# Experimental observation of higher-order anapoles in individual silicon
disks under in-plane illumination

Evelyn Díaz-Escobar1 Nanophotonics Technology Center, Universitat Politècnica
de València, Camino de Vera s/n, 46022, Valencia, Spain  Ángela I. Barreda1
Friedrich Schiller University Jena, Institute of Solid State Physics, Max-
Wien-Platz 1, 07743 Jena, Germany  Amadeu Griol  Nanophotonics Technology
Center, Universitat Politècnica de València, Camino de Vera s/n, 46022,
Valencia, Spain  Alejandro Martínez  Nanophotonics Technology Center,
Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia,
Spain<EMAIL_ADDRESS>
###### Abstract

Anapole states - characterized by a strong suppression of far-field scattering
- naturally arise in high-index nanoparticles as a result of the interference
between certain multipolar moments. Recently, the first-order electric
anapole, resulting from the interference between the electric and toroidal
dipoles, was characterized under in-plane illumination as required in on-chip
photonics. Here, we go a step further and report on the observation of higher-
order (magnetic and second-order electric) anapole states in individual
silicon disks under in-plane illumination. To do so, we increase the disk
dimensions (radius and thickness) so that such anapoles occur at telecom
wavelengths. Experiments show dips in the far-field scattering perpendicular
to the disk plane at the expected wavelengths and the selected polarizations,
which we interpret as a signature of high-order anapoles. Some differences
between normal and in-plane excitation are discussed, in particular the non-
cancellation of the sum of the Cartesian electric and toroidal moments for in-
plane incidence. Our results pave the way towards the use of different anapole
states in photonic integrated circuits, either on silicon or other high-index
dielectric materials.

###### keywords:

American Chemical Society, LaTeX

Nanophotonics Technology Center, Universitat Politècnica de València, Camino
de Vera s/n, 46022, Valencia, Spain IR,NMR,UV

1these authors contributed equally to this work

##  1 Introduction

Interference between different radiative moments supported by high-index
dielectric particles can eventually lead to scattering cancellation in the
far-field, resulting in the so-called anapole states 1, 2, 3. Interestingly,
such non-radiative states are accompanied by a strong field concentration
inside the nanoparticle4, 3, 5, a feature that has been used to boost light-
matter interaction and enhance nonlinear effects such as harmonic generation
6, 7 or Raman scattering 8. One of the most interesting nanostructures to
study anapole states is a high-index disk, since its radius can be defined
lithographically to tune the anapole mode at a given frequency 1, 4, 9. Most
studies of anapole states in high-index dielectric disks have addressed
electric anapoles that arise from the destructive interference between the
Cartesian electric and toroidal dipoles, which can be excited upon normal
illumination 1, 10, 11. Indeed, multiple higher-order electric anapoles arise
when the wavelength-to-radius wavelength is increased 12, 4, 13. But anapoles
can also have a magnetic character when involving the magnetic and toroidal
dipoles 14, 15, 3. The existence of magnetic anapoles in dielectric disks
under normal illumination requires larger aspect ratios, which has led to the
recent observation of these states at microwave frequencies 16.

In Ref. 9, we reported that the first electric anapole state can also be
observed when the disk is illuminated in-plane using an integrated waveguide
propagating the transverse-electric (TE) mode. Interestingly, we found that
the electric anapole, characterized in the far-field by a huge dip in the top
scattering, takes place at a different wavelength than the energy maximum in
the disk, which was attributed to retardation effects in the disk. In
principle, the same approach could be used to observe higher-order anapoles as
long as the disk dimensions are properly tailored to support such states, and
the radiation of the multipoles that contribute to the anapole formation takes
place in the vertical (or out-of-plane) direction, which is the one accessible
in experiments (see details below).

In this work, we analyze both numerically and experimentally the in-plane (on-
chip) realization of the second-order electric and the magnetic anapole states
in an individual silicon disk built in a photonic integrated circuit (PIC).
The second-order electric anapole can be observed following the same procedure
as in 9 \- this is, using the TE waveguide mode to excite the transverse
electric dipole - but making the disk bigger. To this end, we increase the
silicon thickness from the standard 220 nm thickness up to 350 nm, as well as,
the disk radius, to place the anapole at the telecom wavelengths window. To
observe signatures of magnetic anapoles, we have to address excitation of the
magnetic field in the disk along the transversal direction, since an ideal
vertical magnetic (which could be excited using the TE waveguide mode) dipole
does not radiate out-of-plane. This is achieved by using the transverse
magnetic (TM) mode of the driving silicon waveguide. We observe strong dips -
whose central wavelengths depend on the disk radius - in the far-field
scattering spectra for the relevant polarizations, which we consider a
signature of the excitation of the targeted second-order electric and magnetic
anapoles 16. We also observe important differences between normal and in-plan
excitation, remarkably for the second-order electric anapole, which we discuss
in the text. Our results confirm the potential of this integrated approach to
excite complex multipolar states in isolated high-index disks, opening new
avenues to use anapole states in PICs.

##  2 Description of the system

Figure 1(a) displays a sketch of the structure used to observe the higher-
order anapole states: an $x$-axis-oriented silicon waveguide with rectangular
cross-section can propagate either its fundamental TE or TM mode characterized
by its main field components $E_{\mathrm{z}}$ and $H_{\mathrm{y}}$ or
$E_{\mathrm{y}}$ and $H_{\mathrm{z}}$. The waveguide is abruptly terminated,
and the output fields illuminating the disk (separated from the waveguide by a
gap $g$) are essentially the same as in the waveguide 17. Notice that guided
modes are characterized by a main transverse component but they also carry
longitudinal field components 18 that are disregarded in this work for the
sake of simplicity. Thus, the field exiting the waveguide can excite either
the electric dipole or the magnetic dipole along the $y$-axis depending on the
propagating waveguide mode (TE or TM, respectively). Ideally, an $y$-axis
oriented electric (magnetic) dipole will radiate in a doughnut-shape pattern
so that the electric far-field in the out-of-plane direction will be polarized
along $y$ ($x$) 5. Therefore, in analogy to the case of the electric anapole
9, detecting a reduction of the far-field scattering when measuring either the
$E_{\mathrm{y}}$ or the $E_{\mathrm{x}}$ component could be interpreted as a
signature of a second-order electric or a magnetic anapole state,
respectively. Notably, the use of the TM mode also allows to excite the
fundamental $z$-axis electric dipole but, in contrast to 16, this component
will not radiate in the vertical direction, which makes easier to detect the
magnetic anapole.

Figure 1: (a) Sketch of the in-plane excitation of a silicon disk (radius $r$,
thickness $t$) from a waveguide (width $w$) end propagating either the TE or
the TM mode. The main electric field components for each mode are depicted.
The waveguide end and the disk are spaced by a gap $g$. The excited electric
(magnetic) state scatters light mostly polarized along the $y$ ($x$)
direction. (b) SEM image of one of the fabricated circuits, showing the input
waveguide, a Y-splitter, and two waveguides, both ended in an abrupt
termination, and one of them having a disk. (c) SEM image showing in detail
the disk and the waveguide termination. Nominal disk radius $r={550}$ nm,
thickness $t={350}$ nm, waveguide width $w={450}$ nm and gap $g={500}$ nm.

##  3 Numerical simulations

We start our study by calculating the scattering properties of a single
silicon disk surrounded by air. The effect of the substrate in the experiments
will mainly consist on a red-shift of the whole optical response 9. We fix the
disk thickness to $t={350}$ nm, which is thicker than the ${220}$ nm value
typical in silicon-on-insulator photonics but still available in commercial
wafers. We choose a radius $r={550}$ nm and perform numerical simulations of
the disk response upon normal and lateral illumination with the electric field
along the $x$ direction. Therefore, the lateral illumination mimics the scheme
in Fig. 1 for the TE waveguide mode. The contributions to the scattering
cross-section of the main multipole moments (Cartesian electric p, magnetic m,
quadrupolar electric $Q_{\mathrm{ele}}$ and quadrupolar magnetic
$Q_{\mathrm{mag}}$, together with their respective toroidal moments (T) are
shown in Figs. 2(a) and (b) for normal and lateral illumination, respectively.
It can be seen (Fig. 2(a)) that the second electric anapole condition - this
is, $P+T_{P}=0$ \- is satisfied at a wavelength of
$1475\text{\,}\mathrm{n}\mathrm{m}$ under normal incidence. However, this
condition is not met for lateral incidence (Fig. 2(b)). So, if for the first
electric anapole, we identified a major difference between in-plane and normal
incidence (the decoupling of the scattering minimum and energy maximum in the
disk 9), for the second electric anapole, the difference is even more
noticeable, and there is neither a scattering minimum nor a dip in $P+T_{P}$
at the expected (from normal incidence simulations) anapole wavelength.

This behavior can be also attributed to retardation effects: the disk diameter
is comparable to the wavelength which may result in different scattering
properties depending on the illumination direction. However, in on-chip
photonics, the key requirement is to ensure minimum out-of-plane scattering
(so that light remains in the PIC) whilst in-plane scattering can occur and
even used to build - for instance - chains of disks for energy transport 19,
20. To verify if the scattering is reduced in the normal direction (parallel
to the disk axis), we performed finite domain time domain (FDTD) simulations
and calculated the scattered $E_{\mathrm{y}}$ field along different
directions. As shown in Fig. 2(c) there is a dip in the scattering response in
the wavelength region where the electric anapole is predicted from normal
excitation. This result also keeps when the disk is illuminated from the
waveguide propagating the TE mode. As shown in Fig. 2(d), the normalized top
scattering (calculated as the ratio between the scattering with and without
disk to remove the waveguide termination effect 9) also displays a strong
reduction in the expected wavelength region. Notice also that the electric
field inside the disk also displays a maximum at a wavelength slightly red-
shifted compared to the scattering. This means that the decoupling of the
minimum scattering condition and the maximum-energy enhancement (we assume
that the maximum energy takes place when the electric field in the disk center
is maximized, as we observed in Ref.9) also takes place for the second
electric anapole.

