Quantum dynamics in low dimensional Quantum dynamics in low dimensional systems.systems.

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

AFOSRAFOSR

Superconductivity andSuperconductivity andSuperfluidity in Finite Systems,Superfluidity in Finite Systems,U of Wisconsin, Madison 05/2008U of Wisconsin, Madison 05/2008

Roman BarankovRoman Barankov (BU)(BU)Claudia De GrandiClaudia De Grandi (BU)(BU)Vladimir Gritsev Vladimir Gritsev (Harvard)(Harvard)Vadim OganesyanVadim Oganesyan (Yale)(Yale)

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

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

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

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, …

Adiabatic process.Adiabatic process.

Assume no first order phase transitions.Assume no first order phase transitions.

Adiabatic theorem:Adiabatic theorem:

““Proof”:Proof”: thenthen

Adiabatic theorem for isolated systems.Adiabatic theorem for isolated systems.

EEBB(0) is the energy of the state adiabatically connected to (0) is the energy of the state adiabatically connected to

the state A. the state A.

General expectation:General expectation:

Alternative (microcanonical) definition: Alternative (microcanonical) definition: In a cyclic adiabatic In a cyclic adiabatic process the energy of the system process the energy of the system does not change:does not change:no work done on the system and no heating.no work done on the system and no heating.

Integrable systems: density of excitationsIntegrable systems: density of excitations

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:

Is there anything wrong with this picture?Is there anything wrong with this picture?

HHint: low dimensions. Similar to Landau expansion in the int: low dimensions. Similar to Landau expansion in the order parameter.order parameter.

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 Phase space available for these transitions decreases with Hence expectHence expect

More specific reason.More specific reason.

Equilibrium: high density of low-energy states Equilibrium: high density of low-energy states

•strong quantum or thermal fluctuations, strong quantum or thermal fluctuations, •destruction of the long-range order,destruction of the long-range order,•breakdown of mean-field descriptions, breakdown of mean-field descriptions,

Dynamics Dynamics population of the low-energy states due to finite rate population of the low-energy states due to finite rate breakdown of the adiabatic approximation.breakdown of the adiabatic approximation.

This talk: three regimes of response to the slow ramp:This talk: three regimes of response to the slow ramp:

A.A. Analytic (linear response) – high Analytic (linear response) – high dimensions: dimensions:

B.B. Non-analytic – low dimensions (can use Non-analytic – low dimensions (can use perturbation theory in ) perturbation theory in )

C.C. Non-adiabatic – lower dimensionsNon-adiabatic – lower dimensions

A.P. and V. Gritsev, Nature Physics: A.P. and V. Gritsev, Nature Physics: nphys963 (2008).nphys963 (2008).

Some examples.Some examples.

1. Gapless critical phase (superfluid, magnet, crystal, …).1. Gapless critical phase (superfluid, magnet, crystal, …).

LZ condition:LZ condition:

2. Example: crossing a QCP.2. 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; Zurek et. al. 2005 A.P. 2003; Zurek et. al. 2005

~

Possible breakdown of the Fermi-Golden rule (linear Possible breakdown of the Fermi-Golden rule (linear response) scaling due to bunching of bosonic excitations.response) scaling due to bunching of bosonic excitations.

222

2

1

22 xs

xsdxH

Bogoliubov Hamiltonian:Bogoliubov Hamiltonian:

tt )(

In cold atoms: start from free Bose gas and slowly turn on In cold atoms: start from free Bose gas and slowly turn on interactions.interactions.

Hamiltonian of Goldstone modes: superfluids, phonons in solids, Hamiltonian of Goldstone modes: superfluids, phonons in solids, (anti)ferromagnets, …(anti)ferromagnets, …

Zero temperature regime:Zero temperature regime:

Assuming the system thermalizes at a fixed energyAssuming the system thermalizes at a fixed energy

EnergyEnergy

Finite TemperaturesFinite Temperatures

d=1,2d=1,2

d=1;d=1; d=2;d=2;

Artifact of the quadratic approximation or the real result?Artifact of the quadratic approximation or the real result?

Non-adiabatic regime!Non-adiabatic regime!dLTE 3/73/1

TSTE , d=3d=3

Numerical verification (bosons on a lattice).Numerical verification (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

Thermalization at long times.Thermalization at long times.

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

t=0 t=3.2/ t=12.8/ t=28.8/ t=51.2/ t=80/ Thermal

a ja

0

L/ sin(j/L)

Correlation Functions

2D, T=0.22D, T=0.2

3/13/1 LTE

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

(with (with R. Barankov and C. De Grandi, R. Barankov and C. De Grandi, talk by Claudia De Granditalk by Claudia De Grandi))

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

Expansion of quantum dynamics around classical limit.Expansion of quantum dynamics around classical limit.

