Hardwiring Maxwell s Demon Tobias Brandes (Institut für Theoretische Physik, TU Berlin)

Transcription

1 Hardwiring Maxwell s Demon Tobias Brandes (Institut für Theoretische Physik, TU Berlin) Introduction. Feedback loops in transport by hand. by hardwiring : thermoelectric device. Maxwell demon limit. Co-workers: Gernot Schaller, Philipp Strasberg (TU Berlin), Massimiliano Esposito (Luxembourg). Tobias Brandes (Berlin) 1 / 10

2 Introduction: Maxwell s demon Thought experiment (Maxwell 1867) Two volumes of gas in equilibrium seperated by sliding door. Demon opens/closes door in order to separate fast from slow gas molecules. Decrease of entropy.... to show that the second law of thermodynamics has only a statistical certainty. Tobias Brandes (Berlin) 2 / 10

3 Introduction: Maxwell s demon Szilard engine (1929) Figure from M. B. Plenio, V. Vitelli (2001) Completely converts heat k b T ln 2 into work. Landauer (1961): demon has a memory. Colloquium: The physics of Maxwell s demon and information; K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009). Tobias Brandes (Berlin) 2 / 10

4 Introduction: Maxwell s demon Maxwell s demon as a feedback control mechanism Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nature Physics 6, 988 (2010). Non-equilibrium feedback manipulation of a Brownian particle. Tobias Brandes (Berlin) 2 / 10

5 Introduction: feedback control Open loop and closed loop (feedback) control SYSTEM Open-loop control SYSTEM Closed-loop control ('coherent', passive) Measurement Device SYSTEM Closed-loop control (measurement based, active) Tobias Brandes (Berlin) 3 / 10

6 Feedback control of transport through nanostructures Tobias Brandes (Berlin) 4 / 10

7 Feedback control of transport through nanostructures Electron Statistics in Quantum Dots Full Counting Statistics Probability p(n, t) of n electrons after time t. C. Flindt, C. Fricke, F. Hohls, T. Novotný, K. Netocný, T. Brandes, and R. J. Haug; PNAS 106, (2009). Tobias Brandes (Berlin) 4 / 10

8 Feedback controlled tunnel barrier Single electron transistor Modify tunnel rates, e.g. Γ R, depending on dot occupation. G. Schaller, C. Emary, G. Kießlich, TB; Phys. Rev. B 84, (2011). Tobias Brandes (Berlin) 5 / 10

9 Feedback controlled tunnel barrier Single junction Integrate out the dot for Γ L Γ R Γ. Rates γ = Γf L (1 f R ), γ = Γf R (1 f L ). p n (t) probability for n charges transfered to right reservoir. ṗ n = γp n 1 + γp n+1 (γ + γ)p n Tobias Brandes (Berlin) 5 / 10

10 Feedback controlled tunnel barrier Single junction Integrate out the dot for Γ L Γ R Γ. Rates γ = Γf L (1 f R ), γ = Γf R (1 f L ). p n (t) probability for n charges transfered to right reservoir. ṗ n = γp n 1 + γp n+1 (γ + γ)p n Without feedback: Identical microscopic forward and backward rates Γ = Γ. Local detailed balance condition with affinity A, γ γ = ea, A µ L µ R k B T Tobias Brandes (Berlin) 5 / 10

11 Feedback controlled tunnel barrier Single junction Integrate out the dot for Γ L Γ R Γ. Rates γ = Γf L (1 f R ), γ = Γf R (1 f L ). p n (t) probability for n charges transfered to right reservoir. ṗ n = γp n 1 + γp n+1 (γ + γ)p n With feedback: Different forward and backward rates Γ Γ ( by hand ) Violates local detailed balance condition, γ γ = ea+ln Γ Γ, A µ L µ R k B T Tobias Brandes (Berlin) 5 / 10

12 Feedback controlled tunnel barrier Single junction Integrate out the dot for Γ L Γ R Γ. Rates γ = Γf L (1 f R ), γ = Γf R (1 f L ). p n (t) probability for n charges transfered to right reservoir. ṗ n = γp n 1 + γp n+1 (γ + γ)p n Elevate (violated) local detailed balance to modified exchange fluctuation theorem p n (t) p n (t) = e(a+ln Γ Γ )n, A µ L µ R k B T. Tobias Brandes (Berlin) 5 / 10

