We predict a generic signature of quantum interference in many-body bosonic systems resulting in a coherent enhancement of the average return probability in Fock space. This enhancement is robust with respect to variations of external parameters even though it represents a dynamical manifestation of the delicate superposition principle in Fock space. It is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. Using a semiclassical approach based on interfering paths in Fock space, we calculate the magnitude of the backscattering peak and its dependence on gauge fields that break time-reversal invariance. We confirm our predictions by comparing them to exact quantum evolution probabilities in Bose-Hubbard models, and discuss their relevance in the context of many-body thermalization.

We consider the many-body spectra of interacting bosonic quantum fields on a lattice in the semiclassical limit of large particle number N . We show that the many-body density of states can be expressed as a coherent sum over oscillating long-wavelength contributions given by periodic, nonperturbative solutions of the, typically nonlinear, wave equation of the classical (mean-field) limit. To this end, we construct the semiclassical approximation for both the smooth and oscillatory parts of the many-body density of states in terms of a trace formula starting from the exact path integral form of the propagator between many-body quadrature states. We therefore avoid the use of a complexified classical limit characteristic of the coherent state representation. While quantum effects such as vacuum fluctuations and gauge invariance are exactly accounted for, our semiclassical approach captures quantum interference and therefore is valid well beyond the Ehrenfest time where naive quantum-classical correspondence breaks down. Remarkably, due to a special feature of harmonic systems with incommensurable frequencies, our formulas are generically valid also in the free-field case of noninteracting bosons.

Quantum cavities or dots have markedly different properties depending on whether their classical counterparts are chaotic or not. Connecting a superconductor to such a cavity leads to notable proximity effects, particularly the appearance, predicted by random matrix theory, of a hard gap in the excitation spectrum of quantum chaotic systems. Andreev billiards are interesting examples of such structures built with superconductors connected to a ballistic normal metal billiard since each time an electron hits the superconducting part it is retroreflected as a hole (and vice versa). Using a semiclassical framework for systems with chaotic dynamics, we show how this reflection, along with the interference due to subtle correlations between the classical paths of electrons and holes inside the system, is ultimately responsible for the gap formation. The treatment can be extended to include the effects of a symmetry-breaking magnetic field in the normal part of the billiard or an Andreev billiard connected to two phase-shifted superconductors. Therefore, we are able to see how these effects can remold and eventually suppress the gap. Furthermore, the semiclassical framework is able to cover the effect of a finite Ehrenfest time, which also causes the gap to shrink. However, for intermediate values this leads to the appearance of a second hard gap-a clear signature of the Ehrenfest time.

We present a rigorous derivation of a semiclassical propagator for anticommuting (fermionic) degrees of freedom, starting from an exact representation in terms of Grassmann variables. As a key feature of our approach the anticommuting variables are integrated out exactly, and an exact path integral representation of the fermionic propagator in terms of commuting variables is constructed. Since our approach is not based on auxiliary (Hubbard-Stratonovich) fields, it surpasses the calculation of fermionic determinants yielding a standard form D[ψ, ψ * ]e iR[ψ,ψ * ] with real actions for the propagator. These two features allow us to provide a rigorous definition of the classical limit of interacting fermionic fields and therefore to achieve the long-standing goal of a theoretically sound construction of a semiclassical van Vleck-Gutzwiller propagator in fermionic Fock space.As an application, we use our propagator to investigate how the different universality classes (orthogonal, unitary and symplectic) affect generic many-body interference effects in the transition probabilities between Fock states of interacting fermionic systems.

When coupling a superconductor to a normal conducting region the physical properties of the system are highly affected by the superconductor. We investigate the effect of one or two superconductors on the conductance of a ballistic chaotic quantum dot to leading order in the total channel number using trajectory-based semiclassics. The results show that the effect of one superconductor on the conductance is of the order of the number of channels and that the sign of the quantum correction from the Drude conductance depends on the particular ratios of the numbers of channels of the superconducting and normal conducting leads. In the case of two superconductors with the same chemical potential, we additionally study how the conductance and the sign of quantum corrections are affected by their phase difference. As far as random matrix theory results exist these are reproduced by our calculations. Furthermore, in the case that the chemical potential of the superconductors is the same as that of one of the two normal leads the conductance shows, under certain conditions, similar effects as a normal metal-superconductor junction. The semiclassical framework is also able to treat the thermopower of chaotic Andreev billiards consisting of one chaotic dot, two normal leads, and two superconducting islands and shows it to be antisymmetric in the phase difference of the superconductors.