Engineering continuous-variable entanglement in mechanical oscillators with optimal control

Abstract

We demonstrate an optimal quantum control strategy for the deterministic preparation of entangled harmonic oscillator states in trapped ions. The protocol employs dynamical phase modulation of laser-driven Jaynes-Cummings and anti-Jaynes-Cummings interactions. We prepare Two-Mode Squeezed Vacuum (TMSV) states in the mechanical motions of a trapped ion and characterize the states with phase-space tomography. First, we verify continuous-variable entanglement by measuring an Einstein-Podolsky-Rosen entanglement parameter of 0.0132(7), which is below the threshold of 0.25 for Reid's EPR criterion. Second, we perform a continuous-variable Bell test and find a violation of the Clauser-Horne-Shimony-Holt inequality, measuring 2.26(3), which is above the entanglement threshold of 2. We also demonstrate the flexibility of our method by preparing a non-Gaussian entangled oscillator state--a superposition of TMSV states.

Figures (5)

• Figure 1: Preparation of continuous-variable entangled oscillator states using optimal quantum control. a. Energy diagram illustrating the two coherent interactions used by the control scheme, which include a Jaynes-Cummings (JC) interaction coupled to an oscillator with frequency $\omega_1$ (red arrow) and controllable phase $\phi_r$, and an anti-JC interaction coupled to an oscillator with frequency $\omega_2$ (blue arrow) and controllable phase $\phi_b$. Both interactions have equal coupling strength, $\Omega$. b. Simultaneously driving the JC and anti-JC interactions couples both oscillators to a shared ancillary spin. c. CV entanglement is obtained by initializing the spin and oscillators to their ground state, and applying the JC-type interactions with numerically optimized phase-modulations, $\phi_r(t)$ (red function with spotted hatch) and $\phi_b(t)$ (blue function with horizontal hatch), for duration $\mathrm{T}$. Shown here is a two-mode squeezed vacuum state which exhibits non-classical position and momentum correlations between the two oscillators.
• Figure 2: Experimental characteristic function tomography of a two-mode squeezed vacuum state (TMSS) with target squeezing parameter, $r = 1$. All panels show the real component of the joint characteristic function, $\mathrm{Re}\left[\chi(\beta_1, \beta_2)\right]$, along four pairs of quadratures: a.$\{\mathrm{Re}[\beta_1], \mathrm{Re}[\beta_2]\}$, b.$\{\mathrm{Im}[\beta_1], \mathrm{Re}[\beta_2]\}$, c.$\{\mathrm{Re}[\beta_1], \mathrm{Im}[\beta_2]\}$, d.$\{\mathrm{Im}[\beta_1], \mathrm{Im}[\beta_2]\}$. Panels a. and d. exhibit correlation consistent with two-mode squeezing. Dashed lines plot the Gaussian functions that are fitted to the data, from which the variances $V_{\vec{\beta}_{\mathrm{Re}}^-}$ and $V_{\vec{\beta}_{\mathrm{Im}}^+}$ are extracted and used to quantify entanglement with Reid's EPR criterion. Panels b. and c. show negligible correlation, consistent with uncorrelated orthogonal quadratures. All panels are measured for $\beta_1>0$ and $\beta_2>0$, and we use the Hermitian property of the characteristic function, $\chi(\beta_1, \beta_2)^* = \chi(-\beta_1, -\beta_2)$, to determine $\chi(\beta_1, \beta_2)$ for $\beta_1 < 0$ and $\beta_2 < 0$Valahu2023. Insets show the theoretical characteristic function obtained from numerical simulations, and show general agreement with experiments.
