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\begin{document}

\title{Basic operations for a quantum link model with ultracold
  $^{87}$Sr on a lattice}

\author{Michael A. Perlin}

\maketitle

We consider loading ultracold $^{87}$Sr atoms into an isolated
three-site cell of a state-dependent 2-D optical superlattice, with
real on-sites wavefunctions $\phi_\L$ (left) and $\phi_\R$ (right) for
atoms in the ground electronic state ($^1S_0$), and $\phi_\C$ (center)
for atoms in the excited electronic state ($^3P_0$).  The
single-particle Hamiltonian is then
\begin{align}
  H_0 = \sum_{s,\mu} E_s c_{s\mu}^\dag c_{s\mu},
\end{align}
for sites $r,s\in\set{\L,\C,\R}$, nuclear spins
$\mu\in\set{-I,\cdots,I}$, and fermionic annihilation operators
$c_{s\mu}$.  The energy mismatches $\Delta_{pq}\equiv E_p-E_q$ between
sites $p\ne q$ will generally set the largest energy scale throughout
this work.  If there are multiple atoms loaded within the same cell,
these atoms interact through the two-body Hamiltonian
\begin{align}
  H_{\t{int}} = \f12 \sum_{\substack{p,q,r,s\\\mu,\nu}}
  V^{pq}_{rs} c_{r\mu}^\dag c_{s\nu}^\dag c_{q\nu} c_{p\mu}
  \label{eq:H_int_full}
\end{align}
for interaction strengths $V^{pq}_{rs}$ which depend only on the sites
and their associated electronic states; these interaction strengths do
not depend on the nuclear spin states involved.  Defining
\begin{align}
  K^{pq}_{rs} \equiv \int d^3x~ \phi_r^* \phi_s^* \phi_q \phi_p,
  &&
  G_{\sigma=\g,\e} \equiv \f{4\pi}{m_A} a_{\sigma\sigma},
  &&
  G_\pm \equiv \f{2\pi}{m_A} \p{a_{\e\g+} \pm a_{\e\g-}},
\end{align}
\begin{align}
  G^{\alpha\beta}_{\gamma\delta} \equiv \left\{
    \begin{array}{ll}
      G_\alpha & ~ \alpha = \beta = \gamma = \delta \\
      G_+ & ~ \alpha \ne \beta ~ \t{and} ~
            (\alpha,\beta) = (\gamma,\delta) \\
      G_- & ~ \alpha \ne \beta ~ \t{and} ~
            (\alpha,\beta) = (\delta,\gamma) \\
      0 & ~ \t{otherwise}
    \end{array}\right.,
\end{align}
for an atomic mass $m_A$ and scattering lengths $a_X$, the interaction
strengths in \eqref{eq:H_int_full} are
\begin{align}
  V^{pq}_{rs} = K^{pq}_{rs}
  G^{\varepsilon\p{p}\varepsilon\p{q}}_{\varepsilon\p{r}\varepsilon\p{s}},
\end{align}
where $\varepsilon\p{p}\in\set{\g,\e}$ selects the electronic state
associated with site $p$.  As a first approximation, we assume that
the two-body overlap integral $K^{pq}_{rs}$ is negligible unless
$\set{p,q}=\set{r,s}$, i.e. we neglect interaction-assisted tunneling
in $H_{\t{int}}$.  In the presence of only $H_0$ and $H_{\t{int}}$,
this approximation can also be justified by the secular approximation
due to mismatched single-particle on-site energies $E_s$.  The
interaction Hamiltonian then simplifies to
\begin{align}
  H_{\t{int}}
  = \f12 \sum_{p,\mu,\nu} V_p c_{p\mu}^\dag c_{p\nu}^\dag c_{p\nu} c_{p\mu}
  + \f12 \sum_{\substack{p\ne q\\\mu,\nu}} V_{pq}
  \p{c_{p\mu}^\dag c_{q\nu}^\dag c_{q\nu} c_{p\mu}
    + c_{q\mu}^\dag c_{p\nu}^\dag c_{q\nu} c_{p\mu}},
  \label{eq:H_int_two}
\end{align}
where
\begin{align}
  V_p \equiv V^{pp}_{pp},
  &&
  V_{pq} \equiv V^{pq}_{pq} = V^{pq}_{qp},
  &&
  n_{p\mu} \equiv c_{p\mu}^\dag c_{p\mu}.
\end{align}
As made clear by the form in \eqref{eq:H_int_two}, the interaction
Hamiltonian $H_{\t{int}}$ generally induces nuclear spin exchange
between atoms in different orbitals.  In order to suppress nuclear
spin exchange, we further restrict ourselves to a regime in which the
two-body overlap integral $K^{pq}_{rs}$ is negligible unless all atoms
are on the same site (i.e. unless $p=q=r=s$), such that
\begin{align}
  H_{\t{int}}
  = \f12 \sum_{p,\mu,\nu} V_p c_{p\mu}^\dag c_{p\nu}^\dag c_{p\nu} c_{p\mu}
  = \sum_{\substack{p\\\mu<\nu}} V_p n_{p\mu} n_{p\nu}.
  \label{eq:H_int}
\end{align}
Throughout the remainder of this work, to simplify matters we
generally assume that atoms initially in the electronic ground state
on site $\L$ or $\R$ lie in the nuclear spin manifold spanned by
$\set{\ket{-I},\ket{-I+1}}$, while atoms initially in an excited
electronic state on site $\C$ lie in the nuclear spin manifold spanned
by $\set{\ket{I},\ket{I-1}}$.  We also assume that each site is
initially populated with at most one atom.

