Apparent violations of the second law in the quantum-classical dynamics of interacting levitated nanoparticles

Pedro V. Paraguass\'u; Thiago Guerreiro

REVIEW 1 major objections 4 minor 70 references

Apparent violations of the second law in the quantum-classical dynamics of interacting levitated nanoparticles

T0 review · 1 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For a levitated nanoparticle driven by a quantum partner, the probability of an apparent second-law violation is bounded by 50 percent because the work distribution is Gaussian.

desk verdict A credible application of the known Gaussian 50% bound to levitated hybrid quantum-classical dynamics, but the work protocol has a free-energy reference mismatch that needs fixing. read the letter →

arxiv 2501.12305 v2 pith:NNOU4CCX submitted 2025-01-21 quant-ph cond-mat.stat-mech

Pedro V. Paraguassú , Thiago Guerreiro This is my paper · ORCID

classification quant-phcond-mat.stat-mech

keywords stochasticthermodynamicsfreelunchsecondlawviolationslevitatednanoparticlesquantum-classicalinteractionGaussianworkdistributionsqueezed-coherentstatesirreversible

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies apparent violations of the second law of thermodynamics—'free lunches,' where the work done on a particle falls below the free-energy change—in a pair of levitated nanoparticles, one classical and one quantum. It argues that in this system the thermodynamic work is exactly Gaussian distributed, which caps the probability of a free lunch at 50 percent. It then shows how the noise source matters: classical thermal noise produces free lunches more readily than the colored quantum noise induced by the quantum particle, while squeezing or adding phonons lowers the free-lunch probability at local minima even though fluctuations grow. The result matters because it transfers a general stochastic-thermodynamics bound to a concrete quantum-classical levitated platform and provides quantitative predictions that experiments could test.

What carries the argument

The argument is carried by the linear classical Langevin equation (Eq. 4) driven by a deterministic force f(t) plus quantum and thermal noise. Because the position x(t) is a linear functional of Gaussian noises, the thermodynamic work w[x] = -∫₀^τ ḟ(t)x(t)dt is Gaussian; its mean W satisfies W ≥ ΔF by the second law, so the free-lunch probability in Eq. (17), P = 1/2[1 + erf(χ)] with χ = (ΔF − W)/(√2 σ_W) ≤ 0, is capped at 50 percent. The relevant control parameter is the signed ratio I_Wirr = W_irr/σ_W of mean irreversible work to work fluctuation width, which sets the error-function argument; squeezing changes this ratio because the irreversible work scales as $\cosh$(2r) while the variance scales as $\cosh$^{3/2}(r).

What would settle it

Measure the per-trajectory work histogram for a levitated nanoparticle driven by the same oscillating force and compute its third cumulant: if the skewness is measurably nonzero, the Gaussian bound of Eq. (17) does not apply. Alternatively, recompute Eq. (9) with the initial quench term included and check whether the free-lunch probabilities change materially.

Watch

Extended reading notes

Core claim

The central discovery is that the work performed on the classical nanoparticle by its quantum counterpart is a Gaussian random variable, so the probability of a free lunch—an individual trajectory with w < ΔF—cannot exceed 50 percent as long as the mean work obeys the second law W ≥ ΔF. The paper computes this probability for coherent and squeezed-coherent states, showing that thermal white noise yields more free lunches than the colored quantum-induced noise, and that increasing phonon number or squeezing lowers the probability at local minima even though both increase fluctuations. At special process durations the mean irreversible work and the variance vanish together, the work distribution collapses to a delta function, and the free-lunch probability sits exactly at 50 percent.

Load-bearing premise

The calculation assumes the nanoparticle starts in equilibrium without the deterministic force and that the work integral does not include the energy spent switching that force on at t = 0, even though the force is generally nonzero at that instant.

