# Doppler-assisted quantum resonances through swappable excitation pathways in
Potassium vapor

Gourab Pal
Raman Research Institute
Bangalore, India
<EMAIL_ADDRESS>
&Subhasish Dutta Gupta
TIFR Hyderabad, IISER Kolkata
IIT Jodhpur
<EMAIL_ADDRESS>
&Saptarishi Chaudhuri
Raman Research Institute
Bangalore, India
<EMAIL_ADDRESS>

###### Abstract

We report the observation of two additional sub-natural line width quantum
interference in the $D_{2}$ manifold of ${}^{39}K$ vapor, in addition to the
usual single Electromagnetically induced transparency peak. The other two
features appear exclusively because ${}^{39}K$ ground hyperfine splitting is
smaller than the Doppler broadened absorption profile. This allows probe and
control beams to swap their transition pathways. The control beam detuning
captures the nature of the coherence, therefore an unusual phenomenon of
conversion from perfect transparency to enhanced absorption is observed and
explained by utilizing adiabatic elimination of the excited state in the
Master equation. Controlling such dark and bright resonances leads to new
applications in quantum technologies viz. frequency offset laser stabilization
and long-lived quantum memory.

##  1 Introduction

For the past few decades, manipulating and controlling an atomic medium’s
optical responses [1] using quantum interference across excitation channels
has been a versatile field of study. One of the most fascinating quantum
interference phenomena is electromagnetically-induced transparency (EIT) [2,
3, 4], which modifies the medium’s dispersion characteristics dramatically and
opens the door to several cutting-edge uses in quantum information processing
[5], including the creation of high-precision quantum sensors [6], atomic
clocks [7], magnetometers [7], etc. As an application to quantum memory [8],
it is feasible to slow down [9] or store photons [10] for a considerable time
due to the steep anomalous dispersion in an EIT medium. The EIT medium can be
used for creating entangled bi-photons[11] having application in quantum
communication. More recently, exotic effects Goos-Hänchen [12], and Imbert-
Fedorov shifts [13] have been observed employing EIT medium.

Figure 1: (Up) Schematic of experimental setup. Essential optical components
are labeled in the figure. The inset shows a typical three levels
$\Lambda$-system with relevant parameters. (Down) All four $\Lambda$-type
configurations are supported due to small ground state splitting (see text) in
${}^{39}K$ atomic ensemble. Here ground states are $\ket{F=1}$ (indicated by a
thin line) and $\ket{F=2}$ (indicated by a relatively thick line). (color
online)

Numerous experiments have been carried out in diverse atomic systems such as
${}^{87}Rb$[14], ${}^{133}Cs$ [15], ${}^{23}Na$ [16], metastable He [17],
Erbium-doped fiber [18], molecular Lithium [19], EIT with Rydberg atoms [20,
21]. These systems exhibit strong quantum interference phenomena (like EIT)
due to their wide ground state separation and considerable energy spacing
among excited states. Furthermore, in a typical atomic vapor, the Doppler
broadening is much smaller than the ground state separation, leading to the
observation of just one EIT peak. On the other hand, the ground state
separation in ${}^{39}K$ is smaller (461.7MHz [22]) than the Doppler
broadening at room temperature which makes the study of EIT in ${}^{39}K$
intriguing. There exist very few quantum interference-based experiments in
${}^{39}K$, Long et.al. [23] used a chirped waveform electro-optic modulator
to investigate the EIT, A. Sargsyan et al. [24] demonstrated on-resonance EIT
and Gozzini et.al [25] observed EIT-EIA transition in Hanle configuration with
polarization as a tuning parameter. In this article, we experimentally
demonstrate the emergence of three distinct quantum interference line shapes
in a $\Lambda$-type system in ${}^{39}K$ atomic ensemble. The ground-state
hyperfine splitting being smaller than the unresolved two overlapping Doppler
absorption profiles, the excitation pathways of the control and probe beam can
be swapped. We also observe a transition from complete transparency to
enhanced absorption as we vary the control beam detuning from the blue side to
the red side of its resonance point, while the two-photon detuning condition
is satisfied.

Figure 2: (Left) The usual case $C_{1}$ with two-photon detuning
$\Delta_{c}-\Delta_{p}=0$. (Right) Control and probe transition pathways are
swapped. In the swapped frame (case $C_{2}$),
$\Delta^{{}^{\prime}}_{c}-\Delta^{{}^{\prime}}_{p}=0$. See text for details.
(color online)

##  2 Experimental setup and methods

Figure 1 (Upper panel) shows a schematic of the experiment. Two independent
grating stabilized diode lasers (Toptica DL Pro and DL100) are used as the
control and probe beam. The probe and control lasers are tuned near F=1 to F’
and F=2 to F’ respectively in the $D_{2}$ manifold of ${}^{39}K$, with a
typical laser line width of 1MHz. Two separate acousto-optic modulators are
used to independently control the intensity and detuning of the probe and
control beams. The frequency reference is obtained using two saturated
absorption spectroscopy, see supplementary [26]. A 75 mm long and 25mm
diameter spectroscopy-grade commercial vapor cell containing ${}^{39}K$ (in
natural abundance, $\sim 93\%$) is placed inside a three-layered $\mu$-metal
jacket to efficiently shield any stray magnetic field. The probe beam and
control beams are mode-filtered using single mode polarization maintaining
fibers and passed through two independent telescopic arrangements with exit
beam diameters of 1mm and 4mm, respectively. The two beams are prepared in
orthogonal linear polarization states and merged using a polarizing beam
splitter cube before being sent to the atoms. The probe laser is kept in
frequency scan mode whereas the control laser frequency is kept at various
detunings. In the end, the control beam is filtered out using a linear
polarizer (extinction ratio $1:10^{5}$), and the signal is obtained using a
photo-detector (Thorlabs PDA10A2) with 150MHz bandwidth. A high-speed
oscilloscope (2Gs/sec) is used for visualization and data recording.

Figure 3: (Upper panel) Observation of three bright resonances for
$\Delta_{c}=230.85$ MHz. The inset shows the zoomed views. (Lower panel) On-
resonant ($\Delta_{c}=0$) bright resonance line shapes (a,b,c). The orange
line indicates experimental data and the blue line is the theoretical model
fit. Subplot (d) shows the admixture of dark and bright resonance at
$\Delta_{p}=-461.7$ MHz. (color online)

##  3 Theoritical description

A semi-classical approach with density matrix formalism is applied to this
three-level $\Lambda$ system as shown in the inset of figure 1 (Upper panel).
The two ground states are $\ket{F=1}$ and $\ket{F=2}$ hyperfine levels in
${}^{2}S_{1/2}$ and the excited state is taken as unresolved
$\ket{F^{\prime}}$ in ${}^{2}P_{3/2}$ containing F’=0,1,2 and 3 states. The
Master equation that governs the dynamics of a mixed state is,
$\dot{\rho}=-\frac{i}{\hbar}[H,\rho]+\mathcal{L}\rho$. Here $\rho$ is the
density matrix of the system, and H is the EIT Hamiltonian [3]. The last term
$\mathcal{L}\rho$, is the pertinent Lindbladian term that controls all
potential decay processes such as leaky transitions from other excited states,
spontaneous emission, and decoherence resulting from Doppler-broadening. Such
decay terms account for the interactions with the environment and thus make
the system’s dynamics non-unitary. In our case, the dominant decay processes
are due to Doppler broadened spontaneous emission and energy-conserving
dephasing effects like collision of atoms with walls, and atom-atom
collisions.

