Research ArticlePHYSICS

Probing dynamical phase transitions with a superconducting quantum simulator

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Science Advances  17 Jun 2020:
Vol. 6, no. 25, eaba4935
DOI: 10.1126/sciadv.aba4935
  • Fig. 1 Quantum simulator and experimental pulse sequences.

    (A) False-color optical micrograph of the device highlighting various circuit elements such as qubits (red), the resonator bus (black), qubit XY-control lines (blue), and Z-control lines (green). (B) Connectivity graph of the 16-qubit system when all qubits are equally detuned from the resonator bus by Δ/2π ≃ −450 MHz, with the colored straight lines representing the magnitude of the qubit-qubit couplings. Four pairs of qubits (Q3 and Q14), (Q4 and Q13), (Q5 and Q12), and (Q6 and Q11) have relatively small couplings because of their noticeable cross-talk couplings λijc that neutralize the resonator-induced parts. (C) Experimental pulse sequences for simulating the DPT. First, the qubits are initialized in the ∣00…0〉 state at their corresponding idle frequencies. Then, the rectangular pulses and resonant microwave pulses are applied almost simultaneously to realize the quantum quench. Last, the 16-qubit joint readout is executed, yielding the probabilities {P00…0, P00…1, …, P11…1}, from which σjz can be calculated. When necessary, single-qubit rotation pulses Rj(θj,ϕj)=exp[iθj(cosϕjσjx+sinϕjσjy)/2] (in black dotted box) are applied in advance to bring the axis defined by (θj, ϕj + π/2) in the Bloch sphere of Qj to the σz direction before the readout.

  • Fig. 2 Magnetization and spin correlation.

    (A) Experimental (left) and numerical (right) data of the time evolution of the average spin magnetization shown in the Bloch sphere for different strengths of the transverse fields. (B) Time evolution of the magnetization 〈σz(t)〉. (C) Nonequilibrium order parameter σz¯, as a function of hx/2π. (D) Dynamics of the Bloch vector length σ. (E) Averaged spin correlation Czz¯ versus hx/2π. The regions with light red and light blue in (C) and (E) show the DFP and DPP, respectively, separated by a theoretically predicted critical point hcx/2π5.7MHz. The solid curves in (B) to (E) are the numerical results using the Hamiltonian of our experimental system without considering decoherence.

  • Fig. 3 Loschmidt echo.

    (A) Time evolution of the Loschmidt echo L(t) for different transverse field strengths. (B) Earliest minimum point of L(t) during its dynamics, Lmin(1), as a function of hx/2π. It is shown that Lmin(1) is close to zero in the DPP, while it becomes relatively large in the DFP. The behavior of Lmin(1) versus hx is similar to that in the LMG model (see the Supplementary Materials). The solid curves in (A) and (B) are the numerical results using Eq. 1 without considering decoherence.

  • Fig. 4 Quasidistribution Q-function and spin-squeezing parameter.

    (A) Experimental and numerical data of Q(θ, ϕ) in spherical coordinates, when the minimum values of the spin-squeezing parameters are achieved during the time evolutions with the strengths of the transverse fields hx/2π ≃ 3 and 6 MHz, respectively. (B) Time evolution of the spin-squeezing parameters with hx/2π ≃ 3 and 6 MHz, respectively. (C) Minimum spin-squeezing parameter ξmin2 as a function of hx. The solid lines in (B) are the numerical results using the Hamiltonian of our experimental system without considering decoherence. The blue shaded area in (B) is only accessible for entangled states. The dotted line in (C) is the piecewise linear fit, whose minimum point is close to the theoretically predicted critical point hcx/2π5.7MHz (dashed line).

Supplementary Materials

  • Supplementary Materials

    Probing dynamical phase transitions with a superconducting quantum simulator

    Kai Xu, Zheng-Hang Sun, Wuxin Liu, Yu-Ran Zhang, Hekang Li, Hang Dong, Wenhui Ren, Pengfei Zhang, Franco Nori, Dongning Zheng, Heng Fan, H. Wang

    Download Supplement

    This PDF file includes:

    • Device parameters
    • Correction of XY crosstalk
    • Calibration of the transverse field
    • Phase calibration of the rotation pulse
    • Numerical simulation on the effects of disordered couplings
    • Measurement of the spin-squeezing parameter
    • Finite-size effect of the Loschmidt echo in the LMG model
    • Loschmidt echo, rate function, and anomalous dynamical phase: Numerical results and possible signatures
    • Additional experimental data
    • Table S1
    • Figs. S1 to S9
    • References

    Files in this Data Supplement:

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