Probing excited-state quantum phase transitions with trapped cold ions

Abstract

We propose concrete protocols to realize quantum criticality due to excited-state quantum phase transitions (ESQPTs) experimentally in presumably the simplest and most resilient system involving a single trapped ion oscillating in a radio-frequency Paul trap. We identify a specific class of excited states of the Extended Rabi Model (ERM) Hamiltonian, which occur between two critical ESQPT energies of the model in its (anti)Jaynes-Cummings superradiant phase. Properties of these states motivate the definition of several ESQPT witness observables. We study their critical scaling behaviors as well as various distinct state evolutions by driving the system across the quantum criticalities by changing the qubit-phonon coupling strength linearly in time at different finite rates. A mapping of the theoretical control parameters of the ERM to the experimental parameters of a trapped ion setup is provided, and simulations are performed for values referencing existing state-of-the-art setups, addressing both unitary state evolutions as well as relevant open-system corrections.

Figures (11)

• Figure 1: (a) Different qubit-phonon interaction phases of the semiclassical ERM Hamiltonian \ref{['7']} in the $(\lambda, \delta)$ plane. The individual colored regions represent areas with different phonon-vacuum properties. The red horizontal line at $\delta=0.5$ represents our choice of typical parameter-space path. Red point at $(\lambda,\,\delta)=(4,\,0.5)$ corresponds to parameters, for which we plot the Gaussian-smoothed densities of states (b) for different system sizes $\Delta$. Horizontal dashed lines in (b) indicate local maximum (orange), saddle point (black) and global minimum (purple) of the classical Hamiltonian.
• Figure 2: Level dynamics of the ERM Hamiltonian \ref{['6']}, with system size $\Delta=15$ as a function of interaction strength $\lambda$ for (a) $\delta=0$, (b) $\delta=0.5$ and (c) $\delta=1$. Red levels indicate eigenstates with negative parity and gray levels indicate eigenstates with positive parity with respect to the total number of excitations. The orange line represents the energy of the non-interacting ground state. The purple dot at $\lambda_c$ indicates the ground-state QPT, and the subsequent purple line $e_{\rm min}$ represents the ground-state energy. In (b), the black dot at $\lambda_0$ and the subsequent black line $e_{\rm sad}$ indicate the second ESQPT energy, resulting from the transition to the second superradiant phase S2. In panel (c) $\lambda_0 = \lambda_c$ and $e_{\rm sad} = e_\mathrm{min}$.
• Figure 3: Level dynamics as in Fig. 2 (b), but color-coded by the qubit-motion entanglement entropy $S_E$GardinerZollerBooks. $S_E$ is normalized by its maximal value of $\ln(2)$.
• Figure 4: Expectation values of the phonon-number operator $\langle\hat{n}\rangle$ in eigenstates of the Hamiltonian \ref{['6']} with $\Delta=15$, $\lambda=4$ and $\delta=0$ (panel a) and $\delta=0.5$ (b) with six selected states' Wigner functions in panels (c) - (h). The vertical axes in (a) and (b) display the eigenvalues, and the horizontal axes show the mean number of phonons in the corresponding parity-distinguished eigenstate (gray for positive parity, red for negative parity). The orange dashed horizontal line marks the non-interacting ground-state energy [semiclassical saddle point in (a) and local maximum in (b)], the purple horizontal line marks the ground-state energy (semiclassical minimum) and the purple vertical line marks the corresponding phonon number. The black horizontal line in (b) marks the second ESQPT energy (semiclassical saddle point) and its vertical counterpart likewise displays the corresponding phonon number. States shown in (c) - (h) are indicated in (b), and their Wigner functions are displayed along with contours of the classical Hamiltonian $h(x',p',-1/2)$\ref{['7']}.
• Figure 5: Strength functions, Eq. \ref{['StrF']}, in instantaneous quench protocols ($\tau_f=0$) corresponding to a sudden increase of the interaction parameter to $\lambda_{f}=4$ at (a) $\delta=0$ and (b) $\delta=0.5$, for different system sizes $\Delta$. Horizontal dashed lines indicate the semiclassical critical points: local maximum [$e_{\rm vac}$ in (b)], saddle point [$e_{\rm sad}$ in (b) and $e_{\rm vac}$ in (a)], and global minimum ($e_{\rm min}$).