# Localizing genuine multimode entanglement: Asymmetric gains via non-
Gaussianity

Ratul Banerjee1, Saptarshi Roy1, Tamoghna Das2, Aditi Sen(De)1 1Harish-Chandra
Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India
2International Centre for Theory of Quantum Technologies, University of
Gdańsk, 80-952 Gdańsk, Poland.

###### Abstract

Measurement-based quantum correlation mimics several characteristics of
multipartite quantum correlations and at the same time, it reduces the parent
system to a smaller subsystem. On the other hand, genuine multipartite
entanglement measures can capture certain features of a multisite composite
system that are inaccessible via bipartite quantum correlation quantifiers. We
merge these two concepts by introducing localizable genuine multimode
entanglement for continuous variable systems, both for Gaussian and non-
Gaussian multimode parent states. We report a compact form of localizable
generalized geometric measure for multimode Gaussian states when Gaussian
measurements are performed in some of the modes. We show that non-Gaussian
measurements can concentrate more genuine multimode entanglement compared to
the Gaussian ones. For non-Gaussian states with non-Gaussian measurements, we
find that although four-mode squeezed vacuum state has permutation symmetry
with respect to the exchange of first and third modes as well as the second
and the fourth modes, the symmetry can be broken by performing measurements in
one of the modes in case of addition while for subtraction, such symmetry is
preserved, thereby providing a method for distinguishing multimode photon-
added and -subtracted states via localizations.

##  I Introduction

The “weirdness” of quantum physics have puzzled scientists over the years
although these peculiarities, absent in classical resources, turn out to be
boons for designing quantum information processing tasks Einstein et al.
(1935); Bell (2004); Schrdinger (1935); Trimmer (1980); Landsman (2017);
Collins et al. (2002). A conceptual revolution has paved the way for the
discovery of remarkable protocols like quantum teleportation Bennett et al.
(1993); Bouwmeester et al. (1997); Pirandola et al. (2015), dense coding
Bennett and Wiesner (1992); Guo et al. (2019), quantum key distribution (QKD)
Ekert (1991); Bennett et al. (1992), quantum sensing Degen et al. (2017),
quantum computation Huang et al. (2020); Arute et al. (2019) and other quantum
technologies like quantum memories Brennen et al. (2015); Heshami et al.
(2016) and quantum batteries Binder et al. (2018) to name a few. Most of the
protocols, showing better efficiencies than their classical counterparts,
employ some form of quantum resources Chitambar and Gour (2019) which include
quantum coherence Streltsov et al. (2017), entanglement Horodecki et al.
(2009), quantum correlations, independent of entanglement Modi et al. (2012);
Bera et al. (2017). Therefore, classification of nonclassical resources has
utmost importance both from the perspectives of better understanding of
quantum mechanics and utilizing the quantum fuel to build the quantum
technologies.

We will be concentrating here on entanglement for which the resource theory is
well-established in the literature, especially for two spatially separated
systems. In this bipartite regime, the categorization takes a simpler form and
hence the detection as well as quantification are comparatively easier than
the system comprising more than two parties. However, multipartite
entanglement is shown to be useful for information processing tasks like in
quantum networks and in measurement-based quantum computation Epping et al.
(2017); McCutcheon et al. (2016); Raussendorf and Briegel (2001). A multiparty
pure state is said to be genuinely multipartite entangled if it is not product
across every bipartition. Based on the geometry of quantum states, the
generalized geometric measure (GGM) Sen (De) (see also Shimony (1995); Barnum
and Linden (2001); Wei and Goldbart (2003); Blasone et al. (2008)) attempts to
quantify the genuine multiparty entanglement content of a state by computing
the minimum distance of a given state from the set of non-genuinely
multipartite entangled states (cf. Das et al. (2016a); Buchholz et al.
(2016)). On the other hand, multipartite entanglement content can also be
effectively characterized if local measurements are performed on the subset of
all the parties, thereby localizing entanglement in the unmeasured parties,
referred to entanglement of assistance or localizable entanglement DiVincenzo
et al. (1998); Smolin et al. (2005); Popp et al. (2005); Gour and Spekkens
(2006); Sadhukhan et al. (2017); Amaro et al. (2018).

Given the resources, the next thing is to consider the platform on which the
quantum communication and computing tasks can be performed. Several effective
substrates have been identified that include cold atomic systems Mandel et al.
(2003); Bloch (2005); Treutlein et al. (2006); Cramer et al. (2013); Leibfried
et al. (2005); Monz et al. (2011); Barreiro et al. (2013), superconducting
qubits Barends et al. (2014); Arute et al. (2019), nuclear magnetic resonance
(NMR) molecules Negrevergne et al. (2006); Raimond et al. (2001); Leibfried et
al. (2003); Haffner et al. (2008); Duan and Monroe (2010); Singer et al.
(2010), photonic systems Ren et al. (2017); Erhard et al. (2019) and many
others. In this work, we investigate the classification of measurement-based
genuine multimode entanglement, GME, in continuous variable states of light
which can be realized in optical bosonic modes. In contrast to the issues
faced when discrete polarization states of light have been used for realizing
several recent groundbreaking communication protocols, continuous variable
(CV) systems Braunstein and van Loock (2005a); Braunstein and Pati (2003) can
overcome certain difficulties present in photonic qubits, like Bell-basis
distinguishability. In particular, CV states can be prepared with almost unit
probability by using the nonlinear interaction of a crystal with a laser
Braunstein and van Loock (2005a). All these advantages make CV systems a
prominent physical system for building quantum gadgets.

Our aim is to calibrate entanglement present in the multimode CV systems via
measurement process. Specifically, we perform both Gaussian as well as non-
Gaussian local measurements in one or many modes of a multimode Gaussian and
non-Gaussian parent state to localize the GME of the whole state into the
unmeasured modes consisting of multiple modes. The entanglement generated in
the unmeasured modes is quantified by generalized geometric measure Roy et al.
(2020), leading to a localizable genuine multimode entanglement (LGME). In the
CV sector, localization of bipartite entanglement for Gaussian states is
discussed Fiurášek and Mišta (2007); Lami et al. (2020). We systematize our
investigations by splitting our analysis into three parts. First, we restrict
ourselves to the Gaussian regime where we examine LGME of a Gaussian state,
namely a four-mode squeezed vacuum (FMSV) state Das et al. (2016b), by
employing optimal Gaussian measurements in one of the four modes of the FMSV.
Secondly, we replace the Gaussian measurements with the non-Gaussian ones that
involve photon counting schemes. Note that the chosen non-Gaussian measurement
scheme which can be easily performed experimentally, although there can be
other optimal ones. However, our analysis reveals that LGME obtained from non-
Gaussian measurements is always higher than the one obtained with the optimal
Gaussian measurements.

Finally, we take up the most general case where both the states and
measurements are non-Gaussian. A prominent and efficient method to create non-
Gaussian states is to add (subtract) photons in Gaussian states. Moreover, it
was shown that non-Gaussianity, both in the form of states and operations, can
be useful for several quantum information processing tasks like distillation,
error corrections, quantum sensing than Gaussian states and operations Lloyd
and Braunstein (1999); Chuang et al. (1997); Eisert et al. (2002); Giedke and
Ignacio Cirac (2002); Cerf et al. (2005); Niset et al. (2009); Huver et al.
(2008); Adesso et al. (2009); Zhang and van Loock (2011); Sabapathy et al.
(2011). By employing the same non-Gaussian measurements, we report that photon
subtracted FMSV can localize more GME than that of the photon-added state. In
this respect, we first notice that four-mode Gaussian states is symmetric with
the permutation of first and third as well as second and fourth modes. We then
show that if one performs measurement in the fourth mode, the addition of
photons in the second mode which is unmeasured leads to a higher multimode
entanglement than that of the fourth mode, thereby showing symmetry-breaking
nature induced by measurements. Interestingly, in case of subtraction,
measurement is unable to break the permutation symmetry present in the four-
mode non-Gaussian states. Therefore, LGME can act as a method to discriminate
multimode photon-subtracted states from the photon-added ones. Moreover, such
contrasting behavior can furnish a mechanism by which the modes used for
photon addition can be detected.

The paper is organized as follows. In Sec. II, we introduce localizable
genuine multimode entanglement or measurement-based multimode entanglement.
After setting the stage, in Sec. III, we derive a compact form of LGME for
Gaussian states with optimal Gaussian measurement and also show that non-
Gaussian measurements can perform better than the optimal Gaussian measurement
in the process of localization. In Sec. IV, we investigate LGME in the photon-
added and -subtracted states. We finally conclude in Sec. V.

##  II Localizable Genuine Multimode Entanglement

Let us introduce the localizable genuine multimode entanglement for an
arbitrary $N$-mode state. To localize entanglement, local Gaussian or non-
Gaussian measurements are performed on $L$ modes of an $N$-mode state. Since
we are interested in the residual multimode entanglement, we must have
$N-L=k\geq 3$. For a given multimode entanglement measure, $\mathcal{E}$, the
localizable or measurement-based genuine multimode entanglement can then be
defined as

$\displaystyle\mathcal{L}\mathcal{E}=\max_{\\_}{\\{\mathcal{M}\\}}\int
p_{\\_}k\mathcal{E}(|\psi\rangle_{\\_}k),$ (1)

where maximization is performed over the local measurements on $L$ modes, and
the averaging is performed over the measurement outcomes of the resulting
state of $k\geq 3$-modes.

