We use a functional approach to the Casimir effect in order to evaluate the exact vacuum energy for a real scalar field in $d+1$ dimensions, in the presence of backgrounds that, in a particular limit, impose Dirichlet boundary conditions on one or two parallel surfaces. Outside of that limit, the background may be thought of as describing finite-width mirrors with frequency-dependent transmission and reflection coefficients. We provide new explicit results for the Casimir energy in some particular backgrounds
We compute the Casimir energy of a real scalar field in the presence of a pair of partially transparent plane mirrors, modeled by Dirac delta potentials.
We extend our previous work on a derivative expansion for the Casimir energy, to the case of the electromagnetic field coupled to two thin, imperfect mirrors. The latter are described by means of vacuum polarization tensors localized on the mirrors. We apply the results so obtained to compute the first correction to the proximity force approximation to the static Casimir effect.
We investigate the Casimir force between two dissimilar plane mirrors the material properties of which are described by Drude or Lorentz models. We calculate analytically the short and long distance asymptote of the force and relate its behavior to the influence of interacting surface plasmons. In particular we discuss conditions under which Casimir repulsion could be achieved.
We study the Casimir interaction energy due to the vacuum fluctuations of the Electromagnetic (EM) field in the presence of two mirrors, described by $2+1$-dimensional, generally nonlocal actions, which may contain both parity-conserving and parity-breaking terms. We compare the results with the ones corresponding to Chern-Simons boundary conditions, and evaluate the interaction energy for several particular situations.
We consider a real massless scalar field in 1+1 dimensions satisfying time-dependent Robin boundary condition at a static wall. This condition can simulate moving reflecting mirrors whose motions are determined by the time-dependence of the Robin parameter. We show that particles can be created from vacuum, characterizing in this way a dynamical Casimir effect.
We investigate, in the context of a real massless scalar field in $1+1$ dimensions, models of partially reflecting mirrors simulated by Dirac $δ-δ^{\prime}$ point interactions. In the literature, these models do not exhibit full transparency at high frequencies. In order to provide a more realistic feature for these models, we propose a modified $δ-δ^{\prime}$ point interaction that enables full transparency in the limit of high frequencies. Taking this modified $δ-δ^{\prime}$ model into account, we investigate the Casimir force, comparing our results with those found in the literature.
We study a quantum model of dynamical Casimir effect in an optical cavity enclosed by a freely moving mirror attached to a harmonic spring. The quantum fluctuations of the friction force exerted by the dynamical Casimir emission onto the moving mirror are investigated, as well as their consequences on the quantum state of the mirror oscillation. Observable signatures of the interplay of the nonlinear nature of the effective mirror-cavity coupling and of the discreteness of the emitted photons are pointed out, in particular as a fast diffusion of the mirror oscillation phase. These results are …
We study the Dynamical Casimir Effect resulting from the oscillatory motion of either one or two flat semitransparent mirrors, coupled to a quantum real and massless scalar field. Our approach is based on a perturbative evaluation, in the coupling between mirrors and field, of the corresponding effective action, which is used to compute the particle creation rate. The amplitude of the oscillation is not necessarily small. We first obtain results for a single mirror, both for non-relativistic and relativistic motions, showing that only for the latter the effects may be significant. For two mirr…
We calculate the Casimir interaction energy in $d=2$ spatial dimensions between two (zero-width) mirrors, one flat, and the other slightly curved, upon which {\em imperfect\/} conductor boundary conditions are imposed for an Electromagnetic (EM) field. Our main result is a second-order Derivative Expansion (DE) approximation for the Casimir energy, which is studied in different interesting limits. In particular, we focus on the emergence of a non-analyticity beyond the leading-order term in the DE, when approaching the limit of perfectly-conducting mirrors. We also show that the system conside…
