})\right\rvert}^{2}\,e^{-i\bm{r}^{{}^{\prime}}\bm{q}_{+}}.$
(7)

The integrals over momentum space can be now implemented

$\displaystyle\braket{\Psi}{\Psi_{G}}=$ $\displaystyle N\cdot N_{G}\iint
d\bm{r}\>d\bm{r}^{{}^{\prime}}\>\Phi_{F}(\bm{r})\,\mathrm{V}_{F}(\bm{r}^{{}^{\prime}})$
$\displaystyle\times\frac{1}{2\pi}\int
d\bm{q}_{s}\,\,e^{i\bm{q}_{s}(\bm{r}+\bm{r}^{{}^{\prime}})}\frac{1}{2\pi}\int
d\bm{q}_{i}\,e^{i\bm{q}_{i}(\bm{r}-\bm{r}^{{}^{\prime}})},$

that give rise to two delta functions

$\displaystyle\braket{\Psi}{\Psi_{G}}=$ $\displaystyle N\cdot N_{G}\iint
d\bm{r}\>d\bm{r}^{{}^{\prime}}\>\Phi_{F}(\bm{r})\,\mathrm{V}_{F}(\bm{r}^{{}^{\prime}})$
$\displaystyle\times(2\pi)^{2}\,\delta{(\bm{r}+\bm{r}^{{}^{\prime}})}\,\delta{(\bm{r}-\bm{r}^{{}^{\prime}})}.$

The integrals over delta functions are trivial to perform, which gives us:

$\braket{\Psi}{\Psi_{G}}=N\cdot N_{G}\>\Phi_{F}(0)\,\mathrm{V}_{F}(0)\pi^{2}.$
(8)

Note that the expression (8) has been derived independent of the pump beam and
is quite universal. Moreover, we can see from Eq. (7) that $\mathrm{V}_{F}(0)$
equals just to $1/2\pi$, if the pump field $V_{p}(\bm{q}_{p})$ is normalized,
i.e., the fidelity is independent of the spatial distribution of the pump
beam.

The normalization constants from (8) can be calculated in the same way by
using the method of Fourier transform (Eq. (8) for the case
$\braket{\Psi}{\Psi}$ or $\braket{\Psi_{G}}{\Psi_{G}}$), giving
$N=\sqrt{2L/(k_{p}\pi^{2})}$ and $N_{G}=\sqrt{2L\alpha/(k_{p}\pi)}$
respectively. In the last step, we calculate the missing function value
$\Phi_{F}(0)$ of Eq. (8) from Eq. (II.1.1) and come up with the final
expression for the fidelity

$\braket{\Psi}{\Psi_{G}}=2\sqrt{\frac{\alpha}{\pi}}\operatorname{arccot}{\alpha}.$
(9)

The first thing to be remarked is that the fidelity is independent of the pump
and crystal characteristics and depends only on the optimization factor
$\alpha$. Let us now find out the maxima of this function. The expression (9)
as function of $\alpha$ is presented in Fig. 1. The maximum is reached for
$\alpha=0.718$ with the fidelity value $\braket{\Psi}{\Psi_{G}}_{max}=0.9$.
Note that this is very close to the value $\alpha=0.724$ such that the sinc
and Gaussian functions have equal widths at $1/e^{2}$ from the peak intensity.

Thus, the Gaussian approximation delivers the best matching with the sinc-like
state, if $\alpha=0.718$ giving an overlap value of $0.9$.

Figure 1: Dependence of the fidelity of the sinc\- and Gaussian-like states on
the optimization factor $\alpha$. The maximum possible overlap equals to $0.9$
for $\alpha=0.718.$

####  II.1.2 Super-Gaussian

The sinc function can also be approximated to a super-Gaussian function Miatto
_et al._ (2012)

$\displaystyle\operatorname{sinc}{\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}}\approx\exp\biggl{[}-\alpha\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}^{2}\biggr{]}.$
(10)

The derivation of the scalar product $\braket{\Psi}{\Psi_{SG}}$ can be
performed in the same way as for the Gaussian approximation. The normalization
constant for the state with a super-Gaussian phase matching is given by
$N_{SG}=\sqrt{2\alpha/\pi}\,2L/(k_{p}\pi)$. The corresponding fidelity reads
as

