Pith. sign in

REVIEW 3 major objections 5 minor 58 references

Entangling two levitated particles in free space via trap modulation and Bayesian feedback

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper proposes that periodically modulating the trapping beam, together with Kalman-filter Bayesian feedback on continuous homodyne measurements, generates steady entanglement between two levitated nanoparticles with weak Coulomb…

desk verdict Solid conditional-entanglement work undercut by a sign error in the excess-noise equation that the headline unconditional claims depend on. read the letter →

arxiv 2505.12865 v1 pith:H5EEO7H5 submitted 2025-05-19 quant-ph

Jinke Cao , Qi Guo , Huatang Tan This is my paper · ORCID
classification quant-ph
keywords quantumentanglementlevitatednanoparticlesoptomechanicsBayesianfeedbackKalmanfilterparametricmodulationCoulombcouplingcontinuousmeasurement
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

The paper proposes a scheme to entangle two optically levitated nanoparticles in free space by combining periodic modulation of the trapping-beam amplitude with continuous homodyne detection and optimal Bayesian feedback. The central claim is that this combination yields both conditional and unconditional steady-state entanglement at Coulomb coupling strengths and detection efficiencies well below those required by earlier feedback-only proposals. The physical mechanism is that the modulation squeezes both the common and differential normal modes of the two-particle system, while the Coulomb interaction makes the two modes squeeze by different amounts; that squeezing imbalance is the resource that entangles the two particles. The paper supports the claim with simulations of logarithmic negativity and an effective-Hamiltonian analysis showing two parametric resonances.

What carries the argument

The central objects are the two normal modes defined by the beam-splitter transformation, plus the Kalman-filter/LQR feedback loop. After a rotating-wave approximation, the effective Hamiltonian reads $H_{\rm eff} = \Delta_+ \hat{c}_+^\dagger \hat{c}_+ + \frac{\omega_m\alpha}{4}(\hat{c}_+\hat{c}_+ + \hat{c}_+^\dagger\hat{c}_+^\dagger) + \Delta_- \hat{c}_-^\dagger \hat{c}_- + \frac{\omega_m\alpha}{4}(\hat{c}_-\hat{c}_- + \hat{c}_-^\dagger \hat{c}_-^\dagger)$, with detunings $\Delta_+ = \omega_m + \omega_m\alpha^2/4 - \Omega/2$ and $\Delta_- = \Delta_+ + 2g$. This Hamiltonian has two parametric resonances, at $\Omega \simeq 2\omega_m$ and $\Omega \simeq 2\omega_m + 4g$, where the two normal modes are strongly squeezed. The Coulomb interaction shifts one resonance relative to the other, producing a difference in the magnitude or direction of the two modes' squeezing; the homodyne measurement stabilizes the parametric dynamics, and the feedback suppresses the excess noise that would otherwise destroy the entanglement.

What would settle it

Measure the unconditional covariance matrix of two charged levitated particles with $g/\omega_m \approx 0.2$, trap modulation at $\Omega \approx 2\omega_m$, independent homodyne detection near $\eta \approx 0.5$, and LQR feedback: if the logarithmic negativity is not positive in steady state, the central claim is refuted. A cheaper numerical check is to simulate the same protocol with correlated Wiener noise between the two detection channels; if the entanglement vanishes, the independence assumption is doing the work.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a periodically modulated optical trap, combined with continuous homodyne measurement and a Kalman-filter/linear-quadratic-regulator feedback loop, can entangle two Coulomb-coupled levitated nanoparticles in steady state without strong coupling or near-unit detection efficiency. The unconditional entanglement survives for coupling strengths as low as $|g|/\omega_m \approx 0.1$ and detection efficiencies around $\eta \approx 0.5$, while conditional entanglement appears already near $\eta \approx 0.1$, in both repulsive and attractive interaction regimes. The modulation induces strong squeezing in both normal modes, and the Coulomb coupling shifts the differential-mode resonance so that the two modes' squeezing ellipses differ; this difference, not the Coulomb interaction itself, is what generates the entanglement. Parameters in the simulations match recent levitated-particle experiments, including trap frequency 29.6 kHz, room temperature and atmospheric pressure.