Editorial extensions

If this is right

In this levitated setup, no process with Gaussian work can make free lunches more likely than 50 percent; exceeding that cap would require an asymmetric, skewed work distribution.
At special process durations the process becomes reversible (W_irr = 0), the work distribution narrows to a delta function, and the free-lunch probability is exactly 50 percent.
For identical parameters, classical thermal noise yields more free lunches than the quantum-induced colored noise.
Squeezing the quantum state raises fluctuations but lowers the free-lunch probability at local minima, because the irreversible work grows faster than the fluctuation width.
Increasing the phonon number lowers the free-lunch probability, while longer process durations push the quantum case toward the 50 percent ceiling asymptotically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending beyond the paper: the 50 percent cap applies to any linear driven system with Gaussian work, so non-Gaussian noise or nonlinear driving in levitated setups could push free-lunch probabilities above half; measuring the skewness of the work histogram would immediately reveal such a regime.
Extending beyond the paper: including the work required to switch on the deterministic force at t = 0 would shift the computed free-energy difference and could reduce the reported free-lunch probabilities; recomputing with this quench term would separate the physical effect from a protocol convention.
Extending beyond the paper: free-lunch probability could serve as a state-sensitive probe of quantum backaction in hybrid quantum-classical systems, since it distinguishes thermal from quantum noise without measuring the quantum particle directly.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely new: it applies the free-lunch concept from stochastic thermodynamics to a hybrid quantum-classical levitated nanoparticle model and shows how the 50% Gaussian bound plays out for coherent and squeezed-coherent states. The analytical work moments, the separation of thermal versus quantum noise, and the squeezing scalings are all derived cleanly from the linear dynamics. That part deserves credit. The model itself is imported from the authors' prior work, which is fine given that it is published and not silently containing the main result.

But there is a real soft spot, and it is load-bearing. The free energy difference in Eq. (12) is computed in Appendix A from an initial equilibrium distribution with no force, P0(x0) ~ exp(-β mω_c^2 x0^2/2). The work protocol in Eq. (9), however, starts with f(0) generally nonzero: for the coherent force (7) with θ = 0, f(0) = -2√n(ℏg/x0). So the system is not initially in equilibrium for the potential actually in place. Either the initial state should be the tilted distribution including f(0)x0, or the sudden switch-on work -f(0)x(0) must be added to Eq. (9). The stress-test note is right: this changes the reference ΔF itself, and the paper's claim that the phase dependence is small does not address that. Since all reported free-lunch probabilities are P(w < ΔF), the numbers in Figs. 2, 3, and 6 are referenced to the wrong free-energy difference for θ = 0 data. That is a fixable error, but it has to be fixed.

Two smaller issues: the delta-function discussion around Eq. (23) is mathematically sloppy—a Dirac delta at the boundary does not have a well-defined half-mass integral, and the cleaner statement is that the limiting Gaussian CDF at the mean is 1/2. And the scalings like σ ∝ cosh^{3/2}(r) are stated without derivation; a symbolic-checkable appendix line would help.

The known Gaussian bound is not new, but its application to this specific hybrid system is. If the protocol mismatch is corrected, the results are likely worth publishing. I would send it to peer review with a request for revision rather than desk reject it.

Referee Report

1 major / 4 minor

Summary. The paper studies apparent violations of the second law ("free lunches") in a hybrid quantum-classical system of two Coulomb-interacting levitated nanoparticles, using a stochastic optomechanical model developed in the authors' previous work. The classical nanoparticle obeys a Langevin equation driven by thermal white noise, a quantum-induced colored noise, and a deterministic force originating from a coherent or squeezed-coherent state of the quantum nanoparticle. The authors define work via w = -∫ ḟ(t)x(t)dt, show that the resulting work distribution is Gaussian, derive the bound P(w < ΔF) ≤ 50%, identify times at which the mean irreversible work vanishes, and compare free-lunch probabilities for purely classical thermal noise, purely quantum noise, and squeezed-coherent states. They conclude that thermal noise produces more free lunches than quantum noise and that squeezing reduces the free-lunch probability at local minima despite increasing fluctuations.

