1. Josephson Devices with Cold Atoms Andrea Trombettoni (SISSA, Trieste) Perugia, 18 July 2007

2. Outlook -) Bosonic equivalent of superconducting devices: Josephson junction (JJ) arrays of JJ SQUID analogies/differences -) Phase transitions of two-dimensional bosonic arrays: the Berezinskii-Kosterlitz-Thouless transition

3. Superconducting weak links: a Josephson junction Josephson current at T<T BCS

4. A Bose-Einstein condensate in a double well (T<T BEC ) is a bosonic Josephson junction: Bosonic Josephson junctions [Oberthaler et al. (2005)]

5. T=0 Dynamics in a Double Well Gross-Pitaevskii equation Two-mode ansatz: Phases and numbers: tunneling rate / Josephson energy interaction term / charging energy

6. Josephson Hamiltonian Relative phase:Fractional imbalance: Current: [Smerzi et al., 1997][Oberthaler et al., 1997]

7. Ultracold bosons in an optical lattice: an array of bosonic Josephson junctions a 3D lattice It is possible to control: - height barrier (i.e., the Josephson energy) - interaction term (i.e., the charging energy) - the shape of the network - the dimensionality (1D, 2D, …) …

8. Dynamics in Bosonic Arrays When V 0  Discrete Non Linear Schroedinger Equation [Cataliotti et al. (2001)]

9. Quantum Effects in Bosonic Arrays Increasing V, one passes from a superfluid to a Mott insulator [Greiner et al. (2001)] Similar phase transitions studied in superconducting arrays [see e.g. Fazio and van der Zant 2001]: interaction termJosephson term Connection among the two models

10. The route for the bosonic SQUID Ring trap: difficult to experimentally obtain, but significant progress in the groups of Phillips and Stamper-Kurn – as well as in atom chip setups. [Anderson, Dholakia, and Wright (2003)] Or adding two or more barriers and measure rotation…

11. Outlook -) Bosonic equivalent of superconducting devices: Josephson junction (JJ) arrays of JJ SQUID analogies/differences -) Phase transitions of two-dimensional bosonic arrays: the Berezinskii-Kosterlitz-Thouless transition (with P. Sodano and A. Smerzi - very recent experimental results in Cornell’s group at JILA, Boulder)

12. BKT in a nutshell In two dimensions: T<T BKT T>T BKT Few, tightly bound vortex antivortex pairs Free vortices normal state Experiments: liquid He films (e.g. Reppy and coworkers), superconducting Josephson junction arrays (e.g. Tinkham and coworkers), 2D trapped Bose gas: Z. Hadzibabic et al. Nature 441, 1118 (2006), recent work in W. Phillips group, 2D Hydrogen (Turku group, Finland) T=0 superfluid state

13. Signature of the BKT in a 2D array [A. Trombettoni, A. Smerzi and P. Sodano, New J. Phys. (2005)] central peak of the momentum distribution effects of vortex-antivortex pairs on the momentum distribution

14. A two-dimensional Josephson junction array Period 4.7  m J V OL....................................................... T  N well Tunneling energy “Charging energy“ Due to on-site interactions Thermal fluctuations of relative phases! No quantum fluctuations of relative phases ! Initial condensate [Schweikhard, Tung, Cornell, arXiv: 0704.0289]

15. t Lattice strength J V OL Image 10s 2s tr Observation of phase defects J/T=15 J/T=2.2J/T=1.4J/T=1.1 No lattice J/T=0.3J/T=0.8

16. Thermally driven vortex proliferation

17. Ultracold atoms in trapping periodic potentials provide new experimentally realizable quantum devices on which to test well- known paradigms of the statistical mechanics / field theory: -) varying the height potential  quantum phase transitions -) 2D physics -) dilute fermions and boson/fermion mixtures -) interaction can be enhanced/tuned through Feshbach resonances -) inhomogeneity can be tailored – defects/impurities can be added -) effects of the nonlinear interactions on the dynamics -) strong analogies with superconducting and superfluid systems -) quantum coherence / superfluidity on a mesoscopic scale -) quantum vs finite temperature physics -) long-range interactions can be controlled (dipolar atomic gases) … Conclusions

