The Keldysh action of a multi-terminal time-dependent scatterer

0.34 MB·English

by  I. Snyman

The Keldysh action of a multi-terminal time-dependent scatterer.
I. Snyman
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
Y.V. Nazarov
Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
(Dated: February 2, 2008)
WepresentaderivationoftheKeldyshactionofageneralmulti-channeltime-dependentscatterer
8
0 in thecontextof theLandauer-Bu¨ttikerapproach. The action is aconvenientbuildingblock in the
0 theory of quantum transport. This action is shown to take a compact form that only involves
2 the scattering matrix and reservoir Green functions. We derive two special cases of the general
result,onevalidwhenreservoirsarecharacterizedbywell-definedfillingfactors,theotherwhenthe
n
scatterer connects two reservoirs. We illustrate its use by considering Full Counting Statistics and
a
theFermi Edge Singularity.
J

5
PACSnumbers: 73.23.-b,73.50.Td,05.40.-a
1

]
I. INTRODUCTION treated non-perturbatively. A related development was
l
l due to Keldysh21. While being a perturbative diagram-
a
h The pioneering work of Landauer1,2 and Bu¨ttiker3,4 matic technique, it allowed for the treatment of general
- systems and shared the idea of a forward and backward
s lay the foundations for what is now known as the scat-
e tering approach to electron transport. The basic tenet time-contour with Feynman and Vernon. Applications
m involving the scattering approach require both the no-
is that a coherent conductor is characterizedby its scat-
tionofthe non-perturbativeinfluence functionalandthe
. tering matrix. More precisely the transmission matrix
at definesasetoftransparenciesforthevariouschannelsor generality of Keldysh’s formalism. Until now, the com-
m bination of the Feynman-Vernon method with the scat-
modesinwhichtheelectronspropagatethroughthecon-
teringapproachwasdoneonancase-specificbasis. Only
- ductor. As a consequence, conductance is the sum over
d those elements relevant to the particular application un-
transmission probabilities. Subsequently, it was discov-
n derconsiderationweredeveloped. Inthis paperweunify
eredthatthesametransmissionprobabilitiesfullydeter-
o
previous developments by deriving general formulas for
mine the current noise, also outside equilibrium, where
c
[ the fluctuation-dissipation theorem does not hold5. the influence functional, or equivalently the Keldysh ac-
Indeed, as the theory of Full Counting Statistics6,7,8 tionofageneralscattererconnectedtochargereservoirs.
1
v later revealed, the complete probability distribution for The Keldysh action of a general scatterer can be con-
3 outcomesofacurrentmeasurementisentirelycharacter- sidered as a building block. Its “interface” is the set of
9 ized by the transmission probabilities of the conductor. fields χ (t). Throughthis interface,the actions of many
±
2 The fact that the scattering formalismgives suchan ele- conductors can be combined into quantum circuits. As
2 gantandcompletedescriptioninspiredsometorevisites- in the case of classical electronics, a simple set of rules,
.
1 tablishedresults. Thusforinstance interactingproblems applied at the nodes of such a circuit, suffice to describe
0 such as the Fermi Edge Singularity9,10 was recast in the the behavior of the whole network23,24.
8 language of the scattering approach11,12,13,14. The scat-
0 teringapproachhasfurtherbeenemployedsuccessfullyin TheinfluencefunctionalZ[χ±]andtheKeldyshaction
v: problemswhereacoherentconductorinteractswithother A[χ±]=lnZ[χ±] depend on two sets of time-dependent
fields χ (t) and χ (t) corresponding to forward and
i elements, including, but not restricted to, measuring de- + −
X backward evolution in time with different Hamiltoni-
vices and an electromagnetic environment15,16,17,18. It
r is also widely applied to study transport in mesoscopic ans. Recently, the situation was considered where χ±
a
are time-independent but the scatterer was allowed to
superconductors19.
fluctuate in time22. We consider the case where also
Many of these more advanced applications are unified
the fields χ are time-dependent. Functional derivatives
through a method developed by Feynman and Vernon ±
withrespecttothesefieldsgeneratecumulantsofthedis-
for characterizing the effect of one quantum system on
tributionofoutcomesforthemeasurementofthedegrees
another when they are coupled20. The workof Feynman
of freedom coupled to χ . Since these fields enter the
and Vernon dealt with the effect of a bath of oscillators ±
Hamiltonian of the scatterer as a time-dependent poten-
coupledto a quantumsystem. It introducedthe concept
tialenergyterm,theireffectiscapturedbythescattering
of a time-contour describing propagation first forwards
matrix. Since the fields differ for forward and backward
then backwards in time. By using the path-integral for-
evolution, the scattering matrices for forward and back-
malism, it was possible to characterize the bath by an
ward evolution differ.
“influence functional” that did not depend on the sys-
tem that the bath was coupled to. This functional was Our main result is summarized by a formula for this
2
Keldysh action. by a check sign) is now retained in the trace, but the
channel structure is traced out. Thus is obtained
1+Gˆ 1 Gˆ
A[sˆ]=Trln" 2 +sˆ −2 #−Trlnsˆ−. (1) [χ±]= 1 Trln 1+Tn GˇL[χ±],GˇR[χ±] −2 .
A 2 n " (cid:8) 4 (cid:9) #
In this formula, Gˆ is the Keldysh Green function char- X (4)
acterizing the reservoirs connected to the scatterer25. In this expression, the field dependence χ± is shifted to
It is to be viewed as an operator with kernel the Keldysh Green functions GˇL and GˇR of the left and
G(α,α′;c;t,t′)δc,c′ where the Keldysh indices α, α′ right reservoirs. This formula makes it explicit that the
+, refer to time-contour ordering, c, c′ Z refer t∈o conductoriscompletelycharacterizedbyitstransmission
{chan−n}el space, and t, t′ R are continuous t∈ime indices. eigenvalues Tn.
