The dynamical Casimir effect is an intriguing phenomenon in which photons are generated from vacuum due to a non-adiabatic change in some boundary conditions. In particular, it connects the motion of an accelerated mechanical mirror to the generation of photons. While pioneering experiments demonstrating this effect exist, a conclusive measurement involving a mechanical generation is still missing. We show that a hybrid system consisting of a piezoelectric mechanical resonator coupled to a superconducting cavity may allow to electro-mechanically generate measurable photons from vacuum, intrinsically associated to the dynamical Casimir effect. Such an experiment may be achieved with current technology, based on film bulk acoustic resonators directly coupled to a superconducting cavity. Our results predict a measurable photon generation rate, which can be further increased through additional improvements such as using superconducting metamaterials.
We show that the dynamical Casimir effect in an optomechanical system can be achieved under incoherent mechanical pumping. We adopt a fully quantum-mechanical approach for both the cavity field and the oscillating mirror. The dynamics is then evaluated using a recently-developed master equation approach in the dressed picture, including both zero and finite temperature photonic reservoirs. This analysis shows that the dynamical Casimir effect can be observed even when the mean value of the mechanical displacement is zero. This opens up new possibilities for the experimental observation of this…
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 …
We theoretically propose and investigate a feasible experimental scheme for the realization of the dynamical Casimir effect (DCE) in a hybrid optomechanical cavity with a moving end mirror containing an interacting cigar-shaped Bose-Einstein condensate (BEC). We show that in the red-detuned regime of cavity optomechanics together with the weak optomechanical coupling limit by \textit{coherent} modulation of the \textit{s}-wave scattering frequency of the BEC and the mechanical spring coefficient of the mechanical oscillator (MO), the mechanical and atomic quantum vacuum fluctuations are parame…
Quantum fluctuations give rise to van der Waals and Casimir forces that dominate the interaction between electrically neutral objects at sub-micron separations. Under the trend of miniaturization, such quantum electrodynamical effects are expected to play an important role in micro- and nano-mechanical devices. Nevertheless, utilization of Casimir forces on the chip level remains a major challenge because all experiments so far require an external object to be manually positioned close to the mechanical element. Here, by integrating a force-sensing micromechanical beam and an electrostatic act…
We study the dynamical Casimir effect using a fully quantum-mechanical description of both the cavity field and the oscillating mirror. We do not linearize the dynamics, nor do we adopt any parametric or perturbative approximation. By numerically diagonalizing the full optomechanical Hamiltonian, we show that the resonant generation of photons from the vacuum is determined by a ladder of mirror-field {\em vacuum Rabi splittings}. We find that vacuum emission can originate from the free evolution of an initial pure mechanical excited state, in analogy with the spontaneous emission from excited …
The coupling between corrugated surfaces due to the lateral Casimir force is employed to propose a nanoscale mechanical device composed of two racks and a pinion. The noncontact nature of the interaction allows for the system to be made frustrated by choosing the two racks to move in the same direction and forcing the pinion to choose between two opposite directions. This leads to a rich and sensitive phase behavior, which makes the device potentially useful as a mechanical {\em sensor} or {\em amplifier}. The device could also be used to make a mechanical {\em clock} signal of tunable frequen…
In this work the induced van der Waals interaction between a pair of neutral atoms or molecules is considered by use of a statistical mechanical method. Commonly this interaction is obtained by standard quantum mechanical perturbation theory to second order. However, the latter is restricted to electrostatic interactions between charges and dipole moments. So with radiating dipole-dipole interaction where retardation effects are important for large separations of the particles, other methods are needed, and the resulting induced interaction is the Casimir-Polder interaction usually obtained by…