Significance. If the quantitative results are correct, the paper provides a useful bridge between levitated optomechanics and stochastic thermodynamics, with explicit analytic expressions for the work mean and variance, a clean decomposition of the variance into thermal and quantum parts, and falsifiable predictions for how coherent amplitude, phase, phonon number, and squeezing affect apparent second-law violations. The use of realistic experimental parameters from levitated-particle setups and the extension to squeezed-coherent states are strengths, and the Gaussian-work bound is derived in a transparent way. The manuscript does not include code or machine-checked computations, and the symbolic integrations are only summarized, so the numerical figures cannot be fully reproduced from the text alone; nevertheless, the analytical structure is sufficiently clear to be checked by an interested reader.

major comments (1)

[Section V.B, Eq. (23)] The Dirac-delta limiting argument in Section V.B is mathematically fragile. For a Gaussian with mean W = ΔF and positive variance, P(w < ΔF) = 1/2, but in the exact zero-variance limit the work is deterministic and equal to ΔF, so the strict inequality w < ΔF has probability 0, or at best a convention-dependent value if one integrates a delta distribution with a step at the atom. The transition from Eq. (17) to Eq. (23) is therefore not justified, and the assertion that the maximum P = 50% occurs at the variance zeros in the classical case is not supported by the model. This directly affects the interpretation of Fig. 2, where the maxima are identified with the vanishing of σβ^2.

minor comments (4)

[Figure 6 caption and Section VI.D] The caption of Fig. 6b states "with r = 1", while the text fixes the squeezing parameter at r = 0.5; the figure and text should be reconciled.
[Sections II-III, Eqs. (6)-(7), (24), Appendix A] The frequency notation is inconsistent: Eq. (7) and Eq. (24) use ω where the quantum-particle frequency ω_y was defined in Section II, Eq. (6) uses ω_y, and Appendix A uses ω_c for the classical frequency ω_x; the authors should define and use one symbol consistently, and similarly for x0 versus x_zpf.
[Figure captions, Fig. 2 inset and Fig. 6] The figure captions write P(W < ΔF) with a capital W; since W denotes the mean work in Eq. (10), the probability should be written P(w < ΔF) to distinguish the random work w from its mean.
[Reference list] Reference [15] is cited as "Manuscript in preparation"; if it is not publicly available, the authors should cite a published version or remove the citation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the free-lunch probability is a derived consequence of an explicitly imported linear-Langevin model, not a repackaging of the model's output.

full rationale

The paper's derivation chain is not circular. The dynamics (Langevin equation (4), deterministic force (7), noise correlations (6), and squeezed-state generalizations (24)-(26)) is imported from the authors' prior work [53] and [60]. This import is a model premise, not the paper's conclusion: the free-lunch probability (17) is then computed by linear response from the mean and variance of w, with W from Eq. (B2) and sigma^2_W from Eq. (B3). Equation (12)/(A3) for Delta F is derived independently from the equilibrium Boltzmann distributions (A1)-(A2) of a shifted harmonic oscillator; it is not fitted to W and is not defined as (W minus something). The 50% bound follows from the Gaussian form (14) together with W >= Delta F, which is a mathematical consequence of the assumed distribution and not a restatement of any fitted parameter. The self-citation to [53] is load-bearing in the sense that the model is taken from there, but [53] is an externally published prior model and does not contain the present paper's target quantities, so under the independence rule it does not create circularity. Two caveats lie outside the circularity verdict: Appendix B omits the advertised closed-form work moments ('Any symbolic computation software can handle the calculations easily'), and the protocol may be internally inconsistent if f(0) is nonzero because the Delta F of Eq. (12) starts from a force-free equilibrium while the work integral (9) assumes the force is already present at t=0. These are correctness risks, not cases of the derivation reducing to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to produce the central results; the phonon number n and squeezing parameter r are varied as model inputs. The analysis relies on the quantum-classical dynamics of a prior paper by the same authors, on the standard stochastic thermodynamics work definition, and on specific equilibrium assumptions for the initial and final states.

