Nonadiabatic dynamics Nonadiabatic dynamics in closed Hamiltonian systems.in closed Hamiltonian systems.

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

University of Utah, University of Utah, Condensed Matter Condensed Matter

Seminar, 10/27/2009 Seminar, 10/27/2009

R. BarankovR. Barankov BUBUClaudia De Grandi Claudia De Grandi BUBUVladimir GritsevVladimir Gritsev U. of FribourgU. of Fribourg

Cold atoms:Cold atoms:(controlled and tunable Hamiltonians, isolation from environment)(controlled and tunable Hamiltonians, isolation from environment)

1. Equilibrium thermodynamics:1. Equilibrium thermodynamics:Quantum simulations of equilibrium Quantum simulations of equilibrium condensed matter systemscondensed matter systems

Experimental examples:Experimental examples:

Optical Lattices:Optical Lattices:

OL are tunable (in OL are tunable (in real time): from real time): from weak modulations weak modulations to tight binding to tight binding regime.regime.

Can change Can change dimensionality and dimensionality and study 1D, 2D, and study 1D, 2D, and 3D physics.3D physics.

Both fermions and Both fermions and bosons.bosons.

I. Bloch, Nature Physics 1, 23 - 30 (2005)

Superfluid insulator Superfluid insulator phase transition in an phase transition in an optical latticeoptical lattice

Greiner et. al. 2003Greiner et. al. 2003

Repulsive Bose gas.

T. Kinoshita, T. Wenger, D. S. Weiss ., Science 305, 1125, 2004

Also, B. Paredes1, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T.W. Hänsch and I. Bloch, Nature 277 , 429 (2004)

M. Olshanii, 1998M. Olshanii, 1998

interaction parameterinteraction parameter

Lieb-Liniger, complete solution 1963.Lieb-Liniger, complete solution 1963.

Experiments:Experiments:

Local pair correlations.

Kinoshita et. Al., Science 305, 1125, 2004

Cold atoms:Cold atoms:(controlled and tunable Hamiltonians, isolation from environment)(controlled and tunable Hamiltonians, isolation from environment)

1. Equilibrium thermodynamics:1. Equilibrium thermodynamics:Quantum simulations of equilibrium Quantum simulations of equilibrium condensed matter systemscondensed matter systems

2. Quantum dynamics:2. Quantum dynamics:

Coherent and incoherent dynamics, Coherent and incoherent dynamics, integrability, quantum chaos, …integrability, quantum chaos, …

In the continuum this system is equivalent to an integrable KdV In the continuum this system is equivalent to an integrable KdV equation. The solution splits into non-thermalizing solitons Kruskal equation. The solution splits into non-thermalizing solitons Kruskal and Zabusky (1965 ).and Zabusky (1965 ).

Qauntum Newton Craddle.(collisions in 1D interecating Bose gas – Lieb-Liniger model)

T. Kinoshita, T. R. Wenger and D. S. Weiss, Nature 440, 900 – 903 (2006)

No thermalization in1D.

Fast thermalization in 3D.

Quantum analogue of the Fermi-Pasta-Ulam problem.

3. = 1+2 Nonequilibrium thermodynamics?3. = 1+2 Nonequilibrium thermodynamics?

Cold atoms:Cold atoms:(controlled and tunable Hamiltonians, isolation from environment)(controlled and tunable Hamiltonians, isolation from environment)

1. Equilibrium thermodynamics:1. Equilibrium thermodynamics:Quantum simulations of equilibrium Quantum simulations of equilibrium condensed matter systemscondensed matter systems

2. Quantum dynamics:2. Quantum dynamics:

Coherent and incoherent dynamics, Coherent and incoherent dynamics, integrability, quantum chaos, …integrability, quantum chaos, …

Thermalization in classical systems (origin of ergodicity): chaotic Thermalization in classical systems (origin of ergodicity): chaotic many-body dynamics implies exponential in time sensitivity to many-body dynamics implies exponential in time sensitivity to initial fluctuations.initial fluctuations.

Thermalization in quantum systems (EoM are linear in time – no chaos?)Thermalization in quantum systems (EoM are linear in time – no chaos?)

Consider the time average of a certain observable Consider the time average of a certain observable AA in an isolated in an isolated system after a quench. system after a quench.

mn

nmtEEi

mnmn

nmmn AeAttA nm

,,

)(,

,,, )0()(

mn

nnnn

TAdttA

TA

,,,0

)(1

Eignestate thermalization hypothesis (Eignestate thermalization hypothesis (Srednicki 1994; M. Rigol, V. M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854 , 2008Dunjko & M. Olshanii, Nature 452, 854 , 2008.): .): AAn,nn,n~ ~ const const (n) (n) so so

there is no dependence on there is no dependence on nnnn..