In this paper, we take an arbitrary four-mode squeezed vacuum state as the
parent state, and by performing a measurement on a single mode, we obtain the
resulting three-mode state. Depending on the types of measurement and the
initial state, the resulting state is either Gaussian or non-Gaussian in
nature. After localization, multimode entanglement is quantified
geometrically, i.e. via generalized geometric measure Sen (De); Shimony
(1995); Barnum and Linden (2001); Wei and Goldbart (2003); Blasone et al.
(2008); Das et al. (2016a); Buchholz et al. (2016); Roy et al. (2020). For a
$N$-mode state, $|\psi_{\\_}N\rangle$, it is defined via Fubini study metric
Arnold (1989); Kobayashi (1996) as

$\displaystyle{\cal G}(|\psi_{\\_}{N}\rangle)=1-\max_{\\_}{|\chi\rangle\in
nG}|\langle\chi|\psi_{\\_}{N}\rangle|^{2}=1-$
$\displaystyle\max\big{\\{}\lambda_{\\_}{\cal A:B}|{\cal A}\cup{\cal
B}=\\{1,2,\ldots,N\\},{\cal A}\cap{\cal B}=\emptyset\big{\\}}.$ (2)

The maximization is taken over the set of $N$-mode pure states,
$\\{|\chi\rangle\\}$, which are not genuinely multimode entangled and the set
is denoted by $nG$. Interestingly, the optimization can be conquered by using
Schmidt decomposition for continuous variable systems Botero and Reznik
(2003); Herbut (2018) which can be easily extended for any normalizable
infinite dimensional state. For four-mode Gaussian states, the details will be
presented in the succeeding section.

##  III Gaussian genuine Localizable entanglement: Gaussian vs. Non-Gaussian
measurements

Before computing LGME for four-mode squeezed vacuum state, let us first
briefly describe the method to compute GGM for multimode state. Upto local
displacement operations, a Gaussian state is specified uniquely by its
covariance matrix Ren et al. (2017); Erhard et al. (2019). For an arbitrary
$m$-mode Gaussian state, $\rho$, the covariance matrix, $\Lambda$, is a
$2m\times 2m$ matrix, defined as
$\Lambda_{\\_}{ij}=\frac{1}{2}\big{\langle}\\{R_{\\_}i,R_{\\_}j\\}\big{\rangle}-\langle
R_{\\_}i\rangle\langle R_{\\_}j\rangle$, where
$\vec{R}=(q_{\\_}1,p_{\\_}1,q_{\\_}2,p_{\\_}2,...q_{\\_}m,p_{\\_}m)^{\text{T}}$.
Here $q_{\\_}i$s and $p_{\\_}i$s are the usual phase space quadrature
operators which can be given in terms of raising and lowering operators as
$q_{\\_}j=\frac{1}{\sqrt{2}}(a_{\\_}j+a_{\\_}j^{\dagger}),~{}p_{\\_}j=\frac{1}{\sqrt{2}i}(a_{\\_}j-a_{\\_}j^{\dagger})$,
where $i=\sqrt{-1}$. If $\rho$ has to be a valid density matrix, the
covariance matrix in terms of the symplectic matrix, $J$, has to satisfy
$\Lambda+iJ\geq 0,\text{ where
}J=\bigoplus_{\\_}{i=1}^{m}\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$. By using
Williamson’s theorem Braunstein and van Loock (2005b); Ferraro et al. (2005),
the covariance matrix $\Lambda$ can be obtained from $\Lambda^{d}$ by
appropriate symplectic transformation ($\textbf{S}_{\\_}\Lambda$),
$\Lambda=\textbf{S}_{\\_}\Lambda\Lambda^{d}\textbf{S}_{\\_}\Lambda^{\text{T}}$
with $\Lambda^{d}=\bigoplus_{\\_}{i=1}^{m}\nu_{\\_}i\mathbb{I}_{\\_}2$, where
$\\{\nu_{\\_}i\\}$s are the symplectic eigenvalues of $\Lambda$ and
$\mathbb{I}_{\\_}2$ is the $2\times 2$ identity matrix. The GGM
($\mathcal{G}$) of a $N$-mode pure Gaussian state $|\psi_{\\_}{N}\rangle$ in
terms of symplectic eigenvalues takes the compact form as Roy et al. (2020)

$\displaystyle\mathcal{G}(|\psi_{\\_}{N}\rangle)=1-\max\mathcal{P}_{\\_}m\Big{\\{}\prod_{\\_}{i=1}^{m}\frac{2}{1+2\nu_{\\_}i}\Big{\\}}_{\\_}{m=1}^{\big{[}\frac{N}{2}\big{]}}.$
(3)

Here maximization is performed by considering symplectic eigenvalues of all
the reduced states of $|\psi\rangle_{\\_}{N}$ with $m$-modes which are
obtained by applying the permutation operator, $\mathcal{P}_{\\_}m$ while
$[x]$ denotes the integral part of $x$.

###  III.1 Concentrating genuine entanglement of Gaussian states with Gaussian
measurement

To compute LGME with the Gaussian state, we consider the four-mode squeezed
vacuum state Das et al. (2016b) as the parent state on which Gaussian
measurements are performed. The FMSV state with squeezing strength $r$ can be
prepared in the laboratories by using linear optical elements like $50$:$50$
beam splitters and two single-mode squeezed vacuum states and can be written
as

$\displaystyle|FMSV\rangle$ (4) $\displaystyle=$ $\displaystyle
e^{\frac{r}{2}\sum_{\\_}{i=1}^{4}a_{\\_}i^{\dagger}a_{\\_}{i+1}^{\dagger}-a_{\\_}ia_{\\_}{i+1}}|0000\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\cosh
r}\sum_{\\_}{n=0}^{\infty}\left(\frac{1}{2}\tanh
r\right)^{n}\sum_{\\_}{r_{\\_}1=0}^{n}\sum_{\\_}{r_{\\_}2=0}^{n}\sqrt{\binom{n}{r_{\\_}1}}\sqrt{\binom{n}{r_{\\_}2}}$
$\displaystyle\hskip
93.95122pt|n-r_{\\_}1\rangle|n-r_{\\_}2\rangle|r_{\\_}1\rangle|r_{\\_}2\rangle,$

Here for $i=4$, $i+1$ is considered to be $1$. The GGM in this case reads as

$\displaystyle\mathcal{G}(|FMSV\rangle)=1-\max\Big{\\{}\frac{2}{1+\cosh^{2}r},$
$\displaystyle\frac{2}{1+\cosh 2r},\Big{(}\frac{2}{1+\cosh
r}\Big{)}^{2}\Big{\\}}.$ (5)

It is important to note here that the FMSV state has an inherent permutation
symmetry – it remains invariant under the exchange of first and third mode as
well as the second and fourth mode.

Figure 1: Gaussian vs. non-Gaussian measurements. Taking four-mode squeezed
vacuum state, given in Eq. (4), as the parent state, genuine multimode
entanglement (ordinate) is computed with respect to the squeezing parameter,
$\lambda=\tanh r$ (abscissa). The measurement is performed in the fourth mode.
The solid line represents the LGME of the FMSV state after optimal Gaussian
measurements while the dashed line is for the same with non-Gaussian
measurements. Clearly, photon counting non-Gaussian measurements are
advantageous to localize multimode entanglement compared to the optimal
Gaussian ones. Both the axes are dimensionless.

Without loss of generality, let us perform a projective Gaussian measurement
in the fourth mode of the FMSV state. The most general covariance matrix of a
single-mode Gaussian state with which the measurement is performed is the
squeezed coherent state whose covariance matrix can be represented as

$\displaystyle\sigma_{\\_}m=$
$\displaystyle\begin{bmatrix}\cosh{2r^{\prime}}+\cos{\phi}\sinh{2r^{\prime}}-&\sin{\phi}\sinh{2r^{\prime}}\\\
\sin{\phi}\sinh{2r^{\prime}}&\cosh{2r^{\prime}}-\cos{\phi}\sinh{2r^{\prime}}\end{bmatrix},$

with $r^{\prime}$ being the squeezing parameter, and $\phi$ being the angle of
the quadrature. It turns out that the reduced covariance matrices for the
various measurement outcomes are identical while only the displacement vectors
are outcome-dependent. Since the GGM content depends only on the covariance
matrix, we do not require to perform averaging over the different measurement
outcomes involved in the definition (Eq. (1)). The post-measured three-mode
covariance matrix, $(\Lambda_{\\_}m)$, can be computed as

$\Lambda_{\\_}{m}=\sigma_{\\_}{123}-\sigma_{\\_}{c}.((\sigma_{\\_}{m}+\sigma_{\\_}{4})^{-1}).\sigma_{\\_}{c}^{T},$
(7)

where $\sigma_{\\_}{123}$ and $\sigma_{\\_}4$ denote the first three modes and
the fourth mode respectively of the covariance matrix of $|FMSV\rangle$,
$\Lambda^{\text{FMSV}}$, which is given by

$\displaystyle\frac{1}{2}\begin{bmatrix}\cosh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\sinh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z\\\ \frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\cosh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\sinh^{2}r~{}\mathbb{I}_{\\_}2\\\
\sinh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\cosh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z\\\ \frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\sinh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\cosh^{2}r~{}\mathbb{I}_{\\_}2\end{bmatrix},$

with $\mathbb{I}_{\\_}2$ and $\sigma_{\\_}z$ being the identity and Pauli
matrix in the z-direction respectively. Here, $\sigma_{\\_}c$ represents the
correlation matrix between the first three and the fourth mode of FMSV. In
particular, they are given by