Figure 2: Scattering and near-field simulations of a silicon disk with
$t=$350\text{\,}\mathrm{n}\mathrm{m}$$ and
$r=$550\text{\,}\mathrm{n}\mathrm{m}$$ to observe the second electric anapole.
Spectral contribution under (a) normal and (b) lateral illumination of the
dipolar electric (p) in red and magnetic (m) in blue, and quadrupolar electric
($Q_{\mathrm{ele}}$) in purple and magnetic ($Q_{\mathrm{mag}}$) in cyan
moments, together with their respective toroidal moments, represented by $T$,
to the scattering cross-section. Also, the individual contributions of the
electric dipole (green), and its respective toroidal moment (yellow) to the
scattering are represented; (c) Scattering ($E_{\mathrm{y}}$ component) of the
disk illuminated in free space (FS) by a plane wave with the electric field
parallel to the disk axis in the forward, backward and out-of-plane (top)
directions. The top scattering shows a minimum around the wavelength for which
the second electric anapole is predicted for normal illumination; (d)
Normalized top-scattering of the disk (blue curve) when illuminated by the
light exiting from a waveguide propagating the TE mode. The yellow curve shows
the normalized $|E_{\mathrm{y}}|^{2}$ component at the center of the disk; (e)
Near-field pattern of the norm of the electric field at the anapole state
wavelength, normalized to its maximum value, at $\lambda=1510$ nm; (f)
Electric field lines at at $\lambda=1510$ nm under waveguide illumination.
Details about the simulations are given in the Supplementary Information.

Figures 2(e) and (f) show the electric near-field at $\lambda=1510$ nm (the
wavelength showing the top scattering minimum in Fig. 2(c)) on the symmetry
plane of the disk under left-side lateral illumination either from free space
or from a waveguide, respectively (see details in the Supplementary
Information). We observe field patterns similar to those occurring under
normal illumination 4 in the sense of having five field maxima separated by
four nodes (forming four vortices of the electric field, as shown by the
arrows depicted in Fig. 2(f)). The asymmetry in the near-field is due to the
symmetry-breaking illumination conditions9.

Similar results can be obtained for the magnetic anapole (Fig. 3), though now
we consider $r={450}$ nm and, in the illumination, the incident field has the
main component of the magnetic field pointing along the $y$-axis. It can be
seen that the magnetic anapole condition - this is, $M+T_{M}=0$ \- is
satisfied at a wavelength of $1470\text{\,}\mathrm{n}\mathrm{m}$ for both
normal (Fig. 3(a)) and lateral (Fig. 3(b)) illumination. Indeed, in comparison
to the case of the electric anapole, we can see a quite broad region with
reduced scattering under lateral illumination. We also performed FDTD
simulations and calculated the scattered $E_{y}$ field for both free-space
(Fig. 3(c)) and waveguide (Fig. 3(d)) illumination. In both situations, there
is a dip in the top scattering response in the wavelength region where the
magnetic anapole takes place. This confirms that a reduction of the out-of-
plane scattering for the $x$-polarized far-field can be considered a signature
of the existence of a magnetic anapole. Finally, Figs. 3(e) and (f) depict the
magnetic field on the disk plane at the magnetic anapole wavelength
($\lambda=1480$ nm) under free-space and waveguide lateral illumination,
respectively. The in-plane magnetic field in the disk shows three maxima, two
nodes and two vortices of the in-plane magnetic field, as expected for a
magnetic anapole 14.

Figure 3: Scattering and near-field simulations of a silicon disk with
$t=$350\text{\,}\mathrm{n}\mathrm{m}$$ and
$r=$450\text{\,}\mathrm{n}\mathrm{m}$$ to observe a magnetic anapole. Panels
(a) and (b) are as in Fig. 2 but for the new illumination; (c) Top-scattering
($E_{\mathrm{x}}$ component) of the disk illuminated in free space (FS) by a
plane wave with the electric field parallel to the disk axis; (d) Normalized
top-scattering of the disk (blue curve) when illuminated by the light exiting
from a waveguide propagating the TM mode. The yellow curve shows the
normalized $|H_{\mathrm{y}}|^{2}$ field component at the disk center; (e)
Near-field pattern of the norm of the magnetic field at the anapole state
wavelength ($\lambda=1480$ nm), normalized to its maximum value; (f) Magnetic
field lines at at $\lambda=1480$ nm under waveguide illumination. Details
about the simulations are given in the Supplementary Information.

###  3.1 Far-field scattering measurements of the on-chip silicon disk

To confirm our numerical predictions, we fabricated different samples
containing sets of waveguide-disk circuits on a silicon-on-insulator wafer and
using standard fabrication tools (see Supplementary Information). Figure 1(b)
shows a scanning electron microscope (SEM) image of one of the fabricated
circuits, containing an input waveguide, a Y-junction to split up the incoming
guiding signals into two paths and two waveguide terminations, with only one
of them having an adjacent disk (see details in Fig. 1(c)). This circuit
allows us to measure simultaneously the top far-field scattering with and
without disk for normalization purposes using the experimental set-up
described in Ref. 9. The circuit-setup pair also allows us to inject either
the TE or the TM mode into the input waveguide (see details in the
Supplementary Information). In addition, using a polarizer we can filter the
polarization of the scattered field to select either the $E_{\mathrm{y}}$ or
the $E_{\mathrm{x}}$ component that allows to identify the scattering produced
by the transverse electric or magnetic dipoles.

Figure 4: Experimental results of the out-of-plane scattering normalized to
the value for a bare waveguide termination (no disk), recorded for disks with
different radii (shown in the panels) under illumination with the TE (a) and
TM (b) waveguide modes, detecting the $E_{\mathrm{y}}$ and $E_{\mathrm{x}}$
components of the far-field.

First, we characterized the second electric anapole by using the TE mode of
the input waveguide. Figure 4(a) shows the measured normalized out-of-plane
scattering for three different disks having nominal radii of $525$, $550$ and
$575\text{\,}\mathrm{n}\mathrm{m}$. We observe that there is a wavelength
region with reduced scattering (reaching even 0.5 for
$r=$550\text{\,}\mathrm{n}\mathrm{m}$$) which confirm the results of the
numerical simulations presented in Fig. 2. Notice that the overall response is
slightly red-shifted as a result of the presence of the silica substrate. The
top scattering is, indeed, smaller than for the case of the terminated
waveguide without disk (in other words, normalized scattering $<1$ in
agreement with numerical simulations), which means that not only the
scattering is well suppressed, but also that the excitation of the electric
and toroidal dipoles corrects the divergence of the beam exiting the waveguide
termination and contribute to reduce the overall scattering out of the chip.
This was also observed in the case of the first electric anapole 9 and also
occurs in the case of the magnetic anapole shown below. As expected, the
wavelengths at which the scattering minimum is measured increase in proportion
to the disk radius 2. Although the second electric anapole is supposed to
provide a narrower response than the first one 4, we observed similar
bandwidth in the scattering response for both the first 9 and second anapoles.

Finally, we characterized the magnetic anapole by using the TM mode of the
input waveguide, being the results depicted in Fig. 4(b). Again, we observe
well defined minima (reaching normalized scattering values below even 0.5) of
the top scattering at wavelengths that depend on the disk radius. Again, the
minima are red-shifted in comparison to the numerical simulations, which we
attribute to the silica substrate.

##  4 Conclusion

We have shown that the use of integrated waveguides for on-chip excitation of
wavelength-sized silicon disks enables the observation of higher-order
anapoles under in-plane illumination. In particular, for the second-order
electric and the magnetic anapole, a strong reduction in the top scattering is
predicted from numerical simulations and observed in experimental
measurements. Completing the results in Ref. 9, we observe some differences
between normal and in-plane excitation. Remarkably, for the second-order
electric anapole the condition $P+T_{\mathrm{P}}=0$ is not satisfied under in-
plane illumination, which can also be attributed to retardation effects in the
disk. Still, our results show that the out-of-plane scattering is severely
reduced, which is an important feature in the practical use of anapole states
in on-chip integrated photonics. The use of the TM-like mode enables the
observation of the magnetic anapole. Further engineering of the disk could
lead to hybrid anapoles, where the far-field scattering arising from the
spherical electric and magnetic dipoles is simultaneously cancelled 14. Our
results highlight the utility of this approach (on-chip excitation) for
driving complex photonic states. Indeed, by mixing TE and TM modes in the
waveguide, as well as, changing the position between the waveguide termination
and the disk, it could become possible to excite other complex states based on
mode interference21, 22 for on-chip photonics.

##  5 Author contributions

A.M. conceived the idea. E.D.E., A.M., and A.B. performed the numerical
simulations. A.G. fabricated the sample. All authors extensively discussed the
results and contributed to the manuscript.

##  6 Funding

E.D.E. acknowledges funding from Generalitat Valenciana under grant
GRISOLIAP/2018/164. A.B. acknowledges financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the
International Research Training Group (IRTG) 2675 "Meta-ACTIVE", project
number 437527638. A.M. thanks funding from Generalitat Valenciana (Grants No.
PROMETEO/2019/123, IDIFEDER/2020/041 and IDIFEDER/2021/061).

##  7 Notes

The authors declare no competing financial interest.

