- Physikalisches Institut, University of Bonn, Bonn, Germany
This study examines the dynamics of polarizable particles, coupled to a lossy cavity mode, that are transversally driven by a laser. The analysis is performed in a regime where the cavity linewidth exceeds the recoil frequency by several orders of magnitude. Using a two-stage cooling protocol, we show that the particles’ kinetic energy can be reduced to the recoil energy. This cooling protocol relies in its first stage on a high laser power such that the particles cool into a strongly self-organized pattern. This can be seen as a strongly magnetized state. In a second stage, we adiabatically ramp down the laser intensity such that the particles’ kinetic energy is transferred to their potential energy and the particles are “demagnetized”. In this second stage, we optimize the ramping speed, which needs to be fast enough to avoid unwanted heating and slow enough such that the dynamics remains approximately adiabatic.
1 Introduction
The realization of quantum technologies (Ladd et al., 2010; Acín et al., 2018; Barzanjeh et al., 2022) based on polarizable particles such as atoms, ions, molecules, and nanoparticles relies on the precise control of their motional degrees of freedom. One important step for achieving full control of these particles is to reduce their residual motion. A key technique to achieve this is laser cooling (Wineland and Itano, 1979; Chu, 1998; Wieman et al., 1999; Cohen-Tannoudji, 1998; Phillips, 1998; Stenholm, 1986; Metcalf and Van der Straten, 1999), which can be used to achieve temperatures that leave particles close to their zero-point motion. The basic principle behind laser cooling is the enhanced absorption rate of laser photons that lower the particle’s momentum. Subsequently, incoherent scattering of a photon from the particle into free space results in lower kinetic energy of the particle. Despite the great success of laser cooling, one major problem is that it typically relies on closed transitions and the atomic species at hand. This hinders the universal application of conventional laser cooling techniques to more complex systems such as molecules or nanoparticles.
A good candidate for overcoming this problem is cavity cooling, where a particles’ motion is cooled by coherent scattering of laser photons (Horak et al., 1997; Vuletić and Chu, 2000; Domokos et al., 2001; Domokos and Ritsch, 2002; Black et al., 2003; Maunz et al., 2004; Morigi et al., 2007; Schleier-Smith et al., 2011; Wolke et al., 2012; Hosseini et al., 2017). Here, the particles’ kinetic energy is carried away by the scattered cavity photons while the internal state of the particles’ remain almost unaltered. The simplest form of cavity cooling requires driving with a laser frequency that is red-detuned with respect to the cavity resonance. That way the cavity promotes the emission of blue-shifted photons which leaves the atoms in average at lower energy. In such a setup the minimum temperature is typically bounded by the linewidth of the cavity (Domokos et al., 2001). Cavity cooling of single atoms (Maunz et al., 2004) and collective cooling (Black et al., 2003; Hosseini et al., 2017) have been realized in experimental labs. Since cavity cooling does not rely on incoherent scattering from a specific internal state, it is has been proposed for cooling molecules (Morigi et al., 2007) and experimentally realized for cooling nanoparticles (Asenbaum et al., 2013; Delić et al., 2019). Although sub-recoil cooling has been achieved experimentally (Wolke et al., 2012) in an optical cavity with very narrow linewidth, the limit set by the cavity linewidth usually lies well above the recoil limit.
This paper investigates a situation where the cavity linewidth is orders of magnitude larger than the recoil frequency, which is, for instance, the case for the experiment described in Hosseini et al. (2017) but also in several other experiments. We demonstrate that it is theoretically possible to achieve temperatures that are of the order of a single recoil by using a combination of cavity cooling and adiabatic control of optomechanical forces. The key ingredient is that the scattered photons, besides cooling, also mediate collective interactions which allow the particles to self-organize (Domokos and Ritsch, 2002; Asbóth et al., 2005). Self-organization occurs if the driving-laser power exceeds a threshold determined by the cavity parameters and the temperature of the particles. Here, the particles spontaneously form a pattern with a spacing that is determined by the wavelength of the light and allows for the constructive interference of scattered photons. Atomic self-organization has been observed with ultra-cold bosons (Baumann et al., 2010), thermal atoms (Arnold et al., 2012), and ultra-cold fermions (Wu et al., 2023; Helson et al., 2023). The formation of a self-organized pattern can be described as a ferromagnetic phase of a long-range interacting system where the collectively scattered light field can be understood as a parameter that measures the magnetization of the atomic ensemble (Schütz et al., 2015).
