Persistent current noise and electron-electron interactions

0.63 MB·English

by  Andrew G. Semenov

Persistent current noise and electron-electron interactions
Andrew G. Semenov1 and Andrei D. Zaikin2,1
1I.E.Tamm Department of Theoretical Physics, P.N.Lebedev Physics Institute, 119991 Moscow, Russia
2Institut fu¨r Nanotechnologie, Karlsruher Institut fu¨r Technologie (KIT), 76021 Karlsruhe, Germany
We analyze fluctuations of persistent current (PC) produced by a charged quantum particle
moving in a ring and interacting with a dissipative environment formed by diffusive electron gas.
We demonstrate that in the presence of interactions such PC fluctuations persist down to zero
temperature. In the case of weak interactions and/or sufficiently small values of the ring radius
1
1 R PC noise remains coherent and can be tuned by external magnetic flux Φx piercing the ring.
0 In the opposite limit of strong interactions and/or large values of R fluctuations in the electronic
2 bath strongly suppress quantum coherence of the particle down to T = 0 and induce incoherent
Φx-independent current noise in the ring which persists even at Φx = 0 when the average PC is
n
absent.
a
J

6 I. INTRODUCTION such systems. The main goal of this paper is to the-
2 oretically analyze such current noise in the presence of

quantum decoherence produced by electron-electron in-
] Persistent currents (PC) in normal meso- and nanor-
l teractions.
l ings pierced by external magnetic flux represent one of
a Note that the existing theory of quantum decoher-
the fundamental consequences of quantum coherence of
h ence of electrons in disorderedconductors with electron-
s- ealteucrtersonmwayavpeerfusinscttaiotndsiswtahnicchesactosmuffipacrieanbtlelywloitwhttehmepsyers-- electroninteractions7,8 is rathercomplicated, merely be-
e cause of the Pauli principle which needs to be properly
m tem size. While the average value of PC was intensively
accountedforinsuchsystems. Atthesametime,theba-
investigatedboththeoretically1andexperimentally2dur-
. sic physics of this phenomenon can be explained already
t inglastdecades,littlewasknownaboutequilibriumfluc-
a without unnecessarycomplications: It is due to the elec-
tuations of PC.
m
tron interaction with the fluctuating quantum electro-
At non-zero T it is quite natural to expect non-
- magnetic field produced by other electrons moving in a
d vanishing thermal fluctuations of PC3. Somewhat less
disordered conductor. Hence, for the sake of physical
n trivial is the limit T → 0 when the system approaches
transparency it is sometimes useful to employ a simpli-
o
its (non-degenerate) ground state. In this limit no PC
fied model which mimics all key features of the “real”
c
[ fluctuations occur only provided the current operator Iˆ problem of interacting electrons in a disordered conduc-
commutes with the total Hamiltonian Hˆ of the system, tor exceptfor the Pauliexclusionprinciple. This model9
1
otherwisefluctuationsofPCdonotvanishevenprovided dealswithaquantumparticlemovinginaringandinter-
v
1 the system remains exactly in its ground state4. For in- actingwith somequantumdissipativeenvironment. The
2 stance, PC does fluctuate down to T = 0 provided the latter could be, e.g., a bath of Caldeira-Leggett oscilla-
1 ringiscoupledtosomeexternaldissipativeenvironment. tors or electrons in a disordered conductor. In the case
5 This situation is encountered, e.g., within the model5 of Caldeira-Leggett environment decoherence of a quan-
1. where it was demonstrated that, while the averagevalue tum particle on a ring was investigated with the aid of
0 of PC hIˆi in the ground state decreases with increasing imaginary9 and real-time10 approaches which yield sim-
1 coupling with the environment, PC fluctuations remain ilar results, i.e. exponential suppression of quantum co-
1 non-zero and even increase with the coupling constant. herence down to T = 0 at sufficiently large ring radii.
v: In the latter example interaction with the dissipative In fact, the problem9,10 is exactly equivalent to that of
i environment produces quantum decoherence which, in Coulomb blockade in a single electron box where expo-
X
turn, causes suppression of PC hIˆi even at T = 0. On nentialreductionoftheeffectivechargingenergyatlarge
ar the other hand, ground state fluctuations of PC can oc- conductances is also well established11,12. The model of
cur also in the absence of any decoherence. E.g., the a particleina diffusive electrongaswasextensivelyused
operators Iˆand Hˆ do not commute for a quantum par- by different authors9,13–18 in order to investigate the ef-
ticle on a ring in the presence of an external periodic fect of interaction-induced decoherence on the average
potential. This model describes, e.g., the properties of value of PC. Below we will employ the same model in
superconducting nanorings with quantum phase slips6. orderto study aninterplaybetweenPCfluctuations and
In this case PC fluctuations do not vanish even in the quantum decoherence.
