A complete analysis of spin coherence in the full-loop Stern Gerlach interferometer using non-squeezed and squeezed coherent states of the Quantum harmonic oscillator

Abstract

A Stern-Gerlach interferometer uses a magnetic field gradient to split particle wave functions into spatially separated wave-packets according to their respective spin projections. Over the years, quite a few proposals have been put forward by various groups to exploit this effect in order to create stable macroscopic spatial superpositions between micron-sized neutral test masses over appreciably long time scales. One such proposal put forward by Bose et al. and co-workers in 2017 uses this idea to show that two masses cannot be gravitationally entangled if not for the presence of a quantum coherent mediator (i.e., through spin correlation measurements between two quantum spins, each embedded in a test mass, they seek to demonstrate that gravity can act as a quantum coherent mediator [see [1]]. This primarily involves cooling the test mass to the ground state of a harmonic trap, thereby releasing it in a Stern-Gerlach interferometer. A key aspect of this approach involves the measure of the visibility of the SG-interferometer, a quantity that provides an estimate of the degree of spin coherence that is conserved over the total interferometric time after the wavepackets are combined in both, position and momentum space. A successful implementation of this idea however requires the knowledge of several experimental parameters, some of which include the temperature to which the test mass must be cooled initially, the admissible experimental errors in the measure of the phase-space observables (i.e., spatial and momentum splitting between the wave-packets with respect to the initial position and momentum uncertainties of the test mass 1 in the ground state of the harmonic trap) and the total time-of-flight of the wave-packets in the interferometer. To this end, we present a rigorous mathematical analysis for the visibility in a general SG interferometer for non-squeezed and squeezed thermal coherent states of the Quantum harmonic oscillator. Additionally, by considering suitable experimental errors in the measure of the phase-space variables and subject to the desired accuracy in the measure of the visibility, we derive constraints on the temperature of the initially prepared wave-packet of the test mass for both, the non-squeezed and squeezed coherent states. We show that for wave-packet split sizes of the order of microns, masses of the order of 10 −14 -10 −15 kg can be used to realize such a proposal in practice for time intervals as high as 0.5 seconds. Our results show that for the squeezed case, the temperatures required can be scaled up by several orders of magnitude (as opposed to the non-squeezed case) if one considers a squeezing in the momentum space of the initially prepared wave-packet.