assumptions (4)

domain assumption The quantum-classical stochastic dynamics in Eqs. (4), (6), and (7), taken from ref. [53], describes the interaction of the two levitated nanoparticles.
Section II; the entire paper is an application of this model rather than a derivation of it.
standard math The work functional w[x] = -∫ ḟ(t) x(t) dt and the second law bound W ≥ ΔF apply to the process studied.
Section III; standard stochastic thermodynamics, but its validity here depends on a clearly defined force protocol.
domain assumption The equilibrium free energy difference is ΔF = -f(τ)^2/(2mω_c^2) as derived in Appendix A.
Appendix A, Eqs. (A1)-(A3); assumes the initial equilibrium distribution has no deterministic force and the final distribution is equilibrated with force f(τ).
domain assumption Initial conditions x0 and v0 are sampled from the thermal equilibrium distribution of the unperturbed harmonic oscillator.
Section II and Appendix B; this is needed to compute the work mean and variance; other initial preparations would change the free lunch probabilities.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Apparent violations of the second law in the quantum-classical dynamics of interacting levitated nanoparticles." pith.science (2026). https://pith.science/paper/NNOU4CCX

@misc{pith2026250112305,
  author       = {Pith},
  title        = {Pith review of: Apparent violations of the second law in the quantum-classical dynamics of interacting levitated nanoparticles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NNOU4CCX}},
  note         = {Machine review of arXiv:2501.12305}
}

read the original abstract

Random exchanges of energy arise naturally in stochastic systems. As a consequence, apparent violations of the second law of thermodynamics can occur, as it holds true on average. Here we investigate the occurrence of these apparent violations -- termed free lunches -- in a quantum-classical system comprised of levitated nanoparticles exchanging energy via the Coulomb interaction. We consider different initial states for the quantum system, and how these exert work and fluctuations upon the classical particle affecting the probability of free lunches. With that, we initiate the study of hybrid quantum-classical systems through the lens of stochastic thermodynamics.

Figures

Figures reproduced from arXiv: 2501.12305 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 55 canonical work pages

[1]

The average ⟨

After interacting, the quantum particle ex- 9 erts a deterministic force, and in the end of the pro- cess, the final random variable potential energy will be uf = 1 2 mω2 c x2 τ − f (τ )xτ . The average ⟨. . .⟩0 is obtained by considering equilibrium of the variablex0 and xτ . The equilibrium distribution of both quantities are P0(x0) ∼ exp −β mω2 c 2 x2 ...

work page 2024
[2]

Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica A: Statistical Mechanics and its Applications 504, 176 (2018)

U. Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica A: Statistical Mechanics and its Applications 504, 176 (2018)

work page 2018
[3]

Millen, T

J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vami- vakas, Optomechanics with levitated particles, Reports on Progress in Physics 83, 026401 (2020)

2020
[4]

Tom´ e and M

T. Tom´ e and M. J. De Oliveira, Stochastic dynamics and irreversibility (Springer, 2015)

work page 2015
[5]

Sekimoto, Stochastic Energetics, Lecture Notes in Physics, Vol

K. Sekimoto, Stochastic Energetics, Lecture Notes in Physics, Vol. 799 (Springer, 2010)

work page 2010
[6]

Peliti and S

L. Peliti and S. Pigolotti, Stochastic thermodynamics: an introduction (Princeton University Press, 2021)

2021
[7]

P. V. Paraguass´ u, L. Defaveri, S. M. D. Queir´ os, and W. A. Morgado, Probabilities for informational free 10 lunches in stochastic thermodynamics, Journal of Statis- tical Mechanics: Theory and Experiment 2022, 123204 (2022)

work page 2022
[8]

Saha and S

B. Saha and S. Mukherji, Work distribution function for a brownian particle driven by a nonconservative force, The European Physical Journal B 88, 1 (2015)

work page 2015

Show all 70 references

[9]

Ryabov, M

A. Ryabov, M. Dierl, P. Chvosta, M. Einax, and P. Maass, Work distribution in a time-dependent logarithmic–harmonic potential: exact results and asymptotic analysis, Journal of Physics A: Mathemati- cal and Theoretical 46, 075002 (2013)

