Revealing inherent quantum interference and entanglement of a Dirac particle

The Dirac equation, a cornerstone of modern physics, describes spin-1/2 particles relativistically. While its profound impact across various scientific fields is well-established, the underlying physics of its dynamical solutions remains partially understood, particularly concerning Zitterbewegung (ZB), the oscillatory motion stemming from interference between positive and negative energy components. The extremely small amplitude of ZB in free electrons makes direct observation challenging, leading to ongoing debates about its existence and interpretation. Although various quantum systems have been used to simulate ZB, the fact that similar phenomena can appear in classical systems raises questions about the truly quantum origins of ZB in Dirac particles. This paper addresses the fundamental question of whether ZB's quantum origin is deeper than the oscillatory motion itself, exploring whether a more universal quantum interference behavior underlies Dirac particle dynamics even in the absence of ZB. This exploration is crucial for understanding Dirac particle behavior at a fundamental level, a topic lacking in-depth investigation.

The literature extensively explores Zitterbewegung simulations using diverse quantum systems, including circuit quantum electrodynamics, ion traps, ultracold atoms, semiconductor quantum wells, graphene, and moiré excitons. While these studies offer valuable insights into ZB, they also highlight its presence in classical wave systems, thus not uniquely characterizing Dirac particles. This has motivated the current research to look beyond Zitterbewegung to find more fundamental quantum characteristics inherent in the behavior of Dirac particles. Previous studies have predicted phase space quantum interference effects for specific states with both positive and negative energy components, but the presence of such effects without ZB has not been demonstrated.

The study focuses on the one-dimensional Dirac Hamiltonian, where the spinorial nature of the particle is encoded in the two lowest energy levels of a superconducting Xmon qubit, while its position and momentum are mapped to the quadratures of a microwave field in a resonator. The Hamiltonian is experimentally realized by applying two longitudinal parametric modulations and a transverse continuous microwave driving to the qubit. The first modulation controls the qubit-resonator interaction, while the second adjusts the effective mass. The transverse drive simulates the coupling between internal and external degrees of freedom. The initial state preparation involves initializing the resonator to a coherent state and the qubit to a superposition state. The Wigner function (WF), a quasiprobability distribution in phase space, is then measured using Wigner tomography. This involves displacing the resonator and measuring the photon number distribution. The negativity of the WF and the entanglement entropy, quantifying the entanglement between the qubit (internal degree of freedom) and the resonator (external degree of freedom), serve as indicators of nonclassical behavior. Klein tunneling is simulated by adding a linear potential to the Hamiltonian via a continuous microwave drive to the resonator, allowing observation of mesoscopic superpositions in phase space. Numerical simulations are performed using established theoretical frameworks and compared with the experimental results. The analysis focuses on the negativity of the Wigner function and the entanglement entropy to demonstrate the nonclassical nature of the observed quantum interference.

The researchers' theoretical predictions and experimental results demonstrate a universal quantum interference behavior in the position-momentum phase space of Dirac particles. This interference is manifested by the negativity of the measured Wigner function, a signature of nonclassicality. The negativity is observed in the Wigner functions conditioned on the qubit state, indicating a correlation between the internal (pseudospin) and external (spatial) degrees of freedom. The entanglement entropy between these degrees of freedom further confirms the quantum nature of the interference. The experimental results using a superconducting qubit-resonator system show excellent agreement with theoretical predictions, demonstrating the splitting of an initial Gaussian wavepacket into two parts propagating in opposite directions. The negativity of the Wigner function is clearly shown in the experimental results. The evolution of the entanglement entropy is also measured and compared with the numerical simulation, showing good agreement. The simulation of Klein tunneling in a linear potential field reveals a cat-like state, a mesoscopic superposition of two separated wavepackets in phase space, further highlighting the nonclassical interference behavior. Importantly, the study shows that this phase space quantum interference exists even when restricting the system to the positive energy branch, demonstrating that it is a universal inherent characteristic of Dirac particles, and not solely a consequence of Zitterbewegung. The time evolution of the average position (related to Zitterbewegung) is also observed and matches the simulation results, indicating that ZB has a deeper quantum origin rooted in this phase space quantum interference.

The findings of this study significantly advance our understanding of Dirac particle dynamics. The demonstration of phase space quantum interference as a universal inherent characteristic, independent of Zitterbewegung, resolves some of the ambiguity surrounding the quantum nature of Dirac particle behavior. The experimental validation of this phenomenon using a superconducting qubit-resonator system provides strong evidence for the fundamental nonclassical nature of Dirac particle motion. The observed negativity of the Wigner function and the presence of entanglement are clear indicators of quantum effects that go beyond classical analogs. This work not only provides fundamental insights into relativistic quantum mechanics but also opens avenues for exploring quantum technologies. The demonstrated nonclassical effects could serve as valuable resources for quantum-enhanced sensing and other quantum information processing tasks.

This research successfully revealed a universal quantum interference behavior inherent in Dirac particles, manifested as negativity in the phase space Wigner function and pseudospin-momentum entanglement. This behavior, confirmed through both numerical simulations and an on-chip experiment employing a superconducting qubit-resonator setup, surpasses the previously understood Zitterbewegung, providing deeper insights into the fundamental quantum nature of Dirac particle dynamics. The demonstrated nonclassical effects have significant implications for advancing quantum technologies, particularly in quantum sensing applications. Future research could explore the extension of these findings to higher dimensions and more complex systems, investigate the control and manipulation of this phase space interference for quantum information processing, and investigate the connection between this inherent quantum interference and other relativistic quantum phenomena.

The study focuses on a one-dimensional system, which simplifies the experimental setup and theoretical analysis. Extending these findings to higher dimensions would be a valuable future research direction. The experimental setup involves approximations and idealizations of the Dirac Hamiltonian, although the measured results show good agreement with the simulations. The impact of these approximations on the observed phenomena should be further investigated. The experimental setup may require further improvement for finer resolution and more precise measurements.