Dynamics of Quantum Many Body Systems Far From Thermal Equilibrium

Transcription

1 Dynamics of Quantum Many Body Systems Far From Thermal Equilibrium Marco Schiro CNRS-IPhT Saclay Schedule: Friday Jan 22, 29 - Feb 05,12,19. 10h-12h Contacts: marco.schiro@cea.fr Lecture Notes: ipht.cea.fr ( Cours Notes des Cours) mschirophysics.wordpress.com/teaching/ Cours de Physique Théorique

2 Why Quantum Nonequilibrium? Nature Physics Vol 11, No 2, Feb 2015

3 Quantum Control of Light and Matter Condensed Matter Light used to probe phases or (more recently) to induce phases of matter Atomic Physics Light used to trap (optical lattices) to probe or to excite atoms Quantum Optics Quantum Many Body Physics with Light Novel Far From Equilibrium Quantum Regimes Accessible!

4 Watching Electronic Correlations in Real-Time Pump&Probe Spectroscopies Pump: Trigger non-equilibrium electronic excitations Probe: Track relaxation of transient states Typical Electronic time-scales now accessible! L. Perfetti et al, PRL(07) Control Conducting Properties of Materials in Real-Time Hajlaoui et al, Nat Comm (2014)

5 Pump&Probe Spectroscopies D. Fausti et al, Science (2011) Signature of Transient Superconductivity?! Exploring Quantum Matter along non thermal (?) pathways Interplay between Phonons, Electrons, Broken Symmetries

6 Quantum Transport at Nanoscale Grobis, Rau, Potok and Goldhaber-Gordon (2006) Experiments build circuits made by nano-scale systems (artificial atoms) coupled to metallic reservoirs Novel Regime for Quantum Transport (Interaction Effects, Reduced Dimensionality, Non-Equilibrium,..) Example: Out of Equilibrium Kondo Effect in Quantum Dots Research at IPhT: H. Saleur, O. Parcollet

7 Dynamics with Ultracold Atoms in Optical Lattices Trapping Neutral Atoms in Lattices of Light V (x, y, z) =V 0 (sin 2 kx + sin 2 ky + sin 2 kz) Very good isolation from the environment (no phonons, no impurities!) Low energy scales (~Khz) / Long Time Scales Parameters (interaction, hopping) tunable in real-time! M. Greiner et al, Nature (2002)

8 Collapse and Revival of Matter Waves Repulsive Bosonic Particles Hopping on a Lattice H = b i b j + h.c. + U X n i (n i 1) 2 J X ij Equilibrium Phase Diagram M. Fisher et al, PRB (1989) i M. Greiner et al, Nature (2002) Superfluid to Mott Transition

9 Quantum Newton s Cradle Kinoshita et al, Nature (2006) Bosonic Atoms in 1d Optical Trap, Contact Interactions Slow relaxation after thousands of collisions to a non thermal distribution

10 Dynamics of Density Wave Initial State Exp/Theory: Inhomogeneous Initial State S. Trotzky et al, Nat Phys (2012) Fast relaxation toward equilibrium Comparison with numerics (DMRG)

11 oupling Light and Matter at the Quantum Level Circuit QED Cavity QED General idea: (dipole)coupling harmonic cavity mode ( Light ) to strongly anharmonic excitation ( Matter ), i.e. qubits H Rabi = r a a + q + + g a + a + + Entangled Light-Matter excitations under coherent and dissipative ( losses ) dynamics Route to Non-Linear Optics with Single Photons

12 Platforms for Many Body Physics with Light (I) Exciton-Polariton Condensate Circuit QED Arrays Driven-Dissipative Superfluid J. Koch, A. Houck and H. E. Tureci Nat. Phys (2012) Le Hur, Henriet, Petrescu, Plekhanov,Roux, Schiro, CRAS (2015) I. Carusotto, C. Ciuti, RMP 85 (2013) Also: Atoms in Photonic Crystals, NV Centers in Diamonds,

13 Platforms for Many Body Physics with Light (II) Cold-Atoms in Optical Cavities Electronic NanoCircuits Coupled To Resonators SPEC ( Quantronics ) Self-Organization/Dicke Transition M. Delbecq et al PRL (2011) Correlated Materials Coupled To Cavities Baumann, Guerlin, Brennecke, Esslinger Nature (2009) Laplace, Pena, Gariglio, Triscone, Cavalleri ArXiv (2015)

14 Numerical Studies of ETH

15 Numerical Studies of ETH (I) Bose Hubbard Chain X H = J ij Weak Quench, Uf/J =1 b i b j + h.c. Biroli, Kollath, Lauchli PRL 105, (2010) + U 2 X (G 1 ) = h b j b j+1 i i n i (n i 1)

16 Numerical Studies of ETH (I) Biroli, Kollath, Lauchli PRL 105, (2010) Strong Quench, Uf/J =10 (G 1 ) = h b j b j+1 i (G 1 ) Support does not shrink, rare states?

