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arXiv:2408.11909 · Physical Review A

Spin-dependent force and inverted harmonic potential for rapid creation of macroscopic quantum superpositions

Run Zhou, Qian Xiang, Anupam Mazumdar

Published May 12, 2025• 7 citations
arXivHTMLPDF

Abstract

Creating macroscopic spatial superposition states is crucial for investigating matter-wave interferometry and advancing quantum sensor technology. Currently, two potential methods exist to achieve this objective. The first involves using inverted harmonic potential (IHP) to spatially delocalize quantum states through coherent inflation [New J. Phys. 19, 123029 (2017)]. The second method employs a spin-dependent force to separate two massive wave packets spatially [Phys. Rev. Lett. 111, 180403 (2013)]. The disadvantage of the former method is the slow initial coherent inflation, while the latter is hindered by the diamagnetism of spin-embedded nanocrystals, which suppresses spatial separation. In this study, we integrate two methods: first, we use the spin-dependent force to generate initial spatial separation, and second, we use IHP to achieve coherent inflating trajectories of the wave packets. This approach enables the attainment of massive large spatial superposition in minimal time. For instance, a spatial superposition with a mass of ${10}^{\ensuremath{-}15}\phantom{\rule{0.16em}{0ex}}\mathrm{kg}$ and a size of around $50\phantom{\rule{0.16em}{0ex}}\textmu{}\mathrm{m}$ is realized in 0.1 s. We also calculate the evolution of wave packets in both harmonic potential (HP) and IHP using the path integral approach.