## References

  * Miroshnichenko et al. 2015 Miroshnichenko, A. E.; Evlyukhin, A. B.; Yu, Y. F.; Bakker, R. M.; Chipouline, A.; Kuznetsov, A. I.; Luk’yanchuk, B. S.; Chichkov, B. N.; Kivshar, Y. S. Nonradiating anapole modes in dielectric nanoparticles. _Nature Communications_ 2015, _6_ , 8069
  * Wang and Dal Negro 2016 Wang, R.; Dal Negro, L. Engineering non-radiative anapole modes for broadband absorption enhancement of light. _Optics Express_ 2016, _24_ , 19048
  * Baryshnikova et al. 2019 Baryshnikova, K. V.; Smirnova, D. A.; Luk’yanchuk, B. S.; Kivshar, Y. S. Optical Anapoles: Concepts and Applications. _Advanced Optical Materials_ 2019, _7_ , 1801350
  * Yang et al. 2018 Yang, Y.; Zenin, V. A.; Bozhevolnyi, S. I. Anapole-Assisted Strong Field Enhancement in Individual All-Dielectric Nanostructures. _ACS Photonics_ 2018, _5_ , 1960–1966
  * Savinov et al. 2019 Savinov, V.; Papasimakis, N.; Tsai, D. P.; I., Z. N. Optical anapoles. _Communications Physics_ 2019, _2_ , 69
  * Grinblat et al. 2017 Grinblat, G.; Li, Y.; Nielsen, M. P.; Oulton, R. F.; Maier, S. A. Efficient Third Harmonic Generation and Nonlinear Subwavelength Imaging at a Higher-Order Anapole Mode in a Single Germanium Nanodisk. _ACS Nano_ 2017, _11_ , 953–960
  * Timofeeva et al. 2018 Timofeeva, M.; Lang, L.; Timpu, F.; Renaut, C.; Bouravleuv, A.; Shtrom, I.; Cirlin, G.; Grange, R. Anapoles in Free-Standing III–V Nanodisks Enhancing Second-Harmonic Generation. _Nano Letters_ 2018, _18_ , 3695–3702
  * Baranov et al. 2018 Baranov, D. G.; Verre, R.; Karpinski, P.; Käll, M. Anapole-Enhanced Intrinsic Raman Scattering from Silicon Nanodisks. _ACS Photonics_ 2018, _5_ , 2730–2736
  * Diaz-Escobar et al. 2021 Diaz-Escobar, E.; Bauer, T.; Pinilla-Cienfuegos, E.; Barreda, A. I.; Griol, A.; Kuipers, L.; Martínez, A. Radiationless anapole states in on-chip photonics. _Light: Science & Applications_ 2021, _10_ , 204
  * Barreda et al. 2019 Barreda, A. I.; Saiz, J. M.; González, F.; Moreno, F.; Albella, P. Recent advances in high refractive index dielectric nanoantennas: Basics and applications. _AIP Advances_ 2019, _9_ , 040701
  * Barreda et al. 2022 Barreda, A.; Vitale, F.; Minovich, A. E.; Ronning, C.; Staude, I. Applications of Hybrid Metal-Dielectric Nanostructures: State of the Art. _Advanced Photonics Research_ 2022, _3_ , 2100286
  * Zenin et al. 2017 Zenin, V. A.; Evlyukhin, A. B.; Novikov, S. M.; Yang, Y.; Malureanu, R.; Lavrinenko, A. V.; Chichkov, B. N.; Bozhevolnyi, S. I. Direct Amplitude-Phase Near-Field Observation of Higher-Order Anapole States. _Nano Letters_ 2017, _17_ , 7152–7159
  * Tian et al. 2019 Tian, J.; Luo, H.; Yang, Y.; Ding, F.; Qu, Y.; Zhao, D.; Qiu, M.; Bozhevolnyi, S. I. Active control of anapole states by structuring the phase-change alloy Ge2Sb2Te5. _Nature Communications_ 2019, _10_ , 396
  * Luk’yanchuk et al. 2017 Luk’yanchuk, B.; Paniagua-Domínguez, R.; Kuznetsov, A. I.; Miroshnichenko, A. E.; Kivshar, Y. S. Hybrid anapole modes of high-index dielectric nanoparticles. _Phys. Rev. A_ 2017, _95_ , 063820
  * Lamprianidis and Miroshnichenko 2018 Lamprianidis, A. G.; Miroshnichenko, A. E. Excitation of nonradiating magnetic anapole states with azimuthally polarized vector beams. _Beilstein J. Nanotechnol._ 2018, _9_ , 1478–1490
  * Kapitanova et al. 2020 Kapitanova, P.; Zanganeh, E.; Pavlov, N.; Song, M.; Belov, P.; Evlyukhin, A.; Miroshnichenko, A. Seeing the Unseen: Experimental Observation of Magnetic Anapole State Inside a High-Index Dielectric Particle. _Annalen der Physik_ 2020, _532_ , 2000293
  * Espinosa-Soria et al. 2016 Espinosa-Soria, A.; Griol, A.; Martínez, A. Experimental measurement of plasmonic nanostructures embedded in silicon waveguide gaps. _Optics Express_ 2016, _24_ , 9592
  * Espinosa-Soria and Martinez 2016 Espinosa-Soria, A.; Martinez, A. Transverse Spin and Spin-Orbit Coupling in Silicon Waveguides. _IEEE Photonics Technology Letters_ 2016, _28_ , 1561–1564
  * Mazzone et al. 2017 Mazzone, V.; Totero Gongora, J. S.; Fratalocchi, A. Near-Field Coupling and Mode Competition in Multiple Anapole Systems. _Applied Sciences_ 2017, _7_
  * Huang et al. 2021 Huang, T.; Wang, B.; Zhang, W.; Zhao, C. Ultracompact Energy Transfer in Anapole-based Metachains. _Nano Letters_ 2021, _21_ , 6102–6110, PMID: 34269592
  * Rybin et al. 2017 Rybin, M. V.; Koshelev, K. L.; Sadrieva, Z. F.; Samusev, K. B.; Bogdanov, A. A.; Limonov, M. F.; Kivshar, Y. S. High-$Q$ Supercavity Modes in Subwavelength Dielectric Resonators. _Phys. Rev. Lett._ 2017, _119_ , 243901
  * Koshelev et al. 2020 Koshelev, K.; Kruk, S.; Melik-Gaykazyan, E.; Choi, J.-H.; Bogdanov, A.; Park, H.-G.; Kivshar, Y. Subwavelength dielectric resonators for nonlinear nanophotonics. _Science_ 2020, _367_ , 288–292


# Quantum Cosmology in Thick Brane

S. H. Hendi1,2,4 111email address<EMAIL_ADDRESS>N. Riazi3 222email
address<EMAIL_ADDRESS>and S. N. Sajadi1,2333email address:
<EMAIL_ADDRESS>1Department of Physics, School of Science, Shiraz
University, Shiraz 71454, Iran
2Biruni Observatory, School of Science, Shiraz University, Shiraz 71454, Iran
3Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839,
Iran
4Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada

###### Abstract

One of the interesting open problems in the cosmological framework is applying
quantum physics to the whole universe, consistently. Although different
conceptual aspects of quantum cosmology are still very much alive, till now,
all the attempts of obtaining a unique and well-defined version of quantum
cosmology have been unfeasible. Motivated by what was mentioned above, in this
paper, the quantum tunneling of thick brane is investigated through deriving
the Wheeler-DeWitt equation and corresponding solutions. Besides, the
cosmological analysis of the brane is studied based on two special cases of
the scalar field. Finally, it is found that the constant scalar potential
arising from a time-dependent scalar field is supported by the so-called slow-
roll inflation.

##  I Introduction

One of the cornerstones of theoretical physics is the determination of
spacetime dimension. In other words, it is an open question that whether the
spacetime dimension is really $3+1$ or it might be, fundamentally, higher than
four. The Kaluza—Klein (KK) theory is the first serious attempt beyond the
usual four-dimensional spacetime. The fifteen components of the five-
dimensional KK metric are divided into ten components identifying the four-
dimensional spacetime metric, a $U(1)$ gauge field with four components, and
one scalar field called the dilaton. Although it is argued that the approach
of KK theory is not accurate, the concept of extra dimension in such a theory
provided a basis for further hot topics such as superstring theory, brane
cosmology, and AdS/CFT correspondence or its generalization to gauge/gravity
duality.

The key point of the braneworld scenario is the fact that the
$(3+1)$-dimensional brane is embedded in a five-dimensional geometry in which
the extra dimension may be large or compact. Regarding the effective models of
brane cosmology such as the Randall and Sundrum RS  involving an extra-
dimension with non-trivial warp factor, it is possible to discuss the
hierarchy problem to explain the weakness of gravity relative to the other
forces of nature. Indeed, it is supposed that our visible universe is a
membrane inside a higher-dimensional space, and particles and interactions
corresponding to electromagnetic, weak, and strong forces are restricted to
such a membrane.

In this paper, we take into account a five-dimensional space. There are two
known cases in this direction. As the first case, one may consider an
infinitely thin brane-universe along the extra-dimension. The validity of this
option depends on the energy scales that one wishes to examine. The second
case is devoted to the brane with finite thickness. Hereafter, we consider a
thick brane model by assuming that the energy-momentum tensor describing the
brane has a finite distribution over a limited length interval along the fifth
dimension.

The first approach to describe the universe in the context of quantum theory
was reported in $1960$ by Wheeler 13  and DeWitt 14 . They proposed a quantum
gravity equation to describe the wave function of the universe, which is known
as the Wheeler-DeWitt (WD) equation. This equation is analogous to a zero-
energy Schrödinger equation in which the Hamiltonian contains both the
gravitation and scalar fields. However, one of the most important problems for
solving the WD equation is the subject of initial conditions. Unlike a
classical system, for cosmological models, there are no external initial
conditions because there is no external time parameter to the universe. To
solve this problem, two different approaches are known: the Hartle-Hawking no-
boundary 15 -18  and the Vilenkin tunneling proposal 19 -22 . The first
proposal is based on an assumption that the wave function of the universe is
given by a path integral over compact Euclidean geometries, and therefore,
this universe has no boundary in this space. The second one states that the
universe spontaneously nucleates and then evolves along the lines of an
inflationary scenario. The mathematical description of this approach is
closely analogous to that of quantum tunneling through a potential barrier.
Indeed, only the outgoing modes of the wave function should be taken at the
singular boundary of superspace 22 ; 23 .

