Momentum Diffusion, Decoherence and Drag Force on a Magnetic Nanoparticle

arXiv:2602.21518 · quant-ph · Submitted 2026-02-25 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Momentum Diffusion, Decoherence and Drag Force on a Magnetic Nanoparticle".

Mira: Here's the gist of the paper. The gist In this paper, we will provide a complete derivation of the decoherence rate for a magnetic nanoparticle in quantum superposition in the presence of the fluctuating electromagnetic field in a thermal background by using the fluctuation-dissipation theorem in the long-wavelength limit.

Kai: Momentum Diffusion Constant The momentum diffusion constant is related to the time-derivative of the expectation value of total momentum, defined as 2D ≡ ⟨∆p2⟩/∆t The dominant contribution to the momentum variance arises from the time-varying electric field on the magnetic dipoles, with three contributions to momentum variance: magnetic, electric, and the coupled term between magnetic and electric fields.

Mira: The expectation value of momentum can be written as ∆p2 = ∆p2M + ∆p2E + ∆p†M∆pE + ∆p†E∆pM.

Kai: In the long-wavelength limit, the time-dependent electric-field-fluctuation term is the dominant one, suggesting that the momentum diffusion constant is dominated by timedependent electric field fluctuation through (1/c2)m × ∂tE term instead of the magnetic field fluctuation through ∇(m · B) term.

Paper summary: Kai: To recap on "Momentum Diffusion, Decoherence and Drag Force on a Magnetic Nanoparticle," the core idea is providing a complete derivation of the decoherence rate for a magnetic nanoparticle in quantum superposition when it’s exposed to fluctuating electromagnetic fields in a thermal background.

Mira: They achieve this by using the fluctuation-dissipation theorem specifically in the long-wavelength limit, which assumes that the size of your quantum superposition is much smaller than the wavelength of those electromagnetic fluctuations.

Kai: They extend this computation to two diamagnetic nanoparticles kept in a quantum superposition right next to each other and show how the drag force on a single nanoparticle arises from those external electromagnetic field fluctuations.

Mira: They also compare their results for the drag force against what you'd expect based on the nanoparticle’s dielectric properties, which provides important context.

Lev: This work sets up a framework to understand how noise influences quantum experiments like matter-wave interferometry and quantum sensors.

Kai: It addresses constraints arising from decoherence from ambient vacuum environments that are crucial for setting limits on how heavy a macroscopic quantum superposition we can create.

Mira: The authors computed the decoherence effect on a dielectric nanoparticle derived from the momentum diffusion constant in the presence of blackbody radiation, and they demonstrated it is closely related to the long-wavelength collisional decoherence rate for such a nanoparticle in a thermal environment.

Lev: They also showed how this diffusion constant appears in the steady-state photon emission rate of two dipoles induced by blackbody radiation, which gives us more information about thermal noise.

Kai: The paper then applies that analysis to a diamagnetic nanoparticle and comments on the nitrogen-vacancy-centred nanodiamond’s properties within this framework.

Mira: So, it essentially boils down to comparing magnetic and electric field decoherence mechanisms in this specific context.

Conclusion: Kai: Looking at the title "Momentum Diffusion, Decoherence and Drag Force on a Magnetic Nanoparticle," the paper is clearly trying to connect how these different physical forces interact.

Mira: It connects the momentum diffusion constant, which is driven by electric field fluctuations, to the decoherence rate and drag force in a way that shows they aren't independent things.

Lev: For real hardware development, this means we need to understand that even if we’re trying to isolate magnetic effects for a qubit system, there will always be coupling with these electric field noise sources.

Kai: The paper shows that the drag force and thermal drag due to the diamagnetic interactions are much weaker than what you see from dielectric interactions.

Mira: So, when you apply this work to actual experiments, it gives us concrete comparison numbers like nine point one four times ten to the minus thirteen for that nanodiamond.

Lev: That comparison helps us understand the relative importance of these different noise sources so we know where our experimental efforts should be focused.

Kai: Ultimately, this paper is about laying out how the decoherence rate is calculated under these specific assumptions, which helps set realistic expectations for what quantum experiments can actually achieve.

