Thermodynamics of quantum coherences César A. Rodríguez-Rosario Bremen Center of Computational Materials Science Thomas Frauenheim - BCCMS Alán Aspuru-Guzik - Harvard University Quantum Phononics Workshop 2015 arXiv:1308.1245

BCCMS: Molecular Junctions Workshop “Understanding quantum transport in molecular junctions”

Phonons help, Phonons hurt

Need “Quantum Thermodynamics” "The theory of its operation is rudimentary and attempts to improve its performance are still made in an almost haphazard way" -Sadi Carnot Need “Quantum Thermodynamics”

Wishlist Non-equilibrium thermodynamics Quantum coherence without too many details of the dynamics Quantum coherence Easy in intermediate regime Intuition: role of coherences in transport simple validation for computational methods

Outline Laws of quantum thermodynamics (0,1,2) Microreview - Non-equilibrium Thermo Coherence / Decoherence Laws of quantum thermodynamics (0,1,2) Quantum reciprocal relations and coherences Role of coherences in transport

Microreview: Non-equilibrium Thermo Boltzmann

Onsager 1931

DEVIATIONS FROM EQUILIBRIUM Onsager 1931 reciprocal generalized fluxes: generalized forces: DEVIATIONS FROM EQUILIBRIUM “When two or more irreversible transport processes (heat conduction, electrical conduction and diffusion) take place simultaneously in a thermodynamic system, the processes may interfere with each other” Onsager: Nobel Prize 1968

Non-equilibrium thermo Entropy Flux Entropy Rate Entropy Prod. Rate Second Law is: If heat rate: Then: Prigogine: Nobel Prize 1977

Thermo Fluxes and Forces temp/entropy pressure/volume stress/strain chemical potential/number fluxes: Onsager reciprocal relations Regime such that Reciprocal Relations:

Foundation: Stochastic processes Prob. Distribution (vector) rates (matrix)

Stationary Distribution

Fixed Point Fixed Line

Baths in Thermodynamics big (Markovian) interaction described by equilibrium thermo forces describe non-equilibrium deviations equilibrium is a single Gibbs state (?)

‘surprise’ of assuming and learning it was Gibbs Entropy Relative Entropy ‘surprise’ of assuming and learning it was

Derivation Identify: Substitute: Like Prigogine!

Onsager reciprocal relations many baths: assume: NESS, detailed balance, linear approximation reciprocal:

Classical transport Great for macroscopic transport Simple intuition complements computation Unknown for quantum regime

Coherence / Decoherence

Decoherence Bath (phonons!) big (Markovian) interaction described by equilibrium thermo forces describe non-equilibrium deviations equilibrium is NOT single Gibbs state

Fixed Point Fixed Line

Decoherence:

Open Quantum System Dynamics (stochastic) effects Schrödinger Eqn. + Quantum (Stochastic) Master Equation Dynamical Map

Relaxation to Gibbs state (thermalization)

(dephasing, transition to classical, transition to incoherence) Decoherence (dephasing, transition to classical, transition to incoherence)

(finally, new results) Laws of Quantum Thermodynamics

Zeroth Law (equilibrium, ‘thermometer’ describes bath)

First Law (energy conservation) Change in Energy: Alicki ‘78

Heat due to decoherence Bath has no ‘temperature’ defined Heat depends on quantum coherences

Second Law (irreversibility) in general relaxation case: Spohn 1977 Entropy production due to decoherence

Quantum reciprocal relations and coherences

Interplay between baths Decoherence in transport Interplay between baths

Non-Equilibrium Steady State many baths: evolutions stationary distributions Non-Equilibrium Steady State state creates steady-state fluxes to baths

Quantum reciprocal relations force matrix flux matrix assume: NESS , detailed balance, linear approx reciprocal: superoperator relaxation case: Spohn & Lebowitz 1978

Coherences = thermo force Quantum coherences are deviations from the equilibrium for the pure decoherence bath

Role of coherences in quantum transport

Reciprocal relations: Decoherence (phonons) and Transport

Example (clever way) Reciprocal relationship: decoherence and transport Transport is affected by decoherence force: Decoherence is affected by transport force:

Example (clever way) Reciprocal relationship: decoherence and transport Transport is affected by decoherence force: Decoherence is affected by transport force:

Example (clever way) Reciprocal relationship: decoherence and transport Transport is affected by decoherence force: Decoherence is affected by transport force:

Example (clever way) Reciprocal relationship: decoherence and transport Transport is affected by decoherence force: Decoherence is affected by transport force:

Intuition gained Decoherence creates thermo force/flux Reciprocal relationship between decoherence from phonons and transport Like thermo-electrics, but with quantum phonon decoherence! arXiv:1308.1245

Contributions Contributions Onsager relationships are true in quantum regime! Derived quantum thermodynamics to understand decoherence Decoherence from phonons creates heat and changes entropy production, leads to new fluxes and forces! arXiv:1308.1245 Contributions

Thanks Alán Aspuru-Guzik (Harvard) Thomas Frauenheim (BCCMS) Gabriele Penazzi (BCCMS) Vlatko Vedral (Oxford U.) Stephanie Wehner (CQT Singapore) arXiv:1308.1245

Contributions Contributions Onsager relationships are true in quantum regime! Derived quantum thermodynamics to understand decoherence Decoherence from phonons creates heat and changes entropy production, leads to new fluxes and forces! arXiv:1308.1245 Contributions