Numerical simulations of bosons and fermions in three dimensional - PowerPoint PPT Presentation

Dissertation ETH Number 21384 Numerical simulations of bosons and fermions in three dimensional optical lattices Ping Nang MA Ph.D. oral examination (Sept. 27, 2013) Main Collaborators: Prof. Lode POLLET (LMU, Munich), Dr. Sebastiano PILATI (ICTP, Italy) Supervisor: Prof. Matthias TROYER Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Contents 1. Optical lattice - introduction 2. Magnetism in optical lattices Density Functional Theory Reference: P. N. Ma, S. Pilati, M. Troyer, and X. Dai, Density functional theory for atomic Fermi gases, Nature Phys. 8 , 601 (2012) 3. Thermometry in optical lattices Fluctuation-dissipation thermometry Wing thermometry Reference: P. N. Ma, L. Pollet, and M. Troyer, Measuring the equation of state of trapped ultracold bosonic systems in an optical lattice with in-situ density imaging , Phys. Rev. A. 82 , 033627 (2010) 4. Directed worm algorithm (QMC) - optional 5. Conclusion/Outlook September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Optical lattices -- setup by 3 orthogonal pairs of laser beams. Short-range X V 0 sin 2 ( kx i ) V ( x ) = Van der Waals interaction x i = x,y,z Lattice strength: V 0 nvironmen Lattice separation: ng d = λ 2 , Lattice wavevector: tial of an or k = 2 π λ = π d [ s-wave scattering length: a be confined by -- setup by 3 orthogonal pairs of laser beams. Cartoon illustration Immanuel Bloch’s laboratory, Max Planck Institute September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Optical lattices -- Gaussian laser beams induce trapping of atoms. ✓ − 2 r 2 ◆ trapping envelope: V 0 exp w 2 0 -- Current experiments: Atoms are around the vicinity of the center. ~ harmonic trapping: V T ( x ) = V T x 2 ~ waist effects are minimal: Bosons in an optical lattice. QMC-DWA simulation. U/t = 8.11, T/t = 1.00, N = 280,000, w o = 150 μ m (Red) Minimal effect seen due to waist corrections. September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Optical lattices For deep lattices, or large V 0 : 1. bosons in an optical lattice b j + U ˆ ˆ i ˆ X X X b † ( µ � V T x 2 (boson Hubbard model) H = � t n i ( n i � 1) � i ) n i 2 i i h i,j i 2. fermions in an optical lattice ˆ X X c † X ( µ � V T x 2 H = � t ˆ i σ ˆ c j σ + U n i " n i # 1) � i ) n i (Hubbard model) i h i,j i , σ i hopping onsite interaction Easy and convenient conversion within ALPS Python: >>> band Optical lattice: ================ >>> import numpy; V0 [Er] = 8.805 8 8 >>> import pyalps.dwa; lamda [nm] = 765 843 843 >>> Er2nK = 188.086 154.89 154.89 >>> V0 = numpy.array([8.805, 8. , 8. ]); #lattice strength [Er] L = 160 >>> wlen = numpy.array([765., 843., 843.]); #laser wavelength [nm] g = 5.51132 >>> a = 101; #s − wave scattering length [bohr radius] >>> m = 86.99; #mass [a.m.u.] Band structure: >>> L = 160; #lattice of size L^3 =============== >>> t [nK] : 4.77257 4.77051 4.77051 >>> band = pyalps.dwa.bandstructure(V0, wlen, a, m, L); U [nK] : 38.7027 >>> U/t : 8.1094 8.1129 8.1129 September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Bosons in an optical lattice b j + U ˆ ˆ i ˆ X X X b † ( µ � V T x 2 H = � t n i ( n i � 1) � i ) n i 2 i i h i,j i Phase diagram for homogeneous systems: Temperature Large U : incompressible Mott- µ M. Fisher et al , PRB 40 , 546 (1989) S. Trotsky, L. Pollet et al , Nature Phys. 6 , 998-1004 (2010) insulator at Integer filling / U Normal Phase diagram (V=0) = n 2 T c quantum critical region = Quantum phase transition n 1 varying U/t = t / U n 0 Large t : superfluid BEC Superfluid Mott Insulator QCP U / J At zero temperature At finite temperature Quantitative validation: 1. on time-of-flight (tof) images: 2. on density profiles: 0 r c S. Trotsky, L. Pollet et al , Nature Phys. 6 , 998-1004 (2010) S. Fang, et al , PRA 83 , 031605 (2011) a 13.6 nK 18.8 nK 26.5 nK 30.7 nK 