Quantum Physics Basics ========================= Time Independent Two Level Systems ------------------------------------ In general a quantum two level system can be described using Schrodinger equation .. math:: 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}, where :math:`\omega_0` is real. We define the unperturbed Hamiltonian, .. math:: H_0 = \frac{1}{2}\begin{pmatrix} -\omega_0 & 0 \\ 0 & \omega_0 \end{pmatrix}. Introduce perturbation .. math:: W = \frac{1}{2}\begin{pmatrix} -\delta & w \\ w* & \delta \end{pmatrix}, where and identity matrix could have been added to restore the actual potential field. The system Hamiltonian becomes .. math:: 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}, where :math:`\omega = \omega_0 +\delta` is called **detuning** and :math:`w` is the **Rabi frequency**. The Schrodinger equation can be solved which gives us the general solution .. math:: \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} = C_+ e^{\lambda_+ t}\eta_+ + C_- e^{\lambda_- t} \eta_-, where .. math:: \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}. .. admonition:: Rabi Frequency :class: warning Rabi frequency :math:`w` will be generalized to the generalized Rabi frequency .. math:: R=\sqrt{w^2+\omega^2}, at which frequency the final system is oscillating. We can determine the coefficients :math:`C_\pm` using initial condition. Suppose we have the system initially in state .. math:: \begin{pmatrix} 1\\ 0 \end{pmatrix}. The coefficients are .. math:: C_\pm = \pm \frac{1}{2} \frac{w^*}{\sqrt{w^2+\omega^2}}. The final solution is .. math:: \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}. The transition probability is .. math:: 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). Energy Spectrum ~~~~~~~~~~~~~~~~~~ It's important to visualize the energy spectrum. The energy levels are in fact .. math:: E_\pm = \pm \lambda_R. We notice that for zero off-diagonal perturbation :math:`w=0`, we have .. math:: E_\pm = \pm \omega, which means the energy levels are linear to :math:`\omega`. At :math:`\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 :math:`w\neq 0`, the degeneracy is gone. The crossing of the energies is avoided. References of Rabi System ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1. A very short and concise introduction: `Rabi Model `_ by Michael G. Moore at MSU. 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 .. math:: H = \frac{1}{2}\begin{pmatrix} -\omega_0 & w e^{i k t} \\ w e^{-ikt} & \omega_0 \end{pmatrix} . :label: eqn-rabi-time-dependent-two-level-hamiltonian 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 :math:`-k` instead of :math:`\delta`. In other words, in the time dependent Rabi system :math:`-k` is equivalent to :math:`\delta` in the time independent system. Thus the detuning becomes .. math:: \omega = \omega_0 - k. References of Time Dependent Rabi System ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1. `Rabi Oscillations `_ by Thilo Bauch and Goran Johansson. Bloch Sphere and Bloch Vectors -------------------------------------------------- A two level system density matrix can be expanded using identity matrix and Pauli matrix, .. math:: \rho = \frac{1}{2}(1 + \mathbf P \cdot \boldsymbol \sigma), as defined in :ref:`neutrino-flavor-isospin`. The vector :math:`\mathbf P` represents the vector of state in Bloch sphere. We can also project the Hamiltonian into this space, .. math:: H = - \frac{\boldsymbol \sigma}{2} \cdot \mathbf H. We take the :eq:`eqn-rabi-time-dependent-two-level-hamiltonian` as an example. A simple derivation shows that the equation of motion for the Bloch vector or the flavor isospin vector is precession .. math:: \dot {\mathbf P} = \mathbf P \times \mathbf H, where .. math:: \mathbf H = \begin{pmatrix} -w \cos(kt)\\ w\sin(kt)\\ \omega_0 \end{pmatrix}. We identify this vector as a rotation around z axis. It also becomes obvious that the so called time indepedent case is literally :eq:`eqn-rabi-time-dependent-two-level-hamiltonian` in a corotating frame.