# Measurement induced nonclassical states from coherent state heralded by
Knill-Laflamme-Milburn-type SU(3) interference

Xue–xiang Xu† and Shan-jun Ma  Department of Physics, Jiangxi Normal
University, Nanchang 330022, China
†Corresponding author<EMAIL_ADDRESS>
###### Abstract

We theoretically generate nonclassical states from coherent state heralded by
Knill-Laflamme-Milburn (KLM)-type SU(3) interference. Injecting a coherent
state in signal mode and two single-photon sources in other two auxiliary
modes of SU(3) interferometry, a broad class of useful nonclassical states are
obtained in the output signal port after making two single-photon-counting
measurements in the two output auxiliary modes. The nonclassical properties,
in terms of anti-bunching effect and squeezing effect as well as the
negativity of the Wigner function, are studied in detail by adjusting the
interaction parameters. The results show that the input coherent state can be
transformed into non-Gaussian states with higher nonclassicality after
measurement induction. The maximum squeezing of our generated states can be
arrived at about 1.9 dB.

PACS: 03.67.-a, 05.30.-d, 42.50,Dv, 03.65.Wj

Keywords: nonclassical state; non-Gaussian state; SU(3) interference; linear
optics; mesurement-induced; Wigner function

##  I Introduction

Nonclassical optical states, as the basic sources, play essential roles in
continuous variables (CV) quantum information processing (QIP) 1 ; 2 . Quantum
information over CVs has yielded many exciting advances, both theoretically
and experimently, in fields such as quantum teleportation 3 ; 4 , quantum
metrology 5 , and potentially quantum computation 6 ; 7 . In order to enhance
performance to implement various tasks and provide quantum advantages not
attainable classically, a number of world-wide groups has embarked on route
towards understanding, generating, and ultimately manipulating various
nonclassical states of light 8 . In recent years, many types of nonclassical
states have been proposed in theory and even have already been implemented in
laboratories. Agarwal and Tara generated a nonclassical state by operating
photon addition on a coherent state and investigated the non-classicality of
field states 9 . Zavatta et al. had realized the single-photon added coherent
state 10  and single-photon-added thermal state 11 .

Gaussian states play a prominent roles in CV quantum information 12 ; 13 .
However, many quantum technological protocols, such as entanglement
distillation 14 ; 15  or quantum error correction 16 , necessarily require the
use of non-Gaussian state, which can extract ultimate potential of quantum
information theory unaccessible by the Gaussian states. With the increasing
significance of non-Gaussian states, the issues of generating non-Gaussian
states have been naturally arisen. It is well-known that the non-Gaussian
states must exhibit non-Gaussian features in their Wigner distributions in the
phase space 17  and gererating non-Gaussian states normally requires
nonlinearities in the field operators 18 . In order to obtain non-Gaussian
states, there is an alternative idea of measurement-induced scheme (also the
conditional probabilistic operation) to obtain nonlinearity 19 . Some
effective nonlinearity 20  is associated with non-Gaussian measurement such as
homodyne measurements or photon countings 21 ; 22  based on a linear quantum-
optical system 23 , which generally consist of beam splitting, phase shifting,
squeezing, displacement, and various detection 24 .

Many examples of such measurement-induced schemes by linear optics have been
theoretically implemented, where nonlinear operators are obtained and then
non-Gaussian states are generated 25 ; 26 . Dakna et al. generated a
Schrodinger-cat-like state from a single-mode squeezed vacuum state by
subtracting photons with low reflectance beam-splitters (BSs) and photon
counters 27 . Subsequently, Lvovsky and Mlynek proposed the idea of “quantum-
optical catalysis” and generated a coherent superposition state
$t\left|0\right\rangle+\alpha\left|1\right\rangle$ 28 . This state was
generated in one of the BS output channels if a coherent state
$\left|\alpha\right\rangle$ and a single-photon Fock state
$\left|1\right\rangle$ are present in two input ports and a single photon is
registered in the other BS output. They called this transformation as
“quantum-optical catalysis” because the single photon itself remains
unaffected but facilitates the conversion of the target ensemble. Following
Lvovsky and Mlynek’s work and using “quantum-optical catalysis”, Bartley et
al. further generated multiphoton nonclassical states via interference between
coherent and Fock states, which exhibit a wide range of nonclassical phenomena
29 . Indeed, the key step in Bartley’s scheme is to employ “quantum-optical
catalysis” on the input coherent state one time. Recently, we operated
“quantum-optical catalysis” on each mode of the two-mode squeezed vacuum state
and generated a non-Gaussian two-mode quantum state with higher entanglement,
accompanied by a nonlinear operator
$c_{0}\allowbreak+c_{1}a^{{\dagger}}b^{{\dagger}}+c_{2}a^{{\dagger}2}\allowbreak
b^{{\dagger}2}$ 30 . Enlighten by above works, we further consider the
following problem: If we perform “quantum-optical catalysis” on a coherent
state two times, then what states will be generated and what nonclassical
phenomena will happen? This is the kernal focus of our work.

In 2000, Knill, Laflamme, and Milburn (KLM) implemented a nonlinear sign (NS)
change using a optical network composed of three beam splitters in succession,
whose main features are the use of two ancilla modes with one prepared photon
and post-selection based on measuring the ancillas 31 . Because of the SU(3)
algebra property of this setup, we name it as KLM-type SU(3) interferometry.
Following KLM’s program and using KLM-type SU(3) interferometry, Ralph et al.
constructed a conditional quantum control-NOT gate from linear optical
elements 32 . Laterly, Scheel et al. investigated the generation of nonlinear
operators and constructed useful single-mode and two-mode quantum gates
necessary for all-optical QIP 33 . Hence we also use the KLM-type SU(3)
interferometry as our framework to generate nonclassical states.

In this paper, based on the network of KLM-type SU(3) interferometry, we
generate a broad class of nonclassical states from a coherent state by
operating two-fold “quantum-optical catalysis”, a kind of measurement-induced
scheme. Special attention is paid to study the nonclassical properties for the
generated states in terms of anti-bunching effect and squeezing effect as well
as the negativity of the Wigner function. We show that the input coherent
state can be transformed into non-Gaussian states with higher nonclassicality
after measurement induction. The paper is organized as follows. In section II,
We briefly outline the KLM-type SU(3) interferometry and introduce the
measurement-induced scheme for generating nonclassical states. In particular,
the normalization factor (i.e. success probability) is discussed. In section
III, we investigate the nonclassical properties (anti-bunching effect and
squeezing effect) of the generated state and analyze the effect of the
interaction. Then in section IV we consider the negativity of the Wigner
function, another nonclassical indication for a quantum state. We summarize
our main results in Sec.V.

##  II Measurement induced nonclassical state

In this section, we briefly outline the dynamics of KLM SU(3) interferometry
and apply this interferometry to generate nonclassical states by measurement-
induced protocol. The theoretical scheme is proposed and the success
probability is derived.

###  II.1 KLM-type SU(3) interferometry

Linear optical networks have important applications in QIP and can be realized
in experiment 34 . The quantum mechanical description of linear optics can be
found in Ref 35 . Recently, there has been a growing interest in linear
optical networks in the context of quantum information technology. It has been
demonstrated that universal (nondeterministic) quantum computation is
poissible when linear optical networks are combined with single photon
detectors 36 . Among the networks, the KLM-type SU(3) interferometry is an
important instrument in quantum mechanics and quantum technology.

As illustrated in Fig.1, the main framework of KLM-type SU(3) interferometry
is composed of three beam splitters (BSs) in sequence. The three bosonic modes
containing the photons will be described by creation (annhilation) operators
labeled $a^{{\dagger}}\left(a\right)$, $b^{{\dagger}}\left(b\right)$, and
$c^{{\dagger}}\left(c\right)$. The action of this network can be described by
the unitary operator
$U=U_{3}\left(\eta_{3}\right)U_{2}\left(\eta_{2}\right)U_{1}\left(\eta_{1}\right)$,
where
$U_{1}\left(\eta_{1}\right)=e^{\theta_{1}\left(bc^{{\dagger}}-b^{{\dagger}}c\right)}$,
$U_{2}\left(\eta_{2}\right)=e^{\theta_{2}\left(ba^{{\dagger}}-b^{{\dagger}}a\right)}$
and
$U_{3}\left(\eta_{3}\right)=e^{\theta_{3}\left(bc^{{\dagger}}-b^{{\dagger}}c\right)}$
are the corresponding operators of the three BSs with their respective
reflection rates $\cos\theta_{j}=\eta_{j}$ ($j=1,2,3$). One finds that this
effect generates a linear transformation of the mode operators in the
Heisenberg picture. Hence we know the dynamics of the creation operators:

$U\left(\begin{array}[]{c}a^{{\dagger}}\\\ b^{{\dagger}}\\\
c^{{\dagger}}\end{array}\right)U^{{\dagger}}=\left(\begin{array}[]{ccc}S_{11}&S_{12}&S_{13}\\\
S_{21}&S_{22}&S_{23}\\\
S_{31}&S_{32}&S_{33}\end{array}\right)\left(\begin{array}[]{c}a^{{\dagger}}\\\
b^{{\dagger}}\\\ c^{{\dagger}}\end{array}\right),$ (1)

