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Equilibration times in closed long-range quantum spin models - - PowerPoint PPT Presentation

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Closed systems Equilibration Ion trap realization Kinetic theory Equilibration times in closed long-range quantum spin models Michael Kastner Stellenbosch, South Africa New quantum states of matter in and out of equilibrium Firenze,

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Closed systems Equilibration Ion trap realization Kinetic theory

Equilibration times in closed long-range quantum spin models

Michael Kastner

Stellenbosch, South Africa

“New quantum states of matter in and out of equilibrium” Firenze, 30 May 2012

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Cold atoms or ions arranged in lattices

Cold atoms aligned in

  • ptical lattice; or

trapped ions arranged in a Coulomb crystal. Tune interactions via Feshbach resonances, microwave radiation, . . . Controlled engineering

  • f condensed-matter

Hamiltonians.

(from: I. Bloch et al., Rev. Mod. Phys. 80 (2008) 885–964) Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Cold atoms in optical lattices

Statistical description

After the cooling is switched off: conservation of energy and conservation of particle number For pure s-wave scattering: (low temperature, no permanent dipole moment) conservation of magnetization. = ⇒ Closed-system dynamics = ⇒ Equilibration in closed quantum systems? = ⇒ Statistical description in the microcanonical ensemble

It will depend on the type of system studied whether there are significant differences to the standard open-system, canonical situation.

= ⇒ Long-range makes a big difference!

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Long-range Ising model

Chain of N interacting spin-1/2 particles in a magnetic field, HN = NN

N

  • i=1

N/2

  • j=1

σz

i σz i+j

jα − h

N

  • i=1

σz

i .

α > 1 = ⇒

∞

  • j=1

j−α < ∞: short-range 0 < α < 1 = ⇒

∞

  • j=1

j−α = ∞: long-range Normalization NN =

  • 2

N/2

  • j=1

j−α −1 to make energy extensive.

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Time evolution of the long-range Ising model

HN = NN

N

  • i=1

N/2

  • j=1

σz

i σz i+j

jα − h

N

  • i=1

σz

i

Goal: Study time evolution of expectation value A(t), where A(a1, . . . , aN) =

N

  • i=1

ai σx

i ,

ai ∈ R. with respect to initial state operators ̺0 which are diagonal in the σx

i -eigenbasis.

Inspired by G. G. Emch, J. Math. Phys. 7, 1198 (1966), C. Radin, J. Math. Phys. 11, 2945 (1970). Experimental motivation by magnetic resonance experiments.

A(t) = Tr

  • e−iHNtA eiHNtρ(0)
  • = · · · =

= A(0) cos(2ht)

N/2

  • j=1

cos2 2NNt jα

  • Calculation very similar to G. G. Emch, J. Math. Phys. 7, 1198 (1966).

For simplicity: h = 0 = ⇒ no Larmor precession cos(2ht).

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Approach to equilibrium in the long-range Ising model?

A(t) = A(0)

N/2

  • j=1

cos2 2NNt jα

  • N finite: A(t) is quasiperiodic =

⇒ Poincaré recurrences N infinite: Get inspiration from plots. . .

5 10 15 20 25 30 t 0.2 0.4 0.6 0.8 1.0 AtA0 N106 N105 N104

short-range upper bound?

10 100 1000 10000 t 0.2 0.4 0.6 0.8 1.0 AtA0 N106 N105 N104 N103

long-range lower bound?

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Upper bound on A(t) in the thermodynamic limit

A(t) = lim

N→∞A(0) N/2

  • j=1

cos2 2NNt jα

  • A(t) A(0) exp
  • −cN−qt2

with q =      1 for 0 α < 1/2, 2 − 2α for 1/2 < α < 1, for α > 1.

  • M. Kastner, Phys. Rev. Lett. 106, 130601 (2011).

5 10 15 20 25 30 t 0.2 0.4 0.6 0.8 1.0 At A0 N 106 N 105 N 104

short-range

10 100 1000 10000 t 0.2 0.4 0.6 0.8 1.0 AtA0 N106 N105 N104 N103

long-range

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Lower bound on A(t) for α < 1 (long-range)

A(t) = A(0)

N/2

  • j=1

cos2 2NNt jα

  • Proposition: For any fixed time τ and some small δ > 0, there is a

finite N0(τ) such that |A(t) − A(0)| < δ ∀t < τ, N > N0(τ).

