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## Quantum Circuits

The main motivation to study such systems comes from their potential application to quantum computation. They are typically composed of one or several superconducting qubits coupled to discrete quantized modes of the electromagnetic field confined into a cavity. A main challenge is to design them in such a way that they can store quantum information for long enough times, in spite of the various couplings to their environment, which induce decoherence. Almost two decades ago, B. Douçot and L. Ioffe have shown that a rather efficient way to achieve such protection against decoherence is to consider circuits composed of exotic circuit elements that require superconducting Cooper pairs to form pairs in order to be able to tunnel from one element to another. This idea has been revisited recently in a collaboration with the experimental group led by Z. Leghtas at ENS Paris. They measured a tenfold suppression of flux sensitivity of the first transition energy only, implying a twofold increase in the vacuum phase fluctuations and showing that the ground state is delocalized over several Josephson wells. These results provide additional experimental confirmation that exotic elements based on pairs of Cooper pairs constitute a promising route to build protected qubits.

We are also studying topological energy pumps, realized in a system of two cavity modes with mutually incommensurate frequencies coupled to a qubit. The coupling is said to be topological when the qubit instantaneous eigenstates cover the whole Bloch sphere as a function of the two cavity oscillator angle variables. It is known that in this situation, and when the oscillator frequencies are smaller than the coupling strength, a quantized flux of photons from one mode to the other is generated. We have shown that any initial separable state of this system generically evolves into an adiabatic cat state. Such a state is a superposition of two adiabatic states in which the qubit is entangled between the modes. The topological coupling between the qubit and the modes gives rise to the separation in energy between these two components, which evolve into states with distinguishable energy content.

We are currently exploring the generation of quantum chaos in this system at long time scales, such that the Born-Oppenheimer adiabatic approximation breaks down. This may provide an interesting example of a system with a mixed energy spectrum, i.e. where two classes of eigenstates obeying different statistical distributions coexist at the same energies.