Waiting Times and Noise in Transport

Transcription

1 Waiting Times and Noise in Transport Tobias Brandes (Institut für Theoretische Physik, TU Berlin) Main Idea. Some Formalism. Examples. arxiv: (2008); Ann. Phys. (Berlin) 17, 477 (2008). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

2 Basic Idea (Electronic) Transport Current comes in individual charges, I = e i=1 δ(t t i). Measure times t i in leads system properties. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

3 Basic Idea (Electronic) Transport Current comes in individual charges, I = e i=1 δ(t t i). Measure times t i in leads system properties. Statistical Tools I Transfer of electrons between leads and system: stochastic process Bagrets, Nazarov Multi -probabilities p(n 1,t 1,n 2,t 2,...) full counting statistics (FCS). current fluctuation spectra S(ω) dτeiωτ {δî(τ), δî(0)}. higher cumulant spectra Emary, Marcos, Aguado, Brandes Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

4 Basic Idea (Electronic) Transport Current comes in individual charges, I = e i=1 δ(t t i). Measure times t i in leads system properties. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

5 Basic Idea (Electronic) Transport Current comes in individual charges, I = e i=1 δ(t t i). Measure times t i in leads system properties....daß die Gesamtheit der Schwingungsfrequenzen und der die Intensität der Linien bestimmenden Größen... als ein vollwertiger Ersatz der Bahnen gelten könnte. W. Heisenberg, Der Teil und das Ganze Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

6 Basic Idea (Electronic) Transport Current comes in individual charges, I = e i=1 δ(t t i). Measure times t i in leads system properties. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

7 Basic Idea (Electronic) Transport Current comes in individual charges, I = e i=1 δ(t t i). Measure times t i in leads system properties. Statistical Tools II: Waiting Time Distributions Probability density w ij (τ) for time τ between (electron) transfer of type i, followed by transfer of type j. Microscopic definition? Information content? Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

8 Bidirectional Counting of Single Electrons T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama, Science 312, 1634 (2006) Destroys quantum coherence. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

9 Master Equation Methods Main Ingredients Simple microscopic models. (Quantum) system H S + bath(s) H B (electronic leads, phonons). Basis of (many body) system states α. System density operator ˆρ(t). (Dis)Advantages Good for exploring new models, mechanisms, concepts. Perturbative in coupling to the leads: some/much interesting physics lost. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

10 Master Equations Main Ingredients Master equation ρ(t) = Lρ(t). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

11 Master Equations Main Ingredients Master equation ρ(t) = Lρ(t). Example: Single quantum dot with 1 level. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

12 Master Equations Main Ingredients Master equation ρ(t) = Lρ(t). Example: Single quantum dot with 1 level. Density matrix as vector ρ = (p 0,p 1 ) T, superoperator as matrix ( ) ΓL Γ L = L 0 + J = R. Γ L Γ R Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

13 Master Equations Main Ingredients Master equation ρ(t) = Lρ(t). Example: Single quantum dot with 1 level. Density matrix as vector ρ = (p 0,p 1 ) T, superoperator as matrix ( ) ΓL Γ L = L 0 + J = R. Γ L Γ R Notation: column kets ρ stat = 0, ρ empty = 1 = (1,0) T etc. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

14 Master Equations Main Ingredients Master equation ρ(t) = Lρ(t). Example: Single quantum dot with 1 level. Density matrix as vector ρ = (p 0,p 1 ) T, superoperator as matrix ( ) ΓL Γ L = L 0 + J = R. Γ L Γ R Notation: column kets ρ stat = 0, ρ empty = 1 = (1,0) T etc. Jump operators, e.g. J = 1 1 with 1 = (0,Γ R ). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

15 Formal Solution of Master Equation ρ(t) = Lρ(t) L = L 0 + L 1, L 1 = M k=1 J k, J k k k ρ(t) = M n=0 l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ c l n...l 1 (t;t n,...,t 1 ) 0 ρ c l n...l 1 (t;t n,...,t 1 ) e L 0(t t n) J ln e L 0(t n t n 1 ) J ln 1...J l1 e L 0(t 1 ) ρ 0 Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