\section{Global control gates}
\label{sec:global_control}

We wish to build a toolkit of operations to manipulate atoms in our
three-site cell, with the long term goal of combining many such cells
to simulate a quantum link model.  Before we discuss any multi-body
gates, we first need to construct some basic, single-body global
control operations.  These basic operations together with the two-body
interaction Hamiltonian $H_{\t{int}}$ in \eqref{eq:H_int} will be our
building blocks for realizing more complicated gates.

The first simple operation we consider is the application of a static
magnetic field with strength $B$ along the nuclear spin quantization
axis (which we will generally keep fixed), realizing the Hamiltonian
\begin{align}
  H_\B = -\gamma_{\text{n}} B \sum_{s,\mu} \mu n_{s\mu},
\end{align}
where $\gamma_{\text{n}}$ is the nuclear gyromagnetic ratio.  Applying
this Hamiltonian for a time $t=\theta/\gamma_nB$ generates the phase
gate
\begin{align}
  U_\B\p{\theta} = \sum_{s,\mu} e^{i \theta \mu} n_{s\mu}.
\end{align}
As the nuclear spins $\mu$ are half-integers, the operation
$U_\B\p{\theta}$ is periodic in $\theta$ with period $4\pi$.

We can also shine an auxiliary laser with angular frequency $\omega$
to apply the Hamiltonian
\begin{align}
  H_\Omega^{(\sigma)}
  = - \sum_\mu e^{i\omega t}
  \p{\Omega_{\C,\L,\mu}^{(\sigma)} c_{\L,\mu}^\dag
    + \Omega_{\C,\R,\mu}^{(\sigma)} c_{\R,\mu}^\dag} c_{\C,\mu+\sigma}
  + \t{h.c.},
\end{align}
where $\sigma\in\set{-1,0,1}$ is determined by the clock laser
polarization; $\L\leftrightarrow\R$ coupling is forbidden because
$H_\Omega^{(\sigma)}$ is strictly off-diagonal in electronic state;
and $\t{h.c.}$ denotes the Hermitian conjugate, i.e.
$X+\t{h.c.}\equiv X+X^\dag$.  By evaluating the Clebsch-Gordan
coefficients $\bk{I,\mu+\sigma;1,-\sigma|I,\mu}$, we can extract the
$\mu,\sigma,p,q$ dependence of $\Omega_{pq\mu}^{(\sigma)}$ to find
\begin{align}
  \Omega_{pq\mu}^{(0)} = \Omega_{pq}^{(0)} \mu,
  &&
  \Omega_{pq\mu}^{(\pm1)} = \pm \Omega_{pq}^{(1)}
  \sqrt{\p{I\mp\mu}\p{I\pm\mu+1}},
  &&
  \Omega_{pq}^{(\sigma)} \equiv \Omega_\sigma \int d^3x~ \phi_q^* \phi_p,
\end{align}
for experimentally tunable intensities $\Omega_0$ and $\Omega_1$ which
are independent of nuclear spin and lattice site.  In this section, we
will generally assume that these intensities are real and large
compared to all interaction energies,
i.e. $\Omega_0,\Omega_1\gg\max_p\abs{V_p}$.  We also consider the case
of ${}^{87}$Sr with net nuclear spin $I=9/2$ in order to fix the
relative magnitudes of couplings $\Omega_{pq\mu}^{(\sigma)}$ for
different nuclear spins $\mu$.  The Hamiltonians $H_\Omega^{(\sigma)}$
can be thought of diagrammatically as
\begin{align}
  H_\Omega^{(0)} \sim
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0);
    \draw (7,0) -- (8,0);
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$s\in\set{\L,\R}$};
    \draw[trans] (0.5,0) -- (0.5,-2) node[midway,left] {$-9$};
    \draw[trans] (2.5,0) -- (2.5,-2) node[midway,left] {$-7$};
