From unitary dynamics to statistical mechanics in isolated quantum systems

Transcription

1 From unitary dynamics to statistical mechanics in isolated quantum systems Marcos Rigol Department of Physics The Pennsylvania State University The Tony and Pat Houghton Conference on Non-Equilibrium Statistical Mechanics ICERM, Brown University May 4, 2015 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

2 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

3 Experiments with ultracold gases in 1D Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998). where U 1D (x) = g 1D δ(x) g 1D = 2 a s ω 1 Ca s mω 2 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

4 Experiments with ultracold gases in 1D Effective one-dimensional δ potential M. Olshanii, PRL 81, 938 (1998). where U 1D (x) = g 1D δ(x) g 1D = 2 a s ω 1 Ca s mω 2 Girardeau 60, Lieb and Liniger 63 T. Kinoshita, T. Wenger, and D. S. Weiss, Science 305, 1125 (2004). T. Kinoshita, T. Wenger, and D. S. Weiss, Phys. Rev. Lett. 95, (2005). γ eff = mg 1D 2 ρ Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

5 Absence of thermalization in 1D? Density profile Momentum profile density momentum distribution position momentum T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006). MR, A. Muramatsu, and M. Olshanii, Phys. Rev. A 74, (2006). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

6 Absence of thermalization in 1D? Density profile Momentum profile τ=0 τ=0 density momentum distribution position momentum T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006). MR, A. Muramatsu, and M. Olshanii, Phys. Rev. A 74, (2006). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

7 Absence of thermalization in 1D? n k τ 1000 Experiment 2000 Theory π/2 π/4 0 π/4 π/2 k Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

8 Absence of thermalization in 1D? T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006). γ = mg 1D 2 ρ g 1D: Interaction strength ρ: One-dimensional density If γ 1 the system is in the strongly correlated Tonks-Girardeau regime If γ 1 the system is in the weakly interacting regime Gring et al., Science 337, 1318 (2012). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

9 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

10 Exact results from quantum mechanics If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a generic observable O will evolve in time following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ ψ 0. Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

11 Exact results from quantum mechanics If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a generic observable O will evolve in time following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ ψ 0. What is it that we call thermalization? O(τ) = O(E 0 ) = O(T ) = O(T, µ). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

12 Exact results from quantum mechanics If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a generic observable O will evolve in time following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ ψ 0. What is it that we call thermalization? O(τ) = O(E 0 ) = O(T ) = O(T, µ). One can rewrite O(τ) = α,α C α C αe i(e α Eα )τ O α α where ψ 0 = α C α α, Taking the infinite time average (diagonal ensemble ˆρ DE α C α 2 α α ) τ 1 O(τ) = lim dτ Ψ(τ τ τ ) Ô Ψ(τ ) = C α 2 O αα Ô diag, 0 α which depends on the initial conditions through C α = α ψ 0. Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

13 Width of the energy density after a sudden quench Initial state ψ 0 = α Cα α is an eigenstate of Ĥ0. At τ = 0 Ĥ 0 Ĥ = Ĥ0 + Ŵ with Ŵ = j ŵ(j) and Ĥ α = E α α. MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

14 Width of the energy density after a sudden quench Initial state ψ 0 = α Cα α is an eigenstate of Ĥ0. At τ = 0 Ĥ 0 Ĥ = Ĥ0 + Ŵ with Ŵ = j ŵ(j) and Ĥ α = E α α. The width of the weighted energy density E is then E = Eα C 2 α 2 ( E α C α 2 ) 2 = ψ 0 Ŵ 2 ψ 0 ψ 0 Ŵ ψ0 2, α α or E = j 1,j 2 σ [ ψ 0 ŵ(j 1)ŵ(j 2) ψ 0 ψ 0 ŵ(j 1) ψ 0 ψ 0 ŵ(j 2) ψ 0 ] N N, where N is the total number of lattice sites. MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

