FOMO2024 School

Event info: https://www.matterwaveoptics.eu/FOMO2024

2nd September

Introduction to the Summer School and Atom Cooling

By Wolf von Klitzing Outline: Energy Scales, Atom Cooling, Bose-Einstein-Condensate, Interferometry Ingredients: Vacuum (XUV), Lasers, Magnetic Fields, RF

Energy scales

Room Temp: 297 Kelvin = 230 m/s ( … )

How can we convert temperature to speed? Help me ❓

Atoms frequently used

Alkali Atoms: Rubidium, Sodium, Lithium, Potassium Alkaline Earth-like Atoms: Calcium, Strontium, Ytterbium Rare Earth Elements: Erbium, Dysprosium Others: Hydrogen, Helium Molecules: Di-atoms

How do lasers allow us to force an atom from one state to another state? In particular, can we force it to a lower energy state? Help me ❓ We use the Magneto Optic Trap ( MOT Transition)

MOT

Currently we are able to load $10^{10}$ Rubidium atoms on a MOT Trap.

Notation note: $\Gamma$ represents the decay $e^{-\Gamma t}$, and $\gamma$ represents the half line width in angular units ( rad/s )

We use it on the scattering equation ${\Gamma }_{scatt}=\frac{.5\ast \Gamma }{1+\frac{I}{I_{sat}}+{\left(\frac{{\omega }_{atom}-{\omega }_{las}}{.5\ast \Gamma }\right)}^2}$, where:

• $\gamma =\frac{\Gamma }{2}$ Is the line width
• $\delta =\Delta \omega /\gamma$ is the normalized detuning
• $s=\frac{I}{I_{sat}}$ is the saturation parameter This way we can rewrite ${\Gamma }_{scatt}=\frac{s}{1+s+{\delta }^2}\gamma$ .

Keywords: Doppler Effect, Detuning, Molasses Force.

I need to better understand how we can “increase the detuning”. Help me ❓

Bose-Einstein condensate

• The presenter wants to start a website that collects all BECs. Reach out on this.

• $f\propto e^{-\frac{E}{k_{B}T}}$ Maxwell Boltzmann

• $f={\left(e^{\frac{E}{k_{B}T}-1}\right)}^{-1}$ Bose-Einstein statistics for photons

• $f={\left(e^{\frac{K-\mu }{k_{B}T}-1}\right)}^{-1}$ Bose-Einstein statistics for particles

Introduction to Atom Interferometry ( atom optics )

By Enno Giese and Friedrich Alexander

Dipole interaction

We have

• $\hat{d}E$ being the interaction energy
• $d=|q|r$
• $r$ Is the radius
• $q$ is the magnetic charge

We specify a two level atom with a ground state $|g⟩$ and an excited state $|e⟩$. We describe the two level atom = $d_{eg}|e⟩⟨g|+d_{ge}|g⟩⟨e|$.

The electric field is described by:

• $E(z,t)=E_g{e}^{id(z,t)}+h_c$, a chirped plane wave
• $E_g$ is the amplitude
• $d(z,t)={\psi }_e+kz-wt+k\alpha t^2\frac{1}{2}$ is the phase. I need to check how the expression for this is derived. Help me ❓

We get a position dependence: $⟨z|e^{ikz}|p⟩=e^{ikz}⟨z|p⟩\approx e^{ikz}{e}^{i\frac{pz}{\hslash }}=e^{i{\frac{\hslash k+p}{\hslash }}^2}\approx ⟨z|p+\hslash k⟩$. Hence we can thing of this as a momentum displacement. We also get $e^{-ikz}f(\hat{p})e^{ikz}|p⟩=\cdots =f(\hat{p}+\hslash k)|p⟩$.