    \draw[trans] (5.5,0) -- (5.5,-2) node[midway,left] {$7$};
    \draw[trans] (7.5,0) -- (7.5,-2) node[midway,left] {$9$};
  \end{tikzpicture},
\end{align}
where the horizontal lines represent nuclear spin levels from $-9/2$
to $9/2$ (i.e. from left to right) within each orbital state $\C$ or
$s\in\set{\L,\R}$, and the double-headed arrows represent a coupling
between two states.  The numbers next to the arrows represents the
relative magnitudes of the couplings.  Similarly,
\begin{align}
  H_\Omega^{(1)} \sim
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw (4,0) -- (5,0);
    \draw[dotted] (5.5,0) -- (6.5,0);
    \draw (7,0) -- (8,0);
    \draw (9,0) -- (10,0);
    \draw (11,0) -- (12,0);
    \draw (11,0) -- (12,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw (4,-2) -- (5,-2);
    \draw[dotted] (5.5,-2) -- (6.5,-2);
    \draw (7,-2) -- (8,-2);
    \draw (9,-2) -- (10,-2);
    \draw (11,-2) -- (12,-2);
    \draw (11,-2) -- (12,-2) node[right] {$s\in\set{\L,\R}$};
    \draw[trans] (0.5,-2) -- (2.5,0) node[midway,left] {$3$};
    \draw[trans] (2.5,-2) -- (4.5,0) node[midway,left] {$4$};
    \draw[trans] (7.5,-2) -- (9.5,0) node[midway,left] {$4$};
    \draw[trans] (9.5,-2) -- (11.5,0) node[midway,left] {$3$};
  \end{tikzpicture},
  \label{eq:H_O_+}
\end{align}
and
\begin{align}
  H_\Omega^{(-1)} \sim
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw (4,0) -- (5,0);
    \draw[dotted] (5.5,0) -- (6.5,0);
    \draw (7,0) -- (8,0);
    \draw (9,0) -- (10,0);
    \draw (11,0) -- (12,0);
    \draw (11,0) -- (12,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw (4,-2) -- (5,-2);
    \draw[dotted] (5.5,-2) -- (6.5,-2);
    \draw (7,-2) -- (8,-2);
    \draw (9,-2) -- (10,-2);
    \draw (11,-2) -- (12,-2);
    \draw (11,-2) -- (12,-2) node[right] {$s\in\set{\L,\R}$};
    \draw[trans] (0.5,0) -- (2.5,-2) node[midway,right] {$3$};
    \draw[trans] (2.5,0) -- (4.5,-2) node[midway,right] {$4$};
    \draw[trans] (7.5,0) -- (9.5,-2) node[midway,right] {$4$};
    \draw[trans] (9.5,0) -- (11.5,-2) node[midway,right] {$3$};
  \end{tikzpicture},
  \label{eq:H_O_-}
\end{align}
where we neglect couplings for nuclear spins $\mu$ outside of
$\set{-I,-I+1,I-1,I}$.

By turning on $H_\Omega^{(0)}$ for a time $t=\pi/\Omega^{(0)}_{\C,s}$
with an angular frequency $\omega$ resonant on the energy difference
$\Delta_{\C,s}=E_C-E_s$ for $s\in\set{\L,\R}$, we can apply the
$\pi$-pulse gate
\begin{align}
  U_{\pi,s}
  \equiv \sum_\mu e^{i\pi\mu} c_{s\mu}^\dag c_{\C,\mu} + \t{h.c}
  \coloneqq i \times
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0);
    \draw (7,0) -- (8,0);
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$s$};
    \draw[trans] (0.5,0) -- (0.5,-2) node[midway,left] {$-1$};
    \draw[trans] (2.5,0) -- (2.5,-2) node[midway,left] {$1$};
    \draw[trans] (5.5,0) -- (5.5,-2) node[midway,left] {$-1$};
    \draw[trans] (7.5,0) -- (7.5,-2) node[midway,left] {$1$};
  \end{tikzpicture},
  \label{eq:U_pi}
\end{align}
which swaps atoms between orbitals $\C$ and $s$ and applies a
spin-dependent sign.

