General quantum quenches in the transverse field Ising chain Tatjana Puškarov and Dirk Schuricht Institute for Theoretical Physics Utrecht University Novi Sad, 23.12.2015
Non-equilibrium dynamics of isolated quantum systems 1. Consider an isolated many-particle quantum system 2. Take it out of equilibrium 3. Let the system evolve 4. Observe what happens: does the system thermalise/does it relax? how can we describe the steady state? how do the observables behave?
Ultracold atoms in optical lattices Simulation of solid-state models: electrons crystal lattice atoms optical lattice control of tuneable parameters: lattice type, dimensionality, interactions parameters tuneable in time almost perfect isolation from the environment Bloch, Dalibard & Zwerger, RMP, 2008 87 Rb BEC JILA BEC SF to MI transition Greiner et al., Nature, 2002
Quantum Newton s cradle 1D tubes with 40-250 87 Rb atoms initially momentum eigenstate with ±k essentially unitary time evolution Kinoshita, Wenger & Weiss, Nature, 2006
Quantum Newton s cradle 1D = 18 t = 15 3D t = 0 = 3.2 t = 25 t = 2 = 14 t = 25 t = 4 no thermalisation in 1D t = 9 thermalisation after ~3 collisions in 3D Kinoshita, Wenger & Weiss, Nature, 2006
Relaxation following sudden quenches Consider a system H(g) with a tuning parameter g pre quench: H(g i ), (0)i g g f quench: post quench: g i! g f H(g f ), (t)i =e ih(g f )t (0)i g i 0 t (t)i =e ih(g f )t (0)i = X n hn (0)ie ie nt ni decomposition in terms of post-quench Hamiltonian eigenstates Observables: h (t) O (t)i = X m,n h (0) mihn (0)ihn O mie i(e n E m )t interference of oscillating phases
Relaxation following quenches Full system AUB: initial state is a pure state i the system never relaxes as a whole density matrix observables (t) = (t)ih (t) h (t) O (t)i =tr[ (t)o] A Subsystem B: B A (infinite) acts as a bath for B (finite) the system relaxes locally density matrix observables B (t) =tr A (t) lim lim t!1 L!1 h (t) O B (t)i h (t) (t)i = const
Can we express Generic systems Relaxation following quenches B (1) in terms of a statistical ensemble? thermalisation A B energy is the only local integral of motion Gibbs ensemble G = 1 Z G exp H Deutsch, PRA, 1991 Integrable systems relaxation additional local integrals of motion generalised Gibbs ensemble Rigol et al., PRL, 2007! GGE = 1 X exp ni n Z GGE n Lagrange multipliers tr ( GGE I n )=h 0 I n 0i full set of local integrals of motion [H, I n ]=[I m,i n ]=0
General quantum quenches Consider a system H(g) with a tuning parameter g g g f g i 0 t Initial system: t<0 H(g i ) Post-quench system: H(g f ) t> 0i (t)i =e ih(g f )(t ) ( )i
General quantum quenches Consider a system H(g) with a tuning parameter g g g f g i 0 t Initial system: t<0 H(g i ) Post-quench system: H(g f ) t> 0i (t)i =e ih(g f )(t ) ( )i
General quantum quenches Consider a system H(g) with a tuning parameter g g g f g i 0 t Initial system: t<0 H(g i ) Post-quench system: H(g f ) t> 0i (t)i =e ih(g f )(t ) ( )i
General quantum quenches Consider a system H(g) with a tuning parameter g sudden quench g g f linear quench cosine quench g i general quench 0 t Initial system: t<0 H(g i ) Post-quench system: H(g f ) t> 0i (t)i =e ih(g f )(t ) ( )i
Compare the duration of quench General quantum quenches gap in the system, interaction energies! 0!1 to the timescales in the system: g g f g g f g i g i 0 t 0 t sudden limit the change is too fast for the system to react in intermediate times it only notices the final value of the quench parameter adiabatic limit the change is slow enough for the system to follow in the ground state of each intermediate Hamiltonian
