Feedback Control and Counting Statistics in Quantum Transport Tobias Brandes (Institut für Theoretische Physik, TU Berlin)

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1 Feedback Control and Counting Statistics in Quantum Transport Tobias Brandes (Institut für Theoretische Physik, TU Berlin) Quantum Transport Example: particle counting. Moments, cumulants, generalized density operators. Quantum dots. Feedback control Introduction, various kinds of feedback control. Wiseman-Milburn control. Information and thermodynamics, Maxwell demon control. (Talk on Tuesday) Time-versus-number feedback control. A recent experiment with quantum dots. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

2 Feedback control Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

3 Feedback control Centrifugal governor, Boulton and Watt Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

4 Feedback control Centrifugal governor, Boulton and Watt Stochastic cooling of particle collider beam, S. van der Meer (1972), Nobel Prize (1984) - discovery of W and Z bosons. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

5 Feedback control Centrifugal governor, Boulton and Watt Photodetector signal corrects laser diode pump current, S. Machida, Y. Yamamoto (1986). Stochastic cooling of particle collider beam, S. van der Meer (1972), Nobel Prize (1984) - discovery of W and Z bosons. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

6 Feedback control Centrifugal governor, Boulton and Watt Photodetector signal corrects laser diode pump current, S. Machida, Y. Yamamoto (1986). Stochastic cooling of particle collider beam, S. van der Meer (1972), Nobel Prize (1984) - discovery of W and Z bosons. Feedback control of a single-atom trajectory, A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. W. H. Pinkse, K. Murr and G. Rempe (2009). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

7 Feedback control Challenge for Quantum Systems Unitary dynamics versus measurement process. How to model the feedback loop? H. M. Wiseman, G. J. Milburn, Quantum measurement and Control (2009) Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

8 Passive versus active feedback (Passive) Coherent feedback control of quantum transport C. Emary, J. Gough, PRB 90, (2014). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

9 Passive versus active feedback (Passive) Coherent feedback control of quantum transport C. Emary, J. Gough, PRB 90, (2014). (Passive) Cavity in waveguide J. Kabuss, D. O. Krimer, S. Rotter, K. Stannigel, A. Knorr, and A. Carmele, PRA 92, (2015). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

10 Passive versus active feedback (Passive) Coherent feedback control of quantum transport (Active): feedback control of electron counting statistics. C. Emary, J. Gough, PRB 90, (2014). (Passive) Cavity in waveguide TB; Phys. Rev. Lett. 105, (2010); T. Wagner, P. Strasberg, J. C. Bayer, E. P. Rugeramigabo, TB, R. J. Haug; nature J. Kabuss, D. O. Krimer, S. Rotter, K. nanotechnology (2016). Stannigel, A. Knorr, and A. Carmele, PRA 92, (2015). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

11 Active feedback master equation General setup Open system Hamiltonian. H = H S + H res + H T. HS system. Hres reservoir. H T system-reservoir coupling. G. Schaller, Open Quantum Systems Far From Equilibibrium (2014) Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

12 Active feedback master equation Quantum jumps n-resolved density operator ρ(t) n=0 ρ(n) (t) Stochastic trajectories {x n } = x n,..., x 1 : jumps of type l i at time t i. path integral ρ (n) (t) = M l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ(t {x n }). 0 Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

13 Active feedback master equation Without feedback n-resolved density operator ρ(t) n=0 ρ(n) (t) Stochastic trajectories {x n } = x n,..., x 1 : jumps of type l i at time t i. path integral ρ (n) (t) = M l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ(t {x n }). 0 Without feedback: ρ(t {x n }) S t tn J ln S tn t n 1 J ln 1...J l1 S t1 ρ in. Reduced density matrix ρ(t) = Lρ(t), L L 0 + J, S t e L 0t. Moelmer, Zoller, Hegerfeldt, Carmichael, s Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

