Research

Hybrid quantum-classical systems

Hybrid quantum-classical systems are dynamical systems where classical and quantum degrees of freedom evolve coupled to each other. Usually, they are approximations to full quantum models where some of the degrees of freedom correspond to particles with mass much higher than others and which therefore evolve with much longer characteristic times. These heavier and slower particles can be accurately described by classical models and therefore the whole system can be modeled by a combination of these classical objects interacting with the remaining quantum degrees of freedom used for the light and quick particles. The best known example for this type of dynamical system are molecular systems where nuclei and inner electrons can be described as a classical subsystem which interacts with the valence electrons. These valence electrons must keep a quantum description if we want to model accurately some physical phenomena as chemical reactivity or photo-induced processes such as photosynthesis. The interaction of a quantum system and a classical measuring device also fits within this framework, even if the problem becomes extremely complicated by the large difference in scale. In these approaches, the extension to statistical mechanical models based on hybrid dynamics is a natural requirement because of the macroscopic nature of the classical domains.

Hybrid quantum-classical field models

But we can find a similar situation when considering field theoretical models. Indeed, we can also consider a model where a quantum field theory interacts with a classical field. Several interesting examples fit in this framework, the most remarkable one being a quantum field theory which evolves in a given gravitational background. In this situation, models based on Minkowski-backgrounds field theories fail and all ingredients in our model must be reconsidered. Our approach is based on Malliavin stochastic calculus which provides rigorous mathematical tools to operate on spaces of fields modeled as Hida functions or distributions defined on the space of classical fields. These sets admit rigorously defined Gaussian measures and therefore square integrable functionals depending on classical degrees of freedom become meaningful. States become sections of suitable bundles defined on the space of classical objects and such dependence requires of the definition of a suitable connection to capture the effect on the quantum state of a change in the classical parameter. The simple case of scalar quantum field models is being considered now but more general cases will be analyzed in the future. Besides the case of external gravitational classical fields we plan to expand the framework to other cases of quantum-classical interactions such as the Casimir or the Schwinger effects.

Hybrid geometrodynamics

We generalize the Hamiltonian picture of General Relativity coupled to classical matter, known as geometrodynamics, to the case where such matter is described by a Quantum Field Theory in Curved Spacetime, but gravity is still described by a classical metric tensor field over a spatial hypersurface and its associated momentum. Thus, in our approach there is no non-dynamic background structure, apart from the manifold of events, and the gravitational and quantum degrees of freedom have their dynamics inextricably coupled. Given the Hamiltonian natureof the framework, we work with the generators of hypersurface deformations over the manifold of quantum states. The construction relies heavily on the differential geometry of a fibration of the set of quantum states over the set of gravitational variables. An important feature of this work is the use of Gaussian measures over the space of matter fields and of Hida distributions to define a common superspace to all possible Hilbert spaces with different measures, to properly characterize the Schrödinger wave functional picture of QFT in curved spacetime. The most remarkable physical features of the construction are norm conservation of the quantum state (even if the total dynamics are non-unitary), the clear identification of the hybrid conserved quantities and the description of a dynamical backreaction of quantum matter on geometry and vice versa, which shall modify the physical properties the gravitational field would have in the absence of backreaction.

Funding

PROJECT: HQ-DYNCONTROL: Dynamics and control of quantum and hybrid quantum-classical systems
REFERENCE: PID2021-123251NB-I00
PRINCIPAL INVESTIGATOR(S): Alberto Castro, Jesús Clemente-Gallardo
BUDGET: 48.000 €
INSTITUTION: AEI- EU
YEARS: 2022-2026

Research team

• Alberto Castro (co-PI)
• Jesús Clemente-Gallardo (co-PI)
• Carlos Bouthelier-Madre (PhD student)
• David Martínez-Crespo (PhD student)