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

3. 1) The The Kibble Zurek mechanism

4. Cosmology : symmetry breaking during expansion and cooling of the early universe Condensed matter: Vortices in Helium Liquid crystals Superconductors Superfluids 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) Cosmology in the lab Similar free-energy landscape near a critical point

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 What is the size of the domain? The density of defects? Defect = boundary between domains Laguna & Zurek, PRL 78, 2519(1997) Real field in the GL potential Laguna & Zurek, PRL 78, 2519(1997) Real field in the GL potential Langevin dynamics

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

9. adiabatic Non-adiabatic freezing Linear quench The Kibble Zurek mechanism 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 Boulder, USA: Hg + (mercury) Aarhus, Denmark: 40 Ca + (red) and 24 Mg + (blue) Innsbruck, Austria: 40 Ca + Oxford, England: 40 Ca +

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

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

14. Zig Zag movie Wolfgang Lange, Sussex

15. Zig Zag movie Wolfgang Lange, Sussex

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

17. Spectrum structure Dispersion relation for the ion trap and profile of the two most localized states. High localized mode Low localized mode Landa etal, 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 Zig Zag Control parameter Linear

21. Thermodynamic limit Eq. of motion of the slow modes in the presence of Doppler cooling Get the input of the Kibble-Zurek theory Estimate the correlation length and relaxation time Relaxation time: Correlation length:

22. anti-kink kink 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:

23. Dynamics through the structural phase PT

24. 0.256 On the ring Homogeneous system:

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

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

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

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

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

30. Inhomogeneous KZM Inhomogeneous density, spatially dependent critical frequency. Linear quench: Adiabatic dynamics is possible even in the thermodynamic limit when In contrast with the homogeneous KZM Solitons: Zurek, PRL 102 (2009) Causality restricts the effective size of the chain Front satisfying: Versus sound velocity:

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

32. NEW SCALING LAW! 0.995 Theory vs. “experiment” Nice agreement

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

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

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

36. Thanks PhD position available