# Detecting the Néel vector of altermagnet by attaching a topological
insulator
and crystalline valley-edge insulator

Motohiko Ezawa  Department of Applied Physics, The University of Tokyo, 7-3-1
Hongo, Tokyo 113-8656, Japan

###### Abstract

In order to detect the Néel vector of an altermagnet, we investigate
topological phases in a bilayer system composed of an altermagnet and a two-
dimensional topological insulator described by the Bernevig-Hughes-Zhang
model. A topological phase transition occurs from a first-order topological
insulator to a trivial insulator at a certain critical altermagnetization if
the Néel vector of altermagnet is along the $x$ axis or the $y$ axis. It is
intriguing that valley-protected edge states emerge along the Néel vector in
this trivial insulator, which are as stable as the topological edge states. We
name it a crystalline valley-edge insulator. On the other hand, the system
turns out to be a second-order topological insulator when the Néel vector is
along the $z$ axis. The tunneling conductance has a strong dependence on the
Néel vector. In addition, the band gap depends on the Néel vector, which is
measurable by optical absorption. Hence, it is possible experimentally to
detect the Néel vector by measuring tunneling conductance and optical
absorption.

Ferromagnets and antiferromagnets are typical examples of magnets. On one
hand, ferromagnets are used for magnetic memory, where the direction of
magnetization stores the nonvolatile memory. The magnetization direction can
be read out electrically by using the anomalous Hall effect due to the time-
reversal breaking. However, ferromagnets produce stray field due to the
nonzero net magnetization, which prevents ultra high density integrated
memory. On the other hand, antiferromagnet spintronics[1, 2, 3, 4, 5, 6, 7]
have a merit that it is possible to make ultra high density integrated memory
due to the zero net magnetization. In addition, the flipping speed of
antiferromagnets is much faster than that of ferromagnets[8]. However, it is
very hard to read out the Néel vector because the anomalous Hall effect is
absent due to the combination symmetry of time-reversal symmetry and
translational symmetry. One of the solution is multipolar antiferromagnet[9],
where the anomalous Hall effect is observed.

Recently, altermagnets have attracted rapid growth of interest[10, 11, 12]. It
is the third-type of magnets in terms of symmetry. Altermagnets break time-
reversal symmetry. Hence, an anomalous Hall effect emerges[13, 14, 15, 16],
which can be used for readout of the $z$ component of altermagnetization.
Furthermore, there is no net magnetization in altermagnets, which may lead to
an ultra-high density integration and high flipping rate of memory. Namely,
altermagnets have merits of both ferromagnets and antiferromagnets. The
characteristic feature of altermagnets is a momentum-dependent band structure
for each spin [10, 11, 12]. Indeed, momentum dependent band structures are
observed by Angle-Resolved Photo-Emission Spectroscopy (ARPES)[17, 18, 19, 20,
21]. Furthermore, spin current is generated by applying electric field[22, 23,
24, 25] owing to the above-mentioned characteristic band structure. There are
only few works on topological properties[26] induced by altermagnets except
for Majorana states[27, 28, 29].

In this paper, analyzing a bilayer system composed of an altermagnet and a
two-dimensional topological insulator, we construct a topological phase
diagram. It has intriguing features. On one hand, a topological phase
transition occurs from a first-order topological insulator to a trivial
insulator at a certain altermagnetization $J_{\text{cr}}$ when the Néel vector
is along the $x$ axis or the $y$ axis. This trivial insulator is intriguing
since it is characterized by the emergence of edge states parallel to the Néel
vector, which are as robust as topological edges. Furthermore, an edge state
connects two valleys either in the occupied band or in the unoccupied band, as
is a reminiscence of the valley-protected edge states in the valley-Chern
insulator[30]. Hence, we name it the $x$-axis crystalline valley-edge
insulator ($x$-CVEI) or the $y$-axis crystalline valley-edge insulator
($y$-CVEI). On the other hand, the system becomes a second-order topological
insulator (SOTI) when the Néel vector is along the $z$ axis. We show that
tunneling conductance has a sharp dependence on the Néel vector. In addition,
the band gap has a dependence on the Néel vector, which is measured by optical
absorption. Therefore, the Néel vector is experimentally determined by
combining the tunneling conductance and the optical absorption spectra.

Model: We analyze the bilayer system where $d$-wave altermagnet is attached on
a two-dimensional topological insulator. The Hamiltonian is given by

$H=H_{\text{BHZ}}+H_{\text{Alter}}.$ (1)

The topological insulator is described by the Bernevig-Hughes-Zhang (BHZ)
model[31],

$H_{\text{BHZ}}=M\left(k\right)\sigma_{0}\otimes\tau_{z}+\lambda\left(\sin
k_{x}\sigma_{x}\otimes\tau_{x}+\sin k_{y}\sigma_{y}\otimes\tau_{x}\right),$
(2)

with $M\left(\mathbf{k}\right)=m-t\left(\cos k_{x}+\cos k_{y}\right)$, where
$m$ is the mass parameter, $t$ is the hopping parameter, $\lambda$ is the
spin-orbit interaction, $\sigma$ is the Pauli matrix for the spin and $\tau$
is the Pauli matrix for the orbital. The BHZ model describes a topological
insulator for $\left|m/\left(2t\right)\right|<1$.

The effect of $d$-wave altermagnet is described by the Hamiltonian[10, 11, 12,
27, 29, 28]

$H_{\text{Alter}}\left(\mathbf{k}\right)=J\left(\cos k_{x}-\cos
k_{y}\right)\left(\mathbf{s}\cdot\mathbf{\sigma}\right)\otimes\tau_{x},$ (3)

with
$\mathbf{s}=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)$,
where $J$ is the magnitude of altermagnetization and $J\mathbf{s\ }$is the
Néel vector. The Néel vector is controlled by spin-orbit torque[32, 33] or
spin-split torque[34, 35]. The characteristic feature of altermagnet is that
the magnetization has a momentum dependence as in the Hamiltonian (3).

