Steady many-body entanglements in dissipative systems

We propose a dissipative method for the preparation of many-body steady entangled states in spin and fermionic chains. The scheme is accomplished by means of an engineered set of Lindbladians acting over the eigenmodes of the system, whose spectrum is assumed to be resolvable. We apply this idea to prepare a particular entangled state of a spin chain described by the $XY$ model, emphasizing its generality and experimental feasibility. Our results show that our proposal is capable of achieving high fidelities and purities for a given target state, even when dephasing and thermal dissipative processes are taken into account. Moreover, the method exhibits a remarkable robustness against fluctuations in the model parameters.

Figure 1

Schematic representation of the system composed of a spin chain (our system of interest) described by an $XY$ Hamiltonian. The reservoir consists of a set of two-level systems that mediates the dissipation and pumping of the spin chain normal modes. By controlling the system-reservoir couplings one can, in principle, find a Hamiltonian whose spectrum is resolved enough thus through its pump that we can access the target state ($|s_j\rangle$) individually. $|s_i\rangle$ is the $i\mathrm{th}$ eigenstate of the Hamiltonian of the system of interest with ${\omega }_i$ its corresponding eigenenergy. ${\gamma }_{+i}\left({\gamma }_{-i}\right)$ is the pump (cooling) rates of the $i\mathrm{th}$ level of the Hamiltonian, $J_{i,i+1}=J$ is the system coupling between spins, while ${\lambda }_{i,k}$ is the coupling between the $i\mathrm{th}$ system spin and the $k\mathrm{th}$ reservoir spin. In the scheme above $i=k$; the emerging off-resonant couplings can be eliminated by performing the rotating wave approximation (RWA).

Figure 2

(a) Fidelity ${\mathcal{F}}_{\mathrm{steady}}=\sqrt{Tr\left(\left|\varphi \right.〉\left.〈\varphi \right|{\rho }_{ss}\right)}$ as a function of $\gamma /\kappa$ for various temperatures (represented by $\overline{n}$) for the $N=5$ spin chain, with steady state as given by the Eq. (12). Here we consider only one engineered reservoir connected with the pump Liouvillian at ${\gamma }_{{}^{+}}=\gamma$ and ${\gamma }_{{}^{-}}={\kappa }_{\varphi }=0$. (b) Same as in (a) but for five different numbers of reservoirs (as indicated in the figure). Here we set ${\gamma }_{{}^{+}}={\gamma }_{{}^{-}}=\gamma ,{\kappa }_{\varphi }=\kappa$. The temperature of all the reservoirs is assumed to be equal, $\overline{n}=0.001$. (c) Purity ${\mathcal{P}}_{\mathrm{steady}}=Tr\left[{\rho }_{ss}^2\right]$ and (d) concurrence between the second and third spins against $\gamma /\kappa$ for the same parameters of panel (b). (e) Time evolution ($\tau =t/\gamma$) of the fidelity and purity $\mathcal{P}\left(t\right)$ for three different degrees of randomness $(10,20,30)\%$ of the effective decay rates ${\gamma }_{{}^{+}}$ and ${\gamma }_{{}^{-}}$. (f) Fidelity and purity are obtained for the state ${\left|\varphi \right.〉}_N$ against number $N$ of spins in the chain with the other parameters kept the same as in panel (e).