Direct measurement of nonlocal interactions in the many-body localized phase

The interplay of interactions and strong disorder can lead to an exotic quantum many-body localized (MBL) phase of matter. Beyond the absence of transport, the MBL phase has distinctive signatures, such as slow dephasing and logarithmic entanglement growth; they commonly result in slow and subtle modifications of the dynamics, rendering their measurement challenging. Here, we experimentally characterize these properties of the MBL phase in a system of coupled superconducting qubits. By implementing phase sensitive techniques, we map out the structure of local integrals of motion in the MBL phase. Tomographic reconstruction of single and two-qubit density matrices allows us to determine the spatial and temporal entanglement growth between the localized sites. In addition, we study the preservation of entanglement in the MBL phase. The interferometric protocols implemented here detect affirmative quantum correlations and exclude artifacts due to the imperfect isolation of the system. By measuring elusive MBL quantities, our work highlights the advantages of phase sensitive measurements in studying novel phases of matter.

Figure 1

Many-body localization with superconducting qubits. The constituents of the many-body localized phase are localized orbitals (local integrals of motion, LIOMs). Larger disorder yields stronger localized LIOMs with a decreased typical length scale $\xi$. The spatial disorder yields a distribution of these length scales $\xi$. The shaded region indicates effective nonlocal interactions ${\tilde{J}}_{ij}$ between two LIOMs, giving rise to nontrivial dephasing dynamics and logarithmic entanglement growth.

Figure 2

Ergodicity breakdown at strong disorder. (a) Disorder averaged on-site population vs time for $n_{\text{ph}}=2$. In a chain of nine superconducting qubits; two qubits were excited ($Q_6$, $Q_9$). The on-site population of $Q_9$ was measured for various magnitudes of disorder $w/J$, with $J=40\text{MHz}$ (averaged over 50 realizations). For a given disorder realization, we perform several thousand experimental runs. The parameter ${\tau }_{\text{hop}}={\left(2\pi J\right)}^{-1}$ has been introduced to connect the laboratory time $t$ with the hopping energy. $N_{\text{ref}}$ is defined to be the average on-site population across instances of disorder at the reference time $t_{\text{ref}}=100\text{ns}$, after initial transients have been damped. The dashed black line indicates average photon loss for a single qubit measured in isolation. (b) Histograms of $N_{Q_9}\left(t\right)$ at the times and disorders indicated in (a) by numerals i–vi. (c) $N_{\text{ref}}$ vs disorder for $n_{\text{ph}}=1,2,3$. Inset shows which superconducting qubits were initially excited. The error bars denote the standard error of the mean over disorder realizations.

Figure 3

Interferometric signatures of remote entanglement. (a) SE and DEER pulse sequences. DEER differs from SE by the addition of a remote $\pi /2$ pulse simultaneously applied with the SE $\pi$ pulse between the free precession intervals. (b) $\overline{\langle {\sigma }^z\rangle }=\overline{\langle 1-2a^{†}a\rangle }$, and (c) purity of the single qubit for SE (red) and DEER (solid) experiments. The remote DEER pulse induces dephasing, decreasing the purity. The contrast between SE and DEER probes the nonlocal interaction ${\tilde{J}}_{ij}$ between the SE lattice site and the DEER site. Error bars correspond to the standard error of the mean over the different disorder realizations.

Figure 4

Distribution of the couplings between the localized orbitals (a) The pulse sequence for measuring ${\tilde{J}}_{ij}$ showing the evolution of $\langle {\sigma }_i^X\rangle$ on $Q_i$ for $Q_j$ initialized in $|0\rangle$ (red) and in $|1\rangle$ (blue). The rest of the superconducting qubits are initialized in $|0\rangle$. (b) The histogram of ${\tilde{J}}_{ij}$ values measured for 700 instances of disorder vs distance between $Q_i$ and $Q_j$ (seven distinct chains and 100 disorder on each chain). (c) The mean value of ${\tilde{J}}_{ij}$ over all disorder instances in linear (main) and semilogarithmic scale (inset) for two ratios of disorder $w/J=5$ (green) and $w/J=10$ (gray) as a function of distance $|i-j|$. Error bars show the $95\%$ confidence interval based on statistical errors estimated by resampling data via the jack-knife method.

Figure 5

Growth and preservation of entanglement. (a)–(c) Entanglement of formation between superconducting qubits in various two-qubit subsystems ($A,B_i$). To observe the development of entanglement between sites $A$ and $B$ the subsystem is initialized in a product of single-qubit superposition states and the entanglement of formation of the two-qubit density matrix is extracted, for subsystems of (a) $1\times 10$, (b) $2\times 5$, and (c) $3\times 5$ array of qubits with $J=30$ MHz and $w/J=10$. (d), (e) In a two-qubit subsystem ($A,B$) of a $3\times 7$ array of qubits, a Bell pair is created, and the logarithmic negativity (d) and coherent information (e) are extracted from measurements of the subsystem density matrix and averaged over 80 realizations of disorder for $J=30$ MHz with $w/J=12$. Exemplary error bars show the standard error of the mean. We initialize an excitation at a position $C_i$ which via the effective nonlocal interactions contributes to the dephasing of the Bell pair.