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 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mo>∼</a:mo><a:mi mathvariant="script">O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>G</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mo>∼</f:mo><f:mi mathvariant="script">O</f:mi><f:mo stretchy="false">(</f:mo><f:msup><f:mi>c</f:mi><f:mrow><f:mo>−</f:mo><f:mn>2</f:mn></f:mrow></f:msup><f:mo stretchy="false">)</f:mo></f:math>. 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 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo stretchy="false">|</k:mo><k:mn>1</k:mn><k:mo stretchy="false">⟩</k:mo></k:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mo stretchy="false">|</o:mo><o:mn>0</o:mn><o:mo stretchy="false">⟩</o:mo></o:math> to decay via graviton emission. In particular, the superposition of number states <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mfrac><s:mn>1</s:mn><s:msqrt><s:mn>2</s:mn></s:msqrt></s:mfrac><s:mrow><s:mo stretchy="false">[</s:mo><s:mo stretchy="false">|</s:mo><s:mn>0</s:mn><s:mo stretchy="false">⟩</s:mo><s:mo>+</s:mo><s:mo stretchy="false">|</s:mo><s:mn>1</s:mn><s:mo stretchy="false">⟩</s:mo><s:mo stretchy="false">]</s:mo></s:mrow></s:math> is a steady state and never decoheres. Published by the American Physical Society 2024