Photon Generation in an Electromagnetic Cavity with a Time-Dependent Boundary

We report the observation of photon generation in a microwave cavity with a time-dependent boundary condition. Our system is a microfabricated quarter-wave coplanar waveguide cavity. The electrical length of the cavity is varied by using the tunable inductance of a superconducting quantum interference device. It is measured at a temperature significantly less than the resonance frequency. When the length is modulated at approximately twice the static resonance frequency, spontaneous parametric oscillations of the cavity field are observed. Time-resolved measurements of the dynamical state of the cavity show multiple stable states. The behavior is well described by theory. Our results may be considered a preliminary step towards demonstrating the dynamical Casimir effect.

Figure 1

(a) A micrograph of a $\lambda /4$ cavity. Metal layers appear light and the substrate dark. The cavity’s length is $d=5.56\mathrm{mm}$. The center conductor is $13\mu \mathrm{m}$ wide with a gap of $7\mu \mathrm{m}$ to the ground planes. The cavity is probed through a small coupling capacitance (right enlargement) while the other end is terminated to ground through a SQUID (left enlargement). Changing the external magnetic field through the SQUID loop changes the boundary condition of the cavity. (b) Simplified block diagram. The cavity is measured by using a circulator and a cold amplifier. The magnetic field is applied via on-chip control lines, one for high frequency and one for dc.

Figure 2

The output of the pumped cavity. (a) Two output power spectra referred to the cavity output, for different working points–wider (red) peak: $f_p=10.3651\mathrm{GHz}$, $A_p=0.11$; narrower (blue) peak: $f_p=10.3637\mathrm{GHz}$, $A_p=0.23$. The offset frequency is centered at $f_p/2$. The widths of the peaks are the switching rates between different states (see Fig. 3). (b) Integrated output power as a function of $f_p$ for $A_p=0.23$. The amplifier noise power has been subtracted. The predicted linear dependence on detuning from the bifurcation point is clear. (c) Integrated output power as a function of $A_p$ and $f_p$. The increased noise near the left boundary is due to the slow switching of the output power in this region. (d) Theoretical fit from solving Eq. (1). The color scales are the same in both plots.

Figure 3

Exploring the metapotential. 2D histograms of measured in-phase $q_1$ and quadrature $q_2$ voltages for three different working points. (a) Just below threshold near zero detuning, the metapotential softens in one direction and we observe a noise ellipse. (b) Above threshold and red detuned, the system has bifurcated, exhibiting two finite-amplitude oscillating states. These states have equal amplitude but are shifted by 180° in phase. (c) When the pump is far red detuned, a quiet state can coexist with the oscillating states, in agreement with theory. The noise circles in (b) and (c) are dominated by the measurement amplifier. The phase rotations between the three images are also instrumental artifacts.