The Casimir effect is a well-known macroscopic consequence of quantum vacuum fluctuations, but whereas the static effect (Casimir force) has long been observed experimentally, the dynamic Casimir effect is up to now undetected. From an experimental viewpoint a possible detection would imply the vibration of a mirror at gigahertz frequencies. Mechanical motions at such frequencies turn out to be technically unfeasible. Here we present a different experimental scheme where mechanical motions are avoided, and the results of laboratory tests showing that the scheme is practically feasible. We thin…
I consider the quantum electromagnetic field in a coaxial cylindrical waveguide, such that the outer cylindrical surface has a time-dependent radius. The field propagates parallel to the axis, inside the annular region between the two cylindrical surfaces. When the mechanical frequency and the thickness of the annular region are small enough, only Transverse Electromagnetic (TEM) photons may be generated by the dynamical Casimir effect. The photon emission rate is calculated in this regime, and compared with the case of parallel plates in the limit of very short distances between the two cylin…
We propose a superconducting circuit comprising a dc-SQUID with mechanically compliant arm embedded in a coplanar microwave cavity that realizes an optomechanical system with a degenerate or non-degenerate parametric interaction generated via the dynamical Casimir effect. For experimentally feasible parameters, this setup is capable of reaching the single-photon, ultra-strong coupling regime, while simultaneously possessing a parametric coupling strength approaching the renormalized cavity frequency. This opens up the possibility of observing the interplay between these two fundamental nonline…
We consider an atom in its ground state undergoing a non-relativistic oscillation in free space. The interaction with the electromagnetic quantum vacuum leads to two effects to leading order in perturbation theory. When the mechanical frequency is larger than the atomic transition frequency, the dominant effect is the motion-induced transition to an excited state with the emission of a photon carrying the excess energy. We compute the angular distribution of emitted photons and the excitation rate. On the other hand, when the mechanical frequency is smaller than the transition frequency, the l…
We investigate how the dynamical Casimir effect can entangle quantum systems in different coupling regimes of circuit quantum electrodynamics, and show the robustness of such entanglement generation against dissipative effects with current technology. We consider two qubit-resonator systems, which are coupled by a SQUID driven with an external magnetic field, and explore the entire range of coupling regimes between each qubit and its respective resonator. In this scheme, we derive a semianalytic explanation for the entanglement generation between both superconducting qubits when they are coupl…
Thermal effects contributing to the Casimir interaction between objects are usually small at room temperature and they are difficult to separate from quantum mechanical contributions at higher temperatures. We propose that the thermal Casimir force effect can be observed for a graphene flake suspended in a fluid between substrates at the room temperature regime. The properly chosen materials for the substrates and fluid induce a Casimir repulsion. The balance with the other forces, such as gravity and buoyancy, results in a stable temperature dependent equilibrium separation. The suspended gra…
In this paper, we theoretically propose and investigate a feasible experimental scheme for realizing the dynamical Casimir effect (DCE) of phonons in an optomechanical setup formed by a ground-state precooled mechanical oscillator (MO) inside a Fabry-P{é}rot cavity, which is driven by an amplitude-modulated classical laser field in the dispersive (far-detuned) regime. The time modulation of the driving field leads to the parametric amplification of the mechanical vacuum fluctuations of the MO, which results in the generation of Casimir phonons over time scales longer than the cavity lifetime. …
An experimental set-up for the measurement of the Casimir effect at separations larger than a few microns is presented. The apparatus is based on a mechanical resonator and uses a homodyne detection technique to sense the Casimir force in the plane-parallel configuration. First measurements in the 3-10 micrometer range show an unexpected large force probably due to patch effects.
The Casimir effect is a force arising in the macroscopic world as a result of radiation pressure of vacuum fluctuations. It thus plays a key role in the emerging domain of nano-electro-mechanical systems (NEMS). This role is reviewed in the present paper, with discussions of the influence of the material properties of the mirrors, as well as the geometry dependence of the Casimir effect between corrugated mirrors. In particular, the lateral component of the Casimir force and restoring torque between metal plates with misaligned corrugations are evaluated.
