Spatial superposition for a two-dimensional matter-wave interferometer in an inverted harmonic potential with gyroscopic rotational stability

Spatial superposition for a two-dimensional matter-wave interferometer in an inverted harmonic potential with gyroscopic rotational stability presents a mathematical model demonstrating how to create macroscopic quantum spatial superpositions using a Stern-Gerlach Interferometer (SGI) scheme within an inverted harmonic potential (IHP). This study is crucial because generating such superpositions is pivotal for testing the validity of quantum mechanics at larger scales and for investigating fundamental phenomena like wave-function collapse and gravitational decoherence.

The gist: A mathematical model of the spatial and rotational motion of a nanodiamond in an inverted harmonic potential to create a macroscopic quantum spatial superposition.

Model Framework

The study is based on the Stern-Gerlach Interferometer (SGI) scheme, which utilizes linear and quadratic magnetic fields to generate a harmonic potential (linear magnetic field) and a non-linear/quadratic magnetic field (non-linear/quadratic magnetic field). The model incorporates two-dimensional dynamics into the depiction of nanoparticle dynamics in linear and inverted harmonic potentials. Key elements include deriving equations of motion for rotational degrees of freedom, specifically libration, precession, and rotation. The Hamiltonian for the NV nanodiamond is given by:

Hˆ = Pˆ2/2m − χρm 2µ0 B squared + µ(Sˆ · B) + ħDSˆ2z − Htrap.

Magnetic Field Profiles and Stages

The magnetic fields are divided into two types: linear for stages 1, 3, and 5, and non-linear for stages 2 and 4. The switching mechanism is defined by the profile:

B(x, y) = Bl(x, y) for t ≤ t1; Bnl(x, y) for t1 ≤ t ≤ t2; Bl(x, y) for t2 ≤ t ≤ t3; Bnl(x, y) for t3 ≤ t ≤ t4; Bl(x, y) for stage 5.

The linear magnetic field is defined by:

Bl(x, y) = (B0(l) + ηlxˆex − ηlyeˆy).

The non-linear magnetic field is defined by:

Bnl(x, y) = [B0(nl) − ηnl(x squared − y 2)]ˆex + 2ηnlxy eˆy.

Equations of Motion

The two-dimensional equations of motion are formulated based on the applied magnetic fields at each stage:

For the linear part (stages 1, 3, 5):

d2⟨x⟩l/dt2 = −ω 2l⟨x⟩l − sµηl m + χρµ0 B0(l)ηl x̂− (Equation 7).

d2⟨y⟩l/dt2 = -(ω 2l − ω 2y)⟨y⟩ l (Equation 8).

For the non-linear part (stages 2, 4):

d2⟨x⟩nl/dt2 = ω 2nl⟨x⟩nl (Equation 9).

d2⟨y⟩nl/dt2 = (ω 2nl − ω 2y)⟨y⟩nl (Equation 10).

Rotational Dynamics and Stability

Rotational dynamics are incorporated by applying an initial angular rotation along the NV axis, (omega0), as suggested in [62, 63] to stabilise the libration angle, (β), precession angle, (α), and rotation angle, (γ). The rotational kinetic energy is formulated as:

Trot = 1/2Σ i=13Iiomega 2i.

The equations of motion for the rotational modes are derived from the Hamiltonian in Eq. (C1) and lead to:

β¨ = (pα − pγ cos β)(pα cos β − pγ) / 2 sin3 β - µsBNV/I ≈ −omega 2(β − β¯).

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.

Wave-Packet Evolution and Contrast

The spatial trajectory is constructed using Gaussian wave packets, where the evolution of width and center are described by equations (D10) to (D26). The contrast of spatial motion in the x-direction is calculated as:

Cx = exp − (xR − xL) squared / 8σ 2x(t) − σ 2x(t)(bR − bL) squared.

Based on the provided scientific paper, here are specific improvements that can be made to AI systems, categorized by their potential applications:

)1. Quantum Simulation and Modeling of Macroscopic Systems: The paper provides a detailed mathematical model (Hamiltonian and equations of motion) for a nanodiamond in an inverted harmonic potential undergoing spatial superposition.

• The improved AI system can perform high-fidelity simulations of quantum dynamics, including the coupling between translational motion (x, y) and rotational degrees of freedom (libration, precession, rotation).

• It can accurately predict the evolution of wave packets under complex magnetic field profiles (linear and non-linear/quadratic).

• It can simulate the Humpty-Dumpty problem by modeling how initial rotational states affect spin contrast during recombination.

)2. Quantum Gravity and Fundamental Physics Testing: The paper is motivated by testing quantum gravity, specifically the QGEM protocol and the quantum analogue of light-bending tests.

• The improved AI system can be used to model the entanglement between matter (nanodiamond) and photons/gravitons under specific experimental conditions (e.g., high magnetic field gradients).

• It can simulate the effects of massive gravitons on spacetime geometry as probed by matter waves.

)3. Advanced Quantum State Control and Stabilization: The paper demonstrates how initial rotation along the NV axis provides gyroscopic stability to mitigate decoherence and stabilize libration angles.

• The AI system can design optimal initial rotation parameters (e.g., minimizing the required initial angular velocity, optimizing the off-center displacement 'd') needed to maintain coherence across multiple stages of a complex interferometer sequence.

• It can predict the necessary stabilization techniques (like the use of a trap frequency in the y-direction) to minimize unwanted oscillations in rotational modes during rapid potential switching.

)4. Decoherence Mitigation Strategies: The paper analyzes how thermal noise affects spin contrast and provides bounds for quantum states (Eq. 24, 25).

• The AI can develop predictive models to determine the minimum required repetition rate or temperature control necessary to keep decoherence rates below specific thresholds (e.g., controlling them within the range of 10−2–10−3 Hz) for a given spatial superposition size.

• It can simulate the impact of thermal occupation numbers (n) on contrast, allowing experimentalists to predict how temperature fluctuations will degrade the final spin entanglement witness.

)5. Quantum Metrology and Sensing: The paper explores using nanodiamonds as quantum sensors based on their magnetic moment response to external fields.

• The AI can optimize the design of magnetic field profiles (B(x, y)) to maximize the sensitivity of the resulting spatial superposition measurement, effectively acting as a quantum lens for matter waves.

• It can calculate optimal readout strategies by analyzing how different initial conditions affect the final contrast (Eq. 8), guiding experimental setup choices for maximizing measurable quantum effects.

The improved AI system can perform the following specific tasks:

1. Perform high-fidelity, multi-dimensional simulations of nanoparticle dynamics in coupled harmonic/inverted harmonic potentials to generate macroscopic spatial superpositions with predicted fidelity and size metrics (e.g., predicting the trade-off between superposition width and loop time).

2. Design optimal initial rotational conditions (initial angular velocity, off-center displacement) required to maintain gyroscopic stability against external torques across all five stages of the SGI protocol, thereby maximizing spin contrast at recombination.

3. Develop predictive models for decoherence rates based on magnetic field gradients and thermal noise levels, allowing for the optimization of experimental parameters (like magnetic field bias B0 and gradient ηl) to achieve a target quantum coherence time.

4. Simulate the evolution of quantum wave packets in both spatial (x, y) and rotational (precession, libration, rotation) degrees of freedom under dynamic potential switching, providing exact solutions for the Gaussian wave packet parameters in both linear and non-linear regimes.

5. Analyze and predict the final spin contrast for a given experimental setup by calculating the mismatch between left and right arms at recombination time using derived formulas (Eq. 13, 14), allowing researchers to determine if a proposed spatial superposition is experimentally viable before costly fabrication or experimentation.