∈
The dependence on the fields χ is carried by the time- The plan of the text is as follows. After making the
±
dependent scattering matrix sˆ, that also has Keldysh necessary definitions, we derive Eq. (1) from a model
structure,owingtoforwardandbackwardtime-evolution Hamiltonian. The derivation makes use of contour or-
withdifferentHamiltonians. Explicitly,itistobeviewed dered Green functions and the Keldysh technique. Sub-
asanoperatorwithkernels(α;c,c′;t)δα,α′δ(t t′)where sequently, we derive the special cases of Eq. (2) and Eq.
−
indices carry the same meaning as in the kernel of Gˆ. (4). Whiletheformulas(2)and(4)haveappearedinthe
This formula is completely general. literature before, as far as we know, there has not yet
appeared a formal derivation.
1. It holds for multi-terminal devices with more than We conclude by applying the formulas to several
two reservoirs. generic set-ups, and verify that results agree with the
existingliterature. Particularly,weexplainindetailhow
2. It holds for devices such as Hall bars where parti-
thepresentworkisconnectedtothetheoryofFullCount-
cles in a single chiral channel enter and leave the
ing Statistics and to the scattering theory of the Fermi
conductor at different reservoirs.
Edge Singularity.
3. Itholdswhenreservoirscannotbecharacterizedby
stationaryfilling factors. Reservoirsmay be super-
conducting, or contain “counting fields” coupling II. DERIVATION
them to a dynamical electromagnetic environment
or a measuring device.
G 1 3
When the reservoirs can indeed be characterized by in 2 4 Gin
u
filling factors fˆ(ε), the Keldysh structure can explicitly Gout 21 43 Gout
be traced out to yield
[sˆ ,sˆ ]=Trln sˆ (1 fˆ)+sˆ fˆ Trlnsˆ . (2) 1 1
A + − h − − + i− − Gin 32 u 23 Gout
Inthis expressionoperatorsretainchannelstructureand 4 4
time structure. The time-dependent scattering matrices
z− z+
sˆ have kernels s (c,c′;t)δ(t t′) that depend on the
± ±
−
field χ±(t). In “time” representation, fˆ is the Fourier FIG. 1: We consider a general scatterer connected to reser-
transform to time of the reservoir filling factors, and as voirs. The top figure is a diagram of one possible physical
such has a kernelf(c;t,t′)δc,c′ diagonalin channel space realization of a scatterer. Channels carry electrons towards
anddependingontwotimes. Thisformulaisofthesame and away from a scattering region (shaded dark gray) where
type as the Levitov-Lesovik formula for zero frequency inter-channel scattering takes place. Reservoirs are charac-
Full Counting Statistics (FCS)8, but contains informa- terized by Keldysh Green functions Gin(out). These Green
functions also carry a channel index, in order to account for,
tion about finite frequencies due to the arbitrary time-
among other things, voltage biasing. In setups such as the
dependence of sˆ .
± the Quantum Hall experiment where there is a Hall voltage,
Another formula may be derived from Eq. (1), valid
G will differ from G , while in an ordinary QPC, the two
in out
fortwoterminaldevices. Eachterminalmaystillbecon-
will be identical. The bottom figure shows how the physical
nected to the scatterer by an arbitrary number of chan- setup is represented in our model. Channels are unfolded so
nels. Wedenotethe twoterminalsleft(L)andright(R). that all electrons enter at z− and leave at z+.
In this case the reservoir Green function has the form
Gˆ = GˇL 0 (3) We consider a general scatterer connecting a set of
0 Gˇ charge reservoirs. We allow the scatterer to be time-
(cid:18) R (cid:19)channelspace
dependent. A sufficient theoretical description is pro-
where Gˇ have no further channel space structure. videdbysetoftransportchannelsinterruptedbyapoten-
L(R)
Matrix structure in Keldysh and time indices (indicated tialthatcausesinter-channelscattering. Weconsiderthe
3
regimewherethescatteringmatrixisenergy-independent tially separated. This means that the scattering poten-
inthetransportenergywindow. Sincetransportispurely tial u (z) is non-zero only in a region z− < z < z+
mn
determinedbythescatteringmatrix,allmodelsthatpro- while tunneling between the reservoirs and the conduc-
duce the same scattering matrix give identical results. toronlytakesplace outsidethis region. Note thatinour
Regardless of actual microscopic detail, we may there- model,scatteringchannelshavebeen“unfolded”,sothat
fore conveniently take the Hamiltonian of the scatterer insteadofworkingwithachannelthatconfinesparticles
to be intheinterval( ,0]andallowingforpropagationboth
−∞
in the positive and negative directions, we equivalently
=v dz ψ† (z) iδ ∂ +u (z) ψ (z) work with channels in which particles propagate along
H F m {− m,n z m,n } n ( , ), but only in the positive direction. Hence, to
m,nZ
X −∞ ∞
make contact with most physical setups, we consider z
+ res+ T, (5) −
H H andz toreferto the samephysicalpositionina channel,
where represents the reservoirs, and takes ac- but opposite propagation directions.
res T
H H
count of tunneling between the conductor and the reser-
voirs. The scattering region and the reservoirs are spa- We consider the generating functional
t1 t1
=eA =Tr +exp i dt +(t) ρ −exp i dt −(t) (6)
0
Z T − H T H
(cid:20) (cid:26) Zt0 (cid:27) (cid:26) Zt0 (cid:27)(cid:21)
in which ± is obtained from by replacing u (z) Green function g of the conductor. We state the
mn
with arbitHrary time-dependent fHunctions u± (z,t). In equations of motion that g obeys.
