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arXiv:2210.05689 · Physical Review A

Mass-independent scheme for enhancing spatial quantum superpositions

Run Zhou, Ryan J. Marshman, Sougato Bose, Anupam Mazumdar

Published Mar 16, 2023• 26 citations
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Abstract

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.

DOI
10.1103/physreva.107.032212

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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)\textmum][O(10--100)\phantom{\rule{4pt}{0ex}}\textmu{}\mathrm{m}][O(10−−100)\textmum] of mesoscopic masses [m\ensuremath∼O(10\ensuremath−14−−10\ensuremath−15)kg][m\ensuremath{\sim}O({10}^{\ensuremath{-}14}--{10}^{\ensuremath{-}15})\phantom{\rule{0.16em}{0ex}}\mathrm{kg}][m\ensuremath∼O(10\ensuremath−14−−10\ensuremath−15)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)\textmumO(400--600)\phantom{\rule{4pt}{0ex}}\textmu{}\mathrm{m}O(400−−600)\textmum from an initial modest split of the wave function. We provide an analytical and numerical analysis of our scheme.
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.
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 jjj 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+12j+12j+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.
A large spatial quantum superposition of size $O(1\text{--}10)\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\text{m}$ for mass $m\ensuremath{\sim}{10}^{\ensuremath{-}17}\text{--}{10}^{\ensuremath{-}14}$ kg is required to probe the foundations of quantum mechanics and test the classical and quantum nature of gravity via entanglement in a laboratory. In this paper, we will show that it is possible to accelerate the two spin states of a macroscopic nanocrystal sourced by the inhomogeneous nonlinear magnetic field in a Stern-Gerlach-type setup. We will assume that the electronic spin can be embedded at the center of the nanocrystal, such as the nitrogen-vacancy (NV) center of diamond. Our analysis will be generic to any dopant or any material. We will show that we can create a desired superposition size within 1–21\text{--}21–2 s by catapulting the trajectories of the two spin states with a modest magnetic field gradient and then recombine the trajectories for a coherent interference. We will show the demanding nature of the precision required in the magnetic field to recover a 9999%99 spin coherence confidence level at the moment of interference.
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.
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.
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.