Here, the scattering matrix $S$ is a $3\times 3$ one with its elements:
$S_{11}=-\sqrt{\eta_{2}}$, $S_{12}=\sqrt{\eta_{1}\left(1-\eta_{2}\right)}$,
$S_{13}=\sqrt{\left(1-\eta_{1}\right)\left(1-\eta_{2}\right)}$,
$S_{21}=\sqrt{\left(1-\eta_{2}\right)\eta_{3}}$,
$S_{22}=\sqrt{\left(1-\eta_{1}\right)\left(1-\eta_{3}\right)}+\sqrt{\eta_{1}\eta_{2}\eta_{3}}$,
$S_{23}=\sqrt{\left(1-\eta_{1}\right)\eta_{2}\eta_{3}}-\sqrt{\eta_{1}\left(1-\eta_{3}\right)}$,
$S_{31}=\sqrt{\left(1-\eta_{2}\right)\left(1-\eta_{3}\right)}$,
$S_{32}=\sqrt{\eta_{1}\eta_{2}\left(1-\eta_{3}\right)}-\sqrt{\left(1-\eta_{1}\right)\eta_{3}}$,
and
$S_{33}=\sqrt{\eta_{1}\eta_{3}}+\sqrt{\left(1-\eta_{1}\right)\eta_{2}\left(1-\eta_{3}\right)}$.

###  II.2 Scheme for generating nonclassical states

The considered network is actually a three-input and three-output linear
optical system. Three beams of optical field, i.e. a coherent state
$\left|\alpha_{a}\right\rangle$ and two single-photon resources
$\left|1_{b}\right\rangle$ and $\left|1_{c}\right\rangle$, are injected into
the three-input ports in their respective optical modes $a$, $b$, $c$. By the
way, we assusme the amplitude of the coherent state
$\alpha=\left|\alpha\right|e^{i\theta}$ with $\theta=0$ for simplification.
After the interaction in the linear optical system, we make single-photon-
counting measurements in the $b$-mode and $c$-mode output ports. Thus a
conditional quantum state $\left|\psi_{a}\right\rangle$ is obtained
theoretically and given by

$\left|\psi_{a}\right\rangle=\frac{1}{\sqrt{p_{d}}}\left\langle
1_{c}\right|\left\langle
1_{b}\right|U\left|\alpha_{a}\right\rangle\left|1_{b}\right\rangle\left|1_{c}\right\rangle$
(2)

which is generated by measurement induction. The normalization factor $p_{d}$
represents the probability heralded by the successful single-photon detection
in two auxiliary modes. After detailed deviation in appendix A, the explicit
expression of $\left|\psi_{a}\right\rangle$ is given by

$\left|\psi_{a}\right\rangle=\left(c_{0}+c_{1}a^{{\dagger}}+c_{2}a^{{\dagger}2}\right)\left|S_{11}\alpha\right\rangle,$
(3)

with the cofficients $c_{0}=\Pi\left(S_{22}S_{33}+S_{23}S_{32}\right)$,
$c_{1}=\alpha\Pi(S_{12}S_{21}S_{33}+S_{12}S_{31}S_{23}+S_{21}S_{13}S_{32}+S_{13}S_{22}S_{31})$,
$c_{2}=\alpha^{2}\Pi S_{12}S_{21}S_{13}S_{31}$ and
$\Pi=e^{-\left(1-S_{11}^{2}\right)\left|\alpha\right|^{2}/2}/\sqrt{p_{d}}$.
Obviously, a optical operator $c_{0}+c_{1}a^{{\dagger}}+c_{2}a^{{\dagger}2}$
is implemented in this interaction. Note that the coefficients $c_{0}$,
$c_{1}$, and $c_{2}$ are as the functions of all the relevant parameters. The
interaction parameters involve the coherent-state amplitude $\alpha$, the
beam-splitter reflectance rates $\eta_{1}$, $\eta_{2}$, and $\eta_{3}$. The
generated quantum state can be looked as a non-Gaussian state by operating
this operator on another coherent state $\left|S_{11}\alpha\right\rangle$. Not
surprisingly, the input coherent state becomes non-Gaussian after the process.
In particular, the generated state $\left|\psi_{a}\right\rangle$ can be
reduced to the input coherent state $\left|\alpha_{a}\right\rangle$ when
$\eta_{1}=\eta_{2}=\eta_{3}=\eta\rightarrow 1$.

Figure 1: (Colour online) Schematic setup of the KLM circuit for generating a
quantum state $\left|\psi_{a}\right\rangle$ from an inputting coherent state
$\left|\alpha_{a}\right\rangle$ using three beam splitters (with reflection
rates $\eta_{1},\eta_{2},\eta_{3}$ respectively), two single-photon sources,
and two single-photon resolving detectors. In contrast with
$\left|\alpha_{a}\right\rangle$, the generated state
$\left|\psi_{a}\right\rangle$ has a wide range of nonclassical properties.

By tuning the parameters of the interaction, namely, $\left|\alpha\right|$,
$\eta_{1}$, $\eta_{2}$, and $\eta_{3}$, the cofficients may be modulated,
generating abroad class of nonclassical states with a wide range of
nonclassical phenomena, as shown next by us.

###  II.3 Success probability of detection

Normalization is important for discussing the properties of a quantum state.
The normalization factor of the generated states in theory is actually the
probability $p_{d}$ of counting successfully single photons at the two
auxiliary modes in experiment. The density operator of the generated state
$\rho_{a}=\left|\psi_{a}\right\rangle\left\langle\psi_{a}\right|$ is expressed
in Appendix C. According to $\mathrm{Tr}\left(\rho_{a}\right)=1$, the success
probability to get the state $\left|\psi_{a}\right\rangle$ from our proposal
is given by

$\displaystyle p_{d}$ $\displaystyle=$
$\displaystyle(g_{0}+g_{1}\left|\alpha\right|^{2}+g_{2}\left|\alpha\right|^{4}$
(4)
$\displaystyle+g_{3}\left|\alpha\right|^{6}+g_{4}\left|\alpha\right|^{8})e^{-\left(1-S_{11}^{2}\right)\left|\alpha\right|^{2}},$

Here $g_{0}$, $g_{1}$, $g_{2}$, $g_{3}$, and $g_{4}$ are given in appendix D.

In our following numerical works, the quantities under considering are
discussed only in the two special cases: (1) set
$\eta_{1}=\eta_{2}=\eta_{3}=\eta$ and change $\eta$, (2) set
$\eta_{1}=\eta_{3}=1/2$ and change $\eta_{2}$. In order to exhibit numerically
the probability $p_{d}$, we plot the contour of $p_{d}$ in the
$\left(\eta,\left|\alpha\right|\right)$ plain space in Fig.2 (a) and in the
$\left(\eta_{2},\left|\alpha\right|\right)$ plain space in Fig.2 (b).
Additionally, we plot $p_{d}$ as a function of $\eta$ in Fig.2 (c) and as a
function of $\eta_{2}$ in Fig.2 (d) for $\left|\alpha\right|=1,2,3$. For a
large $\left|\alpha\right|$ and small $\eta$ (or $\eta_{2}$), the probability
of detection $p_{d}$ is small and even less than $0.05$. For values closer to
$\eta=1$, the generated state gets closer to the origin input state
$\left|\alpha\right\rangle$ and the probability gets closer to $1$.

Figure 2: (Colour online) Success probability $p_{d}$ of detection (a) contour
plot in ($\eta,\left|\alpha\right|$) space with
$\eta=\eta_{1}=\eta_{2}=\eta_{3}$; (b) contour plot in
($\eta_{2},\left|\alpha\right|$) space with $\eta_{1}=\eta_{3}=1/2$; (c) as a
fuction of $\eta$; (d) as a fuction of $\eta_{2}$. Here the black, red and
brown lines are corresponding to $\left|\alpha\right|=1$,
$\left|\alpha\right|=2$, $\left|\alpha\right|=3$, respectively. For a large
amplitude of the coherent state $\left|\alpha\right|$, the probability is
smaller than $0.05$. When $\eta$ is limit to $1$, the probability is limit to
1.

##  III Nonclassical properties of the generated states

Quantum states of light can be classified according to their statistical
properties. They are usually compared to a reference state, namely, the
coherent state 37 . In comparison with the input coherent state, what
nonclassical properties will exhibit after meaurement induction in our
proposed scheme. Analytical expressions of the expected values $\left\langle
a^{{\dagger}k}a^{l}\right\rangle$ for arbitrary $k,l$ found in appendix F
allow us to study the statistical properties of this generated states in our
following works. Here, we will focus on studying some nonclassical properties
of this generated quantum state, including anti-bunching effect and quadrature
squeezing effect.

