We calculate the number of photons produced by the parametric resonance in a cavity with vibrating walls. We consider the case that the frequency of vibrating wall is $n ω_1 (n=1,2,3,...)$ which is a generalization of other works considering only $2 ω_1$, where $ω_1$ is the fundamental-mode frequency of the electromagnetic field in the cavity. For the calculation of time-evolution of quantum fields, we introduce a new method which is borrowed from the time-dependent perturbation theory of the usual quantum mechanics. This perturbation method makes it possible to calculate the photon number for any $n$ and to observe clearly the effect of the parametric resonance.
The particles produced from the vacuum in the dynamical Casimir effect are highly entangled. In order to quantify the correlations generated by the process of vacuum decay induced by moving mirrors, we study the entanglement evolution in the dynamical Casimir effect by computing the time-dependent Rényi and von Neumann entanglement entropy analytically in arbitrary dimensions. We consider the system at parametric resonance, where the effect is enhanced. We find that, in (1+1) dimensions, the entropies grow logarithmically for large times, $S_A(τ)\sim\frac{1}{2}\log(τ)$, while in higher dimensi…
The Casimir Force, a quantum mechanical effect, has been observed in several microelectromechanical systems (MEMS) platforms. Due to its extreme sensitivity to the separation of two objects, the Casimir Force has been proposed as an excellent avenue for quantum metrology. Practical application, however, is challenging due to attractive forces leading to stiction and failure of the device, called Casimir pull-in. In this work, we design and simulate a Casimir-driven metrology platform, where a time-delay based parametric amplification technique is developed to achieve a steady state and avoid p…
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…
We introduce different strategies to enhance photon generation in a cavity within the Rabi model in the ultrastrong coupling regime. We show that a bang-bang strategy allows to enhance the effect of up to one order of magnitude with respect to simply driving the system in resonance for a fixed time. Moreover, up to about another order of magnitude can be gained exploiting quantum optimal control strategies. Finally, we show that such optimized protocols are robust with respect to systematic errors and noise, paving the way to future experimental implementations of such strategies.
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 show that oscillations of a homogeneous medium with constant material coefficients produce pairs of photons. Classical analysis of an oscillating medium reveals regions of parametric resonance where the electromagnetic waves are exponentially amplified. The quantum counterpart of parametric resonance is an exponentially growing number of photons in the same parameter regions. This process may be viewed as another manifestation of the dynamical Casimir effect. However, in contrast to the standard dynamical Casimir effect, photon production here takes place in the entire volume and is not due…
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…
\emph{Resonance Casimir-Polder interaction(RCPI)} occurs in nature when one or more atoms are in their excited states and the exchange of real photon is involved between them due to vacuum fluctuations of the quantum fields. In recent times, many attempts have been made to show that the curved spacetime such as the \emph{de-Sitter spacetime} can be separated from a \emph{thermal Minkowski spacetime} using \emph{RCPI}. Motivated from these ideas, here we study the \emph{RCPI} between two atoms that interact with a massive scalar field in Schwarzschild spacetime provided the atoms are placed in …
We investigate analytically and numerically the nonstationary circuit QED setup in which $N$ independent qubits interact with a single mode of the Electromagnetic field confined in a resonator. We consider the harmonic time modulation of some parameter (atomic transition frequency or the atom-field coupling strength) and derive the unitary dynamics up to the second order in the modulation depth for $N=1$ and $N\gg 1$. It is shown that all the resonant phenomena that occur for modulation frequencies $\sim 2ω_{0}$ (where $ω_{0}$ is the cavity frequency) also occur for the halved frequencies. How…
We recently proposed a new approach to analyze the parametric resonance in a vibrating cavity based on the analysis of classical optical paths. This approach is used to examine various models of cavities with moving walls. We prove that our method is useful to extract easily basic physical outcome.