18. Trapped ultracold atoms: Fermions Tuning the interactions… … and inducing a fermionic “condensate” A non-interacting Fermi gas

19. Probing the superfluidity for fermionic gases Close to the crossover … … and in a lattice

20. More on the BCS-BEC crossover (I)

21. More on the BCS-BEC crossover (II) Typycal phase-diagram of hole-doped high-Tc superconductors

22. - Dream To induce desired macroscopic coherent behaviors by acting on the topology of networks: Effects induced by the topology (i.e.: not observable on a regular lattice) on bosonic systems: ultracold bosonic gases and Josephson networks Inhomogeneous network = non-translationally invariant network Inhomogeneity due to topology = how the lattice sites are connected and/or to external fields Rather new area: A. Kitaev, quant-ph/9707021 R. Burioni et al., Europh. Lett. 52, 251 (2000) L. B. Ioffe et al., Nature 415, 503 (2002) B. Doucout et al., Phys. Rev. Lett. 90, 107003 (2003) P. Sodano et al., cond-mat/0609639 (to appear on New J. Phys.)

23. Some examples Bosons undergo Bose-Einstein condensation: they localize on the comb’s backbone [R. Burioni et al., EPL 52, 251 (2000)] Ground states with high degeneracy [B. Doucout et al., PRL 90, 107003 (2003)] backbone

24. Creating a comb-shaped network with superconducting Josephson junctions -Nb trilayer technology -Josephson critical currents - capacitance - classical regime P. Silvestrini et al., cond-mat/0512478; P. Sodano et al., cond-mat/0609372

25. Creating a star-shaped network with ultracold bosons Temperatures ~ 0-500 nK Number of particles ~ 1000-10000 Number of wells ~ 100 Corresponding network

26. A theoretical model for bosons in optical networks (Bose-Hubbard model) large Quantum Phase model for superconducting Josephson networks In the following: JJN on discrete structures which are not necessarily regular lattices: Graphs B.E.C. of non interacting bosons on inhomogeneous low-dimensional structures: d<2.

27. Adjacency matrix Coordination number Chemical distance (shortest path from i to j)

28. Star lattice O arms Total number of sites: distance from the center labels arms coordination number of a given site: 2 coordination number of the center: p Spatial Bose-Einstein condensation in the center at

29. Formed by N S states and divided in 3 parts: { E 0,  0, E + } pL-1 delocalized states with 00 E 0 <-2t and E + >2t two bound states(E 0 : localized ground-state) p=2 Linear chain p≠2 Gapped Spectrum Density of states: Energy spectrum

30. Ground-state wavefunction Adding arms enhances localization Exponentially localized around the center, i.e. around the topological defect (~Anderson localization on inhomogeneous media)

31. Ground-State Delocalized Excited Topology effect with the interwell barrier V 0 2π 50 KHz and filling f 200 then: E J 50 nK Typical of one-dimensional condensate [see W. Ketterle and N. J. van Druten, Phys. Rev. A 54, 656 (1996)] Thermodynamics for bosons hopping on a star lattice I. Brunelli et al., J. Phys. B 37, S275 (2004)

32. Boson distribution x far away form the center I. Brunelli et al., J. Phys. B 37, S275 (2004) Signature of the spatial Bose-Einstein condensation: decrease of the Josephson critical currents

33. Mapping with a 1-dimensional system with an “external field” in the centre of the finger Ground-state eigenfunction Bose-Einstein condensation on a comb lattice

34. Superconducting Josephson junctions on a comb lattice On a comb of superconducting Josephson networks, one expects that critical currents along the backbone increase and along the fingers decrease: a)4.2 K b)1.2 K backbone vs. chain finger vs. chain P. Silvestrini et al., cond-mat/0512478, cond-mat/0609372

