high-frequency electromagnetic emission from non-local wavefunctions giovanni modanese资料.pdf

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applied
sciences
Article
High-Frequency Electromagnetic Emission from
Non-Local Wavefunctions
Giovanni Modanese
Faculty of Science and Technology, Free University of Bozen-Bolzano, 39100 Bolzano, Italy;
giovanni.******@




Received: 16 April 2019; Accepted: 9 May 2019; Published: 15 May 2019

Abstract: In systems with non-local potentials or other kinds of non-locality, the Landauer-Büttiker
formula of quantum transport leads to replacing the usual gauge-invariant current densityJ with a
currentJext which has a non-local part and coincides with the current of the extended Aharonov-Bohm
electrodynamics. It follows that the electromagnetic fleld generated by this current can have
some peculiar properties and in particular the electric fleld of an oscillating dipole can have a
long-range longitudinal component. The calculation is complex because it requires the evaluation
of double-retarded integrals. We report the outcome of some numerical integrations with speciflc
parameters for the source: dipole length ~107 cm, frequency 10 GHz. The resulting longitudinal
fleld EL turns out to be of the order of102 to 103 times larger than the transverse component (only
for the non-local part of the current). Possible applications concern the radiation fleld generated by
Josephson tunnelling in thick superconductor-normal-superconductor (SNS) junctions in yttrium
barium oxide (YBCO) and by current flow in molecular nanodevices.
Keywords: extended Aharonov-Bohm electrodynamics; local conservation laws; quantum transport;
schrödinger equation with non-local potential
1. Introduction
The extended Maxwell equations by Aharonov and Bohm [1–10] are employed for the calculation
of electromagnetic flelds generated by sources which violate the local charge conservation condition
¶tr + r
J = 0. Barring exceptional situations in cosmology where such violations may occur at the
macroscopic level, a possible microscopic failure of local conservation has been predicted in quantum
mechanics in the following situations:
1. In systems described by fractional quantum mechanics [11–17].
2. In ordinary quantum mechanics, in the presence of non-local potentials [17–26], and in particular
in flrst-principles calculations of transport properties using density functional theory and
non-equilibrium Green functions [27–29]. The latter approach has been very successful for
the exact description of quantum transport in nano-devices, which is otherwise not viable in
terms of local quantum fleld theories.
3. For the proximity effect in superconductors, especially in thick superconductor-normal-
superconductor (SNS) junctions in cuprates, where the Gorkov equation cannot be properly
approximated by a local Ginzburg-Landau equation [9,17,30,31].
Concerning Point 2, we recall that the Landauer-Büttiker formula for the current in quantum
transport, when applied to wavefunctions in the presence of a non-local potential [27,28], inevitably
leads to the deflnition of a non-local charge densityrext and current densityJext which differ from
Appl. Sci. 2019,9, 1982; doi: Warning : .
Appl. Sci. 2019,9, 1982 2 of 14
the usual gauge-invariant expression, and coincide with those of the extended Aharonov-Bohm
electrodynamics, namely
Z
ext nonloc 1 ¶ 3 I (tret,y)
r = r + r = r 2 d y (1)
4pc ¶t jx yj
Z
ext nonloc 1 3 I (tret,y)
J = J + J = J + r d y (2)
4pc jx yj
where tret = t c1jx yj and the “extra-source”I(t,x) is the function which quantifles the violation
of local current conservation:
¶r
I(t,x) = + r
J (3)
¶t
2 ih¯  
r = jYj ; J = (Y rY YrY ) (4)
2m
The currentJ, which can be interpreted as ~rv in a classical limit, is locally non-conserved and
has in this case “sources and sinks” which are, however, invisible to an electromagnetic probe (this is
the so-called “censorship property” of Aharonov-Bohm electrodynamics and constitutes a safeguard
of the locality of the electromagnetic fleld).
In terms of the extended charge densityrext and extended currentJext the
Aharonov-Bohm-Maxwell equations in CGS units are then written in the familiar form
r
E = 4prext (5)
1 ¶B
r E = (6)
c ¶t
r
B = 0 (7)
1 ¶E 4p ext
r B = J (8)
c ¶t c
The Landauer-Büttiker formula employed in [27] gives the currentIa flowing through a leada
coupled to another leadb as
2e Z  
I = dE(f f )Tr GrG GaG (9)
a h¯ a b a b
where Ga, Gb are the linewidths of the leads,fa, fb their Fermi distributions,Gr is the retarded Green
function of the scattering region andGa the corresponding advanced Green function. The authors
of [27] prove that the current calculated from the surface integral ofJext over the interface between
the scattering region and the leada is equal to the that obtained from Equation (9). For this purpose,
they expressJext in terms of Green functions, generalizing the standard method of [32] to the case of a
non-local potential.
Other authors ([29] and references) deflne the extended current in a different way from
References [27,28], and take into account the possibility of adding to it a solenoidal component.
The correct deflnition of the physical current is still an open question, also regarding the dissipation
properties of the non-local part: should the latter be interpreted as a “virtual” current or as a real : .
Appl. Sci. 2019,9, 1982 3 of 14
current with real dissipation? In this context, a detailed calculation and experimental veriflcation of
the predictions of Aharonov-Bohm extended electrodynamics would clearly be of special interest.
In this work, we are concerned with the computation of the electromagnetic fleld generated by the
non-local part of the current. This fleld is independent from any solenoidal component, and therefore
the ambiguities mentioned above do not directly affect our results. It turns out that the radiation fleld
generated by an oscillating dipole with a failure in local conservation (the most obvious example,
apart from the quasi-static case examined in [9]) has very interesting features: namely, it contains an
anomalous longitudinal electrical component with large strength and long range.
For the frequency considered (10 GHz) we found that the strength of the longitudinal component
at a distance between 3l and 13l is in the order of102 to 103 times the standard transverse component.
This factor must be weighted with a small factor that measures the importance of the non-local current
in comparison to the standard current. According to [27], flrst principles calculations of conventional
current density can give errors for current as large as 20% for molecular devices. However, most
molecular devices do not carry currents large enough to generate macroscopic flelds. An exception
could be graphene [33]. Other materials which exhibit macroscopic quantization, large currents, and
possibly non-local currents are, as mentioned, cuprate superconductors.
The computation of the radiation fleld is technically very difflcult due to the presence of
double-retarded integrals and “secondary sources”rext, Jext extended in space. So we had to
resort to a compl