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 + W irr (F = U TS = free energy) with Σ 0 and W irr 0 (second law) thermodynamics does not allow computation of Σ, W irr Nonequilibrium entropy production: Σ = β(w F) = βw irr 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 J = Σ X with e.g. X = (µ 1 µ 2 )/T Efficiency of thermodynamic devices: η = ( 1 T ) 1 T 1 Σ T 2 Q 2 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) H(λ) = i σ z i σ z i+1 + λσx i Critical point at λ c = 1 paramagnetic (ferromagnetic) phase λ > λ c (λ < λ c ) Global quench of transverse field λ 0 λ 1 Entropy production S irr =β( W 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) P( Σ) = eσ negative fluctuations exponentially small: P( Σ) = P(Σ)e Σ Integrated form: e Σ = 1 implies Σ 0 valid far from equilibrium (beyond linear response)
Generalization of the second law Crooks equality: Crooks PRE (1999) P F (W ) P R ( W ) = eβ(w F ) = e Σ 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 Σ = βw ir = β t 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 Σ > 0 and reversible if Σ = 0 arrow of time (Eddington 1927)
Arrow of time (Quasi) reversible processes Here: entropy production Σ 0 (during duration of experiment) "Lectures on thermodynamics" George Porter 1965 (Nobel 1968)
Arrow of time Irreversible processes Here: entropy production Σ > 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 m d 2 x dt 2 = F(t) Reversal: t t = τ t dt = dt dt 2 = dt 2 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 quasi circular in our solar system all planets orbit in the same direction (law) (initial condition) (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 10 24 particles
Irreversibility in a single-particle system Second law for a driven system: W F (for isothermal process) preferred direction (even for a single particle) (Campisi, Hänggi 2011) reversible for W = F and irreversible for W > 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 H F t [ ] = 2π ν(t) σx C cos φ(t) + σy C sin φ(t)
Experimental observation Nonequilibrium entropy production: Σ = β(w F) experimental proof of Σ 0 for driven quantum system
Experimental observation Mean entropy production: Σ = S(ρ F t ρ B t τ ) = tr[ρ F t ln ρ F t ρ F t ln ρ B t 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).