Experimental characterization of the quantum many-body localization transition

As strength of disorder enhances beyond a threshold value in many-body systems, a fundamental transformation happens through which the entire spectrum localizes, a phenomenon known as many-body localization. This has profound implications as it breaks down fundamental principles of statistical mechanics, such as thermalization and ergodicity. Due to the complexity of the problem, the investigation of the many-body localization transition has remained a big challenge. The experimental exploration of the transition point is even more challenging as most of the proposed quantities for studying such an effect are practically infeasible. Here, we experimentally implement a scalable protocol for detecting the many-body localization transition point, using the dynamics of an $N=12$ superconducting qubit array. We show that the sensitivity of the dynamics to random samples becomes maximum at the transition point, which leaves its fingerprints in all spatial scales. By exploiting three quantities, each with a different spatial resolution, we identify the transition point with an excellent match between simulation and experiment. In addition, one can detect the evidence of a mobility edge through slight variation of the transition point as the initial state varies. The protocol is easily scalable and can be performed across various physical platforms.

Figure 1

Schematics of the experiment. (a) The illustration of the phase diagram of the system across the whole energy spectrum vs disorder strength. Each energy eigenstate localizes at a different disorder strength, resulting in a many-body mobility edge. (b) The 12 transmon qubits, illustrated as red crosses, are arranged in a 1D chain. The direct coupling between them is realized via capacitors. Each qubit has individual $Z$ (yellow) and $XY$ (green) control lines for state manipulation. The 12 readout resonators (blue) are dispersively coupled to their corresponding qubits, and then divided into two groups to couple to the transmission lines (orange). (c) The three steps of the experimental procedure are depicted, namely initialization, evolution, and readout. For state preparation, the corresponding qubits are excited to $|1\rangle$ by applying $X$ gates. After that, for the state evolution, all the qubits are tuned to their working frequencies $\omega +h_{\ell }$ with $h_{\ell }$ being a random number in the interval $[-h,h]$. After time $t$, for readout, the qubits are detuned to stop the evolution and then simultaneously measured in the ${\sigma }_z$ direction. (d) Our protocol can be understood in a simple way. Each circle represents a qubit, and the filling color shows the corresponding average population. Namely, empty, full, and fractional filling colors stand for $\langle \widehat{n_{\ell }}\rangle =0, \langle \widehat{n_{\ell }}\rangle =1$, and $0<\langle \widehat{n_{\ell }}\rangle <1$, respectively. The dynamics start with a given initial state. Each column shows the average population of each site at long time dynamics for a specific random instance of qubit frequencies. In the ergodic regime, the system thermalizes with $\langle \widehat{n_{\ell }}\rangle \sim 1/2$ and thus the variation with respect to different random realizations is small. Deep in the MBL regime, the dynamics is almost frozen and $\langle \widehat{n_{\ell }}\rangle$ remains close to its initial value, leading also to a weak dependence on the different random realizations. Remarkably, around the MBL transition, the long time dynamics shows a strong dependence on the random potential. We exploit this feature to identify the MBL transition point.

Figure 2

Dynamics of the quantities and equilibrium values. The nonequilibrium dynamics of the system, initialized in a Néel state $|0,1,\cdots ,1\rangle$, averaged over five random instances, for the three quantities: (a) the autocorrelation $\mathcal{C}\left(t\right)$; (b) the number entropy of half-chain $S^{(N/2)}\left(t\right)$; and (c) the Hamming distance $\mathcal{D}\left(t\right)$. The solid lines are numerical simulations, and markers are the experimental data. The disorder strength $h$ is chosen to be $h/J_1=1$ (ergodic), $h/J_1=3$ (near the MBL transition), and $h/J_1=7$ (MBL). All three quantities equilibrate after a transient time. To see the long time behavior, we consider the equilibrium values averaged over all given initial states and 60 random realizations as a function of disorder strength $h/J_1$ for (d) the autocorrelation function $\overline{\langle {\mathcal{C}}_{\text{eq}}\rangle }$; (e) the number entropy of half-chain $\overline{\langle S_{\text{eq}}^{(N/2)}\rangle }$; and (f) the Hamming distance $\overline{\langle {\mathcal{D}}_{\text{eq}}\rangle }$. Again, the solid lines are numerical simulations, and markers are the experimental data. All three quantities smoothly change from the ergodic to the MBL phase without revealing the MBL transition point. The standard deviation, depicted as a shadow for numerical simulation and error bars for experimental data, quantifies the uncertainty in our estimations. We note the good agreement between simulation and experiment.