The relative magnitudes of the couplings $\Omega_{pq\mu}^{(\pm1)}$ for
different nuclear spins $\mu$ similarly allows us to turn on
$H_\Omega^{(\pm1)}$ for an appropriately resonant angular frequency
$\omega$ and a time $t=\p{\pi/2}/\Omega_{\C,s}^{(1)}$, thereby
realizing the gates
\begin{align}
  U_{\pi,s}^+
  \equiv -i\p{c_{s,-I+1} c_{\C,-I+1}^\dag + c_{s,I-1}^\dag c_{\C,I}}
  + \t{h.c.}
  \coloneqq -i \times
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0);
    \draw (7,0) -- (8,0);
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$s$};
    \draw[trans] (2.5,0) -- (0.5,-2);
    \draw[trans] (7.5,0) -- (5.5,-2);
  \end{tikzpicture},
  \label{eq:U_pi_+}
\end{align}
and
\begin{align}
  U_{\pi,s}^-
  \equiv i\p{c_{s,-I+1} c_{\C,-I+1}^\dag + c_{s,I}^\dag c_{\C,I-1}}
  + \t{h.c.}
  \coloneqq i \times
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0);
    \draw (7,0) -- (8,0);
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$s$};
    \draw[trans] (0.5,0) -- (2.5,-2);
    \draw[trans] (5.5,0) -- (7.5,-2);
  \end{tikzpicture},
  \label{eq:U_pi_-}
\end{align}
where the remaining states addressed by $H_\Omega^{(\pm1)}$ in
\eqref{eq:H_O_+} and \eqref{eq:H_O_-} are unaffected because they
undergo Rabi cycles which bring them back to their initial states.

All of the above $\pi$-pulse gates can be turned into $\pi/2$-pulse
gates $U_{\pi/2,s}$, $U_{\pi/2,s}^\pm$ simply by applying the
appropriate Hamiltonian for half of the specified time.  Furthermore,
if we are guaranteed that only certain nuclear spin states are
occupied, we can also apply $\pi$-pulse gates by turning on
$H_\Omega^{(\pm1)}$ for a time $t=\p{\pi/6}/\Omega_{\C,s}^{(1)}$,
giving us
\begin{align}
  \tilde U_{\pi,s}^+
  \coloneqq i \times
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (5.5,0);
    \draw (6,0) -- (7,0);
    \draw (8,0) -- (9,0);
    \draw (8,0) -- (9,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (5.5,-2);
    \draw (6,-2) -- (7,-2);
    \draw (8,-2) -- (9,-2);
    \draw (8,-2) -- (9,-2) node[right] {$s$};
    \draw[trans] (2.5,0) -- (0.5,-2);
    \draw[trans, dashed] (4.5,0) -- (2.5,-2);
    \draw[trans, dashed] (6.5,0) -- (4.5,-2);
    \draw[trans] (8.5,0) -- (6.5,-2);
  \end{tikzpicture},
  &&
  \tilde U_{\pi,s}^-
  \coloneqq -i \times
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (5.5,0);
    \draw (6,0) -- (7,0);
    \draw (8,0) -- (9,0);
    \draw (8,0) -- (9,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (5.5,-2);
    \draw (6,-2) -- (7,-2);
    \draw (8,-2) -- (9,-2);
    \draw (8,-2) -- (9,-2) node[right] {$s$};
    \draw[trans] (2.5,-2) -- (0.5,0);
    \draw[trans, dashed] (4.5,-2) -- (2.5,0);
    \draw[trans, dashed] (6.5,-2) -- (4.5,0);
    \draw[trans] (8.5,-2) -- (6.5,0);
  \end{tikzpicture},
\end{align}
where the dashed arrows indicate incomplete Rabi cycles that mix nuclear
spin states in our target subspace with those outside it.  For
simplicity, we will work strictly with the cleaner $\pi$-pulse gates
in \eqref{eq:U_pi}, \eqref{eq:U_pi_+}, and \eqref{eq:U_pi_-}.

