Single-photon limit of dispersive readout of a qubit with a photodetector

We study the dispersive readout of a qubit in the ultimate limit of a single-photon probe. The use of a single-photon probe avoids the errors due to nonorthogonality of coherent states. A photodetector is used in the scheme we consider. The dynamics of the system is studied using the Heisenberg-Langevin equations. We treat the counter-rotating terms in the Hamiltonian perturbatively, which leads to the Bloch-Siegert shift in the resonator frequency. It is shown how the shift can improve the readout. The theory of photon transport through the qubit and the resonator it couples to is provided while taking the effect of the counter-rotating terms into account. To calculate the readout contrast, we use two approaches. The first one neglects the qubit relaxation and allows us to derive a compact expression for the contrast. Also, we obtain simple estimates for the system parameters to maximize the contrast. The second approach accounts for the qubit relaxation, which allows us to further improve the contrast. We demonstrate that for a readout time of $1\mu \mathrm{s}$, a contrast of more than 75% can be achieved for an ideal detector and single-photon source.

Figure 1

Measurement scheme. A single-photon pulse $|1_{\xi }\rangle$ of exponentially damped shape is incident on the first port of a cavity. The cavity resonant frequency is ${\omega }_r$ and its leakage is $\kappa /2$ to each of its ports. The cavity is dispersively coupled to a qubit. The qubit transition frequency is ${\omega }_q$. It is prepared at the initial moment of time. To prepare the excited state $|\uparrow \rangle$, a $\pi$ pulse is used. On the second port there is an on-off photodetector with a quantum efficiency $\eta$. The pulse can be generated by decay of a two-level system with the transition frequency ${\omega }_p$. The two-level system is protected from a reflected photon by a circulator. The dashed circulator indicates there is no backaction on the cavity due to reflection off the detector. The backaction can also be avoided without a circulator [8].

Figure 2

The system as modeled by the Hamiltonian. The semi-infinite waveguides $\mathrm{I}$ and $\mathrm{II}$ couple to the resonator. They model the lack of backaction on the cavity and the qubit: after a photon scatters off the resonator, it does not return. Notations are as in Fig. 1.

Figure 3

Readout contrast as a function of the dimensionless dispersive pull $X$ and cavity leakage $K$ (65). The dashed line shows the position of maximum for each $X$.

Figure 4

Dependence of the readout contrast on the resonator decay rate $\kappa$ and the resonator-qubit coupling strength $g$ for (a) $t_{\mathrm{m}}=1\mu \text{s}$, (b) $t_{\mathrm{m}}=2\mu \text{s}$, (c) $t_{\mathrm{m}}=5\mu \text{s}$, and (d) $t_{\mathrm{m}}=10\mu \text{s}$. Measurement time and pulse duration are related as $t_{\mathrm{m}}=6t_p$. The resonator and the qubit frequencies are ${\omega }_r=4.09\mathrm{GHz}$ and ${\omega }_q=5.0\mathrm{GHz}$, respectively. The photodetector is set to be ideal with $\eta =1$. A star marks the position of the maximal contrast $C_{\max }$, which can be achieved for given qubit-resonator detuning and readout pulse duration.

Figure 5

Dependence of the maximal readout contrast on the readout pulse duration $t_p$ for different qubit-resonator detunings. To satisfy the criterion (87) we choose $\lambda /\mathrm{\Lambda }=10$. The rest of the parameters and notations are the same as in Fig. 4.

Figure 6

Dependence of the maximal readout contrast on the detuning between the qubit and resonator for different readout pulse durations $t_p$. The rest of the parameters are as described in the caption of Fig. 5.