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1. Cold Atoms and Simulation of Bosonic Systems

   Naoki Kawashima (ISSP) Recent Prograss in Many Body Theory OSU, July 28, 2009 [1] Y. Kato, Q. Zhou, N. Kawashima and N. Trivedi, Nature Physics 4 (2008) 617. [2] Y. Kato and N. Kawashima, Phys. Rev. E 79 (2009) 021104. [3] T. Sato, T. Suzuki and N. Kawashima, unpublished.
2. Bose Hubbard Model in 3D Optical Lattice Laser beams create a periodic intensity pattern, which acts as a lattice trapping neutral atoms. M Greiner et al. Nature 415 39 Immanuel Bloch, Nature Physics (2005)
3. Collaborators OSU: Nandini Trivedi Qi Zhou ISSP: Yasuyuki Kato, Takafumi Suzuki Toshihiro Sato
4. World-Line Monte Carlo Before 1995, simulations were done with discretized imaginary time. Interaction world line
5. Naive Sampling ◆Slow ◆Non-ergodic (conserved quantities) ◆Difficulty in computing off-diagonal correlations
6. DLA (Directed-Loop Algorithm) O. F. Syljuasen, A. W. Sandvik: Phys. Rev. E 66, 046701(2002) Worm Cycle
7. DLA for Bose Hubbard Model --- Scattering at U-term High density for large U or nmax. But actually it seldom scatters.
8. Modified DLA Bose Hubbard Model in 3D Y. Kato, T. Suzuki, N. K.: Phys. Rev. E 75 (2007) Two ways to realize the same stochastic (Poisson) process. Step by step construction. One step construction.
9. Bose Hubbard Model in 3D with Harminic Potential Efficiency of the Algorithm Number of Vertices CPU Time for 1MCS t = 1 t/U = 0.25 m/U = 0.5 bt = 10.0 W/t = 0 (homogeneous) nmax = 8 Energy and Error
10. Mott-Superfluid Transition M. Greiner, et al., NATURE 415, 39 (2002) 150,000 lattice sites, 2x105 atoms potential depth "Time-of-flight" experiment: The potential depth increases from panel 'a' to panel 'h'. The interference pattern become obscure as we enter the Mott phase (and lose the coherence). Momentum distribution observed by "time-of-flight"
11. Bose Hubbard Model in 3D Mott - Superfluid Transition 87Rb in Optical Lattice M. Greiner et al. :Nature 415, 39 (2002) . I. Bloch: Nature Phys. 1, 23 (2005) Schematic phase diagram at T=0 Bose-Hubbard Model cubic lattice M. P. A. Fisher et. al.(1989)
12. Bose Hubbard Model in 3D Uniform case Generic Finite-T Phase Diagram T=0.1 t Kato, Zhou, N.K., and Trivedi, Nature Physics 4 617 (2008)
13. Bose Hubbard Model in 3D BNPY Finite-Size Scaling for Classical Systems above d=4 Binder, Nauenberg, Privman and Young: PRB31 1498 (1985) Free energy density However, is singular at u=0. So, we assume it has the form: Consistency with the mean-field critical behavior yields Binder parameter of 5D Ising model (p=0, u-dependence not shown)
14. Bose Hubbard Model in 3D Finite Size Scaling at D = d+z > 4 Free energy density Compressibility Susceptibility Superfluid Density In the present case, d=3, z=2.
15. Example: Compressibility k Bose Hubbard Model in 3D Uniform case k
16. Obtained Phase Diagram at T=0 Bose Hubbard Model in 3D Uniform case The transition point obtained by the Mott gap estimated in the Mott phase. (CF: B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. Svistunov:Phys. Rev. B 75, 134302(2007)) Mott r=3. Finite Size Scaling Mott r=2. SF Mott r=1. r=0.
17. Bose Hubbard Model in 3D with Harminic Potential Optical Lattice Experiment--- Bosons in a trapping potential --- Optical Lattice Experiment ... Markus Greiner, et al. : Nature 415, 39 (2002) Immanuel Bloch, Nature Physics (2005) 2x105 atoms 150,000 lattice sites Direct comparison between numerics and experiments is possible.
