Implementation of an Impedance-Matched $\Lambda$ System by Dressed-State Engineering

In one-dimensional optical setups, light-matter interaction is drastically enhanced by the interference between the incident and scattered fields. Particularly, in the impedance-matched $\Lambda$-type three-level systems, a single photon deterministically induces the Raman transition and switches the electronic state of the system. Here, we show that such a $\Lambda$ system can be implemented by using dressed states of a driven superconducting qubit and a resonator. The input microwave photons are perfectly absorbed and are down-converted into other frequency modes in the same waveguide. The proposed setup is applicable to the detection of single microwave photons and the swapping of the photon and matter qubits.

Figure 1

Interaction between a $\Lambda$ system and a photon propagating in a semi-infinite one-dimensional waveguide: (a) simple reflection, (b) elastic scattering, and (c) inelastic scattering. ${\Gamma }_{ij}$ ($i$, $j=e$, $m$, $g$) denotes the radiative decay rate for the $|i⟩\to |j⟩$ transition.

Figure 2

Schematic of the considered setup. A qubit is coupled dispersively to a resonator, which is further coupled to a semi-infinite waveguide (waveguide 1). The qubit is driven by a microwave field propagating along another waveguide (waveguide 2).

Figure 3

Structure of the four lowest levels of the qubit-resonator system for $E=0$: (a) Nesting and (b) un-nesting regimes. The nesting regime is realized when the drive frequency satisfies ${\omega }_q-3\chi <{\omega }_d<{\omega }_q-\chi$. Arrows indicate the direction of the cavity decay. Oblique decay paths (gray arrows) are generated by the drive field ($E\ne 0$).

Figure 4

(a) Dependences of ${\tilde{\kappa }}_{31}$, ${\tilde{\kappa }}_{32}$, ${\tilde{\kappa }}_{41}$, and ${\tilde{\kappa }}_{42}$ on the drive power. The drive frequency is in the nesting regime (${\omega }_d/2\pi =4.87\mathrm{GHz}$). The drive power is expressed in terms of the Rabi frequency ${\Omega }_R=\sqrt{\gamma }|E|$. The curves for ${\tilde{\kappa }}_{31}$ and ${\tilde{\kappa }}_{42}$ (${\tilde{\kappa }}_{32}$ and ${\tilde{\kappa }}_{41}$) are mostly overlapping. (b) The same plot as (a) in the un-nesting regime (${\omega }_d/2\pi =4.83\mathrm{GHz}$). (c) Reflection coefficient $|r|$ as a function of the drive power and the probe frequency in the nesting regime (${\omega }_d/2\pi =4.87\mathrm{GHz}$). (d) Same plot as (c) in the un-nesting regime (${\omega }_d/2\pi =4.83\mathrm{GHz}$). In (c) and (d), the probe field is weak ($|F{|}^2={10}^2\mathrm{photons}/\mathrm{s}$) and is in the linear-response regime. The following parameters are assumed: ${\omega }_q/2\pi =5\mathrm{GHz}$, ${\omega }_r/2\pi =10\mathrm{GHz}$, $g/2\pi =500\mathrm{MHz}$, $\kappa /2\pi =20\mathrm{MHz}$, and $\gamma /2\pi =0.1\mathrm{MHz}$.

Figure 5

(a) Power spectrum density of the output field (reflected field in waveguide 1) under the impedance-matching condition (${\omega }_d/2\pi =4.87\mathrm{GHz}$ and ${\Omega }_R/2\pi \approx 19\mathrm{MHz}$). The input probe frequency is tuned to ${\tilde{\omega }}_{41}$ ($2\pi \times 10.066\mathrm{GHz}$), whereas the dominant peak appears at ${\tilde{\omega }}_{42}$ ($2\pi \times 9.977\mathrm{GHz}$). The probe power $|F{|}^2$ is ${10}^5$ (solid line), ${10}^6$ (dotted line) and ${10}^7$ (dashed line) photons/s, respectively. (b) Down-conversion efficiency as a function of the probe power.