Saarland University Simulating non-equilibrium quantum relaxation - PowerPoint PPT Presentation

Saarland University Simulating non-equilibrium quantum relaxation in closed interacting quantum many body systems H. Rieger, B. Blass Theoretical physics, Saarland University, Saarbrücken, Germany Simulating quantum processes and devices 624. Heraeus Seminar, Bad Godesberg, 19.-22.9.2019

Quantum thermalization – 2008 (numerics) H = -Σ <ij> J ij (a i + a j + a j + a i ), hard core bosons

Quantum Thermalization – 2016 (experimental) - STATISTICAL PHYSICS - Quantum thermalization through - entanglement in an isolated - many-body system , Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, t Philipp M. Preiss, Markus Greiner * rg SCI E N CE 4 19 AUGUST 2016 • VOL 353 ISSUE 6301 H = -Σ <ij> J ij (a i + a j + a j + a i ) + U Σ i n i (n i -1)

Thermalization in closed systems

Outline • Focus on transverse Ising model • 1d : integrable: Generalized Gibbs ensemble (GGE) • 2d: non-integrable: numerical studies of thermalizaIon aJer quantum quenches

The transverse Ising model X X σ x i σ x σ z H = � J i � h i ( ij ) n.n. i ✓ ◆ ✓ ◆ σ x , σ z Pauli matrices, n.n. = nearest neighbors in d-dimensions, p.b.c. Order parameter: Magne&za&on (not conserved!) m = h σ x i i Equilibrium phase diagram: T 2d T c (=2.25 J) h σ x i i = 0 1d T h σ x i i = 0 FM PM QC QC h σ x i i > 0 FM PM h σ x i i = 0 h σ x i i = 0 h c h σ x h h c h i i > 0 (=J) (=3.04 J)

Non-equilibrium relaxation with heat bath Quench from T=0, h=0 to T>0, h>0; heat bath dynamics, thermalizaIon è ρ~e -H/T log m(t) T 0 1d t è ~ e -t/τ h c h m(t) T 1 T c 2d è ~m rem + ce -t/τ 0 t h c h

Non-equilibrium dynamics in closed system Quench from h=0 to h>0; Schrödinger dynamics (conserved energy E) è Thermaliza&on ?? Temperature T(E) ? log m(t) ? 1d T 0 t ? ? è ? ~ e -t/τ if thermalizing 0 h c h m(t) T ? 2d 1 T c ? ? è if thermalizing ? ~m rem + ce -t/τ 0 0 t h c h

Quantum Quench in the 1d TIM (at T=0) X L X X ε p η + ( J σ x i σ x i +1 + h σ z H = � i ) = p η p ( η p Fermion operators i =1 p X | ψ ( t ) i = e − itH | ψ 0 i Quench: log m(t) |ψ(t=0> GS of H with e.g. h=0, dynamics for H with h>0 (exactly calculable) 0 t L=∞ T è ~ e -t/τ(h) 0 h c h Z π 1 / τ ( h ) = 2 with m i ( t ) = h ψ ( t ) | σ x i | ψ ( t ) i / exp( � t/ τ ( h )) dp v p · f p ( h 0 , h ) ( π 0 [Rieger, Iglói 2011] v p = ∂ε p f p ( h, h 0 ) = h ψ 0 | η + p ( h ) η p ( h ) | ψ 0 i and [Calabrese, Essler, Fagon 2012] ∂ p Does the exponenIal relaxaIon mean that the system is thermalized? (no, because f p ≠ e -βε(p) )

Look at finite system – what‘s going on? 0 → 0.5 ~const m l (t) = <σ l (t)> magne&za&on at site l • Exponential relaxation • Quasi-stationary regime • Exponential recovery

Quasi-particles = kinks (in FM phase: h<J) Kinks = non-interacting fermions Finite system: are created in pairs (+p,-p) move with velocity ± v p … and will flip spins upon arrival! E.g. C(r 1 t 1 ;r 2 t 2 ) = < σ r1 (t 1 ) σ r2 (t 2 )>: Reflection at the boundaries at i=0 and i=L!

