Detection of out-of-time-order correlators and information scrambling in cold atoms: Ladder-$\mathit{XX}$ model

Out-of-time-order correlators (OTOCs), recently at the center of discussion on quantum chaos, are tools to understand the information scrambling in different phases of quantum many-body systems. We propose a disordered ladder spin model, the XX ladder, which can be designed in a scalable cold atom setup to detect OTOCs with a sign reversal protocol alternative to existing approaches for evolution backward in time. We study both the clean and disordered XX ladder and characterize different phases (ergodic and many-body localized phases) of the model based on the decay properties of OTOCs. The emergent effective lightcone shows sublinear behavior, while the butterfly cones drastically differ from the lightcone via demonstrating superlinear behavior. Based on our results, one can observe how the information scrambling changes in the transition from well-studied one-dimensional spin models to unexplored two-dimensional spin models in a local setting.

Figure 1

The average ratio of level spacings ${〈r_{\gamma }^n〉}_{\gamma ,n}$ with respect to disorder strength $h$. Coupling strengths are set to $J_{\perp }=J_{\parallel }$ and ${〈r_{\gamma }^n〉}_{\gamma ,n}$ is averaged over $5\times {10}^3$ to 10 random realizations for single-chain sizes ranging between $L=4\text{and}8$. Inset: ${〈r_{\gamma }^n〉}_{\gamma ,n}$ at $h=1\left[J_{\left|\right|}\right]$ with respect to rung interaction strength $\alpha$ where $J_{\perp }=\alpha J_{\parallel }$ for $L=7$.

Figure 2

The OTOC of the ladder-XX model at $h=1\left[J_{\left|\right|}\right]$ between two distant operators ${\sigma }_1^z$ and ${\sigma }_7^z$ in the first chain with respect to $\alpha$ for $L=7$. $\alpha \sim 1$ corresponds to the interacting limit, whereas the cases $\alpha \ll 1$ and $\alpha \gg 1$ are integrable limits of the ladder-XX model. The OTOC is averaged over 100 different random samples. The plot shows the mean values (see Appendix pp1 for the error bars on the curves).

Figure 3

(a) The exponential and (b) power-law decay of OTOCs for ${\sigma }_1^z$ with ${\sigma }_5^z$ (blue triangles), ${\sigma }_6^z$ (red pentagrams), ${\sigma }_7^z$ (orange diamonds), and ${\sigma }_8^z$ (purple circles) observables in a system size of $L=8$. The inset in (b) shows the Lyapunov-like exponent extracted from exponential fitting for both $L=7$ (black asterisks) and $L=8$ (red triangles). (c) No disorder case: Only power-law decay of OTOCs for ${\sigma }_1^z$ with ${\sigma }_5^z$ (orange diamonds), ${\sigma }_6^z$ (red pentagrams), and ${\sigma }_7^z$ (blue triangles) observables when $L=7$ and $h=0$. (d) Crossover region with $h=5\left[J_{\left|\right|}\right]$ (red pentagrams) and MBL with $h=10$ (blue triangles) for observables ${\sigma }_1^z-{\sigma }_7^z$ with $L=7$.

Figure 4

The dynamical exponent $\gamma$ with respect to the OTOC contour values $\eta$ extracted from analyzing data sets for $L=6$ with observables from ${\sigma }_2^z$ to ${\sigma }_6^z$ (light blue circles), with observables from ${\sigma }_4^z$ to ${\sigma }_6^z$ (dark blue stars), $L=7$ from ${\sigma }_4^z$ to ${\sigma }_7^z$ (green squares), and $L=8$ from ${\sigma }_4^z$ to ${\sigma }_8^z$ (red triangles) for a random disorder strength of $h=1$. We averaged the data over $2\times {10}^2$, $1\times {10}^2$, $1\times {10}^2$, and $1\times {10}^1$ times for the first two $L=6$, 7, and 8 system sizes, respectively. Inset: The rates of the sublinear, linear, and superlinear wavefronts for a system size of $L=7$. The markers are the data points, while the lines are the differentiations of the wavefront curves.

Figure 5

A demonstration of wavefronts for a system size of (a) $L=8$ and (b) $L=6$, where the x axis and y axis are the distance and time, respectively. (a) The fitted wavefronts change from sublinear to superlinear in time between the displacements $\mathrm{\Delta }x=3$ and 7 units. (b) Only the sublinear wavefronts are fitted between $\mathrm{\Delta }x=1$ and 5 units (dotted lines), while the solid lines show irregular wavefronts appearing later in time.