###  III.1 Anti-bunching effect

The second-order autocorrelation function
$g^{\left(2\right)}\left(0\right)=\left\langle a^{\dagger
2}a^{2}\right\rangle/\left\langle a^{\dagger}a\right\rangle^{2}$ determines
whether the source produce effects following antibunching
($g^{\left(2\right)}\left(0\right)<1$), bunching ($1\leqslant
g^{\left(2\right)}\left(0\right)\leqslant 2$), or superbunching
($g^{\left(2\right)}\left(0\right)>2$). Additionally, it also determines
whether the source produce photons following sub-
($g^{\left(2\right)}\left(0\right)<1$), super-
($g^{\left(2\right)}\left(0\right)>1$), or Poissonlike
($g^{\left(2\right)}\left(0\right)=1$) statistics 38 . For a coherent state,
we have $g^{\left(2\right)}\left(0\right)=1$, which shows its character of
Poisson distribution. Here, we shall examine the anti-bunching effect (a
strictly nonclassical character) of the generated states, which describes
whether the photons in the beam tend to stay apart.

The variations of $g^{\left(2\right)}\left(0\right)$ with the interaction
parameters are showed in Fig.3. For several given amplitudes of coherent state
($\left|\alpha\right|=1,2,3$), we plot $g^{\left(2\right)}\left(0\right)$ as a
function of $\eta$ in Fig.3 (a) and as a function of $\eta_{2}$ in Fig.3 (b).
In a extreme case, we verify that $g^{\left(2\right)}\left(0\right)=1$ when
$\eta$ (or $\eta_{2}$) is limit to $0$, i.e., the states corresponding to the
input coherent state $\left|\alpha\right\rangle$. Moreover, the feasibility
regions for antibunching, bunching and superbunching are exhibited in the
($\eta,\left|\alpha\right|$) parameter space in Fig.3 (c) and in the
($\eta_{2},\left|\alpha\right|$) parameter space in Fig.3 (d). It is found
that there may present antibunching effect in a wide range of interaction
parameters. The results show that the generated state
$\left|\psi_{a}\right\rangle$ can exhibit a broad range of nonclassical
features by tuning the interaction parameter.

Figure 3: (Color online) Second-order autocorrelation function
$g^{\left(2\right)}\left(0\right)$ of the generated state. The regions shows
antibunching $g^{\left(2\right)}\left(0\right)<1$ (strictly nonclassical),
bunching $1\leqslant g^{\left(2\right)}\left(0\right)\leqslant 2$, and
superbunching $g^{\left(2\right)}\left(0\right)>2$. (a) as a function of
$\eta$ ($\eta_{1}=\eta_{2}=\eta_{3}=\eta$); (b) as a function of $\eta_{2}$
($\eta_{1}=\eta_{3}=1/2$). Here the black, red and brown lines are
corresponding to $\left|\alpha\right|=1$, $\left|\alpha\right|=2$,
$\left|\alpha\right|=3$, respectively. (c) the feasibility region in ($\eta$,
$\left|\alpha\right|$) plain space.(d) the feasibility region in ($\eta_{2}$,
$\left|\alpha\right|$) plain space. Here the blue, yellow and brown regions
are corresponding to antibunching, bunching, and superbunching cases
respectively.

###  III.2 Quadrature squeezing effect

Squeezed light has come a long way since its first demonstration 30 years 39 .
Significant advancements have been made from the initial 0.3 dB squeezing till
todys near 13 dB squeezing 40 . Hence we will ask two questions: (1) Whether
our generated states are squeezed states? (2) If they are squeezed states,
then how much can the squeezing degree be arrived? Here we will consider the
squeezing effect of these states.

Firstly we make a brief review of quadrature squeezing effect. Many
experiments have been carried out dealing with noise in the quadrature
component of the field, which is defined by two quadrature operators
$X=\left(a+a^{{\dagger}}\right)/\sqrt{2}$ and
$P=\left(a-a^{{\dagger}}\right)/\left(\sqrt{2}i\right)$, analogous to the
position and momentum of a harmonic oscillator 41 . Both quadrature variances,
related with the creation and annhilation operators, can expressed as
$\left\langle\Delta X^{2}\right\rangle=(\left\langle
a^{{\dagger}2}\right\rangle-\left\langle
a^{{\dagger}}\right\rangle^{2}+\left\langle a^{2}\right\rangle-\left\langle
a\right\rangle^{2}+1)/2+\left\langle a^{{\dagger}}a\right\rangle-\left\langle
a^{{\dagger}}\right\rangle\left\langle a\right\rangle$ and $\left\langle\Delta
P^{2}\right\rangle=(-\left\langle a^{{\dagger}2}\right\rangle+\left\langle
a^{{\dagger}}\right\rangle^{2}-\left\langle a^{2}\right\rangle+\left\langle
a\right\rangle^{2}+1)/2+\left\langle a^{{\dagger}}a\right\rangle-\left\langle
a^{{\dagger}}\right\rangle\left\langle a\right\rangle$ respectively, as can be
seen from their definitions. These components cannot be measured
simulatanously because of the commutation relation $[X,P]=i$. It follows that
the product of the variances in the measurements of the two quadratures $X$
and $P$ satisfies $\Delta X^{2}\Delta P^{2}\geq 1/4$ (a Heisenberg
inequality). For a vacuum state $\left|0\right\rangle$ or a coherent state
$\left|\alpha\right\rangle$, the uncertainty relation is satisfied as an
equality, and the two variances are identical: $\Delta
X^{2}|_{\left|0\right\rangle,\left|\alpha\right\rangle}=\Delta
P^{2}|_{\left|0\right\rangle,\left|\alpha\right\rangle}=1/2$. A quantum state
is called squeezed if the variance of a quadrature amplitude is below the
variance of a vacuum or a coherent state, i.e. $\Delta X^{2}<1/2$ or $\Delta
P^{2}<1/2$. The squeezing effect of a light field comes at the expense of
increasing the fluctuations in the other quadrature amplitude. Here we can
adopt quantum squeezing quantified in a dB scale through
$dB[X]=10\log_{10}\left(\Delta X^{2}/\Delta
X^{2}|_{\left|0\right\rangle}\right)$, $dB[P]=10\log_{10}\left(\Delta
P^{2}/\Delta P^{2}|_{\left|0\right\rangle}\right)$. In other words, if $dB[X]$
or $dB[P]$ is negative, this quantum state is a squeezed state.

In Fig.4 (a), the behaviour of $dB[X]$ as a function of $\eta$
($=\eta_{1}=\eta_{2}=\eta_{3}$) for different $\left|\alpha\right|$. By
minimizing the expression of $\Delta X^{2}$, the largest squeezing attained is
around $0.321772$, below the vacuum noise level of $1/2$ by approximately
$1.91422$ dB (using MATHENATICA) for $\left|\alpha\right|=1$. The squeezed
degree is bigger than that ($1.25$ dB) in Ref.29 . In Fig.4 (b), the behaviour
of $dB[X]$ as a function of $\eta_{2}$ for different $\left|\alpha\right|$
with $\eta_{1}=\eta_{3}=1/2$. Moreover, the purple regions show the
feasibility squeezed region of $X$ quadrature component in the
($\eta,\left|\alpha\right|$) parameter space in Fig.4 (c) and in the
($\eta_{2},\left|\alpha\right|$) parameter space in Fig.4 (d). Fig.4 (c) and 4
(d) show a wide range of squeezing for $\left|\alpha\right|$ and
$\eta\left(\eta_{2}\right)$.

Figure 4: (Color online) Quadrature variance of $X$ component relative to the
vacuum (unsqueezed) state in units of dB (a) as a function of $\eta$
($\eta_{1}=\eta_{2}=\eta_{3}=\eta$); (b) as a function of $\eta_{2}$
($\eta_{1}=\eta_{3}=1/2$). Here the black, red and brown solid lines are
corresponding to $\left|\alpha\right|=1$, $\left|\alpha\right|=2$,
$\left|\alpha\right|=3$ in X component, respectively. The feasibility squeezed
(purple) and unsqueezed (brown) region of the $X$ component (c) in ($\eta$,
$\left|\alpha\right|$) plain space; (d) in ($\eta_{2}$, $\left|\alpha\right|$)
plain space.

##  IV Wigner function of the generated states

The negative Wigner function is a witness of the nonclassicality of a quantum
state 42 . Additionaly, we can determinate whether this quantum state is non-
Gaussian state from the form of the Wigner function 43 . Since non-Gaussian
quantum state can provide quantum advantages not attainable classically, it is
necessary to study the Wigner functions of our generated states.

The analytical expression of the Wigner function for the generated states is
derived in appendix F. The results plotted in Fig.5 are obtained for optimal
choices with different parameters
$\left(\left|\alpha\right|,\eta_{1},\eta_{2},\eta_{3}\right)$. Fig.5 shows
that the Wigner functions are negative in some regions of the phase space,
which is a witness of the nonclassicality. As we all know, the coherent state
is a typical Gaussian state whose Wigner function has no negative regions. In
comparison with the input coherent state, the generated states show the non-
Gaussian features with negative regions in moderate parameters range.

Meanwhile, we plot the negative volume $\delta$ as a function of $\eta$ (or
$\eta_{2}$) for $\left|\alpha\right|=1$, $2$, $3$ in Fig.6. It is obvious to
see that for different $\left|\alpha\right|$, the maximun negative volumn
locate at different $\eta$ (or $\eta_{2}$). For instance, when
$\left|\alpha\right|=1$, $\delta_{max}$ is located at $\eta=0$ (or
$\eta_{2}=0$) and $\delta$ is decreaing with $\eta$ (or $\eta_{2}$) increasing
and limit to zero at large $\eta$ (or $\eta_{2}$); For
$\left|\alpha\right|=2$, $\delta_{max}$ is located at around $\eta=0.27$ (or
$\eta_{2}=0.5$); For $\left|\alpha\right|=3$, $\delta_{max}$ is located at
around $\eta=0.7$ (or $\eta_{2}=0.8$).

