Time Crystal in the Nonlinear Phonon Mode of the Trapped Ions

TL;DR

This work addresses realizing a continuous-time crystal (CTC) in a quantum many-body system by engineering dissipative nonlinear dynamics in the vibrational mode of a two-ion chain. It employs adiabatic elimination to derive an effective Lindblad master equation with controllable linear gain $g$ and nonlinear damping $\kappa$ on the phonon mode, and it couples this mode to an external drive $\varepsilon$ and detuning $\Delta$ to realize a Hopf bifurcation and a limit-cycle (time-crystal) phase without periodic driving. The authors validate the scheme with realistic $^{40}$Ca$^{+}$ parameters, compare effective dynamics to full spin-phonon dynamics, and demonstrate robust time-crystal behavior against initial thermal states, phonon heating, spin dephasing, and laser-control errors, including analysis via the Van der Pol limit and Husimi $Q$-function trajectories. The results provide a practical, experimentally feasible route to observe a dissipative continuous-time crystal in a nonlinear phonon mode, advancing the study of temporal order in non-equilibrium quantum systems and offering a platform for exploring dissipative phase transitions in trapped ions.

Abstract

Time crystals constitute a novel phase of matter defined by the spontaneous breaking of timetranslation symmetry. Here we present a scheme to realize a continuous-time crystal of the vibrational phonon in the normal mode of two coupled ultra-cold ions. By utilizing two addressable standing-wave lasers and adiabatic elimination method, we generate a controllable nonlinear phonon mode with the well-designed efficient linear gain and nonlinear damping. By controlling these parameters to satisfy the phase transition conditions of Hopf bifurcation and limit cycle phase, it behaves as a stable dissipative dynamics over timescales significantly longer than the oscillation period, indicating the emergence of discrete time-translation symmetry breaking in the phonon mode, i.e., a phonon time crystal. We further numerically simulate this phonon time crystal by using accessible experimental parameters and also demonstrate a robustness to the initial thermal state and thermalization of phonon mode, spin dephasing, and the control errors of Rabi frequencies. These results provide a practical scheme for observing a time crystal in a nonlinear phonon mode and will advance the research of time crystals.

Figures (7)

