Interferometric Approach to Probing Fast Scrambling

TL;DR

This work proposes and analyzes an interferometric scheme for out-of-time-order correlation functions, using only local quantum control and no reverse time evolution, and utilizes a combination of Ramsey interferometry and the recently demonstrated ability to directly measure Renyi entropies.

Abstract

Out-of-time-order correlation functions provide a proxy for diagnosing chaos in quantum systems. We propose and analyze an interferometric scheme for their measurement, using only local quantum control and no reverse time evolution. Our approach utilizes a combination of Ramsey interferometry and the recently demonstrated ability to directly measure Renyi entropies. To implement our scheme, we present a pair of cold-atom-based experimental blueprints; moreover, we demonstrate that within these systems, one can naturally realize the transverse-field Sherrington-Kirkpatrick (TFSK) model, which exhibits certain similarities with fast scrambling black holes. We perform a detailed numerical study of scrambling in the TFSK model, observing an interesting interplay between the fast scrambling bound and the onset of spin-glass order.

Figures (3)

• Figure 1: (a) Schematic of our interferometric protocol. An ancilla qubit is coupled to two copies of a quantum system (prepared at inverse temperature $\beta/2$) at different locations $i,j$ and $i',j'$. By performing ancilla-state dependent interactions both before and after time evolution, one can measure out-of-time-order correlation functions. (b) Circuit diagram illustrating the interferometric protocol. After preparing the ancilla in $1/\sqrt{2}(\left| \uparrow \right\rangle+ \left| \downarrow \right\rangle$), controlled unitaries are performed, first conditioned on $\left| \uparrow \right\rangle$ ($V$) and then conditioned on $\left| \downarrow \right\rangle$ ($W$). Simultaneous measurement of $\sigma_x$ on the ancilla and SWAP on the systems results in the correlator $F(t)$.
• Figure 2: Numerics on the the Sherrington-Kirkpatrick model in a transverse field. For system sizes $N=10,12$, we perform $10^3$ disorder averages, for $N=14$, we perform $10^2$ disorder averages and for $N=16$, we perform $29$ disorder averages. Throughout the simulations, $J=1$ and $\Gamma = 1.35$. (a) Depicts decay of the autocorrelation function $R(t) = \langle \sigma^z_i (t) \sigma^z_i (0) \rangle_\beta$ and growth of the scrambling correlator $C(t) = -\langle [\sigma^z_i(t), \sigma^z_j(0)]^2 \rangle_\beta$ at various inverse temperatures $\beta = 1.1, 2, 4$. The approximate time scale for dissipation is indicated as $t_d$. (b) Numeric fitting to the phenomenological function $C_f(t)$ where $N_c$ and $\Delta$ are fixed for all curves to extract the Lyapunov exponent, $\lambda$, as a function of temperature. (c) Depicts $\lambda$ as a function of temperature. At intermediate temperatures, the growth of $C(t)$ seems to be approximately linear. The temperature of the spin glass transition as determined from Monte Carlo is indicated as the blue dashed line mukherjee2015classical. The red dashed line depicts the scrambling bound; since the exponent controlling the early time growth of $C_f(t)$ is $2\Delta \lambda$ (see footnote), we plot $2\pi T / 2\Delta$ to compare directly with the extracted $\lambda$. The inset shows the differences in $C(t)$ at low and high temperatures. At low temperatures $C(t)$ does not reach its maximally scrambled value even at late times $\sim 10^2/J$. (d) Comparison between $C(t)$ and $C_2(t)$ for $\beta = 4$. The inset depicts the ratio of $\lambda$ as extracted from $C(t)$ and $C_2(t)$ as a function of system size (black points) and $\lambda_{\beta=4}$ as a function of system size (blue points).
• Figure 3: (a) Schematic of a one-dimensional lattice of ultracold atoms interacting with two impurities. There exists a strong interspecies Feshbach resonance between impurity and atom only when the impurity resides in state $|f\rangle$ . By driving the left impurity to state $|f\rangle$ before time evolution and the right impurity to $|f\rangle$ after time evolution ($\tau$ is the time needed to accumulate the Feshbach interaction), one can implement controlled unitaries $V$ and $W$ at asymmetric positions in time and space. (b) Schematic of a 1D lattice of Rydberg atoms interacting with the ancilla via a long-range blockade-based gate. Each atom consists of two ground state hyperfine levels $\left| \uparrow \right\rangle$, $\left| \downarrow \right\rangle$ coupled to a Rydberg state $\left| r \right\rangle$ via an off-resonant laser field.