mn
this expressions +exp and −exp respectivelyrefer to
T T 2. We define the Keldysh action = ln , and con-
time-ordered(i.e. largesttime tothe left) andanti-time-
A Z
sider its variation δ . We discover that δ can be
ordered (i.e. largest time to the right) exponentials. In
A A
the language of Feynman end Vernon20 this is known as expressed in terms of g.
the influence functional. It gives a complete character-
3. We therefore determine g inside the scattering re-
ization of the effect that the electrons in the scatterer
gion in terms of the scattering matrix of the con-
have on any quantum system that interact with. Fur-
ductor and its value at the edges of the scattering
thermore, the functional generates expectation values
region, where the reservoirs impose boundary con-
Z
of time-ordered products of operators as follows. Let Q
ditions.
be an operator
4. Thisallowsustoexpressthevariationoftheaction
z+
in terms of the reservoir Green functions G
Q= dzψ† (z)q (z)ψ (z). (7) in(out)
m mn n and the scattering matrix s of the conductor.
mnZz−
X
5. The variation δ is then integrated to find the ac-
Choose u±mn(z,t)=umn(z)+χ±(t)qmn(z). Then tion and theAgenerating functional .
A Z
M N
− Q(t ) + Q(t′)
*T  j T k !+ A. Preliminaries: Definition of the Green function
j=1 k=1
Y Y
M  N
δ δ The first step is to move from the Schr¨odinger picture
= i i [χ] (8)
− δχ−(t ) δχ+(t′) Z |χ=0 tothe Heisenbergpicture. Toshortennotationwedefine
j=1(cid:18) j (cid:19)k=1(cid:18) k (cid:19)
Y Y two time-evolution operators:
By merging the power of the Keldysh formalism of
contour-ordered Green functions with that of the Lan- (t ,t )= +exp i tf dt′ ±(t′) . (9)
± f i
dauer scattering formalism for quantum transport, we U T − H
(cid:26) Zti (cid:27)
obtainanexpressionfor intermsoftheKeldyshGreen
Z Associated with every Schr¨odinger picture operator we
functions in the reservoirs and the time dependent scat-
tering matrices associated with uˆ±(z,t). define two Heisenberg operators, one corresponding to
evolution with each of the two Hamiltonians ±.
The argument will proceed in the following steps:
H
1. Firstly we introduce the key object that enables a Q±(t)= ±(tf,ti)†Q ±(tf,ti). (10)
U U
systematicanalysisof ,namelythesingleparticle
Z
4
In order to have the tools of the Keldysh formalism at our disposal, we need to define four Green functions
g++(z,t;z′,t′) = eATr +(t ,t ) + ψ† (z′,t′)ψ (z,t) ρ −(t ,t ) †
m,n − U 1 0 T n+ m+ 0 U 1 0
g+−(z,t;z′,t′) = eATr h+(t ,t )ψ ((cid:16)z,t)ρ ψ† (z′,t′) −((cid:17)t ,t(cid:0)) † (cid:1) i
m,n U 1 0 m+ 0 n− U 1 0
g−+(z,t;z′,t′) = eATrh +(t ,t )ψ† (z′,t′)ρ ψ (z,t)(cid:0) −(t ,t )(cid:1)†i
m,n U 1 0 n+ 0 m− U 1 0
g−−(z,t;z′,t′) = eATrh +(t ,t )ρ − ψ† (z′,t′)ψ ((cid:0)z,t) −(cid:1)(ti,t ) † . (11)
m,n U 1 0 0T n− m− U 1 0
h (cid:16) (cid:17)(cid:0) (cid:1) i
Herethesymbol +ordersoperatorswithlargertimear- formofthe self-energycanbe derivedfromthe following
T
guments to the left. If permutation is requiredto obtain model for the reservoirs: We imagine every point z in a
the time-ordered form, the product is multiplied with channel m outside (z−,z+) to exchange electrons with
( 1)n where n is the parity of the permutation. Sim- an independent Fermion bath with a constant density of
il−arly, − anti-time-orders with the same permutation states ν. The terms and are explicitly
res T
T H H
parity convention.
The Green functions can be grouped into a matrix in
= dEν dzEa† (E,z)a (E,z)
Keldysh space Hres m Z Zz6∈(z−,z+) m m
X
gm,n(z,t;z′,t′)=(cid:18)ggmm−+++,,nn((zz,,tt;;zz′′,,tt′′)) ggmm−+−−,,nn((zz,,tt;;zz′′,,tt′′)) (cid:19). HT = Xm cmZ dEν Zz6∈(z−,z+)dzψm† (z)am(E,z)
(12) +a† (E,z)ψ (z), (16)
Notation can be further shortened by incorporating m m
channel-indices into the matrix structure of the Green
where the tunneling amplitude c characterizesthe cou-
function, thereby defining anobjectg¯(z,t;z′,t′). The el- m
pling between the reservoir and channel m. More gen-
ementofg¯thatislocatedonrowmandcolumnn,isthe
eral reservoir models need not be considered, since, as
2 2 matrix g .
× m,n we shall see shortly, the effect of the reservoirs is con-
The Green function satisfies the equation of motion
tained entirely in a boundary conditions on the Green
function g¯inside the scatterer. This boundary condition
i∂ +v i∂ v u¯(z,t) g¯(z,t;z′,t′)
{ t F z− F } does not depend on microscopic detail, but only on the
dt′′Σ(z;t t′′)g¯(z,t′′;z′t′)=δ(t t′)δ(z z′)¯1. reservoir Green functions G¯in(out).