15 Width of the energy density after a sudden quench Initial state ψ 0 = α Cα α is an eigenstate of Ĥ0. At τ = 0 Ĥ 0 Ĥ = Ĥ0 + Ŵ with Ŵ = j ŵ(j) and Ĥ α = E α α. The width of the weighted energy density E is then E = Eα C 2 α 2 ( E α C α 2 ) 2 = ψ 0 Ŵ 2 ψ 0 ψ 0 Ŵ ψ0 2, α α or E = j 1,j 2 σ [ ψ 0 ŵ(j 1)ŵ(j 2) ψ 0 ψ 0 ŵ(j 1) ψ 0 ψ 0 ŵ(j 2) ψ 0 ] N N, where N is the total number of lattice sites. Since the width W of the full energy spectrum is N ɛ = E W N 1 N, so, as in any thermal ensemble, ɛ vanishes in the thermodynamic limit. MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

16 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

17 Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian Ĥ = J ) (ˆb iˆb j + H.c. + U ˆn iˆn j, i,j i,j MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). ˆb 2 i = ˆb 2 i = 0 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

18 Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian Ĥ = J ) (ˆb iˆb j + H.c. + U ˆn iˆn j, i,j i,j MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). Nonequilibrium dynamics in 2D ˆb 2 i = ˆb 2 i = 0 Initial Weak n.n. U = 0.1J N b = 5 bosons N = 21 lattice sites Hilbert space: D = All states are used! Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

19 Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian Ĥ = J ) (ˆb iˆb j + H.c. + U ˆn iˆn j, i,j i,j MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). ˆb 2 i = ˆb 2 i = 0 One can rewrite O(τ) = α,α C α C αe i(e α Eα )τ O α α where ψ 0 = α C α α, and taking the infinite time average (diagonal ensemble) 1 O(τ) = lim τ τ τ 0 dτ Ψ(τ ) Ô Ψ(τ ) = α C α 2 O αα Ô diag, which depends on the initial conditions through C α = α ψ 0. Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

20 Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian Ĥ = J ) (ˆb iˆb j + H.c. + U ˆn iˆn j, i,j i,j MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). Nonequilibrium dynamics in 2D ˆb 2 i = ˆb 2 i = 0 2 Time evolution of n(k x ) time average tj=000 Weak n.n. U = 0.1J n(k x ) N b = 5 bosons N = 21 lattice sites Hilbert space: D = All states are used! k x [2π/(L x d)] Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

21 Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian Ĥ = J ) (ˆb iˆb j + H.c. + U ˆn iˆn j, i,j i,j MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). Nonequilibrium dynamics in 2D ˆb 2 i = ˆb 2 i = 0 n(k x =0) relaxation dynamics time average Weak n.n. U = 0.1J N b = 5 bosons N = 21 lattice sites Hilbert space: D = All states are used! tj Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

22 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

23 Statistical description after relaxation Canonical calculation } O = Tr {Ôˆρ ( ) ˆρ = Z 1 exp Ĥ/k BT { ( )} Z = Tr exp Ĥ/k BT } E 0 = Tr {Ĥ ˆρ T = 1.9J n(k x ) Momentum distribution initial state time average canonical k x [2π/(L x d)] Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

24 Statistical description after relaxation Canonical calculation } O = Tr {Ôˆρ ( ) ˆρ = Z 1 exp Ĥ/k BT { ( )} Z = Tr exp Ĥ/k BT } E 0 = Tr {Ĥ ˆρ T = 1.9J Microcanonical calculation 1 O = Ψ α N Ô Ψ α states α with E 0 E < E α < E 0 + E N states : # of states in the window n(k x ) n(k x ) Momentum distribution initial state time average canonical k x [2π/(L x d)] initial state time average microcanonical k x [2π/(L x d)] Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