Load-bearing premise

The scheme assumes the two homodyne detectors read out the two particles' positions independently, with independent noise; if the back-scattered fields from the two particles interfere or cannot be separated, the Kalman filter's state estimate is corrupted and the simulated entanglement is not produced.

Editorial extensions

If this is right

  • Unconditional entanglement becomes compatible with coupling strengths an order of magnitude below the mechanical frequency and detection efficiencies near 50%, both within reach of current levitated-particle platforms.
  • Strong conditional entanglement ($\langle E_N^c\rangle > \ln 2$) can be reached by increasing the modulation depth, which the paper argues is difficult in conventional optomechanical schemes because of stability constraints.
  • The feedback can be implemented either with identical forces (requiring unequal charges on the two particles) or with independent forces (allowing equal charges), so the scheme adapts to different trap geometries.
  • Because the entanglement oscillates at the modulation period, any protocol that uses the generated entanglement must be synchronized with the modulation cycle.

Reading between the lines

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

  • Inference: A natural testable extension is to feed correlated noise into the two homodyne channels; if entanglement survives cross-talk between the scattered fields, the scheme is robust, whereas if it disappears, the independence of the two readouts is the truly load-bearing requirement.
  • Inference: The same squeezing-imbalance mechanism could in principle entangle more than two particles in a modulated common trap, as long as individual position readout remains possible, though the paper does not address multi-particle arrays.
  • Inference: The predicted double-‘V' structure of conditional entanglement versus modulation frequency and coupling strength provides a sharp experimental fingerprint: measuring that map would directly confirm the mechanism before attempting the full feedback protocol.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a continuous-measurement control scheme for entangling two optically levitated nanoparticles in free space. The trap frequencies are periodically modulated, the particles interact through a Coulomb potential, and two homodyne detectors monitor their positions. The authors use a Kalman filter to obtain conditional Gaussian states and an LQR/Bayesian feedback law to reduce excess noise, and they compute conditional and unconditional logarithmic negativity from the corresponding covariance matrices. They report that frequency modulation enhances both conditional and unconditional entanglement, allowing detection efficiencies as low as eta ~ 0.1-0.5 and Coulomb couplings |g|/omega_m ~ 0.1-0.2, and they explain the effect through unequal normal-mode squeezing induced by the Coulomb coupling. The parameters are taken from recent levitated-optomechanics experiments.

Significance. If the reported results survive scrutiny, the paper provides a concrete route toward unconditional entanglement of levitated particles with considerably relaxed experimental requirements compared with earlier Markovian-feedback proposals. The mechanism, namely modulation-induced squeezing of the two normal modes with a Coulomb-coupling-induced imbalance, is physically coherent and offers a testable prediction for the dependence of entanglement on g/omega_m, modulation depth, and detection efficiency. The paper applies standard Gaussian filtering and LQG control consistently, uses realistic parameters, and does not fit any free parameter to the target entanglement. No code or raw data is provided, so numerical reproducibility rests on the equations as written.