$\displaystyle\sigma_{\\_}{123}=\frac{1}{2}\begin{bmatrix}\cosh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\sinh^{2}r~{}\mathbb{I}_{\\_}2\\\ \frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\cosh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z\\\ \sinh^{2}r~{}\mathbb{I}_{\\_}2&\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z&\cosh^{2}r~{}\mathbb{I}_{\\_}2\end{bmatrix},$
$\displaystyle\sigma_{\\_}4=\frac{1}{2}\cosh^{2}r~{}\mathbb{I}_{\\_}2,~{}\sigma_{\\_}c=\frac{1}{2}\begin{bmatrix}\frac{1}{2}\sinh
2r~{}\sigma_{\\_}z\\\ \sinh^{2}r~{}\mathbb{I}_{\\_}2\\\ \frac{1}{2}\sinh
2r~{}\sigma_{\\_}z\end{bmatrix}.$

Note that $\Lambda_{\\_}m\equiv\Lambda_{\\_}m(r,r^{\prime},\phi)$, and the
corresponding localizable GME in terms of GGM can be expressed as

$\displaystyle\mathcal{L}\mathcal{G}(\Lambda^{\text{FMSV}}(r))=\max_{\\_}{r^{\prime},\phi}\mathcal{G}(\Lambda_{\\_}m(r,r^{\prime},\phi)).$
(8)

Our analysis reveals that the optimal measurement configuration turns out to
be homodyne detection of either $x$ or $p$ quadrature. Hence, for our
analysis, without loss of generality, we consider the $x$-quadrature.
Therefore, the maximal LGME is obtained for $r^{\prime}\rightarrow\infty$ and
$\phi=0$. In this optimal configuration, $\Lambda_{\\_}m^{opt}$ takes a simple
form and its three single mode reduced covariance matrices,
$R_{\\_}1,R_{\\_}2$, and $R_{\\_}3$, are given by

$\displaystyle R_{\\_}1=R_{\\_}3=\frac{1}{2}\begin{bmatrix}\cosh^{2}r&0\\\
0&1\end{bmatrix},$ (9)

and

$\displaystyle R_{\\_}2=\frac{1}{2}\begin{bmatrix}\cosh^{2}r&0\\\
0&1+\tanh^{2}r\end{bmatrix}.$ (10)

The symplectic eigenvalues of $R_{\\_}{1(3)}$ and $R_{\\_}2$ are computed to
be $\nu_{\\_}1=\frac{1}{2}\cosh r$ and $\nu_{\\_}2=\frac{1}{2}\sqrt{\cosh 2r}$
which leads to the LGME of the FMSV state using Gaussian measurements as

$\displaystyle\mathcal{L}\mathcal{G}(\Lambda^{\text{FMSV}})$ $\displaystyle=$
$\displaystyle\mathcal{G}(\Lambda_{\\_}m^{opt})$ $\displaystyle=$
$\displaystyle 1-\max\Big{\\{}\frac{2}{1+\cosh r},\frac{2}{1+\sqrt{\cosh
2r}}\Big{\\}}.$

However, we find that for all $r>0$, $\cosh r<\sqrt{\cosh 2r}$, and,
therefore, $\frac{2}{1+\cosh r}$ is always larger, thereby providing a compact
form of LGME for FMSV state with optimal local Gaussian measurement as

$\displaystyle\mathcal{L}\mathcal{G}(\Lambda^{\text{FMSV}})=\tanh^{2}\frac{r}{2}.$
(11)

Clearly, it increases monotonically with the variation of the squeezing
parameter, $r$ as shown in Fig. 1.

###  III.2 Non-Gaussian measurement enhances localization

Let us now see whether more entanglement can be localized from a Gaussian
state, when instead of optimal Gaussian measurements, one employs some non-
Gaussian measurement schemes. As non-Gaussian measurements, the photon
counting operations are performed in any one of the modes of the Gaussian
state and can be translated to a measurement in the Fock basis
$\\{|k\rangle\\}$, where $|k\rangle$ denotes the $k$-photon state. When the
fourth mode of $|FMSV\rangle$ is measured in the Fock basis and $|k\rangle$
clicks, the resulting normalized three-mode post-measured state is denoted by
$|\psi_{\\_}k\rangle$, where we have $\langle
k|FMSV\rangle=\sqrt{p_{\\_}k}|\psi_{\\_}k\rangle$ with $p_{\\_}k$ being the
clicking probability.

The LGME content of $|FMSV\rangle$ under the photon counting (non-Gaussian)
measurement procedure reads as

$\displaystyle\mathcal{L}\mathcal{G}(|FMSV\rangle)=\sum_{\\_}kp_{\\_}k\mathcal{G}(|\psi_{\\_}k\rangle).$
(12)

In Fig. 1, we observe that by using the photon counting non-Gaussian
measurement, higher amount of GME can be localized than the obtained via the
optimal Gaussian measurement. Notice for three-mode Gaussian state, such an
advantageous scenario with non-Gaussian measurement was also reported Fiurášek
and Mišta (2007).

Figure 2:  Localizable genuine multimode entanglement (vertical axis) against
the number of photons added and subtracted (horizontal axis), $m$, from a
single mode of the FMSV state. The solid line is for adding photons from a
single mode while the dashed line represents subtraction of photons from the
same. In legend, the subscript, $i=1,2,4$s in $m_{\\_}i$ indicate the mode in
which photons are added (subtracted). First of all, localizable entanglement
gets enhanced due to the addition (subtraction) of photons. We observe that
the LGME for adding or subtracting photons in (from) mode $1$ or $3$ as well
as subtracting photons in mode $2$ or $4$ give rise to the same value while
adding photons in the second and fourth mode leads to a different entanglement
content. Such an observation is interesting since the parent state remains
same with the interchange of mode $1$ and $3$ as well as mode $2$ and $4$.
Both the axes are dimensionless.

##  IV  more localization of Genuine multimode entanglement for photon-
subtracted states

In this section, we investigate the role of non-Gaussianity, both at the level
of states and measurements for concentrating entanglement into smaller
subsystems. Our analysis in the previous section already hinted at the
enhancement properties of LGME with non-Gaussian measurements. Here we examine
the most general case of localization of GME for non-Gaussian states by
employing non-Gaussian measurements. We have already discussed the non-
Gaussian measurement strategy in the previous section. Now we discuss the
procedure for deGaussification of the FMSV state.

Among the various methods by which a state can be deGaussified, we choose to
employ photon addition and subtraction operations. Such a choice is based on
two important features – these photonic operations can be implemented
experimentally, and secondly they have been shown to have enhancing effects in
varied scenarios ranging from bipartite entanglement, violation of Bell
inequality, to genuine multimode entanglement in the continuous variable
sector, thereby showing its possible applications in quantum information
processing tasks Navarrete-Benlloch et al. (2012); Yang (2016); Duc and Dat
(2020); Mojaveri et al. (2019); Roy et al. (2018); Walschaers et al. (2019);
Ra et al. (2019); Roy et al. (2020); Das et al. (2016b). LGME, in this case,
will be denoted as $\mathcal{LG}^{add}$ and $\mathcal{LG}^{subtract}$ to
represent photon-addition and -subtraction respectively.

The photon-added and -subtracted FMSV states with $m_{\\_}i$ number of photons
are added or subtracted in or from the mode $i$ $(i=1,2,3,4)$ respectively
read as

$\displaystyle|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle=\frac{1}{\sqrt{N^{add}}}\sum_{\\_}{n=0}^{\infty}\sum_{\\_}{r_{\\_}1=0}^{n}\sum_{\\_}{r_{\\_}2=0}^{n}\left(\frac{1}{2}\tanh
r\right)^{n}\sqrt{\binom{n}{r_{\\_}1}}\sqrt{\binom{n}{r_{\\_}2}}\sqrt{\frac{(n-r_{\\_}1+m_{\\_}1)!}{(n-r_{\\_}1)!}}\sqrt{\frac{(n-r_{\\_}2+m_{\\_}2)!}{(n-r_{\\_}2)!}}$
$\displaystyle\sqrt{\frac{(r_{\\_}1+m_{\\_}3)!}{r_{\\_}1!}}\sqrt{\frac{(r_{\\_}2+m_{\\_}4)!}{r_{\\_}2!}}|n-r_{\\_}1+m_{\\_}1\rangle_{\\_}1|n-r_{\\_}2+m_{\\_}2\rangle_{\\_}2|r_{\\_}1+m_{\\_}3\rangle_{\\_}3|r_{\\_}2+m_{\\_}4\rangle_{\\_}4,$
(13)
$\displaystyle|\psi^{sub\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle=\frac{1}{\sqrt{N^{sub}}}\sum_{\\_}{n=M}^{\infty}\sum_{\\_}{r_{\\_}1=m_{\\_}3}^{n-m_{\\_}1}\sum_{\\_}{r_{\\_}2=m_{\\_}4}^{n-m_{\\_}2}\left(\frac{1}{2}\tanh
r\right)^{n}\sqrt{\binom{n}{r_{\\_}1}}\sqrt{\binom{n}{r_{\\_}2}}\sqrt{\frac{(n-r_{\\_}1)!}{(n-r_{\\_}1-m_{\\_}1)!}}\sqrt{\frac{(n-r_{\\_}2)!}{(n-r_{\\_}2-m_{\\_}2)!}}$
$\displaystyle\sqrt{\frac{r_{\\_}1!}{(r_{\\_}1-m_{\\_}3)!}}\sqrt{\frac{r_{\\_}2!}{(r_{\\_}2-m_{\\_}4)!}}|n-r_{\\_}1-m_{\\_}1\rangle_{\\_}1|n-r_{\\_}2-m_{\\_}2\rangle_{\\_}2|r_{\\_}1-m_{\\_}3\rangle_{\\_}3|r_{\\_}2-m_{\\_}4\rangle_{\\_}4,$
(14)

where $M=\max[m_{\\_}1+m_{\\_}3,m_{\\_}2+m_{\\_}4]$ and, $N^{add}$ and
$N^{sub}$ are the respective normalization constants and can be expressed as
follows

