NOON States of Nine Quantized Vibrations in Two Radial Modes of a Trapped Ion

We develop a deterministic method to generate and verify arbitrarily high NOON states of quantized vibrations (phonons), through the coupling to the internal state. We experimentally create the entangled states up to $N=9$ phonons in two vibrational modes of a single trapped ${{}^{171}\mathrm{Yb}}^{+}$ ion. We observe an increasing phase sensitivity of the generated NOON state as the number of phonons $N$ increases and obtain the fidelity from the contrast of the phase interference and the population of the phonon states through the two-mode projective measurement, which are significantly above the classical bound. We also measure the quantum Fisher information of the generated state and observe Heisenberg scaling in the lower bounds of phase sensitivity as $N$ increases. Our scheme is generic and applicable to other photonic or phononic systems such as circuit QED systems or nanomechanical oscillators, which have Jaynes-Cummings-type of interactions.

Figure 1

Experimental setup. (a) Side and (b) top view of the trap and laser configuration. The figures are not to scale. The diameter of the dc and rf rods and the center-to-center distance between is are 0.5 mm. The red gradient color in (a) indicates the effective two-dimensional harmonic potential in the Paul trap.

Figure 2

Generation sequence of the NOON state of $N=3$. The 3D grid describes the total Hilbert space of the system that consists of spin and $X$ and $Y$ vibrational modes. The lower and the upper layers represent $|\downarrow ⟩$ and $|\uparrow ⟩$ spin states and $X$ and $Y$ axes stand for the $X$ and $Y$ modes, where the intersection points in the grid represent the basis states as $|\sigma ,n_X,n_Y⟩$. The red arrows indicate carrier transitions, the blue arrows indicate blue-sideband transitions, and the green arrows indicate composite-pulse operations (see Supplemental Material [41], Composite Pulse). The numbers on the arrows denote the order of the operations. For example, the first blue arrow describes the blue-sideband transition on the $X$ mode from $|\downarrow ,0,0⟩$ to $|\downarrow ,1,0⟩$.

Figure 3

Parity oscillations of the generated NOON states from $N=1$ to $N=9$. (a) The blue dots are experimental data, and the red lines are fitting curves with $⟨\mathrm{\Pi }(\phi )⟩=A\cos k\phi +B\sin k\phi +C$, and $C_P=\sqrt{A^2+B^2}$. (b) The time evolution of the blue-sideband transition of the output mode with $\phi =0$ for the NOON state of $N=7$ and its fitting. (c) The corresponding phonon distribution $P_n$. We note that, generally, $\sum _n{P}_n<1$ due to the experimental errors in the generation stage. The error bars are derived from the fitting error with a confidence level of 0.95 throughout the Letter.

Figure 4

Fidelity and quantum Fisher information of the generated states. (a) The experimental results of fidelity as well as $C_P$ and $P_{N,0}+P_{0,N}$. The error bars of $C_P$ are derived from the fitting error and those of $P_{N,0}+P_{0,N}$ from the shot-noise error. (b) The quantum Fisher information of the generated states.