High-frequency electromagnetic emission from non-local wavefunctions

In systems with non-local potentials or other kinds of non-locality, the Landauer-B\"uttiker formula of quantum transport leads to replace the usual gauge-invariant current density $\textbf{J}$ with a current $\textbf{J}^{ext}$ which has a non-local part and coincides with the current of the extended Aharonov-Bohm electrodynamics. It follows that the electromagnetic field generated by this current can have some peculiar properties, and in particular the electric field 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 specific parameters for the source: dipole length $\sim 10^{-7}$ cm, frequency 10 GHz. The resulting longitudinal field $E_L$ turns out to be of the order of $10^2$ to $10^3$ times larger than the transverse component (only for the non-local part of the current). Possible applications concern the radiation field generated by Josephson tunnelling in thick SNS junctions in YBCO and by current flow in molecular nano-devices.

In systems with non-local potentials or other kinds of non-locality, the Landauer-Büttiker formula of quantum transport leads to replace the usual gauge-invariant current density J with a current J ext which has a non-local part and coincides with the current of the extended Aharonov-Bohm electrodynamics. It follows that the electromagnetic field generated by this current can have some peculiar properties, and in particular the electric field 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 specific parameters for the source: dipole length ∼ 10 −7 cm, frequency 10 GHz. The resulting longitudinal field E L turns out to be of the order of 10 2 to 10 3 times larger than the transverse component (only for the non-local part of the current). Possible applications concern the radiation field generated by Josephson tunnelling in thick SNS junctions in YBCO and by current flow in molecular nano-devices.
2. In ordinary quantum mechanics, in the presence of non-local potentials [17][18][19][20][21][22][23][24][25][26], and in particular in first-principles calculations of transport properties using density functional theory and non-equilibrium Green functions [27][28][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 field theories. 3. For the proximity effect in superconductors, especially in thick 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 definition of a non-local charge density ρ ext and current density J ext which differ from the usual gauge-invariant expression, and coincide with those of the extended Aharonov-Bohm electrodynamics, namely where t ret = t − c −1 |x − y| and the "extra-source" I(t, x) is the function which quantifies the violation of local current conservation: The current J, which can be interpreted as ∼ ρv in a classical limit, is locally nonconserved 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 field).
Other authors ( [29] and refs.) define the extended current in a different way from Refs. [27,28], and take into account the possibility of adding to it a solenoidal component. The correct definition 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 current with real dissipation? In this context, a detailed calculation and experimental verification 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 field generated by the non-local part of the current. This field is independent from any solenoidal component, and therefore the ambiguities mentioned above do not directly affect our results. It turns out that the radiation field 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 3λ and 13λ is of the order of 10 2 to 10 3 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], first 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 fields. An exception could be graphene [32]. Other materials which exhibit macroscopic quantization, large currents and possibly non-local currents are, as mentioned, cuprate superconductors.
The computation of the radiation field is technically very difficult due to the presence of double-retarded integrals and "secondary sources" ρ ext , J ext extended in space. So we had to resort to a complex integro-dipolar expansion and to long 6-dimensional Monte Carlo A non−loc = 1 4π In the following the suffix non-loc will be omitted. (or locally) one obtains the transversality condition E 0 · k = 0, where k defines the propagation direction of the wave. The first equation of the extended Aharonov-Bohm theory in vacuum is instead where S is a scalar field which satisfies the equation The "extra-current" I is non zero at the points where the local conservation of charge fails.
If charge is locally conserved everywhere, then the S field is completely decoupled from matter. In this case, even in the extended theory no longitudinal components should be expected.
Eq. (8) can be solved for S, obtaining the first extended Maxwell equation in vacuum with a non-local source term: This shows that the divergence of E in vacuum is equal to a term that we can call "secondary charge density" or "cloud charge", generated in the surrounding space by the local nonconservation of the "primary current". Therefore in a wave solution in vacuum the electric field can have a longitudinal component.
In order to find the concrete predictions of the theory and assess the feasibility of an experimental check, it is necessary to compute exactly the longitudinal electric radiation field E L generated by an appropriate source, compare its magnitude order with that of the transverse field E T and make sure that it does not vanish for some reason not apparent from the general form of the equations. Symmetry can play a crucial role here. We have previously proven in [9], for instance, that in the case of a quasi-stationary extra-source I representing a Josephson weak link with local non-conservation, the anomalous magnetic field generated by I is zero, and there is indeed an observable effect because the corresponding Biot-Savart field is missing. This happens, however, for a source I with spherical symmetry; otherwise the anomalous field partially replaces the missing Biot-Savart field.
A. Steps needed to write the integral expression for the electric field With reference to Fig. 1, consider an oscillating dipolar source with two opposite charges at x = −a and x = +a, of the following form: where f (x) is essentially a regularized double-δ, whose support can be adapted to describe a sphere or a disk (see below, eq. (17)) The absence of current (J = 0) violates local conservation and can be described as the consequence of a "strong-tunnelling" process [17]. In a real source, only a small part of the total charge will oscillate without a current, so we are focussing our attention on the field generated by that part.
