Effective Spin Systems in Coupled Microcavities

We show that atoms trapped in microcavities that interact via the exchange of virtual photons can model an anisotropic Heisenberg spin-$1/2$ lattice in an external magnetic field. All parameters of the effective Hamiltonian can individually be tuned via external lasers. Since the occupations of excited atomic levels and photonic states are strongly suppressed, the effective model is robust against decoherence mechanisms, has a long lifetime, and its implementation is feasible with current experimental technology. The model provides a feasible way to create cluster states in these devices.

Figure 1

Level structure, driving lasers, and relevant couplings to the cavity mode to generate effective ${\sigma }^x{\sigma }^x$- and ${\sigma }^y{\sigma }^y$-couplings for one atom. The cavity mode couples with strengths $g_a$ and $g_b$ to transitions $|a⟩\leftrightarrow |e⟩$ and $|b⟩\leftrightarrow |e⟩$, respectively. One laser with frequency ${\omega }_a$ couples to transition $|a⟩\leftrightarrow |e⟩$ with Rabi frequency ${\Omega }_a$ and another laser with frequency ${\omega }_b$ to $|b⟩\leftrightarrow |e⟩$ with ${\Omega }_b$. The dominant two-photon processes are indicated in faint gray arrows.

Figure 2

Level structure, driving lasers, and relevant couplings to the cavity mode to generate effective ${\sigma }^z{\sigma }^z$-couplings for one atom. The cavity mode couples with strengths $g_a$ and $g_b$ to transitions $|a⟩\leftrightarrow |e⟩$ and $|b⟩\leftrightarrow |e⟩$, respectively. Two lasers with frequencies $\omega$ and $\nu$ couple with Rabi frequencies ${\Omega }_a$, respectively, ${\Lambda }_a$ to transition $|a⟩\leftrightarrow |e⟩$ and ${\Omega }_b$, respectively, ${\Lambda }_b$ to $|b⟩\leftrightarrow |e⟩$. The dominant two-photon processes are indicated in faint gray arrows.

Figure 3

The occupation probability $p(a_1)$ of state $|a_1⟩$ (solid line) and the probability $p({\downarrow }_1)$ of spin 1 to point down (dashed line) for the parameters ${\omega }_e={10}^6\mathrm{GHz}$, ${\omega }_{ab}=30\mathrm{GHz}$, ${\Delta }_a=30\mathrm{GHz}$, ${\Delta }_b=60\mathrm{GHz}$, ${\omega }_C={\omega }_e-{\Delta }_b+2\mathrm{GHz}$, ${\tilde{\Delta }}_a=15\mathrm{GHz}$, ${\Omega }_a={\Omega }_b=2\mathrm{GHz}$, ${\Lambda }_a={\Lambda }_b=0.71\mathrm{GHz}$, $g_a=g_b=1\mathrm{GHz}$, $J_C=0.2\mathrm{GHz}$, and ${\delta }_1=-0.0165\mathrm{GHz}$ (a) respectively ${\delta }_1=-0.0168\mathrm{GHz}$ (b). Both, the occupation of the excited atomic states $⟨|e_j⟩⟨e_j|⟩$ and the photon number $⟨a^{†}a⟩$ are always smaller than 0.03.

Figure 4

(a) The von Neumann entropy $E_{\mathrm{vN}}$ of the reduced density matrix of 1 effective spin in multiples of ln 2 and (b) the purity of the reduced state of the effective spin model for 3 cavities where $J_z=0.021\mathrm{MHz}$. The plots assume that no spontaneous emission took place. For a spontaneous emission rate of 0.1 MHz ($g=1\mathrm{GHz}$), the probability for a decay event in the total time range is 1.5%. Hence, cluster state generation fails with probability $0.005n$ for $n\ll 1/0.005=200$ cavities, irrespective of the lattice dimension.