Quench dynamics of one-dimensional interacting bosons in a disordered potential: Elastic dephasing and critical speeding-up of thermalization Marco Tavora (New York University) Aditi Mitra (New York University) Achim Rosch (University of Cologne)

Supported by: NSF DMR 1303177

t <0

Description of the problem At t = 0, system of 1D bosons is driven out of equilibrium via a quench involving a sudden switching on of a disordered potential 2 2⎤ ⎡ u 1 Hi = d x ⎢ K (πΠ(x) ) + ( ∂ x φ (x) ) ⎥ ∫ 2π K ⎣ ⎦

Hf =

t >0

(t < 0)

2 2⎤ ⎡ ⎡ 1 u 1 * 2iφ −2iφ ⎤ d x K π Π(x) + ∂ φ (x) + d x − η (x)∂ φ + ξ e + ξ e (t > 0) ( ) ( ) ⎢ ⎥ ⎢ ⎥ x x ∫ ∫ 2π K ⎣ ⎦ !###### ⎣ π "######$⎦

(

)

Vdis

Hi is a Luttinger liquid (LL), where the excitations are independent density modes. The parameters u and K contain the information about the interactions: u is the velocity of the density oscillations and K describes the decay of the correlators We investigate the regime where disorder is irrelevant by studying the superfluid side of the superfluid-Bose glass QCP

dD! b = (3− 2K ) D! b dl

D 3 K= 2

Bose Glass

Superfluid

K

Dynamics close to superfluid-Bose glass QCP

⎡ 1 * 2iφ −2iφ ⎤ Vdis = ∫ d x ⎢ − η (x)∂ x φ + ξ e + ξ e ⎥ ⎣ π ⎦

(

Forward scattering

η (x)η ( x ′ ) = D f δ (x − x ′ )

)

Backward scattering

ξ (x)ξ * ( x ′ ) = Dbδ (x − x ′ )

We will study the dynamics of: Density-density correlation function

Rφφ (r,t) = Ψ(t) e2iφ (r ) e−2iφ (0) Ψ(t) Q = 2πρ 0

Single-particle correlation function (superfluidity)

Rθθ (r,t) = Ψ(t) eiθ (r ) e− iθ (0) Ψ(t)

Equilibrium:

⎛ 1⎞ Rφφ (r) ∼ cos ( 2πρ 0 r ) ⎜ ⎟ ⎝ r⎠

Vdis,f

2K

⎛ 1⎞ Rθθ (r) ∼ ⎜ ⎟ ⎝ r⎠

,

Df

Rφφ (r) ⎯⎯→ Rφφ ,dis (r) = Rφφ (r)e

1 = − ∫ d x η (x)∂ x φ π

(

Figures taken from Giamarchi notes (1998)

−( DK 2 /u 2 ) r

D

Vdis,b = ∫ d x ξ *e2iφ + ξ e−2iφ Pins density fluctuations and can suppress superfluidity

1/(2 K )

)

f Rθθ (r) ⎯⎯ → Rθθ ,dis (r) = Rθθ (r)

Imperfect pinning due to randomness of ξ

Superfluidity unchanged

What happens if only forwards scattering is present? 2 2⎤ ⎡ u 1 1 H f (Db = 0) = d x K π Π(x) + ∂ φ (x) − ( ) ( ) ⎥ π ∫ d x η (x)∂ x φ (x) 2π ∫ ⎢⎣ K x ⎦

K x ! φ (x) = φ (x) − ∫ dyη ( y) u 2⎤ 2 ⎡ u 1 ! H f (Db = 0) = d x K π Π(x) + ∂ φ (x) ( ) K( x ) ⎥ 2π ∫ ⎢⎣ ⎦ Ψ(t = 0) = vac

φ

H f (Db = 0) = ∑ u | p | Γ †p Γ p p

= Excited state of φ! -fields ⇒ Γ p quasiparticles out of equilibrium

Rφφ(0) (r,t) = 〈ψ i | e2iφ (r ,t ) e− i2φ (0,t ) |ψ i 〉 D Rθθ(0) (r,t) = 〈ψ i | eiθ (r ,t ) e− iθ (0,t ) |ψ i 〉 D

f

f

=0

e =0

e

−

−

iK ⎡ u ε=± ⎢⎣

i ⎡ 2u ⎢⎣

r+εut

∑ ∫r

r+ut

∫r−ut

d yη ( y )−

d yη ( y )−

εut

∫0

⎤ d yη ( y ) ⎥ ⎦

⎤ d yη ( y ) ⎥ − ut ⎦

∫

ut

The correlators are what they would have been in the absence of the forward scattering disorder but multiplied by random phases.

