Universal Dynamics of a Soliton after a Quantum Quenchffranchi/presentations... · Universal...

Universal Dynamics

of a Soliton

after a

Quantum Quench

Collaborators:

A. Gromov (Stony Brook), M. Kulkarni (Princeton),

A. Trombettoni (CNR-SISSA)

Fabio Franchini

arXiv:1408.3618

• Introduction & Motivations:

Out of equilibrium & Quantum Quenches

• Our Quench Protocol

• Hydrodynamis & KdV Reduction

• Results: Universal splitting

• Conclusions

Outline

Universal Dynamics of Soliton after a Quantum Quench n. 2 Fabio Franchini

• Experimental progresses challenge us with new questions:

Out-of-Equilibrium


Transition from Superfluid

to Mott Insulator

Greiner, Mandel, Esslinger, Haensch & Bloch,

Nature 415 (2002)

Quantum Newton’s Cradle

Kinoshita, Wenger, & Weiss, Nature 440 (2006)

• Quest for recurring structures in out-of-equilibrium

systems:

Aging

(Dynamical) Quantum Phase Transitions

Work Statistics

Preaching to the Choir…


Equilibration?

Thermalization?

Pre-Thermalization?

GGE?

Non-equilibrium State?

GGE?

Yes

No

• Reductionist Approach (universalities?)

• Different Set-ups to be considered

• Typical protocol: Quantum Quench

Initial condition: ground state of local Hamiltonian

Evolution: different Hamiltonian

• Extended excited states also considered

(free fermions)

Out-of-Equilibrium Stat. Mech.?


Bucciantini, Kormos, Calabrese, JPA 47 (2014)

Our question:

What happens if you change the

interaction strength

in a system prepared

in a (moving) localized excitation?

Our Answer:

Short time dynamics is Universal !

Quenching a Soliton


• Take a system in its Ground State

• Let it evolve according to different Hamiltonian

• Unitary evolution:

Quantum Quenches


jª(t)i =X

j

hjjª0i eiEjt jji

jª0i

H 6= H0

Hjji = Ej jji

Does the system reach a

stationary state, in some sense?

• Restricted to local observables, most quantum

quenches result in an effective stationary mixed state

• Moreover, generally:

(Gibbs distribution consequence of

Eigenstate Thermalization Hypothesis)

Gibbs Ensemble


jª(t)i =X

j

cj eiEjt jji

½e® =e¡¯effH

Z

limt!1

hª(t)jOjª(t)i = Tr [½e®: O]

Deutsch, PRA 43 (1991); Srednicki, PRE 50 (1994); Rigol, Dunjko, & Olshanii, Nature 452 (2009)

• If system has local conservation laws (f.i. integrabilty),

these should be included → G.G.E.

• Open problem: find all local charges

Generalized Gibbs Ensemble


½e® =e¡P

l¯lIl

Z

limt!1


Countless efforts from • SISSA (Mussardo, Silva, Gambassi & collaborators); • Pisa (Calabrese & collaborators); • Oxford (Cardy, Essler & collaborators); • Amsterdam (Caux & collaborators); • Many more (Polkovnikov, Mitra, Kehrein, Andrei, Prosen)…

• Quantum dynamics → Unitary Evolution

• A pure sate evolve into a pure state

• However, locally, the asymptotic state can be

approximated by a mixed one:

• Out-of-equilibrium quantum systems act as their own bath

• Locality allows transition from quantum to classical

Unitary Dynamics


limt!1


• Instead of a ground state, let’s start with a

localized excited state in interacting system

• Let it evolve with a different Hamiltonian

• Previously: local quenches or extended excited

states (in free systems)

• Universality emerges for short times!

Our Protocol


• Consider a cold-atom system

• Prepare a single solitonic state

Our Set-Up


x

x

½(x¡ v t)

v

½0

½0

v

½(x¡ v t)

Bright Soliton (v > c)

Dark (gray) Soliton (v < c)

• A localized excitation cannot be eigenstate of translational invariant Hamiltonian

• Nonetheless, long-lived localized excitations are observed in cold atom systems:

Solitons?


Strecker, Partridge, Truscott, & Hulet, Nature 417 (2002)

• Soliton: “Localized excitation that propagates at

constant velocity while maintaining its shape”

• Stable solutions of certain PDE

→ balancing of dispersive and non-linear terms

• Multi-soliton solutions exist only for integrable systems

• Solitonic states are ubiquitous

Solitons


nonlinearity

dispersion Courtesy of A. Abanov

Soliton on Scott Russell Aqueduct


Dugald Duncan/Heriot-Watt University, Edinburgh

https://www.youtube.com/watch?v=SknvLa8qEu0

Morning Glory


•300 m

•2000 m

40-60 km/h

Rolling Clouds in the Gulf of Carpentaria,

Northern Australia

Morning Glory: solitons


Mick Petroff - Wikimedia Commons

Soliton Dynamics


• Soliton-like solutions evolve without deformation

Courtesy of A. Abanov

Soliton Dynamics


D.R. Christie (1988)



Multi-solitons exist only

in integrable dynamics

• Generated from ground state applying phase mask

• It is not clear how to describe them as quantum states

→ Probably some sort of coherent state

for interacting systems

• Emerge naturally from semi-classical hydrodyamic

description

→ Low-entanglement excitations!

