Experimental Demonstration of the Timelike Unruh Effect with a Trapped-Ion System

Abstract

The Unruh effect predicts that an accelerated observer perceives the Minkowski vacuum as a thermal bath, but its direct observation requires extreme accelerations beyond current experimental reach. Foundational theory [Olson & Ralph, Phys. Rev. Lett. 106, 110404 (2011)] shows that an equivalent thermal response, known as the timelike Unruh effect, can occur for detectors following specific timelike trajectories without acceleration, enabling laboratory tests with stationary yet time-dependent detectors. Here, we report a proof-of-principle demonstration of the timelike Unruh effect in a quantum system of trapped ion, where a two-level spin serves as the detector and is temporally coupled to the ambient field encoded in the ion's vibrational motion. Specifically, we study both excitation and emission dynamics of the detector moving along a spacetime trajectory in the future/past light cone, and demonstrate the thermal response of the detector to the Minkowski vacuum that resembles the Unruh effect. This work establishes a controllable tabletop platform for exploring relativistic quantum physics under accessible laboratory conditions.

Figures (6)

• Figure 1: (a) A spacetime diagram separated into four quadrants: the left and right Rindler wedges (L and R), and the future and past light cones (F and P). The red arrow represents the spacetime trajectory of the detector. (b) Sketch of experimental model. It represents the model at the beginning $t=\tau=0$. Spin states are labeled $\{\ket{g},\ket{e}\}$, while photon numbers are labeled $\ket{n=0, 1, …}$ and plotted vertically. The excitation and emission process corresponds to blue and red arrows respectively. The spin frequency $\omega_q$ is bigger than the field frequency $\omega_p$ and the coupling strength is $g_0$. (c) A trapped $^{40}{\rm Ca}^{+}$ ion in the ground state of vibrational level applied with two customized red and blue sideband lasers, could be equalized as a two-level detector, with internal transition frequency $\omega_q$, coupling with a single mode photon field of frequency $\omega_p$ in the vacuum.
• Figure 2: Evolution of transition probability during excitation (top) and emission (bottom) processes. Here, $\alpha = 10~\text{MHz}, \omega_p/(2\pi)=25~\text{kHz}$ and three time periods are set as 0-80, 200-280 and 400-480 µ s. The experimental points at $t>0.8~\text{ms}$ are detected with $10^4$ repetitions while others are detected with $10^3$ repetitions.
• Figure 3: Effective transition probability $P_\text{eff}$ with various $\beta$. Experimental data for the excitation (emission) process are shown as green circles (blue squares), with corresponding error bars indicating one standard deviation. Theoretical predictions from Eq. \ref{['Eq:P_eff']} are plotted as orange solid lines, as separated by the horizontal dashed line $P_\text{eff}=1$. The emission part is implemented with $\omega_p/(2\pi)=50~\text{kHz}$ and $25~\text{kHz}$, labeled with zones I,II respectively. The simulation results considering effects from experimental noise are displayed by yellow bands. The inset shows the transition probability ratio $\eta$ for various values of $\beta$.
• Figure S1: Comparison between two simulation results with parameter $\omega_p/(2\pi)=25~\text{kHz}$ (red) and 50 kHz (blue). Other parameters are shown in Table \ref{['tab:experimental_parameters']}. The solid lines denote theoretical prediction \ref{['Eq:P_final']} while the dots represent simulation results. The vertical dashed line corresponds to zone I/II division in Fig. 3 (main text).
• Figure S2: Comparison of theoretical predictions and calculated results with $C=0$ and $C=1$. The parameter settings of points are the same as those in Fig. 2 (main text).