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 Rubidium atoms on a MOT Trap.

Notation note: represents the decay , and represents the half line width in angular units ( rad/s )

We use it on the scattering equation , where:

  • Is the line width
  • is the normalized detuning
  • is the saturation parameter This way we can rewrite .

Keywords: Doppler Effect, Detuning, Molasses Force.

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

Bose-Einstein condensate

Introduction to Atom Interferometry ( atom optics )

By Enno Giese and Friedrich Alexander

Dipole interaction

We have

  • being the interaction energy
  • Is the radius
  • is the magnetic charge

We specify a two level atom with a ground state and an excited state . We describe the two level atom = .

The electric field is described by:

  • , a chirped plane wave
  • is the amplitude
  • is the phase. I need to check how the expression for this is derived. Help me ❓

We get a position dependence: . Hence we can thing of this as a momentum displacement. We also get .

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 E-field evaluates at , where . Dipole operator := . The Hamiltonian becomes . 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:

In the above, we have:

  • happens with perfect chirping
  • is the gained energy during absorption.

rabi oscillations

We assume a narrow momentum distribution , allowing us to model the system as rabi oscillations. The resulting time independent Hamiltonian is given by . We get the solution as a time evolution operator . It is relevant that the initial phase of light enters the equation.

.

  • should actually be 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 , allowing us to describe the system as

We use dimensionless units in order to remove constants, obtaining

Where

  • is needed for our initial conditions. We start with initial conditions as and . A normalization condition requires us to have . We get express the solution as follows, in order to simplify the resulting solution:
  • .

The above allows us to write

  • Using the above 2 pairs of equations to find differential equations for and we get

( magic )

We end up getting

Allowing us to write

Markov Approximation

By making a Markov Approximation we can assume . With this we get , the standing for the Markov Approximation Differential Equation. The standard solution is , where .

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

Since we designed our system with independent internal and external Hamiltonians, we can use the Baker-Campbell-Hausdorff formula to get . The internal dynamics can be described as . Since this is a diagonal Hamiltonian, we get . The external dynamics are described by a Total Mach-Zehnder sequence of a Beam Splliter, Mirror, and Beam Splitter. Algebraically this becomes . It is unclear why the second time evolution has a time correction. Help me ❓ The associated matrix elements are:

  • , where is a Global Phase.

  • is the description of the Mach-Zehnder interferometer along one of the respective arm . These are Unitary Operators, i.e. .

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

Detection

We use the

  • Projectors .
  • integrated particle number .
  • input state .
  • Ground state population

With the form above and we get . Since the are Unitary Operators, . Writing allows us to get , which is an oscillation around the value . This is our interference signal.

describes our evolution along branches. Focusing on individual operators, we have:

  • ( 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 represent a state in red, and a state in blue, we are able to calculate the transition probability as .

Eletronic State Image

( 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 between two states and . Schrodinger Equation gives us ., where is our Hamiltonian in one dimension. Our is such that and .

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

We will work with:

  • the atomic length scale in m
  • Optical : m

With the above we get . In other words dipoles do not have self interactions.

We are able to write

the off diagonal terms show us the state interaction terms which we need to discover. with . The Hamiltonian then becomes . This is a pain to solve due to the time dependency of . We can get rid of it by applying a rotating unitary transformation in order to remove the time dependency. The relevant Unitary Operator is , creating the Rotating wave approximation hamiltonian .

Experimentally we can take , which gives us as a Taylor expansion of .

Optical Dipole Traps for Neutral Atoms

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

where . We get

  • .

We make the assumption , i.e. our Detuning is much smaller than our oscillations, to get

Optical Lattice

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 , where . By setting we retrieve the Bose-Hubbard Hamiltonian , where

  • .
  • When we get a superfluid state When we get

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:
  • 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 , we can optimize the control without these terms, and then take the Fourier Transform of our control , 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 , which we want to evolve into a desired state . We have a seed which is our initial guess . We apply the method of steepest descent to iterate to a new point by applying the transformation . We split time into intervals through , with , so that

Our time evolution operator . Hence we get

Prelims:

  • is the fidelity of our system.
  • Forward evolution: .
  • Backward evolution:

This allows us to derive

We try to calculate below:

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