The non-unitary Lindbladian term, see supplementary [26], is
$\mathcal{L}\rho=\Gamma_{31}\mathcal{D}[\hat{\sigma}_{1,3}]\rho+\Gamma_{32}\mathcal{D}[\hat{\sigma}_{2,3}]\rho+\gamma_{2}\mathcal{D}[\hat{\sigma}_{2,2}]\rho+\gamma_{3}\mathcal{D}[\hat{\sigma}_{3,3}]\rho$,
where $\Gamma_{ij}$ denotes spontaneous decay rate from state i to state j,
$\gamma_{i}$ is a energy-conserving dephasing term for state i,
$\hat{\sigma}_{i,j}$ is a jump operator defined as
$\hat{\sigma}_{i,j}=\ket{i}\bra{j}$ and $\mathcal{D}$ is coined as Lindblad
super-operator whose action is defined as
$\mathcal{D}[A]B=ABA^{\dagger}-\frac{1}{2}\\{A^{\dagger}A,B\\}$ for any two
operators A and B. In light of such formalism, the susceptibility seen by the
probe $\chi(\omega_{p})$ can be obtained by solving coherence terms under
steady-state conditions as,

$\chi(\omega_{p})=\frac{2n|d_{13}|}{\epsilon_{0}E_{p}}\frac{\Omega_{p}[2i\Delta+\gamma_{2}]}{-4i\Delta_{p}\Delta+i\Omega_{c}^{2}-2\Delta_{p}\gamma_{2}+i\Gamma_{3}[2i\Delta+\gamma_{2}]}$
(1)

Figure 4: (Upper panel) In the red-side of control beam resonance, subplot a
shows complete transparency due to case $C_{2}$, subplot b shows full
absorption due to case $C_{1}$. (Lower panel) The opposite parity line shapes
are seen with blue-detuned control light as shown in subplot c and d.

Here, n is the density of atoms at cell temperature T, $|d_{13}|$ is the
dipole matrix element, $E_{p}$ is probe field
amplitude,$\Gamma_{3}=\Gamma_{32}+\Gamma_{31}+\gamma_{3}$ is the total
decoherence of excited state, $\gamma_{2}$ indicates ground state decoherence,
$\gamma_{3}$ is excited state energy conserving decoherence, $\Omega_{p}$,
$\Omega_{c}$ are probe and control Rabi flopping frequencies, respectively and
$\Delta_{p}$, $\Delta_{c}$ and $\Delta=\Delta_{p}-\Delta_{c}$, are respective
probe detuning, control detuning and two-photon detuning from a given
transition (see figure 1).

Figure 5: Transition from bright resonance to dark resonance as we vary the
control detuning for the case $C_{1}$. The x-axis is the probe detuning,
y-axis represents probe transmission. Inset-box shows the corresponding
$\Delta_{c}$ values. (color online)

##  4 Results and discussions

The ground state splitting of the Potassium atom is 461.7 MHz which is smaller
than the unresolved two Doppler broadened absorption profiles (684 MHz [26] at
a cell temperature of $60^{o}$C). Under this circumstance, the usual EIT peak
with $\Delta_{c}=0$ is observed which is the standard case $C_{1}$ in figure
1. However, since there exists a significant number of atoms with velocities
such that a Doppler shift can be higher than ground-state hyperfine splitting,
the excitation pathways of the control and probe beam can now be swapped. The
swapping mechanism is illustrated in figure 2 where, as an example,
$\Delta_{c}=-923.4$MHz is taken. Before swapping,
$\Delta_{c}=\Delta_{p}=-923.4$ MHz, so the two-photon detuning is satisfied.
However after swapping, $\Delta^{{}^{\prime}}_{c}=\Delta_{c}-461.7=-1385.1$
and $\Delta^{{}^{\prime}}_{p}=\Delta_{p}+461.7$. Since the probe laser is
scanning, the probe frequency is adjusted to $\Delta_{p}=-1846.8$ so that
$\Delta^{{}^{\prime}}_{p}=-1385.1$ and the two-photon detuning condition is
satisfied in this swapped frame. A third line shape is obtained by applying
the same arguments but with control and probe beam making a valid $\Lambda$
from the same hyperfine state either $\ket{2}$ (for the case $C_{3}$) or
$\ket{1}$ (for the case $C_{4})$ [26]. See figure 1 (lower panel) for all four
possible configurations. There are multiple $\Lambda$-system within a given
case, considering all possible magnetic sublevels. We have considered all such
transition probabilities with their corresponding Clebs-Gordon coefficients.
The resulting line shapes and their positions remain largely unaffected as
discussed in [26].

As a representative dataset, figure 3 (Upper panel) shows all three EIT line
shapes riding on the Doppler broadened absorption profile, here the control
detuning is set to $\Delta_{c}=230.85$ MHz. The corresponding background-
subtracted features are zoomed in the insets. The relevant physics is well
captured by introducing correct dephasing terms. The figures (a,b,c) in figure
3 (lower panel) show all possible bright resonances with corresponding
theoretical model fit. The subplot (c) is the usual single EIT peak, which is
also observed in other atoms like Rb, Cs, etc. The characteristic variation of
EIT line width with $\Omega_{c}$ for the case of $\Delta_{c}=-923.4$ MHz is
also studied with theoretical fitting (see supplementary [26]) as presented in
the susceptibility expression. The inclusion of an overlapping Doppler
broadened profile enables us to exchange the excitation pathways of control
and probe which give two additional features (a and b). The asymmetric
mismatch between theory and experiment is accounted for non-zero probe
detuning and the existence of multiple excited states [27, 28] in the case of
${}^{39}K$. The asymmetry primarily originates from the admixture of pure dark
state and pure bright state as can be seen in subplot (d) of figure 2 where
the dip is observed due to bright resonance and the peak is due to dark
resonance.

When the control beam detuning is taken to far-off red detuned,
$\Delta_{c}=-923.4$MHz, a complete absorption and complete transmission line
shapes are observed at probe detunings $\Delta_{p}=-923.4$MHz and
$\Delta_{p}=-1846.8$MHz respectively. For blue-detuning,
$\Delta_{c}=1385.1$MHz, complete absorption, and transmission are seen in
$\Delta_{p}=461.7$MHz and $\Delta_{p}=1385.1$MHz respectively as shown in
figure 4. For the cases of far-off resonant complete absorption, we have
adiabatically eliminated [29] the state $\ket{F^{\prime}}$ from the Master
equation to get the correct line shape by switching off the spontaneous decay
in the Lindblad term. This is a reasonable description because, at far-off
resonance from the one-photon transition the lasers do not significantly
populate the excited state thereby reducing the spontaneous decay from the
excited state. The theoretical model fit gives effective $\gamma_{2}$ values
ranging from 1.2 MHz to 2.5 MHz depending on the line shape position. The
other variable parameters $\gamma_{3},\Gamma_{31},\Gamma_{32}$ are adjusted
according to the probe and control detunings. For a given line shape, if it
appears inside the Doppler, then the dephasing will be controlled by the
spontaneous decay term and if it appears outside the Doppler, then the
dephasing will come primarily from $\gamma_{3}$. As an example, for
$\Delta_{c}=-923.4$ MHz (case $C_{3}$), the values of $\gamma_{2}=1.2$ MHz,
$\Gamma_{31}=\Gamma_{32}=0$ and $\gamma_{3}=342$ MHz.

Figure 6: Ad-mixture parameter $\eta$ as a function of control detuning
$\Delta_{c}$. As $\Delta_{c}$ is swept from the red side to the blue side,
complete transparency is obtained due to the formation of dark resonance,
consequently, the bright state ad-mixture parameter $\eta$ vanishes after
$\Delta_{c}\geq 0$.

