slides.tex

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\hypersetup{%
  pdfauthor={Michael Goerz},
  pdftitle={Optimal Control Techniques for Quantum Interferometry},
  pdfsubject={Use of QuantumControl.jl in the context of quantum interferometry},
  pdfkeywords={quantum control; optimal control; quantum dynamics; interferometry; robustness; Julia},
}

\title{Optimal Control Techniques \\for Quantum Interferometry}
\author[\href{https://michaelgoerz.net}{https://michaelgoerz.net}]{Michael~H.~Goerz}
\institute[Army Research Lab]{DEVCOM Army Research Lab}
\date{IMSI Quantum Hardware Workshop, Chicago, October 30, 2024}

\begin{document}

{%  Title page
  \setbeamertemplate{footline}{}
  \frame{%
    \titlepage
    \note{\dots}
  }
}
\addtocounter{framenumber}{-1}

\begin{frame}{Quantum Interferometry}
  \begin{itemize}
    \item Ensemble of atoms \pause
    \item Create a superposition \pause
    \item Separate components \pause
    \item Accumulate a relative phase \pause
    \item Recombine \pause
    \item Imprint phase on measurement
  \end{itemize}
\end{frame}

\begin{frame}{Atomic Fountain Interferometer}
  \begin{textblock}{3}(1.0,1.5)
    \includegraphics<1,3->[height=6cm]{images/atomic_fountain.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,2.5)
    \hfill\includegraphics<2>[trim=0 0 0 0,clip]{images/afioct/fig1.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,1.5)
    \hfill\includegraphics<4>[trim=8.3cm 0 0 0,clip]{images/afioct/fig2.pdf}
  \end{textblock}
  \begin{textblock}{10}(5.0,7.5)
      \onslide<2,4->{%
        \hfill\footnotesize{--- Goerz, Kasevich, Malinovsky.  Atoms 11, 36 (2023)}
      }
  \end{textblock}
\end{frame}

\begin{frame}{Atomic Fountain Interferometer -- Robustness}
  \begin{textblock}{3}(1.0,1.5)
    \includegraphics[height=6cm]{images/atomic_fountain.pdf}
  \end{textblock}
  \begin{textblock}{13}(1.0,2.25)
    \hfill\includegraphics[trim=0 0 9.2cm 0,clip]{images/afioct/fig4.pdf}
  \end{textblock}
  \begin{textblock}{1}(14.0,2.25)
    \includegraphics[trim=12.6cm 0 0 0,clip]{images/afioct/fig4.pdf} % colorbar
  \end{textblock}
  \begin{textblock}{10}(5.0,7.5)
    \onslide<1->{%
      \hfill\footnotesize{--- Goerz, Kasevich, Malinovsky.  Atoms 11, 36 (2023)}
    }
  \end{textblock}
\end{frame}

\begin{frame}{Atomic Fountain Interferometer -- Rapid Adiabatic Passage (RAP)}
  \begin{textblock}{3}(1.0,1.5)
    \includegraphics<1-2>[height=6cm]{images/atomic_fountain.pdf}
  \end{textblock}
  \begin{textblock}{3}(12.2,5.2)
    \includegraphics<2->[trim=9.0cm 0 0 0,clip,height=3cm]{images/afioct/fig1.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,1.5)
    \hfill\includegraphics<1-2>[trim=8.3cm 0 0 0,clip]{images/afioct/fig2.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,1.5)
    \hfill\includegraphics<3>[trim=6.0cm 0 0 0,clip]{images/afioct/fig3.pdf}
  \end{textblock}
  \begin{textblock}{10}(5.0,7.5)
    \footnotesize{--- Goerz, Kasevich, Malinovsky.  Atoms 11, 36 (2023)}
  \end{textblock}
\end{frame}


\begin{frame}{Atomic Fountain Interferometer -- Robustness RAP}
  \begin{textblock}{3}(1.0,1.5)
    \includegraphics<1-2>[height=6cm]{images/atomic_fountain.pdf}
  \end{textblock}
  \begin{textblock}{13}(1.0,2.25)
    \hfill\includegraphics<1>[trim=0 0 9.2cm 0,clip]{images/afioct/fig4.pdf}
  \end{textblock}
  \begin{textblock}{13}(1.0,2.25)
    \hfill\includegraphics<2>[trim=0 0 5.2cm 0,clip]{images/afioct/fig4.pdf}
  \end{textblock}
  \begin{textblock}{1}(14.0,2.25)
    \includegraphics[trim=12.6cm 0 0 0,clip]{images/afioct/fig4.pdf} % colorbar
  \end{textblock}
  \begin{textblock}{10}(5.0,7.5)
    \onslide<1->{%
      \hfill\footnotesize{--- Goerz, Kasevich, Malinovsky.  Atoms 11, 36 (2023)}
    }
  \end{textblock}
\end{frame}