ground state4 and, at the same time, quantum coher- The structure of our paper is as follows. In Sec. 2 we
ence remains fully preserved. As a result, the magnitude define ourmodelanddescribeourbasicformalismofthe
of PC fluctuations turns out to be sensitive to external influence functional. In Sec. 3 we analyze PC fluctua-
magnetic flux4. This observation implies that quantum tionswithinthe frameworkoftheperturbationtheoryin
coherence and decoherence in meso- and nanorings can theinteraction. InSec. 4wegobeyondthe perturbation
be probed by measuring the equilibrium current noise in theory and evaluate equilibrium current-current correla-
2
tion functions in the limit of strong interactions. Our inverseelectronFermimomentumk−1, i.e. k l ≫1. On
F F
main conclusions are outlined in Sec. 5. Further techni- the other hand, the electron mean free path should re-
cal details of our calculation are specified in Appendices main much smaller than the ring radius l ≪R. We also
A and B. note that Eq. (5) applies at frequencies ω ≪ω ∼v /l.
c F
Fluctuating electrons cause fluctuations of the electric
potential V in the system. Within Gaussian approxima-
II. THE MODEL AND BASIC FORMALISM tion such fluctuations are described by the correlator
In our analysis we will employ a model9,13 of a quan- ω 4π
hVVi =−coth Im . (6)
tumparticlewithmassM ona1dringofradiusRpierced ω,k 2T k2ǫ(ω,k)
by magnetic flux Φ . This quantum particle interacts
x
withadissipativeenvironment(bath) describedbysome Interactionbetweentheparticleonaringandfluctuating
collective variable V representing its degrees of freedom. electronsintheenvironmentisdescribedbythestandard
The total Hamiltonian for this system reads Coulomb term
Hˆ = (φˆ−φx)2 +Hˆ (V)+Hˆ (θ,V), (1) Hˆint =eVˆ, (7)
2MR2 env int
where e denotes the particle charge.
where θ is the angle variable parameterizing the particle
position on the ring, φˆ = −i ∂ is the angular momen- In order to evaluate the correlation function (4) it is
∂θ necessary to describe quantum dynamics of our system.
tum operator, φ = Φ/Φ and Φ is the flux quantum.
x 0 0
For this purpose let us introduce the evolution operator
The first term in Eq. (1) represents the particle kinetic
energy, Hˆ (V) is the Hamiltonian of the bath, and the Uˆ(t,t0) and define the density matrix operator
env
termHˆ (θ,V)describesinteractionbetweentheparticle
int ρˆ(t)=Uˆ(t,0)ρˆUˆ†(t,0), (8)
and the bath. Note that within our model the interac- i
tion term involves only the coordinate (not momentum)
operators. where ρi is the initial density matrix. The kernel of the
Let us define the current operator for a particle on a evolution operator can be expressed as a path integral
ring. It reads over the angle variable. We have
Iˆ= e θˆ˙ = ie[Hˆ,θˆ]= e(φˆ−φx). (2) hθ1|Uˆ(t,0)|θ1′i=eiφx(θ1−θ1′)
2π 2π 2πMR2 ∞
× e2πimφxU(θ +2πm,t;θ′,0), (9)
1 1
EmployingtheHeisenbergrepresentationofthisoperator
m=−∞
X
Iˆ(t)=eitHˆIˆe−itHˆ, (3) where
one can define the current-current correlation function
θ(t)=θ1
hIˆ(t)Iˆ(0)i. Evaluating the symmetrized version of this
U(θ ,t;θ′,0)= DVDθeiS+iSenv, (10)
correlator in equilibrium one arrives at PC noise power4 1 1
θ(0)Z=θ′
1
1 dω
S(t)= hIˆ(t)Iˆ(0)+Iˆ(0)Iˆ(t)i−hIˆi2 = S e−iωt,
ω
2 2π S is the action of the environment and
Z env
(4)
which represents the central object in our subsequent t
θ˙2(t )
analysis. S = 1 −eV(r (t ),t ) dt . (11)
θ 1 1 1
Similarly to Refs. 9,13 we will model the environment Z 4EC !
by 3d diffusive electron gas with the inverse dielectric 0
function
Here E = 1/(2MR2) and r is the vector with compo-
C θ
1 −iω+Dk2 nents (Rcosθ,Rsinθ).
