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Acceleration noise induced decoherence in Stern-Gerlach interferometers for gravity experiments

Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, 9747 AG Groningen, The Netherlands

^*Contact author: mengzhi.wu@rug.nl

Phys. Rev. Research 8, 023115 – Published 4 May, 2026

DOI: https://doi.org/10.1103/9bzl-lfnq

Abstract

Stern-Gerlach interferometer (SGI) is a kind of matter-wave interferometer driven by magnetic field and has been proposed for various gravity experiments. Stochastic noises can lead to decoherence problems of SGI via various mechanisms. In this paper, I will theoretically study several mechanisms including dephasing, contrast loss, and position localization decoherence. I will first present a rigorous proof that the dephasing behaves as a linear response to the noise, with a transfer function given by the Fourier transform of the unperturbed classical trajectories. Then I will demonstrate that stochastic acceleration noise only induces dephasing to the witness operator constructed in spin space, while it does not lead to contrast loss or position localization decoherence due to common mode cancellation. In contrast, higher-order noise can induce both contrast loss and position localization decoherence, contributing a decay factor proportional to the noise power spectrum density at the intrinsic frequency, which can be physically interpreted as the resonance between the noise and test mass. Based on the result, I apply the framework to analyze two typical noise sources, magnetic field noise and quadratic noise.

