††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Optomechanical generation of coherent GHz vibrations in a phononic waveguide

Guilhem Madiot  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
Campus UAB, Bellaterra, 08193 Barcelona, Spain  Ryan C. Ng  Catalan Institute
of Nanoscience and Nanotechnology (ICN2), Campus UAB, Bellaterra, 08193
Barcelona, Spain  Corresponding authors<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>Guillermo Arregui  DTU Electro, Department of
Electrical and Photonics Engineering, Technical University of Denmark, Ørsteds
Plads 343, DK-2800 Kgs. Lyngby, Denmark  Omar Florez  Catalan Institute of
Nanoscience and Nanotechnology (ICN2), Campus UAB, Bellaterra, 08193
Barcelona, Spain  Dept. de Física, Universitat Autonoma de Barcelona, 08193
Bellaterra, Spain  Marcus Albrechtsen  DTU Electro, Department of Electrical
and Photonics Engineering, Technical University of Denmark, Ørsteds Plads 343,
DK-2800 Kgs. Lyngby, Denmark  Søren Stobbe  DTU Electro, Department of
Electrical and Photonics Engineering, Technical University of Denmark, Ørsteds
Plads 343, DK-2800 Kgs. Lyngby, Denmark  NanoPhoton - Center for
Nanophotonics, Technical University of Denmark, Ørsteds Plads 345A, DK-2800
Kgs. Lyngby, Denmark  Pedro D. García  Catalan Institute of Nanoscience and
Nanotechnology (ICN2), Campus UAB, Bellaterra, 08193 Barcelona, Spain
Corresponding authors<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Clivia M.
Sotomayor-Torres  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
Campus UAB, Bellaterra, 08193 Barcelona, Spain  ICREA - Institució Catalana de
Recerca i Estudis Avançats, 08010 Barcelona, Spain

###### Abstract

Nanophononics has the potential for information transfer, in an analogous
manner to its photonic and electronic counterparts. The adoption of phononic
systems has been limited, due to difficulties associated with the generation,
manipulation, and detection of phonons, especially at GHz frequencies.
Existing techniques often require piezoelectric materials with an external
radiofrequency excitation that are not readily integrated into existing CMOS
infrastructures, while non-piezoelectric demonstrations have been inefficient.
In this work, we explore the optomechanical generation of coherent phonons in
a suspended 2D silicon phononic crystal cavity with a guided mode around 6.8
GHz. By incorporating an air-slot into this cavity, we turn the phononic
waveguide into an optomechanical platform that exploits localized photonic
modes resulting from inherent fabrication imperfections for the transduction
of mechanics. Such a platform exhibits very fine control of phonons using
light, and is capable of coherent self-sustained phonon generation via
mechanical lasing around 6.8 GHz. The ability to generate high frequency
coherent mechanical vibrations within such a simple 2D CMOS-compatible system
could be a first step towards the development of sources in phononic circuitry
and the coherent manipulation of other solid-state properties.

##  I Introduction

Phononics has received less attention relative to its photonic and electronic
counterparts, largely due to challenges associated with generating,
transporting, manipulating, and detecting phonons, despite their potential as
carriers of information signals [1, 2, 3, 4, 5, 6]. The utility of phononics
lies in its ability to operate in the MHz and GHz regimes, which connects the
frequency regimes of electronics and optics [7, 8, 9]. In an analogous manner
to photonic crystals, the dispersion relation and mode profiles of acoustic
phonons can be engineered by making structures with periodic elastic
properties. In planar structures, careful design coupled with improvements in
nanofabrication have enabled the realization of mechanical band gaps up the
GHz frequency range [10, 11], while the introduction of linear defects into
these crystals permits phonon waveguiding [12, 13, 14]. However, the
excitation of phonons in phononic circuits still remains a challenge, with
existing approaches relying on electromechanical actuation in piezoelectric
materials [15, 16, 17, 18, 19]. While such an approach does in theory offer
the technical advantages of ultra-compact on-chip integrability with low power
consumption, piezoelectric solutions always rely on an external RF signal that
is not as readily integrated. Furthermore, these approaches are lacking in
their ability to control and manipulate phonons and are restricted to a
limited bandwidth that is often set by the specific interdigitated transducer
design used for excitation. Surface acoustic wave excitation in a non-
piezoelectric platform has recently been shown via thermo-elastic modulation
[20] although this process remains quite inefficient. As an alternative,
cavity optomechanical platforms are able to finely control the phonon signal
via radiation pressure forces while relying on an integrated photonic circuit
rather than an external RF source, circumventing these aforementioned
problems. Within these platforms, self-sustained mechanical oscillations
driven by light, otherwise known as phonon lasing [21, 22, 23, 24, 25], can
potentially be used to generate a phonon signal directly within a phononic
circuit [26, 9, 27, 28]. This is typically achieved by side-coupling an
optomechanical cavity to another phononic waveguide and requires proper
matching of the respective mechanical bandwidths and mode profiles [29, 30]. A
solution that does not require interdigitated transducers or more elaborate
optomechanical architectures is thus an important building block for phononic
circuitry.

FIG. 1: Acoustic waveguide design on a two-dimensional shamrock phononic
crystal. (a) Supercell of the waveguide geometry, with the free boundary at
the silicon-air interface highlighted in red. (b) Associated mechanical band
structure for the center and shield regions showing the symmetric (blue) and
antisymmetric (red) modes. As the modes which are antisymmetric with respect
to the z mid-plane are optomechanically dark, the guided mechanical mode in
the center region is confined by the gap of the shield region. (c) Schematic
illustration of the phononic waveguide incorporating a center cavity region
($n_{a}=0.23a_{\textrm{c}}$, $n_{b}=0.30a_{\textrm{c}}$, $a_{\textrm{c}}=500$
nm) and shield region with slightly larger period along the x direction
($a_{\textrm{s}}=560$ nm) to confine the mechanical mode. The schematic is not
to scale. (d) In-plane displacement spectrum plotted as a function of the
k-vector. This simulation is made accounting for the full structure within one
simulation (i.e., both the center and shield regions together), and the
acoustic Fabry-Pérot fringes can be observed. (e) Displacement mode profiles
for the full structure showing the acoustic confinement within the center
region for the first three harmonics.

Here, we present a 2D silicon phononic crystal waveguide sustaining an
acoustic guided mode around 6.8 GHz and exploit optomechanical interactions to
transduce and generate coherent acoustic phonons directly within the phononic
waveguide itself. To realize this, an optomechanical interface is obtained by
bringing together two identical phononic crystal cavities with mirror
symmetries separated by an air gap defined by their interfaces. This results
in a slot photonic-crystal waveguide that allows for the optomechanical
readout and transduction of mechanics by employing naturally-emerging
disorder-induced localized optical cavities. Confinement of the mechanical
waveguide modes induced by reflections at the ends of the waveguide result in
the observation of acoustic Fabry-Pérot fringes, with the associated strong
optomechanical interaction leading to dynamical back-action and subsequent
mechanical lasing of certain mechanical modes. From a device perspective, the
ability to observe mechanical lasing and coherently generate phonons in such a
platform is a first step towards high-throughput coupling to other mode-
matched acoustic waveguides.

##  II Phononic waveguide design

We design a phononic crystal (PnC) waveguide based on a 220 nm silicon
membrane where shamrock-shaped holes [31] tiled in a triangular lattice are
etched, as schematically illustrated in Figure 1a. The shamrocks are the
result of three overlapping elliptical holes, with each ellipse defined by its
short and long axes given by $n_{\textrm{a}}=0.23a_{\textrm{c}}$ and
$n_{\textrm{b}}=0.30a_{\textrm{c}}$, respectively, where $a_{\textrm{c}}=500$
nm is the PnC lattice period. Such a PnC opens a large phononic band gap
centered around 6.2 GHz, which enables the creation of guided phononic
waveguides through line defects. We achieve this by terminating a crystal with
a straight free-surface, as indicated in red in Figure 1a, resulting in an
acoustic waveguide that can be thought of as a silicon ”wire” with edges
defined by air and by the shamrock PnC [29]. The row of shamrocks closest to
the free surface is shifted away from the edge by
$d_{\textrm{y}}\sqrt{3}a_{\textrm{c}}/2$, where $d_{\textrm{y}}=0.1155$.
Subsequent rows along the $y$ direction are kept at their nominal positions in
the crystal lattice. The distance between the free surface and the center of
the first row of shamrocks sets the width of the waveguide region,
$d_{\textrm{wg}}$, and controls the number of supported modes. We calculate
the band structure of the waveguides using finite-element-method (FEM)
simulations and show two examples of this in Figure 1b. In these band
structures, the bands are associated to $z$-symmetric or $z$-antisymmetric
modes (i.e., symmetric or antisymmetric with respect to the center mid-plane,
shown in blue and red, respectively). Due to the optomechanical transduction
described later, we focus only on the $z$-symmetric modes. By choosing
$d_{\textrm{wg}}=190$ nm, the waveguide possesses a single propagating
$z$-symmetric guided mode, highlighted by a thick blue line in Figure 1b for
two geometries with different horizontal pitches, $a_{x}$, of 500 nm (light
blue border) and 560 nm (yellow border), both with the same shamrock
dimensions. The combination of the above parameters allows for the band of
interest to be flattened at the Brillouin zone edge as well as to be shifted
away from the band gap edge to prevent coupling to bulk modes.

We explore the phononic waveguide with band structure given in the left of
Figure 1b (light blue border), where both the horizontal and vertical pitch
coincide. To improve the temporal confinement of the acoustic modes supported,
we discretize the waveguide spectrum by creating a long acoustic cavity in the
direction of the waveguide ($x$). This cavity is formed by sandwiching a
center region with horizontal pitch $a_{\textrm{c}}$ = 500 nm between two
shield regions with horizontal pitch $a_{\textrm{s}}$ = 560 nm, respectively
shown with blue and yellow in the schematic of Figure 1c. While both the
center and shield regions possess a guided mechanical mode within the band gap
as shown in Figure 1b, the greater horizontal period of the shield region
shifts the symmetric guided mode to a lower frequency and prevents coupling
between the modes of each region. This results in an acoustic Fabry-Pérot
cavity for the mode in the central region, which significantly increases the
mechanical quality factor, $Q_{\text{m}}$, of the acoustic phonons [32]. We
study structures with varying center region lengths $L$, while the shield
region is set to have fixed 8 unit cells along the $x$ direction in all cases.
Figure 1d shows the reconstructed waveguide band obtained after Fourier-
transforming the in-plane $y$-displacement along the free surface for the
modes supported by a structure of length $L$=20$a_{\textrm{c}}$. An identical
mechanical Q-factor of $Q_{m}=700$ is assumed for all modes regardless of the
radiation-limited value obtained through simulation, in accordance with the
experimental observations that follow. The shield structure effectively
generates a long cavity that inherits the dispersion properties of the
underlying guided mode in the center region (Figure 1b, left). This is also
observed when the mode profiles are explored in real space, shown via the
first three harmonics in Figure 1e. The displacement profile results from the
waveguide mode and is thus concentrated at the free surface, dominated by an
in-plane component along the y direction, and decays quickly within the
crystal away from this free surface. Along the waveguide axis, the modes have
an increasing number of nodes and anti-nodes, mimicking a doubly-clamped
string. This simulation is realized with a short cavity for computational
efficiency, but the same physical picture also holds in longer structures.
Unlike the simulations shown in Figure 1d and Figure 1e, a realistic waveguide
is subject to fabrication imperfections such as sidewall roughness. Such
disorder ultimately transforms the spectro-spatial acoustic properties in the
slow sound region (around the $\Gamma$ point) when the length of the waveguide
overcomes the acoustic localization length [33], a feature that we recover
through FEM simulations of long waveguides subject to a minimal positional
disorder model. While this same effect is exploited in the optical domain in
the following section, experimentally distinguishing between Fabry-Pérot
cavity modes and disorder-induced mode localization is a complicated task that
is outside the present scope.