Figure 3

Probability distributions of the equilibrium quantities. By repeating the experiment for $R=60$ random realizations, a set of equilibrium outcomes are obtained, for which we fit a probability distribution $P\left(Q_{\text{eq}}^{\left(\mathbf{s}\right)}\right)$. Averaging over all given initial states leads to an initial state independent probability distribution $\overline{P\left(Q_{\text{eq}}\right)}$. By considering different disorder strengths $h/J_1=1$ (ergodic), $h/J_1=2$ and 3 (near the MBL transition), and $h/J_1=7$ (MBL), we estimate the probability distribution of (a) the autocorrelation function $\overline{P\left({\mathcal{C}}_{\text{eq}}\right)}$; (b) the number entropy of half-chain $\overline{P\left(S_{\text{eq}}^{(N/2)}\right)}$; and (c) the Hamming distance $\overline{P\left({\mathcal{D}}_{\text{eq}}\right)}$. In all the figures, the small rugs are the experimental data, the bars are the histograms, and the solid lines are the fitting distributions. In the ergodic regime, the distribution is narrow as the system thermalizes, and the final outcomes are determined solely by the thermal state. By increasing the disorder strength $h/J_1$, the outcomes can take a larger range of values as the final results depend heavily on the random potential pattern. Consequently, the probability distribution gets wider. As we go into the MBL regime, the dynamics tends to get frozen, so the final outcomes are mainly determined by the initial state and depend weakly on the random potential pattern. Therefore, the distribution gets narrower again.

Figure 4

Standard deviations of the quantities as a function of disorder strength. The standard deviation $\mathrm{\Delta }{Q}_{\text{eq}}^{\left(\mathbf{s}\right)}$ with respect to random samples for two representative initial states $|{\mathbf{s}}_k\rangle$ (see Appendix pp3 for other initial states) as a function of disorder strength $h/J_1$ for (a) the autocorrelation function $\mathrm{\Delta }{\mathcal{C}}_{\text{eq}}^{\left(\mathbf{s}\right)}$; (b) the number entropy of half chain $\mathrm{\Delta }{S}_{\text{eq}}^{(N/2),\left(\mathbf{s}\right)}$; and (c) the Hamming distance $\mathrm{\Delta }{\mathcal{D}}_{\text{eq}}^{\left(\mathbf{s}\right)}$. Remarkably, the $\mathrm{\Delta }{Q}_{\text{eq}}^{\left(\mathbf{s}\right)}$ shows an evident peak around $h/J_1\approx 3$, revealing the elusive MBL transition point. Moreover, the location of the peak slightly changes for each initial state, which is a clear indication of the mobility edge. To present the average behavior with respect to the initial states, we plot the average standard deviation $\overline{\mathrm{\Delta }{Q}_{\text{eq}}}$ as a function of $h_1/J$ for (d) the autocorrelation function $\overline{\mathrm{\Delta }{\mathcal{C}}_{\text{eq}}}$; (e) the number entropy of half-chain $\overline{\mathrm{\Delta }{S}_{\text{eq}}^{(N/2)}}$; and (f) the Hamming distance $\overline{\mathrm{\Delta }{\mathcal{D}}_{\text{eq}}}$. The shadow around the theory simulations represents the behavior for various initial states.

Figure 5

Comparison of the experimental and simulated results in calibration of frequency alignment. For the positive case, the working frequencies of the 12 qubits range from 4.3225 to 4.3775 GHz. For the negative case, they are opposite. In both the positive and negative cases, the $Q_7$ is excited to $|1\rangle$ and then starts the evolution. The time evolutions for other qubits excited are not shown. The differences of the time evolution between the positive and negative cases come from the frequency differences on each sites. After optimizing the frequency differences, the maximum frequency difference is determined as 15.4 MHz on $Q_{11}$.

Figure 6

Data-processing and post-selection of measurement results. In (a)–(c) (upper panel), we plot the equilibrium autocorrelation $\overline{\langle {\mathcal{C}}_{\text{eq}}\rangle }$, the half-chain number entropy $\overline{\langle S_{\text{eq}}^{(N/2)}\rangle }$, and the Hamming distance $\overline{\langle {\mathcal{D}}_{\text{eq}}\rangle }$ as a function of disorder strength $h/J_1$ for the closed (Sim Closed) and open system simulations (Sim Open) as well as the raw (Exp) and post-selected experimental (Exp PS) data. The error bars in the experimental data and shadows around the solid lines in the numerical simulations represent the standard deviation with respect to random realizations. In (a)–(c) (lower panel), we plot the standard deviation for the equilibrium autocorrelation $\overline{\mathrm{\Delta }{\mathcal{C}}_{\text{eq}}}$, the half-chain number entropy $\overline{\mathrm{\Delta }{S}_{\text{eq}}^{(N/2)}}$, and the Hamming distance $\overline{\mathrm{\Delta }{\mathcal{D}}_{\text{eq}}}$ as a function of disorder strength $h/J_1$ for the closed and open system simulations as well as the raw and post-selected experimental data. We note that, for both the averaged values and the standard deviation, the post-selected experimental data follow the simulations from the unitary evolution, while the raw data pursue the open quantum system evolution.