Figure 1:  (a) Bulk band gap in the $J$-$\theta$ plane with $0\leq J\leq t$
and $\pi/2\geq\theta\geq 0$, where we have set $\phi=0$. The band gap closes
at $J=J_{\text{cr}}$ and $\theta=\pi/2$. (b) Topological phase diagram in the
$\phi$-$J$ plane with $0\leq\phi\leq\pi$ and $-2\leq J/t\leq 2$, where we have
set $\theta=\pi/2$. The topological phase boundaries are analytically given by
Eq.(7). A star represents a point at which the band structure in ribbon
geometry is shown in Fig.3. We have set $m=t$ and $\lambda=0.5t$.

Band gap: We study the bulk band gap. If the Néel vector is along the $x$
axis, the energy spectrum is given by

$E^{2}=M^{2}\left(\mathbf{k}\right)+\left(\lambda\sin k_{x}+J\left(\cos
k_{x}-\cos k_{y}\right)\right)^{2}+\lambda^{2}\sin^{2}k_{y}.$ (4)

The minimum of the band gap is taken at $k_{y}=0$. The solution
$M\left(k_{x},0\right)=0$ is $k_{x}=\pm\arccos\left(m/t-1\right)$. The gap
closing condition is

$\left|J_{\text{cr}}/\lambda\right|=1/\sqrt{2t/m-1}.$ (5)

We show the band gap in the $J$-$\theta$ plane at $\phi=0$ in Fig.1(a). The
band gap closes at $J=J_{\text{cr}}$ when $\theta=\pi/2$.

If the Néel vector is in the $x$-$y$ plane with angle $\phi$, the energy
spectrum is given by

$\displaystyle E^{2}$ $\displaystyle=$ $\displaystyle
M^{2}\left(\mathbf{k}\right)+\left(\lambda\sin k_{x}+J\cos^{2}\phi\left(\cos
k_{x}-\cos k_{y}\right)\right)^{2}$ (6) $\displaystyle+\left(\lambda\sin
k_{y}+J\sin^{2}\phi\left(\cos k_{x}-\cos k_{y}\right)\right)^{2}.$

The bulk band gap closes at $J=J_{\text{cr}}$ with

$\left|\frac{J_{\text{cr}}}{\lambda}\right|=\sqrt{\frac{m\left(2m+\sqrt{2}F\right)\sec^{2}2\phi}{2\left(4t^{2}-m^{2}\right)}},$
(7)

and $F\equiv\sqrt{4t^{2}+m^{2}+\left(4t^{2}-m^{2}\right)\cos 4\phi}$. We have
shown the gap-closing curves in Fig.1(b). We later argue that they are
topological phase boundaries when the Néel vector is within the $x$-$y$ plane.

Figure 2: (a1) Bulk band gap in the $m$-$J$ plane, where the Néel vector is
along the $x$ axis. (b1) Topological phase diagram in the $m$-$J$ plane, where
the Néel vector is along the $x$ axis. The phase boundaries are analytically
given by Eq.(5). (a2) Bulk band gap in the $m$-$J$ plane, where the Néel
vector is along the $z$ axis. (b2) Topological phase diagram in the $m$-$J$
plane, where the Néel vector is along the $z$ axis. Red disk indicates the gap
closing point.

We show the band-gap structure in the $m$-$J$ plane when the Néel vector is
along the $x$-axis in Fig.2(a1) and the gap-closing curves in Fig.2(b1). We
later argue that they are topological phase boundaries. The trivial phase of
the BHZ model remains as it is for $\left|m/\left(2t\right)\right|>1$ and
$J\neq$0\. However, a new type of trivial insulator emerge for
$\left|m/\left(2t\right)\right|<1$ and $J\neq 0$, as described below.

If the Néel vector is along the $z$ axis ($\theta=0$), the energy spectrum is
given by

$E^{2}=M^{2}\left(\mathbf{k}\right)+\lambda^{2}\left(\sin^{2}k_{x}+\sin^{2}k_{y}\right)+J^{2}\left(\cos
k_{x}-\cos k_{y}\right)^{2}.$ (8)

The band gap is shown in Fig.2(a2). It implies that the altermagnetization
does not contribute to the gap closing and the gap closing condition is
identical to that of the BHZ model, i.e., $m=\pm 2t$ as in Fig.2(b2). Hence,
the topological phase boundaries remain the same as those of the BHZ model.
However, the topological phase is changed from the first-order one to the
second-order one, as described below.

Figure 3: Band structure in ribbon geometry for (a) no altermagnetization,
(b1)$\sim$(b3) the Néel vector along the $x$ axis, (c1)$\sim$(c3) the Néel
vector along the $y$ axis, (b4) the Néel vector with angle $\phi=\pi/8$, (c4)
the Néel vector with angle $\phi=\pi/4$, and (d) the Néel vector along the $z$
axis. The ribbon is taken along the $x$ axis. The horizontal axis is the
momentum $k_{x}$. (b1) and (c1) $J=0.2t$. (b2) and (c2)
$J=0.5t=J_{\text{cr}}$. (b3), (c3), (b4) and (c4) $J=0.7t$. We have set
$m/t=1$, $\lambda/t=0.5$. Red color indicates the edge state, while cyan color
indicates the bulk state. The point ($\phi$,$J$) of each figure except (d) is
indicated as a star in the phase diagram Fig.1(b).

Ribbon geometry: We analyze the energy spectrum of the total Hamiltonian with
ribbon geometry to see the topological property. In the absence of the
altermagnetization, $J=0$, there are topological edge states of the BHZ model
(2), which cross at the momentum $k_{x}=0$, as in Fig.3(a).

If the altermagnetization is along the $x$ axis,
$J\mathbf{s}_{x}=\left(J,0,0\right)$, the crossing point of the topological
edge states moves away from $k_{x}=0$ for $J<J_{\text{cr}}$ as in Fig.3(b1).
At the critical point $J=J_{\text{cr}}$, the bulk band gap closes and the
system is a semimetal as in Fig.3(b2). For $J>J_{\text{cr}}$, each edge state
connects either the occupied band or the unoccupied band, as in Fig.3(b3). It
indicates the system is trivial. Although the edge states are nontopological,
they are robust against the order of the band gap because the turning point of
the edge state much exceeds the band gap. Indeed, it requires perturbation
much larger than the band gap to remove the trivial edge states. These
nontopological edge states are as robust as topological edges. They are a
reminiscence of the valley-protected edge states with each edge connecting two
valleys either in the occupied band or in the unoccupied band[30].