$H=(H_{external}+\hslash {\omega }_e)|e⟩⟨e|+(H_{external}+\hslash {\omega }_g)|g⟩⟨g|-dE(z,t)$ is the Hamiltonian of the interaction. This is easier to analyse In the Interaction Picture wrt the first two terms. The E-field evaluate at $\varphi (z,t)$ E-field evaluates at $\varphi (z(t),t)$, where $z(t)=z+\frac{\beta t}{m}-\frac{1}{2}g{t}^2$. Dipole operator := $d(t)=d_{e}g|e⟩⟨e|e^{i{\omega }_{at}t}+hc$. The Hamiltonian becomes $H_I=-d_{eg}{E}_0\left[e^i[{\omega }_{at}t+\varphi (z(t),t)]|e⟩⟨g|+hc+e^i[{\omega }_{at}t-\varphi (z(t),t)]|g⟩⟨e|+hc\right]=\frac{\hslash \dots }{z}$. ${\omega }_{at}$ is the Frequency of Oscillation of the atom. We are able to drop the fast rotating terms, a process known as the Rotating wave approximation. Not sure if this is the same as the Rotating wave transformation. Help me ❓

We use the Baker-Campbell-Hausdorff formula on our Rotating wave approximation:

$$\begin{array}{rl}{H}_I& =\frac{\hslash \dots }{2}\left(e^i{\omega }_{at}t+\varphi (z,t)|e⟩⟨g|+hc\right)\\ & =\frac{\hslash \dots }{2}\left(e^{i({\varphi }_0+kz)}{e}^{i({\omega }_{at}+\frac{\hat{p}k)}{m}+\frac{\hslash k^2}{2m}-\omega )t{e}^{i(\dots }}|e⟩⟨g|+hc\right)\end{array}$$

In the above, we have:

• $\alpha =g$ happens with perfect chirping
• $k{\omega }_{at}+{\frac{\hat{p}+\hslash k}{2m}}^2-\frac{{\hat{p}}^2}{2m}$ is the gained energy during absorption.

rabi oscillations

We assume a narrow momentum distribution $\Delta p$, allowing us to model the system as rabi oscillations. The resulting time independent Hamiltonian is given by $H_I=\frac{\hslash {\Omega }_R}{2}\left(e^{i({\varphi }_e+kz)}|e⟩⟨g|+hc\right)$. We get the solution as a time evolution operator ${\hat{S}}_R=e^{-\frac{i}{\hslash }{H}_I\tau }=\cos (\frac{{\Omega }_R\tau }{2})H_{ext}⨂H_{int}-i\sin (\frac{{\Omega }_R\tau }{2})(e^{i({\varphi }_e+kz)}|e⟩⟨g|+hc)$. It is relevant that the initial phase of light ${\varphi }_e$ enters the equation.

$|\varphi ({\Omega }_R\tau )⟩={\hat{S}}_R|p_{res}⟩⨂|g⟩=\cos (\frac{{\Omega }_R\tau }{2})|p+res⟩⨂|g⟩-i\sin (\frac{{\Omega }_R\tau }{2})(e^{i({\varphi }_e+kz)}|p_{res}+\hslash k⟩⟨e|+hc)$.

• ${\varphi }_e$ should actually be ${\varphi }_l$ in all of the above

Diffraction mechanisms

We can use other diffraction mechanisms, such as Raman diffraction, Bragg diffraction

Landau-Zener transitions made simple

By Wolfgang Schleisch

Goal

Do a one line derivation of the transition probability between a two state quantum system, where

• The Hamiltonian is time independent.
• The Energy level difference evolves linearly over time.

We will make use of the Ricatti Equation.

Landau-Zener problem

We take a two level atom, where the 2 levels are moving linearly “past” each other. This can be achieved through a magnetic field, where each state reacts differently to the field. We model this as a time evolving vector ${\left(\begin{array}{c}a\\ b\end{array}\right)}_t$, allowing us to describe the system as

$$i\hslash \frac{d}{dt}\left(\begin{array}{c}{a}_t\\ b_t\end{array}\right)=H_t\left(\begin{array}{c}{a}_t\\ b_t\end{array}\right)=\left(\begin{array}{ccc}\alpha g& & g\\ g& & \alpha g\end{array}\right)\left(\begin{array}{c}{a}_t\\ b_t\end{array}\right)$$

We use dimensionless units in order to remove constants, obtaining

$$i\frac{d}{dt}\left(\begin{array}{c}{a}_{\tau }\\ b_{\tau }\end{array}\right)=\left(\begin{array}{ccc}-ϵ\tau & & 1\\ 1& & -ϵ\tau \end{array}\right)\left(\begin{array}{c}{a}_t\\ b_t\end{array}\right)$$