\begin{frame}{Atomic Fountain Interferometer -- Optimal Control (OCT)}
  \begin{textblock}{14}(1.0,1.5)
    \hfill\includegraphics<1>[trim=6.0cm 0 0 0,clip]{images/afioct/fig3.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,2.0)
    \includegraphics<2>[trim=0 0 9.5cm 0, clip]{images/afioct/fig6.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,2.0)
    \includegraphics<3>[trim=0 0 4.25cm 0, clip]{images/afioct/fig6.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,2.0)
    \includegraphics<4>{images/afioct/fig6.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,1.5)
    \hfill\includegraphics<5>[trim=6.0cm 0 0 0,clip]{images/afioct/fig3.pdf}
  \end{textblock}
  \begin{textblock}{14}(1.0,1.5)
    \hfill\includegraphics<6>[trim=0 0 0 0,clip]{images/afioct/fig7.pdf}
  \end{textblock}
  \begin{textblock}{10}(5.0,7.5)
    \onslide<1->{%
      \hfill\footnotesize{--- Goerz, Kasevich, Malinovsky.  Atoms 11, 36 (2023)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Atomic Fountain Interferometer -- Robustness OCT}
  \begin{textblock}{3}(1.0,1.5)
    \includegraphics<1>[height=6cm]{images/atomic_fountain.pdf}
  \end{textblock}
  \begin{textblock}{13}(1.0,2.25)
    \hfill\includegraphics<1>[trim=0 0 5.2cm 0,clip]{images/afioct/fig4.pdf}
  \end{textblock}
  \begin{textblock}{13}(1.0,2.25)
    \hfill\includegraphics<2>[trim=0 0 1.2cm 0,clip]{images/afioct/fig4.pdf}
  \end{textblock}
  \begin{textblock}{1}(14.0,2.25)
    \includegraphics[trim=12.6cm 0 0 0,clip]{images/afioct/fig4.pdf}
  \end{textblock}
  \begin{textblock}{10}(5.0,7.5)
    \onslide<1->{%
      \hfill\footnotesize{--- Goerz, Kasevich, Malinovsky.  Atoms 11, 36 (2023)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{JuliaQuantumControl}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<2>[width=\textwidth]{images/JuliaQuantumControl}
    % \includegraphics<3>[width=\textwidth]{images/JuliaQuantumControlPackages}
  \end{textblock}
\end{frame}


\begin{frame}
  \begin{center}
    {\Large \color{DarkRed} \bf Why Julia?}
    \vspace{1.5cm}
    \pause
    \begin{itemize}
      \item Performance \pause -- compiles to low-level machine code (matches Fortran) \pause
      \item Flexibility \pause -- multiple dispatch\pause, ecosystem\pause
      \item Expressiveness \pause -- clean syntax, unicode, notebook environment
    \end{itemize}
  \end{center}
\end{frame}


\begin{frame}{JuliaQuantumControl}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<1->[width=\textwidth]{images/JuliaQuantumControl.png}
  \end{textblock}
  \begin{textblock}{14.5}(0.75,5.28)
    \onslide<2>{%
      \begin{block}{Design Principle}
        \begin{center}
          Maximum performance and composability through abstract interfaces
        \end{center}
      \end{block}
    }
  \end{textblock}
\end{frame}

\begin{frame}{Optimization Functional}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics[width=\textwidth]{images/functional.png}
  \end{textblock}
\end{frame}

\begin{frame}{Control Problem and Trajectories}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<1>[width=\textwidth]{images/api_controlproblem.png}
    \includegraphics<2>[width=\textwidth]{images/api_controlproblem2.png}
    \includegraphics<3>[width=\textwidth]{images/api_trajectory.png}
    \includegraphics<4>[width=\textwidth]{images/api_trajectory2.png}
  \end{textblock}
\end{frame}

\begin{frame}{Dynamical Generators}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<1>[width=\textwidth]{images/glossary_generator.png}
    \includegraphics<2>[width=\textwidth]{images/api_check_generator.png}
  \end{textblock}
\end{frame}

\begin{frame}{Optimization schemes}
  \begin{textblock}{15.5}(0.25,1.50)
    \includegraphics<2>[width=\textwidth]{images/schemes_comparison.pdf}
  \end{textblock}
  \begin{textblock}{10}(5.0,7.2)
    \onslide<2>{%
      \hfill\footnotesize{--- Goerz, Carrasco, Malinovsky.  Quantum 6, 871 (2022)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Propagators}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<2>[width=\textwidth]{images/api_propagator.png}
    \includegraphics<3>[width=\textwidth]{images/api_propagator2.png}
    \includegraphics<4>[width=\textwidth]{images/differentialequations.png}
  \end{textblock}
\end{frame}


\begin{frame}{Rotating Tractor Interferometer}
  \begin{textblock}{13.5}(1.25,0.75)
    \begin{center}
      \includegraphics<2>{images/pinwheel}
    \end{center}
  \end{textblock}
  \begin{textblock}{15.5}(1.5,1.50)
    \only<3->{%
      \includegraphics{images/rottai_concept}
    }
  \end{textblock}
  \begin{textblock}{8.0}(6.5,1.75)
    \only<4->{%
      \begin{equation*}
        H_{\pm}(\theta, t) = -\frac{\hbar^2}{2M}\frac{\partial^2}{\partial \theta^2} + V_0 \cos\left(m (\theta + \phi_{\pm}(t) )\right)
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{14.0}(1.0,6.25)
    \only<4->{%
      In co-moving frame:
      \vspace{-1.1cm}
      \begin{equation*}
          \hspace{2.5cm}
          \tilde{H}_{\pm} (t)= -\frac{\hbar^2}{2M}\frac{\partial^2}{\partial \theta^2} + V_0 \cos\left(m \theta\right) - i \hbar \omega_{\pm}(t) \frac{\partial}{\partial \theta}
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{10}(5.0,8.2)
    \onslide<2->{%
      \hfill\footnotesize{--- Dash, Goerz \emph{et al.} AVS Quantum Sci. 6, 014407 (2023)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Project-Specific Data Structures}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics[width=\textwidth]{images/rottai_code}
  \end{textblock}
\end{frame}