$\braket{\Psi}{\Psi_{SG}}=(2\pi\alpha)^{0.25}\operatorname{Erf}\biggl{[}{\frac{1}{2\sqrt{\alpha}}}\biggr{]},$
(11)

which as the expression from (9) is independent of the pump and crystal
characteristics. The fidelity (11) as function of $\alpha$ is presented in
Fig. 2. The maximum overlap value $\braket{\Psi}{\Psi_{G}}_{max}\approx 0.943$
can be achieved if the optimization factor is $\alpha=0.255$. Although the
fidelity $\braket{\Psi}{\Psi_{SG}}$ can be higher than that of the Gaussian
approximation, the super-Gaussian approximation is not advantageous from the
mathematical point of view, since the term
$\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{4}$ with the
fourth power of transverse momenta appears in the exponential and makes it
difficult to derive analytical expressions for SPDC.

Figure 2: The same as in Fig. 1 but for the super-Gaussian-like state. The
maximum possible overlap equals to $0.94$ for $\alpha=0.255.$

####  II.1.3 Cosine-Gaussian approximation

As suggested in Ref.Kolenderski  _et al._ (2009), we can consider instead of a
Gaussian more general cosine-Gaussian approximation

$\displaystyle\operatorname{sinc}{\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}}\approx$
$\displaystyle\exp\biggl{(}-\alpha\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}$
$\displaystyle\times\cos\biggl{(}\beta\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}.$
(12)

This is a better approximation than the usual Gaussian approximation, since
the side oscillation of the sinc function can be optimized with the cosine
function. Moreover, the cosine function can be rewritten by using the Euler’s
formula

$\displaystyle\operatorname{sinc}{\biggl{(}\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}}\approx
Re\biggl{\\{}\exp\biggl{[}-(\alpha-i\beta)\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{]}\biggr{\\}},$
(13)

which brings us back to the simple Gaussian-like function with the
optimization factor $\alpha-i\beta$. Therefore, all formulas derived for the
Gaussian approximation can be easily translated into the cosine-Gaussian case
by the replacement of optimization factors $\alpha\to\alpha-i\beta$ and taking
then the real part of the derived expression. The normalization constant of
the state with phase matching (13) is

$N_{CG}=\sqrt{\frac{4L}{\pi
k_{p}}\,}\sqrt{\frac{\alpha^{3}+\alpha\beta^{2}}{2\alpha^{2}+\beta^{2}}},$

and consequently, the overlap integral takes the form

$\displaystyle\braket{\Psi}{\Psi_{CG}}=$
$\displaystyle\sqrt{\frac{2}{\pi}}\biggr{(}\arctan{\frac{1-\beta}{\alpha}}+\arctan{\frac{1+\beta}{\alpha}}\biggl{)}$
$\displaystyle\times\sqrt{\frac{\alpha^{3}+\alpha\beta^{2}}{2\alpha^{2}+\beta^{2}}}.$

We calculated the maximal fidelity to be
$\braket{\Psi}{\Psi_{CG}}_{max}\approx 0.94$ for the optimization factors
$\alpha=0.49$ and $\beta=0.39$. Thus, the same fidelity of the super-Gaussian
approximation can be achieved also for the cosine-Gaussian approximation by
keeping the Gaussian-like behavior.

In table 1, we summarized the approximations and the corresponding
optimization factors for the spatially-resolved biphoton state. In addition,
we also presented the cosine-super-Gaussian approximations for the spatially-
resolved biphoton state that delivers the highest fidelity $0.97$. However,
again from the mathematical point of view, it is not convenient to deal with a
super-Gaussian function, because of the term
$\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{4}$ in the
exponential.

$\operatorname{sinc}{x^{2}}\approx$ | $\exp{(-\alpha x^{2})}$ | $\exp{(-\alpha x^{4})}$ | $\exp{(-\alpha x^{2})}\cos{\beta x^{2}}$ | $\exp{(-\alpha x^{4})}\cos{\beta x^{2}}$
---|---|---|---|---
$\alpha,\beta$ | $0.72$ | $0.25$ | $0.39,\>0.49$ | $0.07,\>0.5$
fidelity | $0.9$ | $0.94$ | $0.94$ | $0.97$
Table 1: optimization factors of Gaussian, super-Gaussian, cosine-Gaussian and
cosine-super-Gaussian approximations and the corresponding fidelities