If the altermagnetization is along the $y$ axis,
$J\mathbf{s}_{y}=\left(0,J,0\right)$, the topological edge states are
maintained for $J<J_{\text{cr}}$ as in Fig.3(c1). At the critical point
$J=J_{\text{cr}}$, the bulk band gap closes as in Fig.3(c2). Edge states
disappear for $J>J_{\text{cr}}$ as in Fig.3(c3).

We show the band structure when the altermagnetization is given by
$J\mathbf{s}=J\left(\cos\phi,\sin\phi,0\right)$ for the cases $\phi=\pi/8$,
$\pi/4$ in Fig.3(b4), (c4), respectively. The bulk band gap closes in the
vicinity of $\phi=\pi/8$ as in Fig.3(b4) and Fig.1(b). The valley-protected
edge states become the topological edge states as in Fig.3(c4) and Fig.1(b).

If the altermagnetization is along the $z$ axis, $Js_{z}=\left(0,0,J\right)$,
the topological edge states anticross and a finite gap emerges in edge states
as in Fig.3(d). It means that the system is not a first-order topological
insulator. We will soon see that the system is a second-order topological
insulator, which is characterized by the emergence of corner states in square
geometry.

Square geometry: We investigate the energy spectrum of the total Hamiltonian
with square geometry to see the topological property more in detail.

In the case of $J\mathbf{s}_{x}=(J,0,0)$ or
$J\mathbf{s}_{y}=\left(0,J,0\right)$, the edge states emerge along the $x$
axis or the $y$ axis, as in Fig.4(a1) or (a2), respectively. Correspondingly,
the energy spectrum is linear as a function of the eigenindex in the vicinity
of the zero energy, as in Fig.4(b1) or (b2). These edge states are
nontopological as we have argued in ribbon geometry. On the other hand, there
are no edge states along the $y$ axis or the $x$ axis in the case of
$J\mathbf{s}_{x}=(J,0,0)$ or $J\mathbf{s}_{y}=\left(0,J,0\right)$ as in
Fig.4(a1) or (a2), respectively. It is a reminiscence of edge states of
topological crystalline insulator[36]. The system is the $x$-axis crystalline
valley-edge insulator ($x$-CVEI) in Fig.4(a1) and the $y$-axis crystalline
valley-edge insulator ($y$-CVEI) in Fig.4(a2).

Figure 4: (a1)$\sim$(a3) Spatial distribution of the zero-energy states marked
in red. (b1)$\sim$(b3) Energy spectrum as a function of the eigenindex of the
Hamiltonian. The inset in (b3) shows the presence of the four corner states at
zero energy. (c1)$\sim$(c3) Tunneling conductance as a function of the energy.
(d1)$\sim$(d3) Illustration of a square sample with four leads. (a1)$\sim$(d1)
The Néel vector is along the $x$ axis. (a2)$\sim$(d2) The Néel vector is along
the $y$ axis. (a3)$\sim$(d3) The Néel vector is along the $z$ axis. We have
set $m/t=1$, $\lambda/t=0.5$ and $J=0.7t>J_{\text{cr}}$.

Figure 5:  (a1) and (a2) Tunneling conductance in the $\phi$-$\theta$ plane.
(b1) and (b2) Tunneling conductance as a function of $\theta$ at $\phi=0$. The
horizontal axis is $0\leq\theta\leq\pi$. (c1) and (c2) Tunneling conductance
as a function of $\phi$ at $\theta=\pi/2$. The horizontal axis is
$0\leq\phi\leq\pi$. (a1)$\sim$(c1) $J=0.7t>J_{\text{cr}}$. (a2)$\sim$(c2)
$J=0.2t<J_{\text{cr}}$.

In the case of $J\mathbf{s}_{z}=\left(0,0,J\right)$, four topological corner
states emerge as in Fig.4(a3). Correspondingly, there are four zero-energy
corner states within a finite gap as in Fig.4(b3). As a result, the system is
a second-order topological insulator.

Conductance: We show that the Néel vector is detectable by measuring the
conductance in the setup shown in Fig.4(d1)$\sim$(d3), where four leads are
attached to the corners of the square sample. We assume that the leads are
single-atomic chains with semi-infinite length.

The conductance between two leads is calculated based on the Landauer
formalism[37, 38, 39, 40, 41]. The conductance $\sigma(E)$ at energy $E$ is
calculated as[37]

$\sigma(E)=(e^{2}/h)\text{Tr}[\Gamma_{\text{L}}(E)G_{\text{D}}^{{\dagger}}(E)\Gamma_{\text{R}}(E)G_{\text{D}}(E)],$
(9)

where
$\Gamma_{\text{R(L)}}(E)=i[\Sigma_{\text{R(L)}}(E)-\Sigma_{\text{R(L)}}^{{\dagger}}(E)]$
is the line width with the self-energies $\Sigma_{\text{L}}(E)$ and
$\Sigma_{\text{R}}(E)$ for the left and right leads, and
$G_{\text{D}}(E)=[E-H-\Sigma_{\text{L}}(E)-\Sigma_{\text{R}}(E)]^{-1}$ is the
Green function with the Hamiltonian $H$ for the sample. The self energy of a
single-atomic semi-infinite chain is analytically obtained[37] as
$\Sigma_{\text{L}}(E)=\Sigma_{\text{R}}(E)=E-i\sqrt{\left|t^{2}-E^{2}\right|}$.