major comments (3)
  1. [Sec. II, Eq. (11)] Eq. (11) as printed has a sign error in the measurement-induced diffusion term. From Eq. (5a) with M = A - B K_opt, the excess-noise covariance obeys dV_ex/dt = M V_ex + V_ex M^T + 4 V_c C C^T V_c by Ito's lemma, because the stochastic term 2 V_c C dW contributes a positive diffusion. The printed equation contains -4 V_c C C^T V_c. In the open-loop limit K_opt = 0, the sum of Eq. (5b) and Eq. (11) would not reproduce dV_u/dt = A V_u + V_u A^T + N, and V_u can lose positive semidefiniteness. Since the unconditional entanglement results in Fig. 3 are computed from V_u = V_c + V_ex, this is load-bearing. Please correct Eq. (11), state explicitly which sign was used in the numerics, and re-run or confirm all unconditional results.
  2. [Sec. II, Eqs. (3)-(6)] The protocol assumes that the two homodyne detectors measure the back-scattered light of the two particles independently, with block-diagonal C and independent Wiener increments dW_k. If the scattered fields from the two traps overlap or a single detector collects light from both particles, the photocurrents become correlated, C acquires cross terms, and the Kalman-filter estimate and the resulting feedback are no longer those simulated. The paper should state this assumption explicitly in the experimental-implementation discussion and, ideally, provide a sensitivity estimate showing that small cross-detection does not destroy the entanglement.
  3. [Sec. II, Eq. (8)] The LQR controller minimizes the cooling cost in Eq. (8); it is optimal for the chosen P and Q, not for entanglement. The abstract and introduction use the terms 'optimal control scheme' and 'optimal Bayesian feedback,' which are justified only with respect to that cost. If the authors intend to claim that the protocol is optimal for generating entanglement, an optimization over P and Q or a direct derivation of an entanglement-maximizing controller is needed. As written, the term 'optimal' should be qualified.
minor comments (5)
  1. [Sec. II, Eq. (5a)] The term 'B u' in Eq. (5a) should read 'B u dt' to keep the equation dimensionally consistent.
  2. [Eq. (3)] The photocurrent notation I_k(t) = sqrt(eta K_ba) <x_k>_c dt + dW_k(t) is dimensionally awkward; writing I_k dt = sqrt(eta K_ba) <x_k>_c dt + dW_k would be clearer.
  3. [Fig. 3 caption] The caption of Fig. 3 says 'Time-averaged Conditional entanglement', but the text and the panels describe unconditional entanglement; please correct the caption.
  4. [Sec. II, Eq. (9)] The terminal or boundary condition for the backward Riccati equation is not specified, and for a periodically modulated A(t) the method used to obtain the steady-state periodic K_opt should be described.
  5. [Sec. III, Eq. (16)] The effective Hamiltonian in Eq. (16) is derived under the assumptions alpha << 1 and g << omega_m, while the simulations use alpha = 0.2; the text should note that Eq. (16) is only a qualitative explanation and that the numerics solve the full time-dependent equations.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: entanglement is computed from first-principles covariance equations with independent experimental parameters; a possible sign error in Eq. (11) is a correctness risk, not circularity.

full rationale

No self-definitional, fitted-input, or self-citation-load-bearing circularity is present. The entanglement figures are produced by integrating the conditional and unconditional covariance equations (5) and (11), with matrices A, B, C, and N fixed by the Hamiltonian and by independent experimental parameters (omega_m = 29.6 kHz, gamma/omega_m = 1.4e-11, Gamma_th/omega_m = 2.5e-3, Gamma_ba/omega_m = 0.053, g/omega_m = 0.2) taken from refs. [32,33,56]. The Kalman-filter equations are standard and are cross-referenced to the textbook [43]; the self-citations [41,42] are not load-bearing because the same equations are independently established and are derivable from the stochastic master equation (4). The normal-mode effective Hamiltonian (16) is a post-hoc mechanistic explanation that does not enter the covariance simulation as an input. The note-added disclosure about [58] is an overlap remark, not a circular dependence. The only caveat worth flagging is a possible sign inconsistency in the unconditional excess-noise equation: Ito's lemma applied to d<X>_c = (A - B K_opt)<X>_c dt + 2 V_c C dW gives a positive diffusion contribution +4 V_c C C^T V_c, while Eq. (11) prints -4 V_c C C^T V_c. If taken literally this would violate V_u >= V_c and could affect the reported unconditional entanglement; however, this is a correctness and reproducibility risk, not a circularity, because no fitted parameter or self-citation is being disguised as a prediction. Overall, the derivation chain is self-contained and the claims are not equivalent to their inputs by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The scheme introduces no new physical entities; the claimed entanglement depends on standard linear quantum optics plus a set of hand-chosen control parameters (modulation depth, frequency, feedback cost weights). The main assumptions are the quadratic Coulomb coupling and independent homodyne detection.