We study the Casimir energy due to a quantum real scalar field coupled to two planar, infinite, zero-width, parallel mirrors with non-homogeneous properties. These properties are represented, in the model we use, by scalar functions defined on each mirror's plane. Using the Gelfand-Yaglom's theorem, we construct a Lifshitz-like formula for the Casimir energy of such a system. Then we use it to evaluate the energy perturbatively, for the case of almost constant scalar functions, and also implementing a Derivative Expansion, under the assumption that the spatial dependence of the properties is s…
We study the general form of the Casimir interaction between two flat finite-width, parallel, infinite mirrors, described by their respective vacuum polarization tensors, under the assumption of translation invariance along the directions parallel to the planes of the mirrors. Their properties along the normal coordinates are left, on the other hand, rather arbitrary, but constrained by the Ward identity for the vacuum polarization tensors. We construct solutions to those identities, evaluating the corresponding Casimir energy of the system by reducing the problem to a collection of one-dimens…
We calculate the Casimir energy for scalar and gauge fields in interaction with zero-width mirrors, including quantum effects due to the matter fields inside the mirrors. We consider models where those fields are either scalar or fermionic, obtaining general expressions for the energy as a function of the vacuum field 1PI function. We also study, within the frame of a concrete model, the role of the dissipation induced by those degrees of freedom, showing that, after integration of the matter fields, the effective theory for the electromagnetic field contains modes with complex energies. As fo…
The leading radiative correction to the Casimir energy for two parallel penetrable mirrors is calculated within QED perturbation theory. It is found to be of the order $α$ like the known radiative correction for ideally reflecting mirrors from which it differs only by a monotonic, powerlike function of the frequency at which the mirrors become transparent. This shows that the $O(α^2)$ radiative correction calculated recently by Kong and Ravndal for ideally reflecting mirrors on the basis of effective field theory methods remains subleading even for the physical case of penetrable mirrors.
We demonstrate that the Unruh-DeWitt harmonic-oscillator detectors in (1+1) dimensions derivative-coupled with a massless scalar field can mimic the atom mirrors in free space. Without introducing the Dirichlet boundary condition to the field, the reflectivity of our detector/atom mirror is dynamically determined by the interaction of the detector's internal oscillator and the field. When the oscillator-field coupling is strong, a broad frequency range of the quantum field can be mostly reflected by the detector mirror at late times. Constructing a cavity model with two such detector mirrors, …
After reviewing some essential features of the Casimir effect and, specifically, of its regularization by zeta function and Hadamard methods, we consider the dynamical Casimir effect (or Fulling-Davis theory), where related regularization problems appear, with a view to an experimental verification of this theory. We finish with a discussion of the possible contribution of vacuum fluctuations to dark energy, in a Casimir like fashion, that might involve the dynamical version.
We calculate the spectrum and the total rate of created particles for a real massless scalar field in $1+1$ dimensions, in the presence of a partially transparent moving mirror simulated by a Dirac $δ-δ^{\prime}$ point interaction. We show that, for this model, a partially reflecting mirror can produce a larger number of particles in comparison with a perfect one. In the limit of a perfect mirror, our formulas recover those found in the literature for the particle creation by a moving mirror with a Robin boundary condition.
We comment on a recent publication [1] by Fosco, Lombardo and Mazzitelli on Casimir energies for material slabs (`finite width mirrors') and report a discrepancy between results obtained there for a single mirror and some previous calculations. We provide a simple consistency check which proves that the method used in [1] is not reliable when applied to approximations of piecewise constant profile of the mirror. We also present an alternative method for calculation of the Casimir energy in such systems based on our recent work. Our results coincide both with perturbation theory and with some o…
We study the field theoretical model of a scalar field in presence of spacial inhomogeneities in form of one and two finite width mirrors (material slabs). The interaction of the scalar field with the defect is described with position-dependent mass term. Within this model we derive the interaction of two finite width mirrors, establish the correspondence of the model to the Lifshitz formula and construct limiting procedure to obtain finite self-energy of a single mirror without any normalization condition.