Working in the framework of brane’s theory, the main motivation of this paper
is to understand the earliest moments of the universe, where we expect that
quantum effects are dominant. This paper is organized as follows: In the next
section, the canonical quantization is applied on the thick brane, the WD
equation is obtained and its solution is investigated. Section III is devoted
to the cosmology of thick brane and finally in Sec. IV concluding remarks are
presented.

##  II Canonical Quantum Cosmology

As the starting point, we consider the usual five-dimensional Einstein-scalar
field theory with the following bulk action

$S=\int
d^{5}x\sqrt{g^{(5)}}\left(R^{(5)}-\dfrac{1}{2}\partial_{A}\phi\partial^{A}\phi-V(\phi)\right),$
(1)

where the bulk indices $A$ and $B$ runs through $0,1,2,3,4$ i.e. over all
five-dimensional spacetime. In order to obtain the field equations, one may
use the variation of the action with respect to the bulk metric $g_{AB}$ and
scalar field $\phi$ to achieve

$\displaystyle R_{AB}-\dfrac{1}{2}g_{AB}R^{(5)}$ $\displaystyle=$
$\displaystyle T^{\phi}_{AB},$ (2) $\displaystyle\square_{5}\phi$
$\displaystyle=$ $\displaystyle\dfrac{dV}{d\phi},$ (3)

where $T^{\phi}_{AB}=\partial_{A}\phi\partial_{B}\phi-
g_{AB}\left(\dfrac{1}{2}\partial_{C}\phi\partial^{C}\phi+V(\phi)\right)$ and
$\square_{5}\phi=g^{AB}\phi_{;AB}$. At this point, we have to consider the
geometrical properties of the spacetime by introducing a metric. We adopt the
following metric ansatz to describe the five-dimensional spacetime
Hendi:2020qkk

$ds^{2}=a^{2}(w)\left[-dt^{2}+b^{2}(t)\left(\dfrac{dr^{2}}{1-\kappa
r^{2}}+r^{2}\left[d\theta^{2}+sin^{2}(\theta)d\phi^{2}\right]\right)\right]+dw^{2},$
(4)

where $a(w)$ is the warp function of the brane, $b(t)$ is the brane scale
factor and $\kappa$ is related to the scalar curvature. Now, we wish to
quantize the system described by the action (1) through the canonical
quantization. To investigate the canonical quantization, one should promote
the metric $g_{ij}$, the conjugate momenta $\Pi_{ij}$, the Hamiltonian density
$H$ and the momentum density $H_{i}$ to quantum operators satisfying canonical
commutation relations.

At first, we should calculate the Lagrangian term by term. The Ricci scalar of
the mentioned metric is given by

$R^{(5)}=-\dfrac{2\left[4ab^{2}a^{{}^{\prime\prime}}+6b^{2}a^{{}^{\prime}2}-3b\ddot{b}-3\dot{b}^{2}-3\kappa\right]}{a^{2}b^{2}},$
(5)

where $a=a(w)$, $b=b(t)$. Also dot and prim are derivatives with respect to
$t$ and $w$, respectively. By using Eq. (1), the Lagrangian is

$L=-2a^{2}b^{3}\left[\frac{3\dot{b}^{2}}{b^{2}}+6a^{{}^{\prime}2}+4aa^{{}^{\prime\prime}}-\dfrac{1}{4}\dot{\phi}(w,t)^{2}+\right.\\\
\left.\dfrac{1}{4}a^{2}\phi^{{}^{\prime}}(w,t)^{2}+\frac{1}{2}a^{2}V-\frac{3\kappa}{b^{2}}\right].\hskip
170.71652pt$ (6)

In addition, the momenta conjugate to $b$ and $\phi$ are

$\displaystyle\Pi_{b}$ $\displaystyle=$ $\displaystyle\dfrac{\partial
L}{\partial\dot{b}}=-12a^{2}b\dot{b},$ (7) $\displaystyle\Pi_{\phi}$
$\displaystyle=$ $\displaystyle\dfrac{\partial
L}{\partial\dot{\phi}}=a^{2}b^{3}\dot{\phi}.$ (8)

Now, we can calculate the Hamiltonian with the following explicit form

$H=\Pi_{b}\dot{b}+\Pi_{\phi}\dot{\phi}-L=-\Pi_{b}^{2}+\dfrac{12}{b^{2}}\Pi_{\phi}^{2}+U(a,b,\phi)=0,$
(9)

where $U(a,b,\phi)$ is

$U(a,b,\phi)=24a^{4}b^{4}\left(12a^{{}^{\prime}2}+8aa^{{}^{\prime\prime}}+\dfrac{a^{2}\phi^{{}^{\prime}2}}{2}+a^{2}V-\frac{6\kappa}{b^{2}}\right).$
(10)

By taking the replacement $\Pi_{b}\rightarrow-i\dfrac{\partial}{\partial b}$
and $\Pi_{\phi}\rightarrow-i\dfrac{\partial}{\partial\phi}$ and imposing
$H\Psi=0$, one can find the following WD equation

$\left(\dfrac{\partial^{2}}{\partial
b^{2}}-\dfrac{12}{b^{2}}\dfrac{\partial^{2}}{\partial\phi^{2}}+U(a,b,\phi)\right)\Psi(b,\phi)=0,$
(11)

which has been regarded as the wave function equation of universe. The mini-
superspace for this model is a two-dimensional space with coordinates
$(b,\phi)$ with $0<b<\infty$ and $-\infty<\phi<\infty$. To solve the WD
equation, we can follow the separation of variable method by using the
following assumption

$\Psi(b,\phi)=\Phi(\phi)B(b).$ (12)

Taking Eq. (12) into account, it is easy to find that the WD equation becomes

$\dfrac{1}{B}\dfrac{d^{2}B}{db^{2}}-\dfrac{12}{\Phi
b^{2}}\dfrac{d^{2}\Phi}{d\phi^{2}}+\nu b^{4}-\varrho b^{2}=0,$ (13)

where Eq. (10) helps us to obtain

$\displaystyle\nu=$
$\displaystyle\left[96a^{4}(3a^{{}^{\prime}2}+2aa^{{}^{\prime\prime}})+24a^{6}\left(\frac{1}{2}\phi^{{}^{\prime}2}+V(\phi)\right)\right],$
(14) $\displaystyle\varrho=$ $\displaystyle 144a^{4}\kappa.$ (15)

In order to investigate the WD equation, we have to know the properties of the
superpotential $U(b,w)$ and the corresponding wave function $\Psi(b,\phi)$.
Considering Eq. (13), one finds that a regular solution for the wave function
in the limit $b\ll 1$ leads to the fact that the wave function should be
$\phi$-independent since the coefficient of $\partial_{\phi}^{2}$ in the WD
equation diverges. Therefore, the WD equation reduces to

$\dfrac{d^{2}B}{db^{2}}-\varrho b^{2}B=0,$ (16)

with the following exact solution

$B(b\ll
1)=c_{1}\sqrt{b}J_{1/4}\left(\dfrac{\sqrt{-\varrho}b^{2}}{2}\right)+c_{2}\sqrt{b}Y_{1/4}\left(\dfrac{\sqrt{-\varrho}b^{2}}{2}\right),$
(17)

where $c_{1}$ and $c_{2}$ are two integration constants, and $J$ and $Y$ are
two Bessel functions. It is straightforward to find that we have to choose
$c_{1}=0$ and $c_{2}=-1$ through imposing the tunneling condition

$\lim_{b\rightarrow
0}c_{2}\sqrt{b}Y_{1/4}\left(\dfrac{\sqrt{-\varrho}b^{2}}{2}\right)=-\dfrac{2c_{2}}{(-\varrho)^{\frac{1}{8}}\Gamma\left(\frac{3}{4}\right)}+\mathcal{O}(b),$
(18)

where the wave function goes to the maximum values for vanishing $b$. In
addition, for the Hartle-Hawking no boundary, we should choose
$c_{1}=1,c_{2}=0$ since

$\lim_{b\rightarrow
0}c_{2}\sqrt{b}J_{1/4}\left(\dfrac{\sqrt{-\varrho}b^{2}}{2}\right)=\dfrac{c_{1}2^{\frac{3}{2}}3^{\frac{1}{4}}\Gamma\left(\frac{3}{4}\right)}{\pi}b+\mathcal{O}(b^{5}),$
(19)

and when $b$ goes to zero the wave function vanishes.