35. Comparison for the critical currents with the experimental results P. Sodano, A. Trombettoni, P. Silvestrini, R. Russo, and B. Ruggiero, cond-mat/0609639 Using a discretized version of the Bogoliubov-de Gennes equation, one finally gets Contribution of the localized eigenstates of the adjacency matrix

36. Anderson localization for bosons Random Impurities Single/Many Impurities Very active field of research: J. Lye et al., PRL 95, 070401 (2005) C. Fort et al., PRL 95, 170410 (2005) D. Clement et al., PRL 95, 170409 (2005) T. Schulte et al., PRL 95, 170411 (2005) H. Gimperlein et al., PRL 95, 170401 (2005) M. Modugno, cond-mat/0509807 (with or without an optical lattice) General idea: topological defect ~ single impurity

37. Anderson localization around topological defects Analogy with the single impurity problem : ChainChain + single defectStar Delocalized states Exponentially localized states N.B. Also differences: ordering on the chain – no ordering on the star …

38. Localization vs. interactions I: single impurity (impurity in 0) Variational ansatz: Minimization of the energy for U=0 gives the correct result With fixed  the minimization of the energy for finite U gives a critical value for the interaction:

39. adjacency matrix of the star Variational ansatz: (x distance from the star’s center) Similarly to before, the minimization of the energy for finite U gives a critical value for the interaction: (p number of star’s arms) From the point of view of the ground-state localization, the star’s topology corresponds to an equivalent defect  = 4 t (p-2) Localization vs. interactions II: star lattice A. Smerzi, P. Sodano, and A. Trombettoni, in preparation

40. Conclusions and perspectives - Topology role in inducing and control new coeherent states at for bosons on inhomogenoeus networks - Observed enhancement of critical currents in inhomogeneous networks of superconducting Josephson junctions: good agreement with Bogoliubov-de Gennes results - Localization around topological defects: Analogies with single impurity’s localization Competition between topologically induced disorder and interactions - Propagation of wavepackets and solitons in inhomogeneous networks: Controlling soliton dynamics through the topology - Statistical mechanics models (both classical and quantum) on inhomogenous networks: a lot to do...

41. Capacitance matrix Charging energy Josephson coupling Classical XY regime Quantum XY regime Theoretical models I: superconducting Josephson networks (Quantum Phase model)

42. Trapped ultracold atoms: Bosons Bose-Einstein condensation of a dilute bosonic gas Probe of superfluidity: vortices

43. Inhomogeneous Comb Homogeneous Chain Spectrum Planewave solutionsGround state localized around the backbone – “Hidden” spectrum of localized states R. Burioni et al., Europhys. Lett. 52, 251 (2000 ) Bogoliubov-de Gennes Theory for the Critical Current Enhancement in Comb Shaped Josephson Networks

44. Retrieving the standard BCS theory In the homogeneous limit, the quantum number  is the momentum k: Putting U=0 and µ=E F and assuming a BCS point-like interaction, one gets the BCS equation for the gap:

45. Relation between the chemical potential and the Fermi energy Using the equation for the number of particles it follows at T=0 when  <<E F Since U<0, then µ<E F : increasing the attraction, the Hartree-Fock term U increases and the chemical potential µ decreases.

46. Bogoliubov-de Gennes equations: continuous case For an inhomogeneous fermionic systems with attractive interactions BdG Equations Self-consistency conditions

47. Bogoliubov-de Gennes Equations: lattice case Lattice BdG Equations Hopping parameter Encoding the network´s connectivity (=topology) Discretization: Self-consistency condition Lattice chemical potential

48. Lattice Bogoliubov-de Gennes equations for the comb Away from the backbone, the fingers may be regarded as a linear chain (U(i)=U c and  i)=  c ). Setting on the backbone U(i)=U b and  i)=  b, one gets with Contribution of the localized eigenstates of the adjacency matrix At low temperature:

49. Lattice Bogoliubov-de Gennes equations for the chain We have to set One gets the „bulk“ BCS results with  <<t