We apply a perturbative approach to evaluate the Casimir energy for a massless real scalar field in 3+1 dimensions, subject to Dirichlet boundary conditions on two surfaces. One of the surfaces is assumed to be flat, while the other corresponds to a small deformation, described by a single function $η$, of a flat mirror. The perturbative expansion is carried out up to the fourth order in the deformation $η$, and the results are applied to the calculation of the Casimir energy for corrugated mirrors in front of a plane. We also reconsider the proximity force approximation within the context of …
The generation of photons in a three dimensional rectangular cavity with two moving boundaries is studied by using the Multiple Scale Analysis (MSA). It is shown that number of photons are enhanced for the cavity whose walls oscillate symmetrically with respect to the center of the cavity. The non-stationary Casimir effect is also discussed for the cavity which oscillates as a whole.
We consider an optical cavity enclosed by a freely moving mirror attached to a spring and we study the quantum friction effect exerted by the dynamical Casimir emission on the mechanical motion of the mirror. Observable signatures of this simplest example of back-reaction effect are studied in both the ring-down oscillations of the mirror motion and in its steady-state motion under a monochromatic force. Analytical expressions are found in simple yet relevant cases and compared to complete numerical solution of the master equation. A circuit-QED device allowing for experimental observation of …
In this article we compute the Casimir force between two finite-width mirrors at finite temperature, working in a simplified model in 1+1 dimensions. The mirrors, considered as dissipative media, are modeled by a continuous set of harmonic oscillators which in turn are coupled to an external environment at thermal equilibrium. The calculation of the Casimir force is performed in the framework of the theory of quantum open systems. It is shown that the Casimir interaction has two different contributions: the usual radiation pressure from vacuum, which is obtained for ideal mirrors without dissi…
Radiation from accelerating mirrors in a Minkowski spacetime provides insights into the nature of horizons, black holes, and entanglement entropy. We introduce new, simple, symmetric and analytic moving mirror solutions and study their particle, energy, and entropy production. This includes an asymptotically static case with finite emission that is the black hole analog of complete evaporation. The total energy, total entropy, total particles, and spectrum are the same on both sides of the mirror. We also study its asymptotically inertial, drifting analog (which gives a black hole remnant) to …
I show that reflection positivity implies that the force between any mirror pair of charge-conjugate probes of the quantum vacuum is attractive. This generalizes a recent theorem of Kenneth and Klich to interacting quantum fields, to arbitrary semiclassical bodies, and to quantized probes with non-overlapping wavefunctions. I also prove that the torques on charge-conjugate probes tend always to rotate them into a mirror-symmetric position.
We provide a Quantum Field Theory derivation of Lifshitz formula for the Casimir force due to a fluctuating real scalar field in $d+1$ dimensions. The field is coupled to two imperfect, thick, plane mirrors, which are modeled by background potentials localized on their positions. The derivation proceeds from the calculation of the vacuum energy in the Euclidean version of the system, reducing the problem to the evaluation of a functional determinant. The latter is written, via Gelfand-Yaglom's formula, in terms of functions depending on the structure of the potential describing each mirror; th…
We propose a method to separate enantiomers using parity violation in the Casimir-Polder potential between chiral mirrors and chiral molecules. The proposed setup involves a molecular beam composed of chiral molecules passing through a planar cavity consisting of two chiral mirrors. Enantiomers of opposite handedness are deflected differently due to a chiral dependence of the Casimir-Polder potential resulting in the separation of the enantiomers. Our setup provides an alternative experimental tool for enantiomer separation, as well as shedding light on the fundamental properties of the Casimi…
We present an approach to studying the Casimir effects by means of the effective theory. An essential point of our approach is replacing the mirror separation into the size of space S^1 in the adiabatic approximation. It is natural to identify the size of space S^1 with the scale factor of the Robertson-Walker-type metric. This replacement simplifies the construction of a class of effective models to study the Casimir effects. To check the validity of this replacement we construct a model for a scalar field coupling to the two-dimensional gravity and calculate the Casimir effects by the effect…
We consider the quantum radiation of scalar particles from a surface wave excited on a plane surface of a mirror. It is assumed that the field obeys Dirichlet condition on the boundary of the mirror. In both cases of running and standing surface waves the expression is given for the spectral-angular distribution of the number of the radiated quanta.