In the limit $b\gg 1$, we can neglect the derivative with respect to the
scalar field $\phi$. Thus, $\phi$ plays the role of a parameter in the WD
equation, and the problem is reduced to the one-dimensional mini-superspace
model. So, Eq. (13) reduces to

$\dfrac{d^{2}B}{db^{2}}+(\nu b^{4}-\varrho b^{2})B=0,$ (20)

with a semi-classical solution known as the Vilenkin tunneling wave function

$B(b)\approx\dfrac{c\exp{\left(\dfrac{i(\nu
b^{2}-\varrho)^{(3/2)}}{3\nu}\right)}}{b(\nu b^{2}-\varrho)^{(1/4)}},$ (21)

which is an oscillatory solution for $\nu b^{2}-\varrho>0$ with a damped
amplitude while for $\nu b^{2}-\varrho<0$ it is only a damping solution as

$B(b)\approx\dfrac{c\exp{\left(\dfrac{(\varrho-\nu
b^{2})^{(3/2)}}{3\nu}\right)}}{b(\varrho-\nu b^{2})^{(1/4)}}.$ (22)

These are the analogue of a wave-packet peaked about a classical particle
trajectory in the ordinary quantum mechanics.
The exact solution of Eq. (20) in the region of $\nu/\varrho b^{2}>1$ where
superpotential is positive is

$\displaystyle B(b)$ $\displaystyle=$ $\displaystyle
c_{3}e^{f(b,\kappa)}HenuT(\lambda,\delta,\eta,\zeta b)+$ (23) $\displaystyle
c_{4}e^{-f(b,\kappa)}HenuT(\lambda,\delta,\eta,-\zeta b),$

where

$\displaystyle f(b,\kappa)=$ $\displaystyle\dfrac{(-0.33)(-1.5\varrho+\nu
b^{2})b}{\sqrt{-\nu}},$ $\displaystyle\lambda$
$\displaystyle=\dfrac{-(0.16+0.28I)\varrho^{2}}{(-\nu)^{(\frac{4}{3})}},$
$\displaystyle\delta$ $\displaystyle=0,$ $\displaystyle\eta$
$\displaystyle=\dfrac{(-0.57+0.99I)\varrho}{(-\nu)^{\frac{2}{3}}},$
$\displaystyle\zeta=$ $\displaystyle(0.44+0.76I)(-\nu)^{\frac{1}{6}}b.$

It is notable that by converting $\nu\rightarrow-\nu$, $\eta\rightarrow-\eta$
and $f\rightarrow-f$, one can obtain the solution for the case of $\nu/\varrho
b^{2}<1$, where superpotential is positive. Besides for $b\gg 1$ the
superpotential consists of two terms, a curvature term $\varrho b^{2}$ and
$\nu b^{4}-$term as

$U(b)=\nu b^{4}-\varrho b^{2}.$ (24)

To have a quantum tunneling, the superpotential may have a maximum. So, we
should have a barrier with the following conditions

$\dfrac{dU}{db}=0,\hskip 14.22636pt\dfrac{d^{2}U}{db^{2}}<0.$ (25)

The first condition leads to

$\dfrac{dU}{db}=0\hskip 14.22636pt\Rightarrow\hskip
14.22636ptb=b_{bar}=\sqrt{\dfrac{\varrho}{2\nu}}$ (26)

in which by inserting it into the second condition, we find

$\dfrac{d^{2}U}{db^{2}}\mid_{b=b_{bar}}<0\hskip 14.22636pt\Rightarrow\hskip
14.22636pt4\varrho<0,$ (27)

and the height of barrier is

$U(b=b_{bar})=U_{max}=-\dfrac{\varrho^{2}}{4\nu}.$ (28)

In order to have a positive $U_{max}$, we should consider a limitation as
$\nu<0$. Finally, the conditions for having a quantum tunneling are

$\nu<0\text{ \ \ and \ \ }\varrho<0.$ (29)

The condition $\varrho<0$ imposes $\kappa<0$ while for the condition $\nu<0$,
we should look at $\nu$ in more details since it depends on the kind of thick
brane. One may also look at the tunneling rate between the turning points
$b=0$ and $b=\sqrt{\nu/\varrho}$ which is approximately given by

$\displaystyle|T|^{2}$ $\displaystyle\approx$
$\displaystyle\exp{\left(-2\int_{0}^{\sqrt{\nu/\varrho}}\sqrt{\nu
b^{4}-\varrho b^{2}}db\right)}$ (30) $\displaystyle\approx$
$\displaystyle\exp{\left(\dfrac{2(-\varrho)^{3/2}}{3\nu}\left[1-\left(\dfrac{\nu^{2}}{\varrho^{2}}-1\right)^{(3/2)}\right]\right)}.$

In the special cases, we get

$\displaystyle|T|^{2}$
$\displaystyle\approx\exp{\left(\dfrac{2\nu^{2}}{3(-\varrho)^{3/2}}\right)},\hskip
14.22636pt$ $\displaystyle\nu^{2}/\varrho^{2}$ $\displaystyle\gg 1$ (31)
$\displaystyle|T|^{2}$
$\displaystyle\approx\exp{\left(-8\sqrt{-\kappa}\right)},\hskip 14.22636pt$
$\displaystyle\nu^{2}/\varrho^{2}$ $\displaystyle=1$ (32)
$\displaystyle|T|^{2}$
$\displaystyle\approx\exp{\left(\dfrac{2(-\varrho)^{(3/2)}\left(1-i\right)}{3\nu}\right)},\hskip
14.22636pt$ $\displaystyle\nu^{2}/\varrho^{2}$ $\displaystyle\ll 1$ (33)

It is obvious that for the first two cases, the rate of tunneling is real but
it is imaginary for the third case, and therefore, the condition
$\nu^{2}/\varrho^{2}\ll 1$ does not occur.
Now, we will more investigate quantum tunneling by considering thick branes
with two different four-dimensional solutions in the case of
$\phi(w,t)=\phi(w)$.

###  II.1 Model 1: Four-dimensional de-Sitter thick brane solution

The stable de-Sitter thick brane solution with a scalar field potential is
given by Wang:2002pka ,Dzhunushaliev:2009va

$V(\phi)=V_{0}\cos^{2(1-\sigma)}\left(\dfrac{\phi}{\phi_{0}}\right),\hskip
14.22636pt\phi_{0}=\sqrt{3\sigma(1-\sigma)},$ (34)

where $V_{0}$ and $\sigma$ are arbitrary constants. The $Z_{2}$-symmetric
solution is also calculated as

$a=\dfrac{1}{\left(\cosh\left(\dfrac{Hw}{\sigma}\right)\right)^{\sigma}},\hskip
14.22636pt\phi=\phi_{0}\arcsin\left(\tanh\left(\dfrac{Hw}{\sigma}\right)\right),$
(35)

where $H^{2}=\frac{2\sigma V_{0}}{3(1+3\sigma)}$ and $0<\sigma<1$. In the
limit $\sigma\rightarrow 0$, the solution approaches a de-Sitter thin brane
embedded in a five-dimensional Minkowski bulk.
By using Eqs. (14) and (15) in the limit of $w=0$, one can obtain

$\nu=\dfrac{12V_{0}(3\sigma-11)}{3\sigma+1},\hskip
14.22636pt\varrho=144\kappa,$ (36)

where we have to consider the constraint $0<\sigma<1$ to have quantum
tunneling. Accordingly, the effective potential is

$U=144\kappa b^{2}\left({\Theta}-1\right),$ (37)

where
${\Theta}=\dfrac{V_{0}(3\sigma-11)b^{2}}{12\kappa\left(1+3\sigma\right)}$, and
the barrier is characterized as

$\displaystyle b_{bar}$ $\displaystyle=$
$\displaystyle\frac{2.45b}{2}\sqrt{\dfrac{1}{3{\Theta}}},$ $\displaystyle
U_{max}$ $\displaystyle=$
$\displaystyle\dfrac{-(3555.69\kappa+10667.08\kappa\sigma+9504V_{0}-2592\sigma
V_{0})b^{4}}{144\kappa\left(1+3\sigma\right){\Theta}^{2}}.$

As a result, the wave functions are

$\begin{array}[]{cc}\begin{array}[]{c}\psi\approx\dfrac{c_{1}\exp{\left(\dfrac{108ib^{2}\kappa^{\frac{1}{2}}\left({\Theta}-1\right)^{\frac{3}{2}}}{25{\Theta}}\right)}}{b\left[144\kappa\left({\Theta}-1\right)\right]^{(1/4)}},\\\
\end{array}&{\Theta}>1\\\
\begin{array}[]{c}\psi\approx\dfrac{c_{1}\exp{\left(\dfrac{108b^{2}\kappa^{\frac{1}{2}}\left(1-{\Theta}\right)^{\frac{3}{2}}}{25{\Theta}}\right)}}{b\left[144\kappa\left(1-{\Theta}\right)\right]^{(1/4)}},\\\
\end{array}&{\Theta}<1\end{array}.$ (38)

After some manipulations, we can find that the tunneling rate is

$|T|^{2}\approx\exp\left({\dfrac{-0.055b^{2}\left((-144\kappa)^{(3/2)}-\left[144\kappa\left(\dfrac{{\Theta}^{2}}{b^{4}}-1\right)\right]^{(3/2)}\right)}{12\kappa{\Theta}}}\right).$
(39)

in which for the case of ${\Theta}^{2}\gg b^{4}$, we have

$|T|^{2}\approx\exp\left(\dfrac{8V_{0}^{2}(3\sigma-11)^{2}}{(-\kappa)^{(3/2)}}\right).$
(40)

###  II.2 Model 2: Four-dimensional Minkowski thick brane solution

Here, we consider the following superpotential Dzhunushaliev:2009va

$W(\phi)=c\sin(b\phi).$ (41)

Taking such a superpotential into account, one can find the following
$Z_{2}$-symmetric and stable solution for the sine-Gordon scalar potential,
warp function and scalar field

$\displaystyle V(\phi)=$
$\displaystyle\dfrac{3}{2}c^{2}\left[3b^{2}\cos^{2}(b\phi)-4\sin^{2}(b\phi)\right],$
(42) $\displaystyle a(w)=$
$\displaystyle\left[\cosh(cb^{2}w)\right]^{\frac{-1}{3b^{2}}},$ (43)
$\displaystyle\phi(w)=$
$\displaystyle\dfrac{2}{b}\arctan\left[\tanh\left(\dfrac{3cb^{2}w}{2}\right)\right].$
(44)

Notably, in the limit $c\rightarrow 0$, the solution approaches a Minkowski
thin brane embedded in a five-dimensional AdS bulk with an effective
cosmological constant $\Lambda=-6c^{2}$. By using Eqs. (14) and (15) for
$w=0$, we find

$\nu=88c^{2}b^{2},\hskip 14.22636pt\varrho=144\kappa.$ (45)

According to what is obtained above, it is obvious that there is no quantum
tunneling since $\nu$ is positive.