≈ , (5)
ǫ(ω,k) 4πσ As we are interested in the dynamics of the particle
ratherthanthatofthebath,itisconvenienttotraceout
whereσ istheDrudeconductivityofthisgas,D =v l/3 fluctuating potential of the environment V. Making use
F
is the electron diffusion coefficient, v is Fermi velocity of a standard simplifying assumption that at the initial
F
and l is the electron elastic mean free path. As usually, time moment the total density matrix is factorized into
below we will assume disorder to be not very strong im- the product of the equilibrium bath density matrix and
plying this mean free path to be much longer than the some initial particle density matrix ρˆ, we obtain
i
3
2π
∞
ρ(θ1,θ2;t)= ei(θ1+2πm1)φx−i(θ2+2πm2)φx dθ1′dθ2′e−i(θ1′−θ2′)φxρi(θ1′,θ2′)
m1,mX2=−∞ Z0
× θF(t)=θ1+2πm1DθF θB(t)=θ2+2πm2DθBeiR0t[((θ˙F)2−(θ˙B)2)/4EC]dt′F[θF,θB], (12)
θF(0Z)=θ′ θB(0Z)=θ′
1 2
where ρ(θ ,θ ;t)≡hθ |ρˆ(t)|θ i and
1 2 1 2
t
F[θF,θB]= e−iR0(eVF(rθF(t′),t′)−eVB(rθB(t′),t′))dt′ =exp(−iSR−SI), (13)
* +
V
is the influence functional19 which depends on the angle variables on the forwardand backwardparts of the Keldysh
contour, respectively θF and θB. Calculation of this influence functional amounts to averaging over the quantum
variableV whichisalsodefinedontheKeldyshcontour. Thisprocedure7caneasilybeadaptedtoourpresentsituation
of a particle on a ring where no Pauli exclusion principle needs to be taken into account, cf., e.g.14. Introducing the
new variables θ = (θF +θB)/2 and θ = θF −θB, after the standard algebra (see Appendix A for further details)
+ −
we obtain
t
∞
S [θ ,θ ]=πα a n dt′θ˙ (t′)sin(nθ (t′)), (14)
R + − n + −
n=1 Z
X 0
and
t t
∞ πT2 nθ (t′) nθ (t′′)
S [θ ,θ ]=−2πα a dt′ dt′′ cos(n(θ (t′)−θ (t′′)))sin − sin − , (15)
I + − n sinh2(πT(t′−t′′)) + + 2 2
n=1 Z Z
X 0 0
where α = 3/(8k2l2) is the effective coupling constant for −1/2 < φ < 1/2. The result (16) should be pe-
F x
in our problem and a are the Fourier coefficients equal riodically continued outside this interval of flux values.
n
to a = (2/(πr))ln(r/n) for n < r ≡ R/l ≫ 1 and We observe that for α ≪ 1 the first order perturbative
n
to zero a = 0 otherwise. The weak disorder condition correction to the average value of PC is negligibly small
n
k l ≫ 1 obviously implies α ≪ 1, i.e. the coupling except in the immediate vicinity of half-integer flux val-
F
constant always remains small within the applicability ues φ = ±1/2,±3/2,..., where the two lowest energy
x
range of our model. levels become very close to each other and the perturba-
We also note that the above influence functional re- tion theory fails already in the first order.
duces to the Caldeira-Leggett one if we choose9,13 α = Turning now to PC noise let us recallthat inthe limit
ηR2/π (where η is effective viscosity of the Caldeira- α → 0 the current operator (2) commutes with the to-
Leggett bath), a =1 and a =0 for all n>1. tal Hamiltonian. Hence, in the absence of interactions
1 n
PC noise vanishes in the low temperature limit for the
model in question4. On the other hand, in the presence
III. PERTURBATION THEORY ofinteractionsPCnoiseremainsnon-zeroandimportant
contributionstothecurrent-currentcorrelator(4)areex-
pected to occur already at sufficiently small values of α
Let us assume that both effective coupling constant
and R. This limit will be studied perturbatively below
α and ring radius R are sufficiently small and proceed
in this section.
within the perturbationtheory in the interaction. Keep-
ing only the first order correction ∼ α to the average
value of PC, in the low temperature limit T ≪ E one
C
finds13 A. Density matrix and Dyson equation
r
eE α n+2φ
hIˆi= C φ − na ln x , (16) In orderto proceedit will be convenientfor us to pass
x n
π " 2 n=1 (cid:18)n−2φx(cid:19)# tothemomentumrepresentation. PerformingtheFourier
X
4
transformation of the density matrix ν we arrive at the perturbation series for the density
n
matrix which can be expressed in terms of Keldysh dia-
2π 2π gramsconsistingoftwosolidlines(implyingforwardand
dθ dθ
ρ˜(m,n;t)= 1 2ρ(θ ,θ ;t)e−imθ1+inθ2. (17) backward propagators)connected by dashed lines corre-
1 2
Z 2π Z 2π spondingtothe propagatorsΠσ,σ′(t−t′)forthe ν -field.
0 0 n n
One of suchdiagramsis depicted in Fig. 1 and is similar
and making use of the influence functional in the form to Keldysh diagrams encountered, e.g., in the Coulomb
(A11) (see Appendix A), we obtain blockade problem21.
Let us denote the sum of all diagrams contributing
∞
ρ˜(m ,m ;t)= ρ˜(m′,m′) to the evolution kernel for the particle density matrix
1 2 m′1,mX′2=−∞ i 1 2 as Umm12,,mm′1′2(t), where lower (upper) indices correspond to
× K˜(m ,t;m′,0)K˜∗(m ,t;m′,0) , (18) forward (backward) lines. Provided upper and lower in-
1 1 2 2 νn dices coincide, only lower indices will be indicated, i.e.
where D E Umm,,mm′′(t) ≡ Um,m′(t). The latter quantities can also be
considered as elements of the matrix Uˆ(t). In what fol-
∞ 2π lowswewillalsousetheshort-handnotationfortheden-
K˜(m,t;m′,0)= dθ1 d2θπ1′ e−i(m−φx)θ1+i(m′−φx)θ1′ sity matrix Pmm12(t) ≡ ρ˜(m1,m2;t) and denote its diago-
−Z∞ Z0 nal elmements as Pm(t) or asma−knet-vector |P(t)i. In this
× θ(t)=θ1DθeiR0t(cid:18)4Eθ˙2C+nP∞=1(νn(t′)einθ(t′)+c.c.)(cid:19)dt(′19) n k
θ(0)Z=θ′
1
m m−n m−n−k m−n
Let us expand the exponent in Eq. (19) in series in ν
n
and evaluate Gaussian path integrals in all terms of this
FIG.1: TypicalKeldyshdiagram contributingtothedensity
expansion. In the zeroth order (all ν =0) we obtain
n matrix.