free parameters (4)
  • modulation depth alpha = 0.2 (main figures)
    Hand-chosen control parameter; larger alpha enhances squeezing but is constrained by stability.
  • modulation frequency Omega = 2 omega_m (resonance)
    Set to twice the mechanical frequency for parametric resonance; chosen by hand.
  • control effort parameter q/omega_m = 0.1
    Chosen in the LQR cost function; controls feedback strength.
  • cooling cost matrix P scaling = diag(omega_m, omega_m, omega_m, omega_m)
    Picked to target cooling; not derived from entanglement optimization.
assumptions (5)
  • domain assumption Coulomb interaction truncated at second order, X1,2 << d
    Eq. (1) and preceding text; all Gaussian dynamics rely on purely quadratic coupling.
  • domain assumption Independent homodyne detection of each particle's scattered light
    Eq. (3) and discussion of two detectors; filter assumes independent Wiener noises.
  • domain assumption Gaussian initial states and linear dynamics
    Needed for Kalman filter representation, Eqs. (5a,5b).
  • standard math Rotating-wave approximation and alpha<<1, g<<omega_m for effective Hamiltonian Eq. (16)
    Used to explain squeezing enhancement; simulations use alpha=0.2 and g=0.2 omega_m, which are borderline.
  • domain assumption Markovian decoherence with constant rates gamma, K_th and measurement-induced K_ba
    Master equation Eq. (4) assumes standard Lindblad form.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Entangling two levitated particles in free space via trap modulation and Bayesian feedback." pith.science (2026). https://pith.science/paper/H5EEO7H5

@misc{pith2026250512865,
  author       = {Pith},
  title        = {Pith review of: Entangling two levitated particles in free space via trap modulation and Bayesian feedback},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5EEO7H5}},
  note         = {Machine review of arXiv:2505.12865}
}
read the original abstract

We propose an optimal control scheme for generating quantum entanglement between two optically-levitated nanoparticles in free space. Specifically, we consider that the mechanical motion frequencies of the two levitated particles are modulated by adjusting the amplitude of the trapping beam. The two particles are coupled through Coulomb interaction, and the particles' positions are continuously monitored via homodyne detection on the back-scattered light from both particles. By employing an optimal Bayesian feedback scheme, we achieve unconditional entanglement between the two particles in steady states. More precisely, a Kalman filter is used to estimate the states of the two particles and subsequently a linear quadratic regulator is applied to derive the optimal feedback forces exerted on the particles. Physically, periodic modulation enables significant quantum squeezing in both the common mode and the differential mode of the two particles. The Coulomb coupling between the particles introduces a difference in the squeezing of the two normal modes, thereby facilitating the entanglement of the two levitated particles. Our scheme allows for the realization of both conditional and unconditional entanglement at relatively low measurement efficiencies and with low requirements for Coulomb coupling strength, significantly enhancing the feasibility of implementation.

Figures

Figures reproduced from arXiv: 2505.12865 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of a scheme for entangling two [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Time-averaged conditional entanglement [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Time-averaged conditional logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Time-averaged Conditional entanglement [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Time-averaged conditional logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 37 canonical work pages

  1. [58]

    A. N. Poddubny, K. Winkler, B. A. Stickler, U. Deli´ c, M. Aspelmeyer, and A. V. Zasedatelev, Nonequilibrium entanglement between levitated masses under optimal control (2024), arXiv:2408.06251

    arXiv 2024

  2. [1]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009)

    2009

  3. [2]

    S. L. Braunstein and P. van Loock, Quantum informa- tion with continuous variables, Rev. Mod. Phys.77, 513 (2005)

    2005

  4. [3]

    Eisert, M

    J. Eisert, M. B. Plenio, S. Bose, and J. Hartley, To- wards quantum entanglement in nanoelectromechanical devices, Phys. Rev. Lett.93, 190402 (2004)

    work page 2004

  5. [4]

    L. F. Wei, Y.-x. Liu, C. P. Sun, and F. Nori, Probing tiny motions of nanomechanical resonators: Classical or quan- tum mechanical?, Phys. Rev. Lett.97, 237201 (2006)

    work page 2006

  6. [5]