2013
[10]

Chvosta, D

P. Chvosta, D. Lips, V. Holubec, A. Ryabov, and P. Maass, Statistics of work performed by optical tweez- ers with general time-variation of their stiffness, Journal of Physics A: Mathematical and Theoretical 53, 275001 (2020)

2020
[11]

Cohen, Properties of nonequilibrium steady states: a path integral approach, Journal of Statistical Mechanics: Theory and Experiment 2008, P07014 (2008)

E. Cohen, Properties of nonequilibrium steady states: a path integral approach, Journal of Statistical Mechanics: Theory and Experiment 2008, P07014 (2008)

2008
[12]

Seifert, Entropy production along a stochastic trajec- tory and an integral fluctuation theorem, Physical review letters 95, 040602 (2005)

U. Seifert, Entropy production along a stochastic trajec- tory and an integral fluctuation theorem, Physical review letters 95, 040602 (2005)

2005
[13]

D. S. Salazar, Detailed fluctuation theorem bound for apparent violations of the second law, Physical Review E 104, L062101 (2021)

2021
[14]

Merhav and Y

N. Merhav and Y. Kafri, Statistical properties of entropy production derived from fluctuation theorems, Journal of Statistical Mechanics: Theory and Experiment 2010, P12022 (2010)

2010
[15]

Barros, S

N. Barros, S. Ciliberto, and L. Bellon, Probabilistic work extraction on a classical oscillator beyond the second law, Physical Review Letters 133, 057101 (2024)

2024
[16]

Hertz et al, Apparent violations of the second law in the breathing parabola model (2025), (Manuscript in preparation)

A. Hertz et al, Apparent violations of the second law in the breathing parabola model (2025), (Manuscript in preparation)

2025
[17]

Debiossac, M

M. Debiossac, M. L. Rosinberg, E. Lutz, and N. Kiesel, Non-markovian feedback control and acausality: An ex- perimental study, Physical Review Letters 128, 200601 (2022)

2022
[18]

Rademacher, M

M. Rademacher, M. Konopik, M. Debiossac, D. Grass, E. Lutz, and N. Kiesel, Nonequilibrium control of thermal and mechanical changes in a levitated system, Physical Review Letters 128, 070601 (2022)

2022
[19]

Millen and J

J. Millen and J. Gieseler, Single particle thermodynam- ics with levitated nanoparticles, Thermodynamics in the Quantum Regime: Fundamental Aspects and New Di- rections , 853 (2018)

2018
[20]

Gonzalez-Ballestero, M

C. Gonzalez-Ballestero, M. Aspelmeyer, L. Novotny, R. Quidant, and O. Romero-Isart, Levitodynamics: Lev- itation and control of microscopic objects in vacuum, Sci- ence 374, eabg3027 (2021)

2021
[21]

Raynal, T

D. Raynal, T. de Guillebon, D. Gu´ ery-Odelin, E. Trizac, J.-S. Lauret, and L. Rondin, Shortcuts to equilibrium with a levitated particle in the underdamped regime, Physical Review Letters 131, 087101 (2023)

2023
[22]

Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Physical Review X 7, 021051 (2017)

S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Physical Review X 7, 021051 (2017)

2017
[23]

Gieseler, L

J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, Non- equilibrium steady state of a driven levitated particle with feedback cooling, New Journal of Physics17, 045011 (2015)

2015
[24]

Gieseler and J

J. Gieseler and J. Millen, Levitated nanoparticles for mi- croscopic thermodynamics—a review, Entropy 20, 326 (2018)

2018
[25]

Kremer, I

O. Kremer, I. Califrer, D. Tandeitnik, J. P. von der Weid, G. Tempor˜ ao, and T. Guerreiro, All-electrical cooling of an optically levitated nanoparticle, Physical Review Ap- plied 22, 024010 (2024)

2024
[26]

Vanner, J

M. Vanner, J. Hofer, G. Cole, and M. Aspelmeyer, Cooling-by-measurement and mechanical state tomogra- phy via pulsed optomechanics, Nature communications 4, 2295 (2013)

