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Dynamics of phase transitions in ion traps

Dynamics of phase transitions in ion traps. A. Retzker , A. Del Campo, M. Plenio , G. Morigi and G. De Chiara. Quantum Engineering of States and Devices: Theory and Experiments. Contents. Nucleation of defects in phase transition (the Kibble- Zurek mechanism).

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Dynamics of phase transitions in ion traps

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  1. Dynamics of phase transitions in ion traps A. Retzker, A. Del Campo, M. Plenio, G. Morigi and G. De Chiara Quantum Engineering of States and Devices: Theory and Experiments

  2. Contents Nucleation of defects in phase transition (the Kibble-Zurek mechanism) Trapped ions: structural phases &kinks Inhomogeneous vs. Homogeneoussetup Conclusion & Outlook

  3. The Kibble Zurek mechanism 1) The

  4. Cosmology in the lab • Cosmology : symmetry breaking during expansion and cooling of the early universe • Condensed matter: • Vortices in Helium • Liquid crystals • Superconductors • Superfluids Similar free-energy landscape near a critical point T. W. B. Kibble, JPA 9, 1387 (1976); Phys. Rep. 67, 183 (1980) W. H. Zurek, Nature (London) 317, 505 (1985); Acta Phys. Pol. B. 1301 (1993)

  5. Second order phase transition Landau theory: Free energy landscape, changes across a 2PT from single to double well potential, parametrized by a relative temperature

  6. Second order phase transition Universal behaviour of the order parameters: divergence of correlation/healing length dynamical relaxation time

  7. KZM in the Ginzburg-Landau potential Laguna & Zurek, PRL 78, 2519(1997) Real field in the GL potential Langevindynamics Defect = boundary between domains Whatisthesize of thedomain? Thedensity of defects?

  8. The Kibble Zurek mechanism Linear quench Non-adiabatic freezing adiabatic adiabatic Freeze-out time

  9. The Kibble Zurek mechanism Linear quench Non-adiabatic freezing adiabatic adiabatic The average size of a domain is given by the correlation length at the freeze out time

  10. 2) A test-bed: ion chains arXiv:1002.2524

  11. Cold ion crystals Oxford, England: 40Ca+ Innsbruck, Austria: 40Ca+ Boulder, USA: Hg+ (mercury) Aarhus, Denmark: 40Ca+ (red) and 24Mg+ (blue)

  12. Structural phases in trapped ions N ions on a ring with harmonic transverse confinement Birkl et al. Nature 357, 310(1992)

  13. Structuralphases in ion traps N ions on a ring with a harmonic transverse confinement Zig Zag Fishmanet al. PRB 77, 064111 (2008) Morigi et al, PRL 93, 170602 (2004); Phys. Rev. E 70, 066141 (2004).

  14. ZigZag movie Wolfgang Lange, Sussex

  15. ZigZag movie Wolfgang Lange, Sussex

  16. Quantum kink properties Normal modes of the double well model

  17. Spectrum structure High localized mode Low localized mode Dispersion relation for the ion trap and profile of the two most localized states. Landaetal, Phys. Rev. Lett. 104, 043004 (2010)

  18. Kinks T. Schaetz MPQ Left kink Right kink

  19. Kinks T. Schaetz MPQ Two kinks Extended kink

  20. KZM in the Ginzburg-Landau potential Chain of trapped ions on a ring: transverse normal modes, Re/Im parts of Effective Ginzburg-Landau potential for a small amplitude of transverse osc. Continuous description of the field for the transverse modes Long wavelength expansion wrt the zig-zag mode Controlparameter Linear Zig Zag

  21. Thermodynamic limit Eq. of motion of the slow modes in the presence of Doppler cooling Gettheinput of theKibble-Zurektheory Estimatethecorrelationlength and relaxation time Relaxation time: Correlationlength:

  22. Dynamics through the structural phase PT Consider a linear in time quench in the square of the transverse frequency Coupled Langevin equations: After the transition: anti-kink kink

  23. Dynamics through the structural phase PT

  24. On the ring Homogeneous system: 0.256

  25. Inhomogeneous KZM Axial and transverse harmonic potential (instead of a ring trap):

  26. Inhomogeneous KZM Inhomogeneous ground state Dubin, Phys. Rev. D, 55 4017 (1997) Thomas-Fermi Spatially dependant critical frequency (within LDA) Inhomogeneous transition!

  27. On the trap KZM scaling Density of defects as a function of the rate ? Kink interaction 50 ions

  28. On the trap KZM scaling Density of defects as a function of the rate Kink interaction 0.995

  29. On the trap: dynamics Slow quench: adiabatic limit Fast quench: nucleation of kinks

  30. Inhomogeneous KZM Inhomogeneous density, spatiallydependentcriticalfrequency. Linear quench: Causalityrestrictstheeffectivesize of thechain Front satisfying: Versus sound velocity: Adiabatic dynamics is possible even in the thermodynamic limit when In contrast with the homogeneous KZM Solitons: Zurek, PRL 102 (2009)

  31. Inhomogeneous KZM Defects appear if: Effective size of the chain for KZM:

  32. Theory vs. “experiment” 0.995 Nice agreement

  33. Model Adiabatic limit Fishman et al, PRB 77, 064111 (2008) The soft mode renormalizes the double wells Soft mode

  34. Model model Simulations The soft mode stops the double wells Soft mode

  35. Outlook 3D case Kinks dynamics With optical lattice: transverse Kontorova model Full counting statistics Other systems

  36. PhD position available Thanks

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