Radiative process of tripartite entangled probes in inertial motion

TL;DR

This work analyzes how three Unruh–DeWitt qubits arranged in tripartite W states radiate when coupled to a quantum field in flat spacetime. By systematically comparing static and inertial-motion configurations under eternal and Gaussian switching, and by introducing a thermal bath, the study reveals that collective transition rates are strongly dictated by the initial geometry, velocity directions, and environmental temperature. The authors derive explicit expressions for auto, cross, and collective rates, show degeneracy patterns among the four $|\Omega^{-}_{n}\rangle$ states, and discuss how switching and motion can suppress or enhance decoherence effects. These insights provide design principles for multipartite quantum systems with improved resilience to environmental noise and relativistic effects.

Abstract

We study the radiative process of three entangled quantum probes initially prepared in a tripartite W state. As a basic set-up, we consider the probes to be inertial in flat spacetime and investigate how the radiative process is affected by different probe configurations. We take the quantum probes as either static or moving with uniform velocities and consider different switching scenarios. Our main observation confirms that the radiative process depends distinctively on the initial configuration in which the probes are arranged, as well as on the direction of the probe velocity. We also extend our analysis to a thermal environment, thereby simulating a more realistic background. We thoroughly discuss the effects due to different switchings, the thermal background, and probe motion on the radiative process of these tripartite entangled probes. We also comment on how the observations from this work can help prepare a set-up least affected by quantum decoherence.

Figures (11)

