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
Closed systems Equilibration Ion trap realization Kinetic theory Cold atoms or ions arranged in lattices Cold atoms aligned in optical lattice; or trapped ions arranged in a Coulomb crystal. Tune interactions via Feshbach resonances, microwave radiation, . . . Controlled engineering of 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
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
Closed systems Equilibration Ion trap realization Kinetic theory Long-range Ising model Chain of N interacting spin-1 / 2 particles in a magnetic field, N N / 2 σ z i σ z N i + j σ z � � � H N = N N − h i . j α i = 1 j = 1 i = 1 ∞ j − α < ∞ : � α > 1 = ⇒ short-range j = 1 ∞ � j − α = ∞ : 0 < α < 1 = ⇒ long-range j = 1 N / 2 � − 1 � � j − α N N = Normalization 2 to make energy extensive. j = 1 Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Time evolution of the long-range Ising model N N / 2 σ z i σ z N i + j σ z H N = N N � � − h � i j α i = 1 j = 1 i = 1 Goal: Study time evolution of expectation value � A � ( t ) , where N a i σ x A ( a 1 , . . . , a N ) = � a i ∈ R . i , i = 1 with respect to initial state operators ̺ 0 which are diagonal in the σ x 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. e − i H N t A e i H N t ρ ( 0 ) � A � ( t ) � � = · · · = = Tr N / 2 � 2 N N t � � A � ( 0 ) cos ( 2 ht ) � cos 2 = j α j = 1 Calculation very similar to G. G. Emch, J. Math. Phys. 7 , 1198 (1966). For simplicity: h = 0 = ⇒ no Larmor precession cos ( 2 ht ) . Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Approach to equilibrium in the long-range Ising model? N / 2 � 2 N N t � � A � ( t ) = � A � ( 0 ) � cos 2 j α j = 1 N finite: � A � ( t ) is quasiperiodic = ⇒ Poincaré recurrences N infinite: Get inspiration from plots. . . � A �� t ��� A �� 0 � � A �� t ��� A �� 0 � N � 10 4 1.0 1.0 N � 10 3 N � 10 5 N � 10 4 0.8 0.8 N � 10 6 N � 10 5 N � 10 6 0.6 0.6 0.4 0.4 0.2 0.2 t t 5 10 15 20 25 30 10 100 1000 10000 short-range long-range upper bound? lower bound? Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Upper bound on � A � ( t ) in the thermodynamic limit N / 2 � 2 N N t � � A � ( t ) = lim N →∞ � A � ( 0 ) � cos 2 j α j = 1 − cN − q t 2 � � A � ( t ) � � A � ( 0 ) exp � with 1 for 0 � α < 1 / 2 , q = 2 − 2 α for 1 / 2 < α < 1 , for α > 1 . 0 M. Kastner, Phys. Rev. Lett. 106 , 130601 (2011). � A �� t ��� A �� 0 � � A �� t ��� A �� 0 � N � 10 4 1.0 1.0 N � 10 5 N � 10 3 0.8 0.8 N � 10 4 N � 10 6 N � 10 5 0.6 0.6 N � 10 6 0.4 0.4 0.2 0.2 t t 5 10 15 20 25 30 10 100 1000 10000 short-range long-range Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Lower bound on � A � ( t ) for α < 1 (long-range) N / 2 � 2 N N t � � � A � ( t ) = � A � ( 0 ) cos 2 j α j = 1 Proposition: For any fixed time τ and some small δ > 0, there is a finite N 0 ( τ ) such that |� A � ( t ) − � A � ( 0 ) | < δ ∀ t < τ , N > N 0 ( τ ) . M. Kastner, Phys. Rev. Lett. 106 , 130601 (2011). δ : experimental resolution for measurement of A , � A �� t ��� A �� 0 � N � 10 3 1.0 N � 10 4 τ : duration of the experiment. 0.8 N � 10 5 N � 10 6 ⇒ Within experimental resolution and for 0.6 0.4 large enough system size, no deviation of � A � ( t ) from its initial value can be 0.2 t observed for times t � τ . 