Where

• $ϵ=\frac{\alpha }{g^2}$
• $\dot{\alpha }=iϵ\tau \alpha -ib$
• $\dot{\beta }=-ia-iϵ\tau b$
• ${\tau }_0\in {\mathbb{R}}^{+}$ is needed for our initial conditions. We start with initial conditions as $\alpha (-{\tau }_0)=1$ and $\beta (-{\tau }_0)=0$. A normalization condition requires us to have $|\alpha {|}^2+\beta {|}^2=1$. We get express the solution as follows, in order to simplify the resulting solution:
• $\alpha (\tau )=e^{\frac{1}{2}iϵ{\tau }^2}{e}^{-\frac{1}{2}iϵ{\tau }_0^2}\tilde{\alpha }(\tau )$.
• $\beta (\tau )=e^{-\frac{1}{2}iϵ{\tau }^2}{e}^{-i\frac{1}{2}ϵ{\tau }_0^2}\tilde{\alpha }\tilde{\beta }(\tau )$

The above allows us to write

• $\dot{\alpha }=iϵ\tau \alpha +e^{\frac{1}{2}iϵ{\tau }^2}{e}^{i\frac{1}{2}ϵ{\tau }_0^2}\dot{\tilde{\alpha }}$
• $\dot{\beta }=-iϵ\beta +e^{\frac{1}{2}iϵ{\tau }^2}{e}^{i\frac{1}{2}ϵ{\tau }_0^2}\dot{\tilde{\alpha }}$ Using the above 2 pairs of equations to find differential equations for $\tilde{\alpha }$ and $\tilde{\beta }$ we get

$$\{\begin{array}{rl}\dot{\tilde{\alpha }}& =-i{e}^{-\frac{1}{2}iϵ{\tau }^2}{e}^{\frac{1}{2}iϵ{\tau }_0^2}\tilde{\beta }{e}^{-\frac{1}{2}iϵ{\tau }^2}{e}^{\frac{1}{2}iϵ{\tau }^2}\\ \dot{\tilde{\beta }}& =-i{e}^{-\frac{1}{2}iϵ{\tau }^2}{e}^{\frac{1}{2}iϵ{\tau }_0^2}\tilde{\beta }{e}^{-\frac{1}{2}iϵ{\tau }^2}{e}^{\frac{1}{2}iϵ{\tau }^2}\end{array}$$

( magic )

We end up getting

$$\begin{array}{rl}\tilde{\beta }& =-i{\int }_{-{\tau }_0}^{\tau }{e}^{iϵ(\tau {’}^2-{\tau }_0^2)}\tilde{\alpha }(\tau ’)d\tau ’\\ \dot{\tilde{\alpha }}& =-e^{-iϵ(\tau {’}^2-{\tau }_0^2)}{\int }_{-{\tau }_0}^{\tau }{e}^{i({\tau }^2-{\tau }_0^2)}\tilde{\alpha }(\tau ’)d\tau ’\end{array}$$

Allowing us to write $\dot{\tilde{\alpha }}(\tau )=-e^{-iϵ{\tau }^2}{\int }_{-{\tau }_0}^{\tau }{e}^{iϵ\tau {’}^2}\tilde{\alpha }(\tau ’)d\tau ’$

Markov Approximation

By making a Markov Approximation we can assume $\tilde{\alpha }(\tau ’)\approx \alpha (\tau )$. With this we get ${\dot{\tilde{\alpha }}}_M=-{\eta }_M(\tau ){\tilde{\alpha }}_M(\tau )$, the $M$ standing for the Markov Approximation Differential Equation. The standard solution is ${\tilde{\alpha }}_M(\tau )=e^{-{\int }_{{\tau }_0}^{\tau }{\eta }_M(\tau ’)d\tau ’}$, where ${\eta }_M(\tau ):=e^{-iϵ{\tau }^2}{\int }_{-{\tau }_0}^{\tau }{e}^{iϵ\tau {’}^2}d\tau ’$.

Unfinished notes ( comments )

We also made use of Dyson series, Fresnel integral, Stuckelberg oscillations. We can see the complete derivation in https://iopscience.iop.org/article/10.1088/1361-6455/acc774/pdf.