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This conclusion can be generalized to a generic matter-wave interferometer under a general noise. Suppose the interferometer is built up through a Lagrangian $L [x_{i} (t), {\dot{x}}_{i} (t)]$ and the noise is described by a Lagrangian $δ L [x_{i} (t), {\dot{x}}_{i} (t)]$ . Denote the ideal trajectory as $x_{i}^{0} (t)$ , and the ideal phase is $ϕ^{0} = \int L [x_{i}^{0}, {\dot{x}}_{i}^{0}] / ℏ d t$ . Then, the phase under noise at the leading order is $\begin{matrix} ϕ^{0} + δ ϕ & = \frac{1}{ℏ} \int L [x_{i}^{0} + δ x_{i}, {\dot{x}}_{i}^{0} + δ {\dot{x}}_{i}^{0}] + δ L [x_{i}^{0}, {\dot{x}}_{i}^{0}] d t \\ = \frac{1}{ℏ} \int L [x_{i}^{0}, {\dot{x}}_{i}^{0}] + \frac{\partial L}{\partial x_{i}} |_{(x_{i}^{0}, {\dot{x}}_{i}^{0})} δ x_{i} \\ + \frac{\partial L}{\partial {\dot{x}}_{i}} |_{(x_{i}^{0}, {\dot{x}}_{i}^{0})} δ {\dot{x}}_{i} + δ L [x_{i}^{0}, {\dot{x}}_{i}^{0}] d t . \end{matrix}$ According to the Euler-Lagrangian equation $\partial L / \partial x_{i} = d (\partial L / \partial {\dot{x}}_{i}) / d t$ , one can prove an identity $\begin{matrix} \frac{\partial L}{\partial x_{i}} {|_{(x_{i}^{0}, {\dot{x}}_{i}^{0})} δ x_{i} + \frac{\partial L}{\partial {\dot{x}}_{i}} |}_{(x_{i}^{0}, {\dot{x}}_{i}^{0})} δ {\dot{x}}_{i} = \frac{d}{d t} (\frac{\partial L}{\partial {\dot{x}}_{i}} |_{(x_{i}^{0}, {\dot{x}}_{i}^{0})} δ x_{i}) . \end{matrix}$ Thus, the phase fluctuation arising from the fluctuation of the trajectory $δ x_{i}$ and the velocity $δ {\dot{x}}_{i}$ is the integral of a total derivative. Then, one may read $δ ϕ = \int δ L [x_{i}^{0}, {\dot{x}}_{i}^{0}] / ℏ d t$ by neglecting this total derivative term. Note that the same conclusion has been proved by using Feynman's path integral method [76].
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Here, the following identities are used: $\begin{matrix} \sqrt{{\hat{ρ}}_{id} (t)} & = {\hat{U}}_{id} (t) \sqrt{{\hat{ρ}}_{0}} {\hat{U}}_{id}^{†} (t), \\ {\hat{U}}_{id}^{†} (t) {\hat{U}}_{n} (t) & = \hat{D} (δ α (t) e^{i ω_{0} t}) e^{i δ ϕ (t)} . \end{matrix}$
To compute the ensemble average of $〈 x_{1} | {\hat{ρ}}_{n} (t) | x_{2} 〉$ , one can consider the Glauber's $P$ -representation [52] of the initial density matrix $\begin{matrix} {\hat{ρ}}_{0} = \int P (β) | β 〉 〈 β | d^{2} β . \end{matrix}$ According to the identity $\begin{matrix} \hat{D} (δ α (t)) | β 〉 = e^{i ϕ (β, δ α)} | β + δ α (t) 〉, \end{matrix}$ the matrix element of ${\hat{ρ}}_{n} (t)$ becomes $\begin{matrix} 〈 x_{1} | {\hat{ρ}}_{n} (t) | x_{2} 〉 = \int P (β) 〈 x_{1}^{'} (t) | β + δ α (t) 〉 〈 β + δ α (t) | x_{2}^{'} (t) 〉 d^{2} β . \end{matrix}$ Based on the wavefunction of a coherent state, one can obtain the ensemble average of the matrix element in the $P$ -representation satisfying $\begin{matrix} E [〈 x_{1}^{'} (t) | β + δ α (t) 〉 〈 β + δ α (t) | x_{2}^{'} (t) 〉] \\ = 〈 x_{1}^{'} (t) | β 〉 〈 β | x_{2}^{'} (t) 〉 e^{- \frac{m ω_{0}}{ℏ} σ_{α}^{2} (t) {(x_{1}^{'} (t) - x_{2}^{'} (t))}^{2}} . \end{matrix}$ Then, one can obtain the result (52).
If there is a biased magnetic field $B_{0} \neq 0$ , then the amplitudes $A_{\pm}$ defined in Eq. (31) are not opposite to each other. As a result, the trajectories $x_{1} (t)$ and $x_{- 1} (t)$ do not cancel each other. In this case, there is still a dephasing effect, and the transfer function takes a similar form as Eq. (32) but the factor $Δ A$ is replaced by $A_{+} + A_{-}$ .
Suppose the final states before squeezing are $| ψ_{+} 〉$ and $| ψ_{-} 〉$ . After applying the squeezing operator, their overlap becomes $\begin{matrix} 〈 ψ_{+} | {\hat{S}}^{†} (ρ) \hat{S} (ρ) | ψ_{-} 〉 = 〈 ψ_{+} | ψ_{-} 〉, \end{matrix}$ which remains unchanged because $\hat{S} (ρ)$ is unitary.
The trick to compute the purity and the von Neumann entropy is to diagonalize ${\hat{ρ}}_{spin}$ . In particular, suppose the nonzero eigenvalues of ${\hat{ρ}}_{spin}$ are ${λ_{i}}$ , then the purity and the von Neumann entropy of ${\hat{ρ}}_{spin}$ are $\sum_{i} λ_{i}^{2}$ and $- \sum_{i} λ_{i} ln λ_{i}$ , respectively. For the density matrix ${\hat{ρ}}_{spin}$ with a decay factor $e^{- Γ_{tot}}$ on the off-diagonal terms, its eigenvalues are $λ_{1, 2} = (1 \pm e^{- Γ_{tot}}) / 2$ and $λ_{3} = 0$ . Especially, when the total decoherence factor $Γ_{tot}$ is very small, then the entropy can be approximated as $\begin{matrix} \begin{matrix} S & = - [\frac{1 + e^{- Γ_{tot}}}{2} ln \frac{1 + e^{- Γ_{tot}}}{2} + \frac{1 - e^{- Γ_{tot}}}{2} ln \frac{1 - e^{- Γ_{tot}}}{2}] \\ \approx \frac{1 + e^{- Γ_{tot}}}{2} \times \frac{1 - e^{- Γ_{tot}}}{2} + \frac{1 - e^{- Γ_{tot}}}{2} \times \frac{1 + e^{- Γ_{tot}}}{2} \\ = \frac{1}{2} (1 - e^{- 2 Γ_{tot}}) \approx Γ_{tot}, \end{matrix} \end{matrix}$ where the first approximated equation used an approximation $ln ((1 \pm e^{- Γ_{tot}}) / 2) \approx (- 1 \pm e^{Γ_{tot}}) / 2$ .
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Phys. Rev. Research 8, 023115– Published 4 May, 2026

Vol. 8, Iss. 2 — May - July 2026

Received 14 May 2025
Accepted 2 April 2026

DOI: https://doi.org/10.1103/9bzl-lfnq

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Acceleration noise induced decoherence in Stern-Gerlach interferometers for gravity experiments

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