The Casimir mutual free energy F for a system of two dielectric concentric nonmagnetic spherical bodies is calculated, at arbitrary temperatures. Whereas F has recently been evaluated for the special case of metals (refractive index n=\infty), here analogous results are presented for dielectrics, and shown graphically when n=2.0. Our calculational method relies upon quantum statistical mechanics. The Debye expansions for the Riccati-Bessel functions when carried out to a high order are found to be very useful in practice (thereby overflow/underflow problems are easily avoided), and also to giv…
We consider the problem of motion-induced photon creation from quantum vacuum inside closed, perfectly conducting cavities with time-dependent geometries. These include one dimensional Fabry-Perrot resonators with Dirichlet or Neumann boundary conditions, three dimensional cylindrical waveguides, and a spherical shell. The number of Casimir TE, TM and TEM photons is computed. We also present a classical mechanical analogue of the one dimensional dynamical Casimir effect.
We establish an explicit analogy between the dynamical Casimir effect and the photon emission of a thin non-linear crystal pumped inside a cavity. This allows us to propose a system based on a type-I optical parametric oscillator (OPO) to simulate a cavity oscillating in vacuum at optical frequencies. The resulting photon flux is expected to be more easily detectable than with a mechanical excitation of the mirrors. We conclude by comparing different theoretical predictions and suggest that our experimental proposal could help discriminate between them.
We derive a general procedure for finding the electromagnetic normal modes in layered structures. We apply this procedure to planar, spherical and cylindrical structures. These normal modes are important in a variety of applications. They are the only input needed in calculations of Casimir interactions. We present explicit expression for the condition for modes and Casimir energy for a large number of specific geometries. The layers are allowed to be two-dimensional so graphene and graphene-like sheets as well as two-dimensional electron gases can be handled within the formalism. Also forces …
We study the Casimir effect in axion electrodynamics. A finite $θ$-term affects the energy dispersion relation of photon if $θ$ is time and/or space dependent. We focus on a special case with linearly inhomogeneous $θ$ along the $z$-axis. Then we demonstrate that the Casimir force between two parallel plates perpendicular to the $z$-axis can be either attractive or repulsive, dependent on the gradient of $θ$. We call this repulsive component in the Casimir force induced by inhomogeneous $θ$ the anomalous Casimir effect.
The statistical mechanical approach to Casimir problems for dielectrics separated by a vacuum gap turns out to be compact and effective. A central ingredient of this method is the effect of interacting fluctuating dipole moments of the polarizable particles. At arbitrary temperature the path integral formulation of quantized particles, developed by Høye-Stell and others, is needed. At high temperature - the limit considered in the present paper - the classical theory is however sufficient. Our present theory is related to an idea put forward earlier by Jancovici and {Š}amaj (2004), namely to e…
We propose a method for the detection of a dynamical Casimir effect. Assuming that the Casimir photons are being generated in an electromagnetic cavity with a vibrating wall (dynamical Casimir effect), we consider electrons passing through the cavity to be interacting with the intracavity field. We show that the dynamical Casimir effect can be observed via the measurement of the change in the average or in the variance of the electron kinetic energy. We point out that the enhancement of the effect due to finite temperatures makes it easier to detect the Casimir photons.
We discuss the Casimir effect for boundary conditions involving perfect electromagnetic conductors (PEMCs). Based on the corresponding reciprocal Green's tensor we construct the Green's tensor for two perfectly reflecting plates with magnetoelectric coupling (non-reciprocal media) within the framework of macroscopic quantum electrodynamics. We calculate the Casimir force between two PEMC plates in terms of the PEMC parameter M and the duality transformation angle $θ$ resulting in a universal analytic expression that connects the attractive Casimir force with the repulsive Boyer force. We relat…
We study the effect of two non-interacting impurity atoms near by a one-dimensional nanowire, which is modeled as a tight-binding hopping model. The virtual single-electron hopping between two impurities will induce an additional energy depending on the distance of two impurities, which gives a electronic Casimir-Polder effect. We find that the Casimir-Polder force between the two impurities decreases with the impurity-impurity distance exponentially. And the effects of nanowire and finite temperature on the Casimir-Polder force are also discussed in detail, respectively.
