1. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dieter Jaksch (University of Oxford, UK) Bose-Einstein condensates and optical lattices (basics) Dieter Jaksch University of Oxford EU networks: OLAQUI, QIPEST

2. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Motivation

3. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. The System Ultracold atoms Weakly interacting BEC  GPE Atoms in a lattice  strong correlations Fermions & Bosons  quantum statistics Polar interactions  long range Main Properties Adjustable spatial dimension Very low temperatures pK to nK Strong correlations possible  no mean field approach possible Full quantum dynamics  no semiclassical approach 

4. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Aims and Goals Provide the physics background for better understanding current research on: BEC optical lattice physics mathematical methods for strongly correlated quantum systems Explain the physical basis of the Gross-Pitaevskii equation the (Bose)-Hubbard model in optical lattices approximate descriptions of strongly correlated 1D systems Give an overview of a selection of recent work in this field Dynamics of the superfluid-Mott insulator transition Excitation spectrum of the 1D Bose-Hubbard model Loading and Cooling / mixtures of different species

5. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Basics of many particle quantum mechanics

6. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. One species of particles and one motional state only The Fock states are orthogonal and normalized (h.|. i is the scalar product) Since bosons are indistinguishable these Fock states fully describe the state of one species of bosons in a single motional state Fock states (bosons) n

7. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Creation and destruction operators (bosons) This leads to the definition of creation and destruction operators Creation operator Its hermitian conjugate, the destruction operator a Therefore we can write Furthermore we find

8. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Number operator and commutator The commutation relation is Matrix representation in the Fock basis |ni We also define the number operator

9. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Several motional states and species 0 1 a b  +1

10. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Number operators and commutators We define number operators similar to before The commutation relations are

11. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Note: Fermionic particles No two fermions can occupy the same quantum state (Pauli principle). This is reflected by properties of the fermionic creation and destruction operators. The anticommutator relations are so that the square of each creation operator gives zero. No two particles can be created in a single quantum state. These anticommutator relations extend to several species and quantum states like the commutator relations do for bosons.

12. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Note: Bose-Einstein condensates In the case of a Bose-Einstein condensate a large number N of bosonic particles occupy the same quantum state. As a crude approximation (better justified and mathematically more rigorous approaches yielding the same result exist) one assumes that it does not matter physically whether N or N-1 particles exist in the condensate. Therefore This effectively means that the destruction and the creation operators for particles in the Bose-Einstein condensate are replaced by a number The Bose-Einstein condensate is thus described classically by c-numbers instead of a full quantum treatment Note: The macroscopic wave function arises from similar arguments if the spatial degrees of freedom are included in the treatment

13. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Example: Coherent state A coherent state is a superposition of Fock states It is an eigenstate of the destruction operator The expected number of particles is When replacing a   for a BEC this corresponds to assuming a coherent state of the atoms in a motional state described by destruction operator a.

14. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Particle energies (I) The Hamiltonian H which governs the dynamics of the quantum system will be the sum of all energies in our case. There will be several contributions Potential energy Kinetic energy E0E0 E1E1  +1 A particle gains energy by hopping between different states

15. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Particle energies (II) Interaction energy n particles in the same state Interactions between particles in different states E0E0 E1E1 Each particle interactions with all particles in the other state E0E0 E1E1 Each particle interacts with n-1 particles in the same state

16. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Example: Tunnelling (I) Chain of atoms with kinetic energy and periodic boundaries Hamiltonian Introduce discrete Fourier transformed operators q 2  ]- ,(N-1)/N, … 1] with commutation relations   ´ 

17. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Example: Tunnelling (II) Rewrite the Hamiltonian The eigenstates are with single particle eigenenergies q Blochband excitations 4J ground state EqEq

18. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Example: Repulsive Interaction A single quantum state with repulsive interaction and potential energy Apply the Hamiltonian to a Fock state It is thus an eigenstate with eigenenergie (U (n-1)/2 + E 0 ) n. This is the ground state for n g particles given by E0E0 E1E1

19. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Optical lattices and Hubbard models

20. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Optical lattices superimposed on a BEC Interference of standing wave laser beams induces AC-Stark shifts to trap the atoms in a periodic lattice potential

21. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Munich: I. Bloch, T. Haensch et al.

22. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Optical lattices: Basics AC – Stark shift Spontaneous emission    |0i |1i  laser |0i |1i   AC –Stark shift <{ } Spontaneous emission I{ } ¼   À 1À 1  Spontaneous emission rates of less than 1s -1 shift:

23. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Optical lattice: Basics The dominant real part acts as a conservative potential V(x). For a standing wave laser configuration we obtain Spatially periodic potential  realization of a lattice model Very little spontaneous processes  motion described by Schroedinger equation Shape and properties of the potential adjustable by varying laser parameters! Additional background potential by magnetic or optical fields Superlattice potentials by superimposing additional lattice potentials Creation of quasi random patterns using additional incommensurate lasers k … laser wave vector V 0 … lattice depth / laser intensity

24. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Lattice design laser square lattice laser triangular lattice 1D 2D 3D different internal states

25. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Ultracold atoms in an optical lattice Only two particle interactions for a s ¿ a 0 and few particles per lattice sites, i.e. a dilute gas  ¿1 V 0 is varying quickly on the length scale of optical wave lengths ¼ 500nm and cannot be treated as a small perturbation like the trap potential V T and interactions Solve the one particle problem including kinetic term and optical potential Treat trap potential and interaction term as a perturbation Restrict calculations to small temperatures T Hamiltonian of trapped interacting particle

26. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Single particle problem in 1D Mathieu equation for the mode functions (~ dimensionless parameters) Bloch bands with normalizable Bloch wave functions in the stable regions Stable regions a) V 0 = 5 E R b) V 0 = 10 E R c) V 0 = 25 E R Lowest band: E (0) q = -2 J cos(q)

27. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Wannier functions These are mode functions pertaining to a certain Bloch band and localized at a lattice site Note: This definition is not unique because of the arbitrary phase in the Bloch wave functions. The degree of localization depends strongly on their choice. At small temperatures only the lowest Bloch band n=(0,0,0) will be occupied Wannier functions

28. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Optical lattice Described by a Hubbard model Hopping term J and interaction U (s-wave scattering for bosons) are adjustable via the lattice depth … destruction operator for an atom in lattice site  A q are the momentum destruction operators … number of particles in lattice site  defined as U  J V0V0

29. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Hopping and interaction terms Recoil energy: E R = ~ 2 k 2 /2m Na: E R ¼ 25 kHz Rb: E R ¼ 3.8 kHz Validity: only lowest Bloch band occupied n a s 3 ¿ 1, i.e. low density, weak interactions

30. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. U Hubbard picture: trap levels molecular levels Microscopic picture: Two atoms in one well scattering length trap size molecular picture: trap energy 0 internuclear separation 

31. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Changing the lattice potential Shallow lattice: JÀU Deep lattice J ¿ U U 4J U D.J., C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

32. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. The Mott insulator– loading from a BEC BEC phase J À U: Mott insulator J ¿ U (commensurate): Theory: D.J, et al. 1998, Experiment: M. Greiner, et al., 2002 J/U Mott n=1 n=2 n=3 superfluid  /U quantum freezing super fluid Mott melting

33. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Simulation of dynamic and static properties

34. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Questions Static properties of the ground state |Gi Long range correlations On-site fluctuations (Linear) response to external perturbations Dynamic properties for a given initial state |  i Unitary evolution according to Hamiltonian (setting ~ =1) Non-unitary evolution due to interaction with bath or collisions starting from an initial density operator 

35. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Theoretical Methods The GPE cannot describe the MI in the deep optical lattice. It also fails to include correlations between distant particles beyond mean field. Improvements for the time-independent case analytical mean field theory (Gutzwiller) numerical exact diagonalization Quantum Monte Carlo Improvements for the time-dependent case analytical mean field theory (Gutzwiller)... ??? numerical exact time evolution for small systems DMRG in 1D... ??? standard (?) condensed matter ? ?

36. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Allow for non-coherent state in each lattice site Not number conserving ansatz Remarks: not number preserving (i.e. the superfluid will have a phase) number preserving version Variational method Time independent Gutzwiller occupation lattice site   contains the chemical potential to fix the mean number of particles hG|N|Gi.

37. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 0 1 2 3 n superfluid parameter > 0 Mott phase = 0 critical point minimum Variation around the Mott state: Mott insulator UÀJ

38. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Superfluid phase U¿J 0 1 2 3 n superfluid parameter or recover Gross-Pitaevskii equation

39. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Time dependent Gutzwiller Time-dependent ansatz Variational method Resulting equations Only nearest neighbour hopping h ,  i J ,  = J for h ,  i J ,  = 0 otherwise superfluid parameter

40. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. In the superfluid limit JÀU Gross Pitaevskii equation coherent state k Blochband BEC excitations Recover the GPE

41. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Strong quantum correlations (?) The Gutzwiller ansatz describes Limit of small number of particles in many localized weakly coupled modes Suppressed onsite particle fluctuations in the MI regime System in terms of non-number conserving quantum states Still missing Still product state of different lattice sites (similar to GPE)  no correlations beyond mean field Nucleation of the superfluid Critical region Time scale required for build up of coherence Local versus distant coherences What is not described is of particular interest!

42. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Time dependent DMRG (I) System described by a state and fix the maximum occupation as n max. Based on work of Vidal (2003, 2005), Verstraate & Cirac (2004), Werner (1990) for spins Perform successive SD of the system Truncate these to a maximum rank Use the SDs to form tensors and This gives an expansion in matrix product states The tensor  [  ] n replaces f n (  ) from Gutzwiller ansatz

43. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Examples A superposition state is written as with  = 2 -1/2,  1 A =|1i,  2 A =|0i,  1 B =|0i,  2 B =|1i A superposition state is written as with  = 1,  1 A =|0i,  1 B =2 -1/2 (|0i+|1i)

44. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. State in site 2State in site 3 Correlations State index Site Number Time dependent DMRG (II) Applying a series of Schmidt decompositions

45. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Local quantum operations Only the Gamma tensor for the corresponding site needs to be updated (O(  2 ) basic operations)

46. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Only the Gamma tensor for the corresponding sites and the lambda tensor in between need to be updated (O(  3 ) basic operations) Two site operations

47. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Recovering the required form Arrange  into a matrix with row index n k,  1 and column index n m,  3 Perform a singular value decomposition and identify the ~ variables with the new Schmidt decomposition This procedure can be extended to higher numbers of involved sites but the efficiency goes down. Instead we will decompose the evolution of the system into single-site and two- site operations Extensions to operations involving distant sites are possible but not necessary in our case because of the local nature of the interactions and hopping terms.

48. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Trotter expansion The unitary time evolution U according to the Schrödinger equation can be applied via a Trotter expansion (we use a 4 th –order expansion in all calculations) Here we have defined Overview of key advantages of TEBD : ―Efficient in storing a state : ―Efficient update for 1 and 2-local unitaries : ―Inaccuracies grow slowly : ―For 1D systems with 2-local Hamiltonians the maximum grows at worst logarithmically with the size at small energies one and two site operations Trotter parameters

49. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Strong correlations (?) This approach allows to describe systems in 1D where the correlations at long distances are mean field like or scale like Systems at criticality with long range strong correlations of the form require   1 and are thus not appropriately described In higher than one dimension  scales badly with the size of the system The amount of entanglement and thus  scales with the size of the boundary of the system. In 1D this is constant leading to  / log(L) while in 2D and 3D the boundary increases with the system size L A

50. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Towards higher dimensions MPS and MERA: G. Vidal PEPS: F. Verstraete and J.I. Cirac WGS: M. Plenio and H. Briegel

51. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Density operators Every expectation value is obtained by an average of the form Therefore a density operator  = |  ih  | contains all the physical information If a system is not fully prepared (e.g. in a thermal state or in the presence of decoherence) classical uncertainty about the state of the system is present in addition to the quantum nature contained in |  i. In these situations only the (classical) probability p i for the system occupying the state |  i i is known. The expectation value needs to be weighted accordingly The density operator  =  i p i |  i i h  i | can thus describe systems prepared in pure states (a ket |  i) as well as in mixed states (kets |  i i with probability p i )

52. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Simulation of mixed states Arrange the NxN matrix  as a vector with N 2 elements Introduce superoperators L on these matrices of dimension N 2 x N 2 The evolution equation is then formally equivalent to the Schroedinger equation. For a typical master equation of Lindblad type If L decomposes into single site and two site operations the same techniques as discussed for pure states and unitary evolution can be applied