Half-integer vs. integer effects in quantum synchronization of spin systems

Ryan Tan; Christoph Bruder; Martin Koppenhöfer

doi:10.22331/q-2022-12-29-885

Half-integer vs. integer effects in quantum synchronization of spin systems

Ryan Tan¹, Christoph Bruder¹, and Martin Koppenhöfer²

¹Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
²Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA

Published:	2022-12-29, volume 6, page 885
Eprint:	arXiv:2208.12766v2
Doi:	https://doi.org/10.22331/q-2022-12-29-885
Citation:	Quantum 6, 885 (2022).

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Abstract

We study the quantum synchronization of a single spin driven by an external semiclassical signal for spin numbers larger than $S = 1$, the smallest system to host a quantum self-sustained oscillator. The occurrence of interference-based quantum synchronization blockade is found to be qualitatively different for integer vs. half-integer spin number $S$. We explain this phenomenon as the interplay between the external signal and the structure of the limit cycle in the generation of coherence in the system. Moreover, we show that the same dissipative limit-cycle stabilization mechanism leads to very different levels of quantum synchronization for integer vs. half-integer $S$. However, by choosing an appropriate limit cycle for each spin number, comparable levels of quantum synchronization can be achieved for both integer and half-integer spin systems.

Featured image: Maximum level of synchronization for different values of spin $S$ and different limit cycles. By choosing different limit-cycle oscillators depending on the size of the spin system, a monotonic growth of the maximum level of quantum synchronization as a function of the size of the spin of the system is found.

Popular summary

Classical synchronization has been studied since the 17th century and has applications in many areas of our daily lives, such as in time-keeping devices and power grids. Quantum systems can synchronize, too, and they feature a number of genuinely quantum effects in their synchronization behavior. An example is interference-based quantum synchronization blockade in driven quantum limit-cycle oscillators, where a destructive interference effect prevents synchronization even though an external signal is applied. Spin systems are a convenient platform to study quantum synchronization because of their finite (and typically low-dimensional) Hilbert space.

Here, we analyze how quantum synchronization depends on the size of the spin system. For specific combinations of a quantum limit-cycle oscillator and an applied signal, we find qualitative differences in the number of synchronization blockades and strong oscillations in the maximum amount of synchronization, depending whether the spin is integer or half-integer. However, if one chooses different limit-cycle oscillators depending on the size of the spin system, a monotonic growth of the maximum level of quantum synchronization as a function of the size of the spin of the system is found.

Our results shed light on the complex interference effects in quantum synchronization and are a first step towards studying the quantum-to-classical transition in synchronization.

► BibTeX data

@article{Tan2022halfintegervs,
  doi = {10.22331/q-2022-12-29-885},
  url = {https://doi.org/10.22331/q-2022-12-29-885},
  title = {Half-integer vs. integer effects in quantum synchronization of spin systems},
  author = {Tan, Ryan and Bruder, Christoph and Koppenh{\"{o}}fer, Martin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {885},
  month = dec,
  year = {2022}
}

► References

[1] Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths. ``Synchronization: A universal concept in nonlinear sciences''. Cambridge Nonlinear Science Series. Cambridge University Press. (2001).
https://doi.org/10.1017/CBO9780511755743

[2] Steven H Strogatz. ``Sync: How order emerges from chaos in the universe, nature, and daily life''. Hachette UK. (2012). url: https://www.hachettebooks.com/titles/steven-h-strogatz/sync/9781401304461/.
https://www.hachettebooks.com/titles/steven-h-strogatz/sync/9781401304461/

[3] O.V. Zhirov and D.L. Shepelyansky. ``Quantum synchronization''. Eur. Phys. J. D 38, 375 (2006).
https://doi.org/10.1140/epjd/e2006-00011-9

[4] Max Ludwig and Florian Marquardt. ``Quantum many-body dynamics in optomechanical arrays''. Phys. Rev. Lett. 111, 073603 (2013).
https://doi.org/10.1103/PhysRevLett.111.073603

[5] Claire Davis-Tilley and AD Armour. ``Synchronization of micromasers''. Phys. Rev. A 94, 063819 (2016).
https://doi.org/10.1103/PhysRevA.94.063819

[6] Tony E. Lee and H.R. Sadeghpour. ``Quantum synchronization of quantum van der Pol oscillators with trapped Ions''. Phys. Rev. Lett. 111, 234101 (2013).
https://doi.org/10.1103/PhysRevLett.111.234101

[7] Talitha Weiss, Stefan Walter, and Florian Marquardt. ``Quantum-coherent phase oscillations in synchronization''. Phys. Rev. A 95, 041802 (2017).
https://doi.org/10.1103/PhysRevA.95.041802