\begin{frame}{Adiabatic Dynamics of Rotating TAI}
  \begin{textblock}{7.0}(0.5,1.00)
    \includegraphics<1-3>{images/animate_rottai/frame_000.pdf}
    \includegraphics<4>{images/animate_rottai/frame_100.pdf}
    \includegraphics<5>{images/animate_rottai/frame_200.pdf}
    \includegraphics<6->{images/animate_rottai/frame_300.pdf}
  \end{textblock}
  \begin{textblock}{7.0}(0.5,2.00)
    \onslide<2>{%
      \begin{center}
        {\color{DarkRed}$\pi/2$ pulse}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{7.0}(0.5,2.00)
    \onslide<9>{%
      \begin{center}
        {\color{DarkRed}inverse \\ $\pi/2$ pulse}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{7.0}(0.5,5.50)
    \onslide<3->{%
      \begin{equation*}
        \omega(t) = \begin{cases}
          \omega_{0} \sin^2\left(\frac{\pi t}{2 t_r}\right)         & 0 \leq t < t_r            \\
          \omega_{0}                                                & t_r \leq t < t_r + t_{\text{loop}}  \\
          \omega_{0} \cos^2\left(\frac{\pi t^\prime }{2t_r} \right) & T - t_r \leq t \leq T
        \end{cases}
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{7.75}(8.0,2.00)
    \begin{center}
      \includegraphics<3-7>{images/adiabatic_dynamics_50πps_1}
      \includegraphics<8->{images/adiabatic_dynamics_50πps_2}
    \end{center}
  \end{textblock}
  \begin{textblock}{10}(5.0,8.2)
    \onslide<1->{%
      \hfill\footnotesize{--- Dash, Goerz \emph{et al.} AVS Quantum Sci. 6, 014407 (2023)}
    }
  \end{textblock}
\end{frame}

\begin{frame}{Interferometric Response of Rotating TAI}
  \begin{textblock}{15.5}(0.25,1.30)
    \begin{equation*}
      \Delta \Phi_S = \frac{4 m\Omega A}{\hbar}\,,
      \quad
      A
      = \frac{R^2}{2}
        \underbrace{\int_{0}^{T}\omega(t^\prime)dt^\prime}_{=\only<1-4>{n}\only<5>{2}\only<6>{10}\pi}
    \end{equation*}
  \end{textblock}
  \begin{textblock}{15.5}(0.25,3.00)
    \onslide<2->{%
      \begin{equation*}
        |c_{\pm}|^2
        = \frac{1}{2}
          \pm \frac{1}{2} \Re\left[{\color<3>{DarkRed}\eta} e^{-i \Delta\Phi}\right]
        \onslide<4->{%
          \qquad \rightarrow  \qquad
          |c_{-}|^2
          = \frac{1}{2} - \frac{\cos{\Delta\Phi}}{2} = \sin^2\left(\frac{\Delta\Phi}{2}\right)
        }
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{4}(1.85,4.00)
    \onslide<3-4>{%
      \begin{equation*}
        \color{DarkRed}
        \eta = \braket{\Psi_{-}(\theta, T)|\Psi_{+}(\theta, T)}
        = 1 \quad \text{if adiabatic}
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{15.5}(0.25,4.2)
    \onslide<5->{%
      \begin{center}
        \includegraphics<5>{images/cn_sim_results_1.pdf}
        \includegraphics<6>{images/cn_sim_results_2.pdf}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{10}(5.0,8.2)
    \onslide<5->{%
      \hfill\footnotesize{--- Dash, Goerz \emph{et al.} AVS Quantum Sci. 6, 014407 (2023)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Non-Adiabatic Dynamics of Rotating TAI}
  \begin{textblock}{7.75}(0.25,2.00)
    \includegraphics<1>{images/fidelity_map_1}
    \includegraphics<2-8>{images/fidelity_map_2}
  \end{textblock}
  \begin{textblock}{7.0}(8.5,2.25)
    \onslide<1-2>{%
      \begin{equation*}
        \omega(t) = \begin{cases}
          \omega_{0} \sin^2\left(\frac{\pi t}{2 t_r}\right)         & 0 \leq t < t_r            \\
          {\color{gray}\omega_{0}}                                  & {\color{gray}t_r \leq t < t_r + t_{\text{loop}}}  \\
          {\color{gray}\omega_{0} \cos^2\left(\frac{\pi t^\prime }{2t_r} \right)} & {\color{gray}T - t_r \leq t \leq T}
        \end{cases}
      \end{equation*}
      \vspace{2mm}
      \par
      $\ket{\Psi_{\text{tgt}}} = $ ground state of moving potential
    }
  \end{textblock}
  \begin{textblock}{7.75}(8.00,1.00)
    \includegraphics<3>{images/guess_dynamics_1.pdf}
    \includegraphics<4>{images/guess_dynamics_2.pdf}
    \includegraphics<5>{images/guess_dynamics_3.pdf}
    \includegraphics<6>{images/guess_dynamics_4.pdf}
    \includegraphics<7>{images/guess_dynamics_5.pdf}
    \includegraphics<8->{images/guess_dynamics_6.pdf}
  \end{textblock}
  \begin{textblock}{7.75}(0.25,2.00)
    \includegraphics<9-11>{images/guess_sagnac_1.pdf}
    \includegraphics<12>{images/guess_sagnac_2.pdf}
  \end{textblock}
  \begin{textblock}{7.75}(0.25,5.75)
    \onslide<9->{%
      \begin{equation*}
        \Delta \Phi_S = \frac{4 m\Omega A}{\hbar}\,,
        \quad
        A = \frac{R^2}{2} \cdot 10\pi
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{7.75}(0.25,7.00)
    \only<9>{%
      \begin{equation*}
        |c_{-}|^2
        = \frac{1}{2} - \frac{\cos{\Delta\Phi}}{2} = \sin^2\left(\frac{\Delta\Phi}{2}\right)
      \end{equation*}
    }
    \only<10->{%
      \begin{equation*}
        |c_{-}|^2
        = \frac{1}{2} - \frac{1}{2} \Re\left[{\color<11>{DarkRed}\eta} e^{-i \Delta\Phi}\right]
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{10}(5.0,8.2)
    \onslide<1-6>{%
      \hfill\footnotesize{--- Dash, Goerz \emph{et al.} AVS Quantum Sci. 6, 014407 (2023)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Rotating TAI Control Problem}
  \begin{textblock}{15.0}(0.5,1.20)
    \begin{center}
      \begin{equation*}
        \omega(t) = \begin{cases}
          \omega_{\text{opt}}(t)  & 0 \leq t < t_r            \\
          \omega_{0}              & t_r \leq t < t_r + t_{\text{loop}}  \\
          \omega_{\text{opt}}(t') & T - t_r \leq t \leq T
        \end{cases}
      \end{equation*}
      \par
      \vspace{8mm}
      Find $\omega_{\text{opt}}(t)$ for short $t_r$ so that
      \begin{equation*}
        \color{DarkRed}
        \Psi(\theta, t=0) \rightarrow \Psi_{\text{tgt}}(\theta, t=t_r)
      \end{equation*}
      where $\ket{\Psi_{\text{tgt}}} = $ ground state of moving potential
    \end{center}
  \end{textblock}
\end{frame}


