Quantum bath engineering of a high impedance microwave mode through quasiparticle tunneling

We demonstrate a new approach to dissipation engineering in microwave quantum optics. For a single mode, dissipation usually corresponds to quantum jumps, where photons are lost one by one. Here, we are able to tune the minimal number of lost photons per jump to be two (or more) with a simple dc voltage. As a consequence, different quantum states experience different dissipation. Causality implies that the states must also experience different energy shifts. Our measurements of these Lamb shifts are in good agreement with the predictions of the Kramers-Kronig relations for single quantum states in a regime of highly non-linear bath coupling. This work opens new possibilities for quantum state manipulation in circuit QED, without relying on the Josephson effect.