Solitons in cold atoms


• Existence of solitons (and many more experimental

probes) indicates the validity of hydrodynamic

description for cold atoms (f.i. Gross-Pitaevskii Eq.)

• Semi-classical description: only density & velocity

→ single-body reduced

• Valid for superfluids, weakly interacting systems…

→ low entanglement states

Hydrodynamic Approach


1. Excite a solitonic state & let it evolve

2. At some point, change interaction strength of

underlying quantum Hamiltonian (change scattering

length, sound velocity…)

3. Follow evolution immediately after the quench

• Use effective (semi-classical) hydrodynamics,

not unitary evolution

Our proposal


• We consider a one-component, Galilean

invariant, isentropic, inviscid fluid:

Hydrodynamcis


H =

Zdx

·½v2

2+ ½²(½) + A(½)

(@x½)2

4½

¸

_½ + @(½v) = 0

_v + @

µv2

2+ !(½)¡ A0(½)(@p½)2 ¡A(½)

@2p

½p

½

¶= 0 Euler

Continuity

! = @½ [½²(½)]Enthalpy: Quantum pressure

• Short times (qualitatively like wave equation):

• Quench acts as external perturbation: soliton

splits into transmitted and reflected component

• Longer Times: different scenarios

(soliton trains + dispersive waves

vs. dissipation)

What to expect


Vc ! c0

suddenlyVtVr

• Linearizing non-linear PDE: Bogolioubov modes

(phonons, Luttinger Liquid…)

Linearization


H =

Zdx

·½v2

2+ ½²(½) + A(½)

(@x½)2

4½

¸

_½ + @(½v) = 0

_v + @

µv2

2+ !(½)¡ A0(½)(@p½)2 ¡A(½)

@2p

½p

½

¶= 0 Euler

Continuity

• Non-linear behavior for small perturbations:

• Particular scaling of density, velocity, space & time

→ Korteweg-de Vries equation (KdV)

(non-linear fixed point for local interactions) •

• KdV: wave on shallow water surfaces, chiral equation

KdV Reduction


H =

Zdx

·½v2

2+ ½²(½) + A(½)

(@x½)2

4½

¸

½(x; t) = ½0 + ² ½(1)(x; t) + ²2 ½(2)(x; t) + : : :

v(x; t) = ² v(1)(x; t) + ²2 v(2)(x; t) + : : :

Kulkarni & Abanov, PRA 86 (2012)

• KdV scaling:

KdV Reduction


½(x; t) = ½0 + ² ½(1)(x; t) + ²2 ½(2)(x; t) + : : :

v(x; t) = ² v(1)(x; t) + ²2 v(2)(x; t) + : : :

_u§ ¨ @x·cu§ +

³

2u2§ ¡ ®@2xu§

¸= 0

u§ = ½(1) =

c

!00v(1)

c ´p

½0!00

³ ´ c½0

+@c

@½0

® ´ A(½0)4c

Kulkarni, & Abanov, PRA 86 (2012)

_½ + @(½v) = 0

_v + @

µv2

2+ !(½)¡ A0(½)(@p½)2 ¡A(½)

@2p

½p

½

¶= 0 Euler

Continuity

• Lieb-Liniger:

• For weak interaction collective

description by Non-Linear Schrödinger Equation

• Reduce to canonical hydrodynamic form with ansatz

Example: Lieb-Liniger NLSE


Hmicro(c) = ¡NX

j=1

@2

@x2j+ 2g

X

j

KdV Reduction:

Example: NLSE


i¹h@tª(x; t) =

½¡ ¹h

2

2m@xx + c jÃ(x; t)j2

¾Ã(x; t)

c =

rg½0m

; ³ =3

2

c

½0; ® =

¹h2

8m2 c

!(½) =g

m½

A =¹h2

2m2_½ + @(½v) = 0

_v + @

µv2

2+ !(½)¡ A0(½)(@p½)2 ¡A(½)

@2p

½p

½

¶= 0

ª =p

½ eim¹h

R xv(x0)dx0

_u§ ¨ @x·cu§ +

³

2u2§ ¡ ®@2xu§

¸= 0 ; u§ = ±½ =

c

!00±v

±½(x; t) = ½0 ¡ ½(x; t) ¿ ½0

1. Excite a shallow solitonic state & let it evolve

→ Dynamic described by KdV

2. Change interaction strength of underlying quantum

Hamiltonian

3. Describe post quench dynamics by new Kdv,

with parameters modified by quench

• Universality for short time from KdV!