Figure 5 (a-i) shows a transition from bright resonance to dark resonance as
we vary the control beam detuning keeping the probe laser scanning across the
whole $D_{2}$ manifold. The blue side of this case always gives bright
resonance as can be seen from (f-i) of figure 5. The line-plot colors are
chosen to represent the detuning of the control beam. These line shapes are
obtained for the case $C_{1}$. A similar transition can be found in other
cases also. To indicate the extent of mixing of bright and dark resonance, the
ad-mixing parameter $\eta$ is defined as the ratio of bright resonance
contrast (defined as b) and dark resonance contrast (defined as d). The
variation of $\eta$ is plotted against the control beam detuning which gave
exponential crossover from bright resonance to dark resonance as we sweep the
control beam detuning as shown in figure 6.

##  5 Conclusion and outlook

By choosing both the Doppler and exchanging the probe and control beam
excitation routes, we have been able to explain all possible line shapes and
positions and they match well with the density matrix formalism-based
theoretical model. The adiabatic elimination works well to describe the bright
resonance in blue as well as red detuning regions. The exact line shape can be
matched by taking into account the intervention of other excited states as
described in [30]. Going beyond the Gaussian probe and control, a probe beam
carrying orbital angular momentum (Laguerre Gaussian mode) can have a
fundamentally unique response [31] to this EIT medium because of the spatial
variation of Rabi frequency that can lead to line shape narrowing [32] as well
as change in asymmetries. Another novel avenue to explore EIT in Potassium
will be to study cold Potassium atoms [33] in a strong magnetic field where
the external magnetic field can couple the Zeeman sub-levels of different
hyperfine states. The explanation of all these quantum interference line
shapes enriches the understanding of quantum coherence and has applications to
frequency-offset tight laser locking [34] and quantum technology applications
such as optical isolators [35] quantum memory [8], slow light [36].

Appendices

##  Appendix A The Master equation

In the density matrix formalism, the Von Neumann equation governs the time
evolution of mixed-state density matrix $\rho$

$\dot{\rho}=-\frac{i}{\hbar}[H,\rho]$ (2)

Here H is the EIT Hamiltonian [3] for our 3-level $\Lambda$ system as
described in the main text.

$H=-\frac{i}{\hbar}\begin{bmatrix}0&0&\Omega_{p}\\\
0&2(\Delta_{p}-\Delta_{c})&\Omega_{c}\\\
\Omega_{p}&\Omega_{c}&2\Delta_{p}\end{bmatrix}$ (3)

In atomic systems, there are many decay processes such as spontaneous
emission, dephasing, or decoherence due to inhomogeneous magnetic fields,
ground state collisions, spin-exchange collisions, and so on. Such terms make
the above Hamiltonian non-unitary. We can make a Lindblad term that needs to
be added to the above Hamiltonian to correctly describe our present system.
The irreversible Lindblad term can be written as

$\mathcal{L}\rho=\Gamma_{31}\mathcal{D}[\hat{\sigma}_{1,3}]\rho+\Gamma_{32}\mathcal{D}[\hat{\sigma}_{2,3}]\rho+\gamma_{2}\mathcal{D}[\hat{\sigma}_{2,2}]\rho+\gamma_{3}\mathcal{D}[\hat{\sigma}_{3,3}]\rho$
(4)

Figure 7: Representation of overlapping Doppler broadened absorption profiles.
The thick grey line is the experimental data, the red and green are individual
Doppler absorption fitted by Gaussian, and the blue line is the convolution of
two Doppler. (color online)

where $\Gamma_{ij}$ denotes spontaneous decay from state i to state j,
$\gamma_{i}$ is a energy-conserving dephasing term for state i,
$\hat{\sigma}_{i,j}$ is a jump operator defined as
$\hat{\sigma}_{i,j}=\ket{i}\bra{j}$ and $\mathcal{D}$ is coined as Lindblad
super-operator whose action is defined as
$\mathcal{D}[A]B=ABA^{\dagger}-\frac{1}{2}\\{A^{\dagger}A,B\\}$ for any two
operators A and B. Setting the basis as
$\\{[1,0,0]^{T},[0,1,0]^{T},[0,0,1]^{T}\\}$, the jump operators can be easily
constructed. Upon simplification, the Lindblad term becomes

$\mathcal{L}\rho=-\frac{1}{2}\begin{bmatrix}0&\gamma_{2}\rho_{12}&(\Gamma_{31}+\Gamma_{32}+\gamma_{3})\rho_{13}\\\
\gamma_{2}\rho_{12}^{*}&0&(\Gamma_{31}+\Gamma_{32}+\gamma_{3}+\gamma_{2})\rho_{23}\\\
(\Gamma_{31}+\Gamma_{32}+\gamma_{3})\rho_{13}^{*}&(\Gamma_{31}+\Gamma_{32}+\gamma_{3}+\gamma_{2})\rho_{23}^{*}&0\end{bmatrix}$
(5)

Here in $\rho$ matrix, we need to set $\rho_{11}=1$, $\rho_{22}=0$ and
$\rho_{33}=0$.

Figure 8: Saturated Absorption Spectroscopy of ${}^{39}K$ used for the
conversion of time scales in oscilloscope to frequency scale.

##  Appendix B Steady state solution of coherence terms

After adding Lindbladian to the Von Neumann equation, we can read out two
required coherence terms

$\dot{\rho}_{12}=[-\frac{1}{2}\gamma_{2}+i(\Delta_{c}-\Delta_{p})]\rho_{12}-\frac{1}{2}i\Omega_{c}\rho_{13}$
(6)

$\dot{\rho}_{13}=[-\frac{1}{2}(\Gamma_{31}+\Gamma_{32}+\gamma_{3})-i\Delta_{p}]\rho_{13}-\frac{1}{2}i\Omega_{c}\rho_{12}-\frac{1}{2}i\Omega_{p}$
(7)

In steady state conditions, we can set $\dot{\rho}_{ij}=0$, which gives two
linear equations that can be solved analytically. The solution for $\rho_{13}$
is directly related to the susceptibility seen by the probe.

$\chi_{p}\sim\frac{\Omega_{p}[2i(\Delta_{p}-\Delta_{c})+\gamma_{2}]}{-4i\Delta_{p}(\Delta_{p}-\Delta_{c})+i\Omega_{c}^{2}-2\Delta_{p}\gamma_{2}+i(\Gamma_{31}+\Gamma_{32}+\gamma_{3})[2i(\Delta_{p}-\Delta_{c})+\gamma_{2}]}$
(8)

In the main text, we only plot the probe transmission in various detunings
which is the imaginary part of $\chi(p)$.

##  Appendix C Overlapping Doppler broadened absorption profile in ${}^{39}K$

The ground state hyperfine splitting is 461.7MHz. Now, at 333K, the Doppler
width is 342 MHz, calculated using the formula
$\sigma_{D}=\sqrt{\frac{k_{B}T}{mc^{2}}}\omega_{p}$, where $k_{B}$ is
Boltzmann constant, m is atomic mass, c is the velocity of light and
$\omega_{p}$ is the probe frequency in Hz.

Figure 7 shows the convolution of two Doppler broadened absorptions for
$\ket{F=1}\rightarrow\ket{F^{\prime}}$ and
$\ket{F=2}\rightarrow\ket{F^{\prime}}$. For theoretical modeling, we have
taken the Doppler decoherence $\gamma_{3}=342$ along with total spontaneous
decay of excited state $\Gamma_{3}=\Gamma_{32}+\Gamma_{31}=684$ MHz when
dealing with line shapes inside any of the Doppler.