This study aims to present a protocol which can lower the kinetic energy of polarizable particles close to the recoil limit, even if the cavity linewidth is orders of magnitude wider. It thus proposes a two-stage cooling protocol that uses both cavity cooling and self-organization to bring the particles to a final kinetic energy that is of the order of the recoil energy. The first stage uses collective cavity cooling of many particles with high laser power. The final temperature of the particles is here mostly determined by the cavity linewidth while the particles form a strongly self-organized (magnetized) pattern (Figure 1a). For these parameters, while the atoms possess a rather high kinetic energy, they are tightly confined in space in a pattern which supports constructive interference of scattered laser photons. In the second stage, laser power is slowly decreased such that the magnetization of the particles is adiabatically decreased (Figure 1c). Like the magnetocaloric effect and the principle of adiabatic demagnetization (Tishin and Spichkin, 2003), this results in a decrease of the magnetization of the particles and simultaneously lowers their kinetic energy (Figure 1d). In contrast, however, we do not ramp an external magnetic field but the laser driving amplitude which effectively reduces the particle–particle interactions. This principle is also related to so-called release–retrap or adiabatic trap relaxation protocols which are common in optical lattices and are used to achieve low temperatures and high phase-space densities (DePue et al., 1999; Hu et al., 2017). In such protocols, the particles are cooled in tightly confined trapping potentials while a subsequent adiabatic trap relaxation lowers kinetic energy even further. In contrast, however, the demagnetization presented here is of a collective nature and comes from strong cavity-mediated atom–atom interactions instead of deep laser trapping. This is important, since it allows us to perform the demagnetization fast enough such that cavity shot noise does not significantly heat the system while decreasing the driving-laser power. At the end of this ramp, particles reach a final temperature that can be orders of magnitude lower than that of conventional cavity cooling while the particles reach a spatially homogeneous state (Figure 1b).

Figure 1. Particles are transversally driven by a laser with Rabi frequency
This paper is structured as follows. Section 2 introduces the semiclassical equations that are used to simulate the system. Furthermore, we show analytical predictions for final kinetic energy following an adiabatic ramp. In Section 3, we analyze the effects of dissipation and show the actual proposed cooling protocol. Conclusions are drawn in Section 4, and Supplementary Appendix A provides details of the calculations in Section 2.
2 Physical setup
We consider a setup of
2.1 Semiclassical description
We now present a semiclassical description of the particles’ center of mass motion and the cavity field. The coupled equations for the motion of the particles with position
and
Equations 1a, b, c, and d describe the driven-dissipative dynamics of the particles that couple to a dissipative cavity mode.
To better understand the forces that are mediated by the cavity, it is useful to eliminate the cavity degrees of freedom from the dynamics. Here, we work in the limit where
Using this result in Equations 1a, b, c, and d results in
with
We emphasize that
with
where we used Equations 3, 4 and the average runs over different initializations and trajectories.
