Vacuum amplification across a programmable bosonic Bogoliubov stability boundary in a trapped-ion chain

Abstract

A quantum oscillator always carries vacuum fluctuations, and modulating its parameters converts them into correlated pairs of real quanta. This vacuum amplification underlies squeezed light and Hawking radiation. In a closed quadratic system, pair creation competes with the energy splitting of the modes, and growth turns exponential only once pairing overcomes the splitting. Here we realize this competition in two motional modes of a trapped-ion chain. A two-mode-squeezing drive creates the pairs, and a beam-splitter coupling splits the motion into two coupled modes at different frequencies. The two rates are tuned and calibrated independently. Starting near the motional ground state, we cross the stability boundary in two ways. First, with the drive at the sum of the two bare-mode frequencies, the phonon number stays bounded until the pairing rate exceeds the splitting and grows exponentially beyond it. Second, tuning the drive into resonance with one coupled mode lowers the threshold to zero, and only that mode is amplified into a squeezed vacuum. Separate control of the two interactions makes the threshold for vacuum amplification programmable. This control can be used to simulate particle creation and quantum phase transitions, and to operate parametrically driven bosonic devices close to their stability limit.

The paper asks whether independently controlled, coherent mode mixing and pair creation can produce a programmable vacuum-amplification threshold. It implements a quadratic two-mode Hamiltonian in the axial and radial motion of a trapped-ion chain, combining a static beam-splitter coupling $g_{\mathrm{BS}}$ with driven two-mode squeezing $g_{\mathrm{TMS}}$. Both rates are independently calibrated; the modes start near $\bar n=0.1$. At bare-mode degeneracy and zero drive detuning, increasing pairing changes bounded occupation oscillations into exponential growth at $g_{\mathrm{TMS}}=g_{\mathrm{BS}}$, with $g_{\mathrm{BS}}=1.44\pm0.05$ kHz. Pulse-envelope fits resolve the closing Bogoliubov gap and emerging instability rate; the fitted pairing calibration agrees within 3% with an independent calibration. Detuning the drive to twice either coupled-mode frequency instead gives a predicted zero threshold and selectively amplifies that mode. With $g_{\mathrm{BS}}=7.40\pm0.01$ kHz and a 220-$\mu$s pulse, the measured squeezing parameters are $r_-=1.48\pm0.06$ and $r_+=1.50\pm0.06$, versus the symmetric-point bound $r_{\max}=0.155\pm0.007$ at the same coupling ratio. Sideband-flop reconstructions show suppressed odd-number populations and reject thermal statistics. The experiment probes two modes, not the proposed many-mode or nonlinear extensions. Threshold fits exclude occupations $\bar n\geq5$ because readout loses sensitivity. Near resonance, apparent excitation in the unamplified mode is attributed to readout leakage, supported by simulations. Number statistics do not directly establish phase coherence; the paper cites separate interferometry. Heating is negligible during the pulses, while the measured swap-coherence time is $1.43\pm0.10$ ms.