− − − − We do notneedto know the explicit formofthe reser-
Z
(13) voirGreenfunctions yet. Rathertheargumentbelowre-
liesexclusivelyonthepropertyofG¯ thatitsquares
in(out)
The delta-functions on the right of Eq. (13) encode the
to unity25:
factthatdue totime-orderingg++ andg−− havea step-
mn mn
structure
dt′′G¯(t t′′) G¯(t′′ t′) =δ(t t′)¯1. (17)
1 z z′ − in(out) − in(out) −
θ(z z′)δ(t t′ − )δ +f(z,t;z′t′) (14) Z
mn
v − − − v
F F A differential equation similar to Eq. (13) holds for g¯†.
where f is continuous in all its arguments. The self-
energy
B. Varying the action A.
G¯ (τ) G¯ (τ)
Σ(z;τ)= i in θ(z− z) i out θ(z z+) (15)
− 2τ − − 2τ −
c c We are now ready to attack the generating functional
results from the reservoirs and determines how the scat- . For our purposes, it is most convenient to consider
Z
teringchannelsarefilled. ItisamatrixinKeldyshspace. = ln . We will call this object the action. Our
A Z
The time τ is the characteristic time correlations sur- strategy is as follows: We will obtain an expression for
c
vive in the region of the conductor that is connected the variation δ resulting from a variation uˆ(z,t)
A →
to the reservoirs, before the reservoirs scramble them. uˆ(z,t)+ δuˆ(z,t) of the scattering potentials. This ex-
G¯ (τ) is the reservoir Green functions where elec- pression will be in terms of the reservoir filling factors fˆ
in(out)
trons enter (leave) the scattering region, summed over and the scattering matrices associated with uˆ(z,t). We
reservoirlevelsandnormalizedtobedimensionless. This then integrate to find .
A
5
We start by writing
t1
δ = iv eA dt dz δu+ (z,t) ψ† (z)ψ (z) (t) δu− (z,t) ψ† (z)ψ (z) (t) (18)
A − F n,m m n + − n,m m n −
Xm,nZt0 Z (cid:16) (cid:10) (cid:11) (cid:10) (cid:11) (cid:17)
where
ψ† (z)ψ (z) (t)
m n +
(cid:10) (cid:11) t1 t t1
= Tr +exp i dt′ +(t′) ψ† (z)ψ (z) +exp i dt′ +(t′) ρ −exp i dt′ −(t′)
T − H m n T − H 0T H
(cid:20) (cid:26) Zt (cid:27) (cid:26) Zt0 (cid:27) (cid:26) Zt0 (cid:27)(cid:21)
ψ† (z)ψ (z) (t)
m n −
(cid:10) (cid:11) t1 t t1
= Tr +exp i dt′ +(t′) ρ −exp i dt′ −(t′) ψ† (z)ψ (z) −exp i dt′ −(t′) .
T − H 0T H m n T − H
(cid:20) (cid:26) Zt0 (cid:27) (cid:26) Zt0 (cid:27) (cid:26) Zt (cid:27)(cid:21)
(19)
C. Expressing δA in terms of the Green function g. g¯(z,t 0+;z′,t′) for points z andz′ inside the scattering
region−whereu¯ is non-zero,to the value ofg¯atz− where
In terms of the defined Green functions, the variation electrons enter the scatterer. For z z′ and t t′, the
≤ ≤
δ becomes equations of motion give
A
t1 z z− z′ z−
δA = ivF dt dz δu+n,m(z,t) g¯(z,t+ −v −0;z′,t+ −v )
Xm,nZt0 g++Z(z,t (cid:16) 0+;z,t) =s¯(z,t)g¯(zF−,t 0+;z′−,t′)s¯†F(z′,t′), (22)
× m,n − −
+δu− (z,t)g−−(z,t+0+;z,t) where
n,m m,n
= iv t1dt dz Tr δu¯(z,t)g¯(z,t+0k;(cid:17)z,t) . s¯(z,t)= exp i z dz′′u¯(z′′,t+ z′′−z−) . (23)
F Zt0 Z Z (cid:26)− Zz− vF (cid:27)
(cid:2) (cid:3)(20) The symbol indicates that the exponent is ordered
Z
alongthe z-axis,with the largestco-ordinatein the inte-
The object δu¯ is constructed by combining the chan-
grandto the left. Note thatthe potentialu¯ atpositionz
nel and Keldysh indices of the variation of the poten-
is evaluated at the time instant t+(z z−)/v that an
tial. The trace is over both Keldysh and channel in- − F
electron entering the scattering region at time t reaches
dices. The symbol 0k refers to the regularization ex-
z. Often the time-dependence of the potential is slow
plicitly indicated in the first line, i.e. the first time ar-
on the time-scale (z+ z−)/v representing the time a
gument of g++(z,t 0+;z,t) is evaluated an infinitesi- − F
− transported electron spends in the scattering region and
mal time 0+ > 0 before the second argument, while in
g−−(z,t+0−;z,t), the first time argument is evaluated u¯(z,t+ z−vFz−) can be replaced with u¯(z,t). This is how-
an infinitesimal time 0+ after the second. This is done evernotrequiredfortheanalysisthatfollowstobevalid.
Substitution into Eq. (24) yields
sothatthe timeordering(anti-timeordering)operations
give the order of creation and annihilation operators re-
quired in Eq. (18). δA=v dtTr w¯(t)g(z−,t 0+;z−,t)
F
−
Itprovesveryinconvenienttodealwiththe0k regular- Z
(cid:2) 1(cid:3) τˇ
izationof Eq. (20). It is preferable to have the firsttime dt lim δ(t t′)Tr w¯(t)ˆ1 − 3 . (24)
arguments of both g++ and g−− evaluated an infinitesi- − t′→t − 2
Z (cid:20) (cid:18) (cid:19)(cid:21)
mal time 0+ before the second. Taking into account the
step-structure of gˆ++ we have with
g¯(z,t+0k;z′t′)=g¯(z,t 0+;z′,t′) z+ z z−
− w¯(t) = i dzs¯†(z,t)δu¯(z,t+ − )s¯(z,t)
+ v1 δ(t−t′− zv−z′)ˆ1 1−2τˇ3 . (21) = −s¯†(tZ)zδ−s¯(t). vF (25)
F F (cid:18) (cid:19)
1 0 In this equation z+ is located where electrons leave the
Here τˇ is the third Pauli matrix acting in
3 0 1 scatterer. Importantly, here Tr still denotes a trace over
(cid:18) − (cid:19)
Keldyshspace. Theequationsofmotionallowustorelate channel and Keldysh indices. We will later on redefine
6
the symbol to include also a trace over the (continuous) There is no inhomogeneous term on the right-hand side,
time index, at which point the second term in Eq. (24) because we restrict z to be less than z−. We thus find
will(perhapsdeceptively)looklessoffensive,butnotyet.