##  III Optomechanical cavity

FIG. 2:  Optical characterization of a slotted two-dimensional optomechanical
waveguide. (a) False-color SEM image of the shamrock optomechanical cavity
waveguide formed by bringing two PnC waveguides of opposite symmetry from
Figure 1a together, separated by an air-slot region. The air-slot enables the
excitation of mechanical modes via localized optical modes when a tapered
fiber loop is brought into contact with the waveguide. A higher magnification
SEM image focused on the center slot is highlighted in red. The
electromagnetic energy density from FEM simulations shows the slot-confined
light within the thin air gap between both PnC’s (bottom right). (b) A
representative optical transmission spectrum (top) is taken with a tapered
fiber loop, where dips in transmission indicate coupling to a resonant optical
mode in the structure. A spatially varying spectral map (bottom) shows the
transmission spectrum (color scale) as the fiber loop is moved along the
length of the waveguide. The dotted line is the position at which a
representative optical transmission spectrum is taken. In the spectral map, at
lower wavelengths, the periodic dips in transmission indicate the extended
Fabry-Pérot modes, while the modes at higher wavelengths, which are more
irregular in wavelength, are localized modes. (c) Optical band structure
showing the guided optical mode alongside the spectral map and representative
spectrum, where the guided mode flattens at the Brillouin zone edge,
indicating a high group index, or low group velocity and subsequent mode
localization.

The shamrock crystal proposed for the phononic waveguide also supports a
photonic band gap [34]. The acoustic design sustains the guided mechanical
mode of interest at a silicon-air free surface. However, this free surface is
normally an inconvenient location for a photonic waveguide mode as air does
not confine light, which complicates the direct optomechanical transduction of
motion. Instead, we can exploit this otherwise disadvantageous nature of the
mechanical waveguide by adding a mirror-symmetric version of the waveguide a
distance of 40 nm away from the original, which leads to two independent
mechanical membranes separated by a center air-slot region. The resulting
design is shown in the false-color SEM image of Figure 2a. Such an approach
leads to a slot photonic-crystal waveguide for the optics. This allows the
electromagnetic field to be locally enhanced in the slot while guiding light
along its length, creating a cavity-optomechanical system that enables light
to couple to in-plane mechanical motion [35]. A FEM simulation of the
electromagnetic energy density is shown in the bottom right of Figure 2a and
additional details about the slot-guided mode and the associated photonic band
structure are provided in Appendix A.

The suspended silicon structures are fabricated on a commercial silicon-on-
insulator wafer following the fabrication process flow described in appendix
B. The resulting structures are optically characterized with a tapered fiber
loop that is placed in contact with the structure along the waveguide axis,
allowing light that is travelling in the fiber to couple evanescently into
resonant optical modes of the optomechanical cavity structure [32]. The
mechanical shield region plays an analogous role for the slot-guided optical
mode, making the structure a very long optical cavity-waveguide (see appendix
A). A representative optical spectrum taken at a central position along a
waveguide of length $L_{\textrm{total}}\sim$84 $\upmu\textrm{m}$, is shown in
the top panel of Figure 2b, in which resonant optical modes manifest as
spectral transmission dips. To understand the spatial structure of the
observed optical modes, we acquire transmission spectra at different fiber
loop positions, denoted as $x_{\mathrm{\ell}}$, along the waveguide axis.
Spectra are acquired in steps of $\sim 2$ $\upmu\textrm{m}$ and
$x_{\mathrm{\ell}}=0$ and $x_{\mathrm{\ell}}=75$ $\upmu\textrm{m}$ denote the
two bounds of the central region of the waveguide. The resulting spatially
varying spectral map is shown in Figure 2b (bottom). The black dotted line
indicates the specific position at which the representative transmission
spectrum shown in the top panel of Figure 2b is taken. At the shorter
wavelength end of the spectrum, periodic dips in transmission are observed,
which are extended Fabry-Pérot modes spanning the entire waveguide and result
from reflection at the waveguide edges, similar to the mechanical modes shown
in Figure 1e. At higher wavelengths the spectral location of the modes is less
regular and the modes are spatially localized to specific sections of the
waveguide in a manner akin to Anderson localization [36, 35]. This results
from localization of the light field due to multiple scattering off of
nanoscale fabrication imperfections such as sidewall roughness resulting from
etching at the silicon-air interface (see SEM inset of the zoom-in of the slot
in Figure 2a) [37]. Any further mention of optical localization in this work
implicitly refers to disorder-induced localization. This localization occurs
at longer wavelengths where the slot-guided light is slowed down, as indicated
by the slot-mode dispersion in the central region (Figure 2c) where the band
flattens near the Brillouin zone edge. In this regime, light-matter
interactions are enhanced and the system becomes increasingly sensitive to any
nanoscale imperfections leading to localized optical cavities with a mode
volume that decreases with increasing group index [38]. At even longer
wavelengths, after approximately $\lambda$ = 1488 nm, the cut off of the
guided mode is reached and near-unity transmission is observed where there is
no available optical mode to couple into.

FIG. 3: GHz mechanical spectroscopy of an acoustic cavity-waveguide. (a)
Brillouin light scattering (BLS) spectroscopy measurement in which the GHz
mechanical guided mode can be observed within the mechanical band gap. (b) The
same mechanical mode is probed with the tapered fiber loop over the same
frequency range as in the BLS measurement to allow for direct comparison. The
inset shows a magnified spectrum of the same mode, allowing for the different
peaks to be more clearly observed. (c) Optical transmission spectrum when
probing with a tapered fiber loop showing localized optical-modes that are
slightly shifted by the thermo-optic effect. The mechanical spectra that
result from driving 4 of these optical modes (as indicated by dashed lines in
(c)) are shown in (d).

While inherent fabrication imperfections such as these are commonly considered
a hindrance, here we exploit them to localize the optical field into optical
cavities that do not require deliberate adiabatic tuning of the geometry. The
localized optical modes supported by this structure have moderate room-
temperature $Q$-factors below $2\cdot 10^{4}$ (see appendix C), which is much
smaller than other optical cavities used for cavity optomechanics, including
those that incorporate localized modes [39, 40, 41, 35, 42]. However, the
strong field confinement in the slot leads to a considerable optomechanical
coupling to the GHz acoustic modes, enabling their use as transducers of
mechanical motion.

##  IV Mechanical characterization

We characterize the mechanical spectrum of the cavity-waveguides described
above by using two different techniques. First, we use Brillouin Light
Scattering (BLS) spectroscopy, where non-resonant light is focused onto the
sample surface and non-linearly scattered by thermally activated acoustic
phonons, which is then measured. The back-scattered signal from these acoustic
phonons is collected and the vibrations manifest as peaks in the BLS spectrum.
We previously reported on the use of this technique for mechanical dispersion
mapping [14], which we apply here to obtain the mechanical band gap for each
domain (shield and center), as well as the waveguide dispersion in the central
region, as detailed in Appendix D. The strength of this technique lies in its
relative ease of incorporation as it is non-invasive and contactless. Further,
it enables the detection of high frequency GHz guided mechanical modes at room
temperature, without the need to drive an optical mode to observe mechanics
and without the need for incorporating other on-chip architectures that may
add complexity [14, 26, 29, 43]. Figure 3a shows the BLS spectrum of the
central region of the waveguide, measured using a 532 nm laser at an incident
angle of 20.1∘, which corresponds to a wavevector of
$k_{x}=0.7\pi$/$a_{\textrm{c}}$. The spectrum exhibits a broad ($\approx 160$
MHz) peak at $\sim 6.85$ GHz over a relatively flat band gap region spanning
from 4.8 to 7.5 GHz. Comparing the BLS spectrum to the simulated band
(Appendix D) shows quantitative agreement between the probed structure and the
simulated one, indicating that the measured mode at 6.8 GHz corresponds to the
waveguide mode. This is also confirmed by focusing the laser beam within the
crystal away from the waveguide, where the peak corresponding to the guided
mode then vanishes. In principle, the BLS spectrum should reveal spectral
features for all Fabry-Pérot acoustic modes having reciprocal space components
including the $k$-value determined by the excitation angle. This should lead
to a set of peaks over a bandwidth that decreases with waveguide length.
However, only a single broad peak is observed. This is due to the limited
spectral resolution of the BLS measurement ($\sim 100$ MHz) as well as to a
relaxation of the phase-matching condition resulting from focusing of the
laser light, which increases the probed wave-vectors to the full solid-angle
of the incident beam. The measured spectrum is therefore an integration of all
of the Brillouin scattering events occurring over a range $\Delta k_{x}$
around the targeted $k_{x}$.

FIG. 4: Optomechanical amplification of a GHz mechanical mode. (a) Radio-
frequency spectrum of the transmitted light showing the mechanical mode.
Dynamical back-action is observed, which leads to mechanical lasing. The color
scale indicates the laser power in mW. The inset shows the driven optical mode
which exhibits a greater thermo-optic shift as the power is increased. (b)
Schematic illustration of a ”phononic laser”, where acoustic waves (green
arrows) are generated in the cavity which is side-coupled to a waveguide.

The second technique we employ for mechanical characterization is the
measurement of the optomechanically-induced transmittance fluctuations of a
monochromatic light source driving the localized optical cavities. Figure 3b
shows the RF spectrum detected with an electronic spectrum analyzer (ESA) when
a localized mode with central wavelength $\lambda_{L}=1484.5$ nm is driven
with a blue-detuned laser. The signal-to-noise ratio (SNR) is calculated from
the power spectral density by normalizing to the noise level measured when the
laser is off. To allow for a direct comparison with the BLS measurement in
Figure 3a, the spectrum is shown over the same frequency range. The inset in
Figure 3b shows a zoom-in around 6.76 GHz which, unlike the BLS spectrum,
shows overlapping spectral lines corresponding to different mechanical modes
supported by the waveguide over a bandwidth of $\approx 110$ MHz. These are
roughly equi-spaced in frequency, and possibly correspond to acoustic Fabry-
Pérot modes (such as those shown in simulations in Figure 1c). However, the
bandwidth over which they are observed is much smaller than the full frequency
span of the guided mode, indicating that the optomechanical couplings $g_{0}$
to the driven optical mode are too low in many cases. This is further
supported by the relative intensities among the transduced peaks and through
the data shown in Figure 3c and d, which respectively show the optical
transmission spectrum and the associated mechanical RF transduction spectra
when four of the localized modes are driven (as given by the dashed lines in
Figure 3c). The localized nature of the optical modes is revealed in several
ways. First, a strong thermo-optic nonlinearity results in a significant red-
shift and in sawtooth-shaped lineshapes in Figure 3c. It also manifests in the
strength of the optomechanically-transduced mechanical modes. For optical
modes below 1484 nm, in the Fabry-Pérot regime, no mechanical modes are
observed above the noise floor. The transduction intensity generally increases
when the laser drives an optical mode at longer wavelengths. This results from
a combination of the reduced mode volumes, an increase in quality factor of
the localized optical modes closer to the cut-off of the band, and a stronger
optomechanical coupling strength [44]. In addition, a comparison of the
spectra in Figure 3d indicates that part of the observed mechanical peaks span
more than one spectrum but may not always appear due to a different
transduction strength which in turn depends on the driven optical mode.
Similar results were obtained for cavities of varying lengths (see appendix
E).

