Search Results (5)

In the framework of the quantum theory of many-particle systems, we study the compatibility of approximated non-equilibrium Green’s functions (NEGFs) and of approximated solutions of the Dyson equation with a modified continuity equation of the form ${\partial }_t⟨\rho ⟩+(1-\gamma )\nabla ·⟨\mathbf{J}⟩=0$. A continuity equation of this kind allows the e.m. coupling of the system in the extended Aharonov–Bohm electrodynamics, but not in Maxwell electrodynamics. Focusing on the case of molecular junctions simulated numerically with the Density Functional Theory (DFT), we further discuss the re-definition of local current density proposed by Wang et al., which also turns out to be compatible with the extended Aharonov–Bohm electrodynamics. Full article

We reconsider some well-known tunneling processes from the point of view of Aharonov-Bohm electrodynamics, a unique extension of Maxwell’s theory which admits charge-current sources that are not locally conserved. In particular we are interested into tunneling phenomena having relatively long range (otherwise the non-Maxwellian effects become irrelevant, especially at high frequency) and involving macroscopic wavefunctions and coherent matter, for which it makes sense to evaluate the classical e.m. field generated by the tunneling particles. For some condensed-matter systems, admitting discontinuities in the probability current is a possible way of formulating phenomenological models. In such cases, the Aharonov-Bohm theory offers a logically consistent approach and allows to derive observable consequences. Typical e.m. signatures of the failure of local conservation are at high frequency the generation of a longitudinal electric radiation field, and at low frequency a small effect of “missing” magnetic field. Possible causes of this failure are instant tunneling and phase slips in superconductors. For macroscopic quantum systems in which the phase-number uncertainty relation $\mathsf{\Delta }N\mathsf{\Delta }\phi \sim 1$ applies, the expectation value of the anomalous source $I={\partial }_t\rho +\nabla ·\mathbf{j}$ has quantum fluctuations, thus becoming a random source of weak non-Maxwellian fields. Full article

In quantum theory, for a system with macroscopic wavefunction, the charge density and current density are represented by non-commuting operators. It follows that the anomaly $I={\partial }_t\rho +\nabla ·\mathbf{j}$, being essentially a linear combination of these two operators in the frequency-momentum domain, does not admit eigenstates and has a minimum uncertainty fixed by the Heisenberg relation $\mathsf{\Delta }N\mathsf{\Delta }\varphi \simeq 1$, which involves the occupation number and the phase of the wavefunction. We give an estimate of the minimum uncertainty in the case of a tunnel Josephson junction made of Nb. Due to this violation of the local conservation of charge, for the evaluation of the e.m. field generated by the system it is necessary to use the extended Aharonov–Bohm electrodynamics. After recalling its field equations, we compute in general form the energy–momentum tensor and the radiation power flux generated by a localized oscillating source. The physical requirements that the total flux be positive, negative or zero yield some conditions on the dipole moment of the anomaly I. Full article

An ongoing debate in the first-principles description of conduction in molecular devices concerns the correct definition of current in the presence of non-local potentials. If the physical current density $\mathbf{j}=(-ie\hslash /2m)({\Psi }^{*}\nabla \Psi -\Psi \nabla {\Psi }^{*})$ is not locally conserved but can be re-adjusted by a non-local term, which current should be regarded as real? Situations of this kind have been studied for example, for currents in saturated chains of alkanes, silanes and germanes, and in linear carbon wires. We prove that in any case the extended Maxwell equations by Aharonov-Bohm give the e.m. field generated by such currents without any ambiguity. In fact, the wave equations have the same source terms as in Maxwell theory, but the local non-conservation of charge leads to longitudinal radiative contributions of $\mathbf{E}$, as well as to additional transverse radiative terms in both $\mathbf{E}$ and $\mathbf{B}$. For an oscillating dipole we show that the radiated electrical field has a longitudinal component proportional to $\omega \widehat{P}$, where $\widehat{P}$ is the anomalous moment $\int \widehat{I}\left(\mathbf{x}\right)\mathbf{x}{d}^{3}x$ and $\widehat{I}$ is the space-dependent part of the anomaly $I={\partial }_t\rho +\nabla ·\mathbf{j}$. For example, if a fraction $\eta$ of a charge q oscillating over a distance $2a$ lacks a corresponding current, the predicted maximum longitudinal field (along the oscillation axis) is $E_{L,max}=2\eta {\omega }^{2}qa/\left(c^{2}r\right)$. In the case of a stationary current in a molecular device, a failure of local current conservation causes a “missing field” effect that can be experimentally observable, especially if its entity depends on the total current; in this case one should observe at a fixed position changes in the ratio $B/i$ in dependence on i, in contrast with the standard Maxwell equations. The missing field effect is confirmed by numerical solutions of the extended equations, which also show the spatial distribution of the non-local term in the current. Full article

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 density $\mathbf{J}$ with a current ${\mathbf{J}}^{ext}$ 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 superconductor-normal-superconductor (SNS) junctions in yttrium barium oxide (YBCO) and by current flow in molecular nanodevices. Full article