Quantum Parametric Oscillator with Trapped Ions

A strong nonlinear coupling between harmonic oscillators is highly desirable for quantum information processing and quantum simulation, but is difficult to achieve in many physical systems. Here, we exploit the Coulomb interaction between two trapped ions to achieve strong nonlinear coupling between normal modes of motion at the single-phonon level. We experimentally demonstrate phonon up- and down-conversion and apply this coupling to directly measure the parity and Wigner functions of the ions’ motional states. Our results represent the fully quantum operation of a degenerate parametric oscillator and hold promise for quantum computation schemes that involve continuous variables.

Figure 1

(a) Schematic of the experimental setup. One of the ${\mathrm{Yb}}^{+}$ ions is in the $S_{1/2}$ ground state and interacts with Doppler cooling and Raman beams. The other ion is prepared in a dark metastable ${}^2{F}_{7/2}$ state and is sympathetically cooled. Three Raman beams $R1$, $R2$, and $R3$ control the motional states along the axial and radial directions. The red labels show the Raman beam polarizations. The 7.0 G magnetic field $\overrightarrow{B}$ is parallel to the $R1$ Raman beam. (b) Trap electrode configuration. The radial trapping potential is generated by 30 MHz rf signal $V_{\mathrm{rf}}$ and can be fine tuned by adjusting offset voltages on the $x$ electrodes with two identical low-pass filters (LPF). Small additional dc voltage applied to the $y$ electrodes helps to compensate stray electric fields.

Figure 2

Phonon state evolution in the axial (solid dots) and radial (open circles) modes, where either one (blue) or two (red) phonons are initially added to the radial mode. The sinusoidal fit (red line) reveals the coupling rate to be $3.02\pm 0.02\mathrm{kHz}$. The error bars show $1\sigma$ statistical uncertainty. The oscillation’s amplitude after about 10 ms is a factor of 0.39(7) compared to the initial amplitude, limited by the coherence time of the phonons in the radial mode [25].

Figure 3

Avoided crossing observed for the axial mode after sideband cooling. The left plot shows the probability $p_1$ of the transition from a “dark” $|F=0,m_F=0⟩$ to “bright” $|F=1,m_F=0⟩$ state as a function of detuning $\delta$ from the resonance condition, and Raman detuning. We extract the coupling strength from the mode splitting at resonance, as shown on the right panel. The splitting is measured to be $2.97\pm 0.03\mathrm{kHz}$. Dots correspond to the measured frequencies at the peak centers and the solid lines show the eigenvalues of the Hamiltonian [Eq. (2)].

Figure 4

Wigner functions for different quantum states. (a)–(f) Wigner functions (from left to right) for the vacuum state, coherent states $|\alpha {⟩}_r$ for $\alpha =0.87(3)$ and 1.73(6), Schrödinger cat states $(|\alpha {⟩}_r+|-\alpha {⟩}_r)/\sqrt{2}$, $(|\alpha {⟩}_r-|-\alpha {⟩}_r)/\sqrt{2}$, and $(|\alpha {⟩}_r-|i\alpha {⟩}_r)/\sqrt{2}$ for $\alpha =1.73(6)$. The top row corresponds to the experimental data, while the bottom row shows the calculated Wigner functions. (g)–(i) Wigner functions of the Fock states $|n{⟩}_r$ averaged over the phase of $\alpha$ with $n=1$, 2, and 5. The solid lines show the theoretical prediction $W(|\alpha |)=2(-1{)}^n{e}^{-2|\alpha {|}^2}{L}_n(4|\alpha {|}^2)/\pi$; the points are experimentally measured values. The negative measured values of the Wigner functions for the Schrödinger cat and Fock states demonstrate the nonclassical character of these states. Reconstruction of each Wigner function requires up to 60 000 measurements and takes up to 3 h.