2013
[27]

Piotrowski, D

J. Piotrowski, D. Windey, J. Vijayan, C. Gonzalez- Ballestero, A. de los R ´ ıos Sommer, N. Meyer, R. Quidant, O. Romero-Isart, R. Reimann, and L. Novotny, Simulta- neous ground-state cooling of two mechanical modes of a levitated nanoparticle, Nature Physics 19, 1009 (2023)

2023
[28]

Romero-Isart, M

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, Toward quantum superposition of living organisms, New Journal of Physics 12, 033015 (2010)

2010
[29]

Tebbenjohanns, M

F. Tebbenjohanns, M. L. Mattana, M. Rossi, M. Frim- mer, and L. Novotny, Quantum control of a nanoparticle optically levitated in cryogenic free space, Nature 595, 378 (2021)

2021
[30]

Deli´ c, M

U. Deli´ c, M. Reisenbauer, K. Dare, D. Grass, V. Vuleti´ c, N. Kiesel, and M. Aspelmeyer, Cooling of a levitated nanoparticle to the motional quantum ground state, Sci- ence 367, 892 (2020)

2020
[31]

Magrini, P

L. Magrini, P. Rosenzweig, C. Bach, A. Deutschmann- Olek, S. G. Hofer, S. Hong, N. Kiesel, A. Kugi, and M. Aspelmeyer, Real-time optimal quantum control of mechanical motion at room temperature, Nature 595, 373 (2021)

2021
[32]

Pikovski, M

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. Kim, and ˇC. Brukner, Probing planck-scale physics with quantum optics, Nature Physics 8, 393 (2012)

2012
[33]

Neumeier, M

L. Neumeier, M. A. Ciampini, O. Romero-Isart, M. As- pelmeyer, and N. Kiesel, Fast quantum interference of a nanoparticle via optical potential control, Proceedings of the National Academy of Sciences 121, e2306953121 (2024)

2024
[34]

Kamba and K

M. Kamba and K. Aikawa, Revealing the velocity un- certainties of a levitated particle in the quantum ground state, Physical Review Letters 131, 183602 (2023)

2023
[35]

P. T. Grochowski, H. Pichler, C. A. Regal, and O. Romero-Isart, Quantum control of continuous systems via nonharmonic potential modulation, arXiv preprint arXiv:2311.16819 (2023)

2023 arXiv
[36]

Weiss, M

T. Weiss, M. Roda-Llordes, E. Torrontegui, M. As- pelmeyer, and O. Romero-Isart, Large quantum delocal- ization of a levitated nanoparticle using optimal control: Applications for force sensing and entangling via weak forces, Physical Review Letters 127, 023601 (2021)

2021
[37]

D. C. Moore and A. A. Geraci, Searching for new physics using optically levitated sensors, Quantum Science and Technology 6, 014008 (2021)

2021
[38]

Chaste, A

J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Ru- rali, and A. Bachtold, A nanomechanical mass sensor with yoctogram resolution, Nature nanotechnology7, 301 (2012)

2012
[39]

Ranjit, D

G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, Attonewton force detection using mi- crospheres in a dual-beam optical trap in high vacuum, Physical Review A 91, 051805 (2015)

2015
[40]

Ranjit, M

G. Ranjit, M. Cunningham, K. Casey, and A. A. Geraci, Zeptonewton force sensing with nanospheres in an optical lattice, Physical Review A 93, 053801 (2016). 11

2016
[41]

Toroˇ s, G

M. Toroˇ s, G. Gasbarri, and A. Bassi, Colored and dis- sipative continuous spontaneous localization model and bounds from matter-wave interferometry, Physics Letters A 381, 3921 (2017)

2017
[42]

G. C. Ghirardi, P. Pearle, and A. Rimini, Markov pro- cesses in hilbert space and continuous spontaneous local- ization of systems of identical particles, Physical Review A 42, 78 (1990)

1990
[43]