K˜(m,t;m′,0)=δm,m′K0(m;t) (20)
notationstheevolutionofthedensitymatrixisexpressed
with K0(z;t)=e−iEC(z−φx)2t, the term with ν1 =1 and by means of the equation
ν yields the contribution
n6=1
δm−n,m′K0(m;t−t1)K0(m−n;t1), (21) Pmm12(t)= Umm12,,mm′1′2(t)Pmm′1′2(0). (22)
and so on. Representing each term in this expansion di- mX′1,m′2
agrammatically,we can indicate the unperturbed propa-
geaacthoriδnmse,mrt′iKon0(omf;eti)nθ(2(t0))(boyr ae−sionlxid(t)li)naedadnsd(oobrserervmeotvheast) mInatrnondeurcainsgatshuemseolff-aelnleirrgryedSummci12b,,mlme′1′2d(ita−grta′m) isnwaesatrarnivdearadt
the momentum n at a time t. Collecting all contribu- the following Dyson equationfor the kernelof the evolu-
tions to all orders in α in Eq. (18) and averaging over tion operator
t t
Umm12,,mm′1′2(t)=K0(m1;t)K0∗(m2;t)δm1,m′1δm2,m′2 + dt1 dt2
nX1,n2Z0 tZ1
×K0(m1;t−t2)K0∗(m2;t−t2)Smm12,,nn12(t2−t1)Unn12,,mm′1′2(t1). (23)
Suppose that our system was described by diagonal determined by the equation
density matrix at some initial time which tends to −∞.
Then it should remain in the diagonal state at all later
t
times as well. The evolution of such density matrix is ∂|P(t)i
= dt′Sˆ(t−t′)|P(t′)i. (24)
∂t
Z
−∞
5
Atlongenoughtimes|P(t)itendstoitsequilibrium(and, m−n m m m+n
hence, time-independent) value |Peqi which obeys the
equation
n n n n
t
dt′Sˆ(t−t′)|Peqi=0. (25) m m−n m−n m
m+n m m m−n
Z
−∞
Performing the Fourier transformationof the self-energy
n n n n
∞
Sˆ = dteiωtSˆ(t) one can rewrite the above equa-
ω
0 m m+n m+n m
tion as SRˆ |Peqi = 0. In equilibrium one obviously has
0
Pmeq =e−Em/T/Z,whereZ = e−Em/T is the partition FIG. 2: First order self-energy diagrams
m
function and Em =EC(m−φxP)2.
C. Current-current correlator
B. Self-energy
Let us identically rewrite the current noise power (4)
in the form
The evolutionkernel for diagonalelements of the den-
∞
sity matrix can also be expressed via the self-energy as
S =2ℜ dteiωthI (t)I (0)i−2πδ(ω)hI i2, (32)
ω + + +
i
Uˆ = . (26) Z0
ω ω−iSˆ
ω where we defined I = (IF +IB)/2. A typical diagram
+
Since all elements of the ket-vector |P(t)i are real, one contributing to this expressionis depicted in Fig. 3. Let
candemonstratethatthe self-energyremainspurelyreal us recall that in our problem the current and momen-
in the zero frequency limit. Introducing the matrices Σˆ tumoperatorscoincidewitheachotheruptoaconstant.
ω
and Γˆ (which are non-singular at small frequencies) we Hence, inserting the current operator at a time t inside
ω
can write the diagram yields the factor eEC(m−φx)/π, where m
istheparticlemomentumvalueatthistime. Inwhatfol-
Sˆ =iωΣˆ −Γˆ (27) lowswewilldividealldiagramsintotwodifferentclasses,
ω ω ω
those with equal momenta at both upper and lower at
and, hence, t=0andallothers. Summingupalldiagramsineachof
these two classeswe arriveat two different contributions
i
Uˆ = (1+Σˆ )−1. (28) to the current-current correlator:
ω ω+i(1+Σˆ )−1Γˆ ω
ω ω
hI (t)I (0)i=hE|Iˆ(0)Uˆ(t)Iˆ(0)|Peqi
+ +
Here we employ a simple approximation which amounts
0 t
to neglecting Σˆω and keeping only the leading in α cor- + dt′′ dt′hE|Iˆ(0)Uˆ(t−t′)Iˆ(1)(t′,t′′)|Peqi, (33)
rection to Γˆ . Then we obtain
ω
Z Z
−∞ 0
i
Uˆω(0) ≈ ω+iΓˆ(0), (29) where Iˆ(0) is a diagonal matrix with elements Iˆm(0,)n =
ω 2E (m−φ )δ andIˆ(1)(t,t′)isdefinedasasumofall
C x m,n
where the real part of the self-energy is defined as irreducible diagrams containing t=0. After the Fourier
transformation we obtain
παa E −E +ω
[Γˆ(ω0)]m+n,m =− 2|n| eEmm++nn−TEm+mω −1 Sω =2ℜhE|Iˆ(0)Uˆω−(Iˆ2π(0δ)(+ω)IˆhωE(1)|I)ˆ|(P0)e|qPieqi2. (34)
E −E −ω
m+n m
+ , (30) This equation defines a formally exact expression for
eEm+n−TEm−ω −1! PC noise power. Proceedingperturbatively in α one can
check that the contribution containing Iˆ(1) can be ne-
ω
glected as it turns out to be small as ∼ α ≪ 1 as com-
∞
[Γˆ(0)] =− [Γˆ(0)] +[Γˆ(0)] . (31) pared to the terms with Iˆ(0). Then after some manipu-
ω m,m ω m+n,m ω m−n,m ω
nX=1(cid:16) (cid:17) lations (see Appendix B for further details) we get
The corresponding first order self-energy diagrams are i|PeqihE|
S =2ℜhE|Iˆ(0) Uˆ − Iˆ(0)|Peqi. (35)
depicted in Fig. 2. ω ω ω+i0
(cid:18) (cid:19)
6
m 0 m−n S
ω
!&’!"$
!&’!"%
n k !&’!"(
!&’!")