Figure 5: (Color online) Wigner functions of the measurement-induced state
with parameters $\left(\left|\alpha\right|,\eta_{1},\eta_{2},\eta_{3}\right)$:
(a) $\left(1,0.05,0.05,0.05\right)$; (b) $\left(2,0.5,0.05,0.5\right)$; (c)
$\left(2,0.25,0.25,0.25\right)$; (d) $\left(2,0.5,0.5,0.5\right)$. The result
shows that the Wigner function of the generated state has the negative
regions, which is different from that of the input coherent state and also
shows the nonclassical character. Figure 6: (Color online) The negative volumn
$\delta$ of the Wigner functions of the generated states (a) as a function of
$\eta$ ($\eta_{1}=\eta_{2}=\eta_{3}=\eta$); (b) as a function of $\eta_{2}$
($\eta_{1}=\eta_{3}=1/2$). The solid, dotted and dashed line are corresponding
to $\left|\alpha\right|=1$, $\left|\alpha\right|=2$, $\left|\alpha\right|=3$,
respectively.

##  V Discussion and Conclusion

Our results show that there exist optimal nonclassical properites in different
parameter ranges. The optimal performance as required can be obtained by
adjusting the interaction parameters. A prominent character is that the the
maxinum squeezing can be reached to $1.91422$ dB. In comparison with the
states generated in Bartley’s work, the squeezing degree of our generated
states is enhanced. In table I, we numerate some values of the probability of
detection $p_{d}$, the auto-correlation function
$g^{\left(2\right)}\left(0\right)$, the squeezing in $X$ component $dB[X]$,
and the negative volume $\delta$ of the Wigner functions in their
corresponding $\left(\left|\alpha\right|,\eta_{1},\eta_{2},\eta_{3}\right)$
parameters.

Table 1: Values of the probability of detection $p_{d}$, the auto-correlation function $g^{\left(2\right)}\left(0\right)$, the squeezing in $X$ component $dB[X]$, and the negative volume $\delta$ of the WFs in their corresponding interaction parameter values of $\left(\left|\alpha\right|,\eta_{1},\eta_{2},\eta_{3}\right)$.  $\left(\left|\alpha\right|,\eta_{1},\eta_{2},\eta_{3}\right)$ | $p_{d}$ | $g^{\left(2\right)}\left(0\right)$ | $dB\left[X\right]$ | $\delta$
---|---|---|---|---
$\left(1,0.3,0.3,0.3\right)$ | 0.15223 | 26.0347 | -1.16685 | 0.0103
$\left(1,0.6,0.6,0.6\right)$ | 0.27011 | 0.43187 | -0.52375 | 0.0010
$\left(1,0.9,0.9,0.9\right)$ |  0.67275 | 0.97122 | -0.08971 | 0.0009
$\left(1,0.5,0.7,0.5\right)$ | 0.35303 | 0.76685 | -0.31669 | 0.0009
$\left(2,0.5,0.7,0.5\right)$ | 0.03770 | 1.47059 | -0.43680 | 0.0334
$\left(3,0.5,0.7,0.5\right)$ | 0.01392 | 0.99035 | 1.80934 | 0.1283

In summary, this paper employs two-fold quantum-optical catalysis to generate
a class of nonclassical states based on KLM-type SU(3) interferometry. By
measurement induction, we implemente a nonlinear operator
$c_{0}+c_{1}a^{{\dagger}}+c_{2}a^{{\dagger}2}$ and obtain a broad class of
useful non-Gaussian quantum states with higher nonclassicality. We discuss the
success probability of detection and the nonclassical properties in terms of
anti-bunching effect and squeezing effect as well as the negativity of the
Wigner function. In compared with the input coherent state, these generated
nonclassical states exhibit a lot of nonclassical properties. The results show
that the optimal antibunching and the maximum squeezing as well as the maximum
negative volume of the Wigner functions are located in different parameter
points. Hence one can choose the optimal performance of these nonclassical
properties to implement various technological tasks. In addition, the
generation of these nonclassical states is feasible with current technology.
The experiment realization of such states is desired to achieve in the future.
Our results can provid a theoretical reference for experiments.

###### Acknowledgements.

This work was supported by the National Nature Science Foundation of China
(Grants No. 11264018 and No. 11447002) and the Natural Science Foundation of
Jiangxi Province of China (Grants No. 20142BAB202001 and No. 20151BAB202013)

Appendix A: Transformation relation of the SU(3) interferometry

The network can be seen as three multiports in cascade, one for each of the
beam splitters with the unitary operators $U_{j}\left(\eta_{j}\right)$ and the
scattering matrices $S_{j}$ $\left(j=1,2,3\right)$. For each stage, the
dyanamics is
$U_{j}\left(a^{{\dagger}},b^{{\dagger}},c^{{\dagger}}\right)^{T}U_{j}^{{\dagger}}=S_{j}\left(a^{{\dagger}},b^{{\dagger}},c^{{\dagger}}\right)^{T}$,
with the scattering metrix

$S_{1}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\sqrt{\eta_{1}}&\sqrt{1-\eta_{1}}\\\
0&\sqrt{1-\eta_{1}}&-\sqrt{\eta_{1}}\end{array}\right),$
$S_{2}=\left(\begin{array}[]{ccc}-\sqrt{\eta_{2}}&\sqrt{1-\eta_{2}}&0\\\
\sqrt{1-\eta_{2}}&\sqrt{\eta_{2}}&0\\\ 0&0&1\end{array}\right),$
$S_{3}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\sqrt{\eta_{3}}&\sqrt{1-\eta_{3}}\\\
0&\sqrt{1-\eta_{3}}&-\sqrt{\eta_{3}}\end{array}\right),$

So the total operator is
$U=U_{3}\left(\eta_{3}\right)U_{2}\left(\eta_{2}\right)U_{1}\left(\eta_{1}\right)$
and the total scattering matrix is $S=S_{3}S_{2}S_{1}$. Thus Eq.(1) is
obtained. One can also refer the detailed information in Ref.35 .

Appendix B: Explicit expression of the generated state

Substituting
$\left|\alpha_{a}\right\rangle=e^{-\left|\alpha\right|^{2}/2+\alpha
a^{{\dagger}}}\left|0_{a}\right\rangle$,
$\left|1_{b}\right\rangle=\frac{d}{ds_{1}}e^{s_{1}b^{{\dagger}}}\left|0_{b}\right\rangle|_{s_{1}=0}$,
$\left|1_{c}\right\rangle=\frac{d}{ds_{2}}e^{s_{2}c^{{\dagger}}}\left|0_{c}\right\rangle|_{s_{2}=0}$,
$\left\langle 1_{b}\right|=\frac{d}{ds_{3}}\left\langle
0_{b}\right|e^{s_{3}b}|_{s_{3}=0}$ and $\left\langle
1_{c}\right|=\frac{d}{ds_{4}}\left\langle 0_{c}\right|e^{s_{4}c}|_{s_{4}=0}$,
into Eq.(2) and using the transformation in Eq.(1) as well as the fact
$U\left|0_{a}\right\rangle\left|0_{b}\right\rangle\left|0_{c}\right\rangle=\left|0_{a}\right\rangle\left|0_{b}\right\rangle\left|0_{c}\right\rangle$,
we finally arrive at the derivative form of $\left|\psi_{a}\right\rangle$,

$\displaystyle\left|\psi_{a}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{-\left|\alpha\right|^{2}/2}}{\sqrt{p_{d}}}\frac{d^{4}}{ds_{4}ds_{3}ds_{2}ds_{1}}e^{\alpha\left(S_{12}s_{3}+S_{13}s_{4}\right)}$
$\displaystyle\times
e^{S_{22}s_{1}s_{3}+S_{23}s_{1}s_{4}+S_{32}s_{2}\allowbreak
s_{3}+S_{33}s_{2}s_{4}}$ $\displaystyle\times e^{a^{{\dagger}}\left(\alpha
S_{11}+S_{21}s_{1}+S_{31}s_{2}\right)}\left|0_{a}\right\rangle|_{s_{1}=s_{2}=s_{3}=s_{4}=0},$

Therefore the explicit form in Eq.(2) can be obtained after making derivation.

