# 1.3. Quantum Physics Basics¶

## 1.3.1. Time Independent Two Level Systems¶

In general a quantum two level system can be described using Schrodinger equation

$\begin{split}i\partial_t \begin{pmatrix} \psi_1\\ \psi_2 \end{pmatrix}= \frac{1}{2} \begin{pmatrix} -\omega_0 & 0 \\ 0 & \omega_0 \end{pmatrix} \begin{pmatrix} \psi_1\\ \psi_2 \end{pmatrix},\end{split}$

where $$\omega_0$$ is real. We define the unperturbed Hamiltonian,

$\begin{split}H_0 = \frac{1}{2}\begin{pmatrix} -\omega_0 & 0 \\ 0 & \omega_0 \end{pmatrix}.\end{split}$

Introduce perturbation

$\begin{split}W = \frac{1}{2}\begin{pmatrix} -\delta & w \\ w* & \delta \end{pmatrix},\end{split}$

where and identity matrix could have been added to restore the actual potential field. The system Hamiltonian becomes

$\begin{split}H =& \frac{1}{2}\begin{pmatrix} -\omega_0 - \delta & w \\ w^* & \omega_0 +\delta \end{pmatrix} \\ =& \frac{1}{2}\begin{pmatrix} -\omega & w \\ w^* & \omega \end{pmatrix},\end{split}$

where $$\omega = \omega_0 +\delta$$ is called detuning and $$w$$ is the Rabi frequency. The Schrodinger equation can be solved which gives us the general solution

$\begin{split}\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} = C_+ e^{\lambda_+ t}\eta_+ + C_- e^{\lambda_- t} \eta_-,\end{split}$

where

$\begin{split}\lambda_\pm =& \pm \frac{i}{2}\sqrt{w^2+\omega^2} \equiv \pm i \lambda_R\\ \eta_\pm =& \begin{pmatrix} \frac{-\omega \pm \sqrt{w^2+\omega^2}}{w^*} \\ 1 \end{pmatrix}.\end{split}$

Rabi Frequency

Rabi frequency $$w$$ will be generalized to the generalized Rabi frequency

$R=\sqrt{w^2+\omega^2},$

at which frequency the final system is oscillating.

We can determine the coefficients $$C_\pm$$ using initial condition. Suppose we have the system initially in state

$\begin{split}\begin{pmatrix} 1\\ 0 \end{pmatrix}.\end{split}$

The coefficients are

$C_\pm = \pm \frac{1}{2} \frac{w^*}{\sqrt{w^2+\omega^2}}.$

The final solution is

$\begin{split}\begin{pmatrix} \psi_1(t) \\ \psi_2(t) \end{pmatrix} = \frac{w^*}{\sqrt{w^2+\omega^2}} \begin{pmatrix} -i\frac{\omega}{w^*} \sin(\lambda_R t) + \frac{\sqrt{\omega^2+w^2}}{w^*} \cos(\lambda_R t)\\ i \sin(\lambda_R t) \end{pmatrix}.\end{split}$

The transition probability is

$\begin{split}P_{1\to 2} =& \frac{w^2}{w^2+\omega^2} \sin^2(\lambda_R t)\\ =& \frac{w^2}{w^2+\omega^2} \sin^2 \left(\sqrt{w^2+\omega^2}/2 \right).\end{split}$

### 1.3.1.1. Energy Spectrum¶

It’s important to visualize the energy spectrum. The energy levels are in fact

$E_\pm = \pm \lambda_R.$

We notice that for zero off-diagonal perturbation $$w=0$$, we have

$E_\pm = \pm \omega,$

which means the energy levels are linear to $$\omega$$. At $$\omega=0$$ we have degeneracy of the two state, i.e., the two states have the same energy. However, as we introduce non-zero off-diagonal perturbation $$w\neq 0$$, the degeneracy is gone. The crossing of the energies is avoided.

### 1.3.1.2. References of Rabi System¶

1. A very short and concise introduction: Rabi Model by Michael G. Moore at MSU.

## 1.3.2. Time Dependent Two Level System¶

The previous calculations are for time independent perturbations. For time dependent perturbations, the situation would be drastically different.

Here we introduce a system with Hamiltonian

(1)$\begin{split}H = \frac{1}{2}\begin{pmatrix} -\omega_0 & w e^{i k t} \\ w e^{-ikt} & \omega_0 \end{pmatrix} .\end{split}$

To solve the Schrodinger equation, we can go to the corotating frame. The solution to it is very similar to time independent case, with a detuning shifted by $$-k$$ instead of $$\delta$$. In other words, in the time dependent Rabi system $$-k$$ is equivalent to $$\delta$$ in the time independent system. Thus the detuning becomes

$\omega = \omega_0 - k.$

### 1.3.2.1. References of Time Dependent Rabi System¶

1. Rabi Oscillations by Thilo Bauch and Goran Johansson.

## 1.3.3. Bloch Sphere and Bloch Vectors¶

A two level system density matrix can be expanded using identity matrix and Pauli matrix,

$\rho = \frac{1}{2}(1 + \mathbf P \cdot \boldsymbol \sigma),$

as defined in Neutrino Flavour Isospin. The vector $$\mathbf P$$ represents the vector of state in Bloch sphere.

We can also project the Hamiltonian into this space,

$H = - \frac{\boldsymbol \sigma}{2} \cdot \mathbf H.$

We take the (1) as an example. A simple derivation shows that the equation of motion for the Bloch vector or the flavor isospin vector is precession

$\dot {\mathbf P} = \mathbf P \times \mathbf H,$

where

$\begin{split}\mathbf H = \begin{pmatrix} -w \cos(kt)\\ w\sin(kt)\\ \omega_0 \end{pmatrix}.\end{split}$

We identify this vector as a rotation around z axis. It also becomes obvious that the so called time indepedent case is literally (1) in a corotating frame.

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