Spectroscopic signatures of localization with interacting photons in superconducting qubits

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Science  01 Dec 2017:
Vol. 358, Issue 6367, pp. 1175-1179
DOI: 10.1126/science.aao1401
  • Fig. 1 Time-domain spectroscopy.

    (A) Optical micrograph of the device. (B) Pulse sequence used to measure eigenvalues of a time-independent Hamiltonian, Eq. 2, with Embedded Image MHz, Embedded Image, and Embedded Image randomly chosen from [0,100] MHz. Initially, all the qubits are in the Embedded Image state. Using a microwave pulse, one of the qubits is then placed in the superposition of the Embedded Image and Embedded Image states (Embedded Image depicted here). The coefficients in the Hamiltonian are set by applying square pulses on the qubits Embedded Image and couplers Embedded Image. After the evolution, a microwave Embedded Image pulse is applied to the qubit to measure Embedded Image or Embedded Image. (C) Typical data set showing Embedded Image and Embedded Image versus time. (D) The FT of Embedded Image for Embedded Image. The peaks in the FT correspond to the eigenvalues of the Hamiltonian. The probability of a Fock state on Embedded Image being in the ninth eigenstate Embedded Image is highlighted. (E) Average of the square of FT amplitudes shown in (D). Averaging is done to show all nine peaks in one curve.

  • Fig. 2 Hofstadter butterfly.

    In Eq. 3, we set on-site potentials Embedded Image MHz and coupling Embedded Image MHz. (A) Data similar to Fig. 1D, averaged squared FT magnitude, are shown for 100 values of dimensionless magnetic field b ranging from 0 to 1. (B) For each b value, we identify nine peaks and plot their location as a colored dot. The numerically computed eigenvalues of Eq. 2 are shown as gray lines (21). The color of each dot is the absolute value of the difference between the measured eigenvalue and the numerically computed one.

  • Fig. 3 Level statistics in a disordered potential.

    In Eq. 2, we set hopping to Embedded Image MHz, which fixes Embedded Image. To obtain a disordered potential, we set Embedded Image with four different irrational values of Embedded Image chosen and the results averaged over b. (A) The schematic of energy levels shows how Embedded Image is defined. (B) The histogram of Embedded Image measured for various values of disorder Embedded Image is presented as a color plot. (C) The measured histogram Embedded Image of Embedded Image for Embedded Image and 5. The dashed lines are plots of Embedded Image and Embedded Image according to Eq. 4, and the solid lines are numerical simulations (21). The change from the GOE toward the Poisson distribution is indicative of vanishing of level repulsion with increase in Embedded Image.

  • Fig. 4 Participation ratio and mobility edges.

    In Eq. 2, we set Embedded Image, Embedded Image MHz, which results in Embedded Image.We measure the evolution of Embedded Image for all pairs of Embedded Image as a function of time for various strengths of disorder Embedded Image. From the magnitude of the peaks seen in the FT of the data, the probabilities relating the positions of two-photon Fock states to energy eigenstates Embedded Image are extracted. See fig. S3 for details. On the basis of those data, we calculated (A) Embedded Image and (B) Embedded Image using Eq. 5 and plotted the results. The Embedded Image is the width of the energy band at a given Embedded Image.

Supplementary Materials

  • Spectroscopic signatures of localization with interacting photons in superconducting qubits

    P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas,† A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, J. Martinis

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Figs. S1 to S7
    • References

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