From unitary dynamics to statistical mechanics in isolated quantum - PowerPoint PPT Presentation

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, 2015 1 / 29

Outline Introduction 1 Experiments with ultracold gases Unitary evolution and thermalization Generic (nonintegrable) systems 2 Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble Summary 4 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 2 / 29

Experiments with ultracold gases in 1D Effective one-dimensional δ potential M. Olshanii, PRL 81 , 938 (1998). U 1 D ( x ) = g 1 D δ ( x ) where 2 � a s ω ⊥ � mω ⊥ g 1 D = 1 − Ca s 2 � Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 3 / 29

Experiments with ultracold gases in 1D Effective one-dimensional δ potential M. Olshanii, PRL 81 , 938 (1998). U 1 D ( x ) = g 1 D δ ( x ) where 2 � a s ω ⊥ � mω ⊥ g 1 D = 1 − Ca s 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 , 190406 (2005). γ eff = mg 1 D � 2 ρ Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 3 / 29

Absence of thermalization in 1D? Density profile Momentum profile momentum distribution density position momentum T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440 , 900 (2006). MR, A. Muramatsu, and M. Olshanii, Phys. Rev. A 74 , 053616 (2006). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 4 / 29

Absence of thermalization in 1D? Density profile Momentum profile τ =0 τ =0 momentum distribution density position momentum T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440 , 900 (2006). MR, A. Muramatsu, and M. Olshanii, Phys. Rev. A 74 , 053616 (2006). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 4 / 29

Absence of thermalization in 1D? n k 0 0.2 0.4 0.6 0.8 1 0 1000 τ 2000 Experiment Theory 3000 4000 −π/2 −π/4 0 π/4 π/2 k Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 5 / 29

Absence of thermalization in 1D? T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440 , 900 (2006). γ = mg 1 D � 2 ρ g 1 D : 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, 2015 6 / 29

Outline Introduction 1 Experiments with ultracold gases Unitary evolution and thermalization Generic (nonintegrable) systems 2 Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble Summary 4 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 7 / 29

Exact results from quantum mechanics If the initial state is not an eigenstate of � H � E 0 = � ψ 0 | � | ψ 0 � � = | α � where H | α � = E α | α � and H | ψ 0 � , then a generic observable O will evolve in time following | ψ ( τ ) � = e − i � O ( τ ) ≡ � ψ ( τ ) | � Hτ | ψ 0 � . O | ψ ( τ ) � where Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 8 / 29

Exact results from quantum mechanics If the initial state is not an eigenstate of � H � E 0 = � ψ 0 | � | ψ 0 � � = | α � where H | α � = E α | α � and H | ψ 0 � , then a generic observable O will evolve in time following | ψ ( τ ) � = e − i � O ( τ ) ≡ � ψ ( τ ) | � Hτ | ψ 0 � . O | ψ ( τ ) � where 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, 2015 8 / 29

Exact results from quantum mechanics If the initial state is not an eigenstate of � H � E 0 = � ψ 0 | � | ψ 0 � � = | α � where H | α � = E α | α � and H | ψ 0 � , then a generic observable O will evolve in time following | ψ ( τ ) � = e − i � O ( τ ) ≡ � ψ ( τ ) | � Hτ | ψ 0 � . O | ψ ( τ ) � where What is it that we call thermalization? O ( τ ) = O ( E 0 ) = O ( T ) = O ( T, µ ) . One can rewrite � � C ⋆ α ′ C α e i ( E α ′ − E α ) τ O α ′ α O ( τ ) = where | ψ 0 � = C α | α � , α ′ ,α α ρ DE ≡ � α | C α | 2 | α �� α | ) Taking the infinite time average (diagonal ensemble ˆ � τ � 1 dτ ′ � Ψ( τ ′ ) | ˆ | C α | 2 O αα ≡ � ˆ O | Ψ( τ ′ ) � = O ( τ ) = lim O � diag , τ τ →∞ 0 α which depends on the initial conditions through C α = � α | ψ 0 � . Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 8 / 29

