Figure 1

Parametric amplification of pendulum motion by a child standing on a swing. The amplification is driven by changing the center of mass (stars), and thus effective length, of the pendulum at twice the frequency of the unperturbed swing.Reuse & Permissions

Figure 2

Relationships between quantum amplification mechanisms. Counterclockwise from the parametric amplifier: For a single mode of the Minkowski vacuum, the nondegenerate parametric amplifier (NDPA) and Unruh effect (UE) share the same form of Bogoliubov transformations resulting in both exhibiting a two-mode squeezed state. The UE is in turn connected to Hawking radiation (HR) through the equivalence principle relating inertial and gravitational acceleration. The exponential redshifting (Doppler shift) of the field modes near the black hole horizon results in Bogoliubov transformations that are identical to those for the dynamical Casimir effect (DCE), provided the mirror’s trajectory is given by Eq. (66). Here one obtains an identical Doppler shift, leading to a thermal spectrum for the emitted radiation. Finally, the DCE and a degenerate parametric amplifier (DPA) can be related by considering the case of a single-mode cavity with a sinusoidally time-dependent boundary condition.Reuse & Permissions

Figure 3

The principle of a parametric amplifier: a pump photon is down-converted by a nonlinear medium into a signal and an idler photon, whose frequencies add up to that of the pump photon.Reuse & Permissions

Figure 4

Paths of accelerated observers in Rindler coordinates $(c\tau ,\xi )$ with proper time $\tau$ and constant acceleration $a=c^2/\xi$ as viewed in Minkowski spacetime with coordinates $(ct,x)$. Lines (dashed) of constant proper time $\tau$ are also indicated. Observers in the right Rindler wedge (RRW) are out of causal contact with the left Rindler wedge (LRW) due to the presence of a horizon at $ct=\pm x$. Arrows give the direction of increasing proper time in each Rindler wedge.Reuse & Permissions

Figure 5

Formation of a horizon by the gravitational collapse of a spherical object. Before the horizon forms, light rays leaving the surface of the object are free to propagate out to spatial infinity. In contrast, once the mass of the body is within the Schwarzschild radius $r_s=2GM/c^2$, light rays are trapped behind the horizon and eventually encounter the singularity (dashed line). The horizon demarcates the last light ray able to escape from the surface to infinity and the first trapped ray inside the radius $r_s$. Equivalently, the horizon can be characterized by looking at the causal structure of spacetime indicated by light cones that give the direction of propagation for light rays at a given point. As one approaches the horizon, the light cone begins to tilt toward the black hole singularity. On the horizon, the light cone aligns along the $ct$ direction such that a light ray emitted from the horizon is stationary in space. As the time component of the metric vanishes on the horizon, a light ray on the horizon also appears frozen in time. Inside the horizon, even time itself points toward the singularity, so that nothing can escape.Reuse & Permissions

Figure 6

Cartoon of a black hole with vacuum fluctuations. Far from the horizon, vacuum fluctuations result in virtual particles that quickly annihilate each other. At the horizon, however, one particle in a virtual pair may be trapped inside the horizon, allowing its partner to escape to arbitrary large distances, the Hawking effect.Reuse & Permissions

Figure 7

An oscillating mirror in free space generates photons due to its interaction with vacuum fluctuations. This effect is known as the dynamical Casimir effect. The photons are generated in pairs with frequencies that add up to the frequency of the mirror’s oscillation. The photon-pair production can be interpreted as up-conversion of virtual photons of the quantum vacuum fluctuations, or, equivalently, as down-conversion of pump phonons from the oscillatory motion of the mirror.Reuse & Permissions

Figure 8

Mirror trajectories in the original coordinates and the transformed coordinates for (a) a single mirror and (b) a cavity with variable length. The coordinate transformations simplify the boundary-value problem, but finding the correct transformation functions ($f$, $g$, and $R$, respectively) can itself be a difficult problem.Reuse & Permissions

Figure 9

Schemes for superconducting circuit implementations of vacuum amplification processes: (a) The SQUID based parametric amplifier from 35. (b) The parametric amplifier can be approximated as a lumped $LC$ circuit with a current dependent inductance. The small normal current resistance is also depicted. (c) Spectrum of a parametric amplifier. For the nondegenerate case, one has separate peaks for the signal and idler modes satisfying ${\omega }_s+{\omega }_i={\omega }_p$. In contrast, the degenerate amplifier satisfies ${\omega }_s={\omega }_i$. (d) Illustration of a dc-SQUID array transmission line with accompanying bias line and flux pulse used in generating an analog event horizon and Hawking radiation. (e) Lumped circuit model valid for frequencies below the plasma frequency and negligible SQUID self-inductance. (f) One-dimensional blackbody spectrum of emitted Hawking radiation. The characteristic (Hawking) temperature of the distribution is determined by the gradient of the SQUID-array speed of light in a frame comoving with the flux pulse. (g) Circuit diagram of a SQUID-terminated coplanar waveguide used in generating the dynamical Casimir effect. Modulation of the SQUID’s Josephson energy is performed by the time-varying external flux ${\Phi }_{\mathrm{ext}}(t)$. (h) Equivalent lumped-element circuit model for the semi-infinite coplanar waveguide and dc SQUID. (i) Spectrum of photons emitted by the DCE assuming the SQUID is driven by a sinusoidally varying flux.Reuse & Permissions

Figure 10

Effect of a steplike flux bias pulse on the SQUID array speed of light $c_s(x)$ as seen in a frame moving with the pulse. Here velocities have been normalized with respect to the unbiased speed of light $c_s$ $[{\Phi }_{\mathrm{ext}}(x)=0]$. The pulse velocity was chosen to be $u=0.95c_s(0)$. In the comoving frame, the horizon occurs where $c_s(x)=u$. Similar to a black hole, the horizon is a unidirectional surface, and the arrow at the bottom indicates the only permissible direction for a photon to transverse the horizon.Reuse & Permissions