Entanglement witness for combined atom interferometer-mechanical oscillator setup

We investigate how to entangle an atom interferometer and a macroscopic mechanical oscillator to create nonclassical states of the oscillator. This work is done in the context of an experimental-theoretical collaboration that aims to implement the proposal in the paper “Using an Atom Interferometer to Infer Gravitational Entanglement Generation,” written by Carney . []. Here, we propose an entanglement witness (EW), from whose violation the generation of entanglement can be determined. We do this for three scenarios: the noiseless case, when including thermal noise, and with atomic dephasing. Thermal noise can arise from two sources: the first being when the oscillator starts in a thermal state, and the second when a continuous thermal bath is in contact with the oscillator. We find that for the appropriate oscillator quality factor <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>Q</a:mi> </a:math> , violation always exists for any value of coupling <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>λ</b:mi> </b:math> and thermal occupancy <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mover accent="true"> <c:mi>n</c:mi> <c:mo>¯</c:mo> </c:mover> </c:math> . Cooling the oscillator to its ground state provides an <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>O</e:mi> <e:mo>(</e:mo> <e:mover accent="true"> <e:mi>n</e:mi> <e:mo>¯</e:mo> </e:mover> <e:mo>)</e:mo> </e:mrow> </e:math> improvement in the EW violation than when starting in a thermal state; in this case, we find that couplings of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>λ</g:mi> <g:mo>=</g:mo> <g:msup> <g:mn>10</g:mn> <g:mrow> <g:mo>−</g:mo> <g:mn>4.5</g:mn> </g:mrow> </g:msup> </g:mrow> </g:math> require at least <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mo>∼</h:mo> <h:msup> <h:mn>10</h:mn> <h:mrow> <h:mn>5.2</h:mn> </h:mrow> </h:msup> </h:mrow> </h:math> measurements for violation to be observed. For atomic dephasing, we find that EW violation can exist for an atom interferometer with a coherence time of 600 s, which is the parameter our experimental collaborators hope to reach. Finally, we consider how to use magnetic interactions to realize this idea within the context of the torsional pendulum that has been planned for our collaboration.