Reconstruction of the Jaynes-Cummings field state of ionic motion in a harmonic trap

A quantum state is fully characterized by its density matrix or equivalently by its quasiprobabilities in phase space. A scheme to identify the quasiprobabilities of a quantum state is an important tool in the recent development of quantum technologies. One of the most fundamental interaction models in quantum optics is the so-called Jaynes-Cummings model (JCM), which has been massively studied theoretically and experimentally. However, the expected essential dynamics of the field states under the resonant JCM has not been observed experimentally due to the lack of a proper reconstruction scheme. In this paper, we further develop a highly efficient vacuum measurement scheme and study the JCM dynamics in a trapped ion system with the capability of the vacuum measurement to reconstruct its quasiprobability $Q$ function, which is a preferred choice to study the core of the dynamics of a quantum state in phase space. During the JCM dynamics, the Gaussian peak of the initial coherent state bifurcates and rotates around the origin of phase space. They merge at the so-called revival time at the other side of phase space. The measured $Q$ function agrees with the theoretical prediction. Moreover, we reconstruct the Wigner function by deconvoluting the $Q$ function and observe the quantum interference in the Wigner function at half of the revival time, where the vibrational state becomes nearly disentangled from the internal energy states and forms a superposition of two composite states. The scheme can be applied to other physical setups including cavity or circuit-QED and optomechanical systems.

Figure 1

Raman laser schemes and the vacuum measurement. (a) The Hilbert space of the system is composed of the direct product of qubit states $\left\{|\downarrow \rangle ,|\uparrow \rangle \right\}$ and phonon number states $\left\{|n=0\rangle ,|1\rangle ,|2\rangle ,...\right\}$. Raman laser beams, which have ${\sigma }_{-}$ polarization and are detuned by ${\mathrm{\Delta }}_{\mathrm{p}}\approx$ 12.9 THz from the $P_{1/2}$ manifold, perform anti-JCM, displacement operation, and the vacuum measurement by adjusting their beating frequencies. (b) The vacuum component is measured by transferring the population of $|\downarrow ,n\rangle$ to that of $|\uparrow ,n-1\rangle$ for any value of $n$ at the same duration of pulse. The atom remaining in no fluorescence state $|\downarrow \rangle$ indicates the phononic state being in $|0\rangle$. (c) The uniform transfer for any phononic state $|n\rangle$ to $|n-1\rangle$ is accomplished by the scheme of shortcuts to the adiabaticity, where ${\mathrm{\Omega }}_{\mathrm{ut}}=\left(2\pi \right)22.7$ kHz, $\beta =0.075, {\mathrm{\Delta }}_{\mathrm{ut}}=1.9{\mathrm{\Omega }}_{\mathrm{ut}}$, and the total duration $T_{\mathrm{ut}}=198.2\mu \mathrm{s}$. (d) $Q$ function of phononic Fock state $n=0,1,2$ depending on the amount of displacement $\left|\alpha \right|$. The points with error bars are the experimental results while the dashed lines are by the theory. The error bars are obtained by the standard deviation of the quantum projection noise with 100 repetitions.

Figure 2

The time evolution of the $Q$ function for an initial coherent state under anti-JCM interaction. (a) Collapse and revival of the Rabi oscillation signal and (b) experimentally measured and (c) numerically calculated $Q$ functions of the phonon field with the initial coherent state $|\beta =1.62\left(0.05\right)\rangle$ depending on the duration of anti-JCM interaction. (a) $P\left(|\uparrow \rangle \right)$ is the probability of detecting the atom in $|\uparrow \rangle$. The points are obtained after 100 repetitions. The solid line is from fitting the data with ${\sum }_{n=0}\frac{1}{2}\left[1-{\text{e}}^{-\gamma t}cos\left(\sqrt{n+1}\eta \mathrm{\Omega }t\right)\right]$, where $\gamma$ is the empirical decay constant. At (b) and (c), the time for each snapshot of the $Q$ functions are labeled as (i)–(v) in the unit of the revival time $t_{\mathrm{rev}}$, where $t_{\mathrm{rev}}=\frac{4\pi \left|\alpha \right|}{\eta \mathrm{\Omega }}=108.8\mu \mathrm{s}$. In (b), each $Q$ function is obtained from 100 repetitions of the vacuum measurements after 384 different displacements, where the amplitude and the phase of displacement $\left|\alpha \right|{\text{e}}^{i\phi }$ are scanned from zero to 3.0 with the step size of 0.2 and from zero to $2\pi$ with the 24 steps, respectively.

Figure 3

(a) Test of the influence of the optical pumping sequence to the phonon number distribution. The coherent states prepared in $|\downarrow \rangle$ state (red circles) are compared with those prepared in $|\uparrow \rangle$ state by an optical pumping pulse (blue circles). The amount of the displacement $\left|\alpha \right|$ is measured by coherent-state fitting to the phonon number distribution. (b) The relation between Rabi contrast, purity, and atomic coherence. The red curve is the Rabi oscillation signal, the blue dashed line is the purity of the phononic state, and the green dot-dashed line is the atomic coherence, the absolute value of the off-diagonal elements of the atomic density matrix after tracing over the phononic field state. All curves are obtained by the numerical calculations.

Figure 4

(a) Generalized echo-sequence time reversal of anti-JCM evolution for the interaction time $t=t_{\mathrm{rev}}/2$. The $\varphi =\pi$ phase of the anti-JCM Hamiltonian produces the negative sign; $H_{\mathrm{aJC}}\left(\pi \right)=-H_{\mathrm{aJC}}\left(0\right)$, which performs the time-reversal operation. (b) The measured $Q$ function of the phononic state after time-reversal operation of (a). (c) JCM sequence for the time reversal of anti-JCM evolution for the interaction time $t=t_{\mathrm{rev}}/2$. The JCM coupling is realized by a ${\sigma }_{\mathrm{x}}$ pulse and the anti-JCM coupling. The phase $\frac{\pi }{2}$ of the second anti-JCM pulse is chosen to maximize the fidelity of the reverse operation. (d) Measured $Q$ function of the phononic state after the reversal operation. The total number of measurements for the $Q$-function reconstruction is the same as that in Fig. 2.

Figure 5

The Wigner function reconstruction from the $Q$ function at various times of the anti-JCM evolution, $t=\frac{1}{4},\frac{1}{2},\frac{3}{4}{t}_{\mathrm{rev}}$. The negativities of the Wigner functions indicate the emergence of the nonclassical state during the dynamic evolution. (a) The Wigner functions are reconstructed from the density matrix obtained by deconvoluting the experimentally measured values of the $Q$ functions shown in Fig. 2. The density matrices are reconstructed by deconvoluting the $Q$ functions with the convex optimization. We note that we use the data $\left|\alpha \right|\le 1$ for the optimum fidelity and it is necessary to use proper initial guess of the density matrix for the convergence of the deconvolution. (b) The Wigner functions are directly obtained by the numerical calculation of the anti-JCM dynamics.