25 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

26 Eigenstate thermalization hypothesis Paradox? C α 2 O αα = O microcan. (E 0 ) α 1 O αα N E0, E E 0 E α < E Left hand side: Depends on the initial conditions through C α = Ψ α ψ I Right hand side: Depends only on the initial energy Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

27 Eigenstate thermalization hypothesis Paradox? C α 2 O αα = O microcan. (E 0 ) α 1 O αα N E0, E E 0 E α < E Left hand side: Depends on the initial conditions through C α = Ψ α ψ I Right hand side: Depends only on the initial energy i) For physically relevant initial conditions, C α 2 practically do not fluctuate. ii) Large (and uncorrelated) fluctuations occur in both O αα and C α 2. A physically relevant initial state performs an unbiased sampling of O αα. MR and M. Srednicki, PRL 108, (2012). K. He and MR, Phys. Rev. A 87, (2013). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

28 Eigenstate thermalization hypothesis Paradox? C α 2 O αα = O microcan. (E 0 ) α 1 O αα N E0, E E 0 E α < E Left hand side: Depends on the initial conditions through C α = Ψ α ψ I Right hand side: Depends only on the initial energy MR, PRA 82, (2010). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

29 Eigenstate thermalization hypothesis Paradox? C α 2 O αα = O microcan. (E 0 ) α 1 O αα N E0, E E 0 E α < E Left hand side: Depends on the initial conditions through C α = Ψ α ψ I Right hand side: Depends only on the initial energy i) For physically relevant initial conditions, C α 2 practically do not fluctuate. ii) Large (and uncorrelated) fluctuations occur in both O αα and C α 2. A physically relevant initial state performs an unbiased sampling of O αα. MR and M. Srednicki, PRL 108, (2012). K. He and MR, Phys. Rev. A 87, (2013). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

30 Eigenstate thermalization hypothesis Paradox? C α 2 O αα = O microcan. (E 0 ) α 1 O αα N E0, E E 0 E α < E Left hand side: Depends on the initial conditions through C α = Ψ α ψ I Right hand side: Depends only on the initial energy Eigenstate thermalization hypothesis (ETH) [J. M. Deutsch, PRA (1991); M. Srednicki, PRE 50, 888 (1994); MR, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).] iii) The expectation value Ψ α Ô Ψ α of a few-body observable Ô in an eigenstate of the Hamiltonian Ψ α, with energy E α, of a large interacting many-body system equals the thermal average of Ô at the mean energy E α : Ψ α Ô Ψ α = O microcan. (E α ) Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

31 Eigenstate thermalization hypothesis 2 Momentum distribution Eigenstates a d are the ones with energies closest to E 0 n(k x ) time average/microcan. eigenstate a eigenstate b eigenstate c eigenstate d k x [2π/L x a] Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

32 Eigenstate thermalization hypothesis 2 Momentum distribution Eigenstates a d are the ones with energies closest to E 0 3 n(k x ) time average/microcan. eigenstate a eigenstate b eigenstate c eigenstate d k x [2π/L x a] n(k x =0) ρ(e) exact ρ(e) microcan. ρ(e) canonical E[J] 2 1 ρ(e)[j -1 ] n(k x = 0) vs energy ρ(e) = P(E) dens. stat. P (E) exact C α 2 P (E) mic constant P (E) can exp ( E/k B T ) Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

33 One-dimensional integrable case Similar experiment in one dimension Initial 8 sites 13 sites Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

34 One-dimensional integrable case Similar experiment in one dimension Initial 8 sites 13 sites 0.6 Time average vs Stat. Mech. 0.4 No thermalization! n(k x ) 0.2 time average microcan. canonical k x [2π/L x a] Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

