I propose an idealized electron interference experiment with a light-port/dark-port configuration and consider the possible outcomes for the average energy of electric potential that is present in a statistically relevant set of single electron measurements. The changes in energy of electric potential of the electron during its evolution will cause a shift in the kinetic energy or momentum of the electron, and this calculated shift may be interpreted differently for the case of interference and non-interference. I also propose an idealized single electron spatial interference effect and the associated changes in states of electric potential.
I propose an idealized electron interference experiment with a light-port/dark-port configuration and consider the possible outcomes for the average energy of electric potential that is present in a statistically relevant set of single electron measurements. The changes in energy of electric potential of the electron during its evolution will cause a shift in the kinetic energy or momentum of the electron, and this calculated shift may be interpreted differently for the case of interference and non-interference. I also propose an idealized single electron spatial interference effect and the associated changes in states of electric potential.
In the so-called electric Aharonov-Bohm effect, a quantum interference pattern shift is produced when electrons move in an electric field-free region but, at the same time, under the presence of a time-dependent electric potential. Analogous fringe shifts are observed in interference experiments where electrons, travelling through an electrostatic field, suffer a classical lag effect produced by electric forces. Since these experiments are reported mainly in specialized journals, it could be rather difficult and time consuming for a student to attain a comprehensive overview of the subject. Therefore, particular attention has been addressed to reviewing the theory, to describing a couple of experiments and to the interpretation of the results. We believe that students can be suitably introduced to exploring the “paradoxical” aspects of the interaction of electrons with electric potentials and fields.
A quantum free particle is linked with a complex probability for impulse strikes at x of exp(ipx), but the momentum and kinetic energy are p and pp/2m everywhere. This suggests that Newtonian ideas of work are associated with p. Conservation of energy, however, is not only linked with pp/2m (nonrelativistically), but with interaction. If one has statistical inteferfence, i.e. an OR probability scenario, then it seems that average kinetic energy for particular p values does not match classical expectations because translational symmtery is broken. There is a shift in interaction probability in x (no longer a unit modulus) and kinetic energy linked with interaction must shift also. Given the root exp(ipx), this shift must occur in additions and subtractions of interaction in space and hence kinetic energy linked with probabilistic interaction. A simple case is that of a single particle bound state with positive parity. In such a case, pp/2m {a(p)exp(ipx) + a(-p)exp(-ipx}) (with a(p)=a(-p) yields a(p) 2 cos(px) pp/2m. Classically, one expects an average kinetic energy from p and -p of pp/2m, but the quantum result may be 0 any fraction less than 1 of pp/2m or the negative of this. It is only the overal KEave(x) that, when added to V(x), yields E. At the single p value one has unusual results. These seem to be due to interference and so we consider two other cases. We next examine exp(ipx) + exp(i2px). Classically, one would expect an average kinetic energy of ½ ( pp/2m + 4pp/2m) = 2.5 pp/2m, but for px= 3.14, one has 3 (pp/2m) which exceeds the classical average. As a third example, one may consider an equilibrium of a sin(pave x) state in an infinite well potential with a sin(2pave x) state (maintained by a photon bath at a certain temperature. If exp(-pp/2mT) is set to .5, then for pave x= 3.14 + 3.14/4, one has an average kinetic energy of .52 pave* pave/2m which is much less than the minimum classical average of pave*pave/2m. Thus, quantum interference seems to have unusual effects on average kinetic energy with respect to classical expectations.
It is shown that the maximal phase sensitivity of a two-path interferometer with high-intensity coherent light and squeezed vacuum in the input ports can be achieved by photon-number-resolving detection of only a small number of photons in a dark output port. It is then possible to achieve the quantum Cram\'er-Rao bound of the two-path interferometer using only the field displacement dependence of the photon number statistics in the single mode output of the dark port represented by a field-displaced squeezed vacuum state. We find that, at small field displacements, it is not sufficient to use the average photon number as the estimator, indicating that an optimal phase estimation depends critically on measurements of the precise photon number. We therefore analyze the effect of detection efficiency on the Fisher information and show that there is a transition from low robustness against photon losses associated with quantum interference effects at low field displacements to high robustness against photon losses at high field displacements. The transition between the two regimes occurs at field shifts proportional to the third power of the squeezing factor, indicating that squeezing greatly enhances the phase interval in which quantum effects are relevant in optimal phase estimations using photon resolving detectors. The case under study could thus be understood as a ``missing link'' between genuine multiphoton interference and the straightforward suppression of noise usually associated with squeezed light.
In classical mechanics, one assumes one may measure precise momentum and x positions. One may argue, however, that a measurement involves changing the momentum of a particle (i.e. an impulse hit) and that such a change does not occur either in an instant of time or at a point x. For example, a classical particle in a potential V(x) changes its velocity in a region dx, but at the same time is said to have a velocity of v(x) to leading order as dx shrinks to zero. One may argue that dx does not shrink to zero and may not be tiny with respect to an overall length scale. Given that an interaction involves a particle and a potential or another particle, one may argue that a finite dx should be associated with both. Thus a particle with constant momentum should have a tiny dx associated with it, possibly a function of p, but at the same time there is a constant probability for a measurement to take place at any x point.Thus we argue that there are two sets of probability schemes, one for dx and another for a larger length scale and constant probability. This should point to two entropy schemes, each associated with the type of measurement done and its analysis. For example, a two slit system may be used to measure the interaction probability. This gives rise to an interference probability. At the same time, many two slit system may be used to measure the center of mass position of the particle, ignoring the interference pattern. This gives rise to P(x)=1/L and a completely different Shannon’s entropy. Given that there is a dx probability scheme for a constant p associated with interactions, an associated entropy should appear when such an interaction occurs, but this now involves breaking the constant p and considering the associated dx probabilities. Thus there is both momentum and spatial entropy combined as in two slit interference. In the case of a bound state, there are numerous exp(ipx) eigenstates (due to interactions with V(x)), suggesting both internal p and x entropies, but overall one has a single energy eigenstate. In terms of energy one has a fixed state (although exp(-iEn t) suggests there are frequency pockets in time pointing towards a kind of time entropy. If one considers only interactions which change a particle from one En to another, a particle in a single state at T=temperature=0 has entropy 0 (even though the state itself has momentum and spatial entropy). Thus it seems entropy is selective depending on the type of interactions present and what eigenvalues they change. In other words, classically one counts states when computing entropy, but in quantum mechanics one may count states by defining states in terms of quantum numbers. One may consider a bound state to be a single certain state with eigenvalue En and consider entropy being created if interactions (photon/phonon absorption/emission) break the single En. In such a case, one may ignore the breaking of exp(ipx)s within a bound state which implies momentum entropy as well as the spatial entropy linked with even a single exp(ipx). In other words, one may select various different kinds of probabilities linked with different eigenvalues and their measurements when calculating Shannon’s entropy.