First, we study the case $J>J_{\text{cr}}$. We show the conductance as a
function of the energy $E$ in Fig.4(c1), (c2) and (c3). We have calculated the
conductance between the leads A and B. The conductance is nonzero at $E=0$
when the Néel vector is along the $x$ axis as in Fig.4(c1). It is understood
that the current flows through the edge state along the $x$ axis as in
Fig.4(d1). On the other hand, it is zero at $E=0$ when the Néel vector is
along the $y$ axis as in Fig.4(c2). It is because there is no edge state to
carry the current along the $x$ axis as in Fig.4(d2). The conductance is also
zero at $E=0$ when the Néel vector is along the $z$ axis as in Fig.4(c3). It
is because there is no edge state although corner states exist as in
Fig.4(d3). It seems to be difficult to differentiate the cases where the Néel
vector is along the $y$ axis or the $z$ axis by measuring the conductance
because the tunneling conductance is zero for both cases. However, it is
differentiated by mearing the tunneling conductance between the leads A and D,
where the conductance is nonzero if the Néel vector is along the $y$ axis but
zero if the Néel vector is along the $z$ axis.

The conductance in the $\phi$-$\theta$ plane is shown in Fig.5(a1). The
conductance as a function of $\theta$ at $\phi=0$ is shown in Fig.5(b1). The
conductance takes maximum value at $\theta=\pi/2$ and takes minimum value at
$\theta=0$. The conductance as a function of $\phi$ at $\theta=\pi/2$ is shown
in Fig.5(c1). The conductance takes the maximum value at $\phi=0$ and the
minimum value at $\phi=\pi/2$. Hence, we can detect the Néel vector by
measuring the conductance.

Next, we study the case $J<J_{\text{cr}}$. We show the conductance in the
$\phi$-$\theta$ plane in Fig.5(a2). The dependence on the angle $\phi$ is tiny
comparing with the case $J>J_{\text{cr}}$, as in Fig.5(c2). It is because the
system is a first-order topological insulator, where the topological edge
state surround the sample. The conductance is zero for $\theta=0$ and $\pi$ as
in the case of the system $J>J_{\text{cr}}$, as in Fig.5(b2). It is because
there are topological corner states but there are no edge states.

Figure 6: Optical absorption spectra. Red curves indicate the spectra with
$\theta=\pi/2$, while blue curves indicate that with $\theta=0$. We have set
$\phi=0$. The horizontal axis is the photon energy $\hbar\omega$. Insets show
the bulk band gap as a function of $\theta$ at $\phi=0$. (a)
$J=0.2t<J_{\text{cr}}$ and (b) $J=0.7t>J_{\text{cr}}$. The red and blue
circles correspond to the optical absorption colored in red and blue.

Optical absorption: We show that the band gap is observed by examining the
optical absorption spectrum. We analyze the optical inter-band transition from
the state $|u_{\text{v}}(\mathbf{k})\rangle$ in the valence band to the state
$|u_{\text{c}}(\mathbf{k})\rangle$ in the conduction band. We apply a
circularly polarized light, where the electromagnetic potential is given by
$A(t)=(A_{x}\sin\omega t,A_{y}\cos\omega t)$.

The optical absorption is calculated based on the Kubo formula[42, 43, 44, 45,
36],

$\displaystyle\varepsilon\left(\omega\right)=$ $\displaystyle\frac{\pi
e^{2}}{\varepsilon_{0}m_{e}^{2}\omega^{2}}\sum_{i}\int_{BZ}\frac{d\mathbf{k}}{\left(2\pi\right)^{2}}f(\mathbf{k})\left|P_{j}(\mathbf{k})\right|^{2}$
$\displaystyle\times\delta\left[E_{\text{c}}(\mathbf{k})-E_{\text{v}}(\mathbf{k})-\hbar\omega\right],$
(10)

where $P_{j}(\mathbf{k})$ is the optical matrix element
$P_{j}(\mathbf{k})\equiv m_{0}\left\langle
u_{\text{c}}(\mathbf{k})\right|\frac{\partial H}{\partial
k_{j}}\left|u_{\text{v}}(\mathbf{k})\right\rangle$, $E_{\text{c}}(\mathbf{k})$
and $E_{\text{v}}(\mathbf{k})$ are the energies of the conduction and valence
bands, and $f(\mathbf{k})$ is the Fermi distribution function.

We show the optical absorption spectra at $\theta=0$ and $\pi/2$ in Fig.6.
There is a gap in the spectrum, implying the sudden occurrence of the optical
absorption at the moment that the photon energy becomes the same as the band-
gap energy. Hence, the band gap is measured by optical absorption experiments.

In conclusion, it is possible to determine the Néel vector by measuring the
tunneling conductance and the optical absorption.

The author is very much grateful to S. Seki for helpful discussions on the
subject. This work is supported by CREST, JST (Grants No. JPMJCR20T2) and
Grants-in-Aid for Scientific Research from MEXT KAKENHI (Grant No. 23H00171).

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# Broadband biphoton source for quantum optical coherence tomography
based on a Michelson interferometer

Konstantin Katamadze  Anna Romanova<EMAIL_ADDRESS>Denis Chupakhin
Alexander Pashchenko  Sergei Kulik  Quantum Technology Centre, Faculty of
Physics, Lomonosov Moscow State University, Moscow, Russia

###### Abstract

Broadband correlated photon pairs (biphotons) are valuable in quantum
metrology, but current generation methods either involve complex nonlinear
structures or lack sufficient bandwidth and brightness. In this work, we
theoretically describe and experimentally demonstrate a novel technique for
generation of a bright collinear biphoton field with a broad spectrum,
achieved by using a tightly focused pump in a bulk nonlinear crystal. As the
most straightforward application of the source, we employ Michelson
interferometer-based quantum optical coherence tomography (QOCT). Utilizing
the source enables the demonstration of record resolution and dispersion
cancellation for this QOCT scheme.

††preprint: APS/123-QED

##  I  Introduction

Correlated photon pairs (biphotons) obtained through spontaneous parametric
down-conversion (SPDC) [1] are one of the primary tools in quantum optics. The
key role is played by time correlations between the photons of a pair, which
become stronger the broader the spectrum of the biphoton field. A
comprehensive review[2] details various implementations of broadband
biphotons. Some are based on spatially homogeneous nonlinear media, where
bandwidth is limited by phase matching and dispersion relations [3, 4, 5, 6,
7, 8, 9, 10]. Others use inhomogeneous media, where phase matching conditions
vary, allowing broader bandwidth limited by spatial modulation range only [11,
12, 13, 14, 15, 16, 17, 18]. Achieving a biphoton field with a bandwidth up to
150-200 THz is feasible with these techniques. However, achieving higher
values is challenging because, in all the mentioned approaches (with the
exception of [8]), spectral broadening is inherently linked to a decrease in
spectral intensity. In the best-case scenario, integral intensity may be
conserved, but this is typically divergent from real experimental conditions.

