Author Archives: wdengquantum.me

Consider an atomic two-level system with ground state \(|0\rangle\) and excited state \(|1\rangle\). Recall that in the interaction picture, after making the rotating wave approximation and boosting into a steady-state rotating frame, one had the resultant time-independent steady-state Hamiltonian: \[H_{\infty}=\frac{\hbar}{2}\tilde{\boldsymbol{\Omega}}\cdot\boldsymbol{\sigma}\] … Continue reading →

The purpose of this post is to explain the \(2\) key models of classical optics, namely geometrical optics (also known as ray optics) and physical optics (also known as wave optics). Although historically geometrical optics came before physical optics, and … Continue reading →

Problem #\(1\): When someone comes up to you on the street and just says “Ising model”, what should be the first thing you think of? Solution #\(1\): The classical Hamiltonian: \[H=-E_{\text{int}}\sum_{\langle i,j\rangle}\sigma_i\sigma_j-E_{\text{ext}}\sum_i\sigma_i\] (keeping in mind though that there many variants … Continue reading →

The purpose of this post is to study the universal properties of fully developed turbulence \(\text{Re}\gg\text{Re}^*\sim 10^3\). Thanks to direct numerical simulation (DNS), there is strong evidence to suggest that the nonlinear advective term \(\left(\textbf v\cdot\frac{\partial}{\partial\textbf x}\right)\textbf v\) in the … Continue reading →

The purpose of this post is to document the uses of several standard components used in optics experiments. Optical Fibers & APC Connectors An optical fiber is a waveguide for light waves. The idea is to use it to transmit … Continue reading →

In cold atom experiments, one very basic question one can ask is, given some atom cloud, what is the number of atoms \(N\) in the cloud? One way is to basically shine some light on the atom cloud and see … Continue reading →

The purpose of this post is to describe the relevant theory needed to understand the paper “High signal to noise absorption imaging of alkali atoms at moderatemagnetic fields” by Hans et al. In particular, a key paper which they cite … Continue reading →

Consider an non-interacting gas of identical fermions (e.g. electrons \(e^-\)); this is called an ideal Fermi gas. Because the Pauli exclusion principle prohibits identical fermions from occupying the same quantum state, the grand canonical partition function \(\mathcal Z\) for an … Continue reading →

The purpose of this post is to prove several general identities concerning the quantum statistical mechanics of an isolated, ideal Bose gas at equilibrium. Problem #\(1\): Specify the physics (i.e. write down the Hamiltonian \(H\) for an isolated, ideal Bose … Continue reading →

Problem: Consider an isolated atom with time-independent Hamiltonian \(H_0\). Such an atom will have many bound \(H_0\)-eigenstates, but for simplicity focus on just two such bound states (think of it as a qubit) \(|0\rangle\) and \(|1\rangle\) (called the ground state … Continue reading →