The strong confinement of the optical field in the air-slot and of the
mechanical displacement to the waveguide region, enables dynamical back-action
and subsequent mechanical lasing, indicated by the shifting, narrowing, and
increase in amplitude of the mechanical peak with increasing laser power
(Figure 4a). This occurs despite the relatively low values of the measured
optical $Q$-factor. The color scale in Figure 4a indicates the input laser
power and the inset shows the transmittance across the optical resonance as
the laser power is varied, with high powers leading to a greater thermo-optic
shift. The transduced spectra for each power are acquired at a fixed coupling
fraction, approximately corresponding to a detuning $\Delta=-4$ GHz. Even at
relatively low input optical powers of $\sim$1.6 mW, high amplitude mechanical
lasing in the GHz is observed. In fact, depending on the driven optical mode,
different Fabry-Pérot acoustic modes (Figure 3d) can be made to lase. For the
particular optical and mechanical mode of Figure 4a, we extract an
optomechanical coupling strength, $g_{0}/2\pi$, of 135 kHz from a linear fit
to the evolution of the mechanical linewidth with input optical power [45]
(see appendix C). This assumes that optomechanical damping is the only source
of linewidth narrowing, a hypothesis that is confirmed by calibration of the
$g_{0}/2\pi$ using an alternative method relying on the use of an electro-
optic phase modulator ($\phi$-EOM) (see appendix F), the results of which
further confirm the localized nature of the optical modes in this platform.

##  V Conclusion

In summary, we present a 2D suspended silicon phononic crystal waveguide
sustaining a single ($z$-symmetric) acoustic band near 7 GHz that sits within
a 2.3 GHz-wide mechanical band gap, which we confirm using BLS spectroscopy.
To enable the generation of coherent acoustic phonons propagating along this
waveguide mode, we transform the structure into an efficient optomechanical
waveguide by mirroring the PnC and leaving an air-slot in the middle between
them. This results in a slot photonic-crystal waveguide supporting a slot-
guided optical mode at telecom wavelengths. We then exploit localized optical
modes spontaneously formed due to sidewall roughness to couple to the original
phononic waveguides on both sides of the slot. The optomechanical coupling
strengths achieved are enough to enable coherent phonon generation at
relatively low input powers directly within the waveguide. This is beneficial
compared to previously employed optomechanical generation schemes that use a
carefully engineered optomechanical cavity at the edge of the phononic
waveguide of interest. This concept is schematically illustrated in Figure 4b
for the structure presented here. In this case, the generation bandwidth is
not limited to the linewidth of a single mechanical cavity mode but to
multiple mechanical modes which can be selectively driven using specific
localized optical modes and then transferred into a perfectly mode-matched
semi-infinite waveguide.

A peculiarity of the design presented in this work stems from the utilization
of two mechanical waveguides separated by an air-slot. In this case, the
system can be visualized as two individual plates that are separated by an air
gap. The separation of these plates means that they are mechanically-
uncoupled, but sustain mechanical modes that both couple to the same localized
optical mode. Such a mechanical-optical-mechanical modal configuration has
previously been explored in a number of platforms [46, 47, 48] including
zipper nanobeam structures that also exploit field enhancement effects in air
slots [49], although the design here extends this deeper into the GHz regime.
The physical properties that are specifically encountered in multimode
optomechanical systems lead to a wide variety of interesting phenomena such as
synchronization [50], exceptional points [51, 52], or non-reciprocital states
[53, 54], which each may serve to enrich the breadth of integrated phononics.

###### Acknowledgements.

We acknowledge the support from the project LEIT funded by the European
Research Council, H2020 Grant Agreement No. 885689. ICN2 is supported by the
Severo Ochoa program from the Spanish Research Agency (AEI, grant no.
SEV-2017-0706) and by the CERCA Programme / Generalitat de Catalunya. R.C.N.
acknowledges funding from the EU-H2020 research and innovation programme under
the Marie Sklodowska Curie Individual Fellowship (Grant No. 897148). O.F. is
supported by a BIST PhD fellowship under the Marie Sklodowska Curie grant
agreement (No. 754558). M.A. and S.S. gratefully acknowledge funding from the
Villum Foundation Young Investigator Program (Grant No. 13170), the Danish
National Research Foundation (Grant No. DNRF147 - NanoPhoton), Innovation Fund
Denmark (Grant No. 0175-00022 - NEXUS), and Independent Research Fund Denmark
(Grant No. 0135-00315 - VAFL).

##  Appendix A Photonic design

In addition to confining the mechanical modes to the length of the waveguide,
the shield region also confines the optical modes. The optical band structures
for the center and shield regions are calculated with FEM simulations (Figure
5a and Figure 5b, respectively). The guided mode within each region is
highlighted in red. The light cone is shaded in dark grey. The slot-confined
optical guided mode is not perfectly decoupled from the bulk bands, which
results in the relatively poor optical Q-factors for the Fabry-Pérot optical
modes. However, the guided modes found in the two regions do not overlap in
frequency, and thus they cannot couple to one another. The associated mode
profile for this mode for the center region is shown in Figure 5 (top),
showing that the field is confined in the slot. Note that the axes are rotated
by 90o for this mode profile.

FIG. 5: Photonic crystal design. Photonic band structures of (a) the center
region (left) and (b) the shield region (right), with the slot-confined guided
mode highlighted in red, and the light cones shaded in dark grey. The
simulated electromagnetic energy of the guided mode, which shows that it is
confined inside the slot, is shown above the band structures.

##  Appendix B Fabrication

The sample is fabricated in chips cleaved from a commercial 12-inch silicon-
on-insulator (SOI) wafer, which has a nominal 220 nm thick device layer and a
2 $\mu$m buried oxide. The lithography and release etch of the suspended
membrane is carried out as detailed in [55]. To reduce the sidewall roughness,
we improve the mask selectivity by using a multi-layered hard mask, which we
etch sequentially in the same reactor using modified versions of the CORE-
process [56, 55, 57]. In this process, oxygen is used to passivate the etch in
the oxidation-step (O-step), the passivation at the bottom of all features is
removed in a low-pressure SF6 plasma in the remove-step (R-step), and the open
features are etched isotopically in the etch-step (E-step).

We first deposit 30 nm of poly-crystalline chromium followed by 30 nm of poly-
crystalline silicon on the SOI chip without breaking vacuum to avoid oxidation
of the chromium layer. We spin-coat $\sim 50$ nm of chemically semi-amplified
resist (AR-P CSAR6200.09 diluted 1:1 in anisole) and define the patterns with
a 100 keV 100 MHz JEOL-9500FSZ electron-beam writer with a current of 195 pA
and a dose density of 3 aC/nm2 in the center. The pattern is etched into the
poly-crystalline silicon hard mask from the CSAR soft mask using 8 cycles of a
modified CORE process [55]. In this modified process, the time of the E-step
is reduced from 73 s $\rightarrow$ 20 s to reduce sidewall roughness and the
passivation is reduced accordingly by decreasing the time of the O-step from 3
s $\rightarrow$ 2 s. Following this, the soft mask is removed in a 30 W oxygen
plasma at 50 mTorr for 5 min. The gas flows and pressure here are the same as
the O-step in the CORE-process. The pattern is then etched into the poly-
crystalline chromium hard mask from the poly-crystalline silicon hard mask in
65 cycles of the unmodified chromium-etching CORE-process [57], which is an
oxygen-rich plasma at 50 mTorr and 40 W, with 3 sccm of SF6 flow for 1 s every
10 s. Next, the crystalline silicon device-layer (100) is etched from the
chromium hard mask in 25 cycles of another modified CORE-process. For this
modified process, the time of the E-step is reduced from 73 s $\rightarrow$ 20
s to reduce sidewall roughness similar to in the poly-crystalline silicon
etch. Since this silicon layer is etched from a hard mask rather than a soft
mask, this is compensated by increasing the platen power of the R-step from 10
W $\rightarrow$ 18 W, while keeping the passivation O-step at 3 s. This
improves yield but was not possible when etching from a soft mask as the
oxygen plasma of the O-step slowly etches the resist laterally (mask
retraction) causing a change in feature size and therefore should be reduced
as much as possible. Finally, the remaining chromium mask is stripped by
submerging the chip for 10 min in commercial solution, Chrome Etch 18, which
consists of nitric acid (HNO3), perchloric acid (HClO4), and ceric salts
((NH4)2Ce(NO3)6), followed by a 2 min dip in a Piranha solution (4:1 mix of
H2SO4:H2O2). The structures are released by a 3.8 $\upmu$m isotropic underetch
of the buried oxide in anhydrous hydrofluoric acid (99.995 %) over 25 min in a
temperature- and pressure-controlled process [55] using an SPTS uEtch vapor
process.

##  Appendix C Determination of $g_{0}$ by fitting the mechanical linewidth

FIG. 6: Optical Q-factor determination. Localized optical mode transmission
response (blue dots) measured with input power $<0.2$ mW. The dip is fitted
with a Lorentzian function (red line) enabling the extraction of
$\lambda_{c}=1482.510$ nm, $\kappa_{i}/2\pi=4.0$ GHz, and
$\kappa_{e}/2\pi=7.0$ GHz.

We also estimate the optomechanical coupling strength, $g_{0}$, by fitting the
mechanical linewidth [35, 45]. The effective mechanical linewidth is given by:

$\begin{split}\Gamma_{\mathrm{eff}}=\Gamma_{\mathrm{m}}+g_{0}^{2}|\overline{a}|^{2}&\Big{(}\frac{\kappa_{t}}{(\Delta+\Omega_{\mathrm{m}})^{2}+\kappa_{t}^{2}/4}\\\
&-\frac{\kappa_{t}}{(\Delta-\Omega_{\mathrm{m}})^{2}+\kappa_{t}^{2}/4}\Big{)}\end{split}$
(1)

where $\Omega_{\mathrm{m}}/2\pi$ and $\Gamma_{\mathrm{m}}/2\pi$ are the
mechanical resonance frequency and natural linewidth, respectively.
$\Delta=\omega_{L}-\omega_{c}$ is the optical detuning, $\kappa_{t}$ is the
cavity total optical linewidth, and $|\overline{a}|^{2}$ is the intracavity
photon number given by

$|\overline{a}|^{2}=\frac{\kappa_{e}}{\Delta^{2}+\kappa_{t}^{2}/4}\frac{P_{\mathrm{in}}}{\hbar\omega_{L}}$

where $\kappa_{e}$ represents the cavity extrinsic loss rate. From this
equation, the mechanical linewidth $\Gamma_{\mathrm{eff}}$ is directly
proportional to the incident laser power $P_{\mathrm{in}}$. The optomechanical
coupling $g_{0}$ can be extracted from this fit by knowledge of all other
parameters provided that they are kept constant.