!"!%
t
m m−n m−n−k m−n
!"!$#
FIG. 3: Typical diagram contributing to the current-current
correlator (32).
!"!$
S
ω
!&’!"$
!"!% !&’!"% !"!!#
!&’!"(
!&’!")
!"!$# ! ω
! !"# $ $"# % %"#
FIG. 5: PC noise power at T = 0.05EC and different flux
!"!$ values. Units and parameters are thesame as in Fig. 4.
!"!!#
S
ω &’!
&’!"!()*
&’!"!+)*
!"!!%# &’!"!,)*
!! !"# $ $"# % %"# ω &&’’!!""$$%#))**
!"!!%
FIG. 4: PC noise power at T = 0 πα = 0.05, r = 5 and
different flux values. Frequency ω and noise power Sω are
normalized respectively by EC and by e2EC/(4π2). !"!!$#
!"!!$
Note that at small ω the quantity Uˆ tends to
ω
i|PeqihE|/(ω + i0), i.e. PC noise power is regular in !"!!!#
the zero frequency limit. At non-zero ω the difference
between these two terms is proportional to α, thus pro- !
ω
! !"# $ $"# % %"#
viding nonvanishing PC noise at such frequencies in the
presence of interactions.
FIG. 6: PC noise power at φx =0.15 and different tempera-
tures. Units and parameters are thesame as in Fig. 4.
D. Results
The perturbative expression for the current noise
perature PC noise vanishes at φ = 0. In general, how-
x
power (35) was evaluated numerically at different tem-
ever, this feature does not hold, as it can be observed,
peratures and flux values. The results are presented in
e.g., from the expression for PC noise power in terms of
Figs. 4-6. One observes that the noise power strongly
theexacteigenstatesofthetotalHamiltonian4. Non-zero
depends on the magnetic flux φ . This property illus-
x PCnoiseatφ =0willalsobedemonstratedinthenext
trates the coherentnature of PC noise4. PC noise grows x
section where we employ non-perturbative quasiclassical
with increasing φ and diverges as the flux approaches
x analysis of the problem.
the point φ = 0.5. This divergence has the same phys-
x
ical origin as that in Eq. (16). In this limit the dis- Atnon-zeroT there appearsadditionalzerofrequency
tance between the two lowest energy levels δE(φ ) = noise power peak. This peak grows rapidly with in-
x
E (1−2|φ |) becomes small and the system undergoes creasingtemperatureandeventuallyassimilatesallother
C x
rapid transitions between these energy states. As these peaks. As a result, at sufficiently high temperatures
levelscorrespondto differentPC values suchtransitions, only a wide hump remains, and PC noise becomes flux-
in turn, yield strong current fluctuations. independent, i.e incoherent. The dependence of zero fre-
Itis importantto observethatPC noisepersistsdown quency PC noise on temperature is illustrated in Fig. 7.
to T → 0. In this case S remains zero at frequencies Interestingly,atlowenoughT thisdependence turnsout
ω
smaller than the inter-level distance ω < δE(φ ) and to be non-monotonous, while in the high temperature
x
becomes non-zero otherwise. We also note that in the limit it approaches the linear dependence S ∝ T, as it
0
lowest (∼ α) order of the perturbation theory zero tem- will also be demonstrated in the next section.
7
!" hξn(t)λm(t′)iξ,λ =0. (40)
Inthe hightemperaturelimit these correlatorsreduceto
#$!
those describing the white noise
hξ (t)ξ (t′)i =2δ παa n2Tδ(t−t′) (41)
n m ξ,λ m,n n
#!!
and the Langevin equation can be solved exactly. As a
result, we arrive at the high temperature noise power
$!
e2γTE2
S = C . (42)
ω π2(ω2+(γE )2)
C
!