\section{Tunneling gates}
\label{sec:tunneling}

In order to suppress nuclear spin exchange processes, we restricted
ourselves to a regime in which two-body overlap integrals for pairs of
atoms on different sites vanish.  The on-site wavefunctions $\phi_p$
are therefore fairly localized to site $p$, with some overlap in the
support of $\phi_\C$ with $\phi_s$ for $s\in\set{\L,\R}$ to allow for
the control gates $U_{\pi,s}^\pm$.  As a consequence, we expect
vanishing mutual support of $\phi_\L$ and $\phi_\R$, which means that
we cannot induce $\L\leftrightarrow\R$ tunneling directly.
Nonetheless, we can use the $\pi$-pulse gates $U_{\pi,s}$ in
\eqref{eq:U_pi} to induce indirect $\L\leftrightarrow\R$ tunneling
through the central site $\C$ via
\begin{multline}
  U_\pi^\T
  \equiv U_{\pi,\L} U_{\pi,\R} U_{\pi,\L}
  \cong U_{\pi,\R} U_{\pi,\L} U_{\pi,\R} \\
  \cong
  \sp{\begin{tikzpicture}[
      scale=0.5,
      trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
      ]
      \draw (0,0) -- (1,0);
      \draw (2,0) -- (3,0);
      \draw[dotted] (3.5,0) -- (4.5,0);
      \draw (5,0) -- (6,0);
      \draw (7,0) -- (8,0);
      \draw (7,0) -- (8,0) node[right] {$\L$};
      \draw (0,-2) -- (1,-2);
      \draw (2,-2) -- (3,-2);
      \draw[dotted] (3.5,-2) -- (4.5,-2);
      \draw (5,-2) -- (6,-2);
      \draw (7,-2) -- (8,-2);
      \draw (7,-2) -- (8,-2) node[right] {$\C$};
      \draw (0,-4) -- (1,-4);
      \draw (2,-4) -- (3,-4);
      \draw[dotted] (3.5,-4) -- (4.5,-4);
      \draw (5,-4) -- (6,-4);
      \draw (7,-4) -- (8,-4);
      \draw (7,-4) -- (8,-4) node[right] {$\R$};
      \draw[trans] (0.5,0) -- (0.5,-2);
      \draw[trans] (2.5,0) -- (2.5,-2);
      \draw[trans] (5.5,0) -- (5.5,-2);
      \draw[trans] (7.5,0) -- (7.5,-2);
    \end{tikzpicture}}
  \sp{\begin{tikzpicture}[
      scale=0.5,
      trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
      ]
      \draw (0,0) -- (1,0);
      \draw (2,0) -- (3,0);
      \draw[dotted] (3.5,0) -- (4.5,0);
      \draw (5,0) -- (6,0);
      \draw (7,0) -- (8,0);
      \draw (7,0) -- (8,0) node[right] {$\L$};
      \draw (0,-2) -- (1,-2);
      \draw (2,-2) -- (3,-2);
      \draw[dotted] (3.5,-2) -- (4.5,-2);
      \draw (5,-2) -- (6,-2);
      \draw (7,-2) -- (8,-2);
      \draw (7,-2) -- (8,-2) node[right] {$\C$};
      \draw (0,-4) -- (1,-4);
      \draw (2,-4) -- (3,-4);
      \draw[dotted] (3.5,-4) -- (4.5,-4);
      \draw (5,-4) -- (6,-4);