\begin{frame}{Optimization with QuantumControl.jl}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<1-2>[width=\textwidth]{images/optimization_screenshot1}
    \includegraphics<3>[width=\textwidth]{images/optimization_screenshot2}
  \end{textblock}
  \begin{textblock}{12.8}(2.0,3.70)
    \onslide<2>{%
      \begin{block}{Guided Control}
        \vspace{2mm}
        \begin{equation*}
          \omega_{\text{opt}}(t) = \omega(t) + S(t)\delta\omega(t)
        \end{equation*}
        \vspace{2mm}
      \end{block}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Optimized Dynamics or Rotating TAI}
  \begin{textblock}{7.75}(0.25,1.00)
    \includegraphics<1-6>{images/guess_dynamics.pdf}
  \end{textblock}
  \begin{textblock}{7.75}(8.00,1.00)
    \includegraphics<2>{images/opt_dynamics_1.pdf}
    \includegraphics<3>{images/opt_dynamics_2.pdf}
    \includegraphics<4>{images/opt_dynamics_3.pdf}
    \includegraphics<5>{images/opt_dynamics_5.pdf}
    \includegraphics<6->{images/opt_dynamics_6.pdf}
  \end{textblock}
  \begin{textblock}{7.75}(0.25,2.00)
   \includegraphics<7>{images/opt_sagnac_1.pdf}
   \includegraphics<8->{images/opt_sagnac_2.pdf}
  \end{textblock}
  \begin{textblock}{7.75}(0.25,5.75)
    \onslide<7->{%
      \begin{equation*}
        \Delta \Phi_S = \frac{4 m\Omega A}{\hbar}\,,
        \quad
        A = \frac{R^2}{2} \cdot 10\pi
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{7.75}(0.25,7.00)
   \only<7>{%
     \begin{equation*}
       |c_{-}|^2
       = \frac{1}{2} - \frac{1}{2} \Re\left[\eta e^{-i \Delta\Phi}\right]
     \end{equation*}
   }
   \only<8->{%
     \begin{equation*}
       |c_{-}|^2
       = \frac{1}{2} - \frac{\cos{\Delta\Phi}}{2} = \sin^2\left(\frac{\Delta\Phi}{2}\right)
     \end{equation*}
   }
  \end{textblock}
\end{frame}


\begin{frame}{Nuclear Spin Gyroscope}
  \begin{textblock}{15}(0.5,1.0)
    \onslide<2>{%
      \begin{center}
        \includegraphics[width=\textwidth]{images/JarmolaSA2021_Fig2}
      \end{center}
      \hfill {\footnotesize --- Adapted from Fig 2 of Jarmola \textit{et. al.} Sci. Adv. 7, eabl3840 (2021)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{NV Center Hamiltonian with Crosstalk}
  \begin{textblock}{6}(0.5,2.00)
    \input{images/nvcenter_system_diagram.tikz}
    % \includegraphics<1>[trim=0 0 4cm 5mm,clip]{images/nvcenter_system_diagram.pdf}
  \end{textblock}
  \begin{textblock}{8}(7.0,2.00)
    \only<1>{%
      \begin{equation*}
        \Op{H} = \begin{pmatrix}
              0     & \Omega(t) &    0                       \\
          \Omega(t) & \omega_1  & \Omega(t)                  \\
              0     & \Omega(t)  & \omega_1 - \omega_2
        \end{pmatrix}
      \end{equation*}
      \begin{align*}
          \Omega(t) &= \Omega_p(t) \cos\left(\omega_p t + \phi_p(t) + \phi_p^{(i)}\right)  \\
                    &\quad + \Omega_s(t) \cos\left(\omega_s t + \phi_s(t) + \phi_s^{(i)}\right)
      \end{align*}
    }
    \only<2>{%
      \begin{equation*}
        \Op{H}_{\text{RWA}} = \begin{pmatrix}
            -\frac{\delta}{2}  &   \Omega_{1,0}(t)   &    0              \\
            \Omega_{1,0}^*(t)  &      \Delta         & \Omega_{0,-1}(t)  \\
                    0          & \Omega_{0,-1}^*(t)  & \frac{\delta}{2}
        \end{pmatrix}
      \end{equation*}
      \begin{align*}
        \Omega_{1,0}(t) &= \frac{\Omega_p(t)}{2} e^{i \phi_p^{(i)}} + \frac{\Omega_s(t)}{2} e^{i\phi_s^{(i)}} e^{-i \omega_{ps} t} \\
        \Omega_{0,-1}(t) &= \frac{\Omega_s^*(t)}{2} e^{-i \phi_p^{(i)}} + \frac{\Omega_p^*(t)}{2} e^{-i\phi_s^{(i)}} e^{-i \omega_{ps} t}
      \end{align*}
    }
  \end{textblock}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<3>[width=\textwidth]{images/ramsey_ham_code.png}
  \end{textblock}
\end{frame}