We study the vacuum fluctuations of a quantum scalar field in the presence of a thin and inhomogeneous flat mirror, modeled with a delta potential. Using Heat-Kernel techniques, we evaluate the Euclidean effective action perturbatively in the inhomogeneities (nonperturbatively in the constant background). We show that the divergences can be absorbed into a local counterterm, and that the remaining finite part is in general a nonlocal functional of the inhomogeneities, which we compute explicitly for massless fields in $D=4$ dimensions. For time-independent inhomogeneities, the effective action…
Creation of scalar massless particles in two-dimensional Minkowski space-time--as predicted by the dynamical Casimir effect--is studied for the case of a semitransparent mirror initially at rest, then accelerating for some finite time, along a specified trajectory, and finally moving with constant velocity. When the reflection and transmission coefficients are those in the model proposed by Barton, Calogeracos, and Nicolaevici [$r(w)=-iα/(\w+iα)$ and $s(w)=\w/(\w+iα)$, with $α\geq 0$], the Bogoliubov coefficients on the back side of the mirror can be computed exactly. This allows us to prove t…
A Hamiltonian approach is introduced in order to address some severe problems associated with the physical description of the dynamical Casimir effect at all times. For simplicity, the case of a neutral scalar field in a one-dimensional cavity with partially transmitting mirrors (an essential proviso) is considered, but the method can be extended to fields of any kind and higher dimensions. The motional force calculated in our approach contains a reactive term --proportional to the mirrors' acceleration-- which is fundamental in order to obtain (quasi)particles with a positive energy all the t…
A new solution for an analytic spectrum of particle creation by an accelerating mirror (dynamical Casimir effect) is given. It is the first model to simultaneously radiate thermally and emit a finite number of particles.
We calculate the Casimir energy for scalar fields in interaction with finite-width mirrors, described by nonlocal interaction terms. These terms, which include quantum effects due to the matter fields inside the mirrors, are approximated by means of a local expansion procedure. As a result of this expansion, an effective theory for the vacuum field emerges, which can be written in terms of generalized $δ$-potentials. We compute explicitly the Casimir energy for these potentials and show that, for some particular cases, it is possible to reinterpret them as imposing imperfect Dirichlet boundary…
Nonlinear terms in the equations of motion can induce secularly growing loop corrections to correlation functions. Recently such corrections were shown to affect the particle production by a nonuniformly moving ideal mirror. We extend this conclusion to the cases of ideal vibrating cavity and single semitransparent mirror. These models provide natural IR and UV scales and allow a more accurate study of the loop behavior. In both cases we confirm that two-loop correction to the Keldysh propagator quadratically grows with time. This growth indicates a breakdown of the semiclassical approximation…
We derive a general expression for the Casimir energy corresponding to two flat parallel mirrors in d+1 dimensions, described by nonlocal interaction potentials. For a real scalar field, the interaction with the mirrors is implemented by a term which is a quadratic form in the field, with a nonlocal kernel. The resulting expression for the energy is a function of the parameters that define the nonlocal kernel. We show that the general expression has the correct limit in the zero width case, and also present the exact solution for a particular case.
We derive a master equation for a mirror interacting with the vacuum field via radiation pressure. The dynamical Casimir effect leads to decoherence of a 'Schroedinger cat' state in a time scale that depends on the degree of 'macroscopicity' of the state components, and which may be much shorter than the relaxation time scale. Coherent states are selected by the interaction as pointer states.
