13 research outputs found
Casimir-Polder interaction between a polarizable particle and a plate with a hole
We determine exactly the non-retarded Casimir-Polder interaction between a
neutral but polarizable particle and a perfectly reflecting sheet containing a
circular hole. The calculation reveals a strong dependence of the interaction
on the orientation of the particle's electric dipole moment with respect to the
surface. For a dipole moment that is polarized perpendicularly to the surface
the interaction potential has two saddle points lying above and below the plane
of the surface, on a line through the centre of the aperture. For a dipole
moment that is polarized parallel to the surface there is only one saddle point
right in the middle of the aperture. Provided the particles' motion could be
confined to a line through the middle of the aperture, this effect could
potentially be used for population-sensitive trapping of particles.Comment: 4 pages, 5 figure
Interaction of an atom with layered dielectrics
We determine the energy-level shift experienced by a neutral atom due the quantum electromagnetic interaction with a layered dielectric body. We use the technique of normal-mode expansion to quantize the electromagnetic field in the presence of a layered, nondispersive, and nonabsorptive dielectric. We explicitly calculate the equal-time commutation relations between the electric field and vector potential operators. We show that the commutator can be expressed in terms of a generalized transverse d function and that this is a consequence of using the generalized Coulomb gauge to quantize the electromagnetic field. These mathematical tools turn out to be very helpful in the calculation of the energy-level shift of the atom, which can be in its ground state or excited. The results for the shift are then analyzed asymptotically in various regions of the systems parameter space, with a view to providing quick estimates of the influence of a single dielectric layer on the Casimir-Polder interaction between an atom and a dielectric half space. We also investigate the impact of resonances between the wavelength of the atomic transition and the thickness of the top layer
Quantum elecrodynamics near material boundaries
Quantum electrodynamics in free-space is a well-understood and a very successful theory. This is not the case when polarizable boundaries are present, which is a common scenario. The presence of reflective surfaces affects the photon field. Thereby the quantummechanical vacuum fluctuations of the electromagnetic field are constrained leading to changes in the interaction energies of charged particles which are directly measurable. One of the most famous examples of such an effect is the Lamb shift of an atom in front of a perfectly reflecting mirror, which depends on the distance of the atom from the mirror, thus giving rise to an attractive force - the so-called Casimir-Polder force. This thesis touches upon current challenges of quantum electrodynamics with externally applied boundary conditions, which is of increasing importance for nanotechnology and its applications in physics, chemistry and biology. When studying the abovementioned vacuum effects one can use models of various degrees of sophistication for the material properties that need to be taken into account. The simplest is to assume perfect reflectivity. This leads to simple boundary conditions on the electromagnetic field and thereby its quantum fluctuations. The difficulty of such calculations then lies only in the possibly complex geometry of the macroscopic body. The next possible level of sophistication is to allow imperfect reflectivity. The simplest way to achieve this is by considering a material with constant and frequency-independent refractive index. However, for all real material surfaces the reflectivity is frequency-dependent. Causality then requires that dispersion is accompanied by absorption. The aim of this project was twofold: (i) to construct, using well-understood tools of theoretical physics, the microscopic theory of quantum systems, like atoms, interacting with macroscopic polarizable media, which would facilitate relatively simple perturbative calculations of QED corrections due to the presence of boundaries, (ii) to apply the developed formalism to the calculation of the Casimir-Polder force between an atom and a realistic material
Gauge transformation in macroscopic quantum electrodynamics near polarizable surfaces
To describe charged particles interacting with the quantized electromagnetic
field, we point out the differences of working in the so-called generalized and
the true Coulomb gauges. We find an explicit gauge transformation between them
for the case of the electromagnetic field operators quantized near a
macroscopic boundary described by a piece-wise constant dielectric function.
Starting from the generalized Coulomb gauge we transform operators into the
true Coulomb gauge where the vector potential operator is truly transverse
everywhere. We find the generating function of the gauge transformation to
carry out the corresponding unitary transformation of the Hamiltonian and show
that in the true Coulomb gauge the Hamiltonian of a particle near a polarizable
surface contains extra terms due to the fluctuating surface charge density
induced by the vacuum field. This extra term is represented by a
second-quantised operator on equal footing with the vector field operators. We
demonstrate that this term contains part of the electrostatic energy of the
charged particle interacting with the surface and that the gauge invariance of
the theory guarantees that the total interaction energy in all cases equals the
well known result obtainable by the method of images when working in
generalized Coulomb gauge. The mathematical tools we have developed allow us to
work out explicitly the equal-time commutation relations and shed some light on
typical misconceptions regarding issues of whether the presence of the
boundaries should affect the field commutators or not, especially when the
boundaries are modelled as perfect reflectors.Comment: 10 pages, 1 figur
Gauge transformation in macroscopic quantum electrodynamics near polarizable surfaces
To describe charged particles interacting with the quantized electromagnetic
field, we point out the differences of working in the so-called generalized and
the true Coulomb gauges. We find an explicit gauge transformation between them
for the case of the electromagnetic field operators quantized near a
macroscopic boundary described by a piece-wise constant dielectric function.
