Superposed circular motion Unruh effect in (3+1) dimensions

Technical Summary: Superposed Circular Motion Unruh Effect in (3+1) Dimensions

Problem Statement
The Unruh-DeWitt (UDW) detector model is a standard tool for probing relativistic quantum field theories, famously demonstrating that a uniformly accelerated detector coupled to a scalar vacuum responds thermally (the Unruh effect). While the response of detectors undergoing uniform circular motion has been established as approximately thermal, recent interest has focused on extending this framework to detectors undergoing superpositions of classical trajectories. This paper investigates the response of a UDW detector interacting with a massless scalar quantum field in (3+1) dimensions when the detector is placed in a coherent superposition of two distinct circular trajectories. The primary goal is to quantify how quantum interference between these trajectories modifies the transition probability and effective temperature compared to incoherent (classical) mixtures or single trajectories.

Methodology
The authors employ a quantum control model where the detector's trajectory is dictated by a control degree of freedom. The total system is initialized in a state where the field is in the vacuum $|0\rangle$, the detector is in its ground state $|g\rangle$, and the control is in an equal, coherent superposition of two trajectories, $|\psi\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |2\rangle)$.

The interaction is modeled via a Hamiltonian in the interaction picture involving a Gaussian switching function $\eta(\tau) = \exp(-\tau^2/2\sigma^2)$ and a monopole moment operator. The transition probability $P_E$ is calculated to leading order in the coupling constant $\lambda$. Crucially, the calculation includes off-diagonal terms ($i \neq j$) in the density operator summation, which represent non-local contributions from the Wightman function $W_{ji}(\tau_i, \tau'_j)$ pulled back along the distinct trajectories. These terms capture self-interference effects absent in classical mixtures.

The study analyzes four distinct geometric scenarios for the superposed circular trajectories:

1. Vertically displaced: Two concentric circles stacked vertically (separated by distance $H$).
2. Horizontally displaced: Two circles separated within the plane of rotation (separated by distance $L$).
3. Static and orbiting: A superposition of a static point and a circular trajectory orbiting that point.
4. Concentric: Two coplanar circles with different radii but identical angular velocities.

Numerical results are generated for the transition probability and an effective temperature $T_{\text{eff}}$, defined via the ratio of excitation to de-excitation probabilities ($e^{\Omega/T_{\text{eff}}} = P_E(-\Omega)/P_E(\Omega)$), particularly in regimes where the switching function is broad relative to the acceleration timescale.

Key Contributions and Results

• Interference in Displaced Trajectories (Cases a & b): For vertically and horizontally displaced identical circles, the superposition introduces oscillatory behavior in the transition probability as a function of tangential velocity. These oscillations are most prominent near the peak response of the mixed-state case. The interference effects are highly sensitive to the displacement magnitude relative to the interaction timescale; for large separations, the system reverts to the behavior of a single trajectory. In the ultrarelativistic limit, the off-diagonal Wightman function leads to a response function suppressed by a sinc function dependent on the separation.
• Damping in Asymmetric Trajectories (Cases c & d): When the superposed trajectories are physically distinct (e.g., a static point vs. an orbit, or concentric circles of different radii), the interference does not produce oscillations. Instead, the transition probability is significantly dampened compared to the incoherent mixture. Specifically, in the static-orbiting case, the effective temperature is significantly reduced, and its variation with the energy gap $\Omega$ is enhanced.
• Effective Temperature: The study finds that for displaced trajectories, the effective temperature exhibits oscillations similar to the transition probability. In the limit of high velocities, superposed systems generally exhibit higher temperatures than non-superposed cases, though they converge to the mixed-state temperature as $v \to 1$ if the trajectories have different physical parameters. For non-displaced centers (static-orbit and concentric), the temperatures converge to the mixed-state value at low velocities.
• Thermalization Analysis: In the regime of wide switching functions (approaching infinite interaction time), the authors analyze the effective temperature as a function of the energy gap. For vertically stacked identical circles, interference has only a minor effect on the temperature. Conversely, for the static-orbiting superposition, the interference causes a significant reduction in effective temperature and a stronger dependence on the energy gap.

Significance and Claims
The paper concludes that the superposition of trajectories fundamentally alters the detector's response, introducing interference effects that can either oscillate or dampen the transition probability and effective temperature depending on the geometric configuration. The authors claim these results provide potential experimental guidance for analogue implementations of trajectory superpositions.

Specifically, the paper suggests that these effects could be observable in ultracold atom systems, building on recent proposals to realize the circular motion Unruh effect using density fluctuations in Bose-Einstein condensates or height fluctuations in superfluid helium films. The authors note that while these analogue systems are (2+1)-dimensional (unlike the (3+1) model studied), the interference effects identified—particularly in the regime of narrow to moderate switching functions (pulsed lasers) or effectively constant switching (continuous-wave lasers) for the static-orbiting scenario—should be observable. The work serves as a theoretical step toward providing experimental guidance for testing trajectory superposition formalisms in quantum gravity and relativistic quantum information contexts.