Width of the energy density after a sudden quench Initial state | ψ 0 � = � α C α | α � is an eigenstate of � H 0 . At τ = 0 � H 0 → � � H = � H 0 + � � � W with W = w ( j ) ˆ and H | α � = E α | α � . j MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 9 / 29

Width of the energy density after a sudden quench Initial state | ψ 0 � = � α C α | α � is an eigenstate of � H 0 . At τ = 0 � H 0 → � � H = � H 0 + � � � W with W = w ( j ) ˆ and H | α � = E α | α � . j The width of the weighted energy density ∆ E is then �� � � α | C α | 2 − ( E α | C α | 2 ) 2 = � ψ 0 | � W 2 | ψ 0 � − � ψ 0 | � ∆ E = W | ψ 0 � 2 , E 2 α α or � � √ N →∞ ∆ E = [ � ψ 0 | ˆ w ( j 1 ) ˆ w ( j 2 ) | ψ 0 � − � ψ 0 | ˆ w ( j 1 ) | ψ 0 �� ψ 0 | ˆ w ( j 2 ) | ψ 0 � ] ∝ N, j 1 ,j 2 ∈ σ 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, 2015 9 / 29

Width of the energy density after a sudden quench Initial state | ψ 0 � = � α C α | α � is an eigenstate of � H 0 . At τ = 0 � H 0 → � � H = � H 0 + � � � W with W = w ( j ) ˆ and H | α � = E α | α � . j The width of the weighted energy density ∆ E is then �� � � α | C α | 2 − ( E α | C α | 2 ) 2 = � ψ 0 | � W 2 | ψ 0 � − � ψ 0 | � ∆ E = W | ψ 0 � 2 , E 2 α α or � � √ N →∞ ∆ E = [ � ψ 0 | ˆ w ( j 1 ) ˆ w ( j 2 ) | ψ 0 � − � ψ 0 | ˆ w ( j 1 ) | ψ 0 �� ψ 0 | ˆ w ( j 2 ) | ψ 0 � ] ∝ N, j 1 ,j 2 ∈ σ where N is the total number of lattice sites. Since the width W of the full energy spectrum is ∝ N ∆ ǫ = ∆ E 1 N →∞ ∝ √ , W N so, as in any thermal ensemble, ∆ ǫ 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, 2015 9 / 29

Outline Introduction 1 Experiments with ultracold gases Unitary evolution and thermalization Generic (nonintegrable) systems 2 Time evolution vs exact time average Statistical description after relaxation Eigenstate thermalization hypothesis Time fluctuations 3 Integrable systems Time evolution Generalized Gibbs ensemble Summary 4 Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 10 / 29

Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian � � � � � ˆ b † i ˆ ˆ b † 2 = ˆ b 2 H = − J b j + H.c. + U ˆ n i ˆ n j , i = 0 i � i,j � � i,j � MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian � � � � � ˆ b † i ˆ ˆ b † 2 = ˆ b 2 H = − J b j + H.c. + U ˆ n i ˆ n j , i = 0 i � i,j � � i,j � MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). Nonequilibrium dynamics in 2D Weak n.n. U = 0 . 1 J N b = 5 bosons N = 21 lattice sites Hilbert space: D = 20349 Initial All states are used! Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29

Relaxation dynamics of hard-core bosons in 2D Hard-core boson Hamiltonian � � � � � ˆ b † i ˆ ˆ b † 2 = ˆ b 2 H = − J b j + H.c. + U n i ˆ ˆ n j , i = 0 i � i,j � � i,j � MR, V. Dunjko, and M. Olshanii, Nature 452 , 854 (2008). “One can rewrite � � α ′ C α e i ( E α ′ − E α ) τ O α ′ α C ⋆ O ( τ ) = | ψ 0 � = C α | α � , where α ′ ,α α and taking the infinite time average (diagonal ensemble) � τ � 1 dτ ′ � Ψ( τ ′ ) | ˆ O | Ψ( τ ′ ) � = | C α | 2 O αα ≡ � ˆ O ( τ ) = lim O � diag , τ τ →∞ 0 α which depends on the initial conditions through C α = � α | ψ 0 � .” Marcos Rigol (Penn State) Dynamics in quantum systems May 4, 2015 11 / 29