Our approach


• Quantum Quench:

• Initial KdV Soliton:

Quench


H0 = H(c0) ! H = H(c)

s(x; t) = ¡U cosh¡2·

x§ V tW

¸

W = 2

r®

c¡ V

U = 3c¡ V

³

NLSE: Gray Soliton

• Quantum Quench:

Quench


H0 = H(c0) ! H = H(c)

Harmonic Calogero: Bright Soliton

Soliton Splitting


• Quench acts as external perturbation: soliton splits

into transmitted and reflected components

• Continuity and momentum conservation yield

• Here: just kinematics. Need (KdV) dynamics to fix

Soliton Splitting


u(x; t) = R s(x¡ Vrt) + T s(x¡ Vtt)

R(V; Vr; Vt) =Vt ¡ VVt ¡ Vr

T (V; Vr; Vt) =V ¡ VrVt ¡ Vr

Vr;t

• Using KdV we determined

• Completely UNIVERSAL !

Chiral Profiles


u(x; t) = R s(x¡ Vrt) + T s(x¡ Vtt)

Vr = ¡ [c¡ ´ R (c¡ V )]c0

c

Vt = [c¡ ´ T (c¡ V )]c0

c

´ ´1 + ½0

c0@c0

@½0

1 + ½0c@c@½0

R =1

2

·1¡ c

c0V

´ V + (1¡ ´) c

¸

T =1

2

·1 +

c

c0V

´ V + (1¡ ´) c

¸

Vr = ¡ (T c + R V )c0

c

Vt = (R c + T V )c0

c

R =1

2

h1¡ c

c0

i

T =1

2

h1 +

c

c0

i

Same as linear problem, but non-trivial velocities!

c / ½µ0)

´ = 1

Numerical Checks: Amplitudes


R =1

2

h1¡ c

c0

i

T =1

2

h1 +

c

c0

i

Numerical Checks: Velocities


Black: V = 0:96cRed: V = 0:90c

Circles & Squares: NLSEAsteriscs: Con¯ned NLSETriangles: Power-Law NLSE

Vr = ¡ (T c + R V )c0

c

Vt = (R c + T V )c0

c

• Integrable in harmonic confinement!

• Long(ish)-range model: hydrodynamics in Benjamin-Ono

class (not KdV, different dispersion)

• Solitons have longer tails, but

quench prediction still holds

• Microscopic Classical Newtonian evolution

Harmonic Calogero


H =1

2m

NX

j=1

p2j +¹h2

2m

X

j 6=k

¸2

(xj ¡ xk)2+ !

NX

j=1

x2j ;

Harmonic Calogero


• Gamayun & al. considered same set-up

• Large time using integrability of NLSE (ISM)

• If integer ⇒ solitons (no dispersive waves)

Large time asymptotics for NLSE


c0

c= · 2·¡ 1

Gamayun, Bezvershenko, and Cheianov, arXiv:1408.3312

• We studied a quantum quench on localized excited state

using an effective semi-classical hydrodynamics

• Universal dynamics for short time after quench:

predicted shape and velocities of chiral profiles

• Great agreement with numerical simulations

• Experimentally feasible!

• Open questions: quantum nature of a soliton, microscopic

unitary evolution, large time behavior

Conclusions


Thank you!

…but wait, there is more!

Spontaneous Breaking

of U(N) Symmetry

in Invariant Matrix Models

Fabio Franchini

arXiv:1412.xxxx

Really done now,

Thank you!

Further Splitting


Universal Dynamics �of a Soliton �after a �Quantum QuenchOutlineOut-of-EquilibriumPreaching to the Choir…Out-of-Equilibrium Stat. Mech.?Quenching a SolitonQuantum QuenchesGibbs EnsembleGeneralized Gibbs EnsembleUnitary DynamicsOur ProtocolOur Set-UpSolitons?SolitonsSoliton on Scott Russell AqueductMorning GloryMorning Glory: solitonsSoliton DynamicsSoliton DynamicsSolitons in cold atomsSolitons in cold atomsSolitons in cold atomsHydrodynamic ApproachOur proposalHydrodynamcisWhat to expectLinearizationKdV ReductionKdV ReductionExample: Lieb-Liniger NLSEExample: NLSEOur approachQuenchQuenchSoliton SplittingSoliton SplittingChiral ProfilesNumerical Checks: AmplitudesNumerical Checks: VelocitiesHarmonic CalogeroHarmonic CalogeroLarge time asymptotics for NLSEConclusions…but wait, there is more!Spontaneous Breaking �of U(N) Symmetry �in Invariant Matrix ModelsFurther Splitting

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