##  III The Cosmology of Thick Brane

In the following, to have a deep insight for the case of $\kappa<0$, we study
the cosmology in brane. So, by using of the ansatz metric (4), the explicit
form of Einstein field equation (2) and the equation of motion for the scalar
field (3) which are arisen form the variation of action can be written as are
Ahmed:2013lea

$\begin{array}[]{cc}\dfrac{3}{a^{2}}\left(a^{{}^{\prime}2}+aa^{{}^{\prime\prime}}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}\right)=-\dfrac{1}{2a^{2}}\dot{\phi}^{2}-\dfrac{1}{2}\phi^{{}^{\prime}2}-V,&tt-
component\\\
\dfrac{1}{a^{2}}\left(-2\dfrac{\ddot{b}}{b}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}+3a^{{}^{\prime}2}+3aa^{{}^{\prime\prime}}\right)=\dfrac{1}{2a^{2}}\dot{\phi}^{2}-\dfrac{1}{2}\phi^{{}^{\prime}2}-V,&ij-
component\\\
\dfrac{3}{a^{2}}\left(-\dfrac{\ddot{b}}{b}+2a^{{}^{\prime}2}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}\right)=\dfrac{1}{2a^{2}}\dot{\phi}^{2}+\dfrac{1}{2}\phi^{{}^{\prime}2}-V,&ww-
component\\\ \ddot{\phi}+3\dfrac{\dot{b}}{b}\dot{\phi}+a^{2}\dfrac{\partial
V}{\partial\phi}-a^{2}\phi^{{}^{\prime\prime}}-4a^{{}^{\prime}}a\phi^{{}^{\prime}}=0.&scalar\text{
}field\text{ }equation\end{array}$ (46)

Here, we consider two possibility for $\phi$. As the first one, we look at the
case of $\phi(w,t)=\phi(w)$ and then, we consider $\phi(w,t)=\phi(t)$ as the
second case.

###  III.1 Case I: $\phi(w,t)=\phi(w):$

Considering that the scalar field and the warp function depend only on the
extra dimension, one can simplify the relations that are combined in Eq. (46)
with the following forms

$\displaystyle tt-component:\hskip 14.22636pt$
$\displaystyle\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}=\dfrac{a^{2}}{3}\left[3\left(\dfrac{a^{{}^{\prime}2}}{a^{2}}+\dfrac{a^{{}^{\prime\prime}}}{a}\right)+\dfrac{1}{2}\phi^{{}^{\prime}2}+V\right],$
(47) $\displaystyle ij-component:\hskip 14.22636pt$ $\displaystyle
2\dfrac{\ddot{b}}{b}+\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}=a^{2}\left[3\left(\dfrac{a^{{}^{\prime}2}}{a^{2}}+\dfrac{a^{{}^{\prime\prime}}}{a}\right)+\dfrac{1}{2}\phi^{{}^{\prime}2}+V\right],$
(48) $\displaystyle ww-component:\hskip 14.22636pt$
$\displaystyle\dfrac{\ddot{b}}{b}+\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}=\dfrac{a^{2}}{3}\left[6\dfrac{a^{{}^{\prime}2}}{a^{2}}-\dfrac{1}{2}\phi^{{}^{\prime}2}+V\right],$
(49) $\displaystyle scalar\text{ }field\text{ }equation:\hskip 14.22636pt$
$\displaystyle\phi^{{}^{\prime\prime}}+4\dfrac{a^{{}^{\prime}}}{a}\phi^{{}^{\prime}}-\dfrac{\partial
V}{\partial\phi}=0,$ (50)

where the left (right) hand side depends only on $t$ ($w$). So we can obtain
the following set of equations for $b(t)$

$\displaystyle\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}=$
$\displaystyle C_{t},$ (51)
$\displaystyle\dfrac{2\ddot{b}}{b}+\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}=$
$\displaystyle C_{x},$ (52)
$\displaystyle\dfrac{\ddot{b}}{b}+\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}=$
$\displaystyle C_{w},$ (53)

where $C_{t,x,w}$ are unknown constants. The consistency condition for these
equations guides us to find a relation between such unknown constants

$C_{w}=\dfrac{C_{x}+C_{t}}{2}.$ (54)

On the other hand, from the right hand side of these equations, one can obtain

$C_{x}=3C_{t},$ (55)

and consequently, we can rewrite all the constants in term of $C_{w}$ as

$C_{t}=\dfrac{1}{2}C_{w}=\dfrac{1}{2}\lambda,\hskip
28.45274ptC_{x}=\dfrac{3}{2}C_{w}=\dfrac{3}{2}\lambda$ (56)

where $\lambda=C_{w}$ is a constant. Now, combination of Eqs. (51)-(53) leads
to the following differential equations

$\dfrac{\ddot{b}}{b}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}=0,$ (57)
$\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}-\dfrac{\lambda}{2}=0.$ (58)

By solving Eq. (57), one can find

$b(t)=\dfrac{c_{1}}{2}\left(e^{\dfrac{t+c_{2}}{c_{1}}}+\kappa
e^{-\dfrac{t+c_{2}}{c_{1}}}\right).$ (59)

Inserting the scale factor (59) into the conditional equation (58), we find

$c_{1}=\sqrt{\dfrac{2}{\lambda}},$ (60)

and therefore, the scale factor can be rewritten as

$b(t)=\sqrt{\dfrac{2}{\lambda}}\left(\dfrac{e^{\sqrt{\frac{\lambda}{2}}(t+c_{2})}+\kappa
e^{-\sqrt{\frac{\lambda}{2}}(t+c_{2})}}{2}\right),$ (61)

where $c_{2}$ is an integration constant and $\kappa=-1$. It is worth
mentioning that in the case of $\kappa=-1$ and $\lambda<0$, the scale factor
becomes

$b(t)=\sqrt{\dfrac{2}{\lambda}}\sin\left(\sqrt{\dfrac{2}{\lambda}}(t+c_{2})\right)$

which corresponds to the cyclic universe. Also, for $\kappa=-1$ with positive
$\lambda$, we find

$b(t)=\sqrt{\dfrac{2}{\lambda}}\sinh\left(\sqrt{\dfrac{2}{\lambda}}(t+c_{2})\right).$

Now, we want to consider the $w$-dependent part of solutions that are

$\displaystyle\dfrac{a^{2}}{3}\left[3\left(\dfrac{a^{{}^{\prime}2}}{a^{2}}+\dfrac{a^{{}^{\prime\prime}}}{a}\right)+\dfrac{1}{2}\phi^{{}^{\prime}2}+V\right]=$
$\displaystyle C_{t},$ (62) $\displaystyle
a^{2}\left[3\left(\dfrac{a^{{}^{\prime}2}}{a^{2}}+\dfrac{a^{{}^{\prime\prime}}}{a}\right)+\dfrac{1}{2}\phi^{{}^{\prime}2}+V\right]=$
$\displaystyle C_{x},$ (63)
$\displaystyle\dfrac{a^{2}}{3}\left[6\dfrac{a^{{}^{\prime}2}}{a^{2}}-\dfrac{1}{2}\phi^{{}^{\prime}2}+V\right]=$
$\displaystyle C_{w},$ (64)

where by considering Eq. (56), we can rewrite them as

$\displaystyle
3\left(\dfrac{a^{{}^{\prime\prime}}}{a}+\dfrac{a^{{}^{\prime}2}}{a^{2}}\right)-\dfrac{3\lambda}{2a^{2}}=$
$\displaystyle-\dfrac{1}{2}\phi^{{}^{\prime}2}-V,$ (65)
$\displaystyle\dfrac{6a^{{}^{\prime}2}}{a^{2}}-\dfrac{3\lambda}{a^{2}}=$
$\displaystyle\dfrac{1}{2}\phi^{{}^{\prime}2}-V,$ (66)
$\displaystyle\phi^{{}^{\prime\prime}}+4\dfrac{a^{{}^{\prime}}}{a}\phi^{{}^{\prime}}-\dfrac{\partial
V}{\partial\phi}=$ $\displaystyle 0.$ (67)

Now, by using Eqs. (65) and (66) and some algebraic calculations, one can
obtain

$\displaystyle V=$
$\displaystyle-\dfrac{3}{2}\left(\dfrac{a^{{}^{\prime\prime}}}{a}+\dfrac{3a^{{}^{\prime}2}}{a^{2}}-\dfrac{3\lambda}{2a^{2}}\right),$
(68) $\displaystyle\phi^{{}^{\prime}2}=$ $\displaystyle
3\left(\dfrac{a^{{}^{\prime}2}}{a^{2}}-\dfrac{a^{{}^{\prime\prime}}}{a}-\dfrac{\lambda}{2a^{2}}\right).$
(69)

It is obvious that obtaining an exact analytical solution for the above
equations is hard. Therefore, at first we try to obtain a numerical solution
for the warp functions by assuming the following ansatz for kink like scalar
field

$\phi=\tanh(w).$ (70)

Inserting Eq. (70) into Eq. (69) and using ODE plot, one can plot Fig. 1 for
the warp function in the case of negative $\lambda$.

Figure 1: Numerical plot of warp function in terms of $w$ for $\lambda=-1$.

