Measurement of the Casimir force between superconductors

Abstract

The Casimir force follows from quantum fluctuations of the electromagnetic field and yields a nonlinear attractive force between closely spaced conductive objects. Its magnitude depends on the conductivity of the objects up to optical frequencies. Measuring the Casimir force in superconductors should allow to isolate frequency-specific contributions to the Casimir effect, as frequencies below the superconducting gap energy are expected to contribute differently than those above it. There is significant interest in this contribution as it is suspected to contribute to an unexplained discrepancy between predictions and measurements of the Casimir force between normal metals, which questions the basic principles on which estimates of the magnitude are based. Here, we observe the Casimir force between superconducting objects through the nonlinear dynamics it imparts to a superconducting drum resonator in a microwave optomechanical system. There is excellent agreement between the experiment and the Casimir force magnitude computed for this device across three orders of magnitude of displacement. Furthermore, the Casimir nonlinearity is intense enough that, with a modified design, this device type should operate in the single-phonon nonlinear regime. Accessing this regime has been a long-standing goal that would greatly facilitate quantum operations of mechanical resonators.

Figures (29)

• Figure 1: Effects of a Casimir force on a superconducting drum resonator. a: Our device consists of two plates separated by a vacuum gap. The top plate is mechanically compliant and experiences a mechanical restoring potential (orange line) which is minimum for a separation $d$ between plates (see inset). The Casimir potential (blue) adds to this restoring potential, and the sum (black) has a local minimum at separation $d'$ shifted from $d$. The total potential is not harmonic around $d'$ for large amplitudes which causes nonlinear behavior. b: Schematic of the microwave optomechanical measurement scheme. We strongly drive the microwave cavity at its center frequency, with an additional strong drive tone at $\omega_\mathrm{sb}$ which is close to $\omega_\mathrm{c} - \omega_\mathrm{m}$. The sideband drive is swept in frequency, and we read out the emitted signal in a window around $\omega_\mathrm{c} + \omega_\mathrm{m}$. c: Measured displacement response for various nominal sideband drive powers (labels) at $-20$ dBm cavity drive, showing the expected Lorentzian behavior at low drive power that transitions to a strong softening nonlinearity at high drive power. The Casimir force solution (black line) becomes multi-valued, with two stable branches (solid black) and an unstable solution (dotted black/semitransparent). Our measurements follow the stable solutions, leading to a hysteresis depending on the sweep direction. The theory matches well to all curves using only a single fit parameter, $d$, common to all curves. The right panel highlights details of the center region indicated by the dashed box, and the hysteresis depending on the sweep direction is indicated by arrows. The data points at the highest powers are made transparent for visual clarity.
• Figure 2: Responses measured for all combinations of drive parameters for cavity drive powers (a: $-15$ dBm, b: $-30$ dBm, c: $-40$ dBm) and sideband drive powers (colors). All theory curves (black lines) across all panels share only a single fit parameter $d$.
• Figure 3: Optomechanical calibration. a: Theory curves generated with equal maximum mechanical amplitude for various values of $d$ (black & grey lines). For smaller $d$, the Casimir force is stronger and leads to a larger softening nonlinearity. For $d = 18$ nm there is excellent agreement with the measured data (green). b: The drive efficiency decreases sharply from 1 at high amplitude, as most of the time the instantaneous cavity frequency $\omega_\mathrm{c}(t)$ is far away from the drive frequency. This effect is stronger, comparatively, in the bad cavity limit ($\omega_\mathrm{m} \ll \kappa$), but it is noticeable in our experiments at the largest amplitudes. c: The drive efficiency leads to a correction in the Casimir curves at large amplitudes.
• Figure 4: Response observed at higher-order sidebands. a: The full optomechanical measurement consists of six sidebands that are read out sequentially. The first red sideband ($-\omega_\mathrm{m}$) is at the same frequency as our swept sideband drive, so its signal is superposed on a pedestal from the directly reflected sideband drive signal. b: Superimposition of sidebands. All sidebands encode the mechanical displacement with a known proportionality (see main text), so they collapse on top one another when overlaid. The frequency axis of each sideband has been shifted to align the sideband frequencies $\omega_c+n\omega_m$, and the frequency axis of the red sidebands (below the cavity frequency) has been flipped about the sideband frequency. Furthermore the frequency range has been divided by two for second sidebands and three for third sidebands. The vertical axes of $2^\mathrm{nd}$ and $3^\mathrm{rd}$ order sidebands have been squared and cubed respectively, as expected from theory.
• Figure 5: The Casimir pressure. The Casimir pressure for various distances as calculated via the Lifshitz formula and the BCS model, and a fit proportional to $d^{-3.193}$.