Quantum dynamics in ultracold atoms

Transcription

1 Rather don t use Power-Points title Page Use my ypage one instead Quantum dynamics in ultracold atoms Corinna Kollath (Ecole Polytechnique Paris, France) T. Giamarchi (University of Geneva) A. Läuchli (MPI Dresden) I. McCulloch (Queensland) A. Kleine, U. Schollwöck (RWTH Aachen)

2 Quantum dynamics Nanostructures Quantum processing Delft Innsbruck Spintronics Quantum gases Sadler et al (2006)

3 Preparing ultracold atoms laser cooling T~100 μk evaporative cooling T~100nK Boson ( 87 Rb, F=2, m F =2) Fermion ( 40 K, F=9/2, m F =9/2) magnetic trap

4 Bose-Einstein condensate Boulder Dilute gases: n cm -3 Ultracold: T degeneracy 100 nk Weak interactions: n 1/3 a << 1

5 Strong interactions in quantum gases 2004 Fermi gases in optical lattices 2004 BEC- BCS crossover Fermi gases at unitarity 2003 low dimensional gases 2002 superfluid to Mott insulator transition Boulder 2001 Nobelprize BEC: Cornell, Ketterle, Wieman 1999 Quantum degenerate Fermi gases 1995 BEC created in dilute atomic gases 1924 predicted by Bose and Einstein

6 Optical lattices standing wave laser field -> periodic potential for atoms intensity it of laser -> strength th of potential ti wavelength/2 -> lattice spacing different geometries possible 2D lattice:

7 Bosons in an optical lattice kinetic energy interaction energy + H = J ( b b + h. c.) + U / 2 n ( n 1) < ij > i j j j Jaksch et al. (1998) j U and J related to lattice height U tunable by Feshbach resonances

8 Cold gases as quantum simulators superfluid Mott-insulator well tunable in time well decoupled from environment large kinetic energy large interaction -> quantum dynamics in isolated system time-of-flight fli images ~ momentum distribution Greiner et al (2001)

9 Quantum dynamics in a closed system time evolution by Schrödinger equation small time step: ψ ( t + Δt) e i Δ th ( t ) ψ ( t) = n e i Δ te n c n n Hilbert space state ex: quench across superfluid to Mott-insulator transition i methods: exact diagonalization and time-dependent DMRG

10 The idea of DMRG: reduced Hilbert space S. White (1992) problem: too large Hilbert space idea: construct effective Hilbert space Variational method in matrix product state space L l ( Al [ σ l ] L A1 [ σ 1]) σ l Lσ 1 Schmidt decomposition of 'wanted' state ψ = λ α L l 1 σ l R σ α l + 2 l + 1 α α block L exact sites σ l σ l +1 block R keep only M highest λ ψ M α = 1 λ α sweep free sites L l R l α + 1 α new block L new block R

11 idea: time-dependent DMRG static: breakdown after short time (Cazalilla,Marston) enlarged: numerically very expensive (Luo,Xiang,Wang; Sh Schmitteckert) t) adaptive: numerically cheap long times (Vidal; Daley,CK,Schollwöck,Vidal; White,Feiguin)

12 Algorithm: time-step time-evolution (Schrödinger eq) ψ ( t ) ψ ( t + Δt) eff H(t) eff H(t+Δt) t) eff eff Suzuki Trotter decomposition U U l U 1 with U l, l + 1 l, l + l odd + l even exp( ih l, l 1Δt) block L block L exact sites apply U l,l+1 block R block R errors: Trotter-Suzuki error truncation error new block L new block R repeat for all sites new block L new block R

13 Trotter error ~LΔt n S z dominating at short time well controlled by Δt Gobert, CK, et al PRE (2004)

14 Truncation error runaway time: crossover between Trotter error and truncation error errors well controlled

15 Experiment: abrupt change from superfluid to Mott-insulator height of lattic ce potentia al superfluid state t time after quench time-of-flight images ~ momentum distribution Greiner et al. Nature (2002)

16 Theoretical description interaction strength Mott-insulator questions: r response on short times? how do correlations build up/decay? entanglement evolution? superfluid speed? time T? if stationary state: what are its properties? subsystems thermalized?