After adiabatic elimination of the cavity degrees of freedom, we derived a dynamical description from a classical Hamiltonian. This implies that Equations 5a, b conserve the mean energy
In the following, we are interested in changing
2.2 Adiabatic ramp of the interaction strength
We assume that the distribution function of the particles is given by a thermal state which can be seen as the stationary state of the system reached after sufficiently long times. This state is given by
with single-particle kinetic energy
and partition function
We assume a time dependent
Using Equations 5a, b and Equation 7, we obtain the dynamical evolution of the single-particle kinetic energy
Furthermore, we may write
with
Then, using
and the integration of this equation leads to
where
using the notation
where
See Supplementary Appendix A for a detailed derivation. The value for
In Figure 2a we plotted

Figure 2. (a) The stable solution
In the spatially homogeneous phase, for
However, when we assume that the coupling strength is initialized such that the particles are in the self-organized phase and ramped to a value where the particles are distributed spatially homogeneously—that is,
This result of the right-hand side of Equation 11 in Figure 2b is shown as black solid line. It is a monotonous decreasing function with
where
We now discuss how this principle might be applicable to the driven-dissipative dynamics of particles in a cavity.
3 Cooling protocol
In order to apply the results of the previous section, we must first analyze the dissipative effects in the particles’ dynamics. This is done by comparing the results of Equations 5a, b that discard any dissipative effects with the dynamics, including dissipation, in Equations 1a, b, c, d.
3.1 Effects of dissipation
We first initialize the particles in a strongly self-organized thermal state with kinetic energy
for different ramping times

Figure 3. The kinetic energies
While our original assumption was that the ramp is close to adiabatic, we expect this assumption to fail, especially because the system parameters are ramped across a phase transition. An observable to test this is the kurtosis
The kurtosis is
We now analyze the dependence of the minimum achievable temperature on the ramping time

Figure 4. Final kinetic energy
We also find that the result of the kinetic energy for
3.2 Cooling protocol
We now use this gained insight to minimize the kinetic energy of particles that are initially in a spatially homogeneous configuration. We thus assume that the initial state is a thermal state with temperature
In a first stage, we perform a quench in the driving-laser intensity determined by
where
In a second stage, we consider a ramp from
By minimizing Equation 16 with respect to
at an optimum value of
This minimum kinetic energy is of the order of the recoil energy
We now consider simulations that test this prediction. Following the procedure of the first stage, Figure 5a shows the dynamics of the kinetic energy after a quench from

Figure 5. Dynamics of the kinetic energies
Following the second stage, we ramp the coupling strength according to Equation 13 with a ramping time of
4 Conclusion
In this paper, we have studied the possibility of cooling transversally driven particles inside an optical cavity using a combination of cavity cooling and a protocol reminiscent of adiabatic demagnetization. To analyze the effect of dissipation, we have performed simulations of dissipative and conservative dynamical models for this physical setup. We have shown that the particles can reach kinetic energies comparable to the recoil limit for the parameter choice below the typical limit of cavity cooling. To achieve this final kinetic energy, we have tuned the laser power from a value well above the self-organization threshold to below it. The duration of this ramp is chosen sufficiently long such that it seems to be quasi-adiabatic for the coherent dynamics but sufficiently rapid such that dissipation has only a minor effect on the final kinetic energy.
While the results presented here rely on adiabatically changing the coupling or interaction strength that results in a change of the internal magnetization, we expect that similar physics can be achieved by changing an additional external field that simulates an effective magnetic field. This can, for instance, be accomplished by modulating a laser that directly drives the cavity beside the transversal laser field (Niedenzu et al., 2013).
It is important to emphasize that the cooling stage is crucial to achieving the final kinetic energy as it reduces entropy. However, the precise cooling protocol is rather arbitrary. In particular, the choice of cavity cooling, visible in Figure 5, can be replaced by other schemes, such as a ramp of the interaction strength instead of a quench, or even by other laser cooling mechanisms. Importantly, the sole outcome of this first stage is the preparation of a sufficiently cold and highly self-organized (magnetized) particle ensemble.