Introduction to Atom Interferometry ( structures of atom interferometers )

Evolution between pulses

• $H:=1_{ext}⨂H_{int}+H_{ext}⨂1_{int}$ is our decoupled Hamiltonian.
• $|\psi (t)⟩=U(t)|\psi (0)⟩$ is our time evolution operator.
• The above gives us by the Schrodinger Equation that $i\hslash \frac{\partial }{\partial t}U=HU$. I need to flesh out why this holds. Help me ❓

Since we designed our system with independent internal and external Hamiltonians, we can use the Baker-Campbell-Hausdorff formula to get $U(t)=e^{-\frac{i}{\hslash }Ht}=e^{-\frac{i}{\hslash }{H}_{ext}t}{e}^{-\frac{i}{\hslash }{H}_{int}t}=U_{ext}(t)U_{int}(t)$. The internal dynamics can be described as $H_{int}=\hslash {\omega }_e|e⟩⟨e|+\hslash {\omega }_g|g⟩⟨g|$. Since this is a diagonal Hamiltonian, we get $U(t)=e^{-\frac{i}{\hslash }Ht}=e^{-i{\omega }_{e}t}|e⟩⟨e|+e^{-i{\omega }_{g}t}|g⟩⟨g|$. The external dynamics are described by a Total Mach-Zehnder sequence of a Beam Splliter, Mirror, and Beam Splitter. Algebraically this becomes $U_{ext}=S_{\frac{\pi }{2}}^{(2)}U(T+\delta T)S_{\pi }^{(1)}U(T)S_{\frac{\pi }{2}}^{(0)}$. It is unclear why the second time evolution has a $\delta T$ time correction. Help me ❓ The associated matrix elements are:

• $⟨g|U_{ext}|g⟩=\frac{1}{2}{e}^{i{\varphi }_{gl}}[U_1+U_2]$, where ${\varphi }_{gl}$ is a Global Phase.

• $U_j$ is the description of the Mach-Zehnder interferometer along one of the respective arm $j$. These are Unitary Operators, i.e. $U_j^{†}{U}_j=1_{ext}$.

• I am not clear how to think about the $U_j$ only operating on the centre of mass. Help me ❓

Detection

We use the

• Projectors ${\Pi }_{zg}:=|z⟩⟨z|⨂|g⟩⟨g|$.
• integrated particle number ${\Pi }_g=\int |z⟩⟨z|dz⨂|g⟩⟨g|=1_{ext}⨂|g⟩⟨g|$.
• input state $|{\psi }_{in}⟩:=|{\psi }_0⟩⨂|g⟩$.
• Ground state population $I_g:=⟨1_{ext]⨂|g⟩⟨g|}⟩=⟨{\psi }_0|⟨g|U_{total}^{†}|g⟩⟨g|U_{total}|g⟩|{\psi }_0⟩$

With the form above and $⟨g|U_{total}|g⟩=\frac{1}{2}{e}^{i{\varphi }_{gl}}[U_1+U_2]$ we get $I_g=⟨{\psi }_0|[U_1^{†}+U_2^{†}][U_1+U_2]|{\psi }_0⟩$. Since the $U_i$ are Unitary Operators, $I_g=\frac{1}{4}\left(2+⟨{\psi }_0|U_2^{†}{U}_1{\psi }_0⟩+⟨{\psi }_0|U_1^{†}{U}_2{\psi }_0⟩\right)$. Writing $⟨{\psi }_0|U_2^{†}{U}_1{\psi }_0⟩=r{e}^{i\varphi }\in \mathbb{C}$ allows us to get $I_g=\frac{1}{2}(1+r\cos (\varphi ))$, which is an oscillation around the value $\frac{1}{2}$. This is our interference signal.