Broadband biphotons find applications in diverse quantum metrological
problems. Examples include quantum clock synchronization [19, 20, 21, 22, 23],
two-photon microscopy [24, 25, 26], two-photon spectroscopy [27, 28, 29].
Another notable and straightforward application is quantum optical coherence
tomography (QOCT) [30], which extends the capabilities of classical optical
coherence tomography (OCT) [31]. Early OCT systems employed a broadband light
source and a Michelson interferometer (MI), with one arm serving as a scanning
reference and the other containing the test sample. Despite advancements in
OCT’s accuracy through intricate experimental designs and sophisticated data
analysis, challenges persist, particularly in addressing limited scanning
depth and resolution for highly dispersive samples.

A quantum version of OCT utilizes entangled photon pairs and the Hong-Ou-
Mandel effect. This method, being dispersion-tolerant, offers double the
resolution, reaching record values [32]. The initial hurdle was the
requirement for a bright broadband biphoton source, typically produced by non-
collinear or type-II SPDC. However, further research showed that QOCT could
also be achieved using a brighter, broader, and more easily adjustable
collinear type-I SPDC source, coupled with an MI [33, 34, 35, 36].

This work introduces a novel biphoton broadening technique based on tightly
focusing the pump and target modes [37], applied to MI-based QOCT. Ideally,
this broadening approach preserves spectral intensity, leading to an increase
in integral intensity with a broader spectral bandwidth.

The article unfolds as follows. In Section II, we provide a theoretical
description of the source’s operation and detail its experimental
implementation. Section III delves into the theoretical and laboratory aspects
of the interferometer designed for QOCT. Finally, in Section IV, we present
and discuss the experimental results of QOCT experiments.

##  II Bright source of broadband biphoton field

###  II.1 Theory

Figure 1: Experimental setup. a) Broadband biphoton source: $L$ – crystal
length, $W$ – modes waist, $\Theta_{\text{p}}$ – pump walk-off angle, L1, L2 –
lenses. b) QOCT Michelson interferometer: BS – beam splitter, DM – dichroic
mirror splitting biphotons at 1000 nm, M1 – reference mirror, M2 – sample
mirror, standing on the piezo-translator which is attached to a mechanical
translator, D1, D2 – single-photon detectors (SPD), based on Si avalanche
photodiodes (APD) Laser components COUNT NIR, D3 – SPD, based on InGaAs APD
(MPD PDM–IR).

A comprehensive theoretical description of our biphoton source can be found in
[37]. Similar to the listed techniques [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18], we employ SPDC process as a biphoton source. SPDC
involves the decay of pump (p) photons into pairs of signal (s) and idler (i)
photons, satisfying energy conservation and phase-matching conditions:

$\displaystyle\omega_{\text{s}}+\omega_{\text{i}}-\omega_{\text{p}}$
$\displaystyle=0,$ (1)
$\displaystyle\vec{k}_{\text{s}}+\vec{k}_{\text{i}}-\vec{k}_{\text{p}}$
$\displaystyle=\vec{\Delta}_{k},$ (2)

where $\omega_{\text{p,s,i}}$ are frequencies, $k_{\text{p,s,i}}$ are wave
vectors, and $\vec{\Delta}_{k}$ is a phase mismatch, typically $\lesssim$ the
inverse nonlinear media length $1/L$.

Since our biphoton source interfaces with an interferometer, our focus is on
the photon pair rate in a single spatial target mode, filterable by a single-
mode fiber. We assume a Gaussian profile for the pump, signal, and idler modes
propagating along the $z$-axis (top inset in Fig. 1a):

$\begin{gathered}\vec{E}_{j}(\vec{r},t)=\frac{1}{2}\left[\vec{E_{j}}g_{j}(\vec{r})\operatorname{e}^{-i\omega_{j}t}+\text{
c.c. }\right],\\\ \text{where }\quad
g_{j}(\vec{r})=\operatorname{e}^{ik_{j}z}\operatorname{e}^{-\frac{x^{2}+(y-\Theta_{j}z)^{2}}{W^{2}_{j}}},\quad
j=\text{p, s, i}.\end{gathered}$ (3)

Here, $\vec{E_{j}}$ is a field amplitude, $\Theta_{j}$ is a walk-off angle,
and $W_{j}$ is a beam waist. Diffraction terms are omitted under the
assumption that the Rayleigh length is significantly larger than other
longitudinal scales of the system. According to [38], the photon pair rate in
the selected signal and idler modes can be calculated using a mode overlap
integral:

$R=\frac{8d^{2}P}{\pi^{6}W^{2}_{\text{p}}W^{2}_{\text{s}}W^{2}_{\text{i}}c^{3}}\frac{\omega_{\text{s}}\omega_{\text{i}}}{n_{\text{p}}n_{\text{s}}n_{\text{i}}}\int_{-\infty}^{+\infty}\text{d}\omega\\\
\times\left|\int_{-\infty}^{+\infty}\text{d}x\text{d}y\int_{-L/2}^{L/2}\text{d}zg_{\text{p}}(\vec{r})g_{\text{s}}^{*}(\vec{r})g_{\text{i}}^{*}(\vec{r})\right|^{2},$
(4)

where $d$ is an effective second-order nonlinear susceptibility,
$n_{\text{p}},\ n_{\text{s}},\ n_{\text{i}}$ are refractive indices for the
pump, signal, and idler modes, $c$ is the speed of light, and
$P=c\epsilon_{0}n_{\text{p}}\pi
W_{\text{p}}^{2}\left(\left|E_{\text{p}}\right|/2\right)^{2}$ is the pump
power.