13 Feedback controlled tunnel barrier Single junction: Interpretation Rates γ = Γf L (1 f R ), γ = Γf R (1 f L ). p n(t) p n (t) = e(a+ln Γ Γ)n, A µ L µ R k B T. Stationary charge current J = γ γ (set e = 1). With feedback Γ Γ finite J even for zero voltage drop. Tobias Brandes (Berlin) 5 / 10

14 Feedback controlled tunnel barrier Single junction: Interpretation Rates γ = Γf L (1 f R ), γ = Γf R (1 f L ). p n(t) p n (t) = e(a+ln Γ Γ)n, A µ L µ R k B T. Only n (number of transfered charges) thermodyn. relevant. Shannon entropy S n p n ln p n. Decompose Ṡ = Ṡe + Ṡi with Ṡi 0, J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976); M. Esposito, C. Van den Broek, Phys. Rev. E 82, (2010). lim t p n (t) p n (t) = t eṡi, Ṡ i = AJ + ln J, J γ γ Γ Γ Ṡ i = dissipated electric power per k B T plus information current. Tobias Brandes (Berlin) 5 / 10

15 Hardwiring the demon Key idea Microscopic model for larger system : SET + detector. Reduced SET dynamics described by effective model as above. Tobias Brandes (Berlin) 6 / 10

16 Hardwiring the demon Thermoelectric device V 1, T 1 V 2, T 2 I Energy to current converter. Different temperatures in different parts of the system. Energy-dependent tunnel rates. charge conducting quantum dot gate quantum dot V g, T g J g R. Sánchez, M. Büttiker, Phys. Rev. B 83, (2011); Europhys. Lett. 100, (2012). V1 C1 C2 C Vg Cg V2 Tobias Brandes (Berlin) 6 / 10

17 Hardwiring the demon Single level SET (bottom, two reservoirs L/R) and detector (top, one reservoir) States 0E, 0F, 1E, 1F. Energies 0, ɛ s, ɛ d, ɛ s + ɛ d + U. Energy dependent rates. P. Strasberg, G. Schaller, TB, M. Esposito, Phys. Rev. Lett. 110, (2013). Tobias Brandes (Berlin) 6 / 10

18 Hardwiring the demon State of the demon P. Strasberg, G. Schaller, TB, M. Esposito, Phys. Rev. Lett. 110, (2013). E State of the SET Detector requirements: Fast ΓD, Γ U D Γ α, Γ U α. Precise U k B T D. SET requirements: Spatial asymmetry Γ U R Γ U L, Γ L Γ R. F Tobias Brandes (Berlin) 6 / 10

19 Hardwiring the demon Stochastic thermodynamics of bipartite systems J. M. Horowitz and M. Esposito, arxiv: (2014). Tobias Brandes (Berlin) 6 / 10

20 Maxwell demon limit Reduced SET dynamics Full rate equation ρ ij = i j W ij,i j ρ i j Separation of time scales for fast detector Γ D, Γ U D Γ α, Γ U α. Technical trick: use conditional stationary probabilities M. Esposito, Phys. Rev. E 85, (2012). SET rate equation ρ i = i V ii ρ i. Tobias Brandes (Berlin) 7 / 10

21 Maxwell demon limit Reduced SET dynamics Full rate equation ρ ij = i j W ij,i j ρ i j Separation of time scales for fast detector Γ D, Γ U D Γ α, Γ U α. Technical trick: use conditional stationary probabilities M. Esposito, Phys. Rev. E 85, (2012). SET rate equation ρ i = i V ii ρ i. Reduced fluctuation theorem for SET with information current I lim t p n (t) = exp [An + I t], p n (t) I t ln f L Uf RΓ U L Γ R fr Uf LΓ U R Γ L n, A µ L µ R k B T. Tobias Brandes (Berlin) 7 / 10

22 Maxwell demon limit Reduced SET dynamics Full rate equation ρ ij = i j W ij,i j ρ i j Separation of time scales for fast detector Γ D, Γ U D Γ α, Γ U α. Technical trick: use conditional stationary probabilities M. Esposito, Phys. Rev. E 85, (2012). SET rate equation ρ i = i V ii ρ i. Reduced fluctuation theorem for SET with information current I lim t p n (t) = exp [An + I t], p n (t) I t ln f L Uf RΓ U L Γ R fr Uf LΓ U R Γ L n, A µ L µ R k B T. Reduction to previous model ( feedback by hand ): f U α /f α = 1 I t = (ln Γ U L Γ R/Γ U R Γ L)n Tobias Brandes (Berlin) 7 / 10