We first calibrate the optical parameters by fitting the transmission response
with the transmission function
$T(\Delta)=(\Delta^{2}+\kappa_{i}^{2})/(\Delta^{2}+\kappa_{t}^{2})$ where the
internal decay rate $\kappa_{i}$, the external decay rate $\kappa_{e}$, and
the cavity resonance frequency $\omega_{c}$ are used as fitting parameters.
The total decay rate of the cavity is: $\kappa_{t}=\kappa_{i}+\kappa_{e}$. The
measurement (Figure 6) is performed with input laser power
$P_{\mathrm{in}}<0.2$ mW, which is sufficiently low to remove the thermo-
optical nonlinearities. We obtain $\omega_{c}/2\pi=202.36$ THz (i.e.
$\lambda_{c}=1482.510$ nm), $\kappa_{i}/2\pi=4.0$ GHz, and
$\kappa_{e}/2\pi=7.0$ GHz. We deduce the localized optical mode loaded
Q-factor, $Q_{\mathrm{opt}}=18,400$.

FIG. 7: Vacuum optomechanical coupling calibration. a) Mechanical peak power
(blue dots) plotted with the detection noise floor (black line). b) Effective
mechanical linewidth plotted as a function of the input laser power. The data
is extracted from Figure 3a. An optomechanical coupling rate $g_{0}/2\pi=135$
kHz is obtained from the linear fit of the mechanical linewidth (red line).

The same measurement is reproduced for increasing input power, as shown in the
inset of Figure 4a. Higher input power leads to a significant red-shift of the
cavity resonance, which exhibits hysteretic behavior typical of the thermo-
optical nonlinearities encountered in silicon photonic devices [58, 59]. We
assume that Appendix C remains valid despite this deviation from the linear
optical cavity scenario and lock the detuning parameter by always setting the
laser wavelength such that $T(\Delta)=0.5$, which corresponds to
$\Delta\approx-4$ GHz in the linear regime. The RF transmission spectrum is
recorded as shown in Figure 4a. The mechanical resonance is fitted with a
Lorentzian lineshape such that the mechanical peak amplitude
$S_{p}(\Omega_{\mathrm{m}})$ and linewidth $\Gamma_{\mathrm{eff}}$ can be
plotted as a function of the input laser power (see Figure 7a and Figure 7b,
respectively). The detection noise floor is also shown in Figure 7a for
reference. An exponential amplification of the mechanical mode is first
observed, followed by a threshold around 1.58 mW. Above this threshold, the
mode lies in the lasing regime. The mechanical linewidth is fitted below this
threshold with a red line in Figure 7b. We deduce the vacuum optomechanical
coupling rate $g_{0}/2\pi=135$ kHz from the slope and the natural mechanical
linewidth $\Gamma_{\mathrm{m}}/2\pi=5.6$ MHz.

##  Appendix D Phononic crystal characterization with Brillouin light
scattering spectroscopy.

FIG. 8: PnC characterization. FEM simulation of the mechanical band diagram of
the center (a) and the shield (b) domains. Symmetrical (antisymmetrical) modes
are shown in blue (red). The band gap at the K point is highlighted in yellow.
Experimental characterization of the center (c) and shield (d) domains using
BLS spectroscopy performed at the K point.

We compute the band diagram for a PnC for both the center and the shield
domains using FEM simulations, shown in Figure 8a and Figure 8b, respectively.
These simulations do not consider a waveguide, i.e., there is no guided mode.
In both cases, we observe a full mechanical band gap. The band gap of the
center region is centered at 6.2 GHz with a width of 2.3 GHz while that of the
shield region is centered at 6.1 GHz with a width of 1.5 GHz. The band gap is
highlighted in yellow.

We are able to characterize the thermal phonons in both domains by BLS
spectroscopy. The BLS laser is focused in the center domain, away from the
phononic waveguide. The measurement is performed at a specific angle,
$\alpha=20.1^{\circ}$, which corresponds to the wavevector lying at the K
point of the Brillouin zone (see dashed line in Figure 8a and b). The data
shown in Figure 8a was acquired in approximately 12 hours (91,000 photon
counts). The same measurement is performed by focusing the laser in the shield
region (see Figure 8b) with about 29,000 photon counts. This long acquisition
time is required due to the relatively weak Brillouin scattering signal
compared to elastic Rayleigh scattering. The band gap at the K point
(highlighted in yellow) in Figure 8a and b are also placed over the same
frequency range as their associated BLS measurements in Figure 8c and d and
show good agreement for both of the regions between simulation and
measurement.

FIG. 9: BLS waveguide characterization. Experimental measurements of the
guided mode (dark dots) and of the band gap edges (dark grey dots) compared
with the simulated symmetrical and antisymmetrical mechanical modes, shown
with blue and red lines, respectively. Error bars indicate the BLS measured
peak linewidths.

When the laser spot is placed at the center of the waveguide within the center
region of the crystal, the thermally excited mechanical mode that was
identified as the guided acoustic mode can be detected, as shown in fig. 3a.
Different measurements are taken at this position for various wavevectors by
changing the incident angle. For example, an incident angle of 20.1∘
corresponds to a wavevector of $k_{x}=0.7\pi/a$. In doing so, the dispersion
relation can be experimentally mapped (Figure 9, and allow for a direct
comparison of the experimental measurements (points) with the FEM simulations
(lines). The grey and black points correspond to the band gap edges and the
guided mode, respectively. A close quantitative and qualitative correspondence
between the observed phononic elastic dispersion relation and the associated
simulation is observed.

##  Appendix E Cavity length

FIG. 10: Cavity length dependence of optical and mechanical spectra. The
optical transmission spectrum (top) and associated mechanical spectrum
(bottom) for three different cavity lengths of a) L=25 $\upmu\textrm{m}$, b)
L=75 $\upmu\textrm{m}$, and c) L=100 $\upmu\textrm{m}$.

The cavity length plays a crucial role in the spectral distributions of both
photonic and phononic modes. We measure the optical transmission (Figure 10a)
for three different structures of varying length ($L$ = 25 $\mu$m, $L$ = 75
$\mu$m, and $L$ = 100 $\mu$m), but with the same geometric parameters as
specified in Section II. There is a tendency towards a greater optical density
of states and a narrowing of the modes with increasing cavity length. Each of
the spectra are taken with a different input optical power of
$P_{\mathrm{in}}=5$ mW, $P_{\mathrm{in}}=4$ mW, and $P_{\mathrm{in}}=1.4$ mW
from shortest to longest cavity length. For each optical transmission
spectrum, the mechanical noise spectrum is also measured as a function of the
laser wavelength. The spectra are normalized by the frequency-dependent noise
floor of the ESA, leading to the SNR as indicated by the color scale (Figure
10b). Mechanical modes in the phononic band gap are transduced in the laser
field when an optical mode is driven. This transduction increases
significantly when driving localized optical modes, i.e. where the optical
band flattens at the Brillouin zone edge, as discussed in Section IV. Both the
optical and mechanical modes of the $L$ = 75 $\mu$m cavity are spectrally
shifted, with red-shifted optical modes and mechanical modes shifted to lower
frequency, relative to the spectra of the other two cavity lengths. This shift
has gradually increased over the time scale of months over which the sample
has undergone measurement, and likely results from slight modification of the
structure due to long illumination times under high powers in BLS
spectroscopy.

##  Appendix F Determination of $g_{0}$ with a phase electro-optic modulator

To further study the localized nature of the modes in this system, the
$g_{0}/2\pi$ was measured at different positions along the waveguide by
calibration with a phase electro-optic modulator (EOM) (Table 1). The
calibration of the optomechanical coupling using a phase EOM [60] is made in
the absence of dynamical back-action. The four different positions in Table 1
correspond with the three positions indicated in Figure 3b, along with an
additional one at $x_{\mathrm{\ell}}=12$ $\upmu\textrm{m}$. At each position,
a localized optically driven mode with wavelength, $\lambda_{L}$, excites the
mechanical mode at frequency, $\Omega_{\mathrm{m}}$, with a mechanical
dissipation rate, $\Gamma_{\mathrm{m}}$. The different optomechanical
couplings (i.e. varying $g_{0}/2\pi$) at various positions along the waveguide
at slightly different values of $\lambda_{L}$ but roughly the same value of
$\Omega_{\mathrm{m}}$ further suggests localized behavior (notably, excitation
of a mechanical mode driven by a localized optical mode). Furthermore, these
results confirm the previous calibration of $g_{0}$ (appendix C).

$x_{\mathrm{\ell}}$ ($\upmu\textrm{m}$) |  $g_{0}/2\pi$ (kHz) |  $\lambda_{\textrm{L}}$ (nm) |  $\Omega_{\textrm{m}}$ (GHz) |  $\Gamma_{\textrm{m}}$ (MHz)
---|---|---|---|---
12 | 72 $\pm$ 4 | 1484.14 | 6.845 | 4.8
19 | 104 $\pm$ 6 | 1483.47 | 6.849 | 10
35 | 169 $\pm$ 10 | 1484.05 | 6.846 | 2.5
60 | 124 $\pm$ 8 | 1485.85 | 6.846 | 13.4
Table 1: Phase electro-optic modulator measured $g_{0}/2\pi$ at the four
different positions along the waveguide

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#  Feb 2023 Schur-like index of the Klebanov-Witten theory via the AdS/CFT
correspondence

Shota Fujiwara111E-mail<EMAIL_ADDRESS>

Mandelstam Institute for Theoretical Physics,
University of Witwatersrand, Johannesburg 2050, South Africa

We calculate an unrefined limit of the superconformal index (the Schur-like
index) of the Klebanov-Witten theory from the dual AdS side by using the
recently developed giant graviton expansion method. Our formula includes
multiple sums of the contribution of the D3-branes wrapped on three-cycles in
the internal space. We numerically confirm the validity of our formula by
comparing indices obtained by the formula with the gauge theory results.

##  1 Introduction

Ever since the discovery of the AdS/CFT correspondence [1], the duality has
been providing us with new ways to analyze the superconformal field theories.
Among them the superconformal index is a very efficient tool since the
superconformal index is independent of the coupling. Actually, a lot of work
has been done for the superconformal index in relation to the AdS/CFT
correspondence. For example, at the large $N$ limit the superconformal index
of the four-dimensional $\mathop{\mathcal{N}}\nolimits=4$ $U(N)$ SYM theories
were reproduced from type IIB supergravity on $\mathop{\mathrm{AdS_{5}\times
S^{5}}}\nolimits$ [2].

