Analytical Control of Quantum Coherence: Markovian Revival via Basis Engineering and Exact Non-Markovian Criteria

TL;DR

This work challenges the long-held belief that coherence revival requires non-Markovian environments by showing that universal coherence control is achievable through basis engineering in the $σ_x/σ_y$ bases and environmental tuning. It introduces a closed-form universal criterion $ω_0^c$ to design non-Markovian dynamics for any finite window $[0,t_{\max}]$, while also enabling Markovian revival when $ω_k>ω_k^c$ in the same bases. The authors derive exact revival conditions: in the $σ_z$ basis, revival follows $ω_0=n·6.285/t_{\max}$; in the $σ_x/σ_y$ bases, revival requires matching $ω_0$ and $ω_k$ as $ω_k=πω_0/6.285$, yielding periodic full revivals with period $T=6.285/ω_0$. The results provide a predictive toolkit for enhancing quantum memory, sensing, and error mitigation across platforms by jointly tailoring the environment and the system’s basis.

Abstract

The preservation of quantum coherence is besieged by a fundamental dogma: its revival necessitates non-Markovian memory effects from structured environments. This paradigm has constrained quantum control strategies and obscured simpler paths to coherence protection. Here, we shatter this belief by demonstrating unambiguous coherence revival even in strictly Markovian regimes, achieved solely through basis engineering in the $σ_x/σ_y$ bases. We establish a comprehensive analytical framework for predictive coherence control, delivering three universal design principles. First, we derive a minimum critical noise based frequency, $ω_{0}^{c} = 1.57/(0.4996 \cdot t_{\max})$, serving as a universal criterion for engineering non-Markovian dynamics over any interval $[0, t_{\max}]$. Crucially, we show that Markovian environments ($ω_0 < ω_0^c$) can exhibit coherence revival when the Zeeman energy satisfies $ω_k > π/(2t_{\max})$, decoupling revival from environmental memory. Furthermore, for non-Markovian environments, we provide exact conditions for periodic and complete revival: setting $ω_0 = n \cdot 6.285/t_{\max}$ guarantees revival in the $σ_z$ basis, while combining it with $ω_k = πω_0 / 6.285$ ensures perfect revival in the $σ_x/σ_y$ bases. Our results, validated by rigorous quantum simulations, provide a predictive toolkit for coherence control, offering immediate strategies for enhancing quantum memory, sensing, and error mitigation.

Figures (11)

• Figure 1: Decoherence function $\Gamma(t)$ dynamics. (a) Time evolution for $\omega_0\in[0.06285,0.1258,0.2514]$ MHz. (b) Comparison between $\Gamma(t)$ and fitted $\Gamma'(t)$ for $\omega_0\in[0.01,0.03,0.05,0.0629]$ MHz. Parameters: $\alpha=0.5$, $\omega_J=50$ MHz.
• Figure 2: (a)-(c) Non-Markovianity measure $\mathcal{N}$ versus noise based frequency $\omega_0$ for time intervals $t\in[0,50]$ ms, $[0,100]$ ms, and $[0,200]$ ms respectively, with $\omega_J=50$ MHz, $\alpha=0.5$, and $F(j)=1/j$. The dynamical phase transitions occur at $\omega_0^c=0.0629$, $0.0315$, and $0.0158$ MHz for the respective time intervals.
• Figure 3: Temporal evolution of the off-diagonal density matrix elements and the corresponding quantum coherence. (a1, a2) Dynamics under a noise based frequency of $\omega_0=0.03$ MHz and noise strength $\alpha=0.5$. (b1-b3) Dynamics under $\omega_0=0.0314$ MHz with $\alpha=0.5$ (b1, b3) and $\alpha=1.0$ (b2, b3). (c1, c2) Dynamics under $\omega_0=0.2514$ MHz and $\alpha=0.5$. All quantum simulations were performed with an ensemble size of $N=500$.
• Figure 4: In Markovian environment ($\omega_0=0.03$ MHz): (a-d) Time evolution of free oscillation term $\vert\cos(\omega_{k}t)\vert$, decay term $e^{-2\Gamma(t)}$, and quantum coherence $C_{K_{x}}^{l}(\rho_{X}(t))$ for qubit frequencies (a) $\omega_k=0.005$ MHz, (b) $\omega_k=0.0157$ MHz, (c) $\omega_k=0.05$ MHz, and (d) $\omega_k=0.3$ MHz (noise strength $\alpha=0.5$, noise cutoff frequency $\omega_J=50$ MHz, ensemble size $N=500$). Cases (a) and (b) correspond to $\omega_k\leq\omega_k^{c}$, while (c) and (d) demonstrate $\omega_k>\omega_k^{c}$.
• Figure 5: Non-Markovian environment ($\omega_0=0.2514$ MHz $>\omega_0^{c}$): Temporal evolution of coherence oscillation term $\vert\cos(\omega_{k}t)\vert$, decay term $e^{-2\Gamma(t)}$, and quantum coherence $C_{K_{x}}^{l}(\rho_{X}(t))$ for qubit frequencies. (a) $\omega_k=0.0157$ MHz, (b) $\omega_k=0.1257=\frac{\pi\omega_0}{6.285}$ MHz, and (c) $\omega_k=0.3$ MHz. The parameters, $\alpha=0.5$, $\omega_J=50$ MHz, $N=500$.