23 Maxwell demon limit Energetics: first law Cycle between system energies: ɛ d ɛ s + ɛ d + U ɛ s 0 ɛ d. Net energy U transferred from SET to detector. Tobias Brandes (Berlin) 7 / 10

24 Maxwell demon limit Energetics: first law For U ɛ S, modification of first law I E L + I E R = I E D 0 negligible Tobias Brandes (Berlin) 7 / 10

25 Maxwell demon limit Summary of demon conditions Separation of time scales: Fast demon ΓD, Γ U D Γ α, Γ U α. Separation of energy scales: No back-action, precision: k B T U k B T D. Almost no work done: ɛ S U. Spatial/ energy lever condition Γα Γ U α, Γ L Γ (U) R, α = L,R. [( p n (t) lim t p n (t) exp A + ln ΓU L Γ ) ] R Γ U R Γ n, A µ L µ R L k B T. Tobias Brandes (Berlin) 7 / 10

26 Maxwell demon limit Where is the demon? Tobias Brandes (Berlin) 7 / 10

27 Maxwell demon limit Where is the demon? Maxwell s demon A feedback mechanism that shuffles particles against a chemical or thermal gradient by adjusting barriers without requiring work. Hardwiring of the feedback mechanism. Information is really physical: rates in term ln ( Γ U L Γ R/Γ U R Γ L). Tobias Brandes (Berlin) 7 / 10

28 Summary Minimal microscopic model for measurement-based feedback: Based on stochastic thermodynamics of four-state model. Approaches ideal Maxwell demon in well-defined parameter limit. Open questions: Hardwiring of Wiseman-Milburn feedback. Thermodynamics and quantum coherence. Interacting feedback schemes. Tobias Brandes (Berlin) 8 / 10

29 Master Equation General setup Open system Hamiltonian. H = H S + H res + H T. HS system. Hres reservoir. H T system-reservoir coupling. Reduced density matrix ρ(t), Liouvillian L, Born-Markov approximation ρ(t) = Lρ(t), L L 0 + J Tobias Brandes (Berlin) 9 / 10

30 Master Equation Quantum jumps Jump-resolved ( n-resolved ) Master equation ρ(t) = ρ n (t) = n=0 n=0 t 0 t2 dt n... dt 1 ρ c (t; t n,..., t 1 ) 0 ρ c (t; t n,..., t 1 ) e L 0 (t t n) J e L 0 (t n t n 1 ) J...J e L 0 t 1 ρ 0 Non-unitary free time-evolution, interrupted by n quantum jumps at times t i. Tobias Brandes (Berlin) 9 / 10

31 Master Equation Feedback conditioned on quantum jumps Scheme 1 Rotate state vector by upgrading counting field χ in ρ = (L0 + e iχ J )ρ to superoperator K, H. Wiseman, G. Milburn (1990s). Scheme 2 Effectively modulate system reservoir coupling Maxwell demon. Tobias Brandes (Berlin) 9 / 10

32 Alternative scheme Model with phonons 1 0,05 0,5 0 electron current -0,05 0-0,1-0,05 0 0,05 0,1-0, bias voltage T. Krause, G. Schaller, TB, Phys. Rev. B 84, (2011). Electrons and phonons at different temperatures. Incomplete fluctuation theorem with shift term current at zero bias. See also O. Entin-Wohlman, Y. Imry, and A. Aharony, Phys. Rev. B 82, (2010). Tobias Brandes (Berlin) 10 / 10

33 Thermodynamics of Wiseman-Milburn Feedback Feedback conditioned on quantum jumps Scheme 1 Rotate state vector by upgrading counting field χ in ρ = (L0 + e iχ J )ρ to superoperator K, H. Wiseman, G. Milburn (1990s). Scheme 2 Effectively modulate system reservoir coupling Maxwell demon. Tobias Brandes (Berlin) 11 / 10

34 Thermodynamics of Wiseman-Milburn Feedback Qubit model: absorption and emission of photons Master equation ρ = Lρ. Feedback couples coherences to populations. ( ) Lp 0 L = L cp L c Tobias Brandes (Berlin) 11 / 10