###  II.2 Frequency-resolved biphoton state

We assume large area collection modes $\bm{q_{s}}=\bm{q_{i}}\approx 0$
Valencia  _et al._ (2007), in order to remain only with the spectral part of
the right-hand side of Eq. (2). The biphoton state transforms into

$\displaystyle\ket{\Psi}=N\iint$ $\displaystyle
d\Omega_{s}\>d\Omega_{i}\,\mathrm{S}(\Omega_{s}+\Omega_{i})\>\operatorname{sinc}\biggl{(}\frac{L\Delta
k_{z}}{2}\biggr{)}$
$\displaystyle\times\hat{a}^{\dagger}_{s}(\Omega_{s})\>\hat{a}^{\dagger}_{i}(\Omega_{i})\ket{vac}.$
(14)

where the phase-mismatch now is

$\Delta
k_{z}=\frac{\Omega_{s}+\Omega_{i}}{u_{p}}-\frac{\Omega_{s}}{u_{s}}-\frac{\Omega_{i}}{u_{i}}.$

The temporal distribution of the pump with a Gaussian envelope of a pulse
duration $t_{0}$ can be modeled as
$\mathrm{S}_{p}(\omega_{p})=\exp{[-(\omega_{p}-\omega_{p,0})^{2}\,t_{0}^{2}/4]}$.
The corresponding Gaussian-like state is given by

$\displaystyle\ket{\Psi_{G}}=N\iint$ $\displaystyle
d\Omega_{s}\>d\Omega_{i}\,\mathrm{S}(\Omega_{s}+\Omega_{i})\>\exp\biggl{[}-\alpha\biggl{(}\frac{L\Delta
k_{z}}{2}\biggr{)}^{2}\biggl{]}$
$\displaystyle\times\hat{a}^{\dagger}_{s}(\Omega_{s})\>\hat{a}^{\dagger}_{i}(\Omega_{i})\ket{vac}.$
(15)

Unfortunately, we could not find any analytical expression for the fidelity
$\braket{\Psi}{\Psi_{G}}$ and performed the calculations numerically. The
optimum value for $\alpha$ is $0.25$ similar to the super-Gaussian
approximation of the spatially-resolved biphoton state. Although the value
$\alpha=0.25$ is independent of $t_{0}$, the maximum value for the fidelity
$\braket{\Psi}{\Psi_{G}}_{max}$ does depend on the pulse duration, but not on
the crystal length $L$. So, if $t_{0}$ is large enough so that the pump
spectral envelope $\mathrm{S}_{p}(\omega_{p})$ dominates over the sinc
function, the maximum overlap value is close to unity. This means that the
sinc term is negligible. If the sinc function starts to dominate over
$\mathrm{S}_{p}(\omega_{p})$ (small $t_{0}$), the fidelity decreases till the
lowest values $0.95$ or, i.e., $\braket{\Psi}{\Psi_{G}}_{max}(t_{0})\geq 0.95$
for $\alpha=0.25$.

Up to this point, we handled the Gaussian-like state (II.2) as an
approximation to the real sinc-like state (II.2). However, in recent years
domain engineered crystals with Gaussian phase-matching function have
attracted attention. These engineered crystals have been mainly used, in order
to engineer pure spectral states Tambasco  _et al._ (2016); Graffitti  _et
al._ (2017, 2018), where only the spectral DOF of the biphoton state has been
considered. Thus, the state from Eq. (II.2) can be easily generated in
experiments. On the other hand, a general investigation of domain-engineering
techniques for the spatio-temporal state is still lacking and could be an
interesting topic to look at, for instance, in the scope of generating
heralded single spatio-temporal mode.

###  II.3 Spatio-temporal biphoton state

In this section, we investigate, how good the Gaussian approximation preserves
the coupling between the spatial and spectral DOF in SPDC. The description of
the spatio-temporal biphoton state with a Gaussian phase-matching function is
rather rare but has been successfully used to characterize the correlation
strength between these two DOF Osorio  _et al._ (2008). Here, the phase-
matching function $\operatorname{sinc}(L\Delta k_{z}/2)$ is again approximated
by $\exp{[-\alpha(L\Delta k_{z}/2})^{2}]$, but the momentum mismatch is
characterized by the full expression from Eq. (2). Surprisingly, this
approximation holds very well for $\alpha=0.25$ giving a fidelity value of
$0.94$. Similar to the frequency-resolved biphoton state, the fidelity depends
on the pulse duration $t_{0}$, but not on the crystal length. The argument
here is the same: as long as the sinc function dominates over the pump
function, the fidelity remains the same with the value $0.94$. This is the
minimum value for fidelity, and it becomes larger if $t_{0}$ is increased.