$\displaystyle N^{add}=\sum_{\\_}{n=0}^{\infty}\left(\frac{1}{2}\tanh
r\right)^{2n}\sum_{\\_}{r_{\\_}1,r_{\\_}2=0}^{n}{\binom{n}{r_{\\_}1}}{\binom{n}{r_{\\_}2}}$
$\displaystyle\frac{(n-r_{\\_}1+m_{\\_}1)!}{(n-r_{\\_}1)!}\frac{(n-r_{\\_}2+m_{\\_}2)!}{(n-r_{\\_}2)!}\frac{(r_{\\_}1+m_{\\_}3)!}{r_{\\_}1!}\frac{(r_{\\_}2+m_{\\_}4)!}{r_{\\_}2!},~{}~{}~{}$
(15) $\displaystyle N^{sub}=\sum_{\\_}{n=M}^{\infty}\left(\frac{1}{2}\tanh
r\right)^{2n}\sum_{\\_}{r_{\\_}1=m_{\\_}3}^{n-m_{\\_}1}{\binom{n}{r_{\\_}1}}\sum_{\\_}{r_{\\_}2=m_{\\_}4}^{n-m_{\\_}2}{\binom{n}{r_{\\_}2}}$
$\displaystyle\frac{(n-r_{\\_}1)!}{(n-r_{\\_}1-m_{\\_}1)!}\frac{(n-r_{\\_}2)!}{(n-r_{\\_}2-m_{\\_}2)!}\frac{r_{\\_}1!}{(r_{\\_}1-m_{\\_}3)!}\frac{r_{\\_}2!}{(r_{\\_}2-m_{\\_}4)!}.~{}~{}~{}$
(16)

Let us first concentrate on localizing GME in
$|\psi^{sub\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle$ via photon counting
measurements in the fourth mode. When the $k$-photon state clicks in the
fourth mode, the normalized post-measurement state is computed as

$\displaystyle|\psi^{sub\\{m_{\\_}i\\}}_{\\_}{FMSV}(k)\rangle=\frac{1}{\sqrt{p_{\\_}k^{sub}}}{}_{\\_}4\langle
k|\psi^{sub\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle,$
$\displaystyle=\sum_{\\_}{n=n_{\\_}{min}}^{\infty}\sum_{\\_}{r_{\\_}1=0}^{n+M-m_{\\_}1-m_{\\_}3}f^{sub}_{\\_}k(n,r_{\\_}1,\\{m_{\\_}i\\})$
$\displaystyle|n+M-m_{\\_}1-m_{\\_}3-r_{\\_}1\rangle_{\\_}1|n+M-k-
m_{\\_}4-m_{\\_}2\rangle_{\\_}2|r_{\\_}1\rangle_{\\_}3~{}~{}~{}~{}~{}~{}$ (17)

where $n_{\\_}{min}=\max\,[0,k+m_{\\_}2+m_{\\_}4-M]$, and

$\displaystyle f^{sub}_{\\_}k(n,r_{\\_}1,\\{m_{\\_}i\\})$
$\displaystyle=\frac{1}{\sqrt{p_{\\_}k^{sub}N^{sub}}}\left(\frac{1}{2}\tanh
r\right)^{n}(n+M)!\frac{1}{\sqrt{r_{\\_}1!k!}}\times$
$\displaystyle\frac{1}{\sqrt{(n+M-r_{\\_}1-m_{\\_}1-m_{\\_}3)!}}\frac{1}{\sqrt{(n+M-k-
m_{\\_}2-m_{\\_}4)!}},$ (18)

and the subscript in the first line of Eq. (17) indicates that the measurement
is on the fourth mode. It occurs with probability,

$\displaystyle p_{\\_}k^{sub}=|{}_{\\_}4\langle
k|\psi^{sub\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle|^{2}$
$\displaystyle=\frac{1}{N^{sub}}\sum_{\\_}{n=n_{\\_}{min}}^{\infty}\left(\frac{1}{2}\tanh
r\right)^{2n}\sum_{\\_}{r_{\\_}1=0}^{n+M-m_{\\_}1-m_{\\_}3}\frac{1}{r_{\\_}1!}\frac{1}{k!}$
$\displaystyle\frac{(n+M)!}{(n+M-r_{\\_}1-m_{\\_}1-m_{\\_}3)!}\frac{(n+M)!}{(n+M-k-
m_{\\_}2-m_{\\_}4)!}.~{}~{}~{}~{}~{}~{}$ (19)

The LGME content of $|\psi^{sub\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle$ can then be
obtained by using Eq. (12) and the form of the photon-subtracted state leads
to the following proposition.

Figure 3:  Localizable genuine multimode entanglement (ordinate) vs. the
number of photons added to and subtracted from two different modes of the FMSV
state (abscissa), where the total number of photons added or subtracted are
fixed to $20$. Here $m_{\\_}i+m_{\\_}j=20$, where $i,j=1,2,3,4$. . The label
of the horizontal axis is for $m_{\\_}i$ where $i<j$. The dashed lines
represent photon-addition while the solid line is for subtracting photons from
the same. The legend, add${}_{\\_}{ij}$ or sub${}_{\\_}{ij}$ represent
addition of photons in $(i,j)$-pair of modes or subtraction of photons from
the same. When photons are subtracted from first and third modes or second and
fourth modes, LGME remains constant (see text for the proof). Both the axes
are dimensionless.

Proposition 1. The localizable GME of the photon-subtracted state is a
function of $m_{\\_}1+m_{\\_}3$ and $m_{\\_}2+m_{\\_}4$.

###### Proof.

First we note that when the $k$th outcome appears after the photon-counting
measurement of the fourth mode, the Schmidt coefficients of each resulting
three-mode state are functions of $m_{\\_}1+m_{\\_}3$ and $m_{\\_}2+m_{\\_}4$,
as seen in Eqs. (17), (18), and (19). And the GME in Eqs. (II) and (12) is a
function of Schmidt coefficients. Hence the proof. ∎

The two immediate corollaries of the above proposition are following:

Corollary 1. The localizable GME of the photon-subtracted state is invariant
under the interchange of $m_{\\_}1\leftrightarrow m_{\\_}3$ and
$m_{\\_}2\leftrightarrow m_{\\_}4$.

Corollary 2. The localizable GME of the photon-subtracted state remains
constant when photons are subtracted from the first and the third modes in a
constraint manner, i.e., when $m_{\\_}1+m_{\\_}3=n$. Similarly for
$m_{\\_}2+m_{\\_}4=n^{\prime}$.

The post-measurement state of $|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle$,
when $k$th photon is detected in the fourth mode, is given by

$\displaystyle|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}(k)\rangle=\frac{1}{\sqrt{p_{\\_}k^{add}}}{}_{\\_}4\langle
k|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle,$
$\displaystyle=\sum_{\\_}{n=k-m_{\\_}4}^{\infty}\sum_{\\_}{r_{\\_}1=0}^{n}f^{add}_{\\_}k(n,r_{\\_}1,\\{m_{\\_}i\\})$
$\displaystyle\hskip
36.135pt|n+m_{\\_}1-r_{\\_}1\rangle_{\\_}1|n-k+m_{\\_}2+m_{\\_}4\rangle_{\\_}2|r_{\\_}1+m_{\\_}3\rangle_{\\_}3,~{}~{}~{}~{}~{}~{}$
(20)

where

$\displaystyle f^{add}_{\\_}k(n,r_{\\_}1,\\{m_{\\_}i\\})$
$\displaystyle=\frac{1}{\sqrt{p_{\\_}k^{add}N^{add}}}\left(\frac{1}{2}\tanh
r\right)^{n}\sqrt{\binom{n}{r_{\\_}1}}\sqrt{\binom{n}{k}}\sqrt{\frac{(n+m_{\\_}1-r_{\\_}1)!}{(n-r_{\\_}1)!}}$
$\displaystyle\sqrt{\frac{(n-k+m_{\\_}2+m_{\\_}4)!}{(n-k+m_{\\_}4)!}}\sqrt{\frac{(r_{\\_}1+m_{\\_}3)!}{r_{\\_}1!}}\sqrt{\frac{k!}{(k-m_{\\_}4)!}},~{}~{}~{}$
(21)

and the probability of getting $k$th photon in the fourth mode is given by

$\displaystyle p_{\\_}k^{add}=|{}_{\\_}4\langle
k|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle|^{2}$
$\displaystyle=\frac{1}{N^{add}}\sum_{\\_}{n=k-m_{\\_}4}^{\infty}\left(\frac{1}{2}\tanh
r\right)^{2n}\sum_{\\_}{r_{\\_}1=0}^{n}{\binom{n}{r_{\\_}1}}{\binom{n}{k}}\frac{(n+m_{\\_}1-r_{\\_}1)!}{(n-r_{\\_}1)!}$
$\displaystyle\frac{(n-k+m_{\\_}2+m_{\\_}4)!}{(n-k+m_{\\_}4)!}\frac{(r_{\\_}1+m_{\\_}3)!}{r_{\\_}1!}\frac{k!}{(k-m_{\\_}4)!}.$
(22)

The form of $|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}\rangle$ leads to the
computation of $\mathcal{LG}^{add}$ which also possess the following symmetry:
Proposition 2. The localizable GME of the photon-added state is invariant
under the interchange of the number of photons added in mode one and three,
i.e., $m_{\\_}1\leftrightarrow m_{\\_}3$.