Appendix C: Density operator of the generated state

The conjugate state of $\left|\psi_{a}\right\rangle$ can be given by

$\displaystyle\left\langle\psi_{a}\right|$ $\displaystyle=$
$\displaystyle\frac{e^{-\left|\alpha\right|^{2}/2}}{\sqrt{p_{d}}}\frac{d^{4}}{dh_{4}dh_{3}dh_{2}dh_{1}}e^{\alpha^{\ast}\left(S_{12}h_{3}+S_{13}h_{4}\right)}$
$\displaystyle\times
e^{S_{22}h_{1}h_{3}+S_{23}h_{1}h_{4}+S_{32}h_{2}\allowbreak
h_{3}+S_{33}h_{2}h_{4}}$ $\displaystyle\times\left\langle
0_{a}\right|e^{a\left(\alpha^{\ast}S_{11}+S_{21}h_{1}+S_{31}h_{2}\right)}|_{h_{1}=h_{2}=h_{3}=h_{4}=0},$

Then, the density operator is

$\displaystyle\rho_{a}$ $\displaystyle=$
$\displaystyle\frac{e^{-\left|\alpha\right|^{2}}}{p_{d}}\frac{d^{8}}{dh_{4}dh_{3}dh_{2}dh_{1}ds_{4}ds_{3}ds_{2}ds_{1}}$
$\displaystyle\times
e^{\alpha\left(S_{12}s_{3}+S_{13}s_{4}\right)+\alpha^{\ast}\left(S_{12}h_{3}+S_{13}h_{4}\right)}$
$\displaystyle\times
e^{S_{22}s_{1}s_{3}+S_{23}s_{1}s_{4}+S_{32}s_{2}\allowbreak
s_{3}+S_{33}s_{2}s_{4}}$ $\displaystyle\times
e^{S_{22}h_{1}h_{3}+S_{23}h_{1}h_{4}+S_{32}h_{2}\allowbreak
h_{3}+S_{33}h_{2}h_{4}}$ $\displaystyle\times e^{a^{{\dagger}}\left(\alpha
S_{11}+S_{21}s_{1}+S_{31}s_{2}\right)}\left|0_{a}\right\rangle\left\langle
0_{a}\right|e^{a\left(\alpha^{\ast}S_{11}+S_{21}h_{1}+S_{31}h_{2}\right)}$
$\displaystyle|_{s_{1}=s_{2}=s_{3}=s_{4}=h_{1}=h_{2}=h_{3}=h_{4}=0}$

Appendix D: Success probability $p_{d}$ of detection

Due to $\mathrm{tr}\left(\rho_{a}\right)=1$, we have

$\displaystyle p_{d}$ $\displaystyle=$ $\displaystyle
e^{-\left(1-S_{11}^{2}\right)\left|\alpha\right|^{2}}\frac{d^{8}}{dh_{4}dh_{3}dh_{2}dh_{1}ds_{4}ds_{3}ds_{2}ds_{1}}$
$\displaystyle\times e^{\left(S_{11}S_{21}\allowbreak
h_{1}+S_{11}S_{31}h_{2}+S_{12}s_{3}+S_{13}s_{4}\right)\alpha}$
$\displaystyle\times
e^{\left(S_{11}S_{21}s_{1}+S_{11}S_{31}s_{2}+S_{12}h_{3}+S_{13}h_{4}\right)\alpha^{\ast}}$
$\displaystyle\times
e^{S_{22}\left(s_{1}s_{3}+h_{1}h_{3}\right)+S_{23}\left(s_{1}s_{4}+h_{1}h_{4}\right)+S_{21}^{2}s_{1}h_{1}+S_{31}^{2}s_{2}h_{2}}$
$\displaystyle\times e^{S_{32}\left(s_{2}\allowbreak s_{3}+h_{2}\allowbreak
h_{3}\right)+S_{33}\left(s_{2}s_{4}+h_{2}h_{4}\right)+S_{21}S_{31}\left(s_{2}h_{1}+\allowbreak
s_{1}h_{2}\right)}$
$\displaystyle|_{s_{1}=s_{2}=s_{3}=s_{4}=h_{1}=h_{2}=h_{3}=h_{4}=0}$

which lead to the result in Eq.(4) with

$g_{0}=\left(S_{23}S_{32}+S_{22}S_{33}\right)^{2},$ $\displaystyle g_{1}$
$\displaystyle=$
$\displaystyle(S_{13}S_{22}S_{31}+S_{12}S_{23}S_{31}+S_{13}S_{21}S_{32}+S_{12}S_{21}S_{33})$
$\displaystyle\times\left(S_{13}S_{22}S_{31}+S_{12}S_{23}S_{31}+S_{13}S_{21}S_{32}\right.$
$\displaystyle\left.+2S_{11}S_{23}S_{32}+S_{12}S_{21}S_{33}+2S_{11}S_{22}S_{33}\right),$
$\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle
S_{11}^{2}S_{13}^{2}\left(S_{22}S_{31}+S_{21}S_{32}\right)^{2}$
$\displaystyle+4S_{11}S_{12}^{2}S_{12}^{2}S_{13}S_{21}S_{31}\left(S_{23}S_{31}+S_{21}S_{33}\right)$
$\displaystyle+2S_{12}^{2}S_{13}^{2}S_{21}^{2}S_{31}^{2}+S_{11}^{2}S_{12}^{2}\left(S_{23}S_{31}+S_{21}S_{33}\right)^{2}$
$\displaystyle+4S_{11}S_{12}S_{13}S_{13}S_{21}S_{31}\left(S_{22}S_{31}+S_{21}S_{32}\right)$
$\displaystyle+2S_{11}^{2}S_{12}S_{13}S_{21}S_{32}\left(2S_{23}S_{31}+S_{21}S_{33}\right)$
$\displaystyle+2S_{11}^{2}S_{12}S_{13}S_{22}S_{31}\left(S_{23}S_{31}+2S_{21}S_{33}\right),$
$\displaystyle g_{3}$ $\displaystyle=$ $\displaystyle
2S_{11}^{3}S_{12}S_{13}^{2}S_{21}S_{31}\left(S_{22}S_{31}+S_{21}S_{32}\right)$
$\displaystyle+4S_{11}^{2}S_{12}^{2}S_{13}^{2}S_{21}^{2}S_{31}^{2}+2S_{11}^{3}S_{12}^{2}S_{13}^{2}S_{21}S_{31}S_{23}$
$\displaystyle+2S_{11}^{3}S_{12}^{2}S_{13}S_{21}^{2}S_{31}S_{33},$
$g_{4}=S_{11}^{4}S_{12}^{2}S_{13}^{2}S_{21}^{2}S_{31}^{2}.$

Appendix E: General expressions of the expected values $\left\langle
a^{{\dagger}k}a^{l}\right\rangle$

According to $\left\langle
a^{{\dagger}k}a^{l}\right\rangle=\mathrm{tr}\left(a^{{\dagger}k}a^{l}\rho_{a}\right)$
and making detailed calculation, we obtain

$\displaystyle\left\langle a^{{\dagger}k}a^{l}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{-\left(1-S_{11}^{2}\right)\left|\alpha\right|^{2}}}{p_{d}}\frac{d^{8+k+l}}{dh_{4}dh_{3}dh_{2}dh_{1}ds_{4}ds_{3}ds_{2}ds_{1}d\mu^{k}d\nu^{l}}$
$\displaystyle\times
e^{\left(S_{11}S_{21}h_{1}+S_{11}S_{31}h_{2}+S_{12}s_{3}+S_{13}s_{4}\right)\alpha}$
$\displaystyle\times e^{\left(S_{11}S_{21}s_{1}+\allowbreak
S_{11}S_{31}s_{2}+S_{12}h_{3}+S_{13}h_{4}\right)\alpha^{\ast}}$
$\displaystyle\times
e^{+S_{22}\left(s_{1}s_{3}+h_{1}h_{3}\right)+S_{23}\left(s_{1}s_{4}+h_{1}h_{4}\right)+S_{21}^{2}s_{1}h_{1}+S_{31}^{2}s_{2}h_{2}}$
$\displaystyle\times e^{+S_{32}\left(s_{2}\allowbreak s_{3}+h_{2}\allowbreak
h_{3}\right)+S_{33}\left(s_{2}s_{4}+h_{2}h_{4}\right)+S_{21}\allowbreak
S_{31}\left(s_{2}h_{1}+s_{1}h_{2}\right)}$ $\displaystyle\times
e^{S_{11}\left(\mu\alpha^{\ast}+\nu\alpha\right)+\mu\left(S_{21}h_{1}+S_{31}h_{2}\right)+\nu\left(S_{21}\allowbreak
s_{1}+S_{31}s_{2}\right)}$
$\displaystyle|_{s_{1}=s_{2}=s_{3}=s_{4}=h_{1}=h_{2}=h_{3}=h_{4}=\mu=\nu=0}.$

Using this general expression, we can study the statistical properties of our
generated states.