DOI
10.1103/physreva.111.052207

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Creating macroscopic spatial superposition states is crucial for investigating matter-wave interferometry and advancing quantum sensor technology. Currently, two potential methods exist to achieve this objective. The first involves using inverted harmonic potential (IHP) to spatially delocalize quantum states through coherent inflation [New J. Phys. 19, 123029 (2017)]. The second method employs a spin-dependent force to separate two massive wave packets spatially [Phys. Rev. Lett. 111, 180403 (2013)]. The disadvantage of the former method is the slow initial coherent inflation, while the latter is hindered by the diamagnetism of spin-embedded nanocrystals, which suppresses spatial separation. In this study, we integrate two methods: first, we use the spin-dependent force to generate initial spatial separation, and second, we use IHP to achieve coherent inflating trajectories of the wave packets. This approach enables the attainment of massive large spatial superposition in minimal time. For instance, a spatial superposition with a mass of 10\ensuremath−15kg{10}^{\ensuremath{-}15}\phantom{\rule{0.16em}{0ex}}\mathrm{kg}10\ensuremath−15kg and a size of around 50\textmum50\phantom{\rule{0.16em}{0ex}}\textmu{}\mathrm{m}50\textmum is realized in 0.1 s. We also calculate the evolution of wave packets in both harmonic potential (HP) and IHP using the path integral approach.
Abstract Creating a massive spatial quantum superposition, such as the Schrödinger cat state, where the mass and the superposition size within the range 10−19 − 10−14 kg and Δx ∼ 10 nm − 100 μm, is a challenging task. The methods employed so far rely either on wavepacket expansion or on a quantum ancilla, e.g. single spin dependent forces, which scale inversely with mass. In this paper, we present a novel approach that combines gravitational acceleration and diamagnetic repulsion to generate a large spatial superposition in a relatively short time. After first creating a modest initial spatial superposition of 1 μm, achieved through techniques such as the Stern–Gerlach (SG) apparatus, we will show that we can achieve an ∼102−103 fold improvement to the spatial superposition size (1 μm → 980 μm) between the wave packets in less than 0.02 s by using the Earth’s gravitational acceleration and then the diamagnetic repulsive scattering of the nanocrystal, neither of which depend on the object mass. Finally, the wave packet trajectories can be closed so that spatial interference fringes can be observed. Our findings highlight the potential of combining gravitational acceleration and diamagnetic repulsion to create and manipulate large spatial superpositions, offering new insights into creating macroscopic quantum superpositions.
This thesis explores how to create macroscopic matter-wave interferometers (MMWIs) using nanodiamonds with nitrogen-vacancy (NV) centers. MMWIs are important for testing the limits of quantum mechanics and studying the possible quantum nature of gravity. However, building MMWIs with large masses is very challenging because of decoherence and the diamagnetic properties of diamonds. This work presents several experimental schemes to increase the spatial separation of massive quantum objects. The key idea is to use the diamagnetism of diamonds not only as a limitation but also as a tool to help separate wave packets. Methods include catapulting trajectories with nonlinear magnetic fields, using diamagnetic repulsion, combining gravitational acceleration with magnetic fields, and applying inverted harmonic potentials. The results show that it is possible to achieve large spatial superpositions, up to tens of micrometers, in short timescales. While a full-scale MMWI is still difficult to realize, these findings may contribute to future efforts to observe quantum effects in macroscopic systems.
This study presents a mathematical model of the spatial and rotational motion of a nanodiamond in an inverted harmonic potential to create a macroscopic quantum spatial superposition. The model is based on the Stern-Gerlach Interferometer (SGI) scheme, which utilises linear and quadratic magnetic fields to generate a harmonic potential (linear magnetic field) and a non-linear potential (non-linear/quadratic magnetic field). By incorporating two-dimensional dynamics into the model, we provide a more realistic and accurate depiction of nanoparticle dynamics in linear and inverted harmonic potentials and explore the interaction between motion in a two-dimensional plane. Importantly, we derive the equations of motion for the rotational degrees of freedom, i.e. libration, precession, and rotation. The results show that adding a magnetic-field bias term to the magnetic-field profile in the linear stage affects the classical equations of motion but does not affect the width of the wave packet. Moreover, the libration mode always forms a harmonic potential at each stage because the applied initial angular velocity is dominated by the nanoparticle's defect axis, making it more stable in the presence of the trap frequency in the orthogonal direction along the axis that enables the creation of a macroscopic quantum superposition.
We investigate a Stern-Gerlach type matter-wave interferometer where an inhomogeneous magnetic field couples to an embedded spin in a nanoparticle to create spatial superpositions. Employing a sequence of harmonic and inverted harmonic oscillator potentials created by external magnetic fields, we aim to enhance the one-dimensional superposition of a nanodiamond with mass ∼10−15\sim 10^{-15}∼10−15 kg to ∼1μ\sim 1 μ∼1μm. However, random fluctuations of the magnetic field stochastically perturbs the interferometer paths and induce dephasing. We quantitatively estimate the susceptibility of the interferometer to white noise arising from magnetic-field fluctuations. Constraining the dephasing rate ΓΓΓ to be low enough that the final coherence e−Γτ≤0.1e^{-Γτ}\leq 0.1e−Γτ≤0.1 (where τττ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: δηIHP/ηIHP≲10−13δη_\text{IHP}/η_\text{IHP}\lesssim 10^{-13}δηIHP​/ηIHP​≲10−13, where ηIHPη_\text{IHP}ηIHP​ is the magnetic field curvature that gives rise to the inverted harmonic potential, and δηHP/ηHP≲10−6δη_\text{HP}/η_\text{HP}\lesssim 10^{-6}δηHP​/ηHP​≲10−6, where ηHPη_\text{HP}ηHP​ is the linear magnetic field gradient that gives rise to the harmonic potential. For such tiny fluctuations, we demonstrate that the Humpty-Dumpty problem arising from a mismatch in position and momentum does not cause a loss in contrast of the interferometer. Further, we show that constraining the dephasing rate leads to stricter bounds on the noise parameters than enforcing a contrast threshold, indicating that good dephasing control ensures high interferometric contrast.
Placing a large mass in a large spatial superposition, such as a Schr\"odinger cat state, is a significant and important challenge. In particular, the large spatial superposition $[O(10--100)\phantom{\rule{4pt}{0ex}}\textmu{}\mathrm{m}]$ of mesoscopic masses $[m\ensuremath{\sim}O({10}^{\ensuremath{-}14}--{10}^{\ensuremath{-}15})\phantom{\rule{0.16em}{0ex}}\mathrm{kg}]$ makes it possible to test the quantum nature of gravity via entanglement in the laboratory. To date, the proposed methods of achieving this spatial delocalization are to use wave-packet expansions or quantum ancilla- (for example, spin-) dependent forces, all of whose efficacy reduces with mass. Thus increasing the spatial splitting independently of the mass is an important open challenge. In this paper we present a method of achieving a mass-independent enhancement of superposition via diamagnetic repulsion from current-carrying wires. We analyze an example system which uses the Stern-Gerlach effect to creating a small initial splitting and then apply our diamagnetic repulsion method to enhance the superposition size $O(400--600)\phantom{\rule{4pt}{0ex}}\textmu{}\mathrm{m}$ from an initial modest split of the wave function. We provide an analytical and numerical analysis of our scheme.
We propose an interferometric scheme based on an untrapped nano-object subjected to gravity. The motion of the center of mass (c.m.) of the free object is coupled to its internal spin system magnetically, and a free flight scheme is developed based on coherent spin control. The wave packet of the test object, under a spin-dependent force, may then be delocalized to a macroscopic scale. A gravity induced dynamical phase (accrued solely on the spin state, and measured through a Ramsey scheme) is used to reveal the above spatially delocalized superposition of the spin-nano-object composite system that arises during our scheme. We find a remarkable immunity to the motional noise in the c.m. (initially in a thermal state with moderate cooling), and also a dynamical decoupling nature of the scheme itself. Together they secure a high visibility of the resulting Ramsey fringes. The mass independence of our scheme makes it viable for a nano-object selected from an ensemble with a high mass variability. Given these advantages, a quantum superposition with a 100 nm spatial separation for a massive object of 10^{9} amu is achievable experimentally, providing a route to test postulated modifications of quantum theory such as continuous spontaneous localization.