\begin{frame}{Parameterized Pulses}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<2>[width=\textwidth]{images/nvramsey_nb_parameterziation.png}
    \includegraphics<3>[width=\textwidth]{images/pulse_parameterization.png}
  \end{textblock}
\end{frame}


\begin{frame}{Population Response Signal}
  \begin{textblock}{15}(0.5,1.0)
    \begin{center}
      \includegraphics<2>{images/ramsey_1}
      \includegraphics<3>{images/ramsey_2}
      \includegraphics<4>{images/ramsey_3}
      \includegraphics<5->{images/ramsey_4}
    \end{center}
  \end{textblock}
  \begin{textblock}{15}(0.5,5.0)
    \begin{center}
      \includegraphics<2-5>[height=3.5cm]{images/JarmolaSA2021_Fig2}
    \end{center}
  \end{textblock}
  \begin{textblock}{13}(1.5,5.5)
    \onslide<6->{%
      \begin{center}
        \begin{block}{}
          \begin{equation*}
            J_T(\{\ket{\Psi_{\mu,\tau}(T)}\})
            = \sum_{\mu} \left\vert
            \text{FFT}([P_0({\color<7>{DarkRed}\tau}; {\color<7>{DarkRed}\mu})]) - \text{FFT}([P_0({\tau}; {\mu}=1)])
              \right\vert
          \end{equation*}
          Make spectrum for any $\mu$ look like spectrum for $\mu = 1$
        \end{block}
      \end{center}
    }
  \end{textblock}
\end{frame}


\begin{frame}{NV Center Optimization Problem}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<2,6>[width=\textwidth]{images/nvramsey_nb_problem.png}
    \includegraphics<1>[width=\textwidth]{images/nvramsey_nb_problem1.png}
    \includegraphics<3-4>[width=\textwidth]{images/nvramsey_nb_problem2.png}
    \includegraphics<5>[width=\textwidth]{images/nvramsey_nb_problem3.png}
    \includegraphics<7->[width=\textwidth]{images/nvramsey_nb_problem4.png}
  \end{textblock}
  \begin{textblock}{13}(1.5,5.5)
    \onslide<1-3,5-6>{%
      \begin{center}
        \begin{block}{}
          \begin{equation*}
            J_T(\{\ket{\Psi_{\mu,\tau}(T)}\})
            = \sum_{\mu} \left\vert
            {\color<6>{DarkRed}\text{FFT}}([P_0({\color<1-5>{DarkRed}\tau}; {\color<1>{DarkRed}\mu})]) - {\color<6>{DarkRed}\text{FFT}}([P_0({\color<2-5>{DarkRed}\tau}; {\mu}=1)])
              \right\vert
          \end{equation*}
          Make spectrum for any $\mu$ look like spectrum for $\mu = 1$
        \end{block}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{13}(1.5,5.5)
    \onslide<4>{%
      \begin{center}
        \begin{block}{}
          \begin{equation*}
            \Omega_{1,0}(t) = \frac{\Omega_p(t)}{2} e^{i \color{DarkRed}\phi_p^{(i)}} + \frac{\Omega_s(t)}{2} e^{i \color{DarkRed}\phi_s^{(i)}} e^{-i \omega_{ps} t}
          \end{equation*}
          Absorb phase difference in RF pulse
        \end{block}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{13}(1.5,2.05)
    \onslide<7->{%
      \begin{center}
        \begin{block}{}
          \begin{equation*}
            J_T(\{\ket{\Psi_{\mu,\tau}(T)}\})
            = \sum_{\mu} \left\vert
            {\color{DarkRed}\text{FFT}}([P_0({\tau}; {\mu})]) - {\color{DarkRed}\text{FFT}}([P_0({\tau}; {\mu}=1)])
              \right\vert
          \end{equation*}
          Make spectrum for any $\mu$ look like spectrum for $\mu = 1$
        \end{block}
      \end{center}
    }
  \end{textblock}
\end{frame}

\begin{frame}{Semi-automatic differentiation}
  \begin{textblock}{13}(1.5,2.05)
    \onslide<1>{%
      \begin{center}
        \begin{block}{}
          \begin{equation*}
            J_T(\{\ket{\Psi_{\mu,\tau}(T)}\})
            = \sum_{\mu} \left\vert
            {\color{DarkRed}\text{FFT}}([P_0({\tau}; {\mu})]) - {\color{DarkRed}\text{FFT}}([P_0({\tau}; {\mu}=1)])
              \right\vert
          \end{equation*}
          Make spectrum for any $\mu$ look like spectrum for $\mu = 1$
        \end{block}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{7}(8.5,2.25)
    \onslide<3>{%
      Automatic Differentiation:\par
      evaluate $J_T$ inside AD framework
    }
  \end{textblock}
  \begin{textblock}{14}(1.0,4.0)
    \onslide<4,5>{%
      \begin{block}{Semi-AD}
        Use a chain rule to split the gradient into
        \begin{itemize}
          \item a numerically expensive but analytic part
          \item a non-analytic but computationally cheap part
      \end{itemize}
      \end{block}
    }
  \end{textblock}
  \begin{textblock}{7}(1.0,2.0)
    \onslide<2->{%
      \begin{equation*}
        \begin{split}
          \nabla J_T
          &= \frac{\partial J_T\only<5->{(\{\Psi_k(T)\})}}{\partial \epsilon_{nl}}
          \\
          \onslide<6->{%
          &= 2 \Re \Bigg[
            \sum_k
              {\color<6>{white}\underbrace{\color{black}\frac{\partial J_T}{\partial \ket{\Psi_k(T)}}}_{\equiv \bra{\chi_k}}}
              \frac{\partial \ket{\Psi_k(T)}}{\partial \epsilon_{nl}}
            \Bigg]
          }
          \\
          \onslide<8->{%
          &= 2 \Re \Bigg[
            \sum_k
              \frac{\partial}{\partial \epsilon_{nl}}
              {\braket{\chi_k(T) | \Psi_k(T)}}
            \Bigg]
          }
        \end{split}
      \end{equation*}
    }
  \end{textblock}
  \begin{textblock}{7}(8.5,2.5)
    \includegraphics<9->[width=\textwidth]{images/grape_scheme}
  \end{textblock}
  \begin{textblock}{15}(1.0,7.5)
    \onslide<4->{%
      \footnotesize{--- Goerz, Carrasco, Malinovsky.  Quantum 6, 871 (2022)}
    }
  \end{textblock}
\end{frame}