In this work we investigate the dynamical Casimir effect in a nonideal cavity by deriving an effective Hamiltonian. We first compute a general expression for the average number of particle creation, applicable for any law of motion of the cavity boundary. We also compute a general expression for the linear entropy of an arbitrary state prepared in a selected mode, also applicable for any law of motion of the cavity boundary. As an application of our results we have analyzed both the average number of particle creation and linear entropy within a particular oscillatory motion of the cavity boun…
An ultracold atom above a horizontal mirror experiences quantum reflection from the attractive Casimir-Polder interaction, which holds it against gravity and leads to quantum levitation states. We analyze this system by using a Liouville transformation of the Schrödinger equation and a Langer coordinate adapted to problems with a classical turning point. Reflection on the Casimir-Polder attractive well is replaced by reflection on a repulsive wall and the problem is then viewed as an ultracold atom trapped inside a cavity with gravity and Casimir-Polder potentials acting respectively as top an…
We consider here, a two-atom system is uniformly moving through a circular ring at an ultra-relativistic speed and weakly interacting with common external fields. The vacuum fluctuations of the quantum fields generate the entanglement between the atoms. Hence an effective energy shift is originated, which depends on the inter-atomic distance. This is commonly known as resonance Casimir-Polder interaction (RCPI). It is well known that, for a linearly accelerated system coupled with a massless scalar field, we get a thermal response when the local inertial approximation is valid. On the contrary…
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.
The phenomenon of particle creation within a resonantly vibrating lossy cavity is investigated for the example of a massless scalar field at finite temperature. Leakage is provided by insertion of a dispersive mirror into a larger ideal cavity. Via the rotating wave approximation we demonstrate that for the case of parametric resonance the exponential growth of the number of created particles and the strong enhancement at finite temperatures are preserved in the presence of reasonable losses. The relevance for experimental tests of quantum radiation via the dynamical Casimir effect is addresse…
We study a stochastic version of the dynamical Casimir effect, computing the particle creation inside a cavity produced by a random motion of one of its walls. We first present a calculation perturbative in the amplitude of the motion. We compare the stochastic particle creation with the deterministic counterpart. Then we go beyond the perturbative evaluation using a stochastic version of the multiple scale analysis, that takes into account stochastic parametric resonance. We stress the relevance of the coupling between the different modes induced by the stochastic motion. In the single-mode a…
We study the joint probability distribution function of the work and the change of photon number of the nonequilibrium process of driving the electromagnetic (EM) field in a three-dimensional cavity with an oscillating boundary. The system is initially prepared in a grand canonical equilibrium state and we obtain the analytical expressions of the characteristic functions of work distributions in the single-resonance and multiple-resonance conditions. Our study demonstrates the validity of the fluctuation theorems of the grand canonical ensemble in nonequilibrium processes with particle creatio…
Motivated by a recent proposal for an experimental verification of the dynamical Casimir effect, the macroscopic electromagnetic field within a perfect cavity containing a thin slab with a time-dependent dielectric permittivity is quantized in terms of the dual potentials. For the resonance case, the number of photons created out of the vacuum due to the dynamical Casimir effect is calculated for both polarizations (TE and TM). PACS: 42.50.Lc, 03.70.+k, 42.50.Dv, 42.60.Da.
The frequencies of transitions between hyperfine levels of ground-state atoms can be measured with exquisite precision using magnetic-resonance techniques. This makes hyperfine transitions ideal probes of QED effects originating from the interaction of atoms with the quantized electromagnetic field. One of the most remarkable effects predicted by QED is the Casimir-Polder shift experienced by the energy levels of atoms placed near one or more dielectric objects. Here we compute the Casimir-Polder shift and the width of hyperfine transitions between ground-state Zeeman sub-levels of an hydrogen…
We report a theoretical study of the optical properties of a three-level emitter embedded in an optical cavity including the non-rotating wave terms of the light-matter interaction Hamiltonian. Rabi oscillations induced by a continuous wave drive laser are responsible for a periodic time-modulation of the effective cavity resonance, which results in a significant dynamical Casimir emission. A clear signature of the back-action effect of the dynamical Casimir emission onto the drive laser is visible as a sizable suppression of its absorption.
