Loss of coherence and coherence protection from a graviton bath

We consider a quantum harmonic oscillator coupled with a graviton bath. We discuss the loss of coherence in the matter sector due to the matter-graviton vertex interaction, which leads to a loss of coherence provided that the matter-wave system is allowed to emit gravitons by the kinematics. Working in the quantum-field-theory framework, we obtain a master equation by tracing away the gravitational field at the leading order $\sim \mathcal{O}(G)$ and $\sim \mathcal{O}(c^{-2})$. We find that the decoherence rate is proportional to the cube of the harmonic trapping frequency and vanishes for a free particle, as expected for a system without a mass quadrupole. Furthermore, our quantum model of graviton emission recovers the known classical formula for gravitational radiation from a classical harmonic oscillator for coherent states with a large occupation number. In addition, we find that the quantum harmonic oscillator eventually settles in a steady state with a remnant coherence of the ground and first excited states. While classical emission of gravitational waves would make the harmonic system loose all of its energy, our quantum field theory model does not allow the number states $|1⟩$ and $|0⟩$ to decay via graviton emission. In particular, the superposition of number states $\frac{1}{\sqrt{2}}[|0⟩+|1⟩]$ is a steady state and never decoheres.

Figure 1

Graphical illustration of linear quadrupole radiation. (a) The linear quadrupole is generated by the relative motion of two masses, $m_1$ and $m_2$, which are coupled with coupling ${\omega }_{\mathrm{m}}$. The center-of-mass motion (mass $M$) is unperturbed by the emission of gravitational waves (which are of type “$+$”), while the relative motion (mass $\mu$) slowly decays as its energy is converted to gravitational waves and radiated away. The two-particle problem can be always mapped to the problem of a harmonically trapped reduced mass $\mu =\frac{m_1{m}_2}{m_1+m_2}$ with coupling $\sim \mu x^2$, where $x$ is relative distance between the two masses. (b) Case $m_1\sim m_2$. A physical realization consists of a trapped particle (i.e., mass $m_1$) inside a box containing the apparatus to generate a harmonic trap (i.e., mass $m_2$). The recoil of the system/box is on equal footing as $m_1\sim m_2$. (c) Case $m_1\ll m_2$. The problem reduces to the motion of the lighter mass $m_1$ in a harmonic trap with frequency ${\omega }_{\mathrm{m}}$. The recoil of the heavier mass $m_2$ is negligible with respect to the recoil of the lighter mass $m_1$ (which can be approximately identified with the reduced mass $\mu$).