  • M. Kastner, Phys. Rev. Lett. 106, 130601 (2011).

10 100 1000 10000 t 0.2 0.4 0.6 0.8 1.0 AtA0 N106 N105 N104 N103

δ: experimental resolution for measurement of A, τ: duration of the experiment. ⇒ Within experimental resolution and for large enough system size, no deviation

  • f A(t) from its initial value can be
  • bserved for times t τ.

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Spin–spin correlators

Goal: Study time evolution of expectation values of spin–spin correlators σa

i σb j

where a, b ∈ {x, y, z}, with respect to initial state operators ̺0 which are diagonal in the σx

i -eigenbasis.

1 10 100 1000 104 105 t 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

Two-step process Second time scale involved N-scaling different for first and second step

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Generalizations

Higher-dimensional lattices More general couplings, HN = NN

  • i,j

ǫ(|i − j|)σz

i σz j − h N

  • i=1

σz

i

with |ǫ(j)| ∼ cj−α and some c > 0 General observables ? More general (non-integrable) models ? Question: Is quasi-stationary behaviour generic for long-range systems and arbitrary initial conditions

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

N-dependence of the pair interaction strength

HN = NN

N

  • i=1

N/2

  • j=1

σz

i σz i+j

jα − h

N

  • i=1

σz

i

where NN ∼

  • Nα−1

for 0 α < 1, const. for α > 1.

Is this N-dependent prefactor the sole cause of the N-scaling of relaxation times? No!

0 < α < 1/2:

1 10 100 1000 104 105 t 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

0.01 0.1 1 10 100 t 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

1/2 < α < 1:

1 10 100 1000 t 0.0 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

0.01 0.1 1 10 100 t 0.0 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

with prefactor NN without prefactor NN

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Extensivity in physics

HN = NN

N

  • i=1

N/2

  • j=1

σz

i σz i+j

jα − h

N

  • i=1

σz

i

0.2 0.4 0.6 0.8 1.0 1N 0.2 0.4 0.6 0.8 1.0 J

With or without N-dependent prefactor NN: Which one is the physically relevant scenario? Equilibrium properties: Prefactor NN necessary to have a well-defined and non-trivial thermodynamic limit. Nonequilibrium properties: In general unclear. Some dynamical properties seem to have a well-defined limit in the absence of NN.

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Summary / Take-home message

Equilibrium: Nonequilibrium: Prefactor NN necessary No general reason to include NN N-independent relaxation time scale for α > 1 N-independent relaxation time scale for α > 1/2

1 10 100 1000 t 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

0.01 0.1 1 10 100 t 0.0 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

Diverging (with N) relaxation time scale τ ∝ Nq with q = min{1/2, 1 − α} for 0 < α < 1 Diverging (with N) relaxation time scale τ ∝ Nα−1/2 for 0 < α < 1/2

1 10 100 1000 t 0.0 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

0.01 0.1 1 10 100 t 0.2 0.4 0.6 0.8 1.0 Ct C0 Σi

x

Σi

yΣj y

Σi

xΣj x

  • M. Kastner, Diverging equilibration times in long-range quantum spin models, Phys. Rev. Lett. 106, 130601 (2011).

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Experimental realization

Beryllium ions in a Penning trap

  • J. W. Britton et al., Engineered two-dimensional Ising interactions in a

trapped-ion quantum simulator with hundreds of spins, Nature 484, 489 (2012).

2d Coulomb crystal on a triangular lattice Valence-electron spin states as qubits (Ising spins) Spin–spin interactions mediated by crystal’s transverse motional degrees of freedom Effective anti-ferromagnetic Ising Hamiltonian H =

  • i<j

Jijσz

i σz j −

  • i

B · σi Jij ≈ |i − j|−α with 0.05 α 1.4

Michael Kastner Equilibration times in closed long-range quantum spin models

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Closed systems Equilibration Ion trap realization Kinetic theory

Classical Vlasov description of quasi-stationary behaviour

Vlasov equation: Time-evolution equation for 1-particle distribution function Like Boltzmann equation, but without collision integral Important in plasma physics Exact for Curie-Weiss (α = 0) in the thermodynamic limit Quasi-stationary states correspond to stable stationary solutions of Vlasov equation

Campa, Dauxois, Ruffo, Phys. Rep. 480, 57 (2009)

Quantum Vlasov equation for long-range spin systems?

Michael Kastner Equilibration times in closed long-range quantum spin models