We apply the scattering approach to the Casimir interaction between two dielectric half-spaces separated by an electrolyte solution. We take the nonlocal electromagnetic response of the intervening medium into account, which results from the presence of movable ions in solution. In addition to the usual transverse modes, we consider longitudinal channels and their coupling by reflection at the surface of the local dielectric. The Casimir interaction energy is calculated from the matrix describing a round-trip of coupled transverse and longitudinal waves between the interacting surfaces. The no…
The Casimir effect refers to the existence of a macroscopic force between conducting plates in vacuum due to quantum fluctuations of fields. These forces play an important role, among other things, in the design of nano-scale mechanical devices. Accurate experimental observations of this phenomenon have motivated the development of new theoretical approaches in dealing with the effects of different geometries, temperature etc. In this talk, I will focus on a new method we have developed in calculating the contribution to the Casimir effect due to diffraction from edges and holes in different g…
It is well known that residual electrostatic forces create significant difficulties to precise measurements of the Casimir force and to wide use of Casimir-operated microdevices. We experimentally demonstrate that with the help of Ar-ion cleaning of the surfaces it is possible to make electrostatic effects negligibly small as compared to the Casimir interaction. Our experimental setup consists of the dynamic atomic force microscope supplemented with an Ar-ion gun and argon reservoir. The residual potential difference between the Au-coated surfaces of a sphere and a plate was measured both befo…
The dynamics of three interacting objects has been investigated extensively in Newtonian gravitational physics (often termed the three-body problem), and is important for many quantum systems, including nuclei, Efimov states, and frustrated spin systems. However, the dynamics of three macroscopic objects interacting through quantum vacuum fluctuations (virtual photons) is still an unexplored frontier. Here, we report the first observation of Casimir interactions between three isolated macroscopic objects. We propose and demonstrate a three terminal switchable architecture exploiting opto-mecha…
The canonical quantization of macroscopic electromagnetism was recently presented in New J. Phys. 12 (2010) 123008. This theory is here used to derive the Casimir effect, by considering the special case of thermal and zero-point fields. The stress-energy-momentum tensor of the canonical theory follows from Noether's theorem, and its electromagnetic part in thermal equilibrium gives the Casimir energy density and stress tensor. The results hold for arbitrary inhomogeneous magnetodielectrics and are obtained from a rigorous quantization of electromagnetism in dispersive, dissipative media. Conti…
We study the properties of the classical electromagnetic (EM) radiation produced by two physically different yet closely related systems, which may be regarded as classical analogues of the Dynamical Casimir Effect (DCE). They correspond to two flat, infinite, parallel planes, one of them static and imposing perfect conductor boundary conditions, while the other performs a rigid oscillatory motion. The systems differ just in the electrical properties of the oscillating plane: one of them is just a planar dipole layer (representing, for instance, a small-width electret). The other, instead, has…
We study analytically the time-dependent boundary conditions of superconducting microwave circuit experiments in the high plasma frequency limit, in which the conditions are Robin-type and relate the value of the field to the spatial derivative of the field. We give an explicit solution to the field evolution for boundary condition modulations that are small in magnitude but may have arbitrary time dependence, in a formalism that applies both to a semiopen waveguide and to a closed waveguide with two independently adjustable boundaries. The correspondence between the microwave Robin boundary c…
The dynamical Casimir effect (DCE) is the production of photons by the amplification of vacuum fluctuations. In this paper we demonstrate new resonance conditions in DCE that potentially allow the production of optical photons when the mechanical frequency is smaller than the lowest frequency of the cavity field. We consider a cavity with one mirror fixed and the other allowed to oscillate. In order to identify the region where production of photons takes place, we do a linear stability analysis and investigate the dynamic stability of the system under small fluctuations. By using a numerical …
