Single-Shot Measurements of Phonon Number States Using the Autler-Townes Effect

We present a single-shot method to measure motional states in the number basis. The technique can be applied to systems with at least three nondegenerate energy levels which can be coupled to a linear quantum harmonic oscillator. The method relies on probing an Autler-Townes splitting that arises when a phonon-number changing transition is strongly coupled. We demonstrate the method using a single trapped ion and show that it may be used in a nondemolition fashion to prepare phonon number states. We also show how the Autler-Townes splitting can be used to measure phonon number distributions.

Figure 1

Autler-Townes scheme for phonon number detection. (left) The $|D,n⟩\leftrightarrow |S,n⟩$ resonance is weakly probed while levels $|S,n⟩$ and $|D^{\prime },n+1⟩$ are strongly coupled via a laser on a blue sideband transition. (right) The system described in terms of dressed states. The splitting between the dressed states reveals the $|S,n⟩\leftrightarrow |D^{\prime },n+1⟩$ coupling strength ${\mathrm{\Omega }}_C$. The splitting behavior is similar when coupling to a red sideband transition.

Figure 2

Splitting of the Autler-Townes doublet. When the coupling field is resonant to a phonon-number-changing transition the splitting of the Autler-Townes doublet depends on the number of phonons in the system. (a) For the coupling field resonant to a BSB transition the splitting scales with the phonon number $n$ as $\sqrt{n+1}$. (b) For the coupling field resonant to a RSB transition the splitting scales with $\sqrt{n}$. Error bars represent quantum projection noise (68% confidence intervals). The lines are fits to the data with the amplitude and the splitting as a fitting parameters. (c) Splitting of the Autler-Townes doublet extracted from the fitting curves in (a) and (b). The scalings of the splitting are described by $\sqrt{n+1}$ (BSB transition) and $\sqrt{n}$ (RSB transition), respectively. Note that the phonon number presented only represents a subset and not all the consecutive Fock states.

Figure 3

(a) Measurement sequence: The ion is initialized in $|D,n⟩$. In the first step, a $\pi$ pulse is applied on the $|D⟩\leftrightarrow |S⟩$ carrier transition while a phonon-number-changing transition is strongly coupled (in the figure a BSB transition is shown). The probe frequency is set to an Autler-Townes peak resonance, such that the $|D⟩\to |S⟩$ transfer only occurs if the prepared Fock state $|n⟩$ equals the probed Fock state $|m⟩$. In the second step, any population in $|D^{\prime },n+1⟩$ is transferred to $|S^{\prime },n⟩$ before both $S$ states are coupled to the fluorescing state $|f⟩$. Finally, detection of fluorescence indicates $n=m$. (b) Experimental demonstration of the sequence in (a). The ion was prepared in different Fock states, and when the probed Fock state matched the prepared one, fluorescence was detected. (c) Experimental results when the transfer step is instead applied to the $|S,n⟩\leftrightarrow |D,n⟩$ carrier transition, moving all the population into the $D$ manifold, and enabling a nondestructive measurement of the ion motion in the Fock basis.

Figure 4

The Autler-Townes splitting shows multiple peaks for a thermal distribution of the ion motion. The peak positions (indicated by dashed lines) are defined by the phonon number and the amplitudes by the population probability of the respective phonon number. By scanning the Autler-Townes spectrum the thermal distribution of the ion motion can be obtained.