Quantum Simulation of Oscillatory Unruh Effect with Superposed Trajectories

TL;DR

This work addresses the challenge of observing the Unruh effect by using a trapped-ion quantum simulator to emulate an oscillatory detector interacting with a cavity field. The authors implement a time-dependent detector-field coupling through programmable laser pulses and realize both single and superposed detector trajectories, analyzing the resulting JC/anti-JC–type dynamics under Floquet driving. They observe coordinated excitation of the detector and field for oscillatory motion and demonstrate quantum interference between coherently superposed trajectories, revealing nonclassical relativistic effects with potential implications for quantum gravity. The study provides a scalable platform for exploring relativistic quantum field theory phenomena and suggests pathways toward direct experimental observations of the Unruh effect in quantum simulators.

Abstract

The Unruh effect predicts an astonishing phenomenon that an accelerated detector would detect counts despite being in a quantum field vacuum in the rest frame. Since the required detector acceleration for its direct observation is prohibitively large, recent analog studies on quantum simulation platforms help to reveal various properties of the Unruh effect and explore the not-yet-understood physics of quantum gravity. To further reveal the quantum aspect of the Unruh effect, analogous experimental exploration of the correlation between the detector and the field, and the consequences for coherent quantum trajectories of the detector without classical counterparts, are essential steps but are currently missing. Here, we utilize a laser-controlled trapped ion to experimentally simulate an oscillating detector coupled with a cavity field. We observe joint excitation of both the detector and the field in the detector's frame, coincide with the coordinated dynamics predicted by the Unruh effect. Particularly, we simulate the detector moving in single and superposed quantum trajectories, where the latter case shows coherent interference of excitation. Our demonstration reveals properties of quantum coherent superposition of accelerating trajectories associated with quantum gravity theories that have no classical counterparts, and may offer a new avenue to investigate phenomena in quantum field theory and quantum gravity. We also show how a generalization of the method and results in this work may be beneficial for direct observation of the Unruh effect.

Figures (4)

• Figure 1: Illustration of the model and the experimental implementation. (a) The detector oscillating in a cavity of length $L$. The energy levels of the detector and photon of cavity mode are $\omega_q$ and $\omega_p$. The coupling strength $g(x)$ between the detector and the photon is position-dependent due to the stationary wave form in the cavity. (b) The quantum superposition of two trajectories $x_0(t)$ and $x_1(t)$. Trajectory $x_0(t)$ ($x_1(t)$) is labeled with the solid red line (dashed blue line). An additional control qubit $\ket{i_c}$ is introduced, with $i=\{0,1\}$, designating individual paths for the detector. (c) Experimental scheme for quantum simulation of the detector-photon model with superposed trajectories. Four combined states of detector and control qubit are mapped to four internal levels of a $^{40}\rm{Ca}^+$ ion. Transitions between $\ket{g}$ and $\ket{e}$ could be individually driven with $729\text{ nm}$ laser beams in different frequencies. For single trajectory simulation, only the energy levels and transition with respect to $\ket{0_c}$ are utilized.
• Figure 2: Experimental simulation of oscillatory Unruh effect with single trajectory. The evolution of detector excitation $\langle\sigma_z\rangle$ and photon creation $\langle N \rangle$ for different oscillation amplitudes. For $u=0$ ($u=1.5$), data points with error bars represent the experimental results, with dark green dots (light green triangles), yellow squares (red pentagons) corresponding to the measurements of $\langle{\sigma_z}\rangle$ and $\langle{N}\rangle$, respectively. Solid lines represent theoretical results including experimental imperfections (See in Supplement Material). The case with $u = 0$ represents a static detector which observes no excitation and a vacuum photon field. The case with $u=1.5$ represents an oscillating detector. In the detector's frame, the photon field and the detector experience periodic excitation simultaneously, showing coordinated dynamics.
• Figure 3: Experimental simulation of superposed trajectories. (a) and (b) show the evolution of detector excitation $\langle\sigma_z\rangle$. (c) and (d) show the evolution of photon number $\langle N \rangle$. Solid (Dot-dashed) lines represent numerical simulation for a theoretical model with (without) experimental imperfections (See in Supplement Material). Data points with error bars represent the experimental results. Dark blue circles, green blue triangles, orange-yellow squares, and red-orange pentagons represent projectors $\hat{P}_i=\ketbra{i_c}$ with $i=\{0,1,+,-\}$, respectively. For the measurements regarding $\ketbra{0_c}$ and $\ketbra{1_c}$, both the detector and mean photon field excitation experience sinusoidal oscillation with different rates, due to distinguished trajectories of the detector. The difference between the experimental observation of $\langle\ketbra{+_c}\otimes\hat{O}\rangle$ and $\langle\ketbra{-_c}\otimes\hat{O}\rangle$ indicates a coherent superposition for the trajectories rather than a classical mixture, where such difference can be observed with both $\hat{O}=\{\sigma_z,N\}$.
• Figure S1: (a) Experiment data and fitting curves of scanning red and blue sideband pulse length $t_{\text{SB}}$ after evolution. (b) Fitted phonon distribution. Data points with error bars are experimental results. Solid curves represent fit results. The dark and cyan color stands for red sideband and blue sideband. Both main oscillation parts of the red/blue sideband curve could be found. Combining estimation of Lamb-Dicke parameter $\eta=0.065$, red/blue sideband oscillation part could be distinguished as $\ket{e,n=1}\to\ket{g,n=2}$ and $\ket{g,n=0}\to\ket{e,n=1}$ transition. The red sideband oscillates faster than the blue sideband. The fitting result is $\langle N \rangle=0.235\pm0.045$. The phonon mostly distributed in $\ket{g,0}$ and $\ket{e,1}$, which match theoretical expectation.