Two close parallel mirrors attract due to a small force (Casimir effect) originating from the electromagnetic quantum vacuum uctuations of the electromagnetic field. These vacuum uctuations can also induce motional forces exerted upon one mirror when the other one moves. Here we consider an optomechanical system consisting of two vibrating mirrors coupled to an optical resonator. We find that motional forces can determine noticeable coupling rates between the two spatially separated vibrating mirrors. We show that, by tuning the two mechanical oscillators into resonance, energy is exchanged be…
We present a general analytical approach to investigate relativistic corrections in the dynamical Casimir effect (DCE). Particularly, we discuss the behavior of the additional frequency bands that appear in the spectral distribution of the created particles when the first relativistic corrections are taken into account. We do that in the context of circuit QED, by analyzing the setup used in the first measurement of the DCE, where a system with a time-dependent boundary condition simulates the mechanical motion of a single mirror in 1+1 dimensions. Our method is applicable to a large class of …
The electromagnetic vacuum is known to have energy. It has been recently argued that the quantum vacuum can possess momentum, that adds up to the momentum of matter. This ``Casimir momentum'' is closely related to the Casimir effect, in which case energy is exchanged. In previous theory it was treated semi-classically. We present a non-relativistic quantum theory for the linear momentum of electromagnetic zero-point fluctuations, considering an harmonic oscillator subject to crossed, quasi-static magnetic and electric and coupled to the quantum vacuum. We derive a contribution of the quantum v…
We have performed numerical simulations to evaluate the effect on the capacitance of finite size boundaries realistically present in the parallel plane, sphere-plane, and cylinder-plane geometries. The potential impact of edge effects in assessing the accuracy of the parameters obtained in the electrostatic calibrations of Casimir force experiments is then discussed.
The role of the vacuum, in the Casimir Effect, is a matter of some dispute: the Casimir force has been variously described as a phenomenon resulting "from the alteration, by the boundaries, of the zero-point electromagnetic energy", or a "Van der Waals force between the metal plates" that can be "computed without reference to zero point energies". Neither of these descriptions are grounded in a consistently quantum mechanical treatment of matter interacting with the electromagnetic field. However, the Casimir Effect has been canonically described within the framework of macroscopic quantum ele…
Using covariant quantization of the electromagnetic field, the Casimir force per unit area experienced by a long conducting cylindrical shell, under both Dirichlet and Neumann boundary conditions, is calculated. The renormalization procedure is based on the plasma cut-off frequency for real conductors. The real case of a gold (silver) cylindrical shell is considered and the corresponding electromagnetic Casimir pressure is computed. It is discussed that the Dirichlet and Neumann problems should be considered separately without adding their corresponding results.
In this article we investigate the Casimir effect in the presence of a medium by quantizing the Electromagnetic (EM) field in the presence of a magnetodielectric medium by using the path integral formalism. For a given medium with definite electric and magnetic susceptibilities, explicit expressions for the Casimir force are obtained which are in agree with the original Casimir force between two conducting parallel plates immersed in the quantum electromagnetic vacuum.
Vacuum fluctuations provide a fundamental source of dissipation for systems coupled to quantum fields by radiation pressure. In the dynamical Casimir effect, accelerating neutral bodies in free space give rise to the emission of real photons while experiencing a damping force which plays the role of a radiation reaction force. Analog models where non-stationary conditions for the electromagnetic field simulate the presence of moving plates are currently under experimental investigation. A dissipative force might also appear in the case of uniform relative motion between two bodies, thus leadin…
We compute the Casimir energy of the photon field in a de Sitter space and find it to be proportional to the size of the horizon, the same form of the holographic dark energy. We suggest to make metamaterials to mimic de Sitter space in laboratory and measure the predicted Casimir energy.