10 100 1000 10000 Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Spin–spin correlators Goal: Study time evolution of expectation values of spin–spin correlators � σ a i σ b where a , b ∈ { x , y , z } , j � with respect to initial state operators ̺ 0 which are diagonal in the σ x i -eigenbasis. � C �� t ��� C �� 0 � x Σ j x Σ i 1.0 Two-step process y Σ j y Σ i 0.8 x Second time scale Σ i involved 0.6 N -scaling different for 0.4 first and second step 0.2 10 5 t 10 4 1 10 100 1000 Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Generalizations Higher-dimensional lattices More general couplings, N ǫ ( | i − j | ) σ z i σ z σ z H N = N N � j − h � i i = 1 � i , j � 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
Closed systems Equilibration Ion trap realization Kinetic theory N -dependence of the pair interaction strength N N / 2 σ z i σ z N � N α − 1 for 0 � α < 1 , i + j σ z � � � H N = N N − h where N N ∼ i j α const. for α > 1 . i = 1 j = 1 i = 1 Is this N -dependent prefactor the sole cause of the N -scaling of relaxation times? No! � C �� t ��� C �� 0 � � C �� t ��� C �� 0 � x Σ j x x Σ j x Σ i Σ i 1.0 1.0 y Σ j y y Σ j y Σ i Σ i 0 < α < 1 / 2: 0.8 x 0.8 x Σ i Σ i 0.6 0.6 0.4 0.4 0.2 0.2 10 5 t t 10 4 1 10 100 1000 0.01 0.1 1 10 100 � C �� t ��� C �� 0 � � C �� t ��� C �� 0 � x Σ j x Σ j x x Σ i Σ i 1.0 1.0 y Σ j y y Σ j y Σ i Σ i 1 / 2 < α < 1: 0.8 0.8 x x Σ i Σ i 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 t t 1 10 100 1000 0.01 0.1 1 10 100 with prefactor N N without prefactor N N Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Extensivity in physics J 1.0 N N / 2 σ z i σ z N 0.8 i + j σ z H N = N N � � − h � 0.6 i j α 0.4 i = 1 j = 1 i = 1 0.2 1 � N 0.2 0.4 0.6 0.8 1.0 With or without N -dependent prefactor N N : Which one is the physically relevant scenario? Equilibrium properties: Nonequilibrium properties: Prefactor N N necessary to have In general unclear. Some dynamical a well-defined and non-trivial properties seem to have a well-defined thermodynamic limit. limit in the absence of N N . Michael Kastner Equilibration times in closed long-range quantum spin models
Closed systems Equilibration Ion trap realization Kinetic theory Summary / Take-home message Equilibrium: Nonequilibrium: Prefactor N N necessary No general reason to include N N N -independent relaxation time N -independent relaxation time scale for α > 1 scale for α > 1 / 2 � C �� t ��� C �� 0 � � C �� t ��� C �� 0 � x Σ j x x Σ j x Σ i Σ i 1.0 1.0 y Σ j y y Σ j y Σ i Σ i 0.8 0.8 x x Σ i Σ i 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 � 0.2 t t 1 10 100 1000 0.01 0.1 1 10 100 Diverging (with N ) relaxation Diverging (with N ) relaxation time scale τ ∝ N q with q = time scale τ ∝ N α − 1 / 2 for 0 < min { 1 / 2 , 1 − α } for 0 < α < 1 α < 1 / 2 � C �� t ��� C �� 0 � � C �� t ��� C �� 0 � x Σ j x Σ j x x Σ i Σ i 1.0 1.0 y Σ j y y Σ j y Σ i Σ i 0.8 x 0.8 x Σ i Σ i 0.6 0.6 0.4 0.4 0.2 0.2 0.0 t t 1 10 100 1000 0.01 0.1 1 10 100 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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