We theoretically study the dynamical Casimir effect (DCE), i.e., parametric amplification of a quantum vacuum, in an optomechanical cavity interacting with a photonic crystal, which is considered to be an ideal system to study the microscopic dissipation effect on the DCE. Starting from a total Hamiltonian including the photonic band system as well as the optomechanical cavity, we have derived an effective Floquet-Liouvillian by applying the Floquet method and Brillouin-Wigner-Feshbach projection method. The microscopic dissipation effect is rigorously taken into account in terms of the energy…
We investigate the dynamical Casimir effect in a coplanar waveguide (CPW) terminated by a superconducting quantum interference device (SQUID). Changing the magnetic flux through the SQUID parametrically modulates the boundary condition of the CPW, and thereby, its effective length. Effective boundary velocities comparable to the speed of light in the CPW result in broadband photon generation which is identical to the one calculated in the dynamical Casimir effect for a single oscillating mirror. We estimate the power of the radiation for realistic parameters and show that it is experimentally …
Dynamical Casimir effect (DCE) is the name assigned to the process of generating quanta from vacuum due to an accelerated motion of macroscopic neutral bodies (mirrors) or time-modulation of cavity material properties, as well as the simulation of such processes. Here I review the theoretical results on the detection of DCE using intracavity quantum detectors, such as multi-level atoms, atomic networks and harmonic oscillators. I also stress the mathematical equivalence of this problem to the physics of optical parametric oscillators interacting with atoms or quantum wells, studied in Quantum …
We discuss the prospects of employing an NbN superconducting microwave stripline resonator for studying the dynamical Casimir effect experimentally. Preliminary experimental results, in which optical illumination is employed for modulating the resonance frequencies of the resonator, show that such a system is highly promising for this purpose. Moreover, we discuss the undesirable effect of heating which results from the optical illumination, and show that degradation in noise properties can be minimized by employing an appropriate design.
We study analytically and numerically the dynamical Casimir effect in a cavity containing two stationary 2-level atoms that interact with the resonance field mode via the Tavis-Cummings Hamiltonian. We determine the modulation frequencies for which the field and atomic excitations are generated and study the corresponding dynamical behaviors in the absence of damping. It is shown that the two-atom setup allows for monitoring of photon generation without interrupting the growth, and different entangled states can be generated during the process.
We report sensitive detection of parametric resonances in a high-density sample of ultracold $^{87}Rb$ atoms confined to a far-off-resonance optical dipole trap. Fluorescence imaging of the expanded ultracold atom cloud after a period of parametric excitation shows significant modification of the atomic spatial distribution and has high sensitivity compared with traditional measurements of parametrically-driven trap loss. Using this approach, a significant shift of the parametric resonance frequency is observed, and attributed to the anharmonic shape of the dipole trap potential.
We study the peculiarities of the nonstationary Casimir effect (creation of photons in cavities with moving boundaries) in the special case of two resonantly coupled modes with frequencies $ω_0$ and $(3Δ)ω_0$, parametrically excited due to small amplitude oscillations of the ideal cavity wall at the frequency $2ω_0(1δ)$ (with $|δ|,|Δ|\ll 1$). The effects of thermally induced oscillations in time dependences of the mean numbers of created photons and the exchange of quantum purities between the modes are discovered. Squeezing and photon distributions in each modes are calculated for initial vac…
Dynamical Casimir effect of the optomechanical cavity interacting with one-dimensional photonic crystal is theoretically investigated in terms of the complex spectral analysis of Floquet-Liouvillian in the symplectic-Floquet space. The quantum vacuum fluctuation of the intra-cavity mode is parametrically amplified by a periodic motion of the mirror boundary, and the amplified photons are spontaneously emitted to the photonic band. We have derived the non-Hermitian effective Floquet-Liouvillian from the total system Liouvillian with the use of the Brillouin-Wigner-Feshbach projection method in …
We theoretically investigate the dynamical Casimir effect in electrical circuits based on superconducting microfabricated waveguides with tunable boundary conditions. We propose to implement a rapid modulation of the boundary conditions by tuning the applied magnetic flux through superconducting quantum interference devices (SQUIDs) that are embedded in the waveguide circuits. We consider two circuits: (i) An open waveguide circuit that corresponds to a single mirror in free space, and (ii) a resonator coupled to a microfabricated waveguide, which corresponds to a single-sided cavity in free s…
We take a closer look at the fundamental Casimir-Polder interaction between quantum particles and dispersive dielectric surfaces with surface polariton or plasmon resonances. Linear response theory shows that in the near field, van der Waals, regime the free energy shift of a particle contains a thermal component that depends exclusively on the population/excitation of the evanescent surface polariton/plasmon modes. Our work makes evident the link between particle surface interaction and near field thermal emission and demonstrates how this can be used to engineer Casimir-Polder forces. We als…
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 show two examples in which the dynamical Casimir effect can be achieved by modulating the Kerr or higher order nonlinearities. In the first case the cavity field is coupled to an arbitrary number of qubits or an harmonic oscillator via the dipole interaction. In the second case, the modulation of the nonlinearities is accompanied by the off-resonance modulation of the cavity frequency. We present the analytic description of the phenomenon and supplement it with numeric simulations, demonstrating that photons can be created from vacuum and the resulting hyper-Poissonian photon statistics is …
In this paper we develop a quantum optical method to detect the axion-nucleon interaction. We ultilize a levitated optomechanical system consisting of a silica nanosphere and an optical cavity here. We translate the trapping positions of the nanosphere, resulting the shift of its resonance frequency, which can be determined from measuring the resulting resonance shift in the transmission spectrum. Furthermore, The frequency shift can be related to the additional forces due to two-axion exchange via substraction. Based on noise ananlysis, estimation and calculation, we set the stringent prospec…
We present the study of parametric resonance in a one-dimensional cavity based on the analysis of classical optical paths. The recursive formulas for field energy are given. We separate the mechanism of particle production and the resonance amplification of radiation. The production of photons is a purely quantum effect described in terms of quantum anomalies in recursive formulas. The resonance enhancement is a classical phenomenon of focusing and amplifying beams of photons due to Döppler effect.
