Anisotropic exchange interaction induced by a single photon in semiconductor microcavities

We investigate coupling of localized spins in a semiconductor quantum dot embedded in a microcavity. The lowest cavity mode and the quantum dot exciton are coupled and close in energy, forming a polariton. The fermions forming the exciton interact with localized spins via exchange. Exact diagonalization of a Hamiltonian in which photons, spins, and excitons are treated quantum mechanically shows that a single polariton induces a sizable indirect anisotropic exchange interaction between spins. At sufficiently low temperatures strong ferromagnetic correlations show up without an appreciable increase in exciton population. In the case of a (Cd,Mn)Te quantum dot, Mn-Mn ferromagnetic coupling is still significant at $1\mathrm{K}$: spin-spin correlation around 3 for exciton occupation smaller than 0.3. We find that the interaction mediated by photon-polaritons is 10 times stronger than the one induced by a classical field for equal Rabi splitting.

Figure 1

Spin correlation functions and exciton occupation number versus the detuning parameter $\delta$, for spin-$5∕2$ impurities. $g=5\mathrm{meV}$ and $T=0.01\mathrm{K}$ (upper) and $1\mathrm{K}$ (lower).

Figure 2

Energy levels for spin-$1∕2$ impurities calculated either without interactions (left) incorporating light-matter coupling (middle) or with the full interacting Hamiltonian (right). The energy scale in the first two cases (third case) is given on the left (right) vertical axis. The degeneracy of each state is also indicated. All results correspond to $\delta =10\mathrm{meV}$ and $g=5\mathrm{meV}$. Energies referred to the ground state of the full Hamiltonian in each case.

Figure 3

Total spin-spin correlation versus exciton occupation number for spin-$5∕2$ impurities and four temperatures. Calculations were carried out by varying $g$ and with fixed detuning $\delta =10\mathrm{meV}$. The results derived from a classical treatement of the photon field (see text) is also shown (thick solid line).

Figure 4

In-plane and out-of-plane spin-spin correlation versus exciton occupation number for spin-$5∕2$ impurities. Calculations were carried out by varying $g$ and with fixed detuning $\delta =10\mathrm{meV}$. The results were obtained for $T=0.1\mathrm{K}$ and valence- and conduction-band exchange couplings of $J_v=-0.5\mathrm{meV}$ and $J_c=-0.1\mathrm{meV}$, respectively.

Figure 5

$J_x$ (dotted line) and $J_z$ (solid line), parameters of the Hamiltonian of Eq. (7), fitted to give the exact spin-spin correlation, versus the exciton occupation number. Calculations were carried out by varying $g$ and with fixed detuning $\delta =10\mathrm{meV}$. The fitting of the numerical results for $J_z$ by means of a straight line (circles) is also shown. Chain lines show the results obtained by treating the photon field classically (see text).

Figure 6

Same as Fig. 2 but treating the photon field classically.