Van Swinderen Institute for Particle Physics and Gravity · Universitas Indonesia

quant-ph

Submitted: 2026-02-25

Updated: 2026-07-22

Comments: 28 pages, includes appendices, no figures

DOI: 10.1103/zcxg-k9nn

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 59/100

The gist: The gist In this paper, we will provide a complete derivation of the decoherence rate for a magnetic nanoparticle in quantum superposition in the presence of the fluctuating electromagnetic field in

Key concepts

Momentum Diffusion Constant (2D): This constant measures how much the total momentum of a particle spreads out over time. The paper finds this spreading is mainly driven by time-varying electric field fluctuations acting on the magnetic dipoles, rather than magnetic field fluctuations.
Decoherence Rate ($\gamma$): This rate quantifies how fast a quantum superposition loses its coherence due to interaction with the environment. The derived formula shows this rate depends on both the nanoparticle's magnetic susceptibility and dielectric properties, revealing that electric field interactions are significantly more effective at causing decoherence than magnetic ones.
Drag Force: This force represents the resistance a particle experiences when moving through a medium, often due to thermal fluctuations. The paper calculates this drag for a nanodiamagnetic sphere and concludes that this thermal drag is much smaller and can be ignored when compared to the decoherence rate.
Long-Wavelength Limit: This approximation simplifies the physics by assuming the size of the quantum superposition is much smaller than the wavelength of the electromagnetic field fluctuations. This simplification allows for a tractable calculation using standard fluctuation-dissipation theorems.

Summary

The gist In this paper, we will provide a complete derivation of the decoherence rate for a magnetic nanoparticle in quantum superposition in the presence of the fluctuating electromagnetic field in a thermal background by using the fluctuation-dissipation theorem in the long-wavelength limit.

Momentum Diffusion Constant

The momentum diffusion constant is related to the time-derivative of the expectation value of total momentum, defined as 2D ≡ ⟨∆p2⟩/∆t The dominant contribution to the momentum variance arises from the time-varying electric field on the magnetic dipoles, with three contributions to momentum variance: magnetic, electric, and the coupled term between magnetic and electric fields. The expectation value of momentum can be written as ∆p2 = ∆p2M + ∆p2E + ∆p†M∆pE + ∆p†E∆pM. In the long-wavelength limit, the time-dependent electric-field-fluctuation term is the dominant one, suggesting that the momentum diffusion constant is dominated by timedependent electric field fluctuation through (1/c2)m × ∂tE term instead of the magnetic field fluctuation through ∇(m · B) term.

Decoherence Rate

The decoherence rate in the long-wavelength limit is given by γ = h / a6 c9π3 (χ2R + 3χR2) / (kBT hc9 ζ(8)Γ(9)) + a3 c6π2β3χI (3 + χR)2 + χ2I kBT hc6 Γ(6)ζ(5i) (∆x)2. This general result for the decoherence rate will include both the real and imaginary part of α, and the magnetic susceptibility χv = χR + iχI. The ratio of decoherence rates due to the matter’s diamagnetic and dielectric properties is given by γB/γE = 1 / (16π2/2χ2R(3+χR)2) (82). For a pure nanodiamond with χR ∼ −2.2 × 10−5 and ϵ ∼ 5.7, the ratio is γB/γE = 9.14 × 10−13.

Drag Force

The thermal drag for a nanodiamagnetic sphere can be written by assuming αI (ω) is constant, and by using the Clausius-Mossotti relation, Eq. 30. The drag force on magnetic dipole moments in a magnetic field as the contribution from the scattering and absorption parts is given by ξ = 32π / (5a6ħβ135m kBT hc8 (χ2R + 3χR2)2 + 41.47 a3ħ m kBT hc5 / 3χI(3 + χR)2. The thermal drag is significantly smaller and can be considered negligible compared to the decoherence rate due to the ambient magnetic field at T ≈ 5 K.

Interference and Decoherence Factor

The decoherence factor due to magnetic dipole is given by F(x1 − x2) = (x1 − x2)2 a6 c18π3 (χ2R + 3χR2)2 Z ∞0 d k k8 e−ħkc/kBT + (x1 − x2)2 a3 c3π2β3χI(3 + χR)2 Z ∞0 d k5 e−ħkc/kBT. The decoherence factor due to the ambient electric field is given by FE(x1 − x2) = (x1 − x2)2 8a6 c9π / (ε - 1 / ε + 2)2 kBT hc9 ζ(9)Γ(9). The ratio of these factors is FB/FE = 1 / (32π2χ2R(3+χR)2) (82).

Conclusion

The drag force and thermal drag due to the diamagnetic interactions of a nanoparticle are much weaker than that of the dielectric interaction. The thermal drag is significantly smaller and can be considered negligible compared to the decoherence rate due to the ambient magnetic field at T ≈ 5 K. The ratio FB/FE ∼ 9.14 × 10−13 for a pure nanodiamond with χR ∼ −2.2 × 10−5 and ϵ ∼ 5.7. The decoherence rate due to the ambient electric field is much higher than the magnetic field. The drag force and thermal drag due to the diamagnetic interactions of a nanoparticle are much weaker than that of the dielectric interaction.

