Utilizing the Stern-Gerlach apparatus to create matter-wave superposition states is a long-sought-after goal, not only due to its potential applications in the quantum realm, but also because of its fundamental implications for studying the quantum properties of gravity. The main challenge in creating a macroscopic quantum interferometer arises from the loss of coherence, primarily through two channels. One channel involves strong coupling with the environment for macroscopic matter, leading to decoherence. The other channel relates to the precision of wave-packet overlap, which can occur due to external and internal fluctuations of various sources. The latter introduces a unique challenge for larger-scale masses by perturbing the center of mass motion of the macroscopic object. Here we study a particular challenge, namely, the issue of internal degrees of freedom, specifically phonon fluctuations and contrast reduction. This work investigates the contrast reduction caused by spin--magnetic-field--phonon and diamagnetism-phonon interactions in the quantum gravity-induced entanglement of masses protocol configuration.
Utilizing the Stern-Gerlach apparatus to create matter-wave superposition states is a long-sought-after goal, not only due to its potential applications in the quantum realm, but also because of its fundamental implications for studying the quantum properties of gravity. The main challenge in creating a macroscopic quantum interferometer arises from the loss of coherence, primarily through two channels. One channel involves strong coupling with the environment for macroscopic matter, leading to decoherence. The other channel relates to the precision of wave-packet overlap, which can occur due to external and internal fluctuations of various sources. The latter introduces a unique challenge for larger-scale masses by perturbing the center of mass motion of the macroscopic object. Here we study a particular challenge, namely, the issue of internal degrees of freedom, specifically phonon fluctuations and contrast reduction. This work investigates the contrast reduction caused by spin--magnetic-field--phonon and diamagnetism-phonon interactions in the quantum gravity-induced entanglement of masses protocol configuration.
Matter-wave interferometry is a striking quantum phenomenon that differs fundamentally from classical mechanics. In the quantum regime, matter behaves like a wave and can exist in spatial superpositions, allowing a single quantum object to move along multiple trajectories simultaneously. Similar to optical interferometers, such quantum matter systems act as interferometers themselves, with the differential phase that encodes information about external forces experienced during their evolution. This makes matter-wave interferometry a powerful platform for quantum sensing across various applications: (i) One of the most direct applications is precision measurement. Atomic interferometers can measure local acceleration and have already been implemented in devices such as accelerometers, seismometers, gravimeters, and gradiometers. (ii) Atomic interferometers are proposed as complementary tools for future gravitational-wave observatories, including ongoing projects MIGA, MAGIS-100, AION and ZAIGA. (iii) Matter-wave interferometry also offers a promising platform for probing quantum aspects of gravity. Numerous tabletop proposals aim to test quantum gravity models, investigate collapse mechanisms, study gravity-induced decoherence, and search for gravity-mediated quantum entanglement. The Stern-Gerlach interferometer (SGI) is a type of matter-wave interferometer that manipulates spatial superpositions using magnetic-field gradients. The witness of an SGI can be designed via its spin, enabling more precise measurements of the differential phase. Recent developments in atomic chip technology have enabled the generation of highly controlled magnetic fields on compact platforms, allowing for precise manipulation of spin states and making SGIs experimentally realizable. Robustness is a critical challenge for matter-wave interferometers. In real experiments, various sources of classical noises arise from mechanical vibrations, seismic motions, inertial forces, tidal forces, magnetic field fluctuations and so on. These noises can disturb the spin and spatial degrees of freedom of the test mass, resulting in witness loss of the interferometer. While spin decoherence is relatively well understood and can be formulated by the Bloch-Redfield master equation, the impact on spatial degrees of freedom still requires systematic investigation. This thesis therefore aims to explore witness loss in an SGI induced by stochastic noises acting on spatial degrees of freedom. As investigated by this thesis, the witness gets lost via several mechanisms: (i) Dephasing. The spatial phase is fluctuated by stochastic noise, resulting in a random phase factor eiδϕ, whose ensemble average contributes a decay factor e−Γ to the spin density matrix and leads to a loss of witness. As is demonstrated by the thesis, this decay factor is a linear response to the external noise, with a transfer function as the Fourier transform of the unperturbed classical trajectories. (ii) Humpty-Dumpty problem. The classical trajectories are also disturbed by stochastic noise, resulting in a non-closure problem of the interferometer, known as the Humpty-Dumpty problem. The resulting witness loss is usually characterized by the contrast of the final states ⟨ψ1|ψ2⟩, which equals the phase-space distance |α1 − α2|2 when the test mass is in coherent