Several nontrivial phenomena emerge when a quantum field is subjected to dynamical perturbations, with prominent examples including the Hawking and Unruh effects, as well as the dynamical Casimir effect. In this work, we compute the number of particles produced via the dynamical Casimir effect in an ideal cavity, where one of the mirrors is allowed to move under the influence of a gravitational wave. Assuming an oscillatory mirror motion and a plane gravitational wave, we identify the resonance conditions that lead to an exponential increase in the number of created particles through parametric amplification.
Several nontrivial phenomena emerge when a quantum field is subjected to dynamical perturbations, with prominent examples including the Hawking and Unruh effects, as well as the dynamical Casimir effect. In this work, we compute the number of particles produced via the dynamical Casimir effect in an ideal cavity, where one of the mirrors is allowed to move under the influence of a gravitational wave. Assuming an oscillatory mirror motion and a plane gravitational wave, we identify the resonance conditions that lead to an exponential increase in the number of created particles through parametric amplification.
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 R\'enyi 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, ${S}_{A}(\ensuremath{\tau})\ensuremath{\sim}\frac{1}{2}\mathrm{log}(\ensuremath{\tau})$, while in higher dimensions (n+1) the growth is linear, ${S}_{A}(t)\ensuremath{\sim}\ensuremath{\lambda}\ensuremath{\tau}$, where $\ensuremath{\lambda}$ can be identified with the Lyapunov exponent of a classical instability in the system. In (1+1) dimensions, strong interactions among field modes prevent the parametric resonance from manifesting as a Lyapunov instability, leading to a sublinear entropy growth associated with a constant rate of particle production in the resonant mode. Interestingly, the logarithmic growth comes with a prefactor of 1/2 which cannot occur in time-periodic systems with finitely many degrees of freedom and is thus a special property of bosonic field theories.
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 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 superconducting quantum interference device), three remarkable results will show up, namely, (i) a quite different spectral distribution for the created particles, deviating from the typical parabolic shape, (ii) an enhancement by a factor of approximately $5\ifmmode\times\else\texttimes\fi{}1{0}^{3}$ in the number of created particles with half driving frequency of the effective moving mirror, and (iii) an enhancement by a factor of $3\ifmmode\times\else\texttimes\fi{}1{0}^{2}$ in the particle creation rate. These results may guide the experimentalists in their search for alternative routes to observe the DCE in future experiments.
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 a 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 very different from the squeezed vacuum state.
We study theoretically an array of two-level qubits moving relative to a one-dimensional waveguide. This motion can be implemented mechanically or simulated via the modulation of the couplings between the qubits and the waveguide. When the frequency of this motion approaches twice the qubit resonance frequency, it induces parametric generation of photons and excitation of the qubits. The proposed quantum optomechanical system offers a plethora of possibilities for exploring various quantum electrodynamics phenomena. However, their theoretical analysis is challenging due to the presence of quantum nonlinearity, a continuum of propagating photonic modes, and the excitation of strongly nonequilibrium qubit states, which make many conventional analytical tools inapplicable. To address these challenges, we develop a comprehensive general theoretical framework that incorporates both perturbative diagrammatic techniques and a rigorous master-equation approach. Our calculations reveal several intriguing effects, including the directional dynamical Casimir effect, where momenta of emitted photon pairs are correlated, and the waveguide-mediated collective Unruh effect, where motion drives the qubits to a nontrivial steady state that can be entangled and exhibit phase transitions. Additionally, we examine the radiation back-action on the qubit motion, which becomes particularly pronounced when subradiant modes in the qubit array are excited. The back-action can significantly alter the mechanical spectra, potentially leading to the formation of hybrid phonon-biphoton modes.
We evaluate the effective action for the Dynamical Casimir Effect (DCE) for a real scalar field in d+1 dimensions within the worldline formulation of quantum field theory. The scalar field is coupled to a spacetime-dependent mass term, which here plays the role of the moving medium and imposes imperfect boundary conditions on time-dependent surfaces. Expanding in powers of the departure of the geometry from a planar configuration, the worldline path integral factorizes into simpler, lower-dimensional ones. In the limit of a strong coupling to the surface, we recover the Dirichlet result and derive the systematic corrections in inverse powers of the coupling, calculating the imaginary part of the effective action up to arbitrary order of said powers. Finally, we also apply the method to a two-surface configuration.
When one of the parameters in the Euler-Lagrange equations of motion of a sys- tem is modulated, particles can be generated out of the quantum vacuum. This phenomenon is known as the dynamical Casimir efiect, and it was recently realized experimentally in systems of superconducting circuits, for example by using modulated resonators made of coplanar waveg- uides, or arrays of superconducting quantum intereference devices (SQUIDs) forming a Josephson metamaterial. In this paper, we consider a simple electrical circuit model for dynamical Casimir efiects, consisting of an LC resonator, with the inductor modulated externally at 10.8GHz and with the resonant frequency tunable over a range of §400MHz around 5.4GHz. The circuit is analyzed classically using a circuit simulator (APLAC). We demonstrate that if an additional source of classical noise couples to the resonator (on top of the quantum vacuum), for example via dissipative \internal modes, then the resulting spectrum of the photons in the cavity will present two strongly asymmetric branches. However, according to the theory of the dynamical Casimir efiect, these branches should be symmetric, a prediction which is conflrmed by our ex- perimental data. The simulation presented here therefore shows that the origin of the photons generated in our experiment with Josephson metamaterials is the quantum vacuum, and not a spurious classical noise source. The picture about vacuum that emerges from modern quantum fleld theory is very difierent from that ofiered by classical physics. In classical fleld theory, the vacuum is the zero-energy state of the fleld, deflned by the absence of any excitation. In quantum fleld theory, the vacuum state has a flnite zero-point energy associated with it, and the uncertainty principle indicates that ∞uctuations exist even in this state. Due to the existence of ∞uctuations, the quantum vacuum can become unstable under certain perturbations, and the energy of the perturbation is converted into creation of real particles (1). In the Schwinger efiect for example, a static intense electric fleld can create pairs of electrons and positrons. In very strong gravitational flelds, at the event horizon of black holes, the vacuum becomes unstable and energy is radiated away (Hawking efiect). By the equivalence principle, an accelerated observer in the Minkowski vacuum will detect a flnite-temperature fleld (Unruh efiect). In a previous paper, we reported the observation of the dynamical Casimir efiect using a ∞ux- biased Josephson metamaterial embedded in a microwave cavity at 5.4GHz (2). A non-adiabatic change in the index of refraction of the cavity (or in the electrical length) was realized by modulat- ing the ∞ux at values close to double the resonant frequency of the cavity. We measured also the frequency-correlated photons emitted from the cavity (which is kept at a temperature T = 50mK), and we obtained a power spectrum displaying a bimodal, \sparrow-tail distribution, with two sym- metric branches. Furthermore, we demonstrated that the generated photons are squeezed below the vacuum level and that the state of the two-photon fleld is nonseparable. The experimental results are in excellent agreement with the theoretical predictions. An alternative realization of the dynam- ical Casimir efiect consists in modulating an efiective boundary condition, realized experimentally as a SQUID structure placed at the end of a coplanar waveguide resonator (3). Remarkably, the experiments presented in Refs. (2,3) are the only existing measurements of the dynamical Casimir efiect, which was predicted theoretically in 1970 (4).