At lower temperatures the white noise approximation
!"# # #! # (41) becomes inaccurate and Eqs. (39), (40) should be
employed. Inthis casenoisetermsintheLangevinequa-
FIG. 7: Zero frequency PC noise power S0 as a function of tion can be treated perturbatively24. Keeping only the
temperature for φx =0.15, πα=0.05 and r =10. Tempera- zeroth and the first order contributions one gets the so-
ture and PC noise power are normalized respectively by EC
and by e2EC/(4π2). lution of Eq. (37) in the form
θ (t)=θ(0)+θ(1)(t), (43)
+ + +
IV. NON-PERTURBATIVE ANALYSIS
whereθ(0) isanarbitrary(andphysicallyirrelevant)con-
+
Let us now turn to the limit of strong interactions stant and θ(1)(t) obeys the equation
+
in which case the effect of a dissipative environment on
the particlemotionbecomes largesubstantiallyreducing 1 γ ∞
fluctuations of the angle variable θ. It is important to − 2E θ¨+(1)(t)− 2θ˙+(1)(t)= ξn(t). (44)
C
stress that for the model under consideration this situa- n=1
X
tion can be realized even at small values of the effective
Resolving this equation one immediately arrives at the
coupling constant α ≪ 1 provided the ring radius be-
noise power in the form
comes sufficiently large13, i.e.
e2γE2 ω
4παr ≫1. (36) S = C ωcoth , (45)
ω 2π2(ω2+(γE )2) 2T
C
In this limit and provided temperature is not too low it
whichagainreduces to Eq. (42)in the high temperature
sufficestoemploythesemiclassicalapproximationandto
limit T ≫ω. Note that for ω ≪γE the parameter E
expand the effective action (14), (15) up to quadratic in C C
θ terms. As usually22–24, the resulting effective action drops out and the noise power becomes
−
can be exactly rewritten in terms of the quasiclassical
e2ω ω
LFoarngtheveinmoeqduelatsitoundifeodrhtheree”tcheinstLera-nogf-emviansse”quvaatriioanblteaθk+es. Sω = 2π2γ coth2T, (46)
the form
i.e. in this case S ∝ 1/α. For ω → 0 this expression
ω
− 1 θ¨+(t)− γθ˙+(t)= ∞ (ξn(t)cos(nθ+(t)) faulsrothdeerpriecdteudceisntFoiSg.08∝aTt/dγiff.eTrehnetnvoailsueepsoowfeTr.Sω (45)is
2E 2
C
nX=1 Comparing Eqs. (42), (45) with perturbative in the
+λn(t)sin(nθ+(t))), (37) interactionresultsforthe noisepowerderivedinthepre-
vious section we observe a striking difference between
where we defined
them: While in the weak interaction limit PC noise is
∞ sensitive to the externally applied magnetic flux φx, in
γ =2πα a n2 =4παr2 (38) the opposite limit of strong interactions the noise power
n
S turns out to be essentially independent on φ . The
n=1 ω x
X
latter observation implies that in the non-perturbative
and introduced Gaussian stochastic fields ξ (t) with the
n limit(36)quantumcoherenceoftheparticleissuppressed
correlators
by strong interactions with the dissipative environment.
This conclusion is fully consistent with earlier results13
hξ (t)ξ (t′)i =hλ (t)λ (t′)i =
n m ξ,λ n m ξ,λ derived for PC hIi in the limit (36).
πT2 Note that Eq. (45) defines only the dominating con-
=−δ παa n2 , (39)
m,n n sinh2(πT(t−t′)) tribution to the noise power. In addition there alsoexist
8
S to do with the fact that the angle variable θ is compact
ω ’(!
’()* (i.e. definedonaring). Ontheotherhand,theLangevin
%
’(")*
’(+)* equationmethodemployedaboveeffectively”decompact-
’(&!)*
ifies” our problem, thus being able to capture only the
$ φx-independent contributions to Sω. In order to esti-
matetheleadingφ -dependentcorrectiontoEq. (45)we
x
will make use of the approachinitially developed for the
# problem of weak Coulombblockade in metallic quantum
dots25–27. Thisapproachestablishestherelationbetween
thedensitymatricesandexpectationvaluesevaluatedfor
" the problemsdescribed by the same Hamiltonianbut re-
spectivelycompactandnon-compactvariables. Withthe
aid of25–27 for the expectation value of the current oper-
!! " # $ % &! &" &# ω ator one finds
FIG. 8: Noise power at different temperatures for πα=0.05 hIˆi= eEC Ntr(e2πiN(φˆ−φx)φˆρˆnp). (47)
and r=25. Unitsare the same as in Fig. 4. π PNtr(e2πiN(φˆ−φx)ρˆnp)
Here ρˆ stands for thPe reduced equilibrium density ma-
np
smallcorrectionstothisresultwhichdoshowthedepen- trix for a particle described by non-compact (i.e. de-
dence on the external flux φ piercing the ring. Tech- finedonastraightline)variableθ. Analogouslythenoise
x
nically, the existence of this flux-dependent terms has power is given by the autocorrelationfunction
S(t)= (eEC)2 Ntr((φˆρˆnp+ρˆnpφˆ)Uˆn†p(t,0)φˆe2πiN(φˆ−φx)Uˆnp(t,0)), (48)
2π2 P Ntr(e2πiN(φˆ−φx)ρˆnp)
P
where the evolution operator Uˆ (t,0) is again defined In the semiclassical limit averaging in these equations is
np
for a non-compact variable θ. With the aid of the path convenientlyperformedwithintheaboveLangevinequa-
integralsonecanrewritetheaboveequationsrespectively tion technique. With the aid of Eqs. (43) and (44) one
as easily finds
hIˆ(t)i= e Nhθ˙+(t)e2πiN(θ˙+(t)/(2EC)−φx)i (49)
2πP Nhe2πiN(θ˙+(t)/(2EC)−φx)i hIi= ieK(0) NNe−2πiNφx−π2N2K(0)/(2EC2) (51)
and P 2EC P Ne−2πiNφx−π2N2K(0)/(2EC2)
P
S(t)= e2 Nhθ˙+(t)θ˙+(0)e2πiN(θ˙+(t)/(2EC)−φx)i. (50) and
4π2P Nhe2πiN(θ˙+(t)/(2EC)−φx)i
P
e2K(t) π2K(0) N2e−2πiNφx−π2N2K(0)/(2EC2)
S(t)= 1− N , (52)
4π2 EC2 P Ne−2πiNφx−π2N2K(0)/(2EC2) !