18. Bose Hubbard Model in 3D with harmonic trap Numerical Results 1 --- "Big wedding cake" The number of bosons cf: 2.0x105 (Bloch's group experiment 2005) We are getting close to experiments even in size. Kato and N.K., unpublished
19. Worldlines The blue line is actually a single longest loop. It spans the whole sphere indicating large quantum fluctuations. And it sometimes covers both spheres and sometimes only one, indicating the weak correlation between two spheres.
20. Bose Hubbard Model in 3D with harmonic trap Coherence between superfluid spheres NR: the number of sites at distance R. Si∈R: sum over all sites at distance R. MOTT [What’s expected?] If the spheres are isolated…2peaks If the spheres are coherent …4peaks R’ R’ R1 R1 R1 R2 R2 R2 R2 R1 R R2 R1 R
21. Bose Hubbard Model in 3D with Harminic Potential Two spheres are strongly correlated.
22. Bose Hubbard Model in 3D with Harminic Potential Momentum Distribution and Density Profile N=105 Total density ... ρ(r), ρLDA(r) ◆Just having peaks does notimply the existence of the superfluid region． ◆LDA is good. Superfluid density ... ρsLDA(r)
23. Bose Hubbard Model in 3D with Harminic Potential Local compressibility shows the singularity more clearly than the density profile ... can be measured experimentally.
24. Even more detailed comparison has been done ... S. Trotzky, L. Pollet, F. Gerbier, U. Schnorrberger, I. Bloch, N.V. Prokof’ev, B. Svistunov, and M. Troyer (arXiv:0905.4882)
25. Bose Hubbard Model in 3D Simulation of Bose-Hubbard Model (Summary) 1) The mean-field scaling seems to work for the Bose-Hubbard model at a generic (z=2) quantum critical point. 2) Simulation size is now comparable toexperiments. 3) "big wedding cake" structure, and coherence between the superfluid spheres.
26. 2D Continuous System (Gross Pitaevskii Equation)
27. quasi 2D system with Harminic Potential Phase Observation in 2D Stock et al, PRL95 190403 (2005) A 3D ciger is sliced into quasi-2D discs by a 1D optical lattice potential, and two discs are selected to interfere each other. Interference pattern. A vortex line seems to exist in at least one of the disc.
28. Projected Gross-Pitaevskii Equation Projected Gross-Pitaevskii Equation M. J. Davis, R. J. Ballagh and K. Burnett, J. Phys. B 81, 4487 (2001) The equation of motion for the quantum Bose field operator : exernal potential ( as; s-wave scattering length ) At low temperature, relevant fluctuation is the classical one, and classical description may be accurate even at finite temperature.(B. V. Svistunov, 1991) Two Basic Assumptions Modes in this region act only as a heat resovour () Neglect quantum fluctuation energy spectrum for harmonic trap system
29. Projected Gross-Pitaevskii Equation (Somewhat adhoc) separation of the classical and non-classical (incoherent) region at energy cutoffecut ncut (ecut)=3 The equation of motion for classical region C Projected Gross-Piaevskii equation (PGPE) projection onto the classical region C ( 　　 ; eigenstates of the single particle in classical region C )
30. quasi 2D system with Harminic Potential "The superfluid radius" --- Rc "LDA" estimate of Rc Fisher Hohenberg: PRB 37 4936 (1988)
31. quasi 2D system with Harminic Potential Previous Calculation (quasi-2D) Simula and Blakie, PRL96, 020404 (2006) --- projected GP equation criterion for KT transition ... average number of vortex pairs is non-zero 0.25<p 0<p<0.25 p=0 interference at the core Evidence of KT-type transition is observed, but "unarguably 2D character" has not been confirmed. T=0.86T0, N=4x104, Ncl=3x103
32. quasi 2D system with Harminic Potential "Unarguable" evidence of 2D nature? Gaussian Action for Superfluid Semi-classical description: Effective action in terms of the phase variable: +(2D Coulomb interaction between vortices) The vortex contribution is irrelevant in the superfluid region (R < R0). R0 is the superfluid radius defined by K(R0) =2/p.
33. quasi 2D system with Harminic Potential Correlation Function of Non-Uniform 2D Superfluid Semi-classical description: Characteristic to 2D system
34. quasi 2D system with Harminic Potential Correlation Function Radial Correlation Function Effective Exponent at the Boundary LDA estimate of Rc may be systematically overestimate?