In the last line of Eq. (25), s¯(t) = s¯(z+,t) is the (time-
dependent) scattering matrix. We sent the boundaries g¯(z− 0+,ε;z−,ε′)
t and t overwhich we integrate in the definition of the −
0 1 G¯ (ε)
action,to and respectively,whichwillallowusto =eiε∆z/vF exp in ∆z g¯(z− ∆z,ε;z−,ε′).
−∞ ∞ − 2l −
Fourier transformto frequency in a moment. The action (cid:20) c (cid:21)
remainswell-definedaslongasthe potentialsu+ andu− (28)
only differ for a finite time.
Here the correlation length l is the correlation time τ
c c
D. Relating g inside the scattering region to g at multiplied by the Fermi velocity vF. Using the fact that
reservoirs. Imposing boundary conditions implied G¯(ε) squares to unity, it is easy to verify that
in
by reservoirs.
Ourtask isnow tofind g¯(z−,t 0+;z−,t). Becauseof G¯ (ε)
the t t′ dependence of the self-−energy, it is convenient exp in ∆z
− 2l
to tra−nsform to Fourier space, where (cid:26) c (cid:27)
1+G¯ (ε) ∆z 1 G¯ (ε) ∆z
in in
= exp + − exp .
g¯(z,ε;z−,ε′) = dtdt′eiεtg¯(z,t;z−,t′)e−iε′t′ 2 (cid:18)−2lc(cid:19) 2 (cid:18)2lc(cid:19)
(29)
Z
G¯ (ε) = dteiεtG¯(t) . (26)
in(out) in(out)
Z Since spacial correlations decay beyond z−, g¯(z−
In frequency domain, the property that G¯ squares ∆z,ε;z−,ε′) does not blow up as we make ∆z large−r.
in(out)
to unity is expressed as G¯ (ε)2 = ¯1. (Due to the From this we derive the condition
in(out)
standard conventions for Fourier transforms, the matrix
elements of the identity operator in energy domain is
2πδ(ε ε′).) The equation of motion for z <z− reads 1+G¯ (ε) g¯(z− 0+,ε;z−,ε′)=0. (30)
in
− −
G¯ (ε) (cid:2) (cid:3)
iε+v ∂ + in g¯(z,ε;z−,ε′)=0. (27)
F z
− 2τ
(cid:26) c (cid:27)
Transformed back to the time-domain this reads
dt′′ δ(t t′′)+G¯ (t t′′) g¯(z− 0+,t′′;z−,t′)=0. (31)
in
− − −
Z
(cid:2) (cid:3)
We can play the same game at z+ where particles leave the scatterer. The equation of motion reads
G¯ (ε)
iε+v ∂ +θ(z z+) out g¯(z,ε;z+,ε′)=2πδ(z z′)δ(ε ε′). (32)
F z
− − 2τ − −
(cid:26) c (cid:27)
This has the general solution
z z′ G¯ (ε)
g¯(z,ε;z′,ε′) = exp iε − (z z+)θ(z z+) (z′ z+)θ(z′ z+) out
v − − − − − − 2l
(cid:26) F c (cid:27)
(cid:2) 2π (cid:3)
g¯(z′ 0+,ε′;z′,ε′)+ θ(z z′)δ(ε ε′) . (33)
× − v − −
(cid:20) F (cid:21)
We will need to relate the Green function evaluated at z <z+ to the Green function evaluated at z >z+, and so we
explicitly show the inhomogeneous term. The same kind of argument employed at z− then yields the condition
2π
1 G¯ (ε) g¯(z+ 0+,ε;z+,ε′)+ δ(ε ε′) =0, (34)
out
− − v −
(cid:20) F (cid:21)
(cid:2) (cid:3)
where the inhomogeneous term in the equation of motion is responsible for the delta-function. In time-domain this
reads
1
dt′′ δ(t t′′) G¯ (t t′′) g¯(z+ 0+,t′′;z+,t′)+ δ(t′′ t′) =0. (35)
out
− − − − v −
Z (cid:20) F (cid:21)
(cid:2) (cid:3)
7
It remains for us to relate g¯(z+ 0+,t+ z+−z−;z+,t′+ z+−z−) to g¯(z− 0+,t;z−,t′). This is done with the help
− vF vF −
of Eq. (22), from which follows
z+ z− z+ z−
g¯(z+ 0+,t+ − ;z+,t′+ − )=s¯(t)g¯(z− 0+,t;z−,t′)s¯†(t′). (36)
− v v −
F F
We substitute this into Eq. (35), multiply from the right with s¯(t′) and from the left with s¯†(t). If we define
G¯′ (t,t′)=s¯†(t)G¯ (t t′)s¯(t′) the resulting boundary condition is
out out −
1
dt′′ δ(t t′′) G¯′ (t t′′) g¯(z− 0+,t′′;z−,t′)+ δ(t′′ t′) =0. (37)
− − out − − v −
Z (cid:20) F (cid:21)
(cid:2) (cid:3)
E. Finding the variation of the action in terms of where the trace is over time, channel and, in the first
the reservoir Green functions and the scattering term, Keldysh indices. The operator s′ is related to the
matrix. scattering matrix s through s′ =Λs.
