One-dimensional spin-1/2 fermionic gases with two-body losses: Weak dissipation and spin conservation

We present a theoretical analysis of the dynamics of a one-dimensional spin-1/2 fermionic gas subject to weak two-body losses. Our approach highlights the crucial role played by spin conservation in the determination of the full time evolution. We focus in particular on the dynamics of a gas that is initially prepared in a Dicke state with a fully symmetric spin wave function, in a band insulator, or in a Mott insulator. In the latter case, we investigate the emergence of a steady symmetry-resolved purification of the gas. Our results could help with the modelization and understanding of recent experiments with alkaline-earth(-like) gases like ytterbium and fermionic molecules.

Figure 1

Dissipative dynamics of the normalized density of a band insulator for $L=4,6,8$, and 10. The various colors refer to different dissipation strengths, from $\hslash \gamma /J={10}^{-2}$ to $\hslash \gamma /J=10$ (see legend). The thin blue dotted curve is the prediction for the thermodynamic limit in Eq. (18), whereas the thick blue dashed curve is Eq. (20). The latter faithfully describes the weakly dissipative limit even at small sizes. The plot highlights the collapse of the curves for $\hslash \gamma /J={10}^{-2}, {10}^{-1}$, and 1. On the other hand, the appearance of a different behavior in the strongly dissipative Zeno limit is evident.

Figure 2

Time derivative of the population $\dot{N}\left(t\right)$ for $L=8$ and $\hslash \gamma /J={10}^{-1}$. Thick black solid line: calculation of $\dot{N}\left(t\right)$ using the right-hand side of Eq. (6) by running a numerical simulation with $N_{\mathrm{traj}}=2000$ quantum trajectories that computes explicitly all the necessary quantities. Thin red line: numerical derivative of $N\left(t\right)$ computed with $N_{\mathrm{traj}}=2000$ quantum trajectories and using the Euler method.

Figure 3

Lossy dynamics from an initial band insulator with periodic boundary conditions using quantum trajectories ($N_{\mathrm{traj}}={10}^4$ for $L=4,6$ and $N_{\mathrm{traj}}={10}^3$ for $L=8$) and $\hslash \gamma /J=0.1$. Top left: dynamics of the population $N\left(t\right)$. Top right: at finite size, the stationary state is not Dicke, as quantified by the operator ${\widehat{O}}_{\mathrm{ND}}$. Bottom: time derivative of the population computed using the right-hand side of Eq. (6) and with the numerical derivative of $N\left(t\right)$.

Figure 4

Dissipative dynamics of a Néel state for $L=4,6,8$, and 10 for $\hslash \gamma /J=0.1$. The time-dependent population is plotted as a red dashed line. The upper bound to the total population given in Eq. (23). The blue solid curve (theory) is a fit to the dynamics using Eq. (21) and taking $n_{\infty }$ as the only fitting parameter.

Figure 5

Dissipative dynamics of a Néel state for $L=4,6,8$, and 10 for $\hslash \gamma /J=0.1$ and long-time decay of $N\left(t\right)$ to the stationary value $N_{\infty }$. In all cases we observe an exponential decay. The blue solid curve is not a fit, but the decay obtained from the theoretical prediction $N_t-N_{\infty }=e^{-t/\tau }$ with $\tau$ given by Eq. (24), which provides an excellent description of the decay time at all lattice lengths.

Figure 6

Decay time of the population $N\left(t\right)$ approaching the asymptotic number of particles $N_{\infty }$ as a function of $\langle S^2\rangle /{\hslash }^2$ for $L=6$ and $\hslash \gamma /J=0.1$. Red solid line: theoretical curve for $\tau$ predicted in the thermodynamic limit (24). Blue crosses: numerical fits of the decay time $\tau$ performed for 14 random Mott insulators with ${10}^4$ quantum trajectories.

Figure 7

Symmetry-resolved purity for the particle sectors $n=2,4,6$ (solid lines) and purity of the full density matrix (dashed line). Data are obtained for a typical set of parameters, $J/\gamma =10\hslash , L=8$, and $N_{\mathrm{traj}}={10}^3$.

Figure 8

Purity ratio $\mathcal{P}\left({\rho }_n\right)/\mathcal{P}{\left({\rho }_n\right)}_{\min }$ for $n=2,4,6$. Parameters are set as in Fig. 7.

Figure 9

Dissipative dynamics of the normalized density of a band insulator for $L=8$ and ${10}^4$ quantum trajectories. Different colors refer to different dissipation strengths, from $\hslash \gamma /J=0.1$ to $\hslash \gamma /J=10$. Solid lines: simulations with $U=\hslash \gamma$. Dashed lines: corresponding dynamics for $U=0$. The plot highlights the collapse of the curves for $U\ne 0$ and $U=0$ in the weakly dissipative limit.

Figure 10

Dissipative dynamics of the normalized density for a band insulator for $J/\gamma ={10}^2$, 10, 1, and 0.1. Different colors refer to different sizes, from $L=4$ to $L=10$. The blue dotted curve is the prediction for the thermodynamic limit in Eq. (18) and highlights the importance of finite-size effects in our simulations. The appearance of a different behavior in the strongly dissipative limit is evident.

Figure 11

$\mathrm{Var}{N}_t$ (left panel) and ${\langle \widehat{\mathrm{\Pi }}\rangle }_t$ (right panel) for $L=8$ and $J/\hslash \gamma =10$. Black solid line: calculation of the two quantities using 2000 quantum trajectories. Red dashed line: approximation using formula (19a) and using the numerically computed value for $N\left(t\right)$.

Figure 12

Symmetry-resolved purity $\mathcal{P}\left({\widehat{\rho }}_n\right)$ for $n=2,4,6$ for different values of $N_{\mathrm{traj}}$. The blue dashed lines represent the theoretical prediction for the steady-state symmetry-resolved purity $\mathcal{P}\left({\rho }_n\right)=2/\left(\genfrac{}{}{0pt}{}{L}{n}\right)$. Parameters are set as in Fig. 7.