Research ArticlePHYSICS

Observation of generalized Kibble-Zurek mechanism across a first-order quantum phase transition in a spinor condensate

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Science Advances  22 May 2020:
Vol. 6, no. 21, eaba7292
DOI: 10.1126/sciadv.aba7292
  • Fig. 1 (Color online) Theoretical demonstration of the existence of a power-law scaling and impulse and adiabatic regions in the dynamics.

    (A) 〈ρ0〉 for each energy level as a function of q. The existence of an AFM metastable state for q > 0 and a polar metastable state for q < 0 is observed. (B) The scaling of the q onset of the spin excitations qa with respect to the quench rate v. Orange squares, green diamonds, and blue circles are obtained by the numerical simulation, the KZM, and the generalized KZM, respectively. The inset displays the scaling for the two gaps used in the KZM (green diamonds) and the generalized KZM (blue circles) with power-law fitting exponents of ν = 0.521 and ν = 0.371, respectively. (C) The evolution of the maximally occupied level nmax(t) for distinct v when q is varied from positive to negative values. The solid red line depicts the maximally occupied energy level nsmax for the initial state. This line coincides with the metastable polar phase as shown in (A). (D) The evolution of the probability on the maximally occupied level, i.e., Pm=∣〈ψnmax(q)∣ψ(t)〉∣2, for distinct v. In (C) and (D), the diagonal crosses label the position qa where the spin excitations begin appearing, calculated by the numerical simulation. In (C) and (D), the filled light blue region shows the frozen region for v = 260 Hz/s, where the evolving state remains unchanged. We take c2 = 25.4 Hz and the total atom number N = 1.0 × 104 in the numerical simulation with the energy level index of the Hamiltonian varying from 1,2,…,5001.

  • Fig. 2 (Color online) Experimentally measured mean value and SD of

    ρ0. 〈ρ0〉 and Δρ0 (denoted by the error bar) are evaluated with respect to q(t) as q(t) is slowly varied from positive to negative values for a number of ramp rates v with each point repeating 10 times. The horizontal and vertical dashed lines show the 〈ρ0〉 threshold ρ0c = 0.98 and the phase transition point qc, respectively. 〈ρ0〉 remains unchanged in the frozen region until at qa when it begins to change, entering into the adiabatic region. Here, c2 = 25.5 ± 1.5 Hz.

  • Fig. 3 (Color online) Experimentally observed scaling for with respect to the quench rate shown in the logarithmic scale.

    qaqcv In (A) and (B), q is tuned from around 15 Hz to around −38 Hz, and in (C), q is tuned from around −12 to 28 Hz. In (A) to (C), c2 = 25.5 ± 1.5 Hz, c2 = 23.5 ± 0.7 Hz, and c2 = 25.2 ± 0.9 Hz, respectively. The fitting of the experimental data shows the power-law scaling with the exponent of 0.728 ± 0.20 in (A), 0.723 ± 0.25 in (B), and 0.724 ± 0.32 in (C) with 95% confidence boundary. In the insets, we also plot the results of the numerical simulation (orange line), the KZM (green line), and the generalized KZM (blue line). For the numerical simulation, we take N = 1.16 × 104 in (A), N = 0.99 × 104 in (B), and N = 1.07 × 104 in (C) associated with the corresponding c2. The experimentally observed exponents agree well with the exponents of 0.739 in (A), 0.744 in (B), and 0.734 in (C), which are obtained by the numerical calculation. The corresponding exponents predicted by the (generalized) KZM are 0.662 (0.733), 0.657 (0.740), and 0.662 (0.730), respectively. The error of ∣qaqc∣ arises from the onset time errors in experiments. For instance, if ρ0(t1) > 0.98 and ρ0(t2) < 0.98, we take ta = (t1 + t2)/2 with the error of t1t2, leading to the error of qa being v(t1t2). In experiments, the error is smaller than 0.5 Hz, and if v < 52 Hz/s, the error is smaller than 0.2 Hz.

Supplementary Materials

  • Supplementary Materials

    Observation of generalized Kibble-Zurek mechanism across a first-order quantum phase transition in a spinor condensate

    L.-Y. Qiu, H.-Y. Liang, Y.-B. Yang, H.-X. Yang, T. Tian, Y. Xu, L.-M. Duan

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