Quantum Darwinism in Quantum Brownian Motion

Quantum Darwinism—the redundant encoding of information about a decohering system in its environment—was proposed to reconcile the quantum nature of our Universe with apparent classicality. We report the first study of the dynamics of quantum Darwinism in a realistic model of decoherence, quantum Brownian motion. Prepared in a highly squeezed state—a macroscopic superposition—the system leaves records whose redundancy increases rapidly with initial delocalization. Redundancy appears rapidly (on the decoherence time scale) and persists for a long time.

Figure 1

Information about the system ($\mathcal{S}$) is available from fragments of the bath ($\mathcal{E}$). The QBM $\mathcal{E}$ is composed of many bands (${\mathcal{E}}_{\omega }$) each represented by an oscillator with frequency $\omega$. Interaction produces correlated joint states of $\mathcal{S}\mathcal{E}$: $\mathcal{S}$ develops independent correlations with each band (black lines), increasing by $a_k^2$ the phase space volume of $\mathcal{S}$ and ${\mathcal{E}}_{\omega }$. A fragment $\mathcal{F}$ (red) comprises several noncontiguous bands.

Figure 2

Partial information plots (PIPs) show how information about $\mathcal{S}$ is stored in $\mathcal{E}$, plotting ${\mathcal{I}}_{\mathcal{S}\mathrm{\text{:}}\mathcal{F}}$ (information in a random fragment $\mathcal{F}\subset \mathcal{E}$) against $\mathcal{F}$’s size ($f$). PIPs are symmetric when the joint state is pure [5]. We initialized $\mathcal{S}$ in an $x$-squeezed state, which evolves into a superposition of $|x⟩$ states that decohere. Plot (a) shows PIPs at $t=4$ (${\tau }_{\mathrm{decoherence}}\approx 0.1$), for three different values of squeezing. Small fragments provide most of the available information about $\mathcal{S}$; squeezing changes the amount of redundant information, not the overall shape of the PIP. Numerics (dots) agree with a simple theory (lines). Plot (b) tracks one state during decoherence ($t=0.01,\dots ,4$), and confirms the PIPs’ invariant shape.

Figure 3

Delocalized states of a decohering oscillator ($\mathcal{S}$) are redundantly recorded by the environment ($\mathcal{E}$). In plot (a), $\mathcal{S}$ is initially squeezed in $x$ by $s_x=6.3\times {10}^3$, and we see that redundancy [$R_{\delta }(t)$] increases exponentially with the information deficit $\delta$ (as $R_{\delta }\approx s^{2\delta }$). $R_{10\%}\sim 10$ may seem modest, but $\delta =10\%$ implies resolving $x$ to within $\sim 3$ ground-state widths, comparable to experiments in nanomechanical resonators [13]. At $\delta \sim 0.5$, $\mathcal{E}$ resolves $\sim \sqrt{s}$ different locations, and $R_{50\%}\gtrsim {10}^3$ saturates our numerical resolution. Plots (b)–(d) show that $R_{10\%}$—redundancy of 90% of the available information—grows with the initial squeezing ($s_x$ or $s_p$). Dots denote numerics; lines—our theory. Redundancy develops with decoherence: $p$-squeezed states (c) decohere almost instantly, while $x$-squeezed states (a),(b) decohere only as they evolve into $p$-squeezed states. Redundancy persists thereafter (d), and by $t\sim O({\gamma }^{-1})$ dissipation boosts $R_{10\%}$ above our simple theory.