###### Proof.

When $m_{\\_}i$ number of photons are added in mode $i$, the corresponding
state given in Eq. (20) can be expressed as

$\displaystyle|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}(k)\rangle$
$\displaystyle=\sum_{\\_}{n=k-m_{\\_}4}^{\infty}\sum_{\\_}{r_{\\_}1=0}^{n}f^{add}_{\\_}k(n,n-r_{\\_}1,\\{m_{\\_}i\\})$
$\displaystyle\hskip
36.135pt|r_{\\_}1+m_{\\_}1\rangle_{\\_}1|n-k+m_{\\_}2+m_{\\_}4\rangle_{\\_}2|n-r_{\\_}1+m_{\\_}3\rangle_{\\_}3,~{}~{}~{}~{}~{}~{}$
(23)

where we replace $r_{\\_}1\rightarrow n-r_{\\_}1$, and
$f^{add}_{\\_}k(n,n-r_{\\_}1,\\{m_{\\_}i\\})$ is given in Eq. (21). Now if
instead of adding $m_{\\_}1$ number of photons in the first mode, we add
$m_{\\_}3$ in mode one and $m_{\\_}1$ in mode three, i.e., interchanging
$m_{\\_}1\leftrightarrow m_{\\_}3$, while $m_{\\_}2$ and $m_{\\_}4$ remain
same, we have

$\displaystyle|\psi^{add\\{m_{\\_}i\\}}_{\\_}{FMSV}(k)\rangle_{\\_}{m_{\\_}1\leftrightarrow
m_{\\_}3}$
$\displaystyle=\sum_{\\_}{n=k-m_{\\_}4}^{\infty}\sum_{\\_}{r_{\\_}1=0}^{n}f^{add}_{\\_}k(n,n-r_{\\_}1,\\{m_{\\_}i\\})_{\\_}{m_{\\_}1\leftrightarrow
m_{\\_}3}$ $\displaystyle\hskip
14.45377pt|r_{\\_}1+m_{\\_}3\rangle_{\\_}1|n-k+m_{\\_}2+m_{\\_}4\rangle_{\\_}2|n-r_{\\_}1+m_{\\_}1\rangle_{\\_}3,$
$\displaystyle=\sum_{\\_}{n=k-m_{\\_}4}^{\infty}\sum_{\\_}{r_{\\_}1=0}^{n}f^{add}_{\\_}k(n,r_{\\_}1,\\{m_{\\_}i\\})$
$\displaystyle\hskip
14.45377pt|r_{\\_}1+m_{\\_}3\rangle_{\\_}1|n-k+m_{\\_}2+m_{\\_}4\rangle_{\\_}2|n-r_{\\_}1+m_{\\_}1\rangle_{\\_}3.~{}~{}~{}~{}~{}$
(24)

Hence, one can easily see that Eq. (20) is same as Eq. (24) with mode one and
three being interchanged, which shows that the LGME is invariant under the
interchange of $m_{\\_}1\leftrightarrow m_{\\_}3$ in case of photon-addition.
∎

_Single-mode operations._ The observations from the single-mode photon
addition or subtraction can be listed as follows (see Fig. 2):

  1. 1.

As obtained in case of other multimode entanglement measures Navarrete-
Benlloch et al. (2012); Yang (2016); Duc and Dat (2020); Mojaveri et al.
(2019); Roy et al. (2018), we also observe here that the amount of
entanglement localized in modes gets increased due to the addition
(subtraction) of photons in the multimode squeezed vacuum states, irrespective
of choice of the modes on which measurements are performed.

  2. 2.

Unlike two-mode entanglement Navarrete-Benlloch et al. (2012), LGME is higher
in the photon-subtracted state than that of the photon-added state (cf. Roy et
al. (2020); Das et al. (2016b)). Note that the enhancement features of LGME on
addition and subtraction of photons can be explained by the non-Gaussianity
Genoni et al. (2008); Genoni and Paris (2010); Marian and Marian (2013)
induced in the FMSV state by the photonic operations. However, non-Gaussianity
cannot conclusively answer the question that which type of deGaussification
(photon addition or subtraction) induces greater enhancement of LGME.

  3. 3.

As mentioned before, the four-mode squeezed vacuum state has a symmetry in the
exchange of modes, i.e. $1\leftrightarrow 3$ and $2\leftrightarrow 4$ and
hence its entanglement properties remain same if one adds (subtracts) photons
in mode $1$ or in mode $3$. Similarly for mode $2$ or $4$. We show here that
the introduction of measurement breaks the symmetry – we get more entanglement
when photon is added in the unmeasured modes compared to the one when photons
are added to the measured one. Specifically, when the measurement is performed
in mode $4$, we obtain
$\mathcal{LG}^{add}(m_{\\_}2)>\mathcal{LG}^{add}(m_{\\_}4)$. Interestingly,
such a symmetry-breaking does not occur in the photon-subtracted state.
Therefore, we show that the photon-addition procedure can be used as a
detection method to identify the mode in which photons are added.

  4. 4.

With the increase of squeezing parameter, LGME increases as one expects.
Interestingly, the gap between the LGME obtained in different modes decreases
with the moderate number of photons added in a single mode and with a moderate
squeezing strength.

Figure 4:  LGME (y-axis) with respect to the equal number of photons added and
subtracted in two different modes, i.e., $m_{\\_}i=m_{\\_}j$. And
correspondingly the legends are represented as add${}_{\\_}{i,j}$ and
sub${}_{\\_}{i,j}$. All other specifications are same as in Fig. 3.

_Two-mode operations._ Let us now move to the scenario when a fixed number of
photons are added (subtracted) to (from) two modes, i.e. in
$(m_{\\_}i,m_{\\_}j)$-pair. Suppose $m_{\\_}i+m_{\\_}j=n$ is added
(subtracted) with $i\neq j$. Here we can divide the situation into two
categories as depicted in Figs. 3 and 4 \- (1) measurement is performed
neither on $i$th nor on the $j$th mode; (2) measurement is performed on, say
$j$th mode. In our case $j=4$. Like single-mode operations, we again confirm
that twin operations, measuring and adding photons, in the same mode has
adverse effects in localization of multimode entanglement.

_Case 1._ Let us first consider the pair $(m_{\\_}1,m_{\\_}3)$. LGME
monotonically increases till the point with $m_{\\_}i=m_{\\_}j=n/2$, then
decreases and reaches to the same value since LGME coincides when addition
happens either in mode $1$ or $3$. For mode $(1,2)$-pair, LGME clearly
decreases with $m_{\\_}1$, as can be seen from Fig. 2. From Theorem 1, it is
clear that LGME remains constant with $(m_{\\_}i,m_{\\_}{i+2})$-pair for
photon-subtracted state. On the other hand, in case of mode $(1,2)$-pair, we
observe that for $n=20$, the maximum amount of GME can be localized when eight
photons are subtracted from the first mode which reflects the fact that there
is no symmetry in the exchange of $m_{\\_}1$ and $m_{\\_}2$.

_Case 2._ In this picture, one of the parties, in our case, the fourth party
performs the photon counting measurement. Hence we are interested in the pair,
$(m_{\\_}i,m_{\\_}4)$. In case of addition, we observe that in case of
$(m_{\\_}2,m_{\\_}4)$-duo, since the symmetry is broken due to the measurement
on the fourth mode, localized entanglement of photon-added state at
$(m_{\\_}2=0,m_{\\_}4=20)$ is different than that of
$(m_{\\_}2=20,m_{\\_}4=0$). Moreover, from Fig. 2, we find that entanglement
is less when photons are added in the measured mode compared to that of the
unmeasured mode. Hence we have
$\mathcal{LG}(m_{\\_}2=20,m_{\\_}4=0)>\mathcal{LG}(m_{\\_}2=0,m_{\\_}4=20)$
and it is an increasing curve with $m_{\\_}2$ although not monotonic. On the
other hand, photon-subtraction does not break the symmetry of the initial
state and so localized entanglement remains constant with $m_{\\_}2$ or
$m_{\\_}4$ as shown in Fig. 3.

_Special Case._ We now consider the situation where equal number of photons
are added or subtracted from modes. Note that it is a special configuration of
the above mentioned case since the photonic operations are equally distributed
in the two modes. We highlight this because it neatly captures the LGME
enhancement properties of bimodal photonic operations, see Fig. 4 and again
establishes that photon-subtraction is better than that of photon-addition,
i.e. $\mathcal{LG}^{sub}\geq\mathcal{LG}^{add}$.

##  V Conclusion

Characterization and manipulation of quantum resources form integral parts of
research in quantum information science facilitating optimal usage of the
same. In this work, we focus on concentrating genuine multimode entanglement
(GME) in a multimode state from a state with higher number of modes using
local projective measurements. We were interested to quantify the amount of
localizable GME (LGME) both in the Gaussian and non-Gaussian paradigms.

We started off our analysis with a four-mode Gaussian squeezed vacuum (FMSV)
state and found the GME content that can be localized in the remaining three
modes by employing optimal Gaussian measurements in the fourth mode. We showed
that the maximum localization is possible when Gaussian measurement reduces to
homodyne detection in either $x$ or $p$ quadrature and can be easily
implemented experimentally. We also reported that if one adopts a non-Gaussian
measurement strategy like photon counting measurement scheme, it can lead to a
higher LGME in comparison to the optimal Gaussian measurement setting.

