Optical band-gap materials are typically periodic structures composed of at least two kinds of materials. The simplest example is a one-dimensionally periodic structure arising from the ordered stacking of two layers of optically mismatched materials. Surface modes, such as surface plasmons and surface phonon polaritons, can also display band gaps if the planar surface is capped with an ordered array of stripes made from an electronically dense material. Quantum control could involve quantum dots composed of two-level or three-level atoms within a band-gap system or near a structured surface, in addition to the use of external tunable laser fields. We aim to explore the interaction between surface plasmon and quantum dots in one-dimensional periodic nanorod arrays and to explore the potential of such scenarios for realizing scalable quantum information processing.

Firstly, we study two boundary conditions for the electric and magnetic field components of surface plasmons in one-dimensional periodic nanorod arrays located at the interface between vacuum and metal. Furthermore, we use Bloch’s theorem of periodic structure to derive the characteristic dispersion relationship determined by the surface plasmon wave vector and Bloch vector. By adjusting the characteristic parameters of the system, including the nanorod spacing, nanorod material properties, and the type of media involved, we probe the surface plasmon band structure while varying different parameters. Then, we carry out a theoretical analysis of how surface plasmons manipulate the excited states of quantum dots by adjusting the characteristic parameters. Finally, by quantizing the surface plasmon field, we calculate the transition rate of these quantum dots under the action of the surface plasmon.

The results show that the surface plasmon exhibits a distinct band and a band-gap structure. By adjusting the nanorod spacing, nanorod material properties, and the type of media involved, the surface plasmon band structure varies with these parameters. It is also proved that these surface modes (SPPs) are traveling waves. Furthermore, it is shown that when the frequency of SPP is small (photon-like), the corresponding plasmon wave vector k?x is also small, and the coupling is much weaker. In contrast, when the k?x is in the range of 0.6<Ω<0.8, the surface plasmon has a strong coupling effect with the quantum dots near the interface, which means that the coupling is quite strong when the surface mode is phonon-like. Meanwhile, when k?x is larger, the lifetime of the surface modes is longer, which implies that the phonon-like surface modes have longer interaction time with quantum dots. This should provide a better opportunity to control the transition from the excited states to the ground states. The results also show that it may be possible to manipulate quantum states. 1) The transition between excited and ground states can be controlled by turning on or off specific regions of the excitation field frequency. 2) The transition is also stopped by increasing the frequency to the edge of the gap. These two methods, by physical meaning, indicate that they could provide a feasible option for manipulating quantum states flexibly in real-world applications. The calculation results show that the transition rate of quantum states involved in this system is roughly in the order of 10⁵, so the manipulation frequency of the quantum states can reach the order of 10¹³ s^-1. Obviously, such a frequency can provide a theoretical criterion for ultrafast quantum state manipulation. This shows that rapid manipulation and reading of qubits can be achieved by surface plasmon, and the speed and efficiency of quantum computing can be improved. Faster quantum state manipulation means swifter quantum information processing per unit time, which lays a foundation for batch quantum information processing and quantum state manipulation in the field of quantum computing.

In this paper, the characteristic dispersion relationship of surface plasmon, including amplitude, phase velocity, and frequency, is studied in one-dimensional periodic nanorod arrays located at the interface between vacuum and metal. The results show that the SPPs have band gap features. By adjusting the characteristic parameters of the system, the SPPs band structure varies correspondingly. It is verified that the SPPs are traveling waves. Taking advantage of this feature, we discuss a promising scheme for coupling surface mode SPPs with quantum dots within periodic nanorod arrays located near the metal-vacuum interface. Due to the small size of these modes, it is easy to couple them to the quantum dots located in each cell, which leads to a strong coupling effect between the surface plasmons and the quantum dots near the metal-vacuum interface. By adjusting the characteristic parameters, the number of quantum dots can be made to resonate with the SPPs. The transition rate of these quantum states under the influence of SPPs can be calculated, and then the coupling strength and interaction time (lifetime) can also be determined. Theoretically, it is shown that large-scale quantum state manipulation can be achieved via surface plasmons on the periodic nanoarray structure, thus providing a theoretical basis for large-scale quantum information processing. This study extends previous research on the interaction between quantum dots and SPPs, which not only verifies the characteristics of the SPP field on conventional metal-medium surfaces but also provides some easily realizable nanofabrication techniques for subsequent experimental verification, to compare the differences in scalable quantum state manipulation between theoretical calculations and experimental measurements.