Figure 9: All four possible probe-control excitation configurations.

##  Appendix D Setting frequency reference

Doppler-free Saturated Absorption Spectroscopy [37] is performed using an
independent vapor cell. The blue line in figure 8 shows the hyperfine
transition lines where the strongest peak corresponds to
$\ket{F=2}\rightarrow\ket{F^{\prime}}$ transition, the weakest peak
corresponds to $\ket{F=1}\rightarrow\ket{F^{\prime}}$ transition and the
middle dip is the crossover resonance in the spectroscopy of ${}^{39}K$. The
orange line shows a typical dark resonance. The separation between the two
peaks in the spectroscopy refers to the ground hyperfine splitting as

$|T_{\ket{F=2}\rightarrow\ket{F^{\prime}}}-T_{\ket{F=1}\rightarrow\ket{F^{\prime}}}|=461.7MHz$
(9)

From the above equation, all time scales from the oscilloscopes can be
converted to frequency scale.

Figure 10: Demonstration of multiple $\Lambda$ systems for two types of
polarization of light. Blue lines are used for $\pi$ polarized light and red
lines are used for $\sigma$ polarized light. Only a few $\Lambda^{\prime}s$
are shown here for clarity. (color online)

##  Appendix E Three line shapes

Lets denote the probe susceptibility in equation 8 as
$\chi(\Delta_{p},\Delta_{c})$. Case $C_{1}$ is the standard case. For case
$C_{2}$, we can exchange the role of probe and control in the following way
$\chi(\Delta_{p},\Delta_{c})\rightarrow\chi(\Delta_{p}+461.7,\Delta_{c}-461.7)$.
This new configuration still satisfies the two-photon resonance condition. In
the same argument, the susceptibility for case $C_{3}$ is
$\chi(\Delta_{p}+461.7,\Delta_{c})$ and for case $C_{4}$ is
$\chi(\Delta_{p},\Delta_{c}-461.7)$. Finally, the net absorption can be
written as

$\alpha_{p}=\Sigma_{i=1}^{4}\sum_{i=1}^{k}x_{i}\int_{v=-\infty}^{+v=\infty}Im(\chi)_{i}\mathcal{P}(v)\,dv$
(10)

Here i is for case $C_{i}$ and $\mathcal{P}$ is the 1D Maxwell-Boltzmann
velocity distribution given by

$\mathcal{P}=\frac{1}{\sqrt{2\pi}D}e^{-\frac{v^{2}}{D^{2}}}$ (11)

where $D=\sqrt{\frac{2k_{B}T}{m}}$ is the most probable velocity. Figure 9
depicts all four cases.

##  Appendix F Contribution of all possible $\Lambda$-system due to magnetic
sub levels

Apart from having four different case as shown in figure 9, there are multiple
$\Lambda$ systems possible within a given case due to the existence of near-
degenerate magnetic sub-levels of each hyperfine states. Considering the
configuration of case $C_{1}$, few $\Lambda$’s are shown in figure 10 where we
use blue lines for linear polarized light and red lines for circular polarized
light. Since we use linear polarization in our experiment, and a linear
polarization state is a combination of right and left circular polarizations,
all types of probe and control polarization choices need to be taken into
account such as
$Probe(\pi)-Control(\pi),Probe(\sigma^{+})-Control(\sigma^{+}),Probe(\sigma^{-})-Control(\sigma^{-}),Probe(\pi)-Control(\sigma^{+}),Probe(\pi)-Control(\sigma^{-}),Probe(\sigma^{+})-Control(\pi),Probe(\sigma^{-})-Control(\pi),Probe(\sigma^{+})-Control(\sigma^{-})$
and $Probe(\sigma^{-})-Control(\sigma^{+})$. The Rabi frequencies
($\Omega_{p},\Omega_{c}$) are scales as $\Omega_{c}\rightarrow s\Omega_{c}$
and $\Omega_{p}\rightarrow s^{\prime}\Omega_{p}$ when adding contributions
where s and s’ are the Clebs Gordon co-efficient [38] for the respective
transitions. However, there is no significant difference in line shape and
positions after considering all the possible $\Lambda$’s within a given case.

Figure 11: (Left) EIT line width is plotted against control Rabi frequency
$\Omega_{c}$ with error bars. The blue-filled circles are experimental data
points and the orange solid line is the theoretical fit. (Right) EIT contrast
is plotted against $\Omega_{c}$ with error bars. (color online)

##  Appendix G EIT line width characteristics

At $\Delta_{c}=-923.4$, EIT features are recorded for case $C_{1}$ for various
values of control power. The Gaussian background is subtracted and a
Lorentzian fit is performed. Figure 11 shows the line width and contrast
variation with control Rabi frequency $\Omega_{c}$. The contrast increases
with $\Omega_{c}$ at the expense of broader line width. By rearranging the
imaginary part of probe susceptibility, we can read off the EIT line width

$\Gamma_{EIT}=2[\gamma_{2}+\frac{\Omega_{c}^{2}}{\sqrt{OD}\Gamma_{32}}]$ (12)

Here OD is the optical density at temperature T. The expression to calculate
OD [39] is given by

$OD=\frac{n\omega_{p}|d_{13}|L}{\epsilon_{0}c\hbar(\Gamma_{31}+\Gamma_{32})}$
(13)

In the above equation, $n=2.3\times 10^{10}$ atoms/cc is the atomic density at
cell temperature $T=60^{o}$C, $\omega_{p}=391.016$ THz is the probe frequency
in $D_{2}$ line, $|d_{13}|=2.46\times 10^{-29}$ C-m is the dipole matrix
element [22], $L=75$ mm is the cell length, $\epsilon_{0}$, $\hbar$, c are
vacuum permittivity, reduced Planck’s constant, velocity of light respectively
and $\Gamma_{31}$,$\Gamma_{32}$ are Doppler broadened spontaneous decay rates.
The density at temperature T is calculated using [22]

$\log p=7.4077-\frac{4453}{T}$ (14)

##  Appendix H Acknowledgments

The authors thank Dr. Sanjukta Roy for insightful discussions. The authors
also thank S. Majumder and A. Misra for their discussions and assistance. The
authors also thank the RRI IT section and workshop.

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# Symmetry Classification and Enumeration of Square-Tile Sikku Kolams

Venkatraman Gopalan  Department of Materials Science and Engineering, The
Pennsylvania State University, University Park, PA, 16802; email:
<EMAIL_ADDRESS>
## Abstract

Pulli Kolam is an ancient mathematical artform that is still practiced today
in south India by over a quarter million people. _Pulli_ in the Tamizh
language means dots. A specific type of pulli kolam is _sikku_ kolam where a
series of dots are placed using rice flour and lines are drawn around them
with three simple rules: all dots are individually encircled, line crossings
are discrete points, and there are no loose ends in the lines drawn. A
topological approach to viewing a kolam was presented earlier
[arXiv:1503.02130] where a kolam could be broken up into pieces (tiles) of a
puzzle and then assembled together following the above rules. Here we focus on
sikku kolams assembled from square tiles. Kolams are classified by their point
group symmetry, and formulae are derived for counting the number of kolams in
each symmetry type. Some interesting constraint problems are posed that could
help formulate new games based on this elegant artform.