Information about equilibrium is fully contained in diagonal Information about equilibrium is fully contained in diagonal elements of the density matrix.elements of the density matrix.

This is true for all thermodynamic observables: energy, This is true for all thermodynamic observables: energy, pressure, magnetization, …. (pick your favorite). They all are pressure, magnetization, …. (pick your favorite). They all are linear in linear in ..This is not true about von Neumann entropy! This is not true about von Neumann entropy!

)ln( TrSn

Off-diagonal elements do not average to zero.Off-diagonal elements do not average to zero.

The usual way around: coarse-grain density matrix (remove by The usual way around: coarse-grain density matrix (remove by hand fast oscillating off-diagonal elements of hand fast oscillating off-diagonal elements of ..

Problem: not a unique procedure, explicitly violates time Problem: not a unique procedure, explicitly violates time reversibility and Hamiltonian dynamics.reversibility and Hamiltonian dynamics.

Von Neumann entropy: always conserved in time (in isolated Von Neumann entropy: always conserved in time (in isolated systems). More generally it is invariant under arbitrary unitary systems). More generally it is invariant under arbitrary unitary transfomationstransfomations

lnln)(ln)()( TrUUUTrUttTrtSn

Thermodynamics: entropy is conserved only for adiabatic Thermodynamics: entropy is conserved only for adiabatic (slow, reversible) processes. Otherwise it increases.(slow, reversible) processes. Otherwise it increases.

Quantum mechanics: for adiabatic processes there are no Quantum mechanics: for adiabatic processes there are no transitions between energy levels: transitions between energy levels: )(const)( ttnn

If these two adiabatic theorems are related then the entropy If these two adiabatic theorems are related then the entropy should only depend on should only depend on nnnn. . Simple resolution:Simple resolution:

n

nnnndS ln the sum is taken in the the sum is taken in the instantaneous energy basis.instantaneous energy basis.

??????

Connection between two adiabatic theorems allows us to Connection between two adiabatic theorems allows us to define define heat (heat (A.P., Phys. Rev. Lett. A.P., Phys. Rev. Lett. 101101, 220402, 2008, 220402, 2008 ))..

Consider an arbitrary dynamical process and work in the Consider an arbitrary dynamical process and work in the instantaneous energy basis (adiabatic basis).instantaneous energy basis (adiabatic basis).

),()()0()()(

)0()()()(),(

tQEt

ttE

ttadn

nnnntn

nnntn

nnntnt

• Adiabatic energy is the function of the state.Adiabatic energy is the function of the state.

• Heat is the function of the process.Heat is the function of the process.

• Heat vanishes in the adiabatic limit. Heat vanishes in the adiabatic limit. Now this is not the Now this is not the postulate, this is a consequence of the Hamiltonian dynamics!postulate, this is a consequence of the Hamiltonian dynamics!

Isolated systems. Initial stationary state.Isolated systems. Initial stationary state.

nmnnm 0)0(

Unitarity of the evolution:Unitarity of the evolution:

m

nmnmnnn tpt ))(()( 000

m m

mnnm tptp )()(

In general there is no detailed balance even for cyclic In general there is no detailed balance even for cyclic processes (but within the Fremi-Golden rule there is).processes (but within the Fremi-Golden rule there is).

2

,)( nmAnm UUIp

m

nmnmnnn tpt ))(()( 000

yieldsyields

n

nmnmnn

nnn tpttQ )()()()( 00

If there is a detailed balance thenIf there is a detailed balance then

n

nmnmmn tptQ )())((2

1)( 00

Heat is non-negative for cyclic processes if the initial density Heat is non-negative for cyclic processes if the initial density matrix is passive . matrix is passive . Second law of Second law of thermodynamics in Thompson (Kelvin’s form).thermodynamics in Thompson (Kelvin’s form).

0))(( 00 nmmn

The statement is also true without the detailed balance but the proof The statement is also true without the detailed balance but the proof is more complicated is more complicated (Thirring, Quantum Mathematical Physics, Springer (Thirring, Quantum Mathematical Physics, Springer 1999).1999).

Properties of d-entropy (Properties of d-entropy (A. Polkovnikov, A. Polkovnikov, arXiv:0806.2862.arXiv:0806.2862. ) )..

Jensen’s inequality:Jensen’s inequality:

0]ln)[()lnln( dddd TrTr

Therefore if the initial density matrix is stationary (diagonal) thenTherefore if the initial density matrix is stationary (diagonal) then

)0()0()()( dnnd SStStS

Now assume that the initial state is thermal equilibriumNow assume that the initial state is thermal equilibrium

]exp[10

nn Z

Let us consider an infinitesimal change of the system and Let us consider an infinitesimal change of the system and compute energy and entropy change.compute energy and entropy change.

nnn

nnnnd

nnnnnn

TS

E

00

00

1)1(ln

Recover the first law of thermodynamics Recover the first law of thermodynamics (Fundamental Relation).(Fundamental Relation).

dS

STE

Ed

If If stands for the stands for thevolume the we findvolume the we find dSTVPE

Classic example: freely expanding Classic example: freely expanding gasgas

Suddenly remove the wallSuddenly remove the wall

0 GibbsS by Liouville theoremby Liouville theorem

)0(2

1)0( nnnn double number of occupied statesdouble number of occupied states

2lnNSd result of Hamiltonian dynamics!result of Hamiltonian dynamics!