In order to compute the field of the source (10) using the extended Maxwell equations we must write the potentials φ and A as double-retarded integrals like in eqs. (5), (6), and The integrand in eqs. (5), (6) is given by I = ∂ t ρ + ∇ · J; therefore, since J = 0, one has here The steps needed to obtain the contribution E φ i are then the following: • Retardate t → t − k|y − z| in I(t, z), divide I by |y − z| and integrate in d 3 z.
• Differentiate with respect to t and multiply by (−k 2 ).
• Multiply by 1/|x − y| and integrate in d 3 y.
• Differentiate with respect to x i and multiply by (−1).
The steps needed to obtain the contribution E A i are the following: • Differentiate with respect to y i and multiply by (−k).
• Multiply by 1/|x − y| and integrate in d 3 y.
• Differentiate with respect to t and multiply by (−k).
Through these steps one arrives, after long but straightforward manipulations, at the following expression for the electric field, as a double retarded integral: where E φ i (t, x) is the contribution of the scalar potential: and the contribution of the vector potential is Here K is the wavenumber: K = kω = c −1 ω = 2πλ −1 ; the phase Ω is given by and f (z) is a regularized representation of the double δ-function of the dipolar source in eq.
(10) (because, as discussed in [17], extra-sources originating from a non-local wavefunction are smooth): The parameter a represents the length of the dipole and is taken equal to 2.5 · 10 −7 cm (in the following, a 1 = a 2 = 0; a 3 = a). The parameter ε represents the size in the 3-direction of the dipole charges, and d their size in the 1-and 2-directions. At the beginning, the oscillation frequency is set to ω = 2π · 10 10 Hz and d = ε = 10 −7 cm.
The field is computed at the point x = r √ 2 , 0, r √ 2 ; at the beginning we set r = 10 cm (approximately equal to three wavelengths), then r is increased up to 40 cm. The transverse and longitudinal components of the electric field at this position are defined by the expressions In addition to the integral in eq. (14) there is also another contribution to E φ , due to the normal density ρ (not ρ non−loc ) of the source (10). The corresponding φ is given in Sect. IV Let us first consider the case of dipole charges having spherical symmetry, so that d = ε in the definition of f (z). We rewrite the integral for E φ i as the sum of two integrals E φ,a i and E φ,−a i for the sources at a and −a, in which the z variable is shifted by −a and a, respectively: For the first integral, with shift −a, we define a new variable u = z − a. Define a regularized δ-function for a source centered at the origin: The electric field generated by the scalar potential of the source at a can be written as where We have symbolically denoted the integration range of u as "M − a", meaning that it is equal to the integration range M of z (−R z ≤ z i ≤ R z ) shifted by a quantity −a.
The function G φ (t, x, y, u + a) can be expanded as a term of order zero in a = |a| and a term of order 1: Actually, the small quantity in which we make the expansion is a/|y| and we therefore expect that the expansion is accurate where |y| a, which is what we need, as explained above.
Let us expand the factor 1/|y − u − a| to first order in a. Define v = y − u. |v| is of order |y|, because F (u) has range ε < a; therefore |v| a. In the following we denote v = |v|.
we have and we find the following first order approximations: and sin Ω sin Ω 0 + cos Ω 0 ∆Ω where and Similarly, cos Ω cos Ω 0 − sin Ω 0 K∆ a Now we can rewrite the function G φ (t, x, y, u + a) as follows: Therefore in the decomposition of G φ , the part G φ 0 , with the terms independent from a is and the part of first order in a is given by In the sum E φ i = E φ,a i + E φ,−a i the terms with G φ 0 cancel, because the integral over the region "M − a" is equal to an integral over M , due to the short range of the function F (u).
The remaining term of first order in a gives The electric field generated by the vector potential of the source at a can be written as where The function G A i can be approximately decomposed in a part independent from a and a part linear in a, as done before for G φ : In order to find G A 0,i and G A 1,i we expand the factors present in G A i to first order in a. Start For the component i = 1, a i = 0, therefore the factor (y i − u i − a i ) does not have components of order a. We obtain whose first order part is The case of i = 3 is more involved, because a i = a in that case. We write and similarly for the term with |y −u−a| 3 in (39). Expanding to first order in a and keeping the linear terms we obtain Then we proceed as in (35), (36) to obtain by a standard oscillating dipole with the same frequency and amplitude. By standard we mean that its current is locally conserved. A textbook formula for this case is and yields an amplitude E T q · 0.8 · 10 −7 (CGS units), supposing an harmonic oscillation with amplitude a = 2.5 · 10 −7 cm, ω = 2π · 10 10 Hz. Since E L is of the order of q · 10 −4 (see raw data in Tab. II of the Appendix), this shows that the anomalous longitudinal field E L of an oscillating dipole with "full" strong tunnelling (i.e., one in which all charge oscillates between −a and a without an intermediate current) is about 2 or 3 orders of magnitude larger than the regular transverse field E T of a corresponding conserved source.