Now, we try to obtain an analytical approximation solution for the large and
small values of $w$. The corresponding approximate equations are

$\displaystyle
3\dfrac{a^{{}^{\prime\prime}}}{a}+\phi_{s}^{{}^{\prime}}=0,\hskip
28.45274ptw\rightarrow 0$ (71)
$\displaystyle\dfrac{3\lambda}{2a^{2}}+\phi_{l}^{{}^{\prime}}=0,\hskip
28.45274ptw\rightarrow\infty.$ (72)

and

$\displaystyle\dfrac{3a^{{}^{\prime\prime}}}{2a}+V_{s}=0,\hskip
28.45274ptw\rightarrow 0$ (73)
$\displaystyle\dfrac{9\lambda}{4a^{2}}-V_{l}=0,\hskip
28.45274ptw\rightarrow\infty.$ (74)

In order to solve these equations according to Fig. 1 for $\lambda=-1$, we
assume the following ansatz for the warp function

$a(w)=(w^{2}-\alpha^{2})^{2}.$ (75)

By inserting (75) into Eq. (71), one can get

$\displaystyle\phi_{s}$ $\displaystyle=$
$\displaystyle\dfrac{24}{{\alpha}}\arctan\left(\dfrac{w}{{\alpha}}\right),$
(76) $\displaystyle\phi_{l}$ $\displaystyle=$
$\displaystyle\dfrac{\lambda}{\alpha^{6}(w^{2}-\alpha^{2})^{3}}\left(\alpha
w(33\alpha^{4}-40\alpha^{2}w^{2}+15w^{4})\right.$ (77)
$\displaystyle\left.-15(w^{2}-\alpha^{2})^{3}\arctan(\dfrac{w}{\alpha})\right).$

and

$\displaystyle V_{s}=$
$\displaystyle\dfrac{6}{\alpha^{2}}\left[3-2\cosh^{2}(\alpha\phi)\right]\cosh^{2}(\alpha\phi),$
(78) $\displaystyle V_{l}=$
$\displaystyle\dfrac{9\lambda}{4\alpha^{8}}\cosh^{8}(\alpha\phi).$ (79)

In figures (2), we have plotted the behavior of above functions. As can be
seen in the figure (2a), the approximate scalar field has mirror symmetry
$\phi(w)=-\phi(-w)$. The scalar potential plotted in the Fig. (2a). It can be
seen that the potential is an even function of $w$ ($V(w)=V(-w)$), but it is
unbounded from below. This is a consequence of choosing the warp function (75)
as $w^{4}$ when $|w|\to\infty$.

Figure 2: Plot of $\phi_{s}$ (blue line) and $\phi_{l}$ (red line) in terms of
$w$ for $\lambda=-1$ and $\alpha=5$ (left) plot of $V_{s}$(blue line) and
$V_{l}$ (red line) in terms of $w$ for $\lambda=-1,\alpha=5$ (right).

###  III.2 Case II: $\phi(w,t)=\phi(t):$

By considering a time dependent scalar field, it is straightforward to
simplify the relations of Eq. (46) as follow

$\displaystyle tt-component:\hskip 14.22636pt$
$\displaystyle\dfrac{3}{a^{2}}\left(a^{{}^{\prime}2}+aa^{{}^{\prime\prime}}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}\right)=-\dfrac{1}{2a^{2}}\dot{\phi}^{2}-V,$
(80) $\displaystyle ij-component:\hskip 14.22636pt$
$\displaystyle\dfrac{1}{a^{2}}\left(-2\dfrac{\ddot{b}}{b}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}+3a^{{}^{\prime}2}+3aa^{{}^{\prime\prime}}\right)=\dfrac{1}{2a^{2}}\dot{\phi}^{2}-V,$
(81) $\displaystyle ww-component:\hskip 14.22636pt$
$\displaystyle\dfrac{3}{a^{2}}\left(-\dfrac{\ddot{b}}{b}+2a^{{}^{\prime}2}-\dfrac{\dot{b}^{2}}{b^{2}}-\dfrac{\kappa}{b^{2}}\right)=\dfrac{1}{2a^{2}}\dot{\phi}^{2}-V,$
(82) $\displaystyle scalar\text{ }field\text{ }equation:\hskip 14.22636pt$
$\displaystyle\ddot{\phi}+3\dfrac{\dot{b}}{b}\dot{\phi}+a^{2}\dfrac{\partial
V}{\partial\phi}=0.$ (83)

Subtracting Eq. (81) from Eq. (82), we can obtain

$\dfrac{\ddot{b}}{b}+\dfrac{2\dot{b}^{2}}{b^{2}}+\dfrac{2\kappa}{b^{2}}-3a^{{}^{\prime}2}+3aa^{{}^{\prime\prime}}=0.$
(84)

Separation of time-dependent and $w-$dependent terms, we find

$\displaystyle\dfrac{\ddot{b}}{b}+\dfrac{2\dot{b}^{2}}{b^{2}}+\dfrac{2\kappa}{b^{2}}=$
$\displaystyle c,$ (85) $\displaystyle
3a^{{}^{\prime}2}-3aa^{{}^{\prime\prime}}=$ $\displaystyle c,$ (86)

where $c$ is the separation constant. Now, we can solve Eq. (86) to obtain

$a(w)=c_{1}\sqrt{c}\sin\left(\dfrac{w+c_{2}}{\sqrt{3}c_{1}}\right).$ (87)

in which $c_{1}$ and $c_{2}$ are two integration constants.

Considering Eqs. (80), (82) and (85), we can find

$\dot{\phi}^{2}=6\left(\dfrac{\dot{b}^{2}}{b^{2}}+\dfrac{\kappa}{b^{2}}-\dfrac{c}{3}\right).$
(88)

As the final step, we use Eq. (82), to achieve a constant scalar potential as
follows

$V=-\dfrac{6a^{{}^{\prime\prime}}}{a}=\dfrac{2}{3c_{1}}.$ (89)

which is supported for the flat region of slow-roll inflation. By inserting
the above potential in Eq. (83) and applying the slow-roll inflation criteria
($\dot{\phi}\ll V,\ddot{\phi}\ll 3H\dot{\phi}$), we can find the initial value
for the scalar field $\phi=\phi_{0}$.

According to Eqs. (85) and (88), we plot Fig. 3 to show the behavior of both
the scale factor and scalar field in terms of time for $\kappa=-1$ and
different values of $c$. According to Eq. (87), only the case of $c=1$ is
acceptable since $c=0,-1$ leads to zero and imaginary warp functions. As can
be seen in figure (3a), the scale factor in terms of time has an inflationary
behavior ($b\propto t^{2}$). Also, such a behavior one can see in figure (3b)
for scalar field in terms of time. This behavior is similar to the inflaton
field in high energy Choudhury:2011sq .

Figure 3: Plots of scale factor in terms of $t$ for $\kappa=-1$ and $c=-1$(red
line), $c=0$ (black line) and $c=1$ (blue line)(left) plots of $\phi$ in terms
of $t$ for $\kappa=-1,c=1$ (blue line) and $c=-1$ (red line) and in the inset
for the large value has been plotted (right).

##  IV Conclusion

In this paper, we have presented five-dimensional thick brane solutions
supported by a scalar field. We have briefly addressed the formalism of
canonical gravity and the WD equation as applied to the brane. We have seen
that only in the case of $\kappa=-1$, tunneling occurs which means that the
appropriate classical cosmology subject to quantization is the spatially
paraboloid case.

Then, we studied the cosmology of thick brane for $\kappa=-1$ and in the case
of time (in)dependent scalar field. In each case, we have obtained the
corresponding solutions of both the scale factor and scalar field. It is worth
mentioning that considering a time-dependent scalar field leads to a constant
scalar potential which is supported by the known slow-roll inflation.

We also leave the case where $\phi(w,t)=\phi(w)\varphi(t)$ in appendix for
future consideration.

##  Appendix A Case: $\phi(w,t)=\phi(w)\varphi(t)$

Here we consider the case $\phi(w,t)=\phi(w)\varphi(t)$. By using of scalar
field equation (46), we have

$\displaystyle\dfrac{\dot{\varphi}(t)}{\varphi(t)}+\dfrac{3\dot{b}(t)}{b(t)}\dfrac{\dot{\varphi}(t)}{\varphi(t)}-$
$\displaystyle\left(\dfrac{a^{2}\phi^{{}^{\prime\prime}}(w)}{\phi(w)}+4aa^{{}^{\prime}}\dfrac{\phi^{{}^{\prime}}(w)}{\phi(w)}\right)=$
$\displaystyle-\dfrac{a^{2}}{\varphi(t)\phi(w)}\dfrac{\partial
V}{\partial\phi(w,t)}.$ (90)

Also, by adding $tt$ and $ww$ components of field equations (46), one can
obtain

$\dfrac{3}{a^{2}}\left(3a^{{}^{\prime}2}+aa^{{}^{\prime\prime}}-\dfrac{\ddot{b}}{b}-\dfrac{2\dot{b}^{2}}{b^{2}}-\dfrac{2\kappa}{b^{2}}\right)=-2V.$
(91)

In order to solve the field equation (A) and (91), we should separate the
equations in terms of $t$ and $w$. To this end, one should consider
appropriate choices for $V(\phi)$. As a simple case we considered $V=0$. In
this case the field equations reduce to

$\displaystyle\dfrac{\ddot{b}}{b}+\dfrac{2\dot{b}^{2}}{b^{2}}+\dfrac{2\kappa}{b^{2}}-c=0$
$\displaystyle 3a^{{}^{\prime}2}+aa^{{}^{\prime\prime}}-c=0$ (92)

and

$\displaystyle\ddot{\varphi}(t)+\dfrac{3\dot{b}}{b}\dot{\varphi}(t)-c^{{}^{\prime}}\varphi(t)=0$
$\displaystyle\phi^{{}^{\prime\prime}}(w)+\dfrac{4a^{{}^{\prime}}}{a}\phi^{{}^{\prime}}(w)-\dfrac{c^{{}^{\prime}}}{a^{2}}\phi(w)=0.$
(93)

One can solve the equations (A) for warp function and scale factor and then
insert them in to the equations (A) obtain the scalar fields.