With a clear indication that non-Gaussianity aids in more effective
localization of GME, we considered the scenario where we computed LGME of non-
Gaussian states by applying non-Gaussian measurements. The measurement
strategy is kept fixed to be the photon counting operations as before, and the
states under investigation are obtained by deGaussifying the FMSV state by
performing photon addition and subtraction in its various modes. These dual
dose of non-Gaussianity enables localization of substantial GME from the
optimal Gaussian case. In addition to the overall enhancement of LGME, we also
observed that for obtaining LGME, local measurements can break the inherent
symmetry, present in the original state in case of addition while such a
symmetry remains unaffected in the photon-subtracted state even after the
measurement. It also revealed that LGME of the photon-added states can be
applied to identify the modes in which photons are added and works for a
moderate amount of squeezing in the original state, achievable in current
experiments. On the other hand, due to this property in the photon-subtracted
state, there exist a situation when LGME remains constant with a fixed number
of subtraction of photons from two modes. Our analysis indicates that the
local measurements can uncover certain characteristics of states which are
otherwise impossible to highlight.

###### Acknowledgements.

TD acknowledges The ’International Centre for Theory of Quantum Technologies’
project (contract no. 2018/MAB/5). We acknowledge the support from the
Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of
Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme-
1/2019/23.

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††footnotetext: †<EMAIL_ADDRESS>‡<EMAIL_ADDRESS>
# Observation of Quadratic (Charge-2) Weyl Point Splitting
in Near-Infrared Photonic Crystals

Christina Jörg1,2,†,∗, Sachin Vaidya2,‡,∗, Jiho Noh2,3, Alexander Cerjan2,4,5,
Shyam Augustine1, Georg von Freymann1,6, Mikael C. Rechtsman2 1Physics
Department and Research Center OPTIMAS, Technische Universität Kaiserslautern,
67663 Kaiserslautern, Germany
2Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA
3Department of Mechanical Science and Engineering, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA
4Sandia National Laboratories, Albuquerque, New Mexico 87123, USA
5Center for Integrated Nanotechnologies, Sandia National Laboratories,
Albuquerque 87123, New Mexico, USA
6Fraunhofer Institute for Industrial Mathematics ITWM, 67663 Kaiserslautern,
Germany
∗These authors contributed equally to this work.

###### Abstract

Weyl points are point degeneracies that occur in momentum space of three-
dimensional periodic materials and are associated with a quantized topological
charge. We experimentally observe the splitting of a quadratic (charge-2) Weyl
point into two linear (charge-1) Weyl points in a 3D micro-printed photonic
crystal via Fourier-transform infrared spectroscopy. Using a theoretical
analysis rooted in symmetry arguments, we show that this splitting occurs
along high-symmetry directions in the Brillouin zone. This micro-scale
observation and control of Weyl points is important for realizing robust
topological devices in the near-infrared.

##  I Introduction

Weyl materials have drawn significant interest over the last decade for their
ability to support Weyl points (WPs), which are topologically protected
degeneracies in three-dimensional periodic systems. They have been examined in
a wide range of physical systems such as conventional solids Xu  _et al._
(2015); Lv  _et al._ (2015a, b); Arnold  _et al._ (2016); Armitage  _et al._
(2018); Meng  _et al._ (2020); Yan and Felser (2017), photonic systems Lu  _et
al._ (2013, 2015); Yang  _et al._ (2018); Noh  _et al._ (2017); Chang  _et
al._ (2017); Goi  _et al._ (2018); Takahashi  _et al._ (2018); Cerjan  _et
al._ (2019); Vaidya  _et al._ (2020); Wang  _et al._ (2021); Devescovi  _et
al._ (2021), acoustic systems Yang and Zhang (2016); Ma  _et al._ (2019); Deng
_et al._ (2020); Luo  _et al._ (2021), cold atoms Mai  _et al._ (2017); Dubček
_et al._ (2015), electric circuits Li  _et al._ (2020) and coupled resonator
arrays Lin  _et al._ (2016); Zhang and Zhu (2017). In photonics, WPs are
sought for a range of applications such as designing large-volume single-mode
lasers Chua  _et al._ (2014) and mediating long-range interactions between
quantum emitters García-Elcano  _et al._ (2020); Ying  _et al._ (2019), and as
such they have been realized in a variety of optical and photonic systems Lu
_et al._ (2013, 2015); Yang  _et al._ (2018); Noh  _et al._ (2017); Chang  _et
al._ (2017); Goi  _et al._ (2018); Takahashi  _et al._ (2018); Cerjan  _et
al._ (2019); Vaidya  _et al._ (2020); Wang  _et al._ (2021); Devescovi  _et
al._ (2021) spanning orders of magnitude in wavelength. WPs are quantized
sources and sinks of Berry curvature in momentum space, analogous to magnetic
monopoles. Therefore, they can be topologically characterized by the closed
surface integral of the Berry curvature; this “charge” is always an integer
and is identical to the first Chern number. On the 2D surfaces of a Weyl
material, this topological charge manifests through the appearance of surface-
localized conduction channels, so-called Fermi-arc surface states, that
connect two WPs with opposite charges. Since the Berry curvature vanishes
identically in the Brillouin zone for structures with both spatial inversion
and time reversal symmetry, it is necessary to break one or both of these
symmetries to obtain WPs.

Figure 1:  (a) An exploded view of the chiral woodpile PhC (one unit cell in
$z$) and its Brillouin zone. (b) Band structure of the chiral woodpile PhC,
made out of dielectric rods ($\varepsilon=2.31$), with equal widths for all
rods at $k_{y}=k_{z}=0$ showing bands 3 to 8. The quadratic WP at
$\mathbf{\Gamma}$ is marked with a blue circle. The inset shows a plot of the
Berry phase for band 4 around $\mathbf{\Gamma}$. The double winding indicates
that the degeneracy has a charge of 2. (c) The band structure of the same
chiral woodpile PhC in (b) but with increased rod width $w_{0}$ in the blue
layer of each unit cell. The linear WPs that occur due to the splitting of the
quadratic WP in (b) are marked with blue circles. The inset shows a plot of
the Berry phase for band 4 around one of the degeneracies. The single winding
indicates that the degeneracy has a charge of 1.  Figure 2:  False color SEM
image of a typical chiral woodpile photonic crystal with increased rod width
$w_{0}$ in the blue rods. (a) Top view, (b) side view.

A Weyl point can possess an arbitrary integer charge, $m$, and can be
described by the generic three-dimensional Hamiltonian of the form Fang  _et
al._ (2012),

$\displaystyle\mathcal{H}(\mathbf{k})=k_{+}^{|m|}\sigma_{+}+k_{-}^{|m|}\sigma_{-}+k_{z}\sigma_{z}+\omega_{0}I$
(1)

where $k_{\pm}=(k_{x}\pm ik_{y})$, $\sigma_{\pm}=1/2(\sigma_{x}\pm
i\sigma_{y})$, $\sigma_{x,y,z}$ are the Pauli matrices, $I$ is the identity
matrix, and $\omega_{0}$ is the frequency at which the WP occurs in the
spectrum. The eigenvalues of $\mathcal{H}(\mathbf{k})$ describe the two bands
in the vicinity of the WP and are given by

$\displaystyle\lambda_{\pm}=\omega_{0}\pm\sqrt{(k_{x}^{2}+k_{y}^{2})^{|m|}+k_{z}^{2}}.$
(2)

From equation (2) it is evident that the leading order $k_{x,y}$-dependence in
the dispersion around a WP is governed by its charge and henceforth, charge-1
WPs will be referred to as “linear” and charge-2 WPs as “quadratic”.

While charge-1 WPs have been extensively studied, higher-charged WPs have
received relatively less attention. These exotic WPs, sometimes referred to as
unconventional WPs, have markedly different properties than their conventional
counterparts such as higher-order algebraic dispersion, multiple Fermi-arc
surface states and large density of states for high charge ($m\geq 2$).
Higher-charged WPs are known to occur in non-symmorphic crystals due to the
stabilization of multiple linear WPs at high symmetry points in momentum space
Chang  _et al._ (2017); Hirschmann  _et al._ (2021); Chen  _et al._ (2018). In
photonics, quadratic WPs were predicted Takahashi  _et al._ (2018) and
observed in all-dielectric woodpile-like structures Vaidya  _et al._ (2020).
Such structures exhibit several additional symmetries and therefore provide an
ideal platform for tuning the locations of linear WPs that can form from
splitting a quadratic WP due to symmetry breaking Chen  _et al._ (2018); Roy
_et al._ (2017). This tunability of WPs could be of importance in the
realization of three-dimensional large-volume, single-mode lasing devices that
rely on the vanishing photonic density of states at frequency-isolated linear
WPs Chua  _et al._ (2014).

In this article, we experimentally demonstrate that under careful symmetry
breaking, quadratic WPs can be split into two linear WPs of the same charge.
We show by analyzing the underlying symmetries of the structures that for
certain choices of defects this splitting occurs strictly along high symmetry
directions in momentum space. We also show that the momentum-space separation
of the resulting linear WPs can be readily controlled via geometric parameters
of the structure. While WPs have mostly been observed in large-scale
structures with mm- or cm-scale lattice constants Yang  _et al._ (2018); Lu
_et al._ (2015); Li  _et al._ (2018), our platform of choice is a micron-scale
three-dimensional photonic crystal (PhC) with low dielectric contrast which is
fabricated by a two-photon polymerization process Deubel  _et al._ (2004).
This allows for the realization of WPs at near-infrared wavelengths that are
characterized using Fourier transform infrared (FTIR) spectroscopy.

