Separation of relaxation time scales in a quantum Newton s cradle Rianne van den Berg Universiteit van Amsterdam CRM: Beyond integrability
Research team UvA Bram Wouters PhD student BNL Robert Konik PI UvA Sebas Eliëns PhD student UvA Vladimir Gritsev PI UvA Jacopo De Nardis PhD student UvA Jean-Sébastien Caux PI 2
Overview Overview of systems & methods used : Tonks-Girardeau on a ring & trap Methods: The Quench Action approach & the Fermi-Bose mapping The Bragg pulse / Kapitza-Dirac pulse Short time Bragg pulse t=0 momentum distribution function for all interaction strengths Time evolution of density & momentum distribution function On a ring geometry In a trap geometry Conclusions & Outlook 3
Systems & methods 4
Systems & methods Delta potential interacting bosons in 1D: Lieb-Liniger model NX 1 @ 2 X H LL = +2c (x i x j ). 2m i=1 @x 2 i We consider the Tonks-Girardeau model: Different geometries: 1applei<jappleN c!1 B(x 1,...,x N ; t)! 0, x i! x j, i 6= j B(x 1,...,x N ; t) = Y sgn(x i x j ) F (x 1,...,x N ; t) 1applei<jappleN Systems A gas on a Ring / periodic boundary conditions NX 1 A gas in a harmonic trap V(~x) = 2 m!2 x 2 i i=1 5
Systems & methods Quench Action approach (Caux & Essler, PRL 2013) Observables in the thermodynamic limit: sum over a countable number of particle-hole excitations lim th h 0 (t) Ô 0(t)i = 1 Z D e X S QA[ ] 2 e e i! e[ ]t h Ô, ei +c.c. = 1 2 Saddle point is defined as X e e i! e[ sp ]t h sp Ô sp, ei +c.c. 0= S QA [ ] ( ) = sp, S QA [ ] = 2lim th log h 0 i S YY [ ] Saddle point density is compliant with the GGE prediction Z 1 1 lim th N h 0 ˆQ m 0 i = d sp ( ) m, 8 m =1, 2, 3,... 1 6
Systems & methods Fermi Bose mapping [ Girardeau, J. Math. Phys 1960 ] Construct many body bosonic wavefunction from single particle fermionic wavefunctions B(x 1,...,x N ; t) = Y [ Pezer & Buljan, PRL 2007 ] Efficient method to compute one body density matrix h (x; t) (y; t)i = 1applei<jappleN NX i,j=1 sgn(x i x j )det N [ n(x i ; t)] / p N! i (x; t)a ij (x, y; t) j (y; t) A(x, y; t) = P 1 T det P with P ij (x, y; t) = ij 2 Gives acces to momentum distribution function n(k) Z y x dx 0 i (x 0 ; t) j (x 0 ; t) 7
Bragg pulse 8
Bragg/Kapitza-Dirac Pulse Bragg pulse: One body cosine potential with Bragg momentum q and amplitude V0 [ Wang et al., PRL 2005, Wu et al., PRA 2005 ] [ Kinoshita et al., Nat Lett, 2006 ] Time evolution operator " during Bragg Z pulse:! L Û B (V 0,T 0,q)=exp i Ĥ + V 0 dx cos(qx)ˆ (x) #T 0 For very short, and high amplitude pulses neglect dynamics due to kinetic energy terms: Area of the pulse: V (x) =V 0 cos(qx) Û B (A, q) = lim Û B (A/T 0,T 0,q)=exp T 0!0 ial =exp 2 (ˆ q +ˆ q ) with A V 0 T 0 0 ia Z L 0 ˆ q = X k dx cos(qx)ˆ (x) k+q k! 9
t=0 momentum distribution Initial one-body density matrix after short Bragg pulse h (x) (y)i = h e ia R dz cos(qz) (z) (x) (y)e ia R dz 0 cos(qz 0 ) (z 0) i Commutator of Bragg operator & one body reduced density matrix f[ˆ (x)] (y) = (y)f[ˆ (x)+ (x y)] f[ˆ (x)] (y) = (y)f[ˆ (x) (x y)] h (x) (y)i = e ia(cos(qx) cos(qy)) h (x) (y) i Copy of ground state momentum distribution peak of model under consideration to multiples of Bragg momentum q Z L ring: n(k) = 1 dx e ikx I 0 (i2a sin(qx/2)) h (x) (0) i L 0 Finite c influence only comes from the ground state n(k)! 10
t=0 momentum distribution Initial momentum distribution function for different interactions strengths c N = 50, A = 1.5, q = 3π 11
Ring geometry 12