• Figure 1: In the above figure, we have depicted the energy levels of the collective quantum states for a system of three entangled atomic probes through a schematic diagram. All the permissible transitions between different energy levels are also highlighted in this figure. As previously mentioned, one can note that the separable states, i.e., the collective ground $|g\rangle$ and excited $|e\rangle$ states, have no degeneracies. At the same time, the entangled states $|\Omega^{-}_{n}\rangle$ are degenerate in their energy eigenvalues. Similarly, the entangled states $|\Omega^{+}_{n}\rangle$ are also degenerate in their energy eigenvalues. From this figure, one can notice that the transitions from the state $|e\rangle$ to $|\Omega^{-}_{n}\rangle$ and $|g\rangle$ are not possible and vice versa. Similarly, the transition from the state $|\Omega^{+}_{n}\rangle$ to $|g\rangle$ and its reverse are not possible.
• Figure 2: We present a schematic diagram for three probes $A$, $B$, and $C$ that are static. The distances between these probes are respectively $d_{AB}$, $d_{BC}$, and $d_{AC}$. The angle formed between $d_{AB}$ and $d_{AC}$ is $\theta$. By varying these parameters of three distances and the angle one gets all possible probe configurations.
• Figure 3: Top-left: We have plotted the collective transition probability rates $\mathcal{R}_{|\Omega^{-}_{n}\rangle\to|g\rangle}(\omega)$ as functions of the dimensionless probe energy gap $|\omega|d$ for a scenario where the three probes are stationed at the three vertices of an equilateral triangle. In this scenario, the transition rates from the states $|\Omega^{-}_{2}\rangle$, $|\Omega^{-}_{3}\rangle$, and $|\Omega^{-}_{4}\rangle$ are the same. Top-right: We have plotted the collective transition probability rates $\mathcal{R}_{|\Omega^{-}_{n}\rangle\to|g\rangle}(\omega)$ as functions of the dimensionless probe energy gap $|\omega|d$ when the probes are stationed equidistant on a straight line. In this scenario, we have considered $d_{AB}=2\,d_{AC}$, and here the transition probability rates from the states $|\Omega^{-}_{2}\rangle$ and $|\Omega^{-}_{4}\rangle$ are the same. From these plots, one can observe that as the energy gap or the separation between the probes increases, the different transition probability rates tend to reach a fixed value. Bottom-left: The collective transition probabilities are plotted as functions of the angle $\theta$ between the vectors ${\bf d}_{AB}$ and ${\bf d}_{AC}$ for $|\omega|\,d_{AB}=1=|\omega|\,d_{AC}$. From this plot, one can confirm that when $\theta=\pi/3$ or $\theta=5\pi/3$, i.e., when the probes are at the vertices of an equilateral triangle, the transition probability rates from the states $|\Omega^{-}_{2}\rangle$, $|\Omega^{-}_{3}\rangle$, and $|\Omega^{-}_{4}\rangle$ become equal. Bottom-right: The collective transition probabilities are plotted as functions of the angle $\theta$ between the vectors ${\bf d}_{AB}$ and ${\bf d}_{AC}$ for $|\omega|\,d_{AB}=1=2\,|\omega|\,d_{AC}$. From this plot, we confirm that when $\theta=0$ or $\theta=2\pi$, i.e., when the probes are on a straight line placed equidistant, the transition probability rates from the states $|\Omega^{-}_{2}\rangle$ and $|\Omega^{-}_{4}\rangle$ become equal. We would like to mention that all of the above plots in this figure are obtained for eternal switching $\kappa(\tau_{j})=1$.
• Figure 4: Top-left: We have plotted the collective transition probability rates $\mathcal{R}_{|\Omega^{-}_{n}\rangle\to|g\rangle}(\omega)$ as functions of the dimensionless probe energy gap $|\omega|d$ for a scenario where the three probes are stationed at the three vertices of an equilateral triangle and for Gaussian switching. Here also, like the eternal switching scenario, the transition rates from the states $|\Omega^{-}_{2}\rangle$, $|\Omega^{-}_{3}\rangle$, and $|\Omega^{-}_{4}\rangle$ are equal. We have provided two sets of plots corresponding to fixed $|\omega|T=10$ and $|\omega|T=2$. One can observe that with decreasing $T$, the damping-like effect in the collective transition rate is larger. Top-right: We have plotted the collective transition probability rates as functions of the dimensionless probe energy gap $|\omega|d$ when the probes are stationed equidistant on a straight line. We have considered $d_{AB}=2\,d_{AC}$, and here the transition probability rates from the states $|\Omega^{-}_{2}\rangle$ and $|\Omega^{-}_{4}\rangle$ are the same. From these plots, one can also observe that as $T$ increases, we encounter fewer ripples in the collective transition probability rate, and all these rates reach a single fixed value for a smaller value of $d$ as compared to the situation of larger $T$. Bottom-left: The collective transition probability rates are plotted as functions of the dimensionless switching time $|\omega|T$ for $|\omega|d_{AB}=1=2|\omega|d_{AC}$ and $\theta=\pi$. Bottom-right: The collective transition probability rates are plotted as functions of the dimensionless switching time $|\omega|T$ for $|\omega|d_{AB} =10= 2|\omega|d_{AC}$ and $\theta=\pi$. From the bottom-left and bottom-right plots, we observe that for very low switching time $(|\omega|T<1)$, all the transition probability rates increase with decreasing $T$. We also observe that for $|\omega|T\ge 2$, this feature is not the same for all the probability rates. All of these above plots in this figure are obtained for Gaussian switching of $\kappa(\tau_{j})=e^{-\tau_{j}^2/T^2}$.
• Figure 5: On the left, we have plotted the collective transition probability rates $\mathcal{R}_{|\Omega^{-}_{n}\rangle\to |g\rangle}(\omega)$ as functions of the dimensionless probe energy gap $\omega\,d_{BC}$ for $\textsc{v}=0.5$ and $\phi_{BC}=\pi/2$. At the same time, on the right, the collective transition probability rates are plotted as functions of the dimensionless probe energy gap $\omega\,d_{BC}$ for $\textsc{v}=0.5$ and $\phi_{BC}=0$. It is to be noted that $\phi_{BC}$ denotes the angle between the direction of velocity $\textbf{v}$ of the probes ($B$ and $C$) and ${\bf d}_{BC}$. In the above plots, the red curves correspond to the transition probability rates $\mathcal{R}_{|\Omega^{-}_{1}\rangle\to |g\rangle}(\omega)$ and $\mathcal{R}_{|\Omega^{-}_{4}\rangle\to |g\rangle}(\omega)$, which happens to be equal in this scenario. While the blue curves correspond to the transition probability rates $\mathcal{R}_{|\Omega^{-}_{2}\rangle\to |g\rangle}(\omega)$ and $\mathcal{R}_{|\Omega^{-}_{3}\rangle\to |g\rangle}(\omega)$, which are the same. From both the above plots, one can notice that the different maxima or minima corresponding to the red curves appear when the blue curves have minima or maxima, respectively. However, there is a distinguishing difference between the two. For instance, for $\phi_{BC}=\pi/2$, the red and blue curves do not overlap each other at their maxima or minima. In contrast, for $\phi_{BC}=0$, the red and blue curves overlap at some of their maxima or minima.