14 Active feedback master equation With feedback n-resolved density operator ρ(t) n=0 ρ(n) (t) Stochastic trajectories {x n } = x n,..., x 1 : jumps of type l i at time t i. path integral ρ (n) (t) = With feedback: M l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ(t {x n }). 0 ρ(t {x n }) = S(t {x n })J ln (t n {x n 1 })S(t n {x n 1 })... J l2 (t 2 {x 1 })S(t 2 {x 1 })J l1 (t 1 )S(t 1 )ρ in. Future time evolution conditioned upon full trajectory. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

15 Active feedback master equation Feedback protocols 0 Open loop control, J l (t {x n }) = J l (t). 1 Feedback conditioned on the previous jump, J l (t {x n }) = J lln (t t n ). Wiseman-Milburn quantum feedback control, Jlln (t t n ) = e K l J l. Wiseman, Milburn ( 90s); Pöltl, Emary TB (2011); Schaller, Kiesslich, Emary, TB (2011); Kiesslich, Emary, Schaller, TB (2012); Daryanoosh, Wiseman, TB (2016). Waiting time feedback control TB, Emary (2016). Delayed quantum control, Emary (2013). 2 Time-versus-number feedback, J l (t {x n }) = J l (t, n). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

16 Wiseman-Milburn feedback Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

17 Wiseman-Milburn feedback The idea Dissipation originates from quantum jumps. Fight dissipation by acting immediately after each jump: feedback control. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

18 Wiseman-Milburn feedback Counting fields and feedback ρ = (L 0 + e iχ J )ρ, counting field χ. (R. J. Cook 1981). FEEDBACK: complex number e iχ superoperator e K (Wiseman, s). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

19 Wiseman-Milburn feedback Stabilization: the effective Hamiltonian (C. Emary, 2011) Quantum jump ρ J ρ. Non-jump ρ e L 0τ ρ. Mixes up everything. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

20 Wiseman-Milburn feedback Stabilization: the effective Hamiltonian Quantum jump ρ J ρ. Non-jump ρ e L0τ ρ. Mixes up everything. FEEDBACK IDEA: No mixing if L 0 ρ k = λ k ρ k. Compensate quantum jumps by rotation into ρ k via e K J ρ = ρ k. eigenvalue problem to determine K in L 0 e K J ρ k = λ k ρ k. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

21 Wiseman-Milburn feedback Stabilization: the effective Hamiltonian Effective, non-hermitian Hamiltonian H eff via L 0 ρ = i(h eff ρ ρh eff ). Eigenstates ρ k = Ψ k Ψ k of L 0 are pure density matrices. The Ψ k are (right) eigenstates of H eff. (Non-hermitian part of H eff contains the back-action of the measurement process.) Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

22 Wiseman-Milburn feedback: charge qubit C. Pöltl, C. Emary, TB; Phys. Rev. B 84, (2011). Γ L T C Γ R µ L Effective Hamiltonian ǫ L ǫ R H eff H S iγ L iγ R 2 R R µ R Pure eigenstates 0 0, ψ ± ψ ± Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

23 Wiseman-Milburn feedback: charge qubit Control Stabilization Electron tunnels into L which is instantaneously rotated into ψ + (alternatively, ψ ). effective single level system, decay rate γ (±) R 2Iε ± C L Γ L T C Γ R Γ L γ (j) R µ L µ L ǫ L ǫ R Stabilized µ R µ R Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

24 Wiseman-Milburn feedback: charge qubit Controlled master equation C L Γ L T C Γ R ρ = (L 0 + C L J L + J R ) µ L C L e 2iθ C n 0 Σ, Σρ [σ, ρ] ǫ L ǫ R µ R Rotation in Liouville-space about angle θ C around direction n 0 = (sin θ, 0, cos θ), Pauli matrix vector σ. Every qubit rotation as possible control operation, J. Wang and H. M. Wiseman, Phys. Rev. A 64, (2001). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

25 Wiseman-Milburn feedback: charge qubit Controlled master equation C L Γ L T C Γ R ρ = (L 0 + C L J L + J R ) µ L C L e 2iθ C n 0 Σ, Σρ [σ, ρ] ǫ L ǫ R µ R Rotation in Liouville-space about angle θ C around direction n 0 = (sin θ, 0, cos θ), Pauli matrix vector σ. Every qubit rotation as possible control operation, J. Wang and H. M. Wiseman, Phys. Rev. A 64, (2001). Project out empty state (Γ L ). Bloch representation of stationary state ρ stat = ( σ x, σ y, σ z ). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