Figure 6

Initial-state preparation at $h=1\left[J_{\left|\right|}\right]$. (a) The scaling of the mean error ${\epsilon }_1\left(t\right)$ with respect to $M/N$ sampling ratio, where $M$ and $N$ are the number of randomly sampled states and the dimension of the Hilbert space, respectively. The blue triangles, red pentagrams, orange diamonds, and purple circles stand for a single-chain size of $L=3$ to 6, where all have an exponent of $b\sim -2.5$ in the fit $\propto a\text{exp}\left(-bM/N\right)$. (b) The scaling of the mean error for small $M/N$ ratio has power-law scaling $\propto a{\left(M/N\right)}^b$ with $b\sim -0.5$ for all system sizes of $L=4$ (blue triangles), $L=5$ (red pentagrams), $L=6$ (orange diamonds), and $L=7$ (purple circles). (c) The data collapse applied to the scaling of the mean of the error ${\epsilon }_1\left(t\right)$ with respect to the system size for only one randomly sampled Fock state. Each data point is a random realization where the fitted curve gives an exponent of $b\sim -2.26$ in ${\epsilon }_1\left(t\right)\propto {\left(2L\right)}^b$. (d) The scaling of the effective dimension $d_e$ with the Hilbert-space size, $N$, gives linear scaling $d_e=0.3N$, mimicking an infinite-temperature state. (e) The error signal ${\epsilon }_1\left(t\right)$ with respect to time, for an average of $M=7$ (blue dashed), $M=36$ (red dash dotted), $M=133$ (green dotted), $M=178$ (black solid), and $M=748$ (pink circles) randomly sampled Fock states. (f) The error signal ${\epsilon }_2\left(t\right)=\left|\right|F^{\text{ex}}{\left(t\right)|}^2-\frac{1}{M}{\sum }_j^M|F_j\left(t\right){|}^2|$ with respect to time, for an average of $M=7$ (blue dashed), $M=36$ (red dash dotted), $M=133$ (orange dotted), $M=178$ (purple solid), and $M=461$ (green circles) randomly sampled Fock states. Both (e) and (f) have a system size of $L=6$.

Figure 7

(a) The schematic that illustrates the circuit for OTOC measurement with the spin operators ${\sigma }_1^z$ and ${\sigma }_i^z$. The circuit utilizes interference measurements providing $\text{Tr}\left\{\left|{\psi }_{f1}\right.〉〈{\psi }_{f1}|{\psi }_{f2}〉\left.〈{\psi }_{f2}\right|\right\}={\left|F_j\left(\tau \right)\right|}^2$. (b) Schematic for Hamiltonian sign-reversal protocol for evolution backwards in time: red and blue spheres stand for spin-up and -down states, respectively. We simultaneously perform $R_z\left(\pi \right)R_x\left(\pi \right)$ gates for the odd-numbered spins in the first leg and even-numbered spins in the second leg, while only one gate $R_x\left(\pi \right)$ is applied to the rest of the spins. $R_z\left(\pi \right)$ and $R_x\left(\pi \right)$ are denoted by green and purple wiggly lines, meaning that the single-spin gates for cold atom systems could be realized via laser pulses [31, 66] or microwaves [67].

Figure 8

The measurement circuit for the interferometric approach [20] on the ladder-XX model with local spin observables ${\sigma }_1^z$ and ${\sigma }_i^z$ by using an auxiliary spin $\left|{\psi }_c\right.〉$ to measure only the real part of the OTOC.

Figure 9

Error bars of the out-of-time-order correlators with disorder strength of $h=1\left[J_{\left|\right|}\right]$ between two distant operators ${\sigma }_1^z$ and ${\sigma }_7^z$ with respect to different rung interaction strengths $\alpha$ where $J_{\perp }=\alpha J_{\parallel }$ for $L=7$. The OTOC is averaged over 100 different random samples. The curves are $\alpha =0.01$ (blue solid), $\alpha =0.1$ (orange dashed), $\alpha =0.5$ (yellow dotted), $\alpha =1$ (purple solid), $\alpha =1.5$ (green solid), $\alpha =2$ (pink dashed), $\alpha =10$ (crimson dotted), and $\alpha =100$ (black dotted).

Figure 10

The difference $|F_{i=6}^{\text{ex}}\left(t\right)-F_{i=6}^{\sim }\left(t\right)|$ for only one Haar-distributed random state (blue dashed), averaged over 40 random states (red dash dotted), 80 states (green dotted), and 100 states (black solid). Only the real part of $F_i^{\sim }\left(t\right)$ is taken since the imaginary part is practically zero.

Figure 11

(a) Semilogarithmic plot for ${\sigma }_1^z$ with ${\sigma }_5^z$ (blue pentagrams), ${\sigma }_6^z$ (red circles), and ${\sigma }_7^z$ (orange diamonds) observables in a system size of $L=7$. The Lyapunov-like exponents follow as 1.4342 ($R^2=0.9989$), 1.2507 ($R^2=0.9996$), and 1.1767 ($R^2=0.9994$) for ${\sigma }_5^z–{\sigma }_7^z$ with dashed, solid, and dotted lines, respectively. (b) Logarithmic plot for ${\sigma }_1^z$ with ${\sigma }_5^z$ (blue pentagrams), ${\sigma }_6^z$ (red circles), and ${\sigma }_7^z$ (orange diamonds) observables in a system size of $L=7$. The power-law exponents follow as 2.4335 ($R^2=0.9999$), 2.6165 ($R^2=0.9991$), and 2.6565 ($R^2=0.9997$) for ${\sigma }_5^z–{\sigma }_7^z$ with dashed, solid, and dotted lines, respectively. The data are averaged over 100 different realizations of the Hamiltonian at $h=1\left[J_{\left|\right|}\right]$ for both subfigures.

Figure 12

(a) The EON (eigenstate occupation number) distributions $|c_{\beta }{|}^2$ with respect to eigenenergies $E_{\beta }$ for $L=6$ (blue) and $L=7$ (orange) sizes when only one Fock state is randomly set. (b) The scaling of the mean of the error $|F^{\text{ex}}\left(t\right)-\frac{1}{M}{\sum }_j{F}_j\left(t\right)|$ with the system size when we use only one randomly sampled Fock state. Different curves are different random realizations with the legend showing the exponent of the corresponding power-law decay. The error bars stand for $1\sigma$ standard deviation around the mean of the error signal.