\begin{frame}{NV Center Optimized Pulses}
  \begin{textblock}{15.5}(0.25,1.00)
    \includegraphics<2>[width=\textwidth]{images/nvramsey_nb_solution1.png}
    \includegraphics<3>[width=\textwidth]{images/nvramsey_nb_solution2.png}
  \end{textblock}
\end{frame}


\begin{frame}{Optimized Signal Spectrum}
  \begin{textblock}{15}(0.5,1.0)
    \begin{center}
      \includegraphics<1>{images/ramsey_4}
      \includegraphics<2>{images/ramsey_5}
      \includegraphics<3>{images/ramsey_6}
      \includegraphics<4>{images/ramsey_7}
      \includegraphics<5>{images/ramsey_8}
    \end{center}
  \end{textblock}
  \begin{textblock}{13}(1.5,5.5)
    \onslide<1>{%
      \begin{center}
        \begin{block}{}
          \begin{equation*}
            J_T(\{\ket{\Psi_{\mu,\tau}(T)}\})
            = \sum_{\mu} \left\vert
                \text{FFT}([P_0(\tau; \mu)]) - \text{FFT}([P_0(\tau; \mu=1)])
              \right\vert
          \end{equation*}
          Make spectrum for any $\mu$ look like spectrum for $\mu = 1$
        \end{block}
      \end{center}
    }
  \end{textblock}
\end{frame}

\begin{frame}{Conclusion}
  \begin{textblock}{13}(1.5,1.5)
    \onslide<1->{%
      \begin{itemize}
        \item<2-> Quantum Interferometry Implementations
          \begin{itemize}
            \item<3-> Atomic Fountain Interferometer: robust momentum space transfer
            \item<4-> Rotating Tractor Interferometer: non-adiabatic phase space transport
            \item<5-> Nuclear Spin Gyroscope: ``double quantum'' control, spectral optimization
          \end{itemize}
      \end{itemize}
    }
  \end{textblock}
  \begin{textblock}{14}(1.0,3.75)
    \includegraphics<6>[width=\textwidth]{images/attribution.pdf}
  \end{textblock}
  \begin{textblock}{13}(1.5,4.0)
    \onslide<7->{%
      \begin{itemize}
        \item<7-> QuantumControl.jl Framework
          \begin{itemize}
            \item<8-> Define control problems in terms of ``trajectories''
            \item<9-> Define dynamics in terms of ``generators'' and stateful ``propagators''
            \item<10-> Separate ``control amplitudes`` from actual ``controls''
            \item<11-> Efficient project-specific data structures through multiple dispatch
          \end{itemize}
      \end{itemize}
    }
  \end{textblock}
  \begin{textblock}{13}(1.5,7.0)
    \onslide<12>{%
      \begin{center}
        {\color{DarkRed}
        \Large Thank You!
        }
      \end{center}
    }
  \end{textblock}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\backupbegin

\begin{frame}\end{frame}

\begin{frame}{Semi-Automatic Differentiation}
  \begin{textblock}{14}(1.0,1.25)
    \only<1>{{\color{Red}\bf Quantum 6, 871 (2022) --- arXiv:2205.15044}}
  \end{textblock}
  \begin{textblock}{14}(1.0,1.75)
    \only<1>{%
      \frame{\includegraphics[width=\textwidth]{images/SemiAD_firstpage}}
    }
  \end{textblock}
  \begin{textblock}{14}(1.0,6.25)
    \only<4>{{\color{Red}\bf Goerz \emph{at al.} Quantum 6, 871 (2022) --- arXiv:2205.15044}}
  \end{textblock}
  \begin{textblock}{3.0}(12.25,1.75)
    \only<1>{%
      \begin{center}
        \includegraphics[width=1.5cm]{images/profile/seba.jpeg}\\
        \includegraphics[width=1.5cm]{images/profile/vlad.jpg}
      \end{center}
    }
  \end{textblock}
  \begin{textblock}{10}(3.0, 7.02)
    \only<1>{\tiny
      \setbeamercolor{upcol}{fg=black,bg=gray!20}
      \setbeamercolor{lowcol}{fg=black,bg=gray!10}
      \begin{beamerboxesrounded}[upper=upcol,lower=lowcol]{Funding}
       DEVCOM Army Research Laboratory, Cooperative Agreement Numbers W911NF-16-2-0147, \\W911NF-21-2-0037; DIRA-TRC No. DTR19-CI-019
      \end{beamerboxesrounded}
    }
  \end{textblock}
  \begin{textblock}{15}(0.5, 1.0)
    \only<2->{%
      \begin{align*}
        \onslide<3->{\nabla}J(\{\epsilon_{nl}\})
          &= \onslide<3->{\frac{\partial}{\partial_{\epsilon_{nl}}}}
              J_T(\{\ket{\Psi_k(T)}\}) + \dots \\
        \onslide<5->{%
          &= 2\Re \sum_k
            \underbrace<6->{\frac{\partial J_T}{\partial \ket{\Psi_k(T)}}}_{%
              \equiv \bra{\chi_k(T)}
            }
            \frac{\partial \ket{\Psi_k(T)}}{\partial \epsilon_{nl}}
            \onslide<6->{%
              ; \qquad {\color<8->{DarkRed}\ket{\chi_k(T)} = \frac{\partial J_T}{\partial \bra{\Psi_k(T)}}}
            }
            \\
        }
        \onslide<7->{%
          &= 2 \Re \sum_k \frac{\partial}{\partial \epsilon_{nl}} \langle \chi_k(T) \vert \Psi_k(T) \rangle \\
        }
        \onslide<9->{%
          &= 2 \Re \sum_k \frac{\partial}{\partial \epsilon_{nl}} \langle \chi_k(T) \vert
              \Op{U}_N \dots \Op{U}_{n+1} \Op{U}_{n}  \Op{U}_{n-1} \dots \Op{U}_1  \vert
              \Psi_k(t=0) \rangle  \\
        }
        \onslide<10->{%
          &= 2 \Re \sum_k
            \underbrace<11->{%
              \Big\langle \chi_k(T) \Big\vert \Op{U}_N \dots \Op{U}_{n+1} {\color<12>{DarkRed}\frac{\partial \Op{U}_n}{\partial \epsilon_{nl}}}
            }_{\text{backward propagation}}\;
            \underbrace<11->{%
              \Op{U}_{n-1} \dots \Op{U}_1 \Big\vert \Psi_k(t=0) \Big\rangle
            }_{\text{forward propagation}}
        }
      \end{align*}
    }
  \end{textblock}
\end{frame}