Regarding the ultimate limits of this cooling protocol, this analysis is performed with semiclassical equations. This means our approach is only valid for kinetic energies that are above the recoil limit. In addition, we have not included the quantum statistics of the particles, which becomes relevant for low temperatures. Including the latter would be an interesting extension of our work since one might expect different distributions for bosons (Baumann et al., 2010) and fermions (Helson et al., 2023; Zhang et al., 2021). In future research, it might also be interesting to use multi-mode cavities that provide more possibilities to tune the interactions and dissipation (Torggler and Ritsch, 2014; Keller et al., 2017; Keller et al., 2018). The study of such systems is not only interesting for advances in laser and cavity cooling but also as a simulator for classical and quantum thermodynamics (Vinjanampathy and Anders, 2016; Niedenzu et al., 2018). In conclusion, the engineering of interactions and dissipation for particles in optical cavities is a versatile tool for quantum technologies and studying new physics.
Data availability statement
The raw data supporting the conclusions of this article will be made available by the author without undue reservation.
Author contributions
SJ: Writing – original draft and Writing – review and editing.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 277625399—TRR 185 (B4) and under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1—390534769.
Acknowledgments
SJ acknowledges stimulating discussions with Stefan Schütz, John Cooper, and Giovanna Morigi.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no Generative AI was used in the creation of this manuscript.
Publisher’s note
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frqst.2025.1535581/full#supplementary-material
References
Acín, A., Bloch, I., Buhrman, H., Calarco, T., Eichler, C., Eisert, J., et al. (2018). The quantum technologies roadmap: a european community view. New J. Phys. 20, 080201. doi:10.1088/1367-2630/aad1ea
Arnold, K. J., Baden, M. P., and Barrett, M. D. (2012). Self-organization threshold scaling for thermal atoms coupled to a cavity. Phys. Rev. Lett. 109, 153002. doi:10.1103/physrevlett.109.153002
Asbóth, J. K., Domokos, P., Ritsch, H., and Vukics, A. (2005). Self-organization of atoms in a cavity field: threshold, bistability, and scaling laws. Phys. Rev. A 72, 053417. doi:10.1103/physreva.72.053417
Asenbaum, P., Kuhn, S., Nimmrichter, S., Sezer, U., and Arndt, M. (2013). Cavity cooling of free silicon nanoparticles in high vacuum. Nat. Commun. 4, 2743. doi:10.1038/ncomms3743
Barzanjeh, S., Xuereb, A., Gröblacher, S., Paternostro, M., Regal, C. A., and Weig, E. M. (2022). Optomechanics for quantum technologies. Nat. Phys. 18, 15–24. doi:10.1038/s41567-021-01402-0
Baumann, K., Guerlin, C., Brennecke, F., and Esslinger, T. (2010). Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306. doi:10.1038/nature09009