Figure 3:  (a)-(e) Measured angle-resolved FTIR spectra of the chiral woodpile
PhC with varying values of the width $w_{0}$. The locations of the WPs are
marked with arrows. (f)-(j) The corresponding RCWA simulated spectra of the
chiral woodpile PhC. The dashed black lines are the $k_{z}=0$ bulk bands
calculated from MPB.

##  II Results

The particular structure that we employ to realize this phenomenon is a chiral
woodpile PhC whose unit cell consists of stacked and partially overlapping
layers of rods that have a relative $45^{\circ}$ in-plane rotation between
them as shown in Fig. 1 (a). The lattice constant in all three directions is
$a=$2.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The four rods in the unit
cell have height $h$ and widths $w_{0}$, $w_{45}$, $w_{90}$, $w_{135}$ where
the subscripts identify the rods by their angle of orientation with respect to
the $x$-axis. The rods are made out of a nearly lossless, non-magnetic
dielectric material of dielectric constant $\varepsilon=2.31$ (see Methods).
Due to the chirality of this structure, inversion symmetry is broken which
allows WPs to exist. Furthermore, when all rod widths and heights are equal,
this structure belongs to the non-symmorphic space group $P4_{2}22$ (# 93),
which has a screw axis along the $z$ direction. This screw symmetry, defined
by a $90^{\circ}$ rotation in the $x$-$y$ plane and $a/2$ shift along the $z$
direction, results in the presence of a quadratic WP at the Brillouin zone
center ($\mathbf{\Gamma}$), as shown in Fig. 1, (b) and corners
($\mathbf{R}$).

Instead, when the rod width in one layer in the unit cell is taken to be
different from the widths of the other three layers, the screw symmetry is
broken, resulting in a splitting of the quadratic WP into two linear WPs as
shown in Fig. 1 (c). For example, when a width defect is introduced such that
$w_{45}=w_{90}=w_{135}=w$ and $w_{0}\neq w$, the space group of the structure
is reduced to $P222$ (# 16). While this subgroup does not contain the screw
symmetry, it retains $C_{2}$ rotations about the $x$, $y$ and $z$ axes that
restrict the splitting directions of the WPs to high symmetry lines in
momentum space. This can be argued as follows: Let the splitting associated
with this change in space group by a small perturbation result in two linear
WPs, one of which is located at a generic momentum point
$(k_{x},\,k_{y},\,k_{z})$. The location of both WPs can be inferred from the
symmetry operations of the space group $P222$ and time reversal symmetry as
they must either map to themselves or to each other under these operations.
Time reversal symmetry requires a WP to be located at
$(-k_{x},\,-k_{y},\,-k_{z})$. However, the aforementioned $C_{2}$ rotations
require a WP to be located at $(-k_{x},\,k_{y},\,-k_{z})$,
$(k_{x},\,-k_{y},\,-k_{z})$ and $(-k_{x},\,-k_{y},\,k_{z})$. Since there are
only two linear WPs that can occur from the splitting of a single quadratic
WP, at least two components of these momentum vectors must be set to $0$ or
$\pi/a$ such that they map to themselves under a sign flip. As a result, for
the quadratic WP at $\mathbf{\Gamma}$, the symmetry-consistent directions of
splitting are $\mathbf{\Gamma-X}$, $\mathbf{\Gamma-Y}$ or $\mathbf{\Gamma-Z}$,
and for the quadratic WP at $\mathbf{R}$, the symmetry-consistent directions
of splitting are $\mathbf{R-U}$, $\mathbf{R-T}$ or $\mathbf{R-M}$. A similar
analysis shows that the splitting can be made to occur along diagonal
directions in momentum space (e.g. along $\mathbf{\Gamma-M}$ for the WP at
$\mathbf{\Gamma}$) when $w_{45}$ or $w_{135}$ is chosen to be different from
the other three rod widths. In the most general case where all four rod widths
are different, the space group is further reduced to $P2$ (#3) which retains
$C_{2}$ rotation about the $z$ axis. This still restricts the splitting to
occur in the $k_{z}=0$ plane or along $\mathbf{\Gamma-Z}$ for the quadratic WP
at $\mathbf{\Gamma}$ and $k_{z}=\pi/a$ plane or along $\mathbf{R-M}$ for the
quadratic WP at $\mathbf{R}$. For the remainder of this article, we will focus
on the case where a width defect is introduced in a single layer in each unit
cell such that $w_{45}=w_{90}=w_{135}=w$ and $w_{0}\neq w$.

Figure 4:  The angular separation of WPs, $\theta$, as a function of the
symmetry-breaking parameter, $1-w/w_{0}$. All data points correspond to
samples that were fabricated with the same value of $w$.

To demonstrate the topological nature of WPs in the split and unsplit cases,
we directly calculate their topological charges from the electromagnetic
eigenmodes extracted from MIT Photonic Bands (MPB) Johnson and Joannopoulos
(2001). Since WPs act as sources or sinks of Berry curvature, their charge can
be obtained by integrating the Berry curvature over a closed sphere enclosing
the WP in momentum space. Alternatively, this can also be calculated using a
series of line integrals of Berry connection on closed contours that are
sampled from such a sphere. These line integrals represent the geometric
phase, called the Berry phase ($\theta_{B}$), acquired by the eigenmodes as
they are adiabatically transported around the closed contours. In the discrete
form, $\theta_{B}$ on one such contour ($C$) is given by Resta (2000)

$\theta_{B}(C)=-\textrm{Im}\ln\left(\prod_{i=1}^{N}\frac{\braket{\psi(\mathbf{k}_{i})}{\psi(\mathbf{k}_{i+1})}}{\left|\braket{\psi(\mathbf{k}_{i})}{\psi(\mathbf{k}_{i+1})}\right|}\right)$
(3)

where the index $i$ discretizes the contour $C$ into $N$ points and
$\ket{\psi(\mathbf{k}_{i})}$ is the periodic part of the magnetic field
eigenmode of the PhC at the momentum point $\mathbf{k}_{i}$. These contours
can be chosen to lie parallel to the $k_{x}$-$k_{y}$ plane and $\theta_{B}$
can therefore be plotted as a function of $k_{z}$. The topological charge of
the WP is the winding number of $\theta_{B}(k_{z})$. The plots for
$\theta_{B}$ for the split and unsplit cases are shown in the insets of Fig. 1
(b)-(c) which confirm that the degeneracy at $\mathbf{\Gamma}$ is a quadratic
WP of charge $+2$ and the split degeneracies along the $\mathbf{\Gamma-X}$
direction are linear WPs of charge $+1$ each.

For the experiment, we fabricate chiral woodpile PhCs via two-photon
lithography of a liquid negative-tone photoresist (IP-Dip, refractive index =
1.52) using a Nanoscribe professional GT. We print several samples of this PhC
with a lattice constant of $2.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with
varying values of $w_{0}$, while fixing $w$ to
$0.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for the symmetry broken samples.
Due to the voxel’s height in $z$, adjacent layers overlap by approximately 50%
of the rod height. A scanning electron microscope (SEM) image of a typical
sample is shown in Fig. 2. For characterizing the fabricated PhC, we measure
the angle-resolved transmittance via Fourier-transform infrared spectroscopy
(FTIR). Details about the fabrication and measurement setup can be found in
the Methods section.

For the case of all equal rod widths, we observe a quadratic WP at
$2.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength at the
$\mathbf{\Gamma}$ point (Fig. 3 (a)). As previously described, an increase in
$w_{0}$ splits the WP along the $\mathbf{\Gamma}-\mathbf{X}$ direction into
two linear WPs. This splitting can be seen as a spectral feature corresponding
to the two involved bands piercing through each other with the linear WPs
occurring at their intersection points as shown in Fig. 3 (b)-(e).
Furthermore, the angular separation between the two linear WPs increases with
$w_{0}$. Along the orthogonal direction, $\mathbf{\Gamma}-\mathbf{Y}$, the
separation of the same spectral feature indicates that the two bands move
apart and are non-degenerate (see Supplementary Material). To compare our
experimental results with theory, we also perform RCWA (rigorous coupled-wave
analysis) simulations as implemented in Stanford Stratified Structure Solver
(S4) Liu and Fan (2012), to obtain the transmission spectrum of this PhC. The
simulated spectra are plotted in Fig. 3 (f)-(j) and show an excellent
agreement with the experimentally obtained data. Moreover, the sharp spectral
features match the $k_{z}=0$ Weyl bands obtained from MPB (dashed lines in
Fig. 3 (f)-(j)) allowing for a direct observation of the splitting.

We extract the angular separation $\theta$ between the two linear WPs from the
experimental data and plot it as a function of the symmetry-breaking parameter
$1-w/w_{0}$ in Fig. 4, demonstrating the tunable nature of WP splitting in
this structure. In our experiment, the separation of WPs in momentum space is
found to be a monotonic function of the symmetry-breaking parameter, however
we note that this is not true in general. For higher dielectric contrast, the
separation of the WPs can first increase and then decrease to zero for
increasing values of the symmetry-breaking parameter. This can lead to the
formation of accidental quadratic WPs which are not symmetry protected but are
obtained on the fine tuning of parameters but are otherwise identical to the
symmetry-protected quadratic WPs (see Supplementary Material for an example).