###  II.4 Biphoton state in Laguerre Gaussian basis

At the end of this paper, we apply the Gaussian approximation to the results
from our previous work. In Ref. Baghdasaryan  _et al._ (2022), we derived a
general expression for the spatio-temporal biphoton state and successfully
tested it in Ref. Sevilla-Gutiérrez  _et al._ (2022). The expression for the
biphoton state from Ref. Baghdasaryan  _et al._ (2022) still includes one-
dimensional integration over the crystal length, which should be performed
numerically. The Gaussian or cosine-Gaussian approximations would transform
that expression into a fully analytical form.

We construct the Gaussian approximation for the spatially-resolved biphoton
state by ignoring the dispersive properties of interacting beams, so that the
phase matching has the form
$\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}/2k_{p}$. We
consider the biphoton state with an LG pump beam of well-defined radial index
$p$ and projection of orbital angular momentum (OAM) $\ell$

$\displaystyle\ket{\Psi_{p,\ell}}=N\iint$ $\displaystyle
d\bm{q}_{s}\>d\bm{q}_{i}\>\mathrm{LG}_{p}^{\ell}(\bm{q}_{s}+\bm{q}_{i})\,\exp\biggl{(}-\alpha\frac{L\mathinner{\\!\left\lvert{\bm{q}_{s}-\bm{q}_{i}}\right\rvert}^{2}}{4k_{p}}\biggr{)}$
$\displaystyle\times\hat{a}^{\dagger}_{s}(\bm{q}_{s})\>\hat{a}^{\dagger}_{i}(\bm{q}_{i})\ket{vac},$
(16)

where the spatial distribution of the pump in the momentum space is given by

$\displaystyle\mathrm{LG}_{p}^{\ell}(\rho,\varphi)=$ $\displaystyle
e^{\frac{-\rho^{2}\,w^{2}}{4}}\,e^{i\ell\,\varphi}\,\sum_{u=0}^{p}\,T_{u}^{p,\ell}\,\rho^{2u+\mathinner{\\!\left\lvert\ell\right\rvert}}$
(17)

with $T_{u}^{p,\ell}$ being

$\displaystyle T_{u}^{p,\ell}=$
$\displaystyle\sqrt{\frac{p!\,(p+|\ell|)!}{\pi}}\,\biggr{(}\frac{w}{\sqrt{2}}\biggl{)}^{2u+|\ell|+1}\,\frac{(-1)^{p+u}(i)^{\ell}}{(p-u)!\,(\mathinner{\\!\left\lvert\ell\right\rvert}+u)!\,u!}.$

Similar to Ref. Baghdasaryan  _et al._ (2022), we perform a mode decomposition
of the biphoton state in LG basis $\ket{p,\ell}=\int
d\bm{q}\,\mathrm{LG}_{p}^{\ell}(\bm{q})\,\hat{a}^{\dagger}(\bm{q})\ket{vac}$
for the characterization of the biphoton spatial DOF

$\displaystyle\ket{\Psi_{p,\ell}}$ $\displaystyle=$
$\displaystyle\sum_{p_{s},p_{i}=0}^{\infty}\sum^{\infty}_{\ell_{s},\ell_{i}=-\infty}C_{p,p_{s},p_{i}}^{\ell,\ell_{s},\ell_{i}}\ket{p_{s},\ell_{s}}\ket{p_{i},\ell_{i}},$
(18)

where the coincidence amplitudes can be calculated from the overlap integral
$C^{\ell,\ell_{s},\ell_{i}}_{p,p_{s},p_{i}}=\braket{p_{s},\ell_{s};p_{i},\ell_{i}}{\Psi}$:

$\displaystyle C^{\ell,\ell_{s},\ell_{i}}_{p,p_{s},p_{i}}=N\iint
d\bm{q}_{s}\>d\bm{q}_{i}\>\Phi(\bm{q}_{s},\bm{q}_{i})\>[\mathrm{LG}_{p_{s}}^{\ell_{s}}(\bm{q}_{s})]^{*}$
$\displaystyle\times[\mathrm{LG}_{p_{i}}^{\ell_{i}}(\bm{q}_{i})]^{*},$ (19)

and, where the parameters {$p_{s},p_{i}$} and {$\ell_{s},\ell_{i}$} are radial
and OAM mode numbers of signal and idler photons respectively. The derivation
of $C^{\ell,\ell_{s},\ell_{i}}_{p,p_{s},p_{i}}$ is similar to the derivation
from Ref. Baghdasaryan  _et al._ (2022), hence we show only the final
expression that for $\ell\geq 0$ reads as

$\displaystyle C_{p,p_{s},p_{i}}^{\ell,\ell_{s},\ell_{i}}=$ $\displaystyle
N\,\pi^{2}\>\delta_{\ell,\ell_{s}+\ell_{i}}$
$\displaystyle\sum_{u=0}^{p}\sum_{s=0}^{p_{s}}\sum_{i=0}^{p_{i}}T_{u}^{p,\ell}\>(T_{s}^{p_{s},\ell_{s}})^{*}\>(T_{i}^{p_{i},\ell_{i}})^{*}\>\sum_{n=0}^{\mathinner{\\!\left\lvert\ell\right\rvert}}\sum_{m=0}^{u}$
$\displaystyle\binom{\mathinner{\\!\left\lvert\ell\right\rvert}}{n}\binom{u}{m}\sum_{f=0}^{u-m}\sum_{v=0}^{m}\>\binom{u-m}{f}\binom{m}{v}\Gamma[h]\>\Gamma[b]$
$\displaystyle\frac{D^{d}}{H^{h}\>B^{b}}\>{{}_{2}}{\tilde{F}}_{1}\biggl{[}h,b,1+d,\frac{D^{2}}{H\,B}\biggl{]},$
(20)

and
$C_{p,p_{s},p_{i}}^{\ell,\ell_{s},\ell_{i}}=(C_{p,p_{s},p_{i}}^{-\ell,-\ell_{s},-\ell_{i}})^{*}$
for $\ell\leq 0$. The function ${{}_{2}}{\tilde{F}}_{1}$ is known as the
regularized hypergeometric function Inc. , and the missing coefficients in the
expression (II.4) are given by

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{w_{p}^{2}}{4}+\frac{w_{s}^{2}}{4}+\frac{\alpha
L}{4k_{p}},\qquad D=-\frac{w_{p}^{2}}{4}+\frac{\alpha L}{4k_{p}}$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{w_{p}^{2}}{4}+\frac{w_{i}^{2}}{4}+\frac{\alpha
L}{4k_{p}},\qquad d=\ell_{i}+m-n-2v.$ $\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2+2s+\ell+\ell_{i}+2(-f+u)-2n-2v+\mathinner{\\!\left\lvert\ell_{s}\right\rvert}),$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2+2f+2i+\ell_{i}+2m-2v+\mathinner{\\!\left\lvert\ell_{i}\right\rvert}).$

The expression (II.4) is now fully analytical. We can then plug in either
$\alpha=0.72$ with corresponding fidelity of $0.9$ or $\alpha=0.49-0.39i$ for
the cosine-Gaussian approximation with fidelity of $0.94$ by then taking the
real part of the expression (II.4)
$\mathrm{Re}[C_{p,p_{s},p_{i}}^{\ell,\ell_{s},\ell_{i}}]$. This expression can
be used to model experiments involving very high spatial modes, where the
numerical approach could fail or take too much computational power.

##  III Conclusion

The Gaussian-like phase matching functions turn out to deliver very good
approximations to the sinc-like states. For most experiments, we recommend the
cosine-Gaussian approximation with $\alpha=0.39$ and $\beta=0.49$ for
spatially-resolved biphoton state, as this provides analytical results for
many parameters of interest with a high fidelity. For the spatio-temporal and
frequency-resolved biphoton state, we suggest to use the Gaussian
approximation with $\alpha=0.25$. However, with more challenging experiments
aiming to maximize fiber coupling efficiencies or spatial entanglement, we
still recommend to use the general sinc-like state.

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