Criticality of environmental information obtainable by dynamically controlled quantum probes

A universal approach to decoherence control combined with quantum estimation theory reveals a critical behavior, akin to a phase transition, of the information obtainable by a qubit probe concerning the memory time of environmental fluctuations of generalized Ornstein-Uhlenbeck processes. The criticality is intrinsic to the environmental fluctuations but emerges only when the probe is subject to suitable dynamical control aimed at inferring the memory time. A sharp transition is anticipated between two dynamical phases characterized by either a short or long memory time compared to the probing time. This phase transition of the environmental information is a fundamental feature that characterizes open quantum-system dynamics and is important for attaining the highest estimation precision of the environment memory time under experimental limitations.

Figure 1

(a) Estimation of the noise fluctuations by a qubit probing the environment. A dynamically controlled qubit probe undergoes pure dephasing due to the probe-environment interaction $H_{SB}=g{\sigma }_{z}B$. The dephasing is characterized on the qubit observable $〈{\sigma }_x({\tau }_c,t)〉\propto e^{-\mathcal{J}({\tau }_c,t)}$ for an (optimal) initial-state: the symmetric superposition of the spin-up/-down states in the basis ${\sigma }_z$, $\frac{1}{\sqrt{2}}(\left|\uparrow \right.〉+\left|\downarrow \right.〉)=\left|+\right.〉$ [59] (Appendix pp1). Here we focus in estimating ${\tau }_c$. (b) Time dependence of the normalized attenuation factor $\mathcal{J}$ of the qubit state probing an Ornstein-Uhlenbeck process ($\beta =2$) for free evolution (dashed) compared to its counterpart under dynamical control (solid). The latter time dependence exhibits a smooth transition (marked by a circle) between two well-defined dynamical phases (regimes) associated with a long and short memory time of the environment depending on the ratio $\frac{t}{{\tau }_c}$.

Figure 2

Criticality of the probe-extracted information on the environmental correlation (memory) time ${\tau }_c$. (a) Spectral density for the Ornstein-Uhlenbeck process (Lorentzian spectrum, $\beta =2$: red solid-line). The spectrum's derivative $\left|\frac{dG}{d{\tau }_c}\right|$ exhibits a critical behavior $\left|\frac{dG}{d{\tau }_c}\right|\propto \left|\omega -{\omega }_0\right|$ at ${\omega }_0={\tau }_c^{-1}{(\beta -1)}^{-\frac{1}{\beta }}$ (orange, lighter solid-line). Then the two dynamical regimes occur when the narrow filters probe frequency components of $G\left(\omega \right)$ on both sides of the critical point. Two typical CW filter functions $F_t\left(\omega \right)$ (green sinc function curves, in linear scale) scan the spectrum on both sides of the transition ($N=20$). (b) The attainable relative-error ${\varepsilon }_{\mathcal{F}}^{\mathrm{ctrl}}\left({\tau }_c,t=\frac{\pi N}{{\omega }_{\mathrm{ctrl}}}\right)$ on ${\tau }_c$ by the qubit probe under CW control ($\sqrt{2N}g{\tau }_c=1$, $\beta =2$, and ${\omega }_{\mathrm{ctrl}}=\frac{\pi N}{t}$, see Appendices pp2-s2–pp5). The divergence $\propto {\left|{\omega }_{\mathrm{ctrl}}-{\omega }_0\right|}^{-1}$ at the critical point evidences the critical behavior of $\left|\frac{dG}{d{\tau }_c}\right|$.

Figure 3

Critical transition of the minimal error in the environment memory-time estimation determined by a probe under CW control in the narrow-filter approximation. The noise spectrum is a Lorentzian ($\beta =2$). (a) The minimum relative error per measurement ${\varepsilon }_{\mathcal{F}}^{\mathrm{ctrl}}({\tau }_c,t_{\mathrm{opt}})$ of the memory time ${\tau }_c$, as a function of $\sqrt{2N}g{\tau }_c$, obtained by optimizing the control time $t_{\mathrm{opt}}$. It exhibits a critical behavior at $\sqrt{2N}g{\tau }_c\approx 1$. Insets: ${\varepsilon }_{\mathcal{F}}^{\mathrm{ctrl}}\left({\tau }_c,t=\frac{\pi N}{{\omega }_{\mathrm{ctrl}}}\right)$ for global minima located in the SM (left inset) and LM (right inset) regimes. The critical point emerges when both local minima are equal [as in Fig. 2]. (b) The optimal scaled measurement time $\frac{t_{\mathrm{opt}}}{t_0}$ as a function of $\sqrt{2N}g{\tau }_c$. (c),(d) Probability $p_{+}\left(t\right)$ as a function of time. For $\sqrt{2N}g{\tau }_c<1$, the optimal time $t_{\mathrm{opt}}^{SM}$ (red circles) corresponds to a linear attenuation factor ${\mathcal{J}}^{SM}\propto t$ [panel (c) and Fig. 1], while for $\sqrt{2N}g{\tau }_c>1$ the optimal time $t_{\mathrm{opt}}^{LM}$ (blue solid circles) belongs to a regime where ${\mathcal{J}}^{LM}\propto t^{\beta +1}$ [panel (d)]. At $\sqrt{2N}g{\tau }_c\approx 1$ [vertical dashed line in (a) and (b), green solid line in (c) and (d)] the transition between the two regimes is observed. The optimal time jumps between $t_{\mathrm{opt}}^{LM}$ and $t_{\mathrm{opt}}^{SM}$ at the critical point, avoiding the time $t_0=\pi N{\tau }_c{(\beta -1)}^{\frac{1}{\beta }}$ [horizontal dashed line in (b)] where no information about ${\tau }_c$ can be extracted from the probe.

Figure 4

Minimal relative error per measurement in the estimation of ${\tau }_c$ as a function of $g{\tau }_c$ for a Lorentzian environmental spectrum under dynamical control. Practical limitations on the number of pulses of a CPMG sequence, such that $N\le N_{\max }$, may prevent attaining the ultimate bound (dashed line). A sudden change of the dynamical control strategy as a function of $g{\tau }_c$ may help: For $g{\tau }_c$ lower than the critical value, the highest precision is achieved by the single-pulse Hahn echo ($N=1$). However, if $g{\tau }_c$ is larger than the critical value, the CPMG sequence with $N=N_{\max }$ is optimal. This dynamical control strategy under practical limitations reduces the minimal error represented by the shaded area, which is determined by the optimal control on each side of the intersection (critical value) of the Hahn and the CPMG curves.