Exploring quantum signatures of chaos on a Floquet synthetic lattice

Editors' Suggestion

Exploring quantum signatures of chaos on a Floquet synthetic lattice

Eric J. Meier, Jackson Ang'ong'a, Fangzhao Alex An, and Bryce Gadway

Phys. Rev. A 100, 013623 – Published 19 July 2019

Abstract

Ergodicity and chaos play an integral role in the behavior of dynamical systems and are crucial to the formulation of statistical mechanics. Still, a general understanding of how ergodicity and chaos emerge in the dynamical evolution of closed quantum systems remains elusive. Here, we develop an experimental platform for the realization of canonical quantum chaotic Hamiltonians based on quantum simulation with synthetic lattices. We map the angular momentum projection states of an effective quantum spin onto the linear momentum states of a ${}^{87}{Rb}$ Bose-Einstein condensate, which can be alternatively viewed as synthetic lattice sites. This synthetic lattice, with local and dynamical control of tight-binding lattice parameters, enables new capabilities related to the experimental study of quantum chaos. In particular, the capabilities of our system let us tune the effective size of our spin, allowing us to illustrate how classical chaos can emerge from a discrete quantum system. Moreover, spectroscopic control over our synthetic lattice allows us to explore unique aspects of our spin's dynamics by measuring the out-of-time-ordered correlation function and enables future investigations into new symmetry classes of chaotic kicked-top systems.

Received 20 May 2019

DOI:https://doi.org/10.1103/PhysRevA.100.013623

Physics Subject Headings (PhySH)

Atom optics Bose-Einstein condensates Floquet systems Quantum chaos Quantum chaotic transport

Atomic, Molecular & OpticalNonlinear DynamicsGeneral Physics

Authors & Affiliations

Eric J. Meier ^*, Jackson Ang'ong'a^*, Fangzhao Alex An, and Bryce Gadway ^†

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA

^*These authors contributed equally to this work.
^†bgadway@illinois.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 100, Iss. 1 — July 2019

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Experimental scheme. (a) Time-of-flight absorption image (top) and cartoon (bottom) depicting a $J = 2$ lattice where the lattice sites represent the angular momentum sublevels $m_{J}$ . (b) Arbitrary torque vector on the equator of the Bloch sphere (left) emulated in this system through the tunneling links $| t_{n} (ϕ_{n} = ϕ) |$ (right).
Reuse & Permissions
Figure 2
Demonstrations of linear rotations. (a) Evolution of $〈 J_{z} 〉$ for several spin sizes starting in ${| θ = 0, ϕ = 0 〉}_{CSS}$ and evolving under a $J_{x}$ operator. The solid blue lines are results from simulations of Eq. (3) with no free parameters, and the dashed gray lines show the theoretical $π$ pulse times. (b) Expectation value $〈 J_{z} 〉$ for a spin-2 state evolving under a $J_{x}$ operation until the gray dashed line at $\approx 2.2 ℏ / t$ . At this time, the operation is switched to either $J_{- y}$ (red dots and solid red theory line) or $J_{- x}$ (open blue dots and dashed blue theory line). (c) Top: Experimental absorption images showing the evolution of a $J = 5$ spin starting in $| J = 5, m_{J} {= 0 〉}_{z}$ evolving under a $- J_{y}$ operator. Bottom: Simulated absorption images showing the final atomic distribution and the initial state $| J = 5, m_{J} {= 0 〉}_{z}$ . All error bars are one standard error of the mean. The error bars represent only statistical errors and are not visible because they are smaller than the data markers.
Reuse & Permissions
Figure 3
State preparation and measurement. (a) Absorption images (in the $z$ basis) of a $J = 2$ spin rotating from ${| θ = 0, ϕ = 0 〉}_{CSS}$ to ${| θ = π, ϕ = 0 〉}_{CSS}$ under a $J_{y}$ operator. (b) Bloch sphere representation of the state rotation shown in (a). The state vector is depicted by the red arrows, and the $J_{y}$ operator is depicted by the blue arrow. (c) Images (averaged over many shots) of a $J = 2$ spin in the state ${| θ = 0.50 π, ϕ = 0.41 π 〉}_{CSS}$ as measured along the $x, y$ , and $z$ bases. (d) Bloch sphere depiction of the measured vector shown in (c).
Reuse & Permissions
Figure 4
Squeezing of the artificial spin. Absorption images of the $y$ -basis spin projection as a function of the effective squeezing time $α$ when starting in (a) ${| θ = π / 2, ϕ = - π / 2 〉}_{CSS}$ and (b) $| J = 2, m_{J} {= 0 〉}_{z}$ . (c) Density distributions for initial state ${| θ = π / 2, ϕ = - π / 2 〉}_{CSS}$ shown at effective times $α / π = 0.5$ (left) and $α / π = 1.0$ (right). (d) Spin length $L$ versus the effective squeezing time $α$ . The red squares and simulation line are for initial state ${| θ = π / 2, ϕ = - π / 2 〉}_{CSS}$ , and the blue dots and simulation line are for $| J = 2, m_{J} {= 0 〉}_{z}$ . All error bars are one standard error of the mean.
Reuse & Permissions
Figure 5
Out-of-time-ordered correlation function. ${| F |}^{2}$ for initial states (a) ${| θ = π / 2, ϕ = - π / 2 〉}_{CSS}$ and (b) $| J = 2, m_{J} {= 0 〉}_{z}$ as a function of the effective dynamics time $α$ . Shaded regions indicate the results of numerical simulations incorporating the uncertainty in the calibrated tunneling rate. All error bars are one standard error of the mean.
Reuse & Permissions
Figure 6
Chaotic behavior in the kicked-top model. (a) Spin length $L$ for initial state ${| θ = π / 2, ϕ = - π / 2 〉}_{CSS}$ measured after each kick in a set of eight kicks. The open blue dots and dashed blue simulation line are $(ρ, κ / 2 J) = (π / 8, π / 5)$ , and the closed orange dots and solid orange simulation line are $(ρ, κ / 2 J) = (π / 8, π / 2)$ . (b) Simulated spin length of the effective spin for different initial states. The color represents spin length averaged over $N \in {5, 6}$ kicks, with respect to the color bar at right. The seven open black dots represent the measurements taken to calculate the averaged spin length $\bar{L}$ . (c) $\bar{L}$ as a function of the kick strength $κ$ for $(ρ, J) = (π / 8, 2)$ . Shaded regions indicate results from a numerical simulation incorporating the uncertainty in the calibrated tunneling rate. (d) $\bar{L}$ as a function of the size of the spin $J$ for $(ρ, κ / 2 J) = (π / 8, π / 2)$ . The solid line connects points obtained from a numerical simulation. All error bars are one standard error of the mean.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information