26 Wiseman-Milburn feedback: charge qubit Right eigenvalues of H eff : ε 0 = i Γ L 2, ε = 1 ( iγr 16TC 2 4 ) Γ2 R Bloch sphere σ z Eigenstates of the free Liouvillian For Γ R >4T C : σ x =0, σ y = 4T C Γ R, σ z = For Γ R <4T C : 16T 2 σ C Γ 2 R x =, σ y = Γ R, 4T C 4T C Γ 2 R 16T 2 C Γ R σ z =0 ρ σ y ρ wo ρ ++ σ x Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

27 Maxwell demon type feedback Introduction Maxwell s thought experiment Two volumes of gas in equilibrium separated by sliding door. Demon opens/closes door, separates fast from slow gas molecules. Decrease of entropy.... to show that the second law of thermodynamics has only a statistical certainty. Concept of a transport device that acts like Maxwell s demon. 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

28 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

29 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

30 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

31 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

32 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

33 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

34 Maxwell demon type feedback 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

35 Maxwell demon type feedback 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. Information gain via feedback modifies exchange fluctuation relation. General scenario T. Sagawa and M. Ueda ( 08), J. M. P. Parrondo ( 11); D. Abreu, U. Seifert ( 12); T. Munakata, M. L. Rosinberg ( 12); H. Tasaki ( 13); J. M. Horowitz, M. Esposito ( 14); D. Hartich, A. C. Barato, U. Seifert ( 14); J. M. Horowitz ( 15); J. M. Horowitz, K. Jacobs ( 15) Example e β(w F ) I = 1 modified Jarzynski relation. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

36 Feedback controlled tunnel barrier Single electron transistor Rate equation ρ = Lρ, ρ = (ρ 0, ρ 1 ) T. Explicitely break local detailed balance: L = α=l,r ( Γα f α Γ α (1 f α )e iχ Γ α f α Γ α (1 f α ) ) G. Schaller, C. Emary, G. Kießlich, TB; Phys. Rev. B 84, (2011). Fluctuation relation (affinity A V /k B T, voltage V µ L µ R ) lim t p n (t) p n (t) = e ( A+ln Γ R Γ +ln Γ ) L n R Γ L. M. Esposito, G. Schaller; EPL 99, (2012). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

37 Feedback controlled tunnel barrier Single electron transistor Fluctuation relation (affinity A V /k B T, voltage V µ L µ R ) lim t p n (t) p n (t) = e ( A+ln Γ R Γ +ln Γ ) L n R Γ L. M. Esposito, G. Schaller; EPL 99, (2012). Term ln Γ R + ln Γ L Γ R Γ L as offset-voltage = V /k B T G. Schaller, C. Emary, G. Kießlich, TB; Phys. Rev. B 84, (2011). Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

38 Hardwiring the demon Key idea Microscopic model for larger system : SET + detector. Reduced SET dynamics described by effective model as above. Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

39 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

40 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

41 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

42 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

43 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

44 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

45 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

46 Maxwell demon limit Energetics: first law For ɛ S U, modification of first law I E L + I E R = I E D 0 negligible Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

47 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

48 Maxwell demon limit Where is the demon? Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14

49 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) Feedback Control of Quantum Transport Taiwan Winter School / 14

50 Summary Transport master equations Example: particle counting. Moments, cumulants, generalized density operators. Quantum dots. Feedback control Introduction, various kinds of feedback control. Wiseman-Milburn control. Information and thermodynamics, Maxwell demon control. Co-workers: G. Schaller (Berlin); C. Emary (Newcastle); C. Pöltl (Strasbourg); M. Esposito, P. Strasberg (Luxembourg) Tobias Brandes (Berlin) Feedback Control of Quantum Transport Taiwan Winter School / 14