Considering a collinear e-oo type-I phase matching, i.e.,
$\Theta_{\text{s}}=\Theta_{\text{i}}=0$, and assuming equal waists for
simplicity ($W_{\text{p}}=W_{\text{s}}=W_{\text{i}}=W$), we focus on a low-
waist case where an effective overlap length
$L_{\text{eff}}=W/\Theta_{\text{p}}\ll L$. Thus, in (4), we can integrate over
$z$ from $-\infty$ to $+\infty$. After evaluating the integral, we obtain:

$R=\underbrace{\frac{4d^{2}P\omega_{\text{s}}\omega_{\text{i}}}{3\pi^{3}c^{3}\epsilon_{0}n_{\text{p}}n_{\text{s}}n_{\text{i}}\rho^{2}}}_{S_{0}}\underbrace{\int\operatorname{e}^{-3\left(\Delta_{k}\left(\omega_{\text{s}}\right)L_{\text{eff}}/2\right)^{2}}\text{d}\omega_{\text{s}}}_{\Delta_{\omega}}.$
(5)

Here, the factor $S_{0}$ is a spectral coincidence rate, independent of waist
$W$ and crystal length $L$, while the factor $\Delta_{\omega}$ defines the
integral spectral bandwidth and depends on the effective interaction length
$L_{\text{eff}}$ instead of $L$, making both the integral rate $R$ and the
bandwidth $\Delta_{\omega}$ proportional to $1/\sqrt{W}$.

Therefore, the primary idea is to reduce the interaction length
$L_{\text{eff}}$ and compensate for a small rate through high focusing of the
pump and target modes. A similar approach was demonstrated in [9], where
authors used extremely short nonlinear media, achieving a significant biphoton
rate due to high focusing. Conversely, another biphoton broadening technique
using pump focusing [4] is based on a quite different approach and cannot
provide as extensive broadening as our technique.

###  II.2 Implementation

The schematic representation of the source is presented in Fig. 1a. To achieve
dispersion cancellation, as detailed in [39, 40], we employed a narrowband
pump laser (Toptica DLC DL pro HP 405) with a wavelength of 405 nm and a
bandwidth of approximately 100 kHz. The laser radiation passed through a
single-mode fiber and was then tightly focused onto a 1 mm-thick nonlinear
crystal (BBO type-I). The resulting collinear degenerate biphoton field was
collimated using a parabolic mirror and passed through a pump-blocking 450 nm
longpass filter. Subsequently, another parabolic mirror was utilized to couple
the field into a single-mode polarization-maintaining photonic crystal fiber
(NKT Photonics LMA-PM-10), chosen for its ability to guide and preserve the
broadband spectrum. The biphoton field was then directed into the Michelson
interferometer, described in the next section.

The alignment of the pump and target beams was meticulously optimized to
achieve the maximum coincidence count rate, measured directly after the output
of the fiber, reaching a final value of 3 kcps. The pump power, measured in
front of the crystal, was 8 mW. Post-experiment, the crystal was removed, and
the pump waist in free space was measured as $W_{p}=5.7~{}\upmu\rm m$.

##  III Interferometer

###  III.1 Theory

A comprehensive theoretical description of biphoton field interference in a
Michelson interferometer (MI) can be found in [34, 35, 40]. Here, we will
provide the main basis and results. Consider a biphoton field in a single
spatial mode with the following spectral quantum state:

$\left|{{\Psi}_{\text{in}}}\right\rangle=\frac{1}{\sqrt{2}}\iint{\text{d}{{\omega}_{\text{s}}}\text{d}{{\omega}_{\text{i}}}}f\left({{\omega}_{\text{s}}},{{\omega}_{\text{i}}}\right)\hat{a}_{\text{in}}^{\dagger}\left({{\omega}_{\text{s}}}\right)\hat{a}_{\text{in}}^{\dagger}\left({{\omega}_{\text{i}}}\right)\left|\text{vac}\right\rangle.$
(6)

where $f(\omega_{\text{s}},\omega_{\text{i}})$ is a spectral amplitude,
approximated as a double-Gaussian function with the two-photon spectral
distribution [41, 42] (Fig. 2):

${{\left|f({{\omega}_{\text{s}}},{{\omega}_{\text{i}}})\right|}^{2}}=\frac{1}{\pi\delta\Delta}\underbrace{\operatorname{e}^{-\frac{{{\left({{\omega}_{1}}-{{\omega}_{2}}\right)}^{2}}}{2{{\Delta}^{2}}}}}_{\text{Phasematching}}\underbrace{\operatorname{e}^{-\frac{{{\left({{\omega}_{1}}+{{\omega}_{2}}-2{{\omega}_{0}}\right)}^{2}}}{2{{\delta}^{2}}}}}_{\text{Pump}}.$
(7)

Here, $\omega_{0}\equiv\omega_{\text{p}}/2$, $\delta$ is the standard
deviation (StD) of the pump spectral distribution, and
$\Delta\sim\Delta_{\omega}$ is related to the phase matching bandwidth.

Now, let this field pass through an MI with a delay $T$ in a fixed arm and
delay $\tau$ in a scanning arm (see Fig. 1b and Fig. S1 in [40]). The MI
output is split into two paths ending with single-photon detectors. Measuring
the coincidence count rate depending on the MI delay $\tau$ allows obtaining
an interferogram:

Figure 2: Double-Gaussian two-photon spectral distributions
$\absolutevalue{f(\omega_{\text{s}},\omega_{\text{i}})}$.