Transverse Field Ising chain H = J NX i=1 x i x i+1 + g z i FM PM 0 g =0: 1 1 g = 1 : ###...#i!!!!i """..."i g PM to AFM transition in an ultracold atom system Response of the system to a quantum quench Meinert et al., PRL, 2013 Simon et al., Nature, 2011 Sachdev, Sengupta & Girvin, PRB, 2002
Transverse Field Ising chain H = J NX i=1 x i x i+1 + g z i 1) Jordan-Wigner transformation z i =1 2c i c i x i = Y j<i(1 2c j c j )(c i + c i ) 2) Fourier transformation 3) Bogoliubov transformation k = u k c k iv k c k k = u k c k +iv k c k H = X k " k k k 1 2 " k " k =2J p 1+g 2 2g cos k =2J 1 g k
Global general quench in TFI chain H(t) = J NX i=1 x i x i+1 + g(t) z i g g f g i Each instantaneous Hamiltonian can be diagonalised as H(t) = X " k,t k,t 1 k,t 2 k,t = u k,t c k,t k 0 iv k,t c k,t t " k,t =2J p 1+(g(t)) 2 2g(t) cos k How to relate the infinitely many instantaneous Hamiltonians?
Global general quench in TFI chain Cast the dynamics into the Bogoliubov coefficients: c k (t) =u k (t) k +iv k (t) k Explicit time dependance from Heisenberg equations of motion: i @ @t c k(t) =[c k,h] i @ @t c k (t) =[c k,h] i @ @t uk (t) v k (t) = Ak (t) B k B k uk (t) A k (t) v k (t) During quench @ 2 @t 2 y(t)+ Post quench apple A k (t) 2 + B 2 k ± i @ @t A k(t) y(t) =0 @ 2 @t 2 y(t)+!2 ky(t) =0 A k (t) =2 g(t) cos k B k =2sink sudden vs linear quench
During quench Transverse magnetisation Post quench M z (t) =h z (t)i t 3/2 Stationary values behaviour during the quench depends on the quench duration behaviour after the quench general - algebraic decay with internal oscillations GGE constructed from the postquench mode occupation numbers
C(x, t) =ho(x, t)o(0,t)i ho(t)i 2 Two-point correlation functions t 0 x<2vt C(x, t) 6= 0 light-cone x>2vt C(x, t) =0 x Cheneu et al, Nature, 2012 initially correlations only within the system correlation length 0 state 0i acts as a source of entangled quasiparticles quasiparticles propagate classically through the system onset of correlations at t x 2v Calabrese & Cardy, PRL, 2006
C(x, t) =ho(x, t)o(0,t)i ho(t)i 2 Two-point correlation functions Langen et al, Nature Phys 2013 Cheneu et al, Nature, 2012
Transverse two-point correlation function z n(t) = z i (t) z i+n(t) t 3 algebraic decay with internal oscillations light-cone effect and Lieb- Robinson bound observable n = 10 n = 20 n = 30 n = 40 n = 50 n = 60 n = 70 n = 80 n = 90 n = 100 Time evolution of the transverse two-point correlation function for various separations
30 Transverse two-point correlation function z n(t) = z i (t) z i+n(t) 25 t 20 15 10 5 0.0010 0.0008 0.0006 0.0004 0.0002 0 x<2vt C(x, t) 6= 0 x>2vt C(x, t) =0 x 20 40 60 80 100 C(x, t) =ho(x, t)o(0,t)i ho(t)i 2 Modification of the quasiparticle picture: at any time t during the quench the quasiparticles propagate with their instantaneous velocity for general quenches the quasiparticles will not be created just at over the quench time t =0 but
Summary and outlook Isolated 1D systems show non-trivial non-equilibrium dynamics Local observables in the system relax (GGE) Two-point functions allow for observation of the light-cone effect General quenches add additional features compared to the sudden case TFIC: quenching across the critical point (Kibble-Zurek mechanism, DPT) Karrasch & Schuricht, PRB, 2013