Rabi oscillations, Ramsey fringes and spin echoes in an electrical circuit

The state variables of an electrical circuit, like voltages and currents can be made to behave quantum mechanically by minimizing the coupling to external degrees of freedom through a proper design. Circuits based on superconducting tunnel junctions have displayed signatures of macroscopic quantum behavior [1–7], but the level of coherence of the quantum states remained until now much smaller than for isolated atoms or ions. In the “quantronium” circuit presented here, a coherence quality factor of more than 104 has been obtained, thus allowing the coherent manipulation of the state of the system like in atomic physics and NMR experiments [8]. The quantronium consists of a superconducting loop interrupted by two adjacent small Josephson tunnel junctions with capacitance Cj/2 and Josephson energy EJ/2 each, which define a low capacitance superconducting electrode called the “island”, and by a large Josephson junction with large Josephson energy (EJ0 ≈ 20EJ) (see Fig. 1). The island is biased by a voltage source U through a gate capacitance Cg . In addition to EJ , the quantronium has a second energy scale which is the Cooper pair Coulomb energy ECP = (2e)2/2 ( Cg + Cj ) . The temperature T and the superconducting gap ∆ satisfy kBT ∆/ lnN and ECP < ∆ − kBT lnN , where N is the total number of paired electrons in the island. The number of excess electrons is then even [9, 10], and the system has discrete quantum states which are in general quantum superpositions of several charge states with different number N̂ of excess Cooper pairs in the island. Neglecting the loop inductance and the charging energy of the large junction, the Hamiltonian of the circuit is