$⟨g|U_{tot}|g⟩=⟨g|S_{\frac{\pi }{2}}^{(2)}U(T+\delta T)S_{\pi }^{(1)}U(T)S_{\frac{\pi }{2}}^{(0)}|g⟩$ describes our evolution along branches. Focusing on individual operators, we have:

• $S_{\frac{\pi }{2}}^{(2)}=\frac{1}{\sqrt{2}}\left[1_{ext}⨂|e⟩⟨e|-i{e}^{i({\varphi }_l+ph{i}_z)}|e⟩⟨g|-i{e}^{-i({\varphi }_l+{\varphi }_z)}|g⟩⟨e|+1_{ext}⨂|g⟩⟨g|\right]$
• $S_{\frac{\pi }{2}}^{(0)}|g⟩=\frac{1}{\sqrt{2}}\left[-i{e}^{i({\varphi }_l+ph{i}_z)}|e⟩+1_{ext}⨂|g⟩\right]$
• $⟨g|S_{\frac{\pi }{2}}^{(2)}=\frac{1}{\sqrt{2}}\left[-i{e}^{-i({\varphi }_l+{\varphi }_z)}⟨e|+1_{ext}⨂⟨g|\right]$
• ( missing equations from here )

3rd September

Interference in Phase Space

We will use the WKB approximation to solve the Schrodinger Equation. We start with an atom with an heavy nucleus and a light electron. We have 2 different potential curves, as seen below. Letting $|m⟩$ represent a state in red, and $|n⟩$ a state in blue, we are able to calculate the transition probability as $W_{m\leftarrow n}=|⟨m|n⟩{|}^2$.

( remaining notes taken in paper, to be able to diagram )

Introduction to machine learning for data science

By Yannis Pantazis and Grigorios Tsagkatakis

4th September

Quantum Control of Atoms in Optical Lattices

Outline: Trapping atoms in light fields → BEC in an optical lattice

We start with an energy delta of $\hslash {\omega }_0$ between two states $|g⟩$ and $|e⟩$. Schrodinger Equation gives us $|\psi (t)⟩=e^{-iH\frac{t}{\hslash }}|\psi (0)⟩$., where $H=\frac{P^2}{2m}+V(x)$ is our Hamiltonian in one dimension. Our $H$ is such that $H|e⟩=\hslash {\omega }_0|e⟩$ and $H|g⟩=0|g⟩$.

Comment: See the Steck Quantum Optics book for more details on this.

We will work with:

• the atomic length scale $\theta$ in $10^{-10}-10^{-11}$ m
• Optical $\lambda$ : $400-900$ m
• $\bar{F}=-e\bar{E}$
• $\bar{F}=-\Delta U$
• $U=e\bar{r}\bar{E}=-\bar{d}\bar{E}$
• $\bar{E}(t)=\bar{e}{E}_0\cos (\omega t)$

With the above we get $⟨e|\bar{d}\bar{E}|e⟩=0=⟨g|\bar{d}\bar{E}|g⟩$. In other words dipoles do not have self interactions.

We are able to write

$$\left(\begin{array}{cc}0& \\ & \hslash {\omega }_0\end{array}\right)$$

the off diagonal terms show us the state interaction terms which we need to discover. $\hat{d}=d_{eg}{\hat{\sigma }}^{†}+d_{ge}\hat{\sigma }$ with $\hat{\sigma }=|g⟩⟨e|$. The Hamiltonian then becomes $H=\hslash {\omega }_0{\hat{\sigma }}^{†}\hat{\sigma }+\hslash \Omega \cos ({\omega }_{0}t)(\hat{\sigma }+{\hat{\sigma }}^{†})$. This is a pain to solve due to the time dependency of $H$. We can get rid of it by applying a rotating unitary transformation in order to remove the time dependency. The relevant Unitary Operator is $U_R=\exp \left(i{\omega }_l|e⟩⟨e|t\right)$, creating the Rotating wave approximation hamiltonian $H_{RWA}=\left(\begin{array}{cc}0& \frac{n}{2}\\ \frac{n}{2}& 0\end{array}\right)$.

Experimentally we can take $\Delta \gg \Omega$, which gives us ${\lambda }_{\pm }=\pm \frac{\hslash }{2}(\frac{\Delta }{2}+\frac{{\Omega }^2}{4\Delta })$ as a Taylor expansion of ${\lambda }_{\pm }=\pm \sqrt{{\Omega }^2+{\Delta }^2}\frac{\hslash }{2}$.

Optical Dipole Traps for Neutral Atoms

The paper titled as above guides most of the equations below.

