1. 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:

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

3. Phonons help,
Phonons hurt

4. 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”

5. 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

6. 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

7. Microreview: Non-equilibrium Thermo
Boltzmann

8. Onsager 1931

9. 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

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

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

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

13. Stationary Distribution

14. Fixed
Point
Fixed
Line

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

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

17. Derivation
Identify:
Substitute:
Like Prigogine!

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

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

20. Coherence / Decoherence

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

22. Fixed
Point
Fixed
Line

23. Decoherence:

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

25. Relaxation to Gibbs state (thermalization)

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

27. (finally, new results) Laws of Quantum Thermodynamics

28. Zeroth Law (equilibrium, ‘thermometer’ describes bath)

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

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

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

32. Quantum reciprocal relations and coherences

33. Interplay between baths
Decoherence in transport
Interplay between baths

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

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

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

37. Role of coherences in quantum transport

38. Reciprocal relations: Decoherence (phonons) and Transport

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

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

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

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

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

44. 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:
Contributions

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

46. 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:
Contributions