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 kg to ≈1 µm. 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 Γ to be low enough that the final coherence e−Γτ > 0.1 (where τ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: δηIHP/ηIHP ≾ 10−13, where ηIHP is the magnetic-field curvature that gives rise to the inverted harmonic potential, and δηHP/ηHP ≾ 10−6, where 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. Furthermore, 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.
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 kg to ≈1 µm. 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 Γ to be low enough that the final coherence e−Γτ > 0.1 (where τ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: δηIHP/ηIHP ≾ 10−13, where ηIHP is the magnetic-field curvature that gives rise to the inverted harmonic potential, and δηHP/ηHP ≾ 10−6, where 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. Furthermore, 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.
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 kg to ∼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 Γ to be low enough that the final coherence e−Γτ≤0.1 (where τ is the experimental time duration), we obtain the following bounds on the noise to signal ratios: δηIHP/ηIHP≲10−13, where ηIHP is the magnetic field curvature that gives rise to the inverted harmonic potential, and δηHP/ηHP≲10−6, where 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.
In this paper, we will show how random fluctuations in the magnetic field will jitter the paths of a matter-wave interferometer randomly, hence, decohere the quantum superposition. To create a large spatial superposition with nanoparticles, we envisage embedding a spin in a nanoparticle as a defect and applying an inhomogeneous magnetic field as in a Stern-Gerlach type experiment to create a macroscopic quantum superposition. Such matter-wave interferometers are the cornerstone for many new fundamental advancements in physics; particularly, adjacent matter-wave interferometers can use entanglement features to test physics beyond the Standard Model, test the equivalence principle, improve quantum sensors, and test the quantum nature of spacetime in a lab. In particular, we will use white and flicker noise to study the decoherence and constrain the parameters keeping in mind ambient temperatures suitable for superconducting wires embedded on a chip. We will show that to obtain a tiny spatial superposition of a nanometer separation, Δx∼O(10−9)m and to minimize decoherence, Γ≤O(2πω0), where Γ is the decoherence and ω0 is the frequency of the oscillator, we will need current fluctuations to be δI/I≤O(10−8), which is not impossible to obtain in superconducting wire arrangements. For such tiny fluctuations, we demonstrate that the Humpty-Dumpty problem in a matter-wave interferometer arising from a mismatch in position and momentum does not cause a loss in contrast.
We analyze decoherence in a one-loop Stern--Gerlach--type matter-wave interferometer for a massive nanoparticle embedded with a nitrogen vacancy (NV)-centered nanodiamond evolving under an effective harmonic-oscillator dynamics in a magnetic-field gradient. We assume that the Stern-Gerlach interferometer is subjected to an acceleration a external to the system, which is at an angle θ0 with respect to the direction of the superposition. For a one loop interferometer, we quantify dephasing from two noise channels: fluctuations in the external acceleration δa(t) and fluctuations in the tilt angle δθ(t). At the level of the action, we treat these two external noise as stochastic inputs, compute the resulting stochastic phase difference between the interferometer arms, and obtain the dephasing rate Γ. We obtain the transfer function of the interferometer for each of the noise sources. We also show explicit results of constraints on the power spectral density of the noise sources for an interferometer that produces a superposition size of Δx∼1nm of a nanodiamond of mass m=10−15kg by considering white noise statistics and imposing a coherence target Γτ≤1, where τ≃0.015s. We find Saa≲O(10−11)ms−2Hz−1/2 if we take the external acceleration, a=0ms−2 and θ0=0∘ (along the direction of the superposition), and Sθθ≲O(10−10)radHz−1/2 for a=g=9.81ms−2 and θ0=90∘ (superposition direction is perpendicular to the Earth's gravity). We have also found an operating regime where the acceleration noise can be minimized by either varying θ0 or a for a fixed set of other experimental parameters.
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 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.
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.