Black, A. T., Chan, H. W., and Vuletić, V. (2003). Observation of collective friction forces due to spatial self-organization of atoms: from Rayleigh to bragg scattering. Phys. Rev. Lett. 91, 203001. doi:10.1103/physrevlett.91.203001
Chu, S. (1998). Nobel lecture: the manipulation of neutral particles. Rev. Mod. Phys. 70, 685–706. doi:10.1103/revmodphys.70.685
Cohen-Tannoudji, C. N. (1998). Nobel lecture: manipulating atoms with photons. Rev. Mod. Phys. 70, 707–719. doi:10.1103/revmodphys.70.707
Delić, U., Reisenbauer, M., Grass, D., Kiesel, N., Vuletić, V., and Aspelmeyer, M. (2019). Cavity cooling of a levitated nanosphere by coherent scattering. Phys. Rev. Lett. 122, 123602. doi:10.1103/physrevlett.122.123602
DePue, M. T., McCormick, C., Winoto, S. L., Oliver, S., and Weiss, D. S. (1999). Unity occupation of sites in a 3d optical lattice. Phys. Rev. Lett. 82, 2262–2265. doi:10.1103/physrevlett.82.2262
Domokos, P., Horak, P., and Ritsch, H. (2001). Semiclassical theory of cavity-assisted atom cooling. J. Phys. B 34, 187–198. doi:10.1088/0953-4075/34/2/306
Domokos, P., and Ritsch, H. (2002). Collective cooling and self-organization of atoms in a cavity. Phys. Rev. Lett. 89, 253003. doi:10.1103/physrevlett.89.253003
Grießer, T., Niedenzu, W., and Ritsch, H. (2012). Cooperative self-organization and sympathetic cooling of a multispecies gas in a cavity. New J. Phys. 14, 053031. doi:10.1088/1367-2630/14/5/053031
Helson, V., Zwettler, T., Mivehvar, F., Colella, E., Roux, K., Konishi, H., et al. (2023). Density-wave ordering in a unitary fermi gas with photon-mediated interactions. Nature 618, 716–720. doi:10.1038/s41586-023-06018-3
Horak, P., Hechenblaikner, G., Gheri, K. M., Stecher, H., and Ritsch, H. (1997). Cavity-induced atom cooling in the strong coupling regime. Phys. Rev. Lett. 79, 4974–4977. doi:10.1103/physrevlett.79.4974
Hosseini, M., Duan, Y., Beck, K. M., Chen, Y.-T., and Vuletić, V. (2017). Cavity cooling of many atoms. Phys. Rev. Lett. 118, 183601. doi:10.1103/physrevlett.118.183601
Hu, J., Urvoy, A., Vendeiro, Z., Crépel, V., Chen, W., and Vuletić, V. (2017). Creation of a bose-condensed gas of 87rb by laser cooling. Science 358, 1078–1080. doi:10.1126/science.aan5614
Jäger, S. B., Schütz, S., and Morigi, G. (2016). Mean-field theory of atomic self-organization in optical cavities. Phys. Rev. A 94, 023807. doi:10.1103/physreva.94.023807
Keller, T., Jäger, S. B., and Morigi, G. (2017). Phases of cold atoms interacting via photon-mediated long-range forces. J. Stat. Mech. 2017, 064002. doi:10.1088/1742-5468/aa71d7
Keller, T., Torggler, V., Jäger, S. B., Schütz, S., Ritsch, H., and Morigi, G. (2018). Quenches across the self-organization transition in multimode cavities. New J. Phys. 20, 025004. doi:10.1088/1367-2630/aaa161
Ladd, T. D., Jelezko, F., Laflamme, R., Nakamura, Y., Monroe, C., and O’Brien, J. L. (2010). Quantum computers. Nature 464, 45–53. doi:10.1038/nature08812
Maunz, P., Puppe, T., Schuster, I., Syassen, N., Pinkse, P. W. H., and Rempe, G. (2004). Cavity cooling of a single atom. Nature 428, 50–52. doi:10.1038/nature02387
Metcalf, H. J., and Van der Straten, P. (1999). Laser cooling and trapping. Springer Science & Business Media.