Recently, great progress has been achieved toward the calculation of the index
at the finite $N$ region by using the method called the giant graviton
expansion [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. See also [15, 16, 17, 18]
for references about the giant graviton itself. For the AdS${}_{5}\times
S^{5}$ case, even at the finite $N$ the index was reproduced by including the
contribution of D3-branes wrapping on three-cycles in $S^{5}$ [3, 8]. More
precisely, the authors of [3] study the single-wrapping D3-brane contribution
to the index, and the multiple-wrapping D3-branes contribution was also
considered in [8]. The author numerically checked the following giant graviton
expansion formula for small $N$:

$\displaystyle\frac{\mathcal{I}^{\mathrm{SYM}}_{N}}{\mathcal{I}_{\mathrm{KK}}^{S^{5}}}=\sum_{n_{1},n_{2},n_{3}=0}^{\infty}\mathcal{I}_{(n_{1},n_{2},n_{3})}.$
(1)

In this equation $\mathcal{I}^{\mathrm{SYM}}_{N}$ denotes the superconformal
index of the $\mathop{\mathcal{N}}\nolimits=4$ $U(N)$ SYM theory and
$\mathcal{I}_{\mathrm{KK}}^{S^{5}}$ is the Kaluza-Klein index of the dual
AdS${}_{5}\times S^{5}$ supergravity. Also, $n_{i}$ ($i=1,2,3$) denotes the
wrapping number of D3-branes for each three-cycle and
$\mathcal{I}_{(n_{1},n_{2},n_{3})}$ is a superconformal index of the D3-brane
system with the wrapping numbers $(n_{1},n_{2},n_{3})$. Note that this formula
contains multiple sums of the D3-brane configurations, but the authors of [10]
show a similar formula with only a single sum. The relation of these two
formulae was studied in [13].

From the success of the AdS${}_{5}\times S^{5}$ case, it is natural to expect
that similar relations hold for other examples of the AdS/CFT correspondence.
When $S^{5}$ is replaced by a five-dimensional Sasaki-Einstein manifold
($\mathrm{SE}_{5}$), the corresponding gauge theory becomes a toric quiver
gauge theory which is realized on D3-branes placed on the Calabi-Yau three-
folds. The superconformal indices for these toric cases at the large $N$ limit
were studied in [19, 20]. In addition, finite $N$ corrections from the single-
wrapping D3-brane were studied in [5]. However, the multiple-wrapping
D3-branes contribution was not considered in that paper, and it is a remaining
problem. If we include the multiple-wrapping D3-branes the natural extension
of the formula (1) for the toric cases becomes

$\displaystyle\frac{\mathcal{I}^{\mathrm{toric}}_{N}}{\mathcal{I}_{\mathrm{KK}}^{\mathrm{SE}_{5}}}=\sum_{n_{1},n_{2},...,n_{d}=0}^{\infty}\mathcal{I}_{(n_{1},n_{2},...,n_{d})},$
(2)

where $d$ is the number of vertices of the toric diagram.

In this paper, as a simple example of a $\mathrm{SE}_{5}$, we especially study
the $\mathrm{SE}_{5}=T^{1,1}$ case. The dual 4d
$\mathop{\mathcal{N}}\nolimits=1$ $SU(N)_{1}\times SU(N)_{2}$ superconformal
field theory is known as the Klebanov-Witten theory [21]. However, as
discussed in [8] there are some difficulties regarding how we determine the
contours of the gauge integral in $\mathcal{I}_{(n_{1},n_{2},...,n_{d})}$. To
avoid this problem we instead consider an unrefined limit of the
superconformal index like the Schur limit of four-dimensional
$\mathop{\mathcal{N}}\nolimits=2$ theories [22]. (See Subsection 3.1 for our
definition of the limit.) If we take such a limit sometimes a nice
simplification occurs and the analysis becomes easier. For example, giant
graviton expansions with such Schur (like) limits were studied in the
literature for the $N=4$ $U(N)$SYM theory in [6] and for the 6d
$\mathop{\mathcal{N}}\nolimits=(2,0)$ theory in[7]. Then the purpose of this
paper is to confirm the formula (2) for the unrefined index of the Klebanov-
Witten theory.

##  2 Index for the Klebanov-Witten theory

Let us explain the definition of the superconformal index for the Klebanov-
Witten theory. Our convention follows from [5] and we will summarize shortly
here. For a detailed explanation of our convention see Section 2 and
Subsection 4.1 of [5].

The Klebanov-Witten theory is a 4d $\mathop{\mathcal{N}}\nolimits=1$
$SU(N)_{1}\times SU(N)_{2}$ gauge theory described by the quiver diagram shown
in Figure 1.

Figure 1: The quiver diagram of the Klebanov-Witten theory and the toric
diagram of $T^{1,1}$ are shown.

The dual gravity theory is type IIB superstring theory on AdS${}_{5}\times
T^{1,1}$. The toric diagram of $T^{1,1}$ is also shown in Figure 1. We denote
a supersymmetric three-cycle corresponding to a vertex of the toric diagram as
$S_{I}$ ($I=1\sim 4$).

There are four independent charges $R_{I}$ ($I=1\sim 4$) corresponding to the
four vertices of the toric diagram. We can determine the charge assignments of
the fields by using the perfect matching [23, 24, 25]. Our charge assignments
are summarized in Table 1.

Table 1: The charge assignments for the Klebanov-Witten theory is shown.
$\displaystyle\begin{array}[]{c|l|l|l|l|l|l|l|l}\hline\cr&R_{1}&R_{2}&R_{3}&R_{4}&r_{*}&F_{A}&F_{B}&NB\\\
\hline\cr A_{1}&1&0&0&0&1/2&1&0&1\\\ A_{2}&0&1&0&0&1/2&0&1&-1\\\
B_{1}&0&0&1&0&1/2&-1&0&1\\\ B_{2}&0&0&0&1&1/2&0&-1&-1\\\ \hline\cr\end{array}$

Note that on the gravity side $R_{I}$ corresponds to the angular momentum and
a D3-brane wrapped on $S_{I}$ has $R_{I}=N$ and otherwise 0. Then we define
the superconformal index of the Klebanov-Witten theory by

$\displaystyle\mathcal{I}^{\mathrm{KW}}=\mathop{\rm
tr}\nolimits\left[(-1)^{F}q^{3\overline{J}}y^{2J}v_{1}^{R_{1}}v_{2}^{R_{2}}v_{3}^{R_{3}}v_{4}^{R_{4}}\right],\quad
v_{1}v_{2}v_{3}v_{4}=q^{3}.$ (3)

It is also useful to use another basis for the charges. As global symmetries,
the Klebanov-Witten theory has an R-symmetry, two $SU(2)$ mesonic symmetries,
and also one baryonic symmetry. We denote the superconformal $R$-charge
obtained by the a-maximization [26] as $r_{*}$ and the Caltan generator of two
$SU(2)$ as $F_{A}$ and $F_{B}$, and the baryonic charge as $B$. These
generators can be expressed as a linear combination of $R_{I}$ and we take as
follows:

$\displaystyle r_{*}=\frac{1}{2}(R_{1}+R_{2}+R_{3}+R_{4}),$ (4) $\displaystyle
F_{A}=R_{1}-R_{3},\qquad F_{B}=R_{2}-R_{4},$ (5) $\displaystyle
B=\frac{1}{N}(R_{1}-R_{2}+R_{3}-R_{4}).$ (6)

We introduce new fugacities $u,v,\zeta$ corresponding to these generators. In
terms of these fugacities the index is written by

$\displaystyle\mathcal{I}^{\mathrm{KW}}=\mathop{\rm
tr}\nolimits\left[(-1)^{F}q^{\frac{3}{2}r_{*}+3\overline{J}}y^{2J}u^{F_{A}}v^{F_{B}}\zeta^{B}\right],$
(7)

where fugacities in this equation and the previous ones in (3) are related by

$\displaystyle v_{1}=q^{\frac{3}{4}}u\zeta^{\frac{1}{N}},\qquad
v_{2}=\frac{q^{\frac{3}{4}}v}{\zeta^{\frac{1}{N}}},\qquad
v_{3}=\frac{q^{\frac{3}{4}}\zeta^{\frac{1}{N}}}{u},\qquad
v_{4}=\frac{q^{\frac{3}{4}}}{v\zeta^{\frac{1}{N}}}.$ (8)

We will use this expression of the index in numerical calculation.

Notice that the baryonic charge is related to the wrapping number of D3-branes
on the gravity side [27]. Let $n_{I}$ ($I=1\sim 4$) be the wrapping number of
D3-branes for each cycle $S_{I}$. We can easily find a D3-brane configuration
with the wrapping numbers $(n_{1},n_{2},n_{3},n_{4})$ has

$\displaystyle B=n_{1}-n_{2}+n_{3}-n_{4}.$ (9)

##  3 Giant graviton expansion for the Klebanov-Witten theory

Let us discuss the giant graviton expansion formula for the Klebanov-Witten
theory. The formula (2) now becomes

$\displaystyle\frac{\mathcal{I}^{\mathrm{KW}}_{N}}{\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}}=\sum_{n_{1},n_{2},n_{3},n_{4}=0}^{\infty}\mathcal{I}_{(n_{1},n_{2},n_{3},n_{4})},$
(10)

where $\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}$ is the Kaluza-Klein contribution
of supergravity on AdS${}_{5}\times T^{1,1}$ [28] and it is given by

$\displaystyle\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}=\mathop{\rm
Pexp}\nolimits[i_{\mathrm{KK}}^{T^{1,1}}].$ (11)

$\mathop{\rm Pexp}\nolimits$ is the plethystic exponential given by

$\displaystyle\mathop{\rm
Pexp}\nolimits[f(x_{i})]=\mathrm{exp}\left[\sum_{n=1}^{\infty}\frac{1}{n}f(x_{i}^{n})\right],$
(12)

and $i_{\mathrm{KK}}^{T^{1,1}}$ is the single-particle index of the Kaluza-
Klein modes on $T^{1,1}$. It is given by the following sum [19, 20]:

$\displaystyle
i_{\mathrm{KK}}^{T^{1,1}}=\sum_{I=1}^{4}\frac{w_{I+\frac{1}{2}}}{1-w_{I+\frac{1}{2}}},$
(13)

where $w_{I+\frac{1}{2}}$ $(I=1\sim 4)$ are fugacities associated with the
edge between the vertices $I$ and $I+1$ of the toric diagram 222Here we treat
$I$ as a cyclic variable $I\sim I+4$; for example,
$w_{4+\frac{1}{2}}=w_{\frac{1}{2}}$. and explicitly given by [5]

$\displaystyle w_{\frac{1}{2}}=v_{2}v_{3},\qquad
w_{1+\frac{1}{2}}=v_{3}v_{4},\qquad w_{2+\frac{1}{2}}=v_{4}v_{1},\qquad
w_{3+\frac{1}{2}}=v_{1}v_{2}.$ (14)