Appendix F: Wigner function of the generated state

According to the formula of the Wigner function in the coherent state
representation $\left|z\right\rangle$, i.e
$W(\beta)=\frac{2e^{2\left|\beta\right|^{2}}}{\pi}\int\frac{d^{2}z}{\pi}\left\langle-z\right|\rho\left|z\right\rangle
e^{-2\left(z\beta^{\ast}-z^{\ast}\beta\right)}$ with
$\beta=\left(q+ip\right)/\sqrt{2}$, the Wigner function is given by

$\displaystyle W(\beta)$ $\displaystyle=$
$\displaystyle\frac{2e^{-\left(1-S_{11}^{2}\right)\left|\alpha\right|^{2}-2\left|\beta-
S_{11}\alpha\right|^{2}}}{\pi
p_{d}}\frac{d^{8}}{dh_{4}dh_{3}dh_{2}dh_{1}ds_{4}ds_{3}ds_{2}ds_{1}}$
$\displaystyle\times
e^{\alpha\left(-h_{1}S_{11}S_{21}-h_{2}S_{11}S_{31}+S_{12}s_{3}+S_{13}s_{4}\right)}$
$\displaystyle\times e^{\alpha^{\ast}\left(-S_{11}S_{21}s_{1}\allowbreak-
S_{11}S_{31}s_{2}+S_{12}h_{3}+S_{13}h_{4}\right)}$ $\displaystyle\times
e^{+S_{22}\left(s_{1}s_{3}+h_{1}h_{3}\right)+S_{23}\left(s_{1}s_{4}+h_{1}h_{4}\right)-S_{21}^{2}s_{1}h_{1}-S_{31}^{2}s_{2}h_{2}}$
$\displaystyle\times e^{+S_{32}\left(s_{2}\allowbreak s_{3}+h_{2}\allowbreak
h_{3}\right)+S_{33}\left(s_{2}s_{4}+h_{2}h_{4}\right)-S_{21}S_{31}\left(s_{2}h_{1}+s_{1}h_{2}\right)}$
$\displaystyle\times
e^{2\beta\left(h_{1}S_{21}+h_{2}S_{31}\right)+2\beta^{\ast}\left(S_{21}s_{1}+S_{31}\allowbreak
s_{2}\right)}$
$\displaystyle|_{s_{1}=s_{2}=s_{3}=s_{4}=h_{1}=h_{2}=h_{3}=h_{4}=0}.$

After derivative, the analytical expression can be obtained.

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# Raw Radar data based Object Detection and Heading estimation using Cross
Attention ††thanks: 1The author is with the AVL GmbH, 93059 Regensburg,
Germany<EMAIL_ADDRESS>††thanks: 2The author is with Robert-Bosch GmbH,
31139 Hildesheim, Germany<EMAIL_ADDRESS>††thanks: 3The author
is with the e:fs TechHub GmbH, 85080 Gaimersheim, Germany christian5.mayr@efs-
auto.de ††thanks: 4The author is with Institute of Automatic Control, RWTH
Aachen University, 52074 Aachen, Germany<EMAIL_ADDRESS>
Ravi Kothari1, Ali Kariminezhad2, Christian Mayr3, Haoming Zhang4

###### Abstract

Radar is an inevitable part of the perception sensor set for autonomous
driving functions. It plays a gap-filling role to complement the shortcomings
of other sensors in diverse scenarios and weather conditions.

In this paper, we propose a Deep Neural Network (DNN) based end-to-end object
detection and heading estimation framework using raw radar data. To this end,
we approach the problem in both a ”Data-centric” and ”model-centric” manner.
We refine the publicly available CARRADA [1] dataset and introduce Bivariate
norm annotations. Besides, the baseline model is improved by a transformer [2]
inspired cross-attention fusion and further center-offset maps are added to
reduce localisation error. Our proposed model improves the detection mean
Average Precision (mAP) by 5%, while reducing the model complexity by almost
23%. For comprehensive scene understanding purposes, we extend our model for
heading estimation. The improved ground truth and proposed model is available
at Github.

###### Index Terms:

DNN, cross-attention, raw radar data, object detection, heading forecasting

##  I Introduction

As the driving assistant functions mature, challenging requirements are
expected to be fulfilled in the environment perception module. This means
robust perception is anticipated in extreme weather conditions, urban
environments, etc. This robustness requirement should be accompanied by a
boosted perception performance while keeping the latency sufficiently low. To
achieve these demanding goals, the integration of radar systems in the
perception sensor set is indispensable. Current Frequency-Modulated
Continuous-Wave (FMCW) automotive radar systems deliver detections in the form
of a sparse point cloud. These data points are filtered from the raw spectrum
by the known Constant False Alarm Rate (CFAR) [3] algorithm. This intensity-
based filtering process does not capture the complete object features and is
solely triggered by their Radar Cross Section (RCS) and surrounding noise
attributes. That means, the reflections from the obstacles with sufficiently
high RCS are likely to be detected as objects and the rest are considered as
noise. To capitalise on rich radar signature and extend its use for small
objects detection and classification, recently many researchers have explored
machine learning algorithms to directly process Fast Fourier Transform (FFT)
maps.

Inspired by the computer vision community, researchers from radar society have
also employed Convolutional Neural Network (CNN)-based models for detection,
object classification and path prediction [4, 5, 6, 7]. Nevertheless, the past
works have been limited to individual aspects of radar properties, such as
exploiting micro-Doppler for classification or utilizing only range-angle maps
for object detection. This boundation is due to the difficulty in fusing
distinct domains such as Doppler channel with range-angle. On top of that, a
competent radar-based perception module should also provide the object heading
for effective scene understanding. To exploit the full potential of radar, we
have incorporated the radar spectrum, i.e., raw data, in an end to end object
detection and heading forecasting model. In this work, we consider the neural
network proposed in RAMP-CNN [8] as our baseline for fusing all three radar
channels. This baseline is further improved by the proposed cross-attention
mechanism and center-offset integration. Due to limited angular resolution in
radar, regressing tight 2D/3D bounding boxes, especially for Vulnerable Road
User (VRU) is infeasible. Thus for our purpose, a detection is defined as
estimating an object center, similar to [9]. Furthermore, we improve the
ground-truth representation using the Bivariate norm for the open-source
dataset CARRADA [1]. To improve upon the limited position accuracy of the
baseline, we have incorporated a center-offset loss function. Moreover, for
better scene understanding, a heading estimation is inferred with the same
detector head. In summary, our main contributions are the following:

  1. 1.

Cross attention based Doppler feature fusion

  2. 2.

Novel ground truth representation for object detection

  3. 3.

center-offset loss, to reduce distance error

  4. 4.

Detector with heading estimation framework

The paper is organised as follows, in section II the latest results and the
state-of-the-art works are presented. Followed by section III, where we
discuss the chosen dataset and its subsequent processing. In section IV the
network architecture is explained and in section V the proposed model is
compared with the baseline. Finally, section VI is dedicated to conclusion and
future scope of DNN-based radar detection.

##  II Related Works

Previous works [10, 9, 11, 8, 12, 13, 14] on data-driven radar detection
demonstrates the utility of Machine Learning (ML) in bringing out the full
potential of radar sensor. These works generally focus on individual aspects
of radar signals and process them with specific Neural Network (NN) models.
The authors in [15] use Short Time Fourier Transformation (STFT) heat maps as
the input to the CNN to classify human activities. They have utilized the fact
that dynamic objects have unique Doppler signatures, which could be captured
by CNN. Furthermore, the authors in [10] use a different form of micro-Doppler
representation, Range-Doppler (RD) maps as the input. Their model was a
classifier with 3 classes (pedestrian, cyclist and car). These were initial
methods to demonstrate the importance of micro-Doppler signatures in object
classification.

The authors in [16] have shown a DNN-based radar detection using a two-stage
network, where a Region-of-Interest (ROI) is predicted and further processed
through a classifier. It’s an attempt at DNN-based semantic segmentation of a
3D point cloud.

Emulating the single-stage anchor-based object detectors (inspired by image
processing), the authors of [12] have applied You Only Look Once (YOLO) [17]
on RD maps. On the same note [14] have used a ResNet [18] based detector on
Range-Angle (RA) maps. Although these works report improved detection
capabilities, they are restricted in their approach by using individual radar
features as either Doppler or RA map.

The authors in [13] made the first attempt at rationally fusing different
feature encodings. Their proposed model uses an autoencoder, which is capable
of detecting vehicles in 2D Bounding Box (BB) s. This method fuses the three
two-dimensional RA, Angle-Doppler (AD) and RD encoded feature maps by imposing
redundancy in the missing dimension to bring all three feature maps in common
space.

Due to low angular resolution in automotive radar, the accurate 2D BB
annotation of the radar spectrum is challenging. To resolve the problem of
radar box annotation, the authors in [9] introduced a probabilistic map, which
uses the target peaks to create Gaussian labels. This approach leads to a
stable training process and circumvents the need for box annotation [19]. The
authors in [8] have further extended Gaussian maps output to utilize all three
radar channels. However, the adoption of an additional skip loss function for
better utilization of Doppler branch, along with encoder feature replication
similar to [13] indicates this network architecture’s limitations in
efficiently extracting Doppler features.

Heading estimation is critical in scene understanding, especially in urban
scenarios. In the past [6] have shown that orientation estimation of a car is
possible with radar, albeit its based on a point cloud dataset and evaluated
on car class. We believe an object’s doppler signature also contains its
heading information i.e, velocity vector and it could be estimated even for
small objects such as VRU. In this paper, we extend the detection network with
further heading prediction.

##  III Dataset

On one hand, radar points are not dense enough to manually annotate 3D BB. On
the other, reading radar FFT spectrum’s requires expertise. To overcome this
hurdle, dataset benchmarks such as nuScenes [20], Oxford robocar [21] and
Astyx [22], which are all based on 3D sparse point clouds use cross-annotation
with the help of camera and lidar sensors. The dearth of raw radar data could
be attributed to the absence of off the shelf sensors and the vast size of
data to be stored and hosted. Only in the last decade or so, with the advent
for System on Chip (SOC) and high bandwidth storage, it’s now possible for
researchers to record and share raw radar data reasonably.

