This project aims to implement analog quantum simulation of the Bose-Hubbard-Hamiltonian in the driven dissipative regime in the realm of circuit QED. The Bose-Hubbard-Hamiltonian describes interacting bosons in a lattice. For the simulation we use a system of series-connected, capacitively coupled, nonlinear and tunable superconducting resonators. The resonators can each be driven individually by an external microwave source through a capacitively coupled transmission line. To achieve interaction between the particles, we introduce a nonlinearity in the form of galvanically coupled SQUIDs, placed in the current maximum of each resonator. The nonlinearity can be tuned by external coils and on-chip antennas. Thanks to this tunability, the system can be brought into different regimes, e.g. a regime where the coupling between the resonators surpasses the nonlinearity of the resonators or vice versa.
As a first step, we look at a two resonator system which can be scaled up in further experiments. Calculations of the second-order correlation functions of the field inside each resonator and between the fields in the resonators predict two interesting regimes for these correlation functions, which can be reached by tuning the nonlinearity and changing the drive power of the microwave field applied to the entrance of one resonator. In the first regime, the second-order correlation function of the field in one resonator is smaller than one, which means the field is antibunched. At the same time, the cross-correlation between the two resonators shows bunching. By adjusting the parameters named above, we can switch this behavior to the opposite case, where the field inside one resonator is bunched, but there is antibunching between the two resonator fields.
In order to experimantally reproduce this behavior, we use a silicon chip with an aluminum pattern forming this two resonator system with a SQUID in each of them. We are currently setting up the experiment, which is placed inside a dilution refrigerator, as the experiments are performed at around 20 – 30 mK to suppress thermal excitations.
The next step is to perform characterization measurements of the two resonator sample, most importantly of the tunability of the system. After this, a setup to measure the second-order correlation function has to be implemented, which also contains the implementation of a FPGA measurement card, which is already worked on. A challenge here is to be able to resolve the signal which is transmitted through the resonator in the background noise, as the calculations predict only a small signal.
Furthermore we plan to scale the two resonators to a larger chain of resonators and to two dimensional lattices in order to be able to simulate behaviour that cannot be calculated analytically. Larger lattice systems should also show superfluid and mott-insulating phases, as is predicted by the Bose-Hubbard-Hamiltonian.
M. Hartmann, Heriot-Watt University, Edinburgh, UK
Walther-Meissner Strasse 8
85748 Garching, Germany