16 Formal Solution of Master Equation ρ(t) = Lρ(t) L = L 0 + L 1, L 1 = M k=1 J k, J k k k ρ(t) = M n=0 l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ c l n...l 1 (t;t n,...,t 1 ) 0 ρ c l n...l 1 (t;t n,...,t 1 ) e L 0(t t n) J ln e L 0(t n t n 1 ) J ln 1...J l1 e L 0(t 1 ) ρ 0 Non-unitary free time-evolution, interrupted by n quantum jumps of type l i at times t i cf. Carmichael et al Free Propagator matrix element w kl (τ) k e L 0τ l = TrJ ke L 0τ J l ρ 0 I l. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

17 Formal Solution of Master Equation ρ(t) = Lρ(t) L = L 0 + L 1, L 1 = M k=1 J k, J k k k ρ(t) = M n=0 l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ c l n...l 1 (t;t n,...,t 1 ) 0 ρ c l n...l 1 (t;t n,...,t 1 ) e L 0(t t n) J ln e L 0(t n t n 1 ) J ln 1...J l1 e L 0(t 1 ) ρ 0 Analogy: Decomposition of Green s functions, G = G 0 + G 0 VG Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

18 Formal Solution of Master Equation ρ(t) = Lρ(t) L = L 0 + L 1, L 1 = M k=1 J k, J k k k ρ(t) = M n=0 l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ c l n...l 1 (t;t n,...,t 1 ) 0 ρ c l n...l 1 (t;t n,...,t 1 ) e L 0(t t n) J ln e L 0(t n t n 1 ) J ln 1...J l1 e L 0(t 1 ) ρ 0 More complex systems. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

19 Formal Solution of Master Equation ρ(t) = Lρ(t) L = L 0 + L 1, L 1 = M k=1 J k, J k k k ρ(t) = M n=0 l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ c l n...l 1 (t;t n,...,t 1 ) 0 ρ c l n...l 1 (t;t n,...,t 1 ) e L 0(t t n) J ln e L 0(t n t n 1 ) J ln 1...J l1 e L 0(t 1 ) ρ 0 Molecule. J. Koch, M. E. Raikh, F. v. Oppen; PRL (2005). H. Hübener, T. Brandes; PRL (2007). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

20 Formal Solution of Master Equation ρ(t) = Lρ(t) L = L 0 + L 1, L 1 = M k=1 J k, J k k k ρ(t) = M n=0 l 1 =1,...,l n=1 t 0 t2 dt n... dt 1 ρ c l n...l 1 (t;t n,...,t 1 ) 0 ρ c l n...l 1 (t;t n,...,t 1 ) e L 0(t t n) J ln e L 0(t n t n 1 ) J ln 1...J l1 e L 0(t 1 ) ρ 0 emitter Γ e Ω Γ c QD1 QD2 current Double Quantum Dot. G. Kießlich, E. Schöll, T. Brandes, F. Hohls, and R.J. Haug; PRL (2007). collector Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

21 Waiting Time Distribution Definition w kl (τ) TrJ ke L 0τ J l ρ 0 I l = k e L 0τ l. I l TrJ l ρ 0 = l 0 stationary current due to jump processes of type l. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

22 Waiting Time Distribution Definition w kl (τ) TrJ ke L 0τ J l ρ 0 I l = k e L 0τ l. I l TrJ l ρ 0 = l 0 stationary current due to jump processes of type l. Waiting times w kl (τ) do not require knowledge of stationary state ρ0. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

23 Waiting Time Distribution Definition w kl (τ) TrJ ke L 0τ J l ρ 0 I l = k e L 0τ l. I l TrJ l ρ 0 = l 0 stationary current due to jump processes of type l. Waiting times w kl (τ) do not require knowledge of stationary state ρ0. are manifestly real. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

24 Waiting Time Distribution Definition w kl (τ) TrJ ke L 0τ J l ρ 0 I l = k e L 0τ l. I l TrJ l ρ 0 = l 0 stationary current due to jump processes of type l. Waiting times w kl (τ) do not require knowledge of stationary state ρ0. are manifestly real. normalization 0 dt k w kl (t) = k ŵ kl (0) = 0 (L L 0 )L 1 0 J l 0 I l = 1. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

