Theory of microwave single-photon detection using an impedance-matched $\Lambda$ system

By properly driving a qubit-resonator system in the strong dispersive regime, we implement an “impedance-matched” $\Lambda$ system in the dressed states, where a resonant single photon deterministically induces a Raman transition and excites the qubit. Combining this effect and a fast dispersive readout of the qubit, we realize a detector of itinerant microwave photons. We theoretically analyze the single-photon response of the $\Lambda$ system and evaluate its performance as a detector. We achieve a high detection efficiency without relying on precise temporal control of the input pulse shape and under a conservative estimate of the system parameters. The detector can also be reset quickly by applying microwave pulses, which allows a short dead time and a high repetition rate.

Figure 1

Schematic of the setup. A superconducting qubit (two-level atom) is coupled dispersively to a resonator. The resonator is coupled to a semi-infinite waveguide (WG1), through which a signal photon is input. The qubit is coupled to another waveguide (WG2), through which a drive pulse is applied.

Figure 2

(a) I mode of the qubit-resonator system. This is attained when the qubit drive is off. Arrows indicate the directions of radiative decays. (b) $\Lambda$ mode of the qubit-resonator system. The four decay rates are identical. This is attained when a proper drive field is applied to the qubit. (c) Radiative decay rates ${\tilde{\kappa }}_{31},{\tilde{\kappa }}_{32},{\tilde{\kappa }}_{41}$, and ${\tilde{\kappa }}_{42}$ as functions of the drive amplitude. The four rates become identical at ${\Omega }_d^{\mathrm{imp}}/2\pi =13.2\text{MHz}$. (d) Reflection coefficient $\left|r\right|$ of a weak signal field applied through WG1. White dashed lines show ${\tilde{\omega }}_{31}$ and ${\tilde{\omega }}_{41}$. Impedance matching ($\left|r\right|\simeq 0$) occurs at $({\Omega }_d,{\omega }_p)\simeq ({\Omega }_d^{\mathrm{imp}},{\tilde{\omega }}_{31})$ and $({\Omega }_d^{\mathrm{imp}},{\tilde{\omega }}_{41})$.

Figure 3

Pulse sequence for single-photon detection. (i) Capture stage. We simultaneously input a signal photon through WG1 and a classical drive pulse through WG2. (ii) Readout stage. We apply a classical readout pulse through WG1 and switch off a drive pulse through WG2. (iii) Reset stage. We simultaneously apply a classical reset pulse through WG1 and a classical drive pulse through WG2. The carrier frequencies of the three pulses applied through WG1 are different from each other.

Figure 4

Envelopes of the signal photon and the qubit drive pulse. We use a smooth drive pulse in order to switch the I and $\Lambda$ modes adiabatically.

Figure 5

Time evolution of the qubit excitation probability, for a signal pulse containing zero photon: the actual evolution $p_0\left(t\right)$ (red solid) and the adiabatic approximation ${\overline{p}}_0\left(t\right)$ (blue dashed). The parameters used are: ${\Omega }_d/2\pi ={\Omega }_d^{\mathrm{imp}}/2\pi =13.2\text{MHz},l=100\text{ns},\beta =2$, and $w=20\text{ns}$ in (a) and $w=30\text{ns}$ in (b).

Figure 6

(a) Time evolution of the qubit excitation probability $p_1\left(t\right)$ for a signal pulse containing one photon (red solid). The result for zero-photon case, $p_0\left(t\right)$, is also plotted for reference (blue dashed). The parameters used are: $l=100\text{ns},\beta =2,w=30\text{ns}$, and ${\omega }_s/2\pi =10.007\text{GHz}$. (b) Dependence of the detection probability $P_1$ on the signal pulse length $l$ for various values of $\gamma /2\pi$: $0.1$ MHz (red solid), $0.02$ MHz (blue dashed), and 0 MHz (green dotted). (c) Dependence of the detection probability $P_1$ on the drive amplitude ${\Omega }_d$ and the signal photon frequency ${\omega }_s$.

Figure 7

(a) Time evolution of the qubit excitation probabilities, $p_1\left(t\right)$ (red solid) and $p_2\left(t\right)$ (blue dashed). The input photons are on resonance with the $\Lambda$ system for ${\omega }_s/2\pi =10.007\text{GHz}$ and are out of resonance for ${\omega }_s/2\pi =10.030\text{GHz}$. (b) Ratio of the detection probabilities, $P_2/P_1$, as a function of ${\Omega }_d$ and ${\omega }_s$.

Figure 8

Dependence of the detection probability on the pulse length $l$ for various pulse shapes: Gaussian (red solid), square (blue dashed), and exponential (green dotted). The qubit decay rate $\gamma /2\pi$ is 0 MHz in (a), $0.02$ MHz in (b), and $0.1$ MHz in (c).

Figure 9

(a) Time evolution of the qubit excitation probability $p_g\left(t\right)$ in the reset stage, starting from $|g,0\rangle$. The mean photon number in the reset pulse $\langle n\rangle$ is 0 (red solid), 10 (blue dashed), and 20 (green dotted). The three lines are mostly overlapping. The parameters used are: ${\Omega }_d/2\pi =44\text{MHz},{\omega }_r/2\pi =9.860\text{GHz},l=100\text{ns},\beta =2$, and $w=30\text{ns}$. (b) Time evolution of $p_e\left(t\right)$, starting from $|e,0\rangle$. The natural qubit decay is also plotted by the thin line. (c) Qubit excitation probability for $\langle n\rangle =10$ at the end of reset stage starting from $|e,0\rangle$.