At low dielectric contrast, the WPs are embedded inside the $k_{z}$-projected
bands of the PhC which could obscure the dispersion features of the WPs in the
spectrum. However, we observe that the $k_{z}=0$ Weyl bulk bands are clearly
visible as boundaries of sharp features in the spectrum as indicated by the
dashed lines in Fig. 3 (f)-(j). To explain this observation, we perform a
coupling analysis similar to Vaidya  _et al._ (2020); Kao  _et al._ (2015);
Karakasoglu  _et al._ (2018); Saba  _et al._ (2011); Takahashi  _et al._
(2014) which reveals that the relevant projected bands are nearly of either s-
or p- character and hence couple selectively to light of these polarizations
(see Supplementary Material). This selective coupling results in the boundary
of the projected bands, corresponding to $k_{z}=0$ bulk bands, appearing as
sharp drops in the transmission of polarized light allowing for the
observation of WPs despite the lack of a local bandgap.

Perhaps the most direct physical manifestation of the non-trivial topology of
a WP is the presence of surface states that form Fermi arcs connecting WPs of
opposite charges. Interestingly, this property is closely related to that of
unidirectional edge states found in two-dimensional Chern insulators. We now
present a numerical exploration of the surface states associated with both the
quadratic and linear WPs in our structure. Since the WPs of interest occur
near the Brillouin zone center, the surface states lie above the light line of
air. As a result, they are leaky resonances that radiate away from the surface
with a finite lifetime. At low dielectric contrast such as in our experiment,
the lifetime of such resonances can be so low that they are effectively
unobservable in experiment and hard to extract and analyze numerically. To
overcome this difficulty, we consider an interface between two chiral woodpile
PhCs with opposite handedness for our analysis. This leads to a doubling of
the number of surface states due to the presence of WPs on both sides of the
interface. Moreover, the surface states originating from both structures are
degenerate in frequency and momentum and can therefore hybridize to form
states with even and odd symmetry with respect to the mirror plane at the
interface. Nevertheless, the topological origin of these surface states can be
directly confirmed by examining the surface spectrum along a circular contour
that encloses the projection of the WPs.

We consider a system consisting of two chiral woodpile PhCs with opposite
chirality and equal rod widths in all layers that meet at an interface
parallel to the $y=0$ plane. The surface band structure calculated along a
circular loop in the $k_{x}$-$k_{z}$ plane enclosing the quadratic WP at
$\mathbf{\Gamma}$ is shown in Fig. 5 (a). This loop has a radius of 0.05
$(2\pi/a)$ and is parametrized by a single angular variable $0\leq
t/(2\pi)<1$. The surface band structure reveals the existence of two pairs of
hybridized states that cross the non-trivial gap formed along the loop. Due to
the bulk-boundary correspondence principle, the number of surface states
implies that the magnitude of charge of the enclosed WPs is 2. Next, we
consider the same simulation but for an increased rod width $w_{0}$ that
splits the quadratic WP. The surface band structure along a loop of radius
0.033 $(2\pi/a)$ enclosing just one of the linear WPs is shown in Fig. 5 (b)
which reveals only one pair of surface states in the gap formed along the
loop. This indicates that the charge of the enclosed WP has magnitude of 1.

Figure 5:  Surface states associated with the WPs at the interface between two
low-contrast chiral woodpile PhCs with opposite chirality. (a) The
$k_{y}$-projected band structure along the circular loop, parameterized by
$t$, enclosing the quadratic WP at $\mathbf{\Gamma}$. The non-trivial gap
formed along the loop contains two pairs of hybridized surface states (dashed
lines) with even and odd symmetry with respect to the mirror plane at the
interface. The blue solid colors are projections of the bulk bands. (b) The
$k_{y}$-projected band structure along the circular loop enclosing one of the
split linear WPs, showing only a single pair of even and odd surface states.

In conclusion, we have observed the splitting of a quadratic Weyl point into
two linear Weyl points in a low-contrast 3D PhC. We find that the splitting
can be made to occur strictly along high-symmetry directions in momentum
space, a consequence of controlled symmetry breaking and that their separation
can be readily tuned via the geometric parameters of the crystal. The micron-
scale periodicity of our structure allows us to access Weyl points in the
near-infrared optical spectrum. Our approach opens new avenues for designing
large-volume single-mode lasers Chua  _et al._ (2014), using Weyl points and
Fermi arc surface states in microscale photonic structures, relevant to near-
infrared optics.

##  III acknowledgments

C.J. gratefully acknowledges funding from the Alexander von Humboldt
Foundation within the Feodor-Lynen Fellowship program. G.v.F. acknowledges
funding by the Deutsche Forschungsgemeinschaft through CRC/Transregio 185
OSCAR (project No. 277625399). We would like thank the Nano Structuring Center
Kaiserslautern for technical support. M.C.R., S.V. and A.C. acknowledge the
support of the U.S. Office of Naval Research (ONR) Multidisciplinary
University Research Initiative (MURI) under Grant No. N00014-20-1-2325. A.C.
acknowledges support from the Center for Integrated Nanotechnologies, an
Office of Science User Facility operated for the U.S. Department of Energy
(DOE) Office of Science, and the Laboratory Directed Research and Development
program at Sandia National Laboratories. Sandia National Laboratories is a
multimission laboratory managed and operated by National Technology &
Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. DOE’s National Nuclear Security
Administration under contract DE-NA-0003525. The views expressed in the
article do not necessarily represent the views of the U.S. DOE or the United
States Government.

##  IV Methods

### Fabrication

For the fabrication of the chiral photonic woodpile samples in the IP-Dip
resist we use a Nanoscribe Professional GT at a scan speed of
$20\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{s}$ and laser power of 62%,
which corresponds to $34\text{\,}\mathrm{m}\mathrm{W}$ on the entrance lens of
the objective. The structures are printed onto Menzel cover slips
(borosilicate glass). Since we use the dip-in configuration of the Nanoscribe
the cover slips need to be coated with approximately
$13\text{\,}\mathrm{n}\mathrm{m}$ of $\mathrm{Al}_{2}\mathrm{O}_{3}$ in order
to facilitate interface finding. The coated cover slips show a transmission of
greater than 75% for all wavelengths used in our measurements. After printing,
the sample is developed for $10\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ in
PGMEA and $10\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ in isopropanol,
subsequently. It is then transferred to a solution of
$150\text{\,}\mathrm{m}\mathrm{g}$ Irgacure 651 in
$24\text{\,}\mathrm{m}\mathrm{l}$ of isopropanol and illuminated for 60s with
UV light from an Omnicure S2000 with 95% iris opening. This post-print UV
curing Oakdale  _et al._ (2016) increases the stability of the woodpile
structure. In the end, the sample is blow-dried in a stream of nitrogen. The
complete footprint of the structure is approximately
$1\text{\,}\mathrm{m}\mathrm{m}^{2}$ with 20 layers in height. To achieve such
a large footprint within reasonable writing time we use elaborate stitching:
The structure is printed in portions of 4x4 angled blocks between which the
stage is moved for larger travel distance. Inside each block and layer we
print the sample using the galvanometer-scanning in combination with piezo-
stitching for reduced vignetting and more precise positioning. The alignment
of the stage, piezo and galvo axes is ensured by employing the axis
transformation implemented in NanoWrite. The structural parameters are
determined via scanning electron microscopy and are listed in the
Supplementary Material. While the usual rods consist of just one printed line,
the defect rod width is increased by printing multiple lines at a hatching
distance between $10\text{\,}\mathrm{n}\mathrm{m}$ and
$50\text{\,}\mathrm{n}\mathrm{m}$. For the structure in Fig. 3b) the laser
power used for printing the $w_{0}$\- rods was decreased to 55%.

### Measurement

For the measurement of the spectra of the woodpile samples we use the Hyperion
3000 microscope attached to a Bruker Vertex v70 FTIR. The spectra are taken in
transmission mode with a halogen lamp and a MCT detector cooled by liquid
nitrogen. To increase $k$-space resolution the lower 15x Cassegrain objective
is covered except for a pinhole of $1\text{\,}\mathrm{m}\mathrm{m}$ in
diameter, such that we obtain a nearly collimated beam. We estimate the spread
of this beam to be approximately $\pm 0.3$° in the direction along which we
measure (the $k_{x}$ direction in Fig. 3), and approximately $\pm 2$°
perpendicular to that Cerjan  _et al._ (2021). The sample is then tilted with
respect to the beam along its $x$-axis, around perpendicular incidence of the
beam. This is done by tilting the sample holder in steps of approximately
0.4°. As we cannot determine the position of perfect perpendicular incidence
from the sample positioning (within an error of approximately 5°), we take
spectra for both positive and negative tilting angles, and determine $\Gamma$
from the symmetry of the measured angle resolved transmittance spectra. All
spectra are referenced to the transmission of the used substrates, and
individually scaled to their maximum at each angle. For each angle we average
over 64 measurements with an FTIR resolution set to 4/cm in wavelength. The
small dip in transmittance around $2.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$
wavelength, constant across angles, is due to the absorption in the IP-Dip
Fullager  _et al._ (2017). The spectra are post-processed to remove fringes
that appear in the spectra due to multiple reflections in the glass and
$\mathrm{Al}_{2}\mathrm{O}_{3}$.

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## Author contributions

C.J. and S.V. contributed equally to this work. S.V. and J.N. conceived the
main idea of the project. S.V. and J.N. performed the theoretical analysis and
numerical simulations with the assistance of A.C. C.J. fabricated the
structures and performed all measurements with help from S.A. M.C.R. and
G.v.F. supervised the project. All authors discussed the results and
contributed significantly to the preparation of the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to C.J., S.V.,
or M.C.R.