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 …
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically instable solutions. Similarly, the spectrum of qua…
We study the relationship between entanglement and parametric resonance in a system of two coupled time-dependent oscillators. As a measure of bipartite entanglement, we calculate the linear entropy for the reduced density operator, from which we study the entanglement dynamics. In particular, we find that the bipartite entanglement increases in time up to a maximal mixing scenario, when the set of auxiliary dynamical parameters are under parametric resonance. Moreover, we obtain a closed relationship between the correlations in the ground state, the localisation of the Wigner function in phas…
High-fidelity parametric gates have been demonstrated with superconducting qubits via rf flux modulation of the qubit frequency. The modulation however leads to renormalization of the bare qubit-qubit coupling, thereby reducing the gate speed. Here, we realize a parametric-resonance gate, which is activated by bringing the average frequency of the modulated qubit in resonance with a static-frequency qubit while approximately retaining the bare qubit-qubit coupling. The activation of parametric-resonance gates does not depend on the frequency of modulation, allowing us to choose the modulation …
The cooperative resonance for a spin one half system interacting with dc and ac magnetic field is considered. This interaction in the system collective regime can result in parametic resonance and rapid excitation of the excited spin state of the dense atomic gas. The phenomenon is studied using the density matrix approach. We discuss the implementation of this effect and possible applications of the quantum amplification by superradiant emission of radiation.
The phenomenon of particle creation within an almost resonantly vibrating cavity with losses is investigated for the example of a massless scalar field at finite temperature. A leaky cavity is designed via the insertion of a dispersive mirror into a larger ideal cavity (the reservoir). In the case of parametric resonance the rotating wave approximation allows for the construction of an effective Hamiltonian. The number of produced particles is then calculated using response theory as well as a non-perturbative approach. In addition we study the associated master equation and briefly discuss th…
A quantum mechanical version of a classical inverted pendulum is analyzed. The stabilization of the classical motion is reflected in the bounded evolution of the quantum mechanical operators in the Heisenberg picture. Interesting links with the quantum Zeno effect are discussed.