## References

  * (1) T. Kaluza, Sitz. Preuss. Akad. Wiss. Phys. Math. K 1. 966 (1921).
  * (2) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).
  * (3) N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998).
  * (4) I. Antoniadis, S. Dimopoulos and G. Dvali, Nucl. Phys. B 516, 70 (1998).
  * (5) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999).
  * (6) J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000).
  * (7) S. B. Giddings, E. Katz and L. Randall, JHEP 0003, 023 (2000).
  * (8) O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, Phys. Rev. D 62, 046008 (2000).
  * (9) M. Gremm, Phys. Lett. B 478, 434 (2000).
  * (10) C. Csaki, J. Erlich, T. J. Hollowood and Y. Shirman, Nucl. Phys. B 581, 309 (2000).
  * (11) A. Kehagias and K. Tamvakis, Phys. Lett. B 504, 38 (2001).
  * (12) M. Gremm, Phys. Rev. D 62, 044017 (2000).
  * (13) V. Dzhunushaliev, Grav. Cosmol. 13, 302 (2007).
  * (14) V. Dzhunushaliev, V. Folomeev, S. Myrzakul and R. Myrzakulov, Mod. Phys. Lett. A 23, 2811 (2008).
  * (15) M. Gremm, Phys. Lett. B 478, 434 (2000).
  * (16) A. Wang, Phys. Rev. D 66, 024024 (2002).
  * (17) D. Bazeia, F. A. Brito and J.R. Nascimento, Phys. Rev. D 68, 085007 (2003).
  * (18) D. Bazeia, C. Furtado and A. R. Gomes, JCAP 0402, 002 (2004).
  * (19) S. Kobayashi, K. Koyama and J. Soda, Phys. Rev. D 65, 064014 (2002).
  * (20) A. Melfo, N. Pantoja and A. Skirzewski, Phys. Rev. D 67, 105003 (2003).
  * (21) C. Barcel´o, C. Germani and C.F. Sopuerta, Phys. Rev. D 68, 104007 (2003).
  * (22) H. Guo, Y.-X. Liu, S.-W. Wei and C.-E. Fu, Europhys. Lett. 97, 60003 (2012).
  * (23) K.A. Bronnikov and B.E. Meierovich, Grav. Cosmol. 9, 313 (2003).
  * (24) A. Karch, L. Randall, JHEP, 0105, 008 (2001).
  * (25) P. D. Mannheim, Brane-localized Gravity, (World Scientific Publishing Company, Singapore) 2005; SenGupta S., Aspects of warped brane world models, arXiv:0812.1092[hep-th].
  * (26) J. A. Wheeler, Ann. Phys. (N.Y.) 2, 604 (1957).
  * (27) B. S. DeWitt, Phys. Rev. 160, 1113 (1967).
  * (28) J. B. Hartle, S. W. Hawking, Phys. Rev. D 28, 2960 (1983).
  * (29) S. W. Hawking, Nucl. Phys. B 239, 257 (1984).
  * (30) S. W. Hawking, D. N. Page, Nucl. Phys. B 264, 185 (1986).
  * (31) J. J. Halliwell, S. W. Hawking, Phys. Rev. D 31, 1777 (1985).
  * (32) A. Vilenkin, Phys. Lett. B 117, 25 (1982).
  * (33) A. Vilenkin, Phys. Rev. D 27, 2848 (1983).
  * (34) A. Vilenkin, Phys. Rev. D 30, 549 (1984).
  * (35) A. Vilenkin, Phys. Rev. D 37, 888 (1988).
  * (36) F. Darabi, W. N. Sajko and P. S. Wesson, Class. Quant. Grav. 17, 4357 (2000).
  * (37) S. H. Hendi, N. Riazi and S. N. Sajadi, Phys. Rev. D 102, no.12, 124034 (2020).
  * (38) A. Ahmed, B. Grzadkowski and J. Wudka, JHEP 1404, 061 (2014).
  * (39) A. Wang, Phys. Rev. D 66, 024024 (2002).
  * (40) V. Dzhunushaliev, V. Folomeev and M. Minamitsuji, Rept. Prog. Phys. 73, 066901 (2010).
  * (41) S. Choudhury and S. Pal, Phys. Rev. D 85, 043529 (2012).


# Comment on Deterministic Information Bottleneck

Sarah E. Marzen<EMAIL_ADDRESS>W. M. Keck Science Department, Claremont CA
91711

###### Abstract

We make the case that although Deterministic Information Bottleneck may be a
contribution to clustering, it should not be used to aid lossy compression
without the addition of blocklength. We therefore suggest a new objective
function that does so and leave its testing to future work.

–

###### pacs:

02.50.-r 89.70.+c 05.45.Tp

††preprint: arxiv.org:1702.08565 [physics.gen-ph]

In $2017$, the “Deterministic Information Bottleneck” (DIB) method was
published and is now highly cited. The basic idea is that according to the
information bottleneck method [1], an extension of rate-distortion theory [2]
to an informational distortion measure, one should minimize expected
distortion and also minimize the mutual information between the compressed
signal and the original variable at the same time. The information bottleneck
method is great for clustering, e.g. as in Ref. [3], and it also is a
contribution to information theory [4], but if one is not careful, one might
misunderstand where the objective function comes from. In this short arXiv
paper, we simply aim to describe where these rate-distortion objectives come
from and why DIB is a contribution to clustering but not to information theory
as it stands. We then modify the objective so that it might be a contribution
to information theory and leave the testing for later work.

From one of my own papers [5]: “In the classic rate-distortion setup, one
sends a sequence of $n$ letters $x_{0:n}$ to an encoder that chooses one of
$M$ words for those $n$ letters and then sends that word to a decoder which
produces a guess as to what those letters were, $\hat{x}_{0:n}$. The material
constraint is actually $\log M/n$, not a mutual information. This corresponds
to a more intuitive notion of resource constraints in the biological sense–
number of molecules or number of neurons, normalized by “blocklength” $n$.
Some distortion measure is defined, $d(x,\hat{x})$, which in some extensions
can be a distortion of the entire block $x_{0:n}$ relative to $\hat{x}_{0:n}$
rather than letter-by-letter. There are some rates $\log M/n$ and distortions
$\sum d(x_{i},\hat{x}_{i})/n$ that are achievable and some that are
unachievable given any combination of encoder and decoder. A theorem shows
that the curve separating achievable from unachievable is given by replacing
the rate $\log M/n$ with a mutual information $I[X;\hat{X}]$ and the average
distortion with an expected distortion if all is memoryless. This curve is
accurate in the limit that blocklength $n$ goes to infinity. Otherwise, the
rate-distortion curve that separates achievable from unachievable is given by
$R_{n}(D)$ rather than $R(D)$, and $R_{n}(D)$ is horribly difficult to
calculate [2].”

In the information bottleneck method, one replaces $d(x,\hat{x})$ with an
informational distortion measure (a Kullback-Leibler divergence), which does
not correspond to the classic rate-distortion setup, in which $d(\hat{x},x)$
does not depend on $P(Y|\hat{X}=\hat{x})$. As a result, the objective in the
information bottleneck method becomes $I[R;Y]-\beta I[X;R]$, where $R$ is the
compressed variable, $X$ is the variable that we compress, and $Y$ is the
variable that we try to retain information about. There is a rate-distortion
theorem justifying this objective [4].

The wonderful thing about the rate-distortion theorem is that it allows one to
test if a sensor is near-optimal. One calculates the rate and expected
distortion and places it relative to the rate-distortion curve, as in Ref.
[6]. It is also known that the actual soft clusters obtained in order to get
the rate-distortion curve are terrible for coding, as their coding rate is
given by $H[R]$ rather than by $I[R;X]$ [2], although they may be useful for
machine learning as stated earlier. As such, they should not be used as a null
model to decide if an information-theoretic objective is successful at
producing good codes.

DIB seems to be a fix to the need for clusters with a good coding rate. In
DIB, we replace $I[X;R]$ with $H[R]+\lambda H[R|X]$ for some other $\lambda$
and we take the limit of determinism, where $\lambda$ becomes very large.
However, there is no rate-distortion theorem justifying this objective as
revealing sensors that are closer to optimal, and in fact, there should not
be, as the objective that reveals if a sensor is close to the boundary between
achievable and unachievable is the original information bottleneck objective
function.

We might ask– does DIB lead to good information theory codes? The objective
indeed leads to deterministic codes. Deterministic codes are what one would
desire for lossy compression, but it is well-known that the best codes often
require large blocklengths $n$, though there are exceptions. We therefore
simply propose that perhaps the objective function of $\langle
d(\hat{x}_{0:n},x_{0:n})\rangle+\beta H[R]+\alpha H[R|X_{0:n}]-\lambda
H[\hat{X}_{0:n}|R]$ should be investigated for information theorists, despite
the computational overhead that is incurred by including blocklength.

All of this does not mean that DIB is not great at getting interesting
clusters for machine learning and data science purposes.

## References

  * [1] Naftali Tishby, Fernando C Pereira, and William Bialek.  The information bottleneck method.  arXiv preprint physics/0004057, 2000.
  * [2] Thomas Berger.  Rate distortion theory: A mathematical basis for data compression.  Prentice-Hall, Inc., 1971.
  * [3] Susanne Still, William Bialek, and Léon Bottou.  Geometric clustering using the information bottleneck method.  Advances in neural information processing systems, 16, 2003.
  * [4] Michael Gastpar, Bixio Rimoldi, and Martin Vetterli.  To code, or not to code: Lossy source-channel communication revisited.  IEEE Transactions on Information Theory, 49(5):1147–1158, 2003\.
  * [5] Sarah Marzen.  Resource-rational reinforcement learning and sensorimotor causal states.  arXiv preprint arXiv:2404.18775, 2024.
  * [6] Stephanie E Palmer, Olivier Marre, Michael J Berry, and William Bialek.  Predictive information in a sensory population.  Proceedings of the National Academy of Sciences, 112(22):6908–6913, 2015.