CRUW [9] is one of the first datasets delivering annotated RA radar spectrum,
where the labels are derived from image-based object detection and supervision
of the accompanying camera. Improving on CRUW, the CARRADA dataset covers
annotating the complete radar raw data (range, angle and Doppler), where the
labels are provided as BB s, sparse points and dense masks. The objects in the
scenarios consist of multiple pedestrians, cyclists and cars interacting at
different velocities, angles and ranges (5m -50 m). On top of that the data
includes VRU located in a noisy environment (snow). These scenarios are
critical in highlighting the importance of radar where most of the camera and
lidar sensor fails. CARRADA has also been proposed for temporal predictions
[11], which conforms with our goal for object heading estimation.

CARRADA dataset labels are machine-generated (automatically labelled). It
applies object detection (Mask R-CNN [23]) on camera images and further
projects on Bird’s-eye view (BEV) plane to get the annotation for radar raw
data. Since the effectiveness of ML model is dependent on the quality of input
data, we cannot directly rely on automatic labels. We pre-process this data to
improve the annotation quality, create additional labels for heading
estimation and transform 2D BB to Gaussian maps.

###  III-A Dataset Pre-Processing

Auto-labelling with a 2D image detector struggles with small objects, e.g.,
pedestrians at long distances. Such kinds of Mis-annotated ground truth lead
to a sub-optimal network. In a data-centric manner, we first manually cross-
check all the 7000 annotated frames and add additional 2D BB for missed
instances. These annotations are further projected on the RA and RD planes. In
the end, our improved annotation had a 15% more VRU as compared to the
original dataset.

Since, we set our goal not only for object detection, but also heading
estimation, we had to supplement the CARRADA dataset with additional
annotation. The heading ground truth is computed by finding the object’s
position in multiple frames, these points are further joined via a spline and
the heading is computed as the differentiation of the curve at each instance.

###  III-B Bivariate Norm

Replacing the 2D BB s with a 2D Gaussian norms of unique variance has been
found to improve the performance of radar based detector to some extent [9].
However, applying a common Gaussian label for all the dimensions (range,
angle, Doppler) and all the classes is far from optimal. This is due to the
fact that, the radar signature changes with RCS and object location. Hence, an
ideal ground truth map should be flexible to reflect the real radar
signatures. This guarantees that the optimizer will punish the network as per
the signal strength. To achieve this, RODNet [9] tweaked the respective
Gaussian annotations using multiple hyper-parameters that control the variance
based on range, angle and object class. This method improves the ground truth,
but it comes with an extra overhead of finding these hyper-parameters.

To make an adaptive groundtruth, we propose the mapping of object-relevant raw
spectrum with Gaussian maps. These maps required the Gaussian second-order
moments. These moments are derived from the spectrum bounded with the proposed
BB. This process guarantees that the ground truth accurately reflects the
original signal without using any extra parameters. This can be interpreted as
a combination of semantic segmentation and Gaussian representation. Below are
the steps showing the Bivariate norm mapping process:

  1. 1.

Normalise the BB map by its max bin, i.e., in range 0-1.

  2. 2.

Noise separation by masking with a threshold.

  3. 3.

Calculate the covariance and center of the resultant masked bins.

  4. 4.

Project the Bivariate norm as the ground truth using the derived covariance
and mean values.

Given the range and angle first-order moments, $\mu_{r}$ and $\mu_{a}$ as
object location and the derived range-angle covariance matrix (second-order
moments) $\boldsymbol{\Sigma}$, a bivariate Gaussian map is formulated as:

$f(r,a)=\frac{1}{2\pi\sigma_{r}\sigma_{a}\sqrt{1-\rho^{2}}}\cdot\exp\Biggl{(}\Biggl{.}-\frac{1}{2(1-\rho^{2})}\biggl{[}{\Bigl{(}\frac{r-\mu_{r}}{\sigma_{r}}\Bigr{)}}^{2}\\\
-2\rho\Bigl{(}\frac{r-\mu_{r}}{\sigma_{r}}\Bigr{)}\Bigl{(}\frac{a-\mu_{a}}{\sigma_{a}}\Bigr{)}+{\Bigl{(}\frac{a-\mu_{a}}{\sigma_{a}}\Bigr{)}}^{2}\biggr{]}\Biggr{.}\Biggr{)}$
(1)

Figure 1: uniform Gaussian vs adaptive variance Gaussian (Bivariate Norm) of a
car (red) and pedestrian (blue)

Fig 1, shows the variance directly captured from raw data. The angle axis has
a high variance, indicating low angular resolution of the radar sensor.
Similarly, with small objects (Pedestrians) at large distances, the range
variance also increases. Hence, the proposed adaptive variance norm is a
closer reflection of ground truth as compared to a hard-coded Gaussian patch.

For the baseline network, we have considered correlation $\rho$ to be zero and
the range and angle variances are assumed to be equal.

##  IV Network Architecture

We aim to detect objects in RA coordinates, thus the network output will be of
a similar shape as the input RA map. Most of the past works in input-output
mapping have been inspired by medical imaging (semantic segmentation),
denoising networks or some form of BB detectors. Unlike image-based object
detectors, where the network needs to go deep for enlarging the receptive
fields. A radar signature remains fairly localised in RA plane, thus even a
network with small receptive field could perform well.

Convolution-deconvolution (CDC) type backbone belong to a family of
autoencoders and are aimed at providing the degrees of freedom for spatial
decoding of the encoded features in the same input dimension. The authors in
[24] utilised a CDC network for BB detection. Correspondingly, RODNet [9]
applied this approach with radar-based detection on the CRUW dataset, which
delivered a 78% average recall. Owing to CDC’s light architecture and its
adaptability to different task we have chosen CDC backbone for our model.

###  IV-A Fusion

The incompatibility of Doppler and range-angle have driven the past fusion
approaches to directly project Doppler intensities to RA maps [11] or use 2D
Doppler feature replication [13, 8] on a RA axis before passing it through
decoder. The former makes the network more reliant on RA maps and the latter
creates a redundant fusion features. Hence, these models struggle with small
targets, which inherits a consistent Doppler pattern but have low RCS. To
realise the full potential of Doppler signature, we introduce a transformer
[2] based cross attention module, it uses Doppler features to attend on
encoded RA maps.

Given a generalised feature map of size $F\in R^{C\times H\times W}$,
$F_{RA},F_{RD}$ and $F_{AD}$ are the encoded features for cross attention
block. The intermediate feature map $F^{\prime}$ is a matrix multiple of
$F_{RD}$ and transposed $F_{AD}$, as shown in eq. 2.

$F^{\prime}=F_{RD}\times F^{T}_{AD}$ (2)

Where $\times$ denotes matrix multiplication. $F^{\prime}$ is further
normalised with softmax and a Hadamard (element wise) product $\odot$ is
applied with ${F_{RA}}$ feature map.

$F^{\prime\prime}=softmax(F^{\prime})\odot F_{RA}$ (3)
$F_{CA}=\|F^{\prime\prime}+F_{RA}\|_{layernorm}$ (4)

The output of a cross attention block $F_{CA}$ consists of a skip connection
from the original $F_{RA}$ feature map which is added to $F^{\prime\prime}$.
The resultant is normalised over layers, similar to the original attention
block implementation in [2]. This skip connection helps network, with
stationary objects, where Doppler information is absent.

###  IV-B Metrics

The network output, i.e., Gaussian maps from the decoder are passed through a
peak detector. If the detected maximum lies on the center of the kernel and
has a confidence score greater than a pre-defined threshold, e.g., 0.1, it’s
considered a peak. Finally, these peak coordinates and intensity values are
passed through Distance-based non-max Suppression (DNMS) algorithm to remove
overlapping detections. This approach is similar to non-max Suppression (NMS)
from image-based detectors, here we switch the Intersection over Union (IOU)
values with distance among peak pairs. In this work, a prediction is
considered a true positive if it lies within a distance threshold of the
ground truth. We evaluate our models with (2m and 1m) distance threshold. As
compared to radar counter-parts such as more accurate Lidar 3D BB, our
association distance is limited due to low angular resolution and weak radar
signatures of VRU at large distances.

###  IV-C Center-offset

Owning to the fact that the network output is a rough estimate of Gaussian,
the max kernel was sometimes detecting peaks away from the Gaussian center,
this in turn lead to reduction in localization accuracy. To assist the network
in finding Gauss center, we introduced a center-offset mask. A mask of size
($256\times 256$) for range and angle axis is created individually, and the
normalised bin offset values are filled in a region of $9\times 9$ around the
Gauss center. The network is trained to regress this offset patch and during
inference, all the detected peak are first shifted using their respective
range-angle offset maps and then DNMS is applied.

