Maximizing Information on the Environment by Dynamically Controlled Qubit Probes

Analia Zwick, Gonzalo A. Álvarez, and Gershon Kurizki

Phys. Rev. Applied 5, 014007 – Published 25 January 2016

Abstract

We explore the ability of a qubit probe to characterize unknown parameters of its environment. By resorting to the quantum estimation theory, we analytically find the ultimate bound on the precision of estimating key parameters of a broad class of ubiquitous environmental noises (“baths”) which the qubit may probe. These include the probe-bath coupling strength, the correlation time of generic types of bath spectra, and the power laws governing these spectra, as well as their dephasing times $T_{2}$ . Our central result is that by optimizing the dynamical control on the probe under realistic constraints one may attain the maximal accuracy bound on the estimation of these parameters by the least number of measurements possible. Applications of this protocol that combines dynamical control and estimation theory tools to quantum sensing are illustrated for a nitrogen-vacancy center in diamond used as a probe.

Received 29 July 2015

DOI:https://doi.org/10.1103/PhysRevApplied.5.014007

Physics Subject Headings (PhySH)

Open quantum systems Quantum coherence & coherence measures Quantum information with solid state qubits Quantum parameter estimation

Nitrogen vacancy centers in diamond

Quantum Information, Science & Technology

Authors & Affiliations

Analia Zwick, Gonzalo A. Álvarez, and Gershon Kurizki

Weizmann Institute of Science, Rehovot 76100, Israel

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 5, Iss. 1 — January 2016

Subject Areas

Quantum Information

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Scheme of noise-spectra parameter estimation. (a) Scheme of the probing process and its dynamical control: A dynamically controlled qubit probe undergoes dephasing by the environment (bath). The coupling strength $g$ of the probe qubit and the environment affects the energy splitting between the symmetric and antisymmetric qubit states $| \pm ⟩ = (1 / \sqrt{2}) (| ↑ ⟩ \pm | ↓ ⟩)$ . (b) An optimal qubit-probe filter function $F_{t}^{opt} (ω)$ generated by frequent projections is spectrally flat to the extent that it conforms to be quantum Zeno regime and allows the optimal determination of the coupling strength $g$ regardless of the shape of $G (ω) = g^{2} S (ω)$ . (c) A filter function of an optimally controlled qubit probe, $F_{t}^{opt} (ω)$ , for obtaining the ultimate bound on the correlation time ( $τ_{c}$ ) estimation. Such a filter must overlap only with the power-law tail of the Ornstein-Uhlenbeck noise spectrum $S_{β} (τ_{c}, ω)$ , or with $ω > 0$ but not with $ω_{c}$ for the (super-)Ohmic spectrum $S_{s} (τ_{c}, ω)$ . The FID filter function $F_{t}^{free} (ω)$ does not fulfill the requirements for achieving the bound, since it is centered at $ω = 0$ .
Reuse & Permissions
Figure 2
Optimization for achieving the ultimate precision bound. (a) Magnetization $⟨ σ_{x} ⟩$ as a function of the measurement time $t$ of a qubit probe experiencing dephasing due to the Ornstein-Uhlenbeck process, with Lorentzian spectrum $G_{β = 2} (τ_{c} = 10, ω)$ with $g = 1$ under a Carr-Purcell-Meiboom-Gill (CPMG) control sequence with $N = 8$ pulses using the exact analytical expression derived from Eq. (4) [11]. For the particular control, the optimal time $t^{opt}$ that determines the best tradeoff between the signal contrast (almost halfway between 0 and 1) and the highest sensitivity of the decay rate to $τ_{c}$ provides the most accurate estimation of $τ_{c}$ . (b) Minimal relative error $ϵ (τ_{c}, t^{opt})$ per measurement ( $N_{m} = 1$ ) of the estimation of $τ_{c}$ as a function of $g τ_{c}$ for a Lorentzian spectrum $G_{β = 2} (τ_{c} = 10, ω)$ (dashed lines) and a super-Ohmic spectrum $G_{s = 2} (τ_{c} = 10, ω)$ (solid lines) calculated from the integral of Eq. (4), both with $g = 1$ . The minimal relative error under optimal control [CPMG (CW) with $N$ pulses (cycles) satisfying the conditions given in the main text, green lines] achieves the ultimate bounds [ $ϵ_{0} / 3 \sqrt{N_{m}}$ (dashed) and $ϵ_{0} / \sqrt{N_{m}}$ (solid) green lines for $s = 2$ and $β = 2$ , respectively]. It can improve the best precision obtainable under free evolution (blue lines) by several orders of magnitude.
Reuse & Permissions
Figure 3
Simulation of an experimental real-time adaptive-estimation protocol for realistic conditions with a NVC spin probe. (a),(b) Convergence of the real-time adaptive-estimation protocol to the theoretically predicted values for estimating $τ_{c}$ . Free evolution of the probe (blue circles) is contrasted with that of a dynamically controlled probe under a CPMG (green square) sequence with $N = 8$ in the presence of an Ornstein-Uhlenbeck process with Lorentzian spectrum $G_{β = 2} (τ_{c}^{true} = 10 μ s, ω)$ , with $g = 1 MHz$ consistently with the spectral density of a HPHT diamond sample determined in Ref. [54]. The simulated curves derived from exact analytical results of Eq. (4) [11] are averaged over 600 realizations. In (a), the optimal measurement time $t_{m}$ as a function of $N_{m}$ converges to the value $t^{opt}$ for the CPMG case. Similar curves converging to the corresponding $t^{opt}$ are observed for other controls and free evolution. In (b), the minimal relative error $ϵ (τ_{c}, t^{opt})$ converges to the (Cramer-Rao) bound. Under free evolution, the regime where $ϵ \propto (1 / \sqrt{N_{m}})$ is attained for $N_{m} ≫ 100$ . The ultimate bound ( $ϵ_{0} / \sqrt{N_{m}}$ dashed line, $α = β - 1$ ) is attained only by optimal control. (c) Convergence to the minimal relative error $ϵ (g, t_{opt})$ to the (Cramer-Rao) bound for estimating $g$ by $N = 500$ consecutive projective measurements in the Zeno regime (green triangle) compared to the estimation under free evolution (blue circle). In this case, $G_{β = 2} (g = 0.03 MHz, ω)$ , with $τ_{c} = 10 μ s$ , consistently with the spectral density of a ${}^{12}C$ diamond sample determined in Ref. [54]. Here too the ultimate bound ( $ϵ_{0} / 2 \sqrt{N_{m}}$ dashed line, $α = 2$ ) is attained only by optimal control. (d) Proposed scheme for using a NVC as a qubit probe for its environment. The $m_{s} = 0$ ( $| 0 ⟩$ ) state is fully populated by laser irradiation (dashed curly arrow). Microwave (MW) pulses are selectively applied between the states with $m_{s} = 0$ and $- 1$ ( $| 0 ⟩$ and $| - 1 ⟩$ ) of the electronic ground states to initialize the spin probe in a $| + ⟩ = (1 / \sqrt{2}) (| 0 ⟩ + | - 1 ⟩)$ state and effect the $π$ pulse CPMG sequence for estimating $τ_{c}$ . For estimating $g$ , projective measurements are emulated by combining MW $π / 2$ pulses on the $0 \leftrightarrow - 1$ transition and laser-induced relaxation between the ground and exited electronic states that conserve the spin components (solid curly arrows). The readout is done at the end of the $N$ -pulse sequence by detecting the laser-induced fluorescence signal.
Reuse & Permissions

Physical Review Applied