On the right hand side of the the formula,
$\mathcal{I}_{(n_{1},n_{2},n_{3},n_{4})}$ is the contribution of the D3-branes
with wrapping numbers $(n_{1},n_{2},n_{3},n_{4})$:

$\displaystyle\mathcal{I}_{(n_{1},n_{2},n_{3},n_{4})}=v_{1}^{n_{1}}v_{2}^{n_{2}}v_{3}^{n_{3}}v_{4}^{n_{4}}\int
d\mu_{1}d\mu_{2}d\mu_{3}d\mu_{4}\mathop{\rm Pexp}\nolimits[i_{\mathrm{all}}].$
(15)

$d\mu_{I}$ is the Haar measure of $U(n_{I})$ gauge group. $i_{\mathrm{all}}$
is the single-particle index of the quiver theory on the D3-brane system given
by

$\displaystyle
i_{\mathrm{all}}=\sum_{I=1}^{4}\left(i^{v}_{I}\chi_{U(n_{I})}^{\mathrm{adj}}+i^{h}_{I,I+1}(\chi_{U(n_{I})}^{\mathrm{fund}}\chi_{U(n_{I+1})}^{\overline{\mathrm{fund}}}+\chi_{U(n_{I})}^{\overline{\mathrm{fund}}}\chi_{U(n_{I+1})}^{\mathrm{fund}})\right),$
(16)

where $\chi_{U(n_{I})}^{\mathrm{adj}}$ is an adjoint character for the
$U(n_{I})$ and
$\chi_{U(n_{I})}^{\mathrm{fund}}(\chi_{U(n_{I})}^{\overline{\mathrm{fund}}})$
is a (anti-) fundamental charater for the $U(n_{I})$. $i^{v}_{I}$ and
$i^{h}_{I,I+1}$ each denote the single-particle indices for the vector
multiplet and the hyper multiplet. They are explicitly given by [5, 8] 333In
[8], for the $S^{5}$ case the single-particle index for the hypermultiplet was
directly derived by analyzing the modes on D3-branes. It is given by setting
$w_{I+\frac{1}{2}}=qu_{I+2}$ in the definition of $i^{h}_{I,I+1}$ ($I=1,2,3$)
in (18). For the $T^{1,1}$ case we can obtain $i^{h}_{I,I+1}$ by simply
replacing $w_{I+\frac{1}{2}}$ with the one for $T^{1,1}$ as discussed in [5].

$\displaystyle i^{v}_{I}$
$\displaystyle=1-\frac{(1-q^{-3}w_{I-\frac{1}{2}}w_{I+\frac{1}{2}})(1-q^{\frac{3}{2}}y)(1-q^{\frac{3}{2}}y^{-1})}{(1-w_{I-\frac{1}{2}})(1-w_{I+\frac{1}{2}})},$
(17) $\displaystyle i^{h}_{I,I+1}$
$\displaystyle=\frac{w_{I+\frac{1}{2}}^{\frac{1}{2}}}{1-w_{I+\frac{1}{2}}}\frac{(1-q^{\frac{3}{2}}y)(1-q^{\frac{3}{2}}y^{-1})}{q^{\frac{3}{2}}}.$
(18)

Also, for later convenience we denote the integrand in (15) as
$F_{n_{1},n_{2},n_{3},n_{4}}$:

$\displaystyle F_{n_{1},n_{2},n_{3},n_{4}}(q,y,u,v)=\int
d\mu_{1}d\mu_{2}d\mu_{3}d\mu_{4}\mathop{\rm Pexp}\nolimits[i_{\mathrm{all}}].$
(19)

###  3.1 Schur-like limit

As discussed in [8], the determination of the contour in the gauge integral
(19) requires special care. To avoid the difficulty, in this paper we instead
consider an unrefined limit of the superconformal index which we call the
Schur-like limit. 444The Klebanov-Witten theory is related to
$\mathrm{S}^{5}/\mathbb{Z}_{2}$ model by mass deformation and this limit is
related to the Schur limit of the index of that model. Therefore we call this
limit the “Schur-like” limit even though this theory has only
$\mathop{\mathcal{N}}\nolimits=1$ symmetry.  Namely, we take the following
specialization in the definition of the index (7):

$\displaystyle v\rightarrow u,\qquad y\rightarrow 1,$ (20)

In this limit the single-particle indices $i^{h}_{2,3}$ and $i^{h}_{4,1}$
become simple form

$\displaystyle i^{h}_{2,3}=i^{h}_{4,1}=q^{-\frac{3}{4}}-q^{\frac{3}{4}}.$ (21)

Then $F_{n_{1},n_{2},n_{3},n_{4}}$ is factorized to

$\displaystyle
F_{n_{1},n_{2},n_{3},n_{4}}(q,u)=F_{n_{1},n_{2}}(q,u)F_{n_{3},n_{4}}(q,u^{-1})q^{\frac{3}{2}(n_{2}n_{3}+n_{4}n_{1})},$
(22)

where $F_{n_{1},n_{2}}(q,u)$ denotes the contribution of D3-branes with
wrapping numbers $(n_{1},n_{2},0,0)$ and given by
$F_{n_{1},n_{2},0,0}(q,1,u,u)$ in (19). Also, $F_{n_{3},n_{4}}(q,u^{-1})$
corresponds to the contribution of $(0,0,n_{3},n_{4})$ configuration. Moreover
$F_{n_{1},n_{2}}(q,u)$ is invariant under the exchange of the subscripts

$\displaystyle F_{n_{1},n_{2}}(q,u)=F_{n_{2},n_{1}}(q,u).$ (23)

We calculated $F_{n_{1},n_{2}}(q,u)$ up to $n_{1}+n_{2}\leq 4$ and here we
show explicit results up to a certain order of $q$ which is enough to
reproduce all results in Section 4.

$\displaystyle F_{0,0}(q,u)$ $\displaystyle=1$ $\displaystyle F_{1,0}(q,u)$
$\displaystyle=\frac{u^{2}}{u^{2}-1}+(1-\frac{1}{u^{2}})q^{\frac{3}{2}}+(1-\frac{1}{u^{6}})q^{3}+(-\frac{1}{u^{4}}+\frac{1}{u^{6}}-\frac{1}{u^{10}}+1)q^{\frac{9}{2}}$
$\displaystyle+(1-\frac{1}{u^{14}})q^{6}+(-\frac{1}{u^{6}}+\frac{1}{u^{12}}-\frac{1}{u^{18}}+1)q^{\frac{15}{2}}+(\frac{1}{u^{10}}-\frac{1}{u^{12}}-\frac{1}{u^{22}}+1)q^{9}$
$\displaystyle+(-\frac{1}{u^{8}}+\frac{1}{u^{18}}-\frac{1}{u^{26}}+1)q^{\frac{21}{2}}+\cdots,$
$\displaystyle F_{2,0}(q,u)$
$\displaystyle=\frac{u^{6}}{(u^{2}-1)^{2}(u^{2}+1)}+q^{\frac{3}{2}}+(\frac{1}{u^{6}}+2)q^{3}+(\frac{1}{u^{4}}+\frac{1}{u^{12}}+2)q^{\frac{9}{2}}$
$\displaystyle+(\frac{3}{u^{8}}+\frac{1}{u^{18}}+3)q^{6}+(\frac{2}{u^{12}}+\frac{1}{u^{24}}+3)q^{\frac{15}{2}}+(\frac{1}{u^{6}}+\frac{2}{u^{10}}+\frac{4}{u^{16}}+\frac{1}{u^{30}}+4)q^{9}$
$\displaystyle+(\frac{3}{u^{20}}+\frac{1}{u^{36}}+4)q^{\frac{21}{2}}+\cdots,$
$\displaystyle F_{1,1}(q,u)$
$\displaystyle=-\frac{u^{6}(u^{2}-3)q^{\frac{3}{2}}}{(u^{2}-1)^{2}(u^{2}+1)}+(3-u^{4})q^{3}+(-u^{6}+\frac{3}{u^{6}}+4)q^{\frac{9}{2}}$
$\displaystyle+(-u^{8}+\frac{1}{u^{4}}+\frac{3}{u^{12}}+5)q^{6}+(-u^{10}+\frac{6}{u^{8}}+\frac{3}{u^{18}}+6)q^{\frac{15}{2}}$
$\displaystyle+(-u^{12}+\frac{3}{u^{12}}+\frac{3}{u^{24}}+7)q^{9}+(-u^{14}+\frac{1}{u^{6}}+\frac{5}{u^{10}}+\frac{8}{u^{16}}+\frac{3}{u^{30}}+8)q^{\frac{21}{2}}+\cdots,$
$\displaystyle F_{3,0}(q,u)$
$\displaystyle=\frac{u^{12}}{(u^{2}-1)^{3}(u^{6}+2u^{4}+2u^{2}+1)}+\frac{u^{4}q^{\frac{3}{2}}}{u^{4}-1}+\frac{(2u^{8}+u^{6}-u^{4}+1)q^{3}}{u^{4}(u^{4}-1)}$
$\displaystyle+\frac{(3u^{14}-3u^{12}+u^{10}+u^{8}-u^{2}+1)q^{\frac{9}{2}}}{u^{12}(u^{2}-1)}$
$\displaystyle+\frac{(4u^{24}+u^{22}-2u^{20}+u^{18}+2u^{12}+u^{10}-u^{4}+1)q^{6}}{u^{20}(u^{4}-1)}$
$\displaystyle+\frac{(5u^{32}-4u^{28}+2u^{26}+u^{24}+u^{20}-2u^{18}+2u^{14}+u^{12}-u^{4}+1)q^{\frac{15}{2}}}{u^{28}(u^{4}-1)}+\cdots,$
$\displaystyle F_{2,1}(q,u)$
$\displaystyle=\frac{u^{12}(u^{6}-2u^{4}-2u^{2}+6)q^{3}}{(u^{2}-1)^{3}(u^{2}+1)(u^{4}+u^{2}+1)}+\frac{u^{4}(u^{10}-2u^{6}-2u^{4}+6)q^{\frac{9}{2}}}{u^{4}-1}$
$\displaystyle+\frac{(u^{22}-u^{18}-u^{16}-2u^{14}+8u^{8}+4u^{6}-6u^{4}+6)q^{6}}{u^{4}(u^{4}-1)}$
$\displaystyle-\frac{(-u^{32}+u^{30}+u^{24}+u^{22}-u^{20}+u^{18}-14u^{14}+14u^{12}-4u^{10}-4u^{8}+6u^{2}-6)q^{\frac{15}{2}}}{u^{12}(u^{2}-1)}+\cdots,$
$\displaystyle F_{4,0}(q,u)$
$\displaystyle=\frac{u^{20}}{(u^{2}-1)^{4}(u^{2}+1)^{2}(u^{4}+1)(u^{4}+u^{2}+1)}+\cdots,$
$\displaystyle F_{3,1}(q,u)$
$\displaystyle=-\frac{u^{20}(u^{12}-2u^{10}-u^{8}+u^{6}+4u^{4}+3u^{2}-10)q^{\frac{9}{2}}}{(u^{2}-1)^{4}(u^{2}+1)^{2}(u^{4}+1)(u^{4}+u^{2}+1)}+\cdots,$
$\displaystyle F_{2,2}(q,u)$
$\displaystyle=\frac{u^{20}(-3u^{12}+5u^{10}+4u^{8}-3u^{6}-9u^{4}-8u^{2}+20)q^{6}}{(u^{2}-1)^{4}(u^{2}+1)^{2}(u^{4}+1)(u^{4}+u^{2}+1)}+\cdots.$
(24)

##  4 Results and comparisons for $N=2$

In this section we calculate the D3-branes contribution to the indices and
compare them with the gauge theory results to confirm the formula (10). In
addition to the fugacity $q$ we expand the index in terms of $\zeta$,

$\displaystyle\mathcal{I^{\mathrm{KW}}}=\sum_{i\in\mathbb{Z}}\mathcal{I}^{\mathrm{KW}}_{B=i}\
\zeta^{i},$ (25)

and analyze each sector with each baryonic charge.