    Poot and H

    M. Poot and H. S. van der Zant, Mechanical systems in the quantum regime, Phys. Rep.511, 273 (2012)

    work page 2012

  7. [6]

    Barzanjeh, A

    S. Barzanjeh, A. Xuereb, S. Gr¨ oblacher, M. Paternostro, C. A. Regal, and E. M. Weig, Optomechanics for quan- tum technologies, Nat. Phys.18, 15 (2022)

    2022

  8. [7]

    Fr¨ owis, P

    F. Fr¨ owis, P. Sekatski, W. D¨ ur, N. Gisin, and N. San- gouard, Macroscopic quantum states: Measures, fragility, and implementations, Rev. Mod. Phys.90, 025004 (2018)

    2018

Show all 58 references
  1. [8]

    Aspelmeyer, T

    M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys.86, 1391 (2014)

    2014

  2. [9]

    Vitali, S

    D. Vitali, S. Gigan, A. Ferreira, H. R. B¨ ohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. As- pelmeyer, Optomechanical entanglement between a mov- able mirror and a cavity field, Phys. Rev. Lett.98, 030405 (2007)

    2007

  3. [10]

    Genes, A

    C. Genes, A. Mari, P. Tombesi, and D. Vitali, Robust en- tanglement of a micromechanical resonator with output optical fields, Phys. Rev. A78, 032316 (2008)

    2008

  4. [11]

    S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Ham- merer, Quantum entanglement and teleportation in pulsed cavity optomechanics, Phys. Rev. A84, 052327 (2011)

    2011

  5. [12]

    H. Tan, L. F. Buchmann, H. Seok, and G. Li, Achieving steady-state entanglement of remote micromechanical os- cillators by cascaded cavity coupling, Phys. Rev. A87, 022318 (2013)

    2013

  6. [13]

    M. J. Woolley and A. A. Clerk, Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir, Phys. Rev. A89, 063805 (2014)

    2014

  7. [14]

    Ockeloen-Korppi, E

    C. Ockeloen-Korppi, E. Damsk¨ agg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sil- lanp¨ a¨ a, Stabilized entanglement of massive mechanical oscillators, Nature556, 478 (2018)

    2018

  8. [15]

    J. Li, I. M. Haghighi, N. Malossi, S. Zippilli, and D. Vi- tali, Generation and detection of large and robust entan- glement between two different mechanical resonators in cavity optomechanics, New J. Phys.17, 103037 (2015)

    2015

  9. [16]

    Millen, T

    J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas, Optomechanics with levitated particles, Rep. Prog. Phys. 83, 026401 (2020)

    2020

  10. [17]

    D. E. Chang, C. Regal, S. Papp, D. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, Cavity opto- mechanics using an optically levitated nanosphere, Proc. Nat. Acad. Sci.107, 1005 (2010)

    2010

  11. [18]

    Romero-Isart, A

    O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, Optically lev- itating dielectrics in the quantum regime: Theory and protocols, Phys. Rev. A83, 013803 (2011). 8

    2011

  12. [19]

    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- ence374, eabg3027 (2021)

    2021

  13. [20]

    Rashid, T

    M. Rashid, T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston, M. S. Kim, and H. Ulbricht, Experimen- tal realization of a thermal squeezed state of levitated optomechanics, Phys. Rev. Lett.117, 273601 (2016)

    2016

  14. [21]

    Militaru, M

    A. Militaru, M. Rossi, F. Tebbenjohanns, O. Romero- Isart, M. Frimmer, and L. Novotny, Ponderomotive squeezing of light by a levitated nanoparticle in free space, Phys. Rev. Lett.129, 053602 (2022)

    2022

  15. [22]

    Kamba, N

    M. Kamba, N. Hara, and K. Aikawa, Quantum squeez- ing of a levitated nanomechanical oscillator (2025), arXiv:2504.17944

    2025 arXiv

  16. [23]

