1. Fluctuations of Entropy Production in Partially Masked Electric Circuits Chung Yuan Christian University - Physics Seminar, Nov. 4 th, 2015 Yung-Fu Chen Department of Physics National Central University

2. Thermodynamics System of interest is thermalized to its surrounding bath Exchanging energy: heat To what extent this heat-bath scenario is valid?

3. Thermodynamics System of interest is thermalized to its surrounding heat bath Exchanging energy: heat To what extent this heat-bath scenario is valid? Heat bath consists objects with similar scales Validity of Fluctuation Theorem (FT)

4. Nonequilibrium Thermodynamics in Small Systems System under investigation is out of equilibrium Dissipation in observation time span ~ thermal fluctuation Stochastic thermodynamics G. M. Wang et al., PRL 89, 050601 (2002) D. J. Evans et al., PRL 71, 2401 (1993) Fluctuation theorem: extension of 2nd law of thermodynamics The first experimental demonstration: Brownian motion under dragged restored force

5. Fluctuation Theorem Detailed form Integral form Distribution of entropy production fluctuation

6. Entropy Production with Partial Information Theoretical works Only partial transitions are observed Integral FT is preserved Partial degrees of freedom are observed Integral FT is preserved Descriptions of different time-scales Excessive part of entropy production in NESS is an invariance Entropy production by fast variables Hidden if the fast variables are not observed K. Kawaguchi et al., PRE 88, 022147 (2013) N. Shiraishi et al., PRE 91, 012130 (2015) Y. Nakayama et al., PRE 91, 012115 (2015) H. Chun et al., PRE 91, 052128 (2015)

7. Simulation work Two coupled two-level systems Observing dynamics of one system FT-like relation preserved M. Borrelli et al., PRE 91, 012145 (2015) Entropy Production with Partial Degrees of Freedom

8. Entropy Production with Partial Degrees of Freedom J. Mehl et al., PRL 108, 220601 (2012) Experimental work Two coupled magnetic particles Observing dynamics of one particle Apparent entropy production FT-like relation preserved

9. Coupled RC Circuits Entropy production evaluated from partial degrees of freedom Equilibrium and nonequilibrium steady-state (NESS) Equation of state Langevin equation Coupling by C C is linear Fluctuation-dissipation theorem (FDT) analysis is applicable

10. Outline Entropy production in electronic circuits Coupled RC circuits Complete description FT hold Reduce descriptions Naïve description Coarse-grained description Conclusions

11. N. Garnier et al., PRE 71, 060101(R) (2005) Johnson-Nyquist noise Thermal Noise in Single RC Circuit Equation of state (Langevin equation)

12. Entropy Production in Single RC Circuit R. Van Zon et al., PRL 92, 130601 (2004) N. Garnier et al., PRE 71, 060101(R) (2005)

13. Steady-State FT RC circuits S. Joubaud et al., EPL 82, 30007 (2008) FT hold

14. Two RC circuits coupled by a capacitor Driven out-of-equilibrium by temperature difference Heat flow occurs Nonequilibrium Coupled RC Circuits S. Ciliberto et al., PRL 110, 180601 (2013)

15. Entropy Production in Coupled RC Circuits S. Ciliberto et al., PRL 110, 180601 (2013) FT holds in all circumstances

16. Coupled RC Circuits - Quantum Version D. Golubev et al., PRB 92, 085412 (2015) Random scattering of photons : transmission probability of photon Quantum version

17. Coupled RC Circuits Driven by Currents Equation of state uncorrelated and white

18. Circuit parameters Measurement of voltage time traces Amplified to 10 4 Sampling rate: 1024 Hz Samples: 5 × 10 5 Currents Coupled RC Circuits Driven by Currents

19. Distributions of Voltages Gaussian Positive correlation between V 1 and V 2 Hint of heat exchange between two sub-circuits Consistent with FDT analysis

20. Energy and Entropy Definitions

21. Equilibrium: Entropy from Complete Description

22. NESS: Entropy distribution is Gaussian Symmetry function is linear Symmetry function slope σ close to 1: FT is satisfied:

23. Focus of Talk Entropy production in reduced descriptions Intentionally ignore V 2 ; observe dynamics of V 1 only Interpret V 1 as measurement from single RC FT holds? Two reduced descriptions Naïve description Coarse-grained description

24. (B) Coarse-grained description Resort to tracing out V 2 in, identical to those of single RC R 2 contributes circuit parameters explicitly (A) Naïve description R 2 : uncorrelated background noise sources Circuit parameters determined by low frequency white noise, average, and variance of V 1 Reduced Description

25. Equilibrium ( I 1 = 0 ): Entropy Production Inferred by Naïve Reduced Description NESS ( I 1 = 108 fA): All symmetry functions are linear Data supported by FDT σ greatly deviates from 1 for large C C FT holds for small C C and large τ

26. NESS ( I 1 = 108 fA): All symmetry functions are linear Data supported by FDT FT holds for small C C and small τ Equilibrium ( I 1 = 0 ): Entropy Production Inferred by Coarse-Grained Reduced Description

27. Weak coupling ( C C = 98.8 pF ) One exponential relaxation time scale in dynamics of V 1 All expectations (full description, reduced description (A) and (B)) from FDT analysis have little difference; data match expectations Nearly Markovian V 2 serves as a thermal background Autocorrelation and Spectral Density