17 Total revival of the wave function Mottinsulator U U f only interaction term: time evolution operator U ˆ ˆ 2 n n exp[ it 1 h f j( j 1)] j superfluid U i time integer value all Fock states revive latest at T=h/U -> wave function evolves periodically in time T=h/U

18 Relaxation with finite hopping Fouriertransformation <b + j(t)b j+1 (t)> > <b + j (t)b j+1 (t) U i =2 J, U f =40 J frequency ω quasi-particle frequency bands -> beating -> relaxation ~1/(zJ) C. Kollath, A. Läuchli, E. Altman, PRL 98, (2007)

19 Light-cone like evolution to quasi-steady state r density-density correlations n 0n r ( t ) n 0 n r ( t ) distance r distance=velocity x time 6 1 first da amping quasi -steady state 0 2 time/[h/j] A. Läuchli and C. Kollath, J.Stat. Mech. (2008) light cone like evolution in different models: Lieb and Robinson (1972) spin models Igloi and Rieger D. Gobert, CK, U. Schollwoeck, G. Schütz (2005) Calabrese and Cardy (2006) conformal field theory specific exactly solvable models

20 Speed of correlations position of dip in density-density correlation sound velocity velocity of fermion model sound velocity

21 Speed of correlations sound velocity velocity of particle-hole excitations velocity of fermion model sound velocity

22 Entanglement evolution l block A von Neuman entropy of block A S = Tr ρ log ρ A A A A saturation after different times t~ v l (open boundary conditions) P. Calabrese and J.Cardy (2005) for CFT A. Läuchli and C. Kollath (2008)

23 Summary: quench distance r 6 1 first da amping quasi -steady state what determines speed of light-cone? deviations from light-cone? general understanding of speed? Long-time limit? 0 2 time/[h/j] S. Manmana et al. (2007) non-integrable fermionic model specific exactly solvable models (Luttinger model, Ising model, ) M. Rigol et al. PRL 98, (2007), M. Cazalilla PRL (2007), P. Calabrese and Cardy PRL (2006), Barthel and Schollwöck (2008), Roux(2008),Flesch et al (2008)...

24 Dynamic of local excitations single particle excitations density perturbations characteristics of systems transport through nanostructures information transfer here: spin-charge separation in real time

25 Dynamics of single particle excitations 3D Fermi liquid quasi-particle with spin and charge 1D Luttinger liquid separation of spin and charge key feature

26 Spin-charge separation: simple sketch one-dimension two-dimensions me tim are held together holon spinon move separately

27 Condensed matter physics H = H +H +H interation spectral function introducing charge: ρ(x) ~ ρ (x) + bosonization ρ (x) valid at low energy and spin: σ(x) ~ ρ (x) - ρ (x) -q V ρ q V σ q V ρ using bosonic (amplitude and phase) fields Voit (1993) H = H ρ + H σ short times? strong perturbations? no interaction! interfaces?

28 Single particle excitation two component fermions add single particle h / J 0.9 n charge = n +n 0.4 n spin = n -n C.Kollath, U. Schollwöck, and W. Zwerger PRL 95,

29 Single particle excitation 0.9 n charge = n +n 0.4 n spin = n -n C.Kollath, U. Schollwöck, and W. Zwerger PRL 95,

30 Single particle excitation and entropy growth two component bosons n charge = n +n separation of spin and charge strong growth of entropy with time contribution of spin and charge part n spin = n -n A. Kleine, CK, I. McCulloch, U. Schollwöck (2008)

31 Density excitation and its entropy growth

32 Comparison of maximum entropy growth separation of spin and charge strong growth of entropy with time for single particle excitation (numerically difficult) slow growth of entropy with time for density excitation

33 Experimental observations condensed matter: cold atoms: detection in real time measure of density average over several lattice sites Auslaender et al. (2005) tunneling between parallel wires Raman spectroscopy py spectral function Dao et al. PRL 98, (2007) Stewart et al. Nature 454 (2008)

34 Spectral function two component mixture of bosons in one-dimension n charge =0.63, u=3, u 12 =2.1 A. Kleiner, C.Kollath, I.McCulluoch, T. Giamarchi, U. Schollwöck, (2008)

35 Applications of DMRG variants non-equilibrium situations dynamics across quantum phase transition finite temperature thermodynamics in spin-ladders C. Rüegg et al. (2008) A. Laeuchli and CK (2008) local excitations & dynamic properties spin-charge separation spectral function higher h dimensionsi... Kleine et al. (2008)

36 Postdoc position available