Nonequilibrium dynamics of arbitrary-range Ising models with decoherence: An exact analytic solution

The interplay between interactions and decoherence in many-body systems is of fundamental importance in quantum physics. In a step toward understanding this interplay, we obtain an exact analytic solution for the nonequilibrium dynamics of Ising models with arbitrary couplings (and therefore in arbitrary dimension) and subject to local Markovian decoherence. Our solution shows that decoherence significantly degrades the nonclassical correlations developed during coherent Ising spin dynamics, which relax much faster than predicted by treating decoherence and interactions separately. We also show that the competition of decoherence and interactions induces a transition from oscillatory to overdamped dynamics that is absent at the single-particle or mean-field level. These calculations are applicable to ongoing quantum information and emulation efforts using a variety of atomic, molecular, optical, and solid-state systems. In particular, we apply our results to the NIST Penning trapped-ion experiment and show that the current experiment is capable of producing entanglement amongst hundreds of quantum spins.

Figure 1

Schematic illustration of the various decoherence processes: $T_1$ spin relaxation with rates ${\Gamma }_{\mathrm{ud}},{\Gamma }_{\mathrm{du}}$ (red and blue spin, respectively), and $T_2$ dephasing with rate ${\Gamma }_{\text{el}}$ (green spin). In atomic systems, one way this decoherence can arise is due to spontaneous emission from an excited level (top panels) [32].

Figure 2

Series of Raman flips of the spin on site $j$ can be formally accounted for as a magnetic field of strength $2J_{jk}/\mathcal{N}$ acting for a time ${\tau }_j^{\mathrm{up}}-{\tau }_j^{\mathrm{down}}$. In the notation defined below, this series of jumps is represented by the operator ${\widehat{Q}}_j\propto {\widehat{\sigma }}_j^{+}\left(t_1\right){\widehat{\sigma }}_j^{-}\left(t_2\right){\widehat{\sigma }}_j^{+}\left(t_3\right){\widehat{\sigma }}_j^{-}\left(t_4\right)$.

Figure 3

Plots of $\Phi (J,t)$ for $\gamma =0$ and ${\Gamma }_{\mathrm{r}}/{\Gamma }_{\mathrm{r}}^{\mathrm{c}}\in \{0,1/4,1/2,3/4,1\}$, showing a transition from oscillatory to damped behavior.

Figure 4

(a) Example of how spin squeezing is affected by decoherence (dashed lines are $\Gamma =0.06J$; solid lines are for $\Gamma =0$) for long-ranged (red, $\zeta =0$) and short-ranged (blue, $\zeta =3$) interactions.

Figure 5

Transverse-spin relaxation and revivals for $\zeta =0$, with parameters corresponding to expected experimental capabilities in [10] (blue solid line). The dotted blue line is obtained by treating decoherence at the single-particle level, and underestimates the detrimental effect of Raman decoherence by about a factor of 35. Inset: transverse-spin fluctuations peaking at time ${\tau }_{\mathrm{r}}/2$. Experimental parameters, exact treatment (red solid line); experimental parameters, single-particle treatment (red-dotted line); no decoherence (black dashed line).