User:DJIndica/Sandbox

From Wikipedia, the free encyclopedia

< User:DJIndica

My talk

Electromagnetism

Electricity · Magnetism

Tensors in Relativity

This box: view • talk • edit

The table's caption
Column heading 1	Column heading 2	Column heading 3
Row heading 1	Cell 2	Cell 3
Row heading A	Cell B	Cell C

title1

num1

num2

RABI FREQUENCY

The Rabi frequency is an important quantity in semi-classical atomic physics. For a given atomic transition in a given light field, the Rabi frequency represents the coupling strength between the (classical) light and the (quantum) transition. Rabi flopping between the levels of a 2-level system illuminated with resonant light, will occur at the Rabi frequency.

Rabi flopping in a 2-level system Actual size

1 Definition
2 Generalized Rabi frequency
3 Derivation
4 Template test
5 Testing
6 Major edits
7 Other contributions
8 My images
9 Family tree
10 Electromagnetism2

[edit] Definition

$\chi_{ij} = {\vec{d}_{ij}.\vec{E}_0 \over \hbar}$

where

$\vec{d}_{i,j}$ is the transition dipole moment for the $i \rightarrow j$ transition.

$\vec{E}_0 = \hat{\epsilon}E_0$ is the vector electric field amplitude which includes the polarization.

The numerator has dimensions of energy, dividing by $\hbar$ gives an angular frequency. As $\vec{d}_{ij} = \vec{d}_{ji}^*$ , we have $\chi_{ij} = \chi_{ji}^*$ .

By analogy with a classical dipole, it is clear that an atom with a large dipole moment will be more susceptible to perturbation by electric and magnetic fields. The dot product includes a factor of $cosθ$ , where $θ$ is the angle between the polarization of the light and the transition dipole moment. When they are parallel or antiparallel the interaction is strongest, when they are perpendicular there is no interaction at all. The vector electric field amplitude defines both the intensity and the polarization of the light.

[edit] Generalized Rabi frequency

For light that is off resonance with a transition, it is common to define the generalized Rabi frequency $Ω i, j$ . Rabi flopping actually occurs at the generalized Rabi frequency.

$\Omega_{i,j} = \sqrt{|\chi_{i,j}|^2 + \Delta^2}$

where $Δ = ω l i g h t - ω t r a n s i t i o n$ is the detuning, a measure of how far the light is off resonance with the transition.

[edit] Derivation

We start with the Hamiltonian for a 2-level system with frequency splitting $ω 0$ , illuminated by a light field with frequency $ω$

$\hat{H} = \begin{pmatrix} {\hbar \omega_0 \over 2} & \chi_{21}e^{i\omega t/2}\\ \chi_{12}e^{-i\omega t/2} & -{\hbar \omega_0 \over 2} \end{pmatrix}$

with a wavefunction for the system (in the interaction picture) given by

$|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle$

We can write the time-dependent Schrödinger equation in the form

$\dot{c}(t) = {1 \over i\hbar}\hat{H}c(t)$

And solve using the rotating wave approximation, giving a probability of being in states $|1\rangle$ and $|2\rangle$

$P_1 = \left({\chi_{21} \over \Omega_{21}}\right)^2 \sin^2\left({\Omega_{21} \over 2}t\right)$

$P_2 = 1 - \left({\chi_{21} \over \Omega_{21}}\right)^2 \sin^2\left({\Omega_{21} \over 2}t\right)$

[edit] Template test

Electrostatics

Electromagnetism
Electricity · Magnetism

Magnetostatics

Electrodynamics

Electrical Network

Tensors in Relativity

This box: view • talk • edit

Electrostatics

Electromagnetism
Electricity · Magnetism
Electric charge
Coulomb's law
Electric field
Gauss's law
Electric potential
Electric dipole moment
Magnetostatics
Ampère's circuital law
Magnetic field
Magnetic flux
Biot-Savart law
Magnetic dipole moment
Electrodynamics
Electrical current
Lorentz force law
Electromotive force
(EM) Electromagnetic induction
Faraday-Lenz law
Displacement current
Maxwell's equations
(EMF) Electromagnetic field
(EM) Electromagnetic radiation
Electrical Network
Electrical conduction
Electrical resistance
Capacitance
Inductance
Impedance
Resonant cavities
Waveguides
Tensors in Relativity
Electromagnetic tensor
Electromagnetic stress-energy tensor
This box: view • talk • edit

[edit] Testing

hello, what's your name?

[edit] Family tree

Bernhard of Lippe

Armgard of Sierstorpff-Cramm

Hendrik of Mecklenburg-Schwerin

Wilhelmina of the Netherlands

Bernhard of Lippe-Biesterfeld

Juliana of the Netherlands

Beatrix of the Netherlands

“

large quotation marks

”

Help:Formula

Wikipedia:Conditional_tables

ParserFunctions

Help:Parser functions

[edit] Electromagnetism2

Electrostatics

Electromagnetism
Electricity · Magnetism

Magnetostatics

Electrodynamics

Electrical Network

Tensors in Relativity

This box: view • talk • edit

Electrostatics

Electromagnetism
Electricity · Magnetism
Electric charge Coulomb's law Electric field Gauss's law Electric potential Electric dipole moment
Magnetostatics

Electrodynamics

Electrical Network

Tensors in Relativity

This box: view • talk • edit