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
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The presenter wants to start a website that collects all BECs. Reach out on this.
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Bose-Einstein statistics for photons
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Bose-Einstein statistics for particles
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
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
- is our decoupled Hamiltonian.
- is our time evolution operator.
- The above gives us by the Schrodinger Equation that . 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 . 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:
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, where is a Global Phase.
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is the description of the Mach-Zehnder interferometer along one of the respective arm . These are Unitary Operators, i.e. .
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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 .
( 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:
- 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
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Adding noise to both controls and experiment parameters
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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 )
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Using a low-discrepancy sequence ( i.e. Sobol sequence ) to do better than Monte Carlo
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Look up Pink noise
Large Scale Atom Interferometry
By Jason Hogan
Keywords: Ramsey sequence