Dynamics of coupled qubits interacting with an off-resonant cavity

We study a model for a pair of qubits that interact with a single off-resonant cavity mode and, in addition, exhibit a direct interqubit coupling. Possible realizations for such a system include coupled superconducting qubits in a line resonator as well as exciton states or electron spin states of quantum dots in a cavity. The emergent dynamical phenomena are strongly dependent on the relative energy scales of the interqubit coupling strength, the coupling strength between qubits and cavity mode, and the cavity mode detuning. We show that the cavity mode dispersion enables a measurement of the state of the coupled-qubit system in the perturbative regime. We discuss the effect of the direct interqubit interaction on a cavity-mediated two-qubit gate. Further, we show that for asymmetric coupling of the two qubits to the cavity, the direct interqubit coupling can be controlled optically via the ac Stark effect.

Figure 1

(Color online) Energy level spectrum in the subspace $N=1$ as a function of $\Delta$ for $g=0.05\Omega$ and (a) $J_{\perp }=0.2\Omega$, $J_z=0$, and (b) $J_{\perp }=0.3\Omega$, $J_z=0$, respectively. Solid lines indicate the exact energy eigenvalues, dashed lines the approximate analytical values obtained with Eq. (6). The three anticrossings discussed in the text are indicated by arrows and circles in (b).

Figure 2

(Color online) (a) Triplet level scheme in the “dressed state” picture. (b) Magnification of the exact anticrossing (solid lines) of $\mid T_{-};2⟩$ and $\mid T_0;1⟩$ shown in Fig. 1b, in comparison with a two-level description (dashed lines).

Figure 3

Energies of two coupled identical qubits according to Eq. (6) (solid lines) and for the optical interaction without the RWA (dashed lines), see Eq. (B7), of the triplet states (a) $\mid T_{-}⟩$, (b) $\mid T_{+}⟩$, and (c), (d) $\mid T_0⟩$ as a function of the photon number $n$ or the frequency $\omega$ of the cavity mode. The dotted lines indicate the energy levels in the absence of the coupling to the cavity, respectively. The parameters used are $J=0.001\Omega$ and $g=0.0002\Omega$. Further, $\omega =0.96\Omega$ in (a), (b), (c), and $n={10}^4$ in (d).

Figure 4

Energy level scheme of two coupled qubits for the general case. The arrows indicate transitions under photon absorption and emission. Contrary to the symmetric case discussed in Sec. 2, for the general case there are four allowed transitions with coupling constants ${\gamma }_i$, $i=1,\cdots ,4$.

Figure 5

(Color online) Schemes for coupled qubits: (a) Pair of quantum dots interacting with one cavity mode, as discussed in Sec. 4A. (b) Nested quantum shells as discussed in Sec. 4B. (c) Cross section through nanocrystal containing a pair of nested quantum shells. (d) Two Cooper-pair boxes in a line resonator, as discussed in Sec. 4C.