Each of the three archetypes possess a distinct range of Josephson energy to charging energy ratio. Josephson energy refers to the energy stored in Josephson junctions when current passes through, and charging energy is the energy required for one Cooper pair to charge the junction's total capacitance. Josephson energy can be written , where is the critical current parameter of the Josephson junction, is (superconducting) flux quantum, and is the phase difference across the junction. Notice that the term indicates nonlinearity of the Josephson junction. Charge energy is written , where is the junction's capacitance and is electron charge. Of the three archetypes, phase qubits allow the most of Cooper pairs to tunnel through the junction, followed by flux qubits, and charge qubits allow the fewest. Phase qubit The phase qubit possesses a Josephson to charge energy ratio on the order of magnitude . For phase qubits, energy levels correspond to different quantum charge oscillation amplitudes across a Josephson junction, where charge and phase are analogous to momentum and position respectively as analogous to a quantum harmonic oscillator. Note that in this context phase is the complex argument of the superconducting wave function (also known as the superconducting order parameter), not the phase between the different states of the qubit. Flux qubit The flux qubit (also known as a persistent-current qubit) possesses a Josephson to charging energy ratio on the order of magnitude . For flux qubits, the energy levels correspond to different integer numbers of magnetic flux quanta trapped in a superconducting ring. Fluxonium Fluxonium qubits are a specific type of flux qubit operable at relatively low frequencies (approximately 600 MHz). This results in longer qubit lifetime. The fluxonium circuit's Hamiltonian can be written as follows: , consistent with the notation in. Note that . Charge qubit The charge qubit, also known as the Cooper pair box, possesses a Josephson to charging energy ratio on the order of magnitude . For charge qubits, different energy levels correspond to an integer number of Cooper pairs on a superconducting island (a small superconducting area with a controllable number of charge carriers). Indeed, the first experimentally realized qubit was the Cooper pair box, achieved in 1999. Transmon Transmons are a special type of qubit with a shunted capacitor specifically designed to mitigate noise. The transmon qubit model was based on the Cooper pair box (illustrated in the table above in row one