28. Strong coupling ( C C = 9.71 nF ) Two relaxation time scales in dynamics of V 1 revealed in full description and data Reduced white noise; resumed to full magnitude in the 2 nd plateau at low frequency Naïve description (A) captures the data in long time (low frequency) result Coarse-grained description (B) captures the data in short time (high frequency) result

29. FT holds if dynamics of a system is Markovian Dynamics of ( V 1, V 2 ) in full description of coupled RC circuit is Markovian (FT holds) Dynamics of V in authentic single RC circuit is Markovian (FT holds) However, dynamics of V 1 in both reduced descriptions is not Markovian (FT-like hold) and are not enough to determine dynamics of V 1 FT holds in different time regimes in two reduced descriptions Circuit parameters in naïve description (A) inferred from long time behaviors of V 1 FT holds in long time regime (large τ ) Circuit parameters in coarse-grained description (B) inferred from short time behaviors of V 1 FT holds in short time regime (short τ ) Discussions

30. Summary Electric circuits serve as a simplest template to study nonequilibrium thermodynamics in small systems Entropy productions inferred from reduced descriptions fail to obey Fluctuation Theorem (FT) to some extents In weak coupling regime FT holds Future works: experimentally clarify entropy productions in different types of reduced information Entropy production by fast variables Descriptions of different time-scales Y. Nakayama et al., PRE 91, 012115 (2015) H. Chun et al., PRE 91, 052128 (2015)

31. Group: Guan-Hsun Chiang Chia-Wei Chou Department of Physics, National Central University Collaborators: Prof. Chi-Lun Lee Prof. Pik-Yin Lai Department of Physics, National Central University Many Thanks Thank you. Question?

32. Facilities E-beam lithography Line width ~ 100 nm E-beam and thermal evaporation Shadow evaporation with Al oxidation in-situ to form tunnel barrier in single pump down Al, Pb, Cu, Au, Ti, Cr, Pd, … Dilution refrigerator Cryogen-free Base temperature: < 10 mK Cooling power: > 350 µW at 100 mK Large experimental space: 30cm x 30cm x 20cm Flexible wirings 0.1T superconducting magnet

33. Major Apparatus: Dilution Refrigerator System from BlueFors Cryogen free GHScryostat and control panelcompressor for PT

34. Specifications Room T 0.7 K 0.05 K 0.01 K Dilution refrigerator Base temperature: < 10 mK Cooling power: > 350 µW at 100 mK Large space: 30cm x 30cm x 20cm Flexible wirings: heavily-filtered twist pairs coaxial cables magnet leads 0.1T superconducting magnet 8T-3” superconducting magnet 50 K 3 K Two-stage pulse tube cooler Base temperature: 2.8 K Cooling power: 1.0 W at 4.2 K 35 W at 45 K

35. Device images: Cu/Al/Cu box S NN gate CPS in NISIN-SEB Schematic: S N SET N whole circuit gate Manipulating CPS at single-electron level positive charge-transfer correlation in two NIS when CPS exists II

36. Cu/AlO x /Al//AlO x /Cu SEB C.-H. Sun et al., APL 104, 232601 (2014) C N1 = C N2 = 128 aF C S = 4 aF C T2 = C T1 = 50 aF

37. Determination of Stability Diagram C.-H. Sun et al., APL 104, 232601 (2014) Various charge tunneling processes in SEB are indentified

38. Charge Tunneling Processes elastic cotunneling

39. Charge Tunneling Processes elastic cotunneling quasiparticle tunneling or Cooper-pair–electron cotunneling

40. Charge Tunneling Processes elastic cotunneling quasiparticle tunneling

41. Charge Tunneling Processes elastic cotunneling quasiparticle tunneling inelastic cotunneling

42. Charge Tunneling Processes elastic cotunneling quasiparticle tunneling inelastic cotunneling No CPS E c ~ 0.6 meV > Δ ~ 0.2 meV Difficult for S to lose 2e

43. Cotunneling in NISIN-Box C.-H. Sun et al., APL 104, 232601 (2014) Cotunneling rate at degeneracy: 1 Hz Estimated error rate of NSN hybrid SET due to elastic cotunneling: 10 -5

44. Double quantum dots Nonequilibrium Thermodynamics In Electronic Systems B. Küng et al., PRX 2, 011001 (2012)

45. Small System Driven by External Force C. Jarzynski, PRL 78, 2690 (1997) G. E. Crooks, PRE 60, 2721 (1999) Jarzynski equality Crook fluctuation theorem external controlled parameter space i f F R associated fluctuation relations

46. Demonstration of Jarzynski Equality and Crook Fluctuation Relation in SEB O.-P. Saira et al., PRL 109, 180601 (2012)

47. Demonstration of Szilard Engine and Landauer Principle in SEB J. V. Koski et al., PNAS 111,13786 (2014)

48. Fluctuation-Dissipation Theorem in SEB Correlation of charge state spontaneous fluctuation in SEB Relaxation from an non-equilibrium state Fluctuation-dissipation theorem

49. Quantum Version of Szilard Engine S. W. Kim et al., PRL 106, 070401 (2011)