At this point, it is convenientto incorporate time into
the matrix-structureofthe objectsG¯ ,G¯′ andg¯. The F. Integrating the variation to find the action A.
in out
resultingmatriceswillbewrittenwithoutoverbars. Thus
for instance s will denote a matrix diagonal in time- We now have to integrate δ to find . This is most
indices, whose entry (t,t′) is δ(t t′)s¯(t). Similarly the convenientlydone byworkingAinabasiswAhereGisdiag-
(t,t′) entry of G is G¯ −(t t′). Also let g− be onal. Since G2 = 1, every eigenvalue of G is 1. There-
in(out) in(out)
− ±
the matrix whose (t,t′) entry is g¯(z− 0+,t;z−,t′). In fore, there is a basis in which
−
this notation G2 = G′ 2 = I and Eq. (31) and Eq.
in out I 0
(37) read G= . (43)
0 I
(I+G )g− = 0 (cid:18) − (cid:19)
in In this representation s′ can be written as
(I G′ ) g−+1/v = 0. (38)
− out F
s′ s′
These two equations de(cid:0)termine g−(cid:1)uniquely as follows: s′ = 11 12 . (44)
s′ s′
From the first of the two equations we have (cid:18) 21 22 (cid:19)
0 = G′ (I +G )g− Here the two indices of the subscript has the following
out in
meaning: The first refers to a left eigenspace of G, the
= (I G′ )g−+(I+G′ G )g− (39)
− − out out in second to a right eigenspace. A subscript 1 denotes the
In the first term we can make the substitution (I subspace of eigenstates of G with eigenvalue 1. A sub-
G′ )g− = (I G′ )/v which follows from Eq.−(38−). script 2 refers to the subspace of eigenstates of G with
out − out F
Thus we find eigenvalue 1. In this representation,
−
1 1
g− = (I G′ ) 1 0 0
−vF I +G′outGin − out (1−G)Gs′+s′G = 0 (s′ )−1 , (45)
1 1 (cid:18) 22 (cid:19)
= (1 G ) (40)
v − in G′ +G so that
F out in
andthelastlinefollowsfromthefactthatG2in = G′out2 = δA=Tr δs′22 (s−221)′ −Tr δsˆ−(sˆ−)† , (46)
I. We have taken special care here to allow for different
and thus (cid:2) (cid:3) (cid:2) (cid:3)
reservoirGreenfunctions atz− whereparticlesenterthe
conductor and z+ where they leave the conductor. In = Trlns′ Trlns
A 22− −
order to proceed we must now absorb the difference be- eA = (Dets )−1Dets′ (47)
tween the two Green functions in the scattering matrix. − 22
We define Λ through the equation Intheseequations,s isthescatteringmatrixassociated
−
G¯out =Λ−1GinΛ (41) wbeithinHclu−deads dinefitnheedoppreervaiotiuosnlys.oIftstatkimineg stthreucttruarceeisantdo
and drop subscripts on the Green functions by setting determinant.
G G . SubstitutedbackintoEq. (24)forthevariation NotethatintherepresentationwhereGisdiagonal,it
in
≡
of the action yields holds that
1 1+G 1 G I s′
δ =Tr δs′(1 G) Tr δsˆ (sˆ )† , (42) +s′ − = 12 . (48)
A − Gs′+s′G − − − 2 2 0 s′
(cid:20) (cid:21) (cid:18) 22 (cid:19)
(cid:2) (cid:3)
8
Due to the upper-(block)-triangular structure it holds will also assume that electrons enter and leavea channel
that Det s′ = Det 1+G +s′1−G leading to our main from the same reservoir, so that G = G and hence
22 2 2 in out
result s′ = s. We recall as well as that the Keldysh structure
(cid:2) (cid:3)
of the scattering matrix is
1+G 1 G
=Trln +s′ − Trlns . (49)
−
A 2 2 −
(cid:20) (cid:21) sˆ 0
s= + . (52)
where it has to be noted that many matrices have the 0 sˆ−
(cid:18) (cid:19)
same determinant as the above. Some obvious examples
include
Here sˆ have channel and time (or equivalently energy)
±
indices. sˆ is diagonal in time-indices, with the entries
I 0 ±
0 s′ = (1+G)/2+(1−G)s′(1−G)/4 onthe time- diagonalthe time-dependentscatteringma-
(cid:18) 22 (cid:19) trices corresponding to evolution with the Hamiltonians
I 0 = (1+G)/2+(1 G)s′/2. (50) ±.
