Quantum memory coupled to cavity modes

Inspired by spin-electric couplings in molecular magnets, we introduce in the Kitaev honeycomb model a linear modification of the Ising interactions due to the presence of quantized cavity fields. This allows to control the properties of the low-energy toric code Hamiltonian, which can serve as a quantum memory, by tuning the physical parameters of the cavity modes, like frequencies, photon occupations, and coupling strengths. We study the properties of the model perturbatively by making use of the Schrieffer-Wolff transformation and show that, depending on the specific setup, the cavity modes can be useful in several ways. They allow to detect the presence of anyons through frequency shifts and to prolong the lifetime of the memory by enhancing the anyon excitation energy or mediating long-range anyon-anyon interactions with tunable sign. We consider both resonant and largely detuned cavity modes.

Figure 1

Illustration of the Kitaev honeycomb model with cavity modes. (a) Schematic view of the honeycomb model at the center of an optical or microwave cavity. The two shaded areas represent mirrors and the curves in the middle two quantized cavity modes with frequencies ${\omega }_{1,2}$ to which the spin model couples according to Eq. (2). (b) Kitaev honeycomb lattice. The spins are represented by white or black dots respectively belonging to the $A$ or $B$ sublattice. Ising interactions of type ${\sigma }^x{\sigma }^x$ along $x$ links (red), ${\sigma }^y{\sigma }^y$ along $y$ links (green), or ${\sigma }^z{\sigma }^z$ along $z$ links (blue) are represented by colored links with different directions. Strongly interacting $z$ dimers are depicted by thick blue links and form a lattice with basis vectors ${\mathbf{e}}_{1,2}$ while $\mathbf{t}$ is a vector which connects $A$- and $B$-sublattice sites. Cavity modes (here, two of them, with frequencies ${\omega }_{1,2}$) modify the Ising interactions of certain links (as indicated by dashed lines), according to Eq. (2).

Figure 2

Pictorial representation of superexchange interactions. The bridge sites (dots in the middle of each $x$ link) mediate the magnetic ${\sigma }^x{\sigma }^x$ interactions between the spins (dots at each site of the honeycomb lattice). Here, the $x$ links have a nonvanishing dipole moment and can thus be linearly coupled to external electric fields. By controlling the equilibrium position of the bridge sites, the coupling to the cavity modes can in principle be tuned locally.

Figure 3

Mapping of the Kitaev honeycomb model to the toric code. Left-hand side: each $z$ dimer, formed by a black and a white spin [see also Fig. 1b], corresponds to a new lattice site (large blue dots). To these new sites, hard-core bosons and effective spins are associated. A low-energy (zero hard-core bosons) toric code Hamiltonian for the effective spins is obtained by perturbation theory. The resulting toric code model is schematically illustrated on the right-hand side. The dark and white unit cells on the left-hand-side correspond now to plaquette (p) and star (s) operators, respectively.

Figure 4

Honeycomb lattice with two distinct $x$ couplings $K_{x_{1,2}}$ and two distinct $y$ couplings $K_{y_{1,2}}$.

Figure 5

Mean-field gap ${\mu }_{\mathrm{mf}}$, given by Eq. (40) with $A$ as in Eq. (48), plotted for different temperatures (solid curves). The dashed curve is the bare gap $\mu$ of Eq. (55). We used $J_z=1011$ K, $J_{x_{1,2}}=J_{y_{1,2}}=22$ K, $\langle a_x^{†}{a}_x\rangle =100$, $\langle a_y^{†}{a}_y\rangle =0$, ${\delta }_x={\delta }_y=0.21$ K, ${\omega }_x=3991$ K, and ${\omega }_y=3980$ K.

Figure 6

Pictorial representation of the connected and disconnected terms entering the perturbation expansion (B3). Both loops $\mathit{abcd}$ and $\mathit{abba}$ represent the different contributions arising from the connected term $T_{-2}^{+}{T}_0^0{T}_0^0{T}_{+2}^{-}$. We see that only $\mathit{abcd}$ loops lead to terms proportional to $W_{\mathbf{a}}$, since all four links are different. This is not the case for the $\mathit{abba}$ loop and the corresponding terms are constants. The remaining fourth-order terms entering Eq. (B3) can be represented by two disconnected loops $\mathit{aa}$ and $\mathit{bb}$. Since the number of different positions of each $\mathit{aa}$ and $\mathit{bb}$ loop is proportional to $N$, the contribution of the disconnected terms is $~N^2$.