Decay of the survival probability of a local excitation in multi-qubit platforms

Abstract

We present a theoretical study of the survival probability of a state initially prepared in the one-particle sector of a multi-qubit system. The motivation for our work is the ongoing laboratory development of multi-qubit platforms based on superconducting circuits. Using elementary concepts of random matrix theory, we obtain analytic expressions for the survival probability in mathematical models of platforms which, albeit stylized, have been previously shown to provide relevant benchmarks for experimental data. In particular, we show that the decay properties are sensitive to the property of the Hamilton operator to have extended states. The survival probability does not appear instead to depend on whether the interaction between qubits is described by a Gaussian orthogonal ensemble (often interpreted as a model of ''chaotic'' dynamics) or is modeled by an analytically solvable chain. We interpret this phenomenon as a manifestation of a general mechanism for the emergence of equilibration in purely unitary dynamics. Finally, under the same hypothesis of an initial preparation with projection on a large fraction of the extended eigenstates of the Hamilton operator, we show how to extend the classical Kac-Mazur-Montroll estimate of the return time to the quantum survival probability.

Figures (8)

• Figure 1: (\ref{['fig:explicit1']}): Evolution of the survival probability (\ref{['sp-explicit:sp']}) (ordinate) versus time (abscissa) for $N=10,20,30,40$ qubits. For short times all curves are superimposed. (\ref{['fig:explicit2']}): in the solvable chain with $N=100$ no revival occur over the plotted time scale. All plots for $g=1/\sqrt{2}$.
• Figure 2: Evolution of the survival probability (\ref{['sp-explicit:sp']}) (ordinate) versus time (abscissa) for $N=10,20,30,40,100$ contrasted with the asymptotic expression specified by the Bessel function (\ref{['sp-explicit:Bessel']}). For short times all curves are superimposed. The dotted curve is the prediction of the Mandelstam-Tamm lower bound (\ref{['Zeno:lb']}) set identically to zero for times larger than the threshold values (\ref{['Zeno:splim']}). For times large $g t\,\geq\,10$ the curve for $N=10$ differs from the others, in that it displays a first revival. All plots for $g=1/\sqrt{2}$
• Figure 3: (\ref{['fig:Hexp1']}) & (\ref{['fig:Hexp3']}): time evolution of the survival probability (ordinate) of a single particle (central qubit) excitation for the dynamics generated by \ref{['Fock-exe2:H']} versus time (abscissa). As the number of particles $N$ increases oscillations are suppressed and the survival probability appears to decay exponentially (see section \ref{['sec:mqm-Lee']}). The level splittings of all qubits but the central one are sampled according to identical independent uniform distributions in $[\omega-\Delta,\omega+\Delta]$. The mean value $\omega$ is exactly the splitting of the central qubit. In numerics we measure all energies in units of $\omega$, and we set $\omega=1.0$ and $\Delta=0.1$ as in KaWuClPe2025. Off diagonal elements are Gaussian independent identically distributed random variables with mean zero and variance $\sigma^{2}$. In the numerics, we choose $\sigma=\sqrt{1.5\,\Gamma\, \Delta}$ with $\Gamma=10^{-3}$. The plots describe one realization of the random dynamics. We choose the numerical value of $\sigma$ such to obtain curves reproducing the behavior of those of KaWuClPe2025 where off- diagonal elements of the single particle Hamiltonian are also sampled according to a uniform distribution.
• Figure 4: Evolution of the survival probability (ordinate) versus time (abscissa). In both (\ref{['fig:pt1']}) & (\ref{['fig:pt2']}) $\sigma=\sqrt{1.5\,\Gamma\, \Delta}/10$ whereas all other parameters are as in Fig \ref{['fig:Hexp']}. As expected, increasing the number of qubits at fixed value of the coupling constant decreases the accuracy of standard perturbation theory
• Figure 5: Contours for the evaluation of the survival probability of the non-relativistic Lee model.

Theorems & Definitions (1)

• Definition 1