25 Waiting Time Distribution Definition w kl (τ) TrJ ke L 0τ J l ρ 0 I l = k e L 0τ l. I l TrJ l ρ 0 = l 0 stationary current due to jump processes of type l. Waiting times w kl (τ) do not require knowledge of stationary state ρ0. are manifestly real. normalization 0 dt k w kl (t) = k ŵ kl (0) = 0 (L L 0 )L 1 0 J l 0 I l = 1. wkl (τ) 0 probability density. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

26 Waiting Time Distribution Definition w kl (τ) TrJ ke L 0τ J l ρ 0 I l = k e L 0τ l. I l TrJ l ρ 0 = l 0 stationary current due to jump processes of type l. So far only rarely used in Mesoscopics J. H. Davies, P. Hyldgaard, S. Hershfield, and J. W. Wilkins (92); J. Koch, M. E. Raikh, and F. von Oppen (05); S. Welack, M. Esposito, U. Harbola, and S. Mukamel (07); M. Esposito and K. Lindenberg (08). Quantum Optics: resonance fluorescence. H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Phys. Rev. A 39, 1200 (1989). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

27 Example: N-Level Chain (Ring) L 0 + J = Γ Γ N Γ 0 Γ Γ 1 Γ Γ N 1 Γ N, Transitions N 0 at rates Γ i. Single jump operator J = 1 1 with 1 (0,...,Γ N ) and 1 = (1,0,...,0) T. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

28 Example: N-Level Chain (Ring) ŵ(z) (collector lead) ŵ N (z) = Γ N Γ 0 Γ 1..., z + Γ 0 z + Γ 1 z + Γ N Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

29 Example: N-Level Chain (Ring) ŵ(z) (collector lead) ŵ N (z) = Γ N Γ 0 Γ 1..., z + Γ 0 z + Γ 1 z + Γ N Single level dot ŵ 1 (z) = Γ R Γ L z + Γ R z + Γ L Poles ±iγ (Γ Γ R + Γ L ) in noise spectrum, S(ω) 1 ( dte iωt {δi(t),δi(0)} = I 1 2Γ ) LΓ R 2 Γ 2 + ω 2. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

30 Example: N-Level Chain (Ring) ŵ(z) (collector lead) ŵ N (z) = Γ N Γ 0 Γ 1..., z + Γ 0 z + Γ 1 z + Γ N Single level dot J. H. Davies, P. Hyldgaard, S. Hershfield, and J. W. Wilkins, Phys. Rev. B 46, 9620 (1992). w 1 (τ) = Γ R Γ L e Γ Lτ e Γ Rτ Γ R Γ L Vanishing of w 1 (τ) for τ = 0 single reset character (as in resonance fluorescence of single atoms). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

31 Example: N-Level Chain (Ring) ŵ(z) (collector lead) ŵ N (z) = Γ N Γ 0 Γ 1..., z + Γ 0 z + Γ 1 z + Γ N In the time domain, convolution of exponentials. Short time behaviour, w N (τ 0) Γ 0...Γ N N! N + 1 elementary transitions with probabilities Γ i within the system. τ N Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

32 Example: N-Level Chain (Ring) ŵ(z) (collector lead) ŵ N (z) = Γ N Γ 0 Γ 1..., z + Γ 0 z + Γ 1 z + Γ N Now N, Γ i = (N + 1)γ. deterministic transport w (τ) = δ ( ) τ 1 γ, noise S (ω) = 0. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

33 Example: N-Level Chain (Ring) ŵ(z) (collector lead) ŵ N (z) = Γ N Γ 0 Γ 1..., z + Γ 0 z + Γ 1 z + Γ N Now N, Γ i = (N + 1)γ. deterministic transport w (τ) = δ ( ) τ 1 γ, noise S (ω) = 0. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

34 Example: Two Coherently Coupled Quantum Dots L 0 + J = Γ L 0 Γ R 0 0 Γ L T c 0 0 Γ R 0 2T c 0 γ + γ Γ R 2 γ ε 0 T c T c ε Γ R 2 γ Quantum system: occupations and coherences are coupled. Electron-phonon scattering rates γ,γ ± T. Brandes, Phys. Rep. (2005); G. Kießlich et al. PRL (2007); C. Flindt et al. PRL (2008). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

