Superconducting Qubits
Learn basic physics of superconducting qubits here.
Transmon
Before delving into what exactly a transmon is, let’s consider a circuit consisting of a Josephson junction shunted by a capacitor. The quantum Hamiltonian of this circuit can be written as the sum of the charging energy of the capacitor and the potential energy of the Josephson junction.
\[\mathcal{H} = \frac{Q^2}{2C} - E_J\cos(\varphi)\]To understand this better we can expand the cosine about it’s minimum at \(\varphi = 0\).
\[\mathcal{H} = \frac{Q^2}{2C} - E_J\bigg(1 - \frac{\varphi^2}{2!} + \frac{\varphi^4}{4!} + \mathcal{O}(\varphi^6)\bigg)\]Remember that the superconducting phase is related to the magnetic flux across the Josephson junction by this formula
\[\varphi = 2\pi\frac{\Phi}{\Phi_0}\]where \(\Phi_0 = h/2e\) is the magnetic flux quantum. Let’s replace this in the second order term (proportional to \(\varphi^2\)) and rewrite it in a way that is more familiar
\[E_J\frac{\varphi^2}{2!} = E_J\frac{(2\pi\Phi)^2}{2!\Phi^2_0} = \frac{\Phi^2}{2L_J}\]We now have a term that looks like the potential energy of an inductor with an inductance \(L_J\) that is
\[L_J = \frac{\Phi_0^2}{(2\pi)^2 E_J}\]The Hamiltonian is then modified into
\[\mathcal{H} = \frac{Q^2}{2C} + \frac{\Phi^2}{2L_J} + \frac{\varphi^4}{4!} - E_J + \mathcal{O}(\varphi^6).\]Now like any good physicist we make an approximation and discard terms that are equal or higher in order than \(\varphi^6\). Also, we ignore the constant shift \(-E_J\) because constant shifts in energy levels are trivial and we only care about the relative difference between the energy levels. The first two terms of the Hamiltonian are the familiar LC oscillator. These terms describe the electrical and magnetic energies stored in the capacitor and the inductor. As we saw in the case of an LC oscillator, these can be written in second-quantization formalism.
\[H = \hbar\omega_0 (a^\dagger a + \frac{1}{2}) + \frac{(2\pi)^4}{\Phi^4_0}\Phi^4_{ZPF}(a^\dagger + a)^4\]