1d TIM does not thermalize exact semi classical 0 → 0.2 log m l (t) 0 → 0.1, L=256 log m l (t) QP occupation probability: thermal Period: QPs non-interacIng, T period = L/v max ~ L/h è f p conserved True f p è no thermalizaIon towards f p ~e -βε(p)

The 2d transverse Ising model: equil. & quenches Quenches: ( J i ; h i ) ! ( J f ; h f ) Final energy: H = � J R 0 � h λ E f , λ | h Ψ i , 0 | Ψ f , λ i | 2 h E f = P ˆ X X � x � x � z ˆ R ˆ ˆ energy in the system after th R 2 2 Excess energy: < R , R 0 > R E exc = E f � E f , 0 , Effec&ve T eff h ˆ µ x = 1 H f i CGE = E f X � x ˆ ˆ R . temperature : N R 1 X Equilibrium phase diagram p | Ψ i , 0 i = | "" . . . ""i z = | x i 2 N 1 . 5 1 . 0 4 x Interac&on 1 . 5 • L = 4 paramagnetic 0 . 8 • L = 8 3 1 . 0 • L = 16 1 . 0 T/J quenches: • L = 32 0 . 6 T/J <μ x > 0 . 5 T/J ferromagnetic 2 0 . 4 (0,h) -> (J,h) 0 . 0 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 h/J 0 . 2 1 0 . 0 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 0 h/J 0 1 2 3 4 5 6 7 8 9 10 h/J 1 . 5 − 0 . 4 − 0 . 6 1 . 5 − 0 . 8 paramagnetic 1 . 0 field <H> T/J − 1 . 0 1 . 0 − 1 . 2 quenches : T/J 0 . 5 − 1 . 4 − 1 . 6 (J,0) -> (J,h) 0 . 5 − 1 . 8 ferromagnetic 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 h/J 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 h/J 1 n o p | Ψ i , 0 i = | "" . . . ""i x + | ## . . . ##i x 2 5 2 0 2

Time evolution 1 X | Ψ ( t = 0) i = | Ψ i , 0 i | Ψ i , 0 i = | "" . . . ""i z = p | x i 2 N x 1 n o | Ψ i , 0 i = p | "" . . . ""i x + | ## . . . ##i x H f t | Ψ ( t = 0) i | Ψ ( t ) i = e � ı ˆ 2 O h ˆ X X | c f , λ | 2 O λλ + c ⇤ f , λ c f , λ 0 e ı ( E f , λ � E f , λ 0 ) t O λλ 0 O i t = λ 6 = λ 0 λ Z T (15) 1 dt h ˆ X | c f , λ | 2 O λλ . lim O i t = T T !1 0 λ h ˆ O i diag = Tr[ ˆ ρ diag = P h ˆ λ p f , λ | Ψ f , λ ih Ψ f , λ | an O ˆ ρ diag ] or the distributions of the (pos d p f , λ = | c f , λ | 2 .

Techniques for non-integrable systems L x L square lance, H = � J R 0 � h ˆ X X � x � x � z ˆ R ˆ ˆ p.b.c. R 2 2 < R , R 0 > R 2d TFIM Non-integrable (in parIcular not a free fermion model as in 1d) ! | ψ ( t ) i = e − itH | ψ 0 i What can one do to study the (n.eq.) dynamics ? 1) Mean field theory (or truncated hierarchy of correlaIons) 2) Exact diaginalizaIon of snall systems (up L=4 or 5) 3) Time series expansion 4) PerturbaIon theory (e.g. in h) 5) Time dependent variaIonal calculaIons 6) Real Ime Quantum Monte Carlo 7) Quantum Boltzmann equaIon (?) 8) ...