P
where we introduced the correlator K(t) = We also obtain
hθ(1)(t)θ(1)(0)i which Fourier transform equals to ∞
+ + dω ω ω
K(0)=4γE2 coth . (54)
∞ C 2π ω2+(γE )2 2T
2γE2 ω Z C
K ≡ dteiωtK(t)= C ωcoth . (53) 0
ω ω2+(γE )2 2T
Z C Logarithmicdivergencecontainedinthisintegralcaneas-
−∞
9
ily be cured if we recall that (a) our diffusive electron ina ringandinteractingwith anenvironmentformedby
gasmodel(5)is applicableonly atfrequenciesω ≪ω ∼ 3d diffusive electron gas. Specifically, we restricted our
c
v /l (hence, the integral in Eq. (54) should be cut at attention to PC noise and evaluated symmetric current-
F
ω ∼ ω ) and (b) our Langevin equation approach be- current correlation function in two different limits of
c
comes insufficient in the low temperature limit where it weakandstronginteractions. Notethatalthoughwithin
shouldbe supplementedbyothertechniques. Asaresult our model the effective coupling constant α describing
of these considerations we may write Coulomb interaction between the particle and the bath
alwaysremainssmall, α≪1,interactionscanbe treated
K(0)=C+2ECT perturbativelyonlyforsufficientlysmallvaluesofthering
2γE2 γE γE radius R, while for larger R (36) non-perturbative anal-
+ C ln C −ψ 1+ C , (55)
ysis of interaction effects becomes unavoidable.
π 2πT 2πT
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
where ψ(z) is the digamma function and the constant C
In the absence of interactions within our model PC
effectively accounts for the low temperature behavior of
fluctuates only at non-zero T and no such fluctuations
oursystem. Thevalueofthisconstantcanbedetermined
could occur provided the system remains in its ground
if we compare the expression for PC (51) derived here
with the results of the instanton analysis13. Since in the state at T = 0. In the presence of interactions the cur-
limit (36) we have K(0) ≫ E2, it suffices to keep only rent operator does not anymore commute with the total
C
Hamiltonian of the system and fluctuations of PC gen-
the terms with N = 0,±1 in Eqs. (51) and (52). Then
comparing Eq. (51) with the result13 hIi∝exp(−4παr) erally persist down to zero temperature4. In the pertur-
bativeregimeofweakinteractionsandatsufficientlylow
one may identify C as
T quantum coherence of the particle remains preserved,
C ≃ 8αrEC2 . (56) PC noise is coherent and, hence, the noise power Sω can
be tuned by external magnetic flux φ . In contrast, in
π x
the limit of strong interactions (36) fluctuations in the
Finally, for the current noise power we obtain electronic bath strongly suppress quantum coherence of
the particle down to T = 0. In this case the average
S = e2γEC2ωcoth(ω/2T) value of PC hIi gets strongly suppressed as well13, while
ω 2π2(ω2+(γEC)2) the current noise, on the contrary, does not vanish and
2π2K(0) −π2K(0) becomespracticallyflux-independent. Inotherwords,in
× 1− E2 e 2EC2 cos(2πφx) . (57) this regime fluctuations in the environment induce inco-
C ! herent background current noise in the ring which per-
sists even at zero flux φ = 0 when the average PC is
x
As it was already anticipated, in the limit of strong in-
absent hIi=0.
teractions (36) the coherent (flux-dependent) contribu-
tion in Eq. (57) just represents a small correctionto the
main incoherent term (45). In this respect an accurate
We alsopointoutthat, while in the perturbativelimit
evaluation of this small correction may even be consid-
PC noise power S tends to increase with the coupling
ω
ered as exceeding, it suffices to demonstrate that this
constant α, in the non-perturbative regime the depen-
φ -dependent correction remains small in the limit (36).
x dence of S on α becomes more complicated, cf. Eqs.
