When Bob orbits Alice: entanglement harvesting in circular motion

1. Problem Statement

The paper investigates the phenomenon of entanglement harvesting in a relativistic setting involving two qubits (two-level systems) coupled to a massless scalar field in the Minkowski vacuum.

• The Setup: "Alice" is an inertial observer at rest, while "Bob" undergoes uniform circular motion with radius $R$ and angular velocity $\Omega$.
• The Core Question: How does the trajectory-dependent nature of circular motion (specifically the non-inertial rotation) affect the ability of the two qubits to extract entanglement from the vacuum fluctuations?
• Context: While the Unruh effect (linear acceleration) is well-studied, the response of detectors in rotating frames presents theoretical subtleties. Unlike linear acceleration, which yields a thermal spectrum, rotating frames do not possess a global event horizon, and the relationship between the rotating vacuum and the Minkowski vacuum is unitarily equivalent (no particle production in the global sense). However, local detectors can still exhibit non-zero excitation probabilities. The authors aim to quantify how this specific kinematic configuration influences quantum correlations (entanglement) and total correlations (mutual information).

2. Methodology

The authors employ the Unruh-DeWitt detector model within the framework of Relativistic Quantum Information (RQI).

• Field Quantization:

  • They perform canonical quantization of a massless scalar field in a coordinate system adapted to a uniformly rotating frame.
  • They define the positive frequency Wightman functions ($W(x, x')$) for three cases: Alice-Alice (inertial), Bob-Bob (rotating), and Alice-Bob (crossed).
  • Crucially, they derive the Wightman function for Bob's circular trajectory, which depends on the Lorentz factor $\gamma = (1 - \Omega^2 R^2)^{-1/2}$.

• Interaction Model:

  • Two qubits interact with the field via a time-dependent switching function $\chi(\tau)$, modeled as a Gaussian: $\chi(\tau) = e^{-\tau^2/\sigma^2}$, where $\sigma$ characterizes the interaction duration.
  • The interaction Hamiltonian is $H_I(\tau) = \lambda \chi(\tau) \sigma_x(\tau) \otimes \phi(x(\tau))$, where $\lambda$ is a small coupling constant.

• Dynamics and Approximation:

  • The system evolves according to the von Neumann equation.
  • Using perturbation theory up to second order in the coupling constant ($\lambda^2$), they compute the reduced density matrix of the two-qubit system ($\rho_{AB}$) by tracing out the field degrees of freedom.
  • The density matrix elements are expressed in terms of transition probabilities ($L_{AA}, L_{BB}$) and non-local correlation terms ($L_{AB}, M$).

• Quantifiers:

  • Concurrence ($C_{AB}$): Used to measure entanglement. $C_{AB} = 2 \max(0, |M| - \sqrt{L_{AA}L_{BB}})$.
  • Mutual Information ($I_{AB}$): Used to measure total (classical + quantum) correlations.

3. Key Contributions

1. Explicit Analytical Derivations: The paper provides closed-form expressions (in terms of error functions and integrals) for the transition probabilities and non-local terms specifically for a circular trajectory, extending previous work that often focused on linear acceleration or static configurations.
2. Resolution of Rotating Frame Dynamics: The authors rigorously apply the Unruh-DeWitt model to a rotating frame, clarifying the interplay between the unitary equivalence of vacua (no global particles) and the local excitation of a detector (non-zero response function).
3. Parameter Space Mapping: They systematically map the dependence of entanglement harvesting on the dimensionless parameters:
   • $R/\sigma$: Ratio of orbital radius to interaction duration.
   • $E\sigma$: Ratio of qubit energy gap to interaction duration.
   • $\Omega\sigma$: Ratio of angular velocity to interaction duration.

4. Key Results

The numerical and analytical results reveal several distinct behaviors:

• Transition Probabilities:

  • The transition probability for the rotating qubit ($L_{BB}$) is enhanced compared to the inertial case ($L_{AA}$).
  • The difference $(L_{BB} - L_{AA})$ increases in both magnitude and width as the radius $R$ increases (for fixed $\Omega$), indicating that rotational motion significantly amplifies vacuum fluctuations felt by the detector.

• Entanglement (Concurrence):

  • Dependence on Energy Gap: Concurrence decreases as the energy gap $E\sigma$ increases.
  • Dependence on Radius: For a fixed angular velocity, concurrence remains relatively robust (approximately constant) over a wide range of radii $R/\sigma$.
  • Abrupt Vanishing: Entanglement vanishes abruptly when the local noise term $\sqrt{L_{AA}L_{BB}}$ exceeds the non-local correlation term $|M|$. This threshold depends on $E\sigma$ and $\Omega\sigma$.
  • Angular Velocity Effect: Increasing $\Omega$ shifts the regime where entanglement persists. Higher angular velocities allow entanglement to survive for larger energy gaps before the rapid degradation sets in, but eventually, the "flat" behavior of the concurrence curve changes.

• Total Correlations (Mutual Information):

  • Mutual information exhibits greater robustness than concurrence. It remains nearly constant as $R/\sigma$ increases, even in regimes where concurrence has already vanished (indicating the presence of classical correlations).
  • Relativistic Threshold: Total correlations remain stable until the system approaches the relativistic limit where $R\Omega \approx 1$ (tangential velocity approaches $c$). Near this threshold, mutual information drops sharply, signaling that extreme rotational motion suppresses the extraction of any correlations from the field.

5. Significance and Implications

• Non-Thermal Signatures: The study highlights that circular motion does not simply mimic a thermal bath (as linear acceleration does via the Unruh effect). Instead, it introduces unique kinematic signatures where the response depends sensitively on the radius and switching duration, distinct from the idealized thermal response of linear acceleration.
• Robustness of Quantum Resources: The findings suggest that entanglement harvesting is surprisingly robust against changes in orbital radius in non-relativistic regimes. This is relevant for potential applications in satellite-based quantum communications where satellites are in orbit (circular motion).
• Relativistic Limits: The sharp degradation of correlations near the speed of light limit ($R\Omega \to 1$) underscores the fundamental constraints imposed by special relativity on quantum information processing in non-inertial frames.
• Theoretical Clarification: By calculating the concurrence explicitly, the authors resolve the apparent paradox between the unitary equivalence of rotating and inertial vacua (which suggests no particles) and the non-zero detector response (which suggests excitations), showing that local entanglement extraction is a viable probe of these vacuum fluctuations.

In summary, the paper demonstrates that while circular motion enhances local excitation probabilities, it allows for the harvesting of significant quantum and classical correlations, provided the motion remains within non-relativistic limits. The transition from quantum to classical correlations is governed by the delicate balance between local noise and non-local field correlations, which is highly sensitive to the angular velocity and energy gap of the qubits.