      \draw (7,-4) -- (8,-4);
      \draw (7,-4) -- (8,-4) node[right] {$\R$};
      \draw[trans] (0.5,-2) -- (0.5,-4);
      \draw[trans] (2.5,-2) -- (2.5,-4);
      \draw[trans] (5.5,-2) -- (5.5,-4);
      \draw[trans] (7.5,-2) -- (7.5,-4);
    \end{tikzpicture}}
  \sp{\begin{tikzpicture}[
      scale=0.5,
      trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
      ]
      \draw (0,0) -- (1,0);
      \draw (2,0) -- (3,0);
      \draw[dotted] (3.5,0) -- (4.5,0);
      \draw (5,0) -- (6,0);
      \draw (7,0) -- (8,0);
      \draw (7,0) -- (8,0) node[right] {$\L$};
      \draw (0,-2) -- (1,-2);
      \draw (2,-2) -- (3,-2);
      \draw[dotted] (3.5,-2) -- (4.5,-2);
      \draw (5,-2) -- (6,-2);
      \draw (7,-2) -- (8,-2);
      \draw (7,-2) -- (8,-2) node[right] {$\C$};
      \draw (0,-4) -- (1,-4);
      \draw (2,-4) -- (3,-4);
      \draw[dotted] (3.5,-4) -- (4.5,-4);
      \draw (5,-4) -- (6,-4);
      \draw (7,-4) -- (8,-4);
      \draw (7,-4) -- (8,-4) node[right] {$\R$};
      \draw[trans] (0.5,0) -- (0.5,-2);
      \draw[trans] (2.5,0) -- (2.5,-2);
      \draw[trans] (5.5,0) -- (5.5,-2);
      \draw[trans] (7.5,0) -- (7.5,-2);
    \end{tikzpicture}} \\
  \cong
  \begin{tikzpicture}[
    scale=0.5,
    trans/.style={thick,<->,shorten >=2pt,shorten <=2pt,>=stealth}
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0);
    \draw (7,0) -- (8,0);
    \draw (7,0) -- (8,0) node[right] {$\L$};
    \draw (0,-2) -- (1,-2);
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$\R$};
    \draw[trans] (0.5,0) -- (0.5,-2);
    \draw[trans] (2.5,0) -- (2.5,-2);
    \draw[trans] (5.5,0) -- (5.5,-2);
    \draw[trans] (7.5,0) -- (7.5,-2);
  \end{tikzpicture},
  \label{eq:U_pi_T}
\end{multline}
where $\cong$ denotes equality up to an overall phase.  Crucially,
$U_\pi^\T$ swaps atoms between sites $\L$ and $\R$ while leaving atoms
on site $\C$ unaffected.  We can also construct $\pi/2$-pulse gates
for $\L\leftrightarrow\R$ tunneling via
\begin{align}
  U_{\pi/2}^\T
  \equiv U_{\pi,\L} U_{\pi/2,\R} U_{\pi,\L}
  \cong U_{\pi,\R} U_{\pi/2,\L} U_{\pi,\R},
\end{align}
which is identical to the composite gate $U_\pi^\T$ in
\eqref{eq:U_pi_T}, but with a replacement of the second $\pi$-pulse by
a $\pi/2$ pulse.  Finally, we can make use of the
nuclear-spin-selective properties of $U_{\pi,s}^\pm$ to construct
nuclear-spin-selective $\L\leftrightarrow\R$ tunneling gates, simply
by replacing instances of $U_{\pi,s}$ above by $U_{\pi,s}^\pm$.