##  1 Introduction

_Pulli_ kolam has been practiced for thousands of years in the southern India,
presently the states of Tamil Nadu, Andhra Pradesh, Telangana, Kerela and
Karnataka with a combined current population of over 260 million people. A
particular type of pulli kolam is called _sikku_ kolam (which translates in
Tamizh to ‘knot kolam’) or _kambi_ kolam (which translates to ‘wire kolam’).
This article focuses on the sikku kolams constructed from square tiles. We
will refer to them as _square-tile kolams_. However, note that although the
tiles themselves are squares, the assembled kolam need not be square; it can
be any shape you are able to make using these square tiles following the three
rules of kolam making stated in the Abstract.

##  2 Six Basic Kolam Tiles

Figure 1: Six types of square kolam tiles considered in this article. The grey
arrows indicate loose ends. These tiles were fabricated by Christopher
Danielson

The six types of square-tiles considered in this article are shown in Fig. 1.
They have been given descriptive names as indicated. Each tile has a single
dot at its center, and segments of one or more lines encircling it. ‘Loose
ends’ are as indicated in the figure by grey arrows. The _circle_ , _drop_ ,
_eye_ , _door_ , _fan_ , and _diamond_ have respectively, 0, 1, 2, 2, 3, and 4
loose ends. When two loose ends come together, they annihilate each other, and
the corresponding lines cross continuously across the two tiles. Note that
there are many more tile types beyond the above six basic types, indeed
infinitely many. We will cover such an extension of the tile types in future
works.

The tiles are designed in such a way that two of the three rules for a kolam
mentioned in the Abstract are automatically satisfied, namely, (1) all the
dots are individually encircled and (2) lines cross only at discrete points.
Therefore in assembling these tiles, there is only one remaining rule to pay
attention to: (3) _no loose ends_. This is done by arranging the tiles
together to get rid of all loose ends by pairing loose ends that annihilate
each other. Any assembly of these tiles which has no loose ends could be
considered a valid pulli kolam.

##  3 Prior Literature

Many previous pioneering works exist that have provided mathematical insights
into the form of a kolam over the past four decades. These include converting
kolams into numbers and linear diagrams (Yanagisawa & Nagata, 2007), using
graph, picture, and array grammers (Siromoney et al., 1974; Siromoney &
Siromoney 1987; Subramanian & Siromoney, 1987; David et al. 2007; Pradella et
al., 2011; Siromoney et. al., 1976; Siromoney et. al., 1983; Demaine et al.,
2007; Subramanian, 1979), extended pasting schemes (Robinson, 2007), morphism
of monoids (Allouche et al., 2006), L- and P-systems (Prusinkiewicz & Hanan,
1989; Subramanian, 2007),gestural lexicons (Waring, 2012), knot theory
(Ishimoto, 2007), and mirror curves (Gerdes, 1999). Yanagisawa & Nagata, 2007
begin with 5 rules for kolam (instead of three rules here), define square unit
tiles that can be assembled into larger kolams, define two types of nearest
neighbor interactions between dots (line crossing, 1, or uncrossing, 0) and
convert these tiles into binary number arrays. Gopalan & VanLeeuwen presented
a purely topological approach, specifying only 3 mandatory rules for defining
a kolam, and generalized the ideas to any arbitrary arrangement of dots
arranged in any shape (not necessarily square arrays), generalized
‘interactions’ between any two dots (instead of only the nearest or next
nearest neighbors), and to three or more number of ‘bonds’ between an
interacting pair of dots. Their work suggested that for a given number of
dots, N, there are a limited number of parent kolam types from which all other
kolams originate. All parent kolams within a parent kolam type are homotopic
(or topologically equivalent).

The present work is closely related to the approach of Gopalan & VanLeeuwen,
2015 where we restrict here, the type of kolam studied to those composed using
square tiles. The interactions between dots is restricted to only the nearest
neighbor dots. This was also the specific case studied by Yanagisawa & Nagata,
2007 and Nagata, 2015; however they place more than one dot per tile and
discuss the kolams in terms of linear diagrams and knot theory. In
distinction, the present work classifies the square tile kolams in terms of
group theory, assigning _point groups_ , well known in crystallography, to
kolams. The enumeration of different kolam types also follow these symmetry
classifications.

##  4 Eight Symmetries of Square-tile Kolams

Figure 2: Symmetries of all square-tile kolams can be classified into one the
eight _point groups_ shown above. The symmetry elements are depicted to the
right of each example kolam where the grey square represents the entire kolam.
Each kolam is composed from a subset of the six basic tile types shown in Fig.
1. The 4 _m_ _$m_{d}$_ point group for example represents a 4-fold rotational
symmetry (i.e. $360^{\circ}/4$ rotation angle) at the center, two equivalent
mirrors (depicted by _m_) that are parallel to the edges of the kolam, and two
others equivalent diagonal mirrors, (depicted by _$m_{d}$_) that are along the
diagonals of the kolam.

Using the six basic kolam tiles shown in Fig. 1, we can compose kolams,
examples of which are shown in Fig. 2. All such kolams composed from square-
tiles can be classified into one of the eight _point group_ symmetries
depicted in Fig. 2. Groups are standard mathematical sets that obey certain
properties such as closure, associativity, identity, and inverse. In our case,
for a given kolam, a symmetry group is a set of _all_ possible symmetry
operations one can perform on the kolam that will leave the kolam unchanged. A
point group specifies that all the symmetry operations must _at least_ leave
one point in the kolam unmoved. In our case, it will typically be the center
of the kolam about which one can rotate the kolam, and through which all
mirrors pass through.

Rotations and mirrors are the point group symmetry operations that are
relevant to the 2-dimensional kolams. Some helpful terminology:
(1) A _p_ -fold rotation axis means that the kolam can be rotated about this
axis by $360^{\circ}/\emph{p}$ and the kolam would still look the same, where
p is a natural number. In the square-tile kolams, 4-fold ($90^{\circ}$),
2-fold ($180^{\circ}$), and 1-fold ($360^{\circ}$) axes are the only ones that
can be observed.
(2) A mirror plane is represented by _m_ and will flip the coordinate
perpendicular to the mirror while leaving the other coordinates parallel to
the mirror unchanged. All mirrors should pass through the center of a kolam.
In the square tile kolams, there are two types of mirrors: the ones parallel
to the edges of the kolam (denoted in Fig. 2 by black lines labeled _m_) and
the ones parallel to the diagonals of the kolam (denoted in Fig. 2 by
_$m_{d}$_ lines).

The highest possible symmetry point group for a square-tile kolam is
$4\emph{m}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}m_{d}}$
as seen in Fig. 2. The grey box next to the kolam titled
$4\emph{m}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}m_{d}}$
schematically represents the entire kolam, and the various symmetry operations
are symbolically represented within this grey box. The 4-fold rotation axis
projects out of the plane of the paper at the center of the kolam, and allows
a rotation of the entire kolam by $360^{\circ}/4=90^{\circ}$, after which the
kolam would look identical to its state before the rotation. In other words,
four times around a full circle rotation about the 4-fold axis, the kolam
looks identical. The presence of a 4-fold rotation symmetry automatically
implies the presence of a 2-fold axis. All kolams also trivially have 1-fold
rotation. There are two sets of mirrors also passing through the center of the
kolam: the ones indicated in black (_m_ are parallel to the edges of the
kolam, while the ones in blue (namely, _$m_{d}$_) are parallel to the
diagonals. The two black mirror operations are symmetry equivalent (conjugate)
since the 4-fold axis can rotate one of them into the other. Similarly, the
two blue mirrors are equivalent for the same reason. However the black and the
blue mirrors are not equivalent since no other symmetry operation within the
point group can transform a black mirror into a blue one or vice versa. In
crystallography, this point group is indicated by the symbol 4 _m_ _$m_{d}$_.
Note that the presence of any two of these symmetry operations (4-fold, _m_
and _$m_{d}$_) would automatically generate the third.