Figure 7

Convergence with respect to the number of random realizations. The three quantities are investigated for three different number of random realizations, namely $R=60$, 500, and 1000, computed for the initial state $|{\mathbf{s}}_1\rangle$. The upper panel represents the mean values of (a) the autocorrelation $\langle {\mathcal{C}}_{\text{eq}}^{\left({\mathbf{s}}_1\right)}\rangle$; (b) the half-chain number entropy $\langle S_{\text{eq}}^{(N/2),\left({\mathbf{s}}_1\right)}\rangle$; and (c) the Hamming distance $\langle {\mathcal{D}}_{\text{eq}}^{\left({\mathbf{s}}_1\right)}\rangle$ as a function of $h/J_1$. In the lower panel, the standard deviations are plotted as a function of $h/J_1$ for (a) the autocorrelation $\mathrm{\Delta }{\mathcal{C}}_{\text{eq}}^{\left({\mathbf{s}}_1\right)}$; (b) the half-chain number entropy $\mathrm{\Delta }{S}_{\text{eq}}^{(N/2),\left({\mathbf{s}}_1\right)}$; and (c) the Hamming distance $\mathrm{\Delta }{\mathcal{D}}_{\text{eq}}^{\left({\mathbf{s}}_1\right)}$. The good agreement between the curves, especially the location at which the standard deviation peaks, shows that the choice of $R=60$ random realizations can faithfully capture the statistical behavior with respect to disorder in the system.

Figure 8

Initial states dependence. The equilibrium values of the three quantities averaged over random realizations as a function of disorder strength $h/J_1$ for various initial states $|\mathbf{s}\rangle =|{\mathbf{s}}_k\rangle$ (upper panel). The panels, respectively, represent the mean value of (a) the autocorrelation $\langle {\mathcal{C}}_{\text{eq}}^{\left(\mathbf{s}\right)}\rangle$; (b) the number entropy $\langle S_{\text{eq}}^{(N/2),\left(\mathbf{s}\right)}\rangle$; and (c) the Hamming distance $\langle {\mathcal{D}}_{\text{eq}}^{\left(\mathbf{s}\right)}\rangle$. We note that the three quantities smoothly change from their expected value in the ergodic phase to the MBL regime with similar qualitative behavior for all the initial states. The standard deviation of equilibrium values of the three quantities with respect to the $R=60$ random realizations as a function of disorder strength $h/J_1$ for various initial states $|\mathbf{s}\rangle =|{\mathbf{s}}_k\rangle$ is plotted in the lower panels. The lower panels, respectively, represent the standard deviation of (a) the autocorrelation $\mathrm{\Delta }{\mathcal{C}}_{\text{eq}}^{\left(\mathbf{s}\right)}$; (b) the number entropy $\mathrm{\Delta }{S}_{\text{eq}}^{(N/2),\left(\mathbf{s}\right)}$; and (c) the Hamming distance $\mathrm{\Delta }{\mathcal{D}}_{\text{eq}}^{\left(\mathbf{s}\right)}$. We note that for all three quantities, the peak of the standard deviation varies slightly for different initial states, evidencing the existence of the mobility edge.

Figure 9

Number entropy of different subsystems. (a) The equilibrium number entropy $\langle S_{\text{eq}}^{\left(m\right),\left(\mathbf{s}\right)}\rangle$ as a function of disorder strength $h/J_1$, for different subsystem size $m$ and the initial state $|\mathbf{s}\rangle =|{\mathbf{s}}_1\rangle$. As $m$ gets larger, the maximum obtainable number entropy increases, which can be seen in the ergodic phase, where larger subsystems achive higher entropies. (b) The standard deviation of the number entropy with respect to $R=60$ disorder samples $\mathrm{\Delta }{S}_{\text{eq}}^{\left(m\right),\left(\mathbf{s}\right)}$ as a function of disorder strength $h/J_1$ for different subsystem size $m$ and the initial state $|\mathbf{s}\rangle =|{\mathbf{s}}_1\rangle$. As the subsystem size $m$ increases, the location of the peak mostly remains around $h/J_1\sim 2-3$, indicating that the number entropy can characterize the MBL transition better when the subsystem is larger, namely $m\sim N/2$.

Figure 10

Entanglement entropy. (a) Disorder-averaged half-chain entanglement entropy divided by the Page value $ST$ as a function of disorder strength $h/J_1$ for different system size $N$. The curves collapse at the transition point around $h/J_1\approx 3$. (b) Standard deviation of the half-chain entanglement entropy $\mathrm{\Delta }S$ divided by $ST$ as a function of disorder strength $h/J_1$ for different system size $N$. $\mathrm{\Delta }S/ST$ shows a pronounced peak with increase $N$ located at $h_c/J_1\approx 2.7$.