\section{Controlled-phase gates}
\label{sec:controlled_phase}

We now construct a collisional controlled-phase gate on an atom in
site $\C$, conditional on the presence of an atom in site $s$.  We
first prepare an atom on site $\C$ in the nuclear spin manifold
spanned by $\set{\ket{I},\ket{I-1}}$, and consider the possibility of
an atom on site $s$ with nuclear spin state $\ket{-I}$, preparing the
state
\begin{align}
  \ket{\psi_{sm}}
  = \ket{s,-I;m} \otimes \p{\alpha\ket{\C,I} + \beta\ket{\C,I-1}}
  \sim
  \begin{tikzpicture}[
    scale=0.5,
    sim/.style={decorate, decoration=snake},
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0) node[midway, above] {$\beta$};
    \draw (7,0) -- (8,0) node[midway, above] {$\alpha$};
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2) node[midway, above] {$m$};
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$s$};
    \draw[sim] (6,0.5) -- (7,0.5);
  \end{tikzpicture}
  \label{eq:psi_sm}
\end{align}
where $m=0$ ($m=1$) marks the absence (presence) of an atom on site
$s$ with spin $-I$.  The above diagram is merely provided as a sketch
of the state; we will make no attempt at making this sketch a formal
representation.

Applying the gate $U_{\pi,s}^-$ in \eqref{eq:U_pi_-} to the state
$\ket{\psi_{sm}}$ in \eqref{eq:psi_sm} gives us
\begin{align}
  U_{\pi,s}^-\ket{\psi_{sm}}
  \cong \ket{s,-I;m} \otimes \p{\alpha\ket{\C,I} + \beta\ket{s,I}}
  \sim
  \begin{tikzpicture}[
    scale=0.5,
    sim/.style={decorate, decoration=snake},
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0);
    \draw (7,0) -- (8,0) node[midway, above] {$\alpha$};
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2) node[midway, above] {$m$};
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2) node[midway, above] {$\beta$};
    \draw (7,-2) -- (8,-2) node[right] {$s$};
    \draw[sim] (7.5,0.1) -- (7.5,-0.9);
  \end{tikzpicture}.
\end{align}
Holding for a time $t$ (i.e. evolving under $H_0+H_{\t{int}}$) now
makes $\beta$ pick up a phase relative to $\alpha$ as
$\beta\to\tilde\beta_{sm}\p{t}\equiv\beta e^{it\Delta_{\C,s}-itV_sm}$,
where $\Delta_{\C,s}=E_\C-E_s$ is the single-particle energy
difference between the orbitals $\C$ and $s$, and $V_s$ is the
interaction energy of two atoms in orbital $s$.  Subsequently applying
$U_{\pi,s}^-$ then gives us the state
\begin{align}
  U_{\pi,s}^- U_{\t{hold}}\p{t} U_{\pi,s}^- \ket{\psi_{sm}}
  \cong \ket{s,-I;m} \otimes
  \p{\alpha\ket{\C,I} + \tilde\beta_{sm}\p{t}\ket{\C,I-1}}
  \sim
  \begin{tikzpicture}[
    scale=0.5,
    sim/.style={decorate, decoration=snake},
    ]
    \draw (0,0) -- (1,0);
    \draw (2,0) -- (3,0);
    \draw[dotted] (3.5,0) -- (4.5,0);
    \draw (5,0) -- (6,0) node[near start, above]
    {$\tilde\beta_{sm}\p{t}$};
    \draw (7,0) -- (8,0) node[midway, above] {$\alpha$};
    \draw (7,0) -- (8,0) node[right] {$\C$};
    \draw (0,-2) -- (1,-2) node[midway, above] {$m$};
    \draw (2,-2) -- (3,-2);
    \draw[dotted] (3.5,-2) -- (4.5,-2);
    \draw (5,-2) -- (6,-2);
    \draw (7,-2) -- (8,-2);
    \draw (7,-2) -- (8,-2) node[right] {$s$};
    \draw[sim] (6.5,0.5) -- (7,0.5);
  \end{tikzpicture}.
\end{align}
The extra relative phase $e^{it\Delta_{\C,s}}$ on
$\tilde\beta_{sm}\p{t}$ can be cancelled out by the application of a
magnetic field with $U_\B\p{t\Delta_{\C,s}}$. In the basis
$\ket{m}_s\equiv\ket{s,-I;m}$ for site $s$ and
$\ket{n}_\C\equiv\ket{\C,I-n}$ for site $\C$, the net effect of our
sequence is thus the controlled-phase gate
\begin{multline}
  U_{\P,s}^{(0)}\p{t}
  \equiv U_\B\p{t\Delta_{\C,s}}
  U_{\pi,s}^- U_{\t{hold}}\p{t} U_{\pi,s}^- \\
  = \sum_{m,n\in\set{0,1}} \exp\p{-itV_snm} \op{m}_s\otimes\op{n}_\C
  = \op{0}_s \otimes \1_\C + \op{1}_s \otimes
  \begin{pmatrix}
    1 & 0 \\ 0 & e^{-itV_s}
  \end{pmatrix}_\C.
  \label{eq:U_P_0}
\end{multline}
A similar controlled-phase gate $U_{\P,s}^{(1)}\p{t}$ can be realized
conditional on the presence of an atom on site $s$ with nuclear spin
$-I+1$ by using $U_{\pi,s}^+$ in place of $U_{\pi,s}^-$.  Note that if
the atom on site $s$ is in a nontrivial superposition of nuclear spin
states $\ket{-I}$ and $\ket{-I+1}$, there will be two atoms on site
$\C$ during the hold time of $U_{\P,s}^{(0)}\p{t}$ and
$U_{\P,s}^{(1)}\p{t}$, resulting in atom losses from
${}^3P_0$-${}^3P_0$ interactions.