Classical (saddle point) limit: Classical (saddle point) limit:

(i) Newtonian equations for particles, (i) Newtonian equations for particles,

(ii) Gross-Pitaevskii equations for matter waves, (ii) Gross-Pitaevskii equations for matter waves,

(iii) Maxwell equations for classical e/m waves and charged particles, (iii) Maxwell equations for classical e/m waves and charged particles,

(iv) Bloch equations for classical rotators, etc.(iv) Bloch equations for classical rotators, etc.Questions:Questions:

What shall we do with equations of motion?What shall we do with equations of motion?

What shall we do with initial conditions?What shall we do with initial conditions?

Challenge : Challenge : How to reconcile exponential complexity of quantum many body How to reconcile exponential complexity of quantum many body systems and power law complexity of classical systems?systems and power law complexity of classical systems?

Partial answers.Partial answers.

Leading order in Leading order in : equations of motion do not change. Initial : equations of motion do not change. Initial conditions are described by a Wigner “probability’’ distribution:conditions are described by a Wigner “probability’’ distribution:

/)2/()2/(*),( ipexxdpxW

G.S. of a harmonic G.S. of a harmonic oscillator:oscillator:

Quantum-classical correspondence:Quantum-classical correspondence:

;;

Semiclassical (truncated Wigner approximation):Semiclassical (truncated Wigner approximation):

•Exact for harmonic theories! Exact for harmonic theories!

•Not limited to low temperatures and to 1D!Not limited to low temperatures and to 1D!

•Asymptotically exact at short times.Asymptotically exact at short times.

Summary:Summary:

Expectation value is substituted by the average over the initial conditions. Expectation value is substituted by the average over the initial conditions.

Beyond the semiclassical approximation.Beyond the semiclassical approximation.

Quantum jump.Quantum jump.

3 p

Each jump carries an extra factor of Each jump carries an extra factor of 2.2.

Recover sign problem = exponential complexity in exact Recover sign problem = exponential complexity in exact formulation of quantum dynamics. formulation of quantum dynamics.

)cos(

2

1

2

1 22 VdxH x

Illustration: Sine-Grodon model, Illustration: Sine-Grodon model, ββ plays the role of plays the role of

V(t) V(t) = 0.1 tanh (0.2 = 0.1 tanh (0.2 t)t)

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

0.10

cos

t

TWA

1st Correction Linear Response

Tonks 2

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

Semiclassical

1st Correction Linear Response

t

Example (back to FPU problem) .Example (back to FPU problem) .with V. Oganesyanwith V. Oganesyan

m m = 10, = 10, = 1, = 1, = 0.2, = 0.2, L L = 100= 100

Choose initial state corresponding to initial displacement at wave Choose initial state corresponding to initial displacement at wave vector vector k k = 2= 2//L L (first excited mode)(first excited mode)..

Follow the energy in the first excited mode as a function of time.Follow the energy in the first excited mode as a function of time.

Classical simulationClassical simulation

0 10000 20000 30000 40000 500000

5

10

15

20

Ene

rgy

Time

Semiclassical simulationSemiclassical simulation

-1

-3

-5-7

-9-11

10080

6040

20

0.5

1.0

1.5

2.0

Time/1000

Ene

rgy

ln

Similar problem with bosons in an optical lattice.Similar problem with bosons in an optical lattice.

Prepare and release Prepare and release a system of bosons a system of bosons from a single site.from a single site.

Little evidence of thermalization in the classical limit. Strong Little evidence of thermalization in the classical limit. Strong evidence of thermalization in the quantum and semiclassical limits.evidence of thermalization in the quantum and semiclassical limits.

Many-site generalization 60 sites, populate each 10Many-site generalization 60 sites, populate each 10 thth site. site.

Conclusions.Conclusions.

A.A. Mean field (analytic): Mean field (analytic):

B.B. Non-analyticNon-analytic

C.C. Non-adiabaticNon-adiabatic

Three generic regimes of a system response to a slow ramp:Three generic regimes of a system response to a slow ramp:

Many open challenging questions on nonequilibrium Many open challenging questions on nonequilibrium quantum dynamics. quantum dynamics.

Cold atoms should be able to provide unique valuable Cold atoms should be able to provide unique valuable experiments.experiments.

Example: optimal crossing of a QCP.Example: optimal crossing of a QCP.(work with Roman Barankov)(work with Roman Barankov)

tuning parameter tuning parameter

gap

gap

=(=( t)t)rr, , 0 0

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

)1/(])1[(~ rzrdex rzn

)ln(ln1 11 z

ropt

power corresponding to an power corresponding to an optical adiabatic passage optical adiabatic passage through a critical point.through a critical point.