35 Example: Two Coherently Coupled Quantum Dots G. Kießlich, E. Schöll, T. Brandes, F. Hohls, and R.J. Haug; PRL 2007 emitter Γ e QD1 Ω QD2 Γ c collector current InAs Quantum Dot Stacks Asymmetric tunnel rates Γ e > Γ c. Bias voltage V SD changes level difference ε = ε 1 ε 2. P. Barthold, F. Hohls, N. Maire, K. Pierz, and R. J. Haug; PRL First experimental use of current fluctuations as proof of coherent coupling between quantum dots. Model assumption Transport through one stack only: single quantum system. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

36 Example: Two Coherently Coupled Quantum Dots 0.3 a) no phonons T=1.4K T=2.7K 0.2 T=12 K!" #$ b) sequential 0.8 tunneling Eq. (8) ε [mv] current [na] Fano factor Fano Factor F S(0)/2eI (no phonons) Super-Poissonian F > 1 for Γ e > Γ c, ε 0 Ellatari, Gurvitz Needs coherent coupling Ω. Needs strong Coulomb blockade. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

37 Example: Two Coherently Coupled Quantum Dots 0.3 a) no phonons T=1.4K T=2.7K 0.2 T=12 K!" #$ b) sequential 0.8 tunneling Eq. (8) ε [mv] current [na] Fano factor Fano Factor F S(0)/2eI (with phonons) Asymmetry: phonon emission/absorption. Crossover coherent incoherent with increasing temperature. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

38 Example: Two Coherently Coupled Quantum Dots Prediction for waiting time distribution: Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

39 Example: Two Coherently Coupled Quantum Dots Prediction for waiting time distribution: w(τ)/γ R T= 4K T= 8K T= 16K ε=1.0 mev 0.05 ε=0.2 mev Coherent oscillations. w(τ) = 1 3! 2T 2 c Γ R Γ L τ 3 + O(τ 4 ) (coherent tunneling) instead of τ 2 (sequential tunneling). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

40 Example: Two Coherently Coupled Quantum Dots Poles of ŵ(z) (no e-p, R ε 2 + 4T 2 c Γ2 R 4 ): z 1 = Γ L, z 2/3,4/5 = Γ ( ) R 2 ± (i) R ΓR ε R Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

41 Example: Two Coherently Coupled Quantum Dots Poles of ŵ(z) (no e-p, R ε 2 + 4T 2 c Γ2 R 4 ): z 1 = Γ L, z 2/3,4/5 = Γ ( ) R 2 ± (i) R ΓR ε R w R (z): count on right side only, w ij (z): count on both sides; ŵ R (z) = ŵ RR (z) + ŵrl(z)ŵ LR (z) 1 ŵ LL (z) Γ L = ŵ RL (z) z + Γ L no subsequent jumps L and R, ŵ LL (z) = ŵ RR (z) = 0. trivial re-charging of empty Ddot, ŵ LR (z) = Γ L /(z + Γ L ). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

42 Relation with Non-Equilibrium Noise Spectrum Symmetrized Noise Spectrum S kl (ω) dt 2 eiωt {δi k (t),δi l (0)} Fano factor F = S(0) 2eI. Spectrum, τ, τ φ. B. Elattari, S. A. Gurvitz 02; C. Flindt et al. 04; R. Aguado, T. Brandes 04; A. Cottet et al. 04; N. Lambert et al. 05);... Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

43 Relation with Non-Equilibrium Noise Spectrum Symmetrized Noise Spectrum S kl (ω) S kl (ω) = δ kl I l dt 2 eiωt {δi k (t),δi l (0)} { ± Fano factor F = S(0) 2eI. Spectrum, τ, τ φ. B. Elattari, S. A. Gurvitz 02; C. Flindt et al. 04; R. Aguado, T. Brandes 04; A. Cottet et al. 04; N. Lambert et al. 05);... } [ ] (1 W(±iω)) 1 W(±iω) I l + (k l) kl Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