Thermodynamic adiabatic theorem.Thermodynamic adiabatic theorem.

General expectation:General expectation:

In a cyclic adiabatic process the energy of the system In a cyclic adiabatic process the energy of the system does does not change: not change: no work done on the system, no heating, and no no work done on the system, no heating, and no entropy is generated .entropy is generated .

22 )0()(,0)( SSEE

- - is the rate of change of external parameter.is the rate of change of external parameter.

Adiabatic theorem in quantum mechanicsAdiabatic theorem in quantum mechanics

Landau Zener process:Landau Zener process:

In the limit In the limit 0 transitions between 0 transitions between different energy levels are suppressed.different energy levels are suppressed.

This, for example, implies reversibility (no work done) in a This, for example, implies reversibility (no work done) in a cyclic process.cyclic process.

Adiabatic theorem in QM Adiabatic theorem in QM suggestssuggests adiabatic theorem adiabatic theorem in thermodynamics:in thermodynamics:

Breakdown of Taylor expansion in low dimensions, Breakdown of Taylor expansion in low dimensions, especially near singularities (phase transitions).especially near singularities (phase transitions).

1.1. Transitions are unavoidable in large gapless systems.Transitions are unavoidable in large gapless systems.

2.2. Phase space available for these transitions decreases with the Phase space available for these transitions decreases with the raterateHence expectHence expect

22 )0()(,0)( SSEE

Low dimensions: high density of low energy states, breakdown of Low dimensions: high density of low energy states, breakdown of mean-field approaches in equilibriummean-field approaches in equilibrium

Three regimes of response to the slow ramp:Three regimes of response to the slow ramp:A.P. and V.Gritsev, Nature Physics 4, 477 (2008)A.P. and V.Gritsev, Nature Physics 4, 477 (2008)

A.A. Mean field (analytic) – high dimensions: Mean field (analytic) – high dimensions:

B.B. Non-analytic – low dimensions, near singularities like QCPNon-analytic – low dimensions, near singularities like QCP

C.C. Non-adiabatic – low dimensions, bosonic excitationsNon-adiabatic – low dimensions, bosonic excitations

In all three situations quantum and thermodynamic adiabatic In all three situations quantum and thermodynamic adiabatic theorem are smoothly connected.theorem are smoothly connected. The adiabatic theorem in thermodynamics does follow from the The adiabatic theorem in thermodynamics does follow from the adiabatic theorem in quantum mechanics.adiabatic theorem in quantum mechanics.

20)( EE

2,||0)( rEE r

0,2,||0)( rLEE r

Origin of quadratic scaling: adiabatic perturbation theoryOrigin of quadratic scaling: adiabatic perturbation theory

Can we say anything about system response in the limit Can we say anything about system response in the limit 0?0?

Assume that we start in the ground state.Assume that we start in the ground state.

Need to solveNeed to solve

Convenient to work in the adiabatic (co-moving, instantaneous) basisConvenient to work in the adiabatic (co-moving, instantaneous) basis

SchrSchrödinger equation in the adiabatic basisödinger equation in the adiabatic basis

Assume the proximity to the instantaneous ground state (small Assume the proximity to the instantaneous ground state (small excitation probability)excitation probability)

Landau-Zener problemLandau-Zener problem

Now start Now start ttii - -, t, tff – finite (or vice versa) – finite (or vice versa)

For finite interval of excitation the transition probability scales as For finite interval of excitation the transition probability scales as the second power of the rate (not exponentially).the second power of the rate (not exponentially).

Gapless systems with quasi-particle excitationsGapless systems with quasi-particle excitations

Ramping in generic gapless regime (low energy contribution)Ramping in generic gapless regime (low energy contribution)

22 *)(~*

*)(~*

Ed

dEE

dt

dE

EE

E*E* Uniform system:Uniform system: zkcE )()(

d

cdE

ln~*

Density of quasi-particles (entropy):Density of quasi-particles (entropy):zd

E

ex dEEn /*

0

|~|)(~ Absorbed energy density (heating):Absorbed energy density (heating):

zzdQ /)(|~|

High dimensions: high energies dominate dissipation, low-High dimensions: high energies dominate dissipation, low-dimensions – low energies dominate dissipation.dimensions – low energies dominate dissipation.

Low energy contribution:Low energy contribution:

zzdzdex Qsn /)(/ ||,||,

High energy contribution.High energy contribution.

