Local Equivalence of Ensembles M. Cramer Ulm University on work - PowerPoint PPT Presentation

Local Equivalence of Ensembles M. Cramer Ulm University on work with F.G.S.L. Brandão Microsoft Research and University College London M. Guta University of Nottingham

Why Do Systems Thermalize? H/T /Z % T = e − ˆ ˆ

Why Do Systems Thermalize? lack of knowledge, ignorance Jaynes’ principle H/T /Z % T = e − ˆ ˆ

Why Do Systems Thermalize? – Kinematics and Dynamics part of a large (closed) system % C = tr \ C [ˆ ˆ % ] C

Why Do Systems Thermalize? – Kinematics and Dynamics part of a large (closed) system % C = tr \ C [ˆ ˆ % ] H C /T /Z ≈ e − ˆ C

Why Do Systems Thermalize? – Kinematics and Dynamics part of a large (closed) system ˆ % C = tr \ C [ˆ % ] H/T /Z e − ˆ ⇥ ⇤ ≈ tr \ C C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath % C ( ˆ 0 ) ⊗ ˆ % B C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath, unitary evolution e − i t ˆ e i t ˆ H � H � % C ( ˆ 0 ) ⊗ ˆ % B C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath, unitary evolution e − i t ˆ e i t ˆ H � H ⇤ ⇥ � tr \ C % C ( ˆ 0 ) ⊗ ˆ % C ( t ) ˆ % B = C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath, unitary evolution e − i t ˆ e i t ˆ H � H ⇤ ⇥ � tr \ C % C ( ˆ 0 ) ⊗ ˆ % C ( t ) ˆ % B = H/T /Z e − ˆ ⇥ ⇤ t →∞ → tr \ C − − C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ quantum quench e − i t ˆ e i t ˆ H � H ⇤ ⇥ � tr \ C % C ( ˆ 0 ) ⊗ ˆ % C ( t ) ˆ % B = H/T /Z e − ˆ ⇥ ⇤ t →∞ → tr \ C − − C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ % C ≈ tr \ C Canonical Typicality 
 Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091 Entanglement and the foundations of statistical mechanics 
 Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225 Thermalization in Nature and on a Quantum Computer 
 Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389 Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems 
 Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420 Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems 
 Brandão, Cramer, arxiv:1502.03263 H/T /Z e − ˆ quantum quench ⇥ ⇤ % C ( t ) ˆ t →∞ → tr \ C − − Time-dependence of correlation functions following a quantum quench 
 Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225 Relaxation in a Completely Integrable Many-Body Quantum System 
 Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476 E ff ect of suddenly turning on interactions in the Luttinger model 
 Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236 Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems 
 Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314 Thermalization and its mechanism for generic isolated quantum systems 
 Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324 Foundation of Statistical Mechanics under Experimentally Realistic Conditions 
 Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092 Quantum mechanical evolution towards thermal equilibrium 
 Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ % C ≈ tr \ C Canonical Typicality 
 Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091 Entanglement and the foundations of statistical mechanics 
 Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225 Thermalization in Nature and on a Quantum Computer 
 Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389 Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems 
 Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420 Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems 
 Brandão, Cramer, arxiv:1502.03263 H/T /Z e − ˆ quantum quench ⇥ ⇤ % C ( t ) ˆ t →∞ → tr \ C − − Time-dependence of correlation functions following a quantum quench 
 Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225 Relaxation in a Completely Integrable Many-Body Quantum System 
 Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476 E ff ect of suddenly turning on interactions in the Luttinger model 
 Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236 Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems 
 Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314 Thermalization and its mechanism for generic isolated quantum systems 
 Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324 Foundation of Statistical Mechanics under Experimentally Realistic Conditions 
 Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092 Quantum mechanical evolution towards thermal equilibrium 
 Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ % C ≈ tr \ C Canonical Typicality 
 Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091 gen. can. pricniple for random | ψ i 2 H R ⇢ H C ⌦ H B Entanglement and the foundations of statistical mechanics 
 Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225 Thermalization in Nature and on a Quantum Computer 
 with high probability % C ≈ tr \ C [ ˆ R /d R ] Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389 Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems 
 …thermal? Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420 Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems 
 Brandão, Cramer, arxiv:1502.03263 H/T /Z e − ˆ quantum quench ⇥ ⇤ % C ( t ) ˆ t →∞ → tr \ C − − Time-dependence of correlation functions following a quantum quench 
 Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225 Relaxation in a Completely Integrable Many-Body Quantum System 
 Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476 E ff ect of suddenly turning on interactions in the Luttinger model 
 Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236 Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems 
 Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314 Thermalization and its mechanism for generic isolated quantum systems 
 Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324 Foundation of Statistical Mechanics under Experimentally Realistic Conditions 
 Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092 Quantum mechanical evolution towards thermal equilibrium 
 Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ % C ≈ tr \ C Canonical Typicality 
 Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091 gen. can. pricniple for random | ψ i 2 H R ⇢ H C ⌦ H B Entanglement and the foundations of statistical mechanics 
 Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225 Thermalization in Nature and on a Quantum Computer 
 with high probability % C ≈ tr \ C [ ˆ R /d R ] Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389 Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems 
 …thermal? Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420 Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems 
 Brandão, Cramer, arxiv:1502.03263 H/T /Z e − ˆ quantum quench ⇥ ⇤ % C ( t ) ˆ t →∞ → tr \ C − − Time-dependence of correlation functions following a quantum quench 
 no thermalization Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225 Integrable Relaxation in a Completely Integrable Many-Body Quantum System 
 instead: generalized Gibbs Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476 E ff ect of suddenly turning on interactions in the Luttinger model 
 ensemble Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236 Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems 
 Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314 “equilibrium state”, close non-integrable Thermalization and its mechanism for generic isolated quantum systems 
 Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324 to it for most times Foundation of Statistical Mechanics under Experimentally Realistic Conditions 
 Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092 …thermal? time scale? Quantum mechanical evolution towards thermal equilibrium 
 Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385