## Final Report Summary - MUFO (Multiscattering Formalism: Casimir Effect and Related Topics)

The original project had 4 different objectives, I will describe them with the obtained results:

Objective 1 – Casimir effect between spheres and cylinders.

Sub-objective 1.1 - The study of the interaction between a cylinder and a sphere

Sub-objective 1.2 - The study of the Casimir force between two spheres confined in a cylinder.

This part of the project is finished, and has lead to one publication during the grant period and another before it started. This work completes the study of the Casimir effect between the 3 main used geometries: Spheres, cylinders and plates, and with the description of the transformation matrices it opens the door to study the Casimir effect (or any fluctuation induced physical process) between compact objects and axial symmetric objects. In particular, we have studied the Casimir effect between spheres (or atoms instead) and cylinders in the short and long distances limits analytically, and for all distances numerically. We have also studied the modification of the Casimir effect between atoms placed in the cylinder axis, and we have seen that the confinement suppress the Casimir effect with an effective mass proportional to the cylinder radius. This result is completely different if you use a coaxial cylinder instead, because in that case a TEM mode appears and produces a long range Casimir effect between the atoms, as described by E. Shahmoon

(E. Shahmoon, I. Mazets, and G. Kurizki, Giant vacuum forces via transmission lines, Proc. Natl. Acad. Sci. USA 111, 10485 (2014))

Objective 2 – Multiscattering formalism based numerical method for the Casimir effect

This has not been performed since another group proposed a more suitable method before the fellow took the position.

Objective 3 – Generalization of multiscattering formalism to static systems

Sub-objective 3.1 - Interaction energy between charges and dielectrics at any temperature.

Sub-objective 3.2 - Modification of the local density of states (DoS) of the electromagnetic field due to the presence of dielectrics.

The first sub-objective has been seen to be easily solved, but the obtained results were already known in the literature.

The second sub-objective is probably the core topic of the fellowship. It started as the study of the modification of the DoS of the electromagnetic field because of the presence of (isolated) dielectrics in the system, something already done in a particular case by Berry and Howls (M.V. Berry and C. J. Howls, High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, ans the Stokes phenomenon, Proc. R. Soc. Lond. A (1994) 447, 527-555). By studying the formalism, we discovered that it is the same formalism used in the derivation of the PFA and corrections to PFA, and we are generalizing it to study non-equilibrium systems, as the heat emissivity and the non-equilibrium self-force of an isolated object. The problem has demonstrated to be mathematically challenging, and it should be described as a work in progress.

Objective 4 – Generalization of multiscattering formalism to dynamical systems

Sub-objective 4.1 - Casimir effect on systems of dielectrics with relative velocities

Sub-objective 4.2 - Casimir effect of systems with accelerated dielectrics.

The first sub-objective had been performed by an other group before the fellow took the position.

The second sub-objective has leaded to a collaboration with Dr. Eduardo Martin-Martinez of the IQC in University of Waterloo (Ontario, Canada). It consist on the study of the influence of the presence of a plate over a Unruh-DeWitt detector (the pure quantum description of an atom) with an arbitrary given trajectory. We expect this collaboration can be of the interest of two different communities: Casimir physics and Quantum Information.

Objective 1 – Casimir effect between spheres and cylinders.

Sub-objective 1.1 - The study of the interaction between a cylinder and a sphere

Sub-objective 1.2 - The study of the Casimir force between two spheres confined in a cylinder.

This part of the project is finished, and has lead to one publication during the grant period and another before it started. This work completes the study of the Casimir effect between the 3 main used geometries: Spheres, cylinders and plates, and with the description of the transformation matrices it opens the door to study the Casimir effect (or any fluctuation induced physical process) between compact objects and axial symmetric objects. In particular, we have studied the Casimir effect between spheres (or atoms instead) and cylinders in the short and long distances limits analytically, and for all distances numerically. We have also studied the modification of the Casimir effect between atoms placed in the cylinder axis, and we have seen that the confinement suppress the Casimir effect with an effective mass proportional to the cylinder radius. This result is completely different if you use a coaxial cylinder instead, because in that case a TEM mode appears and produces a long range Casimir effect between the atoms, as described by E. Shahmoon

(E. Shahmoon, I. Mazets, and G. Kurizki, Giant vacuum forces via transmission lines, Proc. Natl. Acad. Sci. USA 111, 10485 (2014))

Objective 2 – Multiscattering formalism based numerical method for the Casimir effect

This has not been performed since another group proposed a more suitable method before the fellow took the position.

Objective 3 – Generalization of multiscattering formalism to static systems

Sub-objective 3.1 - Interaction energy between charges and dielectrics at any temperature.

Sub-objective 3.2 - Modification of the local density of states (DoS) of the electromagnetic field due to the presence of dielectrics.

The first sub-objective has been seen to be easily solved, but the obtained results were already known in the literature.

The second sub-objective is probably the core topic of the fellowship. It started as the study of the modification of the DoS of the electromagnetic field because of the presence of (isolated) dielectrics in the system, something already done in a particular case by Berry and Howls (M.V. Berry and C. J. Howls, High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, ans the Stokes phenomenon, Proc. R. Soc. Lond. A (1994) 447, 527-555). By studying the formalism, we discovered that it is the same formalism used in the derivation of the PFA and corrections to PFA, and we are generalizing it to study non-equilibrium systems, as the heat emissivity and the non-equilibrium self-force of an isolated object. The problem has demonstrated to be mathematically challenging, and it should be described as a work in progress.

Objective 4 – Generalization of multiscattering formalism to dynamical systems

Sub-objective 4.1 - Casimir effect on systems of dielectrics with relative velocities

Sub-objective 4.2 - Casimir effect of systems with accelerated dielectrics.

The first sub-objective had been performed by an other group before the fellow took the position.

The second sub-objective has leaded to a collaboration with Dr. Eduardo Martin-Martinez of the IQC in University of Waterloo (Ontario, Canada). It consist on the study of the influence of the presence of a plate over a Unruh-DeWitt detector (the pure quantum description of an atom) with an arbitrary given trajectory. We expect this collaboration can be of the interest of two different communities: Casimir physics and Quantum Information.