Optical perspective on the time-dependent Dirac oscillator

TL;DR

This work extends the Dirac oscillator to a time-dependent frequency within an optical JC/AJC framework, exploring how ω(t) modulations affect angular-momentum observables and spin–orbit entanglement. By analyzing constant, exponential, and sinusoidal frequency schemes, it reveals relativistic Zitterbewegung behavior, analytically tractable dynamics, and, in the sinusoidal case, aperiodic evolution away from the Weyl limit. The study demonstrates controllable relativistic dynamics in simulable optical systems and broadens the DO's applicability to quantum simulations and experimental platforms. Overall, it provides a comprehensive look at how time-dependent parameters reshape the DO's quantum dynamics and entanglement structure.

Abstract

The Dirac oscillator is a relativistic quantum system, characterized by its linearity in both position and momentum. Moreover, considering $(1{+}1)$ and $(2{+}1)$ dimensions, the system can be mapped onto the Jaynes-Cummings and anti-Jaynes-Cummings models, as illustrated in an exact manner by Bermudez \emph{et al.} [\href{ https://doi.org/10.1103/PhysRevA.76.041801}{Phys. Rev. A 76, 041801(R)}]. Using the optical counterparts of the Dirac oscillator, we analyze an extension of the model that incorporates a time-dependent frequency. We focus on the consequences of these time modulations on the angular momentum observables and spin-orbit entanglement. Noticeable changes in the \emph{Zitterbewegung} are found. We show that a specific choice of time dependence yields aperiodic evolution of the observables, whereas an alternative choice allows analytical solutions.

Figures (16)

• Figure 1: We summarize the different coupling signs and dimensions ($(1{+}1)$ or $(2{+}1)$), along with their respective optical mappings. In the $(1{+}1)$ dimensions case, the system corresponds to the interaction of a TLS with a HO, yielding JC interactions for positive coupling and AJC interactions for negative coupling. The conserved quantity is symbolized by $I_1$. In the $(2{+}1)$ case, the system exhibits spin–orbit coupling, with JC interactions associated with right-handed chirality for positive coupling, and AJC interactions associated with left-handed chirality for negative coupling. The conserved quantity is symbolized by $I_2$.
• Figure 2: Expectation value of the spin observable, using units such that $m=c=\hbar=1$. We analyze different relativistic regimes, employing: $g_0=0.5$ (dotted red line), $g_0=1.0$ (dashed green line) and $g_0=2.0$ (solid blue line). In (a), we consider an initial number state and $n_r=0$, while in (b), we consider the initial coherent state with $|\alpha_r|^2=5$.
• Figure 3: Expectation value of the orbital angular momentum observable, using units such that $m=c=\hbar=1$. We analyze different relativistic regimes, employing: $g_0=0.5$ (dotted red line), $g_0=1.0$ (dashed green line) and $g_0=2.0$ (solid blue line). In (a), we consider an initial number state and $n_r=0$, while in (b), we consider the initial coherent state with $|\alpha_r|^2=5$.
• Figure 4: Time evolution of the spin-orbit entanglement, using units such that $m=c=\hbar=1$. We analyze different relativistic regimes, employing: $g_0=0.5$ (dotted red line), $g_0=1.0$ (dashed green line) and $g_0=2.0$ (solid blue line). In (a), we consider an initial number state and $n_r=0$, while in (b), we consider the initial coherent state with $|\alpha_r|^2=5$.
• Figure 5: Time evolution of the energy eigenvalues considering the exponential modulation, Eq. \ref{['eq:energy_2+1_exp']}, using units such that $m=c=\hbar=1$ and $\xi_0=1$, and considering different quanta: $n_r=0$ (solid red line), $n_r=1$ (solid yellow line), $n_r=2$ (solid green line), $n_r=3$ (solid blue line). The rest mass energies, $\pm mc^2$, are shown as dashed black lines, while the constant energy eigenvalues from Eq. \ref{['eq:energy_2+1']} are represented by dashed colored lines, matching the colors of their corresponding time-dependent counterparts.