ω
We also note that exponential dependence of PC on the
(45) and (52). In particular, at sufficiently low frequen-
ring radius hIi ∝ exp(−4παr) applies down to temper- cies we find S ∝ 1/(αR2), i.e. in this regime the noise
ω
atures T ∼ E /(4παr), whereas at even lower T → 0
C power decreases with increasing both α and the ring ra-
it crosses over to a weaker (power law) dependence (cf.
dius R. On the other hand, the average PC value de-
Figs. 1and2inRef. 13)butremainsstronglysuppressed
creases even much stronger and, hence, the ratio S /hIi
ω
in the limit αr ≫ 1. Accordingly, one can expect that increases with increasing α and R.
in the same limit the flux-dependent correction to the
incoherentnoiseterm(45)remainssmalldowntoT =0,
thoughatT ≪EC/(4παr) itmaydeviate fromthe form Perhapsthe mostimportant resultof this paper is the
(57). Unfortunately, quantitative non-perturbative anal- prediction of (i) coherent flux-dependent fluctuations of
ysis of the exact zero temperature limit appears difficult persistent current in sufficiently small rings and (ii) in-
since neither Langevin equation approach nor instanton coherent flux-independent current noise in larger rings.
analysis13 can be trusted in this limit. Thus, quantum coherence and its suppression by inter-
actions in meso- and nanorings can be experimentally
investigated not only by detecting the average PC value
V. CONCLUSIONS (whichcanhappentobeverysmall)butalsobymeasur-
ingPCnoiseanditsdependenceontheexternalmagnetic
In this paper we analyzed fluctuations of persistent flux. We believe it would be interesting to perform such
currentproducedby a chargedquantumparticle moving experiments in the near future.
10
Appendix A: Influence Functional
Let us present the derivation of the influence functional defined by Eqs. (13), (14) and (15). In order to perform
Gaussianaveragingoverfluctuatingelectricpotentialitissufficienttodefineonlythesecondordervoltagecorrelators.
Introducing the variables V+ = (VF +VB)/2 and V− = VF −VB we can express these correlators in terms of the
Green functions
hV+(r (t),t)V+(r (t′),t′)i = iGK(2R|sin((θ−χ)/2)|,t−t′), (A1)
θ χ
hV−(r (t),t)V+(r (t′),t′)i = iGA(2R|sin((θ−χ)/2)|,t−t′), (A2)
θ χ
hV+(r (t),t)V−(r (t′),t′)i = iGR(2R|sin((θ−χ)/2)|,t−t′), (A3)
θ χ
hV−(r (t),t)V−(r (t′),t′)i = 0, (A4)
θ χ
wherethelastequationisadirectconsequenceofcausality. HereGR,GA andGK arerespectivelyretarded,advanced
and Keldysh Green functions related to the dielectric function ǫ(k,ω) of the environment as follows
4π
GR(k,ω)= GA(k,ω)=(GR(k,ω))∗. (A5)
k2ǫ(k,ω)
ω
2GK(k,ω)=coth GR(k,ω)−GA(k,ω) . (A6)
2T
(cid:0) (cid:1)
Combining the above expressions with Eq. (5), in the case of a diffusive metal we obtain
2πα 2πα iπT2
GR(X,t−t′)= δ′(t−t′) GK(X,t−t′)= . (A7)
e2 (X/l)2+1 e2 (X/l)2+1sinh2(πT(t−t′))
Gaussian averaging overpthe V-fields can now easily be performed, cf., e.g.p,7. As a result we arrive at Eq. (13). The
expression for the imaginary part of the action in this equation reads
t t
πT2
S [θ (t),θ (t)]=−πα dt′ dt′′ ζ(θF(t′)−θF(t′′))+
I + − sinh2(πT(t′−t′′))
Z Z
0 0 (cid:0)
+ζ(θB(t′)−θB(t′′))−ζ(θF(t′)−θB(t′′))−ζ(θB(t′)−θF(t′′)) , (A8)
where ζ(x)=[4(R/l)2sin2(x/2)+1]−1/2. Expanding ζ(x) in the Fourier series (cid:1)
∞ ∞
nx 1
ζ(x)=a − a sin2 =c+ a cos(nx) (A9)
0 n n
2 2
nX=1 (cid:16) (cid:17) nX=1
with a =(2/(πr))ln(r/n) for n<r and a =0 for n>r and using the identity
n n
θ θ
cos(θF −θF)+cos(θB −θB)−cos(θF −θB)−cos(θB −θF)=4cos(θ −θ )sin 1− sin 2− (A10)
1 2 1 2 1 2 1 2 1+ 2+ 2 2
one arrives at Eq. (15). Eq. (14) is recovered in a similar manner. Let us also note that the Caldeira-Leggett
environment is described by the function ζ(x) = −2sin2(x/2) = cos(x)−1 which should be employed in that case
instead of Eq. (A9).
Let us also rewrite our influence functional in a somewhat different form, which can be conveniently used in our
perturbative calculations. Employing the definition (−1)F =1 and (−1)B =−1 we obtain
i (−1)σ ∞ t(νσ(t′)einθσ(t′)+νσ∗(t′)e−inθσ(t′))dt′
F[θF,θB]= e σ=PF,B nP=1R0 n n (A11)
* +
νn
Here ν is Gaussian stochastic complex variable described by the correlator
n
hνσ∗(t)νσ′(t′)i= iδmn DK(t−t′)+ (−1)σDR(t′−t)+ (−1)σ′DR(t−t′) . (A12)
n m 2 m 2 m 2 m
!