We show results for the $N=2$ with $B=0,1,2,3,4$ case. For the $B=0,2,4$
sectors we calculated the indices up to the leading contribution of quadruple-
wrapping D3-branes. For the $B=1,3$ sectors which don’t have the quadruple-
wrapping D3-branes contribution we calculated the indices up to $q^{12}$
terms. The gauge theory results are easily obtained by the localization method
and shown in appendix A.

###  4.1 $B=0$

We first analyze the $B=0$ sector. The corresponding D3-brane configurations
for this sector are shown in Table 2. We can easily calculate the D3-branes
contribution to the index by using the explicit results in (3.1).

Table 2: The D3-brane configurations for the $B=0$ sector are shown. A tuple
$(n_{1},n_{2},n_{3},n_{4})$ denotes a configuration of $n_{I}$ D3-branes
wrapped on each three-cycle $S_{I}$.
$\displaystyle\begin{array}[]{c|l|l|l|l}\hline\cr
B&\text{single}&\text{double}&\text{triple}&\text{quadruple}\\\
\hline\cr\hbox{\multirowsetup
0}&&(n_{1},n_{2},n_{3},n_{4})=(1,1,0,0),&&(2,2,0,0),(2,0,0,2),(0,2,2,0),\\\
&&(0,1,1,0),(0,0,1,1),(1,0,0,1)&&(0,0,2,2),(2,1,0,1),(1,2,1,0),\\\
&&&&(1,0,1,2),(0,1,2,1),(1,1,1,1)\\\ \hline\cr\end{array}$

By summing up the contributions of all double-wrapping configurations in Table
2, we get

$\displaystyle\mathcal{I}_{B=0}^{\mathrm{double}}$
$\displaystyle=\mathcal{I}_{(1,1,0,0)}+\mathcal{I}_{(0,1,1,0)}+\mathcal{I}_{(0,0,1,1)}+\mathcal{I}_{(1,0,0,1)}$
$\displaystyle=\left(-\chi_{2}+3\chi_{4}-\chi_{6}+3\right)q^{\frac{9}{2}}+\left(-5\chi_{2}+3\chi_{4}+\chi_{6}-\chi_{8}+2\right)q^{6}$
$\displaystyle+\left(-3\chi_{2}+4\chi_{4}-2\chi_{6}+\chi_{8}-\chi_{10}+5\right)q^{\frac{15}{2}}$
$\displaystyle+\left(-9\chi_{2}+7\chi_{4}-3\chi_{6}+3\chi_{8}-\chi_{10}-\chi_{12}+10\right)q^{9}$
$\displaystyle-2\left(4\chi_{2}-4\chi_{4}+\chi_{6}-2\chi_{8}+\chi_{10}+\chi_{12}-5\right)q^{\frac{21}{2}}$
$\displaystyle+\left(-7\chi_{2}+7\chi_{4}-5\chi_{6}+5\chi_{8}-4\chi_{10}+6\chi_{12}-\chi_{14}-\chi_{16}-5\chi_{18}+3\chi_{20}+6\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}),$
(26)

where $\chi_{n}$ is $SU(2)$ character defined in

$\displaystyle\chi_{n}=\frac{u^{n+1}-u^{-(n+1)}}{u-u^{-1}}.$ (27)

Also, the quadruple-wrapping D3-branes contribution is given by

$\displaystyle\mathcal{I}_{B=0}^{\mathrm{quadruple}}$
$\displaystyle=\mathcal{I}_{(2,2,0,0)}+\mathcal{I}_{(2,0,0,2)}+\mathcal{I}_{(0,2,2,0)}+\mathcal{I}_{(0,0,2,2)}+\mathcal{I}_{(2,1,0,1)}+\mathcal{I}_{(1,2,1,0)}+\mathcal{I}_{(1,0,1,2)}+\mathcal{I}_{(0,1,2,1)}+\mathcal{I}_{(1,1,1,1)}$
$\displaystyle=\left(-\chi_{2}+5\chi_{4}-8\chi_{6}+10\chi_{8}-6\chi_{12}-\chi_{14}+\chi_{16}+5\chi_{18}-3\chi_{20}+15\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}),$
(28)

Then we can find the contribution of D3-branes in the $B=0$ sector

$\displaystyle\mathcal{I}_{B=0}^{\mathrm{D}3}=1+\mathcal{I}_{B=0}^{\mathrm{double}}+\mathcal{I}_{B=0}^{\mathrm{quadruple}},$
(29)

perfectly agrees with the gauge theory result (A). 555The term “1” corresponds
to $\mathcal{I}_{(0,0,0,0)}$.

###  4.2 $B=1$

Next, we consider the $B=1$ sector. The D3-brane configurations in this sector
are summarized in Table 3.

Table 3: The D3-brane configurations for the $B=1$ sector are shown.
$\displaystyle\begin{array}[]{c|l|l|l|l}\hline\cr
B&\text{single}&\text{double}&\text{triple}&\text{quadruple}\\\
\hline\cr\hbox{\multirowsetup
1}&(1,0,0,0),(0,0,1,0)&&(2,1,0,0),(2,0,0,1),(0,1,2,0),&\\\
&&&(0,0,2,1),(1,1,1,0),(1,0,1,1)&\\\ \hline\cr\end{array}$

We can easily calculate the single-wrapping D3 contribution,

$\displaystyle\mathcal{I}_{B=1}^{\mathrm{single}}$
$\displaystyle=\mathcal{I}_{(1,0,0,0)}+\mathcal{I}_{(0,0,1,0)}$
$\displaystyle=\chi_{2}q^{\frac{3}{2}}+\left(\chi_{2}-3\right)q^{3}+\left(2\chi_{2}-\chi_{4}-1\right)q^{\frac{9}{2}}+\left(-\chi_{2}+\chi_{4}+\chi_{6}-\chi_{8}\right)q^{6}$
$\displaystyle+\left(\chi_{2}+\chi_{10}-\chi_{12}-1\right)q^{\frac{15}{2}}+\left(2\chi_{2}-\chi_{4}-\chi_{8}+\chi_{10}+\chi_{14}-\chi_{16}-1\right)q^{9}$
$\displaystyle+\left(\chi_{2}-\chi_{6}+2\chi_{8}-\chi_{10}+\chi_{18}-\chi_{20}-1\right)q^{\frac{21}{2}}$
$\displaystyle+\left(\chi_{2}+\chi_{4}-\chi_{6}-\chi_{14}+\chi_{16}+\chi_{22}-\chi_{24}-1\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}),$
(30)

and also the triple-wrapping D3s contribution is given by

$\displaystyle\mathcal{I}_{B=1}^{\mathrm{triple}}$
$\displaystyle=\mathcal{I}_{(2,1,0,0)}+\mathcal{I}_{(2,0,0,1)}+\mathcal{I}_{(0,1,2,0)}+\mathcal{I}_{(0,0,2,1)}+\mathcal{I}_{(1,1,1,0)}+\mathcal{I}_{(1,0,1,1)}$
$\displaystyle=q^{\frac{15}{2}}\left(4\chi_{2}-3\chi_{4}+5\chi_{6}-\chi_{8}-2\chi_{10}+\chi_{12}+1\right)$
$\displaystyle+\left(10\chi_{2}-9\chi_{4}+6\chi_{6}+\chi_{8}-\chi_{10}-\chi_{12}-\chi_{14}+\chi_{16}-14\right)q^{9}$
$\displaystyle+\left(12\chi_{2}-9\chi_{4}+9\chi_{6}-4\chi_{8}+2\chi_{10}-\chi_{12}-\chi_{14}-\chi_{18}+\chi_{20}-14\right)q^{\frac{21}{2}}$
$\displaystyle+\left(14\chi_{2}-9\chi_{4}+8\chi_{6}-4\chi_{8}+3\chi_{10}-2\chi_{12}-\chi_{16}-\chi_{22}+\chi_{24}-9\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}).$
(31)

We can find the sum of these contributions

$\displaystyle\mathcal{I}_{B=1}^{\mathrm{D}3}$
$\displaystyle=\mathcal{I}_{B=1}^{\mathrm{single}}+\mathcal{I}_{B=1}^{\mathrm{triple}}$
(32)

coincides with the gauge theory result (A).

###  4.3 $B=2$

Then we consider the $B=2$ sector. The D3-brane configurations are shown in
Table 4.

Table 4: The D3-brane configurations for the $B=2$ sector are shown.
$\displaystyle\begin{array}[]{c|l|l|l|l}\hline\cr
B&\text{single}&\text{double}&\text{triple}&\text{quadruple}\\\
\hline\cr\hbox{\multirowsetup
2}&&(2,0,0,0),(0,0,2,0),(1,0,1,0)&&(3,1,0,0),(3,0,0,1),(0,1,3,0)\\\
&&&&(0,0,3,1),(2,0,1,1),(2,1,1,0)\\\ &&&&(1,1,2,0),(1,0,2,1)\\\
\hline\cr\end{array}$

We can get the double-wrapping contribution,

$\displaystyle\mathcal{I}_{B=2}^{\mathrm{double}}$
$\displaystyle=\mathcal{I}_{(2,0,0,0)}+\mathcal{I}_{(0,0,2,0)}+\mathcal{I}_{(1,0,1,0)}$
$\displaystyle=\left(\chi_{4}+1\right)q^{3}+\left(-2\chi_{2}+\chi_{4}+1\right)q^{\frac{9}{2}}+\left(-2\chi_{2}+2\chi_{4}-\chi_{6}+3\right)q^{6}$
$\displaystyle+\left(-4\chi_{2}+3\chi_{4}-\chi_{6}+\chi_{8}-\chi_{10}+6\right)q^{\frac{15}{2}}$
$\displaystyle+\left(-4\chi_{2}+4\chi_{4}-\chi_{6}+2\chi_{8}-\chi_{10}-\chi_{12}+5\right)q^{9}$
$\displaystyle+\left(-3\chi_{2}+3\chi_{4}-3\chi_{6}+3\chi_{8}-2\chi_{10}+3\chi_{12}-\chi_{14}-2\chi_{18}+\chi_{20}+3\right)q^{\frac{21}{2}}+\mathcal{O}(q^{\frac{23}{2}}),$
(33)

and also the quadruple-wrapping contribution

$\displaystyle\mathcal{I}_{B=2}^{\mathrm{quadruple}}$
$\displaystyle=\mathcal{I}_{(3,1,0,0)}+\mathcal{I}_{(3,0,0,1)}+\mathcal{I}_{(0,1,3,0)}+\mathcal{I}_{(0,0,3,1)}+\mathcal{I}_{(2,0,1,1)}+\mathcal{I}_{(2,1,1,0)}+\mathcal{I}_{(1,1,2,0)}+\mathcal{I}_{(1,0,2,1)}$
$\displaystyle=\left(3\chi_{4}-3\chi_{6}+5\chi_{8}-3\chi_{12}+2\chi_{18}-\chi_{20}+7\right)q^{\frac{21}{2}}+\mathcal{O}(q^{\frac{23}{2}}).$
(34)

The sum of these contributions

$\displaystyle\mathcal{I}_{B=2}^{\mathrm{D}3}$
$\displaystyle=\mathcal{I}_{B=2}^{\mathrm{double}}+\mathcal{I}_{B=2}^{\mathrm{quadruple}}$
(35)

is consistent with the gauge theory result (A).