• Figure 3: Continuous-variable Bell test with a TMSS. a. The quantum circuit used to perform the Bell test. Four predefined measurement settings $\{\alpha_k, \gamma_l\}$ with $k, l \in \{0, 1\}$ are randomly selected at each iteration of the experiment. These settings determine which displacements are performed by two subsequent SDF pulses, and the measurement is completed by a $\hat{\sigma}_z$ spin projection. This measurement gives a binary outcome, $c_m(\alpha_k, \gamma_l)$, which averaged over many repetitions corresponds to the correlation measurement, $C(\alpha_k, \gamma_l)$. b. The correlation measurements correspond to values of the joint characteristic function, which are plotted here for a TMSS with $r=1$ in the $\{\mathrm{Re}[\beta_1], \mathrm{Re}[\beta_2]\}$ quadrature. The measurement parameters $\alpha_k$ and $\gamma_l$ are numerically optimized to maximize the Bell signal, $\mathrm{B}$. The optimal measurement parameters for a state with $r=1$ are indicated by black stars and correspond to $\{ \alpha_0, \alpha_1 \} = \{ \gamma_0, \gamma_1 \} = \{-0.1248, 0.4041\}$. c. The Bell signal, $\mathrm{B}$, of Eq. (\ref{['Eq:BellTest']}) is plotted with respect to the total number of measurements, $M$. Values of $\mathrm{B}$ that surpass the classical limit of $2$ (dashed line) violate the CHSH inequality and indicate non-classical correlations. The dotted line at $\mathrm{B}=2.32$ bounds the CHSH inequality for a TMSS in the limit of infinite squeezing. The largest experimentally measured Bell signal with $r=1$ is $\mathrm{B}=2.26(3)$ (red square), while the most statistically significant violation of the CHSH inequality is obtained for $B=2.186(9)$ (blue dot), over 21 $\sigma$ from the classical limit. Error bars correspond to the standard deviation of the measured Bell signal determined by quantum projection noise.
• Figure 4: Joint $\chi$-function tomography of a superposition of TMSS. (Left) The experimentally prepared superposition state. (Right) numerically simulated characteristic function of the ideal target state, $\ket{\Psi_\times}$ with squeezing parameter $r=1$. The experimentally reconstructed characteristic function, $\mathrm{Re}\left[\chi(\mathrm{Re}[\beta_1], \mathrm{Re}[\beta_2])\right]$, shows features of the superposition state, with correlation along both axes, $(\mathrm{Re}[\beta_1] \pm \mathrm{Re}[\beta_2])$. Dashed lines are the fits using Eq. (\ref{['Eq:Superposition State2']}), from which we find squeezing parameters fitted to be $\{0.60(3), 0.69(4)\}$ along two orthogonal directions.
• Figure 5: Experimentally measured $\chi$-function, from which we extract variances of TMSS with increasing amounts of target squeezing. a. Experimentally obtained $\chi$-function values averaged over both correlated quadratures; hollow markers plot reconstructions along the squeezed axis, $\mathrm{Re}\left[\bar{\chi}_\mathrm{s}(\alpha)\right]$ of Eq. \ref{['eq:chi_mean_s']}; filled makers plot reconstructions along the anti-squeezed axis, $\mathrm{Re}\left[\bar{\chi}_\mathrm{as}(\alpha)\right]$ of Eq. \ref{['eq:chi_mean_as']}. The measurements are repeated for TMSS with increasing target squeezing parameters $r=\{0.25, 0.75, 1.0, 1.25\}$. Solid black line plots the theoretical $\chi$-function of a vacuum state. Error bars are calculated from quantum projection noise. The reconstructed $\chi$-functions are fit to Gaussian functions, which are plotted here with (solid) dashed colored lines for fits to the reconstructions along the (anti-)squeezed axis. b. Variances extracted from the fits are used to quantify squeezing as a function of the target squeezing parameter, $r$. Squeezing is plotted in decibels referenced to an ideal vacuum state, with $\mathrm{V}~\mathrm{(dB)} = 10~\mathrm{log}_{10}(\mathrm{V}/\mathrm{V}_\mathrm{vac})$, where $\mathrm{V}_\mathrm{vac} = 1$ is the vacuum variance. Error bars represent three standard deviations of fitting uncertainty from the Gaussian distributions in a. Dashed lines plot the theoretical squeezing of ideal TMSS. The vacuum state variance observed experimentally matches a thermal occupation of $\bar{n}\approx 0.06$ in both modes.