states. (iii) Position localization decoherence. Due to the randomness of the noise, the spatial wavefunction of the test mass is thus also stochastic, of which the ensemble average experiences decoherence characterized by the power spectrum density of external noise. However, as illustrated by this thesis, this type of decoherence does not affect spin-based witnesses in most practical scenarios. The general theoretical framework is mainly constructed in chapter 3 of this thesis. The subsequent chapters analyze several representative noise sources: (i) Gravitational and electrostatic noise. Molecules in the surrounding gas move randomly due to thermal motion, generating stochastic gravitational accelerations and electrostatic forces that can be expanded in multipole series. Since the random motion follows Maxwell’s distribution and approximately has a cylindrical symmetry, the resulting noise power spectrum density takes the form of Bessel functions. (ii) Inertial acceleration noise. The mechanical support of the experimental apparatus is always fluctuated by vibrational noise, resulting in random inertial force on the test mass. In practice, dephasing can be mitigated through destructive interference by introducing appropriate coupling between different directional components. (iii) Inertial torsion noise. In suspended experiments, the experimental apparatus is fluctuated by torsional noise, leading to another inertial noise on the test mass. The power spectrum density of the inertial torsion noise is formulated by the selfconvolution of the original noise, resulting in the peak frequencies that are doubled. Taken together, these analyses build a comprehensive picture of how stochastic noise affects the spatial dynamics of an SGI and leads to the loss of spin-based witnesses. Future work may include more realistic noise sources such as magnetic noise and trap-potential noise. These insights will be valuable for future experimental implementations, particularly in precision sensing for quantum gravity, where preserving coherence and maintaining reliable witness signals are essential.
We discuss the coherent splitting and recombining of a nanoparticle in a mesoscopic “closed-loop” Stern–Gerlach interferometer in which the observable is the spin of a single impurity embedded in the particle. This spin, when interacting with a pulsed magnetic gradient, generates the force on the particle. We calculate the internal decoherence, which arises as the displaced impurity excites internal degrees of freedom (phonons) that may provide WelcherWeg information and preclude interference. We estimate the constraints this decoherence channel puts on future interference experiments with massive objects. We find that for a wide range of masses, forces, and temperatures, phonons do not inhibit Stern–Gerlach interferometry with micro-scale objects. However, phonons do constitute a fundamental limit on the splitting of larger macroscopic objects if the applied force induces phonons.
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.
We investigate a two-dimensional matter-wave interferometer where both spatial and rotational dynamics of a nanoparticle are intertwined in closing the one-loop interferometer in the Stern-Gerlach type setup. We consider the spin-contrast of the nitrogen-vacancy (NV) centred nanodiamond in combination with a two-dimensional magnetic field setup to extend the one-dimensional Stern--Gerlach interferometry. We analyse the dynamical motion along with the rigid rotation under the influence of the external magnetic field. Regarding rotation, we incorporate Euler-angle dynamics to analyse the stability of rotational degrees of freedom and their influence on the spin contrast to address the Humpty-Dumpty problem. We show that by imparting external rotation provides the gyroscopic stability to the liberating mode of the NV-spin and hence helps to improve the contrast. Our scheme creates a tiny spatial superposition of size ∼0.21μm for mass m=10−17kg in less than t∼0.013s.
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.
A recently proposed experimental protocol for quantum gravity induced entanglement of masses (QGEM) requires in principle realizable, but still very ambitious, set of parameters in matter-wave interferometry. Motivated by easing the experimental realization, in this paper, we consider the parameter space allowed by a slightly modified experimental design, which mitigates the Casimir potential between two spherical neutral test masses by separating the two macroscopic interferometers by a thin conducting plate. Although this setup will reintroduce a Casimir potential between the conducting plate and the masses, there are several advantages of this design. First, the quantum gravity induced entanglement between the two superposed masses will have no Casimir background. Secondly, the matter-wave interferometry itself will be greatly facilitated by allowing both the mass ${10}^{\ensuremath{-}16}--{10}^{\ensuremath{-}15}\phantom{\rule{0.28em}{0ex}}\mathrm{kg}$ and the superposition size $\mathrm{\ensuremath{\Delta}}x\ensuremath{\sim}20\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$ to be a one-two order of magnitude smaller than those proposed earlier, and thereby also two orders of magnitude smaller magnetic field gradient of ${10}^{4}\phantom{\rule{0.28em}{0ex}}{\mathrm{Tm}}^{\ensuremath{-}1}$ to create that superposition through the Stern-Gerlach effect. In this context, we will further investigate the collisional decoherences and decoherence due to vibrational modes of the conducting plate.