A blueprint for experiments exploring the Poincaré quantum recurrence theorem

TL;DR

This work analyzes quantum recurrence (Poincaré recurrence) in a quasi-isolated N+1 qubit system, showing that revivals in a central qubit coupled to N environment qubits persist in principle and that the revival time $τ_P$ grows exponentially with the environment size due to dephasing among many detuned environment modes. The authors derive analytic expressions for the revival probability and time in the weak-coupling, single-excitation regime, and establish the distribution of the cumulative detuning $Δ$, yielding $p_N(Δ)$ and $μ_N(δ)$ that control revival likelihood. They extend the framework by including a bath of $M$ two-level systems and demonstrate, with realistic superconducting-qubit parameters, that revival signatures survive for microsecond time scales despite environmental leakage, outlining a concrete experimental blueprint. Overall, the work links quantum recurrence to thermalization in isolated quantum systems and provides a practical route to observe revivals in present-day multi-qubit platforms.

Abstract

The quantum form of the Poincaré recurrence theorem stipulates that a system with a time-independent Hamiltonian and discrete energy levels returns arbitrarily close to its initial state in a finite time. Qubit systems, being highly isolated from their dissipative surroundings, provide a possible experimental testbed for studying this theoretical construct. Here we investigate a $N$-qubit system, weakly coupled to its environment. We present quantitative analytical and numerical results on both the revival probability and time, and demonstrate that the system indeed returns arbitrarily close to its initial state in a time exponential in the number of qubits $N$. The revival times become astronomically large for systems with just a few tens of qubits. Given the lifetimes achievable in present-day superconducting multi-qubit systems, we propose a realistic experimental test of the theory and scaling of Poincaré revivals. Our study of quantum recurrence provides new insight into how thermalization emerges in isolated quantum systems.

Figures (8)

• Figure 1: Concept of revivals. (a) At time $t=0$, a central "test" qubit is initialized in its excited state, while it is coupled to $N$ "environmental" qubits (here $N=4$), all of which are initially in their ground states. The solid line illustrates the resulting time evolution of the excited-state population of the central qubit. The accompanying schematic provides a conceptual visualization of the qubit configurations at various points in time. Here $\tau_\delta$ refers to the first passage time, i.e. the Poincaré time for a given configuration of system parameters, with $\tau_{\rm P}$ the average of $\tau_\delta$ over different configurations. (b) Examples of time traces of the excited-state population $p_e$ of the central qubit for different numbers $N$ of environmental qubits. The parameters are $\Delta\Omega/\Omega_0=0.1$, and $\Gamma_0=0.01\Omega_0$; coupling strengths are distributed uniformly from $0$ up to maximum value determined by $\Gamma_0$. For $N=1$, we see the expected sinusoidal Rabi oscillations, but with increasing $N$ the oscillations become less regular and weaker until for $N=10^6$, we see only exponential decay.
• Figure 2: Probability density distributions $p_N(\Delta)$ of $\Delta\equiv \sum_{i=1}^{N}\Delta_i$ for (a) $N=3$, (b) $N=6$, and (c) $N=30$. For illustration, we set $\mathcal{T}=1$ here. The magenta line is the Gaussian distribution, the dark blue symbols show the numerical stochastic values (histogram), and the aqua line is from Eq. \ref{['final_pNDelta1']}, valid for small $\Delta$. The green symbols show the numerically calculated cumulative histogram $\mu_N(\delta)=\int_{0}^{\delta} p_N(\Delta)d\Delta$. (d) shows the same distribution on a linear scale for four different values of $N$.
• Figure 3: Revival probability $\mu_N(\delta)$ to have $p_e(\tau)>1-\delta$ in the long time limit, and the Poincaré time $\tau_{\rm P}$. (a) Comparison between the full numerical solution of the Schrödinger equation and the first-order approximation given by Eq. \ref{['1st-order-12']}, as well as the analytical solution given by Eq. \ref{['final_prob_mu1']}. The dashed line represents an exponential fit $\mu_N\propto e^{-1.93\,N}$. The parameters for this figure are $\Delta\Omega/\Omega_0=0.1$, $\mathcal{G}=0.001$, and $\delta=0.001$. (b) Numerical results for $\mu_N (\delta)$ for three values of $\delta$. In each case, $\mu_N(\delta)$ exhibits an exponential decay with $N$, with the corresponding fits $e^{-0.35N}$, $e^{-0.48N}$, and $e^{-0.69N}$, ordered from top to bottom and shown by solid lines. The parameters used in this panel are $\Delta\Omega/\Omega_0=0.1$ and $\Gamma_0=0.01\Omega_0$, where $\mathcal{G}_i$ uniformly distributed between $0$ and its maximum value determined by $\Gamma_0$. (c) Comparison of Poincare revival time $\tau_{\rm P}$ from the full numerical results and the analytical solution Eq. \ref{['tau_P_v12']}. Dashed line represents an exponential fit of the form $\tau_{\rm P} \propto \exp(bN)$ with $b=1.91$. Parameters are $\Delta\Omega/\Omega_0=0.1$, $\mathcal{G}=0.001$ and $\delta=0.001$. (d) Numerical results for the dependence of $\tau$ on $N$ for different threshold values of $\delta$. The data demonstrate the exponential increase predicted by the model, for $\Delta\Omega/\Omega_0=0.1$ and $\Gamma_0=0.01\Omega_0$, with uniform distributions of the couplings from $0$ to their maximum value determined by $\Gamma_0$.
• Figure 4: A possible experimental setup consists of $N+1$ qubits (one test qubit and $N$ environment qubits) coupled all-to-all via a superconducting router marked as $R$ within the dashed circle Wu2024. (a) The energies of all the qubits $\hbar\Omega_j$ and their couplings $g_{jk}$ are tunable. (b) A more realistic model of (a), taking into account the decoherence due to the external bath, which here is modeled by $M$ bath TLSs. (c) Numerical results for system in (b). We use a model size $N=5$, $M=10000$, $T_1=10~\mu_N$s, with the parameters $\Delta\Omega/\Omega_0=0.02$ and $\Gamma_0/\Omega_0=4\times 10^{-6}$. Purple and brown lines show the population $p_e(\tau)=|\mathscr{C}_0(t)|^2$ of the test qubit with and without relaxation, respectively. The green line is $\sum_{k=0}^{N}|\mathscr{C}_k(t)|^2$, showing the population in the $N+1$ test and environment qubits including relaxation, where decay results due to leakage to the TLS bath. Light blue line is $\sum_{k=0}^{N}|\mathscr{C}_k(t)|^2+\sum_{j=1}^{M}|D_j(t)|^2$, which remains at unity, demonstrating conservation of probability in the closed system formed by all the qubits and TLSs.
• Figure S1: Time traces of the excited state population $p_e(\tau)$ of the central qubit. (a)-(c) Comparison of the numerical solution of the Schrödinger equation and the weak coupling approximation of Eq. \ref{['1st-order-1']}. Here, $\mathcal{G}_j$ has a uniform distribution over the interval $0<\mathcal{G}_j<10^{-4}$ in (a), $0<\mathcal{G}_j<10^{-3}$ in (b), and $0<\mathcal{G}_j<10^{-2}$ in (c). Equation \ref{['1st-order-1']} yields a good approximation for low values of $\mathcal{G}_j$, but fails when coupling gets stronger. In each case $p_e(\tau)$ represents an almost periodic function Bohr1947Amerio1971, ideally in the longe-time limit.