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Modern Physics Letters A

ON THE ENTANGLED STATE REPRESENTATION, EXCITATION REPRESENTATION AND EFFECTIVE NUMBER-PHASE STATE REPRESENTATIONS IN RINDLER SPACE

Hong-Yi Fan, Hai-Yan He

Published Oct 21, 2004• 2 citations
Publisher

Abstract

We analyze the Unruh effect from the point of view of quantum entanglement. We introduce the entangled state representation in Rindler space and show that the Minkowski vacuum state is an entangled state in Rindler space. The corresponding squeezing operator, which is related to the acceleration of the detector, is obtained naturally. The excitation representation and number state–phase state representations are also introduced in Rindler space. The number-phase commutative relation is established.

DOI
10.1142/s0217732304015014

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We analyze the Unruh effect from the point of view of quantum entanglement. We introduce the entangled state representation in Rindler space and show that the Minkowski vacuum state is an entangled state in Rindler space. The corresponding squeezing operator, which is related to the acceleration of the detector, is obtained naturally. The excitation representation and number state–phase state representations are also introduced in Rindler space. The number-phase commutative relation is established.
While the Unruh effect has traditionally been studied under the assumption of uniform acceleration, a simplification motivated by experimental considerations, it is not necessarily true for all non-inertial motions. We propose a novel approach for the indirect detection of the Unruh effect without relying on the former restriction. Previous studies have shown that probing the decoherence of an Unruh-DeWitt detector can significantly reduce the acceleration required for observing the effect by several orders of magnitude compared to earlier proposals. Building on this idea, we develop a theoretical framework describing a non-inertial observer equipped with a detector undergoing non-uniform, time-dependent acceleration. We show that, in a non-uniformly accelerated Rindler spacetime, the particle distribution perceived in the Minkowski vacuum acquires a time-dependent modification of the standard Unruh spectrum. Furthermore, we demonstrate that the inclusion of quantum entanglement leads to a deformation of the Minkowski vacuum into squeezed states.
We demonstrate that the future and left Rindler wedges of Minkowski spacetime are entangled, leading to the Unruh effect. Similarly, the past and right Rindler wedges are also entangled. We propose a protocol to extract this entanglement using two two-state detectors located in the past and right Rindler wedges. By scaling the detector transition frequencies inversely with Minkowski time, entanglement from the quantum field is transferred to the detectors, suggesting they may support quantum teleportation via the vacuum. Our protocol can be implemented with current quantum systems, such as flux-tunable transmon qubits. This research provides new insights into the entanglement properties of spacetime and hints at practical applications for secure quantum information transfer using the vacuum state of a quantum field.
Emission of photons by atoms can occur into modes which extend into a region causally disconnected with the emitter.For example, a uniformly accelerated ground-state atom emits a photon into the Unruh-Minkowski mode which is exponentially larger in the causally disconnected region.This makes an impression that photon emission is acausal.Here we show that conventional quantum optical analysis yields that a detector atom will not detect the emitted photon in the region noncausally connected with the emitter.However, joint excitation probability of atoms in the causally disconnected regions can be correlated due to entanglement of Minkowski vacuum and be much larger than the product of independent excitation probabilities.Moreover, atoms uniformly accelerated in the same Rindler wedge cannot become simultaneously excited without changing the state of the field, that is, the Unruh-Minkowski photon emitted by one atom cannot be absorbed by the other atom.We discuss examples demonstrating interesting features of Minkowski vacuum entanglement.
We studied the change of the nonlocal correlation of the entanglement in Rindler space-time by showing that the Unruh effect can be interpreted as a noisy quantum channel having a complete positive and trace preserving map with an ``operator sum representation.'' It is shown that the entanglement fidelity is obtained in analytic form from the operator sum representation, which agrees well numerically with the entanglement monotone and the entanglement measure obtained previously. Nonzero entropy exchange between the system QQQ and region II of the Rindler wedge indicates the nonlocal correlation between causally disconnected regions. We have also shown the subadditivity of entropies numerically.
Free quantum field theory in flat space-time is often believed to be well established, holding no surprises. We hope, by the end of this paper, to have demonstrated that surprises still exist. For example, we will show that a uniformly accelerated atom in Minkowski space-time emits entangled photon pairs in a squeezed state which mimics entanglement of the Minkowski vacuum in that the entanglement is between Minkowski modes which are dominantly in opposite causal wedges of the space-time. Similar emission of photon pairs occurs if an atom is held above the black hole event horizon. Namely, a ground-state atom becomes excited by emitting a ``negative'' -energy photon under the horizon and then spontaneously decays back to the ground state by emitting a positive-energy photon outside the horizon, which propagates away from the black hole.