35 Breakdown of eigenstate thermalization 0.6 Momentum distribution Eigenstates a d are the ones with energies closest to E n(k x ) eigenstate a eigenstate b eigenstate c eigenstate d k x [2π/L x a] n(k x =0) ρ(e) exact ρ(e) microcan. ρ(e) canonical E[J] ρ(e)[j -1 ] n(k x = 0) vs energy ρ(e) = P(E) dens. stat. P (E) exact C α 2 P (E) mic constant P (E) can exp ( E/k B T ) Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

36 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

37 Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian Ĥ = J ) (ˆb iˆb j + H.c. + U ˆn iˆn j, i,j i,j ˆb 2 i = ˆb 2 i = 0 Nonequilibrium dynamics in 2D n(k x =0) relaxation dynamics time average Weak n.n. U = 0.1J N b = 5 bosons N = 21 lattice sites Hilbert space: D = All states are used! tj Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

38 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

39 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α 1 O αα Oαα typical N states One needs: O typical α α Otypical αα Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

40 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α 1 O αα Oαα typical N states One needs: O typical α α Otypical αα Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

41 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α 1 O αα Oαα typical N states One needs: O typical α α Otypical αα M. Srednicki, J. Phys. A 29, L75 (1996). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

42 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

43 Bose-Fermi mapping Hard-core boson Hamiltonian in an external potential Ĥ = J ) (ˆb iˆb i+1 + H.c. + v i ˆn i, constraints i i ˆb 2 i = ˆb 2 i = 0 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

44 Bose-Fermi mapping Hard-core boson Hamiltonian in an external potential Ĥ = J ) (ˆb iˆb i+1 + H.c. + 2 v i ˆn i, constraints ˆb i = ˆb 2 i = 0 i i Map to spins and then to fermions (Jordan-Wigner transformation) σ + i = ˆf i i 1 β=1 e iπ ˆf ˆf i 1 β β, σ i = β=1 e iπ ˆf β ˆf β ˆfi Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

45 Bose-Fermi mapping Hard-core boson Hamiltonian in an external potential Ĥ = J ) (ˆb iˆb i+1 + H.c. + 2 v i ˆn i, constraints ˆb i = ˆb 2 i = 0 i i Map to spins and then to fermions (Jordan-Wigner transformation) σ + i Exact Green s function = ˆf i Computation time L 2 N 3 i 1 β=1 e iπ ˆf ˆf i 1 β β, σ i = β=1 [ (P G ij (τ) = det l (τ) ) ] P r (τ) e iπ ˆf β ˆf β ˆfi 3000 lattice sites, 300 particles MR and A. Muramatsu, PRL 93, (2004); PRL 94, (2005). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

46 Relaxation dynamics in an integrable system Density profile Momentum profile n 0.1 n k x/a 0 π π/2 0 π/2 ka π MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, (2007). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

47 Outline 1 Introduction Experiments with ultracold gases Unitary evolution and thermalization 2 Generic (nonintegrable) systems Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble 4 Summary Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

48 Statistical description after relaxation Thermal equilibrium [ ) ] ˆρ = Z 1 exp (Ĥ µ ˆNb /k B T { [ ) ]} Z = Tr exp (Ĥ µ ˆNb /k B T } { } E = Tr {Ĥ ˆρ, N b = Tr ˆNb ˆρ MR, PRA 72, (2005). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

49 Statistical description after relaxation Thermal equilibrium [ ) ] ˆρ = Z 1 exp (Ĥ µ ˆNb /k B T { [ ) ]} Z = Tr exp (Ĥ µ ˆNb /k B T } { } E = Tr {Ĥ ˆρ, N b = Tr ˆNb ˆρ n k= Evolution of n k=0 Time evolution Thermal MR, PRA 72, (2005). n k τ n k after relaxation After relax. (N b =30) After relax. (N b =15) Thermal (N b =30) Thermal (N b =15) 0 -π -π/2 0 π/2 π ka Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