As a final comment, we mention that if we are guaranteed a single atom
on e.g.  site $\L$ and no atom on site $\R$, we can use the
nuclear-spin selective $\L\leftrightarrow\R$ tunneling gates mentioned
at the end of section \ref{sec:tunneling} to construct a
controlled-phase gate conditional on the nuclear spin of an atom on
site $s$, rather than on the mere presence of such an atom.  That is,
we can construct a gate similar to that in \eqref{eq:U_P_0}, but with
the nuclear spin basis $\set{\ket{-I},\ket{-I+1}}$ for site $s$.  We
can provide and explain this construction in more detail upon request;
the basic idea is to
\begin{enumerate*}[label=(\roman*)]
\item move e.g. the $-I+1$ spin component of the electronic
  ground-state atom in site $\L$ to site $\R$,
\item act with the controlled-phase gates $U_{\P,s}^{(0)}$ in
  \eqref{eq:U_P_0}, and then
\item move the $-I+1$ spin component of the electronic ground-state
  atom back to site $\L$.
\end{enumerate*}

\end{document}


\begin{flushleft}
    For each of the following dataset results, my solution was run over 50 independent runs for priori number of \\
    clusters and 50 independent runs for each possible optimal number of clusters in the range from 2 to 15. \\
    Therefore, the results showed below, are averages of those 50 independent runs.\\
    \subsection{Iris Dataset Results}
        \subsubsection{Priori Number of Clusters}
            \begin{itemize}
                \item Number of clusters: 3
                \item Average number of samples per cluster: {50, 52, 48}
            \end{itemize}
            Given that the actual number of samples per cluster is {50, 50, 50}, \\
            my implementation has an error rate of 2.667% (i.e. 4/150 samples were incorrectly clustered). \\
        \subsubsection{Dynamic Optimal Number of Clusters}
            \begin{itemize}
                \item Determined optimal number of clusters: 3
                \item Average number of samples per cluster: {53, 50, 47}
            \end{itemize}
            Given that the actual number of cluster is 3, my implementation was able to determine the correct optimal \\
            number of clusters each time using the approach mentioned in the Implementation section. \\
            For that optimal number of clusters, the average number of samples per cluster over those 50 independent \\
            runs was {54, 50, 47}. Given that the actual number of samples per cluster is {50, 50, 50}, \\
            my implementation has an error rate of 4% (i.e. 6/150 samples were incorrectly clustered). \\

    \subsection{Wine Dataset Results}
        \subsubsection{Priori Number of Clusters}
            \begin{itemize}
                \item Number of clusters: 3
                \item Average number of samples per cluster: {73, 71, 35}
            \end{itemize}
            Given that the actual number of samples per cluster is {59, 71, 48}, \\
            my implementation has an error rate of 15.084% (i.e. 27/179 samples were incorrectly clustered). \\
        \subsubsection{Dynamic Optimal Number of Clusters}
            \begin{itemize}
                \item Determined optimal number of clusters: 4
                \item Average number of samples per cluster: {46, 44, 46, 40}
            \end{itemize}
            Given that the actual number of cluster is 3, my implementation was off by 1 \\
            cluster each time using the approach mentioned in the Implementation section. This is due to \\
            the fact that the dataset was not normalized and thus skewed the results in favour of the attribute \\
            with the highest value. To try and mitigate this effect, one could preprocess the dataset such that all \\
            attributes are on the same scale, i.e. from 0 to 1. Also, one could apply Principal Component Analysis \\
            (PCA) reduction on the dataset such that only "relevant" features of the dataset are utilized.
            For that optimal number of clusters, the average number of samples per cluster over those 50 independent \\
            runs was {46, 44, 46, 40}. Given that the actual number of samples per cluster is {59, 71, 48}, \\
            my implementation has an error rate of 45.810% (i.e. 83/179 samples were incorrectly clustered). \\

    \subsection{Epileptic Seizure Recognition Dataset Results}
        \subsubsection{Priori Number of Clusters}
            \begin{itemize}
                \item Number of clusters: 5
                \item Average number of samples per cluster: {3267, 2202, 3204, 2668, 159}
            \end{itemize}
            Given that the actual number of samples per cluster is {2300, 2300, 2300, 2300, 2300}, \\
            my implementation has an error rate of 38.939% (i.e. 4478/11500 samples were incorrectly clustered). \\
        \subsubsection{Dynamic Optimal Number of Clusters}
            \begin{itemize}
                \item Determined optimal number of clusters: 12
                \item Average number of samples per cluster: {1298, 800, 977, 982, 666, 395, 1198, 934, 736, 1190, 1253, 1066}
            \end{itemize}
            Given that the actual number of samples per cluster is {2300, 2300, 2300, 2300, 2300}, \\
            my implementation has an error rate of 47.783% (i.e. 5495/11500 samples were incorrectly clustered). \\
\end{flushleft}