43.6 nK Experiment experiment 2 hk b 11.9 nK 19.1 nK 26.5 nK 31.8 nK 47.7 nK QMC QMC c V 0 /E R = 8 , U/t = 8.11, N = 280,000 September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Fermions in an optical lattice ˆ X X c † X ( µ � V T x 2 H = � t ˆ i σ ˆ c j σ + U n i " n i # 1) � i ) n i i h i,j i , σ i Quantum Monte Carlo -- negative sign problem M. Troyer, U-J. Weise, PRL 94 , 170201 (2005) � c A ( c ) p ( c ) � A � = � c p ( c ) � c A ( c ) s ( c ) | p ( c ) | / � c | p ( c ) | ≡ � As � ′ = � s � ′ . � c s ( c ) | p ( c ) | / � c | p ( c ) | ~ scales exponentially with 1) inverse temperature β , and � � ∼ e β N ∆ f ∆ s ( � s 2 � − � s � 2 ) /M 1 − � s � 2 2) system size N. √ √ � s � = = . � s � M � s � M Therefore, phase diagram for fermions is not entirely clear in general At half-filling, the Hubbard model exhibits antiferromagnetic ground state. September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Magnetism Time Line Stoner ferromagnetism experimentally 2009 detected in (ultracold) gases. c k + 1 U ˆ X c † X c † c † H = c k 2 # ˆ c k 1 " ✏ k ˆ k ˆ ˆ k 1 + q " ˆ k 2 � q # ˆ N 2 Magnetism in gases k , σ k 1 k 2 q 6 =0 Magnetism in solids R 1965 � ~ 2 ✓ ◆ 2 m r 2 + V ( r ) + V HXC Kohn-Sham DFT: ( ⇢ ↑ , ⇢ ↓ ; r ) � σ n = ✏ σ n � σ n σ 1. many-body ➝ effective single-body quantum problem 2. largely successful in electronic structure problems 1947: Birth of the transistor 3. inadequate to explain (strong) magnetism 1928: Quantum mechanics appiled to solids 1926 Quantum Mechanics Birth of Heisenberg model � ~ 2 i ~ ∂ ✓ ◆ 2 m r 2 + V ( x , t ) ∂ t ψ ( x , t ) = ψ ( x , t ) (Schrodinger equation): 1887 Classical electromagnetism (Maxwell’s equations): 585 B.C. “ ... loadstone attracts iron because it has a soul.” — Thales of Miletus, ∼ 585 B.C. September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Magnetism in (ultracold) gases Interaction strength: k F a Short-range Stoner ferromagnetism: Van der Waals interaction U c k + 1 ˆ X c † X c † c † H = ✏ k ˆ k ˆ ˆ k 1 + q " ˆ k 2 � q # ˆ c k 2 # ˆ c k 1 " N 2 k , σ k 1 k 2 q 6 =0 -- first observed experimentally in an Ultracold atomic gas density ultracold 6 Li gaseous cloud in 2009. G-B. Jo, et al, Itinerant ferromagnetism in a Fermi gas of ultracold atoms, Trapping potential Science 325 , 5947 (2009). V T ( x ) = V T x 2 -- phase diagram 1 Cartoon illustration P = ρ ↑ − ρ ↓ 0.8 ρ ↑ + ρ ↓ 0.6 P PM FM setup an 0.4 optical lattice 0.2 0 Q: Magnetism? 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 k F a S. Pilati, G. Bertaina, S. Giorgini, and M. Troyer, Itinerant ferromagnetism of a repulsive atomic Fermi gas: a quantum Monte Carlo study Phys. Rev. Lett. 105 , 030405 (2010). September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13

Numerical simulations of bosons and fermions in three dimensional optical lattices. Magnetism in optical lattices Ferromagnetism is enhanced by optical lattice (band structure effects): P = ρ ↑ − ρ ↓ ρ ↑ + ρ ↓ 2 2 2 1 1 1 KS-LSDA KS-LSDA KS-LSDA V 0 = 0.5 E R V 0 = 2.0 E R V 0 = 4.0 E R 0.8 0.8 0.8 1.5 1.5 1.5 0.6 0.6 0.6 1 1 1 n n n 0.4 0.4 0.4 0.5 0.5 0.5 0.2 0.2 0.2 (d) (b) (c) 0 0 0 0 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 a / d a / d a / d Kohn-Sham density functional theory (KS-DFT): Q: How valid is KS-DFT? A: KS-DFT works (quantitatively) � ~ 2 ✓ ◆ 2 m r 2 + V ( r ) + V HXC for shallow optical lattice ( ⇢ ↑ , ⇢ ↓ ; r ) n = ✏ σ � σ n � σ n σ 6 KS-LSDA 5 -- KS-DFT is exact only with exact h.x.c. potential: FN-DMC 4 E / N [E R ] 3 1.8 � n(x,0,0) 2 V HXC ( ⇢ ↑ , ⇢ ↓ ; r ) = �⇢ σ ( r ) [ ✏ HXC ( ⇢ ↑ , ⇢ ↓ ; r )] 1.4 σ 1 1 V 0 = 2 E R n = 0.5 -- Local density approximation (LDA) to h.x.c. potential 0.6 0 a/d = 0.04 -0.4 -0.2 0 0.2 0.4 x / d -1 (See appendix B in thesis.) 0 1 2 3 4 5 V 0 [E R ] September 27, 2013 Ping Nang MA - pingnang@phys.ethz.ch PhD oral examination, ETH Zurich Thursday, September 26, 13