Generalized Dicke Nonequilibrium Dynamics in Trapped Ions

We explore trapped ions as a setting to investigate nonequilibrium phases in a generalized Dicke model of dissipative spins coupled to phonon modes. We find a rich dynamical phase diagram including superradiantlike regimes, dynamical phase coexistence, and phonon-lasing behavior. A particular advantage of trapped ions is that these phases and transitions among them can be probed in situ through fluorescence. We demonstrate that the main physical insights are captured by a minimal model and consider an experimental realization with $\mathrm{Ca}{}^{+}$ ions trapped in a linear Paul trap with a dressing scheme to create effective two-level systems with a tunable dissipation rate.

Figure 1

(a) Schematic diagram of a one-dimensional ion crystal in different dynamical phases. Shown are two electronic states of each ion representing states $|\downarrow ⟩$ and $|\uparrow ⟩$ of a fictitious spin. Transitions between electronic (spin) states are driven by a laser with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }$. The state $|\uparrow ⟩$ relaxes to $|\downarrow ⟩$ at a spontaneous decay rate $\gamma$. Photons emitted in this process are detected with spatial and temporal resolution. Electronic states couple to motional degrees of freedom through a far-detuned standing-wave laser field. Upon increasing this coupling there is a dynamical phase transition where the ions become displaced in the trap (dashed circles). The displacement leads to a large effective detuning ${\mathrm{\Delta }}^{\prime }$ such that the electronic state $|\uparrow ⟩$ is rarely populated and the fluorescence signal is greatly reduced. (b) Regions in parameter space where both dynamical phases coexist are revealed by an intermittent fluorescence signal. The data shown correspond to a QJMC trajectory of five ions and display the times and positions of photon emissions (top) and the spin $z$-polarization $J_z$ (bottom).

Figure 2

Dynamical phase diagram for $\mathrm{\Delta }=0$. (a) The MF phase diagram found from the fixed-point analysis of Eqs. (2) as a function of the coupling $V$ and inverse lifetime $\gamma$ for $\omega /N=\mathrm{\Omega }$. Shown are the bright ($B$) and dark ($D$) regimes and a region of phase coexistence ($B+D$). The phases are distinguished by both spin-polarization and ion-displacement order parameters. Inset: QJMC simulations for a five-ion system at parameter values $V$, $\gamma$ as indicated. Shown are quantum trajectories in the steady state, with markers indicating the times at which photon emissions occur for each of the five ions along the ordinate axes. (b,c) Plots of the polarization $J_z$ and phonon displacement $X=(A+A^{*})/2$ at fixed points as a function of $V/\mathrm{\Omega }$ for $\gamma /\mathrm{\Omega }=0.15$ (b) and $\gamma /\mathrm{\Omega }=0.6$ (c). Solid lines indicate stable fixed points.

Figure 3

Dynamical phase diagram for $\mathrm{\Delta }=-0.01\mathrm{\Omega }$ with all other parameters as in Fig. 2. In addition to the bright ($B$), dark ($D$), and coexistence ($B+D$) regimes, points where the bright fixed point disappears at a Hopf bifurcation are shown for $\kappa =0$ (solid blue line) and $\kappa /N=10^{-4}\mathrm{\Omega }$ (dashed blue line). Beyond these birfurcations, limit-cycle solutions corresponding to phonon lasing ($PL$) exist. Beyond the critical coupling there exists a new region ($PL+D$) where the dark fixed point is stable but the bright fixed point bifurcates. The parameters indicated by “$+$” are used for the insets. Shown inset are plots of the MF dynamics with different initial conditions showing the Bloch sphere and oscillator phase plane for limit-cycle (red) and fixed-point (green) steady states. The dark fixed point is labeled “$\times$.”

Figure 4

(a) Oscillator displacements at the fixed points of the three-spin, three-oscillator model as a function of the coupling $V$, for $\gamma =0.1$ and $\kappa =\mathrm{\Delta }=0$. Regions where the fixed points are stable are shown with solid lines with all other fixed-point classifications plotted with dashed lines. Plotted are the center-of-mass mode with frequency ${\omega }_1$ (blue), and the modes (see the Supplemental Material [39]) with frequencies ${\omega }_2$ (green) and ${\omega }_3$ (red). (b,c,d) Sample trajectories with phonon modes $m=1$ and 2 included showing the occupation of the two included phonon modes (top), the polarization $J_z$ (middle), and photon emissions (bottom) of a three-ion simulation for $V=0.1$ (b), $V=1.5$ (c), and $V=3.5$ (d).

Figure 5

(a) Level structure of $\mathrm{Ca}{}^{+}$ ions driven by dressing laser of Rabi frequency ${\mathrm{\Omega }}_D$ with detuning ${\mathrm{\Delta }}_D$, with decay rates ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ shown. The driven transition has $\mathrm{\Omega },\delta \ll {\mathrm{\Omega }}_D\ll {\mathrm{\Delta }}_D,{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$. (b) The effective two-level scheme after projecting out the fast degrees of freedom, with unmodified Rabi frequency $\mathrm{\Omega }$, effective detuning and decay rate $\mathrm{\Delta }$, and $\gamma$. The scheme provides the states $|\downarrow ⟩$ and $|\uparrow ⟩$ which derive from $|4S_{1/2}⟩$ and $|3D_{5/3}⟩$ dressed with $|4P_{3/2}⟩$.