    Kuang, R

    T. Kuang, R. Huang, W. Xiong, Y. Zuo, X. Han, F. Nori, C.-W. Qiu, H. Luo, H. Jing, and G. Xiao, Nonlinear multi-frequency phonon lasers with active levitated op- tomechanics, Nat. Phys.19, 414 (2023)

    2023

  17. [24]

    Gieseler, B

    J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, Subkelvin parametric feedback cooling of a laser-trapped nanoparticle, Phys. Rev. Lett.109, 103603 (2012)

    2012

  18. [25]

    Kiesel, F

    N. Kiesel, F. Blaser, U. Deli´ c, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, Cavity cooling of an optically lev- itated submicron particle, Proc. Nat. Acad. Sci.110, 14180 (2013)

    2013

  19. [26]

    Asenbaum, S

    P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, Cavity cooling of free silicon nanoparticles in high vacuum, Nat. Commun.4, 2743 (2013)

    2013

  20. [27]

    Tebbenjohanns, M

    F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain, and L. Novotny, Cold damping of an optically levitated nanoparticle to microkelvin temperatures, Phys. Rev. Lett.122, 223601 (2019)

    2019

  21. [28]

    J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, Optically levitated nan- odumbbell torsion balance and ghz nanomechanical ro- tor, Phys. Rev. Lett.121, 033603 (2018)

    2018

  22. [29]

    Reimann, M

    R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, Ghz rotation of an optically trapped nanoparticle in vacuum, Phys. Rev. Lett.121, 033602 (2018)

    2018

  23. [30]

    T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.- Q. Yin, and T. Li, Torsional optomechanics of a levitated nonspherical nanoparticle, Phys. Rev. Lett.117, 123604 (2016)

    2016

  24. [31]

    S. Kuhn, A. Kosloff, B. A. Stickler, F. Patolsky, K. Horn- berger, M. Arndt, and J. Millen, Full rotational control of levitated silicon nanorods, Optica4, 356 (2017)

    2017

  25. [32]

    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, Nature595, 373 (2021)

    2021

  26. [33]

    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, Nature595, 378 (2021)

    2021

  27. [34]

    Xu, Y.-H

    Y. Xu, Y.-H. Liu, C. Liu, and J.-Q. Liao, Simultaneous ground-state cooling of two levitated nanoparticles by co- herent scattering, Phys. Rev. A109, 053521 (2024)

    2024

  28. [35]

    Rudolph, U

    H. Rudolph, U. c. v. Deli´ c, M. Aspelmeyer, K. Horn- berger, and B. A. Stickler, Force-gradient sensing and en- tanglement via feedback cooling of interacting nanopar- ticles, Phys. Rev. Lett.129, 193602 (2022)

    2022

  29. [36]

    Winkler, A

    K. Winkler, A. V. Zasedatelev, B. A. Stickler, U. Deli´ c, A. Deutschmann-Olek, and M. Aspelmeyer, Steady-state entanglement of interacting masses in free space through optimal feedback control (2024), arXiv:2408.07492

    2024

  30. [37]

    Bothner, S

    D. Bothner, S. Yanai, A. Iniguez-Rabago, M. Yuan, Y. M. Blanter, and G. A. Steele, Cavity electromechanics with parametric mechanical driving, Nat. commun.11, 1589 (2020)

    2020

  31. [38]

    Ricci, R

    F. Ricci, R. A. Rica, M. Spasenovi´ c, J. Gieseler, L. Rondin, L. Novotny, and R. Quidant, Optically lev- itated nanoparticle as a model system for stochastic bistable dynamics, Nat. Commun.8, 15141 (2017)

    2017

  32. [39]

    O. c. v. ˇCernot´ ık and R. Filip, Strong mechanical squeez- ing for a levitated particle by coherent scattering, Phys. Rev. Res.2, 013052 (2020)

    2020

  33. [40]

    Mashaal, L

    A. Mashaal, L. Stefan, A. Ranfagni, L. Catalini, I. Cher- nobrovkin, T. Capelle, E. C. Langman, and A. Schliesser, Strong thermomechanical noise squeezing stabilized by feedback, Phys. Rev. Res.7, L012071 (2025)