$\alpha (\omega )=6\pi {ϵ}_0{e}^3\frac{\frac{P}{{\omega }_0}}{{\Delta }^2+\frac{i\omega P}{{\omega }_0^2}}$ where $\Delta :={\omega }_e-{\omega }_0$. We get

• $U_{dip}=-\frac{1}{2}⟨\bar{d}\bar{E}⟩=\frac{1}{2{ϵ}_{0}c}\mathrm{Re}[\alpha ]I$
• $P_{abs}=⟨\bar{d}\bar{E}⟩=\frac{{\omega }_e}{{ϵ}_{0}c}\mathrm{Im}[\alpha ]I$.

We make the assumption $|\Delta |\ll {\omega }_0$, i.e. our Detuning is much smaller than our oscillations, to get

• $U_{dip}=\frac{3\pi c^2}{2{\omega }_0^3}\frac{P}{\Delta }I(\bar{r})$
• $R_{scatt}=\frac{3\pi c^2}{2\hslash {\omega }_0^3}{\left(\frac{P}{\Delta }\right)}^{2}I(\bar{r})$

Optical Lattice

$$\begin{array}{rlc}V(x)& =-V_i{\cos }^2(2k_{l}x)& \\ & =-\frac{V_i}{4}(e^{-2i{k}_{l}x}+e^{2i{k}_{l}x}+2)& k_l:=\frac{2\pi }{t_c}\end{array}$$

$\dots$

5th September

Quantum control of atoms in Optical Lattices

By Carrie Weidner Keywords: Bloch band, Bloch state, wannier state, Sagnac effect, Crab algorithm, split state QuTiP Crab link: https://qutip.org/docs/4.0.2/guide/guide-control.html#the-crab-algorithm

In Second Quantization, The Bose-Einstein condensate Hamiltonian is given by $H=\int dx{\psi }^{†}(x)\left[-\frac{{\hslash }^2}{2m}{\Delta }^2+V(x)\right]\psi (x)+g\int dx{\psi }^{†}(x){\psi }^{†}(x)\psi (x)\psi (x)$, where $x=\overrightarrow{x}\in {\mathbb{R}}^3$. By setting $\psi (x)={\sum }_j{a}_j^{†}{w}_0(x-x_j)$ we retrieve the Bose-Hubbard Hamiltonian $H_{BH}=-J{\sum }_{⟨j,j’⟩}{a}_j^{†}{a}_j+{\sum }_j\frac{U}{2}{n}_j(n_j-1)$, where

• $J=\int dx{w}_0(x-x_j)[-\frac{{\hslash }^2}{2m}{\Delta }^2]+V(x)]w_0(x-x_j)$.
• $U=g\int dx|w_0(x-x_j){|}^4$ When $U\ll J$ we get a superfluid state $|{\psi }_{SF}⟩\propto {\prod }_{i=j}^M{\varphi }_q(x)$ When $U\gg J$ we get $|{\psi }_{MI}⟩\propto {\prod }_{j=1}^M{A}_j^{†}|0⟩$

A dictionary of quantum control

• Coherent control ( aka physics first approach ): uses quantum interference to find the path to the desired state
• Optimal control ( aka control first approach ): optimize a quantum process to extract physical insights/mechanisms
• State transfer: $|{\psi }_A⟩\to |{\psi }_B⟩$
• Open-loop control: optimisation of a model
• Closed-loop control: optimising an experimental system
• Landscape: the set of optimisation parameters
• Local traps: local minima of our landscape that do not map to global minima
• functional: the number to be minimised
• Quantum speed limit: fastest possible transfer time

Remark: Instead of regularising the derivative of our control function $u(t)$, we can optimize the control without these terms, and then take the Fourier Transform of our control $u(t)$, remove high mode terms, and take the Fourier Transform inverse, using this result as our solution.

Artificial Gauge Potentials and Maxwell Equations

By Dana Anderson Keywords: Glauber coherence, Coulomb interaction, Van-der Wall interaction

Relevant info can be found in https://arxiv.org/abs/2204.04549, and see https://rdc.reed.edu/v1/resources/7d0f8177-e9c9-42ce-b467-0f4b80700dd8 for Classical Gauge Field Theory.