A boundary undergoing relativistic motion can create particles from quantum vacuum fluctuations in a phenomenon known as the dynamical Casimir effect. We examine the creation of particles, and more generally the transformation of quantum field states, due to boundary motion in curved spacetime. We provide a novel method enabling the calculation of the effect for a wide range of trajectories and spacetimes. We apply this to the experimental scenario used to detect the dynamical Casimir effect, now adopting the Schwarzschild metric, and find novel resonances in particle creation as a result of t…
We study theoretically the non-stationary circuit QED system in which the artificial atom transition frequency has a small periodic modulation in time, prescribed externally. We show that, in the dispersive regime, when the modulation periodicity meets the `resonances', the dynamics may be described by the dynamical Casimir effect, Jaynes-Cummings or Anti-Jaynes-Cummings effective Hamiltonians. In the resonant atom-cavity regime, under the modulation `resonance' the dynamics resembles the behavior of the dynamical Casimir effect in a vibrating cavity containing a resonant two-level atom, and e…
We generate squeezed state of light at 860 nm with a monolithic optical parametric oscillator. The optical parametric oscillator consists of a periodically poled KTiOPO_4 crystal, both ends of which are spherically polished and mirror-coated. We achieve both phase matching and cavity resonance by controlling only the temperature of the crystal. We observe up to -8.0 dB of squeezing with the bandwidth of 142 MHz. Our technique makes it possible to drive many monolithic cavities simultaneously by a single laser. Hence our monolithic optical parametric oscillator is quite suitable to continuous-v…
We study the problem of the behavior of a quantum massless scalar field in the space between two parallel infinite perfectly conducting plates, one of them stationary, the other moving periodically. We reformulate the physical problem into a problem about the asymptotic behavior of the iterates of a map of the circle, and then apply results from theory of dynamical systems to study the properties of the map. Many of the general mathematical properties of maps of the circle translate into properties of the field in the cavity. For example, we give a complete classification of the possible reson…
We study cooperative phenomena in the fluctuation-induced forces between a surface and a system of neutral two-level quantum emitters prepared in a coherent collective state, showing that the total Casimir-Polder force on the emitters can be modified via their mutual correlations. Particularly, we find that a collection of emitters prepared in a super- or subradiant state experiences an enhanced or suppressed collective vacuum-induced force, respectively. The collective nature of dispersion forces can be understood as resulting from the interference between the different processes contributing…
Using the recently derived representation for the polarization tensor in (2+1)-dimensional space-time allowing an analytic continuation to the entire plane of complex frequencies, we obtain simple analytic expressions for the reflection coefficients of graphene at nonzero Matsubara frequencies. In the framework of the Lifshitz theory, these coefficients are shown to lead to nearly exact results for the Casimir free energy and pressure between two graphene sheets. The constituent parts of large thermal effect, arising in the Casimir interaction at short separations due to an explicit parametric…
We demonstrate parametric resonance in Bose-Einstein condensates (BECs) with attractive two-body interaction in a harmonic trap under parametric excitation by periodic modulation of the s-wave scattering length. We obtain nonlinear equations of motion for the widths of the condensate using a Gaussian variational ansatz for the Gross-Pitaevskii condensate wave function. We conduct both linear and nonlinear stability analyses for the equations of motion and find qualitative agreement, thus concluding that the stability of two equilibrium widths of a BEC might be inverted by parametric excitation…
We investigate Floquet dynamics of a cavity-spinor Bose-Einstein condensate coupling system via periodic modulation of the cavity pump laser. Parametric resonances are predicted and we show that due to cavity feedback-induced nonlinearity the spin oscillation can be amplified to all orders of resonance, thus facilitating its detection. Real-time observation on Floquet dynamics via cavity output is also discussed.
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 …
We develop a theory of parametric resonance in tunable superconducting cavities. The nonlinearity introduced by the SQUID attached to the cavity, and damping due to connection of the cavity to a transmission line are taken into consideration. We study in detail the nonlinear classical dynamics of the cavity field below and above the parametric threshold for the degenerate parametric resonance, featuring regimes of multistability and parametric radiation. We investigate the phase-sensitive amplification of external signals on resonance, as well as amplification of detuned signals, and relate th…
Many molecular, quantum-dot, and optomechanical nanocavity-QED systems demonstrate strong nonlinear interactions between electrons, photons, and phonon (vibrational) modes. We show that such systems can be described by a universal model in the vicinity of the nonlinear parametric resonance involving all three degrees of freedom. We solve the nonperturbative quantum dynamics in the strong coupling regime of the nonlinear resonance, taking into account quantization, dissipation, and fluctuations of all fields. We find analytic solutions for quantum states in the rotating wave approximation which…
The evolution of a quantum oscillator, with periodically varying frequency and damping, is studied in the two cases of parametric resonance (PR) producing a limited, or unlimited stretching of the wave function. The different asymptotic behaviors of the energy distribution, show the non trivial interplay between PR and the initial quantum state. In the first case, the oscillator's mean energy tends asymptotically to a fully classical value, independent of the initial state, with vanishing relative quantum fluctuations. In the second case, the memory of the initial state persists over arbitrary…
A fundamental prediction of quantum mechanics is that there are random fluctuations everywhere in a vacuum because of the zero-point energy. Remarkably, quantum electromagnetic fluctuations can induce a measurable force between neutral objects, known as the Casimir effect, which has attracted broad interests. The Casimir effect can dominate the interaction between microstructures at small separations and has been utilized to realize nonlinear oscillation, quantum trapping, phonon transfer, and dissipation dilution. However, a non-reciprocal device based on quantum vacuum fluctuations remains a…
