Entanglement generated by the dispersive interaction: The dressed coherent state

In the dispersive regime of qubit-cavity coupling, a classical cavity drive populates the cavity but leaves the qubit state unaffected. However, the dispersive Hamiltonian is derived after both a frame transformation and an approximation. Therefore, to connect to external experimental devices, the inverse frame transformation from the dispersive frame back to the laboratory frame is necessary. In this work we show that in the laboratory frame the system is best described by an entangled state known as the dressed coherent state, and thus even in the dispersive regime, entanglement is generated between the qubit and the cavity. Also, we show that further qubit evolution depends on both the amplitude and phase of the dressed coherent state and use the dressed coherent state to calculate the measurement contrast of a recently developed dispersive readout protocol.

Figure 1

Schematic diagrams of the dressed coherent states $\overline{\left|g,\alpha \right.〉}$ and $\overline{\left|e,\alpha \right.〉}$ for ${\left|\alpha \right|}^2=4$. The red and blue curves are the Poisson distribution of superposition coefficients, $P_n={\left|\overline{\left.〈g/e,n\right|g/e,\alpha 〉}\right|}^2$.

Figure 2

Comparison of the numerical state created by simulation of Eq. (23) with the dressed coherent state as a function of (a) photon number ${\left|\alpha \right|}^2$, with $\epsilon$ and $\lambda$ constant; (c) $\lambda$, with $\epsilon$ and ${\left|\alpha \right|}^2$ constant, and (d) drive strength $\epsilon$, with ${\left|\alpha \right|}^2$ and $\lambda$ constant. (b) The difference in overlap with the numerical state between the dressed coherent state and the undressed product state as a function of photon number ${\left|\alpha \right|}^2$, with $\epsilon$ and $\lambda$ constant. Values of ${\left|\alpha \right|}^2,\lambda$, and $\epsilon$ chosen to be commensurate with current experiments. Additional phase matching was required to obtain high fidelity (see Appendix pp2).

Figure 3

Qubit excited-state probability as a function of the time of the applied qubit drive, starting in a dressed coherent state with either purely real or purely imaginary $\beta$ for ${\left|\beta \right|}^2=4$. The drive strength is strong, $\left|\eta \right|=0.05{\omega }_{\mathrm{q}}$, so that phase effects are clearly visible. The drive frequency is set to the average shifted qubit frequency, $\omega ={\omega }_{\mathrm{q}}+2\left({\left|\beta \right|}^2+1\right),\eta$ is purely real (i.e., $\left|\eta \right|=\eta$), and $\lambda =0.1$ as elsewhere.