Cross-Kerr Nonlinearity for Phonon Counting

State measurement of a quantum harmonic oscillator is essential in quantum optics and quantum information processing. In a system of trapped ions, we experimentally demonstrate the projective measurement of the state of the ions’ motional mode via an effective cross-Kerr coupling to another motional mode. This coupling is induced by the intrinsic nonlinearity of the Coulomb interaction between the ions. We spectroscopically resolve the frequency shift of the motional sideband of the first mode due to the presence of single phonons in the second mode and use it to reconstruct the phonon number distribution of the second mode.

Figure 1

(a) Energy diagram of the bare motional eigenstates at resonance condition equation (1). The plot on the right shows the coherent energy exchange between the axial and radial modes after we initialize the system in the state $|1_a,0_b⟩$. The blue (red) dots are the probability for the system to be in the state $|1_a,0_b⟩$ ($|0_a,2_b⟩$) and the blue (red) line is the corresponding fit. Further measurements show that the 1/e decay time for these oscillations is 41(14) ms, which sets the bound for the coherence time of both modes. (b)–(e) Probability of finding the bright ion in the state $|\uparrow ⟩$ as a function of Raman detuning and the two-mode detuning $\delta$. Raman detuning is scanned around (b),(c) the first order and (d),(e) the second order motional blue sideband of the axial mode. The duration of the Raman pulse is chosen to match the $\pi$ pulse of the corresponding blue sideband at large $\delta$. Solid lines show the eigenenergies of the Hamiltonian equation (2).

Figure 2

(a) Full frequency scan of the axial blue sideband when the system is prepared in Fock states $|0_a,0_b⟩$ (blue) and $|0_a,10_b⟩$ (green). For the state $|0_a,10_b⟩$, the reduced height of the main peak and the residual population detected in the states with $n_b<10$ are due to imperfections of the state preparation that, according to the fit, populates the state $|0_a,10_b⟩$ with probability $p_{10}=0.80(2)$ and leaves significant populations $p_9=0.06(2)$ and $p_8=0.06(2)$ in the states $|0_a,9_b⟩$ and $|0_a,8_b⟩$. The preparation of state $|0_a,0_b⟩$ populates other Fock states with the probability $<0.04$. (b) Measured position of the axial motional sideband for the Fock states $|0_b⟩$ to $|10_b⟩$ prepared in the radial mode. The axial mode is initially prepared in the vacuum state $|0_a⟩$. Only part of the data around the largest peak is shown for each state. (c) Calculated shift of the axial sideband frequency as a function of detuning $\delta$ for different states ($|0_a,0_b⟩$ to $|0_a,10_b⟩$). (d) Calculated and measured shift of the axial sideband frequency as a function of the number of phonons added to the radial mode for the detuning $\delta /2\pi =14.3\mathrm{kHz}$.

Figure 3

(Top row) Frequency scans of the axial blue sideband for (a) the coherent state, (b) the thermal state, and (c) the squeezed vacuum state. The resolved peaks are clearly visible. Green lines show the fits of the experimental data for the corresponding states. Solid red bars indicate the expected probability $p_n$ to have $n$ phonons for a given quantum state. (Bottom row) Reconstructions of phonon number distribution for (d) the squeezed Fock state and (e),(f) the squeezed thermal states are obtained by the projective measurement as described in the text. The state parameters extracted from the fit of the experimental data (green) are in reasonable agreement with the independent calibration (red). The errors correspond to $1\sigma$ statistical uncertainties.