Phys. Rev. Lett. 130, 113602 (2023) - Quantum Uncertainty Limit for Stern-Gerlach Interferometry with Massive Objects

Quantum Uncertainty Limit for Stern-Gerlach Interferometry with Massive Objects

Yonathan Japha and Ron Folman

Phys. Rev. Lett. 130, 113602 – Published 16 March 2023

Abstract

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.

Received 2 October 2022
Accepted 14 February 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.113602

Physics Subject Headings (PhySH)

Quantum coherence & coherence measures Quantum correlations, foundations & formalism Quantum entanglement

General Physics

Authors & Affiliations

Yonathan Japha^* and Ron Folman

Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel

^*Corresponding author. japhay@bgu.ac.il

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Vol. 130, Iss. 11 — 17 March 2023

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Figure 1
Model: (a) Adiabatic energy eigenvalues of a NV spin as a function of the angle $θ$ of the spin axis relative to the magnetic field. At $θ = 0$ the energy of the weak-field-seeking state $| + ⟩$ (curves labeled by magnetic field magnitudes in units of $| ε | / μ$ ) has a maximum, while the energy of the strong-field-seeking state $| - ⟩$ (lower curves) has a minimum, which is the equilibrium point for the librational dynamics. The gap $2 | ε |$ is due to strain and electric fields in the crystal. The orientation of the spin axis should be initially cooled to the librational ground state or to a low temperature state (both having a Gaussian distribution represented by the black curve). (b) The magnetic field is a sum of a 2D quadrupole (green field lines) and a homogeneous field $B_{0}$ (orange arrow), where the origin of the evolution is at the center of the quadrupole field. If the initial spin axis ${\hat{n}}_{S}$ is parallel to $B_{0}$ then the ND follows a linear trajectory (red arrow) along ${\hat{n}}_{S^{*}}$ (the reflection of ${\hat{n}}_{S}$ about the $x$ -axis), where the local quadrupole field (blue arrows tangent to the field lines) is also parallel to $B_{0}$ . (c) We assume that the spin axis points along ${\hat{n}}_{1}$ , one of the principal axes ${\hat{n}}_{j}$ of the ND, shifted by $θ_{2}$ , $θ_{3}$ from the direction of the magnetic field (orange arrow). For $θ_{2}$ , $θ_{3} ≪ 1$ the angular dynamics is harmonic if the spin state is $| - ⟩$ and free evolution if the spin state is $| 0 ⟩$ , while rotations around the spin axis (angle $θ_{1}$ ) are spin independent.
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Figure 2
Spatial and angular dynamics along the two interferometer paths (labeled $a$ and $b$ ). (a) Spatial trajectories: distance $r (t)$ from the initial point in units of the maximal distance $δ r_{\max}$ between the paths. A longitudinal (1D) SGI where both paths are along the same straight line is achieved if the spin axis is parallel to the magnetic field [see Fig. 1]. If the sequence is precise then both paths recombine at $t = 4 T$ with the same position and momentum, while the wave packet spread is the same along the two paths. (b) Angular evolution. While the expectation value of the spin direction remains parallel to the magnetic field if prepared so, its quantum uncertainty $σ$ is nonzero and diffuses the longitudinal nature of the spatial paths in (a). More importantly, the state-dependent angular potential [quadratic potential for the spin state $| - ⟩$ , Fig. 1] leads to a path-dependent evolution of the angular wave function (characterized by $σ$ and $\dot{σ}$ ). This prevents full recombination at the end of the sequence ( $t = 4 T$ ), unless the librational frequency $ω$ is fixed and $ω T$ is an integer multiple of $π$ , as shown by the solid curve in Fig. 3. Here $ω T = 0.45$ and the different values of $σ$ and $\dot{σ}$ at $t = 4 T$ give rise to a contrast reduced by 2.75% (same solid curve), even if the preparation and the sequence are perfectly precise.
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Figure 3
SGI contrast $| C |$ and rotational phase shift $δ ϕ_{R}$ as a function of $ω T$ , where $ω = \sqrt{μ B_{0} / I}$ is the initial librational frequency and $T$ is the gradient-pulse duration. (a) The contrast is optimal when $ω T ≪ 1$ or at peaks where $ω T$ is near an integer multiple of $π$ (smaller peaks appear in between [32]). The solid curve shows the contrast when the angular d.o.f. are prepared in the ground state of energy $\frac{1}{2} ℏ ω$ , and the librational frequency at the location $r (t)$ of the spin, when it is in the magnetically sensitive state $| - ⟩$ , hardly varies [e.g., in the limit $B^{'} r (t) ≪ B_{0}$ or when $B_{0} (t) = B_{0} (0) - B^{'} r_{-} (t)$ , see text]. The contrast drops as a power law of the rotational temperature $T_{R}$ (dashed-dotted curve for $T_{R} = 3 ℏ ω / k_{B}$ ). If the librational frequency changes along the paths with the magnetic field $B (t) = B_{0} + B^{'} r (t)$ [ $r (t)$ shown in Fig. 2], then $C$ is not peaked at exact multiples of $π$ and does not reach $C = 1$ , as shown by the dashed curve, corresponding to $T_{R} = 0$ and $B^{'} = B_{0} / δ r_{\max}$ . For this value of $B^{'}$ the librational frequency reaches $ω (t) = \sqrt{5} ω (0)$ at $t = 4 T$ and differs between the arms by $| ω_{a} - ω_{b} | = (\sqrt{5 / 2} - \sqrt{3 / 2}) ω (0)$ at $t = 2 T$ , where $δ r (t = 2 T) = δ r_{\max} = ω T \sqrt{I / M} = \sqrt{2 / 5} R ω T$ , $R$ being the ND radius. See numerical example at the end of the Letter for real numbers. (b) Rotational phase shift for $T_{R} = 0$ and fixed $ω$ . The shift is small near $ω T = 0$ and at $ω T = n π$ it is equal to an integer multiple of $2 π$ .
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Physical Review Letters

Quantum Uncertainty Limit for Stern-Gerlach Interferometry with Massive Objects

Yonathan Japha and Ron Folman

Phys. Rev. Lett. 130, 113602 – Published 16 March 2023

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