With that detailed explanation, the other seven symmetry point groups depicted
in Fig. 2 should be clear, namely: 4, $2\emph{mm}$, 2 _$m_{d}m_{d}$_ , 2, _m_
, _$m_{d}$_ , and 1. Note that the point group 2mm has two inequivalent
mirrors since no other symmetry operation within the group can transform one
of those mirrors into the other one. The point group 1 has no symmetry
operations other than the 1-fold, namely a rotation about an axis located at
its center by $360^{\circ}$; the 1-fold axis is not labeled. As an aside, the
pair of point groups, _m_ and _$m_{d}$_ would not be separately distinguished
in crystallography where infinite periodic crystal lattices are considered,
but we choose to distinguish them here in kolams for practical and artistic
reasons. The same is true for the pair $2\emph{mm}$ and 2 _$m_{d}m_{d}$_.

##  5 Kolam count by symmetry group

Of the infinitely many possible arrangement of the tiles in Fig. 1, here we
consider two basic types of square-tile kolam templates for enumeration. These
are (1) Single rectangle (1R) with _k_ _l_ tiles (and hence dots), and (2) Two
rectangles (2R), with _k_ _l_ +(_k_ -1)(_l_ -1) tiles (and hence dots) as
depicted by the examples in Fig. 3.

Figure 3: Two basic kolam templates considered in the article for enumeration.
The single rectangle (1R) template (left two images) has
$\emph{k}\times\emph{l}=3\times 4=12$ dots (shown by white rectangle), while
the Two rectangles (2R) template (right two images) has _k_ $\times$_l_ +(_k_
-1)$\times$(_l_ -1)=$3\times 4+2\times 3=18$ dots (shown by two white
rectangles). The bottom two depictions are outlines in grey of the tiles in
the top two kolams; additionally all edges where lines cross in the top kolams
are indicated by $\times$ symbols in the bottom two depictions. The examples
given here have a symmetry of 2mm (see center image), however all other
symmetries groups in Fig. 2 are also possible. To get square templates, set
k=l.

A scheme to count the number of possible kolams with these two templates (1R
and 2R) is given in the bottom depictions in Fig. 3, where the tile outlines
from the kolams above have been reproduced in grey, and the shared edges where
lines cross have been indicated with the symbol $\times$ placed on those
edges. Thus, all the information needed to construct the two kolams on the top
of Fig. 3 is contained in the bottom map of line crossings. One way to
enumerate the possible kolams is therefore to count the total number of shared
edges and assign either a line crossing, ‘$\times$’ or no crossing to each of
them. Without any symmetry considerations, (i.e. for the point group 1 in Fig.
2), each shared edge can be independently specified. For other symmetry
groups, a subset of these can be independently specified to construct the
whole kolam. Defining the minimum number of shared edges, $E_{s,min}$, that
need to be specified for any point group, one can write the number of kolams
as

$N(PG)=2^{E_{s,min}^{PG}}\hskip 39.83385pt$ (1)

where ‘PG’ labels the point group of the kolams being enumerated. Here, the
superscript PG can be one of the eight point group symmetries shown in Fig. 2.
It is not difficult to find the general expressions for $E_{s,min}^{PG}$ for
the 1R and 2R templates shown in Fig. 3 for each of the eight point groups
illustrated in Fig. 2. These expressions for $E_{s,min}^{PG}$ are given in
Table 1.

Point Group (PG) symmetry of the kolam
---
1 |  _m_ ($\perp\textbf{k}$) |  _m_ ($\perp\textbf{l})$ |  _$m_{d}$_ |  2 |  2 _mm_ |  2 _$m_{d}m_{d}$_ |  4 |  4 _m_ $m_{d}$

$E_{s,min}^{PG}$ for Single Rectangle(1R): ($kl$) tiles

$2kl-k-l$
|  k-even: $kl-\frac{k}{2}$ |  l-even: $kl-\frac{l}{2}$ |  _k=l=n_ : $n(n-1)$ |  $(k+l)$ even:
$kl-\frac{(k+l)}{2}$ |  $(kl)$ even:
$\frac{kl}{2}$ |  _k=l=n_ : $\frac{n(n-1)}{2}$ |  _k=l=n_ : $\frac{n(n-1)}{2}$ |  _k=l=n_ even:
$\frac{n^{2}}{4}$
_k=l=n_ odd:
$\frac{n^{2}-1}{4}$
|  k-odd: $kl-\frac{(k+1)}{2}$
|  l-odd: $kl-\frac{(l+1)}{2}$ |  $k\neq l:$
no kolam |  $(k+l)$ odd:
$kl-\frac{(k+l)}{2}+\frac{1}{2}$ |  $(kl)$ odd:
$\frac{(kl-1)}{2}$ |  $k\neq l:$
no kolam |  $k\neq l:$
no kolam |  $k\neq l:$
no kolam

$E_{s,min}^{PG}$ for Two Rectangle (2R): ($kl+(k-1)(l-1)$) tiles

$4(kl-k-l+1)$
|  2(_k_ -1) (_l_ -1)
|  2(_k_ -1) (_l_ -1) |  _k=l=n_ :
$2(\emph{n}-1)^{2}$
+$(\emph{n}-1)$ |  2(_k_ -1) (_l_ -1) |  (_k_ -1) (_l_ -1) |  _k=l=n_ :
$(n-1)^{2}$\+
$(n-1)$
|  _k=l=n_ :
$(\emph{n}-1)^{2}$
|  _k=l=n_ :
$\frac{(\emph{n}-1)^{2}}{2}+\frac{(\emph{n}-1)}{2}$

|  |  $k\neq l:$
no kolam

|  $k\neq l:$
no kolam
|  $k\neq l:$
no kolam
|  $k\neq l:$
no kolam
Table 1: The number of minimum shared edges, $E_{s,min}^{PG}$ for a kolam of a
given template in Fig. 3 classified by its symmetry group labeled by the point
groups(PG) listed in Fig. 2. The number of Kolams will then be given by Eq. 1.
By k and l is meant the respective edge _directions_ in which _k_ -tiles and
_l_ -tiles are counted in the rectangular templates in Fig. 3.

To get a sense for how many kolams are generated, Table 2 lists the number of
kolams for a few example number of tiles for each template in Fig.3. As one
can see, the number of possible kolams becomes very large very quickly. The
higher the symmetry, the fewer the kolams, and lower the symmetry, the more
the number of kolams. Figure 4 and 5 depicts all the square-tile kolams with
the highest symmetry, 4 _$mm_{d}$_ up to $3\times 3$ for the 2R templates and
up to $4\times 4$ for the 1R templates. The other kolams can be similarly
constructed by the technique shown in Figure 3.