$M(\tau)=\frac{1}{16}\left[M_{\text{c}}+M_{0}(\tau)-M_{1}(\tau)+M_{2}(\tau)\right],$
(8) $\displaystyle\text{where}\quad{{M}_{c}}$
$\displaystyle=\left(1+R\right)^{2},$ (9a) $\displaystyle{{M}_{0}}(\tau)$
$\displaystyle=2R\operatorname{e}^{-\frac{{{\left(T-\tau\right)}^{2}}}{2{{\Delta}^{-2}}}},$
(9b) $\displaystyle{{M}_{1}}(\tau)$
$\displaystyle=4r\left(1+R\right)\operatorname{e}^{-\frac{{{\left(T-\tau\right)}^{2}}}{8\Delta_{+}^{-2}}}\cos\left[{{\omega}_{0}}\left(T-\tau\right)\right],$
(9c) $\displaystyle{{M}_{2}}(\tau)$ $\displaystyle=2R\
\operatorname{e}^{-\frac{{{\left(T-\tau\right)}^{2}}}{2{{\delta}^{-2}}}}\cos\left[2{{\omega}_{0}}\left(T-\tau\right)\right].$
(9d)

Here $\Delta_{+}^{2}\equiv\delta^{2}+\Delta^{2}\approx\Delta^{2}$.

The total interferogram $M(\tau)$ is plotted in Fig. 3a. Its spectrum
$\absolutevalue{\tilde{M}(\omega)}$ in Fig. 3b contains three peaks:
$\tilde{M}_{0}$ centered at $\omega=0$ with StD $\Delta$, $\tilde{M}_{1}$
centered at $\omega_{0}$ with StD $\Delta/2$, and $\tilde{M}_{2}$ centered at
$2\omega_{0}$ with StD $\delta$. The corresponding interferogram terms are
plotted in Fig. 3c. Term $M_{0}$ represents a peak with StD $1/\Delta$,
corresponding to HOM-interference. Term $M_{1}$ has twice the StD and
corresponds to single-photon interference. Finally, $M_{2}$ term with a
$1/\delta$ StD is equivalent to pump interference in an MI.

Figure 3: Total interferogram $M(\tau)$ (a), it’s spectrum (b) with terms
$\tilde{M}_{0}(\omega)$, $\tilde{M}_{1}(\omega)$, $\tilde{M}_{2}(\omega)$ and
corresponding interferogram terms $M_{0}(\tau)$, $M_{1}(\tau)$, $M_{2}(\tau)$
(c).

The filtering of the $M_{0}$ term allows achieving a resolution that is twice
as good as that of the $M_{1}$ term, corresponding to standard OCT. Moreover,
similar to the usual HOM-based QOCT, the $M_{0}$ term provides dispersion
cancellation (see [40] for details). However, we observe that terms $M_{0}$
and $M_{1}$ can be separated only if $\Delta\ll\omega_{\text{p}}/3$.
Otherwise, the spectra $\tilde{M}_{0}(\omega)$ and $\tilde{M}_{1}(\omega)$
overlap, and the terms cannot be well separated anymore.

###  III.2 Implementation

For the basic experiment in QOCT, we constructed the setup illustrated in Fig.
1b. Biphoton pairs were emitted from the fiber, collimated using a parabolic
mirror. The beam was then split on a non-polarized beam splitter (BS) into two
channels: the sample channel and the reference channel. The sample channel
included a mirror and an optional 2-mm wide glass plate to introduce
dispersion. The reference channel consisted of a mirror mounted on a combined
piezo-mechanical translator stage. The reflected beams were recombined on the
subsequent BS.

To broaden the detection bandwidth and cover both the visible (VIS) and
infrared (IR) ranges, we employed a longpass dichroic mirror (DM) with a cut-
on wavelength $\lambda_{\text{c}}=1000~{}\text{nm}$. In both channels,
biphotons were coupled with single-mode fibers using parabolic mirrors. The
visible part was further split by a fiber beam splitter and directed to two
Si-based detectors (Laser Components COUNT NIR), labeled as D1 and D2. The
infrared part was directed to an InGaAs detector (MPD PDM-IR), labeled as D3.
All detector outputs were connected to coincidence circuits, enabling
coincidence measurements between the two visible-range detectors D1&D2 (VIS-
VIS) and between the IR detector and both visible-range detectors: D3&D1 and
D3&D2 (IR-VIS).

##  IV Experimental results and discussion

Initially, we conducted a $z$-scan using a sample comprising only a mirror.
The results are depicted in Fig. 4a–f.

In Fig. 4a we show raw interferogram data for VIS and IR channels,
representing classical OCT. Both interferograms exhibit sine functions with
Gaussian amplitude modulation. The separation of the centers of the two
Gaussian envelopes may indicate slight dispersion in either the sample or
reference arm.

The spectra of the interferograms presented in Fig. 4c exhibit two distinct
peaks, revealing a separation at the cut-on wavelength of the dichroic mirror
($\lambda_{\text{c}}=1000~{}\text{nm}$). Assuming an ideal biphoton spectrum
resembling a Gaussian function, the estimated full width at half-maximum
(FWHM) is approximately $2\pi\times 235~{}\text{THz}$, corresponding to a
wavelength range of about 610–1190 nm.

Notably, this experimental value is nearly twice as large as the theoretical
estimate derived from (5) [37]. Given that, for degenerate type-I phase
matching, the spectral bandwidth $\Delta_{\omega}\propto
1/\sqrt{L_{\text{eff}}}$ [37], this implies an effective mode overlap length
$L_{\text{eff}}$ roughly 4 times smaller than the theoretical value of
$W_{\text{p}}/\Theta_{\text{p}}=85~{}\upmu\rm m$. This deviation can be
attributed to unaccounted aberrations and beam diffraction within the
experimental setup. Simultaneously, for a fixed interaction modes waist $W$,
the source’s spectral generation rate $S_{0}\propto 1/L_{\text{eff}}^{2}$
[37]. Consequently, the experimentally determined $S_{0}$ can be up to 16
times smaller than the theoretically estimated value of
$S_{0}=125~{}\frac{\text{cps}}{\text{mW}\times\text{THz}}$ [37]. However,
considering the spectral efficiency of VIS detectors and the bandwidth of
coincidence detection, the total registered coincidence count rate of 3 kcps
suggests the source’s spectral generation rate of $S_{0}\approx
16~{}\frac{\text{cps}}{\text{mW}\times\text{THz}}$. This value is only 8 times
smaller than the estimated value, which reveals a more efficient experimental
performance – twice as favorable.

Thus, the total source’s generation rate is approximately $4\
\frac{\text{kcps}}{\text{mW}}$, which is brighter than the previous
experimental results (0.8 $\frac{\text{kcps}}{\text{mW}}$) for a bulk crystal
and a waist of 82 $\upmu$m from [38].