50 Statistical description after relaxation Thermal equilibrium [ ) ] ˆρ = Z 1 exp (Ĥ µ ˆNb /k B T { [ ) ]} Z = Tr exp (Ĥ µ ˆNb /k B T } { } E = Tr {Ĥ ˆρ, N b = Tr ˆNb ˆρ n k= Evolution of n k=0 Time evolution Thermal MR, PRA 72, (2005). Integrals of motion (underlying noninteracting fermions) Ĥ F ˆγ m f 0 = E mˆγ m f 0 } } f {Îf m = {ˆγ m ˆγ m f n k τ n k after relaxation After relax. (N b =30) After relax. (N b =15) Thermal (N b =30) Thermal (N b =15) 0 -π -π/2 0 π/2 π ka Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

51 Statistical description after relaxation Thermal equilibrium [ ) ] ˆρ = Z 1 exp (Ĥ µ ˆNb /k B T { [ ) ]} Z = Tr exp (Ĥ µ ˆNb /k B T } { } E = Tr {Ĥ ˆρ, N b = Tr ˆNb ˆρ n k= Evolution of n k=0 Time evolution Thermal GGE MR, PRA 72, (2005). Generalized Gibbs ensemble [ ˆρ c = Zc 1 exp ] λ m Î m m { [ Z c = Tr exp ]} λ m Î m m } Îm τ=0 = Tr {Îm ˆρ c n k τ n k after relaxation After relaxation Thermal GGE 0 -π -π/2 0 π/2 π ka Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

52 Summary Thermalization occurs in generic isolated systems Finite size effects Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

53 Summary Thermalization occurs in generic isolated systems Finite size effects Eigenstate thermalization hypothesis Ψ α Ô Ψ α = O microcan. (E α ) Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

54 Summary Thermalization occurs in generic isolated systems Finite size effects Eigenstate thermalization hypothesis Ψ α Ô Ψ α = O microcan. (E α ) Small time fluctuations smallness of off-diagonal elements Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

55 Summary Thermalization occurs in generic isolated systems Finite size effects Eigenstate thermalization hypothesis Ψ α Ô Ψ α = O microcan. (E α ) Small time fluctuations smallness of off-diagonal elements Time plays only an auxiliary role E a E b E c E a E b E c thermal thermal thermal thermal thermal thermal coherence time dephasing coherence Thermal state Initial state EIGENSTATE THERMALIZATION Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

56 Summary Thermalization occurs in generic isolated systems Finite size effects Eigenstate thermalization hypothesis Ψ α Ô Ψ α = O microcan. (E α ) Small time fluctuations smallness of off-diagonal elements Time plays only an auxiliary role Integrable systems are different (Generalized Gibbs ensemble) E a E b E c E a E b E c thermal thermal thermal thermal thermal thermal coherence time dephasing coherence Thermal state Initial state EIGENSTATE THERMALIZATION Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

57 Summary Thermalization occurs in generic isolated systems Finite size effects Eigenstate thermalization hypothesis Ψ α Ô Ψ α = O microcan. (E α ) Small time fluctuations smallness of off-diagonal elements Time plays only an auxiliary role Integrable systems are different (Generalized Gibbs ensemble) Thermalization and ETH break down close integrability (finite system) Quantum equivalent of KAM? E a E b E c E a E b E c thermal thermal thermal thermal thermal thermal coherence time dephasing coherence EIGENSTATE THERMALIZATION Thermal state Initial state Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

58 Collaborators Vanja Dunjko (U Mass Boston) Alejandro Muramatsu (Stuttgart U) Maxim Olshanii (U Mass Boston) Anatoli Polkovnikov (Boston U) Lea F. Santos (Yeshiva U) Mark Srednicki (UC Santa Barbara) Current group members: Deepak Iyer, Baoming Tang Former group members: Kai He (NOAA), Ehsan Khatami (SJSU) Supported by: Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29

59 Information entropy (S j = D k=1 ck j 2 ln c k j 2 ) L.F. Santos and MR, PRE 81, (2010); PRE 82, (2010). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, / 29