Number, $N(PG)$, of Single Rectangle (1R) Kolams for each Point Group (PG)
---
_k_ $\times$ _l_ $\downarrow$ | 1 |  _m_ ($\perp\textbf{k}$) |  _m_ ($\perp\textbf{l})$ | _$m_{d}$_ | 2 | 2 _mm_ | 2 _$m_{d}m_{d}$_ | 4 | 4 _m_ $m_{d}$
1 $\times$ 1 | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$
1 $\times$ 2 | $2^{1}$ | $2^{1}$ | $2^{1}$ | - | $2^{1}$ | $2^{1}$ | - | - | -
2 $\times$ 2 | $2^{4}$ | $2^{3}$ | $2^{3}$ | $2^{2}$ | $2^{2}$ | $2^{2}$ | $2^{1}$ | $2^{1}$ | $2^{1}$
2 $\times$ 3 | $2^{7}$ | $2^{5}$ | $2^{4}$ | - | $2^{4}$ | $2^{3}$ | - | - | -
3 $\times$ 3 | $2^{12}$ | $2^{7}$ | $2^{7}$ | $2^{6}$ | $2^{6}$ | $2^{4}$ | $2^{3}$ | $2^{3}$ | $2^{2}$
3 $\times$ 4 | $2^{17}$ | $2^{10}$ | $2^{10}$ | - | $2^{8}$ | $2^{6}$ | - | - | -
4 $\times$ 4 | $2^{24}$ | $2^{14}$ | $2^{14}$ | $2^{12}$ | $2^{12}$ | $2^{8}$ | $2^{6}$ | $2^{6}$ | $2^{4}$
4 $\times$ 5 | $2^{31}$ | $2^{18}$ | $2^{17}$ | - | $2^{16}$ | $2^{10}$ | - | - | -
5 $\times$ 5 | $2^{40}$ | $2^{22}$ | $2^{22}$ | $2^{20}$ | $2^{20}$ | $2^{12}$ | $2^{10}$ | $2^{10}$ | $2^{6}$
Number, $N(PG)$, of Two Rectangles (2R) Kolams for each Point Group (PG)
1 $\times$ 1 | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$ | $2^{0}$
1 $\times$ 2 | $2^{0}$ | $2^{0}$ | $2^{0}$ | - | $2^{0}$ | $2^{0}$ | - | - | -
2 $\times$ 2 | $2^{4}$ | $2^{2}$ | $2^{2}$ | $2^{3}$ | $2^{2}$ | $2^{1}$ | $2^{2}$ | $2^{1}$ | $2^{1}$
2 $\times$ 3 | $2^{8}$ | $2^{4}$ | $2^{4}$ | - | $2^{4}$ | $2^{2}$ | - | - | -
3 $\times$ 3 | $2^{12}$ | $2^{8}$ | $2^{8}$ | $2^{10}$ | $2^{8}$ | $2^{4}$ | $2^{6}$ | $2^{4}$ | $2^{3}$
3 $\times$ 4 | $2^{24}$ | $2^{12}$ | $2^{12}$ | - | $2^{12}$ | $2^{6}$ | - | - | -
4 $\times$ 4 | $2^{28}$ | $2^{18}$ | $2^{18}$ | $2^{21}$ | $2^{18}$ | $2^{9}$ | $2^{12}$ | $2^{9}$ | $2^{6}$
4 $\times$ 5 | $2^{40}$ | $2^{24}$ | $2^{24}$ | - | $2^{24}$ | $2^{12}$ | - | - | -
5 $\times$ 5 | $2^{56}$ | $2^{32}$ | $2^{32}$ | $2^{36}$ | $2^{32}$ | $2^{16}$ | $2^{20}$ | $2^{16}$ | $2^{10}$
Table 2: The number of kolams, $N(PG)$ from Eq. 1 for the 1R and 2R templates
depicted in Fig. 3. The expressions for the minimum shared edges in Table 1
are used. Figure 4: All the 1R and 1R template kolams (see Fig. 3) with
$4mm_{d}$ symmetry for (a) $k\times l=1\times 1$. (b) $k\times l=2\times 2$,
and (c) $k\times l=3\times 3$. The count is consistent with Table 2. Figure 5:
All the 1R rectangle kolams (see Fig. 3) with $4mm_{d}$ symmetry for $k\times
l=4\times 4$. The count is consistent with Table 2

##  6 Necessary and Sufficient Conditions for a Kolam to Exist

The sufficient condition for composing a kolam would be the three rules given
in the abstract. Two of these are automatically satisfied by the construction
of the kolam tiles in Fig. 1. Thus for the square-tile kolams, the sufficient
condition is simply _no loose ends_.

We can now identify some necessary conditions for achieving no loose ends. As
an example, let us say we are given the following tiles and asked to compose a
kolam: 1 circle, 3 drops, 2 eyes, 2 fans, four doors and one square. Before we
even get started, we can already tell that no kolam is possible with these
tiles without leaving a loose end (and hence violating one of the basic rules
for a kolam). This is because we have 3 drops and 2 fans in the example above,
for a total of 5, which is an odd number; hence no kolam is possible. The
proof for this necessary condition is straightforward and given next.

Let _$N_{i}$_ represent the number of tiles of type _i_ , where _i_ can be _C_
(circle), _Dr_(drop), _E_(eye), _Do_(door), _F_(fan), or _Di_ (diamond); We
recall that these tiles respectively have 0, 1, 2, 2, 3, and 4 loose ends.
Since the loose ends must occur in even numbers to completely annihilate each
other, we can conclude that:

$\displaystyle 0N_{C}+1N_{Dr}+2(N_{E}+N_{Do})+3N_{f}+4N_{Di}$
$\displaystyle=2m$ $\displaystyle\implies N_{Dr}+N_{f}$
$\displaystyle=2m^{\prime}$ (2) $\displaystyle(\textit{necessary condition for
a kolam})$

where, _m_ and _$m^{\prime}$_ are both natural numbers. Equation 6 implies the
following _necessary but insufficient condition_ : _the total number of drops
plus fans, $N_{Dr}+N_{f}$, should be an even number._

Even if we have an even number of loose ends that we can in principle
annihilate, there need to be a sufficient number of shared edges between tiles
that allow for such annihilation. If $E_{s}$ is the _total_ number of shared
edges in a kolam, and a pair of loose ends annihilate at a shared edge, then
it follows that

$m\leq E_{s}<T\hskip 28.45274pt(\textit{necessary condition for a kolam})\\\ $
(3)

where m is the number of _pairs_ of loose ends given by Eq. 6 and _T_ is the
fixed number of tiles specified for composing a kolam, given by
$T=N_{C}+N_{Dr}+N_{E}+N_{Do}+N_{f}+N_{Di}$. The first equality above asserts
that the total number of pairs of loose ends should be less than or equal to
the number of shared edges, $E_{s}$, which by the second equality should be
less than the total number of tiles, _T_. Thus even before we get started, we
can test for the necessary conditions given by Eqs. 6 and 3.

##  7 Conditions imposed by Symmetry

Additional constraints be imposed by a specific symmetry of the kolam one may
want to compose.

4-fold rotation: If the desired kolam has a 4-fold rotation axis, then for an
even number of tiles, _T_ , there must be four copies of $T/4$ tiles, which
implies that T must be divisible by 4. If _T_ is odd, then there must be four
copies of $(T-1)/4$ tiles; therefore (T-1) should be divisible by 4. Further
for odd _T_ , there must be one odd tile that is placed in the center of the
kolam. The 4-fold axis would pass through this center tile. For $T=1$, only
one tile, namely a circle can form a kolam by itself and has a 4-fold symmetry
(it actually has an $\infty$-fold rotation axis, which includes the 4-fold).
$T=2$ or $T=3$ cannot result in a 4-fold kolam. For $T\geq 4$, and assuming
the above conditions on $T$ are met, a minimum of $(T-1)/4+1$ tiles will need
to be specified for $T$ odd and $T/4$ tiles for $T$ even in order to construct
the whole kolam.

