Nonlinear coupling of continuous variables at the single quantum level

We experimentally investigate nonlinear couplings between individual quanta of the vibrational modes of strings of cold ions stored in linear ion traps. The nonlinearity is caused by the ions’ Coulomb interaction and gives rise to a Kerr-type interaction Hamiltonian $H=\hslash \chi {\widehat{n}}_r{\widehat{n}}_s$, where ${\widehat{n}}_r$, ${\widehat{n}}_s$ are phonon number operators of two interacting vibrational modes. We precisely measure the resulting oscillation frequency shift and observe a collapse and revival of the contrast in a Ramsey experiment over a range of trap frequencies. With $2.97\mathrm{MHz}$ stretch mode and $3.61\mathrm{MHz}$ rocking mode frequencies, we observed a coupling of $20.5\mathrm{Hz}∕\mathrm{phonon}$. Implications for ion trap experiments aiming at high-fidelity quantum gate operations are discussed.

Figure 1

(Color online) Contrast $C\left(\tau \right)$ of a Ramsey experiment on the stretch mode as a function of the Ramsey waiting time $\tau$. The axial trap frequency was ${\omega }_z=\left(2\pi \right)1486\mathrm{kHz}$; the radial modes were cooled close to the Doppler limit. The initial contrast is limited by imperfect sideband cooling, spurious excitation of the second ion by the focused laser beam, and magnetic field noise. The subsequent loss of contrast is caused by a dephasing of the stretch mode oscillation. The inset shows the pulse sequence of the Ramsey experiment with $C$ denoting carrier pulses and $B$ blue sideband pulses.

Figure 2

(Color online) Contrast $C\left(\tau \right)$ of a spin echo experiment on the stretch mode as a function of the Ramsey waiting time $\tau$. Insertion of the spin echo increased the coherence time by nearly two orders of magnitude as compared with a simple Ramsey experiment.

Figure 3

(Color online) $D$-state population as a function of the spin echo phase. The shift $\mathrm{\Delta }\varphi$ of the spin echo phase pattern by a rocking mode phonon is observed by measuring the $D$-state population of ion 1 as function of the phase of the last spin echo pulse. The symbol $(◇)$ labels experiments with an extra phonon in the second spin echo time, $(○)$ experiments with no extra phonon.

Figure 4

(Color online) (Inset) Shift of the stretch-mode echo phase pattern as a function of the waiting time due to a single rocking mode phonon. The frequency shift is inferred from the slope of the function. (Main graph) Shift of the stretch mode frequency by a single rocking mode phonon as a function of the axial trap frequency ${\omega }_z$. The solid line is the frequency shift predicted by Eq. (1); the dotted line is a fit to the data using a power law $d{\nu }_s∕d{n}_r\propto {\omega }_z^{\beta }$ with $\beta =3.25$.

Figure 5

(Color online) Collapse and revival of the contrast in a Ramsey experiment probing the motional coherence of the two lowest stretch mode quantum states. The revival occurs at the time predicted by the spin echo experiment for a frequency ${\omega }_z∕2\pi =1716\mathrm{kHz}$.