We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, nonzero temperatures, and spatial arrangements in which one object is enclosed in another. Our method combines each object's classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details…
In this article, we rigorously analyze the effects of the dispersion forces (Casimir and van der Waals forces) on a nano-optomechanical device based on a silicon waveguide and a silicon dioxide substrate, surrounded by air and driven by optical forces. The dispersion forces are calculated using a modified Lifshitz theory, in order to take into account the device thickness and material s dielectric permittivities, which are obtained from experimental optical data and validated by means of a rigorous 3D FDTD simulation. We also take into account the mechanical nonlinearity of the waveguide, whic…
In this work, we present detailed measurements of the Casimir-Lifshitz force between two gold surfaces (a sphere and a plate) immersed in ethanol and study the effect of residual electrostatic forces, which are dominated by static fields within the apparatus and can be reduced with proper shielding. Electrostatic forces are further reduced by Debye screening through the addition of salt ions to the liquid. Additionally, the salt leads to a reduction of the Casimir-Lifshitz force by screening the zero-frequency contribution to the force; however, the effect is small between gold surfaces at the…
We study the Topological Casimir effect, in which extra vacuum energy emerges as a result of the topological features of the theory, rather than due to the conventional fluctuations of the physical propagating degrees of freedom. We compute the corresponding topological term in quantum Maxwell theory defined on a compact manifold. Numerically, the topological effect is much smaller than the conventional Casimir effect. However, we argue that the Topological Casimir Effect is highly sensitive to an external magnetic field, which may help to discriminate it from the conventional Casimir effect. …
We study the corrections to the Casimir effect in the classical geometry of two parallel metallic plates, separated by a distance $a$, due to the presence of a minimal length ($\hbar\sqrt{(β)}$) arising from quantum mechanical models based on a Generalized Uncertainty Principle (GUP). The approach for the quantization of the electromagnetic field is based on projecting onto the maximally localized states of a few specific GUP models and was previously developed to study the Casimir-Polder effect. For each model we compute the lowest order correction in the minimal length to the Casimir energy …
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…
The microscopic theory of the Casimir effect in the dielectric is studied in the framework when absorption is realized via a reservoir modeled by a set of oscillators with continuously distributed frquencies with the aim to see if the effects depend on the form of interaction with the reservoir. A simple case of the one-dimensional dielectric between two metallic plates is considered. Two possible models are investigated, the direct interaction of the electromagnetic field with the reservoir and indirect interaction via an intermediate oscillator imitating the atom. It is found that with the s…
Electrostatic patch potentials give rise to forces between neutral conductors at distances in the micrometer range and must be accounted for in the analysis of Casimir force experiments. In this paper we develop a quasi-local model for describing random potentials on metallic surfaces. In contrast to some previously published results, we find that patches may provide a significant contribution to the measured signal, and may render the experimental data at distances below 1 micrometer compatible with theoretical predictions based on the Drude model.
Temporal modulation of the quantum vacuum through fast motion of a neutral body or fast changes of its optical properties is known to promote virtual into real photons, the so-called dynamical Casimir effect. Empowering modulation protocols with spatial control could enable to shape the spectral, spatial, spin, and entanglement properties of the emitted photon pairs. Space-time quantum metasurfaces have been proposed as a platform to realize this physics via modulation of their optical properties. Here, we report the mechanical analog of this phenomenon by considering systems whose lattice str…
We study the creation and entanglement of quasiparticle pairs due to a periodic variation of the mode frequencies of a homogeneous quantum system. Depending on the values of the parameters describing the periodic modulation, the number of created pairs either oscillates or, in a narrow resonant frequency interval, grows exponentially in time. For a system initially in a thermal state, we determine in which cases the final state is quantum mechanically entangled, i.e., where the bipartite state is nonseparable. We include some weak dissipation, expected to be found in any experimental setup, an…
We show that Casimir-force calculations for a finite number of non-overlapping obstacles can be mapped onto quantum-mechanical billiard-type problems which are characterized by the scattering of a fictitious point particle off the very same obstacles. With the help of a modified Krein trace formula the genuine/finite part of the Casimir energy is determined as the energy-weighted integral over the log-determinant of the multi-scattering matrix of the analog billiard problem. The formalism is self-regulating and inherently shows that the Casimir energy is governed by the infrared end of the mul…
