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
Over the years, quite a few proposals have been put forward by various groups to exploit the Stern-Gerlach effect 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. 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 wave-packets are combined in both, position and momentum space. A successful implementation of this idea however requires the knowledge of several experimental parameters. 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, we derive constraints on the temperature of the initially prepared wave-packet of the test mass required 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 1.0e-14 - 1.0e-15 kg can be used to realize such a proposal 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.
Over the years, quite a few proposals have been put forward by various groups to exploit the Stern-Gerlach effect 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. 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 wave-packets are combined in both, position and momentum space. A successful implementation of this idea however requires the knowledge of several experimental parameters. 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, we derive constraints on the temperature of the initially prepared wave-packet of the test mass required 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 1.0e-14 - 1.0e-15 kg can be used to realize such a proposal 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.
Stern-Gerlach and/or matter-wave interferometry has garnered significant interest amongst members of the scientific community over the past few decades. Early theoretical results by Schwinger et al. demonstrate the fantastic precision capabilities required to realize a full-loop Stern-Gerlach interferometer, i.e., a Stern-Gerlach setup that houses the capability of recombining the split wave-packets in both, position and momentum space over a certain characteristic interferometric time. Over the years, several proposals have been put forward that seek to use Stern-Gerlach and/or matter-wave interferometry as a tool for a myriad of applications of general interest, some of which include tests for fundamental physics (viz., quantum wave-function collapse, stringent tests for the Einstein equivalence principle at the quantum scale, breaking the Standard Quantum Limit (SQL) barrier, and so forth), precision sensing, quantum metrology, gravitational wave detection and inertial navigation. In addition, a large volume of work in the existing literature has been dedicated to the possibility of using matter-wave interferometry for tests of quantum gravity. Inspired by the developments in this timely research field, this Perspective attempts to provide a general overview of the theory involved, the challenges that are yet to be addressed and a brief outlook on what lays ahead.
Recently, there has been a proposal to test the quantum nature of gravity in the laboratory by witnessing the growth of entanglement between two masses in spatial quantum superpositions. The required superpositions can be created via Stern-Gerlach interferometers, which couple an embedded spin qubit quantum state to the spatial dynamics of each mass. The masses would entangle only if gravity is quantum in nature. Here, we generalise the experiment to an arbitrary spin j or equivalently to an ensemble of uniformly coupled spins. We first exemplify how to create a generalized Stern-Gerlach interferometer, which splits the mass into 2j+1 trajectories. This shows that a controlled protocol can be formulated to encode the amplitudes of any spin state to a spatial superposition. Secondly, two masses in spatial superpositions of the above form are left to interact via gravity, and the entanglement is computed. Different families of initial spin states are varied to find the optimal spin state that maximizes the entanglement. We conclude that larger spins can offer a modest advantage in enhancing gravity-induced entanglement.
We analytically solve the entangling quantum dynamics of two interacting Stern-Gerlach Interferometers~(SGI). Each SGI exploits an operator-valued force applied by a qubit to create and recombine a non-Gaussian state of matter. The entangling phase between the two qubits generated by the leading-order gravitational interaction of the massive degrees of freedom is found to be mass-independent, both for unitary and open dynamics, irrespective of the temperature and squeezing of the initial states. Further, we show that the solution of the four interferometric paths reveals that the mere presence of the interaction does not allow for a perfect recombination of the centre of mass. This second-order effect, alongside higher-order interaction terms, can be used to bound the mass from above and below, thus restricting the experiment's regime to mesoscopic masses. By solving the open dynamics which includes diffusion and dephasing with initial squeezed thermal states, the bounds are tightened by the inclusion of realistic experimental noise. We discuss diamagnetic levitated masses with embedded NV-centres as a specific physical implementation.
The Stern-Gerlach (SG) effect, discovered almost a century ago, has become a paradigm of quantum mechanics. Surprisingly there is little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Specifically, no high-visibility spatial interference pattern has been observed with such a scheme, and furthermore no full-loop SG interferometer has been realized with the scheme as envisioned decades ago. On the contrary, numerous theoretical studies explained why it is a near impossible endeavor. Here we demonstrate for the first time both a high-visibility spatial SG interference pattern and a full-loop SG interferometer, based on an accurate magnetic field, originating from an atom chip, that ensures coherent operation within strict constraints described by previous theoretical analyses. This also allows us to observe the gradual emergence of time-irreversibility as the splitting is increased. Finally, achieving this high level of control over magnetic gradients may facilitate technological applications such as large-momentum-transfer beam splitting for metrology with atom interferometry, ultra-sensitive probing of electron transport down to shot-noise and squeezed currents, as well as nuclear magnetic resonance and compact accelerators.
In this paper, we will show how finite-temperature corrections and spin-dependent/independent noise will affect the contrast in a matter-wave interferometer, especially with massive objects and large spatial superposition sizes. Typically, spin is embedded in a nanoparticle as a defect, which can be manipulated by the external magnetic field to create a macroscopic quantum superposition. These massive matter-wave interferometers are the cornerstone for many new fundamental advancements in physics; particularly, macroscopic quantum superposition can use entanglement features to, e.g., test physics beyond the Standard Model, test the equivalence principle, improve quantum sensors, and test the quantum nature of spacetime in a laboratory. We will consider a Stern-Gerlach-type apparatus to create macroscopic quantum superposition in a harmonic oscillator trap, and figure out the spin contrast loss due to linear spin-independent and spin-dependent noise in a single interferometer. We will show that spin contrast loss due to spin-independent noise does not depend on the initial thermal state of the matter wave function. However, spin contrast loss due to spin-dependent fluctuations is dependent on the initial thermal occupation of the quantum state. We will keep our discussion general as far as the noise parameters are concerned.
In this paper, we will show how finite-temperature corrections and spin-dependent/independent noise will affect the contrast in a matter-wave interferometer, especially with massive objects and large spatial superposition sizes. Typically, spin is embedded in a nanoparticle as a defect, which can be manipulated by the external magnetic field to create a macroscopic quantum superposition. These massive matter-wave interferometers are the cornerstone for many new fundamental advancements in physics; particularly, macroscopic quantum superposition can use entanglement features to, e.g., test physics beyond the Standard Model, test the equivalence principle, improve quantum sensors, and test the quantum nature of spacetime in a lab. We will consider a Stern-Gerlach type apparatus to create macroscopic quantum superposition in a harmonic oscillator trap, and figure out the spin contrast loss due to linear spin-independent and spin-dependent noise in a single interferometer. We will show that spin contrast loss due to spin-independent noise does not depend on the initial thermal state of the matter wave function. However, spin contrast loss due to spin-dependent fluctuations do depend on the initial thermal occupation of the quantum state. We will keep our discussion general as far as the noise parameters are concerned.