メゾスコピック系における電流ゆらぎの普遍性

慶應大学 物理学科 齊藤圭司

I  Brief  survey  on  recent  progress  in  statistical  physics
Equilibrium, near equilibrium, far-from-equilibrium
Jarzynski eq
Fluctuation theorem

Equilibrium,  near  equilibrium,  far  from  equilibrium

V +VV
left  lead

Vlarge	small	
0right  leadfar  from  equilibriumnear  equilibrium
equilibrium

Jarzynski  equality
Fluctuation  theorem  (FT)
Macroscopic  FT etc.

Linear  response  theory
Onsager-Machlup  Theory
Onsager  relation
Fluctuation  dissipation  theoryBrownian  motion
Statistical  mechanics
Themodynamics Second  lawJarzynski  equality

Wee

W

=e

=dx(0)dp(0)e=edx(0)dp(0)Jarzynski PRL (1996)

F[H (x( ),p( )) H0 (x(0),p(0))] e

H0 (x(0),p(0))

Z0H (x( ),p( ))Z0=dx( )dp( )eH (x( ),p( ))Z0

Liouville  theorem
this  implies♦  relation  between  work  and  equilibrium  free  energy♦  detailed  description  of  the  second  law

WF♦  (  this  can  provide  experimental  protocol  to  get  free  energy  )  0=eF

Experimental  demonstration  of  Jarzynski  equality♢ RNA stretching experiment 	

J. Liphardt et al., Science (2002)♢ Roughly this is equivalent to	RNA	♢ measuring force asa function of stretch	♢ extracting
equilibrium free energy
from Jarzynski equality

F =1ln e

W

Reversible
Irreversible
green,
34pN/sRed,
52 pN/sFluctuation  theorem  (FT)P ( S) = P (Evans et al. (1993)
Gallavotti and Cohen (1996)
Lebowitz and Spohn (1999)S)e SP ( S) :  distribution  of  entropy  produced  during  finite  time  this  implies♦relation  between  positive  and  negative  entropy  production♦  detailed  description  of  the  second  law
♦  it  reproduces  Onsager  (-Casimir)  relationS0

Lij = Lji

♦  it  reproduces  linear  response  theory,  fluctuation  dissipation  theory

Entropy  difference  ?
♢system  +  bath  without  current  flow
system	S=♢

heat bath	ln final + ln initial

Qheat

system  +  bath  with  current  flow
particle bath	
system	S=µln final + ln initial( QenergyµQparticle )Example of experiment on FT in classical heat transport	♢experimental  setup

Gomez-Solano, etal., EPL (2010)	

f (t)

TL

TL

External  field  satisfying

TRln P ( S)/P (S)f (t)f (0) = 2 kB TR (t)

S=S

1
TL

1
TRQheatSteady  state  entropy  production  for  electric  conduction

S=♢  entropy  produced  :  Jule  heating

VQ

Q=V +VVdtI(t)0I(t) electric current	

♢  this  expression  and  FT  indicate  that  distribution  of  Q  has  universal  relation    P ( Q) = eVQP (Q)II  Fluctuation  in  mesoscopic  electron  conduction  current noise in mesoscopic transport
counting statistics
universal relation between transport coefficients
experiments

Current noise in mesoscopic transport
if  the  conductor  is  macroscopic、：measurement

Vfluctuation  is  very  small  compared  to  average  current

I(t)t

Small  scale  (e.g.,  mesoscopic  scale)  of  conductor  gives  ♢large  fluctuation  in  current♢making  histogram  is  possible-q+qElecric  transport  has  very  old  history  on  current  fluctuation        i)  Zero  temperature (Schottky    (1918))  Current  noise  S

Average  :    I

V♢  Noise  tells  us  important  information:      elementary  charge

averagenoiseS
F ==e
2I

“e” is obtained‘The noise is the signal (Landauer)’

SF ==e2I(Review, Blanter and Buttiker(1999))

Fractional  quantum  hole  efect  has  1/3  elementary  charge  and  it  was  experimentally  proved  by  noise  measurement.  
This  leads  to  Nobel  prize  !

eSF ==2I3ii)  finite  temperatures♢  expansion  of  average  current  and  voltage  w.r.t.  voltage  ♢  relashionship  Johnson-Nyquist  relation  (FDT)♢Only  this  ??    Counting  statistics

Left electrode

Right electrode♢ measure Q =dtI(t)0♢ then, P (Q) ？
♢ equivalently, cumulant generating function ?

