Decoherence by black holes via holography

Technical Summary: Decoherence by Black Holes via Holography

Problem Statement
The paper addresses the longstanding debate regarding whether black hole evaporation violates the principles of quantum mechanics, specifically focusing on the mechanism of information loss and decoherence. Recent work by Danielson, Satishchandran, and Wald (DSW) proposed that quantum superpositions maintained outside a Killing horizon decohere at a constant rate due to entanglement with soft-mode radiation of long-range fields (electromagnetic or gravitational). While this effect has been analyzed using semiclassical quantum field theory in curved spacetime and interpreted locally as thermal decoherence, a fully quantum-mechanical description of how this manifests in black hole physics remains an open challenge. The authors aim to investigate the DSW effect and the associated decoherence patterns using the holographic principle, which offers a non-perturbative, fully quantum description of black holes.

Methodology
The authors employ the holographic duality (AdS/CFT and Lifshitz holography) to model the decoherence of a quantum system (a moving mirror or particle) interacting with a strongly coupled quantum critical field.

1. Holographic Setup: The boundary quantum system is dual to a probe brane (or string for $n=0$) propagating in an asymptotically Lifshitz geometry. The environment is modeled by two distinct bulk geometries:
   • Zero Temperature: Pure Lifshitz spacetime.
   • Finite Temperature: Lifshitz black hole geometry.
2. Influence Functional Formalism: The authors utilize the Feynman-Vernon influence functional approach within the open quantum systems framework. The decoherence of the boundary system is determined by the on-shell action of the probe string/brane in the bulk.
   • The real part of the influence functional (related to the Hadamard function, $G_H$) encodes decoherence.
   • The imaginary part (related to the retarded Green's function, $G_R$) encodes phase shifts.
   • These functions are derived holographically by solving the equations of motion for small fluctuations of the probe in the bulk and imposing appropriate boundary conditions (infalling at the horizon, fixed at the boundary).
3. Interference Experiment: The study considers an interference experiment where a particle is in a superposition of two spatially separated trajectories ($C_1$ and $C_2$) for a duration $\tau_0$ before recombination. The decoherence functional $W$ is calculated to determine the suppression of the interference term.
4. Extensions: The analysis includes:
   • Stringy Corrections: An order-of-magnitude estimate of $\alpha'$ corrections (finite 't Hooft coupling) to the zero-temperature decoherence.
   • Unruh Effect: A holographic model of an entangled EPR pair (two endpoints of a string) where one or both endpoints accelerate, comparing static and accelerating configurations to probe the role of causality.

Key Results

• Finite Temperature (Lifshitz Black Hole):

  • The interaction with the thermal environment dual to a Lifshitz black hole results in ohmic dissipation.
  • The decoherence functional $W$ scales linearly with the interaction time $\tau_0$, leading to a constant decoherence rate.
  • The decay rate scales with the cube of the black hole temperature ($T^3$) in the relativistic limit ($z=1, n=0$), consistent with previous DSW predictions. This confirms the local thermal interpretation of the DSW effect within a strongly coupled holographic framework.

• Zero Temperature (Pure Lifshitz Spacetime):

  • The system exhibits supraohmic dissipation.
  • In the large-time limit ($\tau_0 \to \infty$), the decoherence functional vanishes, implying that coherence is recovered for adiabatic motion. This aligns with flat-space results where supraohmic environments do not cause persistent decoherence.
  • Dependence on Dynamical Exponent $z$: As the dynamical critical exponent $z$ increases, the recovery of coherence slows down.
  • Extremal Limit ($z \to \infty$): As $z$ approaches infinity, the decoherence functional exhibits a logarithmic decay ($W \propto \ln \tau_0$). This behavior mimics the decoherence patterns associated with extremal black holes, suggesting a continuous spectrum of decoherence behaviors connecting flat space ($z=1$) to extremal black holes ($z \to \infty$).

• Stringy Corrections:

  • The authors estimate $\alpha'$ corrections to the zero-temperature decoherence. They find that for $z > 2$, these corrections can become significant at late times, potentially enhancing or suppressing coherence recovery depending on the sign of the correction coefficient.

• Unruh Effect and EPR Pairs:

  • Accelerating Strings: A holographic model of an accelerating string (dual to an accelerating charged particle) yields a constant decoherence rate proportional to the Unruh temperature, analogous to the finite-temperature black hole case.
  • Static vs. Accelerating EPR Pairs:
    • For a static entangled pair (string with two fixed endpoints), the decoherence of one particle is suppressed at late times. This is attributed to the causal connection between the two entangled particles; once the time scale exceeds the separation distance ($t \gg 1/r_0$), the entangled partner influences the system, suppressing the environmental decoherence.
    • For an accelerating pair (Unruh effect), the two endpoints are causally separated by bifurcate horizons. Consequently, the decoherence of one particle is not suppressed by the other, resulting in a constant decoherence rate similar to the single-particle case.

Significance and Claims
The paper claims that Lifshitz holography provides a unified framework to study decoherence across different spacetime geometries, from flat space to black holes.

1. Validation of DSW: The results generalize the DSW effect to strongly coupled field theories, confirming that black holes (modeled as thermal environments in the dual) induce a constant decoherence rate.
2. Extremal Black Holes: The study suggests that extremal black holes represent a limiting case where coherence recovery is maximally slow (logarithmic decay), bridging the gap between standard flat-space decoherence and black hole physics.
3. Role of Causality: The investigation of holographic EPR pairs highlights that causality is a crucial factor in determining whether entanglement suppresses decoherence. Decoherence is suppressed only when the entangled partners are in causal contact.
4. Quantum Gravity Insights: By utilizing the large-$N$ limit, the authors demonstrate consistency between the holographic description and semiclassical black hole physics. They note that extending this to finite-$N$ corrections (fully quantum black holes) remains an open question, particularly regarding how decoherence terminates when the system's degrees of freedom exceed those of the black hole.

The authors maintain a modest tone, acknowledging that their stringy correction analysis is an order-of-magnitude estimate and that the specific model of the dual quantum field theory for finite-$N$ effects requires further development.