Figure 1

Comparison of the effective system spectra in the interaction picture and Schrödinger picture. The ground state of the effective Hamiltonian $\mid {\psi }_1⟩$ is not the ground state of the total system, ∣3⟩. The figure is not to scale. For optical qubits, $\nu ∕J\approx {10}^8$, and for hyperfine qubits, $\nu ∕J\approx {10}^3$.Reuse & Permissions

Figure 2

Comparison of the system and bath spectra. We assume a physical bath of positive energy modes. (a) The expected spectrum for two qubits naturally coupled by $H_{xy}$. If the temperature is less than the energy gap to the excited state, excitations will be exponentially suppressed. (b) In an interaction frame, where the effective interaction is $H_{xy}$, the system spectrum is the same. However, the bath modes are shifted to effective negative energy. Consequently, a negative energy boson can be emitted, allowing for the spontaneous excitation of the system.Reuse & Permissions

Figure 3

(Color online) The compass model in the interaction picture does not suppress errors due to spontaneous emission. The decay of the population of the logical ∣0⟩ (solid line) is comparable to the decay of the excited state in the physical qubit (dotted line). The same is true if we consider the population of the subspace $A_{++}$ associated with the logical ∣0⟩. The applied compass model Hamiltonian drives the system from the bare ground state (all ions in the $\mid S⟩$ state) and the result to zeroth order in $J∕\nu$ is a completely mixed state.Reuse & Permissions