F ( ) := lndQP (Q)e1
P (Q) =
2d e i Q+Fi QnQc =F( )(i )nn=0(Standard)  Theoretical  protocol  to  get  distribution  H = HS + HL + HR + Hinti) switch off int. & projective measurement onto the number state	P̂i = | ii|

i =LiSiRi

suppose that number of electrons in the left lead 	iteration	ii) switch on int. & free time evolution during τ	iii) switch off int. & projective measurement onto the number state	
j = LjSjRj
P̂j = | jj|suppose that number of electrons in the left lead 	electrons are transmitted 	Pj

i = (Q(ninj ))|2|U||ji♢  take  decoupled  initial  condition  0 =e

(HL µL NL )e

(HR µR NR )S♢  P(Q)  is  expressed  by  the  operator  form

P (Q) =Pji P0,i(ninj ))|P0,i =i| 0| ii,j=

(Qi,j1=
2j |U| i |2i| 0| i

d e i Q Tr U ei NL 0 U † e i NL

♢  cumulant  generating  function

F ( ) :=

1log Tr U ei NL 0 U † e i NL

♢ remark1 : this formulation uses two-times projective measurement

Specific  case:  non-interacting  spinless  electron  :Transmission coefficient of electron with frequency ω:Fermi distribution of Lead α=L or R♢ this reproduces an average current and current noise	the first derivative reproduces average current expression	

cf. Landauer formula	♢ symmetry and fluctuation theorem	

Symmetry for generating function with magnetic field♢ Coulomb Interaction♢ Magnetic Fields	

Measure 1

F ( ; B) = F (B+ iA; B): thermodynamic force (Affinity)
Quantum version of fluctuation theorem with magnetic fields	
Schwinger Keldysh Calculation + Time-reversal arguments

KS, Abhishek Dhar, PRL Vol.99, 180601 (2007)
KS, Yasuhiro Utsumi, PRB vol. 78, 115429(2008)Universal relation between transport coefficientsmeasureB

In general, charge transfer is a nonlinear function ofDefinition of transport coefficientSymmetry reproduces Linear response resultsF ( ; B) = F (+ iA; B)

nnn
F(;B)
=(1)
F ( ; B)
nn
(i )
(i )

A

F ( ; B) =Ai) Johnson-Nyquist Relation	ii) Onsager-Casimir relation	

(i )F ( ; B)

Relationships beyond linear response regime

Symmetrize the coefficients

Universal relations from FT	In more understandable wayBV

Symmetrized  conductanceS,A

G= G(B) ± G( B)S S,A = S(B) ± S( B)

KS  and  A.  Dhar,    PRL  (2007)
KS  and  Y.  Utsumi  PRB  (2008)
KS  and  A.  Dhar,  PRL(2010) 	

SAverage  current ICurrent  fluctuation

Experimental approach♢ classsical case♢ quantum caseExperimental approach in classical regime
T. Fujisawa et al., Scince (2006)♢ coupled quantum-dots	-q+q

♢ state-dependent I_QPC	BHME

Experimental approach in quantum regime
Kobayashi group PRL (2010)	

♢Aharonov-Bhom Interferometer	

Interference of current(evidence of quantum)	

Backgroud
Gate voltage	

Magnetic Field	

♢ full-distribution is difficult to get even in the present-day experiments.
instead, current noise is possible to measure	Results of Experiments	
Check of Johnson-Nyquist Relation (Kubo Formula)	

Results on New elations	Experiment versus theory	まとめと展望♢量子測定に基づく理論の構築と実験との比較を行った。

-recent  studies  ♢ 量子観測の効果♢ 測定による反跳効果♢ 測定の精度限界による効果

Demonstration - 2 terminal interacting Aharonov-Bohm interferometer

Microcopic -Reversibility
preserving Hartree-Levelcalculation