Morigi, G., Pinkse, P. W. H., Kowalewski, M., and de Vivie-Riedle, R. (2007). Cavity cooling of internal molecular motion. Phys. Rev. Lett. 99, 073001. doi:10.1103/physrevlett.99.073001
Niedenzu, W., Grießer, T., and Ritsch, H. (2011). Kinetic theory of cavity cooling and self-organisation of a cold gas. Europhys. Lett. 96, 43001. doi:10.1209/0295-5075/96/43001
Niedenzu, W., Mukherjee, V., Ghosh, A., Kofman, A. G., and Kurizki, G. (2018). Quantum engine efficiency bound beyond the second law of thermodynamics. Nat. Commun. 9, 165. doi:10.1038/s41467-017-01991-6
Niedenzu, W., Schütz, S., Habibian, H., Morigi, G., and Ritsch, H. (2013). Seeding patterns for self-organization of photons and atoms. Phys. Rev. A 88, 033830. doi:10.1103/physreva.88.033830
Phillips, W. D. (1998). Nobel lecture: laser cooling and trapping of neutral atoms. Rev. Mod. Phys. 70, 721–741. doi:10.1103/revmodphys.70.721
Schleier-Smith, M. H., Leroux, I. D., Zhang, H., Van Camp, M. A., and Vuletić, V. (2011). Optomechanical cavity cooling of an atomic ensemble. Phys. Rev. Lett. 107, 143005. doi:10.1103/physrevlett.107.143005
Schütz, S., Habibian, H., and Morigi, G. (2013). Cooling of atomic ensembles in optical cavities: semiclassical limit. Phys. Rev. A 88, 033427. doi:10.1103/physreva.88.033427
Schütz, S., Jäger, S. B., and Morigi, G. (2015). Thermodynamics and dynamics of atomic self-organization in an optical cavity. Phys. Rev. A 92, 063808. doi:10.1103/physreva.92.063808
Schütz, S., Jäger, S. B., and Morigi, G. (2016). Dissipation-assisted prethermalization in long-range interacting atomic ensembles. Phys. Rev. Lett. 117, 083001. doi:10.1103/physrevlett.117.083001
Schütz, S., and Morigi, G. (2014). Prethermalization of atoms due to photon-mediated long-range interactions. Phys. Rev. Lett. 113, 203002. doi:10.1103/physrevlett.113.203002
Stenholm, S. (1986). The semiclassical theory of laser cooling. Rev. Mod. Phys. 58, 699–739. doi:10.1103/revmodphys.58.699
Tishin, A., and Spichkin, Y. (2003). The magnetocaloric effect and its applications. 1st ed. Boca Raton: CRC Press.
Torggler, V., and Ritsch, H. (2014). Adaptive multifrequency light collection by self-ordered mobile scatterers in optical resonators. Optica 1, 336. doi:10.1364/optica.1.000336
Vinjanampathy, S., and Anders, J. (2016). Quantum thermodynamics. Contemp. Phys. 57, 545–579. doi:10.1080/00107514.2016.1201896
Vuletić, V., and Chu, S. (2000). Laser cooling of atoms, ions, or molecules by coherent scattering. Phys. Rev. Lett. 84, 3787–3790. doi:10.1103/physrevlett.84.3787
Wieman, C. E., Pritchard, D. E., and Wineland, D. J. (1999). Atom cooling, trapping, and quantum manipulation. Rev. Mod. Phys. 71, S253–S262. doi:10.1103/revmodphys.71.s253
Wineland, D. J., and Itano, W. M. (1979). Laser cooling of atoms. Phys. Rev. A 20, 1521–1540. doi:10.1103/physreva.20.1521
Wolke, M., Klinner, J., Keßler, H., and Hemmerich, A. (2012). Cavity cooling below the recoil limit. Science 337, 75–78. doi:10.1126/science.1219166
Wu, Z., Fan, J., Zhang, X., Qi, J., and Wu, H. (2023). Signatures of prethermalization in a quenched cavity-mediated long-range interacting fermi gas. Phys. Rev. Lett. 131, 243401. doi:10.1103/physrevlett.131.243401
Keywords: adiabatic demagnetization, cavity QED, semiclassical dynamic simulation, self-organization, stochastic differential equation, cavity cooling
Citation: Jäger SB (2025) Cooling strongly self-organized particles using adiabatic demagnetization. Front. Quantum Sci. Technol. 4:1535581. doi: 10.3389/frqst.2025.1535581
Received: 27 November 2024; Accepted: 08 May 2025;
Published: 19 June 2025.
Edited by:
Jorge Yago Malo, University of Pisa, ItalyReviewed by:
Pietro Lombardi, National Research Council (CNR), ItalyTomohiro Hashizume, University of Hamburg, Germany
Copyright © 2025 Jäger. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Simon B. Jäger, c2phZWdlcjJAdW5pLWJvbm4uZGU=
†ORCID: Simon B. Jäger, orcid.org/0000-0002-2585-5246