In order to obtain a more precise estimate of the benchmark transverse field, we shall next compute it from the standard solution of the Maxwell equations with a source which is exactly equal to the source (10) "completed" with a current which ensures local conservation.
This also makes the entire computation self-contained and yields a consistency check for the formalism employed.
After writing the time derivative of the charge density ρ in (10), we set it equal by definition to −∇J c and obtain in this way the conserved current density J c . It is straightforward to check that from the condition one has The standard Maxwell equations in Lorenz gauge in CGS units for the potentials φ c , A c are (the subscript c stays for "conserved") and their solutions (k = c −1 ) The integral for φ c gives The corresponding contribution to the electric field is obtained from −∇φ.
The integral for A c 3 gives (the other components of A c vanish) The corresponding contribution to the electric field is obtained with −k∂ t and is These formulas allow to obtain the components E c T , E c L , taking into account that we have fixed for simplicity θ = 45 • . Setting the distance at r = 10 cm for comparison with the anomalous fields, we can compute the field components for different values of t. Since all components oscillate at high frequency, we take the root mean square of E c T , E c L over many values of t. With 1000 values we obtain As expected, E c L E c T , since we are at a distance r 3λ. With a further increase in the distance we then pass to r = 25 cm (10 cm + 5λ) and r = 40 cm (10 cm + 10λ). Like for r = 10 cm, we find values of E L close to the maxima of the oscillation, but the oscillation amplitude appears to have increased: we have respectively I change signs.
In order to vary the shape of the sources, we change the parameter d in the Gaussian charge density Ansatz (17); this parameter fixes the size of the source in the directions z 1 and z 2 , i.e. transversally with respect to the oscillation direction of the dipole. The data in Tab. I have been obtained setting d = ε = 10 −7 cm, thus with sources having spherical symmetry.
One observes that the longitudinal emission is independent from d, at least up to d = 20 · 10 −7 cm, which corresponds in practice to having two parallel discs instead of two pointlike sources. For practical applications in superconductors this is important, because wide junctions are more likely to carry a large current, in comparison to pointlike contacts.
This independence from d at high frequency should be contrasted with the behavior of the anomalous magnetic field in the quasi-static case [9]: in that case, increasing d rapidly leads to the suppression of the anomaly.

C. Dependence on the frequency
The choice of the oscillation frequency in the calculation is crucial because it defines the wavelength and therefore the integration regions. The value f = 10 10 Hz seems to be a good compromise, because such a frequency can be easily obtained in the self-oscillation of a Josephson junction and is still accessible as an external bias for a molecular nano-device.
We also made some variations of f in the calculation. Setting for instance f = 0.5 · 10 10 Hz, we evaluated the longitudinal field at distance r = 20 cm, which corresponds to little more than 3λ (like r = 10 cm for f = 10 10 Hz), and similarly with f = 2 · 10 10 Hz. In each case, the value of E L found was compared to the r.m.s. of E c T at the same distance for a standard conserved source. The resulting ratios show only a weak dependence on the frequency in this range.

VI. CONCLUSIONS
At the level of fundamental interactions there are no doubts on the full validity of quantum field theory, and in particular of QED and of the principle of local charge conservation.
Nevertheless, in the presence of non-local interactions (either as an effective descriptive model, or with fundamental motivations like in fractional quantum mechanics), the failure of local conservation of the "ρv current" inevitably leads to a new "emergent" phenomenology, characterized by secondary currents which may extend outside the primary source and generate non-standard fields. The real physical properties of these secondary currents are not yet properly understood. We think that experiments will play a fundamental role in clarifying this issue. In our latest work [9] we proposed a design of a device for the detection of anomalous magnetic fields generated by quasi-stationary non-conserved currents. For the case of an high-frequency oscillating source considered in this paper the choice of the experimental strategy is more obvious, namely a search for longitudinal electric fields in the radiation zone. We plan to discuss this in more details in forthcoming work.
Another crucial question is, for which materials the non-local part of the current is expected to achieve the level sufficient for detection (at least 1 part in 10 5 , if we admit for instance that a longitudinal field of the order of 1% of the transverse field can be safely detected).
The choice of the dipole length a for our numerical solution has been motivated by a possible application to Josephson tunnelling in YBCO. In the case of molecular nano-devices the typical sizes and shapes of current sources and sinks arising in the case of local nonconservation should be estimated through the density functional theory; on the experimental side, trials with, e.g., graphene antennas emitting in the GHz range could give useful insights.
Acknowledgment -This work was supported by the Open Access Publishing Fund of the Free University of Bozen-Bolzano.