17 Numerical Studies of ETH (II) H = LX i=1 J z i Kim, Ikeda, Huse, PRE 90, (2014) Ising Spin Chain with Transverse and Longitudinal Field (non-integrable) z i+1 + h z i + g x i Fixed g,h,j=(0.9,0.8,1.0) O nn ' O(E n )

18 Eigenstate to Eigenstate Fluctuations r = hn +1 O n +1i hn O ni In the thermodynamic limit: P (r)! (r) Kim, Ikeda, Huse, PRE 90, (2014)

19 Outliers and Average Value of r...both (?) exponentially small in L? Kim, Ikeda, Huse, PRE 90, (2014)

20 Work Statistics/Probability Diagonal Ensemble Hard-core Bosons H = X i L. Santos, A. Polkovnikov, M. Rigol PRL 107, (2011) Jb i b i+1 + J 0 b i b i+2 + hc + X (Vn i n i+1 + V 0 n i n i+2 ) i L = 24,N =8 P DE P DE

21 Numerical Studies of ETH (III) - Diag Elements Hard-core bosons L = 24,N =8 Rigol PRL (2009) J = V =1

22 Numerical Studies of ETH (III) - Off Diag Elements Hard-core bosons L = 24,N =8 Rigol PRA (2009) J = V =1 J 0 = V 0 =0 J 0 = V 0 =0.32

23 Numerical Studies of ETH (III) - Off Diag Elements Hard-core bosons, long-range (1/r^3) repulsion Khatami et al PRL (2013) f O (E,!) E fixed in the middle of the spectrum Large Frequency: Intermediate Frequency: Small Frequency: saturation f(!) exp(!) f(!) p L f(!l)

24 Numerical Studies of ETH (III) - Off Diag Elements XXZ Spin Ladder Beugeling, Moessner, Haque PRE91, (2015) Off Diag terms exponentially small!

25 Quench Dynamics in Integrable and Nearly Integrable Systems

26 Quench Dynamics in the TFIC (I) m z (t) Γ 0 =0.5 Γ= L= L= L= L= t C. Lupo Master Thesis (2015)

27 Quench Dynamics in the TFIC (II) Fagotti, Essler, Calabrese(2012) Quench within ordered phase - Order parameter relaxes to zero Z exponentially 1 L X h i x i t exp( t) i = 0 cos k = dk de k dk log k 0 ( + 0 ) cos k +1 E k E k0

28 Quench Dynamics in the Hubbard Model H = t X ij c i c j + U(t) X i (n i 1) 2 U(t) (t = 0) = FS (t) = e iht FS U f O(t) = (t) O (t) t After a transient, expects relaxation to equilibrium (finite T)

29 Quench Dynamics in the Hubbard Model Initial State: Fermi Sea 0i = Y k <k F c k" c k# Switch-on weak two-particle interactions Moeckel&Kehrein(2008) Eckstein, Werner&Kollar(2009) MS&Fabrizio(2010) F U 1? Dynamics of Distribution Function Moeckel&Kehrein PRL (2008) n1(") " Long-lived metastable state: 1/( F U) 2 t 1/( F U) 4 Metastable Fermi Liquid, more correlated than in equilibrium: Z neq (U) <Z eq (U) Energy thermalized while distribution function still far from equilibrium

30 Quench Dynamics in the Hubbard Model Moeckel&Kehrein(2008) Long-lived metastable state: 1/( F U) 2 t 1/( F U) 4 Metastable Fermi Liquid, more correlated than in equilibrium: Z neq (U) <Z eq (U) Energy thermalized while distribution function still far from equilibrium

31 Quench Dynamics in the Hubbard Model (II) Eckstein, Werner, Kollar PRL(10) E pot Results obtained with Nonequilibrium Dynamical Mean Field Theory on longer time scales the escape from the metastable pre-thermal plateaux is visible!

32 Pre-Thermalization Originally introduced in the context of high-energy physics, to explain time scales in heavy-ion collision Berges, Borsanyi, Wetterich PRL (2004) Great interest in the quantum stat-phys community, theory and experiment! Science (2012)

33 Quench Dynamics from weak to strong coupling Weak coupling : F U 1 Moeckel&Kehrein PRL (2008) Trapping into a metastable state at t 1/ F U 2 DMFT: Eckstein, Kollar &Werner PRL (2009) New bottleneck at strong coupling Escape and delayed thermalization Dynamical Transition

34 ?Dynamical Critical Point in the Hubbard Model? DMFT results show a dynamical transition at U? ' 3.3t Energetics gives an effective temperature T eff t T c (U)? Werner, Millis PRB(07) What controls this critical point?

35 Dynamical Transition in the AFM Hubbard Model (I) Tsuji, Eckstein, Werner PRL( 13) Fermi Hubbard Model, Anti Ferromagnetic Phase Diagram Quench of the Hubbard interaction Ui >Uf starting from the ordered phase

36 Dephasing does not wash out the transition. Similar result in O(N) models (Sciolla, Biroli) Dynamical Transition in the AFM Hubbard Model (II) Time-Dependent Hartree Fock results Tsuji, Eckstein, Werner PRL( 13)

37 Dynamical Transition in the AFM Hubbard Model (II) Nonequilibrium DMFT results Tsuji, Eckstein, Werner PRL( 13)

38 Tsuji, Eckstein, Werner PRL( 13)

39 Dynamical Transition in the AFM Hubbard Model (II) Non-Equilibrium Phase Diagram Tsuji, Eckstein, Werner PRL( 13) Ordered Phase survive at high-energy well above the critical temperature