The goal is to derive Maxwell equation duals ( i.e. the Maxwell Wave Equations) based on Gauge Theory. The derivation was not shown during the lecture.

6th September

Large scale atom interferometry

By Jason Hogan Keywords: Galilean transformation operator, Flat spacetime, Minkowski tensor, Lorentzian transformation, tidal force, Compton frequency

Quantum Control of Atoms in Optical Lattices

By Dana Anderson Keywords: Impedance method

Robust Quantum Control

By Carrie Weidner Keywords： Grape algorithm, Crab algorithm, Nelder-Mead optimization

GRAPE Algorithm

QuTiP docs: https://qutip.org/docs/4.0.2/guide/guide-control.html#the-grape-algorithm

We have an initial state $|{\psi }_0⟩$, which we want to evolve into a desired state $|{\psi }_d⟩$. We have a seed $u_o$ which is our initial guess $\mathcal{F}\{u_0\}$. We apply the method of steepest descent to iterate to a new point $u_1$ by applying the transformation $u_1=u_0+{\alpha }_0\Delta \mathcal{F}\{u_0\}$. We split time into intervals through $t_0,t_1,\dots ,t_n$, with $t_n=T$, so that $u(t)=(u(t_1),u(t_2),\dots ,u(t_n))$

Our time evolution operator ${\hat{U}}_j:=\hat{U}(t_j)=\exp \left(i(\hat{T}+\hat{V}(U_i))dt\right)$. Hence we get

$$\begin{array}{rl}\mathcal{F}(u)& =|⟨{\psi }_d|\psi (T)⟩{|}^2\\ & =|⟨{\psi }_d\prod {\hat{U}}_j|{\psi }_0⟩{|}^2\end{array}$$

Prelims:

• $c:=⟨{\psi }_d|\psi (T)⟩$
• $\mathcal{F}=|c{|}^2=c^{\ast }c$ is the fidelity of our system.
• $k\in \mathbb{C}$
• $⟨\chi (T)|:=-i⟨\psi (T)|{\psi }_d⟩⟨{\psi }_d|$
• Forward evolution: $|\psi (t_j)⟩={\prod }_{j’=1}^j{U}_{j’}|{\psi }_0⟩$.
• Backward evolution: $|\psi (t_j)⟩={\prod }_{j’=1}^j{U}_j^{†}|\psi (T)⟩$

This allows us to derive

$$\begin{array}{rl}\frac{d}{d{u}_j}\mathcal{F}u& =\frac{d}{d{u}_j}⟨{\psi }_d|{\prod }_{j’=1}^j{U}_{j’}|{\psi }_0⟩\\ & =\frac{\partial }{\partial u_j}(c^{\ast }c)\\ & =2\mathrm{Re}\left[c^{\ast }\frac{\partial }{\partial u_j}c\right]\end{array}$$

We try to calculate $\frac{dc}{d{u}_j}$ below:

$$\begin{array}{rl}\frac{d}{d{u}_j}⟨{\psi }_d|U_{j’=1}^j{U}_j|{\psi }_0⟩& =⟨{\psi }_d|U_t{U}_{t-1}\dots \frac{d{U}_j}{d{u}_j}\dots U_2{U}_1|{\psi }_0⟩\\ & =\dots \end{array}$$

• Look for implementations of Nelder-Mead optimization , and reach out to Carrie Weidner about it. It might be useful to write Crab algorithm, … before reaching out. I should also ask for the paper recommendation on how open quantum systems lead to dampening of the system, i.e. visualizing the eigenvalues of Schrodinger Equation as having Attractors. See Carrie Weidner’s colleague work under the Python Package QuOCS.

Robust Control

• Adding noise to both controls and experiment parameters

• Monte Carlo methods to estimate the expected fidelity around the best controls with a certain noise level we want to tolerate. This needs to be led by the physical system: some parameters might be impossible to perturb, while others are harder ( like power laser fluctuations )

• Using a low-discrepancy sequence ( i.e. Sobol sequence ) to do better than Monte Carlo

• Look up Pink noise

Large Scale Atom Interferometry

By Jason Hogan

Keywords: Ramsey sequence