###  4.4 $B=3$

Next, let us consider the $B=3$ sector. The brane configurations are shown in
Table 5.

Table 5: The D3-brane configurations for the $B=3$ sector are shown.
$\displaystyle\begin{array}[]{c|l|l|l|l}\hline\cr
B&\text{single}&\text{double}&\text{triple}&\text{quadruple}\\\ \hline\cr
3&&&(3,0,0,0),(0,0,3,0),(2,0,1,0),(1,0,2,0)&\\\ \hline\cr\end{array}$

We obtain the follwing result:

$\displaystyle\mathcal{I}_{B=3}^{\mathrm{D}3}$
$\displaystyle=\mathcal{I}_{B=3}^{\mathrm{triple}}=\mathcal{I}_{(3,0,0,0)}+\mathcal{I}_{(0,0,3,0)}+\mathcal{I}_{(2,0,1,0)}+\mathcal{I}_{(1,0,2,0)}$
$\displaystyle=\left(\chi_{2}+\chi_{6}\right)q^{\frac{9}{2}}+\left(2\chi_{2}-2\chi_{4}+\chi_{6}-3\right)q^{6}+\left(3\chi_{2}-2\chi_{4}+2\chi_{6}-\chi_{8}-4\right)q^{\frac{15}{2}}$
$\displaystyle+\left(5\chi_{2}-3\chi_{4}+2\chi_{6}-\chi_{8}+\chi_{10}-\chi_{12}-3\right)q^{9}$
$\displaystyle+\left(7\chi_{2}-5\chi_{4}+8\chi_{6}-4\chi_{8}+2\chi_{10}-\chi_{12}-\chi_{14}-10\right)q^{\frac{21}{2}}$
$\displaystyle+\left(18\chi_{2}-13\chi_{4}+7\chi_{6}-4\chi_{8}+5\chi_{10}-2\chi_{12}-\chi_{16}-14\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}).$
(36)

This result agrees with the gauge theory result (A).

###  4.5 $B=4$

Finally, we consider the $B=4$ sector. We show the D3-brane configurations for
this sector in Table 6.

Table 6: The D3-brane configurations for the $B=4$ sector are shown.
$\displaystyle\begin{array}[]{c|l|l|l|l}\hline\cr
B&\text{single}&\text{double}&\text{triple}&\text{quadruple}\\\
\hline\cr\hbox{\multirowsetup 4}&&&&(4,0,0,0),(0,0,4,0),(3,0,1,0),\\\
&&&&(1,0,3,0),(2,0,2,0)\\\ \hline\cr\end{array}$

We get the following quadruple-wrapping D3-branes contribution

$\displaystyle\mathcal{I}_{B=4}^{\mathrm{D}3}$
$\displaystyle=\mathcal{I}_{B=4}^{\mathrm{quadruple}}=\mathcal{I}_{(4,0,0,0)}+\mathcal{I}_{(0,0,4,0)}+\mathcal{I}_{(3,0,1,0)}+\mathcal{I}_{(1,0,3,0)}+\mathcal{I}_{(2,0,2,0)}$
$\displaystyle=\left(\chi_{4}+\chi_{8}+1\right)q^{6}+\mathcal{O}(q^{\frac{13}{2}}),$
(37)

and this result agrees with (42).

##  5 Conclusions and discussion

In this paper we calculated the Schur-like index of the Klebanov-Witten theory
from our proposing formula (10) based on the AdS/CFT correspondence. We
numerically checked the matching between the index of the gravity side and the
gauge theory side for $N=2$ regarding each baryonic charge sector. Especially
for $B=0,2,4$ we calculated the index up to the first order term of the
contribution of quadruple-wrapping D3-branes. For $B=1,3$ we calculated up to
$q^{12}$ order terms. In every case we found perfect agreement with the gauge
theory results.

There are many future directions. Our analysis is limited to the Schur-like
index. Hence it is the remaining task to check if the giant graviton expansion
works for the general superconformal index. Also, Our formula (10) has
multiple sums of the wrapped D3-brane contributions, and it is an interesting
problem to discuss whether we can reduce the multiple sums to the simple sum
discussed in [10] by suitably choosing the expansion variable. We will return
to this point in the near future. It is also an interesting problem to check
the giant graviton expansion (2) including multiple-wrapping D3-branes for
other Sasaki-Einstein manifolds. Furthermore, such giant graviton expansion
could be applied to a much broader range of the AdS/CFT correspondence. In
particular, its application to theories with no Lagrangian description is a
very important issue.

## Acknowledgments

The author would like to give his great thanks to Yosuke Imamura for
collaboration in the early stage of this work and excellent advice. The author
also thanks Daisuke Yokoyama, Shuichi Murayama, and Tatsuya Mori for helpful
discussions. SF is supported by the South African Research Chairs Initiative
of the Department of Science and Innovation and the National Research
Foundation grant 78554.

##  Appendix A Results from gauge theory calculation

We show the results of the ratio of the indices of the gauge theory side and
the the Kalzua-Klein index for $N=2$.

$\displaystyle\frac{\mathcal{I}^{\mathrm{KW}}_{B=0}}{\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}}$
$\displaystyle=\left(-\chi_{2}+3\chi_{4}-\chi_{6}+3\right)q^{\frac{9}{2}}+\left(-5\chi_{2}+3\chi_{4}+\chi_{6}-\chi_{8}+2\right)q^{6}$
$\displaystyle+\left(-3\chi_{2}+4\chi_{4}-2\chi_{6}+\chi_{8}-\chi_{10}+5\right)q^{\frac{15}{2}}$
$\displaystyle+\left(-9\chi_{2}+7\chi_{4}-3\chi_{6}+3\chi_{8}-\chi_{10}-\chi_{12}+10\right)q^{9}$
$\displaystyle-2\left(4\chi_{2}-4\chi_{4}+\chi_{6}-2\chi_{8}+\chi_{10}+\chi_{12}-5\right)q^{\frac{21}{2}}$
$\displaystyle+\left(-8\chi_{2}+12\chi_{4}-13\chi_{6}+15\chi_{8}-4\chi_{10}-2\chi_{14}+21\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}),$
(38)
$\displaystyle\frac{\mathcal{I}^{\mathrm{KW}}_{B=1}}{\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}}$
$\displaystyle=\chi_{2}q^{\frac{3}{2}}+\left(\chi_{2}-3\right)q^{3}+\left(2\chi_{2}-\chi_{4}-1\right)q^{\frac{9}{2}}+\left(-\chi_{2}+\chi_{4}+\chi_{6}-\chi_{8}\right)q^{6}$
$\displaystyle+\left(5\chi_{2}-3\chi_{4}+5\chi_{6}-\chi_{8}-\chi_{10}\right)q^{\frac{15}{2}}$
$\displaystyle+\left(12\chi_{2}-10\chi_{4}+6\chi_{6}-\chi_{12}-15\right)q^{9}$
$\displaystyle+\left(13\chi_{2}-9\chi_{4}+8\chi_{6}-2\chi_{8}+\chi_{10}-\chi_{12}-\chi_{14}-15\right)q^{\frac{21}{2}}$
$\displaystyle+\left(15\chi_{2}-8\chi_{4}+7\chi_{6}-4\chi_{8}+3\chi_{10}-2\chi_{12}-\chi_{14}-10\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}),$
(39)
$\displaystyle\frac{\mathcal{I}^{\mathrm{KW}}_{B=2}}{\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}}$
$\displaystyle=\left(\chi_{4}+1\right)q^{3}+\left(-2\chi_{2}+\chi_{4}+1\right)q^{\frac{9}{2}}+\left(-2\chi_{2}+2\chi_{4}-\chi_{6}+3\right)q^{6}$
$\displaystyle+\left(-4\chi_{2}+3\chi_{4}-\chi_{6}+\chi_{8}-\chi_{10}+6\right)q^{\frac{15}{2}}$
$\displaystyle+\left(-4\chi_{2}+4\chi_{4}-\chi_{6}+2\chi_{8}-\chi_{10}-\chi_{12}+5\right)q^{9}$
$\displaystyle+\left(-3\chi_{2}+6\chi_{4}-6\chi_{6}+8\chi_{8}-2\chi_{10}-\chi_{14}+10\right)q^{\frac{21}{2}}+\mathcal{O}(q^{\frac{23}{2}}),$
(40)
$\displaystyle\frac{\mathcal{I}^{\mathrm{KW}}_{B=3}}{\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}}$
$\displaystyle=\left(\chi_{2}+\chi_{6}\right)q^{\frac{9}{2}}+\left(2\chi_{2}-2\chi_{4}+\chi_{6}-3\right)q^{6}+\left(3\chi_{2}-2\chi_{4}+2\chi_{6}-\chi_{8}-4\right)q^{\frac{15}{2}}$
$\displaystyle+\left(5\chi_{2}-3\chi_{4}+2\chi_{6}-\chi_{8}+\chi_{10}-\chi_{12}-3\right)q^{9}$
$\displaystyle+\left(7\chi_{2}-5\chi_{4}+8\chi_{6}-4\chi_{8}+2\chi_{10}-\chi_{12}-\chi_{14}-10\right)q^{\frac{21}{2}}$
$\displaystyle+\left(18\chi_{2}-13\chi_{4}+7\chi_{6}-4\chi_{8}+5\chi_{10}-2\chi_{12}-\chi_{16}-14\right)q^{12}+\mathcal{O}(q^{\frac{25}{2}}),$
(41)
$\displaystyle\frac{\mathcal{I}^{\mathrm{KW}}_{B=4}}{\mathcal{I}_{\mathrm{KK}}^{T^{1,1}}}$
$\displaystyle=\left(\chi_{4}+\chi_{8}+1\right)q^{6}+\mathcal{O}(q^{\frac{13}{2}}),$
(42)

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