The sum of VIS and IR interferograms (Fig. 4a) is plotted in Fig. 4e. A
Gaussian approximation estimates its FWHM as $1.48\ \upmu$m, corresponding to
classical OCT resolution. Interferograms of VIS-VIS and IR-VIS coincidences
are shown in Fig. 4b. Their spectra (Fig. 4d) qualitatively agree with
theoretical spectra, presented in Fig. 3b and also Fig. S3b in [40]. Both VIS-
VIS and IR-VIS spectra exhibit a narrowband peak at $\omega_{\text{p}}$
(corresponding to $M_{2}$ term). The VIS-VIS spectrum exhibits a broadband
peak centered at $\omega_{\text{p}}/2$, corresponding to the $M_{1}$ term
equivalent to single-photon interference in a Michelson Interferometer (MI).
In the case of IR-VIS coincidence detection, which is equivalent to selecting
a non-degenerate part of the biphoton spectrum, the $M_{1}$ term is
represented by two peaks (consistent with Fig. S3b). These peaks are separated
from the VIS-VIS $M_{1}$ peak by the frequency $\omega_{\text{c}}\equiv 2\pi
c/\lambda_{\text{c}}$ and its conjugated frequency
$\omega_{\text{p}}/2+\Delta_{\text{c}}$, where
$\Delta_{\text{c}}\equiv\omega_{\text{p}}/2-\omega_{\text{c}}$. Finally, the
VIS-VIS interferogram exhibits a peak centered at zero frequency,
corresponding to the Hong-Ou-Mandel (HOM) peak $M_{0}$. The same term is
represented in the IR-VIS interferogram as a shifted peak (also in agreement
with Fig. S3b), separated from the VIS-VIS $M_{0}$ peak by the frequency
$2\Delta_{\text{c}}$.

Given that the total biphoton bandwidth is estimated at $2\pi\times
235~{}\text{THz}$ from single-photon interferograms and is close to
$\omega_{\text{p}}/3=2\pi\times 247$ THz, the $M_{0}$ and $M_{1}$ terms
exhibit partial overlap, and the $M_{0}$ term cannot be fully filtered. This
overlap becomes apparent from Fig.4d, where the $M_{0}$ peak coincides with
the left $M_{1}$ peak in the IR-VIS spectrum. To address this, we applied
filtering to separate the $M_{0}$ term: VIS-VIS coincidences were filtered
from $0$ to $2\Delta_{\text{c}}$, and IR-VIS coincidences were filtered from
$2\Delta_{\text{c}}$ to $\omega_{\text{p}}/3$.

The sum of these filtered interferograms is presented in Fig. 4f, illustrating
the HOM peak $M_{0}$ with an additional influence of the sine term $M_{1}$.
Employing a Gaussian approximation, we estimated its FWHM to be $1.23\
\upmu$m, corresponding to the achieved QOCT resolution. The QOCT resolution
enhancement was slightly less than two, as a portion of the $M_{0}$ spectrum
was lost during the filtering procedure.

Figure 4: Experimental results. a) Raw interferogram in single count rate; the
red curve refers to the Si detectors detecting wavelengths of $~{}400-1000$ nm
(VIS) and the brown curve refers to InGaAs detector, which measures
$1000-1500$ (IR). b) Raw interferogram in coincidence count rate; the red
curve refers to the coincidences between two Si detectors (VIS-VIS) and the
brown curve shows coincidences between Si detectors and InGaAs detector (IR-
VIS). c) The red curve refers to the spectrum of the VIS counts, the brown
curve shows the spectrum of the IR counts, and the dotted grey curve shows the
approximate joint spectrum. d) The red solid curve refers to the spectrum of
the VIS-VIS coincidences, the brown solid curve shows the spectrum of the IR-
VIS coincidences, and the dotted curves show the data that were filtered out.
e) The blue curve corresponds to the joint single counts and the dotted red
curve corresponds to the Gaussian approximation. f) The blue curve shows joint
IR-VIS and VIS-VIS coincidences filtered from useless frequencies, the red
dotted curve shows Gaussian approximation. g) Joint IR and VIS data, taken
with the dispersive glass plate placed in the sample arm. Data have a break
since they were taken in two steps because the full interferogram did not fit
into the range of the piezo-stage. h) The blue curve corresponds to the
processed data for the coincidences in the presence of dispersion.

However, a remarkable QOCT resolution enhancement was observed for a sample
with dispersion introduced by a 2-mm glass plate. The combined single-photon
interferogram (Fig. 4g) showcases well-separated IR and VIS terms due to
dispersion. The dispersion-induced broadening was so significant that a single
scan couldn’t cover the interference pattern range, requiring a combination of
two scans for bandwidth estimation ($\approx 50~{}\upmu\rm m$). The processed
coincidence interferogram (Fig. 4h) resembles the interferogram without
dispersion (Fig. 4f), demonstrating excellent dispersion cancellation and a 35
times resolution improvement compared to classical OCT.

##  V Conclusion

In this study, we have introduced a cost-effective technique for generating a
bright (4 kcps/mW pair generation rate) and ultra-broadband (235 THz
bandwidth) biphoton field utilizing a bulk nonlinear crystal and tight
focusing of a pump beam. Demonstrating both theoretical insights and
experimental validation, we applied this technique to quantum optical
coherence tomography (QOCT), achieving an impressive axial resolution of
$1.23\ \upmu$m. Notably, we have established, for the first time, that
Michelson interferometer based QOCT manifests dispersion cancellation,
analogous to those observed in traditional Hong-Ou-Mandel-based
implementation, resulting in a minimum 35-fold enhancement in resolution
compared to classical Optical Coherence Tomography techniques. Furthermore,
our findings indicate that the attained QOCT resolution is predominantly
constrained by interferogram filtering procedures, thereby implying potential
improvement through the investigation of advanced filtering techniques or
modified interference schemes.

###### Acknowledgements.

This research was performed according to the MSU287 Program of Development,
Project No 23-SCH06-05.

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