What happens if only forwards scattering is present? ⎡ ⎤ 1 R ( r,t ) = ⎢ ⎥ 2 2 ⎣ 1+ r Λ ⎦ (0) φφ

2K

e

−

K 2D f u

1/(2 K )

⎡ ⎤ 1 (0) Rθθ ( r,t ) = ⎢ ⎥ 2 2 ⎣ 1+ r Λ ⎦

e

−

[2tΘ(|r|/ u−2t )+( 4t −|r|/ u )Θ(2t −|r|/ u)Θ(|r|/ u−t )+3|r|Θ(t −|r|/ u)]

Df 4u

[2t−(2t −|r|/ u)Θ(2t −|r|/ u)]

Disorder-averaged correlators decay exponentially with time for short times, with a crossover to a steady-state behavior with an exponential decay in position at long times Quench creates left- and right-moving QP which pick up random phases: operator at r is affected by phases picked up in the region ⎡⎣ r − ut,r ⎤⎦ by the right movers, and phases picked up in the region ⎡⎣ r + ut,r ⎤⎦ by the left movers In contrast to equilibrium, when the system is quenched, even forward scattering strongly affects superfluidity due to random dephasing

Different time regimes Quench t=0

t=Λ

Prethermalization −1

Quench generates nonequilibrium quasiparticles

Thermalization

t = η −1 ≫ Λ −1

t→∞

Forward scattering kills superfluidity

Inelastic effects important

Dephasing makes backward scattering more “irrelevant”

Quasiparticles scatter among each other

Quasiparticles almost free

∂n p (t) ∂t

= I( p,t)

What happens when t ≫ η −1 ? At longer times, the interplay of disorder and interactions leads to thermalization which is strongly enhanced close to the superfluid-Bose glass transition.

⎫ u | p | ∂n p (t) iπ K 1 K R A R A ⎡ ⎤ ⎡ ⎤ =− n p (t) ⎣ Σ − Σ ⎦ ( p,t) − ⎣ Σ ( p,t) − (Σ − Σ )( p,t) ⎦ ⎬ 2 ∂t 2 2 Λ ⎭

{

Σ

R,A,K

∼e

2iφ

⇒ Multiparticle scattering

f b Teff ,0 T + T † eff eff n p (t ! 0) = 〈ψ i | Γ p Γ p |ψ i 〉 = = u| p| u| p|

Short 8me dynamics and the long 8me dynamics of the kine8c equa8on smoothly connected by perturba8vely evolving n( p) forward in 8me at short 8mes. We then use this distribu8on as the ini8al condi8on for the kine8c equa8on.

What happens when t ≫ η −1 ?

n p (t ! 0) =

Teff ,0 u| p|

qneq =

q e

u|q|/Teq −1

Equilibrium at long times

At long wavelengths, the distribution n( p) looks like an effective temperature, but for u | p |≥ Teff ,0 , n( p) , instead of being exponentially suppressed, maintains a slow power-law decay with momentum up to p → Λ

What happens when t ≫ η −1 ? Critical speeding up Dramatic reduction of thermalization time on approaching the superfluid Bose-glass QCP is due to the backward scattering disorder becoming more relevant, facilitating thermalization

What happens when t ≫ η −1 ? Out-scattering rate in the long wave limit: ∞ ⎛ π K ⎞ i(Σ R − Σ A ) p→0 γ ( p,t) =⎜ ⎯⎯⎯ → = 4KDb ∫ d(Λτ )sin ⎡⎣ 2K tan −1 Λτ ⎤⎦ ( Λτ ) e− I (t ,τ ) ⎟ −∞ ⎝ 2 ⎠ u| p|

∞

I(t,τ )= 2K ∫ 0

dq −α q e ⎡⎣1+ 2nq (t) ⎤⎦ ⎡⎣1− cos ( quτ ) ⎤⎦ q

Time-scale γ 0−1 for leaving PT regime

(

γ 0 ~ DbTeff ,0 ~ Db D f + bDb

)

Time-scale γ th−1 for final approach to thermal equilibrium

Thermalization rate from long time tail agrees well with analytic estimate

γ th ~ Db ⎡⎣Teff ,0 ⎤⎦

K −1

Conclusions • We investigated the quench dynamics in a system of interacting bosons which is quenched by the switching-on of a disordered potential • We derived a novel quantum kinetic equation accounting for both disorder and interactions • We found that, in contrast to equilibrium, random forward scattering destroys superfluidity by inducing elastic dephasing which are important even at short times • A longer times interplay of disorder and interactions leads to thermalization which is strongly enhanced close to the superfluid-Bose glass QCP

Quench dynamics of one-dimensional interacting bosons in a disordered potential: Elastic dephasing and critical speeding-up of thermalizatio...

Published on Apr 17, 2018

Quench dynamics of one-dimensional interacting bosons in a disordered potential: Elastic dephasing and critical speeding-up of thermalizatio...