s′ s′ − H
(cid:18) 21 22 (cid:19) With this structure in Keldysh space, we find
III. TRACING OUT THE KELDYSH 1+(sˆ 1)fˆ (sˆ 1)fˆ
eA =Det +− − +−
STRUCTURE (sˆ 1)(fˆ 1) sˆ (1 fˆ)+fˆ
(cid:18) −− − − − (cid:19)
1
Up to this point the only property of G that we relied ×Det sˆ−1 . (53)
on was the fact that it squaresto identity. Hence the re- (cid:18) − (cid:19)
sult(Eq. 49)holdsinasettingthatismoregeneralthan
that of a scatterer connected to reservoirs characterized We can remove the Keldysh structure from the determi-
by filling factors. (The reservoirs may for instance be nant with the aid of the general formula
superconducting). In the specific case of reservoirschar-
acterized by filling factors it holds that
A B
Det = Det(AD ACA−1B)
C D −
dε 1 2fˆ(ε) 2fˆ(ε) (cid:18) (cid:19)
G¯(τ)= 2πe−iετ 2−2fˆ(ε) 1+2fˆ(ε) . (51) = Det(DA CA−1BA). (54)
Z (cid:18) − − (cid:19) −
Herefˆ(ǫ)isdiagonalinchannelindices,andf (ǫ)isthe Noting that in our case the matrices B and A commute,
m
fillingfactorinthereservoirconnectedtochannelm. We so that CA−1BA=CB, we have
eA = Det sˆ (1 fˆ)+fˆ 1+(sˆ 1)fˆ sˆ (1 fˆ)+fˆ 1 (sˆ 1)fˆ Det sˆ−1
− − − − − − − +− −
= Deth(cid:16)sˆ (1 fˆ)+sˆ f(cid:17)ˆ(cid:16)Det sˆ−1 . (cid:17) (cid:16) (cid:17) i (cid:0) (cid:1) (55)
− − + −
h i (cid:0) (cid:1)
IV. AN EXAMPLE: FULL COUNTING As a consistency check of our results, we apply our
STATISTICS OF TRANSPORTED CHARGE. analysis to re-derive this formula. We will consider the
most general setup, where every scattering channel is
Adeterminantformulaofthistypeappearsintheliter- connected to a distinct voltage-biased terminal. To ad-
ature of Full Counting Statistics8 of transported charge. dress the situation where leads connect several channels
This formula can be stated as follows: the generat- to the same terminal, the voltages and “counting fields”
ing function for transported charge through a conductor associatedwith channels in the same lead, are set equal.
characterized by a scattering matrix sˆis Let us start by defining number operators
Z[χ]=Det 1+(sˆ†−χsˆχ−1)fˆ (56) Nk = dz [θ(−z0−z)+θ(z−z0)]ψk†(z)ψk(z) (57)
Z
h i
where sˆ is a scattering matrix, modified to depend The operator countsthe number ofparticlesinchan-
χ k
N
on the counting field χ that, in this case, is time- nel k that are located outside the interval [ z ,z ]. The
0 0
−
independent. (The precise definition may be found be- coordinate z is chosen to lie between the reservoirs the
0
low.) scattering region. With the full counting statistics of
9
transported charge, we mean the generating functional V. TRACING OUT THE CHANNEL
(a la Levitov) STRUCTURE.
(χ ,...,χ ,t)
1 N
Z A largeclass ofexperiments anddevices in the field of
= eiPkχkNk(0)/2e−iPkχkNk(t)eiPkχkNk(0)/2 (58) quantum transport is based on two terminal setups. In
such a setup the channel space of the scatterer is natu-
D E
The function (χ,t) generates all moments of the joint
rallypartitionedinto a leftandrightset,eachconnected
Z
distribution function for (n ,n ,...,n ) charges to be
1 2 N toitsownreservoir. Wearegenerallyinterestedintrans-
transported into terminal (1,2,...,N) in the time inter-
port between left and right as opposed to internal dy-
val (0,t). Formally t is sent to infinity, and this causes a
namics on the left- or right-hand sides. The scattering
singularityinallirreducablemomentsofthedistribution
matrices have the general structure
of thransported charge. Dealing with this singularity is
a subtle issue, which we will not concern ourselves with. r t′ X±
The interested reader is referred to the literature26,27. sˆ± =X t r′ X−1, X = L X± . (64)
The dynamics of the the operators (t) are deter- (cid:18) (cid:19) (cid:18) R (cid:19)
k
N
mined by the Hamiltonian Here r(r′) describes left (right) to left (right) reflec-
tion,whilet(t′)describesleft(right)toright(left)trans-
=v dz ψ† (z) i∂ δ +u (z) ψ (z).
H f m {− z m,n m,n } n mission (t is not to be confused with time). These ma-
Xm,nZ trices have no time or Keldysh structure but still have
(59)
matrix structure in the space of left or right channel in-
Asalreadymentioned,wechoosetocountchargesoutside
dices. The operators X±(τ) and X±(τ) have diagonal
the scattering potential region, so that every com- L R
Nk Keldysh structure (denoted by the superscript ) and
mutes with the potential energy term in the Hamilto- ±
diagonaltimestructure(hereindicatedbyτ toavoidcon-
nian. Using the short-hand notation χ =χ. ,
m mNm N fusionwiththetransmissionmatrixt). Theydonothave
we manipulate the definition of (χ,t) to find
Z P internalchannelstructureandasaresulttheKeldyshac-
(χ,t) = eiχ.N/2eiHte−iχ.Ne−iHte−iχ.N/2 tionisinsensitivetothe internaldynamicsontheleft-or
Z right-hand sides. Our shorthand for the Keldysh scat-
= DeiHχte−iH−χt . E (60) tering matrix will be XsX−1 where we remember that s
has no Keldysh structure.
In this equation,(cid:10)the Hamilton(cid:11)ian χ is defined as
H We now consider the square of the generating func-
=eiχ.N/2 e−iχ.N/2 tional and employ the first expressionwe obtained for
χ
H H Z
it (Eq. 49) which retainsKeldysh structure in the deter-
=vf dz ψm† (z) −i∂zδm,n+u(mχ,)n(z) ψn(z) minant.
