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Cavity-QED simulator of slow and fast scrambling

J. Marino and A. M. Rey
Phys. Rev. A 99, 051803(R) – Published 17 May 2019
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    Abstract

    We study information scrambling, as diagnosed by the out-of-time order correlations (OTOCs), in a system of large spins collectively interacting via spatially inhomogeneous and incommensurate exchange couplings. The model is realizable in a cavity QED system in the dispersive regime. Fast scrambling, signaled by an exponential growth of the OTOCs, is observed when the couplings do not factoriZe into the product of a pair of local interaction terms, and at the same time the state of the spins points initially coplanar to the equator of the Bloch sphere. When one of these conditions is not realized, OTOCs grow algebraically with an exponent sensitive to the orientation of the spins in the initial state. The impact of initial conditions on the scrambling dynamics is attributed to the presence of a global conserved quantity, which critically slows down the evolution for initial states close to the poles of the Bloch sphere.

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    • Received 1 October 2018
    • Revised 28 November 2018

    DOI:https://doi.org/10.1103/PhysRevA.99.051803

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Nonequilibrium statistical mechanicsQuantum cavitiesSpin dynamics
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    J. Marino1,2,3 and A. M. Rey1,2

    • 1JILA, NIST and University of Colorado, and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
    • 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA
    • 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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    Issue

    Vol. 99, Iss. 5 — May 2019

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    Images

    • Figure 1
      Figure 1

      Sensitivity of the scrambling dynamics to different initial conditions and to the nature of many-body coupling Vij. For nonseparable interactions Vij≠ViVj, and for initial spin states pointing close to the equator of the Bloch sphere, information is scrambled fast in Cz(t), an OTOC computed from Jiz spin operators [see Eq. (7)], growing exponentially before saturation. In all other instances, the growth of Cz(t) is algebraic in time (α≃5/2 and β≃9/2).

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    • Figure 2
      Figure 2

      Slow scrambling is signaled by the algebraic growth of Cz(t) (blue curve) as a function of time for separable interactions, Vij=ViVj, and initial state pointing along the x̂ direction. Here N=55 sites (N=F10) and R=500. The red curve is the semilogarithmic fit, lnCz(t)≃βln(v2t)+C, with β=4.55 and C=0.15 (in the figure we plot the actual evolution over four decades; the logarithm is in natural basis). At longer times (not shown in the plot), Cz(t) saturates to a constant value.

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    • Figure 3
      Figure 3

      Fast scrambling is signaled by the exponential growth of Cz(t) (blue curve) as a function of time for nonseparable couplings, Vij≠ViVj, and initial state pointing along the x̂ direction. Here N=55 sites (N=F10) and R=500. The red curve is the semilogarithmic fit, lnCz(t)≃λv2t+C, with λ=0.14 and C=11.5 (in the figure we plot the actual evolution over three decades; the logarithm is in natural basis). At longer times Cz(t) saturates to a constant value. Inset: Semilogarithmic plot of the variance ln[(ΔJiz)2](t)≃λ′v2t+C′, with λ′=0.15 and C′=−3.3.

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    • Figure 4
      Figure 4

      Dependence of the scrambling exponent, λ, as a function of the initial value of the rescaled magnetization 〈J̃zi〉 for nonseparable interactions. The red arrow on the Bloch sphere delimits the critical angle ϑc, below which scrambling is slow (the blue line and arrow indicate that identical phenomenology holds in the southern hemisphere). λ is extracted from Cz(t) evaluated on a system of N=55 sites, with R=300.

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