Absorption spectra of superconducting qubits driven by bichromatic microwave fields

We report experimental observation of two distinct quantum interference patterns in the absorption spectra when a transmon superconducting qubit is subjected to a bichromatic microwave field with the same Rabi frequencies. Within the two-mode Floquet formalism with no dissipation processes, we propose a graph-theoretical representation to model the interaction Hamiltonian for each of these observations. This theoretical framework provides a clear visual representation of various underlying physical processes in a systematic way beyond rotating-wave approximation. The presented approach is valuable to gain insights into the behavior of multichromatic field driven quantum two-level systems, such as two-level atoms and superconducting qubits. Each of the observed interference patterns is represented by appropriate graph products on the proposed color-weighted graphs. The underlying mechanisms and the characteristic features of the observed fine structures are identified by the transitions between the graph vertices, which represent the doubly dressed states of the system. The good agreement between the numerical simulation and experimental data confirms the validity of the theoretical method. Such multiphoton interference may be used in manipulating the quantum states and/or generate nonclassical microwave photons.

Figure 1

(a) An aluminum cavity used to hold the sample. Its fundamental TE101 mode is ${\omega }_{\mathrm{cav}}/2\pi =10.678\mathrm{GHz}$. A transmon qubit on a high-resistance Si substrate is located in the middle of the cavity as shown in the right part of the cavity. (b) Optical (left) and SEM (right) images of a transmon qubit. A Josephson junction, the middle part shown in the SEM image, is capacitively shunted by two Al pads indicated by the light gray parts in the optical image. (c) Circuit diagram of setup. Attenuators, filters, and circulators are used to reduce the external noise. The output signal is then amplified and down-converted with a local oscillator and digitized.

Figure 2

Energy level and microwave drive diagrams represent one-photon transition (left) and two-photon transition (right), respectively.

Figure 3

Illustration of (a) transverse, and (b) longitudinal couplings of a two-level system, represented by the $K_2$ complete graph, with the probe, represented by the $P_{\infty }^P$ path graph, and coupler, represented by the $P_{\infty }^C$ path graph, external fields. The direct product (a), and the Cartesian product (b) are presented as a schematic exhibition of subsequent interactions of the two-level system with the bichromatic external field. For clarity, the second production is shown only onto the small part of the first product graph, inside the green box. The lighter blue rectangle box in the final product in (a) is to make the pattern easier to follow, and is not representing any real edges.

Figure 4

(a) Experimental measurement of the single-photon transitions in a transmon superconducting qubit. (b) Two-mode Floquet result of the transverse coupling of a two-level system with the bichromatic external field. (c) Experimental measurement of the double-photon transitions in a transmon superconducting qubit. (d) Two-mode Floquet result of the longitudinal coupling of a two-level system with the bichromatic external field. The dynamics of the system along the diagonal dashed green line will be analyzed later to reveal the mechanism for the central fine pattern. The smaller transition probability of experimental spectra is due to dissipation which was not taken into account in our theoretical calculation.

Figure 5

(a) Schematic illustration of the doubly dressed two-level system. The multiphoton resonances, due to the coupler tone, between the single-dressed states are shown by green edges. The numeric label on each edge indicates the required number of photons to make that resonance happen. (b) The quasienergies, and (c) transition probabilities along with the dashed green line in Fig. 4. This result corresponds to the case of a superconducting qubit with longitudinal coupling to the bichromatic microwave field.