• Figure 1: The scheme to realize the time crystal in the trapped $^{40}\mathrm{Ca}^{+}$ ion system, where an external alternating electric field along the radial direction ($x$-direction) of ions is applied to drive the normal vibrational mode of two ions, two polarized 854 nm lasers couple the metastable state $|e\rangle$ to the excited state $|p\rangle$ with the decay rate $\gamma$ to produce an effective dissipation process from the metastable state to ground state, and two standing wave laser beams of 729 nm with the detuning $\delta_{1,2}$ couple the assistant two-level system, consisting of the states $|g\rangle$ and $|e\rangle$, to the normal phonon mode of two ions to create the linear gain and nonlinear damping on the vibrational mode, respectively.
• Figure 2: The mean deviation $\delta_F$ between the numerical result and the effective dynamics, where the initial state of system is in the metastable state $|e\rangle$ and the insets denote the evolution of fidelity when $\Omega_e/2\pi=[0.5, 2, 4]$ MHz, respectively, and the effective dissipative rate $\Gamma$. Here, the decay rate of excited state $|p\rangle$ is chosen as $\gamma/2\pi=22.4$ MHz for the $^{40}\mathrm{Ca}^{+}$ ion, and the effective decay rate is given by $\Gamma = \Omega_e^2/\gamma$.
• Figure 3: (a)The evolution of average phonon number in phonon mode for the given linear gain rate $g=4$ kHz, where the blue solid curve denotes analytical result, green dash curve is the result of effective dynamics under the master equation of Eq. (\ref{['Eq12']}), the solid curves are the results under the master equation of Eq. (\ref{['Eq9']}) with Hamiltonian given by Eq. (\ref{['Eq5']}), the dot-dash curves are the results under the master equation of Eq. (\ref{['Eq9']}) with Hamiltonian given by Eq. (\ref{['Eq10']}) and the dash curves are obtained with the compensation of third-order term $H_{2,1}^{(2)}$, respectively. The initial state of phonon mode and spin state are corresponding to the Fock state $|0\rangle$ and ground state $|g\rangle$, the parameters are selected as $\Gamma=1.09$ MHz, $\lambda_1=33$ kHz and $\Omega_1=482$ kHz for the cyan curves, and they are selected as $\Gamma=272$ kHz, $\lambda_1=16.5$ kHz and $\Omega_1=241$ kHz for the brown curves. The red curve in the inset denote the results by consider the second term $H_{2,1}^{(2)}$ of $H_{2,1}$. (b) The evolution of average phonon number in phonon mode for the given nonlinear damping rate $\kappa=0.4$ kHz, where the black solid curve denotes the result of effective dynamics under the master equation of Eq. (\ref{['Eq12']}), and other curves are obtained by the master equation of Eq. (\ref{['Eq9']}), where the Hamiltonian is given by Eq. (\ref{['Eq7']}) with $\lambda_2=\eta^2\Omega_2/2$ for the red dash curve, the Hamiltonian given by Eq. (\ref{['Eq11']}) without Stark shift compensation and coupling strength amendment for the purple curve, with only Stark shift compensation for the cyan the dot-dash curve, and with Stark shift compensation and coupling strength amendment for the green dots curve. The initial state of phonon mode and spin state are corresponding to the Fock state $|4\rangle$ and ground state $|g\rangle$, the parameters are selected as $\Gamma_2=272.4$ kHz, $\lambda_2=5.22$ kHz, $\Omega_2=2.22$ MHz and $\bar{\delta}=787$ kHz without coupling strength amendment, and $\lambda_2=5.97$ kHz, $\Omega_2=2.54$ MHz and $\bar{\delta}=1.03$ MHz with the coupling strength amendment. Here the Lamb-Dicke parameter for two ions is $\eta=0.069$ and $\omega_r/2\pi=1$ MHz.
• Figure 4: The evolution of average phonon number $\langle a^{\dagger}a\rangle$ (a) and the state purity $\text{Tr}(\rho^2)$ (b) in radial normal phonon mode of two ions, where the black solid and red solid curves are corresponding to the results under the master equation of Eq. (\ref{['Eq14']}) and Eq. (\ref{['Eq17']}), and cyan dots, brown dot-dash, green dash curves denote the results under the master equation of Eq. (\ref{['Eq16']}) with only the compensation of Stark shift, with the compensation of third-order term $H_{2,1}^{(2)}$ and Stark shift, and with the compensation of third-order term $H_{2,1}^{(2)}$, Stark shift and coupling strength amendment, respectively. The initial state of phonon mode and spin state for the two ions are corresponding to the vacuum state $|0\rangle$ and ground state $|g\rangle_1|g\rangle_2$. The parameters are selected as $\Gamma_1=1.09$ MHz, $\lambda_1=23.3$ kHz, $\Omega_1=341$ kHz, $\Gamma_2=272$ kHz, $\lambda_2=5.22$ kHz, $\Omega_2=2.22$ MHz and $\bar{\delta}=787$ kHz without coupling strength amendment, and $\lambda_2=5.97$ kHz, $\Omega_2=2.54$ MHz and $\bar{\delta}=1.03$ MHz with the coupling strength amendment. Here the Lamb-Dicke parameter for two ions is $\eta=0.069$, the vibrational frequency $\omega_r/2\pi=1$ MHz, the linear gain $g=2$ kHz, the nonlinear damping $\kappa=0.4$ kHz, and the frequency of external electric field is selected as $\omega_e=\omega_r$ and the driving strength $\varepsilon=0$, respectively.
• Figure 5: (a) The evolution of state purity $\text{Tr}(\rho^2)$ under different initial states in the spin states of two ions, where the initial state of phonon mode is in the vacuum state $|0\rangle$, the blue solid curve denotes the effective dynamics under the master equation of Eq. (\ref{['Eq14']}), and other curves are obtained by the master equation of Eq. (\ref{['Eq17']}). (b1) and (b2) The evolution of average phonon number $\langle a^{\dagger}a\rangle$ and state purity $\text{Tr}(\rho^2)$ for the different nonlinear damping rates $\kappa=0.6, 0.4, 0.2$ kHz corresponding to the red, purple and cyan curves, where the dot and solid curves are obtained by the master equation of Eq. (\ref{['Eq14']}) and Eq. (\ref{['Eq17']}), respectively. The insets of (b1-b2) denote the quantum Husimi distribution of phonon states at different evolution time (shown by the square and diamond dots in the cyan curve), where the red dashed line represents the limit cycle. (c) The evolution of the state purity $\text{Tr}(\rho^2)$ for the different thermal state of phonon mode, where the curves from the upper to bottom are corresponding to the initial average phonon number $\langle a^{\dagger}a\rangle_{\rm ini}=0, 1, 2$, the dot and solid curves are obtained by the master equation of Eq. (\ref{['Eq14']}) and Eq. (\ref{['Eq17']}), respectively, and the inset denotes the corresponding evolution of average phonon number $\langle a^{\dagger}a\rangle$. Here, we obtain the nonlinear damping $\kappa=0.6, 0.4, 0.2$ kHz by adjusting the $\Omega_2$ and $\delta_2$ for the given $\Gamma_2=272$ kHz, the parameters are selected as $\Gamma_1=1.09$ MHz, $\Omega_1=341$ kHz and $\lambda_1=23.3$ kHz to obtain the linear gain $g=2$ kHz, the frequency of external electric field is selected as $\Delta=5g$ and the driving strength $\varepsilon=\sqrt{g(g^2+4\Delta^2)}/8\sqrt{\kappa}$, respectively, and the duration of evolution $\tau=20/g$.