Xm,nZ n o 2
(61) 2 =Det 1+G +XsX−11−G Dets† (65)
Z 2 2
The transformed potential is u(χ) (z) = u (z) + (cid:20) (cid:21)
m,n m,n
δm,nχ2m(δ(z−z0)−δ(z+z0)). One way to verify this is Hereweexploitedthefactthatsˆ−actsonhalfofKeldysh
tonotethatformallyHχ isrelatedtoHthroughagauge space together with the fact that sˆ+ = sˆ−, i.e. s has no
transformation where the gauge field in each channel is Keldysh structure, to write exp2Tr lnsˆ = Dets. We
−
proportionaltoθ(z z0)+θ( z z0). Thedelta-functions now shift X to act on G and define
in u(χ) arise as a gr−adient o−f th−e gauge field that appear
in the transformation of the kinetic term in H. Gˇ =X−1GX P = 1+Gˇ Q= 1−Gˇ (66)
The calculation of the full counting statistics has now 2 2
been cast into the form of the trace of a density matrix
The operators P and Q are complementary projection
after forwardand backwardtime evolutioncontrolledby
operators i.e. P2 = P, Q2 = Q, PQ = QP = 0 and
different scattering potentials. Our result, Eq. (55), is
P+Q=I. Becauseofthis, it holdsthat Det(P +sQ)=
therefore applicable, with
Det(P +Qs). Thus we find
z+
sˆ± = Zexp −iZz− dz uˆ(±χ)(z)! Z2 =e2A =Det(Ps†+sQ) (67)
= e∓iχˆ/2s0e±iχˆ/2 =s±χ. (62) Theleftchannelsareallconnectedtoasinglereservoir
while the right channels are all connected to a different
Inthisequation,χˆisadiagonalmatrixinchannelspace,
reservoir. This means that the reservoir Green function
with entries δ χ . Substitution into Eq. (55) gives
m,n m
has channel space structure
(χ)=Det 1+(sˆ† sˆ 1)fˆ , (63)
Z −χ χ− Gˇ
in agreement with the exhisting literature8.i Gˇ = L Gˇ (68)
(cid:18) R (cid:19)
10
where G and G have no further channel space struc- The projection operator Q has the same structure. The
L R
ture. At this point it is worth explicitly stating the scattering matrix s has the structure
structure of operators carefully. In general, an opera-
tor carries Keldysh indices, indices corresponding to left
achnadnrnieglhst,,acnhdantinmeleininddiciceess.wiHthoiwnetvheer Plef,tQorarnidghstaseretsdoi-f s(k,k′;α,α′;c,c′;t,t′)=s(α,α′;c,c′)δk,k′δ(t−t′). (70)
agonalorevenstructureless,i.e. proportionaltoidentity
in some of these indices. Let us denote Keldysh indices
with k,k′ +, , left and right with α,α′ L,R ,
channelind∈ic{esw−it}hinthe leftorrightsetswith∈c{,c′ }Z We now use the formula Det A B =
and time t,t′ R. Then P has the explicit form ∈ C D
∈ Det(A)Det(D CA−1B)toeliminateleft(cid:18)-rightst(cid:19)ructure
P(k,k′;α,α′;c,c′;t,t′)=P(k,k′;α;t,t′)δα,α′δc,c′. (69) from the deter−minant.
P r†+Q r P t†+Q t′
2 = L L L R
Z P t′†+Q t P r′†+Q r′
(cid:18) R L R R (cid:19)
= Det P r†+Q r Det P r′†+Q r′ P t′†+Q t P r†−1+Q r−1 P t†+Q t′
L L R R R L L L L r
−
= Det((cid:0)P r†+Q r)(cid:1) h (cid:16) (cid:17)(cid:16) (cid:17)(cid:0) (cid:1)i
L L
a
| Det P{zR(r′† }PL t′† r†−1t†)+(r′ QLtr−1t′)QR PR(PL t′† r†−1t′+QL t′†r−1t′)QR (71)
× − − −
h i
b
| {z }
Here it is important to recognize that the reflection and P and Q are diagonal to find
L L
transmissionmatricescommutewiththeprojectionoper-
u′†√1 Tu† 0
ators PL,R and QL,R. Furthermore, notice that, in term a = Det 0− u√1 Tu′
b, the projection operator PR always appears on the left (cid:18) − (cid:19)
of any product involving other projectors, while QR al- = Det I√1 T , (75)
ways appears on the right. This means that in the basis −
where Where I =P +Q(cid:16) =P +(cid:17)Q is the identity operator
L L R L
I(k,k′;c,c′;t,t′) = δk,k′δc,c′δ(t t′) in Keldysh, channel
I 0 0 0 −
P = Q = (72) and time indices. For term b we find
R 0 0 R 0 I
(cid:18) (cid:19) (cid:18) (cid:19) T
b=Det P √1 T +P
R L
− √1 T
termbisthedeterminantofanupperblock-diagonalma- h (cid:18) − (cid:19)
trix. As such, it only depends on the diagonalblocks, so T
+ √1 T +Q Q (76)
that the term PR(...)QR may be omitted. Hence − L√1 T R
(cid:18) − (cid:19) i
Combining the expressions for a and b we find
b=Det P (r′† P t′† r†−1t†)
R − L 2 =e2A =Det[1 T(P Q +P Q )]. (77)
R L L R
h Z −
+(r′ QLtr−1t′)QR . (73) Using the fact that P =(1+Gˇ )/2 and Q =
− L(R) L(R) L(R)
i (1 Gˇ )/2andtakingthelogarithmwefinally obtain
L(R)
−
Nowweinvoketheso-calledpolardecompositionofthe the remarkable result
scattering matrix28
1 T
= Trln 1+ n Gˇ ,Gˇ 2 (78)
L R
A 2 4 −
r =u√1 Tu′ t′ =iu√Tv n (cid:20) (cid:21)
− (74) X (cid:0)(cid:8) (cid:9) (cid:1)
t=iv′√Tu′ r′ =v′√1 Tv This formula was used in [15] to study the effects on
−
transport of electromagnetic interactions among elec-
where u, u′, v and v′ are unitary matrices and T is a trons. In [17] the same formula was employed to study
diagonal matrix with the transmission probabilities T theoutputofatwo-levelmeasuringdevicecoupledtothe
n
on the diagonal. We evaluate term a in the basis where radiation emitted by a QPC.