    2025

  34. [41]

    Tan, Genuine photon-magnon-phonon einstein- podolsky-rosen steerable nonlocality in a continuously- monitored cavity magnomechanical system, Phys

    H. Tan, Genuine photon-magnon-phonon einstein- podolsky-rosen steerable nonlocality in a continuously- monitored cavity magnomechanical system, Phys. Rev. Res.1, 033161 (2019)

    2019

  35. [42]

    Tan and H

    H. Tan and H. Zhan, Strong mechanical squeezing and optomechanical steering via continuous monitoring in op- tomechanical systems, Phys. Rev. A100, 023843 (2019)

    2019

  36. [43]

    H. M. Wiseman and G. J. Milburn,Quantum measure- ment and control(Cambridge university press, 2009)

    2009

  37. [44]

    A. C. Doherty and K. Jacobs, Feedback control of quan- tum systems using continuous state estimation, Phys. Rev. A60, 2700 (1999)

    1999

  38. [45]

    H. M. Wiseman, S. Mancini, and J. Wang, Bayesian feedback versus markovian feedback in a two-level atom, Phys. Rev. A66, 013807 (2002)

    2002

  39. [46]

    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, Science377, 987 (2022)

    2022

  40. [47]

    Deplano, A

    Q. Deplano, A. Pontin, A. Ranfagni, F. Marino, and F. Marin, Coulomb coupling between two nanospheres trapped in a bichromatic optical tweezer, Optica11, 1773 (2024)

    2024

  41. [48]

    T. W. Penny, A. Pontin, and P. F. Barker, Sympathetic cooling and squeezing of two colevitated nanoparticles, Phys. Rev. Res.5, 013070 (2023)

    2023

  42. [49]

    D. S. Bykov, L. Dania, F. Goschin, and T. E. Northup, 3d sympathetic cooling and detection of levitated nanopar- ticles, Optica10, 438 (2023)

    2023

  43. [50]

    Vijayan, Z

    J. Vijayan, Z. Zhang, J. Piotrowski, D. Windey, F. van der Laan, M. Frimmer, and L. Novotny, Scalable all-optical cold damping of levitated nanoparticles, Nat. Nanotechnol.18, 49 (2023)

    2023

  44. [51]

    Reisenbauer, H

    M. Reisenbauer, H. Rudolph, L. Egyed, K. Hornberger, A. V. Zasedatelev, M. Abuzarli, B. A. Stickler, and U. Deli´ c, Non-hermitian dynamics and non-reciprocity of optically coupled nanoparticles, Nat. Phys.20, 1629 (2024)

    2024

  45. [52]

    Y. Li, C. Li, J. Zhang, Y. Dong, and H. Hu, Collective- motion-enhanced acceleration sensing via an optically levitated microsphere array, Phys. Rev. Appl.20, 024018 (2023). 9

    2023

  46. [53]

    W. J. Zhang, Y. Zhang, Q. Guo, A. P. Liu, G. Li, and T. Zhang, Strong mechanical squeezing and optomechan- ical entanglement in a dissipative double-cavity system via pump modulation, Phys. Rev. A104, 053506 (2021)

    2021

  47. [54]

    Mari and J

    A. Mari and J. Eisert, Opto-and electro-mechanical en- tanglement improved by modulation, New J. Phys.14, 075014 (2012)

    2012

  48. [55]

    H. Tan, W. Deng, Q. Wu, and G. Li, Steady-state light- mechanical quantum steerable correlations in cavity op- tomechanics, Phys. Rev. A95, 053842 (2017)

    2017

  49. [56]

    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- ence367, 892 (2020)

    2020

  50. [57]

    Gonzalez-Ballestero, P

    C. Gonzalez-Ballestero, P. Maurer, D. Windey, L. Novotny, R. Reimann, and O. Romero-Isart, Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach, Phys. Rev. A100, 013805 (2019)

    2019

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.