\begin{frame}{Aside: Wirtinger derivatives --- derivatives w.r.t. complex numbers}
  %\begin{textblock}{3.0}(0.5,1.0)
    \begin{equation*}
      J_T (\{z_k\}) = J_T(\{\Re[z_k], \Im[z_k]\}); \qquad J_T \in \Reals, \quad z_k \in \Complex
    \end{equation*}
    \pause
    \begin{equation*}
      \frac{\partial J_T (\{z_k\})}{\partial \epsilon_{nl}} =
      \sum_k \left(
        \frac{\partial J_T}{\partial \Re[z_k]}
        \frac{\partial \Re[z_k]}{\partial \epsilon_{nl}}
        + \frac{\partial J_T}{\partial \Im[z_k]}
        \frac{\partial \Im[z_k]}{\partial \epsilon_{nl}}
      \right);
      \qquad \epsilon_{nl} \in \Reals
    \end{equation*}
    \pause
    {
      \color{DarkRed}
      \begin{align*}
        \text{Define} \quad \frac{\partial J_T (\{z_k\})}{\partial z_k}
        & \equiv \frac{1}{2} \left(
          \frac{\partial J_T}{\partial \Re[z_k]}
          - i \frac{\partial J_T}{\partial \Im[z_k]}
        \right) \\
        \frac{\partial J_T (\{z_k\})}{\partial z_k^*}
        & \equiv \frac{1}{2} \left(
          \frac{\partial J_T}{\partial \Re[z_k]}
          + i \frac{\partial J_T}{\partial \Im[z_k]}
        \right)
        = \left(\frac{\partial J_T}{\partial z_k}\right)^*
      \end{align*}
    }
    \vspace{1mm}
    \pause
    \begin{equation*}
       \frac{\partial J_T (\{z_k\})}{\partial \epsilon_{nl}}
       =
       \sum_k \left(
         \frac{\partial J_T}{\partial z_k}
         \frac{\partial z_k}{\partial \epsilon_{nl}}
         + \frac{\partial J_T}{\partial z_k^*}
         \frac{\partial z_k^*}{\partial \epsilon_{nl}}
       \right)
      =
       2 \Re \left[
        \sum_k
        \frac{\partial J_T}{\partial z_k} \frac{\partial z_k}{\partial \epsilon_{nl}}
       \right]
    \end{equation*}
  %\end{textblock}
\end{frame}


\begin{frame}{Gradient of Time Evolution Operator}
    \begin{equation*}%
      \begin{pmatrix}
        \frac{\partial \Op{U}^{\dagger}_n}{\partial \epsilon_{n1}} \ket{\chi_k(t_{n})} \\
          \vdots\\
          \frac{\partial \Op{U}^{\dagger}_n}{\partial \epsilon_{nL}} \ket{\chi_k(t_{n})} \\
          \Op{U}^{\dagger}_n \ket{\chi_k(t_{n})}
        \end{pmatrix}
      = \exp \left[-\ii \begin{pmatrix}
        \Op{H}^\dagger_n & 0 & \dots & 0 &\Op{H}_n^{(1)\dagger} \\
        0 & \Op{H}^\dagger_n & \dots & 0 & \Op{H}_n^{(2)\dagger} \\
        \vdots & & \ddots & & \vdots \\
        0 & 0 & \dots & \Op{H}^\dagger_n & \Op{H}_n^{(L)\dagger} \\
        0 & 0 & \dots & 0 & \Op{H}^\dagger_n\,,
      \end{pmatrix} dt_n\right]
      \begin{pmatrix} 0 \\ \vdots \\ 0 \\ \ket{\chi_k(t_n)} \end{pmatrix}
    \end{equation*}
    \vspace{5mm}
    \begin{equation*}
      \Op{U}_n = \exp[-\ii \Op{H}_n dt_n]\,;
      \qquad
      \Op{H}_n^{(l)} = \frac{\partial \Op{H}_n}{\partial \epsilon_l(t)}
    \end{equation*}
    \par
    \vspace{5mm}
    \hfill \footnotesize{--- Goodwin, Kuprov, J. Chem. Phys. 143, 084113 (2015)}\\
    \vspace{3mm}
    \hfill \footnotesize{\url{https://github.com/JuliaQuantumControl/QuantumGradientGenerators.jl}}
\end{frame}

