Dissipative phase transitions: Independent versus collective decay and spin squeezing

We study the $XY$ model with infinite-range interactions (Lipkin-Meshkov-Glick model) in the presence of dissipation from spontaneous decay. We show that independent and collective decay lead to qualitatively different phase transitions of the steady state, even though the phase boundary is the same. Independent decay leads to a second-order phase transition to a ferromagnet, while collective decay leads to a first-order transition to a time-dependent oscillatory phase. Then we show that the addition of a drive leads to infinite spin squeezing for collective decay in the thermodynamic limit. Our results can be experimentally seen in trapped-ion and cavity-QED experiments.

Figure 1

$\langle J_z\rangle /j$ with $j=N/2$ for (a) independent decay and (b) collective decay. Solid lines are mean-field predictions. Numerical results are shown for different numbers of atoms: $N=10$ (circles), $N=100$ (triangles), and $N=1000$ (squares). The phase transition is at $\left|V\right|=\gamma /2$ for both types of decay.

Figure 2

Examples of mean-field periodic orbits for $V=0.6{\gamma }_c$. (a), (b) Trajectories in time showing $X$ (dashed line), $Y$ (dotted line), and $Z$ (solid line). Panels (a) and (b) are for different initial conditions. (c), (d) Trajectories on the Bloch sphere and projected onto the $xy$ plane. Different colors correspond to different initial conditions. The black arrow shows the direction of the trajectories.

Figure 3

Wigner function of the steady state for $N=50$ atoms. (a) $V=0.4{\gamma }_c$ and (b) $V=0.6{\gamma }_c$. Black denotes 0, while white denotes the maximum value. Due to the $Z_2$ symmetry, the Wigner function in panel (b) has another peak on the opposite side of the Bloch sphere. Compare panel (b) with Fig. 2.

Figure 4

Wigner function of the steady state for $N=50$ atoms. (a) $V_x=2{\gamma }_c$ and $\Omega =0$. (b) $V_x=0$ and $\Omega =0.45{\gamma }_c$. Black denotes 0, while white denotes the maximum value.

Figure 5

Spin-squeezing parameter as a function of drive strength $\Omega$ for $V_x=0$ (blue [gray] squares) and $V_x=\gamma$ (red [gray] triangles). Solid lines are the analytical result of Eq. (29). Dashed lines and squares and triangles are the numerical results for $N=1000$, which deviate from the analytical predictions near the critical point due to finite-size effects.

Figure 6

Minimum spin-squeezing parameter as a function of $N$ for $V_x=0$. (Minimized with respect to $\Omega$.) (a) Linear scale. (b) Log-log scale.