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.
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.
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.
We analyze the fundamental coherence limit of a nano-object with an embedded spin in a Stern-Gerlach interferometer. This limit stems from the which-path information provided by the object's rotational degrees of freedom due to the evolution of their quantum uncertainty. We show that such interferometry is straightforward in a weak magnetic field and short duration. Large wave packet separation is made possible with proper fine-tuning over long durations. This opens the door to fundamental tests of quantum theory and quantum gravity. The results and conclusions are extendable to any type of interferometry with complex objects.
There is a severe impediment in Stern-Gerlach interferometry with a spin-embedded nanoparticle. This is because particle rotation creates a significant mismatch of the wave function of angular degrees of freedom during the creation and recombination of the spatial superposition, resulting in an inferior visibility contrast. We show that imparting angular momentum along the direction of a defect, such as the nitrogen-vacancy (NV) center, during the Stern-Gerlach interferometry provides excellent gyroscopic quantum stability to a nanodiamond, which opens up a way to maintain significant visibility in the quantum interference. The spin contrast improves by many folds for a wide-ranging value of the initial angular momentum, e.g., 10^3–10^6 Hz for a mass of order 10^{-14}–10^{-17} kg nanodiamond. Our scheme is independent of the material and is pertinent to any nanoparticle that can be used to create massive and large spatial superpositions with an embedded spin defect.
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.
The rotation of a nanoparticle in a quantum system has many applications, from theory to experiments. This paper will treat nanoparticle rotational dynamics for spin-embedded nanorotors. We will model it as a rigid body that properly treats the rotation in the coframe of the nanorotor in the presence of external fields. Besides rotation, we will further investigate how to create large spatial superpositions in the inhomogeneous external magnetic field, such as in the case of the Stern-Gerlach apparatus. The spin-embedded nanorotors play a crucial role in creating matter-wave interferometers through their spin and external magnetic field interaction Hamiltonian. We aim to provide a holistic interpretation of the dynamics of three Euler angles, their quantum evolution, and the nanorotor's spatial motion in a Stern-Gerlach-type setup, where we will consider one-full-loop interferometry. We will then study how the quantum evolution of all the Euler angles leads to a spin coherence loss upon interference and what manifests the Einstein-de Haas effect in an external magnetic field. In particular, we show that by imparting rotation along the direction of the magnetic field, we can stabilize the nanorotor's libration mode. We will also extend our analysis to a case where the initial state of the libration mode is thermal and discuss the contrast loss due to interference of the nanorotor upon one-loop completion.
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.