44 Relation with Non-Equilibrium Noise Spectrum Symmetrized Noise Spectrum S kl (ω) S kl (ω) = δ kl I l dt 2 eiωt {δi k (t),δi l (0)} { ± Fano factor F = S(0) 2eI. Spectrum, τ, τ φ. B. Elattari, S. A. Gurvitz 02; C. Flindt et al. 04; R. Aguado, T. Brandes 04; A. Cottet et al. 04; N. Lambert et al. 05);... } [ ] (1 W(±iω)) 1 W(±iω) I l + (k l) kl W(z) S kl (ω) but not vice-versa (cf. single dot example). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

45 Relation with Non-Equilibrium Noise Spectrum Symmetrized Noise Spectrum S kl (ω) S kl (ω) = δ kl I l dt 2 eiωt {δi k (t),δi l (0)} { ± Fano factor F = S(0) 2eI. Spectrum, τ, τ φ. B. Elattari, S. A. Gurvitz 02; C. Flindt et al. 04; R. Aguado, T. Brandes 04; A. Cottet et al. 04; N. Lambert et al. 05);... } [ ] (1 W(±iω)) 1 W(±iω) I l + (k l) kl Matrix W(z) of waiting times ŵ kl (z) distinction single reset system. After any jump, system is in a unique state ( empty ). RF, strong CB dots,... multiple reset system. Molecules Koch, v. Oppen, Raikh 05,... Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

46 Single/Multiple Resets Sum J α = k kα kα of jump operators, e.g. α = R, k = energy. In general entangled. Separable: J α = α k kα α α single reset. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

47 Single/Multiple Resets Sum J α = k kα kα of jump operators, e.g. α = R, k = energy. In general entangled. Separable: J α = α k kα α α single reset. Multi-level single dot γ L Γ 1... Γ N 1 Γ N γ 1 Γ L = γ 2 0 Γ γ N Γ N γ L γ γ N. Single-reset. R (0,Γ 1,Γ 2,...,Γ N ), R = (1,0,...,0) T. L γ 1 L (0,γ 1,γ 2,...,γ N ), L = (γ L,0,...,0) T. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

48 Single/Multiple Resets Sum J α = k kα kα of jump operators, e.g. α = R, k = energy. In general entangled. Separable: J α = α k kα α α single reset. L = Multiple reset. Anderson-impurity (2 electrons) 2γ L γ R 0 2γ L (Γ L + γ R ) 2Γ R 0 Γ L 2Γ R Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

49 Single/Multiple Reset: How to Distinguish Sums of jump operators, J α k J kα. Reduced waiting times ŵ (r) αβ (z) TrJ α(z L 0 ) 1 J β (z)ρ 0 TrJ β ρ 0. Reduced noise spectrum S (r) αβ (ω) = kl S kα,lβ(ω). Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

50 Single/Multiple Reset: How to Distinguish Sums of jump operators, J α k J kα. Reduced waiting times ŵ (r) αβ (z) TrJ α(z L 0 ) 1 J β (z)ρ 0 TrJ β ρ 0. Reduced noise spectrum S (r) αβ (ω) = kl S kα,lβ(ω). Relation S (r) αβ (ω) = F[ŵ(r) αβ ](ω) (single reset). Multiple reset, e.g. Anderson 1.05 impurity: fidelity F(ω) F[ŵ (r) αβ ](ω)/s(r) αβ (ω) S ~ (ω) / S (r) (ω) ω γ R =Γ R =0.1 γ R =Γ R =1.0 γ R =Γ R =10 Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

51 Single/Multiple Reset: How to Distinguish Entropy of Currents E α k I kα I α log I kα I α. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

52 Single/Multiple Reset: How to Distinguish Entropy of Currents E α k I kα I α log I kα I α. 2.5 Waiting times for counting, e.g., in both left and right lead, η ŵ LL (0) = ŵ RR (0). E = E R = E L and η functions of 0.5 γ R /Γ L only (Anderson-impurity) 0 Current Entropy E, η E (three-state) η (three-state) E (classical) η (classical) γ R /Γ L (three-state), Γ L /Γ R (classical) Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

53 Summary: Waiting Time Distribution ŵ(z) Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

54 Summary: Waiting Time Distribution ŵ(z) Fundamental for analysing transport. ŵ(z) noise S(ω), FCS, but not vice versa. Examples: transfer properties of emitter and collector neatly separated. NOT discussed in this talk: FCS. Further examples. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

