Coherent State Description of Gravitational Waves from Binary Black Holes

1. Problem Statement

The paper addresses the fundamental question of how classical gravitational waves (GWs) emitted by macroscopic astrophysical systems, specifically binary black holes (BBHs), should be described within the framework of quantum mechanics.

• Context: While General Relativity successfully describes GWs classically, quantum theory posits that all physical phenomena involve quantized fields (gravitons).
• The Gap: It is commonly assumed that classical GWs correspond to coherent states of gravitons (analogous to coherent states of photons in quantum optics). However, strong gravitational fields and nonlinear interactions in the Einstein-Hilbert action could theoretically generate squeezed states, which are non-classical.
• Objective: The authors aim to rigorously derive the quantum state of gravitons emitted by a BBH during the inspiral phase, determining whether the state is purely coherent or if higher-order nonlinear effects generate significant squeezing. They specifically calculate the squeezing parameter for the historic event GW150914 to assess the detectability of quantum gravity effects.

2. Methodology

The authors employ a perturbative quantum field theory approach in a Minkowski background, treating the binary black hole system as a classical source.

• Action and Expansion:

  • They start with the total action $S = S_{EH} + S_{1} + S_{2}$, comprising the Einstein-Hilbert action and the geodesic actions of two point masses ($m_1, m_2$).
  • The metric is expanded as $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $h_{\mu\nu}$ represents the graviton field in the transverse-traceless (TT) gauge.
  • The interaction Hamiltonian is derived by expanding the action to second order in the metric perturbation $h_{ij}$.
    • Linear Term ($\propto h_{ij}$): Generates a coherent state.
    • Quadratic Term ($\propto h_{ij}h_{kl}$): Generates a squeezed state.

• Quantization:

  • The graviton field is quantized using canonical commutation relations.
  • The time evolution operator $\hat{U}$ is constructed using the interaction Hamiltonian in the interaction picture.
  • Coherent State Derivation: By retaining only the linear interaction term, the authors derive the displacement operator $\hat{D}(\alpha)$ and the coherent parameter $\alpha(k)$.
  • Squeezed State Derivation: By including the quadratic interaction term, they derive the squeeze operator $\hat{S}(\beta)$ and the squeezing parameter $\beta(k, k')$.

• Assumptions:

  • Circular orbital motion for the binary system.
  • Negligible back-reaction on the orbit (valid for the inspiral phase).
  • Negligible environmental decoherence during propagation (justified by the extremely small graviton-matter cross-section and the long-wavelength limit relative to interstellar dust).

3. Key Contributions

1. Formal Derivation of the Quantum State: The paper provides a first-principles derivation showing that the leading-order description of BBH GWs is indeed a coherent state, bridging the gap between classical quadrupole radiation and quantum field theory.
2. Identification of Squeezing Mechanism: The authors explicitly demonstrate that the nonlinear self-interaction of gravity (the $h^2$ term in the interaction Hamiltonian) generates a squeezed state component, moving beyond the standard coherent state approximation.
3. Quantitative Estimation for GW150914: They calculate the specific magnitude of the squeezing parameter for the first detected GW event, providing a concrete benchmark for future observational tests.
4. Resolution of Amplitude Discrepancy: The paper addresses a factor-of-two discrepancy between the coherent state expectation value and the standard quadrupole formula, attributing it to the inclusion of both advanced (ingoing) and retarded (outgoing) solutions in the coherent state construction, whereas classical theory imposes retarded boundary conditions.

4. Results

• Coherent State Amplitude:

  • The expectation value of the metric perturbation $\langle \alpha | h_{ij} | \alpha \rangle$ reproduces the classical gravitational wave signal.
  • For GW150914 (masses $36 M_\odot$ and $29 M_\odot$, reduced mass $\mu \sim 16 M_\odot$, distance 410 Mpc), the calculated amplitude matches the observed strain of $\sim 10^{-21}$.

• Squeezing Parameter ($\zeta$):

  • The squeezing parameter is derived as a function of the reduced mass $\mu$, orbital velocity $a\Omega$, and frequency $f$.
  • The formula is given by:
    $$\zeta \simeq \frac{1}{8\pi M_p^2} \mu (a\Omega)^4 f$$
  • Numerical Estimate: For GW150914, the squeezing parameter is estimated to be:
    $$\zeta \sim 2 \times 10^{-4}$$
  • Scaling: The result indicates that while the Planck mass suppression ($1/M_p^2$) is immense, the macroscopic nature of the source ($\mu \sim 10^{34}$ g) compensates significantly, resulting in a non-zero but small squeezing effect. The parameter depends weakly on the black hole masses because $\mu$ and the frequency $f$ are inversely related for binary systems.

5. Significance and Implications

• Validation of the Coherent State Hypothesis: The results confirm that for current astrophysical sources, the quantum state of gravitational waves is well-approximated by a coherent state. The non-classical (squeezed) component is a higher-order effect ($\sim 10^{-4}$) and does not dominate the signal.
• Quantum Gravity Probes: While the squeezing is small, it is non-zero. This suggests that detecting the quantum nature of gravity (specifically gravitons) via squeezed state signatures is theoretically possible, though challenging.
• Future Detection: The authors propose that Hanbury Brown-Twiss (HBT) interferometry could potentially be used to probe these non-classical correlations in future high-precision experiments.
• Cosmological Connection: The paper notes that if primordial gravitational waves from inflation are present, they could enhance the squeezing, offering a potential indirect window into the early universe and the nature of inflationary quantum fluctuations.

In conclusion, the paper establishes a rigorous quantum mechanical framework for binary black hole inspirals, confirming that while they are predominantly coherent states, the subtle presence of squeezed states offers a potential pathway for future experimental verification of the quantum nature of gravity.