Trapped-Ion Quantum Simulator: Experimental Application to Nonlinear Interferometers

We show how an experimentally realized set of operations on a single trapped ion is sufficient to simulate a wide class of Hamiltonians of a spin-$1/2$ particle in an external potential. This system is also able to simulate other physical dynamics. As a demonstration, we simulate the action of two $n\mathrm{t}\mathrm{h}$ order nonlinear optical beam splitters comprising an interferometer sensitive to phase shift in one of the interferometer beam paths. The sensitivity in determining these phase shifts increases linearly with $n$, and the simulation demonstrates that the use of nonlinear beam splitters ($n=2,3$) enhances this sensitivity compared to the standard quantum limit imposed by a linear beam splitter ($n=1$).

Figure 1

Principle of the Mach-Zehnder interferometer. Modes ${\Psi }_a$ and ${\Psi }_b$ are superposed on the first beam splitter. After the beam splitter has acted, the modes ${\Psi }_{a^{\prime }}$, ${\Psi }_{b^{\prime }}$ are propagated along separate paths to a second beam splitter. Mode ${\Psi }_{b^{\prime }}$ may undergo a variable phase shift induced by the phase shifter $\varphi$. Modes ${\Psi }_{a^{\prime \prime }}$ and ${\Psi }_{b^{\prime \prime }}$ emerge after the second beam splitter and one of the modes is put onto a detector. Varying $\varphi$ will lead to a sinusoidal behavior of the intensity on the detector (fringes).

Figure 2

Interference fringes for simulated interferometers (a) of order $n=1$ (integration time per data point was 0.50 s); (b) $n=2$ (0.53 s); and (c) $n=3$ (0.63 s). $⟨n_{a^{\prime \prime }}⟩$ is the probability to find the ion in $|1{⟩}_{a^{\prime \prime }}$, while $t$ is the time for which the trap frequency was shifted by $\Delta {\omega }_z$, directly proportional to the phase shift $\varphi =\Delta {\omega }_{z}t$. The frequency of the fringes increases linearly with order $n$. $C$ is the observed contrast of the fringes.

Figure 3

Noise-to-signal ratios $\delta \varphi (N_b)$ for the $n=1$ linear interferometer (solid squares), $n=2$ nonlinear interferometer (solid circles), and $n=3$ nonlinear interferometer (open circles) vs the number of measurements $N_b$. The solid line is the theoretical limit for the linear ($n=1$) interferometer, assuming perfect contrast and detection efficiency.