EE

E*E*

6

2||)(

EEpex

2||)()(~ dEEEpn exex

2|~| QSimilarlySimilarly

Example: crossing a QCP.Example: crossing a QCP.

tuning parameter tuning parameter

gap

gap t, t, 0 0

Gap vanishes at the transition. Gap vanishes at the transition. No true adiabatic limit!No true adiabatic limit!

How does the number of excitations scale with How does the number of excitations scale with ? ?

A.P. 2003,A.P. 2003,Zurek, Dorner, Zoller 2005Zurek, Dorner, Zoller 2005

Another Example: loading 1D condensate into an optical Another Example: loading 1D condensate into an optical lattice or merging two 1D condensateslattice or merging two 1D condensates(C. De Grandi, R. Barankov, AP, PRL 2008 )(C. De Grandi, R. Barankov, AP, PRL 2008 )

Relevant sineRelevant sineGordon model:Gordon model:

)cos(

2

1

2

1 22 VdxH x

K 2

K 2

K – Luttinger liquid parameter K – Luttinger liquid parameter

Results:Results:

)3/(1

/

2

2K

ex

zd

ex

nK

nK

K=2 corresponds to a SF-IN K=2 corresponds to a SF-IN transition in an infinitesimal transition in an infinitesimal lattice (H.P. Büchler, et.al. 2003) lattice (H.P. Büchler, et.al. 2003)

1.0 1.5 2.0 2.5 3.0 3.5 4.0K

VOptimal

Sudden and slow quenches starting from the QCPSudden and slow quenches starting from the QCP

Probing quasi-particle statistics in nonlinear dynamical probes.Probing quasi-particle statistics in nonlinear dynamical probes.

KK00 11

massive bosonsmassive bosons massive fermionsmassive fermions(hard core bosons)(hard core bosons)

T=0T=0Lnex ln3/1 2/1exn

T>0T>0 Tnex

More adiabaticMore adiabatic

LTnex3/1

Less adiabaticLess adiabatic

TT

bosonic-likebosonic-like fermionic-likefermionic-like

transition?transition?

ConclusionsConclusions

1.1. Adiabatic theorems in quantum mechanics and Adiabatic theorems in quantum mechanics and thermodynamics are directly connected. thermodynamics are directly connected.

2.2. Diagonal entropy satisfies laws of thermodynamics from Diagonal entropy satisfies laws of thermodynamics from microscopics. Heat and entropy change result from the microscopics. Heat and entropy change result from the nonadiabatic transitions between microscopic energy nonadiabatic transitions between microscopic energy levels.levels.

3.3. Maximum entropy state with Maximum entropy state with nnnn=const is the natural =const is the natural

attractor of the Hamiltonian dynamics.attractor of the Hamiltonian dynamics.

4.4. Universal scaling of density of excitations, heat, entropy Universal scaling of density of excitations, heat, entropy for sudden and slow dynamics near QCP. These scaling for sudden and slow dynamics near QCP. These scaling laws are closely related to scaling of fidelity and other laws are closely related to scaling of fidelity and other susceptibilities. susceptibilities.

M. Rigol, V. Dunjko & M. Olshanii,Nature 452, 854 (2008)

a, Two-dimensional lattice on which five hard-core bosons propagate in time.

b, The corresponding relaxation dynamics of the central component n(kx = 0) of the marginal momentum

distribution, compared with the predictions of the three ensembles

c, Full momentum distribution function in the initial state, after relaxation, and in the different ensembles.

Non-adiabatic regime (bosons on a lattice).Non-adiabatic regime (bosons on a lattice).

Use the fact that quantum Use the fact that quantum fluctuations are weak in the SF fluctuations are weak in the SF phase and expand dynamics in phase and expand dynamics in the effective Planck’s constant:the effective Planck’s constant:

JnU 0/

Nonintegrable model in all spatial dimensions, expect thermalization.Nonintegrable model in all spatial dimensions, expect thermalization.

)tanh()( 0 tUtU

T=0.02T=0.02

3/13/4 LTE

2D, T=0.22D, T=0.2

3/13/1 LTE

Classical systems.Classical systems.

nn probability to occupy an orbit with energy E.probability to occupy an orbit with energy E.

Instead of energy levels we Instead of energy levels we have orbits.have orbits.

])(exp[ ti mnnm describes the motion on describes the motion on this orbits. this orbits.

Classical d-entropyClassical d-entropy

dNSd )(ln)()( )),((),()( qpqpd

The entropy “knows” only about conserved quantities, The entropy “knows” only about conserved quantities, everything else is irrelevant for thermodynamics! Severything else is irrelevant for thermodynamics! Sdd satisfies satisfies

laws of thermodynamics, unlike the usually defined laws of thermodynamics, unlike the usually defined .lnS