Bassi, K

A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ul- bricht, Models of wave-function collapse, underlying the- ories, and experimental tests, Reviews of Modern Physics 85, 471 (2013)

2013
[44]

G. Afek, F. Monteiro, J. Wang, B. Siegel, S. Ghosh, and D. C. Moore, Limits on the abundance of mil- licharged particles bound to matter, Physical Review D 104, 012004 (2021)

2021
[45]

G. Afek, D. Carney, and D. C. Moore, Coherent scat- tering of low mass dark matter from optically trapped sensors, Physical review letters 128, 101301 (2022)

2022
[46]

Monteiro, G

F. Monteiro, G. Afek, D. Carney, G. Krnjaic, J. Wang, and D. C. Moore, Search for composite dark matter with optically levitated sensors, Physical Review Letters 125, 181102 (2020)

2020
[47]

Arvanitaki and A

A. Arvanitaki and A. A. Geraci, Detecting high- frequency gravitational waves with optically levitated sensors, Physical review letters 110, 071105 (2013)

2013
[48]

Carlesso, A

M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht, Testing the gravitational field generated by a quantum superposition, New Journal of Physics 21, 093052 (2019)

2019
[49]

S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroˇ s, M. Paternostro, A. A. Geraci, P. F. Barker, M. Kim, and G. Milburn, Spin entanglement witness for quantum gravity, Physical review letters 119, 240401 (2017)

2017
[50]

T. W. Van De Kamp, R. J. Marshman, S. Bose, and A. Mazumdar, Quantum gravity witness via entangle- ment of masses: Casimir screening, Physical Review A 102, 062807 (2020)

2020
[51]

Gasbarri, A

G. Gasbarri, A. Belenchia, M. Carlesso, S. Donadi, A. Bassi, R. Kaltenbaek, M. Paternostro, and H. Ul- bricht, Testing the foundation of quantum physics in space via interferometric and non-interferometric exper- iments with mesoscopic nanoparticles, Communications Physics 4, ...

2021
[52]

R. J. Marshman, A. Mazumdar, and S. Bose, Locality and entanglement in table-top testing of the quantum na- ture of linearized gravity, Physical Review A101, 052110 (2020)

2020
[53]

Hanif, D

F. Hanif, D. Das, J. Halliwell, D. Home, A. Mazumdar, H. Ulbricht, and S. Bose, Testing whether gravity acts as a quantum entity when measured, Physical Review Letters 133, 180201 (2024)

2024
[54]

P. V. Paraguass´ u, L. Abrah˜ ao, and T. Guerreiro, Quantum-induced stochastic optomechanical dynamics, Journal of the Optical Society of America B 41, 2798 (2024)

2024
[55]

Rieser, M

J. Rieser, M. A. Ciampini, H. Rudolph, N. Kiesel, K. Hornberger, B. A. Stickler, M. Aspelmeyer, and U. Deli´ c, Tunable light-induced dipole-dipole interaction between optically levitated nanoparticles, Science 377, 987 (2022)

2022
[56]

Rudolph, U

H. Rudolph, U. Deli´ c, M. Aspelmeyer, K. Hornberger, and B. A. Stickler, Force-gradient sensing and entangle- ment via feedback cooling of interacting nanoparticles, Physical review letters 129, 193602 (2022)

2022
[57]

A. O. Caldeira and A. J. Leggett, Path integral approach to quantum brownian motion, Physica A: Statistical me- chanics and its Applications 121, 587 (1983)

1983
[58]

Pelargonio and A

S. Pelargonio and A. Zaccone, Generalized langevin equa- tion with shear flow and its fluctuation-dissipation theo- rems derived from a caldeira-leggett hamiltonian, Physi- cal Review E 107, 064102 (2023)

2023
[59]

Janssen and W

N. Janssen and W. Zwerger, Nonlinear transport of po- larons, Physical Review B 52, 9406 (1995)

1995
[60]

R. P. Feynman and F. L. Vernon Jr, The theory of a gen- eral quantum system interacting with a linear dissipative system, Annals of physics 281, 547 (2000)