Real Time Dependent Variational Ansatz Principle: X • Choose set of „physically relevant“ subspace V = span{|Φ 1 >,..., |Φ M >}, M<<2 N M M X X • IniIal state , Ansatz: | ψ ( t ) i = α k ( t ) | φ k i | ψ 0 i = α k | φ k i X X reads in the x -basis k =1 k =1 • At each Ime t minimize distance between � 2 . X � ˙ Ψ exact ( x , t ) � ˙ � � D ( t ) = Ψ var ( x , t ) i ∂ X x X and α k ( t ) | φ k i H α k ( t ) | φ k i ∂ t k k • ResulIng differenIal equaIon for variaIonal parameters α k (t): X i ∂ X ∂ t α k ( t ) = α k 0 ( t ) h φ k | H | φ k 0 i k 0 • Solve and calculate observales

Interaction quenches (paramagnetic phase) Jastrow Ansatz ⇣ X ⌘ α r ( t ) ˆ C xx | Ψ ( t ) i = exp | "" . . . ""i z c.f. Carleo etal as variaIonal funcIons r PRA 2014 r = 1 ˆ X C xx σ x σ x ˆ R ˆ r R + r N r R X h δ ˆ r δ ˆ ( t ) δ ˆ C xx C xx α r 0 ( t ) = � ı h E local C xx EquaIons of r 0 i t ˙ r i t f MoIon for α r r 0 with δ O = O � h O i t h x | ˆ th δ ˆ O = ˆ O � h ˆ gy E local ( x , t ) = H f | Ψ ( t ) i / h x | Ψ ( t ) i . O i t f values ˆ IniIal condiIon α r ( t = 0) = 0 . x | Ψ ( x , t ) | 2 O ( x ) P ExpectaIon values h ˆ O i t = P x | Ψ ( x , t ) | 2 | i | i ⇥ ⇤ A ( x ! x 0 , t ) = min 1 , Q ( x ! x 0 , t ) CalculaIon via Monte Carlo n �o X α R C xx r ( x 0 ) � C xx Q ( x ! x 0 , t ) = exp � 2 r ( t ) r ( x ) r (A5)

Field quenches (ferromagnetic phase) N m,n 1 X | Ψ ( t ) i = α m,n ( t ) | Ψ m,n i X | Ψ k | Ψ m,n i = m,n i p N m,n m,n k =1 Symmetric superposiIon of all states with n spins up and m kinks 4 8 12 16 L number of α m,n 45 848 4551 14834 � � h Eq. Of X t m 0 ,n 0 ; m,n α m 0 ,n 0 ( t ) ı ˙ α m,n ( t ) = � J ( N � n ) α m,n ( t ) moIon: 2 h X m 0 ,n 0 h t m 0 ,n 0 ; m,n = T m 0 ,n 0 ; m,n / p N m 0 ,n 0 N m,n . T er m 0 and n 0 runs over all subspaces th sitions bet = number of transiIons between T m,n ; m 0 ,n 0 . T IniIal valuse: H n,m and H n‘,m‘ via 1 spin flip. ( 1 if ( m, n ) = (0 , 0) or ( N, 0) √ α m,n ( t = 0) = 2 0 else CalculaIon with rare event sampling Monte Carlo

Observables Z Z t 0 + ∆ t O i t = 1 h ˆ dt Tr[ ˆ O ˆ ρ ( t )] th ˆ ρ ( t ) = | Ψ ( t ) ih Ψ ( t ) | an ∆ t ˆ t 0 Time average: Z t 0 + ∆ t p t ( O j ) = 1 dt Tr[ δ ( O j � ˆ O ) ˆ ρ ( t )] ∆ t t 0 1 T eff O e − ˆ h ˆ Tr[ ˆ H/T eff ] O i CGE = Z T eff Thermal average: CGE 1 O ) e − ˆ p T eff Tr[ δ ( O j � ˆ H/T eff ] . CGE ( O j ) = Z CGE Calculated with conInuous imaginary Ime cluster Monte Carlo