\begin{frame}{Generalized GRAPE scheme}
  \begin{textblock}{15}(0.5,1.0)
    \begin{center}
      \input{images/grape.tikz}
    \end{center}
  \end{textblock}
  \begin{textblock}{3}(11.5, 1.4)
    \begin{tikzpicture}
      \draw<4-> (0,0) node[line width = 3pt, draw=black, inner sep = 6pt, fill=gray!20,rounded corners=10] {
        $\tau^{(k)} = \Braket{\chi_k(T)|\Psi_k(T)}$
      };
    \end{tikzpicture}
  \end{textblock}
  \begin{textblock}{3}(0.3, 1.4)
    \begin{tikzpicture}
      \draw<5-> (0,0) node[line width = 3pt, draw=black, inner sep = 6pt, fill=gray!20,rounded corners=10] {
        $\nabla J_T = 2 \Re \sum_k \nabla \tau^{(k)}$
      };
    \end{tikzpicture}
  \end{textblock}
  \begin{textblock}{15}(0.5,8.15)
    \hfill \footnotesize{--- Goerz \emph{et al.} Quantum 6, 871 (2022)}
  \end{textblock}
\end{frame}

\begin{frame}{Semi-Automatic Differentiation}
  \begin{center}
    \begin{equation*}
      {\color{DarkRed}\ket{\chi_k(T)} = \frac{\partial J_T}{\partial \bra{\Psi_k(T)}}}
    \end{equation*}
    \par \vspace{5mm}
    is the only thing evaluated inside AD framework
  \end{center}
  \vspace{8mm}
  \begin{itemize}
    \pause
    \item[$\rightarrow$]  $J_T = J_T(\Op{U})$
    \pause
    \item[$\rightarrow$] $J_T = J_T(\{\tau_k\})$ with $\tau_k = \Braket{\Psi_k(T) | \Psi_k^{\tgt}}$
  \end{itemize}
\end{frame}


\begin{frame}{Gradients of parametrized pulses}
    \begin{equation*}%
      \begin{pmatrix}
        \frac{\partial \Op{U}}{\partial u_1} \ket{\Psi_k} \\
          \vdots\\
          \frac{\partial \Op{U}}{\partial u_N} \ket{\Psi_k} \\
          \Op{U} \ket{\Psi_k}
        \end{pmatrix}
      = \exp \left[-\ii \TimeOrder \int_0^T \begin{pmatrix}
        \Op{H}(t) & 0 & \dots & 0 &\Op{H}^{(1)}(t) \\
        0 & \Op{H}(t) & \dots & 0 & \Op{H}^{(2)}(t) \\
        \vdots & & \ddots & & \vdots \\
        0 & 0 & \dots & \Op{H}(t) & \Op{H}^{(N)}(t) \\
        0 & 0 & \dots & 0 & \Op{H}(t)
      \end{pmatrix} dt\right]
      \begin{pmatrix} 0 \\ \vdots \\ 0 \\ \ket{\Psi_k} \end{pmatrix}
    \end{equation*}
    \par
    \vspace{5mm}
    \hspace{1cm}
    with $\Op{H}^{(n)}(t) = \frac{\partial \Op{H}(t)}{\partial u_n}$
    \par
    \vspace{5mm}
    {\footnotesize
      \hfill --- ``GOAT'': Machnes \textit{et al.}  Phys. Rev. Lett. 120, 150401 (2018)
    }
    \hfill \footnotesize{\url{https://github.com/JuliaQuantumControl/QuantumGradientGenerators.jl}}

\end{frame}


\begin{frame}{Open Quantum Systems}
  \subhead{Lindblad equation:}
  \begin{align*}
  \frac{d}{dt}\Op{\rho}(t)
      & =-i\left[\Op{H},\Op{\rho}(t)\right]+\mathcal{L}_{D}(\Op{\rho}(t))\\
      & =-i\left[\Op{H},\Op{\rho}(t)\right]+\sum_{k}\left(\Op{A}_{k}\Op{\rho}\Op{A}_{k}^{\dagger}-\frac{1}{2}\Op{A}_{k}^{\dagger}\Op{A}_{k}\Op{\rho}-\frac{1}{2}\Op{\rho}\Op{A}_{k}^{\dagger}\Op{A}_{k}\right)
  \end{align*}
  \subhead{Vectorization rule:}
  \begin{equation*}
    \vectorize\left(\Op{A} \Op{\rho} \Op{B} \right)
    = \left(\mat{B}^{T} \otimes \mat{A}\right) \vec{\rho}
  \end{equation*}
  \subhead{Matrix representation of Lindbladian:}
  \begin{equation*}
    \mat{L} =
      -i (\identity \otimes \mat{H}) + i (\mat{H}^T \otimes \identity)
      + \sum_k \left[
        (\mat{A}_k^\dagger)^T \otimes \mat{A}_k
        - \half \left(\identity \otimes \mat{A}_k^\dagger \mat{A}_k\right)
        - \half \left((\mat{A}_k^\dagger \mat{A}_k)^T \otimes \identity\right)
        \right]\
  \end{equation*}
  \hfill {\footnotesize --- Goerz \textit{et. al.} arXiv:1312.0111v2 (2021), Appendix B}
\end{frame}


\begin{frame}{Gradient-free optimization}
  \begin{center}
  \includegraphics{images/nelder_mead}\\
      {\footnotesize\vspace{-2mm}\hspace{-15mm} Doria et al. PRL 106, 190501 (2011)}
  \end{center}
  \vspace{12pt}
  e.g. Nelder-Mead (simplex), genetic algorithms\dots
\end{frame}

\backupend

\end{document}