2000
[61]

Parikh, F

M. Parikh, F. Wilczek, and G. Zahariade, Signatures of the quantization of gravity at gravitational wave detec- tors, Physical Review D 104, 046021 (2021)

2021
[62]

Parikh, F

M. Parikh, F. Wilczek, and G. Zahariade, Quantum me- chanics of gravitational waves, Physical Review Letters 127, 081602 (2021)

2021
[63]

Cho and B.-L

H.-T. Cho and B.-L. Hu, Graviton noise on tidal forces and geodesic congruences, Physical Review D 107, 084005 (2023)

2023
[64]

Chawla and M

S. Chawla and M. Parikh, Quantum gravity corrections to the fall of an apple, Physical Review D 107, 066024 (2023)

2023
[65]

S. Bo, S. H. Lim, and R. Eichhorn, Functionals in stochastic thermodynamics: how to interpret stochastic integrals, Journal of Statistical Mechanics: Theory and Experiment 2019, 084005 (2019)

2019
[66]

Jarzynski, Nonequilibrium equality for free energy dif- ferences, Physical Review Letters 78, 2690 (1997)

C. Jarzynski, Nonequilibrium equality for free energy dif- ferences, Physical Review Letters 78, 2690 (1997)

1997
[67]

E. A. Jackson, Equilibrium statistical mechanics (Courier Corporation, 2000)

2000
[68]

Danger, A

J.-L. Danger, A. Ghazel, E. Boutillon, and H. Laa- mari, Efficient fpga implementation of gaussian noise generator for communication channel emulation, in ICECS 2000. 7th IEEE International Conferenceon Electronics, Circuits and Systems, Vol. 1 (IEEE, 2000) pp. 366–369

2000
[69]

Kremer, D

O. Kremer, D. Tandeitnik, R. Mufato, I. Califrer, B. Calderoni, F. Calliari, B. Melo, G. Tempor˜ ao, and T. Guerreiro, Perturbative nonlinear feedback forces for optical levitation experiments, Physical Review A 109, 023521 (2024)

2024
[70]

W. P. Bowen and G. J. Milburn, Quantum optomechanics (CRC press, 2015)

2015

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

The average ⟨

After interacting, the quantum particle ex- 9 erts a deterministic force, and in the end of the pro- cess, the final random variable potential energy will be uf = 1 2 mω2 c x2 τ − f (τ )xτ . The average ⟨. . .⟩0 is obtained by considering equilibrium of the variablex0 and xτ . The equilibrium distribution of both quantities are P0(x0) ∼ exp −β mω2 c 2 x2 ...

work page 2024

[2] [2]

Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica A: Statistical Mechanics and its Applications 504, 176 (2018)

U. Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica A: Statistical Mechanics and its Applications 504, 176 (2018)

work page 2018

[3] [3]

Millen, T

J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vami- vakas, Optomechanics with levitated particles, Reports on Progress in Physics 83, 026401 (2020)

2020

[4] [4]

Tom´ e and M

T. Tom´ e and M. J. De Oliveira, Stochastic dynamics and irreversibility (Springer, 2015)

work page 2015

[5] [5]

Sekimoto, Stochastic Energetics, Lecture Notes in Physics, Vol

K. Sekimoto, Stochastic Energetics, Lecture Notes in Physics, Vol. 799 (Springer, 2010)

work page 2010

[6] [6]

Peliti and S

L. Peliti and S. Pigolotti, Stochastic thermodynamics: an introduction (Princeton University Press, 2021)

2021

[7] [7]

P. V. Paraguass´ u, L. Defaveri, S. M. D. Queir´ os, and W. A. Morgado, Probabilities for informational free 10 lunches in stochastic thermodynamics, Journal of Statis- tical Mechanics: Theory and Experiment 2022, 123204 (2022)

work page 2022

[8] [8]

Saha and S

B. Saha and S. Mukherji, Work distribution function for a brownian particle driven by a nonconservative force, The European Physical Journal B 88, 1 (2015)

work page 2015