Starting from the generalized Coulomb gauge we transform operators into the
true Coulomb gauge where the vector potential operator is truly transverse
everywhere. We find the generating function of the gauge transformation to
carry out the corresponding unitary transformation of the Hamiltonian and show
that in the true Coulomb gauge the Hamiltonian of a particle near a polarizable
surface contains extra terms due to the fluctuating surface charge density
induced by the vacuum field. This extra term is represented by a
second-quantised operator on equal footing with the vector field operators. We
demonstrate that this term contains part of the electrostatic energy of the
charged particle interacting with the surface and that the gauge invariance of
the theory guarantees that the total interaction energy in all cases equals the
well known result obtainable by the method of images when working in
generalized Coulomb gauge. The mathematical tools we have developed allow us to
work out explicitly the equal-time commutation relations and shed some light on
typical misconceptions regarding issues of whether the presence of the
boundaries should affect the field commutators or not, especially when the
boundaries are modelled as perfect reflectors
Exact Casimir-Polder potentials: interaction of an atom with a conductor-patched dielectric surface
We study the interaction between a neutral atom or molecule and a
conductor-patched dielectric surface. We model this system by a perfectly
reflecting disc lying atop of a non-dispersive dielectric half-space, both
interacting with the neutral atom or molecule. We assume the interaction to be
non-retarded and at zero temperature. We find an exact solution to this
problem. In addition we generate a number of other useful results. For the case
of no substrate we obtain the exact formula for the van der Waals interaction
energy of an atom near a perfectly conducting disc. We show that the
Casimir-Polder force acting on an atom that is polarized in the direction
normal to the surface of the disc displays intricate behaviour. This part of
our results is directly relevant to recent matter-wave experiments in which
cold molecules are scattered by a radially symmetric object in order to study
diffraction patterns and the so-called Poisson spot. Furthermore, we give an
exact expression for the non-retarded limit of the Casimir-Polder interaction
between an atom and a perfectly-conducting bowl.Comment: 9 pages, 9 figure
Quantum electrodynamics near anisotropic polarizable materials: Casimir-Polder shifts near multilayers of graphene
In a recent paper, we formulated a theory of nonrelativistic quantum electrodynamics in the presence of an inhomogeneous Huttner-Barnett dielectric. Here we generalize the formalism to anisotropic materials and show how it may be modified to include conducting surfaces. We start with the derivation of the photon propagator for a slab of material and use it to work out the energy-level shift near a medium whose conductivity in the direction parallel to the surface far exceeds that in the direction perpendicular to the surface. We investigate the influence of the anisotropy of the material's electromagnetic response on the Casimir-Polder shifts, both analytically and numerically, and show that it may have a significant impact on the atom-surface interaction, especially in the nonretarded regime, i.e., for small atom-surface separations. Our results for the energy shift may be used to estimate the Casimir-Polder force acting on quantum objects close to multilayers of graphene or graphite. They are particularly important for the case of trapped cold molecules whose dispersive interactions with surfaces often fall within the nonretarded regime where the anisotropy of the material strongly influences the Casimir-Polder force. We also give a formula for the change in the spontaneous decay rate of an excited atom or molecule near an anisotropically conducting surface
Magnetic moment of an electron near a surface with dispersion
Boundary-dependent radiative corrections that modify the magnetic moment of an electron near a dielectric or conducting surface are investigated. Normal-mode quantization of the electromagnetic field and perturbation theory applied to the Dirac equation for a charged particle in a weak magnetic field yield a general formula for the magnetic moment correction in terms of any choice of electromagnetic mode functions. For two particular models, a non-dispersive dielectric and an undamped plasma, it is shown that, by using contour integration techniques over a complex wave vector, this can be simplified to a formula featuring just integrals over TE and TM reflection coefficients of the surface. Analysing the magnetic moment correction for several models of surfaces, we obtain markedly different results from the previously considered simplistic 'perfect reflector' model, which is due to the inclusion of physically important features of the surface like evanescent field modes and dispersion in the material. Remarkably, for a general dispersive dielectric surface, the magnetic moment correction of an electron nearby has a peak whose position and height can be tuned by choice of material parameters
Force on a neutral atom near conducting microstructures
We derive the non-retarded energy shift of a neutral atom for two different
geometries. For an atom close to a cylindrical wire we find an integral
representation for the energy shift, give asymptotic expressions, and
interpolate numerically. For an atom close to a semi-infinite halfplane we
determine the exact Green's function of the Laplace equation and use it derive
the exact energy shift for an arbitrary position of the atom. These results can
be used to estimate the energy shift of an atom close to etched microstructures
that protrude from substrates.Comment: 7 pages, 5 figure
Retarded Casimir-Polder force on an atom near reflecting microstructures
We derive the fully retarded energy shift of a neutral atom in two different
geometries useful for modelling etched microstructures. First we calculate the
energy shift due to a reflecting cylindrical wire, and then we work out the
energy shift due to a semi-infinite reflecting half-plane. We analyze the
results for the wire in various limits of the wire radius and the distance of
the atom from the wire, and obtain simple asymptotic expressions useful for
estimates. For the half-plane we find an exact representation of the
Casimir-Polder interaction in terms of a single, fast converging integral,
which is easy to evaluate numerically.Comment: 12 pages, 8 figure