55 Summary: Waiting Time Distribution ŵ(z) Fundamental for analysing transport. ŵ(z) noise S(ω), FCS, but not vice versa. Examples: transfer properties of emitter and collector neatly separated. NOT discussed in this talk: FCS. Further examples. Future work? Entanglement structure of composed jump operators, e.g. J = Waiting times for (non-interacting) mesoscopic scatterers. Many Fermions. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

56 Example: Classical Particles Entering and Leaving L = Γ L Γ R Γ L Γ R Γ L 2Γ R Γ L 2Γ R Γ L 3Γ R , Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

57 Example: Classical Particles Entering and Leaving L = Γ L Γ R Γ L Γ R Γ L 2Γ R Γ L 2Γ R Γ L 3Γ R , System with 0, 1, 2,... particles. State ρ = (p 0,p 1,p 2,p 3,...). Average n t with d dt n t = Γ L Γ R n t Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

58 Example: Classical Particles Entering and Leaving L = Γ L Γ R Γ L Γ R Γ L 2Γ R Γ L 2Γ R Γ L 3Γ R Stationary state with p n = αn n! e α, α Γ L Γ R. Stationary current I = I R = I L = Γ L., Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

59 Example: Classical Particles Entering and Leaving Decompose L = L 0 + J R + J L Jump operators [J R ] kl = kγ R δ k,l 1, [J L ] kl = Γ L δ k,l+1. Waiting time distributions ŵ LL (z) = ŵ RR (z) = ŵ LR (z) = ŵ RL (z) = n=0 n=0 n=0 αp n z/γ R + α + n + 1 np n z/γ R + α + n 1 (n + 1)p n z/γ R + α + n + 1. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

60 Example: Classical Particles Entering and Leaving Decompose L = L 0 + J R + J L Jump operators [J R ] kl = kγ R δ k,l 1, [J L ] kl = Γ L δ k,l+1. Waiting time distributions ŵ LL (z) = ŵ RR (z) = ŵ LR (z) = ŵ RL (z) = n=0 n=0 n=0 αp n z/γ R + α + n + 1 np n z/γ R + α + n 1 (n + 1)p n z/γ R + α + n + 1. Waiting time distribution for just counting on the left side, ŵ L (z) = ŵ LL (z) + ŵlr(z)ŵ RL (z) 1 ŵ RR = Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

61 Example: Classical Particles Entering and Leaving Decompose L = L 0 + J R + J L Jump operators [J R ] kl = kγ R δ k,l 1, [J L ] kl = Γ L δ k,l+1. Waiting time distributions ŵ LL (z) = ŵ RR (z) = ŵ LR (z) = ŵ RL (z) = n=0 n=0 n=0 αp n z/γ R + α + n + 1 np n z/γ R + α + n 1 (n + 1)p n z/γ R + α + n + 1. Waiting time distribution for just counting on the left side, ŵ L (z) = ŵ LL (z) + ŵlr(z)ŵ RL (z) 1 ŵ RR = Γ L z + Γ L. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

62 Relation to Full Counting Statistics p(n,t) number of jumps after time t. G(χ,t) = e inχ p(n,t) = Tre (L 0+e iχ J )t ρ 0 n=0 Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

63 Relation to Full Counting Statistics p(n,t) number of jumps after time t. G(χ,t) = e inχ p(n,t) = Tre (L 0+e iχ J )t ρ 0 n=0 Ĝ(χ,z) = 0 (z L 0 e iχ J ) 1 0 =... = 0 (z L 0 ) (z L 0 ) (z L 0 ) 1 0 e iχ ŵ(z) (J 1 1 ) Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15

64 Relation to Full Counting Statistics p(n,t) number of jumps after time t. G(χ,t) = e inχ p(n,t) = Tre (L 0+e iχ J )t ρ 0 n=0 Ĝ(χ,z) = 0 (z L 0 e iχ J ) 1 0 =... = 0 (z L 0 ) (z L 0 ) (z L 0 ) 1 0 e iχ ŵ(z) (J 1 1 ) FCS from G(χ,t ) e z 0(χ)t : solution z 0 (χ) of ŵ(z 0 ) = e iχ, z 0 (χ = 0) = 0. Tobias Brandes (Berlin) Waiting Times Taipeh, 25 June, / 15