Bragg pulse on a ring Hamiltonian after the Bragg pulse: NX 1 @ 2 X H TG = lim +2c c!1 2m i=1 @x 2 i 1applei<jappleN (x i x j ) FB mapping: Fermionic single particle wave functions after Bragg pulse 1X j(x; t) = ( ia) p 1 e ikix e ik2 i t L i= 1 (k j k i )%q,0i k j k i q Overlaps for many body wave functions from real space " # hµ ÛB i = L N det N ( j µ k )%q,0i j µ k ( ia) q j,k 13
Bragg pulse on a ring Time evolution of density Quench action gives exact result for density: lim th hˆ (x; t)i = n +Re 1 X 2Z, 6=0 i I (i2a sin(q 2 t))e ixq sin(2q t F ) 2q t β is refers to the satellite around k = βq Different velocities due to different satellites Density decays as 1/t Typical relaxation times scale as 1/q 14
Bragg pulse on a ring Time evolution of density A = 1 q = 3π Fermi Bose mapping: N=50 Quench Action FB - QA 15
Bragg pulse on a ring FB: N=50 Time evolution of the momentum distribution function Dephasing causes a broadening of n(k) QA Relaxed n(k) has peaks with a slight shift from q finite size effects: delta peaks not reproduced due to discretisation 16 A = 1.4 q = 3π
Trap geometry 17
Bragg pulse in a trap Model Hamiltonian NX 1 @ 2 H TG+trap = lim c!1 2m @x 2 + i=1 i Propagator: Mehler kernel r m! K(x, y; t) = 2 i sin(!t) exp NX i=1 1 2 m!2 x 2 i +2c X 1applei<jappleN m!(x 2 + y 2 ) cos(!t)+2m!xy 2i sin(!t) (x i x j ) Fermionic single particle wave functions after Bragg pulse 1X j(x; t) = ( i) n I n (ia)e in cos(!t) (x+ nq 2m! sin(!t) ) n= 1 n(x) = 1 p 2n n! j (x + nq m! sin(!t))e i!(j+ 1 2 )t m! 1/4 m!x e 2 p 2 H n m!x 18
Bragg pulse in a trap 19 Time evolution of the density Relaxation due to hard core limit is well separated from collective motion in trap Periodicity due to hard core limit, all eigenfunctions are periodic in ω Mechanisms that will break perfect periodicity: Finite c interactions Anharmonic terms in trapping potential N = 50 ω = 10/N A = 1.5 q = 3π
Bragg pulse in a trap Short-time density evolution 20
Bragg pulse in a trap Short-time density evolution 20
Bragg pulse in a trap Local Density Approximation: density evolution Fermi Bose mapping: N=50 Quench action + LDA: density profile of trap ground state n 0 (x) = p 2N!! 2 x 2 21 FB - QALDA A = 1.5 q = π ω = 10/N
Bragg pulse in a trap N = 50 ω = 10/N A = 1.5 q = 3π 22
Bragg pulse in a trap N = 50 ω = 10/N A = 1.5 q = 3π Time evolution of the momentum distribution function Broadening of momentum distribution function happens on very short time scales Suggests initial broadening is mostly due to hard core interaction and can be simulated by bosons on a ring Periodicity due to hard core limit, all eigenfunctions are periodic in ω 23
Bragg pulse in a trap Momentum distribution up to T = 0.06 π/ω for the trap & the ring N = 50, A = 1.5, q = 3π, ω = 10/N 24
Conclusion & Outlook 25
Conclusion & Outlook Main conclusion Well separated time scales for relaxation due to hard core interactions and collective motions in the trap Outlook N = 50 V0 = 100 T = 0.015 q = 3π Simulations of finite duration Bragg pulses Corrections in 1/c for the evolution of the density Anharmonic corrections to trapping potential 26
Conclusion & Outlook Main conclusion Well separated time scales for relaxation due to hard core interactions and collective motions in the trap Thank you Outlook N = 50 V0 = 100 T = 0.015 q = 3π Simulations of finite duration Bragg pulses Corrections in 1/c for the evolution of the density Anharmonic corrections to trapping potential 26