We apply the derivative expansion approach to the Casimir effect for a real scalar field in $d$ spatial dimensions, to calculate the next to leading order term in that expansion, namely, the first correction to the proximity force approximation. The field satisfies either Dirichlet or Neumann boundary conditions on two static mirrors, one of them flat and the other gently curved. We show that, for Dirichlet boundary conditions, the next to leading order term in the Casimir energy is of quadratic order in derivatives, regardless of the number of dimensions. Therefore it is local, and determined…
We study the field theoretical model of a scalar field in presence of spacial inhomogeneities in form of one and two finite width mirrors (material slabs). The interaction of the scalar field with the defect is described with position-dependent mass term. For the single layer system we develop a rigorous calculation method and derive explicitly the propagator of the theory, S-matrix elements and the Casimir self-energy of the slab. Detailed investigation of particular limits of self-energy is presented, and connection to know cases is discussed. The calculation method is found applicable to th…
Recently [J. Haro and E. Elizalde, Phys. Rev. Lett. {\bf 97}, 130401 (2006)], a Hamiltonian formulation has been introduced in order to address some longstanding severe problems associated with the physical description of the dynamical Casimir effect at all times while the mirrors are moving. Here we present the complete calculation providing precise details, in particular, of the regularization procedure, which is decisive for the correct derivation of physically meaningful quantities. A basic difference when comparing with the results previously obtained by other authors is the fact that the…
We use a functional approach to evaluate the Casimir free energy for a self-interacting scalar field in $d+1$ dimensions, satisfying Dirichlet boundary conditions on two parallel planes. When the interaction is turned off, exact results for the free energy in some particular cases may be found, as well as low and high temperature expansions based on a duality relation that involves the inverse temperature $β$ and the distance between the mirrors, $a$. For the interacting theory, we derive and implement two different approaches. The first one is a perturbative expansion built with a thermal pro…
The Casimir Effect in (3+1)-dimensional lattice Yang-Mills theory at finite temperature: the unexpected universality of quarkiton and glueton boundary states
Motivated by experiments in which moving boundaries are simulated by time-dependent properties of static systems, we discuss the model of a massless scalar field submitted to a time-dependent Robin boundary condition (BC) at a static mirror in 1+1 dimensions. Using a perturbative approach, we compute the spectral distribution of the created particles and the total particle creation rate, considering a thermal state as the initial field state.
In this paper we study the one dimensional dynamical Casimir effect. We consider a one dimensional cavity formed by two mirrors, one of which performs an oscillatory motion with a frequency resonant with the cavity. The naive solution, perturbative in powers of the amplitude, contains secular terms. Therefore it is valid only in the short time limit. Using a renormalization group technique to resum these terms, we obtain an improved analytical solution which is valid for longer times. We discuss the generation of peaks in the density energy profile and show that the total energy inside the cav…
We show that the vacuum ground state energy for massive scalars on a 1-dim L-sites periodic lattice can be interpreted as the thermodynamic free energy of particles at temperature 1/L governed by the Arutyunov-Frolov mirror Hamiltonian. Although the obligatory zero-point sum-over-frequencies is finite on the lattice, a renormalization prescription is necessary in order to obtain a physical sensible result for the lattice Casimir energy. The coefficients of every term in the large L expansion of the lattice Casimir energy are provided in terms of modified Bessel functions.
We develop a unified scattering approach to dynamical Casimir problems which can be applied to both accelerating boundaries, as well as dispersive objects in relative motion. A general (trace) formula is derived for the radiation from accelerating boundaries. Applications are provided for objects with different shapes in various dimensions, and undergoing rotational or linear motion. Within this framework, photon generation is discussed in the context of a modulated optical mirror. For dispersive objects, we find general results solely in terms of the scattering matrix. Specifically, we discus…
We extend a recently proposed Quantum Field Theory (QFT) approach to the Lifshitz formula, originally implemented for a real scalar field, to the case of a fluctuating vacuum electromagnetic (EM) field, coupled to two flat, parallel mirrors. The general result is presented in terms of the invariants of the vacuum polarization tensors due to the media on each mirror. We consider mirrors that have small widths, with the zero-width limit as a particular case. We apply the latter to models involving graphene sheets, obtaining results which are consistent with previous ones.
