Quantum metrology leverages quantum effects to amplify estimation sensitivity and is applied to gravity wave detection, measurement standards, and super-resolution. As an important concept in quantum metrology and quantum estimation theory, quantum Fisher information (QFI) has attracted much attention from researchers. In quantum estimation theory, the QFI serves as a reliable quantity for evaluating the precision of parameter estimation. The Cramer-Rao theorem suggests that the lower bound of estimation precision is the inverse of the QFI. Consequently, quantum evolution that can increase the QFI is of paramount interest in quantum metrology. Recently, the non-Markovian dynamics induced by the giant-atom-waveguide system have captured considerable attention, and related advances have been made on quantum entanglement and quantum coherence. The effect of non-Markovianity induced by the giant-atom-waveguide system on QFI is a compelling subject that warrants examination. We aim to theoretically delineate the dynamics of the QFI for a giant-atom state coupling to a one-dimensional waveguide at multiple coupling points. A giant atom coupling to a one-dimensional waveguide at multiple coupling points with different coupling strengths is considered and the dynamics of QFI encoded on the giant atom’s state is analyzed. Specifically, the effect of the time delay, the different coupling strengths, and the number of coupling points on the QFI will be analyzed in detail. By a comparison between a small atom and a giant atom, we discern the quantum enhancement in the dynamics of a giant-atom-waveguide system. The analysis is expected to yield insights into the dynamics of QFI for a giant atom coupling to a one-dimensional waveguide.

We consider the model consisting of a giant atom coupling a one-dimensional waveguide at multiple coupling points and write out the Hamiltonian in position space. Utilizing Fourier transformation, we convert the Hamiltonian into momentum space. The time evolution of the quantum state is derived with the Schrodinger equation by using the Wigner-Weisskopf approximation. While an analytical solution is elusive, the time-delayed differential equation provides the time evolution of the quantum state with numerical simulation. Then, with the resulted density matrix, we calculate both the one-parameter and two-parameter QFI through the QFI matrix based on the quantum estimation theory. After that, the effects of the accumulated phase, the time delay, the coupling strengths, and the number of the multiple coupling points on the QFI are analyzed in detail. In addition, the condition of the occurrence of a bound state is identified from the coefficient of the time-delayed differential equation.

For the zero accumulated phase, the QFI exhibits a monotonic decrease for zero time delays. For the non-zero time delays, the QFI initially diminishes at a uniform rate and decreases faster when the scaled time takes the corresponding time delay. For the accumulated phase, a value of π is taken; for the zero time delay, the QFI remains. While for the nonzero time delay, the QFI initially diminishes and then stabilizes at a value that decreases with increasing time delay. These stable values of QFI correspond to the bound states of the giant atom in the waveguide. Regarding the effect of coupling strengths, a stable QFI is attainable solely under conditions of equal coupling and an accumulated phase of π. As for the effect of the number of coupling points, our results imply that the QFI decreases monotonically for the zero accumulated phase and zero time delay. While for non-zero time delay, the QFI initially decreases and then stabilizes at a value independent of the number of coupling points when the accumulated phase is π. With regard to the two-parameter QFI, the optimal sensitivity is investigated to achieve the best possible optimal sensitivity in the two-parameter measurement under the same conditions as the one-parameter QFI. Through analysis, we delineate the conditions for the emergence of bound states within the giant-atom-waveguide system.

We investigate the dynamics of single-parameter and dual-parameter QFI of a giant-atom coupled to a one-dimensional waveguide at multiple coupling points with an even number. We analyze the effects of accumulated phase, photon propagation time delay, varying coupling strengths, and the number of coupling points on the evolution of QFI. When the accumulated phase takes a value of zero, the single-parameter QFI exhibits a monotonic decline to zero for different time delays. However, when the accumulated phase increases, the single-parameter QFI decreases with some oscillations for the nonzero time delay. In particular, the single-parameter QFI stabilizes when the accumulated phase reaches π. This stable value is time-delay dependent and is indicative of the system transitioning to a bound state. With regard to the effect of the different coupling strengths, the single-parameter QFI decreases rapidly to zero when the accumulated phase is null. Conversely, when the accumulated phase is π, the decrease is notably slower. A stable QFI value is attainable only under conditions of uniform coupling strength and an accumulated phase of π. Regarding the number of coupling points, we find that the single-parameter QFI diminishes over time but at varying rates contingent on the number of coupling points when the accumulated phase is zero. When the accumulated phase is π, the single-parameter QFI stabilizes irrespective of the number of coupling points, with the time delay being the determining factor for the giant-atom state. The conditions for achieving optimal dual-parameter estimation are found to be congruent with those necessary for the single-parameter to maintain stability. With a comparison between a small atom and a giant atom, the dynamics of the QFI can be enhanced by a giant atom due to the occurrence of a bound state when related parameters are satisfied. The occurrence of a bound state is discussed and the condition to achieve a bound state is identified. The case of multiple coupling points with an odd number is also discussed. Our study may contribute to understanding the dynamics of QFI of a giant-atom coupling to a one-dimensional waveguide at multiple coupling points.