Figure 2: Bivariate Cross Attention Network. Object detection purpose (T=1)
and for heading estimation ablation study (T=1/3/5)

###  IV-D Heading

Radar range-angle-Doppler (RAD) signatures are dependent on the objects’
position, orientation and velocity. Thus, theoretically based on these
distinct fingerprints, it’s possible to trace those patterns to determine
objects’ heading with single instance. In this paper, we adopt this
observations and propose estimating objects’ headings with a single RAD frame.
The heading angle ground truth is embedded in $\sin(\theta)$ and
$\cos(\theta)$ maps. Since both the angles are complementary, together they
can cover the complete range of ($-\pi,+\pi$). As the localisation ground
truth is in the form of a Gaussian, similar representation is not feasible in
heading annotation. Thus, we set the heading maps in a $64\times 64$ map,
considering that the network should detect the gauss center within an accuracy
of 4 bins in the RA map. The single-frame results are also further compared
with multi-frames for analysing the impact of past frames in heading
prediction.

###  IV-E Architecture

The proposed network consists of 6 encoder down-sampling layers and 4 layers
of transpose convolutions for the decoder. Each layer consists of PReLU [25]
activation and batch normalization [26]. The network is further boosted by the
cross attention based fusion. Furthermore, the decoder has multiple heads for
Bivariate norm, center-offset, and heading maps. Fig.2 illustrates the network
architecture with the corresponding feature dimensions. As our work is
targeting radar application in the automotive field, it’s crucial to have a
low computation and memory footprint. Thus, we develop and analyse the work
with a single radar frame. As an exception, only the ablation study of heading
estimation is conducted with multiple frames. The encoder’s first layer is
adapted to use stacked past frames (represented by T).

###  IV-F Loss Function

We train Bivariate norm mask with focal-loss [27] $L_{b}$, similarly center-
offset $L_{c}$ uses focal-loss, but with masking over the patch region.
Furthermore, the heading loss $L_{h}$ is determined to be Mean Squared Error
(MSE). Total loss incorporates weighted individual losses.

$L=w_{1}L_{b}+w_{2}L_{c}+w_{3}L_{h}$ (5)

##  V Evaluation

In this section, we present the evaluation results and its analysis.
Furthermore, the baseline is compared with the proposed network. After dataset
pre-processing, almost 9000 instances are delivered. These instances are
further split into train, validate and test sets (70%, 10%, 20%).

###  V-A Baseline

This work aims to showcase the importance of Doppler channels and introduce
methods of effectively fusing doppler features and study the impact of
adaptive ground truth representation via Bivariate norms. The single RA map
baseline model is chosen as RODNet-CDC [9], and for a Range-Angle-Doppler
(RAD) based method we have selected RAMP-CNN [8]. The term baseline model in
this work refers to only the use of model architecture i.e, backbone + header.
The original work consists of many intricacies such as the use of temporal
inception layer in RODNet [9] or data prepossessing and all-perspective loss
function in RAMP [8], which is not under our purview. Thus our results do not
reflect the competency of complete original methods but rather explore the
inefficiency in fusion architectures and ground truth representation. Due to
the absence of benchmark models in heading estimation with raw radar data, we
evaluate our proposed model with different number of input RAD frames.

###  V-B Implementation

PyTorch 1.9.0 framework is utilized for training and validation. All
experiments were run on a Nvidia Quadro RTX 8000 GPU. To separate the model
contributions from the pre-processing step, we first evaluate the baseline
network using cross attention module with plain Gaussian labels and
additionally center-offset and bivariate norms are introduced. The data
augmentation (Gaussian noise, horizontal flip) is same for all the models.
Batch size of 16 is used, with an initial learning rate of 1e-4, it’s further
reduced on plateau (factor=0.1, patience=4). Adam [28] is used as an
optimizer. 80 epochs were found to be sufficient where the evaluation losses
were achieving the minima.

###  V-C Detection Results

Network | Input | Ground truth | AP @ 2m and 1m | Para.
---|---|---|---|---
|  |  | Pedestrian | Cyclist | Car | Overall |
RODNet-CDC [9] | RA | Gauss | 0.88 | 0.80 | 0.84 | 0.80 | 0.93 | 0.87 | 0.88 | 0.82 | 4.6 M
RAMP-CNN [8] | RAD | Gauss | 0.89 | 0.82 | 0.86 | 0.79 | 0.93 | 0.88 | 0.89 | 0.83 | 14.23 M
Cross Atten. | RAD | Gauss | 0.92 | 0.81 | 0.85 | 0.82 | 0.96 | 0.86 | 0.91 | 0.83 | 11.05 M
Cross Atten. + offset loss | RAD | Gauss | 0.91 | 0.82 | 0.86 | 0.84 | 0.94 | 0.86 | 0.90 | 0.84 | 11.05 M
Bivar Cross Atten. + offset loss | RAD | Bivariate | 0.91 | 0.8 | 0.93 | 0.88 | 0.97 | 0.91 | 0.94 | 0.86 | 11.05 M
TABLE I: Model performance and complexity comparison using single radar frame

Cyclists and pedestrians are safety-critical classes hence, the term VRU, on
top of that their detection is challenging due to their low RCS. RODNet [9]
delivers its highest Average Precision (AP) (0.93) (corresponding to class
car). This manifests to network dependence on high radar reflection for
detection (vehicles poses high RCS due to the metallic material and big size).
The additional Doppler channel in RAMP-CNN [8] has improved the mAP by 1% over
RODNet [9], but this performance improvement comes with additional (3x) model
parameters.

To isolate the contribution of Bivariate norm, we first show the cross
attention network trained exclusively with plain Gaussian. Interestingly, our
proposed cross attention module boosts the AP2m for classes car and pedestrian
by 3% each. This shows the importance of the proposed cross attention in
extracting Doppler features. But the same results are not reflected in the 1m
distance threshold, this indicates the low accuracy detections in Gauss based
representation. Moreover, this module is realizable with a low complexity,
which is granted due to applying a single decoder. The proposed fusion network
results in a 23% reduction in the number of parameters compared to RAMP-CNN.
Introducing center-offset loss improves the AP in short distance threshold
(1m) but its performance reduces in 2m association. As the loss function is
sensitive to the size of offset-patch, thus it might be further improved with
an class dependent patch size. We understand the need for a bigger dataset and
new DNN models to improve pedestrian detection performance in 1m distance
threshold.

Combining the proposed ”data-centric” approach of relabeling with Bivariate
Gaussian and ”model-centric” approach by integrating cross attention module
and center-offset patch boosts the class ”Cyclist” AP2m by 7%. Compared to the
baseline, the proposed network improves mAP by 4%. The significance of center-
offset maps could be depicted with the drop of 20% in MSE distance error
(0.15m to 0.12m). The use of Doppler has also restricted the miss-
classification rate below 1%, proving that the network doesn’t exclusively
rely on RA maps signatures.

###  V-D Heading Estimation

We consider the proposed network is deep enough to effectively extract heading
features. This is achieved by adding an additional heading detector and
adapting the encoder for accepting multiple frames. We report the heading
accuracy in $\pm$45°, $\pm$22.5° and $\pm$11.25°regions. These angular ranges
are varied from coarse to fine, which could be further used in scene
understanding with VRUs. Based on CARRADA [11] benchmark, it is observed that
an average human stride could be captured in 5 radar frames. Hence, this is
taken as the maximum number of input frames for the ablation study.

No. of | Heading Accuracy
---|---
input Frames |  $\pm$45° |  $\pm$22.5° |  $\pm$11.25°
1 | 71% | 54% | 35%
3 | 85% | 67% | 42%
5 | 91% | 77% | 50%
TABLE II: Accuracy comparison of the network (Bivariate Cross Attention),
while considering multiple past frames for heading estimation

Table II gives the absolute angular error between ground truth and predicted
heading. As the number of frames increases the accuracy in fine region
drastically goes up, which suggests the network is able to effectively fuse
the maps in temporal space. Fig 3 shows the qualitative result of a 5 frame
network. This example highlights the importance of radar in detecting occluded
objects, where vision based sensors struggle. Furthermore, the network is
capable of heading estimation of VRUs at long ranges without tracking. Proving
our hypothesis, the single-frame network has also achieved an accuracy of 71%
for $\pm$45°. The low performance of the network in the fine heading region of
$\pm$11.25°, suggests the coarseness of features embedded in RAD frame.

Figure 3: Heading estimation and object detection using 5 past frames. The
Black arrow indicates the ground truth heading and the colored arrow is the
class specific predicted heading.

##  VI Conclusion

In this paper, we have proposed ”data-centric” and ”model-centric” novelties
for object detection and heading estimation from radar raw data. In this
regard, we contributed to both data pre-processing and optimized network
design. In the pre-processing phase, the objects are labelled via class-
independent covariance Gaussian masks deduced from the dataset. For optimizing
the network, an extra feature map known as center-offset feature map is
exploited, which guides the network in finding Gaussian center. The combined
model resulted in an overall mAP improvement of 4% compared to the baseline.
This performance improvement comes along 23% reduction in optimization
parameters, i.e., network weights. Moreover, we have generalized the proposed
network for object heading estimation, which eventually extracts the heading
information from Range-Angle-Doppler maps. For future investigation, we would
like to explore the performance gap in pedestrian detection and heading
estimation in fine region. The next generations radar sensors with higher
number of angle channels could be a way to resolve these issues.

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