The paper is based on the Bachelor Thesis defended this year in ITEP, Moscow. This is the extended version of [arXiv:1707.02242] and contains a lot more technical details of the calculations. We consider (1+1)-dimensional massless scalar field theory with Dirichlet boundary conditions on arbitrary time-like curve. It is well known that in this situation there is a non-zero energy flux at the tree-level, if the latter curve corresponds to a non--stationary motion of the boundary. Such a problem is usually referred to as the radiation due to moving mirrors. We calculate quantum loop corrections …
We apply the background field method and the effective action formalism to describe the four-dimensional dynamical Casimir effect. Our picture corresponds to the consideration of quantum cosmology for an expanding FRW universe (the boundary conditions act as a moving mirror) filled by a quantum massless GUT which is conformally invariant. We consider cases in which the static Casimir energy is repulsive and attractive. Inserting the simplest possible inertial term, we find, in the adiabatic (and semiclassical) approximation, the dynamical evolution of the scale factor and the dynamical Casimir…
We calculate quantum loop corrections to the stress-energy flux caused by moving mirrors. We consider massless, self-interacting, $ϕ^4$, real scalar theory. In these calculations we encounter a new and quite unexpected subtleties due to the absence of global hyperbolicity in the presence of mirrors. We attempt to clearly phrase as many as possible hidden assumptions and complications that appear along the course of the solution of the problem in question. On top of that we find that quantum loop corrections to the stress-energy flux are growing with time and are not suppressed in comparison wi…
We consider a real massless scalar field in 3+1 dimensions satisfying a Robin boundary condition at a nonrelativistic moving mirror. Considering vacuum as the initial field state, we compute explicitly the number of particles created per unit frequency and per unit solid angle, exhibiting in this way the angular dependence of the spectral distribution. The well known cases of Dirichlet and Neumann boundary conditions may be reobtained as particular cases from our results. We show that the particle creation rate can be considerably reduced (with respect to the Dirichlet and Neumann cases) for p…
We found new signatures of the dynamical Casimir effect (DCE) in the context of superconducting circuits. We show that if the recent experiment made by Wilson {\it et al}, which brought the DCE into reality for the first time, is repeated with slight modifications (for instance, different values for the capacitance of the SQUID), three remarkable results will show up, namely: {\it (i)} a quite different spectral distribution for the created particles, deviating from the typical parabolic shape; {\it (ii)} an enhancement by a factor of approximately $5 \times 10^3 $ in the number of created par…
The friction force on the moving interface between two different vacuum states of superfluid 3He is considered at low temperature. Since the dominating mechanism of the friction is the Andreev reflection of the massless "relativistic" fermions, which live on the A-phase side of the interface, the results are similar to that for the perfectly reflecting mirror moving in the quantum vacuum.
Dirichlet boundary conditions on a surface can be imposed on a scalar field, by coupling it quadratically to a $δ$-like potential, the strength of which tends to infinity. Neumann conditions, on the other hand, require the introduction of an even more singular term, which renders the reflection and transmission coefficients ill-defined because of UV divergences. We present a possible procedure to tame those divergences, by introducing a minimum length scale, related to the non-zero `width' of a {\em nonlocal} term. We then use this setup to reach (either exact or imperfect) Neumann conditions,…
We study the Dynamical Casimir Effect (DCE) due to an Abelian gauge field in 2+1 dimensions, in the presence of semitransparent, zero-width mirrors, which may move or deform in a time-dependent way. We obtain general expressions for the probability of motion-induced pair creation, which we render in a more explicit form, for some relevant states of motion.
We study the field theoretical model of a real scalar field in presence of spacial inhomogeneity in form of a finite width mirror (material layer). The interaction of the scalar field with the defect is described with position-dependent mass term. We calculate the propagator of the theory, the Casimir energy and the pressure on the boundaries of the layer. We discuss the renormalization procedure for the model in dimensional regularization.