[8] Niels Lörch, Simon E Nigg, Andreas Nunnenkamp, Rakesh P Tiwari, and Christoph Bruder. ``Quantum synchronization blockade: Energy quantization hinders synchronization of identical oscillators''. Phys. Rev. Lett. 118, 243602 (2017).
https://doi.org/10.1103/PhysRevLett.118.243602

[9] Ehud Amitai, Niels Lörch, Andreas Nunnenkamp, Stefan Walter, and Christoph Bruder. ``Synchronization of an optomechanical system to an external drive''. Phys. Rev. A 95, 053858 (2017).
https://doi.org/10.1103/PhysRevA.95.053858

[10] Ehud Amitai, Martin Koppenhöfer, Niels Lörch, and Christoph Bruder. ``Quantum effects in amplitude death of coupled anharmonic self-oscillators''. Phys. Rev. E 97, 052203 (2018).
https://doi.org/10.1103/PhysRevE.97.052203

[11] Najmeh Es'haqi-Sani, Gonzalo Manzano, Roberta Zambrini, and Rosario Fazio. ``Synchronization along quantum trajectories''. Phys. Rev. Research 2, 023101 (2020).
https://doi.org/10.1103/PhysRevResearch.2.023101

[12] Christopher W Wächtler and Gloria Platero. ``Topological synchronization of quantum van der pol oscillators'' (2022). arxiv:2208.01061.
arXiv:2208.01061

[13] Steven H Strogatz. ``Nonlinear dynamics and chaos: with applications to physics''. CRC Press. (2015).

[14] Igor Goychuk, Jesús Casado-Pascual, Manuel Morillo, Jörg Lehmann, and Peter Hänggi. ``Quantum stochastic synchronization''. Phys. Rev. Lett. 97, 210601 (2006).
https://doi.org/10.1103/PhysRevLett.97.210601

[15] O. V. Zhirov and D. L. Shepelyansky. ``Synchronization and bistability of a qubit coupled to a driven dissipative oscillator''. Phys. Rev. Lett. 100, 014101 (2008).
https://doi.org/10.1103/PhysRevLett.100.014101

[16] G. L. Giorgi, F. Plastina, G. Francica, and R. Zambrini. ``Spontaneous synchronization and quantum correlation dynamics of open spin systems''. Phys. Rev. A 88, 042115 (2013).
https://doi.org/10.1103/PhysRevA.88.042115

[17] Minghui Xu, D. A. Tieri, E. C. Fine, James K. Thompson, and M. J. Holland. ``Synchronization of two ensembles of atoms''. Phys. Rev. Lett. 113, 154101 (2014).
https://doi.org/10.1103/PhysRevLett.113.154101

[18] V. Ameri, M. Eghbali-Arani, A. Mari, A. Farace, F. Kheirandish, V. Giovannetti, and R. Fazio. ``Mutual information as an order parameter for quantum synchronization''. Phys. Rev. A 91, 012301 (2015).
https://doi.org/10.1103/PhysRevA.91.012301

[19] Alexandre Roulet and Christoph Bruder. ``Synchronizing the smallest possible system''. Phys. Rev. Lett. 121, 053601 (2018).
https://doi.org/10.1103/PhysRevLett.121.053601

[20] Alexandre Roulet and Christoph Bruder. ``Quantum synchronization and entanglement generation''. Phys. Rev. Lett. 121, 063601 (2018).
https://doi.org/10.1103/PhysRevLett.121.063601

[21] Martin Koppenhöfer and Alexandre Roulet. ``Optimal synchronization deep in the quantum regime: Resource and fundamental limit''. Phys. Rev. A 99, 043804 (2019).
https://doi.org/10.1103/PhysRevA.99.043804

[22] C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen, A. Retzker, and C. Monroe. ``Realization of a quantum integer-spin chain with controllable interactions''. Phys. Rev. X 5, 021026 (2015).
https://doi.org/10.1103/PhysRevX.5.021026

[23] Matthew Neeley, Markus Ansmann, Radoslaw C Bialczak, Max Hofheinz, Erik Lucero, Aaron D O'Connell, Daniel Sank, Haohua Wang, James Wenner, Andrew N Cleland, et al. ``Emulation of a quantum spin with a superconducting phase qudit''. Science 325, 722–725 (2009).
https://doi.org/10.1126/science.1173440

[24] Martin Koppenhöfer, Christoph Bruder, and Alexandre Roulet. ``Quantum synchronization on the ibm q system''. Phys. Rev. Research 2, 023026 (2020).
https://doi.org/10.1103/PhysRevResearch.2.023026

[25] Arif Warsi Laskar, Pratik Adhikary, Suprodip Mondal, Parag Katiyar, Sai Vinjanampathy, and Saikat Ghosh. ``Observation of quantum phase synchronization in spin-1 atoms''. Phys. Rev. Lett. 125, 013601 (2020).
https://doi.org/10.1103/PhysRevLett.125.013601

