Dissipative spin chains: Implementation with cold atoms and steady-state properties

We propose a quantum optical implementation of a class of dissipative spin systems, including the $XXZ$ and Ising model, with ultracold atoms in optical lattices. By employing the motional degree of freedom of the atoms and detuned Raman transitions, we show how to obtain engineerable dissipation and a tunable transversal magnetic field, enabling the study of the dynamics and steady-states of dissipative spin models. As an example of effects made accessible this way, we consider small spin chains and weak dissipation and show by numerical simulation that steady-state expectation values display pronounced peaks at certain critical system parameters. We show that this effect is related to degeneracies in the Hamiltonian and derive a sufficient condition for its occurrence.

Figure 1

Relevant level structure and coupling and decay terms of single atom trapped at a lattice site. The upper-left part shows the internal levels of the atom: $\Lambda$ system $|g\rangle$, $|r\rangle$, $|e\rangle$, off-resonantly driven by lasers. The right part shows motional states in the lattice potential. Lower part: After adiabatic elimination of $|e\rangle$, an effective two-level system with tunable decay rate $\Gamma$ and dephasing rate $\gamma$ is obtained.

Figure 2

Effective two-level system $|g\rangle$-$|r\rangle$ in the optical lattice potential with motional states $|0\rangle$ and $|1\rangle$. Choosing resonance conditions as explained in Sec. 2b, the atoms are selectively excited from $|g\rangle \otimes |1\rangle$ to the state $|r\rangle \otimes |0\rangle$ and spontaneously decay into $|g\rangle \otimes |0\rangle$.

Figure 3

Decay of the effective two-level system $\left\{\right|0\rangle \},\{|1\rangle \}$ as described by the effective master equation derived in Sec. 2b. The dissipation strength $A^{-}$ is given in Eq. (7).

Figure 4

Level scheme and transitions used to implement the transverse magnetic field. Left: A detuned Raman transition couples the internal ground state $|g\rangle$ and the excited state $|e\rangle$ of the atom. Right: Adiabatic elimination of the excited state $|e\rangle$ leads to an effective magnetic field in $x$ direction (see Sec. 2c), which drives transitions between the motional states $|0\rangle$ and $|1\rangle$.

Figure 5

Tunneling between neighboring lattice wells with tunnel amplitudes $t_0$ and $t_1$. States with two atoms per lattice well are treated in perturbation theory in Sec. 2d, as the on-site interaction is much larger than the tunneling amplitudes.

Figure 6

$XXZ$ model with 4 spins, ${\alpha }_1=\frac{1}{4}{\alpha }_2$ and open boundary conditions under local dissipation of form given by Eq. (20). Upper part: Steady-state expectation value $\langle J^x\rangle$ plotted versus the magnetic field $B_x/{\alpha }_2$. Peaks are observed that narrow for decreasing the dissipation strength. Lower part: Spectrum of $XXZ$ chain in the magnetic field $B_x$ plotted versus $B_x/{\alpha }_2$. Peaks in the steady-state expectation value (upper part) appear at crossing points of the Hamiltonian that are marked with black circles.

Figure 7

Ising model with 4 spins with open boundary conditions in a transverse magnetic field $B_x$ and under local dissipation of form given by Eq. (23). Upper part: Steady-state expectation value $\langle J^x\rangle$ plotted versus $B_x/{\alpha }_3$. Peaks are observed that become more narrow for decreasing dissipation strength. Lower part: Spectrum of the Hamiltonian plotted versus $B_x/{\alpha }_3$. Peaks in the steady-state expectation value (upper part) appear at degeneracy points of the Hamiltonian that are marked with black circles.

Figure 8

Ising model with 6 spins with periodic boundary conditions in a magnetic field $B_x$ and under collective dissipation of the form given by Eq. (24) in the translation and reflection symmetric subspace $T=R=1$. Upper part: Steady-state expectation value $\langle J^x\rangle$ plotted versus $B_x/{\alpha }_3$. Lower part: Spectrum of the Hamiltonian plotted versus $B_x/{\alpha }_3$.

Figure 9

Heisenberg model with 4 spins with open boundary conditions in a magnetic field $B_x$ and under local dissipation as given by Eq. (23) with different dissipation strengths ${\gamma }_k$. Upper panel: steady-state expectation value $\langle J^x\rangle$ plotted versus the $B_x/{\alpha }_3$. Lower part: Spectrum of the Hamiltonian plotted versus $B_x/{\alpha }_3$.

Figure 10

(Upper part) Steady-state infidelity $I_{\delta B}\left(B_x\right)$ (see text) for the four-spin $XXZ$ model in transverse field $B_x$ with local dissipation (cf. Fig. 6) for $\delta B=3\times {10}^{-6}$ and weak dissipation $\gamma =0.5\times {10}^{-4}$. (Lower part) Spectrum of $H_{\mathrm{spin}}$, dashed vertical lines indicate degeneracy points (encircled). Peaks in $I_{\delta B}$ line up with degeneracy points of $H\left(B_x\right)$, except for two (colored green) for which condition (28) does not hold. The inset shows the vicinity of $B_x\approx 0.25$, where three crossings of eigenvalues occur: $({\lambda }_5,{\lambda }_6),({\lambda }_1,{\lambda }_2)$, and $({\lambda }_{13},{\lambda }_{14})$. The dash-dotted horizontal lines show the 2-norm (scaled by $7\times {10}^{-7}$) of the left-hand side of Eq. (28) $C_{i,j}={\parallel {\mathrm{P}}^{{\Delta }_{i,j}}{\mathcal{L}}_1{\lim }_{x^{\prime }\to x}{\rho }_{ss}\left(x^{\prime }\right)\parallel }_2$ for the three relevant projectors ${\mathrm{P}}^{{\Delta }_{i,j}},(i,j)=(1,2)$ (green), $=(5,6)$ (blue), and $=(13,14)$ (magenta). $C_{5,6}$ vanishes at the crossing of $({\lambda }_5,{\lambda }_6)$, hence there is no peak in $I_{\delta B}$, while the other two lead to a peak in $I_{\delta B}$, since $C_{i,j}$ is finite.