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Hopf-link semimetals exhibit exotic gapless band structures with fascinating topological properties, which have never been observed in nature. Here we demonstrate nodal lines with topological form of Hopf-link chains in artificial semimetal-bands. Driving superconducting quantum circuits with elaborately designed microwave fields, we mapped the momentum space of a lattice to a parameter space of the Hamiltonian for a Hopf-link semimetal. By measuring the energy spectrum, we directly imaged nodal lines in cubic lattices. By tuning the driving fields, we adjusted various parameters of Hamiltonian. Important topological features, such as link-unlink topological transitions and the robustness of the Hopf-link chain structure were investigated. Moreover, we extracted the linking number by detecting the Berry phase associated with different loops encircling nodal lines. This topological invariant clearly reveals the nontrivial topology of the Hopf-link semimetal. Our results provide knowledge for developing new materials and quantum devices., Comment: 6 pages, 4 figures. arXiv admin note: text overlap with arXiv:1702.01512

We propose and experimentally demonstrate a scheme to simulate the interaction between a two-level system and a classical light field. Under the transversal driving of two microwave tones, the effective Hamiltonian in an appropriate rotating frame is identical to that of the general semi-classical Rabi model. We experimentally realize this Hamiltonian with a superconducting transmon qubit. By tuning the strength, phase, and frequency of the two microwave driving fields, we simulate the quantum dynamics from the weak to extremely strong driving regime. Under these conditions, we observe that, as a function of increased Rabi drive strength, the qubit evolution gradually deviates from the normal sinusoidal Rabi oscillation, in accordance with the predictions of the general semi-classical Rabi model far beyond the weak driving limit. Our scheme provides an effective approach to investigate the extremely strong interaction between a two-level system and a classical light field. Such strong interactions are usually inaccessible in experiments.

We propose a scheme to perform two-qubit state tomography by independently measuring the ensemble average of quantum states of two coupled superconducting transmon qubits. Each qubit is capacitively coupled to its own readout cavity and can be measured separately. In order to obtain the density matrix of a two-qubit state, we apply four two-qubit unitary operations to the initial state and measure the corresponding qubit states, from which the elements of the two-qubit density matrix can be extracted. By using this scheme, we measure the entangled two qubits with high fidelity.

We realize the dark state in a three-dimensional transmon superconducting qutrit that consists of three cascading energy levels: |0⟩, |1⟩, and |2⟩. When the system is simultaneously driven with two tone microwaves on resonance with |0⟩, |1⟩ and |1⟩, |2⟩, respectively, it is found that the state of the system will generally evolve in time domain. However, the qutrit state is frozen-out if it is initialized in a dark state for corresponding drive amplitudes. We observe this freeze-out phenomenon by changing the amplitude of the microwaves while fixing the initial state or vice versa.