[26] V. R. Krithika, Parvinder Solanki, Sai Vinjanampathy, and T. S. Mahesh. ``Observation of quantum phase synchronization in a nuclear-spin system''. Phys. Rev. A 105, 062206 (2022).
https://doi.org/10.1103/PhysRevA.105.062206

[27] G. S. Agarwal and R. R. Puri. ``Nonequilibrium phase transitions in a squeezed cavity and the generation of spin states satisfying uncertainty equality''. Optics Communications 69, 267–270 (1989).
https://doi.org/10.1016/0030-4018(89)90113-2

[28] G. S. Agarwal and R. R. Puri. ``Cooperative behavior of atoms irradiated by broadband squeezed light''. Phys. Rev. A 41, 3782–3791 (1990).
https://doi.org/10.1103/PhysRevA.41.3782

[29] R. G. Unanyan and M. Fleischhauer. ``Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions''. Phys. Rev. Lett. 90, 133601 (2003).
https://doi.org/10.1103/PhysRevLett.90.133601

[30] Julien Vidal, Rémy Mosseri, and Jorge Dukelsky. ``Entanglement in a first-order quantum phase transition''. Phys. Rev. A 69, 054101 (2004).
https://doi.org/10.1103/PhysRevA.69.054101

[31] Klaus Mølmer and Anders Sørensen. ``Multiparticle entanglement of hot trapped ions''. Phys. Rev. Lett. 82, 1835–1838 (1999).
https://doi.org/10.1103/PhysRevLett.82.1835

[32] D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland. ``Toward heisenberg-limited spectroscopy with multiparticle entangled states''. Science 304, 1476–1478 (2004).
https://doi.org/10.1126/science.1097576

[33] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland. ``Creation of a six-atom schrödinger cat state''. Nature 438, 639–642 (2005).
https://doi.org/10.1038/nature04251

[34] Peter Groszkowski, Martin Koppenhöfer, Hoi-Kwan Lau, and A. A. Clerk. ``Reservoir-engineered spin squeezing: Macroscopic even-odd effects and hybrid-systems implementations''. Phys. Rev. X 12, 011015 (2022).
https://doi.org/10.1103/PhysRevX.12.011015

[35] A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio. ``Measures of quantum synchronization in continuous variable systems''. Phys. Rev. Lett. 111, 103605 (2013).
https://doi.org/10.1103/PhysRevLett.111.103605

[36] Tony E. Lee, Ching-Kit Chan, and Shenshen Wang. ``Entanglement tongue and quantum synchronization of disordered oscillators''. Phys. Rev. E 89, 022913 (2014).
https://doi.org/10.1103/PhysRevE.89.022913

[37] Fernando Galve, Gian Luca Giorgi, and Roberta Zambrini. ``Lectures on general quantum correlations and their applications''. Chapter Quantum Correlations and Synchronization Measures, pages 393–420. Springer International Publishing. (2017).
https://doi.org/10.1007/978-3-319-53412-1_18

[38] Noufal Jaseem, Michal Hajdušek, Parvinder Solanki, Leong-Chuan Kwek, Rosario Fazio, and Sai Vinjanampathy. ``Generalized measure of quantum synchronization''. Phys. Rev. Research 2, 043287 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043287

[39] Michael R. Hush, Weibin Li, Sam Genway, Igor Lesanovsky, and Andrew D. Armour. ``Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers''. Phys. Rev. A 91, 061401 (2015).
https://doi.org/10.1103/PhysRevA.91.061401

[40] Talitha Weiss, Andreas Kronwald, and Florian Marquardt. ``Noise-induced transitions in optomechanical synchronization''. New Journal of Physics 18, 013043 (2016).
https://doi.org/10.1088/1367-2630/18/1/013043

[41] J. M. Radcliffe. ``Some properties of coherent spin states''. J. of Phys. A: General Physics 4, 313 (1971).
https://doi.org/10.1088/0305-4470/4/3/009

[42] Berislav Buca, Cameron Booker, and Dieter Jaksch. ``Algebraic theory of quantum synchronization and limit cycles under dissipation''. SciPost Physics 12, 097 (2022).
https://doi.org/10.21468/SciPostPhys.12.3.097

[43] D. M. Brink and G. R. Satchler. ``Angular momentum''. Clarendon Press. (1968).

[44] E. P. Wigner. ``Group theory: And its application to the quantum mechanics of atomic spectra''. Academic Press. (1959).

Cited by

[1] Christopher W. Wächtler and Gloria Platero, "Topological synchronization of quantum van der Pol oscillators", Physical Review Research 5 2, 023021 (2023).

[2] Parvinder Solanki, Faraz Mohd Mehdi, Michal Hajdušek, and Sai Vinjanampathy, "Symmetries and Synchronization Blockade", arXiv:2212.09388, (2022).

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