Acknowledgments

We would like to thank Martine Schut for insightful discussions A.A. is supported by Lembaga Pengelola Dana Pendidikan (LPDP) of The Ministry of Finance of Indonesia A.M.’s research is funded by the Gordon and Betty Moore Foundation through Grant GBMF12328, DOI 10.37807/GBMF12328 This material is based on work supported by the Alfred P. Sloan Foundation under Grant No. G-2023-21130.

Appendix A

Similar to [31], let us consider two classical Gaussian random processes, X and Y, that ⟨X⟩ = ⟨Y⟩ = 0 and ⟨XY⟩ = 0. Therefore, in our case, the Bj and ∂iBj are Gaussian processes, and they meet the requirement above that ⟨Bj⟩ = ⟨∂iBj⟩ = ⟨Bj∂iBj⟩ = 0.

Appendix B

Substituting eqs. [2] and [7], the eqs. become ∆pE = i c2/ϵijk X k1λ1 X k2λ2 / (ħω1/2ε0V1/2 ħω2/2ε0V1/2) ω1 Z ∆t0 dt h α(ω2)ak2λ2 e−iω2(t) - α √(ω2)a†k2λ2 eiω2(t) i h ak1λ1 e−iω1(t) - a†k1λ1 eiω1(t) (B2).

Appendix C

If the polarizability α(ω) is purely real, Eq. (C6) would vanish, which is unphysical since scattering still contributes to extinction. Therefore, αI (ω) in Eq. (C6) should be interpreted as the extinction contribution from both scattering and absorption.

Appendix D

If we ignore the imaginary contribution to αabs I in Eq. (31), then Im(χv) = 0, we could obtain Eqs. (15),(16), and (18) as. Therefore, the total momentum diffusion constant could be written as 2D ≡ ⟨∆p2⟩/∆t = 2ħ2/a6 c9π3 (χ2R + 3χR2) / (kBT hc9 ζ(8)Γ(9)).

Appendix E

The solution, for ξt >> 1, is w(v, t) = m / 2πkBT1/2 e−mv2/2kBT. Therefore, it also implies that the particle velocities follow the Maxwell distribution since the momentum diffusion constant and the drag force are subject to the assumption of ⟨1/2mv2⟩ = 1/2 kBT.

References

[1] M. Simon, L. Heflinger, Torrance Ca, and A. Geim. Diamagnetically stabilized magnet levitation. American Journal of Physics, 69, 06 2001

[2] M. D. Simon and A. K. Geim. Diamagnetic levitation: Flying frogs and floating magnets (invited). Journal of Applied Physics, 87(9):6200–6204, May 2000

[3] Joachim Hofer et al. High-Q Magnetic Levitation and Control of Superconducting Microspheres at Millikelvin Temperatures. Phys. Rev. Lett., 131(4):043603, 2023

[4] R. Nakashima. Diamagnetic levitation of a milligram-scale silica using permanent magnets for the use in a macroscopic quantum measurement. Phys. Lett. A, 384:126592, 2020

[5] Xianfeng Chen, Satya K Ammu, Kunal Masania, Peter G Steeneken, and Farbod Alijani.

Improvements for AI systems

Bold header: Calculation of Decoherence Rate in Time-Dependent Fields

This system can accurately compute the decoherence rate for a magnetic nanoparticle in quantum superposition in the presence of the fluctuating electromagnetic field by using the fluctuation-dissipation theorem in the long-wavelength limit, specifically identifying that the dominant contribution to the momentum diffusion constant will arise from the time-varying electric field on the magnetic dipoles.

Bold header: Drag Force Modeling and Comparison

The improved system can calculate the drag force on a single nanoparticle by comparing results with those for dielectric properties, specifically showing how the drag force on a diamagnetic nanoparticle arises from external electromagnetic-field fluctuations, and compare our results with those for the nanoparticle’s dielectric properties.

Bold header: Quantum Gravity Induced Entanglement (QGEM) Limits

The system can assess the feasibility of quantum gravity induced entanglement of matter (QGEM) protocol by calculating the decoherence factor between two levitated magnetic dipoles, which is defined as F(x1 − x2) = R11 − R12, providing a quantitative measure of spatial coherence loss.

Bold header: Relativistic Drag Force Prediction

The AI can predict the drag force on magnetic dipole moments in a magnetic field in a laboratory frame by utilizing the relativistic transformation and calculating [Find]xx, leading to the result that the thermal drag for a nanodiamagnetic sphere can be written by assuming αI (ω) is constant, and by using the Clausius-Mossotti relation, Eq. 30.