35. GP Equation Summary quasi 2D system with Harminic Potential The finite-temperature behavior of quasi-2D Bose gas trapped in the harmonic potential ●Observed nonclassical rotational inertia (NCRI). ●Power-law behavior (with varying exponent) consistent with the 2D Gaussian model, in particular, the exponent at the superfluid boundary seems to be around h ~ 1/4 at r ~ R0. --- We need an independent estimate of R0.
36. Concluding Remark Quantum Monte Carlo simulation is exact (apart from statistical error) and robust in studying static properties, while it does not work for dynamic properties, such as vortex motion. Projected GP equation seems to produce realistic results comparable to experiments even at finite temperature, while its validity depends upon somewhat adhoc assumptions and systematic study of its accuracy has not been done yet. Comparison between MC and GP on the equal footing has to be done.
37. END
38. Problem in Conventional DLAwhen Applied to BHM Bose Hubbard Model in 3D ... the maximum number of bosons on the same site The problem is essentiallhy due to the process in which the head goes through a vertex unscattered.
39. Gross-Pitaevskii Equation for Finite T? Projected Gross-Pitaevskii Equation M. J. Davis, R. J. Ballagh and K. Burnett, J. Phys. B 81, 4487 (2001) At low temperature, the classical fluctuation is dominating the thermodynamic property of the system, and classical description may be accurate even at finite temperature. (B. V. Svistunov, 1991) So, we could use the GP equation for higher temperature states, not only for T=0. But, high energy states is scarcely populated and classical description is not so good for those states. Better introduce the cut-off energy εcut .
40. Projected Gross-Pitaevskii Equation (Somewhat adhoc) separation of the classical and non-classical (incoherent) region at energy cutoffecut ncut (ecut)=3 The equation of motion for classical region C Projected Gross-Piaevskii equation (PGPE) projection onto the classical region C ( 　　 ; eigenstates of the single particle in classical region C )
41. Nonclassical Rotational Inertia (NCRI) Onset of non-zero rs Trot*~189[nk]
42. quasi 2D system with Harminic Potential Correlation Function Profile T = 178[nK] log G（r) phase 2p 0 r R0 2D homogeneous system (Normal region) 　　　　　　　　　　　 (Superfluid region) Trot* ； Irot begins to appear h(R0)=1/4 (R0: critical radius)
43. 3D Uniform Lattice (Monte Carlo Simulation)
44. Bose Hubbard Model in 3D Optical Lattice Laser beams create a periodic intensity pattern, which acts as a lattice trapping neutral atoms. M Greiner et al. Nature 415 39 Immanuel Bloch, Nature Physics (2005)
45. What if the head goes away and never returns (or it takes too long to return)?
46. Green's function Y Ｘ
47. What if the head goes away and never returns (or it takes too long to return)? It never happens in the normal state. The range of the correlation is finite, the total length of the head's trajectory is O(1), so it is destined to come back soon. It is trickier in the superfluid state, since the path can be O(N). But it is still not as bad as a random walk that usually takes time of O(N2). It is OK to spend this much time because in any way it takes O(N) time to obtain a configuration statistically independent from the previous ones.
48. Creation/Annihilationof a worm Ｘ For example, satisfies the detailed balance condition. This factor comes from the choice between placing a+ a or a a+ at X (In the case shown, it is for a+ above a.)
49. DLA for Bose Hubbard Model --- Scattering at t-term The product of the density and scattering density is always O(1). We can take the limit of large nmax. Then the algorithm does not depend on nmax .
50. 3D Non-Uniform Lattice (Monte Carlo simulation)
51. Bose Hubbard Model in 3D with harmonic trap Numerical Results 2 --- Coherence between superfluid spheres Coherence between the spheres?
52. Bose Hubbard Model in 3D with harmonic trap Kato and N.K., unpublished signal of coherence
53. 3D XY Universality Class M. Campostrini, et al (Phys. Rev. B 63 (2001) 214503) h= 0.0380(4) and n = 0.67155(27) A rather sharp "spike" (with non-diverging peak height)
54. Bose Hubbard Model in 3D Uniform case μ－t Phase Diagram at T>0 At a finite temperature, we have only superfluid and normal states. But, the normal state can be further divided into Mott and disordered states by a crossover around T ~ Δ.
55. Momentum distribution function and time-of-flight images
56. Bose Hubbard Model in 3D with Harminic Potential Bose Hubbard Model