Irreversibility and the arrow of time in a quenched quantum system Eric Lutz Department of Physics University of Erlangen-Nuremberg Outline 1 Physics far from equilibrium Entropy production Fluctuation theorems 2 Thermodynamic arrow of time Characterization of irreversibility Observation in a driven quantum system Introduction Observation: Thermodynamics describes equilibrium transformations Challenge: Generalization to arbitrary nonequilibrium processes Motivation: Far from equilibrium quantum regime accessible in recent cold-atom experiments Thermodynamics: a short reminder Equilibrium (nonequilibrium) processes: Entropy: ∆S = Q=T +Σ Work: W = ∆F + Wirr (F = U − TS = free energy) with hΣi ≥ 0 and hWirr i ≥ 0 (second law) thermodynamics does not allow computation of Σ, Wirr Nonequilibrium entropy production: Σ= β(W − ∆F) = βWirr β = 1=T difference between total work and equilibrium work quantifies irreversibility Nonequilibrium entropy production Generator of nonequilibrium currents: e.g. computation of heat or particle current Σ J = with e.g. X = (µ − µ )=T X 1 2 Efficiency of thermodynamic devices: T T η = 1 − 1 − 1 hΣi T2 Q2 fundamental quantity: zero at equilibrium, small in linear response regime, large far from equilibrium Quantum critical system Quantum Ising ring at finite temperature Dorner et al. PRL (2012) X z z x H(λ) = − σi σi+1 + λσi i Critical point at λc = 1 paramagnetic (ferromagnetic) phase λ > λc (λ < λc) Global quench of transverse field λ0 ! λ1 Entropy production ∆Sirr =β(hW i − ∆F) evaluated by diagonalization Generalization of the second law In small systems: Σ is random because of thermal and quantum fluctuations Fluctuation relation: Evans, Morris and Cohen PRL (1993) P(Σ) = eΣ P(−Σ) negative fluctuations exponentially small: P(−Σ) = P(Σ)e−Σ Integrated form: he−Σi = 1 implies hΣi ≥ 0 valid far from equilibrium (beyond linear response) Generalization of the second law Crooks equality: Crooks PRE (1999) P (W ) F = eβ(W −∆F) = eΣ PR(−W ) general connection between nonequilibrium work and equilibrium free energy difference Classical experiment: fluctuation theorem Colloidal particle in a driven optical trap: Wang et al. PRL (2002) Average over 540 trajectories: t= .01s (black) and t=2s (grey) Latex beads d= 6,3 micron R t Σ = βWir = β 0 ds v:F(s) Quantum experiment: Crooks equality Batalhão et al. PRL (2014) NMR system with B field quench: Quantum work distribution Outline 1 Physics far from equilibrium Entropy production Fluctuation theorems 2 Thermodynamic arrow of time Characterization of irreversibility Observation in a driven quantum system Arrow of time Macroscopic processes have a preferred direction in time "Heat flows from hot to cold" reversed process does not spontaneously occur irreversibility In thermodynamics: mean entropy production is positive irreversible if hΣi> 0 and reversible if hΣi= 0 arrow of time (Eddington 1927) Arrow of time (Quasi) reversible processes Here: entropy production hΣi ' 0 (during duration of experiment) "Lectures on thermodynamics" George Porter 1965 (Nobel 1968) Arrow of time Irreversible processes Here: entropy production hΣi > 0 (reversal not observed) "Lectures on thermodynamics" George Porter 1965 (Nobel 1968) Apparent paradox Macroscopic systems made of microscopic particles (atoms) Microscopic laws of physics are reversible Example: Newton’s law of motion d 2x m = F(t) dt2 Reversal: t ! t0 = τ − t dt0 = −dt dt02 = dt2 also true for Maxwell, Schrödinger, .... Question: how to explain macroscopic irreversibility? Apparent paradox Processes in nature are described by: i) laws of physics fixed ii) initial (boundary) conditions random Newton 1687, Wigner 1963 Examples: planetary orbits = ellipses (law) quasi circular in our solar system (initial condition) all planets orbit in the same direction (initial condition) Resolution: initial conditions break time reversal (Boltzmann 1896) Irreversibility in a many-particle system Bottle of perfume: Initial state breaks time reversal — reversal unlikely for 1024 particles Irreversibility in a single-particle system Second law for a driven system: hW i≥ ∆F (for isothermal process) preferred direction (even for a single particle) (Campisi, Hänggi 2011) reversible for hW i =∆ F and irreversible for hW i > ∆F irreversibility does not only occur in many-body systems Experimental observation Spin-1=2 driven by an external magnetic field: Batalhão et al. PRL 2015 F h C C i Ht = 2π~ ν(t) σx cos φ(t) + σy sin φ(t) Experimental observation Nonequilibrium entropy production: Σ= β(W − ∆F) experimental proof of hΣi ≥ 0 for driven quantum system Experimental observation Mean entropy production: F B F F F B hΣi = S(ρt jjρt−τ ) = tr[ρt ln ρt − ρt ln ρt−τ ] experimental demonstration of the arrow of time Summary The arrow of time is not an abstract, philosophical concept it can be quantified and observed in the lab Batalhão et al. Irreversibility and the arrow of time in a quenched quantum system, Phys. Rev. Lett. 115, 190601 (2015) (Editor’s Suggestion, Featured in Physics)..