2-fold rotation: If the desired kolam has a 2-fold rotation axis, then for an
even number of tiles, _T_ , there must be two copies of $T/2$ tiles which
follows from the fact that T is divisible by 2. If _T_ is odd, then there must
be two copies of $(T-1)/2$ tiles where (T-1) is divisible by 2. Further for
odd _T_ , there must be one odd tile that is placed in the center of the
kolam. The 2-fold axis would pass through this center tile. For $T=1$, only
circle would qualify as a kolam by itself which also has a 2-fold rotation
symmetry. For $T\geq 2$, a minimum of $(T-1)/2+1$ tiles will need to be
specified for $T$ odd and $T/2$ tiles for $T$ even in order to construct the
whole kolam.

One mirror, _m_ : For the presence of one mirror parallel to one of the edge
directions, (k or l), of a 1R template with $T=k\times l$ tiles, $(T-k)$ and
$(T-l)$ must trivially be divisible by 2 if $T$ is odd. In such a case ($T$
odd), only $(T+k)/2$ tiles will need to be specified to create the whole kolam
with a mirror parallel to the $k$-edge. The tiles lying on the mirror line of
the kolam must have mirror symmetry themselves with respect to mirror lines
parallel to the square tile edges, which rules out the door tile. If $T$ is
even and assuming that $l$ is even and $k$ is odd, only $T/2$ tiles need to be
specified if the mirror is parallel to the $k$-edge; further no tiles sit on
the mirror line. For the mirror parallel to the $l$ edge in this case,
$(T+l)/2$ tiles will need to be specified for creating the whole kolam, and
the tiles sitting on the mirror plane cannot be the door tile. The case when
$k$ and $l$ are both even have no tiles sitting on mirrors lines, and only
$T/2$ tiles need to be specified. For the 2R template, the same analysis as
for 1R can be performed except separately for the $\emph{T}=k\times l$ tiles
and then for $(k-1)\times(l-1)$ additional tiles. Distinct from 1R template,
2R kolams will always have tiles lying on the mirror lines, and these tiles
must have mirror symmetry about the diagonal of the individual square tiles.
This restricts the tiles lying on the mirror _m_ in 2R kolams to either
circles, diamonds or doors.

Two mirrors, _m_ : If there are two mirrors in a square-tile 1R kolam, they
must be orthogonal ($90^{o}$ to each other). Further, their intersection must
have a 2-fold rotation axis ; in other words, the point group symmetry will be
2 _mm_ as shown in Fig. 2. (The point group $4mm_{d}$ also has these
symmetries, but in addition possesses a pair of diagonal mirror, $m_{d}$;
hence it is discussed separately below.) The constraints imposed by the 2-fold
and the mirrors, $m$ as described above hold for this 1R kolam as well. The
minimum number of tiles that need to be specified for a 1R kolam with 2 _mm_
symmetry is $(T+l)/4$ tiles if $k$ is odd and $l$ is even, $(T+k)/4$ tiles if
$l$ is odd and $k$ is even, $(T+l+k)/4$ if $k$ and $l$ are both odd; and $T/4$
tiles if $k$ and $l$ are both even. The tiles on the mirror lines of a 1R
kolam cannot be the door tiles. For the 2R kolams with 2 _mm_ symmetry, the
same considerations as a 1R kolam apply, but applied separately to a $k\times
l$ 1R kolam and in addition to a $(k-1)\times(l1-)$ 1R kolam, and then summing
them up for the total minimum number of tiles that need to be specified.
Distinct from 1R, 2R kolams will always have tiles lying on the mirror lines,
and these tiles must have mirror symmetry about the diagonal of the individual
square tiles. This restricts the tiles lying on the mirror _m_ in 2R kolams to
either circles, diamonds or doors.

One diagonal mirror, _$m_{d}$_ : For the presence of one _diagonal_ mirror
parallel to one of the diagonals of a 1R template with $T=k\times l$ tiles,
$k=l$, and hence $T=k^{2}$ must be a square number. In other words, only
square kolams can have diagonal mirrors, _$m_{d}$_. Further, only circles,
doors and diamonds can be along the diagonals of such a 1R kolam. Also,
$(T-k)$ must be an even number, and one needs to specify a minimum of
$(T+k)/2$ tiles to complete the full kolam. For a 2R template with one
_diagonal_ mirror parallel to one of the diagonals of the template, again
$T=k^{2}+(k-1)^{2}$ is a sum of two square numbers. Further, doors cannot be
on the diagonal mirror, $m_{d}$. A 2R kolam with one $m_{d}$ mirror requires
$(T+2k+1)/2$ tiles to be specified to complete the entire kolam.

Two diagonal mirrors, $m_{d}$: The point groups $2m_{d}m_{d}$ and $4mm_{d}$
have a pair of diagonal mirrors that are orthogonal. These kolams need to have
a square number of tiles, $T=k^{2}$, for 1R, and a sum of squares,
$T=k^{2}+(k-1)^{2}$, for the 2R. For the $2m_{d}m_{d}$ and the $4mm_{d}$ 1R
kolams, only circle, diamond and door tiles can be along the diagonals. The
minimum number of tiles that need to be specified to construct the whole
$2m_{d}m_{d}$ 1R kolam ($T=k^{2}$) is given by $(T+2k+1)/4$ if $k$ is odd, and
$(T+2k)/4$ if $k$ is even. For the $4mm_{d}$ 1R kolams, $(T+4k+3)/8$ if $k$ is
odd, and $(T+2k)/8$ if $k$ is even. For the $2m_{d}m_{d}$ and $4mm_{d}$ 2R
kolams which contain $k^{2}+(k-1)^{2}$ tiles, similar analysis can be employed
as the 1R kolams, except once for $k^{2}$ 1R kolam and then for a $(k-1)^{2}$
1R kolam, and then summing them up for the total minimum number of tiles
required to be specified to complete the kolam. 2R kolams will always have
tiles lying on the mirror lines, and these tiles must have mirror symmetry
about the diagonal of the individual square tiles. This restricts the tiles
lying on the mirror _m_ in 2R kolams to either circles, diamonds or doors.

1-fold rotation: All kolams have 1-fold rotation symmetry. If this is the only
symmetry present in a kolam, symmetry does not impose any further constraints
on the kolam beyond the three rules for the kolam.

##  8 Discussion and Conclusions

We have classified all square-tile kolams into 8 symmetry groups, and
presented expressions for enumerating them for two specific templates, namely,
Single rectangles (1R) and two rectangles (2R).

We conclude by noting some future extensions of this study. We have presented
only 6 basic square tiles in Fig. 1; There can be many more in principle by
adding flourishes to the lines that encircle the dots and by considering
connectivity between dots that are not nearest neighbors. There are also
square-tiles with different orientations of the lines within a tile that would
be needed to cover a larger variety of kolams. Secondly, assembling these
square tiles can only generate kolams with a square or rectangular grid of
dots. In general, any arbitrary arrangement of dots should be possible.
Hexagonal tiles would be a straightforward extension to this work which will
be explored in a separate study. An interesting transformation between the 2R
and hexagonal grid templates is that a square 2R template when stretched in
one direction by square root of 3 (or 1.732) turns it into a regular hexagonal
grid. In this sense, the discussion here with the six basic square tiles is
not comprehensive in terms of being able to make all types of kolams with
polygonal shaped tiles. Nonetheless, the number of kolams that are indeed
possible with just these six tiles balloons quickly as a power law with the
number of tiles, as shown in table 2, illustrating the tremendous richness of
this artform.

##  9 Acknowledgements

I would like to thank Dr. Amy Alznauer from Northwestern University who helped
design these square tile kolams. I would like to thank Christopher Danielson
for manufacturing these tiles from maple wood.

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